Datasets:

miwytt

/

arXivEdits_alignments

like 0

1905.06077

{'1905.06077-1-0-0': 'In this work, we provide a simulation framework to perform systematic studies on the effects of spinal joint compliance and actuation on bounding performance of a [MATH]-DOF quadruped spined robot Stoch 2.', '1905.06077-1-0-1': 'Fast quadrupedal locomotion with active spine is an extremely hard problem, and involves a complex coordination between the various degrees of freedom.', '1905.06077-1-0-2': 'Therefore, past attempts at addressing this problem have not seen much success.', '1905.06077-1-0-3': 'Deep-Reinforcement Learning seems to be a promising approach, after its recent success in a variety of robot platforms, and the goal of this paper is to use this approach to realize the aforementioned behaviors.', '1905.06077-1-0-4': 'With this learning framework, the robot reached a bounding speed of [MATH] m/s with a maximum Froude number of [MATH].', '1905.06077-1-0-5': 'Simulation results also show that use of active spine, indeed, increased the stride length, improved the cost of transport, and also reduced the natural frequency to more realistic values.', '1905.06077-1-1-0': 'Key words: Deep-RL, Quadruped, Active spine', '1905.06077-1-2-0': '# Introduction', '1905.06077-1-3-0': 'Speed, for legged animals, is an important aspect in nature for survival.', '1905.06077-1-3-1': 'Animals like cheetah, hounds etc, use their spine to achieve remarkably high speed gaits.', '1905.06077-1-3-2': 'Studies on these animals show that the spine movement increases the effective stride length, provides auxiliary power to the legs and helps to harness energy by storing and releasing it.', '1905.06077-1-3-3': 'From the evolutionary point of view, it is natural to consider the spine as a propulsive engine of the vertebrate body.', '1905.06077-1-3-4': 'All of these observations point to the fact that spine is crucial for locomotion.', '1905.06077-1-4-0': 'There have been quite a few quadruped robots having passive or active spine in their structure, namely, BobCat, Sevel, MIT Cheetah, INU [CITATION].', '1905.06077-1-4-1': 'There is a large body of work over planer spine models with one DOF revolute [CITATION], and prismatic [CITATION] joints.', '1905.06077-1-4-2': 'Apart from the one DOF model, the spines were modeled as point masses in these works.', '1905.06077-1-4-3': 'On the other hand, robots such as Sevel [CITATION], INU [CITATION] contain a two DOF spine.', '1905.06077-1-4-4': 'This feature allows the spine to be longer while maintaining its orientation during bounding.', '1905.06077-1-4-5': 'However, reduced order models were used when constructing the empirical model of the robot.', '1905.06077-1-4-6': 'Moreover, in [CITATION], the legs were assumed to be massless, which result in large reality gaps.', '1905.06077-1-4-7': 'From a practical standpoint, we cannot use the approaches mentioned above, due to the fact that the spine models cannot be reduced or approximated, and the legs have significant masses (by upto [MATH]) in the real hardware.', '1905.06077-1-4-8': 'Apart from these, inaccuracy in modeling, imperfect ground contacts play a significant role while transferring on the real hardware.', '1905.06077-1-5-0': 'To address the above issues and to develop an accurate model of the robot and environment, researchers began to use high fidelity physics engine based simulators [CITATION] to simulate the robot and environment model.', '1905.06077-1-5-1': 'Works such as [CITATION] have pursued this approach to bridge the gap between theory and experiment.', '1905.06077-1-5-2': 'Despite these tools, the results obtained with these simulators still required a noticeable amount of manual tuning and intervention to deploy on hardware.', '1905.06077-1-5-3': 'This is not practically feasible.', '1905.06077-1-6-0': 'The principal challenge to be solved in using spine is that of control, namely, determining what actions should be applied over time in order to produce efficient and natural looking gaits.', '1905.06077-1-6-1': 'It is still an open question as to what type of control strategies can enhance locomotion performance with an active spine.', '1905.06077-1-6-2': 'Spine introduces redundancy, and developing a control strategy is often challenging and time consuming.', '1905.06077-1-6-3': 'Complexities arise due to nonlinear and coupled dynamics of legged systems, and from the existing trade-off between different performance criteria such as gait speed, energy efficiency, and stability.', '1905.06077-1-6-4': 'Therefore, most of the existing quadruped robots of today have completely avoided using spine by featuring a single rigid body with four legs with individually actuated hips and/or knees.', '1905.06077-1-7-0': 'We have previously explored Deep Reinforcement learning (D-RL) based methods to realize efficient quadrupedal walking in Stoch [CITATION], and our goal in this paper is to investigate the efficacy of D-RL for active spine behaviors.', '1905.06077-1-7-1': 'In other words, we want to revisit the problem of spine bounding with this new approach, which has seen a lot of success in recent years [CITATION].', '1905.06077-1-7-2': 'Therefore, in this work, we provide insights into how RL can be successfully applied to such problems.', '1905.06077-1-7-3': 'To the best of our knowledge, this is one of the earliest work which aims to learn complex control policy for a spined quadruped robot.', '1905.06077-1-7-4': 'We also evaluate numerous performance metrics to validate the advantage of spine.', '1905.06077-1-7-5': 'Also, we show that the final control policy produces actions that are both robust and efficient.', '1905.06077-1-8-0': 'We have arranged the paper in the following manner.', '1905.06077-1-8-1': 'Section [REF] describes the design of our simulated robot model and simulation framework used for the training.', '1905.06077-1-8-2': 'Section [REF] describes the D-RL framework used to learn the control policy.', '1905.06077-1-8-3': 'Finally, in section [REF], all the results obtained through various simulations are discussed thoroughly.', '1905.06077-1-9-0': '# Simulation model design', '1905.06077-1-10-0': 'Stoch 2 is a quadrupedal robot designed and developed in-house at the Indian Institute of Science (IISc), Bangalore, India.', '1905.06077-1-10-1': 'It is the second generation robot in the Stoch series.', '1905.06077-1-10-2': 'A detailed description of the previous model can be found in [CITATION].', '1905.06077-1-10-3': 'In this section, we first describe the overview of Stoch 2 robot and the essential hardware details, and then describe the simulation framework used for training.', '1905.06077-1-10-4': 'The scope of this work is limited to the simulation results portraying quadrupedal spine locomotion, whereas the hardware experiments are a part of our future work.', '1905.06077-1-11-0': 'The robot is designed as three modules: two body modules and one spine module.', '1905.06077-1-11-1': 'The body modules are connected via spine, as shown in Fig. [REF]a.', '1905.06077-1-11-2': 'The overall size and form factor of the robot is similar to Stoch [CITATION].', '1905.06077-1-11-3': 'Each body module is composed of two legs.', '1905.06077-1-11-4': 'A single leg contains three degrees of freedom.', '1905.06077-1-11-5': 'Each of them corresponds to the flexor and extension movement of hip, knee and abduction movements.', '1905.06077-1-11-6': 'However, the simulation model uses only hip and knee motion while keeping the abduction locked in position.', '1905.06077-1-11-7': 'Each leg comprises of a five bar linkage mechanism, where two of the links remain actuated.', '1905.06077-1-11-8': 'This enables the leg to follow a given trajectory in a plane.', '1905.06077-1-11-9': 'The central spine is designed as a serial [MATH] DOF mechanism.', '1905.06077-1-11-10': 'Each of the spine joint is actuated.', '1905.06077-1-11-11': 'Overall, the robot simulation model consists of [MATH] actuated degrees of freedom, including four legs and the spine.', '1905.06077-1-11-12': 'The key specifications of the simulation model are summarized in Table [REF].', '1905.06077-1-12-0': 'We used Pybullet [CITATION] simulator, built on top of Bullet3 physics engine.', '1905.06077-1-12-1': 'A three-dimensional computer-aided-design (CAD) model is developed using SolidWorks [CITATION] to capture the kinematics and inertial properties of the robot.', '1905.06077-1-12-2': 'This model is transferred to Pybullet by using a Universal Robot Description Format [CITATION] (URDF) exporter.', '1905.06077-1-12-3': 'In addition, actual mass of all the links, actuator force limits, joint limits and kinematic-loop constraints of the flexural joints are measured and manually updated in the URDF file for a more realistic simulation.', '1905.06077-1-13-0': '# Reinforcement Learning based Controller', '1905.06077-1-14-0': 'In this section, we will outline the deep reinforcement learning framework used for learning spine based locomotion behaviours.', '1905.06077-1-15-0': '## Background', '1905.06077-1-16-0': 'In reinforcement learning setting, the locomotion problem is formulated as a Markov Decision Process (MDP).', '1905.06077-1-16-1': 'An MDP is characterized by a tuple [MATH] where [MATH] is the set of robot states referred to as state space, and [MATH] is the set of feasible actions referred to as the action space.', '1905.06077-1-16-2': 'The transition probability function [MATH] models the evolution of states based on actions, and [MATH] is the scalar value at every transition step.', '1905.06077-1-16-3': '[MATH] is called the discount factor defined in the range [MATH].', '1905.06077-1-17-0': 'In reinforcement learning, the fundamental idea is to discover a policy, denoted as [MATH], that maximizes the expected cumulative reward over time.', '1905.06077-1-17-1': 'A parameterized policy [MATH] with the parameters [MATH] is the probability density of [MATH] given [MATH].', '1905.06077-1-17-2': 'The optimal parameters of the policy yield the maximum sum of the cumulative rewards given by [EQUATION]', '1905.06077-1-17-3': 'Policy gradients [CITATION] is one of the popular methods to solve this optimization problem which takes gradient ascent steps in the direction of increasing cumulative reward.', '1905.06077-1-17-4': 'We discuss more details about the algorithm used in Section [REF].', '1905.06077-1-18-0': '## State and Action Space', '1905.06077-1-19-0': '### State Space', '1905.06077-1-20-0': 'Similar to the work in [CITATION], the state is represented by angles, velocities, torques of the active joints (legs and spine), and body orientation (in quaternions).', '1905.06077-1-20-1': 'The combined representation yields a 34-D state space.', '1905.06077-1-21-0': '### Action Space', '1905.06077-1-22-0': 'The action space consists of the continuous-valued control signal for each active joint.', '1905.06077-1-22-1': "For each leg, the agent learns the legs' end-point positions in polar coordinates represented as [MATH] where [MATH].", '1905.06077-1-22-2': 'This particular choice of action space ensures that the five-bar leg mechanism never encounters a singularity.', '1905.06077-1-22-3': 'We use a custom inverse kinematics solver to compute the joint angles from polar coordinates.', '1905.06077-1-22-4': 'As seen from Fig [REF], the five bar linkage is divided into two 2R Serial linkage and solved for each branch.', '1905.06077-1-22-5': 'The details of the equations for a 2R Serial linkage can be found here [CITATION].', '1905.06077-1-22-6': 'Safety limits are also included in the inverse kinematic solver to avoid singular positions.', '1905.06077-1-22-7': 'However, the agent directly learns the joint angle [MATH] for the spine motors.', '1905.06077-1-22-8': 'The two motors in the spine are coupled with the relation [MATH] where [MATH] represents the joint angle.', '1905.06077-1-22-9': 'During the bound gait, we learn separate end-point trajectories for front and rear leg pairs.', '1905.06077-1-22-10': 'Note that both the legs in front and back module executes the same end-point trajectory during bound.', '1905.06077-1-22-11': 'The polar coordinates for the four legs and joint angle for the spine collectively provide a [MATH] dimensional action space.', '1905.06077-1-22-12': 'The polar coordinates and the spine angle are restricted to a bounding box, thereby indirectly imposing angle limits on the joint angles.', '1905.06077-1-23-0': '## Reward Function', '1905.06077-1-24-0': "We designed a reward function that gives more positive reinforcement as the robot's base speed attains a desired velocity, and simultaneously penalizes high energy consumption.", '1905.06077-1-24-1': 'The agent receives a scalar reward after each action step according to the reward function [EQUATION]', '1905.06077-1-24-2': 'Here [MATH] is the desired velocity along the [MATH]-axis, and [MATH] is manually adjusted for various values of [MATH].', '1905.06077-1-24-3': '[MATH] is the energy spent by the actuators for the current step, and [MATH], [MATH] are the weights corresponding to each term ([MATH] and [MATH] respectively in our experiments).', '1905.06077-1-24-4': '[MATH] is computed as [EQUATION] where [MATH] are the motor torques, and [MATH] are the motor velocities of the [MATH] motor respectively.', '1905.06077-1-25-0': '## Network and Learning Algorithm', '1905.06077-1-26-0': 'We are employing Policy Proximation Optimization (PPO) [CITATION] for learning optimal polices that yield action values in continuous space [CITATION], [CITATION].', '1905.06077-1-26-1': 'The actor and critic network in the learning algorithm consists of two fully connected layers with the first and second layers consisting of 128 and 64 nodes respectively.', '1905.06077-1-26-2': 'Activation units in the actor and critic networks are ReLU [MATH] ReLU [MATH], and ReLU [MATH] ReLU [MATH] linear respectively.', '1905.06077-1-26-3': 'We used the open source implementation of PPO by Tensorflow Agents [CITATION] that creates the symbolic representation of the computation graph.', '1905.06077-1-26-4': 'The implementation is highly parallelized and performs full-batch gradient ascent updates, using Adam [CITATION] optimizer, on the batch of data collected from multiple environment instances.', '1905.06077-1-26-5': 'The pybullet simulator is configured to spawn multiple agents for faster collection of samples and in our case 30 agents were used in parallel threads.', '1905.06077-1-26-6': 'The proposed approach yields walking gaits that are efficient, fast and also robust in simulation.', '1905.06077-1-26-7': 'The learning algorithm runs for a maximum of [MATH] million steps and the observed training time is [MATH] hours on a Intel Core i7 @3.5Ghz[MATH] cores and 32 GB RAM machine.', '1905.06077-1-27-0': 'Having obtained an optimal policy for bounding, we will now discuss the simulation results and also justify how the use of active spine is beneficial to locomotion.', '1905.06077-1-28-0': '# Results', '1905.06077-1-29-0': 'In this section, we will provide simulation results and also make comparisons between spine and rigid models.', '1905.06077-1-29-1': 'We trained multiple bounding gaits with target speeds ranging from [MATH] m/s and [MATH]m/s for both the models.', '1905.06077-1-29-2': 'The maximum speed obtained was [MATH]m/s with an active spine, and [MATH]m/s without spine.', '1905.06077-1-29-3': 'Figure [REF] shows the gait tiles for the target speed [MATH]m/s. Froude number observed was [MATH] at [MATH] body lengths per second.', '1905.06077-1-29-4': 'Figure [REF] shows the end point trajectories and spine angles for the same gait.', '1905.06077-1-29-5': 'We will analyze the gaits obtained based on traits like cost of transport, stride lengths, power, torque profiles etc.', '1905.06077-1-29-6': 'Video results are provided in the following link: https://youtu.be/INp4aa-8z2E.', '1905.06077-1-30-0': '## Cost of Transport (CoT)', '1905.06077-1-31-0': 'In our experiments, we use Cost of Transport (CoT) as a metric of measurement for the gait performance.', '1905.06077-1-31-1': 'CoT is nothing but the mechanical work normalized by the weight and the distance traveled.', '1905.06077-1-31-2': 'To determine CoT, we take the integral of the positive work done by the actuators [CITATION]: [EQUATION] where [MATH] is the torque of the motors.', '1905.06077-1-31-3': '[MATH] is speed of the motor.', '1905.06077-1-31-4': '[MATH] is the average distance covered by the robot along X-axis.', '1905.06077-1-31-5': '[MATH] is the mass of the robot.', '1905.06077-1-31-6': '[MATH] is the gravity.', '1905.06077-1-32-0': 'We pick [MATH] trials and then average the CoT computed over these trials for consistency.', '1905.06077-1-32-1': 'Figure [REF] shows the velocity profiles and center of mass height of the spine and rigid models.', '1905.06077-1-32-2': 'Fig. [REF] shows the CoT comparison between the rigid and spine models.', '1905.06077-1-32-3': 'It can be verified that CoT of the spine model is always less than that of the rigid model for the velocities shown.', '1905.06077-1-32-4': 'Moreover, the CoT for rigid models is almost twice as much as that for the spine models.', '1905.06077-1-32-5': 'This observation is concurrent with the results obtained in [CITATION].', '1905.06077-1-33-0': '## Stride Length Comparison', '1905.06077-1-34-0': 'Stride length provides a measure of how far the robot has walked during each step.', '1905.06077-1-34-1': 'It is the distance between two successive placements of the same foot.', '1905.06077-1-34-2': 'To measure the stride length, heel to heel distance is determined.', '1905.06077-1-34-3': 'It can be verified from Figure [REF] that there is a [MATH] increase in stride length due to active spine.', '1905.06077-1-34-4': 'Moreover, spine helps in reducing the bounding frequency [CITATION], thereby allowing the gaits to be more realistic.', '1905.06077-1-35-0': '## Torque, Power Profile and Gait Diagram', '1905.06077-1-36-0': 'In figure [REF], we compare the torque profile for one gait cycle of the bound gait with both active and rigid models, which are at nearly the same speeds.', '1905.06077-1-36-1': 'It can be observed that the peak torque values have depreciated for the active spine by nearly [MATH].', '1905.06077-1-36-2': 'Apart from this, it can be verified that the torque profile correlates well with the stance and swing phase of the robot.', '1905.06077-1-36-3': 'The average torque experienced by the front leg motors are higher, while the impulses experienced by the back legs were higher.', '1905.06077-1-37-0': '# Conclusion', '1905.06077-1-38-0': 'We showed that spine is a key driving factor for the improvement of speed and COT in quadrupeds.', '1905.06077-1-38-1': 'Unlike existing approaches, we used Deep-Reinforcement Learning to realize spine behaviors.', '1905.06077-1-38-2': 'It is worth noting that the training was from scratch, and did not require prior understanding of the mechanisms involved in using the spine.', '1905.06077-1-38-3': 'We have made comparisons with rigid models, and showed that spine improves cost of transport, power, torque, and stride lengths significantly.', '1905.06077-1-38-4': 'Future work will involve implementation of these policies on hardware.'}

{'1905.06077-2-0-0': 'In this work, we provide a simulation framework to perform systematic studies on the effects of spinal joint compliance and actuation on bounding performance of a [MATH]-DOF quadruped spined robot Stoch 2.', '1905.06077-2-0-1': 'Fast quadrupedal locomotion with active spine is an extremely hard problem, and involves a complex coordination between the various degrees of freedom.', '1905.06077-2-0-2': 'Therefore, past attempts at addressing this problem have not seen much success.', '1905.06077-2-0-3': 'Deep-Reinforcement Learning seems to be a promising approach, after its recent success in a variety of robot platforms, and the goal of this paper is to use this approach to realize the aforementioned behaviors.', '1905.06077-2-0-4': 'With this learning framework, the robot reached a bounding speed of [MATH] m/s with a maximum Froude number of [MATH].', '1905.06077-2-0-5': 'Simulation results also show that use of active spine, indeed, increased the stride length, improved the cost of transport, and also reduced the natural frequency to more realistic values.', '1905.06077-2-1-0': 'Key words: Deep-RL, Quadruped, Active spine', '1905.06077-2-2-0': '# Introduction', '1905.06077-2-3-0': 'Speed, for legged animals, is an important aspect in nature for survival.', '1905.06077-2-3-1': 'Animals like cheetah, hounds etc, use their spine to achieve remarkably high speed gaits.', '1905.06077-2-3-2': 'Studies on these animals show that the spine movement increases the effective stride length, provides auxiliary power to the legs and helps to harness energy by storing and releasing it.', '1905.06077-2-3-3': 'From the evolutionary point of view, it is natural to consider the spine as a propulsive engine of the vertebrate body.', '1905.06077-2-3-4': 'All of these observations point to the fact that spine is crucial for locomotion.', '1905.06077-2-4-0': 'There have been quite a few quadruped robots having passive or active spine in their structure, namely, BobCat, Sevel, MIT Cheetah, INU [CITATION].', '1905.06077-2-4-1': 'There is a large body of work over planer spine models with one DOF revolute [CITATION], and prismatic [CITATION] joints.', '1905.06077-2-4-2': 'Apart from the one DOF model, the spines were modeled as point masses in these works.', '1905.06077-2-4-3': 'On the other hand, robots such as Sevel [CITATION], INU [CITATION] contain a two DOF spine.', '1905.06077-2-4-4': 'This feature allows the spine to be longer while maintaining its orientation during bounding.', '1905.06077-2-4-5': 'However, reduced order models were used when constructing the empirical model of the robot.', '1905.06077-2-4-6': 'Moreover, in [CITATION], the legs were assumed to be massless, which result in large reality gaps.', '1905.06077-2-4-7': 'From a practical standpoint, we cannot use the approaches mentioned above, due to the fact that the spine models cannot be reduced or approximated, and the legs have significant masses (by upto [MATH]) in the real hardware.', '1905.06077-2-4-8': 'Apart from these, inaccuracy in modeling, imperfect ground contacts play a significant role while transferring on the real hardware.', '1905.06077-2-5-0': 'To address the above issues and to develop an accurate model of the robot and environment, researchers began to use high fidelity physics engine based simulators [CITATION] to simulate the robot and environment model.', '1905.06077-2-5-1': 'Works such as [CITATION] have pursued this approach to bridge the gap between theory and experiment.', '1905.06077-2-5-2': 'Despite these tools, the results obtained with these simulators still required a noticeable amount of manual tuning and intervention to deploy on hardware.', '1905.06077-2-5-3': 'This is not practically feasible.', '1905.06077-2-6-0': 'The principal challenge to be solved in using spine is that of control, namely, determining what actions should be applied over time in order to produce efficient and natural looking gaits.', '1905.06077-2-6-1': 'It is still an open question as to what type of control strategies can enhance locomotion performance with an active spine.', '1905.06077-2-6-2': 'Spine introduces redundancy, and developing a control strategy is often challenging and time consuming.', '1905.06077-2-6-3': 'Complexities arise due to nonlinear and coupled dynamics of legged systems, and from the existing trade-off between different performance criteria such as gait speed, energy efficiency, and stability.', '1905.06077-2-6-4': 'Therefore, most of the existing quadruped robots of today have completely avoided using spine by featuring a single rigid body with four legs with individually actuated hips and/or knees.', '1905.06077-2-7-0': 'We have previously explored Deep Reinforcement learning (D-RL) based methods to realize efficient quadrupedal walking in Stoch [CITATION], and our goal in this paper is to investigate the efficacy of D-RL for active spine behaviors.', '1905.06077-2-7-1': 'In other words, we want to revisit the problem of spine bounding with this new approach, which has seen a lot of success in recent years [CITATION].', '1905.06077-2-7-2': 'Therefore, in this work, we provide insights into how RL can be successfully applied to such problems.', '1905.06077-2-7-3': 'To the best of our knowledge, this is one of the earliest work which aims to learn complex control policy for a spined quadruped robot.', '1905.06077-2-7-4': 'We also evaluate numerous performance metrics to validate the advantage of spine.', '1905.06077-2-7-5': 'Also, we show that the final control policy produces actions that are both robust and efficient.', '1905.06077-2-8-0': 'We have arranged the paper in the following manner.', '1905.06077-2-8-1': 'Section [REF] describes the design of our simulated robot model and simulation framework used for the training.', '1905.06077-2-8-2': 'Section [REF] describes the D-RL framework used to learn the control policy.', '1905.06077-2-8-3': 'Finally, in section [REF], all the results obtained through various simulations are discussed thoroughly.', '1905.06077-2-9-0': '# Simulation model design', '1905.06077-2-10-0': 'Stoch 2 is a quadrupedal robot designed and developed in-house at the Indian Institute of Science (IISc), Bangalore, India.', '1905.06077-2-10-1': 'It is the second generation robot in the Stoch series.', '1905.06077-2-10-2': 'A detailed description of the previous model can be found in [CITATION].', '1905.06077-2-10-3': 'In this section, we first describe the overview of Stoch 2 robot and the essential hardware details, and then describe the simulation framework used for training.', '1905.06077-2-10-4': 'The scope of this work is limited to the simulation results portraying quadrupedal spine locomotion, whereas the hardware experiments are a part of our future work.', '1905.06077-2-11-0': 'The robot is designed as three modules: two body modules and one spine module.', '1905.06077-2-11-1': 'The body modules are connected via spine, as shown in Fig. [REF]a.', '1905.06077-2-11-2': 'The overall size and form factor of the robot is similar to Stoch [CITATION].', '1905.06077-2-11-3': 'Each body module is composed of two legs.', '1905.06077-2-11-4': 'A single leg contains three degrees of freedom.', '1905.06077-2-11-5': 'Each of them corresponds to the flexor and extension movement of hip, knee and abduction movements.', '1905.06077-2-11-6': 'However, the simulation model uses only hip and knee motion while keeping the abduction locked in position.', '1905.06077-2-11-7': 'Each leg comprises of a five bar linkage mechanism, where two of the links remain actuated.', '1905.06077-2-11-8': 'This enables the leg to follow a given trajectory in a plane.', '1905.06077-2-11-9': 'The central spine is designed as a serial [MATH] DOF mechanism.', '1905.06077-2-11-10': 'Each of the spine joint is actuated.', '1905.06077-2-11-11': 'Overall, the robot simulation model consists of [MATH] actuated degrees of freedom, including four legs and the spine.', '1905.06077-2-11-12': 'The key specifications of the simulation model are summarized in Table [REF].', '1905.06077-2-12-0': 'We used Pybullet [CITATION] simulator, built on top of Bullet3 physics engine.', '1905.06077-2-12-1': 'A three-dimensional computer-aided-design (CAD) model is developed using SolidWorks [CITATION] to capture the kinematics and inertial properties of the robot.', '1905.06077-2-12-2': 'This model is transferred to Pybullet by using a Universal Robot Description Format [CITATION] (URDF) exporter.', '1905.06077-2-12-3': 'In addition, actual mass of all the links, actuator force limits, joint limits and kinematic-loop constraints of the flexural joints are measured and manually updated in the URDF file for a more realistic simulation.', '1905.06077-2-13-0': '# Reinforcement Learning based Controller', '1905.06077-2-14-0': 'In this section, we will outline the deep reinforcement learning framework used for learning spine based locomotion behaviours.', '1905.06077-2-15-0': '## Background', '1905.06077-2-16-0': 'In reinforcement learning setting, the locomotion problem is formulated as a Markov Decision Process (MDP).', '1905.06077-2-16-1': 'An MDP is characterized by a tuple [MATH] where [MATH] is the set of robot states referred to as state space, and [MATH] is the set of feasible actions referred to as the action space.', '1905.06077-2-16-2': 'The transition probability function [MATH] models the evolution of states based on actions, and [MATH] is the scalar value at every transition step.', '1905.06077-2-16-3': '[MATH] is called the discount factor defined in the range [MATH].', '1905.06077-2-17-0': 'In reinforcement learning, the fundamental idea is to discover a policy, denoted as [MATH], that maximizes the expected cumulative reward over time.', '1905.06077-2-17-1': 'A parameterized policy [MATH] with the parameters [MATH] is the probability density of [MATH] given [MATH].', '1905.06077-2-17-2': 'The optimal parameters of the policy yield the maximum sum of the cumulative rewards given by [EQUATION]', '1905.06077-2-17-3': 'Policy gradients [CITATION] is one of the popular methods to solve this optimization problem which takes gradient ascent steps in the direction of increasing cumulative reward.', '1905.06077-2-17-4': 'We discuss more details about the algorithm used in Section [REF].', '1905.06077-2-18-0': '## State and Action Space', '1905.06077-2-19-0': '### State Space', '1905.06077-2-20-0': 'Similar to the work in [CITATION], the state is represented by angles, velocities, torques of the active joints (legs and spine), and body orientation (in quaternions).', '1905.06077-2-20-1': 'The combined representation yields a 34-D state space.', '1905.06077-2-21-0': '### Action Space', '1905.06077-2-22-0': 'The action space consists of the continuous-valued control signal for each active joint.', '1905.06077-2-22-1': "For each leg, the agent learns the legs' end-point positions in polar coordinates represented as [MATH] where [MATH].", '1905.06077-2-22-2': 'This particular choice of action space ensures that the five-bar leg mechanism never encounters a singularity.', '1905.06077-2-22-3': 'We use a custom inverse kinematics solver to compute the joint angles from polar coordinates.', '1905.06077-2-22-4': 'As seen from Fig [REF], the five bar linkage is divided into two 2R Serial linkage and solved for each branch.', '1905.06077-2-22-5': 'The details of the equations for a 2R Serial linkage can be found here [CITATION].', '1905.06077-2-22-6': 'Safety limits are also included in the inverse kinematic solver to avoid singular positions.', '1905.06077-2-22-7': 'However, the agent directly learns the joint angle [MATH] for the spine motors.', '1905.06077-2-22-8': 'The two motors in the spine are coupled with the relation [MATH] where [MATH] represents the joint angle.', '1905.06077-2-22-9': 'During the bound gait, we learn separate end-point trajectories for front and rear leg pairs.', '1905.06077-2-22-10': 'Note that both the legs in front and back module executes the same end-point trajectory during bound.', '1905.06077-2-22-11': 'The polar coordinates for the four legs and joint angle for the spine collectively provide a [MATH] dimensional action space.', '1905.06077-2-22-12': 'The polar coordinates and the spine angle are restricted to a bounding box, thereby indirectly imposing angle limits on the joint angles.', '1905.06077-2-23-0': '## Reward Function', '1905.06077-2-24-0': "We designed a reward function that gives more positive reinforcement as the robot's base speed attains a desired velocity, and simultaneously penalizes high energy consumption.", '1905.06077-2-24-1': 'The agent receives a scalar reward after each action step according to the reward function [EQUATION]', '1905.06077-2-24-2': 'Here [MATH] is the desired velocity along the [MATH]-axis, and [MATH] is manually adjusted for various values of [MATH].', '1905.06077-2-24-3': '[MATH] is the energy spent by the actuators for the current step, and [MATH], [MATH] are the weights corresponding to each term ([MATH] and [MATH] respectively in our experiments).', '1905.06077-2-24-4': '[MATH] is computed as [EQUATION] where [MATH] are the motor torques, and [MATH] are the motor velocities of the [MATH] motor respectively.', '1905.06077-2-25-0': '## Network and Learning Algorithm', '1905.06077-2-26-0': 'We are employing Proximal Policy Optimization (PPO) [CITATION] for learning optimal polices that yield action values in continuous space [CITATION], [CITATION].', '1905.06077-2-26-1': 'The actor and critic network in the learning algorithm consists of two fully connected layers with the first and second layers consisting of 128 and 64 nodes respectively.', '1905.06077-2-26-2': 'Activation units in the actor and critic networks are ReLU [MATH] ReLU [MATH], and ReLU [MATH] ReLU [MATH] linear respectively.', '1905.06077-2-26-3': 'We used the open source implementation of PPO by Tensorflow Agents [CITATION] that creates the symbolic representation of the computation graph.', '1905.06077-2-26-4': 'The implementation is highly parallelized and performs full-batch gradient ascent updates, using Adam [CITATION] optimizer, on the batch of data collected from multiple environment instances.', '1905.06077-2-26-5': 'The pybullet simulator is configured to spawn multiple agents for faster collection of samples and in our case 30 agents were used in parallel threads.', '1905.06077-2-26-6': 'The proposed approach yields walking gaits that are efficient, fast and also robust in simulation.', '1905.06077-2-26-7': 'The learning algorithm runs for a maximum of [MATH] million steps and the observed training time is [MATH] hours on a Intel Core i7 @3.5Ghz[MATH] cores and 32 GB RAM machine.', '1905.06077-2-27-0': 'Having obtained an optimal policy for bounding, we will now discuss the simulation results and also justify how the use of active spine is beneficial to locomotion.', '1905.06077-2-28-0': '# Results', '1905.06077-2-29-0': 'In this section, we will provide simulation results and also make comparisons between spine and rigid models.', '1905.06077-2-29-1': 'We trained multiple bounding gaits with target speeds ranging from [MATH] m/s and [MATH]m/s for both the models.', '1905.06077-2-29-2': 'The maximum speed obtained was [MATH]m/s with an active spine, and [MATH]m/s without spine.', '1905.06077-2-29-3': 'Figure [REF] shows the gait tiles for the target speed [MATH]m/s. Froude number observed was [MATH] at [MATH] body lengths per second.', '1905.06077-2-29-4': 'Figure [REF] shows the end point trajectories and spine angles for the same gait.', '1905.06077-2-29-5': 'We will analyze the gaits obtained based on traits like cost of transport, stride lengths, power, torque profiles etc.', '1905.06077-2-29-6': 'Video results are provided in the following link: https://youtu.be/INp4aa-8z2E.', '1905.06077-2-30-0': '## Cost of Transport (CoT)', '1905.06077-2-31-0': 'In our experiments, we use Cost of Transport (CoT) as a metric of measurement for the gait performance.', '1905.06077-2-31-1': 'CoT is nothing but the mechanical work normalized by the weight and the distance traveled.', '1905.06077-2-31-2': 'To determine CoT, we take the integral of the positive work done by the actuators [CITATION]: [EQUATION] where [MATH] is the torque of the motors.', '1905.06077-2-31-3': '[MATH] is speed of the motor.', '1905.06077-2-31-4': '[MATH] is the average distance covered by the robot along X-axis.', '1905.06077-2-31-5': '[MATH] is the mass of the robot.', '1905.06077-2-31-6': '[MATH] is the gravity.', '1905.06077-2-32-0': 'We pick [MATH] trials and then average the CoT computed over these trials for consistency.', '1905.06077-2-32-1': 'Figure [REF] shows the velocity profiles and center of mass height of the spine and rigid models.', '1905.06077-2-32-2': 'Fig. [REF] shows the CoT comparison between the rigid and spine models.', '1905.06077-2-32-3': 'It can be verified that CoT of the spine model is always less than that of the rigid model for the velocities shown.', '1905.06077-2-32-4': 'Moreover, the CoT for rigid models is almost twice as much as that for the spine models.', '1905.06077-2-32-5': 'This observation is concurrent with the results obtained in [CITATION].', '1905.06077-2-33-0': '## Stride Length Comparison', '1905.06077-2-34-0': 'Stride length provides a measure of how far the robot has walked during each step.', '1905.06077-2-34-1': 'It is the distance between two successive placements of the same foot.', '1905.06077-2-34-2': 'To measure the stride length, heel to heel distance is determined.', '1905.06077-2-34-3': 'It can be verified from Figure [REF] that there is a [MATH] increase in stride length due to active spine.', '1905.06077-2-34-4': 'Moreover, spine helps in reducing the bounding frequency [CITATION], thereby allowing the gaits to be more realistic.', '1905.06077-2-35-0': '## Torque, Power Profile and Gait Diagram', '1905.06077-2-36-0': 'In figure [REF], we compare the torque profile for one gait cycle of the bound gait with both active and rigid models, which are at nearly the same speeds.', '1905.06077-2-36-1': 'It can be observed that the peak torque values have depreciated for the active spine by nearly [MATH].', '1905.06077-2-36-2': 'Apart from this, it can be verified that the torque profile correlates well with the stance and swing phase of the robot.', '1905.06077-2-36-3': 'The average torque experienced by the front leg motors are higher, while the impulses experienced by the back legs were higher.', '1905.06077-2-37-0': '# Conclusion', '1905.06077-2-38-0': 'We showed that spine is a key driving factor for the improvement of speed and COT in quadrupeds.', '1905.06077-2-38-1': 'Unlike existing approaches, we used Deep-Reinforcement Learning to realize spine behaviors.', '1905.06077-2-38-2': 'It is worth noting that the training was from scratch, and did not require prior understanding of the mechanisms involved in using the spine.', '1905.06077-2-38-3': 'We have made comparisons with rigid models, and showed that spine improves cost of transport, power, torque, and stride lengths significantly.', '1905.06077-2-38-4': 'Future work will involve implementation of these policies on hardware.'}

[['1905.06077-1-6-0', '1905.06077-2-6-0'], ['1905.06077-1-6-1', '1905.06077-2-6-1'], ['1905.06077-1-6-2', '1905.06077-2-6-2'], ['1905.06077-1-6-3', '1905.06077-2-6-3'], ['1905.06077-1-6-4', '1905.06077-2-6-4'], ['1905.06077-1-24-0', '1905.06077-2-24-0'], ['1905.06077-1-24-1', '1905.06077-2-24-1'], ['1905.06077-1-24-2', '1905.06077-2-24-2'], ['1905.06077-1-24-3', '1905.06077-2-24-3'], ['1905.06077-1-24-4', '1905.06077-2-24-4'], ['1905.06077-1-0-0', '1905.06077-2-0-0'], ['1905.06077-1-0-1', '1905.06077-2-0-1'], ['1905.06077-1-0-2', '1905.06077-2-0-2'], ['1905.06077-1-0-3', '1905.06077-2-0-3'], ['1905.06077-1-0-4', '1905.06077-2-0-4'], ['1905.06077-1-0-5', '1905.06077-2-0-5'], ['1905.06077-1-10-0', '1905.06077-2-10-0'], ['1905.06077-1-10-1', '1905.06077-2-10-1'], ['1905.06077-1-10-2', '1905.06077-2-10-2'], ['1905.06077-1-10-3', '1905.06077-2-10-3'], ['1905.06077-1-10-4', '1905.06077-2-10-4'], ['1905.06077-1-11-0', '1905.06077-2-11-0'], ['1905.06077-1-11-1', '1905.06077-2-11-1'], ['1905.06077-1-11-2', '1905.06077-2-11-2'], ['1905.06077-1-11-3', '1905.06077-2-11-3'], ['1905.06077-1-11-4', '1905.06077-2-11-4'], ['1905.06077-1-11-5', '1905.06077-2-11-5'], ['1905.06077-1-11-6', '1905.06077-2-11-6'], ['1905.06077-1-11-7', '1905.06077-2-11-7'], ['1905.06077-1-11-8', '1905.06077-2-11-8'], ['1905.06077-1-11-9', '1905.06077-2-11-9'], ['1905.06077-1-11-10', '1905.06077-2-11-10'], ['1905.06077-1-11-11', '1905.06077-2-11-11'], ['1905.06077-1-11-12', '1905.06077-2-11-12'], ['1905.06077-1-32-0', '1905.06077-2-32-0'], ['1905.06077-1-32-1', '1905.06077-2-32-1'], ['1905.06077-1-32-2', '1905.06077-2-32-2'], ['1905.06077-1-32-3', '1905.06077-2-32-3'], ['1905.06077-1-32-4', '1905.06077-2-32-4'], ['1905.06077-1-32-5', '1905.06077-2-32-5'], ['1905.06077-1-16-0', '1905.06077-2-16-0'], ['1905.06077-1-16-1', '1905.06077-2-16-1'], ['1905.06077-1-16-2', '1905.06077-2-16-2'], ['1905.06077-1-16-3', '1905.06077-2-16-3'], ['1905.06077-1-4-0', '1905.06077-2-4-0'], ['1905.06077-1-4-1', '1905.06077-2-4-1'], ['1905.06077-1-4-2', '1905.06077-2-4-2'], ['1905.06077-1-4-3', '1905.06077-2-4-3'], ['1905.06077-1-4-4', '1905.06077-2-4-4'], ['1905.06077-1-4-5', '1905.06077-2-4-5'], ['1905.06077-1-4-6', '1905.06077-2-4-6'], ['1905.06077-1-4-7', '1905.06077-2-4-7'], ['1905.06077-1-4-8', '1905.06077-2-4-8'], ['1905.06077-1-31-0', '1905.06077-2-31-0'], ['1905.06077-1-31-1', '1905.06077-2-31-1'], ['1905.06077-1-31-2', '1905.06077-2-31-2'], ['1905.06077-1-31-3', '1905.06077-2-31-3'], ['1905.06077-1-31-4', '1905.06077-2-31-4'], ['1905.06077-1-31-5', '1905.06077-2-31-5'], ['1905.06077-1-31-6', '1905.06077-2-31-6'], ['1905.06077-1-17-0', '1905.06077-2-17-0'], ['1905.06077-1-17-1', '1905.06077-2-17-1'], ['1905.06077-1-17-2', '1905.06077-2-17-2'], ['1905.06077-1-17-3', '1905.06077-2-17-3'], ['1905.06077-1-17-4', '1905.06077-2-17-4'], ['1905.06077-1-27-0', '1905.06077-2-27-0'], ['1905.06077-1-38-0', '1905.06077-2-38-0'], ['1905.06077-1-38-1', '1905.06077-2-38-1'], ['1905.06077-1-38-2', '1905.06077-2-38-2'], ['1905.06077-1-38-3', '1905.06077-2-38-3'], ['1905.06077-1-38-4', '1905.06077-2-38-4'], ['1905.06077-1-12-0', '1905.06077-2-12-0'], ['1905.06077-1-12-1', '1905.06077-2-12-1'], ['1905.06077-1-12-2', '1905.06077-2-12-2'], ['1905.06077-1-12-3', '1905.06077-2-12-3'], ['1905.06077-1-8-0', '1905.06077-2-8-0'], ['1905.06077-1-8-1', '1905.06077-2-8-1'], ['1905.06077-1-8-2', '1905.06077-2-8-2'], ['1905.06077-1-8-3', '1905.06077-2-8-3'], ['1905.06077-1-26-1', '1905.06077-2-26-1'], ['1905.06077-1-26-2', '1905.06077-2-26-2'], ['1905.06077-1-26-3', '1905.06077-2-26-3'], ['1905.06077-1-26-4', '1905.06077-2-26-4'], ['1905.06077-1-26-5', '1905.06077-2-26-5'], ['1905.06077-1-26-6', '1905.06077-2-26-6'], ['1905.06077-1-26-7', '1905.06077-2-26-7'], ['1905.06077-1-3-0', '1905.06077-2-3-0'], ['1905.06077-1-3-1', '1905.06077-2-3-1'], ['1905.06077-1-3-2', '1905.06077-2-3-2'], ['1905.06077-1-3-3', '1905.06077-2-3-3'], ['1905.06077-1-3-4', '1905.06077-2-3-4'], ['1905.06077-1-36-0', '1905.06077-2-36-0'], ['1905.06077-1-36-1', '1905.06077-2-36-1'], ['1905.06077-1-36-2', '1905.06077-2-36-2'], ['1905.06077-1-36-3', '1905.06077-2-36-3'], ['1905.06077-1-5-0', '1905.06077-2-5-0'], ['1905.06077-1-5-1', '1905.06077-2-5-1'], ['1905.06077-1-5-2', '1905.06077-2-5-2'], ['1905.06077-1-5-3', '1905.06077-2-5-3'], ['1905.06077-1-14-0', '1905.06077-2-14-0'], ['1905.06077-1-29-0', '1905.06077-2-29-0'], ['1905.06077-1-29-1', '1905.06077-2-29-1'], ['1905.06077-1-29-2', '1905.06077-2-29-2'], ['1905.06077-1-29-3', '1905.06077-2-29-3'], ['1905.06077-1-29-4', '1905.06077-2-29-4'], ['1905.06077-1-29-5', '1905.06077-2-29-5'], ['1905.06077-1-29-6', '1905.06077-2-29-6'], ['1905.06077-1-7-0', '1905.06077-2-7-0'], ['1905.06077-1-7-1', '1905.06077-2-7-1'], ['1905.06077-1-7-2', '1905.06077-2-7-2'], ['1905.06077-1-7-3', '1905.06077-2-7-3'], ['1905.06077-1-7-4', '1905.06077-2-7-4'], ['1905.06077-1-7-5', '1905.06077-2-7-5'], ['1905.06077-1-20-0', '1905.06077-2-20-0'], ['1905.06077-1-20-1', '1905.06077-2-20-1'], ['1905.06077-1-22-0', '1905.06077-2-22-0'], ['1905.06077-1-22-1', '1905.06077-2-22-1'], ['1905.06077-1-22-2', '1905.06077-2-22-2'], ['1905.06077-1-22-3', '1905.06077-2-22-3'], ['1905.06077-1-22-4', '1905.06077-2-22-4'], ['1905.06077-1-22-5', '1905.06077-2-22-5'], ['1905.06077-1-22-6', '1905.06077-2-22-6'], ['1905.06077-1-22-7', '1905.06077-2-22-7'], ['1905.06077-1-22-8', '1905.06077-2-22-8'], ['1905.06077-1-22-9', '1905.06077-2-22-9'], ['1905.06077-1-22-10', '1905.06077-2-22-10'], ['1905.06077-1-22-11', '1905.06077-2-22-11'], ['1905.06077-1-22-12', '1905.06077-2-22-12'], ['1905.06077-1-34-0', '1905.06077-2-34-0'], ['1905.06077-1-34-1', '1905.06077-2-34-1'], ['1905.06077-1-34-2', '1905.06077-2-34-2'], ['1905.06077-1-34-3', '1905.06077-2-34-3'], ['1905.06077-1-34-4', '1905.06077-2-34-4'], ['1905.06077-1-26-0', '1905.06077-2-26-0']]

[['1905.06077-1-6-0', '1905.06077-2-6-0'], ['1905.06077-1-6-1', '1905.06077-2-6-1'], ['1905.06077-1-6-2', '1905.06077-2-6-2'], ['1905.06077-1-6-3', '1905.06077-2-6-3'], ['1905.06077-1-6-4', '1905.06077-2-6-4'], ['1905.06077-1-24-0', '1905.06077-2-24-0'], ['1905.06077-1-24-1', '1905.06077-2-24-1'], ['1905.06077-1-24-2', '1905.06077-2-24-2'], ['1905.06077-1-24-3', '1905.06077-2-24-3'], ['1905.06077-1-24-4', '1905.06077-2-24-4'], ['1905.06077-1-0-0', '1905.06077-2-0-0'], ['1905.06077-1-0-1', '1905.06077-2-0-1'], ['1905.06077-1-0-2', '1905.06077-2-0-2'], ['1905.06077-1-0-3', '1905.06077-2-0-3'], ['1905.06077-1-0-4', '1905.06077-2-0-4'], ['1905.06077-1-0-5', '1905.06077-2-0-5'], ['1905.06077-1-10-0', '1905.06077-2-10-0'], ['1905.06077-1-10-1', '1905.06077-2-10-1'], ['1905.06077-1-10-2', '1905.06077-2-10-2'], ['1905.06077-1-10-3', '1905.06077-2-10-3'], ['1905.06077-1-10-4', '1905.06077-2-10-4'], ['1905.06077-1-11-0', '1905.06077-2-11-0'], ['1905.06077-1-11-1', '1905.06077-2-11-1'], ['1905.06077-1-11-2', '1905.06077-2-11-2'], ['1905.06077-1-11-3', '1905.06077-2-11-3'], ['1905.06077-1-11-4', '1905.06077-2-11-4'], ['1905.06077-1-11-5', '1905.06077-2-11-5'], ['1905.06077-1-11-6', '1905.06077-2-11-6'], ['1905.06077-1-11-7', '1905.06077-2-11-7'], ['1905.06077-1-11-8', '1905.06077-2-11-8'], ['1905.06077-1-11-9', '1905.06077-2-11-9'], ['1905.06077-1-11-10', '1905.06077-2-11-10'], ['1905.06077-1-11-11', '1905.06077-2-11-11'], ['1905.06077-1-11-12', '1905.06077-2-11-12'], ['1905.06077-1-32-0', '1905.06077-2-32-0'], ['1905.06077-1-32-1', '1905.06077-2-32-1'], ['1905.06077-1-32-2', '1905.06077-2-32-2'], ['1905.06077-1-32-3', '1905.06077-2-32-3'], ['1905.06077-1-32-4', '1905.06077-2-32-4'], ['1905.06077-1-32-5', '1905.06077-2-32-5'], ['1905.06077-1-16-0', '1905.06077-2-16-0'], ['1905.06077-1-16-1', '1905.06077-2-16-1'], ['1905.06077-1-16-2', '1905.06077-2-16-2'], ['1905.06077-1-16-3', '1905.06077-2-16-3'], ['1905.06077-1-4-0', '1905.06077-2-4-0'], ['1905.06077-1-4-1', '1905.06077-2-4-1'], ['1905.06077-1-4-2', '1905.06077-2-4-2'], ['1905.06077-1-4-3', '1905.06077-2-4-3'], ['1905.06077-1-4-4', '1905.06077-2-4-4'], ['1905.06077-1-4-5', '1905.06077-2-4-5'], ['1905.06077-1-4-6', '1905.06077-2-4-6'], ['1905.06077-1-4-7', '1905.06077-2-4-7'], ['1905.06077-1-4-8', '1905.06077-2-4-8'], ['1905.06077-1-31-0', '1905.06077-2-31-0'], ['1905.06077-1-31-1', '1905.06077-2-31-1'], ['1905.06077-1-31-2', '1905.06077-2-31-2'], ['1905.06077-1-31-3', '1905.06077-2-31-3'], ['1905.06077-1-31-4', '1905.06077-2-31-4'], ['1905.06077-1-31-5', '1905.06077-2-31-5'], ['1905.06077-1-31-6', '1905.06077-2-31-6'], ['1905.06077-1-17-0', '1905.06077-2-17-0'], ['1905.06077-1-17-1', '1905.06077-2-17-1'], ['1905.06077-1-17-2', '1905.06077-2-17-2'], ['1905.06077-1-17-3', '1905.06077-2-17-3'], ['1905.06077-1-17-4', '1905.06077-2-17-4'], ['1905.06077-1-27-0', '1905.06077-2-27-0'], ['1905.06077-1-38-0', '1905.06077-2-38-0'], ['1905.06077-1-38-1', '1905.06077-2-38-1'], ['1905.06077-1-38-2', '1905.06077-2-38-2'], ['1905.06077-1-38-3', '1905.06077-2-38-3'], ['1905.06077-1-38-4', '1905.06077-2-38-4'], ['1905.06077-1-12-0', '1905.06077-2-12-0'], ['1905.06077-1-12-1', '1905.06077-2-12-1'], ['1905.06077-1-12-2', '1905.06077-2-12-2'], ['1905.06077-1-12-3', '1905.06077-2-12-3'], ['1905.06077-1-8-0', '1905.06077-2-8-0'], ['1905.06077-1-8-1', '1905.06077-2-8-1'], ['1905.06077-1-8-2', '1905.06077-2-8-2'], ['1905.06077-1-8-3', '1905.06077-2-8-3'], ['1905.06077-1-26-1', '1905.06077-2-26-1'], ['1905.06077-1-26-2', '1905.06077-2-26-2'], ['1905.06077-1-26-3', '1905.06077-2-26-3'], ['1905.06077-1-26-4', '1905.06077-2-26-4'], ['1905.06077-1-26-5', '1905.06077-2-26-5'], ['1905.06077-1-26-6', '1905.06077-2-26-6'], ['1905.06077-1-26-7', '1905.06077-2-26-7'], ['1905.06077-1-3-0', '1905.06077-2-3-0'], ['1905.06077-1-3-1', '1905.06077-2-3-1'], ['1905.06077-1-3-2', '1905.06077-2-3-2'], ['1905.06077-1-3-3', '1905.06077-2-3-3'], ['1905.06077-1-3-4', '1905.06077-2-3-4'], ['1905.06077-1-36-0', '1905.06077-2-36-0'], ['1905.06077-1-36-1', '1905.06077-2-36-1'], ['1905.06077-1-36-2', '1905.06077-2-36-2'], ['1905.06077-1-36-3', '1905.06077-2-36-3'], ['1905.06077-1-5-0', '1905.06077-2-5-0'], ['1905.06077-1-5-1', '1905.06077-2-5-1'], ['1905.06077-1-5-2', '1905.06077-2-5-2'], ['1905.06077-1-5-3', '1905.06077-2-5-3'], ['1905.06077-1-14-0', '1905.06077-2-14-0'], ['1905.06077-1-29-0', '1905.06077-2-29-0'], ['1905.06077-1-29-1', '1905.06077-2-29-1'], ['1905.06077-1-29-2', '1905.06077-2-29-2'], ['1905.06077-1-29-3', '1905.06077-2-29-3'], ['1905.06077-1-29-4', '1905.06077-2-29-4'], ['1905.06077-1-29-5', '1905.06077-2-29-5'], ['1905.06077-1-29-6', '1905.06077-2-29-6'], ['1905.06077-1-7-0', '1905.06077-2-7-0'], ['1905.06077-1-7-1', '1905.06077-2-7-1'], ['1905.06077-1-7-2', '1905.06077-2-7-2'], ['1905.06077-1-7-3', '1905.06077-2-7-3'], ['1905.06077-1-7-4', '1905.06077-2-7-4'], ['1905.06077-1-7-5', '1905.06077-2-7-5'], ['1905.06077-1-20-0', '1905.06077-2-20-0'], ['1905.06077-1-20-1', '1905.06077-2-20-1'], ['1905.06077-1-22-0', '1905.06077-2-22-0'], ['1905.06077-1-22-1', '1905.06077-2-22-1'], ['1905.06077-1-22-2', '1905.06077-2-22-2'], ['1905.06077-1-22-3', '1905.06077-2-22-3'], ['1905.06077-1-22-4', '1905.06077-2-22-4'], ['1905.06077-1-22-5', '1905.06077-2-22-5'], ['1905.06077-1-22-6', '1905.06077-2-22-6'], ['1905.06077-1-22-7', '1905.06077-2-22-7'], ['1905.06077-1-22-8', '1905.06077-2-22-8'], ['1905.06077-1-22-9', '1905.06077-2-22-9'], ['1905.06077-1-22-10', '1905.06077-2-22-10'], ['1905.06077-1-22-11', '1905.06077-2-22-11'], ['1905.06077-1-22-12', '1905.06077-2-22-12'], ['1905.06077-1-34-0', '1905.06077-2-34-0'], ['1905.06077-1-34-1', '1905.06077-2-34-1'], ['1905.06077-1-34-2', '1905.06077-2-34-2'], ['1905.06077-1-34-3', '1905.06077-2-34-3'], ['1905.06077-1-34-4', '1905.06077-2-34-4']]

[['1905.06077-1-26-0', '1905.06077-2-26-0']]

[]

[]

[]

['1905.06077-1-1-0', '1905.06077-2-1-0']

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

https://arxiv.org/abs/1905.06077

1512.04485

{'1512.04485-1-0-0': 'We generalize the definition of Yang-Baxter basis of type [MATH] Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math.', '1512.04485-1-0-1': 'Phys., 40 (1997), 75-90) to all the Lie types and prove their duality.', '1512.04485-1-0-2': "As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of [MATH]-adic groups.", '1512.04485-1-1-0': '# Introduction', '1512.04485-1-2-0': 'Yang-Baxter basis of Hecke algebra of type [MATH] was defined in the paper of Lascoux-Leclerc-Thibon [CITATION].', '1512.04485-1-2-1': 'There is also a modified version in [CITATION].', '1512.04485-1-2-2': 'First we generalize the latter version to all the Lie types.', '1512.04485-1-2-3': "Then we will solve the Casselman's problem on the basis of Iwahori fixed vectors using Yang-Baxter basis and Demazure-Lusztig type operator.", '1512.04485-1-2-4': 'This paper is an extended abstract and the detailed proofs will appear in [CITATION].', '1512.04485-1-3-0': '# Generic Hecke algebra', '1512.04485-1-4-0': '## Root system, Weyl group and generic Hecke algebra', '1512.04485-1-5-0': 'Let [MATH] be a (reduced) semisimple root data cf. [CITATION].', '1512.04485-1-5-1': 'More precisely [MATH] is a weight lattice with [MATH].', '1512.04485-1-5-2': 'There is a pairing [MATH].', '1512.04485-1-5-3': '[MATH] is a root system with simple roots [MATH] and positive roots [MATH].', '1512.04485-1-5-4': '[MATH] is the set of coroots, and there is a bijection [MATH], [MATH].', '1512.04485-1-5-5': 'We also denote the coroot [MATH].', '1512.04485-1-5-6': 'Weyl group [MATH] is generated by simple reflections [MATH].', '1512.04485-1-5-7': 'The action of [MATH] on [MATH] is given by [MATH] for [MATH].', '1512.04485-1-5-8': 'We define generic Hecke algebra [MATH] over [MATH] with two parameters [MATH] as follows.', '1512.04485-1-5-9': 'Generators are [MATH], with relations [MATH] for [MATH] and the braid relations [MATH] , where [MATH] is the order of [MATH] for [MATH].', '1512.04485-1-5-10': 'We need to extend the coefficients to the quotient field of the group algebra [MATH].', '1512.04485-1-5-11': 'An element of [MATH] is denoted as [MATH].', '1512.04485-1-5-12': 'The Weyl group acts on [MATH] by [MATH].', '1512.04485-1-5-13': 'We extend the coefficient ring [MATH] of [MATH] to [EQUATION] where [MATH] is the quotient field of [MATH].', '1512.04485-1-5-14': '[EQUATION].', '1512.04485-1-5-15': 'Then [MATH] is a [MATH]-basis.', '1512.04485-1-6-0': '## Yang-Baxter basis and its properties', '1512.04485-1-7-0': 'Yang-Baxter basis was introduced in the paper [CITATION] to investigate the relation with Schubert calculus.', '1512.04485-1-7-1': 'There is also a variant in [CITATION] for type [MATH] case.', '1512.04485-1-7-2': 'We generalize that results to all Lie types.', '1512.04485-1-8-0': 'For [MATH], we define [MATH].', '1512.04485-1-8-1': 'Then [MATH].', '1512.04485-1-8-2': 'In particuar, if [MATH], [MATH].', '1512.04485-1-9-0': 'For [MATH], if [MATH], let [MATH].', '1512.04485-1-9-1': 'Then these satisfy the Yang-Baxter relations, i.e. if we write [MATH] for fixed [MATH], the following equations hold.', '1512.04485-1-9-2': 'We assume all appearance of [MATH] is nonzero.', '1512.04485-1-9-3': '[EQUATION]', '1512.04485-1-9-4': 'We can prove these equations by direct calculations.', '1512.04485-1-10-0': 'Following [CITATION] we define the Yang-Baxter basis [MATH] for [MATH] recursively as follows.', '1512.04485-1-10-1': 'We use the Bruhat order [MATH] on [MATH] (cf.[CITATION]).', '1512.04485-1-11-0': '[MATH], [MATH] if [MATH].', '1512.04485-1-12-0': 'Using the Yang-Baxter relation above it is easy to see that [MATH] does not depend on a reduced decomposition of [MATH].', '1512.04485-1-12-1': 'As the leading term of [MATH] with respect to the Bruhat order is [MATH], they also form a [MATH]-basis [MATH] of [MATH].', '1512.04485-1-12-2': 'We are interested in the transition coefficients [MATH] between the two basis [MATH] and [MATH] , i.e. [EQUATION].', '1512.04485-1-13-0': 'Take a reduced expression of [MATH] e.g. [MATH] where [MATH] is the length of [MATH] (cf. [CITATION]).', '1512.04485-1-13-1': 'Then [MATH] is expressed as follows.', '1512.04485-1-13-2': '[EQUATION] where [MATH] for [MATH].', '1512.04485-1-13-3': 'The set [MATH] is independent of the reduced decomposition of [MATH].', '1512.04485-1-13-4': 'The Yang-Baxter basis defined in [CITATION] is normalized as follows.', '1512.04485-1-13-5': '[EQUATION].', '1512.04485-1-14-0': 'The relation to [MATH]-theory Schubert calculus is as follows.', '1512.04485-1-14-1': 'If we set [MATH] and replacing [MATH] by [MATH], the coefficient of [MATH] in [MATH] is the localization [MATH] at [MATH] of the equivariant [MATH]-theory Schubert class [MATH] (cf. [CITATION]).', '1512.04485-1-15-0': 'Let [MATH] be the longest element in [MATH].', '1512.04485-1-15-1': 'Define [MATH]-algebra homomorphism [MATH] by [MATH].', '1512.04485-1-15-2': 'Let [MATH] be the ring homomorphism on [MATH] induced by [MATH] and extend to [MATH].', '1512.04485-1-16-0': '(Lascoux [CITATION] Lemma 1.8.1 for type [MATH] case) For [MATH], [EQUATION] where [MATH] acts only on the coefficients.', '1512.04485-1-17-0': 'When [MATH] there exists [MATH] such that [MATH].', '1512.04485-1-17-1': 'Using the induction assumption on [MATH], we get the formula for [MATH].', '1512.04485-1-18-0': 'Taking the coefficient of [MATH] in the above equation, we get [EQUATION]', '1512.04485-1-19-0': '## Inner product and orthogonality', '1512.04485-1-20-0': 'Define inner product [MATH] on [MATH] by [MATH] in [MATH], where [MATH] if [MATH].', '1512.04485-1-20-1': 'It is easy to see that [MATH] for [MATH] and [MATH].', '1512.04485-1-20-2': 'There is an involution [MATH] defined by [MATH].', '1512.04485-1-20-3': 'It is easy to see that [MATH] for [MATH].', '1512.04485-1-21-0': 'The following proposition is due to A.Lascoux for the type [MATH] case [CITATION] P.33.', '1512.04485-1-22-0': 'For all [MATH], [EQUATION].', '1512.04485-1-23-0': 'We can use induction on the length [MATH] of [MATH] to prove the equation.', '1512.04485-1-24-0': 'We have another orthogonality between [MATH] and [MATH].', '1512.04485-1-24-1': '(Type [MATH] case was due to [CITATION] Theorem 5.1 , [CITATION] Theorem 1.8.4.)', '1512.04485-1-25-0': 'For all [MATH], [EQUATION].', '1512.04485-1-26-0': 'We use induction on [MATH] and use the fact that if [MATH] and [MATH], then [MATH] for some [MATH].', '1512.04485-1-27-0': '## Duality between the transition coefficients', '1512.04485-1-28-0': 'Recall that we have two transition coeffixients [MATH] defined by the following expansions.', '1512.04485-1-29-0': '[EQUATION]', '1512.04485-1-29-1': 'Below gives a relation between them.', '1512.04485-1-29-2': '(Lascoux [CITATION] Corollary 1.8.5 fot type [MATH] case) For [MATH], [EQUATION].', '1512.04485-1-30-0': 'We will calculate [MATH] in two ways.', '1512.04485-1-30-1': 'As [MATH], [EQUATION] by the orthogonality on [MATH] (Proposition 4).', '1512.04485-1-30-2': 'On the other hand, as [MATH] for [MATH], we can expand [MATH] in terms of [MATH] as follows.', '1512.04485-1-30-3': '[EQUATION]', '1512.04485-1-30-4': 'So we have [EQUATION]', '1512.04485-1-30-5': 'Then using the orthogonality on [MATH] (Proposition 3) and Corollary 1, [EQUATION].', '1512.04485-1-30-6': 'The theorem is proved.', '1512.04485-1-31-0': '## Recurrence relations', '1512.04485-1-32-0': 'Here we give some recurrence relations on [MATH] and [MATH].', '1512.04485-1-33-0': '(left [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-1-33-1': 'By the definition we have [MATH] from which we can deduce the recurrence formula.', '1512.04485-1-34-0': 'We note that by this recurrence we can identify [MATH] as a coefficient of transition between two bases of the space of Iwahori fixed vectors cf.Theorem 3 below.', '1512.04485-1-35-0': '(right [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-1-35-1': 'We can use the equation [MATH] and taking the coefficient of [MATH], we get the formula.', '1512.04485-1-36-0': '(left [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-1-36-1': 'We can prove the recurrence relation using Corollary 2 below.', '1512.04485-1-37-0': '(right [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-1-37-1': 'We can prove the recurrence relation using Corollary 2 below.', '1512.04485-1-38-0': "# Kostant-Kumar's twisted group algebra", '1512.04485-1-39-0': 'Let [MATH] be the (generic) twisted group algebra of Kostant-Kumar.', '1512.04485-1-39-1': 'Its element is of the form [MATH] for [MATH] and the product is defined by [EQUATION].', '1512.04485-1-40-0': 'Define [MATH]) by [EQUATION].', '1512.04485-1-41-0': 'We have the following equations.', '1512.04485-1-42-0': '(1) [MATH] for [MATH].', '1512.04485-1-43-0': '(2) [MATH], where [MATH] is the order of [MATH].', '1512.04485-1-44-0': 'These equations can be shown by direct calculations.', '1512.04485-1-45-0': 'By this proposition we can define [MATH] for a reduced expression [MATH].', '1512.04485-1-45-1': 'These [MATH] become a [MATH]-basis of [MATH].', '1512.04485-1-46-0': 'This operator [MATH] can be seen as a generic Demazure-Lusztig operator.', '1512.04485-1-46-1': "When [MATH], it becomes [MATH] in Kumar's book[CITATION](12.2.E", '1512.04485-1-46-2': 'We can also set [MATH] which satisfies [EQUATION].', '1512.04485-1-46-3': "For example, if we set [MATH] and [MATH] and replace [MATH] by [MATH], it becomes Lusztig's [MATH] [CITATION].", '1512.04485-1-46-4': 'If we set [MATH] and [MATH] and replace [MATH] by [MATH], it becomes [MATH] in [CITATION].', '1512.04485-1-47-0': 'We can define a [MATH]-module isomorphism [MATH] by [MATH].', '1512.04485-1-47-1': 'Let [MATH].', '1512.04485-1-47-2': 'Define [MATH] and [MATH].', '1512.04485-1-47-3': 'Then it becomes that [MATH].', '1512.04485-1-47-4': "In particular [MATH]'s satisfy the braid relations.", '1512.04485-1-47-5': 'We can show below by induction of length [MATH].', '1512.04485-1-48-0': 'For [MATH], we have [EQUATION].', '1512.04485-1-49-0': 'If [MATH], [MATH].', '1512.04485-1-49-1': 'Therefore [MATH].', '1512.04485-1-49-2': 'If [MATH], by induction hypothesis we can assume [MATH].', '1512.04485-1-49-3': 'As [MATH] is a [MATH]-isomorphism, it follows that [MATH].', '1512.04485-1-49-4': 'Then [MATH].', '1512.04485-1-49-5': 'We used the recurrence relation (Proposition 5) for the last equality.', '1512.04485-1-49-6': 'Therefore [MATH].', '1512.04485-1-49-7': 'The theorem is proved.', '1512.04485-1-50-0': '(Explicit formula for [MATH])', '1512.04485-1-51-0': 'Let [MATH] ba a reduced expression.', '1512.04485-1-51-1': 'Then we have [EQUATION] where for [MATH], [MATH].', '1512.04485-1-52-0': 'Taking the inverse image of the map [MATH], the equality [MATH] becomes [EQUATION].', '1512.04485-1-53-0': 'As [MATH] is a reduced expression, [MATH].', '1512.04485-1-53-1': 'By expanding this we get the formula.', '1512.04485-1-54-0': 'Using Theorem 1, we also have a closed form for [MATH].', '1512.04485-1-54-1': 'We have another conjectural formula for [MATH] using [MATH]-chain cf. [CITATION].', '1512.04485-1-55-0': 'Type [MATH].', '1512.04485-1-55-1': 'We use notation [MATH], [MATH], [MATH].', '1512.04485-1-56-0': 'When [MATH], [MATH],then [MATH] and [MATH].', '1512.04485-1-57-0': 'When [MATH], [MATH], then [MATH] and [MATH].', '1512.04485-1-58-0': 'When[MATH], [MATH], then [MATH] and [MATH].', '1512.04485-1-59-0': "# Casselman's problem", '1512.04485-1-60-0': 'In his paper [CITATION] B. Casselman gave a problem concerning transition coefficient of two basis in the space of Iwahori fixed vectors of a principal series representation of a [MATH]-adic group.', '1512.04485-1-60-1': 'We relate the problem with the Yang-Baxter basis and give an answer to the problem.', '1512.04485-1-61-0': '## Principal series representations of [MATH]-adic group and Iwahori fixed vector', '1512.04485-1-62-0': 'We follow the notations of M.Reeder [CITATION].', '1512.04485-1-62-1': 'Let [MATH] be a reductive [MATH]-adic group over a non-archimedian local field [MATH].', '1512.04485-1-62-2': 'For simplicity we restrict to the case of split semisimple [MATH].', '1512.04485-1-62-3': 'Associated to [MATH], there is the ring of integer [MATH], the prime ideal [MATH] with a generator [MATH], and the residue field with [MATH] elements.', '1512.04485-1-62-4': 'Let [MATH] be a minimal parabolic subgroup (Borel) of [MATH], and [MATH] be the maximal split torus of [MATH] so that [MATH] where [MATH] is the rank of [MATH].', '1512.04485-1-62-5': 'For an unramified quasi-character [MATH] of [MATH], i.e. a group homomorphism [MATH] which is trivial on [MATH], where [MATH] is a maximal compact subgtoup of [MATH].', '1512.04485-1-62-6': 'Let [MATH] be the complex torus dual to [MATH], where [MATH] is the group of rational characters on [MATH], i.e. [MATH].', '1512.04485-1-62-7': 'We have a pairing [MATH] given by [MATH].', '1512.04485-1-62-8': 'This gives an identification [MATH] of [MATH] with the set of unramified quasi-characters on [MATH] (cf. [CITATION] Exercise 18,19).', '1512.04485-1-63-0': 'Let [MATH] be the set of roots of [MATH] in [MATH], [MATH] be the set of positive roots corresponding to [MATH] and [MATH] be the set of simple roots .', '1512.04485-1-63-1': 'For a root [MATH], we define [MATH] by [EQUATION] for [MATH] where [MATH] is the one parameter subgroup (coroot) corresponding to [MATH].', '1512.04485-1-64-0': 'As the definition shows, [MATH] is defined using the coroot [MATH].', '1512.04485-1-64-1': 'So it should be parametrized by [MATH], but for convenience we follow the notation of [CITATION].', '1512.04485-1-64-2': 'Later we will identify [MATH] with [MATH] by the map [MATH] of root data.', '1512.04485-1-65-0': '[MATH] acts on right of [MATH] so that [MATH] for [MATH], [MATH] and [MATH].', '1512.04485-1-65-1': 'The action of [MATH] on [MATH] is given by [MATH] for [MATH], [MATH] and [MATH].', '1512.04485-1-66-0': 'The principal series representation [MATH] of [MATH] associated to a unramified quasicharacter [MATH] of [MATH] is defined as follows.', '1512.04485-1-66-1': 'As a vector space over [MATH] it consists of locally constant functions on [MATH] with values in [MATH] which satisfy the left relative invariance properties with respect to [MATH] where [MATH] is extented to [MATH] with trivial value on the unipotent radical [MATH] of [MATH].', '1512.04485-1-66-2': '[EQUATION].', '1512.04485-1-66-3': 'Here [MATH] is the modulus of [MATH].', '1512.04485-1-66-4': 'The action of [MATH] on [MATH] is defined by right translation, i.e. for [MATH] and [MATH], [MATH].', '1512.04485-1-67-0': 'Let [MATH] be the Iwahori subgroup which is the inverse image [MATH] of the Borel subgroup [MATH] of [MATH] by the projection [MATH].', '1512.04485-1-67-1': 'Then we define [MATH] to be the space of Iwahori fixed vectors in [MATH], i.e. [EQUATION].', '1512.04485-1-67-2': 'This space has a natural basis [MATH].', '1512.04485-1-67-3': '[MATH] is supported on [MATH] and satisfies [EQUATION]', '1512.04485-1-68-0': "## Intertwiner and Casselman's basis", '1512.04485-1-69-0': 'From now on we always assume that [MATH] is regular i.e. the stabilizer [MATH] is trivial.', '1512.04485-1-69-1': 'The intertwining operator [MATH] is defined by [EQUATION] where [MATH], where [MATH] is the unipotent radical of opposite parabolic [MATH].', '1512.04485-1-69-2': 'It has the property that for [MATH] with [MATH], then [EQUATION].', '1512.04485-1-70-0': "The Casselman's basis [MATH] of [MATH] is defined as follows.", '1512.04485-1-70-1': '[MATH] and [EQUATION]', '1512.04485-1-70-2': 'M.Reeder characterizes this using the action of affine Hecke algebra (cf. [CITATION] Section 2).', '1512.04485-1-70-3': 'The affine Hecke algebra [MATH] is the convolution algebra of [MATH] bi-invariant locally constant functions on [MATH] with values in [MATH].', '1512.04485-1-70-4': 'By the theorem of Iwahori-Matsumoto it can be described by generators and relations.', '1512.04485-1-70-5': 'The basis [MATH] consists of characteristic functions [MATH] of double coset [MATH].', '1512.04485-1-70-6': 'Let [MATH] be the Hecke algebra of the finite Weyl group [MATH] generated by the simple reflections [MATH] for simple roots [MATH].', '1512.04485-1-70-7': 'As a vector space [MATH] is the tensor product of two subalgebras [MATH].', '1512.04485-1-70-8': 'The subalgebra [MATH] is commutative and isomorphic to the coordinate ring of the complex torus [MATH] with a basis [MATH], where [MATH] is defined as follows (cf. [CITATION]).', '1512.04485-1-70-9': 'Define [MATH].', '1512.04485-1-70-10': 'For [MATH], choose [MATH] such that [MATH].', '1512.04485-1-70-11': 'Then [MATH] where for [MATH], [MATH] is the length function defined by [MATH] and [MATH] is the characteristic function of [MATH].', '1512.04485-1-71-0': 'By Lemma (4.1) of [CITATION], there exists a unique [MATH] for each [MATH] such that', '1512.04485-1-72-0': '[MATH] and', '1512.04485-1-73-0': '[MATH] for all [MATH].', '1512.04485-1-74-0': 'Here [MATH]f[MATH].', '1512.04485-1-75-0': '## Transition coefficients', '1512.04485-1-76-0': 'Let [EQUATION] and [EQUATION]', '1512.04485-1-76-1': "The Casselman's problem is to find an explicit formula for [MATH] and [MATH].", '1512.04485-1-77-0': "To relate the results in Sections 2 and 3 with the Casselman's problem, in this subsection we specialize the parameters [MATH], [MATH] and take tensor product with the complex field [MATH].", '1512.04485-1-77-1': 'For example the Yang-Baxter basis [MATH] will become a [MATH] basis in [MATH].', '1512.04485-1-77-2': 'The generic Demazure-Lusztig operator defined in Section 3 will become [EQUATION]', '1512.04485-1-77-3': 'Then [MATH].', '1512.04485-1-78-0': 'We identify [MATH] with [MATH] (cf. Remark 4).', '1512.04485-1-78-1': "Then, [EQUATION] [MATH]'s satisfy the same recurrence relation (Proposition 5 with [MATH]) as [MATH]'s (cf. [CITATION] Proposition (2.2)).", '1512.04485-1-78-2': 'The initial condition [MATH] leads to the second the equation.', '1512.04485-1-78-3': 'The first equation then also holds.', '1512.04485-1-78-4': 'Note that the [MATH] in [CITATION] is our [MATH].', '1512.04485-1-79-0': 'There is also a direct proof that does not use recurrence relation cf. [CITATION].', '1512.04485-1-80-0': 'We have a closed formula for [MATH] and [MATH] by Corollary 2 and Theorem1.', '1512.04485-1-81-0': 'For [MATH], we have [EQUATION] and [EQUATION].', '1512.04485-1-82-0': 'When [MATH], we can specialize [MATH] to 1 and we get the first equation from the definition of [MATH], since [MATH].', '1512.04485-1-82-1': 'We can also specialize [MATH] to [MATH] and [MATH] gives the second equation.', '1512.04485-1-83-0': 'The left hand side of the first equation in Corollary 4 is [MATH] in [CITATION].', '1512.04485-1-83-1': 'So this gives another proof of Theorem 1.4 in [CITATION].', '1512.04485-1-84-0': '## Whittaker function', '1512.04485-1-85-0': 'M.Reeder [CITATION] specified a formula for the Wittaker function [MATH] and using [MATH], he got a formula for [MATH].', '1512.04485-1-85-1': 'For [MATH], let [MATH] be [EQUATION].', '1512.04485-1-86-0': 'Formally the result of M.Reeder [CITATION] Corollary [MATH] is written as follows.', '1512.04485-1-86-1': 'For [MATH] and [MATH], [EQUATION].', '1512.04485-1-86-2': 'Then using Corollary 3, we have an explicit formula of [MATH].', '1512.04485-1-87-0': "## Relation with Bump-Nakasuji's work", '1512.04485-1-88-0': 'Now we explain the relation between this paper and Bump-Nakasuji [CITATION].', '1512.04485-1-88-1': 'First of all, the notational conventions are slightly different.', '1512.04485-1-88-2': 'Especially in the published [CITATION] the natural base and intertwiner are differently parametrized.', '1512.04485-1-88-3': 'The natural basis [MATH] in [CITATION] is our [MATH].', '1512.04485-1-88-4': 'The intertwiner [MATH] in [CITATION] is our [MATH] so that if [MATH], [MATH] while [MATH].', '1512.04485-1-89-0': 'Conjecture 1.2 and Conjecture 1.3 in [CITATION] are equivalent.', '1512.04485-1-90-0': 'This follows from Theorem 1.', '1512.04485-1-91-0': 'Maki Nakasuji, Graduate School of Science and Technology, Sophia University, Tokyo 102-8554, Japan', '1512.04485-1-92-0': 'email address: nakasuji@sophia.ac.jp', '1512.04485-1-93-0': 'Hiroshi Naruse, Graduate School of Education, University of Yamanashi, Yamanashi 400-8510, Japan', '1512.04485-1-94-0': 'email address: hnaruse@yamanashi.ac.jp'}

{'1512.04485-2-0-0': 'We generalize the definition of Yang-Baxter basis of type [MATH] Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math.', '1512.04485-2-0-1': 'Phys., 40 (1997), 75-90) to all the Lie types and prove their duality.', '1512.04485-2-0-2': "As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of [MATH]-adic groups.", '1512.04485-2-1-0': '# Introduction', '1512.04485-2-2-0': 'Yang-Baxter basis of Hecke algebra of type [MATH] was defined in the paper of Lascoux-Leclerc-Thibon [CITATION].', '1512.04485-2-2-1': 'There is also a modified version in [CITATION].', '1512.04485-2-2-2': 'First we generalize the latter version to all the Lie types.', '1512.04485-2-2-3': "Then we will solve the Casselman's problem on the basis of Iwahori fixed vectors using Yang-Baxter basis and Demazure-Lusztig type operator.", '1512.04485-2-2-4': 'This paper is an extended abstract and the detailed proofs will appear in [CITATION].', '1512.04485-2-3-0': '# Generic Hecke algebra', '1512.04485-2-4-0': '## Root system, Weyl group and generic Hecke algebra', '1512.04485-2-5-0': 'Let [MATH] be a (reduced) semisimple root data cf. [CITATION].', '1512.04485-2-5-1': 'More precisely [MATH] is a weight lattice with [MATH].', '1512.04485-2-5-2': 'There is a pairing [MATH].', '1512.04485-2-5-3': '[MATH] is a root system with simple roots [MATH] and positive roots [MATH].', '1512.04485-2-5-4': '[MATH] is the set of coroots, and there is a bijection [MATH], [MATH].', '1512.04485-2-5-5': 'We also denote the coroot [MATH].', '1512.04485-2-5-6': 'The Weyl group [MATH] of [MATH] is generated by simple reflections [MATH].', '1512.04485-2-5-7': 'The action of [MATH] on [MATH] is given by [MATH] for [MATH].', '1512.04485-2-5-8': 'We define generic Hecke algebra [MATH] over [MATH] with two parameters [MATH] as follows.', '1512.04485-2-5-9': 'Generators are [MATH], with relations [MATH] for [MATH] and the braid relations [MATH] , where [MATH] is the order of [MATH] for [MATH].', '1512.04485-2-5-10': 'We need to extend the coefficients to the quotient field of the group algebra [MATH].', '1512.04485-2-5-11': 'An element of [MATH] is denoted as [MATH].', '1512.04485-2-5-12': 'The Weyl group acts on [MATH] by [MATH].', '1512.04485-2-5-13': 'We extend the coefficient ring [MATH] of [MATH] to [EQUATION] where [MATH] is the quotient field of [MATH].', '1512.04485-2-5-14': '[EQUATION].', '1512.04485-2-5-15': 'For [MATH], an expression of [MATH] with minimal number of generators [MATH] is called a reduced expression in which case we write [MATH] and call it the length of [MATH].', '1512.04485-2-5-16': 'Then [MATH] is well defined and [MATH] forms a [MATH]-basis of [MATH].', '1512.04485-2-6-0': '## Yang-Baxter basis and its properties', '1512.04485-2-7-0': 'Yang-Baxter basis was introduced in the paper [CITATION] to investigate the relation with Schubert calculus.', '1512.04485-2-7-1': 'There is also a variant in [CITATION] for type [MATH] case.', '1512.04485-2-7-2': 'We generalize that results to all Lie types.', '1512.04485-2-8-0': 'For [MATH], we define [MATH].', '1512.04485-2-8-1': 'Then [MATH].', '1512.04485-2-8-2': 'In particuar, if [MATH], [MATH].', '1512.04485-2-9-0': 'For [MATH], if [MATH], let [MATH].', '1512.04485-2-9-1': 'Then these satisfy the Yang-Baxter relations, i.e. if we write [MATH] for fixed [MATH], the following equations hold.', '1512.04485-2-9-2': 'We assume all appearance of [MATH] is nonzero.', '1512.04485-2-9-3': '[EQUATION]', '1512.04485-2-9-4': 'We can prove these equations by direct calculations.', '1512.04485-2-10-0': 'In [CITATION] I. Cherednik treated Yang-Baxter relation in more general setting.', '1512.04485-2-10-1': 'There is also a related work [CITATION] by S. Kato and the proof of Theorem 2.4 in [CITATION] suggests a uniform way to prove Yang-Baxter relations without direct calculations.', '1512.04485-2-11-0': 'We use the Bruhat order [MATH] on elements [MATH] (cf.[CITATION]).', '1512.04485-2-11-1': 'Following [CITATION] we define the Yang-Baxter basis [MATH] for [MATH] recursively as follows.', '1512.04485-2-12-0': '[MATH], [MATH] if [MATH].', '1512.04485-2-13-0': 'Using the Yang-Baxter relation above it is easy to see that [MATH] does not depend on a reduced expression of [MATH].', '1512.04485-2-13-1': 'As the leading term of [MATH] with respect to the Bruhat order is [MATH], they also form a [MATH]-basis [MATH] of [MATH].', '1512.04485-2-13-2': 'We are interested in the transition coefficients [MATH] between the two basis [MATH] and [MATH] , i.e. [EQUATION].', '1512.04485-2-14-0': 'Take a reduced expression of [MATH] e.g. [MATH] where [MATH] is the length of [MATH] (cf. [CITATION]).', '1512.04485-2-14-1': 'Then [MATH] is expressed as follows.', '1512.04485-2-14-2': '[EQUATION] where [MATH] for [MATH].', '1512.04485-2-14-3': 'The set [MATH] is independent of the reduced expression of [MATH].', '1512.04485-2-14-4': 'The Yang-Baxter basis defined in [CITATION] is normalized as follows.', '1512.04485-2-14-5': '[EQUATION].', '1512.04485-2-15-0': 'The relation to [MATH]-theory Schubert calculus is as follows.', '1512.04485-2-15-1': 'If we set [MATH] and replacing [MATH] by [MATH].', '1512.04485-2-15-2': 'Then the coefficient of [MATH] in [MATH] is the localization [MATH] at [MATH] of the equivariant [MATH]-theory Schubert class [MATH] (cf. [CITATION]).', '1512.04485-2-16-0': 'Let [MATH] be the longest element in [MATH].', '1512.04485-2-16-1': 'Define [MATH]-algebra homomorphism [MATH] by [MATH].', '1512.04485-2-16-2': 'Let [MATH] be the ring homomorphism on [MATH] induced by [MATH] and extend to [MATH].', '1512.04485-2-17-0': '(Lascoux [CITATION] Lemma 1.8.1 for type [MATH] case) For [MATH], [EQUATION] where [MATH] acts only on the coefficients.', '1512.04485-2-18-0': 'When [MATH] there exists [MATH] such that [MATH].', '1512.04485-2-18-1': 'Using the induction assumption on [MATH], we get the formula for [MATH].', '1512.04485-2-19-0': 'Taking the coefficient of [MATH] in the above equation, we get [EQUATION]', '1512.04485-2-20-0': '## Inner product and orthogonality', '1512.04485-2-21-0': 'Define inner product [MATH] on [MATH] by [MATH] in [MATH], where [MATH] if [MATH].', '1512.04485-2-21-1': 'It is easy to see that [MATH] for [MATH] and [MATH].', '1512.04485-2-21-2': 'There is an involution [MATH] defined by [MATH].', '1512.04485-2-21-3': 'It is easy to see that [MATH] for [MATH].', '1512.04485-2-22-0': 'The following proposition is due to A.Lascoux for the type [MATH] case [CITATION] P.33.', '1512.04485-2-23-0': 'For all [MATH], [EQUATION].', '1512.04485-2-24-0': 'We can use induction on the length [MATH] of [MATH] to prove the equation.', '1512.04485-2-25-0': 'We have another orthogonality between [MATH] and [MATH].', '1512.04485-2-25-1': '(Type [MATH] case was due to [CITATION] Theorem 5.1 , [CITATION] Theorem 1.8.4.)', '1512.04485-2-26-0': 'For all [MATH], [EQUATION].', '1512.04485-2-27-0': 'We use induction on [MATH] and use the fact that if [MATH] and [MATH], then [MATH] for some [MATH].', '1512.04485-2-28-0': '## Duality between the transition coefficients', '1512.04485-2-29-0': 'Recall that we have two transition coefficients [MATH] defined by the following expansions.', '1512.04485-2-30-0': '[EQUATION]', '1512.04485-2-30-1': 'Below gives a relation between them.', '1512.04485-2-30-2': '(Lascoux [CITATION] Corollary 1.8.5 for type [MATH] case) For [MATH], [EQUATION].', '1512.04485-2-31-0': 'We will calculate [MATH] in two ways.', '1512.04485-2-31-1': 'As [MATH], [EQUATION] by the orthogonality on [MATH] (Proposition 4).', '1512.04485-2-31-2': 'On the other hand, as [MATH] for [MATH], we can expand [MATH] in terms of [MATH] as follows.', '1512.04485-2-31-3': '[EQUATION]', '1512.04485-2-31-4': 'So we have [EQUATION]', '1512.04485-2-31-5': 'Then using the orthogonality on [MATH] (Proposition 3) and Corollary 1, [EQUATION].', '1512.04485-2-31-6': 'The theorem is proved.', '1512.04485-2-32-0': '## Recurrence relations', '1512.04485-2-33-0': 'Here we give some recurrence relations on [MATH] and [MATH].', '1512.04485-2-34-0': '(left [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-2-34-1': 'By the definition we have [MATH] from which we can deduce the recurrence formula.', '1512.04485-2-35-0': 'We note that by this recurrence we can identify [MATH] as a coefficient of transition between two bases of the space of Iwahori fixed vectors cf. Theorem 3 below.', '1512.04485-2-36-0': '(right [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-2-36-1': 'We can use the equation [MATH] and taking the coefficient of [MATH], we get the formula.', '1512.04485-2-37-0': '(left [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-2-37-1': 'We can prove the recurrence relation using Corollary 2 below.', '1512.04485-2-38-0': '(right [MATH]) For [MATH] and [MATH], if [MATH] then [EQUATION]', '1512.04485-2-38-1': 'We can prove the recurrence relation using Corollary 2 below.', '1512.04485-2-39-0': "# Kostant-Kumar's twisted group algebra", '1512.04485-2-40-0': 'Let [MATH] be the (generic) twisted group algebra of Kostant-Kumar.', '1512.04485-2-40-1': 'Its element is of the form [MATH] for [MATH] and the product is defined by [EQUATION].', '1512.04485-2-41-0': 'Define [MATH]) by [EQUATION].', '1512.04485-2-42-0': 'We have the following equations.', '1512.04485-2-43-0': '(1) [MATH] for [MATH].', '1512.04485-2-44-0': '(2) [MATH], where [MATH] is the order of [MATH].', '1512.04485-2-45-0': 'These equations can be shown by direct calculations.', '1512.04485-2-46-0': 'By this proposition we can define [MATH] for a reduced expression [MATH].', '1512.04485-2-46-1': 'These [MATH] become a [MATH]-basis of [MATH].', '1512.04485-2-47-0': 'This operator [MATH] can be seen as a generic Demazure-Lusztig operator.', '1512.04485-2-47-1': "When [MATH], it becomes [MATH] in Kumar's book[CITATION](12.2.E", '1512.04485-2-47-2': 'We can also set [MATH] which satisfies [EQUATION].', '1512.04485-2-47-3': "For example, if we set [MATH] and [MATH] and replace [MATH] by [MATH], it becomes Lusztig's [MATH] [CITATION].", '1512.04485-2-47-4': 'If we set [MATH] and [MATH] and replace [MATH] by [MATH], it becomes [MATH] in [CITATION].', '1512.04485-2-48-0': 'We can define a [MATH]-module isomorphism [MATH] by [MATH].', '1512.04485-2-48-1': 'Let [MATH].', '1512.04485-2-48-2': 'Define [MATH] and [MATH].', '1512.04485-2-48-3': 'Then it becomes that [MATH].', '1512.04485-2-48-4': "In particular, [MATH]'s satisfy the braid relations.", '1512.04485-2-48-5': 'We can show below by induction on length [MATH].', '1512.04485-2-49-0': 'For [MATH], we have [EQUATION].', '1512.04485-2-50-0': 'If [MATH], [MATH].', '1512.04485-2-50-1': 'Therefore [MATH].', '1512.04485-2-50-2': 'If [MATH], by induction hypothesis we can assume [MATH].', '1512.04485-2-50-3': 'As [MATH] is a [MATH]-isomorphism, it follows that [MATH].', '1512.04485-2-50-4': 'Then [MATH].', '1512.04485-2-50-5': 'We used the recurrence relation (Proposition 5) for the last equality.', '1512.04485-2-50-6': 'Therefore [MATH].', '1512.04485-2-50-7': 'The theorem is proved.', '1512.04485-2-51-0': '(Explicit formula for [MATH])', '1512.04485-2-52-0': 'Let [MATH] be a reduced expression.', '1512.04485-2-52-1': 'Then we have [EQUATION] where for [MATH], [MATH].', '1512.04485-2-53-0': 'Taking the inverse image of the map [MATH], the equality [MATH] becomes [EQUATION].', '1512.04485-2-54-0': 'As [MATH] is a reduced expression, [MATH].', '1512.04485-2-54-1': 'By expanding this we get the formula.', '1512.04485-2-55-0': 'Using Theorem 1, we also have a closed form for [MATH].', '1512.04485-2-55-1': 'We have another conjectural formula for [MATH] using [MATH]-chain cf. [CITATION].', '1512.04485-2-56-0': 'Type [MATH].', '1512.04485-2-56-1': 'We use notation [MATH], [MATH], [MATH].', '1512.04485-2-57-0': 'When [MATH], [MATH], then [MATH] and [EQUATION].', '1512.04485-2-58-0': 'When [MATH], [MATH], then [MATH] and [EQUATION].', '1512.04485-2-59-0': 'When [MATH], [MATH], then [MATH] and [EQUATION]', '1512.04485-2-60-0': "# Casselman's problem", '1512.04485-2-61-0': 'In his paper [CITATION] B. Casselman gave a problem concerning transition coefficients between two bases in the space of Iwahori fixed vectors of a principal series representation of a [MATH]-adic group.', '1512.04485-2-61-1': 'We relate the problem with the Yang-Baxter basis and give an answer to the problem.', '1512.04485-2-62-0': '## Principal series representations of [MATH]-adic group and Iwahori fixed vector', '1512.04485-2-63-0': 'We follow the notations of M.Reeder [CITATION].', '1512.04485-2-63-1': 'Let [MATH] be a connected reductive [MATH]-adic group over a non-archimedian local field [MATH].', '1512.04485-2-63-2': 'For simplicity we restrict to the case of split semisimple [MATH].', '1512.04485-2-63-3': 'Associated to [MATH], there is the ring of integer [MATH], the prime ideal [MATH] with a generator [MATH], and the residue field with [MATH] elements.', '1512.04485-2-63-4': 'Let [MATH] be a minimal parabolic subgroup (Borel) of [MATH], and [MATH] be the maximal split torus of [MATH] so that [MATH] where [MATH] is the rank of [MATH].', '1512.04485-2-63-5': 'For an unramified quasi-character [MATH] of [MATH], i.e. a group homomorphism [MATH] which is trivial on [MATH], where [MATH] is a maximal compact subgtoup of [MATH].', '1512.04485-2-63-6': 'Let [MATH] be the complex torus dual to [MATH], where [MATH] is the group of rational characters on [MATH], i.e. [MATH].', '1512.04485-2-63-7': 'We have a pairing [MATH] given by [MATH].', '1512.04485-2-63-8': 'This gives an identification [MATH] of [MATH] with the set of unramified quasi-characters on [MATH] (cf. [CITATION] Exercise 18,19).', '1512.04485-2-64-0': 'Let [MATH] be the set of roots of [MATH] in [MATH], [MATH] be the set of positive roots corresponding to [MATH] and [MATH] be the set of simple roots .', '1512.04485-2-64-1': 'For a root [MATH], we define [MATH] by [EQUATION] for [MATH] where [MATH] is the one parameter subgroup (coroot) corresponding to [MATH].', '1512.04485-2-65-0': 'As the definition shows, [MATH] is defined using the coroot [MATH].', '1512.04485-2-65-1': 'So it should be parametrized by [MATH], but for convenience we follow the notation of [CITATION].', '1512.04485-2-65-2': 'Later we will identify [MATH] with [MATH] by the map [MATH] of root data.', '1512.04485-2-66-0': '[MATH] acts on right of [MATH] so that [MATH] for [MATH], [MATH] and [MATH].', '1512.04485-2-66-1': 'The action of [MATH] on [MATH] is given by [MATH] for [MATH], [MATH] and [MATH].', '1512.04485-2-67-0': 'The principal series representation [MATH] of [MATH] associated to a unramified quasicharacter [MATH] of [MATH] is defined as follows.', '1512.04485-2-67-1': 'As a vector space over [MATH] it consists of locally constant functions on [MATH] with values in [MATH] which satisfy the left relative invariance properties with respect to [MATH] where [MATH] is extended to [MATH] with trivial value on the unipotent radical [MATH] of [MATH].', '1512.04485-2-67-2': '[EQUATION].', '1512.04485-2-67-3': 'Here [MATH] is the modulus of [MATH].', '1512.04485-2-67-4': 'The action of [MATH] on [MATH] is defined by right translation, i.e. for [MATH] and [MATH], [MATH].', '1512.04485-2-68-0': 'Let [MATH] be the Iwahori subgroup which is the inverse image [MATH] of the Borel subgroup [MATH] of [MATH] by the projection [MATH].', '1512.04485-2-68-1': 'Then we define [MATH] to be the space of Iwahori fixed vectors in [MATH], i.e. [EQUATION].', '1512.04485-2-68-2': 'This space has a natural basis [MATH].', '1512.04485-2-68-3': '[MATH] is supported on [MATH] and satisfies [EQUATION]', '1512.04485-2-69-0': "## Intertwiner and Casselman's basis", '1512.04485-2-70-0': 'From now on we always assume that [MATH] is regular i.e. the stabilizer [MATH] is trivial.', '1512.04485-2-70-1': 'The intertwining operator [MATH] is defined by [EQUATION] where [MATH], with [MATH] being the unipotent radical of opposite parabolic [MATH] which corresponds to the negative roots [MATH].', '1512.04485-2-70-2': 'The integral is convergent when [MATH] for all [MATH] such that [MATH] (cf. [CITATION] Proposition 63), and may be meromorphically continued to all [MATH].', '1512.04485-2-70-3': 'It has the property that for [MATH] with [MATH], then [EQUATION].', '1512.04485-2-71-0': "The Casselman's basis [MATH] of [MATH] is defined as follows.", '1512.04485-2-71-1': '[MATH] and [EQUATION]', '1512.04485-2-71-2': 'M.Reeder characterizes this using the action of affine Hecke algebra (cf. [CITATION] Section 2).', '1512.04485-2-71-3': 'The affine Hecke algebra [MATH] is the convolution algebra of [MATH] bi-invariant locally constant functions on [MATH] with values in [MATH].', '1512.04485-2-71-4': 'By the theorem of Iwahori-Matsumoto it can be described by generators and relations.', '1512.04485-2-71-5': 'The basis [MATH] consists of characteristic functions [MATH] of double coset [MATH].', '1512.04485-2-71-6': 'Let [MATH] be the Hecke algebra of the finite Weyl group [MATH] generated by the simple reflections [MATH] for simple roots [MATH].', '1512.04485-2-72-0': 'As a vector space [MATH] is the tensor product of two subalgebras [MATH].', '1512.04485-2-72-1': 'The subalgebra [MATH] is commutative and isomorphic to the coordinate ring of the complex torus [MATH] with a basis [MATH], where [MATH] is defined as follows (cf. [CITATION]).', '1512.04485-2-72-2': 'Define [MATH].', '1512.04485-2-72-3': 'For [MATH], choose [MATH] such that [MATH].', '1512.04485-2-72-4': 'Then [MATH] where for [MATH], [MATH] is the length function defined by [MATH] and [MATH] is the characteristic function of [MATH].', '1512.04485-2-73-0': 'By Lemma (4.1) of [CITATION], there exists a unique [MATH] for each [MATH] such that', '1512.04485-2-74-0': '[MATH] and', '1512.04485-2-75-0': '[MATH] for all [MATH].', '1512.04485-2-76-0': 'Here [MATH]f[MATH].', '1512.04485-2-77-0': '## Transition coefficients', '1512.04485-2-78-0': 'Let [EQUATION] and [EQUATION]', '1512.04485-2-78-1': "The Casselman's problem is to find an explicit formula for [MATH] and [MATH].", '1512.04485-2-79-0': "To relate the results in Sections 2 and 3 with the Casselman's problem, in this subsection we specialize the parameters [MATH], [MATH] and take tensor product with the complex field [MATH].", '1512.04485-2-79-1': 'For example, the Yang-Baxter basis [MATH] will become a [MATH] basis in [MATH].', '1512.04485-2-79-2': 'The generic Demazure-Lusztig operator defined in Section 3 will become [EQUATION]', '1512.04485-2-79-3': 'Then [MATH].', '1512.04485-2-80-0': 'We identify [MATH] with [MATH] (cf. Remark 4).', '1512.04485-2-80-1': "Then, [EQUATION] [MATH]'s satisfy the same recurrence relation (Proposition 5 with [MATH]) as [MATH]'s (cf. [CITATION] Proposition (2.2)).", '1512.04485-2-80-2': 'The initial condition [MATH] leads to the second equation.', '1512.04485-2-80-3': 'The first equation then also holds.', '1512.04485-2-80-4': 'Note that the [MATH] in [CITATION] is our [MATH].', '1512.04485-2-81-0': 'There is also a direct proof that does not use recurrence relation cf. [CITATION].', '1512.04485-2-82-0': 'We have a closed formula for [MATH] and [MATH] by Corollary 2 and Theorem1.', '1512.04485-2-83-0': 'For [MATH], we have [EQUATION] and [EQUATION].', '1512.04485-2-84-0': 'When [MATH], we can specialize [MATH] to 1 and we get the first equation from the definition of [MATH], since [MATH].', '1512.04485-2-84-1': 'We can also specialize [MATH] to [MATH] and [MATH] gives the second equation.', '1512.04485-2-85-0': 'The left hand side of the first equation in Corollary 4 is [MATH] in [CITATION].', '1512.04485-2-85-1': 'So this gives another proof of Theorem 1.4 in [CITATION].', '1512.04485-2-86-0': '## Whittaker function', '1512.04485-2-87-0': 'M.Reeder [CITATION] specified a formula for the Whittaker function [MATH] and using [MATH], he got a formula for [MATH].', '1512.04485-2-87-1': 'For [MATH], let [MATH] be [EQUATION].', '1512.04485-2-88-0': 'Formally the result of M.Reeder [CITATION] Corollary [MATH] is written as follows.', '1512.04485-2-88-1': 'For [MATH] and [MATH], [EQUATION].', '1512.04485-2-88-2': 'Then using Corollary 3, we have an explicit formula of [MATH].', '1512.04485-2-89-0': "## Relation with Bump-Nakasuji's work", '1512.04485-2-90-0': 'Now we explain the relation between this paper and Bump-Nakasuji [CITATION].', '1512.04485-2-90-1': 'First of all, the notational conventions are slightly different.', '1512.04485-2-90-2': 'Especially in the published [CITATION] the natural base and intertwiner are differently parametrized.', '1512.04485-2-90-3': 'The natural basis [MATH] in [CITATION] is our [MATH].', '1512.04485-2-90-4': 'The intertwiner [MATH] in [CITATION] is our [MATH] so that if [MATH], [MATH] while [MATH].', '1512.04485-2-91-0': "In the paper [CITATION], another basis [MATH] for the space [MATH] was defined and comparerd with the Casselman's basis.", '1512.04485-2-91-1': 'They defined [MATH] and expand this as [MATH] and conversely [MATH] .', '1512.04485-2-91-2': 'They observed that the transition coefficients [MATH] and [MATH] factor under certain condition.', '1512.04485-2-91-3': 'Let [MATH] and [MATH].', '1512.04485-2-91-4': 'Then the statements of the conjectures are as follows.', '1512.04485-2-91-5': '([CITATION] Conjecture 1.2) Assume that the root system [MATH] is simply-laced.', '1512.04485-2-91-6': 'Suppose [MATH] and [MATH], then [EQUATION] ([CITATION] Conjecture 1.3) Assume that the root system [MATH] is simply-laced.', '1512.04485-2-91-7': 'Suppose [MATH] and [MATH], then [EQUATION].', '1512.04485-2-92-0': 'Conjecture 1.2 and Conjecture 1.3 in [CITATION] are equivalent.', '1512.04485-2-93-0': 'We can show [MATH] and [MATH].', '1512.04485-2-93-1': 'Then it follows by the Theorem 1 that [MATH].', '1512.04485-2-93-2': 'As [MATH] we get the desiered conclusion.'}

[['1512.04485-1-69-0', '1512.04485-2-70-0'], ['1512.04485-1-69-2', '1512.04485-2-70-3'], ['1512.04485-1-78-0', '1512.04485-2-80-0'], ['1512.04485-1-78-1', '1512.04485-2-80-1'], ['1512.04485-1-78-3', '1512.04485-2-80-3'], ['1512.04485-1-78-4', '1512.04485-2-80-4'], ['1512.04485-1-10-0', '1512.04485-2-11-1'], ['1512.04485-1-2-0', '1512.04485-2-2-0'], ['1512.04485-1-2-1', '1512.04485-2-2-1'], ['1512.04485-1-2-2', '1512.04485-2-2-2'], ['1512.04485-1-2-3', '1512.04485-2-2-3'], ['1512.04485-1-2-4', '1512.04485-2-2-4'], ['1512.04485-1-64-0', '1512.04485-2-65-0'], ['1512.04485-1-64-1', '1512.04485-2-65-1'], ['1512.04485-1-64-2', '1512.04485-2-65-2'], ['1512.04485-1-46-0', '1512.04485-2-47-0'], ['1512.04485-1-46-1', '1512.04485-2-47-1'], ['1512.04485-1-46-2', '1512.04485-2-47-2'], ['1512.04485-1-46-3', '1512.04485-2-47-3'], ['1512.04485-1-46-4', '1512.04485-2-47-4'], ['1512.04485-1-54-0', '1512.04485-2-55-0'], ['1512.04485-1-54-1', '1512.04485-2-55-1'], ['1512.04485-1-23-0', '1512.04485-2-24-0'], ['1512.04485-1-5-0', '1512.04485-2-5-0'], ['1512.04485-1-5-1', '1512.04485-2-5-1'], ['1512.04485-1-5-2', '1512.04485-2-5-2'], ['1512.04485-1-5-3', '1512.04485-2-5-3'], ['1512.04485-1-5-4', '1512.04485-2-5-4'], ['1512.04485-1-5-5', '1512.04485-2-5-5'], ['1512.04485-1-5-7', '1512.04485-2-5-7'], ['1512.04485-1-5-8', '1512.04485-2-5-8'], ['1512.04485-1-5-9', '1512.04485-2-5-9'], ['1512.04485-1-5-10', '1512.04485-2-5-10'], ['1512.04485-1-5-11', '1512.04485-2-5-11'], ['1512.04485-1-5-12', '1512.04485-2-5-12'], ['1512.04485-1-5-13', '1512.04485-2-5-13'], ['1512.04485-1-7-0', '1512.04485-2-7-0'], ['1512.04485-1-7-1', '1512.04485-2-7-1'], ['1512.04485-1-7-2', '1512.04485-2-7-2'], ['1512.04485-1-12-1', '1512.04485-2-13-1'], ['1512.04485-1-12-2', '1512.04485-2-13-2'], ['1512.04485-1-77-0', '1512.04485-2-79-0'], ['1512.04485-1-77-2', '1512.04485-2-79-2'], ['1512.04485-1-53-0', '1512.04485-2-54-0'], ['1512.04485-1-53-1', '1512.04485-2-54-1'], ['1512.04485-1-21-0', '1512.04485-2-22-0'], ['1512.04485-1-26-0', '1512.04485-2-27-0'], ['1512.04485-1-60-1', '1512.04485-2-61-1'], ['1512.04485-1-9-1', '1512.04485-2-9-1'], ['1512.04485-1-9-2', '1512.04485-2-9-2'], ['1512.04485-1-9-4', '1512.04485-2-9-4'], ['1512.04485-1-0-0', '1512.04485-2-0-0'], ['1512.04485-1-0-2', '1512.04485-2-0-2'], ['1512.04485-1-86-0', '1512.04485-2-88-0'], ['1512.04485-1-86-2', '1512.04485-2-88-2'], ['1512.04485-1-49-2', '1512.04485-2-50-2'], ['1512.04485-1-49-3', '1512.04485-2-50-3'], ['1512.04485-1-49-5', '1512.04485-2-50-5'], ['1512.04485-1-49-7', '1512.04485-2-50-7'], ['1512.04485-1-16-0', '1512.04485-2-17-0'], ['1512.04485-1-39-0', '1512.04485-2-40-0'], ['1512.04485-1-39-1', '1512.04485-2-40-1'], ['1512.04485-1-82-0', '1512.04485-2-84-0'], ['1512.04485-1-82-1', '1512.04485-2-84-1'], ['1512.04485-1-51-1', '1512.04485-2-52-1'], ['1512.04485-1-45-0', '1512.04485-2-46-0'], ['1512.04485-1-45-1', '1512.04485-2-46-1'], ['1512.04485-1-17-0', '1512.04485-2-18-0'], ['1512.04485-1-17-1', '1512.04485-2-18-1'], ['1512.04485-1-52-0', '1512.04485-2-53-0'], ['1512.04485-1-13-0', '1512.04485-2-14-0'], ['1512.04485-1-13-1', '1512.04485-2-14-1'], ['1512.04485-1-13-4', '1512.04485-2-14-4'], ['1512.04485-1-47-0', '1512.04485-2-48-0'], ['1512.04485-1-47-2', '1512.04485-2-48-2'], ['1512.04485-1-47-3', '1512.04485-2-48-3'], ['1512.04485-1-66-0', '1512.04485-2-67-0'], ['1512.04485-1-66-3', '1512.04485-2-67-3'], ['1512.04485-1-66-4', '1512.04485-2-67-4'], ['1512.04485-1-30-0', '1512.04485-2-31-0'], ['1512.04485-1-30-1', '1512.04485-2-31-1'], ['1512.04485-1-30-2', '1512.04485-2-31-2'], ['1512.04485-1-30-4', '1512.04485-2-31-4'], ['1512.04485-1-30-5', '1512.04485-2-31-5'], ['1512.04485-1-30-6', '1512.04485-2-31-6'], ['1512.04485-1-76-0', '1512.04485-2-78-0'], ['1512.04485-1-76-1', '1512.04485-2-78-1'], ['1512.04485-1-35-1', '1512.04485-2-36-1'], ['1512.04485-1-62-0', '1512.04485-2-63-0'], ['1512.04485-1-62-2', '1512.04485-2-63-2'], ['1512.04485-1-62-3', '1512.04485-2-63-3'], ['1512.04485-1-62-4', '1512.04485-2-63-4'], ['1512.04485-1-62-5', '1512.04485-2-63-5'], ['1512.04485-1-62-6', '1512.04485-2-63-6'], ['1512.04485-1-62-7', '1512.04485-2-63-7'], ['1512.04485-1-62-8', '1512.04485-2-63-8'], ['1512.04485-1-83-0', '1512.04485-2-85-0'], ['1512.04485-1-83-1', '1512.04485-2-85-1'], ['1512.04485-1-33-1', '1512.04485-2-34-1'], ['1512.04485-1-14-0', '1512.04485-2-15-0'], ['1512.04485-1-80-0', '1512.04485-2-82-0'], ['1512.04485-1-88-0', '1512.04485-2-90-0'], ['1512.04485-1-88-1', '1512.04485-2-90-1'], ['1512.04485-1-88-2', '1512.04485-2-90-2'], ['1512.04485-1-88-3', '1512.04485-2-90-3'], ['1512.04485-1-88-4', '1512.04485-2-90-4'], ['1512.04485-1-67-0', '1512.04485-2-68-0'], ['1512.04485-1-67-1', '1512.04485-2-68-1'], ['1512.04485-1-67-2', '1512.04485-2-68-2'], ['1512.04485-1-67-3', '1512.04485-2-68-3'], ['1512.04485-1-63-0', '1512.04485-2-64-0'], ['1512.04485-1-63-1', '1512.04485-2-64-1'], ['1512.04485-1-79-0', '1512.04485-2-81-0'], ['1512.04485-1-18-0', '1512.04485-2-19-0'], ['1512.04485-1-71-0', '1512.04485-2-73-0'], ['1512.04485-1-70-0', '1512.04485-2-71-0'], ['1512.04485-1-70-2', '1512.04485-2-71-2'], ['1512.04485-1-70-3', '1512.04485-2-71-3'], ['1512.04485-1-70-4', '1512.04485-2-71-4'], ['1512.04485-1-70-5', '1512.04485-2-71-5'], ['1512.04485-1-70-6', '1512.04485-2-71-6'], ['1512.04485-1-70-7', '1512.04485-2-72-0'], ['1512.04485-1-70-8', '1512.04485-2-72-1'], ['1512.04485-1-70-10', '1512.04485-2-72-3'], ['1512.04485-1-70-11', '1512.04485-2-72-4'], ['1512.04485-1-69-1', '1512.04485-2-70-1'], ['1512.04485-1-78-2', '1512.04485-2-80-2'], ['1512.04485-1-10-1', '1512.04485-2-11-0'], ['1512.04485-1-5-6', '1512.04485-2-5-6'], ['1512.04485-1-12-0', '1512.04485-2-13-0'], ['1512.04485-1-77-1', '1512.04485-2-79-1'], ['1512.04485-1-28-0', '1512.04485-2-29-0'], ['1512.04485-1-60-0', '1512.04485-2-61-0'], ['1512.04485-1-34-0', '1512.04485-2-35-0'], ['1512.04485-1-51-0', '1512.04485-2-52-0'], ['1512.04485-1-13-3', '1512.04485-2-14-3'], ['1512.04485-1-47-4', '1512.04485-2-48-4'], ['1512.04485-1-47-5', '1512.04485-2-48-5'], ['1512.04485-1-66-1', '1512.04485-2-67-1'], ['1512.04485-1-62-1', '1512.04485-2-63-1'], ['1512.04485-1-85-0', '1512.04485-2-87-0'], ['1512.04485-1-5-15', '1512.04485-2-5-16'], ['1512.04485-1-14-1', '1512.04485-2-15-1'], ['1512.04485-1-14-1', '1512.04485-2-15-2']]

[['1512.04485-1-69-0', '1512.04485-2-70-0'], ['1512.04485-1-69-2', '1512.04485-2-70-3'], ['1512.04485-1-78-0', '1512.04485-2-80-0'], ['1512.04485-1-78-1', '1512.04485-2-80-1'], ['1512.04485-1-78-3', '1512.04485-2-80-3'], ['1512.04485-1-78-4', '1512.04485-2-80-4'], ['1512.04485-1-10-0', '1512.04485-2-11-1'], ['1512.04485-1-2-0', '1512.04485-2-2-0'], ['1512.04485-1-2-1', '1512.04485-2-2-1'], ['1512.04485-1-2-2', '1512.04485-2-2-2'], ['1512.04485-1-2-3', '1512.04485-2-2-3'], ['1512.04485-1-2-4', '1512.04485-2-2-4'], ['1512.04485-1-64-0', '1512.04485-2-65-0'], ['1512.04485-1-64-1', '1512.04485-2-65-1'], ['1512.04485-1-64-2', '1512.04485-2-65-2'], ['1512.04485-1-46-0', '1512.04485-2-47-0'], ['1512.04485-1-46-1', '1512.04485-2-47-1'], ['1512.04485-1-46-2', '1512.04485-2-47-2'], ['1512.04485-1-46-3', '1512.04485-2-47-3'], ['1512.04485-1-46-4', '1512.04485-2-47-4'], ['1512.04485-1-54-0', '1512.04485-2-55-0'], ['1512.04485-1-54-1', '1512.04485-2-55-1'], ['1512.04485-1-23-0', '1512.04485-2-24-0'], ['1512.04485-1-5-0', '1512.04485-2-5-0'], ['1512.04485-1-5-1', '1512.04485-2-5-1'], ['1512.04485-1-5-2', '1512.04485-2-5-2'], ['1512.04485-1-5-3', '1512.04485-2-5-3'], ['1512.04485-1-5-4', '1512.04485-2-5-4'], ['1512.04485-1-5-5', '1512.04485-2-5-5'], ['1512.04485-1-5-7', '1512.04485-2-5-7'], ['1512.04485-1-5-8', '1512.04485-2-5-8'], ['1512.04485-1-5-9', '1512.04485-2-5-9'], ['1512.04485-1-5-10', '1512.04485-2-5-10'], ['1512.04485-1-5-11', '1512.04485-2-5-11'], ['1512.04485-1-5-12', '1512.04485-2-5-12'], ['1512.04485-1-5-13', '1512.04485-2-5-13'], ['1512.04485-1-7-0', '1512.04485-2-7-0'], ['1512.04485-1-7-1', '1512.04485-2-7-1'], ['1512.04485-1-7-2', '1512.04485-2-7-2'], ['1512.04485-1-12-1', '1512.04485-2-13-1'], ['1512.04485-1-12-2', '1512.04485-2-13-2'], ['1512.04485-1-77-0', '1512.04485-2-79-0'], ['1512.04485-1-77-2', '1512.04485-2-79-2'], ['1512.04485-1-53-0', '1512.04485-2-54-0'], ['1512.04485-1-53-1', '1512.04485-2-54-1'], ['1512.04485-1-21-0', '1512.04485-2-22-0'], ['1512.04485-1-26-0', '1512.04485-2-27-0'], ['1512.04485-1-60-1', '1512.04485-2-61-1'], ['1512.04485-1-9-1', '1512.04485-2-9-1'], ['1512.04485-1-9-2', '1512.04485-2-9-2'], ['1512.04485-1-9-4', '1512.04485-2-9-4'], ['1512.04485-1-0-0', '1512.04485-2-0-0'], ['1512.04485-1-0-2', '1512.04485-2-0-2'], ['1512.04485-1-86-0', '1512.04485-2-88-0'], ['1512.04485-1-86-2', '1512.04485-2-88-2'], ['1512.04485-1-49-2', '1512.04485-2-50-2'], ['1512.04485-1-49-3', '1512.04485-2-50-3'], ['1512.04485-1-49-5', '1512.04485-2-50-5'], ['1512.04485-1-49-7', '1512.04485-2-50-7'], ['1512.04485-1-16-0', '1512.04485-2-17-0'], ['1512.04485-1-39-0', '1512.04485-2-40-0'], ['1512.04485-1-39-1', '1512.04485-2-40-1'], ['1512.04485-1-82-0', '1512.04485-2-84-0'], ['1512.04485-1-82-1', '1512.04485-2-84-1'], ['1512.04485-1-51-1', '1512.04485-2-52-1'], ['1512.04485-1-45-0', '1512.04485-2-46-0'], ['1512.04485-1-45-1', '1512.04485-2-46-1'], ['1512.04485-1-17-0', '1512.04485-2-18-0'], ['1512.04485-1-17-1', '1512.04485-2-18-1'], ['1512.04485-1-52-0', '1512.04485-2-53-0'], ['1512.04485-1-13-0', '1512.04485-2-14-0'], ['1512.04485-1-13-1', '1512.04485-2-14-1'], ['1512.04485-1-13-4', '1512.04485-2-14-4'], ['1512.04485-1-47-0', '1512.04485-2-48-0'], ['1512.04485-1-47-2', '1512.04485-2-48-2'], ['1512.04485-1-47-3', '1512.04485-2-48-3'], ['1512.04485-1-66-0', '1512.04485-2-67-0'], ['1512.04485-1-66-3', '1512.04485-2-67-3'], ['1512.04485-1-66-4', '1512.04485-2-67-4'], ['1512.04485-1-30-0', '1512.04485-2-31-0'], ['1512.04485-1-30-1', '1512.04485-2-31-1'], ['1512.04485-1-30-2', '1512.04485-2-31-2'], ['1512.04485-1-30-4', '1512.04485-2-31-4'], ['1512.04485-1-30-5', '1512.04485-2-31-5'], ['1512.04485-1-30-6', '1512.04485-2-31-6'], ['1512.04485-1-76-0', '1512.04485-2-78-0'], ['1512.04485-1-76-1', '1512.04485-2-78-1'], ['1512.04485-1-35-1', '1512.04485-2-36-1'], ['1512.04485-1-62-0', '1512.04485-2-63-0'], ['1512.04485-1-62-2', '1512.04485-2-63-2'], ['1512.04485-1-62-3', '1512.04485-2-63-3'], ['1512.04485-1-62-4', '1512.04485-2-63-4'], ['1512.04485-1-62-5', '1512.04485-2-63-5'], ['1512.04485-1-62-6', '1512.04485-2-63-6'], ['1512.04485-1-62-7', '1512.04485-2-63-7'], ['1512.04485-1-62-8', '1512.04485-2-63-8'], ['1512.04485-1-83-0', '1512.04485-2-85-0'], ['1512.04485-1-83-1', '1512.04485-2-85-1'], ['1512.04485-1-33-1', '1512.04485-2-34-1'], ['1512.04485-1-14-0', '1512.04485-2-15-0'], ['1512.04485-1-80-0', '1512.04485-2-82-0'], ['1512.04485-1-88-0', '1512.04485-2-90-0'], ['1512.04485-1-88-1', '1512.04485-2-90-1'], ['1512.04485-1-88-2', '1512.04485-2-90-2'], ['1512.04485-1-88-3', '1512.04485-2-90-3'], ['1512.04485-1-88-4', '1512.04485-2-90-4'], ['1512.04485-1-67-0', '1512.04485-2-68-0'], ['1512.04485-1-67-1', '1512.04485-2-68-1'], ['1512.04485-1-67-2', '1512.04485-2-68-2'], ['1512.04485-1-67-3', '1512.04485-2-68-3'], ['1512.04485-1-63-0', '1512.04485-2-64-0'], ['1512.04485-1-63-1', '1512.04485-2-64-1'], ['1512.04485-1-79-0', '1512.04485-2-81-0'], ['1512.04485-1-18-0', '1512.04485-2-19-0'], ['1512.04485-1-71-0', '1512.04485-2-73-0'], ['1512.04485-1-70-0', '1512.04485-2-71-0'], ['1512.04485-1-70-2', '1512.04485-2-71-2'], ['1512.04485-1-70-3', '1512.04485-2-71-3'], ['1512.04485-1-70-4', '1512.04485-2-71-4'], ['1512.04485-1-70-5', '1512.04485-2-71-5'], ['1512.04485-1-70-6', '1512.04485-2-71-6'], ['1512.04485-1-70-7', '1512.04485-2-72-0'], ['1512.04485-1-70-8', '1512.04485-2-72-1'], ['1512.04485-1-70-10', '1512.04485-2-72-3'], ['1512.04485-1-70-11', '1512.04485-2-72-4']]

[['1512.04485-1-69-1', '1512.04485-2-70-1'], ['1512.04485-1-78-2', '1512.04485-2-80-2'], ['1512.04485-1-10-1', '1512.04485-2-11-0'], ['1512.04485-1-5-6', '1512.04485-2-5-6'], ['1512.04485-1-12-0', '1512.04485-2-13-0'], ['1512.04485-1-77-1', '1512.04485-2-79-1'], ['1512.04485-1-28-0', '1512.04485-2-29-0'], ['1512.04485-1-60-0', '1512.04485-2-61-0'], ['1512.04485-1-34-0', '1512.04485-2-35-0'], ['1512.04485-1-51-0', '1512.04485-2-52-0'], ['1512.04485-1-13-3', '1512.04485-2-14-3'], ['1512.04485-1-47-4', '1512.04485-2-48-4'], ['1512.04485-1-47-5', '1512.04485-2-48-5'], ['1512.04485-1-66-1', '1512.04485-2-67-1'], ['1512.04485-1-62-1', '1512.04485-2-63-1'], ['1512.04485-1-85-0', '1512.04485-2-87-0']]

[]

[['1512.04485-1-5-15', '1512.04485-2-5-16'], ['1512.04485-1-14-1', '1512.04485-2-15-1'], ['1512.04485-1-14-1', '1512.04485-2-15-2']]

[]

['1512.04485-1-0-1', '1512.04485-1-5-14', '1512.04485-1-8-0', '1512.04485-1-8-1', '1512.04485-1-8-2', '1512.04485-1-9-0', '1512.04485-1-9-3', '1512.04485-1-11-0', '1512.04485-1-13-2', '1512.04485-1-13-5', '1512.04485-1-15-0', '1512.04485-1-15-1', '1512.04485-1-15-2', '1512.04485-1-20-0', '1512.04485-1-20-1', '1512.04485-1-20-2', '1512.04485-1-20-3', '1512.04485-1-22-0', '1512.04485-1-24-0', '1512.04485-1-24-1', '1512.04485-1-25-0', '1512.04485-1-29-0', '1512.04485-1-29-1', '1512.04485-1-29-2', '1512.04485-1-30-3', '1512.04485-1-32-0', '1512.04485-1-33-0', '1512.04485-1-35-0', '1512.04485-1-36-0', '1512.04485-1-36-1', '1512.04485-1-37-0', '1512.04485-1-37-1', '1512.04485-1-40-0', '1512.04485-1-41-0', '1512.04485-1-42-0', '1512.04485-1-43-0', '1512.04485-1-44-0', '1512.04485-1-47-1', '1512.04485-1-48-0', '1512.04485-1-49-0', '1512.04485-1-49-1', '1512.04485-1-49-4', '1512.04485-1-49-6', '1512.04485-1-50-0', '1512.04485-1-55-0', '1512.04485-1-55-1', '1512.04485-1-56-0', '1512.04485-1-57-0', '1512.04485-1-58-0', '1512.04485-1-65-0', '1512.04485-1-65-1', '1512.04485-1-66-2', '1512.04485-1-70-1', '1512.04485-1-70-9', '1512.04485-1-72-0', '1512.04485-1-73-0', '1512.04485-1-74-0', '1512.04485-1-77-3', '1512.04485-1-81-0', '1512.04485-1-85-1', '1512.04485-1-86-1', '1512.04485-1-89-0', '1512.04485-1-90-0', '1512.04485-1-92-0', '1512.04485-1-94-0', '1512.04485-2-0-1', '1512.04485-2-5-14', '1512.04485-2-8-0', '1512.04485-2-8-1', '1512.04485-2-8-2', '1512.04485-2-9-0', '1512.04485-2-9-3', '1512.04485-2-12-0', '1512.04485-2-14-2', '1512.04485-2-14-5', '1512.04485-2-16-0', '1512.04485-2-16-1', '1512.04485-2-16-2', '1512.04485-2-21-0', '1512.04485-2-21-1', '1512.04485-2-21-2', '1512.04485-2-21-3', '1512.04485-2-23-0', '1512.04485-2-25-0', '1512.04485-2-25-1', '1512.04485-2-26-0', '1512.04485-2-30-0', '1512.04485-2-30-1', '1512.04485-2-30-2', '1512.04485-2-31-3', '1512.04485-2-33-0', '1512.04485-2-34-0', '1512.04485-2-36-0', '1512.04485-2-37-0', '1512.04485-2-37-1', '1512.04485-2-38-0', '1512.04485-2-38-1', '1512.04485-2-41-0', '1512.04485-2-42-0', '1512.04485-2-43-0', '1512.04485-2-44-0', '1512.04485-2-45-0', '1512.04485-2-48-1', '1512.04485-2-49-0', '1512.04485-2-50-0', '1512.04485-2-50-1', '1512.04485-2-50-4', '1512.04485-2-50-6', '1512.04485-2-51-0', '1512.04485-2-56-0', '1512.04485-2-56-1', '1512.04485-2-57-0', '1512.04485-2-58-0', '1512.04485-2-59-0', '1512.04485-2-66-0', '1512.04485-2-66-1', '1512.04485-2-67-2', '1512.04485-2-71-1', '1512.04485-2-72-2', '1512.04485-2-74-0', '1512.04485-2-75-0', '1512.04485-2-76-0', '1512.04485-2-79-3', '1512.04485-2-83-0', '1512.04485-2-87-1', '1512.04485-2-88-1', '1512.04485-2-91-3', '1512.04485-2-92-0']

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

https://arxiv.org/abs/1512.04485

1603.02679

{'1603.02679-1-0-0': 'Near-field cosmology - using detailed observations of the Local Group and its environs to study wide-ranging questions in galaxy formation and dark matter physics - has become a mature and rich field over the past decade.', '1603.02679-1-0-1': 'There are lingering concerns, however, that the relatively small size of the present-day Local Group ([MATH] diameter) imposes insurmountable sample-variance uncertainties, limiting its broader utility.', '1603.02679-1-0-2': 'We consider the evolution of the Local Group with time and show that it reaches [MATH] co-moving Mpc in linear size (a volume of [MATH]) at [MATH].', '1603.02679-1-0-3': 'The Local Group is a representative portion of the Universe at early cosmic epochs according to multiple metrics.', '1603.02679-1-0-4': 'In a sense, the Local Group is therefore the ultimate deep field: its stellar fossil record traces the cosmic evolution for galaxies with [MATH] (reaching [MATH] at [MATH]) over a region that, in terms of size, is comparable to or larger than the Hubble Ultra-Deep Field (HUDF) for the entire history of the Universe.', '1603.02679-1-0-5': 'It is highly complementary to the HUDF, as it probes much fainter galaxies but does not contain the intrinsically rarer, brighter sources that are detectable in the HUDF.', '1603.02679-1-0-6': 'Archaeological studies in the Local Group also provide the ability to trace the evolution of individual galaxies across time as opposed to evaluating statistical connections between temporally distinct populations.', '1603.02679-1-0-7': 'In the era, resolved stellar populations will probe regions larger than the HUDF and any deep fields, further enhancing the value of near-field cosmology.', '1603.02679-1-1-0': '# Introduction', '1603.02679-1-2-0': 'The standard introduction of a paper on near-field cosmology extols the virtues of the Local Group as a cosmic Rosetta Stone that provides archaeological clues left behind by untold generations of stars, clues that may unlock unsolved mysteries in galaxy formation.', '1603.02679-1-2-1': 'A frequent concern is that resolved-star studies are inherently limited to a region that is relatively small and likely biased, however, setting fundamental limits on the broader applicability of results based on near-field studies.', '1603.02679-1-2-2': 'In this Letter, we show that the volume spanned by the high-redshift progenitors of the Local Group was large enough to have been typical in many respects.', '1603.02679-1-2-3': 'The archaeological record imprinted on Local Group galaxies is therefore likely to provide an unbiased view of faint galaxy populations at early times, making near-field observations a powerful complement to direct deep-field studies.', '1603.02679-1-3-0': '# Methods', '1603.02679-1-4-0': 'At present, the Local Group consists of two dark matter halos, each with virial mass of [MATH] (e.g., ), approaching each other for presumably the first time .', '1603.02679-1-4-1': 'The co-moving Lagrangian volume of the matter contained within the Local Group - i.e., the co-moving volume spanned at earlier epochs by the particles in the current-day Local Group - was therefore larger in the past; the same is true for all dark matter halos.', '1603.02679-1-4-2': 'Thus, while the Local Group is a very specific (and non-typical) volume of the Universe today, its properties at earlier times should be closer to an average portion of the Universe.', '1603.02679-1-4-3': 'Precisely how much closer is the topic of this paper.', '1603.02679-1-5-0': 'A starting point for understanding the evolution of the Local Group with time is the spherical collapse model , in which the radius of the Local Group at maximum expansion was approximately twice as large as it is at the present day.', '1603.02679-1-5-1': "However, this underestimates the size of the Local Group's Lagrangian volume at earlier times, predominantly because of the non-spherical nature of gravitational collapse in an expanding Universe.", '1603.02679-1-5-2': 'Cosmological zoom-in simulations, by design, supply a way of following the evolution of Lagrangian regions surrounding specific halos from linear fluctuations to the highly non-linear regime (see for a discussion of this technique, which was originally described in , and for examples of the evolution of Lagrangian volumes with redshift).', '1603.02679-1-5-3': 'The ELVIS suite of [MATH]-body zoom-in simulations provides 12 Local Group analogs simulated from [MATH] to [MATH], each of which is uncontaminated by lower resolution particles over a spherical region having a radius of at least [MATH] centered on the [MATH] barycenter of the Local Group.', '1603.02679-1-5-4': 'In what follows, we use this suite to study the co-moving volume probed by the Local Group at higher redshifts.', '1603.02679-1-6-0': 'In our analysis, we first eliminate three ELVIS pairs that contain a third large, nearby halo, as these would bias any results.', '1603.02679-1-6-1': "For the remaining 9 pairs, we identify all subhalos within [MATH] of the Local Group's [MATH] barycenter and track all of their progenitors back through time.", '1603.02679-1-6-2': 'There are occasionally individual subhalos that come from regions that are distant from the vast majority of the matter that forms the Local Group; such subhalos can artificially increase the inferred volume of the Local Group at earlier times.', '1603.02679-1-6-3': 'To eliminate these objects, we run a friends-of-friends group finder with a large linking length of 400 kpc and retain only the main grouping.', '1603.02679-1-6-4': 'In practice, this removes [MATH] of subhalos at [MATH].', '1603.02679-1-6-5': 'We then identify the positions spanned by the progenitors of the remaining Local Group subhalos above the ELVIS completeness limit of [MATH] at each earlier snapshot; this constitutes the "proto-Local-Group" at each epoch.', '1603.02679-1-6-6': 'It is important to note that the number of galaxies or halos in the proto-Local-Group is much larger than the number in the Local Group at [MATH] owing to mergers and disruption over time.', '1603.02679-1-7-0': 'If we are only interested in understanding the Local Group itself, this would be sufficient.', '1603.02679-1-7-1': 'To place the Local Group in context at higher redshifts, however, we must understand the full environment that the proto-Local-Group occupies.', '1603.02679-1-7-2': 'We therefore compute, at each snapshot, the minimum cuboid volume defined by the proto-Local-Group - i.e., the rectangular cuboid defined by the minimum and maximum co-moving coordinate locations of all proto-Local-Group progenitors at that time, [MATH] - and identify all additional halos in this region (i.e., halos that appear to be part of the proto-Local-Group but that do not end up in the Local Group at [MATH]).', '1603.02679-1-7-3': 'The inclusion of these objects roughly doubles the counts at [MATH] within [MATH], with proto-Local-Group halos dominating the central portion of [MATH] and the additional halos populating the outskirts of the volume.', '1603.02679-1-8-0': 'We define the linear size of the proto-Local-Group, [MATH] as the geometric mean of the three axes defining [MATH]; in other words, [MATH].', '1603.02679-1-8-1': 'At [MATH], the Local Group volume is defined by a sphere of radius [MATH], so [MATH] (the actual number depends on the distribution of halos at [MATH] but can never exceed [MATH]).', '1603.02679-1-8-2': 'At higher redshifts, [MATH] can, in principle, become highly elongated in one or two dimensions.', '1603.02679-1-8-3': 'In practice, however, we find that this is not the case: at [MATH], the median minor-to-major axis ratio is 0.76 and the median intermediate-to-major axis ratio is 0.81, and in only one case is the minor axis smaller than half of the major axis size.', '1603.02679-1-8-4': 'The typical [MATH] is moderately prolate: 6 of 9 simulated proto-Local-Group s have a triaxiality parameter [MATH] (see ) larger than 0.5.', '1603.02679-1-9-0': '# The Local Group Through Time', '1603.02679-1-10-0': 'Figure [REF] shows the co-moving linear size, [MATH], of the proto-Local-Group going back in time to [MATH].', '1603.02679-1-10-1': 'Thin gray lines show the size of individual LG pairs from the ELVIS simulation suite, while the thick black line shows the median value across the ELVIS pairs at each redshift.', '1603.02679-1-10-2': 'The linear size of the proto-Local-Group increases with increasing redshift, reaching [MATH] (co-moving) at [MATH].', '1603.02679-1-10-3': 'Going back in time, therefore, the Local Group probes a significantly larger (co-moving) volume than it does today.', '1603.02679-1-10-4': "To give context to the Local Group's size at earlier epochs, Fig. [REF] also shows the co-moving linear size of the HUDF (, assuming an angular size of [MATH]) as a function of redshift (magenta curve).", '1603.02679-1-10-5': 'At all epochs later than [MATH] (the last 85% of cosmic time), the proto-Local-Group covers a larger area on the sky than the HUDF.', '1603.02679-1-11-0': 'It is important to understand how representative such portions of the Universe are at each cosmological epoch.', '1603.02679-1-11-1': 'One way to do this is to compute the rms amplitude of density fluctuations [MATH] in regions having volumes equal to [MATH].', '1603.02679-1-11-2': 'In classical Press-Schechteryearparpress1974 theory and its extensions, the typical scale [MATH] that is collapsing at a given epoch has [MATH] (where subscript "lin" indicates that the relevant rms amplitude comes from linear theory, extrapolated to the redshift in question).', '1603.02679-1-11-3': 'Roughly speaking, scales with [MATH] have collapsed while those with [MATH] are firmly in the linear regime.', '1603.02679-1-12-0': 'In reality, linear theory underestimates [MATH].', '1603.02679-1-12-1': 'Cosmological simulations account for effects of non-linear growth; accordingly, we use the Illustris suite to compute [MATH].', '1603.02679-1-12-2': 'At each snapshot, we evaluate the density field for Illustris-Dark-1 on a [MATH] grid, compute the overdensity [MATH] for each cell [MATH], then calculate [MATH] by smoothing the gridded overdensity field with a real-space top-hat filter having a volume equal to [MATH] (so the mass contained in the volume is [MATH]).', '1603.02679-1-12-3': 'The rms amplitude of fluctuations is therefore equivalently characterized by [MATH] or [MATH]', '1603.02679-1-13-0': 'The resulting values for [MATH] are plotted as a solid black curve in Figure [REF]; the linear theory value of [MATH] is shown as a dashed black curve.', '1603.02679-1-13-1': 'We also compute the same quantities for the HUDF, [MATH], and plot these with magenta curves.', '1603.02679-1-13-2': 'Both the Local Group and the HUDF have [MATH] for [MATH].', '1603.02679-1-13-3': 'At high redshift, both probe volumes that are well-described by linear theory.', '1603.02679-1-13-4': 'In particular, the volumes probed by the proto-Local-Group and by cubic slices of the HUDF ([MATH]) in the reionization era ([MATH]) have [MATH].', '1603.02679-1-13-5': 'For a broad discussion of variance in deep-field galaxy counts, see [CITATION].', '1603.02679-1-14-0': 'The results above have established that the proto-Local-Group was substantially larger at earlier epochs, large enough to cover a volume that becomes non-linear only after [MATH].', '1603.02679-1-14-1': 'A further, and more stringent, test is to compare the mass function in the region defined by the proto-Local-Group to the cosmological mass function at earlier times.', '1603.02679-1-14-2': 'In Figure [REF], we plot the cumulative co-moving number density [MATH] of halos within [MATH] for each ELVIS pair (thin gray lines), as well as the median across the simulation suite (thick black line), at [MATH].', '1603.02679-1-14-3': 'The cosmological expectation, as encapsulated by the Sheth-Tormenyearparsheth2001 mass function, is plotted as a magenta line.', '1603.02679-1-14-4': 'The mass function in [MATH] matches the cosmological mass function for [MATH], with counts in the proto-Local-Group region falling below Sheth-Tormen at higher masses (smaller number densities) owing to the size of this region.', '1603.02679-1-14-5': 'The volume covered by the proto-Local-Group at [MATH] is therefore a cosmologically representative region for the mass function of halos with [MATH], a remarkable result.', '1603.02679-1-15-0': 'As indicated by the scale on the right side of the figure, there should be [MATH] halos with [MATH] (this approximately corresponds to halos above the atomic cooling threshold) and [MATH] halos with [MATH] in the [MATH] proto-Local-Group region.', '1603.02679-1-15-1': 'It is this large number of low-mass systems, coupled with the small value of [MATH], that makes mass functions in the proto-Local-Group cosmologically representative even in the [MATH] co-moving volume at early times (note the small variance in normalizations of the mass functions in Fig. [REF]).', '1603.02679-1-16-0': 'By [MATH], the proto-Local-Group has become somewhat more dense, and the equivalent plot of mass functions shows that counts in the proto-Local-Group volume exceed Sheth-Tormen expectations by [MATH] at this epoch.', '1603.02679-1-16-1': 'However, the shape of the mass functions still matches that of Sheth-Tormen for cumulative number densities larger than [MATH] (equivalently, [MATH]).', '1603.02679-1-17-0': '# Discussion', '1603.02679-1-18-0': 'The similarity between the (1D) size of the proto-Local-Group and the HUDF suggests an interpretation of the Local Group at eariler epochs: observations of the Local Group can be thought of as providing a (very) narrow slice in time of the HUDF.', '1603.02679-1-18-1': 'Given our ability to measure resolved star formation histories of Local Group galaxies, we can look at the Local Group at a variety of "snapshots" in time.', '1603.02679-1-18-2': 'This is therefore the same as looking at a series of thin transverse slices through the HUDF.', '1603.02679-1-18-3': 'A complete census of galaxies within 1-2 Mpc of the Local Group with depth to reach the oldest main sequence turn-off would therefore allow a continuous look at galaxy formation and evolution - i.e., tracking individual galaxies across cosmic time - in a size equivalent to the HUDF to depths of [MATH]) at [MATH].', '1603.02679-1-19-0': 'We emphasize that resolved-star studies of the Local Group provide an almost perfectly complementary view of galaxy formation to deep blank-field (and lensing) observations of the high-redshift Universe with Hubble.', '1603.02679-1-19-1': 'At [MATH], the faintest galaxies in blank-field HST observations are likely to be more massive than the progenitor of the Milky Way at that time .', '1603.02679-1-19-2': 'Archaeological studies of Local Group galaxies extend the range of galaxies to at least 8 magnitudes fainter .', '1603.02679-1-19-3': 'HST is therefore capable of probing galaxy formation over six decades in mass ([MATH]) and 12-13 Gyr in time over an area comparable to the HUDF via the stellar fossil record.', '1603.02679-1-20-0': 'The power of near-field studies, and their complementarity to direct high-[MATH] observations, is emphasized in Figures [REF] and [REF].', '1603.02679-1-20-1': 'For both figures, we assign UV luminosities at [MATH] to ELVIS dark matter halos via abundance matching based on the global UV luminosity function from [CITATION] and the Sheth-Tormen mass function.', '1603.02679-1-20-2': 'Figure [REF] shows slices through the density distribution of the Illustris simulation at [MATH], along with boxes indicating the approximate size of the proto-Local-Group at each of those redshifts.', '1603.02679-1-20-3': 'The insets at [MATH] show galaxies that can be observed either directly in the proto-Local-Group with the James Webb Space Telescope (; left) or through archaeological studies in the Local Group with (right).', '1603.02679-1-20-4': 'deep fields will have many more galaxies at a range of redshifts, while the stellar fossil record in the Local Group probes a huge number of galaxies that will be unobservably faint at cosmic dawn.', '1603.02679-1-21-0': 'Fig. [REF] shows the cumulative galaxy luminosity function from [CITATION] at [MATH] (thick black curve), extrapolated from the observational limit of [MATH] all the way to [MATH].', '1603.02679-1-21-1': 'Each gray line shows the luminosity function in the proto-Local-Group volume from an individual ELVIS simulation, while the dashed black line shows the mean among all ELVIS pairs.', '1603.02679-1-21-2': 'Note that the agreement in normalization between the global luminosity function and the proto-Local-Group regions from ELVIS was not pre-determined but rather is a result of the agreement between the global halo mass function and the halo mass function within the proto-Local-Group at [MATH].', '1603.02679-1-21-3': 'The approximate reach of HST and is shown in the figure, while the proto-LG covers the entire region of the figure in which the halo mass functions are non-zero.', '1603.02679-1-22-0': 'The high angular resolution of and its exquisite sensitivity to (red) ancient main sequence turn-off stars make it ideally suited to extend the same census to 3-5 Mpc, meaning it is capable of surveying a region within the Local Volume that is markedly larger than the HUDF (and any deep field) at all redshifts.', '1603.02679-1-22-1': 'The complementarity with direct observations of galaxies at early cosmic epochs will also be enhanced, as will likely detect galaxies as faint as [MATH] in blank fields at [MATH] , comparable to progenitors of the Large Magellanic Cloud .', '1603.02679-1-22-2': 'The Wide-Field Infrared Survey Telescope (WFIRST) has potential to be similarly transformative for near-field cosmology.', '1603.02679-1-22-3': 'The addition of an optical filter (e.g., [MATH]-band) would improve its angular resolution (and temperature sensitivity), allowing it to reach the ancient main sequence turn-offs of galaxies out to [MATH]5 Mpc.', '1603.02679-1-22-4': "This modification, combined with WFIRST's extremely wide field of view, would capture full star formation histories over the entire spatial extent of virtually any nearby galaxy in a single pointing, revolutionizing how we study and understand the evolution of low-mass galaxies."}

{'1603.02679-2-0-0': 'Near-field cosmology - using detailed observations of the Local Group and its environs to study wide-ranging questions in galaxy formation and dark matter physics - has become a mature and rich field over the past decade.', '1603.02679-2-0-1': 'There are lingering concerns, however, that the relatively small size of the present-day Local Group ([MATH] diameter) imposes insurmountable sample-variance uncertainties, limiting its broader utility.', '1603.02679-2-0-2': "We consider the region spanned by the Local Group's progenitors at earlier times and show that it reaches [MATH] co-moving Mpc in linear size (a volume of [MATH]) at [MATH].", '1603.02679-2-0-3': 'This size at early cosmic epochs is large enough to be representative in terms of the matter density and counts of dark matter halos with [MATH].', '1603.02679-2-0-4': "The Local Group's stellar fossil record traces the cosmic evolution of galaxies with [MATH] (reaching [MATH] at [MATH]) over a region that is comparable to or larger than the Hubble Ultra-Deep Field (HUDF) for the entire history of the Universe.", '1603.02679-2-0-5': 'It is highly complementary to the HUDF, as it probes much fainter galaxies but does not contain the intrinsically rarer, brighter sources that are detectable in the HUDF.', '1603.02679-2-0-6': 'Archaeological studies in the Local Group also provide the ability to trace the evolution of individual galaxies across time as opposed to evaluating statistical connections between temporally distinct populations.', '1603.02679-2-0-7': 'In the era, resolved stellar populations will probe regions larger than the HUDF and any deep fields, further enhancing the value of near-field cosmology.', '1603.02679-2-1-0': '# Introduction', '1603.02679-2-2-0': 'The standard introduction of a paper on near-field cosmology extols the virtues of the Local Group as a cosmic Rosetta Stone that provides archaeological clues left behind by untold generations of stars, clues that may unlock unsolved mysteries in galaxy formation.', '1603.02679-2-2-1': 'A frequent concern is that resolved-star studies are inherently limited to a region that is relatively small and likely biased, however, setting fundamental limits on the broader applicability of results based on near-field studies.', '1603.02679-2-2-2': 'In this Letter, we show that the volume spanned by the high-redshift progenitors of the Local Group was large enough to have been typical in many respects.', '1603.02679-2-2-3': 'The archaeological record imprinted on Local Group galaxies is therefore likely to provide an unbiased view of faint galaxy populations at early times, making near-field observations a powerful complement to direct deep-field studies.', '1603.02679-2-3-0': '# Methods', '1603.02679-2-4-0': 'At present, the Local Group consists of two dark matter halos, each with virial mass of [MATH] (e.g., ), approaching each other for presumably the first time .', '1603.02679-2-4-1': 'The co-moving Lagrangian volume of the matter contained within the Local Group - i.e., the co-moving volume spanned at earlier epochs by the particles in the current-day Local Group - was therefore larger in the past; the same is true for all dark matter halos.', '1603.02679-2-4-2': 'Thus, while the Local Group is a very specific (and non-typical) volume of the Universe today, its properties at earlier times should be closer to an average portion of the Universe.', '1603.02679-2-4-3': 'Precisely how much closer is the topic of this paper.', '1603.02679-2-5-0': 'The spherical collapse model provides a starting point for understanding the evolution of the Local Group.', '1603.02679-2-5-1': 'In this model, the matter within a virialized dark matter halo of radius [MATH] (defined by an average enclosed density of [MATH] times the critical density of the Universe) came from a spherical region with a co-moving Lagrangian radius [MATH] equal to [MATH].', '1603.02679-2-5-2': 'For flat cosmologies with [MATH], this gives [MATH], indicating that the Local Group must have been significantly larger in co-moving size in the early Universe.', '1603.02679-2-5-3': 'Cosmological zoom-in simulations, by design, follow the evolution of Lagrangian regions surrounding specific halos from linear fluctuations to the highly non-linear regime .', '1603.02679-2-5-4': 'The ELVIS suite of [MATH]-body zoom-in simulations provides 12 Local Group analogs simulated from [MATH] to [MATH], each of which is uncontaminated by lower resolution particles over a spherical region having a radius of at least [MATH] centered on the [MATH] barycenter of the Local Group.', '1603.02679-2-5-5': 'In what follows, we use this suite to study the co-moving volume probed by the Local Group at higher redshifts.', '1603.02679-2-6-0': 'In our analysis, we first eliminate three ELVIS pairs that contain a third large, nearby halo, as these would bias any results.', '1603.02679-2-6-1': "For the remaining 9 pairs, we identify all subhalos within [MATH] of the Local Group's [MATH] barycenter and track all of their progenitors back through time.", '1603.02679-2-6-2': 'There are occasionally individual subhalos that come from regions that are distant from the vast majority of the matter that forms the Local Group; such subhalos can artificially increase the inferred volume of the Local Group at earlier times.', '1603.02679-2-6-3': 'To eliminate these objects, we run a friends-of-friends group finder with a large linking length of 400 kpc and retain only the main grouping.', '1603.02679-2-6-4': 'In practice, this removes [MATH] of subhalos at [MATH].', '1603.02679-2-6-5': 'We then identify the positions spanned by the progenitors of the remaining Local Group subhalos above the ELVIS completeness limit of [MATH] at each earlier snapshot; this constitutes the "proto-Local-Group" at each epoch.', '1603.02679-2-6-6': 'It is important to note that the number of galaxies or halos in the proto-Local-Group is much larger than the number in the Local Group at [MATH] owing to mergers and disruption over time.', '1603.02679-2-7-0': 'If we are only interested in understanding the Local Group itself, this would be sufficient.', '1603.02679-2-7-1': 'To place the Local Group in context at higher redshifts, however, we must understand the full environment that the proto-Local-Group occupies.', '1603.02679-2-7-2': 'We therefore compute, at each snapshot, the minimum cuboid volume defined by the proto-Local-Group - i.e., the rectangular cuboid defined by the minimum and maximum co-moving coordinate locations of all proto-Local-Group progenitors at that time, [MATH] - and identify all additional halos in this region (i.e., halos that appear to be part of the proto-Local-Group but that do not end up in the Local Group at [MATH]).', '1603.02679-2-7-3': 'The inclusion of these objects roughly doubles the counts at [MATH] within [MATH].', '1603.02679-2-7-4': 'This doubling is consistent with the extra volume contained in the cuboid [MATH] region circumscribing a sphere, although the proto-Local-Group progenitors are not confined to a spherical region at [MATH].', '1603.02679-2-7-5': 'Proto-Local-Group halos dominate the central portion of [MATH] and the additional halos populate the outskirts of the volume.', '1603.02679-2-8-0': 'We define the linear size of the proto-Local-Group, [MATH] as the geometric mean of the three axes defining [MATH]; in other words, [MATH].', '1603.02679-2-8-1': 'At [MATH], the Local Group volume is defined by a sphere of radius [MATH], so [MATH] (the actual number depends on the distribution of halos at [MATH] but can never exceed [MATH]).', '1603.02679-2-8-2': 'At higher redshifts, [MATH] can, in principle, become highly elongated in one or two dimensions.', '1603.02679-2-8-3': 'In practice, however, we find that this is not the case: at [MATH], the median minor-to-major axis ratio is 0.76 and the median intermediate-to-major axis ratio is 0.81, and in only one case is the minor axis smaller than half of the major axis size.', '1603.02679-2-8-4': 'The typical [MATH] is moderately prolate: 6 of 9 simulated proto-Local-Group s have a triaxiality parameter [MATH] (see ) larger than 0.5.', '1603.02679-2-9-0': '# The Local Group Through Time', '1603.02679-2-10-0': 'Figure [REF] shows the co-moving linear size, [MATH], of the proto-Local-Group going back in time to [MATH].', '1603.02679-2-10-1': 'Thin gray lines show the size of individual Local Group pairs from the ELVIS simulation suite, while the thick black line shows the median value across the ELVIS pairs at each redshift.', '1603.02679-2-10-2': 'The linear size of the proto-Local-Group increases with increasing redshift, reaching [MATH] (co-moving) at [MATH].', '1603.02679-2-10-3': 'Going back in time, therefore, the Local Group probes a significantly larger (co-moving) volume than it does today.', '1603.02679-2-10-4': "To give context to the Local Group's size at earlier epochs, Fig. [REF] also shows the co-moving linear size of the HUDF (, assuming an angular size of [MATH]) as a function of redshift (magenta curve).", '1603.02679-2-10-5': 'At all epochs later than [MATH] (the last 85% of cosmic time), the proto-Local-Group covers a larger area on the sky than the HUDF.', '1603.02679-2-11-0': 'It is important to understand how representative such portions of the Universe are at each cosmological epoch.', '1603.02679-2-11-1': 'One way to do this is to compute the rms amplitude of density fluctuations [MATH] in regions having volumes equal to [MATH].', '1603.02679-2-11-2': 'In classical Press-Schechteryearparpress1974 theory and its extensions, the typical scale [MATH] that is collapsing at a given epoch has [MATH] (where subscript "lin" indicates that the relevant rms amplitude comes from linear theory, extrapolated to the redshift in question).', '1603.02679-2-11-3': 'Roughly speaking, scales with [MATH] have collapsed while those with [MATH] are firmly in the linear regime.', '1603.02679-2-12-0': 'In reality, linear theory underestimates [MATH].', '1603.02679-2-12-1': 'Cosmological simulations account for effects of non-linear growth; accordingly, we use the Illustris suite to compute [MATH].', '1603.02679-2-12-2': 'At each snapshot, we evaluate the density field for Illustris-Dark-1 on a [MATH] grid, compute the overdensity [MATH] for each cell [MATH], then calculate [MATH] by smoothing the gridded overdensity field with a real-space top-hat filter having a volume equal to [MATH] (so the mass contained in the volume is [MATH]).', '1603.02679-2-12-3': 'The rms amplitude of fluctuations is equivalently characterized by [MATH] or [MATH]', '1603.02679-2-13-0': 'The resulting values for [MATH] are plotted as a solid black curve in Figure [REF]; the linear theory value of [MATH] is shown as a dashed black curve.', '1603.02679-2-13-1': 'We also compute the same quantities for the HUDF, [MATH], and plot these with magenta curves.', '1603.02679-2-13-2': 'Both the Local Group and the HUDF have [MATH] for [MATH].', '1603.02679-2-13-3': 'At high redshift, both probe volumes that are well-described by linear theory.', '1603.02679-2-13-4': 'In particular, the volumes probed by the proto-Local-Group and by cubic slices of the HUDF ([MATH]) in the reionization era ([MATH]) have [MATH].', '1603.02679-2-13-5': 'For a broad discussion of variance in deep-field galaxy counts, see [CITATION].', '1603.02679-2-14-0': 'The results above have established that the proto-Local-Group was substantially larger at earlier epochs, large enough to cover a volume that becomes non-linear only after [MATH].', '1603.02679-2-14-1': 'A further, and more stringent, test is to compare the mass function in the region defined by the proto-Local-Group to the cosmological mass function at earlier times.', '1603.02679-2-14-2': 'In Figure [REF], we plot the cumulative co-moving number density [MATH] of halos within [MATH] for each ELVIS pair (thin gray lines), as well as the median across the simulation suite (thick black line), at [MATH].', '1603.02679-2-14-3': 'The cosmological expectation, as encapsulated by the Sheth-Tormenyearparsheth2001 mass function, is plotted as a magenta line.', '1603.02679-2-14-4': 'The mass function in [MATH] matches the cosmological mass function for [MATH], with counts in the proto-Local-Group region falling below Sheth-Tormen at higher masses (smaller number densities) owing to the size of this region.', '1603.02679-2-14-5': 'The volume covered by the proto-Local-Group at [MATH] is therefore a cosmologically representative region for the mass function of halos with [MATH], a remarkable result.', '1603.02679-2-15-0': 'As indicated by the scale on the right side of the figure, there should be [MATH] halos with [MATH] (this approximately corresponds to halos above the atomic cooling threshold) and [MATH] halos with [MATH] in the [MATH] proto-Local-Group region.', '1603.02679-2-15-1': 'It is this large number of low-mass systems, coupled with the small value of [MATH], that makes mass functions in the proto-Local-Group cosmologically representative even in the [MATH] co-moving volume at early times (note the small variance in normalizations of the mass functions in Fig. [REF]).', '1603.02679-2-16-0': 'By [MATH], the proto-Local-Group has become somewhat more dense, and the equivalent plot of mass functions shows that counts in the proto-Local-Group volume exceed Sheth-Tormen expectations by [MATH] at this epoch.', '1603.02679-2-16-1': 'However, the shape of the mass functions still matches that of Sheth-Tormen for cumulative number densities larger than [MATH] (equivalently, [MATH]).', '1603.02679-2-17-0': '# Discussion', '1603.02679-2-18-0': 'The previous sections have explored a simple yet important question: given regions that end up as Local Group analogs at [MATH], how representative is the volume that their progenitors spanned at earlier times?', '1603.02679-2-18-1': 'The rough estimate given at the start of Sec. [REF] indicates that a collapsed dark matter halo covered a comoving region that was [MATH] times larger in the early Universe than at [MATH], and the results of Sec. [REF] show that the Local Group (which is not a virialized region today) was approximately 3.5 times larger than its present-day size at early times.', '1603.02679-2-18-2': 'The proto-Local-Group is large enough be representative at high redshifts both for the matter density it contains and for counts of dark matter halos with [MATH].', '1603.02679-2-19-0': 'These results are only true when starting with Local-Group-like regions at [MATH] and considering their properties at earlier times; proto-Local-Group-sized regions selected at [MATH] are large enough to be representative at that time but will not generally evolve to be Local Groups at [MATH].', '1603.02679-2-19-1': 'The direct, one-to-one connection between galaxies in the [MATH] Local Group and their ancestors in the proto-Local-Group is complicated by any mergers and disruption of galaxies in the intervening time; this also means that surviving galaxies at [MATH] represent a lower limit on the number of similar galaxies in the proto-Local-Group.', '1603.02679-2-19-2': 'While we plan to address this point in future work, we note that the predicted merger histories of low-mass halos in are such that most present-day dwarfs have not had significant mergers (in terms of stellar mass growth) since the reionization era ; Local Group dwarfs are therefore expected to provide a direct window to low-mass systems in the high-redshift Universe.', '1603.02679-2-20-0': "The representative nature of the proto-Local-Group's volume, coupled with the similarity between the (1D) size of the proto-Local-Group and the HUDF, suggest an interpretation of the Local Group at eariler epochs: observations of the Local Group can be thought of as providing a (very) narrow slice in time of the HUDF.", '1603.02679-2-20-1': 'Given our ability to measure resolved star formation histories of Local Group galaxies, we can look at the Local Group at a variety of "snapshots" in time.', '1603.02679-2-20-2': 'This is the same as looking at a series of thin transverse slices through the HUDF.', '1603.02679-2-20-3': 'A complete census of galaxies within 1-2 Mpc of the Local Group with depth to reach the oldest main sequence turn-off would therefore allow a continuous look at galaxy formation and evolution - i.e., tracking individual galaxies across cosmic time - in a size equivalent to the HUDF to depths of [MATH]) at [MATH].', '1603.02679-2-21-0': 'We emphasize that resolved-star studies of the Local Group provide an almost perfectly complementary view of galaxy formation to deep blank-field (and lensing) observations of the high-redshift Universe with Hubble.', '1603.02679-2-21-1': 'At [MATH], the faintest galaxies in blank-field HST observations are likely to be more massive than the progenitor of the Milky Way at that time .', '1603.02679-2-21-2': 'Archaeological studies of Local Group galaxies extend the range of galaxies to at least 8 magnitudes fainter .', '1603.02679-2-21-3': 'HST is therefore capable of probing galaxy formation over six decades in mass ([MATH]) and 12-13 Gyr in time over an area comparable to the HUDF via the stellar fossil record.', '1603.02679-2-22-0': 'The power of near-field studies, and their complementarity to direct high-[MATH] observations, is emphasized in Figures [REF] and [REF].', '1603.02679-2-22-1': 'For both figures, we assign UV luminosities at [MATH] to ELVIS dark matter halos via abundance matching based on the global UV luminosity function from [CITATION] and the Sheth-Tormen mass function.', '1603.02679-2-22-2': 'Figure [REF] shows slices through the density distribution of the Illustris simulation at [MATH], along with boxes indicating the approximate size of the proto-Local-Group at each of those redshifts.', '1603.02679-2-22-3': 'The insets at [MATH] show galaxies that can be observed either directly in the proto-Local-Group with the James Webb Space Telescope (; left) or through archaeological studies in the Local Group with (right).', '1603.02679-2-22-4': 'deep fields will have many more galaxies at a range of redshifts, while the stellar fossil record in the Local Group probes a huge number of galaxies that will be unobservably faint at cosmic dawn.', '1603.02679-2-23-0': 'Fig. [REF] shows the cumulative galaxy luminosity function from [CITATION] at [MATH] (thick black curve), extrapolated from the observational limit of [MATH] to [MATH].', '1603.02679-2-23-1': 'Each gray line shows the luminosity function in the proto-Local-Group volume from an individual ELVIS simulation, while the dashed black line shows the mean among all ELVIS pairs.', '1603.02679-2-23-2': 'Importantly, the agreement in normalization between the global luminosity function and the proto-Local-Group regions from ELVIS was not pre-determined but rather is a result of the agreement between the global halo mass function and the halo mass function within the proto-Local-Group at [MATH].', '1603.02679-2-23-3': 'The approximate reach of HST and is shown in the figure, while the proto-Local-Group covers the entire region of the figure in which the halo mass functions are non-zero.', '1603.02679-2-24-0': 'The high angular resolution of and its exquisite sensitivity to (red) ancient main sequence turn-off stars make it ideally suited to extend the same census to 3-5 Mpc, meaning it is capable of surveying a region within the Local Volume that is markedly larger than the HUDF (and any deep field) at all redshifts.', '1603.02679-2-24-1': 'The complementarity with direct observations of galaxies at early cosmic epochs will also be enhanced, as will likely detect galaxies as faint as [MATH] in blank fields at [MATH] , comparable to progenitors of the Large Magellanic Cloud (; see for related calculations).', '1603.02679-2-24-2': 'The Wide-Field Infrared Survey Telescope (WFIRST) has potential to be similarly transformative for near-field cosmology.', '1603.02679-2-24-3': 'The addition of an optical filter (e.g., [MATH]-band) would improve its angular resolution (and temperature sensitivity), allowing it to reach the ancient main sequence turn-offs of galaxies out to [MATH]5 Mpc.', '1603.02679-2-24-4': "This modification, combined with WFIRST's extremely wide field of view, would capture full star formation histories over the entire spatial extent of virtually any nearby galaxy in a single pointing, revolutionizing how we study and understand the evolution of low-mass galaxies."}

[['1603.02679-1-10-0', '1603.02679-2-10-0'], ['1603.02679-1-10-2', '1603.02679-2-10-2'], ['1603.02679-1-10-3', '1603.02679-2-10-3'], ['1603.02679-1-10-4', '1603.02679-2-10-4'], ['1603.02679-1-10-5', '1603.02679-2-10-5'], ['1603.02679-1-4-0', '1603.02679-2-4-0'], ['1603.02679-1-4-1', '1603.02679-2-4-1'], ['1603.02679-1-4-2', '1603.02679-2-4-2'], ['1603.02679-1-4-3', '1603.02679-2-4-3'], ['1603.02679-1-11-0', '1603.02679-2-11-0'], ['1603.02679-1-11-1', '1603.02679-2-11-1'], ['1603.02679-1-11-2', '1603.02679-2-11-2'], ['1603.02679-1-11-3', '1603.02679-2-11-3'], ['1603.02679-1-5-3', '1603.02679-2-5-4'], ['1603.02679-1-5-4', '1603.02679-2-5-5'], ['1603.02679-1-21-1', '1603.02679-2-23-1'], ['1603.02679-1-19-0', '1603.02679-2-21-0'], ['1603.02679-1-19-1', '1603.02679-2-21-1'], ['1603.02679-1-19-2', '1603.02679-2-21-2'], ['1603.02679-1-19-3', '1603.02679-2-21-3'], ['1603.02679-1-8-0', '1603.02679-2-8-0'], ['1603.02679-1-8-1', '1603.02679-2-8-1'], ['1603.02679-1-8-2', '1603.02679-2-8-2'], ['1603.02679-1-8-3', '1603.02679-2-8-3'], ['1603.02679-1-8-4', '1603.02679-2-8-4'], ['1603.02679-1-0-0', '1603.02679-2-0-0'], ['1603.02679-1-0-1', '1603.02679-2-0-1'], ['1603.02679-1-0-5', '1603.02679-2-0-5'], ['1603.02679-1-0-6', '1603.02679-2-0-6'], ['1603.02679-1-0-7', '1603.02679-2-0-7'], ['1603.02679-1-16-0', '1603.02679-2-16-0'], ['1603.02679-1-16-1', '1603.02679-2-16-1'], ['1603.02679-1-7-0', '1603.02679-2-7-0'], ['1603.02679-1-7-1', '1603.02679-2-7-1'], ['1603.02679-1-7-2', '1603.02679-2-7-2'], ['1603.02679-1-12-0', '1603.02679-2-12-0'], ['1603.02679-1-12-1', '1603.02679-2-12-1'], ['1603.02679-1-12-2', '1603.02679-2-12-2'], ['1603.02679-1-13-0', '1603.02679-2-13-0'], ['1603.02679-1-13-1', '1603.02679-2-13-1'], ['1603.02679-1-13-2', '1603.02679-2-13-2'], ['1603.02679-1-13-3', '1603.02679-2-13-3'], ['1603.02679-1-13-4', '1603.02679-2-13-4'], ['1603.02679-1-13-5', '1603.02679-2-13-5'], ['1603.02679-1-14-0', '1603.02679-2-14-0'], ['1603.02679-1-14-1', '1603.02679-2-14-1'], ['1603.02679-1-14-2', '1603.02679-2-14-2'], ['1603.02679-1-14-3', '1603.02679-2-14-3'], ['1603.02679-1-14-4', '1603.02679-2-14-4'], ['1603.02679-1-14-5', '1603.02679-2-14-5'], ['1603.02679-1-20-0', '1603.02679-2-22-0'], ['1603.02679-1-20-1', '1603.02679-2-22-1'], ['1603.02679-1-20-2', '1603.02679-2-22-2'], ['1603.02679-1-20-3', '1603.02679-2-22-3'], ['1603.02679-1-20-4', '1603.02679-2-22-4'], ['1603.02679-1-22-0', '1603.02679-2-24-0'], ['1603.02679-1-22-2', '1603.02679-2-24-2'], ['1603.02679-1-22-3', '1603.02679-2-24-3'], ['1603.02679-1-22-4', '1603.02679-2-24-4'], ['1603.02679-1-18-1', '1603.02679-2-20-1'], ['1603.02679-1-18-3', '1603.02679-2-20-3'], ['1603.02679-1-2-0', '1603.02679-2-2-0'], ['1603.02679-1-2-1', '1603.02679-2-2-1'], ['1603.02679-1-2-2', '1603.02679-2-2-2'], ['1603.02679-1-2-3', '1603.02679-2-2-3'], ['1603.02679-1-6-0', '1603.02679-2-6-0'], ['1603.02679-1-6-1', '1603.02679-2-6-1'], ['1603.02679-1-6-2', '1603.02679-2-6-2'], ['1603.02679-1-6-3', '1603.02679-2-6-3'], ['1603.02679-1-6-4', '1603.02679-2-6-4'], ['1603.02679-1-6-5', '1603.02679-2-6-5'], ['1603.02679-1-6-6', '1603.02679-2-6-6'], ['1603.02679-1-15-0', '1603.02679-2-15-0'], ['1603.02679-1-15-1', '1603.02679-2-15-1'], ['1603.02679-1-10-1', '1603.02679-2-10-1'], ['1603.02679-1-21-0', '1603.02679-2-23-0'], ['1603.02679-1-21-2', '1603.02679-2-23-2'], ['1603.02679-1-21-3', '1603.02679-2-23-3'], ['1603.02679-1-0-2', '1603.02679-2-0-2'], ['1603.02679-1-0-4', '1603.02679-2-0-4'], ['1603.02679-1-12-3', '1603.02679-2-12-3'], ['1603.02679-1-22-1', '1603.02679-2-24-1'], ['1603.02679-1-18-0', '1603.02679-2-20-0'], ['1603.02679-1-18-2', '1603.02679-2-20-2'], ['1603.02679-1-5-0', '1603.02679-2-5-0'], ['1603.02679-1-5-2', '1603.02679-2-5-3'], ['1603.02679-1-7-3', '1603.02679-2-7-3'], ['1603.02679-1-7-3', '1603.02679-2-7-5']]

[['1603.02679-1-10-0', '1603.02679-2-10-0'], ['1603.02679-1-10-2', '1603.02679-2-10-2'], ['1603.02679-1-10-3', '1603.02679-2-10-3'], ['1603.02679-1-10-4', '1603.02679-2-10-4'], ['1603.02679-1-10-5', '1603.02679-2-10-5'], ['1603.02679-1-4-0', '1603.02679-2-4-0'], ['1603.02679-1-4-1', '1603.02679-2-4-1'], ['1603.02679-1-4-2', '1603.02679-2-4-2'], ['1603.02679-1-4-3', '1603.02679-2-4-3'], ['1603.02679-1-11-0', '1603.02679-2-11-0'], ['1603.02679-1-11-1', '1603.02679-2-11-1'], ['1603.02679-1-11-2', '1603.02679-2-11-2'], ['1603.02679-1-11-3', '1603.02679-2-11-3'], ['1603.02679-1-5-3', '1603.02679-2-5-4'], ['1603.02679-1-5-4', '1603.02679-2-5-5'], ['1603.02679-1-21-1', '1603.02679-2-23-1'], ['1603.02679-1-19-0', '1603.02679-2-21-0'], ['1603.02679-1-19-1', '1603.02679-2-21-1'], ['1603.02679-1-19-2', '1603.02679-2-21-2'], ['1603.02679-1-19-3', '1603.02679-2-21-3'], ['1603.02679-1-8-0', '1603.02679-2-8-0'], ['1603.02679-1-8-1', '1603.02679-2-8-1'], ['1603.02679-1-8-2', '1603.02679-2-8-2'], ['1603.02679-1-8-3', '1603.02679-2-8-3'], ['1603.02679-1-8-4', '1603.02679-2-8-4'], ['1603.02679-1-0-0', '1603.02679-2-0-0'], ['1603.02679-1-0-1', '1603.02679-2-0-1'], ['1603.02679-1-0-5', '1603.02679-2-0-5'], ['1603.02679-1-0-6', '1603.02679-2-0-6'], ['1603.02679-1-0-7', '1603.02679-2-0-7'], ['1603.02679-1-16-0', '1603.02679-2-16-0'], ['1603.02679-1-16-1', '1603.02679-2-16-1'], ['1603.02679-1-7-0', '1603.02679-2-7-0'], ['1603.02679-1-7-1', '1603.02679-2-7-1'], ['1603.02679-1-7-2', '1603.02679-2-7-2'], ['1603.02679-1-12-0', '1603.02679-2-12-0'], ['1603.02679-1-12-1', '1603.02679-2-12-1'], ['1603.02679-1-12-2', '1603.02679-2-12-2'], ['1603.02679-1-13-0', '1603.02679-2-13-0'], ['1603.02679-1-13-1', '1603.02679-2-13-1'], ['1603.02679-1-13-2', '1603.02679-2-13-2'], ['1603.02679-1-13-3', '1603.02679-2-13-3'], ['1603.02679-1-13-4', '1603.02679-2-13-4'], ['1603.02679-1-13-5', '1603.02679-2-13-5'], ['1603.02679-1-14-0', '1603.02679-2-14-0'], ['1603.02679-1-14-1', '1603.02679-2-14-1'], ['1603.02679-1-14-2', '1603.02679-2-14-2'], ['1603.02679-1-14-3', '1603.02679-2-14-3'], ['1603.02679-1-14-4', '1603.02679-2-14-4'], ['1603.02679-1-14-5', '1603.02679-2-14-5'], ['1603.02679-1-20-0', '1603.02679-2-22-0'], ['1603.02679-1-20-1', '1603.02679-2-22-1'], ['1603.02679-1-20-2', '1603.02679-2-22-2'], ['1603.02679-1-20-3', '1603.02679-2-22-3'], ['1603.02679-1-20-4', '1603.02679-2-22-4'], ['1603.02679-1-22-0', '1603.02679-2-24-0'], ['1603.02679-1-22-2', '1603.02679-2-24-2'], ['1603.02679-1-22-3', '1603.02679-2-24-3'], ['1603.02679-1-22-4', '1603.02679-2-24-4'], ['1603.02679-1-18-1', '1603.02679-2-20-1'], ['1603.02679-1-18-3', '1603.02679-2-20-3'], ['1603.02679-1-2-0', '1603.02679-2-2-0'], ['1603.02679-1-2-1', '1603.02679-2-2-1'], ['1603.02679-1-2-2', '1603.02679-2-2-2'], ['1603.02679-1-2-3', '1603.02679-2-2-3'], ['1603.02679-1-6-0', '1603.02679-2-6-0'], ['1603.02679-1-6-1', '1603.02679-2-6-1'], ['1603.02679-1-6-2', '1603.02679-2-6-2'], ['1603.02679-1-6-3', '1603.02679-2-6-3'], ['1603.02679-1-6-4', '1603.02679-2-6-4'], ['1603.02679-1-6-5', '1603.02679-2-6-5'], ['1603.02679-1-6-6', '1603.02679-2-6-6'], ['1603.02679-1-15-0', '1603.02679-2-15-0'], ['1603.02679-1-15-1', '1603.02679-2-15-1']]

[['1603.02679-1-10-1', '1603.02679-2-10-1'], ['1603.02679-1-21-0', '1603.02679-2-23-0'], ['1603.02679-1-21-2', '1603.02679-2-23-2'], ['1603.02679-1-21-3', '1603.02679-2-23-3'], ['1603.02679-1-0-2', '1603.02679-2-0-2'], ['1603.02679-1-0-4', '1603.02679-2-0-4'], ['1603.02679-1-12-3', '1603.02679-2-12-3'], ['1603.02679-1-22-1', '1603.02679-2-24-1'], ['1603.02679-1-18-0', '1603.02679-2-20-0'], ['1603.02679-1-18-2', '1603.02679-2-20-2']]

[]

[['1603.02679-1-5-0', '1603.02679-2-5-0'], ['1603.02679-1-5-2', '1603.02679-2-5-3'], ['1603.02679-1-7-3', '1603.02679-2-7-3'], ['1603.02679-1-7-3', '1603.02679-2-7-5']]

[]

[]

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

https://arxiv.org/abs/1603.02679

cond-mat-0006293

{'cond-mat-0006293-1-0-0': 'We present a detailed description of a classification scheme for phase transitions in finite systems based on the distribution of Fisher zeros of the canonical partition function in the complex temperature plane.', 'cond-mat-0006293-1-0-1': 'We apply this scheme to finite Bose-systems in power law traps within a semi-analytic approach with a continuous one-particle density of states [MATH] for different values of [MATH] and to a three dimensional harmonically confined ideal Bose-gas with discrete energy levels.', 'cond-mat-0006293-1-0-2': 'Our results indicate that the order of the Bose-Einstein condensation phase transition sensitively depends on the confining potential.', 'cond-mat-0006293-1-1-0': '# Introduction', 'cond-mat-0006293-1-2-0': 'In 1924 S. Bose and A. Einstein predicted that in a system of bosons at temperatures below a certain critical temperature [MATH] the single-particle ground state is macroscopically occupied [CITATION].', 'cond-mat-0006293-1-2-1': 'This effect is commonly referred as Bose-Einstein condensation and a large number of phenomena, among others the superfluidity of [MATH]He and superconductivity, are identified as signatures of this effect.', 'cond-mat-0006293-1-2-2': 'However, the physical situation is very intricate in most experiments.', 'cond-mat-0006293-1-2-3': 'In the case of superconductivity, pairs of electrons are treated as effective bosons and the coupling of these Cooper pairs to the phonons of the solid complicates the theoretical description.', 'cond-mat-0006293-1-2-4': 'Even the explanation of superfluidity of [MATH]He, where the interaction of the noble gas atoms is quite weak, constitutes a very difficult and up to date not completely solved problem of statistical physics.', 'cond-mat-0006293-1-3-0': 'Considerable confusion exists about the order of the phase transition in Bose-Einstein systems even for as simple systems as the 3-dimensional ideal gas.', 'cond-mat-0006293-1-3-1': 'The origin of this confusion might be that the condensate phase is a one-state and thus quite unusual phase and that the standard grand-canonical treatment requires a separate handling of the ground state.', 'cond-mat-0006293-1-4-0': 'Recent experiments with dilute gases of alkali atoms in magnetic [CITATION] and optical [CITATION] traps are in some sense the up to now best experimental approximation of the ideal non-interacting Bose-Einstein system in an external power law potential.', 'cond-mat-0006293-1-4-1': 'The achievement of ultra-low temperatures by laser cooling and evaporative cooling opens the opportunity to study the Bose-Einstein condensation under systematic variation of adjustable external parameters, e.g the trap geometry, the number of trapped atoms, the temperature, and by the choice of the alkali atoms the effective interparticle interactions.', 'cond-mat-0006293-1-4-2': 'Even in the approximation of non-interacting particles the explanation of these experiments requires some care, because the number of bosons in these novel traps is finite and fixed and the standard grand-canonical treatment is not appropriate.', 'cond-mat-0006293-1-4-3': 'The effect of the finite particle numbers on the second moments of the distribution function, e.g. the specific heat and the fluctuation of the ground state occupation number has been addressed in a number of publications [CITATION].', 'cond-mat-0006293-1-4-4': 'In [CITATION] we have presented a recursion method to calculate the canonical partition function for non-interacting bosons and investigated the dependency of the thermodynamic properties of the condensate on the trap geometry.', 'cond-mat-0006293-1-5-0': 'In this paper we address the classification of the phase transition of a finite number of non-interacting bosons in a power law trap with an effective one-particle density of states [MATH] being formally equivalent to a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-1-5-1': 'We use a classification scheme based on the distribution of zeros of the canonical partition function initially developed by Grossman et al. [CITATION], and Fisher et al. [CITATION], which has been extended by us [CITATION] as a classification scheme for finite systems.', 'cond-mat-0006293-1-5-2': 'On the basis of this classification scheme we are able to explain the fundamental difference between Bose-Einstein condensation in 3-dimensional traps [CITATION] and the recently discovered Bose-Einstein condensation of a gas of hydrogen atoms absorbed on the surface of liquid helium by Safonov et al. [CITATION] in two dimensions.', 'cond-mat-0006293-1-6-0': 'We give a detailed review of the classification scheme in Sec. [REF].', 'cond-mat-0006293-1-6-1': 'In Sec. [REF] we present the method for the calculation of the canonical partition function in the complex plane and describe details of the numerical implementation.', 'cond-mat-0006293-1-6-2': 'Our results for [MATH] and particle numbers varying from 10 to 300 are presented in Sec. [REF] as well as calculations for a 3-dimensional parabolically confined Bose-gas.', 'cond-mat-0006293-1-7-0': '# Classification scheme', 'cond-mat-0006293-1-8-0': 'In 1952 Yang and Lee have shown that the grand canonical partition function can be written as a function of its zeros in the complex fugacity plane, which lie for systems with hard-core interactions and for the Ising model on a unit circle [CITATION].', 'cond-mat-0006293-1-9-0': 'Grossmann et al. [CITATION] and Fisher [CITATION] have extended this approach to the canonical ensemble by analytic continuation of the inverse temperature to the complex plane [MATH].', 'cond-mat-0006293-1-9-1': 'Within this treatment all phenomenologically known types of phase transitions in macroscopic systems can be identified from the properties of the distribution of zeros of the canonical partition function.', 'cond-mat-0006293-1-10-0': 'In [CITATION] we have presented a classification scheme for finite systems which has its macroscopic equivalent in the scheme given by Grossmann.', 'cond-mat-0006293-1-10-1': 'As usual the canonical partition function reads [EQUATION] which we write as a product [MATH], where [MATH] describes the limiting behavior of [MATH] for [MATH] imposing that [MATH].', 'cond-mat-0006293-1-10-2': 'This limiting partition function will only depend on the external potential applied to the system, whereas [MATH] will depend on the specific interaction between the system particles.', 'cond-mat-0006293-1-10-3': 'E.g. for a [MATH]-particle system in a [MATH]-dimensional harmonic trap [MATH] and thus the zeros of [MATH] are the same as the zeros of [MATH].', 'cond-mat-0006293-1-10-4': 'Since the partition function is an integral function, the zeros [MATH] are complex conjugated and the partition function reads [EQUATION]', 'cond-mat-0006293-1-10-5': 'The zeros of [MATH] are the poles of the Helmholtz free energy [MATH], i.e. the free energy is analytic everywhere in the complex temperature plane except at the zeros of [MATH].', 'cond-mat-0006293-1-11-0': 'Different phases are represented by regions of holomorphy which are separated by zeros lying dense on lines in the complex temperature plane.', 'cond-mat-0006293-1-11-1': 'In finite systems the zeros do not squeeze on lines which leads to a more blurred separation of different phases.', 'cond-mat-0006293-1-11-2': 'We interpret the zeros as boundary posts between two phases.', 'cond-mat-0006293-1-11-3': 'The distribution of zeros contains the complete thermodynamic information about the system and all thermodynamic properties are derivable from it.', 'cond-mat-0006293-1-11-4': 'Within this picture the interaction part of the specific heat is given by [EQUATION]', 'cond-mat-0006293-1-11-5': 'The zeros of the partition function are poles of [MATH].', 'cond-mat-0006293-1-11-6': 'As can be seen from Eq. ([REF]) a zero approaching the real axis infinitely close causes a divergence at real temperature.', 'cond-mat-0006293-1-11-7': 'The contribution of a zero [MATH] to the specific heat decreases with increasing imaginary part [MATH].', 'cond-mat-0006293-1-11-8': 'Thus, the thermodynamic properties of a system are governed by the zeros of [MATH] close to the real axis.', 'cond-mat-0006293-1-12-0': 'The basic idea of the classification scheme for phase transition in small systems presented in [CITATION] is that the distribution of zeros close to the real axis can approximately be described by three parameters, where two of them reflect the order of the phase transition and the third merely the size of the system.', 'cond-mat-0006293-1-13-0': 'We assume that the zeros lie on straight lines (see Fig. [REF]) with a discrete density of zeros given by [EQUATION] with [MATH], and approximate for small [MATH] the density of zeros by a simple power law [MATH].', 'cond-mat-0006293-1-13-1': 'Considering only the first three zeros the exponent [MATH] can be estimated as [EQUATION]', 'cond-mat-0006293-1-13-2': 'The second parameter to describe the distribution of zeros is given by [MATH] where [MATH] is the crossing angle of the line of zeros with the real axis (see Fig. [REF]).', 'cond-mat-0006293-1-13-3': 'The discreteness of the system is reflected in the imaginary part [MATH] of the zero closest to the real axis.', 'cond-mat-0006293-1-14-0': 'In the thermodynamic limit we have always [MATH].', 'cond-mat-0006293-1-14-1': 'In this case the parameters [MATH] and [MATH] coincide with those defined by Grossmann et al [CITATION], who have shown how different types of phase transitions can be attributed to certain values of [MATH] and [MATH].', 'cond-mat-0006293-1-14-2': 'They claimed that [MATH] and [MATH] corresponds to a first order phase transition, second order transitions correspond to [MATH] with [MATH] or [MATH], and that higher order phase transition correspond to [MATH].', 'cond-mat-0006293-1-14-3': 'For macroscopic systems (with [MATH] cannot be smaller than zero, because this would cause a divergence of the internal energy.', 'cond-mat-0006293-1-14-4': 'However in small systems with a finite [MATH] this is possible.', 'cond-mat-0006293-1-15-0': 'In our classification scheme we therefore define phase transitions in small systems to be of first order for [MATH], while second and higher order transitions are defined in complete analogy to the Grossmann scheme augmented by the third parameter [MATH].', 'cond-mat-0006293-1-15-1': 'The definition of a critical temperature [MATH] in small systems is crucial and ambiguous since no thermodynamic properties diverge.', 'cond-mat-0006293-1-15-2': 'Thus, different definitions are possible.', 'cond-mat-0006293-1-15-3': 'We define the critical temperature as [MATH], i.e. the crossing point of the approximated line of zeros with the real temperature axis.', 'cond-mat-0006293-1-15-4': 'An alternative definition is the real part of the first complex zero [MATH].', 'cond-mat-0006293-1-15-5': 'In the thermodynamic limit both definitions coincide.', 'cond-mat-0006293-1-16-0': 'Comparing the specific heats calculated for different discrete distributions of zeros shows the advantages of this classification scheme.', 'cond-mat-0006293-1-16-1': 'Fig. [REF] shows (a) three distributions of zeros lying on straight lines corresponding to a first order transition ([MATH] and [MATH]), a second order transition ([MATH] and [MATH]), and a third order phase transition ([MATH] and [MATH]) and (b) the pertinent specific heats.', 'cond-mat-0006293-1-16-2': 'In all cases the specific heat exhibit a hump extending over a finite temperature region and cannot be used to classify the phase transition.', 'cond-mat-0006293-1-16-3': 'In contrast, even for very small systems (large [MATH]) the order of the phase transition is extractable from the distribution of zeros.', 'cond-mat-0006293-1-17-0': 'The zeros of the canonical partition function have a distinct geometrical interpretation which explains the smoothed curves of the specific heat and other thermodynamic properties in finite systems.', 'cond-mat-0006293-1-18-0': 'Fig. [REF] shows (a) the ground state occupation number [MATH] in the complex temperature plane and (b) the ground state occupation number at real temperatures for a finite ideal Bose gas of [MATH] particles.', 'cond-mat-0006293-1-19-0': 'Zeros of the partition function are poles of the [MATH] and indicated by dark spots, which influence the value of the ground state occupation number at real temperatures impressively.', 'cond-mat-0006293-1-19-1': 'Every pole seems to radiate onto the real axis and therefore determines the occupation number at real temperatures.', 'cond-mat-0006293-1-19-2': 'This radiation extends over a broad temperature range so that the occupation number for real temperatures does not show a discontinuity but a smoothed curve.', 'cond-mat-0006293-1-19-3': 'A closer look at Eq. ([REF]) gives the mathematical explanation for this effect.', 'cond-mat-0006293-1-19-4': 'The discrete distribution of zeros, i.e. [MATH], inhibits the specific heat and all other thermodynamic properties to show a divergency at some critical temperature because the denominators of the arguments of the sum remain finite.', 'cond-mat-0006293-1-20-0': 'Without going into a detailed analysis we note that in the thermodynamic limit the parameter [MATH] is connected to the critical index for the specific heat by [EQUATION]', 'cond-mat-0006293-1-20-1': 'However, since critical indices are used to describe the shape of a divergency at the critical point an extension to small systems seems to be more or less academical.', 'cond-mat-0006293-1-21-0': 'The introduction of complex temperatures might seem artificial at first sight but, in fact, the imaginary parts [MATH] of the complex zeros [MATH] have an obvious quantum mechanical interpretation.', 'cond-mat-0006293-1-21-1': 'We write the quantum mechanical partition function as [EQUATION] introducing a canonical state as a sum over Boltzmann-weighted eigenstates [MATH].', 'cond-mat-0006293-1-21-2': 'We explicitly write the imaginary part as [MATH] since the dimension is [MATH] and the imaginary part therefore can be interpreted as time.', 'cond-mat-0006293-1-21-3': 'Then the imaginary parts [MATH] of the zeros resemble those times for which the overlap of the initial canonical state with the time evoluted state vanishes.', 'cond-mat-0006293-1-21-4': 'However, they are not connected to a single system but to a whole ensemble of identical systems in a heat bath with an initial Boltzmann distribution.', 'cond-mat-0006293-1-22-0': '# BEC in power law traps', 'cond-mat-0006293-1-23-0': 'In this section we assume a continuous single particle density of states [MATH] as an approximation for a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-1-23-1': 'E.g. for the harmonic oscillator this corresponds to the limit of [MATH] and taking only the leading term of the degeneracy of the single particle energy levels.', 'cond-mat-0006293-1-23-2': 'The one-particle partition function is given by the Laplace transformation [EQUATION]', 'cond-mat-0006293-1-23-3': 'The canonical partition function for [MATH] non-interacting bosons can be calculated by the following recursion [CITATION] [EQUATION] where [MATH] is the one-particle partition function at temperature [MATH] and [MATH].', 'cond-mat-0006293-1-23-4': 'For small particle numbers this recursion works fine, even though its numerical effort grows proportional to [MATH].', 'cond-mat-0006293-1-24-0': 'With ([REF]) as [MATH] Eq. ([REF]) leads to a polynomial of order N in [MATH] for [MATH] which can be easily generated using Maple or Mathematica.', 'cond-mat-0006293-1-24-1': 'The zeros of this polynomial can be found by standard numerical methods.', 'cond-mat-0006293-1-25-0': 'Fig. [REF] displays the zeros of the [MATH]-particle partition function for [MATH] in the complex temperature plane for particle numbers [MATH] and [MATH].', 'cond-mat-0006293-1-25-1': 'For [MATH] the zeros approach the positive real axis with increasing particle number and are shifted to higher temperatures which is already at first sight an indicator of phase transitions.', 'cond-mat-0006293-1-25-2': 'For [MATH] the zeros approach the real axis only at negative temperature.', 'cond-mat-0006293-1-25-3': 'This behavior is consistent with the usual prediction that there is no Bose-Einstein condensation for the one-dimensional harmonic oscillator and the two-dimensional ideal Bose gas [CITATION].', 'cond-mat-0006293-1-26-0': 'The symmetry of the distributions of zeros is due to the fact that [MATH] is a polynomial in [MATH].', 'cond-mat-0006293-1-26-1': 'For this reason it can be inferred that for [MATH] the zeros lie on a perfect circle.', 'cond-mat-0006293-1-27-0': 'Fig. [REF] shows the corresponding specific heats calculated using equation ([REF]).', 'cond-mat-0006293-1-27-1': 'As expected, for [MATH] the specific heat has no hump and approaches with increasing temperature the classical value.', 'cond-mat-0006293-1-27-2': 'We therefore expel the analysis of [MATH] from the discussions below.', 'cond-mat-0006293-1-27-3': 'For [MATH] the specific heats show humps or peaks, which get sharper with increasing [MATH] and increasing particle number.', 'cond-mat-0006293-1-27-4': 'However, from these smooth curves the orders of the phase transition cannot be deduced.', 'cond-mat-0006293-1-28-0': 'In Fig. [REF] the classification parameters [MATH] defined above are plotted for two to six dimensions and particle numbers up to [MATH]100.', 'cond-mat-0006293-1-28-1': 'For all values of [MATH] the parameter [MATH] is only a slightly varying function of [MATH] and approaches very fast an almost constant value.', 'cond-mat-0006293-1-28-2': 'Since [MATH] is the primary classification parameter from Fig. [REF](a) we can directly infer that the [MATH] system exhibits a third order phase transition ([MATH] while the transition for all higher dimensions is of second order ([MATH]).', 'cond-mat-0006293-1-28-3': 'For [MATH] the dependence of [MATH] on [MATH] is plotted in Fig. [REF](a).', 'cond-mat-0006293-1-28-4': 'Since [MATH] decreases rather rapidly with increasing [MATH] it can be speculated that systems corresponding to a large [MATH] exhibit a phase transition which is almost of first order.', 'cond-mat-0006293-1-28-5': 'As mentioned above for finite systems even values [MATH] cannot be excluded by mathematical reasons.', 'cond-mat-0006293-1-28-6': 'We note that two-dimensional Bose-gases are an interesting and growing field of research.', 'cond-mat-0006293-1-28-7': 'As it is well known, the ideal free Bose-gas in two dimensions ([MATH]) does not show a phase transition due to thermal fluctuations which destabilize the condensate [CITATION].', 'cond-mat-0006293-1-28-8': 'Switching on a confining potential greatly influences the properties of the gas, the thermal fluctuations are suppressed and the gas will show Bose-Einstein condensation.', 'cond-mat-0006293-1-28-9': 'Recent experiments [CITATION] have shown that Bose-Einstein condensation is possible even though it is called a quasi-condensate.', 'cond-mat-0006293-1-28-10': 'In our notion the quasi-condensate is just a third order phase transition.', 'cond-mat-0006293-1-28-11': 'Thus, our results are in complete agreement with recent experiments and earlier theoretical work.', 'cond-mat-0006293-1-28-12': 'An interesting question in this respect is whether the order of the transition changes for [MATH] in the limit [MATH].', 'cond-mat-0006293-1-28-13': 'Additional calculation for larger [MATH], which are not printed in Fig. [REF] indicate that [MATH] approaches 1 or might even get smaller.', 'cond-mat-0006293-1-28-14': 'Note that [MATH] is equivalent to a hypothetical 4-dimensional ideal Bose gas or Bosons confined in a 2-dimensional parabolic trap.', 'cond-mat-0006293-1-28-15': 'Our results indicate that the order of the phase transition sensitively depends on [MATH] for values around 2.', 'cond-mat-0006293-1-28-16': 'This might be the reason why phase transitions in three space dimensions are sometimes classified as second and sometimes as third order phase transitions.', 'cond-mat-0006293-1-29-0': 'The parameter [MATH] is a measure of the finite size of the system, i.e. the scaling behavior of [MATH] as a function of [MATH] is a measure of how fast a system approaches a true n-th order phase transition in the Ehrenfest sense.', 'cond-mat-0006293-1-29-1': 'The [MATH] dependence of [MATH] is displayed in Fig. [REF](c).', 'cond-mat-0006293-1-29-2': 'The scaling behavior can be approximated by [MATH] with [MATH] ranging between 1.06 and 1.12 for [MATH].', 'cond-mat-0006293-1-30-0': 'The [MATH] dependence of the classification parameter is visualized in Fig. [REF] for 50 particles.', 'cond-mat-0006293-1-30-1': 'For this system size we found [MATH] and [MATH].', 'cond-mat-0006293-1-31-0': 'The results presented above for continuous single particle densities of states [MATH] are obtained within semi-analytical calculations.', 'cond-mat-0006293-1-31-1': 'In order to compare these results to systems with a discrete level density we adopt as a reference system the 3-dimensional harmonic oscillator with the partition function given by [EQUATION] with [MATH].', 'cond-mat-0006293-1-32-0': 'Fig. [REF](a) displays the zeros of the partition function ([REF]) for [MATH] and [MATH].', 'cond-mat-0006293-1-32-1': 'Fig. [REF](b) displays a contour plot of the absolute value of the ground state occupation number [MATH] with [MATH] given by ([REF]) calculated using an alternative recursion formula [CITATION].', 'cond-mat-0006293-1-32-2': 'The zeros of [MATH] are poles of [MATH] and are indicated by dark spots in this figure.', 'cond-mat-0006293-1-33-0': 'Analyzing the distribution of zeros consolidates our speculation that the order of the phase transition sensitively depends on [MATH].', 'cond-mat-0006293-1-33-1': 'The distribution of zeros behaves like the above calculated values for [MATH] but quantitatively like [MATH].', 'cond-mat-0006293-1-33-2': 'Since the degeneracy of the three-dimensional harmonically confined ideal Bose-gas is a second-order polynomial not only the quadratic term has to be taken into account.', 'cond-mat-0006293-1-33-3': 'The linear term becomes dominant for lower temperatures, so for very low temperatures the best approximation of a continuous one-particle density of states is [MATH].', 'cond-mat-0006293-1-33-4': 'The parameter [MATH] supports this statement [CITATION], i.e. [MATH] resides in a region above 1.', 'cond-mat-0006293-1-33-5': 'Whereas the parameter [MATH] behaves like the [MATH] case.', 'cond-mat-0006293-1-33-6': 'Finally the parameter [MATH] which is a measure for the discreteness of the system shows a [MATH] dependence which is comparable to the one for [MATH].', 'cond-mat-0006293-1-33-7': 'Thus, for small systems the phase transition is of third order, it can be speculated if it becomes a second order transition in the thermodynamic limit.', 'cond-mat-0006293-1-34-0': 'Not only qualitatively but also quantitatively our calculations are in very good agreement with recent theoretical works [CITATION].', 'cond-mat-0006293-1-34-1': 'Comparing the critical temperature which we defined in Sec. [REF] with the usually utilized ones like the temperature of the peak of the specific heat [MATH] or the grand-canonically calculated [MATH] confirms our approach.', 'cond-mat-0006293-1-34-2': 'In Fig. [REF] three possible definitions of the critical temperature are given which all coincide in the thermodynamic limit.', 'cond-mat-0006293-1-34-3': 'All definitions show a [MATH] dependence with [MATH] ranging between [MATH] and [MATH].', 'cond-mat-0006293-1-35-0': '# conclusion', 'cond-mat-0006293-1-36-0': 'Starting with the old ideas of Yang and Lee, and Grossmann et al. we have developed a classification scheme for phase transitions in finite systems.', 'cond-mat-0006293-1-36-1': 'Based on the analytic continuation of the inverse temperature [MATH] into the complex plane we have shown the advantages of this approach.', 'cond-mat-0006293-1-36-2': 'The distribution of the so-called Fisher-zeros [MATH] draws enlightening pictures even for small systems whereas the usually referred thermodynamic properties like the specific heat fail to classify the phase transitions properly.', 'cond-mat-0006293-1-36-3': 'The classification scheme presented in this paper enables us to name the order of the transition in a non-ambiguous way.', 'cond-mat-0006293-1-36-4': 'Further we have given an interpretation of complex temperatures not only as an aiding construction.', 'cond-mat-0006293-1-36-5': 'The complex parts [MATH] of the zeros [MATH] resemble times for which a whole ensemble of identical systems under consideration in a heat bath with an initial Boltzmann-distribution looses its memory.', 'cond-mat-0006293-1-37-0': 'We have applied this to ideal non-interacting Bose-gases confined in power-law traps.', 'cond-mat-0006293-1-37-1': 'We have found that the order of the phase transition sensitively depends on the single particle density of states generated by the confining potential.', 'cond-mat-0006293-1-37-2': 'Our results explain the confusion about the order of the phase transitions in different space dimensions and resolve some of the open questions about the nature of the Bose-Einstein condensation.'}

{'cond-mat-0006293-2-0-0': 'We present a detailed description of a classification scheme for phase transitions in finite systems based on the distribution of Fisher zeros of the canonical partition function in the complex temperature plane.', 'cond-mat-0006293-2-0-1': 'We apply this scheme to finite Bose-systems in power law traps within a semi-analytic approach with a continuous one-particle density of states [MATH] for different values of [MATH] and to a three dimensional harmonically confined ideal Bose-gas with discrete energy levels.', 'cond-mat-0006293-2-0-2': 'Our results indicate that the order of the Bose-Einstein condensation phase transition sensitively depends on the confining potential.', 'cond-mat-0006293-2-1-0': '# Introduction', 'cond-mat-0006293-2-2-0': 'In 1924 S. Bose and A. Einstein predicted that in a system of bosons at temperatures below a certain critical temperature [MATH] the single-particle ground state is macroscopically occupied [CITATION].', 'cond-mat-0006293-2-2-1': 'This effect is commonly referred as Bose-Einstein condensation and a large number of phenomena, among others the superfluidity of [MATH]He and superconductivity, are identified as signatures of this effect.', 'cond-mat-0006293-2-2-2': 'However, the physical situation is very intricate in most experiments.', 'cond-mat-0006293-2-2-3': 'In the case of superconductivity, pairs of electrons are treated as effective bosons and the coupling of these Cooper pairs to the phonons of the solid complicates the theoretical description.', 'cond-mat-0006293-2-2-4': 'Even the explanation of superfluidity of [MATH]He, where the interaction of the noble gas atoms is quite weak, constitutes a very difficult and up to date not completely solved problem of statistical physics.', 'cond-mat-0006293-2-3-0': 'Considerable confusion exists about the order of the phase transition in Bose-Einstein systems even for as simple systems as the 3-dimensional ideal gas.', 'cond-mat-0006293-2-3-1': 'The origin of this confusion might be that the condensate phase is a one-state and thus quite unusual phase and that the standard grand-canonical treatment requires a separate handling of the ground state.', 'cond-mat-0006293-2-4-0': 'Recent experiments with dilute gases of alkali atoms in magnetic [CITATION] and optical [CITATION] traps are in some sense the up to now best experimental approximation of the ideal non-interacting Bose-Einstein system in an external power law potential.', 'cond-mat-0006293-2-4-1': 'The achievement of ultra-low temperatures by laser cooling and evaporative cooling opens the opportunity to study the Bose-Einstein condensation under systematic variation of adjustable external parameters, e.g the trap geometry, the number of trapped atoms, the temperature, and by the choice of the alkali atoms the effective interparticle interactions.', 'cond-mat-0006293-2-4-2': 'Even in the approximation of non-interacting particles the explanation of these experiments requires some care, because the number of bosons in these novel traps is finite and fixed and the standard grand-canonical treatment is not appropriate.', 'cond-mat-0006293-2-4-3': 'The effect of the finite particle numbers on the second moments of the distribution function, e.g. the specific heat and the fluctuation of the ground state occupation number has been addressed in a number of publications [CITATION].', 'cond-mat-0006293-2-4-4': 'In [CITATION] we have presented a recursion method to calculate the canonical partition function for non-interacting bosons and investigated the dependency of the thermodynamic properties of the condensate on the trap geometry.', 'cond-mat-0006293-2-5-0': 'In this paper we address the classification of the phase transition of a finite number of non-interacting bosons in a power law trap with an effective one-particle density of states [MATH] being formally equivalent to a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-2-5-1': 'We use a classification scheme based on the distribution of zeros of the canonical partition function initially developed by Grossman et al. [CITATION], and Fisher et al. [CITATION], which has been extended by us [CITATION] as a classification scheme for finite systems.', 'cond-mat-0006293-2-5-2': 'On the basis of this classification scheme we are able to explain the fundamental difference between Bose-Einstein condensation in 3-dimensional traps [CITATION] and the recently discovered Bose-Einstein condensation of a gas of hydrogen atoms absorbed on the surface of liquid helium by Safonov et al. [CITATION] in two dimensions.', 'cond-mat-0006293-2-6-0': 'We give a detailed review of the classification scheme in Sec. [REF].', 'cond-mat-0006293-2-6-1': 'In Sec. [REF] we present the method for the calculation of the canonical partition function in the complex plane and describe details of the numerical implementation.', 'cond-mat-0006293-2-6-2': 'Our results for [MATH] and particle numbers varying from 10 to 300 are presented in Sec. [REF] as well as calculations for a 3-dimensional parabolically confined Bose-gas.', 'cond-mat-0006293-2-7-0': '# Classification scheme', 'cond-mat-0006293-2-8-0': 'In 1952 Yang and Lee have shown that the grand canonical partition function can be written as a function of its zeros in the complex fugacity plane, which lie for systems with hard-core interactions and for the Ising model on a unit circle [CITATION].', 'cond-mat-0006293-2-9-0': 'Grossmann et al. [CITATION] and Fisher [CITATION] have extended this approach to the canonical ensemble by analytic continuation of the inverse temperature to the complex plane [MATH].', 'cond-mat-0006293-2-9-1': 'Within this treatment all phenomenologically known types of phase transitions in macroscopic systems can be identified from the properties of the distribution of zeros of the canonical partition function.', 'cond-mat-0006293-2-10-0': 'In [CITATION] we have presented a classification scheme for finite systems which has its macroscopic equivalent in the scheme given by Grossmann.', 'cond-mat-0006293-2-10-1': 'As usual the canonical partition function reads [EQUATION] which we write as a product [MATH], where [MATH] describes the limiting behavior of [MATH] for [MATH] imposing that [MATH].', 'cond-mat-0006293-2-10-2': 'This limiting partition function will only depend on the external potential applied to the system, whereas [MATH] will depend on the specific interaction between the system particles.', 'cond-mat-0006293-2-10-3': 'E.g. for a [MATH]-particle system in a [MATH]-dimensional harmonic trap [MATH] and thus the zeros of [MATH] are the same as the zeros of [MATH].', 'cond-mat-0006293-2-10-4': 'Since the partition function is an integral function, the zeros [MATH] are complex conjugated and the partition function reads [EQUATION]', 'cond-mat-0006293-2-10-5': 'The zeros of [MATH] are the poles of the Helmholtz free energy [MATH], i.e. the free energy is analytic everywhere in the complex temperature plane except at the zeros of [MATH].', 'cond-mat-0006293-2-11-0': 'Different phases are represented by regions of holomorphy which are separated by zeros lying dense on lines in the complex temperature plane.', 'cond-mat-0006293-2-11-1': 'In finite systems the zeros do not squeeze on lines which leads to a more blurred separation of different phases.', 'cond-mat-0006293-2-11-2': 'We interpret the zeros as boundary posts between two phases.', 'cond-mat-0006293-2-11-3': 'The distribution of zeros contains the complete thermodynamic information about the system and all thermodynamic properties are derivable from it.', 'cond-mat-0006293-2-11-4': 'Within this picture the interaction part of the specific heat is given by [EQUATION]', 'cond-mat-0006293-2-11-5': 'The zeros of the partition function are poles of [MATH].', 'cond-mat-0006293-2-11-6': 'As can be seen from Eq. ([REF]) a zero approaching the real axis infinitely close causes a divergence at real temperature.', 'cond-mat-0006293-2-11-7': 'The contribution of a zero [MATH] to the specific heat decreases with increasing imaginary part [MATH].', 'cond-mat-0006293-2-11-8': 'Thus, the thermodynamic properties of a system are governed by the zeros of [MATH] close to the real axis.', 'cond-mat-0006293-2-12-0': 'The basic idea of the classification scheme for phase transition in small systems presented in [CITATION] is that the distribution of zeros close to the real axis can approximately be described by three parameters, where two of them reflect the order of the phase transition and the third merely the size of the system.', 'cond-mat-0006293-2-13-0': 'We assume that the zeros lie on straight lines (see Fig. [REF]) with a discrete density of zeros given by [EQUATION] with [MATH], and approximate for small [MATH] the density of zeros by a simple power law [MATH].', 'cond-mat-0006293-2-13-1': 'Considering only the first three zeros the exponent [MATH] can be estimated as [EQUATION]', 'cond-mat-0006293-2-13-2': 'The second parameter to describe the distribution of zeros is given by [MATH] where [MATH] is the crossing angle of the line of zeros with the real axis (see Fig. [REF]).', 'cond-mat-0006293-2-13-3': 'The discreteness of the system is reflected in the imaginary part [MATH] of the zero closest to the real axis.', 'cond-mat-0006293-2-14-0': 'In the thermodynamic limit we have always [MATH].', 'cond-mat-0006293-2-14-1': 'In this case the parameters [MATH] and [MATH] coincide with those defined by Grossmann et al [CITATION], who have shown how different types of phase transitions can be attributed to certain values of [MATH] and [MATH].', 'cond-mat-0006293-2-14-2': 'They claimed that [MATH] and [MATH] corresponds to a first order phase transition, second order transitions correspond to [MATH] with [MATH] or [MATH], and that higher order phase transition correspond to [MATH].', 'cond-mat-0006293-2-14-3': 'For macroscopic systems (with [MATH] cannot be smaller than zero, because this would cause a divergence of the internal energy.', 'cond-mat-0006293-2-14-4': 'However in small systems with a finite [MATH] this is possible.', 'cond-mat-0006293-2-15-0': 'In our classification scheme we therefore define phase transitions in small systems to be of first order for [MATH], while second and higher order transitions are defined in complete analogy to the Grossmann scheme augmented by the third parameter [MATH].', 'cond-mat-0006293-2-15-1': 'The definition of a critical temperature [MATH] in small systems is crucial and ambiguous since no thermodynamic properties diverge.', 'cond-mat-0006293-2-15-2': 'Thus, different definitions are possible.', 'cond-mat-0006293-2-15-3': 'We define the critical temperature as [MATH], i.e. the crossing point of the approximated line of zeros with the real temperature axis.', 'cond-mat-0006293-2-15-4': 'An alternative definition is the real part of the first complex zero [MATH].', 'cond-mat-0006293-2-15-5': 'In the thermodynamic limit both definitions coincide.', 'cond-mat-0006293-2-16-0': 'Comparing the specific heats calculated for different discrete distributions of zeros shows the advantages of this classification scheme.', 'cond-mat-0006293-2-16-1': 'Fig. [REF] shows (a) three distributions of zeros lying on straight lines corresponding to a first order transition ([MATH] and [MATH]), a second order transition ([MATH] and [MATH]), and a third order phase transition ([MATH] and [MATH]) and (b) the pertinent specific heats.', 'cond-mat-0006293-2-16-2': 'In all cases the specific heat exhibit a hump extending over a finite temperature region and cannot be used to classify the phase transition.', 'cond-mat-0006293-2-16-3': 'In contrast, even for very small systems (large [MATH]) the order of the phase transition is extractable from the distribution of zeros.', 'cond-mat-0006293-2-17-0': 'The zeros of the canonical partition function have a distinct geometrical interpretation which explains the smoothed curves of the specific heat and other thermodynamic properties in finite systems.', 'cond-mat-0006293-2-18-0': 'Fig. [REF] shows (a) the ground state occupation number [MATH] in the complex temperature plane and (b) the ground state occupation number at real temperatures for a finite ideal Bose gas of [MATH] particles.', 'cond-mat-0006293-2-19-0': 'Zeros of the partition function are poles of the [MATH] and indicated by dark spots, which influence the value of the ground state occupation number at real temperatures impressively.', 'cond-mat-0006293-2-19-1': 'Every pole seems to radiate onto the real axis and therefore determines the occupation number at real temperatures.', 'cond-mat-0006293-2-19-2': 'This radiation extends over a broad temperature range so that the occupation number for real temperatures does not show a discontinuity but a smoothed curve.', 'cond-mat-0006293-2-19-3': 'A closer look at Eq. ([REF]) gives the mathematical explanation for this effect.', 'cond-mat-0006293-2-19-4': 'The discrete distribution of zeros, i.e. [MATH], inhibits the specific heat and all other thermodynamic properties to show a divergency at some critical temperature because the denominators of the arguments of the sum remain finite.', 'cond-mat-0006293-2-20-0': 'Without going into a detailed analysis we note that in the thermodynamic limit the parameter [MATH] is connected to the critical index for the specific heat by [EQUATION]', 'cond-mat-0006293-2-20-1': 'However, since critical indices are used to describe the shape of a divergency at the critical point an extension to small systems seems to be more or less academical.', 'cond-mat-0006293-2-21-0': 'The introduction of complex temperatures might seem artificial at first sight but, in fact, the imaginary parts [MATH] of the complex zeros [MATH] have an obvious quantum mechanical interpretation.', 'cond-mat-0006293-2-21-1': 'We write the quantum mechanical partition function as [EQUATION] introducing a canonical state as a sum over Boltzmann-weighted eigenstates [MATH].', 'cond-mat-0006293-2-21-2': 'We explicitly write the imaginary part as [MATH] since the dimension is [MATH] and the imaginary part therefore can be interpreted as time.', 'cond-mat-0006293-2-21-3': 'Then the imaginary parts [MATH] of the zeros resemble those times for which the overlap of the initial canonical state with the time evoluted state vanishes.', 'cond-mat-0006293-2-21-4': 'However, they are not connected to a single system but to a whole ensemble of identical systems in a heat bath with an initial Boltzmann distribution.', 'cond-mat-0006293-2-22-0': '# BEC in power law traps', 'cond-mat-0006293-2-23-0': 'In this section we assume a continuous single particle density of states [MATH] as an approximation for a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-2-23-1': 'E.g. for the harmonic oscillator this corresponds to the limit of [MATH] and taking only the leading term of the degeneracy of the single particle energy levels.', 'cond-mat-0006293-2-23-2': 'The one-particle partition function is given by the Laplace transformation [EQUATION]', 'cond-mat-0006293-2-23-3': 'The canonical partition function for [MATH] non-interacting bosons can be calculated by the following recursion [CITATION] [EQUATION] where [MATH] is the one-particle partition function at temperature [MATH] and [MATH].', 'cond-mat-0006293-2-23-4': 'For small particle numbers this recursion works fine, even though its numerical effort grows proportional to [MATH].', 'cond-mat-0006293-2-24-0': 'With ([REF]) as [MATH] Eq. ([REF]) leads to a polynomial of order N in [MATH] for [MATH] which can be easily generated using Maple or Mathematica.', 'cond-mat-0006293-2-24-1': 'The zeros of this polynomial can be found by standard numerical methods.', 'cond-mat-0006293-2-25-0': 'Fig. [REF] displays the zeros of the [MATH]-particle partition function for [MATH] in the complex temperature plane for particle numbers [MATH] and [MATH].', 'cond-mat-0006293-2-25-1': 'For [MATH] the zeros approach the positive real axis with increasing particle number and are shifted to higher temperatures which is already at first sight an indicator of phase transitions.', 'cond-mat-0006293-2-25-2': 'For [MATH] the zeros approach the real axis only at negative temperature.', 'cond-mat-0006293-2-25-3': 'This behavior is consistent with the usual prediction that there is no Bose-Einstein condensation for the one-dimensional harmonic oscillator and the two-dimensional ideal Bose gas [CITATION].', 'cond-mat-0006293-2-26-0': 'The symmetry of the distributions of zeros is due to the fact that [MATH] is a polynomial in [MATH].', 'cond-mat-0006293-2-26-1': 'For this reason it can be inferred that for [MATH] the zeros lie on a perfect circle.', 'cond-mat-0006293-2-27-0': 'Fig. [REF] shows the corresponding specific heats calculated using equation ([REF]).', 'cond-mat-0006293-2-27-1': 'As expected, for [MATH] the specific heat has no hump and approaches with increasing temperature the classical value.', 'cond-mat-0006293-2-27-2': 'We therefore expel the analysis of [MATH] from the discussions below.', 'cond-mat-0006293-2-27-3': 'For [MATH] the specific heats show humps or peaks, which get sharper with increasing [MATH] and increasing particle number.', 'cond-mat-0006293-2-27-4': 'However, from these smooth curves the orders of the phase transition cannot be deduced.', 'cond-mat-0006293-2-28-0': 'In Fig. [REF] the classification parameters [MATH] defined above are plotted for two to six dimensions and particle numbers up to [MATH]100.', 'cond-mat-0006293-2-28-1': 'For all values of [MATH] the parameter [MATH] is only a slightly varying function of [MATH] and approaches very fast an almost constant value.', 'cond-mat-0006293-2-28-2': 'Since [MATH] is the primary classification parameter from Fig. [REF](a) we can directly infer that the [MATH] system exhibits a third order phase transition ([MATH] while the transition for all higher dimensions is of second order ([MATH]).', 'cond-mat-0006293-2-28-3': 'For [MATH] the dependence of [MATH] on [MATH] is plotted in Fig. [REF](a).', 'cond-mat-0006293-2-28-4': 'Since [MATH] decreases rather rapidly with increasing [MATH] it can be speculated that systems corresponding to a large [MATH] exhibit a phase transition which is almost of first order.', 'cond-mat-0006293-2-28-5': 'As mentioned above for finite systems even values [MATH] cannot be excluded by mathematical reasons.', 'cond-mat-0006293-2-28-6': 'We note that two-dimensional Bose-gases are an interesting and growing field of research.', 'cond-mat-0006293-2-28-7': 'As it is well known, the ideal free Bose-gas in two dimensions ([MATH]) does not show a phase transition due to thermal fluctuations which destabilize the condensate [CITATION].', 'cond-mat-0006293-2-28-8': 'Switching on a confining potential greatly influences the properties of the gas, the thermal fluctuations are suppressed and the gas will show Bose-Einstein condensation.', 'cond-mat-0006293-2-28-9': 'Recent experiments [CITATION] have shown that Bose-Einstein condensation is possible even though it is called a quasi-condensate.', 'cond-mat-0006293-2-28-10': 'In our notion the quasi-condensate is just a third order phase transition.', 'cond-mat-0006293-2-28-11': 'Thus, our results are in complete agreement with recent experiments and earlier theoretical work.', 'cond-mat-0006293-2-28-12': 'An interesting question in this respect is whether the order of the transition changes for [MATH] in the limit [MATH].', 'cond-mat-0006293-2-28-13': 'Additional calculation for larger [MATH], which are not printed in Fig. [REF] indicate that [MATH] approaches 1 or might even get smaller.', 'cond-mat-0006293-2-28-14': 'Note that [MATH] is equivalent to a hypothetical 4-dimensional ideal Bose gas or Bosons confined in a 2-dimensional parabolic trap.', 'cond-mat-0006293-2-28-15': 'Our results indicate that the order of the phase transition sensitively depends on [MATH] for values around 2.', 'cond-mat-0006293-2-28-16': 'This might be the reason why phase transitions in three space dimensions are sometimes classified as second and sometimes as third order phase transitions.', 'cond-mat-0006293-2-29-0': 'The parameter [MATH] is a measure of the finite size of the system, i.e. the scaling behavior of [MATH] as a function of [MATH] is a measure of how fast a system approaches a true n-th order phase transition in the Ehrenfest sense.', 'cond-mat-0006293-2-29-1': 'The [MATH] dependence of [MATH] is displayed in Fig. [REF](c).', 'cond-mat-0006293-2-29-2': 'The scaling behavior can be approximated by [MATH] with [MATH] ranging between 1.06 and 1.12 for [MATH].', 'cond-mat-0006293-2-30-0': 'The [MATH] dependence of the classification parameter is visualized in Fig. [REF] for 50 particles.', 'cond-mat-0006293-2-30-1': 'For this system size we found [MATH] and [MATH].', 'cond-mat-0006293-2-31-0': 'The results presented above for continuous single particle densities of states [MATH] are obtained within semi-analytical calculations.', 'cond-mat-0006293-2-31-1': 'In order to compare these results to systems with a discrete level density we adopt as a reference system the 3-dimensional harmonic oscillator with the partition function given by [EQUATION] with [MATH].', 'cond-mat-0006293-2-32-0': 'Fig. [REF](a) displays the zeros of the partition function ([REF]) for [MATH] and [MATH].', 'cond-mat-0006293-2-32-1': 'Fig. [REF](b) displays a contour plot of the absolute value of the ground state occupation number [MATH] with [MATH] given by ([REF]) calculated using an alternative recursion formula [CITATION].', 'cond-mat-0006293-2-32-2': 'The zeros of [MATH] are poles of [MATH] and are indicated by dark spots in this figure.', 'cond-mat-0006293-2-33-0': 'Analyzing the distribution of zeros consolidates our speculation that the order of the phase transition sensitively depends on [MATH].', 'cond-mat-0006293-2-33-1': 'The distribution of zeros behaves like the above calculated values for [MATH] but quantitatively like [MATH].', 'cond-mat-0006293-2-33-2': 'Since the degeneracy of the three-dimensional harmonically confined ideal Bose-gas is a second-order polynomial not only the quadratic term has to be taken into account.', 'cond-mat-0006293-2-33-3': 'The linear term becomes dominant for lower temperatures, so for very low temperatures the best approximation of a continuous one-particle density of states is [MATH].', 'cond-mat-0006293-2-33-4': 'The parameter [MATH] supports this statement [CITATION], i.e. [MATH] resides in a region above 1.', 'cond-mat-0006293-2-33-5': 'Whereas the parameter [MATH] behaves like the [MATH] case.', 'cond-mat-0006293-2-33-6': 'Finally the parameter [MATH] which is a measure for the discreteness of the system shows a [MATH] dependence which is comparable to the one for [MATH].', 'cond-mat-0006293-2-33-7': 'Thus, for small systems the phase transition is of third order, it can be speculated if it becomes a second order transition in the thermodynamic limit.', 'cond-mat-0006293-2-34-0': 'Not only qualitatively but also quantitatively our calculations are in very good agreement with recent theoretical works [CITATION].', 'cond-mat-0006293-2-34-1': 'Comparing the critical temperature which we defined in Sec. [REF] with the usually utilized ones like the temperature of the peak of the specific heat [MATH] or the grand-canonically calculated [MATH] confirms our approach.', 'cond-mat-0006293-2-34-2': 'In Fig. [REF] three possible definitions of the critical temperature are given which all coincide in the thermodynamic limit.', 'cond-mat-0006293-2-34-3': 'All definitions show a [MATH] dependence with [MATH] ranging between [MATH] and [MATH].', 'cond-mat-0006293-2-35-0': '# conclusion', 'cond-mat-0006293-2-36-0': 'Starting with the old ideas of Yang and Lee, and Grossmann et al. we have developed a classification scheme for phase transitions in finite systems.', 'cond-mat-0006293-2-36-1': 'Based on the analytic continuation of the inverse temperature [MATH] into the complex plane we have shown the advantages of this approach.', 'cond-mat-0006293-2-36-2': 'The distribution of the so-called Fisher-zeros [MATH] draws enlightening pictures even for small systems whereas the usually referred thermodynamic properties like the specific heat fail to classify the phase transitions properly.', 'cond-mat-0006293-2-36-3': 'The classification scheme presented in this paper enables us to name the order of the transition in a non-ambiguous way.', 'cond-mat-0006293-2-36-4': 'Further we have given an interpretation of complex temperatures not only as an aiding construction.', 'cond-mat-0006293-2-36-5': 'The complex parts [MATH] of the zeros [MATH] resemble times for which a whole ensemble of identical systems under consideration in a heat bath with an initial Boltzmann-distribution looses its memory.', 'cond-mat-0006293-2-37-0': 'We have applied this to ideal non-interacting Bose-gases confined in power-law traps.', 'cond-mat-0006293-2-37-1': 'We have found that the order of the phase transition sensitively depends on the single particle density of states generated by the confining potential.', 'cond-mat-0006293-2-37-2': 'Our results explain the confusion about the order of the phase transitions in different space dimensions and resolve some of the open questions about the nature of the Bose-Einstein condensation.'}

[['cond-mat-0006293-1-8-0', 'cond-mat-0006293-2-8-0'], ['cond-mat-0006293-1-10-0', 'cond-mat-0006293-2-10-0'], ['cond-mat-0006293-1-10-1', 'cond-mat-0006293-2-10-1'], ['cond-mat-0006293-1-10-2', 'cond-mat-0006293-2-10-2'], ['cond-mat-0006293-1-10-3', 'cond-mat-0006293-2-10-3'], ['cond-mat-0006293-1-10-4', 'cond-mat-0006293-2-10-4'], ['cond-mat-0006293-1-10-5', 'cond-mat-0006293-2-10-5'], ['cond-mat-0006293-1-36-0', 'cond-mat-0006293-2-36-0'], ['cond-mat-0006293-1-36-1', 'cond-mat-0006293-2-36-1'], ['cond-mat-0006293-1-36-2', 'cond-mat-0006293-2-36-2'], ['cond-mat-0006293-1-36-3', 'cond-mat-0006293-2-36-3'], ['cond-mat-0006293-1-36-4', 'cond-mat-0006293-2-36-4'], ['cond-mat-0006293-1-36-5', 'cond-mat-0006293-2-36-5'], ['cond-mat-0006293-1-24-0', 'cond-mat-0006293-2-24-0'], ['cond-mat-0006293-1-24-1', 'cond-mat-0006293-2-24-1'], ['cond-mat-0006293-1-11-0', 'cond-mat-0006293-2-11-0'], ['cond-mat-0006293-1-11-1', 'cond-mat-0006293-2-11-1'], ['cond-mat-0006293-1-11-2', 'cond-mat-0006293-2-11-2'], ['cond-mat-0006293-1-11-3', 'cond-mat-0006293-2-11-3'], ['cond-mat-0006293-1-11-4', 'cond-mat-0006293-2-11-4'], ['cond-mat-0006293-1-11-5', 'cond-mat-0006293-2-11-5'], ['cond-mat-0006293-1-11-6', 'cond-mat-0006293-2-11-6'], ['cond-mat-0006293-1-11-7', 'cond-mat-0006293-2-11-7'], ['cond-mat-0006293-1-11-8', 'cond-mat-0006293-2-11-8'], ['cond-mat-0006293-1-19-0', 'cond-mat-0006293-2-19-0'], ['cond-mat-0006293-1-19-1', 'cond-mat-0006293-2-19-1'], ['cond-mat-0006293-1-19-2', 'cond-mat-0006293-2-19-2'], ['cond-mat-0006293-1-19-3', 'cond-mat-0006293-2-19-3'], ['cond-mat-0006293-1-19-4', 'cond-mat-0006293-2-19-4'], ['cond-mat-0006293-1-6-0', 'cond-mat-0006293-2-6-0'], ['cond-mat-0006293-1-6-1', 'cond-mat-0006293-2-6-1'], ['cond-mat-0006293-1-6-2', 'cond-mat-0006293-2-6-2'], ['cond-mat-0006293-1-26-0', 'cond-mat-0006293-2-26-0'], ['cond-mat-0006293-1-26-1', 'cond-mat-0006293-2-26-1'], ['cond-mat-0006293-1-31-0', 'cond-mat-0006293-2-31-0'], ['cond-mat-0006293-1-31-1', 'cond-mat-0006293-2-31-1'], ['cond-mat-0006293-1-25-0', 'cond-mat-0006293-2-25-0'], ['cond-mat-0006293-1-25-1', 'cond-mat-0006293-2-25-1'], ['cond-mat-0006293-1-25-2', 'cond-mat-0006293-2-25-2'], ['cond-mat-0006293-1-25-3', 'cond-mat-0006293-2-25-3'], ['cond-mat-0006293-1-23-0', 'cond-mat-0006293-2-23-0'], ['cond-mat-0006293-1-23-1', 'cond-mat-0006293-2-23-1'], ['cond-mat-0006293-1-23-2', 'cond-mat-0006293-2-23-2'], ['cond-mat-0006293-1-23-3', 'cond-mat-0006293-2-23-3'], ['cond-mat-0006293-1-23-4', 'cond-mat-0006293-2-23-4'], ['cond-mat-0006293-1-29-0', 'cond-mat-0006293-2-29-0'], ['cond-mat-0006293-1-29-1', 'cond-mat-0006293-2-29-1'], ['cond-mat-0006293-1-29-2', 'cond-mat-0006293-2-29-2'], ['cond-mat-0006293-1-37-0', 'cond-mat-0006293-2-37-0'], ['cond-mat-0006293-1-37-1', 'cond-mat-0006293-2-37-1'], ['cond-mat-0006293-1-37-2', 'cond-mat-0006293-2-37-2'], ['cond-mat-0006293-1-17-0', 'cond-mat-0006293-2-17-0'], ['cond-mat-0006293-1-33-0', 'cond-mat-0006293-2-33-0'], ['cond-mat-0006293-1-33-1', 'cond-mat-0006293-2-33-1'], ['cond-mat-0006293-1-33-2', 'cond-mat-0006293-2-33-2'], ['cond-mat-0006293-1-33-3', 'cond-mat-0006293-2-33-3'], ['cond-mat-0006293-1-33-4', 'cond-mat-0006293-2-33-4'], ['cond-mat-0006293-1-33-5', 'cond-mat-0006293-2-33-5'], ['cond-mat-0006293-1-33-6', 'cond-mat-0006293-2-33-6'], ['cond-mat-0006293-1-33-7', 'cond-mat-0006293-2-33-7'], ['cond-mat-0006293-1-16-0', 'cond-mat-0006293-2-16-0'], ['cond-mat-0006293-1-16-1', 'cond-mat-0006293-2-16-1'], ['cond-mat-0006293-1-16-2', 'cond-mat-0006293-2-16-2'], ['cond-mat-0006293-1-16-3', 'cond-mat-0006293-2-16-3'], ['cond-mat-0006293-1-3-0', 'cond-mat-0006293-2-3-0'], ['cond-mat-0006293-1-3-1', 'cond-mat-0006293-2-3-1'], ['cond-mat-0006293-1-20-0', 'cond-mat-0006293-2-20-0'], ['cond-mat-0006293-1-20-1', 'cond-mat-0006293-2-20-1'], ['cond-mat-0006293-1-32-0', 'cond-mat-0006293-2-32-0'], ['cond-mat-0006293-1-32-1', 'cond-mat-0006293-2-32-1'], ['cond-mat-0006293-1-32-2', 'cond-mat-0006293-2-32-2'], ['cond-mat-0006293-1-14-0', 'cond-mat-0006293-2-14-0'], ['cond-mat-0006293-1-14-1', 'cond-mat-0006293-2-14-1'], ['cond-mat-0006293-1-14-2', 'cond-mat-0006293-2-14-2'], ['cond-mat-0006293-1-14-3', 'cond-mat-0006293-2-14-3'], ['cond-mat-0006293-1-14-4', 'cond-mat-0006293-2-14-4'], ['cond-mat-0006293-1-15-0', 'cond-mat-0006293-2-15-0'], ['cond-mat-0006293-1-15-1', 'cond-mat-0006293-2-15-1'], ['cond-mat-0006293-1-15-2', 'cond-mat-0006293-2-15-2'], ['cond-mat-0006293-1-15-3', 'cond-mat-0006293-2-15-3'], ['cond-mat-0006293-1-15-4', 'cond-mat-0006293-2-15-4'], ['cond-mat-0006293-1-15-5', 'cond-mat-0006293-2-15-5'], ['cond-mat-0006293-1-34-0', 'cond-mat-0006293-2-34-0'], ['cond-mat-0006293-1-34-1', 'cond-mat-0006293-2-34-1'], ['cond-mat-0006293-1-34-2', 'cond-mat-0006293-2-34-2'], ['cond-mat-0006293-1-34-3', 'cond-mat-0006293-2-34-3'], ['cond-mat-0006293-1-12-0', 'cond-mat-0006293-2-12-0'], ['cond-mat-0006293-1-5-0', 'cond-mat-0006293-2-5-0'], ['cond-mat-0006293-1-5-1', 'cond-mat-0006293-2-5-1'], ['cond-mat-0006293-1-5-2', 'cond-mat-0006293-2-5-2'], ['cond-mat-0006293-1-27-0', 'cond-mat-0006293-2-27-0'], ['cond-mat-0006293-1-27-1', 'cond-mat-0006293-2-27-1'], ['cond-mat-0006293-1-27-2', 'cond-mat-0006293-2-27-2'], ['cond-mat-0006293-1-27-3', 'cond-mat-0006293-2-27-3'], ['cond-mat-0006293-1-27-4', 'cond-mat-0006293-2-27-4'], ['cond-mat-0006293-1-21-0', 'cond-mat-0006293-2-21-0'], ['cond-mat-0006293-1-21-1', 'cond-mat-0006293-2-21-1'], ['cond-mat-0006293-1-21-2', 'cond-mat-0006293-2-21-2'], ['cond-mat-0006293-1-21-3', 'cond-mat-0006293-2-21-3'], ['cond-mat-0006293-1-21-4', 'cond-mat-0006293-2-21-4'], ['cond-mat-0006293-1-0-0', 'cond-mat-0006293-2-0-0'], ['cond-mat-0006293-1-0-1', 'cond-mat-0006293-2-0-1'], ['cond-mat-0006293-1-0-2', 'cond-mat-0006293-2-0-2'], ['cond-mat-0006293-1-4-0', 'cond-mat-0006293-2-4-0'], ['cond-mat-0006293-1-4-1', 'cond-mat-0006293-2-4-1'], ['cond-mat-0006293-1-4-2', 'cond-mat-0006293-2-4-2'], ['cond-mat-0006293-1-4-3', 'cond-mat-0006293-2-4-3'], ['cond-mat-0006293-1-4-4', 'cond-mat-0006293-2-4-4'], ['cond-mat-0006293-1-2-0', 'cond-mat-0006293-2-2-0'], ['cond-mat-0006293-1-2-1', 'cond-mat-0006293-2-2-1'], ['cond-mat-0006293-1-2-2', 'cond-mat-0006293-2-2-2'], ['cond-mat-0006293-1-2-3', 'cond-mat-0006293-2-2-3'], ['cond-mat-0006293-1-2-4', 'cond-mat-0006293-2-2-4'], ['cond-mat-0006293-1-9-0', 'cond-mat-0006293-2-9-0'], ['cond-mat-0006293-1-9-1', 'cond-mat-0006293-2-9-1'], ['cond-mat-0006293-1-30-0', 'cond-mat-0006293-2-30-0'], ['cond-mat-0006293-1-30-1', 'cond-mat-0006293-2-30-1'], ['cond-mat-0006293-1-13-0', 'cond-mat-0006293-2-13-0'], ['cond-mat-0006293-1-13-1', 'cond-mat-0006293-2-13-1'], ['cond-mat-0006293-1-13-2', 'cond-mat-0006293-2-13-2'], ['cond-mat-0006293-1-13-3', 'cond-mat-0006293-2-13-3'], ['cond-mat-0006293-1-18-0', 'cond-mat-0006293-2-18-0'], ['cond-mat-0006293-1-28-0', 'cond-mat-0006293-2-28-0'], ['cond-mat-0006293-1-28-1', 'cond-mat-0006293-2-28-1'], ['cond-mat-0006293-1-28-2', 'cond-mat-0006293-2-28-2'], ['cond-mat-0006293-1-28-3', 'cond-mat-0006293-2-28-3'], ['cond-mat-0006293-1-28-4', 'cond-mat-0006293-2-28-4'], ['cond-mat-0006293-1-28-5', 'cond-mat-0006293-2-28-5'], ['cond-mat-0006293-1-28-6', 'cond-mat-0006293-2-28-6'], ['cond-mat-0006293-1-28-7', 'cond-mat-0006293-2-28-7'], ['cond-mat-0006293-1-28-8', 'cond-mat-0006293-2-28-8'], ['cond-mat-0006293-1-28-9', 'cond-mat-0006293-2-28-9'], ['cond-mat-0006293-1-28-10', 'cond-mat-0006293-2-28-10'], ['cond-mat-0006293-1-28-11', 'cond-mat-0006293-2-28-11'], ['cond-mat-0006293-1-28-12', 'cond-mat-0006293-2-28-12'], ['cond-mat-0006293-1-28-13', 'cond-mat-0006293-2-28-13'], ['cond-mat-0006293-1-28-14', 'cond-mat-0006293-2-28-14'], ['cond-mat-0006293-1-28-15', 'cond-mat-0006293-2-28-15'], ['cond-mat-0006293-1-28-16', 'cond-mat-0006293-2-28-16'], ['cond-mat-0006293-2-19-1', 'cond-mat-0006293-3-19-1'], ['cond-mat-0006293-2-19-2', 'cond-mat-0006293-3-19-2'], ['cond-mat-0006293-2-19-3', 'cond-mat-0006293-3-19-3'], ['cond-mat-0006293-2-19-4', 'cond-mat-0006293-3-19-4'], ['cond-mat-0006293-2-34-0', 'cond-mat-0006293-3-34-0'], ['cond-mat-0006293-2-34-1', 'cond-mat-0006293-3-34-1'], ['cond-mat-0006293-2-34-2', 'cond-mat-0006293-3-34-2'], ['cond-mat-0006293-2-34-3', 'cond-mat-0006293-3-34-3'], ['cond-mat-0006293-2-9-0', 'cond-mat-0006293-3-9-0'], ['cond-mat-0006293-2-9-1', 'cond-mat-0006293-3-9-1'], ['cond-mat-0006293-2-31-0', 'cond-mat-0006293-3-31-0'], ['cond-mat-0006293-2-31-1', 'cond-mat-0006293-3-31-1'], ['cond-mat-0006293-2-2-0', 'cond-mat-0006293-3-2-0'], ['cond-mat-0006293-2-2-2', 'cond-mat-0006293-3-2-2'], ['cond-mat-0006293-2-10-0', 'cond-mat-0006293-3-10-0'], ['cond-mat-0006293-2-10-1', 'cond-mat-0006293-3-10-1'], ['cond-mat-0006293-2-10-2', 'cond-mat-0006293-3-10-2'], ['cond-mat-0006293-2-10-3', 'cond-mat-0006293-3-10-3'], ['cond-mat-0006293-2-10-4', 'cond-mat-0006293-3-10-4'], ['cond-mat-0006293-2-10-5', 'cond-mat-0006293-3-10-5'], ['cond-mat-0006293-2-21-0', 'cond-mat-0006293-3-21-0'], ['cond-mat-0006293-2-21-1', 'cond-mat-0006293-3-21-1'], ['cond-mat-0006293-2-21-2', 'cond-mat-0006293-3-21-2'], ['cond-mat-0006293-2-21-3', 'cond-mat-0006293-3-21-3'], ['cond-mat-0006293-2-21-4', 'cond-mat-0006293-3-21-4'], ['cond-mat-0006293-2-33-0', 'cond-mat-0006293-3-33-0'], ['cond-mat-0006293-2-33-1', 'cond-mat-0006293-3-33-1'], ['cond-mat-0006293-2-33-2', 'cond-mat-0006293-3-33-2'], ['cond-mat-0006293-2-33-3', 'cond-mat-0006293-3-33-3'], ['cond-mat-0006293-2-33-4', 'cond-mat-0006293-3-33-4'], ['cond-mat-0006293-2-33-5', 'cond-mat-0006293-3-33-5'], ['cond-mat-0006293-2-33-6', 'cond-mat-0006293-3-33-6'], ['cond-mat-0006293-2-33-7', 'cond-mat-0006293-3-33-7'], ['cond-mat-0006293-2-5-0', 'cond-mat-0006293-3-5-0'], ['cond-mat-0006293-2-5-1', 'cond-mat-0006293-3-5-1'], ['cond-mat-0006293-2-17-0', 'cond-mat-0006293-3-17-0'], ['cond-mat-0006293-2-30-0', 'cond-mat-0006293-3-30-0'], ['cond-mat-0006293-2-30-1', 'cond-mat-0006293-3-30-1'], ['cond-mat-0006293-2-26-0', 'cond-mat-0006293-3-26-0'], ['cond-mat-0006293-2-26-1', 'cond-mat-0006293-3-26-1'], ['cond-mat-0006293-2-8-0', 'cond-mat-0006293-3-8-0'], ['cond-mat-0006293-2-13-0', 'cond-mat-0006293-3-13-0'], ['cond-mat-0006293-2-13-1', 'cond-mat-0006293-3-13-1'], ['cond-mat-0006293-2-13-2', 'cond-mat-0006293-3-13-2'], ['cond-mat-0006293-2-13-3', 'cond-mat-0006293-3-13-3'], ['cond-mat-0006293-2-16-0', 'cond-mat-0006293-3-16-0'], ['cond-mat-0006293-2-16-1', 'cond-mat-0006293-3-16-1'], ['cond-mat-0006293-2-16-2', 'cond-mat-0006293-3-16-2'], ['cond-mat-0006293-2-16-3', 'cond-mat-0006293-3-16-3'], ['cond-mat-0006293-2-4-0', 'cond-mat-0006293-3-3-0'], ['cond-mat-0006293-2-4-1', 'cond-mat-0006293-3-3-1'], ['cond-mat-0006293-2-4-2', 'cond-mat-0006293-3-3-2'], ['cond-mat-0006293-2-4-3', 'cond-mat-0006293-3-3-3'], ['cond-mat-0006293-2-4-4', 'cond-mat-0006293-3-3-4'], ['cond-mat-0006293-2-14-0', 'cond-mat-0006293-3-14-0'], ['cond-mat-0006293-2-14-1', 'cond-mat-0006293-3-14-1'], ['cond-mat-0006293-2-14-3', 'cond-mat-0006293-3-14-3'], ['cond-mat-0006293-2-14-4', 'cond-mat-0006293-3-14-4'], ['cond-mat-0006293-2-37-0', 'cond-mat-0006293-3-37-0'], ['cond-mat-0006293-2-37-1', 'cond-mat-0006293-3-37-1'], ['cond-mat-0006293-2-23-0', 'cond-mat-0006293-3-23-0'], ['cond-mat-0006293-2-23-1', 'cond-mat-0006293-3-23-1'], ['cond-mat-0006293-2-23-2', 'cond-mat-0006293-3-23-2'], ['cond-mat-0006293-2-23-3', 'cond-mat-0006293-3-23-3'], ['cond-mat-0006293-2-23-4', 'cond-mat-0006293-3-23-4'], ['cond-mat-0006293-2-24-0', 'cond-mat-0006293-3-24-0'], ['cond-mat-0006293-2-24-1', 'cond-mat-0006293-3-24-1'], ['cond-mat-0006293-2-32-0', 'cond-mat-0006293-3-32-0'], ['cond-mat-0006293-2-32-1', 'cond-mat-0006293-3-32-1'], ['cond-mat-0006293-2-32-2', 'cond-mat-0006293-3-32-2'], ['cond-mat-0006293-2-11-0', 'cond-mat-0006293-3-11-0'], ['cond-mat-0006293-2-11-1', 'cond-mat-0006293-3-11-1'], ['cond-mat-0006293-2-11-2', 'cond-mat-0006293-3-11-2'], ['cond-mat-0006293-2-11-3', 'cond-mat-0006293-3-11-3'], ['cond-mat-0006293-2-11-4', 'cond-mat-0006293-3-11-4'], ['cond-mat-0006293-2-11-5', 'cond-mat-0006293-3-11-5'], ['cond-mat-0006293-2-11-6', 'cond-mat-0006293-3-11-6'], ['cond-mat-0006293-2-11-7', 'cond-mat-0006293-3-11-7'], ['cond-mat-0006293-2-11-8', 'cond-mat-0006293-3-11-8'], ['cond-mat-0006293-2-29-0', 'cond-mat-0006293-3-29-0'], ['cond-mat-0006293-2-29-1', 'cond-mat-0006293-3-29-1'], ['cond-mat-0006293-2-29-2', 'cond-mat-0006293-3-29-2'], ['cond-mat-0006293-2-0-0', 'cond-mat-0006293-3-0-0'], ['cond-mat-0006293-2-0-1', 'cond-mat-0006293-3-0-1'], ['cond-mat-0006293-2-0-2', 'cond-mat-0006293-3-0-2'], ['cond-mat-0006293-2-25-0', 'cond-mat-0006293-3-25-0'], ['cond-mat-0006293-2-25-1', 'cond-mat-0006293-3-25-1'], ['cond-mat-0006293-2-25-2', 'cond-mat-0006293-3-25-2'], ['cond-mat-0006293-2-25-3', 'cond-mat-0006293-3-25-3'], ['cond-mat-0006293-2-15-0', 'cond-mat-0006293-3-15-0'], ['cond-mat-0006293-2-15-1', 'cond-mat-0006293-3-15-1'], ['cond-mat-0006293-2-15-2', 'cond-mat-0006293-3-15-2'], ['cond-mat-0006293-2-15-3', 'cond-mat-0006293-3-15-3'], ['cond-mat-0006293-2-15-4', 'cond-mat-0006293-3-15-4'], ['cond-mat-0006293-2-15-5', 'cond-mat-0006293-3-15-5'], ['cond-mat-0006293-2-28-1', 'cond-mat-0006293-3-28-1'], ['cond-mat-0006293-2-28-2', 'cond-mat-0006293-3-28-2'], ['cond-mat-0006293-2-28-3', 'cond-mat-0006293-3-28-3'], ['cond-mat-0006293-2-28-4', 'cond-mat-0006293-3-28-4'], ['cond-mat-0006293-2-28-5', 'cond-mat-0006293-3-28-5'], ['cond-mat-0006293-2-28-6', 'cond-mat-0006293-3-28-6'], ['cond-mat-0006293-2-28-7', 'cond-mat-0006293-3-28-7'], ['cond-mat-0006293-2-28-8', 'cond-mat-0006293-3-28-8'], ['cond-mat-0006293-2-28-9', 'cond-mat-0006293-3-28-9'], ['cond-mat-0006293-2-28-10', 'cond-mat-0006293-3-28-10'], ['cond-mat-0006293-2-28-11', 'cond-mat-0006293-3-28-11'], ['cond-mat-0006293-2-28-12', 'cond-mat-0006293-3-28-12'], ['cond-mat-0006293-2-28-13', 'cond-mat-0006293-3-28-13'], ['cond-mat-0006293-2-28-14', 'cond-mat-0006293-3-28-14'], ['cond-mat-0006293-2-28-15', 'cond-mat-0006293-3-28-15'], ['cond-mat-0006293-2-28-16', 'cond-mat-0006293-3-28-16'], ['cond-mat-0006293-2-20-0', 'cond-mat-0006293-3-20-0'], ['cond-mat-0006293-2-20-1', 'cond-mat-0006293-3-20-1'], ['cond-mat-0006293-2-6-0', 'cond-mat-0006293-3-6-0'], ['cond-mat-0006293-2-6-1', 'cond-mat-0006293-3-6-1'], ['cond-mat-0006293-2-6-2', 'cond-mat-0006293-3-6-2'], ['cond-mat-0006293-2-27-0', 'cond-mat-0006293-3-27-0'], ['cond-mat-0006293-2-27-1', 'cond-mat-0006293-3-27-1'], ['cond-mat-0006293-2-27-2', 'cond-mat-0006293-3-27-2'], ['cond-mat-0006293-2-27-3', 'cond-mat-0006293-3-27-3'], ['cond-mat-0006293-2-27-4', 'cond-mat-0006293-3-27-4'], ['cond-mat-0006293-2-36-0', 'cond-mat-0006293-3-36-0'], ['cond-mat-0006293-2-36-1', 'cond-mat-0006293-3-36-1'], ['cond-mat-0006293-2-36-2', 'cond-mat-0006293-3-36-2'], ['cond-mat-0006293-2-36-3', 'cond-mat-0006293-3-36-3'], ['cond-mat-0006293-2-36-5', 'cond-mat-0006293-3-36-4'], ['cond-mat-0006293-2-19-0', 'cond-mat-0006293-3-19-0'], ['cond-mat-0006293-2-2-1', 'cond-mat-0006293-3-2-1'], ['cond-mat-0006293-2-12-0', 'cond-mat-0006293-3-12-0'], ['cond-mat-0006293-2-14-2', 'cond-mat-0006293-3-14-2'], ['cond-mat-0006293-2-28-0', 'cond-mat-0006293-3-28-0'], ['cond-mat-0006293-2-5-2', 'cond-mat-0006293-3-5-2'], ['cond-mat-0006293-2-3-0', 'cond-mat-0006293-3-4-0'], ['cond-mat-0006293-2-18-0', 'cond-mat-0006293-3-18-0']]

[['cond-mat-0006293-1-8-0', 'cond-mat-0006293-2-8-0'], ['cond-mat-0006293-1-10-0', 'cond-mat-0006293-2-10-0'], ['cond-mat-0006293-1-10-1', 'cond-mat-0006293-2-10-1'], ['cond-mat-0006293-1-10-2', 'cond-mat-0006293-2-10-2'], ['cond-mat-0006293-1-10-3', 'cond-mat-0006293-2-10-3'], ['cond-mat-0006293-1-10-4', 'cond-mat-0006293-2-10-4'], ['cond-mat-0006293-1-10-5', 'cond-mat-0006293-2-10-5'], ['cond-mat-0006293-1-36-0', 'cond-mat-0006293-2-36-0'], ['cond-mat-0006293-1-36-1', 'cond-mat-0006293-2-36-1'], ['cond-mat-0006293-1-36-2', 'cond-mat-0006293-2-36-2'], ['cond-mat-0006293-1-36-3', 'cond-mat-0006293-2-36-3'], ['cond-mat-0006293-1-36-4', 'cond-mat-0006293-2-36-4'], ['cond-mat-0006293-1-36-5', 'cond-mat-0006293-2-36-5'], ['cond-mat-0006293-1-24-0', 'cond-mat-0006293-2-24-0'], ['cond-mat-0006293-1-24-1', 'cond-mat-0006293-2-24-1'], ['cond-mat-0006293-1-11-0', 'cond-mat-0006293-2-11-0'], ['cond-mat-0006293-1-11-1', 'cond-mat-0006293-2-11-1'], ['cond-mat-0006293-1-11-2', 'cond-mat-0006293-2-11-2'], ['cond-mat-0006293-1-11-3', 'cond-mat-0006293-2-11-3'], ['cond-mat-0006293-1-11-4', 'cond-mat-0006293-2-11-4'], ['cond-mat-0006293-1-11-5', 'cond-mat-0006293-2-11-5'], ['cond-mat-0006293-1-11-6', 'cond-mat-0006293-2-11-6'], ['cond-mat-0006293-1-11-7', 'cond-mat-0006293-2-11-7'], ['cond-mat-0006293-1-11-8', 'cond-mat-0006293-2-11-8'], ['cond-mat-0006293-1-19-0', 'cond-mat-0006293-2-19-0'], ['cond-mat-0006293-1-19-1', 'cond-mat-0006293-2-19-1'], ['cond-mat-0006293-1-19-2', 'cond-mat-0006293-2-19-2'], ['cond-mat-0006293-1-19-3', 'cond-mat-0006293-2-19-3'], ['cond-mat-0006293-1-19-4', 'cond-mat-0006293-2-19-4'], ['cond-mat-0006293-1-6-0', 'cond-mat-0006293-2-6-0'], ['cond-mat-0006293-1-6-1', 'cond-mat-0006293-2-6-1'], ['cond-mat-0006293-1-6-2', 'cond-mat-0006293-2-6-2'], ['cond-mat-0006293-1-26-0', 'cond-mat-0006293-2-26-0'], ['cond-mat-0006293-1-26-1', 'cond-mat-0006293-2-26-1'], ['cond-mat-0006293-1-31-0', 'cond-mat-0006293-2-31-0'], ['cond-mat-0006293-1-31-1', 'cond-mat-0006293-2-31-1'], ['cond-mat-0006293-1-25-0', 'cond-mat-0006293-2-25-0'], ['cond-mat-0006293-1-25-1', 'cond-mat-0006293-2-25-1'], ['cond-mat-0006293-1-25-2', 'cond-mat-0006293-2-25-2'], ['cond-mat-0006293-1-25-3', 'cond-mat-0006293-2-25-3'], ['cond-mat-0006293-1-23-0', 'cond-mat-0006293-2-23-0'], ['cond-mat-0006293-1-23-1', 'cond-mat-0006293-2-23-1'], ['cond-mat-0006293-1-23-2', 'cond-mat-0006293-2-23-2'], ['cond-mat-0006293-1-23-3', 'cond-mat-0006293-2-23-3'], ['cond-mat-0006293-1-23-4', 'cond-mat-0006293-2-23-4'], ['cond-mat-0006293-1-29-0', 'cond-mat-0006293-2-29-0'], ['cond-mat-0006293-1-29-1', 'cond-mat-0006293-2-29-1'], ['cond-mat-0006293-1-29-2', 'cond-mat-0006293-2-29-2'], ['cond-mat-0006293-1-37-0', 'cond-mat-0006293-2-37-0'], ['cond-mat-0006293-1-37-1', 'cond-mat-0006293-2-37-1'], ['cond-mat-0006293-1-37-2', 'cond-mat-0006293-2-37-2'], ['cond-mat-0006293-1-17-0', 'cond-mat-0006293-2-17-0'], ['cond-mat-0006293-1-33-0', 'cond-mat-0006293-2-33-0'], ['cond-mat-0006293-1-33-1', 'cond-mat-0006293-2-33-1'], ['cond-mat-0006293-1-33-2', 'cond-mat-0006293-2-33-2'], ['cond-mat-0006293-1-33-3', 'cond-mat-0006293-2-33-3'], ['cond-mat-0006293-1-33-4', 'cond-mat-0006293-2-33-4'], ['cond-mat-0006293-1-33-5', 'cond-mat-0006293-2-33-5'], ['cond-mat-0006293-1-33-6', 'cond-mat-0006293-2-33-6'], ['cond-mat-0006293-1-33-7', 'cond-mat-0006293-2-33-7'], ['cond-mat-0006293-1-16-0', 'cond-mat-0006293-2-16-0'], ['cond-mat-0006293-1-16-1', 'cond-mat-0006293-2-16-1'], ['cond-mat-0006293-1-16-2', 'cond-mat-0006293-2-16-2'], ['cond-mat-0006293-1-16-3', 'cond-mat-0006293-2-16-3'], ['cond-mat-0006293-1-3-0', 'cond-mat-0006293-2-3-0'], ['cond-mat-0006293-1-3-1', 'cond-mat-0006293-2-3-1'], ['cond-mat-0006293-1-20-0', 'cond-mat-0006293-2-20-0'], ['cond-mat-0006293-1-20-1', 'cond-mat-0006293-2-20-1'], ['cond-mat-0006293-1-32-0', 'cond-mat-0006293-2-32-0'], ['cond-mat-0006293-1-32-1', 'cond-mat-0006293-2-32-1'], ['cond-mat-0006293-1-32-2', 'cond-mat-0006293-2-32-2'], ['cond-mat-0006293-1-14-0', 'cond-mat-0006293-2-14-0'], ['cond-mat-0006293-1-14-1', 'cond-mat-0006293-2-14-1'], ['cond-mat-0006293-1-14-2', 'cond-mat-0006293-2-14-2'], ['cond-mat-0006293-1-14-3', 'cond-mat-0006293-2-14-3'], ['cond-mat-0006293-1-14-4', 'cond-mat-0006293-2-14-4'], ['cond-mat-0006293-1-15-0', 'cond-mat-0006293-2-15-0'], ['cond-mat-0006293-1-15-1', 'cond-mat-0006293-2-15-1'], ['cond-mat-0006293-1-15-2', 'cond-mat-0006293-2-15-2'], ['cond-mat-0006293-1-15-3', 'cond-mat-0006293-2-15-3'], ['cond-mat-0006293-1-15-4', 'cond-mat-0006293-2-15-4'], ['cond-mat-0006293-1-15-5', 'cond-mat-0006293-2-15-5'], ['cond-mat-0006293-1-34-0', 'cond-mat-0006293-2-34-0'], ['cond-mat-0006293-1-34-1', 'cond-mat-0006293-2-34-1'], ['cond-mat-0006293-1-34-2', 'cond-mat-0006293-2-34-2'], ['cond-mat-0006293-1-34-3', 'cond-mat-0006293-2-34-3'], ['cond-mat-0006293-1-12-0', 'cond-mat-0006293-2-12-0'], ['cond-mat-0006293-1-5-0', 'cond-mat-0006293-2-5-0'], ['cond-mat-0006293-1-5-1', 'cond-mat-0006293-2-5-1'], ['cond-mat-0006293-1-5-2', 'cond-mat-0006293-2-5-2'], ['cond-mat-0006293-1-27-0', 'cond-mat-0006293-2-27-0'], ['cond-mat-0006293-1-27-1', 'cond-mat-0006293-2-27-1'], ['cond-mat-0006293-1-27-2', 'cond-mat-0006293-2-27-2'], ['cond-mat-0006293-1-27-3', 'cond-mat-0006293-2-27-3'], ['cond-mat-0006293-1-27-4', 'cond-mat-0006293-2-27-4'], ['cond-mat-0006293-1-21-0', 'cond-mat-0006293-2-21-0'], ['cond-mat-0006293-1-21-1', 'cond-mat-0006293-2-21-1'], ['cond-mat-0006293-1-21-2', 'cond-mat-0006293-2-21-2'], ['cond-mat-0006293-1-21-3', 'cond-mat-0006293-2-21-3'], ['cond-mat-0006293-1-21-4', 'cond-mat-0006293-2-21-4'], ['cond-mat-0006293-1-0-0', 'cond-mat-0006293-2-0-0'], ['cond-mat-0006293-1-0-1', 'cond-mat-0006293-2-0-1'], ['cond-mat-0006293-1-0-2', 'cond-mat-0006293-2-0-2'], ['cond-mat-0006293-1-4-0', 'cond-mat-0006293-2-4-0'], ['cond-mat-0006293-1-4-1', 'cond-mat-0006293-2-4-1'], ['cond-mat-0006293-1-4-2', 'cond-mat-0006293-2-4-2'], ['cond-mat-0006293-1-4-3', 'cond-mat-0006293-2-4-3'], ['cond-mat-0006293-1-4-4', 'cond-mat-0006293-2-4-4'], ['cond-mat-0006293-1-2-0', 'cond-mat-0006293-2-2-0'], ['cond-mat-0006293-1-2-1', 'cond-mat-0006293-2-2-1'], ['cond-mat-0006293-1-2-2', 'cond-mat-0006293-2-2-2'], ['cond-mat-0006293-1-2-3', 'cond-mat-0006293-2-2-3'], ['cond-mat-0006293-1-2-4', 'cond-mat-0006293-2-2-4'], ['cond-mat-0006293-1-9-0', 'cond-mat-0006293-2-9-0'], ['cond-mat-0006293-1-9-1', 'cond-mat-0006293-2-9-1'], ['cond-mat-0006293-1-30-0', 'cond-mat-0006293-2-30-0'], ['cond-mat-0006293-1-30-1', 'cond-mat-0006293-2-30-1'], ['cond-mat-0006293-1-13-0', 'cond-mat-0006293-2-13-0'], ['cond-mat-0006293-1-13-1', 'cond-mat-0006293-2-13-1'], ['cond-mat-0006293-1-13-2', 'cond-mat-0006293-2-13-2'], ['cond-mat-0006293-1-13-3', 'cond-mat-0006293-2-13-3'], ['cond-mat-0006293-1-18-0', 'cond-mat-0006293-2-18-0'], ['cond-mat-0006293-1-28-0', 'cond-mat-0006293-2-28-0'], ['cond-mat-0006293-1-28-1', 'cond-mat-0006293-2-28-1'], ['cond-mat-0006293-1-28-2', 'cond-mat-0006293-2-28-2'], ['cond-mat-0006293-1-28-3', 'cond-mat-0006293-2-28-3'], ['cond-mat-0006293-1-28-4', 'cond-mat-0006293-2-28-4'], ['cond-mat-0006293-1-28-5', 'cond-mat-0006293-2-28-5'], ['cond-mat-0006293-1-28-6', 'cond-mat-0006293-2-28-6'], ['cond-mat-0006293-1-28-7', 'cond-mat-0006293-2-28-7'], ['cond-mat-0006293-1-28-8', 'cond-mat-0006293-2-28-8'], ['cond-mat-0006293-1-28-9', 'cond-mat-0006293-2-28-9'], ['cond-mat-0006293-1-28-10', 'cond-mat-0006293-2-28-10'], ['cond-mat-0006293-1-28-11', 'cond-mat-0006293-2-28-11'], ['cond-mat-0006293-1-28-12', 'cond-mat-0006293-2-28-12'], ['cond-mat-0006293-1-28-13', 'cond-mat-0006293-2-28-13'], ['cond-mat-0006293-1-28-14', 'cond-mat-0006293-2-28-14'], ['cond-mat-0006293-1-28-15', 'cond-mat-0006293-2-28-15'], ['cond-mat-0006293-1-28-16', 'cond-mat-0006293-2-28-16'], ['cond-mat-0006293-2-19-1', 'cond-mat-0006293-3-19-1'], ['cond-mat-0006293-2-19-2', 'cond-mat-0006293-3-19-2'], ['cond-mat-0006293-2-19-3', 'cond-mat-0006293-3-19-3'], ['cond-mat-0006293-2-19-4', 'cond-mat-0006293-3-19-4'], ['cond-mat-0006293-2-34-0', 'cond-mat-0006293-3-34-0'], ['cond-mat-0006293-2-34-1', 'cond-mat-0006293-3-34-1'], ['cond-mat-0006293-2-34-2', 'cond-mat-0006293-3-34-2'], ['cond-mat-0006293-2-34-3', 'cond-mat-0006293-3-34-3'], ['cond-mat-0006293-2-9-0', 'cond-mat-0006293-3-9-0'], ['cond-mat-0006293-2-9-1', 'cond-mat-0006293-3-9-1'], ['cond-mat-0006293-2-31-0', 'cond-mat-0006293-3-31-0'], ['cond-mat-0006293-2-31-1', 'cond-mat-0006293-3-31-1'], ['cond-mat-0006293-2-2-0', 'cond-mat-0006293-3-2-0'], ['cond-mat-0006293-2-2-2', 'cond-mat-0006293-3-2-2'], ['cond-mat-0006293-2-10-0', 'cond-mat-0006293-3-10-0'], ['cond-mat-0006293-2-10-1', 'cond-mat-0006293-3-10-1'], ['cond-mat-0006293-2-10-2', 'cond-mat-0006293-3-10-2'], ['cond-mat-0006293-2-10-3', 'cond-mat-0006293-3-10-3'], ['cond-mat-0006293-2-10-4', 'cond-mat-0006293-3-10-4'], ['cond-mat-0006293-2-10-5', 'cond-mat-0006293-3-10-5'], ['cond-mat-0006293-2-21-0', 'cond-mat-0006293-3-21-0'], ['cond-mat-0006293-2-21-1', 'cond-mat-0006293-3-21-1'], ['cond-mat-0006293-2-21-2', 'cond-mat-0006293-3-21-2'], ['cond-mat-0006293-2-21-3', 'cond-mat-0006293-3-21-3'], ['cond-mat-0006293-2-21-4', 'cond-mat-0006293-3-21-4'], ['cond-mat-0006293-2-33-0', 'cond-mat-0006293-3-33-0'], ['cond-mat-0006293-2-33-1', 'cond-mat-0006293-3-33-1'], ['cond-mat-0006293-2-33-2', 'cond-mat-0006293-3-33-2'], ['cond-mat-0006293-2-33-3', 'cond-mat-0006293-3-33-3'], ['cond-mat-0006293-2-33-4', 'cond-mat-0006293-3-33-4'], ['cond-mat-0006293-2-33-5', 'cond-mat-0006293-3-33-5'], ['cond-mat-0006293-2-33-6', 'cond-mat-0006293-3-33-6'], ['cond-mat-0006293-2-33-7', 'cond-mat-0006293-3-33-7'], ['cond-mat-0006293-2-5-0', 'cond-mat-0006293-3-5-0'], ['cond-mat-0006293-2-5-1', 'cond-mat-0006293-3-5-1'], ['cond-mat-0006293-2-17-0', 'cond-mat-0006293-3-17-0'], ['cond-mat-0006293-2-30-0', 'cond-mat-0006293-3-30-0'], ['cond-mat-0006293-2-30-1', 'cond-mat-0006293-3-30-1'], ['cond-mat-0006293-2-26-0', 'cond-mat-0006293-3-26-0'], ['cond-mat-0006293-2-26-1', 'cond-mat-0006293-3-26-1'], ['cond-mat-0006293-2-8-0', 'cond-mat-0006293-3-8-0'], ['cond-mat-0006293-2-13-0', 'cond-mat-0006293-3-13-0'], ['cond-mat-0006293-2-13-1', 'cond-mat-0006293-3-13-1'], ['cond-mat-0006293-2-13-2', 'cond-mat-0006293-3-13-2'], ['cond-mat-0006293-2-13-3', 'cond-mat-0006293-3-13-3'], ['cond-mat-0006293-2-16-0', 'cond-mat-0006293-3-16-0'], ['cond-mat-0006293-2-16-1', 'cond-mat-0006293-3-16-1'], ['cond-mat-0006293-2-16-2', 'cond-mat-0006293-3-16-2'], ['cond-mat-0006293-2-16-3', 'cond-mat-0006293-3-16-3'], ['cond-mat-0006293-2-4-0', 'cond-mat-0006293-3-3-0'], ['cond-mat-0006293-2-4-1', 'cond-mat-0006293-3-3-1'], ['cond-mat-0006293-2-4-2', 'cond-mat-0006293-3-3-2'], ['cond-mat-0006293-2-4-3', 'cond-mat-0006293-3-3-3'], ['cond-mat-0006293-2-4-4', 'cond-mat-0006293-3-3-4'], ['cond-mat-0006293-2-14-0', 'cond-mat-0006293-3-14-0'], ['cond-mat-0006293-2-14-1', 'cond-mat-0006293-3-14-1'], ['cond-mat-0006293-2-14-3', 'cond-mat-0006293-3-14-3'], ['cond-mat-0006293-2-14-4', 'cond-mat-0006293-3-14-4'], ['cond-mat-0006293-2-37-0', 'cond-mat-0006293-3-37-0'], ['cond-mat-0006293-2-37-1', 'cond-mat-0006293-3-37-1'], ['cond-mat-0006293-2-23-0', 'cond-mat-0006293-3-23-0'], ['cond-mat-0006293-2-23-1', 'cond-mat-0006293-3-23-1'], ['cond-mat-0006293-2-23-2', 'cond-mat-0006293-3-23-2'], ['cond-mat-0006293-2-23-3', 'cond-mat-0006293-3-23-3'], ['cond-mat-0006293-2-23-4', 'cond-mat-0006293-3-23-4'], ['cond-mat-0006293-2-24-0', 'cond-mat-0006293-3-24-0'], ['cond-mat-0006293-2-24-1', 'cond-mat-0006293-3-24-1'], ['cond-mat-0006293-2-32-0', 'cond-mat-0006293-3-32-0'], ['cond-mat-0006293-2-32-1', 'cond-mat-0006293-3-32-1'], ['cond-mat-0006293-2-32-2', 'cond-mat-0006293-3-32-2'], ['cond-mat-0006293-2-11-0', 'cond-mat-0006293-3-11-0'], ['cond-mat-0006293-2-11-1', 'cond-mat-0006293-3-11-1'], ['cond-mat-0006293-2-11-2', 'cond-mat-0006293-3-11-2'], ['cond-mat-0006293-2-11-3', 'cond-mat-0006293-3-11-3'], ['cond-mat-0006293-2-11-4', 'cond-mat-0006293-3-11-4'], ['cond-mat-0006293-2-11-5', 'cond-mat-0006293-3-11-5'], ['cond-mat-0006293-2-11-6', 'cond-mat-0006293-3-11-6'], ['cond-mat-0006293-2-11-7', 'cond-mat-0006293-3-11-7'], ['cond-mat-0006293-2-11-8', 'cond-mat-0006293-3-11-8'], ['cond-mat-0006293-2-29-0', 'cond-mat-0006293-3-29-0'], ['cond-mat-0006293-2-29-1', 'cond-mat-0006293-3-29-1'], ['cond-mat-0006293-2-29-2', 'cond-mat-0006293-3-29-2'], ['cond-mat-0006293-2-0-0', 'cond-mat-0006293-3-0-0'], ['cond-mat-0006293-2-0-1', 'cond-mat-0006293-3-0-1'], ['cond-mat-0006293-2-0-2', 'cond-mat-0006293-3-0-2'], ['cond-mat-0006293-2-25-0', 'cond-mat-0006293-3-25-0'], ['cond-mat-0006293-2-25-1', 'cond-mat-0006293-3-25-1'], ['cond-mat-0006293-2-25-2', 'cond-mat-0006293-3-25-2'], ['cond-mat-0006293-2-25-3', 'cond-mat-0006293-3-25-3'], ['cond-mat-0006293-2-15-0', 'cond-mat-0006293-3-15-0'], ['cond-mat-0006293-2-15-1', 'cond-mat-0006293-3-15-1'], ['cond-mat-0006293-2-15-2', 'cond-mat-0006293-3-15-2'], ['cond-mat-0006293-2-15-3', 'cond-mat-0006293-3-15-3'], ['cond-mat-0006293-2-15-4', 'cond-mat-0006293-3-15-4'], ['cond-mat-0006293-2-15-5', 'cond-mat-0006293-3-15-5'], ['cond-mat-0006293-2-28-1', 'cond-mat-0006293-3-28-1'], ['cond-mat-0006293-2-28-2', 'cond-mat-0006293-3-28-2'], ['cond-mat-0006293-2-28-3', 'cond-mat-0006293-3-28-3'], ['cond-mat-0006293-2-28-4', 'cond-mat-0006293-3-28-4'], ['cond-mat-0006293-2-28-5', 'cond-mat-0006293-3-28-5'], ['cond-mat-0006293-2-28-6', 'cond-mat-0006293-3-28-6'], ['cond-mat-0006293-2-28-7', 'cond-mat-0006293-3-28-7'], ['cond-mat-0006293-2-28-8', 'cond-mat-0006293-3-28-8'], ['cond-mat-0006293-2-28-9', 'cond-mat-0006293-3-28-9'], ['cond-mat-0006293-2-28-10', 'cond-mat-0006293-3-28-10'], ['cond-mat-0006293-2-28-11', 'cond-mat-0006293-3-28-11'], ['cond-mat-0006293-2-28-12', 'cond-mat-0006293-3-28-12'], ['cond-mat-0006293-2-28-13', 'cond-mat-0006293-3-28-13'], ['cond-mat-0006293-2-28-14', 'cond-mat-0006293-3-28-14'], ['cond-mat-0006293-2-28-15', 'cond-mat-0006293-3-28-15'], ['cond-mat-0006293-2-28-16', 'cond-mat-0006293-3-28-16'], ['cond-mat-0006293-2-20-0', 'cond-mat-0006293-3-20-0'], ['cond-mat-0006293-2-20-1', 'cond-mat-0006293-3-20-1'], ['cond-mat-0006293-2-6-0', 'cond-mat-0006293-3-6-0'], ['cond-mat-0006293-2-6-1', 'cond-mat-0006293-3-6-1'], ['cond-mat-0006293-2-6-2', 'cond-mat-0006293-3-6-2'], ['cond-mat-0006293-2-27-0', 'cond-mat-0006293-3-27-0'], ['cond-mat-0006293-2-27-1', 'cond-mat-0006293-3-27-1'], ['cond-mat-0006293-2-27-2', 'cond-mat-0006293-3-27-2'], ['cond-mat-0006293-2-27-3', 'cond-mat-0006293-3-27-3'], ['cond-mat-0006293-2-27-4', 'cond-mat-0006293-3-27-4'], ['cond-mat-0006293-2-36-0', 'cond-mat-0006293-3-36-0'], ['cond-mat-0006293-2-36-1', 'cond-mat-0006293-3-36-1'], ['cond-mat-0006293-2-36-2', 'cond-mat-0006293-3-36-2'], ['cond-mat-0006293-2-36-3', 'cond-mat-0006293-3-36-3'], ['cond-mat-0006293-2-36-5', 'cond-mat-0006293-3-36-4']]

[['cond-mat-0006293-2-19-0', 'cond-mat-0006293-3-19-0'], ['cond-mat-0006293-2-2-1', 'cond-mat-0006293-3-2-1'], ['cond-mat-0006293-2-12-0', 'cond-mat-0006293-3-12-0'], ['cond-mat-0006293-2-14-2', 'cond-mat-0006293-3-14-2'], ['cond-mat-0006293-2-28-0', 'cond-mat-0006293-3-28-0']]

[]

[['cond-mat-0006293-2-5-2', 'cond-mat-0006293-3-5-2'], ['cond-mat-0006293-2-3-0', 'cond-mat-0006293-3-4-0'], ['cond-mat-0006293-2-18-0', 'cond-mat-0006293-3-18-0']]

[]

[]

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

https://arxiv.org/abs/cond-mat/0006293

{'cond-mat-0006293-3-0-0': 'We present a detailed description of a classification scheme for phase transitions in finite systems based on the distribution of Fisher zeros of the canonical partition function in the complex temperature plane.', 'cond-mat-0006293-3-0-1': 'We apply this scheme to finite Bose-systems in power law traps within a semi-analytic approach with a continuous one-particle density of states [MATH] for different values of [MATH] and to a three dimensional harmonically confined ideal Bose-gas with discrete energy levels.', 'cond-mat-0006293-3-0-2': 'Our results indicate that the order of the Bose-Einstein condensation phase transition sensitively depends on the confining potential.', 'cond-mat-0006293-3-1-0': '# Introduction', 'cond-mat-0006293-3-2-0': 'In 1924 S. Bose and A. Einstein predicted that in a system of bosons at temperatures below a certain critical temperature [MATH] the single-particle ground state is macroscopically occupied [CITATION].', 'cond-mat-0006293-3-2-1': 'This effect is commonly referred as Bose-Einstein condensation and a large number of phenomena, among others the condensation phenomena in alkali atoms, the superfluidity of [MATH]He and the superconductivity, are identified as signatures of this effect.', 'cond-mat-0006293-3-2-2': 'However, the physical situation is very intricate in most experiments.', 'cond-mat-0006293-3-3-0': 'Recent experiments with dilute gases of alkali atoms in magnetic [CITATION] and optical [CITATION] traps are in some sense the up to now best experimental approximation of the ideal non-interacting Bose-Einstein system in an external power law potential.', 'cond-mat-0006293-3-3-1': 'The achievement of ultra-low temperatures by laser cooling and evaporative cooling opens the opportunity to study the Bose-Einstein condensation under systematic variation of adjustable external parameters, e.g the trap geometry, the number of trapped atoms, the temperature, and by the choice of the alkali atoms the effective interparticle interactions.', 'cond-mat-0006293-3-3-2': 'Even in the approximation of non-interacting particles the explanation of these experiments requires some care, because the number of bosons in these novel traps is finite and fixed and the standard grand-canonical treatment is not appropriate.', 'cond-mat-0006293-3-3-3': 'The effect of the finite particle numbers on the second moments of the distribution function, e.g. the specific heat and the fluctuation of the ground state occupation number has been addressed in a number of publications [CITATION].', 'cond-mat-0006293-3-3-4': 'In [CITATION] we have presented a recursion method to calculate the canonical partition function for non-interacting bosons and investigated the dependency of the thermodynamic properties of the condensate on the trap geometry.', 'cond-mat-0006293-3-4-0': 'The order of the phase transition in small systems sensitively depends on finite size effects.', 'cond-mat-0006293-3-4-1': 'Compared to the macroscopic system even for as simple systems as the 3-dimensional ideal gas the order of the phase transition might change for mesoscopic systems where the number of particles is finite or for trapped gases with different trap geometries.', 'cond-mat-0006293-3-5-0': 'In this paper we address the classification of the phase transition of a finite number of non-interacting bosons in a power law trap with an effective one-particle density of states [MATH] being formally equivalent to a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-3-5-1': 'We use a classification scheme based on the distribution of zeros of the canonical partition function initially developed by Grossman et al. [CITATION], and Fisher et al. [CITATION], which has been extended by us [CITATION] as a classification scheme for finite systems.', 'cond-mat-0006293-3-5-2': 'On the basis of this classification scheme we are able to extract a qualitative difference between the order of the phase transition occuring in Bose-Einstein condensates in 3-dimensional traps [CITATION] and in 2-dimensional traps which was recently discovered by Safonov et al. in a gas of hydrogen atoms absorbed on the surface of liquid helium [CITATION].', 'cond-mat-0006293-3-5-3': 'Since we do not consider particle interactions this difference is only due to the difference in the confining potential.', 'cond-mat-0006293-3-6-0': 'We give a detailed review of the classification scheme in Sec. [REF].', 'cond-mat-0006293-3-6-1': 'In Sec. [REF] we present the method for the calculation of the canonical partition function in the complex plane and describe details of the numerical implementation.', 'cond-mat-0006293-3-6-2': 'Our results for [MATH] and particle numbers varying from 10 to 300 are presented in Sec. [REF] as well as calculations for a 3-dimensional parabolically confined Bose-gas.', 'cond-mat-0006293-3-7-0': '# Classification scheme', 'cond-mat-0006293-3-8-0': 'In 1952 Yang and Lee have shown that the grand canonical partition function can be written as a function of its zeros in the complex fugacity plane, which lie for systems with hard-core interactions and for the Ising model on a unit circle [CITATION].', 'cond-mat-0006293-3-9-0': 'Grossmann et al. [CITATION] and Fisher [CITATION] have extended this approach to the canonical ensemble by analytic continuation of the inverse temperature to the complex plane [MATH].', 'cond-mat-0006293-3-9-1': 'Within this treatment all phenomenologically known types of phase transitions in macroscopic systems can be identified from the properties of the distribution of zeros of the canonical partition function.', 'cond-mat-0006293-3-10-0': 'In [CITATION] we have presented a classification scheme for finite systems which has its macroscopic equivalent in the scheme given by Grossmann.', 'cond-mat-0006293-3-10-1': 'As usual the canonical partition function reads [EQUATION] which we write as a product [MATH], where [MATH] describes the limiting behavior of [MATH] for [MATH] imposing that [MATH].', 'cond-mat-0006293-3-10-2': 'This limiting partition function will only depend on the external potential applied to the system, whereas [MATH] will depend on the specific interaction between the system particles.', 'cond-mat-0006293-3-10-3': 'E.g. for a [MATH]-particle system in a [MATH]-dimensional harmonic trap [MATH] and thus the zeros of [MATH] are the same as the zeros of [MATH].', 'cond-mat-0006293-3-10-4': 'Since the partition function is an integral function, the zeros [MATH] are complex conjugated and the partition function reads [EQUATION]', 'cond-mat-0006293-3-10-5': 'The zeros of [MATH] are the poles of the Helmholtz free energy [MATH], i.e. the free energy is analytic everywhere in the complex temperature plane except at the zeros of [MATH].', 'cond-mat-0006293-3-11-0': 'Different phases are represented by regions of holomorphy which are separated by zeros lying dense on lines in the complex temperature plane.', 'cond-mat-0006293-3-11-1': 'In finite systems the zeros do not squeeze on lines which leads to a more blurred separation of different phases.', 'cond-mat-0006293-3-11-2': 'We interpret the zeros as boundary posts between two phases.', 'cond-mat-0006293-3-11-3': 'The distribution of zeros contains the complete thermodynamic information about the system and all thermodynamic properties are derivable from it.', 'cond-mat-0006293-3-11-4': 'Within this picture the interaction part of the specific heat is given by [EQUATION]', 'cond-mat-0006293-3-11-5': 'The zeros of the partition function are poles of [MATH].', 'cond-mat-0006293-3-11-6': 'As can be seen from Eq. ([REF]) a zero approaching the real axis infinitely close causes a divergence at real temperature.', 'cond-mat-0006293-3-11-7': 'The contribution of a zero [MATH] to the specific heat decreases with increasing imaginary part [MATH].', 'cond-mat-0006293-3-11-8': 'Thus, the thermodynamic properties of a system are governed by the zeros of [MATH] close to the real axis.', 'cond-mat-0006293-3-12-0': 'The basic idea of the classification scheme for phase transitions in small systems presented in [CITATION] is that the distribution of zeros close to the real axis can approximately be described by three parameters, where two of them reflect the order of the phase transition and the third merely the size of the system.', 'cond-mat-0006293-3-13-0': 'We assume that the zeros lie on straight lines (see Fig. [REF]) with a discrete density of zeros given by [EQUATION] with [MATH], and approximate for small [MATH] the density of zeros by a simple power law [MATH].', 'cond-mat-0006293-3-13-1': 'Considering only the first three zeros the exponent [MATH] can be estimated as [EQUATION]', 'cond-mat-0006293-3-13-2': 'The second parameter to describe the distribution of zeros is given by [MATH] where [MATH] is the crossing angle of the line of zeros with the real axis (see Fig. [REF]).', 'cond-mat-0006293-3-13-3': 'The discreteness of the system is reflected in the imaginary part [MATH] of the zero closest to the real axis.', 'cond-mat-0006293-3-14-0': 'In the thermodynamic limit we have always [MATH].', 'cond-mat-0006293-3-14-1': 'In this case the parameters [MATH] and [MATH] coincide with those defined by Grossmann et al [CITATION], who have shown how different types of phase transitions can be attributed to certain values of [MATH] and [MATH].', 'cond-mat-0006293-3-14-2': 'They claimed that [MATH] and [MATH] corresponds to a first order phase transition, second order transitions correspond to [MATH] with [MATH] or [MATH], third order transitions to [MATH] with arbitrary values of [MATH], and that all higher order phase transition correspond to [MATH].', 'cond-mat-0006293-3-14-3': 'For macroscopic systems (with [MATH] cannot be smaller than zero, because this would cause a divergence of the internal energy.', 'cond-mat-0006293-3-14-4': 'However in small systems with a finite [MATH] this is possible.', 'cond-mat-0006293-3-15-0': 'In our classification scheme we therefore define phase transitions in small systems to be of first order for [MATH], while second and higher order transitions are defined in complete analogy to the Grossmann scheme augmented by the third parameter [MATH].', 'cond-mat-0006293-3-15-1': 'The definition of a critical temperature [MATH] in small systems is crucial and ambiguous since no thermodynamic properties diverge.', 'cond-mat-0006293-3-15-2': 'Thus, different definitions are possible.', 'cond-mat-0006293-3-15-3': 'We define the critical temperature as [MATH], i.e. the crossing point of the approximated line of zeros with the real temperature axis.', 'cond-mat-0006293-3-15-4': 'An alternative definition is the real part of the first complex zero [MATH].', 'cond-mat-0006293-3-15-5': 'In the thermodynamic limit both definitions coincide.', 'cond-mat-0006293-3-16-0': 'Comparing the specific heats calculated for different discrete distributions of zeros shows the advantages of this classification scheme.', 'cond-mat-0006293-3-16-1': 'Fig. [REF] shows (a) three distributions of zeros lying on straight lines corresponding to a first order transition ([MATH] and [MATH]), a second order transition ([MATH] and [MATH]), and a third order phase transition ([MATH] and [MATH]) and (b) the pertinent specific heats.', 'cond-mat-0006293-3-16-2': 'In all cases the specific heat exhibit a hump extending over a finite temperature region and cannot be used to classify the phase transition.', 'cond-mat-0006293-3-16-3': 'In contrast, even for very small systems (large [MATH]) the order of the phase transition is extractable from the distribution of zeros.', 'cond-mat-0006293-3-17-0': 'The zeros of the canonical partition function have a distinct geometrical interpretation which explains the smoothed curves of the specific heat and other thermodynamic properties in finite systems.', 'cond-mat-0006293-3-18-0': 'Fig. [REF] shows (a) the ground state occupation number [MATH] in the complex temperature plane and (b) the ground state occupation number at real temperatures for a finite ideal Bose gas of [MATH] particles, where [MATH] is given by the derivative of the logarithm of the canonical partition function [MATH] with respect to the ground state energy [MATH], i.e. [MATH].', 'cond-mat-0006293-3-19-0': 'Zeros of the partition function are poles of [MATH] and are indicated by dark spots, which influence the value of the ground state occupation number at real temperatures impressively.', 'cond-mat-0006293-3-19-1': 'Every pole seems to radiate onto the real axis and therefore determines the occupation number at real temperatures.', 'cond-mat-0006293-3-19-2': 'This radiation extends over a broad temperature range so that the occupation number for real temperatures does not show a discontinuity but a smoothed curve.', 'cond-mat-0006293-3-19-3': 'A closer look at Eq. ([REF]) gives the mathematical explanation for this effect.', 'cond-mat-0006293-3-19-4': 'The discrete distribution of zeros, i.e. [MATH], inhibits the specific heat and all other thermodynamic properties to show a divergency at some critical temperature because the denominators of the arguments of the sum remain finite.', 'cond-mat-0006293-3-20-0': 'Without going into a detailed analysis we note that in the thermodynamic limit the parameter [MATH] is connected to the critical index for the specific heat by [EQUATION]', 'cond-mat-0006293-3-20-1': 'However, since critical indices are used to describe the shape of a divergency at the critical point an extension to small systems seems to be more or less academical.', 'cond-mat-0006293-3-21-0': 'The introduction of complex temperatures might seem artificial at first sight but, in fact, the imaginary parts [MATH] of the complex zeros [MATH] have an obvious quantum mechanical interpretation.', 'cond-mat-0006293-3-21-1': 'We write the quantum mechanical partition function as [EQUATION] introducing a canonical state as a sum over Boltzmann-weighted eigenstates [MATH].', 'cond-mat-0006293-3-21-2': 'We explicitly write the imaginary part as [MATH] since the dimension is [MATH] and the imaginary part therefore can be interpreted as time.', 'cond-mat-0006293-3-21-3': 'Then the imaginary parts [MATH] of the zeros resemble those times for which the overlap of the initial canonical state with the time evoluted state vanishes.', 'cond-mat-0006293-3-21-4': 'However, they are not connected to a single system but to a whole ensemble of identical systems in a heat bath with an initial Boltzmann distribution.', 'cond-mat-0006293-3-22-0': '# BEC in power law traps', 'cond-mat-0006293-3-23-0': 'In this section we assume a continuous single particle density of states [MATH] as an approximation for a [MATH]-dimensional harmonic oscillator or a [MATH]-dimensional ideal gas.', 'cond-mat-0006293-3-23-1': 'E.g. for the harmonic oscillator this corresponds to the limit of [MATH] and taking only the leading term of the degeneracy of the single particle energy levels.', 'cond-mat-0006293-3-23-2': 'The one-particle partition function is given by the Laplace transformation [EQUATION]', 'cond-mat-0006293-3-23-3': 'The canonical partition function for [MATH] non-interacting bosons can be calculated by the following recursion [CITATION] [EQUATION] where [MATH] is the one-particle partition function at temperature [MATH] and [MATH].', 'cond-mat-0006293-3-23-4': 'For small particle numbers this recursion works fine, even though its numerical effort grows proportional to [MATH].', 'cond-mat-0006293-3-24-0': 'With ([REF]) as [MATH] Eq. ([REF]) leads to a polynomial of order N in [MATH] for [MATH] which can be easily generated using Maple or Mathematica.', 'cond-mat-0006293-3-24-1': 'The zeros of this polynomial can be found by standard numerical methods.', 'cond-mat-0006293-3-25-0': 'Fig. [REF] displays the zeros of the [MATH]-particle partition function for [MATH] in the complex temperature plane for particle numbers [MATH] and [MATH].', 'cond-mat-0006293-3-25-1': 'For [MATH] the zeros approach the positive real axis with increasing particle number and are shifted to higher temperatures which is already at first sight an indicator of phase transitions.', 'cond-mat-0006293-3-25-2': 'For [MATH] the zeros approach the real axis only at negative temperature.', 'cond-mat-0006293-3-25-3': 'This behavior is consistent with the usual prediction that there is no Bose-Einstein condensation for the one-dimensional harmonic oscillator and the two-dimensional ideal Bose gas [CITATION].', 'cond-mat-0006293-3-26-0': 'The symmetry of the distributions of zeros is due to the fact that [MATH] is a polynomial in [MATH].', 'cond-mat-0006293-3-26-1': 'For this reason it can be inferred that for [MATH] the zeros lie on a perfect circle.', 'cond-mat-0006293-3-27-0': 'Fig. [REF] shows the corresponding specific heats calculated using equation ([REF]).', 'cond-mat-0006293-3-27-1': 'As expected, for [MATH] the specific heat has no hump and approaches with increasing temperature the classical value.', 'cond-mat-0006293-3-27-2': 'We therefore expel the analysis of [MATH] from the discussions below.', 'cond-mat-0006293-3-27-3': 'For [MATH] the specific heats show humps or peaks, which get sharper with increasing [MATH] and increasing particle number.', 'cond-mat-0006293-3-27-4': 'However, from these smooth curves the orders of the phase transition cannot be deduced.', 'cond-mat-0006293-3-28-0': 'In Fig. [REF] the classification parameters [MATH] defined above are plotted for two to six dimensions and particle numbers up to [MATH].', 'cond-mat-0006293-3-28-1': 'For all values of [MATH] the parameter [MATH] is only a slightly varying function of [MATH] and approaches very fast an almost constant value.', 'cond-mat-0006293-3-28-2': 'Since [MATH] is the primary classification parameter from Fig. [REF](a) we can directly infer that the [MATH] system exhibits a third order phase transition ([MATH] while the transition for all higher dimensions is of second order ([MATH]).', 'cond-mat-0006293-3-28-3': 'For [MATH] the dependence of [MATH] on [MATH] is plotted in Fig. [REF](a).', 'cond-mat-0006293-3-28-4': 'Since [MATH] decreases rather rapidly with increasing [MATH] it can be speculated that systems corresponding to a large [MATH] exhibit a phase transition which is almost of first order.', 'cond-mat-0006293-3-28-5': 'As mentioned above for finite systems even values [MATH] cannot be excluded by mathematical reasons.', 'cond-mat-0006293-3-28-6': 'We note that two-dimensional Bose-gases are an interesting and growing field of research.', 'cond-mat-0006293-3-28-7': 'As it is well known, the ideal free Bose-gas in two dimensions ([MATH]) does not show a phase transition due to thermal fluctuations which destabilize the condensate [CITATION].', 'cond-mat-0006293-3-28-8': 'Switching on a confining potential greatly influences the properties of the gas, the thermal fluctuations are suppressed and the gas will show Bose-Einstein condensation.', 'cond-mat-0006293-3-28-9': 'Recent experiments [CITATION] have shown that Bose-Einstein condensation is possible even though it is called a quasi-condensate.', 'cond-mat-0006293-3-28-10': 'In our notion the quasi-condensate is just a third order phase transition.', 'cond-mat-0006293-3-28-11': 'Thus, our results are in complete agreement with recent experiments and earlier theoretical work.', 'cond-mat-0006293-3-28-12': 'An interesting question in this respect is whether the order of the transition changes for [MATH] in the limit [MATH].', 'cond-mat-0006293-3-28-13': 'Additional calculation for larger [MATH], which are not printed in Fig. [REF] indicate that [MATH] approaches 1 or might even get smaller.', 'cond-mat-0006293-3-28-14': 'Note that [MATH] is equivalent to a hypothetical 4-dimensional ideal Bose gas or Bosons confined in a 2-dimensional parabolic trap.', 'cond-mat-0006293-3-28-15': 'Our results indicate that the order of the phase transition sensitively depends on [MATH] for values around 2.', 'cond-mat-0006293-3-28-16': 'This might be the reason why phase transitions in three space dimensions are sometimes classified as second and sometimes as third order phase transitions.', 'cond-mat-0006293-3-29-0': 'The parameter [MATH] is a measure of the finite size of the system, i.e. the scaling behavior of [MATH] as a function of [MATH] is a measure of how fast a system approaches a true n-th order phase transition in the Ehrenfest sense.', 'cond-mat-0006293-3-29-1': 'The [MATH] dependence of [MATH] is displayed in Fig. [REF](c).', 'cond-mat-0006293-3-29-2': 'The scaling behavior can be approximated by [MATH] with [MATH] ranging between 1.06 and 1.12 for [MATH].', 'cond-mat-0006293-3-30-0': 'The [MATH] dependence of the classification parameter is visualized in Fig. [REF] for 50 particles.', 'cond-mat-0006293-3-30-1': 'For this system size we found [MATH] and [MATH].', 'cond-mat-0006293-3-31-0': 'The results presented above for continuous single particle densities of states [MATH] are obtained within semi-analytical calculations.', 'cond-mat-0006293-3-31-1': 'In order to compare these results to systems with a discrete level density we adopt as a reference system the 3-dimensional harmonic oscillator with the partition function given by [EQUATION] with [MATH].', 'cond-mat-0006293-3-32-0': 'Fig. [REF](a) displays the zeros of the partition function ([REF]) for [MATH] and [MATH].', 'cond-mat-0006293-3-32-1': 'Fig. [REF](b) displays a contour plot of the absolute value of the ground state occupation number [MATH] with [MATH] given by ([REF]) calculated using an alternative recursion formula [CITATION].', 'cond-mat-0006293-3-32-2': 'The zeros of [MATH] are poles of [MATH] and are indicated by dark spots in this figure.', 'cond-mat-0006293-3-33-0': 'Analyzing the distribution of zeros consolidates our speculation that the order of the phase transition sensitively depends on [MATH].', 'cond-mat-0006293-3-33-1': 'The distribution of zeros behaves like the above calculated values for [MATH] but quantitatively like [MATH].', 'cond-mat-0006293-3-33-2': 'Since the degeneracy of the three-dimensional harmonically confined ideal Bose-gas is a second-order polynomial not only the quadratic term has to be taken into account.', 'cond-mat-0006293-3-33-3': 'The linear term becomes dominant for lower temperatures, so for very low temperatures the best approximation of a continuous one-particle density of states is [MATH].', 'cond-mat-0006293-3-33-4': 'The parameter [MATH] supports this statement [CITATION], i.e. [MATH] resides in a region above 1.', 'cond-mat-0006293-3-33-5': 'Whereas the parameter [MATH] behaves like the [MATH] case.', 'cond-mat-0006293-3-33-6': 'Finally the parameter [MATH] which is a measure for the discreteness of the system shows a [MATH] dependence which is comparable to the one for [MATH].', 'cond-mat-0006293-3-33-7': 'Thus, for small systems the phase transition is of third order, it can be speculated if it becomes a second order transition in the thermodynamic limit.', 'cond-mat-0006293-3-34-0': 'Not only qualitatively but also quantitatively our calculations are in very good agreement with recent theoretical works [CITATION].', 'cond-mat-0006293-3-34-1': 'Comparing the critical temperature which we defined in Sec. [REF] with the usually utilized ones like the temperature of the peak of the specific heat [MATH] or the grand-canonically calculated [MATH] confirms our approach.', 'cond-mat-0006293-3-34-2': 'In Fig. [REF] three possible definitions of the critical temperature are given which all coincide in the thermodynamic limit.', 'cond-mat-0006293-3-34-3': 'All definitions show a [MATH] dependence with [MATH] ranging between [MATH] and [MATH].', 'cond-mat-0006293-3-35-0': '# conclusion', 'cond-mat-0006293-3-36-0': 'Starting with the old ideas of Yang and Lee, and Grossmann et al. we have developed a classification scheme for phase transitions in finite systems.', 'cond-mat-0006293-3-36-1': 'Based on the analytic continuation of the inverse temperature [MATH] into the complex plane we have shown the advantages of this approach.', 'cond-mat-0006293-3-36-2': 'The distribution of the so-called Fisher-zeros [MATH] draws enlightening pictures even for small systems whereas the usually referred thermodynamic properties like the specific heat fail to classify the phase transitions properly.', 'cond-mat-0006293-3-36-3': 'The classification scheme presented in this paper enables us to name the order of the transition in a non-ambiguous way.', 'cond-mat-0006293-3-36-4': 'The complex parts [MATH] of the zeros [MATH] resemble times for which a whole ensemble of identical systems under consideration in a heat bath with an initial Boltzmann-distribution looses its memory.', 'cond-mat-0006293-3-37-0': 'We have applied this to ideal non-interacting Bose-gases confined in power-law traps.', 'cond-mat-0006293-3-37-1': 'We have found that the order of the phase transition sensitively depends on the single particle density of states generated by the confining potential.', 'cond-mat-0006293-3-37-2': 'The distribution of zeros exactly reveals the order of the phase transition in finite systems.'}

1908.02954

{'1908.02954-1-0-0': 'The "rare type match problem" is the situation in which the suspect\'s DNA profile, matching the DNA profile of the crime stain, is not in the database of reference.', '1908.02954-1-0-1': 'The evaluation of this match in the light of the two competing hypotheses (the crime stain has been left by the suspect or by another person) is based on the calculation of the likelihood ratio and depends on the population proportions of the DNA profiles, that are unknown.', '1908.02954-1-0-2': 'We propose a Bayesian nonparametric method that uses a two-parameter Poisson Dirichlet distribution as a prior over the ranked population proportions, and discards the information about the names of the different DNA profiles.', '1908.02954-1-0-3': 'This fits very well the data coming from European Y-STR DNA profiles, and the calculation of the likelihood ratio becomes quite simple thanks to a justified Empirical Bayes approach.', '1908.02954-1-1-0': '# Introduction', '1908.02954-1-2-0': 'The largely accepted method for evaluating how much some available data [MATH] (typically forensic evidence) helps discriminate between two hypotheses of interest (the prosecution hypothesis [MATH] and the defense hypothesis [MATH]), is the calculation of the likelihood ratio (LR), a statistic that expresses the relative plausibility of the data under these hypotheses, defined as [EQUATION]', '1908.02954-1-2-1': "Widely considered the most appropriate framework to report a measure of the 'probative value' of the evidence regarding the two hypotheses , it indicates the extent to which observed data is favoured by one hypothesis over the other.", '1908.02954-1-2-2': 'Forensic literature presents many approaches to calculate the LR, mostly divided into Bayesian and frequentist methods (see [CITATION] for a careful differentiation between these two approaches).', '1908.02954-1-3-0': 'This paper proposes the first Bayesian nonparametric method to the likelihood ratio assessment in the rare type match case, the challenging situation in which there is a match between some characteristic of the recovered material and of the control material, but this characteristic has not been observed before in previously collected samples (i.e. in the database of reference).', '1908.02954-1-3-1': 'This constitutes a problem because the value of the likelihood ratio depends on the unknown proportion of the matching characteristic in a reference population, and the uncertainty over this proportion is, in standard practice for simpler situations dealt with using the relative frequency of the characteristic in the available database.', '1908.02954-1-3-2': 'In particular, we will focus on Y-STR data, for which the rare type match problem is often recurring .', '1908.02954-1-4-0': 'The use of our Bayesian nonparametric method involves the mathematical assumption that there are infinitely many Y-STR profiles.', '1908.02954-1-4-1': 'Of course, we do not believe this literally to be true.', '1908.02954-1-4-2': 'We do suppose that there are so many profiles that we cannot say anything sensible about how many there are, except that it is a very very large number.', '1908.02954-1-4-3': 'The parameter of the model is the infinite-dimensional vector [MATH], made of the (unknown) sorted population proportions of all possible Y-STR profiles.', '1908.02954-1-4-4': 'As a prior over [MATH] we choose the two-parameter Poisson Dirichlet distribution, and we model the uncertainty over its parameters [MATH] and [MATH] through the use of a hyperprior.', '1908.02954-1-4-5': 'The information contained in the names of the profiles is discarded, thereby reducing the full data [MATH] to a smaller set [MATH].', '1908.02954-1-5-0': 'If compared to traditional Bayesian methods such as those discussed in [CITATION], this method has the advantage of having a prior for the parameter p that is more realistic for the population we want to model.', '1908.02954-1-5-1': 'Moreover, although its technical theoretical background, we empirically derived an approximation that makes the method extremely simple to apply for practical use: indeed, simulation experiments show that a hybrid empirical approach that plugs-in maximum likelihood estimators for the hyperparamenter is justified, at least when using Y-STR data from European populations.', '1908.02954-1-5-2': 'The last point in favor of this model, the choice of the two-parameter Poisson Dirichlet prior over p has the following sufficientness property: the probability of observing a new Y-STR profile only depends on the number of already observed Y-STR profiles and the sample size, while the probability of observing a Y-STR profile that is already in the database only depends on its frequency in the database and on the sample size.', '1908.02954-1-6-0': 'The paper is structured in the following way: Section [REF] introduces to the rare type match problem.', '1908.02954-1-6-1': 'Section [REF] presents the model, with the assumptions and the prior distribution chosen for parameter [MATH] along with some theory on random partitions and Chinese restaurant representation, useful to provide a convenient and compact representation of the reduced data [MATH].', '1908.02954-1-6-2': 'Also, a lemma that facilitates computing the likelihood ratio in a very elegant way is presented and proved.', '1908.02954-1-6-3': 'In Section [REF], the likelihood ratio is derived.', '1908.02954-1-6-4': 'Section [REF] proposes the application of this model to a real database of Y-STR profiles.', '1908.02954-1-6-5': 'We will discuss data-driven choices for the hyperparameters, and the derivation of the frequentist likelihood ratio values obtained both with and without reducing the data to partitions, in the ideal situation in which vector [MATH] is known.', '1908.02954-1-7-0': '# The rare type match problem', '1908.02954-1-8-0': "The evaluation of a match between the profile of a particular piece of evidence and a suspect's profile depends on the proportion of that profile in the population of potential perpetrators.", '1908.02954-1-8-1': 'Indeed, it is intuitive that the rarer the matching profile, the more the suspect is in trouble.', '1908.02954-1-8-2': 'Problems arise when the observed frequency of the profile in a sample from the population of interest (i.e., in a reference database) is 0.', '1908.02954-1-8-3': 'This problem can be named as "the new type match problem", but we decided to use the name "rare type match problem", motivated by the fact that a Y-STR profile that has zero occurrences is likely to be rare, even though it is challenging to quantify how rare it is.', '1908.02954-1-8-4': 'The rare type match problem is particularly important in case a new kind of forensic evidence, such as results from DIP-STR markers (see for instance [CITATION]) is involved, for which the available database size is still limited.', '1908.02954-1-8-5': 'The problem also occurs when more established types of evidence, such as Y-chromosome (or mitochondrial) DNA profiles are used since the set of possible Y-STR profiles is extremely large.', '1908.02954-1-8-6': 'As a consequence, most of the Y-STR haplotypes are not represented in the database.', '1908.02954-1-8-7': 'The Y-STR marker system will thus be retained here as an extreme but in practice common and important way in which the problem of the assessing evidential value of rare type match can arise.', '1908.02954-1-8-8': 'This problem is so substantial that it has been defined "the fundamental problem of forensic mathematics" .', '1908.02954-1-9-0': 'This problem has been addressed in well-know non-forensic statistics, and many solutions have been proposed.', '1908.02954-1-9-1': 'The empirical frequency estimator, also called naive estimator, that uses the frequency of the characteristic in the database, puts unit probability mass on the set of already observed characteristics, and it is thus unprepared for the observation of a new type.', '1908.02954-1-9-2': 'A solution could be the add-constant estimators (in particular the well-known add-one estimator, due to [CITATION], and the add-half estimator of [CITATION]), which add a constant to the count of each type, included the unseen ones.', '1908.02954-1-9-3': 'However, this method requires to know the number of possible unseen types, and it performs badly when this number is large compared to the sample size (see [CITATION] for an additional discussion).', '1908.02954-1-9-4': 'Alternatively, [CITATION], based on an intuition on A.M. Turing, proposed the Good-Turing estimator for the total unobserved probability mass, based on the proportion of singleton observations in the sample.', '1908.02954-1-9-5': 'An extension of this estimator is applied to the frequentist LR assessment in the rare type match case in [CITATION].', '1908.02954-1-9-6': 'More recently, [CITATION] have introduced the high-profile estimator, which extends the tail of the naive estimator to the region of unobserved types.', '1908.02954-1-9-7': '[CITATION] improved this estimator and provided the consistency proof.', '1908.02954-1-9-8': 'Papers that address the rare Y-STR haplotype problem in forensic context are for instance [CITATION], [CITATION], [CITATION] and [CITATION].', '1908.02954-1-9-9': 'The latter applies the classical Bayesian approach (the beta-binomial and the Dirichlet multinomial problem) to the LR assessment in the rare haplotype match case.', '1908.02954-1-9-10': 'All these methods do not take into account genetic information contained in the allelic numbers forming a Y-STR DNA profile.', '1908.02954-1-9-11': 'For instance, due to relatedness, the observation of a particular Y-STR profile increases the probability of observing the same Y-STR profile again or Y-STR profiles that differ only for few alleles.', '1908.02954-1-9-12': 'We refer the reader to [CITATION] for models that use population genetics for coancestry.', '1908.02954-1-9-13': 'These models are not designed to be used for the rare type match case, but the Discrete Laplace method presented in [CITATION] can be successfully applied to that purpose, as shown in [CITATION].', '1908.02954-1-9-14': 'More recently, [CITATION] provided an interesting discussion on the importance of taking into account genetic information.', '1908.02954-1-9-15': "However, we start from the point of view that we don't know how to use that information in a sensible way.", '1908.02954-1-9-16': 'For this reason, we prefer to discard it, and only focussing on the part of the data we are more confident to model.', '1908.02954-1-10-0': 'Bayesian nonparametric estimators for the probability of observing a new type have been proposed by [CITATION] using Dirichlet processes, by [CITATION] using a general Gibbs prior, and by [CITATION] with specific interest to the two-parameter Poisson Dirichlet prior.', '1908.02954-1-10-1': 'However, the LR assessment requires not only the probability of observing a new species but also the probability of observing this same species twice (according to the defense the crime stain profile and the suspect profile are two independent observations): to our knowledge, the present paper is the first one to address the problem of LR assessment in the rare type match case using a Bayesian nonparametric model.', '1908.02954-1-10-2': 'As a prior for [MATH] we will use the two-parameter Poisson Dirichlet distribution, which is proving useful in many discrete domains, in particular language modeling .', '1908.02954-1-10-3': 'Besides, it shows a power-law behavior that describes an incredible variety of phenomena , and seems to characterize the distribution of Y-STR haplotypes, too.', '1908.02954-1-11-0': '# The model', '1908.02954-1-12-0': '## Notation', '1908.02954-1-13-0': 'Throughout the paper the following notation is chosen: random variables and their values are denoted, respectively, with uppercase and lowercase characters: [MATH] is a realization of [MATH].', '1908.02954-1-13-1': 'Random vectors and their values are denoted, respectively, by uppercase and lowercase bold characters: [MATH] is a realization of the random vector [MATH].', '1908.02954-1-13-2': 'Probability is denoted with [MATH], while the density of a continuous random variable [MATH] is denoted alternatively by [MATH] or by [MATH] when the subscript is clear from the context.', '1908.02954-1-13-3': 'For a discrete random variable [MATH], the density notation [MATH] and the discrete one [MATH] will be interchangeably used.', '1908.02954-1-13-4': 'Moreover, we will use shorthand notation like [MATH] to stand for the probability density of Y with respect to the conditional distribution of [MATH] given [MATH].', '1908.02954-1-14-0': 'Notice that in Formula [REF], [MATH] was regarded as the event corresponding to the observation of the available data.', '1908.02954-1-14-1': 'However, later in the paper, [MATH] will be regarded as a random variable generically representing the data.', '1908.02954-1-14-2': 'The particular data at hand will correspond to the value [MATH].', '1908.02954-1-14-3': 'In that case, the following notation will thus be preferred: [EQUATION]', '1908.02954-1-14-4': 'Lastly, notice that "DNA types" is used throughout the paper as a general term to indicate Y-STR profiles.', '1908.02954-1-15-0': '## Model assumptions', '1908.02954-1-16-0': 'Our model is based on the two following assumptions:', '1908.02954-1-17-0': '[Assumption 1] There are infinitely many DNA types in Nature.', '1908.02954-1-18-0': 'This assumption, already used by e.g. [CITATION] in the \'infinite alleles model\', allows the use of Bayesian nonparametric methods and is very useful for instance in "species sampling problems" when you are unable to specify the total number of possible different species in Nature.', '1908.02954-1-18-1': 'This assumption is sensible also in case of Y-STR DNA profiles since the state space of possible different haplotypes is so large that can be considered infinite.', '1908.02954-1-19-0': '[Assumption 2] The names of the different DNA types do not contain relevant information.', '1908.02954-1-20-0': 'Actually, the specific sequence of numbers that forms a DNA profile carries information: if two profiles show few differences this means that they are separated by few mutation drifts, hence the profiles share a relatively recent common ancestor.', '1908.02954-1-20-1': 'However, this information is difficult to exploit and may not be so relevant for the LR assessment.', '1908.02954-1-20-2': 'This is the reason why, as already mentioned, we will treat DNA types as "colors", and only consider the partition into different categories.', '1908.02954-1-20-3': 'Stated otherwise, we put no topological structure on the space of the DNA types.', '1908.02954-1-21-0': "Notice that this assumption makes the model a priori suitable for any characteristic which shows many different possible types, thus the approach described in this paper still holds, in principle, after replacing 'DNA types' with any other category.", '1908.02954-1-21-1': 'However, in this paper, we will only focus on the model with Y-STR profiles as categories, for which the model fits quite well the available data (see Section [REF]).', '1908.02954-1-22-0': '## Prior', '1908.02954-1-23-0': 'In Bayesian statistics, parameters of interest are modeled through random variables.', '1908.02954-1-23-1': 'The (prior) distribution over a parameter should represent the uncertainty about its value.', '1908.02954-1-24-0': 'LR assessment for the rare type match involves two unknown parameters of interest: one is [MATH], representing the unknown true hypothesis, the other is [MATH], the vector of the unknown population frequencies of all DNA profiles in the population of potential perpetrators.', '1908.02954-1-24-1': 'The dichotomous random variable [MATH] is used to model parameter [MATH], and the posterior distribution of this random variable, given the data, is the ultimate aim of the forensic inquiry.', '1908.02954-1-24-2': 'Similarly, a random variable [MATH] is used to model the uncertainty over [MATH].', '1908.02954-1-24-3': 'Because of Assumption 1, [MATH] is an infinite-dimensional parameter, hence the need for Bayesian nonparametric methods .', '1908.02954-1-24-4': 'In particular, [MATH], with [MATH] a countable set of indexes, [MATH], and [MATH].', '1908.02954-1-24-5': 'Moreover, because of Assumption 2, data can be reduced to partitions, as explained in Section [REF], and it will turn out that the distribution of these partitions does not depend on the order of the [MATH].', '1908.02954-1-24-6': 'Hence, we can define the parameter [MATH] as having values in [MATH], the ordered infinite-dimensional simplex.', '1908.02954-1-24-7': 'The uncertainty about its value is expressed by the prior distribution over [MATH], for which we choose the two-parameter Poisson Dirichlet distribution .', '1908.02954-1-25-0': 'As a reason for choosing the two-parameter Poisson-Dirichlet distribution among the possible Bayesian nonparametric priors is that it is the only one that has the following very convenient sufficientness property : the probability of observing a new species only depends on the number of already observed species and on the sample size, and the probability of observing an already seen species only depends on its frequency in the sample on the sample size.', '1908.02954-1-25-1': 'Moreover, this choice will be further validated by model fitting (see Section [REF]).', '1908.02954-1-26-0': 'The two-parameter Poisson-Dirichlet distribution can be defined through the following stick-breaking representation :', '1908.02954-1-27-0': '[two-parameter GEM distribution] Given [MATH] and [MATH] satisfying the following conditions: [EQUATION] the vector [MATH] is said to be distributed according to the GEM([MATH]), if [EQUATION] where [MATH], [MATH],... are independent random variables distributed according to [EQUATION].', '1908.02954-1-27-1': 'It holds that [MATH], and [MATH]', '1908.02954-1-28-0': '[Two-parameter Poisson Dirichlet distribution] Given [MATH] and [MATH] satisfying condition [REF], and a vector [MATH], the random vector [MATH] obtained by ranking [MATH], such that [MATH], is said to be Poisson Dirichlet distributed PD[MATH].', '1908.02954-1-28-1': 'Parameter [MATH] is called the discount parameter, while [MATH] is the concentration parameter.', '1908.02954-1-29-0': 'For our model we will not allow [MATH], hence we will assume [MATH].', '1908.02954-1-29-1': 'Indeed, the discount parameter is necessary if we want a prior that shows a power-law behavior: few Y-STR types observed many times and many Y-STR types observed only once, as observed in available databases, see Section [REF].', '1908.02954-1-30-0': 'Lastly, we point out that, in practice, we cannot assume to know parameters [MATH] and [MATH]: we will resolve this by modeling the uncertainty about them using an hyperprior.', '1908.02954-1-31-0': '## Bayesian network representation of the model', '1908.02954-1-32-0': 'The typical data to evaluate in case of a match is [MATH], where [MATH], and', '1908.02954-1-33-0': "[MATH] = suspect's DNA type,", '1908.02954-1-34-0': "[MATH] = crime stain's DNA type (matching with the suspect's type),", '1908.02954-1-35-0': '[MATH] = a reference database of size [MATH], which is assumed as being a random sample of DNA types from the population of possible perpetrators.', '1908.02954-1-36-0': 'The hypotheses of interest for the case are:', '1908.02954-1-37-0': '[MATH] = The crime stain was left by the suspect,', '1908.02954-1-38-0': '[MATH] = The crime stain was left by someone else.', '1908.02954-1-39-0': 'In agreement with Assumption 2, the model will ignore information about the names of the DNA types: data [MATH] will thus be reduced to [MATH] accordingly.', '1908.02954-1-39-1': 'The Bayesian network of Figure [REF] encapsulates the conditional dependencies of the random variables [MATH] of the proposed model, whose joint distribution is defined above in terms of the conditional distribution, using the factorization implied by the Bayesian network itself.', '1908.02954-1-40-0': '[MATH] is a dichotomous random variable that represents the hypotheses of interest and can take values [MATH], according to the prosecution or the defense, respectively.', '1908.02954-1-40-1': 'A uniform prior on the hypotheses is chosen: [EQUATION].', '1908.02954-1-40-2': 'Notice that this choice is made for mathematical convenience since it will not affect the likelihood ratio, the variable [MATH] being in the conditioning part.', '1908.02954-1-41-0': '([MATH]) is the random vector that represents the hyperparameters [MATH] and [MATH], satisfying condition [REF].', '1908.02954-1-41-1': 'The joint prior density of these two parameters will be generically denoted as [MATH]: [EQUATION].', '1908.02954-1-41-2': "For obvious reasons, this will be called the 'hyperprior' throughout the text.", '1908.02954-1-42-0': 'The random vector [MATH] with values in [MATH] represents the ranked population frequencies of Y-STR profile.', '1908.02954-1-42-1': '[MATH] means that [MATH] is the frequency of the most common DNA type in the population, [MATH] is the frequency of the second most common DNA type, and so on.', '1908.02954-1-42-2': 'As a prior for [MATH] we use the two-parameter Poisson Dirichlet distribution: [EQUATION].', '1908.02954-1-43-0': 'The database is assumed to be a random sample from the population.', '1908.02954-1-43-1': 'Integer-valued random variables [MATH], ..., [MATH] are here used to represent the (unknown) ranks in the population of the frequencies of the DNA types in the database.', '1908.02954-1-43-2': 'For instance, [MATH] means that the third individual in the database has the fifth most common DNA type in the population.', '1908.02954-1-43-3': 'Given [MATH] they are an i.i.d. sample from [MATH]: [EQUATION]', '1908.02954-1-43-4': 'To observe [MATH], ..., [MATH], one would need to know the rank, in terms of population proportions, of the frequency of each DNA types in the database.', '1908.02954-1-43-5': 'This is not known, hence [MATH] are not observed.', '1908.02954-1-44-0': "[MATH] represents the rank in the population ordering of the suspect's DNA type.", '1908.02954-1-44-1': 'It is again an independent draw from [MATH].', '1908.02954-1-44-2': "[EQUATION] [MATH] represents the rank in the population ordering, of the crime stain's DNA type.", '1908.02954-1-44-3': 'According to the prosecution, given [MATH], this random variable is deterministic (it is equal to [MATH] with probability 1).', '1908.02954-1-44-4': 'According to the defense it is another sample from [MATH], independent of the previous ones: [EQUATION].', '1908.02954-1-45-0': 'As already mentioned, [MATH] cannot be observed.', '1908.02954-1-45-1': 'They represent the database, where the names of the DNA types have been replaced by their (unknown) ranks in [MATH], and constitute an intermediate layer.', '1908.02954-1-46-0': "Section [REF] recalls some notions about random partitions, useful before defining node [MATH], the 'reduced' data that we want to evaluate.", '1908.02954-1-47-0': '## Random partitions and database partitions', '1908.02954-1-48-0': 'A partition of a set [MATH] is an unordered collection of nonempty and disjoint subsets of [MATH], the union of which forms [MATH].', '1908.02954-1-48-1': 'Particularly interesting for our model are partitions of the set [MATH], denoted as [MATH].', '1908.02954-1-48-2': 'The set of all partitions of [MATH] will be denoted as [MATH].', '1908.02954-1-48-3': 'Random partitions of [MATH] will be denoted as [MATH].', '1908.02954-1-48-4': 'Also, a partition of [MATH] is a finite nonincreasing sequence of positive integers that sum up to [MATH].', '1908.02954-1-48-5': 'Partitions of [MATH] will be denoted as [MATH], while random partitions as [MATH].', '1908.02954-1-49-0': 'Given a sequence of integer valued random variables [MATH], let [MATH] be the random partition defined by the equivalence classes of their indexes using the random equivalence relation [MATH] if and only if [MATH].', '1908.02954-1-49-1': 'This construction allows to build a "reduction" map from the set of values of [MATH] to the set of the partitions of [MATH] as in the following example ([MATH]): [EQUATION]', '1908.02954-1-49-2': 'Similarly, and in agreement with Assumption 2, in our model we can consider the reduction of data which ignores information about the names of the DNA types: this is achieved, for instance, by retaining from the database only the equivalence classes of the indexes of the individuals, according to the equivalence relation "to have the same DNA type".', '1908.02954-1-49-3': 'Stated otherwise, the database is reduced to the partition [MATH], obtained using these equivalence classes.', '1908.02954-1-49-4': 'However, the database only supplies part of the data.', '1908.02954-1-49-5': 'There are also two new DNA profiles that are equal to one another (and different from the already observed ones in the rare type match case).', '1908.02954-1-49-6': "Considering the suspect's profile we obtain the partition [MATH], where the first [MATH] integers are partitioned as in [MATH], and [MATH] constitutes a class by itself.", '1908.02954-1-49-7': 'Considering the crime stain profile we obtain the partition [MATH] where the first [MATH] integers are partitioned as in [MATH], and [MATH] and [MATH] belong to the same (new) class.', '1908.02954-1-49-8': 'Random variables [MATH], [MATH], and [MATH] are used to model [MATH], [MATH], and [MATH], respectively.', '1908.02954-1-50-0': 'Since prosecution and defense agree on the distribution of [MATH], but not on the distribution of [MATH], they also agree on the distribution of [MATH] but disagree on the distribution of [MATH].', '1908.02954-1-51-0': 'The crucial points of the model are the following:', '1908.02954-1-52-0': 'the random partitions can defined through random variables [MATH], ..., [MATH] and through database are the same.', '1908.02954-1-52-1': '[EQUATION] although [MATH], ..., [MATH] were not observable, the random partitions [MATH], and [MATH] are observable.', '1908.02954-1-53-0': 'To clarify, consider the following example of a database with [MATH] different DNA types, from [MATH] individuals: [EQUATION] where [MATH] is the name of the [MATH]th DNA type according to the order chosen for the database.', '1908.02954-1-53-1': 'This database can be reduced to the partition of [MATH]: [EQUATION].', '1908.02954-1-53-2': 'Then, the part of reduced data whose distribution is agreed on by prosecution and defense is [EQUATION] while the entire reduced data [MATH] can be represented as [EQUATION].', '1908.02954-1-54-0': 'Now, assume that we know the rank in the population of each of the DNA types in the database: we know that [MATH] is, for instance, the second most frequent type, [MATH] is the fourth most frequent type, and so on.', '1908.02954-1-54-1': 'Stated otherwise, we are now assuming that we observe the variables [MATH], ..., [MATH]: for instance, [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '1908.02954-1-54-2': 'It is easy to check that [MATH], [MATH], and [MATH].', '1908.02954-1-55-0': 'Data is then defined as [MATH], obtained partitioning the database enlarged with the two new observations (or partitioning [MATH]).', '1908.02954-1-55-1': 'Node [MATH] of Figure [REF] is defined accordingly.', '1908.02954-1-56-0': 'Notice that, given [MATH], [MATH] is deterministic.', '1908.02954-1-56-1': 'An important result is that, according to Proposition 4 in [CITATION] it is possible to derive directly the distribution of [MATH].', '1908.02954-1-56-2': 'In particular, it holds that if [EQUATION] and [EQUATION] then, for all [MATH], the random partition [MATH] has the following distribution: [EQUATION] where [MATH] is the size of the [MATH]th block of [MATH] (the blocks are here ordered according to the least element), and [MATH], [MATH].', '1908.02954-1-56-3': 'This formula is also known as the Pitman sampling formula, further studied in [CITATION] and shows that [MATH] does not depend on [MATH], but only on the sizes and the number of classes in the partitions.', '1908.02954-1-57-0': 'It follows that we can get rid of the intermediate layer of nodes [MATH], ..., [MATH], and have [MATH], while [MATH].', '1908.02954-1-57-1': 'The model of Figure [REF] can thus be simplified to the one in Figure [REF].', '1908.02954-1-58-0': '## Chinese Restaurant representation', '1908.02954-1-59-0': 'There is an alternative characterization of this model, called "Chinese restaurant process", due to [CITATION] for the one-parameter case, and studied in detail for the two-parameter version in [CITATION].', '1908.02954-1-59-1': 'It is defined as follows: consider a restaurant with infinitely many tables, each one infinitely large.', '1908.02954-1-59-2': 'Let [MATH] be integer-valued random variables that represent the seating plan: tables are ranked in order of occupancy, and [MATH] means that the [MATH]th customer seats at the [MATH]th table to be created.', '1908.02954-1-59-3': 'The process is described by the following transition matrix: [EQUATION] where [MATH] is the number of tables occupied by the first [MATH] customers, and [MATH] is the number of customers that occupy table [MATH].', '1908.02954-1-59-4': 'The process depends on two parameters [MATH] and [MATH] with the same conditions [REF].', '1908.02954-1-59-5': 'From [REF] one can easily see the sufficientness property mentioned in Section [REF].', '1908.02954-1-60-0': '[MATH] are not i.i.d., nor exchangeable, but it holds that [MATH] is distributed as [MATH], with [MATH] defined as in [REF], and they are both distributed according to the Pitman sampling formula [REF] .', '1908.02954-1-61-0': 'Stated otherwise, we can articulate the seating plan of [MATH] customers while indexing the tables according to either their observed occupancy (as in [MATH]) or as in [MATH] and we obtain the same partition [MATH].', '1908.02954-1-61-1': 'Similarly [MATH] is obtained when a new customer has chosen an unoccupied table (remember we are in the rare type match case), and [MATH] is obtained when the [MATH]nd customer goes to the table already chosen by the [MATH]st customer (suspect and crime stain have the same DNA type).', '1908.02954-1-61-2': 'In particular, thanks to [REF], we can write: [EQUATION] since the [MATH]nd customer goes to the same table as the [MATH]st (who was sitting alone).', '1908.02954-1-62-0': '## A useful Lemma', '1908.02954-1-63-0': 'The following lemma can be applied to four general random variables [MATH], [MATH], [MATH], and [MATH] whose conditional dependencies are described by the Bayesian network of Figure [REF].', '1908.02954-1-63-1': 'The importance of this result is due to the possibility of applying it to a very common forensic situation: the prosecution and the defense disagree on the distribution of the entirety of data ([MATH]) but agree on the distribution of a part it ([MATH]), and these distributions depend on parameters ([MATH]).', '1908.02954-1-64-0': 'Given four random variables [MATH], [MATH], [MATH] and [MATH], whose conditional dependencies are represented by the Bayesian network of Figure [REF], the likelihood function for [MATH], given [MATH] and [MATH] satisfies [EQUATION].', '1908.02954-1-65-0': 'The Bayesian representation of the model, in Figure [REF], allow to factor the joint probability density of [MATH], [MATH], [MATH] and [MATH] as [EQUATION]', '1908.02954-1-65-1': 'By Bayes formula, [MATH].', '1908.02954-1-65-2': 'This rewriting corresponds to reversing the direction of the arrow between [MATH] and [MATH]:', '1908.02954-1-66-0': 'The random variable [MATH] is now a root node.', '1908.02954-1-66-1': 'This means that when we probabilistically condition on [MATH], the graphical model changes in a simple way: we can delete the node [MATH], but just insert the value [MATH] as a parameter in the conditional probability tables of the variables [MATH] and [MATH] which formerly had an arrow from node [MATH].', '1908.02954-1-66-2': 'The next graph represents this model:', '1908.02954-1-67-0': 'This tells us, that conditional on [MATH], the joint density of [MATH], [MATH] and [MATH] is equal to [EQUATION].', '1908.02954-1-67-1': 'The joint density of [MATH] and [MATH] given [MATH] is obtained by integrating out the variable [MATH].', '1908.02954-1-67-2': 'It can be expressed as a conditional expectation value since [MATH] is the density of [MATH] given [MATH].', '1908.02954-1-67-3': 'We find: [EQUATION].', '1908.02954-1-68-0': 'Recall that this is the joint density of two of our variables, [MATH] and [MATH], after conditioning on the value [MATH].', '1908.02954-1-68-1': 'Let us now also condition on [MATH].', '1908.02954-1-68-2': 'It follows that the density of [MATH] given [MATH] and [MATH] is proportional (as function of [MATH], for fixed [MATH] and [MATH]) to the same expression, [MATH].', '1908.02954-1-69-0': 'This is a product of the prior for [MATH] with some function of [MATH] and [MATH].', '1908.02954-1-69-1': 'Since posterior odds equals prior odds times likelihood ratio, it follows that the likelihood function for [MATH], given [MATH] and [MATH] satisfies [EQUATION].', '1908.02954-1-70-0': 'Given four random variables [MATH], [MATH], [MATH] and [MATH], whose conditional dependencies are represented by the network of Figure [REF], the likelihood ratio for [MATH] against [MATH] given [MATH] and [MATH] satisfies [EQUATION]', '1908.02954-1-71-0': '## Known results about the two-parameter Poisson Dirichlet distribution', '1908.02954-1-72-0': 'We will now list some theoretical results which will be useful in the forthcoming analysis.', '1908.02954-1-72-1': 'Most of these results can be found in [CITATION].', '1908.02954-1-73-0': 'Denote as [MATH] the random number of blocks of a partition [MATH] distributed according to the Pitman sampling formula with parameters [MATH] and [MATH].', '1908.02954-1-74-0': '# The likelihood ratio', '1908.02954-1-75-0': 'Using the hypotheses and the reduction of data [MATH] defined in Section [REF], the likelihood ratio will be defined as [EQUATION].', '1908.02954-1-75-1': 'The last equality holds due to the fact that [MATH] is a deterministic function of [MATH].', '1908.02954-1-76-0': 'Corollary [REF] can be applied to our model since defense and prosecution agree on the distribution of [MATH], but not on the distribution of [MATH], and data depends on parameters [MATH] and [MATH].', '1908.02954-1-76-1': 'Thus, if [MATH] play the role of [MATH], [MATH], and [MATH], by using [REF] and [REF], we obtain: [EQUATION].', '1908.02954-1-76-2': 'By defining the random variable [MATH] we can write the LR as [EQUATION]', '1908.02954-1-77-0': '# Analysis on a real database', '1908.02954-1-78-0': 'In this section, we present the study we made on a database of 18,925 Y-STR 23-loci profiles from 129 different locations in 51 countries in Europe .', '1908.02954-1-78-1': 'Different analyses are performed by considering only 7 Y-STR loci (DYS19, DYS389 I, DYS389 II, DYS3904, DYS3915, DY3926,DY3937) but similar results have been observed with the use of 10 loci.', '1908.02954-1-79-0': 'First, we calculated the maximum likelihood estimators [MATH] and [MATH] using the entire database.', '1908.02954-1-79-1': 'Their values are [MATH] and [MATH].', '1908.02954-1-80-0': 'To check if the two-parameter Poisson Dirichlet prior is a sensible choice we first compare the ranked frequencies from the database with the relative frequencies of several samples of size [MATH] obtained from realizations of PD([MATH]).', '1908.02954-1-80-1': 'Lastly, we will analyze the log-likelihood function for the hyperparameters, given the data [MATH], in order to perform a data-driven choice for the hyperprior.', '1908.02954-1-81-0': '## Model fitting', '1908.02954-1-82-0': 'In Figure [REF], the ranked frequencies from the database are compared to the relative frequencies of samples of size [MATH] obtained from several realizations of PD([MATH]).', '1908.02954-1-82-1': 'To do so we run several times the Chinese Restaurant seating plan (up to [MATH] customers): each run is used to approximate a new realization [MATH] from the PD([MATH]).', '1908.02954-1-82-2': 'The partition of the customers into tables is the same as the partition obtained from an i.i.d. sample of size [MATH] from [MATH].', '1908.02954-1-82-3': 'The ranked relative sizes of each table (thin lines) are compared to the ranked frequencies of our database (thick line).', '1908.02954-1-82-4': 'One can see that for the most common haplotypes (left part of the plot) there is some discrepancy.', '1908.02954-1-82-5': 'However, we are interested in rare haplotypes, which typically have a frequency belonging to the right part of the plot.', '1908.02954-1-82-6': 'In that region, the two-parameter Poisson Dirichlet follows the distribution of the data quite well.', '1908.02954-1-83-0': 'The dotted line shows in Figure [REF] the asymptotic behavior on the two-parameter Poisson Dirichlet distribution.', '1908.02954-1-83-1': 'Indeed, if [MATH], then [EQUATION] for a random variable [MATH] such that [MATH].', '1908.02954-1-83-2': 'This power-law behavior describes an incredible variety of phenomena .', '1908.02954-1-84-0': 'The thick line in Figure [REF] also seems to have a power-law behavior, and to be honest, we were hoping to get the same asymptotic slope of the prior.', '1908.02954-1-84-1': 'This is not what we observe, but in Figure [REF] it can be seen that for such a big value of [MATH] we would need a bigger database (at least [MATH]).', '1908.02954-1-85-0': '## Loglikelihood', '1908.02954-1-86-0': 'It is also interesting to investigate the shape of the loglikelihood function for [MATH] and [MATH] given [MATH].', '1908.02954-1-86-1': 'It is defined as [EQUATION].', '1908.02954-1-86-2': 'In Figure [REF] the log-likelihood reparametrized using [MATH] instead of [MATH] is displayed.', '1908.02954-1-86-3': 'The Gaussian distribution is also displayed (in dashed lines).', '1908.02954-1-86-4': 'This is not done to show an asymptotic property, but to show the symmetry of the loglikelihood, which allows approximation of [MATH] with the marginal mode [MATH], if the prior [MATH] is flat around [MATH] since it holds that [MATH].', '1908.02954-1-87-0': 'Hence, one can approximate the LR itself in the following way: [EQUATION]', '1908.02954-1-87-1': 'Notice that this is equivalent to a hybrid approach, in which the parameters are estimated through the MLE (frequentist) and their values are plugged into the Bayesian LR.', '1908.02954-1-87-2': 'We would like to reiterate that we are not using maximum likelihood estimates of the parameters because we consider the likelihood ratio from a frequentist point of view.', '1908.02954-1-87-3': 'Our aim is to calculate a Bayesian likelihood ratio, and we have observed empirically that using the maximum likelihood estimates of the parameters we can approximate this value.', '1908.02954-1-88-0': 'Hence, in case of a rare type match problem, and using the YHRD database as the reference database we have [MATH].', '1908.02954-1-89-0': 'The Gaussian behavior of Figure [REF] was unexpected.', '1908.02954-1-89-1': 'We expect that increasing [MATH], [MATH] and [MATH] would become independent, thus the ellipses will rotate.', '1908.02954-1-90-0': '## True LR', '1908.02954-1-91-0': "It is also interesting to study the frequentist likelihood ratio values obtained with [REF], and to compare it with the 'true' ones, meaning the LR values obtained when vector [MATH] is known.", '1908.02954-1-91-1': 'This corresponds to the desirable and unrealistic situation of knowing the ranked list of the frequencies of all the DNA types in the population of interest.', '1908.02954-1-91-2': 'Then, the model can be represented by the Bayesian network of Figure [REF].', '1908.02954-1-92-0': 'The likelihood ratio in this case can be obtained using again Corollary [REF], where now [MATH], ..., [MATH] play the role of [MATH].', '1908.02954-1-92-1': '[EQUATION]', '1908.02954-1-92-2': 'Notice that, in the rare type case, [MATH] is observed only once among the [MATH], ..., [MATH].', '1908.02954-1-92-3': 'Hence, we call it a singleton.', '1908.02954-1-92-4': 'Let [MATH] denote the number of singletons, and [MATH] the set of indexes of singletons observations in the augmented database.', '1908.02954-1-92-5': 'Notice also that the knowledge of [MATH] and [MATH], is not enough to observe [MATH], but given [MATH], both [MATH] and [MATH] are fixed and known.', '1908.02954-1-92-6': 'Given [MATH] and [MATH], it holds that the distribution of [MATH] is the same as the distribution of all other singletons.', '1908.02954-1-92-7': 'This implies that: [EQUATION].', '1908.02954-1-93-0': 'Let us denote as [MATH], .', '1908.02954-1-93-1': ', [MATH] the [MATH] different values taken by [MATH], ..., [MATH], ordered decreasingly according to the frequency of their values.', '1908.02954-1-93-2': 'Stated otherwise, if [MATH] is the frequency of [MATH] among [MATH] then [MATH].', '1908.02954-1-93-3': 'Moreover, in case [MATH] and [MATH] have the same frequency ([MATH]), then they are ordered increasingly according to their values.', '1908.02954-1-93-4': 'For instance, if [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], then [MATH].', '1908.02954-1-94-0': 'By definition, it holds that [EQUATION].', '1908.02954-1-95-0': 'Notice that [MATH] is a partition of [MATH], which will be denoted as [MATH].', '1908.02954-1-95-1': 'In the example, [MATH].', '1908.02954-1-95-2': 'Since the distribution of [MATH] only depends on [MATH], the latter can replace [MATH].', '1908.02954-1-95-3': 'Thus, it holds that [EQUATION]', '1908.02954-1-95-4': 'A more compact representation for [MATH] can be obtained by using two vectors [MATH] and [MATH] where [MATH] are the distinct numbers occurring in the partition, ordered, and each [MATH] is the number of repetitions of [MATH].', '1908.02954-1-95-5': '[MATH] is the length of these two vectors, and it holds that [MATH].', '1908.02954-1-95-6': 'In the example above we have that [MATH] can be represented by [MATH] with [MATH] and [MATH].', '1908.02954-1-96-0': 'There is an unknown map, [MATH], treated here as latent variable, which assigns the ranks of the DNA types, ordered according to their frequency in Nature, to one of the number [MATH] corresponding to the position in [MATH] of its frequency in the sample, or to [MATH] if the type if not observed.', '1908.02954-1-96-1': 'Stated otherwise, [EQUATION]', '1908.02954-1-96-2': 'Given [MATH], [MATH] must satisfy the following conditions: [EQUATION]', '1908.02954-1-96-3': 'The map [MATH] can be represented by a vector [MATH] such that [MATH].', '1908.02954-1-96-4': 'In the example above we have that [MATH].', '1908.02954-1-97-0': 'Notice that, given [MATH], the knowledge of [MATH] implies the knowledge of [MATH], ..., [MATH]: indeed it is enough to consider the position of the ranked positive values of [MATH], and solving ties by considering the positions themselves (if [MATH], than the order is given by [MATH] and [MATH]).', '1908.02954-1-97-1': 'For instance, in the example, if we sort the positive values of [MATH] and we collect their positions we get [MATH]: the reader can notice that we got back to [MATH].', '1908.02954-1-98-0': 'This means that to obtain the distribution of [MATH], which appears in [REF], it is enough to obtain the distribution of [MATH], and since we are only interested in the mean of the sum of singletons in samples of size [MATH] from the distribution of [MATH], we can just simulate samples from the distribution of [MATH] and sum the [MATH] such that [MATH].', '1908.02954-1-99-0': 'To simulate samples from the distribution of [MATH] we use a Metropolis-Hastings algorithm, on the space of the vectors [MATH] satisfying condition [REF].', '1908.02954-1-99-1': 'Notice that for the model we assumed [MATH] to be infinitely long, but for simulations we will use a finite [MATH], of length [MATH].', '1908.02954-1-99-2': 'This is equivalent to assume that only [MATH] elements in the infinite [MATH] are positive, and the remaining infinite tail is made of zeros.', '1908.02954-1-99-3': 'Then the state space of the Metropolis-Hastings Markov chain is made of all vectors of length [MATH] whose elements belong to [MATH], and satisfy the condition [REF].', '1908.02954-1-99-4': 'If we start with an initial point [MATH] which satisfies [REF] and, at each allowed move of the Metropolis-Hastings we swap two different values [MATH] and [MATH] inside the vector, condition [REF] remains satisfied.', '1908.02954-1-99-5': 'The algorithm is based on a similar one proposed in [CITATION].', '1908.02954-1-100-0': "This method allows us to obtain the 'true' LR when the vector [MATH] is known.", '1908.02954-1-100-1': 'This is rarely the case, but we can put ourselves in a fictitious world where we know [MATH], and compare the true values for the LR with the one obtained by applying our model when [MATH] is unknown.', '1908.02954-1-101-0': '## What to do next', '1908.02954-1-102-0': 'A real Bayesian statistician chooses the prior and hyperprior according to his beliefs.', '1908.02954-1-102-1': "Depending on the choice of the hyperprior over [MATH] and [MATH] he may or may not believe in the approximation [REF], but he does not really talk of 'error'.", '1908.02954-1-102-2': 'However, hardliner Bayesian statisticians are a rare species, and most of the time the Bayesian procedure consists in choosing priors (and hyperpriors) which are a compromise between personal beliefs and mathematical convenience.', '1908.02954-1-102-3': 'It is thus interesting to investigate the performance of such priors.', '1908.02954-1-102-4': 'This can be done by comparing the Bayesian likelihood ratio with the likelihood ratio a frequentist would obtain if the vector [MATH] was known, and for the same reduction of data.', '1908.02954-1-102-5': "This is what we call 'error': in other words, at the moment we are considering the Bayesian nonparametric method proposed in this paper as a way to estimate (notice the frequentist terminology) the true [MATH].", '1908.02954-1-102-6': 'If we denote by [MATH] the population proportion of the matching profile, another interesting comparison is the one between the Bayesian likelihood ratio and the frequentist likelihood ratio [MATH] (here denoted as [MATH]) that one would obtain knowing [MATH], but not reducing the data to partition.', '1908.02954-1-102-7': 'This is a sort of benchmark comparison and tells us how much we lose by using the Bayesian nonparametric methodology, and by reducing data.', '1908.02954-1-102-8': 'In order to evaluate how much we lose due to the sole reduction of the data, one can compare [MATH] with [MATH].', '1908.02954-1-102-9': 'In total there are three quantities of interest ([MATH], [MATH], and [MATH]), and three differences of interest, which will be denoted as', '1908.02954-1-103-0': 'However, there is an obstacle.', '1908.02954-1-103-1': 'The Metropolis-Hastings algorithm described in Section [REF] is too slow to be used with the entire European database of size [MATH].', '1908.02954-1-103-2': 'Further investigations can be carried out to find out whether the approximation [REF] is meaningful for smaller databases (for instance subsamples from the YHRD data).', '1908.02954-1-103-3': 'When this is checked one can use the entire European database as a ground-truth population from which many samples of smaller sizes are drawn and used as reference databases available for the case ([MATH]).', '1908.02954-1-103-4': 'This would allow having a distribution for the three differences described above.', '1908.02954-1-104-0': '# Conclusion', '1908.02954-1-105-0': 'This paper discusses the first application of a Bayesian nonparametric method to the likelihood ratio assessment in forensic science, in particular to the challenging situation of the rare type match.', '1908.02954-1-105-1': 'If compared to traditional Bayesian methods such as those described in [CITATION], it presents many advantages.', '1908.02954-1-105-2': 'First of all, the prior chosen for the parameter [MATH] is more realistic for the population whose frequencies we want to model.', '1908.02954-1-105-3': 'Moreover, although the theoretical background on which it lies may seem very technical and difficult, the method is extremely simple to apply for practical use, thanks to a discussed approximation: indeed, simulation experiments show that a empirical Bayes approach is justified, at least using Y-STR data from European populations.', '1908.02954-1-106-0': 'More could be done in the future: for instance, investigate other nonparametric priors, and use more realistic populations.', '1908.02954-1-106-1': 'Also, we have the plan of comparing this Bayesian likelihood ratio to the frequentist likelihood ratio obtained both reducing and not reducing the data, and investigate calibration and validation through the use of CLLR and ECE plots .'}

{'1908.02954-2-0-0': 'The "rare type match problem" is the situation in which the suspect\'s DNA profile, matching the DNA profile of the crime stain, is not in the database of reference.', '1908.02954-2-0-1': 'The evaluation of this match in the light of the two competing hypotheses (the crime stain has been left by the suspect or by another person) is based on the calculation of the likelihood ratio and depends on the population proportions of the DNA profiles, that are unknown.', '1908.02954-2-0-2': 'We propose a Bayesian nonparametric method that uses a two-parameter Poisson Dirichlet distribution as a prior over the ranked population proportions, and discards the information about the names of the different DNA profiles.', '1908.02954-2-0-3': 'This fits very well the data coming from European Y-STR DNA profiles, and the calculation of the likelihood ratio becomes quite simple thanks to a justified Empirical Bayes approach.', '1908.02954-2-1-0': '# Introduction', '1908.02954-2-2-0': 'The largely accepted method for evaluating how much some available data [MATH] (typically forensic evidence) helps discriminate between two hypotheses of interest (the prosecution hypothesis [MATH] and the defense hypothesis [MATH]), is the calculation of the likelihood ratio (LR), a statistic that expresses the relative plausibility of the data under these hypotheses, defined as [EQUATION]', '1908.02954-2-2-1': "Widely considered the most appropriate framework to report a measure of the 'probative value' of the evidence regarding the two hypotheses , it indicates the extent to which observed data is favoured by one hypothesis over the other.", '1908.02954-2-2-2': 'Forensic literature presents many approaches to calculate the LR, mostly divided into Bayesian and frequentist methods (see [CITATION] for a careful differentiation between these two approaches).', '1908.02954-2-3-0': 'This paper proposes the first Bayesian nonparametric method to assess the likelihood ratio in the rare type match case, the challenging situation in which there is a match between some characteristic of the recovered material and of the control material, but this characteristic has not been observed before in previously collected samples (i.e. in the database of reference).', '1908.02954-2-3-1': 'This constitutes a problem because the value of the likelihood ratio depends on the unknown proportion of the matching characteristic in a reference population, and the uncertainty over this proportion, in standard practice for simpler situations, is dealt with using the relative frequency of the characteristic in the available database.', '1908.02954-2-3-2': 'In particular, we will focus on Y-STR data, for which the rare type match problem is recurring .', '1908.02954-2-4-0': 'The use of our Bayesian nonparametric method involves the mathematical assumption that there are infinitely many different Y-STR profiles.', '1908.02954-2-4-1': 'Of course, we do not believe this literally to be true.', '1908.02954-2-4-2': 'We do suppose that there are so many profiles that we cannot say anything sensible about their exact number, except that it is very large.', '1908.02954-2-4-3': 'The parameter of the model is the infinite-dimensional vector [MATH], containing the (unknown) sorted population proportions of all possible Y-STR profiles.', '1908.02954-2-4-4': 'As a prior over [MATH] we choose the two-parameter Poisson Dirichlet distribution, and we model the uncertainty over its parameters [MATH] and [MATH] through the use of a hyperprior.', '1908.02954-2-4-5': 'The information contained in the names of the profiles is discarded, thereby reducing the full data [MATH] to a smaller set [MATH].', '1908.02954-2-5-0': 'If compared to traditional Bayesian methods such as those discussed in [CITATION], this method has the advantage of having a prior for the parameter p that is more realistic for the population we intend to model.', '1908.02954-2-5-1': 'Moreover, although its technical theoretical background, we empirically derived an approximation that makes the method extremely simple to apply for practical use: indeed, simulation experiments show that a hybrid empirical approach that plugs-in maximum likelihood estimators for the hyperparamenter is justified, at least when using Y-STR data from European populations.', '1908.02954-2-5-2': 'The last point in favor of the choice of the two-parameter Poisson Dirichlet prior over p is that it has the following sufficientness property: the probability of observing a new Y-STR profile only depends on the number of already observed Y-STR profiles and on the sample size, while the probability of observing a Y-STR profile that is already in the database only depends on its frequency in the database and on the sample size.', '1908.02954-2-6-0': 'The paper is structured as follows: Section [REF] presents the rare type match problem.', '1908.02954-2-6-1': 'Section [REF] presents the model, with the assumptions and the prior distribution chosen for parameter [MATH] along with some theory on random partitions and the Chinese restaurant representation, useful to provide a prediction rule and convenient and compact of the reduced data [MATH].', '1908.02954-2-6-2': 'Also, a lemma that facilitates computing the likelihood ratio in a very elegant way is presented and proved.', '1908.02954-2-6-3': 'In Section [REF], the likelihood ratio is derived.', '1908.02954-2-6-4': 'Section [REF] proposes the application of this model to a real database of Y-STR profiles.', '1908.02954-2-6-5': 'We will discuss data-driven choices for the hyperparameters, and the derivation of the frequentist likelihood ratio values obtained both with and without reducing the data to partitions, in the ideal situation in which vector [MATH] is known.', '1908.02954-2-6-6': 'Also, distribution of the likelihood ratios for different rare type match cases is analysed, along with the analysis of two different errors.', '1908.02954-2-7-0': '# The rare type match problem', '1908.02954-2-8-0': "The evaluation of a match between the profile of a particular piece of evidence and a suspect's profile depends on the proportion of that profile in the population of potential perpetrators.", '1908.02954-2-8-1': 'Indeed, it is intuitive that the rarer the matching profile, the more the suspect is in trouble.', '1908.02954-2-8-2': 'Problems arise when the observed frequency of the profile in a sample (database) from the population of interest is 0.', '1908.02954-2-8-3': 'This problem can be named as "the new type match problem", but we decided to use the name "rare type match problem", motivated by the fact that a Y-STR profile that has zero occurrences is likely to be rare, even though it is challenging to quantify how rare it is.', '1908.02954-2-8-4': 'The rare type match problem is particularly important for new kinds of forensic evidence, such as results from DIP-STR markers (see for instance [CITATION]) is involved, for which the available database size is still limited.', '1908.02954-2-8-5': 'The problem also occurs when more established types of evidence, such as Y-chromosome (or mitochondrial) DNA profiles are used since the set of possible Y-STR profiles is extremely large.', '1908.02954-2-8-6': 'As a consequence, most of the Y-STR haplotypes are not represented in the database.', '1908.02954-2-8-7': 'The Y-STR marker system will thus be retained here as an extreme but in practice common and important way in which the problem of assessing the evidential value of rare type match can arise.', '1908.02954-2-8-8': 'This problem is so substantial that it has been defined "the fundamental problem of forensic mathematics" .', '1908.02954-2-9-0': 'This problem has been addressed in well-know non-forensic statistics, and many solutions have been proposed.', '1908.02954-2-9-1': 'The empirical frequency estimator, also called naive estimator, that uses the frequency of the characteristic in the database, puts unit probability mass on the set of already observed characteristics, and it is thus unprepared for the observation of a new type.', '1908.02954-2-9-2': 'A solution could be the add-constant estimators (in particular the well-known add-one estimator, due to [CITATION], and the add-half estimator of [CITATION]), which add a constant to the count of each type, included the unseen ones.', '1908.02954-2-9-3': 'However, these methods require to know the number of possible unseen types, and they perform badly when this number is large compared to the sample size (see [CITATION] for an additional discussion).', '1908.02954-2-9-4': 'Alternatively, [CITATION], based on an intuition on A.M. Turing, proposed the Good-Turing estimator for the total unobserved probability mass, based on the proportion of singleton observations in the sample.', '1908.02954-2-9-5': 'An extension of this estimator is applied to the frequentist LR assessment in the rare type match case in [CITATION].', '1908.02954-2-9-6': 'Recently, [CITATION] have introduced the high-profile estimator, which extends the tail of the naive estimator to the region of unobserved types.', '1908.02954-2-9-7': '[CITATION] improved this estimator and provided the consistency proof.', '1908.02954-2-9-8': 'Papers that address the rare Y-STR haplotype problem in forensic context are for instance [CITATION], [CITATION], [CITATION] and [CITATION].', '1908.02954-2-9-9': 'The latter applies the classical Bayesian approach (the beta-binomial and the Dirichlet multinomial problem) to the LR assessment in the rare haplotype match case.', '1908.02954-2-9-10': 'All these methods do not take into account genetic information contained in the allelic numbers forming a Y-STR DNA profile.', '1908.02954-2-9-11': 'For instance, due to relatedness, the observation of a particular Y-STR profile increases the probability of observing the same Y-STR profile again or Y-STR profiles that differ only for few alleles.', '1908.02954-2-9-12': 'We refer the reader to [CITATION] for models that use population genetics for coancestry.', '1908.02954-2-9-13': 'These models are not designed to be used for the rare type match case, but the Discrete Laplace method presented in [CITATION] can be successfully applied to that purpose, as shown in [CITATION].', '1908.02954-2-9-14': 'More recently, [CITATION] provided an interesting discussion on the importance of taking into account genetic information.', '1908.02954-2-9-15': "However, we start from the assumption that we don't know how to use that information in a sensible way.", '1908.02954-2-9-16': 'For this reason, we prefer to discard it, and only focus on the part of the data we are more confident to model.', '1908.02954-2-10-0': 'Bayesian nonparametric estimators for the probability of observing a new type have been proposed by [CITATION] using Dirichlet processes, by [CITATION] using a general Gibbs prior, and by [CITATION] with specific focus on the two-parameter Poisson Dirichlet prior, for which [CITATION] provides large sample asymptotic and credible bands.', '1908.02954-2-10-1': 'In particular, [CITATION] shows the link between the Bayesian nonparametric approach and the Good Turing estimator.', '1908.02954-2-10-2': 'However, the LR assessment requires not only the probability of observing a new species but also the probability of observing this same species twice (according to the defense the crime stain profile and the suspect profile are two independent observations): to our knowledge, the present paper is the first to address the problem of LR assessment in the rare type match case using a Bayesian nonparametric model.', '1908.02954-2-10-3': 'As a prior for [MATH] we will use the two-parameter Poisson Dirichlet distribution, which is proving useful in many discrete domains, in particular language modeling .', '1908.02954-2-10-4': 'Besides, it shows a power-law behavior that describes an incredible variety of phenomena , among which the distribution of Y-STR haplotypes, too.', '1908.02954-2-11-0': '# The model', '1908.02954-2-12-0': '## Notation', '1908.02954-2-13-0': 'Throughout the paper the following notation is chosen: random variables and their values are denoted, respectively, with uppercase and lowercase characters: [MATH] is a realization of [MATH].', '1908.02954-2-13-1': 'Random vectors and their values are denoted, respectively, by uppercase and lowercase bold characters: [MATH] is a realization of the random vector [MATH].', '1908.02954-2-13-2': 'Probability is denoted with [MATH], while the density of a continuous random variable [MATH] is denoted alternatively by [MATH] or by [MATH] when the subscript is clear from the context.', '1908.02954-2-13-3': 'For a discrete random variable [MATH], the density notation [MATH] and the discrete one [MATH] will be interchangeably used.', '1908.02954-2-13-4': 'Moreover, we will use shorthand notation like [MATH] to stand for the probability density of Y with respect to the conditional distribution of [MATH] given [MATH].', '1908.02954-2-14-0': 'Notice that in Formula [REF], [MATH] was regarded as the event corresponding to the observation of the available data.', '1908.02954-2-14-1': 'However, later in the paper, [MATH] will be regarded as a random variable generically representing the data.', '1908.02954-2-14-2': 'The particular data at hand will correspond to the value [MATH].', '1908.02954-2-14-3': 'In that case, the following notation will thus be preferred: [EQUATION]', '1908.02954-2-14-4': 'Lastly, notice that "DNA types" is used throughout the paper as a general term to indicate Y-STR profiles.', '1908.02954-2-15-0': '## Model assumptions', '1908.02954-2-16-0': 'Our model is based on the two following assumptions:', '1908.02954-2-17-0': '[Assumption 1] There are infinitely many different DNA types in Nature.', '1908.02954-2-18-0': "This assumption, already used by e.g. [CITATION] in the 'infinite alleles model', allows the use of Bayesian nonparametric methods and is very useful for instance in 'species sampling problems' when the total number of possible different species in Nature cannot be specified.", '1908.02954-2-18-1': 'This assumption is sensible also in case of Y-STR DNA profiles since the state space of possible different haplotypes is so large that it can be considered infinite.', '1908.02954-2-19-0': '[Assumption 2] The names of the different DNA types do not contain relevant information.', '1908.02954-2-20-0': 'Actually, the specific sequence of numbers that forms a DNA profile carries information: if two profiles show few differences this means that they are separated by few mutation drifts, hence the profiles share a relatively recent common ancestor.', '1908.02954-2-20-1': 'However, this information is difficult to use and may not be so relevant for the LR assessment.', '1908.02954-2-20-2': 'This is the reason why, as already mentioned, we will treat DNA types as "colors", and only consider the partition into different categories.', '1908.02954-2-20-3': 'Stated otherwise, we put no topological structure on the space of the DNA types.', '1908.02954-2-21-0': "Notice that this assumption makes the model a priori suitable for any characteristic which has many different possible types, thus the approach described in this paper still holds, in principle, after replacing 'DNA types' with any other category.", '1908.02954-2-21-1': 'However, in this paper, we will only focus on the model with Y-STR profiles as categories, for which the model fits quite well the available data (see Section [REF]).', '1908.02954-2-22-0': '## Prior', '1908.02954-2-23-0': 'In Bayesian statistics, parameters of interest are modeled through random variables.', '1908.02954-2-23-1': 'The (prior) distribution over a parameter should represent the uncertainty about its value.', '1908.02954-2-24-0': 'LR assessment for the rare type match involves two unknown parameters of interest: one is [MATH], representing the unknown true hypothesis, the other is [MATH], the vector of the unknown population frequencies of all DNA profiles in the population of potential perpetrators.', '1908.02954-2-24-1': 'The dichotomous random variable [MATH] is used to model parameter [MATH], and the posterior distribution of this random variable, given the data, is the ultimate aim of the forensic inquiry.', '1908.02954-2-24-2': 'Similarly, a random variable [MATH] is used to model the uncertainty over [MATH].', '1908.02954-2-24-3': 'Because of Assumption 1, [MATH] is an infinite-dimensional parameter, hence the need for Bayesian nonparametric methods .', '1908.02954-2-24-4': 'In particular, [MATH], with [MATH] a countable set of indexes, [MATH], and [MATH].', '1908.02954-2-24-5': 'Moreover, because of Assumption 2, data can be reduced to partitions, as explained in Section [REF], and it will turn out that the distribution of these partitions does not depend on the order of the [MATH].', '1908.02954-2-24-6': 'Hence, we can define the parameter [MATH] as having values in [MATH], the ordered infinite-dimensional simplex.', '1908.02954-2-24-7': 'The uncertainty about its value is expressed by the two-parameter Poisson Dirichlet prior .', '1908.02954-2-25-0': 'As a reason for choosing the two-parameter Poisson Dirichlet distribution among the possible Bayesian nonparametric priors is that it is the only one that has the following very convenient sufficientness property : the probability of observing a new species only depends on the number of already observed species and on the sample size, and the probability of observing an already seen species only depends on its frequency in the sample and on the sample size.', '1908.02954-2-25-1': 'The second reason is the validation through model fitting (see Section [REF]).', '1908.02954-2-26-0': 'The two-parameter Poisson-Dirichlet distribution can be defined through the following stick-breaking representation :', '1908.02954-2-27-0': '[two-parameter GEM distribution] Given [MATH] and [MATH] satisfying the following conditions: [EQUATION] the vector [MATH] is said to be distributed according to the GEM([MATH]), if [EQUATION] where [MATH], [MATH],... are independent random variables distributed according to [EQUATION].', '1908.02954-2-27-1': 'It holds that [MATH], and [MATH].', '1908.02954-2-28-0': 'The GEM distribution (short for Griffin-Engen-McCloskey distribution\') is well known in the literature as the "stick-breaking prior" since it measures the random sizes in which a stick is broken iteratively.', '1908.02954-2-29-0': '[Two-parameter Poisson Dirichlet distribution] Given [MATH] and [MATH] satisfying condition [REF], and a vector [MATH], the random vector [MATH] obtained by ranking [MATH], such that [MATH], is said to be Poisson Dirichlet distributed PD[MATH].', '1908.02954-2-29-1': 'Parameter [MATH] is called the discount parameter, while [MATH] is the concentration parameter.', '1908.02954-2-30-0': 'For our model we will not allow [MATH], hence we will assume [MATH], in order to have a prior that shows a power-law behavior as the one observed in the available database (see Section [REF]).', '1908.02954-2-31-0': 'Lastly, we point out that, in practice, we cannot assume to know parameters [MATH] and [MATH]: we will resolve this by using an hyperprior.', '1908.02954-2-32-0': '## Bayesian network representation of the model', '1908.02954-2-33-0': 'The typical data to evaluate in case of a match is [MATH], where [MATH], and', '1908.02954-2-34-0': "[MATH] = suspect's DNA type,", '1908.02954-2-35-0': "[MATH] = crime stain's DNA type (matching the suspect's type),", '1908.02954-2-36-0': '[MATH] = a reference database of size [MATH], which is assumed as being a random sample of DNA types from the population of possible perpetrators.', '1908.02954-2-37-0': 'The hypotheses of interest for the case are:', '1908.02954-2-38-0': '[MATH] = The crime stain was left by the suspect,', '1908.02954-2-39-0': '[MATH] = The crime stain was left by someone else.', '1908.02954-2-40-0': 'In agreement with Assumption 2, the model will ignore information about the names of the DNA types: data [MATH] will thus be reduced to [MATH] accordingly.', '1908.02954-2-40-1': 'The Bayesian network of Figure [REF] encapsulates the conditional dependencies of the random variables [MATH], whose joint distribution is defined below in terms of the conditional distribution, using the factorization implied by the Bayesian network itself.', '1908.02954-2-41-0': '[MATH] is a dichotomous random variable that represents the hypotheses of interest and can take values [MATH], according to the prosecution or the defense, respectively.', '1908.02954-2-41-1': 'A uniform prior on the hypotheses is chosen: [EQUATION].', '1908.02954-2-41-2': 'Notice that this choice is made for mathematical convenience since it will not affect the likelihood ratio, the variable [MATH] being in the conditioning part.', '1908.02954-2-42-0': '([MATH]) is the random vector that represents the hyperparameters [MATH] and [MATH], satisfying condition [REF].', '1908.02954-2-42-1': 'The joint prior density of these two parameters will be generically denoted as [MATH]: [EQUATION].', '1908.02954-2-42-2': "For obvious reasons, this will be called the 'hyperprior' throughout the text.", '1908.02954-2-43-0': 'The random vector [MATH] with values in [MATH] represents the ranked population frequencies of Y-STR profile.', '1908.02954-2-43-1': '[MATH] means that [MATH] is the frequency of the most common DNA type in the population, [MATH] is the frequency of the second most common DNA type, and so on.', '1908.02954-2-43-2': 'As a prior for [MATH] we use the two-parameter Poisson Dirichlet distribution: [EQUATION].', '1908.02954-2-44-0': 'The database is assumed to be a random sample from the population.', '1908.02954-2-44-1': 'Integer-valued random variables [MATH], ..., [MATH] are here used to represent the (unknown) ranks in the population of the frequencies of the DNA types in the database.', '1908.02954-2-44-2': 'For instance, [MATH] means that the third individual in the database has the fifth most common DNA type in the population.', '1908.02954-2-44-3': 'Given [MATH] they are an i.i.d. sample from [MATH]: [EQUATION]', '1908.02954-2-44-4': 'To observe [MATH], ..., [MATH], one would need to know the rank, in terms of population proportions, of the frequency of each DNA types in the database.', '1908.02954-2-44-5': 'This is not known, hence [MATH] are not observed.', '1908.02954-2-45-0': "[MATH] represents the rank in the population ordering of the suspect's DNA type.", '1908.02954-2-45-1': 'It is again an independent draw from [MATH].', '1908.02954-2-45-2': "[EQUATION] [MATH] represents the rank in the population ordering, of the crime stain's DNA type.", '1908.02954-2-45-3': 'According to the prosecution, given [MATH], this random variable is deterministic (it is equal to [MATH] with probability 1).', '1908.02954-2-45-4': 'According to the defense it is another sample from [MATH], independent of the previous ones: [EQUATION].', '1908.02954-2-46-0': 'As already mentioned, [MATH] cannot be observed.', '1908.02954-2-46-1': 'They represent the database, where the names of the DNA types have been replaced by their (unknown) ranks in [MATH], and constitute an intermediate layer.', '1908.02954-2-47-0': "Section [REF] recalls some notions about random partitions, useful before defining node [MATH], the 'reduced' data that we want to evaluate.", '1908.02954-2-48-0': '## Random partitions and database partitions', '1908.02954-2-49-0': 'A partition of a set [MATH] is an unordered collection of nonempty and disjoint subsets of [MATH], the union of which forms [MATH].', '1908.02954-2-49-1': 'Particularly interesting for our model are partitions of the set [MATH], denoted as [MATH].', '1908.02954-2-49-2': 'The set of all partitions of [MATH] will be denoted as [MATH].', '1908.02954-2-49-3': 'Random partitions of [MATH] will be denoted as [MATH].', '1908.02954-2-49-4': 'Also, a partition of [MATH] is a finite nonincreasing sequence of positive integers that sum up to [MATH].', '1908.02954-2-49-5': 'Partitions of [MATH] will be denoted as [MATH], while random partitions as [MATH].', '1908.02954-2-50-0': 'Given a sequence of integer valued random variables [MATH], let [MATH] be the random partition defined by the equivalence classes of their indexes using the random equivalence relation [MATH] if and only if [MATH].', '1908.02954-2-50-1': 'This construction allows to build a "reduction" map from the set of values of [MATH] to the set of the partitions of [MATH] as in the following example ([MATH]): [EQUATION]', '1908.02954-2-50-2': 'Similarly, and in agreement with Assumption 2, in our model we can consider the reduction of data which ignores information about the names of the DNA types: this is achieved, for instance, by retaining from the database only the equivalence classes of the indexes of the individuals, according to the equivalence relation "to have the same DNA type".', '1908.02954-2-50-3': 'Stated otherwise, the database is reduced to the partition [MATH], obtained using these equivalence classes.', '1908.02954-2-50-4': 'However, the database only supplies part of the data.', '1908.02954-2-50-5': 'There are also two new DNA profiles that are equal to one another (and different from the already observed ones in the rare type match case).', '1908.02954-2-50-6': "Considering the suspect's profile we obtain the partition [MATH], where the first [MATH] integers are partitioned as in [MATH], and [MATH] constitutes a class by itself.", '1908.02954-2-50-7': 'Considering the crime stain profile we obtain the partition [MATH] where the first [MATH] integers are partitioned as in [MATH], and [MATH] and [MATH] belong to the same (new) class.', '1908.02954-2-50-8': 'Random variables [MATH], [MATH], and [MATH] are used to model [MATH], [MATH], and [MATH], respectively.', '1908.02954-2-51-0': 'Since prosecution and defense agree on the distribution of [MATH], but not on the distribution of [MATH], they also agree on the distribution of [MATH] but disagree on the distribution of [MATH].', '1908.02954-2-52-0': 'The crucial points of the model are the following:', '1908.02954-2-53-0': 'the random partitions can defined through random variables [MATH], ..., [MATH] and through database are the same.', '1908.02954-2-53-1': '[EQUATION] although [MATH], ..., [MATH] were not observable, the random partitions [MATH], and [MATH] are observable.', '1908.02954-2-54-0': 'To clarify, consider the following example of a database with [MATH] different DNA types, from [MATH] individuals: [EQUATION] where [MATH] is the name of the [MATH]th DNA type according to the order chosen for the database.', '1908.02954-2-54-1': 'This database can be reduced to the partition of [MATH]: [EQUATION].', '1908.02954-2-54-2': 'Then, the part of reduced data whose distribution is agreed on by prosecution and defense is [EQUATION] while the entire reduced data [MATH] can be represented as [EQUATION].', '1908.02954-2-55-0': 'Now, assume that we know the rank in the population of each of the DNA types in the database: we know that [MATH] is, for instance, the second most frequent type, [MATH] is the fourth most frequent type, and so on.', '1908.02954-2-55-1': 'Stated otherwise, we are now assuming that we observe the variables [MATH], ..., [MATH]: for instance, [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '1908.02954-2-55-2': 'It is easy to check that [MATH], [MATH], and [MATH].', '1908.02954-2-56-0': 'Data is then defined as [MATH], obtained partitioning the database enlarged with the two new observations (or partitioning [MATH]).', '1908.02954-2-56-1': 'Node [MATH] of Figure [REF] is defined accordingly.', '1908.02954-2-57-0': 'Notice that, given [MATH], [MATH] is deterministic.', '1908.02954-2-57-1': 'An important result is that, according to Proposition 4 in [CITATION] it is possible to derive directly the distribution of [MATH].', '1908.02954-2-57-2': 'In particular, it holds that if [EQUATION] and [EQUATION] then, for all [MATH], the random partition [MATH] has the following distribution: [EQUATION] where [MATH] is the size of the [MATH]th block of [MATH] (the blocks are here ordered according to the least element), and [MATH], [MATH].', '1908.02954-2-57-3': 'This formula is also known as the Pitman sampling formula, further studied in [CITATION] and shows that [MATH] does not depend on [MATH], but only on the sizes and the number of classes in the partitions.', '1908.02954-2-58-0': 'It follows that we can get rid of the intermediate layer of nodes [MATH], ..., [MATH], and have [MATH], while [MATH].', '1908.02954-2-58-1': 'The model of Figure [REF] can thus be simplified to the one in Figure [REF].', '1908.02954-2-59-0': '## Chinese Restaurant representation', '1908.02954-2-60-0': 'There is an alternative characterization of this model, called "Chinese restaurant process", due to [CITATION] for the one-parameter case, and studied in detail for the two-parameter version in [CITATION].', '1908.02954-2-60-1': 'It is defined as follows: consider a restaurant with infinitely many tables, each one infinitely large.', '1908.02954-2-60-2': 'Let [MATH] be integer-valued random variables that represent the seating plan: tables are ranked in order of occupancy, and [MATH] means that the [MATH]th customer seats at the [MATH]th table to be created.', '1908.02954-2-60-3': 'The process is described by the following transition matrix: [EQUATION] where [MATH] is the number of tables occupied by the first [MATH] customers, and [MATH] is the number of customers that occupy table [MATH].', '1908.02954-2-60-4': 'The process depends on two parameters [MATH] and [MATH] with the same conditions [REF].', '1908.02954-2-60-5': 'From [REF] one can easily see the sufficientness property mentioned in Section [REF].', '1908.02954-2-61-0': '[MATH] are not i.i.d., nor exchangeable, but it holds that [MATH] is distributed as [MATH], with [MATH] defined as in [REF], and they are both distributed according to the Pitman sampling formula [REF] .', '1908.02954-2-62-0': 'Stated otherwise, we can articulate the seating plan of [MATH] customers while indexing the tables according to either their observed occupancy (as in [MATH]) or as in [MATH] and we obtain the same partition [MATH].', '1908.02954-2-62-1': 'Similarly [MATH] is obtained when a new customer has chosen an unoccupied table (remember we are in the rare type match case), and [MATH] is obtained when the [MATH]nd customer goes to the table already chosen by the [MATH]st customer (suspect and crime stain have the same DNA type).', '1908.02954-2-62-2': 'In particular, thanks to [REF], we can write: [EQUATION] since the [MATH]nd customer goes to the same table as the [MATH]st (who was sitting alone).', '1908.02954-2-63-0': '## A useful Lemma', '1908.02954-2-64-0': 'The following lemma can be applied to four general random variables [MATH], [MATH], [MATH], and [MATH] whose conditional dependencies are described by the Bayesian network of Figure [REF].', '1908.02954-2-64-1': 'The importance of this result is due to the possibility of applying it to a very common forensic situation: the prosecution and the defense disagree on the distribution of the entirety of data ([MATH]) but agree on the distribution of a part it ([MATH]), and these distributions depend on parameters ([MATH]).', '1908.02954-2-65-0': 'Given four random variables [MATH], [MATH], [MATH] and [MATH], whose conditional dependencies are represented by the Bayesian network of Figure [REF], the likelihood function for [MATH], given [MATH] and [MATH] satisfies [EQUATION].', '1908.02954-2-66-0': 'The Bayesian representation of the model, in Figure [REF], allow to factor the joint probability density of [MATH], [MATH], [MATH] and [MATH] as [EQUATION]', '1908.02954-2-66-1': 'By Bayes formula, [MATH].', '1908.02954-2-66-2': 'This rewriting corresponds to reversing the direction of the arrow between [MATH] and [MATH]:', '1908.02954-2-67-0': 'The random variable [MATH] is now a root node.', '1908.02954-2-67-1': 'This means that when we probabilistically condition on [MATH], the graphical model changes in a simple way: we can delete the node [MATH], but just insert the value [MATH] as a parameter in the conditional probability tables of the variables [MATH] and [MATH] which formerly had an arrow from node [MATH].', '1908.02954-2-67-2': 'The next graph represents this model:', '1908.02954-2-68-0': 'This tells us, that conditional on [MATH], the joint density of [MATH], [MATH] and [MATH] is equal to [EQUATION].', '1908.02954-2-68-1': 'The joint density of [MATH] and [MATH] given [MATH] is obtained by integrating out the variable [MATH].', '1908.02954-2-68-2': 'It can be expressed as a conditional expectation value since [MATH] is the density of [MATH] given [MATH].', '1908.02954-2-68-3': 'We find: [EQUATION].', '1908.02954-2-69-0': 'Recall that this is the joint density of two of our variables, [MATH] and [MATH], after conditioning on the value [MATH].', '1908.02954-2-69-1': 'Let us now also condition on [MATH].', '1908.02954-2-69-2': 'It follows that the density of [MATH] given [MATH] and [MATH] is proportional (as function of [MATH], for fixed [MATH] and [MATH]) to the same expression, [MATH].', '1908.02954-2-70-0': 'This is a product of the prior for [MATH] with some function of [MATH] and [MATH].', '1908.02954-2-70-1': 'Since posterior odds equals prior odds times likelihood ratio, it follows that the likelihood function for [MATH], given [MATH] and [MATH] satisfies [EQUATION].', '1908.02954-2-71-0': 'Given four random variables [MATH], [MATH], [MATH] and [MATH], whose conditional dependencies are represented by the network of Figure [REF], the likelihood ratio for [MATH] against [MATH] given [MATH] and [MATH] satisfies [EQUATION]', '1908.02954-2-72-0': '# The likelihood ratio', '1908.02954-2-73-0': 'Using the hypotheses and the reduction of data [MATH] defined in Section [REF], the likelihood ratio will be defined as [EQUATION].', '1908.02954-2-73-1': 'The last equality holds due to the fact that [MATH] is a deterministic function of [MATH].', '1908.02954-2-74-0': 'Corollary [REF] can be applied to our model since defense and prosecution agree on the distribution of [MATH], but not on the distribution of [MATH], and data depends on parameters [MATH] and [MATH].', '1908.02954-2-74-1': 'Thus, if [MATH] play the role of [MATH], [MATH], and [MATH], by using [REF] and [REF], we obtain: [EQUATION].', '1908.02954-2-74-2': 'The expected value is taken with respect to the posterior distribution of [MATH].', '1908.02954-2-74-3': 'By defining the random variable [MATH] we can write the LR as [EQUATION]', '1908.02954-2-75-0': '# Analysis on a real database', '1908.02954-2-76-0': 'In this section, we present the study we made on a database of 18,925 Y-STR 23-loci profiles from 129 different locations in 51 countries in Europe .', '1908.02954-2-76-1': 'Our analyses are performed by considering only 7 Y-STR loci (DYS19, DYS389 I, DYS389 II, DYS3904, DYS3915, DY3926, DY3937) but similar results have been observed with the use of 10 loci.', '1908.02954-2-77-0': '## Model fitting', '1908.02954-2-78-0': 'First, we calculated the maximum likelihood estimators [MATH] and [MATH] using the entire database and the likelihood defined by [REF].', '1908.02954-2-78-1': 'Their values are [MATH] and [MATH].', '1908.02954-2-79-0': 'In Figure [REF], the ranked frequencies from the database are compared to the relative frequencies of samples of size [MATH] obtained from several realizations of PD([MATH]).', '1908.02954-2-79-1': 'To do so we run several times the Chinese Restaurant seating plan (up to [MATH] customers): each run is used to approximate a new realization [MATH] from the PD([MATH]).', '1908.02954-2-79-2': 'The partition of the customers into tables is the same as the partition obtained from an i.i.d. sample of size [MATH] from [MATH].', '1908.02954-2-79-3': 'The ranked relative sizes of each table (thin lines) are compared to the ranked frequencies of our database (thick line).', '1908.02954-2-79-4': 'We can see that for the most common haplotypes (left part of the plot) there is some discrepancy.', '1908.02954-2-79-5': 'However, we are interested in rare haplotypes, which typically have a frequency belonging to the right part of the plot.', '1908.02954-2-79-6': 'In that region, the two-parameter Poisson Dirichlet follows the distribution of the data quite well.', '1908.02954-2-80-0': 'The dotted line shows in Figure [REF] the asymptotic behavior on the two-parameter Poisson Dirichlet distribution.', '1908.02954-2-80-1': 'Indeed, if [MATH], then [EQUATION] for a random variable [MATH] such that [MATH].', '1908.02954-2-80-2': 'This power-law behavior describes an incredible variety of phenomena .', '1908.02954-2-81-0': 'The thick line in Figure [REF] also seems to have a power-law behavior, and to be honest, we were hoping to get the same asymptotic slope of the prior.', '1908.02954-2-81-1': 'This is not what we observe, but in Figure [REF] it can be seen that for such a big value of [MATH] we would need a bigger database (at least [MATH]) to see the correct slope.', '1908.02954-2-82-0': '## Log-likelihood', '1908.02954-2-83-0': 'It is also interesting to investigate the shape of the log-likelihood function for [MATH] and [MATH] given [MATH].', '1908.02954-2-83-1': 'It is defined as [EQUATION].', '1908.02954-2-83-2': 'In Figure [REF] the log-likelihood reparametrized using [MATH] instead of [MATH] is displayed.', '1908.02954-2-83-3': 'A Gaussian distribution centered in the MLE parameters and with covariance matrix the inverse of the Fisher Information, is also displayed (in dashed lines).', '1908.02954-2-83-4': 'This is not done to show an asymptotic property, but to show the symmetry of the log-likelihood, which allows approximation of [MATH] with the marginal mode [MATH], if the hyperprior [MATH] is flat around [MATH] since it holds that [MATH].', '1908.02954-2-84-0': 'Hence, one can approximate the LR itself in the following way: [EQUATION]', '1908.02954-2-84-1': 'Notice that this is equivalent to a hybrid approach, in which the parameters are estimated through the MLE (frequentist) and their values are plugged into the Bayesian LR.', '1908.02954-2-84-2': 'We would like to reiterate that we are not using maximum likelihood estimates of the parameters because we consider the likelihood ratio from a frequentist point of view.', '1908.02954-2-84-3': 'Our aim is to calculate a Bayesian likelihood ratio, and we have observed empirically that using the maximum likelihood estimates of the parameters we can approximate this value.', '1908.02954-2-85-0': "Hence, in case of a rare type match problem, and using the YHRD database as the reference database, we have [MATH], that corresponds to say that it is approximately 40'000 times more likely to observe the reduced data under the prosecution hypothesis than under the defence hypothesis.", '1908.02954-2-86-0': '## True LR', '1908.02954-2-87-0': "It is also interesting to study the frequentist likelihood ratio values obtained with [REF], and to compare it with the 'true' ones, meaning the LR values obtained when vector [MATH] is known.", '1908.02954-2-87-1': 'This corresponds to the desirable and unrealistic situation of knowing the ranked list of the frequencies of all the DNA types in the population of interest.', '1908.02954-2-87-2': 'Then, the model can be represented by the Bayesian network of Figure [REF].', '1908.02954-2-88-0': 'The likelihood ratio in this case can be obtained using again Corollary [REF], where now [MATH], ..., [MATH] play the role of [MATH].', '1908.02954-2-88-1': '[EQUATION]', '1908.02954-2-88-2': 'Notice that, in the rare type case, [MATH] is observed only once among the [MATH], ..., [MATH].', '1908.02954-2-88-3': 'Hence, we call it a singleton.', '1908.02954-2-88-4': 'Let [MATH] denote the number of singletons, and [MATH] the set of indexes of singletons observations in the augmented database.', '1908.02954-2-88-5': 'Notice also that the knowledge of [MATH] and [MATH], is not enough to observe [MATH], but given [MATH], both [MATH] and [MATH] are fixed and known.', '1908.02954-2-88-6': 'Given [MATH] and [MATH], it holds that the distribution of [MATH] is the same as the distribution of all other singletons.', '1908.02954-2-88-7': 'This implies that: [EQUATION].', '1908.02954-2-89-0': 'Let us denote as [MATH], .', '1908.02954-2-89-1': ', [MATH] the [MATH] different values taken by [MATH], ..., [MATH], ordered decreasingly according to the frequency of their values.', '1908.02954-2-89-2': 'Stated otherwise, if [MATH] is the frequency of [MATH] among [MATH] then [MATH].', '1908.02954-2-89-3': 'Moreover, in case [MATH] and [MATH] have the same frequency ([MATH]), then they are ordered increasingly according to their values.', '1908.02954-2-89-4': 'For instance, if [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], then [MATH].', '1908.02954-2-90-0': 'By definition, it holds that [EQUATION].', '1908.02954-2-91-0': 'Notice that [MATH] is a partition of [MATH], which will be denoted as [MATH].', '1908.02954-2-91-1': 'In the example, [MATH].', '1908.02954-2-91-2': 'Since the distribution of [MATH] only depends on [MATH], the latter can replace [MATH].', '1908.02954-2-91-3': 'Thus, it holds that [EQUATION]', '1908.02954-2-91-4': 'A more compact representation for [MATH] can be obtained by using two vectors [MATH] and [MATH] where [MATH] are the distinct numbers occurring in the partition, increasingly ordered, and each [MATH] is the number of repetitions of [MATH].', '1908.02954-2-91-5': '[MATH] is the length of these two vectors, and it holds that [MATH].', '1908.02954-2-91-6': 'In the example above we have that [MATH] can be represented by [MATH] with [MATH] and [MATH].', '1908.02954-2-92-0': 'There is an unknown map, [MATH], treated here as latent variable, which assigns the ranks of the DNA types, ordered according to their frequency in Nature, to one of the number [MATH] corresponding to the position in [MATH] of its frequency in the sample, or to [MATH] if the type if not observed.', '1908.02954-2-92-1': 'Stated otherwise, [EQUATION]', '1908.02954-2-92-2': 'Given [MATH], [MATH] must satisfy the following conditions: [EQUATION]', '1908.02954-2-92-3': 'The map [MATH] can be represented by a vector [MATH] such that [MATH].', '1908.02954-2-92-4': 'In the example above we have that [MATH].', '1908.02954-2-93-0': 'Notice that, given [MATH], the knowledge of [MATH] implies the knowledge of [MATH], ..., [MATH]: indeed it is enough to consider the position of the ranked positive values of [MATH], and to solve ties by considering the positions themselves (if [MATH], than the order is given by [MATH] and [MATH]).', '1908.02954-2-93-1': 'For instance, in the example if we sort the positive values of [MATH] and we collect their positions we get [MATH]: the reader can notice that we got back to [MATH].', '1908.02954-2-94-0': 'This means that to obtain the distribution of [MATH], which appears in [REF], it is enough to obtain the distribution of [MATH].', '1908.02954-2-94-1': 'Since we are only interested in the mean of the sum of singletons in samples of size [MATH] from the distribution of [MATH], we can just simulate samples from the distribution of [MATH] and sum the [MATH] such that [MATH].', '1908.02954-2-95-0': 'To simulate samples from the distribution of [MATH] we use a Metropolis-Hastings algorithm on the space of the vectors [MATH] satisfying condition [REF].', '1908.02954-2-95-1': 'Notice that for the model we assumed [MATH] to be infinitely long, but for simulations we will use a finite [MATH], of length [MATH].', '1908.02954-2-95-2': 'This is equivalent to assume that only [MATH] elements in the infinite [MATH] are positive, and the remaining infinite tail is made of zeros.', '1908.02954-2-95-3': 'Then the state space of the Metropolis-Hastings Markov chain is made of all vectors of length [MATH] whose elements belong to [MATH], and satisfy the condition [REF].', '1908.02954-2-95-4': 'If we start with an initial point [MATH] which satisfies [REF] and, at each allowed move of the Metropolis-Hastings we swap two different values [MATH] and [MATH] inside the vector, condition [REF] remains satisfied.', '1908.02954-2-95-5': 'The algorithm is based on a similar one proposed in [CITATION].', '1908.02954-2-96-0': "This method allows us to approximate the 'true' LR when the vector [MATH] is known.", '1908.02954-2-96-1': 'This is rarely the case, but we can put ourselves in a fictitious world where we know [MATH] (such as the frequencies in the YHRD database, or as in the followinf section the frequencies from a smaller population) and compare the true values for the LR with the one obtained by applying our Bayesian nonparametric model when [MATH] is unknown.', '1908.02954-2-97-0': '## Frequentist-Bayesian analysis of the error', '1908.02954-2-98-0': 'A real Bayesian statistician chooses the prior and hyperprior according to his beliefs.', '1908.02954-2-98-1': "Depending on the choice of the hyperprior over [MATH] and [MATH] he may or may not believe in the approximation [REF], but he does not really talk of 'error'.", '1908.02954-2-98-2': 'However, hardliner Bayesian statisticians are a rare species, and most of the time the Bayesian procedure consists in choosing priors (and hyperpriors) which are a compromise between personal beliefs and mathematical convenience.', '1908.02954-2-98-3': 'It is thus interesting to investigate the performance of such priors.', '1908.02954-2-98-4': 'This can be done by comparing the Bayesian likelihood ratio with the likelihood ratio a frequentist would obtain if the vector [MATH] was known, and for the same reduction of data.', '1908.02954-2-98-5': "This is what we call 'error': in other words, at the moment we are considering the Bayesian nonparametric method proposed in this paper as a way to estimate (notice the frequentist terminology) the true [MATH].", '1908.02954-2-98-6': 'If we denote by [MATH] the population proportion of the matching profile, another interesting comparison is the one between the Bayesian likelihood ratio and the frequentist likelihood ratio [MATH] (here denoted as [MATH]) that one would obtain knowing [MATH], but not reducing the data to partition.', '1908.02954-2-98-7': 'This is a sort of benchmark comparison and tells us how much we lose by using the Bayesian nonparametric methodology, and by reducing data.', '1908.02954-2-99-0': 'In total there are three quantities of interest ([MATH], [MATH], and [MATH]), and two differences of interest, which will be denoted as', '1908.02954-2-100-0': 'In order to analyse these five quantities, we can study their distribution over different rare type match cases.', '1908.02954-2-100-1': 'However, there is an obstacle.', '1908.02954-2-100-2': 'The Metropolis-Hastings algorithm described in Section [REF] is too slow to be used with the entire European database of [CITATION] of size [MATH].', '1908.02954-2-101-0': 'In order to make the computational effort feasible, we consider the haplotype frequencies for the sole Dutch population (of size [MATH]), and we pretend that they are the frequencies from the entire population of possible perpetrators.', '1908.02954-2-101-1': 'This population is summarised in Table [REF], and the maximum likelihood estimators for [MATH] and [MATH] are [MATH], [MATH].', '1908.02954-2-102-0': 'In this way we can use the Metropolis-Hashting algorithm to simulate [MATH].', '1908.02954-2-102-1': 'The model fitting is still good enough, as shown in Figure [REF] (as a side note, notice that the asymptotic behavior is reached faster for this smaller value of [MATH]).', '1908.02954-2-103-0': 'However, it is important for us to stress that the Gaussian shape and consequently the approximation [REF] is not empirically supported for small databases of size [MATH].', '1908.02954-2-104-0': 'In Table [REF] and Figure [REF] (a) we compare the distribution of [MATH], [MATH], and [MATH] obtained by 96 samples of size 100 from the Dutch population of Table [REF].', '1908.02954-2-104-1': 'Each sample represents a different rare type match case with a specific database of reference of size [MATH].', '1908.02954-2-105-0': 'The distribution of the benchmark likelihood ratio [MATH] has more variation than the distribution of the Bayesian likelihood ratio, while [MATH] appears to be the most concentrated around its mean.', '1908.02954-2-106-0': 'In Table [REF] and Figure [REF] (b) we consider the distribution of the two differences, Diff[MATH] and Diff[MATH] .', '1908.02954-2-106-1': 'Diff[MATH] is the smallest and the most concentrated: it ranges between -0.146 and 0.381 and has a small standard deviation.', '1908.02954-2-106-2': 'It means that the nonparametric Bayesian likelihood ratio obtained as in [REF] can be thought of as a good approximation of the frequentist likelihood ratio for the same reduction of data ([MATH]), even though we have not empirically validated the approximation for small databases of size 100.', '1908.02954-2-106-3': 'This difference is due to three things: the approximation [REF], the MLE estimation of the hyperparameters, and the choice of a prior distribution (two-parameter Poisson Dirichlet) which is quite realistic, as shown in Figure [REF], but not perfectly fitting the actual population.', '1908.02954-2-107-0': 'Notice that the difference increases if the Bayesian nonparametric likelihood is compared to the benchmark likelihood ratio (Diff[MATH]).', '1908.02954-2-107-1': 'However, the difference ranges within one order of magnitude, but most of the time lies between -0.676 and 0.115, thus small.', '1908.02954-2-108-0': '# Conclusion', '1908.02954-2-109-0': 'This paper discusses the first application of a Bayesian nonparametric method to the likelihood ratio assessment in forensic science, in particular to the challenging situation of the rare type match.', '1908.02954-2-109-1': 'If compared to traditional Bayesian methods such as those described in [CITATION], it presents many advantages.', '1908.02954-2-109-2': 'First of all, the prior chosen for the parameter [MATH] is more realistic for the population whose frequencies we want to model.', '1908.02954-2-109-3': 'Moreover, although the theoretical background on which it lies may seem very technical and difficult, the method is extremely simple to apply for practical use, thanks to a discussed approximation: indeed, simulation experiments show that a empirical Bayes approach is justified, at least using YHRD data from European population.', '1908.02954-2-110-0': 'More could be done in the future: in particular regarding approximation [REF].', '1908.02954-2-110-1': 'The posterior expectation in the denominator could, for instance, be treated using MCMC algorithms or ABC algorithms.', '1908.02954-2-110-2': 'Then, we can try to improve the efficiency of the Metropolis Hashting algorithm defined in Section [REF] in order to be used with bigger and more realistic populations.', '1908.02954-2-110-3': 'Also, we plan to compare this Bayesian nonparametric method to other existing methods for the rare type match problem, investigating calibration and validation through the use of ECE plots .'}

[['1908.02954-1-59-0', '1908.02954-2-60-0'], ['1908.02954-1-59-1', '1908.02954-2-60-1'], ['1908.02954-1-59-2', '1908.02954-2-60-2'], ['1908.02954-1-59-3', '1908.02954-2-60-3'], ['1908.02954-1-59-4', '1908.02954-2-60-4'], ['1908.02954-1-59-5', '1908.02954-2-60-5'], ['1908.02954-1-63-0', '1908.02954-2-64-0'], ['1908.02954-1-63-1', '1908.02954-2-64-1'], ['1908.02954-1-84-0', '1908.02954-2-81-0'], ['1908.02954-1-48-0', '1908.02954-2-49-0'], ['1908.02954-1-48-1', '1908.02954-2-49-1'], ['1908.02954-1-48-2', '1908.02954-2-49-2'], ['1908.02954-1-48-3', '1908.02954-2-49-3'], ['1908.02954-1-48-4', '1908.02954-2-49-4'], ['1908.02954-1-48-5', '1908.02954-2-49-5'], ['1908.02954-1-50-0', '1908.02954-2-51-0'], ['1908.02954-1-21-1', '1908.02954-2-21-1'], ['1908.02954-1-69-0', '1908.02954-2-70-0'], ['1908.02954-1-69-1', '1908.02954-2-70-1'], ['1908.02954-1-91-0', '1908.02954-2-87-0'], ['1908.02954-1-91-1', '1908.02954-2-87-1'], ['1908.02954-1-91-2', '1908.02954-2-87-2'], ['1908.02954-1-93-0', '1908.02954-2-89-0'], ['1908.02954-1-93-2', '1908.02954-2-89-2'], ['1908.02954-1-93-3', '1908.02954-2-89-3'], ['1908.02954-1-75-0', '1908.02954-2-73-0'], ['1908.02954-1-75-1', '1908.02954-2-73-1'], ['1908.02954-1-103-0', '1908.02954-2-100-1'], ['1908.02954-1-2-0', '1908.02954-2-2-0'], ['1908.02954-1-2-1', '1908.02954-2-2-1'], ['1908.02954-1-2-2', '1908.02954-2-2-2'], ['1908.02954-1-23-0', '1908.02954-2-23-0'], ['1908.02954-1-23-1', '1908.02954-2-23-1'], ['1908.02954-1-42-0', '1908.02954-2-43-0'], ['1908.02954-1-42-1', '1908.02954-2-43-1'], ['1908.02954-1-42-2', '1908.02954-2-43-2'], ['1908.02954-1-13-0', '1908.02954-2-13-0'], ['1908.02954-1-13-1', '1908.02954-2-13-1'], ['1908.02954-1-13-2', '1908.02954-2-13-2'], ['1908.02954-1-13-3', '1908.02954-2-13-3'], ['1908.02954-1-13-4', '1908.02954-2-13-4'], ['1908.02954-1-52-0', '1908.02954-2-53-0'], ['1908.02954-1-52-1', '1908.02954-2-53-1'], ['1908.02954-1-40-0', '1908.02954-2-41-0'], ['1908.02954-1-40-1', '1908.02954-2-41-1'], ['1908.02954-1-40-2', '1908.02954-2-41-2'], ['1908.02954-1-68-0', '1908.02954-2-69-0'], ['1908.02954-1-68-1', '1908.02954-2-69-1'], ['1908.02954-1-68-2', '1908.02954-2-69-2'], ['1908.02954-1-76-0', '1908.02954-2-74-0'], ['1908.02954-1-76-1', '1908.02954-2-74-1'], ['1908.02954-1-76-2', '1908.02954-2-74-3'], ['1908.02954-1-83-0', '1908.02954-2-80-0'], ['1908.02954-1-83-1', '1908.02954-2-80-1'], ['1908.02954-1-83-2', '1908.02954-2-80-2'], ['1908.02954-1-53-0', '1908.02954-2-54-0'], ['1908.02954-1-53-1', '1908.02954-2-54-1'], ['1908.02954-1-53-2', '1908.02954-2-54-2'], ['1908.02954-1-35-0', '1908.02954-2-36-0'], ['1908.02954-1-49-0', '1908.02954-2-50-0'], ['1908.02954-1-49-1', '1908.02954-2-50-1'], ['1908.02954-1-49-2', '1908.02954-2-50-2'], ['1908.02954-1-49-3', '1908.02954-2-50-3'], ['1908.02954-1-49-4', '1908.02954-2-50-4'], ['1908.02954-1-49-5', '1908.02954-2-50-5'], ['1908.02954-1-49-6', '1908.02954-2-50-6'], ['1908.02954-1-49-7', '1908.02954-2-50-7'], ['1908.02954-1-49-8', '1908.02954-2-50-8'], ['1908.02954-1-10-2', '1908.02954-2-10-3'], ['1908.02954-1-96-0', '1908.02954-2-92-0'], ['1908.02954-1-96-2', '1908.02954-2-92-2'], ['1908.02954-1-96-3', '1908.02954-2-92-3'], ['1908.02954-1-96-4', '1908.02954-2-92-4'], ['1908.02954-1-14-0', '1908.02954-2-14-0'], ['1908.02954-1-14-1', '1908.02954-2-14-1'], ['1908.02954-1-14-2', '1908.02954-2-14-2'], ['1908.02954-1-14-3', '1908.02954-2-14-3'], ['1908.02954-1-14-4', '1908.02954-2-14-4'], ['1908.02954-1-61-0', '1908.02954-2-62-0'], ['1908.02954-1-61-1', '1908.02954-2-62-1'], ['1908.02954-1-61-2', '1908.02954-2-62-2'], ['1908.02954-1-67-0', '1908.02954-2-68-0'], ['1908.02954-1-67-1', '1908.02954-2-68-1'], ['1908.02954-1-67-2', '1908.02954-2-68-2'], ['1908.02954-1-65-0', '1908.02954-2-66-0'], ['1908.02954-1-65-1', '1908.02954-2-66-1'], ['1908.02954-1-54-0', '1908.02954-2-55-0'], ['1908.02954-1-54-2', '1908.02954-2-55-2'], ['1908.02954-1-45-0', '1908.02954-2-46-0'], ['1908.02954-1-45-1', '1908.02954-2-46-1'], ['1908.02954-1-99-1', '1908.02954-2-95-1'], ['1908.02954-1-99-2', '1908.02954-2-95-2'], ['1908.02954-1-99-3', '1908.02954-2-95-3'], ['1908.02954-1-99-4', '1908.02954-2-95-4'], ['1908.02954-1-99-5', '1908.02954-2-95-5'], ['1908.02954-1-19-0', '1908.02954-2-19-0'], ['1908.02954-1-66-0', '1908.02954-2-67-0'], ['1908.02954-1-66-1', '1908.02954-2-67-1'], ['1908.02954-1-95-0', '1908.02954-2-91-0'], ['1908.02954-1-95-1', '1908.02954-2-91-1'], ['1908.02954-1-95-2', '1908.02954-2-91-2'], ['1908.02954-1-95-3', '1908.02954-2-91-3'], ['1908.02954-1-95-5', '1908.02954-2-91-5'], ['1908.02954-1-95-6', '1908.02954-2-91-6'], ['1908.02954-1-6-2', '1908.02954-2-6-2'], ['1908.02954-1-6-3', '1908.02954-2-6-3'], ['1908.02954-1-6-4', '1908.02954-2-6-4'], ['1908.02954-1-6-5', '1908.02954-2-6-5'], ['1908.02954-1-39-0', '1908.02954-2-40-0'], ['1908.02954-1-92-0', '1908.02954-2-88-0'], ['1908.02954-1-92-2', '1908.02954-2-88-2'], ['1908.02954-1-92-3', '1908.02954-2-88-3'], ['1908.02954-1-92-4', '1908.02954-2-88-4'], ['1908.02954-1-92-5', '1908.02954-2-88-5'], ['1908.02954-1-92-6', '1908.02954-2-88-6'], ['1908.02954-1-92-7', '1908.02954-2-88-7'], ['1908.02954-1-28-0', '1908.02954-2-29-0'], ['1908.02954-1-28-1', '1908.02954-2-29-1'], ['1908.02954-1-44-0', '1908.02954-2-45-0'], ['1908.02954-1-44-1', '1908.02954-2-45-1'], ['1908.02954-1-44-2', '1908.02954-2-45-2'], ['1908.02954-1-44-3', '1908.02954-2-45-3'], ['1908.02954-1-44-4', '1908.02954-2-45-4'], ['1908.02954-1-64-0', '1908.02954-2-65-0'], ['1908.02954-1-27-0', '1908.02954-2-27-0'], ['1908.02954-1-87-0', '1908.02954-2-84-0'], ['1908.02954-1-87-1', '1908.02954-2-84-1'], ['1908.02954-1-87-2', '1908.02954-2-84-2'], ['1908.02954-1-87-3', '1908.02954-2-84-3'], ['1908.02954-1-57-0', '1908.02954-2-58-0'], ['1908.02954-1-57-1', '1908.02954-2-58-1'], ['1908.02954-1-32-0', '1908.02954-2-33-0'], ['1908.02954-1-55-0', '1908.02954-2-56-0'], ['1908.02954-1-55-1', '1908.02954-2-56-1'], ['1908.02954-1-56-0', '1908.02954-2-57-0'], ['1908.02954-1-56-1', '1908.02954-2-57-1'], ['1908.02954-1-56-2', '1908.02954-2-57-2'], ['1908.02954-1-56-3', '1908.02954-2-57-3'], ['1908.02954-1-79-1', '1908.02954-2-78-1'], ['1908.02954-1-46-0', '1908.02954-2-47-0'], ['1908.02954-1-4-1', '1908.02954-2-4-1'], ['1908.02954-1-4-4', '1908.02954-2-4-4'], ['1908.02954-1-4-5', '1908.02954-2-4-5'], ['1908.02954-1-43-0', '1908.02954-2-44-0'], ['1908.02954-1-43-1', '1908.02954-2-44-1'], ['1908.02954-1-43-2', '1908.02954-2-44-2'], ['1908.02954-1-43-3', '1908.02954-2-44-3'], ['1908.02954-1-43-4', '1908.02954-2-44-4'], ['1908.02954-1-43-5', '1908.02954-2-44-5'], ['1908.02954-1-60-0', '1908.02954-2-61-0'], ['1908.02954-1-8-0', '1908.02954-2-8-0'], ['1908.02954-1-8-1', '1908.02954-2-8-1'], ['1908.02954-1-8-3', '1908.02954-2-8-3'], ['1908.02954-1-8-5', '1908.02954-2-8-5'], ['1908.02954-1-8-6', '1908.02954-2-8-6'], ['1908.02954-1-8-8', '1908.02954-2-8-8'], ['1908.02954-1-105-0', '1908.02954-2-109-0'], ['1908.02954-1-105-1', '1908.02954-2-109-1'], ['1908.02954-1-105-2', '1908.02954-2-109-2'], ['1908.02954-1-78-0', '1908.02954-2-76-0'], ['1908.02954-1-20-0', '1908.02954-2-20-0'], ['1908.02954-1-20-2', '1908.02954-2-20-2'], ['1908.02954-1-20-3', '1908.02954-2-20-3'], ['1908.02954-1-9-0', '1908.02954-2-9-0'], ['1908.02954-1-9-1', '1908.02954-2-9-1'], ['1908.02954-1-9-2', '1908.02954-2-9-2'], ['1908.02954-1-9-4', '1908.02954-2-9-4'], ['1908.02954-1-9-5', '1908.02954-2-9-5'], ['1908.02954-1-9-7', '1908.02954-2-9-7'], ['1908.02954-1-9-8', '1908.02954-2-9-8'], ['1908.02954-1-9-9', '1908.02954-2-9-9'], ['1908.02954-1-9-10', '1908.02954-2-9-10'], ['1908.02954-1-9-11', '1908.02954-2-9-11'], ['1908.02954-1-9-12', '1908.02954-2-9-12'], ['1908.02954-1-9-13', '1908.02954-2-9-13'], ['1908.02954-1-9-14', '1908.02954-2-9-14'], ['1908.02954-1-0-0', '1908.02954-2-0-0'], ['1908.02954-1-0-1', '1908.02954-2-0-1'], ['1908.02954-1-0-2', '1908.02954-2-0-2'], ['1908.02954-1-0-3', '1908.02954-2-0-3'], ['1908.02954-1-24-0', '1908.02954-2-24-0'], ['1908.02954-1-24-1', '1908.02954-2-24-1'], ['1908.02954-1-24-2', '1908.02954-2-24-2'], ['1908.02954-1-24-3', '1908.02954-2-24-3'], ['1908.02954-1-24-4', '1908.02954-2-24-4'], ['1908.02954-1-24-5', '1908.02954-2-24-5'], ['1908.02954-1-24-6', '1908.02954-2-24-6'], ['1908.02954-1-5-1', '1908.02954-2-5-1'], ['1908.02954-1-41-0', '1908.02954-2-42-0'], ['1908.02954-1-41-1', '1908.02954-2-42-1'], ['1908.02954-1-41-2', '1908.02954-2-42-2'], ['1908.02954-1-86-1', '1908.02954-2-83-1'], ['1908.02954-1-86-2', '1908.02954-2-83-2'], ['1908.02954-1-102-0', '1908.02954-2-98-0'], ['1908.02954-1-102-1', '1908.02954-2-98-1'], ['1908.02954-1-102-2', '1908.02954-2-98-2'], ['1908.02954-1-102-3', '1908.02954-2-98-3'], ['1908.02954-1-102-4', '1908.02954-2-98-4'], ['1908.02954-1-102-5', '1908.02954-2-98-5'], ['1908.02954-1-102-6', '1908.02954-2-98-6'], ['1908.02954-1-102-7', '1908.02954-2-98-7'], ['1908.02954-1-82-0', '1908.02954-2-79-0'], ['1908.02954-1-82-1', '1908.02954-2-79-1'], ['1908.02954-1-82-2', '1908.02954-2-79-2'], ['1908.02954-1-82-3', '1908.02954-2-79-3'], ['1908.02954-1-82-5', '1908.02954-2-79-5'], ['1908.02954-1-82-6', '1908.02954-2-79-6'], ['1908.02954-1-18-0', '1908.02954-2-18-0'], ['1908.02954-1-18-1', '1908.02954-2-18-1'], ['1908.02954-1-98-0', '1908.02954-2-94-1'], ['1908.02954-1-84-1', '1908.02954-2-81-1'], ['1908.02954-1-21-0', '1908.02954-2-21-0'], ['1908.02954-1-25-0', '1908.02954-2-25-0'], ['1908.02954-1-103-1', '1908.02954-2-100-2'], ['1908.02954-1-10-1', '1908.02954-2-10-2'], ['1908.02954-1-10-3', '1908.02954-2-10-4'], ['1908.02954-1-99-0', '1908.02954-2-95-0'], ['1908.02954-1-95-4', '1908.02954-2-91-4'], ['1908.02954-1-6-1', '1908.02954-2-6-1'], ['1908.02954-1-39-1', '1908.02954-2-40-1'], ['1908.02954-1-97-0', '1908.02954-2-93-0'], ['1908.02954-1-97-1', '1908.02954-2-93-1'], ['1908.02954-1-100-0', '1908.02954-2-96-0'], ['1908.02954-1-100-1', '1908.02954-2-96-1'], ['1908.02954-1-27-1', '1908.02954-2-27-1'], ['1908.02954-1-4-0', '1908.02954-2-4-0'], ['1908.02954-1-4-3', '1908.02954-2-4-3'], ['1908.02954-1-8-2', '1908.02954-2-8-2'], ['1908.02954-1-8-4', '1908.02954-2-8-4'], ['1908.02954-1-8-7', '1908.02954-2-8-7'], ['1908.02954-1-105-3', '1908.02954-2-109-3'], ['1908.02954-1-78-1', '1908.02954-2-76-1'], ['1908.02954-1-20-1', '1908.02954-2-20-1'], ['1908.02954-1-9-3', '1908.02954-2-9-3'], ['1908.02954-1-9-6', '1908.02954-2-9-6'], ['1908.02954-1-9-15', '1908.02954-2-9-15'], ['1908.02954-1-9-16', '1908.02954-2-9-16'], ['1908.02954-1-30-0', '1908.02954-2-31-0'], ['1908.02954-1-5-0', '1908.02954-2-5-0'], ['1908.02954-1-5-2', '1908.02954-2-5-2'], ['1908.02954-1-3-0', '1908.02954-2-3-0'], ['1908.02954-1-3-1', '1908.02954-2-3-1'], ['1908.02954-1-3-2', '1908.02954-2-3-2'], ['1908.02954-1-86-0', '1908.02954-2-83-0'], ['1908.02954-1-86-4', '1908.02954-2-83-4'], ['1908.02954-1-102-9', '1908.02954-2-99-0'], ['1908.02954-1-82-4', '1908.02954-2-79-4'], ['1908.02954-1-25-1', '1908.02954-2-25-1'], ['1908.02954-1-10-0', '1908.02954-2-10-0'], ['1908.02954-1-29-0', '1908.02954-2-30-0'], ['1908.02954-1-29-1', '1908.02954-2-30-0'], ['1908.02954-1-6-0', '1908.02954-2-6-0'], ['1908.02954-1-79-0', '1908.02954-2-78-0'], ['1908.02954-1-4-2', '1908.02954-2-4-2'], ['1908.02954-1-106-0', '1908.02954-2-110-0'], ['1908.02954-1-106-1', '1908.02954-2-110-3'], ['1908.02954-1-24-7', '1908.02954-2-24-7'], ['1908.02954-1-88-0', '1908.02954-2-85-0'], ['1908.02954-1-86-3', '1908.02954-2-83-3']]

[['1908.02954-1-59-0', '1908.02954-2-60-0'], ['1908.02954-1-59-1', '1908.02954-2-60-1'], ['1908.02954-1-59-2', '1908.02954-2-60-2'], ['1908.02954-1-59-3', '1908.02954-2-60-3'], ['1908.02954-1-59-4', '1908.02954-2-60-4'], ['1908.02954-1-59-5', '1908.02954-2-60-5'], ['1908.02954-1-63-0', '1908.02954-2-64-0'], ['1908.02954-1-63-1', '1908.02954-2-64-1'], ['1908.02954-1-84-0', '1908.02954-2-81-0'], ['1908.02954-1-48-0', '1908.02954-2-49-0'], ['1908.02954-1-48-1', '1908.02954-2-49-1'], ['1908.02954-1-48-2', '1908.02954-2-49-2'], ['1908.02954-1-48-3', '1908.02954-2-49-3'], ['1908.02954-1-48-4', '1908.02954-2-49-4'], ['1908.02954-1-48-5', '1908.02954-2-49-5'], ['1908.02954-1-50-0', '1908.02954-2-51-0'], ['1908.02954-1-21-1', '1908.02954-2-21-1'], ['1908.02954-1-69-0', '1908.02954-2-70-0'], ['1908.02954-1-69-1', '1908.02954-2-70-1'], ['1908.02954-1-91-0', '1908.02954-2-87-0'], ['1908.02954-1-91-1', '1908.02954-2-87-1'], ['1908.02954-1-91-2', '1908.02954-2-87-2'], ['1908.02954-1-93-0', '1908.02954-2-89-0'], ['1908.02954-1-93-2', '1908.02954-2-89-2'], ['1908.02954-1-93-3', '1908.02954-2-89-3'], ['1908.02954-1-75-0', '1908.02954-2-73-0'], ['1908.02954-1-75-1', '1908.02954-2-73-1'], ['1908.02954-1-103-0', '1908.02954-2-100-1'], ['1908.02954-1-2-0', '1908.02954-2-2-0'], ['1908.02954-1-2-1', '1908.02954-2-2-1'], ['1908.02954-1-2-2', '1908.02954-2-2-2'], ['1908.02954-1-23-0', '1908.02954-2-23-0'], ['1908.02954-1-23-1', '1908.02954-2-23-1'], ['1908.02954-1-42-0', '1908.02954-2-43-0'], ['1908.02954-1-42-1', '1908.02954-2-43-1'], ['1908.02954-1-42-2', '1908.02954-2-43-2'], ['1908.02954-1-13-0', '1908.02954-2-13-0'], ['1908.02954-1-13-1', '1908.02954-2-13-1'], ['1908.02954-1-13-2', '1908.02954-2-13-2'], ['1908.02954-1-13-3', '1908.02954-2-13-3'], ['1908.02954-1-13-4', '1908.02954-2-13-4'], ['1908.02954-1-52-0', '1908.02954-2-53-0'], ['1908.02954-1-52-1', '1908.02954-2-53-1'], ['1908.02954-1-40-0', '1908.02954-2-41-0'], ['1908.02954-1-40-1', '1908.02954-2-41-1'], ['1908.02954-1-40-2', '1908.02954-2-41-2'], ['1908.02954-1-68-0', '1908.02954-2-69-0'], ['1908.02954-1-68-1', '1908.02954-2-69-1'], ['1908.02954-1-68-2', '1908.02954-2-69-2'], ['1908.02954-1-76-0', '1908.02954-2-74-0'], ['1908.02954-1-76-1', '1908.02954-2-74-1'], ['1908.02954-1-76-2', '1908.02954-2-74-3'], ['1908.02954-1-83-0', '1908.02954-2-80-0'], ['1908.02954-1-83-1', '1908.02954-2-80-1'], ['1908.02954-1-83-2', '1908.02954-2-80-2'], ['1908.02954-1-53-0', '1908.02954-2-54-0'], ['1908.02954-1-53-1', '1908.02954-2-54-1'], ['1908.02954-1-53-2', '1908.02954-2-54-2'], ['1908.02954-1-35-0', '1908.02954-2-36-0'], ['1908.02954-1-49-0', '1908.02954-2-50-0'], ['1908.02954-1-49-1', '1908.02954-2-50-1'], ['1908.02954-1-49-2', '1908.02954-2-50-2'], ['1908.02954-1-49-3', '1908.02954-2-50-3'], ['1908.02954-1-49-4', '1908.02954-2-50-4'], ['1908.02954-1-49-5', '1908.02954-2-50-5'], ['1908.02954-1-49-6', '1908.02954-2-50-6'], ['1908.02954-1-49-7', '1908.02954-2-50-7'], ['1908.02954-1-49-8', '1908.02954-2-50-8'], ['1908.02954-1-10-2', '1908.02954-2-10-3'], ['1908.02954-1-96-0', '1908.02954-2-92-0'], ['1908.02954-1-96-2', '1908.02954-2-92-2'], ['1908.02954-1-96-3', '1908.02954-2-92-3'], ['1908.02954-1-96-4', '1908.02954-2-92-4'], ['1908.02954-1-14-0', '1908.02954-2-14-0'], ['1908.02954-1-14-1', '1908.02954-2-14-1'], ['1908.02954-1-14-2', '1908.02954-2-14-2'], ['1908.02954-1-14-3', '1908.02954-2-14-3'], ['1908.02954-1-14-4', '1908.02954-2-14-4'], ['1908.02954-1-61-0', '1908.02954-2-62-0'], ['1908.02954-1-61-1', '1908.02954-2-62-1'], ['1908.02954-1-61-2', '1908.02954-2-62-2'], ['1908.02954-1-67-0', '1908.02954-2-68-0'], ['1908.02954-1-67-1', '1908.02954-2-68-1'], ['1908.02954-1-67-2', '1908.02954-2-68-2'], ['1908.02954-1-65-0', '1908.02954-2-66-0'], ['1908.02954-1-65-1', '1908.02954-2-66-1'], ['1908.02954-1-54-0', '1908.02954-2-55-0'], ['1908.02954-1-54-2', '1908.02954-2-55-2'], ['1908.02954-1-45-0', '1908.02954-2-46-0'], ['1908.02954-1-45-1', '1908.02954-2-46-1'], ['1908.02954-1-99-1', '1908.02954-2-95-1'], ['1908.02954-1-99-2', '1908.02954-2-95-2'], ['1908.02954-1-99-3', '1908.02954-2-95-3'], ['1908.02954-1-99-4', '1908.02954-2-95-4'], ['1908.02954-1-99-5', '1908.02954-2-95-5'], ['1908.02954-1-19-0', '1908.02954-2-19-0'], ['1908.02954-1-66-0', '1908.02954-2-67-0'], ['1908.02954-1-66-1', '1908.02954-2-67-1'], ['1908.02954-1-95-0', '1908.02954-2-91-0'], ['1908.02954-1-95-1', '1908.02954-2-91-1'], ['1908.02954-1-95-2', '1908.02954-2-91-2'], ['1908.02954-1-95-3', '1908.02954-2-91-3'], ['1908.02954-1-95-5', '1908.02954-2-91-5'], ['1908.02954-1-95-6', '1908.02954-2-91-6'], ['1908.02954-1-6-2', '1908.02954-2-6-2'], ['1908.02954-1-6-3', '1908.02954-2-6-3'], ['1908.02954-1-6-4', '1908.02954-2-6-4'], ['1908.02954-1-6-5', '1908.02954-2-6-5'], ['1908.02954-1-39-0', '1908.02954-2-40-0'], ['1908.02954-1-92-0', '1908.02954-2-88-0'], ['1908.02954-1-92-2', '1908.02954-2-88-2'], ['1908.02954-1-92-3', '1908.02954-2-88-3'], ['1908.02954-1-92-4', '1908.02954-2-88-4'], ['1908.02954-1-92-5', '1908.02954-2-88-5'], ['1908.02954-1-92-6', '1908.02954-2-88-6'], ['1908.02954-1-92-7', '1908.02954-2-88-7'], ['1908.02954-1-28-0', '1908.02954-2-29-0'], ['1908.02954-1-28-1', '1908.02954-2-29-1'], ['1908.02954-1-44-0', '1908.02954-2-45-0'], ['1908.02954-1-44-1', '1908.02954-2-45-1'], ['1908.02954-1-44-2', '1908.02954-2-45-2'], ['1908.02954-1-44-3', '1908.02954-2-45-3'], ['1908.02954-1-44-4', '1908.02954-2-45-4'], ['1908.02954-1-64-0', '1908.02954-2-65-0'], ['1908.02954-1-27-0', '1908.02954-2-27-0'], ['1908.02954-1-87-0', '1908.02954-2-84-0'], ['1908.02954-1-87-1', '1908.02954-2-84-1'], ['1908.02954-1-87-2', '1908.02954-2-84-2'], ['1908.02954-1-87-3', '1908.02954-2-84-3'], ['1908.02954-1-57-0', '1908.02954-2-58-0'], ['1908.02954-1-57-1', '1908.02954-2-58-1'], ['1908.02954-1-32-0', '1908.02954-2-33-0'], ['1908.02954-1-55-0', '1908.02954-2-56-0'], ['1908.02954-1-55-1', '1908.02954-2-56-1'], ['1908.02954-1-56-0', '1908.02954-2-57-0'], ['1908.02954-1-56-1', '1908.02954-2-57-1'], ['1908.02954-1-56-2', '1908.02954-2-57-2'], ['1908.02954-1-56-3', '1908.02954-2-57-3'], ['1908.02954-1-79-1', '1908.02954-2-78-1'], ['1908.02954-1-46-0', '1908.02954-2-47-0'], ['1908.02954-1-4-1', '1908.02954-2-4-1'], ['1908.02954-1-4-4', '1908.02954-2-4-4'], ['1908.02954-1-4-5', '1908.02954-2-4-5'], ['1908.02954-1-43-0', '1908.02954-2-44-0'], ['1908.02954-1-43-1', '1908.02954-2-44-1'], ['1908.02954-1-43-2', '1908.02954-2-44-2'], ['1908.02954-1-43-3', '1908.02954-2-44-3'], ['1908.02954-1-43-4', '1908.02954-2-44-4'], ['1908.02954-1-43-5', '1908.02954-2-44-5'], ['1908.02954-1-60-0', '1908.02954-2-61-0'], ['1908.02954-1-8-0', '1908.02954-2-8-0'], ['1908.02954-1-8-1', '1908.02954-2-8-1'], ['1908.02954-1-8-3', '1908.02954-2-8-3'], ['1908.02954-1-8-5', '1908.02954-2-8-5'], ['1908.02954-1-8-6', '1908.02954-2-8-6'], ['1908.02954-1-8-8', '1908.02954-2-8-8'], ['1908.02954-1-105-0', '1908.02954-2-109-0'], ['1908.02954-1-105-1', '1908.02954-2-109-1'], ['1908.02954-1-105-2', '1908.02954-2-109-2'], ['1908.02954-1-78-0', '1908.02954-2-76-0'], ['1908.02954-1-20-0', '1908.02954-2-20-0'], ['1908.02954-1-20-2', '1908.02954-2-20-2'], ['1908.02954-1-20-3', '1908.02954-2-20-3'], ['1908.02954-1-9-0', '1908.02954-2-9-0'], ['1908.02954-1-9-1', '1908.02954-2-9-1'], ['1908.02954-1-9-2', '1908.02954-2-9-2'], ['1908.02954-1-9-4', '1908.02954-2-9-4'], ['1908.02954-1-9-5', '1908.02954-2-9-5'], ['1908.02954-1-9-7', '1908.02954-2-9-7'], ['1908.02954-1-9-8', '1908.02954-2-9-8'], ['1908.02954-1-9-9', '1908.02954-2-9-9'], ['1908.02954-1-9-10', '1908.02954-2-9-10'], ['1908.02954-1-9-11', '1908.02954-2-9-11'], ['1908.02954-1-9-12', '1908.02954-2-9-12'], ['1908.02954-1-9-13', '1908.02954-2-9-13'], ['1908.02954-1-9-14', '1908.02954-2-9-14'], ['1908.02954-1-0-0', '1908.02954-2-0-0'], ['1908.02954-1-0-1', '1908.02954-2-0-1'], ['1908.02954-1-0-2', '1908.02954-2-0-2'], ['1908.02954-1-0-3', '1908.02954-2-0-3'], ['1908.02954-1-24-0', '1908.02954-2-24-0'], ['1908.02954-1-24-1', '1908.02954-2-24-1'], ['1908.02954-1-24-2', '1908.02954-2-24-2'], ['1908.02954-1-24-3', '1908.02954-2-24-3'], ['1908.02954-1-24-4', '1908.02954-2-24-4'], ['1908.02954-1-24-5', '1908.02954-2-24-5'], ['1908.02954-1-24-6', '1908.02954-2-24-6'], ['1908.02954-1-5-1', '1908.02954-2-5-1'], ['1908.02954-1-41-0', '1908.02954-2-42-0'], ['1908.02954-1-41-1', '1908.02954-2-42-1'], ['1908.02954-1-41-2', '1908.02954-2-42-2'], ['1908.02954-1-86-1', '1908.02954-2-83-1'], ['1908.02954-1-86-2', '1908.02954-2-83-2'], ['1908.02954-1-102-0', '1908.02954-2-98-0'], ['1908.02954-1-102-1', '1908.02954-2-98-1'], ['1908.02954-1-102-2', '1908.02954-2-98-2'], ['1908.02954-1-102-3', '1908.02954-2-98-3'], ['1908.02954-1-102-4', '1908.02954-2-98-4'], ['1908.02954-1-102-5', '1908.02954-2-98-5'], ['1908.02954-1-102-6', '1908.02954-2-98-6'], ['1908.02954-1-102-7', '1908.02954-2-98-7'], ['1908.02954-1-82-0', '1908.02954-2-79-0'], ['1908.02954-1-82-1', '1908.02954-2-79-1'], ['1908.02954-1-82-2', '1908.02954-2-79-2'], ['1908.02954-1-82-3', '1908.02954-2-79-3'], ['1908.02954-1-82-5', '1908.02954-2-79-5'], ['1908.02954-1-82-6', '1908.02954-2-79-6']]

[['1908.02954-1-18-0', '1908.02954-2-18-0'], ['1908.02954-1-18-1', '1908.02954-2-18-1'], ['1908.02954-1-98-0', '1908.02954-2-94-1'], ['1908.02954-1-84-1', '1908.02954-2-81-1'], ['1908.02954-1-21-0', '1908.02954-2-21-0'], ['1908.02954-1-25-0', '1908.02954-2-25-0'], ['1908.02954-1-103-1', '1908.02954-2-100-2'], ['1908.02954-1-10-1', '1908.02954-2-10-2'], ['1908.02954-1-10-3', '1908.02954-2-10-4'], ['1908.02954-1-99-0', '1908.02954-2-95-0'], ['1908.02954-1-95-4', '1908.02954-2-91-4'], ['1908.02954-1-6-1', '1908.02954-2-6-1'], ['1908.02954-1-39-1', '1908.02954-2-40-1'], ['1908.02954-1-97-0', '1908.02954-2-93-0'], ['1908.02954-1-97-1', '1908.02954-2-93-1'], ['1908.02954-1-100-0', '1908.02954-2-96-0'], ['1908.02954-1-100-1', '1908.02954-2-96-1'], ['1908.02954-1-27-1', '1908.02954-2-27-1'], ['1908.02954-1-4-0', '1908.02954-2-4-0'], ['1908.02954-1-4-3', '1908.02954-2-4-3'], ['1908.02954-1-8-2', '1908.02954-2-8-2'], ['1908.02954-1-8-4', '1908.02954-2-8-4'], ['1908.02954-1-8-7', '1908.02954-2-8-7'], ['1908.02954-1-105-3', '1908.02954-2-109-3'], ['1908.02954-1-78-1', '1908.02954-2-76-1'], ['1908.02954-1-20-1', '1908.02954-2-20-1'], ['1908.02954-1-9-3', '1908.02954-2-9-3'], ['1908.02954-1-9-6', '1908.02954-2-9-6'], ['1908.02954-1-9-15', '1908.02954-2-9-15'], ['1908.02954-1-9-16', '1908.02954-2-9-16'], ['1908.02954-1-30-0', '1908.02954-2-31-0'], ['1908.02954-1-5-0', '1908.02954-2-5-0'], ['1908.02954-1-5-2', '1908.02954-2-5-2'], ['1908.02954-1-3-0', '1908.02954-2-3-0'], ['1908.02954-1-3-1', '1908.02954-2-3-1'], ['1908.02954-1-3-2', '1908.02954-2-3-2'], ['1908.02954-1-86-0', '1908.02954-2-83-0'], ['1908.02954-1-86-4', '1908.02954-2-83-4'], ['1908.02954-1-102-9', '1908.02954-2-99-0'], ['1908.02954-1-82-4', '1908.02954-2-79-4']]

[]

[['1908.02954-1-25-1', '1908.02954-2-25-1'], ['1908.02954-1-10-0', '1908.02954-2-10-0'], ['1908.02954-1-29-0', '1908.02954-2-30-0'], ['1908.02954-1-29-1', '1908.02954-2-30-0'], ['1908.02954-1-6-0', '1908.02954-2-6-0'], ['1908.02954-1-79-0', '1908.02954-2-78-0'], ['1908.02954-1-4-2', '1908.02954-2-4-2'], ['1908.02954-1-106-0', '1908.02954-2-110-0'], ['1908.02954-1-106-1', '1908.02954-2-110-3'], ['1908.02954-1-24-7', '1908.02954-2-24-7'], ['1908.02954-1-88-0', '1908.02954-2-85-0'], ['1908.02954-1-86-3', '1908.02954-2-83-3']]

[]

['1908.02954-1-16-0', '1908.02954-1-17-0', '1908.02954-1-26-0', '1908.02954-1-33-0', '1908.02954-1-34-0', '1908.02954-1-36-0', '1908.02954-1-37-0', '1908.02954-1-38-0', '1908.02954-1-51-0', '1908.02954-1-54-1', '1908.02954-1-65-2', '1908.02954-1-66-2', '1908.02954-1-67-3', '1908.02954-1-70-0', '1908.02954-1-92-1', '1908.02954-1-93-1', '1908.02954-1-93-4', '1908.02954-1-94-0', '1908.02954-1-96-1', '1908.02954-2-16-0', '1908.02954-2-26-0', '1908.02954-2-34-0', '1908.02954-2-35-0', '1908.02954-2-37-0', '1908.02954-2-38-0', '1908.02954-2-39-0', '1908.02954-2-52-0', '1908.02954-2-55-1', '1908.02954-2-66-2', '1908.02954-2-67-2', '1908.02954-2-68-3', '1908.02954-2-71-0', '1908.02954-2-88-1', '1908.02954-2-89-1', '1908.02954-2-89-4', '1908.02954-2-90-0', '1908.02954-2-92-1']

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

https://arxiv.org/abs/1908.02954

1404.3194

{'1404.3194-1-0-0': 'We perform direct numerical simulations of the equations of magnetohydrodynamics with external random forcing and in the presence of gravity.', '1404.3194-1-0-1': 'The domain is a divided in two parts, in the bottom layer the forcing is helical whereas in the top layer the helicity of the forcing is zero with a smooth transition in the middle.', '1404.3194-1-0-2': 'At early times, a large-scale helical dynamo develops in the bottom layer.', '1404.3194-1-0-3': 'At late times the dynamo saturates, but the vertical magnetic flux continues to develop and rises to form dynamic bipolar structures at the top, which later disappear and reappear again.', '1404.3194-1-0-4': 'Their field strength can exceed three times the equipartition value of the turbulence.', '1404.3194-1-0-5': 'This is the first example of self-consistent generation of bipolar magnetic structures from a dynamo simulation.', '1404.3194-1-1-0': '# Introduction', '1404.3194-1-2-0': 'One of the most striking and also most observed magnetic features of the Sun are the sunspots and active regions.', '1404.3194-1-2-1': 'The number of sunspots, the strength of the magnetic field in sunspots, and the magnetic field calculated at the surface of the Sun are often taken as proxies of the solar magnetic field deep inside.', '1404.3194-1-2-2': 'There is general agreement that the evolution of the solar magnetic field is governed by the solar dynamo which operates in the convection zone of the Sun.', '1404.3194-1-2-3': 'This brings us to the question, how is the magnetic field generated by the solar dynamo related to the magnetic field observed at the surface of the Sun?', '1404.3194-1-2-4': 'At present, this question does not have a clear answer.', '1404.3194-1-3-0': 'The conventional picture is that the solar dynamo generates a strong toroidal magnetic field in the form of flux tubes at the bottom of the convection zone, also called the tachocline [CITATION].', '1404.3194-1-3-1': 'This strong magnetic field is buoyant and hence rises up to eventually penetrate through the surface layers of the Sun to create bipolar regions on the surface.', '1404.3194-1-3-2': "During its rise through the convection zone the magnetic flux tube is twisted by the Coriolis force to give rise to a preferential tilt of the bipolar regions with respect to the equator - which is also known as Joy's law.", '1404.3194-1-4-0': 'The traditional picture is prone to criticism on several counts.', '1404.3194-1-4-1': '(a) Several recent numerical simulations of rotating spherical magneto-convection have shown that a solar-like dynamo can operate without a tachocline in the bulk of the convection zone.', '1404.3194-1-4-2': '(b) It is difficult to reconcile the coherent rise of magnetic flux tubes through the turbulent convection zone; numerical simulations of [CITATION], admittedly at moderate magnetic Reynolds numbers, have found no evidence that this is possible.', '1404.3194-1-4-3': '(c) The traditional picture does not yield any quantitative relation between the strength of the magnetic flux tube and the magnetic field observed in sunspots.', '1404.3194-1-4-4': '(d) A natural corollary of the rising flux tube picture is that the active regions will emerge with preferential orientation on the surface of the Sun, whereas recent observations show that active regions actually emerge with random orientations but get preferentially oriented as time progresses.', '1404.3194-1-5-0': 'In the last decade, an alternative scenario has emerged.', '1404.3194-1-5-1': 'In this scenario, first suggested by [CITATION], the turbulent dynamo generates magnetic field in the bulk of the convection zone.', '1404.3194-1-5-2': 'In the near-surface shear layer, that has been observed in helioseismology , the dynamo-generated magnetic field propagates equatorward, satisfying the Parker-Yoshimura rule.', '1404.3194-1-5-3': "The observed preferential orientation of the active regions, the Joy's law, can be understood as an effect of the shear .", '1404.3194-1-5-4': 'In this scenario, which admittedly is yet to be supported by direct numerical simulations, although mean-field calculations do provide support , the active regions must form from a dynamo-generated large-scale magnetic field by the process of magnetic flux concentration operating at or near the surface of the Sun.', '1404.3194-1-6-0': 'There have been two different, mutually complimentary, approaches to understand this process.', '1404.3194-1-6-1': 'On the one hand lies the numerical simulations by [CITATION], [CITATION], [CITATION], and [CITATION] who solve radiative magneto-convection in a Cartesian domain under a simplified setup (non-rotating, no large-scale shear).', '1404.3194-1-6-2': 'All these simulations develop a bipolar magnetic structure on their top, surface but in all the cases the velocity and the magnetic field at the bottom boundary need to be carefully imposed.', '1404.3194-1-6-3': 'Furthermore, in these simulations, with the exception of [CITATION], the mechanism responsible for formation of magnetic structures has not been elucidated.', '1404.3194-1-6-4': 'Another related example are the magneto-convection simulations of [CITATION], where an imposed vertical field segregates into magnetized and unmagnetized regions.', '1404.3194-1-6-5': 'The authors ascribe this to the effect of flux expulsion, but the actual mechanism might well be another one.', '1404.3194-1-6-6': 'On the other hand lies a volume of work , which have investigated the possibility that the negative effective magnetic pressure instability (NEMPI) is a mechanism of flux concentration and formation of active regions.', '1404.3194-1-6-7': 'In all of them, a small (compared to equipartition) background magnetic field has been imposed in a statistically stationary turbulent magneto-fluid in the presence of gravity; a large-scale instability (namely NEMPI) develops which forms magnetic structures.', '1404.3194-1-7-0': 'The essence of this mechanism is related to a negative contribution of turbulence to the effective magnetic pressure (the sum of non-turbulent and turbulent contributions).', '1404.3194-1-7-1': 'This is caused by a suppression of total (kinetic plus magnetic) turbulent pressure by the large-scale magnetic field.', '1404.3194-1-7-2': 'For large magnetic and fluid Reynolds numbers these turbulent contributions are large enough so that the effective magnetic pressure becomes negative.', '1404.3194-1-7-3': 'This results in the excitation of a large-scale instability, i.e., NEMPI.', '1404.3194-1-7-4': 'The instability is efficient if the background magnetic field is within a specific range, which depends on the relative orientation between gravity and the imposed field.', '1404.3194-1-7-5': 'The maximum flux concentration achievable depends on the nonlinear saturation of NEMPI; unipolar spot-like structures and bipolar active region like structures have been obtained under different circumstances.', '1404.3194-1-7-6': 'We emphasize that turbulence plays a crucial role in formation of those unipolar and bipolar magnetic structures.', '1404.3194-1-7-7': 'This may seem somewhat counterintuitive because in many other cases turbulence increases mixing by enhancing diffusion; but there is no contradiction because there are many examples of pattern formation in reaction-diffusion systems that have been long studied and well understood; see e.g., [CITATION].', '1404.3194-1-8-0': 'A shortcoming, that is common between the NEMPI papers and the radiative magneto-convection papers quoted above is that the magnetic field is imposed externally, either over the whole volume or at the lower boundary.', '1404.3194-1-8-1': 'It is then necessary to investigate how the magnetic flux from dynamo generated magnetic fields can be concentrated to form active regions.', '1404.3194-1-8-2': 'Furthermore, it has been observed that NEMPI is suppressed in the presence of rotation , which is an essential ingredient, together with gravity, to the generation of a large-scale magnetic field by dynamo action.', '1404.3194-1-9-0': 'Hence, it is crucial to study the interaction between NEMPI and large-scale dynamo instabilities.', '1404.3194-1-9-1': 'It turns out that there exists a range of parameters over which it is possible for NEMPI to create magnetic flux concentrations from a dynamo generated magnetic field; evidence in support of this picture has been obtained from both mean-field models and direct numerical simulations .', '1404.3194-1-9-2': 'Particularly interesting features of flux concentration from dynamo generated fields, which has not been studied so far, is the case where the dynamo and the NEMPI does not operate at the same physical location but in different parts of the domain.', '1404.3194-1-9-3': 'In particular, the dynamo operates at the deeper layers of a stratified domain but not in the upper layers whereas in the upper layers NEMPI can operate to produce flux concentrations.', '1404.3194-1-9-4': 'In this paper, we study this problem by direct numerical simulations.', '1404.3194-1-10-0': '# The model', '1404.3194-1-11-0': '## Governing equations', '1404.3194-1-12-0': 'We solve the equations of isothermal magnetohydrodynamics (MHD) for the velocity [MATH], the magnetic vector potential [MATH], and the density [MATH], [EQUATION] where the operator [MATH] denotes the convective derivative, [MATH] is the magnetic field, [MATH] the current density, [MATH] is the traceless rate of strain tensor (the commas denote partial differentiation), [MATH] the kinematic viscosity, [MATH] the magnetic diffusivity, and [MATH] the isothermal sound speed.', '1404.3194-1-12-1': 'In addition, we assume the ideal gas law to hold.', '1404.3194-1-12-2': 'Our domain is a Cartesian box of size [MATH] with [MATH].', '1404.3194-1-12-3': 'Periodic boundary conditions on all dynamical variables are assumed in the horizontal ([MATH]) plane.', '1404.3194-1-12-4': 'The velocity satisfies stress-free, non-penetrating boundary condition at the top and bottom boundaries.', '1404.3194-1-12-5': 'The volume-averaged density is therefore constant in time and equal to its initial value, [MATH].', '1404.3194-1-12-6': 'At the bottom boundary, perfectly conducting boundary conditions are imposed on the magnetic field, which is constrained to have only a vertical component at the top boundary (normal field boundary condition).', '1404.3194-1-12-7': 'The gravitational acceleration [MATH] is chosen such that [MATH], which leads to a density contrast in vertical direction between bottom and top of [MATH]; [MATH] is the density scale height.', '1404.3194-1-13-0': '## Kinetic helicity profile', '1404.3194-1-14-0': 'Turbulence is sustained in the medium by injecting energy through the force [MATH] given by [EQUATION] where [MATH] is the position vector.', '1404.3194-1-14-1': 'On dimensional grounds, we choose [MATH], where [MATH] is a nondimensional forcing amplitude.', '1404.3194-1-14-2': 'At each timestep we select randomly the phase [MATH] and the wavevector [MATH] from many possible wavevectors in a certain range around a given forcing wavenumber, [MATH].', '1404.3194-1-14-3': 'Hence [MATH] is a stochastic process that is white-in-time and is integrated by using the Euler-Marayuma scheme .', '1404.3194-1-14-4': 'The Fourier amplitudes, [EQUATION] where [MATH] characterizes the fractional helicity of [MATH], and [EQUATION] is a non-helical forcing function, and [MATH] is an arbitrary unit vector not aligned with [MATH] and [MATH] is the unit vector along [MATH]; note that [MATH].', '1404.3194-1-14-5': 'By virtue of the helical nature of [MATH], a dynamo develops in the domain .', '1404.3194-1-14-6': 'As we want to separate the domain over which dynamo operates from the domain over which it is possible for magnetic flux concentrations to happen, we choose the fractional helicity of the force [MATH] to go to zero at the top layers of our domain, i.e., for [MATH], v.i.z., [EQUATION]', '1404.3194-1-14-7': 'Here [MATH] is the error function, and [MATH] is a length scale chosen to be [MATH].', '1404.3194-1-14-8': 'We use several different values of [MATH] and [MATH].', '1404.3194-1-15-0': '## Non-dimensional parameters', '1404.3194-1-16-0': 'We choose our units such that [MATH] and [MATH].', '1404.3194-1-16-1': 'Our simulations are characterized by the fluid Reynolds number [MATH], the magnetic Prandtl number [MATH], and the magnetic Reynolds number [MATH].', '1404.3194-1-16-2': 'The magnetic field is expressed in units of [MATH], where [MATH] is the root-mean-square velocity at times much larger than the large-scale eddy turn-over times and [MATH] is the (horizontally and temporally) average density at the middle of the domain.', '1404.3194-1-16-3': 'Time is expressed in eddy turnover times, [MATH].', '1404.3194-1-16-4': 'We often find it useful to consider the turbulent-diffusive timescale, [MATH], where [MATH] is the estimated turbulent magnetic diffusivity.', '1404.3194-1-17-0': 'The simulations are performed with the Pencil Code,', '1404.3194-1-18-0': 'which uses sixth-order explicit finite differences in space and a third-order accurate time stepping method.', '1404.3194-1-18-1': 'We typically use a numerical resolutions of [MATH] mesh points although some representative simulations at higher resolutions are also run.', '1404.3194-1-19-0': '# Results', '1404.3194-1-20-0': 'We have performed a number of runs varying mainly the values of [MATH] and [MATH].', '1404.3194-1-20-1': 'We always used [MATH] and, in most of the cases, we had [MATH] and [MATH], but in one case we also used [MATH] and in another [MATH].', '1404.3194-1-20-2': 'Our runs are summarized in Table [REF].', '1404.3194-1-20-3': 'Let us start by describing in detail one representative simulation among the many we have run; v.i.z., the case of Run B in Table [REF].', '1404.3194-1-20-4': 'In this case, the flow is helically forced up to the height of [MATH] with [MATH].', '1404.3194-1-20-5': 'Above the plane [MATH] the flow is indeed forced, but not helically, i.e., with [MATH].', '1404.3194-1-20-6': 'By virtue of helical forcing from the bottom wall up to the height of [MATH], a dynamo develops.', '1404.3194-1-20-7': 'In fig:ME we show the evolution of the volume averaged magnetic energy, [MATH], defined by [EQUATION]', '1404.3194-1-20-8': 'At short times there is a fast exponential growth of [MATH]; the growth rate, [MATH], is given in Table [REF].', '1404.3194-1-20-9': 'The dynamo saturates at about [MATH], see fig:ME(a).', '1404.3194-1-20-10': 'In fig:ME(b), we show the variation of horizontally averaged (over the [MATH] plane) density, [MATH], kinetic energy, [MATH], magnetic energy, [MATH], and kinetic helicity [MATH] as a function of the height [MATH]; where [MATH] is the vorticity.', '1404.3194-1-20-11': 'It is clear from fig:ME(b) that immediately after dynamo saturation both the kinetic helicity and the magnetic field are largely confined within the domain up to the height [MATH], but not the kinetic energy of turbulence.', '1404.3194-1-21-0': '## Flux emergence at the top surface', '1404.3194-1-22-0': 'As the simulation progresses, at [MATH] magnetic flux of both signs emerges on the top surface.', '1404.3194-1-22-1': 'At first the flux emerges as small-scale fluctuations, but within a time of about [MATH], it self-organizes in a bipolar structure.', '1404.3194-1-22-2': 'The two polarities of the bipolar structure then move away from each other.', '1404.3194-1-22-3': 'This is demonstrated in a series of snapshots shown in fig:emerg.', '1404.3194-1-22-4': 'Here, stratified turbulence gives rise to anti-diffusive properties leading to the formation of bipolar structures.', '1404.3194-1-22-5': 'This is the first remarkable result from our simulations.', '1404.3194-1-23-0': '## Formation of bipolar spot', '1404.3194-1-24-0': 'Due to periodic boundary conditions in the [MATH] and [MATH] directions, the two polarities, while moving away from each other, approach each other across the far end of the periodic domain, come close to each other and form a curious bipolar structure.', '1404.3194-1-24-1': 'This is shown in the series of snapshots in fig:spot, where we have shifted the coordinate system (relative to the one in fig:emerg) so as to have the bipolar spot in the middle of the domain.', '1404.3194-1-25-0': '## Recurrent spot activity', '1404.3194-1-26-0': 'This spot-like structure survives up to [MATH] after which it turns into a bipolar band whose evolution is shown in a series of snapshots in fig:band.', '1404.3194-1-26-1': 'At about [MATH] the band dissolves; i.e., the flux at the top surface is close to zero.', '1404.3194-1-26-2': 'And a little while later the band-like structure reappears at a different position on the top surface and with time evolves to a spot-like structure similar to the one shown in fig:spot; compare the last snapshot shown in fig:band with that of fig:spot.', '1404.3194-1-27-0': '## How generic are the observed magnetic structures?', '1404.3194-1-28-0': 'To summarise, in this simulation, Run B, the normal magnetic field at the top surface shows three principal qualitative features: (a) flux emergence, (b) formation of bipolar structures (spots and bands) and (c) a recurrent but not exactly periodic appearance of the bipolar structures.', '1404.3194-1-28-1': 'How typical are these qualitative behaviours with respect to variation of various parameters of our simulation?', '1404.3194-1-28-2': 'This question is addressed in the following manner: (a) We run a simulation, Run O-02, with the same parameters of Run B but with a different fractional helicity, [MATH].', '1404.3194-1-28-3': 'For this run, the helical dynamo instability is excited at a slower rate and the magnetic flux emergence at the top surface happens at a later time, nevertheless the same qualitative feature of bipolar magnetic structures are observed.', '1404.3194-1-28-4': '(b) Keeping the value of fractional helicity, [MATH], to be constant, we vary the height of the dynamo region, [MATH] from [MATH] (Run B) to [MATH] (Run C), [MATH] (Run O-1), and [MATH] (Run A).', '1404.3194-1-28-5': 'The flux emergence happens at different times; for higher [MATH] the flux emergence is faster.', '1404.3194-1-28-6': 'Other than this quantitative change, there is no qualitative change to our results.', '1404.3194-1-28-7': '(c) We run a simulation Run E with the same parameters as Run C, but with bigger resolution ([MATH]) and higher Reynolds number and obtain the same qualitative behaviour.', '1404.3194-1-28-8': 'In another simulation, Run D, we keep all the parameters the same as Run E, except for the forcing wavenumber, [MATH], and obtain the same qualitative behaviour.', '1404.3194-1-28-9': '(d) Finally, we note that gravity plays a crucial role, in simulations without gravity ([MATH]) or even [MATH] (Run B/2), no sharp magnetic structures are seen.', '1404.3194-1-28-10': 'Instead the magnetic field at the top has the same length scales as the dynamo-generated magnetic field at the bottom part of the domain as demonstrated in fig:runBby2_2surf.', '1404.3194-1-28-11': 'It is also clear from our results that the bipolar magnetic structures are strongly influenced by the periodicity of our domain.', '1404.3194-1-28-12': 'Is it possible to obtain similar structures, but at different length scales (relative to the box size) and in a larger domain?', '1404.3194-1-28-13': 'By running a simulation with double the box size ([MATH]) we have found that the characteristic length scales of the bipolar structures scaled by the box-size remains the same.', '1404.3194-1-29-0': '## Sharp bipolar structures', '1404.3194-1-30-0': 'A particularly interesting aspect of these simulations is the formation of bipolar magnetic structures with sharp edges, examples of which are fig:spot or fig:band.', '1404.3194-1-30-1': 'To document the characteristic length scale appearing in magnetic structures, we plot in fig:ang_spec the shell-averaged Fourier spectrum of [MATH] at the top surface at different times corresponding to the snapshots in fig:band.', '1404.3194-1-30-2': 'The plot demonstrates that, to represent the sharp structures, e.g., in the last snapshot in fig:band, a band of Fourier modes ranging up to [MATH] and [MATH] is necessary.', '1404.3194-1-30-3': 'This also underscores the necessity of having a large scale separation ([MATH]) to see these magnetic structures.', '1404.3194-1-30-4': 'Furthermore, we find that at large [MATH] the spectra can be well approximated by a [MATH] power law.', '1404.3194-1-31-0': 'To take a closer look at the bipolar structure, we show in fig:spot3d the spot-like structure from Run A plotted together with the magnetic field lines in a three-dimensional representation.', '1404.3194-1-31-1': 'The magnetic field lines of opposite orientation approach each other with height and merge into a single sharp spot-like structure.', '1404.3194-1-31-2': 'This magnetic structure leaves a clear signature on the velocity field as we demonstrate in fig:spot_vel by plotting the contours of the vertical component of [MATH] overlaid with the horizontal components of velocity as arrows from Run A.', '1404.3194-1-32-0': '## Can NEMPI describe our numerical results?', '1404.3194-1-33-0': 'Let us now try to understand the flux emergence and the formation of bipolar structure.', '1404.3194-1-33-1': 'This falls in the general class of pattern formation in turbulent systems.', '1404.3194-1-33-2': 'A theoretical technique to describe this general class of problem is the mean-field theory where we average over the turbulent state to derive a set of mean-field equations.', '1404.3194-1-33-3': 'The problem of pattern formation then becomes a problem of studying the instabilities using the mean-field equations.', '1404.3194-1-33-4': 'A well-known example, pioneered by [CITATION] and [CITATION] is that of dynamo theory where the mean-field theory is applied to the induction equation [CITATION].', '1404.3194-1-33-5': 'A recent example of an application of this method to understand magneto-rotational instability in the presence of small scale turbulence is by [CITATION].', '1404.3194-1-34-0': 'For the present problem, we need to average the momentum equation over the statistics of turbulence.', '1404.3194-1-34-1': 'As a result of such an averaging, a new term (describing the turbulent contributions) will be added to the large-scale magnetic pressure term .', '1404.3194-1-34-2': 'It has been shown that the effective magnetic pressure that is the sum of non-turbulent and turbulent (new term) contributions, can be negative in the presence of a background magnetic field which, in this problem, will be provided by the dynamo.', '1404.3194-1-35-0': 'From symmetry arguments, such a term can be constructed by using the background magnetic field and gravity.', '1404.3194-1-35-1': 'In the two extreme cases: one in which the gravity and the background magnetic field are perpendicular to each other , and the second in which gravity and the background magnetic field are parallel to each other , the analysis of the instability simplifies.', '1404.3194-1-35-2': 'Unfortunately, the problem is more complicated in the present case where all the three components of magnetic field are present, and a systematic determination from DNS for the new transport coefficients in the effective magnetic pressure has not yet been performed.', '1404.3194-1-35-3': 'Nevertheless there are two signatures of NEMPI that we look for.', '1404.3194-1-35-4': 'Firstly, we know that NEMPI can operate effectively to form flux concentration if the background magnetic field is neither too large or too small, within [MATH] to [MATH] when normalized by the equipartition magnetic field.', '1404.3194-1-35-5': 'We find that this condition is satisfied near the top surface when the first flux emergence occurs as shown in fig:beq_diff_timesa, but not at later stages as shown in fig:beq_diff_timesb.', '1404.3194-1-35-6': 'What is then the mechanism behind the disappearance and reappearance of the magnetic flux at the top surface?', '1404.3194-1-35-7': 'A clue to this puzzle is the fact that within mean-field theory the dynamo operating in the lower-layers of the computational domain can be interpreted as an [MATH] dynamo, where [MATH], where [MATH] and [MATH] are fluctuations.', '1404.3194-1-35-8': 'An [MATH] dynamo for which [MATH] varies within the domain can give rise to dynamo waves , and indeed such dynamo waves are seen in our simulations as shown in the space-time diagram in fig:bxby_bfly.', '1404.3194-1-36-0': 'The second signature of NEMPI is its ability to generate large-scale flows, especially in the case of vertical magnetic field .', '1404.3194-1-36-1': 'Since NEMPI creates regions of negative effective magnetic pressure, it is often accompanied by a converging flow at the surface and a downward flow on and immediately below the location of flux concentration.', '1404.3194-1-36-2': 'In the present simulations, due to the presence of strong turbulent fluctuations we have not been able to detect any such coherent flow, although some evidence in support of such a flow has been found in Fourier filtered velocity field as shown in fig:fourier_flow.', '1404.3194-1-37-0': '# Conclusion', '1404.3194-1-38-0': 'To conclude, in this paper, we have shown that it is possible to generate intense structures of vertical magnetic flux at the top surface of DNS of a density-stratified turbulent dynamo.', '1404.3194-1-38-1': 'Furthermore, a rich dynamic behaviour of the magnetic flux (v.i.z., bipolar spot-like structures appear, then morph into bipolar band-like structures which disappear and reappear at a different place and at a later stage evolve into spot-like structures), is observed.', '1404.3194-1-38-2': 'The characteristic length and time scales of the magnetic flux formed at the top surface are much smaller than the characteristic length scale (and time scale) of the dynamo-generated magnetic field.', '1404.3194-1-38-3': 'The necessary conditions are strong stratification, presence of turbulence, and large scale separation, which is [MATH] in the DNS we present here.', '1404.3194-1-38-4': 'Clearly, there is a mechanism at work here that can concentrate a weak large-scale magnetic field to strong magnetic flux of smaller scale.', '1404.3194-1-38-5': 'Could this mechanism be NEMPI?', '1404.3194-1-38-6': 'At present, we cannot provide a definitive answer to this question, although we do show that the necessary conditions for NEMPI to operate are satisfied during the first emergence of flux at the top surface.', '1404.3194-1-39-0': 'How relevant are our result in understanding the formation of active regions and sunspots?', '1404.3194-1-39-1': 'Unlike the works by e.g., [CITATION] or [CITATION], our simulations do not include radiative convection; turbulence is generated by external forcing.', '1404.3194-1-39-2': 'This should not necessarily be considered a shortcoming of our simulations as the aim of our work has been to present the simplest model that can show formation of bipolar structures from a large-scale dynamo.', '1404.3194-1-39-3': 'Furthermore, this is the first time bipolar structures are found to appear in simulations where the magnetic field is not imposed - as is the case in [CITATION], [CITATION], or [CITATION] - but it is self-consistently generated from a turbulent dynamo.', '1404.3194-1-40-0': 'To end on a positive note, the most remarkable feature of these simulations is that a minimalistic setup consisting solely of stratification and helically forced turbulence could generate such diverse spatio-temporal behaviour.', '1404.3194-1-40-1': 'Could a mean-field model consisting of both dynamo equations and equations describing NEMPI capture such behaviour?', '1404.3194-1-40-2': 'This question will be the subject of future investigations.'}

{'1404.3194-2-0-0': 'We perform direct numerical simulations of the equations of magnetohydrodynamics with external random forcing and in the presence of gravity.', '1404.3194-2-0-1': 'The domain is divided into two parts: a lower layer where the forcing is helical and an upper layer where the helicity of the forcing is zero with a smooth transition in between.', '1404.3194-2-0-2': 'At early times, a large-scale helical dynamo develops in the bottom layer.', '1404.3194-2-0-3': 'At later times the dynamo saturates, but the vertical magnetic field continues to develop and rises to form dynamic bipolar structures at the top, which later disappear and reappear.', '1404.3194-2-0-4': 'Some of the structures look similar to [MATH] spots observed in the Sun.', '1404.3194-2-0-5': 'This is the first example of magnetic flux concentrations, owing to strong density stratification, from self-consistent dynamo simulations that generate bipolar, super-equipartition strength, magnetic structures whose energy density can exceeds the turbulent kinetic energy by even a factor of ten.', '1404.3194-2-1-0': '# Introduction', '1404.3194-2-2-0': 'The most striking and also the most observed magnetic features of the Sun are the sunspots and active regions.', '1404.3194-2-2-1': 'The number of sunspots, the strength of the magnetic field in sunspots, and the magnetic field calculated at the surface of the Sun are often taken as proxies of the solar magnetic field deep inside.', '1404.3194-2-2-2': 'There is general agreement that the evolution of the solar magnetic field is governed by the solar dynamo which operates in the convection zone of the Sun.', '1404.3194-2-2-3': 'This brings us to the question, how is the magnetic field generated by the solar dynamo related to the magnetic field observed at the surface of the Sun?', '1404.3194-2-2-4': 'At present, this question does not have a clear answer.', '1404.3194-2-3-0': 'The conventional picture [CITATION] is that the solar dynamo generates a strong toroidal magnetic field in the form of flux tubes at the bottom of the convection zone, also called the tachocline.', '1404.3194-2-3-1': 'This strong magnetic field is buoyant and hence rises up to eventually penetrate through the surface layers of the Sun to create bipolar regions at the surface.', '1404.3194-2-3-2': "During its rise through the convection zone, the magnetic flux tube is twisted by the Coriolis force to give rise to a preferential tilt of the bipolar regions with respect to the equator - which is also known as Joy's law.", '1404.3194-2-4-0': 'The traditional picture is prone to criticism on several counts.', '1404.3194-2-4-1': '(a) Recent numerical simulations of rotating spherical magneto-convection have shown that a solar-like dynamo can operate in the bulk of the convection zone, even without a tachocline.', '1404.3194-2-4-2': '(b) Is it possible for a magnetic flux tube to rise coherently through the turbulent convection zone and still remain anchored to the tachocline?', '1404.3194-2-4-3': 'Numerical simulations of [CITATION], admittedly at moderate magnetic Reynolds numbers, have found no evidence that this is possible.', '1404.3194-2-4-4': 'Recent simulations by [CITATION] and [CITATION] do find flux loops rising from mid depths of the convection zone, but in contrast to the traditional picture, they are not anchored at the bottom of the convection zone.', '1404.3194-2-4-5': '(c) As the flux tube rises, the magnetic field weakens, so even the traditional picture must invoke a re-amplification process near the surface.', '1404.3194-2-4-6': 'For example, [CITATION] postulated downdrafts "to operate beneath the sunspot to account for the gathering of flux to form a sunspot."', '1404.3194-2-4-7': 'Furthermore, current flux emergence simulations that include a photosphere do show such re-amplification, but the mechanism responsible for the re-amplification process remains unknown.', '1404.3194-2-4-8': '(d) A natural corollary of the rising flux tube picture is that the active regions will emerge with preferential orientation at the surface of the Sun, whereas recent observational analysis shows that active regions actually emerge with random orientations but get preferentially oriented as time progresses.', '1404.3194-2-4-9': 'Note nevertheless that [CITATION] have attempted to explain this discrepency within the framework of the conventional scenario by arguing that departures from a preferred orientation are due to turbulent convection and are restored past the emergence.', '1404.3194-2-5-0': 'In the last decade, an alternative scenario has emerged.', '1404.3194-2-5-1': 'In this scenario, first suggested by [CITATION], the turbulent dynamo generates magnetic field in the bulk of the convection zone.', '1404.3194-2-5-2': 'In the near-surface shear layer, that has been observed in helioseismology , the dynamo-generated magnetic field propagates equatorward, satisfying the Parker-Yoshimura rule .', '1404.3194-2-5-3': "The observed preferential orientation of the active regions, the Joy's law, can be understood as an effect of the shear .", '1404.3194-2-5-4': 'In this scenario, which admittedly is yet to be supported by direct numerical simulations, although mean-field calculations do provide support , the active regions must form from a dynamo-generated large-scale magnetic field by the process of magnetic flux concentration operating at or near the surface of the Sun.', '1404.3194-2-5-5': 'This process may be the same re-amplification process necessary in the conventional scenario.', '1404.3194-2-6-0': 'There have been two different, mutually complimentary, approaches to understand this process.', '1404.3194-2-6-1': 'On the one hand lies the numerical simulations by [CITATION], [CITATION], [CITATION], and [CITATION] who solve radiative magneto-convection in a Cartesian domain under a simplified setup (non-rotating, no large-scale shear).', '1404.3194-2-6-2': 'All these simulations develop a bipolar magnetic structure at the top surface, but in all the cases the velocity and the magnetic field at the bottom boundary need to be carefully imposed.', '1404.3194-2-6-3': 'Furthermore, in these simulations, with the exception of [CITATION], the mechanism responsible for formation of magnetic structures has not been elucidated.', '1404.3194-2-6-4': 'Another related example are the magneto-convection simulations of [CITATION], where an imposed vertical field segregates into magnetized and unmagnetized regions.', '1404.3194-2-6-5': 'The authors ascribe this to the effect of flux expulsion, but the actual mechanism might well be another one.', '1404.3194-2-6-6': 'On the other hand lies a volume of work , which have investigated the possibility that the negative effective magnetic pressure instability (NEMPI) is a mechanism of flux concentration and formation of active regions.', '1404.3194-2-6-7': 'In all of them, a small (compared to equipartition) background magnetic field has been imposed in a statistically stationary turbulent magneto-fluid in the presence of gravity; a large-scale instability (namely NEMPI) develops which forms magnetic structures.', '1404.3194-2-7-0': 'The essence of this mechanism is related to a negative contribution of turbulence to the effective magnetic pressure (the sum of non-turbulent and turbulent contributions).', '1404.3194-2-7-1': 'This is caused by a suppression of total (kinetic plus magnetic) turbulent pressure by the large-scale magnetic field.', '1404.3194-2-7-2': 'For large magnetic and fluid Reynolds numbers these turbulent contributions are large enough so that the effective magnetic pressure becomes negative.', '1404.3194-2-7-3': 'This results in the excitation of a large-scale instability, i.e., NEMPI.', '1404.3194-2-7-4': 'The instability is efficient if the background magnetic field is within a specific range, which depends on the relative orientation between gravity and the imposed field.', '1404.3194-2-7-5': 'The maximum flux concentration achievable depends on the nonlinear saturation of NEMPI; unipolar spot-like structures and bipolar active region-like structures have been obtained under different circumstances.', '1404.3194-2-7-6': 'We emphasize that turbulence plays a crucial role in the formation of those unipolar and bipolar magnetic structures.', '1404.3194-2-7-7': 'This may seem somewhat counterintuitive because in many other cases turbulence increases mixing by enhancing diffusion.', '1404.3194-2-7-8': 'However, there is no contradiction because there are many examples of pattern formation in reaction-diffusion systems that have been long studied and well understood; see, e.g., [CITATION] for a review.', '1404.3194-2-8-0': 'A shortcoming, that is common between the NEMPI papers and the radiative magneto-convection papers quoted above is that the magnetic field is imposed externally, either over the whole volume or at the lower boundary.', '1404.3194-2-8-1': 'It is then necessary to investigate how the magnetic flux from dynamo-generated magnetic fields can be concentrated to form active regions.', '1404.3194-2-8-2': 'Furthermore, it has been observed that NEMPI is suppressed in the presence of rotation , which is an essential ingredient, together with gravity, to the generation of a large-scale magnetic field by dynamo action.', '1404.3194-2-9-0': 'Hence, it is crucial to study the interaction between NEMPI and large-scale dynamo instabilities.', '1404.3194-2-9-1': 'It turns out that there exists a range of parameters over which it is possible for NEMPI to create magnetic flux concentrations from a dynamo-generated magnetic field; evidence in support of this picture has been obtained from both mean-field models and direct numerical simulations .', '1404.3194-2-9-2': 'Particularly interesting cases of flux concentration from dynamo-generated fields, which have not been studied so far, are those where dynamo and NEMPI do not operate at the same physical location, but in different parts of the domain.', '1404.3194-2-9-3': 'For example, the dynamo may operate in the deeper layers of a stratified domain but not in the upper layers, whereas in the upper layers NEMPI can operate to produce flux concentrations.', '1404.3194-2-9-4': 'In this paper, we study this problem by direct numerical simulations.', '1404.3194-2-10-0': '# The model', '1404.3194-2-11-0': '## Governing equations', '1404.3194-2-12-0': 'We solve the equations of isothermal magnetohydrodynamics (MHD) for the velocity [MATH], the magnetic vector potential [MATH], and the density [MATH], [EQUATION] where the operator [MATH] denotes the convective derivative, [MATH] is the magnetic field, [MATH] the current density, [MATH] is the traceless rate of strain tensor (the commas denote partial differentiation), [MATH] the kinematic viscosity, [MATH] the magnetic diffusivity, and [MATH] the isothermal sound speed.', '1404.3194-2-12-1': 'In addition, we assume the ideal gas law to hold.', '1404.3194-2-12-2': 'Our domain is a Cartesian box of size [MATH] with [MATH].', '1404.3194-2-12-3': 'Periodic boundary conditions on all dynamical variables are assumed in the horizontal ([MATH]) plane.', '1404.3194-2-12-4': 'The velocity satisfies stress-free, non-penetrating boundary condition at the top and bottom boundaries.', '1404.3194-2-12-5': 'The volume-averaged density is therefore constant in time and equal to its initial value.', '1404.3194-2-12-6': 'At the bottom, perfectly conducting boundary conditions are imposed on the magnetic field, which is constrained to have only a vertical component at the top boundary (normal field boundary condition).', '1404.3194-2-12-7': 'The gravitational acceleration [MATH] is chosen such that [MATH], which leads to a density contrast in the vertical direction between bottom and top of [MATH].', '1404.3194-2-12-8': 'Here [MATH] is the density scale height.', '1404.3194-2-13-0': '## Forced turbulence', '1404.3194-2-14-0': 'Turbulence is sustained in the medium by injecting energy through the function [MATH] given by [EQUATION] where [MATH] is the position vector.', '1404.3194-2-14-1': 'On dimensional grounds, we choose [MATH], where [MATH] is a nondimensional forcing amplitude.', '1404.3194-2-14-2': 'At each timestep we select randomly the phase [MATH] and the wavevector [MATH] from many possible wavevectors in a certain range around a given forcing wavenumber, [MATH].', '1404.3194-2-14-3': 'Hence [MATH] is a stochastic process that is white-in-time and is integrated by using the Euler-Marayuma scheme .', '1404.3194-2-14-4': 'The Fourier amplitudes, [EQUATION] where [MATH] characterizes the fractional helicity of [MATH], and [EQUATION] is a non-helical forcing function, and [MATH] is an arbitrary unit vector not aligned with [MATH] and [MATH] is the unit vector along [MATH]; note that [MATH].', '1404.3194-2-14-5': 'By virtue of the helical nature of [MATH], a dynamo develops in the domain .', '1404.3194-2-14-6': 'As we want to separate the domain over which dynamo operates from the domain over which it is possible for magnetic flux concentrations to happen, we choose the fractional helicity of the force [MATH] to go to zero at the top layers of our domain, i.e., for [MATH], v.i.z., [EQUATION]', '1404.3194-2-14-7': 'Here [MATH] is the error function, and [MATH] is a length scale chosen to be [MATH].', '1404.3194-2-14-8': 'We use several different values of [MATH] and [MATH].', '1404.3194-2-15-0': '## Non-dimensional parameters', '1404.3194-2-16-0': 'We choose our units such that [MATH] and [MATH].', '1404.3194-2-16-1': 'Our simulations are characterized by the fluid Reynolds number [MATH], the magnetic Prandtl number [MATH], and the magnetic Reynolds number [MATH].', '1404.3194-2-16-2': 'The magnetic field is expressed in units of [MATH].', '1404.3194-2-16-3': 'As the value of the turbulent velocity is set by the local strength of the forcing, which is uniform, the turbulent velocity is also statistically uniform over depth, and therefore we choose to define [MATH] as the root-mean-square velocity based on a volume average in the statistically steady state.', '1404.3194-2-16-4': 'On the other hand, the density varies over several orders of magnitude as a function of depth and hence we choose [MATH] as the horizontally and temporally average density at [MATH], which is the middle of the domain.', '1404.3194-2-16-5': 'Time is expressed in eddy turnover times, [MATH].', '1404.3194-2-16-6': 'We often find it useful to consider the turbulent-diffusive timescale, [MATH], where [MATH] is the estimated turbulent magnetic diffusivity.', '1404.3194-2-17-0': 'The simulations are performed with the Pencil Code,', '1404.3194-2-18-0': 'which uses sixth-order explicit finite differences in space and a third-order accurate time stepping method.', '1404.3194-2-18-1': 'We typically use a numerical resolutions of [MATH] mesh points, although some representative simulations at higher resolutions are also run.', '1404.3194-2-19-0': '# Results', '1404.3194-2-20-0': 'We have performed a number of runs varying mainly the values of [MATH] and [MATH].', '1404.3194-2-20-1': 'We always used [MATH] and, in most of the cases, we had [MATH] and [MATH], but in one case we also used [MATH] and in another [MATH].', '1404.3194-2-20-2': 'Our runs are summarized in Table [REF].', '1404.3194-2-20-3': 'Let us start by describing in detail one representative simulation among the many we have run; v.i.z., the case of Run B in Table [REF].', '1404.3194-2-20-4': 'In this case, the flow is helically forced up to the height of [MATH] with [MATH].', '1404.3194-2-20-5': 'Above the plane [MATH] the flow is indeed forced, but not helically, i.e., with [MATH].', '1404.3194-2-20-6': 'By virtue of helical forcing from the bottom wall up to the height of [MATH], a dynamo develops.', '1404.3194-2-20-7': 'In fig:ME we show the evolution of the volume averaged magnetic energy, [MATH], defined by [EQUATION]', '1404.3194-2-20-8': 'At short times there is a fast exponential growth of [MATH]; the growth rate, [MATH], is given in Table [REF].', '1404.3194-2-20-9': 'The dynamo saturates at about [MATH], see fig:ME(a).', '1404.3194-2-20-10': 'In fig:ME(b), we show the variation of horizontally averaged (over the [MATH] plane) density [MATH], mean squared velocity [MATH], magnetic energy [MATH], and kinetic helicity [MATH] as a function of the height [MATH], where [MATH] is the vorticity.', '1404.3194-2-20-11': 'It is clear from fig:ME(b) that immediately after dynamo saturation, both the kinetic helicity and the magnetic field are largely confined within the domain up to the height [MATH], but not the kinetic energy of the turbulence.', '1404.3194-2-20-12': 'Furthermore, in the deep parts of the domain, the horizontally averaged magnetic energy density is approximately proportional to density and thus to the local equipartition value, [MATH].', '1404.3194-2-21-0': '## Flux emergence at the top surface', '1404.3194-2-22-0': 'As the simulation progresses, at [MATH], magnetic flux of both signs emerges on the top surface.', '1404.3194-2-22-1': 'At first the flux emerges as small-scale fluctuations, but within a time of about [MATH], it self-organizes to a bipolar structure.', '1404.3194-2-22-2': 'The two polarities of the bipolar structure then move away from each other.', '1404.3194-2-22-3': 'This is demonstrated in a series of snapshots shown in fig:emerg.', '1404.3194-2-22-4': 'Here, stratified turbulence gives rise to anti-diffusive properties leading to the formation of bipolar structures.', '1404.3194-2-22-5': 'This is the first remarkable result from our simulations.', '1404.3194-2-22-6': 'Similar behaviour has been seen by [CITATION], although not in self-consistent dynamo simulations but in simulations where the magnetic field at the bottom boundary was imposed in the upwellings.', '1404.3194-2-22-7': 'Furthermore, the self-organization we observe is not driven by radiative convection, as in the simulations of [CITATION] but by forced isothermal turbulent flows.', '1404.3194-2-23-0': '## Formation of an intense bipolar structure', '1404.3194-2-24-0': 'Due to periodic boundary conditions in the [MATH] and [MATH] directions, the two polarities, while moving away from each other, approach each other across the far end of the periodic domain, come close to each other and form a curious bipolar structure, reminiscent of the so-called [MATH] spots .', '1404.3194-2-24-1': 'The [MATH] component of the magnetic field is close to three times [MATH].', '1404.3194-2-24-2': 'This is shown in a series of snapshots in fig:spot, where we have shifted the coordinate system relative to the one in fig:emerg so as to have the bipolar structure in the middle of the top surface.', '1404.3194-2-24-3': 'As we are using periodic boundary conditions along the horizontal directions, we are free to make such a shift.', '1404.3194-2-24-4': 'To illustrate this, we show in fig:periodic the magnetic field at the top of our computational domain in a box that is extended periodically to three times its originally size in both the [MATH] and [MATH] directions.', '1404.3194-2-25-0': '## Recurrent spot activity', '1404.3194-2-26-0': 'This spot-like structure survives up to [MATH], after which it turns into a bipolar band whose evolution is shown in a series of snapshots in fig:band.', '1404.3194-2-26-1': 'At about [MATH] the band dissolves and the field at the top surface is close to zero.', '1404.3194-2-26-2': 'And a little while later the band-like structure reappears at a different position on the top surface and with time evolves to a spot-like structure similar to the one shown in fig:spot; compare the last snapshot shown in fig:band with that of fig:spot.', '1404.3194-2-27-0': '## How generic are the observed magnetic structures?', '1404.3194-2-28-0': 'To summarise, in this simulation, Run B, the normal magnetic field at the top surface shows three principal qualitative features: (a) flux emergence, (b) formation of bipolar structures (spots and bands) and (c) a recurrent but not exactly periodic appearance of the bipolar structures.', '1404.3194-2-28-1': 'How typical are these qualitative behaviours with respect to variation of various parameters of our simulation?', '1404.3194-2-28-2': 'This question is addressed in the following manner: (a) We run a simulation, Run O-02, with the same parameters of Run B but with a different fractional helicity, [MATH].', '1404.3194-2-28-3': 'For this run, the helical dynamo instability is excited at a slower rate and the magnetic flux emergence at the top surface happens at a later time, nevertheless the same qualitative feature of bipolar magnetic structures are observed.', '1404.3194-2-28-4': '(b) Keeping the value of fractional helicity, [MATH], to be constant, we vary the height of the dynamo region, [MATH] from [MATH] (Run B) to [MATH] (Run C), [MATH] (Run O-1), and [MATH] (Run A).', '1404.3194-2-28-5': 'The flux emergence happens at different times; for higher [MATH] the flux emergence is faster.', '1404.3194-2-28-6': 'Other than this quantitative change, there is no qualitative change to our results.', '1404.3194-2-28-7': '(c) We run a simulation Run E with the same parameters as Run C, but with bigger resolution ([MATH]) and higher Reynolds number and obtain the same qualitative behaviour.', '1404.3194-2-28-8': 'In another simulation, Run D, we keep all the parameters the same as Run E, except for the forcing wavenumber, [MATH], and obtain the same qualitative behaviour.', '1404.3194-2-28-9': '(d) Finally, we note that gravity plays a crucial role.', '1404.3194-2-28-10': 'In simulations without gravity ([MATH]) or even [MATH] (Run B/2), no sharp magnetic structures are seen.', '1404.3194-2-28-11': 'Instead the magnetic field at the top has the same length scales as the dynamo-generated magnetic field at the bottom part of the domain, as demonstrated in fig:runBby2_2surf.', '1404.3194-2-28-12': 'It is also clear from our results that the bipolar magnetic structures are strongly influenced by the periodicity of our domain.', '1404.3194-2-28-13': 'Is it possible to obtain similar structures, but at different length scales (relative to the box size) and in a larger domain?', '1404.3194-2-28-14': 'By running a simulation with double the box size ([MATH]) we have found that the characteristic length scales of the bipolar structures scaled by the box-size remains the same.', '1404.3194-2-28-15': 'This is because in our periodic geometry, the scale of the large-scale dynamo is always the largest possible one that fits into the domain.', '1404.3194-2-28-16': 'In future work, it is therefore important to relax this constraint arising from periodic boundary conditions using, for example, spherical geometry.', '1404.3194-2-29-0': '## Sharp bipolar structures', '1404.3194-2-30-0': 'A particularly interesting aspect of these simulations is the formation of bipolar magnetic structures with sharp edges, examples of which are fig:spot or fig:band.', '1404.3194-2-30-1': 'To document the characteristic length scale appearing in magnetic structures, we plot in fig:ang_spec the angle-averaged Fourier spectrum of [MATH] at the top surface at different times corresponding to the snapshots in fig:band.', '1404.3194-2-30-2': 'The plot demonstrates that, to represent the sharp structures, e.g., in the last snapshot in fig:band, Fourier modes up to [MATH] and [MATH] are necessary.', '1404.3194-2-30-3': 'This also underscores the necessity of having a large scale separation ([MATH]) to see these magnetic structures.', '1404.3194-2-30-4': 'Furthermore, we find that at large [MATH], the spectra can be approximated by a [MATH] power law.', '1404.3194-2-31-0': 'To take a closer look at the bipolar structure, we show in fig:spot3d the spot-like structure from Run A plotted together with the magnetic field lines in a three-dimensional representation.', '1404.3194-2-31-1': 'The magnetic field lines of opposite orientation approach each other with height and merge into a single sharp spot-like structure.', '1404.3194-2-31-2': 'This magnetic structure leaves a clear signature on the velocity field as we demonstrate in fig:spot_vel by plotting the contours of the vertical component of [MATH] overlaid with the horizontal components of velocity as arrows from Run A.', '1404.3194-2-32-0': '## Can NEMPI describe our numerical results?', '1404.3194-2-33-0': 'Let us now try to understand the flux emergence and the formation of bipolar structure.', '1404.3194-2-33-1': 'This falls in the general class of pattern formation in turbulent systems.', '1404.3194-2-33-2': 'A theoretical technique to describe this general class of problems is the mean-field theory where we average over the turbulent state to derive a set of mean-field equations.', '1404.3194-2-33-3': 'The problem of pattern formation then becomes a problem of studying the instabilities using the mean-field equations.', '1404.3194-2-33-4': 'A well-known example, pioneered by [CITATION] and [CITATION] is that of dynamo theory where the mean-field theory is applied to the induction equation [CITATION].', '1404.3194-2-33-5': 'A recent example of an application of this method to understand magneto-rotational instability in the presence of small-scale turbulence is by [CITATION].', '1404.3194-2-34-0': 'For the present problem, we need to average the momentum equation over the statistics of turbulence.', '1404.3194-2-34-1': 'As a result of such an averaging, a new term (describing the turbulent contributions) will be added to the large-scale magnetic pressure term .', '1404.3194-2-34-2': 'It has been shown that the effective magnetic pressure that is the sum of non-turbulent and turbulent (new term) contributions, can be negative in the presence of a background magnetic field which, in this problem, will be provided by the dynamo.', '1404.3194-2-35-0': 'From symmetry arguments, such a term can be constructed using the background magnetic field and gravity.', '1404.3194-2-35-1': 'In the two extreme cases: one in which the gravity and the background magnetic field are perpendicular to each other , and the second in which gravity and the background magnetic field are parallel to each other , the analysis of the instability simplifies.', '1404.3194-2-35-2': 'Unfortunately, the problem is more complicated in the present case where all the three components of magnetic field are present.', '1404.3194-2-35-3': 'In that case, a systematic determination of the new transport coefficients in the effective magnetic pressure, using direct numerical simulations (DNS), has not yet been performed.', '1404.3194-2-35-4': 'Nevertheless there are two signatures of NEMPI that we look for.', '1404.3194-2-35-5': 'Firstly, we know the effective magnetic pressure is negative only when the background magnetic field is neither too large or too small, within [MATH] to [MATH] when normalized by the equipartition magnetic field .', '1404.3194-2-35-6': 'We find that this condition is satisfied near the top surface when the first flux emergence occurs, as shown in fig:beq_diff_timesa, but not at later stages as shown in fig:beq_diff_timesb.', '1404.3194-2-35-7': 'What is then the mechanism behind the disappearance and reappearance of the magnetic flux at the top surface?', '1404.3194-2-35-8': 'A clue to this puzzle is the fact that within mean-field theory the dynamo operating in the lower layers of the computational domain can be interpreted as an [MATH] dynamo, where [MATH], where [MATH] and [MATH] are fluctuations.', '1404.3194-2-35-9': 'An [MATH] dynamo for which [MATH] varies within the domain can give rise to dynamo waves , and indeed such dynamo waves are seen in our simulations as shown in the space-time diagram in fig:bxby_bfly.', '1404.3194-2-36-0': 'The second signature of NEMPI is its ability to generate large-scale flows; since NEMPI creates regions of negative effective magnetic pressure, it is often accompanied by a converging flow at the surface and a downward flow on and immediately below the location of flux concentration.', '1404.3194-2-36-1': 'In our simulations, due to the presence of strong turbulent fluctuations, we have not been able to detect any such coherent flow, although some evidence in support of such a flow has been found in the Fourier filtered velocity field as shown in fig:fourier_flow.', '1404.3194-2-36-2': 'Interestingly, similar downflows are also seen in recent simulations by [CITATION], who inject a [MATH] flux tube at the bottom of a solar convection simulation and let it rise to the surface.', '1404.3194-2-36-3': 'Although the emergence process itself is associated with upflows, their results show downflows at the late stages of the flux concentration process.', '1404.3194-2-36-4': 'In such simulations that attempt to be realistic, it is not possible to attribute the observed downflows to one single mechanism.', '1404.3194-2-36-5': 'By contrast, in our simple setup it is likely that NEMPI is indeed the mechanism responsible for generating the downward flow.', '1404.3194-2-37-0': '# Conclusion', '1404.3194-2-38-0': 'To conclude, in this paper, we have shown that it is possible to generate intense structures of vertical magnetic field at the top surface of DNS of a density-stratified turbulent dynamo.', '1404.3194-2-38-1': 'Furthermore, a rich dynamic behaviour of the magnetic field is observed: bipolar spot-like structures appear, then morph into bipolar band-like structures which disappear and reappear at a different place and at a later stage evolve into spot-like structures.', '1404.3194-2-38-2': 'Such structures are similar to [MATH] spots and tend to show anticlockwise rotation, which is consistent with the fact that the kinetic helicity in our simulations is positive.', '1404.3194-2-39-0': 'The characteristic length and time scales of the magnetic field formed at the top surface are much smaller than the characteristic length scale (and time scale) of the dynamo-generated magnetic field.', '1404.3194-2-39-1': 'The necessary conditions are strong stratification, presence of turbulence, and large scale separation, which is at least [MATH] in the DNS we present here.', '1404.3194-2-39-2': 'Clearly, there is a mechanism at work here that can concentrate a weak large-scale magnetic field to strong magnetic field of smaller scale.', '1404.3194-2-39-3': 'Could this mechanism be NEMPI?', '1404.3194-2-39-4': 'At present, we cannot provide a definitive answer to this question, although we do show that the necessary conditions for NEMPI to operate are satisfied during the first emergence of flux at the top surface.', '1404.3194-2-40-0': 'How relevant are our result in understanding the formation of active regions and sunspots?', '1404.3194-2-40-1': 'Unlike the works by e.g., [CITATION] or [CITATION], our simulations do not include radiative hydrodynamic convection; turbulence is generated by external forcing.', '1404.3194-2-40-2': 'This should not necessarily be considered a shortcoming of our simulations as the aim of our work has been to present the simplest model that can show formation of bipolar structures from a large-scale dynamo.', '1404.3194-2-40-3': 'This is the first time bipolar structures are found to appear in simulations where the magnetic field is not imposed - as is the case in [CITATION], [CITATION], or [CITATION] - but it is self-consistently generated from a dynamo in strongly stratified forced turbulence.', '1404.3194-2-41-0': 'The most remarkable feature of these simulations is that a minimalistic setup consisting solely of stratification and helically forced turbulence can generate such diverse spatio-temporal behaviour.', '1404.3194-2-41-1': 'Could a mean-field model consisting of both dynamo equations and equations describing NEMPI capture such behaviour?', '1404.3194-2-41-2': 'This question will be the subject of future investigations.'}

[['1404.3194-1-22-2', '1404.3194-2-22-2'], ['1404.3194-1-22-3', '1404.3194-2-22-3'], ['1404.3194-1-22-4', '1404.3194-2-22-4'], ['1404.3194-1-22-5', '1404.3194-2-22-5'], ['1404.3194-1-34-0', '1404.3194-2-34-0'], ['1404.3194-1-34-1', '1404.3194-2-34-1'], ['1404.3194-1-34-2', '1404.3194-2-34-2'], ['1404.3194-1-12-0', '1404.3194-2-12-0'], ['1404.3194-1-12-1', '1404.3194-2-12-1'], ['1404.3194-1-12-2', '1404.3194-2-12-2'], ['1404.3194-1-12-3', '1404.3194-2-12-3'], ['1404.3194-1-12-4', '1404.3194-2-12-4'], ['1404.3194-1-35-1', '1404.3194-2-35-1'], ['1404.3194-1-35-3', '1404.3194-2-35-4'], ['1404.3194-1-35-6', '1404.3194-2-35-7'], ['1404.3194-1-35-8', '1404.3194-2-35-9'], ['1404.3194-1-16-0', '1404.3194-2-16-0'], ['1404.3194-1-16-1', '1404.3194-2-16-1'], ['1404.3194-1-16-3', '1404.3194-2-16-5'], ['1404.3194-1-16-4', '1404.3194-2-16-6'], ['1404.3194-1-31-0', '1404.3194-2-31-0'], ['1404.3194-1-31-1', '1404.3194-2-31-1'], ['1404.3194-1-31-2', '1404.3194-2-31-2'], ['1404.3194-1-26-2', '1404.3194-2-26-2'], ['1404.3194-1-6-0', '1404.3194-2-6-0'], ['1404.3194-1-6-1', '1404.3194-2-6-1'], ['1404.3194-1-6-3', '1404.3194-2-6-3'], ['1404.3194-1-6-4', '1404.3194-2-6-4'], ['1404.3194-1-6-5', '1404.3194-2-6-5'], ['1404.3194-1-6-6', '1404.3194-2-6-6'], ['1404.3194-1-6-7', '1404.3194-2-6-7'], ['1404.3194-1-2-1', '1404.3194-2-2-1'], ['1404.3194-1-2-2', '1404.3194-2-2-2'], ['1404.3194-1-2-3', '1404.3194-2-2-3'], ['1404.3194-1-2-4', '1404.3194-2-2-4'], ['1404.3194-1-7-0', '1404.3194-2-7-0'], ['1404.3194-1-7-1', '1404.3194-2-7-1'], ['1404.3194-1-7-2', '1404.3194-2-7-2'], ['1404.3194-1-7-3', '1404.3194-2-7-3'], ['1404.3194-1-7-4', '1404.3194-2-7-4'], ['1404.3194-1-14-1', '1404.3194-2-14-1'], ['1404.3194-1-14-2', '1404.3194-2-14-2'], ['1404.3194-1-14-3', '1404.3194-2-14-3'], ['1404.3194-1-14-4', '1404.3194-2-14-4'], ['1404.3194-1-14-5', '1404.3194-2-14-5'], ['1404.3194-1-14-6', '1404.3194-2-14-6'], ['1404.3194-1-14-7', '1404.3194-2-14-7'], ['1404.3194-1-14-8', '1404.3194-2-14-8'], ['1404.3194-1-8-0', '1404.3194-2-8-0'], ['1404.3194-1-8-2', '1404.3194-2-8-2'], ['1404.3194-1-5-0', '1404.3194-2-5-0'], ['1404.3194-1-5-1', '1404.3194-2-5-1'], ['1404.3194-1-5-3', '1404.3194-2-5-3'], ['1404.3194-1-5-4', '1404.3194-2-5-4'], ['1404.3194-1-39-0', '1404.3194-2-40-0'], ['1404.3194-1-39-2', '1404.3194-2-40-2'], ['1404.3194-1-0-0', '1404.3194-2-0-0'], ['1404.3194-1-0-2', '1404.3194-2-0-2'], ['1404.3194-1-40-1', '1404.3194-2-41-1'], ['1404.3194-1-40-2', '1404.3194-2-41-2'], ['1404.3194-1-4-0', '1404.3194-2-4-0'], ['1404.3194-1-28-0', '1404.3194-2-28-0'], ['1404.3194-1-28-1', '1404.3194-2-28-1'], ['1404.3194-1-28-2', '1404.3194-2-28-2'], ['1404.3194-1-28-3', '1404.3194-2-28-3'], ['1404.3194-1-28-4', '1404.3194-2-28-4'], ['1404.3194-1-28-5', '1404.3194-2-28-5'], ['1404.3194-1-28-6', '1404.3194-2-28-6'], ['1404.3194-1-28-7', '1404.3194-2-28-7'], ['1404.3194-1-28-8', '1404.3194-2-28-8'], ['1404.3194-1-28-11', '1404.3194-2-28-12'], ['1404.3194-1-28-12', '1404.3194-2-28-13'], ['1404.3194-1-28-13', '1404.3194-2-28-14'], ['1404.3194-1-30-0', '1404.3194-2-30-0'], ['1404.3194-1-30-3', '1404.3194-2-30-3'], ['1404.3194-1-18-0', '1404.3194-2-18-0'], ['1404.3194-1-9-0', '1404.3194-2-9-0'], ['1404.3194-1-9-4', '1404.3194-2-9-4'], ['1404.3194-1-33-0', '1404.3194-2-33-0'], ['1404.3194-1-33-1', '1404.3194-2-33-1'], ['1404.3194-1-33-3', '1404.3194-2-33-3'], ['1404.3194-1-33-4', '1404.3194-2-33-4'], ['1404.3194-1-20-0', '1404.3194-2-20-0'], ['1404.3194-1-20-1', '1404.3194-2-20-1'], ['1404.3194-1-20-2', '1404.3194-2-20-2'], ['1404.3194-1-20-3', '1404.3194-2-20-3'], ['1404.3194-1-20-4', '1404.3194-2-20-4'], ['1404.3194-1-20-5', '1404.3194-2-20-5'], ['1404.3194-1-20-6', '1404.3194-2-20-6'], ['1404.3194-1-20-7', '1404.3194-2-20-7'], ['1404.3194-1-20-8', '1404.3194-2-20-8'], ['1404.3194-1-20-9', '1404.3194-2-20-9'], ['1404.3194-1-38-5', '1404.3194-2-39-3'], ['1404.3194-1-38-6', '1404.3194-2-39-4'], ['1404.3194-1-24-0', '1404.3194-2-24-0'], ['1404.3194-1-22-0', '1404.3194-2-22-0'], ['1404.3194-1-22-1', '1404.3194-2-22-1'], ['1404.3194-1-12-5', '1404.3194-2-12-5'], ['1404.3194-1-12-6', '1404.3194-2-12-6'], ['1404.3194-1-12-7', '1404.3194-2-12-7'], ['1404.3194-1-35-0', '1404.3194-2-35-0'], ['1404.3194-1-35-5', '1404.3194-2-35-6'], ['1404.3194-1-35-7', '1404.3194-2-35-8'], ['1404.3194-1-3-0', '1404.3194-2-3-0'], ['1404.3194-1-3-1', '1404.3194-2-3-1'], ['1404.3194-1-3-2', '1404.3194-2-3-2'], ['1404.3194-1-26-0', '1404.3194-2-26-0'], ['1404.3194-1-6-2', '1404.3194-2-6-2'], ['1404.3194-1-2-0', '1404.3194-2-2-0'], ['1404.3194-1-7-5', '1404.3194-2-7-5'], ['1404.3194-1-7-6', '1404.3194-2-7-6'], ['1404.3194-1-14-0', '1404.3194-2-14-0'], ['1404.3194-1-8-1', '1404.3194-2-8-1'], ['1404.3194-1-5-2', '1404.3194-2-5-2'], ['1404.3194-1-36-1', '1404.3194-2-36-0'], ['1404.3194-1-36-2', '1404.3194-2-36-1'], ['1404.3194-1-39-1', '1404.3194-2-40-1'], ['1404.3194-1-39-3', '1404.3194-2-40-3'], ['1404.3194-1-0-3', '1404.3194-2-0-3'], ['1404.3194-1-40-0', '1404.3194-2-41-0'], ['1404.3194-1-4-1', '1404.3194-2-4-1'], ['1404.3194-1-4-4', '1404.3194-2-4-8'], ['1404.3194-1-28-10', '1404.3194-2-28-11'], ['1404.3194-1-30-1', '1404.3194-2-30-1'], ['1404.3194-1-30-2', '1404.3194-2-30-2'], ['1404.3194-1-30-4', '1404.3194-2-30-4'], ['1404.3194-1-18-1', '1404.3194-2-18-1'], ['1404.3194-1-9-1', '1404.3194-2-9-1'], ['1404.3194-1-9-2', '1404.3194-2-9-2'], ['1404.3194-1-9-3', '1404.3194-2-9-3'], ['1404.3194-1-33-2', '1404.3194-2-33-2'], ['1404.3194-1-33-5', '1404.3194-2-33-5'], ['1404.3194-1-20-10', '1404.3194-2-20-10'], ['1404.3194-1-20-11', '1404.3194-2-20-11'], ['1404.3194-1-38-0', '1404.3194-2-38-0'], ['1404.3194-1-38-1', '1404.3194-2-38-1'], ['1404.3194-1-38-2', '1404.3194-2-39-0'], ['1404.3194-1-38-3', '1404.3194-2-39-1'], ['1404.3194-1-38-4', '1404.3194-2-39-2'], ['1404.3194-1-24-1', '1404.3194-2-24-2'], ['1404.3194-1-35-2', '1404.3194-2-35-2'], ['1404.3194-1-35-2', '1404.3194-2-35-3'], ['1404.3194-1-35-4', '1404.3194-2-35-5'], ['1404.3194-1-16-2', '1404.3194-2-16-2'], ['1404.3194-1-26-1', '1404.3194-2-26-1'], ['1404.3194-1-7-7', '1404.3194-2-7-7'], ['1404.3194-1-7-7', '1404.3194-2-7-8'], ['1404.3194-1-0-1', '1404.3194-2-0-1'], ['1404.3194-1-0-5', '1404.3194-2-0-5'], ['1404.3194-1-4-2', '1404.3194-2-4-2'], ['1404.3194-1-4-2', '1404.3194-2-4-3'], ['1404.3194-1-28-9', '1404.3194-2-28-9'], ['1404.3194-1-28-9', '1404.3194-2-28-10']]

[['1404.3194-1-22-2', '1404.3194-2-22-2'], ['1404.3194-1-22-3', '1404.3194-2-22-3'], ['1404.3194-1-22-4', '1404.3194-2-22-4'], ['1404.3194-1-22-5', '1404.3194-2-22-5'], ['1404.3194-1-34-0', '1404.3194-2-34-0'], ['1404.3194-1-34-1', '1404.3194-2-34-1'], ['1404.3194-1-34-2', '1404.3194-2-34-2'], ['1404.3194-1-12-0', '1404.3194-2-12-0'], ['1404.3194-1-12-1', '1404.3194-2-12-1'], ['1404.3194-1-12-2', '1404.3194-2-12-2'], ['1404.3194-1-12-3', '1404.3194-2-12-3'], ['1404.3194-1-12-4', '1404.3194-2-12-4'], ['1404.3194-1-35-1', '1404.3194-2-35-1'], ['1404.3194-1-35-3', '1404.3194-2-35-4'], ['1404.3194-1-35-6', '1404.3194-2-35-7'], ['1404.3194-1-35-8', '1404.3194-2-35-9'], ['1404.3194-1-16-0', '1404.3194-2-16-0'], ['1404.3194-1-16-1', '1404.3194-2-16-1'], ['1404.3194-1-16-3', '1404.3194-2-16-5'], ['1404.3194-1-16-4', '1404.3194-2-16-6'], ['1404.3194-1-31-0', '1404.3194-2-31-0'], ['1404.3194-1-31-1', '1404.3194-2-31-1'], ['1404.3194-1-31-2', '1404.3194-2-31-2'], ['1404.3194-1-26-2', '1404.3194-2-26-2'], ['1404.3194-1-6-0', '1404.3194-2-6-0'], ['1404.3194-1-6-1', '1404.3194-2-6-1'], ['1404.3194-1-6-3', '1404.3194-2-6-3'], ['1404.3194-1-6-4', '1404.3194-2-6-4'], ['1404.3194-1-6-5', '1404.3194-2-6-5'], ['1404.3194-1-6-6', '1404.3194-2-6-6'], ['1404.3194-1-6-7', '1404.3194-2-6-7'], ['1404.3194-1-2-1', '1404.3194-2-2-1'], ['1404.3194-1-2-2', '1404.3194-2-2-2'], ['1404.3194-1-2-3', '1404.3194-2-2-3'], ['1404.3194-1-2-4', '1404.3194-2-2-4'], ['1404.3194-1-7-0', '1404.3194-2-7-0'], ['1404.3194-1-7-1', '1404.3194-2-7-1'], ['1404.3194-1-7-2', '1404.3194-2-7-2'], ['1404.3194-1-7-3', '1404.3194-2-7-3'], ['1404.3194-1-7-4', '1404.3194-2-7-4'], ['1404.3194-1-14-1', '1404.3194-2-14-1'], ['1404.3194-1-14-2', '1404.3194-2-14-2'], ['1404.3194-1-14-3', '1404.3194-2-14-3'], ['1404.3194-1-14-4', '1404.3194-2-14-4'], ['1404.3194-1-14-5', '1404.3194-2-14-5'], ['1404.3194-1-14-6', '1404.3194-2-14-6'], ['1404.3194-1-14-7', '1404.3194-2-14-7'], ['1404.3194-1-14-8', '1404.3194-2-14-8'], ['1404.3194-1-8-0', '1404.3194-2-8-0'], ['1404.3194-1-8-2', '1404.3194-2-8-2'], ['1404.3194-1-5-0', '1404.3194-2-5-0'], ['1404.3194-1-5-1', '1404.3194-2-5-1'], ['1404.3194-1-5-3', '1404.3194-2-5-3'], ['1404.3194-1-5-4', '1404.3194-2-5-4'], ['1404.3194-1-39-0', '1404.3194-2-40-0'], ['1404.3194-1-39-2', '1404.3194-2-40-2'], ['1404.3194-1-0-0', '1404.3194-2-0-0'], ['1404.3194-1-0-2', '1404.3194-2-0-2'], ['1404.3194-1-40-1', '1404.3194-2-41-1'], ['1404.3194-1-40-2', '1404.3194-2-41-2'], ['1404.3194-1-4-0', '1404.3194-2-4-0'], ['1404.3194-1-28-0', '1404.3194-2-28-0'], ['1404.3194-1-28-1', '1404.3194-2-28-1'], ['1404.3194-1-28-2', '1404.3194-2-28-2'], ['1404.3194-1-28-3', '1404.3194-2-28-3'], ['1404.3194-1-28-4', '1404.3194-2-28-4'], ['1404.3194-1-28-5', '1404.3194-2-28-5'], ['1404.3194-1-28-6', '1404.3194-2-28-6'], ['1404.3194-1-28-7', '1404.3194-2-28-7'], ['1404.3194-1-28-8', '1404.3194-2-28-8'], ['1404.3194-1-28-11', '1404.3194-2-28-12'], ['1404.3194-1-28-12', '1404.3194-2-28-13'], ['1404.3194-1-28-13', '1404.3194-2-28-14'], ['1404.3194-1-30-0', '1404.3194-2-30-0'], ['1404.3194-1-30-3', '1404.3194-2-30-3'], ['1404.3194-1-18-0', '1404.3194-2-18-0'], ['1404.3194-1-9-0', '1404.3194-2-9-0'], ['1404.3194-1-9-4', '1404.3194-2-9-4'], ['1404.3194-1-33-0', '1404.3194-2-33-0'], ['1404.3194-1-33-1', '1404.3194-2-33-1'], ['1404.3194-1-33-3', '1404.3194-2-33-3'], ['1404.3194-1-33-4', '1404.3194-2-33-4'], ['1404.3194-1-20-0', '1404.3194-2-20-0'], ['1404.3194-1-20-1', '1404.3194-2-20-1'], ['1404.3194-1-20-2', '1404.3194-2-20-2'], ['1404.3194-1-20-3', '1404.3194-2-20-3'], ['1404.3194-1-20-4', '1404.3194-2-20-4'], ['1404.3194-1-20-5', '1404.3194-2-20-5'], ['1404.3194-1-20-6', '1404.3194-2-20-6'], ['1404.3194-1-20-7', '1404.3194-2-20-7'], ['1404.3194-1-20-8', '1404.3194-2-20-8'], ['1404.3194-1-20-9', '1404.3194-2-20-9'], ['1404.3194-1-38-5', '1404.3194-2-39-3'], ['1404.3194-1-38-6', '1404.3194-2-39-4']]

[['1404.3194-1-24-0', '1404.3194-2-24-0'], ['1404.3194-1-22-0', '1404.3194-2-22-0'], ['1404.3194-1-22-1', '1404.3194-2-22-1'], ['1404.3194-1-12-5', '1404.3194-2-12-5'], ['1404.3194-1-12-6', '1404.3194-2-12-6'], ['1404.3194-1-12-7', '1404.3194-2-12-7'], ['1404.3194-1-35-0', '1404.3194-2-35-0'], ['1404.3194-1-35-5', '1404.3194-2-35-6'], ['1404.3194-1-35-7', '1404.3194-2-35-8'], ['1404.3194-1-3-0', '1404.3194-2-3-0'], ['1404.3194-1-3-1', '1404.3194-2-3-1'], ['1404.3194-1-3-2', '1404.3194-2-3-2'], ['1404.3194-1-26-0', '1404.3194-2-26-0'], ['1404.3194-1-6-2', '1404.3194-2-6-2'], ['1404.3194-1-2-0', '1404.3194-2-2-0'], ['1404.3194-1-7-5', '1404.3194-2-7-5'], ['1404.3194-1-7-6', '1404.3194-2-7-6'], ['1404.3194-1-14-0', '1404.3194-2-14-0'], ['1404.3194-1-8-1', '1404.3194-2-8-1'], ['1404.3194-1-5-2', '1404.3194-2-5-2'], ['1404.3194-1-36-1', '1404.3194-2-36-0'], ['1404.3194-1-36-2', '1404.3194-2-36-1'], ['1404.3194-1-39-1', '1404.3194-2-40-1'], ['1404.3194-1-39-3', '1404.3194-2-40-3'], ['1404.3194-1-0-3', '1404.3194-2-0-3'], ['1404.3194-1-40-0', '1404.3194-2-41-0'], ['1404.3194-1-4-1', '1404.3194-2-4-1'], ['1404.3194-1-4-4', '1404.3194-2-4-8'], ['1404.3194-1-28-10', '1404.3194-2-28-11'], ['1404.3194-1-30-1', '1404.3194-2-30-1'], ['1404.3194-1-30-2', '1404.3194-2-30-2'], ['1404.3194-1-30-4', '1404.3194-2-30-4'], ['1404.3194-1-18-1', '1404.3194-2-18-1'], ['1404.3194-1-9-1', '1404.3194-2-9-1'], ['1404.3194-1-9-2', '1404.3194-2-9-2'], ['1404.3194-1-9-3', '1404.3194-2-9-3'], ['1404.3194-1-33-2', '1404.3194-2-33-2'], ['1404.3194-1-33-5', '1404.3194-2-33-5'], ['1404.3194-1-20-10', '1404.3194-2-20-10'], ['1404.3194-1-20-11', '1404.3194-2-20-11'], ['1404.3194-1-38-0', '1404.3194-2-38-0'], ['1404.3194-1-38-1', '1404.3194-2-38-1'], ['1404.3194-1-38-2', '1404.3194-2-39-0'], ['1404.3194-1-38-3', '1404.3194-2-39-1'], ['1404.3194-1-38-4', '1404.3194-2-39-2']]

[]

[['1404.3194-1-24-1', '1404.3194-2-24-2'], ['1404.3194-1-35-2', '1404.3194-2-35-2'], ['1404.3194-1-35-2', '1404.3194-2-35-3'], ['1404.3194-1-35-4', '1404.3194-2-35-5'], ['1404.3194-1-16-2', '1404.3194-2-16-2'], ['1404.3194-1-26-1', '1404.3194-2-26-1'], ['1404.3194-1-7-7', '1404.3194-2-7-7'], ['1404.3194-1-7-7', '1404.3194-2-7-8'], ['1404.3194-1-0-1', '1404.3194-2-0-1'], ['1404.3194-1-0-5', '1404.3194-2-0-5'], ['1404.3194-1-4-2', '1404.3194-2-4-2'], ['1404.3194-1-4-2', '1404.3194-2-4-3'], ['1404.3194-1-28-9', '1404.3194-2-28-9'], ['1404.3194-1-28-9', '1404.3194-2-28-10']]

[]

['1404.3194-1-17-0', '1404.3194-2-17-0']

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

https://arxiv.org/abs/1404.3194

1712.00969

{'1712.00969-1-0-0': 'We study the non-Breit-Wigner line-shape of the [MATH] resonance, using an unitarized effective Lagrangian approach, including the one-loop effects of the nearby thresholds [MATH] and [MATH].', '1712.00969-1-0-1': 'The model contains a single seed state, corresponding to a standard [MATH] with predominant quantum numbers [MATH].', '1712.00969-1-0-2': 'The loops of [MATH]and [MATH] are responsible for the distortion of the spectral function.', '1712.00969-1-0-3': 'A fit of the theoretical data to the total cross-section [MATH] is performed with 28 data points and 4 parameters: strong and electromagnetic coupling constants, the bare mass of [MATH], and a form-factor parameter, leading to a good description of data ([MATH]/d.o.f. [MATH]).', '1712.00969-1-0-4': 'We then analyze various consequences of the fit, among which pole positions (which were not investigated in previous theoretical works), partial cross-sections, and leptonic decays.', '1712.00969-1-0-5': 'The first pole reads [MATH] MeV, and it is in good agreement with the Particle Data Group (PDG) mass and width estimation for this state.', '1712.00969-1-0-6': 'Quite remarkably, there is also a second pole at [MATH] MeV.', '1712.00969-1-0-7': 'This additional, dynamically generated, companion pole is responsible for the deformation of the left side of the line-shape.', '1712.00969-1-0-8': 'Its existence represents one of the main finding of this work.', '1712.00969-1-0-9': 'Moreover, the [MATH] and [MATH] cross sections turn out to be separately in good agreement with data.', '1712.00969-1-0-10': 'The width for the decay [MATH] is 113 eV, hence smaller than the PDG fit of [MATH] eV, yet in agreement with a recent experimental study.', '1712.00969-1-0-11': 'Finally, the final state rescattering effect is found to be small.', '1712.00969-1-0-12': 'The size of the wave-function is estimated within a Schrodinger model to be around [MATH] fm, a value that we relate to the form-factor of the effective field theoretical framework.', '1712.00969-1-1-0': '# Introduction', '1712.00969-1-2-0': 'The [MATH] resonance is a vector state that was first detected at SPEAR [CITATION] in 1977.', '1712.00969-1-2-1': 'The signal was fitted with a pure [MATH]-wave Breit-Wigner distribution, but with a not insignificant error ([MATH] for 15 degrees of freedom).', '1712.00969-1-2-2': 'More recently, the interest on this state was revived, and the parameters are nowadays fitted by the Particle Data group (PDG) as [MATH] MeV for the mass and [MATH] MeV for the width [CITATION].', '1712.00969-1-2-3': "According to the 'quark model review' of the PDG, the [MATH] resonance is classified as a [MATH] charmonium state, the first one above the [MATH] threshold, which is the reason for its relatively large width.", '1712.00969-1-2-4': 'Also for this reason, it is reasonable to expect, besides kinematic interference, additional nonperturbative effects.', '1712.00969-1-3-0': 'Indeed, as stated above, the line shape of this resonance turned out to be quite anomalous.', '1712.00969-1-3-1': 'Although other experiments have been performed with observations in [MATH] channel [CITATION], the deformation on the line-shape of the [MATH] was made clear by the BES Collaboration data in the [MATH] annihilation to hadrons [CITATION].', '1712.00969-1-3-2': 'The existence of a second resonance was suggested, although possible dynamical effects, generated by the [MATH] threshold were not discarded.', '1712.00969-1-3-3': 'A deformation of a line-shape due to the superposition of two resonances has been discussed before e.g., for the scalar kaon [CITATION], where, besides the dominant [MATH] state, an additional dynamically generated state arises from the continuum, the well known [MATH].', '1712.00969-1-3-4': 'Such effect has not been discussed before for the vector charmonium.', '1712.00969-1-4-0': 'Various analysis of the [MATH] have been performed.', '1712.00969-1-4-1': 'In Refs. [CITATION] fits were computed taking into account not only the [MATH] interference but also the tail of the [MATH].', '1712.00969-1-4-2': 'Such inclusion is natural since the [MATH] should be a mixed state [MATH]-[MATH].', '1712.00969-1-4-3': 'In Ref. [CITATION], the deformation from the right side of the resonance, i.e. a dip structure, is explained by the interference with the [MATH] kinematical background which is higher for larger relative momentum.', '1712.00969-1-4-4': 'The same dip is reproduced in [CITATION], using [MATH] background only, and in [CITATION], where in addition, the continuum of light hadrons is removed.', '1712.00969-1-4-5': 'In [CITATION] BES measured an unexpectedly large branching fraction [MATH] non[MATH] of about [MATH], a result that is not contradicted by CLEO in [CITATION], within errors.', '1712.00969-1-4-6': 'Predictions for such non-[MATH] continuum were made including the tails of [MATH] and [MATH], [MATH] and [MATH] decays [CITATION], other excited [MATH] states [CITATION], the [MATH] channel [CITATION], other hadronic decays and radiative decays [CITATION], and final state interactions [CITATION], which in any case do not sum up to the value of [MATH].', '1712.00969-1-4-7': 'Yet, since the phase space to [MATH] is not excessively large, it is likely that the missing decays are simply the sum of all the many Okubo-Zweig-Iizuka (OZI)-suppressed hadronic decays, that have not been studied systematically in the theory.', '1712.00969-1-4-8': 'Estimations for the [MATH] production via [MATH] annihilation are made in [CITATION], which may possibly be measured at the PANDA experiment [CITATION].', '1712.00969-1-5-0': 'In this work, we study the properties of the [MATH] by analyzing its production through electron-positron fusion, and subsequent decay into [MATH] pairs (for a preliminary study, see Ref. [CITATION]).', '1712.00969-1-5-1': 'Our starting point is a vector charmonium "seed" state, which gets dressed by "clouds" of [MATH] and [MATH] mesons.', '1712.00969-1-5-2': 'Our aim is to study the deformation seen on the left side of the resonance in Ref. [CITATION] with mesonic loops combined with the nearby thresholds.', '1712.00969-1-5-3': 'To this end, we use an effective relativistic Lagrangian approach in which a single vector state [MATH] is coupled to [MATH] and [MATH] as well as to lepton pairs.', '1712.00969-1-5-4': 'The propagator of [MATH] is calculated at the resummed one-loop level and fulfills unitarization requirements.', '1712.00969-1-5-5': 'Then, we perform a fit of the four parameters of our approach, i.e. the couplings of [MATH] to [MATH] and of [MATH] to leptons, the mass of [MATH], and a cutoff responsible for the finite dimension of the [MATH] meson, to the experimental result of Refs. [CITATION] and [CITATION] for the cross-section of the reaction [MATH] in the energy region of about [MATH] MeV above the [MATH] threshold.', '1712.00969-1-5-6': 'We assume that the [MATH] resonance dominates in this energy range.', '1712.00969-1-5-7': 'We obtain a very good description of the data, which in turn allows us to determine in a novel and independent way the mass, width, and branching ratios of the [MATH].', '1712.00969-1-5-8': 'Moreover, we study, to our knowledge for the first time, the poles of this state.', '1712.00969-1-5-9': 'Quite remarkably, we find two pole positions for this resonance, one which roughly corresponds to the peak of the resonance, the seed pole, and one additional dynamically generated pole, responsible for the enhancement left from the peak, which emerges due to the strong coupling between the seed state and the mesonic loops.', '1712.00969-1-5-10': 'This is a companion pole, similarly to the one found in the kaonic system in Ref. [CITATION].', '1712.00969-1-5-11': '(For other works on dynamically generated states in general and companion poles in particular, see Refs. [CITATION] and references therein).', '1712.00969-1-6-0': 'As a consequence of the determined parameters, we also show that the cross sections [MATH] and [MATH] agree separately to data.', '1712.00969-1-6-1': 'Being the resonance not an ideal Breit-Wigner one, different definitions for the partial widths are compared to each other.', '1712.00969-1-6-2': 'As a verification of our theoretical approach, we artificially vary the intensity of the coupling in order to show its effect on the line-shape of the resonance.', '1712.00969-1-6-3': 'Smaller couplings lead to a narrow Breit-Wigner shape, while larger couplings to an even larger deformation, which eventually gives rise to two peaks, as in Ref. [CITATION].', '1712.00969-1-6-4': 'In addition, we also test the role of the rescattering effect due to additional processes of the type [MATH] (such rescattering partly mimics also the effect of the charmonium state below threshold [MATH], otherwise not included in this work).', '1712.00969-1-6-5': 'We find that they affect only marginally the results.', '1712.00969-1-6-6': 'This is in agreement with one of the fits found in Refs. [CITATION], where effective Lagrangians including the [MATH], [MATH] loops, and rescattering have been employed.', '1712.00969-1-7-0': 'The paper is organized as it follows.', '1712.00969-1-7-1': 'In Sec. 2 the model is introduced and various theoretical quantities, such as the resummed propagator, spectral function, cross-section, and rescattering, are presented.', '1712.00969-1-7-2': 'In Sec. 3 we show our results: first, we present a fit to data and then examine its consequences.', '1712.00969-1-7-3': 'In Sec. 4 conclusions are drawn.', '1712.00969-1-7-4': 'Some important technical details are discussed in the Appendices.', '1712.00969-1-8-0': '# An Effective Model', '1712.00969-1-9-0': 'In this section we present the Lagrangian of the model and evaluate the most relevant theoretical quantities, namely the resummed propagator and the spectral functions, needed to calculate the cross section for the processes [MATH] and [MATH], via production of the [MATH], and with [MATH] one-loops, as represented in Fig. [REF].', '1712.00969-1-10-0': '## The strong Lagrangian and dispersion relations', '1712.00969-1-11-0': 'We aim to write down a Lagrangian for the strong interaction between a single vectorial resonance and two pseudoscalar mesons.', '1712.00969-1-11-1': 'The bare resonance [MATH] is identified with a bare quarkonium [MATH] state, with predominant quantum numbers [MATH], but admixtures of other quantum numbers, most notably the [MATH] are possible.', '1712.00969-1-11-2': 'In our model, however, neither the quarkonium structure nor the orbital angular momentum mixing are explicit, since we work with mesonic degrees of freedom in the basis of total angular momentum.', '1712.00969-1-11-3': 'The effective strong Lagrangian density, for the charmonium state [MATH], reads [EQUATION] which corresponds to the simplest interaction among [MATH] and its main decay products, the pseudoscalar pairs [MATH] and [MATH], each vertex represented in Fig. [REF].', '1712.00969-1-11-4': 'In this work, we shall take into account the small mass difference between [MATH] and [MATH], i.e. isospin breaking for the masses, but shall keep the same coupling constant, i.e. isospin symmetry for the decays, and we define [MATH].', '1712.00969-1-11-5': 'The full Lagrangian can be found in [REF].', '1712.00969-1-12-0': 'The free propagator of a vector field [MATH], with mass [MATH] and momentum [MATH], is given by [EQUATION] where the term in parenthesis is the sum over the three polarization states of a vector.', '1712.00969-1-12-1': 'We include the resummed one-loop effect as shown in Fig. [REF], leading to the full propagator of [MATH] [EQUATION] where [MATH] is the loop-function, consisting of two contributions, the [MATH] loops and the [MATH] loops.', '1712.00969-1-12-2': 'In particular, we get [EQUATION] where the function [MATH], with [MATH], reads [EQUATION] where [MATH] is the mass of the meson circulating in the loop (either [MATH] or [MATH]), and [MATH] is the internal momentum of the loop.', '1712.00969-1-12-3': 'Note, in the reference frame of the decaying particle, it holds the relation [EQUATION]', '1712.00969-1-12-4': 'An important element for our discussion is the function [MATH] entering in Eq. [REF].', '1712.00969-1-12-5': 'This is a form-factor that depends on a cutoff parameter [MATH] and on the invariant energy squared [MATH].', '1712.00969-1-12-6': 'It is needed to account for the unknown of the black vertices in Fig. [REF], due to the fact that mesons are not elementary particles [CITATION].', '1712.00969-1-12-7': 'The cutoff function must ensure the convergence of the integral.', '1712.00969-1-12-8': 'We choose the following Gaussian form [EQUATION] where the second term is built for convenience (in the isospin limit, [MATH] reduces to [MATH], being [MATH] the relativistic momentum of the final state).', '1712.00969-1-12-9': 'One can formally introduce the vertex function already at the Lagrangian level by rendering it nonlocal, see the discussion in Refs. [CITATION], where it is also pointed out that the special form of the form factor is not important, as long as fast convergence is guaranteed.', '1712.00969-1-12-10': 'However, the exponential form for the cutoff is very typical and has been used in many different approaches in hadron physics, e.g. Ref. [CITATION].', '1712.00969-1-12-11': 'Finally, as shown in Ref. [CITATION], covariance can be always guaranteed when introducing the form factor.', '1712.00969-1-13-0': 'We now turn back to the study of the propagator.', '1712.00969-1-13-1': 'In [REF], it is shown that the relevant quantity is the transverse part of the propagator that, in the rest frame of the decaying particles, and for a single loop, reads [EQUATION] and similarly for the sum [EQUATION]', '1712.00969-1-13-2': 'The scalar part of the one-loop resummed dress propagator (see [REF] for details) reads [EQUATION]', '1712.00969-1-13-3': 'Note, this expression of the propagator fulfills unitarity (cf. Sec. [REF] below) and it is accurate as long as further contributions to the self-energy are small.', '1712.00969-1-13-4': 'The next contribution would be represented by a loop in which the unstable state [MATH] would be exchanged by [MATH] mesons circulating in the loop.', '1712.00969-1-13-5': 'Indeed, as shown in Ref. [CITATION], such contributions are typically very small in hadron physics and can be safely neglected.', '1712.00969-1-14-0': 'The self-energy in Eq. [REF] can either be computed through the integration (see [REF]), or using dispersion relations.', '1712.00969-1-14-1': 'We follow the latter.', '1712.00969-1-14-2': 'To this end, we first decompose [MATH] into its real and imaginary parts [EQUATION]', '1712.00969-1-14-3': 'According to the optical theorem, the imaginary part [MATH] (dispersive term) is given by [EQUATION] with [MATH] being the center-of-mass momentum.', '1712.00969-1-14-4': 'The partial decay widths of [MATH] and [MATH] are, then, calculated as [EQUATION] where the Lorentz invariant amplitudes squared, computed from [MATH] in Eq. [REF], are given by [EQUATION]', '1712.00969-1-14-5': 'Finally, the on-shell partial and total decay widths are [EQUATION]', '1712.00969-1-14-6': 'Once the imaginary part of the loop is known, the real part, the function [MATH], is computed from the dispersion relation [EQUATION] where [MATH] is zero below threshold.', '1712.00969-1-14-7': 'Convergence is guaranteed by the cutoff function.', '1712.00969-1-14-8': 'As we shall see in Sec. [REF], the real part [MATH] causes a distortion in the line-shape due to the continuous shifting of the physical mass of the resonance with the energy.', '1712.00969-1-14-9': 'Note, when the energy [MATH] is complex, the function [MATH] reads (away from the real axis): [EQUATION]', '1712.00969-1-14-10': 'Furthermore, although it is not strictly needed, since the integral in Eq. [REF] is already convergent, we use the once-subtracted dispersion relation, with subtraction in point [MATH], for our convenience.', '1712.00969-1-14-11': 'Hence, the total loop-function is given by [EQUATION] and, the final dressed propagator of meson [MATH] is [EQUATION]', '1712.00969-1-14-12': 'In this way, the parameter [MATH] in the propagator corresponds to the mass of the particle defined as [EQUATION]', '1712.00969-1-14-13': 'Other definitions for the mass are possible, such as the position of the peak, or the real part of the pole, as we shall see below.', '1712.00969-1-15-0': '## Poles', '1712.00969-1-16-0': 'In order to find poles, the energy [MATH] is analytically continued to the complex plane ([MATH]).', '1712.00969-1-16-1': 'In case of two decay channels, the Riemann surface is composed by four Riemann Sheets (RSs).', '1712.00969-1-16-2': 'Poles are found in the unphysical sheet which results in the physical sheet when the energy is real and the resonance can be seen.', '1712.00969-1-16-3': 'Above both thresholds, this corresponds to the condition [MATH] and [MATH] when [MATH] and [MATH], i.e., the fourth RS.', '1712.00969-1-16-4': 'Poles are given when the denominator of the propagator in Eq. [REF] is zero in the correct RS, i.e., [EQUATION] where [EQUATION] where the subscripts [MATH] and [MATH] stand for the first and fourth RS.', '1712.00969-1-17-0': '## Coupling to leptons', '1712.00969-1-18-0': 'The available experimental data comes from the production process [MATH] (see Fig. [REF]), therefore we need to couple the state [MATH] to leptons.', '1712.00969-1-18-1': 'The corresponding interaction Lagrangian is defined by [EQUATION] which is the simplest interaction among a massive vector field [MATH] and a fermion-antifermion pair.', '1712.00969-1-18-2': 'The coupling [MATH], between [MATH] and the electron-positron pair, is here considered to be the same between [MATH] and all leptonic pairs, and it is the overall strength for the annihilation of the leptonic pair [MATH] into one photon, and further conversion into the [MATH] vector.', '1712.00969-1-18-3': 'For our purposes, it is more convenient to enclose this process in a single effective vertex [MATH], see Fig. [REF].', '1712.00969-1-19-0': 'The corresponding decay into leptons reads [EQUATION] where [MATH] is the leptonic mass.', '1712.00969-1-20-0': 'In principle, the loops of leptons should be included in the total one-loop function of resonance [MATH], obtaining [MATH], where dots refer to other possible but suppressed hadronic decay channels.', '1712.00969-1-20-1': 'However, the loop contribution of [MATH] is definitely negligible w.r.t. [MATH].', '1712.00969-1-20-2': 'Here, we shall simply consider [MATH]', '1712.00969-1-21-0': '## Spectral function and cross section', '1712.00969-1-22-0': 'The unitarized spectral function, as a function of energy [MATH], equivalent to the running mass of the [MATH] resonance, is given by [EQUATION] that has the general shape of a relativistic Breit-Wigner distribution, distorted by the loop-function [MATH] (Eq. [REF]).', '1712.00969-1-22-1': 'When no poles below threshold emerge, the normalization above threshold is guaranteed (unitarity): [EQUATION]', '1712.00969-1-22-2': 'The quantity [MATH] is interpreted as the probability that the state [MATH] has a mass between [MATH] and [MATH].', '1712.00969-1-22-3': 'The cross section for [MATH] takes the form [EQUATION]', '1712.00969-1-22-4': 'Hence, the experimental data for this cross section give us direct access to the imaginary part of the propagator of the meson [MATH].', '1712.00969-1-22-5': 'The corresponding amplitude, leading to [MATH], is depicted in Fig. [REF].', '1712.00969-1-23-0': 'One also defines the partial spectral functions [CITATION]: [EQUATION]', '1712.00969-1-23-1': 'Intuitively, [MATH] is the probability that the unstable state [MATH] has a mass between [MATH] and [MATH] and decays in the channel [MATH] (similarly for [MATH]).', '1712.00969-1-23-2': 'Then, the partial cross sections are given by: [EQUATION]', '1712.00969-1-24-0': '## Rescattering', '1712.00969-1-25-0': 'For completeness, we also study the possibility of final state rescattering, that takes place by additional processes of the type [MATH] see Fig. [REF].', '1712.00969-1-25-1': 'Formally, the interaction Lagrangian responsible for the rescattering reads [EQUATION] where [MATH] is an an addtional rescattering coupling parameter.', '1712.00969-1-25-2': 'Note, such rescattering effect includes various [MATH]-meson exchanges between [MATH], and also [MATH]-channel exchanges of nearby vector resonances, such as [MATH], due to the chain [MATH].', '1712.00969-1-25-3': 'Figure [REF] depicts the rescattering effect over the loop function.', '1712.00969-1-25-4': 'The filled square is the total self-energy, including the rescattering expansion, and reads explicitly: [EQUATION]', '1712.00969-1-25-5': 'The once-subtracted technique may be employed to redefine the self-energy in Eq. [REF], in a similar way to Eq. [REF].', '1712.00969-1-25-6': 'This can be done in two different ways [EQUATION] where the subtraction is applied to each loop, and [EQUATION] where the subtraction constant is applied over the whole self-energy without subtractions.', '1712.00969-1-25-7': 'In either way, we shall show that reasonable choices of [MATH] lead to small variation of the spectral function, hence for the state [MATH], rescattering effects seem to play a minor role.', '1712.00969-1-26-0': '# Line-shapes, poles, and wave-function', '1712.00969-1-27-0': 'In this section, we present our fit to experimental data and its consequences, most notably the position of two poles underlying [MATH].', '1712.00969-1-27-1': 'We also discuss the spatial dimension of the resonance.', '1712.00969-1-28-0': '## Fit to data and consequences', '1712.00969-1-29-0': 'As a first necessary step, we determine the four free parameters of our model defined in Sec. [REF], [EQUATION] by performing a fit to the cross section data for the process [MATH].', '1712.00969-1-29-1': 'We use 28 experimental points published in Refs. [CITATION] and [CITATION], and the theoretical expression in Eq. [REF].', '1712.00969-1-29-2': 'The fit to data is shown in Fig. [REF], and the values of the parameters in [REF] are presented in Table [REF].', '1712.00969-1-29-3': 'We get the value [MATH], which shows that a very good description of the data is achieved.', '1712.00969-1-29-4': 'The errors of the parameters can be obtained in the standard way.', '1712.00969-1-29-5': 'Usually, the error [MATH] of a parameter [MATH] corresponds to an increase of the unity of the [MATH] (see e.g. Ref. [CITATION]).', '1712.00969-1-29-6': 'Yet, being the [MATH], we take here a more conservative determination by increasing the errors by the multiplicative factor[MATH] (hence, we take into account that a larger departure from the value at the minimum is still compatible with the data).', '1712.00969-1-30-0': 'There are various consequences of the fit that we discuss in detail:', '1712.00969-1-31-0': "a) The value of the mass [MATH] MeV (corresponding to the zero of the real part of the inverse's propagator, see Eq. ([REF])) is in well agreement with the present PDG fit of [MATH] MeV [CITATION].", '1712.00969-1-32-0': 'b) In Fig. [REF] we present a comparison between theory and the data for the cross sections [MATH] and [MATH], using the parameters in Table [REF].', '1712.00969-1-32-1': 'The good agreement shows that the theory is able to describe the two partial cross sections separately, without the need of some extra parameter.', '1712.00969-1-33-0': 'c) The branching ratios and partial and total ([MATH]) widths are presented in Table [REF].', '1712.00969-1-33-1': 'The partial widths, evaluated on-shell, are defined in Eqs. [REF]-[REF].', '1712.00969-1-33-2': 'The branching ratios are in rough agreement with those quoted in PDG, but the total [MATH] three-level width is sizably smaller than the one referred in PDG which includes all decay channels.', '1712.00969-1-33-3': 'The ratio [MATH] of [MATH] is compatible with the PDG values, which range from [MATH] to [MATH].', '1712.00969-1-33-4': 'However, as shown in Fig. [REF], the functions [MATH] and [MATH] strongly vary in the region of interest.', '1712.00969-1-33-5': 'Alternatively, one may use a different definition, cf. [CITATION], to obtain the partial decay widths and the full decay width to [MATH], by integrating over the spectral function as [EQUATION]', '1712.00969-1-33-6': 'The distributions [MATH] and [MATH] are depicted in Fig. [REF] where, in addition, one can see clearly the effect of the cutoff.', '1712.00969-1-34-0': 'A somewhat naive but still useful alternative determination of the mass is the value corresponding to the peak and the half-height width: [EQUATION] where we estimated the error of the peak mass as [MATH] MeV, close to the determined error for the parameter [MATH], and the error of the distribution width as [MATH] MeV, coincident to the error for [MATH] (see Table [REF]).', '1712.00969-1-34-1': 'These evaluations show that the resonance [MATH] is far from being an ideal Breit-Wigner.', '1712.00969-1-34-2': 'Namely, all these different approaches coincide for states with a very small width.', '1712.00969-1-34-3': 'Here, the distortions are sizable, hence clear definition of mass and width, as well as branching ratios, are difficult.', '1712.00969-1-34-4': 'A commonly used approach to get a uniform result, makes use of poles in the complex plane, as we show below.', '1712.00969-1-35-0': 'd) Pole positions.', '1712.00969-1-35-1': 'As renowned, a theoretically (and lately experimentally as well, see e.g. the [MATH] in PDG) stable approach to describe unstable states is based on the determination of the position of the corresponding poles.', '1712.00969-1-35-2': 'In the present case, one obtains, quite remarkably, two poles on the IV Riemann Sheet.', '1712.00969-1-35-3': 'As seen in Sec. [REF], this is the sheet relevant for a system with two decay channels at energy above both thresholds.', '1712.00969-1-35-4': 'In the isospin limit, the IV RS reduces to the usual II RS.', '1712.00969-1-35-5': 'In the following, the indeterminacy of the pole masses is estimated to be [MATH] MeV, close to the error obtained for the parameter [MATH], and the indeterminacy of of the pole width(s) as [MATH] MeV, coincident with the error for [MATH] (see Table [REF]).', '1712.00969-1-35-6': 'In fact, the errors of these strictly correlated quantities must have a similar magnitude.', '1712.00969-1-35-7': 'The closest pole to the Riemann axis reads: [EQUATION] hence [EQUATION] which agrees quite well with the PDG values [CITATION].', '1712.00969-1-35-8': 'This pole is closely related to the seed charmonium state and to the position and width of the peak.', '1712.00969-1-35-9': 'Furthermore, a second broader pole appears at lower energy: [EQUATION]', '1712.00969-1-35-10': 'This pole is dynamically generated, and it is related to the the deformation of the signal on the left side.', '1712.00969-1-35-11': 'It is also referred to as a companion pole emergent due to the strong dynamics.', '1712.00969-1-35-12': 'The situation is very much reminiscent of the light scalar [MATH] state at lower energy, which was interpreted as a dynamically generated companion pole of [MATH] in Ref. [CITATION].', '1712.00969-1-35-13': 'For additional recent works on this state, see Refs. [CITATION], and references therein.', '1712.00969-1-36-0': 'e) The coupling [MATH] is an outcome of our fit, then the widths [MATH] can be easily computed from Eq. [REF].', '1712.00969-1-36-1': 'Results are shown in Table [REF].', '1712.00969-1-36-2': 'The value for [MATH] is smaller than the one given by PDG fit of [MATH] eV by a factor two, but it is compatible with the analysis in Ref. [CITATION], that gives [MATH] eV.', '1712.00969-1-36-3': 'However, the mismatch of our result with the PDG estimate, that nevertheless lists a quite broad range of values from different experiments, shows that this decay rate should be further investigated in the future.', '1712.00969-1-36-4': 'Also the experimental identification of the decays into [MATH] and [MATH] could help to distinguish among different theories.', '1712.00969-1-37-0': 'f) In our fit, we consider only the [MATH] resonance as a virtual state of the process [MATH].', '1712.00969-1-37-1': 'As discussed above, a possible mixing between bare [MATH] states with quantum numbers [MATH] and [MATH] is automatically taken into account in our bare field [MATH] entering into the Lagrangian defined in Eq. [REF].', '1712.00969-1-37-2': 'Indeed, the bare field, i.e., the field prior to the dressing by mesonic loops, represents the diagonal quark-antiquark state, while the [MATH] is a mixture of [MATH] and [MATH] configurations, with the [MATH] as its orthogonal state.', '1712.00969-1-37-3': 'Moreover, the effect of [MATH] can be taken into account through rescattering (see next subsection).', '1712.00969-1-37-4': 'Yet, the role of the [MATH] as an additional exchange in the reaction [MATH] was not taken into account.', '1712.00969-1-37-5': 'Being [MATH] off-shell in the energy of interest, the propagator is expected to suppress this amplitude; moreover, also its coupling to channel [MATH] is expected to be small, due to the form-factor.', '1712.00969-1-37-6': 'Indeed, in Refs. [CITATION], the [MATH], [MATH] loops, and [MATH]-[MATH] rescattering have been considered, leading to two fits that are in good agreement with one of the experiments (Ref. [CITATION]).', '1712.00969-1-37-7': 'In one of the fits the [MATH] contribution is very small, in agreement with our results.', '1712.00969-1-37-8': 'On the contrary, a second fit with a sizable role of [MATH], has also been found.', '1712.00969-1-37-9': 'Our results strongly favor the first fit of Ref. [CITATION].', '1712.00969-1-37-10': 'Note, in Refs. [CITATION] the pole structure was not examined.', '1712.00969-1-38-0': '## Variation of the coupling and rescattering', '1712.00969-1-39-0': 'First, we study the effect of varying the coupling constant [MATH].', '1712.00969-1-39-1': 'Results are shown in Fig. [REF] for the cross-section, and in Table [REF] for pole positions.', '1712.00969-1-39-2': 'In Quantum Chromodynamics (QCD), a change in such coupling is equivalent to vary the number of colors according to the scaling [MATH].', '1712.00969-1-39-3': 'As expected, a smaller value [MATH] leads to a Breit-Wigner-like shape.', '1712.00969-1-39-4': 'While the first pole gets closer to the real axis, the second pole moves deeper in the complex plane and approaches the [MATH] threshold.', '1712.00969-1-39-5': 'For a certain critical value, in this case [MATH], the second pole completely disappears.', '1712.00969-1-39-6': 'On the contrary, increasing [MATH] implies that the first pole runs away from the real axis, while the second pole approaches it.', '1712.00969-1-39-7': 'Eventually, the dynamically generated pole can get even closer to the real axis than the seed pole.', '1712.00969-1-40-0': 'Finally, it is interesting to study the final-state rescattering, which we show in Fig. [REF].', '1712.00969-1-40-1': 'It depends on the rescattering coupling [MATH] defined in Sec. [REF], which is a free parameter with dimensions GeV[MATH], according to Eq. [REF].', '1712.00969-1-40-2': 'In Fig. [REF] we use [MATH] GeV[MATH].', '1712.00969-1-40-3': 'The visible effect, comparatively with the result without rescattering, consists in pronouncing slightly the asymmetry of the resonance, which might be an improvement to the fit, but that we cannot disentangle with the current experimental precision.', '1712.00969-1-40-4': 'For larger [MATH] values, the cross section becomes inadequate to describe the data, from where we conclude that the rescattering of final states plays a minor role in the dynamics.', '1712.00969-1-41-0': '## Estimating the size of the wave-function', '1712.00969-1-42-0': 'The size of the [MATH] resonance is intrinsically related to the empirical form factor that one needs to include at the effective vertices.', '1712.00969-1-42-1': 'In the isospin limit the form factor reads [MATH], whose Fourier transform is proportional to [MATH], from where the variance [MATH] of the distribution in the coordinate space is given by [MATH].', '1712.00969-1-42-2': 'Here, we would like to understand if the value [MATH] MeV found in our fit is physically meaningful.', '1712.00969-1-42-3': 'Namely, it is a factor two smaller than the value found in Ref. [CITATION] for the light scalar kaonic system.', '1712.00969-1-42-4': 'For this purpose, we estimate the size of [MATH] by employing the Schrodinger model defined in Ref. [CITATION], to which we refer for all technical details.', '1712.00969-1-43-0': 'We consider the coupled system [MATH], where [MATH] is a charmonium system with quantum numbers [MATH], and [MATH] is a meson-meson decay channel.', '1712.00969-1-43-1': 'The model in Ref. [CITATION] has two parameters that we tune to obtain the pole of [MATH].', '1712.00969-1-43-2': 'For each complex energy, the root mean square (r.m.s.) is given by [EQUATION] where the quantity [MATH] is the radial wave-function.', '1712.00969-1-43-3': 'We find a r.m.s. value that varies, respectively, within [MATH] GeV[MATH] fm) and [MATH] GeV[MATH] fm).', '1712.00969-1-43-4': 'The seemingly large value for a heavy system is essentially due to the fact that the [MATH] is dominantly a [MATH]-wave.', '1712.00969-1-43-5': 'In Fig. [REF] we plot the [MATH] distribution for the two component wave-function.', '1712.00969-1-43-6': 'If we compare it to the standard deviation, i.e. [EQUATION] we get a cutoff between [MATH] MeV, pretty close to the parameter [MATH] of our fit.', '1712.00969-1-43-7': 'The range of r.m.s values we get is yet comparable with the [MATH] parameter in Ref. [CITATION] for the same system, found to be [MATH], the physical meaning of it being an average distance for emerging the meson pair from the interaction region, which is consistent with our interpretation.', '1712.00969-1-44-0': '# Summary and Conclusions', '1712.00969-1-45-0': 'Vector mesons can be produced by a single virtual photon, hence they play a crucial role in the experimental study of QCD.', '1712.00969-1-45-1': 'In the light sector, ground-state vector mesons represent the lightest (almost) ideal nonet of [MATH] states with a mass of about [MATH] where [MATH] is the constituent quark mass of about [MATH] MeV.', '1712.00969-1-45-2': 'The production and decays of the [MATH] and [MATH] meson in [MATH] improved our understanding of QCD, see e.g. the theoretical review in Ref. [CITATION], where also the so-called Vector Meson Dominance is presented, and the experimental review in Ref. [CITATION].', '1712.00969-1-45-3': 'Moreover, also excited vector states below 2 GeV exist (see Ref. [CITATION] and references therein).', '1712.00969-1-45-4': 'The renowned charmonium vector state [MATH] opened a new era in hadron physics and is one of the best studied resonances (cf. [CITATION]).', '1712.00969-1-46-0': 'Here, we have focused on an orbital excitation of [MATH], the meson [MATH], whose mass lies just above the [MATH] threshold.', '1712.00969-1-46-1': 'For this reason, this resonance is extremely interesting.', '1712.00969-1-46-2': 'The cross-section [MATH] encloses nonperturbative phenomena nearby the OZI-allowed open decay channel [MATH].', '1712.00969-1-46-3': 'The production and decay to all hadrons shows evidence of a deformation on the left energy side of the resonance [CITATION] that is not obvious in the [MATH] cross section only, as we have verified from our line-shape results.', '1712.00969-1-46-4': 'Within the presented nonperturbative effective Lagrangian model, with only one bare [MATH] seed state [MATH] at the start, we found not only one, but two poles underlying the enhancement: a standard seed pole, very close to the one determined by BES in Ref. [CITATION], and a broader dynamically generated pole at lower energy, being an additional companion pole (see Eqs. [REF] and [REF] for the numerical values).', '1712.00969-1-46-5': 'The existence of the second pole gives us a new insight over the relevance of dynamical effects, here represented by the [MATH] coupled-channel loops, besides the expected kinematical interferences.', '1712.00969-1-47-0': 'The width of about 24 MeV to channel [MATH], that we obtain from the first pole, is consistent with the experimental measurements to this channel, and with the branching fraction estimated to be about [MATH] in Ref. [CITATION], considering that measurements of the width in decays to all hadrons are a few MeV higher (cf. Ref. [CITATION]).', '1712.00969-1-47-1': 'A natural outlook is the inclusion of the many OZI-suppressed hadronic decay channels and the study of their influence on the pole position and on the line-shape.', '1712.00969-1-47-2': 'Other effects, such as final state rescattering and the contribution of the tail of the [MATH] to the amplitude, are expected to improve the results, but we predict their influence to be small.', '1712.00969-1-47-3': 'Better statistics in the data may disentangle such variations.', '1712.00969-1-47-4': 'A related topic is the evaluation of mixing with the vector glueball, whose mass is predicted to be at about 3.8 GeV by lattice QCD [CITATION] (for a general introduction on glueballs, see Ref. [CITATION]).', '1712.00969-1-47-5': 'Namely, mixing effects can be interesting to study the OZI suppressed decays (for decays of the glueball see Ref. [CITATION]).', '1712.00969-1-48-0': 'Finally, the same theoretical approach used in this work, and tested on the well-known resonance [MATH], can be applied to other resonances whose nature is not yet clarified, the so-called [MATH] and [MATH] states [CITATION].', '1712.00969-1-48-1': 'While for the [MATH] the additional pole is responsible for a deformation on the left side but not for an additional peak, a coupling constant which is larger could generate a peak, as shown in Fig. [REF], and in Ref. [CITATION] as an explanation of the resonance [MATH] in the light scalar sector.', '1712.00969-1-48-2': 'It is well possible that some of the [MATH] and [MATH] resonances arise as companion poles due to a strong coupling of a standard charmonium seed state to lighter mesonic resonances.', '1712.00969-1-48-3': 'The strong dressing of the original state with meson-meson clouds generates new poles and new peaks.', '1712.00969-1-48-4': 'An intriguing possibility is that the additional peak arises at higher energy than the seed state, see Ref. [CITATION] for the phenomenological description of this phenomenon in the light sector.', '1712.00969-1-48-5': 'Namely, if new decay channels open just above the mass of the seed state, companion poles and related peaks may arise on the right side.'}

{'1712.00969-2-0-0': 'We study the non-Breit-Wigner line-shape of the [MATH] resonance, predominantly a [MATH] state, using an unitarized effective Lagrangian approach, including the one-loop effects of the nearby thresholds [MATH] and [MATH].', '1712.00969-2-0-1': 'A fit of the theoretical result to the total cross-section [MATH] is performed, leading to a good description of data ([MATH]).', '1712.00969-2-0-2': 'The partial cross sections [MATH] and [MATH] turn out to be separately in good agreement with the experiment.', '1712.00969-2-0-3': 'We find a pole at [MATH] MeV, that is within the Particle Data Group (PDG) mass and width estimation for this state.', '1712.00969-2-0-4': 'Quite remarkably, we find an additional, dynamically generated, companion pole at [MATH] MeV, which is responsible for the deformation on the lower energy side of the line-shape.', '1712.00969-2-0-5': 'The width for the leptonic decay [MATH] is 112 eV, hence smaller than the PDG fit of [MATH] eV, yet in agreement with a recent experimental study.', '1712.00969-2-1-0': '# Introduction', '1712.00969-2-2-0': 'The [MATH] resonance is a vector state that was first detected at SPEAR [CITATION] in 1977.', '1712.00969-2-2-1': 'The signal was fitted with a pure [MATH]-wave Breit-Wigner distribution.', '1712.00969-2-2-2': 'More recently, the interest on this state was revived, and the parameters are nowadays fitted by the Particle Data group (PDG) as [MATH] MeV for the mass and [MATH] MeV for the width [CITATION].', '1712.00969-2-2-3': "According to the 'quark model review' of the PDG, the [MATH] resonance is classified as a [MATH] charmonium state [CITATION], the first one above the [MATH] threshold, which is the reason for its relatively large width.", '1712.00969-2-2-4': 'As a consequence, besides kinematic interference, additional nonperturbative effects are expected.', '1712.00969-2-3-0': 'Indeed, the line-shape of the [MATH] resonance turned out to be quite anomalous.', '1712.00969-2-3-1': 'Although other experiments have been performed with observations in [MATH] channel [CITATION], the deformation on the line-shape of the [MATH] was made clear by the BES Collaboration data in the [MATH] annihilation to hadrons [CITATION].', '1712.00969-2-3-2': 'The existence of a second resonance was suggested, although possible dynamical effects, generated by the [MATH] threshold were not discarded.', '1712.00969-2-3-3': 'A deformation of a line-shape due to the superposition of two resonances has been discussed before e.g., for the scalar kaon [CITATION], where, besides the dominant [MATH] state, an additional dynamically generated state arises from the continuum, the well known [MATH].', '1712.00969-2-3-4': 'Such effect has not been discussed before for the vector charmonium.', '1712.00969-2-4-0': 'Various analysis of the [MATH] have been performed.', '1712.00969-2-4-1': 'In Refs. [CITATION] fits were computed taking into account not only the [MATH] interference but also the tail of the [MATH].', '1712.00969-2-4-2': 'Such inclusion is natural since the [MATH] should be a mixed state [MATH]-[MATH].', '1712.00969-2-4-3': 'In Ref. [CITATION], the deformation from the right side of the resonance, i.e. a dip structure, is explained by the interference with the [MATH] kinematical background which is higher for larger relative momentum.', '1712.00969-2-4-4': 'The same dip is reproduced in [CITATION], using [MATH] background only, in [CITATION], where in addition, the continuum of light hadrons is removed, and in Refs. [CITATION] and [CITATION], using Fano resonances.', '1712.00969-2-4-5': 'In [CITATION] BES measured an unexpectedly large branching fraction [MATH] non[MATH] of about [MATH], a result that is not contradicted by CLEO in [CITATION], within errors.', '1712.00969-2-4-6': 'Predictions for such non-[MATH] continuum were made including the tails of [MATH] and [MATH], [MATH] and [MATH] decays [CITATION], other excited [MATH] states [CITATION], the [MATH] channel [CITATION], other hadronic decays and radiative decays [CITATION], and final state interactions [CITATION], which in any case do not sum up to the value of [MATH].', '1712.00969-2-4-7': 'Yet, since the phase space to [MATH] is not excessively large, it is likely that the missing decays are simply the sum of all the many Okubo-Zweig-Iizuka (OZI)-suppressed hadronic decays, that have not been studied systematically in the theory.', '1712.00969-2-4-8': 'Estimations for the [MATH] production via [MATH] annihilation are made in [CITATION], which may possibly be measured at the PANDA experiment [CITATION].', '1712.00969-2-4-9': 'Mass estimations for the [MATH] have also been made on the lattice [CITATION].', '1712.00969-2-4-10': 'For a review of the charmonium states, see Ref. [CITATION].', '1712.00969-2-5-0': 'In this work, we study the properties of the [MATH] by analyzing its production through electron-positron annihilation, and subsequent decay into [MATH] pairs (for a preliminary study, see Ref. [CITATION]).', '1712.00969-2-5-1': 'Our starting point is a vector charmonium seed state, which gets dressed by "clouds" of [MATH] and [MATH] mesons.', '1712.00969-2-5-2': 'Our aim is to study the deformation seen on the left side of the resonance in Ref. [CITATION] with mesonic loops combined with the nearby thresholds.', '1712.00969-2-5-3': 'To this end, we use an effective relativistic Lagrangian approach in which a single vector state [MATH] is coupled to channels [MATH] and [MATH], as well as to lepton pairs.', '1712.00969-2-5-4': 'The propagator of [MATH] is calculated at the resummed one-loop level and fulfills unitarization requirements.', '1712.00969-2-5-5': 'Then, we perform a fit of the four parameters of our approach, i.e. the effective couplings of [MATH] to [MATH] and of [MATH] to leptons, the mass of [MATH], and a cutoff responsible for the finite dimension of the [MATH] meson, to the experimental cross-section of the reaction [MATH] in Refs. [CITATION] and [CITATION], in the energy region up to 100 MeV above the [MATH] threshold.', '1712.00969-2-5-6': 'We assume that the [MATH] resonance dominates in this energy range.', '1712.00969-2-5-7': 'We obtain a very good description of the data, which in turn allows us to determine in a novel and independent way the mass, width, and branching ratios of the [MATH].', '1712.00969-2-5-8': 'Moreover, we study in detail, to our knowledge for the first time, the poles of this state.', '1712.00969-2-5-9': 'In fact, a pole was found in Ref. [CITATION], using Fano resonances, at about [MATH] MeV, though in lesser detail.', '1712.00969-2-5-10': 'Quite remarkably, we find two poles for this resonance, one which roughly corresponds to the peak of the resonance, the seed pole, and one additional dynamically generated pole, responsible for the enhancement left from the peak, which emerges due to the strong coupling between the seed state and the mesonic loops.', '1712.00969-2-5-11': 'This is a companion pole, similar to the one found in the kaonic system in Ref. [CITATION].', '1712.00969-2-6-0': 'As a consequence of the determined parameters, we also show that the cross sections [MATH] and [MATH] agree separately to data.', '1712.00969-2-6-1': 'Being the resonance not an ideal Breit-Wigner one, different definitions for the partial widths are compared to each other.', '1712.00969-2-6-2': 'As a verification of our theoretical approach, we artificially vary the intensity of the coupling in order to show its effect on the line-shape of the resonance.', '1712.00969-2-6-3': 'Smaller couplings lead to a narrower Breit-Wigner-like shape, while larger couplings to an even larger deformation, which eventually gives rise to two peaks, as in Ref. [CITATION].', '1712.00969-2-6-4': 'In addition, the trajectory of each pole is analyzed, confirming the effects on the line-shape.', '1712.00969-2-7-0': 'The paper is organized as it follows.', '1712.00969-2-7-1': 'In Sec. [REF] the model is introduced and various theoretical quantities, such as the resummed propagator, spectral function, cross-section, and computation of the poles, are presented.', '1712.00969-2-7-2': 'In Sec. [REF] we show our results: first, we present a fit to data and examine its consequences, then we study the position and trajectory of the poles.', '1712.00969-2-7-3': 'In Sec. [REF] conclusions are drawn.', '1712.00969-2-7-4': 'Some important technical details and comparative results are discussed in the Appendices.', '1712.00969-2-8-0': '# An Effective Model', '1712.00969-2-9-0': 'In this section, we present the Lagrangian of the model and evaluate the most relevant theoretical quantities, namely the resummed propagator and the spectral functions, needed to calculate the cross section for the processes [MATH] and [MATH], via production of the [MATH], and with [MATH] one-loops.', '1712.00969-2-10-0': '## The strong Lagrangian and dispersion relations', '1712.00969-2-11-0': 'We aim to write down a Lagrangian for the strong interaction between a single vectorial resonance and two pseudoscalar mesons.', '1712.00969-2-11-1': 'The bare resonance [MATH] is identified with a bare quarkonium [MATH] state, with predominant quantum numbers [MATH], but admixtures of other quantum numbers, most notably the [MATH] are possible.', '1712.00969-2-11-2': 'In our model, however, neither the quarkonium structure nor the orbital angular momentum mixing are explicit, since we work with mesonic degrees of freedom in the basis of total angular momentum.', '1712.00969-2-11-3': 'The effective strong Lagrangian density, for the charmonium state [MATH], reads [EQUATION] which corresponds to the simplest interaction among [MATH] and its main decay products, the pseudoscalar pairs [MATH] and [MATH], each vertex represented in Fig. [REF].', '1712.00969-2-11-4': 'In this work, we shall take into account the small mass difference between [MATH] and [MATH], i.e. isospin breaking for the masses, but we shall keep the same coupling constant, i.e. isospin symmetry for the decays, and we define [MATH].', '1712.00969-2-11-5': 'The full Lagrangian can be found in [REF].', '1712.00969-2-12-0': 'The free propagator of a vector field [MATH], without any loops, with mass [MATH] and momentum [MATH], is given by [EQUATION] where the term in parenthesis is the sum over the three polarization states of a vector.', '1712.00969-2-12-1': 'We include the resummed one-loop effect as shown in Fig. [REF], leading to the full propagator of [MATH] [EQUATION] where [MATH] is the loop-function, consisting of two contributions, the [MATH] loops and the [MATH] loops.', '1712.00969-2-12-2': 'In particular, we get [EQUATION] where the function [MATH], with [MATH], reads [EQUATION] where [MATH] is the mass of the meson circulating in the loop (either [MATH] or [MATH]), [MATH] is the four-momentum of the loop, and [MATH] (see [REF] for computation details).', '1712.00969-2-12-3': 'Note, in the reference frame of the decaying particle, it holds the relation [EQUATION]', '1712.00969-2-12-4': 'An important element for our discussion is the vertex (or cutoff) function [MATH] entering in Eq. [REF].', '1712.00969-2-12-5': 'This is a form-factor that is needed to account for the unknown black vertices in Fig. [REF], due to the fact that mesons are not elementary particles [CITATION].', '1712.00969-2-12-6': 'The cutoff function must ensure the convergence of the integral.', '1712.00969-2-12-7': 'We choose the following Gaussian form [EQUATION] where the second term is built up for convenience.', '1712.00969-2-12-8': 'In the isospin limit, [MATH] reduces to [MATH].', '1712.00969-2-12-9': 'The parameter [MATH], introduced in Eq. [REF], is on the same level of all the other parameters of our model and will be evaluated through our fit to data.', '1712.00969-2-12-10': 'Moreover, even if our form factor depends on the three-momentum [MATH] only, hence strictly valid in the rest frame of the decaying particle, covariance can be recovered by properly generalizing the vertex function, see details in Ref. [CITATION] .', '1712.00969-2-12-11': 'However, the exponential form for the cutoff is very typical and has been used in many different approaches in hadron physics, e.g. Ref. [CITATION].', '1712.00969-2-12-12': 'For completeness, in [REF] we shall also present the results for a different vertex function.', '1712.00969-2-13-0': 'We now turn back to the study of the propagator.', '1712.00969-2-13-1': 'In [REF], it is shown that the relevant quantity is the transverse part of the propagator that, in the rest frame of the decaying particle, and for a single loop, reads [EQUATION] and similarly for the sum [EQUATION]', '1712.00969-2-13-2': 'The scalar part of the one-loop resummed dressed propagator reads [EQUATION]', '1712.00969-2-13-3': 'Note, this expression of the propagator fulfills unitarity (cf. Sec. [REF] below) and it is accurate as long as further contributions to the self-energy are small .', '1712.00969-2-14-0': 'The self-energy in Eq. [REF] can either be computed through the integration (as in [REF]), or using dispersion relations.', '1712.00969-2-14-1': 'We follow the latter.', '1712.00969-2-14-2': 'To this end, we first decompose [MATH] into its real and imaginary parts [EQUATION]', '1712.00969-2-14-3': 'According to the optical theorem, the imaginary part [MATH] (dispersive term) is given by [EQUATION] with [MATH] being the center-of-mass momentum of the final mesons.', '1712.00969-2-14-4': 'The partial decay widths of [MATH] and [MATH] are then calculated as [EQUATION] where the Lorentz invariant amplitudes squared, computed from [MATH] in Eq. [REF], are given by [EQUATION]', '1712.00969-2-14-5': 'Note, by choosing the form factor as function of the energy [[MATH]], when the imaginary part of the loop in Eq. [REF] is taken, the replacement [MATH] is performed (see Eqs. [REF] and [REF]).', '1712.00969-2-14-6': 'Then, the function [MATH] directly enters in various expressions.', '1712.00969-2-15-0': 'Finally, the on-shell partial and total decay widths are [EQUATION]', '1712.00969-2-15-1': 'Once the imaginary part of the loop is known, the real part, the function [MATH], is computed from the dispersion relation [EQUATION] where [MATH] is zero below threshold.', '1712.00969-2-15-2': 'Convergence is guaranteed by the cutoff function.', '1712.00969-2-15-3': 'As we shall see in Sec. [REF], the real part [MATH] causes a distortion in the line-shape due to the continuous shifting of the physical mass of the resonance with the energy.', '1712.00969-2-15-4': 'Note, when the energy [MATH] is complex, the function [MATH] reads (away from the real axis): [EQUATION]', '1712.00969-2-15-5': 'This formula clearly shows that in the first Riemann Sheet (RS) the complex function [MATH] is regular everywhere on the [MATH]-complex plane, besides a cut from [MATH] to infinity.', '1712.00969-2-15-6': 'In particular, [MATH] in all directions .', '1712.00969-2-15-7': 'Furthermore, although it is not strictly needed, since the integral in Eq. [REF] is already convergent, we use the once-subtracted dispersion relation, with subtraction in point [MATH], for our convenience.', '1712.00969-2-15-8': 'Hence, the total loop-function is given by [EQUATION] and, the final dressed propagator of the [MATH] meson is [EQUATION]', '1712.00969-2-15-9': 'In this way, the parameter [MATH] in the propagator corresponds to the mass of the particle defined as [EQUATION]', '1712.00969-2-15-10': 'Other definitions for the mass are possible, such as the position of the peak, or the real part of the pole, as we shall see below.', '1712.00969-2-16-0': '## Poles', '1712.00969-2-17-0': 'In order to find poles, the energy [MATH] is analytically continued to the complex plane ([MATH]).', '1712.00969-2-17-1': 'In case of two decay channels, the Riemann surface is composed by four RSs.', '1712.00969-2-17-2': 'Poles are found in the unphysical sheet which results in the physical sheet when the energy is real and the resonance can be seen.', '1712.00969-2-17-3': 'Above both thresholds, this corresponds to the condition [MATH] and [MATH] when [MATH] and [MATH], i.e., the third RS.', '1712.00969-2-17-4': 'Poles are given when the denominator of the propagator in Eq. [REF] is zero in the correct RS, i.e., [EQUATION] where [EQUATION] where the subscripts [MATH] and [MATH] stand for the first and third RS .', '1712.00969-2-17-5': 'While in the first RS [MATH] is regular everywhere apart from the cut(s) on the real axis (see Eq. [REF] and subsequent discussion), in the other RSs the imaginary part continued to complex plane, [MATH] appears.', '1712.00969-2-17-6': 'As a consequence, the vertex function [MATH] generates a singular point at the complex infinity.', '1712.00969-2-17-7': 'When other choices for the vertex function are made, such as the one in [REF], other singularity types are generated.', '1712.00969-2-17-8': 'In general, any nontrivial function [MATH] will display singularities appearing in RSs different from the first one.', '1712.00969-2-18-0': '## Coupling to leptons', '1712.00969-2-19-0': 'The available experimental data comes from the production process [MATH] (see Fig. [REF]), therefore we need to couple the state [MATH] to leptons.', '1712.00969-2-19-1': 'The corresponding interaction Lagrangian is defined by [EQUATION] which is the simplest interaction among a massive vector field [MATH] and a fermion ([MATH])-antifermion pair.', '1712.00969-2-19-2': 'The coupling [MATH], between [MATH] and the electron-positron pair, is here considered to be the same between [MATH] and all leptonic pairs.', '1712.00969-2-19-3': 'It is the overall strength for the annihilation of the leptonic pair [MATH] into one photon, and further conversion into the [MATH] vector.', '1712.00969-2-19-4': 'In Eq. [REF], we describe the process [MATH] through a single effective vertex proportional to [MATH], but, as shown in Fig. [REF], this coupling constant emerges via an intermediate virtual photon that converts into a charmonium state.', '1712.00969-2-20-0': 'The corresponding decay into leptons reads [EQUATION] where [MATH] is the leptonic mass.', '1712.00969-2-20-1': 'In principle, the loops of leptons should be included in the total one-loop function of resonance [MATH], obtaining [MATH], where dots refer to other possible but suppressed hadronic decay channels.', '1712.00969-2-20-2': 'However, the loop contribution of [MATH] is definitely negligible w.r.t. [MATH].', '1712.00969-2-20-3': 'Here, we shall simply consider [MATH]', '1712.00969-2-21-0': '## Spectral function and cross section', '1712.00969-2-22-0': 'The unitarized spectral function as a function of energy [MATH], equivalent to the running mass of the [MATH] resonance, is given by [EQUATION] that has the general shape of a relativistic Breit-Wigner distribution, distorted by the loop-function [MATH] (cf. Eq. [REF]).', '1712.00969-2-22-1': 'When no poles below threshold emerge, the normalization above threshold is guaranteed (unitarity): [EQUATION]', '1712.00969-2-22-2': 'The quantity [MATH] is interpreted as the probability that the state [MATH] has a mass between [MATH] and [MATH].', '1712.00969-2-22-3': 'The cross section for [MATH] takes the form [EQUATION]', '1712.00969-2-22-4': 'Hence, the experimental data for this cross section give us direct access to the imaginary part of the propagator of the meson [MATH].', '1712.00969-2-22-5': 'The corresponding amplitude, leading to [MATH], is depicted in Fig. [REF].', '1712.00969-2-23-0': 'One also defines the partial spectral functions as [CITATION]: [EQUATION]', '1712.00969-2-23-1': 'Then, the partial cross sections are given by: [EQUATION]', '1712.00969-2-24-0': '# Line-shapes and poles', '1712.00969-2-25-0': 'In this section, we present our fit of Eq. [REF] to experimental data and its consequences, and the notable existence, position, and trajectory of two poles underlying the [MATH].', '1712.00969-2-26-0': '## Fit to data and consequences', '1712.00969-2-27-0': 'As a first necessary step, we determine the four free parameters of our model [EQUATION] defined in Sec. [REF], by performing a fit to the cross section data for the process [MATH].', '1712.00969-2-27-1': 'We use 14 experimental points published in Ref. [CITATION], and the theoretical expression in Eq. [REF].', '1712.00969-2-27-2': 'Note, we use only the data of Ref. [CITATION] for the fit, since the data of Ref. [CITATION] are contained in the data samples of Ref. [CITATION] (the data sets of March 2001 quoted in these two papers are the same [CITATION]).', '1712.00969-2-27-3': 'The fit to data is shown in Fig. [REF], and the values of the parameters in the set [REF] are presented in Table [REF].', '1712.00969-2-27-4': 'We get the value [MATH], which shows that a very good description of the data is achieved.', '1712.00969-2-28-0': 'The errors of the parameters entering the fit are estimated by the square root of the corresponding Hessian matrix (this is a result of the standard procedure according to in which one defines a new set of parameters that diagonalize the Hessian matrix, see e.g. Ref. [CITATION] for details).', '1712.00969-2-29-0': 'There are various consequences of the fit that we discuss in detail:', '1712.00969-2-30-0': "a) The value of the mass [MATH] MeV (corresponding to the zero of the real part of the inverse's propagator, see Eq. ([REF])) is in well agreement with the current PDG fit of [MATH] MeV [CITATION].", '1712.00969-2-31-0': 'b) In Fig. [REF] we present a comparison between theory and the data for the cross sections [MATH] and [MATH], using the parameters in Table [REF].', '1712.00969-2-31-1': 'The good agreement shows that the theory is able to describe the two partial cross sections separately, without the need of some extra parameter, and in agreement with isospin symmetry.', '1712.00969-2-32-0': 'c) The branching ratios and partial and total ([MATH]) widths are presented in Table [REF].', '1712.00969-2-32-1': 'The partial widths, evaluated on-shell, are defined in Eqs. [REF]-[REF].', '1712.00969-2-32-2': 'The branching ratios are in agreement with those quoted in PDG for the [MATH].', '1712.00969-2-32-3': 'The ratio [MATH] of [MATH] is compatible with the PDG values, which range from [MATH] to [MATH].', '1712.00969-2-32-4': 'However, as shown in Fig. [REF], the functions [MATH] and [MATH] strongly vary in the region of interest.', '1712.00969-2-32-5': 'Alternatively, one may use a different definition, cf. [CITATION], to obtain the partial decay widths and the full decay width to [MATH], by integrating over the spectral function as [EQUATION]', '1712.00969-2-32-6': 'A somewhat naive but still useful alternative determination of the mass is the value corresponding to the peak and the half-height width: [EQUATION] where we estimated the error of the peak mass as [MATH] MeV, close to the determined error for the parameter [MATH], and the error of the distribution width as [MATH] MeV as well, coincident to the error for [MATH] (see Table [REF]).', '1712.00969-2-32-7': 'These evaluations show that the resonance [MATH] is far from being an ideal Breit-Wigner.', '1712.00969-2-32-8': 'While all these different approaches coincide for states with a very small width, here the distortions are sizable, hence clear definitions of mass and width, as well as branching ratios, are difficult.', '1712.00969-2-32-9': 'A commonly used approach to get a uniform result, makes use of poles in the complex plane, as we show in Sec. [REF].', '1712.00969-2-33-0': 'This discussion shows that there is no unique definition of the total and the partial widths.', '1712.00969-2-33-1': 'This fact also renders a direct comparison with the fit of the PDG quite difficult.', '1712.00969-2-33-2': 'Namely, according to PDG, one has [MATH] MeV, [MATH] MeV, and [MATH] MeV [CITATION], hence our results for the partial decay widths evaluated on shell and presented in Table [REF], first column, are somewhat smaller.', '1712.00969-2-33-3': 'However, as visible in Fig. [REF], the corresponding function vary strongly in the energy region of interest.', '1712.00969-2-33-4': 'Moreover, instead of using theoretical partial widths which cannot be uniquely defined, one should stress that our approach correctly describes the data for channels [MATH] and [MATH] separately, as visible in Fig. [REF], and as described in point b) above.', '1712.00969-2-34-0': 'Another commonly used approach to circumvent all these definition problems mentioned above, is to move away from the real axis and to study the pole(s) in the complex plane.', '1712.00969-2-34-1': 'As we shall see in detail in Sec. [REF], the seed pole of [MATH] corresponds to a larger decay width of [MATH] MeV which is in very well agreement with PDG.', '1712.00969-2-35-0': 'd) Using the coupling [MATH] that outcomes from the fit, the widths [MATH] can be easily computed from Eq. [REF].', '1712.00969-2-35-1': 'Results are shown in Table [REF].', '1712.00969-2-35-2': 'The value for [MATH] is smaller than the one given by the PDG fit of [MATH] eV by a factor two, but it is compatible with the analysis in Ref. [CITATION], that gives [MATH] eV.', '1712.00969-2-35-3': 'The mismatch of our result with the PDG estimate, that nevertheless lists a quite broad range of values from different experiments, shows that this decay rate should be further investigated in the future.', '1712.00969-2-35-4': 'Also the experimental identification of the decays into [MATH] and [MATH] could help to distinguish among different models.', '1712.00969-2-36-0': 'e) The Gaussian vertex function together with the cutoff value [MATH] MeV takes into account in a simple way the composite nature of the resonance [MATH] and its nonlocal interaction with the [MATH] mesons.', '1712.00969-2-36-1': 'Microscopically the vertex function emerges from the nonlocal nature of both [MATH] and of the decay products [MATH] (for point-like part interaction, [MATH] goes to infinity), see Refs. [CITATION] for an explicit treatment.', '1712.00969-2-36-2': '(Indeed, the rather small value of [MATH] emerging from our fit is intuitively understandable by the fact that the resonance [MATH] is predominantly a [MATH] wave.', '1712.00969-2-36-3': 'A detailed study of this issue by using a microscopic model such as the one in Ref. [CITATION] will be given elsewhere).', '1712.00969-2-36-4': 'The vertex function is a crucial ingredient that defines our model and, among other properties, guarantees the finiteness of the results.', '1712.00969-2-36-5': 'The exponential form used here emerges naturally from various microscopic approaches, see Refs. [CITATION] and refs. therein.', '1712.00969-2-36-6': 'However, our results do not strongly depend on the precise choice of the vertex function as long as it is smooth and at the same time falls sufficiently fast.', '1712.00969-2-36-7': 'As we show in [REF], a hard cutoff would imply that the spectral function would fall abruptly to zero above a certain threshold; this unphysical behavior does not lead to any satisfactory description of data.', '1712.00969-2-36-8': 'On the other hand, as discussed in [REF], the avoidance of a form factor through a three-times subtracted dispersion relation does not lead to satisfactory results, thus confirming that the need of some vertex function for our treatment of mesonic loops is needed.', '1712.00969-2-36-9': 'Once the vertex function is fixed, the model is mathematically consistent at any energy: in fact, the important normalization condition reported in Eq. [REF] is obtained by formally integrating up to infinity (numerically, the normalization is verified by integrating up to [MATH] GeV).', '1712.00969-2-36-10': 'In turn, this means that, from a mathematical perspective, the value of the momentum of the emitted mesons [MATH] can be larger than [MATH].', '1712.00969-2-36-11': 'A different issue is the maximal energy up to which we shall trust our model.', '1712.00969-2-36-12': 'In fact, even if mathematically consistent, the model is limited because it takes into account only one vector resonance and neglects the contributions of other resonances, most notably [MATH] as well as the background (see the comments h) and i) about these topics) and the next opening thresholds, such as [MATH].', '1712.00969-2-36-13': 'We therefore trust the model in the energy range starting from the lowest threshold, [MATH], up to at most [MATH] GeV, when the line-shape goes down and, as described in Refs. [CITATION], the role of the background becomes important.', '1712.00969-2-37-0': 'f) In our fit, we consider only the [MATH] resonance as a virtual state of the process [MATH].', '1712.00969-2-37-1': 'As discussed above, a possible mixing between bare [MATH] states with quantum numbers [MATH] and [MATH] is automatically taken into account in our bare field [MATH] entering into the Lagrangian defined in Eq. [REF].', '1712.00969-2-37-2': 'Indeed, the bare field, i.e., the field prior to the dressing by mesonic loops, represents the diagonal quark-antiquark state, while the [MATH] is a mixture of [MATH] and [MATH] configurations, with the [MATH] as its orthogonal state.', '1712.00969-2-37-3': 'Yet, the role of the [MATH] as an additional exchange in the reaction [MATH] was not taken into account.', '1712.00969-2-37-4': 'Being [MATH] off-shell in the energy of interest, the propagator is expected to suppress this amplitude; moreover, also its coupling to channel [MATH] is expected to be small, due to the form-factor.', '1712.00969-2-37-5': 'Indeed, in Refs. [CITATION], the [MATH], [MATH] loops, and [MATH] rescattering have been considered, leading to fits that are in good agreement with the experiment in Ref. [CITATION].', '1712.00969-2-37-6': 'In both references the [MATH] contribution is quite small, in agreement with our results.', '1712.00969-2-37-7': 'However in their case the [MATH] was still necessary to obtain a good fit, whereas in our model it is not.', '1712.00969-2-37-8': 'Note, in Refs. [CITATION] the pole structure was not examined.', '1712.00969-2-38-0': 'g) In this work we did not consider a four-leg interaction process of the type [MATH] that can be formally described by a four-vertex proportional to [MATH].', '1712.00969-2-38-1': 'Such a rescattering describes elastic scattering, induced by a direct four-body interaction, and also as an effective description of various [MATH]-channel exchanges between [MATH].', '1712.00969-2-38-2': 'Our model can describe properly the data without this contribution, hence the inclusion of a four-leg interaction is not needed to improve the agreement with the data.', '1712.00969-2-38-3': 'This does not necessarily mean that its magnitude is small, but simply that one can hardly disentangle its role from the [MATH]-loops that we have considered in our approach.', '1712.00969-2-38-4': 'In addition, there is a more formal point that should be taken into account: one can in principle redefine the field [MATH] where [MATH] is a parameter on which no physical quantity should depend on [CITATION].', '1712.00969-2-38-5': 'The redefinition would however lead to some rescattering terms.', '1712.00969-2-38-6': 'Of course, the independence on the field redefinition can be fully accomplished only if one could solve the theory exactly, since in any approximation scheme differences would still persist at a given order.', '1712.00969-2-38-7': 'In conclusion, the study of a four-leg interaction term (as well as other effects such as the mixing with [MATH]) should be performed when more precise data will be available.', '1712.00969-2-39-0': 'h) Here, we discuss more closely the differences between our work and the one in Ref. [CITATION].', '1712.00969-2-39-1': 'The main difference is the treatment of the propagator of the [MATH] meson, in particular for what concerns the self-energy.', '1712.00969-2-39-2': 'In Ref. [CITATION] the imaginary part of the self-energy (i.e., the energy-dependent decay width) grows indefinitely with increasing [MATH].', '1712.00969-2-39-3': 'In this way, the Kallen-Lehmann representation does not lead to a normalized spectral function.', '1712.00969-2-39-4': 'Moreover, the real part of the self-energy depends on an additional subtraction constant [MATH] (linked to the regularization of the loop in Ref. [CITATION], hence ultimately corresponding to our cutoff [MATH]).', '1712.00969-2-39-5': 'When [MATH] is set to zero (their so-called minimal subtraction scheme) the real part of the self energy simply vanishes (hence, it is trivial).', '1712.00969-2-39-6': 'When [MATH] is different from zero, the real part also grows indefinitely for increasing [MATH].', '1712.00969-2-39-7': 'Indeed, in the approach of Ref. [CITATION], the propagator of [MATH] is not enough to describe data, also when the rescattering is present; the inclusion of the [MATH] state is necessary.', '1712.00969-2-39-8': 'Hence, our approach is different under many aspects: since it contains a fully consistent treatment of [MATH] loops, the dispersion relations are fulfilled and the normalization of the spectral function is naturally obtained.', '1712.00969-2-39-9': 'In this way, nontrivial effects which modify the form of the spectral function automatically arise.', '1712.00969-2-39-10': 'Moreover, in our case neither the [MATH] nor the rescattering are needed to describe the data, hence the physical picture of the two models is rather different.', '1712.00969-2-39-11': 'In conclusion, the fit in Ref. [CITATION] is performed with six parameters (repeated for different values of the subtraction constant [MATH], for a total of seven), while we use only 4.', '1712.00969-2-39-12': 'Both approaches can describe data: in the future, better data are needed to understand which effects are more important.', '1712.00969-2-39-13': 'Our work suggests that the nearby [MATH] threshold and [MATH] loops are necessary to correctly understand the [MATH].', '1712.00969-2-39-14': 'Furthermore, the presence of two poles is a stable prediction of our approach, see Secs. [REF] and [REF] below.', '1712.00969-2-40-0': 'i) As a last comment, we compare our result with the approach of Refs. [CITATION].', '1712.00969-2-40-1': 'Interestingly, in Ref. [CITATION] it is shown that a suppression of the cross-section at about [MATH] GeV is obtained by a destructive interference of the resonance [MATH] with the background.', '1712.00969-2-40-2': 'This outcome is useful, since it sets a physical limit for our approach, in which no background is present: our description of data cannot be expected above this upper limit (and indeed we perform the fit below it).', '1712.00969-2-40-3': 'However, for what concerns the treatment of [MATH] our model is quite different: in Refs. [CITATION] rescattering is taken into account.', '1712.00969-2-40-4': 'When their rescattering parameter is set to zero, the real part of the [MATH] loop vanishes.', '1712.00969-2-40-5': 'In contrast, in our approach, even without an explicit rescattering term, the [MATH] loops are taken into account in a way that guarantees the unitarity expressed by Eq. [REF] (its fulfillment is one of the most relevant technical aspects of our approach) and they are found to be important for a proper description of the [MATH].', '1712.00969-2-41-0': '## Pole positions', '1712.00969-2-42-0': 'As renowned, a theoretically (and lately experimentally as well, see e.g. the [MATH] in PDG) stable approach to describe unstable states is based on the determination of the position of the corresponding poles.', '1712.00969-2-42-1': 'In the present case, one obtains, quite remarkably, two poles on the III Riemann Sheet.', '1712.00969-2-42-2': 'As seen in Sec. [REF], this is the sheet relevant for a system with two decay channels at energy above both thresholds.', '1712.00969-2-42-3': 'In the isospin limit, the III RS reduces to the usual II RS.', '1712.00969-2-42-4': 'In the following, the indeterminacy of the pole masses is estimated to be [MATH] MeV, close to the error obtained for the parameter [MATH], and the indeterminacy of the pole width(s) as [MATH] MeV, coincident with the error for [MATH] (see Table [REF]).', '1712.00969-2-42-5': 'In fact, the errors of these strictly correlated quantities must have a similar magnitude.', '1712.00969-2-42-6': 'The closest pole to the Riemann axis reads: [EQUATION] hence [EQUATION] which agrees quite well with the PDG values [CITATION].', '1712.00969-2-42-7': 'This pole is closely related to the seed charmonium state and to the position and width of the peak.', '1712.00969-2-42-8': 'Furthermore, a second broader pole appears at lower energy: [EQUATION]', '1712.00969-2-42-9': 'This pole is dynamically generated, and it is related to the the deformation of the signal on the left side.', '1712.00969-2-42-10': 'It is also referred to as a companion pole emergent due to the strong dynamics.', '1712.00969-2-42-11': 'The situation is very much reminiscent of the light scalar [MATH] state at lower energy, which was interpreted as a dynamically generated companion pole of [MATH] in Refs. [CITATION].', '1712.00969-2-42-12': 'For additional recent works on this state, see Refs. [CITATION], and references therein.', '1712.00969-2-42-13': 'This similarity is interesting also in view of an important difference between the two systems: the decay of a scalar kaon into two pseudoscalars is a [MATH]-wave ([MATH]) while that of a vector charmonium into two pseudoscalars is a [MATH]-wave ([MATH]).', '1712.00969-2-43-0': 'We mention again [REF], where a different form factor is used.', '1712.00969-2-43-1': 'The results turn out to be very similar and also in that case two poles, very close to the ones discussed here, are found.', '1712.00969-2-44-0': 'No other poles could be numerically found in the complex plane.', '1712.00969-2-44-1': 'As discussed in comment e), the energy scale [MATH] does not represent a limit of validly of our approach.', '1712.00969-2-44-2': 'Yet, since the numerical value of [MATH] MeV preferred by the fit is rather small (the value of the emitted momentum [MATH] is comparable to it), it is interesting to investigate if two poles exist also when artificially increasing its value.', '1712.00969-2-44-3': 'Upon fixing [MATH] MeV, one gets [MATH] hence the description of data is worsened ([MATH] ) but still qualitatively acceptable; nevertheless, there are still two poles: the seed one corresponds to [MATH] GeV (somewhat too large) and the broad dynamically generated one to [MATH] GeV.', '1712.00969-2-44-4': 'Upon increasing [MATH] even further to [MATH] MeV, the [MATH] worsens, i.e. [MATH] (which is already beyond the border of a satisfactory description of data), but there are still two poles; the seed pole reads [MATH] GeV and the companion pole [MATH] GeV.', '1712.00969-2-44-5': 'This exercise shows that the presence of two poles is a stable result also when changing the value of [MATH], even to values which are safely larger than the value of the modulus of the outgoing momentum of the [MATH] meson (even if the data description gets worse for increasing [MATH]).', '1712.00969-2-45-0': '## Pole Trajectories', '1712.00969-2-46-0': 'Here we study the effect of varying the coupling constant over the line-shape and over the pole positions.', '1712.00969-2-46-1': 'We define [MATH] as the variable parameter (upon varying the scaling factor [MATH]) and let [MATH] be fixed to the value in Table [REF].', '1712.00969-2-46-2': 'In Quantum Chromodynamics (QCD), a change in such coupling is equivalent to vary the number of colors according to the scaling [MATH].', '1712.00969-2-46-3': 'Results are shown in Fig. [REF] for the cross-section, and in Table [REF] for pole positions.', '1712.00969-2-46-4': 'The pole trajectories, as a function of [MATH], are depicted in Fig. [REF].', '1712.00969-2-46-5': 'As expected, as [MATH] decreases, the cross section approaches a Breit-Wigner-like shape, see case where [MATH] in Fig. [REF].', '1712.00969-2-46-6': 'Figure [REF] shows that, as [MATH] gets smaller, the first pole, from the seed, gets closer to the real axis, while the second pole, dynamically generated, moves deeper in the complex plane until it disappears at the [MATH] threshold for a certain critical value, in this case [MATH] (corresponding to [MATH]).', '1712.00969-2-46-7': 'On the contrary, when [MATH] increases, the first pole runs away from the real axis, while the second pole approaches it.', '1712.00969-2-46-8': 'Eventually, the dynamically generated pole gets even closer to the real axis than the seed pole, as it is shown for the case where [MATH].', '1712.00969-2-46-9': 'For such a value of [MATH] (or larger), the second pole originates a second peak in the line-shape (see Fig. [REF]).', '1712.00969-2-47-0': '# Summary, Conclusions, and Perspectives', '1712.00969-2-48-0': 'Vector mesons can be produced by a single virtual photon, hence they play a crucial role in the experimental study of QCD.', '1712.00969-2-48-1': 'The renowned charmonium vector state [MATH] opened a new era in hadron physics and it is one of the best studied resonances (cf. [CITATION]).', '1712.00969-2-48-2': 'Here, we have focused on an orbital excitation of the [MATH], the meson [MATH], whose mass lies just above the [MATH] threshold.', '1712.00969-2-48-3': 'For this reason, this resonance is extremely interesting.', '1712.00969-2-48-4': 'The cross-section [MATH] encloses nonperturbative phenomena nearby the OZI-allowed open decay channel [MATH].', '1712.00969-2-48-5': 'The production and decay to all hadrons shows evidence of a deformation on the line-shape of the resonance [CITATION].', '1712.00969-2-49-0': 'In this work, we presented an unitarized effective Lagrangian model, with one bare [MATH] orbital seed state [MATH] dressed by [MATH] mesonic loops, that has never been employed before to this system.', '1712.00969-2-49-1': 'In particular, our model accounts for one-loop [MATH] contributions by fulfilling unitarity of the spectral function.', '1712.00969-2-49-2': 'Using four parameters only, we obtain a satisfactory fit to the [MATH] cross-section data ([MATH]).', '1712.00969-2-49-3': 'Moreover, the theoretical cross section is in good agreement with data in channels [MATH] and [MATH] separately.', '1712.00969-2-49-4': 'Partial widths are also in agreement with the PDG values.', '1712.00969-2-49-5': 'The partial decay to leptons is smaller than the average in PDG, but still reasonable (and in agreement with Ref. [CITATION]).', '1712.00969-2-49-6': 'The cutoff parameter [MATH], inversely proportional to the size of the wave-function, turns out to be rather small, possibly due to the fact that the [MATH] is a [MATH]-wave.', '1712.00969-2-49-7': 'Other effects, such as final state rescattering modelled by a four-leg interaction term, or the contribution of the tail of the [MATH] to the amplitude, are not taken into account, since our fit is already very good without them (this represents a difference with Ref. [CITATION], where these terms were necessary to describe data but where the normalization of the spectral function of [MATH] was not guaranteed).', '1712.00969-2-49-8': 'The detailed study of the role of these additional effects within our framework needs more precise data and is left for the future.', '1712.00969-2-50-0': 'An important result of our work is the evaluation of two poles at [MATH] and [MATH], the first is a standard seed pole, very close to the one determined by BES in Ref. [CITATION], and the second broader pole is dynamically generated, being an additional companion pole.', '1712.00969-2-50-1': 'The presence of a second pole explains the deviation from a pure Breit-Wigner line-shape.', '1712.00969-2-50-2': 'By varying the coupling constant [MATH], we studied the pole trajectories and its influence over the line-shape.', '1712.00969-2-50-3': 'For small couplings the dynamical pole disappears and the seed pole approaches the real axis, leading to a Breit-Wigner-like line-shape, while for larger couplings the dynamical pole approaches the real axis while the seed pole moves down and to the right, and a second peak becomes evident (see Figs. [REF] and [REF] for details).', '1712.00969-2-50-4': 'The width of about 24 MeV to channel [MATH], that we obtain from the first pole, is consistent with the experimental measurements to this channel, and with the branching fraction estimated to be about [MATH] in Ref. [CITATION], considering that measurements of the width in decays to all hadrons are a few MeV higher (cf. Ref. [CITATION]).', '1712.00969-2-51-0': 'A natural outlook is the inclusion of the many OZI-suppressed hadronic decay channels.', '1712.00969-2-51-1': 'A related topic is the evaluation of mixing with the vector glueball, whose mass is predicted to be at about 3.8 GeV by lattice QCD [CITATION].', '1712.00969-2-51-2': 'Finally, the same theoretical approach used in this work can be applied to other resonances whose nature is not yet clarified, the so-called [MATH] and [MATH] states [CITATION].', '1712.00969-2-51-3': 'It is well possible that some of these resonances arise as companion poles due to a strong coupling of a standard charmonium seed state to lighter mesonic resonances.', '1712.00969-2-51-4': 'The strong dressing of the original state with meson-meson clouds generates new poles and new peaks.', '1712.00969-2-51-5': 'An intriguing possibility is that the additional peak arises at a higher energy than the seed state, see Refs. [CITATION] for a phenomenological description of this phenomenon in the light sector.', '1712.00969-2-51-6': 'Hence, detailed studies of the dynamics underlying each line-shape, that consider poles and interference with the main hadronic degrees of freedom, may shed light on some newly discovered resonances in the charmonium region.', '1712.00969-2-52-0': 'An interesting outlook should be the description of the whole sector [MATH]above the [MATH] threshold.', '1712.00969-2-52-1': 'To this end, one should develop a unique treatment of the four standard charm-anticharm resonances above the [MATH] threshold [MATH], [MATH], [MATH], and [MATH] (and also the resonance [MATH] below).', '1712.00969-2-52-2': 'In this way, following and extending the discussion in Refs. [CITATION], all quantum mixing terms should be properly taken into account.', '1712.00969-2-52-3': 'The aim of such an ambitious project would be the simultaneous description of the whole sector as well as the description of (some of the) newly observed [MATH] resonances in this energy region, which might emerge as dynamically generated states.'}

[['1712.00969-1-2-0', '1712.00969-2-2-0'], ['1712.00969-1-2-2', '1712.00969-2-2-2'], ['1712.00969-1-4-0', '1712.00969-2-4-0'], ['1712.00969-1-4-1', '1712.00969-2-4-1'], ['1712.00969-1-4-2', '1712.00969-2-4-2'], ['1712.00969-1-4-3', '1712.00969-2-4-3'], ['1712.00969-1-4-5', '1712.00969-2-4-5'], ['1712.00969-1-4-6', '1712.00969-2-4-6'], ['1712.00969-1-4-7', '1712.00969-2-4-7'], ['1712.00969-1-4-8', '1712.00969-2-4-8'], ['1712.00969-1-6-0', '1712.00969-2-6-0'], ['1712.00969-1-6-1', '1712.00969-2-6-1'], ['1712.00969-1-6-2', '1712.00969-2-6-2'], ['1712.00969-1-7-0', '1712.00969-2-7-0'], ['1712.00969-1-5-2', '1712.00969-2-5-2'], ['1712.00969-1-5-4', '1712.00969-2-5-4'], ['1712.00969-1-5-6', '1712.00969-2-5-6'], ['1712.00969-1-5-7', '1712.00969-2-5-7'], ['1712.00969-1-36-1', '1712.00969-2-35-1'], ['1712.00969-1-23-2', '1712.00969-2-23-1'], ['1712.00969-1-16-0', '1712.00969-2-17-0'], ['1712.00969-1-16-2', '1712.00969-2-17-2'], ['1712.00969-1-22-1', '1712.00969-2-22-1'], ['1712.00969-1-22-2', '1712.00969-2-22-2'], ['1712.00969-1-22-3', '1712.00969-2-22-3'], ['1712.00969-1-22-4', '1712.00969-2-22-4'], ['1712.00969-1-22-5', '1712.00969-2-22-5'], ['1712.00969-1-39-1', '1712.00969-2-46-3'], ['1712.00969-1-39-2', '1712.00969-2-46-2'], ['1712.00969-1-12-1', '1712.00969-2-12-1'], ['1712.00969-1-12-3', '1712.00969-2-12-3'], ['1712.00969-1-12-7', '1712.00969-2-12-6'], ['1712.00969-1-12-10', '1712.00969-2-12-11'], ['1712.00969-1-32-0', '1712.00969-2-31-0'], ['1712.00969-1-11-0', '1712.00969-2-11-0'], ['1712.00969-1-11-1', '1712.00969-2-11-1'], ['1712.00969-1-11-2', '1712.00969-2-11-2'], ['1712.00969-1-11-3', '1712.00969-2-11-3'], ['1712.00969-1-11-5', '1712.00969-2-11-5'], ['1712.00969-1-29-3', '1712.00969-2-27-4'], ['1712.00969-1-37-0', '1712.00969-2-37-0'], ['1712.00969-1-37-1', '1712.00969-2-37-1'], ['1712.00969-1-37-2', '1712.00969-2-37-2'], ['1712.00969-1-37-4', '1712.00969-2-37-3'], ['1712.00969-1-37-5', '1712.00969-2-37-4'], ['1712.00969-1-37-10', '1712.00969-2-37-8'], ['1712.00969-1-18-0', '1712.00969-2-19-0'], ['1712.00969-1-13-0', '1712.00969-2-13-0'], ['1712.00969-1-3-1', '1712.00969-2-3-1'], ['1712.00969-1-3-2', '1712.00969-2-3-2'], ['1712.00969-1-3-3', '1712.00969-2-3-3'], ['1712.00969-1-3-4', '1712.00969-2-3-4'], ['1712.00969-1-35-1', '1712.00969-2-42-0'], ['1712.00969-1-35-3', '1712.00969-2-42-2'], ['1712.00969-1-35-6', '1712.00969-2-42-5'], ['1712.00969-1-35-7', '1712.00969-2-42-6'], ['1712.00969-1-35-8', '1712.00969-2-42-7'], ['1712.00969-1-35-9', '1712.00969-2-42-8'], ['1712.00969-1-35-10', '1712.00969-2-42-9'], ['1712.00969-1-35-11', '1712.00969-2-42-10'], ['1712.00969-1-35-13', '1712.00969-2-42-12'], ['1712.00969-1-14-1', '1712.00969-2-14-1'], ['1712.00969-1-14-2', '1712.00969-2-14-2'], ['1712.00969-1-14-5', '1712.00969-2-15-0'], ['1712.00969-1-14-6', '1712.00969-2-15-1'], ['1712.00969-1-14-7', '1712.00969-2-15-2'], ['1712.00969-1-14-8', '1712.00969-2-15-3'], ['1712.00969-1-14-9', '1712.00969-2-15-4'], ['1712.00969-1-14-10', '1712.00969-2-15-7'], ['1712.00969-1-14-12', '1712.00969-2-15-9'], ['1712.00969-1-14-13', '1712.00969-2-15-10'], ['1712.00969-1-19-0', '1712.00969-2-20-0'], ['1712.00969-1-20-0', '1712.00969-2-20-1'], ['1712.00969-1-20-1', '1712.00969-2-20-2'], ['1712.00969-1-20-2', '1712.00969-2-20-3'], ['1712.00969-1-45-0', '1712.00969-2-48-0'], ['1712.00969-1-46-1', '1712.00969-2-48-3'], ['1712.00969-1-46-2', '1712.00969-2-48-4'], ['1712.00969-1-34-1', '1712.00969-2-32-7'], ['1712.00969-1-33-0', '1712.00969-2-32-0'], ['1712.00969-1-33-1', '1712.00969-2-32-1'], ['1712.00969-1-33-3', '1712.00969-2-32-3'], ['1712.00969-1-33-4', '1712.00969-2-32-4'], ['1712.00969-1-33-5', '1712.00969-2-32-5'], ['1712.00969-1-47-0', '1712.00969-2-50-4'], ['1712.00969-1-48-3', '1712.00969-2-51-4'], ['1712.00969-1-2-3', '1712.00969-2-2-3'], ['1712.00969-1-4-4', '1712.00969-2-4-4'], ['1712.00969-1-6-3', '1712.00969-2-6-3'], ['1712.00969-1-31-0', '1712.00969-2-30-0'], ['1712.00969-1-7-1', '1712.00969-2-7-1'], ['1712.00969-1-7-3', '1712.00969-2-7-3'], ['1712.00969-1-7-4', '1712.00969-2-7-4'], ['1712.00969-1-5-0', '1712.00969-2-5-0'], ['1712.00969-1-5-1', '1712.00969-2-5-1'], ['1712.00969-1-5-3', '1712.00969-2-5-3'], ['1712.00969-1-5-5', '1712.00969-2-5-5'], ['1712.00969-1-5-8', '1712.00969-2-5-8'], ['1712.00969-1-5-9', '1712.00969-2-5-10'], ['1712.00969-1-5-10', '1712.00969-2-5-11'], ['1712.00969-1-36-2', '1712.00969-2-35-2'], ['1712.00969-1-36-3', '1712.00969-2-35-3'], ['1712.00969-1-36-4', '1712.00969-2-35-4'], ['1712.00969-1-9-0', '1712.00969-2-9-0'], ['1712.00969-1-23-0', '1712.00969-2-23-0'], ['1712.00969-1-16-1', '1712.00969-2-17-1'], ['1712.00969-1-16-3', '1712.00969-2-17-3'], ['1712.00969-1-16-4', '1712.00969-2-17-4'], ['1712.00969-1-22-0', '1712.00969-2-22-0'], ['1712.00969-1-39-6', '1712.00969-2-46-7'], ['1712.00969-1-12-0', '1712.00969-2-12-0'], ['1712.00969-1-12-2', '1712.00969-2-12-2'], ['1712.00969-1-12-4', '1712.00969-2-12-4'], ['1712.00969-1-12-6', '1712.00969-2-12-5'], ['1712.00969-1-32-1', '1712.00969-2-31-1'], ['1712.00969-1-11-4', '1712.00969-2-11-4'], ['1712.00969-1-29-0', '1712.00969-2-27-0'], ['1712.00969-1-29-1', '1712.00969-2-27-1'], ['1712.00969-1-29-2', '1712.00969-2-27-3'], ['1712.00969-1-37-6', '1712.00969-2-37-5'], ['1712.00969-1-0-0', '1712.00969-2-0-0'], ['1712.00969-1-0-9', '1712.00969-2-0-2'], ['1712.00969-1-0-10', '1712.00969-2-0-5'], ['1712.00969-1-18-1', '1712.00969-2-19-1'], ['1712.00969-1-13-1', '1712.00969-2-13-1'], ['1712.00969-1-13-2', '1712.00969-2-13-2'], ['1712.00969-1-13-3', '1712.00969-2-13-3'], ['1712.00969-1-3-0', '1712.00969-2-3-0'], ['1712.00969-1-35-2', '1712.00969-2-42-1'], ['1712.00969-1-35-4', '1712.00969-2-42-3'], ['1712.00969-1-35-5', '1712.00969-2-42-4'], ['1712.00969-1-35-12', '1712.00969-2-42-11'], ['1712.00969-1-14-0', '1712.00969-2-14-0'], ['1712.00969-1-14-3', '1712.00969-2-14-3'], ['1712.00969-1-14-4', '1712.00969-2-14-4'], ['1712.00969-1-14-11', '1712.00969-2-15-8'], ['1712.00969-1-45-4', '1712.00969-2-48-1'], ['1712.00969-1-46-0', '1712.00969-2-48-2'], ['1712.00969-1-34-0', '1712.00969-2-32-6'], ['1712.00969-1-34-4', '1712.00969-2-32-9'], ['1712.00969-1-48-0', '1712.00969-2-51-2'], ['1712.00969-1-48-2', '1712.00969-2-51-3'], ['1712.00969-1-48-4', '1712.00969-2-51-5'], ['1712.00969-1-2-1', '1712.00969-2-2-1'], ['1712.00969-1-2-4', '1712.00969-2-2-4'], ['1712.00969-1-7-2', '1712.00969-2-7-2'], ['1712.00969-1-36-0', '1712.00969-2-35-0'], ['1712.00969-1-39-0', '1712.00969-2-46-0'], ['1712.00969-1-39-3', '1712.00969-2-46-5'], ['1712.00969-1-39-4', '1712.00969-2-46-6'], ['1712.00969-1-39-7', '1712.00969-2-46-8'], ['1712.00969-1-27-0', '1712.00969-2-25-0'], ['1712.00969-1-12-8', '1712.00969-2-12-7'], ['1712.00969-1-12-8', '1712.00969-2-12-8'], ['1712.00969-1-37-7', '1712.00969-2-37-6'], ['1712.00969-1-0-3', '1712.00969-2-0-1'], ['1712.00969-1-0-5', '1712.00969-2-0-3'], ['1712.00969-1-0-7', '1712.00969-2-0-4'], ['1712.00969-1-18-2', '1712.00969-2-19-2'], ['1712.00969-1-18-2', '1712.00969-2-19-3'], ['1712.00969-1-46-3', '1712.00969-2-48-5'], ['1712.00969-1-34-2', '1712.00969-2-32-8'], ['1712.00969-1-34-3', '1712.00969-2-32-8'], ['1712.00969-1-33-4', '1712.00969-2-33-3'], ['1712.00969-1-33-2', '1712.00969-2-32-2'], ['1712.00969-1-47-1', '1712.00969-2-50-2'], ['1712.00969-1-47-1', '1712.00969-2-51-0'], ['1712.00969-1-47-4', '1712.00969-2-51-1']]

[['1712.00969-1-2-0', '1712.00969-2-2-0'], ['1712.00969-1-2-2', '1712.00969-2-2-2'], ['1712.00969-1-4-0', '1712.00969-2-4-0'], ['1712.00969-1-4-1', '1712.00969-2-4-1'], ['1712.00969-1-4-2', '1712.00969-2-4-2'], ['1712.00969-1-4-3', '1712.00969-2-4-3'], ['1712.00969-1-4-5', '1712.00969-2-4-5'], ['1712.00969-1-4-6', '1712.00969-2-4-6'], ['1712.00969-1-4-7', '1712.00969-2-4-7'], ['1712.00969-1-4-8', '1712.00969-2-4-8'], ['1712.00969-1-6-0', '1712.00969-2-6-0'], ['1712.00969-1-6-1', '1712.00969-2-6-1'], ['1712.00969-1-6-2', '1712.00969-2-6-2'], ['1712.00969-1-7-0', '1712.00969-2-7-0'], ['1712.00969-1-5-2', '1712.00969-2-5-2'], ['1712.00969-1-5-4', '1712.00969-2-5-4'], ['1712.00969-1-5-6', '1712.00969-2-5-6'], ['1712.00969-1-5-7', '1712.00969-2-5-7'], ['1712.00969-1-36-1', '1712.00969-2-35-1'], ['1712.00969-1-23-2', '1712.00969-2-23-1'], ['1712.00969-1-16-0', '1712.00969-2-17-0'], ['1712.00969-1-16-2', '1712.00969-2-17-2'], ['1712.00969-1-22-1', '1712.00969-2-22-1'], ['1712.00969-1-22-2', '1712.00969-2-22-2'], ['1712.00969-1-22-3', '1712.00969-2-22-3'], ['1712.00969-1-22-4', '1712.00969-2-22-4'], ['1712.00969-1-22-5', '1712.00969-2-22-5'], ['1712.00969-1-39-1', '1712.00969-2-46-3'], ['1712.00969-1-39-2', '1712.00969-2-46-2'], ['1712.00969-1-12-1', '1712.00969-2-12-1'], ['1712.00969-1-12-3', '1712.00969-2-12-3'], ['1712.00969-1-12-7', '1712.00969-2-12-6'], ['1712.00969-1-12-10', '1712.00969-2-12-11'], ['1712.00969-1-32-0', '1712.00969-2-31-0'], ['1712.00969-1-11-0', '1712.00969-2-11-0'], ['1712.00969-1-11-1', '1712.00969-2-11-1'], ['1712.00969-1-11-2', '1712.00969-2-11-2'], ['1712.00969-1-11-3', '1712.00969-2-11-3'], ['1712.00969-1-11-5', '1712.00969-2-11-5'], ['1712.00969-1-29-3', '1712.00969-2-27-4'], ['1712.00969-1-37-0', '1712.00969-2-37-0'], ['1712.00969-1-37-1', '1712.00969-2-37-1'], ['1712.00969-1-37-2', '1712.00969-2-37-2'], ['1712.00969-1-37-4', '1712.00969-2-37-3'], ['1712.00969-1-37-5', '1712.00969-2-37-4'], ['1712.00969-1-37-10', '1712.00969-2-37-8'], ['1712.00969-1-18-0', '1712.00969-2-19-0'], ['1712.00969-1-13-0', '1712.00969-2-13-0'], ['1712.00969-1-3-1', '1712.00969-2-3-1'], ['1712.00969-1-3-2', '1712.00969-2-3-2'], ['1712.00969-1-3-3', '1712.00969-2-3-3'], ['1712.00969-1-3-4', '1712.00969-2-3-4'], ['1712.00969-1-35-1', '1712.00969-2-42-0'], ['1712.00969-1-35-3', '1712.00969-2-42-2'], ['1712.00969-1-35-6', '1712.00969-2-42-5'], ['1712.00969-1-35-7', '1712.00969-2-42-6'], ['1712.00969-1-35-8', '1712.00969-2-42-7'], ['1712.00969-1-35-9', '1712.00969-2-42-8'], ['1712.00969-1-35-10', '1712.00969-2-42-9'], ['1712.00969-1-35-11', '1712.00969-2-42-10'], ['1712.00969-1-35-13', '1712.00969-2-42-12'], ['1712.00969-1-14-1', '1712.00969-2-14-1'], ['1712.00969-1-14-2', '1712.00969-2-14-2'], ['1712.00969-1-14-5', '1712.00969-2-15-0'], ['1712.00969-1-14-6', '1712.00969-2-15-1'], ['1712.00969-1-14-7', '1712.00969-2-15-2'], ['1712.00969-1-14-8', '1712.00969-2-15-3'], ['1712.00969-1-14-9', '1712.00969-2-15-4'], ['1712.00969-1-14-10', '1712.00969-2-15-7'], ['1712.00969-1-14-12', '1712.00969-2-15-9'], ['1712.00969-1-14-13', '1712.00969-2-15-10'], ['1712.00969-1-19-0', '1712.00969-2-20-0'], ['1712.00969-1-20-0', '1712.00969-2-20-1'], ['1712.00969-1-20-1', '1712.00969-2-20-2'], ['1712.00969-1-20-2', '1712.00969-2-20-3'], ['1712.00969-1-45-0', '1712.00969-2-48-0'], ['1712.00969-1-46-1', '1712.00969-2-48-3'], ['1712.00969-1-46-2', '1712.00969-2-48-4'], ['1712.00969-1-34-1', '1712.00969-2-32-7'], ['1712.00969-1-33-0', '1712.00969-2-32-0'], ['1712.00969-1-33-1', '1712.00969-2-32-1'], ['1712.00969-1-33-3', '1712.00969-2-32-3'], ['1712.00969-1-33-4', '1712.00969-2-32-4'], ['1712.00969-1-33-5', '1712.00969-2-32-5'], ['1712.00969-1-47-0', '1712.00969-2-50-4'], ['1712.00969-1-48-3', '1712.00969-2-51-4']]

[['1712.00969-1-2-3', '1712.00969-2-2-3'], ['1712.00969-1-4-4', '1712.00969-2-4-4'], ['1712.00969-1-6-3', '1712.00969-2-6-3'], ['1712.00969-1-31-0', '1712.00969-2-30-0'], ['1712.00969-1-7-1', '1712.00969-2-7-1'], ['1712.00969-1-7-3', '1712.00969-2-7-3'], ['1712.00969-1-7-4', '1712.00969-2-7-4'], ['1712.00969-1-5-0', '1712.00969-2-5-0'], ['1712.00969-1-5-1', '1712.00969-2-5-1'], ['1712.00969-1-5-3', '1712.00969-2-5-3'], ['1712.00969-1-5-5', '1712.00969-2-5-5'], ['1712.00969-1-5-8', '1712.00969-2-5-8'], ['1712.00969-1-5-9', '1712.00969-2-5-10'], ['1712.00969-1-5-10', '1712.00969-2-5-11'], ['1712.00969-1-36-2', '1712.00969-2-35-2'], ['1712.00969-1-36-3', '1712.00969-2-35-3'], ['1712.00969-1-36-4', '1712.00969-2-35-4'], ['1712.00969-1-9-0', '1712.00969-2-9-0'], ['1712.00969-1-23-0', '1712.00969-2-23-0'], ['1712.00969-1-16-1', '1712.00969-2-17-1'], ['1712.00969-1-16-3', '1712.00969-2-17-3'], ['1712.00969-1-16-4', '1712.00969-2-17-4'], ['1712.00969-1-22-0', '1712.00969-2-22-0'], ['1712.00969-1-39-6', '1712.00969-2-46-7'], ['1712.00969-1-12-0', '1712.00969-2-12-0'], ['1712.00969-1-12-2', '1712.00969-2-12-2'], ['1712.00969-1-12-4', '1712.00969-2-12-4'], ['1712.00969-1-12-6', '1712.00969-2-12-5'], ['1712.00969-1-32-1', '1712.00969-2-31-1'], ['1712.00969-1-11-4', '1712.00969-2-11-4'], ['1712.00969-1-29-0', '1712.00969-2-27-0'], ['1712.00969-1-29-1', '1712.00969-2-27-1'], ['1712.00969-1-29-2', '1712.00969-2-27-3'], ['1712.00969-1-37-6', '1712.00969-2-37-5'], ['1712.00969-1-0-0', '1712.00969-2-0-0'], ['1712.00969-1-0-9', '1712.00969-2-0-2'], ['1712.00969-1-0-10', '1712.00969-2-0-5'], ['1712.00969-1-18-1', '1712.00969-2-19-1'], ['1712.00969-1-13-1', '1712.00969-2-13-1'], ['1712.00969-1-13-2', '1712.00969-2-13-2'], ['1712.00969-1-13-3', '1712.00969-2-13-3'], ['1712.00969-1-3-0', '1712.00969-2-3-0'], ['1712.00969-1-35-2', '1712.00969-2-42-1'], ['1712.00969-1-35-4', '1712.00969-2-42-3'], ['1712.00969-1-35-5', '1712.00969-2-42-4'], ['1712.00969-1-35-12', '1712.00969-2-42-11'], ['1712.00969-1-14-0', '1712.00969-2-14-0'], ['1712.00969-1-14-3', '1712.00969-2-14-3'], ['1712.00969-1-14-4', '1712.00969-2-14-4'], ['1712.00969-1-14-11', '1712.00969-2-15-8'], ['1712.00969-1-45-4', '1712.00969-2-48-1'], ['1712.00969-1-46-0', '1712.00969-2-48-2'], ['1712.00969-1-34-0', '1712.00969-2-32-6'], ['1712.00969-1-34-4', '1712.00969-2-32-9'], ['1712.00969-1-48-0', '1712.00969-2-51-2'], ['1712.00969-1-48-2', '1712.00969-2-51-3'], ['1712.00969-1-48-4', '1712.00969-2-51-5']]

[]

[['1712.00969-1-2-1', '1712.00969-2-2-1'], ['1712.00969-1-2-4', '1712.00969-2-2-4'], ['1712.00969-1-7-2', '1712.00969-2-7-2'], ['1712.00969-1-36-0', '1712.00969-2-35-0'], ['1712.00969-1-39-0', '1712.00969-2-46-0'], ['1712.00969-1-39-3', '1712.00969-2-46-5'], ['1712.00969-1-39-4', '1712.00969-2-46-6'], ['1712.00969-1-39-7', '1712.00969-2-46-8'], ['1712.00969-1-27-0', '1712.00969-2-25-0'], ['1712.00969-1-12-8', '1712.00969-2-12-7'], ['1712.00969-1-12-8', '1712.00969-2-12-8'], ['1712.00969-1-37-7', '1712.00969-2-37-6'], ['1712.00969-1-0-3', '1712.00969-2-0-1'], ['1712.00969-1-0-5', '1712.00969-2-0-3'], ['1712.00969-1-0-7', '1712.00969-2-0-4'], ['1712.00969-1-18-2', '1712.00969-2-19-2'], ['1712.00969-1-18-2', '1712.00969-2-19-3'], ['1712.00969-1-46-3', '1712.00969-2-48-5'], ['1712.00969-1-34-2', '1712.00969-2-32-8'], ['1712.00969-1-34-3', '1712.00969-2-32-8'], ['1712.00969-1-33-4', '1712.00969-2-33-3'], ['1712.00969-1-33-2', '1712.00969-2-32-2'], ['1712.00969-1-47-1', '1712.00969-2-50-2'], ['1712.00969-1-47-1', '1712.00969-2-51-0'], ['1712.00969-1-47-4', '1712.00969-2-51-1']]

[]

['1712.00969-1-30-0', '1712.00969-1-35-0', '1712.00969-2-29-0']

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

https://arxiv.org/abs/1712.00969

1510.08634

{'1510.08634-1-0-0': 'Markovian dynamics of open quantum systems are described by the L-GKS equation, known also as the Lindblad equation.', '1510.08634-1-0-1': 'The equation is expressed by means of left and right matrix multiplications.', '1510.08634-1-0-2': 'This formulation hampers numerical implementations.', '1510.08634-1-0-3': 'Representing the dynamics by a matrix-vector notation overcomes this problem.', '1510.08634-1-0-4': 'We review three approaches to obtain such a representation.', '1510.08634-1-0-5': 'The methods are demonstrated for a driven two-level system subject to spontaneous emission.', '1510.08634-1-1-0': '# Introduction', '1510.08634-1-2-0': 'An open system is a system that interacts with its environment.', '1510.08634-1-2-1': 'A full description has to account for all the degrees of freedom (DOF) of the entire system and its environment.', '1510.08634-1-2-2': 'Usually, only the system DOF are of interest.', '1510.08634-1-2-3': 'A reduced description attempts to describe only the system DOF explicitly, while the environment DOF are integrated out and affect the description implicitly.', '1510.08634-1-2-4': 'The goal is to reduce the description to a small number of variables and obtain a practical way to treat the system [CITATION].', '1510.08634-1-3-0': 'Open systems are often described by a stochastic process which in many cases becomes a simple Markov process.', '1510.08634-1-3-1': 'In brief, a Markov process is a stochastic process with a short time memory, i.e., the process state depends solely on the present state.', '1510.08634-1-3-2': 'Mathematically it can be constructed as the Chapman-Kolmagorov equation for the conditional joint probability [CITATION].', '1510.08634-1-3-3': 'For a Markov process the probability distribution [MATH] on a certain space, either real-space or phase-space, which could be continuous or discrete, follows the differential equation [EQUATION]', '1510.08634-1-3-4': 'The formal solution for Eq. ([REF]) is given by, [EQUATION] where, without loss of generality, we define the initial time to be zero.', '1510.08634-1-3-5': 'The one-parameter family of maps[MATH] is a semigroup with the generator [MATH].', '1510.08634-1-3-6': 'The term semigroup implies that this family of maps does not form a full group.', '1510.08634-1-3-7': 'It lacks the negative range of the parameter [MATH], which implies that the inverse property required by a group is missing.', '1510.08634-1-3-8': 'Physically, this property is the manifestation of irreversible dynamics which allows us to distinguish the future from the past.', '1510.08634-1-3-9': 'The map [MATH] is a positive map that satisfies the composition rule (Markov property) [MATH], and preserves normalization of the probability density.', '1510.08634-1-4-0': 'In the quantum scenario several modifications have to be made.', '1510.08634-1-4-1': 'The probability distribution [MATH] is replaced by the density matrix [MATH].', '1510.08634-1-4-2': 'The property of positivity has to be strengthened to complete positivity.', '1510.08634-1-4-3': 'The dynamics follows the quantum master equation: [EQUATION]', '1510.08634-1-4-4': 'This is a direct consequence of the presence of entangled states [CITATION].', '1510.08634-1-4-5': 'To summarize, the quantum dynamical semigroup is a continuous one-parameter family of maps [MATH], that satisfies [CITATION]:', '1510.08634-1-5-0': '[MATH] is complete positive; [MATH] is trace preserving; [MATH] semigroup (Markov) property; [MATH] is strongly continuous.', '1510.08634-1-6-0': 'Lindblad as well as Gorini, Kossakowski and Sudarshan (L-GKS) introduced the most general form of the quantum dynamical semigroup generator [MATH] that satisfies these requirements [CITATION].', '1510.08634-1-6-1': 'In the Lindblad form the Markovian master equation reads: [EQUATION]', '1510.08634-1-6-2': 'Here, [MATH] is the effective Hamiltonian of the system, [MATH] are positive rates, and [MATH] are operators belonging to the Hilbert space of the system.', '1510.08634-1-6-3': 'We use the notation [MATH] to represent the unitary part of the dynamics, and [MATH] to represent the dissipative part.', '1510.08634-1-6-4': '[MATH], [MATH] and [MATH] are linear operators that operate on the density matrix, usually referred to as super-operators.', '1510.08634-1-7-0': 'The operation of the super-operator [MATH] on the density matrix could be understood as repetitive operations of the super-operator [MATH] as in the Taylor expansion: [EQUATION]', '1510.08634-1-7-1': 'Typically, the resulting dynamics of the system observables (expectation values and other correlation functions) [MATH] will have the analytical form of sum of decaying oscillations: [EQUATION]', '1510.08634-1-7-2': 'Here, [MATH] are the exponential coefficients and [MATH] are the associated amplitudes, both can be complex.', '1510.08634-1-7-3': 'We may divide [MATH] into its real and imaginary parts, [MATH], with [MATH] as the decay rates and [MATH] as the oscillation frequencies.', '1510.08634-1-7-4': 'The coefficients [MATH] are the eigenvalues of the super-operator [MATH], obtained by the eigenvalue equation: [EQUATION]', '1510.08634-1-7-5': 'These eigenvalues characterize the dynamics, and the amplitudes [MATH] are determined by the initial conditions.', '1510.08634-1-8-0': 'As noted above, the dynamics can be investigated by exponentiation of the super-operator [MATH], Eq. ([REF]), or by its eigenvalues, Eq. ([REF]).', '1510.08634-1-8-1': 'The exponentiation and the eigenvalue problem of the (linear) super-operator [MATH] are well defined.', '1510.08634-1-8-2': 'However, they are not suitable for numerical calculations.', '1510.08634-1-8-3': 'Calculations of the exponentiation and the eigenvalue equation of linear operators can be done by common numerical techniques if the linear operator is represented by a matrix.', '1510.08634-1-8-4': 'Therefore, a preferred representation of the dynamics, Eq. ([REF]), is in a matrix-vector notation.', '1510.08634-1-8-5': 'This means that we are looking for a matrix [MATH] and a vector [MATH] such that the dynamics are expressed as [EQUATION]', '1510.08634-1-8-6': 'In this representation, the vector [MATH] represents the state of the system, or some information about it, e.g. a set of expectation values.', '1510.08634-1-8-7': 'Next, we describe three approaches for such a representation, and demonstrate them for a case of a driven two-level system with relaxation.', '1510.08634-1-9-0': '# Matrix-vector representations', '1510.08634-1-10-0': 'Suppose the density matrix [MATH] is an [MATH] matrix (if [MATH] is a function of continuous variables, e.g. [MATH], these variables have to be discretized).', '1510.08634-1-10-1': 'The set of all [MATH] matrices form a linear space of dimension [MATH].', '1510.08634-1-10-2': 'Under appropriate conditions, this linear space can have a Hilbert space construction, using the scalar product defined as [EQUATION].', '1510.08634-1-10-3': 'Such a Hilbert space is called a Liouville space (also known as the Hilbert-Schmidt space).', '1510.08634-1-10-4': 'With such a construction we consider [MATH] as an [MATH] vector.', '1510.08634-1-10-5': 'Similarly, we consider the super-operator [MATH], which is an operator operating on elements in this linear space, as an [MATH] matrix.', '1510.08634-1-11-0': 'The above observation is the first step towards the representation we seek.', '1510.08634-1-11-1': 'In the following, we describe three approaches that use this concept to introduce such representation:', '1510.08634-1-12-0': 'Vec-ing the density matrix is the most natural way to construct an [MATH] vector for the density matrix, and a suitable [MATH] matrix for the super-operator.', '1510.08634-1-12-1': 'The Arnoldi method approximates a large matrix in smaller dimensions, enabling simpler numerical calculations.', '1510.08634-1-12-2': 'With the Heisenberg picture of the L-GKS equation we can search for a representation with a dimension smaller than [MATH].', '1510.08634-1-13-0': 'In the following, we describe these three approaches.', '1510.08634-1-13-1': 'Each of these approaches will be demonstrated in the case of the two-level system.', '1510.08634-1-14-0': '## Vec-ing the density matrix', '1510.08634-1-15-0': 'In this method, known as vec-ing [CITATION], the [MATH] density matrix [MATH] is flattened into an [MATH] vector [MATH] .', '1510.08634-1-15-1': 'This flattening is done by ordering the columns of [MATH] one below the other, so the [MATH] entry of the matrix [MATH] is the [MATH] entry of the vector [MATH].', '1510.08634-1-15-2': 'This is equivalent to choosing the representation basis as the set of matrices with all-zero entries, except one.', '1510.08634-1-16-0': 'The next task is to find the suitable matrix that will represent the operation of the super-operator [MATH] on the density matrix.', '1510.08634-1-16-1': 'We make the following observations [CITATION]:', '1510.08634-1-17-0': 'A left multiplication of the matrix [MATH] by an [MATH] matrix [MATH], i.e. [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH], where [MATH] is the [MATH] identity matrix, and [MATH] is the Kronecker direct product.', '1510.08634-1-17-1': 'Similarly, a right multiplication of the matrix [MATH] by an [MATH] matrix [MATH], i.e. [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH].', '1510.08634-1-17-2': 'Here [MATH] denotes the transpose of the matrix.', '1510.08634-1-17-3': 'Finally, a combination of left and right matrices multiplication, [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH].', '1510.08634-1-18-0': 'The L-GKS super-operator is a sum of such right and left multiplications.', '1510.08634-1-18-1': 'Therefore, the construction of the [MATH] matrix representation for the L-GKS generator has the parts as follows; for the commutator: [EQUATION].', '1510.08634-1-18-2': 'For the dissipative part: [EQUATION]', '1510.08634-1-18-3': 'Then we write [EQUATION] and represent Eq. ([REF]) as [EQUATION] as desired.', '1510.08634-1-19-0': 'The mapping of the density matrix into a density vector yields in a dramatic increment in the dimension of the problem, which becomes [MATH] instead of [MATH].', '1510.08634-1-19-1': 'This yields unfavorable scaling of the desired computations with [MATH]:', '1510.08634-1-20-0': 'For systems larger than a few degrees of freedom, such computations are expensive, and become practically impossible for systems larger than a few hundreds DOF.', '1510.08634-1-21-0': 'The scaling problem suggests that we have to look for approaches that use a smaller number of dimensions.', '1510.08634-1-21-1': 'The following two approaches address this issue.', '1510.08634-1-21-2': 'The Arnoldi method uses a small-dimension approximation of a large matrix.', '1510.08634-1-21-3': 'The operator representation seeks for a small subset of variables that are sufficient to describe the quantities of interest.', '1510.08634-1-21-4': 'These two approaches are described in the next two sections.', '1510.08634-1-22-0': 'Remark: The density matrix [MATH] is hermitian.', '1510.08634-1-22-1': 'Therefore there are only [MATH] unique entries and not [MATH].', '1510.08634-1-22-2': 'This fact can be used to reduce the size of the vectors and matrices, known as a half-vectorization [CITATION].', '1510.08634-1-22-3': 'However, we will not discuss this here.', '1510.08634-1-23-0': '## Arnoldi method', '1510.08634-1-24-0': 'The Arnoldi method is a method to approximate a large matrix [MATH] in a smaller dimension [CITATION].', '1510.08634-1-24-1': 'This is done by choosing an appropriate set of a small number of vectors, which should be representative of the relevant subspace for a specific problem.', '1510.08634-1-24-2': 'Then the desired matrix is represented in the reduced subspace which is spanned by the chosen vectors.', '1510.08634-1-24-3': 'The method starts with an initial vector [MATH] and creates set of [MATH] vectors by the repetitive operation of the matrix [MATH]: [MATH].', '1510.08634-1-24-4': 'Then an orthonormal set is generated from this set by the Gram-Schmidt process.', '1510.08634-1-24-5': 'This orthonormal vectors set spans a subspace with dimension [MATH], and the matrix [MATH] is represented in this subspace by a [MATH] matrix.', '1510.08634-1-24-6': 'This smaller matrix can be used for the efficient evaluation of functions of the matrix [MATH], e.g. the exponential [CITATION] or the eigenvalues [CITATION].', '1510.08634-1-25-0': 'In our case we try to approximate the linear super-operator [MATH] by a matrix which is smaller than [MATH].', '1510.08634-1-25-1': 'Conceptually, we start with the initial density matrix [MATH], and operate [MATH] times with [MATH] to get the set [MATH] which is the starting point for orthogonalization and [MATH]-dimension matrix representation of [MATH].', '1510.08634-1-25-2': 'We note that the operation of [MATH] involves [MATH] matrix-matrix multiplications, which scales as [MATH].', '1510.08634-1-25-3': 'Therefore, it is more efficient to use the operation of [MATH] for the procedure than to use the vec-ing matrix [MATH] (Eq. ([REF])) described in Sec. [REF] above.', '1510.08634-1-26-0': 'The actual procedure follows, adapted to the notation of a super-operator and density matrices:', '1510.08634-1-27-0': 'Begin with the normalized density matrix [MATH].', '1510.08634-1-27-1': 'for [MATH] to [MATH]', '1510.08634-1-28-0': 'Compute a non-orthonormalized new density matrix by setting: [MATH] for [MATH] to [MATH]', '1510.08634-1-29-0': 'Set: [MATH] Subtract the projection on [MATH]: [MATH]', '1510.08634-1-30-0': 'end for Set: [MATH] Normalize [MATH] by setting [MATH]', '1510.08634-1-31-0': 'end for', '1510.08634-1-32-0': 'The procedure yields [EQUATION]', '1510.08634-1-32-1': 'For [MATH], the expression in the RHS vanishes.', '1510.08634-1-32-2': 'Thus, we can define a [MATH] matrix which its general element is given by a matrix element of [MATH] in the Liouville space: [EQUATION] (Note that the procedure also yields [MATH] and [MATH] which are not necessary for our purposes).', '1510.08634-1-32-3': '[MATH] represents the operation of the super-operator [MATH] on the subspace that is spanned by the density matrices [MATH].', '1510.08634-1-32-4': 'The matrix [MATH] is referred to as the Hessenberg matrix of [MATH].', '1510.08634-1-32-5': 'The density matrix has to be approximated by its projection on the subspace: [MATH].', '1510.08634-1-32-6': 'The vector [EQUATION] is the representation of the density matrix in this subspace.', '1510.08634-1-32-7': 'The dynamics of the vector [MATH] is generated by the matrix [MATH] that was constructed in step (2) of the above procedure: [EQUATION].', '1510.08634-1-32-8': 'Exponentiation and eigenvalue calculations of the matrix [MATH] can be done by common numerical techniques [CITATION].', '1510.08634-1-33-0': 'The Arnoldi algorithm usually becomes problematic when a large dimension approximation is required, i.e. when [MATH] is large.', '1510.08634-1-33-1': 'In such a case, a restarted Arnoldi algorithm should be used instead (see, for example, [CITATION]).', '1510.08634-1-33-2': 'This topic is beyond the scope of this paper.', '1510.08634-1-34-0': '## The Heisenberg representation', '1510.08634-1-35-0': 'Not always the full state of the system will be of concern.', '1510.08634-1-35-1': 'In most cases we will be interested only in the expectation values of some measured quantities.', '1510.08634-1-35-2': 'This fact can reduce significantly the dimensions of the problem.', '1510.08634-1-35-3': 'For example, in the standard thermalizing master equation the population and the coherences are decoupled, and the population of a certain level is given by solving a single differential equation [CITATION].', '1510.08634-1-35-4': 'The full state of the system can be reconstructed by calculating all the expectation values of the Lie algebra of the system.', '1510.08634-1-35-5': 'Generally, a full reconstruction of the state will scale as the Vec-ing of the density matrix introduced in Sec. [REF].', '1510.08634-1-35-6': 'Nevertheless, in many cases we can use symmetries to reduce the dimensions of the problem.', '1510.08634-1-35-7': 'For example, if the initial state of harmonic oscillator is a Gaussian state, then it will stay Gaussian along the dynamics and only the first two moments are necessary to retrieve the full state [CITATION].', '1510.08634-1-35-8': 'Another example is coupled two qubits in which the full dimension of the system is 16, but only 3 operators are sufficient to define the energy and coherence of the system [CITATION].', '1510.08634-1-36-0': 'To describe the dynamics of the expectation values, it is common to use the master equation in the Heisenberg representation.', '1510.08634-1-36-1': 'The operator [MATH] belonging to dual Hilbert space of the system follows the dynamics [CITATION]: [EQUATION] which in its differential form is written explicitly as [EQUATION]', '1510.08634-1-36-2': 'If there is a a set of operators [MATH], [MATH], that forms a closed set under the operation of [MATH], meaning [EQUATION] then we can write a closed linear system of coupled differential equations.', '1510.08634-1-36-3': 'The expectation values [MATH] will have the corresponding set of coupled differential equations.', '1510.08634-1-36-4': 'The analytical form of their dynamics will follow the form of Eq. ([REF]).', '1510.08634-1-36-5': 'We define the vector of expectation values [MATH].', '1510.08634-1-36-6': 'This system can be represented in a matrix-vector notation,', '1510.08634-1-37-0': '[EQUATION] where the matrix [MATH] is defined by the equation set Eq. ([REF]), [MATH].', '1510.08634-1-37-1': 'The dimension of this matrix is [MATH].', '1510.08634-1-37-2': 'Note that eigenvalues of the matrix [MATH] are complex conjugates of a subset of the eigenvalues of the super-operator [MATH] of Eq. ([REF]).', '1510.08634-1-38-0': '# Example: The two-level system master equation', '1510.08634-1-39-0': '## The model', '1510.08634-1-40-0': 'As an example, we consider a driven two-level system (TLS) with spontaneous emission [CITATION].', '1510.08634-1-41-0': 'We use the following definitions:', '1510.08634-1-42-0': '[EQUATION].', '1510.08634-1-42-1': 'In addition: [EQUATION].', '1510.08634-1-42-2': 'The commutation relations are: [EQUATION] where [MATH] is the Levi-Civita symbol, defined as: [EQUATION].', '1510.08634-1-43-0': 'The system Hamiltonian is [EQUATION] with [MATH] as the system transition frequency.', '1510.08634-1-43-1': 'The system is driven by the external field [MATH], with the carrier frequency [MATH] and amplitude [MATH].', '1510.08634-1-43-2': 'The coupling to the driving field is expressed by the matrix [EQUATION].', '1510.08634-1-43-3': 'In order to work within a time-independent Hamiltonian, we move to an interaction picture according to [MATH], obtaining [EQUATION] where [MATH] is the detuning between the system and the field frequencies.', '1510.08634-1-44-0': 'The spontaneous emission is expressed by a dissipative term, and the L-GKS master equation takes the form [CITATION]: [EQUATION] where [MATH] is the spontaneous emission rate.', '1510.08634-1-45-0': '## Matrix-vector representation of the TLS dynamics', '1510.08634-1-46-0': 'In the following we will implement the different approaches discussed above, for the relaxed driven TLS.', '1510.08634-1-47-0': '### Vec-ing the density matrix', '1510.08634-1-48-0': 'We use the procedure presented in Sec. ([REF]) for the L-GKS of the two-level system with relaxation, Eq. ([REF]).', '1510.08634-1-49-0': 'The basis for the representation is the trivial set of matrices: [EQUATION].', '1510.08634-1-50-0': 'To represent the commutator as a [MATH] matrix, we use the procedure to get: [EQUATION]', '1510.08634-1-50-1': 'For the dissipator we have: [EQUATION].', '1510.08634-1-50-2': 'We combine all parts to get: [EQUATION]', '1510.08634-1-51-0': '### Arnoldi method', '1510.08634-1-52-0': 'The Arnoldi method that was presented in Sec. [REF], and we use it here to find a representation of the two-level system super-operator.', '1510.08634-1-52-1': 'Note that generally the Arnoldi method is used for the approximation of large matrices.', '1510.08634-1-52-2': 'Here we have a small-size problem ([MATH]), and we create an exact matrix, which represents the two-level system super-operator on a basis that spans the entire space.', '1510.08634-1-53-0': 'We start with an initial density matrix [MATH] (chosen arbitrary): [EQUATION].', '1510.08634-1-53-1': 'Then we follow the Arnoldi iteration procedure to get the basis [MATH]: [EQUATION] with the definition: [MATH].', '1510.08634-1-53-2': 'This implies that the initial vector in this basis is [MATH].', '1510.08634-1-54-0': 'The representation of the super-operator [MATH] in this basis is the matrix obtained by the procedure: [EQUATION]', '1510.08634-1-55-0': '### The Heisenberg representation', '1510.08634-1-56-0': 'For the system described above, the L-GKS equation in the Heisenberg picture is:', '1510.08634-1-57-0': '[EQUATION].', '1510.08634-1-57-1': 'As a set of basis operators we choose the set: [EQUATION].', '1510.08634-1-57-2': 'The L-GKS equation for the operators of the basis gives: [EQUATION] and, of course: [EQUATION].', '1510.08634-1-58-0': 'The expectation values of these operators, will follow the same dynamics.', '1510.08634-1-58-1': 'We denote the vector of this expectation values as [MATH].', '1510.08634-1-58-2': 'The dynamics of this vector is given by [EQUATION] with the matrix: [EQUATION]', '1510.08634-1-59-0': '## Comparison', '1510.08634-1-60-0': 'We demonstrated the three approaches for representing the L-GKS dynamics with a matrix-vector notation.', '1510.08634-1-60-1': 'For the relaxed two-level system we obtained three different matrices.', '1510.08634-1-60-2': 'However, they are all equivalent, representing the dynamics by different bases or in different spaces.', '1510.08634-1-60-3': 'A simple verification is calculating the eigenvalues for different values of the parameters.', '1510.08634-1-60-4': 'As a neat example, for the values of [MATH], [MATH], we know from a previous study [CITATION] that we get a third order non-hermitian degeneracy of the eigenvalue [MATH].', '1510.08634-1-60-5': 'Substituting these values to the three matrices yield the same non-hermitian degeneracy.', '1510.08634-1-61-0': '# Discussion', '1510.08634-1-62-0': 'There is a heavy conceptual and computational price for the reduced description of open quantum systems in Liouville space.', '1510.08634-1-62-1': 'To pave the way to overcome this difficulty, it is desirable to represent the dynamics in the more familiar matrix vector notation.', '1510.08634-1-63-0': 'Significant simplification can be identified in the Heisenberg representation, when a set of operators which is closed to the equation of motion is found.', '1510.08634-1-63-1': 'For the Hamiltonian part, a closed set is obtained when the operators form a closed compact Lie algebra and the Hamiltonian is a linear combination of these operators [CITATION].', '1510.08634-1-63-2': 'Additional requirements are needed for the set to be also closed to the dissipative part [CITATION].', '1510.08634-1-64-0': 'When a closed set of operators cannot be found one has to resort to approximate methods.', '1510.08634-1-64-1': 'The idea is to construct a representative subset of operators.', '1510.08634-1-64-2': 'This set is generated from the initial state with successive applications of the dynamical generator [MATH].', '1510.08634-1-64-3': 'The initial idea can be traced to Lanczos [CITATION] who applied it to obtain iterative solutions to eigenvalue problems of a hermitian operator [MATH].', '1510.08634-1-64-4': 'Since the eigenvalues of [MATH] are complex, the iterative approach is modified to the Arnoldi method [CITATION].', '1510.08634-1-64-5': 'The Arnoldi approach is effective in the reduction of a large scale problem into a relatively small approximation space.', '1510.08634-1-64-6': 'Therefore, it should be considered as a standard approach for the treatment of large-scale L-GKS problems.'}

{'1510.08634-2-0-0': 'Markovian dynamics of open quantum systems are described by the L-GKS equation, known also as the Lindblad equation.', '1510.08634-2-0-1': 'The equation is expressed by means of left and right matrix multiplications.', '1510.08634-2-0-2': 'This formulation hampers numerical implementations.', '1510.08634-2-0-3': 'Representing the dynamics by a matrix-vector notation overcomes this problem.', '1510.08634-2-0-4': 'We review three approaches to obtain such a representation.', '1510.08634-2-0-5': 'The methods are demonstrated for a driven two-level system subject to spontaneous emission.', '1510.08634-2-1-0': '# Introduction', '1510.08634-2-2-0': 'An open system is a system that interacts with its environment.', '1510.08634-2-2-1': 'A full description has to account for all the degrees of freedom (DOF) of the entire system and its environment.', '1510.08634-2-2-2': 'Usually, only the system DOF are of interest.', '1510.08634-2-2-3': 'A reduced description attempts to describe only the system DOF explicitly, while the environment DOF are integrated out and affect the description implicitly.', '1510.08634-2-2-4': 'The goal is to reduce the description to a small number of variables and obtain a practical way to treat the system [CITATION].', '1510.08634-2-3-0': 'Open systems are often described by a stochastic process which in many cases becomes a simple Markov process.', '1510.08634-2-3-1': 'In brief, a Markov process is a stochastic process with a short time memory, i.e., the process state depends solely on the present state.', '1510.08634-2-3-2': 'Mathematically it can be constructed as the Chapman-Kolmagorov equation for the conditional joint probability [CITATION].', '1510.08634-2-3-3': 'For a Markov process the probability distribution [MATH] on a certain space, either real-space or phase-space, which could be continuous or discrete, follows the differential equation [EQUATION]', '1510.08634-2-3-4': 'The formal solution for Eq. ([REF]) is given by, [EQUATION] where, without loss of generality, we define the initial time to be zero.', '1510.08634-2-3-5': 'The one-parameter family of maps[MATH] is a semigroup with the generator [MATH].', '1510.08634-2-3-6': 'The term semigroup implies that this family of maps does not form a full group.', '1510.08634-2-3-7': 'It lacks the negative range of the parameter [MATH], which implies that the inverse property required by a group is missing.', '1510.08634-2-3-8': 'Physically, this property is the manifestation of irreversible dynamics which allows us to distinguish the future from the past.', '1510.08634-2-3-9': 'The map [MATH] is a positive map that satisfies the composition rule (Markov property) [MATH], and preserves normalization of the probability density.', '1510.08634-2-4-0': 'In the quantum scenario several modifications have to be made.', '1510.08634-2-4-1': 'The probability distribution [MATH] is replaced by the density matrix [MATH].', '1510.08634-2-4-2': 'The property of positivity has to be strengthened to complete positivity.', '1510.08634-2-4-3': 'The dynamics follows the quantum master equation: [EQUATION]', '1510.08634-2-4-4': 'This is a direct consequence of the presence of entangled states [CITATION].', '1510.08634-2-4-5': 'To summarize, the quantum dynamical semigroup is a continuous one-parameter family of maps [MATH], that satisfies [CITATION]:', '1510.08634-2-5-0': '[MATH] is complete positive; [MATH] is trace preserving; [MATH] semigroup (Markov) property; [MATH] is strongly continuous.', '1510.08634-2-6-0': 'Lindblad as well as Gorini, Kossakowski and Sudarshan (L-GKS) introduced the most general form of the quantum dynamical semigroup generator [MATH] that satisfies these requirements [CITATION].', '1510.08634-2-6-1': 'In the Lindblad form the Markovian master equation reads: [EQUATION]', '1510.08634-2-6-2': 'Here, [MATH] is the effective Hamiltonian of the system, [MATH] are positive rates, and [MATH] are operators belonging to the Hilbert space of the system.', '1510.08634-2-6-3': 'We use the notation [MATH] to represent the unitary part of the dynamics, and [MATH] to represent the dissipative part.', '1510.08634-2-6-4': '[MATH], [MATH] and [MATH] are linear operators that operate on the density matrix, usually referred to as super-operators.', '1510.08634-2-7-0': 'The operation of the super-operator [MATH] on the density matrix could be understood as repetitive operations of the super-operator [MATH] as in the Taylor expansion: [EQUATION]', '1510.08634-2-7-1': 'Typically, the resulting dynamics of the system observables (expectation values and other correlation functions) [MATH] will have the analytical form of sum of decaying oscillations: [EQUATION]', '1510.08634-2-7-2': 'Here, [MATH] are the exponential coefficients and [MATH] are the associated amplitudes, both can be complex.', '1510.08634-2-7-3': 'We may divide [MATH] into its real and imaginary parts, [MATH], with [MATH] as the decay rates and [MATH] as the oscillation frequencies.', '1510.08634-2-7-4': 'The coefficients [MATH] are the eigenvalues of the super-operator [MATH], obtained by the eigenvalue equation: [EQUATION]', '1510.08634-2-7-5': 'These eigenvalues can be used for the analysis of the L-GKS dynamics.', '1510.08634-2-8-0': 'As noted above, the dynamics can be investigated by exponentiation of the super-operator [MATH], Eq. ([REF]), or by its eigenvalues, Eq. ([REF]).', '1510.08634-2-8-1': 'The exponentiation and the eigenvalue problem of the (linear) super-operator [MATH] are well defined.', '1510.08634-2-8-2': 'However, they are not suitable for numerical calculations.', '1510.08634-2-8-3': 'Calculations of the exponentiation and the eigenvalue equation of linear operators can be done by common numerical techniques if the linear operator is represented by a matrix.', '1510.08634-2-8-4': 'Therefore, a preferred representation of the dynamics, Eq. ([REF]), is in a matrix-vector notation.', '1510.08634-2-8-5': 'This means that we are looking for a matrix [MATH] and a vector [MATH] such that the dynamics are expressed as [EQUATION]', '1510.08634-2-8-6': 'In this representation, the vector [MATH] represents the state of the system, or some information about it, e.g. a set of expectation values.', '1510.08634-2-8-7': 'Next, we describe three approaches for such a representation, and demonstrate them for a case of a driven two-level system with relaxation.', '1510.08634-2-9-0': '# Matrix-vector representations', '1510.08634-2-10-0': 'Suppose the density matrix [MATH] is an [MATH] matrix (if [MATH] is a function of continuous variables, e.g. [MATH], these variables have to be discretized).', '1510.08634-2-10-1': 'The set of all [MATH] matrices form a linear space of dimension [MATH].', '1510.08634-2-10-2': 'Under appropriate conditions, this linear space can have a Hilbert space construction, using the scalar product defined as [EQUATION].', '1510.08634-2-10-3': 'Such a Hilbert space is called a Liouville space (also known as the Hilbert-Schmidt space).', '1510.08634-2-10-4': 'With such a construction we consider [MATH] as an [MATH] vector.', '1510.08634-2-10-5': 'Similarly, we consider the super-operator [MATH], which is an operator operating on elements in this linear space, as an [MATH] matrix.', '1510.08634-2-11-0': 'The above observation is the first step towards the representation we seek.', '1510.08634-2-11-1': 'In the following, we describe three approaches that use this concept to introduce such representation:', '1510.08634-2-12-0': 'Vec-ing the density matrix is the most natural way to construct an [MATH] vector for the density matrix, and a suitable [MATH] matrix for the super-operator.', '1510.08634-2-12-1': 'The Arnoldi method approximates a large matrix in smaller dimensions, enabling simpler numerical calculations.', '1510.08634-2-12-2': 'With the Heisenberg picture of the L-GKS equation we can search for a representation with a dimension smaller than [MATH].', '1510.08634-2-13-0': 'In the following, we describe these three approaches.', '1510.08634-2-13-1': 'Each of these approaches will be demonstrated in the case of the two-level system.', '1510.08634-2-14-0': '## Vec-ing the density matrix', '1510.08634-2-15-0': 'In this method, known as vec-ing [CITATION], the [MATH] density matrix [MATH] is flattened into an [MATH] vector [MATH] .', '1510.08634-2-15-1': 'This flattening is done by ordering the columns of [MATH] one below the other, so the [MATH] entry of the matrix [MATH] is the [MATH] entry of the vector [MATH].', '1510.08634-2-15-2': 'This is equivalent to choosing the representation basis as the set of matrices with all-zero entries, except one.', '1510.08634-2-16-0': 'The next task is to find the suitable matrix that will represent the operation of the super-operator [MATH] on the density matrix.', '1510.08634-2-16-1': 'We make the following observations [CITATION]:', '1510.08634-2-17-0': 'A left multiplication of the matrix [MATH] by an [MATH] matrix [MATH], i.e. [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH], where [MATH] is the [MATH] identity matrix, and [MATH] is the Kronecker direct product.', '1510.08634-2-17-1': 'Similarly, a right multiplication of the matrix [MATH] by an [MATH] matrix [MATH], i.e. [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH].', '1510.08634-2-17-2': 'Here [MATH] denotes the transpose of the matrix.', '1510.08634-2-17-3': 'Finally, a combination of left and right matrices multiplication, [MATH], is equivalent to an operation on the vector [MATH] by the [MATH] matrix [MATH].', '1510.08634-2-18-0': 'The L-GKS super-operator is a sum of such right and left multiplications.', '1510.08634-2-18-1': 'Therefore, the construction of the [MATH] matrix representation for the L-GKS generator has the parts as follows; for the commutator: [EQUATION].', '1510.08634-2-18-2': 'For the dissipative part: [EQUATION]', '1510.08634-2-18-3': 'Then we write [EQUATION] and represent Eq. ([REF]) as [EQUATION] as desired.', '1510.08634-2-19-0': 'The mapping of the density matrix into a density vector yields in a dramatic increment in the dimension of the problem, which becomes [MATH] instead of [MATH].', '1510.08634-2-19-1': 'This yields unfavorable scaling of the desired computations with [MATH]:', '1510.08634-2-20-0': 'For systems larger than a few degrees of freedom, such computations are expensive, and become practically impossible for systems larger than a few hundreds DOF.', '1510.08634-2-21-0': 'The scaling problem suggests that we have to look for approaches that use a smaller number of dimensions.', '1510.08634-2-21-1': 'The following two approaches address this issue.', '1510.08634-2-21-2': 'The Arnoldi method uses a small-dimension approximation of a large matrix.', '1510.08634-2-21-3': 'The operator representation seeks for a small subset of variables that are sufficient to describe the quantities of interest.', '1510.08634-2-21-4': 'These two approaches are described in the next two sections.', '1510.08634-2-22-0': 'Remark: The density matrix [MATH] is hermitian.', '1510.08634-2-22-1': 'Therefore there are only [MATH] unique entries and not [MATH].', '1510.08634-2-22-2': 'This fact can be used to reduce the size of the vectors and matrices, known as a half-vectorization [CITATION].', '1510.08634-2-22-3': 'However, we will not discuss this here.', '1510.08634-2-23-0': '## Arnoldi method', '1510.08634-2-24-0': 'The Arnoldi method is a method to approximate a large matrix [MATH] in a smaller dimension [CITATION].', '1510.08634-2-24-1': 'This is done by choosing an appropriate set of a small number of vectors, which should be representative of the relevant subspace for a specific problem.', '1510.08634-2-24-2': 'Then the desired matrix is represented in the reduced subspace which is spanned by the chosen vectors.', '1510.08634-2-24-3': 'The method starts with an initial vector [MATH] and creates set of [MATH] vectors by the repetitive operation of the matrix [MATH]: [MATH].', '1510.08634-2-24-4': 'Then an orthonormal set is generated from this set by the Gram-Schmidt process.', '1510.08634-2-24-5': 'This orthonormal vectors set spans a subspace with dimension [MATH], and the matrix [MATH] is represented in this subspace by a [MATH] matrix.', '1510.08634-2-24-6': 'This smaller matrix can be used for the efficient evaluation of functions of the matrix [MATH], e.g. the exponential [CITATION] or the eigenvalues [CITATION].', '1510.08634-2-25-0': 'In our case we try to approximate the linear super-operator [MATH] by a matrix which is smaller than [MATH].', '1510.08634-2-25-1': 'Conceptually, we start with the initial density matrix [MATH], and operate [MATH] times with [MATH] to get the set [MATH] which is the starting point for orthogonalization and [MATH]-dimension matrix representation of [MATH].', '1510.08634-2-25-2': 'We note that the operation of [MATH] involves [MATH] matrix-matrix multiplications, which scales as [MATH].', '1510.08634-2-25-3': 'Therefore, it is more efficient to use the operation of [MATH] for the procedure than to use the vec-ing matrix [MATH] (Eq. ([REF])) described in Sec. [REF] above.', '1510.08634-2-26-0': 'The actual procedure follows, adapted to the notation of a super-operator and density matrices:', '1510.08634-2-27-0': 'Begin with the normalized density matrix [MATH].', '1510.08634-2-27-1': 'for [MATH] to [MATH]', '1510.08634-2-28-0': 'Compute a non-orthonormalized new density matrix by setting: [MATH] for [MATH] to [MATH]', '1510.08634-2-29-0': 'Set: [MATH] Subtract the projection on [MATH]: [MATH]', '1510.08634-2-30-0': 'end for Set: [MATH] Normalize [MATH] by setting [MATH]', '1510.08634-2-31-0': 'end for', '1510.08634-2-32-0': 'The procedure yields [EQUATION]', '1510.08634-2-32-1': 'For [MATH], the expression in the RHS vanishes.', '1510.08634-2-32-2': 'Thus, we can define a [MATH] matrix which its general element is given by a matrix element of [MATH] in the Liouville space: [EQUATION] (Note that the procedure also yields [MATH] and [MATH] which are not necessary for our purposes).', '1510.08634-2-32-3': '[MATH] represents the operation of the super-operator [MATH] on the subspace that is spanned by the density matrices [MATH].', '1510.08634-2-32-4': 'The matrix [MATH] is referred to as the Hessenberg matrix of [MATH].', '1510.08634-2-32-5': 'The density matrix has to be approximated by its projection on the subspace: [MATH].', '1510.08634-2-32-6': 'The vector [EQUATION] is the representation of the density matrix in this subspace.', '1510.08634-2-32-7': 'The dynamics of the vector [MATH] is generated by the matrix [MATH] that was constructed in step (2) of the above procedure: [EQUATION].', '1510.08634-2-32-8': 'Exponentiation and eigenvalue calculations of the matrix [MATH] can be done by common numerical techniques [CITATION].', '1510.08634-2-33-0': 'The Arnoldi algorithm usually becomes problematic when a large dimension approximation is required, i.e. when [MATH] is large.', '1510.08634-2-33-1': 'In such a case, a restarted Arnoldi algorithm should be used instead (see, for example, [CITATION]).', '1510.08634-2-33-2': 'This topic is beyond the scope of this paper.', '1510.08634-2-34-0': '## The Heisenberg representation', '1510.08634-2-35-0': 'Not always the full state of the system will be of concern.', '1510.08634-2-35-1': 'In most cases we will be interested only in the expectation values of some measured quantities.', '1510.08634-2-35-2': 'This fact can reduce significantly the dimensions of the problem.', '1510.08634-2-35-3': 'For example, in the standard thermalizing master equation the population and the coherences are decoupled, and the population of a certain level is given by solving a single differential equation [CITATION].', '1510.08634-2-35-4': 'The full state of the system can be reconstructed by calculating all the expectation values of the Lie algebra of the system.', '1510.08634-2-35-5': 'Generally, a full reconstruction of the state will scale as the Vec-ing of the density matrix introduced in Sec. [REF].', '1510.08634-2-35-6': 'Nevertheless, in many cases we can use symmetries to reduce the dimensions of the problem.', '1510.08634-2-35-7': 'For example, if the initial state of harmonic oscillator is a Gaussian state, then it will stay Gaussian along the dynamics and only the first two moments are necessary to retrieve the full state [CITATION].', '1510.08634-2-35-8': 'Another example is coupled two qubits in which the full dimension of the system is 16, but only 3 operators are sufficient to define the energy and coherence of the system [CITATION].', '1510.08634-2-36-0': 'To describe the dynamics of the expectation values, it is common to use the master equation in the Heisenberg representation.', '1510.08634-2-36-1': 'The operator [MATH] belonging to dual Hilbert space of the system follows the dynamics [CITATION]: [EQUATION] which in its differential form is written explicitly as [EQUATION]', '1510.08634-2-36-2': 'If there is a a set of operators [MATH], [MATH], that forms a closed set under the operation of [MATH], meaning [EQUATION] then we can write a closed linear system of coupled differential equations.', '1510.08634-2-36-3': 'The expectation values [MATH] will have the corresponding set of coupled differential equations.', '1510.08634-2-36-4': 'The analytical form of their dynamics will follow the form of Eq. ([REF]).', '1510.08634-2-36-5': 'We define the vector of expectation values [MATH].', '1510.08634-2-36-6': 'This system can be represented in a matrix-vector notation,', '1510.08634-2-37-0': '[EQUATION] where the matrix [MATH] is defined by the equation set Eq. ([REF]), [MATH].', '1510.08634-2-37-1': 'The dimension of this matrix is [MATH].', '1510.08634-2-37-2': 'Note that eigenvalues of the matrix [MATH] are complex conjugates of a subset of the eigenvalues of the super-operator [MATH] of Eq. ([REF]).', '1510.08634-2-38-0': '# Example: The two-level system master equation', '1510.08634-2-39-0': '## The model', '1510.08634-2-40-0': 'As an example, we consider a driven two-level system (TLS) with spontaneous emission [CITATION].', '1510.08634-2-41-0': 'We use the following definitions:', '1510.08634-2-42-0': '[EQUATION].', '1510.08634-2-42-1': 'In addition: [EQUATION].', '1510.08634-2-42-2': 'The commutation relations are: [EQUATION] where [MATH] is the Levi-Civita symbol, defined as:', '1510.08634-2-43-0': '[EQUATION].', '1510.08634-2-44-0': 'The system Hamiltonian is [EQUATION] with [MATH] as the system transition frequency.', '1510.08634-2-44-1': 'The system is driven by the external field [MATH], with the carrier frequency [MATH] and amplitude [MATH].', '1510.08634-2-44-2': 'The coupling to the driving field is expressed by the matrix [EQUATION].', '1510.08634-2-44-3': 'In order to work within a time-independent Hamiltonian, we move to an interaction picture according to [MATH], obtaining [EQUATION] where [MATH] is the detuning between the system and the field frequencies.', '1510.08634-2-45-0': 'The spontaneous emission is expressed by a dissipative term, and the L-GKS master equation takes the form [CITATION]: [EQUATION] where [MATH] is the spontaneous emission rate.', '1510.08634-2-46-0': '## Matrix-vector representation of the TLS dynamics', '1510.08634-2-47-0': 'In the following we will implement the different approaches discussed above, for the relaxed driven TLS.', '1510.08634-2-48-0': '### Vec-ing the density matrix', '1510.08634-2-49-0': 'We use the procedure presented in Sec. ([REF]) for the L-GKS of the two-level system with relaxation, Eq. ([REF]).', '1510.08634-2-50-0': 'The basis for the representation is the trivial set of matrices: [EQUATION].', '1510.08634-2-51-0': 'To represent the commutator as a [MATH] matrix, we use the procedure to get: [EQUATION]', '1510.08634-2-51-1': 'For the dissipator we have: [EQUATION].', '1510.08634-2-51-2': 'We combine all parts to get: [EQUATION]', '1510.08634-2-52-0': '### Arnoldi method', '1510.08634-2-53-0': 'The Arnoldi method that was presented in Sec. [REF], and we use it here to find a representation of the two-level system super-operator.', '1510.08634-2-53-1': 'Note that generally the Arnoldi method is used for the approximation of large matrices.', '1510.08634-2-53-2': 'Here we have a small-size problem ([MATH]), and we create an exact matrix, which represents the two-level system super-operator on a basis that spans the entire space.', '1510.08634-2-54-0': 'We start with an initial density matrix [MATH] (chosen arbitrary): [EQUATION].', '1510.08634-2-54-1': 'Then we follow the Arnoldi iteration procedure to get the basis [MATH]: [EQUATION] with the definition: [MATH].', '1510.08634-2-54-2': 'This implies that the initial vector in this basis is [MATH].', '1510.08634-2-55-0': 'The representation of the super-operator [MATH] in this basis is the matrix obtained by the procedure: [EQUATION]', '1510.08634-2-56-0': '### The Heisenberg representation', '1510.08634-2-57-0': 'For the system described above, the L-GKS equation in the Heisenberg picture is:', '1510.08634-2-58-0': '[EQUATION].', '1510.08634-2-58-1': 'As a set of basis operators we choose the set: [EQUATION].', '1510.08634-2-58-2': 'The L-GKS equation for the operators of the basis gives: [EQUATION] and, of course: [EQUATION].', '1510.08634-2-59-0': 'The expectation values of these operators, will follow the same dynamics.', '1510.08634-2-59-1': 'We denote the vector of this expectation values as [MATH].', '1510.08634-2-59-2': 'The dynamics of this vector is given by [EQUATION] with the matrix: [EQUATION]', '1510.08634-2-60-0': '## Comparison', '1510.08634-2-61-0': 'We demonstrated the three approaches for representing the L-GKS dynamics with a matrix-vector notation.', '1510.08634-2-61-1': 'For the relaxed two-level system we obtained three different matrices.', '1510.08634-2-61-2': 'However, they are all equivalent, representing the dynamics by different bases or in different spaces.', '1510.08634-2-61-3': 'A simple verification is calculating the eigenvalues for different values of the parameters.', '1510.08634-2-61-4': 'As a neat example, for the values of [MATH], [MATH], we know from a previous study [CITATION] that we get a third order non-hermitian degeneracy of the eigenvalue [MATH].', '1510.08634-2-61-5': 'Substituting these values to the three matrices yield the same non-hermitian degeneracy.', '1510.08634-2-62-0': '# Discussion', '1510.08634-2-63-0': 'There is a heavy conceptual and computational price for the reduced description of open quantum systems in Liouville space.', '1510.08634-2-63-1': 'To pave the way to overcome this difficulty, it is desirable to represent the dynamics in the more familiar matrix vector notation.', '1510.08634-2-64-0': 'Significant simplification can be identified in the Heisenberg representation, when a set of operators which is closed to the equation of motion is found.', '1510.08634-2-64-1': 'For the Hamiltonian part, a closed set is obtained when the operators form a closed compact Lie algebra and the Hamiltonian is a linear combination of these operators [CITATION].', '1510.08634-2-64-2': 'Additional requirements are needed for the set to be also closed to the dissipative part [CITATION].', '1510.08634-2-65-0': 'When a closed set of operators cannot be found one has to resort to approximate methods.', '1510.08634-2-65-1': 'The idea is to construct a representative subset of operators.', '1510.08634-2-65-2': 'This set is generated from the initial state with successive applications of the dynamical generator [MATH].', '1510.08634-2-65-3': 'The initial idea can be traced to Lanczos [CITATION] who applied it to obtain iterative solutions to eigenvalue problems of a hermitian operator [MATH].', '1510.08634-2-65-4': 'Since the eigenvalues of [MATH] are complex, the iterative approach is modified to the Arnoldi method [CITATION].', '1510.08634-2-65-5': 'The Arnoldi approach is effective in the reduction of a large scale problem into a relatively small approximation space.', '1510.08634-2-65-6': 'Therefore, it should be considered as a standard approach for the treatment of large-scale L-GKS problems.', '1510.08634-2-66-0': '## Bibliography'}

[['1510.08634-1-35-0', '1510.08634-2-35-0'], ['1510.08634-1-35-1', '1510.08634-2-35-1'], ['1510.08634-1-35-2', '1510.08634-2-35-2'], ['1510.08634-1-35-3', '1510.08634-2-35-3'], ['1510.08634-1-35-4', '1510.08634-2-35-4'], ['1510.08634-1-35-5', '1510.08634-2-35-5'], ['1510.08634-1-35-6', '1510.08634-2-35-6'], ['1510.08634-1-35-7', '1510.08634-2-35-7'], ['1510.08634-1-35-8', '1510.08634-2-35-8'], ['1510.08634-1-19-0', '1510.08634-2-19-0'], ['1510.08634-1-21-0', '1510.08634-2-21-0'], ['1510.08634-1-21-1', '1510.08634-2-21-1'], ['1510.08634-1-21-2', '1510.08634-2-21-2'], ['1510.08634-1-21-3', '1510.08634-2-21-3'], ['1510.08634-1-21-4', '1510.08634-2-21-4'], ['1510.08634-1-58-0', '1510.08634-2-59-0'], ['1510.08634-1-58-1', '1510.08634-2-59-1'], ['1510.08634-1-58-2', '1510.08634-2-59-2'], ['1510.08634-1-46-0', '1510.08634-2-47-0'], ['1510.08634-1-50-0', '1510.08634-2-51-0'], ['1510.08634-1-50-1', '1510.08634-2-51-1'], ['1510.08634-1-50-2', '1510.08634-2-51-2'], ['1510.08634-1-10-0', '1510.08634-2-10-0'], ['1510.08634-1-10-1', '1510.08634-2-10-1'], ['1510.08634-1-10-2', '1510.08634-2-10-2'], ['1510.08634-1-10-3', '1510.08634-2-10-3'], ['1510.08634-1-10-4', '1510.08634-2-10-4'], ['1510.08634-1-10-5', '1510.08634-2-10-5'], ['1510.08634-1-43-0', '1510.08634-2-44-0'], ['1510.08634-1-43-1', '1510.08634-2-44-1'], ['1510.08634-1-43-2', '1510.08634-2-44-2'], ['1510.08634-1-43-3', '1510.08634-2-44-3'], ['1510.08634-1-13-0', '1510.08634-2-13-0'], ['1510.08634-1-13-1', '1510.08634-2-13-1'], ['1510.08634-1-24-0', '1510.08634-2-24-0'], ['1510.08634-1-24-1', '1510.08634-2-24-1'], ['1510.08634-1-24-2', '1510.08634-2-24-2'], ['1510.08634-1-24-3', '1510.08634-2-24-3'], ['1510.08634-1-24-4', '1510.08634-2-24-4'], ['1510.08634-1-24-5', '1510.08634-2-24-5'], ['1510.08634-1-24-6', '1510.08634-2-24-6'], ['1510.08634-1-0-0', '1510.08634-2-0-0'], ['1510.08634-1-0-1', '1510.08634-2-0-1'], ['1510.08634-1-0-2', '1510.08634-2-0-2'], ['1510.08634-1-0-3', '1510.08634-2-0-3'], ['1510.08634-1-0-4', '1510.08634-2-0-4'], ['1510.08634-1-0-5', '1510.08634-2-0-5'], ['1510.08634-1-49-0', '1510.08634-2-50-0'], ['1510.08634-1-52-0', '1510.08634-2-53-0'], ['1510.08634-1-52-1', '1510.08634-2-53-1'], ['1510.08634-1-52-2', '1510.08634-2-53-2'], ['1510.08634-1-16-0', '1510.08634-2-16-0'], ['1510.08634-1-22-0', '1510.08634-2-22-0'], ['1510.08634-1-22-1', '1510.08634-2-22-1'], ['1510.08634-1-22-2', '1510.08634-2-22-2'], ['1510.08634-1-22-3', '1510.08634-2-22-3'], ['1510.08634-1-53-0', '1510.08634-2-54-0'], ['1510.08634-1-53-1', '1510.08634-2-54-1'], ['1510.08634-1-53-2', '1510.08634-2-54-2'], ['1510.08634-1-54-0', '1510.08634-2-55-0'], ['1510.08634-1-7-0', '1510.08634-2-7-0'], ['1510.08634-1-7-1', '1510.08634-2-7-1'], ['1510.08634-1-7-2', '1510.08634-2-7-2'], ['1510.08634-1-7-3', '1510.08634-2-7-3'], ['1510.08634-1-7-4', '1510.08634-2-7-4'], ['1510.08634-1-64-0', '1510.08634-2-65-0'], ['1510.08634-1-64-1', '1510.08634-2-65-1'], ['1510.08634-1-64-2', '1510.08634-2-65-2'], ['1510.08634-1-64-3', '1510.08634-2-65-3'], ['1510.08634-1-64-4', '1510.08634-2-65-4'], ['1510.08634-1-64-5', '1510.08634-2-65-5'], ['1510.08634-1-64-6', '1510.08634-2-65-6'], ['1510.08634-1-3-0', '1510.08634-2-3-0'], ['1510.08634-1-3-1', '1510.08634-2-3-1'], ['1510.08634-1-3-2', '1510.08634-2-3-2'], ['1510.08634-1-3-3', '1510.08634-2-3-3'], ['1510.08634-1-3-4', '1510.08634-2-3-4'], ['1510.08634-1-3-5', '1510.08634-2-3-5'], ['1510.08634-1-3-6', '1510.08634-2-3-6'], ['1510.08634-1-3-7', '1510.08634-2-3-7'], ['1510.08634-1-3-8', '1510.08634-2-3-8'], ['1510.08634-1-3-9', '1510.08634-2-3-9'], ['1510.08634-1-18-0', '1510.08634-2-18-0'], ['1510.08634-1-18-1', '1510.08634-2-18-1'], ['1510.08634-1-18-2', '1510.08634-2-18-2'], ['1510.08634-1-18-3', '1510.08634-2-18-3'], ['1510.08634-1-15-0', '1510.08634-2-15-0'], ['1510.08634-1-15-1', '1510.08634-2-15-1'], ['1510.08634-1-15-2', '1510.08634-2-15-2'], ['1510.08634-1-48-0', '1510.08634-2-49-0'], ['1510.08634-1-62-0', '1510.08634-2-63-0'], ['1510.08634-1-62-1', '1510.08634-2-63-1'], ['1510.08634-1-36-0', '1510.08634-2-36-0'], ['1510.08634-1-36-1', '1510.08634-2-36-1'], ['1510.08634-1-36-2', '1510.08634-2-36-2'], ['1510.08634-1-36-3', '1510.08634-2-36-3'], ['1510.08634-1-36-4', '1510.08634-2-36-4'], ['1510.08634-1-36-5', '1510.08634-2-36-5'], ['1510.08634-1-8-0', '1510.08634-2-8-0'], ['1510.08634-1-8-1', '1510.08634-2-8-1'], ['1510.08634-1-8-2', '1510.08634-2-8-2'], ['1510.08634-1-8-3', '1510.08634-2-8-3'], ['1510.08634-1-8-4', '1510.08634-2-8-4'], ['1510.08634-1-8-5', '1510.08634-2-8-5'], ['1510.08634-1-8-6', '1510.08634-2-8-6'], ['1510.08634-1-8-7', '1510.08634-2-8-7'], ['1510.08634-1-33-0', '1510.08634-2-33-0'], ['1510.08634-1-33-1', '1510.08634-2-33-1'], ['1510.08634-1-33-2', '1510.08634-2-33-2'], ['1510.08634-1-20-0', '1510.08634-2-20-0'], ['1510.08634-1-28-0', '1510.08634-2-28-0'], ['1510.08634-1-32-0', '1510.08634-2-32-0'], ['1510.08634-1-32-1', '1510.08634-2-32-1'], ['1510.08634-1-32-2', '1510.08634-2-32-2'], ['1510.08634-1-32-3', '1510.08634-2-32-3'], ['1510.08634-1-32-4', '1510.08634-2-32-4'], ['1510.08634-1-32-5', '1510.08634-2-32-5'], ['1510.08634-1-32-6', '1510.08634-2-32-6'], ['1510.08634-1-32-7', '1510.08634-2-32-7'], ['1510.08634-1-32-8', '1510.08634-2-32-8'], ['1510.08634-1-27-0', '1510.08634-2-27-0'], ['1510.08634-1-27-1', '1510.08634-2-27-1'], ['1510.08634-1-2-0', '1510.08634-2-2-0'], ['1510.08634-1-2-1', '1510.08634-2-2-1'], ['1510.08634-1-2-2', '1510.08634-2-2-2'], ['1510.08634-1-2-3', '1510.08634-2-2-3'], ['1510.08634-1-2-4', '1510.08634-2-2-4'], ['1510.08634-1-11-0', '1510.08634-2-11-0'], ['1510.08634-1-4-0', '1510.08634-2-4-0'], ['1510.08634-1-4-1', '1510.08634-2-4-1'], ['1510.08634-1-4-2', '1510.08634-2-4-2'], ['1510.08634-1-4-3', '1510.08634-2-4-3'], ['1510.08634-1-4-4', '1510.08634-2-4-4'], ['1510.08634-1-60-0', '1510.08634-2-61-0'], ['1510.08634-1-60-1', '1510.08634-2-61-1'], ['1510.08634-1-60-2', '1510.08634-2-61-2'], ['1510.08634-1-60-3', '1510.08634-2-61-3'], ['1510.08634-1-60-4', '1510.08634-2-61-4'], ['1510.08634-1-60-5', '1510.08634-2-61-5'], ['1510.08634-1-6-0', '1510.08634-2-6-0'], ['1510.08634-1-6-1', '1510.08634-2-6-1'], ['1510.08634-1-6-2', '1510.08634-2-6-2'], ['1510.08634-1-6-3', '1510.08634-2-6-3'], ['1510.08634-1-6-4', '1510.08634-2-6-4'], ['1510.08634-1-57-1', '1510.08634-2-58-1'], ['1510.08634-1-57-2', '1510.08634-2-58-2'], ['1510.08634-1-5-0', '1510.08634-2-5-0'], ['1510.08634-1-63-0', '1510.08634-2-64-0'], ['1510.08634-1-63-1', '1510.08634-2-64-1'], ['1510.08634-1-63-2', '1510.08634-2-64-2'], ['1510.08634-1-17-0', '1510.08634-2-17-0'], ['1510.08634-1-17-1', '1510.08634-2-17-1'], ['1510.08634-1-17-2', '1510.08634-2-17-2'], ['1510.08634-1-17-3', '1510.08634-2-17-3'], ['1510.08634-1-12-0', '1510.08634-2-12-0'], ['1510.08634-1-12-1', '1510.08634-2-12-1'], ['1510.08634-1-12-2', '1510.08634-2-12-2'], ['1510.08634-1-40-0', '1510.08634-2-40-0'], ['1510.08634-1-44-0', '1510.08634-2-45-0'], ['1510.08634-1-25-0', '1510.08634-2-25-0'], ['1510.08634-1-25-1', '1510.08634-2-25-1'], ['1510.08634-1-25-2', '1510.08634-2-25-2'], ['1510.08634-1-25-3', '1510.08634-2-25-3'], ['1510.08634-1-37-0', '1510.08634-2-37-0'], ['1510.08634-1-37-1', '1510.08634-2-37-1'], ['1510.08634-1-37-2', '1510.08634-2-37-2']]

[['1510.08634-1-35-0', '1510.08634-2-35-0'], ['1510.08634-1-35-1', '1510.08634-2-35-1'], ['1510.08634-1-35-2', '1510.08634-2-35-2'], ['1510.08634-1-35-3', '1510.08634-2-35-3'], ['1510.08634-1-35-4', '1510.08634-2-35-4'], ['1510.08634-1-35-5', '1510.08634-2-35-5'], ['1510.08634-1-35-6', '1510.08634-2-35-6'], ['1510.08634-1-35-7', '1510.08634-2-35-7'], ['1510.08634-1-35-8', '1510.08634-2-35-8'], ['1510.08634-1-19-0', '1510.08634-2-19-0'], ['1510.08634-1-21-0', '1510.08634-2-21-0'], ['1510.08634-1-21-1', '1510.08634-2-21-1'], ['1510.08634-1-21-2', '1510.08634-2-21-2'], ['1510.08634-1-21-3', '1510.08634-2-21-3'], ['1510.08634-1-21-4', '1510.08634-2-21-4'], ['1510.08634-1-58-0', '1510.08634-2-59-0'], ['1510.08634-1-58-1', '1510.08634-2-59-1'], ['1510.08634-1-58-2', '1510.08634-2-59-2'], ['1510.08634-1-46-0', '1510.08634-2-47-0'], ['1510.08634-1-50-0', '1510.08634-2-51-0'], ['1510.08634-1-50-1', '1510.08634-2-51-1'], ['1510.08634-1-50-2', '1510.08634-2-51-2'], ['1510.08634-1-10-0', '1510.08634-2-10-0'], ['1510.08634-1-10-1', '1510.08634-2-10-1'], ['1510.08634-1-10-2', '1510.08634-2-10-2'], ['1510.08634-1-10-3', '1510.08634-2-10-3'], ['1510.08634-1-10-4', '1510.08634-2-10-4'], ['1510.08634-1-10-5', '1510.08634-2-10-5'], ['1510.08634-1-43-0', '1510.08634-2-44-0'], ['1510.08634-1-43-1', '1510.08634-2-44-1'], ['1510.08634-1-43-2', '1510.08634-2-44-2'], ['1510.08634-1-43-3', '1510.08634-2-44-3'], ['1510.08634-1-13-0', '1510.08634-2-13-0'], ['1510.08634-1-13-1', '1510.08634-2-13-1'], ['1510.08634-1-24-0', '1510.08634-2-24-0'], ['1510.08634-1-24-1', '1510.08634-2-24-1'], ['1510.08634-1-24-2', '1510.08634-2-24-2'], ['1510.08634-1-24-3', '1510.08634-2-24-3'], ['1510.08634-1-24-4', '1510.08634-2-24-4'], ['1510.08634-1-24-5', '1510.08634-2-24-5'], ['1510.08634-1-24-6', '1510.08634-2-24-6'], ['1510.08634-1-0-0', '1510.08634-2-0-0'], ['1510.08634-1-0-1', '1510.08634-2-0-1'], ['1510.08634-1-0-2', '1510.08634-2-0-2'], ['1510.08634-1-0-3', '1510.08634-2-0-3'], ['1510.08634-1-0-4', '1510.08634-2-0-4'], ['1510.08634-1-0-5', '1510.08634-2-0-5'], ['1510.08634-1-49-0', '1510.08634-2-50-0'], ['1510.08634-1-52-0', '1510.08634-2-53-0'], ['1510.08634-1-52-1', '1510.08634-2-53-1'], ['1510.08634-1-52-2', '1510.08634-2-53-2'], ['1510.08634-1-16-0', '1510.08634-2-16-0'], ['1510.08634-1-22-0', '1510.08634-2-22-0'], ['1510.08634-1-22-1', '1510.08634-2-22-1'], ['1510.08634-1-22-2', '1510.08634-2-22-2'], ['1510.08634-1-22-3', '1510.08634-2-22-3'], ['1510.08634-1-53-0', '1510.08634-2-54-0'], ['1510.08634-1-53-1', '1510.08634-2-54-1'], ['1510.08634-1-53-2', '1510.08634-2-54-2'], ['1510.08634-1-54-0', '1510.08634-2-55-0'], ['1510.08634-1-7-0', '1510.08634-2-7-0'], ['1510.08634-1-7-1', '1510.08634-2-7-1'], ['1510.08634-1-7-2', '1510.08634-2-7-2'], ['1510.08634-1-7-3', '1510.08634-2-7-3'], ['1510.08634-1-7-4', '1510.08634-2-7-4'], ['1510.08634-1-64-0', '1510.08634-2-65-0'], ['1510.08634-1-64-1', '1510.08634-2-65-1'], ['1510.08634-1-64-2', '1510.08634-2-65-2'], ['1510.08634-1-64-3', '1510.08634-2-65-3'], ['1510.08634-1-64-4', '1510.08634-2-65-4'], ['1510.08634-1-64-5', '1510.08634-2-65-5'], ['1510.08634-1-64-6', '1510.08634-2-65-6'], ['1510.08634-1-3-0', '1510.08634-2-3-0'], ['1510.08634-1-3-1', '1510.08634-2-3-1'], ['1510.08634-1-3-2', '1510.08634-2-3-2'], ['1510.08634-1-3-3', '1510.08634-2-3-3'], ['1510.08634-1-3-4', '1510.08634-2-3-4'], ['1510.08634-1-3-5', '1510.08634-2-3-5'], ['1510.08634-1-3-6', '1510.08634-2-3-6'], ['1510.08634-1-3-7', '1510.08634-2-3-7'], ['1510.08634-1-3-8', '1510.08634-2-3-8'], ['1510.08634-1-3-9', '1510.08634-2-3-9'], ['1510.08634-1-18-0', '1510.08634-2-18-0'], ['1510.08634-1-18-1', '1510.08634-2-18-1'], ['1510.08634-1-18-2', '1510.08634-2-18-2'], ['1510.08634-1-18-3', '1510.08634-2-18-3'], ['1510.08634-1-15-0', '1510.08634-2-15-0'], ['1510.08634-1-15-1', '1510.08634-2-15-1'], ['1510.08634-1-15-2', '1510.08634-2-15-2'], ['1510.08634-1-48-0', '1510.08634-2-49-0'], ['1510.08634-1-62-0', '1510.08634-2-63-0'], ['1510.08634-1-62-1', '1510.08634-2-63-1'], ['1510.08634-1-36-0', '1510.08634-2-36-0'], ['1510.08634-1-36-1', '1510.08634-2-36-1'], ['1510.08634-1-36-2', '1510.08634-2-36-2'], ['1510.08634-1-36-3', '1510.08634-2-36-3'], ['1510.08634-1-36-4', '1510.08634-2-36-4'], ['1510.08634-1-36-5', '1510.08634-2-36-5'], ['1510.08634-1-8-0', '1510.08634-2-8-0'], ['1510.08634-1-8-1', '1510.08634-2-8-1'], ['1510.08634-1-8-2', '1510.08634-2-8-2'], ['1510.08634-1-8-3', '1510.08634-2-8-3'], ['1510.08634-1-8-4', '1510.08634-2-8-4'], ['1510.08634-1-8-5', '1510.08634-2-8-5'], ['1510.08634-1-8-6', '1510.08634-2-8-6'], ['1510.08634-1-8-7', '1510.08634-2-8-7'], ['1510.08634-1-33-0', '1510.08634-2-33-0'], ['1510.08634-1-33-1', '1510.08634-2-33-1'], ['1510.08634-1-33-2', '1510.08634-2-33-2'], ['1510.08634-1-20-0', '1510.08634-2-20-0'], ['1510.08634-1-28-0', '1510.08634-2-28-0'], ['1510.08634-1-32-0', '1510.08634-2-32-0'], ['1510.08634-1-32-1', '1510.08634-2-32-1'], ['1510.08634-1-32-2', '1510.08634-2-32-2'], ['1510.08634-1-32-3', '1510.08634-2-32-3'], ['1510.08634-1-32-4', '1510.08634-2-32-4'], ['1510.08634-1-32-5', '1510.08634-2-32-5'], ['1510.08634-1-32-6', '1510.08634-2-32-6'], ['1510.08634-1-32-7', '1510.08634-2-32-7'], ['1510.08634-1-32-8', '1510.08634-2-32-8'], ['1510.08634-1-27-0', '1510.08634-2-27-0'], ['1510.08634-1-27-1', '1510.08634-2-27-1'], ['1510.08634-1-2-0', '1510.08634-2-2-0'], ['1510.08634-1-2-1', '1510.08634-2-2-1'], ['1510.08634-1-2-2', '1510.08634-2-2-2'], ['1510.08634-1-2-3', '1510.08634-2-2-3'], ['1510.08634-1-2-4', '1510.08634-2-2-4'], ['1510.08634-1-11-0', '1510.08634-2-11-0'], ['1510.08634-1-4-0', '1510.08634-2-4-0'], ['1510.08634-1-4-1', '1510.08634-2-4-1'], ['1510.08634-1-4-2', '1510.08634-2-4-2'], ['1510.08634-1-4-3', '1510.08634-2-4-3'], ['1510.08634-1-4-4', '1510.08634-2-4-4'], ['1510.08634-1-60-0', '1510.08634-2-61-0'], ['1510.08634-1-60-1', '1510.08634-2-61-1'], ['1510.08634-1-60-2', '1510.08634-2-61-2'], ['1510.08634-1-60-3', '1510.08634-2-61-3'], ['1510.08634-1-60-4', '1510.08634-2-61-4'], ['1510.08634-1-60-5', '1510.08634-2-61-5'], ['1510.08634-1-6-0', '1510.08634-2-6-0'], ['1510.08634-1-6-1', '1510.08634-2-6-1'], ['1510.08634-1-6-2', '1510.08634-2-6-2'], ['1510.08634-1-6-3', '1510.08634-2-6-3'], ['1510.08634-1-6-4', '1510.08634-2-6-4'], ['1510.08634-1-57-1', '1510.08634-2-58-1'], ['1510.08634-1-57-2', '1510.08634-2-58-2'], ['1510.08634-1-5-0', '1510.08634-2-5-0'], ['1510.08634-1-63-0', '1510.08634-2-64-0'], ['1510.08634-1-63-1', '1510.08634-2-64-1'], ['1510.08634-1-63-2', '1510.08634-2-64-2'], ['1510.08634-1-17-0', '1510.08634-2-17-0'], ['1510.08634-1-17-1', '1510.08634-2-17-1'], ['1510.08634-1-17-2', '1510.08634-2-17-2'], ['1510.08634-1-17-3', '1510.08634-2-17-3'], ['1510.08634-1-12-0', '1510.08634-2-12-0'], ['1510.08634-1-12-1', '1510.08634-2-12-1'], ['1510.08634-1-12-2', '1510.08634-2-12-2'], ['1510.08634-1-40-0', '1510.08634-2-40-0'], ['1510.08634-1-44-0', '1510.08634-2-45-0'], ['1510.08634-1-25-0', '1510.08634-2-25-0'], ['1510.08634-1-25-1', '1510.08634-2-25-1'], ['1510.08634-1-25-2', '1510.08634-2-25-2'], ['1510.08634-1-25-3', '1510.08634-2-25-3'], ['1510.08634-1-37-0', '1510.08634-2-37-0'], ['1510.08634-1-37-1', '1510.08634-2-37-1'], ['1510.08634-1-37-2', '1510.08634-2-37-2']]

[]

[]

[]

[]

['1510.08634-1-4-5', '1510.08634-1-11-1', '1510.08634-1-16-1', '1510.08634-1-19-1', '1510.08634-1-26-0', '1510.08634-1-29-0', '1510.08634-1-30-0', '1510.08634-1-31-0', '1510.08634-1-36-6', '1510.08634-1-41-0', '1510.08634-1-42-0', '1510.08634-1-42-1', '1510.08634-1-56-0', '1510.08634-1-57-0', '1510.08634-2-4-5', '1510.08634-2-11-1', '1510.08634-2-16-1', '1510.08634-2-19-1', '1510.08634-2-26-0', '1510.08634-2-29-0', '1510.08634-2-30-0', '1510.08634-2-31-0', '1510.08634-2-36-6', '1510.08634-2-41-0', '1510.08634-2-42-0', '1510.08634-2-42-1', '1510.08634-2-42-2', '1510.08634-2-43-0', '1510.08634-2-57-0', '1510.08634-2-58-0']

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

https://arxiv.org/abs/1510.08634

hep-th-0604103

{'hep-th-0604103-1-0-0': 'We study matrix models as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-1-0-1': 'As a first step in this direction, we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of a matrix model.', 'hep-th-0604103-1-1-0': '# Introduction', 'hep-th-0604103-1-2-0': 'It has been known that there are problems in the construction of consistent interactions for massless higher spin gauge fields, though there are physically acceptable free field Lagrangian for them.', 'hep-th-0604103-1-2-1': 'Free field Lagrangian for higher spin fields, in which these particles are expressed by totally symmetric tensor or tensor-spinor fields, was originally derived by Fronsdal for bosons [CITATION] and Fang-Fronsdal for fermions [CITATION].', 'hep-th-0604103-1-2-2': 'There are some approaches for the construction of free field Lagrangian [CITATION] [CITATION].', 'hep-th-0604103-1-2-3': 'Interaction problems appear when one tries to couple massless higher spin fields to an electromagnetic field [CITATION], to gravity [CITATION] [CITATION] [CITATION] or to construct self-interactions [CITATION] [CITATION].', 'hep-th-0604103-1-2-4': 'For a review of massless higher spin gauge field theory, see [CITATION].', 'hep-th-0604103-1-3-0': 'At present, there exist various approaches to the theory, which solve the interaction problems in some cases.', 'hep-th-0604103-1-3-1': 'For example, an approach, called the unfolded formalism, was developed by Vasiliev et al [CITATION] [CITATION].', 'hep-th-0604103-1-3-2': 'They succeeded in the construction of interacting higher spin gauge theory with a nonzero cosmological constant [CITATION].', 'hep-th-0604103-1-3-3': 'An approach, called BRST approach, was initiated by development of string field theory on the basis of BRST techniques [CITATION] [CITATION] [CITATION] [CITATION].', 'hep-th-0604103-1-4-0': 'In this paper, we study matrix models as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-1-4-1': 'Recently, it has been shown that the Einstein equation can be obtained from the equation of motion of a matrix model by introducing a new interpretation of the matrix model, in which matrices represent differential operators on a curved spacetime [CITATION].', 'hep-th-0604103-1-4-2': 'Furthermore, it was pointed out that there is a possibility that matrix models include the degrees of freedom of massless higher spin gauge fields.', 'hep-th-0604103-1-4-3': 'An advantage of this formalism is that the matrix model possesses a gauge invariance manifestly, embedded in the [MATH] symmetry.', 'hep-th-0604103-1-4-4': 'Therefore it is interesting to analyze interacting massless higher spin gauge field theory using the matrix model.', 'hep-th-0604103-1-5-0': 'A first step towards constructing massless higher spin gauge field theory is the formulation of the free theory.', 'hep-th-0604103-1-5-1': 'Therefore, in this paper we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of the matrix model.', 'hep-th-0604103-1-6-0': 'There is another motivation for our study.', 'hep-th-0604103-1-6-1': 'Massless higher spin fields are expected to appear in the tensionless limit of string theory, since mass squared of them are all proportional to the string tension.', 'hep-th-0604103-1-6-2': 'Thus in this limit, one should observe an enhancement of gauge symmetry of string theory by that of the massless higher spin field theory.', 'hep-th-0604103-1-6-3': 'On the other hand, matrix models are expected to be a nonperturbative formulation of string theory.', 'hep-th-0604103-1-6-4': 'Therefore our study may be useful for better understanding of gauge symmetry of string theory and can lead to further understanding of nonperturbative aspects of string theory.', 'hep-th-0604103-1-7-0': 'The organization of this paper is as follows.', 'hep-th-0604103-1-7-1': 'In section 2, we briefly review the results of [CITATION].', 'hep-th-0604103-1-7-2': 'In section 3, we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of the matrix model.', 'hep-th-0604103-1-7-3': 'Section 4 is devoted to conclusions and future works.', 'hep-th-0604103-1-8-0': '# Matrix Model', 'hep-th-0604103-1-9-0': 'In this section, we briefly review the results of [CITATION].', 'hep-th-0604103-1-9-1': 'Introducing a new interpretation of a matrix model, we will see the following three facts.', 'hep-th-0604103-1-10-0': 'In this paper, we consider the large [MATH] reduced model of [MATH]-dimensional pure Yang-Mills theory with [MATH] gauge symmetry as the matrix model : [EQUATION] where [MATH] are [MATH] hermitian bosonic matrices.', 'hep-th-0604103-1-10-1': 'Latin indices denote Euclidean spacetime directions.', 'hep-th-0604103-1-10-2': 'This action has the [MATH] Lorentz symmetry and [MATH] unitary matrix symmetry.', 'hep-th-0604103-1-10-3': 'We can also consider supersymmetric version of this model, but in this paper we consider only the bosonic action ([REF]).', 'hep-th-0604103-1-11-0': 'The basic idea of [CITATION] is that matrices represent differential operators on a curved spacetime.', 'hep-th-0604103-1-11-1': 'There are several problems with this identification.', 'hep-th-0604103-1-11-2': 'For example, matrices act as [MATH] on a vector space, which means matrices map a vector space to itself.', 'hep-th-0604103-1-11-3': 'On the contrary, covariant derivatives map a tensor field of rank-[MATH] to a tensor field of rank-[MATH].', 'hep-th-0604103-1-11-4': 'Therefore we should prepare a vector space [MATH] which contain at least tensor fields of any rank and prepare an object [MATH] which is equivalent to a covariant derivative [MATH] such that each component of [MATH] is expressed as an [MATH] on [MATH], in order to interpret covariant derivatives as matrices.', 'hep-th-0604103-1-11-5': 'In [CITATION] the authors showed that [MATH] can be given by the space of smooth functions on the principal [MATH] bundle over a manifold [MATH].', 'hep-th-0604103-1-11-6': 'A smooth function [MATH] on it is defined as the mapping [EQUATION] where [MATH] denotes a patch on [MATH].', 'hep-th-0604103-1-11-7': 'Thus, [MATH] depends on the local coordinate [MATH], where [MATH] and [MATH], of the principal [MATH] bundle over [MATH].', 'hep-th-0604103-1-11-8': '[MATH] can be given by [EQUATION] where [MATH] is the vector representation of [MATH].', 'hep-th-0604103-1-11-9': 'The covariant derivative [MATH] is defined as [EQUATION] where [MATH] is the generator of the local Lorentz group [MATH], [MATH] is the [MATH] and [MATH] is the spin-connection.', 'hep-th-0604103-1-11-10': 'Here, Latin indices denote the local Lorentz indices.', 'hep-th-0604103-1-11-11': 'Notice that [MATH] maps a rank-[MATH] tensor to a rank-[MATH] tensor and [MATH] acts on the local Lorentz indices of these tensors.', 'hep-th-0604103-1-11-12': 'Therefore we have [EQUATION] in this setting, which will be used later.', 'hep-th-0604103-1-12-0': 'Let us see how the Einstein equation can be derived from the equation of motion of the matrix model by applying this interpretation to the matrix model.', 'hep-th-0604103-1-12-1': 'From the action ([REF]), we obtain the following equation of motion : [EQUATION]', 'hep-th-0604103-1-12-2': 'We substitute [MATH] into the equation of motion ([REF]), where [MATH] is introduced to make [MATH] hermitian.', 'hep-th-0604103-1-12-3': 'The commutator of [MATH] becomes [EQUATION] where [MATH] is the Riemann tensor and [MATH] is the torsion tensor.', 'hep-th-0604103-1-12-4': 'Here, we put the torsionless constraint [MATH] to solve the spin-connection in terms of the [MATH].', 'hep-th-0604103-1-12-5': 'Under the torsionless constraint, the equation of motion becomes [EQUATION] and we obtain the following two equations : [EQUATION]', 'hep-th-0604103-1-12-6': 'The first equation is the vacuum Einstein equation.', 'hep-th-0604103-1-12-7': 'The second equation can be derived from the first one and the Bianchi identity : [MATH].', 'hep-th-0604103-1-12-8': 'Thus, we have obtained the Einstein equation from the equation of motion of the matrix model.', 'hep-th-0604103-1-13-0': 'Since each component of matrices [MATH] acts functions on the principal [MATH] bundle over [MATH] as an [MATH], in general, [MATH] may be expanded as [EQUATION] where [MATH] and anticommutator [MATH] are introduced to make [MATH] hermitian.', 'hep-th-0604103-1-13-1': 'The coefficient [MATH] can be taken to be symmetric under exchange of the indices [MATH] because antisymmetric part can be absorbed in the term that is the first order in [MATH].', 'hep-th-0604103-1-13-2': 'Higher order terms expanded in terms of the operators [MATH] and [MATH] also can be taken to be symmetric under permutations of the operators.', 'hep-th-0604103-1-13-3': 'We consider the expansion as a sum of homogeneous polynomials in [MATH] and [MATH], whose coefficients are identified with massless higher spin gauge fields.', 'hep-th-0604103-1-13-4': 'Coefficients of the first order homogeneous polynomial express spin-2 gauge fields, and those of the second order one express spin-3 gauge fields and so on.', 'hep-th-0604103-1-13-5': 'Thus, the number of independent components of higher spin gauge fields grows rapidly with degree in [MATH] and [MATH].', 'hep-th-0604103-1-14-0': 'Here, we mention how gauge symmetries are embedded in the [MATH] symmetry of the matrix model.', 'hep-th-0604103-1-14-1': 'Originally, the [MATH] symmetry of the matrix model is written as [EQUATION] where [MATH] is a [MATH] hermitian matrix.', 'hep-th-0604103-1-14-2': 'In the new interpretation of the matrix model, [MATH] becomes a scalar operator expanded in terms of [MATH] and [MATH].', 'hep-th-0604103-1-15-0': 'Let us see how gauge transformations are generated by [MATH] in the case of spin-3.', 'hep-th-0604103-1-15-1': 'In order to deal with this case, we need to keep track of only the first and the fifth term in ([REF]), [EQUATION]', 'hep-th-0604103-1-15-2': 'We take [MATH] as [EQUATION]', 'hep-th-0604103-1-15-3': 'Then ([REF]) becomes [EQUATION]', 'hep-th-0604103-1-15-4': 'Thus [MATH] transforms as [EQUATION]', 'hep-th-0604103-1-15-5': 'Therefore, [MATH] transforms as a rank-[MATH] symmetric tensor field.', 'hep-th-0604103-1-15-6': 'Other gauge transformations related to higher spin gauge fields are realized in terms of other [MATH].', 'hep-th-0604103-1-15-7': 'Thus, this formulation possesses the gauge invariance related to higher spin gauge fields manifestly.', 'hep-th-0604103-1-15-8': 'This is an advantage of this formulation.', 'hep-th-0604103-1-16-0': '# Free Higher Spin Field Equation in Flat Spacetime', 'hep-th-0604103-1-17-0': 'The free equation of motion for the rank-[MATH] totally symmetric tensor field [MATH] in [MATH]-dimensional flat spacetime, the so-called "Fronsdal equation", is given by [EQUATION] where we use notations [MATH] and [MATH].', 'hep-th-0604103-1-17-1': 'Greek indices run from 1 to [MATH] and denote flat spacetime directions.', 'hep-th-0604103-1-17-2': 'This equation of motion possesses the gauge symmetry, the so-called Fronsdal symmetry, [EQUATION] where the bracket [MATH] denotes symmetrization of the flat spacetime indices and the gauge parameter [MATH] is symmetric under permutations of the indices.', 'hep-th-0604103-1-17-3': 'The conventional formulation for free totally symmetric tensor gauge fields was originally derived by Fronsdal [CITATION].', 'hep-th-0604103-1-17-4': 'The key feature of this formulation is the need for a pair of constraints, one on the parameter [MATH], whose trace [MATH] is required to vanish, and one on the gauge field itself, whose double trace [MATH] is also required to vanish.', 'hep-th-0604103-1-18-0': 'In this section, we show that the free equation of motion of higher spin gauge fields ([REF]) can be derived from that of the matrix model ([REF]).', 'hep-th-0604103-1-18-1': 'In our formulation, as we will see in the next subsection, the constraints on [MATH] are achieved by putting traceless constraints on the fields in [MATH].', 'hep-th-0604103-1-18-2': 'Through the analysis in this section, we can understand how higher spin gauge fields are included in the matrix model.', 'hep-th-0604103-1-19-0': 'We show the case of spin-3 in subsection 3.1 and spin-[MATH] in subsection 3.2.', 'hep-th-0604103-1-20-0': '## Free Spin-3 Field Equation in Flat Spacetime', 'hep-th-0604103-1-21-0': 'Let us first consider the spin-3 case as an example.', 'hep-th-0604103-1-21-1': 'The equation of motion of spin-3 field [MATH] is given by [EQUATION]', 'hep-th-0604103-1-21-2': 'In this and the next subsection, in order to derive free field equations we keep only terms linear with respect to the component fields and we use the ordinary differential operator [MATH] instead of the covariant derivative [MATH].', 'hep-th-0604103-1-21-3': 'Notice that the commutation relation of the operators is given by [EQUATION]', 'hep-th-0604103-1-21-4': 'In order to deal with the spin-3 case, we keep track of the second order homogeneous polynomial of the operators [MATH] and [MATH] in [MATH] : [EQUATION]', 'hep-th-0604103-1-21-5': 'Here, we do not write the anticommutator appeared in ([REF]) explicitly because terms like [MATH] are not needed to derive spin-3 field equation.', 'hep-th-0604103-1-22-0': 'Based on the analogy of the frame formulation of gravity, we can regard that the fields as [MATH] and [MATH] are generalizations of the [MATH] and the spin-connection, respectively.', 'hep-th-0604103-1-22-1': 'We assume that the fields [MATH] and [MATH] satisfy the traceless conditions [EQUATION]', 'hep-th-0604103-1-22-2': 'The totally symmetric tensor field [MATH] is defined in terms of [MATH] as [EQUATION] and [MATH] and [MATH] are auxiliary fields.', 'hep-th-0604103-1-22-3': 'As we will see shortly, the relation ([REF]) can be understood from the gauge transformations which are embedded in the [MATH] symmetry of the matrix model.', 'hep-th-0604103-1-23-0': 'Let us summarize the gauge transformations.', 'hep-th-0604103-1-23-1': 'There are three kinds of gauge transformations in the case of spin-3 :', 'hep-th-0604103-1-24-0': '[(i)] Generalized coordinate transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-1-25-0': '[(ii)] Generalized local Lorentz transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-1-26-0': '[(iii)] Auxiliary gauge transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-1-27-0': 'Based on the analogy of the frame formulation of gravity, the gauge transformation (i), which corresponds to the Fronsdal symmetry, is an extension of generalized coordinate transformation.', 'hep-th-0604103-1-27-1': 'The gauge transformation (ii), which removes the part that is not totally symmetric in the indices of [MATH], is an extension of local Lorentz transformation.', 'hep-th-0604103-1-27-2': 'The gauge symmetry (iii), which does not act on dynamical fields, appears only for spins larger than 2.', 'hep-th-0604103-1-28-0': 'Under the gauge transformation (i), the rank-3 totally symmetric tensor field [MATH] defined in ([REF]) transforms as follows : [EQUATION]', 'hep-th-0604103-1-28-1': 'This is consistent with the identification ([REF]).', 'hep-th-0604103-1-29-0': 'Since the gauge transformation (i) corresponds to the Fronsdal one, in order to derive higher spin field equations we should fix the gauge symmetries (ii) and (iii).', 'hep-th-0604103-1-29-1': 'However, gauge symmetries cannot remove all extra degrees of freedom.', 'hep-th-0604103-1-29-2': 'Therefore, we must impose some constraints that can be used to determine the auxiliary fields [MATH] and [MATH] in terms of the dynamical field [MATH].', 'hep-th-0604103-1-29-3': 'In the rest of this subsection, we will perform these procedures in order.', 'hep-th-0604103-1-30-0': 'Constraints', 'hep-th-0604103-1-31-0': 'Based on the analogy of the frame formulation of gravity, we impose constraints on the field strengths in order to solve the auxiliary the fields [MATH] and [MATH] in terms of [MATH].', 'hep-th-0604103-1-31-1': 'The field strengths are coefficients of the operators in the commutator of [MATH].', 'hep-th-0604103-1-31-2': 'The commutator is calculated as follows : [EQUATION] where the bracket [MATH] denotes antisymmetrization of indices.', 'hep-th-0604103-1-31-3': 'We impose constraints that the coefficients of [MATH] and [MATH] are equal to 0 : [EQUATION]', 'hep-th-0604103-1-31-4': 'From ([REF]), [MATH] is solved in terms of the first order derivatives in [MATH] : [EQUATION]', 'hep-th-0604103-1-31-5': 'This is similar to the torsionless constraint of gravity.', 'hep-th-0604103-1-31-6': 'The constraint on [MATH] ([REF]) implies that [MATH] can be written as a "pure gauge" configuration, [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-1-32-0': 'Imposing these constraints, we obtain [EQUATION]', 'hep-th-0604103-1-32-1': 'Therefore, we can obtain the following equations of motion : [EQUATION] where we define [MATH] as [EQUATION]', 'hep-th-0604103-1-32-2': 'The equation ([REF]) follows from ([REF]).', 'hep-th-0604103-1-32-3': 'Therefore, dynamical field equation of motion is ([REF]).', 'hep-th-0604103-1-32-4': 'In order to make the equation ([REF]) to be symmetric under permutations of indices [MATH] we should impose the following constraint on [MATH], [EQUATION]', 'hep-th-0604103-1-32-5': 'Imposing this constraint, we find that the equation ([REF]) is symmetric under permutations of indices and is second order derivatives in [MATH].', 'hep-th-0604103-1-33-0': 'Here, we summarize the constraints we imposed in this subsection as follows :', 'hep-th-0604103-1-34-0': 'Imposing these constraints, the auxiliary fields [MATH] and [MATH] are expressed in terms of the dynamical field [MATH] and only a symmetric part remains in the equation of motion ([REF]).', 'hep-th-0604103-1-35-0': 'The constraints imposed on the spin-[MATH] fields are the traceless constraints ([REF]) and the field strength constraints ([REF]), ([REF]) and ([REF]).', 'hep-th-0604103-1-35-1': 'These constraints have been imposed in order to express the equation ([REF]) in terms of the dynamical field [MATH].', 'hep-th-0604103-1-35-2': 'However, viewed from the matrix model, field strengths should be introduced as independent degrees of freedom.', 'hep-th-0604103-1-35-3': 'There is a possibility that "the higher spin field strengths" propagate as asymmetric tensor fields.', 'hep-th-0604103-1-36-0': 'Gauge fixing', 'hep-th-0604103-1-37-0': 'So far, we analyzed the elimination of the extra degrees of freedom by imposing the constraints ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-th-0604103-1-37-1': 'Combining these constraints and the equation of motion ([REF]) we find that the equation ([REF]) is expressed in terms of second order derivatives in [MATH] and is symmetric under permutations of indices.', 'hep-th-0604103-1-37-2': 'However, these constraints cannot eliminate all extra degrees of freedom.', 'hep-th-0604103-1-37-3': 'The Fronsdal equation ([REF]) is expressed in terms of the rank-[MATH] totally symmetric tensor field [MATH], but the equation ([REF]) is expressed in terms of "the spin-3 [MATH]" [MATH], which have the part that is not totally symmetric.', 'hep-th-0604103-1-37-4': 'Last remaining extra degrees of freedom is the part that is not totally symmetric in the indices of [MATH].', 'hep-th-0604103-1-37-5': 'Thus, we should eliminate this extra degrees of freedom and express dynamical variable in terms of [MATH] , in order to show that the equation ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-1-37-6': 'This can be done by fixing the gauge symmetries.', 'hep-th-0604103-1-37-7': 'Recall that there are three kinds of the gauge transformations (i), (ii) and (iii)', 'hep-th-0604103-1-38-0': 'and the gauge transformation (i) corresponds to the Fronsdal gauge transformation.', 'hep-th-0604103-1-38-1': 'Therefore it seems that we should fix the gauge symmetries (ii) and (iii).', 'hep-th-0604103-1-38-2': 'As we see, the gauge symmetries (ii) and (iii) can eliminate the part that is not totally symmetric in the indices of [MATH] and we can express dynamical variable in terms of the rank-3 totally symmetric tensor field : [MATH].', 'hep-th-0604103-1-39-0': 'First, we fix the gauge symmetry (iii).', 'hep-th-0604103-1-39-1': 'Gauge fixing can be done by transforming [MATH], choosing the parameter [MATH] as [EQUATION] where we define [MATH], [MATH] and [MATH] as follows : [EQUATION] [MATH] satisfies the following properties : [EQUATION] [MATH] is determined by the traceless condition ([REF]).', 'hep-th-0604103-1-40-0': 'Next, we fix the gauge symmetry (ii) by transforming [MATH] with [EQUATION] [MATH] satisfies the following properties : [EQUATION] where [MATH], [MATH].', 'hep-th-0604103-1-40-1': '[MATH] is determined by the traceless condition ([REF]).', 'hep-th-0604103-1-40-2': 'Carrying out these transformation, we can remove the part that is not totally symmetric in the indices of [MATH] and we have [MATH].', 'hep-th-0604103-1-40-3': 'Substituting [MATH] and [MATH] into the equation ([REF]), we can show that the equation ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-1-41-0': 'It is worth noting the relation between our formulation and the unfolded formalism of higher spin gauge field theory due to Vasiliev [CITATION].', 'hep-th-0604103-1-41-1': 'Vasiliev constructed free field Lagrangian using similar method to the one we have employed in this paper [CITATION].', 'hep-th-0604103-1-41-2': 'Only difference between the unfolded formalism and our formulation is the appearance of [MATH] in free theory.', 'hep-th-0604103-1-41-3': 'In the unfolded formalism, [MATH] appears in interacting theories and contribute to higher derivative interactions.', 'hep-th-0604103-1-41-4': 'It is interesting to investigate the relation between our formulation and the unfolded formalism by analyzing higher spin interactions.', 'hep-th-0604103-1-42-0': '## Free Spin-[MATH] Equation in Flat Spacetime', 'hep-th-0604103-1-43-0': 'In this subsection, using the same method as the one we have employed in the previous subsection, we derive the free equation of motion of the rank-[MATH] totally symmetric tensor field in [MATH]-dimensional flat spacetime ([REF]) from that of the matrix model.', 'hep-th-0604103-1-44-0': 'In order to deal with the spin-[MATH] case, we keep track of the [MATH]-th order homogeneous polynomial of the operators [MATH] and [MATH] : [EQUATION] [MATH] is the "spin-s [MATH]" and [MATH] is "the spin-s connection".', 'hep-th-0604103-1-44-1': 'From the discussion in the previous subsection, it seems that [MATH] are not necessary to derive the equation of motion.', 'hep-th-0604103-1-44-2': 'We set these auxiliary fields to zero : [MATH].', 'hep-th-0604103-1-44-3': 'Therefore, [MATH] becomes [EQUATION]', 'hep-th-0604103-1-44-4': 'Here we assume that the spin-s fields satisfy the traceless condition, [EQUATION]', 'hep-th-0604103-1-44-5': 'The rank-s symmetric tensor field [MATH] is defined as [MATH].', 'hep-th-0604103-1-44-6': '[MATH] satisfies the double traceless condition [MATH] as a consequence of the traceless condition ([REF]).', 'hep-th-0604103-1-45-0': 'Let us summarize the gauge transformations.', 'hep-th-0604103-1-45-1': 'In the case of spin-s, there are [MATH] kinds of gauge transformation.', 'hep-th-0604103-1-45-2': 'Since auxiliary fields [MATH] are set to be zero, the following three kinds of gauge transformations remain :', 'hep-th-0604103-1-46-0': '[(i)] Generalized coordinate transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH].', 'hep-th-0604103-1-47-0': '[(ii)] Generalized local Lorentz transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH] and satisfies [MATH].', 'hep-th-0604103-1-48-0': '[(iii)] Auxiliary gauge transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH] and satisfies [MATH].', 'hep-th-0604103-1-49-0': 'Therefore, [MATH] are transformed under (i) as [EQUATION]', 'hep-th-0604103-1-49-1': 'This is consistent with the identification [MATH].', 'hep-th-0604103-1-49-2': 'In order to derive the equation of motion ([REF]) from that of the matrix model, what we must do is to impose constraints on the auxiliary fields and fix the gauge symmetries (ii) and (iii), as in the spin-3 case.', 'hep-th-0604103-1-50-0': 'Constraints', 'hep-th-0604103-1-51-0': 'Constraints are imposed on the coefficients of [MATH] and [MATH] in the commutator of [MATH].', 'hep-th-0604103-1-51-1': 'We obtain [EQUATION]', 'hep-th-0604103-1-51-2': 'The constraint ([REF]) implies that [MATH] can be written as a "pure gauge" configuration : [EQUATION]', 'hep-th-0604103-1-51-3': 'We impose constraints on [MATH] as [EQUATION]', 'hep-th-0604103-1-51-4': 'Imposing these constraints, the commutators of [MATH] become [EQUATION] and [EQUATION]', 'hep-th-0604103-1-51-5': 'Therefore we obtain the equations of motion [EQUATION] where [MATH] is defined as [EQUATION]', 'hep-th-0604103-1-51-6': 'The second equation ([REF]) can be derived by using the first one ([REF]).', 'hep-th-0604103-1-51-7': 'Therefore, dynamical field equation is the first one.', 'hep-th-0604103-1-51-8': 'Owing to the constraint ([REF]), the equation of motion ([REF]) are symmetric under permutations of indices.', 'hep-th-0604103-1-52-0': 'Here, we summarize the constraints we imposed in this subsection as follows :', 'hep-th-0604103-1-53-0': 'Imposing these constraints, the auxiliary fields [MATH] and [MATH] are expressed in terms of the dynamical field [MATH] and only a symmetric part remains in the equation of motion ([REF]).', 'hep-th-0604103-1-54-0': 'Gauge fixing', 'hep-th-0604103-1-55-0': 'First, we fix the gauge symmetry (iii) by transforming [MATH], choosing the parameter [MATH] [EQUATION] where [MATH] denotes the symmetrization of indices [MATH].', 'hep-th-0604103-1-55-1': 'We define [MATH], [MATH], [MATH] and [MATH] as follows : [EQUATION]', 'hep-th-0604103-1-55-2': 'The parameter [MATH] is symmetric under permutations of the indices [MATH] and satisfies the following properties : [EQUATION] [MATH] and [MATH] are determined by the traceless conditions ([REF]).', 'hep-th-0604103-1-56-0': 'Next, we fix the gauge symmetry (ii) by transforming [MATH] with [EQUATION] where [MATH].', 'hep-th-0604103-1-56-1': '[MATH] is symmetric under permutations of indices [MATH] and satisfies the traceless condition [EQUATION]', 'hep-th-0604103-1-56-2': 'Carrying out these transformation, we can remove the part that is not totally symmetric in the indices of [MATH] and we have [MATH].', 'hep-th-0604103-1-56-3': 'Substituting [MATH] and [MATH] into the equation ([REF]), we can show that ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-1-57-0': '# Conclusions and Future Works', 'hep-th-0604103-1-58-0': 'In this paper, we have shown that the free equation of motion of bosonic massless higher spin gauge fields in [MATH]-dimensional flat spacetime can be derived from that of the matrix model based on the new interpretation of the matrix model.', 'hep-th-0604103-1-58-1': 'In order to derive higher spin field equations, we have done the two things : 1) imposing constraints 2) performing gauge fixing procedures.', 'hep-th-0604103-1-58-2': 'The results of this paper suggest that bosonic massless higher spin fields can be embedded in the matrix model.', 'hep-th-0604103-1-58-3': 'This is a first step towards construction of interacting massless higher spin gauge field theory by using the matrix model.', 'hep-th-0604103-1-58-4': 'Therefore, the matrix model can be used as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-1-59-0': 'There are several things which should be studied further.', 'hep-th-0604103-1-59-1': 'One is the derivation of free fermionic massless higher spin gauge field equations.', 'hep-th-0604103-1-59-2': 'Recently, it was shown that supergravity can be embedded in the supermatrix model [CITATION].', 'hep-th-0604103-1-59-3': 'There is a possibility that fermionic higher spin fields are embedded in the supermatrix model.', 'hep-th-0604103-1-59-4': 'Another one is to construct the interacting massless higher spin gauge field theory.', 'hep-th-0604103-1-59-5': 'As mentioned in Introduction, it is difficult to construct interacting theory.', 'hep-th-0604103-1-59-6': 'Difficulties associated with the requirement of gauge invariance can be overcome by using the matrix model because it has gauge invariance manifestly.'}

{'hep-th-0604103-2-0-0': 'We study matrix models as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-2-0-1': 'As a first step in this direction, we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of a matrix model.', 'hep-th-0604103-2-1-0': '# Introduction', 'hep-th-0604103-2-2-0': 'It has been known that there are problems in the construction of consistent interactions for massless higher spin gauge fields, though there are physically acceptable free field Lagrangian for them.', 'hep-th-0604103-2-2-1': 'Free field Lagrangian for higher spin fields, in which these particles are expressed by totally symmetric tensor or tensor-spinor fields, was originally derived by Fronsdal for bosons [CITATION] and Fang-Fronsdal for fermions [CITATION].', 'hep-th-0604103-2-2-2': 'There are some approaches for the construction of free field Lagrangian [CITATION] [CITATION].', 'hep-th-0604103-2-2-3': 'Interaction problems appear when one tries to couple massless higher spin fields to an electromagnetic field [CITATION], to gravity [CITATION] [CITATION] [CITATION] or to construct self-interactions [CITATION] [CITATION].', 'hep-th-0604103-2-2-4': 'For a review of massless higher spin gauge field theory, see [CITATION].', 'hep-th-0604103-2-3-0': 'At present, there exist various approaches to the theory, which solve the interaction problems in some cases.', 'hep-th-0604103-2-3-1': 'For example, an approach, called the unfolded formalism, was developed by Vasiliev et al [CITATION] [CITATION].', 'hep-th-0604103-2-3-2': 'They succeeded in the construction of interacting higher spin gauge theory with a nonzero cosmological constant [CITATION].', 'hep-th-0604103-2-3-3': 'An approach, called BRST approach, was initiated by development of string field theory on the basis of BRST techniques [CITATION] [CITATION] [CITATION] [CITATION] [CITATION].', 'hep-th-0604103-2-4-0': 'In this paper, we study matrix models as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-2-4-1': 'Recently, it has been shown that the Einstein equation can be obtained from the equation of motion of a matrix model by introducing a new interpretation of the matrix model, in which matrices represent differential operators on a curved spacetime [CITATION].', 'hep-th-0604103-2-4-2': 'Furthermore, it was pointed out that there is a possibility that matrix models include the degrees of freedom of massless higher spin gauge fields.', 'hep-th-0604103-2-4-3': 'An advantage of this formalism is that the matrix model possesses a gauge invariance manifestly, embedded in the [MATH] symmetry.', 'hep-th-0604103-2-4-4': 'Therefore it is interesting to analyze interacting massless higher spin gauge field theory using the matrix model.', 'hep-th-0604103-2-5-0': 'A first step towards constructing massless higher spin gauge field theory is the formulation of the free theory.', 'hep-th-0604103-2-5-1': 'Therefore, in this paper we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of the matrix model.', 'hep-th-0604103-2-6-0': 'There is another motivation for our study.', 'hep-th-0604103-2-6-1': 'Massless higher spin fields are expected to appear in the tensionless limit of string theory, since mass squared of them are all proportional to the string tension.', 'hep-th-0604103-2-6-2': 'Thus in this limit, one should observe an enhancement of gauge symmetry of string theory by that of the massless higher spin field theory.', 'hep-th-0604103-2-6-3': 'On the other hand, matrix models are expected to be a nonperturbative formulation of string theory.', 'hep-th-0604103-2-6-4': 'Therefore our study may be useful for better understanding of gauge symmetry of string theory and can lead to further understanding of nonperturbative aspects of string theory.', 'hep-th-0604103-2-7-0': 'The organization of this paper is as follows.', 'hep-th-0604103-2-7-1': 'In section 2, we briefly review the results of [CITATION].', 'hep-th-0604103-2-7-2': 'In section 3, we show that the free equation of motion of bosonic massless higher spin gauge fields can be derived from that of the matrix model.', 'hep-th-0604103-2-7-3': 'Section 4 is devoted to conclusions and future works.', 'hep-th-0604103-2-8-0': '# Matrix Model', 'hep-th-0604103-2-9-0': 'In this section, we briefly review the results of [CITATION].', 'hep-th-0604103-2-9-1': 'Introducing a new interpretation of a matrix model, we will see the following three facts.', 'hep-th-0604103-2-10-0': 'In this paper, we consider the large [MATH] reduced model of [MATH]-dimensional pure Yang-Mills theory with [MATH] gauge symmetry as the matrix model : [EQUATION] where [MATH] are [MATH] hermitian bosonic matrices.', 'hep-th-0604103-2-10-1': 'Latin indices denote Euclidean spacetime directions.', 'hep-th-0604103-2-10-2': 'This action has the [MATH] Lorentz symmetry and [MATH] unitary matrix symmetry.', 'hep-th-0604103-2-10-3': 'We can also consider supersymmetric version of this model, but in this paper we consider only the bosonic action ([REF]).', 'hep-th-0604103-2-11-0': 'The basic idea of [CITATION] is that matrices represent differential operators on a curved spacetime.', 'hep-th-0604103-2-11-1': 'There are several problems with this identification.', 'hep-th-0604103-2-11-2': 'For example, matrices act as [MATH] on a vector space, which means matrices map a vector space to itself.', 'hep-th-0604103-2-11-3': 'On the contrary, covariant derivatives map a tensor field of rank-[MATH] to a tensor field of rank-[MATH].', 'hep-th-0604103-2-11-4': 'Therefore we should prepare a vector space [MATH] which contain at least tensor fields of any rank and prepare an object [MATH] which is equivalent to a covariant derivative [MATH] such that each component of [MATH] is expressed as an [MATH] on [MATH], in order to interpret covariant derivatives as matrices.', 'hep-th-0604103-2-11-5': 'In [CITATION] the authors showed that [MATH] can be given by the space of smooth functions on the principal [MATH] bundle over a manifold [MATH].', 'hep-th-0604103-2-11-6': 'A smooth function [MATH] on it is defined as the mapping [EQUATION] where [MATH] denotes a patch on [MATH].', 'hep-th-0604103-2-11-7': 'Thus, [MATH] depends on the local coordinate [MATH], where [MATH] and [MATH], of the principal [MATH] bundle over [MATH].', 'hep-th-0604103-2-11-8': '[MATH] can be given by [EQUATION] where [MATH] is the vector representation of [MATH].', 'hep-th-0604103-2-11-9': 'The covariant derivative [MATH] is defined as [EQUATION] where [MATH] is the generator of the local Lorentz group [MATH], [MATH] is the [MATH] and [MATH] is the spin-connection.', 'hep-th-0604103-2-11-10': 'Here, Latin indices denote the local Lorentz indices.', 'hep-th-0604103-2-11-11': 'Notice that [MATH] maps a rank-[MATH] tensor to a rank-[MATH] tensor and [MATH] acts on the local Lorentz indices of these tensors.', 'hep-th-0604103-2-11-12': 'Therefore we have [EQUATION] in this setting, which will be used later.', 'hep-th-0604103-2-12-0': 'Let us see how the Einstein equation can be derived from the equation of motion of the matrix model by applying this interpretation to the matrix model.', 'hep-th-0604103-2-12-1': 'From the action ([REF]), we obtain the following equation of motion : [EQUATION]', 'hep-th-0604103-2-12-2': 'We substitute [MATH] into the equation of motion ([REF]), where [MATH] is introduced to make [MATH] hermitian.', 'hep-th-0604103-2-12-3': 'The commutator of [MATH] becomes [EQUATION] where [MATH] is the Riemann tensor and [MATH] is the torsion tensor.', 'hep-th-0604103-2-12-4': 'Here, we put the torsionless constraint [MATH] to solve the spin-connection in terms of the [MATH].', 'hep-th-0604103-2-12-5': 'Under the torsionless constraint, the equation of motion becomes [EQUATION] and we obtain the following two equations : [EQUATION]', 'hep-th-0604103-2-12-6': 'The first equation is the vacuum Einstein equation.', 'hep-th-0604103-2-12-7': 'The second equation can be derived from the first one and the Bianchi identity : [MATH].', 'hep-th-0604103-2-12-8': 'Thus, we have obtained the Einstein equation from the equation of motion of the matrix model.', 'hep-th-0604103-2-13-0': 'Since each component of matrices [MATH] acts functions on the principal [MATH] bundle over [MATH] as an [MATH], in general, [MATH] may be expanded as [EQUATION] where [MATH] and anticommutator [MATH] are introduced to make [MATH] hermitian.', 'hep-th-0604103-2-13-1': 'The coefficient [MATH] can be taken to be symmetric under exchange of the indices [MATH] because antisymmetric part can be absorbed in the term that is the first order in [MATH].', 'hep-th-0604103-2-13-2': 'Higher order terms expanded in terms of the operators [MATH] and [MATH] also can be taken to be symmetric under permutations of the operators.', 'hep-th-0604103-2-13-3': 'We consider the expansion as a sum of homogeneous polynomials in [MATH] and [MATH], whose coefficients are identified with massless higher spin gauge fields.', 'hep-th-0604103-2-13-4': 'Coefficients of the first order homogeneous polynomial express spin-2 gauge fields, and those of the second order one express spin-3 gauge fields and so on.', 'hep-th-0604103-2-13-5': 'Thus, the number of independent components of higher spin gauge fields grows rapidly with degree in [MATH] and [MATH].', 'hep-th-0604103-2-14-0': 'Here, we mention how gauge symmetries are embedded in the [MATH] symmetry of the matrix model.', 'hep-th-0604103-2-14-1': 'Originally, the [MATH] symmetry of the matrix model is written as [EQUATION] where [MATH] is a [MATH] hermitian matrix.', 'hep-th-0604103-2-14-2': 'In the new interpretation of the matrix model, [MATH] becomes a scalar operator expanded in terms of [MATH] and [MATH].', 'hep-th-0604103-2-15-0': 'Let us see how gauge transformations are generated by [MATH] in the case of spin-3.', 'hep-th-0604103-2-15-1': 'In order to deal with this case, we need to keep track of only the first and the fifth term in ([REF]), [EQUATION]', 'hep-th-0604103-2-15-2': 'We take [MATH] as [EQUATION]', 'hep-th-0604103-2-15-3': 'Then ([REF]) becomes [EQUATION]', 'hep-th-0604103-2-15-4': 'Thus [MATH] transforms as [EQUATION]', 'hep-th-0604103-2-15-5': 'Therefore, [MATH] transforms as a rank-[MATH] symmetric tensor field.', 'hep-th-0604103-2-15-6': 'Other gauge transformations related to higher spin gauge fields are realized in terms of other [MATH].', 'hep-th-0604103-2-15-7': 'Thus, this formulation possesses the gauge invariance related to higher spin gauge fields manifestly.', 'hep-th-0604103-2-15-8': 'This is an advantage of this formulation.', 'hep-th-0604103-2-16-0': '# Free Higher Spin Field Equation in Flat Spacetime', 'hep-th-0604103-2-17-0': 'The free equation of motion for the rank-[MATH] totally symmetric tensor field [MATH] in [MATH]-dimensional flat spacetime, the so-called "Fronsdal equation", is given by [EQUATION] where we use notations [MATH] and [MATH].', 'hep-th-0604103-2-17-1': 'Greek indices run from 1 to [MATH] and denote flat spacetime directions.', 'hep-th-0604103-2-17-2': 'This equation of motion possesses the gauge symmetry, the so-called Fronsdal symmetry, [EQUATION] where the bracket [MATH] denotes symmetrization of the flat spacetime indices and the gauge parameter [MATH] is symmetric under permutations of the indices.', 'hep-th-0604103-2-17-3': 'The conventional formulation for free totally symmetric tensor gauge fields was originally derived by Fronsdal [CITATION].', 'hep-th-0604103-2-17-4': 'The key feature of this formulation is the need for a pair of constraints, one on the parameter [MATH], whose trace [MATH] is required to vanish, and one on the gauge field itself, whose double trace [MATH] is also required to vanish.', 'hep-th-0604103-2-18-0': 'In this section, we show that the free equation of motion of higher spin gauge fields ([REF]) can be derived from that of the matrix model ([REF]).', 'hep-th-0604103-2-18-1': 'In our formulation, as we will see in the next subsection, the constraints on [MATH] are achieved by putting traceless constraints on the fields in [MATH].', 'hep-th-0604103-2-18-2': 'Through the analysis in this section, we can understand how higher spin gauge fields are included in the matrix model.', 'hep-th-0604103-2-19-0': 'We show the case of spin-3 in subsection 3.1 and spin-[MATH] in subsection 3.2.', 'hep-th-0604103-2-20-0': '## Free Spin-3 Field Equation in Flat Spacetime', 'hep-th-0604103-2-21-0': 'Let us first consider the spin-3 case as an example.', 'hep-th-0604103-2-21-1': 'The equation of motion of spin-3 field [MATH] is given by [EQUATION]', 'hep-th-0604103-2-21-2': 'In this and the next subsection, in order to derive free field equations we keep only terms linear with respect to the component fields and we use the ordinary differential operator [MATH] instead of the covariant derivative [MATH].', 'hep-th-0604103-2-21-3': 'Notice that the commutation relation of the operators is given by [EQUATION]', 'hep-th-0604103-2-21-4': 'In order to deal with the spin-3 case, we keep track of the second order homogeneous polynomial of the operators [MATH] and [MATH] in [MATH] : [EQUATION]', 'hep-th-0604103-2-21-5': 'Here, we do not write the anticommutator appeared in ([REF]) explicitly because terms like [MATH] are not needed to derive spin-3 field equation.', 'hep-th-0604103-2-22-0': 'Based on the analogy of the frame formulation of gravity, we can regard that the fields as [MATH] and [MATH] are generalizations of the [MATH] and the spin-connection, respectively.', 'hep-th-0604103-2-22-1': 'We assume that the fields [MATH] and [MATH] satisfy the traceless conditions [EQUATION]', 'hep-th-0604103-2-22-2': 'The totally symmetric tensor field [MATH] is defined in terms of [MATH] as [EQUATION] and [MATH] and [MATH] are auxiliary fields.', 'hep-th-0604103-2-22-3': 'As we will see shortly, the relation ([REF]) can be understood from the gauge transformations which are embedded in the [MATH] symmetry of the matrix model.', 'hep-th-0604103-2-23-0': 'Let us summarize the gauge transformations.', 'hep-th-0604103-2-23-1': 'There are three kinds of gauge transformations in the case of spin-3 :', 'hep-th-0604103-2-24-0': '[(i)] Generalized coordinate transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-2-25-0': '[(ii)] Generalized local Lorentz transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-2-26-0': '[(iii)] Auxiliary gauge transformation, generated by [MATH], [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-2-27-0': 'Based on the analogy of the frame formulation of gravity, the gauge transformation (i), which corresponds to the Fronsdal symmetry, is an extension of generalized coordinate transformation.', 'hep-th-0604103-2-27-1': 'The gauge transformation (ii), which removes the part that is not totally symmetric in the indices of [MATH], is an extension of local Lorentz transformation.', 'hep-th-0604103-2-27-2': 'The gauge symmetry (iii), which does not act on dynamical fields, appears only for spins larger than 2.', 'hep-th-0604103-2-28-0': 'Under the gauge transformation (i), the rank-3 totally symmetric tensor field [MATH] defined in ([REF]) transforms as follows : [EQUATION]', 'hep-th-0604103-2-28-1': 'This is consistent with the identification ([REF]).', 'hep-th-0604103-2-29-0': 'Since the gauge transformation (i) corresponds to the Fronsdal one, in order to derive higher spin field equations we should fix the gauge symmetries (ii) and (iii).', 'hep-th-0604103-2-29-1': 'However, gauge symmetries cannot remove all extra degrees of freedom.', 'hep-th-0604103-2-29-2': 'Therefore, we must impose some constraints that can be used to determine the auxiliary fields [MATH] and [MATH] in terms of the dynamical field [MATH].', 'hep-th-0604103-2-29-3': 'In the rest of this subsection, we will perform these procedures in order.', 'hep-th-0604103-2-30-0': 'Constraints', 'hep-th-0604103-2-31-0': 'Based on the analogy of the frame formulation of gravity, we impose constraints on the field strengths in order to solve the auxiliary the fields [MATH] and [MATH] in terms of [MATH].', 'hep-th-0604103-2-31-1': 'The field strengths are coefficients of the operators in the commutator of [MATH].', 'hep-th-0604103-2-31-2': 'The commutator is calculated as follows : [EQUATION] where the bracket [MATH] denotes antisymmetrization of indices.', 'hep-th-0604103-2-31-3': 'We impose constraints that the coefficients of [MATH] and [MATH] are equal to 0 : [EQUATION]', 'hep-th-0604103-2-31-4': 'From ([REF]), [MATH] is solved in terms of the first order derivatives in [MATH] : [EQUATION]', 'hep-th-0604103-2-31-5': 'This is similar to the torsionless constraint of gravity.', 'hep-th-0604103-2-31-6': 'The constraint on [MATH] ([REF]) implies that [MATH] can be written as a "pure gauge" configuration, [EQUATION] where the parameter [MATH] satisfies [MATH].', 'hep-th-0604103-2-32-0': 'Imposing these constraints, we obtain [EQUATION]', 'hep-th-0604103-2-32-1': 'Therefore, we can obtain the following equations of motion : [EQUATION] where we define [MATH] as [EQUATION]', 'hep-th-0604103-2-32-2': 'The equation ([REF]) follows from ([REF]).', 'hep-th-0604103-2-32-3': 'Therefore, dynamical field equation of motion is ([REF]).', 'hep-th-0604103-2-32-4': 'In order to make the equation ([REF]) to be symmetric under permutations of indices [MATH] we should impose the following constraint on [MATH], [EQUATION]', 'hep-th-0604103-2-32-5': 'Imposing this constraint, we find that the equation ([REF]) is symmetric under permutations of indices and is second order derivatives in [MATH].', 'hep-th-0604103-2-33-0': 'Here, we summarize the constraints we imposed in this subsection as follows :', 'hep-th-0604103-2-34-0': 'Imposing these constraints, the auxiliary fields [MATH] and [MATH] are expressed in terms of the dynamical field [MATH] and only a symmetric part remains in the equation of motion ([REF]).', 'hep-th-0604103-2-35-0': 'The constraints imposed on the spin-[MATH] fields are the traceless constraints ([REF]) and the field strength constraints ([REF]), ([REF]) and ([REF]).', 'hep-th-0604103-2-35-1': 'These constraints have been imposed in order to express the equation ([REF]) in terms of the dynamical field [MATH].', 'hep-th-0604103-2-35-2': 'However, viewed from the matrix model, field strengths should be introduced as independent degrees of freedom.', 'hep-th-0604103-2-35-3': 'There is a possibility that "the higher spin field strengths" propagate as asymmetric tensor fields.', 'hep-th-0604103-2-36-0': 'Gauge fixing', 'hep-th-0604103-2-37-0': 'So far, we analyzed the elimination of the extra degrees of freedom by imposing the constraints ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-th-0604103-2-37-1': 'Combining these constraints and the equation of motion ([REF]) we find that the equation ([REF]) is expressed in terms of second order derivatives in [MATH] and is symmetric under permutations of indices.', 'hep-th-0604103-2-37-2': 'However, these constraints cannot eliminate all extra degrees of freedom.', 'hep-th-0604103-2-37-3': 'The Fronsdal equation ([REF]) is expressed in terms of the rank-[MATH] totally symmetric tensor field [MATH], but the equation ([REF]) is expressed in terms of "the spin-3 [MATH]" [MATH], which have the part that is not totally symmetric.', 'hep-th-0604103-2-37-4': 'Last remaining extra degrees of freedom is the part that is not totally symmetric in the indices of [MATH].', 'hep-th-0604103-2-37-5': 'Thus, we should eliminate this extra degrees of freedom and express dynamical variable in terms of [MATH] , in order to show that the equation ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-2-37-6': 'This can be done by fixing the gauge symmetries.', 'hep-th-0604103-2-37-7': 'Recall that there are three kinds of the gauge transformations (i), (ii) and (iii)', 'hep-th-0604103-2-38-0': 'and the gauge transformation (i) corresponds to the Fronsdal gauge transformation.', 'hep-th-0604103-2-38-1': 'Therefore it seems that we should fix the gauge symmetries (ii) and (iii).', 'hep-th-0604103-2-38-2': 'As we see, the gauge symmetries (ii) and (iii) can eliminate the part that is not totally symmetric in the indices of [MATH] and we can express dynamical variable in terms of the rank-3 totally symmetric tensor field : [MATH].', 'hep-th-0604103-2-39-0': 'First, we fix the gauge symmetry (iii).', 'hep-th-0604103-2-39-1': 'Gauge fixing can be done by transforming [MATH], choosing the parameter [MATH] as [EQUATION] where we define [MATH], [MATH] and [MATH] as follows : [EQUATION] [MATH] satisfies the following properties : [EQUATION] [MATH] is determined by the traceless condition ([REF]).', 'hep-th-0604103-2-40-0': 'Next, we fix the gauge symmetry (ii) by transforming [MATH] with [EQUATION] [MATH] satisfies the following properties : [EQUATION] where [MATH], [MATH].', 'hep-th-0604103-2-40-1': '[MATH] is determined by the traceless condition ([REF]).', 'hep-th-0604103-2-40-2': 'Carrying out these transformation, we can remove the part that is not totally symmetric in the indices of [MATH] and we have [MATH].', 'hep-th-0604103-2-40-3': 'Substituting [MATH] and [MATH] into the equation ([REF]), we can show that the equation ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-2-41-0': 'It is worth noting the relation between our formulation and the unfolded formalism of higher spin gauge field theory due to Vasiliev [CITATION].', 'hep-th-0604103-2-41-1': 'Vasiliev constructed free field Lagrangian using similar method to the one we have employed in this paper [CITATION].', 'hep-th-0604103-2-41-2': 'Only difference between the unfolded formalism and our formulation is the appearance of [MATH] in free theory.', 'hep-th-0604103-2-41-3': 'In the unfolded formalism, [MATH] appears in interacting theories and contribute to higher derivative interactions.', 'hep-th-0604103-2-41-4': 'It is interesting to investigate the relation between our formulation and the unfolded formalism by analyzing higher spin interactions.', 'hep-th-0604103-2-42-0': '## Free Spin-[MATH] Equation in Flat Spacetime', 'hep-th-0604103-2-43-0': 'In this subsection, using the same method as the one we have employed in the previous subsection, we derive the free equation of motion of the rank-[MATH] totally symmetric tensor field in [MATH]-dimensional flat spacetime ([REF]) from that of the matrix model.', 'hep-th-0604103-2-44-0': 'In order to deal with the spin-[MATH] case, we keep track of the [MATH]-th order homogeneous polynomial of the operators [MATH] and [MATH] : [EQUATION] [MATH] is the "spin-s [MATH]" and [MATH] is "the spin-s connection".', 'hep-th-0604103-2-44-1': 'From the discussion in the previous subsection, it seems that [MATH] are not necessary to derive the equation of motion.', 'hep-th-0604103-2-44-2': 'We set these auxiliary fields to zero : [MATH].', 'hep-th-0604103-2-44-3': 'Therefore, [MATH] becomes [EQUATION]', 'hep-th-0604103-2-44-4': 'Here we assume that the spin-s fields satisfy the traceless condition, [EQUATION]', 'hep-th-0604103-2-44-5': 'The rank-s symmetric tensor field [MATH] is defined as [MATH].', 'hep-th-0604103-2-44-6': '[MATH] satisfies the double traceless condition [MATH] as a consequence of the traceless condition ([REF]).', 'hep-th-0604103-2-45-0': 'Let us summarize the gauge transformations.', 'hep-th-0604103-2-45-1': 'In the case of spin-s, there are [MATH] kinds of gauge transformation.', 'hep-th-0604103-2-45-2': 'Since auxiliary fields [MATH] are set to be zero, the following three kinds of gauge transformations remain :', 'hep-th-0604103-2-46-0': '[(i)] Generalized coordinate transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH].', 'hep-th-0604103-2-47-0': '[(ii)] Generalized local Lorentz transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH] and satisfies [MATH].', 'hep-th-0604103-2-48-0': '[(iii)] Auxiliary gauge transformation, generated by [MATH], [EQUATION] where [MATH] is symmetric under permutations of the indices [MATH] and satisfies [MATH].', 'hep-th-0604103-2-49-0': 'Therefore, [MATH] are transformed under (i) as [EQUATION]', 'hep-th-0604103-2-49-1': 'This is consistent with the identification [MATH].', 'hep-th-0604103-2-49-2': 'In order to derive the equation of motion ([REF]) from that of the matrix model, what we must do is to impose constraints on the auxiliary fields and fix the gauge symmetries (ii) and (iii), as in the spin-3 case.', 'hep-th-0604103-2-50-0': 'Constraints', 'hep-th-0604103-2-51-0': 'Constraints are imposed on the coefficients of [MATH] and [MATH] in the commutator of [MATH].', 'hep-th-0604103-2-51-1': 'We obtain [EQUATION]', 'hep-th-0604103-2-51-2': 'The constraint ([REF]) implies that [MATH] can be written as a "pure gauge" configuration : [EQUATION]', 'hep-th-0604103-2-51-3': 'We impose constraints on [MATH] as [EQUATION]', 'hep-th-0604103-2-51-4': 'Imposing these constraints, the commutators of [MATH] become [EQUATION] and [EQUATION]', 'hep-th-0604103-2-51-5': 'Therefore we obtain the equations of motion [EQUATION] where [MATH] is defined as [EQUATION]', 'hep-th-0604103-2-51-6': 'The second equation ([REF]) can be derived by using the first one ([REF]).', 'hep-th-0604103-2-51-7': 'Therefore, dynamical field equation is the first one.', 'hep-th-0604103-2-51-8': 'Owing to the constraint ([REF]), the equation of motion ([REF]) are symmetric under permutations of indices.', 'hep-th-0604103-2-52-0': 'Here, we summarize the constraints we imposed in this subsection as follows :', 'hep-th-0604103-2-53-0': 'Imposing these constraints, the auxiliary fields [MATH] and [MATH] are expressed in terms of the dynamical field [MATH] and only a symmetric part remains in the equation of motion ([REF]).', 'hep-th-0604103-2-54-0': 'Gauge fixing', 'hep-th-0604103-2-55-0': 'First, we fix the gauge symmetry (iii) by transforming [MATH], choosing the parameter [MATH] [EQUATION] where [MATH] denotes the symmetrization of indices [MATH].', 'hep-th-0604103-2-55-1': 'We define [MATH], [MATH], [MATH] and [MATH] as follows : [EQUATION]', 'hep-th-0604103-2-55-2': 'The parameter [MATH] is symmetric under permutations of the indices [MATH] and satisfies the following properties : [EQUATION] [MATH] and [MATH] are determined by the traceless conditions ([REF]).', 'hep-th-0604103-2-56-0': 'Next, we fix the gauge symmetry (ii) by transforming [MATH] with [EQUATION] where [MATH].', 'hep-th-0604103-2-56-1': '[MATH] is symmetric under permutations of indices [MATH] and satisfies the traceless condition [EQUATION]', 'hep-th-0604103-2-56-2': 'Carrying out these transformation, we can remove the part that is not totally symmetric in the indices of [MATH] and we have [MATH].', 'hep-th-0604103-2-56-3': 'Substituting [MATH] and [MATH] into the equation ([REF]), we can show that ([REF]) coincides with the Fronsdal equation ([REF]).', 'hep-th-0604103-2-57-0': '# Conclusions and Future Works', 'hep-th-0604103-2-58-0': 'In this paper, we have shown that the free equation of motion of bosonic massless higher spin gauge fields in [MATH]-dimensional flat spacetime can be derived from that of the matrix model based on the new interpretation of the matrix model.', 'hep-th-0604103-2-58-1': 'In order to derive higher spin field equations, we have done the two things : 1) imposing constraints 2) performing gauge fixing procedures.', 'hep-th-0604103-2-58-2': 'The results of this paper suggest that bosonic massless higher spin fields can be embedded in the matrix model.', 'hep-th-0604103-2-58-3': 'This is a first step towards construction of interacting massless higher spin gauge field theory by using the matrix model.', 'hep-th-0604103-2-58-4': 'Therefore, the matrix model can be used as a new approach to formulate massless higher spin gauge field theory.', 'hep-th-0604103-2-59-0': 'There are several things which should be studied further.', 'hep-th-0604103-2-59-1': 'One is the derivation of free fermionic massless higher spin gauge field equations.', 'hep-th-0604103-2-59-2': 'Recently, it was shown that supergravity can be embedded in the supermatrix model [CITATION].', 'hep-th-0604103-2-59-3': 'There is a possibility that fermionic higher spin fields are embedded in the supermatrix model.', 'hep-th-0604103-2-59-4': 'Another one is to construct the interacting massless higher spin gauge field theory.', 'hep-th-0604103-2-59-5': 'As mentioned in Introduction, it is difficult to construct interacting theory.', 'hep-th-0604103-2-59-6': 'Difficulties associated with the requirement of gauge invariance can be overcome by using the matrix model because it has gauge invariance manifestly.'}

[['hep-th-0604103-1-29-0', 'hep-th-0604103-2-29-0'], ['hep-th-0604103-1-29-1', 'hep-th-0604103-2-29-1'], ['hep-th-0604103-1-29-2', 'hep-th-0604103-2-29-2'], ['hep-th-0604103-1-29-3', 'hep-th-0604103-2-29-3'], ['hep-th-0604103-1-9-0', 'hep-th-0604103-2-9-0'], ['hep-th-0604103-1-9-1', 'hep-th-0604103-2-9-1'], ['hep-th-0604103-1-32-0', 'hep-th-0604103-2-32-0'], ['hep-th-0604103-1-32-1', 'hep-th-0604103-2-32-1'], ['hep-th-0604103-1-32-2', 'hep-th-0604103-2-32-2'], ['hep-th-0604103-1-32-3', 'hep-th-0604103-2-32-3'], ['hep-th-0604103-1-32-4', 'hep-th-0604103-2-32-4'], ['hep-th-0604103-1-32-5', 'hep-th-0604103-2-32-5'], ['hep-th-0604103-1-22-0', 'hep-th-0604103-2-22-0'], ['hep-th-0604103-1-22-1', 'hep-th-0604103-2-22-1'], ['hep-th-0604103-1-22-2', 'hep-th-0604103-2-22-2'], ['hep-th-0604103-1-22-3', 'hep-th-0604103-2-22-3'], ['hep-th-0604103-1-13-0', 'hep-th-0604103-2-13-0'], ['hep-th-0604103-1-13-1', 'hep-th-0604103-2-13-1'], ['hep-th-0604103-1-13-2', 'hep-th-0604103-2-13-2'], ['hep-th-0604103-1-13-3', 'hep-th-0604103-2-13-3'], ['hep-th-0604103-1-13-4', 'hep-th-0604103-2-13-4'], ['hep-th-0604103-1-13-5', 'hep-th-0604103-2-13-5'], ['hep-th-0604103-1-21-0', 'hep-th-0604103-2-21-0'], ['hep-th-0604103-1-21-1', 'hep-th-0604103-2-21-1'], ['hep-th-0604103-1-21-2', 'hep-th-0604103-2-21-2'], ['hep-th-0604103-1-21-3', 'hep-th-0604103-2-21-3'], ['hep-th-0604103-1-21-4', 'hep-th-0604103-2-21-4'], ['hep-th-0604103-1-21-5', 'hep-th-0604103-2-21-5'], ['hep-th-0604103-1-37-0', 'hep-th-0604103-2-37-0'], ['hep-th-0604103-1-37-1', 'hep-th-0604103-2-37-1'], ['hep-th-0604103-1-37-2', 'hep-th-0604103-2-37-2'], ['hep-th-0604103-1-37-3', 'hep-th-0604103-2-37-3'], ['hep-th-0604103-1-37-4', 'hep-th-0604103-2-37-4'], ['hep-th-0604103-1-37-5', 'hep-th-0604103-2-37-5'], ['hep-th-0604103-1-37-6', 'hep-th-0604103-2-37-6'], ['hep-th-0604103-1-37-7', 'hep-th-0604103-2-37-7'], ['hep-th-0604103-1-51-0', 'hep-th-0604103-2-51-0'], ['hep-th-0604103-1-51-2', 'hep-th-0604103-2-51-2'], ['hep-th-0604103-1-51-3', 'hep-th-0604103-2-51-3'], ['hep-th-0604103-1-51-4', 'hep-th-0604103-2-51-4'], ['hep-th-0604103-1-51-5', 'hep-th-0604103-2-51-5'], ['hep-th-0604103-1-51-6', 'hep-th-0604103-2-51-6'], ['hep-th-0604103-1-51-7', 'hep-th-0604103-2-51-7'], ['hep-th-0604103-1-51-8', 'hep-th-0604103-2-51-8'], ['hep-th-0604103-1-17-0', 'hep-th-0604103-2-17-0'], ['hep-th-0604103-1-17-1', 'hep-th-0604103-2-17-1'], ['hep-th-0604103-1-17-2', 'hep-th-0604103-2-17-2'], ['hep-th-0604103-1-17-3', 'hep-th-0604103-2-17-3'], ['hep-th-0604103-1-17-4', 'hep-th-0604103-2-17-4'], ['hep-th-0604103-1-59-0', 'hep-th-0604103-2-59-0'], ['hep-th-0604103-1-59-1', 'hep-th-0604103-2-59-1'], ['hep-th-0604103-1-59-2', 'hep-th-0604103-2-59-2'], ['hep-th-0604103-1-59-3', 'hep-th-0604103-2-59-3'], ['hep-th-0604103-1-59-4', 'hep-th-0604103-2-59-4'], ['hep-th-0604103-1-59-5', 'hep-th-0604103-2-59-5'], ['hep-th-0604103-1-59-6', 'hep-th-0604103-2-59-6'], ['hep-th-0604103-1-40-0', 'hep-th-0604103-2-40-0'], ['hep-th-0604103-1-40-1', 'hep-th-0604103-2-40-1'], ['hep-th-0604103-1-40-2', 'hep-th-0604103-2-40-2'], ['hep-th-0604103-1-40-3', 'hep-th-0604103-2-40-3'], ['hep-th-0604103-1-34-0', 'hep-th-0604103-2-34-0'], ['hep-th-0604103-1-47-0', 'hep-th-0604103-2-47-0'], ['hep-th-0604103-1-35-0', 'hep-th-0604103-2-35-0'], ['hep-th-0604103-1-35-1', 'hep-th-0604103-2-35-1'], ['hep-th-0604103-1-35-2', 'hep-th-0604103-2-35-2'], ['hep-th-0604103-1-35-3', 'hep-th-0604103-2-35-3'], ['hep-th-0604103-1-18-0', 'hep-th-0604103-2-18-0'], ['hep-th-0604103-1-18-1', 'hep-th-0604103-2-18-1'], ['hep-th-0604103-1-18-2', 'hep-th-0604103-2-18-2'], ['hep-th-0604103-1-55-0', 'hep-th-0604103-2-55-0'], ['hep-th-0604103-1-55-1', 'hep-th-0604103-2-55-1'], ['hep-th-0604103-1-55-2', 'hep-th-0604103-2-55-2'], ['hep-th-0604103-1-49-0', 'hep-th-0604103-2-49-0'], ['hep-th-0604103-1-49-1', 'hep-th-0604103-2-49-1'], ['hep-th-0604103-1-49-2', 'hep-th-0604103-2-49-2'], ['hep-th-0604103-1-4-0', 'hep-th-0604103-2-4-0'], ['hep-th-0604103-1-4-1', 'hep-th-0604103-2-4-1'], ['hep-th-0604103-1-4-2', 'hep-th-0604103-2-4-2'], ['hep-th-0604103-1-4-3', 'hep-th-0604103-2-4-3'], ['hep-th-0604103-1-4-4', 'hep-th-0604103-2-4-4'], ['hep-th-0604103-1-43-0', 'hep-th-0604103-2-43-0'], ['hep-th-0604103-1-5-0', 'hep-th-0604103-2-5-0'], ['hep-th-0604103-1-5-1', 'hep-th-0604103-2-5-1'], ['hep-th-0604103-1-25-0', 'hep-th-0604103-2-25-0'], ['hep-th-0604103-1-53-0', 'hep-th-0604103-2-53-0'], ['hep-th-0604103-1-2-0', 'hep-th-0604103-2-2-0'], ['hep-th-0604103-1-2-1', 'hep-th-0604103-2-2-1'], ['hep-th-0604103-1-2-2', 'hep-th-0604103-2-2-2'], ['hep-th-0604103-1-2-3', 'hep-th-0604103-2-2-3'], ['hep-th-0604103-1-2-4', 'hep-th-0604103-2-2-4'], ['hep-th-0604103-1-39-0', 'hep-th-0604103-2-39-0'], ['hep-th-0604103-1-39-1', 'hep-th-0604103-2-39-1'], ['hep-th-0604103-1-41-0', 'hep-th-0604103-2-41-0'], ['hep-th-0604103-1-41-1', 'hep-th-0604103-2-41-1'], ['hep-th-0604103-1-41-2', 'hep-th-0604103-2-41-2'], ['hep-th-0604103-1-41-3', 'hep-th-0604103-2-41-3'], ['hep-th-0604103-1-41-4', 'hep-th-0604103-2-41-4'], ['hep-th-0604103-1-12-0', 'hep-th-0604103-2-12-0'], ['hep-th-0604103-1-12-1', 'hep-th-0604103-2-12-1'], ['hep-th-0604103-1-12-2', 'hep-th-0604103-2-12-2'], ['hep-th-0604103-1-12-3', 'hep-th-0604103-2-12-3'], ['hep-th-0604103-1-12-4', 'hep-th-0604103-2-12-4'], ['hep-th-0604103-1-12-5', 'hep-th-0604103-2-12-5'], ['hep-th-0604103-1-12-6', 'hep-th-0604103-2-12-6'], ['hep-th-0604103-1-12-7', 'hep-th-0604103-2-12-7'], ['hep-th-0604103-1-12-8', 'hep-th-0604103-2-12-8'], ['hep-th-0604103-1-11-0', 'hep-th-0604103-2-11-0'], ['hep-th-0604103-1-11-1', 'hep-th-0604103-2-11-1'], ['hep-th-0604103-1-11-2', 'hep-th-0604103-2-11-2'], ['hep-th-0604103-1-11-3', 'hep-th-0604103-2-11-3'], ['hep-th-0604103-1-11-4', 'hep-th-0604103-2-11-4'], ['hep-th-0604103-1-11-5', 'hep-th-0604103-2-11-5'], ['hep-th-0604103-1-11-6', 'hep-th-0604103-2-11-6'], ['hep-th-0604103-1-11-7', 'hep-th-0604103-2-11-7'], ['hep-th-0604103-1-11-8', 'hep-th-0604103-2-11-8'], ['hep-th-0604103-1-11-9', 'hep-th-0604103-2-11-9'], ['hep-th-0604103-1-11-10', 'hep-th-0604103-2-11-10'], ['hep-th-0604103-1-11-11', 'hep-th-0604103-2-11-11'], ['hep-th-0604103-1-11-12', 'hep-th-0604103-2-11-12'], ['hep-th-0604103-1-3-0', 'hep-th-0604103-2-3-0'], ['hep-th-0604103-1-3-1', 'hep-th-0604103-2-3-1'], ['hep-th-0604103-1-3-2', 'hep-th-0604103-2-3-2'], ['hep-th-0604103-1-14-0', 'hep-th-0604103-2-14-0'], ['hep-th-0604103-1-14-1', 'hep-th-0604103-2-14-1'], ['hep-th-0604103-1-14-2', 'hep-th-0604103-2-14-2'], ['hep-th-0604103-1-48-0', 'hep-th-0604103-2-48-0'], ['hep-th-0604103-1-6-0', 'hep-th-0604103-2-6-0'], ['hep-th-0604103-1-6-1', 'hep-th-0604103-2-6-1'], ['hep-th-0604103-1-6-2', 'hep-th-0604103-2-6-2'], ['hep-th-0604103-1-6-3', 'hep-th-0604103-2-6-3'], ['hep-th-0604103-1-6-4', 'hep-th-0604103-2-6-4'], ['hep-th-0604103-1-44-0', 'hep-th-0604103-2-44-0'], ['hep-th-0604103-1-44-1', 'hep-th-0604103-2-44-1'], ['hep-th-0604103-1-44-2', 'hep-th-0604103-2-44-2'], ['hep-th-0604103-1-44-3', 'hep-th-0604103-2-44-3'], ['hep-th-0604103-1-44-4', 'hep-th-0604103-2-44-4'], ['hep-th-0604103-1-44-5', 'hep-th-0604103-2-44-5'], ['hep-th-0604103-1-44-6', 'hep-th-0604103-2-44-6'], ['hep-th-0604103-1-15-0', 'hep-th-0604103-2-15-0'], ['hep-th-0604103-1-15-1', 'hep-th-0604103-2-15-1'], ['hep-th-0604103-1-15-2', 'hep-th-0604103-2-15-2'], ['hep-th-0604103-1-15-3', 'hep-th-0604103-2-15-3'], ['hep-th-0604103-1-15-4', 'hep-th-0604103-2-15-4'], ['hep-th-0604103-1-15-5', 'hep-th-0604103-2-15-5'], ['hep-th-0604103-1-15-6', 'hep-th-0604103-2-15-6'], ['hep-th-0604103-1-15-7', 'hep-th-0604103-2-15-7'], ['hep-th-0604103-1-15-8', 'hep-th-0604103-2-15-8'], ['hep-th-0604103-1-10-0', 'hep-th-0604103-2-10-0'], ['hep-th-0604103-1-10-1', 'hep-th-0604103-2-10-1'], ['hep-th-0604103-1-10-2', 'hep-th-0604103-2-10-2'], ['hep-th-0604103-1-10-3', 'hep-th-0604103-2-10-3'], ['hep-th-0604103-1-45-0', 'hep-th-0604103-2-45-0'], ['hep-th-0604103-1-45-1', 'hep-th-0604103-2-45-1'], ['hep-th-0604103-1-0-0', 'hep-th-0604103-2-0-0'], ['hep-th-0604103-1-0-1', 'hep-th-0604103-2-0-1'], ['hep-th-0604103-1-27-0', 'hep-th-0604103-2-27-0'], ['hep-th-0604103-1-27-1', 'hep-th-0604103-2-27-1'], ['hep-th-0604103-1-27-2', 'hep-th-0604103-2-27-2'], ['hep-th-0604103-1-24-0', 'hep-th-0604103-2-24-0'], ['hep-th-0604103-1-28-0', 'hep-th-0604103-2-28-0'], ['hep-th-0604103-1-28-1', 'hep-th-0604103-2-28-1'], ['hep-th-0604103-1-58-0', 'hep-th-0604103-2-58-0'], ['hep-th-0604103-1-58-1', 'hep-th-0604103-2-58-1'], ['hep-th-0604103-1-58-2', 'hep-th-0604103-2-58-2'], ['hep-th-0604103-1-58-3', 'hep-th-0604103-2-58-3'], ['hep-th-0604103-1-58-4', 'hep-th-0604103-2-58-4'], ['hep-th-0604103-1-38-0', 'hep-th-0604103-2-38-0'], ['hep-th-0604103-1-38-1', 'hep-th-0604103-2-38-1'], ['hep-th-0604103-1-38-2', 'hep-th-0604103-2-38-2'], ['hep-th-0604103-1-7-0', 'hep-th-0604103-2-7-0'], ['hep-th-0604103-1-7-1', 'hep-th-0604103-2-7-1'], ['hep-th-0604103-1-7-2', 'hep-th-0604103-2-7-2'], ['hep-th-0604103-1-7-3', 'hep-th-0604103-2-7-3'], ['hep-th-0604103-1-56-0', 'hep-th-0604103-2-56-0'], ['hep-th-0604103-1-56-1', 'hep-th-0604103-2-56-1'], ['hep-th-0604103-1-56-2', 'hep-th-0604103-2-56-2'], ['hep-th-0604103-1-56-3', 'hep-th-0604103-2-56-3'], ['hep-th-0604103-1-19-0', 'hep-th-0604103-2-19-0'], ['hep-th-0604103-1-31-0', 'hep-th-0604103-2-31-0'], ['hep-th-0604103-1-31-1', 'hep-th-0604103-2-31-1'], ['hep-th-0604103-1-31-2', 'hep-th-0604103-2-31-2'], ['hep-th-0604103-1-31-3', 'hep-th-0604103-2-31-3'], ['hep-th-0604103-1-31-4', 'hep-th-0604103-2-31-4'], ['hep-th-0604103-1-31-5', 'hep-th-0604103-2-31-5'], ['hep-th-0604103-1-31-6', 'hep-th-0604103-2-31-6'], ['hep-th-0604103-1-26-0', 'hep-th-0604103-2-26-0'], ['hep-th-0604103-1-46-0', 'hep-th-0604103-2-46-0'], ['hep-th-0604103-1-3-3', 'hep-th-0604103-2-3-3']]

[['hep-th-0604103-1-29-0', 'hep-th-0604103-2-29-0'], ['hep-th-0604103-1-29-1', 'hep-th-0604103-2-29-1'], ['hep-th-0604103-1-29-2', 'hep-th-0604103-2-29-2'], ['hep-th-0604103-1-29-3', 'hep-th-0604103-2-29-3'], ['hep-th-0604103-1-9-0', 'hep-th-0604103-2-9-0'], ['hep-th-0604103-1-9-1', 'hep-th-0604103-2-9-1'], ['hep-th-0604103-1-32-0', 'hep-th-0604103-2-32-0'], ['hep-th-0604103-1-32-1', 'hep-th-0604103-2-32-1'], ['hep-th-0604103-1-32-2', 'hep-th-0604103-2-32-2'], ['hep-th-0604103-1-32-3', 'hep-th-0604103-2-32-3'], ['hep-th-0604103-1-32-4', 'hep-th-0604103-2-32-4'], ['hep-th-0604103-1-32-5', 'hep-th-0604103-2-32-5'], ['hep-th-0604103-1-22-0', 'hep-th-0604103-2-22-0'], ['hep-th-0604103-1-22-1', 'hep-th-0604103-2-22-1'], ['hep-th-0604103-1-22-2', 'hep-th-0604103-2-22-2'], ['hep-th-0604103-1-22-3', 'hep-th-0604103-2-22-3'], ['hep-th-0604103-1-13-0', 'hep-th-0604103-2-13-0'], ['hep-th-0604103-1-13-1', 'hep-th-0604103-2-13-1'], ['hep-th-0604103-1-13-2', 'hep-th-0604103-2-13-2'], ['hep-th-0604103-1-13-3', 'hep-th-0604103-2-13-3'], ['hep-th-0604103-1-13-4', 'hep-th-0604103-2-13-4'], ['hep-th-0604103-1-13-5', 'hep-th-0604103-2-13-5'], ['hep-th-0604103-1-21-0', 'hep-th-0604103-2-21-0'], ['hep-th-0604103-1-21-1', 'hep-th-0604103-2-21-1'], ['hep-th-0604103-1-21-2', 'hep-th-0604103-2-21-2'], ['hep-th-0604103-1-21-3', 'hep-th-0604103-2-21-3'], ['hep-th-0604103-1-21-4', 'hep-th-0604103-2-21-4'], ['hep-th-0604103-1-21-5', 'hep-th-0604103-2-21-5'], ['hep-th-0604103-1-37-0', 'hep-th-0604103-2-37-0'], ['hep-th-0604103-1-37-1', 'hep-th-0604103-2-37-1'], ['hep-th-0604103-1-37-2', 'hep-th-0604103-2-37-2'], ['hep-th-0604103-1-37-3', 'hep-th-0604103-2-37-3'], ['hep-th-0604103-1-37-4', 'hep-th-0604103-2-37-4'], ['hep-th-0604103-1-37-5', 'hep-th-0604103-2-37-5'], ['hep-th-0604103-1-37-6', 'hep-th-0604103-2-37-6'], ['hep-th-0604103-1-37-7', 'hep-th-0604103-2-37-7'], ['hep-th-0604103-1-51-0', 'hep-th-0604103-2-51-0'], ['hep-th-0604103-1-51-2', 'hep-th-0604103-2-51-2'], ['hep-th-0604103-1-51-3', 'hep-th-0604103-2-51-3'], ['hep-th-0604103-1-51-4', 'hep-th-0604103-2-51-4'], ['hep-th-0604103-1-51-5', 'hep-th-0604103-2-51-5'], ['hep-th-0604103-1-51-6', 'hep-th-0604103-2-51-6'], ['hep-th-0604103-1-51-7', 'hep-th-0604103-2-51-7'], ['hep-th-0604103-1-51-8', 'hep-th-0604103-2-51-8'], ['hep-th-0604103-1-17-0', 'hep-th-0604103-2-17-0'], ['hep-th-0604103-1-17-1', 'hep-th-0604103-2-17-1'], ['hep-th-0604103-1-17-2', 'hep-th-0604103-2-17-2'], ['hep-th-0604103-1-17-3', 'hep-th-0604103-2-17-3'], ['hep-th-0604103-1-17-4', 'hep-th-0604103-2-17-4'], ['hep-th-0604103-1-59-0', 'hep-th-0604103-2-59-0'], ['hep-th-0604103-1-59-1', 'hep-th-0604103-2-59-1'], ['hep-th-0604103-1-59-2', 'hep-th-0604103-2-59-2'], ['hep-th-0604103-1-59-3', 'hep-th-0604103-2-59-3'], ['hep-th-0604103-1-59-4', 'hep-th-0604103-2-59-4'], ['hep-th-0604103-1-59-5', 'hep-th-0604103-2-59-5'], ['hep-th-0604103-1-59-6', 'hep-th-0604103-2-59-6'], ['hep-th-0604103-1-40-0', 'hep-th-0604103-2-40-0'], ['hep-th-0604103-1-40-1', 'hep-th-0604103-2-40-1'], ['hep-th-0604103-1-40-2', 'hep-th-0604103-2-40-2'], ['hep-th-0604103-1-40-3', 'hep-th-0604103-2-40-3'], ['hep-th-0604103-1-34-0', 'hep-th-0604103-2-34-0'], ['hep-th-0604103-1-47-0', 'hep-th-0604103-2-47-0'], ['hep-th-0604103-1-35-0', 'hep-th-0604103-2-35-0'], ['hep-th-0604103-1-35-1', 'hep-th-0604103-2-35-1'], ['hep-th-0604103-1-35-2', 'hep-th-0604103-2-35-2'], ['hep-th-0604103-1-35-3', 'hep-th-0604103-2-35-3'], ['hep-th-0604103-1-18-0', 'hep-th-0604103-2-18-0'], ['hep-th-0604103-1-18-1', 'hep-th-0604103-2-18-1'], ['hep-th-0604103-1-18-2', 'hep-th-0604103-2-18-2'], ['hep-th-0604103-1-55-0', 'hep-th-0604103-2-55-0'], ['hep-th-0604103-1-55-1', 'hep-th-0604103-2-55-1'], ['hep-th-0604103-1-55-2', 'hep-th-0604103-2-55-2'], ['hep-th-0604103-1-49-0', 'hep-th-0604103-2-49-0'], ['hep-th-0604103-1-49-1', 'hep-th-0604103-2-49-1'], ['hep-th-0604103-1-49-2', 'hep-th-0604103-2-49-2'], ['hep-th-0604103-1-4-0', 'hep-th-0604103-2-4-0'], ['hep-th-0604103-1-4-1', 'hep-th-0604103-2-4-1'], ['hep-th-0604103-1-4-2', 'hep-th-0604103-2-4-2'], ['hep-th-0604103-1-4-3', 'hep-th-0604103-2-4-3'], ['hep-th-0604103-1-4-4', 'hep-th-0604103-2-4-4'], ['hep-th-0604103-1-43-0', 'hep-th-0604103-2-43-0'], ['hep-th-0604103-1-5-0', 'hep-th-0604103-2-5-0'], ['hep-th-0604103-1-5-1', 'hep-th-0604103-2-5-1'], ['hep-th-0604103-1-25-0', 'hep-th-0604103-2-25-0'], ['hep-th-0604103-1-53-0', 'hep-th-0604103-2-53-0'], ['hep-th-0604103-1-2-0', 'hep-th-0604103-2-2-0'], ['hep-th-0604103-1-2-1', 'hep-th-0604103-2-2-1'], ['hep-th-0604103-1-2-2', 'hep-th-0604103-2-2-2'], ['hep-th-0604103-1-2-3', 'hep-th-0604103-2-2-3'], ['hep-th-0604103-1-2-4', 'hep-th-0604103-2-2-4'], ['hep-th-0604103-1-39-0', 'hep-th-0604103-2-39-0'], ['hep-th-0604103-1-39-1', 'hep-th-0604103-2-39-1'], ['hep-th-0604103-1-41-0', 'hep-th-0604103-2-41-0'], ['hep-th-0604103-1-41-1', 'hep-th-0604103-2-41-1'], ['hep-th-0604103-1-41-2', 'hep-th-0604103-2-41-2'], ['hep-th-0604103-1-41-3', 'hep-th-0604103-2-41-3'], ['hep-th-0604103-1-41-4', 'hep-th-0604103-2-41-4'], ['hep-th-0604103-1-12-0', 'hep-th-0604103-2-12-0'], ['hep-th-0604103-1-12-1', 'hep-th-0604103-2-12-1'], ['hep-th-0604103-1-12-2', 'hep-th-0604103-2-12-2'], ['hep-th-0604103-1-12-3', 'hep-th-0604103-2-12-3'], ['hep-th-0604103-1-12-4', 'hep-th-0604103-2-12-4'], ['hep-th-0604103-1-12-5', 'hep-th-0604103-2-12-5'], ['hep-th-0604103-1-12-6', 'hep-th-0604103-2-12-6'], ['hep-th-0604103-1-12-7', 'hep-th-0604103-2-12-7'], ['hep-th-0604103-1-12-8', 'hep-th-0604103-2-12-8'], ['hep-th-0604103-1-11-0', 'hep-th-0604103-2-11-0'], ['hep-th-0604103-1-11-1', 'hep-th-0604103-2-11-1'], ['hep-th-0604103-1-11-2', 'hep-th-0604103-2-11-2'], ['hep-th-0604103-1-11-3', 'hep-th-0604103-2-11-3'], ['hep-th-0604103-1-11-4', 'hep-th-0604103-2-11-4'], ['hep-th-0604103-1-11-5', 'hep-th-0604103-2-11-5'], ['hep-th-0604103-1-11-6', 'hep-th-0604103-2-11-6'], ['hep-th-0604103-1-11-7', 'hep-th-0604103-2-11-7'], ['hep-th-0604103-1-11-8', 'hep-th-0604103-2-11-8'], ['hep-th-0604103-1-11-9', 'hep-th-0604103-2-11-9'], ['hep-th-0604103-1-11-10', 'hep-th-0604103-2-11-10'], ['hep-th-0604103-1-11-11', 'hep-th-0604103-2-11-11'], ['hep-th-0604103-1-11-12', 'hep-th-0604103-2-11-12'], ['hep-th-0604103-1-3-0', 'hep-th-0604103-2-3-0'], ['hep-th-0604103-1-3-1', 'hep-th-0604103-2-3-1'], ['hep-th-0604103-1-3-2', 'hep-th-0604103-2-3-2'], ['hep-th-0604103-1-14-0', 'hep-th-0604103-2-14-0'], ['hep-th-0604103-1-14-1', 'hep-th-0604103-2-14-1'], ['hep-th-0604103-1-14-2', 'hep-th-0604103-2-14-2'], ['hep-th-0604103-1-48-0', 'hep-th-0604103-2-48-0'], ['hep-th-0604103-1-6-0', 'hep-th-0604103-2-6-0'], ['hep-th-0604103-1-6-1', 'hep-th-0604103-2-6-1'], ['hep-th-0604103-1-6-2', 'hep-th-0604103-2-6-2'], ['hep-th-0604103-1-6-3', 'hep-th-0604103-2-6-3'], ['hep-th-0604103-1-6-4', 'hep-th-0604103-2-6-4'], ['hep-th-0604103-1-44-0', 'hep-th-0604103-2-44-0'], ['hep-th-0604103-1-44-1', 'hep-th-0604103-2-44-1'], ['hep-th-0604103-1-44-2', 'hep-th-0604103-2-44-2'], ['hep-th-0604103-1-44-3', 'hep-th-0604103-2-44-3'], ['hep-th-0604103-1-44-4', 'hep-th-0604103-2-44-4'], ['hep-th-0604103-1-44-5', 'hep-th-0604103-2-44-5'], ['hep-th-0604103-1-44-6', 'hep-th-0604103-2-44-6'], ['hep-th-0604103-1-15-0', 'hep-th-0604103-2-15-0'], ['hep-th-0604103-1-15-1', 'hep-th-0604103-2-15-1'], ['hep-th-0604103-1-15-2', 'hep-th-0604103-2-15-2'], ['hep-th-0604103-1-15-3', 'hep-th-0604103-2-15-3'], ['hep-th-0604103-1-15-4', 'hep-th-0604103-2-15-4'], ['hep-th-0604103-1-15-5', 'hep-th-0604103-2-15-5'], ['hep-th-0604103-1-15-6', 'hep-th-0604103-2-15-6'], ['hep-th-0604103-1-15-7', 'hep-th-0604103-2-15-7'], ['hep-th-0604103-1-15-8', 'hep-th-0604103-2-15-8'], ['hep-th-0604103-1-10-0', 'hep-th-0604103-2-10-0'], ['hep-th-0604103-1-10-1', 'hep-th-0604103-2-10-1'], ['hep-th-0604103-1-10-2', 'hep-th-0604103-2-10-2'], ['hep-th-0604103-1-10-3', 'hep-th-0604103-2-10-3'], ['hep-th-0604103-1-45-0', 'hep-th-0604103-2-45-0'], ['hep-th-0604103-1-45-1', 'hep-th-0604103-2-45-1'], ['hep-th-0604103-1-0-0', 'hep-th-0604103-2-0-0'], ['hep-th-0604103-1-0-1', 'hep-th-0604103-2-0-1'], ['hep-th-0604103-1-27-0', 'hep-th-0604103-2-27-0'], ['hep-th-0604103-1-27-1', 'hep-th-0604103-2-27-1'], ['hep-th-0604103-1-27-2', 'hep-th-0604103-2-27-2'], ['hep-th-0604103-1-24-0', 'hep-th-0604103-2-24-0'], ['hep-th-0604103-1-28-0', 'hep-th-0604103-2-28-0'], ['hep-th-0604103-1-28-1', 'hep-th-0604103-2-28-1'], ['hep-th-0604103-1-58-0', 'hep-th-0604103-2-58-0'], ['hep-th-0604103-1-58-1', 'hep-th-0604103-2-58-1'], ['hep-th-0604103-1-58-2', 'hep-th-0604103-2-58-2'], ['hep-th-0604103-1-58-3', 'hep-th-0604103-2-58-3'], ['hep-th-0604103-1-58-4', 'hep-th-0604103-2-58-4'], ['hep-th-0604103-1-38-0', 'hep-th-0604103-2-38-0'], ['hep-th-0604103-1-38-1', 'hep-th-0604103-2-38-1'], ['hep-th-0604103-1-38-2', 'hep-th-0604103-2-38-2'], ['hep-th-0604103-1-7-0', 'hep-th-0604103-2-7-0'], ['hep-th-0604103-1-7-1', 'hep-th-0604103-2-7-1'], ['hep-th-0604103-1-7-2', 'hep-th-0604103-2-7-2'], ['hep-th-0604103-1-7-3', 'hep-th-0604103-2-7-3'], ['hep-th-0604103-1-56-0', 'hep-th-0604103-2-56-0'], ['hep-th-0604103-1-56-1', 'hep-th-0604103-2-56-1'], ['hep-th-0604103-1-56-2', 'hep-th-0604103-2-56-2'], ['hep-th-0604103-1-56-3', 'hep-th-0604103-2-56-3'], ['hep-th-0604103-1-19-0', 'hep-th-0604103-2-19-0'], ['hep-th-0604103-1-31-0', 'hep-th-0604103-2-31-0'], ['hep-th-0604103-1-31-1', 'hep-th-0604103-2-31-1'], ['hep-th-0604103-1-31-2', 'hep-th-0604103-2-31-2'], ['hep-th-0604103-1-31-3', 'hep-th-0604103-2-31-3'], ['hep-th-0604103-1-31-4', 'hep-th-0604103-2-31-4'], ['hep-th-0604103-1-31-5', 'hep-th-0604103-2-31-5'], ['hep-th-0604103-1-31-6', 'hep-th-0604103-2-31-6'], ['hep-th-0604103-1-26-0', 'hep-th-0604103-2-26-0'], ['hep-th-0604103-1-46-0', 'hep-th-0604103-2-46-0']]

[['hep-th-0604103-1-3-3', 'hep-th-0604103-2-3-3']]

[]

[]

[]

['hep-th-0604103-1-23-0', 'hep-th-0604103-1-23-1', 'hep-th-0604103-1-30-0', 'hep-th-0604103-1-33-0', 'hep-th-0604103-1-36-0', 'hep-th-0604103-1-45-2', 'hep-th-0604103-1-50-0', 'hep-th-0604103-1-51-1', 'hep-th-0604103-1-52-0', 'hep-th-0604103-1-54-0', 'hep-th-0604103-2-23-0', 'hep-th-0604103-2-23-1', 'hep-th-0604103-2-30-0', 'hep-th-0604103-2-33-0', 'hep-th-0604103-2-36-0', 'hep-th-0604103-2-45-2', 'hep-th-0604103-2-50-0', 'hep-th-0604103-2-51-1', 'hep-th-0604103-2-52-0', 'hep-th-0604103-2-54-0']

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

https://arxiv.org/abs/hep-th/0604103

nucl-th-0302010

{'nucl-th-0302010-1-0-0': "A self-consistent version of the Thermal Random Phase Approximation (TSCRPA) is developed within the Matsubara Green's Function (GF) formalism.", 'nucl-th-0302010-1-0-1': 'The TSCRPA is applied to the many level pairing model.', 'nucl-th-0302010-1-0-2': 'The normal phase of the system is considered.', 'nucl-th-0302010-1-0-3': 'The TSCRPA results are compared with the exact ones calculated for the Grand Canonical Ensemble.', 'nucl-th-0302010-1-0-4': 'Advantages of the TSCRPA over the Thermal Mean Field Approximation (TMFA) and the standard Thermal Random Phase Approximation (TRPA) are demonstrated.', 'nucl-th-0302010-1-0-5': 'Results for correlation functions, excitation energies, single particle level densities, etc., as a function of temperature are presented.', 'nucl-th-0302010-1-1-0': '# Introduction', 'nucl-th-0302010-1-2-0': 'Pairing properties of finite Fermi systems such as ultrasmall metallic grains have recently received a great deal of attention.', 'nucl-th-0302010-1-2-1': 'This has been spurred by a series of spectacular experiments of Ralph, Black and Tinkham [CITATION].', 'nucl-th-0302010-1-2-2': 'In order to correctly describe pairing properties it has been recognized that the finiteness of the systems (grains) needs to consider quantum fluctuations, good particle number, number parity, etc. seriously, since the coherence length may be of the order of the system size.', 'nucl-th-0302010-1-2-3': 'The situation for metallic grains has in the meanwhile been well described in several review articles [CITATION] (see also [CITATION]).', 'nucl-th-0302010-1-2-4': 'Another system where the finiteness is at the forefront of the theoretical investigation since several decades is the superfluid atomic nucleus.', 'nucl-th-0302010-1-2-5': 'As a matter of fact many of the theoretical tools such as particle number projection, even-odd effects, number parity, blocking effect, particle - particle Random Phase Approximation (pp-RPA), etc. have first been developed in nuclear physics [CITATION] before finding their application to finite systems of condensed matter.', 'nucl-th-0302010-1-2-6': 'Also the schematic pairing model with which we will mostly deal in this paper, namely the Picket Fence Model (PFM), whose exact solution has been found by Richardson and Sherman [CITATION], has essentially been developed in the context of nuclear physics for the description of deformed superfluid nuclei.', 'nucl-th-0302010-1-2-7': 'For finite condensed matter systems an early theoretical description was proposed by Muhlschlegel, Scalapino, and Denton [CITATION] using the Static Path Approximation (SPA) to the partition function.', 'nucl-th-0302010-1-2-8': 'This work stayed rather singular for a long time but the SPA has recently been applied successfully to the PFM both in the condensed matter [CITATION] and nuclear [CITATION] contexts.', 'nucl-th-0302010-1-2-9': 'A further standard method to treat quantum fluctuations namely the well known RPA has quite extensively been used for nuclear systems [CITATION] but equally for condensed matter problems [CITATION].', 'nucl-th-0302010-1-3-0': 'In this work we will further elaborate on the RPA approach.', 'nucl-th-0302010-1-3-1': 'We indeed have recently had quite remarkable success with a self consistent extension of the pp-RPA, which we called Self-Consistent RPA (SCRPA), by reproducing very accurately groundstate and excitation energies of the PFM [CITATION] at zero temperature.', 'nucl-th-0302010-1-3-2': 'This formalism was also developed independently by Ropke and collaborators who called it Cluster - Hartree - Fock (CHF) [CITATION].', 'nucl-th-0302010-1-3-3': 'Such type of generalization of the RPA theory grew out of the works of K.-J. Hara and D. Rowe [CITATION] several decades ago.', 'nucl-th-0302010-1-3-4': 'Shortly afterwards the theory was rederived using the method of many body Green functions [CITATION].', 'nucl-th-0302010-1-3-5': 'The success of the theory motivates us to develop the SCRPA formalism also for the finite temperature case and to study the thermodynamic properties of the BCS Hamiltonian using the PFM as an example.', 'nucl-th-0302010-1-3-6': 'For the extension of SCRPA to finite temperature we use the Matsubara Green functions approach [CITATION].', 'nucl-th-0302010-1-3-7': 'It appeared that the approximation scheme is very effective in treating two-body correlations in the particle-particle (pp) channel as well as the Pauli principle effects.', 'nucl-th-0302010-1-3-8': 'We should mention that we will work with real particles and not with quasiparticles what should limit our approach to temperatures above the critical temperature [MATH] (i.e. to the normal phase).', 'nucl-th-0302010-1-3-9': 'However, as we will see below, the definition of [MATH] in SCRPA is not so clear and we will be able to continue our calculation quite deeply into the superfluid regime.', 'nucl-th-0302010-1-4-0': 'We organize the paper in the following way.', 'nucl-th-0302010-1-4-1': 'In Section 2, the approach is outlined in general.', 'nucl-th-0302010-1-4-2': 'Then, in Section 3, the formalism is applied to the PFM.', 'nucl-th-0302010-1-4-3': 'A comparison with the exact solutions as well as with the results of other approximations is made in Section 4.', 'nucl-th-0302010-1-4-4': 'Section 5 is devoted to comparison with other recent works.', 'nucl-th-0302010-1-4-5': 'In section 6, we will summarize the results and draw some conclusions.', 'nucl-th-0302010-1-4-6': 'In an appendix a variant for the calculation of the occupation numbers is proposed.', 'nucl-th-0302010-1-5-0': '# General formalism', 'nucl-th-0302010-1-6-0': 'In treating a finite many-body system at finite temperature, it is convenient to use the grand canonical ensemble although it violates the number conservation.', 'nucl-th-0302010-1-6-1': 'With the definition [EQUATION] the grand partition function and statistical operator read [EQUATION] where [MATH].', 'nucl-th-0302010-1-6-2': "Then for any Schrodinger operator [MATH] the modified Heisenberg picture can be introduced [EQUATION] and the temperature (or the Matsubara) Green's Function (GF) is defined as [CITATION] [EQUATION]", 'nucl-th-0302010-1-6-3': 'Here, the brackets [MATH] mean the thermodynamic average; [MATH] is a [MATH] ordering operator, which arranges operators with the earliest [MATH] (the closest to [MATH]) to the right.', 'nucl-th-0302010-1-7-0': 'Let us consider the two-body Hamiltonian [EQUATION] where [MATH] are fermion annihilation and creation operators; [MATH] and [MATH] are the kinetic energy and the antisymmetrized matrix element of the two-body interaction.', 'nucl-th-0302010-1-7-1': "The Green's function [MATH] for an arbitrary operator [MATH] obeys the following equation of motion: [EQUATION]", 'nucl-th-0302010-1-7-2': 'In this expression it is possible to split the effective Hamiltonian [MATH] into an instantaneous and a dynamic (frequency dependent) part [CITATION] [EQUATION]', 'nucl-th-0302010-1-7-3': 'In the approximation of the instantaneous effective Hamiltonian i.e. neglecting [MATH], the Dyson equation for the two-body Matsubara GF [MATH] can be written as [EQUATION]', 'nucl-th-0302010-1-7-4': 'In the treatment of two particle correlations let us specify the arbitrary operator [MATH] as [MATH].', 'nucl-th-0302010-1-7-5': 'In this case the Dyson equation (3) takes the following form in the frequency representation [EQUATION] where, in supposing that the single particle density matrix is diagonal in the basis used (this is for example the case inhomogeneous matter): [EQUATION] and [EQUATION]', 'nucl-th-0302010-1-7-6': 'Here [MATH] is the antisymmetrized Kronecker symbol and [MATH] are the single particle occupation numbers which can be found from the single-particle Matsubara GF', 'nucl-th-0302010-1-8-0': '[EQUATION] as [EQUATION]', 'nucl-th-0302010-1-8-1': "In general the single-particle Matsubara Green's function [MATH] obeys the following Dyson equation: [EQUATION] or in the frequency representation [EQUATION] where [EQUATION]", 'nucl-th-0302010-1-8-2': 'Here [MATH] contains already the usual (instantaneous) mean field so that [MATH] denotes only the dynamical part of the mass operator.', 'nucl-th-0302010-1-9-0': 'Now the problem is to find an approximation for the mass operator [MATH] consistent with the SCRPA.', 'nucl-th-0302010-1-9-1': 'A solution to this problem has been proposed in [CITATION], which goes via the two body [MATH] - matrix representation of the single particle mass operator [CITATION], evaluating [MATH] within SCRPA.', 'nucl-th-0302010-1-10-0': 'Let us add at this point a word of physical interpretation of the mean-field operator (6) [CITATION].', 'nucl-th-0302010-1-10-1': 'A quick look allows to realize that it contains no higher than two body correlation functions and therefore for their determination, with (4), one obtains a selfconsistency problem.', 'nucl-th-0302010-1-10-2': 'Furthermore one can consider the nucleus as a gas of zero point pair fluctuations.', 'nucl-th-0302010-1-10-3': 'These fluctuations create their own mean field, i.e. one pair fluctuation moves in the average potential created by all the other pair fluctuations.', 'nucl-th-0302010-1-10-4': 'This average pair fluctuation field is graphically represented in Fig. 1.', 'nucl-th-0302010-1-10-5': 'It gives rise, as usual, to a nonlinear problem.', 'nucl-th-0302010-1-10-6': 'Of course, the single particle mean field introduced in (9) and further developed below is coupled to the selfconsistent pair potential.', 'nucl-th-0302010-1-10-7': 'This is the deeper meaning of [MATH] of (6).', 'nucl-th-0302010-1-11-0': '# Application to the Picket Fence Model', 'nucl-th-0302010-1-12-0': 'The model consists of an equidistant multilevel pairing Hamiltonian with each level two fold degenerate, i.e. only spin up/down fermions of one kind can occupy one level.', 'nucl-th-0302010-1-12-1': 'The corresponding Hamiltonian is given by [EQUATION] with [EQUATION] where [MATH] means the time reversed of [MATH], single particle energies are [MATH] with level spacing [MATH] chosen to be equal to [MATH], and [MATH] stands for the number of levels.', 'nucl-th-0302010-1-12-2': 'The chemical potential [MATH] will be chosen such as to conserve the average number of particles [MATH] of the system.', 'nucl-th-0302010-1-12-3': 'The operators defined in (12) form an SU(2) algebra for each level [MATH] and obey the following commutation relations [EQUATION]', 'nucl-th-0302010-1-13-0': '## SCRPA equations', 'nucl-th-0302010-1-14-0': 'To study the model at finite temperature we define in analogy to (1) the following set of two-body Matsubara GFs [EQUATION] where [EQUATION]', 'nucl-th-0302010-1-14-1': "Applying the instantaneous approximation for the mass operator we obtain the expressions for the two body SCRPA GF's: [EQUATION] with [EQUATION]", 'nucl-th-0302010-1-14-2': 'To find the correlation functions of the form [MATH] we will use the following approximation:', 'nucl-th-0302010-1-15-0': 'when [MATH] it is a simple factorization procedure, which has turned out to be accurate in the zero temperature limit: [EQUATION] but when [MATH] we use the following exact relation [EQUATION] which can easily be obtained taking into account that [MATH] and [MATH] (here [MATH]).', 'nucl-th-0302010-1-15-1': 'It should be noted that in [CITATION] the factorization (16a) was also used for the diagonal part (16b) and quite accurate results were obtained.', 'nucl-th-0302010-1-15-2': 'We will show below that with (16b) one obtains still improved results.', 'nucl-th-0302010-1-15-3': 'As shown in [CITATION] it is possible to avoid above approximation.', 'nucl-th-0302010-1-15-4': 'However, this is at the cost of a considerable numerical complication.', 'nucl-th-0302010-1-15-5': 'We refrain from this here because it brings only very little improvement of results.', 'nucl-th-0302010-1-16-0': 'With this ansatz a particle-particle RPA-like equation is obtained [EQUATION] where [EQUATION] and [EQUATION]', 'nucl-th-0302010-1-16-1': 'From this one easily can find the excitation spectrum of the model in equating the denominator of (17) to zero [EQUATION]', 'nucl-th-0302010-1-16-2': "Knowing the poles of the Green's function (17), one can write down its spectral representation (we here give it as a function of imaginary time), with the corresponding residua: [EQUATION] where the index [MATH] refers to the states above Fermi level and the index [MATH] to the ones below.", 'nucl-th-0302010-1-16-3': 'The following amplitudes were introduced in these formulas: [EQUATION] with [EQUATION]', 'nucl-th-0302010-1-16-4': 'These amplitudes obey the usual normalization conditions [EQUATION]', 'nucl-th-0302010-1-16-5': "Two-body correlation functions can be obtained from the Green's function (19) as follows [EQUATION]", 'nucl-th-0302010-1-17-0': '## Occupation numbers in the SCRPA', 'nucl-th-0302010-1-18-0': 'In order to close the set of the SCRPA equations, it is necessary to find the so far unknown occupation numbers [MATH] .', 'nucl-th-0302010-1-18-1': "For this, we should find the single particle Green's function [MATH] consistent with the SCRPA scheme.", 'nucl-th-0302010-1-18-2': 'As discussed in sect. II, the single particle mass operator [MATH] has in general the exact representation in terms of the two body [MATH]-matrix [CITATION] and then an appropriate approximation for the [MATH] can be obtained.', 'nucl-th-0302010-1-18-3': 'It consists in using the mass operator [MATH] calculated through the [MATH]-matrix found in the framework of SCRPA.', 'nucl-th-0302010-1-18-4': 'As the relation between the [MATH]-matrix and the sum of the all irreducible Feynman graphs in the pp-chanel is also known then the following expression for the single particle mass operator can be obtained [CITATION]: [EQUATION] [MATH] is expressed through the effective Hamiltonian (15) without the disconnected part [EQUATION] where [MATH] is defined below in (30).', 'nucl-th-0302010-1-19-0': 'In addition to this transparent scheme there also exists an additional consistency requirement [CITATION].', 'nucl-th-0302010-1-19-1': 'It follows from the possibility to calculate the average value of the Hamiltonian [MATH] in two ways.', 'nucl-th-0302010-1-19-2': "On the one hand one has the following relation between the single particle Green's function (9) and [MATH] [CITATION]: [EQUATION]", 'nucl-th-0302010-1-19-3': "On the other hand there exists the straightforward calculation of [MATH] through the two body Green's functions (22) [EQUATION] where we used the particle-hole symmetry of the model [11] and reduced sums over p and h only to the one over the particle states.", 'nucl-th-0302010-1-20-0': 'The additional consistency condition lies in the requirement that both expressions (25) and (26) should give exactly the same results.', 'nucl-th-0302010-1-20-1': 'This only is satisfied if the single particle GF is expanded to first order in the renormalized single particle mass operator (23) (one may verify that this is in analogy to the standard RPA scheme, i.e. the standard RPA average energy is obtained via (25) using a single particle GF with only perturbative renormalization from the RPA-modes): [EQUATION] with [EQUATION] and [EQUATION]', 'nucl-th-0302010-1-20-2': 'Finally we get the following expression for the SCRPA single particle mass operator [MATH]: [EQUATION] where [EQUATION]', 'nucl-th-0302010-1-20-3': 'The corresponding single particle occupation numbers [MATH], found from (7), is the following [EQUATION]', 'nucl-th-0302010-1-20-4': "Let us demonstrate now that using the single particle Green's function (27) with the mass operator (31) and occupation numbers (32) in the calculation of the average energy (25) indeed leads to the equation (26).", 'nucl-th-0302010-1-20-5': 'At first one finds the derivative of the single particle GF (27) [EQUATION]', 'nucl-th-0302010-1-20-6': 'Inserting this expression in (25) we obtain [EQUATION]', 'nucl-th-0302010-1-20-7': 'This is exactly equal to (26).', 'nucl-th-0302010-1-21-0': 'The system of the SCRPA equations is fully closed now.', 'nucl-th-0302010-1-21-1': 'Together with (17), (19) and (22) this represents a self-consistent problem for pair fluctuations.', 'nucl-th-0302010-1-22-0': 'We want to indicate at this point that the above way to determine the single particle occupancies is not the only possibility.', 'nucl-th-0302010-1-22-1': 'In the Appendix we will give another variant which, however, yields results close to the ones with the method of this section.', 'nucl-th-0302010-1-22-2': 'The non uniqueness of the occupation numbers reflects the fact that with the truncated ansatz (3), at zero temperature, no corresponding ground state wave function can be found, as explained in [CITATION].', 'nucl-th-0302010-1-22-3': 'For a wave function to exist, the ansatz (3) must be extended.', 'nucl-th-0302010-1-22-4': 'It can, however, be shown that the correction terms are small [CITATION].', 'nucl-th-0302010-1-23-0': '## Exact statistical treatment of the PFM', 'nucl-th-0302010-1-24-0': 'For an exact statistical treatment of the Picket Fence Model we have to find all exact eigenvalues and eigenstates of the Hamiltonian (11).', 'nucl-th-0302010-1-24-1': 'Since singly-occupied levels do not participate in the pair scattering, eigenstates can be classified according to the number of unpaired particles [MATH] (seniority).', 'nucl-th-0302010-1-24-2': 'There are [MATH] different multiplets of this type, each of dimension [MATH] and degeneracy [MATH].', 'nucl-th-0302010-1-24-3': 'If we define the following set of basis states for each multiplet: [EQUATION] where [MATH] for singly-occupied levels and [MATH] or [MATH] for remaining levels ([MATH]), the Hamiltonian matrix will have the following diagonal and off-diagonal elements [EQUATION]', 'nucl-th-0302010-1-24-4': 'The exact eigenvalues and eigenstates can be calculated by diagonalization of this matrix in each multiplet.', 'nucl-th-0302010-1-24-5': 'The exact grand canonical average [MATH] of any operator can then be obtained with the help of the grand partition function [MATH] and the statistical operator [MATH] as [EQUATION]', 'nucl-th-0302010-1-25-0': '# Results and Discussion', 'nucl-th-0302010-1-26-0': 'In order to check the accuracy of our theory and of the different approximations schemes we first calculate the average energy of the system [MATH] as a function of the particle number [MATH] and temperature [MATH].', 'nucl-th-0302010-1-26-1': 'The results of the different calculations are presented in Figures 2 - 4.', 'nucl-th-0302010-1-26-2': 'Calculations were made for a value of the pairing constant [MATH] which is smaller but close to the critical value [MATH] at [MATH].', 'nucl-th-0302010-1-26-3': 'The phase transition from the normal to the superfluid phase occurs in the system when [MATH] .', 'nucl-th-0302010-1-26-4': 'We compare the SCRPA results with the exact ones for the Grand Canonical Ensemble (GCE) as well as results of the standard Thermal RPA (TRPA) and Thermal MFA (TMFA).', 'nucl-th-0302010-1-26-5': 'One can see that when the number of levels [MATH] (and number of particles [MATH]) increases the description of the intrinsic energy becomes better and at [MATH] the TSCRPA results practically coincide with the exact ones.', 'nucl-th-0302010-1-26-6': 'Especially the last case will be considered below more carefully.', 'nucl-th-0302010-1-27-0': 'Let us now come to the discussion of the behavior of the excitation energies.', 'nucl-th-0302010-1-27-1': 'The dependence of the excitation energies of the addition mode (see [CITATION]) as a function of [MATH] is shown in Fig. 5 at zero temperature.', 'nucl-th-0302010-1-27-2': 'The SCRPA (solid lines) is compared with the standard RPA calculations (dashed lines) and the exact ones (open circles).', 'nucl-th-0302010-1-27-3': 'Increasing the interaction constant, the lowest energy in the RPA goes to zero and at [MATH] the collapse takes place which is connected with the transition from the normal to the superfluid phase.', 'nucl-th-0302010-1-27-4': 'At finite temperature (see Fig. 6 where the dependence of the lowest addition mode is presented as a function of [MATH] at [MATH]) this collapse occurs at a higher value of the interaction constant ([MATH]) what is due to the reduced intensity of the residual interaction because of the thermal factors.', 'nucl-th-0302010-1-27-5': 'This collapse is absent in the exact calculations at zero temperature and also in the SCRPA calculations at zero and finite temperatures.', 'nucl-th-0302010-1-27-6': 'It is also remarkable that the SCRPA yields a rise of all excitation energies with increasing [MATH] in contrast with RPA and in very good agreement with the exact results.', 'nucl-th-0302010-1-27-7': "This comes from the fact that in the PFM with the Kramer's degeneracy of levels the Pauli repulsion is extremely strong overruling the original attractive interaction.", 'nucl-th-0302010-1-27-8': 'In this model, therefore, standard RPA gives qualitatively wrong result.', 'nucl-th-0302010-1-28-0': 'We next consider the behavior of the system near the phase transition point.', 'nucl-th-0302010-1-28-1': 'To make distinctions between different results more apparent we not only show the full intrinsic energy [MATH] but also the correlation energy [MATH] which is defined as [EQUATION] where [MATH] is the average energy calculated in Mean Field Approximation.', 'nucl-th-0302010-1-28-2': 'In Figures 7 and 8, the average energy [MATH] and correlation energy [MATH] as a function of [MATH] are displayed for the interaction constant [MATH] (at [MATH] this value of [MATH] is larger than [MATH]).', 'nucl-th-0302010-1-28-3': 'With increasing [MATH] the mean field rearrangement occurs and the system goes from the superfluid phase to the normal one at [MATH].', 'nucl-th-0302010-1-28-4': 'Note, that within the TRPA the lowest excitation energy [MATH] vanishes when [MATH], whereas within the TSCRPA [MATH] stays finite.', 'nucl-th-0302010-1-28-5': 'For both the correlation energy and the intrinsic energy the TSCRPA gives more precise results as compared to the other approximations.', 'nucl-th-0302010-1-28-6': 'It is remarkable that the TSCRPA results are accurate down to practical zero temperature, in spite of the fact that within standard BCS theory one enters the superfluid regime.', 'nucl-th-0302010-1-28-7': 'A quasiparticle formulation of SCRPA [CITATION] will only be necessary for stronger [MATH] values driving the system more deeply into the symmetry broken phase.', 'nucl-th-0302010-1-29-0': 'To analyze the region near the phase transition point in more detail, the heat capacity is calculated as a partial derivative of the intrinsic energy with respect to [MATH] [EQUATION]', 'nucl-th-0302010-1-29-1': 'The results are shown in Fig. 9.', 'nucl-th-0302010-1-29-2': 'The TRPA and TMFA give discontinuities of [MATH] at [MATH] (we recall again that our results are obtained using a normal fluid approach and not transforming to quasiparticles).', 'nucl-th-0302010-1-29-3': 'The heat capacity calculated in the TSCRPA has some kink near [MATH] but has no discontinuities and is quite similar to the exact result through out the whole range of temperature.', 'nucl-th-0302010-1-30-0': 'Nevertheless, the TSCRPA and also the exact solution feel the phase transition to the superfluid phase.', 'nucl-th-0302010-1-30-1': 'It can be seen in Fig. 8 where both the TSCRPA and the exact correlation energies show a depression near [MATH].', 'nucl-th-0302010-1-30-2': 'This originates from strong pair fluctuations leading to the BCS phase transition in TMFA with the critical temperature [MATH] for [MATH].', 'nucl-th-0302010-1-30-3': 'However, one notices (see also [CITATION]) that the sharp phase transition of mean field is in reality completely smeared out and only a faint, though clearly visible, signal survives.', 'nucl-th-0302010-1-31-0': 'To investigate the formation of such fluctuations in more detail, it is useful to consider the spectral function [CITATION] [EQUATION] which includes two-body correlations through the self-consistent treatment of the mass operator [MATH].', 'nucl-th-0302010-1-31-1': 'Based on the spectral function the density of states can be calculated as [CITATION] [EQUATION]', 'nucl-th-0302010-1-31-2': 'The results of the calculations of [MATH] with [MATH] of (27) in (39) are shown in Fig. 10 at different values of [MATH] and for [MATH] = 0.4.', 'nucl-th-0302010-1-31-3': 'It is clearly seen that the distance between the two quasiparticle peaks around the Fermi energy [MATH]) increases with decreasing temperature.', 'nucl-th-0302010-1-31-4': 'This process sets in even above the BCS transition temperature [MATH].', 'nucl-th-0302010-1-31-5': "This rarefaction of the level density around [MATH] above [MATH] is not avoid of similarity with the situation in high [MATH] - superconductors where a so-called 'pseudo gap' in the level density appears already above [MATH] [CITATION].", 'nucl-th-0302010-1-31-6': "This 'pseudo gap' also is often attributed to a decrease in the level density around [MATH] due to pair fluctuations [CITATION].", 'nucl-th-0302010-1-31-7': 'Apparently it is a quite generic feature that pair correlations diminish the density of levels around [MATH] whereas particle-hole correlations give rise to an increase.', 'nucl-th-0302010-1-32-0': 'In order to make the temperature dependence of the gap more transparent let us introduce an effective (or canonical) gap which recently was proposed in Eq. (22) of Ref. [CITATION] [EQUATION]', 'nucl-th-0302010-1-32-1': 'In the BCS approximation, the effective gap [MATH] coincides with the usual grand canonical BCS gap [EQUATION]', 'nucl-th-0302010-1-32-2': 'The dependencies of the effective gap on the interaction constant [MATH] at zero temperature and on temperature [MATH] at [MATH] are shown in Figures 11 and 12.', 'nucl-th-0302010-1-32-3': 'The SCRPA results give a very good description of the gap at zero and non zero temperatures.', 'nucl-th-0302010-1-32-4': 'It is clearly seen that the SCRPA and exact calculations do not display the phase transition at the point where the BSC gap vanishes.', 'nucl-th-0302010-1-32-5': 'Notice that in Fig. 11 the SCRPA result deteriorates for values of [MATH] well beyond the critical value.', 'nucl-th-0302010-1-32-6': 'In this regime a quasiparticle generalization of the SCRPA is necessary [CITATION] because one enters deeply in the superfluid region.', 'nucl-th-0302010-1-33-0': '# Comparison with other works', 'nucl-th-0302010-1-34-0': 'The PFM has recently widely been used for the study of quantum pair fluctuations at finite temperature both in the context of nuclear physics [CITATION] and of ultrasmall metallic grains [CITATION].', 'nucl-th-0302010-1-34-1': 'In both fields the SPA approach where one additionally takes into account number parity projection and quantum (RPA) fluctuation around mean field was employed.', 'nucl-th-0302010-1-35-0': 'To compare our results with the above mentioned formalism let us introduce the relevant energy scales.', 'nucl-th-0302010-1-35-1': 'These are the average level spacing [MATH] and the BCS energy [MATH] [CITATION].', 'nucl-th-0302010-1-35-2': 'The properties of the system described by the pairing Hamiltonian can be calculated as universal functions of the single scaling parameter [MATH].', 'nucl-th-0302010-1-35-3': 'As long as the grain is not too small [MATH], the fluctuation region around [MATH] is narrow, and the mean field (BCS) description of superconductivity is appropriate.', 'nucl-th-0302010-1-35-4': 'It is the case in ref. [CITATION], where the PFM is investigated with characteristic values of [MATH].', 'nucl-th-0302010-1-35-5': 'The result of that work shows that the domain where the parity projection is important lies in a small region near [MATH].', 'nucl-th-0302010-1-35-6': 'At temperatures higher than [MATH] the role of the fluctuation is decreased and usual SPA becomes rather good in reproducing the exact canonical results.', 'nucl-th-0302010-1-36-0': 'When the size of the system is decreased, fluctuations start to smear the normal - superconducting transition.', 'nucl-th-0302010-1-36-1': 'The finite level spacing suppresses the BCS gap and when [MATH] becomes of the order of [MATH], the fluctuation region becomes of order [MATH] and the BCS description of superconductivity breaks down even at zero temperature.', 'nucl-th-0302010-1-36-2': 'Returning to our calculations, we can see that SCRPA yields the best results for [MATH] (see Fig. 11, 12).', 'nucl-th-0302010-1-36-3': 'This region corresponds to ultrasmall grains where strong pairing fluctuations are dominant.', 'nucl-th-0302010-1-36-4': 'In this sense our results can be compared (at least qualitatively) with the results of [CITATION].', 'nucl-th-0302010-1-36-5': 'To make this comparison more accurate we calculated average energy and specific heat for a system with 50 fermions and interaction constant [MATH] and [MATH], what corresponds to [MATH] and [MATH] respectively.', 'nucl-th-0302010-1-36-6': 'In general our results correlate well with [CITATION].', 'nucl-th-0302010-1-36-7': 'From Fig. 13, where [MATH] is displayed as a function of [MATH], we can see that when [MATH] the specific heat approaches a linear dependence with temperature while when [MATH] a bump structure arises at low temperature which is a sign of the presence of strong pairing fluctuations.', 'nucl-th-0302010-1-36-8': 'As it has been seen from our previous discussion, the SCRPA gives a better description with increasing particles number.', 'nucl-th-0302010-1-36-9': 'And while we did not perform exact GCE calculations for the system with 50 particles, we can expect from our studies above and in [CITATION] that our result should be very close to the exact one.', 'nucl-th-0302010-1-37-0': 'In conclusion of this section we can say that the results of TSCRPA are at least as good as the ones of SPA with extensions.', 'nucl-th-0302010-1-37-1': 'However, contrary to SPA, TSCRPA has no problem at low temperature and excitation energies and correlation functions can be calculated directly as a function of temperature.', 'nucl-th-0302010-1-38-0': '# Conclusion', 'nucl-th-0302010-1-39-0': 'In this work we generalized our recent work [CITATION] on the multilevel pairing model (PFM) within the SCRPA approach to finite temperature (TSCRPA).', 'nucl-th-0302010-1-39-1': 'The PFM has been recognised to account in many respects for the physics of (superconducting) metallic nano-grains.', 'nucl-th-0302010-1-39-2': 'In our context SCRPA can be viewed as a self-consistent mean field theory for pair fluctuatuations.', 'nucl-th-0302010-1-39-3': 'The results at [MATH] in [CITATION] are in very close agreement with the exact ones obtained from the Richardson procedure [CITATION].', 'nucl-th-0302010-1-39-4': 'It is therefore an important issue to also exploit SCRPA at finite [MATH] and to assess its accuracy with respect to exact results in this case.', 'nucl-th-0302010-1-39-5': 'Our comparison is mostly done for the case of ten levels with [MATH] particles where it is still of some ease to establish the exact partition function.', 'nucl-th-0302010-1-39-6': 'We, however, also considered with TSCRPA the case of 50 particles in [MATH] levels assuming that the results are of equal quality or even better as the ones obtained for [MATH] particles.', 'nucl-th-0302010-1-39-7': "We base our studies on the Matsubara [MATH] and [MATH] particle Green's functions which allows us to calculate correlation and excitation energies, specific heat, level densities, etc.", 'nucl-th-0302010-1-39-8': 'It can be considered as a general advantage of our approach that all these quantities are directly accessible in the whole range of temperatures and coupling constants.', 'nucl-th-0302010-1-39-9': 'For the latter this holds in this work only true for interaction values not driving the system deeply into the superfluid regime, since in this work we only have been working with normal particles and not with quasiparticles.', 'nucl-th-0302010-1-39-10': 'For [MATH] we have to employ the Self Consistent Quasiparticle RPA (SCQRPA).', 'nucl-th-0302010-1-39-11': 'It has recently been demonstrated in the two level pairing model that also SCQRPA gives very promising results [CITATION].', 'nucl-th-0302010-1-39-12': 'The quality of our results for the above mentioned quantities turns out to be excellent and it does not fail in any qualitative nor quantitative aspect.', 'nucl-th-0302010-1-39-13': 'Most of the time the agreement with the exact solution is within the couple of percent level.', 'nucl-th-0302010-1-39-14': "One particularly interesting feature of our investigation is the fact that we achieved to calculate the single particle Green's function consistently within TSCRPA.", 'nucl-th-0302010-1-39-15': 'This enabled us to give, for the first time, for the PFM the evolution of the single particle level density with temperature.', 'nucl-th-0302010-1-39-16': 'The construction of the exact solution for this quantity is very cumbersome and we refrained from working this out here.', 'nucl-th-0302010-1-39-17': 'However, backed with the positive experience for all other above mentioned quantities we believe that also the level density is reasonably accurate.', 'nucl-th-0302010-1-39-18': 'The interesting aspect of our calculation is that with decreasing temperature the density of single particle states around the Fermi level decreases even above the critical temperature as defined by BCS - theory.', 'nucl-th-0302010-1-39-19': 'It is suggestive to see this feature in analogy to the appearance of a so called pseudogap in high [MATH] superconductors where also a depression in the level density is observed approaching [MATH] from above [CITATION].', 'nucl-th-0302010-1-39-20': 'It would be interesting to attempt an experimental verification with metallic nanograins of our prediction that indeed the density of levels rarefies with decreasing temperature already in the non superfluid regime.', 'nucl-th-0302010-1-40-0': 'We also gave a short comparison of TSCRPA with results at finite [MATH] obtained with other approaches like the static path approximation (SPA) to the partition function.', 'nucl-th-0302010-1-40-1': 'Though the results seem generally comparable, we think that TSCRPA is more versatile, giving direct access to correlation functions, level densities, excitation energies, etc. in the whole temperature range, quantities which are otherwise difficult to obtain.', 'nucl-th-0302010-1-41-0': 'In the present work we restricted ourselves to values of the coupling which are below or slightly above the critical value.', 'nucl-th-0302010-1-41-1': 'In the future we shall elaborate on the SCRPA for quasiparticles at finite [MATH] (TSCQRPA) which will allow us to consider the system deeply in the superfluid phase and to study the transition from one phase to the other in more detail.', 'nucl-th-0302010-1-42-0': 'ACKNOWLEDGMENTS', 'nucl-th-0302010-1-43-0': 'We acknowledge useful discussions with P. Bozek, F. Hekking and J. Hirsch.', 'nucl-th-0302010-1-44-0': 'APPENDIX: OCCUPATION NUMBER - VARIANT', 'nucl-th-0302010-1-45-0': 'Let us give another possible way to find the occupation numbers at finite temperature.', 'nucl-th-0302010-1-45-1': 'Before to complete this task we firstly derive the SCRPA equations with the Equation of Motion method and find some useful relations between phonon amplitudes at zero temperature.', 'nucl-th-0302010-1-45-2': 'To attain this let us introduce the pair addition and removal operators (phonons) [CITATION] as [EQUATION] and apply the variational procedure where all expectation values are found with respect to the vacuum of phonons (A1,A2).', 'nucl-th-0302010-1-45-3': '[EQUATION]', 'nucl-th-0302010-1-45-4': 'If we use the factorisation procedure (16) the following system of equations for the phonon amplitudes is obtained [EQUATION] where [EQUATION] where the single-particle energies are introduced as [EQUATION]', 'nucl-th-0302010-1-45-5': 'Thereafter the expressions for the phonon amplitudes are obtained as [EQUATION]', 'nucl-th-0302010-1-45-6': 'Due to the particle-hole symmetry, the removal mode satisfy exactly the same equations.', 'nucl-th-0302010-1-45-7': 'It means that [EQUATION]', 'nucl-th-0302010-1-45-8': 'Below we will use the notation [MATH] for both modes.', 'nucl-th-0302010-1-46-0': 'Returning to the single particle occupation numbers [MATH] let us remind that in the picket fence model the following exact relation between one body operator [MATH] and two body operator [MATH] is verified for any non singly-occupied level [MATH]: [EQUATION]', 'nucl-th-0302010-1-46-1': 'Taking the average of both parts of this relation with respect to the SCRPA vacuum state we obtain [EQUATION]', 'nucl-th-0302010-1-46-2': 'On the other hand, using relations (A7) we can rewrite this expression as [EQUATION]', 'nucl-th-0302010-1-46-3': 'It is easy to show that the fraction in this expression can be expressed through the GFs as [EQUATION] where [EQUATION]', 'nucl-th-0302010-1-46-4': 'If we now take into account the spectral representation of the pair operator [MATH] [EQUATION] and define the antichronological (time reversed) single particle GF as [EQUATION] we can pass to the expression for the sp GF which gives occupation numbers which are consistent within the frame of the SCRPA with exact relation (A9) [EQUATION] where [MATH] is a single particle mass operator [EQUATION] and [MATH] is the renormalised effective Hamiltonian which has the following form in terms of RPA matrixes [MATH] and [MATH] [EQUATION]', 'nucl-th-0302010-1-46-5': 'From (A16) and (A17) we then can define the occupation numbers as usual.', 'nucl-th-0302010-1-46-6': 'In general, the occupation numbers from (32) and (A16), (A17) have slightly different values.', 'nucl-th-0302010-1-46-7': 'This is due to the fact that the ansatz (A1), (A2) for the RPA operators is too restricted for a groundstate fulfilling [MATH] to exist.', 'nucl-th-0302010-1-46-8': 'It can, however, be shown [CITATION] that the necessary corrections to (A1), (A2) are small.', 'nucl-th-0302010-1-46-9': 'Anyhow, the differences in occupation numbers obtained from the two methods are a measure of the importance of the terms neglected in (A1), (A2).', 'nucl-th-0302010-1-46-10': 'The difference of the mass operators (23) and (A17) is that in the latter both vertices are dressed, whereas in (23) one vertex remains at the unrenormalized value (G).', 'nucl-th-0302010-1-47-0': 'To find occupation numbers [MATH] at finite temperature we adopt the expressions obtained for single particle GF at zero temperature (eq. (A16)-(A18)).', 'nucl-th-0302010-1-47-1': 'To do it, it is necessary to change all zero temperature GFs to the Matsubara ones and use for the vertices the renormalised effective Hamiltonian (15) [EQUATION] where [EQUATION] and [EQUATION]', 'nucl-th-0302010-1-47-2': 'Then we get for the single particle Matsubara GF [EQUATION] where [EQUATION]', 'nucl-th-0302010-1-47-3': 'Taking this integral in the limit [MATH] we obtain the occupation numbers [MATH] in the SCRPA at finite temperature [EQUATION]', 'nucl-th-0302010-1-47-4': 'One should note here one thing about the direct use of the exact relation (A9) for the definition of the [MATH].', 'nucl-th-0302010-1-47-5': 'In reality the identity (A9) is only valid for the collective subspace (spanned by the non singly-occupied levels) or, in other words, for levels which have partial seniority [MATH].', 'nucl-th-0302010-1-47-6': 'But when we work in the grand canonical ensemble, seniority of the level [MATH] is not a good quantum number, since averaging procedure over GCE mixes all seniorities.', 'nucl-th-0302010-1-47-7': 'This fact is reflected in the above expression (A21) where we can see that level occupation numbers [MATH] due to thermal factors [MATH] (Fermi-Dirac distribution) are not equal to [MATH].', 'nucl-th-0302010-1-48-0': 'Before coming to a numerical example let us make a further comment.', 'nucl-th-0302010-1-48-1': 'Knowing the occupation numbers as a function of the amplitudes [MATH] and [MATH] as in (A21) or (32), a natural idea would be to use this to express the ground state energy entirely as a function of [MATH] and [MATH].', 'nucl-th-0302010-1-48-2': 'Then minimizing under the constraint of the normalization conditions (21) leads to an equation determining [MATH] amplitudes.', 'nucl-th-0302010-1-48-3': 'Such a procedure has been proposed in the past by Jolos and Rybarska [CITATION].', 'nucl-th-0302010-1-48-4': 'However, already in our example where the occupation numbers are exactly known (at zero temperature) as a function of [MATH] amplitudes, we have checked that this leads to worse results than with the approach advanced in the main text.', 'nucl-th-0302010-1-48-5': 'In other examples the functional [MATH] may only be known approximately and then the minimization of the ground state leads to deteriorated results, as we also have checked.', 'nucl-th-0302010-1-48-6': "In the Green's function approach where one first calculates excitation energies, i.e. energy differences, before calculating the ground state energy, such uncertainties like the precise knowledge of [MATH] are minimized and the solution of the SCRPA equations is therefore much more stable.", 'nucl-th-0302010-1-49-0': 'Let us now compare the results based on the two different definitions of the occupation numbers (eq. (32) and (A21)).', 'nucl-th-0302010-1-49-1': 'We will denote results with eq. (32) by TSCRPA and results obtained with the second definition (eq. (A21)) by TSCRPA1.', 'nucl-th-0302010-1-49-2': 'All calculations are made for the system with 10 particles on 10 levels.', 'nucl-th-0302010-1-49-3': 'Results for the correlation energy (37) as a function of the coupling constant [MATH] are given in Tab. 1 and Tab. 2 for [MATH] and [MATH], respectively.', 'nucl-th-0302010-1-49-4': 'In these tables we denote results obtained with (A21) and (26) by TSCRPA1-1 and the ones obtained with (A21) and (25) by TSCRPA1-2.', 'nucl-th-0302010-1-49-5': 'We can see that TSCRPA systematically gives underestimated results with respect to the exact ones.', 'nucl-th-0302010-1-49-6': 'Using definition (A21) we then can see that TSCRPA1-1 at zero temperature slightly overestimates the exact ground state energy (this corresponds to the results given in [CITATION]) while TSCRPA1-2 always underestimates it.', 'nucl-th-0302010-1-49-7': 'At finite temperature all approximations (TSCRPA and TSCRPA1) give close results which underestimate the exact ones.', 'nucl-th-0302010-1-49-8': 'In Tab. 3 and 4 we show excitation energies of the first addition mode for [MATH] and [MATH].', 'nucl-th-0302010-1-49-9': 'In both cases, TSCRPA and TSCRPA1, excitation energies grow with increasing [MATH] for zero and nonzero temperatures reproducing quite well the exact results.', 'nucl-th-0302010-1-49-10': 'We can see that for the calculated quantities (correlation and excitation energies) TSCRPA1-1 is slightly closer to the exact results than TSCRPA.', 'nucl-th-0302010-1-50-0': 'Table 1.', 'nucl-th-0302010-1-50-1': 'Correlation energy as function of [MATH] at zero temperature.', 'nucl-th-0302010-1-51-0': 'Table 2.', 'nucl-th-0302010-1-51-1': 'Correlation energy as function of [MATH] at [MATH].', 'nucl-th-0302010-1-52-0': 'Table 3.', 'nucl-th-0302010-1-52-1': 'Excitation energy of the first addition mode as function of [MATH] at [MATH].', 'nucl-th-0302010-1-53-0': 'Table 4.', 'nucl-th-0302010-1-53-1': 'Excitation energy of the first addition mode as function of [MATH] at [MATH].'}

{'nucl-th-0302010-2-0-0': "A self-consistent version of the Thermal Random Phase Approximation (TSCRPA) is developed within the Matsubara Green's Function (GF) formalism.", 'nucl-th-0302010-2-0-1': 'The TSCRPA is applied to the many level pairing model.', 'nucl-th-0302010-2-0-2': 'The normal phase of the system is considered.', 'nucl-th-0302010-2-0-3': 'The TSCRPA results are compared with the exact ones calculated for the Grand Canonical Ensemble.', 'nucl-th-0302010-2-0-4': 'Advantages of the TSCRPA over the Thermal Mean Field Approximation (TMFA) and the standard Thermal Random Phase Approximation (TRPA) are demonstrated.', 'nucl-th-0302010-2-0-5': 'Results for correlation functions, excitation energies, single particle level densities, etc., as a function of temperature are presented.', 'nucl-th-0302010-2-1-0': '# Introduction', 'nucl-th-0302010-2-2-0': 'Pairing properties of finite Fermi systems such as ultrasmall metallic grains have recently received a great deal of attention.', 'nucl-th-0302010-2-2-1': 'This has been spurred by a series of spectacular experiments of Ralph, Black and Tinkham [CITATION].', 'nucl-th-0302010-2-2-2': 'In order to correctly describe pairing properties it has been recognized that the finiteness of the systems (grains) needs to consider quantum fluctuations, good particle number, number parity, etc. seriously, since the coherence length may be of the order of the system size.', 'nucl-th-0302010-2-2-3': 'The situation for metallic grains has in the meanwhile been well described in several review articles [CITATION] (see also [CITATION]).', 'nucl-th-0302010-2-2-4': 'Another system where the finiteness is at the forefront of the theoretical investigation since several decades is the superfluid atomic nucleus.', 'nucl-th-0302010-2-2-5': 'As a matter of fact many of the theoretical tools such as particle number projection, even-odd effects, number parity, blocking effect, particle - particle Random Phase Approximation (pp-RPA), etc. have first been developed in nuclear physics [CITATION] before finding their application to finite systems of condensed matter.', 'nucl-th-0302010-2-2-6': 'Also the schematic pairing model with which we will mostly deal in this paper, namely the Picket Fence Model (PFM), whose exact solution has been found by Richardson and Sherman [CITATION], has essentially been developed in the context of nuclear physics for the description of deformed superfluid nuclei.', 'nucl-th-0302010-2-2-7': 'For finite condensed matter systems an early theoretical description was proposed by Muhlschlegel, Scalapino, and Denton [CITATION] using the Static Path Approximation (SPA) to the partition function.', 'nucl-th-0302010-2-2-8': 'This work stayed rather singular for a long time but the SPA has recently been applied successfully to the PFM both in the condensed matter [CITATION] and nuclear [CITATION] contexts.', 'nucl-th-0302010-2-2-9': 'A further standard method to treat quantum fluctuations namely the well known RPA has quite extensively been used for nuclear systems [CITATION] but equally for condensed matter problems [CITATION].', 'nucl-th-0302010-2-3-0': 'In this work we will further elaborate on the RPA approach.', 'nucl-th-0302010-2-3-1': 'We indeed have recently had quite remarkable success with a self consistent extension of the pp-RPA, which we called Self-Consistent RPA (SCRPA), by reproducing very accurately groundstate and excitation energies of the PFM [CITATION] at zero temperature.', 'nucl-th-0302010-2-3-2': 'This formalism was also developed independently by Ropke and collaborators who called it Cluster - Hartree - Fock (CHF) [CITATION].', 'nucl-th-0302010-2-3-3': 'Such type of generalization of the RPA theory grew out of the works of K.-J. Hara and D. Rowe [CITATION] several decades ago.', 'nucl-th-0302010-2-3-4': 'Shortly afterwards the theory was rederived using the method of many body Green functions [CITATION].', 'nucl-th-0302010-2-3-5': 'The success of the theory motivates us to develop the SCRPA formalism also for the finite temperature case and to study the thermodynamic properties of the BCS Hamiltonian using the PFM as an example.', 'nucl-th-0302010-2-3-6': 'For the extension of SCRPA to finite temperature we use the Matsubara Green functions approach [CITATION].', 'nucl-th-0302010-2-3-7': 'It appeared that the approximation scheme is very effective in treating two-body correlations in the particle-particle (pp) channel as well as the Pauli principle effects.', 'nucl-th-0302010-2-3-8': 'We should mention that we will work with real particles and not with quasiparticles what should limit our approach to temperatures above the critical temperature [MATH] (i.e. to the normal phase).', 'nucl-th-0302010-2-3-9': 'However, as we will see below, the definition of [MATH] in SCRPA is not so clear and we will be able to continue our calculation quite deeply into the superfluid regime.', 'nucl-th-0302010-2-4-0': 'We organize the paper in the following way.', 'nucl-th-0302010-2-4-1': 'In Section 2, the approach is outlined in general.', 'nucl-th-0302010-2-4-2': 'Then, in Section 3, the formalism is applied to the PFM.', 'nucl-th-0302010-2-4-3': 'A comparison with the exact solutions as well as with the results of other approximations is made in Section 4.', 'nucl-th-0302010-2-4-4': 'Section 5 is devoted to comparison with other recent works.', 'nucl-th-0302010-2-4-5': 'In section 6, we will summarize the results and draw some conclusions.', 'nucl-th-0302010-2-4-6': 'In an appendix a variant for the calculation of the occupation numbers is proposed.', 'nucl-th-0302010-2-5-0': '# General formalism', 'nucl-th-0302010-2-6-0': 'In treating a finite many-body system at finite temperature, it is convenient to use the grand canonical ensemble although it violates the number conservation.', 'nucl-th-0302010-2-6-1': 'With the definition [EQUATION] the grand partition function and statistical operator read [EQUATION] where [MATH].', 'nucl-th-0302010-2-6-2': "Then for any Schrodinger operator [MATH] the modified Heisenberg picture can be introduced [EQUATION] and the temperature (or the Matsubara) Green's Function (GF) is defined as [CITATION] [EQUATION]", 'nucl-th-0302010-2-6-3': 'Here, the brackets [MATH] mean the thermodynamic average; [MATH] is a [MATH] ordering operator, which arranges operators with the earliest [MATH] (the closest to [MATH]) to the right.', 'nucl-th-0302010-2-7-0': 'Let us consider the two-body Hamiltonian [EQUATION] where [MATH] are fermion annihilation and creation operators; [MATH] and [MATH] are the kinetic energy and the antisymmetrized matrix element of the two-body interaction.', 'nucl-th-0302010-2-7-1': "The Green's function [MATH] for an arbitrary operator [MATH] obeys the following equation of motion: [EQUATION]", 'nucl-th-0302010-2-7-2': 'In this expression it is possible to split the effective Hamiltonian [MATH] into an instantaneous and a dynamic (frequency dependent) part [CITATION] [EQUATION]', 'nucl-th-0302010-2-7-3': 'In the approximation of the instantaneous effective Hamiltonian i.e. neglecting [MATH], the Dyson equation for the two-body Matsubara GF [MATH] can be written as [EQUATION]', 'nucl-th-0302010-2-7-4': 'In the treatment of two particle correlations let us specify the arbitrary operator [MATH] as [MATH].', 'nucl-th-0302010-2-7-5': 'In this case the Dyson equation (3) takes the following form in the frequency representation [EQUATION] where, in supposing that the single particle density matrix is diagonal in the basis used (this is for example the case inhomogeneous matter): [EQUATION] and [EQUATION]', 'nucl-th-0302010-2-7-6': 'Here [MATH] is the antisymmetrized Kronecker symbol and [MATH] are the single particle occupation numbers which can be found from the single-particle Matsubara GF', 'nucl-th-0302010-2-8-0': '[EQUATION] as [EQUATION]', 'nucl-th-0302010-2-8-1': "In general the single-particle Matsubara Green's function [MATH] obeys the following Dyson equation: [EQUATION] or in the frequency representation [EQUATION] where [EQUATION]", 'nucl-th-0302010-2-8-2': 'Here [MATH] contains already the usual (instantaneous) mean field so that [MATH] denotes only the dynamical part of the mass operator.', 'nucl-th-0302010-2-9-0': 'Now the problem is to find an approximation for the mass operator [MATH] consistent with the SCRPA.', 'nucl-th-0302010-2-9-1': 'A solution to this problem has been proposed in [CITATION], which goes via the two body [MATH] - matrix representation of the single particle mass operator [CITATION], evaluating [MATH] within SCRPA.', 'nucl-th-0302010-2-10-0': 'Let us add at this point a word of physical interpretation of the mean-field operator (6) [CITATION].', 'nucl-th-0302010-2-10-1': 'A quick look allows to realize that it contains no higher than two body correlation functions and therefore for their determination, with (4), one obtains a selfconsistency problem.', 'nucl-th-0302010-2-10-2': 'Furthermore one can consider the nucleus as a gas of zero point pair fluctuations.', 'nucl-th-0302010-2-10-3': 'These fluctuations create their own mean field, i.e. one pair fluctuation moves in the average potential created by all the other pair fluctuations.', 'nucl-th-0302010-2-10-4': 'This average pair fluctuation field is graphically represented in Fig. 1.', 'nucl-th-0302010-2-10-5': 'It gives rise, as usual, to a nonlinear problem.', 'nucl-th-0302010-2-10-6': 'Of course, the single particle mean field introduced in (9) and further developed below is coupled to the selfconsistent pair potential.', 'nucl-th-0302010-2-10-7': 'This is the deeper meaning of [MATH] of (6).', 'nucl-th-0302010-2-11-0': '# Application to the Picket Fence Model', 'nucl-th-0302010-2-12-0': 'The model consists of an equidistant multilevel pairing Hamiltonian with each level two fold degenerate, i.e. only spin up/down fermions of one kind can occupy one level.', 'nucl-th-0302010-2-12-1': 'The corresponding Hamiltonian is given by [EQUATION] with [EQUATION] where [MATH] means the time reversed of [MATH], single particle energies are [MATH] with level spacing [MATH] chosen to be equal to [MATH], and [MATH] stands for the number of levels.', 'nucl-th-0302010-2-12-2': 'The chemical potential [MATH] will be chosen such as to conserve the average number of particles [MATH] of the system.', 'nucl-th-0302010-2-12-3': 'The operators defined in (12) form an SU(2) algebra for each level [MATH] and obey the following commutation relations [EQUATION]', 'nucl-th-0302010-2-13-0': '## SCRPA equations', 'nucl-th-0302010-2-14-0': 'To study the model at finite temperature we define in analogy to (1) the following set of two-body Matsubara GFs [EQUATION] where [EQUATION]', 'nucl-th-0302010-2-14-1': "Applying the instantaneous approximation for the mass operator we obtain the expressions for the two body SCRPA GF's: [EQUATION] with [EQUATION]", 'nucl-th-0302010-2-14-2': 'To find the correlation functions of the form [MATH] we will use the following approximation:', 'nucl-th-0302010-2-15-0': 'when [MATH] it is a simple factorization procedure, which has turned out to be accurate in the zero temperature limit: [EQUATION] but when [MATH] we use the following exact relation [EQUATION] which can easily be obtained taking into account that [MATH] and [MATH] (here [MATH]).', 'nucl-th-0302010-2-15-1': 'It should be noted that in [CITATION] the factorization (16a) was also used for the diagonal part (16b) and quite accurate results were obtained.', 'nucl-th-0302010-2-15-2': 'We will show below that with (16b) one obtains still improved results.', 'nucl-th-0302010-2-15-3': 'As shown in [CITATION] it is possible to avoid above approximation.', 'nucl-th-0302010-2-15-4': 'However, this is at the cost of a considerable numerical complication.', 'nucl-th-0302010-2-15-5': 'We refrain from this here because it brings only very little improvement of results.', 'nucl-th-0302010-2-16-0': 'With this ansatz a particle-particle RPA-like equation is obtained [EQUATION] where [EQUATION] and [EQUATION]', 'nucl-th-0302010-2-16-1': 'From this one easily can find the excitation spectrum of the model in equating the denominator of (17) to zero [EQUATION]', 'nucl-th-0302010-2-16-2': "Knowing the poles of the Green's function (17), one can write down its spectral representation (we here give it as a function of imaginary time), with the corresponding residua: [EQUATION] where the index [MATH] refers to the states above Fermi level and the index [MATH] to the ones below.", 'nucl-th-0302010-2-16-3': 'The following amplitudes were introduced in these formulas: [EQUATION] with [EQUATION]', 'nucl-th-0302010-2-16-4': 'These amplitudes obey the usual normalization conditions [EQUATION]', 'nucl-th-0302010-2-16-5': "Two-body correlation functions can be obtained from the Green's function (19) as follows [EQUATION]", 'nucl-th-0302010-2-17-0': '## Occupation numbers in the SCRPA', 'nucl-th-0302010-2-18-0': 'In order to close the set of the SCRPA equations, it is necessary to find the so far unknown occupation numbers [MATH] .', 'nucl-th-0302010-2-18-1': "For this, we should find the single particle Green's function [MATH] consistent with the SCRPA scheme.", 'nucl-th-0302010-2-18-2': 'As discussed in sect. II, the single particle mass operator [MATH] has in general the exact representation in terms of the two body [MATH]-matrix [CITATION] and then an appropriate approximation for the [MATH] can be obtained.', 'nucl-th-0302010-2-18-3': 'It consists in using the mass operator [MATH] calculated through the [MATH]-matrix found in the framework of SCRPA.', 'nucl-th-0302010-2-18-4': 'As the relation between the [MATH]-matrix and the sum of the all irreducible Feynman graphs in the pp-chanel is also known then the following expression for the single particle mass operator can be obtained [CITATION]: [EQUATION] [MATH] is expressed through the effective Hamiltonian (15) without the disconnected part [EQUATION] where [MATH] is defined below in (30).', 'nucl-th-0302010-2-19-0': 'In addition to this transparent scheme there also exists an additional consistency requirement [CITATION].', 'nucl-th-0302010-2-19-1': 'It follows from the possibility to calculate the average value of the Hamiltonian [MATH] in two ways.', 'nucl-th-0302010-2-19-2': "On the one hand one has the following relation between the single particle Green's function (9) and [MATH] [CITATION]: [EQUATION]", 'nucl-th-0302010-2-19-3': "On the other hand there exists the straightforward calculation of [MATH] through the two body Green's functions (22) [EQUATION] where we used the particle-hole symmetry of the model [11] and reduced sums over p and h only to the one over the particle states.", 'nucl-th-0302010-2-20-0': 'The additional consistency condition lies in the requirement that both expressions (25) and (26) should give exactly the same results.', 'nucl-th-0302010-2-20-1': 'This only is satisfied if the single particle GF is expanded to first order in the renormalized single particle mass operator (23) (one may verify that this is in analogy to the standard RPA scheme, i.e. the standard RPA average energy is obtained via (25) using a single particle GF with only perturbative renormalization from the RPA-modes): [EQUATION] with [EQUATION] and [EQUATION]', 'nucl-th-0302010-2-20-2': 'Finally we get the following expression for the SCRPA single particle mass operator [MATH]: [EQUATION] where [EQUATION]', 'nucl-th-0302010-2-20-3': 'The corresponding single particle occupation numbers [MATH], found from (7), is the following [EQUATION]', 'nucl-th-0302010-2-20-4': "Let us demonstrate now that using the single particle Green's function (27) with the mass operator (31) and occupation numbers (32) in the calculation of the average energy (25) indeed leads to the equation (26).", 'nucl-th-0302010-2-20-5': 'At first one finds the derivative of the single particle GF (27) [EQUATION]', 'nucl-th-0302010-2-20-6': 'Inserting this expression in (25) we obtain [EQUATION]', 'nucl-th-0302010-2-20-7': 'This is exactly equal to (26).', 'nucl-th-0302010-2-21-0': 'The system of the SCRPA equations is fully closed now.', 'nucl-th-0302010-2-21-1': 'Together with (17), (19) and (22) this represents a self-consistent problem for pair fluctuations.', 'nucl-th-0302010-2-22-0': 'We want to indicate at this point that the above way to determine the single particle occupancies is not the only possibility.', 'nucl-th-0302010-2-22-1': 'In the Appendix we will give another variant which, however, yields results close to the ones with the method of this section.', 'nucl-th-0302010-2-22-2': 'The non uniqueness of the occupation numbers reflects the fact that with the truncated ansatz (3), at zero temperature, no corresponding ground state wave function can be found, as explained in [CITATION].', 'nucl-th-0302010-2-22-3': 'For a wave function to exist, the ansatz (3) must be extended.', 'nucl-th-0302010-2-22-4': 'It can, however, be shown that the correction terms are small [CITATION].', 'nucl-th-0302010-2-23-0': '## Exact statistical treatment of the PFM', 'nucl-th-0302010-2-24-0': 'For an exact statistical treatment of the Picket Fence Model we have to find all exact eigenvalues and eigenstates of the Hamiltonian (11).', 'nucl-th-0302010-2-24-1': 'Since singly-occupied levels do not participate in the pair scattering, eigenstates can be classified according to the number of unpaired particles [MATH] (seniority).', 'nucl-th-0302010-2-24-2': 'There are [MATH] different multiplets of this type, each of dimension [MATH] and degeneracy [MATH].', 'nucl-th-0302010-2-24-3': 'If we define the following set of basis states for each multiplet: [EQUATION] where [MATH] for singly-occupied levels and [MATH] or [MATH] for remaining levels ([MATH]), the Hamiltonian matrix will have the following diagonal and off-diagonal elements [EQUATION]', 'nucl-th-0302010-2-24-4': 'The exact eigenvalues and eigenstates can be calculated by diagonalization of this matrix in each multiplet.', 'nucl-th-0302010-2-24-5': 'The exact grand canonical average [MATH] of any operator can then be obtained with the help of the grand partition function [MATH] and the statistical operator [MATH] as [EQUATION]', 'nucl-th-0302010-2-25-0': '# Results and Discussion', 'nucl-th-0302010-2-26-0': 'In order to check the accuracy of our theory and of the different approximations schemes we first calculate the average energy of the system [MATH] as a function of the particle number [MATH] and temperature [MATH].', 'nucl-th-0302010-2-26-1': 'The results of the different calculations are presented in Figures 2 - 4.', 'nucl-th-0302010-2-26-2': 'Calculations were made for a value of the pairing constant [MATH] which is smaller but close to the critical value [MATH] at [MATH].', 'nucl-th-0302010-2-26-3': 'The phase transition from the normal to the superfluid phase occurs in the system when [MATH] .', 'nucl-th-0302010-2-26-4': 'We compare the SCRPA results with the exact ones for the Grand Canonical Ensemble (GCE) as well as results of the standard Thermal RPA (TRPA) and Thermal MFA (TMFA).', 'nucl-th-0302010-2-26-5': 'One can see that when the number of levels [MATH] (and number of particles [MATH]) increases the description of the intrinsic energy becomes better and at [MATH] the TSCRPA results practically coincide with the exact ones.', 'nucl-th-0302010-2-26-6': 'Especially the last case will be considered below more carefully.', 'nucl-th-0302010-2-27-0': 'Let us now come to the discussion of the behavior of the excitation energies.', 'nucl-th-0302010-2-27-1': 'The dependence of the excitation energies of the addition mode (see [CITATION]) as a function of [MATH] is shown in Fig. 5 at zero temperature.', 'nucl-th-0302010-2-27-2': 'The SCRPA (solid lines) is compared with the standard RPA calculations (dashed lines) and the exact ones (open circles).', 'nucl-th-0302010-2-27-3': 'Increasing the interaction constant, the lowest energy in the RPA goes to zero and at [MATH] the collapse takes place which is connected with the transition from the normal to the superfluid phase.', 'nucl-th-0302010-2-27-4': 'At finite temperature (see Fig. 6 where the dependence of the lowest addition mode is presented as a function of [MATH] at [MATH]) this collapse occurs at a higher value of the interaction constant ([MATH]) what is due to the reduced intensity of the residual interaction because of the thermal factors.', 'nucl-th-0302010-2-27-5': 'This collapse is absent in the exact calculations at zero temperature and also in the SCRPA calculations at zero and finite temperatures.', 'nucl-th-0302010-2-27-6': 'It is also remarkable that the SCRPA yields a rise of all excitation energies with increasing [MATH] in contrast with RPA and in very good agreement with the exact results.', 'nucl-th-0302010-2-27-7': "This comes from the fact that in the PFM with the Kramer's degeneracy of levels the Pauli repulsion is extremely strong overruling the original attractive interaction.", 'nucl-th-0302010-2-27-8': 'In this model, therefore, standard RPA gives qualitatively wrong result.', 'nucl-th-0302010-2-28-0': 'We next consider the behavior of the system near the phase transition point.', 'nucl-th-0302010-2-28-1': 'To make distinctions between different results more apparent we not only show the full intrinsic energy [MATH] but also the correlation energy [MATH] which is defined as [EQUATION] where [MATH] is the average energy calculated in Mean Field Approximation.', 'nucl-th-0302010-2-28-2': 'In Figures 7 and 8, the average energy [MATH] and correlation energy [MATH] as a function of [MATH] are displayed for the interaction constant [MATH] (at [MATH] this value of [MATH] is larger than [MATH]).', 'nucl-th-0302010-2-28-3': 'With increasing [MATH] the mean field rearrangement occurs and the system goes from the superfluid phase to the normal one at [MATH].', 'nucl-th-0302010-2-28-4': 'Note, that within the TRPA the lowest excitation energy [MATH] vanishes when [MATH], whereas within the TSCRPA [MATH] stays finite.', 'nucl-th-0302010-2-28-5': 'For both the correlation energy and the intrinsic energy the TSCRPA gives more precise results as compared to the other approximations.', 'nucl-th-0302010-2-28-6': 'It is remarkable that the TSCRPA results are accurate down to practical zero temperature, in spite of the fact that within standard BCS theory one enters the superfluid regime.', 'nucl-th-0302010-2-28-7': 'A quasiparticle formulation of SCRPA [CITATION] will only be necessary for stronger [MATH] values driving the system more deeply into the symmetry broken phase.', 'nucl-th-0302010-2-29-0': 'To analyze the region near the phase transition point in more detail, the heat capacity is calculated as a partial derivative of the intrinsic energy with respect to [MATH] [EQUATION]', 'nucl-th-0302010-2-29-1': 'The results are shown in Fig. 9.', 'nucl-th-0302010-2-29-2': 'The TRPA and TMFA give discontinuities of [MATH] at [MATH] (we recall again that our results are obtained using a normal fluid approach and not transforming to quasiparticles).', 'nucl-th-0302010-2-29-3': 'The heat capacity calculated in the TSCRPA has some kink near [MATH] but has no discontinuities and is quite similar to the exact result through out the whole range of temperature.', 'nucl-th-0302010-2-30-0': 'Nevertheless, the TSCRPA and also the exact solution feel the phase transition to the superfluid phase.', 'nucl-th-0302010-2-30-1': 'It can be seen in Fig. 8 where both the TSCRPA and the exact correlation energies show a depression near [MATH].', 'nucl-th-0302010-2-30-2': 'This originates from strong pair fluctuations leading to the BCS phase transition in TMFA with the critical temperature [MATH] for [MATH].', 'nucl-th-0302010-2-30-3': 'However, one notices (see also [CITATION]) that the sharp phase transition of mean field is in reality completely smeared out and only a faint, though clearly visible, signal survives.', 'nucl-th-0302010-2-31-0': 'To investigate the formation of such fluctuations in more detail, it is useful to consider the spectral function [CITATION] [EQUATION] which includes two-body correlations through the self-consistent treatment of the mass operator [MATH].', 'nucl-th-0302010-2-31-1': 'Based on the spectral function the density of states can be calculated as [CITATION] [EQUATION]', 'nucl-th-0302010-2-31-2': 'The results of the calculations of [MATH] with [MATH] of (27) in (39) are shown in Fig. 10 at different values of [MATH] and for [MATH] = 0.4.', 'nucl-th-0302010-2-31-3': 'It is clearly seen that the distance between the two quasiparticle peaks around the Fermi energy [MATH]) increases with decreasing temperature.', 'nucl-th-0302010-2-31-4': 'This process sets in even above the BCS transition temperature [MATH].', 'nucl-th-0302010-2-31-5': "This rarefaction of the level density around [MATH] above [MATH] is not avoid of similarity with the situation in high [MATH] - superconductors where a so-called 'pseudo gap' in the level density appears already above [MATH] [CITATION].", 'nucl-th-0302010-2-31-6': "This 'pseudo gap' also is often attributed to a decrease in the level density around [MATH] due to pair fluctuations [CITATION].", 'nucl-th-0302010-2-31-7': 'Apparently it is a quite generic feature that pair correlations diminish the density of levels around [MATH] whereas particle-hole correlations give rise to an increase.', 'nucl-th-0302010-2-32-0': 'In order to make the temperature dependence of the gap more transparent let us introduce an effective (or canonical) gap which recently was proposed in Eq. (22) of Ref. [CITATION] [EQUATION]', 'nucl-th-0302010-2-32-1': 'In the BCS approximation, the effective gap [MATH] coincides with the usual grand canonical BCS gap [EQUATION]', 'nucl-th-0302010-2-32-2': 'The dependencies of the effective gap on the interaction constant [MATH] at zero temperature and on temperature [MATH] at [MATH] are shown in Figures 11 and 12.', 'nucl-th-0302010-2-32-3': 'The SCRPA results give a very good description of the gap at zero and non zero temperatures.', 'nucl-th-0302010-2-32-4': 'It is clearly seen that the SCRPA and exact calculations do not display the phase transition at the point where the BSC gap vanishes.', 'nucl-th-0302010-2-32-5': 'Notice that in Fig. 11 the SCRPA result deteriorates for values of [MATH] well beyond the critical value.', 'nucl-th-0302010-2-32-6': 'In this regime a quasiparticle generalization of the SCRPA is necessary [CITATION] because one enters deeply in the superfluid region.', 'nucl-th-0302010-2-33-0': '# Comparison with other works', 'nucl-th-0302010-2-34-0': 'The PFM has recently widely been used for the study of quantum pair fluctuations at finite temperature both in the context of nuclear physics [CITATION] and of ultrasmall metallic grains [CITATION].', 'nucl-th-0302010-2-34-1': 'In both fields the SPA approach where one additionally takes into account number parity projection and quantum (RPA) fluctuation around mean field was employed.', 'nucl-th-0302010-2-35-0': 'To compare our results with the above mentioned formalism let us introduce the relevant energy scales.', 'nucl-th-0302010-2-35-1': 'These are the average level spacing [MATH] and the BCS energy [MATH] [CITATION].', 'nucl-th-0302010-2-35-2': 'The properties of the system described by the pairing Hamiltonian can be calculated as universal functions of the single scaling parameter [MATH].', 'nucl-th-0302010-2-35-3': 'As long as the grain is not too small [MATH], the fluctuation region around [MATH] is narrow, and the mean field (BCS) description of superconductivity is appropriate.', 'nucl-th-0302010-2-35-4': 'It is the case in ref. [CITATION], where the PFM is investigated with characteristic values of [MATH].', 'nucl-th-0302010-2-35-5': 'The result of that work shows that the domain where the parity projection is important lies in a small region near [MATH].', 'nucl-th-0302010-2-35-6': 'At temperatures higher than [MATH] the role of the fluctuation is decreased and usual SPA becomes rather good in reproducing the exact canonical results.', 'nucl-th-0302010-2-36-0': 'When the size of the system is decreased, fluctuations start to smear the normal - superconducting transition.', 'nucl-th-0302010-2-36-1': 'The finite level spacing suppresses the BCS gap and when [MATH] becomes of the order of [MATH], the fluctuation region becomes of order [MATH] and the BCS description of superconductivity breaks down even at zero temperature.', 'nucl-th-0302010-2-36-2': 'Returning to our calculations, we can see that SCRPA yields the best results for [MATH] (see Fig. 11, 12).', 'nucl-th-0302010-2-36-3': 'This region corresponds to ultrasmall grains where strong pairing fluctuations are dominant.', 'nucl-th-0302010-2-36-4': 'In this sense our results can be compared (at least qualitatively) with the results of [CITATION].', 'nucl-th-0302010-2-36-5': 'To make this comparison more accurate we calculated average energy and specific heat for a system with 50 fermions and interaction constant [MATH] and [MATH], what corresponds to [MATH] and [MATH] respectively.', 'nucl-th-0302010-2-36-6': 'In general our results correlate well with [CITATION].', 'nucl-th-0302010-2-36-7': 'From Fig. 13, where [MATH] is displayed as a function of [MATH], we can see that when [MATH] the specific heat approaches a linear dependence with temperature while when [MATH] a bump structure arises at low temperature which is a sign of the presence of strong pairing fluctuations.', 'nucl-th-0302010-2-36-8': 'As it has been seen from our previous discussion, the SCRPA gives a better description with increasing particles number.', 'nucl-th-0302010-2-36-9': 'And while we did not perform exact GCE calculations for the system with 50 particles, we can expect from our studies above and in [CITATION] that our result should be very close to the exact one.', 'nucl-th-0302010-2-37-0': 'In conclusion of this section we can say that the results of TSCRPA are at least as good as the ones of SPA with extensions.', 'nucl-th-0302010-2-37-1': 'However, contrary to SPA, TSCRPA has no problem at low temperature and excitation energies and correlation functions can be calculated directly as a function of temperature.', 'nucl-th-0302010-2-38-0': '# Conclusion', 'nucl-th-0302010-2-39-0': 'In this work we generalized our recent work [CITATION] on the multilevel pairing model (PFM) within the SCRPA approach to finite temperature (TSCRPA).', 'nucl-th-0302010-2-39-1': 'The PFM has been recognised to account in many respects for the physics of (superconducting) metallic nano-grains.', 'nucl-th-0302010-2-39-2': 'In our context SCRPA can be viewed as a self-consistent mean field theory for pair fluctuatuations.', 'nucl-th-0302010-2-39-3': 'The results at [MATH] in [CITATION] are in very close agreement with the exact ones obtained from the Richardson procedure [CITATION].', 'nucl-th-0302010-2-39-4': 'It is therefore an important issue to also exploit SCRPA at finite [MATH] and to assess its accuracy with respect to exact results in this case.', 'nucl-th-0302010-2-39-5': 'Our comparison is mostly done for the case of ten levels with [MATH] particles where it is still of some ease to establish the exact partition function.', 'nucl-th-0302010-2-39-6': 'We, however, also considered with TSCRPA the case of 50 particles in [MATH] levels assuming that the results are of equal quality or even better as the ones obtained for [MATH] particles.', 'nucl-th-0302010-2-39-7': "We base our studies on the Matsubara [MATH] and [MATH] particle Green's functions which allows us to calculate correlation and excitation energies, specific heat, level densities, etc.", 'nucl-th-0302010-2-39-8': 'It can be considered as a general advantage of our approach that all these quantities are directly accessible in the whole range of temperatures and coupling constants.', 'nucl-th-0302010-2-39-9': 'For the latter this holds in this work only true for interaction values not driving the system deeply into the superfluid regime, since in this work we only have been working with normal particles and not with quasiparticles.', 'nucl-th-0302010-2-39-10': 'For [MATH] we have to employ the Self Consistent Quasiparticle RPA (SCQRPA).', 'nucl-th-0302010-2-39-11': 'It has recently been demonstrated in the two level pairing model that also SCQRPA gives very promising results [CITATION].', 'nucl-th-0302010-2-39-12': 'The quality of our results for the above mentioned quantities turns out to be excellent and it does not fail in any qualitative nor quantitative aspect.', 'nucl-th-0302010-2-39-13': 'Most of the time the agreement with the exact solution is within the couple of percent level.', 'nucl-th-0302010-2-39-14': "One particularly interesting feature of our investigation is the fact that we achieved to calculate the single particle Green's function consistently within TSCRPA.", 'nucl-th-0302010-2-39-15': 'This enabled us to give, for the first time, for the PFM the evolution of the single particle level density with temperature.', 'nucl-th-0302010-2-39-16': 'The construction of the exact solution for this quantity is very cumbersome and we refrained from working this out here.', 'nucl-th-0302010-2-39-17': 'However, backed with the positive experience for all other above mentioned quantities we believe that also the level density is reasonably accurate.', 'nucl-th-0302010-2-39-18': 'The interesting aspect of our calculation is that with decreasing temperature the density of single particle states around the Fermi level decreases even above the critical temperature as defined by BCS - theory.', 'nucl-th-0302010-2-39-19': 'It is suggestive to see this feature in analogy to the appearance of a so called pseudogap in high [MATH] superconductors where also a depression in the level density is observed approaching [MATH] from above [CITATION].', 'nucl-th-0302010-2-39-20': 'It would be interesting to attempt an experimental verification with metallic nanograins of our prediction that indeed the density of levels rarefies with decreasing temperature already in the non superfluid regime.', 'nucl-th-0302010-2-40-0': 'We also gave a short comparison of TSCRPA with results at finite [MATH] obtained with other approaches like the static path approximation (SPA) to the partition function.', 'nucl-th-0302010-2-40-1': 'Though the results seem generally comparable, we think that TSCRPA is more versatile, giving direct access to correlation functions, level densities, excitation energies, etc. in the whole temperature range, quantities which are otherwise difficult to obtain.', 'nucl-th-0302010-2-41-0': 'In the present work we restricted ourselves to values of the coupling which are below or slightly above the critical value.', 'nucl-th-0302010-2-41-1': 'In the future we shall elaborate on the SCRPA for quasiparticles at finite [MATH] (TSCQRPA) which will allow us to consider the system deeply in the superfluid phase and to study the transition from one phase to the other in more detail.', 'nucl-th-0302010-2-42-0': 'ACKNOWLEDGMENTS', 'nucl-th-0302010-2-43-0': 'We acknowledge useful discussions with P. Bozek, F. Hekking and J. Hirsch.', 'nucl-th-0302010-2-44-0': 'APPENDIX: OCCUPATION NUMBER - VARIANT', 'nucl-th-0302010-2-45-0': 'Let us give another possible way to find the occupation numbers at finite temperature.', 'nucl-th-0302010-2-45-1': 'Before to complete this task we firstly derive the SCRPA equations with the Equation of Motion method and find some useful relations between phonon amplitudes at zero temperature.', 'nucl-th-0302010-2-45-2': 'To attain this let us introduce the pair addition and removal operators (phonons) [CITATION] as [EQUATION] and apply the variational procedure where all expectation values are found with respect to the vacuum of phonons (A1,A2).', 'nucl-th-0302010-2-45-3': '[EQUATION]', 'nucl-th-0302010-2-45-4': 'If we use the factorisation procedure (16) the following system of equations for the phonon amplitudes is obtained [EQUATION] where [EQUATION] where the single-particle energies are introduced as [EQUATION]', 'nucl-th-0302010-2-45-5': 'Thereafter the expressions for the phonon amplitudes are obtained as [EQUATION]', 'nucl-th-0302010-2-45-6': 'Due to the particle-hole symmetry, the removal mode satisfy exactly the same equations.', 'nucl-th-0302010-2-45-7': 'It means that [EQUATION]', 'nucl-th-0302010-2-45-8': 'Below we will use the notation [MATH] for both modes.', 'nucl-th-0302010-2-46-0': 'Returning to the single particle occupation numbers [MATH] let us remind that in the picket fence model the following exact relation between one body operator [MATH] and two body operator [MATH] is verified for any non singly-occupied level [MATH]: [EQUATION]', 'nucl-th-0302010-2-46-1': 'Taking the average of both parts of this relation with respect to the SCRPA vacuum state we obtain [EQUATION]', 'nucl-th-0302010-2-46-2': 'On the other hand, using relations (A7) we can rewrite this expression as [EQUATION]', 'nucl-th-0302010-2-46-3': 'It is easy to show that the fraction in this expression can be expressed through the GFs as [EQUATION] where [EQUATION]', 'nucl-th-0302010-2-46-4': 'If we now take into account the spectral representation of the pair operator [MATH] [EQUATION] and define the antichronological (time reversed) single particle GF as [EQUATION] we can pass to the expression for the sp GF which gives occupation numbers which are consistent within the frame of the SCRPA with exact relation (A9) [EQUATION] where [MATH] is a single particle mass operator [EQUATION] and [MATH] is the renormalised effective Hamiltonian which has the following form in terms of RPA matrixes [MATH] and [MATH] [EQUATION]', 'nucl-th-0302010-2-46-5': 'From (A16) and (A17) we then can define the occupation numbers as usual.', 'nucl-th-0302010-2-46-6': 'In general, the occupation numbers from (32) and (A16), (A17) have slightly different values.', 'nucl-th-0302010-2-46-7': 'This is due to the fact that the ansatz (A1), (A2) for the RPA operators is too restricted for a groundstate fulfilling [MATH] to exist.', 'nucl-th-0302010-2-46-8': 'It can, however, be shown [CITATION] that the necessary corrections to (A1), (A2) are small.', 'nucl-th-0302010-2-46-9': 'Anyhow, the differences in occupation numbers obtained from the two methods are a measure of the importance of the terms neglected in (A1), (A2).', 'nucl-th-0302010-2-46-10': 'The difference of the mass operators (23) and (A17) is that in the latter both vertices are dressed, whereas in (23) one vertex remains at the unrenormalized value (G).', 'nucl-th-0302010-2-47-0': 'To find occupation numbers [MATH] at finite temperature we adopt the expressions obtained for single particle GF at zero temperature (eq. (A16)-(A18)).', 'nucl-th-0302010-2-47-1': 'To do it, it is necessary to change all zero temperature GFs to the Matsubara ones and use for the vertices the renormalised effective Hamiltonian (15) [EQUATION] where [EQUATION] and [EQUATION]', 'nucl-th-0302010-2-47-2': 'Then we get for the single particle Matsubara GF [EQUATION] where [EQUATION]', 'nucl-th-0302010-2-47-3': 'Taking this integral in the limit [MATH] we obtain the occupation numbers [MATH] in the SCRPA at finite temperature [EQUATION]', 'nucl-th-0302010-2-47-4': 'One should note here one thing about the direct use of the exact relation (A9) for the definition of the [MATH].', 'nucl-th-0302010-2-47-5': 'In reality the identity (A9) is only valid for the collective subspace (spanned by the non singly-occupied levels) or, in other words, for levels which have partial seniority [MATH].', 'nucl-th-0302010-2-47-6': 'But when we work in the grand canonical ensemble, seniority of the level [MATH] is not a good quantum number, since averaging procedure over GCE mixes all seniorities.', 'nucl-th-0302010-2-47-7': 'This fact is reflected in the above expression (A21) where we can see that level occupation numbers [MATH] due to thermal factors [MATH] (Fermi-Dirac distribution) are not equal to [MATH].', 'nucl-th-0302010-2-48-0': 'Before coming to a numerical example let us make a further comment.', 'nucl-th-0302010-2-48-1': 'Knowing the occupation numbers as a function of the amplitudes [MATH] and [MATH] as in (A21) or (32), a natural idea would be to use this to express the ground state energy entirely as a function of [MATH] and [MATH].', 'nucl-th-0302010-2-48-2': 'Then minimizing under the constraint of the normalization conditions (21) leads to an equation determining [MATH] amplitudes.', 'nucl-th-0302010-2-48-3': 'Such a procedure has been proposed in the past by Jolos and Rybarska [CITATION].', 'nucl-th-0302010-2-48-4': 'However, already in our example where the occupation numbers are exactly known (at zero temperature) as a function of [MATH] amplitudes, we have checked that this leads to worse results than with the approach advanced in the main text.', 'nucl-th-0302010-2-48-5': 'In other examples the functional [MATH] may only be known approximately and then the minimization of the ground state leads to deteriorated results, as we also have checked.', 'nucl-th-0302010-2-48-6': "In the Green's function approach where one first calculates excitation energies, i.e. energy differences, before calculating the ground state energy, such uncertainties like the precise knowledge of [MATH] are minimized and the solution of the SCRPA equations is therefore much more stable.", 'nucl-th-0302010-2-49-0': 'Let us now compare the results based on the two different definitions of the occupation numbers (eq. (32) and (A21)).', 'nucl-th-0302010-2-49-1': 'We will denote results with eq. (32) by TSCRPA and results obtained with the second definition (eq. (A21)) by TSCRPA1.', 'nucl-th-0302010-2-49-2': 'All calculations are made for the system with 10 particles on 10 levels.', 'nucl-th-0302010-2-49-3': 'Results for the correlation energy (37) as a function of the coupling constant [MATH] are given in Tab. 1 and Tab. 2 for [MATH] and [MATH], respectively.', 'nucl-th-0302010-2-49-4': 'In these tables we denote results obtained with (A21) and (26) by TSCRPA1-1 and the ones obtained with (A21) and (25) by TSCRPA1-2.', 'nucl-th-0302010-2-49-5': 'We can see that TSCRPA systematically gives underestimated results with respect to the exact ones.', 'nucl-th-0302010-2-49-6': 'Using definition (A21) we then can see that TSCRPA1-1 at zero temperature slightly overestimates the exact ground state energy (this corresponds to the results given in [CITATION]) while TSCRPA1-2 always underestimates it.', 'nucl-th-0302010-2-49-7': 'At finite temperature all approximations (TSCRPA and TSCRPA1) give close results which underestimate the exact ones.', 'nucl-th-0302010-2-49-8': 'In Tab. 3 and 4 we show excitation energies of the first addition mode for [MATH] and [MATH].', 'nucl-th-0302010-2-49-9': 'In both cases, TSCRPA and TSCRPA1, excitation energies grow with increasing [MATH] for zero and nonzero temperatures reproducing quite well the exact results.', 'nucl-th-0302010-2-49-10': 'We can see that for the calculated quantities (correlation and excitation energies) TSCRPA1-1 is slightly closer to the exact results than TSCRPA.', 'nucl-th-0302010-2-50-0': 'Table 1.', 'nucl-th-0302010-2-50-1': 'Correlation energy as function of [MATH] at zero temperature.', 'nucl-th-0302010-2-51-0': 'Table 2.', 'nucl-th-0302010-2-51-1': 'Correlation energy as function of [MATH] at [MATH].', 'nucl-th-0302010-2-52-0': 'Table 3.', 'nucl-th-0302010-2-52-1': 'Excitation energy of the first addition mode as function of [MATH] at [MATH].', 'nucl-th-0302010-2-53-0': 'Table 4.', 'nucl-th-0302010-2-53-1': 'Excitation energy of the first addition mode as function of [MATH] at [MATH].'}

[['nucl-th-0302010-1-16-0', 'nucl-th-0302010-2-16-0'], ['nucl-th-0302010-1-16-1', 'nucl-th-0302010-2-16-1'], ['nucl-th-0302010-1-16-2', 'nucl-th-0302010-2-16-2'], ['nucl-th-0302010-1-16-3', 'nucl-th-0302010-2-16-3'], ['nucl-th-0302010-1-16-4', 'nucl-th-0302010-2-16-4'], ['nucl-th-0302010-1-16-5', 'nucl-th-0302010-2-16-5'], ['nucl-th-0302010-1-6-0', 'nucl-th-0302010-2-6-0'], ['nucl-th-0302010-1-6-1', 'nucl-th-0302010-2-6-1'], ['nucl-th-0302010-1-6-2', 'nucl-th-0302010-2-6-2'], ['nucl-th-0302010-1-6-3', 'nucl-th-0302010-2-6-3'], ['nucl-th-0302010-1-7-0', 'nucl-th-0302010-2-7-0'], ['nucl-th-0302010-1-7-1', 'nucl-th-0302010-2-7-1'], ['nucl-th-0302010-1-7-2', 'nucl-th-0302010-2-7-2'], ['nucl-th-0302010-1-7-3', 'nucl-th-0302010-2-7-3'], ['nucl-th-0302010-1-7-4', 'nucl-th-0302010-2-7-4'], ['nucl-th-0302010-1-7-5', 'nucl-th-0302010-2-7-5'], ['nucl-th-0302010-1-7-6', 'nucl-th-0302010-2-7-6'], ['nucl-th-0302010-1-29-0', 'nucl-th-0302010-2-29-0'], ['nucl-th-0302010-1-29-1', 'nucl-th-0302010-2-29-1'], ['nucl-th-0302010-1-29-2', 'nucl-th-0302010-2-29-2'], ['nucl-th-0302010-1-29-3', 'nucl-th-0302010-2-29-3'], ['nucl-th-0302010-1-49-0', 'nucl-th-0302010-2-49-0'], ['nucl-th-0302010-1-49-1', 'nucl-th-0302010-2-49-1'], ['nucl-th-0302010-1-49-2', 'nucl-th-0302010-2-49-2'], ['nucl-th-0302010-1-49-3', 'nucl-th-0302010-2-49-3'], ['nucl-th-0302010-1-49-4', 'nucl-th-0302010-2-49-4'], ['nucl-th-0302010-1-49-5', 'nucl-th-0302010-2-49-5'], ['nucl-th-0302010-1-49-6', 'nucl-th-0302010-2-49-6'], ['nucl-th-0302010-1-49-7', 'nucl-th-0302010-2-49-7'], ['nucl-th-0302010-1-49-8', 'nucl-th-0302010-2-49-8'], ['nucl-th-0302010-1-49-9', 'nucl-th-0302010-2-49-9'], ['nucl-th-0302010-1-49-10', 'nucl-th-0302010-2-49-10'], ['nucl-th-0302010-1-19-0', 'nucl-th-0302010-2-19-0'], ['nucl-th-0302010-1-19-1', 'nucl-th-0302010-2-19-1'], ['nucl-th-0302010-1-19-2', 'nucl-th-0302010-2-19-2'], ['nucl-th-0302010-1-19-3', 'nucl-th-0302010-2-19-3'], ['nucl-th-0302010-1-20-0', 'nucl-th-0302010-2-20-0'], ['nucl-th-0302010-1-20-1', 'nucl-th-0302010-2-20-1'], ['nucl-th-0302010-1-20-2', 'nucl-th-0302010-2-20-2'], ['nucl-th-0302010-1-20-3', 'nucl-th-0302010-2-20-3'], ['nucl-th-0302010-1-20-4', 'nucl-th-0302010-2-20-4'], ['nucl-th-0302010-1-20-5', 'nucl-th-0302010-2-20-5'], ['nucl-th-0302010-1-20-6', 'nucl-th-0302010-2-20-6'], ['nucl-th-0302010-1-20-7', 'nucl-th-0302010-2-20-7'], ['nucl-th-0302010-1-34-0', 'nucl-th-0302010-2-34-0'], ['nucl-th-0302010-1-34-1', 'nucl-th-0302010-2-34-1'], ['nucl-th-0302010-1-48-0', 'nucl-th-0302010-2-48-0'], ['nucl-th-0302010-1-48-1', 'nucl-th-0302010-2-48-1'], ['nucl-th-0302010-1-48-2', 'nucl-th-0302010-2-48-2'], ['nucl-th-0302010-1-48-3', 'nucl-th-0302010-2-48-3'], ['nucl-th-0302010-1-48-4', 'nucl-th-0302010-2-48-4'], ['nucl-th-0302010-1-48-5', 'nucl-th-0302010-2-48-5'], ['nucl-th-0302010-1-48-6', 'nucl-th-0302010-2-48-6'], ['nucl-th-0302010-1-12-0', 'nucl-th-0302010-2-12-0'], ['nucl-th-0302010-1-12-1', 'nucl-th-0302010-2-12-1'], ['nucl-th-0302010-1-12-2', 'nucl-th-0302010-2-12-2'], ['nucl-th-0302010-1-12-3', 'nucl-th-0302010-2-12-3'], ['nucl-th-0302010-1-3-0', 'nucl-th-0302010-2-3-0'], ['nucl-th-0302010-1-3-1', 'nucl-th-0302010-2-3-1'], ['nucl-th-0302010-1-3-2', 'nucl-th-0302010-2-3-2'], ['nucl-th-0302010-1-3-4', 'nucl-th-0302010-2-3-4'], ['nucl-th-0302010-1-3-5', 'nucl-th-0302010-2-3-5'], ['nucl-th-0302010-1-3-6', 'nucl-th-0302010-2-3-6'], ['nucl-th-0302010-1-3-7', 'nucl-th-0302010-2-3-7'], ['nucl-th-0302010-1-3-8', 'nucl-th-0302010-2-3-8'], ['nucl-th-0302010-1-3-9', 'nucl-th-0302010-2-3-9'], ['nucl-th-0302010-1-18-0', 'nucl-th-0302010-2-18-0'], ['nucl-th-0302010-1-18-1', 'nucl-th-0302010-2-18-1'], ['nucl-th-0302010-1-18-2', 'nucl-th-0302010-2-18-2'], ['nucl-th-0302010-1-18-3', 'nucl-th-0302010-2-18-3'], ['nucl-th-0302010-1-18-4', 'nucl-th-0302010-2-18-4'], ['nucl-th-0302010-1-8-1', 'nucl-th-0302010-2-8-1'], ['nucl-th-0302010-1-8-2', 'nucl-th-0302010-2-8-2'], ['nucl-th-0302010-1-31-0', 'nucl-th-0302010-2-31-0'], ['nucl-th-0302010-1-31-1', 'nucl-th-0302010-2-31-1'], ['nucl-th-0302010-1-31-2', 'nucl-th-0302010-2-31-2'], ['nucl-th-0302010-1-31-3', 'nucl-th-0302010-2-31-3'], ['nucl-th-0302010-1-31-4', 'nucl-th-0302010-2-31-4'], ['nucl-th-0302010-1-31-5', 'nucl-th-0302010-2-31-5'], ['nucl-th-0302010-1-31-6', 'nucl-th-0302010-2-31-6'], ['nucl-th-0302010-1-31-7', 'nucl-th-0302010-2-31-7'], ['nucl-th-0302010-1-53-1', 'nucl-th-0302010-2-53-1'], ['nucl-th-0302010-1-4-0', 'nucl-th-0302010-2-4-0'], ['nucl-th-0302010-1-4-1', 'nucl-th-0302010-2-4-1'], ['nucl-th-0302010-1-4-2', 'nucl-th-0302010-2-4-2'], ['nucl-th-0302010-1-4-3', 'nucl-th-0302010-2-4-3'], ['nucl-th-0302010-1-4-4', 'nucl-th-0302010-2-4-4'], ['nucl-th-0302010-1-4-5', 'nucl-th-0302010-2-4-5'], ['nucl-th-0302010-1-4-6', 'nucl-th-0302010-2-4-6'], ['nucl-th-0302010-1-14-0', 'nucl-th-0302010-2-14-0'], ['nucl-th-0302010-1-14-1', 'nucl-th-0302010-2-14-1'], ['nucl-th-0302010-1-10-0', 'nucl-th-0302010-2-10-0'], ['nucl-th-0302010-1-10-1', 'nucl-th-0302010-2-10-1'], ['nucl-th-0302010-1-10-2', 'nucl-th-0302010-2-10-2'], ['nucl-th-0302010-1-10-3', 'nucl-th-0302010-2-10-3'], ['nucl-th-0302010-1-10-4', 'nucl-th-0302010-2-10-4'], ['nucl-th-0302010-1-10-5', 'nucl-th-0302010-2-10-5'], ['nucl-th-0302010-1-10-6', 'nucl-th-0302010-2-10-6'], ['nucl-th-0302010-1-10-7', 'nucl-th-0302010-2-10-7'], ['nucl-th-0302010-1-41-0', 'nucl-th-0302010-2-41-0'], ['nucl-th-0302010-1-41-1', 'nucl-th-0302010-2-41-1'], ['nucl-th-0302010-1-52-1', 'nucl-th-0302010-2-52-1'], ['nucl-th-0302010-1-24-0', 'nucl-th-0302010-2-24-0'], ['nucl-th-0302010-1-24-1', 'nucl-th-0302010-2-24-1'], ['nucl-th-0302010-1-24-2', 'nucl-th-0302010-2-24-2'], ['nucl-th-0302010-1-24-3', 'nucl-th-0302010-2-24-3'], ['nucl-th-0302010-1-24-4', 'nucl-th-0302010-2-24-4'], ['nucl-th-0302010-1-24-5', 'nucl-th-0302010-2-24-5'], ['nucl-th-0302010-1-9-0', 'nucl-th-0302010-2-9-0'], ['nucl-th-0302010-1-9-1', 'nucl-th-0302010-2-9-1'], ['nucl-th-0302010-1-37-0', 'nucl-th-0302010-2-37-0'], ['nucl-th-0302010-1-37-1', 'nucl-th-0302010-2-37-1'], ['nucl-th-0302010-1-39-0', 'nucl-th-0302010-2-39-0'], ['nucl-th-0302010-1-39-1', 'nucl-th-0302010-2-39-1'], ['nucl-th-0302010-1-39-2', 'nucl-th-0302010-2-39-2'], ['nucl-th-0302010-1-39-3', 'nucl-th-0302010-2-39-3'], ['nucl-th-0302010-1-39-4', 'nucl-th-0302010-2-39-4'], ['nucl-th-0302010-1-39-5', 'nucl-th-0302010-2-39-5'], ['nucl-th-0302010-1-39-6', 'nucl-th-0302010-2-39-6'], ['nucl-th-0302010-1-39-7', 'nucl-th-0302010-2-39-7'], ['nucl-th-0302010-1-39-8', 'nucl-th-0302010-2-39-8'], ['nucl-th-0302010-1-39-9', 'nucl-th-0302010-2-39-9'], ['nucl-th-0302010-1-39-10', 'nucl-th-0302010-2-39-10'], ['nucl-th-0302010-1-39-11', 'nucl-th-0302010-2-39-11'], ['nucl-th-0302010-1-39-12', 'nucl-th-0302010-2-39-12'], ['nucl-th-0302010-1-39-13', 'nucl-th-0302010-2-39-13'], ['nucl-th-0302010-1-39-14', 'nucl-th-0302010-2-39-14'], ['nucl-th-0302010-1-39-15', 'nucl-th-0302010-2-39-15'], ['nucl-th-0302010-1-39-16', 'nucl-th-0302010-2-39-16'], ['nucl-th-0302010-1-39-17', 'nucl-th-0302010-2-39-17'], ['nucl-th-0302010-1-39-18', 'nucl-th-0302010-2-39-18'], ['nucl-th-0302010-1-39-19', 'nucl-th-0302010-2-39-19'], ['nucl-th-0302010-1-39-20', 'nucl-th-0302010-2-39-20'], ['nucl-th-0302010-1-43-0', 'nucl-th-0302010-2-43-0'], ['nucl-th-0302010-1-15-0', 'nucl-th-0302010-2-15-0'], ['nucl-th-0302010-1-15-1', 'nucl-th-0302010-2-15-1'], ['nucl-th-0302010-1-15-2', 'nucl-th-0302010-2-15-2'], ['nucl-th-0302010-1-15-3', 'nucl-th-0302010-2-15-3'], ['nucl-th-0302010-1-15-4', 'nucl-th-0302010-2-15-4'], ['nucl-th-0302010-1-15-5', 'nucl-th-0302010-2-15-5'], ['nucl-th-0302010-1-21-0', 'nucl-th-0302010-2-21-0'], ['nucl-th-0302010-1-21-1', 'nucl-th-0302010-2-21-1'], ['nucl-th-0302010-1-28-0', 'nucl-th-0302010-2-28-0'], ['nucl-th-0302010-1-28-1', 'nucl-th-0302010-2-28-1'], ['nucl-th-0302010-1-28-2', 'nucl-th-0302010-2-28-2'], ['nucl-th-0302010-1-28-3', 'nucl-th-0302010-2-28-3'], ['nucl-th-0302010-1-28-4', 'nucl-th-0302010-2-28-4'], ['nucl-th-0302010-1-28-5', 'nucl-th-0302010-2-28-5'], ['nucl-th-0302010-1-28-6', 'nucl-th-0302010-2-28-6'], ['nucl-th-0302010-1-28-7', 'nucl-th-0302010-2-28-7'], ['nucl-th-0302010-1-32-0', 'nucl-th-0302010-2-32-0'], ['nucl-th-0302010-1-32-1', 'nucl-th-0302010-2-32-1'], ['nucl-th-0302010-1-32-2', 'nucl-th-0302010-2-32-2'], ['nucl-th-0302010-1-32-3', 'nucl-th-0302010-2-32-3'], ['nucl-th-0302010-1-32-4', 'nucl-th-0302010-2-32-4'], ['nucl-th-0302010-1-32-5', 'nucl-th-0302010-2-32-5'], ['nucl-th-0302010-1-32-6', 'nucl-th-0302010-2-32-6'], ['nucl-th-0302010-1-36-0', 'nucl-th-0302010-2-36-0'], ['nucl-th-0302010-1-36-1', 'nucl-th-0302010-2-36-1'], ['nucl-th-0302010-1-36-2', 'nucl-th-0302010-2-36-2'], ['nucl-th-0302010-1-36-3', 'nucl-th-0302010-2-36-3'], ['nucl-th-0302010-1-36-4', 'nucl-th-0302010-2-36-4'], ['nucl-th-0302010-1-36-5', 'nucl-th-0302010-2-36-5'], ['nucl-th-0302010-1-36-6', 'nucl-th-0302010-2-36-6'], ['nucl-th-0302010-1-36-7', 'nucl-th-0302010-2-36-7'], ['nucl-th-0302010-1-36-8', 'nucl-th-0302010-2-36-8'], ['nucl-th-0302010-1-36-9', 'nucl-th-0302010-2-36-9'], ['nucl-th-0302010-1-45-0', 'nucl-th-0302010-2-45-0'], ['nucl-th-0302010-1-45-1', 'nucl-th-0302010-2-45-1'], ['nucl-th-0302010-1-45-2', 'nucl-th-0302010-2-45-2'], ['nucl-th-0302010-1-45-4', 'nucl-th-0302010-2-45-4'], ['nucl-th-0302010-1-45-5', 'nucl-th-0302010-2-45-5'], ['nucl-th-0302010-1-45-6', 'nucl-th-0302010-2-45-6'], ['nucl-th-0302010-1-45-7', 'nucl-th-0302010-2-45-7'], ['nucl-th-0302010-1-45-8', 'nucl-th-0302010-2-45-8'], ['nucl-th-0302010-1-27-0', 'nucl-th-0302010-2-27-0'], ['nucl-th-0302010-1-27-1', 'nucl-th-0302010-2-27-1'], ['nucl-th-0302010-1-27-2', 'nucl-th-0302010-2-27-2'], ['nucl-th-0302010-1-27-3', 'nucl-th-0302010-2-27-3'], ['nucl-th-0302010-1-27-4', 'nucl-th-0302010-2-27-4'], ['nucl-th-0302010-1-27-5', 'nucl-th-0302010-2-27-5'], ['nucl-th-0302010-1-27-6', 'nucl-th-0302010-2-27-6'], ['nucl-th-0302010-1-27-7', 'nucl-th-0302010-2-27-7'], ['nucl-th-0302010-1-27-8', 'nucl-th-0302010-2-27-8'], ['nucl-th-0302010-1-35-0', 'nucl-th-0302010-2-35-0'], ['nucl-th-0302010-1-35-1', 'nucl-th-0302010-2-35-1'], ['nucl-th-0302010-1-35-2', 'nucl-th-0302010-2-35-2'], ['nucl-th-0302010-1-35-3', 'nucl-th-0302010-2-35-3'], ['nucl-th-0302010-1-35-4', 'nucl-th-0302010-2-35-4'], ['nucl-th-0302010-1-35-5', 'nucl-th-0302010-2-35-5'], ['nucl-th-0302010-1-35-6', 'nucl-th-0302010-2-35-6'], ['nucl-th-0302010-1-46-0', 'nucl-th-0302010-2-46-0'], ['nucl-th-0302010-1-46-1', 'nucl-th-0302010-2-46-1'], ['nucl-th-0302010-1-46-2', 'nucl-th-0302010-2-46-2'], ['nucl-th-0302010-1-46-3', 'nucl-th-0302010-2-46-3'], ['nucl-th-0302010-1-46-4', 'nucl-th-0302010-2-46-4'], ['nucl-th-0302010-1-46-5', 'nucl-th-0302010-2-46-5'], ['nucl-th-0302010-1-46-6', 'nucl-th-0302010-2-46-6'], ['nucl-th-0302010-1-46-7', 'nucl-th-0302010-2-46-7'], ['nucl-th-0302010-1-46-8', 'nucl-th-0302010-2-46-8'], ['nucl-th-0302010-1-46-9', 'nucl-th-0302010-2-46-9'], ['nucl-th-0302010-1-46-10', 'nucl-th-0302010-2-46-10'], ['nucl-th-0302010-1-47-0', 'nucl-th-0302010-2-47-0'], ['nucl-th-0302010-1-47-1', 'nucl-th-0302010-2-47-1'], ['nucl-th-0302010-1-47-2', 'nucl-th-0302010-2-47-2'], ['nucl-th-0302010-1-47-3', 'nucl-th-0302010-2-47-3'], ['nucl-th-0302010-1-47-4', 'nucl-th-0302010-2-47-4'], ['nucl-th-0302010-1-47-5', 'nucl-th-0302010-2-47-5'], ['nucl-th-0302010-1-47-6', 'nucl-th-0302010-2-47-6'], ['nucl-th-0302010-1-47-7', 'nucl-th-0302010-2-47-7'], ['nucl-th-0302010-1-22-0', 'nucl-th-0302010-2-22-0'], ['nucl-th-0302010-1-22-1', 'nucl-th-0302010-2-22-1'], ['nucl-th-0302010-1-22-2', 'nucl-th-0302010-2-22-2'], ['nucl-th-0302010-1-22-3', 'nucl-th-0302010-2-22-3'], ['nucl-th-0302010-1-22-4', 'nucl-th-0302010-2-22-4'], ['nucl-th-0302010-1-30-0', 'nucl-th-0302010-2-30-0'], ['nucl-th-0302010-1-30-1', 'nucl-th-0302010-2-30-1'], ['nucl-th-0302010-1-30-2', 'nucl-th-0302010-2-30-2'], ['nucl-th-0302010-1-30-3', 'nucl-th-0302010-2-30-3'], ['nucl-th-0302010-1-40-0', 'nucl-th-0302010-2-40-0'], ['nucl-th-0302010-1-40-1', 'nucl-th-0302010-2-40-1'], ['nucl-th-0302010-1-0-0', 'nucl-th-0302010-2-0-0'], ['nucl-th-0302010-1-0-1', 'nucl-th-0302010-2-0-1'], ['nucl-th-0302010-1-0-2', 'nucl-th-0302010-2-0-2'], ['nucl-th-0302010-1-0-3', 'nucl-th-0302010-2-0-3'], ['nucl-th-0302010-1-0-4', 'nucl-th-0302010-2-0-4'], ['nucl-th-0302010-1-0-5', 'nucl-th-0302010-2-0-5'], ['nucl-th-0302010-1-2-0', 'nucl-th-0302010-2-2-0'], ['nucl-th-0302010-1-2-1', 'nucl-th-0302010-2-2-1'], ['nucl-th-0302010-1-2-2', 'nucl-th-0302010-2-2-2'], ['nucl-th-0302010-1-2-3', 'nucl-th-0302010-2-2-3'], ['nucl-th-0302010-1-2-4', 'nucl-th-0302010-2-2-4'], ['nucl-th-0302010-1-2-5', 'nucl-th-0302010-2-2-5'], ['nucl-th-0302010-1-2-6', 'nucl-th-0302010-2-2-6'], ['nucl-th-0302010-1-2-7', 'nucl-th-0302010-2-2-7'], ['nucl-th-0302010-1-2-8', 'nucl-th-0302010-2-2-8'], ['nucl-th-0302010-1-2-9', 'nucl-th-0302010-2-2-9'], ['nucl-th-0302010-1-26-0', 'nucl-th-0302010-2-26-0'], ['nucl-th-0302010-1-26-1', 'nucl-th-0302010-2-26-1'], ['nucl-th-0302010-1-26-2', 'nucl-th-0302010-2-26-2'], ['nucl-th-0302010-1-26-3', 'nucl-th-0302010-2-26-3'], ['nucl-th-0302010-1-26-4', 'nucl-th-0302010-2-26-4'], ['nucl-th-0302010-1-26-5', 'nucl-th-0302010-2-26-5'], ['nucl-th-0302010-1-26-6', 'nucl-th-0302010-2-26-6']]

[['nucl-th-0302010-1-16-0', 'nucl-th-0302010-2-16-0'], ['nucl-th-0302010-1-16-1', 'nucl-th-0302010-2-16-1'], ['nucl-th-0302010-1-16-2', 'nucl-th-0302010-2-16-2'], ['nucl-th-0302010-1-16-3', 'nucl-th-0302010-2-16-3'], ['nucl-th-0302010-1-16-4', 'nucl-th-0302010-2-16-4'], ['nucl-th-0302010-1-16-5', 'nucl-th-0302010-2-16-5'], ['nucl-th-0302010-1-6-0', 'nucl-th-0302010-2-6-0'], ['nucl-th-0302010-1-6-1', 'nucl-th-0302010-2-6-1'], ['nucl-th-0302010-1-6-2', 'nucl-th-0302010-2-6-2'], ['nucl-th-0302010-1-6-3', 'nucl-th-0302010-2-6-3'], ['nucl-th-0302010-1-7-0', 'nucl-th-0302010-2-7-0'], ['nucl-th-0302010-1-7-1', 'nucl-th-0302010-2-7-1'], ['nucl-th-0302010-1-7-2', 'nucl-th-0302010-2-7-2'], ['nucl-th-0302010-1-7-3', 'nucl-th-0302010-2-7-3'], ['nucl-th-0302010-1-7-4', 'nucl-th-0302010-2-7-4'], ['nucl-th-0302010-1-7-5', 'nucl-th-0302010-2-7-5'], ['nucl-th-0302010-1-7-6', 'nucl-th-0302010-2-7-6'], ['nucl-th-0302010-1-29-0', 'nucl-th-0302010-2-29-0'], ['nucl-th-0302010-1-29-1', 'nucl-th-0302010-2-29-1'], ['nucl-th-0302010-1-29-2', 'nucl-th-0302010-2-29-2'], ['nucl-th-0302010-1-29-3', 'nucl-th-0302010-2-29-3'], ['nucl-th-0302010-1-49-0', 'nucl-th-0302010-2-49-0'], ['nucl-th-0302010-1-49-1', 'nucl-th-0302010-2-49-1'], ['nucl-th-0302010-1-49-2', 'nucl-th-0302010-2-49-2'], ['nucl-th-0302010-1-49-3', 'nucl-th-0302010-2-49-3'], ['nucl-th-0302010-1-49-4', 'nucl-th-0302010-2-49-4'], ['nucl-th-0302010-1-49-5', 'nucl-th-0302010-2-49-5'], ['nucl-th-0302010-1-49-6', 'nucl-th-0302010-2-49-6'], ['nucl-th-0302010-1-49-7', 'nucl-th-0302010-2-49-7'], ['nucl-th-0302010-1-49-8', 'nucl-th-0302010-2-49-8'], ['nucl-th-0302010-1-49-9', 'nucl-th-0302010-2-49-9'], ['nucl-th-0302010-1-49-10', 'nucl-th-0302010-2-49-10'], ['nucl-th-0302010-1-19-0', 'nucl-th-0302010-2-19-0'], ['nucl-th-0302010-1-19-1', 'nucl-th-0302010-2-19-1'], ['nucl-th-0302010-1-19-2', 'nucl-th-0302010-2-19-2'], ['nucl-th-0302010-1-19-3', 'nucl-th-0302010-2-19-3'], ['nucl-th-0302010-1-20-0', 'nucl-th-0302010-2-20-0'], ['nucl-th-0302010-1-20-1', 'nucl-th-0302010-2-20-1'], ['nucl-th-0302010-1-20-2', 'nucl-th-0302010-2-20-2'], ['nucl-th-0302010-1-20-3', 'nucl-th-0302010-2-20-3'], ['nucl-th-0302010-1-20-4', 'nucl-th-0302010-2-20-4'], ['nucl-th-0302010-1-20-5', 'nucl-th-0302010-2-20-5'], ['nucl-th-0302010-1-20-6', 'nucl-th-0302010-2-20-6'], ['nucl-th-0302010-1-20-7', 'nucl-th-0302010-2-20-7'], ['nucl-th-0302010-1-34-0', 'nucl-th-0302010-2-34-0'], ['nucl-th-0302010-1-34-1', 'nucl-th-0302010-2-34-1'], ['nucl-th-0302010-1-48-0', 'nucl-th-0302010-2-48-0'], ['nucl-th-0302010-1-48-1', 'nucl-th-0302010-2-48-1'], ['nucl-th-0302010-1-48-2', 'nucl-th-0302010-2-48-2'], ['nucl-th-0302010-1-48-3', 'nucl-th-0302010-2-48-3'], ['nucl-th-0302010-1-48-4', 'nucl-th-0302010-2-48-4'], ['nucl-th-0302010-1-48-5', 'nucl-th-0302010-2-48-5'], ['nucl-th-0302010-1-48-6', 'nucl-th-0302010-2-48-6'], ['nucl-th-0302010-1-12-0', 'nucl-th-0302010-2-12-0'], ['nucl-th-0302010-1-12-1', 'nucl-th-0302010-2-12-1'], ['nucl-th-0302010-1-12-2', 'nucl-th-0302010-2-12-2'], ['nucl-th-0302010-1-12-3', 'nucl-th-0302010-2-12-3'], ['nucl-th-0302010-1-3-0', 'nucl-th-0302010-2-3-0'], ['nucl-th-0302010-1-3-1', 'nucl-th-0302010-2-3-1'], ['nucl-th-0302010-1-3-2', 'nucl-th-0302010-2-3-2'], ['nucl-th-0302010-1-3-4', 'nucl-th-0302010-2-3-4'], ['nucl-th-0302010-1-3-5', 'nucl-th-0302010-2-3-5'], ['nucl-th-0302010-1-3-6', 'nucl-th-0302010-2-3-6'], ['nucl-th-0302010-1-3-7', 'nucl-th-0302010-2-3-7'], ['nucl-th-0302010-1-3-8', 'nucl-th-0302010-2-3-8'], ['nucl-th-0302010-1-3-9', 'nucl-th-0302010-2-3-9'], ['nucl-th-0302010-1-18-0', 'nucl-th-0302010-2-18-0'], ['nucl-th-0302010-1-18-1', 'nucl-th-0302010-2-18-1'], ['nucl-th-0302010-1-18-2', 'nucl-th-0302010-2-18-2'], ['nucl-th-0302010-1-18-3', 'nucl-th-0302010-2-18-3'], ['nucl-th-0302010-1-18-4', 'nucl-th-0302010-2-18-4'], ['nucl-th-0302010-1-8-1', 'nucl-th-0302010-2-8-1'], ['nucl-th-0302010-1-8-2', 'nucl-th-0302010-2-8-2'], ['nucl-th-0302010-1-31-0', 'nucl-th-0302010-2-31-0'], ['nucl-th-0302010-1-31-1', 'nucl-th-0302010-2-31-1'], ['nucl-th-0302010-1-31-2', 'nucl-th-0302010-2-31-2'], ['nucl-th-0302010-1-31-3', 'nucl-th-0302010-2-31-3'], ['nucl-th-0302010-1-31-4', 'nucl-th-0302010-2-31-4'], ['nucl-th-0302010-1-31-5', 'nucl-th-0302010-2-31-5'], ['nucl-th-0302010-1-31-6', 'nucl-th-0302010-2-31-6'], ['nucl-th-0302010-1-31-7', 'nucl-th-0302010-2-31-7'], ['nucl-th-0302010-1-53-1', 'nucl-th-0302010-2-53-1'], ['nucl-th-0302010-1-4-0', 'nucl-th-0302010-2-4-0'], ['nucl-th-0302010-1-4-1', 'nucl-th-0302010-2-4-1'], ['nucl-th-0302010-1-4-2', 'nucl-th-0302010-2-4-2'], ['nucl-th-0302010-1-4-3', 'nucl-th-0302010-2-4-3'], ['nucl-th-0302010-1-4-4', 'nucl-th-0302010-2-4-4'], ['nucl-th-0302010-1-4-5', 'nucl-th-0302010-2-4-5'], ['nucl-th-0302010-1-4-6', 'nucl-th-0302010-2-4-6'], ['nucl-th-0302010-1-14-0', 'nucl-th-0302010-2-14-0'], ['nucl-th-0302010-1-14-1', 'nucl-th-0302010-2-14-1'], ['nucl-th-0302010-1-10-0', 'nucl-th-0302010-2-10-0'], ['nucl-th-0302010-1-10-1', 'nucl-th-0302010-2-10-1'], ['nucl-th-0302010-1-10-2', 'nucl-th-0302010-2-10-2'], ['nucl-th-0302010-1-10-3', 'nucl-th-0302010-2-10-3'], ['nucl-th-0302010-1-10-4', 'nucl-th-0302010-2-10-4'], ['nucl-th-0302010-1-10-5', 'nucl-th-0302010-2-10-5'], ['nucl-th-0302010-1-10-6', 'nucl-th-0302010-2-10-6'], ['nucl-th-0302010-1-10-7', 'nucl-th-0302010-2-10-7'], ['nucl-th-0302010-1-41-0', 'nucl-th-0302010-2-41-0'], ['nucl-th-0302010-1-41-1', 'nucl-th-0302010-2-41-1'], ['nucl-th-0302010-1-52-1', 'nucl-th-0302010-2-52-1'], ['nucl-th-0302010-1-24-0', 'nucl-th-0302010-2-24-0'], ['nucl-th-0302010-1-24-1', 'nucl-th-0302010-2-24-1'], ['nucl-th-0302010-1-24-2', 'nucl-th-0302010-2-24-2'], ['nucl-th-0302010-1-24-3', 'nucl-th-0302010-2-24-3'], ['nucl-th-0302010-1-24-4', 'nucl-th-0302010-2-24-4'], ['nucl-th-0302010-1-24-5', 'nucl-th-0302010-2-24-5'], ['nucl-th-0302010-1-9-0', 'nucl-th-0302010-2-9-0'], ['nucl-th-0302010-1-9-1', 'nucl-th-0302010-2-9-1'], ['nucl-th-0302010-1-37-0', 'nucl-th-0302010-2-37-0'], ['nucl-th-0302010-1-37-1', 'nucl-th-0302010-2-37-1'], ['nucl-th-0302010-1-39-0', 'nucl-th-0302010-2-39-0'], ['nucl-th-0302010-1-39-1', 'nucl-th-0302010-2-39-1'], ['nucl-th-0302010-1-39-2', 'nucl-th-0302010-2-39-2'], ['nucl-th-0302010-1-39-3', 'nucl-th-0302010-2-39-3'], ['nucl-th-0302010-1-39-4', 'nucl-th-0302010-2-39-4'], ['nucl-th-0302010-1-39-5', 'nucl-th-0302010-2-39-5'], ['nucl-th-0302010-1-39-6', 'nucl-th-0302010-2-39-6'], ['nucl-th-0302010-1-39-7', 'nucl-th-0302010-2-39-7'], ['nucl-th-0302010-1-39-8', 'nucl-th-0302010-2-39-8'], ['nucl-th-0302010-1-39-9', 'nucl-th-0302010-2-39-9'], ['nucl-th-0302010-1-39-10', 'nucl-th-0302010-2-39-10'], ['nucl-th-0302010-1-39-11', 'nucl-th-0302010-2-39-11'], ['nucl-th-0302010-1-39-12', 'nucl-th-0302010-2-39-12'], ['nucl-th-0302010-1-39-13', 'nucl-th-0302010-2-39-13'], ['nucl-th-0302010-1-39-14', 'nucl-th-0302010-2-39-14'], ['nucl-th-0302010-1-39-15', 'nucl-th-0302010-2-39-15'], ['nucl-th-0302010-1-39-16', 'nucl-th-0302010-2-39-16'], ['nucl-th-0302010-1-39-17', 'nucl-th-0302010-2-39-17'], ['nucl-th-0302010-1-39-18', 'nucl-th-0302010-2-39-18'], ['nucl-th-0302010-1-39-19', 'nucl-th-0302010-2-39-19'], ['nucl-th-0302010-1-39-20', 'nucl-th-0302010-2-39-20'], ['nucl-th-0302010-1-43-0', 'nucl-th-0302010-2-43-0'], ['nucl-th-0302010-1-15-0', 'nucl-th-0302010-2-15-0'], ['nucl-th-0302010-1-15-1', 'nucl-th-0302010-2-15-1'], ['nucl-th-0302010-1-15-2', 'nucl-th-0302010-2-15-2'], ['nucl-th-0302010-1-15-3', 'nucl-th-0302010-2-15-3'], ['nucl-th-0302010-1-15-4', 'nucl-th-0302010-2-15-4'], ['nucl-th-0302010-1-15-5', 'nucl-th-0302010-2-15-5'], ['nucl-th-0302010-1-21-0', 'nucl-th-0302010-2-21-0'], ['nucl-th-0302010-1-21-1', 'nucl-th-0302010-2-21-1'], ['nucl-th-0302010-1-28-0', 'nucl-th-0302010-2-28-0'], ['nucl-th-0302010-1-28-1', 'nucl-th-0302010-2-28-1'], ['nucl-th-0302010-1-28-2', 'nucl-th-0302010-2-28-2'], ['nucl-th-0302010-1-28-3', 'nucl-th-0302010-2-28-3'], ['nucl-th-0302010-1-28-4', 'nucl-th-0302010-2-28-4'], ['nucl-th-0302010-1-28-5', 'nucl-th-0302010-2-28-5'], ['nucl-th-0302010-1-28-6', 'nucl-th-0302010-2-28-6'], ['nucl-th-0302010-1-28-7', 'nucl-th-0302010-2-28-7'], ['nucl-th-0302010-1-32-0', 'nucl-th-0302010-2-32-0'], ['nucl-th-0302010-1-32-1', 'nucl-th-0302010-2-32-1'], ['nucl-th-0302010-1-32-2', 'nucl-th-0302010-2-32-2'], ['nucl-th-0302010-1-32-3', 'nucl-th-0302010-2-32-3'], ['nucl-th-0302010-1-32-4', 'nucl-th-0302010-2-32-4'], ['nucl-th-0302010-1-32-5', 'nucl-th-0302010-2-32-5'], ['nucl-th-0302010-1-32-6', 'nucl-th-0302010-2-32-6'], ['nucl-th-0302010-1-36-0', 'nucl-th-0302010-2-36-0'], ['nucl-th-0302010-1-36-1', 'nucl-th-0302010-2-36-1'], ['nucl-th-0302010-1-36-2', 'nucl-th-0302010-2-36-2'], ['nucl-th-0302010-1-36-3', 'nucl-th-0302010-2-36-3'], ['nucl-th-0302010-1-36-4', 'nucl-th-0302010-2-36-4'], ['nucl-th-0302010-1-36-5', 'nucl-th-0302010-2-36-5'], ['nucl-th-0302010-1-36-6', 'nucl-th-0302010-2-36-6'], ['nucl-th-0302010-1-36-7', 'nucl-th-0302010-2-36-7'], ['nucl-th-0302010-1-36-8', 'nucl-th-0302010-2-36-8'], ['nucl-th-0302010-1-36-9', 'nucl-th-0302010-2-36-9'], ['nucl-th-0302010-1-45-0', 'nucl-th-0302010-2-45-0'], ['nucl-th-0302010-1-45-1', 'nucl-th-0302010-2-45-1'], ['nucl-th-0302010-1-45-2', 'nucl-th-0302010-2-45-2'], ['nucl-th-0302010-1-45-4', 'nucl-th-0302010-2-45-4'], ['nucl-th-0302010-1-45-5', 'nucl-th-0302010-2-45-5'], ['nucl-th-0302010-1-45-6', 'nucl-th-0302010-2-45-6'], ['nucl-th-0302010-1-45-7', 'nucl-th-0302010-2-45-7'], ['nucl-th-0302010-1-45-8', 'nucl-th-0302010-2-45-8'], ['nucl-th-0302010-1-27-0', 'nucl-th-0302010-2-27-0'], ['nucl-th-0302010-1-27-1', 'nucl-th-0302010-2-27-1'], ['nucl-th-0302010-1-27-2', 'nucl-th-0302010-2-27-2'], ['nucl-th-0302010-1-27-3', 'nucl-th-0302010-2-27-3'], ['nucl-th-0302010-1-27-4', 'nucl-th-0302010-2-27-4'], ['nucl-th-0302010-1-27-5', 'nucl-th-0302010-2-27-5'], ['nucl-th-0302010-1-27-6', 'nucl-th-0302010-2-27-6'], ['nucl-th-0302010-1-27-7', 'nucl-th-0302010-2-27-7'], ['nucl-th-0302010-1-27-8', 'nucl-th-0302010-2-27-8'], ['nucl-th-0302010-1-35-0', 'nucl-th-0302010-2-35-0'], ['nucl-th-0302010-1-35-1', 'nucl-th-0302010-2-35-1'], ['nucl-th-0302010-1-35-2', 'nucl-th-0302010-2-35-2'], ['nucl-th-0302010-1-35-3', 'nucl-th-0302010-2-35-3'], ['nucl-th-0302010-1-35-4', 'nucl-th-0302010-2-35-4'], ['nucl-th-0302010-1-35-5', 'nucl-th-0302010-2-35-5'], ['nucl-th-0302010-1-35-6', 'nucl-th-0302010-2-35-6'], ['nucl-th-0302010-1-46-0', 'nucl-th-0302010-2-46-0'], ['nucl-th-0302010-1-46-1', 'nucl-th-0302010-2-46-1'], ['nucl-th-0302010-1-46-2', 'nucl-th-0302010-2-46-2'], ['nucl-th-0302010-1-46-3', 'nucl-th-0302010-2-46-3'], ['nucl-th-0302010-1-46-4', 'nucl-th-0302010-2-46-4'], ['nucl-th-0302010-1-46-5', 'nucl-th-0302010-2-46-5'], ['nucl-th-0302010-1-46-6', 'nucl-th-0302010-2-46-6'], ['nucl-th-0302010-1-46-7', 'nucl-th-0302010-2-46-7'], ['nucl-th-0302010-1-46-8', 'nucl-th-0302010-2-46-8'], ['nucl-th-0302010-1-46-9', 'nucl-th-0302010-2-46-9'], ['nucl-th-0302010-1-46-10', 'nucl-th-0302010-2-46-10'], ['nucl-th-0302010-1-47-0', 'nucl-th-0302010-2-47-0'], ['nucl-th-0302010-1-47-1', 'nucl-th-0302010-2-47-1'], ['nucl-th-0302010-1-47-2', 'nucl-th-0302010-2-47-2'], ['nucl-th-0302010-1-47-3', 'nucl-th-0302010-2-47-3'], ['nucl-th-0302010-1-47-4', 'nucl-th-0302010-2-47-4'], ['nucl-th-0302010-1-47-5', 'nucl-th-0302010-2-47-5'], ['nucl-th-0302010-1-47-6', 'nucl-th-0302010-2-47-6'], ['nucl-th-0302010-1-47-7', 'nucl-th-0302010-2-47-7'], ['nucl-th-0302010-1-22-0', 'nucl-th-0302010-2-22-0'], ['nucl-th-0302010-1-22-1', 'nucl-th-0302010-2-22-1'], ['nucl-th-0302010-1-22-2', 'nucl-th-0302010-2-22-2'], ['nucl-th-0302010-1-22-3', 'nucl-th-0302010-2-22-3'], ['nucl-th-0302010-1-22-4', 'nucl-th-0302010-2-22-4'], ['nucl-th-0302010-1-30-0', 'nucl-th-0302010-2-30-0'], ['nucl-th-0302010-1-30-1', 'nucl-th-0302010-2-30-1'], ['nucl-th-0302010-1-30-2', 'nucl-th-0302010-2-30-2'], ['nucl-th-0302010-1-30-3', 'nucl-th-0302010-2-30-3'], ['nucl-th-0302010-1-40-0', 'nucl-th-0302010-2-40-0'], ['nucl-th-0302010-1-40-1', 'nucl-th-0302010-2-40-1'], ['nucl-th-0302010-1-0-0', 'nucl-th-0302010-2-0-0'], ['nucl-th-0302010-1-0-1', 'nucl-th-0302010-2-0-1'], ['nucl-th-0302010-1-0-2', 'nucl-th-0302010-2-0-2'], ['nucl-th-0302010-1-0-3', 'nucl-th-0302010-2-0-3'], ['nucl-th-0302010-1-0-4', 'nucl-th-0302010-2-0-4'], ['nucl-th-0302010-1-0-5', 'nucl-th-0302010-2-0-5'], ['nucl-th-0302010-1-2-0', 'nucl-th-0302010-2-2-0'], ['nucl-th-0302010-1-2-1', 'nucl-th-0302010-2-2-1'], ['nucl-th-0302010-1-2-2', 'nucl-th-0302010-2-2-2'], ['nucl-th-0302010-1-2-3', 'nucl-th-0302010-2-2-3'], ['nucl-th-0302010-1-2-4', 'nucl-th-0302010-2-2-4'], ['nucl-th-0302010-1-2-5', 'nucl-th-0302010-2-2-5'], ['nucl-th-0302010-1-2-6', 'nucl-th-0302010-2-2-6'], ['nucl-th-0302010-1-2-7', 'nucl-th-0302010-2-2-7'], ['nucl-th-0302010-1-2-8', 'nucl-th-0302010-2-2-8'], ['nucl-th-0302010-1-2-9', 'nucl-th-0302010-2-2-9'], ['nucl-th-0302010-1-26-0', 'nucl-th-0302010-2-26-0'], ['nucl-th-0302010-1-26-1', 'nucl-th-0302010-2-26-1'], ['nucl-th-0302010-1-26-2', 'nucl-th-0302010-2-26-2'], ['nucl-th-0302010-1-26-3', 'nucl-th-0302010-2-26-3'], ['nucl-th-0302010-1-26-4', 'nucl-th-0302010-2-26-4'], ['nucl-th-0302010-1-26-5', 'nucl-th-0302010-2-26-5'], ['nucl-th-0302010-1-26-6', 'nucl-th-0302010-2-26-6']]

[]

[]

[]

[]

['nucl-th-0302010-1-3-3', 'nucl-th-0302010-1-8-0', 'nucl-th-0302010-1-14-2', 'nucl-th-0302010-1-42-0', 'nucl-th-0302010-1-44-0', 'nucl-th-0302010-1-45-3', 'nucl-th-0302010-1-50-0', 'nucl-th-0302010-1-50-1', 'nucl-th-0302010-1-51-0', 'nucl-th-0302010-1-51-1', 'nucl-th-0302010-1-52-0', 'nucl-th-0302010-1-53-0', 'nucl-th-0302010-2-3-3', 'nucl-th-0302010-2-8-0', 'nucl-th-0302010-2-14-2', 'nucl-th-0302010-2-42-0', 'nucl-th-0302010-2-44-0', 'nucl-th-0302010-2-45-3', 'nucl-th-0302010-2-50-0', 'nucl-th-0302010-2-50-1', 'nucl-th-0302010-2-51-0', 'nucl-th-0302010-2-51-1', 'nucl-th-0302010-2-52-0', 'nucl-th-0302010-2-53-0']

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

https://arxiv.org/abs/nucl-th/0302010

0910.5029

{'0910.5029-1-0-0': 'Small neutrino masses and their large mixing angles can be generated at the TeV scale by augmenting the Standard Model with an additional generation dependent, anomaly-free [MATH] symmetry, in the presence of three right-handed neutrinos.', '0910.5029-1-0-1': 'The [MATH] gauge boson associated with the breaking of the [MATH] symmetry can be produced at the LHC.', '0910.5029-1-0-2': "The flavorful nature of the [MATH] can be established by measuring its non-universal couplings to the charged leptons as determined by the lepton's [MATH] charges, which also govern the neutrino flavor structure.", '0910.5029-1-0-3': 'While the LHC has the potential of discovering the [MATH] up to [MATH] TeV with 100 fb[MATH] data at [MATH] TeV, to establish the flavorful nature of the [MATH] requires much higher integrated luminosity.', '0910.5029-1-0-4': 'For our bench mark parameters that are consistent with neutrino oscillation data, at the center of mass energy [MATH] TeV, a [MATH] distinction between the dielectron and dimuon channels for [MATH] TeV requires 500 fb[MATH] of data.', '0910.5029-1-0-5': 'We find that the forward backward asymmetry distributions can also be useful in distinguishing the dielectron and dimuon channels in the low invariant mass and transverse momentum regions.', '0910.5029-1-1-0': '# Introduction', '0910.5029-1-2-0': 'The [MATH] gauge boson [CITATION] associated with the breaking of a [MATH] symmetry is predicted in many extensions of the Standard Model, such as the left-right symmetric model [CITATION], Grand Unified Theories (GUT) based on SO(10) [CITATION] and E6 [CITATION], and string inspired models [CITATION].', '0910.5029-1-2-1': 'If the [MATH] is flavor universal, it can only be the [MATH] in order to ensure the cancellation of all gauge anomalies.', '0910.5029-1-2-2': 'By allowing the SM fermions to have generation dependent [MATH] charges, the [MATH], which differs from [MATH], may play the role of a family symmetry which gives rise to fermion mass hierarchy and mixing patterns a la the Froggatt-Nielson mechanism [CITATION].', '0910.5029-1-2-3': 'While [CITATION] found that to have realistic fermion mass and mixing patterns the [MATH] must be anomalous (and thus the [MATH] scale must be close to the string scale), more recent studies [CITATION] have shown that realistic fermion mass and mixing patterns can arise from a non-anomalous [MATH] symmetry, and thus allowing the [MATH] scale to be as low as a TeV.', '0910.5029-1-2-4': 'In addition, all gauge anomalies can be cancelled with no exotic fields other than the right-handed neutrinos.', '0910.5029-1-3-0': 'In the [MATH] model [CITATION], which is based on the SM augmented with an additional generation-dependent, non- anomalous [MATH], small neutrino masses and their large mixing angles are generated at the TeV scale with all Yukawa coupling constants of the order of unity.', '0910.5029-1-3-1': 'Since the leptons are allowed to have generation dependent [MATH] charges in this model, the [MATH] gauge boson, which can be produced at the LHC, couples to the leptons non-universally.', '0910.5029-1-3-2': 'Even though the [MATH] couplings to the fermions are non-universal, all flavor changing neutral currents (FCNCs) [CITATION] can still be satisfied.', '0910.5029-1-3-3': '(More details are given in later sections.)', '0910.5029-1-3-4': "In this paper, we study the LHC's discovery potential of the [MATH] in the [MATH] model.", '0910.5029-1-3-5': 'Specifically we concentrate on the capability of the LHC to distinguishing the [MATH] charges of the leptons, which is important in differentiating the generation-dependent [MATH] from the flavor universal case [CITATION].', '0910.5029-1-3-6': 'In addition to the searches of the resonance in the invariant mass distribution of the dilepton channels [MATH]), we also study the prospects of using the forward backward asymmetry distributions to distinguish different dilepton channels.', '0910.5029-1-4-0': 'The paper is organized as follows.', '0910.5029-1-4-1': 'In Section [REF], we briefly review the anomaly-free [MATH] model.', '0910.5029-1-4-2': 'Section [REF] shows the discovery potential of the [MATH] in the [MATH] model at the LHC as well as the capability of the LHC to distinguish different dilepton decay channels using the forwar backward asymmetry distributions.', '0910.5029-1-4-3': 'In Section [REF], a more general allowed parameter space for a generation dependent anomaly-free [MATH] model is given.', '0910.5029-1-4-4': 'Section [REF] concludes the paper.', '0910.5029-1-5-0': '# The TeV Scale Seesaw Model With A Non-anomalous [MATH] Symmetry', '0910.5029-1-6-0': 'The [MATH] model extends the SM gauge group by an additional [MATH] symmetry.', '0910.5029-1-6-1': 'Because the SM fermions are allowed to have generation dependent [MATH] charges, the non-anomalous [MATH] symmetry can be different from [MATH].', '0910.5029-1-6-2': 'All gauge anomalies are cancelled in the model with no exotic fermions other than the three right-handed neutrinos, [MATH]).', '0910.5029-1-6-3': 'The [MATH] symmetry is broken at the TeV scale by the vacuum expectation value (VEV) of the scalar field, [MATH], which has [MATH] charge of [MATH] and it is a singlet under the SM gauge group.', '0910.5029-1-7-0': 'In the conventional seesaw mechanism [CITATION], the seesaw scale is generally on the order of [MATH] GeV, which is close to the grand unification scale, making it inaccessible to the collider experiments.', '0910.5029-1-7-1': 'In the TeV scale seesaw model [CITATION] we consider, due to the [MATH] symmetry, the usual dimension-4 and dimension-5 operators for neutrino masses are forbidden; neutrino masses are generated by operators with higher dimensionality, thus allowing the seesaw scale to be lowered to a TeV.', '0910.5029-1-7-2': 'The complete Yukawa Lagrangian in the lepton sector is given as follows, [EQUATION] where [MATH] and [MATH] denote the left-handed and right-handed charged leptons, respectively, [MATH] is the SM Higgs, and [MATH].', '0910.5029-1-7-3': 'The matrices [MATH] are the Yukawa couplings with [MATH] and [MATH] being the generation indices for leptons, while [MATH] is the cut-off scale of [MATH].', '0910.5029-1-7-4': 'When the [MATH] field gets a VEV, the [MATH] symmetry is spontaneously broken, and all lepton masses are generated.', '0910.5029-1-7-5': 'Neutrino masses are suppressed by powers of [MATH].', '0910.5029-1-7-6': 'If the powers are large enough, with [MATH], the cutoff scale [MATH] can be as low as a TeV.', '0910.5029-1-8-0': "The resulting mass matrices in the lepton sector are characterized by the exponent matrices, which are determined by the fermions' [MATH] charges, [MATH].", '0910.5029-1-8-1': 'Assuming that the quark and charged lepton masses are generated through dimension-4 operators, using gauge invariance and the six anomaly cancellation conditions, these mass matrices are determined by five independent parameters, [MATH] and [MATH] [CITATION].', '0910.5029-1-8-2': 'We further require that the second and third generations of charged leptons have the same charges and similarly for the right-handed neutrinos ([MATH]), as motivated by the maximal mixing between [MATH] and [MATH].', '0910.5029-1-8-3': 'Since the [MATH] field is the only source of the [MATH] breaking, it follows that the Higgs charge vanishes, [MATH].', '0910.5029-1-8-4': 'Consequently, there is no mixing between the SM [MATH] and the [MATH], and it naturally satisfies the experimental constraints from the electroweak precision measurements, which requires the mixing between [MATH] and [MATH] to be smaller than [MATH] [CITATION].', '0910.5029-1-8-5': 'The [MATH] charges of various fields are summarized in Table [REF].', '0910.5029-1-9-0': 'The exponent matrices that parametrize the neutrino left-handed Majorana, Dirac, and right-handed Majorana mass matrices are given by, [EQUATION] respectively, where [MATH].', '0910.5029-1-9-1': 'The exponent matrix for the charged lepton mass matrix is given by, [EQUATION]', '0910.5029-1-9-2': 'Note that only elements which are integers are allowed in the exponent matrices, since the numbers of flavon field involved in the Froggatt-Nielson diagrams must be integers [CITATION].', '0910.5029-1-9-3': 'Also for the case of negative integer elements, we insert [MATH] field which has the [MATH] charge of -1 instead of [MATH] field defined in the Eq. [REF].', '0910.5029-1-9-4': 'By having non-integer exponents thus provides a way to generate texture zeros in the mass matrices.', '0910.5029-1-9-5': 'With this, neutrinos in this model can either be Dirac or Majorana fermions and both normal and inverted mass orderings can be accommodated.', '0910.5029-1-10-0': 'By allowing generation dependent lepton charges, the FCNCs may arise through the (1, 2) and (1, 3) off-diagonal elements in the charged lepton mass matrix.', '0910.5029-1-10-1': 'However, we can avoid the FCNCs by having non-integer [MATH].', '0910.5029-1-10-2': 'In fact, integer [MATH] is allowed as long as it is large enough to satisfy all of the experimental constraints on FCNCs as the branching fractions of these processes are roughly proportional to [MATH].', '0910.5029-1-11-0': 'The gauge boson [MATH] couples to the fermions through [EQUATION] in which [MATH] is the [MATH] gauge coupling constant.', '0910.5029-1-11-1': 'And the gauge couplings of the [MATH] to the fermions are determined by the [MATH] charges of the fermions.', '0910.5029-1-11-2': 'In the lepton sector, these charges also dictate the neutrino mixing patterns.', '0910.5029-1-11-3': 'Given that its mass [MATH] is on the order of a TeV, the [MATH] can be produced at the LHC.', '0910.5029-1-11-4': 'The [MATH] model can thus be tested at the collider experiments through the dilepton decay channels.', '0910.5029-1-12-0': '# Collider Signatures of the [MATH] Model', '0910.5029-1-13-0': 'In the [MATH] model, the SM leptons can have generation dependent [MATH] charges, leading to non-universal couplings to the [MATH].', '0910.5029-1-13-1': 'We investigate the capability of various kinematic variables to distinguish the charged lepton [MATH] charges through the dilepton decay channels, [MATH].', '0910.5029-1-13-2': 'As a sample model point, we set [MATH] and [MATH], which gives quark and lepton [MATH] charges as summarized in Table [REF].', '0910.5029-1-13-3': 'In addition to being free of all gauge anomalies, these charges lead to realistic neutrino mass and mixing patterns, and satisfy all FCNC constraints.', '0910.5029-1-13-4': 'Furthermore, we take the gauge coupling parameter [MATH] in the analyses presented hereafter.', '0910.5029-1-13-5': 'In the example given above, neutrinos are Dirac particles since only the terms related to the matrix p contribute to the masses of neutrinos.', '0910.5029-1-13-6': 'The predicted absolute neutrino masses are, [MATH], [MATH] eV, [MATH] eV with [MATH].', '0910.5029-1-13-7': 'These masses satisfy both experimental limits on [MATH] and [MATH], and they are of the inverted hierarchical ordering.', '0910.5029-1-14-0': '## The [MATH] Discovery Potential at the LHC', '0910.5029-1-15-0': 'The [MATH] may be discovered by detecting excess signals from backgrounds near its resonance in the dilepton invariant mass distribution.', '0910.5029-1-15-1': 'We first calculate the leading order (LO) cross section of the exclusive [MATH] production, [MATH], at the LHC with the center of mass energy [MATH] TeV.', '0910.5029-1-15-2': 'Since the [MATH] decay width is narrow, we neglect the interference term between the [MATH] and the SM gauge bosons.', '0910.5029-1-15-3': 'The cross section [MATH] is given by, [EQUATION]', '0910.5029-1-15-4': 'Taking [MATH] and integrating out [MATH], with [MATH] being the angle between the incoming annihilating quark and outgoing negatively charged lepton in the dilepton center of mass frame, the total cross section is simplified to [EQUATION] and the total [MATH] decay width becomes [EQUATION] where [MATH] and [MATH] is the top quark mass.', '0910.5029-1-15-5': 'In Eq. [REF] and [REF], [MATH] and [MATH] are parton distribution functions for the protons with [MATH].', '0910.5029-1-15-6': 'For our choice of parameters, neutrinos are Dirac particles and thus the right-handed neutrinos in this model can be very light.', '0910.5029-1-15-7': 'This allows the [MATH] to decay into right-handed neutrino pairs; this decay channel has been included, along with the decay channel of [MATH] into the SM fermions, in the total decay width given by Eq. [REF].', '0910.5029-1-16-0': 'The current experimental limit for [MATH] is [MATH] (900 GeV-1 TeV) [CITATION].', '0910.5029-1-16-1': 'While CDF Run II has discovered at [MATH] significance an excess at the dielectron invariant mass of 240 GeV [CITATION], this excess has not been confirmed by D[MATH] nor has a similar excess been discovered in the dimuon channel.', '0910.5029-1-16-2': 'To be conservative, we consider in our analyses [MATH] TeV.', '0910.5029-1-17-0': 'The main backgrounds which can mimic the signal events [MATH] and [MATH], can be categorized into two types, the reducible and irreducible backgrounds.', '0910.5029-1-17-1': 'Without any selection cuts, the dominant QCD backgrounds are inclusive jets, [MATH]jets, [MATH], [MATH]jets, [MATH], [MATH]jets, and [MATH], which are caused by jets or photons faking the electrons or muons.', '0910.5029-1-17-2': 'These backgrounds can be reduced by applying lepton identification criteria and an isolation cut, which reject fake electrons with a rejection factor [MATH] in the case of jets and [MATH] in the case of photons [CITATION].', '0910.5029-1-17-3': 'The QCD jet backgrounds can be further reduced by applying additional geometrical acceptance of [MATH] [CITATION] and by imposing the transverse momentum cuts [CITATION] which require [MATH] GeV for electrons and [MATH] GeV for muons.', '0910.5029-1-17-4': 'Since the two leptons are more likely to fly back to back in the detector, we also require the azimuthal angle between the two outgoing leptons to be [MATH] (though this does not affect either the signals or backgrounds most of the time).', '0910.5029-1-17-5': 'After these selection cuts, which are summarized in Table [REF], the dominant backgrounds are the irreducible backgrounds, which mainly come from the SM Drell-Yan processes.', '0910.5029-1-17-6': 'Other processes like decay products from [MATH], [MATH], [MATH] and [MATH] etc., can be ignored since their cross sections are very small.', '0910.5029-1-17-7': 'Hence, in our study, [MATH] and [MATH] and only the SM Drell-Yan processes are simulated in PYTHIA [CITATION].', '0910.5029-1-17-8': 'To generate the SM Drell-Yan processes more efficiently, we also require [MATH] GeV.', '0910.5029-1-18-0': 'With these selection cuts, Fig. [REF] shows the the cross sections of [MATH] and [MATH] in the [MATH] model as a function of the [MATH] mass.', '0910.5029-1-18-1': 'The dilepton invariant mass distributions in the [MATH] and [MATH] channels for [MATH] TeV are shown in Fig [REF].', '0910.5029-1-18-2': 'The SM backgrounds, which are almost identical for both channels at the tree level since the gauge couplings of [MATH] and [MATH] to the SM fermions are universal, are highly suppressed in the [MATH] resonance region, allowing a clear distinction between the signals from the backgrounds.', '0910.5029-1-19-0': 'To deduce the [MATH] discovery potential, we use the usual counting method.', '0910.5029-1-19-1': 'Let [MATH] be the number of [MATH] signal events, which satisfies [MATH] and [MATH] be the number of SM Drell-Yan background events.', '0910.5029-1-19-2': 'Both [MATH] and [MATH] satisfy the aforementioned selection cuts.', '0910.5029-1-19-3': 'One can then define the variable [MATH], where [MATH] and [MATH] respectively are the total cross sections of the signal and background events, and [MATH] is the integrated luminosity.', '0910.5029-1-19-4': 'A [MATH] discovery corresponds to [MATH] while a [MATH] exclusion corresponds to [MATH].', '0910.5029-1-19-5': 'With [MATH] fb[MATH] of integrated luminosity and center of mass energy [MATH] TeV, there are 4151 signal events and 102 Drell-Yan background events for the case of [MATH] TeV in the dielectron channel.', '0910.5029-1-19-6': 'For the dimuon channel, the numbers of signal events and background events are 3104 and 108, respectively.', '0910.5029-1-19-7': 'In the case of [MATH] TeV and [MATH] TeV, with [MATH] pb[MATH] of data, there are 20 signal events in the dielectron channel and [MATH] signal events in the dimuon channel, both with less than 1 event from the SM backgrounds.', '0910.5029-1-19-8': 'Fig. [REF] shows the integrated luminosity required for a [MATH] discovery through both the electron and muon channels at [MATH] TeV.', '0910.5029-1-19-9': 'Since the counting method works for a large amount of events which satisfy the Gaussian distribution, we further require the number of signal events [MATH].', '0910.5029-1-19-10': 'Therefore, with [MATH] fb[MATH] of data, the LHC can discover a [MATH] with a mass up to 4.5 TeV at [MATH] TeV.', '0910.5029-1-20-0': '## Establishing the [MATH] Flavor Dependence with Decay Branching Fractions', '0910.5029-1-21-0': 'While there is a great potential for the [MATH] discovery at the LHC during its early low luminosity running as shown in the previous section, to establish the flavorful nature of the [MATH], after it has been discovered in the dilepton channels, turns out to be more difficult.', '0910.5029-1-21-1': 'Since in the [MATH] model, the electron and muon are allowed to have different [MATH] charges, the ratio of the decay branching fractions, [MATH], can in general differ from [MATH].', '0910.5029-1-21-2': 'The ratio [MATH] depends on the parameter, [MATH], which is given in terms of the charge parameters [MATH] and [MATH], as [EQUATION]', '0910.5029-1-21-3': 'If the [MATH] is also discovered in the [MATH] channel, by measuring [MATH] and the ratio of the decay branching fractions of [MATH] to [MATH] channels, [EQUATION] a unique determination of [MATH] can be obtained (see Fig. [REF]).', '0910.5029-1-22-0': 'It is to be noted that when [MATH], the branching fraction of [MATH] vanishes, as the electron is not charged under the [MATH] in this case.', '0910.5029-1-22-1': 'Consequently, if the [MATH] is discovered in the dimuon and (or) [MATH] channels, and if there is no evidence in the dielectron channel, this then suggests that the [MATH] symmetry is flavor dependent.', '0910.5029-1-22-2': 'Similarly, if [MATH], then the left-handed muon is not charged under the [MATH].', '0910.5029-1-22-3': 'Thus one would expect a discovery in the dielectron channel, but not in the dimuon channel.', '0910.5029-1-22-4': 'In principle, if we can detect and measure all [MATH] decay channels precisely enough, we can determine the parameters [MATH] and [MATH], which are the parameters that predict neutrino mixing angles, mass ordering and their Dirac versus Majorana nature.', '0910.5029-1-23-0': 'Our bench mark point, [MATH] and [MATH], predicts a ratio of the branching fractions of [MATH].', '0910.5029-1-23-1': 'To study the integrated luminosity required for distinguishing the [MATH] and [MATH] channels at [MATH] using the counting method, we define the following variable, [EQUATION] where [MATH] and [MATH] are the cross sections of the [MATH] and [MATH] events.', '0910.5029-1-23-2': 'The parameter [MATH] thus gives the difference in the numbers of events between [MATH] and [MATH].', '0910.5029-1-23-3': 'If [MATH], the difference between the dielectron and dimuon events is above [MATH] significance.', '0910.5029-1-23-4': 'Similar to the requirement of [MATH] discussed previously, we also require [MATH].', '0910.5029-1-23-5': 'The integrated luminosity required as a function of the [MATH] mass is shown in Fig. [REF].', '0910.5029-1-23-6': 'At [MATH] TeV with [MATH] fb[MATH] of data, a statistically significant distinction between the branching fractions for [MATH] and [MATH] channels can be obtained up to [MATH] TeV.', '0910.5029-1-23-7': 'For the case of [MATH] TeV, to distinguish the dielectron and dimuon channels, more data are needed.', '0910.5029-1-23-8': 'For [MATH] TeV, to get [MATH] and [MATH], an integrated luminosity of [MATH] fb[MATH] is needed.', '0910.5029-1-24-0': '## Establishing the [MATH] Flavor Dependence with Forward Backward Asymmetry Distributions', '0910.5029-1-25-0': 'Even though the flavorful nature of the [MATH] can be determined by measuring the ratio of branching fractions, [MATH], as discussed above, since the ratio of branching fractions depends on the square of the charged lepton [MATH] charges, the measurement of [MATH] cannot tell the relative sign between [MATH] and [MATH].', '0910.5029-1-25-1': 'On the other hand, the sign of the [MATH] charge can be determined by measuring the forward backward asymmetry distribution, [MATH], as [MATH] is due to the interference between the [MATH] and the SM gauge bosons, Z and [MATH], which can be small, especially near the [MATH] resonance as discussed in Section [REF].', '0910.5029-1-25-2': 'Hence, this measurement requires a large amount of data, which may be possible at the SLHC [CITATION] which allows an instantaneous luminosity of [MATH] cm[MATH]s[MATH].', '0910.5029-1-26-0': 'The forward backward asymmetry, [MATH], is defined as, [EQUATION] where the total cross sections of the forward and backward events are given by [EQUATION] and the angle [MATH] is defined as the angle between the direction of the outgoing negatively charged lepton with respect to the direction of the incoming annihilating quark in the lab frame.', '0910.5029-1-26-1': 'Note that this is different from the usual [MATH] angle which is the angle defined in the center of mass frame, and the angles [MATH] and [MATH] are related by [EQUATION]', '0910.5029-1-26-2': 'We can thus calculate the cross section in the center of mass frame (the general formula can be found in [CITATION]) and transform it from the center of mass frame to the lab frame (the formula can be found in [CITATION]).', '0910.5029-1-27-0': 'Due to an intrinsic ambiguity in deciding which proton the quark comes from, it is still tricky to determine [MATH].', '0910.5029-1-27-1': 'Inspired by [CITATION], we therefore define [MATH] with [MATH] being the angle between the direction of the negatively charged lepton with respect to the direction of the proton which contributes the annihilating quark in the lab frame.', '0910.5029-1-27-2': 'Since experimentally, it is hard to determine which proton contributes to the quark, due to the fact that a quark in the proton typically carries a larger momentum fraction than does an antiquark, we assume that the longitudinal motion of the dilepton system is in the direction of the proton which contributes the annililating quark in the lab frame [CITATION].', '0910.5029-1-28-0': 'To understand how well [MATH] describes the variable [MATH], we compare the lego plot of [MATH] versus [MATH] (Fig. [REF]) and the lego plot of [MATH] versus [MATH] (Fig. [REF]), where [MATH] is the Monte Calo true angle between the direction of the negatively charged lepton and that of the annihilating quark.', '0910.5029-1-28-1': 'In these two figures, the events generated include the full interference structure of [MATH]/Z/[MATH] and satisfy the transverse momentum cut [MATH] GeV or [MATH] GeV, back to back selection cut [MATH], and the requirement that the invariant mass of the dilepton system is within the range of 500 GeV to 2000 GeV.', '0910.5029-1-28-2': 'We find that [MATH] can be used as a good approximation for [MATH].', '0910.5029-1-28-3': 'Although those two figures look very similar, the mistagging rate (that is, [MATH] having a wrong sign compared to [MATH]) is still high at the low [MATH] region, as shown in Fig. [REF].', '0910.5029-1-28-4': 'With an additional cut of [MATH], the mistagging rate can be reduced to a reasonable level ([MATH]) without losing too much data.', '0910.5029-1-28-5': 'Therefore, we use [MATH] to approximate [MATH] when extracting the forward backward asymmetry in the simulations with the requirement of [MATH].', '0910.5029-1-29-0': 'In the case of [MATH] TeV with 500 fb[MATH] of data, Fig. [REF] shows the forward backward asymmetry distribution as a function of the dilepton transverse momentum [MATH] for [MATH] TeV, while Fig. [REF] shows the forward backward asymmetry as a function of the dilepton invariant mass [MATH].', '0910.5029-1-29-1': 'The uncertainties correspond to the statistical errors normalized to [MATH] fb[MATH] of data.', '0910.5029-1-29-2': 'In the case of [MATH] TeV with 500 fb[MATH] of data, Fig. [REF] and [REF] show the forward backward asymmetry as a function of the transverse momentum and invariant mass, respectively, for [MATH] mass of 1 TeV.', '0910.5029-1-29-3': 'By using the forward backward asymmetry distributions, a clear distinction between the dielectron and dimuon channels can be obtained in the low invariant mass and low transverse momentum regions.', '0910.5029-1-29-4': 'As expected, the asymmetry distribution measurements require a large amount of data and thus can only be achieved in the later stages of the LHC operation.', '0910.5029-1-30-0': '# Allowed Region of Parameters [MATH] and [MATH]', '0910.5029-1-31-0': 'In the above analyses, we concentrate on only one bench mark point in the parameter space, which gives rise to realistic neutrino mass and mixing patterns.', '0910.5029-1-31-1': 'Without imposing the constraints from the neutrino sector, it is possible to have a more general generation dependent [MATH] model.', '0910.5029-1-31-2': 'Fig. [REF] shows the parameter space of [MATH] and [MATH] for which the [MATH] discovery is possible.', '0910.5029-1-31-3': 'Specifically, the values of [MATH] and [MATH] in the allowed region gives, [MATH].', '0910.5029-1-31-4': 'We require the [MATH] decay width to be larger than [MATH], so that the decay width is wide enough compared to the detector resolution, and thus enabling the [MATH] resonance to be observable.', '0910.5029-1-31-5': 'The upper limit of 0.1 for the ratio of the decay width to [MATH] is imposed to ensure the narrow width approximation is valid in our analysis.', '0910.5029-1-31-6': 'Generally, any TeV scale generation dependent [MATH] model suffer severe constraints from flavor changing neutral currents.', '0910.5029-1-31-7': 'This is easily avoided by choosing [MATH] to be a non-integer, which naturally gives the texture zeros in the (12), (13), (21) and (31) entries of the charged lepton mass matrix given by Eq. [REF].', '0910.5029-1-31-8': 'The FCNC constraints can also be avoided by having a large integral value for [MATH], so that the off diagonal elements in the charged lepton mass matrix are highly suppressed.', '0910.5029-1-31-9': 'A large parameter space remains even after removing points for which [MATH] is an integer.', '0910.5029-1-32-0': '# Conclusion', '0910.5029-1-33-0': 'We investigate the collider signatures of a TeV scale non-anomalous [MATH] model, which generates at the TeV scale small neutrino masses and their mixing angles.', '0910.5029-1-33-1': 'Since the [MATH] symmetry, which is different from [MATH], is generation dependent, all gauge anomalies are cancelled with no exotic fields other than the three right-handed neutrinos.', '0910.5029-1-33-2': "Specifically, we have investigated the LHC's discovery potential of such a flavorful [MATH].", '0910.5029-1-33-3': 'Through the excess in the dilepton invariant mass distribution, at the center of mass energy of [MATH] TeV, the [MATH] of mass up to 4.5 TeV can be discovered at [MATH] with 100 fb[MATH] of data.', '0910.5029-1-33-4': 'To establish the flavorful nature of [MATH] requires a distinction between the [MATH] and [MATH] channels.', '0910.5029-1-33-5': 'While it requires a much higher integrated luminosity, it is possible to distinguish these two decay channels at sufficient significance level.', '0910.5029-1-33-6': 'For the bench mark point we consider in our analysis, at [MATH] = 14 TeV, a [MATH] distinction between the [MATH] and [MATH] channels can be obtained with 500 fb[MATH] of data, for [MATH] up to 3 TeV.', '0910.5029-1-33-7': 'At [MATH] TeV, 8.32 fb[MATH] of integrated luminosity is required for [MATH] = 1 TeV.', '0910.5029-1-33-8': 'We have also studied the possibility of measuring the [MATH] charges of the leptons using the forward backward asymmetry distributions of the dilepton channels.', '0910.5029-1-33-9': 'With a sufficient integrated luminosity, it is possible to observe the different asymmetry distributions for the dieletron and dimuon channels in the low invariant mass and low transverse momentum regions.', '0910.5029-1-33-10': 'This thus allows to establish the generation-dependent nature of the [MATH] model.', '0910.5029-1-34-0': 'We thank Thomas G. Rizzo for the useful discussion.', '0910.5029-1-34-1': 'The work of M-CC was supported, in part, by the National Science Foundation under grant no.', '0910.5029-1-34-2': 'PHY-0709742.'}

{'0910.5029-2-0-0': 'Small neutrino masses and their large mixing angles can be generated at the TeV scale by augmenting the Standard Model with an additional generation dependent, anomaly-free [MATH] symmetry, in the presence of three right-handed neutrinos.', '0910.5029-2-0-1': 'The [MATH] gauge boson associated with the breaking of the [MATH] symmetry can be produced at the LHC.', '0910.5029-2-0-2': "The flavorful nature of the [MATH] can be established by measuring its non-universal couplings to the charged leptons as determined by the lepton's [MATH] charges, which also govern the neutrino flavor structure.", '0910.5029-2-0-3': 'While the LHC has the potential of discovering the [MATH] up to [MATH] TeV with 100 fb[MATH] data at the center of mass energy [MATH] TeV, to establish the flavorful nature of the [MATH] requires much higher integrated luminosity.', '0910.5029-2-0-4': 'For our bench mark parameters that are consistent with neutrino oscillation data, at [MATH] TeV, a [MATH] distinction between the dielectron and dimuon channels for [MATH] TeV requires 500 fb[MATH] of data.', '0910.5029-2-0-5': 'We find that the forward backward asymmetry distributions can also be useful in distinguishing the dielectron and dimuon channels in the low invariant mass and transverse momentum regions.', '0910.5029-2-1-0': '# Introduction', '0910.5029-2-2-0': 'The [MATH] gauge boson [CITATION] associated with the breaking of a [MATH] symmetry is predicted in many extensions of the Standard Model, such as the left-right symmetric model [CITATION], Grand Unified Theories (GUT) based on SO(10) [CITATION] and E6 [CITATION], and string inspired models [CITATION].', '0910.5029-2-2-1': 'If the [MATH] is flavor universal, it can only be the [MATH] in order to ensure the cancellation of all gauge anomalies.', '0910.5029-2-2-2': 'By allowing the SM fermions to have generation dependent [MATH] charges, the [MATH], which differs from [MATH], may play the role of a family symmetry which gives rise to fermion mass hierarchy and mixing patterns a la the Froggatt-Nielson mechanism [CITATION].', '0910.5029-2-2-3': 'While [CITATION] found that to have realistic fermion mass and mixing patterns the [MATH] must be anomalous (and thus the [MATH] scale must be close to the string scale), more recent studies [CITATION] have shown that realistic fermion mass and mixing patterns can arise from a non-anomalous [MATH] symmetry, and thus allowing the [MATH] scale to be as low as a TeV.', '0910.5029-2-2-4': 'In addition, all gauge anomalies can be cancelled with no exotic fields other than the right-handed neutrinos.', '0910.5029-2-3-0': 'In the [MATH] model [CITATION], which is based on the SM augmented with an additional generation-dependent, non- anomalous [MATH], small neutrino masses and their large mixing angles are generated at the TeV scale with all Yukawa coupling constants of the order of unity.', '0910.5029-2-3-1': 'Since the leptons are allowed to have generation dependent [MATH] charges in this model, the [MATH] gauge boson, which can be produced at the LHC, couples to the leptons non-universally.', '0910.5029-2-3-2': 'Even though the [MATH] couplings to the fermions are non-universal, all flavor changing neutral currents (FCNCs) [CITATION] can still be satisfied.', '0910.5029-2-3-3': '(More details are given in later sections.)', '0910.5029-2-3-4': "In this paper, we study the LHC's discovery potential of the [MATH] in the [MATH] model.", '0910.5029-2-3-5': 'Specifically we concentrate on the capability of the LHC to distinguishing the [MATH] charges of the leptons, which is important in differentiating the generation-dependent [MATH] from the flavor universal case [CITATION].', '0910.5029-2-3-6': 'In addition to the searches of the resonance in the invariant mass distribution of the dilepton channels [MATH]), we also study the prospects of using the forward backward asymmetry distributions to distinguish different dilepton channels.', '0910.5029-2-4-0': 'The paper is organized as follows.', '0910.5029-2-4-1': 'In Section [REF], we briefly review the anomaly-free [MATH] model.', '0910.5029-2-4-2': 'Section [REF] shows the discovery potential of the [MATH] in the [MATH] model at the LHC as well as the capability of the LHC to distinguish different dilepton decay channels using the forward backward asymmetry distributions.', '0910.5029-2-4-3': 'In Section [REF], a more general allowed parameter space for a generation dependent anomaly-free [MATH] model is given.', '0910.5029-2-4-4': 'Section [REF] concludes the paper.', '0910.5029-2-5-0': '# The TeV Scale Seesaw Model With A Non-anomalous [MATH] Symmetry', '0910.5029-2-6-0': 'The [MATH] model extends the SM gauge group by an additional [MATH] symmetry.', '0910.5029-2-6-1': 'Because the SM fermions are allowed to have generation dependent [MATH] charges, the non-anomalous [MATH] symmetry can be different from [MATH].', '0910.5029-2-6-2': 'All gauge anomalies are cancelled in the model with no exotic fermions other than the three right-handed neutrinos, [MATH]).', '0910.5029-2-6-3': 'The [MATH] symmetry is broken at the TeV scale by the vacuum expectation value (VEV) of the scalar field, [MATH], which has [MATH] charge of [MATH] and it is a singlet under the SM gauge group.', '0910.5029-2-7-0': 'In the conventional seesaw mechanism [CITATION], the seesaw scale is generally on the order of [MATH] GeV, which is close to the grand unification scale, making it inaccessible to the collider experiments.', '0910.5029-2-7-1': 'In the TeV scale seesaw model [CITATION] we consider, due to the [MATH] symmetry, the usual dimension-4 and dimension-5 operators for neutrino masses are forbidden; neutrino masses are generated by operators with higher dimensionality, thus allowing the seesaw scale to be lowered to a TeV.', '0910.5029-2-7-2': 'The complete Yukawa Lagrangian in the lepton sector is given as follows, [EQUATION] where [MATH] and [MATH] denote the left-handed and right-handed charged leptons, respectively, [MATH] is the SM Higgs, and [MATH].', '0910.5029-2-7-3': 'The matrices [MATH] are the Yukawa couplings with [MATH] and [MATH] being the generation indices for leptons, while [MATH] is the cut-off scale of [MATH].', '0910.5029-2-7-4': 'When the [MATH] field gets a VEV, the [MATH] symmetry is spontaneously broken, and all lepton masses are generated.', '0910.5029-2-7-5': 'Neutrino masses are suppressed by powers of [MATH].', '0910.5029-2-7-6': 'If the powers are large enough, with [MATH], the cutoff scale [MATH] can be as low as a TeV.', '0910.5029-2-8-0': "The resulting mass matrices in the lepton sector are characterized by the exponent matrices, which are determined by the fermions' [MATH] charges, [MATH].", '0910.5029-2-8-1': 'Assuming that the quark and charged lepton masses are generated through dimension-4 operators, using gauge invariance and the six anomaly cancellation conditions, these mass matrices are determined by five independent parameters, [MATH] and [MATH] [CITATION].', '0910.5029-2-8-2': 'We further require that the second and third generations of charged leptons have the same charges and similarly for the right-handed neutrinos ([MATH]), as motivated by the maximal mixing between [MATH] and [MATH].', '0910.5029-2-8-3': 'Since the [MATH] field is the only source of the [MATH] breaking, it follows that the Higgs charge [MATH] vanishes, [MATH].', '0910.5029-2-8-4': 'Consequently, there is no mixing between the SM [MATH] and the [MATH] at the tree level.', '0910.5029-2-8-5': 'The [MATH] mixing can be induced at one-loop through the the self-energy diagram, [MATH].', '0910.5029-2-8-6': 'This RG-induced [MATH] gives rise to the following contribution [CITATION] to the [MATH] parameter [EQUATION]', '0910.5029-2-8-7': 'Here the term [MATH] arises due to [MATH]; it depends on the fermionic [MATH] couplings, [MATH], the fermionic [MATH] coupling, [MATH], and the mass of the loop fermion, [MATH].', '0910.5029-2-8-8': 'The electroweak precision measurements require extra contribution to [MATH] parameter to be smaller than [MATH] [CITATION].', '0910.5029-2-8-9': 'While this in turn constrains our parameter space for [MATH] and [MATH], we have checked explicitly as shown in the subsequent sections that for [MATH] TeV, all parameters considered in our model can satisfy the experimental precision electroweak constraints.', '0910.5029-2-8-10': 'The [MATH] charges of various fields are summarized in Table [REF].', '0910.5029-2-9-0': 'The exponent matrices that parametrize the neutrino left-handed Majorana, Dirac, and right-handed Majorana mass matrices are given by, [EQUATION] respectively, where [MATH].', '0910.5029-2-9-1': 'The exponent matrix for the charged lepton mass matrix is given by, [EQUATION]', '0910.5029-2-9-2': 'Note that only elements which are integers are allowed in the exponent matrices, since the numbers of flavon field involved in the Froggatt-Nielson diagrams must be integers [CITATION].', '0910.5029-2-9-3': 'Also for the case of negative integer elements, we insert [MATH] field which has the [MATH] charge of -1 instead of [MATH] field defined in the Eq. [REF].', '0910.5029-2-9-4': 'By having non-integer exponents thus provides a way to generate texture zeros in the mass matrices.', '0910.5029-2-9-5': 'With this, neutrinos in this model can either be Dirac or Majorana fermions and both normal and inverted mass orderings can be accommodated.', '0910.5029-2-10-0': 'By allowing generation dependent lepton charges, tree-level FCNCs may arise through the (1, 2) and (1, 3) off-diagonal elements in the charged lepton mass matrix.', '0910.5029-2-10-1': 'In the case of [MATH] being an non-integer, all off-diagonal elements vanish and thus all tree-level FCNCs are avoided albeit having generation dependent lepton charges under the [MATH].', '0910.5029-2-10-2': 'Furthermore, an integral value for [MATH] is allowed as long as it is large enough to satisfy all experimental constraints on FCNCs, as the branching fractions of these processes, including [MATH], [MATH] conversion in nuclei, [MATH], and [MATH]hadron(s), are roughly proportional to [MATH].', '0910.5029-2-10-3': 'The gauge boson [MATH] couples to the fermions through [EQUATION] in which [MATH] is the [MATH] gauge coupling constant.', '0910.5029-2-10-4': 'And the gauge couplings of the [MATH] to the fermions are determined by the [MATH] charges of the fermions.', '0910.5029-2-10-5': 'In the lepton sector, these charges also dictate the neutrino mixing patterns.', '0910.5029-2-10-6': 'Given that its mass [MATH] is on the order of a TeV, the [MATH] can be produced at the LHC.', '0910.5029-2-10-7': 'The [MATH] model can thus be tested at the collider experiments through the dilepton decay channels.', '0910.5029-2-11-0': '# Collider Signatures of the [MATH] Model', '0910.5029-2-12-0': 'In the [MATH] model, the SM leptons can have generation dependent [MATH] charges, leading to non-universal couplings to the [MATH].', '0910.5029-2-12-1': 'We investigate the capability of various kinematic variables to distinguish the charged lepton [MATH] charges through the dilepton decay channels, [MATH].', '0910.5029-2-12-2': 'As a sample model point, we set [MATH] and [MATH], which gives quark and lepton [MATH] charges as summarized in Table [REF].', '0910.5029-2-12-3': 'In addition to being free of all gauge anomalies, these charges lead to realistic neutrino mass and mixing patterns, and satisfy all FCNC and electroweak precision measurement constraints.', '0910.5029-2-12-4': 'Furthermore, we take the gauge coupling parameter [MATH] in the analyses presented hereafter.', '0910.5029-2-12-5': 'In the example given above, neutrinos are Dirac particles since only the terms related to the matrix p contribute to the masses of neutrinos.', '0910.5029-2-12-6': 'The predicted absolute neutrino masses are, [MATH], [MATH] eV, [MATH] eV with [MATH].', '0910.5029-2-12-7': 'These masses satisfy both experimental limits on [MATH] and [MATH], and they are of the inverted hierarchical ordering.', '0910.5029-2-12-8': 'All tree-level FCNC processes are highly suppressed as these sample model parameters lead to a highly suppressed value for [MATH].', '0910.5029-2-12-9': 'For [MATH] TeV and [MATH], the one-loop contribution to [MATH] mixing for our bench mark model parameters is on the order of [MATH].', '0910.5029-2-12-10': 'This gives rise to [MATH] approximately [MATH], which satisfies the current experimental constraints.', '0910.5029-2-13-0': '## The [MATH] Discovery Potential at the LHC', '0910.5029-2-14-0': 'The [MATH] may be discovered by detecting excess signals from backgrounds near its resonance in the dilepton invariant mass distribution.', '0910.5029-2-14-1': 'We first calculate the leading order (LO) cross section of the exclusive [MATH] production, [MATH], at the LHC with the center of mass energy [MATH] TeV.', '0910.5029-2-14-2': 'Since the [MATH] decay width is narrow, we neglect the interference term between the [MATH] and the SM gauge bosons.', '0910.5029-2-14-3': 'The cross section [MATH] is given by, [EQUATION]', '0910.5029-2-14-4': 'Taking [MATH] and integrating out [MATH], with [MATH] being the angle between the incoming annihilating quark and outgoing negatively charged lepton in the dilepton center of mass frame, the total cross section is simplified to [EQUATION] and the total [MATH] decay width becomes [EQUATION] where [MATH] and [MATH] is the top quark mass.', '0910.5029-2-14-5': 'In Eq. [REF] and [REF], [MATH] and [MATH] are parton distribution functions for the protons with [MATH].', '0910.5029-2-14-6': 'For our choice of parameters, neutrinos are Dirac particles and thus the right-handed neutrinos in this model can be very light.', '0910.5029-2-14-7': 'This allows the [MATH] to decay into right-handed neutrino pairs; this decay channel has been included, along with the decay channel of [MATH] into the SM fermions, in the total decay width given by Eq. [REF].', '0910.5029-2-15-0': 'The current experimental limit for [MATH] is [MATH] (900 GeV-1 TeV) [CITATION].', '0910.5029-2-15-1': 'While CDF Run II has discovered at [MATH] significance an excess at the dielectron invariant mass of 240 GeV [CITATION], this excess has not been confirmed by D[MATH] nor has a similar excess been discovered in the dimuon channel.', '0910.5029-2-15-2': 'To be conservative, we consider in our analyses [MATH] TeV.', '0910.5029-2-16-0': 'The main backgrounds which can mimic the signal events [MATH] and [MATH], can be categorized into two types, the reducible and irreducible backgrounds.', '0910.5029-2-16-1': 'Without any selection cuts, the dominant QCD backgrounds are inclusive jets, [MATH]jets, [MATH], [MATH]jets, [MATH], [MATH]jets, and [MATH], which are caused by jets or photons faking the electrons or muons.', '0910.5029-2-16-2': 'These backgrounds can be reduced by applying lepton identification criteria and an isolation cut, which reject fake electrons with a rejection factor [MATH] in the case of jets and [MATH] in the case of photons [CITATION].', '0910.5029-2-16-3': 'The QCD jet backgrounds can be further reduced by applying additional geometrical acceptance of [MATH] [CITATION] and by imposing the transverse momentum cuts [CITATION] which require [MATH] GeV for electrons and [MATH] GeV for muons.', '0910.5029-2-16-4': 'Since the two leptons are more likely to fly back to back in the detector, we also require the azimuthal angle between the two outgoing leptons to be [MATH] (though this does not affect either the signals or backgrounds most of the time).', '0910.5029-2-16-5': 'After these selection cuts, which are summarized in Table [REF], the dominant backgrounds are the irreducible backgrounds, which mainly come from the SM Drell-Yan processes.', '0910.5029-2-16-6': 'Other processes like decay products from [MATH], [MATH], [MATH] and [MATH] etc., can be ignored since their cross sections are very small.', '0910.5029-2-16-7': 'Hence, in our study, [MATH] and [MATH] and only the SM Drell-Yan processes are simulated in PYTHIA [CITATION].', '0910.5029-2-16-8': 'To generate the SM Drell-Yan processes more efficiently, we also require [MATH] GeV.', '0910.5029-2-17-0': 'With these selection cuts, Fig. [REF] shows the the cross sections of [MATH] and [MATH] in the [MATH] model as a function of the [MATH] mass.', '0910.5029-2-17-1': 'The dilepton invariant mass distributions in the [MATH] and [MATH] channels for [MATH] TeV are shown in Fig [REF].', '0910.5029-2-17-2': 'The SM backgrounds, which are almost identical for both channels at the tree level since the gauge couplings of [MATH] and [MATH] to the SM fermions are universal, are highly suppressed in the [MATH] resonance region, allowing a clear distinction between the signals from the backgrounds.', '0910.5029-2-18-0': 'To deduce the [MATH] discovery potential, we use the usual counting method.', '0910.5029-2-18-1': 'Let [MATH] be the number of [MATH] signal events, which satisfies [MATH] and [MATH] be the number of SM Drell-Yan background events.', '0910.5029-2-18-2': 'Both [MATH] and [MATH] satisfy the aforementioned selection cuts.', '0910.5029-2-18-3': 'One can then define the variable [MATH], where [MATH] and [MATH] respectively are the total cross sections of the signal and background events, and [MATH] is the integrated luminosity.', '0910.5029-2-18-4': 'A [MATH] discovery corresponds to [MATH] while a [MATH] exclusion corresponds to [MATH].', '0910.5029-2-18-5': 'With [MATH] fb[MATH] of integrated luminosity and center of mass energy [MATH] TeV, there are 4151 signal events and 102 Drell-Yan background events for the case of [MATH] TeV in the dielectron channel.', '0910.5029-2-18-6': 'For the dimuon channel, the numbers of signal events and background events are 3104 and 108, respectively.', '0910.5029-2-18-7': 'In the case of [MATH] TeV and [MATH] TeV, with [MATH] pb[MATH] of data, there are 20 signal events in the dielectron channel and [MATH] signal events in the dimuon channel, both with less than 1 event from the SM backgrounds.', '0910.5029-2-18-8': 'Fig. [REF] shows the integrated luminosity required for a [MATH] discovery through both the electron and muon channels at [MATH] TeV.', '0910.5029-2-18-9': 'Since the counting method works for a large amount of events which satisfy the Gaussian distribution, we further require the number of signal events [MATH].', '0910.5029-2-18-10': 'Therefore, with [MATH] fb[MATH] of data, the LHC can discover a [MATH] with a mass up to 4.5 TeV at [MATH] TeV.', '0910.5029-2-19-0': '## Establishing the [MATH] Flavor Dependence with Decay Branching Fractions', '0910.5029-2-20-0': 'While there is a great potential for the [MATH] discovery at the LHC during its early low luminosity running as shown in the previous section, to establish the flavorful nature of the [MATH], after it has been discovered in the dilepton channels, turns out to be more difficult.', '0910.5029-2-20-1': 'Since in the [MATH] model, the electron and muon are allowed to have different [MATH] charges, the ratio of the decay branching fractions, [MATH], can in general differ from [MATH].', '0910.5029-2-20-2': 'The ratio [MATH] depends on the parameter, [MATH], which is given in terms of the charge parameters [MATH] and [MATH], as [EQUATION]', '0910.5029-2-20-3': 'If the [MATH] is also discovered in the [MATH] channel, by measuring [MATH] and the ratio of the decay branching fractions of [MATH] to [MATH] channels, [EQUATION] a unique determination of [MATH] can be obtained (see Fig. [REF]).', '0910.5029-2-21-0': 'It is to be noted that when [MATH], the branching fraction of [MATH] vanishes, as the electron is not charged under the [MATH] in this case.', '0910.5029-2-21-1': 'Consequently, if the [MATH] is discovered in the dimuon and (or) [MATH] channels, and if there is no evidence in the dielectron channel, this then suggests that the [MATH] symmetry is flavor dependent.', '0910.5029-2-21-2': 'Similarly, if [MATH], then the left-handed muon is not charged under the [MATH].', '0910.5029-2-21-3': 'Thus one would expect a discovery in the dielectron channel, but not in the dimuon channel.', '0910.5029-2-21-4': 'In principle, if we can detect and measure all [MATH] decay channels precisely enough, we can determine the parameters [MATH] and [MATH], which are the parameters that predict neutrino mixing angles, mass ordering and their Dirac versus Majorana nature.', '0910.5029-2-22-0': 'Our bench mark point, [MATH] and [MATH], predicts a ratio of the branching fractions of [MATH].', '0910.5029-2-22-1': 'To study the integrated luminosity required for distinguishing the [MATH] and [MATH] channels at [MATH] using the counting method, we define the following variable, [EQUATION] where [MATH] and [MATH] are the cross sections of the [MATH] and [MATH] events.', '0910.5029-2-22-2': 'The parameter [MATH] thus gives the difference in the numbers of events between [MATH] and [MATH].', '0910.5029-2-22-3': 'If [MATH], the difference between the dielectron and dimuon events is above [MATH] significance.', '0910.5029-2-22-4': 'Similar to the requirement of [MATH] discussed previously, we also require [MATH].', '0910.5029-2-22-5': 'The integrated luminosity required as a function of the [MATH] mass is shown in Fig. [REF].', '0910.5029-2-22-6': 'At [MATH] TeV with [MATH] fb[MATH] of data, a statistically significant distinction between the branching fractions for [MATH] and [MATH] channels can be obtained up to [MATH] TeV.', '0910.5029-2-22-7': 'For the case of [MATH] TeV, to distinguish the dielectron and dimuon channels, more data are needed.', '0910.5029-2-22-8': 'For [MATH] TeV, to get [MATH] and [MATH], an integrated luminosity of [MATH] fb[MATH] is needed.', '0910.5029-2-23-0': '## Establishing the [MATH] Flavor Dependence with Forward Backward Asymmetry Distributions', '0910.5029-2-24-0': 'Even though the flavorful nature of the [MATH] can be determined by measuring the ratio of branching fractions, [MATH], as discussed above, since the ratio of branching fractions depends on the square of the charged lepton [MATH] charges, the measurement of [MATH] cannot tell the relative sign between [MATH] and [MATH].', '0910.5029-2-24-1': 'On the other hand, the sign of the [MATH] charge can be determined by measuring the forward backward asymmetry distribution, [MATH], as [MATH] is due to the interference between the [MATH] and the SM gauge bosons, Z and [MATH], which can be small, especially near the [MATH] resonance as discussed in Section [REF].', '0910.5029-2-24-2': 'Hence, this measurement requires a large amount of data, which may be possible at the SLHC [CITATION] allowing an instantaneous luminosity of [MATH] cm[MATH]s[MATH].', '0910.5029-2-25-0': 'The forward backward asymmetry, [MATH], is defined as, [EQUATION] where the total cross sections of the forward and backward events are given by [EQUATION] and the angle [MATH] is defined as the angle between the direction of the outgoing negatively charged lepton with respect to the direction of the incoming annihilating quark in the lab frame.', '0910.5029-2-25-1': 'Note that this is different from the usual [MATH] angle which is the angle defined in the center of mass frame, and the angles [MATH] and [MATH] are related by [EQUATION]', '0910.5029-2-25-2': 'We can thus calculate the cross section in the center of mass frame (the general formula can be found in [CITATION]) and transform it from the center of mass frame to the lab frame (the formula can be found in [CITATION]).', '0910.5029-2-26-0': 'Due to an intrinsic ambiguity in deciding which proton the quark comes from, it is still tricky to determine [MATH].', '0910.5029-2-26-1': 'Inspired by [CITATION], we therefore define [MATH] with [MATH] being the angle between the direction of the negatively charged lepton with respect to the direction of the proton which contributes the annihilating quark in the lab frame.', '0910.5029-2-26-2': 'Since experimentally, it is hard to determine which proton contributes to the quark, due to the fact that a quark in the proton typically carries a larger momentum fraction than does an antiquark, we assume that the longitudinal motion of the dilepton system is in the direction of the proton which contributes the annililating quark in the lab frame [CITATION].', '0910.5029-2-27-0': 'To understand how well [MATH] describes the variable [MATH], we compare the lego plot of [MATH] versus [MATH] (Fig. [REF]) and the lego plot of [MATH] versus [MATH] (Fig. [REF]), where [MATH] is the Monte Calo true angle between the direction of the negatively charged lepton and that of the annihilating quark.', '0910.5029-2-27-1': 'In these two figures, the events generated include the full interference structure of [MATH]/Z/[MATH] and satisfy the transverse momentum cut [MATH] GeV or [MATH] GeV, back to back selection cut [MATH], and the requirement that the invariant mass of the dilepton system is within the range of 500 GeV to 2000 GeV.', '0910.5029-2-27-2': 'We find that [MATH] can be used as a good approximation for [MATH].', '0910.5029-2-27-3': 'Although those two figures look very similar, the mistagging rate (that is, [MATH] having a wrong sign compared to [MATH]) is still high at the low [MATH] region, as shown in Fig. [REF].', '0910.5029-2-27-4': 'With an additional cut of [MATH], the mistagging rate can be reduced to a reasonable level ([MATH]) without losing too much data.', '0910.5029-2-27-5': 'Therefore, we use [MATH] to approximate [MATH] when extracting the forward backward asymmetry in the simulations with the requirement of [MATH].', '0910.5029-2-28-0': 'In the case of [MATH] TeV with 500 fb[MATH] of data, Fig. [REF] shows the forward backward asymmetry distribution as a function of the dilepton transverse momentum [MATH] for [MATH] TeV, while Fig. [REF] shows the forward backward asymmetry as a function of the dilepton invariant mass [MATH].', '0910.5029-2-28-1': 'The uncertainties correspond to the statistical errors normalized to [MATH] fb[MATH] of data.', '0910.5029-2-28-2': 'In the case of [MATH] TeV with 500 fb[MATH] of data, Fig. [REF] and [REF] show the forward backward asymmetry as a function of the transverse momentum and invariant mass, respectively, for [MATH] mass of 1 TeV.', '0910.5029-2-28-3': 'By using the forward backward asymmetry distributions, a clear distinction between the dielectron and dimuon channels can be obtained in the low invariant mass and low transverse momentum regions.', '0910.5029-2-28-4': 'As expected, the asymmetry distribution measurements require a large amount of data and thus can only be achieved in the later stages of the LHC operation.', '0910.5029-2-29-0': '# Allowed Region of Parameters [MATH] and [MATH]', '0910.5029-2-30-0': 'In the above analyses, we concentrate on only one bench mark point in the parameter space, which gives rise to realistic neutrino mass and mixing patterns.', '0910.5029-2-30-1': 'Without imposing the constraints from the neutrino sector, it is possible to have a more general generation dependent, non-amolaous [MATH] model.', '0910.5029-2-30-2': 'Fig. [REF] shows the parameter space of [MATH] and [MATH] for which the [MATH] discovery is possible.', '0910.5029-2-30-3': 'Specifically, the values of [MATH] and [MATH] in the allowed region gives, [MATH].', '0910.5029-2-30-4': 'We require the [MATH] decay width to be larger than [MATH], so that the decay width is wide enough compared to the detector resolution, and thus enabling the [MATH] resonance to be observable.', '0910.5029-2-30-5': 'The upper limit of 0.1 for the ratio of the decay width to [MATH] is imposed to ensure the narrow width approximation is valid in our analysis.', '0910.5029-2-30-6': 'The direct search limits from the Tevatron can be translated into constraints on parameters [MATH] and [MATH] (for the definition of [MATH], see Ref. [CITATION].)', '0910.5029-2-30-7': 'In the class of non-anomalous models we consider, [MATH], and thus the Tevatron constraints for [MATH] = 900 GeV [CITATION] [EQUATION] translate into [MATH], which are equivalent to.', '0910.5029-2-30-8': '[EQUATION] for the dielectron and dimuon channels respectively in our model, with [MATH].', '0910.5029-2-30-9': 'Furthermore, to satisfy the electroweak precision constraints, the parameters (a, b) must be chosen such that [MATH].', '0910.5029-2-30-10': 'Generally, any TeV scale generation dependent [MATH] model suffer severe constraints from flavor changing neutral currents.', '0910.5029-2-30-11': 'This is easily avoided by choosing [MATH] to be a non-integer, which naturally gives the texture zeros in the (12), (13), (21) and (31) entries of the charged lepton mass matrix given by Eq. [REF].', '0910.5029-2-30-12': 'The FCNC constraints can also be avoided by having a large integral value for [MATH], so that the off diagonal elements in the charged lepton mass matrix are highly suppressed.', '0910.5029-2-30-13': 'A large parameter space remains even after removing points for which [MATH] is an integer.', '0910.5029-2-31-0': '# Conclusion', '0910.5029-2-32-0': 'We investigate the collider signatures of a TeV scale non-anomalous [MATH] model, which generates at the TeV scale small neutrino masses and their mixing angles.', '0910.5029-2-32-1': 'Since the [MATH] symmetry, which is different from [MATH], is generation dependent, all gauge anomalies are cancelled with no exotic fields other than the three right-handed neutrinos.', '0910.5029-2-32-2': "Specifically, we have investigated the LHC's discovery potential of such a flavorful [MATH].", '0910.5029-2-32-3': 'Through the excess in the dilepton invariant mass distribution, at the center of mass energy of [MATH] TeV, the [MATH] of mass up to 4.5 TeV can be discovered at [MATH] with 100 fb[MATH] of data.', '0910.5029-2-32-4': 'To establish the flavorful nature of [MATH] requires a distinction between the [MATH] and [MATH] channels.', '0910.5029-2-32-5': 'While it requires a much higher integrated luminosity, it is possible to distinguish these two decay channels at sufficient significance level.', '0910.5029-2-32-6': 'For the bench mark point we consider in our analysis, at [MATH] = 14 TeV, a [MATH] distinction between the [MATH] and [MATH] channels can be obtained with 500 fb[MATH] of data, for [MATH] up to 3 TeV.', '0910.5029-2-32-7': 'At [MATH] TeV, 8.32 fb[MATH] of integrated luminosity is required for [MATH] = 1 TeV.', '0910.5029-2-32-8': 'We have also studied the possibility of measuring the [MATH] charges of the leptons using the forward backward asymmetry distributions of the dilepton channels.', '0910.5029-2-32-9': 'With a sufficient integrated luminosity, it is possible to observe the different asymmetry distributions for the dieletron and dimuon channels in the low invariant mass and low transverse momentum regions.', '0910.5029-2-32-10': 'This thus allows to establish the generation-dependent nature of the [MATH] model.', '0910.5029-2-33-0': 'We thank Thomas G. Rizzo for the useful discussion.', '0910.5029-2-33-1': 'The work of M-CC was supported, in part, by the National Science Foundation under grant no.', '0910.5029-2-33-2': 'PHY-0709742.'}

[['0910.5029-1-18-0', '0910.5029-2-17-0'], ['0910.5029-1-18-1', '0910.5029-2-17-1'], ['0910.5029-1-18-2', '0910.5029-2-17-2'], ['0910.5029-1-21-0', '0910.5029-2-20-0'], ['0910.5029-1-21-1', '0910.5029-2-20-1'], ['0910.5029-1-21-2', '0910.5029-2-20-2'], ['0910.5029-1-21-3', '0910.5029-2-20-3'], ['0910.5029-1-7-0', '0910.5029-2-7-0'], ['0910.5029-1-7-1', '0910.5029-2-7-1'], ['0910.5029-1-7-2', '0910.5029-2-7-2'], ['0910.5029-1-7-3', '0910.5029-2-7-3'], ['0910.5029-1-7-4', '0910.5029-2-7-4'], ['0910.5029-1-7-5', '0910.5029-2-7-5'], ['0910.5029-1-7-6', '0910.5029-2-7-6'], ['0910.5029-1-34-0', '0910.5029-2-33-0'], ['0910.5029-1-34-1', '0910.5029-2-33-1'], ['0910.5029-1-22-0', '0910.5029-2-21-0'], ['0910.5029-1-22-1', '0910.5029-2-21-1'], ['0910.5029-1-22-2', '0910.5029-2-21-2'], ['0910.5029-1-22-3', '0910.5029-2-21-3'], ['0910.5029-1-22-4', '0910.5029-2-21-4'], ['0910.5029-1-17-0', '0910.5029-2-16-0'], ['0910.5029-1-17-1', '0910.5029-2-16-1'], ['0910.5029-1-17-2', '0910.5029-2-16-2'], ['0910.5029-1-17-3', '0910.5029-2-16-3'], ['0910.5029-1-17-4', '0910.5029-2-16-4'], ['0910.5029-1-17-5', '0910.5029-2-16-5'], ['0910.5029-1-17-6', '0910.5029-2-16-6'], ['0910.5029-1-17-7', '0910.5029-2-16-7'], ['0910.5029-1-17-8', '0910.5029-2-16-8'], ['0910.5029-1-29-0', '0910.5029-2-28-0'], ['0910.5029-1-29-1', '0910.5029-2-28-1'], ['0910.5029-1-29-2', '0910.5029-2-28-2'], ['0910.5029-1-29-3', '0910.5029-2-28-3'], ['0910.5029-1-29-4', '0910.5029-2-28-4'], ['0910.5029-1-6-0', '0910.5029-2-6-0'], ['0910.5029-1-6-1', '0910.5029-2-6-1'], ['0910.5029-1-6-2', '0910.5029-2-6-2'], ['0910.5029-1-6-3', '0910.5029-2-6-3'], ['0910.5029-1-31-0', '0910.5029-2-30-0'], ['0910.5029-1-31-2', '0910.5029-2-30-2'], ['0910.5029-1-31-3', '0910.5029-2-30-3'], ['0910.5029-1-31-4', '0910.5029-2-30-4'], ['0910.5029-1-31-5', '0910.5029-2-30-5'], ['0910.5029-1-31-6', '0910.5029-2-30-10'], ['0910.5029-1-31-7', '0910.5029-2-30-11'], ['0910.5029-1-31-8', '0910.5029-2-30-12'], ['0910.5029-1-31-9', '0910.5029-2-30-13'], ['0910.5029-1-16-0', '0910.5029-2-15-0'], ['0910.5029-1-16-1', '0910.5029-2-15-1'], ['0910.5029-1-16-2', '0910.5029-2-15-2'], ['0910.5029-1-9-0', '0910.5029-2-9-0'], ['0910.5029-1-9-1', '0910.5029-2-9-1'], ['0910.5029-1-9-2', '0910.5029-2-9-2'], ['0910.5029-1-9-3', '0910.5029-2-9-3'], ['0910.5029-1-9-4', '0910.5029-2-9-4'], ['0910.5029-1-9-5', '0910.5029-2-9-5'], ['0910.5029-1-23-0', '0910.5029-2-22-0'], ['0910.5029-1-23-1', '0910.5029-2-22-1'], ['0910.5029-1-23-2', '0910.5029-2-22-2'], ['0910.5029-1-23-3', '0910.5029-2-22-3'], ['0910.5029-1-23-4', '0910.5029-2-22-4'], ['0910.5029-1-23-5', '0910.5029-2-22-5'], ['0910.5029-1-23-6', '0910.5029-2-22-6'], ['0910.5029-1-23-7', '0910.5029-2-22-7'], ['0910.5029-1-23-8', '0910.5029-2-22-8'], ['0910.5029-1-19-0', '0910.5029-2-18-0'], ['0910.5029-1-19-1', '0910.5029-2-18-1'], ['0910.5029-1-19-2', '0910.5029-2-18-2'], ['0910.5029-1-19-3', '0910.5029-2-18-3'], ['0910.5029-1-19-4', '0910.5029-2-18-4'], ['0910.5029-1-19-5', '0910.5029-2-18-5'], ['0910.5029-1-19-6', '0910.5029-2-18-6'], ['0910.5029-1-19-7', '0910.5029-2-18-7'], ['0910.5029-1-19-8', '0910.5029-2-18-8'], ['0910.5029-1-19-9', '0910.5029-2-18-9'], ['0910.5029-1-19-10', '0910.5029-2-18-10'], ['0910.5029-1-15-0', '0910.5029-2-14-0'], ['0910.5029-1-15-1', '0910.5029-2-14-1'], ['0910.5029-1-15-2', '0910.5029-2-14-2'], ['0910.5029-1-15-3', '0910.5029-2-14-3'], ['0910.5029-1-15-4', '0910.5029-2-14-4'], ['0910.5029-1-15-5', '0910.5029-2-14-5'], ['0910.5029-1-15-6', '0910.5029-2-14-6'], ['0910.5029-1-15-7', '0910.5029-2-14-7'], ['0910.5029-1-4-0', '0910.5029-2-4-0'], ['0910.5029-1-4-1', '0910.5029-2-4-1'], ['0910.5029-1-4-3', '0910.5029-2-4-3'], ['0910.5029-1-4-4', '0910.5029-2-4-4'], ['0910.5029-1-0-0', '0910.5029-2-0-0'], ['0910.5029-1-0-1', '0910.5029-2-0-1'], ['0910.5029-1-0-2', '0910.5029-2-0-2'], ['0910.5029-1-0-5', '0910.5029-2-0-5'], ['0910.5029-1-27-0', '0910.5029-2-26-0'], ['0910.5029-1-27-1', '0910.5029-2-26-1'], ['0910.5029-1-27-2', '0910.5029-2-26-2'], ['0910.5029-1-8-0', '0910.5029-2-8-0'], ['0910.5029-1-8-1', '0910.5029-2-8-1'], ['0910.5029-1-8-2', '0910.5029-2-8-2'], ['0910.5029-1-8-5', '0910.5029-2-8-10'], ['0910.5029-1-33-0', '0910.5029-2-32-0'], ['0910.5029-1-33-1', '0910.5029-2-32-1'], ['0910.5029-1-33-2', '0910.5029-2-32-2'], ['0910.5029-1-33-3', '0910.5029-2-32-3'], ['0910.5029-1-33-4', '0910.5029-2-32-4'], ['0910.5029-1-33-5', '0910.5029-2-32-5'], ['0910.5029-1-33-6', '0910.5029-2-32-6'], ['0910.5029-1-33-7', '0910.5029-2-32-7'], ['0910.5029-1-33-8', '0910.5029-2-32-8'], ['0910.5029-1-33-9', '0910.5029-2-32-9'], ['0910.5029-1-33-10', '0910.5029-2-32-10'], ['0910.5029-1-13-0', '0910.5029-2-12-0'], ['0910.5029-1-13-1', '0910.5029-2-12-1'], ['0910.5029-1-13-2', '0910.5029-2-12-2'], ['0910.5029-1-13-4', '0910.5029-2-12-4'], ['0910.5029-1-13-5', '0910.5029-2-12-5'], ['0910.5029-1-13-6', '0910.5029-2-12-6'], ['0910.5029-1-13-7', '0910.5029-2-12-7'], ['0910.5029-1-3-0', '0910.5029-2-3-0'], ['0910.5029-1-3-1', '0910.5029-2-3-1'], ['0910.5029-1-3-2', '0910.5029-2-3-2'], ['0910.5029-1-3-3', '0910.5029-2-3-3'], ['0910.5029-1-3-4', '0910.5029-2-3-4'], ['0910.5029-1-3-5', '0910.5029-2-3-5'], ['0910.5029-1-3-6', '0910.5029-2-3-6'], ['0910.5029-1-25-0', '0910.5029-2-24-0'], ['0910.5029-1-25-1', '0910.5029-2-24-1'], ['0910.5029-1-2-0', '0910.5029-2-2-0'], ['0910.5029-1-2-1', '0910.5029-2-2-1'], ['0910.5029-1-2-2', '0910.5029-2-2-2'], ['0910.5029-1-2-3', '0910.5029-2-2-3'], ['0910.5029-1-2-4', '0910.5029-2-2-4'], ['0910.5029-1-26-0', '0910.5029-2-25-0'], ['0910.5029-1-26-1', '0910.5029-2-25-1'], ['0910.5029-1-26-2', '0910.5029-2-25-2'], ['0910.5029-1-28-0', '0910.5029-2-27-0'], ['0910.5029-1-28-1', '0910.5029-2-27-1'], ['0910.5029-1-28-2', '0910.5029-2-27-2'], ['0910.5029-1-28-3', '0910.5029-2-27-3'], ['0910.5029-1-28-4', '0910.5029-2-27-4'], ['0910.5029-1-28-5', '0910.5029-2-27-5'], ['0910.5029-1-11-0', '0910.5029-2-10-3'], ['0910.5029-1-11-1', '0910.5029-2-10-4'], ['0910.5029-1-11-2', '0910.5029-2-10-5'], ['0910.5029-1-11-3', '0910.5029-2-10-6'], ['0910.5029-1-11-4', '0910.5029-2-10-7'], ['0910.5029-1-31-1', '0910.5029-2-30-1'], ['0910.5029-1-4-2', '0910.5029-2-4-2'], ['0910.5029-1-0-3', '0910.5029-2-0-3'], ['0910.5029-1-0-4', '0910.5029-2-0-4'], ['0910.5029-1-8-3', '0910.5029-2-8-3'], ['0910.5029-1-13-3', '0910.5029-2-12-3'], ['0910.5029-1-25-2', '0910.5029-2-24-2'], ['0910.5029-1-10-0', '0910.5029-2-10-0'], ['0910.5029-1-8-4', '0910.5029-2-8-4'], ['0910.5029-1-8-4', '0910.5029-2-8-8'], ['0910.5029-1-10-2', '0910.5029-2-10-2']]

[['0910.5029-1-18-0', '0910.5029-2-17-0'], ['0910.5029-1-18-1', '0910.5029-2-17-1'], ['0910.5029-1-18-2', '0910.5029-2-17-2'], ['0910.5029-1-21-0', '0910.5029-2-20-0'], ['0910.5029-1-21-1', '0910.5029-2-20-1'], ['0910.5029-1-21-2', '0910.5029-2-20-2'], ['0910.5029-1-21-3', '0910.5029-2-20-3'], ['0910.5029-1-7-0', '0910.5029-2-7-0'], ['0910.5029-1-7-1', '0910.5029-2-7-1'], ['0910.5029-1-7-2', '0910.5029-2-7-2'], ['0910.5029-1-7-3', '0910.5029-2-7-3'], ['0910.5029-1-7-4', '0910.5029-2-7-4'], ['0910.5029-1-7-5', '0910.5029-2-7-5'], ['0910.5029-1-7-6', '0910.5029-2-7-6'], ['0910.5029-1-34-0', '0910.5029-2-33-0'], ['0910.5029-1-34-1', '0910.5029-2-33-1'], ['0910.5029-1-22-0', '0910.5029-2-21-0'], ['0910.5029-1-22-1', '0910.5029-2-21-1'], ['0910.5029-1-22-2', '0910.5029-2-21-2'], ['0910.5029-1-22-3', '0910.5029-2-21-3'], ['0910.5029-1-22-4', '0910.5029-2-21-4'], ['0910.5029-1-17-0', '0910.5029-2-16-0'], ['0910.5029-1-17-1', '0910.5029-2-16-1'], ['0910.5029-1-17-2', '0910.5029-2-16-2'], ['0910.5029-1-17-3', '0910.5029-2-16-3'], ['0910.5029-1-17-4', '0910.5029-2-16-4'], ['0910.5029-1-17-5', '0910.5029-2-16-5'], ['0910.5029-1-17-6', '0910.5029-2-16-6'], ['0910.5029-1-17-7', '0910.5029-2-16-7'], ['0910.5029-1-17-8', '0910.5029-2-16-8'], ['0910.5029-1-29-0', '0910.5029-2-28-0'], ['0910.5029-1-29-1', '0910.5029-2-28-1'], ['0910.5029-1-29-2', '0910.5029-2-28-2'], ['0910.5029-1-29-3', '0910.5029-2-28-3'], ['0910.5029-1-29-4', '0910.5029-2-28-4'], ['0910.5029-1-6-0', '0910.5029-2-6-0'], ['0910.5029-1-6-1', '0910.5029-2-6-1'], ['0910.5029-1-6-2', '0910.5029-2-6-2'], ['0910.5029-1-6-3', '0910.5029-2-6-3'], ['0910.5029-1-31-0', '0910.5029-2-30-0'], ['0910.5029-1-31-2', '0910.5029-2-30-2'], ['0910.5029-1-31-3', '0910.5029-2-30-3'], ['0910.5029-1-31-4', '0910.5029-2-30-4'], ['0910.5029-1-31-5', '0910.5029-2-30-5'], ['0910.5029-1-31-6', '0910.5029-2-30-10'], ['0910.5029-1-31-7', '0910.5029-2-30-11'], ['0910.5029-1-31-8', '0910.5029-2-30-12'], ['0910.5029-1-31-9', '0910.5029-2-30-13'], ['0910.5029-1-16-0', '0910.5029-2-15-0'], ['0910.5029-1-16-1', '0910.5029-2-15-1'], ['0910.5029-1-16-2', '0910.5029-2-15-2'], ['0910.5029-1-9-0', '0910.5029-2-9-0'], ['0910.5029-1-9-1', '0910.5029-2-9-1'], ['0910.5029-1-9-2', '0910.5029-2-9-2'], ['0910.5029-1-9-3', '0910.5029-2-9-3'], ['0910.5029-1-9-4', '0910.5029-2-9-4'], ['0910.5029-1-9-5', '0910.5029-2-9-5'], ['0910.5029-1-23-0', '0910.5029-2-22-0'], ['0910.5029-1-23-1', '0910.5029-2-22-1'], ['0910.5029-1-23-2', '0910.5029-2-22-2'], ['0910.5029-1-23-3', '0910.5029-2-22-3'], ['0910.5029-1-23-4', '0910.5029-2-22-4'], ['0910.5029-1-23-5', '0910.5029-2-22-5'], ['0910.5029-1-23-6', '0910.5029-2-22-6'], ['0910.5029-1-23-7', '0910.5029-2-22-7'], ['0910.5029-1-23-8', '0910.5029-2-22-8'], ['0910.5029-1-19-0', '0910.5029-2-18-0'], ['0910.5029-1-19-1', '0910.5029-2-18-1'], ['0910.5029-1-19-2', '0910.5029-2-18-2'], ['0910.5029-1-19-3', '0910.5029-2-18-3'], ['0910.5029-1-19-4', '0910.5029-2-18-4'], ['0910.5029-1-19-5', '0910.5029-2-18-5'], ['0910.5029-1-19-6', '0910.5029-2-18-6'], ['0910.5029-1-19-7', '0910.5029-2-18-7'], ['0910.5029-1-19-8', '0910.5029-2-18-8'], ['0910.5029-1-19-9', '0910.5029-2-18-9'], ['0910.5029-1-19-10', '0910.5029-2-18-10'], ['0910.5029-1-15-0', '0910.5029-2-14-0'], ['0910.5029-1-15-1', '0910.5029-2-14-1'], ['0910.5029-1-15-2', '0910.5029-2-14-2'], ['0910.5029-1-15-3', '0910.5029-2-14-3'], ['0910.5029-1-15-4', '0910.5029-2-14-4'], ['0910.5029-1-15-5', '0910.5029-2-14-5'], ['0910.5029-1-15-6', '0910.5029-2-14-6'], ['0910.5029-1-15-7', '0910.5029-2-14-7'], ['0910.5029-1-4-0', '0910.5029-2-4-0'], ['0910.5029-1-4-1', '0910.5029-2-4-1'], ['0910.5029-1-4-3', '0910.5029-2-4-3'], ['0910.5029-1-4-4', '0910.5029-2-4-4'], ['0910.5029-1-0-0', '0910.5029-2-0-0'], ['0910.5029-1-0-1', '0910.5029-2-0-1'], ['0910.5029-1-0-2', '0910.5029-2-0-2'], ['0910.5029-1-0-5', '0910.5029-2-0-5'], ['0910.5029-1-27-0', '0910.5029-2-26-0'], ['0910.5029-1-27-1', '0910.5029-2-26-1'], ['0910.5029-1-27-2', '0910.5029-2-26-2'], ['0910.5029-1-8-0', '0910.5029-2-8-0'], ['0910.5029-1-8-1', '0910.5029-2-8-1'], ['0910.5029-1-8-2', '0910.5029-2-8-2'], ['0910.5029-1-8-5', '0910.5029-2-8-10'], ['0910.5029-1-33-0', '0910.5029-2-32-0'], ['0910.5029-1-33-1', '0910.5029-2-32-1'], ['0910.5029-1-33-2', '0910.5029-2-32-2'], ['0910.5029-1-33-3', '0910.5029-2-32-3'], ['0910.5029-1-33-4', '0910.5029-2-32-4'], ['0910.5029-1-33-5', '0910.5029-2-32-5'], ['0910.5029-1-33-6', '0910.5029-2-32-6'], ['0910.5029-1-33-7', '0910.5029-2-32-7'], ['0910.5029-1-33-8', '0910.5029-2-32-8'], ['0910.5029-1-33-9', '0910.5029-2-32-9'], ['0910.5029-1-33-10', '0910.5029-2-32-10'], ['0910.5029-1-13-0', '0910.5029-2-12-0'], ['0910.5029-1-13-1', '0910.5029-2-12-1'], ['0910.5029-1-13-2', '0910.5029-2-12-2'], ['0910.5029-1-13-4', '0910.5029-2-12-4'], ['0910.5029-1-13-5', '0910.5029-2-12-5'], ['0910.5029-1-13-6', '0910.5029-2-12-6'], ['0910.5029-1-13-7', '0910.5029-2-12-7'], ['0910.5029-1-3-0', '0910.5029-2-3-0'], ['0910.5029-1-3-1', '0910.5029-2-3-1'], ['0910.5029-1-3-2', '0910.5029-2-3-2'], ['0910.5029-1-3-3', '0910.5029-2-3-3'], ['0910.5029-1-3-4', '0910.5029-2-3-4'], ['0910.5029-1-3-5', '0910.5029-2-3-5'], ['0910.5029-1-3-6', '0910.5029-2-3-6'], ['0910.5029-1-25-0', '0910.5029-2-24-0'], ['0910.5029-1-25-1', '0910.5029-2-24-1'], ['0910.5029-1-2-0', '0910.5029-2-2-0'], ['0910.5029-1-2-1', '0910.5029-2-2-1'], ['0910.5029-1-2-2', '0910.5029-2-2-2'], ['0910.5029-1-2-3', '0910.5029-2-2-3'], ['0910.5029-1-2-4', '0910.5029-2-2-4'], ['0910.5029-1-26-0', '0910.5029-2-25-0'], ['0910.5029-1-26-1', '0910.5029-2-25-1'], ['0910.5029-1-26-2', '0910.5029-2-25-2'], ['0910.5029-1-28-0', '0910.5029-2-27-0'], ['0910.5029-1-28-1', '0910.5029-2-27-1'], ['0910.5029-1-28-2', '0910.5029-2-27-2'], ['0910.5029-1-28-3', '0910.5029-2-27-3'], ['0910.5029-1-28-4', '0910.5029-2-27-4'], ['0910.5029-1-28-5', '0910.5029-2-27-5'], ['0910.5029-1-11-0', '0910.5029-2-10-3'], ['0910.5029-1-11-1', '0910.5029-2-10-4'], ['0910.5029-1-11-2', '0910.5029-2-10-5'], ['0910.5029-1-11-3', '0910.5029-2-10-6'], ['0910.5029-1-11-4', '0910.5029-2-10-7']]

[['0910.5029-1-31-1', '0910.5029-2-30-1'], ['0910.5029-1-4-2', '0910.5029-2-4-2'], ['0910.5029-1-0-3', '0910.5029-2-0-3'], ['0910.5029-1-0-4', '0910.5029-2-0-4'], ['0910.5029-1-8-3', '0910.5029-2-8-3'], ['0910.5029-1-13-3', '0910.5029-2-12-3'], ['0910.5029-1-25-2', '0910.5029-2-24-2'], ['0910.5029-1-10-0', '0910.5029-2-10-0']]

[]

[['0910.5029-1-8-4', '0910.5029-2-8-4'], ['0910.5029-1-8-4', '0910.5029-2-8-8'], ['0910.5029-1-10-2', '0910.5029-2-10-2']]

[]

['0910.5029-1-34-2', '0910.5029-2-33-2']

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

https://arxiv.org/abs/0910.5029

1511.06530

{'1511.06530-1-0-0': 'Although the latest high-end smartphone has powerful CPU and GPU, running deeper convolutional neural networks (CNNs) for complex tasks such as ImageNet classification on mobile devices is challenging.', '1511.06530-1-0-1': 'To deploy deep CNNs on mobile devices, we present a simple and effective scheme to compress the entire CNN, which we call one-shot whole network compression.', '1511.06530-1-0-2': 'The proposed scheme consists of three steps: (1) rank selection with variational Bayesian matrix factorization, (2) Tucker decomposition on kernel tensor, and (3) fine-tuning to recover accumulated loss of accuracy, and each step can be easily implemented using publicly available tools.', '1511.06530-1-0-3': 'We demonstrate the effectiveness of the proposed scheme by testing the performance of various compressed CNNs (AlexNet, VGG-S, GoogLeNet, and VGG-16) on the smartphone.', '1511.06530-1-0-4': 'Significant reductions in model size, runtime, and energy consumption are obtained, at the cost of small loss in accuracy.', '1511.06530-1-0-5': 'In addition, we address the important implementation level issue on [MATH] convolution, which is a key operation of inception module of GoogLeNet as well as CNNs compressed by our proposed scheme.', '1511.06530-1-1-0': '# Introduction', '1511.06530-1-2-0': 'Deployment of convolutional neural networks (CNNs) for computer vision tasks on mobile devices is gaining more and more attention.', '1511.06530-1-2-1': 'On mobile applications, it is typically assumed that training is performed on the server and test or inference is executed on the mobile devices.', '1511.06530-1-2-2': 'One of the most critical issues in mobile applications of CNNs is that mobile devices have strict constraints in terms of computing power, battery, and memory capacity.', '1511.06530-1-2-3': 'Thus, it is imperative to obtain CNNs tailored to the limited resources of mobile devices.', '1511.06530-1-3-0': 'Deep neural networks are known to be over-parameterized, which facilitates convergence to good local minima of the loss function during training .', '1511.06530-1-3-1': 'To improve test-time performance on mobile devices, such redundancy can be removed from the trained networks without noticeable impact on accuracy.', '1511.06530-1-3-2': 'Recently, there are several studies to apply low-rank approximations to compress CNNs by exploiting redundancy .', '1511.06530-1-3-3': 'Such compressions typically focus on convolution layers since they dominate total computation cost especially in deep neural networks .', '1511.06530-1-3-4': 'Existing methods, though effective in reducing the computation cost of a single convolutional layer, introduce a new challenge called whole network compression which aims at compressing the entire network.', '1511.06530-1-4-0': 'Whole network compression: It is nontrivial to compress whole and very deep CNNs for complex tasks such as ImageNet classification.', '1511.06530-1-4-1': 'Recently, [CITATION] showed that entire convolutional layers can be accelerated with "asymmetric (3d)" decomposition.', '1511.06530-1-4-2': 'In addition, they also presented the effective rank selection and optimization method.', '1511.06530-1-4-3': 'Although their proposed decomposition of layers can be easily implemented in popular development tools (e.g. Caffe, Torch, and Theano), the rank selection and optimization parts still require because they consist of multiple steps and depend on the output of previous layers.', '1511.06530-1-4-4': 'In this paper, we present much simpler but still powerful whole network compression scheme which takes entire convolutional and fully-connected layers into account.', '1511.06530-1-5-0': 'Contribution: This paper makes the following major contributions.', '1511.06530-1-6-0': 'This paper is organized as follows.', '1511.06530-1-6-1': 'Section 2 reviews related work.', '1511.06530-1-6-2': 'Section 3 explains our proposed scheme.', '1511.06530-1-6-3': 'Section 4 gives experimental results.', '1511.06530-1-6-4': 'Section 5 summarizes the paper.', '1511.06530-1-7-0': '# Related Work', '1511.06530-1-8-0': '## CNN Compression', '1511.06530-1-9-0': 'CNN usually consists of convolutional layers and fully-connected layers which dominate computation cost and memory consumption respectively.', '1511.06530-1-9-1': 'After [CITATION] showed the possibility of removing the redundancy of neural networks, several CNN compression techniques have been proposed.', '1511.06530-1-9-2': 'A recent study showed that the weight matrix of a fully-connected layer can be compressed by applying truncated singular value decomposition (SVD) without significant drop in the prediction accuracy.', '1511.06530-1-9-3': 'More recently, various methods based on vector quantization , hashing techniques , circulant projection , and tensor train decomposition were proposed and showed better compression capability than SVD.', '1511.06530-1-9-4': 'To speed up the convolutional layers, several methods based on low-rank decomposition of convolutional kernel tensor were proposed , but they compress only single or a few layers.', '1511.06530-1-10-0': 'Concurrent with our work, [CITATION] presented "asymmetric (3d) decomposition" to accelerate the entire convolutional layers, where the original [MATH] convolution is decomposed to [MATH], [MATH], and [MATH] convolution.', '1511.06530-1-10-1': 'In addition, they also present a rank selection method based on PCA accumulated energy and an optimization method which minimizes the reconstruction error of non-linear responses.', '1511.06530-1-10-2': 'In the extended version , the additional fine-tuning of entire network was considered for further improvement.', '1511.06530-1-10-3': 'Compared with these works, our proposed scheme is different in that (1) Tucker decomposition is adopted to compress the entire convolutional and fully-connected layers, (2) the kernel tensor reconstruction error is minimized instead of non-linear response, (3) a global analytic solution of VBMF is applied to determine the rank of each layer, and (4) a single run of fine-tuning is performed to account for the accumulation of errors.', '1511.06530-1-11-0': 'A pruning approach also aims at reducing the total amount of parameters and operations in the entire network.', '1511.06530-1-11-1': 'Pruning based approaches can give significant reductions in parameter size and computation workload.', '1511.06530-1-11-2': 'However, it is challenging to achieve runtime speed-up with conventional GPU implementation as mentioned in .', '1511.06530-1-12-0': 'Orthogonal to model level compression, implementation level approaches were also proposed.', '1511.06530-1-12-1': 'The FFT method was used to speed-up convolution .', '1511.06530-1-12-2': 'In , CPU code optimizations to speed-up the execution of CNN are extensively explored.', '1511.06530-1-13-0': '## Tensor Decomposition', '1511.06530-1-14-0': 'A tensor is a multi-way array of data.', '1511.06530-1-14-1': 'For example, a vector is 1-way tensor and a matrix is 2-way tensor.', '1511.06530-1-14-2': 'Two of the most popular tensor decomposition models are CANDECOMP/PARAFAC model and Tucker model .', '1511.06530-1-14-3': 'In this paper, we extensively use Tucker model for whole network compression.', '1511.06530-1-14-4': 'Tucker decomposition is a higher order extension of the singular value decomposition (SVD) of matrix, in the perspective of computing the orthonormal spaces associated with the different modes of a tensor.', '1511.06530-1-14-5': 'It simultaneously analyzes mode-[MATH] matricizations of the original tensor, and merges them with the core tensor as illustrated in Fig. [REF].', '1511.06530-1-15-0': 'In our whole network compression scheme, we apply Tucker-2 decomposition, which is also known as GLRAM , from the second convolutional layer to the first fully connected layers.', '1511.06530-1-15-1': 'For the other layers, we apply Tucker-1 decomposition, which is equivalent to SVD.', '1511.06530-1-15-2': 'For more information on the tensor decomposition, the reader is referred to the survey paper .', '1511.06530-1-16-0': '# Proposed Method', '1511.06530-1-17-0': 'Fig. [REF] illustrates our one-shot whole network compression scheme which consists of three steps: (1) rank selection; (2) Tucker decomposition; (3) fine-tuning.', '1511.06530-1-17-1': "In the first step, we analyze principal subspace of mode-3 and mode-4 matricization of each layer's kernel tensor with global analytic variational Bayesian matrix factorization.", '1511.06530-1-17-2': "Then we apply Tucker decomposition on each layer's kernel tensor with previously determined rank.", '1511.06530-1-17-3': 'Finally, we fine-tune the entire network with standard back-propagation.', '1511.06530-1-18-0': '## Tucker Decomposition on Kernel Tensor', '1511.06530-1-19-0': 'Convolution kernel tensor: In CNNs, the convolution operation maps an input (source) tensor [MATH] of size [MATH] into output (target) tensor [MATH] of size [MATH] using the following linear mapping: [EQUATION] where [MATH] is a 4-way kernel tensor of size [MATH], [MATH] is stride, and [MATH] is zero-padding size.', '1511.06530-1-20-0': 'Tucker Decomposition: The rank-[MATH] Tucker decomposition of 4-way kernel tensor [MATH] has the form: [EQUATION] where [MATH] is a core tensor of size [MATH] and [MATH], [MATH], [MATH], and [MATH] are factor matrices of sizes [MATH], [MATH], [MATH], and [MATH], respectively.', '1511.06530-1-21-0': 'In the Tucker decomposition, every mode does not have to be decomposed.', '1511.06530-1-21-1': 'For example, we do not decompose mode-1 and mode-2 which are associated with spatial dimensions because they are already quite small ([MATH] is typically 3 or 5).', '1511.06530-1-21-2': 'Under this variant called Tucker-2 decomposition , the kernel tensor is decomposed to: [EQUATION] where [MATH] is a core tensor of size [MATH].', '1511.06530-1-21-3': 'After substituting [REF] into [REF], performing rearrangements and grouping summands, we obtain the following three consecutive expressions for the approximate evaluation of the convolution [REF]: [EQUATION] where [MATH] and [MATH] are intermediate tensors of sizes [MATH] and [MATH], respectively.', '1511.06530-1-22-0': '[MATH] convolution: As illustrated in Fig. [REF], computing [MATH] from [MATH] in [REF] as well as [MATH] from [MATH] in [REF] is [MATH] convolutions that essentially perform pixel-wise linear re-combination of input maps.', '1511.06530-1-22-1': 'It is introduced in network-in-network and extensively used in inception module of GoogLeNet .', '1511.06530-1-22-2': 'Note that computing [REF] is similar to inception module in the sense that [MATH] convolution is applied after dimensional reduction with [MATH] convolution, but different in the sense that there is no non-linear ReLU function between [REF] and [REF].', '1511.06530-1-22-3': 'In addition, similar to , we compute smaller intermediate output tensor [MATH] in [REF] and then recover its size in [REF].', '1511.06530-1-22-4': 'The Tucker-2 decomposition naturally integrates two compression techniques.', '1511.06530-1-23-0': 'Complexity analysis: The convolution operation in [REF] requires [MATH] parameters and [MATH] multiplication-addition operations.', '1511.06530-1-23-1': 'With Tucker decomposition, compression ratio [MATH] and speed-up ratio [MATH] are given by: [EQUATION] and these are bounded by [MATH].', '1511.06530-1-24-0': 'Tucker vs CP: Recently, CP decomposition is applied to approximate the convolution layers of CNNs for ImageNet which consist of 8 layers .', '1511.06530-1-24-1': 'However it cannot be applied to the entire layers and the instability issue of low-rank CP decomposition is reported .', '1511.06530-1-24-2': 'On the other hand, our kernel tensor approximation with Tucker decomposition can be successfully applied to the entire layers of AlexNet, VGG-S, GoogLeNet, and VGG-16', '1511.06530-1-25-0': '## Rank Selection with Global Analytic VBMF', '1511.06530-1-26-0': 'The rank-[MATH] are very important hyper-parameters which control the trade-off between performance (memory, speed, energy) improvement and accuracy loss.', '1511.06530-1-26-1': 'Instead of selecting the rank-[MATH] by time consuming trial-and-error, we considered data-driven one-shot decision via empirical Bayes with automatic relevance determination (ARD) prior .', '1511.06530-1-27-0': 'At the first time, we designed probabilistic Tucker model which is similar to , and applied empirical variational Bayesian learning.', '1511.06530-1-27-1': 'However, the rank selection results were severely unreliable because they heavily depend on (1) initial condition, (2) noise variance estimation policy, and (3) threshold setting for pruning.', '1511.06530-1-27-2': 'For this reason, we decided to use a sub-optimal but highly reproducible approach.', '1511.06530-1-28-0': 'We employed recently developed global analytic solutions for variational Bayesian matrix factorization (VBMF) .', '1511.06530-1-28-1': 'The global analytic VBMF is a very promising tool because it can automatically find noise variance, rank and even provide theoretical condition for perfect rank recovery .', '1511.06530-1-28-2': 'We determined the rank [MATH] and [MATH] by applying global analytic VBMF on mode-3 matricization (of size [MATH]) and mode-4 matricization (of size [MATH]) of kernel tensor [MATH], respectively.', '1511.06530-1-29-0': '## Fine-Tuning', '1511.06530-1-30-0': 'Because we minimize the reconstruction error of linear kernel tensors instead of non-linear responses, the accuracy is significantly dropped after whole network compression (e.g. more than 50[MATH] in the case of AlexNet).', '1511.06530-1-30-1': 'However, as shown in Fig. [REF], we can easily recover the accuracy by using fine-tuning with ImageNet training dataset.', '1511.06530-1-30-2': 'We observed that accuracy is recovered quickly in one epoch.', '1511.06530-1-30-3': 'However, more than 10 epochs are required to recover the original accuracy.', '1511.06530-1-31-0': 'While reported difficulty on finding a good SGD learning rate, our single learning rate scheduling rule works well for various compressed CNNs.', '1511.06530-1-31-1': 'In our experiment, we set the base learning [MATH] and decrease it by a factor of 10 every 5 epochs.', '1511.06530-1-31-2': 'Because of GPU memory limitation, we set the batch size: 128, 128, 64, and 32 for AlexNet, VGG-S, GoogLeNet, and VGG-16, respectively.', '1511.06530-1-32-0': 'We also tried to train the architecture of the approximated model from scratch on the ImageNet training dataset.', '1511.06530-1-32-1': 'At this time, we only tested the Gaussian random initialization and it did not work.', '1511.06530-1-32-2': 'We leave the use of other initialization methods and batch normalization as future work.', '1511.06530-1-33-0': '# Experiments', '1511.06530-1-34-0': "We used four representative CNNs, AlexNet, VGG-S, GoogLeNet, and VGG-16, which can be downloaded on Berkeley's Caffe model zoo.", '1511.06530-1-34-1': 'In the case of inception module of GoogLeNet, we only compressed the [MATH] convolution kernel which is the main computational part.', '1511.06530-1-34-2': 'In the case of VGG-16, we only compressed the convolutional layers as done in .', '1511.06530-1-34-3': 'Top-5 single-view accuracy is measured using 50,000 validation images from the ImageNet2012 dataset.', '1511.06530-1-35-0': 'We performed experiments on Nvidia Titan X (for fine-tuning and runtime comparison on Caffe+cuDNN2) and a smartphone, Samsung Galaxy S6 (for the comparison of runtime and energy consumption).', '1511.06530-1-35-1': 'The application processor of the smartphone (Exynos 7420) is equipped with a mobile GPU, ARM Mali T760.', '1511.06530-1-35-2': 'Compared with the GPU used on Titan X, the mobile GPU gives 35 times (6.6TFlops vs 190GFlops) lower computing capability and 13 times (336.5GBps vs 25.6GBps) smaller memory bandwidth.', '1511.06530-1-36-0': 'In order to run Caffe models on the mobile GPU, we developed a mobile version of Caffe called S-Caffe (Caffe for Smart mobile devices) where all the Caffe models can run on our target mobile devices (for the moment, Samsung smartphones) without modification.', '1511.06530-1-36-1': 'We also developed an Android App which performs image classification by running each of the four CNNs (AlexNet, VGG-S, GoogLeNet , and VGG-16) on the smartphone.', '1511.06530-1-37-0': 'We measured the power consumption of whole smartphone which is decomposed into the power consumption of GPU, main memory, and the other components of smartphone, e.g., ARM CPU, display, modem, etc. and give component-level analysis, especially, the power consumption of GPU and main memory (see supplementary material for details of measurement environment).', '1511.06530-1-37-1': 'The measurement results of runtime and energy consumption are the average of 50 runs.', '1511.06530-1-38-0': '## Overall Results', '1511.06530-1-39-0': 'Table [REF] shows the overall results for the three CNNs.', '1511.06530-1-39-1': 'Our proposed scheme gives [MATH] (AlexNet), [MATH] (VGG-S), [MATH] (GoogLeNet), and [MATH] (VGG-16) reductions in total weights and FLOPs, respectively.', '1511.06530-1-39-2': 'Such reductions offer [MATH]) runtime improvements on the smartphone (Titan X).', '1511.06530-1-39-3': 'We report the energy consumption of mobile GPU and main memory.', '1511.06530-1-39-4': 'The smartphone gives larger reduction ratios (e.g., [MATH] vs. [MATH] for AlexNet) for energy consumption than runtime.', '1511.06530-1-39-5': 'We will give a detailed analysis in the following subsection.', '1511.06530-1-40-0': "Comparison with [CITATION]'s method: The accuracy of our compressed VGG-16 is 89.40[MATH] for theoretical [MATH] speed-up, and it is comparable to the 89.6[MATH] (88.9[MATH]) for theoretical [MATH]) speed-up in .", '1511.06530-1-41-0': '## Layerwise Analysis', '1511.06530-1-42-0': 'Tables [REF], [REF], [REF] and [REF]', '1511.06530-1-43-0': 'show the detailed comparisons.', '1511.06530-1-43-1': 'Each row has two results (the above one for the original uncompressed CNN and the other one for the compressed CNN), and improvements.', '1511.06530-1-43-2': 'For instance, in Table [REF], the second convolutional layer having the input and output channel dimensions of [MATH] and [MATH] is compressed to give the Tucker-2 ranks of [MATH] and [MATH], which reduces the amount of weights from [MATH] to [MATH].', '1511.06530-1-43-3': 'After compression, a layer in the compressed network performs three matrix multiplications.', '1511.06530-1-43-4': 'We give the details of three matrix multiplications for each of weights, FLOPs, and runtime.', '1511.06530-1-43-5': 'For instance, on the smartphone (column S6 in Table [REF]), the second convolutional layer of compressed AlexNet takes 10.53ms which is decomposed to 0.8ms, 7.43ms and 2.3ms for the three matrix multiplications.', '1511.06530-1-44-0': 'In Tables [REF], [REF], [REF] and [REF] we have two observations.', '1511.06530-1-45-0': 'Observation 1: Given a compressed network, the smartphone tends to give larger performance gain than the Titan X.', '1511.06530-1-45-1': 'It is mainly because the mobile GPU on the smartphone lacks in thread-level parallelism.', '1511.06530-1-45-2': 'It has 24 times less number of threads (2K vs. 48K in terms of maximum number of threads) than that in Titan X. Compression reduces the amount of weights thereby reducing cache conflicts and memory latency.', '1511.06530-1-45-3': 'Due to the small thread-level parallelism, the reduced latency has more impact on the performance of threads on the mobile GPU than that on Titan X.', '1511.06530-1-46-0': 'Observation 2: Given the same compression rate, the smartphone tends to exhibit larger performance gain at fully-connected layers than at convolutional layers.', '1511.06530-1-46-1': 'We think it is also due to the reduced cache conflicts enabled by network compression as explained above.', '1511.06530-1-46-2': 'Especially, in the case of fully-connected layers, the effect of weight reduction can give more significant impact because the weights at the fully-connected layers are utilized only once, often called dead-on-arrival (DoA) data.', '1511.06530-1-46-3': 'In terms of cache performance, such DoA data are much more harmful than convolution kernel weights (which are reused multiple times).', '1511.06530-1-46-4': 'Thus, weight reduction at the fully connected layer can give more significant impact on cache performance thereby exhibiting more performance improvement than in the case of weight reduction at convolutional layers.', '1511.06530-1-47-0': '## Energy Consumption Analysis', '1511.06530-1-48-0': 'Fig. [REF] compares power consumption on the smartphone.', '1511.06530-1-48-1': 'Each network gives the power consumption of GPU and main memory.', '1511.06530-1-48-2': 'Note that we enlarged the time axis of compressed networks for a better comparison.', '1511.06530-1-48-3': 'We omitted VGG-16 since VGG-16 gives similar trend.', '1511.06530-1-49-0': 'The figure shows that the compression reduces power consumption (Y axis) as well as runtime (X axis), which explains why the reduction in energy consumption is larger than that in runtime in Table [REF].', '1511.06530-1-49-1': 'Fig. [REF] also shows that the GPU power consumption of compressed CNN is smaller than that of uncompressed CNN.', '1511.06530-1-49-2': 'We analyze this due to the extensive usage of [MATH] convolutions in the compressed CNN.', '1511.06530-1-49-3': 'When executing convolutions, we apply optimization techniques such as Caffeinated convolution.', '1511.06530-1-49-4': 'In such a case, in terms of cache efficiency, [MATH] convolutions are inferior to the other convolutions, e.g., [MATH], [MATH], etc.', '1511.06530-1-49-5': 'since the amount of data reuse is proportional to the total size of convolution kernel.', '1511.06530-1-49-6': 'Thus, [MATH] convolutions tend to incur more cache misses than the other larger convolutions.', '1511.06530-1-49-7': 'Cache misses on the mobile GPU without sufficient thread level parallelism often incur stall cycles, i.e., make GPU cores idle consuming less power, which reduces the power consumption of GPU core during the execution of [MATH] convolution.', '1511.06530-1-50-0': 'As mentioned earlier, our proposed method improves cache efficiency by reducing the amount of weights.', '1511.06530-1-50-1': 'However, [MATH] convolutions have negative impacts on cache efficiency and GPU core utilization.', '1511.06530-1-50-2': 'Fig. [REF] shows the combined effects.', '1511.06530-1-50-3': 'In the compressed networks, the power consumption of GPU core is reduced by [MATH] convolutions and tends to change more frequently due to frequent executions of [MATH] convolution while, in the case of uncompressed networks, especially for AlexNet and VGG-S, the power consumption of GPU core tends to be stable during the execution of convolutional layers.', '1511.06530-1-50-4': 'In the case of uncompressed GoogLeNet, the power consumption tends to fluctuate.', '1511.06530-1-50-5': 'It is mainly because (1) GoogLeNet consists of many small layers (about 100 building blocks), and (2) [MATH] convolutions are heavily utilized.', '1511.06530-1-51-0': 'The three compressed networks show similar behavior of frequent fluctuations in power consumption mostly due to [MATH] convolutions.', '1511.06530-1-51-1': 'Fig. [REF] also shows that, in the uncompressed networks, fully connected layers incur significant amount of power consumption in main memory.', '1511.06530-1-51-2': 'It is because the uncompressed networks, especially AlexNet and VGG-S have large numbers (more than tens of mega-bytes) of weights in fully connected layers which incur significant amount of memory accesses.', '1511.06530-1-51-3': 'As shown in Fig. [REF], the proposed scheme reduces the amount of weights at fully connected layers thereby reducing the power consumption in main memory.', '1511.06530-1-52-0': '# Discussion', '1511.06530-1-53-0': 'Although we can obtain very promising results with one-shot rank selection, it is not fully investigated yet whether the selected rank is really optimal or not.', '1511.06530-1-53-1': 'As future work, we will investigate the optimality of our proposed scheme.', '1511.06530-1-53-2': 'The [MATH] convolution is a key operation in our compressed model as well as in inception module of GoogLeNet.', '1511.06530-1-53-3': 'Due to its characteristics, e.g. channel compression and computation reduction, we expect that [MATH] convolutions will become more and more popular in the future.', '1511.06530-1-53-4': 'However, as shown in our experimental results, it lacks in cache efficiency.', '1511.06530-1-53-5': 'We expect further investigations are required to make best use of 1x1 convolutions.', '1511.06530-1-54-0': 'Whole network compression is challenging due to the large design space and associated long design time.', '1511.06530-1-54-1': 'In order to address this problem, we propose a one-shot compression scheme which applies a single general low-rank approximation method and a global rank selection method.', '1511.06530-1-54-2': 'Our one-shot compression enables fast design and easy implementation with publicly available tools.', '1511.06530-1-54-3': 'We evaluated the effectiveness of the proposed scheme on a smartphone and Titan X.', '1511.06530-1-54-4': 'The experiments show that the proposed scheme gives, for four CNNs (AlexNet, VGG-S, GoogLeNet, and VGG-16) average [MATH]), [MATH]), [MATH]), and [MATH]) improvements in runtime (energy consumption) on the smartphone.', '1511.06530-1-55-0': '# Appendices', '1511.06530-1-56-0': '## Experimental Setup', '1511.06530-1-57-0': 'This section describes the details of experimental setup including the measurement system for power consumption and exemplifies the measured data.', '1511.06530-1-58-0': '### Measurement System', '1511.06530-1-59-0': 'Fig. [REF] shows the power measurement system.', '1511.06530-1-59-1': 'As the figure shows, it consists of a probe board (left) having a Samsung Galaxy S6 smartphone and power probes and a monitor board (right).', '1511.06530-1-59-2': 'The probe board provides 8 probes which are connected to the power pins of application processor (to be introduced below).', '1511.06530-1-59-3': 'The power profiling monitor samples, for each power probe, the electric current every 0.1ms and gives power consumption data with time stamps.', '1511.06530-1-60-0': 'Fig. [REF] illustrates the main board of the smartphone (Fig. [REF] (a)), the application processor chip package (red rectangle in Fig. 2 (a)) consisting of the application processor and main memory (LPDDR4 DRAM) in the smartphone (Fig. [REF] (b)), and a simplified block diagram of the application processor (Fig. [REF] (c)).', '1511.06530-1-60-1': 'The power measurement system provides the probes connected to the power pins for mobile GPU (ARM Mali T760 in Fig. [REF] (c)) and main memory (LPDDR4 DRAM in Fig. [REF] (b)).', '1511.06530-1-61-0': '### Measured Data Example: GoogLeNet Case Fig. [REF] shows the power consumption data for the uncompressed GoogLeNet.', '1511.06530-1-61-1': 'We also identified the period of each layer, e.g., the first convolutional layer (Conv 1 in the figure), and the first Inception module (i3a).', '1511.06530-1-61-2': 'As mentioned in our submission, the profile of power consumption shows more frequent fluctuations in Inception modules than in the convolutional layers.', '1511.06530-1-61-3': 'The figure also shows that the first two convolutional layers (Conv 1 and Conv 2) occupy about 1/4 of total energy consumption while Inception modules consume about 3/4 of total energy consumption.', '1511.06530-1-62-0': '## Layerwise Analysis', '1511.06530-1-63-0': 'We report detailed comparison results VGG-S, GoogLeNet, and VGG-16.'}

{'1511.06530-2-0-0': 'Although the latest high-end smartphone has powerful CPU and GPU, running deeper convolutional neural networks (CNNs) for complex tasks such as ImageNet classification on mobile devices is challenging.', '1511.06530-2-0-1': 'To deploy deep CNNs on mobile devices, we present a simple and effective scheme to compress the entire CNN, which we call one-shot whole network compression.', '1511.06530-2-0-2': 'The proposed scheme consists of three steps: (1) rank selection with variational Bayesian matrix factorization, (2) Tucker decomposition on kernel tensor, and (3) fine-tuning to recover accumulated loss of accuracy, and each step can be easily implemented using publicly available tools.', '1511.06530-2-0-3': 'We demonstrate the effectiveness of the proposed scheme by testing the performance of various compressed CNNs (AlexNet, VGG-S, GoogLeNet, and VGG-16) on the smartphone.', '1511.06530-2-0-4': 'Significant reductions in model size, runtime, and energy consumption are obtained, at the cost of small loss in accuracy.', '1511.06530-2-0-5': 'In addition, we address the important implementation level issue on [MATH] convolution, which is a key operation of inception module of GoogLeNet as well as CNNs compressed by our proposed scheme.', '1511.06530-2-1-0': '# Introduction', '1511.06530-2-2-0': 'Deployment of convolutional neural networks (CNNs) for computer vision tasks on mobile devices is gaining more and more attention.', '1511.06530-2-2-1': 'On mobile applications, it is typically assumed that training is performed on the server and test or inference is executed on the mobile devices.', '1511.06530-2-2-2': 'One of the most critical issues in mobile applications of CNNs is that mobile devices have strict constraints in terms of computing power, battery, and memory capacity.', '1511.06530-2-2-3': 'Thus, it is imperative to obtain CNNs tailored to the limited resources of mobile devices.', '1511.06530-2-3-0': 'Deep neural networks are known to be over-parameterized, which facilitates convergence to good local minima of the loss function during training .', '1511.06530-2-3-1': 'To improve test-time performance on mobile devices, such redundancy can be removed from the trained networks without noticeable impact on accuracy.', '1511.06530-2-3-2': 'Recently, there are several studies to apply low-rank approximations to compress CNNs by exploiting redundancy .', '1511.06530-2-3-3': 'Such compressions typically focus on convolution layers since they dominate total computation cost especially in deep neural networks .', '1511.06530-2-3-4': 'Existing methods, though effective in reducing the computation cost of a single convolutional layer, introduce a new challenge called whole network compression which aims at compressing the entire network.', '1511.06530-2-4-0': 'Whole network compression: It is nontrivial to compress whole and very deep CNNs for complex tasks such as ImageNet classification.', '1511.06530-2-4-1': 'Recently, [CITATION] showed that entire convolutional layers can be accelerated with "asymmetric (3d)" decomposition.', '1511.06530-2-4-2': 'In addition, they also presented the effective rank selection and optimization method.', '1511.06530-2-4-3': 'Although their proposed decomposition of layers can be easily implemented in popular development tools (e.g. Caffe, Torch, and Theano), the rank selection and optimization parts still require because they consist of multiple steps and depend on the output of previous layers.', '1511.06530-2-4-4': 'In this paper, we present much simpler but still powerful whole network compression scheme which takes entire convolutional and fully-connected layers into account.', '1511.06530-2-5-0': 'Contribution: This paper makes the following major contributions.', '1511.06530-2-6-0': 'This paper is organized as follows.', '1511.06530-2-6-1': 'Section 2 reviews related work.', '1511.06530-2-6-2': 'Section 3 explains our proposed scheme.', '1511.06530-2-6-3': 'Section 4 gives experimental results.', '1511.06530-2-6-4': 'Section 5 summarizes the paper.', '1511.06530-2-7-0': '# Related Work', '1511.06530-2-8-0': '## CNN Compression', '1511.06530-2-9-0': 'CNN usually consists of convolutional layers and fully-connected layers which dominate computation cost and memory consumption respectively.', '1511.06530-2-9-1': 'After [CITATION] showed the possibility of removing the redundancy of neural networks, several CNN compression techniques have been proposed.', '1511.06530-2-9-2': 'A recent study showed that the weight matrix of a fully-connected layer can be compressed by applying truncated singular value decomposition (SVD) without significant drop in the prediction accuracy.', '1511.06530-2-9-3': 'More recently, various methods based on vector quantization , hashing techniques , circulant projection , and tensor train decomposition were proposed and showed better compression capability than SVD.', '1511.06530-2-9-4': 'To speed up the convolutional layers, several methods based on low-rank decomposition of convolutional kernel tensor were proposed , but they compress only single or a few layers.', '1511.06530-2-10-0': 'Concurrent with our work, [CITATION] presented "asymmetric (3d) decomposition" to accelerate the entire convolutional layers, where the original [MATH] convolution is decomposed to [MATH], [MATH], and [MATH] convolution.', '1511.06530-2-10-1': 'In addition, they also present a rank selection method based on PCA accumulated energy and an optimization method which minimizes the reconstruction error of non-linear responses.', '1511.06530-2-10-2': 'In the extended version , the additional fine-tuning of entire network was considered for further improvement.', '1511.06530-2-10-3': 'Compared with these works, our proposed scheme is different in that (1) Tucker decomposition is adopted to compress the entire convolutional and fully-connected layers, (2) the kernel tensor reconstruction error is minimized instead of non-linear response, (3) a global analytic solution of VBMF is applied to determine the rank of each layer, and (4) a single run of fine-tuning is performed to account for the accumulation of errors.', '1511.06530-2-11-0': 'A pruning approach also aims at reducing the total amount of parameters and operations in the entire network.', '1511.06530-2-11-1': 'Pruning based approaches can give significant reductions in parameter size and computation workload.', '1511.06530-2-11-2': 'However, it is challenging to achieve runtime speed-up with conventional GPU implementation as mentioned in .', '1511.06530-2-12-0': 'Orthogonal to model level compression, implementation level approaches were also proposed.', '1511.06530-2-12-1': 'The FFT method was used to speed-up convolution .', '1511.06530-2-12-2': 'In , CPU code optimizations to speed-up the execution of CNN are extensively explored.', '1511.06530-2-13-0': '## Tensor Decomposition', '1511.06530-2-14-0': 'A tensor is a multi-way array of data.', '1511.06530-2-14-1': 'For example, a vector is 1-way tensor and a matrix is 2-way tensor.', '1511.06530-2-14-2': 'Two of the most popular tensor decomposition models are CANDECOMP/PARAFAC model and Tucker model .', '1511.06530-2-14-3': 'In this paper, we extensively use Tucker model for whole network compression.', '1511.06530-2-14-4': 'Tucker decomposition is a higher order extension of the singular value decomposition (SVD) of matrix, in the perspective of computing the orthonormal spaces associated with the different modes of a tensor.', '1511.06530-2-14-5': 'It simultaneously analyzes mode-[MATH] matricizations of the original tensor, and merges them with the core tensor as illustrated in Fig. [REF].', '1511.06530-2-15-0': 'In our whole network compression scheme, we apply Tucker-2 decomposition, which is also known as GLRAM , from the second convolutional layer to the first fully connected layers.', '1511.06530-2-15-1': 'For the other layers, we apply Tucker-1 decomposition, which is equivalent to SVD.', '1511.06530-2-15-2': 'For more information on the tensor decomposition, the reader is referred to the survey paper .', '1511.06530-2-16-0': '# Proposed Method', '1511.06530-2-17-0': 'Fig. [REF] illustrates our one-shot whole network compression scheme which consists of three steps: (1) rank selection; (2) Tucker decomposition; (3) fine-tuning.', '1511.06530-2-17-1': "In the first step, we analyze principal subspace of mode-3 and mode-4 matricization of each layer's kernel tensor with global analytic variational Bayesian matrix factorization.", '1511.06530-2-17-2': "Then we apply Tucker decomposition on each layer's kernel tensor with previously determined rank.", '1511.06530-2-17-3': 'Finally, we fine-tune the entire network with standard back-propagation.', '1511.06530-2-18-0': '## Tucker Decomposition on Kernel Tensor', '1511.06530-2-19-0': 'Convolution kernel tensor: In CNNs, the convolution operation maps an input (source) tensor [MATH] of size [MATH] into output (target) tensor [MATH] of size [MATH] using the following linear mapping: [EQUATION] where [MATH] is a 4-way kernel tensor of size [MATH], [MATH] is stride, and [MATH] is zero-padding size.', '1511.06530-2-20-0': 'Tucker Decomposition: The rank-[MATH] Tucker decomposition of 4-way kernel tensor [MATH] has the form: [EQUATION] where [MATH] is a core tensor of size [MATH] and [MATH], [MATH], [MATH], and [MATH] are factor matrices of sizes [MATH], [MATH], [MATH], and [MATH], respectively.', '1511.06530-2-21-0': 'In the Tucker decomposition, every mode does not have to be decomposed.', '1511.06530-2-21-1': 'For example, we do not decompose mode-1 and mode-2 which are associated with spatial dimensions because they are already quite small ([MATH] is typically 3 or 5).', '1511.06530-2-21-2': 'Under this variant called Tucker-2 decomposition , the kernel tensor is decomposed to: [EQUATION] where [MATH] is a core tensor of size [MATH].', '1511.06530-2-21-3': 'After substituting [REF] into [REF], performing rearrangements and grouping summands, we obtain the following three consecutive expressions for the approximate evaluation of the convolution [REF]: [EQUATION] where [MATH] and [MATH] are intermediate tensors of sizes [MATH] and [MATH], respectively.', '1511.06530-2-22-0': '[MATH] convolution: As illustrated in Fig. [REF], computing [MATH] from [MATH] in [REF] as well as [MATH] from [MATH] in [REF] is [MATH] convolutions that essentially perform pixel-wise linear re-combination of input maps.', '1511.06530-2-22-1': 'It is introduced in network-in-network and extensively used in inception module of GoogLeNet .', '1511.06530-2-22-2': 'Note that computing [REF] is similar to inception module in the sense that [MATH] convolution is applied after dimensional reduction with [MATH] convolution, but different in the sense that there is no non-linear ReLU function between [REF] and [REF].', '1511.06530-2-22-3': 'In addition, similar to , we compute smaller intermediate output tensor [MATH] in [REF] and then recover its size in [REF].', '1511.06530-2-22-4': 'The Tucker-2 decomposition naturally integrates two compression techniques.', '1511.06530-2-23-0': 'Complexity analysis: The convolution operation in [REF] requires [MATH] parameters and [MATH] multiplication-addition operations.', '1511.06530-2-23-1': 'With Tucker decomposition, compression ratio [MATH] and speed-up ratio [MATH] are given by: [EQUATION] and these are bounded by [MATH].', '1511.06530-2-24-0': 'Tucker vs CP: Recently, CP decomposition is applied to approximate the convolution layers of CNNs for ImageNet which consist of 8 layers .', '1511.06530-2-24-1': 'However it cannot be applied to the entire layers and the instability issue of low-rank CP decomposition is reported .', '1511.06530-2-24-2': 'On the other hand, our kernel tensor approximation with Tucker decomposition can be successfully applied to the entire layers of AlexNet, VGG-S, GoogLeNet, and VGG-16', '1511.06530-2-25-0': '## Rank Selection with Global Analytic VBMF', '1511.06530-2-26-0': 'The rank-[MATH] are very important hyper-parameters which control the trade-off between performance (memory, speed, energy) improvement and accuracy loss.', '1511.06530-2-26-1': 'Instead of selecting the rank-[MATH] by time consuming trial-and-error, we considered data-driven one-shot decision via empirical Bayes with automatic relevance determination (ARD) prior .', '1511.06530-2-27-0': 'At the first time, we designed probabilistic Tucker model which is similar to , and applied empirical variational Bayesian learning.', '1511.06530-2-27-1': 'However, the rank selection results were severely unreliable because they heavily depend on (1) initial condition, (2) noise variance estimation policy, and (3) threshold setting for pruning.', '1511.06530-2-27-2': 'For this reason, we decided to use a sub-optimal but highly reproducible approach.', '1511.06530-2-28-0': 'We employed recently developed global analytic solutions for variational Bayesian matrix factorization (VBMF) .', '1511.06530-2-28-1': 'The global analytic VBMF is a very promising tool because it can automatically find noise variance, rank and even provide theoretical condition for perfect rank recovery .', '1511.06530-2-28-2': 'We determined the rank [MATH] and [MATH] by applying global analytic VBMF on mode-3 matricization (of size [MATH]) and mode-4 matricization (of size [MATH]) of kernel tensor [MATH], respectively.', '1511.06530-2-29-0': '## Fine-Tuning', '1511.06530-2-30-0': 'Because we minimize the reconstruction error of linear kernel tensors instead of non-linear responses, the accuracy is significantly dropped after whole network compression (e.g. more than 50[MATH] in the case of AlexNet).', '1511.06530-2-30-1': 'However, as shown in Fig. [REF], we can easily recover the accuracy by using fine-tuning with ImageNet training dataset.', '1511.06530-2-30-2': 'We observed that accuracy is recovered quickly in one epoch.', '1511.06530-2-30-3': 'However, more than 10 epochs are required to recover the original accuracy.', '1511.06530-2-31-0': 'While reported difficulty on finding a good SGD learning rate, our single learning rate scheduling rule works well for various compressed CNNs.', '1511.06530-2-31-1': 'In our experiment, we set the base learning [MATH] and decrease it by a factor of 10 every 5 epochs.', '1511.06530-2-31-2': 'Because of GPU memory limitation, we set the batch size: 128, 128, 64, and 32 for AlexNet, VGG-S, GoogLeNet, and VGG-16, respectively.', '1511.06530-2-32-0': 'We also tried to train the architecture of the approximated model from scratch on the ImageNet training dataset.', '1511.06530-2-32-1': 'At this time, we only tested the Gaussian random initialization and it did not work.', '1511.06530-2-32-2': 'We leave the use of other initialization methods and batch normalization as future work.', '1511.06530-2-33-0': '# Experiments', '1511.06530-2-34-0': "We used four representative CNNs, AlexNet, VGG-S, GoogLeNet, and VGG-16, which can be downloaded on Berkeley's Caffe model zoo.", '1511.06530-2-34-1': 'In the case of inception module of GoogLeNet, we only compressed the [MATH] convolution kernel which is the main computational part.', '1511.06530-2-34-2': 'In the case of VGG-16, we only compressed the convolutional layers as done in .', '1511.06530-2-34-3': 'Top-5 single-view accuracy is measured using 50,000 validation images from the ImageNet2012 dataset.', '1511.06530-2-35-0': 'We performed experiments on Nvidia Titan X (for fine-tuning and runtime comparison on Caffe+cuDNN2) and a smartphone, Samsung Galaxy S6 (for the comparison of runtime and energy consumption).', '1511.06530-2-35-1': 'The application processor of the smartphone (Exynos 7420) is equipped with a mobile GPU, ARM Mali T760.', '1511.06530-2-35-2': 'Compared with the GPU used on Titan X, the mobile GPU gives 35 times (6.6TFlops vs 190GFlops) lower computing capability and 13 times (336.5GBps vs 25.6GBps) smaller memory bandwidth.', '1511.06530-2-36-0': 'In order to run Caffe models on the mobile GPU, we developed a mobile version of Caffe called S-Caffe (Caffe for Smart mobile devices) where all the Caffe models can run on our target mobile devices (for the moment, Samsung smartphones) without modification.', '1511.06530-2-36-1': 'We also developed an Android App which performs image classification by running each of the four CNNs (AlexNet, VGG-S, GoogLeNet , and VGG-16) on the smartphone.', '1511.06530-2-37-0': 'We measured the power consumption of whole smartphone which is decomposed into the power consumption of GPU, main memory, and the other components of smartphone, e.g., ARM CPU, display, modem, etc. and give component-level analysis, especially, the power consumption of GPU and main memory (see supplementary material for details of measurement environment).', '1511.06530-2-37-1': 'The measurement results of runtime and energy consumption are the average of 50 runs.', '1511.06530-2-38-0': '## Overall Results', '1511.06530-2-39-0': 'Table [REF] shows the overall results for the three CNNs.', '1511.06530-2-39-1': 'Our proposed scheme gives [MATH] (AlexNet), [MATH] (VGG-S), [MATH] (GoogLeNet), and [MATH] (VGG-16) reductions in total weights and FLOPs, respectively.', '1511.06530-2-39-2': 'Such reductions offer [MATH]) runtime improvements on the smartphone (Titan X).', '1511.06530-2-39-3': 'We report the energy consumption of mobile GPU and main memory.', '1511.06530-2-39-4': 'The smartphone gives larger reduction ratios (e.g., [MATH] vs. [MATH] for AlexNet) for energy consumption than runtime.', '1511.06530-2-39-5': 'We will give a detailed analysis in the following subsection.', '1511.06530-2-40-0': "Comparison with [CITATION]'s method: The accuracy of our compressed VGG-16 is 89.40[MATH] for theoretical [MATH] speed-up, and it is comparable to the 89.6[MATH] (88.9[MATH]) for theoretical [MATH]) speed-up in .", '1511.06530-2-41-0': '## Layerwise Analysis', '1511.06530-2-42-0': 'Tables [REF], [REF], [REF] and [REF]', '1511.06530-2-43-0': 'show the detailed comparisons.', '1511.06530-2-43-1': 'Each row has two results (the above one for the original uncompressed CNN and the other one for the compressed CNN), and improvements.', '1511.06530-2-43-2': 'For instance, in Table [REF], the second convolutional layer having the input and output channel dimensions of [MATH] and [MATH] is compressed to give the Tucker-2 ranks of [MATH] and [MATH], which reduces the amount of weights from [MATH] to [MATH].', '1511.06530-2-43-3': 'After compression, a layer in the compressed network performs three matrix multiplications.', '1511.06530-2-43-4': 'We give the details of three matrix multiplications for each of weights, FLOPs, and runtime.', '1511.06530-2-43-5': 'For instance, on the smartphone (column S6 in Table [REF]), the second convolutional layer of compressed AlexNet takes 10.53ms which is decomposed to 0.8ms, 7.43ms and 2.3ms for the three matrix multiplications.', '1511.06530-2-44-0': 'In Tables [REF], [REF], [REF] and [REF] we have two observations.', '1511.06530-2-45-0': 'Observation 1: Given a compressed network, the smartphone tends to give larger performance gain than the Titan X.', '1511.06530-2-45-1': 'It is mainly because the mobile GPU on the smartphone lacks in thread-level parallelism.', '1511.06530-2-45-2': 'It has 24 times less number of threads (2K vs. 48K in terms of maximum number of threads) than that in Titan X. Compression reduces the amount of weights thereby reducing cache conflicts and memory latency.', '1511.06530-2-45-3': 'Due to the small thread-level parallelism, the reduced latency has more impact on the performance of threads on the mobile GPU than that on Titan X.', '1511.06530-2-46-0': 'Observation 2: Given the same compression rate, the smartphone tends to exhibit larger performance gain at fully-connected layers than at convolutional layers.', '1511.06530-2-46-1': 'We think it is also due to the reduced cache conflicts enabled by network compression as explained above.', '1511.06530-2-46-2': 'Especially, in the case of fully-connected layers, the effect of weight reduction can give more significant impact because the weights at the fully-connected layers are utilized only once, often called dead-on-arrival (DoA) data.', '1511.06530-2-46-3': 'In terms of cache performance, such DoA data are much more harmful than convolution kernel weights (which are reused multiple times).', '1511.06530-2-46-4': 'Thus, weight reduction at the fully connected layer can give more significant impact on cache performance thereby exhibiting more performance improvement than in the case of weight reduction at convolutional layers.', '1511.06530-2-47-0': '## Energy Consumption Analysis', '1511.06530-2-48-0': 'Fig. [REF] compares power consumption on the smartphone.', '1511.06530-2-48-1': 'Each network gives the power consumption of GPU and main memory.', '1511.06530-2-48-2': 'Note that we enlarged the time axis of compressed networks for a better comparison.', '1511.06530-2-48-3': 'We omitted VGG-16 since VGG-16 gives similar trend.', '1511.06530-2-49-0': 'The figure shows that the compression reduces power consumption (Y axis) as well as runtime (X axis), which explains why the reduction in energy consumption is larger than that in runtime in Table [REF].', '1511.06530-2-49-1': 'Fig. [REF] also shows that the GPU power consumption of compressed CNN is smaller than that of uncompressed CNN.', '1511.06530-2-49-2': 'We analyze this due to the extensive usage of [MATH] convolutions in the compressed CNN.', '1511.06530-2-49-3': 'When executing convolutions, we apply optimization techniques such as Caffeinated convolution.', '1511.06530-2-49-4': 'In such a case, in terms of cache efficiency, [MATH] convolutions are inferior to the other convolutions, e.g., [MATH], [MATH], etc.', '1511.06530-2-49-5': 'since the amount of data reuse is proportional to the total size of convolution kernel.', '1511.06530-2-49-6': 'Thus, [MATH] convolutions tend to incur more cache misses than the other larger convolutions.', '1511.06530-2-49-7': 'Cache misses on the mobile GPU without sufficient thread level parallelism often incur stall cycles, i.e., make GPU cores idle consuming less power, which reduces the power consumption of GPU core during the execution of [MATH] convolution.', '1511.06530-2-50-0': 'As mentioned earlier, our proposed method improves cache efficiency by reducing the amount of weights.', '1511.06530-2-50-1': 'However, [MATH] convolutions have negative impacts on cache efficiency and GPU core utilization.', '1511.06530-2-50-2': 'Fig. [REF] shows the combined effects.', '1511.06530-2-50-3': 'In the compressed networks, the power consumption of GPU core is reduced by [MATH] convolutions and tends to change more frequently due to frequent executions of [MATH] convolution while, in the case of uncompressed networks, especially for AlexNet and VGG-S, the power consumption of GPU core tends to be stable during the execution of convolutional layers.', '1511.06530-2-50-4': 'In the case of uncompressed GoogLeNet, the power consumption tends to fluctuate.', '1511.06530-2-50-5': 'It is mainly because (1) GoogLeNet consists of many small layers (about 100 building blocks), and (2) [MATH] convolutions are heavily utilized.', '1511.06530-2-51-0': 'The three compressed networks show similar behavior of frequent fluctuations in power consumption mostly due to [MATH] convolutions.', '1511.06530-2-51-1': 'Fig. [REF] also shows that, in the uncompressed networks, fully connected layers incur significant amount of power consumption in main memory.', '1511.06530-2-51-2': 'It is because the uncompressed networks, especially AlexNet and VGG-S have large numbers (more than tens of mega-bytes) of weights in fully connected layers which incur significant amount of memory accesses.', '1511.06530-2-51-3': 'As shown in Fig. [REF], the proposed scheme reduces the amount of weights at fully connected layers thereby reducing the power consumption in main memory.', '1511.06530-2-52-0': '# Discussion', '1511.06530-2-53-0': 'Although we can obtain very promising results with one-shot rank selection, it is not fully investigated yet whether the selected rank is really optimal or not.', '1511.06530-2-53-1': 'As future work, we will investigate the optimality of our proposed scheme.', '1511.06530-2-53-2': 'The [MATH] convolution is a key operation in our compressed model as well as in inception module of GoogLeNet.', '1511.06530-2-53-3': 'Due to its characteristics, e.g. channel compression and computation reduction, we expect that [MATH] convolutions will become more and more popular in the future.', '1511.06530-2-53-4': 'However, as shown in our experimental results, it lacks in cache efficiency.', '1511.06530-2-53-5': 'We expect further investigations are required to make best use of 1x1 convolutions.', '1511.06530-2-54-0': 'Whole network compression is challenging due to the large design space and associated long design time.', '1511.06530-2-54-1': 'In order to address this problem, we propose a one-shot compression scheme which applies a single general low-rank approximation method and a global rank selection method.', '1511.06530-2-54-2': 'Our one-shot compression enables fast design and easy implementation with publicly available tools.', '1511.06530-2-54-3': 'We evaluated the effectiveness of the proposed scheme on a smartphone and Titan X.', '1511.06530-2-54-4': 'The experiments show that the proposed scheme gives, for four CNNs (AlexNet, VGG-S, GoogLeNet, and VGG-16) average [MATH]), [MATH]), [MATH]), and [MATH]) improvements in runtime (energy consumption) on the smartphone.', '1511.06530-2-55-0': '# Appendices', '1511.06530-2-56-0': '## Experimental Setup', '1511.06530-2-57-0': 'This section describes the details of experimental setup including the measurement system for power consumption and exemplifies the measured data.', '1511.06530-2-58-0': '### Measurement System', '1511.06530-2-59-0': 'Fig. [REF] shows the power measurement system.', '1511.06530-2-59-1': 'As the figure shows, it consists of a probe board (left) having a Samsung Galaxy S6 smartphone and power probes and a monitor board (right).', '1511.06530-2-59-2': 'The probe board provides 8 probes which are connected to the power pins of application processor (to be introduced below).', '1511.06530-2-59-3': 'The power profiling monitor samples, for each power probe, the electric current every 0.1ms and gives power consumption data with time stamps.', '1511.06530-2-60-0': 'Fig. [REF] illustrates the main board of the smartphone (Fig. [REF] (a)), the application processor chip package (red rectangle in Fig. 2 (a)) consisting of the application processor and main memory (LPDDR4 DRAM) in the smartphone (Fig. [REF] (b)), and a simplified block diagram of the application processor (Fig. [REF] (c)).', '1511.06530-2-60-1': 'The power measurement system provides the probes connected to the power pins for mobile GPU (ARM Mali T760 in Fig. [REF] (c)) and main memory (LPDDR4 DRAM in Fig. [REF] (b)).', '1511.06530-2-61-0': '### Measured Data Example: GoogLeNet Case Fig. [REF] shows the power consumption data for the uncompressed GoogLeNet.', '1511.06530-2-61-1': 'We also identified the period of each layer, e.g., the first convolutional layer (Conv 1 in the figure), and the first Inception module (i3a).', '1511.06530-2-61-2': 'As mentioned in our submission, the profile of power consumption shows more frequent fluctuations in Inception modules than in the convolutional layers.', '1511.06530-2-61-3': 'The figure also shows that the first two convolutional layers (Conv 1 and Conv 2) occupy about 1/4 of total energy consumption while Inception modules consume about 3/4 of total energy consumption.', '1511.06530-2-62-0': '## Layerwise Analysis', '1511.06530-2-63-0': 'We report detailed comparison results VGG-S, GoogLeNet, and VGG-16.'}

[['1511.06530-1-31-0', '1511.06530-2-31-0'], ['1511.06530-1-31-1', '1511.06530-2-31-1'], ['1511.06530-1-31-2', '1511.06530-2-31-2'], ['1511.06530-1-34-0', '1511.06530-2-34-0'], ['1511.06530-1-34-1', '1511.06530-2-34-1'], ['1511.06530-1-34-2', '1511.06530-2-34-2'], ['1511.06530-1-34-3', '1511.06530-2-34-3'], ['1511.06530-1-14-0', '1511.06530-2-14-0'], ['1511.06530-1-14-1', '1511.06530-2-14-1'], ['1511.06530-1-14-2', '1511.06530-2-14-2'], ['1511.06530-1-14-3', '1511.06530-2-14-3'], ['1511.06530-1-14-4', '1511.06530-2-14-4'], ['1511.06530-1-14-5', '1511.06530-2-14-5'], ['1511.06530-1-10-0', '1511.06530-2-10-0'], ['1511.06530-1-10-1', '1511.06530-2-10-1'], ['1511.06530-1-10-2', '1511.06530-2-10-2'], ['1511.06530-1-10-3', '1511.06530-2-10-3'], ['1511.06530-1-24-0', '1511.06530-2-24-0'], ['1511.06530-1-24-1', '1511.06530-2-24-1'], ['1511.06530-1-24-2', '1511.06530-2-24-2'], ['1511.06530-1-39-0', '1511.06530-2-39-0'], ['1511.06530-1-39-1', '1511.06530-2-39-1'], ['1511.06530-1-39-2', '1511.06530-2-39-2'], ['1511.06530-1-39-3', '1511.06530-2-39-3'], ['1511.06530-1-39-4', '1511.06530-2-39-4'], ['1511.06530-1-39-5', '1511.06530-2-39-5'], ['1511.06530-1-19-0', '1511.06530-2-19-0'], ['1511.06530-1-15-0', '1511.06530-2-15-0'], ['1511.06530-1-15-1', '1511.06530-2-15-1'], ['1511.06530-1-15-2', '1511.06530-2-15-2'], ['1511.06530-1-49-0', '1511.06530-2-49-0'], ['1511.06530-1-49-1', '1511.06530-2-49-1'], ['1511.06530-1-49-2', '1511.06530-2-49-2'], ['1511.06530-1-49-3', '1511.06530-2-49-3'], ['1511.06530-1-49-4', '1511.06530-2-49-4'], ['1511.06530-1-49-5', '1511.06530-2-49-5'], ['1511.06530-1-49-6', '1511.06530-2-49-6'], ['1511.06530-1-49-7', '1511.06530-2-49-7'], ['1511.06530-1-46-0', '1511.06530-2-46-0'], ['1511.06530-1-46-1', '1511.06530-2-46-1'], ['1511.06530-1-46-2', '1511.06530-2-46-2'], ['1511.06530-1-46-3', '1511.06530-2-46-3'], ['1511.06530-1-46-4', '1511.06530-2-46-4'], ['1511.06530-1-9-0', '1511.06530-2-9-0'], ['1511.06530-1-9-1', '1511.06530-2-9-1'], ['1511.06530-1-9-2', '1511.06530-2-9-2'], ['1511.06530-1-9-3', '1511.06530-2-9-3'], ['1511.06530-1-9-4', '1511.06530-2-9-4'], ['1511.06530-1-48-0', '1511.06530-2-48-0'], ['1511.06530-1-48-1', '1511.06530-2-48-1'], ['1511.06530-1-48-2', '1511.06530-2-48-2'], ['1511.06530-1-48-3', '1511.06530-2-48-3'], ['1511.06530-1-36-0', '1511.06530-2-36-0'], ['1511.06530-1-36-1', '1511.06530-2-36-1'], ['1511.06530-1-53-0', '1511.06530-2-53-0'], ['1511.06530-1-53-1', '1511.06530-2-53-1'], ['1511.06530-1-53-2', '1511.06530-2-53-2'], ['1511.06530-1-53-3', '1511.06530-2-53-3'], ['1511.06530-1-53-4', '1511.06530-2-53-4'], ['1511.06530-1-53-5', '1511.06530-2-53-5'], ['1511.06530-1-28-0', '1511.06530-2-28-0'], ['1511.06530-1-28-1', '1511.06530-2-28-1'], ['1511.06530-1-28-2', '1511.06530-2-28-2'], ['1511.06530-1-57-0', '1511.06530-2-57-0'], ['1511.06530-1-43-0', '1511.06530-2-43-0'], ['1511.06530-1-43-1', '1511.06530-2-43-1'], ['1511.06530-1-43-2', '1511.06530-2-43-2'], ['1511.06530-1-43-3', '1511.06530-2-43-3'], ['1511.06530-1-43-4', '1511.06530-2-43-4'], ['1511.06530-1-43-5', '1511.06530-2-43-5'], ['1511.06530-1-12-0', '1511.06530-2-12-0'], ['1511.06530-1-12-1', '1511.06530-2-12-1'], ['1511.06530-1-12-2', '1511.06530-2-12-2'], ['1511.06530-1-17-0', '1511.06530-2-17-0'], ['1511.06530-1-17-1', '1511.06530-2-17-1'], ['1511.06530-1-17-2', '1511.06530-2-17-2'], ['1511.06530-1-17-3', '1511.06530-2-17-3'], ['1511.06530-1-2-0', '1511.06530-2-2-0'], ['1511.06530-1-2-1', '1511.06530-2-2-1'], ['1511.06530-1-2-2', '1511.06530-2-2-2'], ['1511.06530-1-2-3', '1511.06530-2-2-3'], ['1511.06530-1-50-0', '1511.06530-2-50-0'], ['1511.06530-1-50-1', '1511.06530-2-50-1'], ['1511.06530-1-50-2', '1511.06530-2-50-2'], ['1511.06530-1-50-3', '1511.06530-2-50-3'], ['1511.06530-1-50-4', '1511.06530-2-50-4'], ['1511.06530-1-50-5', '1511.06530-2-50-5'], ['1511.06530-1-60-0', '1511.06530-2-60-0'], ['1511.06530-1-60-1', '1511.06530-2-60-1'], ['1511.06530-1-11-0', '1511.06530-2-11-0'], ['1511.06530-1-11-1', '1511.06530-2-11-1'], ['1511.06530-1-11-2', '1511.06530-2-11-2'], ['1511.06530-1-32-0', '1511.06530-2-32-0'], ['1511.06530-1-32-1', '1511.06530-2-32-1'], ['1511.06530-1-32-2', '1511.06530-2-32-2'], ['1511.06530-1-40-0', '1511.06530-2-40-0'], ['1511.06530-1-3-0', '1511.06530-2-3-0'], ['1511.06530-1-3-1', '1511.06530-2-3-1'], ['1511.06530-1-3-2', '1511.06530-2-3-2'], ['1511.06530-1-3-3', '1511.06530-2-3-3'], ['1511.06530-1-3-4', '1511.06530-2-3-4'], ['1511.06530-1-23-0', '1511.06530-2-23-0'], ['1511.06530-1-23-1', '1511.06530-2-23-1'], ['1511.06530-1-27-0', '1511.06530-2-27-0'], ['1511.06530-1-27-1', '1511.06530-2-27-1'], ['1511.06530-1-27-2', '1511.06530-2-27-2'], ['1511.06530-1-61-1', '1511.06530-2-61-1'], ['1511.06530-1-61-2', '1511.06530-2-61-2'], ['1511.06530-1-61-3', '1511.06530-2-61-3'], ['1511.06530-1-22-0', '1511.06530-2-22-0'], ['1511.06530-1-22-1', '1511.06530-2-22-1'], ['1511.06530-1-22-2', '1511.06530-2-22-2'], ['1511.06530-1-22-3', '1511.06530-2-22-3'], ['1511.06530-1-22-4', '1511.06530-2-22-4'], ['1511.06530-1-0-0', '1511.06530-2-0-0'], ['1511.06530-1-0-1', '1511.06530-2-0-1'], ['1511.06530-1-0-2', '1511.06530-2-0-2'], ['1511.06530-1-0-3', '1511.06530-2-0-3'], ['1511.06530-1-0-4', '1511.06530-2-0-4'], ['1511.06530-1-0-5', '1511.06530-2-0-5'], ['1511.06530-1-21-0', '1511.06530-2-21-0'], ['1511.06530-1-21-1', '1511.06530-2-21-1'], ['1511.06530-1-21-2', '1511.06530-2-21-2'], ['1511.06530-1-21-3', '1511.06530-2-21-3'], ['1511.06530-1-59-0', '1511.06530-2-59-0'], ['1511.06530-1-59-1', '1511.06530-2-59-1'], ['1511.06530-1-59-2', '1511.06530-2-59-2'], ['1511.06530-1-59-3', '1511.06530-2-59-3'], ['1511.06530-1-6-0', '1511.06530-2-6-0'], ['1511.06530-1-6-1', '1511.06530-2-6-1'], ['1511.06530-1-6-2', '1511.06530-2-6-2'], ['1511.06530-1-6-3', '1511.06530-2-6-3'], ['1511.06530-1-6-4', '1511.06530-2-6-4'], ['1511.06530-1-51-0', '1511.06530-2-51-0'], ['1511.06530-1-51-1', '1511.06530-2-51-1'], ['1511.06530-1-51-2', '1511.06530-2-51-2'], ['1511.06530-1-51-3', '1511.06530-2-51-3'], ['1511.06530-1-54-0', '1511.06530-2-54-0'], ['1511.06530-1-54-1', '1511.06530-2-54-1'], ['1511.06530-1-54-2', '1511.06530-2-54-2'], ['1511.06530-1-54-3', '1511.06530-2-54-3'], ['1511.06530-1-54-4', '1511.06530-2-54-4'], ['1511.06530-1-4-0', '1511.06530-2-4-0'], ['1511.06530-1-4-1', '1511.06530-2-4-1'], ['1511.06530-1-4-2', '1511.06530-2-4-2'], ['1511.06530-1-4-3', '1511.06530-2-4-3'], ['1511.06530-1-4-4', '1511.06530-2-4-4'], ['1511.06530-1-37-0', '1511.06530-2-37-0'], ['1511.06530-1-37-1', '1511.06530-2-37-1'], ['1511.06530-1-30-0', '1511.06530-2-30-0'], ['1511.06530-1-30-1', '1511.06530-2-30-1'], ['1511.06530-1-30-2', '1511.06530-2-30-2'], ['1511.06530-1-30-3', '1511.06530-2-30-3'], ['1511.06530-1-45-0', '1511.06530-2-45-0'], ['1511.06530-1-45-1', '1511.06530-2-45-1'], ['1511.06530-1-45-2', '1511.06530-2-45-2'], ['1511.06530-1-45-3', '1511.06530-2-45-3'], ['1511.06530-1-26-0', '1511.06530-2-26-0'], ['1511.06530-1-26-1', '1511.06530-2-26-1'], ['1511.06530-1-35-0', '1511.06530-2-35-0'], ['1511.06530-1-35-1', '1511.06530-2-35-1'], ['1511.06530-1-35-2', '1511.06530-2-35-2']]

[['1511.06530-1-31-0', '1511.06530-2-31-0'], ['1511.06530-1-31-1', '1511.06530-2-31-1'], ['1511.06530-1-31-2', '1511.06530-2-31-2'], ['1511.06530-1-34-0', '1511.06530-2-34-0'], ['1511.06530-1-34-1', '1511.06530-2-34-1'], ['1511.06530-1-34-2', '1511.06530-2-34-2'], ['1511.06530-1-34-3', '1511.06530-2-34-3'], ['1511.06530-1-14-0', '1511.06530-2-14-0'], ['1511.06530-1-14-1', '1511.06530-2-14-1'], ['1511.06530-1-14-2', '1511.06530-2-14-2'], ['1511.06530-1-14-3', '1511.06530-2-14-3'], ['1511.06530-1-14-4', '1511.06530-2-14-4'], ['1511.06530-1-14-5', '1511.06530-2-14-5'], ['1511.06530-1-10-0', '1511.06530-2-10-0'], ['1511.06530-1-10-1', '1511.06530-2-10-1'], ['1511.06530-1-10-2', '1511.06530-2-10-2'], ['1511.06530-1-10-3', '1511.06530-2-10-3'], ['1511.06530-1-24-0', '1511.06530-2-24-0'], ['1511.06530-1-24-1', '1511.06530-2-24-1'], ['1511.06530-1-24-2', '1511.06530-2-24-2'], ['1511.06530-1-39-0', '1511.06530-2-39-0'], ['1511.06530-1-39-1', '1511.06530-2-39-1'], ['1511.06530-1-39-2', '1511.06530-2-39-2'], ['1511.06530-1-39-3', '1511.06530-2-39-3'], ['1511.06530-1-39-4', '1511.06530-2-39-4'], ['1511.06530-1-39-5', '1511.06530-2-39-5'], ['1511.06530-1-19-0', '1511.06530-2-19-0'], ['1511.06530-1-15-0', '1511.06530-2-15-0'], ['1511.06530-1-15-1', '1511.06530-2-15-1'], ['1511.06530-1-15-2', '1511.06530-2-15-2'], ['1511.06530-1-49-0', '1511.06530-2-49-0'], ['1511.06530-1-49-1', '1511.06530-2-49-1'], ['1511.06530-1-49-2', '1511.06530-2-49-2'], ['1511.06530-1-49-3', '1511.06530-2-49-3'], ['1511.06530-1-49-4', '1511.06530-2-49-4'], ['1511.06530-1-49-5', '1511.06530-2-49-5'], ['1511.06530-1-49-6', '1511.06530-2-49-6'], ['1511.06530-1-49-7', '1511.06530-2-49-7'], ['1511.06530-1-46-0', '1511.06530-2-46-0'], ['1511.06530-1-46-1', '1511.06530-2-46-1'], ['1511.06530-1-46-2', '1511.06530-2-46-2'], ['1511.06530-1-46-3', '1511.06530-2-46-3'], ['1511.06530-1-46-4', '1511.06530-2-46-4'], ['1511.06530-1-9-0', '1511.06530-2-9-0'], ['1511.06530-1-9-1', '1511.06530-2-9-1'], ['1511.06530-1-9-2', '1511.06530-2-9-2'], ['1511.06530-1-9-3', '1511.06530-2-9-3'], ['1511.06530-1-9-4', '1511.06530-2-9-4'], ['1511.06530-1-48-0', '1511.06530-2-48-0'], ['1511.06530-1-48-1', '1511.06530-2-48-1'], ['1511.06530-1-48-2', '1511.06530-2-48-2'], ['1511.06530-1-48-3', '1511.06530-2-48-3'], ['1511.06530-1-36-0', '1511.06530-2-36-0'], ['1511.06530-1-36-1', '1511.06530-2-36-1'], ['1511.06530-1-53-0', '1511.06530-2-53-0'], ['1511.06530-1-53-1', '1511.06530-2-53-1'], ['1511.06530-1-53-2', '1511.06530-2-53-2'], ['1511.06530-1-53-3', '1511.06530-2-53-3'], ['1511.06530-1-53-4', '1511.06530-2-53-4'], ['1511.06530-1-53-5', '1511.06530-2-53-5'], ['1511.06530-1-28-0', '1511.06530-2-28-0'], ['1511.06530-1-28-1', '1511.06530-2-28-1'], ['1511.06530-1-28-2', '1511.06530-2-28-2'], ['1511.06530-1-57-0', '1511.06530-2-57-0'], ['1511.06530-1-43-0', '1511.06530-2-43-0'], ['1511.06530-1-43-1', '1511.06530-2-43-1'], ['1511.06530-1-43-2', '1511.06530-2-43-2'], ['1511.06530-1-43-3', '1511.06530-2-43-3'], ['1511.06530-1-43-4', '1511.06530-2-43-4'], ['1511.06530-1-43-5', '1511.06530-2-43-5'], ['1511.06530-1-12-0', '1511.06530-2-12-0'], ['1511.06530-1-12-1', '1511.06530-2-12-1'], ['1511.06530-1-12-2', '1511.06530-2-12-2'], ['1511.06530-1-17-0', '1511.06530-2-17-0'], ['1511.06530-1-17-1', '1511.06530-2-17-1'], ['1511.06530-1-17-2', '1511.06530-2-17-2'], ['1511.06530-1-17-3', '1511.06530-2-17-3'], ['1511.06530-1-2-0', '1511.06530-2-2-0'], ['1511.06530-1-2-1', '1511.06530-2-2-1'], ['1511.06530-1-2-2', '1511.06530-2-2-2'], ['1511.06530-1-2-3', '1511.06530-2-2-3'], ['1511.06530-1-50-0', '1511.06530-2-50-0'], ['1511.06530-1-50-1', '1511.06530-2-50-1'], ['1511.06530-1-50-2', '1511.06530-2-50-2'], ['1511.06530-1-50-3', '1511.06530-2-50-3'], ['1511.06530-1-50-4', '1511.06530-2-50-4'], ['1511.06530-1-50-5', '1511.06530-2-50-5'], ['1511.06530-1-60-0', '1511.06530-2-60-0'], ['1511.06530-1-60-1', '1511.06530-2-60-1'], ['1511.06530-1-11-0', '1511.06530-2-11-0'], ['1511.06530-1-11-1', '1511.06530-2-11-1'], ['1511.06530-1-11-2', '1511.06530-2-11-2'], ['1511.06530-1-32-0', '1511.06530-2-32-0'], ['1511.06530-1-32-1', '1511.06530-2-32-1'], ['1511.06530-1-32-2', '1511.06530-2-32-2'], ['1511.06530-1-40-0', '1511.06530-2-40-0'], ['1511.06530-1-3-0', '1511.06530-2-3-0'], ['1511.06530-1-3-1', '1511.06530-2-3-1'], ['1511.06530-1-3-2', '1511.06530-2-3-2'], ['1511.06530-1-3-3', '1511.06530-2-3-3'], ['1511.06530-1-3-4', '1511.06530-2-3-4'], ['1511.06530-1-23-0', '1511.06530-2-23-0'], ['1511.06530-1-23-1', '1511.06530-2-23-1'], ['1511.06530-1-27-0', '1511.06530-2-27-0'], ['1511.06530-1-27-1', '1511.06530-2-27-1'], ['1511.06530-1-27-2', '1511.06530-2-27-2'], ['1511.06530-1-61-1', '1511.06530-2-61-1'], ['1511.06530-1-61-2', '1511.06530-2-61-2'], ['1511.06530-1-61-3', '1511.06530-2-61-3'], ['1511.06530-1-22-0', '1511.06530-2-22-0'], ['1511.06530-1-22-1', '1511.06530-2-22-1'], ['1511.06530-1-22-2', '1511.06530-2-22-2'], ['1511.06530-1-22-3', '1511.06530-2-22-3'], ['1511.06530-1-22-4', '1511.06530-2-22-4'], ['1511.06530-1-0-0', '1511.06530-2-0-0'], ['1511.06530-1-0-1', '1511.06530-2-0-1'], ['1511.06530-1-0-2', '1511.06530-2-0-2'], ['1511.06530-1-0-3', '1511.06530-2-0-3'], ['1511.06530-1-0-4', '1511.06530-2-0-4'], ['1511.06530-1-0-5', '1511.06530-2-0-5'], ['1511.06530-1-21-0', '1511.06530-2-21-0'], ['1511.06530-1-21-1', '1511.06530-2-21-1'], ['1511.06530-1-21-2', '1511.06530-2-21-2'], ['1511.06530-1-21-3', '1511.06530-2-21-3'], ['1511.06530-1-59-0', '1511.06530-2-59-0'], ['1511.06530-1-59-1', '1511.06530-2-59-1'], ['1511.06530-1-59-2', '1511.06530-2-59-2'], ['1511.06530-1-59-3', '1511.06530-2-59-3'], ['1511.06530-1-6-0', '1511.06530-2-6-0'], ['1511.06530-1-6-1', '1511.06530-2-6-1'], ['1511.06530-1-6-2', '1511.06530-2-6-2'], ['1511.06530-1-6-3', '1511.06530-2-6-3'], ['1511.06530-1-6-4', '1511.06530-2-6-4'], ['1511.06530-1-51-0', '1511.06530-2-51-0'], ['1511.06530-1-51-1', '1511.06530-2-51-1'], ['1511.06530-1-51-2', '1511.06530-2-51-2'], ['1511.06530-1-51-3', '1511.06530-2-51-3'], ['1511.06530-1-54-0', '1511.06530-2-54-0'], ['1511.06530-1-54-1', '1511.06530-2-54-1'], ['1511.06530-1-54-2', '1511.06530-2-54-2'], ['1511.06530-1-54-3', '1511.06530-2-54-3'], ['1511.06530-1-54-4', '1511.06530-2-54-4'], ['1511.06530-1-4-0', '1511.06530-2-4-0'], ['1511.06530-1-4-1', '1511.06530-2-4-1'], ['1511.06530-1-4-2', '1511.06530-2-4-2'], ['1511.06530-1-4-3', '1511.06530-2-4-3'], ['1511.06530-1-4-4', '1511.06530-2-4-4'], ['1511.06530-1-37-0', '1511.06530-2-37-0'], ['1511.06530-1-37-1', '1511.06530-2-37-1'], ['1511.06530-1-30-0', '1511.06530-2-30-0'], ['1511.06530-1-30-1', '1511.06530-2-30-1'], ['1511.06530-1-30-2', '1511.06530-2-30-2'], ['1511.06530-1-30-3', '1511.06530-2-30-3'], ['1511.06530-1-45-0', '1511.06530-2-45-0'], ['1511.06530-1-45-1', '1511.06530-2-45-1'], ['1511.06530-1-45-2', '1511.06530-2-45-2'], ['1511.06530-1-45-3', '1511.06530-2-45-3'], ['1511.06530-1-26-0', '1511.06530-2-26-0'], ['1511.06530-1-26-1', '1511.06530-2-26-1'], ['1511.06530-1-35-0', '1511.06530-2-35-0'], ['1511.06530-1-35-1', '1511.06530-2-35-1'], ['1511.06530-1-35-2', '1511.06530-2-35-2']]

[]

[]

[]

[]

['1511.06530-1-5-0', '1511.06530-1-20-0', '1511.06530-1-42-0', '1511.06530-1-44-0', '1511.06530-1-63-0', '1511.06530-2-5-0', '1511.06530-2-20-0', '1511.06530-2-42-0', '1511.06530-2-44-0', '1511.06530-2-63-0']

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

https://arxiv.org/abs/1511.06530

cond-mat-0506045

{'cond-mat-0506045-1-0-0': 'We report on the effect of elastic intervalley scattering on the energy transport between electrons and phonons in many-valley semiconductors.', 'cond-mat-0506045-1-0-1': 'We derive a general expression for the electron-phonon energy flow rate at the limit where elastic intervalley scattering dominates over diffusion.', 'cond-mat-0506045-1-0-2': 'Electron heating experiments on heavily doped n-type Si samples with electron concentration in the range [MATH] m[MATH] are performed at sub-1 K temperatures.', 'cond-mat-0506045-1-0-3': 'We find a good agreement between the theory and the experiment.', 'cond-mat-0506045-1-1-0': 'Since the low temperature hot electron experiments by Roukes et al. [CITATION], the energy transport between electrons and phonons has continued to be a topical subject.', 'cond-mat-0506045-1-1-1': 'Recently, there has been significant experimental and theoretical interest in the electron-phonon (e-ph) energy relaxation in metals and semiconductors at low temperatures [CITATION] .', 'cond-mat-0506045-1-1-2': 'The understanding of thermal e-ph coupling is important for several low temperature devices such as microbolometers, calorimeters and on chip refrigerators [CITATION].', 'cond-mat-0506045-1-1-3': 'This coupling plays also an important role in correct interpretation of low temperature experiments [CITATION] and the e-ph energy relaxation rate gives direct information about phonon mediated electron dephasing [CITATION].', 'cond-mat-0506045-1-2-0': 'Interaction between electrons and phonons is strongly affected by the disorder of the electron system and, therefore, the problem is commonly divided into two special cases: pure and impure (or diffusive) limit of e-ph interaction.', 'cond-mat-0506045-1-2-1': 'The cross-over between these two regions is defined as [MATH] , where [MATH] is the phonon wavevector and [MATH] the electron mean free path.', 'cond-mat-0506045-1-2-2': 'If the whole phonon system is to be considered then the phonon wavevector can be conveniently replaced by the thermal phonon wave vector [MATH], where [MATH] is the temperature of the lattice and [MATH] the sound velocity.', 'cond-mat-0506045-1-2-3': 'Recent theory for single-valley semiconductors [CITATION] predicts that the e-ph energy relaxation is strongly enhanced when the system enters from the pure limit ([MATH]) to the diffusive limit ([MATH]).', 'cond-mat-0506045-1-2-4': 'The behavior is the opposite in comparison to metals where it is well known, since the pioneering work by A. B. Pippard [CITATION], that the disorder of the electron system tends to suppress the e-ph energy relaxation (see also Ref. [CITATION]).', 'cond-mat-0506045-1-2-5': 'In semiconductors, due to small electron density, the e-ph interacion can be described by deformation potential coupling constants, which do not depend on the electronic variables, while in metals the coupling strongly depends on the electron momentum [CITATION].', 'cond-mat-0506045-1-2-6': 'This fundamental difference eventually leads to disorder enhancement of the relaxation in the diffusive limit in single-valley semiconductors [CITATION].', 'cond-mat-0506045-1-3-0': 'In many-valley semiconductors the situation is further altered due to intervalley scattering, which is the topic of our work.', 'cond-mat-0506045-1-3-1': 'We approach the e-ph energy transport problem by first considering the phonon energy attenuation rate due to electrons (or phonon-electron energy relaxation rate).', 'cond-mat-0506045-1-3-2': 'This procedure is attractive, because it enables straigth forward comparison between our work and previous literature, which has concentrated mainly on ultrasonic attenuation [CITATION].', 'cond-mat-0506045-1-3-3': 'We derive expression for the total e-ph energy flow rate (by using the phonon energy attenuation rate) and perform low temperature electron heating experiments to heavily doped n-type silicon samples.', 'cond-mat-0506045-1-3-4': 'We find excellent agreement between the theoretical and the experimental e-ph temperature responses.', 'cond-mat-0506045-1-4-0': 'As discussed above the electron-phonon coupling in semiconductors can be described through deformation potential coupling constants, which do not depend on the electron variables (in a single valley).', 'cond-mat-0506045-1-4-1': 'For example, in Si the strain induced conduction band energy shift is [CITATION] [EQUATION] where [MATH] is the valley index (see Fig. [REF]), [MATH] is the symmetric strain component of displacement [MATH] is the component along the valley axis), and [MATH]) is the dilatational (uniaxial) deformation potential constant.', 'cond-mat-0506045-1-4-2': 'When an acoustic phonon propagates through a many-valley electron system the strain field associated with the phonon makes the band edges of the valley minima oscillate according to Eq. ([REF]).', 'cond-mat-0506045-1-4-3': 'In the diffusive long wavelength limit the phonon momentum itself cannot transfer the electrons from one minima to another, because this process would require [MATH] is the lattice constant).', 'cond-mat-0506045-1-4-4': 'Then the electron nonequilibrium, generated by the acoustic field, relaxes towards local equilibrium by two processes: diffusion and elastic intervalley impurity scattering.', 'cond-mat-0506045-1-4-5': 'When the strain lifts the valley degeneracy elastic intervalley scattering provides a path for the electron system to relax towards local equilibrium.', 'cond-mat-0506045-1-4-6': 'This path is favorable if the time scale related to diffusion over length [MATH]is sufficiently large, i.e., when [MATH] where [MATH] is the elastic intervalley transition rate ([MATH] is the diffusion coefficient).', 'cond-mat-0506045-1-4-7': 'In the case of Si the coaxial valleys that are coupled via rate [MATH] are always equivalent.', 'cond-mat-0506045-1-4-8': 'Thus in the place of [MATH] we should use rate [MATH] which couples the perpendicular valleys (see Fig. [REF]).', 'cond-mat-0506045-1-5-0': 'By using a Boltzmann equation approach - described, e.g., in Ref. [CITATION] - and using Eq. ([REF]) we find that phonon-electron energy relaxation rate is [CITATION] [EQUATION] where we have assumed linear dispersion relations [MATH] is the mode index).', 'cond-mat-0506045-1-5-1': 'In the case of Si the effective relaxation rate [MATH] and the factor [MATH], where [MATH] and [MATH] ) is the phonon polarization ( [MATH])[CITATION].', 'cond-mat-0506045-1-5-2': '[MATH] and [MATH] are the (single spin and valley) density of states at Fermi level and density of the crystal, respectively.', 'cond-mat-0506045-1-5-3': 'We have especially chosen to write [MATH] in the form of Eq. ([REF]), because this form is actually fully general.', 'cond-mat-0506045-1-5-4': 'In the general case [MATH] depends on the details of [MATH] (and on the number of valleys) and for the effective relaxation rate we have [MATH].', 'cond-mat-0506045-1-5-5': 'There is also a single valley contribution arising from electron number density fluctuation in the valleys, but this term is small due to strong screening: the ratio between single valley and many-valley phonon-electron energy relaxation rate scales roughly as [MATH], where [MATH] is the inverse of the screening length [CITATION].', 'cond-mat-0506045-1-6-0': 'We can describe a degenerate electron system by an equilibrium distribution at temperature [MATH].', 'cond-mat-0506045-1-6-1': 'This holds even in the presence of net heat flow between electrons and phonons.', 'cond-mat-0506045-1-6-2': 'The heat flow only creates a non-equilibrium between the electrons and phonons, which relaxes towards equilibrium at rate [MATH] per single phonon mode.', 'cond-mat-0506045-1-6-3': 'By following Perrin and Budd [CITATION] this non-equilibrium can be expressed using the relaxation time approximation of the phonon-electron collision integral [EQUATION] where [MATH] and [MATH] are the nonequilibrium and equilibrium phonon distribution functions, respectively.', 'cond-mat-0506045-1-6-4': 'The total stationary heat flow [MATH] through the coupled electron-phonon system is the energy average of the collision integral:', 'cond-mat-0506045-1-7-0': '[EQUATION] where the summation is performed over the acoustic eigenmodes of the crystal.', 'cond-mat-0506045-1-7-1': 'The only experimentally meaningful situation is such that the phonon system is coupled to some thermalizing bath, which is at temperature [MATH].', 'cond-mat-0506045-1-7-2': 'If the coupling is strong or [MATH] is small we can approximate [MATH], where [MATH] is the (possibly local) phonon temperature, and Eq. ([REF]) reduces to the familiar form: [EQUATION] where [MATH] is the energy flow rate control function.', 'cond-mat-0506045-1-7-3': 'Using Eqs. ([REF])-([REF]) and assuming that [MATH] is clearly below unity the energy flow rate control function can be expressed in a closed form [EQUATION] where the first equality is valid for arbitrary many-valley system.', 'cond-mat-0506045-1-7-4': 'The constant [MATH] and [MATH] stands for average over a solid angle.', 'cond-mat-0506045-1-7-5': 'The second equality applies for silicon and there we have further assumed that the phonon eigenmodes are isotropic and purely longitudinal ([MATH] ) or transversal ([MATH]) and that they are described by the respective sound velocities [MATH] and [MATH].', 'cond-mat-0506045-1-8-0': 'Eq. ([REF]) is valid when [MATH] and [MATH].', 'cond-mat-0506045-1-8-1': 'At low temperatures the dominating condition is the latter and can be written also as [MATH], where [MATH] is the momentum relaxation time.', 'cond-mat-0506045-1-8-2': 'Condition [MATH] defines the crossover temperature below which elastic intervalley scattering induced electron-phonon relaxation dominates over diffusion.', 'cond-mat-0506045-1-8-3': 'If [MATH] is not several orders of magnitude larger than [MATH] this differs very little from the impure-pure threshold [MATH].', 'cond-mat-0506045-1-9-0': 'Eq. ([REF]) suggests that intervalley scattering induced electron-phonon energy relaxation rate [MATH], which can be seen from approximate rate equation [MATH], where [MATH] is the electron heat capacity.', 'cond-mat-0506045-1-9-1': 'As the phonon mediated dephasing rate [MATH][CITATION] we find an important relation [MATH].', 'cond-mat-0506045-1-10-0': 'Note that the single-valley contribution to the energy flow rate, calculated by Sergeev et al. [CITATION], is several orders of magnitude smaller than Eq. ([REF]) at small [MATH], due to strong screening, even at modest electron densities.', 'cond-mat-0506045-1-10-1': 'Thus the many-valley effect is expected to dominate at low temperatures.', 'cond-mat-0506045-1-10-2': 'We have tested Eq. ([REF]) experimentally in the case of n[MATH] Si:', 'cond-mat-0506045-1-11-0': 'The n[MATH] Si samples were fabricated on unibond silicon-on-insulator substrates.', 'cond-mat-0506045-1-11-1': 'Properties of the samples are listed in Table [REF] and a detailed description about the sample fabrication can be found in [CITATION].', 'cond-mat-0506045-1-11-2': 'The sample geometry and the experiment is depicted in Fig. [REF](a).', 'cond-mat-0506045-1-11-3': 'In the experiments the samples were mounted on a sample holder of a dilution refrigerator.', 'cond-mat-0506045-1-11-4': 'The electron and phonon temperatures were simultaneously measured by utilizing the superconductor-semiconductor-superconductor (S-Sm-S) thermometry [CITATION] while the electron gas in the Si film was heated with a DC power density [MATH] created by electric current density [MATH].', 'cond-mat-0506045-1-11-5': 'Note that as the electronic coupling to the n[MATH] Si film is made via superconducting Al the heat flow in the experiment follows accurately a path electrons[MATH]phonons[MATH]substrate/sample holder (phonons) and, therefore, the experimental [MATH] is equal to the left-hand-side of Eq. ([REF]).', 'cond-mat-0506045-1-11-6': 'Heating of the electron gas can cause a substantial increase in the temperature of the phonon thermometer, as reported recently for a similar n+ Si sample as discussed here [CITATION].', 'cond-mat-0506045-1-11-7': 'To assure that the nonequilibrium phonon distribution (of the phonons that interact with the electrons in the Si layer) can be reasonably described with an equilibrium distribution function we consider heating power range where [MATH] is clearly below unity.', 'cond-mat-0506045-1-12-0': 'Fig. [REF](b) shows the experimental power density vs. [MATH] at bath temperature [MATH] mK.', 'cond-mat-0506045-1-12-1': 'The solid curves are least of square fits to [MATH] with the slope [MATH] as a single fitting parameter.', 'cond-mat-0506045-1-12-2': 'We observe that the electron-phonon temperature response predicted by Eq. ([REF]) describes all the samples extremely well.', 'cond-mat-0506045-1-12-3': 'The slopes [MATH] are plotted against the electron density in Fig. [REF] (left vertical axis).', 'cond-mat-0506045-1-12-4': '[MATH] increases as function of [MATH], which is expected result, because [MATH].', 'cond-mat-0506045-1-13-0': 'In order to perform more quantitative comparison between the theory and experiment we estimate the density of states from free electron gas expression [MATH], where we use valley degeneracy [MATH] and Si density of states mass [MATH] is the free electron mass).', 'cond-mat-0506045-1-13-1': 'For the other parameters we use the typical values for Si: [MATH] eV, [MATH] kgm[MATH] m/s.', 'cond-mat-0506045-1-13-2': 'Now the intervalley scattering time [MATH] can be determined from [MATH] [see Eq. ([REF])] and it is plotted on the right vertical axis of Fig. [REF].', 'cond-mat-0506045-1-13-3': 'The cross-over temperature from the condition [MATH] is found to be [MATH] 2 K (average from all the samples).', 'cond-mat-0506045-1-13-4': 'Thus we are at [MATH] limit.', 'cond-mat-0506045-1-14-0': 'Eq. ([REF]) gives also the phonon or ultrasonic attenuation constant [MATH].', 'cond-mat-0506045-1-14-1': 'Using this result and the ultrasonic attenuation data obtained by M. Dutoit [CITATION] from n[MATH] Si with [MATH] m[MATH] at temperature of 2 K [MATH]at [MATH] 1/s we find [MATH] ps.', 'cond-mat-0506045-1-14-2': 'This fits to our measurements extremely well, which is an important result: experiment that probes heat transport between electrons and one coherent acoustic mode [CITATION] coincides with our experiment that probes heat transport between electrons and phonon gas obeying quantum statistics.', 'cond-mat-0506045-1-15-0': 'At high [MATH] one would expect slowly decreasing or a roughly constant [MATH] while our results show a weak increase as a function of [MATH].', 'cond-mat-0506045-1-15-1': 'This unexpected result could be explained by noting that our samples are in the limit of strong disorder ([MATH] on average from Table [REF]).', 'cond-mat-0506045-1-15-2': 'Whereas, Eq. ([REF]) is essentially based on a semiclassical free electron gas model, at least finally when the approximation [MATH] is made.', 'cond-mat-0506045-1-15-3': 'Correction terms arising from interaction and quantum interference effects can be included to our model in the spirit of Ref. [CITATION].', 'cond-mat-0506045-1-15-4': 'As similar terms appear in the conductivity the magnitude of these quantum corrections can be estimated from low field magnetoresistance and temperature dependency of resistivity.', 'cond-mat-0506045-1-15-5': 'At the moment such data for thin film n[MATH] Si is not available.', 'cond-mat-0506045-1-16-0': 'Finally, we point out that the intervalley scattering induced electron-phonon energy relaxation can be observed also in several other material systems than n[MATH] Si.', 'cond-mat-0506045-1-16-1': 'Canonical examples would be n[MATH] Ge and two dimensional electron gas in [MATH] Si inversion layer.', 'cond-mat-0506045-1-16-2': 'As the [MATH]-point in the valence band of elemental semiconductors is divided into heavy hole, light hole and split-off bands the effect should be particularly strong in various hole systems.', 'cond-mat-0506045-1-16-3': 'However, due to complicated nature of the valence band maximum and effectively zero distance of the different bands in k-space the theory, which is valid for conduction band electrons, should be modified[CITATION].', 'cond-mat-0506045-1-17-0': 'In summary, we have studied the effect of elastic intervalley transitions on the electron-phonon energy relaxation rate in many-valley semiconductors in the diffusive limit.', 'cond-mat-0506045-1-17-1': 'We derived a general expression for the electron-phonon energy flow rate [Eq. ([REF])] and discussed the special case of n [MATH] silicon.', 'cond-mat-0506045-1-17-2': 'Low temperature experiments on heavily doped Si samples were performed and good agreement between the theory and the experiment was found.', 'cond-mat-0506045-1-18-0': 'We want to acknowledge the skillful contribution of M. Markkanen in the sample fabrication.', 'cond-mat-0506045-1-18-1': 'This work has been partially funded by the Academy of Finland (project numbers 46804, 205470, 205467 and 53903).', 'cond-mat-0506045-1-18-2': 'PK also acknowledges financial support of Ulla Tuominen and Emil Aaltonen Foundations.'}

{'cond-mat-0506045-2-0-0': 'We report on the effect of elastic intervalley scattering on the energy transport between electrons and phonons in many-valley semiconductors.', 'cond-mat-0506045-2-0-1': 'We derive a general expression for the electron-phonon energy flow rate at the limit where elastic intervalley scattering dominates over diffusion.', 'cond-mat-0506045-2-0-2': 'Electron heating experiments on heavily doped n-type Si samples with electron concentration in the range [MATH] m[MATH] are performed at sub-1 K temperatures.', 'cond-mat-0506045-2-0-3': 'We find a good agreement between the theory and the experiment.', 'cond-mat-0506045-2-1-0': 'Since the low temperature hot electron experiments by Roukes et al. [CITATION], the energy transport between electrons and phonons has continued to be a topical subject.', 'cond-mat-0506045-2-1-1': 'Recently, there has been significant experimental and theoretical interest in the electron-phonon (e-ph) energy relaxation in metals and semiconductors at low temperatures [CITATION] .', 'cond-mat-0506045-2-1-2': 'The understanding of thermal e-ph coupling is important for several low temperature devices such as microbolometers, calorimeters and on chip refrigerators [CITATION].', 'cond-mat-0506045-2-1-3': 'This coupling plays also an important role in correct interpretation of low temperature experiments [CITATION] and the e-ph energy relaxation rate gives direct information about phonon mediated electron dephasing [CITATION].', 'cond-mat-0506045-2-2-0': 'Interaction between electrons and phonons is strongly affected by the disorder of the electron system and, therefore, the problem is commonly divided into two special cases: pure and impure (or diffusive) limit of e-ph interaction.', 'cond-mat-0506045-2-2-1': 'The cross-over between these two regions is defined as [MATH] , where [MATH] is the phonon wavevector and [MATH] the electron mean free path.', 'cond-mat-0506045-2-2-2': 'If the whole phonon system is to be considered then the phonon wavevector can be conveniently replaced by the thermal phonon wave vector [MATH], where [MATH] is the temperature of the lattice and [MATH] the sound velocity.', 'cond-mat-0506045-2-2-3': 'Recent theory for single-valley semiconductors [CITATION] predicts that the e-ph energy relaxation is strongly enhanced when the system enters from the pure limit ([MATH]) to the diffusive limit ([MATH]).', 'cond-mat-0506045-2-2-4': 'The behavior is the opposite in comparison to metals where it is well known, since the pioneering work by A. B. Pippard [CITATION], that the disorder of the electron system tends to suppress the e-ph energy relaxation (see also Ref. [CITATION]).', 'cond-mat-0506045-2-2-5': 'In semiconductors, due to small electron density, the e-ph interaction can be described by deformation potential coupling constants, which do not depend on the electronic variables, while in metals the coupling strongly depends on the electron momentum [CITATION].', 'cond-mat-0506045-2-2-6': 'This fundamental difference eventually leads to disorder enhancement of the relaxation in the diffusive limit in single-valley semiconductors [CITATION].', 'cond-mat-0506045-2-3-0': 'In many-valley semiconductors the situation is further altered due to intervalley scattering, which is the topic of our work.', 'cond-mat-0506045-2-3-1': 'Due to lack of screening the e-ph energy flow rate is strongly enhanced in many valley semiconductors in comparison to single valley ones at diffusive low temperature limit.', 'cond-mat-0506045-2-3-2': 'We approach the e-ph energy transport problem by first considering the phonon energy attenuation rate due to electrons (or phonon-electron energy relaxation rate).', 'cond-mat-0506045-2-3-3': 'This procedure is attractive, because it enables straigth forward comparison between our work and previous literature, which has concentrated mainly on ultrasonic attenuation [CITATION].', 'cond-mat-0506045-2-3-4': 'We derive expression for the total e-ph energy flow rate (by using the phonon energy attenuation rate) and perform low temperature electron heating experiments to heavily doped n-type silicon samples.', 'cond-mat-0506045-2-3-5': 'We find excellent agreement between the theoretical and the experimental e-ph temperature responses.', 'cond-mat-0506045-2-4-0': 'As discussed above the electron-phonon coupling in semiconductors can be described through deformation potential coupling constants, which do not depend on the electron variables (in a single valley).', 'cond-mat-0506045-2-4-1': 'The strain induced conduction band energy shifts [MATH], [MATH] is the number of valleys) can be written conveniently in matrix notation as [MATH], where [MATH] and [MATH] is the deformation potential [MATH] matrix (containing the deformation potential coupling constants).', 'cond-mat-0506045-2-4-2': '[MATH] is the strain component vector and [MATH] are the symmetric strain components of displacement [MATH].', 'cond-mat-0506045-2-4-3': 'For example, for the six Si conduction band minima (see Fig. [REF]) we have [MATH] [CITATION], where [MATH]) is the dilatational (uniaxial) deformation potential constant.', 'cond-mat-0506045-2-5-0': 'Here we deal with long wavelength limit where the phonon field can be identified with a classical acoustic wave [MATH] with polarization [MATH]).', 'cond-mat-0506045-2-5-1': 'The strain now reduces to [MATH]), which can be expressed in matrix form as [MATH] and we find equation [EQUATION] which describes how the displacement makes the band edges oscillate in a many valley system.', 'cond-mat-0506045-2-5-2': 'In the diffusive long wavelength limit the phonon momentum itself cannot transfer the electrons from one minima to another, because this process would require large momentum [MATH] is the lattice constant).', 'cond-mat-0506045-2-5-3': 'Then the electron nonequilibrium, generated by the acoustic field, relaxes towards local equilibrium by two processes: diffusion and elastic intervalley impurity scattering.', 'cond-mat-0506045-2-5-4': 'When the strain lifts the valley degeneracy elastic intervalley scattering provides a path for the electron system to relax towards local equilibrium.', 'cond-mat-0506045-2-5-5': 'This path is favorable if the time scale related to diffusion over length [MATH]is sufficiently large, i.e., when [MATH] where [MATH] is the total elastic intervalley transition rate ([MATH] is the diffusion coefficient).', 'cond-mat-0506045-2-5-6': 'In this limit the linearized many-valley relaxation-time Boltzmann equation [CITATION] reduces to a simple rate equation, which couples the change in the electron density [MATH] of valley [MATH] to that of valley [MATH] via intervalley scattering rate [MATH]: [EQUATION]', 'cond-mat-0506045-2-5-7': 'Here [MATH] is the single spin and valley density of states at Fermi level [MATH].', 'cond-mat-0506045-2-5-8': 'We assume that strain equivalent valleys are coupled with rate [MATH] and that the valleys whose degeneracy can be lifted with strain are coupled with rate [MATH].', 'cond-mat-0506045-2-5-9': 'In the case of Si the coaxial valleys are always equivalent whereas the degeneracy of the perpendicular valleys can be lifted (see Fig. [REF]).', 'cond-mat-0506045-2-5-10': 'Now the solution of Eq. ([REF]) is [MATH], where [MATH] and [MATH].', 'cond-mat-0506045-2-5-11': 'The phonon-electron relaxation rate [MATH] is related to the dissipated heat [MATH] of the acoustic field through standard relation [MATH], where [MATH] is the acoustic energy flux density and [MATH] stands for time average.', 'cond-mat-0506045-2-5-12': 'Using this relation and Eq. ([REF]) we find [EQUATION] where we have used linear dispersion relations [MATH] is the mode index).', 'cond-mat-0506045-2-5-13': 'The factor [MATH] and it obviously depends only on the polarization [MATH], on the direction of propagation [MATH] and on the deformation potential coupling constants.', 'cond-mat-0506045-2-5-14': 'In the case of Si we have [MATH] and [MATH].', 'cond-mat-0506045-2-5-15': 'Note that Eq. ([REF] ) does not depend on screening, because there are no total electron density fluctuations, i.e., [MATH].', 'cond-mat-0506045-2-6-0': 'We can describe a degenerate electron system by an equilibrium distribution at temperature [MATH].', 'cond-mat-0506045-2-6-1': 'This holds even in the presence of net heat flow between electrons and phonons.', 'cond-mat-0506045-2-6-2': 'The heat flow only creates a non-equilibrium between the electrons and phonons, which relaxes towards equilibrium at rate [MATH] per single phonon mode.', 'cond-mat-0506045-2-6-3': 'By following Perrin and Budd [CITATION] this non-equilibrium can be expressed using the relaxation time approximation of the phonon-electron collision integral [EQUATION] where [MATH] and [MATH] are the nonequilibrium and equilibrium phonon distribution functions, respectively.', 'cond-mat-0506045-2-6-4': 'The total stationary heat flow [MATH] through the coupled electron-phonon system is the energy average of the collision integral: [EQUATION] where the summation is performed over the acoustic eigenmodes of the crystal.', 'cond-mat-0506045-2-6-5': 'The only experimentally meaningful situation is such that the phonon system is coupled to some thermalizing bath, which is at temperature [MATH].', 'cond-mat-0506045-2-6-6': 'If the coupling is strong or [MATH] is small we can approximate [MATH], where [MATH] is the (possibly local) phonon temperature, and Eq. ([REF]) reduces to the familiar form: [EQUATION] where [MATH] is the energy flow rate control function.', 'cond-mat-0506045-2-6-7': 'Using Eqs. ([REF])-([REF]) and assuming that [MATH] is clearly below unity the energy flow rate control function can be expressed in a closed form [EQUATION] where the first equality is valid for arbitrary many-valley system.', 'cond-mat-0506045-2-6-8': 'The constant [MATH] and [MATH] stands for average over a solid angle.', 'cond-mat-0506045-2-6-9': 'The second equality applies for silicon and there we have further assumed that the phonon eigenmodes are isotropic and that they are described by the longitudinal and transversal sound velocities [MATH] and [MATH].', 'cond-mat-0506045-2-7-0': 'Eq. ([REF]) is valid when [MATH] and [MATH].', 'cond-mat-0506045-2-7-1': 'At low temperatures the dominating condition is the latter and can be written also as [MATH], where [MATH] is the momentum relaxation time.', 'cond-mat-0506045-2-7-2': 'Condition [MATH] defines the crossover temperature below which elastic intervalley scattering induced electron-phonon relaxation dominates over diffusion.', 'cond-mat-0506045-2-7-3': 'If [MATH] is not orders of magnitude larger than [MATH] this differs very little from the impure-pure threshold [MATH].', 'cond-mat-0506045-2-8-0': 'Eq. ([REF]) suggests that intervalley scattering induced electron-phonon energy relaxation rate [MATH], which can be seen from approximate rate equation [MATH], where [MATH] is the electron heat capacity.', 'cond-mat-0506045-2-8-1': 'As the phonon mediated dephasing rate [MATH][CITATION] we find an important relation [MATH].', 'cond-mat-0506045-2-9-0': 'As already pointed above screening plays no role in [MATH] and as a result intervalley scattering induced electron-phonon energy flow rate in Eq. ([REF]) does not include any screening parameters, like for example screening wave vector [MATH].', 'cond-mat-0506045-2-9-1': 'Note, however, that there exists also single-valley contribution to the energy relaxation which is due to number density fluctuations in a single valley, but this conribution is strongly screened in doped semiconductors [CITATION].', 'cond-mat-0506045-2-9-2': 'By using the single valley result calculated by Sergeev et al. [CITATION] and Eq. ([REF]) we find that the ratio between many-valley and single valley energy flow rate scales roughly as [MATH], where [MATH] nm[MATH]and [MATH]=K. Thus the many-valley effect is expected to fully dominate in the diffusive limit at high electron densities and low temperatures.', 'cond-mat-0506045-2-9-3': 'We have tested Eq. ([REF]) experimentally in the case of n[MATH] Si:', 'cond-mat-0506045-2-10-0': 'The n[MATH] Si samples were fabricated on unibond silicon-on-insulator substrates.', 'cond-mat-0506045-2-10-1': 'Properties of the samples are listed in Table [REF] and a detailed description about the sample fabrication can be found in [CITATION].', 'cond-mat-0506045-2-10-2': 'The sample geometry and the experiment is depicted in Fig. [REF](a).', 'cond-mat-0506045-2-10-3': 'In the experiments the samples were mounted on a sample holder of a dilution refrigerator.', 'cond-mat-0506045-2-10-4': 'The electron and phonon temperatures were simultaneously measured by utilizing the superconductor-semiconductor-superconductor (S-Sm-S) thermometry [CITATION] while the electron gas in the Si film was heated with a DC power density [MATH] created by electric current density [MATH].', 'cond-mat-0506045-2-10-5': 'Note that as the electronic coupling to the n[MATH] Si film is made via superconducting Al the heat flow in the experiment follows accurately a path electrons[MATH]phonons[MATH]substrate/sample holder (phonons) and, therefore, the experimental [MATH] is equal to the left-hand-side of Eq. ([REF]).', 'cond-mat-0506045-2-10-6': 'Heating of the electron gas can cause a substantial increase in the temperature of the phonon thermometer, as reported recently for a similar n+ Si sample as discussed here [CITATION].', 'cond-mat-0506045-2-10-7': 'To assure that the nonequilibrium phonon distribution (of the phonons that interact with the electrons in the Si layer) can be reasonably described with an equilibrium distribution function we consider heating power range where [MATH] is clearly below unity.', 'cond-mat-0506045-2-11-0': 'Fig. [REF](b) shows the experimental power density vs. [MATH] at bath temperature [MATH] mK.', 'cond-mat-0506045-2-11-1': 'The solid curves are least of square fits to [MATH] with the slope [MATH] as a single fitting parameter.', 'cond-mat-0506045-2-11-2': 'We observe that the electron-phonon temperature response predicted by Eq. ([REF]) describes all the samples extremely well.', 'cond-mat-0506045-2-11-3': 'The slopes [MATH] are plotted against the electron density in Fig. [REF] (left vertical axis).', 'cond-mat-0506045-2-11-4': '[MATH] increases as function of [MATH], which is expected result, because [MATH].', 'cond-mat-0506045-2-12-0': 'In order to perform more quantitative comparison between the theory and experiment we estimate the density of states from free electron gas expression [MATH], where we use Si density of states mass [MATH] is the free electron mass).', 'cond-mat-0506045-2-12-1': 'For the other parameters we use the typical values for Si: [MATH] eV, [MATH] kgm[MATH] m/s.', 'cond-mat-0506045-2-12-2': 'Now the intervalley scattering time [MATH] can be determined from [MATH] [see Eq. ([REF])] and it is plotted on the right vertical axis of Fig. [REF].', 'cond-mat-0506045-2-12-3': 'The cross-over temperature from the condition [MATH] is found to be [MATH] 5 K (average from all the samples).', 'cond-mat-0506045-2-12-4': 'Thus we are at [MATH] limit.', 'cond-mat-0506045-2-13-0': 'Eq. ([REF]) gives also the phonon or ultrasonic attenuation constant [MATH].', 'cond-mat-0506045-2-13-1': 'Using this result and the ultrasonic attenuation data obtained by M. Dutoit [CITATION] from n[MATH] Si with [MATH] m[MATH] at temperature of 2 K [MATH]at [MATH] 1/s we find [MATH] m[MATH] ps.', 'cond-mat-0506045-2-13-2': 'This fits to our measurements extremely well, which is an important result: experiment that probes heat transport between electrons and one coherent acoustic mode [CITATION] coincides with our experiment that probes heat transport between electrons and phonon gas obeying quantum statistics.', 'cond-mat-0506045-2-14-0': 'At high [MATH] one would expect slowly decreasing or a roughly constant [MATH] while our results show a weak increase as a function of [MATH].', 'cond-mat-0506045-2-14-1': 'This unexpected result could be explained by noting that our samples are in the limit of strong disorder ([MATH] on average from Table [REF]).', 'cond-mat-0506045-2-14-2': 'Whereas, Eq. ([REF]) is essentially based on a semiclassical free electron gas model, at least finally when the approximation [MATH] is made.', 'cond-mat-0506045-2-14-3': 'Correction terms arising from interaction and quantum interference effects can be included to our model in the spirit of Ref. [CITATION].', 'cond-mat-0506045-2-14-4': 'As similar terms appear in the conductivity the magnitude of these quantum corrections can be estimated from low field magnetoresistance and temperature dependency of resistivity.', 'cond-mat-0506045-2-14-5': 'At the moment such data for thin film n[MATH] Si is not available.', 'cond-mat-0506045-2-15-0': 'Finally, we point out that the intervalley scattering induced electron-phonon energy relaxation can be observed also in several other material systems than n[MATH] Si.', 'cond-mat-0506045-2-15-1': 'Canonical examples would be n[MATH] Ge and two dimensional electron gas in [MATH] Si inversion layer.', 'cond-mat-0506045-2-15-2': 'As the [MATH]-point in the valence band of elemental semiconductors is divided into heavy hole, light hole and split-off bands the effect should be particularly strong in various hole systems.', 'cond-mat-0506045-2-15-3': 'However, due to complicated nature of the valence band maximum and effectively zero distance of the different bands in k-space the theory, which is valid for conduction band electrons, should be modified[CITATION].', 'cond-mat-0506045-2-16-0': 'In summary, we have studied the effect of elastic intervalley transitions on the electron-phonon energy relaxation rate in many-valley semiconductors in the diffusive limit.', 'cond-mat-0506045-2-16-1': 'We derived a general expression for the electron-phonon energy flow rate [Eq. ([REF])] and discussed the special case of n [MATH] silicon.', 'cond-mat-0506045-2-16-2': 'Low temperature experiments on heavily doped Si samples were performed and good agreement between the theory and the experiment was found.', 'cond-mat-0506045-2-17-0': 'We want to acknowledge the skillful contribution of M. Markkanen in the sample fabrication.', 'cond-mat-0506045-2-17-1': 'This work has been partially funded by the Academy of Finland (project numbers 46804, 205470, 205467 and 53903).', 'cond-mat-0506045-2-17-2': 'PK also acknowledges financial support of Ulla Tuominen and Emil Aaltonen Foundations.'}

[['cond-mat-0506045-1-11-0', 'cond-mat-0506045-2-10-0'], ['cond-mat-0506045-1-11-1', 'cond-mat-0506045-2-10-1'], ['cond-mat-0506045-1-11-2', 'cond-mat-0506045-2-10-2'], ['cond-mat-0506045-1-11-3', 'cond-mat-0506045-2-10-3'], ['cond-mat-0506045-1-11-4', 'cond-mat-0506045-2-10-4'], ['cond-mat-0506045-1-11-5', 'cond-mat-0506045-2-10-5'], ['cond-mat-0506045-1-11-6', 'cond-mat-0506045-2-10-6'], ['cond-mat-0506045-1-11-7', 'cond-mat-0506045-2-10-7'], ['cond-mat-0506045-1-17-0', 'cond-mat-0506045-2-16-0'], ['cond-mat-0506045-1-17-1', 'cond-mat-0506045-2-16-1'], ['cond-mat-0506045-1-17-2', 'cond-mat-0506045-2-16-2'], ['cond-mat-0506045-1-3-0', 'cond-mat-0506045-2-3-0'], ['cond-mat-0506045-1-3-1', 'cond-mat-0506045-2-3-2'], ['cond-mat-0506045-1-3-2', 'cond-mat-0506045-2-3-3'], ['cond-mat-0506045-1-3-3', 'cond-mat-0506045-2-3-4'], ['cond-mat-0506045-1-3-4', 'cond-mat-0506045-2-3-5'], ['cond-mat-0506045-1-8-1', 'cond-mat-0506045-2-7-1'], ['cond-mat-0506045-1-8-2', 'cond-mat-0506045-2-7-2'], ['cond-mat-0506045-1-18-0', 'cond-mat-0506045-2-17-0'], ['cond-mat-0506045-1-18-1', 'cond-mat-0506045-2-17-1'], ['cond-mat-0506045-1-18-2', 'cond-mat-0506045-2-17-2'], ['cond-mat-0506045-1-1-0', 'cond-mat-0506045-2-1-0'], ['cond-mat-0506045-1-1-1', 'cond-mat-0506045-2-1-1'], ['cond-mat-0506045-1-1-2', 'cond-mat-0506045-2-1-2'], ['cond-mat-0506045-1-1-3', 'cond-mat-0506045-2-1-3'], ['cond-mat-0506045-1-12-0', 'cond-mat-0506045-2-11-0'], ['cond-mat-0506045-1-12-1', 'cond-mat-0506045-2-11-1'], ['cond-mat-0506045-1-12-2', 'cond-mat-0506045-2-11-2'], ['cond-mat-0506045-1-12-3', 'cond-mat-0506045-2-11-3'], ['cond-mat-0506045-1-12-4', 'cond-mat-0506045-2-11-4'], ['cond-mat-0506045-1-15-0', 'cond-mat-0506045-2-14-0'], ['cond-mat-0506045-1-15-1', 'cond-mat-0506045-2-14-1'], ['cond-mat-0506045-1-15-2', 'cond-mat-0506045-2-14-2'], ['cond-mat-0506045-1-15-3', 'cond-mat-0506045-2-14-3'], ['cond-mat-0506045-1-15-4', 'cond-mat-0506045-2-14-4'], ['cond-mat-0506045-1-15-5', 'cond-mat-0506045-2-14-5'], ['cond-mat-0506045-1-0-0', 'cond-mat-0506045-2-0-0'], ['cond-mat-0506045-1-0-1', 'cond-mat-0506045-2-0-1'], ['cond-mat-0506045-1-0-2', 'cond-mat-0506045-2-0-2'], ['cond-mat-0506045-1-0-3', 'cond-mat-0506045-2-0-3'], ['cond-mat-0506045-1-14-0', 'cond-mat-0506045-2-13-0'], ['cond-mat-0506045-1-14-2', 'cond-mat-0506045-2-13-2'], ['cond-mat-0506045-1-2-0', 'cond-mat-0506045-2-2-0'], ['cond-mat-0506045-1-2-1', 'cond-mat-0506045-2-2-1'], ['cond-mat-0506045-1-2-2', 'cond-mat-0506045-2-2-2'], ['cond-mat-0506045-1-2-3', 'cond-mat-0506045-2-2-3'], ['cond-mat-0506045-1-2-6', 'cond-mat-0506045-2-2-6'], ['cond-mat-0506045-1-16-0', 'cond-mat-0506045-2-15-0'], ['cond-mat-0506045-1-16-1', 'cond-mat-0506045-2-15-1'], ['cond-mat-0506045-1-16-2', 'cond-mat-0506045-2-15-2'], ['cond-mat-0506045-1-16-3', 'cond-mat-0506045-2-15-3'], ['cond-mat-0506045-1-13-1', 'cond-mat-0506045-2-12-1'], ['cond-mat-0506045-1-13-2', 'cond-mat-0506045-2-12-2'], ['cond-mat-0506045-1-13-4', 'cond-mat-0506045-2-12-4'], ['cond-mat-0506045-1-9-0', 'cond-mat-0506045-2-8-0'], ['cond-mat-0506045-1-9-1', 'cond-mat-0506045-2-8-1'], ['cond-mat-0506045-1-6-0', 'cond-mat-0506045-2-6-0'], ['cond-mat-0506045-1-6-1', 'cond-mat-0506045-2-6-1'], ['cond-mat-0506045-1-6-2', 'cond-mat-0506045-2-6-2'], ['cond-mat-0506045-1-6-3', 'cond-mat-0506045-2-6-3'], ['cond-mat-0506045-1-7-1', 'cond-mat-0506045-2-6-5'], ['cond-mat-0506045-1-7-2', 'cond-mat-0506045-2-6-6'], ['cond-mat-0506045-1-7-3', 'cond-mat-0506045-2-6-7'], ['cond-mat-0506045-1-7-4', 'cond-mat-0506045-2-6-8'], ['cond-mat-0506045-1-4-0', 'cond-mat-0506045-2-4-0'], ['cond-mat-0506045-1-4-4', 'cond-mat-0506045-2-5-3'], ['cond-mat-0506045-1-4-5', 'cond-mat-0506045-2-5-4'], ['cond-mat-0506045-1-8-3', 'cond-mat-0506045-2-7-3'], ['cond-mat-0506045-1-14-1', 'cond-mat-0506045-2-13-1'], ['cond-mat-0506045-1-2-4', 'cond-mat-0506045-2-2-4'], ['cond-mat-0506045-1-2-5', 'cond-mat-0506045-2-2-5'], ['cond-mat-0506045-1-13-0', 'cond-mat-0506045-2-12-0'], ['cond-mat-0506045-1-13-3', 'cond-mat-0506045-2-12-3'], ['cond-mat-0506045-1-7-5', 'cond-mat-0506045-2-6-9'], ['cond-mat-0506045-1-4-3', 'cond-mat-0506045-2-5-2'], ['cond-mat-0506045-1-4-6', 'cond-mat-0506045-2-5-5'], ['cond-mat-0506045-1-7-0', 'cond-mat-0506045-2-6-4'], ['cond-mat-0506045-1-4-1', 'cond-mat-0506045-2-4-1'], ['cond-mat-0506045-1-4-1', 'cond-mat-0506045-2-4-3'], ['cond-mat-0506045-1-5-0', 'cond-mat-0506045-2-5-6'], ['cond-mat-0506045-1-5-0', 'cond-mat-0506045-2-5-12'], ['cond-mat-0506045-1-5-1', 'cond-mat-0506045-2-5-11'], ['cond-mat-0506045-1-5-1', 'cond-mat-0506045-2-5-14'], ['cond-mat-0506045-1-5-2', 'cond-mat-0506045-2-5-7'], ['cond-mat-0506045-1-5-5', 'cond-mat-0506045-2-5-6'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-8'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-9'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-14']]

[['cond-mat-0506045-1-11-0', 'cond-mat-0506045-2-10-0'], ['cond-mat-0506045-1-11-1', 'cond-mat-0506045-2-10-1'], ['cond-mat-0506045-1-11-2', 'cond-mat-0506045-2-10-2'], ['cond-mat-0506045-1-11-3', 'cond-mat-0506045-2-10-3'], ['cond-mat-0506045-1-11-4', 'cond-mat-0506045-2-10-4'], ['cond-mat-0506045-1-11-5', 'cond-mat-0506045-2-10-5'], ['cond-mat-0506045-1-11-6', 'cond-mat-0506045-2-10-6'], ['cond-mat-0506045-1-11-7', 'cond-mat-0506045-2-10-7'], ['cond-mat-0506045-1-17-0', 'cond-mat-0506045-2-16-0'], ['cond-mat-0506045-1-17-1', 'cond-mat-0506045-2-16-1'], ['cond-mat-0506045-1-17-2', 'cond-mat-0506045-2-16-2'], ['cond-mat-0506045-1-3-0', 'cond-mat-0506045-2-3-0'], ['cond-mat-0506045-1-3-1', 'cond-mat-0506045-2-3-2'], ['cond-mat-0506045-1-3-2', 'cond-mat-0506045-2-3-3'], ['cond-mat-0506045-1-3-3', 'cond-mat-0506045-2-3-4'], ['cond-mat-0506045-1-3-4', 'cond-mat-0506045-2-3-5'], ['cond-mat-0506045-1-8-1', 'cond-mat-0506045-2-7-1'], ['cond-mat-0506045-1-8-2', 'cond-mat-0506045-2-7-2'], ['cond-mat-0506045-1-18-0', 'cond-mat-0506045-2-17-0'], ['cond-mat-0506045-1-18-1', 'cond-mat-0506045-2-17-1'], ['cond-mat-0506045-1-18-2', 'cond-mat-0506045-2-17-2'], ['cond-mat-0506045-1-1-0', 'cond-mat-0506045-2-1-0'], ['cond-mat-0506045-1-1-1', 'cond-mat-0506045-2-1-1'], ['cond-mat-0506045-1-1-2', 'cond-mat-0506045-2-1-2'], ['cond-mat-0506045-1-1-3', 'cond-mat-0506045-2-1-3'], ['cond-mat-0506045-1-12-0', 'cond-mat-0506045-2-11-0'], ['cond-mat-0506045-1-12-1', 'cond-mat-0506045-2-11-1'], ['cond-mat-0506045-1-12-2', 'cond-mat-0506045-2-11-2'], ['cond-mat-0506045-1-12-3', 'cond-mat-0506045-2-11-3'], ['cond-mat-0506045-1-12-4', 'cond-mat-0506045-2-11-4'], ['cond-mat-0506045-1-15-0', 'cond-mat-0506045-2-14-0'], ['cond-mat-0506045-1-15-1', 'cond-mat-0506045-2-14-1'], ['cond-mat-0506045-1-15-2', 'cond-mat-0506045-2-14-2'], ['cond-mat-0506045-1-15-3', 'cond-mat-0506045-2-14-3'], ['cond-mat-0506045-1-15-4', 'cond-mat-0506045-2-14-4'], ['cond-mat-0506045-1-15-5', 'cond-mat-0506045-2-14-5'], ['cond-mat-0506045-1-0-0', 'cond-mat-0506045-2-0-0'], ['cond-mat-0506045-1-0-1', 'cond-mat-0506045-2-0-1'], ['cond-mat-0506045-1-0-2', 'cond-mat-0506045-2-0-2'], ['cond-mat-0506045-1-0-3', 'cond-mat-0506045-2-0-3'], ['cond-mat-0506045-1-14-0', 'cond-mat-0506045-2-13-0'], ['cond-mat-0506045-1-14-2', 'cond-mat-0506045-2-13-2'], ['cond-mat-0506045-1-2-0', 'cond-mat-0506045-2-2-0'], ['cond-mat-0506045-1-2-1', 'cond-mat-0506045-2-2-1'], ['cond-mat-0506045-1-2-2', 'cond-mat-0506045-2-2-2'], ['cond-mat-0506045-1-2-3', 'cond-mat-0506045-2-2-3'], ['cond-mat-0506045-1-2-6', 'cond-mat-0506045-2-2-6'], ['cond-mat-0506045-1-16-0', 'cond-mat-0506045-2-15-0'], ['cond-mat-0506045-1-16-1', 'cond-mat-0506045-2-15-1'], ['cond-mat-0506045-1-16-2', 'cond-mat-0506045-2-15-2'], ['cond-mat-0506045-1-16-3', 'cond-mat-0506045-2-15-3'], ['cond-mat-0506045-1-13-1', 'cond-mat-0506045-2-12-1'], ['cond-mat-0506045-1-13-2', 'cond-mat-0506045-2-12-2'], ['cond-mat-0506045-1-13-4', 'cond-mat-0506045-2-12-4'], ['cond-mat-0506045-1-9-0', 'cond-mat-0506045-2-8-0'], ['cond-mat-0506045-1-9-1', 'cond-mat-0506045-2-8-1'], ['cond-mat-0506045-1-6-0', 'cond-mat-0506045-2-6-0'], ['cond-mat-0506045-1-6-1', 'cond-mat-0506045-2-6-1'], ['cond-mat-0506045-1-6-2', 'cond-mat-0506045-2-6-2'], ['cond-mat-0506045-1-6-3', 'cond-mat-0506045-2-6-3'], ['cond-mat-0506045-1-7-1', 'cond-mat-0506045-2-6-5'], ['cond-mat-0506045-1-7-2', 'cond-mat-0506045-2-6-6'], ['cond-mat-0506045-1-7-3', 'cond-mat-0506045-2-6-7'], ['cond-mat-0506045-1-7-4', 'cond-mat-0506045-2-6-8'], ['cond-mat-0506045-1-4-0', 'cond-mat-0506045-2-4-0'], ['cond-mat-0506045-1-4-4', 'cond-mat-0506045-2-5-3'], ['cond-mat-0506045-1-4-5', 'cond-mat-0506045-2-5-4'], ['cond-mat-0506045-1-2-4', 'cond-mat-0506045-2-2-4']]

[['cond-mat-0506045-1-8-3', 'cond-mat-0506045-2-7-3'], ['cond-mat-0506045-1-14-1', 'cond-mat-0506045-2-13-1'], ['cond-mat-0506045-1-2-5', 'cond-mat-0506045-2-2-5'], ['cond-mat-0506045-1-13-0', 'cond-mat-0506045-2-12-0'], ['cond-mat-0506045-1-13-3', 'cond-mat-0506045-2-12-3'], ['cond-mat-0506045-1-7-5', 'cond-mat-0506045-2-6-9'], ['cond-mat-0506045-1-4-3', 'cond-mat-0506045-2-5-2'], ['cond-mat-0506045-1-4-6', 'cond-mat-0506045-2-5-5']]

[]

[['cond-mat-0506045-1-7-0', 'cond-mat-0506045-2-6-4'], ['cond-mat-0506045-1-4-1', 'cond-mat-0506045-2-4-1'], ['cond-mat-0506045-1-4-1', 'cond-mat-0506045-2-4-3'], ['cond-mat-0506045-1-5-0', 'cond-mat-0506045-2-5-6'], ['cond-mat-0506045-1-5-0', 'cond-mat-0506045-2-5-12'], ['cond-mat-0506045-1-5-1', 'cond-mat-0506045-2-5-11'], ['cond-mat-0506045-1-5-1', 'cond-mat-0506045-2-5-14'], ['cond-mat-0506045-1-5-2', 'cond-mat-0506045-2-5-7'], ['cond-mat-0506045-1-5-5', 'cond-mat-0506045-2-5-6'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-8'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-9'], ['cond-mat-0506045-1-4-7', 'cond-mat-0506045-2-5-14']]

[]

['cond-mat-0506045-1-6-4', 'cond-mat-0506045-1-8-0', 'cond-mat-0506045-1-10-2', 'cond-mat-0506045-2-7-0', 'cond-mat-0506045-2-9-3']

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

https://arxiv.org/abs/cond-mat/0506045

1611.04664

{'1611.04664-1-0-0': 'For every [MATH], and every [MATH], we prove that there are a computable function [EQUATION] and a finite union [MATH] of proper torsion cosets [MATH] such that, for every [MATH], [MATH] contains all but at most [MATH] of the torsion points [MATH] satisfying [MATH].', '1611.04664-1-0-1': 'This extends a well known structural theorem from torsion points lying exactly on a subvariety to torsion points lying very near to the subvariety.', '1611.04664-1-0-2': 'As a consequence, we prove that the averages of [MATH] over [MATH] converge as [MATH] to the Mahler measure of [MATH]: [EQUATION].', '1611.04664-1-0-3': 'By the work of B. Kitchens, D. Lind, K. Schmidt and T. Ward, this convergence consequence amounts to the following statement in dynamics: For every Noetherian [MATH]-action [MATH] by automorphisms of a compact abelian group [MATH] having a finite topological entropy [MATH], the exponential growth rate [MATH] of the number [MATH] of connected components of the group [MATH] of [MATH]-periodic points of [MATH] exists as [MATH], and equals the topological entropy [MATH].', '1611.04664-1-0-4': 'Moreover, it follows that all weak-[MATH] limit measures of the push-forwards of the Haar measures on [MATH], under any a sequence of positive integers [MATH], are measures of maximum entropy [MATH].', '1611.04664-1-1-0': 'The convergence issue solved here had been recently raised by J. H. Silverman, as a conjecture.', '1611.04664-1-1-1': "We give some indication on how an alternative, very different route to Silverman's conjecture and to its dynamical implication is also possible through Habegger's recent work on Diophantine approximation to definable sets.", '1611.04664-1-1-2': 'Our own solution given in this paper is direct and self-contained, while our main result on the uniform boundedness of the number of exceptional points extends to Diophantine approximations by points of small canonical height, and is best possible in that context.', '1611.04664-1-2-0': '# Introduction', '1611.04664-1-3-0': 'Ergodic and dynamical systems theory have been marked over the past three decades by an extensive interaction with central problems of classical Diophantine analysis.', '1611.04664-1-3-1': 'This continues a much older tradition begun by Hermann Weyl in a 1914 paper [CITATION] that he conceived as "an application of number theory" to Boltzmann\'s ergodic hypothesis (Sur une application de la theorie des nombres a la mecanique statistique et la theorie des perturbations).', '1611.04664-1-3-2': "In more recent times, which have witnessed ergodic methods penetrating deeply into all aspects of number theory, the direction of implications is usually reversed, and another line of inquiry, emphasizing higher rank and rigidity phenomena with their relation to Diophantine approximations, was started with Furstenberg's influential 1966 paper [CITATION], where the famous [MATH] problem was formulated.", '1611.04664-1-3-3': "Our present paper owes something to both traditions, by using number theory to verify a hypothesis on the distribution of periodic trajectories for a class of higher rank dynamical systems that contains Furstenberg's [MATH] system as a very special case: the Noetherian [MATH]-actions by automorphisms of a compact abelian group.", '1611.04664-1-3-4': 'In the rank one case this problem was solved long ago, by Lind in [CITATION].', '1611.04664-1-3-5': 'Our result addresses the higher rank situation, where the number theory is less understood, and could, conceivably, be expected to behave differently.', '1611.04664-1-4-0': 'A superb introduction to ergodic ideas in number theory is in the book [CITATION] by Einsiedler and Ward.', '1611.04664-1-4-1': 'The reader will find in our final section [REF] an overview of Diophantine/dynamical pairs of mathematically equivalent problems, placing our result in a broad yet, we hope, sufficiently focused context.', '1611.04664-1-5-0': 'In this paper, we treat an equivalent pair of questions about algebraic [MATH]-actions: the [MATH]-actions [MATH] by automorphisms of a compact group [MATH].', '1611.04664-1-5-1': "This addresses the problem left open in T. Ward's thesis [CITATION] and then in K. Schmidt's book [CITATION], and the papers [CITATION] of D. Lind, K. Schmidt and E. Verbitskiy: when [MATH] is abelian, and under an appropriate - and necessary - Noetherianness assumption (the descending chain condition on closed invariant subgroups), does the exponential growth rate of the size of the component group of the subgroup [MATH] of [MATH]-periodic points, indexed by an order sublattice [MATH], converge to the topological entropy of the system, assuming the latter is finite?", '1611.04664-1-5-2': 'We answer this in the affirmative in the symmetric case [MATH] of the cubical sublattices.', '1611.04664-1-5-3': 'On applying another paper [CITATION] of Ward, this furthermore proves that, when [MATH] has a completely positive entropy, the subgroups [MATH] become equidistributed in the Haar measure of the group [MATH], as [MATH].', '1611.04664-1-5-4': "Completely positive entropy is the natural condition as it means equivalently that the Haar measure is the unique probability measure of maximum entropy: this is Berg's theorem, proved in this situation by Lind, Schmidt and Ward [CITATION].", '1611.04664-1-5-5': 'In the general case, when [MATH] supports more than one maximum entropy measure, the conclusion is that all weak-[MATH] limits over some such sequence of levels [MATH] are measures of maximum entropy.', '1611.04664-1-6-0': "As we indicate in sections [REF] and [REF] below, an entirely different solution, coming again from the number theory side, is implicit in Habegger's very recent manuscript [CITATION].", '1611.04664-1-6-1': 'Habegger does not note the dynamical connection, but the corresponding convergence problem follows easily from his general estimate on the number of rational points lying very near to a subset of [MATH] definable in a polynomially bounded [MATH]-minimal expansion of the real numbers, in conjunction with the Ax and Koksma-Hlawka theorems.', '1611.04664-1-6-2': "Habegger's and ours Diophantine results are quite different, and generalize in completely disjoint directions.", '1611.04664-1-6-3': 'Neither of the two appears capable of addressing the growth rate and equidistribution of the general groups [MATH], and even less the equidistribution of individual periodic trajectories in [MATH] that are "long enough" with respect to a certain other characteristic the orbit, that we shall not discuss in this paper, which following McMullen could be called the "denominator" of the orbit (see chapter 21 of [CITATION] for the case of the doubling map on the circle).', '1611.04664-1-6-4': 'On the other hand, as the groups [MATH] considered exhaust the periodic points of the [MATH]-action, our result can be seen as an averaged form of the growth and equidistribution of periodic trajectories of an algebraic [MATH]-action.', '1611.04664-1-7-0': 'In the case that presently concerns us, the substance of the problem lies in the systems that exhibit only a partially hyperbolic behavior; the case of a strictly hyperbolic dynamics (expansive [MATH]-actions) has no Diophantine content, and was settled already in Ward [CITATION].', '1611.04664-1-7-1': 'The Diophantine problem was recognized by Lind in [CITATION] to emerge from the dual description (or standard symbolic model) of an algebraic [MATH]-action, by means of commutative algebra.', '1611.04664-1-7-2': 'A [MATH]-action [MATH] by automorphisms of a compact abelian group [MATH] is given by a module [MATH] over the Laurent series ring [MATH].', '1611.04664-1-7-3': 'The mentioned descending chain condition amounts precisely to the Noetherianness condition of the [MATH]-module [MATH].', '1611.04664-1-7-4': 'When this is the case (but not without the condition, nor when the group [MATH] is non-abelian), Kitchens and Schmidt (see [CITATION], Cor.', '1611.04664-1-7-5': '4.8) proved that the periodic points of [MATH] are dense.', '1611.04664-1-7-6': "One then expects the situation to be similar to the basic case of a toral automorphism, where the dynamics is given by an invertible integer matrix [MATH]; its entropy [MATH] equals the Mahler measure [MATH] of the companion polynomial [MATH]; the number of points with finite period [MATH] equals [MATH]; and the convergence [MATH] amounts precisely to A. O. Gelfond's theorem that an algebraic integer [MATH] of unit modulus [MATH] may not be exponentially approached by an [MATH]-th root of unity [MATH]: [MATH], as [MATH].", '1611.04664-1-7-7': 'The extension of this to solenoids ([MATH]) covers the case of an arbitrary algebraic [MATH] of unit modulus.', '1611.04664-1-7-8': "For a lucid and leisurely treatment of the [MATH] case, including Yuzvinskii's calculation of the entropy, we refer the reader to Everest and Ward's book [CITATION], in addition to Lind's original paper [CITATION].", '1611.04664-1-8-0': 'That the general situation is indeed similar, only much more complicated, emerged from the work of Kitchens and Schmidt [CITATION], Ward [CITATION] and Lind, Schmidt and Ward [CITATION].', '1611.04664-1-8-1': 'By an addition formula for the entropy, due to Yuzvinskii and Thomas in the rank one case, a devissage reduces the problem to the case that [MATH] is an ideal of the Laurent ring: the case of a cyclic action.', '1611.04664-1-8-2': 'Then the entropy of [MATH] turns out to be zero unless the ideal [MATH] is principal, in which case it equals the Mahler measure [MATH] (if [MATH], and infinity if [MATH]).', '1611.04664-1-8-3': 'In such a way, Lind, Schmidt and Ward (see also the introduction in Lind, Schmidt and Verbitskiy [CITATION]) obtain the equivalence of the following pair of a dynamical and a Diophantine statements.', '1611.04664-1-8-4': 'With [MATH] ranging over all finite index subgroups of [MATH], let [MATH], and denote [MATH] the number of connected components of the group of [MATH]-periodic points for [MATH].', '1611.04664-1-8-5': 'Let [MATH] be the topological entropy of the system [MATH], which coincides with the metric entropy for the Haar measure.', '1611.04664-1-8-6': 'On the dynamical side we expect:', '1611.04664-1-9-0': '(A) Assume [MATH] has finite topological entropy and satisfies the d.c.c.: every descending chain of closed invariant subgroups of [MATH] is stationary.', '1611.04664-1-9-1': 'Then, [EQUATION]', '1611.04664-1-10-0': 'In general, as explained in [CITATION] on page 619, the Noetherian and finite entropy conditions are certainly necessary in such a statement; nothing could be said for non-Noetherian systems, where the growth rate need not converge and any rate between [MATH] and [MATH] may occur.', '1611.04664-1-10-1': 'Under the Noetherian and finite entropy conditions, Schmidt proves in section 21 of [CITATION] the weaker statement identifying [MATH] with the limit supremum over sublattices [MATH].', '1611.04664-1-10-2': "He first proves the upper bound in a direct application of the topological definition of [MATH], and then uses Gelfond's theorem and the formula for the entropy to exhibit a particular sequence [MATH] along which [MATH].", '1611.04664-1-10-3': 'This sequence is very special; in particular, the finite subgroup [MATH] is of the form [MATH], where [MATH] with respect to [MATH]; see section [REF] for more on this.', '1611.04664-1-11-0': 'The general problem (A) thus amounts to establishing a lower bound on the growth of periodic points, and as already mentioned, the same applies to equidistribution.', '1611.04664-1-11-1': 'Similarly to the Brauer-Siegel theorem and to questions on orbit growth in the arithmetic dynamics of rational maps (for the latter, see Silverman [CITATION]; Kawaguchi-Silverman [CITATION]; Dimitrov [CITATION]), the upper bounds come easily, and the question of the lower bound proves to be a subtle problem related intrinsically to Diophantine Approximations.', '1611.04664-1-12-0': "In the case at hand, the cited work of Kitchens, Schmidt, Lind and Ward renders (A) precisely equivalent to the following multidimensional extension of Gelfond's result.", '1611.04664-1-12-1': 'Note that as [MATH], the finite group [MATH] is equidistributed in the Haar measure [MATH] of the torus [MATH].', '1611.04664-1-13-0': '(B) Let [MATH] be a non-zero Laurent polynomial.', '1611.04664-1-13-1': 'Then [EQUATION]', '1611.04664-1-14-0': 'This is obvious in the case that the hypersurface [MATH] has empty intersection with the real torus [MATH].', '1611.04664-1-14-1': 'From the dynamical point of view, this happens precisely when the cyclic [MATH]-action dual to the [MATH]-module [MATH] is expansive.', '1611.04664-1-14-2': 'By the alluded devissage procedure, Lind, Schmidt and Ward were thus able to prove (A) for all expansive systems [MATH].', '1611.04664-1-14-3': 'We note that, for expansive systems, both the finiteness of the entropy and the Noetherianness conditions are automatic: see Kitchens and Schmidt [CITATION], Th.', '1611.04664-1-14-4': '5.2.', '1611.04664-1-15-0': 'A further progress was made by Lind, Schmidt and Verbitskiy in [CITATION], who gave two proofs of (B) in the case that the zero locus [MATH] intersects the torus [MATH] at a finite set.', '1611.04664-1-15-1': "The first is Diophantine, and consists of the observation that since the finitely many intersection points are necessarily algebraic, an application of Gelfond's (one-dimensional) theorem suffices to cover that case.", '1611.04664-1-15-2': 'More significant is their second proof, purely dynamical, by means of a construction of rapidly decaying homoclinic points.', '1611.04664-1-15-3': 'In a subsequent paper [CITATION], the same authors then extended their dynamical method to prove (B) in the "generic" case that the intersection locus [MATH] has real codimension at least two in the torus.', '1611.04664-1-15-4': 'The dynamical equivalent of this "atoral" hypothesis turns out to be precisely the existence of summable homoclinic points, and that is what allowed the authors to bypass the delicate Diophantine issues about torsion points getting extremely close to the zero locus [MATH].', '1611.04664-1-16-0': 'In the present paper, we solve (B) for all Laurent polynomials [MATH] in the case that [MATH] runs over the cubical sublattices [MATH], so that the average is taken over all [MATH]-tuples of [MATH]-th roots of unity.', '1611.04664-1-16-1': 'It follows from the work of Kitchens, Lind, Schmidt and Ward cited above that (A) is true when all the [MATH]-periodic points are taken together.', '1611.04664-1-17-0': 'The precise statement of the equidistribution consequence about periodic trajectories is as follows.', '1611.04664-1-17-1': "The proof of the implication can be found in Ward's paper [CITATION] as well as in section 22 of Schmidt's book [CITATION].", '1611.04664-1-18-0': 'For [MATH], let [MATH] be the push-forward on [MATH] of the Haar measure of the subgroup [MATH] of [MATH]-periodic points of [MATH].', '1611.04664-1-18-1': 'If [MATH] is a probability measure having [MATH] in the weak-[MATH] topology for some sequence [MATH], then [MATH] is a measure of maximum entropy: [MATH].', '1611.04664-1-19-0': 'In particular, if the system [MATH] has a completely positive entropy, then the measures [MATH] converge to the Haar measure of [MATH]: the periodic points equidisitribute in the Haar measure.', '1611.04664-1-20-0': 'One may wish to refine the equidistribution corollary to individual periodic trajectories.', '1611.04664-1-20-1': 'However, some care is needed in the statement as, already for the case of the doubling map on the circle [MATH], it is not true that individual periodic trajectories are equidistributed in Haar measure as [MATH] approaches infinity.', '1611.04664-1-20-2': "This is similar to Duke's theorem, where an individual closed geodesic may be poorly distributed.", '1611.04664-1-20-3': 'The parallel with Duke\'s theorem appears to run rather deep, including the idea of equidistribution emerging from a lower bound on total length in terms of the discriminant of the quadratic order (respectively, what could be called with McMullen the "denominator" [MATH] of the periodic orbit).', '1611.04664-1-20-4': 'An elaboration of the case of the doubling map on the circle can be found in the final section 21 ("The discriminant-regulator paradox") of McMullen\'s course [CITATION], where the exponential sum bounds of Bourgain, Glibichuk and Konyagin [CITATION] are interpreted as giving equidistribution of individual periodic orbits having length [MATH] exceeding [MATH] for a fixed [MATH].', '1611.04664-1-20-5': 'This refined equidistribution conjecture now extends to arbitrary Noetherian algebraic [MATH]-actions of finite entropy.', '1611.04664-1-20-6': "In this generality, equidistribution appears to be a very difficult open problem, as is the corresponding conjecture in Duke's theorem and the higher rank generalizations thereof.", '1611.04664-1-21-0': 'We will focus our exposition on (B) and suppress the details of its equivalence with (A), and the consequence on equidistribution.', '1611.04664-1-21-1': "Those details are well known in the [MATH]-actions community, and may be read, respectively, from the introduction following Theorem 1.3 of [CITATION], and from Ward's paper [CITATION] (exposed in chapter 22 of Schmidt's book [CITATION]).", '1611.04664-1-21-2': 'Our proof of Theorem [REF] is effective, in the sense of yielding a quantitative estimate on the convergence in (B) in terms of [MATH], [MATH] and the degree and height of the polynomial [MATH].', '1611.04664-1-21-3': 'Correspondingly, the equidistribution in the Corollary is also effective.', '1611.04664-1-22-0': 'The convergence problem (B) was also raised as Conjecture 15 in a recent paper [CITATION] of Silverman, whose motivation was a study of higher rank divisibility sequences having a linear torus as the underlying algebraic group.', '1611.04664-1-22-1': "The cubical case of [MATH] solved by Theorem [REF] was given a separate attention as Conjecture 2 in Silverman's paper.", '1611.04664-1-23-0': '# Diophantine approximation by torsion points', '1611.04664-1-24-0': 'By the classical Koksma-Hlawka inequality on numerical integration, Theorem [REF] would follow at once if we could show [EQUATION]', '1611.04664-1-24-1': 'This is not what we do, however, and ([REF]) remains wide open, as does (B) for arbitrary sublattices [MATH].', '1611.04664-1-24-2': 'It is important here to stress that (B) and ([REF]) are both arithmetic statements, in that they could only be true for polynomials [MATH] having algebraic coefficients.', '1611.04664-1-24-3': 'For polynomials with complex transcendental coefficients, an exceptional set of [MATH] is clearly needed in ([REF]).', '1611.04664-1-25-0': 'Taking account of such an exceptional set, a result of such a type does indeed hold, as Habbeger shows in a recently released manuscript [CITATION].', '1611.04664-1-25-1': 'Working in the much wider context of rational approximations to a subset of [MATH] definable in a fixed polynomially bounded [MATH]-minimal expansion of the real numbers, Habegger accomplishes this by continuing the determinental method of Bombieri, Pila and Wilkie; technically speaking, he does an extrapolation with an auxiliary function, and not an interpolation determinant.', '1611.04664-1-25-2': 'This continues an extensive literature on limiting the number of rational points lying in a definable set, which Habegger extends to rational points lying very near to the set.', '1611.04664-1-25-3': "An overview of the method, in increasing order of technical detail, is presented in Scanlon's paper [CITATION], Zannier's monograph [CITATION], and the recently released volume [CITATION] edited by Jones and Wilkie.", '1611.04664-1-26-0': 'Habegger proves a general estimate of the same quality as the Pila-Wilkie counting theorem, in which closeness to the definable set is measured in terms of a Diophantine exponent depending on the definable set as well as the "[MATH]" of the counting theorem.', '1611.04664-1-26-1': "The following result is implicit in Habegger's manuscript; it follows from his Theorem 2, using Ax's theorem as in section 8 of loc. cit. to treat the algebraic locus [MATH], and applying induction on [MATH].", '1611.04664-1-27-0': 'Recalling the standard terminology, a torsion coset in [MATH] is a translate [MATH] by a torsion point [MATH] of a connected algebraic subgroup [MATH] of [MATH] (an algebraic subtorus).', '1611.04664-1-27-1': 'The torsion coset is proper if it is strictly contained in [MATH], i.e. [MATH].', '1611.04664-1-28-0': '[Habegger [CITATION]] Fix a non-zero complex Laurent polynomial [MATH] and an [MATH].', '1611.04664-1-28-1': 'Then, there exists a finite union [MATH] of proper torsion cosets [MATH] such that, for every [MATH], all but at most [MATH] points [MATH] having a finite order [MATH] and not lying in [MATH] fulfil [EQUATION]', '1611.04664-1-28-2': 'In particular, for a set of [MATH] of full density, ([REF]) and ([REF]) hold for all [MATH], and for all [MATH] they fail for at most [MATH] values [MATH].', '1611.04664-1-29-0': 'Compared to this we give the following result, which is limited to the [MATH]-case and is stronger regarding the number of exceptions of a fixed order [MATH].', '1611.04664-1-29-1': 'In contrast, it is less precise regarding bounded orders [MATH].', '1611.04664-1-29-2': 'Though the method allows for a refined bound, we only consider an [MATH] rather than an [MATH] bound, as it is precisely the "sub-Liouville" quality of the bound that matters in our application to Theorem [REF].', '1611.04664-1-30-0': 'For a non-zero multivariate Laurent polynomial [MATH], let [MATH] be the logarithm of the maximum absolute value of a coefficient of [MATH].', '1611.04664-1-30-1': 'We take the degree [MATH] under the standard embedding [MATH].', '1611.04664-1-30-2': "We write [MATH] for Euler's function.", '1611.04664-1-31-0': 'The number of torsion cosets [MATH] in [MATH], as well as the degrees of the subtori [MATH] and orders of the torsion points [MATH], are also bounded by an effectively computable function of [MATH] and [MATH].', '1611.04664-1-32-0': 'The restriction to [MATH]-coefficients as opposed to [MATH]-coefficients is no loss of generality.', '1611.04664-1-32-1': 'Liouville\'s "trivial" Diophantine bound gives ([REF]) with some [MATH], depending on [MATH] and [MATH].', '1611.04664-1-32-2': "Thus ([REF]) amounts to the subexponential improvement over Liouville's bound.", '1611.04664-1-32-3': "In that regard it is a typical Diophantine statement, although we conjecture that the conclusion extends to complex polynomials, as with Habegger's theorem.", '1611.04664-1-33-0': 'We record an easy corollary of Theorem [REF] applied to the case [MATH].', '1611.04664-1-34-0': 'For every [MATH] and [MATH], there is a finite computable constant [MATH] such that, for all [MATH], all but at most [MATH] sums [MATH] of [MATH] roots of unity of orders dividing [MATH] either fulfil [EQUATION] or else have a non-empty vanishing subsum.', '1611.04664-1-35-0': "Either of the two methods, Habegger's and ours, seems intrinsically incapable of reaching the other's result.", '1611.04664-1-35-1': 'In our case, this is because we work from the product formula in the level [MATH] cyclotomic field, doing Diophantine approximations on a power of the ambient linear torus where we need the torsion points to lie in a common cyclotomic field.', '1611.04664-1-35-2': "In Habegger's case, it is due to working with the theory of the exponential function in a general context of rational approximation to sets definable in a polynomially bounded [MATH]-minimal expansion of the reals, where the transposition of our result towards ([REF]) does not hold.", '1611.04664-1-36-0': 'If a point [MATH] of order [MATH] lies exactly on the hypersurface ([MATH]), then so does its Galois orbit, whose size [MATH].', '1611.04664-1-36-1': 'Hence, neglecting the uniformity part concerning the height of the hypersurface, both Theorems [REF] and [REF] are generalizations of the classical structural theorem about torsion points satisfying a polynomial relation, conjectured by Lang and first proved, independently, by Laurent [CITATION] and Sarnak (in an unpublished manuscript, later revisited by Sarnak and Adams as [CITATION]):', '1611.04664-1-37-0': 'The set of torsion points lying in a fixed algebraic subvariety [MATH] over [MATH] is covered by finitely many non-trivial monomial relations [MATH], and the number and degrees of these monomial relations is bounded by a certain explicit function of [MATH] and [MATH] alone.', '1611.04664-1-38-0': 'This theorem, also known as "toral Manin-Mumford," has now a multitude of different proofs and generalizations, and we refer to Zannier [CITATION] for a thorough discussion.', '1611.04664-1-38-1': "A particularly simple proof of the uniformity statement is in Bombieri and Zannier [CITATION], also reproduced in chapter 4 of Bombieri and Gubler's book [CITATION].", '1611.04664-1-38-2': 'For a sharper quantitative bound on the number of monomial relations, we refer to Aliev and Smyth [CITATION].', '1611.04664-1-39-0': 'Theorem [REF] does not however give a new proof of the Laurent-Sarnak theorem, which instead is used in a crucial point of the proof.', '1611.04664-1-39-1': 'The argument may be viewed as doing Diophantine approximation to extrapolate from a general structural result about points lying exactly on a subvariety to a general structural result about points lying very near to a subvariety.', '1611.04664-1-40-0': 'An easy consequence of the Laurent-Sarnak theorem is that, for a fixed non-zero Laurent polynomial [MATH] and [MATH] running over the finite index subgroups, the number of [MATH] having [MATH] is [MATH], effectively, as [MATH].', '1611.04664-1-40-1': 'This means that a negligible proportion of points is excluded by the sum in (B).', '1611.04664-1-40-2': 'Taking now [MATH], the Koksma-Hlawka inequality shows that the truncation of the sum to the complement of the [MATH]-tubular neighborhood of the hypersurface [MATH] converges to the integral as [MATH] and then [MATH].', '1611.04664-1-40-3': "A simple estimate shows that not more than [MATH] of the [MATH]-torsion points are left behind in the tubular neighborhood (indeed, this is even for the [MATH]-tubular neighborhood), and hence, applying the trivial Liouville bound on each of the exceptional points, Theorem [REF] follows for [MATH] from from either of Habegger's Theorem [REF] (with [MATH]), or from our more elementary Theorem [REF].", '1611.04664-1-40-4': 'We give the details in the short section [REF] below.', '1611.04664-1-40-5': "This leads to two independent proofs of Theorem [REF] and Silverman's Conjecture 2 in [CITATION], one of which is presented in the present paper in full detail.", '1611.04664-1-41-0': 'One may wonder whether or not Theorem [REF] should in fact hold for all [MATH] having [MATH], with an empty exceptional set.', '1611.04664-1-41-1': 'In this regard it may be worth remarking that Theorem [REF] and its proof does generalize immediately to a boundedness result on Diophantine approximation by small points of [MATH], and an exceptional set is definitely required in this generality.', '1611.04664-1-41-2': 'See section [REF] below.', '1611.04664-1-42-0': 'Our method for Theorem [REF] will be that of Thue, Siegel, Roth and Schmidt.', '1611.04664-1-42-1': 'The novel point in this context is a use of Cartesian sets for the non-vanishing of the auxiliary construction at the special points.', '1611.04664-1-42-2': "Of course, non-vanishing at some point of a Cartesian product can be regarded as one of the oldest and simplest zero estimates, going back to Lang's multidimensional extension of the Schneider-Lang theorem.", '1611.04664-1-42-3': '(A genuinely multidimensional Schneider-Lang theorem was subsequently achieved by Bombieri [CITATION], without involving a Cartesian hypothesis.)', '1611.04664-1-42-4': 'It is also the basis of the polynomial method of Algebraic Combinatorics; see Alon [CITATION].', '1611.04664-1-43-0': 'Nonetheless, this simple idea appears not to have been exploited in the context of a Thue-Siegel method.', '1611.04664-1-43-1': "Such a way of getting non-vanishing of the auxiliary construction at some special point, without using derivations, could also be used for Diophantine approximations over positive characteristic, and leads to a substitute of Schmidt's Subspace theorem over a global function field even in the case that the divisor is not defined over the field of constants.", '1611.04664-1-43-2': 'The case of a constant divisor was solved by Julie Wang [CITATION] (we discuss this in relation to mixing in the final section [REF]), whereas for non-constant divisors the Vojta conjectures would appear to fail beyond hope over function fields of positive characteristic.', '1611.04664-1-43-3': 'We suggest that a finiteness statement should hold even then, concerning the best possible bound on sets of solutions of clustering heights.', '1611.04664-1-43-4': "This would be non-trivial in the higher dimensional case, as opposed to the classical situation of Roth's theorem.", '1611.04664-1-43-5': 'We refer to Kim, Thakur and Voloch [CITATION] for a discussion of the one-dimensional case.', '1611.04664-1-44-0': 'We hope to devote a subsequent paper to a weak Subspace theorem over function fields of positive characteristic.', '1611.04664-1-45-0': 'An outline, and organization.', '1611.04664-1-45-1': 'For inductive reasons, it turns out convenient to prove the thesis of Theorem [REF] step by step, inductively from [MATH] to [MATH], by proving the existence of an effectively computable function [MATH] and a finite union [MATH] of proper torsion cosets [MATH], such that [MATH] holds for any set [MATH] of [MATH]-torsion points that has [MATH], [EQUATION] and such that [MATH] maps to a single point under the projection [MATH] onto the last [MATH] coordinates vector.', '1611.04664-1-45-2': 'This holds vacuously for [MATH], with [MATH] and [MATH], and we aim to prove the statement for the maximal (unrestricted) case [MATH].', '1611.04664-1-46-0': 'Applying induction on [MATH], we assume that [MATH] and that the statement is true for lower [MATH] and all pairs [MATH].', '1611.04664-1-46-1': 'Also, the statement is trivial for [MATH], and we may and do assume upon decreasing the dimension [MATH] of the ambient torus that no component of the hypersurface [MATH] is contained by a proper torus coset (a translate of a lower dimensional subtorus).', '1611.04664-1-47-0': 'Let us write [MATH], where [MATH] stands for the first [MATH] coordinates and [MATH] stands for the last [MATH] coordinates.', '1611.04664-1-47-1': 'Introducing a parameter [MATH], to be taken sufficiently large with respect to [MATH] and [MATH], and then a further large parameter [MATH], the idea is to create a polynomial identity in [MATH] and in the [MATH] blocks [MATH] of variables [MATH], [MATH]: [EQUATION] with partial degrees less than [MATH] in all [MATH] and [MATH], with order of vanishing at least [EQUATION] along the variety [MATH], and with integral coefficients [MATH] having absolute values bounded by [MATH] as [MATH].', '1611.04664-1-48-0': 'Notice that simply raising [MATH] to a power leads to coefficients with exponential size.', '1611.04664-1-48-1': 'The crucial subexponential estimate is obtained by a typical application of the Thue-Siegel lemma for a linear system in the unknown coefficients [MATH], an application that requires having [MATH] as [MATH].', '1611.04664-1-48-2': 'This is carried out in section 2, by separating a variable.', '1611.04664-1-48-3': 'We note that in the basic case of [MATH] and a single variable polynomial [MATH], and most particularly for [MATH], much more refined constructions via this technique appear in the literature, see Bombieri and Vaaler [CITATION].', '1611.04664-1-49-0': 'With this construction at hand, consider now an [MATH]-tuple [EQUATION] of [MATH]-torsion points in [MATH].', '1611.04664-1-49-1': 'Let [MATH] denote the Galois group of the level [MATH] cyclotomic field [MATH].', '1611.04664-1-49-2': 'By assumption, all these points [MATH] have a common vector of last [MATH] coordinates.', '1611.04664-1-49-3': 'We wish to specialize the right-hand side ([REF]) of the identity to [MATH] for [MATH], and the left-hand side ([REF]) to all [MATH] for [MATH].', '1611.04664-1-49-4': 'If [MATH], then [MATH] is a non-zero rational integer, hence at least [MATH] in absolute value.', '1611.04664-1-49-5': 'On taking [MATH] and then [MATH] this will imply [EQUATION] for at least one [MATH], contradicting the defining assumption of the set [MATH].', '1611.04664-1-50-0': 'Suppose then that all points of [MATH] are constrained by the non-zero algebraic relation [MATH], that depends on [MATH] and the original [MATH], but not on [MATH].', '1611.04664-1-50-1': 'By the Laurent-Sarnak structural theorem ("Lang\'s [MATH] conjecture"), the set of all such [MATH]-tuples is covered by finitely many non-trivial multiplicative relations [MATH].', '1611.04664-1-50-2': 'Those relations that do not depend on the [MATH] variables must involve the [MATH] variable with a non-zero multi-index in [MATH].', '1611.04664-1-50-3': 'On enlarging [MATH] to a [MATH] by adding all these multiplicative [MATH] relations, we may assume that no point of [MATH] fulfils any of the [MATH]-independent relations.', '1611.04664-1-50-4': 'Then all points of the Cartesian power [MATH] are covered by finitely many fixed multiplicative relations, each of which involves at least one of the [MATH]-variables.', '1611.04664-1-50-5': 'A positive proportion of points in [MATH] will belong to a fixed such relation, and a monoidal change of the [MATH]-coordinates brings us easily within the scope of the induction hypothesis.', '1611.04664-1-51-0': '# The auxiliary construction', '1611.04664-1-52-0': 'It will be convenient to consider the notion of size on a logarithmic scale.', '1611.04664-1-52-1': 'Hence, for non-zero [MATH] (we may assume for Theorem [REF] that [MATH] is a polynomial), we write [MATH] for the maximum of the logarithms of absolute values of coefficients of [MATH].', '1611.04664-1-53-0': 'We write [MATH] for the length [MATH] variables block [MATH] and [MATH] for the length [MATH] variables [MATH] and consider the ideal [MATH] generated by the polynomials [MATH].', '1611.04664-1-53-1': 'The goal of this section is the following', '1611.04664-1-54-0': "We will need only the simplest possible version of Siegel's lemma; this is Lemma 2.9.1 in [CITATION].", '1611.04664-1-55-0': "Siegel's lemma.", '1611.04664-1-55-1': 'Let [MATH] be an [MATH] matrix, [MATH], whose entries are rational integers bounded in magnitude by [MATH].', '1611.04664-1-55-2': 'Then the homogeneous linear system [MATH] has a non-zero solution [MATH], satisfying [EQUATION] width 5pt height 5pt depth 0pt', '1611.04664-1-56-0': "The next type of lemma, of probabilistic nature, is familiar from the parameter count in the proof of Roth's theorem (cf. Bombieri-Gubler [CITATION], Lemma 6.3.5).", '1611.04664-1-57-0': 'The polytope [EQUATION] has volume satisfying [EQUATION]', '1611.04664-1-57-1': 'Proof.', '1611.04664-1-57-2': 'For any [MATH], the characteristic function of [MATH] is bounded by [MATH]; let us choose [MATH].', '1611.04664-1-57-3': 'Since the condition [MATH] takes the expression [MATH] in the coordinates [MATH], we see that [MATH] is bounded by the integral of [MATH] over the box [MATH].', '1611.04664-1-57-4': 'This is just the [MATH]-th power of the one-dimensional integral [EQUATION] proving our claim.', '1611.04664-1-57-5': 'width 5pt height 5pt depth 0pt', '1611.04664-1-58-0': 'The construction now proceeds as follows.', '1611.04664-1-58-1': 'We assume without loss of generality (the [MATH] statement of the theorem being trivial) that no component of the zero locus of [MATH] is contained by a proper torus coset of [MATH], and write [MATH] and [EQUATION] where [MATH] and the [MATH] are integer polynomials without common factor.', '1611.04664-1-59-0': 'We look for a polynomial [EQUATION] in [MATH] variables [MATH], [MATH] and [MATH], in which all components of the multi-indices [MATH] and [MATH] take values in [MATH].', '1611.04664-1-59-1': 'Then the polynomial [EQUATION] still has its partial degrees in the "first variables" [MATH] smaller than [MATH].', '1611.04664-1-60-0': 'Consider the following finite procedure applied iteratively starting from [MATH]:', '1611.04664-1-61-0': 'At each stage, every occurrence of an [MATH] will be accompanied by a corresponding [MATH].', '1611.04664-1-61-1': 'Replace [MATH] by [MATH], and repeat until the partial degrees in the [MATH] are all [MATH].', '1611.04664-1-62-0': 'Owing to the exponential boundedness of linear recursions and multinomial coefficients, at the end of our procedure we have presented [MATH] in the form [EQUATION] where the [MATH] are linear forms in the [MATH] with coefficients rational integers of logarithmic size [MATH].', '1611.04664-1-62-1': 'Here, [MATH] ranges over [MATH]-tuples [MATH]; [MATH] ranges over [MATH]; [MATH] as before stands for the block [MATH], and similarly [MATH]; and [MATH] is an abbreviation for [MATH].', '1611.04664-1-63-0': 'Proof of Proposition [REF].', '1611.04664-1-63-1': "We apply Siegel's lemma to the linear system [EQUATION] in the unknowns [MATH].", '1611.04664-1-63-2': 'As each [MATH] ranges over [MATH] and [MATH] ranges over [MATH], the number of free parameters is [MATH].', '1611.04664-1-63-3': 'We use Lemma 2 to estimate the number [MATH] of equations ([REF]).', '1611.04664-1-63-4': 'We will take [MATH] with a suitable sufficiently large constant [MATH] depending only on [MATH].', '1611.04664-1-63-5': 'Then, in the asymptotic that [MATH] is fixed), the number of lattice points in the dilation by [MATH] of the polytope [MATH] is [MATH].', '1611.04664-1-63-6': 'The multi-index [MATH] takes [MATH] possibilities, [MATH] takes [MATH] possibilities, and [MATH] takes [MATH] possibilities.', '1611.04664-1-63-7': 'We conclude that for [MATH] large, there are fewer than [EQUATION] equations to solve.', '1611.04664-1-64-0': 'We choose [EQUATION] making [EQUATION]', '1611.04664-1-64-1': 'The Dirichlet exponent of our linear system is [MATH], while the logarithmic size [MATH] is bounded by [MATH].', '1611.04664-1-64-2': "Consequently, Siegel's lemma supplies a non-zero solution [MATH], of logarithmic size [MATH] satisfying [EQUATION] such that ([REF]) hold in ([REF]) with [MATH].", '1611.04664-1-64-3': 'On augmenting the choice of [MATH] in ([REF]) to subsume the [MATH] constant, we gain (i).', '1611.04664-1-65-0': 'For (ii), we have insured that the associated polynomial [MATH] in ([REF]) lies in [MATH].', '1611.04664-1-65-1': 'It remains to prove that this in fact implies the a priori stronger conclusion [MATH].', '1611.04664-1-65-2': 'For this, noting the co-primality of [MATH] with the [MATH] in ([REF]) and that none of these polynomials involves [MATH], it suffices to remark that none of the [MATH] belong to any of the associated primes of the ideal [MATH].', '1611.04664-1-65-3': 'width 5pt height 5pt depth 0pt', '1611.04664-1-66-0': '# Non-vanishing of the auxiliary construction at some special point', '1611.04664-1-67-0': 'We fix an [MATH] and set out to prove by induction on [MATH] the successively stronger statements that there exists a finite union set [MATH] of non-trivial multiplicative relations [MATH], [MATH], whose number and degrees [MATH] are bounded by a computable function of [MATH] and [MATH], such that the following is true for every [MATH]:', '1611.04664-1-68-0': 'If all elements of a set [MATH] of [MATH]-torsion points satisfy ([REF]) but not any of the multiplicative relations from [MATH], and share a common vector for their last [MATH] coordinates, then [MATH] is bounded by a computable function of [MATH] and [MATH].', '1611.04664-1-69-0': 'This trivially holds for [MATH], with [MATH] and [MATH].', '1611.04664-1-69-1': 'We proceed to the induction step, taking [MATH] and assuming the statement holds for lower values of [MATH].', '1611.04664-1-69-2': 'Choose [MATH] in Proposition [REF] to be large enough subject to [EQUATION] and fix it.', '1611.04664-1-69-3': 'Fixing then a [MATH] as in the proposition, assumed for later reference to satisfy [EQUATION] we obtain a non-zero polynomial [MATH] of partial degrees bounded by [MATH] and fulfilling (i) and (ii) of Proposition [REF].', '1611.04664-1-70-0': 'Assume first that [MATH] vanishes, for some [MATH] and [MATH] as above, at all points of the Cartesian power [MATH].', '1611.04664-1-70-1': 'The Laurent-Sarnak structural theorem then implies that each point of this Cartesian power fulfils one of finitely many non-zero monomial relations [EQUATION].', '1611.04664-1-70-2': 'We will include in [MATH] at least the union [MATH] of the finitely many non-zero multiplicative relations [MATH] (or [MATH] in terms of [MATH]) from [MATH] that have [MATH] (so necessarily [MATH]).', '1611.04664-1-70-3': 'This allows us to assume that the Cartesian power [MATH] is covered by finitely many relations [EQUATION] where [MATH] are [MATH]-th roots of unity while [MATH] are non-zero vectors in [MATH] that depend on [MATH] and not on [MATH].', '1611.04664-1-70-4': 'Moreover, the number and degrees of these relations can be taken to be bounded by a universal function [MATH].', '1611.04664-1-71-0': 'By the pigeonhole principle, one of the relations ([REF]) is fulfilled by at least [MATH] of the [MATH] points of the Cartesian power.', '1611.04664-1-71-1': 'If [MATH] is this relation and [MATH] an index where [MATH], the pigeonhole principle again implies that for some point [MATH], the set of solutions to [MATH] in [MATH] projecting to [MATH] has cardinality at least [MATH].', '1611.04664-1-71-2': 'Specializing the [MATH] coordinates to [MATH], this then gives the existence of another [MATH]-th root of unity, [MATH], such that the fixed non-zero multiplicative relation [MATH] is fulfilled by the vectors of first [MATH] coordinates [MATH] of all points in a subset [MATH] of cardinality at least [MATH].', '1611.04664-1-72-0': 'Upon relabeling the coordinates [MATH], we may assume [MATH] and let [MATH] and [MATH].', '1611.04664-1-72-1': 'Then [EQUATION] is a non-zero integer Laurent polynomial in [MATH], [MATH] and [MATH], and hence a non-zero integer Laurent polynomial in [MATH] and [MATH].', '1611.04664-1-72-2': 'By construction, all points [MATH] satisfy [MATH] and [EQUATION] for [MATH], with implied constant depending effectively on [MATH] and [MATH], and hence ultimately on [MATH] and [MATH].', '1611.04664-1-73-0': 'The points in [MATH] thus fulfil ([REF]) and have a common vector for their last [MATH] coordinates.', '1611.04664-1-73-1': 'Applying the induction hypothesis with [MATH] replaced by the expression of [MATH] in the [MATH] coordinates, and with [MATH] in place of [MATH], we define [MATH] and conclude that [MATH] is bounded by a computable function of [MATH] and [MATH], and hence by a computable function of [MATH] and [MATH].', '1611.04664-1-73-2': 'Hence the same is true for [MATH].', '1611.04664-1-74-0': 'This brings the vanishing assumption to the range of the induction hypothesis, and hence allows us to assume [MATH] for some [MATH].', '1611.04664-1-75-0': '# Proof of Theorem [REF]', '1611.04664-1-76-0': 'We are reduced to proving that at least one of [MATH] has to satisfy ([REF]) if [MATH] and [MATH].', '1611.04664-1-76-1': 'We argue by contradiction.', '1611.04664-1-77-0': 'Writing [MATH] for the Galois group of [MATH] and noting that the number of terms in [MATH] is bounded by [MATH], clause (i) in Proposition [REF] along with our choices ([REF]) and ([REF]) insure the bound [EQUATION]', '1611.04664-1-77-1': 'We use this bound for [MATH], while for [MATH] we use instead the right-hand side of the identity ([REF]) and the defining assumption that [MATH] for all [MATH]: [EQUATION]', '1611.04664-1-77-2': 'We may now conclude the proof of Theorem [REF].', '1611.04664-1-77-3': 'Since [MATH] is a non-zero rational integer, we have [MATH].', '1611.04664-1-77-4': 'Summing the negative of ([REF]) over [MATH] and ([REF]), we reach a contradiction as soon as [MATH].', '1611.04664-1-77-5': 'width 5pt height 5pt depth 0pt', '1611.04664-1-78-0': '# Proof of Theorem [REF]', '1611.04664-1-79-0': "We appeal to the classical Koksma-Hlawka inequality, a standard reference for which is chapter 5 of Kuipers and Niederreiter's monograph [CITATION].", '1611.04664-1-79-1': 'The function [MATH] on the torus [MATH] has Hardy-Krause total variation [MATH] as [MATH], and the set (sequence) [MATH] has discrepancy [MATH].', '1611.04664-1-79-2': 'The complement of any subset of [MATH] of cardinality [MATH] still has discrepancy [MATH].', '1611.04664-1-79-3': 'As [MATH] (integrability of the log singularity), the integrals of [MATH] and [MATH] over the torus differ by less than [MATH] for [MATH] and so, by the Koksma-Hlawka inequality, all [MATH] and [MATH] and [MATH] as above satisfy [EQUATION] where the implied constant depends only on the polynomial [MATH].', '1611.04664-1-80-0': 'Choose [MATH] and [MATH] the set of [MATH] with [MATH].', '1611.04664-1-80-1': 'Then the right hand side of the Koksma-Hlawka bound approaches zero as first [MATH] and then [MATH].', '1611.04664-1-80-2': 'We have to show two points under this asymptotics:', '1611.04664-1-81-0': 'In fact (1) even holds for the number of [MATH]-points in the [MATH] tubular neighborhood of [MATH].', '1611.04664-1-81-1': 'For, the volume of the [MATH] tubular neighborhood is [MATH], and that tubular neighborhood contains the disjoint union of the [MATH] hypercubes having for Southwest corner a [MATH]-point in the [MATH]-tubular neighborhood, using the identification [MATH].', '1611.04664-1-81-2': 'Similarly, as the exceptional set [MATH] in Theorem [REF] is a union of finitely many lower dimensional torsion cosets independent of [MATH], we have [MATH].', '1611.04664-1-82-0': 'Now (2) also follows immediately for [MATH] from Theorem [REF], applying ([REF]) to each of the [MATH] points in [MATH] lying in neither [MATH] nor the exceptional set of [MATH] elements, and the trivial Liouville [MATH] bound at each of the remaining [MATH] points.', '1611.04664-1-82-1': "This completes the proof of Theorem [REF] in the [MATH] case, whereas the [MATH] case follows from Gelfond's theorem, see Lind [CITATION].", '1611.04664-1-83-0': "The same conclusion in the [MATH] case follows just as well from Habegger's Theorem [REF], on taking [MATH] with the argument just given.", '1611.04664-1-83-1': 'width 5pt height 5pt depth 0pt', '1611.04664-1-84-0': '# Complements, generalizations and related problems', '1611.04664-1-85-0': "## Subsets of complex embeddings With Liouville's general inequality, the trivial bound extends to any subset of the places: uniformly over all [MATH], it holds either [EQUATION].", '1611.04664-1-85-1': 'Already with the polynomial [MATH] this bound does not extend to an [MATH], because a positive proportion of the primitive [MATH]-th roots of unity belong to the [MATH] neighborhood of [MATH].', '1611.04664-1-85-2': 'However, if [MATH] is constrained to satisfy [MATH] as [MATH] then the proof shows immediately that all but [MATH] points [MATH] fulfil either [MATH] or [MATH].', '1611.04664-1-86-0': 'We see no evidence neither for nor against the natural question of whether the same should persist for all [MATH], [MATH], without exception.', '1611.04664-1-86-1': 'If true, this would imply that the Galois equidistribution of torsion points of [MATH] extends to the singular test functions of the form [MATH], where [MATH], with the convention as in (B) that all [MATH] values are dropped from the average over the Galois orbit.', '1611.04664-1-86-2': "In turn, this would imply Silverman's general Conjecture 15 in [CITATION].", '1611.04664-1-86-3': "Such a statement does hold for [MATH], by Gelfond's theorem and the theory of logarithmic linear forms, but it remains wide open for higher dimensions.", '1611.04664-1-87-0': 'In the case of [MATH], the problem of no exceptional set reduces to the bound ([REF]).', '1611.04664-1-87-1': 'This had been raised as an open ended question already in the 1980s, by Myerson [CITATION].', '1611.04664-1-87-2': "As in Feldman's refinement of Baker's theorem in the theory of logarithmic linear forms, Myerson asks ([REF]) even in the stronger form of an [MATH] bound.", '1611.04664-1-87-3': 'If true, a simple packing argument shows this type of refinement of ([REF]) to be best possible; see, for example, Theorem 2 in Konyagin and Lev [CITATION].', '1611.04664-1-87-4': "Habbeger's Theorem [REF] shows that such a bound does indeed hold for all but at most [MATH] primes [MATH].", '1611.04664-1-88-0': "## Myerson's conjecture", '1611.04664-1-89-0': "The refined convergence hypothesis in [REF], if true with an empty exceptional set, would also imply Myerson's conjecture [CITATION] about the asymptotic growth, as [MATH] for a fixed [MATH], of the norm from [MATH] to [MATH] of the 'Gaussian period' [MATH].", '1611.04664-1-89-1': 'For [MATH] the asymptotic formula was proved by Myerson and refined by Duke [CITATION] to an apparently optimal error term.', '1611.04664-1-89-2': "Myerson's conjecture is also discussed in chapter 10 of Konyagin and Shparlinski's book [CITATION].", '1611.04664-1-90-0': '## Small points We have been concerned in this paper with Diophantine approximations by roots of unity.', '1611.04664-1-90-1': 'As is customary in the subject, one could ask if the hypothesis that the point [MATH] is torsion could be weakened to a hypothesis that [MATH] is an algebraic point with sufficiently small canonical height.', '1611.04664-1-90-2': 'A straightforward modification of the proof, with Shou-wu Zhang\'s structural theorem on "Bogomolov [MATH]" replacing the Laurent-Sarnak "Lang [MATH]" theorem (see Theorem 4.2.2 in Bombieri and Gubler\'s book [CITATION]), yields the following simultaneous generalization of both Theorem [REF] and Zhang\'s theorem.', '1611.04664-1-91-0': 'For every finite extension [MATH], and every subset [MATH] having [MATH], all but at most [MATH] points [MATH] of canonical height [MATH] satisfy either [EQUATION].', '1611.04664-1-92-0': '## An example The following example shows that, unlike possibly for the roots of unity case, an exceptional set is definitely needed in the general Theorem [REF].', '1611.04664-1-92-1': 'It is taken from the paper [CITATION] of Baker, Ih and Rumely, with a similar consideration appearing in Autissier [CITATION].', '1611.04664-1-93-0': 'Consider the equation [MATH].', '1611.04664-1-93-1': 'It is usually irreducible over [MATH].', '1611.04664-1-93-2': 'For example, with [MATH] replaced by [MATH], it is Eisenstein at [MATH] if [MATH], showing that it certainly is irreducible for arbitrarily large [MATH]; fix a sequence of such [MATH] going to infinity.', '1611.04664-1-93-3': "By Rouche's theorem, there is one root [MATH] close to [MATH], and then the equation shows [EQUATION].", '1611.04664-1-93-4': 'On the other hand, the equation has bounded length and hence certainly [MATH] as [MATH]; cf. Bombieri-Gubler [CITATION], Prop. 1.6.6 and Lemma 1.6.7.', '1611.04664-1-93-5': "(Alternatively, note that the remaining conjugates of [MATH] are close to the unit circle, again by Rouche's theorem.)", '1611.04664-1-93-6': 'Thus, already with [MATH] and a single place, an [MATH] (non-empty exceptional set) is required for infinitely many [MATH] in Theorem [REF].', '1611.04664-1-94-0': 'We note also that, by a theorem of Mignotte [CITATION], the statement of Theorem [REF] does hold with empty exceptional set ([MATH]) for the case [MATH], as [MATH].', '1611.04664-1-94-1': 'Sub-Liouville bounds of this type form a small subject of its own, which has come to be known as algebraic points close to one; see [CITATION].', '1611.04664-1-95-0': '## [MATH]-adic metrics Theorem [REF] and its proof extends also to the [MATH]-adic places, and a similar example to the above, using the equation [MATH], shows that even for [MATH], the [MATH]-adic version too is best possible in this extension to small algebraic points.', '1611.04664-1-95-1': 'This is in stark contrast with the case of [MATH]-adic roots of unity (the analog of the [MATH] case of Theorem [REF]), where Tate and Voloch [CITATION] have shown that a torsion point not lying in a subvariety of [MATH] is [MATH]-adically bounded away from the subvariety.', '1611.04664-1-95-2': "For general [MATH] having [MATH], Tate and Voloch's theorem shows that the [MATH]-adic variant of Theorem [REF] does hold with a best possible bound and an empty exceptional set for the case of [MATH]-adic roots of unity.", '1611.04664-1-96-0': '## Algebraic dynamics, preperiodic points, and convergence to the Mahler measure A generalization of Theorem [REF] could be expected in the framework of algebraic dynamics.', '1611.04664-1-96-1': 'Let [MATH] be rational maps over [MATH] of degrees [MATH], and consider [MATH] the dynamical height function [MATH] of Call and Silverman; it is non-negative and vanishes exactly on the preperiodic points.', '1611.04664-1-96-2': 'One could ask about extending Theorem [REF] to Diophantine approximation by points [MATH] having a small enough dynamical height [MATH], in particular, to Diophantine approximation by the preperiodic points of [MATH].', '1611.04664-1-97-0': "In this direction, Szpiro and Tucker [CITATION] have used Roth's theorem to prove that the dynamical Mahler measure [MATH] of an integer univariate polynomial [MATH] is approached, as [MATH], by the averages of [MATH] over either of the sets [MATH] and [MATH], for any given [MATH] that is not one of [MATH] exceptional point for the rational iteration [MATH].", '1611.04664-1-97-1': 'Here, [MATH] is the Brolin-Lyubich measure: the unique invariant probability measure of maximum entropy for [MATH].', '1611.04664-1-97-2': 'It could be interesting to explore a higher dimensional case of this problem.', '1611.04664-1-97-3': 'Note that, by the example in [REF], convergence to the Mahler measure can fail for a sequence of small points, already for [MATH] and the Weil canonical height.', '1611.04664-1-98-0': '## Arakelov theory One might also expect a more general conceptual framework for the Diophantine result of this paper.', '1611.04664-1-98-1': "Given any non-negative height function [MATH] arising by Arakelov's intersection theory from a non-trivial nef semipositive adelically metrized line bundle on a projective variety [MATH], it could be asked whether for every subvariety [MATH] and [MATH] there are [MATH], and a proper closed algebraic subset [MATH], such that, for every number field [MATH], all but at most [MATH] points [MATH] having [MATH] are of distance at least [MATH] from [MATH].", '1611.04664-1-99-0': '## CM points Instead of torsion points, one could consider these problems for different sets of special points in special varieties.', '1611.04664-1-99-1': 'In this direction, Habegger [CITATION] has extended the Tate-Voloch theorem to the case of CM points in a power of the modular curve that are ordinary at a fixed finite prime [MATH].', '1611.04664-1-99-2': 'He notes that the Tate-Voloch statement fails for supersingular CM points; however, one could expect the analog of ([REF]) to hold also for the supersingular points, as well as for the Archimedean valuations.', '1611.04664-1-99-3': 'Indeed, it is the [MATH]-ordinary points that are similar to [MATH]-adic roots of unity, by means of the Serre-Tate theory of canonical liftings which Habegger exploits in his paper.', '1611.04664-1-99-4': 'In contrast, the [MATH]-supersingular points are not discrete, and should behave similarly to the complex roots of unity.', '1611.04664-1-100-0': 'It could be interesting to establish a version of Theorem [REF] for singular moduli.', '1611.04664-1-100-1': "Results in this direction are again implied by Habegger's recent work [CITATION], at least in the Archimedean situation.", '1611.04664-1-101-0': '## Diophantine approximation by closed orbits The [MATH] case of Theorems [REF] and [REF] are trivial within a fixed Galois orbit, and so these results may be regarded as bounding the number of Galois orbits of torsion or small points that get very near to a fixed subvariety.', '1611.04664-1-101-1': 'Likewise, [REF] outlines a similar hypothesis on how closely may a subvariety be approached by the periodic trajectories of a rational iteration on a power of the projective line.', '1611.04664-1-101-2': 'One could also ask for the metric forms of these results, in a wider dynamical context.', '1611.04664-1-101-3': "For instance, what are the analogs of Dirichlet's and Khintchin's theorems for Diophantine approximation by preperiodic points in the Julia set of a rational function?", '1611.04664-1-101-4': 'For a generic point [MATH] on a complete finite area surface of negative curvature, how closely is [MATH] approached by a closed geodesic or a closed horocycle, in terms of the length of the geodesic or the horocycle?', '1611.04664-1-102-0': "## Schmidt's Subspace theorem Finally, the strong parallel with the proofs of the theorems of Roth and Schmidt lead us to ask whether our Theorem [REF] here could be merged into a common generalization with Schmidt's Subspace theorem or some of its noteworthy consequences.", '1611.04664-1-102-1': 'For example, in view of Laurent\'s theorem [CITATION] on Diophantine approximation from a given finitely generated subgroup [MATH], we could inquire about the existence of an [MATH] an [MATH], and a finite union [MATH] of proper torus cosets, such that, for all number fields [MATH], the inequality [EQUATION] has at most [MATH]-rational solutions [MATH] from Habegger\'s "truncated cone" (from [CITATION]) [EQUATION] around [MATH].', '1611.04664-1-102-2': 'This is in the spirit of Poonen\'s "Mordell-Lang plus Bogomolov" [CITATION], now in the Diophantine approximation context of our paper.', '1611.04664-1-103-0': '# Logarithmic forms', '1611.04664-1-104-0': 'In this section we comment on the shortcomings of logarithmic linear forms theory towards verifying (B) over all product subgroups [MATH].', '1611.04664-1-104-1': "The best general estimates for logarithmic linear forms are obtained from Baker's method, with refinements from the theory of zero multiplicity estimates on commutative group varieties.", '1611.04664-1-104-2': "Baker and Wustholz's 1993 theorem [CITATION] remains the state of the art in this subject, crowning a long sequence of step by step improvements over Baker's original 1966 result.", '1611.04664-1-104-3': 'An account of the Baker-Wustholz theorem is exposed in the short monograph [CITATION] by the same authors, which is now the authoritative reference for logarithmic linear forms theory.', '1611.04664-1-104-4': 'An improvement of the constants in the rational case, particularly regarding the exponential dependence in the number of logarithms, was further achieved by Matveev [CITATION].', '1611.04664-1-105-0': 'One could naively hope that Theorems [REF] and [REF] might combine with logarithmic linear forms theory to solve all [MATH] cases of (B), with the former covering the case that the group [MATH] is not almost cyclic and the latter covering the case that the group [MATH] is almost cyclic.', '1611.04664-1-105-1': 'This turns out to be not possible, in particular the state of the art results in logarithmic linear forms theory are not nearly as strong as we need in the aspect of primary relevance to our problem, which is precisely the degree aspect of the number field generated by the bases of the logarithmic forms.', '1611.04664-1-106-0': 'What concerns us from logarithmic linear forms theory is a sub-Liouville upper bound on a quantity of the form [MATH] (equivalently, on the rational linear form [MATH] in two logarithms [MATH] and [MATH]), for [MATH] and for certain non-torsion algebraic points [MATH] of degree [MATH] having a bounded absolute (logarithmic) height [MATH].', '1611.04664-1-106-1': 'In this particular case, the Baker-Wustholz theorem amounts to a bound [EQUATION] provided the left-hand side is finite, where [MATH], and the implied constant is absolute (and small).', '1611.04664-1-106-2': 'The logarithmic dependence on [MATH] is best possible as far as it goes, but the dependence on [MATH] leaves a lot to be desired.', '1611.04664-1-106-3': "Liouville's trivial bound is [EQUATION] and ([REF]) is not a sub-Liouville bound unless [MATH].", '1611.04664-1-106-4': 'While ([REF]) does solve (B) in the case, for instance, of the finite groups of the form [MATH] having where [MATH], the results of logarithmic forms theory do not appear capable of reaching Theorem [REF] proved in this paper.', '1611.04664-1-107-0': "The fact is that, in the preset stage of the subject, the [MATH] power stands as a crucial barrier for any of Gelfond's, Schneider's or Baker's methods to succeed, with any [MATH] at all.", '1611.04664-1-107-1': "As Matveev notes in [CITATION] (in the introduction of the second paper), a similar [MATH] dependence had arisen already Gelfond's work, as typified by Theorem 3.4.", '1611.04664-1-107-2': 'III of his 1952 book [CITATION].', '1611.04664-1-107-3': 'A bound like ([REF]) was first reached by Waldschmmidt [CITATION] in 1980, and since unimproved in the [MATH] aspect.', '1611.04664-1-107-4': "Mignotte and Waldschmidt [CITATION] have shown that Schneider's method leads to a [MATH] version of ([REF]), which Laurent [CITATION] then refined to a rather small numerical constant, using an interpolation determinant.", '1611.04664-1-108-0': 'One could conceivably expect the bound ([REF]) to refine to an [MATH], for [MATH] non-torsion.', '1611.04664-1-108-1': 'Clearly, such an optimistic statement would be best possible jointly in [MATH] and [MATH].', '1611.04664-1-108-2': "Moreover, in view of Dirichlet's Approximation theorem, even the [MATH] asymptotic form of this hypothetical best possible bound would force Salem numbers to be bounded away from [MATH].", '1611.04664-1-108-3': 'This indicates that a refinement like this is hopelessly out of reach.', '1611.04664-1-108-4': "It could nonetheless be worth remarking that a completely uniform statement of this strength would combine with Habegger's Theorem [REF] solve (B) in all [MATH] cases, for arbitrary [MATH].", '1611.04664-1-108-5': 'Of course, even then many cases of finite subgroups would remain.', '1611.04664-1-109-0': '# Diophantine/Dynamical pairs', '1611.04664-1-110-0': 'We conclude the paper with an overview of mathematically equivalent pairs of Diophantine and dynamical problems.', '1611.04664-1-111-0': '##', '1611.04664-1-112-0': "The brilliant example of this has been Margulis's solution of the longstanding Oppenheim conjecture in the arithmetic theory of real indefinite quadratic forms.", '1611.04664-1-112-1': 'This began with a recasting of the problem by Raghunathan in terms of orbit closures in homogeneous dynamics (an [MATH]-orbit in [MATH] is either closed or dense), to which Margulis could apply a well developed ergodic arsenal of unipotent flows.', '1611.04664-1-112-2': "Margulis's proof found its proper setting with Ratner's ensuing theorems on measure rigidity, algebraicity of orbit closures and equidistribution in the homogeneous dynamics of flows generated by ad-unipotent one-parameter subgroups of a Lie group (real or [MATH]-adic).", '1611.04664-1-112-3': "These fundamental developments on the ergodic side have since found diverse applications back to number theory, including, for two notable instances, to Mazur's anticyclotomic conjecture in elliptic curves Iwasawa theory (Vatsal [CITATION]), and the non-Poisson distribution of gaps in the sequence [MATH] (Elkies-McMullen [CITATION]).", '1611.04664-1-113-0': "While Ratner's theorems exclude the case of diagonalizable (torus) flows - indeed, they patently fail for geodesic flows, - such (hyperbolic) flows and actions are of a still more basic Diophantine character, beginning with E. Artin's symbolic coding of modular geodesics via the continued fraction expansion.", '1611.04664-1-113-1': "Furthermore, like in Furstenberg's [MATH] problem, higher rank hyperbolic homogeneous dynamics is believed to regain a similar rigidity to the unipotent case.", '1611.04664-1-113-2': "For the case of the Weyl chamber flow on [MATH], the orbit closure Margulis conjecture amounts precisely to another old problem of Diophantine approximations: Cassels and Swinnerton-Dyer's refinement [CITATION] of Littlewood's conjecture that [MATH] for all [MATH].", '1611.04664-1-113-3': 'This problem has remained unsolved from both its Diophantine and ergodic sides, even in the basic case [MATH].', '1611.04664-1-113-4': "Cassels and Swinnerton-Dyer gave in their paper [CITATION] a Diophantine solution of the case that [MATH] is a cubic extension of [MATH], while much more recently Einsiedler, Katok and Lindenstrauss [CITATION] developed an entropy method by which they could establish Littlewood's conjecture for all [MATH] outside of a set of Hausdorff dimension zero.", '1611.04664-1-114-0': '## With homogeneous flows, the applications so far have been from dynamics to number theory: one finds a dynamical formulation of a Diophantine problem, which one seeks to approach by ergodic theory methods.', '1611.04664-1-114-1': '(There are also a few cases where alternative proofs are available from either the ergodic or Diophantine sides.)', '1611.04664-1-114-2': 'The interaction, however, is genuinely organic, and does not limit to using ergodic theory as a tool for number theory.', '1611.04664-1-114-3': 'We leave aside here the "intrinsic joins" such as the Diophantine character of KAM theory or the currently developing subject of Arithmetic Dynamics, and focus purely on pairs of an a priori dynamical and an a priori Diophantine problems that turn out to be mathematically equivalent.', '1611.04664-1-114-4': "Besides homogeneous dynamics, another setting for this is supplied by the robust class of higher rank systems studied in Klaus Schmidt's book [CITATION]: the ones said to be of algebraic origin, namely, the [MATH]-actions by automorphisms of a compact group.", '1611.04664-1-114-5': 'This is the context that our paper addresses.', '1611.04664-1-115-0': "Here, the implications thus far discovered tend to go with Weyl's direction, from number theory to dynamics; and here too, the Diophantine questions were formulated and studied, for their intrinsic interest, long before their dynamical significance could be found.", '1611.04664-1-115-1': 'The celebrated Lehmer problem, on the spectral gap in the Mahler measure (see Bombieri [CITATION] for a perspective in Diophantine Geometry), was shown by Lind to be equivalent to either of the following dynamical questions: [CITATION] does the infinite torus [MATH] admit an ergodic automorphism with finite entropy?', '1611.04664-1-115-2': '[CITATION] are all Bernoulli shifts algebraizable, in the form of a measurable equivalence with an automorphism of a compact group?', '1611.04664-1-115-3': "Lind, Schmidt and Ward's paper [CITATION] places this relation precisely in the general context of [MATH]-actions by automorphisms of a compact group, asking whether such dynamical systems are general or negligible from the measurable point of view.", '1611.04664-1-115-4': "Lehmer's problem equates to an alternative for the set of entropies of such systems: either it is countable, or it is the full continuum [MATH].", '1611.04664-1-116-0': '## These questions are, it seems, completely open ended.', '1611.04664-1-116-1': "A better understood indication of the depth of the interaction, which regards strictly the higher rank case, is the equivalence of W.M. Schmidt, A.J. Van der Poorten and H.P. Schlickewei's theorem of the finiteness of non-degenerate solutions to the equation [MATH] in a finitely generated group [MATH] (see Evertse and Gyory's book [CITATION] for a comprehensive overview of this subject), and K. Schmidt and T. Ward's theorem [CITATION] about higher order mixing, see also chapter VIII of Schmidt's [CITATION] or chapter 8.2 of Einsiedler and Ward's books [CITATION]: a mixing [MATH]-action by automorphisms of a compact connected abelian group is mixing of all orders.", '1611.04664-1-116-2': "The only known proof of this pair of equivalent results comes from number theory (via Weyl's criterion), with the Thue-Siegel-Roth-Schmidt method that we too employ in this paper.", '1611.04664-1-116-3': 'A dynamical approach to higher order mixing, should it exist in a form amenable to a completely quantitative estimate, could be expected to shed a light on a notorious Diophantine problem: to give an effective solution to the multivariate [MATH]-unit equation.', '1611.04664-1-117-0': '## The Diophantine/dynamical pair (A) and (B) reflects the growth and distribution of periodic points of algebraic [MATH]-actions.', '1611.04664-1-117-1': 'Theorem [REF] addresses this in the averaged sense of the cubical lattices [MATH].', '1611.04664-1-117-2': 'Refining this to arbitrary [MATH] and, furthermore, to long orbits in [MATH], still eludes a proof, and Theorem [REF] can be seen as an averaged form of growth rate and equidistribution.', '1611.04664-1-118-0': 'Such "averaged" growth and equidistribution properties are, as is well known, typical of dynamical systems that exhibit at least a partial hyperbolicity.', '1611.04664-1-118-1': "They hold for all weakly topologically mixing flows or transformations that satisfy Smale's Axiom A.", '1611.04664-1-118-2': 'We refer to Margulis [CITATION], Bowen [CITATION] and Parry and Pollicott [CITATION] for a detailed study of the growth and equidistribution of the periodic trajectories of such flows.', '1611.04664-1-118-3': "This subject was also inspired by number theory, but this time mainly on the level of analogy and methodology, the inspiration coming from Selberg's trace formula vis a vis the equidistribution of prime ideals in algebraic number fields [MATH] over ray class fields or as points of the space of lattices in [MATH].", '1611.04664-1-118-4': "Very recently, inspired by Sullivan's dictionary between Kleinian groups and the rational maps of complex dynamics, H. Oh and D. Winter [CITATION] obtained a completely new case of hyperbolic equidistribution lying outside of the framework of Axiom A systems.", '1611.04664-1-118-5': 'Let us also mention that, in arithmetically relevant cases, a refined equidistribution is sometimes available in which only the orbits of a fixed length are considered.', '1611.04664-1-118-6': "Duke's theorem [CITATION] is a fine example of this, powered by a lower bound on the total length [MATH] of the union of [MATH] equal length closed geodesics corresponding to a given discriminant, that amounts to Siegel's ineffective theorem [MATH] in prime number theory.", '1611.04664-1-118-7': 'In our application, equidistribution emerges analogously from a lower bound in terms of [MATH] on the size of [MATH].', '1611.04664-1-118-8': "The mechanism for this inference is in Ward's paper [CITATION] already quoted in the introduction, by means of commutative algebra and harmonic analysis on the group [MATH].", '1611.04664-1-119-0': "## The results of this paper appear to unify naturally into a common generalization with W. M. Schmidt's Subspace theorem.", '1611.04664-1-119-1': 'All these Diophantine results are semi-effective, regarding only the number of solutions to a Diophantine inequality.', '1611.04664-1-119-2': 'Section [REF] indicates some elements of such a unification, in addition to a few other extensions concerning points with sufficiently small canonical height.', '1611.04664-1-119-3': "Our motivation in doing this is an attempt to unravel the Diophantine nature of ([REF]), as well as the difficulties inherent in the quest for a higher dimensional Baker theorem - a problem whose central importance to number theory is stressed at the end of section 1.2 of Waldschmidt's book [CITATION].", '1611.04664-1-119-4': 'In dynamics, we have seen the two sides of the unification in, respectively, higher order mixing and the distribution of periodic points.', '1611.04664-1-120-0': '## Besides the refinement to subgroups and long orbits in [MATH], an obvious problem meriting a further study would be to extend the results on the distribution of periodic trajectories to algebraic actions or flows of other amenable groups, such as [MATH].', '1611.04664-1-120-1': "More interesting but completely open ended is the question of reversing the direction of the application, by having a general dynamical paradigm brought to bear on the unsolved problems in the Diophantine unification around A. Baker's and W. M. Schmidt's theorems.", '1611.04664-1-120-2': 'This concerns higher dimensions in the former and effectivity in the latter, and would be especially rewarding with regard to mixing and effectivity.', '1611.04664-1-120-3': "Halmos and Rokhlin's general question, of whether mixing yields an automatic higher order mixing for measure preserving transformations ([MATH]-actions), has turned out to reflect a surprisingly general phenomenon while remaining one of the oldest unsolved problems of classical ergodic theory.", '1611.04664-1-120-4': "The version for the weak mixing property was obtained by Furstenberg over his ergodic proof of Szemeredi's theorem, see [CITATION], Th.", '1611.04664-1-120-5': '3.1.', '1611.04664-1-120-6': "It could be also interesting to pursue the extension of Furstenberg's theorem to higher rank amenable group actions, and then the implications on the [MATH]-unit equation.", '1611.04664-1-121-0': 'An apparently fatal obstruction to a dynamical solution of the [MATH]-unit equation is the existence of counterexamples to automatic higher order strong mixing in the higher rank case.', '1611.04664-1-121-1': 'See Ledrappier [CITATION] for the example that, according to Schmidt [CITATION], has historically played a pivot role in the development of the whole subject of algebraic [MATH]-actions.', '1611.04664-1-121-2': "But a closer examination of Ledrappier's system actually bolsters the case for viewing this classical number theory problem as embedded in a completely general context of mixing in ergodic theory.", '1611.04664-1-121-3': "Ledrappier's example, [MATH], of a mixing algebraic [MATH]-action where a higher order mixing breaks down, has, along with its generalizations, been now understood to reflect a theorem in Diophantine Geometry over global function fields.", '1611.04664-1-121-4': 'Taking account of the effects of Frobenius in positive characteristic, similar structural theorems are available over function fields, leading in turn to corresponding structural theorems about mixing shapes: see Masser [CITATION], Derksen and Masser [CITATION], as well as section VIII.28 of Schmidt [CITATION] and section 7.6 of Evertse and Gyory [CITATION].', '1611.04664-1-121-5': 'The combined Diophantine methods then answer completely the Halmos-Rokhlin problem for algebraic [MATH]-actions on any compact abelian group.', '1611.04664-1-122-0': "As a matter of fact, zero dimensional groups [MATH] (the case in Ledrappier's system) are understood much more satisfactorily in this regard.", '1611.04664-1-122-1': "Replacing Schmidt's Subspace theorem, Julie Wang's effective truncated Second Main theorem [CITATION], for projective space and a constant divisor over a function field of any characteristic, combines with Derksen and Masser's results to solve the higher order mixing problem effectively.", '1611.04664-1-122-2': 'Contrastingly, the Diophantine prototypes over number fields have remained ineffective and much more mysterious.', '1611.04664-1-122-3': 'It remains to be seen whether a deepening of the dynamical connection would eventually be brought to bear, in this direction as well, on such outstanding questions of number theory.'}

{'1611.04664-2-0-0': 'We prove a sub-Liouville bound, up to a uniformly bounded exceptional set, on the distance from an [MATH]-torsion point of [MATH] to an algebraic subset, under a fixed Archimedean place [MATH].', '1611.04664-2-0-1': 'As a consequence, for all non-zero integer Laurent polynomials [MATH] in [MATH] commuting variables, we prove that the averages of [MATH] over [MATH], [MATH], converge as [MATH] to the Mahler measure of [MATH].', '1611.04664-2-1-0': 'By the work of B. Kitchens, D. Lind, K. Schmidt and T. Ward, this convergence consequence amounts to the following statement in dynamics: For every Noetherian [MATH]-action [MATH] by automorphisms of a compact abelian group [MATH] having a finite topological entropy [MATH], the annihilator [MATH] of [MATH] has [MATH] connected components, as [MATH].', '1611.04664-2-1-1': 'Moreover, it follows that all weak-[MATH] limit measures of the push-forwards of the Haar measures on [MATH], under any a sequence of positive integers [MATH], are measures of maximum entropy [MATH].', '1611.04664-2-2-0': 'Combined with work of Thang Le, this solves the abelian case of the problem of the asymptotic growth of torsion in the homology of congruence covers of a fixed finite simplicial complex.', '1611.04664-2-2-1': "We also give an indication on how an alternative, completely different route to the convergence result and its consequences is also possible through Habegger's recent work on Diophantine approximation to definable sets.", '1611.04664-2-3-0': '# Introduction', '1611.04664-2-4-0': 'Ergodic and dynamical systems theory have been marked over the past three decades by an extensive interaction with central problems of classical Diophantine analysis.', '1611.04664-2-4-1': 'This continues a much older tradition begun by Hermann Weyl in a 1914 paper [CITATION] that he conceived as "an application of number theory" to Boltzmann\'s ergodic hypothesis (Sur une application de la theorie des nombres a la mecanique statistique et la theorie des perturbations).', '1611.04664-2-4-2': "In more recent times, which have witnessed ergodic methods penetrating deeply into all aspects of number theory, the direction of implications is usually reversed, and another line of inquiry, emphasizing higher rank and rigidity phenomena with their relation to Diophantine approximations, was started with Furstenberg's influential 1966 paper [CITATION], where the famous [MATH] problem was formulated.", '1611.04664-2-4-3': "Our present paper owes something to both traditions, by using number theory to verify a hypothesis on the distribution of periodic trajectories for a class of higher rank dynamical systems that contains Furstenberg's [MATH] system as a very special case: the Noetherian [MATH]-actions by automorphisms of a compact abelian group.", '1611.04664-2-4-4': 'In the rank one case this problem was solved long ago, by Lind in [CITATION].', '1611.04664-2-4-5': 'Our result addresses the higher rank situation, where the number theory is less understood, and could, conceivably, turn out to behave differently.', '1611.04664-2-5-0': 'A superb introduction to ergodic ideas in number theory is in the book [CITATION] by Einsiedler and Ward.', '1611.04664-2-5-1': 'The reader will find in our final section [REF] an overview of Diophantine/dynamical pairs of mathematically equivalent problems, placing our result in a broad yet, we hope, sufficiently focused context.', '1611.04664-2-6-0': 'In this paper, we treat a pair of equivalent questions about algebraic [MATH]-actions: the [MATH]-actions [MATH] by automorphisms of a compact group [MATH].', '1611.04664-2-6-1': "This addresses the problem left open in T. Ward's thesis [CITATION] and then in K. Schmidt's book [CITATION], and the papers [CITATION] of D. Lind, K. Schmidt and E. Verbitskiy: when [MATH] is abelian, and under an appropriate - and necessary - Noetherianness assumption (the descending chain condition on closed invariant subgroups), does the exponential growth rate of the size of the component group of the subgroup [MATH] of [MATH]-periodic points, indexed by an order sublattice [MATH], converge to the topological entropy of the system, assuming the latter is finite?", '1611.04664-2-6-2': 'We answer this in the affirmative in the symmetric case [MATH] of the cubical sublattices.', '1611.04664-2-6-3': 'On applying another paper [CITATION] of Ward, this furthermore proves that, when [MATH] has a completely positive entropy, the subgroups [MATH] become equidistributed in the Haar measure of the group [MATH], as [MATH].', '1611.04664-2-6-4': "Completely positive (finite) entropy is the natural condition as it means equivalently that the Haar measure is the unique probability measure of maximum entropy: this is Berg's theorem, proved in this situation by Lind, Schmidt and Ward [CITATION].", '1611.04664-2-6-5': 'In the general case, when [MATH] supports more than one maximum entropy measure, the conclusion is that all weak-[MATH] limits over some such sequence of levels [MATH] are measures of maximum entropy.', '1611.04664-2-7-0': "As we indicate in sections [REF] and [REF] below, an entirely different solution, coming again from the number theory side, is implicit in Habegger's very recent manuscript [CITATION].", '1611.04664-2-7-1': 'Habegger does not note the dynamical connection, but the corresponding convergence problem follows easily from his general estimate on the number of rational points lying very near to a subset of [MATH] definable in a polynomially bounded [MATH]-minimal expansion of the real numbers, in conjunction with the Ax and Koksma-Hlawka theorems.', '1611.04664-2-7-2': "Habegger's and our Diophantine results are quite different, and generalize in completely disjoint directions.", '1611.04664-2-7-3': 'Neither of the two appears capable of addressing the growth rate and equidistribution of the general groups [MATH], and even less the equidistribution of individual periodic trajectories in [MATH] that are "long enough" with respect to a certain other characteristic of the orbit, that we shall not discuss in this paper, which following Einsiedler, Lindenstrauss, Michel and Venkatesh [CITATION] could be called the "discriminant" (or "denominator") of the orbit; see also chapter 21 of McMullen [CITATION] for a discussion of the basic case of the doubling map of the circle.', '1611.04664-2-7-4': 'On the other hand, as the groups [MATH] considered exhaust the periodic points of the [MATH]-action, our result can be seen as an averaged form of the growth and equidistribution of periodic trajectories of an algebraic [MATH]-action.', '1611.04664-2-8-0': 'In the case that presently concerns us, the substance of the problem lies in the systems that exhibit only a partially hyperbolic behavior; the case of expansive [MATH]-actions has no Diophantine content, and was settled already in Ward [CITATION].', '1611.04664-2-8-1': 'The Diophantine problem was recognized by Lind in [CITATION] to emerge from the dual description (or standard model) of an algebraic [MATH]-action, by means of commutative algebra.', '1611.04664-2-8-2': 'A [MATH]-action [MATH] by automorphisms of a compact abelian group [MATH] is given by a module [MATH] over the Laurent series ring [MATH].', '1611.04664-2-8-3': 'The descending chain condition mentioned amounts precisely to the Noetherianness condition of the [MATH]-module [MATH].', '1611.04664-2-8-4': 'When this is the case (but not without the condition, nor when the group [MATH] is non-abelian), Kitchens and Schmidt (see [CITATION], Cor.', '1611.04664-2-8-5': '4.8) proved that the periodic points of [MATH] are dense.', '1611.04664-2-8-6': "One then expects the situation to be similar to the basic case of a toral automorphism, where the dynamics is given by an invertible integer matrix [MATH]; its entropy [MATH] equals the Mahler measure [MATH] of the companion polynomial [MATH]; the number of points with finite period [MATH] equals [MATH]; and the convergence [MATH] amounts precisely to A. O. Gelfond's theorem that an algebraic integer [MATH] of unit modulus [MATH] may not be exponentially approached by an [MATH]-th root of unity [MATH]: [MATH], as [MATH].", '1611.04664-2-8-7': 'The extension of this to solenoids ([MATH]) covers the case of an arbitrary algebraic [MATH] of unit modulus.', '1611.04664-2-8-8': "For a lucid and leisurely treatment of the [MATH] case, including Yuzvinskii's calculation of the entropy, we refer the reader to Everest and Ward's book [CITATION], in addition to Lind's original paper [CITATION].", '1611.04664-2-9-0': 'That the general situation is indeed similar, only much more complicated, emerged from the work of Kitchens and Schmidt [CITATION], Ward [CITATION] and Lind, Schmidt and Ward [CITATION].', '1611.04664-2-9-1': 'By an addition formula for the entropy, due to Yuzvinskii and Thomas in the rank one case, a devissage reduces the problem to the case that [MATH], where [MATH] is an ideal of the Laurent ring: the case of a cyclic action.', '1611.04664-2-9-2': 'Then the entropy of [MATH] turns out to be zero unless the ideal [MATH] is principal, in which case it equals the Mahler measure [MATH] (if [MATH], and infinity if [MATH]).', '1611.04664-2-9-3': 'In such a way, Lind, Schmidt and Ward (see also the introduction in Lind, Schmidt and Verbitskiy [CITATION]) obtain the equivalence of the following dynamical and Diophantine statements.', '1611.04664-2-9-4': 'With [MATH] ranging over all finite index subgroups of [MATH], let [MATH], and denote by [MATH] the number of connected components of the group of [MATH]-periodic points for [MATH].', '1611.04664-2-9-5': 'Let [MATH] be the topological entropy of the system [MATH], which coincides with the metric entropy for the Haar measure.', '1611.04664-2-9-6': 'On the dynamical side we expect:', '1611.04664-2-10-0': '(A) Assume [MATH] has finite topological entropy and satisfies the d.c.c.: every descending chain of closed invariant subgroups of [MATH] is stationary.', '1611.04664-2-10-1': 'Then, [EQUATION]', '1611.04664-2-11-0': 'In general, as explained in [CITATION] on page 619, the Noetherian and finite entropy conditions are certainly necessary in such a statement; nothing could be said for non-Noetherian systems, where the growth rate need not converge and any rate between [MATH] and [MATH] may occur (see, however, Baier, Jaidee, Stevens and Ward [CITATION]).', '1611.04664-2-11-1': 'Under the Noetherian and finite entropy conditions, Schmidt proves in section 21 of [CITATION] a weaker statement identifying [MATH] with the limit supremum over sublattices [MATH].', '1611.04664-2-11-2': "He first proves the upper bound in a direct application of the topological definition of [MATH], and then uses Gelfond's theorem and the formula for the entropy to exhibit a particular sequence [MATH] along which [MATH].", '1611.04664-2-11-3': 'This sequence is very special; in particular, the finite subgroup [MATH] is of the form [MATH], where [MATH] with respect to [MATH]; see section [REF] for more on this.', '1611.04664-2-12-0': 'The general problem (A) thus amounts to establishing a lower bound on the growth of periodic points, and as already mentioned, the same applies to equidistribution.', '1611.04664-2-12-1': 'Similarly to the Brauer-Siegel theorem and to questions on orbit growth in the arithmetic dynamics of rational maps (for the latter, see Silverman [CITATION]; Kawaguchi-Silverman [CITATION]; Dimitrov [CITATION]), the upper bounds come easily, and the question of the lower bound proves to be a subtle problem related intrinsically to Diophantine Approximations.', '1611.04664-2-13-0': "In the case at hand, the cited work of Kitchens, Schmidt, Lind and Ward renders (A) precisely equivalent to the following multidimensional extension of Gelfond's result.", '1611.04664-2-13-1': 'Note that as [MATH], the finite group [MATH] is equidistributed in the Haar measure [MATH] of the torus [MATH].', '1611.04664-2-14-0': '(B) Let [MATH] be a non-zero integer Laurent polynomial.', '1611.04664-2-14-1': 'Then [EQUATION]', '1611.04664-2-15-0': 'This is obvious in the case that the hypersurface [MATH] has empty intersection with the real torus [MATH].', '1611.04664-2-15-1': 'From the dynamical point of view, this happens precisely when the cyclic [MATH]-action dual to the [MATH]-module [MATH] is expansive.', '1611.04664-2-15-2': 'By the alluded devissage procedure, Lind, Schmidt and Ward were thus able to prove (A) for all expansive systems [MATH].', '1611.04664-2-15-3': 'We note that, for expansive systems, both the finiteness of the entropy and the Noetherianness conditions are automatic: see Kitchens and Schmidt [CITATION], Th.', '1611.04664-2-15-4': '5.2.', '1611.04664-2-16-0': 'Further progress was made by Lind, Schmidt and Verbitskiy in [CITATION], who gave two proofs of (B) in the case that the zero locus [MATH] intersects the torus [MATH] in a finite set.', '1611.04664-2-16-1': "The first is Diophantine, and consists of the observation that since the finitely many intersection points are necessarily algebraic, an application of Gelfond's (one-dimensional) theorem suffices to cover that case.", '1611.04664-2-16-2': 'More significant is their second proof, purely dynamical, by means of a construction of rapidly decaying homoclinic points.', '1611.04664-2-16-3': 'In a subsequent paper [CITATION], the same authors then extended their dynamical method to prove (B) in the "generic" case that the intersection locus [MATH] has real codimension at least two in the torus.', '1611.04664-2-16-4': 'The dynamical equivalent of this "atoral" hypothesis turns out to be precisely the existence of summable homoclinic points, and that is what allowed the authors to bypass the delicate Diophantine issues about torsion points getting extremely close to the zero locus [MATH].', '1611.04664-2-17-0': 'In the present paper, we solve (B) for all Laurent polynomials [MATH] in the case that [MATH] runs over the cubical sublattices [MATH], so that the average is taken over all [MATH]-tuples of [MATH]-th roots of unity.', '1611.04664-2-17-1': 'It follows from the work of Kitchens, Lind, Schmidt and Ward cited above that (A) is true when all the [MATH]-periodic points are taken together.', '1611.04664-2-18-0': 'The precise statement of the equidistribution consequence about periodic trajectories is as follows.', '1611.04664-2-18-1': "The proof of the implication can be found in Ward's paper [CITATION] as well as in section 22 of Schmidt's book [CITATION].", '1611.04664-2-19-0': 'For [MATH], let [MATH] be the push-forward on [MATH] of the Haar measure of the subgroup [MATH] of [MATH]-periodic points of [MATH].', '1611.04664-2-19-1': 'If [MATH] is a probability measure having [MATH] in the weak-[MATH] topology for some sequence [MATH], then [MATH] is a measure of maximum entropy: [MATH].', '1611.04664-2-20-0': 'In particular, if the system [MATH] has a completely positive entropy, then the measures [MATH] converge to the Haar measure of [MATH]: the periodic points equidisitribute in the Haar measure.', '1611.04664-2-21-0': 'One may wish to refine the equidistribution corollary to individual periodic trajectories.', '1611.04664-2-21-1': 'However, at least for rank [MATH], some care is needed in such a statement.', '1611.04664-2-21-2': 'Already for the case of the doubling map on the circle [MATH], it is not true that individual periodic trajectories are equidistributed in Haar measure as [MATH] approaches infinity.', '1611.04664-2-21-3': "This is similar to Duke's theorem, where an individual closed geodesic may be distributed in a practically arbitrary way.", '1611.04664-2-21-4': "The parallel with Duke's theorem is discussed in Einsiedler, Lindenstrauss, Michel and Venkatesh [CITATION], and appears to run rather deep.", '1611.04664-2-21-5': 'It includes the idea of equidistribution emerging from a lower bound on total length in terms of the discriminant of the quadratic order; respectively, what could be called following these authors the "discriminant" (or "denominator") [MATH] of the periodic orbit.', '1611.04664-2-21-6': 'We shall not define this quantity here, but refer to section 1.7 of [CITATION] or to section 21 ("The discriminant-regulator paradox") of McMullen [CITATION] for the basic case of the doubling map of the circle, where the exponential sum bounds of Bourgain, Glibichuk and Konyagin [CITATION] are interpreted as giving equidistribution of individual periodic orbits having length [MATH] exceeding [MATH] for a fixed [MATH].', '1611.04664-2-21-7': 'This refined equidistribution conjecture now extends to arbitrary Noetherian algebraic [MATH]-actions of finite entropy.', '1611.04664-2-21-8': "In this generality, equidistribution appears to be a very difficult open problem, as is the corresponding conjecture in Duke's theorem and the higher rank generalizations thereof.", '1611.04664-2-21-9': 'A partial progress in the case of torus actions was made by Aka and Einsiedler [CITATION].', '1611.04664-2-22-0': 'We will focus our exposition on (B) and suppress the details of its equivalence with (A), and the consequence on equidistribution.', '1611.04664-2-22-1': "Those details are well known in the [MATH]-actions community, and may be read, respectively, from the introduction following Theorem 1.3 of [CITATION], and from Ward's paper [CITATION] (exposed in chapter 22 of Schmidt's book [CITATION]).", '1611.04664-2-22-2': 'Our proof of Theorem [REF] is effective, in the sense of yielding a quantitative estimate on the convergence in (B) in terms of [MATH], [MATH] and the degree and height of the polynomial [MATH].', '1611.04664-2-22-3': 'Correspondingly, the equidistribution in the Corollary is also effective.', '1611.04664-2-23-0': 'The convergence problem (B) was also raised as Conjecture 15 in a recent paper [CITATION] of Silverman, whose motivation was a study of higher rank divisibility sequences having a linear torus as the underlying algebraic group.', '1611.04664-2-23-1': "The cubical case of [MATH] solved by Theorem [REF] was given a separate attention as Conjecture 2 in Silverman's paper."}

[['1611.04664-1-11-0', '1611.04664-2-12-0'], ['1611.04664-1-11-1', '1611.04664-2-12-1'], ['1611.04664-1-16-0', '1611.04664-2-17-0'], ['1611.04664-1-16-1', '1611.04664-2-17-1'], ['1611.04664-1-19-0', '1611.04664-2-20-0'], ['1611.04664-1-3-0', '1611.04664-2-4-0'], ['1611.04664-1-3-1', '1611.04664-2-4-1'], ['1611.04664-1-3-2', '1611.04664-2-4-2'], ['1611.04664-1-3-3', '1611.04664-2-4-3'], ['1611.04664-1-3-4', '1611.04664-2-4-4'], ['1611.04664-1-15-1', '1611.04664-2-16-1'], ['1611.04664-1-15-2', '1611.04664-2-16-2'], ['1611.04664-1-15-3', '1611.04664-2-16-3'], ['1611.04664-1-15-4', '1611.04664-2-16-4'], ['1611.04664-1-22-0', '1611.04664-2-23-0'], ['1611.04664-1-22-1', '1611.04664-2-23-1'], ['1611.04664-1-5-1', '1611.04664-2-6-1'], ['1611.04664-1-5-2', '1611.04664-2-6-2'], ['1611.04664-1-5-3', '1611.04664-2-6-3'], ['1611.04664-1-5-5', '1611.04664-2-6-5'], ['1611.04664-1-18-0', '1611.04664-2-19-0'], ['1611.04664-1-18-1', '1611.04664-2-19-1'], ['1611.04664-1-6-0', '1611.04664-2-7-0'], ['1611.04664-1-6-1', '1611.04664-2-7-1'], ['1611.04664-1-6-4', '1611.04664-2-7-4'], ['1611.04664-1-17-0', '1611.04664-2-18-0'], ['1611.04664-1-17-1', '1611.04664-2-18-1'], ['1611.04664-1-8-0', '1611.04664-2-9-0'], ['1611.04664-1-8-2', '1611.04664-2-9-2'], ['1611.04664-1-8-5', '1611.04664-2-9-5'], ['1611.04664-1-14-0', '1611.04664-2-15-0'], ['1611.04664-1-14-1', '1611.04664-2-15-1'], ['1611.04664-1-14-2', '1611.04664-2-15-2'], ['1611.04664-1-14-3', '1611.04664-2-15-3'], ['1611.04664-1-10-2', '1611.04664-2-11-2'], ['1611.04664-1-10-3', '1611.04664-2-11-3'], ['1611.04664-1-12-0', '1611.04664-2-13-0'], ['1611.04664-1-12-1', '1611.04664-2-13-1'], ['1611.04664-1-4-0', '1611.04664-2-5-0'], ['1611.04664-1-4-1', '1611.04664-2-5-1'], ['1611.04664-1-7-2', '1611.04664-2-8-2'], ['1611.04664-1-7-4', '1611.04664-2-8-4'], ['1611.04664-1-7-5', '1611.04664-2-8-5'], ['1611.04664-1-7-6', '1611.04664-2-8-6'], ['1611.04664-1-7-7', '1611.04664-2-8-7'], ['1611.04664-1-7-8', '1611.04664-2-8-8'], ['1611.04664-1-21-0', '1611.04664-2-22-0'], ['1611.04664-1-21-1', '1611.04664-2-22-1'], ['1611.04664-1-21-2', '1611.04664-2-22-2'], ['1611.04664-1-21-3', '1611.04664-2-22-3'], ['1611.04664-1-9-0', '1611.04664-2-10-0'], ['1611.04664-1-0-4', '1611.04664-2-1-1'], ['1611.04664-1-3-5', '1611.04664-2-4-5'], ['1611.04664-1-15-0', '1611.04664-2-16-0'], ['1611.04664-1-5-0', '1611.04664-2-6-0'], ['1611.04664-1-5-4', '1611.04664-2-6-4'], ['1611.04664-1-6-2', '1611.04664-2-7-2'], ['1611.04664-1-6-3', '1611.04664-2-7-3'], ['1611.04664-1-8-1', '1611.04664-2-9-1'], ['1611.04664-1-8-3', '1611.04664-2-9-3'], ['1611.04664-1-8-4', '1611.04664-2-9-4'], ['1611.04664-1-10-0', '1611.04664-2-11-0'], ['1611.04664-1-10-1', '1611.04664-2-11-1'], ['1611.04664-1-7-0', '1611.04664-2-8-0'], ['1611.04664-1-7-1', '1611.04664-2-8-1'], ['1611.04664-1-7-3', '1611.04664-2-8-3'], ['1611.04664-1-0-3', '1611.04664-2-1-0'], ['1611.04664-1-1-1', '1611.04664-2-2-1'], ['1611.04664-1-0-2', '1611.04664-2-0-1'], ['1611.04664-1-20-0', '1611.04664-2-21-0'], ['1611.04664-1-20-1', '1611.04664-2-21-2'], ['1611.04664-1-20-2', '1611.04664-2-21-3'], ['1611.04664-1-20-3', '1611.04664-2-21-5'], ['1611.04664-1-20-4', '1611.04664-2-21-6'], ['1611.04664-1-20-5', '1611.04664-2-21-7'], ['1611.04664-1-20-6', '1611.04664-2-21-8']]

[['1611.04664-1-11-0', '1611.04664-2-12-0'], ['1611.04664-1-11-1', '1611.04664-2-12-1'], ['1611.04664-1-16-0', '1611.04664-2-17-0'], ['1611.04664-1-16-1', '1611.04664-2-17-1'], ['1611.04664-1-19-0', '1611.04664-2-20-0'], ['1611.04664-1-3-0', '1611.04664-2-4-0'], ['1611.04664-1-3-1', '1611.04664-2-4-1'], ['1611.04664-1-3-2', '1611.04664-2-4-2'], ['1611.04664-1-3-3', '1611.04664-2-4-3'], ['1611.04664-1-3-4', '1611.04664-2-4-4'], ['1611.04664-1-15-1', '1611.04664-2-16-1'], ['1611.04664-1-15-2', '1611.04664-2-16-2'], ['1611.04664-1-15-3', '1611.04664-2-16-3'], ['1611.04664-1-15-4', '1611.04664-2-16-4'], ['1611.04664-1-22-0', '1611.04664-2-23-0'], ['1611.04664-1-22-1', '1611.04664-2-23-1'], ['1611.04664-1-5-1', '1611.04664-2-6-1'], ['1611.04664-1-5-2', '1611.04664-2-6-2'], ['1611.04664-1-5-3', '1611.04664-2-6-3'], ['1611.04664-1-5-5', '1611.04664-2-6-5'], ['1611.04664-1-18-0', '1611.04664-2-19-0'], ['1611.04664-1-18-1', '1611.04664-2-19-1'], ['1611.04664-1-6-0', '1611.04664-2-7-0'], ['1611.04664-1-6-1', '1611.04664-2-7-1'], ['1611.04664-1-6-4', '1611.04664-2-7-4'], ['1611.04664-1-17-0', '1611.04664-2-18-0'], ['1611.04664-1-17-1', '1611.04664-2-18-1'], ['1611.04664-1-8-0', '1611.04664-2-9-0'], ['1611.04664-1-8-2', '1611.04664-2-9-2'], ['1611.04664-1-8-5', '1611.04664-2-9-5'], ['1611.04664-1-14-0', '1611.04664-2-15-0'], ['1611.04664-1-14-1', '1611.04664-2-15-1'], ['1611.04664-1-14-2', '1611.04664-2-15-2'], ['1611.04664-1-14-3', '1611.04664-2-15-3'], ['1611.04664-1-10-2', '1611.04664-2-11-2'], ['1611.04664-1-10-3', '1611.04664-2-11-3'], ['1611.04664-1-12-0', '1611.04664-2-13-0'], ['1611.04664-1-12-1', '1611.04664-2-13-1'], ['1611.04664-1-4-0', '1611.04664-2-5-0'], ['1611.04664-1-4-1', '1611.04664-2-5-1'], ['1611.04664-1-7-2', '1611.04664-2-8-2'], ['1611.04664-1-7-4', '1611.04664-2-8-4'], ['1611.04664-1-7-5', '1611.04664-2-8-5'], ['1611.04664-1-7-6', '1611.04664-2-8-6'], ['1611.04664-1-7-7', '1611.04664-2-8-7'], ['1611.04664-1-7-8', '1611.04664-2-8-8'], ['1611.04664-1-21-0', '1611.04664-2-22-0'], ['1611.04664-1-21-1', '1611.04664-2-22-1'], ['1611.04664-1-21-2', '1611.04664-2-22-2'], ['1611.04664-1-21-3', '1611.04664-2-22-3'], ['1611.04664-1-9-0', '1611.04664-2-10-0'], ['1611.04664-1-0-4', '1611.04664-2-1-1']]

[['1611.04664-1-3-5', '1611.04664-2-4-5'], ['1611.04664-1-15-0', '1611.04664-2-16-0'], ['1611.04664-1-5-0', '1611.04664-2-6-0'], ['1611.04664-1-5-4', '1611.04664-2-6-4'], ['1611.04664-1-6-2', '1611.04664-2-7-2'], ['1611.04664-1-6-3', '1611.04664-2-7-3'], ['1611.04664-1-8-1', '1611.04664-2-9-1'], ['1611.04664-1-8-3', '1611.04664-2-9-3'], ['1611.04664-1-8-4', '1611.04664-2-9-4'], ['1611.04664-1-10-0', '1611.04664-2-11-0'], ['1611.04664-1-10-1', '1611.04664-2-11-1'], ['1611.04664-1-7-0', '1611.04664-2-8-0'], ['1611.04664-1-7-1', '1611.04664-2-8-1'], ['1611.04664-1-7-3', '1611.04664-2-8-3'], ['1611.04664-1-0-3', '1611.04664-2-1-0']]

[]

[['1611.04664-1-1-1', '1611.04664-2-2-1'], ['1611.04664-1-0-2', '1611.04664-2-0-1']]

[['1611.04664-1-20-0', '1611.04664-2-21-0'], ['1611.04664-1-20-1', '1611.04664-2-21-2'], ['1611.04664-1-20-2', '1611.04664-2-21-3'], ['1611.04664-1-20-3', '1611.04664-2-21-5'], ['1611.04664-1-20-4', '1611.04664-2-21-6'], ['1611.04664-1-20-5', '1611.04664-2-21-7'], ['1611.04664-1-20-6', '1611.04664-2-21-8']]

['1611.04664-1-8-6', '1611.04664-1-9-1', '1611.04664-1-13-0', '1611.04664-1-13-1', '1611.04664-1-14-4', '1611.04664-1-36-1', '1611.04664-1-55-0', '1611.04664-1-57-1', '1611.04664-1-60-0', '1611.04664-1-67-0', '1611.04664-1-80-2', '1611.04664-1-107-2', '1611.04664-1-111-0', '1611.04664-1-120-5', '1611.04664-2-9-6', '1611.04664-2-10-1', '1611.04664-2-14-0', '1611.04664-2-14-1', '1611.04664-2-15-4']

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

https://arxiv.org/abs/1611.04664

1409.6021

{'1409.6021-1-0-0': 'Random [MATH]-intersection graphs have recently received much interest [CITATION].', '1409.6021-1-0-1': 'In a random [MATH]-intersection graph, each vertex is assigned to a set of items in some manner, and two vertices have an edge in between if and only if they share at least [MATH] items.', '1409.6021-1-0-2': 'In particular, in a uniform random [MATH]-intersection graph, each vertex independently selects the same number of items uniformly at random from a common item pool, while in a binomial random [MATH]-intersection graph, each item in some item pool is independently attached to each vertex with the same probability.', '1409.6021-1-0-3': 'These two graph models have numerous applications; e.g., using uniform random [MATH]-intersection graph for cryptanalysis [CITATION], and to model secure wireless sensor networks [CITATION] and online social networks [CITATION], and using a binomial random [MATH]-intersection graph for clustering analysis [CITATION], classification [CITATION] and the design of integrated circuits [CITATION].', '1409.6021-1-1-0': 'For binomial/uniform random [MATH]-intersection graphs, we present results related to [MATH]-connectivity and minimum vertex degree.', '1409.6021-1-1-1': 'Specifically, we derive the asymptotically exact probabilities and asymptotic zero-one laws for the following three properties: (i) [MATH]-vertex-connectivity, (ii) [MATH]-edge-connectivity and (iii) the property of minimum vertex degree being at least [MATH].', '1409.6021-1-1-2': 'Recently, similar results have been obtained by Bloznelis and Rybarczyk [CITATION], but their findings are only for uniform random [MATH]-intersection graphs, not for binomial random [MATH]-intersection graphs, and require parameter conditions which are disjoint from ours - our parameter conditions are useful in practical sensor network applications of the graphs while theirs are not.', '1409.6021-1-1-3': 'In terms of binomial [MATH]-intersection graphs, for the three properties above, our paper reports the first result on the exact probabilities as well as the first result on the zero-one laws.', '1409.6021-1-2-0': 'Keywords-Random intersection graph, vertex connectivity, edge connectivity, vertex degree.', '1409.6021-1-3-0': '# Introduction', '1409.6021-1-4-0': 'Random [MATH]-intersection graphs have received considerable attention recently [CITATION].', '1409.6021-1-4-1': 'In such a graph, each vertex is equipped with a set of items in some manner, and two vertices establish an undirected edge in between if and only if they have at least [MATH] items in common.', '1409.6021-1-4-2': 'A large number of work [CITATION] study the case of [MATH] being [MATH], under which the graphs are simply referred to as random intersection graphs.', '1409.6021-1-5-0': 'Random ([MATH]-)intersection graphs have been used to model secure wireless sensor networks [CITATION], wireless frequency hopping [CITATION], epidemics in human populations [CITATION], social networks [CITATION] such as collaboration networks [CITATION] and common-interest networks [CITATION].', '1409.6021-1-5-1': 'Random intersection graphs also motivated Beer et al. [CITATION] to introduce a general concept of vertex random graphs that subsumes any graph model where random features are assigned to vertices, and edges are drawn based on deterministic relations between the features of the vertices.', '1409.6021-1-6-0': 'Among different models of random [MATH]-intersection graphs, two widely studied models are the so-called uniform random [MATH]-intersection graph and binomial random [MATH]-intersection graph, which are defined in detail below.', '1409.6021-1-7-0': '## Graph models', '1409.6021-1-8-0': 'Uniform random [MATH]-intersection graph.', '1409.6021-1-8-1': 'A uniform random [MATH]-intersection graph, denoted by [MATH], is defined on [MATH] vertices as follows.', '1409.6021-1-8-2': 'Each vertex independently selects [MATH] different items uniformly at random from a pool of [MATH] distinct items.', '1409.6021-1-8-3': 'Two vertices have an edge in between if and only if they share at least [MATH] items.', '1409.6021-1-8-4': 'The notion "uniform" means that all vertices have the same number of items (but likely different sets of items).', '1409.6021-1-8-5': '[MATH] and [MATH] are both functions of [MATH], while [MATH] does not scale with [MATH].', '1409.6021-1-8-6': 'It holds that [MATH].', '1409.6021-1-9-0': 'Binomial random [MATH]-intersection graph.', '1409.6021-1-9-1': 'A binomial random [MATH]-intersection graph, denoted by [MATH], is defined on [MATH] vertices as follows.', '1409.6021-1-9-2': 'Each item from a pool of [MATH] distinct items is assigned to each vertex independently with probability [MATH].', '1409.6021-1-9-3': 'Two vertices establish an edge in between if and only if they have at least [MATH] items in common.', '1409.6021-1-9-4': 'The term "binomial" is used since the number of items assigned to each vertex follows a binomial distribution with parameters [MATH] (the number of trials) and [MATH] (the success probability in each trial).', '1409.6021-1-9-5': '[MATH] and [MATH] are both functions of [MATH], while [MATH] does not scale with [MATH].', '1409.6021-1-9-6': 'Also it holds that [MATH].', '1409.6021-1-10-0': '## Problem statement', '1409.6021-1-11-0': 'Our main goal in this paper is to investigate properties related to [MATH]-connectivity and minimum vertex degree of random [MATH]-intersection graphs ([MATH]-vertex connectivity and [MATH]-edge connectivity are called together as [MATH]-connectivity for convenience).', '1409.6021-1-11-1': 'In particular, we wish to answer the following question:', '1409.6021-1-12-0': 'For uniform random [MATH]-intersection graph [MATH] (resp., binomial random [MATH]-intersection graph [MATH]), with parameters [MATH] (resp., [MATH]) and [MATH] scaling with the number of vertices [MATH], what is the asymptotic behavior of the probabilities that [MATH] (resp., [MATH]) is (i) [MATH]-vertex-connected, (ii) [MATH]-edge-connected, and (iii) has a minimum vertex degree at least [MATH], respectively, as [MATH] grows large?', '1409.6021-1-12-1': 'A graph is said to be [MATH]-vertex-connected if the remaining graph is still connected despite the deletion of at most [MATH] arbitrary vertices, and [MATH]-edge-connectivity is defined similarly for the deletion of edges [CITATION]; with [MATH], these definitions reduce to the standard notion of graph connectivity [CITATION].', '1409.6021-1-12-2': 'The degree of a vertex is defined as the number of edges incident on it.', '1409.6021-1-12-3': 'The three graph properties considered here are related to each other in that [MATH]-vertex-connectivity implies [MATH]-edge-connectivity, which in turn implies that the minimum vertex degree is at least [MATH] [CITATION].', '1409.6021-1-13-0': '## Summary of Results', '1409.6021-1-14-0': 'We summarize our results below, first for a uniform random [MATH]-intersection graph and then for a binomial random [MATH]-intersection graph.', '1409.6021-1-14-1': 'Throughout the paper, both [MATH] and [MATH] are positive integers and do not scale with [MATH].', '1409.6021-1-14-2': 'Also, naturally we consider [MATH] for graph [MATH] and [MATH] for graph [MATH].', '1409.6021-1-14-3': 'We use the standard Landau asymptotic notation [MATH].', '1409.6021-1-14-4': '[MATH] denotes the probability that event [MATH] happens.', '1409.6021-1-15-0': '[MATH]-Connectivity and minimum vertex degree in a uniform random [MATH]-intersection graph.', '1409.6021-1-16-0': 'For uniform random [MATH]-intersection graph [MATH] under [MATH], with sequence [MATH] defined by [EQUATION] if [MATH], then the following convergence results hold:', '1409.6021-1-17-0': '[MATH]-Connectivity and minimum vertex degree in a binomial random [MATH]-intersection graph.', '1409.6021-1-18-0': 'For binomial random [MATH]-intersection graph [MATH] under [MATH] for [MATH] or [MATH] for [MATH] with some constant [MATH], with sequence [MATH] defined by [EQUATION] if [MATH], then the following convergence results hold:', '1409.6021-1-19-0': '## Comparison with related work', '1409.6021-1-20-0': 'Table [REF] on the next page summarizes relevant work in the literature on uniform/binomial random [MATH]-intersection graphs in terms of [MATH]-vertex-connectivity, [MATH]-edge connectivity, and the property of minimum vertex degree being at least [MATH].', '1409.6021-1-20-1': 'As noted in Footnote [REF], a zero-one law [CITATION] means that the probability that the graph has certain property asymptotically converges to [MATH] under some conditions and converges to [MATH] under some other conditions.', '1409.6021-1-21-0': 'Among the related work, Bloznelis and Rybarczyk [CITATION] recently derived the asymptotically exact probabilities of uniform random [MATH]-intersection graphs for the three properties above.', '1409.6021-1-21-1': 'Yet, their result applies only under the condition of [MATH] as [MATH]; specifically, their result holds only on the range [MATH], where [MATH] and [MATH] are some positive constants.', '1409.6021-1-21-2': 'However, in the secure wireless sensor network application of uniform random [MATH]-intersection graphs, [MATH] is expected to be at least on the order of [MATH] to ensure that the network have reasonable resiliency against sensor capture attacks [CITATION]; namely, to obtain results that are useful in practice, it holds that [MATH].', '1409.6021-1-21-3': 'Clearly, because of [MATH], the range of [MATH] assumed by Bloznelis and Rybarczyk [CITATION] does not cover the practical range of [MATH].', '1409.6021-1-21-4': 'The results reported in this paper cover the practical range [MATH] and fill this gap in the literature.', '1409.6021-1-22-0': 'In addition to extending the literature on uniform random [MATH]-intersection graphs, as shown in Table [REF], we also present novel results for binomial random [MATH]-intersection graphs, including asymptotically exact probabilities for [MATH]-connectivity and the property of minimum vertex degree being at least [MATH], under (i) general [MATH], and [MATH] being general or [MATH], or (ii) [MATH] being [MATH], and [MATH] being general.', '1409.6021-1-23-0': '## Roadmap', '1409.6021-1-24-0': 'We organize the rest of the paper as follows.', '1409.6021-1-24-1': 'We detail the main results as theorems in Section [REF].', '1409.6021-1-24-2': 'Sections [REF] and [REF] detail the steps of establishing the theorems.', '1409.6021-1-24-3': 'We conclude the paper in Section [REF].', '1409.6021-1-24-4': 'The Appendix provides additional arguments used in proving the theorems.', '1409.6021-1-25-0': 'Second, to show [MATH] can be indeed confined as [MATH], similar to Lemma [REF], we show the following: For graph [MATH] under [MATH] and edge probability [EQUATION] with [MATH], there exists graph [MATH] under [MATH] and edge probability [EQUATION] with [MATH] and [MATH], such that there exists a graph coupling under which [MATH] is a spanning subgraph of [MATH].', '1409.6021-1-26-0': 'The proof of the above confining argument is similar to property (a) of Lemma [REF], as detailed below.', '1409.6021-1-26-1': 'We define [MATH] by [EQUATION] and define [MATH] such that the edge probability [MATH] of graph [MATH] [EQUATION]', '1409.6021-1-26-2': 'We set [EQUATION] and [EQUATION]', '1409.6021-1-26-3': 'Then by ([REF]) ([REF]) and the fact that [MATH] and [MATH] are both integers, it follows that [EQUATION]', '1409.6021-1-26-4': 'From ([REF]) and ([REF]), by [CITATION], there exists a graph coupling under which [MATH] is a spanning subgraph of [MATH].', '1409.6021-1-26-5': 'Therefore, the proof of property (a) is completed once we show [MATH] defined in [MATH] satisfies [EQUATION]', '1409.6021-1-26-6': 'We first prove ([REF]).', '1409.6021-1-27-0': 'Now we establish ([REF]).', '1409.6021-1-27-1': 'First, by ([REF]), the edge probability in graph [MATH] with [MATH] satisfies [EQUATION]', '1409.6021-1-27-2': 'From ([REF]), we have [MATH], which along with ([REF]) above yields [EQUATION]', '1409.6021-1-27-3': 'Then from ([REF]) and ([REF]), it holds that [EQUATION]', '1409.6021-1-27-4': 'As mentioned, by [MATH], it holds that [MATH] for all [MATH] sufficiently large.', '1409.6021-1-27-5': 'Then from ([REF]), it follows that [EQUATION] which along with Lemmas [REF] and [REF], equation ([REF]) and condition [MATH] induces [EQUATION]', '1409.6021-1-27-6': 'Hence, we have [MATH] and it further holds for all [MATH] sufficient large that [EQUATION]', '1409.6021-1-27-7': 'Applying ([REF]) to ([REF]) and then using ([REF]), Lemma [REF] and [MATH], it follows that [EQUATION]', '1409.6021-1-27-8': 'As noted at the beginning of Section [REF], our proof is for [MATH] since the case of [MATH] already is proved by us [CITATION].', '1409.6021-1-27-9': 'Using [MATH] in ([REF]), it holds that [MATH], which along with ([REF]) and ([REF]) yields ([REF]).', '1409.6021-1-28-0': '## Proof of Lemma [REF]', '1409.6021-1-29-0': 'We first show that the result is true when [MATH] is confined as [MATH] and then demonstrate that [MATH] can be indeed confined as [MATH].', '1409.6021-1-30-0': 'First, when [MATH] is confined as [MATH].', '1409.6021-1-30-1': 'From [MATH] and [MATH], we use Lemma [REF] to obtain [MATH].', '1409.6021-1-30-2': 'Then we apply Theorem [REF] and the result clearly follows in view of [MATH].', '1409.6021-1-31-0': '# Conclusion'}

{'1409.6021-2-0-0': 'Random [MATH]-intersection graphs have recently received much interest in a wide range of application areas.', '1409.6021-2-0-1': 'Broadly speaking, a random [MATH]-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least [MATH] items (the graph is called a random intersection graph when [MATH]).', '1409.6021-2-0-2': 'A special case of particular interest is a uniform random [MATH]-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool.', '1409.6021-2-0-3': 'Another important case is a binomial random [MATH]-intersection graph, where each item from a pool is independently assigned to each vertex with the same probability.', '1409.6021-2-0-4': 'Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random [MATH]-intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random [MATH]-intersection graphs).', '1409.6021-2-1-0': 'In this paper, for binomial/uniform random [MATH]-intersection graphs, we present results related to [MATH]-connectivity and minimum vertex degree.', '1409.6021-2-1-1': 'Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) [MATH]-vertex-connectivity, (ii) [MATH]-edge-connectivity and (iii) the property of minimum vertex degree being at least [MATH].', '1409.6021-2-2-0': 'Keywords-Random intersection graph, random key graph, connectivity, secure sensor network.', '1409.6021-2-3-0': '# Introduction', '1409.6021-2-4-0': 'Random [MATH]-intersection graphs have received considerable attention recently [CITATION].', '1409.6021-2-4-1': 'In such a graph, each vertex is equipped with a set of items in some random manner, and two vertices establish an undirected edge in between if and only if they have at least [MATH] items in common.', '1409.6021-2-4-2': 'A large amount of work [CITATION] study the case of [MATH] being [MATH], under which the graphs are simply referred to as random intersection graphs.', '1409.6021-2-5-0': 'Random ([MATH]-)intersection graphs have been used to model secure wireless sensor networks [CITATION], wireless frequency hopping [CITATION], epidemics in human populations [CITATION], small-world networks [CITATION], trust networks [CITATION], social networks [CITATION] such as collaboration networks [CITATION] and common-interest networks [CITATION].', '1409.6021-2-5-1': 'Random intersection graphs also motivated Beer et al. [CITATION] to introduce a general concept of vertex random graphs that subsumes any graph model where random features are assigned to vertices, and edges are drawn based on deterministic relations between the features of the vertices.', '1409.6021-2-6-0': 'Among different models of random [MATH]-intersection graphs, two widely studied models are the so-called uniform random [MATH]-intersection graph and binomial random [MATH]-intersection graph defined in detail below.', '1409.6021-2-7-0': '## Graph models', '1409.6021-2-8-0': 'Uniform random [MATH]-intersection graph.', '1409.6021-2-8-1': 'A uniform random [MATH]-intersection graph, denoted by [MATH], is defined on [MATH] vertices as follows.', '1409.6021-2-8-2': 'Each vertex independently selects [MATH] different items uniformly at random from a pool of [MATH] distinct items.', '1409.6021-2-8-3': 'Two vertices have an edge in between if and only if they share at least [MATH] items.', '1409.6021-2-8-4': 'The notion "uniform" means that all vertices have the same number of items (but likely different sets of items).', '1409.6021-2-8-5': 'Here [MATH] and [MATH] are both functions of [MATH], while [MATH] does not scale with [MATH].', '1409.6021-2-8-6': 'It holds that [MATH].', '1409.6021-2-8-7': 'Under [MATH], the graph is also known as a random key graph [CITATION].', '1409.6021-2-9-0': 'Binomial random [MATH]-intersection graph.', '1409.6021-2-9-1': 'A binomial random [MATH]-intersection graph, denoted by [MATH], is defined on [MATH] vertices as follows.', '1409.6021-2-9-2': 'Each item from a pool of [MATH] distinct items is assigned to each vertex independently with probability [MATH].', '1409.6021-2-9-3': 'Two vertices establish an edge in between if and only if they have at least [MATH] items in common.', '1409.6021-2-9-4': 'The term "binomial" is used since the number of items assigned to each vertex follows a binomial distribution with parameters [MATH] (the number of trials) and [MATH] (the success probability in each trial).', '1409.6021-2-9-5': 'Here [MATH] and [MATH] are both functions of [MATH], while [MATH] does not scale with [MATH].', '1409.6021-2-9-6': 'Also it holds that [MATH].', '1409.6021-2-10-0': '## Problem Statement.', '1409.6021-2-11-0': 'Our goal in this paper is to investigate properties related to [MATH]-connectivity and minimum vertex degree of random [MATH]-intersection graphs ([MATH]-vertex-connectivity and [MATH]-edge-connectivity are called together as [MATH]-connectivity for convenience).', '1409.6021-2-11-1': 'In particular, we wish to answer the following question:', '1409.6021-2-12-0': 'For a uniform random [MATH]-intersection graph [MATH] (resp., a binomial random [MATH]-intersection graph [MATH]), with parameters [MATH] (resp., [MATH]) and [MATH] scaling with the number of vertices [MATH], what is the asymptotic behavior of the probabilities for [MATH] (resp., [MATH]) (i) being [MATH]-vertex-connected, (ii) being [MATH]-edge-connected, and (iii) having a minimum vertex degree at least [MATH], respectively, as [MATH] grows large?', '1409.6021-2-12-1': 'A graph is said to be [MATH]-vertex-connected if the remaining graph is still connected despite the deletion of at most [MATH] arbitrary vertices, and [MATH]-edge-connectivity is defined similarly for the deletion of edges [CITATION]; with [MATH], these definitions reduce to the standard notion of graph connectivity [CITATION].', '1409.6021-2-12-2': 'The degree of a vertex is defined as the number of edges incident on it.', '1409.6021-2-12-3': 'The three graph properties considered here are related to each other in that [MATH]-vertex-connectivity implies [MATH]-edge-connectivity, which in turn implies that the minimum vertex degree is at least [MATH] [CITATION].', '1409.6021-2-13-0': '## Summary of Results.', '1409.6021-2-14-0': 'We summarize our results below, first for a uniform random [MATH]-intersection graph and then for a binomial random [MATH]-intersection graph.', '1409.6021-2-14-1': 'Throughout the paper, both [MATH] and [MATH] are positive integers and do not scale with [MATH].', '1409.6021-2-14-2': 'Also, naturally we consider [MATH] for graph [MATH] and [MATH] for graph [MATH].', '1409.6021-2-14-3': 'We use the standard Landau asymptotic notation [MATH].', '1409.6021-2-14-4': '[MATH] denotes the probability that event [MATH] happens.', '1409.6021-2-15-0': '[MATH]-Connectivity minimum vertex degree in uniform random [MATH]-intersection graphs:', '1409.6021-2-16-0': 'For a uniform random [MATH]-intersection graph [MATH] under [MATH], with sequence [MATH] defined by [EQUATION] then as [MATH], if [MATH], the following convergence results hold: [MATH].', '1409.6021-2-17-0': '# Conclusion', '1409.6021-2-18-0': 'In this paper, we derive'}

[['1409.6021-1-5-1', '1409.6021-2-5-1'], ['1409.6021-1-12-1', '1409.6021-2-12-1'], ['1409.6021-1-12-2', '1409.6021-2-12-2'], ['1409.6021-1-12-3', '1409.6021-2-12-3'], ['1409.6021-1-14-0', '1409.6021-2-14-0'], ['1409.6021-1-14-1', '1409.6021-2-14-1'], ['1409.6021-1-14-2', '1409.6021-2-14-2'], ['1409.6021-1-14-3', '1409.6021-2-14-3'], ['1409.6021-1-14-4', '1409.6021-2-14-4'], ['1409.6021-1-4-0', '1409.6021-2-4-0'], ['1409.6021-1-9-0', '1409.6021-2-9-0'], ['1409.6021-1-9-1', '1409.6021-2-9-1'], ['1409.6021-1-9-2', '1409.6021-2-9-2'], ['1409.6021-1-9-3', '1409.6021-2-9-3'], ['1409.6021-1-9-4', '1409.6021-2-9-4'], ['1409.6021-1-9-6', '1409.6021-2-9-6'], ['1409.6021-1-8-0', '1409.6021-2-8-0'], ['1409.6021-1-8-1', '1409.6021-2-8-1'], ['1409.6021-1-8-2', '1409.6021-2-8-2'], ['1409.6021-1-8-3', '1409.6021-2-8-3'], ['1409.6021-1-8-4', '1409.6021-2-8-4'], ['1409.6021-1-8-6', '1409.6021-2-8-6'], ['1409.6021-1-5-0', '1409.6021-2-5-0'], ['1409.6021-1-6-0', '1409.6021-2-6-0'], ['1409.6021-1-1-0', '1409.6021-2-1-0'], ['1409.6021-1-1-1', '1409.6021-2-1-1'], ['1409.6021-1-11-0', '1409.6021-2-11-0'], ['1409.6021-1-12-0', '1409.6021-2-12-0'], ['1409.6021-1-4-1', '1409.6021-2-4-1'], ['1409.6021-1-4-2', '1409.6021-2-4-2'], ['1409.6021-1-9-5', '1409.6021-2-9-5'], ['1409.6021-1-8-5', '1409.6021-2-8-5'], ['1409.6021-1-0-0', '1409.6021-2-0-0'], ['1409.6021-1-0-1', '1409.6021-2-0-1'], ['1409.6021-1-0-2', '1409.6021-2-0-1'], ['1409.6021-1-0-2', '1409.6021-2-0-2'], ['1409.6021-1-0-2', '1409.6021-2-0-3'], ['1409.6021-1-0-3', '1409.6021-2-0-4']]

[['1409.6021-1-5-1', '1409.6021-2-5-1'], ['1409.6021-1-12-1', '1409.6021-2-12-1'], ['1409.6021-1-12-2', '1409.6021-2-12-2'], ['1409.6021-1-12-3', '1409.6021-2-12-3'], ['1409.6021-1-14-0', '1409.6021-2-14-0'], ['1409.6021-1-14-1', '1409.6021-2-14-1'], ['1409.6021-1-14-2', '1409.6021-2-14-2'], ['1409.6021-1-14-3', '1409.6021-2-14-3'], ['1409.6021-1-14-4', '1409.6021-2-14-4'], ['1409.6021-1-4-0', '1409.6021-2-4-0'], ['1409.6021-1-9-0', '1409.6021-2-9-0'], ['1409.6021-1-9-1', '1409.6021-2-9-1'], ['1409.6021-1-9-2', '1409.6021-2-9-2'], ['1409.6021-1-9-3', '1409.6021-2-9-3'], ['1409.6021-1-9-4', '1409.6021-2-9-4'], ['1409.6021-1-9-6', '1409.6021-2-9-6'], ['1409.6021-1-8-0', '1409.6021-2-8-0'], ['1409.6021-1-8-1', '1409.6021-2-8-1'], ['1409.6021-1-8-2', '1409.6021-2-8-2'], ['1409.6021-1-8-3', '1409.6021-2-8-3'], ['1409.6021-1-8-4', '1409.6021-2-8-4'], ['1409.6021-1-8-6', '1409.6021-2-8-6']]

[['1409.6021-1-5-0', '1409.6021-2-5-0'], ['1409.6021-1-6-0', '1409.6021-2-6-0'], ['1409.6021-1-1-0', '1409.6021-2-1-0'], ['1409.6021-1-1-1', '1409.6021-2-1-1'], ['1409.6021-1-11-0', '1409.6021-2-11-0'], ['1409.6021-1-12-0', '1409.6021-2-12-0'], ['1409.6021-1-4-1', '1409.6021-2-4-1'], ['1409.6021-1-4-2', '1409.6021-2-4-2'], ['1409.6021-1-9-5', '1409.6021-2-9-5'], ['1409.6021-1-8-5', '1409.6021-2-8-5']]

[]

[['1409.6021-1-0-0', '1409.6021-2-0-0'], ['1409.6021-1-0-1', '1409.6021-2-0-1'], ['1409.6021-1-0-2', '1409.6021-2-0-1'], ['1409.6021-1-0-2', '1409.6021-2-0-2'], ['1409.6021-1-0-2', '1409.6021-2-0-3'], ['1409.6021-1-0-3', '1409.6021-2-0-4']]

[]

['1409.6021-1-2-0', '1409.6021-1-11-1', '1409.6021-1-16-0', '1409.6021-1-18-0', '1409.6021-2-2-0', '1409.6021-2-11-1', '1409.6021-2-15-0', '1409.6021-2-16-0', '1409.6021-2-18-0']

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

https://arxiv.org/abs/1409.6021

1005.0500

{'1005.0500-1-0-0': 'We study dilepton production in proton-proton, [MATH] as well as in [MATH] collisions at the center-of-mass energy [MATH]= 200 GeV per participating nucleon pair within an extended statistical hadronization model.', '1005.0500-1-0-1': 'While the invariant mass spectrum of dielectrons is well understood in the [MATH] collisions, severe discrepancies among different model scenarios based on hadronic degrees of freedom and recent data from the PHENIX Collaboration are found in heavy-ion collisions in the low mass region from 0.15 to 0.6 GeV as well as in the intermediate mass regime from 1.1 to 3 GeV.', '1005.0500-1-0-2': 'Since the experimental yields are underestimated in both regimes we attributed the missing dilepton yield to partonic sources prior to hadronization.', '1005.0500-1-1-0': '# Introduction', '1005.0500-1-2-0': 'The goal of this work is to study the production of electron-positron pairs in proton-proton and heavy-ion collisions at [MATH]=200 GeV.', '1005.0500-1-2-1': 'We will evaluate the invariant mass spectrum [MATH] within an extended hadronization model up to invariant masses of 4 GeV thus covering the low mass as well as the intermediate mass and charmonium regime.', '1005.0500-1-2-2': 'We confront our calculations with corresponding measurements of the PHENIX collaboration (within the experimental acceptance) which has taken data [CITATION] up to 4 GeV in [MATH] at mid-rapidity.', '1005.0500-1-2-3': 'In the case of proton-proton collisions the PHENIX collaboration has found out that the measured spectrum can be described very well up to masses of 4 GeV, if one takes into account all relevant hadronic sources of dileptons in the analysis.', '1005.0500-1-2-4': 'This was done by a simultaneous measurement of all hadron rapidity densities and transverse momentum spectra around mid-rapidity; by using these experimental rapidity densities one can then estimate the dilepton yields at different invariant masses due to the known hadronic decays to [MATH] pairs.', '1005.0500-1-2-5': 'Independently, microscopic transport calculations within the Hadron-String-Dynamics (HSD) framework [CITATION] have come to the same conclusion when incorporating the measured cross section for [MATH] pairs from the PHENIX collaboration [CITATION].', '1005.0500-1-3-0': 'In contrast to the proton-proton collisions, the measured invariant mass spectrum of dileptons in [MATH] collisions have so far not been properly understood theoretically.', '1005.0500-1-3-1': 'Instead, theoretical estimates are found to deviate up to a factor of 4 or 5 from the PHENIX data in the low mass regime [CITATION] (cf. also Fig. 42 in Ref. [CITATION]) and by up to a factor 2 to 3 for intermediate masses.', '1005.0500-1-3-2': 'Such discrepancies among (hadronic) models and experimental data have not been observed at lower Super-Proton-Synchrotron (SPS) beam energies and in different collision systems [CITATION] where a major broadening of the vector-meson resonances is reported.', '1005.0500-1-3-3': 'This might suggest that non-hadronic dilepton channels could be responsible for the discrepancies observed so far.', '1005.0500-1-3-4': 'This issue needs further independent investigations.', '1005.0500-1-4-0': 'In this work, we will partly repeat the analysis by the PHENIX collaboration in the intermediate mass range especially with respect to the contribution from charmed meson decays but instead of using the measured yields as input for the dilepton sources, we wish to implement different models in order to calculate the rapidity densities of different hadrons and subsequently estimate the emission of dileptons from their decays.', '1005.0500-1-4-1': 'These models are not chosen arbitrary but are controlled by the PHENIX data for [MATH] collisions.', '1005.0500-1-5-0': 'We recall that the invariant mass spectrum of dileptons can be divided in three fairly distinct regions in each of which different physics processes are dominant.', '1005.0500-1-5-1': 'Below the [MATH] meson mass, the region referred hereafter as the low mass region (LMR: [MATH][0.0 ; 1.1] GeV), the dilepton production is dominated by the decays of non-charmed mesons, i.e. mesons with essentially light quark content ([MATH]).', '1005.0500-1-5-2': 'In the intermediate mass region (IMR: [MATH][1.1 ; 3.2] GeV), i.e. in between the [MATH] meson and J/[MATH] mass, the invariant mass spectrum of electron-positron pairs is dominated by the semileptonic decay products of open charm mesons.', '1005.0500-1-5-3': 'Strictly speaking this is just a background for the "true" dilepton sources, but since this component is always present in the analysis, one needs to carefully evaluate the contribution from open charm as well.', '1005.0500-1-5-4': 'Furthermore, above about 3 GeV of invariant mass the direct decays of charmonia become dominant and provide a constraint on the number of produced [MATH]-pairs - forming bound states - once the charmonium suppression is controlled independently.', '1005.0500-1-5-5': 'We concentrate in this article on studying the LMR and IMR thus exceeding previous approaches that focused on the LMR and extrapolated to the IMR.', '1005.0500-1-5-6': 'The high mass region (HMR:[MATH]3.8 GeV) of the dilepton invariant mass spectrum is dominated by the Drell-Yan process and [MATH]-meson decays which we will not address here.', '1005.0500-1-6-0': 'In describing the yields or ratios of particle yields of hadrons consisting of [MATH] and [MATH] quarks phenomenological models, in particular "thermal models", have proven to be very useful due to their simplicity and low number of adjustable parameters.', '1005.0500-1-6-1': 'In this work, we will evaluate the yields of light mesons within the statistical hadronization model which has been applied to high-energy elementary [CITATION] and especially heavy-ion [CITATION] collision experiments in order to calculate the yields of different hadron species with fairly a lot of success.', '1005.0500-1-6-2': 'Unlike for the bulk meson production in the LMR, the statistical hadronization model can not be applied to estimate the yields of charmed hadrons in the IMR.', '1005.0500-1-6-3': 'Instead, we need to rely on experimental information here in order to estimate the differential yields of charmed hadrons.', '1005.0500-1-7-0': '# An extended statistical hadronization model', '1005.0500-1-8-0': 'The statistical hadronization model (SHM) has been applied successfully in calculating the number of emitted hadrons in high-energy collision systems [CITATION].', '1005.0500-1-8-1': 'This model is well documented in the references given above and accordingly we will introduce the main concepts only.', '1005.0500-1-8-2': 'We evaluate the hadron yields in the grand-canonical ensemble because the calculations simplify substantially if one does not require exact conservation of Abelian charges and/or energy-momentum.', '1005.0500-1-8-3': 'Our choice is motivated by two reasons: First of all experimental observations show that the grand-canonical ensemble is sufficient enough, i.e. that at RHIC energies the data are well described under the approximations we have chosen.', '1005.0500-1-8-4': 'The other reason is that in order to evaluate the hadron yields in the canonical ensembles, one needs to know the volume as well as the exact (integer) charges on an event by event basis.', '1005.0500-1-8-5': 'However, the PHENIX collaboration has measured only a small fraction of the emitted hadrons while a large part of the system is never observed.', '1005.0500-1-8-6': 'We would need to make severe assumptions for the part of the system not measured, if we were to implement the canonical formalism for the calculation.', '1005.0500-1-8-7': 'We also note that the canonical effects are most pronounced for heavy and exotic hadrons, while the bulk of the dilepton emission arises from the low-mass mesons, which are produced abundantly and do not suffer from canonical suppression effects.', '1005.0500-1-8-8': 'Thus, we deem that performing the analysis in the grand-canonical ensemble should be good enough for our purposes and indeed this seems to be confirmed by our work (see below).', '1005.0500-1-9-0': 'In the SHM, the primary hadron multiplicity of hadron type [MATH] is calculated (in the on-shell Boltzmann approximation) according to [EQUATION]', '1005.0500-1-9-1': 'In Eq. ([REF]) [MATH] denotes the spin, [MATH] the momentum and [MATH] the mass of the particle while [MATH] is a vector consisting of the baryon, electric and strangeness charges of the hadron species [MATH].', '1005.0500-1-9-2': 'The state of the "thermal" fireball is specified by its temperature [MATH], volume [MATH] and chemical potentials (collected in the vector [MATH]) for baryon, electric and strangeness charges.', '1005.0500-1-10-0': 'Several independent experimental measurements have verified that the mid-rapidity region is actually almost net charge free in proton-proton and [MATH] collisions at [MATH]=200 GeV [CITATION].', '1005.0500-1-10-1': 'Thus, we can safely assume that the system is neutral at mid-rapidity and drop the fugacities in the analysis.', '1005.0500-1-10-2': 'In this case we are left with three free parameters characterizing our system: the temperature, the strangeness under-saturation parameter [MATH] and the overall normalization volume [MATH].', '1005.0500-1-10-3': 'The auxiliary parameter [MATH] is necessary to include in our analysis in order to take into account the empirical fact that the strange particle yields are strongly suppressed with respect to SHM estimates in elementary particle collisions.', '1005.0500-1-10-4': 'We have chosen [MATH]=0.6 in our analysis in accordance with statistical hadronization model fits [CITATION] to proton-proton collisions at this beam energy.', '1005.0500-1-10-5': 'In [MATH] collisions, the [MATH] parameter increases monotonically from the [MATH] value to unity as a function of centrality [CITATION] and the effect will be discussed in detail in the forthcoming sections.', '1005.0500-1-10-6': 'The common normalization factor [MATH] for all hadron species is chosen such that the [MATH] rapidity density [MATH] 1.065 - measured by the PHENIX collaboration - is reproduced while for the temperature we have chosen the value [MATH] MeV based on the SHM fits to proton-proton and [MATH] data at this beam energy.', '1005.0500-1-11-0': 'We have included the same collection of hadron species in our analysis as has been included in the works quoted above from which we have taken the thermal parameters.', '1005.0500-1-11-1': 'The primary hadron and resonance yields of each of the hadron species included in the analysis are calculated according to Eq. ([REF]).', '1005.0500-1-11-2': 'For resonances with width larger than 2 MeV, Eq. ([REF]) is convoluted with the relativistic Breit-Wigner distribution and integration over the mass and momentum is enforced.', '1005.0500-1-11-3': 'All unstable resonances are then allowed to decay according to the most recent branching fractions taken from the particle data group tables [CITATION].', '1005.0500-1-12-0': 'We note here that the statistical hadronization model is useful only for evaluating the relative yields of different hadron species since the rapidity and transverse momentum spectra of all hadrons emitted in the high energy collision experiments do not resemble thermal distributions.', '1005.0500-1-12-1': 'This is not a problem for us as long as our results do not depend explicitly on the details of the spectra.', '1005.0500-1-12-2': 'Indeed, this is the case for the "true" dilepton emission from a single decaying hadron, i.e. the invariant mass of the lepton pair does not depend on the momentum of the parent hadron and neither does the (Lorentz invariant) number of hadrons.', '1005.0500-1-12-3': 'Thus, we may evaluate the number of produced hadrons and dileptons within the statistical hadronization model even though the spectra are not correct.', '1005.0500-1-13-0': 'Unfortunately, the discussion above holds true only if the measurement is performed in [MATH], i.e. if all hadrons are measured or at least if the experiment extrapolates the measured kinematical region to the unmeasured regions as well.', '1005.0500-1-13-1': 'The PHENIX dilepton measurement is carried out in a narrow rapidity window around [MATH] and only leptons with [MATH] 200 MeV are taken into account.', '1005.0500-1-13-2': 'The limited acceptance in rapidity is not a severe problem due to the approximate boost invariance around mid-rapidity.', '1005.0500-1-13-3': 'In order to make the rapidity distributions of non-charmed hadrons wider, we have randomly boosted (event by event) our "fireball" along the beam axis so that the rapidity distributions of pions become compatible with the BRAHMS measurements [CITATION].', '1005.0500-1-14-0': 'On the other hand, the limited acceptance in [MATH] raises some problems, because the statistical hadronization model tends to over-populate the low [MATH] part of the spectrum compared with the experimental distributions and thus too few leptons hit the PHENIX acceptance window of [MATH] 200 MeV.', '1005.0500-1-14-1': "We have solved this problem by assuming that the created clusters' transverse momentum is normally distributed (with mean [MATH]=0 but [MATH]) and fitted the width of the clusters' [MATH] distribution so that the measured [MATH] transverse momentum distribution is roughly reproduced.", '1005.0500-1-14-2': 'The resulting pion transverse momentum spectrum is compared with experimental data in Fig. [REF].', '1005.0500-1-14-3': "One can see that once the clusters's transverse momentum profile is fixed to pions, the other light meson transverse momentum spectra are fairly well described with the same set of thermal parameters, too.", '1005.0500-1-15-0': '# Decay widths', '1005.0500-1-16-0': 'In the low invariant mass region the dominant sources of (correlated) dileptons are the direct and Dalitz decays of light mesons.', '1005.0500-1-16-1': 'The dielectron decay channels taken into account in this analysis are listed in the Table [REF].', '1005.0500-1-16-2': 'Each of the direct decays results in a sharp peak in the mass spectrum at the meson nominal mass while the Dalitz decays yield a continuum spectrum from zero invariant mass up to the mass of the decaying meson.', '1005.0500-1-16-3': 'Let us note here that there are many other hadrons and their resonances decaying radiatively into dileptons than the ones listed in Table [REF].', '1005.0500-1-16-4': 'These could be (and are) important in different kinds of collision systems.', '1005.0500-1-16-5': 'For example in heavy-ion collisions at low beam energies, the Dalitz decay of [MATH] resonances dominate the low mass region of the dilepton invariant mass spectrum (see e.g. [CITATION]).', '1005.0500-1-16-6': 'Above mid SPS beam energies, however, the number of mesons exceed the number of baryons even in heavy-ion collisions and at RHIC beam energies, the emission of dileptons from baryons is overwhelmed by orders of magnitude by the mesonic sources.', '1005.0500-1-16-7': 'Thus, we do not consider the dileptons stemming from decays of baryons in this work because the contribution is completely negligible at all invariant masses.', '1005.0500-1-17-0': 'The decay probability of a meson into a pair of leptons depends on the invariant mass of the lepton pair.', '1005.0500-1-17-1': 'A generic expression for the decay probability is known from Ref. [CITATION] [EQUATION]', '1005.0500-1-17-2': 'Here [MATH], [MATH] and [MATH] are the masses of the lepton, the decaying meson and the invariant mass of the dilepton pair, respectively.', '1005.0500-1-17-3': 'The form factors [MATH] have been studied extensively both experimentally and within different models.', '1005.0500-1-17-4': 'In this work we have employed the form factors arising from the vector-meson dominance model considerations.', '1005.0500-1-17-5': 'Explicit expressions of the form factors as well as extensive discussions can be found in Refs. [CITATION] and [CITATION].', '1005.0500-1-18-0': 'The decay widths for the direct decays of vector mesons depend on the mass of the decaying resonance.', '1005.0500-1-18-1': 'In practice this matters for [MATH] direct decays only, since all other mesons are sufficiently narrow that the decay width can be considered constant.', '1005.0500-1-18-2': 'For the [MATH] direct decay, the decay width reads [EQUATION] in which [MATH] is the [MATH] -meson mass and [MATH] denotes the pole mass.', '1005.0500-1-19-0': 'Above the [MATH] meson mass, the dilepton invariant mass spectrum attains contributions mainly from the decays of charmed hadrons.', '1005.0500-1-19-1': "Radiative decays of J/[MATH] 's into dileptons have been studied in detail in [CITATION].", '1005.0500-1-19-2': 'Since it is not possible to (completely) disentangle the direct ([MATH]) and Dalitz decays ([MATH]) of J/[MATH] in the collision experiments, one needs to take into account also the Dalitz decay of J/[MATH] in the analysis.', '1005.0500-1-19-3': 'This will modify somewhat the shape of the invariant mass spectrum of the dileptons stemming from decays of charmonia.', '1005.0500-1-19-4': 'We have implemented the analytical formula [CITATION] [EQUATION] which has been used successfully in describing the spectral shape of the radiative decays of J/[MATH] measured both in DESY as well as in PHENIX [MATH] collisions [CITATION].', '1005.0500-1-19-5': 'In Eq. ([REF]) [MATH] is the mass of the decaying particle, [MATH] the invariant mass of the dilepton pair and [MATH].', '1005.0500-1-19-6': 'This distribution diverges when [MATH]) and thus a cut in energy must be introduced.', '1005.0500-1-19-7': 'A suitable value for the mass cut-off has been found [CITATION] to be around [MATH] 10 MeV, which we have also employed.', '1005.0500-1-19-8': 'Integrating Eq. ([REF]) gives us the widths of the radiative charmonia decays: [MATH] and [MATH] = 0.34 [MATH].', '1005.0500-1-19-9': 'We have used these widths in evaluating the branching ratios for the charmonia Dalitz decays in our calculations.', '1005.0500-1-19-10': 'It is worth mentioning that our estimated branching fraction BR(J/[MATH]) is about twice the value listed in the most recent PDG book [CITATION].', '1005.0500-1-19-11': 'However, our choice agrees somewhat better with the shape of the dielectron spectrum near the J/[MATH] peak than in the case of the PDG value for the J/[MATH] Dalitz decay branching ratio.', '1005.0500-1-19-12': 'The branching fraction for the [MATH] Dalitz decay is not yet measured and thus a comparison is not possible.', '1005.0500-1-20-0': 'For all other hadrons - not explicitly mentioned above - we have assumed a relativistic Breit-Wigner spectral function and the partial widths are then evaluated in a simplified procedure taking into account only trivial mass threshold effects for the different decay channels to correct for the available phase space.', '1005.0500-1-21-0': '# Charmonium and continuum background of dileptons from heavy quark decays', '1005.0500-1-22-0': 'The weak decays of [MATH] mesons [MATH], in which [MATH] denotes one or two non-charmed hadrons, constitute the main source of the "dilepton continuum" in the intermediate mass region at RHIC energies.', '1005.0500-1-22-1': 'At the RHIC energies in proton-proton collisions, there is most often zero or a single charmed quark-anti-quark pair created.', '1005.0500-1-22-2': 'When this pair of charmed quarks hadronizes, the most likely result is that each of the charmed quarks end up in a [MATH] and [MATH] meson.', '1005.0500-1-22-3': 'When the [MATH] mesons subsequently decay into leptons and hadrons, we may have an extra lepton pair stemming from the [MATH] meson decays in the final state.', '1005.0500-1-22-4': 'It is very difficult to subtract the leptons stemming from the [MATH] meson decays in the collision experiments and thus the measured dilepton invariant mass spectrum usually includes this contribution.', '1005.0500-1-22-5': "The contribution of the [MATH] meson decays is only significant far away from the true sources (i.e. the peaks in the dilepton mass spectrum) of dileptons and dominates the spectral shape between the 'peaks' of the [MATH] and J/[MATH] mesons.", '1005.0500-1-22-6': 'Thus, one needs to carefully consider the [MATH] mesons as dilepton emitting sources in high-energy collision experiments.', '1005.0500-1-23-0': 'The slope of the dilepton continuum in the IMR arises as a superposition of the momentum distributions of the measured leptons coming from the individual [MATH] meson decays.', '1005.0500-1-23-1': 'Since the measurement is carried out at mid-rapidity and we are interested in invariant masses larger than 1 GeV, it is clear that the major contribution to the invariant mass of the dilepton continuum in the IMR arises from the transverse momenta of the decaying open charm mesons.', '1005.0500-1-23-2': 'Accordingly, it is important to model the transverse momentum spectrum of the [MATH] mesons more carefully than in the longitudinal direction, i.e. the rapidity distribution.', '1005.0500-1-24-0': 'We will employ a very simple model to evaluate the rapidity distributions of charmed hadrons.', '1005.0500-1-24-1': 'Namely, we assume that all charmed quark-anti-quark pairs are produced via splitting of a hard gluon created in the initial hard collisions.', '1005.0500-1-24-2': 'We also assume that the hard gluons - from which all [MATH]-pairs originate - are created via gluon-gluon fusion processes in the initial hard scatterings of the gluons from the target and projectile.', '1005.0500-1-24-3': 'In this case, the final charmed hadron rapidity distributions will closely follow the rapidity distribution of the hard gluons emitted in the collision experiment.', '1005.0500-1-25-0': 'In order to fix the rapidity distributions of charmed hadrons we harness the idea of limiting fragmentation [CITATION], which has been verified experimentally at ultra-relativistic beam energies both in hadron-hadron [CITATION] as well as in heavy-ion collisions [CITATION].', '1005.0500-1-25-1': 'Let us consider a collision of two gluons with momentum fractions of [MATH] and [MATH] of the colliding projectile and target.', '1005.0500-1-25-2': 'One can show (see e.g. [CITATION]) that the resulting parton rapidity distribution at large momentum fraction (i.e. [MATH] or vice versa) is proportional to the parton distribution function itself [EQUATION] and is approximately independent on the [MATH] scale due to Bjorken scaling.', '1005.0500-1-25-3': 'In Eq. ([REF]) [MATH] and [MATH] are the transverse momentum and rapidity of the produced gluon, [MATH] is the mass of the beam particle (in this case the proton).', '1005.0500-1-25-4': 'Thus, in order to estimate the rapidity distribution of the charmed hadrons, we have to adopt a suitable parameterization for the parton distribution function in Eq. ([REF]).', '1005.0500-1-25-5': 'We have chosen the following NNLO pQCD best fit parameterization from Ref. [CITATION] for our gluon distribution [EQUATION]', '1005.0500-1-25-6': 'This parameterization is given at [MATH]=9 GeV[MATH] and we approximate the gluon distributions at higher [MATH] with the same parameterization.', '1005.0500-1-26-0': 'We can now evaluate the rapidity distribution of charmed hadrons with a modified Brodsky-Gunion-Kuhn (BGK) model [CITATION] introduced in Ref. [CITATION], in which the parton number density of produced partons along the beam axis is proportional to a triangle defined by the momentum fractions [MATH] and [MATH] as follows: The center of mass of the colliding partons move with rapidity [MATH].', '1005.0500-1-26-1': 'We assume that the probability along the rapidity axis to find the hard gluon - fragmenting into a charmed quark pair - is defined by a triangle whose maximum is at [MATH] and which goes linearly to zero at [MATH] and [MATH].', '1005.0500-1-26-2': 'The area of this triangle is set to unity so that it represents a proper probability.', '1005.0500-1-27-0': 'We have estimated the charmonium cross sections by the expression (taken from Ref. [CITATION]) [EQUATION] in which [MATH] is the center-of-mass collision energy (per nucleon pair) and [MATH] is the mass of the charmonium state [MATH].', '1005.0500-1-27-1': 'The parameters [MATH]=10, [MATH]=0.775 and [MATH] are common for all charmonia and fitted to experimental data.', '1005.0500-1-27-2': "The threshold factors read [MATH] while the parameters [MATH] were fitted separately in [CITATION] for each of the states ([MATH]=0.636, 0.581 and 0.21 for [MATH], J/[MATH] and [MATH]' , respectively).", '1005.0500-1-27-3': 'Above, the multiplicity label [MATH] denotes the sum of the three [MATH], [MATH] and [MATH] states.', '1005.0500-1-27-4': 'All these states decay into J/[MATH] and we have taken the sum of their branching ratios (0.55) into J/[MATH] as our branching ratio for the generic "[MATH]".', '1005.0500-1-27-5': 'We have slightly re-adjusted the common normalization factor [MATH] from 0.16 mb to 0.133 mb in order to reproduce exactly the total J/[MATH] production cross section [MATH] measured by the PHENIX collaboration [CITATION].', '1005.0500-1-27-6': "The rapidity distribution of J/[MATH] 's in proton-proton collisions at [MATH]=200 GeV, evaluated according to Eqs. ([REF]), ([REF]) and ([REF]) is compared with the PHENIX measurement in Fig. [REF].", '1005.0500-1-27-7': 'The agreement appears good enough so that we can estimate both the rapidity density at mid-rapidity as well as the total cross section in our simple model.', '1005.0500-1-28-0': 'The reason we have taken the trouble to set up a model that can describe the J/[MATH] rapidity distribution in [MATH] collisions is that we need this model to evaluate the rapidity distribution of [MATH] mesons.', '1005.0500-1-28-1': 'The open charm rapidity distribution is not yet measured at this beam energy and thus we need to calculate it.', '1005.0500-1-28-2': 'Since we have seen that our model can describe the J/[MATH] data in this collision system, we can fairly safely assume that the same model will, at least approximately, describe the longitudinal part of the open charm momentum distribution as well.', '1005.0500-1-29-0': 'Let us turn our attention to the transverse directions now.', '1005.0500-1-29-1': 'As we discussed earlier, the transverse direction contributes most to the invariant mass of the dileptons from open charm decays at mid-rapidity.', '1005.0500-1-29-2': 'This is why we will rely on experimental data here.', '1005.0500-1-29-3': 'The transverse momentum distributions of [MATH] mesons are experimentally not well known, though.', '1005.0500-1-29-4': "What is much better known is the rapidity and transverse mass distribution of J/[MATH] 's in proton-proton collisions at RHIC.", '1005.0500-1-29-5': 'We deem that the momentum distributions of [MATH] mesons resemble the corresponding ones for J/[MATH] since the shape of the distribution - especially in the beam direction - should be primarily determined by the dynamics of the hadronizing charmed quarks.', '1005.0500-1-29-6': "Thus, we assume that the [MATH] meson transverse momentum distribution has a similar form as that for J/[MATH] 's in the same collision system.", '1005.0500-1-29-7': 'The [MATH] spectrum of J/[MATH] [CITATION] (measured by the PHENIX collaboration) can be described well with the power-law function [EQUATION] see Figure [REF].', '1005.0500-1-29-8': 'Here we have used the published data and fitted [MATH]=3.74 and [MATH]=5.11.', '1005.0500-1-30-0': 'According to our best knowledge, the transverse momentum distribution of [MATH] mesons have not been measured in proton-proton collisions at RHIC beam energies.', '1005.0500-1-30-1': 'Preliminary data [CITATION] in [MATH] collisions do exist as well as already published data [CITATION] in [MATH] collisions at [MATH]=200 GeV.', '1005.0500-1-30-2': 'The transverse momentum spectra of [MATH] mesons in these collision systems - divided by the number of binary collisions - along the corresponding J/[MATH] data in [MATH] collisions are shown in Fig. [REF].', '1005.0500-1-30-3': 'The lines shown have the functional from of Eq. ([REF]) and are fitted to the data.', '1005.0500-1-30-4': 'The [MATH] meson data do not allow to reliably fit both [MATH] and [MATH] (as well as the normalization) and so we have chosen to fix the parameter [MATH] as in the case of J/[MATH] and re-fitted [MATH] in order to describe the STAR data for [MATH] collisions.', '1005.0500-1-31-0': 'The dilepton continuum stemming from [MATH] meson decays attains an extra feature compared with the dilepton emission from other sources.', '1005.0500-1-31-1': 'Namely, since the electron and positron are emitted by two different hadrons, the angular correlation as well as re-scattering effects can strongly alter the invariant mass spectrum of the final state dileptons.', '1005.0500-1-31-2': 'We assume here that the emitted leptons themselves always escape the collision zone unscathed and (re-)scattering effects can only take place on the hadronic level.', '1005.0500-1-31-3': 'The angular correlations of the open charm hadrons are experimentally not well known and thus we need to rely on theoretical estimates.', '1005.0500-1-32-0': 'We deem that the two charmed quarks are always emitted in 180[MATH] angle in their respective center-of-mass frame while this angle is typically much smaller in the laboratory frame due to large Lorentz boosts especially in the longitudinal direction.', '1005.0500-1-32-1': 'We assume here that the longitudinal direction is no different from the transverse directions in the CM frame of the fragmenting [MATH]-pair, i.e. that Eq. ([REF]) describes the joint distribution of any two momentum components "[MATH]"=[MATH] in the CM frame of the charmed hadrons.', '1005.0500-1-32-2': 'The angular distribution among the produced [MATH] mesons is then taken exactly back-to-back correlated in their respective CM frame and we need to boost the momenta of the produced [MATH] mesons into the laboratory frame in order to evaluate the angular correlations in the laboratory frame.', '1005.0500-1-32-3': 'We have cross-checked our approach and verified that our angular correlations - evaluated as described above - agree well with correlations evaluated with the PYTHIA [CITATION] event generator.', '1005.0500-1-33-0': 'We have taken into account the 12 lightest [MATH] meson states ([MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH]) all of whose mass is around 2 GeV.', '1005.0500-1-33-1': 'We have assumed that the relative multiplicity of these 12 states is determined purely by their spin -degeneracy while the total number of them is taken from the parameterizations of Ref. [CITATION].', '1005.0500-1-33-2': 'On top of this, we have taken into account the empirical fact that in a jet fragmentation process, hadrons that include a strange or anti-strange quark suffer further suppression.', '1005.0500-1-33-3': 'We have used the canonical value 0.3 for such a strangeness suppression factor for the [MATH] and [MATH] states in our analysis as in Ref. [CITATION] which stems from PYTHIA calculations.', '1005.0500-1-33-4': 'One should not confuse this factor with the [MATH] parameter included in the thermal model analysis.', '1005.0500-1-33-5': 'The factor 0.3 here concerns the hard scatterings only, while the [MATH] parameter takes into account also some of the soft physics on top of the suppression on the hard scattering level.', '1005.0500-1-34-0': 'The kinematics of the decays of the excited [MATH] mesons can be neglected because of the small mass difference between the [MATH] and [MATH] states.', '1005.0500-1-34-1': 'In each of the cases, the excited state relaxes itself by emitting a very soft pion with a momentum of the order of few MeV and thus the daughter [MATH] inherits practically the momentum of the decaying parent state.', '1005.0500-1-34-2': 'We deem that by properly considering all of the lightest [MATH] mesons (and their relative abundances), we can then hope to extract the total charm production cross section in our analysis.', '1005.0500-1-34-3': 'Alternatively we could have just included the lowest lying [MATH] meson states, since these are the only ones decaying into leptons, but in this case the normalization factor, the number of final state [MATH] mesons would not have a clear physical interpretation as a cross section.', '1005.0500-1-34-4': 'Rather than taking the rapidity densities of different charmed hadrons at mid-rapidity as free parameters, we wish to estimate the total cross sections of the different states and suitably distribute the produced charmed hadrons at different rapidities.', '1005.0500-1-34-5': 'In order to estimate the rapidity density of [MATH] mesons around mid-rapidity, we take the total cross section for open charm production from the parameterization given in [CITATION] and distribute these along rapidity as described earlier.', '1005.0500-1-35-0': '# Results for dileptons', '1005.0500-1-36-0': '## Proton-proton collisions', '1005.0500-1-37-0': 'Let us first look at the proton-proton collisions.', '1005.0500-1-37-1': 'Our calculated dielectron yields are compared with the experimental data in Fig. [REF].', '1005.0500-1-37-2': 'The upper panel of Figure [REF] shows the low invariant mass part where the emission is dominated by the decays of light mesons.', '1005.0500-1-37-3': 'One can see that the spectrum can be reproduced very well within the statistical hadronization ansatz described in previous sections.', '1005.0500-1-37-4': 'We mention that the [MATH] dilepton mass spectra are also well reproduced within the HSD transport approach [CITATION] where the charm production and angular correlations are evaluated within PYTHIA.', '1005.0500-1-38-0': 'The shapes of the two peaks arising from the direct decays of [MATH] and [MATH] mesons are essentially determined by the experimental mass resolution.', '1005.0500-1-38-1': 'The natural width of these peaks would be only about 8 and 4 MeV, respectively.', '1005.0500-1-38-2': 'The mass resolution of the experiments, however, exceeds the natural width of these hadrons and thus we have taken this into account by smearing the peaks according to Gaussian distributions with a width corresponding to an experimental mass resolution of 10 MeV.', '1005.0500-1-38-3': 'By doing so, especially the shape of the [MATH] meson peak is strongly modified and becomes compatible with the experimental data as seen in Fig. [REF].', '1005.0500-1-38-4': 'We have not considered in-medium modifications of the spectral functions in this work and thus any apparent change in the vacuum spectral functions is solely due to experimental acceptance cuts.', '1005.0500-1-39-0': 'We find no discrepancy between the data and our model in the low invariant mass region and can conclude that the data is very well described within the statistical hadronization model assuming a completely neutral and thermalized fireball.', '1005.0500-1-39-1': 'The only non-equilibrium feature we have taken into account here is the strangeness suppression factor [MATH].', '1005.0500-1-39-2': 'By this factor we introduce a new - partly free but correlated - normalization for hadrons consisting of one or more strange quarks.', '1005.0500-1-39-3': 'In practice, the action of the [MATH] parameter is best visible in the [MATH] meson peak.', '1005.0500-1-39-4': 'Since there are no resonances decaying into the [MATH] meson the sole rapidity density of the [MATH] meson is calculated according to Eq. ([REF]).', '1005.0500-1-39-5': 'With our choice for the value of [MATH] the [MATH] meson rapidity density is multiplied by a factor [MATH]=0.36.', '1005.0500-1-39-6': 'Obviously, without this extra strangeness suppression the production of dileptons from the decays of [MATH] mesons would be dramatically over-estimated.', '1005.0500-1-39-7': 'The [MATH] parameter affects also the yields of [MATH] and [MATH]\' mesons, whose primary thermal production rates are multiplied by [MATH] in order to take into account the fact that these mesons are considered to carry "hidden strangeness".', '1005.0500-1-39-8': 'Thus the value of [MATH] is mostly (but not solely) determined by the [MATH] meson.', '1005.0500-1-39-9': 'We remind the reader here that, besides the overall normalization, we have not fine tuned any of the thermal parameters in this work, instead, we have used the values fitted in Ref. [CITATION] to STAR data.', '1005.0500-1-40-0': 'The whole invariant mass spectrum is shown in the lower panel of Figure [REF].', '1005.0500-1-40-1': 'Let us look at the IMR between the [MATH] and J/[MATH] peaks now.', '1005.0500-1-40-2': 'The solid and dashed black lines are evaluated as explained in the last Section implementing the transverse momentum profile fitted to the [MATH] meson transverse momentum spectrum in [MATH] collisions.', '1005.0500-1-40-3': 'The only difference between these two lines is that the upper solid line is evaluated assuming back-to-back angular correlations in the center-of-mass frame between the two fragmenting [MATH] mesons while the lower dashed line is the same with random correlations.', '1005.0500-1-40-4': 'From the figure it is clear that the model with random correlations does not describe the IMR spectrum properly for [MATH] reactions.', '1005.0500-1-41-0': 'From this we can conclude, in accordance with the original PHENIX publication [CITATION], that a model with strong correlations (no final state interactions) among the produced [MATH] mesons seems to be favored by the data over the random correlation in case of proton-proton collisions.', '1005.0500-1-41-1': 'We mention that in the extreme case of exact back-to-back correlations in the laboratory frame among the [MATH] mesons, the slope of the dilepton continuum in the IMR is reproduced but in this case too few dileptons are emitted in the LMR.', '1005.0500-1-41-2': 'We do not explicitly show these results but note that the continuum between the [MATH] and [MATH] would be under-estimated by about an order of magnitude in this over-simplified scenario.', '1005.0500-1-42-0': 'We may conclude that once all relevant kinematical as well as acceptance effects are taken into account, the electron-positron invariant mass spectrum can be understood very well in both the LMR and IMR in proton-proton collisions within our simple model and parameterizations.', '1005.0500-1-42-1': 'This will serve as a baseline for our comparative analysis in heavy-ion collisions for different centralities.', '1005.0500-1-43-0': '## Heavy-ion collisions', '1005.0500-1-44-0': 'Let us turn to heavy-ion collisions now.', '1005.0500-1-44-1': 'We have seen in the previous Section that the dilepton invariant mass spectrum can be well understood in proton-proton collisions and we will now try to extrapolate our results from [MATH] collisions to heavy-ion collisions to identify possible additional production channels from partonic sources as suggested e.g. in Ref. [CITATION] for SPS energies.', '1005.0500-1-45-0': 'The non-charmed hadron yields are expected to scale with the participant number ([MATH]) when comparing [MATH] and heavy-ion collisions while - assuming that the charmed quark pairs are created solely in the initial hard scatterings - the number of binary collisions ([MATH]) should be the correct scaling variable for the charm sector.', '1005.0500-1-45-1': 'Both [MATH] and [MATH] can be estimated from the Glauber model and are conventionally used also in the experimental analyses.', '1005.0500-1-45-2': 'We have used the same values for [MATH] and [MATH] that are employed by the experiments at RHIC in order to be consistent with similar previous analyses.', '1005.0500-1-45-3': 'For completeness, we show the numerical values [CITATION] of [MATH] and [MATH] as a function of centrality in [MATH] and [MATH] collisions in Fig. [REF].', '1005.0500-1-46-0': 'We have scaled the dilepton yields from proton-proton collisions to [MATH] and [MATH] collisions as described above, see Figs. [REF] and [REF].', '1005.0500-1-46-1': 'In the lower panel of Fig. [REF] we compare our results scaled from [MATH] collisions to minimum bias [MATH] collisions [CITATION] in the whole invariant mass range while in the top panel a zoom to the LMR is presented.', '1005.0500-1-46-2': 'In Fig. [REF] the experimental results from Ref. [CITATION] in different centrality bins both in [MATH] (upper panel) and [MATH] (lower panel) collisions are shown.', '1005.0500-1-46-3': 'The data in different centrality bins are still preliminary and the error bars are not yet available.', '1005.0500-1-47-0': 'In the following we will concentrate on discussing the [MATH] collisions.', '1005.0500-1-47-1': 'Essentially the same conclusions will, however, hold also for the [MATH] collisions.', '1005.0500-1-47-2': 'Let us take a closer look at the LMR first.', '1005.0500-1-47-3': 'From the top panel of Fig. [REF] one can see that the data can be described well in the most peripheral centrality bin in [MATH] collisions in the LMR by simply scaling the results from [MATH] collisions with the number of participants.', '1005.0500-1-47-4': 'This centrality class is special among the centrality classes in [MATH] collisions because in all other centrality classes the relative strangeness production is found to be nearly in chemical equilibrium with [MATH] [CITATION].', '1005.0500-1-47-5': 'We have taken the increase in relative strangeness production into account in our analysis by setting the [MATH] parameter to unity in every other centrality bins except in the most peripheral collisions, i.e. in the centrality bin labeled with "5" in Fig. [REF]a and "4" in Fig. [REF]b.', '1005.0500-1-47-6': 'The results with both [MATH] (solid) as well as [MATH] (dashed) are shown for the minimum bias [MATH] collisions in the top panel of Figure [REF].', '1005.0500-1-47-7': 'One can see that the increase in relative strangeness production as a function of centrality is not strong enough to explain the excess in the LMR in minimum bias [MATH] collisions.', '1005.0500-1-48-0': 'From Figure [REF], one can see that the LMR data in [MATH] collisions in peripheral and in semi-central bins can be reasonably well described within our scaling scheme.', '1005.0500-1-48-1': 'On the other hand, in the two most central bins as well as in the min bias collisions the increase in strangeness production can not explain the excess of dileptons in the low invariant mass region from 0.15 to 0.6 GeV.', '1005.0500-1-48-2': 'In [MATH] collisions, it seems that there is significant excess in the LMR over the hadronic cocktail only in the most central collision bin while the LMR is fairly well described for the other centralities.', '1005.0500-1-49-0': 'As explained above, we have estimated the charmed hadron yields in heavy-ion collisions by multiplying the corresponding yields in proton-proton collisions with the number of binary collisions for the contributions from charm mesons.', '1005.0500-1-49-1': 'This procedure should hold for open charm hadrons but actually we know that the production of charmonium states suffer suppression in the hot and dense environment of partonic and hadronic nature.', '1005.0500-1-49-2': 'This suppression is usually expressed by the ratio [MATH] which parameterizes the deviation from the simple [MATH] scaling.', '1005.0500-1-49-3': "PHENIX has measured this quantity for J/[MATH] 's both in [MATH] [CITATION] as well as in [MATH] [CITATION] collisions.", '1005.0500-1-49-4': 'So far there is no fully convincing model calculation that explains the observed [MATH] on a satisfactory level and thus we have taken the simplest approach and assumed that [MATH] is a linear function of centrality.', '1005.0500-1-49-5': 'This approximation seems to hold sufficiently far away from the ends of the whole centrality range.', '1005.0500-1-49-6': 'For alternative curves for the [MATH] with centrality we refer the reader to the review [CITATION].', '1005.0500-1-50-0': 'The different model results in Figs. [REF] and [REF] take into account the charmonium suppression.', '1005.0500-1-50-1': 'It is clear from Fig. [REF] that it is necessary and sufficient to include the charmonium suppression effects in order to describe the J/[MATH] peak correctly at all centralities.', '1005.0500-1-50-2': 'Since the [MATH] ratios of the excited charmonium states are not yet measured, we have not implemented any correction for the [MATH].', '1005.0500-1-51-0': 'Let us now turn our attention to the slope between the [MATH] and J/[MATH] peaks.', '1005.0500-1-51-1': 'From the discussion above it is clear that both the very low invariant mass ([MATH] Dalitz decay) as well as high invariant mass ([MATH]) regions are fairly well described by the model.', '1005.0500-1-51-2': 'This is a good starting point to address the physics of the IMR in between.', '1005.0500-1-52-0': 'In [MATH] collisions there is no medium that would distort the correlation among the emitted open charm mesons and indeed, we have seen that the [MATH] data can be best described by assuming strong correlations among the [MATH] mesons.', '1005.0500-1-52-1': 'In central heavy-ion collisions on the other hand, there are thousands of hadrons emitted in each event and one would expect that the produced charmed hadrons interact with the surrounding medium thus destroying the initial correlations.', '1005.0500-1-52-2': 'We have studied the reinteractions of the charm mesons quantitatively within the HSD transport approach (cf. Ref. [CITATION]) and have calculated the probability that neither of the two [MATH] mesons interacts with the surrounding medium.', '1005.0500-1-52-3': 'In this and only in this case the angular correlations among the emitted electrons and positrons would be preserved and remain similar to the [MATH] case.', '1005.0500-1-53-0': 'Our results from the HSD calculations are shown in Fig. [REF].', '1005.0500-1-53-1': 'The probability that the angular correlations remain the same as in [MATH] collisions is calculated as a function of collision centrality in [MATH] (open circles) and in [MATH] (filled circles) collisions at [MATH]=200 GeV.', '1005.0500-1-53-2': 'For practical purposes, we have parameterized these probabilities with the functions shown in the Figure.', '1005.0500-1-53-3': 'In the case of minimum bias [MATH] collisions, a proper weighted average over the whole centrality range (denoted by the filled square in Fig. [REF]) is used in our calculation instead of the explicit parameterization.', '1005.0500-1-54-0': 'We can now study the invariant mass spectrum of the dilepton continuum in heavy-ion collisions in a more realistic scenario in which the slope of the IMR arises as a superposition of correlated and un-correlated open charm decays.', '1005.0500-1-54-1': 'All the three cases are shown for the minimum bias [MATH] collisions in the lower panel of Fig. [REF].', '1005.0500-1-54-2': 'The result retaining the correlations like in [MATH] collisions is shown by the double dotted line in Fig. [REF]b, while the random correlation case is represented by a dashed line.', '1005.0500-1-54-3': 'The solid line in between shows the invariant mass spectrum assuming that in 33% of the cases the dilepton pair stemming from the open charm decays retains the initial correlations while in 67% of the cases at least one of the [MATH] mesons has scattered and thus the correlation is destroyed.', '1005.0500-1-54-4': 'One can see that the most realistic case naturally interpolates between the two extreme cases and seems to describe the experimental data best (within the error bars).', '1005.0500-1-55-0': 'Some further information on the IMR is gained by having an explicit look at the centrality dependence of the dilepton yield for [MATH] and [MATH] collisions at the top RHIC energy.', '1005.0500-1-55-1': 'The solid lines in Figure [REF] are evaluated in this mixed scenario - based on the HSD rescattering calculations - and one can see that this approach practically underestimates the experimental spectra in the IMR for both systems.', '1005.0500-1-55-2': 'Presently, we may only speculate that there seem to be further channels of essentially partonic nature.', '1005.0500-1-55-3': 'Proper conclusions will be possible in near future when the final experimental data with proper error bars become available.', '1005.0500-1-56-0': '# Discussion', '1005.0500-1-57-0': 'As we have seen in the previous Section a scaling of the dielectron yields from [MATH] to heavy-ion collisions can lead to a surprisingly good description of the data in the peripheral collision systems but fails in the more central collisions.', '1005.0500-1-57-1': 'The question we wish to address here is that if we can, nevertheless, understand the observed excess in terms of hadronic degrees of freedom or if additional partonic productions channels have to be incorporated.', '1005.0500-1-58-0': 'According to the statistical hadronization model fits to [MATH] and [MATH] collisions at RHIC, the intensive thermal characteristics of these systems seem very similar at mid-rapidity.', '1005.0500-1-58-1': 'All of the light mesons decaying into dileptons are completely neutral and so the dilepton production rate does not actually depend on the chemical potentials at all and thus, besides the [MATH] parameter discussed before already, the temperature is the only intensive parameter left in the model that could lead to a non-trivial scaling behavior seen in the data.', '1005.0500-1-58-2': 'However, one of the lessons we have learned from the SHM fits to RHIC data is that the temperature is the same in heavy-ion collisions at all centralities and this temperature coincides with the one extracted from the [MATH] collision data.', '1005.0500-1-58-3': 'We have, nevertheless, checked that a moderate increase ([MATH] MeV) in temperature can not explain the observed excess in the LMR.', '1005.0500-1-58-4': 'One should notice that due to the momentum cut [MATH] GeV the increase in temperature affects (relatively speaking) more the dilepton yields from the vector mesons than the emission from [MATH] and thus a change in temperature affects different regions in the invariant mass spectrum with different strength.', '1005.0500-1-58-5': 'One would nevertheless need un-realistically large temperatures of [MATH] MeV if one attempts to assign the dilepton excess to the increase of the fireball temperature.', '1005.0500-1-58-6': 'We rule out such a possibility.', '1005.0500-1-59-0': 'It seems clear that the total multiplicities of hadrons can not increase so dramatically as a function of centrality that one could understand the excess in terms of increased meson production.', '1005.0500-1-59-1': 'On the other hand changes in the dynamics and/or acceptance effects could, in principle, be responsible for the discrepancy between data and the model.', '1005.0500-1-59-2': 'Our analysis is based on hadronic degrees of freedom in statistical equilibrium and one can conclude with relatively high certainty that the observed excess in the LMR in central heavy-ion collisions can not arise from the hadronic freeze-out cocktail studied in this work.', '1005.0500-1-59-3': 'Even though the PHENIX detector rejects most of the produced dileptons, it seems that there is no mechanism that could "transport" so many of the dileptons that are emitted outside the PHENIX acceptance in peripheral collisions inside the acceptance in central collisions.', '1005.0500-1-59-4': 'In principle, one could imagine that the participants would be more strongly stopped in the central collisions leading to larger rapidity density near [MATH] of light mesons which would then translate into an "excess" of dileptons in the central rapidity region.', '1005.0500-1-59-5': 'However, this mechanism is essentially excluded because of the PHOBOS measurement in Ref. [CITATION] from which we know that the rapidity density of charged hadrons near mid-rapidity increases only mildly stronger than linearly as a function of centrality (see Figure [REF]).', '1005.0500-1-60-0': 'The other trivial acceptance driven mechanism could arise in the case that in central heavy-ion collisions the transverse momentum distributions of light mesons would be modified so that a larger fraction of the emitted single electrons would hit the detector.', '1005.0500-1-60-1': 'Since transverse boosts do not change the invariant mass of the emitted lepton pair, only mesons heavier than pions could enhance the dielectron yield in the interesting region in the LMR and even in this case, only the invariant mass region [[MATH] ; 0.6] GeV would be affected, because above 0.6 GeV, virtually all emitted dileptons are accepted and thus acceptance corrections (in [MATH]) do not play any role at invariant masses larger than 0.6 GeV.', '1005.0500-1-60-2': 'The excess region, however, reaches the [MATH] meson mass and beyond.', '1005.0500-1-61-0': 'Independent previous model calculations [CITATION] have been compared with the PHENIX data in the original publication [CITATION] and we refer the reader to Figs. 41 and 42 of Ref. [CITATION] for details.', '1005.0500-1-61-1': 'Our results agree qualitatively and also quantitatively with the previous model calculations in that the proton-proton collisions are well described in the whole invariant mass range while none of the analyses can explain the excess in the central heavy-ion collisions in the low invariant mass region.', '1005.0500-1-62-0': 'The charmed sector or the IMR (if considered) has been treated essentially in a similar fashion as in this work and no solid conclusions have been possible, so far.', '1005.0500-1-62-1': 'In order to go beyond the previous attempts we have calculated the rescattering probabilities of charm mesons dynamically (within HSD) which allows to estimate the amount of uncorrelated electron + positron pairs from [MATH] meson decays as a function of the centrality of the reaction.', '1005.0500-1-62-2': 'Our final results for [MATH] and [MATH] suggest that we clearly underestimate the preliminary yield from PHENIX which might point towards partonic sources [CITATION] also in the intermediate mass regime.', '1005.0500-1-63-0': 'So far we have considered only events with exactly one charmed quark pair.', '1005.0500-1-63-1': 'Processes leading to un-even amounts of charmed quarks are possible but they are more rare than the case we have studied and the corrections are probably not very large.', '1005.0500-1-63-2': 'Events with, e.g. 3 [MATH] mesons tend to populate the low invariant mass region, in which the open charm contribution is insignificant, because in that case only one of the possible two "dilepton"-pairs is (strongly) correlated.', '1005.0500-1-63-3': 'To finally clear up the situation we are going to carry out non-perturbative calculations on correlated charm dynamics within the PHSD transport approach [CITATION] that also includes the dynamics of charm quarks in the partonic phase.'}

{'1005.0500-2-0-0': 'We study dilepton production in proton-proton, [MATH] as well as in [MATH] collisions at the center-of-mass energy [MATH]= 200 GeV per participating nucleon pair within an extended statistical hadronization model.', '1005.0500-2-0-1': 'In extension to earlier studies we incorporate transport calculations for an estimate of uncorrelated [MATH] -pairs from semileptonic [MATH] meson decays.', '1005.0500-2-0-2': 'While the invariant mass spectrum of dielectrons is well understood in the [MATH] collisions, severe discrepancies among different model scenarios based on hadronic degrees of freedom and recent data from the PHENIX Collaboration are found in heavy-ion collisions in the low mass region from 0.15 to 0.6 GeV as well as in the intermediate mass regime from 1.1 to 3 GeV when employing the standard dilepton sources.', '1005.0500-2-0-3': 'We investigate, furthermore, the background from correlated dileptons that are not emitted as a pair from a parent hadron but emerge from semileptonic decays of two correlated daughter hadrons.', '1005.0500-2-0-4': 'Our calculations suggest a sizeable contribution of such sources in central heavy-ion collisions in the low mass region.', '1005.0500-2-0-5': 'However, even the upper limits of our calculations are found to be far below the dilepton mass spectra of the PHENIX Collaboration.', '1005.0500-2-1-0': '# Introduction', '1005.0500-2-2-0': 'The goal of this work is to study the production of correlated electron-positron pairs in proton-proton and heavy-ion collisions at [MATH]=200 GeV.', '1005.0500-2-2-1': 'We will evaluate the invariant mass spectrum [MATH] within an extended hadronization model up to invariant masses of 4 GeV thus covering the low mass as well as the intermediate mass and charmonium regime.', '1005.0500-2-2-2': 'We confront our calculations with corresponding measurements of the PHENIX collaboration (within the experimental acceptance) which has taken data [CITATION] up to 4 GeV in [MATH] at mid-rapidity.', '1005.0500-2-2-3': 'In the case of proton-proton collisions the PHENIX collaboration has found out that the measured spectrum can be described very well up to masses of 4 GeV, if one takes into account all relevant hadronic sources of dileptons in the analysis.', '1005.0500-2-2-4': 'This was done by a simultaneous measurement of all hadron rapidity densities and transverse momentum spectra around mid-rapidity; by using these experimental rapidity densities one can then estimate the dilepton yields at different invariant masses due to the known hadronic decays to [MATH] pairs.', '1005.0500-2-2-5': 'Independently, microscopic transport calculations within the Hadron-String-Dynamics (HSD) framework [CITATION] have come to the same conclusion when incorporating the measured cross section for [MATH] pairs from the PHENIX collaboration [CITATION].', '1005.0500-2-3-0': 'In contrast to the proton-proton collisions, the measured invariant mass spectrum of dileptons in [MATH] collisions have so far not been properly understood theoretically.', '1005.0500-2-3-1': 'Instead, theoretical estimates are found to deviate up to a factor of 4 or 5 from the PHENIX data for central Au+Au collisions in the low mass regime [CITATION] (cf. also Fig. 42 in Ref. [CITATION]) and by up to a factor 2 to 3 for intermediate masses.', '1005.0500-2-3-2': 'Such discrepancies among (hadronic) models and experimental data have not been observed at lower Super-Proton-Synchrotron (SPS) beam energies and in different collision systems [CITATION] where a major broadening of the vector-meson resonances is reported.', '1005.0500-2-3-3': 'This might suggest that non-hadronic dilepton channels could be responsible for the discrepancies observed so far.', '1005.0500-2-3-4': 'This issue needs further independent investigations.', '1005.0500-2-4-0': 'In this work, we will partly repeat the analysis by the PHENIX collaboration in the intermediate mass range especially with respect to the contribution from charmed meson decays but instead of using the measured yields as input for the dilepton sources, we wish to implement different models in order to calculate the rapidity densities of different hadrons and subsequently estimate the emission of dileptons from their decays.', '1005.0500-2-4-1': 'These models are controlled by the PHENIX data for [MATH] collisions.', '1005.0500-2-5-0': 'We recall that the invariant mass spectrum of dileptons can be divided in three fairly distinct regions in each of which different physics processes are dominant.', '1005.0500-2-5-1': 'Below the [MATH] meson mass, the region referred hereafter as the low mass region (LMR: [MATH] [0.0 ; 1.1] GeV), the dilepton production is dominated by the decays of non-charmed mesons, i.e. mesons with essentially light quark content ([MATH]).', '1005.0500-2-5-2': 'In the intermediate mass region (IMR: [MATH][1.1 ; 3.2] GeV), i.e. in between the [MATH] meson and J/[MATH] mass, the invariant mass spectrum of electron-positron pairs is dominated by the semileptonic decay products of open charm mesons.', '1005.0500-2-5-3': 'Strictly speaking this is just a background for the "true" dilepton sources, but since this component is always present in the analysis, one needs to carefully evaluate the contribution from open charm as well.', '1005.0500-2-5-4': 'Furthermore, above about 3 GeV of invariant mass the direct decays of charmonia become dominant and provide a constraint on the number of produced [MATH]-pairs - forming bound states - once the charmonium suppression is controlled independently.', '1005.0500-2-5-5': 'We concentrate in this article on studying the LMR and IMR thus exceeding previous approaches that focused on the LMR and extrapolated to the IMR.', '1005.0500-2-5-6': 'The high mass region (HMR:[MATH]3.8 GeV) of the dilepton invariant mass spectrum is dominated by the Drell-Yan process and [MATH] meson decays which we will not address here.', '1005.0500-2-6-0': 'In describing the yields or ratios of particle yields of hadrons consisting of [MATH] and [MATH] quarks phenomenological models, in particular "thermal models", have proven to be very useful due to their simplicity and low number of adjustable parameters.', '1005.0500-2-6-1': 'In this work, we will evaluate the yields of light mesons within the statistical hadronization model which has been applied to high-energy elementary [CITATION] and especially heavy-ion [CITATION] collision experiments in order to calculate the yields of different hadron species with fairly a lot of success.', '1005.0500-2-6-2': 'We mention that the CERES Collaboration has also analyzed their dilepton data in the LMR on the basis of the statistical hadronization model with some success at Super-Proton-Synchrotron (SPS) energies for [MATH] collisions [CITATION].', '1005.0500-2-7-0': 'Unlike for the bulk meson production in the LMR, the statistical hadronization model can not be applied to estimate the yields of charmed hadrons in the IMR.', '1005.0500-2-7-1': 'Instead, we need to rely on experimental information here in order to estimate the differential yields of charmed hadrons.', '1005.0500-2-7-2': "To this aim we will formulate an 'extended statistical model' including early charm-pair production, collective flow of the hot and dense matter as well as correlated and uncorrelated semileptonic decays of [MATH] mesons.", '1005.0500-2-7-3': 'The amount of [MATH] meson rescattering will be followed up by the HSD transport approach [CITATION] in order to estimate the amount of surviving correlated semileptonic decays in the heavy-ion collisions.', '1005.0500-2-8-0': 'In extension to earlier studies we will, furthermore, explore the contribution of correlated semileptonic decays stemming from kaon-antikaon pairs that emerge from the decay of heavy parent hadrons.', '1005.0500-2-8-1': 'The most important of these sources are proportional to [MATH], where [MATH] denotes the strangeness suppression factor which is low in proton-proton reactions but close to unity in nucleus-nucleus collisions.', '1005.0500-2-8-2': 'Our upper limits for these contribution will be compared to the PHENIX data within the proper acceptance windows as well as the conventional dilepton sources mentioned above.', '1005.0500-2-9-0': '# An extended statistical hadronization model', '1005.0500-2-10-0': 'The statistical hadronization model (SHM) has been applied successfully in calculating the number of emitted hadrons in high-energy collision systems [CITATION].', '1005.0500-2-10-1': 'This model is well documented in the references given above and accordingly we will introduce the main concepts only.', '1005.0500-2-10-2': 'We evaluate the hadron yields in the grand-canonical ensemble because the calculations simplify substantially if one does not require exact conservation of Abelian charges and/or energy-momentum.', '1005.0500-2-10-3': 'Our choice is motivated by two reasons: First of all experimental observations show that the grand-canonical ensemble is sufficient enough, i.e. that at RHIC energies the data are well described under the approximations we have chosen.', '1005.0500-2-10-4': 'The other reason is that in order to evaluate the hadron yields in the canonical ensembles, one needs to know the volume as well as the exact (integer) charges on an event by event basis.', '1005.0500-2-10-5': 'However, the PHENIX collaboration has measured only a small fraction of the emitted hadrons while a large part of the system is never observed.', '1005.0500-2-10-6': 'We would need to make severe assumptions for the part of the system not measured, if we were to implement the canonical formalism for the calculation.', '1005.0500-2-10-7': 'We also note that the canonical effects are most pronounced for heavy and exotic hadrons, while the bulk of the dilepton emission arises from the low-mass mesons, which are produced abundantly and do not suffer from canonical suppression effects.', '1005.0500-2-10-8': 'Thus, we deem that performing the analysis in the grand-canonical ensemble should be good enough for our purposes and indeed this seems to be confirmed by our work (see below).', '1005.0500-2-11-0': 'In the SHM, the primary hadron multiplicity of hadron type [MATH] is calculated (in the on-shell Boltzmann approximation) according to [EQUATION]', '1005.0500-2-11-1': 'In Eq. ([REF]) [MATH] denotes the spin, [MATH] the momentum and [MATH] the mass of the particle while [MATH] is a vector consisting of the baryon, electric and strangeness charges of the hadron species [MATH].', '1005.0500-2-11-2': 'The state of the "thermal" fireball is specified by its temperature [MATH], volume [MATH] and chemical potentials (collected in the vector [MATH]) for baryon, electric and strangeness charges.', '1005.0500-2-12-0': 'Several independent experimental measurements have verified that the mid-rapidity region is actually almost net charge free in proton-proton and [MATH] collisions at [MATH]=200 GeV [CITATION].', '1005.0500-2-12-1': 'The baryon chemical potential is expected to be of the order of 30 MeV [CITATION] in central [MATH] collisions and we have used this value throughout at all centralities while the [MATH] and [MATH] are set to zero.', '1005.0500-2-13-0': 'Once the chemical potentials are fixed, there are three free parameters characterizing our system: the temperature, the strangeness under-saturation parameter [MATH] and the overall normalization volume [MATH].', '1005.0500-2-13-1': 'The auxiliary parameter [MATH] is necessary to include in our analysis in order to take into account the empirical fact that the strange particle yields are strongly suppressed with respect to SHM estimates in elementary particle collisions.', '1005.0500-2-13-2': 'We have chosen [MATH]=0.6 in our analysis in accordance with statistical hadronization model fits [CITATION] to proton-proton collisions at this beam energy.', '1005.0500-2-13-3': 'In [MATH] collisions, the [MATH] parameter increases monotonically from the [MATH] value to unity as a function of centrality [CITATION] and the effect will be discussed in detail in the forthcoming sections.', '1005.0500-2-13-4': 'The common normalization factor [MATH] for all hadron species is determined in different collision systems from the measurements (see below) while for the temperature we have chosen the value [MATH] MeV in all systems based on the SHM fits to proton-proton and [MATH] data at this beam energy.', '1005.0500-2-14-0': 'We have included the same collection of hadron species in our analysis as has been included in the works quoted above from which we have taken the thermal parameters.', '1005.0500-2-14-1': 'The mean primary hadron and resonance yields of each of the hadron species included in the analysis are calculated according to Eq. ([REF]).', '1005.0500-2-14-2': 'For resonances with width larger than 2 MeV, Eq. ([REF]) is convoluted with the relativistic Breit-Wigner distribution and integration over the mass and momentum is enforced.', '1005.0500-2-14-3': 'Once the mean primary yields are known, we assume that, from event-to-event, the multiplicity distribution of each species is governed by the Poisson distribution characterized by the mean multiplicities evaluated with Eq. ([REF]).', '1005.0500-2-14-4': 'We then sample the Poisson distributions for each of the hadron species in each event to obtain the primary multiplicities and choose momenta for every hadron according to the Boltzmann distribution.', '1005.0500-2-14-5': 'All unstable resonances are then allowed to decay according to the most recent branching fractions taken from the particle data group tables [CITATION].', '1005.0500-2-14-6': 'This way we know the momenta of every final state particle and thus it is straightforward to take into account any geometrical and kinematical experimental cuts.', '1005.0500-2-15-0': 'We note here that the statistical hadronization model is useful only for evaluating the relative yields of different hadron species since the rapidity and transverse momentum spectra of all hadrons emitted in the high energy collision experiments do not resemble thermal distributions.', '1005.0500-2-15-1': 'This is not a problem for us as long as our results do not depend explicitly on the details of the spectra.', '1005.0500-2-15-2': 'Indeed, this is the case for the "true" dilepton emission from a single decaying hadron, i.e. the invariant mass of the lepton pair does not depend on the momentum of the parent hadron and neither does the (Lorentz invariant) number of hadrons.', '1005.0500-2-15-3': 'Thus, we may evaluate the number of produced hadrons and dileptons within the statistical hadronization model even though the spectra are not correct.', '1005.0500-2-16-0': 'Unfortunately, the discussion above holds true only if the measurement is performed in [MATH], i.e. if all hadrons are measured or at least if the experiment extrapolates the measured kinematical region to the unmeasured regions as well.', '1005.0500-2-16-1': 'The PHENIX dilepton measurement is carried out in a narrow rapidity window around [MATH] and only leptons with [MATH] 200 MeV are taken into account.', '1005.0500-2-16-2': 'We have also taken into account the azimuthal geometry and effects of the magnetic field on the charged leptons of the PHENIX detector [CITATION] , namely, that electrons and positrons (of charge [MATH]) are accepted in case both of the conditions [EQUATION] are simultaneously fulfilled.', '1005.0500-2-16-3': 'The PHENIX detector consists of two arms with the angular coverage [MATH], [MATH] and [MATH], [MATH].', '1005.0500-2-17-0': 'The limited acceptance in rapidity is not a severe problem here due to the approximate boost invariance of the systems around mid-rapidity at top RHIC energies.', '1005.0500-2-17-1': 'In order to make the rapidity distributions of non-charmed hadrons wider, we have randomly boosted (event by event) our "fireball" along the beam axis so that the rapidity distributions of pions become compatible with the BRAHMS measurements [CITATION].', '1005.0500-2-18-0': 'On the other hand, the limited acceptance in [MATH] raises some problems, because the statistical hadronization model tends to over-populate the low [MATH] part of the spectrum compared with the experimental distributions and thus too few leptons hit the PHENIX acceptance window of [MATH] 200 MeV.', '1005.0500-2-18-1': "We have solved this problem by assuming that the created clusters' transverse momentum is normally distributed (with mean [MATH]=0 but [MATH]) and fitted the width of the clusters' [MATH] distribution together with the system volume [MATH] to the PHENIX data [CITATION] in [MATH] collisions and in 11 different centrality classes in the case of [MATH] collisions.", '1005.0500-2-18-2': 'Comparison of the data and model calculations are shown in Figure [REF] for the [MATH] (top panel) and central [MATH] collisions (bottom panel).', '1005.0500-2-18-3': 'The description of the data is similar at all centralities.', '1005.0500-2-18-4': "The resulting widths for the clusters' transverse momentum distributions are shown in Figure [REF] while the scaling volumes (divided by the volume in proton-proton collisions) are shown in Figure [REF].", '1005.0500-2-19-0': 'The dilepton yields are measured in wider centrality classes than the [MATH] spectra from which we have determined the [MATH] and [MATH] and thus in the following calculations for dielectrons we have used interpolated values for [MATH] and [MATH] based on the values shown in Figures [REF] and [REF].', '1005.0500-2-20-0': '# Decay widths', '1005.0500-2-21-0': 'In the low invariant mass region the dominant sources of (correlated) dileptons are the direct and Dalitz decays of light mesons.', '1005.0500-2-21-1': 'The dielectron decay channels taken into account in this analysis are listed in the Table [REF].', '1005.0500-2-21-2': 'Each of the direct decays results in a sharp peak in the mass spectrum at the meson nominal mass while the Dalitz decays yield a continuum spectrum from zero invariant mass up to the mass of the decaying meson.', '1005.0500-2-21-3': 'Let us note here that there are many other hadrons and their resonances decaying radiatively into dileptons than the ones listed in Table [REF].', '1005.0500-2-21-4': 'These could be (and are) important in different kinds of collision systems.', '1005.0500-2-21-5': 'For example in heavy-ion collisions at low beam energies, the Dalitz decay of [MATH] resonances dominate the low mass region of the dilepton invariant mass spectrum (see e.g. [CITATION]).', '1005.0500-2-21-6': 'Above mid SPS beam energies, however, the number of mesons exceed the number of baryons in heavy-ion collisions and at RHIC beam energies, the emission of dileptons from baryons is overwhelmed by orders of magnitude by the mesonic sources.', '1005.0500-2-21-7': 'Thus, we do not consider the dileptons stemming from decays of baryons in this work because the contribution is completely negligible at all invariant masses.', '1005.0500-2-22-0': 'The decay probability of a meson into a pair of leptons depends on the invariant mass of the lepton pair.', '1005.0500-2-22-1': 'A generic expression for the decay probability is known from Ref. [CITATION] [EQUATION]', '1005.0500-2-22-2': 'Here [MATH], [MATH] and [MATH] are the masses of the lepton, the decaying meson and the invariant mass of the dilepton pair, respectively.', '1005.0500-2-22-3': 'The form factors [MATH] have been studied extensively both experimentally and within different models.', '1005.0500-2-22-4': 'In this work we have employed the form factors arising from the vector-meson dominance model considerations [CITATION] [EQUATION]', '1005.0500-2-22-5': 'The [MATH] channel is calculated from [EQUATION] with the form factor [EQUATION]', '1005.0500-2-22-6': 'The decay widths for the direct decays of vector mesons depend on the mass of the decaying resonance.', '1005.0500-2-22-7': 'In practice this matters for [MATH] direct decays only, since all other mesons are sufficiently narrow that the decay width can be considered constant.', '1005.0500-2-22-8': 'For the [MATH] direct decay, the decay width reads [CITATION] [EQUATION] in which [MATH] is the [MATH] -meson mass and [MATH] denotes the pole mass.', '1005.0500-2-22-9': 'The mass dependent branching fraction of the [MATH] meson into a pair of leptons is obtained from Eq. ([REF]) by dividing it with the mass dependent total width of the [MATH] meson: [EQUATION]', '1005.0500-2-22-10': 'Above the [MATH] meson mass, the dilepton invariant mass spectrum attains contributions mainly from the decays of charmed hadrons.', '1005.0500-2-22-11': "Radiative decays of J/[MATH] 's into dileptons have been studied in detail in [CITATION].", '1005.0500-2-22-12': 'Since it is not possible to (completely) disentangle the direct ([MATH]) and Dalitz decays ([MATH]) of J/[MATH] in the collision experiments, one needs to take into account also the Dalitz decay of J/[MATH] in the analysis.', '1005.0500-2-22-13': 'This will modify somewhat the shape of the invariant mass spectrum of the dileptons stemming from decays of charmonia.', '1005.0500-2-22-14': 'We have implemented the analytical formula [CITATION] [EQUATION] which has been used successfully in describing the spectral shape of the radiative decays of J/[MATH] measured both in DESY as well as in PHENIX [MATH] collisions [CITATION].', '1005.0500-2-22-15': 'In Eq. ([REF]) [MATH] is the mass of the decaying particle, [MATH] the invariant mass of the dilepton pair and [MATH].', '1005.0500-2-22-16': 'This distribution diverges when [MATH]) and thus a cut in energy must be introduced.', '1005.0500-2-22-17': 'A suitable value for the mass cut-off has been found [CITATION] to be around [MATH] 10 MeV, which we have also employed.', '1005.0500-2-22-18': 'Integrating Eq. ([REF]) gives us the widths of the radiative charmonia decays: [MATH] and [MATH] = 0.34 [MATH].', '1005.0500-2-22-19': 'We have used these widths in evaluating the branching ratios for the charmonia Dalitz decays in our calculations.', '1005.0500-2-22-20': 'It is worth mentioning that our estimated branching fraction BR(J/[MATH]) is about twice the value listed in the most recent PDG book [CITATION].', '1005.0500-2-22-21': 'However, our choice agrees somewhat better with the shape of the dielectron spectrum near the J/[MATH] peak than in the case of the PDG value for the J/[MATH] Dalitz decay branching ratio.', '1005.0500-2-22-22': 'The branching fraction for the [MATH] Dalitz decay is not yet measured and thus a comparison is not possible.', '1005.0500-2-23-0': 'For all other hadrons - not explicitly mentioned above - we have assumed a relativistic Breit-Wigner spectral function and the partial widths are then evaluated in a simplified procedure taking into account only trivial mass threshold effects for the different decay channels to correct for the available phase space.', '1005.0500-2-24-0': '# Charmonium and continuum background of dileptons from heavy quark decays', '1005.0500-2-25-0': 'The weak decays of [MATH] mesons [MATH], in which [MATH] denotes one or two non-charmed hadrons, constitute the main source of the "dilepton continuum" in the intermediate mass region at RHIC energies.', '1005.0500-2-25-1': 'At RHIC energies in proton-proton collisions, there is most often zero or a single charmed quark-anti-quark pair created.', '1005.0500-2-25-2': 'When this pair of charmed quarks hadronizes, the most likely result is that each of the charmed quarks end up in a [MATH] and [MATH] meson.', '1005.0500-2-25-3': 'When the [MATH] mesons subsequently decay into leptons and hadrons, we may have an extra lepton pair stemming from the [MATH] meson decays in the final state.', '1005.0500-2-25-4': 'It is very difficult to subtract the leptons originating from the [MATH] meson decays in the collision experiments and thus the measured dilepton invariant mass spectrum usually includes this contribution.', '1005.0500-2-25-5': "The contribution of the [MATH] meson decays is only significant far away from the true sources of dileptons and dominates the spectral shape between the 'peaks' of the [MATH] and J/[MATH] mesons.", '1005.0500-2-25-6': 'Thus, one needs to carefully consider the [MATH] mesons as dilepton emitting sources in high-energy collision experiments.', '1005.0500-2-26-0': 'The slope of the dilepton continuum in the IMR arises as a superposition of the momentum distributions of the measured leptons coming from the individual [MATH] meson decays.', '1005.0500-2-26-1': 'Since the measurement is carried out at mid-rapidity and we are interested in invariant masses larger than 1 GeV, it is clear that the major contribution to the invariant mass of the dilepton continuum in the IMR arises from the transverse momenta of the decaying open charm mesons.', '1005.0500-2-26-2': 'Accordingly, it is important to model the transverse momentum spectrum of the [MATH] mesons more carefully than in the longitudinal direction, i.e. the rapidity distribution.', '1005.0500-2-27-0': 'We will employ a very simple model to evaluate the rapidity distributions of charmed hadrons.', '1005.0500-2-27-1': 'Namely, we assume that all charmed quark-anti-quark pairs are produced via splitting of a hard gluon created in the initial hard collisions.', '1005.0500-2-27-2': 'We also assume that the hard gluons - from which all [MATH]-pairs originate - are created via gluon-gluon fusion processes in the initial hard scatterings of the gluons from the target and projectile.', '1005.0500-2-27-3': 'In this case, the final charmed hadron rapidity distributions will closely follow the rapidity distribution of the hard gluons emitted in the collision experiment.', '1005.0500-2-28-0': 'In order to fix the rapidity distributions of charmed hadrons we harness the idea of limiting fragmentation [CITATION], which has been verified experimentally at ultra-relativistic beam energies both in hadron-hadron [CITATION] as well as in heavy-ion collisions [CITATION].', '1005.0500-2-28-1': 'Let us consider a collision of two gluons with momentum fractions of [MATH] and [MATH] of the colliding projectile and target.', '1005.0500-2-28-2': 'One can show (see e.g. [CITATION]) that the resulting parton rapidity distribution at large momentum fraction (i.e. [MATH] or vice versa) is proportional to the parton distribution function itself [EQUATION] and is approximately independent on the [MATH] scale due to Bjorken scaling.', '1005.0500-2-28-3': 'In Eq. ([REF]) [MATH] and [MATH] are the transverse momentum and rapidity of the produced gluon, [MATH] is the mass of the beam particle (in this case the proton).', '1005.0500-2-28-4': 'Thus, in order to estimate the rapidity distribution of the charmed hadrons, we have to adopt a suitable parametrization for the parton distribution function in Eq. ([REF]).', '1005.0500-2-28-5': 'We have chosen the following NNLO pQCD best fit parametrization from Ref. [CITATION] for our gluon distribution [EQUATION]', '1005.0500-2-28-6': 'This parametrization is given at [MATH]=9 GeV[MATH] and we approximate the gluon distributions at higher [MATH] with the same parametrization.', '1005.0500-2-29-0': 'We can now evaluate the rapidity distribution of charmed hadrons with a modified Brodsky-Gunion-Kuhn (BGK) model [CITATION] introduced in Ref. [CITATION], in which the parton number density of produced partons along the beam axis is proportional to a triangle defined by the momentum fractions [MATH] and [MATH] as follows: The center of mass of the colliding partons move with rapidity [MATH].', '1005.0500-2-29-1': 'We assume that the probability along the rapidity axis to find the hard gluon - fragmenting into a charmed quark pair - is defined by a triangle whose maximum is at [MATH] and which goes linearly to zero at [MATH] and [MATH].', '1005.0500-2-29-2': 'The area of this triangle is set to unity so that it represents a proper probability.', '1005.0500-2-30-0': 'We have estimated the charmonium cross sections by the expression (taken from Ref. [CITATION]) [EQUATION] in which [MATH] is the center-of-mass collision energy (per nucleon pair) and [MATH] is the mass of the charmonium state [MATH].', '1005.0500-2-30-1': 'The parameters [MATH]=10, [MATH]=0.775 and [MATH] are common for all charmonia and fitted to experimental data.', '1005.0500-2-30-2': "The threshold factors read [MATH] while the parameters [MATH] were fitted separately in [CITATION] for each of the states ([MATH]=0.636, 0.581 and 0.21 for [MATH], J/[MATH] and [MATH]' , respectively).", '1005.0500-2-30-3': 'Above, the multiplicity label [MATH] denotes the sum of the three [MATH], [MATH] and [MATH] states.', '1005.0500-2-30-4': 'All these states decay into J/[MATH] and we have taken the sum of their branching ratios (0.55) into J/[MATH] as our branching ratio for the generic "[MATH]".', '1005.0500-2-30-5': 'We have slightly re-adjusted the common normalization factor [MATH] from 0.16 mb to 0.133 mb in order to reproduce exactly the total J/[MATH] production cross section [MATH] measured by the PHENIX collaboration [CITATION].', '1005.0500-2-30-6': "The rapidity distribution of J/[MATH] 's in proton-proton collisions at [MATH]=200 GeV, evaluated according to Eqs. ([REF]), ([REF]) and ([REF]) is compared with the PHENIX measurement in Fig. [REF].", '1005.0500-2-30-7': 'The agreement appears good enough so that we can estimate both the rapidity density at mid-rapidity as well as the total cross section in our simple model.', '1005.0500-2-31-0': 'The reason we have taken the trouble to set up a model that can describe the J/[MATH] rapidity distribution in [MATH] collisions is that we need this model to evaluate the rapidity distribution of [MATH] mesons.', '1005.0500-2-31-1': 'The open charm rapidity distribution is not yet measured at this beam energy and thus we need to calculate it.', '1005.0500-2-31-2': 'Since we have seen that our model can describe the J/[MATH] data in this collision system, we can fairly safely assume that the same model will, at least approximately, describe the longitudinal part of the open charm momentum distribution as well.', '1005.0500-2-32-0': 'Let us turn our attention to the transverse directions now.', '1005.0500-2-32-1': 'As we discussed earlier, the transverse direction contributes most to the invariant mass of the dileptons from open charm decays at mid-rapidity.', '1005.0500-2-32-2': 'This is why we will rely on experimental data here.', '1005.0500-2-32-3': 'The transverse momentum distributions of [MATH] mesons are experimentally not well known, though.', '1005.0500-2-32-4': "What is much better known is the rapidity and transverse mass distribution of J/[MATH] 's in proton-proton collisions at RHIC.", '1005.0500-2-32-5': 'We deem that the momentum distributions of [MATH] mesons resemble the corresponding ones for J/[MATH] since the shape of the distribution - especially in the beam direction - should be primarily determined by the dynamics of the hadronizing charmed quarks.', '1005.0500-2-32-6': "Thus, we assume that the [MATH] meson transverse momentum distribution has a similar form as that for J/[MATH] 's in the same collision system.", '1005.0500-2-32-7': 'The [MATH] spectrum of J/[MATH] [CITATION] (measured by the PHENIX collaboration) can be described well with the power-law function [EQUATION] see Figure [REF].', '1005.0500-2-32-8': 'Here we have used the published data and fitted [MATH]=3.74 and [MATH]=5.11.', '1005.0500-2-33-0': 'According to our best knowledge, the transverse momentum distribution of [MATH] mesons have not been measured in proton-proton collisions at RHIC beam energies.', '1005.0500-2-33-1': 'Preliminary data [CITATION] in [MATH] collisions do exist as well as already published data [CITATION] in [MATH] collisions at [MATH]=200 GeV.', '1005.0500-2-33-2': 'The transverse momentum spectra of [MATH] mesons in these collision systems - divided by the number of binary collisions - along the corresponding J/[MATH] data in [MATH] collisions are shown in Fig. [REF].', '1005.0500-2-33-3': 'The lines shown have the functional from of Eq. ([REF]) and are fitted to the data.', '1005.0500-2-33-4': 'The [MATH] meson data do not allow to reliably fit both [MATH] and [MATH] (as well as the normalization) and so we have chosen to fix the parameter [MATH] as in the case of J/[MATH] and re-fitted [MATH] in order to describe the STAR data for [MATH] collisions.', '1005.0500-2-34-0': 'The dilepton continuum stemming from [MATH] meson decays attains an extra feature compared with the dilepton emission from other sources.', '1005.0500-2-34-1': 'Namely, since the electron and positron are emitted by two different hadrons, the angular correlation as well as re-scattering effects can strongly alter the invariant mass spectrum of the final state dileptons.', '1005.0500-2-34-2': 'We assume here that the emitted leptons themselves always escape the collision zone unscathed and (re-)scattering effects can only take place on the hadronic level.', '1005.0500-2-34-3': 'The angular correlations of the open charm hadrons are experimentally not well known and thus we will employ theoretical estimates.', '1005.0500-2-35-0': 'We deem that the two charmed quarks are always emitted in 180[MATH] angle in their respective center-of-mass frame while this angle is typically much smaller in the laboratory frame due to large Lorentz boosts in the longitudinal direction, especially at large forward and backward rapidities.', '1005.0500-2-35-1': 'We assume here that the longitudinal direction is not different from the transverse directions in the CM frame of the fragmenting [MATH]-pair, i.e. that Eq. ([REF]) describes the joint distribution of any two momentum components "[MATH]"=[MATH] in the CM frame of the charmed hadrons.', '1005.0500-2-35-2': 'The angular distribution among the produced [MATH] mesons is then taken exactly back-to-back correlated in their respective CM frame and we need to boost the momenta of the produced [MATH] mesons into the laboratory frame in order to evaluate the angular correlations in that particular frame.', '1005.0500-2-35-3': 'We have cross-checked our approach and verified that our angular correlations - evaluated as described above - agree well with correlations evaluated with the PYTHIA [CITATION] event generator.', '1005.0500-2-35-4': 'Alternatively, we might also have adopted the angular correlations from the PYTHIA simulations.', '1005.0500-2-36-0': 'We have taken into account the 12 lightest [MATH] meson states ([MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH]) all of whose mass is around 2 GeV.', '1005.0500-2-36-1': 'We have assumed that the relative primary multiplicity of these 12 states is determined purely by their spin -degeneracy while the total number of them is taken from the parametrization of Ref. [CITATION].', '1005.0500-2-36-2': 'On top of this, we have taken into account the empirical fact that in a jet fragmentation process, hadrons that include a strange or anti-strange quark suffer further suppression.', '1005.0500-2-36-3': 'We have used the canonical value 0.3 for such a strangeness suppression factor for the [MATH] and [MATH] states in our analysis as in Ref. [CITATION] which stems from PYTHIA calculations.', '1005.0500-2-36-4': 'One should not confuse this factor with the [MATH] parameter included in the thermal model analysis.', '1005.0500-2-36-5': 'The factor 0.3 here concerns the hard scatterings only, while the [MATH] parameter takes into account also some of the soft physics on top of the suppression on the hard scattering level.', '1005.0500-2-37-0': 'The kinematics of the decays of the excited [MATH] mesons can be neglected because of the small mass difference between the [MATH] and [MATH] states.', '1005.0500-2-37-1': 'In each of the cases, the excited state relaxes itself by emitting a very soft pion and thus the daughter [MATH] inherits practically the momentum of the decaying parent state.', '1005.0500-2-37-2': 'We deem that by properly considering all of the lightest [MATH] mesons (and their relative abundances), we can then hope to extract the total charm production cross section in our analysis.', '1005.0500-2-37-3': 'Alternatively we could have just included the lowest lying [MATH] meson states, since these are the only ones decaying into leptons, but in this case the normalization factor, the number of final state [MATH] mesons would not have a clear physical interpretation as a cross section.', '1005.0500-2-37-4': 'Rather than taking the rapidity densities of different charmed hadrons at mid-rapidity as free parameters, we wish to estimate the total cross sections of the different states and suitably distribute the produced charmed hadrons at different rapidities.', '1005.0500-2-37-5': 'In order to estimate the rapidity density of [MATH] mesons around mid-rapidity, we take the total cross section for open charm production from the parametrization given in [CITATION] and distribute these along rapidity as described earlier.', '1005.0500-2-38-0': '# Results for dileptons', '1005.0500-2-39-0': '## Proton-proton collisions', '1005.0500-2-40-0': 'Let us first look at the proton-proton collisions.', '1005.0500-2-40-1': 'Our calculated dielectron yields are compared with the experimental data in Fig. [REF].', '1005.0500-2-40-2': 'The top panel of Figure [REF] shows the low mass region where the emission is dominated by the decays of light mesons.', '1005.0500-2-40-3': 'One can see that the spectrum can be reproduced very well within the statistical hadronization ansatz described in the previous sections.', '1005.0500-2-40-4': 'We mention that the [MATH] dilepton mass spectra are also well reproduced within the HSD transport approach [CITATION] where the charm production and angular correlations have been evaluated within PYTHIA.', '1005.0500-2-41-0': 'The shapes of the two peaks arising from the direct decays of [MATH] and [MATH] mesons are essentially determined by the experimental mass resolution.', '1005.0500-2-41-1': 'The natural width of these peaks would be only about 8 and 4 MeV, respectively.', '1005.0500-2-41-2': 'The mass resolution of the experiments, however, exceeds the vacuum width of these hadrons and thus we have taken this into account by smearing the peaks according to Gaussian distributions with a width corresponding to an experimental mass resolution of 10 MeV for the light mesons except for the [MATH] -meson direct decay whose resolution was taken to be 8.1 MeV [CITATION].', '1005.0500-2-41-3': 'The mass resolution of dielectrons from the charmed hadron decays was taken to be 2% of [MATH].', '1005.0500-2-41-4': 'By doing so, especially the shape of the [MATH] meson peak is strongly modified and becomes compatible with the experimental data as seen in Fig. [REF].', '1005.0500-2-41-5': 'We have not considered in-medium modifications of the spectral functions in this work and thus any apparent change in the vacuum spectral functions is solely due to experimental acceptance cuts.', '1005.0500-2-42-0': 'We find no discrepancy between the data and our model in the low invariant mass region and can conclude that the data are very well described within the statistical hadronization model assuming a thermalized fireball.', '1005.0500-2-42-1': 'The only non-equilibrium feature we have taken into account here is the strangeness suppression factor [MATH].', '1005.0500-2-42-2': 'By this factor we introduce a new - partly free but correlated - normalization for hadrons consisting of one or more strange quarks.', '1005.0500-2-42-3': 'In practice, the action of the [MATH] parameter is best visible in the [MATH] meson peak.', '1005.0500-2-42-4': 'Since there are no resonances decaying into the [MATH] meson the sole rapidity density of the [MATH] meson is calculated according to Eq. ([REF]).', '1005.0500-2-42-5': 'With our choice for the value of [MATH] the [MATH] meson rapidity density is multiplied by a factor [MATH]=0.36.', '1005.0500-2-42-6': 'Obviously, without this extra strangeness suppression the production of dileptons from the decays of [MATH] mesons would be dramatically over-estimated.', '1005.0500-2-42-7': 'The [MATH] parameter affects also the yields of [MATH] and [MATH]\' mesons, whose primary thermal production rates are multiplied by [MATH] in order to take into account the fact that these mesons are considered to carry "hidden strangeness".', '1005.0500-2-42-8': 'Thus the value of [MATH] is mostly (but not solely) determined by the [MATH] meson.', '1005.0500-2-42-9': 'We remind the reader here that, besides the overall normalization, we have not fine tuned any of the thermal parameters in this work, instead, we have used the values fitted in Ref. [CITATION] to STAR data.', '1005.0500-2-43-0': 'The whole invariant mass spectrum is shown in the lower panel of Figure [REF].', '1005.0500-2-43-1': 'Let us look at the IMR between the [MATH] and J/[MATH] peaks now.', '1005.0500-2-43-2': 'The solid and dashed black lines are evaluated as explained in the last Section implementing the transverse momentum profile fitted to the [MATH] meson transverse momentum spectrum in [MATH] collisions.', '1005.0500-2-43-3': 'The only difference between these two lines is that the upper solid line is evaluated assuming back-to-back angular correlations in the center-of-mass frame between the two fragmenting [MATH] mesons while the lower dashed line is the same with random correlations.', '1005.0500-2-43-4': 'From the figure it is clear that the model with random correlations does not describe the IMR spectrum properly for [MATH] reactions.', '1005.0500-2-44-0': 'From this we can conclude, in accordance with the original PHENIX publication [CITATION], that a model with strong correlations (no final state interactions) among the produced [MATH] mesons seems to be favored by the data over the random correlation in case of proton-proton collisions.', '1005.0500-2-44-1': 'We mention that in the extreme case of exact back-to-back correlations in the laboratory frame among the [MATH] mesons, the slope of the dilepton continuum in the IMR is reproduced but in this case too few dileptons are emitted in the LMR.', '1005.0500-2-44-2': 'We do not explicitly show these results but note that the continuum between the [MATH] and [MATH] would be under-estimated by about an order of magnitude in this over-simplified scenario.', '1005.0500-2-45-0': 'We may conclude that once all relevant kinematical as well as acceptance effects are taken into account, the electron-positron invariant mass spectrum can be understood very well in both the LMR and IMR in proton-proton collisions within our simple model and parametrization.', '1005.0500-2-45-1': 'This will serve as a baseline for our comparative analysis in heavy-ion collisions for different centralities.', '1005.0500-2-45-2': 'We mention in passing that the dilepton mass spectra from our simplified model agree with those from the HSD approach [CITATION] on the 20% level when using vacuum spectral functions for the hadrons in the transport approach.', '1005.0500-2-46-0': '## Heavy-ion collisions', '1005.0500-2-47-0': 'Let us turn to heavy-ion collisions now.', '1005.0500-2-47-1': 'We have seen in the previous Section that the dilepton invariant mass spectrum can be well understood in proton-proton collisions and we will now try to extrapolate our results from [MATH] collisions to heavy-ion collisions to identify the magnitude of possible additional production channels from partonic sources as suggested e.g. in Ref. [CITATION] for SPS energies.', '1005.0500-2-48-0': 'The non-charmed hadron yields are expected to scale with the participant number ([MATH]) when comparing [MATH] and heavy-ion collisions while - assuming that the charmed quark pairs are created solely in the initial hard scatterings - the number of binary collisions ([MATH]) should be the correct scaling variable for the charm sector.', '1005.0500-2-48-1': 'Both [MATH] and [MATH] can be estimated within the Glauber model and are conventionally used also in the experimental analyses.', '1005.0500-2-48-2': 'We have used the same values for [MATH] and [MATH] [CITATION] that are employed by the experiments at RHIC in order to be consistent with similar previous analyses.', '1005.0500-2-49-0': '### Low invariant mass region', '1005.0500-2-50-0': 'The [MATH] and [MATH] collisions are treated different in this paper.', '1005.0500-2-50-1': "For the [MATH] collisions we have fitted the width of the clusters' transverse momentum distribution and the system volume to the PHENIX data, see Figures [REF] and [REF], and evaluated the dilepton invariant mass spectrum in [MATH] collisions implementing the 'optimized' parameters.", '1005.0500-2-50-2': 'Detailed measurements of identified hadron transverse momentum spectra are not yet available in the [MATH] collisions and thus we have evaluated the yields of dielectrons in the LMR in [MATH] collisions such that we use the transverse momentum profile fitted to the [MATH] collisions also in [MATH] collisions at the same [MATH].', '1005.0500-2-50-3': 'The volume in [MATH] collisions at different centralities is evaluated by scaling the system volume in [MATH] collisions with the number of participants.', '1005.0500-2-50-4': 'As one can see from the Figure [REF], the fitted volumes scale with [MATH] in the [MATH] collisions in the [MATH] region relevant for [MATH] collisions and thus we deem that the [MATH] scaling should be a good approximation in the [MATH] case, at least in the non-central collisions.', '1005.0500-2-51-0': 'We have evaluated the dilepton yields in [MATH] and [MATH] collisions as described above, see Figs. [REF] and [REF].', '1005.0500-2-51-1': 'In the lower panel of Fig. [REF] we show our results together with the experimental data [CITATION] in minimum bias [MATH] collisions in the whole invariant mass range while in the top panel a zoom to the LMR is presented.', '1005.0500-2-51-2': 'In Fig. [REF] the experimental results from Refs. [CITATION] in different centrality bins both in [MATH] (top panel) and [MATH] (bottom panel) collisions are shown.', '1005.0500-2-51-3': 'The [MATH] data in different centrality bins are still preliminary and the error bars are not yet available.', '1005.0500-2-52-0': 'In the following we will concentrate on discussing the [MATH] collisions.', '1005.0500-2-52-1': 'Essentially the same conclusions will, however, hold also for the [MATH] collisions.', '1005.0500-2-52-2': 'Let us take a closer look at the LMR first.', '1005.0500-2-52-3': 'From the top panel of Fig. [REF] one can see that the data can be described well in the most peripheral centrality bin in [MATH] collisions in the LMR.', '1005.0500-2-52-4': 'This centrality class is special among the centrality classes in [MATH] collisions because in all other centrality classes the relative strangeness production is found to be nearly in chemical equilibrium with [MATH] [CITATION].', '1005.0500-2-52-5': 'We have taken the increase in relative strangeness production into account in our analysis by setting the [MATH] parameter to unity in every other centrality bins except in the most peripheral collisions, i.e. in the centrality bin labeled with "5" in Fig. [REF]a and "4" in Fig. [REF]b.', '1005.0500-2-52-6': 'The results with both [MATH] (solid) as well as [MATH] (dashed) are shown for the minimum bias [MATH] collisions in the top panel of Figure [REF].', '1005.0500-2-52-7': 'One can see that the increase in relative strangeness production as a function of centrality is not strong enough to explain the excess in the LMR in minimum bias [MATH] collisions.', '1005.0500-2-53-0': 'From Figure [REF], one can see that the LMR data in [MATH] collisions in peripheral and in semi-central bins can be reasonably well described within the statistical hadronization model.', '1005.0500-2-53-1': 'On the other hand, in the two most central bins as well as in the minimum bias collisions the increase in strangeness production can not explain the excess of dileptons in the low invariant mass region from 0.15 to 0.6 GeV.', '1005.0500-2-53-2': 'In [MATH] collisions, it seems that there is significant excess in the LMR over the hadronic cocktail only in the most central collision bin while the LMR is fairly well described for the other centralities.', '1005.0500-2-54-0': "We have, furthermore, studied the effect of the transverse flow on the dielectron invariant mass spectrum in [MATH] collisions by comparing the results evaluated with the maximum and minimum width for the clusters' transverse momentum distribution.", '1005.0500-2-54-1': 'The largest (see Fig. [REF]) width, [MATH]=13.5 GeV, was fitted in (15-20%) most central collisions while for the [MATH] and (60-92%) most central collisions we use [MATH]=10.0 GeV.', '1005.0500-2-54-2': 'We have evaluated the dielectron invariant mass spectrum with these two widths keeping all their parameters fixed ([MATH] MeV, [MATH] and [MATH]) and calculated the ratio of the resulting invariant mass spectra of dielectrons in the PHENIX acceptance, see Figure [REF].', '1005.0500-2-55-0': 'As expected, the increase in the transverse flow enhances the dielectron yields in the PHENIX acceptance especially at very low invariant masses while above the [MATH] meson peak equal amounts of dileptons hit the PHENIX acceptance with both transverse flow profiles.', '1005.0500-2-55-1': 'In general the effect is moderate and the increase in flow can increase the dielectron yields up to 30% in the [MATH] region, much less than the observed excess by factors up to 4-5.', '1005.0500-2-55-2': 'Thus, one can conclude that the broadening of transverse momentum distributions as a function of centrality plays only a minor role in the dielectron radiation from the decays of light hadrons.', '1005.0500-2-56-0': '### Further correlated sources in the LMR', '1005.0500-2-57-0': 'Besides the decays of light mesons, there are other correlated sources for dielectron production in high-energy nuclear collisions.', '1005.0500-2-57-1': 'In this Section, we will address some but not all of such processes.', '1005.0500-2-57-2': 'The contribution from most of these channels is very small and thus we concentrate here on the dominant channels, only.', '1005.0500-2-57-3': 'These additional channels, similarly to the the [MATH] meson case, arise in correlated decays of light hadrons that do not directly decay into electrons but produce single electrons via intermediate hadrons.', '1005.0500-2-57-4': 'An example of such a process is the [MATH]-meson decay [MATH].', '1005.0500-2-57-5': 'The [MATH] meson has a long lifetime and a small hadronic cross section and thus many of the [MATH] meson decays take place outside the fireball.', '1005.0500-2-57-6': 'Now it can happen that the kaons from the [MATH] meson undergo semileptonic decays like [MATH] and charge conjugate for the [MATH].', '1005.0500-2-57-7': 'If the [MATH] meson decays outside the fireball, the correlations are preserved and the experiment measures essentially additional correlated dielectron radiation from the [MATH] meson which survives the experimental subtraction procedure from uncorrelated [MATH] decays.', '1005.0500-2-58-0': 'Unlike the [MATH] meson, most of the short living resonances are expected to decay inside the fireball thus destroying the correlated signal.', '1005.0500-2-58-1': 'However, some of the interactions do take place in the dilute corona of the fireball in which case the correlated signal can be preserved even in central heavy-ion collisions.', '1005.0500-2-58-2': 'We do not perform a precise calculation for these correlations and make an estimate for the relative magnitude of core and corona emission, instead.', '1005.0500-2-58-3': 'Our results thus have to be considered as an upper limit estimate for a correlated background emission.', '1005.0500-2-59-0': 'We have considered this type of correlated dielectron radiation from a selection of fairly light strange and neutral hadrons whose branching fractions into [MATH] are sizeable and reasonably well known.', '1005.0500-2-59-1': 'The additional channels we have studied in this work are listed in Table [REF].', '1005.0500-2-59-2': 'We consider the kaonic channels only in this work, even though most of the processes could proceed via the [MATH] channel as well.', '1005.0500-2-59-3': 'The pionic channels are found to be sub-leading compared to the kaonic channels and thus we have omitted pionic channels in this work.', '1005.0500-2-60-0': "We have considered all [MATH]'s as fixed 50% - 50% mixtures of [MATH] and [MATH] states and ignored the time dependent neutral kaon oscillations.", '1005.0500-2-60-1': 'Making a distinction between the long and short living states is important since the [MATH] has roughly a factor of 100 larger probability to decay semileptonically than the [MATH].', '1005.0500-2-60-2': 'Electrons and positrons stemming from [MATH] decays are taken into account only if the [MATH] has decayed before the first detector (2 meters from the primary vertex) which reduces significantly the di-electron yields from the [MATH] decays.', '1005.0500-2-61-0': 'The contributions from the additional correlated channels in (0-10%) most central [MATH] collisions are shown in top panel of Figure [REF].', '1005.0500-2-61-1': 'We have added some of the channels in Figure [REF] for clarity.', '1005.0500-2-61-2': 'The [MATH] denotes the sum of [MATH], [MATH] and [MATH] while [MATH] is the sum of [MATH], [MATH] and [MATH] and similarly [MATH] denotes the sum of the [MATH], [MATH] and [MATH].', '1005.0500-2-61-3': 'The contribution from the [MATH] is small and is not shown.', '1005.0500-2-61-4': 'For comparison, also the hadronic cocktail contribution from [MATH] and [MATH] mesons are shown by the thick solid lines.', '1005.0500-2-61-5': 'One can see that the correlated channels might indeed give a sizeable contribution to the dielectron invariant mass spectrum in the low invariant mass region and might even over-shine the standard hadronic cocktail emission precisely in the invariant mass region where the large excess was measured by PHENIX in central nucleus-nucleus collisions.', '1005.0500-2-62-0': 'In the bottom panel of Figure [REF] the standard hadronic cocktail result (solid) and the cocktail + additional correlated emission (dashed) are compared with the PHENIX data in the most central bin.', '1005.0500-2-62-1': 'Our results show that the correlated background from the exotic mesonic states may result in a large enhancement of the dielectron yields in the LMR, although re-scattering effects could significantly alter the results at least for some of the correlated channels.', '1005.0500-2-62-2': 'The sizeable contribution - relative to proton-proton reactions - is due to a factor [MATH] which increases roughly by a factor of three from peripheral to central nucleus-nucleus collisions.', '1005.0500-2-62-3': 'Note, however, that we have addressed upper limits, only, and that a decorrelation by interactions in the hot medium is expected.', '1005.0500-2-62-4': 'Nevertheless, even our upper limit is clearly below the PHENIX signal for central nucleus-nucleus collisions and we may conclude that the additional channels considered here should not be responsible for the dilepton excess seen experimentally.', '1005.0500-2-63-0': '### Intermediate mass region', '1005.0500-2-64-0': 'We have estimated the charmed hadron yields in heavy-ion collisions by multiplying the corresponding yields in proton-proton collisions with the number of binary collisions for the contributions from charm mesons.', '1005.0500-2-64-1': 'This procedure should hold for open charm hadrons but actually we know that the production of charmonium states suffer suppression in the hot and dense environment of partonic and hadronic nature.', '1005.0500-2-64-2': 'This suppression is usually expressed by the ratio [MATH] which parametrizes the deviation from the simple [MATH] scaling.', '1005.0500-2-64-3': "PHENIX has measured this quantity for J/[MATH] 's both in [MATH] [CITATION] as well as in [MATH] [CITATION] collisions.", '1005.0500-2-64-4': 'So far there is no fully convincing model calculation that explains the observed [MATH] on a satisfactory level and thus we have taken the simplest approach and assumed that [MATH] is a linear function of centrality.', '1005.0500-2-64-5': 'This approximation seems to hold sufficiently far away from the ends of the whole centrality range.', '1005.0500-2-64-6': 'For alternative curves for the [MATH] with centrality we refer the reader to the review [CITATION].', '1005.0500-2-65-0': 'The different model results in Figs. [REF] and [REF] take into account the charmonium suppression.', '1005.0500-2-65-1': 'It is clear from Fig. [REF] that it is necessary and sufficient to include the charmonium suppression effects in order to describe the J/[MATH] peak correctly at all centralities.', '1005.0500-2-65-2': 'Since the [MATH] ratios of the excited charmonium states are not yet measured, we have not implemented any correction for the [MATH].', '1005.0500-2-66-0': 'Let us now turn our attention to the slope between the [MATH] and J/[MATH] peaks.', '1005.0500-2-66-1': 'From the discussion above it is clear that both the very low invariant mass ([MATH] Dalitz decay) as well as high invariant mass ([MATH]) regions are fairly well described by the model.', '1005.0500-2-66-2': 'This is a good starting point to address the physics of the IMR in between.', '1005.0500-2-67-0': 'In [MATH] collisions there is no medium that would distort the correlation among the emitted open charm mesons and indeed, we have seen that the [MATH] data can be best described by assuming strong correlations among the [MATH] mesons.', '1005.0500-2-67-1': 'In central heavy-ion collisions on the other hand, there are several hundreds of hadrons emitted in each event and one would expect that the produced charmed hadrons interact with the surrounding medium thus destroying the initial correlations.', '1005.0500-2-67-2': 'We have studied the reinteractions of the charm mesons quantitatively within the HSD transport approach (cf. Ref. [CITATION]) and have calculated the probability that neither of the two [MATH] mesons interacts with the surrounding medium.', '1005.0500-2-67-3': 'In this and only in this case the angular correlations among the emitted electrons and positrons would be preserved and remain similar to the [MATH] case.', '1005.0500-2-68-0': 'Our results from the HSD calculations are shown in Fig. [REF].', '1005.0500-2-68-1': 'The probability that the angular correlations remain the same as in [MATH] collisions is calculated as a function of collision centrality in [MATH] (open circles) and in [MATH] (filled circles) collisions at [MATH]=200 GeV.', '1005.0500-2-68-2': 'For practical purposes, we have parametrized these probabilities by the functions shown in the Figure.', '1005.0500-2-68-3': 'In the case of minimum bias [MATH] collisions, a proper weighted average over the whole centrality range (denoted by the filled square in Fig. [REF]) is used in our calculation instead of the explicit parametrization.', '1005.0500-2-69-0': 'We can now study the invariant mass spectrum of the dilepton continuum in heavy-ion collisions in a more realistic scenario in which the slope of the IMR arises as a superposition of correlated and un-correlated open charm decays.', '1005.0500-2-69-1': 'All the three cases are shown for the minimum bias [MATH] collisions in the lower panel of Fig. [REF].', '1005.0500-2-69-2': 'The result retaining the correlations like in [MATH] collisions is shown by the double dotted line in Fig. [REF]b, while the random correlation case is represented by a dashed line.', '1005.0500-2-69-3': 'The solid line in between shows the invariant mass spectrum assuming that in 33% of the cases the dilepton pair stemming from the open charm decays retains the initial correlations while in 67% of the cases at least one of the [MATH] mesons has scattered and thus the correlation is destroyed.', '1005.0500-2-69-4': 'One can see that the most realistic case naturally interpolates between the two extreme cases and seems to describe the experimental data best (within the error bars).', '1005.0500-2-70-0': 'Some further information on the IMR is gained by having an explicit look at the centrality dependence of the dilepton yield for [MATH] and [MATH] collisions at the top RHIC energy.', '1005.0500-2-70-1': 'The solid lines in Figure [REF] are evaluated in this mixed scenario - based on the HSD rescattering calculations - and one can see that this approach practically underestimates the experimental spectra in the IMR for both systems.', '1005.0500-2-71-0': 'Presently, we may only speculate that there seem to be further channels of possibly partonic nature in the IMR as suggested by several groups independently.', '1005.0500-2-71-1': 'Within thermal models this excess might be addressed as thermal dilepton radiation from the QGP [CITATION] while also hadronic [MATH] or [MATH] scattering might contribute as suggested by van Hees and Rapp [CITATION].', '1005.0500-2-71-2': 'The studies by one of the authors on explicit partonic reaction channels in Refs. [CITATION] allow for an implementation in the PHSD transport approach [CITATION] which hopefully might clarify this issue in the near future.', '1005.0500-2-72-0': 'In addition to the change in the angular correlations among the open charm mesons, also the magnitude of the (transverse) momentum of the [MATH] mesons can change due to interactions in the fireball.', '1005.0500-2-72-1': 'The PHENIX collaboration has measured the [MATH] of single electrons coming from heavy quark decays [CITATION] in [MATH] collisions and found that this [MATH] is compatible with unity up to [MATH] of 2 GeV at all centralities.', '1005.0500-2-72-2': 'Thus, heavy-quark energy loss can (significantly) modify the di-electron yields in the IMR coming from [MATH] meson decays only at transverse momenta larger than 2 GeV.', '1005.0500-2-73-0': 'In order to estimate how this affects our results, we have calculated the single electron + positron transverse momentum spectrum within the PHENIX acceptance and divided the results in four classes according to the corresponding invariant mass of the di-electron pair [MATH].', '1005.0500-2-73-1': 'The [MATH] spectra are shown in mass windows of [MATH] [0,1] ; [1,2] ; [2,3] and [3,4] GeV in Figure [REF].', '1005.0500-2-73-2': 'The solid lines indicate the results in the fully correlated case while the dashed lines are evaluated in the random correlation picture.', '1005.0500-2-73-3': 'All curves are normalized to unity.', '1005.0500-2-73-4': 'One can see that a significant fraction of electrons and positrons with [MATH]2 GeV contributes to the invariant mass spectrum at invariant masses larger than 3 GeV, only.', '1005.0500-2-73-5': 'Our calculations concentrate on the region [MATH] [0,[MATH]] and thus we can conclude that our results are only little affected by the heavy-quark energy-loss effects in the medium.', '1005.0500-2-73-6': 'In particular, the invariant mass region [MATH] [1,2] GeV - in which our calculations under-estimate the measurements - appear little affected by the heavy-quark energy loss.', '1005.0500-2-74-0': '# Discussion', '1005.0500-2-75-0': 'As we have seen in the previous Section a scaling of the dielectron yields from [MATH] to heavy-ion collisions can lead to a surprisingly good description of the data in the peripheral collision systems but fails for the more central collisions.', '1005.0500-2-75-1': 'The question we wish to address here is that if we can, nevertheless, understand the observed excess in terms of hadronic degrees of freedom or if additional partonic productions channels have to be incorporated.', '1005.0500-2-76-0': 'According to the statistical hadronization model fits to [MATH] and [MATH] collisions at RHIC, the intensive thermal characteristics of these systems seem very similar at mid-rapidity.', '1005.0500-2-76-1': 'All of the light mesons decaying into dileptons are completely neutral and so the dilepton production rate does not actually depend on the chemical potentials at all and thus, besides the [MATH] parameter discussed before, the temperature is the only intensive parameter left in the model that could lead to a non-trivial scaling behavior seen in the data.', '1005.0500-2-76-2': 'However, one of the lessons we have learned from the SHM fits to RHIC data is that the temperature is the same in heavy-ion collisions at all centralities and this temperature coincides with the one extracted from the [MATH] collision data.', '1005.0500-2-76-3': 'We have, nevertheless, checked that a moderate increase ([MATH] MeV) in temperature can not explain the observed excess in the LMR.', '1005.0500-2-76-4': 'One should notice that due to the momentum cut [MATH] GeV the increase in temperature affects more prominently the dilepton yields from the vector mesons than the emission from [MATH] and thus a change in temperature affects different regions in the invariant mass spectrum with different strength.', '1005.0500-2-76-5': 'One would nevertheless need unrealistically large temperatures of [MATH] MeV if one attempts to assign the dilepton excess (seen by the PHENIX Collaboration) to an increase of the fireball temperature.', '1005.0500-2-76-6': 'We rule out such a possibility.', '1005.0500-2-77-0': 'Independent previous model calculations [CITATION] have been compared with the PHENIX data in the original publication [CITATION] and we refer the reader to Figs. 41 and 42 of Ref. [CITATION] for details.', '1005.0500-2-77-1': 'Our results agree qualitatively and also quantitatively with the previous model calculations in that the proton-proton collisions are well described in the whole invariant mass range while none of the analyses can explain the excess in the central heavy-ion collisions in the low invariant mass region.', '1005.0500-2-77-2': 'In extension of the previous studies we have, furthermore, investigated the possibility that the observed excess might stem from further semileptonic correlated kaon decays which are enhanced in central nucleus-nucleus collisions relative to proton-proton reactions by roughly a factor of three.', '1005.0500-2-77-3': "Our upper limits for the dominant channels considered here clearly show that also these additional 'background sources' are not responsible for the large excess seen by the PHENIX Collaboration in central heavy-ion reactions.", '1005.0500-2-78-0': 'The charmed sector or the IMR (if considered) has been treated essentially in a similar fashion by the PHENIX Collaboration as in this work and no solid conclusions have been possible, so far.', '1005.0500-2-78-1': 'In order to go beyond the previous attempts we have calculated the rescattering probabilities of charm mesons dynamically (within HSD) which allows to estimate the amount of uncorrelated electron + positron pairs from [MATH] meson decays as a function of the centrality of the reaction.', '1005.0500-2-78-2': 'Our final results for [MATH] and [MATH] suggest that we clearly underestimate the preliminary yield from PHENIX which might point towards partonic sources - as suggested in Ref. [CITATION] - in the intermediate mass regime.', '1005.0500-2-79-0': 'So far we have considered only events with exactly one charmed quark pair.', '1005.0500-2-79-1': 'Processes leading to un-even amounts of charmed quarks are possible but they are more rare than the case we have studied and the corrections are probably not very large.', '1005.0500-2-79-2': 'Events with, e.g. 3 [MATH] mesons tend to populate the low invariant mass region, in which the open charm contribution is insignificant, because in that case only one of the possible two "dilepton"-pairs is (strongly) correlated.', '1005.0500-2-79-3': 'To finally clear up the situation we are going to carry out non-perturbative calculations on correlated charm dynamics within the PHSD transport approach [CITATION] that also includes the dynamics of charm quarks in the partonic phase.'}

[['1005.0500-1-26-0', '1005.0500-2-29-0'], ['1005.0500-1-26-1', '1005.0500-2-29-1'], ['1005.0500-1-26-2', '1005.0500-2-29-2'], ['1005.0500-1-25-0', '1005.0500-2-28-0'], ['1005.0500-1-25-1', '1005.0500-2-28-1'], ['1005.0500-1-25-2', '1005.0500-2-28-2'], ['1005.0500-1-25-3', '1005.0500-2-28-3'], ['1005.0500-1-30-0', '1005.0500-2-33-0'], ['1005.0500-1-30-1', '1005.0500-2-33-1'], ['1005.0500-1-30-2', '1005.0500-2-33-2'], ['1005.0500-1-30-3', '1005.0500-2-33-3'], ['1005.0500-1-30-4', '1005.0500-2-33-4'], ['1005.0500-1-4-0', '1005.0500-2-4-0'], ['1005.0500-1-54-0', '1005.0500-2-69-0'], ['1005.0500-1-54-1', '1005.0500-2-69-1'], ['1005.0500-1-54-2', '1005.0500-2-69-2'], ['1005.0500-1-54-3', '1005.0500-2-69-3'], ['1005.0500-1-54-4', '1005.0500-2-69-4'], ['1005.0500-1-55-0', '1005.0500-2-70-0'], ['1005.0500-1-55-1', '1005.0500-2-70-1'], ['1005.0500-1-41-0', '1005.0500-2-44-0'], ['1005.0500-1-41-1', '1005.0500-2-44-1'], ['1005.0500-1-41-2', '1005.0500-2-44-2'], ['1005.0500-1-40-0', '1005.0500-2-43-0'], ['1005.0500-1-40-1', '1005.0500-2-43-1'], ['1005.0500-1-40-2', '1005.0500-2-43-2'], ['1005.0500-1-40-3', '1005.0500-2-43-3'], ['1005.0500-1-40-4', '1005.0500-2-43-4'], ['1005.0500-1-2-1', '1005.0500-2-2-1'], ['1005.0500-1-2-2', '1005.0500-2-2-2'], ['1005.0500-1-2-3', '1005.0500-2-2-3'], ['1005.0500-1-2-4', '1005.0500-2-2-4'], ['1005.0500-1-2-5', '1005.0500-2-2-5'], ['1005.0500-1-38-0', '1005.0500-2-41-0'], ['1005.0500-1-38-1', '1005.0500-2-41-1'], ['1005.0500-1-38-3', '1005.0500-2-41-4'], ['1005.0500-1-38-4', '1005.0500-2-41-5'], ['1005.0500-1-20-0', '1005.0500-2-23-0'], ['1005.0500-1-8-0', '1005.0500-2-10-0'], ['1005.0500-1-8-1', '1005.0500-2-10-1'], ['1005.0500-1-8-2', '1005.0500-2-10-2'], ['1005.0500-1-8-3', '1005.0500-2-10-3'], ['1005.0500-1-8-4', '1005.0500-2-10-4'], ['1005.0500-1-8-5', '1005.0500-2-10-5'], ['1005.0500-1-8-6', '1005.0500-2-10-6'], ['1005.0500-1-8-7', '1005.0500-2-10-7'], ['1005.0500-1-8-8', '1005.0500-2-10-8'], ['1005.0500-1-22-0', '1005.0500-2-25-0'], ['1005.0500-1-22-2', '1005.0500-2-25-2'], ['1005.0500-1-22-3', '1005.0500-2-25-3'], ['1005.0500-1-22-6', '1005.0500-2-25-6'], ['1005.0500-1-0-0', '1005.0500-2-0-0'], ['1005.0500-1-31-0', '1005.0500-2-34-0'], ['1005.0500-1-31-1', '1005.0500-2-34-1'], ['1005.0500-1-31-2', '1005.0500-2-34-2'], ['1005.0500-1-32-3', '1005.0500-2-35-3'], ['1005.0500-1-45-0', '1005.0500-2-48-0'], ['1005.0500-1-48-2', '1005.0500-2-53-2'], ['1005.0500-1-39-1', '1005.0500-2-42-1'], ['1005.0500-1-39-2', '1005.0500-2-42-2'], ['1005.0500-1-39-3', '1005.0500-2-42-3'], ['1005.0500-1-39-4', '1005.0500-2-42-4'], ['1005.0500-1-39-5', '1005.0500-2-42-5'], ['1005.0500-1-39-6', '1005.0500-2-42-6'], ['1005.0500-1-39-7', '1005.0500-2-42-7'], ['1005.0500-1-39-8', '1005.0500-2-42-8'], ['1005.0500-1-39-9', '1005.0500-2-42-9'], ['1005.0500-1-63-0', '1005.0500-2-79-0'], ['1005.0500-1-63-1', '1005.0500-2-79-1'], ['1005.0500-1-63-2', '1005.0500-2-79-2'], ['1005.0500-1-63-3', '1005.0500-2-79-3'], ['1005.0500-1-37-0', '1005.0500-2-40-0'], ['1005.0500-1-37-1', '1005.0500-2-40-1'], ['1005.0500-1-9-0', '1005.0500-2-11-0'], ['1005.0500-1-9-1', '1005.0500-2-11-1'], ['1005.0500-1-9-2', '1005.0500-2-11-2'], ['1005.0500-1-34-0', '1005.0500-2-37-0'], ['1005.0500-1-34-2', '1005.0500-2-37-2'], ['1005.0500-1-34-3', '1005.0500-2-37-3'], ['1005.0500-1-34-4', '1005.0500-2-37-4'], ['1005.0500-1-23-0', '1005.0500-2-26-0'], ['1005.0500-1-23-1', '1005.0500-2-26-1'], ['1005.0500-1-23-2', '1005.0500-2-26-2'], ['1005.0500-1-50-0', '1005.0500-2-65-0'], ['1005.0500-1-50-1', '1005.0500-2-65-1'], ['1005.0500-1-50-2', '1005.0500-2-65-2'], ['1005.0500-1-5-0', '1005.0500-2-5-0'], ['1005.0500-1-5-2', '1005.0500-2-5-2'], ['1005.0500-1-5-3', '1005.0500-2-5-3'], ['1005.0500-1-5-4', '1005.0500-2-5-4'], ['1005.0500-1-5-5', '1005.0500-2-5-5'], ['1005.0500-1-11-0', '1005.0500-2-14-0'], ['1005.0500-1-11-2', '1005.0500-2-14-2'], ['1005.0500-1-11-3', '1005.0500-2-14-5'], ['1005.0500-1-47-0', '1005.0500-2-52-0'], ['1005.0500-1-47-1', '1005.0500-2-52-1'], ['1005.0500-1-47-2', '1005.0500-2-52-2'], ['1005.0500-1-47-4', '1005.0500-2-52-4'], ['1005.0500-1-47-5', '1005.0500-2-52-5'], ['1005.0500-1-47-6', '1005.0500-2-52-6'], ['1005.0500-1-47-7', '1005.0500-2-52-7'], ['1005.0500-1-14-0', '1005.0500-2-18-0'], ['1005.0500-1-16-0', '1005.0500-2-21-0'], ['1005.0500-1-16-1', '1005.0500-2-21-1'], ['1005.0500-1-16-2', '1005.0500-2-21-2'], ['1005.0500-1-16-3', '1005.0500-2-21-3'], ['1005.0500-1-16-4', '1005.0500-2-21-4'], ['1005.0500-1-16-5', '1005.0500-2-21-5'], ['1005.0500-1-16-7', '1005.0500-2-21-7'], ['1005.0500-1-24-0', '1005.0500-2-27-0'], ['1005.0500-1-24-1', '1005.0500-2-27-1'], ['1005.0500-1-24-2', '1005.0500-2-27-2'], ['1005.0500-1-24-3', '1005.0500-2-27-3'], ['1005.0500-1-52-0', '1005.0500-2-67-0'], ['1005.0500-1-52-2', '1005.0500-2-67-2'], ['1005.0500-1-52-3', '1005.0500-2-67-3'], ['1005.0500-1-53-0', '1005.0500-2-68-0'], ['1005.0500-1-53-1', '1005.0500-2-68-1'], ['1005.0500-1-57-1', '1005.0500-2-75-1'], ['1005.0500-1-3-0', '1005.0500-2-3-0'], ['1005.0500-1-3-2', '1005.0500-2-3-2'], ['1005.0500-1-3-3', '1005.0500-2-3-3'], ['1005.0500-1-3-4', '1005.0500-2-3-4'], ['1005.0500-1-33-0', '1005.0500-2-36-0'], ['1005.0500-1-33-2', '1005.0500-2-36-2'], ['1005.0500-1-33-3', '1005.0500-2-36-3'], ['1005.0500-1-33-4', '1005.0500-2-36-4'], ['1005.0500-1-33-5', '1005.0500-2-36-5'], ['1005.0500-1-29-0', '1005.0500-2-32-0'], ['1005.0500-1-29-1', '1005.0500-2-32-1'], ['1005.0500-1-29-2', '1005.0500-2-32-2'], ['1005.0500-1-29-3', '1005.0500-2-32-3'], ['1005.0500-1-29-4', '1005.0500-2-32-4'], ['1005.0500-1-29-5', '1005.0500-2-32-5'], ['1005.0500-1-29-6', '1005.0500-2-32-6'], ['1005.0500-1-29-7', '1005.0500-2-32-7'], ['1005.0500-1-29-8', '1005.0500-2-32-8'], ['1005.0500-1-49-1', '1005.0500-2-64-1'], ['1005.0500-1-49-3', '1005.0500-2-64-3'], ['1005.0500-1-49-4', '1005.0500-2-64-4'], ['1005.0500-1-49-5', '1005.0500-2-64-5'], ['1005.0500-1-49-6', '1005.0500-2-64-6'], ['1005.0500-1-61-0', '1005.0500-2-77-0'], ['1005.0500-1-61-1', '1005.0500-2-77-1'], ['1005.0500-1-27-0', '1005.0500-2-30-0'], ['1005.0500-1-27-1', '1005.0500-2-30-1'], ['1005.0500-1-27-2', '1005.0500-2-30-2'], ['1005.0500-1-27-3', '1005.0500-2-30-3'], ['1005.0500-1-27-4', '1005.0500-2-30-4'], ['1005.0500-1-27-5', '1005.0500-2-30-5'], ['1005.0500-1-27-6', '1005.0500-2-30-6'], ['1005.0500-1-27-7', '1005.0500-2-30-7'], ['1005.0500-1-44-0', '1005.0500-2-47-0'], ['1005.0500-1-28-0', '1005.0500-2-31-0'], ['1005.0500-1-28-1', '1005.0500-2-31-1'], ['1005.0500-1-28-2', '1005.0500-2-31-2'], ['1005.0500-1-58-0', '1005.0500-2-76-0'], ['1005.0500-1-58-2', '1005.0500-2-76-2'], ['1005.0500-1-58-3', '1005.0500-2-76-3'], ['1005.0500-1-58-6', '1005.0500-2-76-6'], ['1005.0500-1-62-1', '1005.0500-2-78-1'], ['1005.0500-1-12-0', '1005.0500-2-15-0'], ['1005.0500-1-12-1', '1005.0500-2-15-1'], ['1005.0500-1-12-2', '1005.0500-2-15-2'], ['1005.0500-1-12-3', '1005.0500-2-15-3'], ['1005.0500-1-51-0', '1005.0500-2-66-0'], ['1005.0500-1-51-1', '1005.0500-2-66-1'], ['1005.0500-1-51-2', '1005.0500-2-66-2'], ['1005.0500-1-42-1', '1005.0500-2-45-1'], ['1005.0500-1-6-0', '1005.0500-2-6-0'], ['1005.0500-1-6-1', '1005.0500-2-6-1'], ['1005.0500-1-6-2', '1005.0500-2-7-0'], ['1005.0500-1-6-3', '1005.0500-2-7-1'], ['1005.0500-1-10-0', '1005.0500-2-12-0'], ['1005.0500-1-10-3', '1005.0500-2-13-1'], ['1005.0500-1-10-4', '1005.0500-2-13-2'], ['1005.0500-1-10-5', '1005.0500-2-13-3'], ['1005.0500-1-13-0', '1005.0500-2-16-0'], ['1005.0500-1-13-1', '1005.0500-2-16-1'], ['1005.0500-1-13-3', '1005.0500-2-17-1'], ['1005.0500-1-17-0', '1005.0500-2-22-0'], ['1005.0500-1-17-1', '1005.0500-2-22-1'], ['1005.0500-1-17-2', '1005.0500-2-22-2'], ['1005.0500-1-17-3', '1005.0500-2-22-3'], ['1005.0500-1-18-0', '1005.0500-2-22-6'], ['1005.0500-1-18-1', '1005.0500-2-22-7'], ['1005.0500-1-19-0', '1005.0500-2-22-10'], ['1005.0500-1-19-1', '1005.0500-2-22-11'], ['1005.0500-1-19-2', '1005.0500-2-22-12'], ['1005.0500-1-19-3', '1005.0500-2-22-13'], ['1005.0500-1-19-4', '1005.0500-2-22-14'], ['1005.0500-1-19-5', '1005.0500-2-22-15'], ['1005.0500-1-19-6', '1005.0500-2-22-16'], ['1005.0500-1-19-7', '1005.0500-2-22-17'], ['1005.0500-1-19-8', '1005.0500-2-22-18'], ['1005.0500-1-19-9', '1005.0500-2-22-19'], ['1005.0500-1-19-10', '1005.0500-2-22-20'], ['1005.0500-1-19-11', '1005.0500-2-22-21'], ['1005.0500-1-19-12', '1005.0500-2-22-22'], ['1005.0500-1-25-4', '1005.0500-2-28-4'], ['1005.0500-1-25-5', '1005.0500-2-28-5'], ['1005.0500-1-25-6', '1005.0500-2-28-6'], ['1005.0500-1-4-1', '1005.0500-2-4-1'], ['1005.0500-1-2-0', '1005.0500-2-2-0'], ['1005.0500-1-22-1', '1005.0500-2-25-1'], ['1005.0500-1-22-4', '1005.0500-2-25-4'], ['1005.0500-1-22-5', '1005.0500-2-25-5'], ['1005.0500-1-0-1', '1005.0500-2-0-2'], ['1005.0500-1-31-3', '1005.0500-2-34-3'], ['1005.0500-1-32-0', '1005.0500-2-35-0'], ['1005.0500-1-32-1', '1005.0500-2-35-1'], ['1005.0500-1-32-2', '1005.0500-2-35-2'], ['1005.0500-1-45-1', '1005.0500-2-48-1'], ['1005.0500-1-45-2', '1005.0500-2-48-2'], ['1005.0500-1-48-0', '1005.0500-2-53-0'], ['1005.0500-1-48-1', '1005.0500-2-53-1'], ['1005.0500-1-39-0', '1005.0500-2-42-0'], ['1005.0500-1-37-2', '1005.0500-2-40-2'], ['1005.0500-1-37-3', '1005.0500-2-40-3'], ['1005.0500-1-37-4', '1005.0500-2-40-4'], ['1005.0500-1-34-1', '1005.0500-2-37-1'], ['1005.0500-1-34-5', '1005.0500-2-37-5'], ['1005.0500-1-5-1', '1005.0500-2-5-1'], ['1005.0500-1-5-6', '1005.0500-2-5-6'], ['1005.0500-1-11-1', '1005.0500-2-14-1'], ['1005.0500-1-47-3', '1005.0500-2-52-3'], ['1005.0500-1-16-6', '1005.0500-2-21-6'], ['1005.0500-1-52-1', '1005.0500-2-67-1'], ['1005.0500-1-53-2', '1005.0500-2-68-2'], ['1005.0500-1-53-3', '1005.0500-2-68-3'], ['1005.0500-1-57-0', '1005.0500-2-75-0'], ['1005.0500-1-3-1', '1005.0500-2-3-1'], ['1005.0500-1-33-1', '1005.0500-2-36-1'], ['1005.0500-1-49-0', '1005.0500-2-64-0'], ['1005.0500-1-49-2', '1005.0500-2-64-2'], ['1005.0500-1-46-0', '1005.0500-2-51-0'], ['1005.0500-1-46-1', '1005.0500-2-51-1'], ['1005.0500-1-46-2', '1005.0500-2-51-2'], ['1005.0500-1-46-3', '1005.0500-2-51-3'], ['1005.0500-1-44-1', '1005.0500-2-47-1'], ['1005.0500-1-58-1', '1005.0500-2-76-1'], ['1005.0500-1-58-4', '1005.0500-2-76-4'], ['1005.0500-1-58-5', '1005.0500-2-76-5'], ['1005.0500-1-62-0', '1005.0500-2-78-0'], ['1005.0500-1-62-2', '1005.0500-2-78-2'], ['1005.0500-1-42-0', '1005.0500-2-45-0'], ['1005.0500-1-17-4', '1005.0500-2-22-4'], ['1005.0500-1-18-2', '1005.0500-2-22-8'], ['1005.0500-1-38-2', '1005.0500-2-41-2'], ['1005.0500-1-14-1', '1005.0500-2-18-1'], ['1005.0500-1-14-2', '1005.0500-2-18-4'], ['1005.0500-1-10-2', '1005.0500-2-13-0'], ['1005.0500-1-10-6', '1005.0500-2-13-4'], ['1005.0500-1-13-2', '1005.0500-2-17-0']]

[['1005.0500-1-26-0', '1005.0500-2-29-0'], ['1005.0500-1-26-1', '1005.0500-2-29-1'], ['1005.0500-1-26-2', '1005.0500-2-29-2'], ['1005.0500-1-25-0', '1005.0500-2-28-0'], ['1005.0500-1-25-1', '1005.0500-2-28-1'], ['1005.0500-1-25-2', '1005.0500-2-28-2'], ['1005.0500-1-25-3', '1005.0500-2-28-3'], ['1005.0500-1-30-0', '1005.0500-2-33-0'], ['1005.0500-1-30-1', '1005.0500-2-33-1'], ['1005.0500-1-30-2', '1005.0500-2-33-2'], ['1005.0500-1-30-3', '1005.0500-2-33-3'], ['1005.0500-1-30-4', '1005.0500-2-33-4'], ['1005.0500-1-4-0', '1005.0500-2-4-0'], ['1005.0500-1-54-0', '1005.0500-2-69-0'], ['1005.0500-1-54-1', '1005.0500-2-69-1'], ['1005.0500-1-54-2', '1005.0500-2-69-2'], ['1005.0500-1-54-3', '1005.0500-2-69-3'], ['1005.0500-1-54-4', '1005.0500-2-69-4'], ['1005.0500-1-55-0', '1005.0500-2-70-0'], ['1005.0500-1-55-1', '1005.0500-2-70-1'], ['1005.0500-1-41-0', '1005.0500-2-44-0'], ['1005.0500-1-41-1', '1005.0500-2-44-1'], ['1005.0500-1-41-2', '1005.0500-2-44-2'], ['1005.0500-1-40-0', '1005.0500-2-43-0'], ['1005.0500-1-40-1', '1005.0500-2-43-1'], ['1005.0500-1-40-2', '1005.0500-2-43-2'], ['1005.0500-1-40-3', '1005.0500-2-43-3'], ['1005.0500-1-40-4', '1005.0500-2-43-4'], ['1005.0500-1-2-1', '1005.0500-2-2-1'], ['1005.0500-1-2-2', '1005.0500-2-2-2'], ['1005.0500-1-2-3', '1005.0500-2-2-3'], ['1005.0500-1-2-4', '1005.0500-2-2-4'], ['1005.0500-1-2-5', '1005.0500-2-2-5'], ['1005.0500-1-38-0', '1005.0500-2-41-0'], ['1005.0500-1-38-1', '1005.0500-2-41-1'], ['1005.0500-1-38-3', '1005.0500-2-41-4'], ['1005.0500-1-38-4', '1005.0500-2-41-5'], ['1005.0500-1-20-0', '1005.0500-2-23-0'], ['1005.0500-1-8-0', '1005.0500-2-10-0'], ['1005.0500-1-8-1', '1005.0500-2-10-1'], ['1005.0500-1-8-2', '1005.0500-2-10-2'], ['1005.0500-1-8-3', '1005.0500-2-10-3'], ['1005.0500-1-8-4', '1005.0500-2-10-4'], ['1005.0500-1-8-5', '1005.0500-2-10-5'], ['1005.0500-1-8-6', '1005.0500-2-10-6'], ['1005.0500-1-8-7', '1005.0500-2-10-7'], ['1005.0500-1-8-8', '1005.0500-2-10-8'], ['1005.0500-1-22-0', '1005.0500-2-25-0'], ['1005.0500-1-22-2', '1005.0500-2-25-2'], ['1005.0500-1-22-3', '1005.0500-2-25-3'], ['1005.0500-1-22-6', '1005.0500-2-25-6'], ['1005.0500-1-0-0', '1005.0500-2-0-0'], ['1005.0500-1-31-0', '1005.0500-2-34-0'], ['1005.0500-1-31-1', '1005.0500-2-34-1'], ['1005.0500-1-31-2', '1005.0500-2-34-2'], ['1005.0500-1-32-3', '1005.0500-2-35-3'], ['1005.0500-1-45-0', '1005.0500-2-48-0'], ['1005.0500-1-48-2', '1005.0500-2-53-2'], ['1005.0500-1-39-1', '1005.0500-2-42-1'], ['1005.0500-1-39-2', '1005.0500-2-42-2'], ['1005.0500-1-39-3', '1005.0500-2-42-3'], ['1005.0500-1-39-4', '1005.0500-2-42-4'], ['1005.0500-1-39-5', '1005.0500-2-42-5'], ['1005.0500-1-39-6', '1005.0500-2-42-6'], ['1005.0500-1-39-7', '1005.0500-2-42-7'], ['1005.0500-1-39-8', '1005.0500-2-42-8'], ['1005.0500-1-39-9', '1005.0500-2-42-9'], ['1005.0500-1-63-0', '1005.0500-2-79-0'], ['1005.0500-1-63-1', '1005.0500-2-79-1'], ['1005.0500-1-63-2', '1005.0500-2-79-2'], ['1005.0500-1-63-3', '1005.0500-2-79-3'], ['1005.0500-1-37-0', '1005.0500-2-40-0'], ['1005.0500-1-37-1', '1005.0500-2-40-1'], ['1005.0500-1-9-0', '1005.0500-2-11-0'], ['1005.0500-1-9-1', '1005.0500-2-11-1'], ['1005.0500-1-9-2', '1005.0500-2-11-2'], ['1005.0500-1-34-0', '1005.0500-2-37-0'], ['1005.0500-1-34-2', '1005.0500-2-37-2'], ['1005.0500-1-34-3', '1005.0500-2-37-3'], ['1005.0500-1-34-4', '1005.0500-2-37-4'], ['1005.0500-1-23-0', '1005.0500-2-26-0'], ['1005.0500-1-23-1', '1005.0500-2-26-1'], ['1005.0500-1-23-2', '1005.0500-2-26-2'], ['1005.0500-1-50-0', '1005.0500-2-65-0'], ['1005.0500-1-50-1', '1005.0500-2-65-1'], ['1005.0500-1-50-2', '1005.0500-2-65-2'], ['1005.0500-1-5-0', '1005.0500-2-5-0'], ['1005.0500-1-5-2', '1005.0500-2-5-2'], ['1005.0500-1-5-3', '1005.0500-2-5-3'], ['1005.0500-1-5-4', '1005.0500-2-5-4'], ['1005.0500-1-5-5', '1005.0500-2-5-5'], ['1005.0500-1-11-0', '1005.0500-2-14-0'], ['1005.0500-1-11-2', '1005.0500-2-14-2'], ['1005.0500-1-11-3', '1005.0500-2-14-5'], ['1005.0500-1-47-0', '1005.0500-2-52-0'], ['1005.0500-1-47-1', '1005.0500-2-52-1'], ['1005.0500-1-47-2', '1005.0500-2-52-2'], ['1005.0500-1-47-4', '1005.0500-2-52-4'], ['1005.0500-1-47-5', '1005.0500-2-52-5'], ['1005.0500-1-47-6', '1005.0500-2-52-6'], ['1005.0500-1-47-7', '1005.0500-2-52-7'], ['1005.0500-1-14-0', '1005.0500-2-18-0'], ['1005.0500-1-16-0', '1005.0500-2-21-0'], ['1005.0500-1-16-1', '1005.0500-2-21-1'], ['1005.0500-1-16-2', '1005.0500-2-21-2'], ['1005.0500-1-16-3', '1005.0500-2-21-3'], ['1005.0500-1-16-4', '1005.0500-2-21-4'], ['1005.0500-1-16-5', '1005.0500-2-21-5'], ['1005.0500-1-16-7', '1005.0500-2-21-7'], ['1005.0500-1-24-0', '1005.0500-2-27-0'], ['1005.0500-1-24-1', '1005.0500-2-27-1'], ['1005.0500-1-24-2', '1005.0500-2-27-2'], ['1005.0500-1-24-3', '1005.0500-2-27-3'], ['1005.0500-1-52-0', '1005.0500-2-67-0'], ['1005.0500-1-52-2', '1005.0500-2-67-2'], ['1005.0500-1-52-3', '1005.0500-2-67-3'], ['1005.0500-1-53-0', '1005.0500-2-68-0'], ['1005.0500-1-53-1', '1005.0500-2-68-1'], ['1005.0500-1-57-1', '1005.0500-2-75-1'], ['1005.0500-1-3-0', '1005.0500-2-3-0'], ['1005.0500-1-3-2', '1005.0500-2-3-2'], ['1005.0500-1-3-3', '1005.0500-2-3-3'], ['1005.0500-1-3-4', '1005.0500-2-3-4'], ['1005.0500-1-33-0', '1005.0500-2-36-0'], ['1005.0500-1-33-2', '1005.0500-2-36-2'], ['1005.0500-1-33-3', '1005.0500-2-36-3'], ['1005.0500-1-33-4', '1005.0500-2-36-4'], ['1005.0500-1-33-5', '1005.0500-2-36-5'], ['1005.0500-1-29-0', '1005.0500-2-32-0'], ['1005.0500-1-29-1', '1005.0500-2-32-1'], ['1005.0500-1-29-2', '1005.0500-2-32-2'], ['1005.0500-1-29-3', '1005.0500-2-32-3'], ['1005.0500-1-29-4', '1005.0500-2-32-4'], ['1005.0500-1-29-5', '1005.0500-2-32-5'], ['1005.0500-1-29-6', '1005.0500-2-32-6'], ['1005.0500-1-29-7', '1005.0500-2-32-7'], ['1005.0500-1-29-8', '1005.0500-2-32-8'], ['1005.0500-1-49-1', '1005.0500-2-64-1'], ['1005.0500-1-49-3', '1005.0500-2-64-3'], ['1005.0500-1-49-4', '1005.0500-2-64-4'], ['1005.0500-1-49-5', '1005.0500-2-64-5'], ['1005.0500-1-49-6', '1005.0500-2-64-6'], ['1005.0500-1-61-0', '1005.0500-2-77-0'], ['1005.0500-1-61-1', '1005.0500-2-77-1'], ['1005.0500-1-27-0', '1005.0500-2-30-0'], ['1005.0500-1-27-1', '1005.0500-2-30-1'], ['1005.0500-1-27-2', '1005.0500-2-30-2'], ['1005.0500-1-27-3', '1005.0500-2-30-3'], ['1005.0500-1-27-4', '1005.0500-2-30-4'], ['1005.0500-1-27-5', '1005.0500-2-30-5'], ['1005.0500-1-27-6', '1005.0500-2-30-6'], ['1005.0500-1-27-7', '1005.0500-2-30-7'], ['1005.0500-1-44-0', '1005.0500-2-47-0'], ['1005.0500-1-28-0', '1005.0500-2-31-0'], ['1005.0500-1-28-1', '1005.0500-2-31-1'], ['1005.0500-1-28-2', '1005.0500-2-31-2'], ['1005.0500-1-58-0', '1005.0500-2-76-0'], ['1005.0500-1-58-2', '1005.0500-2-76-2'], ['1005.0500-1-58-3', '1005.0500-2-76-3'], ['1005.0500-1-58-6', '1005.0500-2-76-6'], ['1005.0500-1-62-1', '1005.0500-2-78-1'], ['1005.0500-1-12-0', '1005.0500-2-15-0'], ['1005.0500-1-12-1', '1005.0500-2-15-1'], ['1005.0500-1-12-2', '1005.0500-2-15-2'], ['1005.0500-1-12-3', '1005.0500-2-15-3'], ['1005.0500-1-51-0', '1005.0500-2-66-0'], ['1005.0500-1-51-1', '1005.0500-2-66-1'], ['1005.0500-1-51-2', '1005.0500-2-66-2'], ['1005.0500-1-42-1', '1005.0500-2-45-1'], ['1005.0500-1-6-0', '1005.0500-2-6-0'], ['1005.0500-1-6-1', '1005.0500-2-6-1'], ['1005.0500-1-6-2', '1005.0500-2-7-0'], ['1005.0500-1-6-3', '1005.0500-2-7-1'], ['1005.0500-1-10-0', '1005.0500-2-12-0'], ['1005.0500-1-10-3', '1005.0500-2-13-1'], ['1005.0500-1-10-4', '1005.0500-2-13-2'], ['1005.0500-1-10-5', '1005.0500-2-13-3'], ['1005.0500-1-13-0', '1005.0500-2-16-0'], ['1005.0500-1-13-1', '1005.0500-2-16-1'], ['1005.0500-1-13-3', '1005.0500-2-17-1'], ['1005.0500-1-17-0', '1005.0500-2-22-0'], ['1005.0500-1-17-1', '1005.0500-2-22-1'], ['1005.0500-1-17-2', '1005.0500-2-22-2'], ['1005.0500-1-17-3', '1005.0500-2-22-3'], ['1005.0500-1-18-0', '1005.0500-2-22-6'], ['1005.0500-1-18-1', '1005.0500-2-22-7'], ['1005.0500-1-19-0', '1005.0500-2-22-10'], ['1005.0500-1-19-1', '1005.0500-2-22-11'], ['1005.0500-1-19-2', '1005.0500-2-22-12'], ['1005.0500-1-19-3', '1005.0500-2-22-13'], ['1005.0500-1-19-4', '1005.0500-2-22-14'], ['1005.0500-1-19-5', '1005.0500-2-22-15'], ['1005.0500-1-19-6', '1005.0500-2-22-16'], ['1005.0500-1-19-7', '1005.0500-2-22-17'], ['1005.0500-1-19-8', '1005.0500-2-22-18'], ['1005.0500-1-19-9', '1005.0500-2-22-19'], ['1005.0500-1-19-10', '1005.0500-2-22-20'], ['1005.0500-1-19-11', '1005.0500-2-22-21'], ['1005.0500-1-19-12', '1005.0500-2-22-22']]

[['1005.0500-1-25-4', '1005.0500-2-28-4'], ['1005.0500-1-25-5', '1005.0500-2-28-5'], ['1005.0500-1-25-6', '1005.0500-2-28-6'], ['1005.0500-1-4-1', '1005.0500-2-4-1'], ['1005.0500-1-2-0', '1005.0500-2-2-0'], ['1005.0500-1-22-1', '1005.0500-2-25-1'], ['1005.0500-1-22-4', '1005.0500-2-25-4'], ['1005.0500-1-22-5', '1005.0500-2-25-5'], ['1005.0500-1-0-1', '1005.0500-2-0-2'], ['1005.0500-1-31-3', '1005.0500-2-34-3'], ['1005.0500-1-32-0', '1005.0500-2-35-0'], ['1005.0500-1-32-1', '1005.0500-2-35-1'], ['1005.0500-1-32-2', '1005.0500-2-35-2'], ['1005.0500-1-45-1', '1005.0500-2-48-1'], ['1005.0500-1-45-2', '1005.0500-2-48-2'], ['1005.0500-1-48-0', '1005.0500-2-53-0'], ['1005.0500-1-48-1', '1005.0500-2-53-1'], ['1005.0500-1-39-0', '1005.0500-2-42-0'], ['1005.0500-1-37-2', '1005.0500-2-40-2'], ['1005.0500-1-37-3', '1005.0500-2-40-3'], ['1005.0500-1-37-4', '1005.0500-2-40-4'], ['1005.0500-1-34-1', '1005.0500-2-37-1'], ['1005.0500-1-34-5', '1005.0500-2-37-5'], ['1005.0500-1-5-1', '1005.0500-2-5-1'], ['1005.0500-1-5-6', '1005.0500-2-5-6'], ['1005.0500-1-11-1', '1005.0500-2-14-1'], ['1005.0500-1-47-3', '1005.0500-2-52-3'], ['1005.0500-1-16-6', '1005.0500-2-21-6'], ['1005.0500-1-52-1', '1005.0500-2-67-1'], ['1005.0500-1-53-2', '1005.0500-2-68-2'], ['1005.0500-1-53-3', '1005.0500-2-68-3'], ['1005.0500-1-57-0', '1005.0500-2-75-0'], ['1005.0500-1-3-1', '1005.0500-2-3-1'], ['1005.0500-1-33-1', '1005.0500-2-36-1'], ['1005.0500-1-49-0', '1005.0500-2-64-0'], ['1005.0500-1-49-2', '1005.0500-2-64-2'], ['1005.0500-1-46-0', '1005.0500-2-51-0'], ['1005.0500-1-46-1', '1005.0500-2-51-1'], ['1005.0500-1-46-2', '1005.0500-2-51-2'], ['1005.0500-1-46-3', '1005.0500-2-51-3'], ['1005.0500-1-44-1', '1005.0500-2-47-1'], ['1005.0500-1-58-1', '1005.0500-2-76-1'], ['1005.0500-1-58-4', '1005.0500-2-76-4'], ['1005.0500-1-58-5', '1005.0500-2-76-5'], ['1005.0500-1-62-0', '1005.0500-2-78-0'], ['1005.0500-1-62-2', '1005.0500-2-78-2'], ['1005.0500-1-42-0', '1005.0500-2-45-0'], ['1005.0500-1-17-4', '1005.0500-2-22-4'], ['1005.0500-1-18-2', '1005.0500-2-22-8']]

[]

[['1005.0500-1-38-2', '1005.0500-2-41-2'], ['1005.0500-1-14-1', '1005.0500-2-18-1'], ['1005.0500-1-14-2', '1005.0500-2-18-4'], ['1005.0500-1-10-2', '1005.0500-2-13-0'], ['1005.0500-1-10-6', '1005.0500-2-13-4'], ['1005.0500-1-13-2', '1005.0500-2-17-0']]

[]

['1005.0500-2-73-1']

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

https://arxiv.org/abs/1005.0500

astro-ph-9906062

{'astro-ph-9906062-1-0-0': '[MATH] Department of Astronomy, Nanjing University, Nanjing 210093, P. R. China', 'astro-ph-9906062-1-1-0': '[MATH] LCRHEA, IHEP, CAS, Beijing 100039, P. R. China', 'astro-ph-9906062-1-2-0': 'Send offprint request to:', 'astro-ph-9906062-1-3-0': 'T. Lu[MATH], Dept. of Astronomy, Nanjing University, Nanjing 210093, P. R. China', 'astro-ph-9906062-1-4-0': '[MATH]Telephone: 86 25 3592507', 'astro-ph-9906062-1-5-0': '[MATH]Telefax: 86 25 3326467', 'astro-ph-9906062-1-6-0': '[MATH]E-mail: tlu@nju.edu.cn', 'astro-ph-9906062-1-7-0': 'All correspondence please send to: Prof. T. Lu', 'astro-ph-9906062-1-8-0': '22pt', 'astro-ph-9906062-1-9-0': 'Intrinsic Parameters of GRB990123 from Its Prompt Optical Flash and Afterglow', 'astro-ph-9906062-1-10-0': 'X. Y. Wang[MATH], Z. G. Dai[MATH] and T. Lu[MATH] Department of Astronomy, Nanjing University, Nanjing 210093, P. R. China', 'astro-ph-9906062-1-11-0': '[MATH] LCRHEA, IHEP, CAS, Beijing 100039, P. R. China', 'astro-ph-9906062-1-12-0': '22pt ABSTRACT', 'astro-ph-9906062-1-13-0': 'We have computed the magnetic energy density fraction ([MATH]) and the electron energy density fraction ([MATH]) of GRB990123 in terms of the optical flash information (reverse shock model) and compared them with those determined independently by the afterglow information (forward shock model).', 'astro-ph-9906062-1-13-1': 'Our result shows: 1) [MATH] and [MATH] are nearly constant from 50 seconds to [MATH] seconds after the gamma-ray trigger, supporting the hypothesis adopted in the standard afterglow model that the magnetic and electron energy densities are constant during the evolution of the external shocks; 2) the electron energy density fraction of GRB990123 is nearly the same as those of GRB970508 and GRB971214, suggesting that it may be a universal parameter.', 'astro-ph-9906062-1-13-2': 'However, the magnetic energy density fraction of GRB990123 is much lower than those of the other two bursts.', 'astro-ph-9906062-1-13-3': 'We speculate that this value may depend on the origins of magnetic fields in the energy flow of GRB.', 'astro-ph-9906062-1-13-4': 'Other instrinsic parameters of GRB990123, such as energy contained in the forward shock [MATH], the initial Lorentz factor [MATH] of the ejecta, are also determined and discussed in this paper.', 'astro-ph-9906062-1-13-5': 'These parameters imply that the reverse external shock may have become ultrarelativistic before it passed through the ejecta shell.', 'astro-ph-9906062-1-14-0': 'Key words: gamma-rays: bursts-shock waves-magnetic fields-optical radiation', 'astro-ph-9906062-1-15-0': '# Introduction', 'astro-ph-9906062-1-16-0': 'The current standard model for gamma-ray bursts (GRBs) and their afterglows is the fireball-plus-shock model (see Piran 1998 for a review).', 'astro-ph-9906062-1-16-1': 'It involves that a large amount of energy, [MATH] ergs, is released within a few seconds in a small volume with negligible baryonic load, [MATH].', 'astro-ph-9906062-1-16-2': 'This leads to a fireball that expands ultra-relativistically with a Lorentz factor [MATH] required to avoid the attenuation of hard [MATH]-rays due to pair production (e.g. Woods [MATH] Loeb 1995; Fenimore, Epstein [MATH] Ho 1993).', 'astro-ph-9906062-1-16-3': 'A substantial fraction of the kinetic energy of the baryons is transferred to a non-thermal population of relativistic electrons through Fermi acceletation in the shock ([MATH] Rees 1993).', 'astro-ph-9906062-1-16-4': 'The accelerated electrons cool via synchrotron emission and inverse Compton scattering in the post-shock magnetic fields and produce the radiation observed in GRBs and their afterglows (e.g. Katz 1994; Sari et al. 1996; Vietri 1997; Waxman 1997a; Wijers et al. 1997).', 'astro-ph-9906062-1-16-5': 'The shock could be either [MATH] due to collisions between fireball shells caused by outflow variability (Paczy[MATH]ski [MATH] Xu 1994; Rees [MATH] 1994), or [MATH] due to the interaction of the fireball with the surrounding interstellar medium (ISM; [MATH] Rees 1993).', 'astro-ph-9906062-1-16-6': 'The radiation from internal shocks can explain the spectra (Pilla [MATH] Loeb 1998) and the fast irregular variability of GRBs (Sari [MATH] Piran 1997), while the synchrotron emission from the external shocks provides a successful model for the broken power law spectra and the power law decay of afterglow light curves (e.g. Waxman 1997a,b; Wijers, Rees [MATH] 1997; Vietri 1997; Dai [MATH] Lu 1998a,b,c).', 'astro-ph-9906062-1-17-0': 'The properties of the synchrotron emission from GRB shocks are determined by the magnetic field strength, [MATH], and the electron energy distribution behind the shock.', 'astro-ph-9906062-1-17-1': 'Both of them are difficult to estimate from first principles, and so the following dimensionless parameters are often used to incorporate modeling uncertainties (Sari et al. 1996), [MATH], [MATH].', 'astro-ph-9906062-1-17-2': 'Here [MATH] and [MATH] are the magnetic and electron energy densities and [MATH] is the total thermal energy density behind the shocks, where [MATH] is the proton mass, [MATH] is the proton number density, and [MATH] is the mean thermal Lorentz factor of the protons.', 'astro-ph-9906062-1-17-3': 'In spite of these uncertainties, an important assumption that [MATH] and [MATH] do not change with time, has been made in the standard external shock model.', 'astro-ph-9906062-1-17-4': 'It is important to note that the constancy of [MATH] and [MATH] can not be deduced directly from the power law decay of the afterglow light curve.', 'astro-ph-9906062-1-17-5': 'In fact, if the magnetic field in the shock is frozen-in, in which case [MATH] varies with time as [MATH], rather than turbulent, it will also lead to a power law decay.', 'astro-ph-9906062-1-17-6': 'So the correctness of this assumption still needs testing and examining.', 'astro-ph-9906062-1-18-0': 'The BeppoSAX satellite ushered in 1999 with the discovery of GRB990123 (Heise et al. 1999), the brightest GRB seen by BeppoSAX to date.', 'astro-ph-9906062-1-18-1': 'This is a very strong burst.', 'astro-ph-9906062-1-18-2': "Its fluence (energy [MATH] keV) of [MATH] erg cm[MATH] (Kippen et al. GCN224) places it at the top [MATH] of the BATSE's bursts.", 'astro-ph-9906062-1-18-3': "An assumption of isotropic emission and the detection of the source's redshift [MATH] (Kulkarni et al. 1999a), lead to a huge energy release about [MATH]ergs in [MATH]-rays alone.", 'astro-ph-9906062-1-18-4': 'GRB990123 would have been amongest the most exciting GRBs even just on the basis of these facts.', 'astro-ph-9906062-1-18-5': 'Furthermore, ROTSE discovered a prompt optical flash of 9-th magnitude (Akerlof et al. 1999).', 'astro-ph-9906062-1-18-6': 'It is the first time that a prompt emission in another wavelength apart from [MATH]-rays has been detected from GRB.', 'astro-ph-9906062-1-18-7': 'Such a strong optical flash was predicted to arise from a reverse external shock propagating into the relativistic ejecta ([MATH] Rees 1997; Sari [MATH] Piran 1998a,b, hereafter SP99a,b).', 'astro-ph-9906062-1-18-8': 'This is the so called "early afterglow".', 'astro-ph-9906062-1-18-9': 'The five last exposures of ROTSE show a power law decay with a slope of [MATH]2.0, which can also be explained by the reverse shock model (SP99b; [MATH] Rees 1999).', 'astro-ph-9906062-1-18-10': 'The usual afterglows in X-ray, optical, IR and radio bands were also detected after the burst.', 'astro-ph-9906062-1-18-11': 'They have two distinguishing features: 1) the radio emission is unique both due to its very early appearance and its rapid decline; 2) the temporal decaying index of the [MATH]-band light curve after two days steepens from about [MATH] to [MATH] (Kulkarni et al. 1999a; Fruchter et al. 1999; Casrto-Tirado et al. 1999), and this steepening might be due to a jet which has transited from a spherical-like phase to sideways expansion phase (Rhoads 1999; Sari et al. 1999) or a dense cloud which has slowed down the relativistic expansion of shock quickly to a non-relativistic one (Dai [MATH] Lu 1999).', 'astro-ph-9906062-1-18-12': 'The usual afterglow is considered to be produced by the forward external shock that propagates into the ISM (see e.g. Piran 1999).', 'astro-ph-9906062-1-19-0': 'Galama et al. (1999) reconstructed the radio-to-X-ray afterglow spectrum on January 24.65 UT.', 'astro-ph-9906062-1-19-1': 'By combining this spectrum with the radio light curve, they estimated the four key quantities (self-absorption frequency [MATH], peak frequency [MATH], cooling frequency [MATH] and peak flux [MATH]) required to compute the intrinsic parameters of the burst (e.g. the magnetic energy density fraction [MATH] and electron energy density fraction [MATH]).', 'astro-ph-9906062-1-19-2': 'They argue that both a higher cooling frequency and a lower peak frequency can be explained by a low magnetic field.', 'astro-ph-9906062-1-19-3': 'They even estimated the field energy density for the afterglow of GRB990123 to be as low as [MATH] times equipartition, much lower than that of well-studied GRB970508 (Wijers [MATH] Galama 1998; hereafter WG98).', 'astro-ph-9906062-1-19-4': 'For previous bursts, we have no other information apart from the afterglow to infer the intrinsic parameters of the external shocks.', 'astro-ph-9906062-1-19-5': 'But now the optical flash of GRB990123 has been fortunately detected, which enables us to determine the intrinsic parameters according to the reverse external shock model and compare them with those determined from the afterglow.', 'astro-ph-9906062-1-19-6': 'WG98 computed the intrinsic parameters of GRB970508 and GRB971214 in terms of their afterglow data, and found that [MATH] is nearly the same for these two bursts, suggesting it may be a universal parameter.', 'astro-ph-9906062-1-19-7': 'Here we attempt to examine whether [MATH] and [MATH] are constant during the evolution of the external shock waves based on the information of the two aspects of GRB990123-the optical flash and the afterglow.', 'astro-ph-9906062-1-20-0': 'The initial Lorentz factor [MATH] is also an important physical parameter of GRBs.', 'astro-ph-9906062-1-20-1': 'It is a crucial ingredient for constraining models of the source itself, since it specifies how "clean" the fireball is as the baryonic load is [MATH].', 'astro-ph-9906062-1-20-2': 'Unfortunately, the spectrum of GRBs can provide only a lower limit to this Lorentz factor ([MATH]).', 'astro-ph-9906062-1-20-3': 'Moreover, the current afterglow observations, which detect radiation from several hours after the burst, do not provide a verification of the initial extreme relativistic motion.', 'astro-ph-9906062-1-20-4': 'A possible method to estimate [MATH] of GRBs has been suggested by Sari [MATH] Piran (1999a), based on identifying the "early afterglow" peak time.', 'astro-ph-9906062-1-20-5': 'In this paper, the initial Lorentz factor has been inferred more precisely from the full set of equations describing the reverse shock region.', 'astro-ph-9906062-1-21-0': 'In section 2, we first compute the intrinsic parameters from the afterglow according to the forward external shock model and then compare them with those determined from the optical flash according to the reverse external shock model (section 3).', 'astro-ph-9906062-1-21-1': 'In the final section, we give our conclusions and discussions.', 'astro-ph-9906062-1-22-0': '# Parameters from afterglow', 'astro-ph-9906062-1-23-0': '## Broad-band spectrum and the cooling frequency', 'astro-ph-9906062-1-24-0': 'Intrinsic parameters like the magnetic energy density fraction [MATH], electron energy density fraction [MATH], energy in the forward external shock [MATH]ergs and ambient density [MATH] can be determined from the afterglow spectrum (WG98; Granot et al. 1998), i. e. if we know all three break frequencies (not necessary at the same time) and the peak flux of the afterglow, we can infer all these parameters.', 'astro-ph-9906062-1-24-1': 'From the unique behaviour of the radio emission and the broad-band spectrum of GRB990123, Galama et al. (1999) estimated the quantities of the afterglow required.', 'astro-ph-9906062-1-24-2': 'We summarize them here: at [MATH] days after the trigger, the self-absorption frequency [MATH]GHz, the peak frequency [MATH]GHz, and the peak flux [MATH]mJy.', 'astro-ph-9906062-1-24-3': 'However, for the cooling frequency [MATH], we only know that it is located at or above the X-ray frequencies from the spectrum.', 'astro-ph-9906062-1-24-4': 'Galama et al. (1999) fitted the optical to X-ray spectral flux distribution with a power law [MATH], where the slope [MATH] is [MATH].', 'astro-ph-9906062-1-24-5': 'So the electron index ( defined as [MATH].', 'astro-ph-9906062-1-24-6': 'This value is consistent with that inferred by Kulkarni et al. (1999a).', 'astro-ph-9906062-1-25-0': 'Recently, Kulkarni et al. (1999b) argued that the radio flare of GRB990123 is not caused by the forward external shock model.', 'astro-ph-9906062-1-25-1': 'They considered that when [MATH], the former evolves as [MATH], which will cause the radio flux to increase and only when [MATH] falls below 8.46GHz (corresponding time is 10 days after the burst), the flux starts to decay.', 'astro-ph-9906062-1-25-2': 'We think that the jet-like geometry of GRB990123 energy flow favoured by Kulkarni et al. (1999a), may help to solve the disagreement.', 'astro-ph-9906062-1-25-3': 'Due to a jet, [MATH] for [MATH] (for [MATH], [MATH] (Rhoads 1999)), decaying faster than the spherical case.', 'astro-ph-9906062-1-25-4': 'So when [MATH] ) , the flux start to decay very steeply as [MATH].', 'astro-ph-9906062-1-25-5': 'Moreover, we conjecture the peak break should not be so abrupt.', 'astro-ph-9906062-1-25-6': 'The realistic light curve may be much rounder at the peak than the simple broken power law (Rhoads 1999), which likely caused the non-detection of radio emission even three days after the burst.', 'astro-ph-9906062-1-25-7': 'So in our work, we take the values of [MATH] and [MATH] inferred by Galama et al.', 'astro-ph-9906062-1-26-0': 'The cooling frequency [MATH] cannot be seen from the radio-to-X-ray spectrum.', 'astro-ph-9906062-1-26-1': 'This indicates that [MATH] is at or above the X-ray frequencies.', 'astro-ph-9906062-1-26-2': 'We need to determine it more precisely.', 'astro-ph-9906062-1-26-3': 'The X-ray afterglow, observed 6 hours after the burst, decayed with [MATH] ( Heise et al. 1999), while the optical afterglow with [MATH] (Kulkarni et al., 1999).', 'astro-ph-9906062-1-26-4': 'An X-ray afterglow decay slope steeper by [MATH] than an optical decay, which seems to be the case in this burst, is predicted by Sari, Piran and Narayan (1998), if the cooling frequency is between the X-rays and the optical.', 'astro-ph-9906062-1-26-5': 'So at the time 6 hours after the burst , [MATH]Hz.', 'astro-ph-9906062-1-26-6': 'Extrapolating it to the time [MATH] as [MATH], we get [MATH]Hz[MATH]Hz.', 'astro-ph-9906062-1-26-7': 'Another speculative constraint on [MATH] is obtained from the GRB spectrum itself by Sari [MATH] Piran (1999b), who constrained [MATH]Hz at the time [MATH] sec. Extrapolating it to [MATH], we get [MATH].', 'astro-ph-9906062-1-26-8': 'Now [MATH] is almost determined, and we take the middle value [MATH]Hz.', 'astro-ph-9906062-1-26-9': 'This result is in agreement with the estimate of Galama et al (1999).', 'astro-ph-9906062-1-26-10': 'So we now have all three break frequencies and the peak flux required to calculate the intrinsic parameters of the afterglow: [EQUATION]', 'astro-ph-9906062-1-26-11': 'We can see that our inferred value of [MATH] is rather low in contrast with that of the well-studied GRB970508, which is [MATH] (see WG98).', 'astro-ph-9906062-1-26-12': 'Just as Galama et al.(1999) have pointed out, the higher cooling frequency and lower peak frequency are caused by a lower field in the forward shock region.', 'astro-ph-9906062-1-26-13': 'We can also find that the electron density fraction [MATH] is almost the same as those of GRB970508 ([MATH])and GRB971214 ([MATH]) (see WG98).', 'astro-ph-9906062-1-26-14': 'We conjecture that [MATH] may be a universal constant, i.e. a constant for every GRB afterglow, while the magnetic energy density fraction [MATH] may depend on the origins of magnetic fields in the shock region, which are possibly different for different GRBs (see e.g. Medvedev [MATH] Loeb 1999).', 'astro-ph-9906062-1-26-15': 'However, for a certain GRB, we speculate that the [MATH] should remain constant during the evolution of the shock waves.', 'astro-ph-9906062-1-26-16': 'This is just the argument we try to demonstrate in the next section.', 'astro-ph-9906062-1-26-17': 'The ambient density given in Eq. (11) is much higher than for GRB970508 and GRB971214.', 'astro-ph-9906062-1-26-18': 'A higher ambient density has been also inferred by Dai [MATH] Lu (1999) and Shi [MATH] Gyuk (1999) to explain the light curve break of the optical afterglow and the radio flare, respectively.', 'astro-ph-9906062-1-26-19': 'We suggest that the denser part of ISM may be ejecta from the GRB source site.', 'astro-ph-9906062-1-27-0': '# Parameters from the optical flash', 'astro-ph-9906062-1-28-0': 'We have obtained [MATH] of GRB990123 afterglow in the above section.', 'astro-ph-9906062-1-28-1': 'It is much lower than those of GRB970508 and GRB971214.', 'astro-ph-9906062-1-28-2': 'On the other hand, the prompt optical flash information of GRB990123 provides an excellent opportunity for us to examine this value.', 'astro-ph-9906062-1-28-3': 'Moreover, the optical flash took place at very early time ([MATH]sec).', 'astro-ph-9906062-1-28-4': 'It also provides us a chance to examine whether [MATH] and [MATH] are constants during the evolution of the external shocks, a key hypothesis of the well-accepted external shock model.', 'astro-ph-9906062-1-29-0': "An optical flash is considered to be produced by the reverse external shock, which heats up the shell's matter and accelerates its electrons (SP99b; M[MATH]sz[MATH]ros [MATH] Rees 1999).", 'astro-ph-9906062-1-29-1': "BATSE's observations triggered ROTSE via BACODINE system (Akerlof et al. 1999).", 'astro-ph-9906062-1-29-2': 'An 11.82 magnitude optical flash was detected on the first 5 seconds exposure, 22.18 seconds after the onset of the burst.', 'astro-ph-9906062-1-29-3': 'Then the optical emission peaked in the following 5 seconds exposure, 25 seconds later, which revealed an 8.95 magnitude signal ([MATH]1Jy).', 'astro-ph-9906062-1-29-4': 'The optical signal decayed to 10.08 magnitude 25 seconds later and continued to decay down to 14.53 magnitude in the subsequent three 75 seconds exposures that took place up to 10 minutes after the burst.', 'astro-ph-9906062-1-29-5': 'The five last exposures depict a power law decay with a slope [MATH] (Akerlof et al. 1999; SP99b).', 'astro-ph-9906062-1-29-6': 'Sari [MATH] Piran (1999b) and M[MATH]sz[MATH]ros [MATH] Rees (1999) assumed that the ejecta shell follows the Blandford-McKee (1976) self-similar solution after the reverse shock has passed through it and explained the decay of [MATH].', 'astro-ph-9906062-1-30-0': 'So we assume at the optical emission peak time ([MATH]sec) that the reverse shock had just passed through the ejecta shell.', 'astro-ph-9906062-1-30-1': 'At this time, the Lorentz factor of the reverse shock [MATH] is approximately given by [EQUATION]', 'astro-ph-9906062-1-30-2': 'Please note that this formula always holds whether the ejecta is jet-like or spherical, because the beaming factor in Eq. (16) and (17) will cancel out each other in the jet-like case.', 'astro-ph-9906062-1-30-3': 'Now we have three equations: (14), (15) and (18) with four unknowns: [MATH], [MATH], [MATH] and [MATH] (the value of ambient density [MATH] has been determined in the above section).', 'astro-ph-9906062-1-30-4': 'Another condition can be obtained from the jump condition of the shock.', 'astro-ph-9906062-1-30-5': 'According to this condition, the Lorentz factor of the shocked shell should be approximately equal to that of the shocked ISM (Piran 1998).', 'astro-ph-9906062-1-30-6': 'The Lorentz factor of the forward shocked ISM can be obtained from the standard afterglow model (e. g. WG98): [EQUATION]', 'astro-ph-9906062-1-30-7': 'Comparing them with those in Eq. (11), we astonishingly find that the values of [MATH] and [MATH] determined from the optical flash information are nearly the same as those from the afterglow, though [MATH] is quite low.', 'astro-ph-9906062-1-30-8': 'Considering the roughness of the reverse shock model used, this result shows the assumption that [MATH] and [MATH] are set by microphysics behind the shocks and should be constants during the evolution of the external shock, is reasonable, supporting the standard afterglow model and the reverse shock model.', 'astro-ph-9906062-1-30-9': 'Moreover, the value of [MATH] is also nearly the same as those of GRB970508 and GRB971214, suggesting the electron energy density fraction may be a universal constant.', 'astro-ph-9906062-1-30-10': 'On the other hand, the value [MATH] of GRB990123 is much lower than those of GRB970508 and GRB971214, which indicates that the magnetic energy density may depend on the origins of magnetic fields in the shock region and that the afterglow of GRB990123 is really a low-field one.', 'astro-ph-9906062-1-31-0': 'We also obtain two by-products, [MATH] and [MATH], which represent the initial Lorentz factor of the ejecta and the Lorentz factor of the ejecta at the optical flash peak time, respectively.', 'astro-ph-9906062-1-31-1': 'Our inferred initial Lorentz factor [MATH] is six times larger than that obtained by Sari [MATH] Piran (1999b), who estimated it in terms of the parameter values inferred for GRB970508.', 'astro-ph-9906062-1-31-2': 'Consequently, at the time the reverse shock has just passed through the ejecta shell, its Lorentz factor was [MATH].', 'astro-ph-9906062-1-31-3': 'This indicates that the reverse shock had become ultrarelativistic before it crossed the entire shell.', 'astro-ph-9906062-1-31-4': 'This result is also different from that obtained by Sari [MATH] Piran (1999b), who found the reverse shock of GRB990123 was only mildly relativistic.', 'astro-ph-9906062-1-31-5': 'However we argue that our result is reasonable according to the criterion presented by Sari [MATH] Piran (1995) (also see Kobayashi et al. 1998).', 'astro-ph-9906062-1-31-6': 'They defined a dimensionless parameter [MATH] constructed from [MATH], [MATH] and [MATH]: [EQUATION] where [MATH] is the Sedov length, [MATH] is the width of the shell ([MATH] is the duration of GRB) and [MATH] is the initial Lorentz factor of the ejecta.', 'astro-ph-9906062-1-31-7': 'If [MATH], the reverse shock becomes relativistic before it crosses the shell; otherwise ([MATH]), the reverse shock remains Newtonian or at best mildly relativistic during the whole energy extraction process.', 'astro-ph-9906062-1-31-8': 'For GRB990123, we find [MATH].', 'astro-ph-9906062-1-31-9': 'So the reverse shock of GRB990123 had become untrarelativistic before it crossed the shell, consistent with our calculated result.', 'astro-ph-9906062-1-32-0': '# Conclusions and Discussions', 'astro-ph-9906062-1-33-0': 'Motivated by checking whether the electron energy density fraction [MATH] and the field energy density fraction [MATH] remain constant during the evolution of the external shock, we have computed these two values from both the afterglow and the optical flash of GRB990123.', 'astro-ph-9906062-1-33-1': 'In order to do this, we present a full set of equations to determine the intrinsic parameters of the reverse shock by combining with the property of the forward shock.', 'astro-ph-9906062-1-33-2': 'Moreover, due to the particular case [MATH] in our calculation, we first derived the formula of [MATH] in this case from the basic synchrotron radiation theory.', 'astro-ph-9906062-1-33-3': 'WG98 have determined the intrinsic parameters of GRB970508 and GRB971214 using the afterglow information.', 'astro-ph-9906062-1-33-4': 'They found [MATH] to be nearly the same for these two bursts.', 'astro-ph-9906062-1-33-5': 'We now choose the same burst-GRB990123, but different physical regions of the burst: the forward shock region (afterglow) and the reverse shock region (optical flash).', 'astro-ph-9906062-1-33-6': 'These two phenomena took place at different time- a few tens of seconds and [MATH] seconds, respectively.', 'astro-ph-9906062-1-33-7': 'We find that [MATH] and [MATH] are nearly constant during the evolution of the external shock from 50 seconds to [MATH] seconds, directly confirming the correctness of the key hypothesis adopted in the standard afterglow model for the first time.', 'astro-ph-9906062-1-33-8': 'The two values are: [MATH], [MATH].', 'astro-ph-9906062-1-33-9': 'We can see that the value of [MATH] is almost equal to those of GRB970508 and GRB971214 obtained in WG98, suggesting the electron energy density fraction may be a universal constant.', 'astro-ph-9906062-1-33-10': 'However, [MATH] of GRB990123 is much lower than those of GRB970508 and GRB971214.', 'astro-ph-9906062-1-33-11': 'We suggest that the magnetic energy density fraction may depend on the origins of magnetic fields in the shock region, which is different for different GRB energy flow.', 'astro-ph-9906062-1-33-12': 'But for the same burst, it should be a constant during the evolution of the external shock waves.', 'astro-ph-9906062-1-34-0': 'From the afterglow data, we also obtain the value of [MATH], the energy contained in the forward shock.', 'astro-ph-9906062-1-34-1': 'The assumed isotropic energy in [MATH]-rays is [MATH], almost an order of magnitude higher than [MATH].', 'astro-ph-9906062-1-34-2': 'The case of [MATH] has also been found in GRB971214.', 'astro-ph-9906062-1-34-3': 'In WG98, two possible interpretations were presented.', 'astro-ph-9906062-1-34-4': 'One is that there is a rather long radiative phase in GRB971214, causing it to emit more of the initial explosion energy in [MATH]-rays and leaving less for the adiabatic phase.', 'astro-ph-9906062-1-34-5': 'Another interpretation is that the beaming of GRB is stronger in [MATH]-rays than in optical, because the former come from the fastest ejecta.', 'astro-ph-9906062-1-34-6': 'But for GRB990123, we argue that the first interpretation is not plausible, because the transition of the blast wave from the radiative phase to adiabatic one took place very early due to a low [MATH]: [EQUATION]'}

{'astro-ph-9906062-2-0-0': 'We have constrained the intrinsic parameters, such as the magnetic energy density fraction ([MATH]), the electron energy density fraction ([MATH]), the initial Lorentz factor ([MATH]) and the Lorentz factor of the reverse external shock ([MATH]), of GRB990123, in terms of the afterglow information (forward shock model) and the optical flash information (reverse shock model).', 'astro-ph-9906062-2-0-1': 'Our result shows: 1) the inferred values of [MATH] and [MATH] are consistent with the suggestion that they may be universal parameters, comparing to those inferred for GRB970508; 2) the reverse external shock may have become relativistic before it passed through the ejecta shell.', 'astro-ph-9906062-2-0-2': 'Other instrinsic parameters of GRB990123, such as energy contained in the forward shock [MATH] and the ambient density [MATH] are also determined and discussed in this paper.', 'astro-ph-9906062-2-1-0': '# Introduction', 'astro-ph-9906062-2-2-0': 'The current standard model for gamma-ray bursts (GRBs) and their afterglows is the fireball-plus-shock model (see Piran 1999 for a review).', 'astro-ph-9906062-2-2-1': 'It involves that a large amount of energy, [MATH] ergs, is released within a few seconds in a small volume with negligible baryonic load, [MATH].', 'astro-ph-9906062-2-2-2': 'This leads to a fireball that expands ultra-relativistically with a Lorentz factor [MATH] required to avoid the attenuation of hard [MATH]-rays due to pair production (e.g. Woods [MATH] Loeb 1995; Fenimore, Epstein [MATH] Ho 1993).', 'astro-ph-9906062-2-2-3': 'A substantial fraction of the kinetic energy of the baryons is transferred to a non-thermal population of relativistic electrons through Fermi acceleration in the shock ([MATH] Rees 1993).', 'astro-ph-9906062-2-2-4': 'The accelerated electrons cool via synchrotron emission and inverse Compton scattering in the post-shock magnetic fields and produce the radiation observed in GRBs and their afterglows (e.g. Katz 1994; Sari et al. 1996; Vietri 1997; Waxman 1997a; Wijers et al. 1997).', 'astro-ph-9906062-2-2-5': 'The shock could be either [MATH] due to collisions between fireball shells caused by outflow variability (Paczy[MATH]ski [MATH] Xu 1994; Rees [MATH] 1994), or [MATH] due to the interaction of the fireball with the surrounding interstellar medium (ISM; [MATH] Rees 1993).', 'astro-ph-9906062-2-2-6': 'The radiation from internal shocks can explain the spectra (Pilla [MATH] Loeb 1998) and the fast irregular variability of GRBs (Sari [MATH] Piran 1997), while the synchrotron emission from the external shocks provides a successful model for the broken power law spectra and the power law decay of afterglow light curves (e.g. Waxman 1997a,b; Wijers, Rees [MATH] 1997; Vietri 1997; Dai [MATH] Lu 1998a,b,c; Dai et al. 1999; Wang et al. 1999a,b; Huang et al. 1998a,b; 1999a,b).', 'astro-ph-9906062-2-3-0': 'The properties of the synchrotron emission from GRB shocks are determined by the magnetic field strength, [MATH], and the electron energy distribution behind the shock.', 'astro-ph-9906062-2-3-1': 'Both of them are difficult to estimate from first principles, and so the following dimensionless parameters are often used to incorporate modeling uncertainties (Sari et al. 1996), [MATH], [MATH].', 'astro-ph-9906062-2-3-2': 'Here [MATH] and [MATH] are the magnetic and electron energy densities and [MATH] is the total thermal energy density behind the shocks, where [MATH] is the proton mass, [MATH] is the proton number density, and [MATH] is the mean thermal Lorentz factor of the protons.', 'astro-ph-9906062-2-3-3': 'In spite of these uncertainties, an important assumption that [MATH] and [MATH] do not change with time, has been made in the standard external shock model.', 'astro-ph-9906062-2-3-4': 'Through the computation, Wijers [MATH] Galama (1998; hereafter WG99) even suggested that they may be universal parameters, i.e. the same for different bursts.', 'astro-ph-9906062-2-4-0': 'The BeppoSAX satellite ushered in 1999 with the discovery of GRB990123 (Heise et al. 1999), the brightest GRB seen by BeppoSAX to date.', 'astro-ph-9906062-2-4-1': 'This is a very strong burst.', 'astro-ph-9906062-2-4-2': "An assumption of isotropic emission and the detection of the source's redshift [MATH] , lead to a huge energy release about [MATH]ergs (Briggs et al. 1999; Kulkarni et al. 1999a) in [MATH]-rays alone.", 'astro-ph-9906062-2-4-3': 'GRB990123 would have been amongest the most exciting GRBs even just on the basis of these facts.', 'astro-ph-9906062-2-4-4': 'Furthermore, ROTSE discovered a prompt optical flash of 9-th magnitude (Akerlof et al. 1999).', 'astro-ph-9906062-2-4-5': 'It is the first time that a prompt emission in another wavelength apart from [MATH]-rays has been detected from GRB.', 'astro-ph-9906062-2-4-6': 'Such a strong optical flash was predicted to arise from a reverse external shock propagating into the relativistic ejecta ([MATH] Rees 1997; Sari [MATH] Piran 1999a,b, hereafter SP99a,b).', 'astro-ph-9906062-2-4-7': 'This is the so called "early afterglow".', 'astro-ph-9906062-2-4-8': 'The five last exposures of ROTSE show a power law decay with a slope of [MATH]2.0, which can also be explained by the reverse shock model (SP99b; [MATH] Rees 1999).', 'astro-ph-9906062-2-4-9': 'The usual afterglows in X-ray, optical, IR and radio bands were also detected after the burst.', 'astro-ph-9906062-2-4-10': 'They have two distinguishing features: 1) the radio emission is unique both due to its very early appearance and its rapid decline; 2) the temporal decaying behaviour of the [MATH]-band light curve after two days steepens from about [MATH] to [MATH] (Kulkarni et al. 1999a; Fruchter et al. 1999; Casrto-Tirado et al. 1999), and this steepening might be due to a jet which has transited from a spherical-like phase to sideways expansion phase (Rhoads 1999; Sari et al. 1999; Wei [MATH] Lu 1999) or a dense medium which has slowed down the relativistic expansion of shock quickly to a non-relativistic one (Dai [MATH] Lu 1999a,b).', 'astro-ph-9906062-2-5-0': 'Galama et al. (1999) assumed that the radio emission is produced by the forward shock as usual and then reconstructed the radio-to-X-ray afterglow spectrum on January 24.65 UT.', 'astro-ph-9906062-2-5-1': 'However, later work shows that the simplest interpretation of this "radio flare" is that it arises in the reverse shock and that such radio emission is an inevitable consequence of the prompt bright optical flash seen by ROTSE (Kulkarni et al. 1999b; Sari [MATH] Piran 1999b).', 'astro-ph-9906062-2-5-2': 'Kulkarni et al. (1999b) also constrained two key parameters of the forward shock, the peak flux [MATH] and the peak frequency [MATH], to within a factor of two.', 'astro-ph-9906062-2-5-3': 'For previous bursts, we have no other information apart from the afterglow to infer the intrinsic parameters of the external shocks.', 'astro-ph-9906062-2-5-4': 'But now the optical flash of GRB990123 has been fortunately detected, which enables us to determine another two key parameters, the initial Lorentz factor [MATH] and the Lorentz factor of the reverse shock [MATH].', 'astro-ph-9906062-2-5-5': 'WG99 computed the intrinsic parameters of GRB970508 and GRB971214 in terms of their afterglow data, and found that [MATH] is nearly the same for these two bursts, suggesting it may be a universal parameter.', 'astro-ph-9906062-2-5-6': 'Granot, Piran [MATH] Sari (1998a,b; hereafter GPS99a,b) modified the set of equations derived by WG99, and inferred the electron energy density fraction and the magnetic energy density fraction of GRB970508 to be: [MATH]; [MATH].', 'astro-ph-9906062-2-5-7': 'Here we apply the set of equations of GPS99a to GRB990123 and try to determine some intrinsic parameters based on the information of the two aspects of GRB990123-the optical flash and the afterglow.', 'astro-ph-9906062-2-6-0': 'The initial Lorentz factor [MATH] is also an important physical parameter of GRBs.', 'astro-ph-9906062-2-6-1': 'It is a crucial ingredient for constraining models of the source itself, since it specifies how "clean" the fireball is as the baryonic load is [MATH].', 'astro-ph-9906062-2-6-2': 'Unfortunately, the spectrum of GRBs can provide only a lower limit to this Lorentz factor ([MATH]).', 'astro-ph-9906062-2-6-3': 'Moreover, the current afterglow observations, which detect radiation from several hours after the burst, do not provide a verification of the initial extreme relativistic motion.', 'astro-ph-9906062-2-6-4': 'A possible method to estimate [MATH] of GRBs has been suggested by Sari [MATH] Piran (1999a), based on identifying the "early afterglow" peak time.', 'astro-ph-9906062-2-6-5': 'In this paper, the initial Lorentz factor has been inferred more precisely from the full set of equations describing the reverse shock region.', 'astro-ph-9906062-2-7-0': 'In section 2 and 3, we compute the intrinsic parameters of GRB990123 from its afterglow and optical flash information.', 'astro-ph-9906062-2-7-1': 'In the final section, we give our conclusions.', 'astro-ph-9906062-2-8-0': '# Parameters from the afterglow', 'astro-ph-9906062-2-9-0': 'Intrinsic parameters like the magnetic energy density fraction [MATH], electron energy density fraction [MATH], energy in the forward external shock [MATH] ergs and ambient density [MATH] can be determined from the afterglow spectrum (GPS99a,b; WG99), i. e. if we know all three break frequencies (not necessary at the same time) and the peak flux of the afterglow, we can infer all these parameters.', 'astro-ph-9906062-2-10-0': '>From the observations of the afterglow of GRB990123, Kulkarni et al. (1999b) have estimated two key parameters of the forward shock: [MATH] and [MATH] at the time [MATH] after the burst.', 'astro-ph-9906062-2-10-1': 'The cooling frequency [MATH] cannot be seen from the radio-to-X-ray spectrum obtained by Galama et al (1999).', 'astro-ph-9906062-2-10-2': 'This indicates that [MATH] is at or above the X-ray frequencies.', 'astro-ph-9906062-2-10-3': 'We need to determine it more precisely.', 'astro-ph-9906062-2-10-4': 'The X-ray afterglow, observed 6 hours after the burst, decayed with [MATH] ( Heise et al. 1999), while the optical afterglow with [MATH] (Kulkarni et al., 1999).', 'astro-ph-9906062-2-10-5': 'An X-ray afterglow decay slope steeper by [MATH] than an optical decay, which seems to be the case in this burst, is predicted by Sari, Piran and Narayan (1998), if the cooling frequency is between the X-rays and the optical.', 'astro-ph-9906062-2-10-6': 'So at the time 6 hours after the burst , [MATH]Hz.', 'astro-ph-9906062-2-10-7': 'Extrapolating it to the time [MATH] as [MATH], we get [MATH]Hz[MATH]Hz.', 'astro-ph-9906062-2-10-8': 'Another speculative constraint on [MATH] is obtained from the GRB spectrum itself by Sari [MATH] Piran (1999b), who constrained [MATH]Hz at the time [MATH] sec. Extrapolating it to [MATH], we get [MATH].', 'astro-ph-9906062-2-10-9': 'Now [MATH] is almost determined, and we take the approximate value [MATH]Hz, which is in agreement with the estimate of Galama et al (1999).', 'astro-ph-9906062-2-10-10': 'In addition, Kulkarni et al. (1999a) have inferred the electron index [MATH] (defined as [MATH]) to be [MATH].', 'astro-ph-9906062-2-10-11': 'Now, apart from the self-absorption frequency [MATH], we have all other three quantities of the afterglow spectrum required to calculate the intrinsic parameters: [EQUATION] where [MATH] is the redshift of the burst and [MATH] is the luminosity distance.', 'astro-ph-9906062-2-11-0': 'Now we have three equations (2), (3) and (4) with four unknowns: [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9906062-2-11-1': 'To solve these equations, we here assume that the electron density fraction [MATH] is the same for different bursts, just an argument of WG99, though in which a different set of equations are used.', 'astro-ph-9906062-2-11-2': 'Since we here use the formulas of GPS99a, we adopt the value of [MATH] from that of GRB970508 inferred according to the above formulas, i.e. [MATH](GPS99b).', 'astro-ph-9906062-2-12-0': 'By combining Eqs. (1) and (2)-(4), we get the values of four intrinsic parameters of the forward shock region: [EQUATION]', 'astro-ph-9906062-2-12-1': 'We astonishingly find that the value [MATH] inferred is very close to that of GRB970508 ([MATH]; GPS99b), considering the uncertainties in the [MATH] and [MATH] of a factor of two.', 'astro-ph-9906062-2-12-2': 'This result supports our above adoption of the value of [MATH] and implies that [MATH] and [MATH] may be universal parameters (i.e. constants for every GRBs), favouring the argument of WG99.', 'astro-ph-9906062-2-13-0': 'The ambient density [MATH] inferred for GRB990123 is [MATH].', 'astro-ph-9906062-2-13-1': 'This density is on the low side of normal for a disc of galaxy but definitely higher than expected for a halo, lending a further support to the notion that bursts occur in gas-rich environment.', 'astro-ph-9906062-2-13-2': 'The inferred isotropic energy left in the adiabatic forward shock is [MATH]ergs, about thirty times less than the isotropic energy in [MATH]-rays.', 'astro-ph-9906062-2-13-3': 'This case is very similar to GRB971214 (WG99).', 'astro-ph-9906062-2-13-4': 'We think, for GRB990123, there are two processes causing [MATH].', 'astro-ph-9906062-2-13-5': 'One is that there is a radiative evolution phase before the adiabatic phase, causing it to emit more of the initial explosion energy and leaving less for the adiabatic phase.', 'astro-ph-9906062-2-13-6': 'According to Sari, Piran [MATH] Narayan (1998), we can estimate the reduced energy of the forward shock in the self-similar deceleration stage, i.e. [EQUATION] where [MATH] denotes the initial isotropic energy of the forward shock during the self-similar stage and [MATH] denotes the final energy after the radiative phase.', 'astro-ph-9906062-2-13-7': 'The value of [MATH] is difficult to be determined precisely and it may be less than [MATH] for GRB990123, according to Freedman Waxman (2000).', 'astro-ph-9906062-2-13-8': 'If we use [MATH] , [MATH], [MATH] (instead of [MATH], since the latter is less reliable than the former, considering the uncertainties in the [MATH] and [MATH] of a factor of two for GRB990123.)', 'astro-ph-9906062-2-13-9': ', [MATH] (see the next section) and [MATH], we then get [MATH].', 'astro-ph-9906062-2-13-10': 'The other process is the sideways expansion of the fireball jet (Kulkarni et al.1999a,b), which can also reduce the energy per solid angle, hence the isotropic energy [MATH].', 'astro-ph-9906062-2-13-11': 'Since the opening angle of the jet is [MATH] (Sari, Piran [MATH] Halpern 1999; Rhoads 1999), at the time [MATH]days (very near the break time of the jet evolution [MATH]days), [MATH] (note that [MATH]), then [MATH].', 'astro-ph-9906062-2-13-12': 'Therefore, the real isotropic energy [MATH] left in the late adiabatic forward shock should be [MATH] , in rough agreement with the above value inferred from the afterglow spectrum.', 'astro-ph-9906062-2-13-13': 'An additional possible loss of energy may be the reverse shock itself if it is radiative.', 'astro-ph-9906062-2-14-0': '# Parameters from the optical flash', 'astro-ph-9906062-2-15-0': "An optical flash is considered to be produced by the reverse external shock, which heats up the shell's matter and accelerates its electrons (SP99b; M[MATH]sz[MATH]ros [MATH] Rees 1999).", 'astro-ph-9906062-2-15-1': "BATSE's observations triggered ROTSE via BACODINE system (Akerlof et al. 1999).", 'astro-ph-9906062-2-15-2': 'An 11.82 magnitude optical flash was detected on the first 5 seconds exposure, 22.18 seconds after the onset of the burst.', 'astro-ph-9906062-2-15-3': 'Then the optical emission peaked in the following 5 seconds exposure, 25 seconds later, which revealed an 8.95 magnitude signal ([MATH]1Jy).', 'astro-ph-9906062-2-15-4': 'The optical signal decayed to 10.08 magnitude 25 seconds later and continued to decay down to 14.53 magnitude in the subsequent three 75 seconds exposures that took place up to 10 minutes after the burst.', 'astro-ph-9906062-2-15-5': 'The five last exposures depict a power law decay with a slope [MATH] (Akerlof et al. 1999; SP99b).', 'astro-ph-9906062-2-15-6': 'Sari [MATH] Piran (1999b) and M[MATH]sz[MATH]ros [MATH] Rees (1999) assumed that the ejecta shell follows the Blandford-McKee (1976) self-similar solution after the reverse shock has passed through it and explained the decay of [MATH].', 'astro-ph-9906062-2-16-0': 'So we assume that at the optical emission peak time ([MATH] sec) the reverse shock had just passed through the ejecta shell.', 'astro-ph-9906062-2-16-1': 'At this time, the Lorentz factor of the reverse shock [MATH] is approximately given by [EQUATION]', 'astro-ph-9906062-2-16-2': 'The observed flux at [MATH] can be obtained by assuming that all the electrons in the reverse shock region contribute the same average power per unit frequency [MATH] at [MATH], which is given by [MATH], where [MATH].', 'astro-ph-9906062-2-16-3': 'Adding one factor of [MATH] to transform to the observer frame and accounting for the redshift, we have: [EQUATION]', 'astro-ph-9906062-2-16-4': 'Please note that this formula always holds whether the ejecta is jet-like or spherical, because the beaming factor in Eq. (10) and (11) will cancel out each other in the jet-like case.', 'astro-ph-9906062-2-16-5': 'According to the jump condition of the shock, the Lorentz factor of the shocked shell should be approximately equal to that of the shocked ISM (Piran 1999).', 'astro-ph-9906062-2-16-6': 'The Lorentz factor of the forward shocked ISM can be obtained from the standard afterglow model (e. g. Sari, Piran [MATH] Narayan 1998): [EQUATION]', 'astro-ph-9906062-2-16-7': 'Please note that here the value [MATH] inferred from the optical flash data is consistent with that inferred independently from the afterglow information.', 'astro-ph-9906062-2-16-8': 'On the other hand, if we substitute the value [MATH] into Eq. (9), we find that the peak frequency of the reverse shock [MATH] is almost located at the optical band.', 'astro-ph-9906062-2-16-9': 'In addition, our inferred initial Lorentz factor [MATH] is six times larger than that obtained by Sari [MATH] Piran (1999b), who have used the ambient density [MATH] of GRB970508.', 'astro-ph-9906062-2-16-10': 'Consequently, at the time the reverse shock has just passed through the ejecta shell, its Lorentz factor was [MATH].', 'astro-ph-9906062-2-16-11': 'This indicates that the reverse shock had become relativistic before it crossed the entire shell.', 'astro-ph-9906062-2-16-12': 'This result is also different from that obtained by Sari [MATH] Piran (1999b), who found the Lorentz factor of the reverse shock of GRB990123 was only near one.', 'astro-ph-9906062-2-16-13': 'However we argue that our result is reasonable according to the criterion presented by Sari [MATH] Piran (1995) (also see Kobayashi et al. 1998).', 'astro-ph-9906062-2-16-14': 'They defined a dimensionless parameter [MATH] constructed from [MATH], [MATH] and [MATH]: [EQUATION] where [MATH] is the Sedov length, [MATH] is the width of the shell ([MATH] is the duration of GRB) and [MATH] is the initial Lorentz factor of the ejecta.', 'astro-ph-9906062-2-16-15': 'If [MATH], the reverse shock becomes relativistic before it crosses the shell; otherwise ([MATH]), the reverse shock remains Newtonian or at best mildly relativistic during the whole energy extraction process.', 'astro-ph-9906062-2-16-16': 'For GRB990123, we find [MATH].', 'astro-ph-9906062-2-16-17': 'So the reverse shock of GRB990123 had become relativistic before it crossed the shell, consistent with our calculated result.', 'astro-ph-9906062-2-17-0': '# Conclusions and Discussions', 'astro-ph-9906062-2-18-0': 'We have constrained some intrinsic parameters, such as the magnetic energy density fraction ([MATH]), the electron energy density fraction ([MATH]), the isotropic energy in the adiabatic forward shock [MATH] and the ambient density [MATH].', 'astro-ph-9906062-2-18-1': 'Due to the lack of the value of the self-absorption frequency [MATH], we made an assumption that [MATH] of GRB990123 is the same as that of GRB970508, then astonishingly find that the inferred value of [MATH] is also nearly equal to that of GRB970508.', 'astro-ph-9906062-2-18-2': 'This result favours the argument proposed by WG99 that the magnetic energy fraction and the electron density fraction may be universal parameters.', 'astro-ph-9906062-2-19-0': 'Another two important intrinsic parameters of GRB990123 are also inferred from the optical flash information: the initial Lorentz factor [MATH] and the Lorentz factor [MATH] at the prompt optical emission peak time of the ejecta.', 'astro-ph-9906062-2-19-1': 'They are: [MATH], [MATH].', 'astro-ph-9906062-2-19-2': 'Our inferred value of the [MATH] is six times larger than that obtained by Sari [MATH] Piran (1999b), who used the ambient density [MATH] inferred for GRB970508.', 'astro-ph-9906062-2-19-3': 'A larger initial Lorentz factor is reasonable in consideration of the huge energy of this burst.', 'astro-ph-9906062-2-19-4': 'The Lorentz factor of the reverse shock at the optical flash peak time is [MATH], which shows that the reverse shock had become relativistic rather than mildly relativistic before it crossed the entire ejecta shell.', 'astro-ph-9906062-2-19-5': 'This result is in agreement with the criterion presented by Sari [MATH] Piran (1995) to judge the RRS case or NRS case.', 'astro-ph-9906062-2-20-0': 'Prompt optical flash has added another dimension to GRB astronomy.', 'astro-ph-9906062-2-20-1': 'Prompt observations in the optical band during and immediately after GRB may provide more and more events of optical flash in the near future, and they will enable us to make more detailed analyses, make more precise determination of intrinsic parameters and test the reverse-forward external shock model.'}

[['astro-ph-9906062-1-29-0', 'astro-ph-9906062-2-15-0'], ['astro-ph-9906062-1-29-1', 'astro-ph-9906062-2-15-1'], ['astro-ph-9906062-1-29-2', 'astro-ph-9906062-2-15-2'], ['astro-ph-9906062-1-29-3', 'astro-ph-9906062-2-15-3'], ['astro-ph-9906062-1-29-4', 'astro-ph-9906062-2-15-4'], ['astro-ph-9906062-1-29-5', 'astro-ph-9906062-2-15-5'], ['astro-ph-9906062-1-29-6', 'astro-ph-9906062-2-15-6'], ['astro-ph-9906062-1-18-0', 'astro-ph-9906062-2-4-0'], ['astro-ph-9906062-1-18-1', 'astro-ph-9906062-2-4-1'], ['astro-ph-9906062-1-18-4', 'astro-ph-9906062-2-4-3'], ['astro-ph-9906062-1-18-5', 'astro-ph-9906062-2-4-4'], ['astro-ph-9906062-1-18-6', 'astro-ph-9906062-2-4-5'], ['astro-ph-9906062-1-18-8', 'astro-ph-9906062-2-4-7'], ['astro-ph-9906062-1-18-9', 'astro-ph-9906062-2-4-8'], ['astro-ph-9906062-1-18-10', 'astro-ph-9906062-2-4-9'], ['astro-ph-9906062-1-19-4', 'astro-ph-9906062-2-5-3'], ['astro-ph-9906062-1-26-1', 'astro-ph-9906062-2-10-2'], ['astro-ph-9906062-1-26-2', 'astro-ph-9906062-2-10-3'], ['astro-ph-9906062-1-26-3', 'astro-ph-9906062-2-10-4'], ['astro-ph-9906062-1-26-4', 'astro-ph-9906062-2-10-5'], ['astro-ph-9906062-1-26-5', 'astro-ph-9906062-2-10-6'], ['astro-ph-9906062-1-26-6', 'astro-ph-9906062-2-10-7'], ['astro-ph-9906062-1-26-7', 'astro-ph-9906062-2-10-8'], ['astro-ph-9906062-1-16-1', 'astro-ph-9906062-2-2-1'], ['astro-ph-9906062-1-16-2', 'astro-ph-9906062-2-2-2'], ['astro-ph-9906062-1-16-4', 'astro-ph-9906062-2-2-4'], ['astro-ph-9906062-1-16-5', 'astro-ph-9906062-2-2-5'], ['astro-ph-9906062-1-17-0', 'astro-ph-9906062-2-3-0'], ['astro-ph-9906062-1-17-1', 'astro-ph-9906062-2-3-1'], ['astro-ph-9906062-1-17-2', 'astro-ph-9906062-2-3-2'], ['astro-ph-9906062-1-17-3', 'astro-ph-9906062-2-3-3'], ['astro-ph-9906062-1-20-0', 'astro-ph-9906062-2-6-0'], ['astro-ph-9906062-1-20-1', 'astro-ph-9906062-2-6-1'], ['astro-ph-9906062-1-20-2', 'astro-ph-9906062-2-6-2'], ['astro-ph-9906062-1-20-3', 'astro-ph-9906062-2-6-3'], ['astro-ph-9906062-1-20-4', 'astro-ph-9906062-2-6-4'], ['astro-ph-9906062-1-20-5', 'astro-ph-9906062-2-6-5'], ['astro-ph-9906062-1-30-1', 'astro-ph-9906062-2-16-1'], ['astro-ph-9906062-1-31-2', 'astro-ph-9906062-2-16-10'], ['astro-ph-9906062-1-31-5', 'astro-ph-9906062-2-16-13'], ['astro-ph-9906062-1-31-6', 'astro-ph-9906062-2-16-14'], ['astro-ph-9906062-1-31-7', 'astro-ph-9906062-2-16-15'], ['astro-ph-9906062-1-18-3', 'astro-ph-9906062-2-4-2'], ['astro-ph-9906062-1-18-7', 'astro-ph-9906062-2-4-6'], ['astro-ph-9906062-1-18-11', 'astro-ph-9906062-2-4-10'], ['astro-ph-9906062-1-24-0', 'astro-ph-9906062-2-9-0'], ['astro-ph-9906062-1-19-6', 'astro-ph-9906062-2-5-5'], ['astro-ph-9906062-1-16-0', 'astro-ph-9906062-2-2-0'], ['astro-ph-9906062-1-16-3', 'astro-ph-9906062-2-2-3'], ['astro-ph-9906062-1-16-6', 'astro-ph-9906062-2-2-6'], ['astro-ph-9906062-1-30-0', 'astro-ph-9906062-2-16-0'], ['astro-ph-9906062-1-30-2', 'astro-ph-9906062-2-16-4'], ['astro-ph-9906062-1-30-6', 'astro-ph-9906062-2-16-6'], ['astro-ph-9906062-1-31-3', 'astro-ph-9906062-2-16-11'], ['astro-ph-9906062-1-31-4', 'astro-ph-9906062-2-16-12'], ['astro-ph-9906062-1-31-9', 'astro-ph-9906062-2-16-17'], ['astro-ph-9906062-1-21-0', 'astro-ph-9906062-2-7-0'], ['astro-ph-9906062-1-21-1', 'astro-ph-9906062-2-7-1'], ['astro-ph-9906062-1-19-0', 'astro-ph-9906062-2-5-0'], ['astro-ph-9906062-1-19-5', 'astro-ph-9906062-2-5-4'], ['astro-ph-9906062-1-19-5', 'astro-ph-9906062-2-5-7'], ['astro-ph-9906062-1-19-7', 'astro-ph-9906062-2-5-7'], ['astro-ph-9906062-1-26-0', 'astro-ph-9906062-2-10-1'], ['astro-ph-9906062-1-26-8', 'astro-ph-9906062-2-10-9'], ['astro-ph-9906062-1-26-9', 'astro-ph-9906062-2-10-9'], ['astro-ph-9906062-1-26-10', 'astro-ph-9906062-2-10-11'], ['astro-ph-9906062-1-30-4', 'astro-ph-9906062-2-16-5'], ['astro-ph-9906062-1-30-5', 'astro-ph-9906062-2-16-5'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-19-2'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-19-3'], ['astro-ph-9906062-1-31-2', 'astro-ph-9906062-2-19-4'], ['astro-ph-9906062-1-31-3', 'astro-ph-9906062-2-19-4'], ['astro-ph-9906062-1-31-5', 'astro-ph-9906062-2-19-5'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-16-9'], ['astro-ph-9906062-1-13-0', 'astro-ph-9906062-2-0-0'], ['astro-ph-9906062-1-13-4', 'astro-ph-9906062-2-0-2'], ['astro-ph-9906062-1-34-4', 'astro-ph-9906062-2-13-5']]

[['astro-ph-9906062-1-29-0', 'astro-ph-9906062-2-15-0'], ['astro-ph-9906062-1-29-1', 'astro-ph-9906062-2-15-1'], ['astro-ph-9906062-1-29-2', 'astro-ph-9906062-2-15-2'], ['astro-ph-9906062-1-29-3', 'astro-ph-9906062-2-15-3'], ['astro-ph-9906062-1-29-4', 'astro-ph-9906062-2-15-4'], ['astro-ph-9906062-1-29-5', 'astro-ph-9906062-2-15-5'], ['astro-ph-9906062-1-29-6', 'astro-ph-9906062-2-15-6'], ['astro-ph-9906062-1-18-0', 'astro-ph-9906062-2-4-0'], ['astro-ph-9906062-1-18-1', 'astro-ph-9906062-2-4-1'], ['astro-ph-9906062-1-18-4', 'astro-ph-9906062-2-4-3'], ['astro-ph-9906062-1-18-5', 'astro-ph-9906062-2-4-4'], ['astro-ph-9906062-1-18-6', 'astro-ph-9906062-2-4-5'], ['astro-ph-9906062-1-18-8', 'astro-ph-9906062-2-4-7'], ['astro-ph-9906062-1-18-9', 'astro-ph-9906062-2-4-8'], ['astro-ph-9906062-1-18-10', 'astro-ph-9906062-2-4-9'], ['astro-ph-9906062-1-19-4', 'astro-ph-9906062-2-5-3'], ['astro-ph-9906062-1-26-1', 'astro-ph-9906062-2-10-2'], ['astro-ph-9906062-1-26-2', 'astro-ph-9906062-2-10-3'], ['astro-ph-9906062-1-26-3', 'astro-ph-9906062-2-10-4'], ['astro-ph-9906062-1-26-4', 'astro-ph-9906062-2-10-5'], ['astro-ph-9906062-1-26-5', 'astro-ph-9906062-2-10-6'], ['astro-ph-9906062-1-26-6', 'astro-ph-9906062-2-10-7'], ['astro-ph-9906062-1-26-7', 'astro-ph-9906062-2-10-8'], ['astro-ph-9906062-1-16-1', 'astro-ph-9906062-2-2-1'], ['astro-ph-9906062-1-16-2', 'astro-ph-9906062-2-2-2'], ['astro-ph-9906062-1-16-4', 'astro-ph-9906062-2-2-4'], ['astro-ph-9906062-1-16-5', 'astro-ph-9906062-2-2-5'], ['astro-ph-9906062-1-17-0', 'astro-ph-9906062-2-3-0'], ['astro-ph-9906062-1-17-1', 'astro-ph-9906062-2-3-1'], ['astro-ph-9906062-1-17-2', 'astro-ph-9906062-2-3-2'], ['astro-ph-9906062-1-17-3', 'astro-ph-9906062-2-3-3'], ['astro-ph-9906062-1-20-0', 'astro-ph-9906062-2-6-0'], ['astro-ph-9906062-1-20-1', 'astro-ph-9906062-2-6-1'], ['astro-ph-9906062-1-20-2', 'astro-ph-9906062-2-6-2'], ['astro-ph-9906062-1-20-3', 'astro-ph-9906062-2-6-3'], ['astro-ph-9906062-1-20-4', 'astro-ph-9906062-2-6-4'], ['astro-ph-9906062-1-20-5', 'astro-ph-9906062-2-6-5'], ['astro-ph-9906062-1-30-1', 'astro-ph-9906062-2-16-1'], ['astro-ph-9906062-1-31-2', 'astro-ph-9906062-2-16-10'], ['astro-ph-9906062-1-31-5', 'astro-ph-9906062-2-16-13'], ['astro-ph-9906062-1-31-6', 'astro-ph-9906062-2-16-14'], ['astro-ph-9906062-1-31-7', 'astro-ph-9906062-2-16-15']]

[['astro-ph-9906062-1-18-3', 'astro-ph-9906062-2-4-2'], ['astro-ph-9906062-1-18-7', 'astro-ph-9906062-2-4-6'], ['astro-ph-9906062-1-18-11', 'astro-ph-9906062-2-4-10'], ['astro-ph-9906062-1-24-0', 'astro-ph-9906062-2-9-0'], ['astro-ph-9906062-1-19-6', 'astro-ph-9906062-2-5-5'], ['astro-ph-9906062-1-16-0', 'astro-ph-9906062-2-2-0'], ['astro-ph-9906062-1-16-3', 'astro-ph-9906062-2-2-3'], ['astro-ph-9906062-1-16-6', 'astro-ph-9906062-2-2-6'], ['astro-ph-9906062-1-30-0', 'astro-ph-9906062-2-16-0'], ['astro-ph-9906062-1-30-2', 'astro-ph-9906062-2-16-4'], ['astro-ph-9906062-1-30-6', 'astro-ph-9906062-2-16-6'], ['astro-ph-9906062-1-31-3', 'astro-ph-9906062-2-16-11'], ['astro-ph-9906062-1-31-4', 'astro-ph-9906062-2-16-12'], ['astro-ph-9906062-1-31-9', 'astro-ph-9906062-2-16-17']]

[]

[['astro-ph-9906062-1-21-0', 'astro-ph-9906062-2-7-0'], ['astro-ph-9906062-1-21-1', 'astro-ph-9906062-2-7-1'], ['astro-ph-9906062-1-19-0', 'astro-ph-9906062-2-5-0'], ['astro-ph-9906062-1-19-5', 'astro-ph-9906062-2-5-4'], ['astro-ph-9906062-1-19-5', 'astro-ph-9906062-2-5-7'], ['astro-ph-9906062-1-19-7', 'astro-ph-9906062-2-5-7'], ['astro-ph-9906062-1-26-0', 'astro-ph-9906062-2-10-1'], ['astro-ph-9906062-1-26-8', 'astro-ph-9906062-2-10-9'], ['astro-ph-9906062-1-26-9', 'astro-ph-9906062-2-10-9'], ['astro-ph-9906062-1-26-10', 'astro-ph-9906062-2-10-11'], ['astro-ph-9906062-1-30-4', 'astro-ph-9906062-2-16-5'], ['astro-ph-9906062-1-30-5', 'astro-ph-9906062-2-16-5'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-19-2'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-19-3'], ['astro-ph-9906062-1-31-2', 'astro-ph-9906062-2-19-4'], ['astro-ph-9906062-1-31-3', 'astro-ph-9906062-2-19-4'], ['astro-ph-9906062-1-31-5', 'astro-ph-9906062-2-19-5'], ['astro-ph-9906062-1-31-1', 'astro-ph-9906062-2-16-9']]

[['astro-ph-9906062-1-13-0', 'astro-ph-9906062-2-0-0'], ['astro-ph-9906062-1-13-4', 'astro-ph-9906062-2-0-2'], ['astro-ph-9906062-1-34-4', 'astro-ph-9906062-2-13-5']]

['astro-ph-9906062-1-1-0', 'astro-ph-9906062-1-2-0', 'astro-ph-9906062-1-4-0', 'astro-ph-9906062-1-5-0', 'astro-ph-9906062-1-6-0', 'astro-ph-9906062-1-7-0', 'astro-ph-9906062-1-8-0', 'astro-ph-9906062-1-11-0', 'astro-ph-9906062-1-12-0', 'astro-ph-9906062-1-14-0', 'astro-ph-9906062-1-31-8', 'astro-ph-9906062-1-33-10', 'astro-ph-9906062-2-13-8', 'astro-ph-9906062-2-13-9', 'astro-ph-9906062-2-16-16', 'astro-ph-9906062-2-19-1']

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

https://arxiv.org/abs/astro-ph/9906062

1510.00219

{'1510.00219-1-0-0': 'We propose a method to detect lower bounds to quantum capacities of a noisy quantum communication channel by means of few measurements.', '1510.00219-1-0-1': 'The method is easily implementable and does not require any knowledge about the channel.', '1510.00219-1-0-2': 'We test its efficiency by studying its performance for most well known single qubit noisy channels and for the generalised Pauli channel in arbitrary finite dimension.', '1510.00219-1-1-0': 'Noise is unavoidably present in any communication channel.', '1510.00219-1-1-1': 'In this case the ability of the channel to convey information is lower than in the ideal noiseless case and it can be quantified in terms of channel capacities.', '1510.00219-1-1-2': 'Depending on the task to be performed and on the resources available, several kinds of capacities can be defined.', '1510.00219-1-1-3': 'The ability of the channel to convey classical information is quantified in terms of the classical channel capacity [MATH] [CITATION], defined as the maximum number of classical bits that can be reliably transmitted per channel use.', '1510.00219-1-1-4': 'If the sender and the receiver share unlimited prior entanglement, the capacity of transmitting classical information is quantified in terms of the entanglement-assisted classical capacity [MATH] [CITATION].', '1510.00219-1-1-5': 'The ability of the channel to convey classical information privately is quantified in terms of the private channel capacity [MATH], defined as the maximum number of classical bits that can be reliably transmitted per channel use in such a way that negligible information can be obtained by a third party [CITATION].', '1510.00219-1-1-6': 'The ability of the channel to convey quantum information is quantified in terms of the quantum capacity [MATH] [CITATION], defined as the maximum number of qubits that can be reliably transmitted per channel use.', '1510.00219-1-1-7': 'In many practical situations a complete knowledge of the kind of noise present along the channel is not available, and sometimes noise can be completely unknown.', '1510.00219-1-1-8': 'It is then important to develop efficient means to establish whether in these situations the channel can still be profitably employed for information transmission.', '1510.00219-1-1-9': 'A standard method to establish this relies on quantum process tomography, where a complete reconstruction of the CP map describing the action of the channel can be achieved, and therefore all its communication properties can be estimated.', '1510.00219-1-1-10': 'This, however, is a demanding procedure in terms of the number of different measurement settings needed, since it scales as [MATH] for a finite [MATH]-dimensional quantum system.', '1510.00219-1-1-11': 'In this Letter we address the situation where we want to gain some information on the channel ability to transmit quantum information by employing a smaller number of measurements, that scales as [MATH].', '1510.00219-1-1-12': 'We derive a lower bound on the channel capacities that can be experimentally accessed with a simple procedure and can be applied to an unknown quantum communication channel.', '1510.00219-1-1-13': 'The efficiency of the method is then studied for many examples of single qubit channels, and for the generalised Pauli channel in arbitrary finite dimension.', '1510.00219-1-2-0': 'In the following we focus on memoryless channels.', '1510.00219-1-2-1': 'We denote the action of a generic quantum channel on a single system as [MATH] and define [MATH], where [MATH] represents the number of channel uses.', '1510.00219-1-2-2': 'The quantum capacity [MATH] measured in qubits per channel use is defined as [CITATION] [EQUATION] where [MATH], and [MATH] denotes the coherent information [CITATION] [EQUATION]', '1510.00219-1-2-3': 'In Eq. ([REF]), [MATH] is the von Neumann entropy, and [MATH] represents the entropy exchange [CITATION], i.e. [MATH], where [MATH] is any purification of [MATH] by means of a reference quantum system [MATH], namely [MATH].', '1510.00219-1-3-0': 'We will now derive a lower bound for the quantum capacity [MATH] that can be easily accessed without requiring full process tomography of the quantum channel.', '1510.00219-1-3-1': 'Since for any complete set of orthogonal projectors [MATH] one has [CITATION] [MATH], then for any orthonormal basis [MATH] in the tensor product of the reference and the system Hilbert spaces one has the following bound to the entropy exchange [EQUATION] where [MATH] denotes the Shannon entropy for the vector of the probabilities [MATH], with [EQUATION]', '1510.00219-1-3-2': 'We will also use the notation [MATH] for the binary Shannon entropy, i.e. [MATH].', '1510.00219-1-4-0': 'From Eq. ([REF]) it follows that for any [MATH] and [MATH] one has the following chain of bounds [EQUATION]', '1510.00219-1-4-1': 'A lower bound [MATH] to the quantum capacity of an unknown channel can then be detected by the following prescription: prepare a bipartite pure state [MATH] and send it through the channel [MATH], where the unknown channel [MATH] acts on one of the two subsystems.', '1510.00219-1-4-2': 'Then measure suitable local observables on the joint output state to estimate [MATH] and [MATH] in order to compute [MATH].', '1510.00219-1-4-3': 'Notice that for a fixed measurement setting, one can infer different vectors of probabilities pertaining to different sets of orthogonal projectors, as will be clarified in the following.', '1510.00219-1-4-4': 'In principle, one can even adopt an adaptive detection scheme to improve the bound ([REF]) by varying the input state [MATH].', '1510.00219-1-5-0': 'We will now consider a fixed input maximally entangled state [MATH], where [MATH] is the finite dimension of each subsystem, and a Bell basis [EQUATION] with [MATH] unitary and [MATH].', '1510.00219-1-5-1': 'The detectable bound takes the form [EQUATION] with [MATH], where [MATH] denotes the Kraus operators of the channel [MATH].', '1510.00219-1-6-0': 'For qubit channels, the probability vector for the Bell basis can be estimated by measuring locally the observables [MATH], [MATH], and [MATH] on the system and on the reference qubits, thus allowing to detect the bound of Eq. ([REF]).', '1510.00219-1-6-1': 'Indeed, by denoting the Bell states as [EQUATION] it can be straightforwardly proved that such a local measurement setting allows to estimate the vector [MATH] pertaining to the projectors onto the following inequivalent bases [EQUATION] with [MATH] real and such that [MATH].', '1510.00219-1-7-0': 'Therefore, in order to obtain the tightest bound in ([REF]) given the fixed local measurements [MATH], the Shannon entropy [MATH] will be minimised as a function of the bases ([REF]-[REF]), by varying the coefficients [MATH] over the three sets.', '1510.00219-1-7-1': 'This optimisation step corresponds to classical processing of the measurement outcomes of [MATH].', '1510.00219-1-8-0': 'This procedure can be generalised for arbitrary finite dimension [MATH].', '1510.00219-1-8-1': 'The bound of Eq. ([REF]) in this case can be detected by measuring [MATH] observables via a local setting and classical processing of the measurement outcomes.', '1510.00219-1-8-2': 'Actually, a set of generalised Bell projectors can be written as follows [CITATION] [EQUATION] where [MATH], and [MATH] represents the unitary operator [EQUATION]', '1510.00219-1-8-3': 'Hence, a set of measurements on the eigenstates of [MATH] allows to estimate [MATH] in Eq. ([REF]).', '1510.00219-1-9-0': 'As mentioned above, the advantage of this procedure is to require [MATH] measurement settings with respect to a complete process tomography, where [MATH] observables have to be measured.', '1510.00219-1-10-0': 'We want to point out that all detectable bounds we are providing also give lower bounds to the private information [MATH], since [MATH] [CITATION].', '1510.00219-1-10-1': 'Moreover, we can also derive a detectable lower bound to the entanglement-assisted classical capacity.', '1510.00219-1-10-2': 'Actually, this is defined as [MATH], where [MATH].', '1510.00219-1-10-3': 'By considering the procedure outlined above, where a maximally entangled state [MATH] is considered as input, we then have the lower bound [MATH].', '1510.00219-1-10-4': 'In the following we will analyse explicit examples of noisy quantum channels, that are typically encountered in practical implementations, and analyse in detail the efficiency of the proposed procedure.', '1510.00219-1-10-5': 'We will consider in particular the most well known channels for qubits, i.e. the dephasing, the depolarising and the more general Pauli channel, the erasure and the amplitude damping channel.', '1510.00219-1-10-6': 'The Pauli channels are also generalised to arbitrary dimension.', '1510.00219-1-10-7': 'We finally consider a family of qubit channels with two Kraus operators.', '1510.00219-1-11-0': 'A dephasing channel for qubits with unknown probability [MATH] can be written as [MATH].', '1510.00219-1-11-1': 'Since it is a degradable channel, its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [MATH].', '1510.00219-1-11-2': 'More generally, we can consider the channel [MATH], for any dimension [MATH], with [MATH] as unitary and traceless operator.', '1510.00219-1-11-3': 'The von Neumann entropy of the output state [MATH] is given by [MATH].', '1510.00219-1-11-4': 'Using the Bell basis ([REF]) one obtains the detectable bound [EQUATION]', '1510.00219-1-11-5': 'This detectable quantum capacity clearly coincides with the quantum channel capacity for [MATH].', '1510.00219-1-12-0': 'The depolarising channel with probability [MATH] for qubits is given by [CITATION] [MATH].', '1510.00219-1-12-1': 'The quantum capacity is still unknown, although one has the upper bound [CITATION] [MATH], thus showing that [MATH] for [MATH].', '1510.00219-1-12-2': 'On the other hand, by random coding the following hashing bound [CITATION] has been proved [EQUATION]', '1510.00219-1-12-3': 'This lower bound coincides with our detectable bound of Eq. ([REF]).', '1510.00219-1-12-4': 'In Fig. [REF] we plot the lower bound ([REF]), along with the upper bound [MATH], versus the probability [MATH].', '1510.00219-1-12-5': 'As we can see, our procedure allows to detect [MATH] as long as [MATH].', '1510.00219-1-13-0': 'In arbitrary dimension [MATH] the depolarising channel takes the form [EQUATION] and the detectable bound is simply generalised to [EQUATION] that can indeed be detected by estimating [MATH] pertaining to the Bell projectors ([REF]).', '1510.00219-1-14-0': 'Similarly to the depolarising channel, for a generic Pauli channel [MATH] the hashing bound [CITATION] provides a lower bound to the quantum capacity [MATH], which coincides with our detectable bound ([REF]) by using a maximally entangled input state and estimating [MATH] for the Bell basis.', '1510.00219-1-14-1': 'In dimension [MATH] one can consider the generalised channel [MATH], in terms of the unitary operators ([REF]), and one has [EQUATION] [MATH] being now the [MATH]-dimensional vector of probabilities [MATH] pertaining to the generalised Bell projectors in Eq. ([REF]).', '1510.00219-1-15-0': 'We consider now an erasure channel [CITATION] with erasure probability [MATH], defined as [EQUATION] where [MATH] denotes the erasure flag which is orthogonal to the system Hilbert space.', '1510.00219-1-15-1': 'Since it is a degradable channel, its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [CITATION] [EQUATION] for [MATH], and [MATH] for [MATH].', '1510.00219-1-16-0': 'The output of any maximally entangled state [MATH] can be written as [EQUATION]', '1510.00219-1-16-1': 'A basis constructed by the union of the projectors on [MATH] (with [MATH]) and Bell projectors (where one of them corresponds to [MATH]) gives a vector of probability ([REF]) with [MATH] elements equal to [MATH] and one element [MATH], while all other terms are vanishing.', '1510.00219-1-16-2': 'We then have [MATH].', '1510.00219-1-16-3': 'The von Neumann entropy of the reduced output state [MATH] is given by [MATH].', '1510.00219-1-16-4': 'It then follows that the detectable bound [MATH] for the erasure channel coincides with [MATH] in Eq. ([REF]).', '1510.00219-1-17-0': 'The amplitude damping channel for qubits has the form [CITATION] [EQUATION] where [MATH] and [MATH].', '1510.00219-1-17-1': 'Since it is a degradable channel [CITATION], its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [EQUATION] for [MATH], and [MATH] for [MATH].', '1510.00219-1-18-0': 'For an input Bell state [MATH] the output is given by [EQUATION]', '1510.00219-1-18-1': 'The reduced output state is given by [MATH], hence it has von Neumann entropy [MATH].', '1510.00219-1-18-2': 'By performing the local measurement of [MATH], [MATH], and [MATH] and optimising [MATH] over the bases ([REF]-[REF]), one can detect the bound [EQUATION] where the optimal vector of probabilities is given by [MATH], and it corresponds to the basis in Eq. ([REF]), with [MATH], [MATH], and [MATH].', '1510.00219-1-18-3': 'This basis is clearly made of projectors on the eigenstates of the output state ([REF]).', '1510.00219-1-18-4': 'It turns out that, as long as [MATH], a non-vanishing quantum capacity is detected.', '1510.00219-1-18-5': 'Indeed the difference [MATH] never exceeds [MATH].', '1510.00219-1-18-6': 'We notice that the Bell basis ([REF]) does not provide the minimum value of [MATH].', '1510.00219-1-18-7': 'Actually, in such case one has [EQUATION] and by using this value of [MATH] a non-vanishing quantum capacity is detected only for [MATH].', '1510.00219-1-18-8': 'In Fig. [REF] we plot the detectable bound from Eq. ([REF]) [which is indistinguishable from the quantum capacity ([REF])], along with the bound obtained by the probability vector ([REF]) pertaining to the Bell projectors, versus the damping parameter [MATH].', '1510.00219-1-18-9': 'The difference of the curves shows how the optimisation of [MATH] over the bases ([REF]-[REF]) is crucial to achieve the optimal bound.', '1510.00219-1-19-0': 'We finally consider the following set of channels, characterised by just two Kraus operators [CITATION], [EQUATION] with [MATH], and [EQUATION]', '1510.00219-1-19-1': 'These channels represent the normal form of equivalence classes, since two channels have the same capacity if they differ merely by unitaries acting on input and output.', '1510.00219-1-19-2': 'Notice that for [MATH] the channel is dephasing, and for [MATH] it is amplitude damping.', '1510.00219-1-19-3': 'These channels are shown to be degradable [CITATION] for [MATH], hence [MATH].', '1510.00219-1-19-4': 'On the other hand, they are antidegradable for [MATH], thus with [MATH].', '1510.00219-1-20-0': 'The coherent information is maximised by diagonal input states and in the region [MATH] the quantum capacity is given by [CITATION] [EQUATION]', '1510.00219-1-20-1': 'Let us consider a detection scheme with the maximally entangled input state [MATH].', '1510.00219-1-20-2': 'The output state can be shown to be diagonal on the basis in Eq. ([REF]), with [EQUATION] and eigenvalues [MATH].', '1510.00219-1-20-3': 'The optimal vector of probabilities [MATH] for the detectable bound [MATH] corresponds to these eigenvalues.', '1510.00219-1-20-4': 'The output entropy of the reduced state is then given by [MATH], hence our detectable quantum capacity can be written as [EQUATION]', '1510.00219-1-20-5': 'We checked numerically that [MATH] for all values of [MATH] and [MATH].', '1510.00219-1-20-6': 'The positive region of the detected capacity [MATH] is plotted in Fig. [REF].', '1510.00219-1-21-0': 'In conclusion, we have proposed a method to detect lower bounds to capacities of quantum communication channels, specifically to the quantum capacity, the entanglement assisted capacity, and the private capacity.', '1510.00219-1-21-1': 'The procedure does not require any a priori knowledge about the quantum channel, which can be also completely unknown, and relies on a number of measurement settings that scales as [MATH].', '1510.00219-1-21-2': 'It is therefore much cheaper than full process tomography and it can be easily accessed in the lab without posing any particular restriction on the nature of the physical system under consideration.', '1510.00219-1-21-3': 'In particular, for quantum optical systems it is easily implementable with present-day technologies [CITATION].', '1510.00219-1-21-4': 'We tested the method for significant qubit channels and it turned out to give extremely good results for various forms of noise.', '1510.00219-1-21-5': 'The method can be successfully applied also to correlated Pauli and amplitude damping channels acting on two qubits [CITATION].', '1510.00219-1-22-0': "We thank Antonio D'Arrigo for valuable suggestions."}

{'1510.00219-2-0-0': 'We propose a method to detect lower bounds to quantum capacities of a noisy quantum communication channel by means of few measurements.', '1510.00219-2-0-1': 'The method is easily implementable and does not require any knowledge about the channel.', '1510.00219-2-0-2': 'We test its efficiency by studying its performance for most well known single qubit noisy channels and for the generalised Pauli channel in arbitrary finite dimension.', '1510.00219-2-1-0': 'Noise is unavoidably present in any communication channel.', '1510.00219-2-1-1': 'In this case the ability of the channel to convey information is lower than in the ideal noiseless case and it can be quantified in terms of channel capacities.', '1510.00219-2-1-2': 'Depending on the task to be performed and on the resources available, several kinds of capacities can be defined.', '1510.00219-2-1-3': 'The ability of the channel to convey classical information is quantified in terms of the classical channel capacity [MATH] [CITATION], defined as the maximum number of classical bits that can be reliably transmitted per channel use.', '1510.00219-2-1-4': 'If the sender and the receiver share unlimited prior entanglement, the capacity of transmitting classical information is quantified in terms of the entanglement-assisted classical capacity [MATH] [CITATION].', '1510.00219-2-1-5': 'The ability of the channel to convey classical information privately is quantified in terms of the private channel capacity [MATH], defined as the maximum number of classical bits that can be reliably transmitted per channel use in such a way that negligible information can be obtained by a third party [CITATION].', '1510.00219-2-1-6': 'The ability of the channel to convey quantum information is quantified in terms of the quantum capacity [MATH] [CITATION], defined as the maximum number of qubits that can be reliably transmitted per channel use.', '1510.00219-2-1-7': 'In many practical situations a complete knowledge of the kind of noise present along the channel is not available, and sometimes noise can be completely unknown.', '1510.00219-2-1-8': 'It is then important to develop efficient means to establish whether in these situations the channel can still be profitably employed for information transmission.', '1510.00219-2-1-9': 'A standard method to establish this relies on quantum process tomography, where a complete reconstruction of the CP map describing the action of the channel can be achieved, and therefore all its communication properties can be estimated.', '1510.00219-2-1-10': 'This, however, is a demanding procedure in terms of the number of different measurement settings needed, since it scales as [MATH] for a finite [MATH]-dimensional quantum system.', '1510.00219-2-1-11': 'In this Letter we address the situation where we want to gain some information on the channel ability to transmit quantum information by employing a smaller number of measurements, that scales as [MATH].', '1510.00219-2-1-12': 'We derive a lower bound on the channel capacities that can be experimentally accessed with a simple procedure and can be applied to an unknown quantum communication channel.', '1510.00219-2-1-13': 'The efficiency of the method is then studied for many examples of single qubit channels, and for the generalised Pauli channel in arbitrary finite dimension.', '1510.00219-2-2-0': 'In the following we focus on memoryless channels.', '1510.00219-2-2-1': 'We denote the action of a generic quantum channel on a single system as [MATH] and define [MATH], where [MATH] represents the number of channel uses.', '1510.00219-2-2-2': 'The quantum capacity [MATH] measured in qubits per channel use is defined as [CITATION] [EQUATION] where [MATH], and [MATH] denotes the coherent information [CITATION] [EQUATION]', '1510.00219-2-2-3': 'In Eq. ([REF]), [MATH] is the von Neumann entropy, and [MATH] represents the entropy exchange [CITATION], i.e. [MATH], where [MATH] is any purification of [MATH] by means of a reference quantum system [MATH], namely [MATH].', '1510.00219-2-3-0': 'We will now derive a lower bound for the quantum capacity [MATH] that can be easily accessed without requiring full process tomography of the quantum channel.', '1510.00219-2-3-1': 'Since for any complete set of orthogonal projectors [MATH] one has [CITATION] [MATH], then for any orthonormal basis [MATH] in the tensor product of the reference and the system Hilbert spaces one has the following bound to the entropy exchange [EQUATION] where [MATH] denotes the Shannon entropy for the vector of the probabilities [MATH], with [EQUATION]', '1510.00219-2-3-2': 'From Eq. ([REF]) it follows that for any [MATH] and [MATH] one has the following chain of bounds [EQUATION]', '1510.00219-2-3-3': 'A lower bound [MATH] to the quantum capacity of an unknown channel can then be detected by the following prescription: prepare a bipartite pure state [MATH] and send it through the channel [MATH], where the unknown channel [MATH] acts on one of the two subsystems.', '1510.00219-2-3-4': 'Then measure suitable local observables on the joint output state to estimate [MATH] and [MATH] in order to compute [MATH].', '1510.00219-2-3-5': 'Notice that for a fixed measurement setting, one can infer different vectors of probabilities pertaining to different sets of orthogonal projectors, as will be clarified in the following.', '1510.00219-2-3-6': 'In principle, one can even adopt an adaptive detection scheme to improve the bound ([REF]) by varying the input state [MATH].', '1510.00219-2-4-0': 'We will now be more specific and consider first the case of qubit channels.', '1510.00219-2-4-1': 'We assume that only the local observables [MATH], [MATH], and [MATH] on the system and on the reference qubits are measured, and we want to optimise the bound [MATH] given these resources.', '1510.00219-2-4-2': 'First, we notice that the above measurements allow to measure [MATH], [MATH] and [MATH] on the system qubit alone, by ignoring the statistics of the measurement results on the reference qubit.', '1510.00219-2-4-3': 'In this way, a complete tomography of the system output state can be performed, and therefore the term [MATH] in Eq. ([REF]) can be estimated exactly.', '1510.00219-2-4-4': 'Furthermore, by denoting the Bell states as [EQUATION] it can be straightforwardly proved that the local measurement settings [MATH] allow to estimate the vector [MATH] pertaining to the projectors onto the following inequivalent bases [EQUATION] with [MATH] real and such that [MATH].', '1510.00219-2-4-5': 'Actually, the measurements corresponding to the above three bases are achieved by orthogonal projectors of the form [EQUATION] where [MATH] denotes the projector onto the state [MATH], and analogously for the other projectors.', '1510.00219-2-4-6': 'The probability vector [MATH] for each choice of basis is then evaluated according to Eq. ([REF]).', '1510.00219-2-4-7': 'The expectation values for terms of the form [MATH] (or [MATH]) can be measured from the outcomes of the observable [MATH] by ignoring the measurement results on the second (or first) qubit, and analogously for the other similar terms in the above projectors.', '1510.00219-2-5-0': 'Therefore, in order to obtain the tightest bound in ([REF]) given the fixed local measurements [MATH], the Shannon entropy [MATH] will be minimised as a function of the bases ([REF]-[REF]), by varying the coefficients [MATH] over the three sets.', '1510.00219-2-5-1': 'In an experimental scenario, after collecting the outcomes of the measurements [MATH], this optimisation step corresponds to classical processing of the measurement outcomes.', '1510.00219-2-6-0': 'Our procedure can be generalised for arbitrary finite dimension [MATH].', '1510.00219-2-6-1': 'For simplicity, we will now consider a fixed input maximally entangled state [MATH], where [MATH] is the finite dimension of each subsystem, and a Bell basis [EQUATION] with [MATH] unitary and [MATH].', '1510.00219-2-6-2': 'The detectable bound takes the form [EQUATION] with [MATH], where [MATH] denotes the Kraus operators of the channel [MATH].', '1510.00219-2-6-3': 'The bound of Eq. ([REF]) in this case can be detected by measuring [MATH] observables via a local setting and classical processing of the measurement outcomes.', '1510.00219-2-6-4': 'Actually, a set of generalised Bell projectors can be written as follows [CITATION] [EQUATION] where [MATH], and [MATH] represents the unitary operator [MATH].', '1510.00219-2-6-5': 'Hence, a set of measurements on the eigenstates of [MATH] allows to estimate [MATH] in Eq. ([REF]).', '1510.00219-2-7-0': 'As mentioned above, the advantage of this procedure is to require [MATH] measurement settings with respect to a complete process tomography, where [MATH] observables have to be measured.', '1510.00219-2-8-0': 'We want to point out that all detectable bounds we are providing also give lower bounds to the private information [MATH], since [MATH] [CITATION].', '1510.00219-2-8-1': 'Moreover, we can also derive a detectable lower bound to the entanglement-assisted classical capacity.', '1510.00219-2-8-2': 'Actually, this is defined as [MATH], where [MATH].', '1510.00219-2-8-3': 'By considering the procedure outlined above, where a maximally entangled state [MATH] is considered as input, we then have the lower bound [MATH].', '1510.00219-2-8-4': 'In the following we will analyse explicit examples of noisy quantum channels that are typically encountered in practical implementations, and analyse in detail the efficiency of the proposed procedure.', '1510.00219-2-8-5': 'We will consider in particular the most well known channels for qubits, i.e. the dephasing, the depolarising and the more general Pauli channel, the erasure and the amplitude damping channel.', '1510.00219-2-8-6': 'The Pauli channels are also generalised to arbitrary dimension.', '1510.00219-2-8-7': 'We finally consider a family of qubit channels with two Kraus operators.', '1510.00219-2-8-8': 'We will always consider input states with maximal entanglement between system and reference.', '1510.00219-2-9-0': 'A dephasing channel for qubits with unknown probability [MATH] can be written as [MATH].', '1510.00219-2-9-1': 'Since it is a degradable channel, its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [MATH], where [MATH] is the binary Shannon entropy.', '1510.00219-2-9-2': 'More generally, we can consider the channel [MATH], for any dimension [MATH], with [MATH] as unitary and traceless operator.', '1510.00219-2-9-3': 'The von Neumann entropy of the output state [MATH] is given by [MATH].', '1510.00219-2-9-4': 'Using the Bell basis ([REF]) one obtains the detectable bound [EQUATION]', '1510.00219-2-9-5': 'This detectable quantum capacity clearly coincides with the quantum channel capacity for [MATH].', '1510.00219-2-10-0': 'The depolarising channel with probability [MATH] for qubits is given by [CITATION] [MATH].', '1510.00219-2-10-1': 'The quantum capacity is still unknown, although one has the upper bound [CITATION] [MATH], thus showing that [MATH] for [MATH].', '1510.00219-2-10-2': 'On the other hand, by random coding the following hashing bound [CITATION] has been proved [EQUATION]', '1510.00219-2-10-3': 'This lower bound coincides with our detectable bound of Eq. ([REF]).', '1510.00219-2-10-4': 'In Fig. [REF] we plot the lower bound ([REF]), along with the upper bound [MATH], versus the probability [MATH].', '1510.00219-2-10-5': 'As we can see, our procedure allows to detect [MATH] as long as [MATH].', '1510.00219-2-11-0': 'In arbitrary dimension [MATH] the depolarising channel takes the form [EQUATION]', '1510.00219-2-11-1': 'Hence, the detectable bound is simply generalised to [EQUATION] and can be detected by estimating [MATH] pertaining to the Bell projectors ([REF]).', '1510.00219-2-12-0': 'Similarly to the depolarising channel, for a generic Pauli channel [MATH] the hashing bound [CITATION] provides a lower bound to the quantum capacity [MATH], which coincides with our detectable bound ([REF]) by using a maximally entangled input state and estimating [MATH] for the Bell basis.', '1510.00219-2-12-1': 'In dimension [MATH] one can consider the generalised channel [MATH] and one has [EQUATION] [MATH] being now the [MATH]-dimensional vector of probabilities [MATH] pertaining to the generalised Bell projectors in Eq. ([REF]).', '1510.00219-2-13-0': 'We consider now an erasure channel [CITATION] with erasure probability [MATH], defined as [EQUATION] where [MATH] denotes the erasure flag which is orthogonal to the system Hilbert space.', '1510.00219-2-13-1': 'Since it is a degradable channel, its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [CITATION] [EQUATION] for [MATH], and [MATH] for [MATH].', '1510.00219-2-13-2': 'The output of any maximally entangled state [MATH] can be written as [EQUATION]', '1510.00219-2-13-3': 'A basis constructed by the union of the projectors on [MATH] (with [MATH]) and Bell projectors (where one of them corresponds to [MATH]) gives a vector of probability ([REF]) with [MATH] elements equal to [MATH] and one element [MATH], while all other terms are vanishing.', '1510.00219-2-13-4': 'We then have [MATH].', '1510.00219-2-13-5': 'The von Neumann entropy of the reduced output state [MATH] is given by [MATH].', '1510.00219-2-13-6': 'It then follows that the detectable bound [MATH] for the erasure channel coincides with [MATH] in Eq. ([REF]).', '1510.00219-2-14-0': 'The amplitude damping channel for qubits has the form [CITATION] [EQUATION] where [MATH] and [MATH].', '1510.00219-2-14-1': 'Since it is a degradable channel [CITATION], its quantum capacity coincides with the one-shot single-letter quantum capacity [MATH], and one has [EQUATION] for [MATH], and [MATH] for [MATH].', '1510.00219-2-14-2': 'For an input Bell state [MATH] the output is given by [EQUATION]', '1510.00219-2-14-3': 'The reduced output state is given by [MATH], hence it has von Neumann entropy [MATH].', '1510.00219-2-14-4': 'By performing the local measurement of [MATH], [MATH], and [MATH] and optimising [MATH] over the bases ([REF]-[REF]), one can detect the bound [EQUATION] where the optimal vector of probabilities is given by [MATH], and it corresponds to the basis in Eq. ([REF]), with [MATH], [MATH], and [MATH].', '1510.00219-2-14-5': 'This basis is clearly made of projectors on the eigenstates of the output state ([REF]).', '1510.00219-2-14-6': 'It turns out that, as long as [MATH], a non-vanishing quantum capacity is detected.', '1510.00219-2-14-7': 'Indeed the difference [MATH] never exceeds [MATH].', '1510.00219-2-14-8': 'We notice that the Bell basis ([REF]) does not provide the minimum value of [MATH].', '1510.00219-2-14-9': 'Actually, in such case one has [EQUATION] and by using this value of [MATH] a non-vanishing quantum capacity is detected only for [MATH].', '1510.00219-2-14-10': 'In Fig. [REF] we plot the detectable bound from Eq. ([REF]) [which is indistinguishable from the quantum capacity ([REF])], along with the bound obtained by the probability vector ([REF]) pertaining to the Bell projectors, versus the damping parameter [MATH].', '1510.00219-2-14-11': 'The difference of the curves shows how the optimisation of [MATH] over the bases ([REF]-[REF]) is crucial to achieve the optimal bound.', '1510.00219-2-15-0': 'We finally consider the following set of channels, characterised by just two Kraus operators [CITATION], [EQUATION] where [MATH] and [MATH], with [MATH].', '1510.00219-2-15-1': 'Details on these channels are given in the Supplemental Material [CITATION].', '1510.00219-2-15-2': 'Our detectable quantum capacity can be written as [EQUATION]', '1510.00219-2-15-3': 'We checked numerically that [MATH] for all values of [MATH] and [MATH].', '1510.00219-2-15-4': 'The positive region of the detected capacity [MATH] is plotted in Fig. [REF].', '1510.00219-2-16-0': 'In conclusion, we have proposed a method to detect lower bounds to capacities of quantum communication channels, specifically to the quantum capacity, the entanglement assisted capacity, and the private capacity.', '1510.00219-2-16-1': 'The procedure does not require any a priori knowledge about the quantum channel and relies on a number of measurement settings that scales as [MATH].', '1510.00219-2-16-2': 'It is therefore much cheaper than full process tomography and it can be easily accessed in the lab without posing any particular restriction on the nature of the physical system under consideration.', '1510.00219-2-16-3': 'In particular, for quantum optical systems it is easily implementable with present-day technologies [CITATION].', '1510.00219-2-16-4': 'We tested the method for significant qubit channels and it turned out to give extremely good results for various forms of noise.', '1510.00219-2-16-5': 'The method can be successfully applied also to correlated Pauli and amplitude damping channels acting on two qubits [CITATION].', '1510.00219-2-17-0': "We thank Antonio D'Arrigo for valuable suggestions.", '1510.00219-2-18-0': '*', '1510.00219-2-19-0': '# Appendix: Supplemental material', '1510.00219-2-20-0': 'The following set of channels [EQUATION] where [MATH] and [MATH], with [MATH].', '1510.00219-2-20-1': 'is characterised by just two Kraus operators [CITATION], and represents the normal form of equivalence classes, since two channels have the same capacity if they differ merely by unitaries acting on input and output.', '1510.00219-2-20-2': 'Notice that for [MATH] the channel is dephasing, and for [MATH] it is amplitude damping.', '1510.00219-2-20-3': 'These channels are shown to be degradable [CITATION] for [MATH], hence [MATH].', '1510.00219-2-20-4': 'On the other hand, they are antidegradable for [MATH], thus with [MATH].', '1510.00219-2-21-0': 'The coherent information is maximised by diagonal input states, and in the region [MATH] the quantum capacity is given by [CITATION] [EQUATION]', '1510.00219-2-21-1': 'For a detection scheme with the maximally entangled input state [MATH], the output state can be shown to be diagonal on the basis in Eq. ([REF]), with [EQUATION] and eigenvalues [MATH].', '1510.00219-2-21-2': 'The optimal vector of probabilities [MATH] for the detectable bound [MATH] corresponds to these eigenvalues.', '1510.00219-2-21-3': 'The output entropy of the reduced state is then given by [MATH], hence the detectable quantum capacity can be written as [EQUATION]'}

[['1510.00219-1-10-0', '1510.00219-2-8-0'], ['1510.00219-1-10-1', '1510.00219-2-8-1'], ['1510.00219-1-10-2', '1510.00219-2-8-2'], ['1510.00219-1-10-3', '1510.00219-2-8-3'], ['1510.00219-1-10-5', '1510.00219-2-8-5'], ['1510.00219-1-10-6', '1510.00219-2-8-6'], ['1510.00219-1-10-7', '1510.00219-2-8-7'], ['1510.00219-1-12-0', '1510.00219-2-10-0'], ['1510.00219-1-12-1', '1510.00219-2-10-1'], ['1510.00219-1-12-2', '1510.00219-2-10-2'], ['1510.00219-1-12-3', '1510.00219-2-10-3'], ['1510.00219-1-12-4', '1510.00219-2-10-4'], ['1510.00219-1-12-5', '1510.00219-2-10-5'], ['1510.00219-1-7-0', '1510.00219-2-5-0'], ['1510.00219-1-14-0', '1510.00219-2-12-0'], ['1510.00219-1-0-0', '1510.00219-2-0-0'], ['1510.00219-1-0-1', '1510.00219-2-0-1'], ['1510.00219-1-0-2', '1510.00219-2-0-2'], ['1510.00219-1-21-0', '1510.00219-2-16-0'], ['1510.00219-1-21-2', '1510.00219-2-16-2'], ['1510.00219-1-21-3', '1510.00219-2-16-3'], ['1510.00219-1-21-4', '1510.00219-2-16-4'], ['1510.00219-1-21-5', '1510.00219-2-16-5'], ['1510.00219-1-1-0', '1510.00219-2-1-0'], ['1510.00219-1-1-1', '1510.00219-2-1-1'], ['1510.00219-1-1-2', '1510.00219-2-1-2'], ['1510.00219-1-1-3', '1510.00219-2-1-3'], ['1510.00219-1-1-4', '1510.00219-2-1-4'], ['1510.00219-1-1-5', '1510.00219-2-1-5'], ['1510.00219-1-1-6', '1510.00219-2-1-6'], ['1510.00219-1-1-7', '1510.00219-2-1-7'], ['1510.00219-1-1-8', '1510.00219-2-1-8'], ['1510.00219-1-1-9', '1510.00219-2-1-9'], ['1510.00219-1-1-10', '1510.00219-2-1-10'], ['1510.00219-1-1-11', '1510.00219-2-1-11'], ['1510.00219-1-1-12', '1510.00219-2-1-12'], ['1510.00219-1-1-13', '1510.00219-2-1-13'], ['1510.00219-1-2-0', '1510.00219-2-2-0'], ['1510.00219-1-2-1', '1510.00219-2-2-1'], ['1510.00219-1-2-2', '1510.00219-2-2-2'], ['1510.00219-1-2-3', '1510.00219-2-2-3'], ['1510.00219-1-19-2', '1510.00219-2-20-2'], ['1510.00219-1-19-3', '1510.00219-2-20-3'], ['1510.00219-1-19-4', '1510.00219-2-20-4'], ['1510.00219-1-9-0', '1510.00219-2-7-0'], ['1510.00219-1-11-0', '1510.00219-2-9-0'], ['1510.00219-1-11-2', '1510.00219-2-9-2'], ['1510.00219-1-11-3', '1510.00219-2-9-3'], ['1510.00219-1-11-4', '1510.00219-2-9-4'], ['1510.00219-1-11-5', '1510.00219-2-9-5'], ['1510.00219-1-20-3', '1510.00219-2-21-2'], ['1510.00219-1-20-5', '1510.00219-2-15-3'], ['1510.00219-1-20-6', '1510.00219-2-15-4'], ['1510.00219-1-17-0', '1510.00219-2-14-0'], ['1510.00219-1-17-1', '1510.00219-2-14-1'], ['1510.00219-1-18-0', '1510.00219-2-14-2'], ['1510.00219-1-18-1', '1510.00219-2-14-3'], ['1510.00219-1-18-2', '1510.00219-2-14-4'], ['1510.00219-1-18-3', '1510.00219-2-14-5'], ['1510.00219-1-18-4', '1510.00219-2-14-6'], ['1510.00219-1-18-5', '1510.00219-2-14-7'], ['1510.00219-1-18-6', '1510.00219-2-14-8'], ['1510.00219-1-18-7', '1510.00219-2-14-9'], ['1510.00219-1-18-8', '1510.00219-2-14-10'], ['1510.00219-1-18-9', '1510.00219-2-14-11'], ['1510.00219-1-5-1', '1510.00219-2-6-2'], ['1510.00219-1-8-1', '1510.00219-2-6-3'], ['1510.00219-1-8-3', '1510.00219-2-6-5'], ['1510.00219-1-15-0', '1510.00219-2-13-0'], ['1510.00219-1-15-1', '1510.00219-2-13-1'], ['1510.00219-1-16-0', '1510.00219-2-13-2'], ['1510.00219-1-16-1', '1510.00219-2-13-3'], ['1510.00219-1-16-2', '1510.00219-2-13-4'], ['1510.00219-1-16-3', '1510.00219-2-13-5'], ['1510.00219-1-16-4', '1510.00219-2-13-6'], ['1510.00219-1-3-0', '1510.00219-2-3-0'], ['1510.00219-1-3-1', '1510.00219-2-3-1'], ['1510.00219-1-4-0', '1510.00219-2-3-2'], ['1510.00219-1-4-1', '1510.00219-2-3-3'], ['1510.00219-1-4-2', '1510.00219-2-3-4'], ['1510.00219-1-4-3', '1510.00219-2-3-5'], ['1510.00219-1-4-4', '1510.00219-2-3-6'], ['1510.00219-1-10-4', '1510.00219-2-8-4'], ['1510.00219-1-14-1', '1510.00219-2-12-1'], ['1510.00219-1-21-1', '1510.00219-2-16-1'], ['1510.00219-1-19-1', '1510.00219-2-20-1'], ['1510.00219-1-6-1', '1510.00219-2-4-4'], ['1510.00219-1-11-1', '1510.00219-2-9-1'], ['1510.00219-1-20-0', '1510.00219-2-21-0'], ['1510.00219-1-20-4', '1510.00219-2-21-3'], ['1510.00219-1-5-0', '1510.00219-2-6-1'], ['1510.00219-1-8-0', '1510.00219-2-6-0'], ['1510.00219-1-8-2', '1510.00219-2-6-4'], ['1510.00219-1-7-1', '1510.00219-2-5-1'], ['1510.00219-1-13-0', '1510.00219-2-11-0'], ['1510.00219-1-13-0', '1510.00219-2-11-1'], ['1510.00219-1-20-1', '1510.00219-2-21-1'], ['1510.00219-1-20-2', '1510.00219-2-21-1']]

[['1510.00219-1-10-0', '1510.00219-2-8-0'], ['1510.00219-1-10-1', '1510.00219-2-8-1'], ['1510.00219-1-10-2', '1510.00219-2-8-2'], ['1510.00219-1-10-3', '1510.00219-2-8-3'], ['1510.00219-1-10-5', '1510.00219-2-8-5'], ['1510.00219-1-10-6', '1510.00219-2-8-6'], ['1510.00219-1-10-7', '1510.00219-2-8-7'], ['1510.00219-1-12-0', '1510.00219-2-10-0'], ['1510.00219-1-12-1', '1510.00219-2-10-1'], ['1510.00219-1-12-2', '1510.00219-2-10-2'], ['1510.00219-1-12-3', '1510.00219-2-10-3'], ['1510.00219-1-12-4', '1510.00219-2-10-4'], ['1510.00219-1-12-5', '1510.00219-2-10-5'], ['1510.00219-1-7-0', '1510.00219-2-5-0'], ['1510.00219-1-14-0', '1510.00219-2-12-0'], ['1510.00219-1-0-0', '1510.00219-2-0-0'], ['1510.00219-1-0-1', '1510.00219-2-0-1'], ['1510.00219-1-0-2', '1510.00219-2-0-2'], ['1510.00219-1-21-0', '1510.00219-2-16-0'], ['1510.00219-1-21-2', '1510.00219-2-16-2'], ['1510.00219-1-21-3', '1510.00219-2-16-3'], ['1510.00219-1-21-4', '1510.00219-2-16-4'], ['1510.00219-1-21-5', '1510.00219-2-16-5'], ['1510.00219-1-1-0', '1510.00219-2-1-0'], ['1510.00219-1-1-1', '1510.00219-2-1-1'], ['1510.00219-1-1-2', '1510.00219-2-1-2'], ['1510.00219-1-1-3', '1510.00219-2-1-3'], ['1510.00219-1-1-4', '1510.00219-2-1-4'], ['1510.00219-1-1-5', '1510.00219-2-1-5'], ['1510.00219-1-1-6', '1510.00219-2-1-6'], ['1510.00219-1-1-7', '1510.00219-2-1-7'], ['1510.00219-1-1-8', '1510.00219-2-1-8'], ['1510.00219-1-1-9', '1510.00219-2-1-9'], ['1510.00219-1-1-10', '1510.00219-2-1-10'], ['1510.00219-1-1-11', '1510.00219-2-1-11'], ['1510.00219-1-1-12', '1510.00219-2-1-12'], ['1510.00219-1-1-13', '1510.00219-2-1-13'], ['1510.00219-1-2-0', '1510.00219-2-2-0'], ['1510.00219-1-2-1', '1510.00219-2-2-1'], ['1510.00219-1-2-2', '1510.00219-2-2-2'], ['1510.00219-1-2-3', '1510.00219-2-2-3'], ['1510.00219-1-19-2', '1510.00219-2-20-2'], ['1510.00219-1-19-3', '1510.00219-2-20-3'], ['1510.00219-1-19-4', '1510.00219-2-20-4'], ['1510.00219-1-9-0', '1510.00219-2-7-0'], ['1510.00219-1-11-0', '1510.00219-2-9-0'], ['1510.00219-1-11-2', '1510.00219-2-9-2'], ['1510.00219-1-11-3', '1510.00219-2-9-3'], ['1510.00219-1-11-4', '1510.00219-2-9-4'], ['1510.00219-1-11-5', '1510.00219-2-9-5'], ['1510.00219-1-20-3', '1510.00219-2-21-2'], ['1510.00219-1-20-5', '1510.00219-2-15-3'], ['1510.00219-1-20-6', '1510.00219-2-15-4'], ['1510.00219-1-17-0', '1510.00219-2-14-0'], ['1510.00219-1-17-1', '1510.00219-2-14-1'], ['1510.00219-1-18-0', '1510.00219-2-14-2'], ['1510.00219-1-18-1', '1510.00219-2-14-3'], ['1510.00219-1-18-2', '1510.00219-2-14-4'], ['1510.00219-1-18-3', '1510.00219-2-14-5'], ['1510.00219-1-18-4', '1510.00219-2-14-6'], ['1510.00219-1-18-5', '1510.00219-2-14-7'], ['1510.00219-1-18-6', '1510.00219-2-14-8'], ['1510.00219-1-18-7', '1510.00219-2-14-9'], ['1510.00219-1-18-8', '1510.00219-2-14-10'], ['1510.00219-1-18-9', '1510.00219-2-14-11'], ['1510.00219-1-5-1', '1510.00219-2-6-2'], ['1510.00219-1-8-1', '1510.00219-2-6-3'], ['1510.00219-1-8-3', '1510.00219-2-6-5'], ['1510.00219-1-15-0', '1510.00219-2-13-0'], ['1510.00219-1-15-1', '1510.00219-2-13-1'], ['1510.00219-1-16-0', '1510.00219-2-13-2'], ['1510.00219-1-16-1', '1510.00219-2-13-3'], ['1510.00219-1-16-2', '1510.00219-2-13-4'], ['1510.00219-1-16-3', '1510.00219-2-13-5'], ['1510.00219-1-16-4', '1510.00219-2-13-6'], ['1510.00219-1-3-0', '1510.00219-2-3-0'], ['1510.00219-1-3-1', '1510.00219-2-3-1'], ['1510.00219-1-4-0', '1510.00219-2-3-2'], ['1510.00219-1-4-1', '1510.00219-2-3-3'], ['1510.00219-1-4-2', '1510.00219-2-3-4'], ['1510.00219-1-4-3', '1510.00219-2-3-5'], ['1510.00219-1-4-4', '1510.00219-2-3-6']]

[['1510.00219-1-10-4', '1510.00219-2-8-4'], ['1510.00219-1-14-1', '1510.00219-2-12-1'], ['1510.00219-1-21-1', '1510.00219-2-16-1'], ['1510.00219-1-19-1', '1510.00219-2-20-1'], ['1510.00219-1-6-1', '1510.00219-2-4-4'], ['1510.00219-1-11-1', '1510.00219-2-9-1'], ['1510.00219-1-20-0', '1510.00219-2-21-0'], ['1510.00219-1-20-4', '1510.00219-2-21-3'], ['1510.00219-1-5-0', '1510.00219-2-6-1'], ['1510.00219-1-8-0', '1510.00219-2-6-0'], ['1510.00219-1-8-2', '1510.00219-2-6-4']]

[]

[['1510.00219-1-7-1', '1510.00219-2-5-1'], ['1510.00219-1-13-0', '1510.00219-2-11-0'], ['1510.00219-1-13-0', '1510.00219-2-11-1'], ['1510.00219-1-20-1', '1510.00219-2-21-1'], ['1510.00219-1-20-2', '1510.00219-2-21-1']]

[]

['1510.00219-1-22-0', '1510.00219-2-17-0', '1510.00219-2-18-0']

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

https://arxiv.org/abs/1510.00219

1802.09113

{'1802.09113-1-0-0': 'First order methods, which solely rely on gradient information, are commonly used in diverse machine learning (ML) and data analysis (DA) applications.', '1802.09113-1-0-1': 'This is attributed to the simplicity of their implementations, as well as low per-iteration computational/storage costs.', '1802.09113-1-0-2': 'However, they suffer from significant disadvantages; most notably, their performance degrades with increasing problem ill-conditioning.', '1802.09113-1-0-3': 'Furthermore, they often involve a large number of hyper-parameters, and are notoriously sensitive to parameters such as the step-size.', '1802.09113-1-0-4': 'By incorporating additional information from the Hessian, second-order methods, have been shown to be resilient to many such adversarial effects.', '1802.09113-1-0-5': 'However, these advantages of using curvature information come at the cost of higher per-iteration costs, which in big data regimes, can be computationally prohibitive.', '1802.09113-1-1-0': 'In this paper, we show that, contrary to conventional belief, second-order methods, when implemented appropriately, can be more efficient than first-order alternatives in many large-scale ML/ DA applications.', '1802.09113-1-1-1': "In particular, in convex settings, we consider variants of classical Newton's method in which the Hessian and/or the gradient are randomly sub-sampled.", '1802.09113-1-1-2': 'We show that by effectively leveraging the power of GPUs, such randomized Newton-type algorithms can be significantly accelerated, and can easily outperform state of the art implementations of existing techniques in popular ML/ DA software packages such as TensorFlow.', '1802.09113-1-1-3': 'Additionally these randomized methods incur a small memory overhead compared to first-order methods.', '1802.09113-1-1-4': "In particular, we show that for million-dimensional problems, our GPU accelerated sub-sampled Newton's method achieves a higher test accuracy in milliseconds as compared with tens of seconds for first order alternatives.", '1802.09113-1-2-0': '# Introduction', '1802.09113-1-3-0': 'Optimization techniques are at the core of many ML/DA applications.', '1802.09113-1-3-1': 'First-order methods that rely solely on gradient of the objective function, have been methods of choice in these applications.', '1802.09113-1-3-2': 'The scale of commonly encountered problems in typical applications necessitates optimization techniques that are fast, i.e., have low per-iteration cost and require few overall iterations, as well as robust to adversarial effects such as problem ill-conditioning and hyper-parameter tuning.', '1802.09113-1-3-3': 'First-order methods such as stochastic gradient descent (SGD) are widely known to have low per-iteration costs.', '1802.09113-1-3-4': 'However, they often require many iterations before suitable results are obtained, and their performance can deteriorate for moderately to ill-conditioned problems.', '1802.09113-1-3-5': 'Contrary to popular belief, ill-conditioned problems often arise in machine learning applications.', '1802.09113-1-3-6': 'For example, the "vanishing and exploding gradient problem" encountered in training deep neural nets [CITATION], is a well-known and important issue.', '1802.09113-1-3-7': 'What is less known is that this is a consequence of the highly ill-conditioned nature of the problem.', '1802.09113-1-3-8': 'Other examples include low-rank matrix approximation and spectral clustering involving radial basis function (RBF) kernels when the scale parameter is large [CITATION].', '1802.09113-1-3-9': 'A subtle, yet potentially more serious, disadvantage of most first-order methods is the large number of hyper-parameters, as well as their high sensitivity to parameter-tuning, which can significantly slow down the training procedure and often necessitate many trial and error steps [CITATION].', '1802.09113-1-4-0': 'Newton-type methods use curvature information in the form of the Hessian matrix, in addition to the to gradient.', '1802.09113-1-4-1': 'This family of methods has not been commonly used in the ML/ DA community because of their high per-iteration costs, in spite of the fact that second-order methods offer a range of benefits.', '1802.09113-1-4-2': 'Unlike first-order methods, Newton-type methods have been shown to be highly resilient to increasing problem ill-conditioning [CITATION].', '1802.09113-1-4-3': 'Furthermore, second-order methods typically require fewer parameters (e.g., inexactness tolerance for the sub-problem solver or line-search parameters), and are less sensitive to their specific settings [CITATION].', '1802.09113-1-4-4': 'By incorporating curvature information at each iteration, Newton-type methods scale the gradient such that it is a more suitable direction to follow.', '1802.09113-1-4-5': 'Consequently, although their iterations may be more expensive than those of the first-order counterparts, second-order methods typically require much fewer iterations.', '1802.09113-1-5-0': 'In this context, by reducing the cost of each iteration through efficient approximation of curvature, coupled with hardware specific acceleration, one can obtain methods that are fast and robust.', '1802.09113-1-5-1': 'In most ML applications, this typically translates to achieving a high test-accuracy early on in the iterative process and without significant parameter tuning; see Section [REF].', '1802.09113-1-5-2': 'This is in sharp contrast with slow-ramping trends typically observed in training with first-order methods, which is often preceded by a lengthy trial and error procedure for parameter tuning.', '1802.09113-1-5-3': 'Indeed, the aforementioned properties, coupled with efficiency obtained from algorithmic innovations and implementations that effectively utilize all available hardware resources, hold promise for significantly changing the landscape of optimization techniques used in ML/DA applications.', '1802.09113-1-6-0': 'With the long-term goal of achieving this paradigm shift, we focus on the commonly encountered finite-sum optimization problem', '1802.09113-1-7-0': '[EQUATION] where each [MATH] is a smooth convex function, representing a loss (or misfit) corresponding to [MATH] observation (or measurement) [CITATION].', '1802.09113-1-7-1': 'In many ML applications, [MATH] in eq. [REF] corresponds to the empirical risk [CITATION], and the goal of solving eq. [REF] is to obtain a solution with small generalization error, i.e., high predictive accuracy on "unseen" data.', '1802.09113-1-7-2': 'We consider eq. [REF] at scale, where the values of [MATH] and [MATH] are large - millions and beyond.', '1802.09113-1-7-3': 'In such settings, the mere computation of the Hessian and the gradient of [MATH] increases linearly in [MATH].', '1802.09113-1-7-4': 'Indeed, for large-scale problems, operations on the Hessian, e.g., matrix-vector products involved in the (approximate) solution of the sub-problems of most Newton-type methods, typically constitute the main computational bottleneck.', '1802.09113-1-7-5': 'In such cases, randomized sub-sampling has been shown to be highly successful in reducing computational and memory costs to be effectively independent of [MATH].', '1802.09113-1-7-6': "For example, a simple instance of eq. [REF] is when the functions [MATH]'s are quadratics, in which case one has an over-constrained least squares problem.", '1802.09113-1-7-7': 'For these problems, randomized numerical linear algebra (RandNLA) techniques rely on random sampling, which is used to compute a data-aware or data-oblivious subspace embedding that preserves the geometry of the entire subspace [CITATION].', '1802.09113-1-7-8': 'Furthermore, non-trivial practical implementations of algorithms based on these ideas have been shown to beat state-of-the-art numerical techniques [CITATION].', '1802.09113-1-7-9': 'For more general problems, theoretical properties of sub-sampled Newton-type methods, for both convex and non-convex problems of the form in eq. [REF], have been recently studied in a series of efforts [CITATION].', '1802.09113-1-7-10': 'However, for real ML/ DA applications beyond least squares, practical and hardware-specific implementations that can effectively draw upon all available computing resources, are lacking.', '1802.09113-1-8-0': 'Contributions: Our contributions in this paper can be summarized as follows:', '1802.09113-1-9-0': "Through a judicious mix of statistical techniques, algorithmic innovations, and highly optimized GPU implementations, we develop an accelerated variant of the classical Newton's method that has low per-iteration cost, fast convergence, and minimal memory overhead.", '1802.09113-1-9-1': 'In the process, we show that, for solving eq. [REF], our accelerated randomized method significantly outperforms state of the art implementations of existing techniques in popular ML/DA software packages such as TensorFlow [CITATION], in terms of improved training time, generalization error, and robustness to various adversarial effects.', '1802.09113-1-10-0': 'This paper is organized as follows.', '1802.09113-1-10-1': 'Section [REF] provides an overview of related literature.', '1802.09113-1-10-2': 'Section [REF] presents technical background regarding sub-sampled Newton-type methods, Softmax classifier as a practical instance of eq. [REF], along with a description of the algorithms and their implementation.', '1802.09113-1-10-3': 'Section [REF] compares and contrasts GPU based implementations of sub-sampled Newton-type methods with first order methods available in TensorFlow.', '1802.09113-1-10-4': 'Conclusions and avenues for future work are presented in Section [REF].', '1802.09113-1-11-0': '# Related Work', '1802.09113-1-12-0': 'The class of first-order methods includes a number of techniques that are commonly used in diverse ML/DA applications.', '1802.09113-1-12-1': 'Many of these techniques have been efficiently implemented in popular software packages.', '1802.09113-1-12-2': 'For example, TensorFlow, [CITATION], has enjoyed considerable success among ML practitioners.', '1802.09113-1-12-3': 'Among first-order methods implemented in TensorFlow for solving [REF] are Adagrad [CITATION], RMSProp [CITATION], Adam [CITATION], Adadelta [CITATION], and SGD with/ without momentum [CITATION].', '1802.09113-1-12-4': 'Excluding SGD, the rest of these methods are adaptive, in that they incorporate prior gradients to choose a preconditioner at each gradient step.', '1802.09113-1-12-5': 'Through the use of gradient history from previous iterations, these adaptive methods non-uniformly scale the current gradient to obtain an update direction that takes larger steps along the coordinates with smaller derivatives and, conversely, smaller steps along those with larger derivatives.', '1802.09113-1-12-6': "At a high level, these methods aim to capture non-uniform scaling of Newton's method, albeit, using limited curvature information.", '1802.09113-1-13-0': 'Theoretical properties of a variety of randomized Newton-type methods, for both convex and non-convex problems of the form eq. [REF], have been recently studied in a series of results, both in the context of ML applications [CITATION], as well as scientific computing applications [CITATION].', '1802.09113-1-14-0': 'GPUs have been successfully used in a variety of ML applications to speed up computations [CITATION].', '1802.09113-1-14-1': 'In particular, Raina et al. [CITATION] demonstrate that modern GPUs can far surpass the computational capabilities of multi-core CPUs, and have the potential to address many of the computational challenges encountered in training large-scale learning models.', '1802.09113-1-14-2': 'Most relevant to this paper, Ngiam et al. [CITATION] show that off-the-shelf optimization methods such as Limited memory BFGS (L-BFGS) and Conjugate Gradient (CG), have the potential to outperform variants of SGD in deep learning applications.', '1802.09113-1-14-3': 'It was further demonstrated that the difference in performance between LBFGS/CG and SGD is more pronounced if one considers hardware accelerators such as GPUs.', '1802.09113-1-14-4': "Extending similar results to full-fledged second-order algorithms, such Newton's method, is a major motivating factor for our work here.", '1802.09113-1-15-0': '# Theory, Algorithms and Implementation Details', '1802.09113-1-16-0': '## Notation', '1802.09113-1-17-0': 'Vectors, [MATH], and matrices, [MATH], are denoted by bold lower and upper case letters, respectively.', '1802.09113-1-17-1': '[MATH] and [MATH] represent the gradient and the Hessian of [MATH] at [MATH], respectively.', '1802.09113-1-17-2': 'The superscript, e.g., [MATH], denotes iteration count.', '1802.09113-1-17-3': '[MATH] denotes a collection of indices drawn from the set [MATH], with potentially repeated items, and its cardinality is denoted by [MATH].', '1802.09113-1-17-4': 'Following Matlab notation, [MATH] denotes vertical stacking of two column vectors [MATH], whereas [MATH] denotes a [MATH] matrix whose columns are formed from the vectors [MATH] and [MATH].', '1802.09113-1-17-5': 'Vector [MATH] norm is denoted by [MATH].', '1802.09113-1-17-6': 'For a boolean variable, [MATH], the indicator function [MATH] evaluates to one if [MATH], and zero otherwise.', '1802.09113-1-17-7': '[MATH] denotes the dot product of vectors [MATH] and [MATH], and [MATH] represents element-wise multiplication of matrices [MATH] and [MATH].', '1802.09113-1-18-0': "## Sub-Sampled Newton's Method", '1802.09113-1-19-0': 'For the optimization problem eq. [REF], in each iteration, consider selecting two sample sets of indices from [MATH], uniformly at random with or without replacement.', '1802.09113-1-19-1': 'Let [MATH] and [MATH] denote the sample collections, and define [MATH] and [MATH] as [EQUATION] to be the sub-sampled gradient and Hessian, respectively.', '1802.09113-1-20-0': 'It has been shown that, under certain bounds on the size of the samples, [MATH] and [MATH], one can, with high probability, ensure that [MATH] and [MATH] are "suitable" approximations to the full gradient and Hessian, in an algorithmic sense [CITATION].', '1802.09113-1-20-1': 'For each iterate [MATH], using the corresponding sub-sampled approximations of the full gradient, [MATH], and the full Hessian, [MATH], we consider inexact Newton-type iterations of the form [EQUATION]', '1802.09113-1-20-2': "The requirement in eq. [REF] is often referred to as Armijo-type line-search [CITATION], and [REF] is the [MATH]-relative error approximation condition of the exact solution to the linear system [EQUATION] which is similar to that arising in classical Newton's Method.", '1802.09113-1-20-3': 'Note that in (strictly) convex settings, where the sub-sampled Hessian matrix is symmetric positive definite (SPD), conjugate gradient (CG) with early stopping can be used to obtain an approximate solution to eq. [REF] satisfying eq. [REF].', '1802.09113-1-20-4': 'It has also been shown [CITATION], that to inherit the convergence properties of the, rather expensive, algorithm that employs the exact solution to eq. [REF], the inexactness tolerance, [MATH], in eq. [REF] can only be chosen in the order of the inverse of the square root of the problem condition number.', '1802.09113-1-20-5': 'As a result, even for ill-conditioned problems, only a relatively moderate tolerance for CG ensures that we indeed maintain convergence properties of the exact update (see also examples in Section [REF]).', '1802.09113-1-20-6': 'Putting all of these together, we obtain Algorithm [REF], which under specific assumptions, has been shown [CITATION] to be globally linearly convergent with problem-independent local convergence rate .', '1802.09113-1-21-0': '[!', '1802.09113-1-21-1': 'htb] Sub-Sampled Newton Method', '1802.09113-1-22-0': 'InputInput ParameterParameters Initial iterate, [MATH] Form [MATH] as in eq. [REF]', '1802.09113-1-23-0': 'Form [MATH] as in eq. [REF]', '1802.09113-1-24-0': '[MATH] STOP', '1802.09113-1-25-0': 'Update [MATH] as in eq. [REF]', '1802.09113-1-26-0': '## Multi-Class classification', '1802.09113-1-27-0': 'For completeness, we now briefly review multi-class classification using softmax and cross-entropy loss function, as an important instance of the problems of the form described in eq. [REF].', '1802.09113-1-27-1': 'Consider a [MATH] dimensional feature vector [MATH], with corresponding labels [MATH], which can belong to one of [MATH] classes.', '1802.09113-1-27-2': 'In such a classifier, the probability that [MATH] belongs to a class [MATH] is given by [MATH], where [MATH] is the weight vector corresponding to class [MATH].', '1802.09113-1-27-3': 'Since probabilities must sum to one, there are in fact only [MATH] degrees of freedom.', '1802.09113-1-27-4': 'Consequently, by defining [MATH], for training data [MATH], the cross-entropy loss function for [MATH] can be written as [EQUATION]', '1802.09113-1-27-5': 'Note that here, [MATH].', '1802.09113-1-27-6': 'It then follows that the full gradient of [MATH] with respect to [MATH] is [EQUATION]', '1802.09113-1-27-7': 'Similarly, for the full Hessian of [MATH], we have [EQUATION]', '1802.09113-1-27-8': 'Sub-sampled variants of the gradient and Hessian are obtained similarly.', '1802.09113-1-27-9': 'Finally, after training phase, a new data [MATH] is classified as [EQUATION]', '1802.09113-1-28-0': '### Numerical Stability', '1802.09113-1-29-0': 'To avoid over-flow in the evaluation of exponential functions in [REF], we use the "Log-Sum-Exp" trick [CITATION].', '1802.09113-1-29-1': 'Specifically, for each data point [MATH], we first find the maximum value among [MATH].', '1802.09113-1-29-2': 'Define [EQUATION] and [EQUATION]', '1802.09113-1-29-3': 'Note that [MATH].', '1802.09113-1-29-4': 'Now, we have [MATH].', '1802.09113-1-29-5': 'For computing [REF], we use [MATH].', '1802.09113-1-29-6': 'Similarly, for [REF] and [REF], we use [EQUATION]', '1802.09113-1-29-7': 'Note that in all these computations, we are guaranteed to have all the exponents appearing in all the exponential functions to be negative, hence avoiding numerical over-flow.', '1802.09113-1-30-0': '### Hessian Vector Product', '1802.09113-1-31-0': 'Given a vector [MATH], we can compute the Hessian-vector product without explicitly forming the Hessian.', '1802.09113-1-31-1': 'For notational simplicity, define [EQUATION] where [MATH] and [MATH] were defined in eqs. [REF] and [REF], respectively.', '1802.09113-1-31-2': "Now using matrices [EQUATION] and [EQUATION] we compute [EQUATION] to get [EQUATION] where [MATH], [MATH], [MATH] is a vector of all [MATH]'s, and each row of the matrix [MATH] is a row vector corresponding to the [MATH] data point, i.e, [MATH].", '1802.09113-1-32-0': "Note that the memory overhead of our accelerated randomized sub-sampled Newton's method is determined by matrices [MATH], [MATH], and [MATH], whose sizes are dictated by the Hessian sample size, [MATH], which is much less than [MATH].", '1802.09113-1-32-1': 'This small memory overhead enables our Newton-type method to scale to large problems, inaccessible to traditional second order methods.', '1802.09113-1-33-0': '## Implementation Details', '1802.09113-1-34-0': 'We present a brief overview of the algorithmic machinery involved in the implementation of iterations described in eq. [REF] and applied to the function defined in eq. [REF] with an added [MATH] regularization term, i.e., [MATH].', '1802.09113-1-34-1': 'Here, [MATH] is the regularization parameter.', '1802.09113-1-34-2': 'We note that for all the algorithms in this section, we assume that matrices are stored in column-major ordering.', '1802.09113-1-35-0': 'Conjugate Gradient', '1802.09113-1-36-0': '[!', '1802.09113-1-36-1': 'htb] Conjugate-Gradient', '1802.09113-1-37-0': 'InputInput ParameterParameters', '1802.09113-1-38-0': '[MATH] - Pointer to Algorithm [REF] to compute Hessian-vector product, [MATH] - Gradient', '1802.09113-1-39-0': '[MATH] - Relative residual tolerance', '1802.09113-1-40-0': '[MATH] - Maximum no.', '1802.09113-1-40-1': 'of iterations [MATH], an approximate solution to [MATH] cg:1[MATH]', '1802.09113-1-41-0': 'cg:2[MATH] initial residual vector cg:3[MATH] initial search direction cg:4[MATH] best solution so far cg:5[MATH]', '1802.09113-1-42-0': 'cg:6[MATH] cg:7 [MATH]', '1802.09113-1-43-0': 'cg:8 [MATH]', '1802.09113-1-44-0': 'cg:9 [MATH]', '1802.09113-1-45-0': 'cg:10[MATH]', '1802.09113-1-46-0': 'cg:11 [MATH] break', '1802.09113-1-47-0': 'cg:12 [MATH]', '1802.09113-1-48-0': 'For the sake of self-containment, in Algorithm [REF], we depict a slightly modified implementation of the classical CG, to approximately solve the linear system in eq. [REF], i.e., [MATH], to satisfy eq. [REF].', '1802.09113-1-48-1': 'This routine takes a function (pointer), [MATH], which computes the matrix-vector product as [MATH], as well as the right-hand side vector, [MATH].', '1802.09113-1-48-2': 'Lines [REF], and [REF] initializes the residual vector [MATH], and search direction [MATH], respectively, while the best residual is initialized on line [REF].', '1802.09113-1-48-3': 'Iterations start on line [REF], which maintains a counter for maximum allowed iterates to compute.', '1802.09113-1-48-4': 'Step-size [MATH] for CG iterations is computed on line [REF], which is used to update the solution vector, [MATH] and residual vector, [MATH].', '1802.09113-1-48-5': 'The minor modification comes from line [REF], which stores the best solution vector thus far.', '1802.09113-1-48-6': 'The termination condition eq. [REF] is evaluated on line [REF].', '1802.09113-1-48-7': 'Finally, the search direction, [MATH], is updated in line [REF].', '1802.09113-1-49-0': 'Line Search method', '1802.09113-1-50-0': '[!', '1802.09113-1-50-1': 'htb] Line Search', '1802.09113-1-51-0': 'InputInput ParameterParameters', '1802.09113-1-52-0': '[MATH] - Current point', '1802.09113-1-53-0': "[MATH] - Newton's direction", '1802.09113-1-54-0': '[MATH] - Function pointer', '1802.09113-1-55-0': '[MATH] - Gradient', '1802.09113-1-56-0': '[MATH] - Initial step size', '1802.09113-1-57-0': '[MATH] - Cost function reduction constant', '1802.09113-1-58-0': '[MATH] - back-tracking parameter', '1802.09113-1-59-0': '[MATH] - maximum line search iterations', '1802.09113-1-60-0': 'ls:1[MATH]', '1802.09113-1-61-0': 'ls:3 [MATH]', '1802.09113-1-62-0': 'ls:4 [MATH] ls:5 [MATH] ls:6 break', '1802.09113-1-63-0': 'ls:7[MATH]', '1802.09113-1-64-0': 'ls:8[MATH]', '1802.09113-1-65-0': 'We use a simple back-tracking line search, shown in Algorithm [REF] for computing the step size in eq. [REF].', '1802.09113-1-65-1': 'Step size, [MATH], is initialized in line [REF], which is typically set to the "natural" step-size of Newton\'s method, i.e., [MATH].', '1802.09113-1-65-2': 'Iterations start at line [REF] by checking the exit criteria, and if required, successively decreasing the step size until the "loose" termination condition is met.', '1802.09113-1-65-3': 'In each of these iterations, if the objective function does not reduce by a specified amount, [MATH], step size is reduced by a fraction, [MATH], of its current value, until the termination condition is met or specified iterations have been exceeded.', '1802.09113-1-65-4': 'It has been shown [CITATION] that this process will terminate after a certain number of iterations, i.e., we are always guaranteed to have [MATH] for some fixed [MATH].', '1802.09113-1-65-5': 'CUDA utility functions', '1802.09113-1-66-0': '[!', '1802.09113-1-66-1': 'htb] ComputeExp', '1802.09113-1-67-0': 'Inputinput Outputoutput IDXidx WarpIDwarp-id LaneIDlane-id BlockIDblock-id IPartlinearPart MAmaxPart EXPsumExpPart', '1802.09113-1-68-0': '[MATH] - where [MATH] - Training classes', '1802.09113-1-69-0': 'maxPart - memory pointer to store eq. [REF]', '1802.09113-1-70-0': 'sumExpPart - memory pointer to store eq. [REF]', '1802.09113-1-71-0': 'linearPart - memory pointer to store eq. [REF]', '1802.09113-1-72-0': 'n - no.', '1802.09113-1-72-1': 'of rows in [MATH]', '1802.09113-1-73-0': 'C - no.', '1802.09113-1-73-1': 'of classes', '1802.09113-1-74-0': ', , compute-exp:1Init.', '1802.09113-1-74-1': '*thread-id [MATH] n compute-exp:2 i [MATH] % n *row no.', '1802.09113-1-74-2': 'compute-exp:3 [MATH] = 0', '1802.09113-1-75-0': 'compute-exp:4 [MATH] in [MATH]', '1802.09113-1-76-0': 'compute-exp:5 [MATH] in [MATH] compute-exp:6[MATH] j [MATH]', '1802.09113-1-77-0': 'compute-exp:7[MATH] += exp ([MATH] - [MATH] )', '1802.09113-1-78-0': 'Bulk of the work in evaluating the softmax function is done by ComputeExp subroutine, shown in Algorithm [REF].', '1802.09113-1-78-1': 'This function takes a matrix, as an input, and computes the following data structures: "[MATH]" stores the maximum component in each of the rows of the input matrix, "[MATH]" stores the partial summation of the term [MATH], and "[MATH]" stores the summation in eq. [REF].', '1802.09113-1-78-2': 'Input matrix, [MATH]A[MATH] , is the product of [MATH] and [MATH] matrices, where [MATH] is a matrix whose [MATH] column is [MATH], i.e., [MATH], and [MATH] is as in eq. [REF].', '1802.09113-1-78-3': 'Line [REF] initializes the idx, thread-id of a given thread.', '1802.09113-1-78-4': 'In the for loop in line [REF], we compute the maximum coordinate per row of the input matrix, and the result is stored in array "maxPart".', '1802.09113-1-78-5': 'Line [REF] computes "linearPart" and "sumExpPart" arrays, which are later used by functions invoking this algorithm.', '1802.09113-1-79-0': 'Softmax function evaluation', '1802.09113-1-80-0': '[!', '1802.09113-1-80-1': 'htb] ComputeFX', '1802.09113-1-81-0': 'Inputinput Outputoutput IDXidx IPartlinearPart MXmaxPart EXPsumExpPart', '1802.09113-1-82-0': 'A - Training features', '1802.09113-1-83-0': '[MATH] - Training classes', '1802.09113-1-84-0': '[MATH] - Weights vector', '1802.09113-1-85-0': '[MATH] - Regularization', '1802.09113-1-86-0': 'n - no.', '1802.09113-1-86-1': 'of rows in A', '1802.09113-1-87-0': 'p - no.', '1802.09113-1-87-1': 'of cols in A', '1802.09113-1-88-0': 'C - no.', '1802.09113-1-88-1': 'of classes', '1802.09113-1-89-0': '[MATH] - Objective function evaluated at [MATH]', '1802.09113-1-90-0': 'fx:1Initialize , , to store eqs. [REF]-[REF],', '1802.09113-1-91-0': 'fx:1 Form [MATH]', '1802.09113-1-92-0': 'fx:2[MATH] *matrix-matrix multiplication fx:3ComputeExp( [MATH], b, , , , n, C)', '1802.09113-1-93-0': 'fx:4Reduce( , pLin, n, [MATH] )', '1802.09113-1-94-0': 'fx:5Reduce( , pMax, n, [MATH] )', '1802.09113-1-95-0': 'fx:6Reduce( , pExp, n, [MATH] )', '1802.09113-1-96-0': 'fx:7temp [MATH] +', '1802.09113-1-97-0': 'fx:8Reduce( temp, pLog, n, [MATH] )', '1802.09113-1-98-0': 'fx:9[MATH] (pMax + pLog - pLin ) + [MATH]', '1802.09113-1-99-0': 'Subroutine ComputeFX, shown in Algorithm [REF], describes the evaluation of objective function at a given point, [MATH].', '1802.09113-1-99-1': 'Line [REF] initializes the memory to store partial results, and line [REF] computes the matrix-matrix product between training set, A, and weight matrix, X. By invoking the CUDA function, ComputeExp, we compute the partial results, maxPart, sumExpPart, linearPart, as described in exp-part,m-part,i-part.', '1802.09113-1-99-2': 'Lines [REF], [REF] and, [REF] compute the sum of the temporary arrays, and store the partial results in pLin, pMax, pExp, respectively.', '1802.09113-1-99-3': 'Reduce operation takes a transformation function, [MATH], which is applied to the input argument before performing the summation.', '1802.09113-1-99-4': 'Reduce is a well known function and many highly optimized implementations are readily available.', '1802.09113-1-99-5': 'We use a variation of the algorithm described in [CITATION].', '1802.09113-1-99-6': 'pLog is computed at line [REF].', '1802.09113-1-99-7': 'Finally, the objective function value is computed at line [REF], by adding intermediate results, pLin, pMax, pExp, pLog and the regularization term, i.e., [EQUATION] where [EQUATION]', '1802.09113-1-99-8': 'Softmax gradient evaluation', '1802.09113-1-100-0': '[!', '1802.09113-1-100-1': 'htb] Compute [MATH]', '1802.09113-1-101-0': 'Inputinput Outputoutput', '1802.09113-1-102-0': 'A - Training features', '1802.09113-1-103-0': '[MATH] - Training classes', '1802.09113-1-104-0': '[MATH] - Weights vector', '1802.09113-1-105-0': '[MATH] - Regularization', '1802.09113-1-106-0': '[MATH] - gradient evaluated at [MATH]', '1802.09113-1-107-0': 'gx:1 Initialize [MATH]', '1802.09113-1-108-0': 'gx:2 Form [MATH]', '1802.09113-1-109-0': 'gx:3 Compute [MATH], similar to Alg. [REF]', '1802.09113-1-110-0': 'gx:4[MATH] BInd + [MATH] X )', '1802.09113-1-111-0': 'Subroutine Compute [MATH], shown in Algorithm [REF], describes the computation of [MATH].', '1802.09113-1-111-1': 'Line [REF] initializes the memory to store temporary results.', '1802.09113-1-111-2': 'Algorithm [REF] can be easily modified to compute BInd.', '1802.09113-1-111-3': 'Line [REF] computes the gradient of the objective function by matrix multiplication and addition of the regularization term.', '1802.09113-1-112-0': 'Softmax Hessian-vector evaluation', '1802.09113-1-113-0': '[!', '1802.09113-1-113-1': 'htb] Compute Hessian-Vector Product, [MATH]', '1802.09113-1-114-0': 'Inputinput Outputoutput IDXidx', '1802.09113-1-115-0': 'A - Training dataset', '1802.09113-1-116-0': '[MATH] - Regularization', '1802.09113-1-117-0': '[MATH] - Weights vector', '1802.09113-1-118-0': '[MATH] - Vector to compute [MATH]', '1802.09113-1-119-0': 'n - no.', '1802.09113-1-119-1': 'of sample points', '1802.09113-1-120-0': 'p - no.', '1802.09113-1-120-1': 'of features', '1802.09113-1-121-0': 'C - no.', '1802.09113-1-121-1': 'of classes', '1802.09113-1-122-0': 'Hq : [MATH], Hessian-vector product', '1802.09113-1-123-0': 'hx:1Init.', '1802.09113-1-123-1': '*thread-id hx:2 Form [MATH]', '1802.09113-1-124-0': 'hx:3[MATH]', '1802.09113-1-125-0': 'hx:4 [MATH] compute as shown in [REF], similar to kernel Alg. [REF]', '1802.09113-1-126-0': 'hx:5[MATH] ComputeU (V, W, n, p, C )', '1802.09113-1-127-0': 'hx:6[MATH] vec( [MATH]U + [MATH]Q )', '1802.09113-1-128-0': '[!', '1802.09113-1-128-1': 'htb] ComputeU', '1802.09113-1-129-0': 'Inputinput Outputoutput IDXidx', '1802.09113-1-130-0': 'V - matrix V as in eq. [REF]', '1802.09113-1-131-0': 'W - matrix W as in eq. [REF]', '1802.09113-1-132-0': 'n - no.', '1802.09113-1-132-1': 'of sample points', '1802.09113-1-133-0': 'p - no.', '1802.09113-1-133-1': 'of features', '1802.09113-1-134-0': 'C - no.', '1802.09113-1-134-1': 'of classes', '1802.09113-1-135-0': 'U : matrix U as shown in [REF]', '1802.09113-1-136-0': 'Initialize *thread-id sum = 0', '1802.09113-1-137-0': '[MATH] n i = % n *row no.', '1802.09113-1-137-1': '[MATH] in [MATH] sum += [MATH] in [MATH] sum', '1802.09113-1-138-0': 'For a given vector, [MATH], Algorithm [REF], computes the Hessian-vector product, [MATH].', '1802.09113-1-138-1': 'Algorithm [REF] is heavily used in CG to solve the linear system [MATH].', '1802.09113-1-138-2': 'Line [REF] computes [MATH], as shown in eq. [REF], a matrix multiplication operation.', '1802.09113-1-138-3': 'Line [REF] computes [MATH] using a function similar to Algorithm [REF], and [MATH] is computed using Alg. [REF] at line [REF].', '1802.09113-1-138-4': 'Finally [MATH] is computed by multiplying [MATH] and [MATH], and adding the regularization term in line [REF].', '1802.09113-1-139-0': '# Experimental Results', '1802.09113-1-140-0': 'We present a comprehensive evaluations of the performance of Newton-type methods presented in this paper.', '1802.09113-1-140-1': 'We compare our methods to various first-order methods - SGD with momentum (henceforth referred to as Momentum) [CITATION], Adagrad [CITATION], Adadelta [CITATION], Adam [CITATION] and RMSProp [CITATION] as implemented in Tensorflow [CITATION].', '1802.09113-1-140-2': 'We describe our benchmarking setup, software used for development, and provide a detailed analysis of the results.', '1802.09113-1-140-3': 'The code used in this work along with the processed datasets are publicly available [CITATION].', '1802.09113-1-140-4': 'Additionally, raw datasets are also available from the UCI Machine Learning Repository [CITATION].', '1802.09113-1-141-0': '## Experimental Setup and Data', '1802.09113-1-142-0': 'Newton-type methods are implemented in C/C++ using CUDA/8.0 toolkit.', '1802.09113-1-142-1': 'For matrix operations, matrix-vector, and matrix-matrix operations, we use cuBLAS and cuSparse libraries.', '1802.09113-1-142-2': 'First order-methods are implemented using Tensorflow/1.2.1 python scripts.', '1802.09113-1-142-3': 'All results are generated using an Ubuntu server with 256GB RAM, 48-core Intel Xeon E5-2650 processors, and Tesla P100 GPU cards.', '1802.09113-1-142-4': 'For all of our experiments, we consider the [MATH]-regularized objective [MATH], where [MATH] is as in eq. [REF] and [MATH] is the regularization parameter.', '1802.09113-1-142-5': 'Seven real datasets are used for performance comparisons.', '1802.09113-1-142-6': 'Table [REF] presents the datasets used, along with the Lipschitz continuity constant of [MATH], denoted by [MATH].', '1802.09113-1-142-7': 'Recall that, an (over-estimate) of the condition-number of the problem, as defined in [CITATION], can be obtained by [MATH].', '1802.09113-1-142-8': 'As it is often done in practice, we first normalize the datasets such that each column of the data matrix [MATH] (as defined in Section [REF]), has Euclidean norm one.', '1802.09113-1-142-9': 'This helps with the conditioning of the problem.', '1802.09113-1-142-10': 'The resulting dataset is, then, split into training and testing sets, as shown in the Table [REF].', '1802.09113-1-143-0': '## Parameterization of Various Methods', '1802.09113-1-144-0': 'The Lipschitz constant, [MATH], is used to estimate the learning rate (step-size) for first order methods.', '1802.09113-1-144-1': 'For each dataset, we use a range of learning rates from [MATH] to [MATH], in increments of [MATH], a total of 13 step sizes, to determine the best performing learning rate (one that yields the maximum test accuracy).', '1802.09113-1-144-2': 'Rest of the hyper-parameters required by first-order methods are set to the default values, as recommended in Tensorflow.', '1802.09113-1-144-3': 'Two batch sizes are used for first-order methods: a small batch size of 128 (empirically, it has been argued that smaller batch sizes might lead to better performance [CITATION]), and a larger batch size of 20% of the dataset.', '1802.09113-1-144-4': 'For Newton-type methods, when the gradient is sampled, its sample size is set to [MATH].', '1802.09113-1-145-0': 'We present results for two implementations of second-order methods: (a) FullNewton, the classical Newton-CG algorithm [CITATION], which uses the exact gradient and Hessian, and (b) SubsampledNewton, sub-sampled variant of Newton-CG using uniform sub-sampling for gradient/Hessian approximations.', '1802.09113-1-145-1': 'When compared with first-order methods that use batch size of 128, SubsampledNewton uses full gradient and 5 for Hessian sample size, referred to as SubsampledNewton-100.', '1802.09113-1-145-2': "When first-order methods' batch size is set to 20, SubsampledNewton uses 20 for gradient and 5 for Hessian sampling, referred to as SubsampledNewton-20.", '1802.09113-1-145-3': 'CG-tolerance is set to [MATH].', '1802.09113-1-145-4': 'Maximum CG iterations is 10 for all of the datasets except Drive Diagnostics and Gisette, for which it is 1000.', '1802.09113-1-145-5': '[MATH] is set to [MATH] and we perform 100 iterations (epochs) for each dataset.', '1802.09113-1-146-0': '## Computing Platforms', '1802.09113-1-147-0': 'For benchmarking first order methods with batch size 128, we use CPU-cores only and for the larger batch size 1-GPU and 1-CPU-core are used.', '1802.09113-1-147-1': 'For brevity we only present the best performance results (lowest time-per-epochs); see [REF] for more detailed discussion on performance results on various compute platforms.', '1802.09113-1-147-2': 'Newton-type methods always use 1-GPU and 1-CPU-core for computations.', '1802.09113-1-148-0': '## Performance Comparisons', '1802.09113-1-149-0': 'Table [REF] presents all the performance results.', '1802.09113-1-149-1': 'Columns 1 and 3 show the plots for cumulative-time vs. test-accuracy and columns 2 and 4 plot the numbers for cumulative-time vs. objective function (training).', '1802.09113-1-149-2': 'Please note that x-axis in all the plots is in "log-scale".', '1802.09113-1-150-0': '### Covertype Dataset', '1802.09113-1-151-0': 'The first row in Table [REF] shows the plots for Covertype dataset.', '1802.09113-1-151-1': 'From the first two columns (batch size 128), we note the following: (i) Newton-type methods minimize the objective function to [MATH] in a smaller time interval (FullNewton: 0.9 secs, SubsampledNewton-20: 0.24 secs ), compared to first-order alternatives (Adadelta - 91 secs, Adagrad - 183 secs, Adam - 57 secs, Momentum - 285 secs, RMSProp - 40 secs); (ii) Compared to first order algorithms, Newton-type methods achieve equivalent test accuracy, [MATH], in a significantly shorter time interval, i.e., 0.9 secs compared with tens of seconds for first order methods (Adadelta: 201 secs, Adagrad: 72 secs, Adam: 285 secs, Momentum: 128 secs, RMSProp: 111 secs); (iii) SubsampledNewton-100 achieves relatively higher test accuracy earlier compared to the FullNewton method in a relatively short time interval (FullNewton: 68 in 1.5 secs, SubsampledNewton-100: 68 in 204 millisecs).', '1802.09113-1-151-2': 'For well-conditioned problems (such as this one), a relaxed CG-tolerance and small sample sizes (5 Hessian sample size) yield desirable results quickly.', '1802.09113-1-152-0': 'Columns 3 and 4 present the performance of first-order methods with batch size 20.', '1802.09113-1-152-1': 'Randomized Newton method, SubsampledNewton-20, achieves higher test accuracy, 68, in a very short time, 1.05 secs, compared to any of the first order methods as shown in column 3 (Adadelta: 65 in 21 secs, Adagrad: 65 in 19 secs, Adam: 68 in 20 secs, Momentum: 68 in 18 secs, RMSProp: 65 in 21 secs).', '1802.09113-1-152-2': 'First order methods, with batch size 20, are executed on GPUs resulting in smaller time-per-epoch; see [REF].', '1802.09113-1-152-3': 'This can be attributed to processing larger batches of the dataset by the GPU-cores, yielding higher efficiency.', '1802.09113-1-153-0': '### Drive Diagnostics Dataset', '1802.09113-1-154-0': 'Results for the Drive Diagnostics dataset are shown in the second row of Table [REF].', '1802.09113-1-154-1': 'These plots clearly indicate that Newton-type methods achieve their lowest objective function value , 3.75e4, much earlier compared to first order methods (FullNewton - 1.3 secs, SubsampledNewton-20 - 0.8 secs, SubsampledNewton-100 - 0.2 secs).', '1802.09113-1-154-2': 'Corresponding times for batch size 128 for first order methods are : Adadelta - 16 secs, Adagrad - 34 secs, Adam - 25 secs, Momentum - 32 secs, RMSProp - 35 secs (lowest objective function value for these methods are [MATH] 3.8e5).', '1802.09113-1-154-3': 'For batch size 20%, except for Adadelta and Momentum, other first order methods achieve their lowest objective function values, which are significantly higher compared to Newton-type methods, in [MATH] 3 seconds.', '1802.09113-1-154-4': 'Momentum is the only first order method that achieves almost equivalent objective function value, 3.8e5 in 0.6 seconds, as Newton-type methods.', '1802.09113-1-155-0': 'All first order methods, with batch size 128, achieve test accuracy of 87% which is same as Newton-type methods but take much longer: FullNewton - 0.2 secs, SubsampledNewton-20 - 0.3 secs, SubsampledNewton-100 - 0.15 secs vs. Adadelta - 30 secs, Adagrad - 36 secs, Adam - 7 secs, Momentum - 32 secs, RMSProp - 7 secs. Here, except Momentum, none of the first order methods with batch size 20% achieve 87% test accuracy in 100 epochs.', '1802.09113-1-156-0': '### MNIST and CIFAR-10 Datasets', '1802.09113-1-157-0': 'Rows 3 and 4 in Table [REF] present plots for MNIST and CIFAR-10 datasets, respectively.', '1802.09113-1-157-1': 'Regardless of the batch size, Newton-type methods clearly outperform first-order methods.', '1802.09113-1-157-2': 'For example, with MNIST dataset, all the methods achieve a test accuracy of 92%.', '1802.09113-1-157-3': 'However, Newton-type methods do so in [MATH] seconds, compared to [MATH] seconds for first order methods with batch size of 128.', '1802.09113-1-158-0': 'CIFAR results are shown in row 4 of Table [REF].', '1802.09113-1-158-1': 'We clearly notice that first order methods, with batch size 128, make slow progress towards achieving their lowest objective function value (and test accuracy) taking almost 100 seconds to reach 8.4e4 (40% test accuracy).', '1802.09113-1-158-2': 'Newton-type methods achieve these values in significantly shorter time (FullNewton - 10 seconds, SubsampledNewton-20 - 4.2 seconds, SubsampledNewton-100 - 2.6 seconds).', '1802.09113-1-158-3': 'The slow progress of first order methods is much more pronounced when batch size is set to 20.', '1802.09113-1-158-4': 'Only Adam and Momentum methods achieve a test accuracy of [MATH] 40 in 100 epochs (taking [MATH] 60 seconds).', '1802.09113-1-158-5': 'Note that CIFAR-10 represents a relatively ill-conditioned problem.', '1802.09113-1-158-6': 'As a result, in terms of lowering the objective function on CIFAR-10, first-order methods are negatively affected by the ill-conditioning, whereas all Newton-type methods show a great degree of robustness.', '1802.09113-1-158-7': 'This demonstrates the versatility of Newton-type methods for solving problems with various degrees of ill-conditioning.', '1802.09113-1-159-0': '### Newsgroups20 Dataset', '1802.09113-1-160-0': 'Plots in row 5 of Table [REF] correspond to Newsgroups20 dataset.', '1802.09113-1-160-1': 'This is a sparse dataset, and the largest in the scope of this work (the Hessian is [MATH] 1e6 [MATH] 1e6).', '1802.09113-1-160-2': 'Here, FullNewton and SubsampledNewton-100 achieve, respectively, 87.22 and 88.46 test accuracy in the first few iterations.', '1802.09113-1-160-3': 'Smaller batch sized first order methods can only achieve a maximum test accuracy of 85 in 100 epochs.', '1802.09113-1-160-4': 'Note that average time per epoch for first order methods is [MATH] 1 sec compared to 75 millisecs for SubsampledNewton-100 iteration.', '1802.09113-1-160-5': 'When 20% gradient is used, as shown in column 3, we notice that the SubsampledNewton-20 method starts with a lower test accuracy of [MATH] 80% in the 5th iteration and slowly ramps up to 85.4% as we near the allotted number of iterations.', '1802.09113-1-160-6': 'This can be attributed to a smaller gradient sample size, and sparse nature of this dataset.', '1802.09113-1-161-0': '### Gisette and Real-Sim Datasets', '1802.09113-1-162-0': 'Rows 6 and 7 in Table [REF] show results for Gisette and Real-Sim datasets, respectively.', '1802.09113-1-162-1': 'FullNewton method for Gisette dataset converges in 11 iterations and yields 98.3 test accuracy in 0.6 seconds.', '1802.09113-1-162-2': 'SubsampledNewton-100 takes 34 iterations to reach 98 test accuracy, whereas first order counterparts, except Momentum method, can achieve 97 test accuracy in 100 iterations.', '1802.09113-1-162-3': 'When batch size is set to 20, we notice that all first order methods make slow progress towards achieving lower objective function values.', '1802.09113-1-162-4': 'Noticeably, none of the first order methods can lower the objective function value to a level achieved by Newton-type methods, which can be attributed to the ill-conditioning of this problem; see Table [REF].', '1802.09113-1-163-0': 'For Real-Sim dataset, relative to first order methods and regardless of batch size, we clearly notice that Newton-type methods achieve similar or lower objective function values, in a comparable or lower time interval.', '1802.09113-1-163-1': 'Further, FullNewton achieves 97.3% in the [MATH] iteration whereas it takes 11 iterations for SubsampledNewton-20.', '1802.09113-1-164-0': '## Sensitivity to Hyper-Parameter Tuning', '1802.09113-1-165-0': 'The "biggest elephant in the room" in optimization using, almost all, first-order methods is that of fine-tuning of various underlying hyper-parameters, most notably, the step-size [CITATION].', '1802.09113-1-165-1': 'Indeed, the success of most such methods is tightly intertwined with many trial and error steps to find a proper parameter settings.', '1802.09113-1-165-2': 'It is highly unusual for these methods to exhibit acceptable performance on the first try, and it often takes many trials and errors before one can see reasonable results.', '1802.09113-1-165-3': 'In fact, the "true training time", which almost always includes the time it takes to appropriately tune these parameters, can be frustratingly long.', '1802.09113-1-165-4': 'In contrast, second-order optimization methods involve much less parameter tuning, and are less sensitive to specific choices of their hyper-parameters [CITATION].', '1802.09113-1-166-0': 'Here, to further highlight such issues, we demonstrate the sensitivity of several first-order methods with respect to their learning rate.', '1802.09113-1-166-1': 'Figure [REF] shows the results of multiple runs of SGD with Momentum, Adagrad, RMSProp and Adam on Newsgroups20 dataset with several choices of step-size.', '1802.09113-1-166-2': 'Each method is run 13 times using step-sizes in the range [MATH] to [MATH], in increments of [MATH], where [MATH] is the Lipschitz constant; see Table [REF].', '1802.09113-1-167-0': 'It is clear that small step-sizes can result in stagnation, whereas large step sizes can cause the method to diverge.', '1802.09113-1-167-1': 'Only if the step-size is within a particular and often narrow range, which greatly varies across various methods, one can see reasonable performance.', '1802.09113-1-168-0': 'For some first-order methods, e.g., momentum based, line-search type techniques simply cannot be used.', '1802.09113-1-168-1': 'For others, the starting step-size for line-search is, almost always, a priori unknown.', '1802.09113-1-168-2': 'This is sharp contrast with randomized Newton-type methods considered here, which come with a priori "natural" step-size, i.e., [MATH] , and furthermore, only occasionally require the line-search to intervene; see [CITATION] for theoretical guarantees in this regard.', '1802.09113-1-169-0': '# Conclusions And Future Work', '1802.09113-1-170-0': "In this paper, we demonstrate that sampled variants of Newton's method, when implemented appropriately, present compelling alternatives to popular first-order methods for solving convex optimization problems in machine learning and data analysis applications.", '1802.09113-1-170-1': 'We discussed, in detail, the GPU-specific implementation of Newton-type methods to achieve similar per-iteration costs as first-order methods.', '1802.09113-1-170-2': 'We experimentally showcased their advantages, including robustness to ill-conditioning and higher predictive performance.', '1802.09113-1-170-3': 'We also highlighted the sensitivity of various first-order methods with respect to their learning-rate.', '1802.09113-1-171-0': 'Extending our results and implementations to non-convex optimization problems and targeting broad classes of machine learning applications, is an important avenue for future work.', '1802.09113-1-172-0': '# More Details On Softmax Function [REF]', '1802.09113-1-173-0': '## Relationship to Logistic Regression with [MATH]-labels', '1802.09113-1-174-0': 'Sometimes, in the literature, for the two-class classification problem, instead of [MATH] the labels are marked as [MATH].', '1802.09113-1-174-1': 'In this case, the corresponding logistic regression is written as [EQUATION]', '1802.09113-1-174-2': 'In this case, we have [EQUATION] where [MATH].', '1802.09113-1-174-3': 'Hence this formulation co-incides with [REF].', '1802.09113-1-175-0': '### Softmax Multi-Class problem is (strictly) convex', '1802.09113-1-176-0': 'Consider the data matrix [MATH] where each row, [MATH], is a row vector corresponding to the [MATH] data point.', '1802.09113-1-176-1': 'The Hessian matrix can be written as [EQUATION] where [EQUATION] and each [MATH] and [MATH] is a [MATH] diagonal matrix corresponding to [REF] and [REF], respectively.', '1802.09113-1-176-2': 'Note that since [EQUATION] the matrix [MATH] is strictly diagonally dominant, and hence it is symmetric positive definite.', '1802.09113-1-176-3': 'So the problem is convex (in fact it is strictly-convex if the data matrix [MATH] is full column rank).', '1802.09113-1-177-0': "# Tensorflow's Performance Comparison on Various Compute Platforms", '1802.09113-1-178-0': 'Columns 1 and 2 of table [REF] plots the results for covertype dataset, when batch size is set to 128, using CPU-only cores (row 1) and 1-GPU-1-CPU-core (row 2) for first-order tensorflow implementations.', '1802.09113-1-178-1': 'Note that newton-type methods always use 1-GPU-1-CPU-core as the compute platform irrespective of any of the hyper-parameter settings.', '1802.09113-1-178-2': 'We clearly notice that the first-order methods takes [MATH] 600 seconds when GPU cores are used compared to [MATH] 350 seconds when CPU cores are used.', '1802.09113-1-178-3': 'This can be attributed to the small batch size used for first-order methods.', '1802.09113-1-178-4': 'Smaller batch size results in computing the gradient, a compute-intensive operation, much more frequently compared to a large batch size.', '1802.09113-1-178-5': 'For the plots shown in table [REF] training size for covertype is set to 450,000.', '1802.09113-1-178-6': 'This means gradient is computed [MATH] 3516 times to complete each of the training epochs in this instance.', '1802.09113-1-178-7': 'Since the batch size is very small most of the GPU cores are idle during every computation of the gradient resulting in low GPU occupancy (which is the ratio of active warps on an SM and maximum allowed warps).', '1802.09113-1-178-8': 'Also with each invocation of gradient computation there is CUDA kernel instantiation overhead which accumulates as well.', '1802.09113-1-178-9': 'Because of above reasons small batch sizes yield high time per epoch for first-order methods.', '1802.09113-1-179-0': 'Columns 3 and 4 of table [REF] plots for the results for covertype dataset using a large batch size, of 20% of the dataset.', '1802.09113-1-179-1': 'Note that batch size for first-order methods is same as the gradient sample size for newton-type methods for these plots.', '1802.09113-1-179-2': 'We clearly notice that first-order tensorflow methods takes [MATH] 55 seconds when CPU-only cores are used as the compute platform compared to [MATH] 22.5 seconds when 1-GPU-1-CPU-core is used, a speedup of [MATH] over CPU only compute platform.', '1802.09113-1-179-3': 'In this instance, during each epoch of first-order methods gradient is evaluated only 5 times.', '1802.09113-1-179-4': 'Because of the large batch size, [MATH] 90,000 points, are processed by the GPU resulting in higher utilization of the GPU cores (compared to the same computation using smaller batch size).', '1802.09113-1-179-5': 'This explains why GPU-cores yield shorter time per epoch when large batch size are used for first-order methods.'}

{'1802.09113-2-0-0': 'First order methods, which solely rely on gradient information, are commonly used in diverse machine learning (ML) and data analysis (DA) applications.', '1802.09113-2-0-1': 'This is attributed to the simplicity of their implementations, as well as low per-iteration computational/storage costs.', '1802.09113-2-0-2': 'However, they suffer from significant disadvantages; most notably, their performance degrades with increasing problem ill-conditioning.', '1802.09113-2-0-3': 'Furthermore, they often involve a large number of hyper-parameters, and are notoriously sensitive to parameters such as the step-size.', '1802.09113-2-0-4': 'By incorporating additional information from the Hessian, second-order methods, have been shown to be resilient to many such adversarial effects.', '1802.09113-2-0-5': 'However, these advantages of using curvature information come at the cost of higher per-iteration costs, which in big data regimes, can be computationally prohibitive.', '1802.09113-2-1-0': 'In this paper, we show that, contrary to conventional belief, second-order methods, when implemented appropriately, can be more efficient than first-order alternatives in many large-scale ML/ DA applications.', '1802.09113-2-1-1': "In particular, in convex settings, we consider variants of classical Newton's method in which the Hessian and/or the gradient are randomly sub-sampled.", '1802.09113-2-1-2': 'We show that by effectively leveraging the power of GPUs, such randomized Newton-type algorithms can be significantly accelerated, and can easily outperform state of the art implementations of existing techniques in popular ML/ DA software packages such as TensorFlow.', '1802.09113-2-1-3': 'Additionally these randomized methods incur a small memory overhead compared to first-order methods.', '1802.09113-2-1-4': "In particular, we show that for million-dimensional problems, our GPU accelerated sub-sampled Newton's method achieves a higher test accuracy in milliseconds as compared with tens of seconds for first order alternatives.", '1802.09113-2-2-0': '# Introduction', '1802.09113-2-3-0': 'Optimization techniques are at the core of many ML/DA applications.', '1802.09113-2-3-1': 'First-order methods that rely solely on gradient of the objective function, have been methods of choice in these applications.', '1802.09113-2-3-2': 'The scale of commonly encountered problems in typical applications necessitates optimization techniques that are fast, i.e., have low per-iteration cost and require few overall iterations, as well as robust to adversarial effects such as problem ill-conditioning and hyper-parameter tuning.', '1802.09113-2-3-3': 'First-order methods such as stochastic gradient descent (SGD) are widely known to have low per-iteration costs.', '1802.09113-2-3-4': 'However, they often require many iterations before suitable results are obtained, and their performance can deteriorate for moderately to ill-conditioned problems.', '1802.09113-2-3-5': 'Contrary to popular belief, ill-conditioned problems often arise in machine learning applications.', '1802.09113-2-3-6': 'For example, the "vanishing and exploding gradient problem" encountered in training deep neural nets [CITATION], is a well-known and important issue.', '1802.09113-2-3-7': 'What is less known is that this is a consequence of the highly ill-conditioned nature of the problem.', '1802.09113-2-3-8': 'Other examples include low-rank matrix approximation and spectral clustering involving radial basis function (RBF) kernels when the scale parameter is large [CITATION].', '1802.09113-2-3-9': 'A subtle, yet potentially more serious, disadvantage of most first-order methods is the large number of hyper-parameters, as well as their high sensitivity to parameter-tuning, which can significantly slow down the training procedure and often necessitate many trial and error steps [CITATION].', '1802.09113-2-4-0': 'Newton-type methods use curvature information in the form of the Hessian matrix, in addition to the to gradient.', '1802.09113-2-4-1': 'This family of methods has not been commonly used in the ML/ DA community because of their high per-iteration costs, in spite of the fact that second-order methods offer a range of benefits.', '1802.09113-2-4-2': 'Unlike first-order methods, Newton-type methods have been shown to be highly resilient to increasing problem ill-conditioning [CITATION].', '1802.09113-2-4-3': 'Furthermore, second-order methods typically require fewer parameters (e.g., inexactness tolerance for the sub-problem solver or line-search parameters), and are less sensitive to their specific settings [CITATION].', '1802.09113-2-4-4': 'By incorporating curvature information at each iteration, Newton-type methods scale the gradient such that it is a more suitable direction to follow.', '1802.09113-2-4-5': 'Consequently, although their iterations may be more expensive than those of the first-order counterparts, second-order methods typically require much fewer iterations.', '1802.09113-2-5-0': 'In this context, by reducing the cost of each iteration through efficient approximation of curvature, coupled with hardware specific acceleration, one can obtain methods that are fast and robust.', '1802.09113-2-5-1': 'In most ML applications, this typically translates to achieving a high test-accuracy early on in the iterative process and without significant parameter tuning; see Section [REF].', '1802.09113-2-5-2': 'This is in sharp contrast with slow-ramping trends typically observed in training with first-order methods, which is often preceded by a lengthy trial and error procedure for parameter tuning.', '1802.09113-2-5-3': 'Indeed, the aforementioned properties, coupled with efficiency obtained from algorithmic innovations and implementations that effectively utilize all available hardware resources, hold promise for significantly changing the landscape of optimization techniques used in ML/DA applications.', '1802.09113-2-6-0': 'With the long-term goal of achieving this paradigm shift, we focus on the commonly encountered finite-sum optimization problem', '1802.09113-2-7-0': '[EQUATION] where each [MATH] is a smooth convex function, representing a loss (or misfit) corresponding to [MATH] observation (or measurement) [CITATION].', '1802.09113-2-7-1': 'In many ML applications, [MATH] in eq. [REF] corresponds to the empirical risk [CITATION], and the goal of solving eq. [REF] is to obtain a solution with small generalization error, i.e., high predictive accuracy on "unseen" data.', '1802.09113-2-7-2': 'We consider eq. [REF] at scale, where the values of [MATH] and [MATH] are large - millions and beyond.', '1802.09113-2-7-3': 'In such settings, the mere computation of the Hessian and the gradient of [MATH] increases linearly in [MATH].', '1802.09113-2-7-4': 'Indeed, for large-scale problems, operations on the Hessian, e.g., matrix-vector products involved in the (approximate) solution of the sub-problems of most Newton-type methods, typically constitute the main computational bottleneck.', '1802.09113-2-7-5': 'In such cases, randomized sub-sampling has been shown to be highly successful in reducing computational and memory costs to be effectively independent of [MATH].', '1802.09113-2-7-6': "For example, a simple instance of eq. [REF] is when the functions [MATH]'s are quadratics, in which case one has an over-constrained least squares problem.", '1802.09113-2-7-7': 'For these problems, randomized numerical linear algebra (RandNLA) techniques rely on random sampling, which is used to compute a data-aware or data-oblivious subspace embedding that preserves the geometry of the entire subspace [CITATION].', '1802.09113-2-7-8': 'Furthermore, non-trivial practical implementations of algorithms based on these ideas have been shown to beat state-of-the-art numerical techniques [CITATION].', '1802.09113-2-7-9': 'For more general problems, theoretical properties of sub-sampled Newton-type methods, for both convex and non-convex problems of the form in eq. [REF], have been recently studied in a series of efforts [CITATION].', '1802.09113-2-7-10': 'However, for real ML/ DA applications beyond least squares, practical and hardware-specific implementations that can effectively draw upon all available computing resources, are lacking.', '1802.09113-2-8-0': 'Contributions: Our contributions in this paper can be summarized as follows:', '1802.09113-2-9-0': "Through a judicious mix of statistical techniques, algorithmic innovations, and highly optimized GPU implementations, we develop an accelerated variant of the classical Newton's method that has low per-iteration cost, fast convergence, and minimal memory overhead.", '1802.09113-2-9-1': 'In the process, we show that, for solving eq. [REF], our accelerated randomized method significantly outperforms state of the art implementations of existing techniques in popular ML/DA software packages such as TensorFlow [CITATION], in terms of improved training time, generalization error, and robustness to various adversarial effects.', '1802.09113-2-10-0': 'This paper is organized as follows.', '1802.09113-2-10-1': 'Section [REF] provides an overview of related literature.', '1802.09113-2-10-2': 'Section [REF] presents technical background regarding sub-sampled Newton-type methods, Softmax classifier as a practical instance of eq. [REF], along with a description of the algorithms and their implementation.', '1802.09113-2-10-3': 'Section [REF] compares and contrasts GPU based implementations of sub-sampled Newton-type methods with first order methods available in TensorFlow.', '1802.09113-2-10-4': 'Conclusions and avenues for future work are presented in Section [REF].', '1802.09113-2-11-0': '# Related Work', '1802.09113-2-12-0': 'The class of first-order methods includes a number of techniques that are commonly used in diverse ML/DA applications.', '1802.09113-2-12-1': 'Many of these techniques have been efficiently implemented in popular software packages.', '1802.09113-2-12-2': 'For example, TensorFlow, [CITATION], has enjoyed considerable success among ML practitioners.', '1802.09113-2-12-3': 'Among first-order methods implemented in TensorFlow for solving [REF] are Adagrad [CITATION], RMSProp [CITATION], Adam [CITATION], Adadelta [CITATION], and SGD with/ without momentum [CITATION].', '1802.09113-2-12-4': 'Excluding SGD, the rest of these methods are adaptive, in that they incorporate prior gradients to choose a preconditioner at each gradient step.', '1802.09113-2-12-5': 'Through the use of gradient history from previous iterations, these adaptive methods non-uniformly scale the current gradient to obtain an update direction that takes larger steps along the coordinates with smaller derivatives and, conversely, smaller steps along those with larger derivatives.', '1802.09113-2-12-6': "At a high level, these methods aim to capture non-uniform scaling of Newton's method, albeit, using limited curvature information.", '1802.09113-2-13-0': 'Theoretical properties of a variety of randomized Newton-type methods, for both convex and non-convex problems of the form eq. [REF], have been recently studied in a series of results, both in the context of ML applications [CITATION], as well as scientific computing applications [CITATION].', '1802.09113-2-14-0': 'GPUs have been successfully used in a variety of ML applications to speed up computations [CITATION].', '1802.09113-2-14-1': 'In particular, Raina et al. [CITATION] demonstrate that modern GPUs can far surpass the computational capabilities of multi-core CPUs, and have the potential to address many of the computational challenges encountered in training large-scale learning models.', '1802.09113-2-14-2': 'Most relevant to this paper, Ngiam et al. [CITATION] show that off-the-shelf optimization methods such as Limited memory BFGS (L-BFGS) and Conjugate Gradient (CG), have the potential to outperform variants of SGD in deep learning applications.', '1802.09113-2-14-3': 'It was further demonstrated that the difference in performance between LBFGS/CG and SGD is more pronounced if one considers hardware accelerators such as GPUs.', '1802.09113-2-14-4': "Extending similar results to full-fledged second-order algorithms, such Newton's method, is a major motivating factor for our work here.", '1802.09113-2-15-0': '# Theory, Algorithms and Implementation Details', '1802.09113-2-16-0': '## Notation', '1802.09113-2-17-0': 'Vectors, [MATH], and matrices, [MATH], are denoted by bold lower and upper case letters, respectively.', '1802.09113-2-17-1': '[MATH] and [MATH] represent the gradient and the Hessian of [MATH] at [MATH], respectively.', '1802.09113-2-17-2': 'The superscript, e.g., [MATH], denotes iteration count.', '1802.09113-2-17-3': '[MATH] denotes a collection of indices drawn from the set [MATH], with potentially repeated items, and its cardinality is denoted by [MATH].', '1802.09113-2-17-4': 'Following Matlab notation, [MATH] denotes vertical stacking of two column vectors [MATH], whereas [MATH] denotes a [MATH] matrix whose columns are formed from the vectors [MATH] and [MATH].', '1802.09113-2-17-5': 'Vector [MATH] norm is denoted by [MATH].', '1802.09113-2-17-6': 'For a boolean variable, [MATH], the indicator function [MATH] evaluates to one if [MATH], and zero otherwise.', '1802.09113-2-17-7': '[MATH] denotes the dot product of vectors [MATH] and [MATH], and [MATH] represents element-wise multiplication of matrices [MATH] and [MATH].', '1802.09113-2-18-0': "## Sub-Sampled Newton's Method", '1802.09113-2-19-0': 'For the optimization problem eq. [REF], in each iteration, consider selecting two sample sets of indices from [MATH], uniformly at random with or without replacement.', '1802.09113-2-19-1': 'Let [MATH] and [MATH] denote the sample collections, and define [MATH] and [MATH] as [EQUATION] to be the sub-sampled gradient and Hessian, respectively.', '1802.09113-2-20-0': 'It has been shown that, under certain bounds on the size of the samples, [MATH] and [MATH], one can, with high probability, ensure that [MATH] and [MATH] are "suitable" approximations to the full gradient and Hessian, in an algorithmic sense [CITATION].', '1802.09113-2-20-1': 'For each iterate [MATH], using the corresponding sub-sampled approximations of the full gradient, [MATH], and the full Hessian, [MATH], we consider inexact Newton-type iterations of the form [EQUATION]', '1802.09113-2-20-2': "The requirement in eq. [REF] is often referred to as Armijo-type line-search [CITATION], and [REF] is the [MATH]-relative error approximation condition of the exact solution to the linear system [EQUATION] which is similar to that arising in classical Newton's Method.", '1802.09113-2-20-3': 'Note that in (strictly) convex settings, where the sub-sampled Hessian matrix is symmetric positive definite (SPD), conjugate gradient (CG) with early stopping can be used to obtain an approximate solution to eq. [REF] satisfying eq. [REF].', '1802.09113-2-20-4': 'It has also been shown [CITATION], that to inherit the convergence properties of the, rather expensive, algorithm that employs the exact solution to eq. [REF], the inexactness tolerance, [MATH], in eq. [REF] can only be chosen in the order of the inverse of the square root of the problem condition number.', '1802.09113-2-20-5': 'As a result, even for ill-conditioned problems, only a relatively moderate tolerance for CG ensures that we indeed maintain convergence properties of the exact update (see also examples in Section [REF]).', '1802.09113-2-20-6': 'Putting all of these together, we obtain Algorithm [REF], which under specific assumptions, has been shown [CITATION] to be globally linearly convergent with problem-independent local convergence rate .', '1802.09113-2-21-0': '[!', '1802.09113-2-21-1': 'htb] Sub-Sampled Newton Method', '1802.09113-2-22-0': 'InputInput ParameterParameters Initial iterate, [MATH] Form [MATH] as in eq. [REF]', '1802.09113-2-23-0': 'Form [MATH] as in eq. [REF]', '1802.09113-2-24-0': '[MATH] STOP', '1802.09113-2-25-0': 'Update [MATH] as in eq. [REF]', '1802.09113-2-26-0': '## Multi-Class classification', '1802.09113-2-27-0': 'For completeness, we now briefly review multi-class classification using softmax and cross-entropy loss function, as an important instance of the problems of the form described in eq. [REF].', '1802.09113-2-27-1': 'Consider a [MATH] dimensional feature vector [MATH], with corresponding labels [MATH], which can belong to one of [MATH] classes.', '1802.09113-2-27-2': 'In such a classifier, the probability that [MATH] belongs to a class [MATH] is given by [MATH], where [MATH] is the weight vector corresponding to class [MATH].', '1802.09113-2-27-3': 'Since probabilities must sum to one, there are in fact only [MATH] degrees of freedom.', '1802.09113-2-27-4': 'Consequently, by defining [MATH], for training data [MATH], the cross-entropy loss function for [MATH] can be written as [EQUATION]', '1802.09113-2-27-5': 'Note that here, [MATH].', '1802.09113-2-27-6': 'It then follows that the full gradient of [MATH] with respect to [MATH] is [EQUATION]', '1802.09113-2-27-7': 'Similarly, for the full Hessian of [MATH], we have [EQUATION]', '1802.09113-2-27-8': 'Sub-sampled variants of the gradient and Hessian are obtained similarly.', '1802.09113-2-27-9': 'Finally, after training phase, a new data [MATH] is classified as [EQUATION]', '1802.09113-2-28-0': '### Numerical Stability', '1802.09113-2-29-0': 'To avoid over-flow in the evaluation of exponential functions in [REF], we use the "Log-Sum-Exp" trick [CITATION].', '1802.09113-2-29-1': 'Specifically, for each data point [MATH], we first find the maximum value among [MATH].', '1802.09113-2-29-2': 'Define [EQUATION] and [EQUATION]', '1802.09113-2-29-3': 'Note that [MATH].', '1802.09113-2-29-4': 'Now, we have [MATH].', '1802.09113-2-29-5': 'For computing [REF], we use [MATH].', '1802.09113-2-29-6': 'Similarly, for [REF] and [REF], we use [EQUATION]', '1802.09113-2-29-7': 'Note that in all these computations, we are guaranteed to have all the exponents appearing in all the exponential functions to be negative, hence avoiding numerical over-flow.', '1802.09113-2-30-0': '### Hessian Vector Product', '1802.09113-2-31-0': 'Given a vector [MATH], we can compute the Hessian-vector product without explicitly forming the Hessian.', '1802.09113-2-31-1': 'For notational simplicity, define [EQUATION] where [MATH] and [MATH] were defined in eqs. [REF] and [REF], respectively.', '1802.09113-2-31-2': "Now using matrices [EQUATION] and [EQUATION] we compute [EQUATION] to get [EQUATION] where [MATH], [MATH], [MATH] is a vector of all [MATH]'s, and each row of the matrix [MATH] is a row vector corresponding to the [MATH] data point, i.e, [MATH].", '1802.09113-2-32-0': "Note that the memory overhead of our accelerated randomized sub-sampled Newton's method is determined by matrices [MATH], [MATH], and [MATH], whose sizes are dictated by the Hessian sample size, [MATH], which is much less than [MATH].", '1802.09113-2-32-1': 'This small memory overhead enables our Newton-type method to scale to large problems, inaccessible to traditional second order methods.', '1802.09113-2-33-0': '## Implementation Details', '1802.09113-2-34-0': 'We present a brief overview of the algorithmic machinery involved in the implementation of iterations described in eq. [REF] and applied to the function defined in eq. [REF] with an added [MATH] regularization term, i.e., [MATH].', '1802.09113-2-34-1': 'Here, [MATH] is the regularization parameter.', '1802.09113-2-34-2': 'We note that for all the algorithms in this section, we assume that matrices are stored in column-major ordering.', '1802.09113-2-35-0': 'Conjugate Gradient', '1802.09113-2-36-0': '[!', '1802.09113-2-36-1': 'htb] Conjugate-Gradient', '1802.09113-2-37-0': 'InputInput ParameterParameters', '1802.09113-2-38-0': '[MATH] - Pointer to Algorithm [REF] to compute Hessian-vector product, [MATH] - Gradient', '1802.09113-2-39-0': '[MATH] - Relative residual tolerance', '1802.09113-2-40-0': '[MATH] - Maximum no.', '1802.09113-2-40-1': 'of iterations [MATH], an approximate solution to [MATH] cg:1[MATH]', '1802.09113-2-41-0': 'cg:2[MATH] initial residual vector cg:3[MATH] initial search direction cg:4[MATH] best solution so far cg:5[MATH]', '1802.09113-2-42-0': 'cg:6[MATH] cg:7 [MATH]', '1802.09113-2-43-0': 'cg:8 [MATH]', '1802.09113-2-44-0': 'cg:9 [MATH]', '1802.09113-2-45-0': 'cg:10[MATH]', '1802.09113-2-46-0': 'cg:11 [MATH] break', '1802.09113-2-47-0': 'cg:12 [MATH]', '1802.09113-2-48-0': 'For the sake of self-containment, in Algorithm [REF], we depict a slightly modified implementation of the classical CG, to approximately solve the linear system in eq. [REF], i.e., [MATH], to satisfy eq. [REF].', '1802.09113-2-48-1': 'This routine takes a function (pointer), [MATH], which computes the matrix-vector product as [MATH], as well as the right-hand side vector, [MATH].', '1802.09113-2-48-2': 'Lines [REF], and [REF] initializes the residual vector [MATH], and search direction [MATH], respectively, while the best residual is initialized on line [REF].', '1802.09113-2-48-3': 'Iterations start on line [REF], which maintains a counter for maximum allowed iterates to compute.', '1802.09113-2-48-4': 'Step-size [MATH] for CG iterations is computed on line [REF], which is used to update the solution vector, [MATH] and residual vector, [MATH].', '1802.09113-2-48-5': 'The minor modification comes from line [REF], which stores the best solution vector thus far.', '1802.09113-2-48-6': 'The termination condition eq. [REF] is evaluated on line [REF].', '1802.09113-2-48-7': 'Finally, the search direction, [MATH], is updated in line [REF].', '1802.09113-2-49-0': 'Line Search method', '1802.09113-2-50-0': '[!', '1802.09113-2-50-1': 'htb] Line Search', '1802.09113-2-51-0': 'InputInput ParameterParameters', '1802.09113-2-52-0': '[MATH] - Current point', '1802.09113-2-53-0': "[MATH] - Newton's direction", '1802.09113-2-54-0': '[MATH] - Function pointer', '1802.09113-2-55-0': '[MATH] - Gradient', '1802.09113-2-56-0': '[MATH] - Initial step size', '1802.09113-2-57-0': '[MATH] - Cost function reduction constant', '1802.09113-2-58-0': '[MATH] - back-tracking parameter', '1802.09113-2-59-0': '[MATH] - maximum line search iterations', '1802.09113-2-60-0': 'ls:1[MATH]', '1802.09113-2-61-0': 'ls:3 [MATH]', '1802.09113-2-62-0': 'ls:4 [MATH] ls:5 [MATH] ls:6 break', '1802.09113-2-63-0': 'ls:7[MATH]', '1802.09113-2-64-0': 'ls:8[MATH]', '1802.09113-2-65-0': 'We use a simple back-tracking line search, shown in Algorithm [REF] for computing the step size in eq. [REF].', '1802.09113-2-65-1': 'Step size, [MATH], is initialized in line [REF], which is typically set to the "natural" step-size of Newton\'s method, i.e., [MATH].', '1802.09113-2-65-2': 'Iterations start at line [REF] by checking the exit criteria, and if required, successively decreasing the step size until the "loose" termination condition is met.', '1802.09113-2-65-3': 'In each of these iterations, if the objective function does not reduce by a specified amount, [MATH], step size is reduced by a fraction, [MATH], of its current value, until the termination condition is met or specified iterations have been exceeded.', '1802.09113-2-65-4': 'It has been shown [CITATION] that this process will terminate after a certain number of iterations, i.e., we are always guaranteed to have [MATH] for some fixed [MATH].', '1802.09113-2-65-5': 'CUDA utility functions', '1802.09113-2-66-0': '[!', '1802.09113-2-66-1': 'htb] ComputeExp', '1802.09113-2-67-0': 'Inputinput Outputoutput IDXidx WarpIDwarp-id LaneIDlane-id BlockIDblock-id IPartlinearPart MAmaxPart EXPsumExpPart', '1802.09113-2-68-0': '[MATH] - where [MATH] - Training classes', '1802.09113-2-69-0': 'maxPart - memory pointer to store eq. [REF]', '1802.09113-2-70-0': 'sumExpPart - memory pointer to store eq. [REF]', '1802.09113-2-71-0': 'linearPart - memory pointer to store eq. [REF]', '1802.09113-2-72-0': 'n - no.', '1802.09113-2-72-1': 'of rows in [MATH]', '1802.09113-2-73-0': 'C - no.', '1802.09113-2-73-1': 'of classes', '1802.09113-2-74-0': ', , compute-exp:1Init.', '1802.09113-2-74-1': '*thread-id [MATH] n compute-exp:2 i [MATH] % n *row no.', '1802.09113-2-74-2': 'compute-exp:3 [MATH] = 0', '1802.09113-2-75-0': 'compute-exp:4 [MATH] in [MATH]', '1802.09113-2-76-0': 'compute-exp:5 [MATH] in [MATH] compute-exp:6[MATH] j [MATH]', '1802.09113-2-77-0': 'compute-exp:7[MATH] += exp ([MATH] - [MATH] )', '1802.09113-2-78-0': 'Bulk of the work in evaluating the softmax function is done by ComputeExp subroutine, shown in Algorithm [REF].', '1802.09113-2-78-1': 'This function takes a matrix, as an input, and computes the following data structures: "[MATH]" stores the maximum component in each of the rows of the input matrix, "[MATH]" stores the partial summation of the term [MATH], and "[MATH]" stores the summation in eq. [REF].', '1802.09113-2-78-2': 'Input matrix, [MATH]A[MATH] , is the product of [MATH] and [MATH] matrices, where [MATH] is a matrix whose [MATH] column is [MATH], i.e., [MATH], and [MATH] is as in eq. [REF].', '1802.09113-2-78-3': 'Line [REF] initializes the idx, thread-id of a given thread.', '1802.09113-2-78-4': 'In the for loop in line [REF], we compute the maximum coordinate per row of the input matrix, and the result is stored in array "maxPart".', '1802.09113-2-78-5': 'Line [REF] computes "linearPart" and "sumExpPart" arrays, which are later used by functions invoking this algorithm.', '1802.09113-2-79-0': 'Softmax function evaluation', '1802.09113-2-80-0': '[!', '1802.09113-2-80-1': 'htb] ComputeFX', '1802.09113-2-81-0': 'Inputinput Outputoutput IDXidx IPartlinearPart MXmaxPart EXPsumExpPart', '1802.09113-2-82-0': 'A - Training features', '1802.09113-2-83-0': '[MATH] - Training classes', '1802.09113-2-84-0': '[MATH] - Weights vector', '1802.09113-2-85-0': '[MATH] - Regularization', '1802.09113-2-86-0': 'n - no.', '1802.09113-2-86-1': 'of rows in A', '1802.09113-2-87-0': 'p - no.', '1802.09113-2-87-1': 'of cols in A', '1802.09113-2-88-0': 'C - no.', '1802.09113-2-88-1': 'of classes', '1802.09113-2-89-0': '[MATH] - Objective function evaluated at [MATH]', '1802.09113-2-90-0': 'fx:1Initialize , , to store eqs. [REF]-[REF],', '1802.09113-2-91-0': 'fx:1 Form [MATH]', '1802.09113-2-92-0': 'fx:2[MATH] *matrix-matrix multiplication fx:3ComputeExp( [MATH], b, , , , n, C)', '1802.09113-2-93-0': 'fx:4Reduce( , pLin, n, [MATH] )', '1802.09113-2-94-0': 'fx:5Reduce( , pMax, n, [MATH] )', '1802.09113-2-95-0': 'fx:6Reduce( , pExp, n, [MATH] )', '1802.09113-2-96-0': 'fx:7temp [MATH] +', '1802.09113-2-97-0': 'fx:8Reduce( temp, pLog, n, [MATH] )', '1802.09113-2-98-0': 'fx:9[MATH] (pMax + pLog - pLin ) + [MATH]', '1802.09113-2-99-0': 'Subroutine ComputeFX, shown in Algorithm [REF], describes the evaluation of objective function at a given point, [MATH].', '1802.09113-2-99-1': 'Line [REF] initializes the memory to store partial results, and line [REF] computes the matrix-matrix product between training set, A, and weight matrix, X. By invoking the CUDA function, ComputeExp, we compute the partial results, maxPart, sumExpPart, linearPart, as described in exp-part,m-part,i-part.', '1802.09113-2-99-2': 'Lines [REF], [REF] and, [REF] compute the sum of the temporary arrays, and store the partial results in pLin, pMax, pExp, respectively.', '1802.09113-2-99-3': 'Reduce operation takes a transformation function, [MATH], which is applied to the input argument before performing the summation.', '1802.09113-2-99-4': 'Reduce is a well known function and many highly optimized implementations are readily available.', '1802.09113-2-99-5': 'We use a variation of the algorithm described in [CITATION].', '1802.09113-2-99-6': 'pLog is computed at line [REF].', '1802.09113-2-99-7': 'Finally, the objective function value is computed at line [REF], by adding intermediate results, pLin, pMax, pExp, pLog and the regularization term, i.e., [EQUATION] where [EQUATION]', '1802.09113-2-99-8': 'Softmax gradient evaluation', '1802.09113-2-100-0': '[!', '1802.09113-2-100-1': 'htb] Compute [MATH]', '1802.09113-2-101-0': 'Inputinput Outputoutput', '1802.09113-2-102-0': 'A - Training features', '1802.09113-2-103-0': '[MATH] - Training classes', '1802.09113-2-104-0': '[MATH] - Weights vector', '1802.09113-2-105-0': '[MATH] - Regularization', '1802.09113-2-106-0': '[MATH] - gradient evaluated at [MATH]', '1802.09113-2-107-0': 'gx:1 Initialize [MATH]', '1802.09113-2-108-0': 'gx:2 Form [MATH]', '1802.09113-2-109-0': 'gx:3 Compute [MATH], similar to Alg. [REF]', '1802.09113-2-110-0': 'gx:4[MATH] BInd + [MATH] X )', '1802.09113-2-111-0': 'Subroutine Compute [MATH], shown in Algorithm [REF], describes the computation of [MATH].', '1802.09113-2-111-1': 'Line [REF] initializes the memory to store temporary results.', '1802.09113-2-111-2': 'Algorithm [REF] can be easily modified to compute BInd.', '1802.09113-2-111-3': 'Line [REF] computes the gradient of the objective function by matrix multiplication and addition of the regularization term.', '1802.09113-2-112-0': 'Softmax Hessian-vector evaluation', '1802.09113-2-113-0': '[!', '1802.09113-2-113-1': 'htb] Compute Hessian-Vector Product, [MATH]', '1802.09113-2-114-0': 'Inputinput Outputoutput IDXidx', '1802.09113-2-115-0': 'A - Training dataset', '1802.09113-2-116-0': '[MATH] - Regularization', '1802.09113-2-117-0': '[MATH] - Weights vector', '1802.09113-2-118-0': '[MATH] - Vector to compute [MATH]', '1802.09113-2-119-0': 'n - no.', '1802.09113-2-119-1': 'of sample points', '1802.09113-2-120-0': 'p - no.', '1802.09113-2-120-1': 'of features', '1802.09113-2-121-0': 'C - no.', '1802.09113-2-121-1': 'of classes', '1802.09113-2-122-0': 'Hq : [MATH], Hessian-vector product', '1802.09113-2-123-0': 'hx:1Init.', '1802.09113-2-123-1': '*thread-id hx:2 Form [MATH]', '1802.09113-2-124-0': 'hx:3[MATH]', '1802.09113-2-125-0': 'hx:4 [MATH] compute as shown in [REF], similar to kernel Alg. [REF]', '1802.09113-2-126-0': 'hx:5[MATH] ComputeU (V, W, n, p, C )', '1802.09113-2-127-0': 'hx:6[MATH] vec( [MATH]U + [MATH]Q )', '1802.09113-2-128-0': '[!', '1802.09113-2-128-1': 'htb] ComputeU', '1802.09113-2-129-0': 'Inputinput Outputoutput IDXidx', '1802.09113-2-130-0': 'V - matrix V as in eq. [REF]', '1802.09113-2-131-0': 'W - matrix W as in eq. [REF]', '1802.09113-2-132-0': 'n - no.', '1802.09113-2-132-1': 'of sample points', '1802.09113-2-133-0': 'p - no.', '1802.09113-2-133-1': 'of features', '1802.09113-2-134-0': 'C - no.', '1802.09113-2-134-1': 'of classes', '1802.09113-2-135-0': 'U : matrix U as shown in [REF]', '1802.09113-2-136-0': 'Initialize *thread-id sum = 0', '1802.09113-2-137-0': '[MATH] n i = % n *row no.', '1802.09113-2-137-1': '[MATH] in [MATH] sum += [MATH] in [MATH] sum', '1802.09113-2-138-0': 'For a given vector, [MATH], Algorithm [REF], computes the Hessian-vector product, [MATH].', '1802.09113-2-138-1': 'Algorithm [REF] is heavily used in CG to solve the linear system [MATH].', '1802.09113-2-138-2': 'Line [REF] computes [MATH], as shown in eq. [REF], a matrix multiplication operation.', '1802.09113-2-138-3': 'Line [REF] computes [MATH] using a function similar to Algorithm [REF], and [MATH] is computed using Alg. [REF] at line [REF].', '1802.09113-2-138-4': 'Finally [MATH] is computed by multiplying [MATH] and [MATH], and adding the regularization term in line [REF].', '1802.09113-2-139-0': '# Experimental Results', '1802.09113-2-140-0': 'We present a comprehensive evaluations of the performance of Newton-type methods presented in this paper.', '1802.09113-2-140-1': 'We compare our methods to various first-order methods - SGD with momentum (henceforth referred to as Momentum) [CITATION], Adagrad [CITATION], Adadelta [CITATION], Adam [CITATION] and RMSProp [CITATION] as implemented in Tensorflow [CITATION].', '1802.09113-2-140-2': 'We describe our benchmarking setup, software used for development, and provide a detailed analysis of the results.', '1802.09113-2-140-3': 'The code used in this work along with the processed datasets are publicly available [CITATION].', '1802.09113-2-140-4': 'Additionally, raw datasets are also available from the UCI Machine Learning Repository [CITATION].', '1802.09113-2-141-0': '## Experimental Setup and Data', '1802.09113-2-142-0': 'Newton-type methods are implemented in C/C++ using CUDA/8.0 toolkit.', '1802.09113-2-142-1': 'For matrix operations, matrix-vector, and matrix-matrix operations, we use cuBLAS and cuSparse libraries.', '1802.09113-2-142-2': 'First order-methods are implemented using Tensorflow/1.2.1 python scripts.', '1802.09113-2-142-3': 'All results are generated using an Ubuntu server with 256GB RAM, 48-core Intel Xeon E5-2650 processors, and Tesla P100 GPU cards.', '1802.09113-2-142-4': 'For all of our experiments, we consider the [MATH]-regularized objective [MATH], where [MATH] is as in eq. [REF] and [MATH] is the regularization parameter.', '1802.09113-2-142-5': 'Seven real datasets are used for performance comparisons.', '1802.09113-2-142-6': 'Table [REF] presents the datasets used, along with the Lipschitz continuity constant of [MATH], denoted by [MATH].', '1802.09113-2-142-7': 'Recall that, an (over-estimate) of the condition-number of the problem, as defined in [CITATION], can be obtained by [MATH].', '1802.09113-2-142-8': 'As it is often done in practice, we first normalize the datasets such that each column of the data matrix [MATH] (as defined in Section [REF]), has Euclidean norm one.', '1802.09113-2-142-9': 'This helps with the conditioning of the problem.', '1802.09113-2-142-10': 'The resulting dataset is, then, split into training and testing sets, as shown in the Table [REF].', '1802.09113-2-143-0': '## Parameterization of Various Methods', '1802.09113-2-144-0': 'The Lipschitz constant, [MATH], is used to estimate the learning rate (step-size) for first order methods.', '1802.09113-2-144-1': 'For each dataset, we use a range of learning rates from [MATH] to [MATH], in increments of [MATH], a total of 13 step sizes, to determine the best performing learning rate (one that yields the maximum test accuracy).', '1802.09113-2-144-2': 'Rest of the hyper-parameters required by first-order methods are set to the default values, as recommended in Tensorflow.', '1802.09113-2-144-3': 'Two batch sizes are used for first-order methods: a small batch size of 128 (empirically, it has been argued that smaller batch sizes might lead to better performance [CITATION]), and a larger batch size of 20% of the dataset.', '1802.09113-2-144-4': 'For Newton-type methods, when the gradient is sampled, its sample size is set to [MATH].', '1802.09113-2-145-0': 'We present results for two implementations of second-order methods: (a) FullNewton, the classical Newton-CG algorithm [CITATION], which uses the exact gradient and Hessian, and (b) SubsampledNewton, sub-sampled variant of Newton-CG using uniform sub-sampling for gradient/Hessian approximations.', '1802.09113-2-145-1': 'When compared with first-order methods that use batch size of 128, SubsampledNewton uses full gradient and 5 for Hessian sample size, referred to as SubsampledNewton-100.', '1802.09113-2-145-2': "When first-order methods' batch size is set to 20, SubsampledNewton uses 20 for gradient and 5 for Hessian sampling, referred to as SubsampledNewton-20.", '1802.09113-2-145-3': 'CG-tolerance is set to [MATH].', '1802.09113-2-145-4': 'Maximum CG iterations is 10 for all of the datasets except Drive Diagnostics and Gisette, for which it is 1000.', '1802.09113-2-145-5': '[MATH] is set to [MATH] and we perform 100 iterations (epochs) for each dataset.', '1802.09113-2-146-0': '## Computing Platforms', '1802.09113-2-147-0': 'For benchmarking first order methods with batch size 128, we use CPU-cores only and for the larger batch size 1-GPU and 1-CPU-core are used.', '1802.09113-2-147-1': 'For brevity we only present the best performance results (lowest time-per-epochs); see [REF] for more detailed discussion on performance results on various compute platforms.', '1802.09113-2-147-2': 'Newton-type methods always use 1-GPU and 1-CPU-core for computations.', '1802.09113-2-148-0': '## Performance Comparisons', '1802.09113-2-149-0': 'Table [REF] presents all the performance results.', '1802.09113-2-149-1': 'Columns 1 and 3 show the plots for cumulative-time vs. test-accuracy and columns 2 and 4 plot the numbers for cumulative-time vs. objective function (training).', '1802.09113-2-149-2': 'Please note that x-axis in all the plots is in "log-scale".', '1802.09113-2-150-0': '### Covertype Dataset', '1802.09113-2-151-0': 'The first row in Table [REF] shows the plots for Covertype dataset.', '1802.09113-2-151-1': 'From the first two columns (batch size 128), we note the following: (i) Newton-type methods minimize the objective function to [MATH] in a smaller time interval (FullNewton: 0.9 secs, SubsampledNewton-20: 0.24 secs ), compared to first-order alternatives (Adadelta - 91 secs, Adagrad - 183 secs, Adam - 57 secs, Momentum - 285 secs, RMSProp - 40 secs); (ii) Compared to first order algorithms, Newton-type methods achieve equivalent test accuracy, [MATH], in a significantly shorter time interval, i.e., 0.9 secs compared with tens of seconds for first order methods (Adadelta: 201 secs, Adagrad: 72 secs, Adam: 285 secs, Momentum: 128 secs, RMSProp: 111 secs); (iii) SubsampledNewton-100 achieves relatively higher test accuracy earlier compared to the FullNewton method in a relatively short time interval (FullNewton: 68 in 1.5 secs, SubsampledNewton-100: 68 in 204 millisecs).', '1802.09113-2-151-2': 'For well-conditioned problems (such as this one), a relaxed CG-tolerance and small sample sizes (5 Hessian sample size) yield desirable results quickly.', '1802.09113-2-152-0': 'Columns 3 and 4 present the performance of first-order methods with batch size 20.', '1802.09113-2-152-1': 'Randomized Newton method, SubsampledNewton-20, achieves higher test accuracy, 68, in a very short time, 1.05 secs, compared to any of the first order methods as shown in column 3 (Adadelta: 65 in 21 secs, Adagrad: 65 in 19 secs, Adam: 68 in 20 secs, Momentum: 68 in 18 secs, RMSProp: 65 in 21 secs).', '1802.09113-2-152-2': 'First order methods, with batch size 20, are executed on GPUs resulting in smaller time-per-epoch; see [REF].', '1802.09113-2-152-3': 'This can be attributed to processing larger batches of the dataset by the GPU-cores, yielding higher efficiency.', '1802.09113-2-153-0': '### Drive Diagnostics Dataset', '1802.09113-2-154-0': 'Results for the Drive Diagnostics dataset are shown in the second row of Table [REF].', '1802.09113-2-154-1': 'These plots clearly indicate that Newton-type methods achieve their lowest objective function value , 3.75e4, much earlier compared to first order methods (FullNewton - 1.3 secs, SubsampledNewton-20 - 0.8 secs, SubsampledNewton-100 - 0.2 secs).', '1802.09113-2-154-2': 'Corresponding times for batch size 128 for first order methods are : Adadelta - 16 secs, Adagrad - 34 secs, Adam - 25 secs, Momentum - 32 secs, RMSProp - 35 secs (lowest objective function value for these methods are [MATH] 3.8e5).', '1802.09113-2-154-3': 'For batch size 20%, except for Adadelta and Momentum, other first order methods achieve their lowest objective function values, which are significantly higher compared to Newton-type methods, in [MATH] 3 seconds.', '1802.09113-2-154-4': 'Momentum is the only first order method that achieves almost equivalent objective function value, 3.8e5 in 0.6 seconds, as Newton-type methods.', '1802.09113-2-155-0': 'All first order methods, with batch size 128, achieve test accuracy of 87% which is same as Newton-type methods but take much longer: FullNewton - 0.2 secs, SubsampledNewton-20 - 0.3 secs, SubsampledNewton-100 - 0.15 secs vs. Adadelta - 30 secs, Adagrad - 36 secs, Adam - 7 secs, Momentum - 32 secs, RMSProp - 7 secs. Here, except Momentum, none of the first order methods with batch size 20% achieve 87% test accuracy in 100 epochs.', '1802.09113-2-156-0': '### MNIST and CIFAR-10 Datasets', '1802.09113-2-157-0': 'Rows 3 and 4 in Table [REF] present plots for MNIST and CIFAR-10 datasets, respectively.', '1802.09113-2-157-1': 'Regardless of the batch size, Newton-type methods clearly outperform first-order methods.', '1802.09113-2-157-2': 'For example, with MNIST dataset, all the methods achieve a test accuracy of 92%.', '1802.09113-2-157-3': 'However, Newton-type methods do so in [MATH] seconds, compared to [MATH] seconds for first order methods with batch size of 128.', '1802.09113-2-158-0': 'CIFAR results are shown in row 4 of Table [REF].', '1802.09113-2-158-1': 'We clearly notice that first order methods, with batch size 128, make slow progress towards achieving their lowest objective function value (and test accuracy) taking almost 100 seconds to reach 8.4e4 (40% test accuracy).', '1802.09113-2-158-2': 'Newton-type methods achieve these values in significantly shorter time (FullNewton - 10 seconds, SubsampledNewton-20 - 4.2 seconds, SubsampledNewton-100 - 2.6 seconds).', '1802.09113-2-158-3': 'The slow progress of first order methods is much more pronounced when batch size is set to 20.', '1802.09113-2-158-4': 'Only Adam and Momentum methods achieve a test accuracy of [MATH] 40 in 100 epochs (taking [MATH] 60 seconds).', '1802.09113-2-158-5': 'Note that CIFAR-10 represents a relatively ill-conditioned problem.', '1802.09113-2-158-6': 'As a result, in terms of lowering the objective function on CIFAR-10, first-order methods are negatively affected by the ill-conditioning, whereas all Newton-type methods show a great degree of robustness.', '1802.09113-2-158-7': 'This demonstrates the versatility of Newton-type methods for solving problems with various degrees of ill-conditioning.', '1802.09113-2-159-0': '### Newsgroups20 Dataset', '1802.09113-2-160-0': 'Plots in row 5 of Table [REF] correspond to Newsgroups20 dataset.', '1802.09113-2-160-1': 'This is a sparse dataset, and the largest in the scope of this work (the Hessian is [MATH] 1e6 [MATH] 1e6).', '1802.09113-2-160-2': 'Here, FullNewton and SubsampledNewton-100 achieve, respectively, 87.22 and 88.46 test accuracy in the first few iterations.', '1802.09113-2-160-3': 'Smaller batch sized first order methods can only achieve a maximum test accuracy of 85 in 100 epochs.', '1802.09113-2-160-4': 'Note that average time per epoch for first order methods is [MATH] 1 sec compared to 75 millisecs for SubsampledNewton-100 iteration.', '1802.09113-2-160-5': 'When 20% gradient is used, as shown in column 3, we notice that the SubsampledNewton-20 method starts with a lower test accuracy of [MATH] 80% in the 5th iteration and slowly ramps up to 85.4% as we near the allotted number of iterations.', '1802.09113-2-160-6': 'This can be attributed to a smaller gradient sample size, and sparse nature of this dataset.', '1802.09113-2-161-0': '### Gisette and Real-Sim Datasets', '1802.09113-2-162-0': 'Rows 6 and 7 in Table [REF] show results for Gisette and Real-Sim datasets, respectively.', '1802.09113-2-162-1': 'FullNewton method for Gisette dataset converges in 11 iterations and yields 98.3 test accuracy in 0.6 seconds.', '1802.09113-2-162-2': 'SubsampledNewton-100 takes 34 iterations to reach 98 test accuracy, whereas first order counterparts, except Momentum method, can achieve 97 test accuracy in 100 iterations.', '1802.09113-2-162-3': 'When batch size is set to 20, we notice that all first order methods make slow progress towards achieving lower objective function values.', '1802.09113-2-162-4': 'Noticeably, none of the first order methods can lower the objective function value to a level achieved by Newton-type methods, which can be attributed to the ill-conditioning of this problem; see Table [REF].', '1802.09113-2-163-0': 'For Real-Sim dataset, relative to first order methods and regardless of batch size, we clearly notice that Newton-type methods achieve similar or lower objective function values, in a comparable or lower time interval.', '1802.09113-2-163-1': 'Further, FullNewton achieves 97.3% in the [MATH] iteration whereas it takes 11 iterations for SubsampledNewton-20.', '1802.09113-2-164-0': '## Sensitivity to Hyper-Parameter Tuning', '1802.09113-2-165-0': 'The "biggest elephant in the room" in optimization using, almost all, first-order methods is that of fine-tuning of various underlying hyper-parameters, most notably, the step-size [CITATION].', '1802.09113-2-165-1': 'Indeed, the success of most such methods is tightly intertwined with many trial and error steps to find a proper parameter settings.', '1802.09113-2-165-2': 'It is highly unusual for these methods to exhibit acceptable performance on the first try, and it often takes many trials and errors before one can see reasonable results.', '1802.09113-2-165-3': 'In fact, the "true training time", which almost always includes the time it takes to appropriately tune these parameters, can be frustratingly long.', '1802.09113-2-165-4': 'In contrast, second-order optimization methods involve much less parameter tuning, and are less sensitive to specific choices of their hyper-parameters [CITATION].', '1802.09113-2-166-0': 'Here, to further highlight such issues, we demonstrate the sensitivity of several first-order methods with respect to their learning rate.', '1802.09113-2-166-1': 'Figure [REF] shows the results of multiple runs of SGD with Momentum, Adagrad, RMSProp and Adam on Newsgroups20 dataset with several choices of step-size.', '1802.09113-2-166-2': 'Each method is run 13 times using step-sizes in the range [MATH] to [MATH], in increments of [MATH], where [MATH] is the Lipschitz constant; see Table [REF].', '1802.09113-2-167-0': 'It is clear that small step-sizes can result in stagnation, whereas large step sizes can cause the method to diverge.', '1802.09113-2-167-1': 'Only if the step-size is within a particular and often narrow range, which greatly varies across various methods, one can see reasonable performance.', '1802.09113-2-168-0': 'For some first-order methods, e.g., momentum based, line-search type techniques simply cannot be used.', '1802.09113-2-168-1': 'For others, the starting step-size for line-search is, almost always, a priori unknown.', '1802.09113-2-168-2': 'This is sharp contrast with randomized Newton-type methods considered here, which come with a priori "natural" step-size, i.e., [MATH] , and furthermore, only occasionally require the line-search to intervene; see [CITATION] for theoretical guarantees in this regard.', '1802.09113-2-169-0': '# Conclusions And Future Work', '1802.09113-2-170-0': "In this paper, we demonstrate that sampled variants of Newton's method, when implemented appropriately, present compelling alternatives to popular first-order methods for solving convex optimization problems in machine learning and data analysis applications.", '1802.09113-2-170-1': 'We discussed, in detail, the GPU-specific implementation of Newton-type methods to achieve similar per-iteration costs as first-order methods.', '1802.09113-2-170-2': 'We experimentally showcased their advantages, including robustness to ill-conditioning and higher predictive performance.', '1802.09113-2-170-3': 'We also highlighted the sensitivity of various first-order methods with respect to their learning-rate.', '1802.09113-2-171-0': 'Extending our results and implementations to non-convex optimization problems and targeting broad classes of machine learning applications, is an important avenue for future work.', '1802.09113-2-172-0': '# More Details On Softmax Function [REF]', '1802.09113-2-173-0': '## Relationship to Logistic Regression with [MATH]-labels', '1802.09113-2-174-0': 'Sometimes, in the literature, for the two-class classification problem, instead of [MATH] the labels are marked as [MATH].', '1802.09113-2-174-1': 'In this case, the corresponding logistic regression is written as [EQUATION]', '1802.09113-2-174-2': 'In this case, we have [EQUATION] where [MATH].', '1802.09113-2-174-3': 'Hence this formulation co-incides with [REF].', '1802.09113-2-175-0': '### Softmax Multi-Class problem is (strictly) convex', '1802.09113-2-176-0': 'Consider the data matrix [MATH] where each row, [MATH], is a row vector corresponding to the [MATH] data point.', '1802.09113-2-176-1': 'The Hessian matrix can be written as [EQUATION] where [EQUATION] and each [MATH] and [MATH] is a [MATH] diagonal matrix corresponding to [REF] and [REF], respectively.', '1802.09113-2-176-2': 'Note that since [EQUATION] the matrix [MATH] is strictly diagonally dominant, and hence it is symmetric positive definite.', '1802.09113-2-176-3': 'So the problem is convex (in fact it is strictly-convex if the data matrix [MATH] is full column rank).', '1802.09113-2-177-0': "# Tensorflow's Performance Comparison on Various Compute Platforms", '1802.09113-2-178-0': 'Columns 1 and 2 of table [REF] plots the results for covertype dataset, when batch size is set to 128, using CPU-only cores (row 1) and 1-GPU-1-CPU-core (row 2) for first-order tensorflow implementations.', '1802.09113-2-178-1': 'Note that newton-type methods always use 1-GPU-1-CPU-core as the compute platform irrespective of any of the hyper-parameter settings.', '1802.09113-2-178-2': 'We clearly notice that the first-order methods takes [MATH] 600 seconds when GPU cores are used compared to [MATH] 350 seconds when CPU cores are used.', '1802.09113-2-178-3': 'This can be attributed to the small batch size used for first-order methods.', '1802.09113-2-178-4': 'Smaller batch size results in computing the gradient, a compute-intensive operation, much more frequently compared to a large batch size.', '1802.09113-2-178-5': 'For the plots shown in table [REF] training size for covertype is set to 450,000.', '1802.09113-2-178-6': 'This means gradient is computed [MATH] 3516 times to complete each of the training epochs in this instance.', '1802.09113-2-178-7': 'Since the batch size is very small most of the GPU cores are idle during every computation of the gradient resulting in low GPU occupancy (which is the ratio of active warps on an SM and maximum allowed warps).', '1802.09113-2-178-8': 'Also with each invocation of gradient computation there is CUDA kernel instantiation overhead which accumulates as well.', '1802.09113-2-178-9': 'Because of above reasons small batch sizes yield high time per epoch for first-order methods.', '1802.09113-2-179-0': 'Columns 3 and 4 of table [REF] plots for the results for covertype dataset using a large batch size, of 20% of the dataset.', '1802.09113-2-179-1': 'Note that batch size for first-order methods is same as the gradient sample size for newton-type methods for these plots.', '1802.09113-2-179-2': 'We clearly notice that first-order tensorflow methods takes [MATH] 55 seconds when CPU-only cores are used as the compute platform compared to [MATH] 22.5 seconds when 1-GPU-1-CPU-core is used, a speedup of [MATH] over CPU only compute platform.', '1802.09113-2-179-3': 'In this instance, during each epoch of first-order methods gradient is evaluated only 5 times.', '1802.09113-2-179-4': 'Because of the large batch size, [MATH] 90,000 points, are processed by the GPU resulting in higher utilization of the GPU cores (compared to the same computation using smaller batch size).', '1802.09113-2-179-5': 'This explains why GPU-cores yield shorter time per epoch when large batch size are used for first-order methods.'}

[['1802.09113-1-3-0', '1802.09113-2-3-0'], ['1802.09113-1-3-1', '1802.09113-2-3-1'], ['1802.09113-1-3-2', '1802.09113-2-3-2'], ['1802.09113-1-3-3', '1802.09113-2-3-3'], ['1802.09113-1-3-4', '1802.09113-2-3-4'], ['1802.09113-1-3-5', '1802.09113-2-3-5'], ['1802.09113-1-3-6', '1802.09113-2-3-6'], ['1802.09113-1-3-7', '1802.09113-2-3-7'], ['1802.09113-1-3-8', '1802.09113-2-3-8'], ['1802.09113-1-3-9', '1802.09113-2-3-9'], ['1802.09113-1-176-0', '1802.09113-2-176-0'], ['1802.09113-1-176-1', '1802.09113-2-176-1'], ['1802.09113-1-176-2', '1802.09113-2-176-2'], ['1802.09113-1-176-3', '1802.09113-2-176-3'], ['1802.09113-1-174-0', '1802.09113-2-174-0'], ['1802.09113-1-174-1', '1802.09113-2-174-1'], ['1802.09113-1-174-2', '1802.09113-2-174-2'], ['1802.09113-1-174-3', '1802.09113-2-174-3'], ['1802.09113-1-178-0', '1802.09113-2-178-0'], ['1802.09113-1-178-1', '1802.09113-2-178-1'], ['1802.09113-1-178-2', '1802.09113-2-178-2'], ['1802.09113-1-178-3', '1802.09113-2-178-3'], ['1802.09113-1-178-4', '1802.09113-2-178-4'], ['1802.09113-1-178-5', '1802.09113-2-178-5'], ['1802.09113-1-178-6', '1802.09113-2-178-6'], ['1802.09113-1-178-7', '1802.09113-2-178-7'], ['1802.09113-1-178-8', '1802.09113-2-178-8'], ['1802.09113-1-178-9', '1802.09113-2-178-9'], ['1802.09113-1-145-0', '1802.09113-2-145-0'], ['1802.09113-1-145-1', '1802.09113-2-145-1'], ['1802.09113-1-145-2', '1802.09113-2-145-2'], ['1802.09113-1-145-3', '1802.09113-2-145-3'], ['1802.09113-1-145-4', '1802.09113-2-145-4'], ['1802.09113-1-145-5', '1802.09113-2-145-5'], ['1802.09113-1-14-0', '1802.09113-2-14-0'], ['1802.09113-1-14-1', '1802.09113-2-14-1'], ['1802.09113-1-14-2', '1802.09113-2-14-2'], ['1802.09113-1-14-3', '1802.09113-2-14-3'], ['1802.09113-1-14-4', '1802.09113-2-14-4'], ['1802.09113-1-4-0', '1802.09113-2-4-0'], ['1802.09113-1-4-1', '1802.09113-2-4-1'], ['1802.09113-1-4-2', '1802.09113-2-4-2'], ['1802.09113-1-4-3', '1802.09113-2-4-3'], ['1802.09113-1-4-4', '1802.09113-2-4-4'], ['1802.09113-1-4-5', '1802.09113-2-4-5'], ['1802.09113-1-154-0', '1802.09113-2-154-0'], ['1802.09113-1-154-1', '1802.09113-2-154-1'], ['1802.09113-1-154-2', '1802.09113-2-154-2'], ['1802.09113-1-154-3', '1802.09113-2-154-3'], ['1802.09113-1-154-4', '1802.09113-2-154-4'], ['1802.09113-1-142-0', '1802.09113-2-142-0'], ['1802.09113-1-142-1', '1802.09113-2-142-1'], ['1802.09113-1-142-2', '1802.09113-2-142-2'], ['1802.09113-1-142-3', '1802.09113-2-142-3'], ['1802.09113-1-142-4', '1802.09113-2-142-4'], ['1802.09113-1-142-5', '1802.09113-2-142-5'], ['1802.09113-1-142-6', '1802.09113-2-142-6'], ['1802.09113-1-142-7', '1802.09113-2-142-7'], ['1802.09113-1-142-8', '1802.09113-2-142-8'], ['1802.09113-1-142-9', '1802.09113-2-142-9'], ['1802.09113-1-142-10', '1802.09113-2-142-10'], ['1802.09113-1-144-0', '1802.09113-2-144-0'], ['1802.09113-1-144-1', '1802.09113-2-144-1'], ['1802.09113-1-144-2', '1802.09113-2-144-2'], ['1802.09113-1-144-3', '1802.09113-2-144-3'], ['1802.09113-1-144-4', '1802.09113-2-144-4'], ['1802.09113-1-160-0', '1802.09113-2-160-0'], ['1802.09113-1-160-1', '1802.09113-2-160-1'], ['1802.09113-1-160-2', '1802.09113-2-160-2'], ['1802.09113-1-160-3', '1802.09113-2-160-3'], ['1802.09113-1-160-4', '1802.09113-2-160-4'], ['1802.09113-1-160-5', '1802.09113-2-160-5'], ['1802.09113-1-160-6', '1802.09113-2-160-6'], ['1802.09113-1-165-0', '1802.09113-2-165-0'], ['1802.09113-1-165-1', '1802.09113-2-165-1'], ['1802.09113-1-165-2', '1802.09113-2-165-2'], ['1802.09113-1-165-3', '1802.09113-2-165-3'], ['1802.09113-1-165-4', '1802.09113-2-165-4'], ['1802.09113-1-17-0', '1802.09113-2-17-0'], ['1802.09113-1-17-1', '1802.09113-2-17-1'], ['1802.09113-1-17-2', '1802.09113-2-17-2'], ['1802.09113-1-17-3', '1802.09113-2-17-3'], ['1802.09113-1-17-4', '1802.09113-2-17-4'], ['1802.09113-1-17-5', '1802.09113-2-17-5'], ['1802.09113-1-17-6', '1802.09113-2-17-6'], ['1802.09113-1-17-7', '1802.09113-2-17-7'], ['1802.09113-1-13-0', '1802.09113-2-13-0'], ['1802.09113-1-157-0', '1802.09113-2-157-0'], ['1802.09113-1-157-1', '1802.09113-2-157-1'], ['1802.09113-1-157-2', '1802.09113-2-157-2'], ['1802.09113-1-157-3', '1802.09113-2-157-3'], ['1802.09113-1-111-0', '1802.09113-2-111-0'], ['1802.09113-1-111-1', '1802.09113-2-111-1'], ['1802.09113-1-111-2', '1802.09113-2-111-2'], ['1802.09113-1-111-3', '1802.09113-2-111-3'], ['1802.09113-1-5-0', '1802.09113-2-5-0'], ['1802.09113-1-5-1', '1802.09113-2-5-1'], ['1802.09113-1-5-2', '1802.09113-2-5-2'], ['1802.09113-1-5-3', '1802.09113-2-5-3'], ['1802.09113-1-20-0', '1802.09113-2-20-0'], ['1802.09113-1-20-1', '1802.09113-2-20-1'], ['1802.09113-1-20-2', '1802.09113-2-20-2'], ['1802.09113-1-20-3', '1802.09113-2-20-3'], ['1802.09113-1-20-4', '1802.09113-2-20-4'], ['1802.09113-1-20-5', '1802.09113-2-20-5'], ['1802.09113-1-20-6', '1802.09113-2-20-6'], ['1802.09113-1-99-0', '1802.09113-2-99-0'], ['1802.09113-1-99-1', '1802.09113-2-99-1'], ['1802.09113-1-99-2', '1802.09113-2-99-2'], ['1802.09113-1-99-3', '1802.09113-2-99-3'], ['1802.09113-1-99-4', '1802.09113-2-99-4'], ['1802.09113-1-99-5', '1802.09113-2-99-5'], ['1802.09113-1-99-6', '1802.09113-2-99-6'], ['1802.09113-1-99-7', '1802.09113-2-99-7'], ['1802.09113-1-155-0', '1802.09113-2-155-0'], ['1802.09113-1-147-0', '1802.09113-2-147-0'], ['1802.09113-1-147-1', '1802.09113-2-147-1'], ['1802.09113-1-147-2', '1802.09113-2-147-2'], ['1802.09113-1-151-0', '1802.09113-2-151-0'], ['1802.09113-1-151-1', '1802.09113-2-151-1'], ['1802.09113-1-151-2', '1802.09113-2-151-2'], ['1802.09113-1-125-0', '1802.09113-2-125-0'], ['1802.09113-1-170-0', '1802.09113-2-170-0'], ['1802.09113-1-170-1', '1802.09113-2-170-1'], ['1802.09113-1-170-2', '1802.09113-2-170-2'], ['1802.09113-1-170-3', '1802.09113-2-170-3'], ['1802.09113-1-27-0', '1802.09113-2-27-0'], ['1802.09113-1-27-1', '1802.09113-2-27-1'], ['1802.09113-1-27-2', '1802.09113-2-27-2'], ['1802.09113-1-27-3', '1802.09113-2-27-3'], ['1802.09113-1-27-4', '1802.09113-2-27-4'], ['1802.09113-1-27-5', '1802.09113-2-27-5'], ['1802.09113-1-27-6', '1802.09113-2-27-6'], ['1802.09113-1-27-7', '1802.09113-2-27-7'], ['1802.09113-1-27-8', '1802.09113-2-27-8'], ['1802.09113-1-27-9', '1802.09113-2-27-9'], ['1802.09113-1-167-0', '1802.09113-2-167-0'], ['1802.09113-1-167-1', '1802.09113-2-167-1'], ['1802.09113-1-149-0', '1802.09113-2-149-0'], ['1802.09113-1-149-1', '1802.09113-2-149-1'], ['1802.09113-1-149-2', '1802.09113-2-149-2'], ['1802.09113-1-168-0', '1802.09113-2-168-0'], ['1802.09113-1-168-1', '1802.09113-2-168-1'], ['1802.09113-1-168-2', '1802.09113-2-168-2'], ['1802.09113-1-158-0', '1802.09113-2-158-0'], ['1802.09113-1-158-1', '1802.09113-2-158-1'], ['1802.09113-1-158-2', '1802.09113-2-158-2'], ['1802.09113-1-158-3', '1802.09113-2-158-3'], ['1802.09113-1-158-4', '1802.09113-2-158-4'], ['1802.09113-1-158-5', '1802.09113-2-158-5'], ['1802.09113-1-158-6', '1802.09113-2-158-6'], ['1802.09113-1-158-7', '1802.09113-2-158-7'], ['1802.09113-1-140-0', '1802.09113-2-140-0'], ['1802.09113-1-140-1', '1802.09113-2-140-1'], ['1802.09113-1-140-2', '1802.09113-2-140-2'], ['1802.09113-1-140-3', '1802.09113-2-140-3'], ['1802.09113-1-140-4', '1802.09113-2-140-4'], ['1802.09113-1-31-0', '1802.09113-2-31-0'], ['1802.09113-1-31-1', '1802.09113-2-31-1'], ['1802.09113-1-31-2', '1802.09113-2-31-2'], ['1802.09113-1-22-0', '1802.09113-2-22-0'], ['1802.09113-1-65-0', '1802.09113-2-65-0'], ['1802.09113-1-65-1', '1802.09113-2-65-1'], ['1802.09113-1-65-2', '1802.09113-2-65-2'], ['1802.09113-1-65-3', '1802.09113-2-65-3'], ['1802.09113-1-65-4', '1802.09113-2-65-4'], ['1802.09113-1-78-0', '1802.09113-2-78-0'], ['1802.09113-1-78-1', '1802.09113-2-78-1'], ['1802.09113-1-78-2', '1802.09113-2-78-2'], ['1802.09113-1-78-3', '1802.09113-2-78-3'], ['1802.09113-1-78-4', '1802.09113-2-78-4'], ['1802.09113-1-78-5', '1802.09113-2-78-5'], ['1802.09113-1-10-0', '1802.09113-2-10-0'], ['1802.09113-1-10-1', '1802.09113-2-10-1'], ['1802.09113-1-10-2', '1802.09113-2-10-2'], ['1802.09113-1-10-3', '1802.09113-2-10-3'], ['1802.09113-1-10-4', '1802.09113-2-10-4'], ['1802.09113-1-152-0', '1802.09113-2-152-0'], ['1802.09113-1-152-1', '1802.09113-2-152-1'], ['1802.09113-1-152-2', '1802.09113-2-152-2'], ['1802.09113-1-152-3', '1802.09113-2-152-3'], ['1802.09113-1-34-0', '1802.09113-2-34-0'], ['1802.09113-1-34-1', '1802.09113-2-34-1'], ['1802.09113-1-34-2', '1802.09113-2-34-2'], ['1802.09113-1-9-0', '1802.09113-2-9-0'], ['1802.09113-1-9-1', '1802.09113-2-9-1'], ['1802.09113-1-171-0', '1802.09113-2-171-0'], ['1802.09113-1-6-0', '1802.09113-2-6-0'], ['1802.09113-1-48-0', '1802.09113-2-48-0'], ['1802.09113-1-48-1', '1802.09113-2-48-1'], ['1802.09113-1-48-2', '1802.09113-2-48-2'], ['1802.09113-1-48-3', '1802.09113-2-48-3'], ['1802.09113-1-48-4', '1802.09113-2-48-4'], ['1802.09113-1-48-5', '1802.09113-2-48-5'], ['1802.09113-1-48-6', '1802.09113-2-48-6'], ['1802.09113-1-48-7', '1802.09113-2-48-7'], ['1802.09113-1-32-0', '1802.09113-2-32-0'], ['1802.09113-1-32-1', '1802.09113-2-32-1'], ['1802.09113-1-0-0', '1802.09113-2-0-0'], ['1802.09113-1-0-1', '1802.09113-2-0-1'], ['1802.09113-1-0-2', '1802.09113-2-0-2'], ['1802.09113-1-0-3', '1802.09113-2-0-3'], ['1802.09113-1-0-4', '1802.09113-2-0-4'], ['1802.09113-1-0-5', '1802.09113-2-0-5'], ['1802.09113-1-38-0', '1802.09113-2-38-0'], ['1802.09113-1-163-0', '1802.09113-2-163-0'], ['1802.09113-1-163-1', '1802.09113-2-163-1'], ['1802.09113-1-12-0', '1802.09113-2-12-0'], ['1802.09113-1-12-1', '1802.09113-2-12-1'], ['1802.09113-1-12-2', '1802.09113-2-12-2'], ['1802.09113-1-12-3', '1802.09113-2-12-3'], ['1802.09113-1-12-4', '1802.09113-2-12-4'], ['1802.09113-1-12-5', '1802.09113-2-12-5'], ['1802.09113-1-12-6', '1802.09113-2-12-6'], ['1802.09113-1-29-0', '1802.09113-2-29-0'], ['1802.09113-1-29-1', '1802.09113-2-29-1'], ['1802.09113-1-29-2', '1802.09113-2-29-2'], ['1802.09113-1-29-4', '1802.09113-2-29-4'], ['1802.09113-1-29-5', '1802.09113-2-29-5'], ['1802.09113-1-29-6', '1802.09113-2-29-6'], ['1802.09113-1-29-7', '1802.09113-2-29-7'], ['1802.09113-1-179-0', '1802.09113-2-179-0'], ['1802.09113-1-179-1', '1802.09113-2-179-1'], ['1802.09113-1-179-2', '1802.09113-2-179-2'], ['1802.09113-1-179-3', '1802.09113-2-179-3'], ['1802.09113-1-179-4', '1802.09113-2-179-4'], ['1802.09113-1-179-5', '1802.09113-2-179-5'], ['1802.09113-1-166-0', '1802.09113-2-166-0'], ['1802.09113-1-166-1', '1802.09113-2-166-1'], ['1802.09113-1-166-2', '1802.09113-2-166-2'], ['1802.09113-1-138-0', '1802.09113-2-138-0'], ['1802.09113-1-138-1', '1802.09113-2-138-1'], ['1802.09113-1-138-2', '1802.09113-2-138-2'], ['1802.09113-1-138-3', '1802.09113-2-138-3'], ['1802.09113-1-138-4', '1802.09113-2-138-4'], ['1802.09113-1-162-0', '1802.09113-2-162-0'], ['1802.09113-1-162-1', '1802.09113-2-162-1'], ['1802.09113-1-162-2', '1802.09113-2-162-2'], ['1802.09113-1-162-3', '1802.09113-2-162-3'], ['1802.09113-1-162-4', '1802.09113-2-162-4'], ['1802.09113-1-7-0', '1802.09113-2-7-0'], ['1802.09113-1-7-1', '1802.09113-2-7-1'], ['1802.09113-1-7-2', '1802.09113-2-7-2'], ['1802.09113-1-7-3', '1802.09113-2-7-3'], ['1802.09113-1-7-4', '1802.09113-2-7-4'], ['1802.09113-1-7-5', '1802.09113-2-7-5'], ['1802.09113-1-7-6', '1802.09113-2-7-6'], ['1802.09113-1-7-7', '1802.09113-2-7-7'], ['1802.09113-1-7-8', '1802.09113-2-7-8'], ['1802.09113-1-7-9', '1802.09113-2-7-9'], ['1802.09113-1-7-10', '1802.09113-2-7-10'], ['1802.09113-1-92-0', '1802.09113-2-92-0'], ['1802.09113-1-1-0', '1802.09113-2-1-0'], ['1802.09113-1-1-1', '1802.09113-2-1-1'], ['1802.09113-1-1-2', '1802.09113-2-1-2'], ['1802.09113-1-1-3', '1802.09113-2-1-3'], ['1802.09113-1-1-4', '1802.09113-2-1-4'], ['1802.09113-1-41-0', '1802.09113-2-41-0'], ['1802.09113-1-19-0', '1802.09113-2-19-0'], ['1802.09113-1-19-1', '1802.09113-2-19-1']]

[['1802.09113-1-3-0', '1802.09113-2-3-0'], ['1802.09113-1-3-1', '1802.09113-2-3-1'], ['1802.09113-1-3-2', '1802.09113-2-3-2'], ['1802.09113-1-3-3', '1802.09113-2-3-3'], ['1802.09113-1-3-4', '1802.09113-2-3-4'], ['1802.09113-1-3-5', '1802.09113-2-3-5'], ['1802.09113-1-3-6', '1802.09113-2-3-6'], ['1802.09113-1-3-7', '1802.09113-2-3-7'], ['1802.09113-1-3-8', '1802.09113-2-3-8'], ['1802.09113-1-3-9', '1802.09113-2-3-9'], ['1802.09113-1-176-0', '1802.09113-2-176-0'], ['1802.09113-1-176-1', '1802.09113-2-176-1'], ['1802.09113-1-176-2', '1802.09113-2-176-2'], ['1802.09113-1-176-3', '1802.09113-2-176-3'], ['1802.09113-1-174-0', '1802.09113-2-174-0'], ['1802.09113-1-174-1', '1802.09113-2-174-1'], ['1802.09113-1-174-2', '1802.09113-2-174-2'], ['1802.09113-1-174-3', '1802.09113-2-174-3'], ['1802.09113-1-178-0', '1802.09113-2-178-0'], ['1802.09113-1-178-1', '1802.09113-2-178-1'], ['1802.09113-1-178-2', '1802.09113-2-178-2'], ['1802.09113-1-178-3', '1802.09113-2-178-3'], ['1802.09113-1-178-4', '1802.09113-2-178-4'], ['1802.09113-1-178-5', '1802.09113-2-178-5'], ['1802.09113-1-178-6', '1802.09113-2-178-6'], ['1802.09113-1-178-7', '1802.09113-2-178-7'], ['1802.09113-1-178-8', '1802.09113-2-178-8'], ['1802.09113-1-178-9', '1802.09113-2-178-9'], ['1802.09113-1-145-0', '1802.09113-2-145-0'], ['1802.09113-1-145-1', '1802.09113-2-145-1'], ['1802.09113-1-145-2', '1802.09113-2-145-2'], ['1802.09113-1-145-3', '1802.09113-2-145-3'], ['1802.09113-1-145-4', '1802.09113-2-145-4'], ['1802.09113-1-145-5', '1802.09113-2-145-5'], ['1802.09113-1-14-0', '1802.09113-2-14-0'], ['1802.09113-1-14-1', '1802.09113-2-14-1'], ['1802.09113-1-14-2', '1802.09113-2-14-2'], ['1802.09113-1-14-3', '1802.09113-2-14-3'], ['1802.09113-1-14-4', '1802.09113-2-14-4'], ['1802.09113-1-4-0', '1802.09113-2-4-0'], ['1802.09113-1-4-1', '1802.09113-2-4-1'], ['1802.09113-1-4-2', '1802.09113-2-4-2'], ['1802.09113-1-4-3', '1802.09113-2-4-3'], ['1802.09113-1-4-4', '1802.09113-2-4-4'], ['1802.09113-1-4-5', '1802.09113-2-4-5'], ['1802.09113-1-154-0', '1802.09113-2-154-0'], ['1802.09113-1-154-1', '1802.09113-2-154-1'], ['1802.09113-1-154-2', '1802.09113-2-154-2'], ['1802.09113-1-154-3', '1802.09113-2-154-3'], ['1802.09113-1-154-4', '1802.09113-2-154-4'], ['1802.09113-1-142-0', '1802.09113-2-142-0'], ['1802.09113-1-142-1', '1802.09113-2-142-1'], ['1802.09113-1-142-2', '1802.09113-2-142-2'], ['1802.09113-1-142-3', '1802.09113-2-142-3'], ['1802.09113-1-142-4', '1802.09113-2-142-4'], ['1802.09113-1-142-5', '1802.09113-2-142-5'], ['1802.09113-1-142-6', '1802.09113-2-142-6'], ['1802.09113-1-142-7', '1802.09113-2-142-7'], ['1802.09113-1-142-8', '1802.09113-2-142-8'], ['1802.09113-1-142-9', '1802.09113-2-142-9'], ['1802.09113-1-142-10', '1802.09113-2-142-10'], ['1802.09113-1-144-0', '1802.09113-2-144-0'], ['1802.09113-1-144-1', '1802.09113-2-144-1'], ['1802.09113-1-144-2', '1802.09113-2-144-2'], ['1802.09113-1-144-3', '1802.09113-2-144-3'], ['1802.09113-1-144-4', '1802.09113-2-144-4'], ['1802.09113-1-160-0', '1802.09113-2-160-0'], ['1802.09113-1-160-1', '1802.09113-2-160-1'], ['1802.09113-1-160-2', '1802.09113-2-160-2'], ['1802.09113-1-160-3', '1802.09113-2-160-3'], ['1802.09113-1-160-4', '1802.09113-2-160-4'], ['1802.09113-1-160-5', '1802.09113-2-160-5'], ['1802.09113-1-160-6', '1802.09113-2-160-6'], ['1802.09113-1-165-0', '1802.09113-2-165-0'], ['1802.09113-1-165-1', '1802.09113-2-165-1'], ['1802.09113-1-165-2', '1802.09113-2-165-2'], ['1802.09113-1-165-3', '1802.09113-2-165-3'], ['1802.09113-1-165-4', '1802.09113-2-165-4'], ['1802.09113-1-17-0', '1802.09113-2-17-0'], ['1802.09113-1-17-1', '1802.09113-2-17-1'], ['1802.09113-1-17-2', '1802.09113-2-17-2'], ['1802.09113-1-17-3', '1802.09113-2-17-3'], ['1802.09113-1-17-4', '1802.09113-2-17-4'], ['1802.09113-1-17-5', '1802.09113-2-17-5'], ['1802.09113-1-17-6', '1802.09113-2-17-6'], ['1802.09113-1-17-7', '1802.09113-2-17-7'], ['1802.09113-1-13-0', '1802.09113-2-13-0'], ['1802.09113-1-157-0', '1802.09113-2-157-0'], ['1802.09113-1-157-1', '1802.09113-2-157-1'], ['1802.09113-1-157-2', '1802.09113-2-157-2'], ['1802.09113-1-157-3', '1802.09113-2-157-3'], ['1802.09113-1-111-0', '1802.09113-2-111-0'], ['1802.09113-1-111-1', '1802.09113-2-111-1'], ['1802.09113-1-111-2', '1802.09113-2-111-2'], ['1802.09113-1-111-3', '1802.09113-2-111-3'], ['1802.09113-1-5-0', '1802.09113-2-5-0'], ['1802.09113-1-5-1', '1802.09113-2-5-1'], ['1802.09113-1-5-2', '1802.09113-2-5-2'], ['1802.09113-1-5-3', '1802.09113-2-5-3'], ['1802.09113-1-20-0', '1802.09113-2-20-0'], ['1802.09113-1-20-1', '1802.09113-2-20-1'], ['1802.09113-1-20-2', '1802.09113-2-20-2'], ['1802.09113-1-20-3', '1802.09113-2-20-3'], ['1802.09113-1-20-4', '1802.09113-2-20-4'], ['1802.09113-1-20-5', '1802.09113-2-20-5'], ['1802.09113-1-20-6', '1802.09113-2-20-6'], ['1802.09113-1-99-0', '1802.09113-2-99-0'], ['1802.09113-1-99-1', '1802.09113-2-99-1'], ['1802.09113-1-99-2', '1802.09113-2-99-2'], ['1802.09113-1-99-3', '1802.09113-2-99-3'], ['1802.09113-1-99-4', '1802.09113-2-99-4'], ['1802.09113-1-99-5', '1802.09113-2-99-5'], ['1802.09113-1-99-6', '1802.09113-2-99-6'], ['1802.09113-1-99-7', '1802.09113-2-99-7'], ['1802.09113-1-155-0', '1802.09113-2-155-0'], ['1802.09113-1-147-0', '1802.09113-2-147-0'], ['1802.09113-1-147-1', '1802.09113-2-147-1'], ['1802.09113-1-147-2', '1802.09113-2-147-2'], ['1802.09113-1-151-0', '1802.09113-2-151-0'], ['1802.09113-1-151-1', '1802.09113-2-151-1'], ['1802.09113-1-151-2', '1802.09113-2-151-2'], ['1802.09113-1-125-0', '1802.09113-2-125-0'], ['1802.09113-1-170-0', '1802.09113-2-170-0'], ['1802.09113-1-170-1', '1802.09113-2-170-1'], ['1802.09113-1-170-2', '1802.09113-2-170-2'], ['1802.09113-1-170-3', '1802.09113-2-170-3'], ['1802.09113-1-27-0', '1802.09113-2-27-0'], ['1802.09113-1-27-1', '1802.09113-2-27-1'], ['1802.09113-1-27-2', '1802.09113-2-27-2'], ['1802.09113-1-27-3', '1802.09113-2-27-3'], ['1802.09113-1-27-4', '1802.09113-2-27-4'], ['1802.09113-1-27-5', '1802.09113-2-27-5'], ['1802.09113-1-27-6', '1802.09113-2-27-6'], ['1802.09113-1-27-7', '1802.09113-2-27-7'], ['1802.09113-1-27-8', '1802.09113-2-27-8'], ['1802.09113-1-27-9', '1802.09113-2-27-9'], ['1802.09113-1-167-0', '1802.09113-2-167-0'], ['1802.09113-1-167-1', '1802.09113-2-167-1'], ['1802.09113-1-149-0', '1802.09113-2-149-0'], ['1802.09113-1-149-1', '1802.09113-2-149-1'], ['1802.09113-1-149-2', '1802.09113-2-149-2'], ['1802.09113-1-168-0', '1802.09113-2-168-0'], ['1802.09113-1-168-1', '1802.09113-2-168-1'], ['1802.09113-1-168-2', '1802.09113-2-168-2'], ['1802.09113-1-158-0', '1802.09113-2-158-0'], ['1802.09113-1-158-1', '1802.09113-2-158-1'], ['1802.09113-1-158-2', '1802.09113-2-158-2'], ['1802.09113-1-158-3', '1802.09113-2-158-3'], ['1802.09113-1-158-4', '1802.09113-2-158-4'], ['1802.09113-1-158-5', '1802.09113-2-158-5'], ['1802.09113-1-158-6', '1802.09113-2-158-6'], ['1802.09113-1-158-7', '1802.09113-2-158-7'], ['1802.09113-1-140-0', '1802.09113-2-140-0'], ['1802.09113-1-140-1', '1802.09113-2-140-1'], ['1802.09113-1-140-2', '1802.09113-2-140-2'], ['1802.09113-1-140-3', '1802.09113-2-140-3'], ['1802.09113-1-140-4', '1802.09113-2-140-4'], ['1802.09113-1-31-0', '1802.09113-2-31-0'], ['1802.09113-1-31-1', '1802.09113-2-31-1'], ['1802.09113-1-31-2', '1802.09113-2-31-2'], ['1802.09113-1-22-0', '1802.09113-2-22-0'], ['1802.09113-1-65-0', '1802.09113-2-65-0'], ['1802.09113-1-65-1', '1802.09113-2-65-1'], ['1802.09113-1-65-2', '1802.09113-2-65-2'], ['1802.09113-1-65-3', '1802.09113-2-65-3'], ['1802.09113-1-65-4', '1802.09113-2-65-4'], ['1802.09113-1-78-0', '1802.09113-2-78-0'], ['1802.09113-1-78-1', '1802.09113-2-78-1'], ['1802.09113-1-78-2', '1802.09113-2-78-2'], ['1802.09113-1-78-3', '1802.09113-2-78-3'], ['1802.09113-1-78-4', '1802.09113-2-78-4'], ['1802.09113-1-78-5', '1802.09113-2-78-5'], ['1802.09113-1-10-0', '1802.09113-2-10-0'], ['1802.09113-1-10-1', '1802.09113-2-10-1'], ['1802.09113-1-10-2', '1802.09113-2-10-2'], ['1802.09113-1-10-3', '1802.09113-2-10-3'], ['1802.09113-1-10-4', '1802.09113-2-10-4'], ['1802.09113-1-152-0', '1802.09113-2-152-0'], ['1802.09113-1-152-1', '1802.09113-2-152-1'], ['1802.09113-1-152-2', '1802.09113-2-152-2'], ['1802.09113-1-152-3', '1802.09113-2-152-3'], ['1802.09113-1-34-0', '1802.09113-2-34-0'], ['1802.09113-1-34-1', '1802.09113-2-34-1'], ['1802.09113-1-34-2', '1802.09113-2-34-2'], ['1802.09113-1-9-0', '1802.09113-2-9-0'], ['1802.09113-1-9-1', '1802.09113-2-9-1'], ['1802.09113-1-171-0', '1802.09113-2-171-0'], ['1802.09113-1-6-0', '1802.09113-2-6-0'], ['1802.09113-1-48-0', '1802.09113-2-48-0'], ['1802.09113-1-48-1', '1802.09113-2-48-1'], ['1802.09113-1-48-2', '1802.09113-2-48-2'], ['1802.09113-1-48-3', '1802.09113-2-48-3'], ['1802.09113-1-48-4', '1802.09113-2-48-4'], ['1802.09113-1-48-5', '1802.09113-2-48-5'], ['1802.09113-1-48-6', '1802.09113-2-48-6'], ['1802.09113-1-48-7', '1802.09113-2-48-7'], ['1802.09113-1-32-0', '1802.09113-2-32-0'], ['1802.09113-1-32-1', '1802.09113-2-32-1'], ['1802.09113-1-0-0', '1802.09113-2-0-0'], ['1802.09113-1-0-1', '1802.09113-2-0-1'], ['1802.09113-1-0-2', '1802.09113-2-0-2'], ['1802.09113-1-0-3', '1802.09113-2-0-3'], ['1802.09113-1-0-4', '1802.09113-2-0-4'], ['1802.09113-1-0-5', '1802.09113-2-0-5'], ['1802.09113-1-38-0', '1802.09113-2-38-0'], ['1802.09113-1-163-0', '1802.09113-2-163-0'], ['1802.09113-1-163-1', '1802.09113-2-163-1'], ['1802.09113-1-12-0', '1802.09113-2-12-0'], ['1802.09113-1-12-1', '1802.09113-2-12-1'], ['1802.09113-1-12-2', '1802.09113-2-12-2'], ['1802.09113-1-12-3', '1802.09113-2-12-3'], ['1802.09113-1-12-4', '1802.09113-2-12-4'], ['1802.09113-1-12-5', '1802.09113-2-12-5'], ['1802.09113-1-12-6', '1802.09113-2-12-6'], ['1802.09113-1-29-0', '1802.09113-2-29-0'], ['1802.09113-1-29-1', '1802.09113-2-29-1'], ['1802.09113-1-29-2', '1802.09113-2-29-2'], ['1802.09113-1-29-4', '1802.09113-2-29-4'], ['1802.09113-1-29-5', '1802.09113-2-29-5'], ['1802.09113-1-29-6', '1802.09113-2-29-6'], ['1802.09113-1-29-7', '1802.09113-2-29-7'], ['1802.09113-1-179-0', '1802.09113-2-179-0'], ['1802.09113-1-179-1', '1802.09113-2-179-1'], ['1802.09113-1-179-2', '1802.09113-2-179-2'], ['1802.09113-1-179-3', '1802.09113-2-179-3'], ['1802.09113-1-179-4', '1802.09113-2-179-4'], ['1802.09113-1-179-5', '1802.09113-2-179-5'], ['1802.09113-1-166-0', '1802.09113-2-166-0'], ['1802.09113-1-166-1', '1802.09113-2-166-1'], ['1802.09113-1-166-2', '1802.09113-2-166-2'], ['1802.09113-1-138-0', '1802.09113-2-138-0'], ['1802.09113-1-138-1', '1802.09113-2-138-1'], ['1802.09113-1-138-2', '1802.09113-2-138-2'], ['1802.09113-1-138-3', '1802.09113-2-138-3'], ['1802.09113-1-138-4', '1802.09113-2-138-4'], ['1802.09113-1-162-0', '1802.09113-2-162-0'], ['1802.09113-1-162-1', '1802.09113-2-162-1'], ['1802.09113-1-162-2', '1802.09113-2-162-2'], ['1802.09113-1-162-3', '1802.09113-2-162-3'], ['1802.09113-1-162-4', '1802.09113-2-162-4'], ['1802.09113-1-7-0', '1802.09113-2-7-0'], ['1802.09113-1-7-1', '1802.09113-2-7-1'], ['1802.09113-1-7-2', '1802.09113-2-7-2'], ['1802.09113-1-7-3', '1802.09113-2-7-3'], ['1802.09113-1-7-4', '1802.09113-2-7-4'], ['1802.09113-1-7-5', '1802.09113-2-7-5'], ['1802.09113-1-7-6', '1802.09113-2-7-6'], ['1802.09113-1-7-7', '1802.09113-2-7-7'], ['1802.09113-1-7-8', '1802.09113-2-7-8'], ['1802.09113-1-7-9', '1802.09113-2-7-9'], ['1802.09113-1-7-10', '1802.09113-2-7-10'], ['1802.09113-1-92-0', '1802.09113-2-92-0'], ['1802.09113-1-1-0', '1802.09113-2-1-0'], ['1802.09113-1-1-1', '1802.09113-2-1-1'], ['1802.09113-1-1-2', '1802.09113-2-1-2'], ['1802.09113-1-1-3', '1802.09113-2-1-3'], ['1802.09113-1-1-4', '1802.09113-2-1-4'], ['1802.09113-1-41-0', '1802.09113-2-41-0'], ['1802.09113-1-19-0', '1802.09113-2-19-0'], ['1802.09113-1-19-1', '1802.09113-2-19-1']]

[]

[]

[]

[]

['1802.09113-1-8-0', '1802.09113-1-21-0', '1802.09113-1-21-1', '1802.09113-1-23-0', '1802.09113-1-24-0', '1802.09113-1-25-0', '1802.09113-1-29-3', '1802.09113-1-35-0', '1802.09113-1-36-0', '1802.09113-1-36-1', '1802.09113-1-37-0', '1802.09113-1-39-0', '1802.09113-1-40-0', '1802.09113-1-40-1', '1802.09113-1-42-0', '1802.09113-1-43-0', '1802.09113-1-44-0', '1802.09113-1-45-0', '1802.09113-1-46-0', '1802.09113-1-47-0', '1802.09113-1-49-0', '1802.09113-1-50-0', '1802.09113-1-50-1', '1802.09113-1-51-0', '1802.09113-1-52-0', '1802.09113-1-53-0', '1802.09113-1-54-0', '1802.09113-1-55-0', '1802.09113-1-56-0', '1802.09113-1-57-0', '1802.09113-1-58-0', '1802.09113-1-59-0', '1802.09113-1-60-0', '1802.09113-1-61-0', '1802.09113-1-62-0', '1802.09113-1-63-0', '1802.09113-1-64-0', '1802.09113-1-65-5', '1802.09113-1-66-0', '1802.09113-1-66-1', '1802.09113-1-67-0', '1802.09113-1-68-0', '1802.09113-1-69-0', '1802.09113-1-70-0', '1802.09113-1-71-0', '1802.09113-1-72-0', '1802.09113-1-72-1', '1802.09113-1-73-0', '1802.09113-1-73-1', '1802.09113-1-74-0', '1802.09113-1-74-1', '1802.09113-1-74-2', '1802.09113-1-75-0', '1802.09113-1-76-0', '1802.09113-1-77-0', '1802.09113-1-79-0', '1802.09113-1-80-0', '1802.09113-1-80-1', '1802.09113-1-81-0', '1802.09113-1-82-0', '1802.09113-1-83-0', '1802.09113-1-84-0', '1802.09113-1-85-0', '1802.09113-1-86-0', '1802.09113-1-86-1', '1802.09113-1-87-0', '1802.09113-1-87-1', '1802.09113-1-88-0', '1802.09113-1-88-1', '1802.09113-1-89-0', '1802.09113-1-90-0', '1802.09113-1-91-0', '1802.09113-1-93-0', '1802.09113-1-94-0', '1802.09113-1-95-0', '1802.09113-1-96-0', '1802.09113-1-97-0', '1802.09113-1-98-0', '1802.09113-1-99-8', '1802.09113-1-100-0', '1802.09113-1-100-1', '1802.09113-1-101-0', '1802.09113-1-102-0', '1802.09113-1-103-0', '1802.09113-1-104-0', '1802.09113-1-105-0', '1802.09113-1-106-0', '1802.09113-1-107-0', '1802.09113-1-108-0', '1802.09113-1-109-0', '1802.09113-1-110-0', '1802.09113-1-112-0', '1802.09113-1-113-0', '1802.09113-1-113-1', '1802.09113-1-114-0', '1802.09113-1-115-0', '1802.09113-1-116-0', '1802.09113-1-117-0', '1802.09113-1-118-0', '1802.09113-1-119-0', '1802.09113-1-119-1', '1802.09113-1-120-0', '1802.09113-1-120-1', '1802.09113-1-121-0', '1802.09113-1-121-1', '1802.09113-1-122-0', '1802.09113-1-123-0', '1802.09113-1-123-1', '1802.09113-1-124-0', '1802.09113-1-126-0', '1802.09113-1-127-0', '1802.09113-1-128-0', '1802.09113-1-128-1', '1802.09113-1-129-0', '1802.09113-1-130-0', '1802.09113-1-131-0', '1802.09113-1-132-0', '1802.09113-1-132-1', '1802.09113-1-133-0', '1802.09113-1-133-1', '1802.09113-1-134-0', '1802.09113-1-134-1', '1802.09113-1-135-0', '1802.09113-1-136-0', '1802.09113-1-137-0', '1802.09113-1-137-1', '1802.09113-2-8-0', '1802.09113-2-21-0', '1802.09113-2-21-1', '1802.09113-2-23-0', '1802.09113-2-24-0', '1802.09113-2-25-0', '1802.09113-2-29-3', '1802.09113-2-35-0', '1802.09113-2-36-0', '1802.09113-2-36-1', '1802.09113-2-37-0', '1802.09113-2-39-0', '1802.09113-2-40-0', '1802.09113-2-40-1', '1802.09113-2-42-0', '1802.09113-2-43-0', '1802.09113-2-44-0', '1802.09113-2-45-0', '1802.09113-2-46-0', '1802.09113-2-47-0', '1802.09113-2-49-0', '1802.09113-2-50-0', '1802.09113-2-50-1', '1802.09113-2-51-0', '1802.09113-2-52-0', '1802.09113-2-53-0', '1802.09113-2-54-0', '1802.09113-2-55-0', '1802.09113-2-56-0', '1802.09113-2-57-0', '1802.09113-2-58-0', '1802.09113-2-59-0', '1802.09113-2-60-0', '1802.09113-2-61-0', '1802.09113-2-62-0', '1802.09113-2-63-0', '1802.09113-2-64-0', '1802.09113-2-65-5', '1802.09113-2-66-0', '1802.09113-2-66-1', '1802.09113-2-67-0', '1802.09113-2-68-0', '1802.09113-2-69-0', '1802.09113-2-70-0', '1802.09113-2-71-0', '1802.09113-2-72-0', '1802.09113-2-72-1', '1802.09113-2-73-0', '1802.09113-2-73-1', '1802.09113-2-74-0', '1802.09113-2-74-1', '1802.09113-2-74-2', '1802.09113-2-75-0', '1802.09113-2-76-0', '1802.09113-2-77-0', '1802.09113-2-79-0', '1802.09113-2-80-0', '1802.09113-2-80-1', '1802.09113-2-81-0', '1802.09113-2-82-0', '1802.09113-2-83-0', '1802.09113-2-84-0', '1802.09113-2-85-0', '1802.09113-2-86-0', '1802.09113-2-86-1', '1802.09113-2-87-0', '1802.09113-2-87-1', '1802.09113-2-88-0', '1802.09113-2-88-1', '1802.09113-2-89-0', '1802.09113-2-90-0', '1802.09113-2-91-0', '1802.09113-2-93-0', '1802.09113-2-94-0', '1802.09113-2-95-0', '1802.09113-2-96-0', '1802.09113-2-97-0', '1802.09113-2-98-0', '1802.09113-2-99-8', '1802.09113-2-100-0', '1802.09113-2-100-1', '1802.09113-2-101-0', '1802.09113-2-102-0', '1802.09113-2-103-0', '1802.09113-2-104-0', '1802.09113-2-105-0', '1802.09113-2-106-0', '1802.09113-2-107-0', '1802.09113-2-108-0', '1802.09113-2-109-0', '1802.09113-2-110-0', '1802.09113-2-112-0', '1802.09113-2-113-0', '1802.09113-2-113-1', '1802.09113-2-114-0', '1802.09113-2-115-0', '1802.09113-2-116-0', '1802.09113-2-117-0', '1802.09113-2-118-0', '1802.09113-2-119-0', '1802.09113-2-119-1', '1802.09113-2-120-0', '1802.09113-2-120-1', '1802.09113-2-121-0', '1802.09113-2-121-1', '1802.09113-2-122-0', '1802.09113-2-123-0', '1802.09113-2-123-1', '1802.09113-2-124-0', '1802.09113-2-126-0', '1802.09113-2-127-0', '1802.09113-2-128-0', '1802.09113-2-128-1', '1802.09113-2-129-0', '1802.09113-2-130-0', '1802.09113-2-131-0', '1802.09113-2-132-0', '1802.09113-2-132-1', '1802.09113-2-133-0', '1802.09113-2-133-1', '1802.09113-2-134-0', '1802.09113-2-134-1', '1802.09113-2-135-0', '1802.09113-2-136-0', '1802.09113-2-137-0', '1802.09113-2-137-1']

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

https://arxiv.org/abs/1802.09113

1611.02287

{'1611.02287-1-0-0': 'We study the effect of the elastic scattering on the non-thermally produced WIMP dark matter and its phenomenological consequences.', '1611.02287-1-0-1': 'The non-thermal WIMP becomes important when the reheating temperature is well below the freeze-out temperature.', '1611.02287-1-0-2': 'In the usual paradigm, the produced high energetic dark matter particles are quickly thermalized due to the elastic scattering with background radiations.', '1611.02287-1-0-3': 'The relic abundance is determined by the thermally averaged annihilation cross-section times velocity at the reheating temperature.', '1611.02287-1-0-4': 'In the opposite limit, the initial abundance is too small for the dark matter to annihilate so that the relic density is determined by the branching fraction of the heavy particle.', '1611.02287-1-0-5': 'We study the regions between these two limits, and show that the relic density depends not only on the annihilation rate, but also on the elastic scattering rate.', '1611.02287-1-0-6': 'Especially, the relic abundance of the [MATH]-wave annihilating dark matter crucially relies on the elastic scattering rate because the annihilation cross-section is sensitive to the dark matter velocity.', '1611.02287-1-0-7': 'We categorize the parameter space into several regions where each region has distinctive mechanism for determining the relic abundance of the dark matter at the present Universe.', '1611.02287-1-0-8': 'The consequence on the (in)direct detection is also studied.', '1611.02287-1-1-0': 'APCTP Pre2016-021 CTPU-16-32', '1611.02287-1-2-0': 'IPMU16-0164', '1611.02287-1-3-0': '# Introduction', '1611.02287-1-4-0': 'The weakly interacting massive particle (WIMP) is one of the promising dark matter (DM) candidates because it can be naturally incorporated in new physics beyond the Standard Model, and it gives interesting observable consequences.', '1611.02287-1-5-0': 'In the standard thermal history, the most important quantity to determine the relic density is a thermal averaged pair annihilation cross-section, [MATH].', '1611.02287-1-5-1': 'Taking "thermal average" is justified because the elastic scattering rate between the WIMP dark matter and the background radiation is much bigger than the annihilation rate, so the kinetic decoupling happens well after the dark matter freeze-out [CITATION].', '1611.02287-1-5-2': 'Usually, the elastic scattering does not have the special role to determine the relic density, but it is crucial for small scale structures since the interaction suppresses the growth of the dark matter density perturbation [CITATION].', '1611.02287-1-5-3': 'When the reheating temperature is low, its effect is more interesting depending on whether the kinetic decoupling happens before or after the end of reheating [CITATION].', '1611.02287-1-6-0': 'In this letter, we study the possibility that the relic abundance of WIMP explicitly depends on the elastic scattering rate.', '1611.02287-1-6-1': 'In the context of self interacting dark matter, such possibility is realized by noting that the dominant annihilation rate from [MATH] scattering and the elastic scattering between the DM and thermal bath can be independent so that we can take suitable parameters to show such behavior [CITATION].', '1611.02287-1-6-2': 'This assumption is not usually valid in the case of WIMP because the annihilation and elastic scattering cannot be treated independently.', '1611.02287-1-7-0': 'However, when the reheating temperature is well below the dark matter mass, a new possibility emerges.', '1611.02287-1-7-1': 'At the end of reheating, the dark matter is produced by the direct decay of heavy particles.', '1611.02287-1-7-2': 'Such non-thermally produced dark matter particles have very high energies.', '1611.02287-1-7-3': 'Evolution of the dark matter momentum gives a strong effect on the annihilation cross-section, and such evolution is determined by the elastic scattering rate.', '1611.02287-1-7-4': 'Consequently, the relative size of the annihilation rate, the elastic scattering rate, and the Hubble rate at the end of reheating can give various mechanisms to determine the final relic density of the dark matter.', '1611.02287-1-7-5': 'We classify the parameter space into the regions where each region has distinctive mechanism to determine the relic density of the dark matter.', '1611.02287-1-7-6': 'We also provide analytic expressions and numerical results for each of those mechanisms.', '1611.02287-1-7-7': 'Especially, we find that the [MATH]-wave annihilating dark matter has more interesting property because the cross-section highly depends on the expectation value of the dark matter momentum.', '1611.02287-1-8-0': 'In section [REF], we present our basic set-up.', '1611.02287-1-8-1': 'In section [REF], we compute the momentum evolution of the dark matter after its production at the end of reheating.', '1611.02287-1-8-2': 'The effect of the momentum evolution on the annihilation cross-section and the corresponding final abundance of the dark matter are discussed in section [REF].', '1611.02287-1-8-3': 'We discuss the constraints from (in)direct detection experiments in section [REF], and conclude in section [REF].', '1611.02287-1-9-0': '# Thermal History of the Non-thermal WIMP', '1611.02287-1-10-0': 'In our set-up, there is the early stage of the matter dominated Universe maintained by a long lived heavy particle, [MATH].', '1611.02287-1-10-1': 'After most of [MATH] decay, the Universe is "reheated" and radiation ([MATH]) starts to dominate the energy density of the Universe with a reheating temperature, [MATH].', '1611.02287-1-10-2': 'On one hand, the dark matter ([MATH]) can be produced either from the scattering of the radiation background, or from the direct decay of [MATH] with a branching fraction [MATH].', '1611.02287-1-11-0': 'Ignoring the sub-leading contributions, the corresponding Boltzmann equations of each components are given as _=&-3 H _-__,', '1611.02287-1-12-0': '_= & - 4 H _+ Br___,', '1611.02287-1-13-0': 'n_= &-3 H n_+Br___m_ -_ann v_rel_n_^2,', '1611.02287-1-14-0': '& + _ann v_rel_T (n_^eq)^2,', '1611.02287-1-15-0': 'H=& _+ _+ _3M_Pl^2, where [MATH] is the reduced Planck mass.', '1611.02287-1-15-1': 'Here we consider a situation that the reheating temperature is lower than the thermal freeze-out temperature of the dark matter ([MATH]).', '1611.02287-1-15-2': 'Before the end of the reheating, [MATH], there are several sources for the dark matter density.', '1611.02287-1-15-3': 'First of all, a usual freeze-out mechanism can work with [MATH], while the resulting abundance is subsequently diluted by continuous entropy injection.', '1611.02287-1-15-4': 'If [MATH] is big, quasi-static equilibrium state can persist until the end of reheating ([MATH]) [CITATION].', '1611.02287-1-15-5': 'Also if [MATH] is large enough to satisfy [MATH], production from an inelastic scattering between thermal bath and a boosted radiation produced by [MATH] decays becomes important [CITATION].', '1611.02287-1-15-6': 'Here we take a rather moderate hierarchy between the mass of [MATH] and [MATH] as [MATH], and a sub GeV reheating temperature so that the inelastic scattering is subdominant.', '1611.02287-1-15-7': 'For [MATH] and a sizable [MATH], the most important source of the late time dark matter abundance is the direct decay of [MATH] at the end of reheating.', '1611.02287-1-15-8': 'It is known that such non-thermal production of the DM can be simplified by assuming that the dark matter is instantaneously produced from the heavy particle decay at [MATH] with an initial amount of the DM given as [MATH] [CITATION].', '1611.02287-1-15-9': 'In summary, we are interested in the following range of parameters: T_rehT_fr m_m_O(10-100)m_.', '1611.02287-1-15-10': 'Because of the hierarchy between masses, the initial energy of the DM is much greater than [MATH].', '1611.02287-1-15-11': 'So we first consider the evolution of the dark matter momentum and then consider its effect on the annihilation rate.', '1611.02287-1-16-0': '# Evolution of the DM momentum', '1611.02287-1-17-0': 'After the dark matter is produced, it experiences two types of interactions.', '1611.02287-1-17-1': 'One is the elastic scattering by the background radiations ([MATH]).', '1611.02287-1-17-2': 'The other one is pair annihilation of the dark matter into the radiations ([MATH]).', '1611.02287-1-17-3': 'The effect of pair production from thermal bath [MATH] is negligible if the DM abundance at [MATH] is much larger than the equilibrium value.', '1611.02287-1-17-4': 'This fact makes the analysis much easier.', '1611.02287-1-17-5': 'Assuming that the annihilation does not crucially modify the overall shape of momentum distribution, the evolution of the dark matter momentum, [MATH], is determined by the elastic scattering.', '1611.02287-1-17-6': 'The corresponding Boltzmann equation is [CITATION]', '1611.02287-1-18-0': 'd p_dt + H p_= - _el v_rel p__,T n_,', '1611.02287-1-19-0': 'where [MATH] is the average over the initial distribution of [MATH] and [MATH] in the rest frame of the thermal plasma, [MATH] is the change of the dark matter momentum from single event of the elastic scattering, and [MATH] is the number density of the background radiation, [MATH].', '1611.02287-1-20-0': 'The right hand side of [REF] can be simplified as [MATH] times _el v_rel p_p__, T .', '1611.02287-1-21-0': 'for different ranges of the dark matter momentum, (I)& m_^2 p_T,', '1611.02287-1-22-0': '(II)& p_T m_^2 p_^2,', '1611.02287-1-23-0': '(III)& p_T p_^2 m_^2.', '1611.02287-1-23-1': 'In ([MATH]), the DM is relativistic in the plasma rest frame, and in the center of mass (cm) frames.', '1611.02287-1-23-2': 'In ([MATH]), the DM is non-relativistic in the cm frame, whereas still relativistic in the plasma rest frame.', '1611.02287-1-23-3': 'In ([MATH]), the DM is non-relativistic in both frames.', '1611.02287-1-23-4': 'In the last case, the additional factor in the elastic scattering rate drives [MATH] to the equilibrium value, [MATH].', '1611.02287-1-23-5': 'We can understand the additional factors of the scattering rate, [MATH], [MATH], [MATH] from the fact that the allowed phase space becomes wider as the collision energies become higher.', '1611.02287-1-23-6': 'Since the common factor [MATH] depends on the temperature, the DM can quickly arrive at kinetic equilibrium or it can just decouple relativistically depending on [MATH].', '1611.02287-1-24-0': 'When the initial momentum of the dark matter is much greater than [MATH], Fig. [REF] shows possible evolution of the momentum for different reheating temperatures.', '1611.02287-1-24-1': 'At a relatively high reheating temperature, the elastic scattering rate is large enough to make the dark matter in kinetic equilibrium instantaneously after its production.', '1611.02287-1-24-2': 'As the temperature goes down, the momentum follows the equilibrium value ([MATH]) until the kinetic decoupling.', '1611.02287-1-24-3': 'If the reheating temperature is relatively low, after the dark matter momentum experiences a small sharp suppression around [MATH], it slowly decreases as [MATH], and it could become non-relativistic well after reheating (magenta line).', '1611.02287-1-25-0': 'There is a natural connection between the momentum evolution and the dark matter pair annihilation rate.', '1611.02287-1-25-1': 'For example, if they are highly relativistic, the cross-section becomes [MATH], and it will increase as the energy of the particle decreases.', '1611.02287-1-25-2': 'Therefore, if the dark matter is not instantaneously thermalized, the annihilation of the dark matter could happen later when the annihilation cross-section becomes large enough to start the annihilation.', '1611.02287-1-26-0': 'Since the corresponding dark matter abundance is affected by the evolution of the annihilation cross-section, we can find the connection between the final yield of the dark matter, and the elastic scattering rate.', '1611.02287-1-27-0': '# Evolution of the pair annihilation cross-section', '1611.02287-1-28-0': 'When the initial abundance ([MATH]) is much greater than [MATH], the production of [MATH] from thermal bath can be ignored, and the corresponding Boltzmann equation for the dark matter number density is simplified as n_+ 3 H n_= -_ann v_rel_ n_^2.', '1611.02287-1-28-1': 'Solving the above equation, we find the yield of the dark matter at the present time, [MATH], as Y_(t_0) =& Y_(t_reh) (1+ n^reh_H_reh_0^1 d u _ann v_rel_p_(u) )^-1,where [MATH], [MATH] is the Hubble rate, and [MATH] is the entropy of the Universe at [MATH].', '1611.02287-1-28-2': 'The yields are denoted by [MATH], [MATH].', '1611.02287-1-28-3': 'The time dependence of [MATH] is determined by that of [MATH] governed by Eq. ([REF]).', '1611.02287-1-28-4': 'Two limiting cases are familiar.', '1611.02287-1-28-5': 'One is that [MATH] quickly arrives at its equilibrium value within the period much shorter than the Hubble time as given in Fig. [REF] with blue color.', '1611.02287-1-28-6': 'The dark matter annihilation happens after its thermalization but still much faster than the Hubble expansion rate.', '1611.02287-1-28-7': 'Therefore, [MATH].', '1611.02287-1-28-8': 'The other limit is that the initial abundance is too small so that [MATH].', '1611.02287-1-28-9': 'Annihilation barely happens, and the yield is preserved; [MATH].', '1611.02287-1-28-10': 'In both cases, the final yields do not explicitly depend on the elastic scattering cross-section.', '1611.02287-1-29-0': 'There is an intermediate domain between these two limiting cases.', '1611.02287-1-29-1': 'Including the above examples, we identify three mechanisms for the relic density of the DM.', '1611.02287-1-29-2': 'After the production of the DM from the direct decay of the heavy particles, the relic abundance is determined by one of the following mechanisms:', '1611.02287-1-30-0': 'The rates are given by _ann(T, E_) &= n__ann v_rel_,', '1611.02287-1-31-0': '_el(T, E_)&= n__el v_relp_p__,T .', '1611.02287-1-32-0': 'Fig. [REF] shows the corresponding domains, heuristically.', '1611.02287-1-32-1': 'The red color denotes the region where the elastic scattering rate is greater than the Hubble parameter for given [MATH] and [MATH].', '1611.02287-1-32-2': 'In this region, the dark matter momentum evolves nearly along the vertical direction.', '1611.02287-1-32-3': 'It quickly approaches to [MATH], which is defined as [MATH] with [MATH].', '1611.02287-1-32-4': 'If [MATH], then the dark matter momentum redshifts as [MATH].', '1611.02287-1-32-5': 'The region (C.A.) is bounded from below by the condition that the dark matter is decoupled with a relativistic energy.', '1611.02287-1-33-0': 'For each regions, we can obtain the approximate formula for the final yield value by solving the Eq. ([REF]).', '1611.02287-1-33-1': 'Since we are interested in the case where [MATH], we parameterize the annihilation and elastic scattering cross-sections in the following way: _ann v_rel_& = _ann^2E_^2 (2p_^2E_^2)^k_ann,', '1611.02287-1-34-0': '_el v_rel_, T &= _el^2m_^2 ( E_^2 T^2m_^4)^k_el.', '1611.02287-1-34-1': '[MATH] and [MATH] are the integers determined by the nature of the interactions, such as spin of the initial and final particles, and CP violating effects, etc.', '1611.02287-1-34-2': 'When the dark matter is non-relativistic, for [MATH], the [MATH]-wave annihilation dominates.', '1611.02287-1-34-3': 'For [MATH], the [MATH]-wave annihilation dominates.', '1611.02287-1-34-4': 'It is common that [MATH] for the elastic scattering.', '1611.02287-1-34-5': 'If the elastic scattering is mediated by a vector boson, [MATH] is also possible.', '1611.02287-1-34-6': 'In this paper, we focus on the cases with [MATH] and [MATH].', '1611.02287-1-35-0': 'Before moving forward, let us define useful quantities that are independent of the dark matter momentum; _ann v_rel_0 &_ann^2m_^2, _ann_0 _ann v_rel_0 n_^reh,', '1611.02287-1-36-0': '_el v_rel_0 &_el^2 T_reh^2m_^4, _el_0 _el v_rel_0 T_reh n_^rehm_.', '1611.02287-1-37-0': 'In order to obtain the final yield value, the Eq. ([REF]) should be evaluated.', '1611.02287-1-37-1': 'The integral part of Eq. ([REF]) can be written as _ann_0H_reh _0^1 du _ann v_rel__ann v_rel_0.', '1611.02287-1-37-2': 'In a naive estimation, comparing [MATH] with [MATH] is the only important criterion.', '1611.02287-1-37-3': 'Fig. [REF] shows the time dependence of the integrand, [MATH].', '1611.02287-1-37-4': 'For [MATH], the annihilation cross-section approaches to [MATH] as the momentum of the dark matter decreases.', '1611.02287-1-37-5': 'However, for [MATH], there is a sharp peak around [MATH] in the region (I.A.), whose the height is [MATH] and the width is [MATH].', '1611.02287-1-37-6': 'Therefore, its contribution to the Eq. ([REF]) is of [MATH].', '1611.02287-1-37-7': 'A simple interpretation is as follows.', '1611.02287-1-37-8': 'If the elastic scattering rate is large enough, the dark matter is quickly thermalized before the dark matter starts to annihilate, so that the peak contribution is small, and most of annihilation happens with a thermal averaged annihilation cross-section as Y_(t_0) & H_reh_ann v_rel _T s_reh', '1611.02287-1-38-0': '=& H_reh_ann v_rel _0 s_reh m_6T_reh.', '1611.02287-1-38-1': 'In the opposite limit, large pair annihilation can happen before the dark matter is completely thermalized.', '1611.02287-1-38-2': 'The corresponding yield is dominantly determined by the peak contribution as Y_(t_0) & _el_0_ann_0 Y_(t_reh)', '1611.02287-1-39-0': '=&_el _0_ann v_rel _0 s_reh.', '1611.02287-1-40-0': 'If [MATH], the production mechanism is lying in either the domain (N.A.) or (C.A.) with the yield value, Y_(t_0) min[ Y_(t_reh) , H_reh_ann v_rel_0 s_reh (c_0 H_reh_el_0)^1/3 ],', '1611.02287-1-41-0': 'where [MATH] is an [MATH] numerical constant.', '1611.02287-1-41-1': 'The enhancement factor [MATH] is interpreted as [MATH], where [MATH] is the decoupling energy at [MATH].', '1611.02287-1-41-2': 'The reason of this factor is that the number density is mostly determined by [MATH], where [MATH] is the temperature at which the dark matter becomes non-relativistic.', '1611.02287-1-42-0': 'If [MATH], the dark matter is completely thermalized at [MATH], and the yield is specified by either (N.A.) or (I.A.).', '1611.02287-1-42-1': 'For [MATH], the yield is simply Y_(t_0) =min[ Y_(t_reh), H_reh_ann v_rel_0 s_reh ] .', '1611.02287-1-42-2': 'However, for [MATH], the formula is rather complicated because the annihilation rate is highly sensitive to the momentum evolution even for the non-relativistic dark matter.', '1611.02287-1-42-3': 'The yield value is', '1611.02287-1-43-0': '&Y_(t_0) min[ Y_(t_reh),.', '1611.02287-1-44-0': '&.', '1611.02287-1-44-1': 'H_reh_ann v_rel_0 s_reh [c_0 H_reh _el_0 + (3 -T_kd^2T_reh^2) T_rehm_ ]^-1].', '1611.02287-1-44-2': 'In the expression, the contribution of [MATH] is coming from the peak around [MATH].', '1611.02287-1-44-3': 'This also can be rephrased in terms of the kinetic decoupling temperature.', '1611.02287-1-44-4': 'After [MATH], the elastic scattering rate scales as [MATH], while the Hubble rate scales as [MATH].', '1611.02287-1-44-5': 'Therefore, from [MATH], we find H_reh_el_0(T_kdT_reh)^4.', '1611.02287-1-44-6': 'As the reheating temperature is lower, the peak contribution becomes more important because [MATH] is nearly independent of [MATH].', '1611.02287-1-44-7': 'The remaining contribution of [MATH] is for [MATH], which gives [MATH] if the peak contribution is neglected.', '1611.02287-1-45-0': 'The analytic formulae are matched with each other at a naive boundary between (C.A.) and (I.A.), [MATH].', '1611.02287-1-45-1': 'The value [MATH] is numerically determined to be [MATH], as it is shown in Fig. [REF].', '1611.02287-1-46-0': '# Dark matter Constraints', '1611.02287-1-47-0': '## Relic density', '1611.02287-1-48-0': 'Now we try to fit the above results to the present dark matter relic abundance [CITATION], _ h^2=&0.11(m_100GeV) (Y_(t_0)4 10^-12), for [MATH], and for different choices of [MATH] and [MATH].', '1611.02287-1-48-1': 'For the WIMP dark matter, taking [MATH] is a reasonable assumption.', '1611.02287-1-48-2': '[MATH] has an upper bound from unitarity and perturbativity condition.', '1611.02287-1-48-3': 'Here we take [MATH] as the criterion for both conditions.', '1611.02287-1-48-4': 'The initial yield [MATH] also has an upper bound.', '1611.02287-1-48-5': 'The direct production from heavy particle decays gives [MATH], so that Y_(t_reh) = 3g_*(T_reh)4g_*S(T_reh) Br_Br_ T_rehE_^reh.For [MATH], [MATH] is bounded by [MATH].', '1611.02287-1-49-0': 'In Fig. [REF] and [REF], we study the allowed parameter space in the plane of [MATH] for different choices of reheating temperature.', '1611.02287-1-49-1': 'For each figures, the green dotted lines stand for the contour to satisfy the present relic density with the condition, [MATH].', '1611.02287-1-49-2': 'For [MATH], the present dark matter abundance is proportional to [MATH] in the (C.A.) region that corresponds to the diagonal line above the boundary ([MATH]).', '1611.02287-1-49-3': 'Below the boundary line, the production mechanism is in the (I.A.) region, and the corresponding [MATH] is proportional to [MATH].', '1611.02287-1-49-4': 'Therefore the slope is slightly changed.', '1611.02287-1-50-0': 'For [MATH], the diagonal line on the right hand side is the same as that of the region (C.A.) with [MATH].', '1611.02287-1-50-1': 'However there is a drastic change around the boundary.', '1611.02287-1-50-2': 'The vertical line corresponds to the (I.A.) region where the contribution is dominated by the term [MATH] in Eq. ([REF]).', '1611.02287-1-50-3': 'Consequently, [MATH], and does not explicit depend on the cross-section.', '1611.02287-1-50-4': 'For a quite small [MATH], [MATH] is proportional to [MATH], so that the slope is changed again.', '1611.02287-1-50-5': 'The numerical calculation smooth the analytic lines around the boundary between (C.A.) and (I.A.).', '1611.02287-1-50-6': 'The more correct boundary line is given as [MATH].', '1611.02287-1-51-0': '## Direct/Indirect Detection', '1611.02287-1-52-0': 'At a low temperature below GeV, the quarks are no longer light degrees of freedom, and the interactions between the dark matter and leptons are more crucial to determine the dark matter density.', '1611.02287-1-52-1': 'On the other hand, the interaction between the dark matter and quarks are more important for the direct/indirect detection.', '1611.02287-1-52-2': 'In order to give a rather strong correlation, here we take leptophilic dark matter.', '1611.02287-1-53-0': 'For [MATH], the present dark matter annihilation is dominated by the [MATH]-wave contribution, which is strongly constrainted by the Fermi-LAT data [CITATION] as in Fig. [REF].', '1611.02287-1-53-1': 'Therefore [MATH] is more viable because the annihilation rate at the present Universe is quite suppressed by the square of the present dark matter velocity ([MATH]) compared to [MATH].', '1611.02287-1-54-0': 'For the direct detection of the dark matter, the 1-loop or 2-loop induced interaction between the dark matter and nucleus can generate the sizable signal.', '1611.02287-1-54-1': 'One can think various effective operators for [MATH] with complex couplings and various spin structure.', '1611.02287-1-54-2': 'As a benchmark example, we assume that the dark matter is a Majorana fermion, and that the interaction is mediated by a real scalar.', '1611.02287-1-54-3': 'Using four component spinor notation, the relevant effective operator is given as', '1611.02287-1-55-0': 'L_eff= ()(l l)^2.', '1611.02287-1-56-0': 'We consider lepton flavor universal couplings in order not to generate any flavor problem.', '1611.02287-1-56-1': 'After matching the effective operator of Eq. ([REF]) to that for the scattering cross-section of Eq. ([REF]), we can apply the constraints from direct detection experiments [CITATION].', '1611.02287-1-57-0': 'For Eq. ([REF]), the first non-vanishing dark matter nucleus ([MATH]) elastic scattering cross-section is generated at the two-loop level; [CITATION] _N =& O(1) (_N^2 Z^2 )(e^4 Z192^2 ^2)^2 (_N v_0m_l)^2', '1611.02287-1-58-0': '& _N^2_n^2 A^2 _n, where [MATH] is the nucleon.', '1611.02287-1-58-1': 'The [MATH] uncertainties are coming from the two-loop induced nucleus form factor, whose evaluation is beyond the scope of this paper.', '1611.02287-1-58-2': '[MATH] is the atomic number, [MATH] is the mass number of the target nucleus.', '1611.02287-1-58-3': '[MATH] for [MATH]Xe.', '1611.02287-1-58-4': '[MATH] is the reduced mass for the dark matter and the nucleus, and [MATH] is the reduced mass for the the dark matter and a nucleon.', '1611.02287-1-58-5': '[MATH] is the typical recoil momentum of the nucleus, and the formula is valid for [MATH].', '1611.02287-1-58-6': 'Taking all those uncertainties as a factor [MATH], in Fig. [REF] we get the excluded region of the cross-section (red color) for a given dark matter mass.', '1611.02287-1-58-7': 'Even though it is generated at a two-loop level, the strong constraint exists for the range that satisfies the dark matter density.', '1611.02287-1-58-8': 'More accurate constraints considering all the [MATH] coefficients correctly will be discussed in future work.', '1611.02287-1-59-0': '# Conclusion', '1611.02287-1-60-0': 'Non-thermal history of the early Universe can be naturally obtained in new physics beyond the Standard Model, and it also provides various interesting effects which cannot be simply captured by the standard thermal history of the Universe with a high reheating temperature.', '1611.02287-1-61-0': 'In this work, we have studied the effect of the elastic scattering between the WIMP dark matter and background radiations when the Universe is reheated at a low temperature.', '1611.02287-1-61-1': 'This effect is crucial if the amount of non-thermally produced dark matter is sizable, and the reheating temperature is well below the freeze-out temperature.', '1611.02287-1-61-2': 'We specified the three conceptual domains for the determination of the dark matter abundance, and presented the analytic and numerical solutions to the Boltzmann equation.', '1611.02287-1-62-0': 'When the reheating temperature is low enough, the elastic scattering rate is not effective to completely thermalize the dark matter.', '1611.02287-1-62-1': 'The dark matter particles decouple from thermal plasma when they are still relativistic, and the annihilation could persist until they become non-relativistic.', '1611.02287-1-62-2': 'In this case, we show that the final abundance of the dark matter could depend on the elastic scattering rate.', '1611.02287-1-62-3': 'Even in the case of instantaneous thermalization, the relative size between the elastic and annihilation rates can change the final abundance for the [MATH]-wave annihilating dark matter.', '1611.02287-1-62-4': 'On the other hand, the non-thermal WIMP mechanism requires large annihilation cross-section to explain the present dark matter relic density.', '1611.02287-1-62-5': 'We studied the constraints from direct/indirect detection experiments by considering the leptophilic dark matter model as a specific example, and showed that wide range of parameter space is severely constrained.', '1611.02287-1-63-0': 'Those strong constraints can be avoided if the dark matter is "Dark WIMP" in which the dark matter is thermalized by and annihilates to dark radiations.', '1611.02287-1-63-1': 'The mechanisms that we have discussed can also be generalized to the dark WIMP scenario.', '1611.02287-1-63-2': 'In such a case, there could be more interesting connection between the history of the early Universe and the signatures imprinted on the cosmic microwave background and large scale structure.'}

{'1611.02287-2-0-0': 'We study the effect of the elastic scattering on the non-thermal WIMP, which is produced by direct decay of heavy particles at the end of reheating.', '1611.02287-2-0-1': 'The non-thermal WIMP becomes important when the reheating temperature is well below the freeze-out temperature.', '1611.02287-2-0-2': 'Usually, two limiting cases have been considered.', '1611.02287-2-0-3': 'One is that the produced high energetic dark matter particles are quickly thermalized due to the elastic scattering with background radiations.', '1611.02287-2-0-4': 'The corresponding relic abundance is determined by the thermally averaged annihilation cross-section at the reheating temperature.', '1611.02287-2-0-5': 'The other one is that the initial abundance is too small for the dark matter to annihilate so that the final relic is determined by the initial amount itself.', '1611.02287-2-0-6': 'We study the regions between these two limits, and show that the relic density depends not only on the annihilation rate, but also on the elastic scattering rate.', '1611.02287-2-0-7': 'Especially, the relic abundance of the [MATH]-wave annihilating dark matter crucially relies on the elastic scattering rate because the annihilation cross-section is sensitive to the dark matter velocity.', '1611.02287-2-0-8': 'We categorize the parameter space into several regions where each region has distinctive mechanism for determining the relic abundance of the dark matter at the present Universe.', '1611.02287-2-0-9': 'The consequence on the (in)direct detection is also studied.', '1611.02287-2-1-0': 'APCTP Pre2016-021 CTPU-16-32', '1611.02287-2-2-0': 'IPMU16-0164', '1611.02287-2-3-0': '# Introduction', '1611.02287-2-4-0': 'The weakly interacting massive particle (WIMP) is one of the promising dark matter (DM) candidates because it can be naturally incorporated in new physics beyond the Standard Model, and it gives interesting observable consequences.', '1611.02287-2-5-0': 'In the standard thermal history, the most important quantity to determine the relic density is a thermal averaged pair annihilation cross-section, [MATH].', '1611.02287-2-5-1': 'Taking "thermal average" is justified because the elastic scattering rate between the WIMP dark matter and the background radiation is much bigger than the annihilation rate, so the kinetic decoupling happens well after the dark matter freeze-out [CITATION].', '1611.02287-2-5-2': 'Usually, the elastic scattering does not have the special role to determine the relic density, but it is crucial for small scale structures since the interaction suppresses the growth of the dark matter density perturbation [CITATION].', '1611.02287-2-5-3': 'When the reheating temperature is low, its effect is more interesting depending on whether the kinetic decoupling happens before or after the end of reheating [CITATION].', '1611.02287-2-6-0': 'In this letter, we study the possibility that the relic abundance of WIMP explicitly depends on the elastic scattering rate.', '1611.02287-2-6-1': 'In the context of self interacting dark matter, such possibility is realized by noting that the dominant annihilation rate from [MATH] scattering and the elastic scattering between the DM and thermal bath can be independent so that we can take suitable parameters to show such behavior [CITATION].', '1611.02287-2-6-2': 'This assumption is not usually valid in the case of WIMP because the annihilation and elastic scattering cannot be treated independently.', '1611.02287-2-7-0': 'However, when the reheating temperature is well below the dark matter mass, a new possibility emerges.', '1611.02287-2-7-1': 'At the end of reheating, the dark matter is produced by the direct decay of heavy particles.', '1611.02287-2-7-2': 'Such non-thermally produced dark matter particles have very high energies.', '1611.02287-2-7-3': 'Evolution of the dark matter momentum gives a strong effect on the annihilation cross-section, and such evolution is determined by the elastic scattering rate.', '1611.02287-2-7-4': 'Consequently, the relative size of the annihilation rate, the elastic scattering rate, and the Hubble rate at the end of reheating can give various mechanisms to determine the final relic density of the dark matter.', '1611.02287-2-7-5': 'We classify the parameter space into the regions where each region has distinctive mechanism to determine the relic density of the dark matter.', '1611.02287-2-7-6': 'We also provide analytic expressions and numerical results for each of those mechanisms.', '1611.02287-2-7-7': 'Especially, we find that the [MATH]-wave annihilating dark matter has more interesting property because the cross-section highly depends on the expectation value of the dark matter momentum.', '1611.02287-2-8-0': 'In section [REF], we present our basic set-up.', '1611.02287-2-8-1': 'In section [REF], we compute the momentum evolution of the dark matter after its production at the end of reheating.', '1611.02287-2-8-2': 'The effect of the momentum evolution on the annihilation cross-section and the corresponding final abundance of the dark matter are discussed in section [REF].', '1611.02287-2-8-3': 'We discuss the constraints from (in)direct detection experiments in section [REF], and conclude in section [REF].', '1611.02287-2-9-0': '# Thermal History of the Non-thermal WIMP', '1611.02287-2-10-0': 'In our set-up, there is the early stage of the matter dominated Universe maintained by a long lived heavy particle, [MATH].', '1611.02287-2-10-1': 'After most of [MATH] decay, the Universe is "reheated" and radiation ([MATH]) starts to dominate the energy density of the Universe with a reheating temperature, [MATH].', '1611.02287-2-10-2': 'On one hand, the dark matter ([MATH]) can be produced either from the scattering of the radiation background, or from the direct decay of [MATH] with a branching fraction [MATH].', '1611.02287-2-11-0': 'Ignoring the sub-leading contributions, the corresponding Boltzmann equations of each components are given as _=&-3 H _-__,', '1611.02287-2-12-0': '_= & - 4 H _+ Br___,', '1611.02287-2-13-0': 'n_= &-3 H n_+Br___m_ -_ann v_rel_n_^2,', '1611.02287-2-14-0': '& + _ann v_rel_T (n_^eq)^2,', '1611.02287-2-15-0': 'H=& _+ _+ _3M_Pl^2, where [MATH] is the reduced Planck mass.', '1611.02287-2-15-1': 'Here we consider a situation that the reheating temperature is lower than the thermal freeze-out temperature of the dark matter ([MATH]).', '1611.02287-2-15-2': 'Before the end of the reheating, [MATH], there are several sources for the dark matter density.', '1611.02287-2-15-3': 'First of all, a usual freeze-out mechanism can work with [MATH], while the resulting abundance is subsequently diluted by continuous entropy injection.', '1611.02287-2-15-4': 'If [MATH] is big, quasi-static equilibrium state can persist until the end of reheating ([MATH]) [CITATION].', '1611.02287-2-15-5': 'Also if [MATH] is large enough to satisfy [MATH], production from an inelastic scattering between thermal bath and a boosted radiation produced by [MATH] decays becomes important [CITATION].', '1611.02287-2-15-6': 'Here we take a rather moderate hierarchy between the mass of [MATH] and [MATH] as [MATH], and a sub GeV reheating temperature so that the inelastic scattering is subdominant.', '1611.02287-2-15-7': 'For [MATH] and a sizable [MATH], the most important source of the late time dark matter abundance is the direct decay of [MATH] at the end of reheating.', '1611.02287-2-15-8': 'It is known that such non-thermal production of the DM can be simplified by assuming that the dark matter is instantaneously produced from the heavy particle decay at [MATH] with an initial amount of the DM given as [MATH] [CITATION].', '1611.02287-2-15-9': 'In summary, we are interested in the following range of parameters: T_rehT_fr m_m_O(10-100)m_.', '1611.02287-2-15-10': 'Because of the hierarchy between masses, the initial energy of the DM is much greater than [MATH].', '1611.02287-2-15-11': 'So we first consider the evolution of the dark matter momentum and then consider its effect on the annihilation rate.', '1611.02287-2-16-0': '# Evolution of the DM momentum', '1611.02287-2-17-0': 'After the dark matter is produced, it experiences two types of interactions.', '1611.02287-2-17-1': 'One is the elastic scattering by the background radiations ([MATH]).', '1611.02287-2-17-2': 'The other one is pair annihilation of the dark matter into the radiations ([MATH]).', '1611.02287-2-17-3': 'The effect of pair production from thermal bath [MATH] is negligible if the DM abundance at [MATH] is much larger than the equilibrium value.', '1611.02287-2-17-4': 'In principle, both of the elastic scattering and annihilation are relevant for the dark matter momentum evolution.', '1611.02287-2-17-5': 'However, as we discuss in appendix [REF], the contribution from annihilation can be safely ignored when the momentum distribution of [MATH] has a narrow width compared to its mean value.', '1611.02287-2-17-6': 'Initially, dark matters are produced from the decays of [MATH], so the width [MATH] is naturally smaller than the mean value of momentum, [MATH].', '1611.02287-2-17-7': 'Until the relaxation time when the momentum distribution arrives at its equilibrium one up to the normalization factor, the width of the distribution is still small and our simplification is justified.', '1611.02287-2-17-8': 'Then the momentum evolution is governed by the following equation [CITATION], d p_dt + H p_&= -n_d(_elv_rel) p _,T', '1611.02287-2-18-0': '& - n__el v_rel p__,T ,', '1611.02287-2-19-0': 'where [MATH], [MATH] is the average over the distribution of [MATH] and [MATH] in the rest frame of the thermal plasma, [MATH] is the elastic scattering cross-section, [MATH] is the change of the dark matter momentum from single event of the elastic scattering, and [MATH] is the number density of the background radiation, [MATH].', '1611.02287-2-20-0': 'The right hand side of [REF] can be simplified as [MATH] times _el v_rel p__, T p_ .', '1611.02287-2-21-0': 'for different ranges of the dark matter momentum, (I)& m_^2 p_T,', '1611.02287-2-22-0': '(II)& p_T m_^2 p_^2,', '1611.02287-2-23-0': '(III)& p_T p_^2 m_^2.', '1611.02287-2-23-1': 'In ([MATH]), the DM is relativistic in the plasma rest frame, and in the center of mass (cm) frames.', '1611.02287-2-23-2': 'In ([MATH]), the DM is non-relativistic in the cm frame, whereas still relativistic in the plasma rest frame.', '1611.02287-2-23-3': 'In ([MATH]), the DM is non-relativistic in both frames.', '1611.02287-2-23-4': 'In the last case, the additional factor in the elastic scattering rate drives [MATH] to the equilibrium value, [MATH].', '1611.02287-2-23-5': 'For the order of magnitude estimation, we obtain [MATH].', '1611.02287-2-23-6': 'Thus the additional factors of the scattering rate, [MATH], [MATH], [MATH] are easily understood from the fact that the allowed phase space, [MATH], becomes wider as the collision energies become higher.', '1611.02287-2-23-7': 'Since the common factor [MATH] depends on the temperature, the DM can quickly arrive at kinetic equilibrium or it can just decouple relativistically depending on [MATH].', '1611.02287-2-24-0': 'When the initial momentum of the dark matter is much greater than [MATH], Fig. [REF] shows possible evolution of the momentum for different reheating temperatures.', '1611.02287-2-24-1': 'At a relatively high reheating temperature, the elastic scattering rate is large enough to make the dark matter in kinetic equilibrium instantaneously after its production.', '1611.02287-2-24-2': 'As the temperature goes down, the momentum follows the equilibrium value ([MATH]) until the kinetic decoupling.', '1611.02287-2-24-3': 'If the reheating temperature is relatively low, after the dark matter momentum experiences a small sharp suppression around [MATH], it slowly decreases as [MATH], and it could become non-relativistic well after reheating (magenta line).', '1611.02287-2-25-0': 'There is a natural connection between the momentum evolution and the dark matter pair annihilation rate.', '1611.02287-2-25-1': 'For example, if they are highly relativistic, the cross-section becomes [MATH], and it will increase as the energy of the particle decreases.', '1611.02287-2-25-2': 'Therefore, if the dark matter is not instantaneously thermalized, the annihilation of the dark matter could happen later when the annihilation cross-section becomes large enough to start the annihilation.', '1611.02287-2-26-0': 'Since the corresponding dark matter abundance is affected by the evolution of the annihilation cross-section, we can find the connection between the final yield of the dark matter, and the elastic scattering rate.', '1611.02287-2-27-0': '# Evolution of the pair annihilation cross-section', '1611.02287-2-28-0': 'When the initial abundance ([MATH]) is much greater than [MATH], the production of [MATH] from thermal bath can be ignored, and the corresponding Boltzmann equation for the dark matter number density is simplified as n_+ 3 H n_= -_ann v_rel_ n_^2.', '1611.02287-2-28-1': 'Solving the above equation, we find the yield of the dark matter at the present time, [MATH], as Y_(t_0) =& Y_(t_reh) (1+ n^reh_H_reh_0^1 d u _ann v_rel_ )^-1,where [MATH], [MATH] is the Hubble rate, and [MATH] is the entropy of the Universe at [MATH].', '1611.02287-2-28-2': 'The yields are denoted by [MATH], [MATH].', '1611.02287-2-28-3': 'The time dependence of [MATH] is determined by that of [MATH] governed by Eq. ([REF]).', '1611.02287-2-28-4': 'More precisely, we have to evaluate the annihilation cross-section that is averaged over the full time dependent momentum distribution function of dark matter.', '1611.02287-2-28-5': 'However, when dark matters are non-thermally produced by two-body decays, the width of the distribution would be small, and for [MATH], [MATH].', '1611.02287-2-28-6': 'Thus it is a good approximation to take [MATH] as the function of [MATH] until the relaxation time, [MATH].', '1611.02287-2-28-7': 'After relaxation, the momentum distribution will be proportional to the equilibrium value, in which the standard deviation and mean value are in the same order.', '1611.02287-2-28-8': 'This leads to [MATH] difference between [MATH] and [MATH], but this does not change our result qualitatively.', '1611.02287-2-28-9': 'Solving the full Boltzmann equations will be discussed in future work.', '1611.02287-2-29-0': 'Two limiting cases are familiar.', '1611.02287-2-29-1': 'One is that [MATH] quickly arrives at its equilibrium value within the period much shorter than the Hubble time as given in Fig. [REF] with blue color.', '1611.02287-2-29-2': 'The dark matter annihilation happens after its thermalization but still much faster than the Hubble expansion rate.', '1611.02287-2-29-3': 'Therefore, [MATH].', '1611.02287-2-29-4': 'The other limit is that the initial abundance is too small so that [MATH].', '1611.02287-2-29-5': 'Annihilation barely happens, and the yield is preserved; [MATH].', '1611.02287-2-29-6': 'In both cases, the final yields do not explicitly depend on the elastic scattering cross-section.', '1611.02287-2-30-0': 'There is an intermediate domain between these two limiting cases.', '1611.02287-2-30-1': 'Including the above examples, we identify three mechanisms for the relic density of the DM.', '1611.02287-2-30-2': 'After the production of the DM from the direct decay of the heavy particles, the relic abundance is determined by one of the following mechanisms:', '1611.02287-2-31-0': 'The rates are given by _ann(T, E_) &= n__ann v_rel_,', '1611.02287-2-32-0': '_el(T, E_)&= n__el v_rel p__,Tp_ .', '1611.02287-2-32-1': 'where [MATH].', '1611.02287-2-33-0': 'Fig. [REF] shows the corresponding domains, heuristically.', '1611.02287-2-33-1': 'The red color denotes the region where the elastic scattering rate is greater than the Hubble parameter for given [MATH] and [MATH].', '1611.02287-2-33-2': 'In this region, the dark matter momentum evolves nearly along the vertical direction.', '1611.02287-2-33-3': 'It quickly approaches to [MATH], which is defined as [MATH].', '1611.02287-2-33-4': 'If [MATH], then the dark matter momentum redshifts as [MATH].', '1611.02287-2-33-5': 'The region (C.A.) is bounded from below by the condition that the dark matter is decoupled with a relativistic energy.', '1611.02287-2-34-0': 'For each regions, we can obtain the approximate formula for the final yield value by solving the Eq. ([REF]).', '1611.02287-2-34-1': 'Since we are interested in the case where [MATH], we parameterize the annihilation and elastic scattering cross-sections in the following way: _ann v_rel_& = _ann^2E_^2 (2p_^2E_^2)^k_ann,', '1611.02287-2-35-0': '_el v_rel_, T &= _el^2m_^2 ( E_^2 T^2m_^4)^k_el.', '1611.02287-2-35-1': '[MATH] and [MATH] are the integers determined by the nature of the interactions, such as spin of the initial and final particles, and CP violating effects, etc.', '1611.02287-2-35-2': 'When the dark matter is non-relativistic, for [MATH], the [MATH]-wave annihilation dominates.', '1611.02287-2-35-3': 'For [MATH], the [MATH]-wave annihilation dominates.', '1611.02287-2-35-4': 'It is common that [MATH] for the elastic scattering.', '1611.02287-2-35-5': 'If the elastic scattering is mediated by a vector boson, [MATH] is also possible.', '1611.02287-2-35-6': 'In this paper, we focus on the cases with [MATH] and [MATH].', '1611.02287-2-36-0': 'Before moving forward, let us define useful quantities that are independent of the dark matter momentum; _ann v_rel_0 &_ann^2m_^2, _ann_0 _ann v_rel_0 n_^reh,', '1611.02287-2-37-0': '_el v_rel_0 &_el^2 T_reh^2m_^4, _el_0 _el v_rel_0 T_reh n_^rehm_.', '1611.02287-2-38-0': 'In order to obtain the final yield value, the Eq. ([REF]) should be evaluated.', '1611.02287-2-38-1': 'The integral part of Eq. ([REF]) can be written as _ann_0H_reh _0^1 du _ann v_rel__ann v_rel_0.', '1611.02287-2-38-2': 'In a naive estimation, comparing [MATH] with [MATH] is the only important criterion.', '1611.02287-2-38-3': 'Fig. [REF] shows the time dependence of the integrand, [MATH].', '1611.02287-2-38-4': 'For [MATH], the annihilation cross-section approaches to [MATH] as the momentum of the dark matter decreases.', '1611.02287-2-38-5': 'However, for [MATH], there is a sharp peak around [MATH] in the region (I.A.), whose the height is [MATH] and the width is [MATH].', '1611.02287-2-38-6': 'Therefore, its contribution to the Eq. ([REF]) is of [MATH].', '1611.02287-2-38-7': 'A simple interpretation is as follows.', '1611.02287-2-38-8': 'If the elastic scattering rate is large enough, the dark matter is quickly thermalized before the dark matter starts to annihilate, so that the peak contribution is small, and most of annihilation happens with a thermal averaged annihilation cross-section as Y_(t_0) & H_reh_ann v_rel _T s_reh', '1611.02287-2-39-0': '=& H_reh_ann v_rel _0 s_reh m_6T_reh.', '1611.02287-2-39-1': 'In the opposite limit, large pair annihilation can happen before the dark matter is completely thermalized.', '1611.02287-2-39-2': 'The corresponding yield is dominantly determined by the peak contribution as Y_(t_0) & _el_0_ann_0 Y_(t_reh)', '1611.02287-2-40-0': '=&_el _0_ann v_rel _0 s_reh.', '1611.02287-2-41-0': 'If [MATH], the production mechanism is lying in either the domain (N.A.) or (C.A.) with the yield value, Y_(t_0) min[ Y_(t_reh) , H_reh_ann v_rel_0 s_reh (c_0 H_reh_el_0)^1/3 ],', '1611.02287-2-42-0': 'where [MATH] is an [MATH] numerical constant.', '1611.02287-2-42-1': 'The enhancement factor [MATH] is interpreted as [MATH], where [MATH] is the decoupling energy at [MATH].', '1611.02287-2-42-2': 'The reason of this factor is that the number density is mostly determined by [MATH], where [MATH] is the temperature at which the dark matter becomes non-relativistic.', '1611.02287-2-43-0': 'If [MATH], the dark matter is completely thermalized at [MATH], and the yield is specified by either (N.A.) or (I.A.).', '1611.02287-2-43-1': 'For [MATH], the yield is simply Y_(t_0) =min[ Y_(t_reh), H_reh_ann v_rel_0 s_reh ] .', '1611.02287-2-43-2': 'However, for [MATH], the formula is rather complicated because the annihilation rate is highly sensitive to the momentum evolution even for the non-relativistic dark matter.', '1611.02287-2-43-3': 'The yield value is', '1611.02287-2-44-0': '&Y_(t_0) min[ Y_(t_reh),.', '1611.02287-2-45-0': '&.', '1611.02287-2-45-1': 'H_reh_ann v_rel_0 s_reh [c_0 H_reh _el_0 + (3 -T_kd^2T_reh^2) T_rehm_ ]^-1].', '1611.02287-2-45-2': 'In the expression, the contribution of [MATH] is coming from the peak around [MATH].', '1611.02287-2-45-3': 'This also can be rephrased in terms of the kinetic decoupling temperature.', '1611.02287-2-45-4': 'After [MATH], the elastic scattering rate scales as [MATH], while the Hubble rate scales as [MATH].', '1611.02287-2-45-5': 'Therefore, from [MATH], we find H_reh_el_0(T_kdT_reh)^4.', '1611.02287-2-45-6': 'As the reheating temperature is lower, the peak contribution becomes more important because [MATH] is nearly independent of [MATH].', '1611.02287-2-45-7': 'The remaining contribution of [MATH] is for [MATH], which gives [MATH] if the peak contribution is neglected.', '1611.02287-2-46-0': 'The analytic formulae are matched with each other at a naive boundary between (C.A.) and (I.A.), [MATH].', '1611.02287-2-46-1': 'The value [MATH] is numerically determined to be [MATH], as it is shown in Fig. [REF].', '1611.02287-2-47-0': '# Dark matter Constraints', '1611.02287-2-48-0': '## Relic density', '1611.02287-2-49-0': 'Now we try to fit the above results to the present dark matter relic abundance [CITATION], _ h^2=&0.11(m_100GeV) (Y_(t_0)4 10^-12), for [MATH], and for different choices of [MATH] and [MATH].', '1611.02287-2-49-1': 'For the WIMP dark matter, taking [MATH] is a reasonable assumption.', '1611.02287-2-49-2': '[MATH] has an upper bound from unitarity and perturbativity condition.', '1611.02287-2-49-3': 'Here we take [MATH] as the criterion for both conditions.', '1611.02287-2-49-4': 'The initial yield [MATH] also has an upper bound.', '1611.02287-2-49-5': 'The direct production from heavy particle decays gives [MATH], so that Y_(t_reh) = 3g_*(T_reh)4g_*S(T_reh) Br_Br_ T_rehE_^reh.For [MATH], [MATH] is bounded by [MATH].', '1611.02287-2-50-0': 'In Fig. [REF] and [REF], we study the allowed parameter space in the plane of [MATH] for different choices of reheating temperature.', '1611.02287-2-50-1': 'For each figures, the green dotted lines stand for the contour to satisfy the present relic density with the condition, [MATH].', '1611.02287-2-50-2': 'For [MATH], the present dark matter abundance is proportional to [MATH] in the (C.A.) region that corresponds to the diagonal line above the boundary ([MATH]).', '1611.02287-2-50-3': 'Below the boundary line, the production mechanism is in the (I.A.) region, and the corresponding [MATH] is proportional to [MATH].', '1611.02287-2-50-4': 'Therefore the slope is slightly changed.', '1611.02287-2-51-0': 'For [MATH], the diagonal line on the right hand side is the same as that of the region (C.A.) with [MATH].', '1611.02287-2-51-1': 'However there is a drastic change around the boundary.', '1611.02287-2-51-2': 'The vertical line corresponds to the (I.A.) region where the contribution is dominated by the term [MATH] in Eq. ([REF]).', '1611.02287-2-51-3': 'Consequently, [MATH], and does not explicit depend on the cross-section.', '1611.02287-2-51-4': 'For a quite small [MATH], [MATH] is proportional to [MATH], so that the slope is changed again.', '1611.02287-2-51-5': 'The numerical calculation smooth the analytic lines around the boundary between (C.A.) and (I.A.).', '1611.02287-2-51-6': 'The more correct boundary line is given as [MATH].', '1611.02287-2-52-0': '## Direct/Indirect Detection', '1611.02287-2-53-0': 'At a low temperature below GeV, the quarks are no longer light degrees of freedom, and the interactions between the dark matter and leptons are more crucial to determine the dark matter density.', '1611.02287-2-53-1': 'On the other hand, the interaction between the dark matter and quarks are more important for the direct/indirect detection.', '1611.02287-2-53-2': 'In order to give a rather strong correlation, here we take leptophilic dark matter.', '1611.02287-2-54-0': 'For [MATH], the present dark matter annihilation is dominated by the [MATH]-wave contribution, which is strongly constrainted by the Fermi-LAT data [CITATION] as in Fig. [REF].', '1611.02287-2-54-1': 'Therefore [MATH] is more viable because the annihilation rate at the present Universe is quite suppressed by the square of the present dark matter velocity ([MATH]) compared to [MATH].', '1611.02287-2-55-0': 'For the direct detection of the dark matter, the 1-loop or 2-loop induced interaction between the dark matter and nucleus can generate the sizable signal.', '1611.02287-2-55-1': 'One can think various effective operators for [MATH] with complex couplings and various spin structure.', '1611.02287-2-55-2': 'As a benchmark example, we assume that the dark matter is a Majorana fermion, and that the interaction is mediated by a real scalar.', '1611.02287-2-55-3': 'Using four component spinor notation, the relevant effective operator is given as', '1611.02287-2-56-0': 'L_eff= ()(l l)^2.', '1611.02287-2-57-0': 'We consider lepton flavor universal couplings in order not to generate any flavor problem.', '1611.02287-2-57-1': 'After matching the effective operator of Eq. ([REF]) to that for the scattering cross-section of Eq. ([REF]), we can apply the constraints from direct detection experiments [CITATION].', '1611.02287-2-58-0': 'For Eq. ([REF]), the first non-vanishing dark matter nucleus ([MATH]) elastic scattering cross-section is generated at the two-loop level; [CITATION] _N =& O(1) (_N^2 Z^2 )(e^4 Z192^2 ^2)^2 (_N v_0m_l)^2', '1611.02287-2-59-0': '& _N^2_n^2 A^2 _n, where [MATH] is the nucleon.', '1611.02287-2-59-1': 'The [MATH] uncertainties are coming from the two-loop induced nucleus form factor, whose evaluation is beyond the scope of this paper.', '1611.02287-2-59-2': '[MATH] is the atomic number, [MATH] is the mass number of the target nucleus.', '1611.02287-2-59-3': '[MATH] for [MATH]Xe.', '1611.02287-2-59-4': '[MATH] is the reduced mass for the dark matter and the nucleus, and [MATH] is the reduced mass for the the dark matter and a nucleon.', '1611.02287-2-59-5': '[MATH] is the typical recoil momentum of the nucleus, and the formula is valid for [MATH].', '1611.02287-2-59-6': 'Taking all those uncertainties as a factor [MATH], in Fig. [REF] we get the excluded region of the cross-section (red color) for a given dark matter mass.', '1611.02287-2-59-7': 'Even though it is generated at a two-loop level, the strong constraint exists for the range that satisfies the dark matter density.', '1611.02287-2-59-8': 'More accurate constraints considering all the [MATH] coefficients correctly will be discussed in future work.', '1611.02287-2-60-0': '# Conclusions', '1611.02287-2-61-0': 'Non-thermal history of the early Universe can be naturally obtained in new physics beyond the Standard Model, and it also provides various interesting effects which cannot be simply captured by the standard thermal history of the Universe with a high reheating temperature, including the works [CITATION].', '1611.02287-2-62-0': 'In this work, we have studied the effect of the elastic scattering between the WIMP dark matter and background radiations when the Universe is reheated at a low temperature.', '1611.02287-2-62-1': 'This effect is crucial if the amount of non-thermally produced dark matter is sizable, and the reheating temperature is well below the freeze-out temperature.', '1611.02287-2-62-2': 'We specified the three conceptual domains for the determination of the dark matter abundance, and presented the analytic and numerical solutions to the Boltzmann equation.', '1611.02287-2-63-0': 'When the reheating temperature is low enough, the elastic scattering rate is not effective to completely thermalize the dark matter.', '1611.02287-2-63-1': 'The dark matter particles decouple from thermal plasma when they are still relativistic, and the annihilation could persist until they become non-relativistic.', '1611.02287-2-63-2': 'In this case, we show that the final abundance of the dark matter could depend on the elastic scattering rate.', '1611.02287-2-63-3': 'Even in the case of instantaneous thermalization, the relative size between the elastic and annihilation rates can change the final abundance for the [MATH]-wave annihilating dark matter.', '1611.02287-2-63-4': 'On the other hand, the non-thermal WIMP mechanism requires large annihilation cross-section to explain the present dark matter relic density.', '1611.02287-2-63-5': 'We studied the constraints from direct/indirect detection experiments by considering the leptophilic dark matter model as a specific example, and showed that wide range of parameter space is severely constrained.', '1611.02287-2-64-0': 'Those strong constraints can be avoided if the dark matter is "Dark WIMP" in which the dark matter is thermalized by and annihilates to dark radiations.', '1611.02287-2-64-1': 'The mechanisms that we have discussed can also be generalized to the dark WIMP scenario.', '1611.02287-2-64-2': 'In such a case, there could be more interesting connection between the history of the early Universe and the signatures imprinted on the cosmic microwave background and large scale structure.'}

[['1611.02287-1-7-0', '1611.02287-2-7-0'], ['1611.02287-1-7-1', '1611.02287-2-7-1'], ['1611.02287-1-7-2', '1611.02287-2-7-2'], ['1611.02287-1-7-3', '1611.02287-2-7-3'], ['1611.02287-1-7-4', '1611.02287-2-7-4'], ['1611.02287-1-7-5', '1611.02287-2-7-5'], ['1611.02287-1-7-6', '1611.02287-2-7-6'], ['1611.02287-1-7-7', '1611.02287-2-7-7'], ['1611.02287-1-45-0', '1611.02287-2-46-0'], ['1611.02287-1-45-1', '1611.02287-2-46-1'], ['1611.02287-1-26-0', '1611.02287-2-26-0'], ['1611.02287-1-0-1', '1611.02287-2-0-1'], ['1611.02287-1-0-5', '1611.02287-2-0-6'], ['1611.02287-1-0-6', '1611.02287-2-0-7'], ['1611.02287-1-0-7', '1611.02287-2-0-8'], ['1611.02287-1-0-8', '1611.02287-2-0-9'], ['1611.02287-1-4-0', '1611.02287-2-4-0'], ['1611.02287-1-8-0', '1611.02287-2-8-0'], ['1611.02287-1-8-1', '1611.02287-2-8-1'], ['1611.02287-1-8-2', '1611.02287-2-8-2'], ['1611.02287-1-8-3', '1611.02287-2-8-3'], ['1611.02287-1-38-1', '1611.02287-2-39-1'], ['1611.02287-1-38-2', '1611.02287-2-39-2'], ['1611.02287-1-41-0', '1611.02287-2-42-0'], ['1611.02287-1-41-1', '1611.02287-2-42-1'], ['1611.02287-1-41-2', '1611.02287-2-42-2'], ['1611.02287-1-50-0', '1611.02287-2-51-0'], ['1611.02287-1-50-1', '1611.02287-2-51-1'], ['1611.02287-1-50-2', '1611.02287-2-51-2'], ['1611.02287-1-50-3', '1611.02287-2-51-3'], ['1611.02287-1-50-4', '1611.02287-2-51-4'], ['1611.02287-1-50-5', '1611.02287-2-51-5'], ['1611.02287-1-50-6', '1611.02287-2-51-6'], ['1611.02287-1-32-0', '1611.02287-2-33-0'], ['1611.02287-1-32-1', '1611.02287-2-33-1'], ['1611.02287-1-32-2', '1611.02287-2-33-2'], ['1611.02287-1-32-4', '1611.02287-2-33-4'], ['1611.02287-1-32-5', '1611.02287-2-33-5'], ['1611.02287-1-53-0', '1611.02287-2-54-0'], ['1611.02287-1-53-1', '1611.02287-2-54-1'], ['1611.02287-1-49-0', '1611.02287-2-50-0'], ['1611.02287-1-49-1', '1611.02287-2-50-1'], ['1611.02287-1-49-2', '1611.02287-2-50-2'], ['1611.02287-1-49-3', '1611.02287-2-50-3'], ['1611.02287-1-49-4', '1611.02287-2-50-4'], ['1611.02287-1-5-0', '1611.02287-2-5-0'], ['1611.02287-1-5-1', '1611.02287-2-5-1'], ['1611.02287-1-5-2', '1611.02287-2-5-2'], ['1611.02287-1-5-3', '1611.02287-2-5-3'], ['1611.02287-1-10-0', '1611.02287-2-10-0'], ['1611.02287-1-10-1', '1611.02287-2-10-1'], ['1611.02287-1-10-2', '1611.02287-2-10-2'], ['1611.02287-1-24-0', '1611.02287-2-24-0'], ['1611.02287-1-24-1', '1611.02287-2-24-1'], ['1611.02287-1-24-2', '1611.02287-2-24-2'], ['1611.02287-1-24-3', '1611.02287-2-24-3'], ['1611.02287-1-23-1', '1611.02287-2-23-1'], ['1611.02287-1-23-2', '1611.02287-2-23-2'], ['1611.02287-1-23-3', '1611.02287-2-23-3'], ['1611.02287-1-23-4', '1611.02287-2-23-4'], ['1611.02287-1-23-6', '1611.02287-2-23-7'], ['1611.02287-1-54-0', '1611.02287-2-55-0'], ['1611.02287-1-54-1', '1611.02287-2-55-1'], ['1611.02287-1-54-2', '1611.02287-2-55-2'], ['1611.02287-1-54-3', '1611.02287-2-55-3'], ['1611.02287-1-29-0', '1611.02287-2-30-0'], ['1611.02287-1-29-1', '1611.02287-2-30-1'], ['1611.02287-1-61-0', '1611.02287-2-62-0'], ['1611.02287-1-61-1', '1611.02287-2-62-1'], ['1611.02287-1-61-2', '1611.02287-2-62-2'], ['1611.02287-1-17-0', '1611.02287-2-17-0'], ['1611.02287-1-17-1', '1611.02287-2-17-1'], ['1611.02287-1-17-2', '1611.02287-2-17-2'], ['1611.02287-1-17-3', '1611.02287-2-17-3'], ['1611.02287-1-33-0', '1611.02287-2-34-0'], ['1611.02287-1-62-0', '1611.02287-2-63-0'], ['1611.02287-1-62-1', '1611.02287-2-63-1'], ['1611.02287-1-62-2', '1611.02287-2-63-2'], ['1611.02287-1-62-3', '1611.02287-2-63-3'], ['1611.02287-1-62-4', '1611.02287-2-63-4'], ['1611.02287-1-62-5', '1611.02287-2-63-5'], ['1611.02287-1-44-2', '1611.02287-2-45-2'], ['1611.02287-1-44-3', '1611.02287-2-45-3'], ['1611.02287-1-44-4', '1611.02287-2-45-4'], ['1611.02287-1-44-6', '1611.02287-2-45-6'], ['1611.02287-1-44-7', '1611.02287-2-45-7'], ['1611.02287-1-37-0', '1611.02287-2-38-0'], ['1611.02287-1-37-1', '1611.02287-2-38-1'], ['1611.02287-1-37-2', '1611.02287-2-38-2'], ['1611.02287-1-37-3', '1611.02287-2-38-3'], ['1611.02287-1-37-4', '1611.02287-2-38-4'], ['1611.02287-1-37-5', '1611.02287-2-38-5'], ['1611.02287-1-37-6', '1611.02287-2-38-6'], ['1611.02287-1-37-7', '1611.02287-2-38-7'], ['1611.02287-1-37-8', '1611.02287-2-38-8'], ['1611.02287-1-15-1', '1611.02287-2-15-1'], ['1611.02287-1-15-2', '1611.02287-2-15-2'], ['1611.02287-1-15-3', '1611.02287-2-15-3'], ['1611.02287-1-15-4', '1611.02287-2-15-4'], ['1611.02287-1-15-5', '1611.02287-2-15-5'], ['1611.02287-1-15-6', '1611.02287-2-15-6'], ['1611.02287-1-15-7', '1611.02287-2-15-7'], ['1611.02287-1-15-8', '1611.02287-2-15-8'], ['1611.02287-1-15-9', '1611.02287-2-15-9'], ['1611.02287-1-15-10', '1611.02287-2-15-10'], ['1611.02287-1-15-11', '1611.02287-2-15-11'], ['1611.02287-1-34-1', '1611.02287-2-35-1'], ['1611.02287-1-34-2', '1611.02287-2-35-2'], ['1611.02287-1-34-3', '1611.02287-2-35-3'], ['1611.02287-1-34-4', '1611.02287-2-35-4'], ['1611.02287-1-34-5', '1611.02287-2-35-5'], ['1611.02287-1-34-6', '1611.02287-2-35-6'], ['1611.02287-1-63-0', '1611.02287-2-64-0'], ['1611.02287-1-63-1', '1611.02287-2-64-1'], ['1611.02287-1-63-2', '1611.02287-2-64-2'], ['1611.02287-1-56-0', '1611.02287-2-57-0'], ['1611.02287-1-56-1', '1611.02287-2-57-1'], ['1611.02287-1-25-0', '1611.02287-2-25-0'], ['1611.02287-1-25-1', '1611.02287-2-25-1'], ['1611.02287-1-25-2', '1611.02287-2-25-2'], ['1611.02287-1-58-1', '1611.02287-2-59-1'], ['1611.02287-1-58-2', '1611.02287-2-59-2'], ['1611.02287-1-58-4', '1611.02287-2-59-4'], ['1611.02287-1-58-5', '1611.02287-2-59-5'], ['1611.02287-1-58-6', '1611.02287-2-59-6'], ['1611.02287-1-58-7', '1611.02287-2-59-7'], ['1611.02287-1-58-8', '1611.02287-2-59-8'], ['1611.02287-1-48-0', '1611.02287-2-49-0'], ['1611.02287-1-48-1', '1611.02287-2-49-1'], ['1611.02287-1-48-2', '1611.02287-2-49-2'], ['1611.02287-1-48-3', '1611.02287-2-49-3'], ['1611.02287-1-48-4', '1611.02287-2-49-4'], ['1611.02287-1-48-5', '1611.02287-2-49-5'], ['1611.02287-1-6-0', '1611.02287-2-6-0'], ['1611.02287-1-6-1', '1611.02287-2-6-1'], ['1611.02287-1-6-2', '1611.02287-2-6-2'], ['1611.02287-1-52-0', '1611.02287-2-53-0'], ['1611.02287-1-52-1', '1611.02287-2-53-1'], ['1611.02287-1-52-2', '1611.02287-2-53-2'], ['1611.02287-1-42-0', '1611.02287-2-43-0'], ['1611.02287-1-42-2', '1611.02287-2-43-2'], ['1611.02287-1-42-3', '1611.02287-2-43-3'], ['1611.02287-1-28-0', '1611.02287-2-28-0'], ['1611.02287-1-28-2', '1611.02287-2-28-2'], ['1611.02287-1-28-3', '1611.02287-2-28-3'], ['1611.02287-1-28-4', '1611.02287-2-29-0'], ['1611.02287-1-28-5', '1611.02287-2-29-1'], ['1611.02287-1-28-6', '1611.02287-2-29-2'], ['1611.02287-1-28-8', '1611.02287-2-29-4'], ['1611.02287-1-28-9', '1611.02287-2-29-5'], ['1611.02287-1-28-10', '1611.02287-2-29-6'], ['1611.02287-1-0-2', '1611.02287-2-0-3'], ['1611.02287-1-0-3', '1611.02287-2-0-4'], ['1611.02287-1-32-3', '1611.02287-2-33-3'], ['1611.02287-1-23-5', '1611.02287-2-23-6'], ['1611.02287-1-19-0', '1611.02287-2-19-0'], ['1611.02287-1-60-0', '1611.02287-2-61-0'], ['1611.02287-1-20-0', '1611.02287-2-20-0'], ['1611.02287-1-28-1', '1611.02287-2-28-1'], ['1611.02287-1-0-0', '1611.02287-2-0-0'], ['1611.02287-1-0-4', '1611.02287-2-0-5']]

[['1611.02287-1-7-0', '1611.02287-2-7-0'], ['1611.02287-1-7-1', '1611.02287-2-7-1'], ['1611.02287-1-7-2', '1611.02287-2-7-2'], ['1611.02287-1-7-3', '1611.02287-2-7-3'], ['1611.02287-1-7-4', '1611.02287-2-7-4'], ['1611.02287-1-7-5', '1611.02287-2-7-5'], ['1611.02287-1-7-6', '1611.02287-2-7-6'], ['1611.02287-1-7-7', '1611.02287-2-7-7'], ['1611.02287-1-45-0', '1611.02287-2-46-0'], ['1611.02287-1-45-1', '1611.02287-2-46-1'], ['1611.02287-1-26-0', '1611.02287-2-26-0'], ['1611.02287-1-0-1', '1611.02287-2-0-1'], ['1611.02287-1-0-5', '1611.02287-2-0-6'], ['1611.02287-1-0-6', '1611.02287-2-0-7'], ['1611.02287-1-0-7', '1611.02287-2-0-8'], ['1611.02287-1-0-8', '1611.02287-2-0-9'], ['1611.02287-1-4-0', '1611.02287-2-4-0'], ['1611.02287-1-8-0', '1611.02287-2-8-0'], ['1611.02287-1-8-1', '1611.02287-2-8-1'], ['1611.02287-1-8-2', '1611.02287-2-8-2'], ['1611.02287-1-8-3', '1611.02287-2-8-3'], ['1611.02287-1-38-1', '1611.02287-2-39-1'], ['1611.02287-1-38-2', '1611.02287-2-39-2'], ['1611.02287-1-41-0', '1611.02287-2-42-0'], ['1611.02287-1-41-1', '1611.02287-2-42-1'], ['1611.02287-1-41-2', '1611.02287-2-42-2'], ['1611.02287-1-50-0', '1611.02287-2-51-0'], ['1611.02287-1-50-1', '1611.02287-2-51-1'], ['1611.02287-1-50-2', '1611.02287-2-51-2'], ['1611.02287-1-50-3', '1611.02287-2-51-3'], ['1611.02287-1-50-4', '1611.02287-2-51-4'], ['1611.02287-1-50-5', '1611.02287-2-51-5'], ['1611.02287-1-50-6', '1611.02287-2-51-6'], ['1611.02287-1-32-0', '1611.02287-2-33-0'], ['1611.02287-1-32-1', '1611.02287-2-33-1'], ['1611.02287-1-32-2', '1611.02287-2-33-2'], ['1611.02287-1-32-4', '1611.02287-2-33-4'], ['1611.02287-1-32-5', '1611.02287-2-33-5'], ['1611.02287-1-53-0', '1611.02287-2-54-0'], ['1611.02287-1-53-1', '1611.02287-2-54-1'], ['1611.02287-1-49-0', '1611.02287-2-50-0'], ['1611.02287-1-49-1', '1611.02287-2-50-1'], ['1611.02287-1-49-2', '1611.02287-2-50-2'], ['1611.02287-1-49-3', '1611.02287-2-50-3'], ['1611.02287-1-49-4', '1611.02287-2-50-4'], ['1611.02287-1-5-0', '1611.02287-2-5-0'], ['1611.02287-1-5-1', '1611.02287-2-5-1'], ['1611.02287-1-5-2', '1611.02287-2-5-2'], ['1611.02287-1-5-3', '1611.02287-2-5-3'], ['1611.02287-1-10-0', '1611.02287-2-10-0'], ['1611.02287-1-10-1', '1611.02287-2-10-1'], ['1611.02287-1-10-2', '1611.02287-2-10-2'], ['1611.02287-1-24-0', '1611.02287-2-24-0'], ['1611.02287-1-24-1', '1611.02287-2-24-1'], ['1611.02287-1-24-2', '1611.02287-2-24-2'], ['1611.02287-1-24-3', '1611.02287-2-24-3'], ['1611.02287-1-23-1', '1611.02287-2-23-1'], ['1611.02287-1-23-2', '1611.02287-2-23-2'], ['1611.02287-1-23-3', '1611.02287-2-23-3'], ['1611.02287-1-23-4', '1611.02287-2-23-4'], ['1611.02287-1-23-6', '1611.02287-2-23-7'], ['1611.02287-1-54-0', '1611.02287-2-55-0'], ['1611.02287-1-54-1', '1611.02287-2-55-1'], ['1611.02287-1-54-2', '1611.02287-2-55-2'], ['1611.02287-1-54-3', '1611.02287-2-55-3'], ['1611.02287-1-29-0', '1611.02287-2-30-0'], ['1611.02287-1-29-1', '1611.02287-2-30-1'], ['1611.02287-1-61-0', '1611.02287-2-62-0'], ['1611.02287-1-61-1', '1611.02287-2-62-1'], ['1611.02287-1-61-2', '1611.02287-2-62-2'], ['1611.02287-1-17-0', '1611.02287-2-17-0'], ['1611.02287-1-17-1', '1611.02287-2-17-1'], ['1611.02287-1-17-2', '1611.02287-2-17-2'], ['1611.02287-1-17-3', '1611.02287-2-17-3'], ['1611.02287-1-33-0', '1611.02287-2-34-0'], ['1611.02287-1-62-0', '1611.02287-2-63-0'], ['1611.02287-1-62-1', '1611.02287-2-63-1'], ['1611.02287-1-62-2', '1611.02287-2-63-2'], ['1611.02287-1-62-3', '1611.02287-2-63-3'], ['1611.02287-1-62-4', '1611.02287-2-63-4'], ['1611.02287-1-62-5', '1611.02287-2-63-5'], ['1611.02287-1-44-2', '1611.02287-2-45-2'], ['1611.02287-1-44-3', '1611.02287-2-45-3'], ['1611.02287-1-44-4', '1611.02287-2-45-4'], ['1611.02287-1-44-6', '1611.02287-2-45-6'], ['1611.02287-1-44-7', '1611.02287-2-45-7'], ['1611.02287-1-37-0', '1611.02287-2-38-0'], ['1611.02287-1-37-1', '1611.02287-2-38-1'], ['1611.02287-1-37-2', '1611.02287-2-38-2'], ['1611.02287-1-37-3', '1611.02287-2-38-3'], ['1611.02287-1-37-4', '1611.02287-2-38-4'], ['1611.02287-1-37-5', '1611.02287-2-38-5'], ['1611.02287-1-37-6', '1611.02287-2-38-6'], ['1611.02287-1-37-7', '1611.02287-2-38-7'], ['1611.02287-1-37-8', '1611.02287-2-38-8'], ['1611.02287-1-15-1', '1611.02287-2-15-1'], ['1611.02287-1-15-2', '1611.02287-2-15-2'], ['1611.02287-1-15-3', '1611.02287-2-15-3'], ['1611.02287-1-15-4', '1611.02287-2-15-4'], ['1611.02287-1-15-5', '1611.02287-2-15-5'], ['1611.02287-1-15-6', '1611.02287-2-15-6'], ['1611.02287-1-15-7', '1611.02287-2-15-7'], ['1611.02287-1-15-8', '1611.02287-2-15-8'], ['1611.02287-1-15-9', '1611.02287-2-15-9'], ['1611.02287-1-15-10', '1611.02287-2-15-10'], ['1611.02287-1-15-11', '1611.02287-2-15-11'], ['1611.02287-1-34-1', '1611.02287-2-35-1'], ['1611.02287-1-34-2', '1611.02287-2-35-2'], ['1611.02287-1-34-3', '1611.02287-2-35-3'], ['1611.02287-1-34-4', '1611.02287-2-35-4'], ['1611.02287-1-34-5', '1611.02287-2-35-5'], ['1611.02287-1-34-6', '1611.02287-2-35-6'], ['1611.02287-1-63-0', '1611.02287-2-64-0'], ['1611.02287-1-63-1', '1611.02287-2-64-1'], ['1611.02287-1-63-2', '1611.02287-2-64-2'], ['1611.02287-1-56-0', '1611.02287-2-57-0'], ['1611.02287-1-56-1', '1611.02287-2-57-1'], ['1611.02287-1-25-0', '1611.02287-2-25-0'], ['1611.02287-1-25-1', '1611.02287-2-25-1'], ['1611.02287-1-25-2', '1611.02287-2-25-2'], ['1611.02287-1-58-1', '1611.02287-2-59-1'], ['1611.02287-1-58-2', '1611.02287-2-59-2'], ['1611.02287-1-58-4', '1611.02287-2-59-4'], ['1611.02287-1-58-5', '1611.02287-2-59-5'], ['1611.02287-1-58-6', '1611.02287-2-59-6'], ['1611.02287-1-58-7', '1611.02287-2-59-7'], ['1611.02287-1-58-8', '1611.02287-2-59-8'], ['1611.02287-1-48-0', '1611.02287-2-49-0'], ['1611.02287-1-48-1', '1611.02287-2-49-1'], ['1611.02287-1-48-2', '1611.02287-2-49-2'], ['1611.02287-1-48-3', '1611.02287-2-49-3'], ['1611.02287-1-48-4', '1611.02287-2-49-4'], ['1611.02287-1-48-5', '1611.02287-2-49-5'], ['1611.02287-1-6-0', '1611.02287-2-6-0'], ['1611.02287-1-6-1', '1611.02287-2-6-1'], ['1611.02287-1-6-2', '1611.02287-2-6-2'], ['1611.02287-1-52-0', '1611.02287-2-53-0'], ['1611.02287-1-52-1', '1611.02287-2-53-1'], ['1611.02287-1-52-2', '1611.02287-2-53-2'], ['1611.02287-1-42-0', '1611.02287-2-43-0'], ['1611.02287-1-42-2', '1611.02287-2-43-2'], ['1611.02287-1-42-3', '1611.02287-2-43-3'], ['1611.02287-1-28-0', '1611.02287-2-28-0'], ['1611.02287-1-28-2', '1611.02287-2-28-2'], ['1611.02287-1-28-3', '1611.02287-2-28-3'], ['1611.02287-1-28-4', '1611.02287-2-29-0'], ['1611.02287-1-28-5', '1611.02287-2-29-1'], ['1611.02287-1-28-6', '1611.02287-2-29-2'], ['1611.02287-1-28-8', '1611.02287-2-29-4'], ['1611.02287-1-28-9', '1611.02287-2-29-5'], ['1611.02287-1-28-10', '1611.02287-2-29-6']]

[['1611.02287-1-0-2', '1611.02287-2-0-3'], ['1611.02287-1-0-3', '1611.02287-2-0-4'], ['1611.02287-1-32-3', '1611.02287-2-33-3'], ['1611.02287-1-23-5', '1611.02287-2-23-6'], ['1611.02287-1-19-0', '1611.02287-2-19-0'], ['1611.02287-1-60-0', '1611.02287-2-61-0'], ['1611.02287-1-20-0', '1611.02287-2-20-0'], ['1611.02287-1-28-1', '1611.02287-2-28-1']]

[]

[['1611.02287-1-0-0', '1611.02287-2-0-0'], ['1611.02287-1-0-4', '1611.02287-2-0-5']]

[]

['1611.02287-1-1-0', '1611.02287-1-2-0', '1611.02287-1-11-0', '1611.02287-1-12-0', '1611.02287-1-13-0', '1611.02287-1-14-0', '1611.02287-1-15-0', '1611.02287-1-18-0', '1611.02287-1-21-0', '1611.02287-1-22-0', '1611.02287-1-23-0', '1611.02287-1-28-7', '1611.02287-1-29-2', '1611.02287-1-30-0', '1611.02287-1-31-0', '1611.02287-1-33-1', '1611.02287-1-34-0', '1611.02287-1-35-0', '1611.02287-1-36-0', '1611.02287-1-38-0', '1611.02287-1-39-0', '1611.02287-1-40-0', '1611.02287-1-42-1', '1611.02287-1-43-0', '1611.02287-1-44-0', '1611.02287-1-44-1', '1611.02287-1-44-5', '1611.02287-1-55-0', '1611.02287-1-57-0', '1611.02287-1-58-0', '1611.02287-1-58-3', '1611.02287-2-1-0', '1611.02287-2-2-0', '1611.02287-2-11-0', '1611.02287-2-12-0', '1611.02287-2-13-0', '1611.02287-2-14-0', '1611.02287-2-15-0', '1611.02287-2-18-0', '1611.02287-2-21-0', '1611.02287-2-22-0', '1611.02287-2-23-0', '1611.02287-2-29-3', '1611.02287-2-30-2', '1611.02287-2-31-0', '1611.02287-2-32-0', '1611.02287-2-32-1', '1611.02287-2-34-1', '1611.02287-2-35-0', '1611.02287-2-36-0', '1611.02287-2-37-0', '1611.02287-2-39-0', '1611.02287-2-40-0', '1611.02287-2-41-0', '1611.02287-2-43-1', '1611.02287-2-44-0', '1611.02287-2-45-0', '1611.02287-2-45-1', '1611.02287-2-45-5', '1611.02287-2-56-0', '1611.02287-2-58-0', '1611.02287-2-59-0', '1611.02287-2-59-3']

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

https://arxiv.org/abs/1611.02287

1905.05243

{'1905.05243-1-0-0': 'Face obscuration is often needed by law enforcement or mass media outlets to provide privacy protection.', '1905.05243-1-0-1': 'Sharing sensitive content where the obscuration or redaction technique may have failed to completely remove all identifiable traces can lead to life-threatening consequences.', '1905.05243-1-0-2': 'Hence, it is critical to be able to systematically measure the face obscuration performance of a given technique.', '1905.05243-1-0-3': 'In this paper we propose to measure the effectiveness of three obscuration techniques: Gaussian blurring, median blurring, and pixelation.', '1905.05243-1-0-4': 'We do so by identifying the redacted faces under two scenarios: classifying an obscured face into a group of identities and comparing the similarity of an obscured face with a clear face.', '1905.05243-1-0-5': 'Threat modeling is also considered to provide a vulnerability analysis for each studied obscuration technique.', '1905.05243-1-0-6': 'Based on our evaluation, we show that pixelation-based face obscuration approaches are the most effective.', '1905.05243-1-1-0': '# Introduction', '1905.05243-1-2-0': 'From TV news to Google StreetView, object obscuration has been used in many applications to provide privacy protection.', '1905.05243-1-2-1': 'Law enforcement agencies use obscuration techniques to avoid exposing the identities of bystanders or officers.', '1905.05243-1-2-2': 'To remove this identifiable information, Gaussian blurring or pixelation methods are commonly used.', '1905.05243-1-2-3': 'Median filtering is also used due to its simple implementation and its non-linearity, which translates to higher information distortion when compared to linear filters such as the Gaussian filter.', '1905.05243-1-2-4': 'These obscuration techniques are able to successfully prevent humans from recognizing the obscured objects.', '1905.05243-1-2-5': 'However, machine learning approaches can identify these objects using the subtle information left in the obscured images.', '1905.05243-1-2-6': 'In this paper, we focus on the performance analysis of the most common obscuration techniques for face redaction.', '1905.05243-1-2-7': 'Specifically, we study Gaussian blurring, median blurring, and pixelation to answer the following question: "How effective are these methods at concealing identity?"', '1905.05243-1-3-0': 'Although these approaches are widely used by Internet news outlets, social media platforms, and government agencies, their performance has not been objectively measured.', '1905.05243-1-3-1': 'The lack of a formal study of these obscuration techniques makes them vulnerable to attacks.', '1905.05243-1-3-2': 'As shown by McPherson [CITATION], a deep learning model with a simple structure is able to identify individuals by analyzing their highly pixelated and blurred faces.', '1905.05243-1-3-3': 'This indicates that human perception is no longer the gold standard to examine the effectiveness of the obscuration methods.', '1905.05243-1-3-4': 'Therefore, we need to consider them under scenarios that allow us to see if we can extract the identifiable information from the obscured face, as shown in Figure [REF].', '1905.05243-1-3-5': 'In particular, we want to test whether face identification is possible by comparing a distorted face against a set of undistorted reference faces and finding the closest match.', '1905.05243-1-3-6': "In order to analyze the vulnerability of these obscuration methods, we design multiple threat models based on the attacker's knowledge of the obscuration method used.", '1905.05243-1-3-7': 'Our simplest threat model assumes that the attacker has no information of any obscuration methods.', '1905.05243-1-3-8': 'In the most challenging threat scenario, we consider that the attacker knows the exact type of the obscuration method and its hyperparameters used.', '1905.05243-1-3-9': 'These unexplored threat models are necessary to offer a complete vulnerability analysis under realistic situations.', '1905.05243-1-4-0': 'The main contributions of this paper are summarized as follows.', '1905.05243-1-4-1': 'First, we design a principal component analysis (PCA) based method to identify redacted faces under the face identification scenario.', '1905.05243-1-4-2': 'We also examine if deep learning based methods can be extended to the face verification scenario.', '1905.05243-1-4-3': 'Finally, we provide a comprehensive analysis of the obscuration performance of Gaussian blurring, median blurring, and pixelation under different threat models.', '1905.05243-1-5-0': '# Related Work', '1905.05243-1-6-0': 'Face Obscuration Methods.', '1905.05243-1-6-1': 'The goal of face obscuration is to remove all the identifiable facial information to prevent identification.', '1905.05243-1-6-2': 'As previously mentioned, Gaussian blurring and pixelation are frequently used in many commercial applications and social media sites.', '1905.05243-1-6-3': 'However, these techniques are not reliable.', '1905.05243-1-6-4': 'As we will show in Section [REF], a poor choice for the kernel size of the Gaussian filter is not able to remove all identifiable information.', '1905.05243-1-6-5': 'An extreme resort to prevent information leaking is to simply gray out the entire facial region by setting all pixels in the facial area to a fixed value.', '1905.05243-1-6-6': 'However, this approach is rarely used because its visual effect is unpleasant, especially if there are many faces to be redacted.', '1905.05243-1-6-7': 'To overcome this issue, a variety of approaches have been proposed that try to balance the removal of identifiable information while preserving some facial features.', '1905.05243-1-7-0': 'The first set of approaches, known as [MATH]-same methods [CITATION], attempt to group faces into clusters based on personal attributes such as age, gender, or facial expression.', '1905.05243-1-7-1': 'Then, a template face for each cluster is generated.', '1905.05243-1-7-2': 'These methods are able to guarantee that any face recognition system cannot do better than [MATH] in recognizing who a particular image corresponds to, where [MATH] is the minimum number of faces among all clusters [CITATION].', '1905.05243-1-7-3': 'In Newton [CITATION] and Gross [CITATION], they simply compute the average face for each cluster.', '1905.05243-1-7-4': 'Therefore, the obscured faces are blurry and cannot handle various facial poses.', '1905.05243-1-7-5': 'Du [CITATION] use the active appearance model [CITATION] to learn the shape and appearance of faces.', '1905.05243-1-7-6': 'Then, they generate a template face for each cluster to produce obscured faces with better visual quality.', '1905.05243-1-8-0': 'Generative adversarial network (GAN) methods [CITATION] are able to produce more realistic faces, since their discriminator is designed to guide the generator by distinguishing real faces from generated faces.', '1905.05243-1-8-1': 'Wu [CITATION] propose a model that generates an obscured face directly from the original face based on conditional adversarial networks [CITATION].', '1905.05243-1-8-2': 'A contrastive loss is used to enforce that the obscured face is different from the input face and a structure similarity loss is used to maintain the correspondence between the two faces.', '1905.05243-1-8-3': 'However, because they need to directly input the original faces, their obscuration performance is not guaranteed.', '1905.05243-1-8-4': 'To overcome this, Sun [CITATION] propose a model that is able to generate an obscured head without the original face region.', '1905.05243-1-9-0': 'Privacy Analysis of Obscuration Methods.', '1905.05243-1-9-1': 'As mentioned in Section [REF], although Gaussian blurring and pixelation are widely used, these methods might still leak sensitive information.', '1905.05243-1-9-2': 'Dufaux and Ebrahimi, and Sah [CITATION] provide an analysis of the obscuration performance of common face identifiers and show the ineffectiveness of current obscuration methods.', '1905.05243-1-9-3': 'By using a simple deep learning model, McPherson [CITATION] also show that obscured images still contain enough information to perform accurate identification.', '1905.05243-1-9-4': 'They uncover the identity of images obscured with Gaussian blurring, pixelation, and privacy-preserving photo sharing (P3) [CITATION] methods.', '1905.05243-1-9-5': 'Oh [CITATION] also propose a semi-supervised model that is able to identify the face under large variations in pose.', '1905.05243-1-10-0': 'To extend the previous literature [CITATION], we first consider the face identification scenario.', '1905.05243-1-10-1': 'By mapping faces to known identities under different threat models, we analyze the vulnerability of each obscuration method.', '1905.05243-1-10-2': 'However, the requirement of known identities weakens this kind of analysis, since query faces usually come from unknown identities.', '1905.05243-1-10-3': 'To overcome this, we also provide a threat analysis under a more challenging and realistic setup: the face verification scenario.', '1905.05243-1-10-4': 'Specifically, we want to measure the similarity of an unknown redacted face to clear target faces.', '1905.05243-1-10-5': 'Since it allows the exposure of unknown identities, this scenario is more realistic than the face identification scenario.', '1905.05243-1-11-0': '# Proposed Method', '1905.05243-1-12-0': 'To evaluate the performance of the obscuration methods, we introduce the threat scenarios and face identification models.', '1905.05243-1-13-0': '## Threat Model', '1905.05243-1-14-0': 'In our model, the attacker aims to identify the redacted faces based on the information still present in the obscured images.', '1905.05243-1-14-1': 'To be clear, we define attacker as a face recognition system that tries to reveal the identity of the obscured faces.', '1905.05243-1-14-2': 'We design three threat models, which vary on how much information about the used obscuration approach is available to the attacker.', '1905.05243-1-15-0': '## Obscured Face Identification', '1905.05243-1-16-0': 'For the obscured face identification problem, we assume a fixed number of identities.', '1905.05243-1-16-1': 'We treat this identification problem as a classification problem where the number of classes is equal to the number of identities.', '1905.05243-1-16-2': 'Compared to the verification task, this identification problem is easier to solve.', '1905.05243-1-16-3': 'We can design a simple identifier to compare the performance of different obscuration methods.', '1905.05243-1-17-0': 'PCA is used for reducing data dimensionality and one of its application is facial recognition (also known as "Eigenfaces") introduced by Turk [CITATION].', '1905.05243-1-17-1': 'The eigenfaces representation develops a fixed linear basis for the facial appearance with low dimensionality.', '1905.05243-1-17-2': 'It represents any face with the vector of coefficients of the linear combination.', '1905.05243-1-17-3': 'Figure [REF] shows an example of eigenfaces.', '1905.05243-1-17-4': 'We will briefly formulate the PCA approach as introduced by Turk [CITATION].', '1905.05243-1-17-5': 'Then, we will present our model for the obscured face identification problem based on PCA.', '1905.05243-1-18-0': 'We first denote the training images as [MATH] and [MATH], where [MATH] is the dimension of image and [MATH] is the number of images in the training set.', '1905.05243-1-19-0': 'The average face shown in Figure [REF] is given by [EQUATION]', '1905.05243-1-19-1': 'The [MATH]-th eigenface (eigenvector) [MATH] can be obtained from the sample covariance matrix [MATH] which is defined as [EQUATION] where [EQUATION]', '1905.05243-1-19-2': 'Since the covariance matrix [MATH] is in [MATH], it is not practical to compute its eigenvectors directly.', '1905.05243-1-19-3': 'Instead, we compute the eigenvectors [MATH] of the matrix [MATH] and then compute [MATH] based on the following relationship: [EQUATION]', '1905.05243-1-19-4': 'Note that the proof of Equation [REF] can be found in [CITATION].', '1905.05243-1-19-5': 'Afterwards, we can find a linear combination of these eigenfaces to approximately reconstruct a new face image by [EQUATION] where [MATH] is the number of eigenfaces we select ([MATH]) and we further denote the projection weights as [EQUATION]', '1905.05243-1-20-0': 'We can then perform face identification given a query face using a classifier based on its projection weights [MATH] from the PCA approach, which can be defined as [EQUATION] where [MATH] is the identity label set when the total number of identities is [MATH].', '1905.05243-1-20-1': 'Since the number of identities cannot be infinite, this method can only be applied to the obscured face identification problem.', '1905.05243-1-20-2': 'In our paper, we use a multi-layer perceptron (MLP) with one hidden layer as the face classifier.', '1905.05243-1-21-0': '## Obscured Face Verification', '1905.05243-1-22-0': 'The obscured face verification problem is defined as: given an obscured face and a clear face, decide if the two faces come from the same person or not.', '1905.05243-1-23-0': 'In order to solve this verification problem, we project the image vector [MATH] to a lower-dimension latent vector in [MATH], where faces from the same person are closer together than faces from different people.', '1905.05243-1-23-1': 'Based on this idea, Schroff [CITATION] design a deep learning model, known as FaceNet, directly learning the mapping function transferring from the image space [MATH] to the compact Euclidean space [MATH].', '1905.05243-1-23-2': 'The distance from each point in the Euclidean space represents the facial similarity.', '1905.05243-1-23-3': 'The model is trained with the triplet loss, which enforces a margin between the faces from the same identity and other faces from different identities in the Euclidean space.', '1905.05243-1-23-4': 'Define the mapping function as [MATH] and given an image [MATH], the corresponding projected point in the Euclidean space is [MATH].', '1905.05243-1-23-5': 'Denote the anchor face as [MATH], positive face as [MATH] (with the same identity as [MATH]) and negative face as [MATH] (with a different identity than [MATH]).', '1905.05243-1-23-6': 'The objective of the triplet loss can be formulated as [EQUATION] where [MATH] is a predefined margin.', '1905.05243-1-23-7': 'By rearranging the terms in Equation [REF], we can define the triplet loss function as [EQUATION] where [MATH] is the function that clips all negative values to 0.', '1905.05243-1-23-8': 'Selecting challenging triplets is crucial for fast convergence.', '1905.05243-1-23-9': 'Therefore, instead of randomly sampling the valid triplets, as proposed by Hermans [CITATION], we input a batch of images with multiple identities into the model and select all valid triplets to compute the triplet loss.', '1905.05243-1-23-10': 'The valid triplet has three faces with two of them coming from the same person and the surplus one coming from the different identity.', '1905.05243-1-23-11': 'The workflow of the FaceNet model in the training phase is shown in Figure [REF], which inputs a batch of faces, computes the encoded vectors and calculates the triplet loss to train the model.', '1905.05243-1-24-0': 'With the knowledge of triplet loss in hand, we can design our model as shown in Figure [REF].', '1905.05243-1-24-1': 'The objective for our model is to ensure that the distance from the obscured face to the clear face from the same identity is closer than the distance between two different identities.', '1905.05243-1-24-2': 'We use two separate deep learning models with the same structure to extract the facial features from the obscured faces [MATH] and clear faces [MATH], which are formulated as [MATH] and [MATH].', '1905.05243-1-24-3': 'Therefore, the projected obscured face in the Euclidean space is [MATH], while the projected clear face in the Euclidean space is [MATH].', '1905.05243-1-24-4': 'We can further simplify the notation of the triplet loss function as follows.', '1905.05243-1-24-5': 'Use [MATH] for the loss of all valid triplets from the obscured faces, use [MATH] for the loss of all valid triplets from clear faces and use [MATH] for the loss of all valid triplets of the combination of obscured and clear faces.', '1905.05243-1-24-6': 'Then, we can define the loss function of this obscured face identification model as [EQUATION]', '1905.05243-1-24-7': 'For the inference phase, we use obscured face and a clear face as inputs to the two mapping functions, [MATH] and [MATH] and compare the [MATH] distance given the predefined threshold value to determine if the two faces come from the same person or not.', '1905.05243-1-24-8': 'The distance threshold value can be obtained based on the value that maximizes the identification accuracy of the validation set.', '1905.05243-1-25-0': '# Experiments', '1905.05243-1-26-0': 'In this section, we describe our experiments.', '1905.05243-1-26-1': 'First, we design a set of experiments for the identification task to provide a quantified analysis of different obscuration methods.', '1905.05243-1-26-2': 'Then, we extend them to the verification problem using the deep learning model to test these obscuration methods in this more realistic scenario.', '1905.05243-1-27-0': '## Datasets', '1905.05243-1-28-0': 'For the identification experiments, we use two datasets: the ATT dataset [CITATION] and the labeled face in the wild (LFW) dataset [CITATION] and for the verification experiments, we use the YouTube face dataset [CITATION].', '1905.05243-1-28-1': 'The ATT dataset provides 400 images of size [MATH] from 40 identities under different lighting conditions and facial expressions.', '1905.05243-1-28-2': 'We choose this dataset in order to compare the results from previous work [CITATION].', '1905.05243-1-28-3': 'Moreover, the LFW dataset contains 13,000 images from 1,680 identities collected from the internet, and compared to the ATT dataset it is more challenging in terms of scales and image quality.', '1905.05243-1-28-4': 'We use this dataset to provide a comprehensive threat analysis of the obscuration methods.', '1905.05243-1-28-5': 'Lastly, the YouTube face dataset contains 621,126 images of 1,595 identities (about 389 images per identity) which is much larger than the LFW dataset.', '1905.05243-1-29-0': '## Obscured Face Identification Experiment', '1905.05243-1-30-0': 'Experimental Design.', '1905.05243-1-30-1': 'We design several experiments to quantify the performance of the three obscuration methods: Gaussian blurring, pixelation and median blurring.', '1905.05243-1-30-2': 'In order to measure the identification accuracy, based on the design in Section [REF], we train a PCA model with 150 eigenfaces and a 3-layer perceptron (the dimension of the hidden layer is 1024) as a classifier.', '1905.05243-1-31-0': 'As mentioned in Section [REF], we first train and test our model using the ATT dataset to analyze the performance of Gaussian and median blurring with kernel sizes: 5, 15, 25 and 45, and pixelation with pixel sizes: 2, 4, 8, 16.', '1905.05243-1-31-1': 'Since there are 10 images for each identity, similar to [CITATION], we use 8 faces for training and 2 for testing.', '1905.05243-1-31-2': 'In order to compare the results, we use the top-1 accuracy as our metric.', '1905.05243-1-32-0': 'We use the more challenging LFW dataset to provide a comprehensive analysis based on the aforementioned threat models.', '1905.05243-1-32-1': 'We remove the identities that have fewer than 25 faces in the dataset, which provides us with 2588 images from 42 people.', '1905.05243-1-32-2': 'We also convert the images into grayscale and use the Viola-Jones face detector [CITATION] to detect faces.', '1905.05243-1-32-3': 'Afterward, we resize the cropped faces as [MATH], which is the same size as the ATT dataset and choose the kernel (pixel) sizes as 5, 15, 25 and 45.', '1905.05243-1-32-4': 'In this experiment, we choose larger pixel sizes for the pixelation method than the experiments using the ATT dataset to provide a more challenging situation.', '1905.05243-1-32-5': 'To evaluate the effectiveness of the obscuration methods, we use the cumulative match characteristic (CMC) curve as suggested by Dufaux and Ebrahimi [CITATION] to compare the identification accuracy with respect to the identification rank.', '1905.05243-1-33-0': 'We designed three experiments based on the three threat models to evaluate the obscuration methods.', '1905.05243-1-33-1': 'In the first experiment which is designed for [MATH], the identifier is trained with the set of clear images and tested with obscured images.', '1905.05243-1-33-2': 'In the second experiment which is designed for [MATH], the identifier is trained on both clear and obscured images and tested with the obscured images of the obscuration methods not used in the training set.', '1905.05243-1-33-3': 'In the third experiment which is designed for [MATH], the identifier is trained on both clear and obscured images and tested with the obscured images employing the same obscuration methods.', '1905.05243-1-34-0': 'Result.', '1905.05243-1-34-1': 'Table [REF] shows the results of top-1 accuracy for face recognition on the different obscured images of the ATT dataset.', '1905.05243-1-34-2': 'Note that McPherson [CITATION] used a deep learning model and Sah [CITATION] used a support vector machine classifier based on the features from the PCA approach.', '1905.05243-1-34-3': 'The Original column indicates that the deep learning approach yields the best identification accuracy, which shows that the CNN model is able to handle the facial features better than PCA-based approaches including ours and Sah [CITATION].', '1905.05243-1-34-4': 'For the Gaussian blurring and median blurring, the results indicate that our model is able to identify the faces even when a significant amount of facial information has been obscured.', '1905.05243-1-34-5': 'As shown in Figure [REF], when we increase the kernel size of the Gaussian and median filter, the facial features such as eyes or noses no longer exist in the images.', '1905.05243-1-34-6': 'Since there are only 10 images for each identity and only two of them are used for testing, the identifier is still able to identify the highly obscured face only from its remaining boundary of the facial region.', '1905.05243-1-34-7': 'For the pixelation method, the results from McPherson [CITATION] yield the best identification accuracy with the increasing pixel size.', '1905.05243-1-34-8': 'This shows that the CNN model is able to identify the faces even without noticeable facial features because of the small scale of the dataset.', '1905.05243-1-35-0': 'The pixelation results from our model show that it has a better obscuration performance as the pixel size increases.', '1905.05243-1-35-1': 'This is because pixelation is not only able to remove the facial features, but also the boundary of the facial region.', '1905.05243-1-35-2': 'Comparing the results from pixelation to the two blurring methods, we can see that the pixelation method is able to remove more identifiable information than Gaussian and median blurring.', '1905.05243-1-36-0': 'Figure [REF] shows the results for our PCA method for the three experiments with different obscuration methods and different kernel sizes for the LFW dataset.', '1905.05243-1-36-1': 'As a baseline, the results in Figure [REF] indicate that without any obscuration the rank-1 and rank-5 accuracies are about 62% and 85%, respectively, based on the curve marked as Original.', '1905.05243-1-37-0': 'For treat model [MATH] in experiment I, comparing the curves with different kernel sizes, all three methods have a better obscuration performance (, lower identification precision) as the kernel size increases.', '1905.05243-1-37-1': 'Comparing the curves of different methods, the Gaussian and median filters with a kernel size of 5 almost have no difference from the results without any obscuration in the testing set.', '1905.05243-1-37-2': 'As shown in Figure [REF] their blurring effect is really trivial.', '1905.05243-1-37-3': 'The pixelation method yields the best obscuration performance, especially for the pixel size of 45.', '1905.05243-1-37-4': 'Moreover, the Gaussian and median filters with a kernel size of 45 also have a good performance, but it is still worse than the pixelation method with a pixel size of 15, 25 and 45.', '1905.05243-1-38-0': 'For treat model [MATH] in experiment II, after adding the obscured images to the training set the identification precision of Gaussian and median filters increases, while the results of the pixelation method almost remain the same.', '1905.05243-1-38-1': 'This is caused by the pixelation with a large pixel size, which not only removes the identifiable facial features like eyes and mouth, but also the outline of the face as mentioned in the previous experiment.', '1905.05243-1-38-2': 'Although the Gaussian and median filters can also obscure the identifiable information, the remaining facial outline provides the classifier information for identification.', '1905.05243-1-39-0': 'For treat model [MATH] in experiment III, if we train with the obscured images with the same obscuration operation used in the testing set, the obscuration performance for all methods will decrease.', '1905.05243-1-39-1': 'This can be especially seen for the pixelation method with a pixel size of 45, although it still achieves the best performance.', '1905.05243-1-40-0': 'In summary, based on the results of obscured face identification, the pixelation method achieves the best obscuration performance compared to the Gaussian and median blurring methods.', '1905.05243-1-41-0': '## Obscured Face Verification Experiment', '1905.05243-1-42-0': 'Experimental Design.', '1905.05243-1-42-1': 'As previously mentioned, the deep learning method [CITATION] yields the best performance even with the biggest pixel size.', '1905.05243-1-42-2': 'However, since the ATT dataset experiment only has two images per identity for testing, the performance could be different when dealing with a larger dataset.', '1905.05243-1-42-3': 'Therefore, in order to further examine the obscuration performance of a larger dataset, we design a verification experiment.', '1905.05243-1-42-4': 'As mentioned in Section [REF], we use the YouTube face dataset for this verification experiment and we split the dataset into training, validation, and testing with ratios of 0.6, 0.2 and 0.2 with cropped faces of [MATH].', '1905.05243-1-42-5': 'We also do random rotation ranging from [MATH] for data augmentation to create more variation of facial poses.', '1905.05243-1-42-6': 'As suggested by Hermans [CITATION], we empirically choose a batch size of 128 with 4 images per identity.', '1905.05243-1-42-7': 'For the deep learning structure, we choose the VGG [CITATION] model with output dimension 128 for both the obscured mapping function [MATH] and the clear mapping function [MATH], since this model has been successfully implemented for facial recognition [CITATION].', '1905.05243-1-42-8': 'For the experiment design, we continue the third experiment (threat model [MATH]) in Section [REF], which assumes that we know the obscuration type and train/test on the same obscuration method, including Gaussian blurring, median blurring, and pixelation.', '1905.05243-1-42-9': 'As shown in Figure [REF], we still choose the same kernel/pixel sizes as in the previous identification experiments (5, 15, 25 and 45).', '1905.05243-1-42-10': 'During the testing phase, we sample face pairs from the same identity and different identities evenly.', '1905.05243-1-42-11': 'For the performance metric, since the face verification problem is just a binary classification problem, we choose the receiver operating characteristic (ROC) curve to examine the performance and use its area under the curve (AUC) as the numerical metric.', '1905.05243-1-43-0': 'Result.', '1905.05243-1-43-1': 'As shown in Figure [REF], similar to the results in Section [REF], with the increase of kernel/pixel size, the three obscuration methods have a better obscuration performance.', '1905.05243-1-43-2': 'Observing the different obscuration methods with the same kernel/pixel sizes, the pixelation method yields the best obscuration performance, which confirms the results from Section [REF].', '1905.05243-1-43-3': 'The performance of pixelation with a pixel size of 5 is even better than the Gaussian and median blurring with a kernel size of 15.', '1905.05243-1-43-4': 'Based on the visual effect from Figure [REF], the abrupt edges generated from the pixelation method remove more identifiable facial information.', '1905.05243-1-43-5': 'Comparing the results of Gaussian and median blurring, the median blurring with a kernel size of 45 is slightly better than Gaussian blurring, since the facial details (eye region or nose) are completely removed when the kernel size increases.', '1905.05243-1-44-0': 'As shown in Figure [REF], we also produce visualization results to illustrate the performance of each obscuration methods in a more realistic scenario.', '1905.05243-1-44-1': 'We pick 6 clear faces from the YouTube face dataset (as shown on the right side of each image in Figure [REF]) and select an obscured image from the testing set.', '1905.05243-1-44-2': 'We intentionally add several challenging reference faces among the 6 faces in order to create extra difficulty for the model, like the first, fourth and fifth faces.', '1905.05243-1-44-3': 'The bar next to each clear face indicates the distance between the clear face and the obscured face to its immediate left and the highlighted green bar represents the face with the minimum distance.', '1905.05243-1-44-4': 'Note that the correct match is the second face from the top.', '1905.05243-1-44-5': 'Therefore, our model can correctly detect the obscured face from the Gaussian and median blurring with a kernel size of 15 (as shown in Figure [REF] and [REF]) and fails on the pixelation method with a pixel size of 5 (as shown in Figure [REF]).', '1905.05243-1-45-0': 'In summary, based on the results, the pixelation method achieves the best obscuration performance, which is similar to the results in the identification experiments.', '1905.05243-1-46-0': '# Conclusion', '1905.05243-1-47-0': 'In this paper, we provide a comprehensive analysis of three obscuration methods: Gaussian blurring, median blurring, and pixelation.', '1905.05243-1-47-1': 'We design a set of experiments to examine if we are able to identify the obscured faces using a PCA based method and a deep learning based method.', '1905.05243-1-47-2': 'Based on the experiment results, we show that the pixelation method achieves the best performance since it brings abrupt edges, which improve the effectiveness of the face obscuration.', '1905.05243-1-47-3': 'Obscured faces using Gaussian and median blurring with a big kernel size are still able to be identified by our methods, although they are already unrecognizable by a human.', '1905.05243-1-47-4': 'Therefore, the designers of redaction systems need to be aware of choosing an effective obscuration method like pixelation and cannot only rely on human perception to determine a successful obscuration.'}

{'1905.05243-2-0-0': 'Face obscuration is needed by law enforcement and mass media outlets to guarantee privacy.', '1905.05243-2-0-1': 'Sharing sensitive content where obscuration or redaction techniques have failed to completely remove all identifiable traces can lead to many legal and social issues.', '1905.05243-2-0-2': 'Hence, we need to be able to systematically measure the face obscuration performance of a given technique.', '1905.05243-2-0-3': 'In this paper we propose to measure the effectiveness of eight obscuration techniques.', '1905.05243-2-0-4': 'We do so by attacking the redacted faces in three scenarios: obscured face identification, verification, and reconstruction.', '1905.05243-2-0-5': 'Threat modeling is also considered to provide a vulnerability analysis for each studied obscuration technique.', '1905.05243-2-0-6': 'Based on our evaluation, we show that the [MATH]-same based methods are the most effective.', '1905.05243-2-1-0': '# Introduction', '1905.05243-2-2-0': 'From TV news to Google StreetView, object obscuration has been used in many applications to provide privacy protection.', '1905.05243-2-2-1': 'Law enforcement agencies use obscuration techniques to avoid exposing the identities of bystanders or officers.', '1905.05243-2-2-2': 'To remove identifiable information, Gaussian blurring or pixelation methods are commonly used.', '1905.05243-2-2-3': 'Median filtering is also used due to its simple implementation and its non-linearity, which translates into higher information distortion when compared to linear filters such as the Gaussian filter.', '1905.05243-2-2-4': 'These simple obscuration techniques are able to successfully prevent humans from recognizing the obscured objects.', '1905.05243-2-2-5': 'Previous work [CITATION] shows that machine learning approaches can still identify these objects using the subtle information left in the obscured images.', '1905.05243-2-2-6': 'More robust and effective techniques have been described including [MATH]-same methods [CITATION] which are able to provide a secured obscuration while preserving non-identifiable information.', '1905.05243-2-2-7': 'Reversible obscuration [CITATION] is another type of method to prevent the leakage of privacy information from unauthorized viewers when sharing an image on social media.', '1905.05243-2-2-8': 'This type of methods is designed to achieve privacy-preserving image sharing by encrypting the images published on the Internet.', '1905.05243-2-2-9': 'Only the viewer with the correct decoding key is able to access the image.', '1905.05243-2-2-10': 'In this paper, we focus on the robustness analysis of several obscuration techniques for face redaction.', '1905.05243-2-2-11': 'We study these obscuration methods to answer the following question: "Is there any remaining identifiable information from the obscured faces to enable re-identification?"', '1905.05243-2-3-0': 'Although several of these approaches are widely used by news outlets, social media platforms, and government agencies, their performance has not been objectively measured.', '1905.05243-2-3-1': 'The lack of a formal study of these obscuration techniques makes it hard to evaluate the quality of redaction systems.', '1905.05243-2-3-2': 'As shown by McPherson et al.[CITATION], a deep learning model with a simple structure is able to identify individuals from their highly pixelated and blurred faces.', '1905.05243-2-3-3': 'This indicates that human perception is no longer the gold standard to examine the effectiveness of obscuration methods.', '1905.05243-2-3-4': 'To provide a better way to examine a given obscuration method, we need to consider it in a controlled environment that can determine how well identifiable information can be extracted from the obscured face.', '1905.05243-2-3-5': 'We design three scenarios: obscured face identification, verification, and reconstruction.', '1905.05243-2-3-6': 'Figure [REF] shows the results from the reconstruction attack for the eight studied obscuration methods.', '1905.05243-2-3-7': "To analyze the vulnerability of these obscuration methods, we also examine multiple threat models based on an attacker's knowledge of the obscuration method used.", '1905.05243-2-3-8': 'Our simplest threat model assumes that the attacker has no information of these obscuration methods.', '1905.05243-2-3-9': 'In the most challenging threat scenario, we consider that the attacker knows the exact type of the obscuration method and its hyperparameters.', '1905.05243-2-3-10': 'These previously unexplored threat models are necessary to offer a complete vulnerability analysis under realistic situations.', '1905.05243-2-4-0': 'The main contributions of this paper are summarized as follows.', '1905.05243-2-4-1': 'First, we design three attack scenarios: obscured face identification, verification, and reconstruction.', '1905.05243-2-4-2': 'We also analyze these attacks based on two widely used deep learning models, VGG19[CITATION] and ResNet50[CITATION] in different threat model conditions.', '1905.05243-2-4-3': 'Finally, we provide a comprehensive robustness analysis of eight obscuration methods.', '1905.05243-2-4-4': 'These methods include three traditional methods (Gaussian blurring, median blurring, and pixelation), three [MATH]-same based methods ([MATH]-same [CITATION], [MATH]-same-net [CITATION], and UP-GAN [CITATION]) and two privacy-preserving image sharing methods (P3 [CITATION] and scrambling [CITATION]).', '1905.05243-2-5-0': '# Related Work', '1905.05243-2-6-0': 'Face Obscuration Methods.', '1905.05243-2-6-1': 'As previously mentioned, Gaussian blurring and pixelation are frequently used in many applications.', '1905.05243-2-6-2': 'However, these techniques are not reliable.', '1905.05243-2-6-3': 'As we will show in Section [REF], Gaussian blurring even with a large kernel size is still not able to defend against the some of our attacks.', '1905.05243-2-6-4': 'An extreme example of blurring to prevent information leaking is to simply gray out the entire facial region by setting all pixels in the facial area to a fixed value.', '1905.05243-2-6-5': 'This approach is rarely used because its visual effect is unpleasant, especially if there are many faces in the scene that need to be redacted.', '1905.05243-2-7-0': 'To address some of these issues, [MATH]-same methods [CITATION] have been proposed to balance the removal of identifiable information while preserving non-identifiable facial features.', '1905.05243-2-7-1': 'These methods attempt to group faces into clusters based on personal attributes such as age, gender, or facial expression.', '1905.05243-2-7-2': 'Then, a template face for each cluster is generated.', '1905.05243-2-7-3': 'These methods can fulfill the requirement of [MATH]-anonymity [CITATION].', '1905.05243-2-7-4': 'More specifically, they are able to guarantee that any face recognition system cannot do better than [MATH] in recognizing to whom a particular image corresponds, where [MATH] is the minimum number of faces among all clusters [CITATION].', '1905.05243-2-7-5': 'In Newton et al.[CITATION] and Gross et al.[CITATION], they simply compute the average face for each cluster.', '1905.05243-2-7-6': 'Therefore, the obscured faces are blurry and cannot handle various facial poses.', '1905.05243-2-7-7': 'Du et al.[CITATION] use the active appearance model [CITATION] to learn the shape and appearance of faces.', '1905.05243-2-7-8': 'Then, they generate a template face for each cluster to produce obscured faces with better visual quality.', '1905.05243-2-7-9': 'A generative neural network, k-same-net, that directly generates faces based on the cluster attributes is described in [CITATION].', '1905.05243-2-7-10': 'To produce more realistic faces, generative adversarial network (GAN) [CITATION] have been used, since its discriminator is designed to guide the generator by distinguishing real faces from generated faces.', '1905.05243-2-7-11': 'Hao et al.[CITATION] propose a method based on conditional GAN [CITATION] that can generate a synthetic face given the facial landmarks and cluster attributes without the original image.', '1905.05243-2-8-0': 'Besides the methods above that permanently remove the identifiable information, reversible obscuration methods [CITATION] are also needed for the purposes of privacy-preserving image sharing.', '1905.05243-2-8-1': 'These reversible obscuration methods split the image information into two parts: 1) the public part which contains most volume, but not meaningful content and 2) a secret part that stores the image decoding key.', '1905.05243-2-8-2': 'Therefore, when publishing an image to social media, the public and secret parts can be stored separately to avoid the leakage of images to unauthorized viewers.', '1905.05243-2-8-3': 'Ra et al.[CITATION] propose a method, P3, which is based on the JPEG encoding framework.', '1905.05243-2-8-4': 'They separate the DCT coefficients in the JPEG encoding process based on a predefined threshold value to generate the public and secret images.', '1905.05243-2-8-5': 'Yuan et al.[CITATION] propose a scrambling method that further reduces the data storage in the secret part.', '1905.05243-2-8-6': 'Instead of thresholding, they randomly flip the the sign of DCT coefficients and store the result as the public image.', '1905.05243-2-8-7': 'For the secret part, they only need to store the random seed to recover the original image.', '1905.05243-2-9-0': 'Privacy Analysis of Obscuration Methods.', '1905.05243-2-9-1': 'Dufaux and Ebrahimi, and Sah et al. [CITATION] provide an analysis of the obscuration performance of simple identifiers and show the ineffectiveness of current obscuration methods.', '1905.05243-2-9-2': 'By using a simple deep learning model, McPherson et al.[CITATION] also show that obscured images still contain enough information to perform accurate identification.', '1905.05243-2-9-3': 'They uncover the identity obscured with blurring, pixelation, and P3 methods.', '1905.05243-2-9-4': 'Oh et al.[CITATION] also propose a semi-supervised model that is able to identify the face under large variations in pose.', '1905.05243-2-10-0': 'To extend the previous literature [CITATION], we first consider the face identification scenario.', '1905.05243-2-10-1': 'By mapping faces to known identities in different threat models, we analyze the vulnerability of each obscuration method using advanced deep learning identification methods.', '1905.05243-2-10-2': 'However, the requirement of known identities weakens this type of analysis, since query faces usually come from unknown identities.', '1905.05243-2-10-3': 'To overcome this, we provide a threat analysis under a more realistic setup: the face verification scenario.', '1905.05243-2-10-4': 'Specifically, we want to measure the similarity of an unknown redacted face to clear target faces.', '1905.05243-2-10-5': 'Since it allows recognizing unseen identities, this scenario is more realistic.', '1905.05243-2-10-6': 'Lastly, a reconstruction scenario is proposed to visualize how well we can recover the true identity using the remaining information from the obscured images.', '1905.05243-2-11-0': '# Proposed Method', '1905.05243-2-12-0': 'To evaluate the performance of the obscuration methods, we first introduce the three threat models based on the amount of knowledge about the obscuration method that is available to the attackers.', '1905.05243-2-12-1': 'Then we describe the three attacks: obscured face identification, verification, and reconstruction.', '1905.05243-2-13-0': '## Threat Modeling', '1905.05243-2-14-0': 'In our model, the attacker aims to identify the redacted faces based on the information still present in the obscured images.', '1905.05243-2-14-1': 'We design three threat models, which vary on how much information about the used obscuration approach is available to the attacker.', '1905.05243-2-15-0': '## Obscured Face Identification Attack', '1905.05243-2-16-0': 'For the obscured face identification attack, we assume a fixed number of identities.', '1905.05243-2-16-1': 'We treat this identification problem as a classification problem where the number of classes is equal to the number of identities.', '1905.05243-2-16-2': 'In this paper, we evaluate the performance of different obscuration methods based on different backbone deep learning models, such as VGG19 or ResNet50 in order to have a more generalizable conclusion.', '1905.05243-2-17-0': '## Obscured Face Verification Attack', '1905.05243-2-18-0': 'The obscured face verification attack is defined as: given an obscured face and a clear face, decide if the two faces come from the same person or not.', '1905.05243-2-18-1': 'Previous work [CITATION] only considers the identification scenario, which assumes all identities are in the dataset.', '1905.05243-2-18-2': 'However, in many cases, we cannot assume the obscured identity is in any dataset.', '1905.05243-2-18-3': 'For example, the attackers may want to find out if the obscured face from a TV news is a person they know.', '1905.05243-2-18-4': 'Therefore, face verification attack is more stringent.', '1905.05243-2-19-0': 'In order to solve this verification problem, we project the image into a low-dimension latent vector, where faces from the same person are closer together than faces from different people.', '1905.05243-2-19-1': 'Therefore, by comparing the distance of the latent vectors, we can determine if the two faces are from the same person or not.', '1905.05243-2-19-2': 'To improve the accuracy, we use the Additive Angular Margin loss (also known as ArcFace) [CITATION] to obtain highly discriminative features for face recognition.', '1905.05243-2-19-3': 'ArcFace simultaneously reduces intra-class difference and enlarge inter-class difference of the embedding vectors.', '1905.05243-2-19-4': 'We choose ArcFace because it yields the best facial recognition performance among the traditional softmax loss [CITATION], contrastive loss [CITATION], triplet loss [CITATION], and other angular space losses, like SphereFace [CITATION] and CosFace [CITATION].', '1905.05243-2-19-5': 'Specifically, ArcFace is designed to enforce a margin between the distance of the sample to its class center and the distances of the sample to the other centers from different classes in angular space.', '1905.05243-2-19-6': 'Given an input image (either clear image or obscured image), we first embed it as a low-dimension vector [MATH] using a deep learning model.', '1905.05243-2-19-7': 'Define an auxiliary projection weight [MATH], where [MATH] is the number of unique identities in the dataset.', '1905.05243-2-19-8': 'We further normalize the embedding vector and projection weight as [MATH] and [MATH], respectively.', '1905.05243-2-19-9': 'The normalized embedding vector then is projected onto [MATH] as follows [EQUATION] where [MATH] is a vector of angular distance from [MATH] to [MATH].', '1905.05243-2-19-10': 'The normalized embedding vector is then re-scaled by multiplying a scalar [MATH] to make it distributed on a hypersphere with a radius of [MATH].', '1905.05243-2-19-11': 'The ArcFace loss function of a single sample is then calculated using softmax cross entropy as follows [EQUATION] where [MATH] is the additive angular margin penalty between [MATH] and [MATH], [MATH] is the angle of the target class of the input image.', '1905.05243-2-19-12': 'Note that the computation of the ArcFace loss is only used to aid the training process.', '1905.05243-2-19-13': 'For inference, we compute the embedding vectors from the clear face [MATH] and obscured face [MATH] using the same deep learning model.', '1905.05243-2-19-14': 'We then compare the angular distance after normalization to a predefined threshold value to determine the verification result.', '1905.05243-2-19-15': 'The threshold value can be obtained based on the value that maximizes the verification accuracy on the validation set.', '1905.05243-2-20-0': '## Obscured Face Reconstruction', '1905.05243-2-21-0': 'As we will show in Section [REF], highly obscured images still contain identifiable information.', '1905.05243-2-21-1': 'To examine the amount of remaining information in obscured images, we design a reconstruction attack to visualize how well we can recover the original image.', '1905.05243-2-21-2': 'We apply a conditional generative adversarial network, Pix2Pix [CITATION], to perform this image reconstruction attack.', '1905.05243-2-21-3': 'Given the obscured images, the generator is trained to reconstruct the clear image guided by the discriminator and the [MATH] distance loss.', '1905.05243-2-21-4': 'To quantify the reconstruction performance, we compute the mean square error (MSE) over pixel-wise differences.', '1905.05243-2-21-5': 'We also compute the identification accuracy based on a face recognition model which is pretrained with clear images.', '1905.05243-2-21-6': 'This test provides us a way to quantify and visualize the amount of identifiable information leaked from the obscuration methods.', '1905.05243-2-22-0': '# Experiments', '1905.05243-2-23-0': 'In this section, we first briefly describe the obscuration methods to be evaluated.', '1905.05243-2-23-1': 'Then, we provide the design and result of the aforementioned attacking scenarios.', '1905.05243-2-24-0': '## Evaluated Methods', '1905.05243-2-25-0': 'In this work, we propose to analyze eight obscuration methods.', '1905.05243-2-25-1': 'These methods include three traditional methods (Gaussian blurring, median blurring, and pixelation), three [MATH]-same based methods ([MATH]-same, [MATH]-same-net, and UP-GAN) and two privacy-preserving image sharing methods (P3 and scrambling).', '1905.05243-2-25-2': 'Examples of obscured faces using these methods are shown in Figure [REF].', '1905.05243-2-25-3': 'We use Gaussian-5 representing the experiment of Gaussian blurring with kernel size of 5.', '1905.05243-2-26-0': 'Traditional obscuration methods.', '1905.05243-2-26-1': 'We evaluate the three obscuration methods including Gaussian blurring, median blurring and pixelation methods for four different kernel (pixel) sizes of 5, 15, 25, and 35.', '1905.05243-2-26-2': 'We use the OpenCV function cv2.getGaussianKernel to compute the kernel of Gaussian blurring.', '1905.05243-2-26-3': 'Note that the Gaussian standard deviation is defined as [EQUATION] where [MATH] is the kernel size.', '1905.05243-2-26-4': 'The pixelation method is implemented by image downsampling and upsampling using nearest-neighbor interpolation.', '1905.05243-2-27-0': '[MATH]-same based obscuration methods.', '1905.05243-2-27-1': '[MATH]-same based methods aim to obscure identifiable information while preserving the non-identifiable information (also known as utility information).', '1905.05243-2-27-2': 'Algorithm [REF] shows the workflow of the [MATH]-same based methods, which is based on [CITATION].', '1905.05243-2-27-3': 'In this work, we choose [MATH].', '1905.05243-2-27-4': 'We evaluate three [MATH]-same based methods: the original [MATH]-same method [CITATION], [MATH]-same-net [CITATION], and UP-GAN [CITATION].', '1905.05243-2-27-5': 'We model the obscuration process as follows.', '1905.05243-2-28-0': 'Suppose we have a clear face dataset [MATH] and an obscuration function [MATH] mapping the clear image [MATH] to the obscured image [MATH] by [MATH].', '1905.05243-2-28-1': 'We use this mapping function building an obscured face dataset [MATH] based on [MATH].', '1905.05243-2-28-2': 'Based on [CITATION], we also need to assume the dataset [MATH] has no two images coming from the same identity to make Algorithm [REF] [MATH]-anonymous.', '1905.05243-2-28-3': 'The [MATH]-same based methods require the function [MATH] mapping [MATH] nearest neighbors from the clear images to a single obscured image.', '1905.05243-2-28-4': 'For example, considering the original [MATH]-same method, the obscured face is obtained by averaging the [MATH] nearest neighbors in the image space.', '1905.05243-2-28-5': 'Therefore, the [MATH] from Algorithm [REF] in this case are the clear images.', '1905.05243-2-29-0': '[MATH]-same-net is a generative deep learning model that generates fake faces given the cluster attributes.', '1905.05243-2-29-1': 'UP-GAN has similar generator architecture to [MATH]-same-net with the same input cluster attributes.', '1905.05243-2-29-2': 'However, it improves the generated image quality using its discriminator and the perceptual loss constraint.', '1905.05243-2-29-3': 'For both [MATH]-same-net and UP-GAN, the [MATH] from Algorithm [REF] are the cluster attributes.', '1905.05243-2-29-4': 'Therefore, the input attribute to the models is the average of the [MATH] nearest neighbors in the attribute space.', '1905.05243-2-30-0': 'As proposed by [CITATION], we choose UTKFace dataset [CITATION], which contains the required utility values (age, gender, and skin tone) and facial landmarks to train [MATH]-same-net and UP-GAN.', '1905.05243-2-30-1': 'The utility values are defined as facial features that do not reveal identity, such as age, gender, skin tone, pose, and expression [CITATION].', '1905.05243-2-30-2': 'For the purpose of obscuration evaluation, we test these two methods on the FaceScrub dataset [CITATION], with a fixed utility values (26 years old, male, and white) and 7-point facial landmarks obtained by Dlib toolkit [CITATION].', '1905.05243-2-30-3': 'These points include the centers of the eyes, the center of the nose, and four points around the mouth.', '1905.05243-2-30-4': 'Note that since the FaceScrub dataset contains different faces from the same identity, the [MATH]-anonymity property in this case may not hold.', '1905.05243-2-30-5': '[tb] Clear face dataset [MATH], privacy constant [MATH] with [MATH] Obscured face dataset [MATH]', '1905.05243-2-31-0': 'Select the [MATH] nearest neighbors [MATH] Add [MATH] copies of [MATH] to [MATH] Remove [MATH] from [MATH]', '1905.05243-2-32-0': 'Workflow of the [MATH]-same based methods.', '1905.05243-2-33-0': 'Privacy-preserving image sharing methods.', '1905.05243-2-33-1': 'Privacy-preserving image sharing methods are designed to encrypt the content of the original image when publishing to social media.', '1905.05243-2-33-2': 'To recover the original images, the encrypted images need a key to decrypt the content.', '1905.05243-2-33-3': 'We evaluate two methods: P3 [CITATION] and scrambling [CITATION].', '1905.05243-2-33-4': 'Both of them are based on the manipulation of DCT coefficients in the JPEG framework.', '1905.05243-2-33-5': 'After obtaining the DCT coefficients from [MATH] image patches, P3 separates the AC coefficients given a predefined threshold value.', '1905.05243-2-33-6': 'It then stores the coefficients that are smaller than the threshold value as the public image.', '1905.05243-2-33-7': 'The secret image contains the DC coefficients and the AC coefficients that are higher than the threshold value.', '1905.05243-2-33-8': 'In this paper, we choose the threshold value as 10.', '1905.05243-2-33-9': 'For the scrambling method, it first evenly and randomly flips the DCT coefficients and stores the result as the public image.', '1905.05243-2-33-10': 'For the secret part, it only stores the random seed.', '1905.05243-2-33-11': 'Therefore, it can restore the image by undoing the flipping process based on the random seed.', '1905.05243-2-33-12': 'In this paper, we scramble both DC and AC DCT coefficients for all YUV components, which is the high-level scrambling as proposed by [CITATION].', '1905.05243-2-34-0': '## Datasets', '1905.05243-2-35-0': 'We use the FaceScrub dataset [CITATION] which contains 106,863 face images from 530 identities.', '1905.05243-2-35-1': 'Therefore, the classification accuracy of randomly guessing is about 0.002.', '1905.05243-2-35-2': 'For the identification and verification attacks, we split the images from each identity into training, validation, and testing sets with the ratio of [MATH].', '1905.05243-2-35-3': 'For the reconstruction attack, we split the identities into three groups for the purpose of training, validation, and testing with the same ratio.', '1905.05243-2-35-4': 'We do so to verify if the reconstruction model is able to recover unknown identity instead of just memorizing faces.', '1905.05243-2-36-0': '## Obscured Face Identification Attack', '1905.05243-2-37-0': 'Experimental Design.', '1905.05243-2-37-1': 'This attack is designed to quantify the obscuration performance in the face identification scenario.', '1905.05243-2-37-2': 'To have a more generalizable conclusion, we run the experiments based on two widely used backbone models, VGG19 and ResNet50.', '1905.05243-2-37-3': 'The input images are resized to [MATH] and the output is the softmax score for classification.', '1905.05243-2-38-0': 'Based on the three threat models, we design the experiments as follows.', '1905.05243-2-38-1': 'In the first experiment for [MATH], the identifier is trained with the set of clear images and tested with obscured images.', '1905.05243-2-38-2': 'In the second experiment for [MATH], the identifier is trained on both clear and obscured images and tested with the obscured images of the obscuration method not used in the training set.', '1905.05243-2-38-3': 'The intuition of threat model [MATH] is to verify if we can enforce the attacker to learn more robust features from this complex dataset.', '1905.05243-2-38-4': 'This can be seen as data augmentation.', '1905.05243-2-38-5': 'Specifically for the three traditional methods, we use the obscured images from two methods during training and use the other one for testing.', '1905.05243-2-38-6': 'For the [MATH]-same based methods and privacy-preserving image sharing methods, we train on all three traditional methods.', '1905.05243-2-38-7': 'Jointly training on clear and obscured images provides a better accuracy compared to learning from the obscured images themselves.', '1905.05243-2-38-8': 'In the third experiment for [MATH], each identifier is trained on both clear and obscured images and tested with the obscured images using the same obscuration method.', '1905.05243-2-39-0': 'Result.', '1905.05243-2-39-1': 'Table [REF] shows the identification accuracy from different obscuration methods and threat models.', '1905.05243-2-39-2': 'The lower the identification accuracy, the better the performance of the obscuration method.', '1905.05243-2-39-3': 'The results of the clear images under [MATH] and [MATH] are obtained by training on all three traditional methods and testing on the clear images.', '1905.05243-2-40-0': 'We first compare the same method and same backbone model with different threat models.', '1905.05243-2-40-1': 'As the attackers get more information (i.e.from [MATH] to [MATH]), the identification accuracy increases.', '1905.05243-2-40-2': 'This means that the identifiable information left in the obscured images can still be learned by the attackers given proper training data.', '1905.05243-2-40-3': 'For example, the accuracy of Gaussian-35 with VGG19 increases from 0.007 to 0.811 for [MATH] and [MATH], respectively.', '1905.05243-2-40-4': 'Therefore, Gaussian blurring completely fails to provide privacy for [MATH], although visually speaking a human is not able to identify someone from the obscured images.', '1905.05243-2-40-5': 'A similar conclusion can be drawn for median blurring.', '1905.05243-2-40-6': 'Although pixelation with a large pixel size can achieve a relatively good performance, comparing the results from [MATH] to [MATH], the attacking accuracy still improves a lot.', '1905.05243-2-40-7': 'e.g.for pixelation-35 with VGG19, the accuracy increases from 0.004 to 0.373, for [MATH] and [MATH], respectively.', '1905.05243-2-40-8': 'The three [MATH]-same based methods achieve a good obscuration performance even for [MATH].', '1905.05243-2-40-9': 'For the privacy-preserving image sharing methods, although they achieve the best performance under [MATH] and [MATH], they still fail to provide a good obscuration under [MATH].', '1905.05243-2-40-10': 'Surprisingly, even for the scrambling method which involves a random flipping process, the attackers can still extract useful features for accurate identification.', '1905.05243-2-40-11': 'Note that these conclusions do not change for different backbone models.', '1905.05243-2-41-0': 'Considering [MATH] itself, besides Gaussian-5 and median-5, all methods achieve an effective obscuration on both VGG19 and ResNet50 models.', '1905.05243-2-41-1': 'This means that the attackers fail to extract identifiable information from the obscured images if they solely learn from the clear image.', '1905.05243-2-41-2': 'For the three traditional methods, the obscuration performance gets better (i.e.identification accuracy gets lower) as the kernel size increases.', '1905.05243-2-41-3': 'The original [MATH]-same method achieves the best obscuration performance among the three [MATH]-same based methods.', '1905.05243-2-41-4': 'For [MATH]-same-net and UP-GAN, since they allow the input of utility information to generate obscured faces, their obscuration performance is a little bit worse than the original [MATH]-same method.', '1905.05243-2-41-5': 'Both of the privacy-preserving image sharing methods achieve the performance of randomly guessing, which means the attackers cannot extract any identifiable information from the obscured images.', '1905.05243-2-42-0': 'For [MATH], by introducing more informative training set, all traditional methods have worse performance, besides pixelation-25 and pixelation-35, which are relatively close to the results obtained from [MATH].', '1905.05243-2-42-1': 'The obscuration performance of Gaussian and median blurring drops significantly (i.e.the identification accuracy greatly increases).', '1905.05243-2-42-2': 'Because the two methods share similar blurring effects, the attackers can learn more robust features from the augmented training set.', '1905.05243-2-42-3': 'For the [MATH]-same based methods and privacy-preserving image sharing methods, compared to [MATH], the augmented training set still does not provide useful knowledge for the attackers.', '1905.05243-2-43-0': 'For [MATH], both attackers achieve the strongest attack for all cases.', '1905.05243-2-43-1': 'Even for pixelation-35, which only contains 9 distinct pixel values, both attackers can still achieve a identification accuracy over 0.5, which is much bigger than the accuracy of randomly guessing (0.002).', '1905.05243-2-43-2': 'The three [MATH]-same based methods achieve the best obscuration performance by a great margin when compared to other methods.', '1905.05243-2-43-3': 'Surprisingly, the two privacy-preserving image sharing methods have a much worse performance compared to their performance in [MATH] and [MATH].', '1905.05243-2-43-4': 'Even successfully concealing the identifiable information in terms of human perception, both methods fail to provide effective obscuration.', '1905.05243-2-44-0': 'Therefore, based on the results from the identification attack, the [MATH]-same based methods achieve the best obscuration performance.', '1905.05243-2-45-0': '## Obscured Face Verification Attack', '1905.05243-2-46-0': 'Experimental Design.', '1905.05243-2-46-1': 'Similarly to the experiment setting in the identification task, we resize the input image to [MATH].', '1905.05243-2-46-2': 'According to [CITATION], we choose the dimension of the embedding vector as 512 and margin [MATH] as 0.5.', '1905.05243-2-46-3': 'However, if we use the re-scale factor [MATH] as suggested by the original paper, we are not able to obtain a stable result.', '1905.05243-2-46-4': 'Therefore, after several experiments, we empirically choose the re-scale factor as [MATH] for VGG19 and [MATH] for ResNet50, which provides the best performance according to the validation set.', '1905.05243-2-46-5': 'The batch size is chosen as 128.', '1905.05243-2-46-6': 'We choose the stochastic gradient descent (SGD) as optimizer with a weight decay of [MATH].', '1905.05243-2-46-7': 'The learning rate starts at 0.1 and is divided by 10 at the epochs of 6, 11, and 16.', '1905.05243-2-46-8': 'For the training of P3 and scrambling, we reduce the starting learning rate to 0.05 due to convergence issues.', '1905.05243-2-46-9': 'Assuming the identification task, we implement the experiments based on the three threat models.', '1905.05243-2-46-10': 'For the performance metric, since the face verification problem is just a binary classification problem, we choose the area under the curve (AUC) of the receiver operating characteristic (ROC) curve to examine the performance.', '1905.05243-2-47-0': 'During testing we need to obtain pairs of faces with the same identity and pairs of faces with different identities.', '1905.05243-2-47-1': 'Due to the large number of combinations of valid pairs from the testing set, in our implementation, we only compute all valid pairs within each mini-batch (128 images which are coming from 64 identities).', '1905.05243-2-47-2': 'Furthermore, we run testing 10 times with different combinations of image pairs.', '1905.05243-2-47-3': 'The average AUC is been reported in Table [REF].', '1905.05243-2-47-4': 'The standard deviation for the tests ranges from [MATH].', '1905.05243-2-47-5': 'Therefore, we can directly use the average AUC to compare different experiments because of the small variation.', '1905.05243-2-48-0': 'Result.', '1905.05243-2-48-1': 'Table [REF] shows the verification AUC from different obscuration methods, threat models and backbone models.', '1905.05243-2-48-2': 'The lower the AUC, the better the performance of the obscuration method.', '1905.05243-2-48-3': 'We first compare the same method and same backbone model to different threat models.', '1905.05243-2-48-4': 'As the attackers get more information (from [MATH] to [MATH]), the verification AUC increases.', '1905.05243-2-48-5': 'Take Gaussian-35 with VGG19 as an instance again.', '1905.05243-2-48-6': 'The AUC increases from 0.561 to 0.963 for [MATH] and [MATH], respectively.', '1905.05243-2-48-7': 'Note that the AUC for randomly guessing is 0.5.', '1905.05243-2-48-8': 'This means that although Gaussian-35 can successfully defend from the attack under [MATH], after introducing the obscured data in the training set, the attackers can still extract enough identifiable information to achieve a high accuracy verification.', '1905.05243-2-48-9': 'For the [MATH]-same based methods, similar to the identification attack, they achieve a robust obscuration performance even for [MATH].', '1905.05243-2-48-10': 'For the privacy-preserving image sharing methods, both of them succeed in [MATH] and [MATH], but fail to obscure the identities under [MATH].', '1905.05243-2-48-11': 'Note that for different backbone models, although there is a small performance difference, choosing different models does not affect the conclusions reached above.', '1905.05243-2-49-0': 'Consider different methods with the same threat model and backbone model.', '1905.05243-2-49-1': 'For the traditional methods, a similar conclusion to the identification attack can be drawn.', '1905.05243-2-49-2': 'As the kernel (pixel) size increases, the AUC decreases for all cases, especially for pixelation-35 with VGG19 which achieves the best performance among the traditional methods.', '1905.05243-2-49-3': 'The [MATH]-same based methods achieve good results for all threat models and both attackers, which agrees with the conclusion from the identification attack.', '1905.05243-2-49-4': 'Although the privacy-preserving image sharing methods can conceal identities well under [MATH] and [MATH], for the stronger [MATH], both of them fail to provide effective obscuration.', '1905.05243-2-50-0': 'Therefore, based on the results from the verification attack, the [MATH]-same based methods achieve the best obscuration performance.', '1905.05243-2-51-0': '## Obscured Face Reconstruction Attack', '1905.05243-2-52-0': 'Experimental Design.', '1905.05243-2-52-1': 'In the previous sections we show that most of the obscuration methods fail to remove all identifiable information.', '1905.05243-2-52-2': 'In this reconstruction attack, we try to use the remaining information from these obscured images to recover the clear image.', '1905.05243-2-52-3': 'If the remaining information has a strong correlation with the information from the clear image, we should be able to reconstruct the original face with a high accuracy.', '1905.05243-2-52-4': 'In this implementation, we choose Pix2Pix [CITATION] which is a GAN model designed for image-to-image translation as our reconstruction model.', '1905.05243-2-53-0': 'Assume that the obscured images and clear images come from two distinct distributions.', '1905.05243-2-53-1': 'The reconstruction model aims to find a mapping function from the obscured image distribution to the clear image distribution.', '1905.05243-2-53-2': 'To quantify the reconstruction performance, we choose mean square error (MSE) as the metric to calculate pixel-wise distance between the clear image and the reconstructed image.', '1905.05243-2-53-3': 'The value range of the clear image and reconstructed image is [MATH].', '1905.05243-2-53-4': 'To evaluate similarity of the identifiable information from the reconstructed image and the clear image, we use the identification accuracy obtained from the ResNet50 model which is pretrained on the clear images.', '1905.05243-2-53-5': 'This is the same setting as [MATH], since the attacker is trained with clear images and tested with obscured images.', '1905.05243-2-54-0': 'Result.', '1905.05243-2-54-1': 'Figure [REF] shows the reconstruction results from Gaussian-25, median-25, pixelation-25, P3, scrambling, [MATH]-same, [MATH]-same-net, and UP-GAN.', '1905.05243-2-54-2': 'Visually, the three [MATH]-same based methods can successfully prevent reconstruction compared with other methods.', '1905.05243-2-54-3': 'Although the privacy-preserving image sharing methods can prevent identification in terms of human perception, the reconstruction model can still recover the images fairly accurately, especially for P3.', '1905.05243-2-54-4': 'For the three traditional methods, pixelation-25 achieves a better obscuration performance compared to Gaussian-25 and median-25.', '1905.05243-2-55-0': 'Table [REF] shows the results of the face reconstruction attack.', '1905.05243-2-55-1': 'Note that setting Clear means we input clear images to Pix2Pix model to achieve an identity mapping.', '1905.05243-2-55-2': 'The exact MSE for the clear image is 0.000144 and the exact MSE for Gaussian-5 is 0.000289.', '1905.05243-2-55-3': 'For the three traditional methods, with the kernel (pixel) size increases, the reconstruction MSE increases and the identification accuracy decreases.', '1905.05243-2-55-4': 'Compared to the identification attack of [MATH], this reconstruction process can help the attackers achieve a stronger attack, since the accuracy from the reconstructed images is higher than the obscured images for most cases.', '1905.05243-2-55-5': 'The [MATH]-same based methods achieve both high MSE and low identification accuracy.', '1905.05243-2-55-6': 'Compared to the three [MATH]-same methods, the two privacy-preserving image sharing methods are vulnerable to the reconstruction attack, because of their low MSE.', '1905.05243-2-55-7': 'Therefore, as with the conclusion in the identification and verification attack, these two methods also fail to conceal identity on this reconstruction attack.', '1905.05243-2-56-0': '# Conclusion', '1905.05243-2-57-0': 'In this paper, we propose a set of experiments to analyze the robustness of face obscuration methods.', '1905.05243-2-57-1': 'We provide a comprehensive analysis of eight obscuration methods: Gaussian blurring, median blurring, pixelation, [MATH]-same, [MATH]-same-net, UP-GAN, P3, and scrambling.', '1905.05243-2-57-2': 'We examine the robustness of these methods under different attacking scenarios including identification, verification, and reconstruction with two widely used deep learning models, VGG19 and ResNet50.', '1905.05243-2-57-3': 'Threat modeling is also considered to evaluate the obscuration methods under different strength of attacks.', '1905.05243-2-57-4': 'Methods such as Gaussian blurring, median blurring, P3, and scrambling fail to provide an effective obscuration under the designed attackers, although they successfully defeat human perception.', '1905.05243-2-57-5': 'We also show that the [MATH]-same based methods can provide a secured privacy protection.', '1905.05243-2-57-6': 'Hence, since relying on human perception is no longer an option to guarantee privacy, the proposed set of experiments should be used to quantify and benchmark the effectiveness of any future face obscuration method.'}

[['1905.05243-1-4-0', '1905.05243-2-4-0'], ['1905.05243-1-10-0', '1905.05243-2-10-0'], ['1905.05243-1-10-4', '1905.05243-2-10-4'], ['1905.05243-1-0-5', '1905.05243-2-0-5'], ['1905.05243-1-2-0', '1905.05243-2-2-0'], ['1905.05243-1-2-1', '1905.05243-2-2-1'], ['1905.05243-1-16-1', '1905.05243-2-16-1'], ['1905.05243-1-9-0', '1905.05243-2-9-0'], ['1905.05243-1-6-3', '1905.05243-2-6-2'], ['1905.05243-1-14-0', '1905.05243-2-14-0'], ['1905.05243-1-14-2', '1905.05243-2-14-1'], ['1905.05243-1-7-1', '1905.05243-2-7-2'], ['1905.05243-1-7-4', '1905.05243-2-7-6'], ['1905.05243-1-7-6', '1905.05243-2-7-8'], ['1905.05243-1-10-2', '1905.05243-2-10-2'], ['1905.05243-1-10-3', '1905.05243-2-10-3'], ['1905.05243-1-3-0', '1905.05243-2-3-0'], ['1905.05243-1-3-2', '1905.05243-2-3-2'], ['1905.05243-1-3-3', '1905.05243-2-3-3'], ['1905.05243-1-3-6', '1905.05243-2-3-7'], ['1905.05243-1-3-7', '1905.05243-2-3-8'], ['1905.05243-1-3-8', '1905.05243-2-3-9'], ['1905.05243-1-3-9', '1905.05243-2-3-10'], ['1905.05243-1-0-2', '1905.05243-2-0-2'], ['1905.05243-1-22-0', '1905.05243-2-18-0'], ['1905.05243-1-24-8', '1905.05243-2-19-15'], ['1905.05243-1-2-2', '1905.05243-2-2-2'], ['1905.05243-1-2-3', '1905.05243-2-2-3'], ['1905.05243-1-2-4', '1905.05243-2-2-4'], ['1905.05243-1-2-6', '1905.05243-2-2-10'], ['1905.05243-1-16-0', '1905.05243-2-16-0'], ['1905.05243-1-33-1', '1905.05243-2-38-1'], ['1905.05243-1-33-2', '1905.05243-2-38-2'], ['1905.05243-1-33-3', '1905.05243-2-38-8'], ['1905.05243-1-9-2', '1905.05243-2-9-1'], ['1905.05243-1-9-3', '1905.05243-2-9-2'], ['1905.05243-1-9-5', '1905.05243-2-9-4'], ['1905.05243-1-6-5', '1905.05243-2-6-4'], ['1905.05243-1-6-6', '1905.05243-2-6-5'], ['1905.05243-1-7-2', '1905.05243-2-7-4'], ['1905.05243-1-7-3', '1905.05243-2-7-5'], ['1905.05243-1-7-5', '1905.05243-2-7-7'], ['1905.05243-1-8-0', '1905.05243-2-7-10'], ['1905.05243-1-4-3', '1905.05243-2-4-3'], ['1905.05243-1-10-1', '1905.05243-2-10-1'], ['1905.05243-1-10-5', '1905.05243-2-10-5'], ['1905.05243-1-3-1', '1905.05243-2-3-1'], ['1905.05243-1-3-4', '1905.05243-2-3-4'], ['1905.05243-1-0-0', '1905.05243-2-0-0'], ['1905.05243-1-0-1', '1905.05243-2-0-1'], ['1905.05243-1-0-3', '1905.05243-2-0-3'], ['1905.05243-1-0-4', '1905.05243-2-0-4'], ['1905.05243-1-0-6', '1905.05243-2-0-6'], ['1905.05243-1-24-7', '1905.05243-2-19-1'], ['1905.05243-1-24-7', '1905.05243-2-19-13'], ['1905.05243-1-45-0', '1905.05243-2-50-0'], ['1905.05243-1-2-5', '1905.05243-2-2-5'], ['1905.05243-1-12-0', '1905.05243-2-12-0'], ['1905.05243-1-40-0', '1905.05243-2-54-4'], ['1905.05243-1-33-0', '1905.05243-2-38-0'], ['1905.05243-1-9-4', '1905.05243-2-9-3'], ['1905.05243-1-6-2', '1905.05243-2-6-1'], ['1905.05243-1-6-4', '1905.05243-2-6-3'], ['1905.05243-1-7-0', '1905.05243-2-7-1'], ['1905.05243-1-23-0', '1905.05243-2-19-0'], ['1905.05243-1-28-4', '1905.05243-2-4-3'], ['1905.05243-1-42-10', '1905.05243-2-47-0'], ['1905.05243-1-42-11', '1905.05243-2-46-10'], ['1905.05243-1-47-0', '1905.05243-2-57-1']]

[['1905.05243-1-4-0', '1905.05243-2-4-0'], ['1905.05243-1-10-0', '1905.05243-2-10-0'], ['1905.05243-1-10-4', '1905.05243-2-10-4'], ['1905.05243-1-0-5', '1905.05243-2-0-5'], ['1905.05243-1-2-0', '1905.05243-2-2-0'], ['1905.05243-1-2-1', '1905.05243-2-2-1'], ['1905.05243-1-16-1', '1905.05243-2-16-1'], ['1905.05243-1-9-0', '1905.05243-2-9-0'], ['1905.05243-1-6-3', '1905.05243-2-6-2'], ['1905.05243-1-14-0', '1905.05243-2-14-0'], ['1905.05243-1-14-2', '1905.05243-2-14-1'], ['1905.05243-1-7-1', '1905.05243-2-7-2'], ['1905.05243-1-7-4', '1905.05243-2-7-6'], ['1905.05243-1-7-6', '1905.05243-2-7-8']]

[['1905.05243-1-10-2', '1905.05243-2-10-2'], ['1905.05243-1-10-3', '1905.05243-2-10-3'], ['1905.05243-1-3-0', '1905.05243-2-3-0'], ['1905.05243-1-3-2', '1905.05243-2-3-2'], ['1905.05243-1-3-3', '1905.05243-2-3-3'], ['1905.05243-1-3-6', '1905.05243-2-3-7'], ['1905.05243-1-3-7', '1905.05243-2-3-8'], ['1905.05243-1-3-8', '1905.05243-2-3-9'], ['1905.05243-1-3-9', '1905.05243-2-3-10'], ['1905.05243-1-0-2', '1905.05243-2-0-2'], ['1905.05243-1-22-0', '1905.05243-2-18-0'], ['1905.05243-1-24-8', '1905.05243-2-19-15'], ['1905.05243-1-2-2', '1905.05243-2-2-2'], ['1905.05243-1-2-3', '1905.05243-2-2-3'], ['1905.05243-1-2-4', '1905.05243-2-2-4'], ['1905.05243-1-2-6', '1905.05243-2-2-10'], ['1905.05243-1-16-0', '1905.05243-2-16-0'], ['1905.05243-1-33-1', '1905.05243-2-38-1'], ['1905.05243-1-33-2', '1905.05243-2-38-2'], ['1905.05243-1-33-3', '1905.05243-2-38-8'], ['1905.05243-1-9-2', '1905.05243-2-9-1'], ['1905.05243-1-9-3', '1905.05243-2-9-2'], ['1905.05243-1-9-5', '1905.05243-2-9-4'], ['1905.05243-1-6-5', '1905.05243-2-6-4'], ['1905.05243-1-6-6', '1905.05243-2-6-5'], ['1905.05243-1-7-2', '1905.05243-2-7-4'], ['1905.05243-1-7-3', '1905.05243-2-7-5'], ['1905.05243-1-7-5', '1905.05243-2-7-7'], ['1905.05243-1-8-0', '1905.05243-2-7-10']]

[]

[['1905.05243-1-4-3', '1905.05243-2-4-3'], ['1905.05243-1-10-1', '1905.05243-2-10-1'], ['1905.05243-1-10-5', '1905.05243-2-10-5'], ['1905.05243-1-3-1', '1905.05243-2-3-1'], ['1905.05243-1-3-4', '1905.05243-2-3-4'], ['1905.05243-1-0-0', '1905.05243-2-0-0'], ['1905.05243-1-0-1', '1905.05243-2-0-1'], ['1905.05243-1-0-3', '1905.05243-2-0-3'], ['1905.05243-1-0-4', '1905.05243-2-0-4'], ['1905.05243-1-0-6', '1905.05243-2-0-6'], ['1905.05243-1-24-7', '1905.05243-2-19-1'], ['1905.05243-1-24-7', '1905.05243-2-19-13'], ['1905.05243-1-45-0', '1905.05243-2-50-0'], ['1905.05243-1-2-5', '1905.05243-2-2-5'], ['1905.05243-1-12-0', '1905.05243-2-12-0'], ['1905.05243-1-40-0', '1905.05243-2-54-4'], ['1905.05243-1-33-0', '1905.05243-2-38-0'], ['1905.05243-1-9-4', '1905.05243-2-9-3'], ['1905.05243-1-6-2', '1905.05243-2-6-1'], ['1905.05243-1-6-4', '1905.05243-2-6-3'], ['1905.05243-1-7-0', '1905.05243-2-7-1']]

[['1905.05243-1-23-0', '1905.05243-2-19-0'], ['1905.05243-1-28-4', '1905.05243-2-4-3'], ['1905.05243-1-42-10', '1905.05243-2-47-0'], ['1905.05243-1-42-11', '1905.05243-2-46-10'], ['1905.05243-1-47-0', '1905.05243-2-57-1']]

['1905.05243-1-6-0', '1905.05243-1-30-0', '1905.05243-1-34-0', '1905.05243-1-42-0', '1905.05243-1-43-0', '1905.05243-2-6-0', '1905.05243-2-26-0', '1905.05243-2-31-0', '1905.05243-2-32-0', '1905.05243-2-37-0', '1905.05243-2-39-0', '1905.05243-2-46-0', '1905.05243-2-48-0', '1905.05243-2-52-0', '1905.05243-2-54-0']

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

https://arxiv.org/abs/1905.05243

1612.04737

{'1612.04737-1-0-0': 'We discuss weak kaon decays in a scenario in which the Standard Model is extended by massive sterile fermions.', '1612.04737-1-0-1': 'After revisiting the analytical expressions for leptonic and semileptonic decays we derive the expressions for decay rates with two neutrinos in the final state.', '1612.04737-1-0-2': 'By using a simple effective model with only one sterile neutrino, compatible with all current experimental bounds and general theoretical constraints, we conduct a thorough numerical analysis which reveals that the impact of the presence of massive sterile neutrinos on kaon weak decays is very small, less than [MATH] on decay rates.', '1612.04737-1-0-3': 'The only exception is [MATH], which can go up to [MATH], thus possibly within the reach of the KOTO experiment.', '1612.04737-1-0-4': 'In other words, if all the future measurements of weak kaon decays turn out to be compatible with the Standard Model predictions, this would not rule out the existence of massive light sterile neutrinos with non-negligible active-sterile mixing.', '1612.04737-1-0-5': 'Instead, for a sterile neutrino of mass below [MATH], one might obtain a huge enhancement of [MATH], otherwise negligibly small in the Standard Model.', '1612.04737-1-1-0': '# Introduction', '1612.04737-1-2-0': 'The Standard Model predicts the strict conservation of lepton flavor to all orders.', '1612.04737-1-2-1': 'The fact that neutrinos oscillate provides a clear evidence of the existence of physics beyond the Standard Model.', '1612.04737-1-2-2': 'New Physics models accommodating massive neutrinos and their mixing open the door to many phenomena which basically have no Standard Model background, such as lepton number violation, violation of lepton flavor universality, or lepton flavor violation.', '1612.04737-1-3-0': 'The experimental effort associated with the search of new physics using observables involving leptons is impressive, and is currently being pursued on all experimental fronts: i) neutrino dedicated experiments which aim at determining neutrino properties, such as the Majorana/Dirac nature, the absolute neutrino masses, the hierarchy of their mass spectrum, leptonic mixing and the CP-violating phases; ii) high-intensity facilities that are studying several low-energy processes such as [MATH], [MATH], [MATH] conversion in atoms, [MATH] and meson [MATH] decays ([MATH], [MATH], [MATH], [MATH], ); iii) the Large Hadron Collider (LHC), which is the privileged discovery ground of new particles, may also allow to probe leptonic mixing in the production and/or decay of the new states.', '1612.04737-1-3-1': 'On the other hand, the recent cosmological data has put constraints on the sum of the light neutrino masses, which is especially restrictive when considering new light neutral states.', '1612.04737-1-4-0': 'Among the several minimal possible scenarios, extending the SM with sterile fermions - which are singlets under the Standard Model gauge group - is a very appealing hypothesis, as their unique (indirect) interaction with the Standard Model fields occur through their mixing with the active neutrinos [MATH] (via their Yukawa couplings).', '1612.04737-1-4-1': 'Due to their very unique nature, there is no bound on their number, and a priori no limits regarding their mass regimes.', '1612.04737-1-4-2': 'Interestingly, sterile fermions (like right-handed neutrinos) are present in many frameworks accounting for neutrino masses and the observed mixing (as is the case for fermion seesaw mechanisms).', '1612.04737-1-4-3': 'The interest in sterile neutrinos and their impact on observables strongly depend on their masses.', '1612.04737-1-4-4': 'Sterile neutrinos at the eV scales were proposed to solve neutrino oscillation anomalies in reactors [CITATION], accelerators [CITATION], as well as in the calibration of gallium-target solar neutrino experiments [CITATION], all suggesting beyond the [MATH]-neutrino paradigm.', '1612.04737-1-4-5': 'The keV scale for sterile neutrinos offers warm dark matter candidates [CITATION], and explanations of some astrophysical issues such as kicks in pulsar velocities [CITATION].', '1612.04737-1-4-6': 'Sterile fermion states with masses above [MATH] GeV have moderate motivation other than their theoretical appeal (GUT theories) and the possibility to have a scenario for baryogenesis via the high-scale leptogenesis [CITATION].', '1612.04737-1-4-7': 'Finally, the appeal for sterile neutrinos in the range [MATH] (and even TeV) strongly resides in their experimental testability due to the many direct and indirect effects in both high-energy (e.g. LHC) [CITATION] and high-intensity (e.g., NA62) experiments.', '1612.04737-1-5-0': 'In order to illustrate the phenomenological effect of a scenario which involves sterile fermions, we focus on a minimal model which extends the Standard Model with an arbitrary number of sterile states with masses in the above-mentioned ranges, known as [MATH] models, without making assumptions concerning the neutrino mass (and/or the lepton mixing) generation mechanism, but allowing to access the degrees of freedom of the sterile neutrinos (their masses, their mixing with the active neutrinos and the new CP-violating phases).', '1612.04737-1-5-1': 'In this work, we address the phenomenological imprints of such sterile fermions on observables involving kaon meson rare decays into lepton final states including neutrinos, which can either be used to set (or update) constraints to model building, or provide interesting observables that could be used as tests of various scenarios of New Physics.', '1612.04737-1-5-2': 'These rare and forbidden decays are being searched for at CERN (NA62) in the charged decay modes [CITATION] and at J-PARC facility (KOTO) in the neutral ones [CITATION].', '1612.04737-1-5-3': 'This study might also be useful for the TREK/E36 experiment at J-PARC, where the data analysis is currently underway [CITATION].', '1612.04737-1-5-4': 'It will further test the lepton universality in kaon two-body decays ([MATH]) and search for a heavy neutrino [CITATION].', '1612.04737-1-6-0': 'Having sterile neutrinos that are sufficiently light to be produced, with non-negligible active sterile mixing angles, may induce important impact on electroweak precision and many other observables.', '1612.04737-1-6-1': 'Our analysis must therefore comply with abundant direct and indirect searches that have already allowed to put constraints on the sterile neutrino masses and their mixing with the active neutrinos.', '1612.04737-1-7-0': 'In this work, we assume the effective case in which the Standard Model is extended by one sterile fermion (3 +1 case) and revisit weak kaon decays such as the leptonic [MATH]), as well as the semileptonic ones [MATH]).', '1612.04737-1-7-1': 'Furthermore we consider the loop-induced weak decays [MATH] and [MATH], derive analytical expressions for their decay rates, which are new results.', '1612.04737-1-7-2': 'In doing the numerical analysis, we (re)derive the expressions for various processes which are used as constraints ([MATH], [MATH], [MATH], ).', '1612.04737-1-7-3': 'We have chosen to present the analytical formulas and numerical results assuming here neutrinos to be Majorana fermions.', '1612.04737-1-7-4': 'A detailed discussion and comparison between the Dirac and Majorana cases is also displayed (in Appendix [REF]).', '1612.04737-1-8-0': 'Our study reveals that the influence of the presence of massive sterile neutrinos on the [MATH], [MATH] and [MATH] decay rates is less than [MATH], thus fully compatible with the Standard Model predictions.', '1612.04737-1-8-1': 'Interestingly, however, we find that [MATH], which is zero in the Standard Model, can be as high as [MATH] and thus possibly within the reach of the KOTO experiment.', '1612.04737-1-8-2': 'In other words, if all the future measurements of weak kaon decays turn out to be compatible with the Standard Model, the paradigm of the existence of sterile neutrinos would still remain valid.', '1612.04737-1-8-3': 'Instead, we show that sterile neutrinos with mass below [MATH] could generate a huge enhancement of [MATH].', '1612.04737-1-9-0': 'Our work is organized as follows: in Sec. [REF] we present a generic model with one sterile neutrino added to the Standard Model and present details concerning the parametrization used in the ensuing analysis.', '1612.04737-1-9-1': 'Sec. [REF] is devoted to a discussion of quantities which are used to constrain the parameter space followed by the actual scan.', '1612.04737-1-9-2': 'The expressions for various weak decay processes of kaons are scrutinized in Sec. [REF], and the sensitivity of these processes on the presence of massive sterile neutrinos is examined and discussed in Sec. [REF].', '1612.04737-1-9-3': 'Our concluding remarks are given in Sec. [REF], while the Feynman rules for the case of Majorana neutrino have been relegated to Appendix [REF].', '1612.04737-1-9-4': 'Appendix [REF] contains a comparison of the analytical expressions for [MATH] when neutrinos are Majorana or Dirac fermions.', '1612.04737-1-10-0': '# Extending the Standard Model with Sterile Fermions', '1612.04737-1-11-0': '## Models with sterile fermions', '1612.04737-1-12-0': 'To discuss the phenomenological consequences of sterile fermion states on low energy physics observables, it is important to have an idea of the underlying framework which involves sterile neutrinos, and among those the testable ones in particular.', '1612.04737-1-12-1': 'Such models are for instance based on low-scale seesaw mechanism, e.g. extension of the Standard Model by exclusively right-handed (RH) neutrinos, like in the usual type I seesaw, which is realized at the TeV scale [CITATION], and for which the Yukawa couplings are small ([MATH]).', '1612.04737-1-12-2': 'They can nevertheless be made higher if one assumes some extra input like the minimal flavor violation [CITATION]), or tiny as it is the case in [MATH]MSM [CITATION].', '1612.04737-1-12-3': 'Other than RH neutrinos, [MATH], one can also consider additional sterile fermions with the opposite lepton numbers of the RH ones, like it is done in the case of the linear [CITATION] or the inverse [CITATION] seesaw mechanisms.', '1612.04737-1-12-4': 'The two latter scenarios are (theoretically and phenomenogically) very appealing as they provide an extra suppression factor, which is linked to a small violation of the total lepton number, allowing to explain the tininess of neutrino masses while having large Yukawa couplings and a comparatively low seesaw scale.', '1612.04737-1-12-5': 'Having relatively light sterile fermions which do not decouple, since they can have non-negligible active-sterile mixing, certainly leads to important consequences and as a result to numerous constraints.', '1612.04737-1-12-6': 'The most important and direct consequence is the modification of the charged and neutral currents as [EQUATION] where [MATH] is the weak coupling constant, and [MATH] is the number of sterile fermions.', '1612.04737-1-12-7': 'The modified lepton mixing matrix, obviously non unitary, also encodes the active-sterile mixing.', '1612.04737-1-12-8': 'In the limit in which the sterile fermions decouple the matrix [MATH] corresponds to the usual Pontecorvo-Maki-Nakagawa-Sakata [MATH] unitary matrix, i.e., [MATH].', '1612.04737-1-12-9': 'Moreover, if sufficiently light, the sterile neutrinos can be produced as decay products.', '1612.04737-1-12-10': 'Both these points might induce huge impact on numerous observables, which in turn can provide abundant constraints on the sterile fermions (masses and the active-sterile mixing angles including the new CP violating phases).', '1612.04737-1-13-0': 'A useful first approach to study the impact of sterile neutrinos on the low energy processes relies on addition of only one sterile fermion to the Standard Model with no hypothesis regarding the origin of the light neutrino masses and the observed lepton mixing ([MATH]).', '1612.04737-1-14-0': '## Effective Approach: Standard Model + one sterile fermion', '1612.04737-1-15-0': 'In essence, since no seesaw hypothesis is made, the physical parameters correspond to the [MATH] mostly active neutrino masses, the mass of the mostly sterile neutrino, and finally the mixing angles and the CP-violating phases encoded in the mixing matrix which relates the physical neutrino to the weak interaction basis.', '1612.04737-1-15-1': 'Due to the modification of the charged current in [REF] the lepton mixing matrix is defined as [EQUATION] where, [MATH] and [MATH] are the unitary transformations that relate the physical charged and neutral lepton states [MATH] and [MATH] to the gauge eigenstates [MATH] and [MATH] as [EQUATION]', '1612.04737-1-15-2': 'In the [MATH] model, the mixing matrix [MATH] includes [MATH] rotation angles, [MATH] Dirac CP-violating phases, in addition to the [MATH] Majorana phases.', '1612.04737-1-15-3': 'It can thus be parametrized as follows: [EQUATION] where [MATH] is the rotation matrix between [MATH] and [MATH], which includes the mixing angle [MATH] and the Dirac CP-violating phase [MATH].', '1612.04737-1-15-4': 'The Majorana CP-violating phases are factorized in the last term of Eq. ([REF]), where [MATH].', '1612.04737-1-15-5': '[MATH] is the [MATH] matrix which encodes the mixing among the active leptons as [EQUATION]', '1612.04737-1-15-6': 'The upper [MATH] submatrix of [MATH] is non-unitary due to the presence of a sterile neutrino and includes the usual Dirac CP phase actively searched for in neutrino oscillation facilities.', '1612.04737-1-15-7': 'In the case where the sterile neutrino decouples, this submatrix would correspond to the usual unitary PMNS lepton mixing matrix, [MATH].', '1612.04737-1-15-8': 'The active-sterile mixing is described by the rotation matrices [MATH] which are defined as: [EQUATION]', '1612.04737-1-16-0': '# Scan of the parameter space in the [MATH] effective approach', '1612.04737-1-17-0': 'In this section we list the quantities that are used in order to constrain the parameter space of the scenario with three active and one (effective) sterile neutrino.', '1612.04737-1-17-1': 'In addition to the current limits on the neutrino data [CITATION], the presence of an extra sterile neutrino requires the introduction of new parameters: its mass, three new (active-sterile) mixing angles and two extra CP-violating phases.', '1612.04737-1-17-2': 'Furthermore, since we assume that neutrinos are Majorana fermions, there are also three Majorana phases which, however, do not play a significant role in the setup discussed in this paper.', '1612.04737-1-18-0': 'Before we list the observables used to constrain the parameter space, we need to emphasize that a price to pay for adding massive sterile neutrinos is that the Fermi constant, extracted from the muon decay, should be redefined according to [MATH].', '1612.04737-1-18-1': 'We checked, however, that for the model used in this paper [MATH] remains an excellent approximation and thus it will be used in the following.', '1612.04737-1-19-0': '[MATH]: An important constraint comes from the combination of the recently established experimental bound [MATH] [CITATION].', '1612.04737-1-19-1': 'In our setup, the branching fraction of this decay is given by the following expression [CITATION]: [EQUATION] where [MATH] and [MATH].', '1612.04737-1-19-2': 'We also use the above expression, mutatis mutandis, to derive additional constraints stemming from the experimental limits, [MATH] and [MATH] [CITATION].', '1612.04737-1-20-0': '[MATH]: Combining the measured [MATH] and [MATH], with the expression [EQUATION] yields useful constraints in the parameter space.', '1612.04737-1-20-1': 'In the above formula [MATH].', '1612.04737-1-20-2': 'Since we do not include the electroweak radiative corrections to this formula we will use in our scan the experimental results with [MATH] uncertainties.', '1612.04737-1-20-3': 'Notice also that unlike [MATH] and [MATH], which have also been recently measured at the LHC [CITATION], the LEP result for [MATH] has not been measured at the LHC.', '1612.04737-1-20-4': 'For that reason, and despite the fact that the LEP result for [MATH] differs from the Standard Model value at the [MATH] level, we prefer not to include [MATH] in our scan.', '1612.04737-1-21-0': '[MATH]: The ratio [MATH] provides an efficient constraint, as recently argued in Ref. [CITATION].', '1612.04737-1-21-1': 'To that end one combines the Standard Model expression for the decay rate with the experimental values to obtain [MATH], the result which is then compared with the formula relevant to the scenario discussed in this paper (cf. next section), namely, [EQUATION]', '1612.04737-1-21-2': 'In this way one gains another interesting constraint to the parameter space.', '1612.04737-1-22-0': '[MATH]: In addition to the active neutrinos, the sterile ones can be used to saturate the experimental [MATH] invisible decay width, [MATH] GeV [CITATION].', '1612.04737-1-22-1': 'The corresponding expression, which we compute by using the Feynman rules derived in Appendix [REF], reads [EQUATION] where [EQUATION]', '1612.04737-1-23-0': '[MATH]: To implement this constraint we compare the experimental limit, [MATH] [CITATION], to the theoretical prediction derived in Ref. [CITATION]: [EQUATION] where the explicit forms of the loop functions [MATH] can be found in Ref. [CITATION].', '1612.04737-1-23-1': 'Similarly, we implement in our scan the bounds arising from the experimental limits [MATH], and [MATH] [CITATION].', '1612.04737-1-24-0': '[MATH] The leptonic decays [MATH]) represent very useful constraints as well.', '1612.04737-1-24-1': 'We derived the relevant expression for this process and found, [EQUATION]', '1612.04737-1-24-2': 'The above formula is then combined with the average of experimental results, summarized in Ref. [CITATION], namely [MATH], and [MATH].', '1612.04737-1-25-0': 'In performing our scan we impose [MATH] eV ([MATH]) and vary the sterile-active mixing angles, as well as all CP-violating phases, within [MATH].', '1612.04737-1-25-1': 'We then impose all of the above constraints, in addition to those arising from the direct searches [CITATION], and require the perturbative unitarity condition [CITATION] which can be written as [EQUATION]', '1612.04737-1-25-2': 'That last condition is important when the sterile neutrino is very heavy as it leads to its decoupling, which can be seen in Fig. [REF].', '1612.04737-1-26-0': 'For the purpose of this paper, in which we study the effects of an additional sterile neutrino on the kaon physics observables, the most interesting region is the one corresponding to [MATH], which we show in Fig. [REF].', '1612.04737-1-26-1': 'As expected, the limits coming from the [MATH] leptonic decays are the most constraining in the plane [MATH].', '1612.04737-1-26-2': 'Notice also that the sharp exclusion of parameters around [MATH] comes from the direct searches discussed in Ref. [CITATION].', '1612.04737-1-27-0': 'In summary, we selected the points in the parameter space which are compatible with a number of constraints discussed in the body of this section.', '1612.04737-1-27-1': 'We will use the results of the above scan to test the sensitivity of the kaon physics observables on the presence of an effective massive sterile neutrino with a mass [MATH] GeV.', '1612.04737-1-28-0': '# Kaon Physics Phenomenology', '1612.04737-1-29-0': 'Before discussing the results stemming from our scan, we will introduce the kaon decays at the heart of our study and present their analytical expressions in our effective model.', '1612.04737-1-30-0': '## Leptonic decays, [MATH]', '1612.04737-1-31-0': 'The effective Hamiltonian we will be working with reads [EQUATION] where [MATH] and [MATH] are the generic couplings to physics beyond the Standard Model, which in our case are the couplings to the massive sterile neutrino.', '1612.04737-1-31-1': 'The relevant hadronic matrix element for this decay is parametrized in terms of the decay constant [MATH] via [EQUATION] so that the decay amplitude becomes [EQUATION] where [MATH] and [MATH] are the momenta of the lepton and neutrino, respectively, and [MATH].', '1612.04737-1-31-2': 'Multiplying this amplitude by its conjugate and after summing over the spins we then get, [EQUATION] so that the final expression for the decay rate reads, [EQUATION]', '1612.04737-1-31-3': 'More explicitly, and after adapting the above formula to the scenario with an extra sterile neutrino, we have [EQUATION]', '1612.04737-1-31-4': 'Similarly, for the process [MATH] we get [EQUATION]', '1612.04737-1-31-5': 'The above expressions can be trivially extended to the case of the pion leptonic decay by simply replacing [MATH], and [MATH].', '1612.04737-1-31-6': 'Modern day lattice QCD computations of the decay constants [MATH], [MATH], and especially of [MATH], have already reached a sub-percent accuracy [CITATION] so that comparing the theoretical expressions (in which the effects of New Physics are included) with the experimental measurements can result in stringent constraints on the New Physics couplings.', '1612.04737-1-32-0': '## Semileptonic decays, [MATH]', '1612.04737-1-33-0': 'To discuss the semileptonic decays [MATH], we again rely on the effective Hamiltonian [REF] and keep the neutrinos massive.', '1612.04737-1-33-1': 'Due to parity, only the vector current contributes on the hadronic side and the relevant hadronic matrix matrix element is parametrized as [EQUATION] where the form factors [MATH] are functions of [MATH] which can take the values [MATH].', '1612.04737-1-33-2': 'Notice that the hadronic matrix element of the decay to a neutral pion is related to the above one by the Clebsch-Gordan coefficient, i.e. [MATH].', '1612.04737-1-33-3': 'This decay is suitably described by its helicity amplitudes.', '1612.04737-1-33-4': 'To that end one first defines the polarization vectors of the virtual vector boson ([MATH]) as [EQUATION] so that the only nonzero helicity amplitudes will be [MATH], or explicitly [EQUATION]', '1612.04737-1-33-5': 'In terms of these functions the decay amplitude reads, [EQUATION]', '1612.04737-1-33-6': 'In the rest frame of the lepton pair the components of the vectors [MATH] and [MATH] of the final leptons are [EQUATION] where [EQUATION] and [MATH] is the angle between [MATH] and the flight direction of the leptonic pair in the lepton-pair rest frame.', '1612.04737-1-33-7': 'The decay rate can then be written as [EQUATION] where the [MATH]-dependent functions are given by, [EQUATION]', '1612.04737-1-33-8': 'After integrating over [MATH] we obtain the usual expression for the differential branching fraction, which is shortly written as, [EQUATION]', '1612.04737-1-33-9': 'Finally, after integrating in [MATH] and splitting up the pieces with contributions of massless and massive neutrino in the final state, we have [EQUATION]', '1612.04737-1-33-10': 'Another observable relevant to [MATH] decays can be easily obtained after subtracting the number of events in the backward from the forward hemispheres.', '1612.04737-1-33-11': 'The resulting forward-backward asymmetry is given by [EQUATION]', '1612.04737-1-33-12': 'Since there are three independent functions in the angular decay distribution [REF] we can define one more linearly independent observable, in addition to [MATH] and [MATH].', '1612.04737-1-33-13': 'We choose the third observable to be the lepton polarization asymmetry.', '1612.04737-1-33-14': 'For that purpose we define the projectors [MATH] where the projection is made along the lepton polarization vector, [EQUATION]', '1612.04737-1-33-15': 'The differential branching fraction can be separated into the positive lepton helicity and the negative one, i.e. [EQUATION] or, for short, [MATH].', '1612.04737-1-33-16': 'The lepton polarization asymmetry is then defined as [EQUATION] or, in terms of form factors and by explicitly displaying the sum over the neutrino species, we write [EQUATION]', '1612.04737-1-33-17': 'Measuring [MATH] and [MATH] is hardly possible, but measuring the integrated characteristics might be feasible.', '1612.04737-1-33-18': 'This is why in the phenomenological application we will be using [MATH] and [MATH], which are obtained by separately integrating the numerator and the denominator in both Eq. [REF] and Eq. [REF].', '1612.04737-1-34-0': '## Loop induced weak decay [MATH]', '1612.04737-1-35-0': 'Details of the derivation of the expressions for this decay rate can be found in the Appendix [REF] of the present paper.', '1612.04737-1-35-1': 'Here we only quote the corresponding effective Hamiltonian that we use, namely, [EQUATION] where [EQUATION] with [MATH], [MATH].', '1612.04737-1-35-2': 'The loop contribution arising from the top quark amounts to [MATH] [CITATION], while the box diagram with the propagating charm depends on the lepton also in the loop, and yields [MATH], [MATH] [CITATION].', '1612.04737-1-35-3': 'Notice also that the sum in the Wilson coefficient [MATH] runs over the charged lepton species and the one in Eq. [REF] over the neutrino mass eigenstates.', '1612.04737-1-35-4': 'Using the same decomposition of the matrix element in terms of the hadronic form factors, already defined in Eq. [REF], and assuming all neutrinos to be of Majorana nature, we have [EQUATION] where [EQUATION]', '1612.04737-1-35-5': 'One should be particularly careful when using the above formula because the leptonic mixing matrix elements are in general complex, and while the functions [MATH] and [MATH] are real, the CKM couplings have both real and imaginary parts.', '1612.04737-1-35-6': 'More specifically, and by using the CKMfitter results [CITATION], we obtain [EQUATION]', '1612.04737-1-35-7': 'The above formula reduces to the Standard Model one after setting [MATH], and by using the unitarity of the [MATH] matrix.', '1612.04737-1-36-0': 'If this decay occurs between the neutral mesons, the situation is slightly more delicate.', '1612.04737-1-36-1': 'When considering [MATH], one should first keep in mind that [MATH], which then means that the effective Hamiltonian [REF] between the initial and the final hadrons will result in two hadronic matrix elements which are related to each other by CP symmetry, namely, [EQUATION]', '1612.04737-1-36-2': 'Furthermore, after invoking the isospin symmetry, we have [EQUATION] where the last matrix element (to a charged pion) is the one defined in Eq. [REF].', '1612.04737-1-36-3': 'With this, we can compute the decay rate and we obtain [EQUATION] where [EQUATION]', '1612.04737-1-36-4': 'Like before, if we set [MATH] and use the lepton mixing matrix unitarity, the above formula will lead to the familiar Standard Model expression (see eg. [CITATION]).', '1612.04737-1-37-0': '## "Invisible decay" [MATH]', '1612.04737-1-38-0': 'One might also look for an "invisible decay", such as the decay of a kaon to neutrinos only.', '1612.04737-1-38-1': 'We use the effective Hamiltonian [REF], and express the hadronic matrix element as, [EQUATION] consistent with Eq. [REF], and derive the expression for the decay rate by keeping in mind that [MATH].', '1612.04737-1-38-2': 'We obtain, [EQUATION]', '1612.04737-1-38-3': 'Since we consider the neutrinos to be Majorana fermions, the processes [MATH] and [MATH] can be viewed as lepton number violating, and as such they can be used to probe the Majorana phases via the last term in Eqs. ([REF],[REF]).', '1612.04737-1-38-4': 'Notice, however, that this term is multiplied by the product of neutrino masses [MATH], and since in our scenario only one neutrino can be massive the other ones are extremely light so that the product of masses will be negligibly small.', '1612.04737-1-38-5': 'The only non-zero possibility is then [MATH], but in this case the Majorana phases cancel out in the product [MATH].', '1612.04737-1-38-6': 'For this reason, the Majorana phases will not be discussed in what follows.', '1612.04737-1-38-7': 'We should also note that Eq. [REF], with the appropriate simplifications, agrees with the one presented in Ref. [CITATION].', '1612.04737-1-39-0': '# Results and discussion', '1612.04737-1-40-0': 'In this section we use the points selected by the constraints discussed in Sec. [REF] and evaluate the sensitivity of the kaon decay observables on the presence of a massive sterile neutrino.', '1612.04737-1-40-1': 'Whenever possible, and to make the situation clearer, for a given observable [MATH] we will consider the ratio [EQUATION] where in the numerator we compute a given observable in the scenario with [MATH] active and [MATH] massive sterile neutrino and divide it by its Standard Model prediction.', '1612.04737-1-40-2': 'Whenever possible, those results will be compared with experimental values, [MATH].', '1612.04737-1-40-3': '[MATH] We first examine the effects of sterile neutrinos on the leptonic decays of a charged kaon.', '1612.04737-1-40-4': 'To that end we define, [EQUATION] and compute its value by employing the expressions derived in the previous section.', '1612.04737-1-40-5': 'To estimate [MATH] we need an estimate of [MATH], which we compute by using [MATH] [CITATION], [MATH] MeV computed in lattice QCD [CITATION], and by adding the electroweak and radiative corrections [CITATION].', '1612.04737-1-40-6': 'We thus end up with, [EQUATION] where, in evaluating the [MATH] ratios, we used the average of the experimental results collected in Ref. [CITATION].', '1612.04737-1-41-0': 'Adding a massive sterile neutrino with parameters selected in a way discussed in Sec. [REF], results in values of the branching fractions which always fall within the experimental bounds except in the case of the mode [MATH], where some points get outside the range allowed by experiment.', '1612.04737-1-41-1': 'This situation is depicted in Fig. [REF] where we see that requiring an agreement with the experimental bound on [MATH] amounts to a new constraint in the region of [MATH].', '1612.04737-1-41-2': 'In Fig. [REF] we also show the impact of [MATH] on the corresponding active-sterile mixing angle, or better [MATH].', '1612.04737-1-42-0': 'This finding is actually equivalent to what has been discussed in Ref. [CITATION] where it has been shown that [MATH] (defined analogously to Eq. [REF] of the present paper) provides a useful constraint when building a viable extension of the Standard Model by including one (or more) sterile neutrino(s).', '1612.04737-1-42-1': 'Knowing that [EQUATION] and since [MATH] is currently constraining while [MATH] is not, it is clear that the two constraints are indeed equivalent.', '1612.04737-1-42-2': '[MATH] As for the semileptonic decays, we focus onto the decays of [MATH] in order to avoid the uncertainties related to the isospin corrections which are present in the decays of charged kaons.', '1612.04737-1-42-3': 'The main remaining worry is to handle the hadronic uncertainties, i.e. those associated with the form factors [MATH].', '1612.04737-1-42-4': 'Those uncertainties are nowadays under control thanks to the recent precision lattice QCD computation with [MATH] dynamical quark flavors, presented in Ref. [CITATION].', '1612.04737-1-42-5': "In that paper the authors computed the form factors at several [MATH]'s which are then fitted to the dispersive parametrization of Ref. [CITATION].", '1612.04737-1-42-6': 'We use those results in our computation and obtain, [EQUATION] thus about [MATH] away from the Standard Model prediction.', '1612.04737-1-42-7': 'Those bounds, however, remain far too above the results we obtain after including an extra sterile neutrino, which is also shown in Fig. [REF].', '1612.04737-1-42-8': 'In other words, the presence of a massive sterile neutrino has a very little impact on the branching fractions of the semileptonic kaon decays [MATH].', '1612.04737-1-42-9': 'Even a significantly increased precision of those measurements is very unlikely to unveil the presence of a sterile neutrino in these decay modes.', '1612.04737-1-43-0': 'As for the other two observables, we first computed them in the Standard Model and obtained [EQUATION]', '1612.04737-1-43-1': 'We then checked their values in our scenario with one massive sterile neutrino and found that they change by a completely insignificant amount (at the one per-mil level).', '1612.04737-1-43-2': 'For example, we get [EQUATION]', '1612.04737-1-43-3': 'To understand why these quantities remain so insensitive to the presence of a heavy sterile neutrino, we checked all the constraints employed in our scan of parameters, and found that the most severe constraints come from the direct searches, i.e. those we took from Ref. [CITATION].', '1612.04737-1-43-4': 'Once taken into account, these constraints prevent the kaon physics observables deviating from their Standard Model values.', '1612.04737-1-43-5': '[MATH] The most interesting decay modes are expected to be the ones with two neutrinos in the final state.', '1612.04737-1-43-6': 'In the Standard Model, we have [CITATION], [EQUATION] where a control over the remaining long distance hadronic contribution to the charged mode can be achieved through numerical simulations of QCD on the lattice for which a strategy has been recently developed in Ref. [CITATION].', '1612.04737-1-43-7': 'These two decay modes are also subjects of an intense experimental research at CERN (NA62) for the charged mode [CITATION], and at J-PARC (KOTO) for the neutral one [CITATION].', '1612.04737-1-43-8': 'We therefore find it important to examine in which way their rates could be affected if the Standard Model is extended by an extra sterile neutrino.', '1612.04737-1-43-9': 'It turns out that experimental constraints limit the deviation from the Standard Model prediction to less than [MATH], which in view of the Standard Model uncertainties [cf. Eq. [REF]] means that the [MATH] decay modes remain blind to the presence of an extra sterile neutrino.', '1612.04737-1-43-10': 'This is illustrated in Fig. [REF].', '1612.04737-1-44-0': 'More specifically, we find [EQUATION]', '1612.04737-1-44-1': 'In other words, measuring [MATH] and [MATH] consistent with the Standard Model predictions would be perfectly consistent with a scenario in which the Standard Model is extended by an extra sterile neutrino.', '1612.04737-1-44-2': 'Notice again that the cut into the parameter space in the region around [MATH] GeV, shown in Fig. [REF], comes from the direct searches [CITATION], implemented in our scan.', '1612.04737-1-44-3': '[MATH] Finally, a similar analysis of the "invisible kaon decay", [MATH], shows that this mode can be largely enhanced if the sterile neutrino is massive.', '1612.04737-1-44-4': 'Due to the available phase space, this decay can be studied for [MATH], and the result is shown in Fig. [REF].', '1612.04737-1-45-0': 'Knowing that in the Standard Model [MATH], the enhancement we observe is indeed substantial and since its decay rate can be comparable to [MATH] its experimental research becomes highly important.', '1612.04737-1-45-1': 'From our analysis we find the upper bound, [EQUATION] which could be within the reach of the KOTO experiment [CITATION] even if the above bound is by an order of magnitude lower.', '1612.04737-1-45-2': 'In any case, an experimental bound on this decay mode would be of great importance for studying the effects of physics beyond the Standard Model in the leptonic sector.', '1612.04737-1-45-3': 'Obviously, a non-zero measurement of [MATH] would be a clean signal of the non-Standard Model physics.', '1612.04737-1-46-0': 'Before closing this section, we should make a brief comment on the lepton flavor violating kaon decays, which in our scenario would be generated by the heavy neutrino running in the loop.', '1612.04737-1-46-1': 'By using the formulas given in Ref. [CITATION] trivially adapted to the kaon decays, and the result of the scan of Sec. [REF], we obtain that these modes are completely negligible, i.e. the branching fractions of all these modes are under [MATH].', '1612.04737-1-47-0': '# Summary', '1612.04737-1-48-0': 'In this paper we presented the results of our study concerning the impact of a massive sterile neutrino on the weak kaon decays such as the leptonic, semileptonic and the decay of a kaon to neutrinos.', '1612.04737-1-48-1': 'In the effective approach, adopted in this work, one sterile neutrino is supposed to mimic the effect of a more realistic model in which the neutrino sector is extended to include one or more sterile neutrinos.', '1612.04737-1-49-0': 'Although the mass of the sterile neutrino, [MATH], can in principle have any value, we focused on the mass range [MATH] GeV, and in particular on [MATH], when the sterile neutrino is kinematically accessible.', '1612.04737-1-49-1': 'In order to constrain six new parameters ([MATH], the three sterile-active neutrino mixing angles and two new phases) we used a number of quantities discussed in the body of the paper, together with the perturbative unitarity requirement, as well as the constraints arising from the direct searches [CITATION].', '1612.04737-1-49-2': 'After combining such selected parameters with the expressions for the leptonic and semileptonic decays we derive here, we find that only [MATH] can significantly deviate from the current experimental value.', '1612.04737-1-49-3': 'That conflict with the data is present in the interval [MATH] GeV.', '1612.04737-1-49-4': 'The other quantities, including the forward backward and the lepton polarization asymmetries, remain unchanged with respect to their Standard Model values with the effect of the massive sterile neutrino remaining at the level of less than [MATH].', '1612.04737-1-50-0': 'We also derived the expressions for the kaon decays to two (Majorana) neutrinos in the final state, namely [MATH] and [MATH].', '1612.04737-1-50-1': 'Our expressions are generic and can be used when studying a New Physics scenario in which heavy neutrinos with no new gauge couplings are involved.', '1612.04737-1-50-2': 'This will be increasingly relevant with the ongoing experimental effort at CERN (NA62) and J-PARC (KOTO) targeting [MATH] and [MATH], respectively.', '1612.04737-1-50-3': 'These two decays, however, appear to be insensitive to the massive sterile neutrino once the experimental and theoretical constraints are taken into account.', '1612.04737-1-50-4': 'In other words, if the experimental results of the weak kaon decays turn out to be consistent with the Standard Model predictions to a [MATH] uncertainty, this would not be in contradiction with the neutrino sector extended by a massive and relatively light sterile neutrino(s).', '1612.04737-1-50-5': 'The only kaon decay mode which appears to be sensitive to the presence of a massive sterile neutrino is [MATH], the branching fraction of which can go up to [MATH], thus within reach of the KOTO experiment.', '1612.04737-1-50-6': 'Knowing that the Standard Model value of this mode is zero, its observation would be a clean signal of New Physics.', '1612.04737-1-51-0': 'Notice also that [MATH] and [MATH] could be used to probe the Majorana phases in the models in which more than one massive neutrino is considered.', '1612.04737-1-51-1': 'In the approach adopted in this paper only one neutrino can be heavy and therefore such a study is prohibited.', '1612.04737-1-51-2': 'If instead one considers a realistic model with more than one heavy neutrino then a study of the Majorana phases becomes possible too [CITATION].', '1612.04737-1-52-0': 'We should mention that one can also consider the situation with a very heavy sterile neutrino [MATH].', '1612.04737-1-52-1': 'In that case the processes discussed in this paper could be modified by the effects of violation of the mixing matrix unitarity.', '1612.04737-1-52-2': 'We checked that possibility in the explicit computation and found that such effects are indeed tiny.', '1612.04737-1-52-3': 'Importantly, however, a heavy sterile neutrino can propagate in the loop induced processes and shift the values of [MATH] and [MATH].', '1612.04737-1-52-4': 'We checked that the corresponding effect remains small and completely drowned in the large (long distance QCD) uncertainties already present in the Standard Model estimates of [MATH] and [MATH] [CITATION].', '1612.04737-1-53-0': 'Finally, the expressions presented in this paper can be easily extended to other similar decays, such as [MATH]-, [MATH]-, [MATH]- and [MATH]-meson decays.', '1612.04737-1-53-1': 'We decided to focus onto the kaon decays because of the recent theoretical developments in taming the hadronic uncertainties and because of the better experimental precision.'}

{'1612.04737-2-0-0': 'We discuss weak kaon decays in a scenario in which the Standard Model is extended by massive sterile fermions.', '1612.04737-2-0-1': 'After revisiting the analytical expressions for leptonic and semileptonic decays we derive the expressions for decay rates with two neutrinos in the final state.', '1612.04737-2-0-2': 'By using a simple effective model with only one sterile neutrino, compatible with all current experimental bounds and general theoretical constraints, we conduct a thorough numerical analysis which reveals that the impact of the presence of massive sterile neutrinos on kaon weak decays is very small, less than [MATH] on decay rates.', '1612.04737-2-0-3': 'The only exception is [MATH], which can go up to [MATH], thus possibly within the reach of the KOTO, NA62 and SHIP experiments.', '1612.04737-2-0-4': 'Plans have also been proposed to search for this decay at the NA64 experiment.', '1612.04737-2-0-5': 'In other words, if all the future measurements of weak kaon decays turn out to be compatible with the Standard Model predictions, this will not rule out the existence of massive light sterile neutrinos with non-negligible active-sterile mixing.', '1612.04737-2-0-6': 'Instead, for a sterile neutrino of mass below [MATH], one might obtain a huge enhancement of [MATH], otherwise negligibly small in the Standard Model.', '1612.04737-2-1-0': '# Introduction', '1612.04737-2-2-0': 'The Standard Model (SM) predicts the strict conservation of lepton flavor to all orders.', '1612.04737-2-2-1': 'The fact that neutrinos oscillate provides clear evidence of the existence of physics beyond the Standard Model.', '1612.04737-2-2-2': 'New physics models accommodating massive neutrinos and their mixing open the door to many phenomena which basically have no Standard Model background, such as lepton number violation, violation of lepton flavor universality, or lepton flavor violation.', '1612.04737-2-3-0': 'The experimental effort associated with the search of new physics using observables involving leptons is impressive, and is currently being pursued on all experimental fronts: (i) neutrino dedicated experiments which aim to determine neutrino properties, such as the Majorana/Dirac nature, the absolute neutrino masses, the hierarchy of their mass spectrum, leptonic mixing and the CP-violating phases; (ii) high-intensity facilities that are studying several low-energy processes such as [MATH], [MATH], [MATH] conversion in atoms, [MATH] and meson [MATH] decays ([MATH], [MATH], [MATH], [MATH], ); (iii) the Large Hadron Collider (LHC), which is the privileged discovery ground of new particles, may also allow one to probe leptonic mixing in the production and/or decay of the new states.', '1612.04737-2-3-1': 'On the other hand, the recent cosmological data have put constraints on the sum of the light neutrino masses, which is especially restrictive when considering new light neutral states.', '1612.04737-2-4-0': 'Among the several minimal possible scenarios, extending the SM with sterile fermions - which are singlets under the Standard Model gauge group - is a very appealing hypothesis, as their unique (indirect) interaction with the Standard Model fields occurs through their mixing with the active neutrinos [MATH] (via their Yukawa couplings).', '1612.04737-2-4-1': 'Due to their very unique nature, there is no bound on their number, and a priori no limits regarding their mass regimes.', '1612.04737-2-4-2': 'Interestingly, sterile fermions (like right-handed neutrinos) are present in many frameworks accounting for neutrino masses and the observed mixing (as is the case for fermion seesaw mechanisms).', '1612.04737-2-4-3': 'The interest in sterile neutrinos and their impact on observables strongly depend on their masses.', '1612.04737-2-4-4': 'Sterile neutrinos at the eV scales were proposed to solve neutrino oscillation anomalies in reactors [CITATION], accelerators [CITATION], as well as in the calibration of gallium-target solar neutrino experiments [CITATION], all suggesting beyond the three-neutrino paradigm.', '1612.04737-2-4-5': 'The keV scale for sterile neutrinos offers warm dark matter candidates [CITATION], and explanations of some astrophysical issues such as kicks in pulsar velocities [CITATION].', '1612.04737-2-4-6': 'Sterile fermion states with masses above [MATH] GeV have moderate motivation other than their theoretical appeal (grand unified theories) and the possibility to have a scenario for baryogenesis via the high-scale leptogenesis [CITATION].', '1612.04737-2-4-7': 'Finally, the appeal for sterile neutrinos in the range [MATH] (and even TeV) strongly resides in their experimental testability due to the many direct and indirect effects in both high-energy (e.g. LHC) [CITATION] and high-intensity (e.g., NA62, NA64) experiments.', '1612.04737-2-5-0': 'In order to illustrate the phenomenological effect of a scenario which involves sterile fermions, we focus on a minimal model which extends the Standard Model with an arbitrary number of sterile states with masses in the above-mentioned ranges, known as [MATH] models, without making assumptions concerning the neutrino mass (and/or the lepton mixing) generation mechanism, but allowing one to access the degrees of freedom of the sterile neutrinos (their masses, their mixing with the active neutrinos and the new CP-violating phases).', '1612.04737-2-5-1': 'In this work, we address the phenomenological imprints of such sterile fermions on observables involving kaon meson rare decays into lepton final states including neutrinos, which can either be used to set (or update) constraints to model building, or provide interesting observables that could be used as tests of various scenarios of new physics.', '1612.04737-2-5-2': 'Notice that the first study devoted to probe massive neutrinos and lepton mixing using leptonic pseudoscalar light meson decays was done in Refs. [CITATION].', '1612.04737-2-5-3': 'These rare and forbidden decays are being searched for at CERN (NA62) in the charged decay modes [CITATION] and at the J-PARC facility (KOTO) [CITATION] and at CERN (NA64) [CITATION] in the neutral ones.', '1612.04737-2-5-4': 'This study might also be useful for the TREK/E36 experiment at J-PARC, where the data analysis is currently under way [CITATION].', '1612.04737-2-5-5': 'It will further test the lepton universality in kaon two-body decays ([MATH]) and search for a heavy neutrino [CITATION].', '1612.04737-2-5-6': 'Finally, this study could also be of use to the proposed SHIP experiment [CITATION], where we propose to use the large number of kaons produced in [MATH]-meson decays to search for forbidden decays.', '1612.04737-2-6-0': 'Having sterile neutrinos that are sufficiently light to be produced with non-negligible active-sterile mixing angles may induce important impact on electroweak precision and many other observables.', '1612.04737-2-6-1': 'Our analysis must therefore comply with abundant direct and indirect searches that have already allowed to put constraints or bounds on the sterile neutrino masses and their mixing with the active neutrinos.', '1612.04737-2-7-0': 'In this work, we assume the effective case in which the Standard Model is extended by one sterile fermion (3 +1 case) and revisit weak kaon decays such as the leptonic [MATH]), as well as the semileptonic ones [MATH]).', '1612.04737-2-7-1': 'Furthermore we consider the loop-induced weak decays [MATH] and [MATH] and derive analytical expressions for their decay rates, which are new results.', '1612.04737-2-7-2': 'In doing the numerical analysis, we (re)derive the expressions for various processes which are used as constraints ([MATH], [MATH], [MATH], ).', '1612.04737-2-7-3': 'We have chosen to present the analytical formulas and numerical results assuming here neutrinos to be Majorana fermions.', '1612.04737-2-7-4': 'A detailed discussion and comparison between the Dirac and Majorana cases is also displayed (in Appendix [REF]).', '1612.04737-2-8-0': 'Our study reveals that the influence of the presence of massive sterile neutrinos on the [MATH], [MATH] and [MATH] decay rates is less than [MATH], thus fully compatible with the Standard Model predictions.', '1612.04737-2-8-1': 'Interestingly, however, we find that [MATH], which is zero in the Standard Model, can be as high as [MATH] and thus possibly within the reach of the NA62(-KLEVER), SHIP as well as the KOTO experiments.', '1612.04737-2-8-2': 'In other words, if all the future measurements of weak kaon decays turn out to be compatible with the Standard Model, the paradigm of the existence of sterile neutrinos would still remain valid.', '1612.04737-2-8-3': 'Instead, we show that sterile neutrinos with mass below [MATH] could generate a huge enhancement of [MATH].', '1612.04737-2-9-0': 'Our work is organized as follows: in Sec. [REF] we present a generic model with one sterile neutrino added to the Standard Model and present details concerning the parametrization used in the ensuing analysis.', '1612.04737-2-9-1': 'Sec. [REF] is devoted to a discussion of quantities which are used to constrain the parameter space followed by the actual scan.', '1612.04737-2-9-2': 'The expressions for various weak decay processes of kaons are scrutinized in Sec. [REF], and the sensitivity of these processes on the presence of massive sterile neutrinos is examined and discussed in Sec. [REF].', '1612.04737-2-9-3': 'Our concluding remarks are given in Sec. [REF], while the Feynman rules for the case of Majorana neutrino have been relegated to Appendix [REF].', '1612.04737-2-9-4': 'Appendix [REF] contains a comparison of the analytical expressions for [MATH] when neutrinos are Majorana or Dirac fermions.', '1612.04737-2-10-0': '# Extending the Standard Model with Sterile Fermions', '1612.04737-2-11-0': '## Models with sterile fermions', '1612.04737-2-12-0': 'To discuss the phenomenological consequences of sterile fermion states on low-energy physics observables, it is important to have an idea of the underlying framework which involves sterile neutrinos, and among those the testable ones in particular.', '1612.04737-2-12-1': 'Such models are for instance based on low-scale seesaw mechanism, e.g. extension of the Standard Model by exclusively right-handed (RH) neutrinos, like in the usual type I seesaw, which is realized at the TeV scale [CITATION], and for which the Yukawa couplings are small ([MATH]).', '1612.04737-2-12-2': 'They can nevertheless be made higher if one assumes some extra input like the minimal flavor violation [CITATION]), or tiny as it is the case in [MATH]MSM [CITATION].', '1612.04737-2-12-3': 'Other than RH neutrinos, [MATH], one can also consider additional sterile fermions with the opposite lepton numbers of the RH ones, like it is done in the case of the linear [CITATION] or the inverse [CITATION] seesaw mechanisms.', '1612.04737-2-12-4': 'The two latter scenarios are (theoretically and phenomenologically) very appealing as they provide an extra suppression factor, which is linked to a small violation of the total lepton number, allowing one to explain the tininess of neutrino masses while having large Yukawa couplings and a comparatively low seesaw scale.', '1612.04737-2-12-5': 'Having relatively light sterile fermions which do not decouple, since they can have non-negligible active-sterile mixing, certainly leads to important consequences and as a result to numerous constraints.', '1612.04737-2-12-6': 'The most important and direct consequence is the modification of the charged and neutral currents as [EQUATION] where [MATH] is the weak coupling constant, and [MATH] is the number of sterile fermions.', '1612.04737-2-12-7': 'The modified lepton mixing matrix, obviously nonunitary, also encodes the active-sterile mixing.', '1612.04737-2-12-8': 'In the limit in which the sterile fermions decouple the matrix [MATH] corresponds to the usual Pontecorvo-Maki-Nakagawa-Sakata [MATH] unitary matrix, i.e., [MATH].', '1612.04737-2-12-9': 'Moreover, if sufficiently light, the sterile neutrinos can be produced as decay products.', '1612.04737-2-12-10': 'Both these points might induce a huge impact on numerous observables, which in turn can provide abundant constraints on the sterile fermions (masses and the active-sterile mixing angles including the new CP-violating phases).', '1612.04737-2-13-0': 'A useful first approach to study the impact of sterile neutrinos on the low-energy processes relies on addition of only one sterile fermion to the Standard Model with no hypothesis regarding the origin of the light neutrino masses and the observed lepton mixing ([MATH]).', '1612.04737-2-14-0': '## Effective Approach: Standard Model + one sterile fermion', '1612.04737-2-15-0': 'In essence, since no seesaw hypothesis is made, the physical parameters correspond to the three mostly active neutrino masses, the mass of the mostly sterile neutrino, and finally the mixing angles and the CP-violating phases encoded in the mixing matrix which relates the physical neutrino to the weak interaction basis.', '1612.04737-2-15-1': 'Due to the modification of the charged current in [REF] the lepton mixing matrix is defined as [EQUATION] where, [MATH] and [MATH] are the unitary transformations that relate the physical charged and neutral lepton states [MATH] and [MATH] to the gauge eigenstates [MATH] and [MATH] as [EQUATION]', '1612.04737-2-15-2': 'In the [MATH] model, the mixing matrix [MATH] includes six rotation angles, three Dirac CP-violating phases, in addition to the three Majorana phases.', '1612.04737-2-15-3': 'It can thus be parametrized as follows: [EQUATION] where [MATH] is the rotation matrix between [MATH] and [MATH], which includes the mixing angle [MATH] and the Dirac CP-violating phase [MATH].', '1612.04737-2-15-4': 'The Majorana CP-violating phases are factorized in the last term of Eq. ([REF]), where [MATH].', '1612.04737-2-15-5': '[MATH] is the [MATH] matrix which encodes the mixing among the active leptons as [EQUATION]', '1612.04737-2-15-6': 'The upper [MATH] submatrix of [MATH] is nonunitary due to the presence of a sterile neutrino and includes the usual Dirac CP phase actively searched for in neutrino oscillation facilities.', '1612.04737-2-15-7': 'In the case where the sterile neutrino decouples, this submatrix would correspond to the usual unitary PMNS lepton mixing matrix, [MATH].', '1612.04737-2-15-8': 'The active-sterile mixing is described by the rotation matrices [MATH] which are defined as [EQUATION]', '1612.04737-2-16-0': '# Scan of the parameter space in the [MATH] effective approach', '1612.04737-2-17-0': 'In this section we list the quantities that are used in order to constrain the parameter space of the scenario with three active and one (effective) sterile neutrino.', '1612.04737-2-17-1': 'In addition to the current limits on the neutrino data [CITATION], the presence of an extra sterile neutrino requires the introduction of new parameters: its mass, three new (active-sterile) mixing angles and two extra CP-violating phases.', '1612.04737-2-17-2': 'Furthermore, since we assume that neutrinos are Majorana fermions, there are also three Majorana phases which, however, do not play a significant role in the setup discussed in this paper.', '1612.04737-2-18-0': 'Before we list the observables used to constrain the parameter space, we need to emphasize that a price to pay for adding massive sterile neutrinos is that the Fermi constant - extracted from the muon decay - should be redefined according to [MATH], where the sum runs over kinematically accessible neutrinos.', '1612.04737-2-18-1': 'We checked, however, that for the model used in this paper [MATH] remains an excellent approximation and thus it will be used in the following.', '1612.04737-2-19-0': '[MATH]: An important constraint comes from the combination of the recently established experimental bound [MATH] [CITATION].', '1612.04737-2-19-1': 'In our setup, the branching fraction of this decay is given by the following expression [CITATION]: [EQUATION] where [MATH] and [MATH].', '1612.04737-2-19-2': 'We also use the above expression, mutatis mutandis, to derive additional constraints stemming from the experimental limits, [MATH] and [MATH] [CITATION].', '1612.04737-2-20-0': '[MATH]: Combining the measured [MATH] and [MATH], with the expression [EQUATION] yields useful constraints in the parameter space.', '1612.04737-2-20-1': 'In the above formula [MATH].', '1612.04737-2-20-2': 'Since we do not include the electroweak radiative corrections to this formula we will use in our scan the experimental results with [MATH] uncertainties.', '1612.04737-2-20-3': 'Notice also that unlike [MATH] and [MATH], which have also been recently measured at the LHC [CITATION], the LEP result for [MATH] has not been measured at the LHC.', '1612.04737-2-20-4': 'For that reason, and despite the fact that the LEP result for [MATH] differs from the Standard Model value at the [MATH] level, we prefer not to include [MATH] in our scan.', '1612.04737-2-21-0': '[MATH]: The ratio [MATH] provides an efficient constraint, as recently argued in Ref. [CITATION].', '1612.04737-2-21-1': 'To that end one combines the Standard Model expression for the decay rate with the experimental values to obtain [MATH], the result which is then compared with the formula relevant to the scenario discussed in this paper (cf. next section), namely, [EQUATION]', '1612.04737-2-21-2': 'In this way one gains another interesting constraint to the parameter space.', '1612.04737-2-22-0': '[MATH]: In addition to the active neutrinos, the sterile ones can be used to saturate the experimental [MATH] invisible decay width, [MATH] GeV [CITATION].', '1612.04737-2-22-1': 'The corresponding expression, which we compute by using the Feynman rules derived in Appendix [REF], reads [EQUATION] where [EQUATION]', '1612.04737-2-23-0': '[MATH]: To implement this constraint we compare the experimental limit, [MATH] [CITATION], to the theoretical prediction derived in Ref. [CITATION]: [EQUATION] where the explicit forms of the loop functions [MATH] can be found in Refs. [CITATION].', '1612.04737-2-23-1': 'Similarly, we implement in our scan the bounds arising from the experimental limits [MATH], and [MATH] [CITATION].', '1612.04737-2-24-0': '[MATH] The leptonic decays [MATH]) represent very useful constraints as well.', '1612.04737-2-24-1': 'We derived the relevant expression for this process and found [EQUATION]', '1612.04737-2-24-2': 'The above formula is then combined with the average of experimental results summarized in Ref. [CITATION], namely [MATH], and [MATH].', '1612.04737-2-25-0': 'We perform a first random scan of [MATH] points using flat priors on the Dirac CP phases and logarithmic priors on all other scan parameters, which are chosen in the following ranges [EQUATION]', '1612.04737-2-25-1': 'We then perform a second random focusing on the window where the heavy neutrino mass is comparable to the kaon mass, using flat priors on the Dirac CP phases and logarithmic priors on all other scan parameters.', '1612.04737-2-25-2': 'We first generate a sample of [MATH] points with parameters chosen as [EQUATION] to which we add [MATH] points with paramaters chosen in the ranges [EQUATION]', '1612.04737-2-25-3': 'The other parameters are fixed from the best fit point in [CITATION],i.e. [EQUATION]', '1612.04737-2-25-4': 'We then impose all of the above constraints, in addition to those arising from the direct searches [CITATION], and require the perturbative unitarity condition [CITATION] which can be written as [EQUATION]', '1612.04737-2-25-5': 'That last condition is important when the sterile neutrino is very heavy as it leads to its decoupling, which can be seen in Fig. [REF].', '1612.04737-2-26-0': 'For the purpose of this paper, in which we study the effects of an additional sterile neutrino on the kaon physics observables, the most interesting region is the one corresponding to [MATH], which we show in Fig. [REF].', '1612.04737-2-26-1': 'As expected, the limits coming from the [MATH] leptonic decays are the most constraining in the plane [MATH].', '1612.04737-2-26-2': 'Notice also that the sharp exclusion of parameters around [MATH] comes from the direct searches discussed in Ref. [CITATION].', '1612.04737-2-26-3': 'It is worth noticing that the bounds shown in Fig. [REF] are in agreement with those provided in Ref. [CITATION] in the considered mass regime, although slightly improved as most of the constraints discussed above have been updated.', '1612.04737-2-27-0': 'In summary, we selected the points in the parameter space which are compatible with a number of constraints discussed in the body of this section.', '1612.04737-2-27-1': 'We will use the results of the above scan to test the sensitivity of the kaon physics observables on the presence of an effective massive sterile neutrino with a mass [MATH] GeV.', '1612.04737-2-28-0': '# Kaon Physics Phenomenology', '1612.04737-2-29-0': 'Before discussing the results stemming from our scan, we will introduce the kaon decays at the heart of our study and present their analytical expressions in our effective model.', '1612.04737-2-29-1': 'In our phenomenological discussion, we will consider the processes for which hadronic uncertainties are under full theoretical control by means of numerical simulations of QCD on the lattice.', '1612.04737-2-29-2': 'Processes such as [MATH] and [MATH] will not be considered, since the corresponding Standard Model predictions depend on large (long-distance QCD) uncertainties.', '1612.04737-2-30-0': '## Leptonic decays, [MATH]', '1612.04737-2-31-0': 'Sterile neutrinos in kaon and pion leptonic decays were first studied and analyzed in [CITATION] with the aim to probe massive neutrinos via lepton mixing; correspondingly, associated tests allowed one to set bounds on neutrino masses and lepton mixing matrix elements [CITATION].', '1612.04737-2-31-1': 'Here we revisit the [MATH] decays in light of the existing data on neutrinos and in the framework of the simple extension of the Standard Model by one sterile fermion with the aim to update the latter obtained results.', '1612.04737-2-32-0': 'The effective Hamiltonian we will be working with reads [EQUATION] where [MATH] and [MATH] are the generic couplings to physics beyond the Standard Model, which in our case are the couplings to the massive sterile neutrino.', '1612.04737-2-32-1': 'The leptonic bilinear [MATH] in Eq. ([REF]) should be understood as [MATH], as in Eq. ([REF]).', '1612.04737-2-32-2': 'In the effective approach, the effect of the active-sterile mixing is encoded in the effective couplings [MATH] and [MATH].', '1612.04737-2-32-3': 'The relevant hadronic matrix element for this decay is parametrized in terms of the decay constant [MATH] via [EQUATION] so that the decay amplitude becomes [EQUATION] where [MATH] and [MATH] are the momenta of the lepton and neutrino, respectively, and [MATH].', '1612.04737-2-32-4': 'Multiplying this amplitude by its conjugate and after summing over the spins we then get [EQUATION] so that the final expression for the decay rate reads [EQUATION]', '1612.04737-2-32-5': 'One can immediately see from the above equation (and the subsequent ones for the observables under this study) that the presence of the sterile state can have two consequences, a phase space effect if its mass is kinematically allowed and a modification of the coupling due to the active-sterile mixing encoded in [MATH].', '1612.04737-2-32-6': 'More explicitly, and after adapting the above formula to the scenario with an extra sterile neutrino, we have [EQUATION]', '1612.04737-2-32-7': 'Similarly, for the process [MATH] we get [EQUATION]', '1612.04737-2-32-8': 'The above expressions can be trivially extended to the case of the pion leptonic decay by simply replacing [MATH], and [MATH].', '1612.04737-2-32-9': 'Modern day lattice QCD computations of the decay constants [MATH], [MATH], and especially of [MATH], have already reached a subpercent accuracy [CITATION] so that comparing the theoretical expressions (in which the effects of new physics are included) with the experimental measurements can result in stringent constraints on the new physics couplings.', '1612.04737-2-33-0': '## Semileptonic decays, [MATH]', '1612.04737-2-34-0': 'To discuss the semileptonic decays [MATH], we again rely on the effective Hamiltonian [REF] and keep the neutrinos massive.', '1612.04737-2-34-1': 'Due to parity, only the vector current contributes on the hadronic side and the relevant hadronic matrix matrix element is parametrized as [EQUATION] where the form factors [MATH] are functions of [MATH] which can take the values [MATH].', '1612.04737-2-34-2': 'Notice that the hadronic matrix element of the decay to a neutral pion is related to the above one by isospin symmetry, i.e. [MATH].', '1612.04737-2-34-3': 'This decay is suitably described by its helicity amplitudes.', '1612.04737-2-34-4': 'To that end one first defines the polarization vectors of the virtual vector boson ([MATH]) as [EQUATION] so that the only nonzero helicity amplitudes will be [MATH], or explicitly [EQUATION]', '1612.04737-2-34-5': 'In terms of these functions the decay amplitude reads [EQUATION]', '1612.04737-2-34-6': 'In the rest frame of the lepton pair the components of the vectors [MATH] and [MATH] of the final leptons are [EQUATION] where [EQUATION] and [MATH] is the angle between [MATH] (in the lepton pair rest frame) and the flight direction of the leptonic pair (opposite to the pion direction) in the kaon rest frame.', '1612.04737-2-34-7': 'The decay rate can then be written as [EQUATION] where the [MATH]-dependent functions are given by [EQUATION]', '1612.04737-2-34-8': 'After integrating over [MATH] we obtain the usual expression for the differential branching fraction, which is shortly written as [EQUATION]', '1612.04737-2-34-9': 'Finally, after integrating in [MATH] and splitting up the pieces with contributions of massless and massive neutrinos in the final state, we have [EQUATION]', '1612.04737-2-34-10': 'Another observable relevant to [MATH] decays can be easily obtained after subtracting the number of events in the backward from the forward hemispheres.', '1612.04737-2-34-11': 'The resulting forward-backward asymmetry is given by [EQUATION]', '1612.04737-2-34-12': 'Since there are three independent functions in the angular decay distribution [REF] we can define one more linearly independent observable, in addition to [MATH] and [MATH].', '1612.04737-2-34-13': 'We choose the third observable to be the charged lepton polarization asymmetry.', '1612.04737-2-34-14': 'For that purpose we define the projectors [MATH] where the projection is made along the lepton polarization vector, [EQUATION]', '1612.04737-2-34-15': 'The differential branching fraction can be separated into the positive lepton helicity and the negative one, i.e. [EQUATION] or, for short, [MATH].', '1612.04737-2-34-16': 'The lepton polarization asymmetry is then defined as [EQUATION] or, in terms of form factors and by explicitly displaying the sum over the neutrino species, we write [EQUATION]', '1612.04737-2-34-17': 'Measuring [MATH] and [MATH] is hardly possible, but measuring the integrated characteristics might be feasible.', '1612.04737-2-34-18': 'This is why in the phenomenological application we will be using [MATH] and [MATH], which are obtained by separately integrating the numerator and the denominator in both Eqs. [REF] and [REF].', '1612.04737-2-35-0': '## Loop-induced weak decay [MATH]', '1612.04737-2-36-0': 'Details of the derivation of the expressions for this decay rate can be found in Appendix [REF] of the present paper.', '1612.04737-2-36-1': 'Here we only quote the corresponding effective Hamiltonian that we use, namely, [EQUATION] where [EQUATION] with [MATH], [MATH].', '1612.04737-2-36-2': 'The loop contribution arising from the top quark amounts to [MATH] [CITATION], while the box diagram with the propagating charm depends on the lepton also in the loop, and yields [MATH], [MATH] [CITATION].', '1612.04737-2-36-3': 'Notice also that the sum in the Wilson coefficient [MATH] runs over the charged lepton species and the one in Eq. [REF] over the neutrino mass eigenstates.', '1612.04737-2-36-4': 'Using the same decomposition of the matrix element in terms of the hadronic form factors, already defined in Eq. [REF], and assuming all neutrinos to be of Majorana nature, we have [EQUATION] where [EQUATION]', '1612.04737-2-36-5': 'One should be particularly careful when using the above formula because the leptonic mixing matrix elements are in general complex, and while the functions [MATH] and [MATH] are real, the Cabibbo-Kobayashi-Maskawa couplings have both real and imaginary parts.', '1612.04737-2-36-6': 'More specifically, and by using the CKMfitter results [CITATION], we obtain [EQUATION]', '1612.04737-2-36-7': 'The above formula reduces to the Standard Model one after setting [MATH], and by using the unitarity of the [MATH] matrix.', '1612.04737-2-37-0': 'If this decay occurs between the neutral mesons, the situation is slightly more delicate.', '1612.04737-2-37-1': 'When considering [MATH], one should first keep in mind that [MATH], which then means that the effective Hamiltonian [REF] between the initial and the final hadrons will result in two hadronic matrix elements which are related to each other by CP symmetry, namely, [EQUATION]', '1612.04737-2-37-2': 'Furthermore, after invoking the isospin symmetry, we have [EQUATION] where the last matrix element (to a charged pion) is the one defined in Eq. [REF].', '1612.04737-2-37-3': 'With this, we can compute the decay rate and we obtain [EQUATION] where [EQUATION]', '1612.04737-2-37-4': 'Like before, if we set [MATH] and use the [MATH] matrix unitarity, the above formula will lead to the familiar Standard Model expression (see e.g. [CITATION]).', '1612.04737-2-38-0': '## "Invisible decay" [MATH]', '1612.04737-2-39-0': 'One might also look for an "invisible decay", such as the decay of a kaon to neutrinos only.', '1612.04737-2-39-1': 'We use the effective Hamiltonian [REF], and express the hadronic matrix element as [EQUATION] consistent with Eq. [REF], and derive the expression for the decay rate by keeping in mind that [MATH].', '1612.04737-2-39-2': 'We obtain [EQUATION]', '1612.04737-2-39-3': 'Since we consider the neutrinos to be Majorana fermions, the processes [MATH] and [MATH] can be viewed as lepton number violating, and as such they can be used to probe the Majorana phases via the last term in Eqs. ([REF]) and ([REF]).', '1612.04737-2-39-4': 'Notice, however, that this term is multiplied by the product of neutrino masses [MATH], and since in our scenario only one neutrino can be massive the other ones are extremely light so that the product of masses will be negligibly small.', '1612.04737-2-39-5': 'The only nonzero possibility is then [MATH], but in this case the Majorana phases cancel out in the product [MATH].', '1612.04737-2-39-6': 'For this reason, the Majorana phases will not be discussed in what follows.', '1612.04737-2-39-7': 'We should also note that Eq. [REF], with the appropriate simplifications, agrees with the one presented in Ref. [CITATION].', '1612.04737-2-40-0': '# Results and discussion', '1612.04737-2-41-0': 'In this section we use the points selected by the constraints discussed in Sec. [REF] and evaluate the sensitivity of the kaon decay observables on the presence of a massive sterile neutrino.', '1612.04737-2-41-1': 'Whenever possible, and to make the situation clearer, for a given observable [MATH] we will consider the ratio [EQUATION] where in the numerator we compute a given observable in the scenario with three active and one massive sterile neutrino and divide it by its Standard Model prediction.', '1612.04737-2-41-2': 'Whenever possible, those results will be compared with experimental values, [MATH].', '1612.04737-2-41-3': '[MATH] We first examine the effects of sterile neutrinos on the leptonic decays of a charged kaon.', '1612.04737-2-41-4': 'To that end we define [EQUATION] and compute its value by employing the expressions derived in the previous section.', '1612.04737-2-41-5': 'To estimate [MATH] we need an estimate of [MATH], which we compute by using [MATH] [CITATION], [MATH] MeV computed in lattice QCD [CITATION], and by adding the electroweak and radiative corrections [CITATION].', '1612.04737-2-41-6': 'We thus end up with [EQUATION] where, in evaluating the [MATH] ratios, we used the average of the experimental results collected in Ref. [CITATION].', '1612.04737-2-42-0': 'Adding a massive sterile neutrino with parameters selected in a way discussed in Sec. [REF], results in values of the branching fractions which always fall within the experimental bounds except in the case of the mode [MATH], where some points get outside the range allowed by experiment.', '1612.04737-2-42-1': 'This situation is depicted in Fig. [REF] where we see that requiring an agreement with the experimental bound on [MATH] amounts to a new constraint in the region of [MATH].', '1612.04737-2-42-2': 'In Fig. [REF] we also show the impact of [MATH] on the corresponding active-sterile mixing angle, or better [MATH].', '1612.04737-2-43-0': 'This finding is actually equivalent to what has been discussed in Ref. [CITATION] where it has been shown that [MATH] (defined analogously to Eq. [REF] of the present paper) provides a useful constraint when building a viable extension of the Standard Model by including one (or more) sterile neutrino(s).', '1612.04737-2-43-1': 'Knowing that [EQUATION] and since [MATH] is currently constraining while [MATH] is not, it is clear that the two constraints are indeed equivalent.', '1612.04737-2-43-2': '[MATH] As for the semileptonic decays, we focus on the decays of [MATH] in order to avoid the uncertainties related to the isospin corrections which are present in the decays of charged kaons.', '1612.04737-2-43-3': 'The main remaining worry is to handle the hadronic uncertainties, i.e. those associated with the form factors [MATH].', '1612.04737-2-43-4': 'Those uncertainties are nowadays under control thanks to the recent precision lattice QCD computation with [MATH] dynamical quark flavors, presented in Ref. [CITATION].', '1612.04737-2-43-5': "In that paper the authors computed the form factors at several [MATH]'s which are then fitted to the dispersive parametrization of Ref. [CITATION].", '1612.04737-2-43-6': 'We use those results in our computation and obtain, [EQUATION] thus about [MATH] away from the Standard Model prediction.', '1612.04737-2-43-7': 'Those bounds, however, remain far too above the results we obtain after including an extra sterile neutrino, which is also shown in Fig. [REF].', '1612.04737-2-43-8': 'In other words, the presence of a massive sterile neutrino has a very little impact on the branching fractions of the semileptonic kaon decays [MATH].', '1612.04737-2-43-9': 'Even a significantly increased precision of those measurements is very unlikely to unveil the presence of a sterile neutrino in these decay modes.', '1612.04737-2-44-0': 'As for the other two observables, we first computed them in the Standard Model and obtained [EQUATION]', '1612.04737-2-44-1': 'We then checked their values in our scenario with one massive sterile neutrino and found that they change by a completely insignificant amount (at the one per-mil level).', '1612.04737-2-44-2': 'For example, we get [EQUATION]', '1612.04737-2-44-3': 'To understand why these quantities remain so insensitive to the presence of a heavy sterile neutrino, we checked all the constraints employed in our scan of parameters, and found that the most severe constraints come from the direct searches, i.e. those we took from Ref. [CITATION].', '1612.04737-2-44-4': 'Once taken into account, these constraints prevent the kaon physics observables from deviating from their Standard Model values.', '1612.04737-2-44-5': '[MATH] The most interesting decay modes are expected to be the ones with two neutrinos in the final state.', '1612.04737-2-44-6': 'In the Standard Model, we have [CITATION] [EQUATION] where a control over the remaining long-distance hadronic contribution to the charged mode can be achieved through numerical simulations of QCD on the lattice for which a strategy has been recently developed in Ref. [CITATION].', '1612.04737-2-44-7': 'These two decay modes are also subjects of an intense experimental research at CERN (NA62) for the charged mode [CITATION], and at J-PARC (KOTO) for the neutral one [CITATION].', '1612.04737-2-44-8': 'We therefore find it important to examine in which way their rates could be affected if the Standard Model is extended by an extra sterile neutrino.', '1612.04737-2-44-9': 'It turns out that experimental constraints limit the deviation from the Standard Model prediction to less than [MATH], which in view of the Standard Model uncertainties [cf. Eq. [REF]] means that the [MATH] decay modes remain blind to the presence of an extra sterile neutrino.', '1612.04737-2-44-10': 'This is illustrated in Fig. [REF].', '1612.04737-2-45-0': 'More specifically, we find [EQUATION]', '1612.04737-2-45-1': 'In other words, measuring [MATH] and [MATH] consistent with the Standard Model predictions would be perfectly consistent with a scenario in which the Standard Model is extended by an extra sterile neutrino.', '1612.04737-2-45-2': 'Notice again that the cut into the parameter space in the region around [MATH] GeV - shown in Fig. [REF] - comes from the direct searches [CITATION], implemented in our scan.', '1612.04737-2-45-3': '[MATH] Finally, a similar analysis of the "invisible kaon decay" [MATH] shows that this mode can be largely enhanced if the sterile neutrino is massive.', '1612.04737-2-45-4': 'Due to the available phase space, this decay can be studied for [MATH], and the result is shown in Fig. [REF].', '1612.04737-2-46-0': 'Knowing that in the Standard Model [MATH], the enhancement we observe is indeed substantial and since its decay rate can be comparable to [MATH] its experimental research becomes highly important.', '1612.04737-2-46-1': 'It has been proposed to search for this decay at the NA64 experiment using [MATH] produced from a [MATH] beam hitting a target [CITATION].', '1612.04737-2-46-2': 'From our analysis we find the upper bound, [EQUATION] which could be within the reach of the KOTO, NA62(-KLEVER) and SHIP experiments even if the above bound is by an order of magnitude lower.', '1612.04737-2-46-3': 'The KOTO experiment aims to reach a sensitivity of [MATH] to [MATH] in its first phase [CITATION] and to have [MATH] at the entrance of the detector in phase 2 [CITATION].', '1612.04737-2-46-4': 'NA62-KLEVER is a project that would succeed NA62 and would aim to produce [MATH] [CITATION].', '1612.04737-2-46-5': 'However, we would like to point out that the decay [MATH] could also be searched for in [MATH]-meson decays, making use of the relatively large branching ratio [MATH] and tagging the [MATH] via the pion.', '1612.04737-2-46-6': 'Beam dumps experiments at the CERN SPS like SHIP or a possible run of NA62 in a beam dump configuration would produce copious amounts of [MATH] mesons.', '1612.04737-2-46-7': 'A year of running in beam dump mode for NA62 would produce [MATH] mesons [CITATION], which would correspond to roughly [MATH] tagged [MATH].', '1612.04737-2-46-8': 'SHIP would accumulate even more data, producing [MATH] mesons [CITATION], which would translate into more than [MATH] tagged [MATH].', '1612.04737-2-46-9': 'In any case, an experimental bound on this decay mode would be of great importance for studying the effects of physics beyond the Standard Model in the leptonic sector.', '1612.04737-2-46-10': 'Obviously, a nonzero measurement of [MATH] would be a clean signal of the non-Standard Model physics.', '1612.04737-2-47-0': 'Before closing this section, we should make a brief comment on the lepton flavor violating kaon decays, which in our scenario would be generated by the heavy neutrino running in the loop.', '1612.04737-2-47-1': 'By using the formulas given in Ref. [CITATION] trivially adapted to the kaon decays, and the result of the scan of Sec. [REF], we obtain that these modes are completely negligible, i.e. the branching fractions of all these modes are under [MATH].', '1612.04737-2-48-0': '# Summary', '1612.04737-2-49-0': 'In this paper we presented the results of our study concerning the impact of a massive sterile neutrino on the weak kaon decays such as the leptonic, semileptonic and the decay of a kaon to neutrinos.', '1612.04737-2-49-1': 'In the effective approach adopted in this work, one sterile neutrino is supposed to mimic the effect of a more realistic model in which the neutrino sector is extended to include one or more sterile neutrinos.', '1612.04737-2-50-0': 'Although the mass of the sterile neutrino [MATH] can in principle have any value, we focused on the mass range [MATH] GeV, and in particular on [MATH], when the sterile neutrino is kinematically accessible.', '1612.04737-2-50-1': 'In order to constrain six new parameters ([MATH], the three sterile-active neutrino mixing angles and two new phases) we used a number of quantities discussed in the body of the paper, together with the perturbative unitarity requirement, as well as the constraints arising from the direct searches [CITATION].', '1612.04737-2-50-2': 'After combining such selected parameters with the expressions for the leptonic and semileptonic decays we derive here, we found that only [MATH] can significantly deviate from the current experimental value.', '1612.04737-2-50-3': 'That conflict with the data is present in the interval [MATH] GeV.', '1612.04737-2-50-4': 'The other quantities, including the forward backward and the lepton polarization asymmetries, remain unchanged with respect to their Standard Model values with the effect of the massive sterile neutrino remaining at the level of less than [MATH].', '1612.04737-2-51-0': 'We also derived the expressions for the kaon decays to two (Majorana) neutrinos in the final state, namely [MATH] and [MATH].', '1612.04737-2-51-1': 'Our expressions are generic and can be used when studying a new physics scenario in which heavy neutrinos with no new gauge couplings are involved.', '1612.04737-2-51-2': 'This will be increasingly relevant with the ongoing experimental effort at CERN (NA62, NA64, SHIP) and J-PARC (KOTO) targeting [MATH] and [MATH], respectively.', '1612.04737-2-51-3': 'These two decays, however, appear to be insensitive to the massive sterile neutrino once the experimental and theoretical constraints are taken into account.', '1612.04737-2-51-4': 'In other words, if the experimental results of the weak kaon decays turn out to be consistent with the Standard Model predictions to a [MATH] uncertainty, this would not be in contradiction with the neutrino sector extended by a massive and relatively light sterile neutrino(s).', '1612.04737-2-51-5': 'The only kaon decay mode which appears to be sensitive to the presence of a massive sterile neutrino is [MATH], the branching fraction of which can go up to [MATH], thus possibly within reach of the NA62(-KLEVER), SHIP and KOTO experiments.', '1612.04737-2-51-6': 'Knowing that the Standard Model value of this mode is zero, its observation would be a clean signal of new physics.', '1612.04737-2-52-0': 'Notice also that [MATH] and [MATH] could be used to probe the Majorana phases in the models in which more than one massive neutrino is considered.', '1612.04737-2-52-1': 'In the approach adopted in this paper only one neutrino can be heavy and therefore such a study is prohibited.', '1612.04737-2-52-2': 'If instead one considers a realistic model with more than one heavy neutrino then a study of the Majorana phases becomes possible too [CITATION].', '1612.04737-2-53-0': 'We should mention that one can also consider the situation with a very heavy sterile neutrino [MATH].', '1612.04737-2-53-1': 'In that case the processes discussed in this paper could be modified by the effects of violation of the mixing matrix unitarity; see for instance[CITATION].', '1612.04737-2-53-2': 'We checked that possibility in the explicit computation and found that such effects are indeed tiny.', '1612.04737-2-53-3': 'Importantly, however, a heavy sterile neutrino can propagate in the loop-induced processes and shift the values of [MATH] and [MATH].', '1612.04737-2-53-4': 'We checked that the corresponding effect remains small and completely drowned in the large (long-distance QCD) uncertainties already present in the Standard Model estimates of [MATH] and [MATH] [CITATION].', '1612.04737-2-54-0': 'Finally, the expressions presented in this paper can be easily extended to other similar decays, such as [MATH]-, [MATH]-, [MATH]- and [MATH]-meson decays.', '1612.04737-2-54-1': 'We decided to focus on the kaon decays because of the recent theoretical developments in taming the hadronic uncertainties and because of the better experimental precision.'}

[['1612.04737-1-49-1', '1612.04737-2-50-1'], ['1612.04737-1-49-3', '1612.04737-2-50-3'], ['1612.04737-1-49-4', '1612.04737-2-50-4'], ['1612.04737-1-21-0', '1612.04737-2-21-0'], ['1612.04737-1-21-1', '1612.04737-2-21-1'], ['1612.04737-1-21-2', '1612.04737-2-21-2'], ['1612.04737-1-12-1', '1612.04737-2-12-1'], ['1612.04737-1-12-2', '1612.04737-2-12-2'], ['1612.04737-1-12-3', '1612.04737-2-12-3'], ['1612.04737-1-12-5', '1612.04737-2-12-5'], ['1612.04737-1-12-6', '1612.04737-2-12-6'], ['1612.04737-1-12-8', '1612.04737-2-12-8'], ['1612.04737-1-12-9', '1612.04737-2-12-9'], ['1612.04737-1-0-0', '1612.04737-2-0-0'], ['1612.04737-1-0-1', '1612.04737-2-0-1'], ['1612.04737-1-0-2', '1612.04737-2-0-2'], ['1612.04737-1-0-5', '1612.04737-2-0-6'], ['1612.04737-1-23-1', '1612.04737-2-23-1'], ['1612.04737-1-26-0', '1612.04737-2-26-0'], ['1612.04737-1-26-1', '1612.04737-2-26-1'], ['1612.04737-1-26-2', '1612.04737-2-26-2'], ['1612.04737-1-5-4', '1612.04737-2-5-5'], ['1612.04737-1-22-0', '1612.04737-2-22-0'], ['1612.04737-1-22-1', '1612.04737-2-22-1'], ['1612.04737-1-24-0', '1612.04737-2-24-0'], ['1612.04737-1-38-0', '1612.04737-2-39-0'], ['1612.04737-1-38-4', '1612.04737-2-39-4'], ['1612.04737-1-38-6', '1612.04737-2-39-6'], ['1612.04737-1-38-7', '1612.04737-2-39-7'], ['1612.04737-1-20-0', '1612.04737-2-20-0'], ['1612.04737-1-20-1', '1612.04737-2-20-1'], ['1612.04737-1-20-2', '1612.04737-2-20-2'], ['1612.04737-1-20-3', '1612.04737-2-20-3'], ['1612.04737-1-20-4', '1612.04737-2-20-4'], ['1612.04737-1-31-0', '1612.04737-2-32-0'], ['1612.04737-1-31-1', '1612.04737-2-32-3'], ['1612.04737-1-31-3', '1612.04737-2-32-6'], ['1612.04737-1-31-4', '1612.04737-2-32-7'], ['1612.04737-1-31-5', '1612.04737-2-32-8'], ['1612.04737-1-48-0', '1612.04737-2-49-0'], ['1612.04737-1-51-0', '1612.04737-2-52-0'], ['1612.04737-1-51-1', '1612.04737-2-52-1'], ['1612.04737-1-51-2', '1612.04737-2-52-2'], ['1612.04737-1-25-1', '1612.04737-2-25-4'], ['1612.04737-1-25-2', '1612.04737-2-25-5'], ['1612.04737-1-46-0', '1612.04737-2-47-0'], ['1612.04737-1-46-1', '1612.04737-2-47-1'], ['1612.04737-1-8-0', '1612.04737-2-8-0'], ['1612.04737-1-8-2', '1612.04737-2-8-2'], ['1612.04737-1-8-3', '1612.04737-2-8-3'], ['1612.04737-1-36-0', '1612.04737-2-37-0'], ['1612.04737-1-36-1', '1612.04737-2-37-1'], ['1612.04737-1-36-2', '1612.04737-2-37-2'], ['1612.04737-1-36-3', '1612.04737-2-37-3'], ['1612.04737-1-4-1', '1612.04737-2-4-1'], ['1612.04737-1-4-2', '1612.04737-2-4-2'], ['1612.04737-1-4-3', '1612.04737-2-4-3'], ['1612.04737-1-4-5', '1612.04737-2-4-5'], ['1612.04737-1-15-1', '1612.04737-2-15-1'], ['1612.04737-1-15-3', '1612.04737-2-15-3'], ['1612.04737-1-15-4', '1612.04737-2-15-4'], ['1612.04737-1-15-5', '1612.04737-2-15-5'], ['1612.04737-1-15-7', '1612.04737-2-15-7'], ['1612.04737-1-18-1', '1612.04737-2-18-1'], ['1612.04737-1-43-0', '1612.04737-2-44-0'], ['1612.04737-1-43-1', '1612.04737-2-44-1'], ['1612.04737-1-43-2', '1612.04737-2-44-2'], ['1612.04737-1-43-3', '1612.04737-2-44-3'], ['1612.04737-1-43-5', '1612.04737-2-44-5'], ['1612.04737-1-43-7', '1612.04737-2-44-7'], ['1612.04737-1-43-8', '1612.04737-2-44-8'], ['1612.04737-1-43-9', '1612.04737-2-44-9'], ['1612.04737-1-43-10', '1612.04737-2-44-10'], ['1612.04737-1-44-0', '1612.04737-2-45-0'], ['1612.04737-1-44-1', '1612.04737-2-45-1'], ['1612.04737-1-44-4', '1612.04737-2-45-4'], ['1612.04737-1-45-0', '1612.04737-2-46-0'], ['1612.04737-1-45-2', '1612.04737-2-46-9'], ['1612.04737-1-9-0', '1612.04737-2-9-0'], ['1612.04737-1-9-1', '1612.04737-2-9-1'], ['1612.04737-1-9-2', '1612.04737-2-9-2'], ['1612.04737-1-9-3', '1612.04737-2-9-3'], ['1612.04737-1-9-4', '1612.04737-2-9-4'], ['1612.04737-1-19-0', '1612.04737-2-19-0'], ['1612.04737-1-19-1', '1612.04737-2-19-1'], ['1612.04737-1-19-2', '1612.04737-2-19-2'], ['1612.04737-1-50-0', '1612.04737-2-51-0'], ['1612.04737-1-50-3', '1612.04737-2-51-3'], ['1612.04737-1-50-4', '1612.04737-2-51-4'], ['1612.04737-1-53-0', '1612.04737-2-54-0'], ['1612.04737-1-7-0', '1612.04737-2-7-0'], ['1612.04737-1-7-2', '1612.04737-2-7-2'], ['1612.04737-1-7-3', '1612.04737-2-7-3'], ['1612.04737-1-7-4', '1612.04737-2-7-4'], ['1612.04737-1-42-0', '1612.04737-2-43-0'], ['1612.04737-1-42-1', '1612.04737-2-43-1'], ['1612.04737-1-42-3', '1612.04737-2-43-3'], ['1612.04737-1-42-4', '1612.04737-2-43-4'], ['1612.04737-1-42-5', '1612.04737-2-43-5'], ['1612.04737-1-42-6', '1612.04737-2-43-6'], ['1612.04737-1-42-7', '1612.04737-2-43-7'], ['1612.04737-1-42-8', '1612.04737-2-43-8'], ['1612.04737-1-42-9', '1612.04737-2-43-9'], ['1612.04737-1-33-0', '1612.04737-2-34-0'], ['1612.04737-1-33-1', '1612.04737-2-34-1'], ['1612.04737-1-33-3', '1612.04737-2-34-3'], ['1612.04737-1-33-4', '1612.04737-2-34-4'], ['1612.04737-1-33-10', '1612.04737-2-34-10'], ['1612.04737-1-33-11', '1612.04737-2-34-11'], ['1612.04737-1-33-12', '1612.04737-2-34-12'], ['1612.04737-1-33-14', '1612.04737-2-34-14'], ['1612.04737-1-33-15', '1612.04737-2-34-15'], ['1612.04737-1-33-16', '1612.04737-2-34-16'], ['1612.04737-1-33-17', '1612.04737-2-34-17'], ['1612.04737-1-35-1', '1612.04737-2-36-1'], ['1612.04737-1-35-2', '1612.04737-2-36-2'], ['1612.04737-1-35-3', '1612.04737-2-36-3'], ['1612.04737-1-35-4', '1612.04737-2-36-4'], ['1612.04737-1-35-6', '1612.04737-2-36-6'], ['1612.04737-1-35-7', '1612.04737-2-36-7'], ['1612.04737-1-41-0', '1612.04737-2-42-0'], ['1612.04737-1-41-1', '1612.04737-2-42-1'], ['1612.04737-1-41-2', '1612.04737-2-42-2'], ['1612.04737-1-40-0', '1612.04737-2-41-0'], ['1612.04737-1-40-2', '1612.04737-2-41-2'], ['1612.04737-1-40-3', '1612.04737-2-41-3'], ['1612.04737-1-40-5', '1612.04737-2-41-5'], ['1612.04737-1-29-0', '1612.04737-2-29-0'], ['1612.04737-1-27-0', '1612.04737-2-27-0'], ['1612.04737-1-27-1', '1612.04737-2-27-1'], ['1612.04737-1-52-0', '1612.04737-2-53-0'], ['1612.04737-1-52-2', '1612.04737-2-53-2'], ['1612.04737-1-17-0', '1612.04737-2-17-0'], ['1612.04737-1-17-1', '1612.04737-2-17-1'], ['1612.04737-1-17-2', '1612.04737-2-17-2'], ['1612.04737-1-49-0', '1612.04737-2-50-0'], ['1612.04737-1-49-2', '1612.04737-2-50-2'], ['1612.04737-1-12-0', '1612.04737-2-12-0'], ['1612.04737-1-12-4', '1612.04737-2-12-4'], ['1612.04737-1-12-7', '1612.04737-2-12-7'], ['1612.04737-1-12-10', '1612.04737-2-12-10'], ['1612.04737-1-0-4', '1612.04737-2-0-5'], ['1612.04737-1-23-0', '1612.04737-2-23-0'], ['1612.04737-1-5-0', '1612.04737-2-5-0'], ['1612.04737-1-5-1', '1612.04737-2-5-1'], ['1612.04737-1-5-2', '1612.04737-2-5-3'], ['1612.04737-1-5-3', '1612.04737-2-5-4'], ['1612.04737-1-24-1', '1612.04737-2-24-1'], ['1612.04737-1-24-2', '1612.04737-2-24-2'], ['1612.04737-1-13-0', '1612.04737-2-13-0'], ['1612.04737-1-38-1', '1612.04737-2-39-1'], ['1612.04737-1-38-3', '1612.04737-2-39-3'], ['1612.04737-1-38-5', '1612.04737-2-39-5'], ['1612.04737-1-31-2', '1612.04737-2-32-4'], ['1612.04737-1-31-6', '1612.04737-2-32-9'], ['1612.04737-1-48-1', '1612.04737-2-49-1'], ['1612.04737-1-8-1', '1612.04737-2-8-1'], ['1612.04737-1-36-4', '1612.04737-2-37-4'], ['1612.04737-1-4-0', '1612.04737-2-4-0'], ['1612.04737-1-4-4', '1612.04737-2-4-4'], ['1612.04737-1-4-6', '1612.04737-2-4-6'], ['1612.04737-1-4-7', '1612.04737-2-4-7'], ['1612.04737-1-15-0', '1612.04737-2-15-0'], ['1612.04737-1-15-2', '1612.04737-2-15-2'], ['1612.04737-1-15-6', '1612.04737-2-15-6'], ['1612.04737-1-15-8', '1612.04737-2-15-8'], ['1612.04737-1-43-4', '1612.04737-2-44-4'], ['1612.04737-1-43-6', '1612.04737-2-44-6'], ['1612.04737-1-44-2', '1612.04737-2-45-2'], ['1612.04737-1-44-3', '1612.04737-2-45-3'], ['1612.04737-1-45-1', '1612.04737-2-46-2'], ['1612.04737-1-45-3', '1612.04737-2-46-10'], ['1612.04737-1-2-0', '1612.04737-2-2-0'], ['1612.04737-1-2-1', '1612.04737-2-2-1'], ['1612.04737-1-2-2', '1612.04737-2-2-2'], ['1612.04737-1-50-1', '1612.04737-2-51-1'], ['1612.04737-1-50-2', '1612.04737-2-51-2'], ['1612.04737-1-50-5', '1612.04737-2-51-5'], ['1612.04737-1-50-6', '1612.04737-2-51-6'], ['1612.04737-1-53-1', '1612.04737-2-54-1'], ['1612.04737-1-7-1', '1612.04737-2-7-1'], ['1612.04737-1-42-2', '1612.04737-2-43-2'], ['1612.04737-1-33-2', '1612.04737-2-34-2'], ['1612.04737-1-33-5', '1612.04737-2-34-5'], ['1612.04737-1-33-6', '1612.04737-2-34-6'], ['1612.04737-1-33-7', '1612.04737-2-34-7'], ['1612.04737-1-33-8', '1612.04737-2-34-8'], ['1612.04737-1-33-9', '1612.04737-2-34-9'], ['1612.04737-1-33-13', '1612.04737-2-34-13'], ['1612.04737-1-33-18', '1612.04737-2-34-18'], ['1612.04737-1-35-0', '1612.04737-2-36-0'], ['1612.04737-1-35-5', '1612.04737-2-36-5'], ['1612.04737-1-40-1', '1612.04737-2-41-1'], ['1612.04737-1-40-4', '1612.04737-2-41-4'], ['1612.04737-1-40-6', '1612.04737-2-41-6'], ['1612.04737-1-3-0', '1612.04737-2-3-0'], ['1612.04737-1-3-1', '1612.04737-2-3-1'], ['1612.04737-1-52-1', '1612.04737-2-53-1'], ['1612.04737-1-52-3', '1612.04737-2-53-3'], ['1612.04737-1-52-4', '1612.04737-2-53-4'], ['1612.04737-1-6-0', '1612.04737-2-6-0'], ['1612.04737-1-6-1', '1612.04737-2-6-1'], ['1612.04737-1-0-3', '1612.04737-2-0-3'], ['1612.04737-1-18-0', '1612.04737-2-18-0']]

[['1612.04737-1-49-1', '1612.04737-2-50-1'], ['1612.04737-1-49-3', '1612.04737-2-50-3'], ['1612.04737-1-49-4', '1612.04737-2-50-4'], ['1612.04737-1-21-0', '1612.04737-2-21-0'], ['1612.04737-1-21-1', '1612.04737-2-21-1'], ['1612.04737-1-21-2', '1612.04737-2-21-2'], ['1612.04737-1-12-1', '1612.04737-2-12-1'], ['1612.04737-1-12-2', '1612.04737-2-12-2'], ['1612.04737-1-12-3', '1612.04737-2-12-3'], ['1612.04737-1-12-5', '1612.04737-2-12-5'], ['1612.04737-1-12-6', '1612.04737-2-12-6'], ['1612.04737-1-12-8', '1612.04737-2-12-8'], ['1612.04737-1-12-9', '1612.04737-2-12-9'], ['1612.04737-1-0-0', '1612.04737-2-0-0'], ['1612.04737-1-0-1', '1612.04737-2-0-1'], ['1612.04737-1-0-2', '1612.04737-2-0-2'], ['1612.04737-1-0-5', '1612.04737-2-0-6'], ['1612.04737-1-23-1', '1612.04737-2-23-1'], ['1612.04737-1-26-0', '1612.04737-2-26-0'], ['1612.04737-1-26-1', '1612.04737-2-26-1'], ['1612.04737-1-26-2', '1612.04737-2-26-2'], ['1612.04737-1-5-4', '1612.04737-2-5-5'], ['1612.04737-1-22-0', '1612.04737-2-22-0'], ['1612.04737-1-22-1', '1612.04737-2-22-1'], ['1612.04737-1-24-0', '1612.04737-2-24-0'], ['1612.04737-1-38-0', '1612.04737-2-39-0'], ['1612.04737-1-38-4', '1612.04737-2-39-4'], ['1612.04737-1-38-6', '1612.04737-2-39-6'], ['1612.04737-1-38-7', '1612.04737-2-39-7'], ['1612.04737-1-20-0', '1612.04737-2-20-0'], ['1612.04737-1-20-1', '1612.04737-2-20-1'], ['1612.04737-1-20-2', '1612.04737-2-20-2'], ['1612.04737-1-20-3', '1612.04737-2-20-3'], ['1612.04737-1-20-4', '1612.04737-2-20-4'], ['1612.04737-1-31-0', '1612.04737-2-32-0'], ['1612.04737-1-31-1', '1612.04737-2-32-3'], ['1612.04737-1-31-3', '1612.04737-2-32-6'], ['1612.04737-1-31-4', '1612.04737-2-32-7'], ['1612.04737-1-31-5', '1612.04737-2-32-8'], ['1612.04737-1-48-0', '1612.04737-2-49-0'], ['1612.04737-1-51-0', '1612.04737-2-52-0'], ['1612.04737-1-51-1', '1612.04737-2-52-1'], ['1612.04737-1-51-2', '1612.04737-2-52-2'], ['1612.04737-1-25-1', '1612.04737-2-25-4'], ['1612.04737-1-25-2', '1612.04737-2-25-5'], ['1612.04737-1-46-0', '1612.04737-2-47-0'], ['1612.04737-1-46-1', '1612.04737-2-47-1'], ['1612.04737-1-8-0', '1612.04737-2-8-0'], ['1612.04737-1-8-2', '1612.04737-2-8-2'], ['1612.04737-1-8-3', '1612.04737-2-8-3'], ['1612.04737-1-36-0', '1612.04737-2-37-0'], ['1612.04737-1-36-1', '1612.04737-2-37-1'], ['1612.04737-1-36-2', '1612.04737-2-37-2'], ['1612.04737-1-36-3', '1612.04737-2-37-3'], ['1612.04737-1-4-1', '1612.04737-2-4-1'], ['1612.04737-1-4-2', '1612.04737-2-4-2'], ['1612.04737-1-4-3', '1612.04737-2-4-3'], ['1612.04737-1-4-5', '1612.04737-2-4-5'], ['1612.04737-1-15-1', '1612.04737-2-15-1'], ['1612.04737-1-15-3', '1612.04737-2-15-3'], ['1612.04737-1-15-4', '1612.04737-2-15-4'], ['1612.04737-1-15-5', '1612.04737-2-15-5'], ['1612.04737-1-15-7', '1612.04737-2-15-7'], ['1612.04737-1-18-1', '1612.04737-2-18-1'], ['1612.04737-1-43-0', '1612.04737-2-44-0'], ['1612.04737-1-43-1', '1612.04737-2-44-1'], ['1612.04737-1-43-2', '1612.04737-2-44-2'], ['1612.04737-1-43-3', '1612.04737-2-44-3'], ['1612.04737-1-43-5', '1612.04737-2-44-5'], ['1612.04737-1-43-7', '1612.04737-2-44-7'], ['1612.04737-1-43-8', '1612.04737-2-44-8'], ['1612.04737-1-43-9', '1612.04737-2-44-9'], ['1612.04737-1-43-10', '1612.04737-2-44-10'], ['1612.04737-1-44-0', '1612.04737-2-45-0'], ['1612.04737-1-44-1', '1612.04737-2-45-1'], ['1612.04737-1-44-4', '1612.04737-2-45-4'], ['1612.04737-1-45-0', '1612.04737-2-46-0'], ['1612.04737-1-45-2', '1612.04737-2-46-9'], ['1612.04737-1-9-0', '1612.04737-2-9-0'], ['1612.04737-1-9-1', '1612.04737-2-9-1'], ['1612.04737-1-9-2', '1612.04737-2-9-2'], ['1612.04737-1-9-3', '1612.04737-2-9-3'], ['1612.04737-1-9-4', '1612.04737-2-9-4'], ['1612.04737-1-19-0', '1612.04737-2-19-0'], ['1612.04737-1-19-1', '1612.04737-2-19-1'], ['1612.04737-1-19-2', '1612.04737-2-19-2'], ['1612.04737-1-50-0', '1612.04737-2-51-0'], ['1612.04737-1-50-3', '1612.04737-2-51-3'], ['1612.04737-1-50-4', '1612.04737-2-51-4'], ['1612.04737-1-53-0', '1612.04737-2-54-0'], ['1612.04737-1-7-0', '1612.04737-2-7-0'], ['1612.04737-1-7-2', '1612.04737-2-7-2'], ['1612.04737-1-7-3', '1612.04737-2-7-3'], ['1612.04737-1-7-4', '1612.04737-2-7-4'], ['1612.04737-1-42-0', '1612.04737-2-43-0'], ['1612.04737-1-42-1', '1612.04737-2-43-1'], ['1612.04737-1-42-3', '1612.04737-2-43-3'], ['1612.04737-1-42-4', '1612.04737-2-43-4'], ['1612.04737-1-42-5', '1612.04737-2-43-5'], ['1612.04737-1-42-6', '1612.04737-2-43-6'], ['1612.04737-1-42-7', '1612.04737-2-43-7'], ['1612.04737-1-42-8', '1612.04737-2-43-8'], ['1612.04737-1-42-9', '1612.04737-2-43-9'], ['1612.04737-1-33-0', '1612.04737-2-34-0'], ['1612.04737-1-33-1', '1612.04737-2-34-1'], ['1612.04737-1-33-3', '1612.04737-2-34-3'], ['1612.04737-1-33-4', '1612.04737-2-34-4'], ['1612.04737-1-33-10', '1612.04737-2-34-10'], ['1612.04737-1-33-11', '1612.04737-2-34-11'], ['1612.04737-1-33-12', '1612.04737-2-34-12'], ['1612.04737-1-33-14', '1612.04737-2-34-14'], ['1612.04737-1-33-15', '1612.04737-2-34-15'], ['1612.04737-1-33-16', '1612.04737-2-34-16'], ['1612.04737-1-33-17', '1612.04737-2-34-17'], ['1612.04737-1-35-1', '1612.04737-2-36-1'], ['1612.04737-1-35-2', '1612.04737-2-36-2'], ['1612.04737-1-35-3', '1612.04737-2-36-3'], ['1612.04737-1-35-4', '1612.04737-2-36-4'], ['1612.04737-1-35-6', '1612.04737-2-36-6'], ['1612.04737-1-35-7', '1612.04737-2-36-7'], ['1612.04737-1-41-0', '1612.04737-2-42-0'], ['1612.04737-1-41-1', '1612.04737-2-42-1'], ['1612.04737-1-41-2', '1612.04737-2-42-2'], ['1612.04737-1-40-0', '1612.04737-2-41-0'], ['1612.04737-1-40-2', '1612.04737-2-41-2'], ['1612.04737-1-40-3', '1612.04737-2-41-3'], ['1612.04737-1-40-5', '1612.04737-2-41-5'], ['1612.04737-1-29-0', '1612.04737-2-29-0'], ['1612.04737-1-27-0', '1612.04737-2-27-0'], ['1612.04737-1-27-1', '1612.04737-2-27-1'], ['1612.04737-1-52-0', '1612.04737-2-53-0'], ['1612.04737-1-52-2', '1612.04737-2-53-2'], ['1612.04737-1-17-0', '1612.04737-2-17-0'], ['1612.04737-1-17-1', '1612.04737-2-17-1'], ['1612.04737-1-17-2', '1612.04737-2-17-2']]

[['1612.04737-1-49-0', '1612.04737-2-50-0'], ['1612.04737-1-49-2', '1612.04737-2-50-2'], ['1612.04737-1-12-0', '1612.04737-2-12-0'], ['1612.04737-1-12-4', '1612.04737-2-12-4'], ['1612.04737-1-12-7', '1612.04737-2-12-7'], ['1612.04737-1-12-10', '1612.04737-2-12-10'], ['1612.04737-1-0-4', '1612.04737-2-0-5'], ['1612.04737-1-23-0', '1612.04737-2-23-0'], ['1612.04737-1-5-0', '1612.04737-2-5-0'], ['1612.04737-1-5-1', '1612.04737-2-5-1'], ['1612.04737-1-5-2', '1612.04737-2-5-3'], ['1612.04737-1-5-3', '1612.04737-2-5-4'], ['1612.04737-1-24-1', '1612.04737-2-24-1'], ['1612.04737-1-24-2', '1612.04737-2-24-2'], ['1612.04737-1-13-0', '1612.04737-2-13-0'], ['1612.04737-1-38-1', '1612.04737-2-39-1'], ['1612.04737-1-38-3', '1612.04737-2-39-3'], ['1612.04737-1-38-5', '1612.04737-2-39-5'], ['1612.04737-1-31-2', '1612.04737-2-32-4'], ['1612.04737-1-31-6', '1612.04737-2-32-9'], ['1612.04737-1-48-1', '1612.04737-2-49-1'], ['1612.04737-1-8-1', '1612.04737-2-8-1'], ['1612.04737-1-36-4', '1612.04737-2-37-4'], ['1612.04737-1-4-0', '1612.04737-2-4-0'], ['1612.04737-1-4-4', '1612.04737-2-4-4'], ['1612.04737-1-4-6', '1612.04737-2-4-6'], ['1612.04737-1-4-7', '1612.04737-2-4-7'], ['1612.04737-1-15-0', '1612.04737-2-15-0'], ['1612.04737-1-15-2', '1612.04737-2-15-2'], ['1612.04737-1-15-6', '1612.04737-2-15-6'], ['1612.04737-1-15-8', '1612.04737-2-15-8'], ['1612.04737-1-43-4', '1612.04737-2-44-4'], ['1612.04737-1-43-6', '1612.04737-2-44-6'], ['1612.04737-1-44-2', '1612.04737-2-45-2'], ['1612.04737-1-44-3', '1612.04737-2-45-3'], ['1612.04737-1-45-1', '1612.04737-2-46-2'], ['1612.04737-1-45-3', '1612.04737-2-46-10'], ['1612.04737-1-2-0', '1612.04737-2-2-0'], ['1612.04737-1-2-1', '1612.04737-2-2-1'], ['1612.04737-1-2-2', '1612.04737-2-2-2'], ['1612.04737-1-50-1', '1612.04737-2-51-1'], ['1612.04737-1-50-2', '1612.04737-2-51-2'], ['1612.04737-1-50-5', '1612.04737-2-51-5'], ['1612.04737-1-50-6', '1612.04737-2-51-6'], ['1612.04737-1-53-1', '1612.04737-2-54-1'], ['1612.04737-1-7-1', '1612.04737-2-7-1'], ['1612.04737-1-42-2', '1612.04737-2-43-2'], ['1612.04737-1-33-2', '1612.04737-2-34-2'], ['1612.04737-1-33-5', '1612.04737-2-34-5'], ['1612.04737-1-33-6', '1612.04737-2-34-6'], ['1612.04737-1-33-7', '1612.04737-2-34-7'], ['1612.04737-1-33-8', '1612.04737-2-34-8'], ['1612.04737-1-33-9', '1612.04737-2-34-9'], ['1612.04737-1-33-13', '1612.04737-2-34-13'], ['1612.04737-1-33-18', '1612.04737-2-34-18'], ['1612.04737-1-35-0', '1612.04737-2-36-0'], ['1612.04737-1-35-5', '1612.04737-2-36-5'], ['1612.04737-1-40-1', '1612.04737-2-41-1'], ['1612.04737-1-40-4', '1612.04737-2-41-4'], ['1612.04737-1-40-6', '1612.04737-2-41-6'], ['1612.04737-1-3-0', '1612.04737-2-3-0'], ['1612.04737-1-3-1', '1612.04737-2-3-1'], ['1612.04737-1-52-1', '1612.04737-2-53-1'], ['1612.04737-1-52-3', '1612.04737-2-53-3'], ['1612.04737-1-52-4', '1612.04737-2-53-4'], ['1612.04737-1-6-0', '1612.04737-2-6-0'], ['1612.04737-1-6-1', '1612.04737-2-6-1']]

[]

[['1612.04737-1-0-3', '1612.04737-2-0-3'], ['1612.04737-1-18-0', '1612.04737-2-18-0']]

[]

['1612.04737-1-38-2', '1612.04737-2-39-2']

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

https://arxiv.org/abs/1612.04737

cond-mat-0305301

{'cond-mat-0305301-1-0-0': 'We solve the coupled gap equations for the [MATH]- and [MATH]-bands of MgB[MATH] in the vortex state and calculate the resulting field dependencies of the specific heat coefficient [MATH] and the thermal conductivity [MATH].', 'cond-mat-0305301-1-0-1': 'The crucial parameters of the theory are the interband pairing interaction [MATH] and the ratio [MATH] of the coherence lengths.', 'cond-mat-0305301-1-0-2': 'For reasonably small [MATH] and s, the small gap [MATH] decreases with increasing magnetic field [MATH] much faster than the large gap [MATH].', 'cond-mat-0305301-1-0-3': 'This gives rise to the observed rapid increase of [MATH] and [MATH] for small fields while [MATH] and [MATH] exhibit conventional field dependencies.', 'cond-mat-0305301-1-0-4': 'Inclusion of intraband impurity scattering yields fairly good agreement with experiments for applied fields along the c axis.', 'cond-mat-0305301-1-1-0': 'Evidence for the for the existence of two superconducting gaps in MgB[MATH] [CITATION] is provided by the rapid rise of the specific heat coefficient [MATH] [CITATION] and the thermal conductivity [MATH] [CITATION] at very low fields.', 'cond-mat-0305301-1-1-1': 'These measured field dependencies can be explained qualitatively by assuming two independent bands where the large s-wave pairing gap [MATH] is associated with the two-dimensional [MATH]-band and the small s-wave gap [MATH] is associated with the three-dimensional [MATH]-band.', 'cond-mat-0305301-1-1-2': '[CITATION] The steep rise of [MATH] and [MATH] can be explained qualitatively by assuming that the "virtual" upper critical field for [MATH] is much smaller than that of [MATH].', 'cond-mat-0305301-1-1-3': '[CITATION] In the present paper we improve the theory of Ref. [CITATION] by taking into account the interband pairing interaction while neglecting the interband impurity scattering.', 'cond-mat-0305301-1-1-4': '[CITATION] We first have to solve the two-band gap equations in the presence of the vortex lattices produced by a magnetic field.', 'cond-mat-0305301-1-1-5': 'Generalization of the linearized gap equations near the upper critical field [MATH] [CITATION] to all averaged fields [MATH] between [MATH] and [MATH] yields, instead of the single gap equation of Ref. [CITATION], the following coupled gap equations for the gaps [MATH] at [MATH]: [EQUATION]', 'cond-mat-0305301-1-1-6': 'Here, the [MATH] are the intra- and interband pairing interactions multiplied by the densities of states [MATH], and the [MATH] are the spectral functions of the anomalous propagators for the Abrikosov vortex lattice: [CITATION] [EQUATION]', 'cond-mat-0305301-1-1-7': 'The [MATH] are the normal state impurity scattering rates and the [MATH] are the field-dependent densities of states.', 'cond-mat-0305301-1-1-8': 'For band 2 (the [MATH]-band) we assume a spherical Fermi surface.', 'cond-mat-0305301-1-1-9': 'Then [MATH] is replaced by [MATH] and [MATH] and [MATH] are averaged over the polar angle [MATH] with respect to the direction of [MATH].', 'cond-mat-0305301-1-1-10': 'For brevity we omit here, and in the following, the terms containing [MATH] and the integrations over [MATH] from 0 to [MATH].', 'cond-mat-0305301-1-1-11': 'For band 1 (the [MATH]-band) and the field along the c axis, [MATH] and thus [MATH].', 'cond-mat-0305301-1-2-0': 'In the limit [MATH] the vortex lattice constant [MATH] tends to infinity.', 'cond-mat-0305301-1-2-1': 'Making use of the asymptotic expansion [MATH], the gap equations, Eq.([REF]), become [EQUATION] where the [MATH] are the gap values at [MATH] in zero field.', 'cond-mat-0305301-1-2-2': 'Here we have made use of the relation [EQUATION] which can be derived with the help of the expression for [MATH] in Eq.([REF]).', 'cond-mat-0305301-1-2-3': 'However, this relation only holds in the absence of interband impurity scattering.', 'cond-mat-0305301-1-2-4': '[CITATION]', 'cond-mat-0305301-1-3-0': 'With the help of the Abrikosov parameters, denoted by [MATH], we now express [MATH] in terms of the reduced field [MATH] and the zero-field gap [MATH]: [CITATION] [EQUATION]', 'cond-mat-0305301-1-3-1': 'Employing these relations we can express the gaps [MATH] and the scattering rates [MATH] in Eq.([REF]) by their ratios with respect to the [MATH], and we can convert the integrations over [MATH] in Eq.([REF]) to integrations over the new variables [MATH].', 'cond-mat-0305301-1-3-2': 'We then divide Eq.([REF]) by [MATH] and Eq.([REF]) by [MATH] and subtract the latter from the former.', 'cond-mat-0305301-1-3-3': 'In this way we obtain two coupled equations for the two unknown functions [MATH] and [MATH] for given values of the parameters [MATH], and [MATH].', 'cond-mat-0305301-1-3-4': 'The quantities [MATH], and [MATH] are defined by [EQUATION]', 'cond-mat-0305301-1-3-5': 'In Fig. (1) we show the reduced gap functions [MATH] and [MATH] for parameter values [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], with [MATH]1/3, 1/4, and 1/5.', 'cond-mat-0305301-1-3-6': 'These values for [MATH], [MATH], and [MATH] agree approximately with the values obtained from band structure calculations.', 'cond-mat-0305301-1-3-7': '[CITATION] We have, however, taken a much smaller [MATH] than the value of [MATH] given in Ref. [CITATION].', 'cond-mat-0305301-1-3-8': 'The reduced impurity scattering rates [MATH] and the values for the ratios [MATH] ans [MATH] agree qualitatively with the values estimated from experiments on MgB[MATH].', 'cond-mat-0305301-1-3-9': 'While the function [MATH] is rather independent of the choice of parameters, the function [MATH] depends sensitively on the value of the coupling constant [MATH].', 'cond-mat-0305301-1-3-10': 'As [MATH] tends to the theoretical value of 0.091 of Ref. Liu, [MATH] is rather close to [MATH] and consequently the functions [MATH] and [MATH] for the two bands are not very different.', 'cond-mat-0305301-1-3-11': 'For vanishing interband coupling [MATH], however, we obtain approximately [MATH] which goes to zero at a smaller effective upper critical field [MATH] where [MATH] (see Eq.([REF])).', 'cond-mat-0305301-1-3-12': 'Thus we see that [MATH] corresponds to the "virtual" upper critical field for the [MATH]-band which was introduced in Ref. Bouquet as the field above which the overlap of the vortex cores with large radius [MATH] (see the STS measurements of Ref. Esk) drives the majority of the [MATH]-band electrons normal.', 'cond-mat-0305301-1-3-13': 'In Fig.(1) we show [MATH] for [MATH] and for the smaller values [MATH] and 1/5 which take into account that, in Eq.([REF]), [MATH].', 'cond-mat-0305301-1-3-14': '[CITATION] The gap [MATH] is seen to decrease more rapidly relative to [MATH] as [MATH] is decreased, remaining, however, finite up to [MATH].', 'cond-mat-0305301-1-4-0': 'In Fig. (2) we have plotted our results for the zero energy density of states [MATH] obtained from the expression for [MATH] (given by Eq. ([REF]) with the numerator set equal to 1) by inserting the previously calculated gap ratios [MATH] together with Eqs. ([REF]) for the functions [MATH].', 'cond-mat-0305301-1-4-1': '[MATH] is obtained by averaging [MATH] over the polar angle [MATH].', 'cond-mat-0305301-1-4-2': 'We note that it is important to calculate the impurity scattering rates [MATH] self-consistently.', 'cond-mat-0305301-1-4-3': 'One sees from Fig. (2) that [MATH] rises steeply for small fields [MATH] and then becomes almost constant above [MATH].', 'cond-mat-0305301-1-4-4': 'The slope at [MATH] and the downward curvature for low fields increase as [MATH] is decreased from 1/3 to 1/4 to 1/5.', 'cond-mat-0305301-1-4-5': 'The function [MATH] is very similar to the function obtained previously for a single band.', 'cond-mat-0305301-1-4-6': '[CITATION] The initial steep rise of [MATH] qualitatively fits the data points for the contribution of the [MATH]-band to the specific heat coefficient [MATH].', 'cond-mat-0305301-1-4-7': '[CITATION] The function [MATH] corresponds to the straight line assumed in Ref. Bouquet for the [MATH]-band contribution to [MATH] for fields applied along the c axis.', 'cond-mat-0305301-1-5-0': 'We turn now to the calculation of the in-plane electronic thermal conductivity [MATH] which is given at [MATH] by the expression in Ref. TF2.', 'cond-mat-0305301-1-5-1': 'Again it is important to take into account the renormalization of the gap by the function [MATH] in the presence of impurity scattering (see Eq.(13) of Ref. TF1).', 'cond-mat-0305301-1-5-2': 'By inserting the functions [MATH] obtained from the gap equations, Eq.([REF]), and the functions [MATH] from Eq.([REF]) into the expresions for the ratios [MATH] we obtain, for applied fields [MATH] along the c axis, the thermal conductivity ratios [MATH] shown in Fig.3.', 'cond-mat-0305301-1-5-3': 'It should be noted that the [MATH]-band contribution [MATH] has been obtained as an average over the polar angle [MATH] by including the factor [MATH] which arises from the square of the group velocity in the ab plane.', 'cond-mat-0305301-1-5-4': 'The curve for the [MATH]-band conductivity [MATH] turns out to be very similar to the curve obtained in Ref. [CITATION] for a single band with the same impurity scattering rate [MATH].', 'cond-mat-0305301-1-5-5': 'The curve [MATH] for the [MATH]-band contribution with [MATH] rises almost linearly with [MATH] where the slope near [MATH] and the downward curvature for low fields increase as [MATH] is decreased from 1/3 to 1/4 to 1/5.', 'cond-mat-0305301-1-5-6': 'For applied fields perpendicular to the c axis, the measured thermal conductivity [MATH] first rises steeply for small fields and then saturates while, for fields along the c axis, it exhibits an upward curvature towards [MATH].', 'cond-mat-0305301-1-5-7': '[CITATION] These different behaviors have been explained by separating the individual contributions of the [MATH]- and [MATH]-bands.', 'cond-mat-0305301-1-5-8': 'Then [MATH] rises steeply with [MATH] and approximately attains its normal-state value at a small field and [MATH] first rises very slowly and then curves upward towards [MATH].', 'cond-mat-0305301-1-5-9': '[CITATION] These experimental curves are similar to our results for [MATH] shown in Fig. (3).', 'cond-mat-0305301-1-6-0': 'We now briefly discuss the parameter values and approximations that have been used to derive our results.', 'cond-mat-0305301-1-6-1': 'We have seen that the rapid increase of the specific heat coefficient [MATH] and the thermal conductivity ratio [MATH] with increasing field [MATH] is due mainly to the [MATH]-band contribution.', 'cond-mat-0305301-1-6-2': 'The reason is that the gap [MATH] associated with the [MATH]-band almost closes at the so-called "virtual" upper critical field [CITATION] [MATH] because the ratio [MATH] of the coherence lengths of the [MATH]- and the [MATH]-bands is much smaller than 1.', 'cond-mat-0305301-1-6-3': 'For the most important parameters entering our gap equations, [MATH] and [MATH], we have used the values [MATH] and [MATH] 1/3, 1/4, and 1/5 which are based on various experiments on MgB[MATH].', 'cond-mat-0305301-1-6-4': '[CITATION] The field dependence of the gap ratio [MATH], and thus of the contributions to [MATH] and [MATH] arising from the [MATH]-band, are nearly the same as those obtained for an independent single [MATH]-band, [CITATION] indicating that the effect of the interband coupling [MATH] is rather small.', 'cond-mat-0305301-1-6-5': 'However, the field dependence of [MATH] differs substantially from that for the independent single [MATH]-band with an effective upper critical field [MATH] as can be seen from Fig. (1).', 'cond-mat-0305301-1-6-6': 'This is because [MATH] is non-zero between [MATH] and [MATH] due to the effect of the interband coupling [MATH].', 'cond-mat-0305301-1-6-7': 'This shows that superconductivity survives even though the vortex cores for the [MATH]-band with giant radius [MATH] [CITATION] start to overlap for [MATH].', 'cond-mat-0305301-1-6-8': 'As can be seen in Fig. (1), the curve for [MATH] is very sensitive to the values of [MATH] and [MATH].', 'cond-mat-0305301-1-6-9': 'We find that the experimental contributions to [MATH] and [MATH] arising from the [MATH]-band [CITATION] can only be fitted by assuming rather small values of [MATH], while band structure calculations yield a much larger value, [MATH].', 'cond-mat-0305301-1-6-10': '[CITATION] The other crucial parameter value needed to obtain good fits of the data is [MATH].', 'cond-mat-0305301-1-6-11': 'Taking a smaller value than [MATH] is reasonable since the ratio of velocities [MATH] in Eq. ([REF]) is smaller than 1.', 'cond-mat-0305301-1-6-12': '[CITATION] The other parameter values used in our numerical calculations are the reduced impurity scattering rates [MATH] and [MATH] which have been estimated from the relevant experiments.', 'cond-mat-0305301-1-6-13': 'It turns out that even for these moderately large impurity scattering rates it is very important to take into account the renormalization of the gap (see Eq. ([REF])) which leads to a large reduction of the effect of impurity scattering in comparison to that calculated without the function [MATH].', 'cond-mat-0305301-1-6-14': 'It is also important that the calculation of the scattering rate [MATH] in the Born limit be carried out self-consistently together with the calculation of the zero-energy density of states [MATH].', 'cond-mat-0305301-1-6-15': 'This yields [MATH], as it should.', 'cond-mat-0305301-1-6-16': 'The shape of the Fermi surface (FS) and the direction of the applied field play an important role because the spectral functions [MATH] in Eq. ([REF]) have to be averaged over the corresponding FS where the velocity [MATH] denotes the component [MATH] perpendicular to [MATH].', 'cond-mat-0305301-1-6-17': 'For the spherical FS we have assumed for the [MATH]-band, [MATH], where [MATH][MATH][MATH].', 'cond-mat-0305301-1-6-18': 'In the limit [MATH] the function [MATH] approaches the BCS spectral function of the anomalous propagator.', 'cond-mat-0305301-1-6-19': 'We find that the results for the averages over the polar angle [MATH] do not differ significantly from the results obtained by setting [MATH].', 'cond-mat-0305301-1-6-20': 'This means that the quasiparticles moving perpendicular to the vortex axes yield the dominant contributions to [MATH] and [MATH].', 'cond-mat-0305301-1-6-21': 'Finally it should be pointed out that we have employed the Abrikosov ground state of the vortex lattice although, in particular at lower fields, a Landau-level expansion or a variational expression [CITATION] is needed to describe the distorted vortex lattice.', 'cond-mat-0305301-1-6-22': 'The results for [MATH] and [MATH] shown in Figs. (2) and (3) for the [MATH]- and [MATH]-bands should still be added by weighting them with the corresponding density of states.', 'cond-mat-0305301-1-6-23': 'One sees that qualitative fits of the measured field dependencies of the specific heat [CITATION] and the thermal conductivity [CITATION] for applied fields along the c axis can be obtained by adjusting the crucial parameters [MATH] for the interband pairing interaction and the ratio [MATH] of the coherence lengths.', 'cond-mat-0305301-1-7-0': 'In conclusion we can say that our two-band theory for the vortex state in MgB[MATH] can satisfactorily account for the observed field dependence of the specific heat coefficient [MATH] and the thermal conductivity [MATH].', 'cond-mat-0305301-1-7-1': 'The small gap [MATH] associated with the [MATH]-band decreases with increasing field [MATH] much faster than the large [MATH]-band gap [MATH] which shows conventional field dependence.', 'cond-mat-0305301-1-7-2': 'This gives rise to the rapid increase of [MATH] and [MATH] at small fields.', 'cond-mat-0305301-1-7-3': 'Due to a small interband pairing interaction [MATH], the gap [MATH] remains finite even in the field region where the large [MATH]-band vortex cores of radius [MATH] overlap.', 'cond-mat-0305301-1-7-4': 'This leads to smooth evolution of the [MATH]-band contributions to [MATH] and [MATH] to their normal state values near a "virtual" upper critical field [MATH] which is much smaller than [MATH] because the ratio [MATH] is much smaller than 1.', 'cond-mat-0305301-1-7-5': 'The Fermi surface topology and the impurity scattering have relatively small influence on the field dependence of [MATH] and [MATH] for fields applied along the c axis.', 'cond-mat-0305301-1-8-0': 'We would like to thank T. Dahm and K. Scharnberg for valuable discussions.', 'cond-mat-0305301-1-9-0': 'Nagamatsu, N. Nakagawa, T. Muranaka,Y. Zenitani, and J. Akimitsu, Nature 410, 63 (2001).', 'cond-mat-0305301-1-9-1': 'Bouquet, Y. Wang, I. Sheikin, T. Plackowski, A. Junod, S. Lee, and S. Tajima, Phys.', 'cond-mat-0305301-1-9-2': 'V. Sologubenko, J. Jun, S. M. Kazakov, J. Karpinski, and H. R. Ott, Phys.', 'cond-mat-0305301-1-9-3': 'Tewordt and D. Fay, Phys.', 'cond-mat-0305301-1-9-4': 'Rev. B (in press).', 'cond-mat-0305301-1-9-5': 'Schopohl and K. Scharnberg, Solid State Commun.', 'cond-mat-0305301-1-9-6': 'Y. Liu,I. I. Mazin, and J. Kortus, Phys.', 'cond-mat-0305301-1-9-7': 'R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M. Kazakov, J. Karpinski, and .', 'cond-mat-0305301-1-9-8': 'Brinkman et al, Phys.', 'cond-mat-0305301-1-9-9': 'Tewordt and D. Fay, Phys.'}

{'cond-mat-0305301-2-0-0': 'We solve the coupled gap equations for the [MATH]- and [MATH]-bands of MgB[MATH] in the vortex state and calculate the resulting field dependencies of the specific heat coefficient [MATH] and the thermal conductivity [MATH].', 'cond-mat-0305301-2-0-1': 'The crucial parameters of the theory are the interband pairing interaction [MATH] and the ratio [MATH] of the coherence lengths.', 'cond-mat-0305301-2-0-2': 'For reasonably small [MATH] and s, the small gap [MATH] decreases with increasing magnetic field [MATH] much faster than the large gap [MATH].', 'cond-mat-0305301-2-0-3': 'This gives rise to the observed rapid increase of [MATH] and [MATH] for small fields while [MATH] and [MATH] exhibit conventional field dependencies.', 'cond-mat-0305301-2-0-4': 'Inclusion of intraband impurity scattering yields fairly good agreement with experiments for applied fields along the c axis.', 'cond-mat-0305301-2-1-0': 'Evidence for the for the existence of two superconducting gaps in MgB[MATH] [CITATION] is provided by the rapid rise of the specific heat coefficient [MATH] [CITATION] and the thermal conductivity [MATH] [CITATION] at very low fields.', 'cond-mat-0305301-2-1-1': 'These measured field dependencies can be explained qualitatively by assuming two independent bands where the large s-wave pairing gap [MATH] is associated with the two-dimensional [MATH]-band and the small s-wave gap [MATH] is associated with the three-dimensional [MATH]-band.', 'cond-mat-0305301-2-1-2': '[CITATION] The steep rise of [MATH] and [MATH] can be explained qualitatively by assuming that the "virtual" upper critical field for [MATH] is much smaller than that of [MATH].', 'cond-mat-0305301-2-1-3': '[CITATION] In the present paper we improve the theory of Ref. [CITATION] by taking into account the interband pairing interaction while neglecting the interband impurity scattering [CITATION] which has been shown to be small.', 'cond-mat-0305301-2-1-4': '[CITATION] We first have to solve the two-band gap equations in the presence of the vortex lattices produced by a magnetic field.', 'cond-mat-0305301-2-1-5': 'Generalization of the linearized gap equations near the upper critical field [MATH] [CITATION] to all averaged fields [MATH] between [MATH] and [MATH] yields, instead of the single gap equation of Ref. [CITATION], the following coupled gap equations for the gaps [MATH] at [MATH]: [EQUATION]', 'cond-mat-0305301-2-1-6': 'Here, the [MATH] are the intra- and interband pairing interactions multiplied by the densities of states [MATH], and the [MATH] are the spectral functions of the anomalous propagators for the Abrikosov vortex lattice: [CITATION] [EQUATION]', 'cond-mat-0305301-2-1-7': 'The [MATH] are the normal state impurity scattering rates, the [MATH] are the field-dependent densities of states, the [MATH] are the Fermi velocities perperdicular to the field, and the function [MATH] is defined in Ref. [CITATION].', 'cond-mat-0305301-2-1-8': 'For band 2 (the [MATH]-band) we assume a spherical Fermi surface.', 'cond-mat-0305301-2-1-9': 'Then [MATH] is replaced by [MATH] and [MATH] and [MATH] are averaged over the polar angle [MATH] with respect to the direction of [MATH].', 'cond-mat-0305301-2-1-10': 'For brevity we omit here, and in the following, the terms containing [MATH] and the integrations over [MATH] from 0 to [MATH].', 'cond-mat-0305301-2-1-11': 'For band 1 (the [MATH]-band) and the field along the c axis, [MATH] and thus [MATH].', 'cond-mat-0305301-2-2-0': 'In the limit [MATH] the vortex lattice constant [MATH] tends to infinity.', 'cond-mat-0305301-2-2-1': 'Making use of the asymptotic expansion [MATH], the gap equations, Eq.([REF]), become [EQUATION] where the [MATH] are the gap values at [MATH] in zero field.', 'cond-mat-0305301-2-2-2': 'Here we have made use of the relation [EQUATION] which can be derived with the help of the expression for [MATH] in Eq.([REF]).', 'cond-mat-0305301-2-2-3': 'However, this relation only holds in the absence of interband impurity scattering.', 'cond-mat-0305301-2-2-4': '[CITATION]', 'cond-mat-0305301-2-3-0': 'With the help of the Abrikosov parameters, denoted by [MATH], we now express [MATH] in terms of the reduced field [MATH] and the zero-field gap [MATH]: [CITATION] [EQUATION]', 'cond-mat-0305301-2-3-1': 'Employing these relations we can express the gaps [MATH] and the scattering rates [MATH] in Eq.([REF]) by their ratios with respect to the [MATH], and we can convert the integrations over [MATH] in Eq.([REF]) to integrations over the new variables [MATH].', 'cond-mat-0305301-2-3-2': 'We then divide Eq.([REF]) by [MATH] and Eq.([REF]) by [MATH] and subtract the latter from the former.', 'cond-mat-0305301-2-3-3': 'In this way we obtain two coupled equations for the two unknown functions [MATH] and [MATH] for given values of the parameters [MATH], and [MATH].', 'cond-mat-0305301-2-3-4': 'The quantities [MATH], and [MATH] are defined by [EQUATION]', 'cond-mat-0305301-2-3-5': 'In Fig. (1) we show the reduced gap functions [MATH] and [MATH] for 3 sets of parameter values: I) [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH]; Ia) [MATH] as in set I, [MATH], [MATH], [MATH]; II) [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', 'cond-mat-0305301-2-3-6': 'The gap ratio has been taken to be [MATH] and the reduced impurity scattering rates [MATH] are [MATH] and [MATH].', 'cond-mat-0305301-2-3-7': 'The [MATH]-matrix elements in I) and II) have been obtained from band structure calculations (Refs. Liu and Goletal, see the review, Ref. MazinAnt).', 'cond-mat-0305301-2-3-8': 'The ratio of Fermi velocities is about [MATH], [CITATION] and the gap ratio [MATH] ranges between about 1/3 and 0.44 [CITATION] which yields a range of the ratio [MATH] of coherence lengths (see Eq.([REF])) between 0.18 and 0.24.', 'cond-mat-0305301-2-3-9': 'While the function [MATH] is rather independent of the choice of parameters, the function [MATH] depends sensitively on the values of [MATH] and [MATH].', 'cond-mat-0305301-2-3-10': 'For vanishing interband coupling [MATH] we obtain approximately [MATH] which goes to zero at a smaller effective upper critical field [MATH] where [MATH] (see Eq.([REF])).', 'cond-mat-0305301-2-3-11': 'Thus we see that [MATH] corresponds to the "virtual" upper critical field for the [MATH]-band which was introduced in Ref. Bouquet as the field above which the overlap of the vortex cores with large radius [MATH] (see the STS measurements of Ref. Esk) drives the majority of the [MATH]-band electrons normal.', 'cond-mat-0305301-2-4-0': 'In Fig.(2) we have plotted our results for the zero energy density of states [MATH] obtained from the expression for [MATH] (given by Eq.([REF]) with the numerator set equal to 1) by inserting the previously calculated gap ratios [MATH] together with Eqs.([REF]) for the functions [MATH].', 'cond-mat-0305301-2-4-1': '[MATH] is obtained by averaging [MATH] over the polar angle [MATH].', 'cond-mat-0305301-2-4-2': 'We note that it is important to calculate the impurity scattering rates [MATH] self-consistently.', 'cond-mat-0305301-2-4-3': 'One sees from Fig. (2) that [MATH] rises steeply for small fields [MATH] and then becomes almost constant above [MATH].', 'cond-mat-0305301-2-4-4': 'The slope at [MATH] and the downward curvature for low fields increase as [MATH] is decreased from 1/4 to 1/6.', 'cond-mat-0305301-2-4-5': 'The function [MATH] is very similar to the function obtained previously for a single band.', 'cond-mat-0305301-2-4-6': '[CITATION] The initial steep rise of [MATH] for [MATH] qualitatively fits the data points for the contribution of the [MATH]-band to the specific heat coefficient [MATH].', 'cond-mat-0305301-2-4-7': '[CITATION] The function [MATH] corresponds to the straight line assumed in Ref. Bouquet for the [MATH]-band contribution to [MATH] for fields applied along the c axis.', 'cond-mat-0305301-2-5-0': 'We turn now to the calculation of the in-plane electronic thermal conductivity [MATH] which is given at [MATH] by the expression in Ref. TF2.', 'cond-mat-0305301-2-5-1': 'Again it is important to take into account the renormalization of the gap by the function [MATH] in the presence of impurity scattering (see Eq.(13) of Ref. TF1).', 'cond-mat-0305301-2-5-2': 'By inserting the functions [MATH] obtained from the gap equations, Eq.([REF]), and the functions [MATH] from Eq.([REF]) into the expresions for the ratios [MATH] we obtain, for applied fields [MATH] along the c axis, the thermal conductivity ratios [MATH] shown in Fig.3.', 'cond-mat-0305301-2-5-3': 'It should be noted that the [MATH]-band contribution [MATH] has been obtained as an average over the polar angle [MATH] by including the factor [MATH] which arises from the square of the group velocity in the ab plane.', 'cond-mat-0305301-2-5-4': 'The curve for the [MATH]-band conductivity [MATH] turns out to be very similar to the curve obtained in Ref. [CITATION] for a single band with the same impurity scattering rate [MATH].', 'cond-mat-0305301-2-5-5': 'The curve [MATH] for the [MATH]-band contribution with [MATH] rises almost linearly with [MATH] where the slope near [MATH] and the downward curvature for low fields increase as [MATH] is decreased from 1/4 to 1/6.', 'cond-mat-0305301-2-5-6': 'For applied fields perpendicular to the c axis, the measured thermal conductivity [MATH] first rises steeply for small fields and then saturates while, for fields along the c axis, it exhibits an upward curvature towards [MATH].', 'cond-mat-0305301-2-5-7': '[CITATION] These different behaviors have been explained by separating the individual contributions of the [MATH]- and [MATH]-bands.', 'cond-mat-0305301-2-5-8': 'Then [MATH] rises steeply with [MATH] and approximately attains its normal-state value at a small field and [MATH] first rises very slowly and then curves upward towards [MATH].', 'cond-mat-0305301-2-5-9': '[CITATION] These experimental curves are similar to our results for [MATH] shown in Fig. (3).', 'cond-mat-0305301-2-6-0': 'We now briefly discuss the parameter values and approximations that have been used to derive our results.', 'cond-mat-0305301-2-6-1': 'We have seen that the rapid increase of the specific heat coefficient [MATH] and the thermal conductivity ratio [MATH] with increasing field [MATH] is due mainly to the [MATH]-band contribution.', 'cond-mat-0305301-2-6-2': 'The reason is that the gap [MATH] associated with the [MATH]-band almost closes at the so-called "virtual" upper critical field [CITATION] [MATH] because the ratio [MATH] of the coherence lengths of the [MATH]- and the [MATH]-bands is much smaller than 1.', 'cond-mat-0305301-2-6-3': 'For the most important parameters entering our gap equations, [MATH] and [MATH], we have used the values [MATH] and [MATH] 1/4, and 1/6 which are based on various experiments on MgB[MATH].', 'cond-mat-0305301-2-6-4': '[CITATION] The field dependence of the gap ratio [MATH], and thus of the contributions to [MATH] and [MATH] arising from the [MATH]-band, are nearly the same as those obtained for an independent single [MATH]-band, [CITATION] indicating that the effect of the interband coupling [MATH] is rather small.', 'cond-mat-0305301-2-6-5': 'However, the field dependence of [MATH] differs substantially from that for the independent single [MATH]-band with an effective upper critical field [MATH] as can be seen from Fig. (1).', 'cond-mat-0305301-2-6-6': 'This is because [MATH] is non-zero between [MATH] and [MATH] due to the effect of the interband coupling [MATH].', 'cond-mat-0305301-2-6-7': 'This shows that superconductivity survives even though the vortex cores for the [MATH]-band with giant radius [MATH] [CITATION] start to overlap for [MATH].', 'cond-mat-0305301-2-6-8': 'As can be seen in Fig. (1), the curve for [MATH] is very sensitive to the values of [MATH] and [MATH].', 'cond-mat-0305301-2-6-9': 'We find that the experimental contributions to [MATH] and [MATH] arising from the [MATH]-band [CITATION] can be fitted by taking the [MATH] given by band structure calculations.', 'cond-mat-0305301-2-6-10': '[CITATION] The other crucial parameter values needed to obtain good fits of the data are [MATH] and [MATH] which lie in the ranges obtained from experiment.', 'cond-mat-0305301-2-6-11': 'The other parameter values used in our numerical calculations are the reduced impurity scattering rates [MATH] and [MATH] which have been estimated from the relevant experiments.', 'cond-mat-0305301-2-6-12': 'It turns out that even for these moderately large impurity scattering rates it is very important to take into account the renormalization of the gap (see Eq. ([REF])) which leads to a large reduction of the effect of impurity scattering in comparison to that calculated without the function [MATH].', 'cond-mat-0305301-2-6-13': 'It is also important that the calculation of the scattering rate [MATH] in the Born limit be carried out self-consistently together with the calculation of the zero-energy density of states [MATH].', 'cond-mat-0305301-2-6-14': 'This yields [MATH], as it should.', 'cond-mat-0305301-2-6-15': 'The shape of the Fermi surface (FS) and the direction of the applied field play an important role because the spectral functions [MATH] in Eq. ([REF]) have to be averaged over the corresponding FS where the velocity [MATH] denotes the component [MATH] perpendicular to [MATH].', 'cond-mat-0305301-2-6-16': 'For the spherical FS we have assumed for the [MATH]-band, [MATH], where [MATH][MATH][MATH].', 'cond-mat-0305301-2-6-17': 'In the limit [MATH] the function [MATH] approaches the BCS spectral function of the anomalous propagator.', 'cond-mat-0305301-2-6-18': 'We find that the results for the averages over the polar angle [MATH] do not differ significantly from the results obtained by setting [MATH].', 'cond-mat-0305301-2-6-19': 'This means that the quasiparticles moving perpendicular to the vortex axes yield the dominant contributions to [MATH] and [MATH].', 'cond-mat-0305301-2-6-20': 'We have approximated the [MATH]-band FS by a sphere whereas the [MATH] have been calculated for the actual FS.', 'cond-mat-0305301-2-6-21': 'This actual FS can be modeled by a half-torus [CITATION] which yields, with the [MATH]-matrix of Ref. [CITATION] and small [MATH], results [CITATION] which agree qualitatively with ours shown in Figs. 1 and 2 for [MATH].', 'cond-mat-0305301-2-6-22': 'Finally it should be pointed out that we have employed the Abrikosov ground state of the vortex lattice although, in particular at lower fields, a Landau-level expansion or a variational expression [CITATION] is needed to describe the distorted vortex lattice.', 'cond-mat-0305301-2-6-23': 'The results for [MATH] and [MATH] shown in Figs. (2) and (3) for the [MATH]- and [MATH]-bands should still be added by weighting them with the corresponding density of states.', 'cond-mat-0305301-2-7-0': 'In conclusion we can say that our two-band theory for the vortex state in MgB[MATH] can satisfactorily account for the observed field dependence of the specific heat coefficient [MATH] and the thermal conductivity [MATH].', 'cond-mat-0305301-2-7-1': 'The small gap [MATH] associated with the [MATH]-band decreases with increasing field [MATH] much faster than the large [MATH]-band gap [MATH] which shows conventional field dependence.', 'cond-mat-0305301-2-7-2': 'This gives rise to the rapid increase of [MATH] and [MATH] at small fields.', 'cond-mat-0305301-2-7-3': 'Due to a small interband pairing interaction [MATH], the gap [MATH] remains finite even in the field region where the large [MATH]-band vortex cores of radius [MATH] overlap.', 'cond-mat-0305301-2-7-4': 'This leads to smooth evolution of the [MATH]-band contributions to [MATH] and [MATH] to their normal state values near a "virtual" upper critical field [MATH] which is much smaller than [MATH] because the ratio [MATH] is much smaller than 1.', 'cond-mat-0305301-2-7-5': 'The Fermi surface topology and the impurity scattering have relatively small influence on the field dependence of [MATH] and [MATH] for fields applied along the c axis.', 'cond-mat-0305301-2-8-0': 'Nagamatsu, N. Nakagawa, T. Muranaka,Y. Zenitani, and J. Akimitsu, Nature 410, 63 (2001).', 'cond-mat-0305301-2-8-1': 'Bouquet, Y. Wang, I. Sheikin, T. Plackowski, A. Junod, S. Lee, and S. Tajima, Phys.', 'cond-mat-0305301-2-8-2': 'V. Sologubenko, J. Jun, S. M. Kazakov, J. Karpinski, and H. R. Ott, Phys.', 'cond-mat-0305301-2-8-3': 'Tewordt and D. Fay, Phys.', 'cond-mat-0305301-2-8-4': 'Schopohl and K. Scharnberg, Solid State Commun.', 'cond-mat-0305301-2-8-5': 'I. Mazin et al., Phys.', 'cond-mat-0305301-2-8-6': 'Y. Liu, I. I. Mazin, and J. Kortus, Phys.', 'cond-mat-0305301-2-8-7': 'A. Golubov et al., J. Phys.', 'cond-mat-0305301-2-8-8': 'Brinkman et al, Phys.', 'cond-mat-0305301-2-8-9': 'R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M. Kazakov, J. Karpinski, and .', 'cond-mat-0305301-2-8-10': 'Iavarone et al., Phys.', 'cond-mat-0305301-2-8-11': 'Tewordt and D. Fay, Phys.', 'cond-mat-0305301-2-8-12': 'Dahm, S. Graser, and N. Schopohl, Physica C (Proceedings of the M2S-Rio, in press).'}

[['cond-mat-0305301-1-0-0', 'cond-mat-0305301-2-0-0'], ['cond-mat-0305301-1-0-1', 'cond-mat-0305301-2-0-1'], ['cond-mat-0305301-1-0-2', 'cond-mat-0305301-2-0-2'], ['cond-mat-0305301-1-0-3', 'cond-mat-0305301-2-0-3'], ['cond-mat-0305301-1-0-4', 'cond-mat-0305301-2-0-4'], ['cond-mat-0305301-1-7-0', 'cond-mat-0305301-2-7-0'], ['cond-mat-0305301-1-7-1', 'cond-mat-0305301-2-7-1'], ['cond-mat-0305301-1-7-2', 'cond-mat-0305301-2-7-2'], ['cond-mat-0305301-1-7-3', 'cond-mat-0305301-2-7-3'], ['cond-mat-0305301-1-7-4', 'cond-mat-0305301-2-7-4'], ['cond-mat-0305301-1-7-5', 'cond-mat-0305301-2-7-5'], ['cond-mat-0305301-1-5-0', 'cond-mat-0305301-2-5-0'], ['cond-mat-0305301-1-5-1', 'cond-mat-0305301-2-5-1'], ['cond-mat-0305301-1-5-2', 'cond-mat-0305301-2-5-2'], ['cond-mat-0305301-1-5-3', 'cond-mat-0305301-2-5-3'], ['cond-mat-0305301-1-5-4', 'cond-mat-0305301-2-5-4'], ['cond-mat-0305301-1-5-6', 'cond-mat-0305301-2-5-6'], ['cond-mat-0305301-1-5-7', 'cond-mat-0305301-2-5-7'], ['cond-mat-0305301-1-5-8', 'cond-mat-0305301-2-5-8'], ['cond-mat-0305301-1-5-9', 'cond-mat-0305301-2-5-9'], ['cond-mat-0305301-1-3-0', 'cond-mat-0305301-2-3-0'], ['cond-mat-0305301-1-3-1', 'cond-mat-0305301-2-3-1'], ['cond-mat-0305301-1-3-2', 'cond-mat-0305301-2-3-2'], ['cond-mat-0305301-1-3-3', 'cond-mat-0305301-2-3-3'], ['cond-mat-0305301-1-3-4', 'cond-mat-0305301-2-3-4'], ['cond-mat-0305301-1-3-12', 'cond-mat-0305301-2-3-11'], ['cond-mat-0305301-1-9-0', 'cond-mat-0305301-2-8-0'], ['cond-mat-0305301-1-9-1', 'cond-mat-0305301-2-8-1'], ['cond-mat-0305301-1-9-2', 'cond-mat-0305301-2-8-2'], ['cond-mat-0305301-1-9-3', 'cond-mat-0305301-2-8-3'], ['cond-mat-0305301-1-9-5', 'cond-mat-0305301-2-8-4'], ['cond-mat-0305301-1-9-7', 'cond-mat-0305301-2-8-9'], ['cond-mat-0305301-1-9-8', 'cond-mat-0305301-2-8-8'], ['cond-mat-0305301-1-9-9', 'cond-mat-0305301-2-8-11'], ['cond-mat-0305301-1-2-0', 'cond-mat-0305301-2-2-0'], ['cond-mat-0305301-1-2-1', 'cond-mat-0305301-2-2-1'], ['cond-mat-0305301-1-2-2', 'cond-mat-0305301-2-2-2'], ['cond-mat-0305301-1-2-3', 'cond-mat-0305301-2-2-3'], ['cond-mat-0305301-1-6-0', 'cond-mat-0305301-2-6-0'], ['cond-mat-0305301-1-6-1', 'cond-mat-0305301-2-6-1'], ['cond-mat-0305301-1-6-2', 'cond-mat-0305301-2-6-2'], ['cond-mat-0305301-1-6-4', 'cond-mat-0305301-2-6-4'], ['cond-mat-0305301-1-6-5', 'cond-mat-0305301-2-6-5'], ['cond-mat-0305301-1-6-6', 'cond-mat-0305301-2-6-6'], ['cond-mat-0305301-1-6-7', 'cond-mat-0305301-2-6-7'], ['cond-mat-0305301-1-6-8', 'cond-mat-0305301-2-6-8'], ['cond-mat-0305301-1-6-13', 'cond-mat-0305301-2-6-12'], ['cond-mat-0305301-1-6-14', 'cond-mat-0305301-2-6-13'], ['cond-mat-0305301-1-6-15', 'cond-mat-0305301-2-6-14'], ['cond-mat-0305301-1-6-16', 'cond-mat-0305301-2-6-15'], ['cond-mat-0305301-1-6-17', 'cond-mat-0305301-2-6-16'], ['cond-mat-0305301-1-6-18', 'cond-mat-0305301-2-6-17'], ['cond-mat-0305301-1-6-19', 'cond-mat-0305301-2-6-18'], ['cond-mat-0305301-1-6-20', 'cond-mat-0305301-2-6-19'], ['cond-mat-0305301-1-6-21', 'cond-mat-0305301-2-6-22'], ['cond-mat-0305301-1-6-22', 'cond-mat-0305301-2-6-23'], ['cond-mat-0305301-1-1-0', 'cond-mat-0305301-2-1-0'], ['cond-mat-0305301-1-1-1', 'cond-mat-0305301-2-1-1'], ['cond-mat-0305301-1-1-2', 'cond-mat-0305301-2-1-2'], ['cond-mat-0305301-1-1-4', 'cond-mat-0305301-2-1-4'], ['cond-mat-0305301-1-1-5', 'cond-mat-0305301-2-1-5'], ['cond-mat-0305301-1-1-6', 'cond-mat-0305301-2-1-6'], ['cond-mat-0305301-1-1-8', 'cond-mat-0305301-2-1-8'], ['cond-mat-0305301-1-1-9', 'cond-mat-0305301-2-1-9'], ['cond-mat-0305301-1-1-10', 'cond-mat-0305301-2-1-10'], ['cond-mat-0305301-1-1-11', 'cond-mat-0305301-2-1-11'], ['cond-mat-0305301-1-4-0', 'cond-mat-0305301-2-4-0'], ['cond-mat-0305301-1-4-1', 'cond-mat-0305301-2-4-1'], ['cond-mat-0305301-1-4-2', 'cond-mat-0305301-2-4-2'], ['cond-mat-0305301-1-4-3', 'cond-mat-0305301-2-4-3'], ['cond-mat-0305301-1-4-5', 'cond-mat-0305301-2-4-5'], ['cond-mat-0305301-1-4-7', 'cond-mat-0305301-2-4-7'], ['cond-mat-0305301-1-5-5', 'cond-mat-0305301-2-5-5'], ['cond-mat-0305301-1-3-9', 'cond-mat-0305301-2-3-9'], ['cond-mat-0305301-1-3-11', 'cond-mat-0305301-2-3-10'], ['cond-mat-0305301-1-9-6', 'cond-mat-0305301-2-8-6'], ['cond-mat-0305301-1-6-3', 'cond-mat-0305301-2-6-3'], ['cond-mat-0305301-1-6-12', 'cond-mat-0305301-2-6-11'], ['cond-mat-0305301-1-1-3', 'cond-mat-0305301-2-1-3'], ['cond-mat-0305301-1-4-4', 'cond-mat-0305301-2-4-4'], ['cond-mat-0305301-1-4-6', 'cond-mat-0305301-2-4-6'], ['cond-mat-0305301-1-6-9', 'cond-mat-0305301-2-6-9'], ['cond-mat-0305301-1-6-9', 'cond-mat-0305301-2-6-10'], ['cond-mat-0305301-1-6-10', 'cond-mat-0305301-2-6-10'], ['cond-mat-0305301-1-1-7', 'cond-mat-0305301-2-1-7']]

[['cond-mat-0305301-1-0-0', 'cond-mat-0305301-2-0-0'], ['cond-mat-0305301-1-0-1', 'cond-mat-0305301-2-0-1'], ['cond-mat-0305301-1-0-2', 'cond-mat-0305301-2-0-2'], ['cond-mat-0305301-1-0-3', 'cond-mat-0305301-2-0-3'], ['cond-mat-0305301-1-0-4', 'cond-mat-0305301-2-0-4'], ['cond-mat-0305301-1-7-0', 'cond-mat-0305301-2-7-0'], ['cond-mat-0305301-1-7-1', 'cond-mat-0305301-2-7-1'], ['cond-mat-0305301-1-7-2', 'cond-mat-0305301-2-7-2'], ['cond-mat-0305301-1-7-3', 'cond-mat-0305301-2-7-3'], ['cond-mat-0305301-1-7-4', 'cond-mat-0305301-2-7-4'], ['cond-mat-0305301-1-7-5', 'cond-mat-0305301-2-7-5'], ['cond-mat-0305301-1-5-0', 'cond-mat-0305301-2-5-0'], ['cond-mat-0305301-1-5-1', 'cond-mat-0305301-2-5-1'], ['cond-mat-0305301-1-5-2', 'cond-mat-0305301-2-5-2'], ['cond-mat-0305301-1-5-3', 'cond-mat-0305301-2-5-3'], ['cond-mat-0305301-1-5-4', 'cond-mat-0305301-2-5-4'], ['cond-mat-0305301-1-5-6', 'cond-mat-0305301-2-5-6'], ['cond-mat-0305301-1-5-7', 'cond-mat-0305301-2-5-7'], ['cond-mat-0305301-1-5-8', 'cond-mat-0305301-2-5-8'], ['cond-mat-0305301-1-5-9', 'cond-mat-0305301-2-5-9'], ['cond-mat-0305301-1-3-0', 'cond-mat-0305301-2-3-0'], ['cond-mat-0305301-1-3-1', 'cond-mat-0305301-2-3-1'], ['cond-mat-0305301-1-3-2', 'cond-mat-0305301-2-3-2'], ['cond-mat-0305301-1-3-3', 'cond-mat-0305301-2-3-3'], ['cond-mat-0305301-1-3-4', 'cond-mat-0305301-2-3-4'], ['cond-mat-0305301-1-3-12', 'cond-mat-0305301-2-3-11'], ['cond-mat-0305301-1-9-0', 'cond-mat-0305301-2-8-0'], ['cond-mat-0305301-1-9-1', 'cond-mat-0305301-2-8-1'], ['cond-mat-0305301-1-9-2', 'cond-mat-0305301-2-8-2'], ['cond-mat-0305301-1-9-3', 'cond-mat-0305301-2-8-3'], ['cond-mat-0305301-1-9-5', 'cond-mat-0305301-2-8-4'], ['cond-mat-0305301-1-9-7', 'cond-mat-0305301-2-8-9'], ['cond-mat-0305301-1-9-8', 'cond-mat-0305301-2-8-8'], ['cond-mat-0305301-1-9-9', 'cond-mat-0305301-2-8-11'], ['cond-mat-0305301-1-2-0', 'cond-mat-0305301-2-2-0'], ['cond-mat-0305301-1-2-1', 'cond-mat-0305301-2-2-1'], ['cond-mat-0305301-1-2-2', 'cond-mat-0305301-2-2-2'], ['cond-mat-0305301-1-2-3', 'cond-mat-0305301-2-2-3'], ['cond-mat-0305301-1-6-0', 'cond-mat-0305301-2-6-0'], ['cond-mat-0305301-1-6-1', 'cond-mat-0305301-2-6-1'], ['cond-mat-0305301-1-6-2', 'cond-mat-0305301-2-6-2'], ['cond-mat-0305301-1-6-4', 'cond-mat-0305301-2-6-4'], ['cond-mat-0305301-1-6-5', 'cond-mat-0305301-2-6-5'], ['cond-mat-0305301-1-6-6', 'cond-mat-0305301-2-6-6'], ['cond-mat-0305301-1-6-7', 'cond-mat-0305301-2-6-7'], ['cond-mat-0305301-1-6-8', 'cond-mat-0305301-2-6-8'], ['cond-mat-0305301-1-6-13', 'cond-mat-0305301-2-6-12'], ['cond-mat-0305301-1-6-14', 'cond-mat-0305301-2-6-13'], ['cond-mat-0305301-1-6-15', 'cond-mat-0305301-2-6-14'], ['cond-mat-0305301-1-6-16', 'cond-mat-0305301-2-6-15'], ['cond-mat-0305301-1-6-17', 'cond-mat-0305301-2-6-16'], ['cond-mat-0305301-1-6-18', 'cond-mat-0305301-2-6-17'], ['cond-mat-0305301-1-6-19', 'cond-mat-0305301-2-6-18'], ['cond-mat-0305301-1-6-20', 'cond-mat-0305301-2-6-19'], ['cond-mat-0305301-1-6-21', 'cond-mat-0305301-2-6-22'], ['cond-mat-0305301-1-6-22', 'cond-mat-0305301-2-6-23'], ['cond-mat-0305301-1-1-0', 'cond-mat-0305301-2-1-0'], ['cond-mat-0305301-1-1-1', 'cond-mat-0305301-2-1-1'], ['cond-mat-0305301-1-1-2', 'cond-mat-0305301-2-1-2'], ['cond-mat-0305301-1-1-4', 'cond-mat-0305301-2-1-4'], ['cond-mat-0305301-1-1-5', 'cond-mat-0305301-2-1-5'], ['cond-mat-0305301-1-1-6', 'cond-mat-0305301-2-1-6'], ['cond-mat-0305301-1-1-8', 'cond-mat-0305301-2-1-8'], ['cond-mat-0305301-1-1-9', 'cond-mat-0305301-2-1-9'], ['cond-mat-0305301-1-1-10', 'cond-mat-0305301-2-1-10'], ['cond-mat-0305301-1-1-11', 'cond-mat-0305301-2-1-11'], ['cond-mat-0305301-1-4-1', 'cond-mat-0305301-2-4-1'], ['cond-mat-0305301-1-4-2', 'cond-mat-0305301-2-4-2'], ['cond-mat-0305301-1-4-3', 'cond-mat-0305301-2-4-3'], ['cond-mat-0305301-1-4-5', 'cond-mat-0305301-2-4-5'], ['cond-mat-0305301-1-4-7', 'cond-mat-0305301-2-4-7']]

[['cond-mat-0305301-1-4-0', 'cond-mat-0305301-2-4-0'], ['cond-mat-0305301-1-5-5', 'cond-mat-0305301-2-5-5'], ['cond-mat-0305301-1-3-9', 'cond-mat-0305301-2-3-9'], ['cond-mat-0305301-1-3-11', 'cond-mat-0305301-2-3-10'], ['cond-mat-0305301-1-9-6', 'cond-mat-0305301-2-8-6'], ['cond-mat-0305301-1-6-3', 'cond-mat-0305301-2-6-3'], ['cond-mat-0305301-1-6-12', 'cond-mat-0305301-2-6-11'], ['cond-mat-0305301-1-1-3', 'cond-mat-0305301-2-1-3'], ['cond-mat-0305301-1-4-4', 'cond-mat-0305301-2-4-4'], ['cond-mat-0305301-1-4-6', 'cond-mat-0305301-2-4-6']]

[]

[['cond-mat-0305301-1-6-9', 'cond-mat-0305301-2-6-9'], ['cond-mat-0305301-1-6-9', 'cond-mat-0305301-2-6-10'], ['cond-mat-0305301-1-6-10', 'cond-mat-0305301-2-6-10'], ['cond-mat-0305301-1-1-7', 'cond-mat-0305301-2-1-7']]

[]

['cond-mat-0305301-1-2-4', 'cond-mat-0305301-1-3-5', 'cond-mat-0305301-2-2-4', 'cond-mat-0305301-2-3-5']

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

https://arxiv.org/abs/cond-mat/0305301

1511.02060

{'1511.02060-1-0-0': 'Current template-based gravitational wave searches for compact binary coalescences (CBC) use waveform models that neglect the higher order modes content of the gravitational radiation emitted, considering only the quadrupolar [MATH] modes.', '1511.02060-1-0-1': 'We study the effect of such a neglection for the case of aligned-spin CBC searches for equal-spin (and non-spinning) binary black holes in the context of two versions of Advanced LIGO: the upcoming 2015 version, known as early Advanced LIGO (eaLIGO) and its Zero-Detuned High Energy Power version, that we will refer to as Advanced LIGO (AdvLIGO).', '1511.02060-1-0-2': 'In addition, we study the case of a non-spinning search for initial LIGO (iLIGO).', '1511.02060-1-0-3': 'We do this via computing the effectualness of the aligned-spin SEOBNRv1 ROM waveform family, which only considers quadrupolar modes, towards hybrid post-Newtonian/Numerical Relativity waveforms which contain higher order modes.', '1511.02060-1-0-4': 'We find that for all LIGO versions, losses of more than [MATH] of events occur in the case of AdvLIGO for mass ratio [MATH] and total mass [MATH] due to the neglection of higher modes, this region of the parameter space being larger for eaLIGO and iLIGO.', '1511.02060-1-0-5': 'Moreover, while the maximum event loss observed over the explored parameter space for AdvLIGO is of [MATH] of events, for iLIGO and eaLIGO this increases up to [MATH].', '1511.02060-1-0-6': 'We find that neglection of higher modes leads to observation-averaged systematic parameter biases towards lower spin, total mass and chirp mass.', '1511.02060-1-0-7': 'For completeness, we perform a preliminar, non-exhaustive comparison of systematic biases to statistical errors.', '1511.02060-1-0-8': 'We find that, for a given SNR, systematic biases dominate over statistical errors at much lower total mass for eaLIGO than for AdvLIGO.', '1511.02060-1-1-0': '# Introduction', '1511.02060-1-2-0': 'Compact Binary Coalescences (CBC) are the most promising candidates for a first direct detection of gravitational waves (GW).', '1511.02060-1-2-1': 'Starting in September 2015, the next generation of GW detectors, Advanced LIGO [CITATION], Advanced Virgo[CITATION] and KAGRA[CITATION] will come online with and eventually reach sensitivities [MATH] times higher than the previous one, increasing by a factor of [MATH] the volume to which they are sensitive.', '1511.02060-1-2-2': 'This generates high expectations for imminent first GW detection[CITATION].', '1511.02060-1-2-3': "The core of searches for CBC's is the so called matched filter (MF)[CITATION].", '1511.02060-1-2-4': 'The MF technique allows for GW signals to be extracted from background noise provided that a correct model (waveform in our case) of the expected signal is used as a filter of the incoming signal.', '1511.02060-1-2-5': 'Otherwise the filter will be suboptimal and the GW signal could be lost or its parameters misidentified.', '1511.02060-1-2-6': "Current GW searches for CBC's implement template banks whose waveforms do only contain the quadrupolar [MATH] modes of the GW emission, known as quadrupolar waveforms.", '1511.02060-1-2-7': 'These neglect the higher order mode (HM) content of the incoming signal.', '1511.02060-1-2-8': 'This is justified by the fact that, in the non-precessing case, most of the power emitted by the source is carried by these two modes.', '1511.02060-1-3-0': 'The goal of this paper is to study the consequences of this neglection in current and future GW searches, both in terms of loss of detections and systematic biases caused in the estimation of the parameters (PE) of the source.', '1511.02060-1-3-1': 'We will focus on the case of aligned-spin template banks and non-precessing equal aligned-spin BBH within the mass range [MATH].', '1511.02060-1-3-2': "As target waveforms, we consider equal aligned-spin systems, and as bank waveforms we use the SEOBNRv1 ROM model [CITATION], which describes the quadrupolar modes of equal aligned spins CBC's using a single effective [MATH] spin parameter .", '1511.02060-1-3-3': 'In particular, we choose as targets four non-spinning systems with mass ratio [MATH] and four spinning cases: a [MATH] system with [MATH] and a [MATH] system with [MATH].', '1511.02060-1-3-4': 'The latter correspond to those systems available in the public NR SXS catalogue [CITATION] having equal spins for which the HM are the strongest and which lie within the parameter space covered by SEOBNRv1 ROM model.', '1511.02060-1-3-5': 'Also, in Appendix 1 we consider the case of a [MATH] system with unequal spins [MATH] and effective spin [MATH].', '1511.02060-1-3-6': 'These are summarized in Table. [REF].', '1511.02060-1-3-7': 'We will consider the case of a SEOBNRv1 ROM template bank including a single effective spin parameter [MATH] for the case of two Advanced LIGO predicted noise curves: the early version (eaLIGO) [CITATION] with a lower frequency cutoff [MATH]Hz and the design Zero-Detuned-High-Energy-Power version (AdvLIGO) [CITATION], with [MATH]Hz.', '1511.02060-1-3-8': 'Since no detections were made in initial LIGO (iLIGO) data, for which searches have been performed using a non-spinning template bank [CITATION], we will also pay attention to the corresponding sensitivity curve [CITATION], for which we will consider non-spinning targets, a non spinning template bank and [MATH]Hz.', '1511.02060-1-4-0': 'The case of non-spinning targets and a non-spinning template bank for the case of AdvLIGO has been widely studied.', '1511.02060-1-4-1': 'Pekowsky et.', '1511.02060-1-4-2': 'al., [CITATION] explored the mass range [MATH] and noted that the match between BBH NR waveforms including HM and the corresponding ones including only quadrupolar modes is [MATH] for most of the orientations of the binary.', '1511.02060-1-4-3': 'They also noticed that however, these orientations coincide with those for which the SNR is the lowest, mitigating the effect of HM when average over orientations is considered.', '1511.02060-1-4-4': 'More recently, Brown et.al., [CITATION] and Capano et.', '1511.02060-1-4-5': 'al., [CITATION] studied respectively the fitting factor (FF) [CITATION] of a non-spinning quadrupolar template bank towards non-spinning waveforms including HM for the total mass range [MATH] and [MATH].', '1511.02060-1-4-6': 'The result is that for total masses [MATH] and mass ratios [MATH] one does not expect event losses larger than [MATH], which is within the commonly accepted limit in GW searches.', '1511.02060-1-4-7': 'Furthermore, [CITATION] also computed the [MATH] [CITATION] and [MATH] [CITATION] of their target signals towards their non-spinning bank, simuating the effect of HM in a full search neglecting HM and estimated the false alarm rate (FAR) of a search including them.', '1511.02060-1-4-8': 'This allowed them to compare the sensitivity of both searches to signals including HM.', '1511.02060-1-4-9': 'They concluded that inclusion of HM in current template banks would only be advantageous for certain regions of the parameter space for which the FF of the bank towards their target signals were particularly low.', '1511.02060-1-4-10': 'In particular for [MATH] and [MATH].', '1511.02060-1-4-11': 'These event loss results widely agree with those presented by Varma et al., [CITATION], who also studied the systematic parameter bias caused by the neglection of HM and compared it with the statistical uncertainty due to the presence of Gaussian noise in the data stream.', '1511.02060-1-4-12': 'They concluded that the former dominate the latter for mass ratio [MATH] and total masses [MATH] for a SNR [MATH].', '1511.02060-1-4-13': 'This study was based on the Fisher information matrix formalism, which allowed them to study a large number of points in the parameter space.', '1511.02060-1-4-14': 'In contrast Littenberg et.', '1511.02060-1-4-15': 'al., [CITATION] studied the presence of systematic biases in the estimated parameters of the CBC but compared them against the expected statistical errors using Markov-Chain Monte-Carlo (MCMC) techniques.', '1511.02060-1-4-16': 'However, the large computational cost of this study restricted it to a few points of the parameter space.', '1511.02060-1-4-17': 'They obtained that, for binaries such that [MATH] and [MATH] and fixing inclination angle to [MATH], systematic errors introduced by the neglection of HM are smaller than the expected statistical errors at SNR [MATH].', '1511.02060-1-4-18': 'However, for larger masses ([MATH], [MATH]= 6), systematic biases will dominate statistical errors at SNR [MATH].', '1511.02060-1-4-19': 'Finally, during the preparation of this paper we became aware that Graff et.', '1511.02060-1-4-20': 'al., [CITATION] shown that higher modes are required for parameter estimation and detection of non-spinning high-mass binaries with an SNR [MATH].', '1511.02060-1-5-0': 'While the above summarized work has considered non-spinning searches and the design Zero-Detuned High Energy Power Advanced LIGO sensitivity curve (AdvLIGO) [CITATION], we extend their studies to the case of aligned spin searches [CITATION] for the early 2015 Advanced LIGO (eaLIGO) sensitivity curve and [CITATION].', '1511.02060-1-5-1': 'We also revisit the case of the initial LIGO (iLIGO) [CITATION] sensitivity curve using a non-spinning template bank and targets.', '1511.02060-1-5-2': 'There are various reasons that motivate these choices: the first is that it is expected that aligned-spin searches like [CITATION] will be implemented in the upcoming Advanced LIGO science runs.', '1511.02060-1-5-3': 'The extra degree of freedom that the spin parameter [MATH] provides could reduce the losses observed for non-spinning targets due to the neglection of HM when non-spinning template banks are considered and of course, we want to test what the effect for spinning systems is.', '1511.02060-1-5-4': 'Also, the different sensitivity curves considered and in particular their different frequency cutoff [MATH], will translate into very different event losses and parameter biases produced.', '1511.02060-1-5-5': 'The fact of including an effective spin parameter [MATH] in our template bank will lead to lower event losses for non-spinning targets than those found in [CITATION] and [CITATION], we will pay the price of important biases in the estimated spin.', '1511.02060-1-5-6': 'This extends the study of Veitch et al.,[CITATION] who concluded that the spin of non-spinning BBH (lacking HM) cannot be accurately measured using a single-effective spin parameter template bank.', '1511.02060-1-5-7': 'Finally, we will see that the value of the spin has a secondary effect in the impact of HM compared to that of the total mass and mass ratio.', '1511.02060-1-6-0': '# Higher Order Modes', '1511.02060-1-7-0': 'Consider a non-precessing CBC with total mass, mass ratio and effective spin collectively denoted by [MATH].', '1511.02060-1-7-1': 'Denoting by [MATH] the luminosity distance between source and detector, consider a frame of reference centered on the source and described by standard spherical coordinates [MATH] such that the [MATH] axis coincides with the total angular momentum of the binary.', '1511.02060-1-7-2': 'Then, the strain [MATH] produced by an emitted GW with effective polarization [MATH] [CITATION] at a given point [MATH] on its sky can be decomposed as a sum of modes [MATH] weighted by spin -2 weighted spherical harmonics[CITATION] [MATH] as: [EQUATION]', '1511.02060-1-8-0': 'where [MATH] and [MATH] denote the real and imaginary part operators, [MATH], [MATH] being real, and the factor [MATH] encodes the amplitude of the antenna pattern of the detector [CITATION].', '1511.02060-1-8-1': 'Fig. [REF] shows the amplitude of the most dominant modes for a non-spinning [MATH] binary.', '1511.02060-1-9-0': 'The effect that HM have in the observed signal depends on three main factors.', '1511.02060-1-9-1': 'First, regarding the source, post-Newtonian results yield that the larger the mass ratio, the larger the ratio [MATH] is [CITATION], as can be noticed in the top row of Fig. [REF], where [MATH] represents the frequency of the [MATH] mode, i.e., [MATH].', '1511.02060-1-9-2': 'Note how in the frequency range shown in these plots, the post-Newtonian amplitude of the [MATH] mode is about 1 order of magnitude larger than that of the next most dominant mode (typically the [MATH], when present) for all the sources shown.', '1511.02060-1-9-3': 'However, although this behaviour is qualitatively kept through the late inspiral and merger (at [MATH] in the bottom plots.)', '1511.02060-1-9-4': ', the NR amplitudes shown in the bottom row of Fig. [REF], show that this ratio can get up to [MATH] for the case of a [MATH] non-spinning system.', '1511.02060-1-9-5': 'As a general trend, the larger [MATH] is, the larger the contribution from HM will be.', '1511.02060-1-9-6': 'This will translate into larger event losses due to neglection of HM for larger [MATH].', '1511.02060-1-10-0': 'The effect of the spin is a bit more intricate since the contribution of the different modes as a function of the spin is mode-dependent.', '1511.02060-1-10-1': 'As an example, top and bottom rows of Fig. [REF] show respectively the PN and NR ratio between the amplitudes of the [MATH] and [MATH] modes wrt., that of the [MATH]: while the relative amplitude of the [MATH] mode grows as the spin gets more positive, the behavior of the [MATH] is the opposite in the PN case.', '1511.02060-1-10-2': 'Note however that the range of variation of the ratios shown in these plots is much tinier than that in Fig. [REF], which suggests that spin should have a sub-dominant effect compared to that of the mass ratio.', '1511.02060-1-11-0': 'The location of the detector on the sky of the source adds a second factor: the [MATH] spherical harmonic is weaker at close to edge-on orientations [MATH], where some higher ones have their maximums.', '1511.02060-1-11-1': 'This implies that signals from edge-on systems will have a larger HM content.', '1511.02060-1-12-0': 'Finally, as noted in [CITATION], there is a combined effect of the detector sensitivity curve and the total mass [MATH] of the CBC: the frequency of each mode roughly scales with the orbital frequency as [MATH] and as the total mass [MATH] increases, [MATH] falls off as [MATH].', '1511.02060-1-12-1': 'When the total mass [MATH] of the source is such that the frequency of the [MATH] mode is below the detector lower frequency cutoff ([MATH]), larger [MATH] modes will dominate the incoming signal in band.', '1511.02060-1-12-2': 'This will make the observed signal be very different from a quadrupolar waveform.', '1511.02060-1-12-3': 'In particular, the lower the seismic wall (the lower the frequency cutoff), the longer PN inspiral (strongly dominated by the [MATH] mode) the detector will be sensitive to.', '1511.02060-1-12-4': 'For this reason we decided to study both the cases of AdvLIGO with a [MATH] Hz frequency cutoff and eaLIGO and iLIGO with [MATH] Hz.', '1511.02060-1-12-5': 'As we will see, the different [MATH] generates notably different event losses.', '1511.02060-1-12-6': 'This effect is visualized in Fig. [REF], where the upper panels show the absolute value of the Fourier transform of the three most dominant modes of a [MATH] non-spinning binary for the cases of [MATH] and [MATH], and the bottom ones the corresponding whitened versions, [MATH], for both eaLIGO and AdvLIGO.', '1511.02060-1-12-7': 'Notice here how the larger flatness and lower frequency cutoff of AdvLIGO makes the [MATH] mode clearly dominate in all the plots shown (particularly at the sweet-spot of the noise curve), while for the case of eaLIGO contributions from HM get comparable to that of the [MATH] for high-mass cases.', '1511.02060-1-12-8': 'In order to estimate how important the contribution of HM will be as a function of the total mass and the detector curve, Fig. [REF] shows the value of the ratio [MATH], where [MATH], as a function of the total mass of the binary.', '1511.02060-1-12-9': 'Note how this ratio grows for the case of eaLIGO.', '1511.02060-1-13-0': '# Data Analysis', '1511.02060-1-14-0': 'Given two real waveforms, [MATH] and [MATH], and the one sided power spectral density curve [MATH] of a detector, the inner product [MATH] can be expressed as [EQUATION] [MATH] being the lower frequency cutoff of the considered noise curve.', '1511.02060-1-14-1': 'The overlap of [MATH] and [MATH] is then defined as [EQUATION]', '1511.02060-1-14-2': 'The signal-to-noise-ratio (SNR) of a signal [MATH] when filtered with a template [MATH] is then given by [EQUATION]', '1511.02060-1-14-3': 'An output signal [MATH] is in general a combination of a GW signal [MATH] and background noise [MATH].', '1511.02060-1-14-4': 'If one assumes the background noise to be Gaussian and with zero-mean, as we will do in this paper, the SNR is directly related to the probability that [MATH] is buried in [MATH] and to the distance at which it can be detected.', '1511.02060-1-14-5': 'Let us denote [MATH].', '1511.02060-1-14-6': 'We then define the match [MATH] as the overlap [MATH] maximized over relative time-shifts and the extrinsic parameters of [MATH], [MATH].', '1511.02060-1-14-7': 'The fitting factor (or effectualness) [MATH] of a bank [MATH] containing waveforms [MATH] with intrinsic parameters [MATH] to a waveform [MATH] is then defined as [CITATION] [EQUATION] and represents the fraction of SNR that the bank [MATH] can recover from the waveform [MATH] at the cost, in general, of a bias [MATH] in the estimation of the intrinsic parameters [MATH] of [MATH].', '1511.02060-1-14-8': 'This is, if [MATH] is the waveform of [MATH] which has the best overlap with [MATH], then in general, [MATH].', '1511.02060-1-15-0': '# Analysis set up', '1511.02060-1-16-0': 'We use as target signals hybrid PN/NR waveforms containing HM as built in [CITATION].', '1511.02060-1-16-1': 'The early inspiral part of the hybrids is built post-Newtonian data computed via the TaylorT1 approximant including 3.5 PN non-spinning [CITATION] and spin-orbit [CITATION] and 2PN spin-spin [CITATION] phase corrections.', '1511.02060-1-16-2': 'We include 3PN non-spinning amplitude corrections for the HM [CITATION] and 3.5PN for the 22 mode [CITATION].', '1511.02060-1-16-3': 'Spin corrections to the amplitudes are used up to 2PN [CITATION].', '1511.02060-1-16-4': 'The late inspiral and merger are described by NR waveforms extrapolated to null infinity to polynomial order [MATH].', '1511.02060-1-16-5': 'The latter have been obtained from the publicly available SXS catalogue [CITATION].', '1511.02060-1-16-6': 'The [MATH] mode of all target waveforms starts at 10Hz for [MATH].', '1511.02060-1-16-7': 'The cases [MATH] included the [MATH] modes while [MATH] cases included the [MATH] modes.', '1511.02060-1-17-0': 'For each hybrid waveform [MATH] in Table [REF] we construct all the signals [MATH] for all the values of [MATH] and [MATH] in Table. [REF]', '1511.02060-1-18-0': 'The described grid suffices for describing all the possible [MATH] since in the non-precessing case it holds [EQUATION]', '1511.02060-1-18-1': 'For the bank templates we use an equal-spin [MATH] reduced order model (ROM) [CITATION] of SEOBNRv1 [CITATION].', '1511.02060-1-18-2': 'The ROM is constructed in the frequency domain and agrees with SEOBNRv1 waveforms to a mismatch of [MATH] for low mass and [MATH] at high mass.', '1511.02060-1-18-3': 'The mismatch can reach [MATH] in isolated regions, for very high mass-ratios and/or high anti-aligned spins.', '1511.02060-1-18-4': 'This behavior is due to the undersampling of non-quasicircular coefficients in SEOBNRv1.', '1511.02060-1-18-5': 'Its range of validity in terms of spin is [MATH].', '1511.02060-1-19-0': 'For each target waveform [MATH] we compute [MATH], the corresponding recovered intrinsic parameters [MATH] and the optimal SNR [MATH].', '1511.02060-1-19-1': 'Maximization of the fitting factor over [MATH], is performed running several Nelder Mead Simplex algorithms as implemented in [CITATION].', '1511.02060-1-19-2': 'We let each of the runs start at different initial regions of the parameter space and the highest result is chosen as the true fitting factor [MATH].', '1511.02060-1-19-3': 'We then compute the fraction of the optimal and suboptimal volumes in which a system [MATH] with parameters [MATH] can be detected as [EQUATION] and the effective fitting factor as [MATH].', '1511.02060-1-19-4': 'The observation-averaged recovered parameters are computed as [EQUATION] and the corresponding averaged parameter bias as [EQUATION] where [MATH] are the recovered parameters for the case that the target waveform does only contain the [MATH] mode.', '1511.02060-1-19-5': 'This accounts for intrinsic biases of the template bank towards the quadrupolar modes of our targets and allows to isolate the effect of HM.', '1511.02060-1-19-6': 'We note that unlike studies like [CITATION], which quote the absolute value of the parameter bias, we prefer to keep track of its sign, as this can be then compared with a-priory estimates.', '1511.02060-1-19-7': 'For instance, since low mass systems have larger frequency content than large mass ones, we expect that the higher mode content of large mass systems will produce averaged-biases to lower masses.', '1511.02060-1-20-0': 'In order to asses the significance of these biases, we compare them to the corresponding statistical uncertainty that searches are affected by due to the presence of Gaussian noise in the data.', '1511.02060-1-20-1': 'For doing so, we employ the indistinguishability criterion for two waveforms [MATH] and [MATH] with mismatch [MATH] given by [CITATION] and used in [CITATION].', '1511.02060-1-20-2': 'Two waveforms are indistinguishable at a given SNR [MATH] if [MATH].', '1511.02060-1-20-3': 'We will thus consider that parameter estimation is not compromised due to systematic biases produced by the presence of HM in the target waveform if the best matching template [MATH] and the one best matching the injection with no HM [MATH] are insdistinguishable.', '1511.02060-1-20-4': 'We stress that this method does not provide a complete parameter estimation study, as, for instance, a bayesian MCMC study [CITATION], would do, but provides a fast first guess of the significance of the systematic parameter bias we find, which we get for free as a result of the fitting factor calculation.', '1511.02060-1-21-0': '# Effect on detection', '1511.02060-1-22-0': 'In general, as [MATH] and [MATH] increase, the larger contribution from HM to the target signal makes [MATH] decrease, which is expected from PN theory.', '1511.02060-1-22-1': 'For AdvLIGO losses do never reach [MATH] for any of the studied cases and [MATH] is reached for high mass [MATH] systems.', '1511.02060-1-22-2': 'In contrast, mainly due to their higher [MATH], for both eaLIGO (and iLIGO) losses reach values of [MATH]) for the highest [MATH] studied.', '1511.02060-1-22-3': 'Losses of [MATH] occur for all the targets with mass parameters [MATH], except for [MATH] (which are not shown in any plot due to the negligible losses found) and losses of [MATH] are present for iLIGO for [MATH], as can be seen in Fig. [REF].', '1511.02060-1-22-4': 'The fact that the seismic wall determines the different behavior of eaLIGO and AdvLIGO is clear from the fact that both detectors have similar losses up to masses of [MATH], when the [MATH] mode of the target waveform dominates the full signal content in the band of both detectors and can be well filtered by a bank that only contains quadrupolar modes.', '1511.02060-1-22-5': 'However, after that point, the [MATH] mode starts to get out of band for eaLIGO while it remains in for AdvLIGO.', '1511.02060-1-23-0': 'We note that our predicted losses for AdvLIGO are a a bit lower than those shown in [CITATION] due to the inclusion of the effective spin parameter [MATH] in our template waveforms.', '1511.02060-1-23-1': 'This provides an extra degree of freedom that can be exploited by quadrupolar waveforms to filter signals containing HM.', '1511.02060-1-23-2': 'This is also the main reason for the different results obtained for iLIGO and eaLIGO.', '1511.02060-1-24-0': 'Regarding the effect of spin, no [MATH] case reached even [MATH] losses.', '1511.02060-1-24-1': 'For the [MATH] case, losses are very similar to the ones for [MATH] (see Fig. [REF], right panel) which is consistent with the statement that spin should be secondary in terms of the impact of HM.', '1511.02060-1-24-2': 'Note however, how losses are a bit larger for the positive spin case than for the negative one for low mass.', '1511.02060-1-24-3': 'This could be however due to the fact that [MATH] lies in the limit of validity of the SEOBNRv1-ROM model.', '1511.02060-1-24-4': 'For high mass, results show that contributions from HM become equally important in terms of [MATH].', '1511.02060-1-24-5': 'Furthermore, the losses observed for [MATH] seem a good guess of those observed for the spinning cases, particularly for the highest masses.', '1511.02060-1-24-6': 'We note that it would have been interesting to study cases with spins closer to [MATH] and higher mass ratios.', '1511.02060-1-24-7': 'However, the only case with reasonably high spins and mass ratio available in the SXS catalogue was the [MATH] used here.', '1511.02060-1-25-0': '# Parameter Bias', '1511.02060-1-26-0': 'Due to its importance in GW data analysis, we will express results not as a function of [MATH] but rather consider the so called chirp mass parameter [MATH] and the total mass [MATH].', '1511.02060-1-26-1': 'Before discussing the averaged systematic errors measured due to the neglection of HM, we want to note that the intrinsic parameter bias [MATH] of the SEOBNRv1-ROM model towards our hybrids containing only the quadrupolar modes were never larger than [MATH] for all the total mass range, except for the [MATH] case, for which these reached maximum values of [MATH].', '1511.02060-1-27-0': 'The main effect of the HM is introducing large frequencies in the detector band, thus one should expect that the quadrupolar SEOBNRv1 waveform best matching a target waveform [MATH] with parameters [MATH] should have a larger frequency content than that corresponding to the quadrupolar template [MATH] having the intrinsic parameters [MATH] of the target.', '1511.02060-1-27-1': 'Intuitively, this can be achieved via introducing biases towards lower total mass and larger positive spin.', '1511.02060-1-27-2': 'Fig. [REF] shows the biases in total mass and spin obtained for all values of [MATH] (thus averaged over [MATH]) for a [MATH] non-spinning system for the cases of eaLIGO and AdvLIGO.', '1511.02060-1-27-3': 'Note that [MATH] corresponds to the center of the plot while its perimeter corresponds to [MATH].', '1511.02060-1-27-4': 'We see how the two different ways of increasing the template frequency (lowering mass and raising spin) compete along the different [MATH].', '1511.02060-1-27-5': 'As expected, the absolute value of the bias grows as [MATH] does.', '1511.02060-1-27-6': 'Also, the different interaction of the modes as a function of [MATH] generates a sort of dipolar pattern where biases vary from positive to negative.', '1511.02060-1-27-7': 'It is remarkable that while averaged biases shown in Fig. [REF] for the systems in Fig. [REF] are of [MATH] for eaLIGO and [MATH] for AdvLIGO, biases for particular edge-on orientations can be much larger, up to [MATH] for the case shown for eaLIGO and [MATH] for the one shown for AdvLIGO.', '1511.02060-1-27-8': 'Note also that even though the total mass chosen for the eaLIGO example is almost a half of that chosen for AdvLIGO, systematic biases are much lower for the latter case due to the lower [MATH] of AdvLIGO, which makes it much more sensitive to the long PN inspiral dominated by the quadrupolar modes.', '1511.02060-1-28-0': 'Fig. [REF] shows the averaged parameter bias over the observable volume, given by Eq. [REF], for the studied targets.', '1511.02060-1-28-1': 'As a general trend, neglection of HM causes observation-averaged biases towards lower ([MATH], [MATH], [MATH]) which increase as [MATH] and [MATH] do.', '1511.02060-1-28-2': 'As expected, biases are much larger for iLIGO and eaLIGO than for Adv.LIGO.', '1511.02060-1-28-3': 'In particular, note that the lower [MATH] of Adv.LIGO allows for an excellent recovery of [MATH] for most of the [MATH] range.', '1511.02060-1-28-4': 'This is due to the larger weight of the PN inspiral in the detector band.', '1511.02060-1-28-5': 'Regarding spinning cases, systematic biases are larger for negative spin cases than for positive spin ones.', '1511.02060-1-28-6': 'For [MATH] we only show the eaLIGO cases, which were the only ones having systematic biases comparable to those of the other cases.', '1511.02060-1-29-0': 'We now compare the observation-averaged biases to the statistical uncertainty we expect for each detector via computing the minimum SNR [MATH] at which PE would be dominated by the systematic biases.', '1511.02060-1-29-1': 'We note that, unlike the volume loss [MATH], the quantity [MATH] is extremely sensitive to tiny variations in the parameters recovered by the Nelder-Mead algorithm, which has the risk of settling in a local maximum.', '1511.02060-1-29-2': 'In particular, for an error [MATH] in the estimation of [MATH], one gets a variation for [MATH] of [MATH].', '1511.02060-1-29-3': 'This will specially affect regions of the parameter space where systematic biases are lower and where the parameter space is denser : so for low mass, large mass ratio, positive spin and AdvLIGO.', '1511.02060-1-29-4': 'Due to this, although we run up to 15 times some of the Nelder-Meads, Fig. [REF] shows several peaks that do only allow us to give a rough estimate of [MATH].', '1511.02060-1-29-5': 'Also, for the same reason, for AdvLIGO we only show results for [MATH].', '1511.02060-1-29-6': 'Results suggest that for AdvLIGO, PE at SNR [MATH] would be affected by HM for [MATH].', '1511.02060-1-29-7': 'However, for the case of eaLIGO, this limit gets reduced to [MATH] due to the larger systematic biases.', '1511.02060-1-30-0': '## On the usage of the SEOBNRv2 waveform model.', '1511.02060-1-31-0': 'We note that during this study, the SEOBNRv2 ROM waveform model [CITATION] became available.', '1511.02060-1-31-1': 'This model does not only supersede SEOBNRv1 ROM but also covers a wider spin range, namely [MATH] while [MATH].', '1511.02060-1-31-2': 'For this reason, we suspected that qualitatively, our results for the [MATH] case might be different when using SEOBNRv2 as quadrupolar model.', '1511.02060-1-31-3': 'As a sanity check, we re-computed the event loss and parameter bias using the SEOBNRv2-ROM family as quadrupolar template model for the cases of the non-spinning [MATH] target and for the [MATH], [MATH] one.', '1511.02060-1-31-4': 'Note that the latter is close to the limit of validity of SEOBNRv1-ROM but well inside the one of SEOBNRv2-ROM.', '1511.02060-1-31-5': 'Qualitatively, both models yielded the same trend in terms of event loss and parameter bias: larger losses as [MATH] and [MATH] increase and observation-averaged biases towards lower [MATH], [MATH] and [MATH].', '1511.02060-1-31-6': 'Quantitatively, both studies (using v1 and v2) yielded very similar results.', '1511.02060-1-31-7': 'The exception to this was the bias of the chirp mass, which differed by up to a [MATH] for both the [MATH] and the [MATH] cases when considering the eaLIGO noise curve.', '1511.02060-1-31-8': 'Also, SEOBNRv1 shown to be better at recovering the HM content of the spinning system.', '1511.02060-1-31-9': "Since none of the models are expected to model HM, we don't find any reason why we should expect the converse to happen.", '1511.02060-1-32-0': '# Conclusions', '1511.02060-1-33-0': 'In this paper we have studied the impact of the current neglection of HM in GW searches for binary black holes.', '1511.02060-1-33-1': 'We have extended previous studies, which focused in non-spinning searches, non-spinning target signals and AdvLIGO to the case of single-aligned spin searches and targets and to the case of AdvLIGO and the up coming eaLIGO.', '1511.02060-1-33-2': 'We have also considered the case of a non-spinning search and targets for the case of iLIGO.', '1511.02060-1-33-3': 'The main results of this study are the following.', '1511.02060-1-34-0': 'In more detail, we have shown that when an effective spin parameter is included in the template bank, neglection of HM in CBC searches is likely to generate losses [MATH] for the [MATH],[MATH] regions of the explored parameter space in the case of AdvLIGO.', '1511.02060-1-34-1': 'This region is tinier than that obtained in [CITATION] ([MATH]), due to the fact that they used a non-spinning template bank.', '1511.02060-1-34-2': 'However, for the case of eaLIGO (and a non-spinning search for iLIGO) we have found potential losses of up to [MATH] due to such a neglection.', '1511.02060-1-34-3': 'Losses of [MATH] happen for eaLIGO for the [MATH], [MATH] and [MATH] for [MATH].', '1511.02060-1-34-4': 'Furthermore, for the eaLIGO case, averaged systematic biases affecting parameter estimation are normally above [MATH] for the most part of the explored parameter space and reach values of [MATH] for the highest [MATH] cases.', '1511.02060-1-34-5': 'We compared the systematic biases to the corresponding statistical uncertainties.', '1511.02060-1-34-6': 'Results for eaLIGO suggest that measurements with SNR[MATH] would be affected by the presence of HM at [MATH] for the largest [MATH] considered.', '1511.02060-1-34-7': 'In the case of AdvLIGO, we estimate that PE is likely to be affected at [MATH] for [MATH] for the largest [MATH] studied.', '1511.02060-1-34-8': 'These value is larger than that obtained by [CITATION], however comparing the two results is intricate since they used non-spinning templates.', '1511.02060-1-35-0': 'The study of the FAR of a GW search including higher modes is out of the scope of this work.', '1511.02060-1-35-1': 'This is however is a crucial instrument for assessing the real significance of the losses we find and for assessing the need of such a search.', '1511.02060-1-35-2': 'Capano et al., [CITATION] demonstrated that the threshold SNR needed for claiming a trigger would have to be raised by roughly [MATH] due to the larger number of templates needed for such a search, which roughly means that the event losses of a search non-including HM w.r.t., a one including them would roughly be [MATH] of those obtained in this paper.', '1511.02060-1-35-3': 'Also, this paper has not considered the effect of signal-based vetoes as the [MATH] [CITATION], used in GW searches [CITATION] for discriminating real signals from background noise transients, known as glitches.', '1511.02060-1-35-4': 'This would especially punish signals for which we found poor fitting factors (which would be treated as glitches), leading to larger event losses.', '1511.02060-1-35-5': 'An obvious limitation of this work is the low number of spinning cases considered.', '1511.02060-1-35-6': 'This is due to the lack of public aligned spin NR waveforms with high HM content.', '1511.02060-1-35-7': 'We chose for this study an SXS case where HM were expected to be weak [MATH] and the one were HM where the strongest possible while having equal spins [MATH].', '1511.02060-1-35-8': 'We aim to extend this study to general unequal spin targets and unequal spin template bank.', '1511.02060-1-35-9': 'We end pointing that another interesting extension of this work would be to consider the case of precessing targets.'}

{'1511.02060-2-0-0': 'Current template-based gravitational wave searches for compact binary coalescences (CBC) use waveform models that neglect the higher order modes content of the gravitational radiation emitted, considering only the quadrupolar [MATH] modes.', '1511.02060-2-0-1': 'We study the effect of such a neglection for the case of aligned-spin CBC searches for equal-spin (and non-spinning) binary black holes in the context of two versions of Advanced LIGO: the upcoming 2015 version, known as early Advanced LIGO (eaLIGO) and its Zero-Detuned High Energy Power version, that we will refer to as Advanced LIGO (AdvLIGO).', '1511.02060-2-0-2': 'In addition, we study the case of a non-spinning search for initial LIGO (iLIGO).', '1511.02060-2-0-3': 'We do this via computing the effectualness of the aligned-spin SEOBNRv1 ROM waveform family, which only considers quadrupolar modes, towards hybrid post-Newtonian/Numerical Relativity waveforms which contain higher order modes.', '1511.02060-2-0-4': 'We find that for all LIGO versions, losses of more than [MATH] of events occur in the case of AdvLIGO for mass ratio [MATH] and total mass [MATH] due to the neglection of higher modes, this region of the parameter space being larger for eaLIGO and iLIGO.', '1511.02060-2-0-5': 'Moreover, while the maximum event loss observed over the explored parameter space for AdvLIGO is of [MATH] of events, for iLIGO and eaLIGO this increases up to [MATH].', '1511.02060-2-0-6': 'We find that neglection of higher modes leads to observation-averaged systematic parameter biases towards lower spin, total mass and chirp mass.', '1511.02060-2-0-7': 'For completeness, we perform a preliminar, non-exhaustive comparison of systematic biases to statistical errors.', '1511.02060-2-0-8': 'We find that, for a given SNR, systematic biases dominate over statistical errors at much lower total mass for eaLIGO than for AdvLIGO.', '1511.02060-2-1-0': '# Introduction', '1511.02060-2-2-0': 'Compact Binary Coalescences (CBC) are the most promising candidates for a first direct detection of gravitational waves (GW).', '1511.02060-2-2-1': 'Starting in September 2015, the next generation of GW detectors, Advanced LIGO [CITATION], Advanced Virgo[CITATION] and KAGRA[CITATION] will come online with and eventually reach sensitivities [MATH] times higher than the previous one, increasing by a factor of [MATH] the volume to which they are sensitive.', '1511.02060-2-2-2': 'This generates high expectations for imminent first GW detection[CITATION].', '1511.02060-2-2-3': "The core of searches for CBC's is the so called matched filter (MF)[CITATION].", '1511.02060-2-2-4': 'The MF technique allows for GW signals to be extracted from background noise provided that a correct model (waveform in our case) of the expected signal is used as a filter of the incoming signal.', '1511.02060-2-2-5': 'Otherwise the filter will be suboptimal and the GW signal could be lost or its parameters misidentified.', '1511.02060-2-2-6': "Current GW searches for CBC's implement template banks whose waveforms do only contain the quadrupolar [MATH] modes of the GW emission, known as quadrupolar waveforms.", '1511.02060-2-2-7': 'These neglect the higher order mode (HM) content of the incoming signal.', '1511.02060-2-2-8': 'This is justified by the fact that, in the non-precessing case, most of the power emitted by the source is carried by these two modes.', '1511.02060-2-3-0': 'The goal of this paper is to study the consequences of this neglection in current and future GW searches, both in terms of loss of detections and systematic biases caused in the estimation of the parameters (PE) of the source.', '1511.02060-2-3-1': 'We will focus on the case of aligned-spin template banks and non-precessing equal aligned-spin BBH within the mass range [MATH].', '1511.02060-2-3-2': "As target waveforms, we consider equal aligned-spin systems, and as bank waveforms we use the SEOBNRv1 ROM model [CITATION], which describes the quadrupolar modes of equal aligned spins CBC's using a single effective [MATH] spin parameter [CITATION].", '1511.02060-2-3-3': 'In particular, we choose as targets four non-spinning systems with mass ratio [MATH] and four spinning cases: a [MATH] system with [MATH] and a [MATH] system with [MATH].', '1511.02060-2-3-4': 'The latter correspond to those systems available in the public NR SXS catalogue [CITATION] having equal spins for which the HM are the strongest and which lie within the parameter space covered by SEOBNRv1 ROM model.', '1511.02060-2-3-5': 'Also, in Appendix 1 we consider the case of a [MATH] system with unequal spins [MATH] and effective spin [MATH].', '1511.02060-2-3-6': 'These are summarized in Table. [REF].', '1511.02060-2-3-7': 'We will consider the case of a SEOBNRv1 ROM template bank including a single effective spin parameter [MATH] for the case of two Advanced LIGO predicted noise curves: the early version (eaLIGO) [CITATION] with a lower frequency cutoff [MATH]Hz and the design Zero-Detuned-High-Energy-Power version (AdvLIGO) [CITATION], with [MATH]Hz.', '1511.02060-2-3-8': 'Since no detections were made in initial LIGO (iLIGO) data, for which searches have been performed using a non-spinning template bank [CITATION], we will also pay attention to the corresponding sensitivity curve [CITATION], for which we will consider non-spinning targets, a non spinning template bank and [MATH]Hz.', '1511.02060-2-4-0': 'The case of non-spinning targets and a non-spinning template bank for the case of AdvLIGO has been widely studied.', '1511.02060-2-4-1': 'Pekowsky et.', '1511.02060-2-4-2': 'al., [CITATION] explored the mass range [MATH] and noted that the match between BBH NR waveforms including HM and the corresponding ones including only quadrupolar modes is [MATH] for most of the orientations of the binary.', '1511.02060-2-4-3': 'They also noticed that however, these orientations coincide with those for which the SNR is the lowest, mitigating the effect of HM when average over orientations is considered.', '1511.02060-2-4-4': 'More recently, Brown et.al., [CITATION] and Capano et.', '1511.02060-2-4-5': 'al., [CITATION] studied respectively the fitting factor (FF) [CITATION] of a non-spinning quadrupolar template bank towards non-spinning waveforms including HM for the total mass range [MATH] and [MATH].', '1511.02060-2-4-6': 'The result is that for total masses [MATH] and mass ratios [MATH] one does not expect event losses larger than [MATH], which is within the commonly accepted limit in GW searches.', '1511.02060-2-4-7': 'Furthermore, [CITATION] also computed the [MATH] [CITATION] and [MATH] [CITATION] of their target signals towards their non-spinning bank, simuating the effect of HM in a full search neglecting HM and estimated the false alarm rate (FAR) of a search including them.', '1511.02060-2-4-8': 'This allowed them to compare the sensitivity of both searches to signals including HM.', '1511.02060-2-4-9': 'They concluded that inclusion of HM in current template banks would only be advantageous for certain regions of the parameter space for which the FF of the bank towards their target signals were particularly low.', '1511.02060-2-4-10': 'In particular for [MATH] and [MATH].', '1511.02060-2-4-11': 'These event loss results widely agree with those presented by Varma et al., [CITATION], who also studied the systematic parameter bias caused by the neglection of HM and compared it with the statistical uncertainty due to the presence of Gaussian noise in the data stream.', '1511.02060-2-4-12': 'They concluded that the former dominate the latter for mass ratio [MATH] and total masses [MATH] for a SNR [MATH].', '1511.02060-2-4-13': 'This study was based on the Fisher information matrix formalism, which allowed them to study a large number of points in the parameter space.', '1511.02060-2-4-14': 'In contrast Littenberg et.', '1511.02060-2-4-15': 'al., [CITATION] studied the presence of systematic biases in the estimated parameters of the CBC but compared them against the expected statistical errors using Markov-Chain Monte-Carlo (MCMC) techniques.', '1511.02060-2-4-16': 'However, the large computational cost of this study restricted it to a few points of the parameter space.', '1511.02060-2-4-17': 'They obtained that, for binaries such that [MATH] and [MATH] and fixing inclination angle to [MATH], systematic errors introduced by the neglection of HM are smaller than the expected statistical errors at SNR [MATH].', '1511.02060-2-4-18': 'However, for larger masses ([MATH], [MATH]= 6), systematic biases will dominate statistical errors at SNR [MATH].', '1511.02060-2-4-19': 'Finally, during the preparation of this paper we became aware that Graff et.', '1511.02060-2-4-20': 'al., [CITATION] shown that higher modes are required for parameter estimation and detection of non-spinning high-mass binaries with an SNR [MATH].', '1511.02060-2-5-0': 'While the above summarized work has considered non-spinning searches and the design Zero-Detuned High Energy Power Advanced LIGO sensitivity curve (AdvLIGO) [CITATION], we extend their studies to the case of aligned spin searches [CITATION] for the early 2015 Advanced LIGO (eaLIGO) sensitivity curve and [CITATION].', '1511.02060-2-5-1': 'We also revisit the case of the initial LIGO (iLIGO) [CITATION] sensitivity curve using a non-spinning template bank and targets.', '1511.02060-2-5-2': 'There are various reasons that motivate these choices: the first is that it is expected that aligned-spin searches like [CITATION] will be implemented in the upcoming Advanced LIGO science runs.', '1511.02060-2-5-3': 'The extra degree of freedom that the spin parameter [MATH] provides could reduce the losses observed for non-spinning targets due to the neglection of HM when non-spinning template banks are considered and of course, we want to test what the effect for spinning systems is.', '1511.02060-2-5-4': 'Also, the different sensitivity curves considered and in particular their different frequency cutoff [MATH], will translate into very different event losses and parameter biases produced.', '1511.02060-2-5-5': 'The fact of including an effective spin parameter [MATH] in our template bank will lead to lower event losses for non-spinning targets than those found in [CITATION] and [CITATION], we will pay the price of important biases in the estimated spin.', '1511.02060-2-5-6': 'This extends the study of Veitch et al.,[CITATION] who concluded that the spin of non-spinning BBH (lacking HM) cannot be accurately measured using a single-effective spin parameter template bank.', '1511.02060-2-5-7': 'Finally, we will see that the value of the spin has a secondary effect in the impact of HM compared to that of the total mass and mass ratio.', '1511.02060-2-6-0': '# Higher Order Modes', '1511.02060-2-7-0': 'Consider a non-precessing CBC with total mass, mass ratio and effective spin collectively denoted by [MATH].', '1511.02060-2-7-1': 'Denoting by [MATH] the luminosity distance between source and detector, consider a frame of reference centered on the source and described by standard spherical coordinates [MATH] such that the [MATH] axis coincides with the total angular momentum of the binary.', '1511.02060-2-7-2': 'Then, the strain [MATH] produced by an emitted GW with effective polarization [MATH] [CITATION] at a given point [MATH] on its sky can be decomposed as a sum of modes [MATH] weighted by spin -2 weighted spherical harmonics[CITATION] [MATH] as: [EQUATION]', '1511.02060-2-8-0': 'where [MATH] and [MATH] denote the real and imaginary part operators, [MATH], [MATH] being real, and the factor [MATH] encodes the amplitude of the antenna pattern of the detector [CITATION].', '1511.02060-2-8-1': 'Fig. [REF] shows the amplitude of the most dominant modes for a non-spinning [MATH] binary.', '1511.02060-2-9-0': 'The effect that HM have in the observed signal depends on three main factors.', '1511.02060-2-9-1': 'First, regarding the source, post-Newtonian results yield that the larger the mass ratio, the larger the ratio [MATH] is [CITATION], as can be noticed in the top row of Fig. [REF], where [MATH] represents the frequency of the [MATH] mode, i.e., [MATH].', '1511.02060-2-9-2': 'Note how in the frequency range shown in these plots, the post-Newtonian amplitude of the [MATH] mode is about 1 order of magnitude larger than that of the next most dominant mode (typically the [MATH], when present) for all the sources shown.', '1511.02060-2-9-3': 'However, although this behaviour is qualitatively kept through the late inspiral and merger (at [MATH] in the bottom plots.)', '1511.02060-2-9-4': ', the NR amplitudes shown in the bottom row of Fig. [REF], show that this ratio can get up to [MATH] for the case of a [MATH] non-spinning system.', '1511.02060-2-9-5': 'As a general trend, the larger [MATH] is, the larger the contribution from HM will be.', '1511.02060-2-9-6': 'This will translate into larger event losses due to neglection of HM for larger [MATH].', '1511.02060-2-10-0': 'The effect of the spin is a bit more intricate since the contribution of the different modes as a function of the spin is mode-dependent.', '1511.02060-2-10-1': 'As an example, top and bottom rows of Fig. [REF] show respectively the PN and NR ratio between the amplitudes of the [MATH] and [MATH] modes wrt., that of the [MATH]: while the relative amplitude of the [MATH] mode grows as the spin gets more positive, the behavior of the [MATH] is the opposite in the PN case.', '1511.02060-2-10-2': 'Note however that the range of variation of the ratios shown in these plots is much tinier than that in Fig. [REF], which suggests that spin should have a sub-dominant effect compared to that of the mass ratio.', '1511.02060-2-11-0': 'The location of the detector on the sky of the source adds a second factor: the [MATH] spherical harmonic is weaker at close to edge-on orientations [MATH], where some higher ones have their maximums.', '1511.02060-2-11-1': 'This implies that signals from edge-on systems will have a larger HM content.', '1511.02060-2-12-0': 'Finally, as noted in [CITATION], there is a combined effect of the detector sensitivity curve and the total mass [MATH] of the CBC: the frequency of each mode roughly scales with the orbital frequency as [MATH] and as the total mass [MATH] increases, [MATH] falls off as [MATH].', '1511.02060-2-12-1': 'When the total mass [MATH] of the source is such that the frequency of the [MATH] mode is below the detector lower frequency cutoff ([MATH]), larger [MATH] modes will dominate the incoming signal in band.', '1511.02060-2-12-2': 'This will make the observed signal be very different from a quadrupolar waveform.', '1511.02060-2-12-3': 'In particular, the lower the seismic wall (the lower the frequency cutoff), the longer PN inspiral (strongly dominated by the [MATH] mode) the detector will be sensitive to.', '1511.02060-2-12-4': 'For this reason we decided to study both the cases of AdvLIGO with a [MATH] Hz frequency cutoff and eaLIGO and iLIGO with [MATH] Hz.', '1511.02060-2-12-5': 'As we will see, the different [MATH] generates notably different event losses.', '1511.02060-2-12-6': 'This effect is visualized in Fig. [REF], where the upper panels show the absolute value of the Fourier transform of the three most dominant modes of a [MATH] non-spinning binary for the cases of [MATH] and [MATH], and the bottom ones the corresponding whitened versions, [MATH], for both eaLIGO and AdvLIGO.', '1511.02060-2-12-7': 'Notice here how the larger flatness and lower frequency cutoff of AdvLIGO makes the [MATH] mode clearly dominate in all the plots shown (particularly at the sweet-spot of the noise curve), while for the case of eaLIGO contributions from HM get comparable to that of the [MATH] for high-mass cases.', '1511.02060-2-12-8': 'In order to estimate how important the contribution of HM will be as a function of the total mass and the detector curve, Fig. [REF] shows the value of the ratio [MATH], where [MATH], as a function of the total mass of the binary.', '1511.02060-2-12-9': 'Note how this ratio grows for the case of eaLIGO.', '1511.02060-2-13-0': '# Data Analysis', '1511.02060-2-14-0': 'Given two real waveforms, [MATH] and [MATH], and the one sided power spectral density curve [MATH] of a detector, the inner product [MATH] can be expressed as [EQUATION] [MATH] being the lower frequency cutoff of the considered noise curve.', '1511.02060-2-14-1': 'The overlap of [MATH] and [MATH] is then defined as [EQUATION]', '1511.02060-2-14-2': 'The signal-to-noise-ratio (SNR) of a signal [MATH] when filtered with a template [MATH] is then given by [EQUATION]', '1511.02060-2-14-3': 'An output signal [MATH] is in general a combination of a GW signal [MATH] and background noise [MATH].', '1511.02060-2-14-4': 'If one assumes the background noise to be Gaussian and with zero-mean, as we will do in this paper, the SNR is directly related to the probability that [MATH] is buried in [MATH] and to the distance at which it can be detected.', '1511.02060-2-14-5': 'Let us denote [MATH].', '1511.02060-2-14-6': 'We then define the match [MATH] as the overlap [MATH] maximized over relative time-shifts and the extrinsic parameters of [MATH], [MATH].', '1511.02060-2-14-7': 'The fitting factor (or effectualness) [MATH] of a bank [MATH] containing waveforms [MATH] with intrinsic parameters [MATH] to a waveform [MATH] is then defined as [CITATION] [EQUATION] and represents the fraction of SNR that the bank [MATH] can recover from the waveform [MATH] at the cost, in general, of a bias [MATH] in the estimation of the intrinsic parameters [MATH] of [MATH].', '1511.02060-2-14-8': 'This is, if [MATH] is the waveform of [MATH] which has the best overlap with [MATH], then in general, [MATH].', '1511.02060-2-15-0': '# Analysis set up', '1511.02060-2-16-0': 'We use as target signals hybrid PN/NR waveforms containing HM as built in [CITATION].', '1511.02060-2-16-1': 'The early inspiral part of the hybrids is built post-Newtonian data computed via the TaylorT1 approximant including 3.5 PN non-spinning [CITATION] and spin-orbit [CITATION] and 2PN spin-spin [CITATION] phase corrections.', '1511.02060-2-16-2': 'We include 3PN non-spinning amplitude corrections for the HM [CITATION] and 3.5PN for the 22 mode [CITATION].', '1511.02060-2-16-3': 'Spin corrections to the amplitudes are used up to 2PN [CITATION].', '1511.02060-2-16-4': 'The late inspiral and merger are described by NR waveforms extrapolated to null infinity to polynomial order [MATH].', '1511.02060-2-16-5': 'The latter have been obtained from the publicly available SXS catalogue [CITATION].', '1511.02060-2-16-6': 'The [MATH] mode of all target waveforms starts at 10Hz for [MATH].', '1511.02060-2-16-7': 'The cases [MATH] included the [MATH] modes while [MATH] cases included the [MATH] modes.', '1511.02060-2-17-0': 'For each hybrid waveform [MATH] in Table [REF] we construct all the signals [MATH] for all the values of [MATH] and [MATH] in Table. [REF]', '1511.02060-2-18-0': 'The described grid suffices for describing all the possible [MATH] since in the non-precessing case it holds [EQUATION]', '1511.02060-2-18-1': 'For the bank templates we use an equal-spin [MATH] reduced order model (ROM) [CITATION] of SEOBNRv1 [CITATION].', '1511.02060-2-18-2': 'The ROM is constructed in the frequency domain and agrees with SEOBNRv1 waveforms to a mismatch of [MATH] for low mass and [MATH] at high mass.', '1511.02060-2-18-3': 'The mismatch can reach [MATH] in isolated regions, for very high mass-ratios and/or high anti-aligned spins.', '1511.02060-2-18-4': 'This behavior is due to the undersampling of non-quasicircular coefficients in SEOBNRv1.', '1511.02060-2-18-5': 'Its range of validity in terms of spin is [MATH].', '1511.02060-2-19-0': 'For each target waveform [MATH] we compute [MATH], the corresponding recovered intrinsic parameters [MATH] and the optimal SNR [MATH].', '1511.02060-2-19-1': 'Maximization of the fitting factor over [MATH], is performed running several Nelder Mead Simplex algorithms as implemented in [CITATION].', '1511.02060-2-19-2': 'We let each of the runs start at different initial regions of the parameter space and the highest result is chosen as the true fitting factor [MATH].', '1511.02060-2-19-3': 'We then compute the fraction of the optimal and suboptimal volumes in which a system [MATH] with parameters [MATH] can be detected as [EQUATION] and the effective fitting factor as [MATH].', '1511.02060-2-19-4': 'The observation-averaged recovered parameters are computed as [EQUATION] and the corresponding averaged parameter bias as [EQUATION] where [MATH] are the recovered parameters for the case that the target waveform does only contain the [MATH] mode.', '1511.02060-2-19-5': 'This accounts for intrinsic biases of the template bank towards the quadrupolar modes of our targets and allows to isolate the effect of HM.', '1511.02060-2-19-6': 'We note that unlike studies like [CITATION], which quote the absolute value of the parameter bias, we prefer to keep track of its sign, as this can be then compared with a-priory estimates.', '1511.02060-2-19-7': 'For instance, since low mass systems have larger frequency content than large mass ones, we expect that the higher mode content of large mass systems will produce averaged-biases to lower masses.', '1511.02060-2-20-0': 'In order to asses the significance of these biases, we compare them to the corresponding statistical uncertainty that searches are affected by due to the presence of Gaussian noise in the data.', '1511.02060-2-20-1': 'For doing so, we employ the indistinguishability criterion for two waveforms [MATH] and [MATH] with mismatch [MATH] given by [CITATION] and used in [CITATION].', '1511.02060-2-20-2': 'Two waveforms are indistinguishable at a given SNR [MATH] if [MATH].', '1511.02060-2-20-3': 'We will thus consider that parameter estimation is not compromised due to systematic biases produced by the presence of HM in the target waveform if the best matching template [MATH] and the one best matching the injection with no HM [MATH] are insdistinguishable.', '1511.02060-2-20-4': 'We stress that this method does not provide a complete parameter estimation study, as, for instance, a bayesian MCMC study [CITATION], would do, but provides a fast first guess of the significance of the systematic parameter bias we find, which we get for free as a result of the fitting factor calculation.', '1511.02060-2-21-0': '# Effect on detection', '1511.02060-2-22-0': 'In general, as [MATH] and [MATH] increase, the larger contribution from HM to the target signal makes [MATH] decrease, which is expected from PN theory.', '1511.02060-2-22-1': 'For AdvLIGO losses do never reach [MATH] for any of the studied cases and [MATH] is reached for high mass [MATH] systems.', '1511.02060-2-22-2': 'In contrast, mainly due to their higher [MATH], for both eaLIGO (and iLIGO) losses reach values of [MATH]) for the highest [MATH] studied.', '1511.02060-2-22-3': 'Losses of [MATH] occur for all the targets with mass parameters [MATH], except for [MATH] (which are not shown in any plot due to the negligible losses found) and losses of [MATH] are present for iLIGO for [MATH], as can be seen in Fig. [REF].', '1511.02060-2-22-4': 'The fact that the seismic wall determines the different behavior of eaLIGO and AdvLIGO is clear from the fact that both detectors have similar losses up to masses of [MATH], when the [MATH] mode of the target waveform dominates the full signal content in the band of both detectors and can be well filtered by a bank that only contains quadrupolar modes.', '1511.02060-2-22-5': 'However, after that point, the [MATH] mode starts to get out of band for eaLIGO while it remains in for AdvLIGO.', '1511.02060-2-23-0': 'We note that our predicted losses for AdvLIGO are a a bit lower than those shown in [CITATION] due to the inclusion of the effective spin parameter [MATH] in our template waveforms.', '1511.02060-2-23-1': 'This provides an extra degree of freedom that can be exploited by quadrupolar waveforms to filter signals containing HM.', '1511.02060-2-23-2': 'This is also the main reason for the different results obtained for iLIGO and eaLIGO.', '1511.02060-2-24-0': 'Regarding the effect of spin, no [MATH] case reached even [MATH] losses.', '1511.02060-2-24-1': 'For the [MATH] case, losses are very similar to the ones for [MATH] (see Fig. [REF], right panel) which is consistent with the statement that spin should be secondary in terms of the impact of HM.', '1511.02060-2-24-2': 'Note however, how losses are a bit larger for the positive spin case than for the negative one for low mass.', '1511.02060-2-24-3': 'This could be however due to the fact that [MATH] lies in the limit of validity of the SEOBNRv1-ROM model.', '1511.02060-2-24-4': 'For high mass, results show that contributions from HM become equally important in terms of [MATH].', '1511.02060-2-24-5': 'Furthermore, the losses observed for [MATH] seem a good guess of those observed for the spinning cases, particularly for the highest masses.', '1511.02060-2-24-6': 'We note that it would have been interesting to study cases with spins closer to [MATH] and higher mass ratios.', '1511.02060-2-24-7': 'However, the only case with reasonably high spins and mass ratio available in the SXS catalogue was the [MATH] used here.', '1511.02060-2-25-0': '# Parameter Bias', '1511.02060-2-26-0': 'Due to its importance in GW data analysis, we will express results not as a function of [MATH] but rather consider the so called chirp mass parameter [MATH] and the total mass [MATH].', '1511.02060-2-26-1': 'Before discussing the averaged systematic errors measured due to the neglection of HM, we want to note that the intrinsic parameter bias [MATH] of the SEOBNRv1-ROM model towards our hybrids containing only the quadrupolar modes were never larger than [MATH] for all the total mass range, except for the [MATH] case, for which these reached maximum values of [MATH].', '1511.02060-2-27-0': 'The main effect of the HM is introducing large frequencies in the detector band, thus one should expect that the quadrupolar SEOBNRv1 waveform best matching a target waveform [MATH] with parameters [MATH] should have a larger frequency content than that corresponding to the quadrupolar template [MATH] having the intrinsic parameters [MATH] of the target.', '1511.02060-2-27-1': 'Intuitively, this can be achieved via introducing biases towards lower total mass and larger positive spin.', '1511.02060-2-27-2': 'Fig. [REF] shows the biases in total mass and spin obtained for all values of [MATH] (thus averaged over [MATH]) for a [MATH] non-spinning system for the cases of eaLIGO and AdvLIGO.', '1511.02060-2-27-3': 'Note that [MATH] corresponds to the center of the plot while its perimeter corresponds to [MATH].', '1511.02060-2-27-4': 'We see how the two different ways of increasing the template frequency (lowering mass and raising spin) compete along the different [MATH].', '1511.02060-2-27-5': 'As expected, the absolute value of the bias grows as [MATH] does.', '1511.02060-2-27-6': 'Also, the different interaction of the modes as a function of [MATH] generates a sort of dipolar pattern where biases vary from positive to negative.', '1511.02060-2-27-7': 'It is remarkable that while averaged biases shown in Fig. [REF] for the systems in Fig. [REF] are of [MATH] for eaLIGO and [MATH] for AdvLIGO, biases for particular edge-on orientations can be much larger, up to [MATH] for the case shown for eaLIGO and [MATH] for the one shown for AdvLIGO.', '1511.02060-2-27-8': 'Note also that even though the total mass chosen for the eaLIGO example is almost a half of that chosen for AdvLIGO, systematic biases are much lower for the latter case due to the lower [MATH] of AdvLIGO, which makes it much more sensitive to the long PN inspiral dominated by the quadrupolar modes.', '1511.02060-2-28-0': 'Fig. [REF] shows the averaged parameter bias over the observable volume, given by Eq. [REF], for the studied targets.', '1511.02060-2-28-1': 'As a general trend, neglection of HM causes observation-averaged biases towards lower ([MATH], [MATH], [MATH]) which increase as [MATH] and [MATH] do.', '1511.02060-2-28-2': 'As expected, biases are much larger for iLIGO and eaLIGO than for Adv.LIGO.', '1511.02060-2-28-3': 'In particular, note that the lower [MATH] of Adv.LIGO allows for an excellent recovery of [MATH] for most of the [MATH] range.', '1511.02060-2-28-4': 'This is due to the larger weight of the PN inspiral in the detector band.', '1511.02060-2-28-5': 'Regarding spinning cases, systematic biases are larger for negative spin cases than for positive spin ones.', '1511.02060-2-28-6': 'For [MATH] we only show the eaLIGO cases, which were the only ones having systematic biases comparable to those of the other cases.', '1511.02060-2-29-0': 'We now compare the observation-averaged biases to the statistical uncertainty we expect for each detector via computing the minimum SNR [MATH] at which PE would be dominated by the systematic biases.', '1511.02060-2-29-1': 'We note that, unlike the volume loss [MATH], the quantity [MATH] is extremely sensitive to tiny variations in the parameters recovered by the Nelder-Mead algorithm, which has the risk of settling in a local maximum.', '1511.02060-2-29-2': 'In particular, for an error [MATH] in the estimation of [MATH], one gets a variation for [MATH] of [MATH].', '1511.02060-2-29-3': 'This will specially affect regions of the parameter space where systematic biases are lower and where the parameter space is denser : so for low mass, large mass ratio, positive spin and AdvLIGO.', '1511.02060-2-29-4': 'Due to this, although we run up to 15 times some of the Nelder-Meads, Fig. [REF] shows several peaks that do only allow us to give a rough estimate of [MATH].', '1511.02060-2-29-5': 'Also, for the same reason, for AdvLIGO we only show results for [MATH].', '1511.02060-2-29-6': 'Results suggest that for AdvLIGO, PE at SNR [MATH] would be affected by HM for [MATH].', '1511.02060-2-29-7': 'However, for the case of eaLIGO, this limit gets reduced to [MATH] due to the larger systematic biases.', '1511.02060-2-30-0': '## On the usage of the SEOBNRv2 waveform model.', '1511.02060-2-31-0': 'We note that during this study, the SEOBNRv2 ROM waveform model [CITATION] became available.', '1511.02060-2-31-1': 'This model does not only supersede SEOBNRv1 ROM but also covers a wider spin range, namely [MATH] while [MATH].', '1511.02060-2-31-2': 'For this reason, we suspected that qualitatively, our results for the [MATH] case might be different when using SEOBNRv2 as quadrupolar model.', '1511.02060-2-31-3': 'As a sanity check, we re-computed the event loss and parameter bias using the SEOBNRv2-ROM family as quadrupolar template model for the cases of the non-spinning [MATH] target and for the [MATH], [MATH] one.', '1511.02060-2-31-4': 'Note that the latter is close to the limit of validity of SEOBNRv1-ROM but well inside the one of SEOBNRv2-ROM.', '1511.02060-2-31-5': 'Qualitatively, both models yielded the same trend in terms of event loss and parameter bias: larger losses as [MATH] and [MATH] increase and observation-averaged biases towards lower [MATH], [MATH] and [MATH].', '1511.02060-2-31-6': 'Quantitatively, both studies (using v1 and v2) yielded very similar results.', '1511.02060-2-31-7': 'The exception to this was the bias of the chirp mass, which differed by up to a [MATH] for both the [MATH] and the [MATH] cases when considering the eaLIGO noise curve.', '1511.02060-2-31-8': 'Also, SEOBNRv1 shown to be better at recovering the HM content of the spinning system.', '1511.02060-2-31-9': "Since none of the models are expected to model HM, we don't find any reason why we should expect the converse to happen.", '1511.02060-2-32-0': '# Conclusions', '1511.02060-2-33-0': 'In this paper we have studied the impact of the current neglection of HM in GW searches for binary black holes.', '1511.02060-2-33-1': 'We have extended previous studies, which focused in non-spinning searches, non-spinning target signals and AdvLIGO to the case of single-aligned spin searches and targets and to the case of AdvLIGO and the upcoming eaLIGO.', '1511.02060-2-33-2': 'We have also considered the case of a non-spinning search and targets for the case of iLIGO.', '1511.02060-2-33-3': 'The main results of this study are the following.', '1511.02060-2-34-0': 'In more detail, we have shown that when an effective spin parameter is included in the template bank, neglection of HM in CBC searches is likely to generate losses [MATH] for the [MATH],[MATH] regions of the explored parameter space in the case of AdvLIGO.', '1511.02060-2-34-1': 'This region is tinier than that obtained in [CITATION] ([MATH]), due to the fact that they used a non-spinning template bank.', '1511.02060-2-34-2': 'However, for the case of eaLIGO (and a non-spinning search for iLIGO) we have found potential losses of up to [MATH] due to such a neglection.', '1511.02060-2-34-3': 'Losses of [MATH] happen for eaLIGO for the [MATH], [MATH] and [MATH] for [MATH].', '1511.02060-2-34-4': 'Furthermore, for the eaLIGO case, averaged systematic biases affecting parameter estimation are normally above [MATH] for the most part of the explored parameter space and reach values of [MATH] for the highest [MATH] cases.', '1511.02060-2-34-5': 'We compared the systematic biases to the corresponding statistical uncertainties.', '1511.02060-2-34-6': 'Results for eaLIGO suggest that measurements with SNR[MATH] would be affected by the presence of HM at [MATH] for the largest [MATH] considered.', '1511.02060-2-34-7': 'In the case of AdvLIGO, we estimate that PE is likely to be affected at [MATH] for [MATH] for the largest [MATH] studied.', '1511.02060-2-34-8': 'These value is larger than that obtained by [CITATION], however comparing the two results is intricate since they used non-spinning templates.', '1511.02060-2-35-0': 'The study of the FAR of a GW search including higher modes is out of the scope of this work.', '1511.02060-2-35-1': 'This is however is a crucial instrument for assessing the real significance of the losses we find and for assessing the need of such a search.', '1511.02060-2-35-2': 'Capano et al., [CITATION] demonstrated that the threshold SNR needed for claiming a trigger would have to be raised by roughly [MATH] due to the larger number of templates needed for such a search, which roughly means that the event losses of a search non-including HM w.r.t., a one including them would roughly be [MATH] of those obtained in this paper.', '1511.02060-2-35-3': 'Also, this paper has not considered the effect of signal-based vetoes as the [MATH] [CITATION], used in GW searches [CITATION] for discriminating real signals from background noise transients, known as glitches.', '1511.02060-2-35-4': 'This would especially punish signals for which we found poor fitting factors (which would be treated as glitches), leading to larger event losses.', '1511.02060-2-35-5': 'An obvious limitation of this work is the low number of spinning cases considered.', '1511.02060-2-35-6': 'This is due to the lack of public aligned spin NR waveforms with high HM content.', '1511.02060-2-35-7': 'We chose for this study an SXS case where HM were expected to be weak [MATH] and the one were HM where the strongest possible while having equal spins [MATH].', '1511.02060-2-35-8': 'We aim to extend this study to general unequal spin targets and unequal spin template bank.', '1511.02060-2-35-9': 'We end pointing that another interesting extension of this work would be to consider the case of precessing targets.'}

[['1511.02060-1-11-0', '1511.02060-2-11-0'], ['1511.02060-1-11-1', '1511.02060-2-11-1'], ['1511.02060-1-12-0', '1511.02060-2-12-0'], ['1511.02060-1-12-1', '1511.02060-2-12-1'], ['1511.02060-1-12-2', '1511.02060-2-12-2'], ['1511.02060-1-12-3', '1511.02060-2-12-3'], ['1511.02060-1-12-4', '1511.02060-2-12-4'], ['1511.02060-1-12-5', '1511.02060-2-12-5'], ['1511.02060-1-12-6', '1511.02060-2-12-6'], ['1511.02060-1-12-7', '1511.02060-2-12-7'], ['1511.02060-1-12-8', '1511.02060-2-12-8'], ['1511.02060-1-12-9', '1511.02060-2-12-9'], ['1511.02060-1-22-0', '1511.02060-2-22-0'], ['1511.02060-1-22-1', '1511.02060-2-22-1'], ['1511.02060-1-22-2', '1511.02060-2-22-2'], ['1511.02060-1-22-3', '1511.02060-2-22-3'], ['1511.02060-1-22-4', '1511.02060-2-22-4'], ['1511.02060-1-22-5', '1511.02060-2-22-5'], ['1511.02060-1-19-0', '1511.02060-2-19-0'], ['1511.02060-1-19-1', '1511.02060-2-19-1'], ['1511.02060-1-19-2', '1511.02060-2-19-2'], ['1511.02060-1-19-3', '1511.02060-2-19-3'], ['1511.02060-1-19-4', '1511.02060-2-19-4'], ['1511.02060-1-19-5', '1511.02060-2-19-5'], ['1511.02060-1-19-6', '1511.02060-2-19-6'], ['1511.02060-1-19-7', '1511.02060-2-19-7'], ['1511.02060-1-27-0', '1511.02060-2-27-0'], ['1511.02060-1-27-1', '1511.02060-2-27-1'], ['1511.02060-1-27-2', '1511.02060-2-27-2'], ['1511.02060-1-27-3', '1511.02060-2-27-3'], ['1511.02060-1-27-4', '1511.02060-2-27-4'], ['1511.02060-1-27-5', '1511.02060-2-27-5'], ['1511.02060-1-27-6', '1511.02060-2-27-6'], ['1511.02060-1-27-7', '1511.02060-2-27-7'], ['1511.02060-1-27-8', '1511.02060-2-27-8'], ['1511.02060-1-20-0', '1511.02060-2-20-0'], ['1511.02060-1-20-1', '1511.02060-2-20-1'], ['1511.02060-1-20-2', '1511.02060-2-20-2'], ['1511.02060-1-20-3', '1511.02060-2-20-3'], ['1511.02060-1-20-4', '1511.02060-2-20-4'], ['1511.02060-1-9-0', '1511.02060-2-9-0'], ['1511.02060-1-9-1', '1511.02060-2-9-1'], ['1511.02060-1-9-2', '1511.02060-2-9-2'], ['1511.02060-1-9-5', '1511.02060-2-9-5'], ['1511.02060-1-9-6', '1511.02060-2-9-6'], ['1511.02060-1-34-0', '1511.02060-2-34-0'], ['1511.02060-1-34-1', '1511.02060-2-34-1'], ['1511.02060-1-34-2', '1511.02060-2-34-2'], ['1511.02060-1-34-3', '1511.02060-2-34-3'], ['1511.02060-1-34-4', '1511.02060-2-34-4'], ['1511.02060-1-34-5', '1511.02060-2-34-5'], ['1511.02060-1-34-6', '1511.02060-2-34-6'], ['1511.02060-1-34-7', '1511.02060-2-34-7'], ['1511.02060-1-34-8', '1511.02060-2-34-8'], ['1511.02060-1-5-0', '1511.02060-2-5-0'], ['1511.02060-1-5-1', '1511.02060-2-5-1'], ['1511.02060-1-5-2', '1511.02060-2-5-2'], ['1511.02060-1-5-3', '1511.02060-2-5-3'], ['1511.02060-1-5-4', '1511.02060-2-5-4'], ['1511.02060-1-5-5', '1511.02060-2-5-5'], ['1511.02060-1-5-6', '1511.02060-2-5-6'], ['1511.02060-1-5-7', '1511.02060-2-5-7'], ['1511.02060-1-8-0', '1511.02060-2-8-0'], ['1511.02060-1-8-1', '1511.02060-2-8-1'], ['1511.02060-1-26-0', '1511.02060-2-26-0'], ['1511.02060-1-26-1', '1511.02060-2-26-1'], ['1511.02060-1-7-0', '1511.02060-2-7-0'], ['1511.02060-1-7-1', '1511.02060-2-7-1'], ['1511.02060-1-7-2', '1511.02060-2-7-2'], ['1511.02060-1-3-0', '1511.02060-2-3-0'], ['1511.02060-1-3-1', '1511.02060-2-3-1'], ['1511.02060-1-3-3', '1511.02060-2-3-3'], ['1511.02060-1-3-4', '1511.02060-2-3-4'], ['1511.02060-1-3-5', '1511.02060-2-3-5'], ['1511.02060-1-3-6', '1511.02060-2-3-6'], ['1511.02060-1-3-7', '1511.02060-2-3-7'], ['1511.02060-1-3-8', '1511.02060-2-3-8'], ['1511.02060-1-29-0', '1511.02060-2-29-0'], ['1511.02060-1-29-1', '1511.02060-2-29-1'], ['1511.02060-1-29-2', '1511.02060-2-29-2'], ['1511.02060-1-29-3', '1511.02060-2-29-3'], ['1511.02060-1-29-4', '1511.02060-2-29-4'], ['1511.02060-1-29-5', '1511.02060-2-29-5'], ['1511.02060-1-29-6', '1511.02060-2-29-6'], ['1511.02060-1-29-7', '1511.02060-2-29-7'], ['1511.02060-1-18-0', '1511.02060-2-18-0'], ['1511.02060-1-18-1', '1511.02060-2-18-1'], ['1511.02060-1-18-2', '1511.02060-2-18-2'], ['1511.02060-1-18-3', '1511.02060-2-18-3'], ['1511.02060-1-18-4', '1511.02060-2-18-4'], ['1511.02060-1-18-5', '1511.02060-2-18-5'], ['1511.02060-1-33-0', '1511.02060-2-33-0'], ['1511.02060-1-33-2', '1511.02060-2-33-2'], ['1511.02060-1-33-3', '1511.02060-2-33-3'], ['1511.02060-1-14-0', '1511.02060-2-14-0'], ['1511.02060-1-14-1', '1511.02060-2-14-1'], ['1511.02060-1-14-2', '1511.02060-2-14-2'], ['1511.02060-1-14-3', '1511.02060-2-14-3'], ['1511.02060-1-14-4', '1511.02060-2-14-4'], ['1511.02060-1-14-5', '1511.02060-2-14-5'], ['1511.02060-1-14-6', '1511.02060-2-14-6'], ['1511.02060-1-14-7', '1511.02060-2-14-7'], ['1511.02060-1-14-8', '1511.02060-2-14-8'], ['1511.02060-1-35-0', '1511.02060-2-35-0'], ['1511.02060-1-35-1', '1511.02060-2-35-1'], ['1511.02060-1-35-2', '1511.02060-2-35-2'], ['1511.02060-1-35-3', '1511.02060-2-35-3'], ['1511.02060-1-35-4', '1511.02060-2-35-4'], ['1511.02060-1-35-5', '1511.02060-2-35-5'], ['1511.02060-1-35-6', '1511.02060-2-35-6'], ['1511.02060-1-35-7', '1511.02060-2-35-7'], ['1511.02060-1-35-8', '1511.02060-2-35-8'], ['1511.02060-1-35-9', '1511.02060-2-35-9'], ['1511.02060-1-24-0', '1511.02060-2-24-0'], ['1511.02060-1-24-1', '1511.02060-2-24-1'], ['1511.02060-1-24-2', '1511.02060-2-24-2'], ['1511.02060-1-24-3', '1511.02060-2-24-3'], ['1511.02060-1-24-4', '1511.02060-2-24-4'], ['1511.02060-1-24-5', '1511.02060-2-24-5'], ['1511.02060-1-24-6', '1511.02060-2-24-6'], ['1511.02060-1-24-7', '1511.02060-2-24-7'], ['1511.02060-1-0-0', '1511.02060-2-0-0'], ['1511.02060-1-0-1', '1511.02060-2-0-1'], ['1511.02060-1-0-2', '1511.02060-2-0-2'], ['1511.02060-1-0-3', '1511.02060-2-0-3'], ['1511.02060-1-0-4', '1511.02060-2-0-4'], ['1511.02060-1-0-5', '1511.02060-2-0-5'], ['1511.02060-1-0-6', '1511.02060-2-0-6'], ['1511.02060-1-0-7', '1511.02060-2-0-7'], ['1511.02060-1-0-8', '1511.02060-2-0-8'], ['1511.02060-1-2-0', '1511.02060-2-2-0'], ['1511.02060-1-2-1', '1511.02060-2-2-1'], ['1511.02060-1-2-2', '1511.02060-2-2-2'], ['1511.02060-1-2-3', '1511.02060-2-2-3'], ['1511.02060-1-2-4', '1511.02060-2-2-4'], ['1511.02060-1-2-5', '1511.02060-2-2-5'], ['1511.02060-1-2-6', '1511.02060-2-2-6'], ['1511.02060-1-2-7', '1511.02060-2-2-7'], ['1511.02060-1-2-8', '1511.02060-2-2-8'], ['1511.02060-1-16-0', '1511.02060-2-16-0'], ['1511.02060-1-16-1', '1511.02060-2-16-1'], ['1511.02060-1-16-2', '1511.02060-2-16-2'], ['1511.02060-1-16-3', '1511.02060-2-16-3'], ['1511.02060-1-16-4', '1511.02060-2-16-4'], ['1511.02060-1-16-5', '1511.02060-2-16-5'], ['1511.02060-1-16-6', '1511.02060-2-16-6'], ['1511.02060-1-16-7', '1511.02060-2-16-7'], ['1511.02060-1-4-0', '1511.02060-2-4-0'], ['1511.02060-1-4-2', '1511.02060-2-4-2'], ['1511.02060-1-4-3', '1511.02060-2-4-3'], ['1511.02060-1-4-4', '1511.02060-2-4-4'], ['1511.02060-1-4-5', '1511.02060-2-4-5'], ['1511.02060-1-4-6', '1511.02060-2-4-6'], ['1511.02060-1-4-7', '1511.02060-2-4-7'], ['1511.02060-1-4-8', '1511.02060-2-4-8'], ['1511.02060-1-4-9', '1511.02060-2-4-9'], ['1511.02060-1-4-10', '1511.02060-2-4-10'], ['1511.02060-1-4-11', '1511.02060-2-4-11'], ['1511.02060-1-4-12', '1511.02060-2-4-12'], ['1511.02060-1-4-13', '1511.02060-2-4-13'], ['1511.02060-1-4-14', '1511.02060-2-4-14'], ['1511.02060-1-4-15', '1511.02060-2-4-15'], ['1511.02060-1-4-16', '1511.02060-2-4-16'], ['1511.02060-1-4-17', '1511.02060-2-4-17'], ['1511.02060-1-4-18', '1511.02060-2-4-18'], ['1511.02060-1-4-19', '1511.02060-2-4-19'], ['1511.02060-1-4-20', '1511.02060-2-4-20'], ['1511.02060-1-31-0', '1511.02060-2-31-0'], ['1511.02060-1-31-1', '1511.02060-2-31-1'], ['1511.02060-1-31-2', '1511.02060-2-31-2'], ['1511.02060-1-31-3', '1511.02060-2-31-3'], ['1511.02060-1-31-4', '1511.02060-2-31-4'], ['1511.02060-1-31-5', '1511.02060-2-31-5'], ['1511.02060-1-31-6', '1511.02060-2-31-6'], ['1511.02060-1-31-7', '1511.02060-2-31-7'], ['1511.02060-1-31-8', '1511.02060-2-31-8'], ['1511.02060-1-31-9', '1511.02060-2-31-9'], ['1511.02060-1-17-0', '1511.02060-2-17-0'], ['1511.02060-1-10-0', '1511.02060-2-10-0'], ['1511.02060-1-10-1', '1511.02060-2-10-1'], ['1511.02060-1-10-2', '1511.02060-2-10-2'], ['1511.02060-1-28-0', '1511.02060-2-28-0'], ['1511.02060-1-28-1', '1511.02060-2-28-1'], ['1511.02060-1-28-2', '1511.02060-2-28-2'], ['1511.02060-1-28-3', '1511.02060-2-28-3'], ['1511.02060-1-28-4', '1511.02060-2-28-4'], ['1511.02060-1-28-5', '1511.02060-2-28-5'], ['1511.02060-1-28-6', '1511.02060-2-28-6'], ['1511.02060-1-23-0', '1511.02060-2-23-0'], ['1511.02060-1-23-1', '1511.02060-2-23-1'], ['1511.02060-1-23-2', '1511.02060-2-23-2'], ['1511.02060-1-3-2', '1511.02060-2-3-2'], ['1511.02060-1-33-1', '1511.02060-2-33-1']]

[['1511.02060-1-11-0', '1511.02060-2-11-0'], ['1511.02060-1-11-1', '1511.02060-2-11-1'], ['1511.02060-1-12-0', '1511.02060-2-12-0'], ['1511.02060-1-12-1', '1511.02060-2-12-1'], ['1511.02060-1-12-2', '1511.02060-2-12-2'], ['1511.02060-1-12-3', '1511.02060-2-12-3'], ['1511.02060-1-12-4', '1511.02060-2-12-4'], ['1511.02060-1-12-5', '1511.02060-2-12-5'], ['1511.02060-1-12-6', '1511.02060-2-12-6'], ['1511.02060-1-12-7', '1511.02060-2-12-7'], ['1511.02060-1-12-8', '1511.02060-2-12-8'], ['1511.02060-1-12-9', '1511.02060-2-12-9'], ['1511.02060-1-22-0', '1511.02060-2-22-0'], ['1511.02060-1-22-1', '1511.02060-2-22-1'], ['1511.02060-1-22-2', '1511.02060-2-22-2'], ['1511.02060-1-22-3', '1511.02060-2-22-3'], ['1511.02060-1-22-4', '1511.02060-2-22-4'], ['1511.02060-1-22-5', '1511.02060-2-22-5'], ['1511.02060-1-19-0', '1511.02060-2-19-0'], ['1511.02060-1-19-1', '1511.02060-2-19-1'], ['1511.02060-1-19-2', '1511.02060-2-19-2'], ['1511.02060-1-19-3', '1511.02060-2-19-3'], ['1511.02060-1-19-4', '1511.02060-2-19-4'], ['1511.02060-1-19-5', '1511.02060-2-19-5'], ['1511.02060-1-19-6', '1511.02060-2-19-6'], ['1511.02060-1-19-7', '1511.02060-2-19-7'], ['1511.02060-1-27-0', '1511.02060-2-27-0'], ['1511.02060-1-27-1', '1511.02060-2-27-1'], ['1511.02060-1-27-2', '1511.02060-2-27-2'], ['1511.02060-1-27-3', '1511.02060-2-27-3'], ['1511.02060-1-27-4', '1511.02060-2-27-4'], ['1511.02060-1-27-5', '1511.02060-2-27-5'], ['1511.02060-1-27-6', '1511.02060-2-27-6'], ['1511.02060-1-27-7', '1511.02060-2-27-7'], ['1511.02060-1-27-8', '1511.02060-2-27-8'], ['1511.02060-1-20-0', '1511.02060-2-20-0'], ['1511.02060-1-20-1', '1511.02060-2-20-1'], ['1511.02060-1-20-2', '1511.02060-2-20-2'], ['1511.02060-1-20-3', '1511.02060-2-20-3'], ['1511.02060-1-20-4', '1511.02060-2-20-4'], ['1511.02060-1-9-0', '1511.02060-2-9-0'], ['1511.02060-1-9-1', '1511.02060-2-9-1'], ['1511.02060-1-9-2', '1511.02060-2-9-2'], ['1511.02060-1-9-5', '1511.02060-2-9-5'], ['1511.02060-1-9-6', '1511.02060-2-9-6'], ['1511.02060-1-34-0', '1511.02060-2-34-0'], ['1511.02060-1-34-1', '1511.02060-2-34-1'], ['1511.02060-1-34-2', '1511.02060-2-34-2'], ['1511.02060-1-34-3', '1511.02060-2-34-3'], ['1511.02060-1-34-4', '1511.02060-2-34-4'], ['1511.02060-1-34-5', '1511.02060-2-34-5'], ['1511.02060-1-34-6', '1511.02060-2-34-6'], ['1511.02060-1-34-7', '1511.02060-2-34-7'], ['1511.02060-1-34-8', '1511.02060-2-34-8'], ['1511.02060-1-5-0', '1511.02060-2-5-0'], ['1511.02060-1-5-1', '1511.02060-2-5-1'], ['1511.02060-1-5-2', '1511.02060-2-5-2'], ['1511.02060-1-5-3', '1511.02060-2-5-3'], ['1511.02060-1-5-4', '1511.02060-2-5-4'], ['1511.02060-1-5-5', '1511.02060-2-5-5'], ['1511.02060-1-5-6', '1511.02060-2-5-6'], ['1511.02060-1-5-7', '1511.02060-2-5-7'], ['1511.02060-1-8-0', '1511.02060-2-8-0'], ['1511.02060-1-8-1', '1511.02060-2-8-1'], ['1511.02060-1-26-0', '1511.02060-2-26-0'], ['1511.02060-1-26-1', '1511.02060-2-26-1'], ['1511.02060-1-7-0', '1511.02060-2-7-0'], ['1511.02060-1-7-1', '1511.02060-2-7-1'], ['1511.02060-1-7-2', '1511.02060-2-7-2'], ['1511.02060-1-3-0', '1511.02060-2-3-0'], ['1511.02060-1-3-1', '1511.02060-2-3-1'], ['1511.02060-1-3-3', '1511.02060-2-3-3'], ['1511.02060-1-3-4', '1511.02060-2-3-4'], ['1511.02060-1-3-5', '1511.02060-2-3-5'], ['1511.02060-1-3-6', '1511.02060-2-3-6'], ['1511.02060-1-3-7', '1511.02060-2-3-7'], ['1511.02060-1-3-8', '1511.02060-2-3-8'], ['1511.02060-1-29-0', '1511.02060-2-29-0'], ['1511.02060-1-29-1', '1511.02060-2-29-1'], ['1511.02060-1-29-2', '1511.02060-2-29-2'], ['1511.02060-1-29-3', '1511.02060-2-29-3'], ['1511.02060-1-29-4', '1511.02060-2-29-4'], ['1511.02060-1-29-5', '1511.02060-2-29-5'], ['1511.02060-1-29-6', '1511.02060-2-29-6'], ['1511.02060-1-29-7', '1511.02060-2-29-7'], ['1511.02060-1-18-0', '1511.02060-2-18-0'], ['1511.02060-1-18-1', '1511.02060-2-18-1'], ['1511.02060-1-18-2', '1511.02060-2-18-2'], ['1511.02060-1-18-3', '1511.02060-2-18-3'], ['1511.02060-1-18-4', '1511.02060-2-18-4'], ['1511.02060-1-18-5', '1511.02060-2-18-5'], ['1511.02060-1-33-0', '1511.02060-2-33-0'], ['1511.02060-1-33-2', '1511.02060-2-33-2'], ['1511.02060-1-33-3', '1511.02060-2-33-3'], ['1511.02060-1-14-0', '1511.02060-2-14-0'], ['1511.02060-1-14-1', '1511.02060-2-14-1'], ['1511.02060-1-14-2', '1511.02060-2-14-2'], ['1511.02060-1-14-3', '1511.02060-2-14-3'], ['1511.02060-1-14-4', '1511.02060-2-14-4'], ['1511.02060-1-14-5', '1511.02060-2-14-5'], ['1511.02060-1-14-6', '1511.02060-2-14-6'], ['1511.02060-1-14-7', '1511.02060-2-14-7'], ['1511.02060-1-14-8', '1511.02060-2-14-8'], ['1511.02060-1-35-0', '1511.02060-2-35-0'], ['1511.02060-1-35-1', '1511.02060-2-35-1'], ['1511.02060-1-35-2', '1511.02060-2-35-2'], ['1511.02060-1-35-3', '1511.02060-2-35-3'], ['1511.02060-1-35-4', '1511.02060-2-35-4'], ['1511.02060-1-35-5', '1511.02060-2-35-5'], ['1511.02060-1-35-6', '1511.02060-2-35-6'], ['1511.02060-1-35-7', '1511.02060-2-35-7'], ['1511.02060-1-35-8', '1511.02060-2-35-8'], ['1511.02060-1-35-9', '1511.02060-2-35-9'], ['1511.02060-1-24-0', '1511.02060-2-24-0'], ['1511.02060-1-24-1', '1511.02060-2-24-1'], ['1511.02060-1-24-2', '1511.02060-2-24-2'], ['1511.02060-1-24-3', '1511.02060-2-24-3'], ['1511.02060-1-24-4', '1511.02060-2-24-4'], ['1511.02060-1-24-5', '1511.02060-2-24-5'], ['1511.02060-1-24-6', '1511.02060-2-24-6'], ['1511.02060-1-24-7', '1511.02060-2-24-7'], ['1511.02060-1-0-0', '1511.02060-2-0-0'], ['1511.02060-1-0-1', '1511.02060-2-0-1'], ['1511.02060-1-0-2', '1511.02060-2-0-2'], ['1511.02060-1-0-3', '1511.02060-2-0-3'], ['1511.02060-1-0-4', '1511.02060-2-0-4'], ['1511.02060-1-0-5', '1511.02060-2-0-5'], ['1511.02060-1-0-6', '1511.02060-2-0-6'], ['1511.02060-1-0-7', '1511.02060-2-0-7'], ['1511.02060-1-0-8', '1511.02060-2-0-8'], ['1511.02060-1-2-0', '1511.02060-2-2-0'], ['1511.02060-1-2-1', '1511.02060-2-2-1'], ['1511.02060-1-2-2', '1511.02060-2-2-2'], ['1511.02060-1-2-3', '1511.02060-2-2-3'], ['1511.02060-1-2-4', '1511.02060-2-2-4'], ['1511.02060-1-2-5', '1511.02060-2-2-5'], ['1511.02060-1-2-6', '1511.02060-2-2-6'], ['1511.02060-1-2-7', '1511.02060-2-2-7'], ['1511.02060-1-2-8', '1511.02060-2-2-8'], ['1511.02060-1-16-0', '1511.02060-2-16-0'], ['1511.02060-1-16-1', '1511.02060-2-16-1'], ['1511.02060-1-16-2', '1511.02060-2-16-2'], ['1511.02060-1-16-3', '1511.02060-2-16-3'], ['1511.02060-1-16-4', '1511.02060-2-16-4'], ['1511.02060-1-16-5', '1511.02060-2-16-5'], ['1511.02060-1-16-6', '1511.02060-2-16-6'], ['1511.02060-1-16-7', '1511.02060-2-16-7'], ['1511.02060-1-4-0', '1511.02060-2-4-0'], ['1511.02060-1-4-2', '1511.02060-2-4-2'], ['1511.02060-1-4-3', '1511.02060-2-4-3'], ['1511.02060-1-4-4', '1511.02060-2-4-4'], ['1511.02060-1-4-5', '1511.02060-2-4-5'], ['1511.02060-1-4-6', '1511.02060-2-4-6'], ['1511.02060-1-4-7', '1511.02060-2-4-7'], ['1511.02060-1-4-8', '1511.02060-2-4-8'], ['1511.02060-1-4-9', '1511.02060-2-4-9'], ['1511.02060-1-4-10', '1511.02060-2-4-10'], ['1511.02060-1-4-11', '1511.02060-2-4-11'], ['1511.02060-1-4-12', '1511.02060-2-4-12'], ['1511.02060-1-4-13', '1511.02060-2-4-13'], ['1511.02060-1-4-14', '1511.02060-2-4-14'], ['1511.02060-1-4-15', '1511.02060-2-4-15'], ['1511.02060-1-4-16', '1511.02060-2-4-16'], ['1511.02060-1-4-17', '1511.02060-2-4-17'], ['1511.02060-1-4-18', '1511.02060-2-4-18'], ['1511.02060-1-4-19', '1511.02060-2-4-19'], ['1511.02060-1-4-20', '1511.02060-2-4-20'], ['1511.02060-1-31-0', '1511.02060-2-31-0'], ['1511.02060-1-31-1', '1511.02060-2-31-1'], ['1511.02060-1-31-2', '1511.02060-2-31-2'], ['1511.02060-1-31-3', '1511.02060-2-31-3'], ['1511.02060-1-31-4', '1511.02060-2-31-4'], ['1511.02060-1-31-5', '1511.02060-2-31-5'], ['1511.02060-1-31-6', '1511.02060-2-31-6'], ['1511.02060-1-31-7', '1511.02060-2-31-7'], ['1511.02060-1-31-8', '1511.02060-2-31-8'], ['1511.02060-1-31-9', '1511.02060-2-31-9'], ['1511.02060-1-17-0', '1511.02060-2-17-0'], ['1511.02060-1-10-0', '1511.02060-2-10-0'], ['1511.02060-1-10-1', '1511.02060-2-10-1'], ['1511.02060-1-10-2', '1511.02060-2-10-2'], ['1511.02060-1-28-0', '1511.02060-2-28-0'], ['1511.02060-1-28-1', '1511.02060-2-28-1'], ['1511.02060-1-28-2', '1511.02060-2-28-2'], ['1511.02060-1-28-3', '1511.02060-2-28-3'], ['1511.02060-1-28-4', '1511.02060-2-28-4'], ['1511.02060-1-28-5', '1511.02060-2-28-5'], ['1511.02060-1-28-6', '1511.02060-2-28-6'], ['1511.02060-1-23-0', '1511.02060-2-23-0'], ['1511.02060-1-23-1', '1511.02060-2-23-1'], ['1511.02060-1-23-2', '1511.02060-2-23-2']]

[['1511.02060-1-3-2', '1511.02060-2-3-2'], ['1511.02060-1-33-1', '1511.02060-2-33-1']]

[]

[]

[]

['1511.02060-1-4-1', '1511.02060-1-9-3', '1511.02060-1-9-4', '1511.02060-2-4-1', '1511.02060-2-9-3', '1511.02060-2-9-4']

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

https://arxiv.org/abs/1511.02060

1704.00339

{'1704.00339-1-0-0': 'Weak antilocalization (WAL) effects in Bi[MATH]Te[MATH] single crystals have been investigated at high and low bulk charge carrier concentrations.', '1704.00339-1-0-1': 'At low charge carrier density the WAL curves scale with the normal component of the magnetic field, demonstrating the dominance of topological surface states in magnetoconductivity.', '1704.00339-1-0-2': 'At high charge carrier density the WAL curves scale with neither the applied field nor its normal component, implying a mixture of bulk and surface conduction.', '1704.00339-1-0-3': 'WAL due to topological surface states shows no dependence on the nature (electrons or holes) of the bulk charge carriers.', '1704.00339-1-0-4': 'The observations of an extremely large, non-saturating magnetoresistance, and ultrahigh mobility in the samples with lower carrier density further support the presence of surface states.', '1704.00339-1-0-5': 'The physical parameters characterizing the WAL effects are calculated using the Hikami-Larkin-Nagaoka formula.', '1704.00339-1-0-6': 'At high charge carrier concentrations, there is a greater number of conduction channels and a decrease in the phase coherence length compared to low charge carrier concentrations.', '1704.00339-1-0-7': 'The extremely large magnetoresistance and high mobility of topological insulators have great technological value and can be exploited in magneto-electric sensors and memory devices.', '1704.00339-1-1-0': '# Introduction', '1704.00339-1-2-0': 'The quantum interference of electrons in solids experiencing a backscattering event, which limits the electrical conductivity, has been of interest for many years.', '1704.00339-1-2-1': 'The resistance in systems with scattering centers increases with decreasing temperature due to increasing phase coherence in backscattering processes where the electron moves through a loop of scattering sites clockwise, as well as counter-clockwise, and the constructive superposition of the scattering amplitudes results in an enhancement of the resistivity.', '1704.00339-1-2-2': 'The phase coherence in this additive scattering process, coined weak localization (WL), is destroyed in an external magnetic field, resulting in a characteristic negative magnetoresistance[CITATION].', '1704.00339-1-2-3': 'On the other hand, in systems with strong spin-orbit interaction, the spin of the electron is tied to the momentum.', '1704.00339-1-2-4': 'This results in a rotation of the spins along the backscattering path and the phase difference between clockwise and counter-clockwise trajectories amounts to 360[MATH].', '1704.00339-1-2-5': 'Rotating a spin by 360[MATH] changes the sign of the scattering amplitude and the two scattering events experience a destructive interference, reducing the resistance in materials with strong spin-orbit coupling [weak antilocalization (WAL)].', '1704.00339-1-2-6': 'The destructive interference effects are diminished in magnetic fields resulting in a positive magnetoresistance, as observed in many metallic compounds with large spin-orbit interaction[CITATION].', '1704.00339-1-3-0': 'WL and WAL are more pronounced in two-dimensional (2D) systems since the probability of two interfering scattering pathways, distinguished only by time reversal, is larger in lower dimensions.', '1704.00339-1-3-1': 'Therefore, surface states of topological insulators (TIs) have been studied recently with respect to the WAL effects.', '1704.00339-1-3-2': 'The surface conduction can be easily studied when the bulk of the TI is insulating and does not contribute to the conductivity.', '1704.00339-1-3-3': 'In many TIs the study of transport from topological surface states is impeded by a significant bulk contribution to the overall conduction due to defects, impurities, or intersite occupancy.', '1704.00339-1-3-4': 'This makes it difficult to extract the topological properties of the surface states by employing magnetotransport methods.', '1704.00339-1-3-5': 'However, in a recent investigation of metallic, hole-doped Bi[MATH]Se[MATH]Te[MATH] and Sb[MATH]Te[MATH]Se, we have shown that quantum oscillations from topological surface states can be resolved well despite the interference with bulk conduction[CITATION][MATH][CITATION][MATH][CITATION].', '1704.00339-1-3-6': 'This motivated us to search for other physical phenomena related to topological surface states in TIs with bulk metallic properties.', '1704.00339-1-4-0': 'Bismuth telluride Bi[MATH]Te[MATH] is shown to be a three dimensional topological insulator by both theoretical and experimental studies[CITATION][MATH][CITATION][CITATION].', '1704.00339-1-4-1': 'Recently Qu [MATH][CITATION] observed that every single crystal of Bi[MATH]Te[MATH] can have different physical properties such as bulk carrier concentrations, carrier types, etc. even if they are grown in the same batch.', '1704.00339-1-4-2': 'They reported that this might be due to a weak or critical composition gradient during the crystal growth process.', '1704.00339-1-4-3': 'In this work, we report the WAL effect in Bi[MATH]Te[MATH] single crystals with metallic bulk conductivity and different carrier densities with hole as well as electron nature.', '1704.00339-1-4-4': 'Three single crystals, S1, S2, and S3, are selected from the same boule of crystals.', '1704.00339-1-4-5': 'People have used either angle dependence of quantum oscillations or WAL curves to identify the presence of topological surface states[CITATION][MATH][CITATION].', '1704.00339-1-4-6': 'In our study, even though samples S1 and S2 show quantum oscillations under fields above 7 T, S3 did not show any signature of quantum oscillations in the field up to 31 T. However, all of those samples show WAL effect at low temperature.', '1704.00339-1-4-7': 'That is why we have used angle dependence of WAL curves to detect the presence of topological surface states in all samples.', '1704.00339-1-4-8': 'For two crystals with lower bulk carrier density (S1, electron-, and S3, hole-doped) we find the clear signature of WAL from 2D surface states.', '1704.00339-1-4-9': "The dependence of the magnetoresistance on the magnetic field angle with the sample's surface clearly proves the 2D character of the WAL effect.", '1704.00339-1-4-10': 'Another single crystal, S2, with higher bulk carrier density does not show a decisive dependence on the field angle, which indicates that WAL effects from surface and bulk states interfere with one another and cannot be unequivocally separated.', '1704.00339-1-5-0': '# Experimental', '1704.00339-1-6-0': 'Single crystals of Bi[MATH]Te[MATH] were grown by a home-made resistance-heated floating zone furnace (RHFZ).', '1704.00339-1-6-1': 'The starting materials of Bi[MATH]Te[MATH] were prepared in the stoichiometric mixture of high purity elements, Bi (99.995%) and Te (99.995%).', '1704.00339-1-6-2': 'The mixture was sintered at 820[MATH]C for 20 hours and then slowly cooled down to room temperature in an evacuated quartz tube.', '1704.00339-1-6-3': 'This material was then used as a feeding rod for the RHFZ experiment.', '1704.00339-1-6-4': 'The as-grown crystals were cleaved naturally with a silvery shining mirror-like surface.', '1704.00339-1-6-5': 'RHFZ is a method similar to directional solidification, in which a small region of the feeding rod gets melted, and this molten zone is moved along the feeding rod.', '1704.00339-1-6-6': 'The molten region will move impurities to one end of the feeding rod and as it moves through the ingot, and this leaves a wake of purer material solidified behind it.', '1704.00339-1-6-7': 'The impurities concentrated in the melt have an appreciable concentration difference between the solid and liquid phases at equilibrium.', '1704.00339-1-6-8': 'It was found that a topological insulator crystal grown using the RHFZ method has a better crystalline uniformity than one grown by the traditional vertical Bridgman method.', '1704.00339-1-6-9': 'All three single crystals (S1, S2, and S3) studied here are selected from the same batch.', '1704.00339-1-7-0': 'Magnetoresistance measurements in magnetic fields up to B=7 T were carried out using the ac-transport option of the physical property measurement system (PPMS, Quantum Design).', '1704.00339-1-7-1': 'A very thin, rectangular piece of Bi[MATH]Te[MATH] was peeled using Scotch tape and attached to a magnesium oxide (MgO) substrate using GE-varnish.', '1704.00339-1-7-2': 'Typical dimensions for these crystals are [MATH] 3mm[MATH]2mm[MATH]0.05mm.', '1704.00339-1-7-3': 'The sample was covered with a plastic mask and six gold contacts were sputtered on the freshly cleaved sample surface.', '1704.00339-1-7-4': 'The sample was then mounted on a rotation platform for measuring longitudinal and Hall resistances.', '1704.00339-1-7-5': 'High field MR measurements at the National High Magnetic Field Laboratory (NHMFL) were performed using conventional lock-in techniques.', '1704.00339-1-7-6': 'A sample was mounted onto the rotating platform of a probe designed at NHMFL.', '1704.00339-1-7-7': 'The probe was inserted into the sample space of a [MATH]He Oxford cryostat, which is installed in a bore of a resistive magnet with a maximum field of 35 T.', '1704.00339-1-7-8': 'A Keithley (6221) current source excites the sample at a fixed frequency and a lock-in amplifier (SR-830) was used to measure longitudinal and Hall voltages at the same frequency.', '1704.00339-1-7-9': 'The sample position with respect to applied field was calibrated by using a Hall sensor.', '1704.00339-1-8-0': '# Results and Discussion', '1704.00339-1-9-0': 'Figure [1] shows the temperature dependence of longitudinal resistivity, [MATH], for S1 and S2.', '1704.00339-1-9-1': 'Both samples show a metallic behavior below room temperature.', '1704.00339-1-9-2': 'The resistivity of S1 is higher than that of S2 throughout the entire temperature range.', '1704.00339-1-9-3': 'The residual resistivity ratios, defined as RRR=[MATH](300 K)/[MATH](5 K), where [MATH](300 K) and [MATH](5 K) are the resistivity values at T=300 and 5 K, respectively, of S1 and S2 are calculated to be 25 and 23, respectively.', '1704.00339-1-9-4': 'Such large RRR values reflect the high crystalline quality of S1 and S2.', '1704.00339-1-9-5': 'The inset to figure [1] displays the Hall resistance, [MATH], versus [MATH] of S1 and S2 at T=5 K. Non-linear field dependence of [MATH] at [MATH]=0 suggests the existence of a multiband effect (electron and hole bands), as observed in other bismuth-based topological systems [CITATION][MATH][CITATION].', '1704.00339-1-9-6': 'The positive slope of the [MATH] curve implies the dominance of [MATH]-type bulk charge carriers in S1 and S2.', '1704.00339-1-9-7': "For the sake of simplicity, we have used the single-carrier Drude band model ([MATH], where [MATH] is the Hall resistivity and [MATH] is electron's charge) for the calculation of the bulk charge carrier density.", '1704.00339-1-9-8': 'At T=5 K, we have estimated bulk charge carriers of 6[MATH] and 3[MATH] cm[MATH] for S1 and S2, respectively, giving sample S2 almost 5 times as many bulk charge carriers as S1.', '1704.00339-1-10-0': 'For the magnetoresistance (MR) measurements of S1 and S2, we have calculated MR as a percentage, defined as MR=[[MATH]/[MATH]-1][MATH]100%, where [MATH] and [MATH] are resistances at zero and [MATH] applied field, respectively.', '1704.00339-1-10-1': 'Figure [2(a)] shows MR of S1 with [MATH] perpendicular to the sample surface (a-b plane).', '1704.00339-1-10-2': 'At [MATH]=2 K, the MR curve of S1 increases linearly with [MATH] and reaches 540 at 7 T.', '1704.00339-1-10-3': 'At given field [MATH]=7 T, this MR value is significantly higher than the previous reports of MR=240 by Wang [MATH],[CITATION] and 90 by Qu [MATH][CITATION].', '1704.00339-1-10-4': 'With increasing temperature, MR remains unchanged up to 12 K, and then decreases rapidly with a further increase in temperature.', '1704.00339-1-10-5': 'At [MATH]=60 K, MR reaches 250% at 7 T, which is almost [MATH] of the value at [MATH]=2 K.', '1704.00339-1-10-6': 'The MR curve shows the parabolic field dependence at higher temperature and low magnetic fields (see the MR curve at [MATH]=70 K in figure [2(a)]).', '1704.00339-1-10-7': 'MR of S2 is displayed in figure [2(b)].', '1704.00339-1-10-8': 'MR increases to 370% at [MATH]=2 K under 7 T.', '1704.00339-1-10-9': 'This value is relatively lower (almost 1.5 times smaller) than that of S1.', '1704.00339-1-10-10': 'The MR curves of S2 display temperature dependence similar to that of S1.', '1704.00339-1-10-11': 'Furthermore, MR of S1 and S2 shows a sharp cusp-like feature at low temperature, which indicates the existence of the WAL effect [CITATION][MATH] [CITATION].', '1704.00339-1-10-12': 'The cusp-like feature of S1 is sharper than that of S2, as shown in figure [2(c)].', '1704.00339-1-10-13': 'This could be due to the contribution of more bulk states[CITATION] to the MR of S2, which we will later discuss in detail.', '1704.00339-1-11-0': 'MR of S1 and S2 increases with magnetic field and does not show any sign of saturation.', '1704.00339-1-11-1': 'In order to investigate further, we have carried out MR measurements in dc magnetic fields up to 34 T at the National High Magnetic Field Laboratory (NHMFL), in Tallahassee, Florida.', '1704.00339-1-11-2': 'S1 shows a massive increase in MR, i.e. 3300% at 0.4 K under 34 T, and still displays no signature of saturation, as shown in figure [2(d)].', '1704.00339-1-11-3': 'This is the first time such a large MR value is observed in bismuth-based topological systems, and it is comparable with that of Dirac semimetal Cd[MATH]As[MATH][CITATION][MATH][CITATION].', '1704.00339-1-11-4': 'However, MR of S2 reaches 1850% under 31 T at 0.4 K, which is almost half that of S1.', '1704.00339-1-11-5': 'Large MR is usually linked with high mobility, as observed in many Weyl and Dirac materials [CITATION][MATH] [CITATION].', '1704.00339-1-11-6': 'We have used the simple Drude model [[MATH], where [MATH] is the Hall coefficient at temperature T] for estimating the effective mobility of S1 and S2.', '1704.00339-1-11-7': 'From our calculations, we have found [MATH]=4.5[MATH] and 3.6[MATH] cm[MATH]V[MATH]S[MATH] for S1 and S2 at 5 K, respectively.', '1704.00339-1-11-8': 'The high mobility of S1 is comparable to that of Bi[MATH]Te[MATH] samples that show topological surface states[CITATION], and even to that of a Cd[MATH]As[MATH] sample[CITATION].', '1704.00339-1-11-9': 'The mobility of S2 is one order of magnitude lower than that of S1.', '1704.00339-1-11-10': 'The linear non-saturating MR was proposed by Abrikosov[CITATION] in systems that show a linear dispersion relation.', '1704.00339-1-11-11': 'Due to the linear dispersion of surface states, a large non-saturating MR is expected to be seen.', '1704.00339-1-11-12': 'Also, due to symmetry protection, the surface electrons are robust against impurity and have ultrahigh mobility.', '1704.00339-1-11-13': 'Thus, observation of these properties in S1 suggests that the WAL effect could be a result of topological surface states.', '1704.00339-1-11-14': 'In order to clarify the origin of WAL in S1 and S2, we have performed the magnetoresistance measurements at different tilt angles.', '1704.00339-1-12-0': 'Figure [3(a)] shows magnetoconductivity (MC), defined as [MATH]/[MATH], measured along different tilt angles, [MATH], at T=0.4 K. Here, [MATH] is defined as the angle between magnetic field and current directions (see the inset to figure 3 (a)).', '1704.00339-1-12-1': 'All of the MC curves of S1 merge together when they are plotted as a function of the normal component of field, [MATH], as shown in figure [3(b)].', '1704.00339-1-12-2': 'This provides strong evidence that topological surface states dominate over the bulk states [CITATION][MATH][CITATION] in the MC of S1.', '1704.00339-1-13-0': 'We have also carried out similar angle-dependent MC experiments on S2 under fields up to 7 T. Figure [3(c)] shows the MC curves of S2 in the field range (-7 to 7 T) at T = 5 K. Initially, MC shows a strong angle dependence while increasing the tilt angle from [MATH] to 30[MATH]; however, it weakly changes with further increase in [MATH].', '1704.00339-1-13-1': 'Also, the MC curves disperse when plotted as a function of the normal component [MATH], as shown in figure [3(d)].', '1704.00339-1-13-2': 'If WAL is caused mainly by the spin-orbit coupling in a 3D bulk channel, MC is independent of [MATH].', '1704.00339-1-13-3': 'This scenario of the MC curves, which scale neither with [MATH] nor normal component [MATH], strongly suggests the presence and superposition of two contributions to MC, one from surface and one from bulk states, which is reasonable since S2 has a higher bulk carrier density than S1.', '1704.00339-1-14-0': 'From the above discussion, we have confirmed that the WAL effect in S1 is due to topological surface states.', '1704.00339-1-14-1': 'Here, both S1 and S2 show metallic behavior, and have [MATH]-type bulk charge carriers.', '1704.00339-1-14-2': 'The only difference is that S1 has fewer bulk charge carriers than S2.', '1704.00339-1-14-3': 'This indicates that the carrier density is an important factor for the observation of topological surface states in MC measurements.', '1704.00339-1-14-4': 'However, whether the domination of topological surface states over the bulk states in MC depends on the nature of the bulk carriers remains a question.', '1704.00339-1-14-5': 'To answer this, we have selected a third single crystal, S3, with electron-like carriers for a comparative study.', '1704.00339-1-15-0': 'The resistivity of S3 also shows metallic behavior below room temperature, as shown in figure 4.', '1704.00339-1-15-1': 'At T=150 K, there is a slight upturn in the resistivity, but it decreases with further cooling.', '1704.00339-1-15-2': 'Qu [MATH][CITATION] have also observed a resistivity increase starting at T=150 K in Bi[MATH]Te[MATH] single crystals.', '1704.00339-1-15-3': 'The RRR value, 18, of S3 is comparable to those of S1 and S2.', '1704.00339-1-15-4': 'The negative slope of [MATH] versus [MATH] (see inset to figure [SP2] in the Supplemental Materials) confirms the presence of [MATH]-type bulk charge carriers.', '1704.00339-1-15-5': 'From the Hall data analyses, the bulk carrier density is estimated to be 1.0[MATH] cm[MATH].', '1704.00339-1-15-6': 'This value lies in between the carrier densities of S1 and S2.', '1704.00339-1-16-0': 'Figure [5(a)] shows the MC curves of S3 along different [MATH] at T=0.4 K.', '1704.00339-1-16-1': 'A cusp-like feature at low magnetic field shows the existence of the WAL effect in S3 as well.', '1704.00339-1-16-2': 'The WAL curves collapse together with the normal component [MATH] (figure [5(b)]), confirming the dominance of topological surface states in the MC of S3.', '1704.00339-1-16-3': 'Similar to the S1 and S2 samples, we have also determined MR and mobility of S3.', '1704.00339-1-16-4': 'MR of S3 increases to 2700% under 31 T at T=0.4 K, and does not show any indication of saturation, as shown in figure [5(c)].', '1704.00339-1-16-5': 'Using the Hall coefficient and resistivity data, we have estimated mobility [MATH]=1.3[MATH] cm[MATH]V[MATH]S[MATH].', '1704.00339-1-16-6': 'These observations in sample S3 confirm that the domination of topological surface states in MC depends only on the carrier density and is independent of the nature of the charge carriers.', '1704.00339-1-17-0': 'We have also estimated several physical parameters that characterize the WAL effect by employing the Hikami-Larkin-Nagaoka (HLN) formula [CITATION].', '1704.00339-1-17-1': 'The HLN formula has already been used to describe the WAL effect in topological systems, for example topological thin films[CITATION] and single crystals[CITATION].', '1704.00339-1-17-2': 'According to the HLN formula, MC can be expressed as,', '1704.00339-1-18-0': '[EQUATION]', '1704.00339-1-19-0': 'Here [MATH] is the digamma function, and [MATH] is the phase coherence length, which is the distance traveled by an electron before its phase is changed.', '1704.00339-1-19-1': 'The parameter [MATH] with [MATH]=1/2 per conduction channel.', '1704.00339-1-19-2': 'Thus, [MATH] represents the number of conduction channels present in a sample.', '1704.00339-1-19-3': 'Using equation (1) with our experimental data, the fitting parameters [MATH] and [MATH] can be determined.', '1704.00339-1-20-0': 'Figure [5(d)] shows the HLN fitting to the MC curve of S1 in low field range (-1 to 1 T).', '1704.00339-1-20-1': 'Similar HLN fitting was performed for the MC data of S2 and S3 to determine [MATH] and [MATH].', '1704.00339-1-20-2': 'The [MATH] values of S1, S2, and S3 at different temperatures are presented in figure [4(e)].', '1704.00339-1-20-3': 'Temperature dependence of [MATH] can be described as[CITATION][MATH][CITATION] [EQUATION] where, [MATH](0) is the zero-temperature phase coherence length, and [MATH] and [MATH] represent the contributions from the electron-electron and electron-phonon interactions, respectively.', '1704.00339-1-20-4': 'Equation (2) is fitted to the temperature dependence of [MATH] with [MATH]=1 and [MATH]=2, shown by solid curves in figure [4(e)].', '1704.00339-1-20-5': 'The fitting parameters [MATH](0), [MATH], and [MATH] of S1, S2, and S3 are presented in table [1].', '1704.00339-1-20-6': 'The [MATH](0) values of S1 and S3 are comparable, but are almost 3 times as large as that of S2.', '1704.00339-1-20-7': 'The larger [MATH](0) values in S1 and S3 are comparable to those of other topological systems[CITATION][MATH][CITATION].', '1704.00339-1-21-0': 'Figure [5(f)] shows the parameter [MATH] as a function of temperature.', '1704.00339-1-21-1': 'The [MATH] values of S1, S2, and S3 are on the order of 10[MATH], which is nearly 10[MATH] times larger than that for two dimensional systems.', '1704.00339-1-21-2': 'These observations are also seen in other topological single crystals[CITATION][MATH] [CITATION] and this discrepancy could be due to the contribution from the dominant bulk channels.', '1704.00339-1-21-3': 'The [MATH] values remain nearly constant with temperature up to 45 K, suggesting the presence of a fixed number of conduction channels in S1, S2, and S3.', '1704.00339-1-21-4': 'Moreover, the value of [MATH] for S1 is comparable to that for S3, but is almost half of the value for S2.', '1704.00339-1-21-5': 'Since the value of [MATH] is a measure of the number of conduction channels present in a sample, S1 and S3 have a smaller number of conduction channels as compared to S2.', '1704.00339-1-22-0': '# Summary', '1704.00339-1-23-0': 'In summary, we have studied the magnetoresistance of three Bi[MATH]Te[MATH] single crystals, all having bulk metallic properties but with different concentrations and nature of charge carriers.', '1704.00339-1-23-1': 'Whereas all of the samples show very large MR and high mobility, the two crystals with lower carrier density (one electron and one hole like) exhibit the largest MR and mobility, comparable with values observed in the Dirac semimetal Cd[MATH]As[MATH].', '1704.00339-1-23-2': 'We have also demonstrated that large MR and high mobility in Bi[MATH]Te[MATH] depend only on the bulk carrier concentration and are independent of the nature of the charge carriers.', '1704.00339-1-23-3': 'The cusp of MR for the samples with low carrier density reflects the characteristics of the WAL effects.', '1704.00339-1-23-4': "The dependence of MC on the angle of the magnetic field with the sample's surface demonstrates that MR is dominated by topological surface conduction in the low carrier density samples.", '1704.00339-1-23-5': 'The third sample with higher carrier density shows an interference of surface and bulk effects on MR, as can be expected with increasing bulk carrier number.', '1704.00339-1-23-6': 'The Hikami-Larkin-Nagaoka formula is used to calculate different physical quantities that characterize the observed WAL effects.', '1704.00339-1-23-7': 'The larger phase coherence length and higher electrical mobility of, and the smaller number of conduction channels present in, the crystals with lower carrier density, confirm that topological surface states dominate the magnetic field effect on the overall conduction in these samples.', '1704.00339-1-24-0': '# acknowledgements', '1704.00339-1-25-0': 'This work is supported in part by the U.S. Air Force Office of Scientific Research, the T. L. L. Temple Foundation, the J. J. and R. Moores Endowment, and the State of Texas through the TCSUH.', '1704.00339-1-25-1': 'The work at NHMFL is supported by the NSF co-operative Agreement No. DMR-1157490 and the State of Florida.', '1704.00339-1-25-2': 'K.S. acknowledges the Department of Energy, Office of Basic Energy Sciences, Materials Sciences, and Engineering Division and through grant DOE FG02-01ER45872.'}

{'1704.00339-2-0-0': 'Weak antilocalization (WAL) effects in Bi[MATH]Te[MATH] single crystals have been investigated at high and low bulk charge carrier concentrations.', '1704.00339-2-0-1': 'At low charge carrier density the WAL curves scale with the normal component of the magnetic field, demonstrating the dominance of topological surface states in magnetoconductivity.', '1704.00339-2-0-2': 'At high charge carrier density the WAL curves scale with neither the applied field nor its normal component, implying a mixture of bulk and surface conduction.', '1704.00339-2-0-3': 'WAL due to topological surface states shows no dependence on the nature (electrons or holes) of the bulk charge carriers.', '1704.00339-2-0-4': 'The observations of an extremely large, non-saturating magnetoresistance, and ultrahigh mobility in the samples with lower carrier density further support the presence of surface states.', '1704.00339-2-0-5': 'The physical parameters characterizing the WAL effects are calculated using the Hikami-Larkin-Nagaoka formula.', '1704.00339-2-0-6': 'At high charge carrier concentrations, there is a greater number of conduction channels and a decrease in the phase coherence length compared to low charge carrier concentrations.', '1704.00339-2-0-7': 'The extremely large magnetoresistance and high mobility of topological insulators have great technological value and can be exploited in magneto-electric sensors and memory devices.', '1704.00339-2-1-0': '# Introduction', '1704.00339-2-2-0': 'The quantum interference of electrons in solids experiencing a backscattering event, which limits the electrical conductivity, has been of interest for many years.', '1704.00339-2-2-1': 'The resistance in systems with scattering centers increases with decreasing temperature due to increasing phase coherence in backscattering processes where the electron moves through a loop of scattering sites clockwise, as well as counter-clockwise, and the constructive superposition of the scattering amplitudes results in an enhancement of the resistivity.', '1704.00339-2-2-2': 'The phase coherence in this additive scattering process, coined weak localization (WL), is destroyed in an external magnetic field, resulting in a characteristic negative magnetoresistance[CITATION].', '1704.00339-2-2-3': 'On the other hand, in systems with strong spin-orbit interaction, the spin of the electron is tied to the momentum.', '1704.00339-2-2-4': 'This results in a rotation of the spins along the backscattering path and the phase difference between clockwise and counter-clockwise trajectories amounts to 360[MATH].', '1704.00339-2-2-5': 'Rotating a spin by 360[MATH] changes the sign of the scattering amplitude and the two scattering events experience a destructive interference, reducing the resistance in materials with strong spin-orbit coupling [weak antilocalization (WAL)].', '1704.00339-2-2-6': 'The destructive interference effects are diminished in magnetic fields resulting in a positive magnetoresistance, as observed in many metallic compounds with large spin-orbit interaction[CITATION].', '1704.00339-2-3-0': 'WL and WAL are more pronounced in two-dimensional (2D) systems since the probability of two interfering scattering pathways, distinguished only by time reversal, is larger in lower dimensions.', '1704.00339-2-3-1': 'Therefore, surface states of topological insulators (TIs) have been studied recently with respect to the WAL effects.', '1704.00339-2-3-2': 'The surface conduction can be easily studied when the bulk of the TI is insulating and does not contribute to the conductivity.', '1704.00339-2-3-3': 'In many TIs the study of transport from topological surface states is impeded by a significant bulk contribution to the overall conduction due to defects, impurities, or intersite occupancy.', '1704.00339-2-3-4': 'This makes it difficult to extract the topological properties of the surface states by employing magnetotransport methods.', '1704.00339-2-3-5': 'However, in a recent investigation of metallic, hole-doped Bi[MATH]Se[MATH]Te[MATH] and Sb[MATH]Te[MATH]Se, we have shown that quantum oscillations from topological surface states can be resolved well despite the interference with bulk conduction[CITATION][MATH][CITATION][MATH][CITATION].', '1704.00339-2-3-6': 'This motivated us to search for other physical phenomena related to topological surface states in TIs with bulk metallic properties.', '1704.00339-2-4-0': 'Bismuth telluride Bi[MATH]Te[MATH] is shown to be a three dimensional topological insulator by both theoretical and experimental studies[CITATION][MATH][CITATION][CITATION].', '1704.00339-2-4-1': 'Recently Qu [MATH][CITATION] observed that every single crystal of Bi[MATH]Te[MATH] can have different physical properties such as bulk carrier concentrations, carrier types, etc. even if they are grown in the same batch.', '1704.00339-2-4-2': 'They reported that this might be due to a weak or critical composition gradient during the crystal growth process.', '1704.00339-2-4-3': 'In this work, we report the WAL effect in Bi[MATH]Te[MATH] single crystals with metallic bulk conductivity and different carrier densities with hole as well as electron nature.', '1704.00339-2-4-4': 'Three single crystals, S1, S2, and S3, are selected from the same boule of crystals.', '1704.00339-2-4-5': 'People have used either angle dependence of quantum oscillations or WAL curves to identify the presence of topological surface states[CITATION][MATH][CITATION].', '1704.00339-2-4-6': 'In our study, even though samples S1 and S2 show quantum oscillations under fields above 7 T, S3 did not show any signature of quantum oscillations in the field up to 31 T. However, all of those samples show WAL effect at low temperature.', '1704.00339-2-4-7': 'That is why we have used angle dependence of WAL curves to detect the presence of topological surface states in all samples.', '1704.00339-2-4-8': 'For two crystals with lower bulk carrier density (S1, electron-, and S3, hole-doped) we find the clear signature of WAL from 2D surface states.', '1704.00339-2-4-9': "The dependence of the magnetoresistance on the magnetic field angle with the sample's surface clearly proves the 2D character of the WAL effect.", '1704.00339-2-4-10': 'Another single crystal, S2, with higher bulk carrier density does not show a decisive dependence on the field angle, which indicates that WAL effects from surface and bulk states interfere with one another and cannot be unequivocally separated.', '1704.00339-2-5-0': '# Experimental', '1704.00339-2-6-0': 'Single crystals of Bi[MATH]Te[MATH] were grown by a home-made resistance-heated floating zone furnace (RHFZ).', '1704.00339-2-6-1': 'The starting materials of Bi[MATH]Te[MATH] were prepared in the stoichiometric mixture of high purity elements, Bi (99.995%) and Te (99.995%).', '1704.00339-2-6-2': 'The mixture was sintered at 820[MATH]C for 20 hours and then slowly cooled down to room temperature in an evacuated quartz tube.', '1704.00339-2-6-3': 'This material was then used as a feeding rod for the RHFZ experiment.', '1704.00339-2-6-4': 'The as-grown crystals were cleaved naturally with a silvery shining mirror-like surface.', '1704.00339-2-6-5': 'RHFZ is a method similar to directional solidification, in which a small region of the feeding rod gets melted, and this molten zone is moved along the feeding rod.', '1704.00339-2-6-6': 'The molten region will move impurities to one end of the feeding rod and as it moves through the ingot, and this leaves a wake of purer material solidified behind it.', '1704.00339-2-6-7': 'The impurities concentrated in the melt have an appreciable concentration difference between the solid and liquid phases at equilibrium.', '1704.00339-2-6-8': 'It was found that a topological insulator crystal grown using the RHFZ method has a better crystalline uniformity than one grown by the traditional vertical Bridgman method.', '1704.00339-2-6-9': 'All three single crystals (S1, S2, and S3) studied here are selected from the same batch.', '1704.00339-2-7-0': 'Magnetoresistance measurements in magnetic fields up to B=7 T were carried out using the ac-transport option of the physical property measurement system (PPMS, Quantum Design).', '1704.00339-2-7-1': 'A very thin, rectangular piece of Bi[MATH]Te[MATH] was peeled using Scotch tape and attached to a magnesium oxide (MgO) substrate using GE-varnish.', '1704.00339-2-7-2': 'Typical dimensions for these crystals are [MATH] 3mm[MATH]2mm[MATH]0.05mm.', '1704.00339-2-7-3': 'The sample was covered with a plastic mask and six gold contacts were sputtered on the freshly cleaved sample surface.', '1704.00339-2-7-4': 'The sample was then mounted on a rotation platform for measuring longitudinal and Hall resistances.', '1704.00339-2-7-5': 'High field MR measurements at the National High Magnetic Field Laboratory (NHMFL) were performed using conventional lock-in techniques.', '1704.00339-2-7-6': 'A sample was mounted onto the rotating platform of a probe designed at NHMFL.', '1704.00339-2-7-7': 'The probe was inserted into the sample space of a [MATH]He Oxford cryostat, which is installed in a bore of a resistive magnet with a maximum field of 35 T.', '1704.00339-2-7-8': 'A Keithley (6221) current source excites the sample at a fixed frequency and a lock-in amplifier (SR-830) was used to measure longitudinal and Hall voltages at the same frequency.', '1704.00339-2-7-9': 'The sample position with respect to applied field was calibrated by using a Hall sensor.', '1704.00339-2-8-0': '# Results and Discussion', '1704.00339-2-9-0': 'Figure [1] shows the temperature dependence of longitudinal resistivity, [MATH], for S1 and S2.', '1704.00339-2-9-1': 'Both samples show a metallic behavior below room temperature.', '1704.00339-2-9-2': 'The resistivity of S1 is higher than that of S2 throughout the entire temperature range.', '1704.00339-2-9-3': 'The residual resistivity ratios, defined as RRR=[MATH](300 K)/[MATH](5 K), where [MATH](300 K) and [MATH](5 K) are the resistivity values at T=300 and 5 K, respectively, of S1 and S2 are calculated to be 25 and 23, respectively.', '1704.00339-2-9-4': 'Such large RRR values reflect the high crystalline quality of S1 and S2.', '1704.00339-2-9-5': 'The inset to figure [1] displays the Hall resistance, [MATH], versus [MATH] of S1 and S2 at T=5 K. Non-linear field dependence of [MATH] at [MATH]=0 suggests the existence of a multiband effect (electron and hole bands), as observed in other bismuth-based topological systems [CITATION][MATH][CITATION].', '1704.00339-2-9-6': 'The positive slope of the [MATH] curve implies the dominance of [MATH]-type bulk charge carriers in S1 and S2.', '1704.00339-2-9-7': "For the sake of simplicity, we have used the single-carrier Drude band model ([MATH], where [MATH] is the Hall resistivity and [MATH] is electron's charge) for the calculation of the bulk charge carrier density.", '1704.00339-2-9-8': 'At T=5 K, we have estimated bulk charge carriers of 6[MATH] and 3[MATH] cm[MATH] for S1 and S2, respectively, giving sample S2 almost 5 times as many bulk charge carriers as S1.', '1704.00339-2-10-0': 'For the magnetoresistance (MR) measurements of S1 and S2, we have calculated MR as a percentage, defined as MR=[[MATH]/[MATH]-1][MATH]100%, where [MATH] and [MATH] are resistances at zero and [MATH] applied field, respectively.', '1704.00339-2-10-1': 'Figure [2(a)] shows MR of S1 with [MATH] perpendicular to the sample surface (a-b plane).', '1704.00339-2-10-2': 'At [MATH]=2 K, the MR curve of S1 increases linearly with [MATH] and reaches 540 at 7 T.', '1704.00339-2-10-3': 'At given field [MATH]=7 T, this MR value is significantly higher than the previous reports of MR=240 by Wang [MATH],[CITATION] and 90 by Qu [MATH][CITATION].', '1704.00339-2-10-4': 'With increasing temperature, MR remains unchanged up to 12 K, and then decreases rapidly with a further increase in temperature.', '1704.00339-2-10-5': 'At [MATH]=60 K, MR reaches 250% at 7 T, which is almost [MATH] of the value at [MATH]=2 K.', '1704.00339-2-10-6': 'The MR curve shows the parabolic field dependence at higher temperature and low magnetic fields (see the MR curve at [MATH]=70 K in figure [2(a)]).', '1704.00339-2-10-7': 'MR of S2 is displayed in figure [2(b)].', '1704.00339-2-10-8': 'MR increases to 370% at [MATH]=2 K under 7 T.', '1704.00339-2-10-9': 'This value is relatively lower (almost 1.5 times smaller) than that of S1.', '1704.00339-2-10-10': 'The MR curves of S2 display temperature dependence similar to that of S1.', '1704.00339-2-10-11': 'Furthermore, MR of S1 and S2 shows a sharp cusp-like feature at low temperature, which indicates the existence of the WAL effect [CITATION][MATH] [CITATION].', '1704.00339-2-10-12': 'The cusp-like feature of S1 is sharper than that of S2, as shown in figure [2(c)].', '1704.00339-2-10-13': 'This could be due to the contribution of more bulk states[CITATION] to the MR of S2, which we will later discuss in detail.', '1704.00339-2-11-0': 'MR of S1 and S2 increases with magnetic field and does not show any sign of saturation.', '1704.00339-2-11-1': 'In order to investigate further, we have carried out MR measurements in dc magnetic fields up to 34 T at the National High Magnetic Field Laboratory (NHMFL), in Tallahassee, Florida.', '1704.00339-2-11-2': 'S1 shows a massive increase in MR, i.e. 3300% at 0.4 K under 34 T, and still displays no signature of saturation, as shown in figure [2(d)].', '1704.00339-2-11-3': 'This is the first time such a large MR value is observed in bismuth-based topological systems, and it is comparable with that of Dirac semimetal Cd[MATH]As[MATH][CITATION][MATH][CITATION].', '1704.00339-2-11-4': 'However, MR of S2 reaches 1850% under 31 T at 0.4 K, which is almost half that of S1.', '1704.00339-2-11-5': 'Large MR is usually linked with high mobility, as observed in many Weyl and Dirac materials [CITATION][MATH] [CITATION].', '1704.00339-2-11-6': 'We have used the simple Drude model [[MATH], where [MATH] is the Hall coefficient at temperature T] for estimating the effective mobility of S1 and S2.', '1704.00339-2-11-7': 'From our calculations, we have found [MATH]=4.5[MATH] and 3.6[MATH] cm[MATH]V[MATH]S[MATH] for S1 and S2 at 5 K, respectively.', '1704.00339-2-11-8': 'The high mobility of S1 is comparable to that of Bi[MATH]Te[MATH] samples that show topological surface states[CITATION], and even to that of a Cd[MATH]As[MATH] sample[CITATION].', '1704.00339-2-11-9': 'The mobility of S2 is one order of magnitude lower than that of S1.', '1704.00339-2-11-10': 'The linear non-saturating MR was proposed by Abrikosov[CITATION] in systems that show a linear dispersion relation.', '1704.00339-2-11-11': 'Due to the linear dispersion of surface states, a large non-saturating MR is expected to be seen.', '1704.00339-2-11-12': 'Also, due to symmetry protection, the surface electrons are robust against impurity and have ultrahigh mobility.', '1704.00339-2-11-13': 'Thus, observation of these properties in S1 suggests that the WAL effect could be a result of topological surface states.', '1704.00339-2-11-14': 'In order to clarify the origin of WAL in S1 and S2, we have performed the magnetoresistance measurements at different tilt angles.', '1704.00339-2-12-0': 'Figure [3(a)] shows magnetoconductivity (MC), defined as [MATH]/[MATH], measured along different tilt angles, [MATH], at T=0.4 K. Here, [MATH] is defined as the angle between magnetic field and current directions (see the inset to figure 3 (a)).', '1704.00339-2-12-1': 'All of the MC curves of S1 merge together when they are plotted as a function of the normal component of field, [MATH], as shown in figure [3(b)].', '1704.00339-2-12-2': 'This provides strong evidence that topological surface states dominate over the bulk states [CITATION][MATH][CITATION] in the MC of S1.', '1704.00339-2-13-0': 'We have also carried out similar angle-dependent MC experiments on S2 under fields up to 7 T. Figure [3(c)] shows the MC curves of S2 in the field range (-7 to 7 T) at T = 5 K. Initially, MC shows a strong angle dependence while increasing the tilt angle from [MATH] to 30[MATH]; however, it weakly changes with further increase in [MATH].', '1704.00339-2-13-1': 'Also, the MC curves disperse when plotted as a function of the normal component [MATH], as shown in figure [3(d)].', '1704.00339-2-13-2': 'If WAL is caused mainly by the spin-orbit coupling in a 3D bulk channel, MC is independent of [MATH].', '1704.00339-2-13-3': 'This scenario of the MC curves, which scale neither with [MATH] nor normal component [MATH], strongly suggests the presence and superposition of two contributions to MC, one from surface and one from bulk states, which is reasonable since S2 has a higher bulk carrier density than S1.', '1704.00339-2-14-0': 'From the above discussion, we have confirmed that the WAL effect in S1 is due to topological surface states.', '1704.00339-2-14-1': 'Here, both S1 and S2 show metallic behavior, and have [MATH]-type bulk charge carriers.', '1704.00339-2-14-2': 'The only difference is that S1 has fewer bulk charge carriers than S2.', '1704.00339-2-14-3': 'This indicates that the carrier density is an important factor for the observation of topological surface states in MC measurements.', '1704.00339-2-14-4': 'However, whether the domination of topological surface states over the bulk states in MC depends on the nature of the bulk carriers remains a question.', '1704.00339-2-14-5': 'To answer this, we have selected a third single crystal, S3, with electron-like carriers for a comparative study.', '1704.00339-2-15-0': 'The resistivity of S3 also shows metallic behavior below room temperature, as shown in figure 4.', '1704.00339-2-15-1': 'At T=150 K, there is a slight upturn in the resistivity, but it decreases with further cooling.', '1704.00339-2-15-2': 'Qu [MATH][CITATION] have also observed a resistivity increase starting at T=150 K in Bi[MATH]Te[MATH] single crystals.', '1704.00339-2-15-3': 'The RRR value, 18, of S3 is comparable to those of S1 and S2.', '1704.00339-2-15-4': 'The negative slope of [MATH] versus [MATH] (see inset to figure [SP2] in the Supplemental Materials) confirms the presence of [MATH]-type bulk charge carriers.', '1704.00339-2-15-5': 'From the Hall data analyses, the bulk carrier density is estimated to be 1.0[MATH] cm[MATH].', '1704.00339-2-15-6': 'This value lies in between the carrier densities of S1 and S2.', '1704.00339-2-16-0': 'Figure [5(a)] shows the MC curves of S3 along different [MATH] at T=0.4 K.', '1704.00339-2-16-1': 'A cusp-like feature at low magnetic field shows the existence of the WAL effect in S3 as well.', '1704.00339-2-16-2': 'The WAL curves collapse together with the normal component [MATH] (figure [5(b)]), confirming the dominance of topological surface states in the MC of S3.', '1704.00339-2-16-3': 'Similar to the S1 and S2 samples, we have also determined MR and mobility of S3.', '1704.00339-2-16-4': 'MR of S3 increases to 2700% under 31 T at T=0.4 K, and does not show any indication of saturation, as shown in figure [5(c)].', '1704.00339-2-16-5': 'Using the Hall coefficient and resistivity data, we have estimated mobility [MATH]=1.3[MATH] cm[MATH]V[MATH]S[MATH].', '1704.00339-2-16-6': 'These observations in sample S3 confirm that the domination of topological surface states in MC depends only on the carrier density and is independent of the nature of the charge carriers.', '1704.00339-2-17-0': 'We have also estimated several physical parameters that characterize the WAL effect by employing the Hikami-Larkin-Nagaoka (HLN) formula [CITATION].', '1704.00339-2-17-1': 'The HLN formula has already been used to describe the WAL effect in topological systems, for example topological thin films[CITATION] and single crystals[CITATION].', '1704.00339-2-17-2': 'According to the HLN formula, MC can be expressed as,', '1704.00339-2-18-0': '[EQUATION]', '1704.00339-2-19-0': 'Here [MATH] is the digamma function, and [MATH] is the phase coherence length, which is the distance traveled by an electron before its phase is changed.', '1704.00339-2-19-1': 'The parameter [MATH] with [MATH]=1/2 per conduction channel.', '1704.00339-2-19-2': 'Thus, [MATH] represents the number of conduction channels present in a sample.', '1704.00339-2-19-3': 'Using equation (1) with our experimental data, the fitting parameters [MATH] and [MATH] can be determined.', '1704.00339-2-20-0': 'Figure [5(d)] shows the HLN fitting to the MC curve of S1 in low field range (-1 to 1 T).', '1704.00339-2-20-1': 'Similar HLN fitting was performed for the MC data of S2 and S3 to determine [MATH] and [MATH].', '1704.00339-2-20-2': 'The [MATH] values of S1, S2, and S3 at different temperatures are presented in figure [4(e)].', '1704.00339-2-20-3': 'Temperature dependence of [MATH] can be described as[CITATION][MATH][CITATION] [EQUATION] where, [MATH](0) is the zero-temperature phase coherence length, and [MATH] and [MATH] represent the contributions from the electron-electron and electron-phonon interactions, respectively.', '1704.00339-2-20-4': 'Equation (2) is fitted to the temperature dependence of [MATH] with [MATH]=1 and [MATH]=2, shown by solid curves in figure [4(e)].', '1704.00339-2-20-5': 'The fitting parameters [MATH](0), [MATH], and [MATH] of S1, S2, and S3 are presented in table [1].', '1704.00339-2-20-6': 'The [MATH](0) values of S1 and S3 are comparable, but are almost 3 times as large as that of S2.', '1704.00339-2-20-7': 'The larger [MATH](0) values in S1 and S3 are comparable to those of other topological systems[CITATION][MATH][CITATION].', '1704.00339-2-21-0': 'Figure [5(f)] shows the parameter [MATH] as a function of temperature.', '1704.00339-2-21-1': 'The [MATH] values of S1, S2, and S3 are on the order of 10[MATH], which is nearly 10[MATH] times larger than that for two dimensional systems.', '1704.00339-2-21-2': 'These observations are also seen in other topological single crystals[CITATION][MATH] [CITATION] and this discrepancy could be due to the contribution from the dominant bulk channels.', '1704.00339-2-21-3': 'The [MATH] values remain nearly constant with temperature up to 45 K, suggesting the presence of a fixed number of conduction channels in S1, S2, and S3.', '1704.00339-2-21-4': 'Moreover, the value of [MATH] for S1 is comparable to that for S3, but is almost half of the value for S2.', '1704.00339-2-21-5': 'Since the value of [MATH] is a measure of the number of conduction channels present in a sample, S1 and S3 have a smaller number of conduction channels as compared to S2.', '1704.00339-2-22-0': '# Summary', '1704.00339-2-23-0': 'In summary, we have studied the magnetoresistance of three Bi[MATH]Te[MATH] single crystals, all having bulk metallic properties but with different concentrations and nature of charge carriers.', '1704.00339-2-23-1': 'Whereas all of the samples show very large MR and high mobility, the two crystals with lower carrier density (one electron and one hole like) exhibit the largest MR and mobility, comparable with values observed in the Dirac semimetal Cd[MATH]As[MATH].', '1704.00339-2-23-2': 'We have also demonstrated that large MR and high mobility in Bi[MATH]Te[MATH] depend only on the bulk carrier concentration and are independent of the nature of the charge carriers.', '1704.00339-2-23-3': 'The cusp of MR for the samples with low carrier density reflects the characteristics of the WAL effects.', '1704.00339-2-23-4': "The dependence of MC on the angle of the magnetic field with the sample's surface demonstrates that MR is dominated by topological surface conduction in the low carrier density samples.", '1704.00339-2-23-5': 'The third sample with higher carrier density shows an interference of surface and bulk effects on MR, as can be expected with increasing bulk carrier number.', '1704.00339-2-23-6': 'The Hikami-Larkin-Nagaoka formula is used to calculate different physical quantities that characterize the observed WAL effects.', '1704.00339-2-23-7': 'The larger phase coherence length and higher electrical mobility of, and the smaller number of conduction channels present in, the crystals with lower carrier density, confirm that topological surface states dominate the magnetic field effect on the overall conduction in these samples.', '1704.00339-2-24-0': '# acknowledgements', '1704.00339-2-25-0': 'This work is supported in part by the U.S. Air Force Office of Scientific Research, the T. L. L. Temple Foundation, the J. J. and R. Moores Endowment, and the State of Texas through the TCSUH.', '1704.00339-2-25-1': 'The work at NHMFL is supported by the NSF co-operative Agreement No. DMR-1157490 and the State of Florida.', '1704.00339-2-25-2': 'K.S. acknowledges the Department of Energy, Office of Basic Energy Sciences, Materials Sciences, and Engineering Division and through grant DOE FG02-01ER45872.'}

[['1704.00339-1-11-0', '1704.00339-2-11-0'], ['1704.00339-1-11-1', '1704.00339-2-11-1'], ['1704.00339-1-11-2', '1704.00339-2-11-2'], ['1704.00339-1-11-3', '1704.00339-2-11-3'], ['1704.00339-1-11-4', '1704.00339-2-11-4'], ['1704.00339-1-11-5', '1704.00339-2-11-5'], ['1704.00339-1-11-6', '1704.00339-2-11-6'], ['1704.00339-1-11-7', '1704.00339-2-11-7'], ['1704.00339-1-11-8', '1704.00339-2-11-8'], ['1704.00339-1-11-9', '1704.00339-2-11-9'], ['1704.00339-1-11-10', '1704.00339-2-11-10'], ['1704.00339-1-11-11', '1704.00339-2-11-11'], ['1704.00339-1-11-12', '1704.00339-2-11-12'], ['1704.00339-1-11-13', '1704.00339-2-11-13'], ['1704.00339-1-11-14', '1704.00339-2-11-14'], ['1704.00339-1-4-0', '1704.00339-2-4-0'], ['1704.00339-1-4-1', '1704.00339-2-4-1'], ['1704.00339-1-4-2', '1704.00339-2-4-2'], ['1704.00339-1-4-3', '1704.00339-2-4-3'], ['1704.00339-1-4-4', '1704.00339-2-4-4'], ['1704.00339-1-4-5', '1704.00339-2-4-5'], ['1704.00339-1-4-6', '1704.00339-2-4-6'], ['1704.00339-1-4-7', '1704.00339-2-4-7'], ['1704.00339-1-4-8', '1704.00339-2-4-8'], ['1704.00339-1-4-9', '1704.00339-2-4-9'], ['1704.00339-1-4-10', '1704.00339-2-4-10'], ['1704.00339-1-19-0', '1704.00339-2-19-0'], ['1704.00339-1-19-1', '1704.00339-2-19-1'], ['1704.00339-1-19-2', '1704.00339-2-19-2'], ['1704.00339-1-19-3', '1704.00339-2-19-3'], ['1704.00339-1-3-0', '1704.00339-2-3-0'], ['1704.00339-1-3-1', '1704.00339-2-3-1'], ['1704.00339-1-3-2', '1704.00339-2-3-2'], ['1704.00339-1-3-3', '1704.00339-2-3-3'], ['1704.00339-1-3-4', '1704.00339-2-3-4'], ['1704.00339-1-3-5', '1704.00339-2-3-5'], ['1704.00339-1-3-6', '1704.00339-2-3-6'], ['1704.00339-1-16-0', '1704.00339-2-16-0'], ['1704.00339-1-16-1', '1704.00339-2-16-1'], ['1704.00339-1-16-2', '1704.00339-2-16-2'], ['1704.00339-1-16-3', '1704.00339-2-16-3'], ['1704.00339-1-16-4', '1704.00339-2-16-4'], ['1704.00339-1-16-5', '1704.00339-2-16-5'], ['1704.00339-1-16-6', '1704.00339-2-16-6'], ['1704.00339-1-14-0', '1704.00339-2-14-0'], ['1704.00339-1-14-1', '1704.00339-2-14-1'], ['1704.00339-1-14-2', '1704.00339-2-14-2'], ['1704.00339-1-14-3', '1704.00339-2-14-3'], ['1704.00339-1-14-4', '1704.00339-2-14-4'], ['1704.00339-1-14-5', '1704.00339-2-14-5'], ['1704.00339-1-23-0', '1704.00339-2-23-0'], ['1704.00339-1-23-1', '1704.00339-2-23-1'], ['1704.00339-1-23-2', '1704.00339-2-23-2'], ['1704.00339-1-23-3', '1704.00339-2-23-3'], ['1704.00339-1-23-4', '1704.00339-2-23-4'], ['1704.00339-1-23-5', '1704.00339-2-23-5'], ['1704.00339-1-23-6', '1704.00339-2-23-6'], ['1704.00339-1-23-7', '1704.00339-2-23-7'], ['1704.00339-1-12-0', '1704.00339-2-12-0'], ['1704.00339-1-12-1', '1704.00339-2-12-1'], ['1704.00339-1-12-2', '1704.00339-2-12-2'], ['1704.00339-1-17-0', '1704.00339-2-17-0'], ['1704.00339-1-17-1', '1704.00339-2-17-1'], ['1704.00339-1-21-0', '1704.00339-2-21-0'], ['1704.00339-1-21-1', '1704.00339-2-21-1'], ['1704.00339-1-21-2', '1704.00339-2-21-2'], ['1704.00339-1-21-3', '1704.00339-2-21-3'], ['1704.00339-1-21-4', '1704.00339-2-21-4'], ['1704.00339-1-21-5', '1704.00339-2-21-5'], ['1704.00339-1-6-0', '1704.00339-2-6-0'], ['1704.00339-1-6-1', '1704.00339-2-6-1'], ['1704.00339-1-6-2', '1704.00339-2-6-2'], ['1704.00339-1-6-3', '1704.00339-2-6-3'], ['1704.00339-1-6-4', '1704.00339-2-6-4'], ['1704.00339-1-6-5', '1704.00339-2-6-5'], ['1704.00339-1-6-6', '1704.00339-2-6-6'], ['1704.00339-1-6-7', '1704.00339-2-6-7'], ['1704.00339-1-6-8', '1704.00339-2-6-8'], ['1704.00339-1-6-9', '1704.00339-2-6-9'], ['1704.00339-1-7-0', '1704.00339-2-7-0'], ['1704.00339-1-7-1', '1704.00339-2-7-1'], ['1704.00339-1-7-2', '1704.00339-2-7-2'], ['1704.00339-1-7-3', '1704.00339-2-7-3'], ['1704.00339-1-7-4', '1704.00339-2-7-4'], ['1704.00339-1-7-5', '1704.00339-2-7-5'], ['1704.00339-1-7-6', '1704.00339-2-7-6'], ['1704.00339-1-7-7', '1704.00339-2-7-7'], ['1704.00339-1-7-8', '1704.00339-2-7-8'], ['1704.00339-1-7-9', '1704.00339-2-7-9'], ['1704.00339-1-9-0', '1704.00339-2-9-0'], ['1704.00339-1-9-1', '1704.00339-2-9-1'], ['1704.00339-1-9-2', '1704.00339-2-9-2'], ['1704.00339-1-9-3', '1704.00339-2-9-3'], ['1704.00339-1-9-4', '1704.00339-2-9-4'], ['1704.00339-1-9-5', '1704.00339-2-9-5'], ['1704.00339-1-9-6', '1704.00339-2-9-6'], ['1704.00339-1-9-7', '1704.00339-2-9-7'], ['1704.00339-1-9-8', '1704.00339-2-9-8'], ['1704.00339-1-10-0', '1704.00339-2-10-0'], ['1704.00339-1-10-1', '1704.00339-2-10-1'], ['1704.00339-1-10-2', '1704.00339-2-10-2'], ['1704.00339-1-10-3', '1704.00339-2-10-3'], ['1704.00339-1-10-4', '1704.00339-2-10-4'], ['1704.00339-1-10-5', '1704.00339-2-10-5'], ['1704.00339-1-10-6', '1704.00339-2-10-6'], ['1704.00339-1-10-7', '1704.00339-2-10-7'], ['1704.00339-1-10-8', '1704.00339-2-10-8'], ['1704.00339-1-10-9', '1704.00339-2-10-9'], ['1704.00339-1-10-10', '1704.00339-2-10-10'], ['1704.00339-1-10-11', '1704.00339-2-10-11'], ['1704.00339-1-10-12', '1704.00339-2-10-12'], ['1704.00339-1-10-13', '1704.00339-2-10-13'], ['1704.00339-1-25-0', '1704.00339-2-25-0'], ['1704.00339-1-25-2', '1704.00339-2-25-2'], ['1704.00339-1-2-0', '1704.00339-2-2-0'], ['1704.00339-1-2-1', '1704.00339-2-2-1'], ['1704.00339-1-2-2', '1704.00339-2-2-2'], ['1704.00339-1-2-3', '1704.00339-2-2-3'], ['1704.00339-1-2-4', '1704.00339-2-2-4'], ['1704.00339-1-2-5', '1704.00339-2-2-5'], ['1704.00339-1-2-6', '1704.00339-2-2-6'], ['1704.00339-1-20-0', '1704.00339-2-20-0'], ['1704.00339-1-20-1', '1704.00339-2-20-1'], ['1704.00339-1-20-2', '1704.00339-2-20-2'], ['1704.00339-1-20-3', '1704.00339-2-20-3'], ['1704.00339-1-20-4', '1704.00339-2-20-4'], ['1704.00339-1-20-5', '1704.00339-2-20-5'], ['1704.00339-1-20-6', '1704.00339-2-20-6'], ['1704.00339-1-20-7', '1704.00339-2-20-7'], ['1704.00339-1-15-0', '1704.00339-2-15-0'], ['1704.00339-1-15-1', '1704.00339-2-15-1'], ['1704.00339-1-15-2', '1704.00339-2-15-2'], ['1704.00339-1-15-3', '1704.00339-2-15-3'], ['1704.00339-1-15-4', '1704.00339-2-15-4'], ['1704.00339-1-15-5', '1704.00339-2-15-5'], ['1704.00339-1-15-6', '1704.00339-2-15-6'], ['1704.00339-1-0-0', '1704.00339-2-0-0'], ['1704.00339-1-0-1', '1704.00339-2-0-1'], ['1704.00339-1-0-2', '1704.00339-2-0-2'], ['1704.00339-1-0-3', '1704.00339-2-0-3'], ['1704.00339-1-0-4', '1704.00339-2-0-4'], ['1704.00339-1-0-5', '1704.00339-2-0-5'], ['1704.00339-1-0-6', '1704.00339-2-0-6'], ['1704.00339-1-0-7', '1704.00339-2-0-7'], ['1704.00339-1-13-0', '1704.00339-2-13-0'], ['1704.00339-1-13-1', '1704.00339-2-13-1'], ['1704.00339-1-13-2', '1704.00339-2-13-2'], ['1704.00339-1-13-3', '1704.00339-2-13-3']]

[['1704.00339-1-11-0', '1704.00339-2-11-0'], ['1704.00339-1-11-1', '1704.00339-2-11-1'], ['1704.00339-1-11-2', '1704.00339-2-11-2'], ['1704.00339-1-11-3', '1704.00339-2-11-3'], ['1704.00339-1-11-4', '1704.00339-2-11-4'], ['1704.00339-1-11-5', '1704.00339-2-11-5'], ['1704.00339-1-11-6', '1704.00339-2-11-6'], ['1704.00339-1-11-7', '1704.00339-2-11-7'], ['1704.00339-1-11-8', '1704.00339-2-11-8'], ['1704.00339-1-11-9', '1704.00339-2-11-9'], ['1704.00339-1-11-10', '1704.00339-2-11-10'], ['1704.00339-1-11-11', '1704.00339-2-11-11'], ['1704.00339-1-11-12', '1704.00339-2-11-12'], ['1704.00339-1-11-13', '1704.00339-2-11-13'], ['1704.00339-1-11-14', '1704.00339-2-11-14'], ['1704.00339-1-4-0', '1704.00339-2-4-0'], ['1704.00339-1-4-1', '1704.00339-2-4-1'], ['1704.00339-1-4-2', '1704.00339-2-4-2'], ['1704.00339-1-4-3', '1704.00339-2-4-3'], ['1704.00339-1-4-4', '1704.00339-2-4-4'], ['1704.00339-1-4-5', '1704.00339-2-4-5'], ['1704.00339-1-4-6', '1704.00339-2-4-6'], ['1704.00339-1-4-7', '1704.00339-2-4-7'], ['1704.00339-1-4-8', '1704.00339-2-4-8'], ['1704.00339-1-4-9', '1704.00339-2-4-9'], ['1704.00339-1-4-10', '1704.00339-2-4-10'], ['1704.00339-1-19-0', '1704.00339-2-19-0'], ['1704.00339-1-19-1', '1704.00339-2-19-1'], ['1704.00339-1-19-2', '1704.00339-2-19-2'], ['1704.00339-1-19-3', '1704.00339-2-19-3'], ['1704.00339-1-3-0', '1704.00339-2-3-0'], ['1704.00339-1-3-1', '1704.00339-2-3-1'], ['1704.00339-1-3-2', '1704.00339-2-3-2'], ['1704.00339-1-3-3', '1704.00339-2-3-3'], ['1704.00339-1-3-4', '1704.00339-2-3-4'], ['1704.00339-1-3-5', '1704.00339-2-3-5'], ['1704.00339-1-3-6', '1704.00339-2-3-6'], ['1704.00339-1-16-0', '1704.00339-2-16-0'], ['1704.00339-1-16-1', '1704.00339-2-16-1'], ['1704.00339-1-16-2', '1704.00339-2-16-2'], ['1704.00339-1-16-3', '1704.00339-2-16-3'], ['1704.00339-1-16-4', '1704.00339-2-16-4'], ['1704.00339-1-16-5', '1704.00339-2-16-5'], ['1704.00339-1-16-6', '1704.00339-2-16-6'], ['1704.00339-1-14-0', '1704.00339-2-14-0'], ['1704.00339-1-14-1', '1704.00339-2-14-1'], ['1704.00339-1-14-2', '1704.00339-2-14-2'], ['1704.00339-1-14-3', '1704.00339-2-14-3'], ['1704.00339-1-14-4', '1704.00339-2-14-4'], ['1704.00339-1-14-5', '1704.00339-2-14-5'], ['1704.00339-1-23-0', '1704.00339-2-23-0'], ['1704.00339-1-23-1', '1704.00339-2-23-1'], ['1704.00339-1-23-2', '1704.00339-2-23-2'], ['1704.00339-1-23-3', '1704.00339-2-23-3'], ['1704.00339-1-23-4', '1704.00339-2-23-4'], ['1704.00339-1-23-5', '1704.00339-2-23-5'], ['1704.00339-1-23-6', '1704.00339-2-23-6'], ['1704.00339-1-23-7', '1704.00339-2-23-7'], ['1704.00339-1-12-0', '1704.00339-2-12-0'], ['1704.00339-1-12-1', '1704.00339-2-12-1'], ['1704.00339-1-12-2', '1704.00339-2-12-2'], ['1704.00339-1-17-0', '1704.00339-2-17-0'], ['1704.00339-1-17-1', '1704.00339-2-17-1'], ['1704.00339-1-21-0', '1704.00339-2-21-0'], ['1704.00339-1-21-1', '1704.00339-2-21-1'], ['1704.00339-1-21-2', '1704.00339-2-21-2'], ['1704.00339-1-21-3', '1704.00339-2-21-3'], ['1704.00339-1-21-4', '1704.00339-2-21-4'], ['1704.00339-1-21-5', '1704.00339-2-21-5'], ['1704.00339-1-6-0', '1704.00339-2-6-0'], ['1704.00339-1-6-1', '1704.00339-2-6-1'], ['1704.00339-1-6-2', '1704.00339-2-6-2'], ['1704.00339-1-6-3', '1704.00339-2-6-3'], ['1704.00339-1-6-4', '1704.00339-2-6-4'], ['1704.00339-1-6-5', '1704.00339-2-6-5'], ['1704.00339-1-6-6', '1704.00339-2-6-6'], ['1704.00339-1-6-7', '1704.00339-2-6-7'], ['1704.00339-1-6-8', '1704.00339-2-6-8'], ['1704.00339-1-6-9', '1704.00339-2-6-9'], ['1704.00339-1-7-0', '1704.00339-2-7-0'], ['1704.00339-1-7-1', '1704.00339-2-7-1'], ['1704.00339-1-7-2', '1704.00339-2-7-2'], ['1704.00339-1-7-3', '1704.00339-2-7-3'], ['1704.00339-1-7-4', '1704.00339-2-7-4'], ['1704.00339-1-7-5', '1704.00339-2-7-5'], ['1704.00339-1-7-6', '1704.00339-2-7-6'], ['1704.00339-1-7-7', '1704.00339-2-7-7'], ['1704.00339-1-7-8', '1704.00339-2-7-8'], ['1704.00339-1-7-9', '1704.00339-2-7-9'], ['1704.00339-1-9-0', '1704.00339-2-9-0'], ['1704.00339-1-9-1', '1704.00339-2-9-1'], ['1704.00339-1-9-2', '1704.00339-2-9-2'], ['1704.00339-1-9-3', '1704.00339-2-9-3'], ['1704.00339-1-9-4', '1704.00339-2-9-4'], ['1704.00339-1-9-5', '1704.00339-2-9-5'], ['1704.00339-1-9-6', '1704.00339-2-9-6'], ['1704.00339-1-9-7', '1704.00339-2-9-7'], ['1704.00339-1-9-8', '1704.00339-2-9-8'], ['1704.00339-1-10-0', '1704.00339-2-10-0'], ['1704.00339-1-10-1', '1704.00339-2-10-1'], ['1704.00339-1-10-2', '1704.00339-2-10-2'], ['1704.00339-1-10-3', '1704.00339-2-10-3'], ['1704.00339-1-10-4', '1704.00339-2-10-4'], ['1704.00339-1-10-5', '1704.00339-2-10-5'], ['1704.00339-1-10-6', '1704.00339-2-10-6'], ['1704.00339-1-10-7', '1704.00339-2-10-7'], ['1704.00339-1-10-8', '1704.00339-2-10-8'], ['1704.00339-1-10-9', '1704.00339-2-10-9'], ['1704.00339-1-10-10', '1704.00339-2-10-10'], ['1704.00339-1-10-11', '1704.00339-2-10-11'], ['1704.00339-1-10-12', '1704.00339-2-10-12'], ['1704.00339-1-10-13', '1704.00339-2-10-13'], ['1704.00339-1-25-0', '1704.00339-2-25-0'], ['1704.00339-1-25-2', '1704.00339-2-25-2'], ['1704.00339-1-2-0', '1704.00339-2-2-0'], ['1704.00339-1-2-1', '1704.00339-2-2-1'], ['1704.00339-1-2-2', '1704.00339-2-2-2'], ['1704.00339-1-2-3', '1704.00339-2-2-3'], ['1704.00339-1-2-4', '1704.00339-2-2-4'], ['1704.00339-1-2-5', '1704.00339-2-2-5'], ['1704.00339-1-2-6', '1704.00339-2-2-6'], ['1704.00339-1-20-0', '1704.00339-2-20-0'], ['1704.00339-1-20-1', '1704.00339-2-20-1'], ['1704.00339-1-20-2', '1704.00339-2-20-2'], ['1704.00339-1-20-3', '1704.00339-2-20-3'], ['1704.00339-1-20-4', '1704.00339-2-20-4'], ['1704.00339-1-20-5', '1704.00339-2-20-5'], ['1704.00339-1-20-6', '1704.00339-2-20-6'], ['1704.00339-1-20-7', '1704.00339-2-20-7'], ['1704.00339-1-15-0', '1704.00339-2-15-0'], ['1704.00339-1-15-1', '1704.00339-2-15-1'], ['1704.00339-1-15-2', '1704.00339-2-15-2'], ['1704.00339-1-15-3', '1704.00339-2-15-3'], ['1704.00339-1-15-4', '1704.00339-2-15-4'], ['1704.00339-1-15-5', '1704.00339-2-15-5'], ['1704.00339-1-15-6', '1704.00339-2-15-6'], ['1704.00339-1-0-0', '1704.00339-2-0-0'], ['1704.00339-1-0-1', '1704.00339-2-0-1'], ['1704.00339-1-0-2', '1704.00339-2-0-2'], ['1704.00339-1-0-3', '1704.00339-2-0-3'], ['1704.00339-1-0-4', '1704.00339-2-0-4'], ['1704.00339-1-0-5', '1704.00339-2-0-5'], ['1704.00339-1-0-6', '1704.00339-2-0-6'], ['1704.00339-1-0-7', '1704.00339-2-0-7'], ['1704.00339-1-13-0', '1704.00339-2-13-0'], ['1704.00339-1-13-1', '1704.00339-2-13-1'], ['1704.00339-1-13-2', '1704.00339-2-13-2'], ['1704.00339-1-13-3', '1704.00339-2-13-3']]

[]

[]

[]

[]

['1704.00339-1-17-2', '1704.00339-1-18-0', '1704.00339-1-25-1', '1704.00339-2-17-2', '1704.00339-2-18-0', '1704.00339-2-25-1']

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

https://arxiv.org/abs/1704.00339

1501.05084

{'1501.05084-1-0-0': 'We study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules.', '1501.05084-1-0-1': 'We calculate the corresponding strong coupling form factors defining these vertices using a three point correlation function.', '1501.05084-1-0-2': 'We obtain the numerical values of the corresponding strong coupling constants via different Dirac structures entering the calculations.', '1501.05084-1-1-0': 'PACS number(s): 13.30.-a, 13.30.', '1501.05084-1-1-1': 'Eg, 11.55.', '1501.05084-1-1-2': 'Hx', '1501.05084-1-2-0': '# Introduction', '1501.05084-1-3-0': 'In the last few years, significant experimental progresses have been made on the spectroscopic and decay properties of heavy hadrons, which accompanied by theoretical studies on various properties of these hadrons.', '1501.05084-1-3-1': 'The mass spectrum of the baryons containing heavy quark has been studied via various methods (see for instance [CITATION] and references therein).', '1501.05084-1-3-2': 'The necessity of a deeper understanding of heavy flavor physics requires a comprehensive study on the processes of baryons containing a heavy quark such as their radiative, strong and weak decays (for some related studies see [CITATION] and references therein).', '1501.05084-1-4-0': 'The investigation of the strong decays of heavy baryons can help us get valuable information on the perturbative and non-perturbative natures of QCD.', '1501.05084-1-4-1': 'The strong coupling constants defining such decays play important role in describing the strong interaction among the heavy baryons and other participated particles.', '1501.05084-1-4-2': 'Therefore, accurate determination of these coupling constants improves our understanding on the interactions as well as the nature and structure of the participated particles.', '1501.05084-1-4-3': 'The present work is an extension of our previous study on the coupling constants [MATH] and [MATH] [CITATION].', '1501.05084-1-4-4': 'Here, we study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules [CITATION].', '1501.05084-1-4-5': 'In particular, we calculate the strong coupling constants [MATH] and [MATH].', '1501.05084-1-4-6': 'These coupling constants together with the [MATH] and [MATH] discussed in our previous work, may also be used in the bottom and charmed mesons clouds description of the nucleon which can be used to explain the exotic events observed by different Collaborations.', '1501.05084-1-4-7': 'In addition, the determination of the properties of the [MATH] and [MATH] mesons in nuclear medium requires the consideration of their interactions with the nucleons, i.e. [EQUATION]', '1501.05084-1-4-8': 'Therefore, to determine the modifications on the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium, one needs to consider the contributions of the baryons [MATH] together with the [MATH] and have the values of the strong coupling constants [MATH] and [MATH] besides the couplings [MATH] and [MATH] [CITATION].', '1501.05084-1-4-9': 'In the literature, one can unfortunately find only a few works on the strong couplings of the heavy baryons with the nucleon and heavy mesons.', '1501.05084-1-4-10': 'One approximate prediction for the strong coupling [MATH] was made at zero transferred momentum squared taking the Borel masses in the initial and final channels as the same [CITATION].', '1501.05084-1-4-11': 'The strong couplings of the charmed baryons with the nucleon and [MATH] meson were also discussed in [CITATION] in the framework of light cone QCD sum rules.', '1501.05084-1-5-0': 'This paper is organized in three sections as follows.', '1501.05084-1-5-1': 'In the next section, we present the details of the calculations of the strong coupling form factors among the particles under consideration.', '1501.05084-1-5-2': 'In section 3, the numerical analysis of the obtained sum rules and discussions about the results are presented.', '1501.05084-1-6-0': '# Theoretical framework', '1501.05084-1-7-0': 'This section is devoted to the details of the calculations of the strong coupling form factors [MATH] and [MATH] from which the strong coupling constants among the participating particles are obtained at [MATH], subsequently.', '1501.05084-1-7-1': 'In order to accomplish this purpose, the following three-point correlation function is used as a starting point: [EQUATION] where [MATH] denotes the time ordering operator and [MATH] is the transferred momentum.', '1501.05084-1-7-2': 'The interpolating currents included in the three-point correlation function can be written in terms of the quark field operators as: [EQUATION] where [MATH] is the charge conjugation operator; and [MATH], [MATH] and [MATH] are color indices.', '1501.05084-1-8-0': 'In the course of calculation of the three-point correlation function one follows two different ways.', '1501.05084-1-8-1': 'In the first way, called as OPE side, the calculation is made in terms of quark and gluon degrees of freedom using the operator product expansion in deep Euclidean region.', '1501.05084-1-8-2': 'In the second way, called as hadronic side, one considers the hadronic degrees of freedom to calculate it.', '1501.05084-1-8-3': 'The QCD sum rules for the coupling form factors are attained via the match of these two sides.', '1501.05084-1-8-4': 'To suppress the contributions of the higher states and continuum a double Borel transformation with respect to the variables [MATH] and [MATH] are applied to both sides.', '1501.05084-1-9-0': '## OPE Side', '1501.05084-1-10-0': 'The OPE side of the correlation function which is calculated in deep Euclidean region, where [MATH] and [MATH], requires the insertion of the explicit expressions of the interpolating currents into the correlation function, Eq. ([REF]).', '1501.05084-1-10-1': "Possible contraction of all quark pairs via Wick's theorem leads to [EQUATION] where [MATH] represents the heavy quark propagator which is given by [CITATION] [EQUATION] and [MATH] and [MATH] are the light quark propagators and are given by [EQUATION]", '1501.05084-1-10-2': 'After some straightforward calculations (for details refer to the Ref. [CITATION]), the correlation function in OPE side comes out in terms of different Dirac structures as [EQUATION] where each [MATH] function includes the perturbative and non-perturbative parts and is written as [EQUATION]', '1501.05084-1-10-3': 'The spectral densities, [MATH], appearing in Eq. ([REF]) are obtained from the imaginary parts of the [MATH] functions, viz. [MATH].', '1501.05084-1-10-4': 'Here to provide examples of the explicit forms of the spectral densities, among the Dirac structures presented above, we only present the results obtained for the Dirac structure [MATH], that is [MATH] and [MATH], which are obtained as [EQUATION] and [EQUATION] where [MATH] stands for the unit-step function and [MATH] and [MATH] are defined as [EQUATION]', '1501.05084-1-11-0': '## Hadronic Side', '1501.05084-1-12-0': 'On the hadronic side, the correlation function is saturated with the complete sets of intermediate [MATH], [MATH] and [MATH] hadronic states having the same quantum numbers as their interpolating currents.', '1501.05084-1-12-1': 'After performing the four-integrals, we get [EQUATION] where [MATH] represents the contributions coming from the higher states and continuum.', '1501.05084-1-12-2': 'We use the following parameterizations for the matrix elements appearing in the above equation [EQUATION] where [MATH] and [MATH] are residues of the [MATH] and [MATH] baryons, respectively, [MATH] is the leptonic decay constant of [MATH] meson and [MATH] is the strong coupling form factor among [MATH], [MATH] and [MATH] particles.', '1501.05084-1-12-3': 'Using Eq. ([REF]) in Eq. ([REF]) and summing over the spins of the particles, we obtain [EQUATION]', '1501.05084-1-12-4': 'To achieve the final form of the hadronic side of the correlation function we apply the double Borel transformation with respect to the initial and final momenta squared, viz. [EQUATION] where [MATH] and [MATH] are Borel mass parameters.', '1501.05084-1-13-0': 'As it was already stated, the match of the hadronic and OPE sides of the correlation function in Borel scheme provide us with the QCD sum rules for the strong form factors.', '1501.05084-1-13-1': 'The consequence of that match for [MATH] structure leads us to [EQUATION] where [MATH] and [MATH] are continuum thresholds in [MATH] and [MATH] channels, respectively.', '1501.05084-1-14-0': '# Numerical results', '1501.05084-1-15-0': 'Having obtained the QCD sum rules for the strong coupling form factors, in this section we present the numerical analysis of our results and discuss the behavior of the strong coupling form factors under consideration with respect to [MATH].', '1501.05084-1-15-1': 'To this aim, beside the input parameters given in table 1, one needs to determine the working intervals of four auxiliary parameters [MATH], [MATH], [MATH] and [MATH].', '1501.05084-1-15-2': 'These parameters originate from the double Borel transformation and continuum subtraction.', '1501.05084-1-15-3': 'The determination of the working regions of them are made on the basis that the results obtained for the strong coupling form factors be roughly independent of these helping parameters.', '1501.05084-1-16-0': 'The continuum thresholds [MATH] and [MATH] depend on the energies of the first excited states in the initial and final states.', '1501.05084-1-16-1': 'They are the energy squares which characterize the beginning of the continuum in the initial and final channels.', '1501.05084-1-16-2': 'If we show the ground states masses in the initial and final channels by [MATH] and [MATH], respectively, the quantities [MATH] and [MATH] are the energies needed to excite the initial and final particles to their first excited states with the same quantum numbers.', '1501.05084-1-16-3': 'These quantities are well known for the states under consideration [CITATION], where they take place roughly between [MATH] and [MATH].', '1501.05084-1-16-4': 'These values lead to the working intervals of the continuum thresholds as [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the strong vertex [MATH].', '1501.05084-1-17-0': 'To determine the Borel parameters [MATH] and [MATH], there are two main consideration which are pole dominance and convergence of the OPE.', '1501.05084-1-17-1': 'The working intervals for these parameters are established on the requirement that the pole contribution exceeds the contributions of the higher states and continuum, and that the contribution of the perturbative part exceeds the non-perturbative contributions.', '1501.05084-1-17-2': 'These considerations lead to the windows [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the Borel mass parameters of the strong vertex [MATH] and for these intervals our results have weak dependencies on the Borel mass parameters (see figures 1-2).', '1501.05084-1-18-0': 'Subsequent to the determination of the auxiliary parameters, their windows together with the other input parameters are used to find out the dependency of the strong coupling form factors on [MATH].', '1501.05084-1-18-1': 'From our analysis we observe that the dependency of the strong coupling form factors on [MATH] is well described by the following fit function: [EQUATION] where the values of the parameters [MATH], [MATH] and [MATH] for different structures are presented in tables [REF] and [REF] for [MATH] and [MATH], respectively.', '1501.05084-1-18-2': 'Considering the average values of the continuum thresholds and Borel mass parameters we demonstrate the dependence of the strong coupling form factors on [MATH] for both the QCD sum rules and fitting results in figure 3.', '1501.05084-1-18-3': 'The figure indicates the truncation of the QCD sum rules at some points at negative values of [MATH] and the overlap between QCD sum rules and fitting results up to these points are well.', '1501.05084-1-18-4': 'The fit function is used to determine the value of the strong coupling constant at [MATH] for all structures, and the results are presented in table [REF].', '1501.05084-1-18-5': 'The errors existing in these results arise from the uncertainties of the input parameters together with the uncertainties coming from the determination of the working regions of the auxiliary parameters.', '1501.05084-1-18-6': 'Table [REF] also contains the average of the coupling constants under consideration, obtained from all the structures used.', '1501.05084-1-19-0': 'To summarize, in this work, the strong coupling constants among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson, namely [MATH] and [MATH] have been calculated in the framework of the three-point QCD sum rules.', '1501.05084-1-19-1': 'The obtained results can be used in the analysis of the related experimental results at LHC.', '1501.05084-1-19-2': 'The predictions can also be used in the bottom and charmed mesons clouds description of the nucleon that may be applied for the explanation of the exotic events observed by different experiments.', '1501.05084-1-19-3': 'These results may also serve the purpose of analyzing of the results of heavy ion collision experiments like [MATH] at FAIR.', '1501.05084-1-19-4': 'Obtained results may also come in handy in the exact determinations of the modifications in the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium.'}

{'1501.05084-2-0-0': 'We study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules.', '1501.05084-2-0-1': 'We calculate the corresponding strong coupling form factors defining these vertices using a three point correlation function.', '1501.05084-2-0-2': 'We obtain the numerical values of the corresponding strong coupling constants via different Dirac structures entering the calculations.', '1501.05084-2-1-0': 'PACS number(s): 13.30.-a, 13.30.', '1501.05084-2-1-1': 'Eg, 11.55.', '1501.05084-2-1-2': 'Hx', '1501.05084-2-2-0': '# Introduction', '1501.05084-2-3-0': 'In the recent years, substantial experimental improvements have been made on the spectroscopic and decay properties of heavy hadrons, which were accompanied by theoretical studies on various properties of these hadrons.', '1501.05084-2-3-1': 'The mass spectrum of the baryons containing heavy quark has been studied using different methods.', '1501.05084-2-3-2': 'The necessity of a deeper understanding of heavy flavor physics requires a comprehensive study on the processes of these baryons such as their radiative, strong and weak decays.', '1501.05084-2-3-3': 'For some of related studies one can refer to references [CITATION].', '1501.05084-2-4-0': 'The investigation of the strong decays of heavy baryons can help us get valuable information on the perturbative and non-perturbative natures of QCD.', '1501.05084-2-4-1': 'The strong coupling constants defining such decays play important role in describing the strong interaction among the heavy baryons and other participated particles.', '1501.05084-2-4-2': 'Therefore, accurate determination of these coupling constants enhance our understanding on the interactions as well as the nature and structure of the participated particles.', '1501.05084-2-4-3': 'The present work is an extension of our previous study on the coupling constants [MATH] and [MATH] [CITATION].', '1501.05084-2-4-4': 'Here, we study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules.', '1501.05084-2-4-5': 'In particular, we calculate the strong coupling constants [MATH] and [MATH].', '1501.05084-2-4-6': 'These coupling constants together with the [MATH] and [MATH] discussed in our previous work, may also be used in the bottom and charmed mesons clouds description of the nucleon which can be used to explain the exotic events observed by different Collaborations.', '1501.05084-2-4-7': 'In addition, the determination of the properties of the [MATH] and [MATH] mesons in nuclear medium requires the consideration of their interactions with the nucleons from which the [MATH] and [MATH] are produced.', '1501.05084-2-4-8': 'Therefore, to determine the modifications on the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium, one needs to consider the contributions of the baryons [MATH] together with the [MATH] and have the values of the strong coupling constants [MATH] and [MATH] besides the couplings [MATH] and [MATH] [CITATION].', '1501.05084-2-4-9': 'In the literature, one can unfortunately find only a few works on the strong couplings of the heavy baryons with the nucleon and heavy mesons.', '1501.05084-2-4-10': 'One approximate prediction for the strong coupling [MATH] was made at zero transferred momentum squared [CITATION].', '1501.05084-2-4-11': 'The strong couplings of the charmed baryons with the nucleon and [MATH] meson were also discussed in [CITATION] in the framework of light cone QCD sum rules.', '1501.05084-2-5-0': 'This paper is organized in three sections as follows.', '1501.05084-2-5-1': 'In the next section, we present the details of the calculations of the strong coupling form factors among the particles under consideration.', '1501.05084-2-5-2': 'In section 3, the numerical analysis of the obtained sum rules and discussions about the results are presented.', '1501.05084-2-6-0': '# Theoretical framework', '1501.05084-2-7-0': 'This section is devoted to the details of the calculations of the strong coupling form factors [MATH] and [MATH] from which the strong coupling constants among the participating particles are obtained at [MATH], subsequently.', '1501.05084-2-7-1': 'In order to accomplish this purpose, the following three-point correlation function is used: [EQUATION] whith [MATH] being the time ordering operator and [MATH] is the transferred momentum.', '1501.05084-2-7-2': 'The currents [MATH], [MATH] and [MATH] presented in Eq. ([REF]) correspond to the the interpolating currents of the [MATH], [MATH] and [MATH], respectively and their explicit expressions can be given in terms of the quark field operators as [EQUATION] where [MATH] denotes the charge conjugation operator; and [MATH], [MATH] and [MATH] are color indices.', '1501.05084-2-8-0': 'In the course of calculation of the three-point correlation function one follows two different ways.', '1501.05084-2-8-1': 'The first way is called as OPE side and the calculation is made in deep Euclidean region in terms of quark and gluon degrees of freedom using the operator product expansion.', '1501.05084-2-8-2': 'The second way is called as hadronic side and the hadronic degrees of freedoms are considered to perform this side of the calculation.', '1501.05084-2-8-3': 'The QCD sum rules for the coupling form factors are attained via the match of these two sides.', '1501.05084-2-8-4': 'The contributions of the higher states and continuum are suppressed by a double Borel transformation applied to both sides with respect to the variables [MATH] and [MATH].', '1501.05084-2-9-0': '## OPE Side', '1501.05084-2-10-0': 'For the calculation of the OPE side of the correlation function which is done in deep Euclidean region, where [MATH] and [MATH], one puts the interpolating currents given in Eq. ([REF]) into the correlation function, Eq. ([REF]).', '1501.05084-2-10-1': "Possible contractions of all quark pairs via Wick's theorem leads to [EQUATION] where [MATH] and [MATH] are the heavy and light quark propagators whose explicit forms can be found in Refs. [CITATION].", '1501.05084-2-11-0': 'After some straightforward calculations (for details refer to the Ref. [CITATION]), the correlation function in OPE side comes out in terms of different Dirac structures as [EQUATION]', '1501.05084-2-11-1': 'Each [MATH] function involves the perturbative and non-perturbative parts and is written as [EQUATION]', '1501.05084-2-11-2': 'The spectral densities, [MATH], appearing in Eq. ([REF]) are obtained from the imaginary parts of the [MATH] functions, i.e., [MATH].', '1501.05084-2-11-3': 'Here to provide examples of the explicit forms of the spectral densities, among the Dirac structures presented above, we only present the results obtained for the Dirac structure [MATH], that is [MATH] and [MATH], which are obtained as [EQUATION] and [EQUATION] where [MATH] stands for the unit-step function and [MATH] and [MATH] are defined as [EQUATION]', '1501.05084-2-12-0': '## Hadronic Side', '1501.05084-2-13-0': 'On the hadronic side, considering the quantum numbers of the interpolating fields one place the complete sets of intermediate [MATH], [MATH] and [MATH] hadronic states into the correlation function.', '1501.05084-2-13-1': 'After carrying out the four-integrals, we get [EQUATION]', '1501.05084-2-13-2': 'In the above equation, the contributions of the higher states and continuum are denoted by [MATH] and the matrix elements are represented in terms of the hadronic parameters as follows: [EQUATION]', '1501.05084-2-13-3': 'Here [MATH] and [MATH] are residues of the [MATH] and [MATH] baryons, respectively, [MATH] is the leptonic decay constant of [MATH] meson and [MATH] is the strong coupling form factor among [MATH], [MATH] and [MATH] particles.', '1501.05084-2-13-4': 'Using Eq. ([REF]) in Eq. ([REF]) and summing over the spins of the particles, we obtain [EQUATION]', '1501.05084-2-13-5': 'To acquire the final form of the hadronic side of the correlation function we perform the double Borel transformation with respect to the initial and final momenta squared, [EQUATION] where [MATH] and [MATH] are Borel mass parameters.', '1501.05084-2-14-0': 'As it was already stated, the match of the hadronic and OPE sides of the correlation function in Borel scheme provides us with the QCD sum rules for the strong form factors.', '1501.05084-2-14-1': 'The consequence of that match for [MATH] structure leads us to [EQUATION] where [MATH] and [MATH] are continuum thresholds in [MATH] and [MATH] channels, respectively.', '1501.05084-2-15-0': '# Numerical analysis', '1501.05084-2-16-0': 'Having obtained the QCD sum rules for the strong coupling form factors, in this section we present the numerical analysis of our results and discuss the dependence of the strong coupling form factors under consideration on [MATH].', '1501.05084-2-16-1': 'To this aim, beside the input parameters given in table 1, one needs to determine the working intervals of four auxiliary parameters [MATH], [MATH], [MATH] and [MATH].', '1501.05084-2-16-2': 'These parameters originate from the double Borel transformation and continuum subtraction.', '1501.05084-2-16-3': 'The determination of the working regions of them is made on the basis of that the results obtained for the strong coupling form factors be roughly independent of these helping parameters.', '1501.05084-2-17-0': 'The continuum thresholds [MATH] and [MATH] are the parameters related to the beginning of the continuum in the initial and final channels.', '1501.05084-2-17-1': 'If the ground state masses are given by [MATH] and [MATH] for the initial and final channels, respectively, to excite the particle to the first excited state having the same quantum numbers with them one needs to provide the energies [MATH] and [MATH].', '1501.05084-2-17-2': 'For the considered transitions, these quantities can be determined from well known excited states of the initial and final states [CITATION] which are roughly in between [MATH] and [MATH].', '1501.05084-2-17-3': 'From these intervals, the working regions for the continuum thresholds are determined as [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the vertex [MATH].', '1501.05084-2-18-0': 'To determine the Borel parameters [MATH] and [MATH], there are two main factors that are to be considered.', '1501.05084-2-18-1': 'These are pole dominance and convergence of the OPE.', '1501.05084-2-18-2': 'The working intervals for these parameters are established on the requirement that the pole contribution exceeds the contributions of the higher states and continuum, and that the contribution of the perturbative part exceeds the non-perturbative contributions.', '1501.05084-2-18-3': 'These considerations lead to the windows [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the Borel mass parameters of the strong vertex [MATH] and for these intervals our results have weak dependence on the Borel mass parameters (see figures 1-2).', '1501.05084-2-19-0': 'Subsequent to the determination of the auxiliary parameters, their working windows together with the other input parameters are used to ascertain the dependency of the strong coupling form factors on [MATH].', '1501.05084-2-19-1': 'From our analysis we observe that the dependency of the strong coupling form factors on [MATH] is well characterized by the following fit function: [EQUATION]', '1501.05084-2-19-2': 'The values of the parameters [MATH], [MATH] and [MATH] corresponding to different structures for [MATH] and [MATH] can be seen in tables tables [REF] and [REF], respectively.', '1501.05084-2-19-3': 'Considering the average values of the continuum thresholds and Borel mass parameters we demonstrate the variation of the strong coupling form factors due to [MATH] for the QCD sum rules as well as the fitting results in figure 3.', '1501.05084-2-19-4': 'The figure indicates the truncation of the QCD sum rules at some points at negative values of [MATH].', '1501.05084-2-19-5': 'It can be seen from the figure that there is a good consistency among the results obtained from the QCD sum rules and fit function up to these points.', '1501.05084-2-19-6': 'The fit function is used to determine the value of the strong coupling constant at [MATH] for all structures, and the results are presented in table [REF].', '1501.05084-2-19-7': 'The present errors in these results originate from the uncertainties of the input parameters together with the uncertainties coming from the determination of the working regions of the auxiliary parameters.', '1501.05084-2-19-8': 'From this table we also conclude that the values of the coupling constants depend on the selected structures, considerably.', '1501.05084-2-19-9': 'We also present the average of the coupling constants under consideration in this table.', '1501.05084-2-20-0': 'To sum up, in this work, the strong coupling constants among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson, namely [MATH] and [MATH], have been calculated in the framework of the three-point QCD sum rules.', '1501.05084-2-20-1': 'The obtained results can be applied in the analysis of the related experimental results at LHC.', '1501.05084-2-20-2': 'The predictions can also be used in the bottom and charmed mesons clouds description of the nucleon that may be applied for the explanation of the exotic events observed by different experiments.', '1501.05084-2-20-3': 'These results may also serve the purpose of analyzing of the results of heavy ion collision experiments like [MATH] at FAIR.', '1501.05084-2-20-4': 'The obtained results may also come in handy in the determinations of the changes in the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium.'}

[['1501.05084-1-4-0', '1501.05084-2-4-0'], ['1501.05084-1-4-1', '1501.05084-2-4-1'], ['1501.05084-1-4-3', '1501.05084-2-4-3'], ['1501.05084-1-4-5', '1501.05084-2-4-5'], ['1501.05084-1-4-6', '1501.05084-2-4-6'], ['1501.05084-1-4-8', '1501.05084-2-4-8'], ['1501.05084-1-4-9', '1501.05084-2-4-9'], ['1501.05084-1-4-11', '1501.05084-2-4-11'], ['1501.05084-1-8-0', '1501.05084-2-8-0'], ['1501.05084-1-8-3', '1501.05084-2-8-3'], ['1501.05084-1-13-1', '1501.05084-2-14-1'], ['1501.05084-1-5-0', '1501.05084-2-5-0'], ['1501.05084-1-5-1', '1501.05084-2-5-1'], ['1501.05084-1-5-2', '1501.05084-2-5-2'], ['1501.05084-1-7-0', '1501.05084-2-7-0'], ['1501.05084-1-12-3', '1501.05084-2-13-4'], ['1501.05084-1-19-2', '1501.05084-2-20-2'], ['1501.05084-1-19-3', '1501.05084-2-20-3'], ['1501.05084-1-18-4', '1501.05084-2-19-6'], ['1501.05084-1-15-1', '1501.05084-2-16-1'], ['1501.05084-1-15-2', '1501.05084-2-16-2'], ['1501.05084-1-0-0', '1501.05084-2-0-0'], ['1501.05084-1-0-1', '1501.05084-2-0-1'], ['1501.05084-1-0-2', '1501.05084-2-0-2'], ['1501.05084-1-17-1', '1501.05084-2-18-2'], ['1501.05084-2-0-0', '1501.05084-3-0-0'], ['1501.05084-2-7-0', '1501.05084-3-7-0'], ['1501.05084-2-7-1', '1501.05084-3-7-1'], ['1501.05084-2-7-2', '1501.05084-3-7-2'], ['1501.05084-2-16-1', '1501.05084-3-16-1'], ['1501.05084-2-16-2', '1501.05084-3-16-2'], ['1501.05084-2-16-3', '1501.05084-3-16-3'], ['1501.05084-2-20-0', '1501.05084-3-22-0'], ['1501.05084-2-20-1', '1501.05084-3-22-1'], ['1501.05084-2-20-2', '1501.05084-3-22-2'], ['1501.05084-2-20-3', '1501.05084-3-22-3'], ['1501.05084-2-20-4', '1501.05084-3-22-4'], ['1501.05084-2-10-0', '1501.05084-3-10-0'], ['1501.05084-2-10-1', '1501.05084-3-10-1'], ['1501.05084-2-4-0', '1501.05084-3-4-0'], ['1501.05084-2-4-1', '1501.05084-3-4-1'], ['1501.05084-2-4-2', '1501.05084-3-4-2'], ['1501.05084-2-4-3', '1501.05084-3-4-3'], ['1501.05084-2-4-4', '1501.05084-3-4-4'], ['1501.05084-2-4-5', '1501.05084-3-4-5'], ['1501.05084-2-4-6', '1501.05084-3-4-6'], ['1501.05084-2-4-7', '1501.05084-3-4-7'], ['1501.05084-2-4-8', '1501.05084-3-4-8'], ['1501.05084-2-4-9', '1501.05084-3-4-9'], ['1501.05084-2-4-10', '1501.05084-3-4-10'], ['1501.05084-2-4-11', '1501.05084-3-4-11'], ['1501.05084-2-19-0', '1501.05084-3-21-0'], ['1501.05084-2-19-1', '1501.05084-3-21-1'], ['1501.05084-2-19-4', '1501.05084-3-21-4'], ['1501.05084-2-19-5', '1501.05084-3-21-5'], ['1501.05084-2-13-0', '1501.05084-3-13-0'], ['1501.05084-2-13-1', '1501.05084-3-13-1'], ['1501.05084-2-13-2', '1501.05084-3-13-2'], ['1501.05084-2-13-3', '1501.05084-3-13-3'], ['1501.05084-2-13-4', '1501.05084-3-13-4'], ['1501.05084-2-13-5', '1501.05084-3-13-5'], ['1501.05084-2-17-0', '1501.05084-3-17-0'], ['1501.05084-2-17-1', '1501.05084-3-17-1'], ['1501.05084-2-17-2', '1501.05084-3-17-2'], ['1501.05084-2-17-3', '1501.05084-3-17-3'], ['1501.05084-2-14-0', '1501.05084-3-14-0'], ['1501.05084-2-14-1', '1501.05084-3-14-1'], ['1501.05084-2-5-0', '1501.05084-3-5-0'], ['1501.05084-2-5-1', '1501.05084-3-5-1'], ['1501.05084-2-5-2', '1501.05084-3-5-2'], ['1501.05084-2-11-0', '1501.05084-3-11-0'], ['1501.05084-2-11-1', '1501.05084-3-11-1'], ['1501.05084-2-11-2', '1501.05084-3-11-2'], ['1501.05084-2-11-3', '1501.05084-3-11-3'], ['1501.05084-2-3-0', '1501.05084-3-3-0'], ['1501.05084-2-3-1', '1501.05084-3-3-1'], ['1501.05084-2-3-2', '1501.05084-3-3-2'], ['1501.05084-2-3-3', '1501.05084-3-3-3'], ['1501.05084-2-8-0', '1501.05084-3-8-0'], ['1501.05084-2-8-1', '1501.05084-3-8-1'], ['1501.05084-2-8-2', '1501.05084-3-8-2'], ['1501.05084-2-8-3', '1501.05084-3-8-3'], ['1501.05084-2-8-4', '1501.05084-3-8-4'], ['1501.05084-1-10-4', '1501.05084-2-11-3'], ['1501.05084-1-4-2', '1501.05084-2-4-2'], ['1501.05084-1-4-4', '1501.05084-2-4-4'], ['1501.05084-1-8-1', '1501.05084-2-8-1'], ['1501.05084-1-8-4', '1501.05084-2-8-4'], ['1501.05084-1-13-0', '1501.05084-2-14-0'], ['1501.05084-1-7-1', '1501.05084-2-7-1'], ['1501.05084-1-12-4', '1501.05084-2-13-5'], ['1501.05084-1-19-0', '1501.05084-2-20-0'], ['1501.05084-1-19-1', '1501.05084-2-20-1'], ['1501.05084-1-19-4', '1501.05084-2-20-4'], ['1501.05084-1-3-0', '1501.05084-2-3-0'], ['1501.05084-1-18-0', '1501.05084-2-19-0'], ['1501.05084-1-18-2', '1501.05084-2-19-3'], ['1501.05084-1-18-5', '1501.05084-2-19-7'], ['1501.05084-1-15-0', '1501.05084-2-16-0'], ['1501.05084-1-15-3', '1501.05084-2-16-3'], ['1501.05084-1-17-2', '1501.05084-2-18-3'], ['1501.05084-2-0-1', '1501.05084-3-0-1'], ['1501.05084-2-0-2', '1501.05084-3-0-2'], ['1501.05084-2-16-0', '1501.05084-3-16-0'], ['1501.05084-2-19-2', '1501.05084-3-21-2'], ['1501.05084-2-19-3', '1501.05084-3-21-3'], ['1501.05084-2-19-6', '1501.05084-3-21-6'], ['1501.05084-2-19-7', '1501.05084-3-21-7'], ['1501.05084-1-10-2', '1501.05084-2-11-0'], ['1501.05084-1-10-3', '1501.05084-2-11-2'], ['1501.05084-1-4-7', '1501.05084-2-4-7'], ['1501.05084-1-4-10', '1501.05084-2-4-10'], ['1501.05084-1-8-2', '1501.05084-2-8-2'], ['1501.05084-1-7-2', '1501.05084-2-7-2'], ['1501.05084-1-12-0', '1501.05084-2-13-0'], ['1501.05084-1-12-1', '1501.05084-2-13-1'], ['1501.05084-1-12-2', '1501.05084-2-13-3'], ['1501.05084-1-3-1', '1501.05084-2-3-1'], ['1501.05084-1-3-2', '1501.05084-2-3-2'], ['1501.05084-1-16-0', '1501.05084-2-17-0'], ['1501.05084-1-16-2', '1501.05084-2-17-1'], ['1501.05084-1-16-3', '1501.05084-2-17-2'], ['1501.05084-1-16-4', '1501.05084-2-17-3'], ['1501.05084-1-18-1', '1501.05084-2-19-1'], ['1501.05084-1-18-3', '1501.05084-2-19-4'], ['1501.05084-1-18-3', '1501.05084-2-19-5'], ['1501.05084-1-18-6', '1501.05084-2-19-8'], ['1501.05084-1-18-6', '1501.05084-2-19-9'], ['1501.05084-1-17-0', '1501.05084-2-18-0'], ['1501.05084-1-17-0', '1501.05084-2-18-1'], ['1501.05084-2-18-1', '1501.05084-3-20-1'], ['1501.05084-2-18-2', '1501.05084-3-20-1'], ['1501.05084-2-18-2', '1501.05084-3-20-2'], ['1501.05084-2-18-3', '1501.05084-3-20-2'], ['1501.05084-2-18-3', '1501.05084-3-20-3'], ['1501.05084-1-10-0', '1501.05084-2-10-0'], ['1501.05084-1-10-1', '1501.05084-2-10-1']]

[['1501.05084-1-4-0', '1501.05084-2-4-0'], ['1501.05084-1-4-1', '1501.05084-2-4-1'], ['1501.05084-1-4-3', '1501.05084-2-4-3'], ['1501.05084-1-4-5', '1501.05084-2-4-5'], ['1501.05084-1-4-6', '1501.05084-2-4-6'], ['1501.05084-1-4-8', '1501.05084-2-4-8'], ['1501.05084-1-4-9', '1501.05084-2-4-9'], ['1501.05084-1-4-11', '1501.05084-2-4-11'], ['1501.05084-1-8-0', '1501.05084-2-8-0'], ['1501.05084-1-8-3', '1501.05084-2-8-3'], ['1501.05084-1-13-1', '1501.05084-2-14-1'], ['1501.05084-1-5-0', '1501.05084-2-5-0'], ['1501.05084-1-5-1', '1501.05084-2-5-1'], ['1501.05084-1-5-2', '1501.05084-2-5-2'], ['1501.05084-1-7-0', '1501.05084-2-7-0'], ['1501.05084-1-12-3', '1501.05084-2-13-4'], ['1501.05084-1-19-2', '1501.05084-2-20-2'], ['1501.05084-1-19-3', '1501.05084-2-20-3'], ['1501.05084-1-18-4', '1501.05084-2-19-6'], ['1501.05084-1-15-1', '1501.05084-2-16-1'], ['1501.05084-1-15-2', '1501.05084-2-16-2'], ['1501.05084-1-0-0', '1501.05084-2-0-0'], ['1501.05084-1-0-1', '1501.05084-2-0-1'], ['1501.05084-1-0-2', '1501.05084-2-0-2'], ['1501.05084-1-17-1', '1501.05084-2-18-2'], ['1501.05084-2-0-0', '1501.05084-3-0-0'], ['1501.05084-2-7-0', '1501.05084-3-7-0'], ['1501.05084-2-7-1', '1501.05084-3-7-1'], ['1501.05084-2-7-2', '1501.05084-3-7-2'], ['1501.05084-2-16-1', '1501.05084-3-16-1'], ['1501.05084-2-16-2', '1501.05084-3-16-2'], ['1501.05084-2-16-3', '1501.05084-3-16-3'], ['1501.05084-2-20-0', '1501.05084-3-22-0'], ['1501.05084-2-20-1', '1501.05084-3-22-1'], ['1501.05084-2-20-2', '1501.05084-3-22-2'], ['1501.05084-2-20-3', '1501.05084-3-22-3'], ['1501.05084-2-20-4', '1501.05084-3-22-4'], ['1501.05084-2-10-0', '1501.05084-3-10-0'], ['1501.05084-2-10-1', '1501.05084-3-10-1'], ['1501.05084-2-4-0', '1501.05084-3-4-0'], ['1501.05084-2-4-1', '1501.05084-3-4-1'], ['1501.05084-2-4-2', '1501.05084-3-4-2'], ['1501.05084-2-4-3', '1501.05084-3-4-3'], ['1501.05084-2-4-4', '1501.05084-3-4-4'], ['1501.05084-2-4-5', '1501.05084-3-4-5'], ['1501.05084-2-4-6', '1501.05084-3-4-6'], ['1501.05084-2-4-7', '1501.05084-3-4-7'], ['1501.05084-2-4-8', '1501.05084-3-4-8'], ['1501.05084-2-4-9', '1501.05084-3-4-9'], ['1501.05084-2-4-10', '1501.05084-3-4-10'], ['1501.05084-2-4-11', '1501.05084-3-4-11'], ['1501.05084-2-19-0', '1501.05084-3-21-0'], ['1501.05084-2-19-1', '1501.05084-3-21-1'], ['1501.05084-2-19-4', '1501.05084-3-21-4'], ['1501.05084-2-19-5', '1501.05084-3-21-5'], ['1501.05084-2-13-0', '1501.05084-3-13-0'], ['1501.05084-2-13-1', '1501.05084-3-13-1'], ['1501.05084-2-13-2', '1501.05084-3-13-2'], ['1501.05084-2-13-3', '1501.05084-3-13-3'], ['1501.05084-2-13-4', '1501.05084-3-13-4'], ['1501.05084-2-13-5', '1501.05084-3-13-5'], ['1501.05084-2-17-0', '1501.05084-3-17-0'], ['1501.05084-2-17-1', '1501.05084-3-17-1'], ['1501.05084-2-17-2', '1501.05084-3-17-2'], ['1501.05084-2-17-3', '1501.05084-3-17-3'], ['1501.05084-2-14-0', '1501.05084-3-14-0'], ['1501.05084-2-14-1', '1501.05084-3-14-1'], ['1501.05084-2-5-0', '1501.05084-3-5-0'], ['1501.05084-2-5-1', '1501.05084-3-5-1'], ['1501.05084-2-5-2', '1501.05084-3-5-2'], ['1501.05084-2-11-0', '1501.05084-3-11-0'], ['1501.05084-2-11-1', '1501.05084-3-11-1'], ['1501.05084-2-11-2', '1501.05084-3-11-2'], ['1501.05084-2-11-3', '1501.05084-3-11-3'], ['1501.05084-2-3-0', '1501.05084-3-3-0'], ['1501.05084-2-3-1', '1501.05084-3-3-1'], ['1501.05084-2-3-2', '1501.05084-3-3-2'], ['1501.05084-2-3-3', '1501.05084-3-3-3'], ['1501.05084-2-8-0', '1501.05084-3-8-0'], ['1501.05084-2-8-1', '1501.05084-3-8-1'], ['1501.05084-2-8-2', '1501.05084-3-8-2'], ['1501.05084-2-8-3', '1501.05084-3-8-3'], ['1501.05084-2-8-4', '1501.05084-3-8-4'], ['1501.05084-1-10-4', '1501.05084-2-11-3']]

[['1501.05084-1-4-2', '1501.05084-2-4-2'], ['1501.05084-1-4-4', '1501.05084-2-4-4'], ['1501.05084-1-8-1', '1501.05084-2-8-1'], ['1501.05084-1-8-4', '1501.05084-2-8-4'], ['1501.05084-1-13-0', '1501.05084-2-14-0'], ['1501.05084-1-7-1', '1501.05084-2-7-1'], ['1501.05084-1-12-4', '1501.05084-2-13-5'], ['1501.05084-1-19-0', '1501.05084-2-20-0'], ['1501.05084-1-19-1', '1501.05084-2-20-1'], ['1501.05084-1-19-4', '1501.05084-2-20-4'], ['1501.05084-1-3-0', '1501.05084-2-3-0'], ['1501.05084-1-18-0', '1501.05084-2-19-0'], ['1501.05084-1-18-2', '1501.05084-2-19-3'], ['1501.05084-1-18-5', '1501.05084-2-19-7'], ['1501.05084-1-15-0', '1501.05084-2-16-0'], ['1501.05084-1-15-3', '1501.05084-2-16-3'], ['1501.05084-1-17-2', '1501.05084-2-18-3'], ['1501.05084-2-0-1', '1501.05084-3-0-1'], ['1501.05084-2-0-2', '1501.05084-3-0-2'], ['1501.05084-2-16-0', '1501.05084-3-16-0'], ['1501.05084-2-19-2', '1501.05084-3-21-2'], ['1501.05084-2-19-3', '1501.05084-3-21-3'], ['1501.05084-2-19-6', '1501.05084-3-21-6'], ['1501.05084-2-19-7', '1501.05084-3-21-7'], ['1501.05084-1-10-2', '1501.05084-2-11-0'], ['1501.05084-1-10-3', '1501.05084-2-11-2']]

[]

[['1501.05084-1-4-7', '1501.05084-2-4-7'], ['1501.05084-1-4-10', '1501.05084-2-4-10'], ['1501.05084-1-8-2', '1501.05084-2-8-2'], ['1501.05084-1-7-2', '1501.05084-2-7-2'], ['1501.05084-1-12-0', '1501.05084-2-13-0'], ['1501.05084-1-12-1', '1501.05084-2-13-1'], ['1501.05084-1-12-2', '1501.05084-2-13-3'], ['1501.05084-1-3-1', '1501.05084-2-3-1'], ['1501.05084-1-3-2', '1501.05084-2-3-2'], ['1501.05084-1-16-0', '1501.05084-2-17-0'], ['1501.05084-1-16-2', '1501.05084-2-17-1'], ['1501.05084-1-16-3', '1501.05084-2-17-2'], ['1501.05084-1-16-4', '1501.05084-2-17-3'], ['1501.05084-1-18-1', '1501.05084-2-19-1'], ['1501.05084-1-18-3', '1501.05084-2-19-4'], ['1501.05084-1-18-3', '1501.05084-2-19-5'], ['1501.05084-1-18-6', '1501.05084-2-19-8'], ['1501.05084-1-18-6', '1501.05084-2-19-9'], ['1501.05084-1-17-0', '1501.05084-2-18-0'], ['1501.05084-1-17-0', '1501.05084-2-18-1'], ['1501.05084-2-18-1', '1501.05084-3-20-1'], ['1501.05084-2-18-2', '1501.05084-3-20-1'], ['1501.05084-2-18-2', '1501.05084-3-20-2'], ['1501.05084-2-18-3', '1501.05084-3-20-2'], ['1501.05084-2-18-3', '1501.05084-3-20-3'], ['1501.05084-1-10-0', '1501.05084-2-10-0'], ['1501.05084-1-10-1', '1501.05084-2-10-1']]

[]

['1501.05084-1-1-0', '1501.05084-1-1-1', '1501.05084-1-1-2', '1501.05084-2-1-0', '1501.05084-2-1-1', '1501.05084-2-1-2', '1501.05084-3-1-0', '1501.05084-3-1-1', '1501.05084-3-1-2', '1501.05084-3-18-2']

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

https://arxiv.org/abs/1501.05084

{'1501.05084-3-0-0': 'We study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules.', '1501.05084-3-0-1': 'We calculate the corresponding strong coupling form factors defining these vertices by using a three point correlation function.', '1501.05084-3-0-2': 'We obtain the numerical values of the corresponding strong coupling constants via the most prominent structure entering the calculations.', '1501.05084-3-1-0': 'PACS number(s): 13.30.-a, 13.30.', '1501.05084-3-1-1': 'Eg, 11.55.', '1501.05084-3-1-2': 'Hx', '1501.05084-3-2-0': '# Introduction', '1501.05084-3-3-0': 'In the recent years, substantial experimental improvements have been made on the spectroscopic and decay properties of heavy hadrons, which were accompanied by theoretical studies on various properties of these hadrons.', '1501.05084-3-3-1': 'The mass spectrum of the baryons containing heavy quark has been studied using different methods.', '1501.05084-3-3-2': 'The necessity of a deeper understanding of heavy flavor physics requires a comprehensive study on the processes of these baryons such as their radiative, strong and weak decays.', '1501.05084-3-3-3': 'For some of related studies one can refer to references [CITATION].', '1501.05084-3-4-0': 'The investigation of the strong decays of heavy baryons can help us get valuable information on the perturbative and non-perturbative natures of QCD.', '1501.05084-3-4-1': 'The strong coupling constants defining such decays play important role in describing the strong interaction among the heavy baryons and other participated particles.', '1501.05084-3-4-2': 'Therefore, accurate determination of these coupling constants enhance our understanding on the interactions as well as the nature and structure of the participated particles.', '1501.05084-3-4-3': 'The present work is an extension of our previous study on the coupling constants [MATH] and [MATH] [CITATION].', '1501.05084-3-4-4': 'Here, we study the strong interactions among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson in the context of QCD sum rules.', '1501.05084-3-4-5': 'In particular, we calculate the strong coupling constants [MATH] and [MATH].', '1501.05084-3-4-6': 'These coupling constants together with the [MATH] and [MATH] discussed in our previous work, may also be used in the bottom and charmed mesons clouds description of the nucleon which can be used to explain the exotic events observed by different Collaborations.', '1501.05084-3-4-7': 'In addition, the determination of the properties of the [MATH] and [MATH] mesons in nuclear medium requires the consideration of their interactions with the nucleons from which the [MATH] and [MATH] are produced.', '1501.05084-3-4-8': 'Therefore, to determine the modifications on the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium, one needs to consider the contributions of the baryons [MATH] together with the [MATH] and have the values of the strong coupling constants [MATH] and [MATH] besides the couplings [MATH] and [MATH] [CITATION].', '1501.05084-3-4-9': 'In the literature, one can unfortunately find only a few works on the strong couplings of the heavy baryons with the nucleon and heavy mesons.', '1501.05084-3-4-10': 'One approximate prediction for the strong coupling [MATH] was made at zero transferred momentum squared [CITATION].', '1501.05084-3-4-11': 'The strong couplings of the charmed baryons with the nucleon and [MATH] meson were also discussed in [CITATION] in the framework of light cone QCD sum rules.', '1501.05084-3-5-0': 'This paper is organized in three sections as follows.', '1501.05084-3-5-1': 'In the next section, we present the details of the calculations of the strong coupling form factors among the particles under consideration.', '1501.05084-3-5-2': 'In section 3, the numerical analysis of the obtained sum rules and discussions about the results are presented.', '1501.05084-3-6-0': '# Theoretical framework', '1501.05084-3-7-0': 'This section is devoted to the details of the calculations of the strong coupling form factors [MATH] and [MATH] from which the strong coupling constants among the participating particles are obtained at [MATH], subsequently.', '1501.05084-3-7-1': 'In order to accomplish this purpose, the following three-point correlation function is used: [EQUATION] whith [MATH] being the time ordering operator and [MATH] is the transferred momentum.', '1501.05084-3-7-2': 'The currents [MATH], [MATH] and [MATH] presented in Eq. ([REF]) correspond to the the interpolating currents of the [MATH], [MATH] and [MATH], respectively and their explicit expressions can be given in terms of the quark field operators as [EQUATION] where [MATH] denotes the charge conjugation operator; and [MATH], [MATH] and [MATH] are color indices.', '1501.05084-3-8-0': 'In the course of calculation of the three-point correlation function one follows two different ways.', '1501.05084-3-8-1': 'The first way is called as OPE side and the calculation is made in deep Euclidean region in terms of quark and gluon degrees of freedom using the operator product expansion.', '1501.05084-3-8-2': 'The second way is called as hadronic side and the hadronic degrees of freedoms are considered to perform this side of the calculation.', '1501.05084-3-8-3': 'The QCD sum rules for the coupling form factors are attained via the match of these two sides.', '1501.05084-3-8-4': 'The contributions of the higher states and continuum are suppressed by a double Borel transformation applied to both sides with respect to the variables [MATH] and [MATH].', '1501.05084-3-9-0': '## OPE Side', '1501.05084-3-10-0': 'For the calculation of the OPE side of the correlation function which is done in deep Euclidean region, where [MATH] and [MATH], one puts the interpolating currents given in Eq. ([REF]) into the correlation function, Eq. ([REF]).', '1501.05084-3-10-1': "Possible contractions of all quark pairs via Wick's theorem leads to [EQUATION] where [MATH] and [MATH] are the heavy and light quark propagators whose explicit forms can be found in Refs. [CITATION].", '1501.05084-3-11-0': 'After some straightforward calculations (for details refer to the Ref. [CITATION]), the correlation function in OPE side comes out in terms of different Dirac structures as [EQUATION]', '1501.05084-3-11-1': 'Each [MATH] function involves the perturbative and non-perturbative parts and is written as [EQUATION]', '1501.05084-3-11-2': 'The spectral densities, [MATH], appearing in Eq. ([REF]) are obtained from the imaginary parts of the [MATH] functions, i.e., [MATH].', '1501.05084-3-11-3': 'Here to provide examples of the explicit forms of the spectral densities, among the Dirac structures presented above, we only present the results obtained for the Dirac structure [MATH], that is [MATH] and [MATH], which are obtained as [EQUATION] and [EQUATION] where [MATH] stands for the unit-step function and [MATH] and [MATH] are defined as [EQUATION]', '1501.05084-3-12-0': '## Hadronic Side', '1501.05084-3-13-0': 'On the hadronic side, considering the quantum numbers of the interpolating fields one place the complete sets of intermediate [MATH], [MATH] and [MATH] hadronic states into the correlation function.', '1501.05084-3-13-1': 'After carrying out the four-integrals, we get [EQUATION]', '1501.05084-3-13-2': 'In the above equation, the contributions of the higher states and continuum are denoted by [MATH] and the matrix elements are represented in terms of the hadronic parameters as follows: [EQUATION]', '1501.05084-3-13-3': 'Here [MATH] and [MATH] are residues of the [MATH] and [MATH] baryons, respectively, [MATH] is the leptonic decay constant of [MATH] meson and [MATH] is the strong coupling form factor among [MATH], [MATH] and [MATH] particles.', '1501.05084-3-13-4': 'Using Eq. ([REF]) in Eq. ([REF]) and summing over the spins of the particles, we obtain [EQUATION]', '1501.05084-3-13-5': 'To acquire the final form of the hadronic side of the correlation function we perform the double Borel transformation with respect to the initial and final momenta squared, [EQUATION] where [MATH] and [MATH] are Borel mass parameters.', '1501.05084-3-14-0': 'As it was already stated, the match of the hadronic and OPE sides of the correlation function in Borel scheme provides us with the QCD sum rules for the strong form factors.', '1501.05084-3-14-1': 'The consequence of that match for [MATH] structure leads us to [EQUATION] where [MATH] and [MATH] are continuum thresholds in [MATH] and [MATH] channels, respectively.', '1501.05084-3-15-0': '# Numerical analysis', '1501.05084-3-16-0': 'Having obtained the QCD sum rules for the strong coupling form factors, in this section, we present the numerical analysis of our results and discuss the dependence of the strong coupling form factors under consideration on [MATH].', '1501.05084-3-16-1': 'To this aim, beside the input parameters given in table 1, one needs to determine the working intervals of four auxiliary parameters [MATH], [MATH], [MATH] and [MATH].', '1501.05084-3-16-2': 'These parameters originate from the double Borel transformation and continuum subtraction.', '1501.05084-3-16-3': 'The determination of the working regions of them is made on the basis of that the results obtained for the strong coupling form factors be roughly independent of these helping parameters.', '1501.05084-3-17-0': 'The continuum thresholds [MATH] and [MATH] are the parameters related to the beginning of the continuum in the initial and final channels.', '1501.05084-3-17-1': 'If the ground state masses are given by [MATH] and [MATH] for the initial and final channels, respectively, to excite the particle to the first excited state having the same quantum numbers with them one needs to provide the energies [MATH] and [MATH].', '1501.05084-3-17-2': 'For the considered transitions, these quantities can be determined from well known excited states of the initial and final states [CITATION] which are roughly in between [MATH] and [MATH].', '1501.05084-3-17-3': 'From these intervals, the working regions for the continuum thresholds are determined as [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the vertex [MATH].', '1501.05084-3-18-0': 'Here, we shall comment on the selection of the most prominent Dirac structure to determine the corresponding strong coupling form factors.', '1501.05084-3-18-1': 'In principle, one can choose any structure for determination of these strong coupling form factors.', '1501.05084-3-18-2': 'However, we should choose the most reliable one considering the following criteria:', '1501.05084-3-19-0': 'Our numerical calculations show that these conditions lead to choose the structure [MATH] as the most prominent structure.', '1501.05084-3-19-1': 'In the following, we will use this structure to numerically analyze the obtained sum rules.', '1501.05084-3-20-0': 'Now, we proceed to present the Borel windows considering the selected structure.', '1501.05084-3-20-1': 'The working windows for the Borel parameters [MATH] and [MATH] are determined considering again the pole dominance and convergence of the OPE.', '1501.05084-3-20-2': 'By requirement that the pole contribution exceeds the contributions of the higher states and continuum, and that the contribution of the perturbative part exceeds the non-perturbative contributions we find the windows [MATH]^2[MATH]^2[MATH] and [MATH]^2[MATH]^2[MATH] for the Borel mass parameters of the strong vertex [MATH].', '1501.05084-3-20-3': 'For these intervals, our results show weak dependence on the Borel mass parameters (see figures 1-2).', '1501.05084-3-21-0': 'Subsequent to the determination of the auxiliary parameters, their working windows together with the other input parameters are used to ascertain the dependency of the strong coupling form factors on [MATH].', '1501.05084-3-21-1': 'From our analysis we observe that the dependency of the strong coupling form factors on [MATH] is well characterized by the following fit function: [EQUATION]', '1501.05084-3-21-2': 'The values of the parameters [MATH], [MATH] and [MATH] for [MATH] and [MATH] can be seen in tables [REF] and [REF], respectively.', '1501.05084-3-21-3': 'Considering the average values of the continuum thresholds and Borel mass parameters we demonstrate the variation of the strong coupling form factors with respect to [MATH] for the QCD sum rules as well as the fitting results in figure 3.', '1501.05084-3-21-4': 'The figure indicates the truncation of the QCD sum rules at some points at negative values of [MATH].', '1501.05084-3-21-5': 'It can be seen from the figure that there is a good consistency among the results obtained from the QCD sum rules and fit function up to these points.', '1501.05084-3-21-6': 'The fit function is used to determine the value of the strong coupling constant at [MATH], and the results are presented in table [REF].', '1501.05084-3-21-7': 'The presented errors in these results originate from the uncertainties of the input parameters together with the uncertainties coming from the determination of the working regions of the auxiliary parameters.', '1501.05084-3-22-0': 'To sum up, in this work, the strong coupling constants among the heavy bottom spin-1/2 [MATH] baryon, nucleon and [MATH] meson as well as the heavy charmed spin-1/2 [MATH] baryon, nucleon and [MATH] meson, namely [MATH] and [MATH], have been calculated in the framework of the three-point QCD sum rules.', '1501.05084-3-22-1': 'The obtained results can be applied in the analysis of the related experimental results at LHC.', '1501.05084-3-22-2': 'The predictions can also be used in the bottom and charmed mesons clouds description of the nucleon that may be applied for the explanation of the exotic events observed by different experiments.', '1501.05084-3-22-3': 'These results may also serve the purpose of analyzing of the results of heavy ion collision experiments like [MATH] at FAIR.', '1501.05084-3-22-4': 'The obtained results may also come in handy in the determinations of the changes in the masses, decay constants and other parameters of the [MATH] and [MATH] mesons in nuclear medium.'}

1609.03261

{'1609.03261-1-0-0': 'We develop and analyze a procedure for gradient-based optimization that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-1-0-1': 'As a member of the SVRG family of algorithms, SCSG makes use of gradient estimates at two scales.', '1609.03261-1-0-2': 'Unlike most existing algorithms in this family, both the computation cost and the communication cost of SCSG do not necessarily scale linearly with the sample size [MATH]; indeed, these costs are independent of [MATH] when the target accuracy is low.', '1609.03261-1-0-3': 'An experimental evaluation of SCSG on the MNIST dataset shows that it can yield accurate results on this dataset on a single commodity machine with a memory footprint of only 2.6MB and only eight disk accesses.', '1609.03261-1-1-0': '# Introduction', '1609.03261-1-2-0': 'Optimization problems in machine learning are often solved by algorithms that either make use of full gradients (obtained by processing the entire dataset) or stochastic gradients (obtained by processing single data points or mini-batches of data points).', '1609.03261-1-2-1': 'The use of the former provides guarantees of eventual convergence and the latter yield advantages in terms of rate of convergence rate, scalability and simplicity of implementation [CITATION].', '1609.03261-1-2-2': 'An impactful recent line of research has shown that a hybrid methodology that makes use of both full gradients and stochastic gradients can obtain the best of both worlds-guaranteed convergence at favorable rates, e.g. [CITATION].', '1609.03261-1-2-3': 'The full gradients provide variance control for the stochastic gradients.', '1609.03261-1-3-0': 'While this line of research represents significant progress towards the goal of designing scalable, autonomous learning algorithms, there remain some inefficiencies in terms of computation.', '1609.03261-1-3-1': 'With the definition of computation and communication cost in Section 2.1, the methods referred to above require [MATH] computation to achieve an [MATH]-approximate solution, where [MATH] is the number of data points, [MATH] is a target accuracy and [MATH] is the dimension of the parameter vector.', '1609.03261-1-3-2': 'Some methods incur a [MATH] storage cost [CITATION].', '1609.03261-1-3-3': 'The linear dependence on [MATH] is problematic in general.', '1609.03261-1-3-4': 'Clearly there will be situations in which accurate solutions can be obtained with less than a single pass through the data; indeed, some problems will require a notionally constant number of steps.', '1609.03261-1-3-5': 'This will be the case, for example, if the data in a regression problem consist of a fixed number of pairs repeated a large number of times.', '1609.03261-1-3-6': 'For deterministic algorithms, the worst case analysis in [CITATION] shows that scanning at least a fixed proportion of the data is necessary; however, learning algorithms are generally stochastic and real-world learning problems are generally not worst case.', '1609.03261-1-3-7': 'Recently, [CITATION] establishes a lower bound for the computational cost of randomized algorithms in minimizing the finite sums.', '1609.03261-1-3-8': 'However, their result only applies to the case in which [MATH] and the summands are smooth, leaving open the possibility that smaller computational complexity can be achieved when these assumptions do not hold.', '1609.03261-1-4-0': 'An equally important bottleneck for learning algorithms is the cost of communication.', '1609.03261-1-4-1': 'For large data sets that must be stored on disk or distributed across many computing nodes, the communication cost can be significant, even dominating the computation cost.', '1609.03261-1-4-2': 'For example, classical stochastic gradient descent (SGD) makes use of random sampling which can incur prohibitive communication cost.', '1609.03261-1-4-3': 'There is an active line of research that focuses on communication costs; see, e.g. [CITATION].', '1609.03261-1-5-0': 'In this article, we present a variant of the stochastic variance reduced gradient (SVRG) method that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-1-5-1': 'In contradistinction to SVRG, the theoretical convergence rate of SCSG has a sublinear regime in terms of both computation and communication.', '1609.03261-1-5-2': 'The basic idea behind SCSG-that of approximating the full gradient in SVRG via a subsample-has been explored by others, but we present several innovations that yield significant improvements both in theory and in practice.', '1609.03261-1-5-3': 'The performance of SCSG is superior to related algorithms when [MATH].', '1609.03261-1-5-4': 'This regime is important in machine learning problems, notably in the common situation in which the sample size is large, ([MATH]), while the required accuracy is low, [MATH].', '1609.03261-1-5-5': 'The analysis in this article shows that SCSG is able to achieve the target accuracy in this regime with less than a single pass through the data.', '1609.03261-1-6-0': 'Although SGD can be shown to incur the same computation cost as SCSG for non-strongly-convex functions (for strongly-convex functions we will show that SCSG has a better rate than SGD), it requires tuning the stepsize based on the information that is unknown in practice, and it requires fixing the number of steps prior to the implementation.', '1609.03261-1-6-1': 'In other words, without intensive tuning, it is not guaranteed to reach an [MATH]-approximate solution for general convex functions.', '1609.03261-1-6-2': 'In contrast, we propose a simple automatic tuning procedure which guarantees that SCSG finds an [MATH]-approximate solution.', '1609.03261-1-7-0': 'Another line of work explores the convergence of first-order methods within one pass of data under a stochastic framework; e.g., streaming SVRG [CITATION] and dynaSAGA[CITATION].', '1609.03261-1-7-1': 'The analyses in this framework assume that data points are independent and identically distributed, an assumption that is often unrealistic in the streaming setting.', '1609.03261-1-7-2': 'In contrast, SCSG does not make such an i.i.d. assumption; indeed, it does not assume any randomness for the observed data.', '1609.03261-1-7-3': 'Admittedly, it is interesting to explore SCSG under the i.i.d. framework to understand implications for generalization error, but such an analysis is beyond the scope of the current paper.', '1609.03261-1-8-0': 'The remainder of the paper is organized as follows.', '1609.03261-1-8-1': 'In Section 2, we review SVRG, discuss several of its variants and we describe the SCSG algorithm.', '1609.03261-1-8-2': 'We provide a theoretical convergence analysis in Section 3.', '1609.03261-1-8-3': 'Our analysis involves a data-dependent quantity, [MATH], which is defined at the beginning of Section 3.', '1609.03261-1-8-4': 'We show in Section 4 that [MATH] is estimable for generalized linear models.', '1609.03261-1-8-5': 'We present an experimental evaluation in Section 5.', '1609.03261-1-8-6': 'All technical proofs are relegated to the Appendices.', '1609.03261-1-9-0': '# Notation, Assumptions and Algorithm', '1609.03261-1-10-0': 'Throughout this paper we consider the following minimization problem: [EQUATION] where [MATH] are convex functions.', '1609.03261-1-10-1': 'Let [MATH] denote a minimizer of [MATH], [MATH] denote the initial value and [MATH].', '1609.03261-1-10-2': 'A point [MATH], possibly random, is called an [MATH]-approximate solution if [EQUATION].', '1609.03261-1-10-3': 'The following assumptions will be used in various contexts in the paper: [A1] [MATH] is convex with [MATH]-Lipschitz gradient, [EQUATION] for some [MATH] and all [MATH]; [MATH] is strongly convex with [EQUATION] for some [MATH] and all [MATH]; [MATH] is strongly convex with [EQUATION] for some [MATH].', '1609.03261-1-11-0': 'If Assumption A2 holds, then Assumption A3 also holds with [MATH].', '1609.03261-1-11-1': 'For unpenalized linear regression where [MATH], Assumption A2 fails to hold for any [MATH] if [MATH].', '1609.03261-1-11-2': 'However, [MATH] is strongly convex if [MATH] is positive definite.', '1609.03261-1-11-3': 'On the other hand, a [MATH]-regularization term is often added to [REF] such that the problem is to minimize [MATH] for some [MATH], in which case [EQUATION] and [MATH] satisfies Assumption A2.', '1609.03261-1-11-4': 'Thus, Assumption A2 is also reasonable for practitioners and it results in tighter convergence analysis and a simpler tuning procedure than Assumption A3 (as shown in Section 3).', '1609.03261-1-12-0': '## Computation and Communication Cost', '1609.03261-1-13-0': 'To formally compare the algorithms, we need to define the computation cost and the communication cost.', '1609.03261-1-13-1': 'In this article we adopt the IFO framework [CITATION] under which the sampling of an index [MATH] and the evaluation of the pair [MATH] incurs one unit of computational cost.', '1609.03261-1-13-2': 'Note that we ignore the effect of dimension, treating it as a constant; if desired, we could multiply the IFO complexity by dimension to track dimension dependence.', '1609.03261-1-14-0': 'To define the communication cost, we need to specify a computational model.', '1609.03261-1-14-1': 'Here we consider two popular models.', '1609.03261-1-14-2': 'In the first model, we have a single machine and the data is too large to fit into the memory and has to be stored on the disk.', '1609.03261-1-14-3': 'We assume that accessing one data point from the disk incurs one unit of communication cost.', '1609.03261-1-14-4': 'In the second model, we consider a distributed system with a datacenter and multiple worker machines.', '1609.03261-1-14-5': 'The data are stored remotely in the workers and the main computation tasks are implemented in the datacenter.', '1609.03261-1-14-6': 'This setting has been considered in various recent papers; see, e.g., [CITATION].', '1609.03261-1-14-7': 'Similar to the first model, we assume that accessing or sampling one data point from a node incurs one unit of communication cost.', '1609.03261-1-14-8': 'It turns out that these two models give the same communication cost for algorithms we considere in this paper and hence we do not distinguish them in the following discussion.', '1609.03261-1-15-0': '[h] Stochastic Variance Reduced Gradient (SVRG) Method', '1609.03261-1-16-0': 'Inputs: Stepsize [MATH], number of stages [MATH], initial iterate [MATH], number of SGD steps [MATH].', '1609.03261-1-17-0': 'Procedure [1] [MATH] Randomly pick [MATH]', '1609.03261-1-18-0': 'Output: (Option 1): [MATH] (Option 2): [MATH]', '1609.03261-1-19-0': '## SVRG and Other Related Works', '1609.03261-1-20-0': 'The stochastic variance reduced gradient (SVRG) method blends gradient descent and stochastic gradient descent, using the former to control the effect of the variance of the latter [CITATION].', '1609.03261-1-20-1': 'We summarize SVRG in Algorithm [REF].', '1609.03261-1-21-0': 'Using the definition from Section 2.1, it is easy to see that the computation cost of SVRG is [MATH].', '1609.03261-1-21-1': 'Each step of SVRG involves accessing all data points to compute the full gradient, which incurs a communication cost of [MATH], and sampling [MATH] data points for stochastic gradient descent, which incurs a communication cost of [MATH].', '1609.03261-1-21-2': 'Thus, the total communication cost is [MATH].', '1609.03261-1-21-3': 'As shown in the convergence analysis of [CITATION], [MATH] is required to be [MATH] to guarantee convergence.', '1609.03261-1-21-4': 'Thus, the computation and communication cost of SVRG are both [MATH].', '1609.03261-1-21-5': 'The costs of the other algorithms considered in Table [REF] can be obtain in a similar fashion.', '1609.03261-1-22-0': 'A number of variants of SVRG have been studied.', '1609.03261-1-22-1': 'For example, a constrained form of SVRG can be obtained by replacing line 8 with a projected gradient descent step [CITATION].', '1609.03261-1-22-2': 'A mini-batch variant of SVRG arises when one samples a subset of indices instead of a single index in line 6 and updates the iterates by the average gradient in this batch in line 7 [CITATION].', '1609.03261-1-22-3': 'Similarly, we can consider implementing the full gradient computation in line 2 using a subsample.', '1609.03261-1-22-4': 'This is proposed in [CITATION], which calculates [MATH] as [MATH] where [MATH] is a subset of size [MATH] uniformly sampled from [MATH].', '1609.03261-1-22-5': '[CITATION] heuristically show the potential for significant complexity reduction, but they only prove convergence for [MATH] and under the stringent condition that [MATH] is uniformly bounded for all [MATH].', '1609.03261-1-22-6': "Similar to Nesterov's acceleration for gradient descent, momentum terms can be added to the SGD steps to accelerate SVRG [CITATION].", '1609.03261-1-22-7': 'Table [REF] summarizes the characteristics of 11 existing popular algorithms as well as SCSG.', '1609.03261-1-22-8': 'The table includes the computation cost of optimizing non-strongly-convex functions (column 1) and strongly convex functions (column 2).', '1609.03261-1-22-9': 'Here strong convexity is only assumed on [MATH] instead of individual [MATH].', '1609.03261-1-22-10': 'In practice, the amount of tuning is of major concern.', '1609.03261-1-22-11': 'For this reason, a fixed stepsize is usually preferred to a complicated stepsize scheme and it is better that the tuning parameter does not depend on unknown quantities; e.g., [MATH].', '1609.03261-1-22-12': 'These issues are documented in column 3 and column 4.', '1609.03261-1-22-13': 'Moreover, many algorithms requires [MATH] to be bounded or at least [MATH] to be Lipschitz.', '1609.03261-1-22-14': 'However, this assumption is not realistic in many cases and it is better to discard it.', '1609.03261-1-22-15': 'To address this issue, we document it in column 5.', '1609.03261-1-23-0': 'Much of this work focuses on the strongly convex case.', '1609.03261-1-23-1': 'In the non-strongly convex setting one way to proceed is to add a [MATH] regularizer [MATH].', '1609.03261-1-23-2': 'Tuning [MATH], however, is subtle and requires multiple runs of the algorithm on a grid of [MATH] [CITATION].', '1609.03261-1-23-3': 'For general convex functions an alternative approach has been presented by [CITATION] (they generate [MATH] by a different scheme in line 4), who also establish a computation rate [MATH].', '1609.03261-1-23-4': 'Another approach is discussed by [CITATION], who establish convergence for non-strongly convex functions (and non-convex functions with Lipschitz gradient) by considering a weaker stopping criterion based on [MATH].', '1609.03261-1-23-5': 'However, their algorithm relies on calculating a full gradient, involves a complicated stepsize-setting scheme and requires a strong assumption that the [MATH] are uniformly bounded.', '1609.03261-1-23-6': 'Other variants of SVRG have been proposed in the distributed computing setting [CITATION] and in the stochastic setting [CITATION].', '1609.03261-1-24-0': '## SCSG', '1609.03261-1-25-0': 'SCSG is similar to [CITATION] in that it implements the gradient computation on a subsample [MATH] of size [MATH].', '1609.03261-1-25-1': 'However, there are three key differences:', '1609.03261-1-26-0': 'The final point is of particular importance in a practical implementation.', '1609.03261-1-26-1': 'By restricting the stochastic gradient step to the batch [MATH], an iteration of SCSG needs only to load that batch into the memory; this can be feasible on a laptop even for large datasets.', '1609.03261-1-26-2': 'We also wish to highlight the use of the geometric distribution; surprisingly this choice yields a much tighter analysis than other choices.', '1609.03261-1-27-0': 'We present the SCSG procedure in Algorithm [REF] and present the method for setting parameters in Algorithm [REF].', '1609.03261-1-28-0': '[htp] Stochastically Controlled Stochastic Gradient (SCSG) Method', '1609.03261-1-29-0': 'Inputs: Stepsize [MATH], batch size [MATH], number of stages [MATH], initial iterate [MATH]', '1609.03261-1-30-0': 'Initialization: Get distribution [MATH] by Algorithm [REF]', '1609.03261-1-31-0': 'Procedure [1] [MATH] Uniformly sample a batch [MATH] with [MATH] Generate [MATH] Randomly pick [MATH]', '1609.03261-1-32-0': 'Output: (Strongly convex case): [MATH] (Non-strongly convex case): [MATH]', '1609.03261-1-33-0': 'The average computation cost of SCSG is [MATH].', '1609.03261-1-33-1': 'By the law of large numbers, this is close to [MATH], which has a similar form to SVRG.', '1609.03261-1-33-2': 'As seen from Table [REF], SCSG and SGD are the only two methods which are able to reach an [MATH]-approximate solution with less than a single pass through the data; moreover, the number of accesses of the data is independent of the sample size [MATH].', '1609.03261-1-33-3': 'For the general convex case, as discussed in Section 1, SCSG requires much less tuning than SGD and, as seen from column 5, SCSG does not require [MATH] to be Lipschitz.', '1609.03261-1-33-4': 'For the strongly convex case (Assumption A3), SCSG has a complexity of [MATH], which is strictly less than that of SGD, which is [MATH].', '1609.03261-1-33-5': 'In both cases, SCSG outperforms SGD in terms of computation and amount of tuning.', '1609.03261-1-34-0': 'On the other hand, all of the other methods in Table [REF] have a communication cost that is the same as the computation cost since they all sample the stochastic gradient index uniformly from [MATH] and hence needs to communicate with disk/worker machines.', '1609.03261-1-34-1': 'In contrast, SCSG is communication-avoiding since it only needs to communicate when calculating [MATH] (line 3) and the ensuing SGD steps do not incur communication costs since the batch is fit into the memory.', '1609.03261-1-34-2': 'As shown in Table [REF] the communication cost of SCSG is free of [MATH] and even free of the condition number [MATH].', '1609.03261-1-34-3': 'In other words, in contrast to most other methods, ill-conditioned problems do not overload the communication.', '1609.03261-1-35-0': 'One might argue that the methods listed in Table [REF] are not designed for communication efficiency.', '1609.03261-1-35-1': 'To make a comparison to algorithms that explicitly aim for communication efficiency, we consider CoCoA [CITATION], DANE [CITATION] and DiSCO [CITATION], and present the results in Table [REF].', '1609.03261-1-35-2': '(Note that CoCoA has an additional factor [MATH] [CITATION] determining the iteration complexity [MATH] and hence the tradeoff between computation and communication.', '1609.03261-1-35-3': 'We discuss the details in Appendix B.1).', '1609.03261-1-35-4': 'These methods are considered in the datacenter/workers model in which there are [MATH] worker machines with an (almost) equal number of data stored in each.', '1609.03261-1-35-5': 'In contrast to SCSG, instead of sub-sampling, these methods save on communication cost by performing updates locally.', '1609.03261-1-35-6': 'Specifically, the iterate is updated in each node in parallel and sent to the datacenter; then the datacenter sums or averages the updates and broadcasts the result to each node.', '1609.03261-1-35-7': 'As a consequence, under our computational model, the communication cost is the product of the number of nodes and the iteration complexity, namely [MATH].', '1609.03261-1-35-8': 'Among these methods, DANE and DiSCO are second-order methods while SCSG and CoCoA are first-order methods.', '1609.03261-1-35-9': 'We record this in column 3 to emphasize the dependence on dimension.', '1609.03261-1-35-10': '(Another first-order method which is similar to SVRG has been developed by [CITATION]; we do not consider this method in our comparison due to the lack of theoretical results.)', '1609.03261-1-35-11': 'In addition, we show the computation cost in column 4.', '1609.03261-1-35-12': 'As seen from Table [REF], SCSG is the only method whose communication cost is free of the number of nodes.', '1609.03261-1-35-13': 'This suggests that SCSG is more scalable in the distributed setting where a large number of worker machines exist; e.g., the mobile device system.', '1609.03261-1-35-14': 'Comparing SCSG with CoCoA, we find that with the same amount of computation, SCSG is more communication-efficient when [MATH], in which case the former has a cost [MATH] while the latter has a cost at least [MATH] (see Appendix B.1 for details).', '1609.03261-1-35-15': 'Further, if the problem is well-conditioned in the sense that [MATH], then the communication cost of CoCoA is [MATH] which could be much larger than that of SCSG once [MATH].', '1609.03261-1-35-16': 'On the other hand, both DANE and DiSCO depend on the condition number in terms of communication and depend on sample size in terms of computation if [MATH].', '1609.03261-1-35-17': 'We notice that the computation cost of DANE and DiSCO match that of SCSG only when [MATH], in which case the communication cost depends on the sample size.', '1609.03261-1-35-18': 'In contrast, this tradeoff does not appear in SCSG.', '1609.03261-1-36-0': 'Finally, in contrast to many existing proposals, our tuning procedure is straightforward.', '1609.03261-1-36-1': 'Generally, only [MATH] and [MATH] need to be tuned and a default stepsize is given in Algorithm [REF].', '1609.03261-1-36-2': 'For certain problems such as generalized linear models, the block size [MATH] can also be set automatically.', '1609.03261-1-36-3': 'Furthermore, it will be shown in the following section that Algorithm [REF] is guaranteed to reach an [MATH]-approximate solution with our recommended set of parameters, regardless of the initial iterate [MATH] (a better initial value will accelerate the algorithm by a constant factor).', '1609.03261-1-36-4': 'In addition, the stepsize [MATH] does not depend on the number of stages [MATH] and a larger [MATH] always produces better results.', '1609.03261-1-36-5': 'Thus, the parameters [MATH] and [MATH] are not essential.', '1609.03261-1-36-6': 'This is in contradistinction to SGD, which depends crucially on these parameters to achieve the same convergence rate as SCSG (for the general convex case).', '1609.03261-1-37-0': '# Convergence Analysis', '1609.03261-1-38-0': 'In this section we present a convergence analysis of SCSG.', '1609.03261-1-38-1': 'Note that SCSG is similar to SVRG when [MATH].', '1609.03261-1-38-2': 'Our theoretical results are similar to those for SVRG in this case; our focus, however, is the case in which [MATH], where additional sampling variation enters in.', '1609.03261-1-38-3': 'To capture the effect of such such variation we define the following quantity: [EQUATION] where [MATH] denotes the [MATH] norm.', '1609.03261-1-38-4': 'We then have the following lemma.', '1609.03261-1-39-0': 'Let [MATH] be a random subset of size [MATH], and define the random variable [MATH].', '1609.03261-1-39-1': 'Then [MATH] and [EQUATION].', '1609.03261-1-40-0': 'The proof, which appears in Appendix A.1, involves a standard technique for analyzing sampling without replacement.', '1609.03261-1-40-1': 'Note that the extra variation vanishes when [MATH] and in general is inversely proportional to the batch size.', '1609.03261-1-40-2': 'To further control the variance, we need to bound [MATH].', '1609.03261-1-40-3': 'Obviously, [MATH] is bounded if [MATH] is uniformly bounded as is often assumed in the literature.', '1609.03261-1-40-4': 'However, via a more refined analysis, we can discard this stringent condition and only require Assumption A1.', '1609.03261-1-40-5': 'The next lemma provides a bound on [MATH]; the proof appears in Appendix A.1).', '1609.03261-1-41-0': 'Under Assumption A1, for any [MATH] [EQUATION].', '1609.03261-1-41-1': 'If further [MATH] is bounded below by [MATH], then [EQUATION] [htp] Initialization of SCSG', '1609.03261-1-42-0': 'Inputs: Stepsize [MATH], batch size [MATH]', '1609.03261-1-43-0': 'Optional Inputs: gradient Lipschitz constant [MATH], strong convexity modulus [MATH]', '1609.03261-1-44-0': 'Procedure:', '1609.03261-1-45-0': '[MATH] is unknown [MATH]; [MATH] geometric distribution with [MATH] is known [MATH]; [MATH]; [MATH] truncated geometric distribution supported on [MATH] with [MATH]', '1609.03261-1-46-0': 'Output: Distribution [MATH]', '1609.03261-1-47-0': '## Non-Strongly Convex Case', '1609.03261-1-48-0': 'Our convergence result for non-strongly convex functions is stated in Theorem [REF].', '1609.03261-1-48-1': 'This theorem is stated for arbitrary [MATH]; the choice in Algorithm [REF] is a special case.', '1609.03261-1-49-0': 'Assume A1 holds.', '1609.03261-1-49-1': 'Let [MATH] and [MATH],', '1609.03261-1-50-0': 'If [MATH] and [MATH], then [EQUATION]', '1609.03261-1-50-1': 'If [MATH] and [MATH], then [EQUATION] for some [MATH] and [MATH], which only depend on [MATH].', '1609.03261-1-50-2': 'In particular, when [MATH], we have [MATH].', '1609.03261-1-51-0': 'Assume A1 holds and select the batch size [MATH], number of stages [MATH] and decay rate [MATH] such that [EQUATION].', '1609.03261-1-51-1': 'Letting [MATH] with [MATH], then [MATH] with [EQUATION]', '1609.03261-1-52-0': '## Strongly Convex Case With Assumption A 2 First we assume that [MATH] is strongly convex in the sense that the Assumption A2 holds.', '1609.03261-1-52-1': 'Our main result in this case is stated in Theorem [REF] for arbitrary decay rate [MATH] and truncation parameter [MATH] (again the specific choices in Algorithm [REF] are a special case).', '1609.03261-1-53-0': 'Assume A1 and A2.', '1609.03261-1-53-1': 'Let [MATH] and [MATH].', '1609.03261-1-53-2': 'Assume that one of the following conditions holds: [EQUATION].', '1609.03261-1-53-3': 'Then', '1609.03261-1-54-0': 'if [MATH] and [MATH], [EQUATION] if [MATH] and [MATH], [EQUATION] for some [MATH], which only depends on [MATH].', '1609.03261-1-54-1': 'In particular, when [MATH], [MATH].', '1609.03261-1-55-0': 'Similar to the non-strongly convex case, we can derive the computation and communication cost of SCSG.', '1609.03261-1-55-1': 'Corollary [REF] summarizes the result for fixed [MATH], in which case the non-uniform sampling ([MATH]) has an additional [MATH] factor and hence is suboptimal when compared to uniform sampling ([MATH]).', '1609.03261-1-55-2': 'However, in the well-conditioned case where [MATH] is small compared to [MATH], we can tune [MATH] intelligently and achieve a much better rate as shown in Table 1.', '1609.03261-1-55-3': 'The results are stated in Corollary [REF].', '1609.03261-1-56-0': 'Assume A1 and A2.', '1609.03261-1-56-1': 'Let [MATH] denote the condition number and [MATH].', '1609.03261-1-56-2': 'Select the parameters such that [EQUATION].', '1609.03261-1-56-3': 'Then [MATH] and [EQUATION].', '1609.03261-1-57-0': 'Assume A1 and A2.', '1609.03261-1-57-1': 'Suppose [MATH] for some universal constant [MATH].', '1609.03261-1-57-2': 'Let [MATH] where [MATH], which guarantees [MATH], and set [MATH] and [MATH] as [EQUATION].', '1609.03261-1-57-3': 'If we set [MATH] and [MATH] as in Corollary [REF], then [EQUATION]', '1609.03261-1-58-0': '## Strongly Convex Case With Assumption A 3 In some applications, Assumption A2 might not be valid but Assumption A3 is.', '1609.03261-1-58-1': 'Since [MATH] is generally hard to estimate in cases without a [MATH]-regularization term, for which Assumption A2 holds and better results can be obtained from Theorem [REF], we sample [MATH] from a geometric distribution as in the general convex case.', '1609.03261-1-58-2': 'The following theorem provides a similar result to Theorem [REF], assuming only A1 and A3 but requiring more stringent conditions on the parameters.', '1609.03261-1-59-0': 'Assume A1 and A3.', '1609.03261-1-59-1': 'Let [MATH] [EQUATION]', '1609.03261-1-59-2': 'If [MATH] and [MATH], then [EQUATION]', '1609.03261-1-59-3': 'If [MATH] and [MATH] for some [MATH], then [EQUATION] where [MATH].', '1609.03261-1-60-0': 'The requirement that [MATH] implies that [EQUATION] and these results together imply that [EQUATION].', '1609.03261-1-60-1': 'Denote by [MATH] the condition number of [MATH], then [EQUATION].', '1609.03261-1-60-2': 'This plays a similar role to [MATH] as in the previous subsection with [MATH] replaced by [MATH].', '1609.03261-1-60-3': 'Similar to Corollary [REF], we can derive the computation and communication cost as follows.', '1609.03261-1-61-0': 'Assume A1 and A3.', '1609.03261-1-61-1': 'Let [MATH] denote the condition number of [MATH].', '1609.03261-1-61-2': 'Set [MATH] to satisfy [EQUATION] for some [MATH] and set [MATH] for any [MATH] with [EQUATION].', '1609.03261-1-61-3': 'Further, select the batch size [MATH] and number of stages [MATH] such that [EQUATION] for some large enough constants [MATH] which only depend on [MATH] and [MATH].', '1609.03261-1-61-4': 'Then [MATH] and [EQUATION].', '1609.03261-1-62-0': 'Unlike our analysis in Section 3.1 and Section 3.2, in this case the parameters are complicated and depend on [MATH] which is hard to obtain in practice.', '1609.03261-1-62-1': 'We should emphasize that SVRG also faces this problem.', '1609.03261-1-62-2': 'In particular, SVRG requires [EQUATION] which implies that [MATH].', '1609.03261-1-62-3': 'Thus intensive tuning is required to achieve the theoretical rate.', '1609.03261-1-62-4': 'If a case arises in practice in which only Assumption A3 is satisfied, we recommend treating it as a non-strongly convex function and setting parameters as in Section 3.1.', '1609.03261-1-63-0': '# Estimating [MATH] and [MATH]', '1609.03261-1-64-0': 'It has been shown in Section 3 that [MATH] and [MATH] determine the block size [MATH], which is the key to reducing both the computation and the communication costs in SCSG.', '1609.03261-1-64-1': 'For this reason, good estimates of [MATH] and [MATH] are needed for efficiency.', '1609.03261-1-64-2': 'Lemma [REF] presents a generic bound for [MATH], provided [MATH] can be estimated.', '1609.03261-1-64-3': 'However, we found that more accurate estimates of [MATH] and [MATH] are possible for generalized linear models (GLMs) [CITATION], which include a broad class of models that are often used in practical applications.', '1609.03261-1-64-4': 'Estimation in the GLM setting reduces to solving ([REF]) with [MATH], where [MATH] is the vector of predictors, [MATH] is the response and [MATH] is a loss function.', '1609.03261-1-64-5': 'This includes popular models such as linear regression, logistic regression, Huber regression and others.', '1609.03261-1-64-6': 'For multi-class problem in which [MATH] with [MATH], [MATH] could take the form [MATH] where [MATH] is the concatenation of [MATH], e.g. the multi-class logistic regression.', '1609.03261-1-64-7': 'First consider the case with only one linear component.', '1609.03261-1-64-8': 'Let [MATH] denote [MATH] and [MATH] denote [MATH].', '1609.03261-1-64-9': 'Then [MATH] can be expressed as [EQUATION] and [MATH] can be set as [EQUATION].', '1609.03261-1-64-10': 'In many cases [MATH] and [MATH] are uniformly bounded; e.g., logistic regression.', '1609.03261-1-64-11': 'In the context of robust statistics [CITATION], the loss function has a form [MATH] where [MATH] has bounded first-order and second-order derivatives, in which cases the above conditions are satisfied.', '1609.03261-1-65-0': 'Thus for GLMs with bounded [MATH] and [MATH], [MATH] and [MATH] can be set as [EQUATION]', '1609.03261-1-65-1': 'The same argument applies to multi-class GLMs.', '1609.03261-1-65-2': 'In particular, via some algebra, it can be shown that [MATH], [MATH] for logistic and multi-class logistic regression; see Appendix B.2 for details.', '1609.03261-1-65-3': 'For moderately large [MATH], we can calculate [MATH] and [MATH] directly.', '1609.03261-1-65-4': 'If [MATH] is prohibitively large, then we can estimate [MATH] and [MATH] based on a batch of data.', '1609.03261-1-66-0': 'In some cases such as least squares regression, [MATH] is bounded but [MATH] is not.', '1609.03261-1-66-1': 'Here we can again estimate [MATH] by [REF].', '1609.03261-1-66-2': 'Although [MATH] cannot be estimated as above, we can apply the second inequality of Lemma [REF] directly since [MATH] is usually nonnegative in most cases.', '1609.03261-1-66-3': 'For least squares regression, we can sharpen this bound by taking advantage of the explicit form of the solution.', '1609.03261-1-66-4': 'In particular, we have [MATH] and hence [EQUATION]', '1609.03261-1-67-0': '# Experiments', '1609.03261-1-68-0': 'In this section, we illustrative the performance of SGSC by implementing it for multi-class logistic regression on the MNIST dataset We normalize the data into the range [MATH] by dividing each entry by [MATH].', '1609.03261-1-68-1': 'No regularization term is added and so the function to be minimized is [EQUATION] where [MATH], [MATH], [MATH] including [MATH] pixels plus an intercept term [MATH] and [MATH].', '1609.03261-1-68-2': 'Direct computation shows that [MATH] and [MATH].', '1609.03261-1-68-3': 'Thu the highest stepsize with a convergence guarantee is [MATH].', '1609.03261-1-68-4': 'We treat this stepsize as a benchmark and also try three more aggressive stepsizes, [MATH].', '1609.03261-1-69-0': 'Suppose that our target accuracy is [MATH].', '1609.03261-1-69-1': 'Corollary [REF] then implies that [MATH].', '1609.03261-1-69-2': 'Based on this benchmark, we try three batch sizes [MATH] as well as SVRG with [MATH].', '1609.03261-1-69-3': 'We record the iterates [MATH] every half pass through the data where [MATH] denotes the number of passes.', '1609.03261-1-69-4': 'Although the theory provides a guarantee for [MATH], the optimal value [MATH] is unknown in practice.', '1609.03261-1-69-5': 'Instead we report [MATH] as an accuracy measure.', '1609.03261-1-69-6': 'This value generally has the same order as [MATH].', '1609.03261-1-69-7': 'To illustrate the average performance, we run the algorithm for 20 times in each case and report the norm of the gradient evaluated at the average of iterates.', '1609.03261-1-69-8': 'In all cases, the initial value is set to be a zero vector.', '1609.03261-1-69-9': 'The top row of Figure 1 displays the results.', '1609.03261-1-69-10': 'It is clear that SCSG with a batch size as small as [MATH] converges faster than SVRG in the first 50 passes.', '1609.03261-1-69-11': 'To achieve an accuracy of [MATH], SCSG requires less than five passes of data while SVRG requires around ten passes through the data.', '1609.03261-1-69-12': 'In the best-tuned case ([MATH]), SCSG only requires two passes.', '1609.03261-1-69-13': 'Moreover, it is seen that performance degrades for [MATH].', '1609.03261-1-69-14': 'This implies that our benchmark stepsize is valid in this case; in practice, we suggest starting from the benchmark stepsize [MATH] and trying multiples of [MATH] in [MATH].', '1609.03261-1-70-0': 'Since our theory indicates that SCSG is able to achieve good accuracy even with less than a single pass through the dataset, we repeat the above procedure while recording the iterates [MATH] every 0.1 of a pass through the data, where the batch size [MATH] is selected from [MATH].', '1609.03261-1-70-1': 'The bottom row of Figure 1 displays the results.', '1609.03261-1-70-2': 'It is seen that [MATH] yields the fastest convergence in the first pass and in the best tuned case, [MATH], an accuracy of [MATH] is achieved within only 0.25 passes through the data for [MATH] and [MATH].', '1609.03261-1-70-3': 'Note that a batch of MNIST dataset with size [MATH] requires no more than 2.6MB of memory and it involves roughly [MATH] accesses of the disk.', '1609.03261-1-70-4': 'In other words, a commodity machine with a very modest memory suffices to achieve reasonable accuracy.', '1609.03261-1-71-0': '# Discussion', '1609.03261-1-72-0': 'We propose SCSG as a member of the SVRG family of algorithms, proving its superior performance in terms of both computation and communication cost.', '1609.03261-1-72-1': 'Both complexities are independent of sample size when the required accuracy is low, for generalized linear models which are widely used in practice.', '1609.03261-1-72-2': 'The real data example also validates our theory.', '1609.03261-1-73-0': 'For practical use, we develop an automatic parameter-tuning procedure.', '1609.03261-1-73-1': 'For a dataset for which the scanning is possible, we recommend using the benchmark stepsize and batch size produced by automatic tuning and running SCSG for certain steps.', '1609.03261-1-73-2': 'If the desired accuracy is not achieved, then double the stepsize or the batch size.', '1609.03261-1-73-3': 'The doubling continues until the batch cannot fit into memory.', '1609.03261-1-73-4': 'For massive data of which the scanning is too costly, we recommend estimating [MATH] and [MATH] on a random batch and repeating the above procedure.', '1609.03261-1-74-0': 'We plan to explore several variants of SCSG in future work.', '1609.03261-1-74-1': 'For example, a non-uniform sampling scheme can be applied to SGD steps to leverage the Lipschitz constants [MATH] as in SVRG.', '1609.03261-1-74-2': 'More interestingly, we can consider a better sampling scheme for [MATH] by putting more weight on influential observations.', '1609.03261-1-74-3': 'The mini-batch settings and proximal settings are also straightforward extensions of our current work.', '1609.03261-1-75-0': 'As a final comment, we found that the previous complexity analysis focuses on the high-accuracy computation for which the dependence on the sample size [MATH] and condition number [MATH] is of major concern.', '1609.03261-1-75-1': 'The low-accuracy regime is unfortunately under-studied theoretically even though it is commonly encountered in practice.', '1609.03261-1-75-2': 'We advocate taking all three parameters, namely [MATH], [MATH] and [MATH], into consideration and distinguishing the analyses for high-accuracy computation and low-accuracy computation as standard practice in the literature.', '1609.03261-1-76-0': '# Technical Proofs', '1609.03261-1-77-0': '## Lemmas', '1609.03261-1-78-0': '[Lemma [REF]] Let [MATH], then it is easy to see that [EQUATION]', '1609.03261-1-78-1': 'Then [MATH] can be reformulated as [EQUATION].', '1609.03261-1-78-2': 'Since [MATH] is the optimum of [MATH], we have [EQUATION] and [EQUATION]', '1609.03261-1-78-3': 'Under Assumption [MATH] and A2 with [MATH] possibly equal to [MATH], [EQUATION]', '1609.03261-1-78-4': 'The same bound holds if we interchange [MATH] and [MATH] on right-hand side.', '1609.03261-1-78-5': 'In particular, when [MATH], [EQUATION] [Lemma [REF]] By Lemma A1 of [CITATION], for any [MATH], [EQUATION] which proves the lemma.', '1609.03261-1-79-0': '[Lemma [REF]] We prove that for any [MATH], [EQUATION].', '1609.03261-1-79-1': 'In fact, by Lemma [REF], [EQUATION].', '1609.03261-1-79-2': 'Summing the above inequality for all [MATH] results in [EQUATION]', '1609.03261-1-79-3': 'Since [MATH] is a minimizer, we know that [MATH] and thus [EQUATION].', '1609.03261-1-79-4': 'On the other hand, note that for any [MATH], [EQUATION].', '1609.03261-1-79-5': 'Thus, [EQUATION].', '1609.03261-1-79-6': 'The first part of the lemma is then proved by setting [MATH].', '1609.03261-1-79-7': 'For the second part, we exchange [MATH] and [MATH] in equation [REF] and obtain that [EQUATION].', '1609.03261-1-79-8': 'Apply the same argument as above we prove the second inequality of the lemma.', '1609.03261-1-80-0': 'Let [MATH] be a random subset with size [MATH], [MATH] be a random element of [MATH] and [EQUATION] thus for any [MATH], under Assumption [MATH] and [MATH] with [MATH] possibly equal to 0, it holds that [EQUATION]', '1609.03261-1-80-1': 'In particular when [MATH], [EQUATION] [Lemma [REF]] Notice that for any [MATH], [EQUATION] then for [MATH] we have [EQUATION] where the last inequality uses the fact that [EQUATION] and the trivial inequality [MATH].', '1609.03261-1-80-2': 'For the first term, by Lemma [REF] we obtain that [EQUATION]', '1609.03261-1-80-3': 'Note that [MATH] and [MATH], the above expression can be simplified as [EQUATION]', '1609.03261-1-80-4': 'For the second term, we simply use Lemma [REF] with [MATH] and obtain that [EQUATION]', '1609.03261-1-80-5': 'Combining the above results and Lemma [REF], we conclude that [EQUATION]', '1609.03261-1-81-0': '## Convergence Analysis of Non-Strongly Convex Case', '1609.03261-1-82-0': '[Theorem [REF]] We prove Theorem [REF] with [EQUATION].', '1609.03261-1-82-1': 'It is easy to show [MATH] when [MATH] via numerical calculation.', '1609.03261-1-83-0': 'Now we state the main proof.', '1609.03261-1-83-1': 'In stage [MATH], [MATH], and we have [MATH].', '1609.03261-1-83-2': 'In the following argument, we omit the subscript [MATH] for brevity.', '1609.03261-1-83-3': '[EQUATION] where the last inequality uses Lemma [REF].', '1609.03261-1-83-4': 'Here we restrict [MATH] such that [EQUATION]', '1609.03261-1-83-5': 'Noticing that ([REF]) implies that [MATH] and the convexity of [MATH] implies [EQUATION] we have [EQUATION]', '1609.03261-1-83-6': 'By definition of [MATH]), [EQUATION] and [EQUATION] where [MATH] is the normalization factor.', '1609.03261-1-83-7': 'In order to be concise, let [EQUATION]', '1609.03261-1-83-8': 'Setting [MATH] in ([REF]), [EQUATION]', '1609.03261-1-83-9': 'It then follows from ([REF]) and ([REF]) that [EQUATION]', '1609.03261-1-83-10': 'This implies that [EQUATION]', '1609.03261-1-83-11': 'By convexity of [MATH], we have [EQUATION] where the third inequality uses the smoothness of [MATH], i.e., [EQUATION] and the fourth inequality uses the fact that [EQUATION].', '1609.03261-1-83-12': 'Then [EQUATION]', '1609.03261-1-83-13': 'We distinguish two cases:', '1609.03261-1-84-0': 'If [MATH], then [MATH].', '1609.03261-1-84-1': 'Setting [MATH] implies that [EQUATION]', '1609.03261-1-84-2': 'If [MATH], then optimizing [MATH] over the set [MATH] produces [MATH], in which case ([REF]) is satisfied and [EQUATION] and [EQUATION].', '1609.03261-1-85-0': 'Based on Theorem [REF], we can obtain the following result.', '1609.03261-1-86-0': 'Under the settings of Theorem [REF], assume [MATH] and [EQUATION] and [EQUATION]', '1609.03261-1-86-1': 'Then [EQUATION] [Corollary [REF]] It is easy to verify that under ([REF]) and ([REF]), [EQUATION] and [EQUATION].', '1609.03261-1-86-2': 'Therefore, [EQUATION] .', '1609.03261-1-87-0': 'As a direct consequence, we obtain the computation and communication complexity of SCSG for non-strongly convex case.', '1609.03261-1-88-0': '[Corollary [REF]] It is known from Corollary [REF] that [EQUATION].', '1609.03261-1-88-1': 'The computation cost is [EQUATION].', '1609.03261-1-88-2': 'By Lemma [REF], we know that [EQUATION] and as a result, [EQUATION].', '1609.03261-1-88-3': 'Similarly the communication cost is [EQUATION]', '1609.03261-1-89-0': '## Convergence Analysis of Strongly Convex Case With Assumption A 2 Similarly to the last section, we first establish a slightly more accurate version of Theorem [REF] as follows.', '1609.03261-1-90-0': 'Let [MATH].', '1609.03261-1-90-1': 'If [MATH] for some [MATH], and one of the following assumptions hold: [(i)] [MATH], [MATH], [MATH]', '1609.03261-1-91-0': 'where [MATH] is the condition number.', '1609.03261-1-91-1': 'Then', '1609.03261-1-92-0': 'If [MATH], [EQUATION]', '1609.03261-1-92-1': 'If [MATH], [EQUATION].', '1609.03261-1-93-0': 'Note that when [MATH], [EQUATION] and thus Part 1 can be simplified by a slightly weaker bound as in Theorem [REF]: [EQUATION].', '1609.03261-1-93-1': 'Similarly, [EQUATION] and thus Part 2 can be simplified by a slightly weaker bound as in Theorem [REF] [EQUATION].', '1609.03261-1-93-2': 'Furthermore, Part 2 of the Theorem implies that the constant [MATH] in Theorem [REF] is [EQUATION] and it is easy to see that [MATH] if [MATH].', '1609.03261-1-94-0': '[Theorem [REF]] Similar to the proof of Theorem [REF], if [MATH] satisfies ([REF]), then [MATH] and [EQUATION] where the last inequality is from the strong convexity of [MATH], i.e. [EQUATION].', '1609.03261-1-94-1': 'Setting [MATH], we have [EQUATION] where [MATH] is defined by ([REF]).', '1609.03261-1-94-2': 'By definition of [MATH] (the stage index [MATH] is omitted for brevity), [EQUATION] and [EQUATION] where [MATH] is the normalization factor such that [EQUATION].', '1609.03261-1-94-3': 'It then follows from ([REF]) and ([REF]) that [EQUATION]', '1609.03261-1-94-4': 'The above inequality can be rewritten as [EQUATION]', '1609.03261-1-94-5': 'Define [MATH] as [EQUATION] then [EQUATION]', '1609.03261-1-94-6': 'This implies that [EQUATION] where the last inequality uses the fact that [MATH] and hence [EQUATION]', '1609.03261-1-94-7': 'Now we prove that under both conditions (i) and (ii), [EQUATION] [(i)] Since [EQUATION] - (1 - )1 - 1 - (1 + )L(1 + ) + 1(frac- (1 - )1 - + 1)frac1 - (1 + )L(1 + ) = L1 - (1 + )(1 + )(1 - )12(1 - ), [EQUATION] m12L^2 (1 - 2)(1 - )2L^2 [EQUATION] = L(1 + )1 - (1 + ) [EQUATION] = 2L1 - 2 [EQUATION]', '1609.03261-1-94-8': 'J_1L2(1 + (1 - )(1 - )L(1 - 2))D_0.', '1609.03261-1-94-9': '[EQUATION]', '1609.03261-1-94-10': 'J_1L2( 1 + 2(1 - )) D_05LD_0.', '1609.03261-1-94-11': '[EQUATION]', '1609.03261-1-94-12': 'EFkfx_T ( 2L1 - 2)^T5LD_0 = (1 - (L + )(1 - 2)L + - 2)^T5LD_0.', '1609.03261-1-94-13': '[EQUATION]', '1609.03261-1-94-14': 'J_24L1 - (L + )(1 + )- 1(n - B)(n - 1)BG_nL41 - (1 + )- 1(n - B)(n - 1)BG_nL.', '1609.03261-1-94-15': '[EQUATION]', '1609.03261-1-94-16': 'J_24(1 - - )^2(n - B)(n - 1)BG_nL.', '1609.03261-1-94-17': '[EQUATION]', '1609.03261-1-94-18': 'J_1L2(1 + (1 - )(1 - )L1 - (1 + (1 - ) / ))D_0.', '1609.03261-1-94-19': '[EQUATION]', '1609.03261-1-94-20': 'J_1L2(1 + 4(1 - )(1 - ))D_0L2(1 + 4)D_05LD_0, [EQUATION] 1 - (1 + (1 - ) / )1 - L + 2(1 + (1 - ) / ) = 1 - + (1 - )212 [EQUATION]', '1609.03261-1-94-21': 'EFkfx_T.', '1609.03261-1-94-22': '[EQUATION] 5LD_0(1 - (L + )1 - (1 - - )L + - (+ (1-)))^T2, [EQUATION] 4(1 - - )^2(n - B)(n - 1)BG_nL2.', '1609.03261-1-94-23': '[EQUATION]', '1609.03261-1-94-24': 'EFkfx_T.', '1609.03261-1-94-25': '[EQUATION]', '1609.03261-1-94-26': 'EFkfx_T.', '1609.03261-1-94-27': '[EQUATION]', '1609.03261-1-94-28': 'E C_comp = Esum_j=1^T(B + N_j) T(B + m).', '1609.03261-1-94-29': '[EQUATION]', '1609.03261-1-94-30': 'E C_comp = O(n + ^2, G_nL + ^2D_0L).', '1609.03261-1-94-31': '[EQUATION]', '1609.03261-1-94-32': 'G_nL^22L^21n_i=1^nf_i(x_0)^2 + 4D_0, [EQUATION]', '1609.03261-1-94-33': 'E C_comp = O( n + , L + L).', '1609.03261-1-94-34': '[EQUATION]', '1609.03261-1-94-35': 'E C_comm = E BT = O( ( nL)log L).', '1609.03261-1-94-36': '[EQUATION] m22 + 1 = O().', '1609.03261-1-94-37': '[EQUATION]', '1609.03261-1-94-38': 'T = O(fracLL) = O(1), m = O( ).', '1609.03261-1-94-39': '[EQUATION]', '1609.03261-1-94-40': 'E C_comp E(B + m)T = O(nL + ) [EQUATION]', '1609.03261-1-94-41': 'E C_commE BT = O( nL).', '1609.03261-1-94-42': '[EQUATION] 2EFkfx_j1 - (EXkx_j-1 - EXkx_j) + 2^2L(1 + )EFkfx_j - 1 + 1(, , B) [EQUATION] 2EFkfx_j1 - 2EFkfx_j - 1 + 2^2L(1 + )EFkfx_j - 1 + 1(, , B).', '1609.03261-1-94-43': '[EQUATION]', '1609.03261-1-94-44': 'EFkfx_j( + (- 1)L) EFkfx_j - 1 + 12(, , B).', '1609.03261-1-94-45': '[EQUATION]', '1609.03261-1-94-46': 'EFkfx_jEFkfx_j-1 [EQUATION]', '1609.03261-1-94-47': 'EFkfx_j EFkfx_j - 1 + 1 + 2G_nn - B(n - 1)B. [EQUATION]', '1609.03261-1-94-48': 'EFkfx_T ^T (f(x_0) - f(x^*)) + 11 - 1 + 2G_nn - B(n - 1)B. [EQUATION] 1c + 2.', '1609.03261-1-94-49': '[EQUATION] : 1c + (2 + )< 1 [EQUATION]', '1609.03261-1-94-50': 'E C_comp = O(( n L + )log L), E C_comm = O(lb n L) L).', '1609.03261-1-94-51': '[EQUATION]', '1609.03261-1-94-52': 'T = ( m1 - n + n1) = ( m1).', '1609.03261-1-94-53': '[EQUATION]', '1609.03261-1-94-54': 'H = O( 1m( n1 + )rb.', '1609.03261-1-94-55': '[EQUATION] ( 1 - nn + mn)^H-mn1 + n + = (lb 1 - mn + )^n + m)^n1 + n + .', '1609.03261-1-94-56': '[EQUATION] (lb 1 - )^1)^n + n1 + -cn1 + n + ] where [MATH].', '1609.03261-1-94-57': 'Note that for any [MATH], [MATH], we conclude that [EQUATION] and hence [EQUATION].', '1609.03261-1-94-58': 'As a consequence, we obtain that the communication cost of CoCoA is at least [EQUATION]', '1609.03261-1-95-0': '## Bounding [MATH] and [MATH] for (Multi-Class) Logistic Regression (Equation [REF])', '1609.03261-1-96-0': 'In Section 4 we claim that [MATH] for (multi-class) logistic regression.', '1609.03261-1-96-1': 'Here we establish this claim.', '1609.03261-1-96-2': 'Denote by [MATH] the concatenation of [MATH] as in Section 5.', '1609.03261-1-96-3': 'Then [EQUATION].', '1609.03261-1-96-4': 'For any [MATH], [EQUATION] and thus [EQUATION] where [EQUATION].', '1609.03261-1-96-5': 'It is easy to see that for any [MATH] and [MATH] [EQUATION].', '1609.03261-1-96-6': 'This entails that [EQUATION].', '1609.03261-1-96-7': 'On the other hand, for any [MATH], [EQUATION] and for any [MATH], [EQUATION].', '1609.03261-1-96-8': 'Thus, [EQUATION].', '1609.03261-1-96-9': 'As a consequence, [EQUATION]'}

{'1609.03261-2-0-0': 'We develop and analyze a procedure for gradient-based optimization that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-2-0-1': 'As a member of the SVRG family of algorithms, SCSG makes use of gradient estimates at two scales, with the number of updates at the faster scale being governed by a geometric random variable.', '1609.03261-2-0-2': 'Unlike most existing algorithms in this family, both the computation cost and the communication cost of SCSG do not necessarily scale linearly with the sample size [MATH]; indeed, these costs are independent of [MATH] when the target accuracy is low.', '1609.03261-2-0-3': 'An experimental evaluation on real datasets confirms the effectiveness of SCSG.', '1609.03261-2-1-0': '# Introduction', '1609.03261-2-2-0': 'The problem of optimizing a finite-sum convex objective: [EQUATION] where each [MATH] is a convex function, is ubiquitous in statistical machine learning.', '1609.03261-2-2-1': 'This problem is often solved by algorithms that either make use of full gradients (obtained by processing the entire dataset) or stochastic gradients (obtained by processing single data points or mini-batches of data points).', '1609.03261-2-2-2': 'The use of the former provides guarantees of eventual convergence and convergence rate as measured by iteration count, while the latter yields advantages in per-iteration complexity, scalability and simplicity of implementation [CITATION].', '1609.03261-2-2-3': 'An impactful recent line of research has shown that a hybrid methodology that makes use of both full gradients and stochastic gradients can obtain the best of both worlds-guaranteed convergence at favorable rates, e.g. [CITATION].', '1609.03261-2-2-4': 'The full gradients provide variance control for the stochastic gradients.', '1609.03261-2-3-0': 'While this line of research represents significant progress towards the goal of designing scalable, autonomous learning algorithms, there remain some inefficiencies in terms of computation.', '1609.03261-2-3-1': 'With the definition of computation and communication cost in Section 2.1, the methods referred to above require [MATH] computation to achieve an [MATH]-approximate solution, where [MATH] is the number of data points, [MATH] is a target accuracy and [MATH] is the dimension of the parameter vector.', '1609.03261-2-3-2': 'Some methods incur a [MATH] storage cost [CITATION].', '1609.03261-2-3-3': 'The linear dependence on [MATH] is problematic in general.', '1609.03261-2-3-4': 'Clearly there will be situations in which accurate solutions can be obtained with less than a single pass through the data; indeed, some problems will require a constant number of steps.', '1609.03261-2-3-5': 'This will be the case, for example, if the data in a regression problem consist of a fixed number of pairs repeated a large number of times.', '1609.03261-2-3-6': 'For deterministic algorithms, the worst case analysis in [CITATION] shows that scanning at least a fixed proportion of the data is necessary; however, learning algorithms are generally stochastic and real-world learning problems are generally not worst case.', '1609.03261-2-4-0': 'An equally important bottleneck for learning algorithms is the cost of communication.', '1609.03261-2-4-1': 'For large data sets that must be stored on disk or distributed across many computing nodes, the communication cost can be significant, even dominating the computation cost.', '1609.03261-2-4-2': 'For example, SVRG makes use of a full gradient over the whole dataset which can incur a prohibitive communication cost.', '1609.03261-2-4-3': 'There is an active line of research that focuses on communication costs; see, e.g. [CITATION].', '1609.03261-2-5-0': 'In this article, we present a variant of the stochastic variance reduced gradient (SVRG) method that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-2-5-1': 'The basic idea behind SCSG-that of approximating the full gradient in SVRG via a subsample-has been explored by others, but we present several innovations that yield significant improvements both in theory and in practice.', '1609.03261-2-5-2': 'In contradistinction to SVRG, the theoretical convergence rate of SCSG has a sublinear regime in terms of both computation and communication.', '1609.03261-2-5-3': 'This regime is important in machine learning problems, notably in the common situation in which the sample size is large, ([MATH]), while the required accuracy is low, [MATH].', '1609.03261-2-5-4': 'The analysis in this article shows that SCSG is able to achieve the target accuracy in this regime with potentially less than a single pass through the data.', '1609.03261-2-6-0': 'In the regime of low accuracy, SCSG is never worse than classical stochastic gradient descent (SGD).', '1609.03261-2-6-1': 'Although SCSG has the same dependence on the target accuracy as SGD, it has a potentially much smaller constant factor.', '1609.03261-2-6-2': 'In fact, the theoretical complexity of SGD depends on the uniform bound of [MATH] over the domain and the component index.', '1609.03261-2-6-3': 'This might be infinite even in the most common least-squares problems.', '1609.03261-2-6-4': 'By contrast, the complexity of SCSG depends on a new measure [MATH], defined in Section [REF] and discussed in Section [REF], which is finite and small for a large class of practical problems.', '1609.03261-2-6-5': 'In particular, [MATH] in many cases where SGD does not have a theoretical convergence guarantee.', '1609.03261-2-6-6': 'The measure [MATH] helps characterize the difficulty of optimization problems in the form of a finite sum and sheds light on some of the intrinsic differences between finite-sum optimization and stochastic approximation; see also [CITATION] and [CITATION].', '1609.03261-2-7-0': 'The remainder of the paper is organized as follows.', '1609.03261-2-7-1': 'In Section 2, we review SVRG, discuss several of its variants and we describe the SCSG algorithm.', '1609.03261-2-7-2': 'We provide a theoretical convergence analysis in Section 3.', '1609.03261-2-7-3': 'In Section 4, we present a comprehensive discussion of the difficulty measure [MATH].', '1609.03261-2-7-4': 'Empirical results on real datasets are presented in Section [REF].', '1609.03261-2-7-5': 'Finally, we conclude our work and discuss potential extensions in Section [REF].', '1609.03261-2-7-6': 'All technical proofs are relegated to the Appendices.', '1609.03261-2-8-0': '# Notation, Assumptions and Algorithm', '1609.03261-2-9-0': 'We write [MATH] as [MATH] and [MATH] as [MATH] for brevity and use [MATH] to denote the Euclidean norm throughout the paper.', '1609.03261-2-9-1': 'We adopt the standard Landau notation ([MATH]).', '1609.03261-2-9-2': 'In some cases, we use [MATH] to hide terms which are polynomial in parameters.', '1609.03261-2-9-3': 'The notation [MATH] will only be used to maximize the readibility in discussions but will not be used in the formal analysis.', '1609.03261-2-9-4': 'For convenience, we use [MATH] to denote the set [MATH] and for any subset [MATH], we write [MATH] for the batch gradient [MATH].', '1609.03261-2-9-5': 'Finally, given random variables [MATH] and [MATH] and a random variable [MATH], denote by [MATH] the conditional expectation of [MATH] given [MATH]; i.e., [MATH].', '1609.03261-2-9-6': 'Note that when [MATH] is independent of [MATH], then [MATH] is equivalent to the expectation of [MATH] holding [MATH] fixed.', '1609.03261-2-9-7': 'Furthermore, we use the symbol [MATH], without the subscript, to denote the expectation over all randomness.', '1609.03261-2-10-0': 'Assumption A1 on the smoothness of individual functions will be used throughout this paper.', '1609.03261-2-10-1': '[A1] [MATH] is convex with [MATH]-Lipschitz gradient [EQUATION] for some [MATH] and all [MATH];', '1609.03261-2-11-0': 'The following assumption will be used in the context of strongly-convex objectives.', '1609.03261-2-11-1': '[A2] [MATH] is strongly-convex with [EQUATION] for some [MATH].', '1609.03261-2-12-0': 'Note that we only require the strong convexity of [MATH] instead of each component.', '1609.03261-2-13-0': 'We define the complexity measure [MATH] that we will use to motivate and analyze SCSG as follows: [EQUATION]', '1609.03261-2-13-1': 'We will abbreviate [MATH] as [MATH] when no confusion can arise.', '1609.03261-2-13-2': 'Note that [MATH] is unique in many situations where [MATH].', '1609.03261-2-13-3': 'When there are multiple minima, we select [MATH] to be the one that minimizes the RHS of [REF].', '1609.03261-2-13-4': 'Further let [MATH] denote the initial value (possibly random) and [EQUATION]', '1609.03261-2-13-5': 'Then [MATH] under Assumptions A1 and A2.', '1609.03261-2-13-6': 'A point [MATH], possibly random, is called an [MATH]-approximate solution if [EQUATION].', '1609.03261-2-14-0': 'To measure computational complexity, we assume that sampling an index [MATH] and computing the pair [MATH] incurs a unit of cost.', '1609.03261-2-14-1': 'This is conventional and called the "IFO framework" in the literature ([CITATION]).', '1609.03261-2-14-2': 'We use use [MATH] to denote the cost to achieve an [MATH]-approximate solution.', '1609.03261-2-14-3': 'In some contexts we also consider [MATH] as the cost to reach a solution [MATH] with [MATH].', '1609.03261-2-15-0': 'Finally, since our analysis relies heavily on the geometric distribution, we formally define it here.', '1609.03261-2-15-1': 'We say a random variable [MATH] if [MATH] is supported on the nonnegative integers with [EQUATION].', '1609.03261-2-15-2': 'The expectation of such a random variable is given by [EQUATION] [h] Stochastic Variance Reduced Gradient (SVRG) Method', '1609.03261-2-16-0': 'Inputs: Stepsize [MATH], number of stages [MATH], initial iterate [MATH], number of SGD steps [MATH].', '1609.03261-2-17-0': 'Procedure [1] [MATH] Randomly pick [MATH]', '1609.03261-2-18-0': 'Output: (Option 1): [MATH] (Option 2): [MATH].', '1609.03261-2-19-0': '## SVRG and other related work', '1609.03261-2-20-0': 'The stochastic variance reduced gradient (SVRG) method blends gradient descent and stochastic gradient descent, using the former to control the effect of the variance of the latter [CITATION].', '1609.03261-2-20-1': 'We summarize SVRG in Algorithm [REF].', '1609.03261-2-21-0': 'Using the definition from Section 2.1, it is easy to see that the computation cost of SVRG is [MATH].', '1609.03261-2-21-1': 'As shown in the convergence analysis of [CITATION], [MATH] is required to be [MATH] to guarantee convergence.', '1609.03261-2-21-2': 'Thus, the computation cost of SVRG is [MATH].', '1609.03261-2-21-3': 'The costs of the other algorithms considered in Table [REF] can be obtain in a similar fashion.', '1609.03261-2-21-4': 'For comparison, we only present the results for the smooth case (Assumption A1).', '1609.03261-2-22-0': 'A number of variants of SVRG have been studied.', '1609.03261-2-22-1': 'For example, a constrained form of SVRG can be obtained by replacing line 8 with a projected gradient descent step [CITATION].', '1609.03261-2-22-2': 'A mini-batch variant of SVRG arises when one samples a subset of indices instead of a single index in line 6 and updates the iterates by the average gradient in this batch in line 7 [CITATION].', '1609.03261-2-22-3': 'Similarly, we can consider implementing the full gradient computation in line 2 using a subsample.', '1609.03261-2-22-4': 'This is proposed in [CITATION], which calculates [MATH] as [MATH] where [MATH] is a subset of size [MATH] uniformly sampled from [MATH].', '1609.03261-2-22-5': '[CITATION] heuristically show the potential for significant complexity reduction, but they only prove convergence for [MATH] under the stringent condition that [MATH] is uniformly bounded for all [MATH] and that all iterates are uniformly bounded.', '1609.03261-2-22-6': "Similar to Nesterov's acceleration for gradient descent, momentum terms can be added to the SGD steps to accelerate SVRG [CITATION].", '1609.03261-2-23-0': 'Much of this work focuses on the strongly convex case.', '1609.03261-2-23-1': 'In the non-strongly convex setting one way to proceed is to add a [MATH] regularizer [MATH].', '1609.03261-2-23-2': 'Tuning [MATH], however, is subtle and requires multiple runs of the algorithm on a grid of [MATH] [CITATION].', '1609.03261-2-23-3': 'For general convex functions an alternative approach has been presented by [CITATION] (they generate [MATH] by a different scheme in line 4), which proves a computation complexity [MATH].', '1609.03261-2-23-4': 'Another approach is discussed by [CITATION], who improve the complexity to [MATH] by scaling the stepsize as [MATH].', '1609.03261-2-23-5': 'However, their algorithm still relies on calculating a full gradient.', '1609.03261-2-23-6': 'Other variants of SVRG have been proposed in the distributed computing setting [CITATION] and in the stochastic setting [CITATION].', '1609.03261-2-24-0': '## SCSG', '1609.03261-2-25-0': '[htp] Stochastically Controlled Stochastic Gradient (SCSG) Method', '1609.03261-2-26-0': 'Inputs: Stepsize [MATH], batch size [MATH], number of stages [MATH], initial iterate [MATH].', '1609.03261-2-27-0': 'Procedure [1] [MATH] Uniformly sample a batch [MATH] with [MATH] Generate [MATH] Randomly pick [MATH]', '1609.03261-2-28-0': 'Output: (Strongly convex case): [MATH] (Non-strongly convex case): [MATH].', '1609.03261-2-29-0': 'SCSG is similar to [CITATION] in that it implements the gradient computation on a subsample [MATH] of size [MATH]; see Algorithm [REF].', '1609.03261-2-29-1': 'However, instead of being fixed, the number of SGD updates in SCSG is a geometrically distributed random variable (line 5).', '1609.03261-2-29-2': 'Surprisingly, this seemingly small technical modification enables the analysis in the non-strongly convex case and yields a much tighter convergence analysis without the need to impose unrealistic assumptions such as the boundedness of iterates produced by the algorithm.', '1609.03261-2-29-3': '(See Section [REF] for details.)', '1609.03261-2-29-4': 'Note that [CITATION] also implicitly use a geometric size for the inner loop.', '1609.03261-2-29-5': 'However, crucially they do not use the iterate at the end of each epoch-i.e., [MATH]-and hence they cannot provide an analysis for the non-strongly convex case.', '1609.03261-2-30-0': 'The average computation cost of SCSG is [MATH].', '1609.03261-2-30-1': 'By the law of large numbers and the expectation formula [REF], this is close to [MATH].', '1609.03261-2-30-2': 'Table [REF] summarizes the computation complexity as well as some other details of SCSG and 11 other existing popular algorithms.', '1609.03261-2-30-3': 'The table includes the computation cost of optimizing non-strongly convex functions (column 1) and strongly convex functions (column 2).', '1609.03261-2-30-4': 'In practice, the amount of tuning is of major concern.', '1609.03261-2-30-5': 'For this reason, a fixed stepsize is usually preferred to a complicated stepsize scheme and it is better that the tuning parameter does not depend on unknown quantities; e.g., [MATH] or the total number of epochs [MATH].', '1609.03261-2-30-6': 'These issues are documented in column 3 and column 4.', '1609.03261-2-30-7': 'Moreover, many algorithms requires [MATH] to be bounded; i.e., [MATH] to be Lipschitz.', '1609.03261-2-30-8': 'However, this assumption is not realistic in many cases and it is better to discard it.', '1609.03261-2-30-9': 'To address this issue, we document it in column 5.', '1609.03261-2-30-10': 'To highlight the dependence on [MATH] and [MATH] (or [MATH]), we implicitly assume that other parameters, e.g., [MATH], are [MATH] as a convention.', '1609.03261-2-31-0': 'As seen from Table [REF], SCSG and SGD are the only two methods which are able to reach an [MATH]-approximate solution with potentially less than a single pass through the data; moreover, the number of accesses of the data is independent of the sample size [MATH].', '1609.03261-2-31-1': 'Comparing to SCSG, SGD requires each [MATH] to be Lipschitz, which is not satisfied by least-square objectives.', '1609.03261-2-31-2': 'By contrast, as will be shown in Section [REF], the computation cost of SCSG only depends on the quantity [MATH], which is relatively small in many cases.', '1609.03261-2-31-3': 'Furthermore, SGD either sets the stepsize based on unknown quantities like the total number of epochs [MATH] or needs to use a time-varying sequence of stepsizes.', '1609.03261-2-31-4': 'This involves intensive tuning as opposed to a fixed stepsize.', '1609.03261-2-32-0': 'Moreover, SCSG is communication-efficient since it only needs to operate on mini-batches (as is the case with SGD).', '1609.03261-2-32-1': 'This is particularly important in modern large-scale problems.', '1609.03261-2-32-2': 'By contrast, those algorithms that require full gradient evaluations either incur an extra communication cost for synchronization or incur an extra computational cost for the asynchronous version to converge; see, e.g., [CITATION].', '1609.03261-2-33-0': '# Convergence Analysis', '1609.03261-2-34-0': 'In this section we present a convergence analysis of SCSG.', '1609.03261-2-34-1': 'We first state the following key lemma that connects our algorithm with the measure [MATH] defined in [REF].', '1609.03261-2-35-0': 'Let [MATH] be a random subset of size [MATH], and define the random variable [MATH].', '1609.03261-2-35-1': 'Then [MATH] and [EQUATION].', '1609.03261-2-36-0': 'The proof, which appears in Appendix [REF], involves a standard technique for analyzing sampling without replacement.', '1609.03261-2-36-1': 'Obviously, [MATH] if [MATH] is uniformly bounded as is often assumed in the literature.', '1609.03261-2-36-2': 'In section [REF] we will present various other situations where [MATH].', '1609.03261-2-37-0': 'Note that the extra variation vanishes when [MATH] and in general is inversely proportional to the batch size.', '1609.03261-2-37-1': 'In the rest of this section, we will first discuss the case [MATH], which we refer to as R-SVRG (Randomized SVRG), to compare with the original SVRG.', '1609.03261-2-37-2': 'Later we will discuss the general case.', '1609.03261-2-38-0': '## Analysis of R-SVRG', '1609.03261-2-39-0': 'We start by deriving the sub-optimality bound for [MATH] and [MATH] respectively.', '1609.03261-2-40-0': 'Let [MATH] and assume that [MATH], then [(1)] under Assumption A1, [EQUATION] under Assumptions A1 and A2, [EQUATION] where [EQUATION].', '1609.03261-2-41-0': 'Based on Theorem [REF], we first consider a constant stepsize [MATH] scaled as [MATH].', '1609.03261-2-42-0': 'Let [MATH] with [MATH].', '1609.03261-2-42-1': 'Then under Assumption A1, with the output [MATH], [EQUATION]', '1609.03261-2-42-2': 'If further Assumption A2 is satisfied, then the output [MATH] satisfies [EQUATION]', '1609.03261-2-42-3': 'The above theorem is appealing in three aspects: 1) in the strongly convex case, no parameter depends on [MATH].', '1609.03261-2-42-4': 'This is in contrast to the original SVRG where the number of SGD updates needs to be proportional to [MATH] in order to guarantee convergence [CITATION].', '1609.03261-2-42-5': 'Being agnostic to [MATH] is useful in that [MATH] is hard to estimate in practice; 2) the same setup also guarantees the convergence of [MATH] in the strongly convex case with an almost identical cost up to a [MATH] factor.', '1609.03261-2-42-6': 'This is important especially in statistical problems and is not covered in existing literature to the best of our knowledge; 3) the same stepsize guarantees the convergence in both the non-strongly convex and the strongly convex case and the only requirement is [MATH], which is quite mild.', '1609.03261-2-42-7': 'Note that the requirement for the convergence of gradient descent is [MATH].', '1609.03261-2-43-0': 'By scaling [MATH] as [MATH], R-SVRG is able to achieve the same complexity as in [CITATION], which is the best bound in the class of SVRG-type algorithms without acceleration techniques.', '1609.03261-2-44-0': 'Let [MATH] with [MATH].', '1609.03261-2-44-1': 'Then under Assumption A1, with the output [MATH], [EQUATION]', '1609.03261-2-45-0': '## Analysis of SCSG', '1609.03261-2-46-0': 'We first discuss the non-strongly convex case and then turn to the strongly convex case.', '1609.03261-2-46-1': 'As in our analysis of R-SVRG, we first derive the sub-optimality bound for [MATH].', '1609.03261-2-47-0': 'Assume that [MATH].', '1609.03261-2-47-1': 'Under Assumption A1, [EQUATION].', '1609.03261-2-48-0': 'Note that the bound in Theorem [REF] can be simplified to [MATH], while the bound in Theorem [REF] can be simplified to [MATH].', '1609.03261-2-48-1': 'Despite the more stringent requirement on [MATH]), these two bounds have two qualitative differences: 1) SCSG has an extra term, [MATH], which characterizes the sampling variance of the mini-batch gradients; 2) SCSG loses an [MATH] in the first term, which is due to the bias of [MATH].', '1609.03261-2-48-2': 'In fact, recalling the definition of [MATH] at the beginning of Section [REF], a simple calculation shows that [EQUATION] which does not equal [MATH] in general.', '1609.03261-2-48-3': 'Most of the novelty of our analysis lies in dealing with the extra bias.', '1609.03261-2-48-4': 'Fortunately, we found that the extra terms do not worsen the complexity by scaling [MATH] as [MATH].', '1609.03261-2-49-0': 'Assume A1 holds.', '1609.03261-2-49-1': 'Set [EQUATION].', '1609.03261-2-49-2': 'Assume that [MATH], then with the output [MATH], [EQUATION]', '1609.03261-2-49-3': 'Corollary [REF] shows that SCSG is never worse than SGD and SVRG (with constant stepsize scaled as [MATH]).', '1609.03261-2-49-4': 'Compared with SGD whose complexity is [MATH] [CITATION] where [EQUATION]', '1609.03261-2-49-5': 'SCSG has a factor [MATH] which is strictly smaller than [MATH].', '1609.03261-2-49-6': 'It will be shown in Section [REF], [MATH] can be much smaller than [MATH] even in the case where [MATH].', '1609.03261-2-50-0': 'Next we consider the strongly convex case.', '1609.03261-2-50-1': 'We again start by deriving a bound for the sub-optimality of the output [MATH].', '1609.03261-2-51-0': 'Assume that [EQUATION].', '1609.03261-2-51-1': 'Under Assumption A1 and Assumption A2 with [MATH], the last iterate [MATH] satisfies [EQUATION] and [EQUATION]', '1609.03261-2-51-2': 'Unlike R-SVRG which guarantees the convergence of [MATH] and [MATH] simultaneously, SCSG needs to use different batch sizes for the two purposes since the second term in [REF] and that in [REF] are on different scales.', '1609.03261-2-51-3': 'The following two corollaries show the setups for these two purposes.', '1609.03261-2-52-0': 'Assume A1 and A2 hold.', '1609.03261-2-52-1': 'Set [EQUATION].', '1609.03261-2-52-2': 'Assume that [MATH], then with the output [MATH], [EQUATION]', '1609.03261-2-52-3': 'Under the same settings of Corollary [REF] except letting [EQUATION] we have that [EQUATION]', '1609.03261-2-52-4': 'For large [MATH], ignoring the log factors, the complexity results [REF] and [REF] can be simplified as [MATH] and [MATH].', '1609.03261-2-52-5': 'By contrast, the complexity results of SGD are [MATH] and [MATH], respectively [CITATION].', '1609.03261-2-52-6': 'Thus, SCSG is not worse than SGD up to a log factor and can significantly outperform SGD when [MATH] in terms of the theoretical complexity.', '1609.03261-2-52-7': 'For small [MATH], SCSG is equivalent to SVRG provided [MATH], which is usually the case in practice.', '1609.03261-2-53-0': '# Additional Discussion of [MATH]', '1609.03261-2-54-0': 'The problem [REF] we consider in this paper is a finite-sum optimization.', '1609.03261-2-54-1': 'It is popular to view it under the framework of stochastic approximation (SA) [CITATION] by rewriting [MATH] as [MATH] where [MATH] is a uniform index on [MATH] and viewing the first-order oracle as drawing [MATH] in every step.', '1609.03261-2-54-2': 'Then it is necessary to assume that [MATH], the variance of the oracle output, is uniformly bounded over the domain.', '1609.03261-2-54-3': 'However, one should expect that the finite-sum optimization is strictly easier than the general SA due to the special structure.', '1609.03261-2-54-4': 'This paper confirms this intuition by introducing a new measure [MATH] to characterize the difficulty of a generic finite-sum optimization problem.', '1609.03261-2-54-5': 'The SCSG algorithm can be viewed as adapted to this measure.', '1609.03261-2-55-0': 'Before delving into the details of [MATH], we briefly review the existing difficulty measures for problem [REF].', '1609.03261-2-55-1': 'We classify existing measures into four categories: initialization, curvature, gradient regularity and heterogeneity; see Table [REF] for the corresponding measures.', '1609.03261-2-55-2': 'The first three categories of measures are used in almost all types of problems while the heterogeneity measures are specific to the form [REF].', '1609.03261-2-55-3': 'To illustrate the importance of heterogeneity, consider a toy example where [MATH] with [MATH].', '1609.03261-2-55-4': "Now consider two classes of problem where the first class assumes the prior knowledge that all [MATH]'s are equal and the second class assumes that the [MATH] are free parameters.", '1609.03261-2-55-5': 'A simple calculation shows that [MATH] are all equal for both classes of problems.', '1609.03261-2-55-6': 'However, it is clear that the second class of problems are much easier using stochastic gradient methods since each single function has exactly the same behavior as the global function.', '1609.03261-2-55-7': 'In fact, [MATH] and [MATH] are zero for the second class of problems while they are non-zero for the first class.', '1609.03261-2-55-8': 'This suggests that heterogeneity between single functions and the global function increases the difficulty of problem [REF].', '1609.03261-2-56-0': 'The first attempt to describe the heterogeneity is through the strong growth condition [CITATION], which requires [EQUATION]', '1609.03261-2-56-1': 'Under [REF], [CITATION] proves that the stochastic gradient methods have the same convergence rate as the full gradient methods.', '1609.03261-2-56-2': 'However, [REF] is unrealistic since it implies for any minimizer [MATH] of [MATH], [MATH] is the stationary point of all individual loss functions.', '1609.03261-2-57-0': 'Later [CITATION] proposed a more realistic measure [EQUATION] and proved that (mini-batch) SGD is adaptive to [MATH].', '1609.03261-2-57-1': 'The condition [MATH] is always weaker than assuming [MATH] are uniformly bounded in that [MATH].', '1609.03261-2-57-2': 'However, in many applications where the domain of [MATH] is non-compact, [MATH].', '1609.03261-2-57-3': 'This can be observed even in our toy example when the domain of [MATH] is [MATH].', '1609.03261-2-57-4': 'One might argue that a projection step may be involved to ensure the boundedness of the iterates.', '1609.03261-2-57-5': 'However this argument is quite weak in that 1) the right size of the set that is projected onto is unknown; 2) the projection step is rarely implemented in practice.', '1609.03261-2-57-6': 'Therefore, [MATH] is still not a desirable measure.', '1609.03261-2-58-0': 'By contrast, our proposed measure [MATH] is well-behaved in most applications without awkward assumptions such as a bounded domain.', '1609.03261-2-58-1': 'Recall that [EQUATION]', '1609.03261-2-58-2': 'This can be viewed as a version of [MATH] which replaces the supremum by the value at a single point, when the optimum of [MATH] is unique.', '1609.03261-2-58-3': 'As a consequence, [MATH].', '1609.03261-2-58-4': 'In addition, when the strong growth condition [REF] holds, [MATH] for all [MATH] and hence [MATH].', '1609.03261-2-58-5': 'These simple facts show that [MATH] is strictly better than [MATH] and [MATH] as a measure of difficulty.', '1609.03261-2-58-6': 'We will show in the next two subsections that [MATH] can be controlled and estimated in almost all problems and is well-behaved in a wide range of applications.', '1609.03261-2-59-0': '## Bounding [MATH] in general cases', '1609.03261-2-60-0': 'It is often assumed that [MATH] is uniformly bounded over the domain.', '1609.03261-2-60-1': 'This is an unrealistic assumption in general, but it implies the boundedness of [MATH] directly and hence provides an example where the problem [REF] is "easy."', '1609.03261-2-61-0': 'With [MATH] and [MATH] defined as in Table [REF], we have [EQUATION].', '1609.03261-2-62-0': 'Surprisingly, [MATH] can be bounded even without any assumption other than A1 by using an arbitrary reference point.', '1609.03261-2-63-0': 'Under Assumption A1, for any [MATH] [EQUATION]', '1609.03261-2-63-1': 'A natural choice is to set the reference point [MATH].', '1609.03261-2-63-2': 'In the streaming setting where the [MATH] are i.i.d. functions with [MATH] and [MATH] for [MATH], the strong law of large number implies that [EQUATION]', '1609.03261-2-63-3': 'This entails that problem [REF] with i.i.d. individual functions is "easy."', '1609.03261-2-63-4': 'This is heuristically reasonable since the i.i.d. assumption, together with the moment conditions, forces the data to be highly homogeneous.', '1609.03261-2-64-0': 'In fact, [REF] can be proved under much broader settings.', '1609.03261-2-64-1': 'For example, when solving a linear equation [MATH], [MATH] can be set as [MATH] and no randomness is involved.', '1609.03261-2-64-2': 'If we set [MATH], then [REF] implies that [EQUATION].', '1609.03261-2-64-3': 'Then [MATH] provided [MATH].', '1609.03261-2-65-0': 'Another type of problems with [MATH] involves pairwise comparisons: [EQUATION] where [MATH] are independent samples.', '1609.03261-2-65-1': 'For example, in preference elicitation or sporting competitions where the data is collected as pairwise comparisons, one can fit a Bradley-Terry model to obtain the underlying "score" that represents the quality of each unit.', '1609.03261-2-65-2': 'The objective function of the Bradley-Terry model is [MATH] where [MATH] is the number of times that unit [MATH] beats unit [MATH] ([CITATION], [CITATION]).', '1609.03261-2-65-3': 'Other examples that involve a similar structure are metric learning ([CITATION], [CITATION]) and convex relaxation of graph cuts ([CITATION]).', '1609.03261-2-65-4': 'In these cases, we can also bound [MATH] under mild conditions.', '1609.03261-2-66-0': 'Let [MATH].', '1609.03261-2-66-1': 'Then [EQUATION].', '1609.03261-2-67-0': 'Finally, it is worth mentioning that [REF] cannot be established for [MATH] unless the domain is compact and regularity conditions beyond the existence of a second moment are imposed to ensure that some version of the uniform law of large numbers can be applied.', '1609.03261-2-68-0': '## Estimating [MATH] in generalized linear models', '1609.03261-2-69-0': 'Optimzation problems in machine learning often take the form of a generalized linear model in which [MATH], with [MATH] being the covariates and [MATH] being the responses, for some convex loss function [MATH].', '1609.03261-2-69-1': 'Let [MATH].', '1609.03261-2-69-2': 'Then by definition [EQUATION].', '1609.03261-2-69-3': 'If [MATH] is uniformly bounded with [MATH], then [EQUATION].', '1609.03261-2-69-4': 'We will show in appendix [REF] that [MATH] (Proposition [REF]) for multi-class logistic regression, regardless of the number of classes.', '1609.03261-2-69-5': 'The same bound can also be derived for Huber regression [CITATION] and the probit model [CITATION].', '1609.03261-2-69-6': 'When the domain is unbounded, the (penalized) least-squares problem has an unbounded [MATH].', '1609.03261-2-69-7': 'However given that [MATH], where [MATH], one can easily show that [EQUATION]', '1609.03261-2-70-0': '## [MATH] in pathological cases', '1609.03261-2-71-0': 'The last two subsections exhibit various examples in which [MATH] is well controlled.', '1609.03261-2-71-1': 'There also exist pathological cases in which [MATH] is large.', '1609.03261-2-71-2': 'For instance, let [MATH] be an even number and let [MATH], where [MATH] with [MATH]) and all other elements are equal to zero.', '1609.03261-2-71-3': 'In this case, [MATH] and by symmetry [MATH].', '1609.03261-2-71-4': 'Another example is a quadratic function with [MATH], in which case [MATH] and hence [MATH].', '1609.03261-2-72-0': 'The first example is due to the high dimension.', '1609.03261-2-72-1': 'When the dimension is comparable to [MATH], even the i.i.d. assumption cannot guarantee good behavior of [MATH], without further conditions, since the law of large numbers fails.', '1609.03261-2-72-2': 'The second example is due to the severe heterogeneity of the components.', '1609.03261-2-72-3': 'In fact the [MATH]-th component reaches its minimum at [MATH] while the global function reaches its minimum at [MATH] and thus most components behave completely differently from the global function.', '1609.03261-2-73-0': 'Nevertheless, it is worth emphasizing that SGD also faces with the same issue in these two cases.', '1609.03261-2-73-1': 'More importantly, SCSG does not suffer from these undesirable properties since it will choose [MATH] automatically; wee Corollary [REF] to Corollary [REF].', '1609.03261-2-74-0': '# Experiments', '1609.03261-2-75-0': 'In this section, we illustrate the performance of SCSG by implementing it for multi-class logistic regression on the MNIST dataset.', '1609.03261-2-75-1': 'We normalize the data into the range [MATH] by dividing each entry by [MATH].', '1609.03261-2-75-2': 'No regularization term is added and so the function to be minimized is [EQUATION] where [MATH], [MATH], [MATH] including [MATH] pixels plus an intercept term [MATH] and [MATH].', '1609.03261-2-75-3': 'Direct computation shows that [MATH] while [MATH].', '1609.03261-2-76-0': 'We measured performance by [MATH] versus the number of passes of data.', '1609.03261-2-76-1': 'For each algorithm, we selected the best-tuned stepsize and then implemented the algorithm ten times and computed average performance.', '1609.03261-2-77-0': 'We compared SCSG with mini-batch SGD, with the batch size [MATH], and SVRG.', '1609.03261-2-77-1': 'Moreover, we considered three variants of SCSG: [(1)] SCSGFixN: Set [MATH], instead of generated from a geometric distribution; SCSGNew: Randomly pick [MATH], instead of from the whole dataset [MATH]; SCSGNewFixN: Set [MATH] and randomly pick [MATH].', '1609.03261-2-78-0': 'The first variant allows us to check whether the use of geometric random variable is essential in practice; the second variant allows checking whether sampling SGD updates from the whole dataset is necessary; and the third variant checks the combination.', '1609.03261-2-79-0': 'For all the variants of SCSG and SGD, we considered three batch sizes [MATH].', '1609.03261-2-79-1': 'The results are plotted in Figure [REF], from which we make the following observations: [1)] SCSG is able to reach an accurate solution very fast; indeed, all versions of SCSG are more efficient than SGD and SVRG in the first five passes.', '1609.03261-2-79-2': 'This confirms our theoretical results.', '1609.03261-2-79-3': 'SCSG with fixed [MATH] is slightly more effective than the original SCSG.', '1609.03261-2-79-4': 'Thus the geometric random variable may not be essential in practice.', '1609.03261-2-79-5': 'It makes no difference whether we sample from the whole dataset or sample from the mini-batch when running the SGD steps in SCSG.', '1609.03261-2-80-0': 'Based on these observations, we recommend implementing SCSGNewFixN as a default since 1) the fixed number of SGD steps stabilizes the procedure; 2) sampling from the mini-batch greatly reduces the communication costs incurred by accessing data from the whole dataset.', '1609.03261-2-81-0': '# Discussion', '1609.03261-2-82-0': 'We have presented SCSG, a member of the SVRG family of algorithms, proving its superior performance in terms of both computation and communication cost relative to other variance-reduction schemes.', '1609.03261-2-82-1': 'Both complexities are independent of sample size when the required accuracy is low, for various functions which are widely optimized in practice.', '1609.03261-2-82-2': 'Experiments on real data supported our theoretical results.', '1609.03261-2-83-0': 'We plan to explore several variants of SCSG in future work.', '1609.03261-2-83-1': 'For example, a non-uniform sampling scheme can be applied to SGD steps to leverage the Lipschitz constants [MATH] as in SVRG.', '1609.03261-2-83-2': 'More interestingly, we can consider a better sampling scheme for [MATH] by putting more weight on influential observations.', '1609.03261-2-83-3': 'The proximal settings are also straightforward extensions of our current work.', '1609.03261-2-84-0': 'As a final comment, we note that previous complexity analyses tend to focus on high-accuracy computation for which the dependence on the sample size [MATH] and condition number [MATH] is of major concern.', '1609.03261-2-84-1': 'The low-accuracy regime appears to be under-studied theoretically even though it is commonly encountered in practice.', '1609.03261-2-84-2': 'We advocate taking all three parameters, namely [MATH], [MATH] and [MATH], into consideration and distinguishing the analyses for high-accuracy computation and low-accuracy computation.'}

[['1609.03261-1-20-0', '1609.03261-2-20-0'], ['1609.03261-1-20-1', '1609.03261-2-20-1'], ['1609.03261-1-16-0', '1609.03261-2-16-0'], ['1609.03261-1-8-0', '1609.03261-2-7-0'], ['1609.03261-1-8-1', '1609.03261-2-7-1'], ['1609.03261-1-8-2', '1609.03261-2-7-2'], ['1609.03261-1-8-6', '1609.03261-2-7-6'], ['1609.03261-1-4-0', '1609.03261-2-4-0'], ['1609.03261-1-4-1', '1609.03261-2-4-1'], ['1609.03261-1-4-3', '1609.03261-2-4-3'], ['1609.03261-1-0-0', '1609.03261-2-0-0'], ['1609.03261-1-0-2', '1609.03261-2-0-2'], ['1609.03261-1-68-1', '1609.03261-2-75-2'], ['1609.03261-1-38-0', '1609.03261-2-34-0'], ['1609.03261-1-21-0', '1609.03261-2-21-0'], ['1609.03261-1-21-3', '1609.03261-2-21-1'], ['1609.03261-1-21-5', '1609.03261-2-21-3'], ['1609.03261-1-5-0', '1609.03261-2-5-0'], ['1609.03261-1-5-1', '1609.03261-2-5-2'], ['1609.03261-1-5-2', '1609.03261-2-5-1'], ['1609.03261-1-5-4', '1609.03261-2-5-3'], ['1609.03261-1-2-2', '1609.03261-2-2-3'], ['1609.03261-1-2-3', '1609.03261-2-2-4'], ['1609.03261-1-3-0', '1609.03261-2-3-0'], ['1609.03261-1-3-1', '1609.03261-2-3-1'], ['1609.03261-1-3-2', '1609.03261-2-3-2'], ['1609.03261-1-3-3', '1609.03261-2-3-3'], ['1609.03261-1-3-5', '1609.03261-2-3-5'], ['1609.03261-1-3-6', '1609.03261-2-3-6'], ['1609.03261-1-39-0', '1609.03261-2-35-0'], ['1609.03261-1-74-0', '1609.03261-2-83-0'], ['1609.03261-1-74-1', '1609.03261-2-83-1'], ['1609.03261-1-74-2', '1609.03261-2-83-2'], ['1609.03261-1-23-0', '1609.03261-2-23-0'], ['1609.03261-1-23-1', '1609.03261-2-23-1'], ['1609.03261-1-23-2', '1609.03261-2-23-2'], ['1609.03261-1-23-6', '1609.03261-2-23-6'], ['1609.03261-2-4-0', '1609.03261-3-4-0'], ['1609.03261-2-4-1', '1609.03261-3-4-1'], ['1609.03261-2-4-3', '1609.03261-3-4-3'], ['1609.03261-2-20-0', '1609.03261-3-20-0'], ['1609.03261-2-20-1', '1609.03261-3-20-1'], ['1609.03261-2-21-0', '1609.03261-3-21-0'], ['1609.03261-2-21-1', '1609.03261-3-21-1'], ['1609.03261-2-21-2', '1609.03261-3-21-2'], ['1609.03261-2-21-3', '1609.03261-3-21-3'], ['1609.03261-2-3-0', '1609.03261-3-3-0'], ['1609.03261-2-3-1', '1609.03261-3-3-1'], ['1609.03261-2-3-2', '1609.03261-3-3-2'], ['1609.03261-2-3-3', '1609.03261-3-3-3'], ['1609.03261-2-3-4', '1609.03261-3-3-4'], ['1609.03261-2-3-5', '1609.03261-3-3-5'], ['1609.03261-2-3-6', '1609.03261-3-3-6'], ['1609.03261-2-37-0', '1609.03261-3-37-0'], ['1609.03261-2-37-1', '1609.03261-3-37-1'], ['1609.03261-2-37-2', '1609.03261-3-37-2'], ['1609.03261-2-2-3', '1609.03261-3-2-3'], ['1609.03261-2-2-4', '1609.03261-3-2-4'], ['1609.03261-2-22-0', '1609.03261-3-22-0'], ['1609.03261-2-22-1', '1609.03261-3-22-1'], ['1609.03261-2-22-2', '1609.03261-3-22-2'], ['1609.03261-2-22-3', '1609.03261-3-22-3'], ['1609.03261-2-22-4', '1609.03261-3-22-4'], ['1609.03261-2-22-5', '1609.03261-3-22-5'], ['1609.03261-2-22-6', '1609.03261-3-22-6'], ['1609.03261-2-5-0', '1609.03261-3-5-0'], ['1609.03261-2-5-1', '1609.03261-3-5-1'], ['1609.03261-2-5-2', '1609.03261-3-5-2'], ['1609.03261-2-5-3', '1609.03261-3-5-3'], ['1609.03261-2-5-4', '1609.03261-3-5-4'], ['1609.03261-2-11-0', '1609.03261-3-11-0'], ['1609.03261-2-11-1', '1609.03261-3-11-1'], ['1609.03261-2-6-2', '1609.03261-3-6-2'], ['1609.03261-2-6-4', '1609.03261-3-6-4'], ['1609.03261-2-23-0', '1609.03261-3-23-0'], ['1609.03261-2-23-1', '1609.03261-3-23-1'], ['1609.03261-2-23-2', '1609.03261-3-23-2'], ['1609.03261-2-23-3', '1609.03261-3-23-3'], ['1609.03261-2-23-4', '1609.03261-3-23-4'], ['1609.03261-2-23-5', '1609.03261-3-23-5'], ['1609.03261-2-23-6', '1609.03261-3-23-6'], ['1609.03261-2-16-0', '1609.03261-3-16-0'], ['1609.03261-2-30-0', '1609.03261-3-30-0'], ['1609.03261-2-30-1', '1609.03261-3-30-1'], ['1609.03261-2-30-2', '1609.03261-3-30-2'], ['1609.03261-2-30-4', '1609.03261-3-30-4'], ['1609.03261-2-30-5', '1609.03261-3-30-5'], ['1609.03261-2-30-6', '1609.03261-3-30-6'], ['1609.03261-2-30-8', '1609.03261-3-30-8'], ['1609.03261-2-30-9', '1609.03261-3-30-9'], ['1609.03261-2-35-0', '1609.03261-3-35-0'], ['1609.03261-2-0-0', '1609.03261-3-0-0'], ['1609.03261-2-0-2', '1609.03261-3-0-2'], ['1609.03261-2-10-1', '1609.03261-3-10-1'], ['1609.03261-2-42-3', '1609.03261-3-42-3'], ['1609.03261-2-42-5', '1609.03261-3-42-5'], ['1609.03261-2-42-7', '1609.03261-3-42-7'], ['1609.03261-2-13-1', '1609.03261-3-13-1'], ['1609.03261-2-13-2', '1609.03261-3-13-2'], ['1609.03261-2-13-4', '1609.03261-3-13-4'], ['1609.03261-2-41-0', '1609.03261-3-41-0'], ['1609.03261-2-7-0', '1609.03261-3-7-0'], ['1609.03261-2-7-1', '1609.03261-3-7-1'], ['1609.03261-2-7-2', '1609.03261-3-7-2'], ['1609.03261-2-7-5', '1609.03261-3-7-5'], ['1609.03261-2-7-6', '1609.03261-3-7-6'], ['1609.03261-2-9-0', '1609.03261-3-9-0'], ['1609.03261-2-9-2', '1609.03261-3-9-2'], ['1609.03261-2-9-7', '1609.03261-3-9-7'], ['1609.03261-2-31-0', '1609.03261-3-31-0'], ['1609.03261-2-31-1', '1609.03261-3-31-1'], ['1609.03261-2-31-2', '1609.03261-3-31-2'], ['1609.03261-2-31-3', '1609.03261-3-31-3'], ['1609.03261-2-31-4', '1609.03261-3-31-4'], ['1609.03261-2-34-0', '1609.03261-3-34-0'], ['1609.03261-2-34-1', '1609.03261-3-34-1'], ['1609.03261-2-12-0', '1609.03261-3-12-0'], ['1609.03261-1-10-2', '1609.03261-2-13-6'], ['1609.03261-1-40-1', '1609.03261-2-37-0'], ['1609.03261-2-36-0', '1609.03261-3-36-0'], ['1609.03261-2-36-1', '1609.03261-3-36-1'], ['1609.03261-2-36-2', '1609.03261-3-36-2'], ['1609.03261-1-22-10', '1609.03261-2-30-4'], ['1609.03261-1-22-12', '1609.03261-2-30-6'], ['1609.03261-1-22-14', '1609.03261-2-30-8'], ['1609.03261-1-22-15', '1609.03261-2-30-9'], ['1609.03261-1-22-0', '1609.03261-2-22-0'], ['1609.03261-1-22-1', '1609.03261-2-22-1'], ['1609.03261-1-22-2', '1609.03261-2-22-2'], ['1609.03261-1-22-3', '1609.03261-2-22-3'], ['1609.03261-1-22-4', '1609.03261-2-22-4'], ['1609.03261-1-22-6', '1609.03261-2-22-6'], ['1609.03261-1-33-0', '1609.03261-2-30-0'], ['1609.03261-1-72-1', '1609.03261-2-82-1'], ['1609.03261-1-68-2', '1609.03261-2-75-3'], ['1609.03261-1-21-4', '1609.03261-2-21-2'], ['1609.03261-1-5-5', '1609.03261-2-5-4'], ['1609.03261-1-2-0', '1609.03261-2-2-1'], ['1609.03261-1-3-4', '1609.03261-2-3-4'], ['1609.03261-1-74-3', '1609.03261-2-83-3'], ['1609.03261-1-75-0', '1609.03261-2-84-0'], ['1609.03261-1-75-1', '1609.03261-2-84-1'], ['1609.03261-1-75-2', '1609.03261-2-84-2'], ['1609.03261-1-25-0', '1609.03261-2-29-0'], ['1609.03261-1-23-3', '1609.03261-2-23-3'], ['1609.03261-2-4-2', '1609.03261-3-4-2'], ['1609.03261-2-14-0', '1609.03261-3-14-0'], ['1609.03261-2-14-2', '1609.03261-3-14-2'], ['1609.03261-2-14-3', '1609.03261-3-14-3'], ['1609.03261-2-21-4', '1609.03261-3-21-4'], ['1609.03261-2-40-0', '1609.03261-3-40-0'], ['1609.03261-2-2-1', '1609.03261-3-2-1'], ['1609.03261-2-15-0', '1609.03261-3-15-0'], ['1609.03261-2-6-0', '1609.03261-3-6-0'], ['1609.03261-2-6-1', '1609.03261-3-6-1'], ['1609.03261-2-6-3', '1609.03261-3-6-3'], ['1609.03261-2-30-3', '1609.03261-3-30-3'], ['1609.03261-2-30-10', '1609.03261-3-30-10'], ['1609.03261-2-0-1', '1609.03261-3-0-1'], ['1609.03261-2-0-3', '1609.03261-3-0-3'], ['1609.03261-2-10-0', '1609.03261-3-10-0'], ['1609.03261-2-42-1', '1609.03261-3-42-1'], ['1609.03261-2-42-2', '1609.03261-3-42-2'], ['1609.03261-2-42-4', '1609.03261-3-42-4'], ['1609.03261-2-42-6', '1609.03261-3-42-6'], ['1609.03261-2-13-3', '1609.03261-3-13-3'], ['1609.03261-2-13-5', '1609.03261-3-13-5'], ['1609.03261-2-13-6', '1609.03261-3-13-6'], ['1609.03261-2-7-3', '1609.03261-3-7-3'], ['1609.03261-2-7-4', '1609.03261-3-7-4'], ['1609.03261-2-43-0', '1609.03261-3-43-0'], ['1609.03261-2-9-1', '1609.03261-3-9-1'], ['1609.03261-2-9-3', '1609.03261-3-9-3'], ['1609.03261-2-9-4', '1609.03261-3-9-4'], ['1609.03261-2-9-5', '1609.03261-3-9-5'], ['1609.03261-2-9-6', '1609.03261-3-9-6'], ['1609.03261-1-40-0', '1609.03261-2-36-0'], ['1609.03261-1-40-3', '1609.03261-2-36-1'], ['1609.03261-1-33-2', '1609.03261-2-31-0'], ['1609.03261-1-22-8', '1609.03261-2-30-3'], ['1609.03261-1-22-11', '1609.03261-2-30-5'], ['1609.03261-1-22-5', '1609.03261-2-22-5'], ['1609.03261-1-72-0', '1609.03261-2-82-0'], ['1609.03261-1-72-2', '1609.03261-2-82-2'], ['1609.03261-1-0-1', '1609.03261-2-0-1'], ['1609.03261-1-68-0', '1609.03261-2-75-0'], ['1609.03261-1-68-0', '1609.03261-2-75-1'], ['1609.03261-1-2-1', '1609.03261-2-2-2'], ['1609.03261-1-23-5', '1609.03261-2-23-5'], ['1609.03261-2-14-1', '1609.03261-3-14-1'], ['1609.03261-2-2-0', '1609.03261-3-2-0'], ['1609.03261-2-2-2', '1609.03261-3-2-2'], ['1609.03261-2-15-1', '1609.03261-3-15-1'], ['1609.03261-2-15-2', '1609.03261-3-15-2'], ['1609.03261-2-6-5', '1609.03261-3-6-5'], ['1609.03261-2-6-6', '1609.03261-3-6-6'], ['1609.03261-2-30-7', '1609.03261-3-30-7'], ['1609.03261-1-10-1', '1609.03261-2-13-4'], ['1609.03261-1-10-3', '1609.03261-2-11-0'], ['1609.03261-1-10-3', '1609.03261-2-11-1'], ['1609.03261-1-22-9', '1609.03261-2-12-0'], ['1609.03261-1-33-0', '1609.03261-2-31-2'], ['1609.03261-1-22-7', '1609.03261-2-30-2'], ['1609.03261-1-22-13', '1609.03261-2-30-7'], ['1609.03261-1-33-1', '1609.03261-2-30-1'], ['1609.03261-1-78-3', '1609.03261-2-51-1'], ['1609.03261-1-41-0', '1609.03261-2-63-0'], ['1609.03261-2-29-0', '1609.03261-3-29-0'], ['1609.03261-2-29-1', '1609.03261-3-29-1'], ['1609.03261-2-29-2', '1609.03261-3-29-2'], ['1609.03261-2-29-4', '1609.03261-3-29-3'], ['1609.03261-2-29-5', '1609.03261-3-29-4'], ['1609.03261-2-32-0', '1609.03261-3-32-0'], ['1609.03261-2-32-1', '1609.03261-3-32-1'], ['1609.03261-2-32-2', '1609.03261-3-32-2'], ['1609.03261-2-39-0', '1609.03261-3-39-0'], ['1609.03261-2-50-1', '1609.03261-3-39-0'], ['1609.03261-2-46-0', '1609.03261-3-46-0'], ['1609.03261-2-48-0', '1609.03261-3-48-0'], ['1609.03261-2-48-1', '1609.03261-3-48-1'], ['1609.03261-2-48-2', '1609.03261-3-48-2']]

[['1609.03261-1-20-0', '1609.03261-2-20-0'], ['1609.03261-1-20-1', '1609.03261-2-20-1'], ['1609.03261-1-16-0', '1609.03261-2-16-0'], ['1609.03261-1-8-0', '1609.03261-2-7-0'], ['1609.03261-1-8-1', '1609.03261-2-7-1'], ['1609.03261-1-8-2', '1609.03261-2-7-2'], ['1609.03261-1-8-6', '1609.03261-2-7-6'], ['1609.03261-1-4-0', '1609.03261-2-4-0'], ['1609.03261-1-4-1', '1609.03261-2-4-1'], ['1609.03261-1-4-3', '1609.03261-2-4-3'], ['1609.03261-1-0-0', '1609.03261-2-0-0'], ['1609.03261-1-0-2', '1609.03261-2-0-2'], ['1609.03261-1-68-1', '1609.03261-2-75-2'], ['1609.03261-1-38-0', '1609.03261-2-34-0'], ['1609.03261-1-21-0', '1609.03261-2-21-0'], ['1609.03261-1-21-3', '1609.03261-2-21-1'], ['1609.03261-1-21-5', '1609.03261-2-21-3'], ['1609.03261-1-5-0', '1609.03261-2-5-0'], ['1609.03261-1-5-1', '1609.03261-2-5-2'], ['1609.03261-1-5-2', '1609.03261-2-5-1'], ['1609.03261-1-5-4', '1609.03261-2-5-3'], ['1609.03261-1-2-2', '1609.03261-2-2-3'], ['1609.03261-1-2-3', '1609.03261-2-2-4'], ['1609.03261-1-3-0', '1609.03261-2-3-0'], ['1609.03261-1-3-1', '1609.03261-2-3-1'], ['1609.03261-1-3-2', '1609.03261-2-3-2'], ['1609.03261-1-3-3', '1609.03261-2-3-3'], ['1609.03261-1-3-5', '1609.03261-2-3-5'], ['1609.03261-1-3-6', '1609.03261-2-3-6'], ['1609.03261-1-39-0', '1609.03261-2-35-0'], ['1609.03261-1-74-0', '1609.03261-2-83-0'], ['1609.03261-1-74-1', '1609.03261-2-83-1'], ['1609.03261-1-74-2', '1609.03261-2-83-2'], ['1609.03261-1-23-0', '1609.03261-2-23-0'], ['1609.03261-1-23-1', '1609.03261-2-23-1'], ['1609.03261-1-23-2', '1609.03261-2-23-2'], ['1609.03261-1-23-6', '1609.03261-2-23-6'], ['1609.03261-2-4-0', '1609.03261-3-4-0'], ['1609.03261-2-4-1', '1609.03261-3-4-1'], ['1609.03261-2-4-3', '1609.03261-3-4-3'], ['1609.03261-2-20-0', '1609.03261-3-20-0'], ['1609.03261-2-20-1', '1609.03261-3-20-1'], ['1609.03261-2-21-0', '1609.03261-3-21-0'], ['1609.03261-2-21-1', '1609.03261-3-21-1'], ['1609.03261-2-21-2', '1609.03261-3-21-2'], ['1609.03261-2-21-3', '1609.03261-3-21-3'], ['1609.03261-2-3-0', '1609.03261-3-3-0'], ['1609.03261-2-3-1', '1609.03261-3-3-1'], ['1609.03261-2-3-2', '1609.03261-3-3-2'], ['1609.03261-2-3-3', '1609.03261-3-3-3'], ['1609.03261-2-3-4', '1609.03261-3-3-4'], ['1609.03261-2-3-5', '1609.03261-3-3-5'], ['1609.03261-2-3-6', '1609.03261-3-3-6'], ['1609.03261-2-37-0', '1609.03261-3-37-0'], ['1609.03261-2-37-1', '1609.03261-3-37-1'], ['1609.03261-2-37-2', '1609.03261-3-37-2'], ['1609.03261-2-2-3', '1609.03261-3-2-3'], ['1609.03261-2-2-4', '1609.03261-3-2-4'], ['1609.03261-2-22-0', '1609.03261-3-22-0'], ['1609.03261-2-22-1', '1609.03261-3-22-1'], ['1609.03261-2-22-2', '1609.03261-3-22-2'], ['1609.03261-2-22-3', '1609.03261-3-22-3'], ['1609.03261-2-22-4', '1609.03261-3-22-4'], ['1609.03261-2-22-5', '1609.03261-3-22-5'], ['1609.03261-2-22-6', '1609.03261-3-22-6'], ['1609.03261-2-5-0', '1609.03261-3-5-0'], ['1609.03261-2-5-1', '1609.03261-3-5-1'], ['1609.03261-2-5-2', '1609.03261-3-5-2'], ['1609.03261-2-5-3', '1609.03261-3-5-3'], ['1609.03261-2-5-4', '1609.03261-3-5-4'], ['1609.03261-2-11-0', '1609.03261-3-11-0'], ['1609.03261-2-11-1', '1609.03261-3-11-1'], ['1609.03261-2-6-2', '1609.03261-3-6-2'], ['1609.03261-2-6-4', '1609.03261-3-6-4'], ['1609.03261-2-23-0', '1609.03261-3-23-0'], ['1609.03261-2-23-1', '1609.03261-3-23-1'], ['1609.03261-2-23-2', '1609.03261-3-23-2'], ['1609.03261-2-23-3', '1609.03261-3-23-3'], ['1609.03261-2-23-4', '1609.03261-3-23-4'], ['1609.03261-2-23-5', '1609.03261-3-23-5'], ['1609.03261-2-23-6', '1609.03261-3-23-6'], ['1609.03261-2-16-0', '1609.03261-3-16-0'], ['1609.03261-2-30-0', '1609.03261-3-30-0'], ['1609.03261-2-30-1', '1609.03261-3-30-1'], ['1609.03261-2-30-2', '1609.03261-3-30-2'], ['1609.03261-2-30-4', '1609.03261-3-30-4'], ['1609.03261-2-30-5', '1609.03261-3-30-5'], ['1609.03261-2-30-6', '1609.03261-3-30-6'], ['1609.03261-2-30-8', '1609.03261-3-30-8'], ['1609.03261-2-30-9', '1609.03261-3-30-9'], ['1609.03261-2-35-0', '1609.03261-3-35-0'], ['1609.03261-2-0-0', '1609.03261-3-0-0'], ['1609.03261-2-0-2', '1609.03261-3-0-2'], ['1609.03261-2-10-1', '1609.03261-3-10-1'], ['1609.03261-2-42-3', '1609.03261-3-42-3'], ['1609.03261-2-42-5', '1609.03261-3-42-5'], ['1609.03261-2-42-7', '1609.03261-3-42-7'], ['1609.03261-2-13-1', '1609.03261-3-13-1'], ['1609.03261-2-13-2', '1609.03261-3-13-2'], ['1609.03261-2-13-4', '1609.03261-3-13-4'], ['1609.03261-2-41-0', '1609.03261-3-41-0'], ['1609.03261-2-7-0', '1609.03261-3-7-0'], ['1609.03261-2-7-1', '1609.03261-3-7-1'], ['1609.03261-2-7-2', '1609.03261-3-7-2'], ['1609.03261-2-7-5', '1609.03261-3-7-5'], ['1609.03261-2-7-6', '1609.03261-3-7-6'], ['1609.03261-2-9-0', '1609.03261-3-9-0'], ['1609.03261-2-9-2', '1609.03261-3-9-2'], ['1609.03261-2-9-7', '1609.03261-3-9-7'], ['1609.03261-2-31-0', '1609.03261-3-31-0'], ['1609.03261-2-31-1', '1609.03261-3-31-1'], ['1609.03261-2-31-2', '1609.03261-3-31-2'], ['1609.03261-2-31-3', '1609.03261-3-31-3'], ['1609.03261-2-31-4', '1609.03261-3-31-4'], ['1609.03261-2-34-0', '1609.03261-3-34-0'], ['1609.03261-2-34-1', '1609.03261-3-34-1'], ['1609.03261-2-12-0', '1609.03261-3-12-0'], ['1609.03261-1-10-2', '1609.03261-2-13-6'], ['1609.03261-1-40-1', '1609.03261-2-37-0'], ['1609.03261-2-36-0', '1609.03261-3-36-0'], ['1609.03261-2-36-1', '1609.03261-3-36-1'], ['1609.03261-2-36-2', '1609.03261-3-36-2'], ['1609.03261-1-22-10', '1609.03261-2-30-4'], ['1609.03261-1-22-12', '1609.03261-2-30-6'], ['1609.03261-1-22-14', '1609.03261-2-30-8'], ['1609.03261-1-22-15', '1609.03261-2-30-9'], ['1609.03261-1-22-0', '1609.03261-2-22-0'], ['1609.03261-1-22-1', '1609.03261-2-22-1'], ['1609.03261-1-22-2', '1609.03261-2-22-2'], ['1609.03261-1-22-3', '1609.03261-2-22-3'], ['1609.03261-1-22-4', '1609.03261-2-22-4'], ['1609.03261-1-22-6', '1609.03261-2-22-6'], ['1609.03261-1-33-0', '1609.03261-2-30-0']]

[['1609.03261-1-72-1', '1609.03261-2-82-1'], ['1609.03261-1-68-2', '1609.03261-2-75-3'], ['1609.03261-1-21-4', '1609.03261-2-21-2'], ['1609.03261-1-5-5', '1609.03261-2-5-4'], ['1609.03261-1-2-0', '1609.03261-2-2-1'], ['1609.03261-1-3-4', '1609.03261-2-3-4'], ['1609.03261-1-74-3', '1609.03261-2-83-3'], ['1609.03261-1-75-0', '1609.03261-2-84-0'], ['1609.03261-1-75-1', '1609.03261-2-84-1'], ['1609.03261-1-75-2', '1609.03261-2-84-2'], ['1609.03261-1-25-0', '1609.03261-2-29-0'], ['1609.03261-1-23-3', '1609.03261-2-23-3'], ['1609.03261-2-4-2', '1609.03261-3-4-2'], ['1609.03261-2-14-0', '1609.03261-3-14-0'], ['1609.03261-2-14-2', '1609.03261-3-14-2'], ['1609.03261-2-14-3', '1609.03261-3-14-3'], ['1609.03261-2-21-4', '1609.03261-3-21-4'], ['1609.03261-2-40-0', '1609.03261-3-40-0'], ['1609.03261-2-2-1', '1609.03261-3-2-1'], ['1609.03261-2-15-0', '1609.03261-3-15-0'], ['1609.03261-2-6-0', '1609.03261-3-6-0'], ['1609.03261-2-6-1', '1609.03261-3-6-1'], ['1609.03261-2-6-3', '1609.03261-3-6-3'], ['1609.03261-2-30-3', '1609.03261-3-30-3'], ['1609.03261-2-30-10', '1609.03261-3-30-10'], ['1609.03261-2-0-1', '1609.03261-3-0-1'], ['1609.03261-2-0-3', '1609.03261-3-0-3'], ['1609.03261-2-10-0', '1609.03261-3-10-0'], ['1609.03261-2-42-1', '1609.03261-3-42-1'], ['1609.03261-2-42-2', '1609.03261-3-42-2'], ['1609.03261-2-42-4', '1609.03261-3-42-4'], ['1609.03261-2-42-6', '1609.03261-3-42-6'], ['1609.03261-2-13-3', '1609.03261-3-13-3'], ['1609.03261-2-13-5', '1609.03261-3-13-5'], ['1609.03261-2-13-6', '1609.03261-3-13-6'], ['1609.03261-2-7-3', '1609.03261-3-7-3'], ['1609.03261-2-7-4', '1609.03261-3-7-4'], ['1609.03261-2-43-0', '1609.03261-3-43-0'], ['1609.03261-2-9-1', '1609.03261-3-9-1'], ['1609.03261-2-9-3', '1609.03261-3-9-3'], ['1609.03261-2-9-4', '1609.03261-3-9-4'], ['1609.03261-2-9-5', '1609.03261-3-9-5'], ['1609.03261-2-9-6', '1609.03261-3-9-6'], ['1609.03261-1-40-0', '1609.03261-2-36-0'], ['1609.03261-1-40-3', '1609.03261-2-36-1'], ['1609.03261-1-33-2', '1609.03261-2-31-0'], ['1609.03261-1-22-8', '1609.03261-2-30-3'], ['1609.03261-1-22-11', '1609.03261-2-30-5'], ['1609.03261-1-22-5', '1609.03261-2-22-5']]

[]

[['1609.03261-1-72-0', '1609.03261-2-82-0'], ['1609.03261-1-72-2', '1609.03261-2-82-2'], ['1609.03261-1-0-1', '1609.03261-2-0-1'], ['1609.03261-1-68-0', '1609.03261-2-75-0'], ['1609.03261-1-68-0', '1609.03261-2-75-1'], ['1609.03261-1-2-1', '1609.03261-2-2-2'], ['1609.03261-1-23-5', '1609.03261-2-23-5'], ['1609.03261-2-14-1', '1609.03261-3-14-1'], ['1609.03261-2-2-0', '1609.03261-3-2-0'], ['1609.03261-2-2-2', '1609.03261-3-2-2'], ['1609.03261-2-15-1', '1609.03261-3-15-1'], ['1609.03261-2-15-2', '1609.03261-3-15-2'], ['1609.03261-2-6-5', '1609.03261-3-6-5'], ['1609.03261-2-6-6', '1609.03261-3-6-6'], ['1609.03261-2-30-7', '1609.03261-3-30-7'], ['1609.03261-1-10-1', '1609.03261-2-13-4'], ['1609.03261-1-10-3', '1609.03261-2-11-0'], ['1609.03261-1-10-3', '1609.03261-2-11-1'], ['1609.03261-1-22-9', '1609.03261-2-12-0'], ['1609.03261-1-33-0', '1609.03261-2-31-2'], ['1609.03261-1-22-7', '1609.03261-2-30-2'], ['1609.03261-1-22-13', '1609.03261-2-30-7'], ['1609.03261-1-33-1', '1609.03261-2-30-1']]

[['1609.03261-1-78-3', '1609.03261-2-51-1'], ['1609.03261-1-41-0', '1609.03261-2-63-0'], ['1609.03261-2-29-0', '1609.03261-3-29-0'], ['1609.03261-2-29-1', '1609.03261-3-29-1'], ['1609.03261-2-29-2', '1609.03261-3-29-2'], ['1609.03261-2-29-4', '1609.03261-3-29-3'], ['1609.03261-2-29-5', '1609.03261-3-29-4'], ['1609.03261-2-32-0', '1609.03261-3-32-0'], ['1609.03261-2-32-1', '1609.03261-3-32-1'], ['1609.03261-2-32-2', '1609.03261-3-32-2'], ['1609.03261-2-39-0', '1609.03261-3-39-0'], ['1609.03261-2-50-1', '1609.03261-3-39-0'], ['1609.03261-2-46-0', '1609.03261-3-46-0'], ['1609.03261-2-48-0', '1609.03261-3-48-0'], ['1609.03261-2-48-1', '1609.03261-3-48-1'], ['1609.03261-2-48-2', '1609.03261-3-48-2']]

['1609.03261-1-15-0', '1609.03261-1-17-0', '1609.03261-1-18-0', '1609.03261-1-25-1', '1609.03261-1-28-0', '1609.03261-1-29-0', '1609.03261-1-30-0', '1609.03261-1-31-0', '1609.03261-1-32-0', '1609.03261-1-39-1', '1609.03261-1-42-0', '1609.03261-1-43-0', '1609.03261-1-44-0', '1609.03261-1-45-0', '1609.03261-1-46-0', '1609.03261-1-49-0', '1609.03261-1-49-1', '1609.03261-1-50-0', '1609.03261-1-50-1', '1609.03261-1-50-2', '1609.03261-1-53-1', '1609.03261-1-53-3', '1609.03261-1-54-0', '1609.03261-1-54-1', '1609.03261-1-56-3', '1609.03261-1-57-0', '1609.03261-1-57-1', '1609.03261-1-57-2', '1609.03261-1-57-3', '1609.03261-1-59-0', '1609.03261-1-59-1', '1609.03261-1-59-2', '1609.03261-1-59-3', '1609.03261-1-61-4', '1609.03261-1-79-5', '1609.03261-1-83-8', '1609.03261-1-83-12', '1609.03261-1-83-13', '1609.03261-1-84-0', '1609.03261-1-84-1', '1609.03261-1-84-2', '1609.03261-1-85-0', '1609.03261-1-86-0', '1609.03261-1-86-1', '1609.03261-1-86-2', '1609.03261-1-90-0', '1609.03261-1-90-1', '1609.03261-1-91-0', '1609.03261-1-91-1', '1609.03261-1-92-0', '1609.03261-1-92-1', '1609.03261-1-94-7', '1609.03261-1-94-8', '1609.03261-1-94-9', '1609.03261-1-94-10', '1609.03261-1-94-11', '1609.03261-1-94-12', '1609.03261-1-94-13', '1609.03261-1-94-14', '1609.03261-1-94-15', '1609.03261-1-94-16', '1609.03261-1-94-17', '1609.03261-1-94-18', '1609.03261-1-94-19', '1609.03261-1-94-20', '1609.03261-1-94-21', '1609.03261-1-94-22', '1609.03261-1-94-23', '1609.03261-1-94-24', '1609.03261-1-94-25', '1609.03261-1-94-26', '1609.03261-1-94-27', '1609.03261-1-94-28', '1609.03261-1-94-29', '1609.03261-1-94-30', '1609.03261-1-94-31', '1609.03261-1-94-32', '1609.03261-1-94-33', '1609.03261-1-94-34', '1609.03261-1-94-35', '1609.03261-1-94-36', '1609.03261-1-94-37', '1609.03261-1-94-38', '1609.03261-1-94-39', '1609.03261-1-94-40', '1609.03261-1-94-41', '1609.03261-1-94-42', '1609.03261-1-94-43', '1609.03261-1-94-44', '1609.03261-1-94-45', '1609.03261-1-94-46', '1609.03261-1-94-47', '1609.03261-1-94-48', '1609.03261-1-94-49', '1609.03261-1-94-50', '1609.03261-1-94-51', '1609.03261-1-94-52', '1609.03261-1-94-53', '1609.03261-1-94-54', '1609.03261-1-94-55', '1609.03261-1-94-56', '1609.03261-1-96-3', '1609.03261-1-96-8', '1609.03261-2-17-0', '1609.03261-2-18-0', '1609.03261-2-25-0', '1609.03261-2-26-0', '1609.03261-2-27-0', '1609.03261-2-28-0', '1609.03261-2-35-1', '1609.03261-2-42-0', '1609.03261-2-44-0', '1609.03261-2-44-1', '1609.03261-2-47-0', '1609.03261-2-47-1', '1609.03261-2-49-0', '1609.03261-2-49-1', '1609.03261-2-51-0', '1609.03261-2-52-1', '1609.03261-2-58-1', '1609.03261-2-61-0', '1609.03261-2-66-0', '1609.03261-2-66-1', '1609.03261-2-69-1', '1609.03261-3-17-0', '1609.03261-3-18-0', '1609.03261-3-25-0', '1609.03261-3-26-0', '1609.03261-3-27-0', '1609.03261-3-28-0', '1609.03261-3-35-1', '1609.03261-3-42-0', '1609.03261-3-44-0', '1609.03261-3-44-1', '1609.03261-3-47-0', '1609.03261-3-47-1', '1609.03261-3-48-3', '1609.03261-3-48-4', '1609.03261-3-48-5', '1609.03261-3-48-6', '1609.03261-3-48-7', '1609.03261-3-48-8', '1609.03261-3-48-9', '1609.03261-3-48-10', '1609.03261-3-48-11', '1609.03261-3-48-12', '1609.03261-3-48-13', '1609.03261-3-48-14', '1609.03261-3-48-15', '1609.03261-3-48-16', '1609.03261-3-48-17', '1609.03261-3-48-18', '1609.03261-3-48-19', '1609.03261-3-48-20', '1609.03261-3-48-21', '1609.03261-3-48-22', '1609.03261-3-48-23', '1609.03261-3-48-24', '1609.03261-3-48-25', '1609.03261-3-48-26', '1609.03261-3-48-27', '1609.03261-3-48-28', '1609.03261-3-48-29', '1609.03261-3-48-30', '1609.03261-3-48-31', '1609.03261-3-48-32', '1609.03261-3-48-33', '1609.03261-3-48-34', '1609.03261-3-48-35', '1609.03261-3-48-36', '1609.03261-3-48-37', '1609.03261-3-48-38', '1609.03261-3-48-39', '1609.03261-3-48-40', '1609.03261-3-48-41', '1609.03261-3-48-42', '1609.03261-3-48-43', '1609.03261-3-48-44', '1609.03261-3-48-45', '1609.03261-3-48-46', '1609.03261-3-48-47', '1609.03261-3-48-48', '1609.03261-3-48-49', '1609.03261-3-48-50', '1609.03261-3-48-51', '1609.03261-3-48-52', '1609.03261-3-48-53', '1609.03261-3-48-54', '1609.03261-3-48-55', '1609.03261-3-48-56', '1609.03261-3-48-57', '1609.03261-3-48-58', '1609.03261-3-48-59', '1609.03261-3-48-60', '1609.03261-3-48-61', '1609.03261-3-48-62', '1609.03261-3-48-63', '1609.03261-3-48-64', '1609.03261-3-48-65', '1609.03261-3-48-66', '1609.03261-3-48-67', '1609.03261-3-48-68', '1609.03261-3-48-69', '1609.03261-3-48-70', '1609.03261-3-48-71', '1609.03261-3-48-72', '1609.03261-3-48-73', '1609.03261-3-48-74', '1609.03261-3-48-75', '1609.03261-3-48-76', '1609.03261-3-48-77', '1609.03261-3-48-82', '1609.03261-3-51-2', '1609.03261-3-51-3', '1609.03261-3-53-0', '1609.03261-3-53-1', '1609.03261-3-54-0', '1609.03261-3-54-1', '1609.03261-3-54-2', '1609.03261-3-54-3', '1609.03261-3-54-4', '1609.03261-3-54-5', '1609.03261-3-54-6', '1609.03261-3-54-7', '1609.03261-3-54-8', '1609.03261-3-54-9', '1609.03261-3-54-10', '1609.03261-3-54-11', '1609.03261-3-54-12', '1609.03261-3-54-13', '1609.03261-3-54-14', '1609.03261-3-54-15', '1609.03261-3-54-16', '1609.03261-3-54-17', '1609.03261-3-54-18', '1609.03261-3-54-19', '1609.03261-3-54-20', '1609.03261-3-54-21', '1609.03261-3-54-22', '1609.03261-3-54-23', '1609.03261-3-54-24', '1609.03261-3-54-25', '1609.03261-3-54-26', '1609.03261-3-54-27', '1609.03261-3-54-28', '1609.03261-3-54-29', '1609.03261-3-54-30', '1609.03261-3-54-31', '1609.03261-3-54-32', '1609.03261-3-54-33', '1609.03261-3-54-34', '1609.03261-3-54-35', '1609.03261-3-54-36', '1609.03261-3-54-37', '1609.03261-3-54-38', '1609.03261-3-54-39', '1609.03261-3-54-40', '1609.03261-3-54-41']

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

https://arxiv.org/abs/1609.03261

{'1609.03261-3-0-0': 'We develop and analyze a procedure for gradient-based optimization that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-3-0-1': 'As a member of the SVRG family of algorithms, SCSG makes use of gradient estimates at two scales and the number of updates is governed by a geometric random variable.', '1609.03261-3-0-2': 'Unlike most existing algorithms in this family, both the computation cost and the communication cost of SCSG do not necessarily scale linearly with the sample size [MATH]; indeed, these costs are independent of [MATH] when the target accuracy is low.', '1609.03261-3-0-3': 'The experimental evaluation on real datasets confirms the effectiveness of SCSG.', '1609.03261-3-1-0': '# Introduction', '1609.03261-3-2-0': 'Optimizing the finite-sum convex objectives is ubiquitous in different areas: [EQUATION] where each [MATH] is a convex function.', '1609.03261-3-2-1': 'These problems are often solved by algorithms that either make use of full gradients (obtained by processing the entire dataset) or stochastic gradients (obtained by processing single data points or mini-batches of data points).', '1609.03261-3-2-2': 'The use of the former provides guarantees of eventual convergence and the latter yields advantages in terms of rate of convergence rate, scalability and simplicity of implementation [CITATION].', '1609.03261-3-2-3': 'An impactful recent line of research has shown that a hybrid methodology that makes use of both full gradients and stochastic gradients can obtain the best of both worlds-guaranteed convergence at favorable rates, e.g. [CITATION].', '1609.03261-3-2-4': 'The full gradients provide variance control for the stochastic gradients.', '1609.03261-3-3-0': 'While this line of research represents significant progress towards the goal of designing scalable, autonomous learning algorithms, there remain some inefficiencies in terms of computation.', '1609.03261-3-3-1': 'With the definition of computation and communication cost in Section 2.1, the methods referred to above require [MATH] computation to achieve an [MATH]-approximate solution, where [MATH] is the number of data points, [MATH] is a target accuracy and [MATH] is the dimension of the parameter vector.', '1609.03261-3-3-2': 'Some methods incur a [MATH] storage cost [CITATION].', '1609.03261-3-3-3': 'The linear dependence on [MATH] is problematic in general.', '1609.03261-3-3-4': 'Clearly there will be situations in which accurate solutions can be obtained with less than a single pass through the data; indeed, some problems will require a constant number of steps.', '1609.03261-3-3-5': 'This will be the case, for example, if the data in a regression problem consist of a fixed number of pairs repeated a large number of times.', '1609.03261-3-3-6': 'For deterministic algorithms, the worst case analysis in [CITATION] shows that scanning at least a fixed proportion of the data is necessary; however, learning algorithms are generally stochastic and real-world learning problems are generally not worst case.', '1609.03261-3-4-0': 'An equally important bottleneck for learning algorithms is the cost of communication.', '1609.03261-3-4-1': 'For large data sets that must be stored on disk or distributed across many computing nodes, the communication cost can be significant, even dominating the computation cost.', '1609.03261-3-4-2': 'For example, SVRG makes use of full gradient over the whole dataset which can incur prohibitive communication cost.', '1609.03261-3-4-3': 'There is an active line of research that focuses on communication costs; see, e.g. [CITATION].', '1609.03261-3-5-0': 'In this article, we present a variant of the stochastic variance reduced gradient (SVRG) method that we refer to as stochastically controlled stochastic gradient (SCSG).', '1609.03261-3-5-1': 'The basic idea behind SCSG-that of approximating the full gradient in SVRG via a subsample-has been explored by others, but we present several innovations that yield significant improvements both in theory and in practice.', '1609.03261-3-5-2': 'In contradistinction to SVRG, the theoretical convergence rate of SCSG has a sublinear regime in terms of both computation and communication.', '1609.03261-3-5-3': 'This regime is important in machine learning problems, notably in the common situation in which the sample size is large, ([MATH]), while the required accuracy is low, [MATH].', '1609.03261-3-5-4': 'The analysis in this article shows that SCSG is able to achieve the target accuracy in this regime with potentially less than a single pass through the data.', '1609.03261-3-6-0': 'In the regime of low accuracy, SCSG is never worse than the classical stochastic gradient descent (SGD).', '1609.03261-3-6-1': 'Although SCSG has the same dependence on the target accuracy as SGD, it has a potentially much smaller factor.', '1609.03261-3-6-2': 'In fact, the theoretical complexity of SGD depends on the uniform bound of [MATH] over the domain and the component index.', '1609.03261-3-6-3': 'This might be infinite even in the most common least square problems.', '1609.03261-3-6-4': 'By contrast, the complexity of SCSG depends on a new measure [MATH], defined in Section [REF] and discussed in Section [REF], which is finite and small for a large class of practical problems.', '1609.03261-3-6-5': 'In particular, [MATH] in many cases where SGD does not have theoretical guarantees to converge.', '1609.03261-3-6-6': 'The measure [MATH] sheds light upon characterizing the difficulty of optimization problems in the form of a finite sum and reveals some intrinsic difference between finite-sum optimization and stochastic approximation, which is considered by other relevant works; e.g., streaming SVRG [CITATION] and dynaSAGA[CITATION].', '1609.03261-3-7-0': 'The remainder of the paper is organized as follows.', '1609.03261-3-7-1': 'In Section 2, we review SVRG, discuss several of its variants and we describe the SCSG algorithm.', '1609.03261-3-7-2': 'We provide a theoretical convergence analysis in Section 3.', '1609.03261-3-7-3': 'In Section 4, we give a comprehensive discussion on the difficulty measure [MATH].', '1609.03261-3-7-4': 'The empirical results on real datasets are presented in Section [REF].', '1609.03261-3-7-5': 'Finally, we conclude our work and discuss potential extensions in Section [REF].', '1609.03261-3-7-6': 'All technical proofs are relegated to the Appendices.', '1609.03261-3-8-0': '# Notation, Assumptions and Algorithm', '1609.03261-3-9-0': 'We write [MATH] as [MATH] and [MATH] as [MATH] for brevity and use [MATH] to denote the Euclidean norm throughout the paper.', '1609.03261-3-9-1': "We adopt the standard Landau's notation ([MATH]).", '1609.03261-3-9-2': 'In some cases, we use [MATH] to hide terms which are polynomial in parameters.', '1609.03261-3-9-3': 'The notation [MATH] will only be used to maximize the readibility in discussions but not be used in the formal analysis.', '1609.03261-3-9-4': 'For convenience, we use [MATH] to denote the set [MATH] and for any subset [MATH], we write [MATH] the batch gradient [MATH] for short.', '1609.03261-3-9-5': 'Finally, given random variables [MATH] and [MATH] and a random variable [MATH], denote by [MATH] the conditional expectation of [MATH] given [MATH], i.e. [MATH].', '1609.03261-3-9-6': 'Note that when [MATH] is independent of [MATH], then [MATH] is equivalent to the the expectation of [MATH] holding [MATH] fixed.', '1609.03261-3-9-7': 'Furthermore, we use the symbol [MATH], without the subscript, to denote the expectation over all randomness.', '1609.03261-3-10-0': 'The assumption A1 on the smoothness of individual functions will be used throughout this paper.', '1609.03261-3-10-1': '[A1] [MATH] is convex with [MATH]-Lipschitz gradient [EQUATION] for some [MATH] and all [MATH];', '1609.03261-3-11-0': 'The following assumption will be used in the context of strongly-convex objectives.', '1609.03261-3-11-1': '[A2] [MATH] is strongly-convex with [EQUATION] for some [MATH].', '1609.03261-3-12-0': 'Note that we only require the strong convexity of [MATH] instead of each component.', '1609.03261-3-13-0': 'Let [MATH] denote the minimizer of [MATH] that minimizes [MATH] in [REF], then [MATH] can be written as [EQUATION]', '1609.03261-3-13-1': 'We will abbreviate [MATH] as [MATH] when no confusion can arise.', '1609.03261-3-13-2': 'Note that [MATH] is unique in many situations where [MATH].', '1609.03261-3-13-3': 'When there are multiple minimum, we select [MATH] be the one that minimizes the RHS of [REF].', '1609.03261-3-13-4': 'Further let [MATH] denote the initial value (possibly random) and [EQUATION]', '1609.03261-3-13-5': 'Then [MATH] under assumption A1 and A2.', '1609.03261-3-13-6': 'A point [MATH], possibly random, is called an [MATH]-approximated solution if [EQUATION].', '1609.03261-3-14-0': 'In terms of the computation complexity, we assume that sampling an index [MATH] and computing the pair [MATH] incurs a unit of cost.', '1609.03261-3-14-1': 'This is conventional and called IFO framework in literature ([CITATION]).', '1609.03261-3-14-2': 'We use use [MATH] to denote the cost to achieve an [MATH]-accurate solution.', '1609.03261-3-14-3': 'In some contexts we also consider [MATH] as the cost to reach a solution [MATH] with [MATH] .', '1609.03261-3-15-0': 'Finally, since our analysis heavily relies on geometric distributions, we formally define them here.', '1609.03261-3-15-1': 'We say a random variable [MATH] if [MATH] is supported on non-negative integers with [EQUATION]', '1609.03261-3-15-2': 'The expectation of the above distributions satisfy that [EQUATION] [h] Stochastic Variance Reduced Gradient (SVRG) Method', '1609.03261-3-16-0': 'Inputs: Stepsize [MATH], number of stages [MATH], initial iterate [MATH], number of SGD steps [MATH].', '1609.03261-3-17-0': 'Procedure [1] [MATH] Randomly pick [MATH]', '1609.03261-3-18-0': 'Output: (Option 1): [MATH] (Option 2): [MATH].', '1609.03261-3-19-0': '## SVRG and Other Related Works', '1609.03261-3-20-0': 'The stochastic variance reduced gradient (SVRG) method blends gradient descent and stochastic gradient descent, using the former to control the effect of the variance of the latter [CITATION].', '1609.03261-3-20-1': 'We summarize SVRG in Algorithm [REF].', '1609.03261-3-21-0': 'Using the definition from Section 2.1, it is easy to see that the computation cost of SVRG is [MATH].', '1609.03261-3-21-1': 'As shown in the convergence analysis of [CITATION], [MATH] is required to be [MATH] to guarantee convergence.', '1609.03261-3-21-2': 'Thus, the computation cost of SVRG is [MATH].', '1609.03261-3-21-3': 'The costs of the other algorithms considered in Table [REF] can be obtain in a similar fashion.', '1609.03261-3-21-4': 'For comparison, we only present the results for smooth case (assumption A1).', '1609.03261-3-22-0': 'A number of variants of SVRG have been studied.', '1609.03261-3-22-1': 'For example, a constrained form of SVRG can be obtained by replacing line 8 with a projected gradient descent step [CITATION].', '1609.03261-3-22-2': 'A mini-batch variant of SVRG arises when one samples a subset of indices instead of a single index in line 6 and updates the iterates by the average gradient in this batch in line 7 [CITATION].', '1609.03261-3-22-3': 'Similarly, we can consider implementing the full gradient computation in line 2 using a subsample.', '1609.03261-3-22-4': 'This is proposed in [CITATION], which calculates [MATH] as [MATH] where [MATH] is a subset of size [MATH] uniformly sampled from [MATH].', '1609.03261-3-22-5': '[CITATION] heuristically show the potential for significant complexity reduction, but they only prove convergence for [MATH] under the stringent condition that [MATH] is uniformly bounded for all [MATH] and that all iterates are uniformly bounded.', '1609.03261-3-22-6': "Similar to Nesterov's acceleration for gradient descent, momentum terms can be added to the SGD steps to accelerate SVRG [CITATION].", '1609.03261-3-23-0': 'Much of this work focuses on the strongly convex case.', '1609.03261-3-23-1': 'In the non-strongly convex setting one way to proceed is to add a [MATH] regularizer [MATH].', '1609.03261-3-23-2': 'Tuning [MATH], however, is subtle and requires multiple runs of the algorithm on a grid of [MATH] [CITATION].', '1609.03261-3-23-3': 'For general convex functions an alternative approach has been presented by [CITATION] (they generate [MATH] by a different scheme in line 4), which proves a computation complexity [MATH].', '1609.03261-3-23-4': 'Another approach is discussed by [CITATION], who improve the complexity to [MATH] by scaling the stepsize as [MATH].', '1609.03261-3-23-5': 'However, their algorithm still relies on calculating a full gradient.', '1609.03261-3-23-6': 'Other variants of SVRG have been proposed in the distributed computing setting [CITATION] and in the stochastic setting [CITATION].', '1609.03261-3-24-0': '## SCSG', '1609.03261-3-25-0': '[htp] Stochastically Controlled Stochastic Gradient (SCSG) Method', '1609.03261-3-26-0': 'Inputs: Stepsize [MATH], batch size [MATH], number of stages [MATH], initial iterate [MATH].', '1609.03261-3-27-0': 'Procedure [1] [MATH] Uniformly sample a batch [MATH] with [MATH] Generate [MATH] Randomly pick [MATH]', '1609.03261-3-28-0': 'Output: (Strongly convex case): [MATH] (Non-strongly convex case): [MATH].', '1609.03261-3-29-0': 'SCSG is similar to [CITATION] in that it implements the gradient computation on a subsample [MATH] of size [MATH]; See Algorithm [REF].', '1609.03261-3-29-1': 'However, instead of being fixed, the number of SGD updates of SCSG is a geometrically distributed random variable (line 5).', '1609.03261-3-29-2': 'Surprisingly, this seemingly technical modification enables the analysis in the non-strongly convex case and a much tighter convergence analysis without imposing unrealistic assumptions like the boundedness of iterates produced by the algorithm; See Section [REF] for details.', '1609.03261-3-29-3': 'Recently we found that [CITATION] also implicitly uses the geometric size of the inner loop.', '1609.03261-3-29-4': 'However, they do not use the iterate at the end of each epoch, i.e. [MATH] and hence cannot prove the non-strongly convex case.', '1609.03261-3-30-0': 'The average computation cost of SCSG is [MATH].', '1609.03261-3-30-1': 'By the law of large numbers and the expectation formula [REF], this is close to [MATH].', '1609.03261-3-30-2': 'Table [REF] summarizes the computation complexity as well as some other details of SCSG and 11 other existing popular algorithms.', '1609.03261-3-30-3': 'The table includes the computation cost of optimizing non-strongly-convex functions (column 1) and strongly convex functions (column 2).', '1609.03261-3-30-4': 'In practice, the amount of tuning is of major concern.', '1609.03261-3-30-5': 'For this reason, a fixed stepsize is usually preferred to a complicated stepsize scheme and it is better that the tuning parameter does not depend on unknown quantities; e.g., [MATH] or the total number of epochs [MATH].', '1609.03261-3-30-6': 'These issues are documented in column 3 and column 4.', '1609.03261-3-30-7': 'Moreover, many algorithms requires [MATH] to be bounded, i.e. [MATH] to be Lipschitz.', '1609.03261-3-30-8': 'However, this assumption is not realistic in many cases and it is better to discard it.', '1609.03261-3-30-9': 'To address this issue, we document it in column 5.', '1609.03261-3-30-10': 'To highlight the dependence on [MATH] and [MATH] (or [MATH]), we implicitly assume that other parameters, e.g. [MATH], are [MATH] as a convention.', '1609.03261-3-31-0': 'As seen from Table [REF], SCSG and SGD are the only two methods which are able to reach an [MATH]-approximate solution with potentially less than a single pass through the data; moreover, the number of accesses of the data is independent of the sample size [MATH].', '1609.03261-3-31-1': 'Comparing to SCSG, SGD requires each [MATH] to be Lipschitz, which is not satisfied by least-square objectives.', '1609.03261-3-31-2': 'By contrast, as will be shown in Section [REF], the computation cost of SCSG only depends on the quantity [MATH], which is relatively small in many cases.', '1609.03261-3-31-3': 'Furthermore, SGD either sets the stepsize based on unknown quantities like the total number of epochs [MATH] or needs to use a time-varying sequence of stepsizes.', '1609.03261-3-31-4': 'This involves intensive tuning as opposed to a fixed stepsize.', '1609.03261-3-32-0': 'On the other hand, SCSG is communication-efficient since it only needs to operate on mini-batches as SGD.', '1609.03261-3-32-1': 'This is particularly important in modern large-scale tasks.', '1609.03261-3-32-2': 'By contrast, those algorithms that require full gradients evaluation either need extra communication for synchronization or need extra computational cost for the asynchronous version to converge; See e.g. [CITATION].', '1609.03261-3-33-0': '# Convergence Analysis', '1609.03261-3-34-0': 'In this section we present a convergence analysis of SCSG.', '1609.03261-3-34-1': 'We first state the following key lemma that connects our algorithm with the measure [MATH] defined in [REF].', '1609.03261-3-35-0': 'Let [MATH] be a random subset of size [MATH], and define the random variable [MATH].', '1609.03261-3-35-1': 'Then [MATH] and [EQUATION].', '1609.03261-3-36-0': 'The proof, which appears in Appendix [REF], involves a standard technique for analyzing sampling without replacement.', '1609.03261-3-36-1': 'Obviously, [MATH] if [MATH] is uniformly bounded as is often assumed in the literature.', '1609.03261-3-36-2': 'In section [REF] we will present various other situations where [MATH].', '1609.03261-3-37-0': 'Note that the extra variation vanishes when [MATH] and in general is inversely proportional to the batch size.', '1609.03261-3-37-1': 'In the rest of this section, we will first discuss the case [MATH], which we refer to as R-SVRG (Randomized SVRG), to compare with the original SVRG.', '1609.03261-3-37-2': 'Later we will discuss the general case.', '1609.03261-3-38-0': '## Analysis of R-SVRG', '1609.03261-3-39-0': 'We start from deriving the sub-optimality bound for [MATH] and [MATH] respectively.', '1609.03261-3-40-0': 'Let [MATH] and assume that [MATH], then [(1)] under the assumption A1, [EQUATION] under the assumption A1 and A2, [EQUATION] where [EQUATION].', '1609.03261-3-41-0': 'Based on Theorem [REF], we first consider a constant stepsize [MATH] scaled as [MATH].', '1609.03261-3-42-0': 'Let [MATH] with [MATH].', '1609.03261-3-42-1': 'Then under the assumption A1, with the output [MATH], [EQUATION]', '1609.03261-3-42-2': 'If further the assumption A2 is satisfied, then the output [MATH] satisfies that [EQUATION]', '1609.03261-3-42-3': 'The above theorem is appealing in three aspects: 1) in the strongly convex case, no parameter depends on [MATH].', '1609.03261-3-42-4': 'This is in contrast to the original SVRG where the number of SGD updates should be proportional to [MATH] in order to guarantee the theoretical convergence [CITATION].', '1609.03261-3-42-5': 'Being agnostic to [MATH] is useful in that [MATH] is hard to estimate in practice; 2) the same setup also guarantees the convergence of [MATH] in the strongly convex case with an almost identical cost up to a [MATH] factor.', '1609.03261-3-42-6': 'This is important especially in statistical problems but unfortunately not covered in existing literature to the best of our knowledge; 3) the same stepsize guarantees the convergence in both the non-strongly convex and the strongly convex case and the only requirement is [MATH], which is quite mild.', '1609.03261-3-42-7': 'Note that the requirement for the convergence of gradient descent is [MATH].', '1609.03261-3-43-0': 'By scaling [MATH] as [MATH], R-SVRG is able to achieve the same complexity of [CITATION], which is the best bound in the class of SVRG-type algorithms without acceleration techniques.', '1609.03261-3-44-0': 'Let [MATH] with [MATH].', '1609.03261-3-44-1': 'Then under the assumption A1, with the output [MATH], [EQUATION]', '1609.03261-3-45-0': '## Analysis of SCSG', '1609.03261-3-46-0': 'Due to the technical complications, we discuss the non-strongly convex case and the strongly convex cases separately in the general case.', '1609.03261-3-46-1': 'Similar to R-SVRG, we first derive the sub-optimality bound for [MATH].', '1609.03261-3-47-0': 'Assume that [MATH].', '1609.03261-3-47-1': 'Under the assumption A1, [EQUATION].', '1609.03261-3-48-0': 'Note that the bound in Theorem [REF] can be simplified as [MATH] while the bound in Theorem [REF] can be simplified as [MATH].', '1609.03261-3-48-1': 'Despite the more stringent requirement on [MATH]), these two bounds have two qualitative difference: 1) SCSG has an extra term [MATH], which characterizes the sampling variance of the mini-batch gradients; 2) SCSG loses an [MATH] in the first term, which is due to the bias of [MATH].', '1609.03261-3-48-2': 'In fact, recall the definition of [MATH] at the beginning of Section [REF], a simple calculation shows that [EQUATION]', '1609.03261-3-48-3': 'B = Ln and = L B, [EQUATION] ^* = _x1n_i=1^nf_i(x) - f(x)^2, [EQUATION] 2 - 8L - (1 + 13L )(1 + B) > 0 [EQUATION]', '1609.03261-3-48-4': 'B = (L)n and = (B).', '1609.03261-3-48-5': '[EQUATION]', '1609.03261-3-48-6': 'B = (L)^2n[EQUATION]', '1609.03261-3-48-7': 'G^2, ^*.', '1609.03261-3-48-8': '[EQUATION] 2n_i=1^n(2b_i^2 - f_i(x^*))4b^2n.', '1609.03261-3-48-9': '[EQUATION] f(x) = _j,k=1^mf_jk(Z_j, Z_k; x) [EQUATION] _j,kE U_jk^2 = O(1)= O_p(1).', '1609.03261-3-48-10': '[EQUATION] = 1n_i=1^n_2(y_i, a_i^Tx^*)^2a_i^2.', '1609.03261-3-48-11': '[EQUATION]', '1609.03261-3-48-12': 'M_11n_i=1^na_i^2.', '1609.03261-3-48-13': '[EQUATION] _ia_i^2y^2n.', '1609.03261-3-48-14': '[EQUATION] f(x) = 1n_i=1^n( ( 1 + _k=1^K - 1e^a_i^Tx_k) - _k=1^K - 1I(y_i = k)a_i^Tx_k), [EQUATION]', '1609.03261-3-48-15': 'E (D_N - D_N + 1) = (frac1 - 1)left( D_0 - E D_N).', '1609.03261-3-48-16': '[EQUATION] ax^2 bx + cax^2 b^2a + 2c.', '1609.03261-3-48-17': '[EQUATION] ax^2 bx + cxb2a + ca + b^24a^2.', '1609.03261-3-48-18': '[EQUATION] x^22(fracb2a)^2 + 2(fracca + b^24a^2) = b^2a^2 + 2ca.', '1609.03261-3-48-19': 'ax^2 b^2a + 2c.', '1609.03261-3-48-20': '[EQUATION] _j=1^Mz_j = 0.', '1609.03261-3-48-21': '[EQUATION]', '1609.03261-3-48-22': 'E 1m_jJz_j^2 = M - m(M - 1)m1M_j=1^Mz_j^2I(m < M)m1M_j=1^Mz_j^2.', '1609.03261-3-48-23': '[EQUATION] 1m_jJz_j = 1m_i=1^nW_jz_j.', '1609.03261-3-48-24': '[EQUATION] _i=1^nz_i = nf(x^*) = 0.', '1609.03261-3-48-25': '[EQUATION]', '1609.03261-3-48-26': 'E_i_k^(j)_k = f(x^(j)_k) + e_j.', '1609.03261-3-48-27': '[EQUATION]', '1609.03261-3-48-28': 'E x_j - x^*^2 < , E ^(j)_N_j_2^2 < , E e_j, x_j - x_j-1 < .', '1609.03261-3-48-29': '[EQUATION] _jx^(j)_k+1 - u^2 = _jx^(j)_k - u^2 - 2f(x^(j)_k), x^(j)_k - u -2_jlangle e_j, x^(j)_k - u+ ^2 E_j^(j)_k^2.', '1609.03261-3-48-30': '[EQUATION] _jlangle e_j, u= _jmathbbE_I_je_j, u= _j E_I_je_j, u= 0.', '1609.03261-3-48-31': '[EQUATION] _j , x^(j)_0= 0.', '1609.03261-3-48-32': '[EQUATION] _jlangle e_j, x^(j)_k - u= _jlangleej, x^(j)_k - x^(j)_0.', '1609.03261-3-48-33': '[EQUATION] -2BmathbbEla e_j, x_j - x_j-12Ee_j^2Ex_j - x_j-1^2.', '1609.03261-3-48-34': '[EQUATION]', '1609.03261-3-48-35': 'E x_j - x_j-1^22Ee_j^2Ex_j - x_j-1^2 - 2E f(x_j), x_j - x_j-1+ ^2BmathbbE ^(j)_N_j^2.', '1609.03261-3-48-36': '[EQUATION]', '1609.03261-3-48-37': 'E x_j - x_j-1^2- 4E f(x_j), x_j - x_j-1+ 2^2BmathbbE ^(j)_N_j^2 + 4^2B^2Ee_j^2.', '1609.03261-3-48-38': '[EQUATION] f(x_j), x_j - x_j-1f(x_j) - f(x_j-1) = (f(x_j) - f(x^*)) - (f(x_j-1) - f(x^*)).', '1609.03261-3-48-39': '[EQUATION]', '1609.03261-3-48-40': 'E e_j^24LI(B < n)BE (f(x_j-1) - f(x^*)) + 2I(B < n)B. [EQUATION]', '1609.03261-3-48-41': 'E 1B_i_jf_i(x^*)^2I(B < n)B. [EQUATION] f(x_j)^22L( f(x^*) - f(x_j) + f(x_j), x_j - x^*).', '1609.03261-3-48-42': '[EQUATION] 2(1 - 8L)Ela f(x_j), x_j - x^*2(1 - 8L) (E (f(x_j) - f(x^*)) + 2Ex_j - x^*^2).', '1609.03261-3-48-43': '[EQUATION]', '1609.03261-3-48-44': 'Ex^(j)_0 - x^*_2^2 = E x_j-1 - x^*_2^2 < .', '1609.03261-3-48-45': '[EQUATION] y^(j)_0 = x^(j)_0, y^(j)_k = y^(j)_k-1 - ^(j)_k, where ^(j) = f_i_k-1(y^(j)_k-1) - f_i_k-1(y^(j)_0) + f(y^(j)_0).', '1609.03261-3-48-46': '[EQUATION] x^(j)_k - y^(j)_k = (mathrmId - f_i_k-1)(x^(j)_k-1) - (mathrmId - f_i_k-1)(y^(j)_k-1) - e_j.', '1609.03261-3-48-47': '[EQUATION]', '1609.03261-3-48-48': 'E y^(j)_k+1 - x^*_2^2 4^2L E (f(y^(j)_0) - f(x^*)) + E y^(j)_k - x^*_2^2.', '1609.03261-3-48-49': '[EQUATION]', '1609.03261-3-48-50': 'Ex^(j)_k - x^*_2^22k^2( Ex^(j)_0 - x^*_2^2 + ^2).', '1609.03261-3-48-51': '[EQUATION]', '1609.03261-3-48-52': 'Ex_j - x^*_2^2 2E N_j^2 ( Ex^(j)_0 - x^*_2^2 + ^2) < , [EQUATION]', '1609.03261-3-48-53': 'E e_j_2^2 < .', '1609.03261-3-48-54': '[EQUATION]', '1609.03261-3-48-55': 'E ^(j)_N_j_2^24L^2 E x_j - x^*_2^2 + 2L^2 E x_j-1 - x^*_2^2 + 2Ee_j_2^2 < .', '1609.03261-3-48-56': '[EQUATION] f(x_j)^22L( f(x^*) - f(x_j) + f(x_j), x_j -x^*).', '1609.03261-3-48-57': '[EQUATION] 2n (1 - 3L)Ela f(x_j), x_j - x^* 2n (1 - 3L)( E (f(x_j) - f(x^*)) + 2x_j - x^*^2).', '1609.03261-3-48-58': '[EQUATION] 2n E (f(x_j) - f(x^*))4^2L n E (f(x_j-1) - f(x^*)) + E x_j-1 - x^*^2 - E x_j - x^*^2.', '1609.03261-3-48-59': '[EQUATION] 2n (1 - 2L)_j=1^TE (f(x_j) - f(x^*)) 4^2L n _f + _x. [EQUATION] _j=1^TE (f(x_j) - f(x^*))T(E (f(x_T) - f(x^*))).', '1609.03261-3-48-60': '[EQUATION]', '1609.03261-3-48-61': 'E (f(x_T) - f(x^*))1T4^2L n _f + _x2n (1 - 2L).', '1609.03261-3-48-62': '[EQUATION]', '1609.03261-3-48-63': 'Q_j2L, 11 + n(1 - 3L)Q_j-1 = Q_j-1.', '1609.03261-3-48-64': '[EQUATION]', '1609.03261-3-48-65': 'Q_T^TQ_0.', '1609.03261-3-48-66': '[EQUATION]', '1609.03261-3-48-67': 'Q_TE x_T - x^*^2 + 2n E (f(x_T) - f(x^*)) [EQUATION]', '1609.03261-3-48-68': 'Q_0 E x_0 - x^*^2 + 2n (2 - 3L)E (f(x_0) - f(x^*))E x_0 - x^*^2 + 4n E (f(x_0) - f(x^*)).', '1609.03261-3-48-69': '[EQUATION]', '1609.03261-3-48-70': 'E (f(x_T) - f(x^*))1T4(L)^2 n _f + L_x2(L) n (1 - 2L) = O(frac_fT + L_xTn).', '1609.03261-3-48-71': '[EQUATION]', '1609.03261-3-48-72': 'T() = O( _f + L_xn), [EQUATION]', '1609.03261-3-48-73': 'E Comp() = O( n T()) = O( n_f + L_x).', '1609.03261-3-48-74': '[EQUATION]', '1609.03261-3-48-75': 'E x_j - x^*^2 + 2n E(f(x_j) - f(x^*))^TQ_0.', '1609.03261-3-48-76': '[EQUATION]', '1609.03261-3-48-77': 'T() = O( LQ_0n / 1), and T_x() = O( Q_0 / 1)] By definition, [EQUATION].', '1609.03261-3-48-78': 'This implies that [EQUATION].', '1609.03261-3-48-79': 'Note that [MATH], we obtain that [EQUATION]', '1609.03261-3-48-80': 'Similarly, [EQUATION] [Corollary [REF]] By part (1) of Theorem [REF], we have [EQUATION].', '1609.03261-3-48-81': 'The assumption A1, [EQUATION].', '1609.03261-3-48-82': 'Thus, [EQUATION].', '1609.03261-3-48-83': 'This implies that [EQUATION] and hence [EQUATION]', '1609.03261-3-49-0': '# Analysis of SCSG', '1609.03261-3-50-0': '## Convergence Analysis for Non-Strongly Convex Objectives', '1609.03261-3-51-0': '[Theorem [REF]] By Theorem [REF], we have [EQUATION]', '1609.03261-3-51-1': 'Telescoping the above inequality for [MATH], we have [EQUATION] where the last inequality uses [MATH].', '1609.03261-3-51-2': 'By convexity, [EQUATION].', '1609.03261-3-51-3': 'Therefore, [EQUATION].', '1609.03261-3-52-0': 'Before proving the results in Section [REF], we derive the computation complexity for arbitrary batch size [MATH] with an appropriately scaled stepsize [MATH] in the non-strongly convex case.', '1609.03261-3-53-0': 'Assume A1 holds.', '1609.03261-3-53-1': 'Set [MATH] with [EQUATION] then with the output [MATH], [EQUATION].', '1609.03261-3-54-0': 'Let [EQUATION].', '1609.03261-3-54-1': 'By part (1) of Theorem [REF], [EQUATION]', '1609.03261-3-54-2': 'Under these conditions, [MATH] is bounded by [EQUATION].', '1609.03261-3-54-3': 'Let [EQUATION] then for any [MATH], [EQUATION].', '1609.03261-3-54-4': 'This implies that [EQUATION].', '1609.03261-3-54-5': 'By definition, [EQUATION].', '1609.03261-3-54-6': 'Therefore, [EQUATION] [Corollary [REF]] Let [MATH].', '1609.03261-3-54-7': 'Then [EQUATION].', '1609.03261-3-54-8': 'By Corollary [REF], [EQUATION] where the last equality uses the fact that [MATH] and [MATH].', '1609.03261-3-54-9': 'As a consequence, [EQUATION] _e = 9^2I(B < n), = B, = L] By Theorem [REF], we have [EQUATION]', '1609.03261-3-54-10': 'The assumption A2 implies that [MATH].', '1609.03261-3-54-11': 'Thus, [EQUATION]', '1609.03261-3-54-12': 'On the other hand, [EQUATION]', '1609.03261-3-54-13': 'Putting [REF] and [REF], we obtain that [EQUATION]', '1609.03261-3-54-14': 'Multiplying both sides of [REF] by [MATH] and summing over [MATH], we obtain that [EQUATION] [Corollary [REF]] By definition, [MATH], and [EQUATION].', '1609.03261-3-54-15': 'As a result, [EQUATION].', '1609.03261-3-54-16': 'By [REF] in Theorem [REF], [EQUATION] where [MATH].', '1609.03261-3-54-17': 'Then [EQUATION].', '1609.03261-3-54-18': 'Define [MATH] as [EQUATION]', '1609.03261-3-54-19': 'E (f(x_T) - f(x^*))( 1 - 92)eps + 92 = .', '1609.03261-3-54-20': '[EQUATION]', '1609.03261-3-54-21': 'T() T() = O(log (frac_f)bigg/ (1)).', '1609.03261-3-54-22': '[EQUATION] = BB 11 = O( 1 + B), [EQUATION] 11 = O( 1 + n) = O( nn)] Therefore, [EQUATION] [Corollary [REF]] Using the same argument as in the proof of Corollary [REF], [MATH].', '1609.03261-3-54-23': 'By [REF] in Theorem [REF], [EQUATION] where [MATH].', '1609.03261-3-54-24': 'Then [EQUATION].', '1609.03261-3-54-25': 'Define [MATH] as [EQUATION]', '1609.03261-3-54-26': 'E (f(x_T) - f(x^*))( 1 - 92)eps + 92 = .', '1609.03261-3-54-27': '[EQUATION]', '1609.03261-3-54-28': 'T() T() = O(log (frac_f)bigg/ (1)).', '1609.03261-3-54-29': '[EQUATION]', '1609.03261-3-54-30': 'E Comp() = O( (frac^2 n + )log (frac_f)rb.', '1609.03261-3-54-31': '[EQUATION] f_i(x) - f_i(x^*)f_i(x^*), x - x^*+ 12Lf_i(x^*) - f_i(x)^2.', '1609.03261-3-54-32': '[EQUATION] a - b^2 = a^2 + b^2 - 2 a, b= 12a^2 - b^2 + 12a - 2b^2 12a^2 - b^2, [EQUATION] 14n_i=1^nf_i(x^*)^2 - 12n_i=1^nf_i(x)^212n_i=1^nf_i(x^*) - f_i(x)^2L(f(x) - f(x^*)).', '1609.03261-3-54-33': '[EQUATION] 2m^2_j,k=1^nU_jk.', '1609.03261-3-54-34': '[EQUATION]', '1609.03261-3-54-35': 'E 2m^2_j,k=1^mU_jk = O(1), [EQUATION] (y_i; a_i^Tx) = ( 1 + _k=1^K - 1e^a_i^Tx_k) - _k=1^K - 1I(y_i = k)a_i^Tx_k, [EQUATION]', '1609.03261-3-54-36': 'M_1_x_2(y_i;a_i^Tx)^22.', '1609.03261-3-54-37': '[EQUATION] f_i(x)x_k = (frace^a_i^Tx_k1 + _k=1^K - 1e^a_i^Tx_k - I(y_i = k))cdot a_i [EQUATION] f_i(x) = H_i(x) a_if_i(x)^2 = H_i(x)^2a_i^2, [EQUATION]', '1609.03261-3-54-38': 'H_i(x) = ( e^a_i^Tx_11 + _k=1^K - 1e^a_i^Tx_k - I(y_i = 1), , e^a_i^Tx_K - 11 + _k=1^K - 1e^a_i^Tx_k - I(y_i = K - 1))^T. [EQUATION]', '1609.03261-3-54-39': 'H_i(x)^2H_i(x)_12.', '1609.03261-3-54-40': '[EQUATION] 2n_i=1^na_i^2.', '1609.03261-3-54-41': '[MATH]'}

1802.00450

{'1802.00450-1-0-0': 'Topological semimetals is a new class of condensed matter systems with nontrivial electronic structure topology.', '1802.00450-1-0-1': 'Their unusual observable properties may often be understood in terms of quantum anomalies.', '1802.00450-1-0-2': 'In particular, Weyl and Dirac semimetals, which have point band touching nodes, are characterized by the chiral anomaly, which leads to the Fermi arc surface states, anomalous Hall effect, negative longitudinal magnetoresistance and planar Hall effect.', '1802.00450-1-0-3': 'In this paper we explore analogous phenomena in nodal line semimetals.', '1802.00450-1-0-4': 'We demonstrate that such semimetals realize a three dimensional analog of the parity anomaly, which is a known property of two dimensional Dirac semimetals arising, for example, on the surface of a three dimensional topological insulator.', '1802.00450-1-0-5': 'We relate one of the characteristic properties of nodal line semimetals, namely the drumhead surface states, to this anomaly, and derive the field theory, which encodes the corresponding anomalous response.', '1802.00450-1-1-0': '# Introduction', '1802.00450-1-2-0': 'Topological (semi)metal is a new phase of matter, which is characterized by a nontrivial electronic structure topology, yet is not an insulator with a gap in the spectrum.', '1802.00450-1-2-1': '[CITATION] Integer momentum space invariants, which characterize topologically nontrivial states of matter, are defined in this case on the Fermi surface, rather than in the whole Brillouin zone (BZ).', '1802.00450-1-2-2': '[CITATION] Such Fermi surface invariants arise from singularities in the electronic structure, in the simplest case isolated points, at which different bands touch.', '1802.00450-1-2-3': 'The significance of such points was emphasized early on by Volovik, [CITATION] and also pointed out by Murakami.', '1802.00450-1-2-4': '[CITATION]', '1802.00450-1-3-0': 'Nontrivial electronic structure topology has both spectroscopic manifestations, in the form of localized edge states, and manifestations in response, in the form of quantized, or just insensitive to perturbations and microscopic details, transport or other response properties.', '1802.00450-1-3-1': 'In the context of Weyl and Dirac semimetals this topological response may have different manifestations, such as negative longitudinal magnetoresistance, [CITATION] giant planar Hall effect, [CITATION] and anomalous Hall effect.', '1802.00450-1-3-2': '[CITATION]', '1802.00450-1-4-0': 'The edge states and the topological response are closely related and one way to understand this relation is in terms of the concept of quantum anomalies.', '1802.00450-1-4-1': 'Quantum anomaly is often described as violation of a classical symmetry (i.e. symmetry of the action) by quantum effects in the presence of external (say electromagnetic) fields.', '1802.00450-1-4-2': "[CITATION] This violation of a symmetry then leads to nonconservation of a current, which should be conserved classically by Noether's theorem.", '1802.00450-1-4-3': 'This viewpoint, however, is strictly applicable only to quantum anomalies in the particle physics context, since condensed matter systems typically do not possess the corresponding symmetries, such as the chiral symmetry, to begin with.', '1802.00450-1-5-0': 'Another, more useful in the condensed matter context, way to understand the anomalies is in terms of failure of gauge invariance.', '1802.00450-1-5-1': 'Current nonconservation may be represented in terms of an anomalous term in the action for the electromagnetic (or some other) field, which is not gauge invariant in the presence of a boundary.', '1802.00450-1-5-2': 'What restores the overall gauge invariance is the contribution of edge states, which precisely cancels the gauge invariance violating part of the bulk action.', '1802.00450-1-5-3': '[CITATION]', '1802.00450-1-6-0': 'It is instructive to see how this works in the case of the simplest topological metal system: a Weyl semimetal with two nodes.', '1802.00450-1-6-1': '[CITATION] The chiral anomaly in this system may be expressed in terms of the following action for the electromagnetic field [CITATION] [EQUATION] where [MATH] describes the momentum space separation between the Weyl nodes ([MATH]) along the [MATH]-axis and [MATH] units are used here and throughout this paper, except in some of the final formulas.', '1802.00450-1-6-2': 'We have also ignored the chiral magnetic effect [CITATION] and related phenomena in Eq. [REF] for simplicity (these effects exist only away from equilibrium and require certain modifications of Eq. [REF], which we do not want to discuss here; they have been discussed in detail, for example in Refs. [CITATION]).', '1802.00450-1-7-0': 'Consider a gauge transformation [MATH].', '1802.00450-1-7-1': 'This changes the action in Eq. [REF] by [EQUATION] [MATH] vanishes identically in the absence of boundaries (i.e. in a system with periodic boundary conditions), but does not vanish in a system with a boundary.', '1802.00450-1-7-2': 'Indeed, suppose [MATH], where [MATH] is the Heaviside step function.', '1802.00450-1-7-3': 'This may represent, for example, a contact between a Weyl semimetal in the [MATH] half-space and vacuum in the [MATH] half-space.', '1802.00450-1-7-4': 'Then we obtain [EQUATION]', '1802.00450-1-7-5': 'Eq. [REF] clearly does not vanish in general and this means that Eq. [REF] fails to be gauge invariant in the presence of a boundary.', '1802.00450-1-7-6': 'This failure of gauge invariance is a symptom that Eq. [REF] is incomplete.', '1802.00450-1-7-7': 'Indeed, what it does not take into account is the Fermi arc edge states, which are present in the Weyl semimetal system, described by Eq. [REF].', '1802.00450-1-7-8': 'For a Weyl semimetal with two nodes, separated along the [MATH]-axis and with a boundary, perpendicular to the [MATH]-axis, these edge states have the form of a chiral sheet, which disperses along the [MATH]-direction and extends between the projections of the Weyl node locations onto the surface BZ.', '1802.00450-1-7-9': 'This chiral sheet may be viewed as [MATH] (where [MATH] is the size of the system in the [MATH]-direction) one dimensional chiral modes, which upon a gauge transformation generate a contribution to the action, which is equal to [MATH], thus cancelling the non-gauge-invariant part of the bulk action, as described in detail in Ref. [CITATION].', '1802.00450-1-8-0': 'In this paper we explore the connection between the edge states and the quantum anomalies in the context of nodal line semimetals.', '1802.00450-1-8-1': '[CITATION] Nodal line semimetals possess surface states that have the form of a drumhead: the surface states are weakly dispersing and exist in a two dimensional (2D) region in the crystal momentum space, bounded by the projection of the bulk nodal line onto the surface BZ.', '1802.00450-1-8-2': '[CITATION] A natural question one may ask is if there exists a bulk anomaly, such as the chiral anomaly described above, which is associated with the drumhead surface states?', '1802.00450-1-9-0': 'We answer this question in the affirmative and demonstrate that the quantum anomaly, associated with the drumhead surface states of nodal line semimetals is closely related to the parity anomaly of (2+1)-dimensional relativistic Dirac fermions.', '1802.00450-1-9-1': 'Somewhat related ideas were put forward recently in Ref. Schnyder17, while an entirely different viewpoint was presented earlier in Ref. Hughes_linenode.', '1802.00450-1-9-2': 'Closely related phenomena in Dirac semimetals in an external magnetic field have been discussed in Ref. Burkov18-1.', '1802.00450-1-10-0': 'The rest of this paper is organized as follows.', '1802.00450-1-10-1': 'In Section [REF] we introduce a model of a thin three dimensional (3D) topological insulator (TI) film, doped with magnetic impurities, and analyze the behavior of this system as a function of the magnetization direction, in particular focusing on a quantum phase transition between two quantum anomalous Hall states of the film with opposite signs of the Hall conductivity.', '1802.00450-1-10-2': 'We demonstrate that at the critical point this system exhibits two massless Dirac fermions, separated in momentum space, and 1D edge states, connecting the two Dirac fermions.', '1802.00450-1-10-3': 'Stacking such layers in the growth direction, in Section [REF] we construct a 3D system, exhibiting a nodal line, which, in a close analogy to the 2D film case, exists at a critical point, separating two distinct Weyl semimetal phases with opposite signs of the anomalous Hall conductivity.', '1802.00450-1-10-4': 'This construction allows us to make an explicit connection between a nodal line in a 3D system and a pair of massless Dirac fermions, separated in momentum space, in a 2D film.', '1802.00450-1-10-5': 'We also make a connection between the drumhead surface state of the 3D nodal line semimetal and the edge states of the double 2D Dirac fermion system.', '1802.00450-1-10-6': 'In Section [REF] we describe analogous physics in PT-symmetric nodal line semimetals.', '1802.00450-1-10-7': 'Based on the results of Sections [REF], [REF] and [REF], in Section [REF] we construct a field theory, which describes anomalous electromagnetic response of nodal line semimetals.', '1802.00450-1-10-8': 'We demonstrate that a necessary ingredient in this field theory is a vielbein field determinant, which encodes the chirality of the Weyl fermions.', '1802.00450-1-10-9': 'The vielbein determinant changes sign at the quantum phase transition, at which the chirality of the Weyl fermions changes sign, requiring the appearance of a nodal line.', '1802.00450-1-10-10': 'We conclude with a brief discussion of the experimental observability of the proposed phenomena in Section [REF].', '1802.00450-1-11-0': '# Thin TI film in an external field', '1802.00450-1-12-0': 'We start by examining a simple, yet realistic, system, which will allow us to most clearly demonstrate the connection that exists between 2D Dirac fermions and 3D nodal lines.', '1802.00450-1-12-1': 'Let us consider a thin film of a 3D topological insulator material.', '1802.00450-1-12-2': 'Assuming the bulk material is a good insulator with a significant bandgap, the low-energy physics may be described by focusing on the 2D Dirac surface states only.', '1802.00450-1-12-3': 'The corresponding Hamiltonian, assuming a single 2D Dirac fermion per surface of the film at a time reversal invariant momentum (TRIM), which we will take to be at the [MATH]-point [MATH] of the BZ of the film for simplicity, is given by [EQUATION]', '1802.00450-1-12-4': 'Here [MATH] is the spin, [MATH] describes the top and bottom surface pseudospin degree of freedom, and [MATH] is the growth direction of the film.', '1802.00450-1-12-5': 'The [MATH] term describes tunneling between the top and the bottom surface, which is nonnegligible when the film is sufficiently thin, which means the thickness of a few unit cells in practice.', '1802.00450-1-12-6': 'The cubic in the crystal momentum term in the square brackets describes hexagonal warping of the surface states, present in Bi[MATH]Te[MATH] and related 3D TI compounds.', '1802.00450-1-12-7': '[CITATION] This term is often omitted when discussing TI surface states, but it (or its analogs in materials with non-hexagonal crystal symmetry) is always present and will play an important role in what follows.', '1802.00450-1-12-8': 'In particular this term breaks the continuous [MATH] symmetry with respect to arbitrary rotations around the [MATH]-axis, which the linearized Hamiltonian of the TI film would possess, down to the physical [MATH] symmetry.', '1802.00450-1-13-0': 'We now imagine doping the TI film with magnetic impurities, or placing it in proximity to a ferromagnetic insulator film, which induces a Zeeman spin-splitting term in Eq. [REF] [EQUATION] where the direction of the vector [MATH] is arbitrary and may be parametrized in the standard way using the polar and azimuthal angles as [MATH].', '1802.00450-1-13-1': 'This sort of a system has been created experimentally, most notably in the context of realizing the 2D quantum anomalous Hall effect (QAHE).', '1802.00450-1-13-2': '[CITATION]', '1802.00450-1-14-0': 'We want to understand the phase diagram of this system as a function of the magnitude and the direction of the spin-splitting field [MATH].', '1802.00450-1-15-0': 'Let us start from the case when [MATH] is along the growth direction of the film.', '1802.00450-1-15-1': 'A similarity transformation [EQUATION] brings the Hamiltonian to the form (the warping term may be ignored here) [EQUATION]', '1802.00450-1-15-2': 'This in turn may be brought to the following block-diagonal form by diagonalizing the [MATH] matrix [EQUATION] where [MATH] are the two eigenvalues of [MATH] and [MATH].', '1802.00450-1-15-3': 'Each of the two [MATH] Hamiltonians [MATH] describes a 2D Dirac fermion of "mass" [MATH].', '1802.00450-1-15-4': 'The [MATH] "mass" goes to zero and changes sign at [MATH], marking a quantum Hall plateau transition between a normal insulator state with [MATH] when [MATH] to a quantum anomalous Hall insulator state with [MATH] when [MATH].', '1802.00450-1-15-5': 'The critical point is a described by a massless 2D Dirac fermion, centered at the [MATH]-point (or any TRIM more generally) in the first BZ.', '1802.00450-1-15-6': '[CITATION]', '1802.00450-1-16-0': 'Now suppose we rotate [MATH] away from the [MATH]-axis.', '1802.00450-1-16-1': 'Here the response of the system depends crucially on the azimuthal angle [MATH], i.e. it depends on the plane in which the field is rotated away from the [MATH]-axis.', '1802.00450-1-16-2': 'This angular dependence exists due to the presence of the hexagonal warping term.', '1802.00450-1-16-3': 'Let us start by rotating the field in the [MATH]-plane, which corresponds to [MATH].', '1802.00450-1-16-4': 'We note that when the field is rotated all the way to the [MATH]-axis, which corresponds to [MATH], the Hamiltonian of the film Eq. [REF] possesses the following symmetry [EQUATION]', '1802.00450-1-16-5': 'Physically this symmetry is simply the mirror reflection symmetry with respect to the [MATH]-plane, which exists even in the presence of the hexagonal warping term since [MATH].', '1802.00450-1-16-6': 'This symmetry has important consequences for how the system responds to the field [MATH], when it is rotated in the [MATH]-plane and is thus normal to the mirror reflection plane [MATH] when [MATH].', '1802.00450-1-17-0': 'To see what happens we again block-diagonalize Eq. [REF] by first rotating the spin quantization axis along the direction of [MATH] and then performing the similarity transformation Eq. [REF].', '1802.00450-1-17-1': 'This brings the Hamiltonian to the form [EQUATION] where [EQUATION] and [MATH] as before.', '1802.00450-1-17-2': 'For all [MATH] the spectrum of [MATH] has a full gap.', '1802.00450-1-17-3': 'When [MATH], however, there are two Dirac band-touching points on the [MATH]-axis, whose coordinates are given by the solution of the equation [EQUATION] which gives [EQUATION] assuming [MATH].', '1802.00450-1-18-0': 'There are two different ways to understand this result.', '1802.00450-1-18-1': 'First, if we take both [MATH] and [MATH] to zero in Eq. [REF], it is clear that the field in the [MATH]-direction simply shifts the gapless top and bottom surface states of the TI film to different points on the [MATH]-axis with coordinates [MATH] in the BZ.', '1802.00450-1-18-2': 'When [MATH], which produces a NI state in the absence of the spin-splitting field, a transition to a semimetallic state with two Dirac points happens at [MATH], [CITATION] with the Dirac point locations given by Eq. [REF].', '1802.00450-1-18-3': 'Crucially, even when the hexagonal warping term is included, mirror reflection symmetry with respect to the [MATH]-plane forces it to vanish everywhere on the [MATH]-axis and protects the gaplessness of the Dirac points, even though time reversal symmetry is violated by the nonzero [MATH] field.', '1802.00450-1-19-0': 'Another way to view the reappearance of the gapless Dirac points when [MATH] is rotated along the [MATH]-direction, is revealed when one considers the behavior of the anomalous Hall conductivity of the TI film, [MATH].', '1802.00450-1-19-1': 'As discussed above, [MATH] when [MATH] and [MATH].', '1802.00450-1-19-2': 'If the field is rotated to the negative [MATH]-direction, the sign of the Hall conductivity will change to [MATH].', '1802.00450-1-19-3': 'The transition between the two quantized values, as the field is rotated from [MATH] to [MATH] direction, can in general happen at any value of the polar angle [MATH].', '1802.00450-1-19-4': 'However, when the field is rotated in the [MATH]-plane, the transition is forced to happen at [MATH] by the mirror reflection symmetry with respect to the [MATH]-plane that exists only at this angle.', '1802.00450-1-19-5': 'This is because this mirror symmetry also forces [MATH] to vanish at [MATH].', '1802.00450-1-19-6': '[CITATION] Indeed, if [MATH], mirror reflection with respect to the [MATH]-plane changes [MATH], while [MATH] does not change.', '1802.00450-1-19-7': 'Thus [MATH] in this case.', '1802.00450-1-19-8': 'The reason that two gapless Dirac points must appear at the transition is that [MATH] changes by [MATH] as [MATH] is rotated through [MATH], each 2D Dirac fermion contributing a quantum of Hall conductance [MATH] when its "mass" changes sign.', '1802.00450-1-19-9': 'The phase diagram of this system as a function of the angle [MATH] is shown in Fig. [REF].', '1802.00450-1-20-0': 'Let us now demonstrate explicitly that when the field is rotated in a different plane, which is not perpendicular to a mirror plane, the transition from [MATH] to [MATH] happens at a nonuniversal angle, which depends on details of the Hamiltonian of the TI film.', '1802.00450-1-20-1': 'Suppose we rotate [MATH] in the [MATH]-plane now, which corresponds to [MATH].', '1802.00450-1-20-2': 'The block-diagonalized Hamiltonian in this case is given by [EQUATION] where [EQUATION]', '1802.00450-1-20-3': 'The gap in this case may close only on the [MATH]-axis at two points with the coordinates [EQUATION] which means that [MATH] when [MATH].', '1802.00450-1-20-4': 'The critical polar angle [MATH], at which the plateau transition from [MATH] to [MATH] happens, satisfies the following equation [EQUATION]', '1802.00450-1-20-5': 'If [MATH], we may ignore [MATH] above and obtain [EQUATION]', '1802.00450-1-20-6': 'In this case the locations of the Dirac points on the [MATH]-axis are given by [EQUATION]', '1802.00450-1-20-7': 'Now let us go back to the case of the field rotated in the [MATH]-plane (the mirror symmetry that exists in this case will play a crucial role later, when we discuss the nodal line semimetals) and establish a connection between the anomalous response and the edge states, along the lines of the discussion in the Introduction.', '1802.00450-1-20-8': 'As established above, the anomalous Hall conductivity as a function of the polar angle [MATH] has the following form in this case [EQUATION]', '1802.00450-1-20-9': 'This singular dependence on the angle may be viewed as being a consequence of the parity anomaly [CITATION] of the two massless 2D Dirac fermions, which appear at [MATH] in the presence of the mirror reflection symmetry.', '1802.00450-1-20-10': 'The quantized [MATH] is associated with chiral 1D edge states, whose chirality determines the sign of [MATH].', '1802.00450-1-20-11': 'This means that at [MATH] the edge states must switch their chirality.', '1802.00450-1-20-12': 'This implies that the critical state with two Dirac points separated in momentum space, realized at [MATH], must itself have edge states, which connect the two Dirac points, as shown in Fig. [REF].', '1802.00450-1-21-0': 'These edge states may naively appear similar to the Fermi arcs of Weyl semimetals.', '1802.00450-1-21-1': 'However, there is an important difference between them.', '1802.00450-1-21-2': 'The Fermi arcs of 3D Weyl semimetals are chiral and connect the conduction and valence bands, apart from connecting the Weyl nodes.', '1802.00450-1-21-3': 'This means that both their location within the gap and their connection to the Weyl nodes are topologically protected.', '1802.00450-1-21-4': 'In contrast, the 1D edge states that connect the 2D Dirac points in Fig. [REF]b are not chiral and may be pushed out of the gap and made to merge with the bulk states by a sufficiently strong perturbation.', '1802.00450-1-21-5': 'The only topologically protected property they have is that they always connect to the bulk Dirac points.', '1802.00450-1-22-0': '# From 2D Dirac points to 3D nodal line', '1802.00450-1-23-0': 'We now extend the physics, discussed in Section [REF], to 3D, by making a multilayer stack of thin TI films.', '1802.00450-1-23-1': '[CITATION] The Hamiltonian of the stack is given by [EQUATION] where [MATH] and [MATH] describe tunneling between the 2D Dirac surface states within the same (S) or neighboring (D) TI layers and we will take the period of the heterostructure in the growth direction to be the unit of length.', '1802.00450-1-23-2': 'As before, let us assume that we may rotate the spin-splitting field [MATH] in two planes: [MATH]-plane, corresponding to [MATH] and [MATH]-plane, corresponding to [MATH].', '1802.00450-1-24-0': 'Let us start with the [MATH]-plane.', '1802.00450-1-24-1': 'The Hamiltonian is block-diagonalized by exactly the same transformations as in Section [REF] and we obtain [EQUATION] where [EQUATION] and [EQUATION]', '1802.00450-1-24-2': 'This is identical to the result obtained before in the single TI film case, Eqs. [REF] and [REF], except the constant tunneling amplitude [MATH] is replaced by the function [MATH].', '1802.00450-1-24-3': 'Thus we may use the results of the previous section by treating every value of [MATH] as parametrizing an effective 2D TI film system with the tunneling amplitude [MATH].', '1802.00450-1-24-4': 'In particular, for all [MATH], every value of [MATH] corresponds to a gapped 2D insulator with either [MATH] or [MATH], except when [MATH], which corresponds to a critical point between the two insulators.', '1802.00450-1-24-5': 'The solution of this equation gives the locations of two Weyl points on the [MATH]-axis, given by [EQUATION]', '1802.00450-1-24-6': 'This corresponds to the anomalous Hall conductivity, given by [EQUATION] which is obtained by summing contributions of effective 2D systems for every value of [MATH] in between the Weyl node locations.', '1802.00450-1-24-7': '[CITATION]', '1802.00450-1-25-0': 'When [MATH], the field is normal to a mirror plane [MATH] and at every [MATH] we get two 2D Dirac points, signifying a transition, at which the sign of [MATH] changes.', '1802.00450-1-25-1': 'The locations of the Dirac points at every [MATH] satisfy the equation [EQUATION]', '1802.00450-1-25-2': 'This equation clearly defines a closed curve in the [MATH]-plane in momentum space (the mirror plane).', '1802.00450-1-25-3': 'This closed curve is a nodal line, at which the two bands, corresponding to the eigenvalue [MATH] touch.', '1802.00450-1-25-4': 'This line exists because of the mirror symmetry with respect to the [MATH]-plane when [MATH] is perpendicular to this plane (i.e. is along the [MATH]-axis).', '1802.00450-1-25-5': 'The nodal line may thus be regarded as a critical state, separating two Weyl semimetals with opposite sign of the anomalous Hall conductivity.', '1802.00450-1-26-0': 'Now let us see what happens when [MATH] is rotated in the [MATH]-plane, so that the mirror symmetry is never present.', '1802.00450-1-26-1': 'In this case we have [EQUATION] where [EQUATION]', '1802.00450-1-26-2': 'The gap in this case may only close on the [MATH] plane, when the following equations are satisfied [EQUATION] i.e. the critical angle in this case is actually a function of [MATH].', '1802.00450-1-26-3': 'This implies that the nodal line does not exist.', '1802.00450-1-26-4': 'The critical point, at which the sign of the 3D Hall conductivity Eq. [REF] flips, also does not exist and is replaced by a smooth crossover, which is possible since a 3D Hall conductivity, unlike a 2D one, is not quantized.', '1802.00450-1-27-0': 'Specifically, the way the crossover occurs is as follows.', '1802.00450-1-27-1': 'The first of Eq. [REF] clearly implies that [MATH].', '1802.00450-1-27-2': 'Then both equations first acquire a real solution when [MATH], [MATH] and [MATH].', '1802.00450-1-27-3': 'This corresponds simply to the locations of the two Weyl points on the [MATH]-axis, [MATH] of Eq. [REF].', '1802.00450-1-27-4': 'At this moment two pairs of extra Weyl points appear at these locations, which carry a topological charge, equal to the topological charge of the original Weyl points.', '1802.00450-1-27-5': 'The topological charge of the original Weyl points themselves changes sign at this moment, so that the total charge at each [MATH] is still the same.', '1802.00450-1-27-6': 'As [MATH] increases past [MATH], the extra two pairs of Weyl nodes shift away from the [MATH]-axis and move towards the [MATH]-axis (see Fig. [REF]), mutually annihilating at a critical angle, given by the solution of the equation [EQUATION] which, taking the last term in Eq. [REF] to be small, so that the [MATH] term can be neglected, gives [EQUATION] where [EQUATION] is a parameter that determines the strength of the mirror symmetry violation by the hexagonal warping term.', '1802.00450-1-27-7': 'Namely, the anomalous Hall conductivity changes smoothly (see Fig. [REF]) from [MATH] to [MATH] in the interval [MATH], whose width is determined by the parameter [MATH]: [EQUATION] which is calculated as in Eq. [REF].', '1802.00450-1-28-0': 'Finally, let us come back to the case of the field rotated in the [MATH]-plane.', '1802.00450-1-28-1': 'It is well known that the nodal line state, realized in this system when the field is rotated along the [MATH]-axis, i.e. is normal to the [MATH] mirror plane, is associated with the drumhead surface states.', '1802.00450-1-28-2': '[CITATION] These surface states fill the projection of the nodal line onto the surface BZ and are dispersionless if particle-hole symmetry violating terms are ignored.', '1802.00450-1-28-3': 'As explained above, a 3D nodal line may be thought of as consisting of pairs of massless 2D Dirac fermions, separated in momentum space.', '1802.00450-1-28-4': 'We demonstrated in Section [REF] that such pairs of massless Dirac fermions lead to 1D edge states, which connect them.', '1802.00450-1-28-5': 'This follows directly from the fact that a pair of massless 2D Dirac fermions describes the critical point between two quantum anomalous Hall insulators with opposite sign of the Hall conductivity.', '1802.00450-1-28-6': 'This implies that the drumhead surface states of 3D nodal line semimetals may be regarded as families of such 1D edge states, parametrized by the second momentum component.', '1802.00450-1-28-7': 'This also implies that, alternatively, the drumhead surface states may be regarded as chiral Fermi arcs of one of the two Weyl semimetal phases, separated by the critical point, described by the nodal line, in the limit when the chirality vanishes and switches sign.', '1802.00450-1-29-0': 'By an exact analogy with the argument given at the end of Section [REF] we observe that the drumhead surface states, unlike the chiral Fermi arcs, are not fully topologically protected in the absence of the particle-hole symmetry.', '1802.00450-1-29-1': '[CITATION] They may be pushed out of the gap and made to merge with the bulk states, but the protected property that remains is that they always connect to the bulk nodal line.', '1802.00450-1-30-0': '# Nodal line in PT-symmetric semimetals', '1802.00450-1-31-0': 'So far we have been discussing the case of Weyl nodal lines, that is nodal lines in a material with broken time reversal symmetry, when two nondegenerate bands touch along the line.', '1802.00450-1-31-1': 'In this section we will demonstrate that similar ideas are applicable to the case of parity and time-reversal (PT) symmetric materials in the absence of the spin-orbit (SO) interactions.', '1802.00450-1-31-2': 'In this case all bands are doubly degenerate with respect to the spin and touching is between two pairs of doubly-degenerate bands.', '1802.00450-1-32-0': 'This case is sufficiently simple that it may be understood without reference to a specific model.', '1802.00450-1-32-1': 'We may start from the most general Hamiltonian of a time reversal and parity invariant system with four degress of freedom per unit cell [CITATION] [EQUATION] where [MATH] are five gamma-matrices, obeying the Clifford algebra, which are even under the product of parity and time reversal.', '1802.00450-1-32-2': 'Taking Pauli matrices [MATH] to represent the spin degree of freedom, and the eigenvalues of [MATH] of another set of Pauli matrices [MATH] to represent two orbital states in the unit cell, related to each other by parity (parity operator is [MATH]), they are given by [EQUATION]', '1802.00450-1-32-3': 'In the absence of the SO interactions, only [MATH] and [MATH] may be present in the Hamiltonian.', '1802.00450-1-32-4': 'Ignoring the term, proportional to the unit matrix, for simplicity, we obtain [EQUATION]', '1802.00450-1-32-5': 'Parity and time reversal symmetry require [MATH] and [MATH].', '1802.00450-1-32-6': 'The electronic structure, described by [MATH], exhibits a nodal line when [MATH].', '1802.00450-1-33-0': 'Now suppose we want to break time reversal symmetry, but keep the parity symmetry.', '1802.00450-1-33-1': 'This can be accomplished by adding a spin-splitting term [MATH], but the effect of this term in the absence of the SO interactions will be trivial, simply lifting the spin degeneracy.', '1802.00450-1-33-2': 'The only way to break time reversal symmetry but keep parity without involving the spin is to add a term [MATH] to the Hamiltonian, where [MATH].', '1802.00450-1-33-3': '[CITATION] Microscopically, this would arise from a magnetic flux within the unit cell of the crystal, leading to Aharonov-Bohm phases of the hopping amplitudes, as in the Haldane Chern insulator model.', '1802.00450-1-33-4': '[CITATION] To see the effect of this term on the nodal line, let us specify the functions [MATH].', '1802.00450-1-33-5': 'Let us take [EQUATION]', '1802.00450-1-33-6': 'The energy eigenvalues are [EQUATION]', '1802.00450-1-33-7': 'The two bands touch along a nodal line in the [MATH]-plane, given by the equation [EQUATION]', '1802.00450-1-33-8': 'Now let us add the time reversal breaking perturbation [EQUATION]', '1802.00450-1-33-9': 'The band energies become [EQUATION]', '1802.00450-1-33-10': 'We see that the nodal line has been gapped out except at two Weyl points on the [MATH]-axis at [MATH].', '1802.00450-1-33-11': 'Taking [MATH], the chirality of the two Weyl points is given by [EQUATION] and thus interchanges as [MATH] is tuned through zero.', '1802.00450-1-33-12': 'Thus the nodal line in PT-symmetric systems has the same meaning as in systems with SO interactions and broken time reversal symmetry: this is a critical state that separates two Weyl semimetal states with opposite signs of the anomalous Hall conductivity.', '1802.00450-1-34-0': '# Field theory of nodal line semimetals', '1802.00450-1-35-0': 'We will now summarize the above analysis of nodal line semimetals in terms of a field theory, which expresses their anomalous response, in the spirit of Eq. [REF], describing the chiral anomaly of Weyl semimetals.', '1802.00450-1-35-1': 'What we have established thus far is that the nodal line may be thought of as a state arising at a critical point between two Weyl semimetal states with interchanged chirality of the Weyl points.', '1802.00450-1-35-2': 'This means that in addition to the vector [MATH], which acts as a chiral gauge field, determining the separation of two Weyl nodes with fixed opposite chirality in momentum space, we need to take into account the possibility of chirality of the Weyl nodes changing sign, without changing their momentum-space location.', '1802.00450-1-35-3': 'This may be expressed with the help of the vielbein fields [MATH], which encode both the effective metric and the chirality, characterizing the Weyl points.', '1802.00450-1-35-4': 'A low-energy Hamiltonian of two Weyl nodes, separated in momentum space, may then be written as [EQUATION] which is a generalization of the ordinary massless Dirac Hamiltonian to a curved space-time with an arbitrary metric.', '1802.00450-1-35-5': 'The chirality-changing transition, with the appearance of a nodal line at the critical point, may be described as one of the vielbein vectors [MATH] flipping its direction and vanishing at the critical point.', '1802.00450-1-35-6': '[CITATION] For example, in the case of the magnetized TI-NI multilayer, described by Eq. [REF], we may take [EQUATION] which vanishes and changes direction at [MATH].', '1802.00450-1-35-7': 'In the case of a PT-symmetric semimetal, described by Eq. [REF], we may take [EQUATION] which again vanishes and changes direction when [MATH].', '1802.00450-1-35-8': 'Defining chirality as [EQUATION] where [MATH] is the determinant of the matrix [MATH], we may then write down the following action, which describes the anomalous response of this system [EQUATION] where we have taken the (fixed) direction of the vector [MATH], which determines the separation between the Weyl nodes, to be the [MATH]-direction.', '1802.00450-1-35-9': 'This action bears a strong resemblance to the Chern-Simons action [EQUATION] which expresses the parity anomaly of a 2D Dirac fermion of mass [MATH].', '1802.00450-1-35-10': 'This is not unexpected of course, given the connection between 3D nodal line and 2D Dirac fermions, which was established in Sections [REF] and [REF].', '1802.00450-1-36-0': 'The information, contained in Eq. [REF] may be stated as follows: this equation is telling us that a massless 2D Dirac fermion describes a direct transition between nsulators with [MATH] and [MATH].', '1802.00450-1-36-1': 'Analogously, Eq. [REF] is telling us that the nodal line, which appears when one of the vielbein vectors [MATH] vanishes, describes a direct transition between two Weyl semimetal states with [MATH].', '1802.00450-1-37-0': '# Discussion and conclusions', '1802.00450-1-38-0': 'In this paper we have demonstrated that the anomalous response of 3D nodal line semimetals is closely related to the parity anomaly of 2D Dirac fermions.', '1802.00450-1-38-1': 'The role of the mass of a 2D Dirac fermion, whose sign enters into the topological Chern-Simons action for the electromagnetic field, when the fermions are integrated out, is played by the determinant of the vielbein matrix [MATH].', '1802.00450-1-39-0': 'Perhaps the most straightforward way to observe these phenomena, given the currently available materials, is to look at magnetic response of type-II Dirac semimetals.', '1802.00450-1-39-1': '[CITATION] Type-II Dirac semimetals possess one or several symmetry-related Dirac points at TRIM in the first BZ.', '1802.00450-1-39-2': 'Material realizations include TlBi(S[MATH] Se[MATH], [CITATION] (Bi[MATH]In[MATH]Se[MATH], [CITATION] and ZrTe[MATH].', '1802.00450-1-39-3': '[CITATION] As was demonstrated in Ref. [CITATION], magnetic response of a type-II Dirac point is always strongly anisotropic: while one of the Zeeman field components acts as a chiral gauge field, splitting the Dirac point into two Weyl points, the other two components instead create nodal lines.', '1802.00450-1-39-4': 'The problem then maps exactly onto the magnetic multilayer system, described in Section [REF] and exactly the same conclusions follow.', '1802.00450-1-39-5': 'The anomaly may then be detected as a step-function-like singularity of the anomalous Hall conductivity of the Dirac semimetal in the presence of an applied magnetic field, as the field is rotated.', '1802.00450-1-39-6': 'The anomalous Hall conductivity in this case is defined as part of the Hall conductivity, which arises from the Zeeman response.', '1802.00450-1-39-7': 'It may be obtained by subtracting off the linear high-field part of the Hall resistivity, as the anomalous Hall signal is usually isolated.', '1802.00450-1-40-0': 'Another possibility is the recently discovered magnetic Weyl semimetal Co[MATH]Sn[MATH]S[MATH].', '1802.00450-1-40-1': '[CITATION] In this material, six pairs of Weyl nodes arise out of nodal lines, gapped by the SO interactions, as revealed by the electronic structure calculations.', '1802.00450-1-40-2': '[CITATION] Thus Co[MATH]Sn[MATH]S[MATH] may naturally reside close to the phase transition at which the sign of the anomalous Hall conductivity changes, however a detailed investigation of how it responds to rotating the direction of magnetization is necessary to understand if this really is the case.', '1802.00450-1-41-0': 'In conclusion, we have presented a theory of anomalous response (quantum anomaly) in nodal line semimetals, which can be related to the existence of drumhead surface states in these systems.', '1802.00450-1-41-1': 'We have shown that both the surface states and the anomalous response are closely analogous to the parity anomaly of (2+1)-dimensional relativistic Dirac fermions, which in the condensed matter physics context is realized as the 2D QAHE.', '1802.00450-1-41-2': 'We have derived a field theory, describing the anomalous response of nodal semimetals, and shown that a crucial ingredient in this field theory is the sign of the determinant of the vielbein fields, describing both the 3D Weyl fermion chirality and the effective low-energy metric, which emerges in Weyl semimetals.', '1802.00450-1-41-3': 'This sign changes at a critical point at which one of the three vielbein vectors vanishes, leading to the emergence of a nodal line.', '1802.00450-1-42-0': 'Financial support was provided by NSERC of Canada.'}

{'1802.00450-2-0-0': 'Topological semimetals is a new class of condensed matter systems with nontrivial electronic structure topology.', '1802.00450-2-0-1': 'Their unusual observable properties may often be understood in terms of quantum anomalies.', '1802.00450-2-0-2': 'In particular, Weyl and Dirac semimetals, which have point band touching nodes, are characterized by the chiral anomaly, which leads to the Fermi arc surface states, anomalous Hall effect, negative longitudinal magnetoresistance and planar Hall effect.', '1802.00450-2-0-3': 'In this paper we explore analogous phenomena in nodal line semimetals.', '1802.00450-2-0-4': 'We demonstrate that such semimetals realize a three dimensional analog of the parity anomaly, which is a known property of two dimensional Dirac semimetals arising, for example, on the surface of a three dimensional topological insulator.', '1802.00450-2-0-5': 'We relate one of the characteristic properties of nodal line semimetals, namely the drumhead surface states, to this anomaly, and derive the field theory, which encodes the corresponding anomalous response.', '1802.00450-2-1-0': '# Introduction', '1802.00450-2-2-0': 'Topological (semi)metal is a new phase of matter, which is characterized by a nontrivial electronic structure topology, yet is not an insulator with a gap in the spectrum.', '1802.00450-2-2-1': '[CITATION] Integer momentum space invariants, which characterize topologically nontrivial states of matter, are defined in this case on the Fermi surface, rather than in the whole Brillouin zone (BZ).', '1802.00450-2-2-2': '[CITATION] Such Fermi surface invariants arise from singularities in the electronic structure, in the simplest case isolated points, at which different bands touch.', '1802.00450-2-2-3': 'The significance of such points was emphasized early on by Volovik, [CITATION] and also pointed out by Murakami.', '1802.00450-2-2-4': '[CITATION]', '1802.00450-2-3-0': 'Nontrivial electronic structure topology has both spectroscopic manifestations, in the form of localized edge states, and manifestations in response, in the form of quantized, or just insensitive to perturbations and microscopic details, transport or other response properties.', '1802.00450-2-3-1': 'In the context of Weyl and Dirac semimetals this topological response may have different manifestations, such as negative longitudinal magnetoresistance, [CITATION] giant planar Hall effect, [CITATION] and anomalous Hall effect.', '1802.00450-2-3-2': '[CITATION]', '1802.00450-2-4-0': 'The edge states and the topological response are closely related and one way to understand this relation is in terms of the concept of quantum anomalies.', '1802.00450-2-4-1': 'Quantum anomaly is often described as violation of a classical symmetry (i.e. symmetry of the action) by quantum effects in the presence of external (say electromagnetic) fields.', '1802.00450-2-4-2': "[CITATION] This violation of a symmetry then leads to nonconservation of a current, which should be conserved classically by Noether's theorem.", '1802.00450-2-4-3': 'This viewpoint, however, is strictly applicable only to quantum anomalies in the particle physics context, since condensed matter systems typically do not possess the corresponding symmetries, such as the chiral symmetry, to begin with.', '1802.00450-2-5-0': 'Another, more useful in the condensed matter context, way to understand the anomalies is in terms of failure of gauge invariance.', '1802.00450-2-5-1': 'Current nonconservation may be represented in terms of an anomalous term in the action for the electromagnetic (or some other) field, which is not gauge invariant in the presence of a boundary.', '1802.00450-2-5-2': 'What restores the overall gauge invariance is the contribution of edge states, which precisely cancels the gauge invariance violating part of the bulk action.', '1802.00450-2-5-3': '[CITATION]', '1802.00450-2-6-0': 'It is instructive to see how this works in the case of the simplest topological metal system: a Weyl semimetal with two nodes.', '1802.00450-2-6-1': '[CITATION] The chiral anomaly in this system may be expressed in terms of the following action for the electromagnetic field [CITATION] [EQUATION] where [MATH] describes the momentum space separation between the Weyl nodes ([MATH]) along the [MATH]-axis and [MATH] units are used here and throughout this paper, except in some of the final formulas.', '1802.00450-2-6-2': 'We have also ignored the chiral magnetic effect [CITATION] and related phenomena in Eq. [REF] for simplicity (these effects exist only away from equilibrium and require certain modifications of Eq. [REF], which we do not want to discuss here; they have been discussed in detail, for example in Refs. [CITATION]).', '1802.00450-2-7-0': 'Consider a gauge transformation [MATH].', '1802.00450-2-7-1': 'This changes the action in Eq. [REF] by [EQUATION] [MATH] vanishes identically in the absence of boundaries (i.e. in a system with periodic boundary conditions), but does not vanish in a system with a boundary.', '1802.00450-2-7-2': 'Indeed, suppose [MATH], where [MATH] is the Heaviside step function.', '1802.00450-2-7-3': 'This may represent, for example, a contact between a Weyl semimetal in the [MATH] half-space and vacuum in the [MATH] half-space.', '1802.00450-2-7-4': 'Then we obtain [EQUATION]', '1802.00450-2-7-5': 'Eq. [REF] clearly does not vanish in general and this means that Eq. [REF] fails to be gauge invariant in the presence of a boundary.', '1802.00450-2-7-6': 'This failure of gauge invariance is a symptom that Eq. [REF] is incomplete.', '1802.00450-2-7-7': 'Indeed, what it does not take into account is the Fermi arc edge states, which are present in the Weyl semimetal system, described by Eq. [REF].', '1802.00450-2-7-8': 'For a Weyl semimetal with two nodes, separated along the [MATH]-axis and with a boundary, perpendicular to the [MATH]-axis, these edge states have the form of a chiral sheet, which disperses along the [MATH]-direction and extends between the projections of the Weyl node locations onto the surface BZ.', '1802.00450-2-7-9': 'This chiral sheet may be viewed as [MATH] (where [MATH] is the size of the system in the [MATH]-direction) one dimensional chiral modes, which upon a gauge transformation generate a contribution to the action, which is equal to [MATH], thus cancelling the non-gauge-invariant part of the bulk action, as described in detail in Ref. [CITATION].', '1802.00450-2-8-0': 'In this paper we explore the connection between the edge states and the quantum anomalies in the context of nodal line semimetals.', '1802.00450-2-8-1': '[CITATION] Nodal line semimetals possess surface states that have the form of a drumhead: the surface states are weakly dispersing and exist in a two dimensional (2D) region in the crystal momentum space, bounded by the projection of the bulk nodal line onto the surface BZ.', '1802.00450-2-8-2': '[CITATION] A natural question one may ask is if there exists a bulk anomaly, such as the chiral anomaly described above, which is associated with the drumhead surface states?', '1802.00450-2-9-0': 'We answer this question in the affirmative and demonstrate that the quantum anomaly, associated with the drumhead surface states of nodal line semimetals is closely related to the parity anomaly of (2+1)-dimensional relativistic Dirac fermions.', '1802.00450-2-9-1': 'Somewhat related ideas were put forward recently in Ref. Schnyder17, while an entirely different viewpoint was presented earlier in Ref. Hughes_linenode.', '1802.00450-2-9-2': 'Closely related phenomena in Dirac semimetals in an external magnetic field have been discussed in Ref. Burkov18-1.', '1802.00450-2-10-0': 'The rest of this paper is organized as follows.', '1802.00450-2-10-1': 'In Section [REF] we introduce a model of a thin three dimensional (3D) topological insulator (TI) film, doped with magnetic impurities, and analyze the behavior of this system as a function of the magnetization direction, in particular focusing on a quantum phase transition between two quantum anomalous Hall states of the film with opposite signs of the Hall conductivity.', '1802.00450-2-10-2': 'We demonstrate that at the critical point this system exhibits two massless Dirac fermions, separated in momentum space, and 1D edge states, connecting the two Dirac fermions.', '1802.00450-2-10-3': 'Stacking such layers in the growth direction, in Section [REF] we construct a 3D system, exhibiting a nodal line, which, in a close analogy to the 2D film case, exists at a critical point, separating two distinct Weyl semimetal phases with opposite signs of the anomalous Hall conductivity.', '1802.00450-2-10-4': 'This construction allows us to make an explicit connection between a nodal line in a 3D system and a pair of massless Dirac fermions, separated in momentum space, in a 2D film.', '1802.00450-2-10-5': 'We also make a connection between the drumhead surface state of the 3D nodal line semimetal and the edge states of the double 2D Dirac fermion system.', '1802.00450-2-10-6': 'In Section [REF] we describe analogous physics in PT-symmetric nodal line semimetals.', '1802.00450-2-10-7': 'Based on the results of Sections [REF], [REF] and [REF], in Section [REF] we construct a field theory, which describes anomalous electromagnetic response of nodal line semimetals.', '1802.00450-2-10-8': 'We demonstrate that a necessary ingredient in this field theory is a vielbein field determinant, which encodes the chirality of the Weyl fermions.', '1802.00450-2-10-9': 'The vielbein determinant changes sign at the quantum phase transition, at which the chirality of the Weyl fermions changes sign, requiring the appearance of a nodal line.', '1802.00450-2-10-10': 'We conclude with a brief discussion of the experimental observability of the proposed phenomena in Section [REF].', '1802.00450-2-11-0': '# Thin TI film in an external field', '1802.00450-2-12-0': 'We start by examining a simple, yet realistic, system, which will allow us to most clearly demonstrate the connection that exists between 2D Dirac fermions and 3D nodal lines.', '1802.00450-2-12-1': 'Let us consider a thin film of a 3D topological insulator material.', '1802.00450-2-12-2': 'Assuming the bulk material is a good insulator with a significant bandgap, the low-energy physics may be described by focusing on the 2D Dirac surface states only.', '1802.00450-2-12-3': 'The corresponding Hamiltonian, assuming a single 2D Dirac fermion per surface of the film at a time reversal invariant momentum (TRIM), which we will take to be at the [MATH]-point [MATH] of the BZ of the film for simplicity, is given by [EQUATION]', '1802.00450-2-12-4': 'Here [MATH] is the spin, [MATH] describes the top and bottom surface pseudospin degree of freedom, and [MATH] is the growth direction of the film.', '1802.00450-2-12-5': 'The [MATH] term describes tunneling between the top and the bottom surface, which is nonnegligible when the film is sufficiently thin, which means the thickness of a few unit cells in practice.', '1802.00450-2-12-6': 'The cubic in the crystal momentum term in the square brackets describes hexagonal warping of the surface states, present in Bi[MATH]Te[MATH] and related 3D TI compounds.', '1802.00450-2-12-7': '[CITATION] This term is often omitted when discussing TI surface states, but it (or its analogs in materials with non-hexagonal crystal symmetry) is always present and will play an important role in what follows.', '1802.00450-2-12-8': 'In particular this term breaks the continuous [MATH] symmetry with respect to arbitrary rotations around the [MATH]-axis, which the linearized Hamiltonian of the TI film would possess, down to the physical [MATH] symmetry.', '1802.00450-2-13-0': 'We now imagine doping the TI film with magnetic impurities, or placing it in proximity to a ferromagnetic insulator film, which induces a Zeeman spin-splitting term in Eq. [REF] [EQUATION] where the direction of the vector [MATH] is arbitrary and may be parametrized in the standard way using the polar and azimuthal angles as [MATH].', '1802.00450-2-13-1': 'This sort of a system has been created experimentally, most notably in the context of realizing the 2D quantum anomalous Hall effect (QAHE).', '1802.00450-2-13-2': '[CITATION]', '1802.00450-2-14-0': 'We want to understand the phase diagram of this system as a function of the magnitude and the direction of the spin-splitting field [MATH].', '1802.00450-2-15-0': 'Let us start from the case when [MATH] is along the growth direction of the film.', '1802.00450-2-15-1': 'A similarity transformation [EQUATION] brings the Hamiltonian to the form (the warping term may be ignored here) [EQUATION]', '1802.00450-2-15-2': 'This in turn may be brought to the following block-diagonal form by diagonalizing the [MATH] matrix [EQUATION] where [MATH] are the two eigenvalues of [MATH] and [MATH].', '1802.00450-2-15-3': 'Each of the two [MATH] Hamiltonians [MATH] describes a 2D Dirac fermion of "mass" [MATH].', '1802.00450-2-15-4': 'The [MATH] "mass" goes to zero and changes sign at [MATH], marking a quantum Hall plateau transition between a normal insulator state with [MATH] when [MATH] to a quantum anomalous Hall insulator state with [MATH] when [MATH].', '1802.00450-2-15-5': 'The critical point is a described by a massless 2D Dirac fermion, centered at the [MATH]-point (or any TRIM more generally) in the first BZ.', '1802.00450-2-15-6': '[CITATION]', '1802.00450-2-16-0': 'Now suppose we rotate [MATH] away from the [MATH]-axis.', '1802.00450-2-16-1': 'Here the response of the system depends crucially on the azimuthal angle [MATH], i.e. it depends on the plane in which the field is rotated away from the [MATH]-axis.', '1802.00450-2-16-2': 'This angular dependence exists due to the presence of the hexagonal warping term.', '1802.00450-2-16-3': 'Let us start by rotating the field in the [MATH]-plane, which corresponds to [MATH].', '1802.00450-2-16-4': 'We note that when the field is rotated all the way to the [MATH]-axis, which corresponds to [MATH], the Hamiltonian of the film Eq. [REF] possesses the following symmetry [EQUATION]', '1802.00450-2-16-5': 'Physically this symmetry is simply the mirror reflection symmetry with respect to the [MATH]-plane, which exists even in the presence of the hexagonal warping term since [MATH].', '1802.00450-2-16-6': 'This symmetry has important consequences for how the system responds to the field [MATH], when it is rotated in the [MATH]-plane and is thus normal to the mirror reflection plane [MATH] when [MATH].', '1802.00450-2-17-0': 'To see what happens we again block-diagonalize Eq. [REF] by first rotating the spin quantization axis along the direction of [MATH] and then performing the similarity transformation Eq. [REF].', '1802.00450-2-17-1': 'This brings the Hamiltonian to the form [EQUATION] where [EQUATION] and [MATH] as before.', '1802.00450-2-17-2': 'For all [MATH] the spectrum of [MATH] has a full gap.', '1802.00450-2-17-3': 'When [MATH], however, there are two Dirac band-touching points on the [MATH]-axis, whose coordinates are given by the solution of the equation [EQUATION] which gives [EQUATION] assuming [MATH].', '1802.00450-2-18-0': 'There are two different ways to understand this result.', '1802.00450-2-18-1': 'First, if we take both [MATH] and [MATH] to zero in Eq. [REF], it is clear that the field in the [MATH]-direction simply shifts the gapless top and bottom surface states of the TI film to different points on the [MATH]-axis with coordinates [MATH] in the BZ.', '1802.00450-2-18-2': 'When [MATH], which produces a NI state in the absence of the spin-splitting field, a transition to a semimetallic state with two Dirac points happens at [MATH], [CITATION] with the Dirac point locations given by Eq. [REF].', '1802.00450-2-18-3': 'Crucially, even when the hexagonal warping term is included, mirror reflection symmetry with respect to the [MATH]-plane forces it to vanish everywhere on the [MATH]-axis and protects the gaplessness of the Dirac points, even though time reversal symmetry is violated by the nonzero [MATH] field.', '1802.00450-2-19-0': 'Another way to view the reappearance of the gapless Dirac points when [MATH] is rotated along the [MATH]-direction, is revealed when one considers the behavior of the anomalous Hall conductivity of the TI film, [MATH].', '1802.00450-2-19-1': 'As discussed above, [MATH] when [MATH] and [MATH].', '1802.00450-2-19-2': 'If the field is rotated to the negative [MATH]-direction, the sign of the Hall conductivity will change to [MATH].', '1802.00450-2-19-3': 'The transition between the two quantized values, as the field is rotated from [MATH] to [MATH] direction, can in general happen at any value of the polar angle [MATH].', '1802.00450-2-19-4': 'However, when the field is rotated in the [MATH]-plane, the transition is forced to happen at [MATH] by the mirror reflection symmetry with respect to the [MATH]-plane that exists only at this angle.', '1802.00450-2-19-5': 'This is because this mirror symmetry also forces [MATH] to vanish at [MATH].', '1802.00450-2-19-6': '[CITATION] Indeed, if [MATH], mirror reflection with respect to the [MATH]-plane changes [MATH], while [MATH] does not change.', '1802.00450-2-19-7': 'Thus [MATH] in this case.', '1802.00450-2-19-8': 'The reason that two gapless Dirac points must appear at the transition is that [MATH] changes by [MATH] as [MATH] is rotated through [MATH], each 2D Dirac fermion contributing a quantum of Hall conductance [MATH] when its "mass" changes sign.', '1802.00450-2-19-9': 'The phase diagram of this system as a function of the angle [MATH] is shown in Fig. [REF].', '1802.00450-2-20-0': 'Let us now demonstrate explicitly that when the field is rotated in a different plane, which is not perpendicular to a mirror plane, the transition from [MATH] to [MATH] happens at a nonuniversal angle, which depends on details of the Hamiltonian of the TI film.', '1802.00450-2-20-1': 'Suppose we rotate [MATH] in the [MATH]-plane now, which corresponds to [MATH].', '1802.00450-2-20-2': 'The block-diagonalized Hamiltonian in this case is given by [EQUATION] where [EQUATION]', '1802.00450-2-20-3': 'The gap in this case may close only on the [MATH]-axis at two points with the coordinates [EQUATION] which means that [MATH] when [MATH].', '1802.00450-2-20-4': 'The critical polar angle [MATH], at which the plateau transition from [MATH] to [MATH] happens, satisfies the following equation [EQUATION]', '1802.00450-2-20-5': 'If [MATH], we may ignore [MATH] above and obtain [EQUATION]', '1802.00450-2-20-6': 'In this case the locations of the Dirac points on the [MATH]-axis are given by [EQUATION]', '1802.00450-2-20-7': 'Now let us go back to the case of the field rotated in the [MATH]-plane (the mirror symmetry that exists in this case will play a crucial role later, when we discuss the nodal line semimetals) and establish a connection between the anomalous response and the edge states, along the lines of the discussion in the Introduction.', '1802.00450-2-20-8': 'As established above, the anomalous Hall conductivity as a function of the polar angle [MATH] has the following form in this case [EQUATION]', '1802.00450-2-20-9': 'This singular dependence on the angle may be viewed as being a consequence of the parity anomaly [CITATION] of the two massless 2D Dirac fermions, which appear at [MATH] in the presence of the mirror reflection symmetry.', '1802.00450-2-20-10': 'The quantized [MATH] is associated with chiral 1D edge states, whose chirality determines the sign of [MATH].', '1802.00450-2-20-11': 'This means that at [MATH] the edge states must switch their chirality.', '1802.00450-2-20-12': 'This implies that the critical state with two Dirac points separated in momentum space, realized at [MATH], must itself have edge states, which connect the two Dirac points, as shown in Fig. [REF].', '1802.00450-2-21-0': 'These edge states may naively appear similar to the Fermi arcs of Weyl semimetals.', '1802.00450-2-21-1': 'However, there is an important difference between them.', '1802.00450-2-21-2': 'The Fermi arcs of 3D Weyl semimetals are chiral and connect the conduction and valence bands, apart from connecting the Weyl nodes.', '1802.00450-2-21-3': 'This means that both their location within the gap and their connection to the Weyl nodes are topologically protected.', '1802.00450-2-21-4': 'In contrast, the 1D edge states that connect the 2D Dirac points in Fig. [REF]b are not chiral and may be pushed out of the gap and made to merge with the bulk states by a sufficiently strong perturbation.', '1802.00450-2-21-5': 'The only topologically protected property they have is that they always connect to the bulk Dirac points.', '1802.00450-2-22-0': '# From 2D Dirac points to 3D nodal line', '1802.00450-2-23-0': 'We now extend the physics, discussed in Section [REF], to 3D, by making a multilayer stack of thin TI films.', '1802.00450-2-23-1': '[CITATION] The Hamiltonian of the stack is given by [EQUATION] where [MATH] and [MATH] describe tunneling between the 2D Dirac surface states within the same (S) or neighboring (D) TI layers and we will take the period of the heterostructure in the growth direction to be the unit of length.', '1802.00450-2-23-2': 'As before, let us assume that we may rotate the spin-splitting field [MATH] in two planes: [MATH]-plane, corresponding to [MATH] and [MATH]-plane, corresponding to [MATH].', '1802.00450-2-23-3': 'The difference between the two planes is that [MATH] is a mirror-symmetric plane, while [MATH] is not, but is normal to a mirror symmetric plane instead.', '1802.00450-2-24-0': '## Mirror symmetric case', '1802.00450-2-25-0': 'Let us start with the case of the field rotated in the [MATH]-plane.', '1802.00450-2-25-1': 'In the case, the multilayer will have mirror symmetry when the field is along the [MATH]-axis and is thus perpendicular to a mirror plane ([MATH]-plane).', '1802.00450-2-25-2': 'The Hamiltonian is block-diagonalized by exactly the same transformations as in Section [REF] and we obtain [EQUATION] where [EQUATION] and [EQUATION]', '1802.00450-2-25-3': 'This is identical to the result obtained before in the single TI film case, Eqs. [REF] and [REF], except the constant tunneling amplitude [MATH] is replaced by the function [MATH].', '1802.00450-2-25-4': 'Thus we may use the results of the previous section by treating every value of [MATH] as parametrizing an effective 2D TI film system with the tunneling amplitude [MATH].', '1802.00450-2-25-5': 'In particular, for all [MATH], every value of [MATH] corresponds to a gapped 2D insulator with either [MATH] or [MATH], except when [MATH], which corresponds to a critical point between the two insulators.', '1802.00450-2-25-6': 'The solution of this equation gives the locations of two Weyl points on the [MATH]-axis, given by [EQUATION]', '1802.00450-2-25-7': 'This corresponds to the anomalous Hall conductivity, given by [EQUATION] which is obtained by summing contributions of effective 2D systems for every value of [MATH] in between the Weyl node locations.', '1802.00450-2-25-8': '[CITATION]', '1802.00450-2-26-0': 'When [MATH], the field is normal to a mirror plane [MATH] and at every [MATH] we get two 2D Dirac points, signifying a transition, at which the sign of [MATH] changes.', '1802.00450-2-26-1': 'The locations of the Dirac points at every [MATH] satisfy the equation [EQUATION]', '1802.00450-2-26-2': 'This equation clearly defines a closed curve in the [MATH]-plane in momentum space (the mirror plane).', '1802.00450-2-26-3': 'This closed curve is a nodal line, at which the two bands, corresponding to the eigenvalue [MATH] touch.', '1802.00450-2-26-4': 'This line exists because of the mirror symmetry with respect to the [MATH]-plane when [MATH] is perpendicular to this plane (i.e. is along the [MATH]-axis).', '1802.00450-2-26-5': 'The nodal line may thus be regarded as a critical state, separating two Weyl semimetals with opposite sign of the anomalous Hall conductivity.', '1802.00450-2-27-0': '## Case without mirror symmetry', '1802.00450-2-28-0': 'Now let us see what happens when [MATH] is rotated in a mirror ([MATH]) plane, so that the mirror symmetry is never present, since it is always broken by the field.', '1802.00450-2-28-1': 'In this case we have [EQUATION] where [EQUATION]', '1802.00450-2-28-2': 'The gap in this case may only close on the [MATH] plane, when the following equations are satisfied [EQUATION] i.e. the critical angle in this case is actually a function of [MATH].', '1802.00450-2-28-3': 'This implies that the nodal line does not exist.', '1802.00450-2-28-4': 'The critical point, at which the sign of the 3D Hall conductivity Eq. [REF] flips, also does not exist and is replaced by a smooth crossover, which is possible since a 3D Hall conductivity, unlike a 2D one, is not quantized.', '1802.00450-2-29-0': 'Specifically, the way the crossover occurs is as follows.', '1802.00450-2-29-1': 'The first of Eq. [REF] clearly implies that [MATH].', '1802.00450-2-29-2': 'Then both equations first acquire a real solution when [MATH], [MATH] and [MATH].', '1802.00450-2-29-3': 'This corresponds simply to the locations of the two Weyl points on the [MATH]-axis, [MATH] of Eq. [REF].', '1802.00450-2-29-4': 'At this moment two pairs of extra Weyl points appear at these locations, which carry a topological charge, equal to the topological charge of the original Weyl points.', '1802.00450-2-29-5': 'The topological charge of the original Weyl points themselves changes sign at this moment, so that the total charge at each [MATH] is still the same.', '1802.00450-2-29-6': 'As [MATH] increases past [MATH], the extra two pairs of Weyl nodes shift away from the [MATH]-axis and move towards the [MATH]-axis (see Fig. [REF]), mutually annihilating at a critical angle, given by the solution of the equation [EQUATION] which, taking the last term in Eq. [REF] to be small, so that the [MATH] term can be neglected, gives [EQUATION] where [EQUATION] is a parameter that determines the strength of the mirror symmetry violation by the hexagonal warping term.', '1802.00450-2-29-7': 'Namely, the anomalous Hall conductivity changes smoothly (see Fig. [REF]) from [MATH] to [MATH] in the interval [MATH], whose width is determined by the parameter [MATH]: [EQUATION] which is calculated as in Eq. [REF].', '1802.00450-2-30-0': '## Drumhead surface states', '1802.00450-2-31-0': 'Finally, let us come back to the case of the field rotated in the [MATH]-plane.', '1802.00450-2-31-1': 'It is well known that the nodal line state, realized in this system when the field is rotated along the [MATH]-axis, i.e. is normal to the [MATH] mirror plane, is associated with the drumhead surface states.', '1802.00450-2-31-2': '[CITATION] These surface states fill the projection of the nodal line onto the surface BZ and are dispersionless if particle-hole symmetry violating terms are ignored.', '1802.00450-2-31-3': 'As explained above, a 3D nodal line may be thought of as consisting of pairs of massless 2D Dirac fermions, separated in momentum space.', '1802.00450-2-31-4': 'We demonstrated in Section [REF] that such pairs of massless Dirac fermions lead to 1D edge states, which connect them.', '1802.00450-2-31-5': 'This follows directly from the fact that a pair of massless 2D Dirac fermions describes the critical point between two quantum anomalous Hall insulators with opposite sign of the Hall conductivity.', '1802.00450-2-31-6': 'This implies that the drumhead surface states of 3D nodal line semimetals may be regarded as families of such 1D edge states, parametrized by the second momentum component.', '1802.00450-2-31-7': 'This also implies that, alternatively, the drumhead surface states may be regarded as chiral Fermi arcs of one of the two Weyl semimetal phases, separated by the critical point, described by the nodal line, in the limit when the chirality vanishes and switches sign.', '1802.00450-2-32-0': 'By an exact analogy with the argument given at the end of Section [REF] we observe that the drumhead surface states, unlike the chiral Fermi arcs, are not fully topologically protected in the absence of the particle-hole symmetry.', '1802.00450-2-32-1': '[CITATION] They may be pushed out of the gap and made to merge with the bulk states, but the protected property that remains is that they always connect to the bulk nodal line.', '1802.00450-2-33-0': '# Nodal line in PT-symmetric semimetals', '1802.00450-2-34-0': 'So far we have been discussing the case of Weyl nodal lines, that is nodal lines in a material with broken time reversal symmetry, when two nondegenerate bands touch along the line.', '1802.00450-2-34-1': 'In this section we will demonstrate that similar ideas are applicable to the case of parity and time-reversal (PT) symmetric materials in the absence of the spin-orbit (SO) interactions.', '1802.00450-2-34-2': 'In this case all bands are doubly degenerate with respect to the spin and touching is between two pairs of doubly-degenerate bands.', '1802.00450-2-35-0': 'This case is sufficiently simple that it may be understood without reference to a specific model.', '1802.00450-2-35-1': 'We may start from the most general Hamiltonian of a time reversal and parity invariant system with four degress of freedom per unit cell [CITATION] [EQUATION] where [MATH] are five gamma-matrices, obeying the Clifford algebra, which are even under the product of parity and time reversal.', '1802.00450-2-35-2': 'Taking Pauli matrices [MATH] to represent the spin degree of freedom, and the eigenvalues of [MATH] of another set of Pauli matrices [MATH] to represent two orbital states in the unit cell, related to each other by parity (parity operator is [MATH]), they are given by [EQUATION]', '1802.00450-2-35-3': 'In the absence of the SO interactions, only [MATH] and [MATH] may be present in the Hamiltonian.', '1802.00450-2-35-4': 'Ignoring the term, proportional to the unit matrix, for simplicity, we obtain [EQUATION]', '1802.00450-2-35-5': 'Parity and time reversal symmetry require [MATH] and [MATH].', '1802.00450-2-35-6': 'The electronic structure, described by [MATH], exhibits a nodal line when [MATH].', '1802.00450-2-36-0': 'Now suppose we want to break time reversal symmetry, but keep the parity symmetry.', '1802.00450-2-36-1': 'This can be accomplished by adding a spin-splitting term [MATH], but the effect of this term in the absence of the SO interactions will be trivial, simply lifting the spin degeneracy.', '1802.00450-2-36-2': 'The only way to break time reversal symmetry but keep parity without involving the spin is to add a term [MATH] to the Hamiltonian, where [MATH].', '1802.00450-2-36-3': '[CITATION] Microscopically, this would arise from a magnetic flux within the unit cell of the crystal, leading to Aharonov-Bohm phases of the hopping amplitudes, as in the Haldane Chern insulator model.', '1802.00450-2-36-4': '[CITATION] To see the effect of this term on the nodal line, let us specify the functions [MATH].', '1802.00450-2-36-5': 'Let us take [EQUATION]', '1802.00450-2-36-6': 'The energy eigenvalues are [EQUATION]', '1802.00450-2-36-7': 'The two bands touch along a nodal line in the [MATH]-plane, given by the equation [EQUATION]', '1802.00450-2-36-8': 'Now let us add the time reversal breaking perturbation [EQUATION]', '1802.00450-2-36-9': 'The band energies become [EQUATION]', '1802.00450-2-36-10': 'We see that the nodal line has been gapped out except at two Weyl points on the [MATH]-axis at [MATH].', '1802.00450-2-36-11': 'Taking [MATH], the chirality of the two Weyl points is given by [EQUATION] and thus interchanges as [MATH] is tuned through zero.', '1802.00450-2-36-12': 'Thus the nodal line in PT-symmetric systems has the same meaning as in systems with SO interactions and broken time reversal symmetry: this is a critical state that separates two Weyl semimetal states with opposite signs of the anomalous Hall conductivity.', '1802.00450-2-37-0': '# Field theory of nodal line semimetals', '1802.00450-2-38-0': 'We will now summarize the above analysis of nodal line semimetals in terms of a field theory, which expresses their anomalous response, in the spirit of Eq. [REF], describing the chiral anomaly of Weyl semimetals.', '1802.00450-2-38-1': 'What we have established thus far is that the nodal line may be thought of as a state arising at a critical point between two Weyl semimetal states with interchanged chirality of the Weyl points.', '1802.00450-2-38-2': 'This means that in addition to the vector [MATH], which acts as a chiral gauge field, determining the separation of two Weyl nodes with fixed opposite chirality in momentum space, we need to take into account the possibility of chirality of the Weyl nodes changing sign, without changing their momentum-space location.', '1802.00450-2-38-3': 'This may be expressed with the help of the vielbein fields [MATH], which encode both the effective metric and the chirality, characterizing the Weyl points.', '1802.00450-2-38-4': 'A low-energy Hamiltonian of two Weyl nodes, separated in momentum space, may then be written as [EQUATION] which is a generalization of the ordinary massless Dirac Hamiltonian to a curved space-time with an arbitrary metric.', '1802.00450-2-38-5': 'The chirality-changing transition, with the appearance of a nodal line at the critical point, may be described as one of the vielbein vectors [MATH] flipping its direction and vanishing at the critical point.', '1802.00450-2-38-6': '[CITATION] For example, in the case of the magnetized TI-NI multilayer, described by Eq. [REF], we may take [EQUATION] which vanishes and changes direction at [MATH].', '1802.00450-2-38-7': 'In the case of a PT-symmetric semimetal, described by Eq. [REF], we may take [EQUATION] which again vanishes and changes direction when [MATH].', '1802.00450-2-38-8': 'Defining chirality as [EQUATION] where [MATH] is the determinant of the matrix [MATH], we may then write down the following action, which describes the anomalous response of this system [EQUATION] where we have taken the (fixed) direction of the vector [MATH], which determines the separation between the Weyl nodes, to be the [MATH]-direction.', '1802.00450-2-38-9': 'This action bears a strong resemblance to the Chern-Simons action [EQUATION] which expresses the parity anomaly of a 2D Dirac fermion of mass [MATH].', '1802.00450-2-38-10': 'This is not unexpected of course, given the connection between 3D nodal line and 2D Dirac fermions, which was established in Sections [REF] and [REF].', '1802.00450-2-39-0': 'The information, contained in Eq. [REF] may be stated as follows: this equation is telling us that a massless 2D Dirac fermion describes a direct transition between nsulators with [MATH] and [MATH].', '1802.00450-2-39-1': 'Analogously, Eq. [REF] is telling us that the nodal line, which appears when one of the vielbein vectors [MATH] vanishes, describes a direct transition between two Weyl semimetal states with [MATH].', '1802.00450-2-40-0': '# Discussion and conclusions', '1802.00450-2-41-0': 'In this paper we have demonstrated that the anomalous response of 3D nodal line semimetals is closely related to the parity anomaly of 2D Dirac fermions.', '1802.00450-2-41-1': 'The role of the mass of a 2D Dirac fermion, whose sign enters into the topological Chern-Simons action for the electromagnetic field, when the fermions are integrated out, is played by the determinant of the vielbein matrix [MATH].', '1802.00450-2-42-0': 'Perhaps the most straightforward way to observe these phenomena, given the currently available materials, is to look at magnetic response of type-II Dirac semimetals.', '1802.00450-2-42-1': '[CITATION] Type-II Dirac semimetals possess one or several symmetry-related Dirac points at TRIM in the first BZ.', '1802.00450-2-42-2': 'Material realizations include TlBi(S[MATH] Se[MATH], [CITATION] (Bi[MATH]In[MATH]Se[MATH], [CITATION] and ZrTe[MATH].', '1802.00450-2-42-3': '[CITATION] As was demonstrated in Ref. [CITATION], magnetic response of a type-II Dirac point is always strongly anisotropic: while one of the Zeeman field components acts as a chiral gauge field, splitting the Dirac point into two Weyl points, the other two components instead create nodal lines.', '1802.00450-2-42-4': 'The problem then maps exactly onto the magnetic multilayer system, described in Section [REF] and exactly the same conclusions follow.', '1802.00450-2-42-5': 'The anomaly may then be detected as a step-function-like singularity of the anomalous Hall conductivity of the Dirac semimetal in the presence of an applied magnetic field, as the field is rotated.', '1802.00450-2-42-6': 'The anomalous Hall conductivity in this case is defined as part of the Hall conductivity, which arises from the Zeeman response.', '1802.00450-2-42-7': 'It may be obtained by subtracting off the linear high-field part of the Hall resistivity, as the anomalous Hall signal is usually isolated.', '1802.00450-2-43-0': 'Another possibility is the recently discovered magnetic Weyl semimetal Co[MATH]Sn[MATH]S[MATH].', '1802.00450-2-43-1': '[CITATION] In this material, six pairs of Weyl nodes arise out of nodal lines, gapped by the SO interactions, as revealed by the electronic structure calculations.', '1802.00450-2-43-2': '[CITATION] Thus Co[MATH]Sn[MATH]S[MATH] may naturally reside close to the phase transition at which the sign of the anomalous Hall conductivity changes, however a detailed investigation of how it responds to rotating the direction of magnetization is necessary to understand if this really is the case.', '1802.00450-2-44-0': 'In conclusion, we have presented a theory of anomalous response (quantum anomaly) in nodal line semimetals, which can be related to the existence of drumhead surface states in these systems.', '1802.00450-2-44-1': 'We have shown that both the surface states and the anomalous response are closely analogous to the parity anomaly of (2+1)-dimensional relativistic Dirac fermions, which in the condensed matter physics context is realized as the 2D QAHE.', '1802.00450-2-44-2': 'We have derived a field theory, describing the anomalous response of nodal semimetals, and shown that a crucial ingredient in this field theory is the sign of the determinant of the vielbein fields, describing both the 3D Weyl fermion chirality and the effective low-energy metric, which emerges in Weyl semimetals.', '1802.00450-2-44-3': 'This sign changes at a critical point at which one of the three vielbein vectors vanishes, leading to the emergence of a nodal line.', '1802.00450-2-45-0': 'Financial support was provided by NSERC of Canada.'}

[['1802.00450-1-38-0', '1802.00450-2-41-0'], ['1802.00450-1-38-1', '1802.00450-2-41-1'], ['1802.00450-1-2-0', '1802.00450-2-2-0'], ['1802.00450-1-2-1', '1802.00450-2-2-1'], ['1802.00450-1-2-2', '1802.00450-2-2-2'], ['1802.00450-1-2-3', '1802.00450-2-2-3'], ['1802.00450-1-8-0', '1802.00450-2-8-0'], ['1802.00450-1-8-1', '1802.00450-2-8-1'], ['1802.00450-1-8-2', '1802.00450-2-8-2'], ['1802.00450-1-33-0', '1802.00450-2-36-0'], ['1802.00450-1-33-1', '1802.00450-2-36-1'], ['1802.00450-1-33-2', '1802.00450-2-36-2'], ['1802.00450-1-33-3', '1802.00450-2-36-3'], ['1802.00450-1-33-4', '1802.00450-2-36-4'], ['1802.00450-1-33-5', '1802.00450-2-36-5'], ['1802.00450-1-33-6', '1802.00450-2-36-6'], ['1802.00450-1-33-7', '1802.00450-2-36-7'], ['1802.00450-1-33-8', '1802.00450-2-36-8'], ['1802.00450-1-33-9', '1802.00450-2-36-9'], ['1802.00450-1-33-10', '1802.00450-2-36-10'], ['1802.00450-1-33-11', '1802.00450-2-36-11'], ['1802.00450-1-33-12', '1802.00450-2-36-12'], ['1802.00450-1-3-0', '1802.00450-2-3-0'], ['1802.00450-1-3-1', '1802.00450-2-3-1'], ['1802.00450-1-27-0', '1802.00450-2-29-0'], ['1802.00450-1-27-1', '1802.00450-2-29-1'], ['1802.00450-1-27-2', '1802.00450-2-29-2'], ['1802.00450-1-27-3', '1802.00450-2-29-3'], ['1802.00450-1-27-4', '1802.00450-2-29-4'], ['1802.00450-1-27-5', '1802.00450-2-29-5'], ['1802.00450-1-27-6', '1802.00450-2-29-6'], ['1802.00450-1-27-7', '1802.00450-2-29-7'], ['1802.00450-1-17-0', '1802.00450-2-17-0'], ['1802.00450-1-17-1', '1802.00450-2-17-1'], ['1802.00450-1-17-2', '1802.00450-2-17-2'], ['1802.00450-1-17-3', '1802.00450-2-17-3'], ['1802.00450-1-31-0', '1802.00450-2-34-0'], ['1802.00450-1-31-1', '1802.00450-2-34-1'], ['1802.00450-1-31-2', '1802.00450-2-34-2'], ['1802.00450-1-13-0', '1802.00450-2-13-0'], ['1802.00450-1-13-1', '1802.00450-2-13-1'], ['1802.00450-1-0-0', '1802.00450-2-0-0'], ['1802.00450-1-0-1', '1802.00450-2-0-1'], ['1802.00450-1-0-2', '1802.00450-2-0-2'], ['1802.00450-1-0-3', '1802.00450-2-0-3'], ['1802.00450-1-0-4', '1802.00450-2-0-4'], ['1802.00450-1-0-5', '1802.00450-2-0-5'], ['1802.00450-1-7-0', '1802.00450-2-7-0'], ['1802.00450-1-7-1', '1802.00450-2-7-1'], ['1802.00450-1-7-2', '1802.00450-2-7-2'], ['1802.00450-1-7-3', '1802.00450-2-7-3'], ['1802.00450-1-7-4', '1802.00450-2-7-4'], ['1802.00450-1-7-5', '1802.00450-2-7-5'], ['1802.00450-1-7-6', '1802.00450-2-7-6'], ['1802.00450-1-7-7', '1802.00450-2-7-7'], ['1802.00450-1-7-8', '1802.00450-2-7-8'], ['1802.00450-1-7-9', '1802.00450-2-7-9'], ['1802.00450-1-21-0', '1802.00450-2-21-0'], ['1802.00450-1-21-1', '1802.00450-2-21-1'], ['1802.00450-1-21-2', '1802.00450-2-21-2'], ['1802.00450-1-21-3', '1802.00450-2-21-3'], ['1802.00450-1-21-4', '1802.00450-2-21-4'], ['1802.00450-1-21-5', '1802.00450-2-21-5'], ['1802.00450-1-35-0', '1802.00450-2-38-0'], ['1802.00450-1-35-1', '1802.00450-2-38-1'], ['1802.00450-1-35-2', '1802.00450-2-38-2'], ['1802.00450-1-35-3', '1802.00450-2-38-3'], ['1802.00450-1-35-4', '1802.00450-2-38-4'], ['1802.00450-1-35-5', '1802.00450-2-38-5'], ['1802.00450-1-35-6', '1802.00450-2-38-6'], ['1802.00450-1-35-7', '1802.00450-2-38-7'], ['1802.00450-1-35-8', '1802.00450-2-38-8'], ['1802.00450-1-35-9', '1802.00450-2-38-9'], ['1802.00450-1-35-10', '1802.00450-2-38-10'], ['1802.00450-1-12-0', '1802.00450-2-12-0'], ['1802.00450-1-12-1', '1802.00450-2-12-1'], ['1802.00450-1-12-2', '1802.00450-2-12-2'], ['1802.00450-1-12-3', '1802.00450-2-12-3'], ['1802.00450-1-12-4', '1802.00450-2-12-4'], ['1802.00450-1-12-5', '1802.00450-2-12-5'], ['1802.00450-1-12-6', '1802.00450-2-12-6'], ['1802.00450-1-12-7', '1802.00450-2-12-7'], ['1802.00450-1-12-8', '1802.00450-2-12-8'], ['1802.00450-1-28-0', '1802.00450-2-31-0'], ['1802.00450-1-28-1', '1802.00450-2-31-1'], ['1802.00450-1-28-2', '1802.00450-2-31-2'], ['1802.00450-1-28-3', '1802.00450-2-31-3'], ['1802.00450-1-28-4', '1802.00450-2-31-4'], ['1802.00450-1-28-5', '1802.00450-2-31-5'], ['1802.00450-1-28-6', '1802.00450-2-31-6'], ['1802.00450-1-28-7', '1802.00450-2-31-7'], ['1802.00450-1-15-0', '1802.00450-2-15-0'], ['1802.00450-1-15-1', '1802.00450-2-15-1'], ['1802.00450-1-15-2', '1802.00450-2-15-2'], ['1802.00450-1-15-3', '1802.00450-2-15-3'], ['1802.00450-1-15-4', '1802.00450-2-15-4'], ['1802.00450-1-15-5', '1802.00450-2-15-5'], ['1802.00450-1-5-0', '1802.00450-2-5-0'], ['1802.00450-1-5-1', '1802.00450-2-5-1'], ['1802.00450-1-5-2', '1802.00450-2-5-2'], ['1802.00450-1-19-0', '1802.00450-2-19-0'], ['1802.00450-1-19-1', '1802.00450-2-19-1'], ['1802.00450-1-19-2', '1802.00450-2-19-2'], ['1802.00450-1-19-3', '1802.00450-2-19-3'], ['1802.00450-1-19-4', '1802.00450-2-19-4'], ['1802.00450-1-19-5', '1802.00450-2-19-5'], ['1802.00450-1-19-6', '1802.00450-2-19-6'], ['1802.00450-1-19-7', '1802.00450-2-19-7'], ['1802.00450-1-19-8', '1802.00450-2-19-8'], ['1802.00450-1-19-9', '1802.00450-2-19-9'], ['1802.00450-1-40-0', '1802.00450-2-43-0'], ['1802.00450-1-40-1', '1802.00450-2-43-1'], ['1802.00450-1-40-2', '1802.00450-2-43-2'], ['1802.00450-1-24-1', '1802.00450-2-25-2'], ['1802.00450-1-24-2', '1802.00450-2-25-3'], ['1802.00450-1-24-3', '1802.00450-2-25-4'], ['1802.00450-1-24-4', '1802.00450-2-25-5'], ['1802.00450-1-24-5', '1802.00450-2-25-6'], ['1802.00450-1-24-6', '1802.00450-2-25-7'], ['1802.00450-1-25-0', '1802.00450-2-26-0'], ['1802.00450-1-25-1', '1802.00450-2-26-1'], ['1802.00450-1-25-2', '1802.00450-2-26-2'], ['1802.00450-1-25-3', '1802.00450-2-26-3'], ['1802.00450-1-25-4', '1802.00450-2-26-4'], ['1802.00450-1-25-5', '1802.00450-2-26-5'], ['1802.00450-1-26-1', '1802.00450-2-28-1'], ['1802.00450-1-26-2', '1802.00450-2-28-2'], ['1802.00450-1-26-3', '1802.00450-2-28-3'], ['1802.00450-1-26-4', '1802.00450-2-28-4'], ['1802.00450-1-23-0', '1802.00450-2-23-0'], ['1802.00450-1-23-1', '1802.00450-2-23-1'], ['1802.00450-1-23-2', '1802.00450-2-23-2'], ['1802.00450-1-36-0', '1802.00450-2-39-0'], ['1802.00450-1-36-1', '1802.00450-2-39-1'], ['1802.00450-1-32-0', '1802.00450-2-35-0'], ['1802.00450-1-32-1', '1802.00450-2-35-1'], ['1802.00450-1-32-2', '1802.00450-2-35-2'], ['1802.00450-1-32-3', '1802.00450-2-35-3'], ['1802.00450-1-32-4', '1802.00450-2-35-4'], ['1802.00450-1-32-5', '1802.00450-2-35-5'], ['1802.00450-1-32-6', '1802.00450-2-35-6'], ['1802.00450-1-16-0', '1802.00450-2-16-0'], ['1802.00450-1-16-1', '1802.00450-2-16-1'], ['1802.00450-1-16-2', '1802.00450-2-16-2'], ['1802.00450-1-16-3', '1802.00450-2-16-3'], ['1802.00450-1-16-4', '1802.00450-2-16-4'], ['1802.00450-1-16-5', '1802.00450-2-16-5'], ['1802.00450-1-16-6', '1802.00450-2-16-6'], ['1802.00450-1-4-0', '1802.00450-2-4-0'], ['1802.00450-1-4-1', '1802.00450-2-4-1'], ['1802.00450-1-4-2', '1802.00450-2-4-2'], ['1802.00450-1-4-3', '1802.00450-2-4-3'], ['1802.00450-1-29-0', '1802.00450-2-32-0'], ['1802.00450-1-29-1', '1802.00450-2-32-1'], ['1802.00450-1-9-0', '1802.00450-2-9-0'], ['1802.00450-1-9-1', '1802.00450-2-9-1'], ['1802.00450-1-9-2', '1802.00450-2-9-2'], ['1802.00450-1-10-0', '1802.00450-2-10-0'], ['1802.00450-1-10-1', '1802.00450-2-10-1'], ['1802.00450-1-10-2', '1802.00450-2-10-2'], ['1802.00450-1-10-3', '1802.00450-2-10-3'], ['1802.00450-1-10-4', '1802.00450-2-10-4'], ['1802.00450-1-10-5', '1802.00450-2-10-5'], ['1802.00450-1-10-6', '1802.00450-2-10-6'], ['1802.00450-1-10-7', '1802.00450-2-10-7'], ['1802.00450-1-10-8', '1802.00450-2-10-8'], ['1802.00450-1-10-9', '1802.00450-2-10-9'], ['1802.00450-1-10-10', '1802.00450-2-10-10'], ['1802.00450-1-41-0', '1802.00450-2-44-0'], ['1802.00450-1-41-1', '1802.00450-2-44-1'], ['1802.00450-1-41-2', '1802.00450-2-44-2'], ['1802.00450-1-41-3', '1802.00450-2-44-3'], ['1802.00450-1-6-0', '1802.00450-2-6-0'], ['1802.00450-1-6-1', '1802.00450-2-6-1'], ['1802.00450-1-6-2', '1802.00450-2-6-2'], ['1802.00450-1-18-0', '1802.00450-2-18-0'], ['1802.00450-1-18-1', '1802.00450-2-18-1'], ['1802.00450-1-18-2', '1802.00450-2-18-2'], ['1802.00450-1-18-3', '1802.00450-2-18-3'], ['1802.00450-1-14-0', '1802.00450-2-14-0'], ['1802.00450-1-39-0', '1802.00450-2-42-0'], ['1802.00450-1-39-1', '1802.00450-2-42-1'], ['1802.00450-1-39-3', '1802.00450-2-42-3'], ['1802.00450-1-39-4', '1802.00450-2-42-4'], ['1802.00450-1-39-5', '1802.00450-2-42-5'], ['1802.00450-1-39-6', '1802.00450-2-42-6'], ['1802.00450-1-39-7', '1802.00450-2-42-7'], ['1802.00450-1-20-0', '1802.00450-2-20-0'], ['1802.00450-1-20-1', '1802.00450-2-20-1'], ['1802.00450-1-20-2', '1802.00450-2-20-2'], ['1802.00450-1-20-3', '1802.00450-2-20-3'], ['1802.00450-1-20-4', '1802.00450-2-20-4'], ['1802.00450-1-20-5', '1802.00450-2-20-5'], ['1802.00450-1-20-6', '1802.00450-2-20-6'], ['1802.00450-1-20-7', '1802.00450-2-20-7'], ['1802.00450-1-20-8', '1802.00450-2-20-8'], ['1802.00450-1-20-9', '1802.00450-2-20-9'], ['1802.00450-1-20-10', '1802.00450-2-20-10'], ['1802.00450-1-20-11', '1802.00450-2-20-11'], ['1802.00450-1-20-12', '1802.00450-2-20-12'], ['1802.00450-1-24-0', '1802.00450-2-25-0'], ['1802.00450-1-26-0', '1802.00450-2-28-0']]

[['1802.00450-1-38-0', '1802.00450-2-41-0'], ['1802.00450-1-38-1', '1802.00450-2-41-1'], ['1802.00450-1-2-0', '1802.00450-2-2-0'], ['1802.00450-1-2-1', '1802.00450-2-2-1'], ['1802.00450-1-2-2', '1802.00450-2-2-2'], ['1802.00450-1-2-3', '1802.00450-2-2-3'], ['1802.00450-1-8-0', '1802.00450-2-8-0'], ['1802.00450-1-8-1', '1802.00450-2-8-1'], ['1802.00450-1-8-2', '1802.00450-2-8-2'], ['1802.00450-1-33-0', '1802.00450-2-36-0'], ['1802.00450-1-33-1', '1802.00450-2-36-1'], ['1802.00450-1-33-2', '1802.00450-2-36-2'], ['1802.00450-1-33-3', '1802.00450-2-36-3'], ['1802.00450-1-33-4', '1802.00450-2-36-4'], ['1802.00450-1-33-5', '1802.00450-2-36-5'], ['1802.00450-1-33-6', '1802.00450-2-36-6'], ['1802.00450-1-33-7', '1802.00450-2-36-7'], ['1802.00450-1-33-8', '1802.00450-2-36-8'], ['1802.00450-1-33-9', '1802.00450-2-36-9'], ['1802.00450-1-33-10', '1802.00450-2-36-10'], ['1802.00450-1-33-11', '1802.00450-2-36-11'], ['1802.00450-1-33-12', '1802.00450-2-36-12'], ['1802.00450-1-3-0', '1802.00450-2-3-0'], ['1802.00450-1-3-1', '1802.00450-2-3-1'], ['1802.00450-1-27-0', '1802.00450-2-29-0'], ['1802.00450-1-27-1', '1802.00450-2-29-1'], ['1802.00450-1-27-2', '1802.00450-2-29-2'], ['1802.00450-1-27-3', '1802.00450-2-29-3'], ['1802.00450-1-27-4', '1802.00450-2-29-4'], ['1802.00450-1-27-5', '1802.00450-2-29-5'], ['1802.00450-1-27-6', '1802.00450-2-29-6'], ['1802.00450-1-27-7', '1802.00450-2-29-7'], ['1802.00450-1-17-0', '1802.00450-2-17-0'], ['1802.00450-1-17-1', '1802.00450-2-17-1'], ['1802.00450-1-17-2', '1802.00450-2-17-2'], ['1802.00450-1-17-3', '1802.00450-2-17-3'], ['1802.00450-1-31-0', '1802.00450-2-34-0'], ['1802.00450-1-31-1', '1802.00450-2-34-1'], ['1802.00450-1-31-2', '1802.00450-2-34-2'], ['1802.00450-1-13-0', '1802.00450-2-13-0'], ['1802.00450-1-13-1', '1802.00450-2-13-1'], ['1802.00450-1-0-0', '1802.00450-2-0-0'], ['1802.00450-1-0-1', '1802.00450-2-0-1'], ['1802.00450-1-0-2', '1802.00450-2-0-2'], ['1802.00450-1-0-3', '1802.00450-2-0-3'], ['1802.00450-1-0-4', '1802.00450-2-0-4'], ['1802.00450-1-0-5', '1802.00450-2-0-5'], ['1802.00450-1-7-0', '1802.00450-2-7-0'], ['1802.00450-1-7-1', '1802.00450-2-7-1'], ['1802.00450-1-7-2', '1802.00450-2-7-2'], ['1802.00450-1-7-3', '1802.00450-2-7-3'], ['1802.00450-1-7-4', '1802.00450-2-7-4'], ['1802.00450-1-7-5', '1802.00450-2-7-5'], ['1802.00450-1-7-6', '1802.00450-2-7-6'], ['1802.00450-1-7-7', '1802.00450-2-7-7'], ['1802.00450-1-7-8', '1802.00450-2-7-8'], ['1802.00450-1-7-9', '1802.00450-2-7-9'], ['1802.00450-1-21-0', '1802.00450-2-21-0'], ['1802.00450-1-21-1', '1802.00450-2-21-1'], ['1802.00450-1-21-2', '1802.00450-2-21-2'], ['1802.00450-1-21-3', '1802.00450-2-21-3'], ['1802.00450-1-21-4', '1802.00450-2-21-4'], ['1802.00450-1-21-5', '1802.00450-2-21-5'], ['1802.00450-1-35-0', '1802.00450-2-38-0'], ['1802.00450-1-35-1', '1802.00450-2-38-1'], ['1802.00450-1-35-2', '1802.00450-2-38-2'], ['1802.00450-1-35-3', '1802.00450-2-38-3'], ['1802.00450-1-35-4', '1802.00450-2-38-4'], ['1802.00450-1-35-5', '1802.00450-2-38-5'], ['1802.00450-1-35-6', '1802.00450-2-38-6'], ['1802.00450-1-35-7', '1802.00450-2-38-7'], ['1802.00450-1-35-8', '1802.00450-2-38-8'], ['1802.00450-1-35-9', '1802.00450-2-38-9'], ['1802.00450-1-35-10', '1802.00450-2-38-10'], ['1802.00450-1-12-0', '1802.00450-2-12-0'], ['1802.00450-1-12-1', '1802.00450-2-12-1'], ['1802.00450-1-12-2', '1802.00450-2-12-2'], ['1802.00450-1-12-3', '1802.00450-2-12-3'], ['1802.00450-1-12-4', '1802.00450-2-12-4'], ['1802.00450-1-12-5', '1802.00450-2-12-5'], ['1802.00450-1-12-6', '1802.00450-2-12-6'], ['1802.00450-1-12-7', '1802.00450-2-12-7'], ['1802.00450-1-12-8', '1802.00450-2-12-8'], ['1802.00450-1-28-0', '1802.00450-2-31-0'], ['1802.00450-1-28-1', '1802.00450-2-31-1'], ['1802.00450-1-28-2', '1802.00450-2-31-2'], ['1802.00450-1-28-3', '1802.00450-2-31-3'], ['1802.00450-1-28-4', '1802.00450-2-31-4'], ['1802.00450-1-28-5', '1802.00450-2-31-5'], ['1802.00450-1-28-6', '1802.00450-2-31-6'], ['1802.00450-1-28-7', '1802.00450-2-31-7'], ['1802.00450-1-15-0', '1802.00450-2-15-0'], ['1802.00450-1-15-1', '1802.00450-2-15-1'], ['1802.00450-1-15-2', '1802.00450-2-15-2'], ['1802.00450-1-15-3', '1802.00450-2-15-3'], ['1802.00450-1-15-4', '1802.00450-2-15-4'], ['1802.00450-1-15-5', '1802.00450-2-15-5'], ['1802.00450-1-5-0', '1802.00450-2-5-0'], ['1802.00450-1-5-1', '1802.00450-2-5-1'], ['1802.00450-1-5-2', '1802.00450-2-5-2'], ['1802.00450-1-19-0', '1802.00450-2-19-0'], ['1802.00450-1-19-1', '1802.00450-2-19-1'], ['1802.00450-1-19-2', '1802.00450-2-19-2'], ['1802.00450-1-19-3', '1802.00450-2-19-3'], ['1802.00450-1-19-4', '1802.00450-2-19-4'], ['1802.00450-1-19-5', '1802.00450-2-19-5'], ['1802.00450-1-19-6', '1802.00450-2-19-6'], ['1802.00450-1-19-7', '1802.00450-2-19-7'], ['1802.00450-1-19-8', '1802.00450-2-19-8'], ['1802.00450-1-19-9', '1802.00450-2-19-9'], ['1802.00450-1-40-0', '1802.00450-2-43-0'], ['1802.00450-1-40-1', '1802.00450-2-43-1'], ['1802.00450-1-40-2', '1802.00450-2-43-2'], ['1802.00450-1-24-1', '1802.00450-2-25-2'], ['1802.00450-1-24-2', '1802.00450-2-25-3'], ['1802.00450-1-24-3', '1802.00450-2-25-4'], ['1802.00450-1-24-4', '1802.00450-2-25-5'], ['1802.00450-1-24-5', '1802.00450-2-25-6'], ['1802.00450-1-24-6', '1802.00450-2-25-7'], ['1802.00450-1-25-0', '1802.00450-2-26-0'], ['1802.00450-1-25-1', '1802.00450-2-26-1'], ['1802.00450-1-25-2', '1802.00450-2-26-2'], ['1802.00450-1-25-3', '1802.00450-2-26-3'], ['1802.00450-1-25-4', '1802.00450-2-26-4'], ['1802.00450-1-25-5', '1802.00450-2-26-5'], ['1802.00450-1-26-1', '1802.00450-2-28-1'], ['1802.00450-1-26-2', '1802.00450-2-28-2'], ['1802.00450-1-26-3', '1802.00450-2-28-3'], ['1802.00450-1-26-4', '1802.00450-2-28-4'], ['1802.00450-1-23-0', '1802.00450-2-23-0'], ['1802.00450-1-23-1', '1802.00450-2-23-1'], ['1802.00450-1-23-2', '1802.00450-2-23-2'], ['1802.00450-1-36-0', '1802.00450-2-39-0'], ['1802.00450-1-36-1', '1802.00450-2-39-1'], ['1802.00450-1-32-0', '1802.00450-2-35-0'], ['1802.00450-1-32-1', '1802.00450-2-35-1'], ['1802.00450-1-32-2', '1802.00450-2-35-2'], ['1802.00450-1-32-3', '1802.00450-2-35-3'], ['1802.00450-1-32-4', '1802.00450-2-35-4'], ['1802.00450-1-32-5', '1802.00450-2-35-5'], ['1802.00450-1-32-6', '1802.00450-2-35-6'], ['1802.00450-1-16-0', '1802.00450-2-16-0'], ['1802.00450-1-16-1', '1802.00450-2-16-1'], ['1802.00450-1-16-2', '1802.00450-2-16-2'], ['1802.00450-1-16-3', '1802.00450-2-16-3'], ['1802.00450-1-16-4', '1802.00450-2-16-4'], ['1802.00450-1-16-5', '1802.00450-2-16-5'], ['1802.00450-1-16-6', '1802.00450-2-16-6'], ['1802.00450-1-4-0', '1802.00450-2-4-0'], ['1802.00450-1-4-1', '1802.00450-2-4-1'], ['1802.00450-1-4-2', '1802.00450-2-4-2'], ['1802.00450-1-4-3', '1802.00450-2-4-3'], ['1802.00450-1-29-0', '1802.00450-2-32-0'], ['1802.00450-1-29-1', '1802.00450-2-32-1'], ['1802.00450-1-9-0', '1802.00450-2-9-0'], ['1802.00450-1-9-1', '1802.00450-2-9-1'], ['1802.00450-1-9-2', '1802.00450-2-9-2'], ['1802.00450-1-10-0', '1802.00450-2-10-0'], ['1802.00450-1-10-1', '1802.00450-2-10-1'], ['1802.00450-1-10-2', '1802.00450-2-10-2'], ['1802.00450-1-10-3', '1802.00450-2-10-3'], ['1802.00450-1-10-4', '1802.00450-2-10-4'], ['1802.00450-1-10-5', '1802.00450-2-10-5'], ['1802.00450-1-10-6', '1802.00450-2-10-6'], ['1802.00450-1-10-7', '1802.00450-2-10-7'], ['1802.00450-1-10-8', '1802.00450-2-10-8'], ['1802.00450-1-10-9', '1802.00450-2-10-9'], ['1802.00450-1-10-10', '1802.00450-2-10-10'], ['1802.00450-1-41-0', '1802.00450-2-44-0'], ['1802.00450-1-41-1', '1802.00450-2-44-1'], ['1802.00450-1-41-2', '1802.00450-2-44-2'], ['1802.00450-1-41-3', '1802.00450-2-44-3'], ['1802.00450-1-6-0', '1802.00450-2-6-0'], ['1802.00450-1-6-1', '1802.00450-2-6-1'], ['1802.00450-1-6-2', '1802.00450-2-6-2'], ['1802.00450-1-18-0', '1802.00450-2-18-0'], ['1802.00450-1-18-1', '1802.00450-2-18-1'], ['1802.00450-1-18-2', '1802.00450-2-18-2'], ['1802.00450-1-18-3', '1802.00450-2-18-3'], ['1802.00450-1-14-0', '1802.00450-2-14-0'], ['1802.00450-1-39-0', '1802.00450-2-42-0'], ['1802.00450-1-39-1', '1802.00450-2-42-1'], ['1802.00450-1-39-3', '1802.00450-2-42-3'], ['1802.00450-1-39-4', '1802.00450-2-42-4'], ['1802.00450-1-39-5', '1802.00450-2-42-5'], ['1802.00450-1-39-6', '1802.00450-2-42-6'], ['1802.00450-1-39-7', '1802.00450-2-42-7'], ['1802.00450-1-20-0', '1802.00450-2-20-0'], ['1802.00450-1-20-1', '1802.00450-2-20-1'], ['1802.00450-1-20-2', '1802.00450-2-20-2'], ['1802.00450-1-20-3', '1802.00450-2-20-3'], ['1802.00450-1-20-4', '1802.00450-2-20-4'], ['1802.00450-1-20-5', '1802.00450-2-20-5'], ['1802.00450-1-20-6', '1802.00450-2-20-6'], ['1802.00450-1-20-7', '1802.00450-2-20-7'], ['1802.00450-1-20-8', '1802.00450-2-20-8'], ['1802.00450-1-20-9', '1802.00450-2-20-9'], ['1802.00450-1-20-10', '1802.00450-2-20-10'], ['1802.00450-1-20-11', '1802.00450-2-20-11'], ['1802.00450-1-20-12', '1802.00450-2-20-12']]

[]

[]

[['1802.00450-1-24-0', '1802.00450-2-25-0'], ['1802.00450-1-26-0', '1802.00450-2-28-0']]

[]

['1802.00450-1-2-4', '1802.00450-1-3-2', '1802.00450-1-5-3', '1802.00450-1-13-2', '1802.00450-1-15-6', '1802.00450-1-24-7', '1802.00450-1-39-2', '1802.00450-1-42-0', '1802.00450-2-2-4', '1802.00450-2-3-2', '1802.00450-2-5-3', '1802.00450-2-13-2', '1802.00450-2-15-6', '1802.00450-2-25-8', '1802.00450-2-42-2', '1802.00450-2-45-0']

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

https://arxiv.org/abs/1802.00450

1006.0407

{'1006.0407-1-0-0': 'Given a matrix [MATH], we present a simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements of [MATH] and then retains some of the remaining elements with probabilities proportional to the square of their magnitudes.', '1006.0407-1-0-1': 'We analyze the approximation accuracy of the proposed algorithm using a recent, elegant matrix-valued Chernoff bound.', '1006.0407-1-0-2': 'As a result, we obtain the best known bounds for element-wise matrix sparsification.', '1006.0407-1-1-0': '# Introduction', '1006.0407-1-2-0': 'Element-wise matrix sparsification was pioneered by Achlioptas and McSherry in 2001 [CITATION], who described sampling-based algorithms to select a small number of elements from an input matrix [MATH] in order to construct a sparse sketch [MATH].', '1006.0407-1-2-1': 'Such sketches were used in approximate eigenvector computations [CITATION], semi-definite programming solvers [CITATION], and matrix completion problems [CITATION].', '1006.0407-1-2-2': 'Motivated by their work, we present a simple matrix sparsification algorithm that achieves the tightest known bounds for element-wise matrix sparsification.', '1006.0407-1-2-3': 'Our main algorithm (Algorithm 1) zeroes out "small" elements of [MATH] and randomly samples the remaining elements of [MATH] with respect to a probability distribution that favors "larger" entries.', '1006.0407-1-3-0': 'Matrix Sparsification Algorithm [1] Input: [MATH], accuracy parameter [MATH].', '1006.0407-1-4-0': 'Let [MATH] and zero-out all entries of [MATH] that are smaller (in absolute value) than [MATH].', '1006.0407-1-5-0': 'Set [MATH] as in Eqn. ([REF]).', '1006.0407-1-6-0': 'For [MATH] (i.i.d. trials with replacement) randomly sample indices [MATH] (entries of [MATH]), with', '1006.0407-1-7-0': '[EQUATION]', '1006.0407-1-7-1': 'Output: [EQUATION].', '1006.0407-1-8-0': 'In Algorithm [MATH] we let [MATH] denote the standard basis vectors for [MATH] (see Section [REF] for more notation).', '1006.0407-1-8-1': 'Our sampling procedure selects [MATH] entries from [MATH] (note that [MATH] is simply [MATH], but with elements less than or equal to [MATH] zeroed out) in [MATH] i.i.d. trials with replacement.', '1006.0407-1-8-2': 'In each trial, elements of [MATH] are retained with probability proportional to their magnitude.', '1006.0407-1-8-3': 'Note that the same element of [MATH] could be selected multiple times and that [MATH] contains at most [MATH] non-zero entries.', '1006.0407-1-8-4': 'Theorem [REF] is our main quality-of-approximation result for Algorithm [MATH].', '1006.0407-1-9-0': 'Let [MATH] be any matrix (assume [MATH]) and let [MATH] be the sparse sketch of [MATH] constructed via Algorithm 1.', '1006.0407-1-9-1': 'If [EQUATION] then, with probability at least [MATH], [EQUATION] [MATH] has at most [MATH] non-zero entries and the construction of [MATH] can be implemented in one pass over the input matrix [MATH] (see Section [REF]).', '1006.0407-1-10-0': 'Our result outperforms prior work in the sense that, using the same accuracy parameter [MATH], the resulting matrix [MATH] has fewer non-zero elements.', '1006.0407-1-10-1': 'In [CITATION] the authors presented a sampling method that requires at least [MATH] non-zero entries in [MATH] to achieve the same accuracy guarantee; our result reduces the sampling complexity by an [MATH] factor.', '1006.0407-1-10-2': 'It is harder to compare our method to the work of [CITATION], which depends on the [MATH].', '1006.0407-1-10-3': 'The latter quantity is, in general, upper bounded only by [MATH], in which case the sampling complexity of [CITATION] is much worse, namely [MATH].', '1006.0407-1-10-4': 'However, it is worth noting that the result of [CITATION] is appropriate for matrices whose "energy" is focused only on a small number of entries, as well as that their bound holds with much higher probability than ours.', '1006.0407-1-10-5': 'Finally, in [CITATION], the authors study the [MATH] and [MATH] norms in the matrix sparsification context.', '1006.0407-1-10-6': 'The authors also present a sampling scheme analogous to ours, but their theoretical analysis is not directly comparable to our results, since the sampling complexity depends on the average of the ratios [MATH].', '1006.0407-1-11-0': 'We conclude this section with Corollary [REF], which is a re-statement of Theorem [REF] involving the stable rank [MATH] of [MATH].', '1006.0407-1-11-1': '(Recall that the stable rank of any matrix [MATH] is defined as the ratio [MATH], which is upper bounded by the rank of [MATH].)', '1006.0407-1-11-2': 'The corollary guarantees relative error approximations for matrices of - say - constant stable rank, such as the ones that arise in [CITATION].', '1006.0407-1-12-0': 'Let [MATH] be any matrix (assume [MATH]) with stable rank [MATH].', '1006.0407-1-12-1': 'Let [MATH] be the sparse sketch of [MATH] constructed via Algorithm 1 (with [MATH]).', '1006.0407-1-12-2': 'If', '1006.0407-1-13-0': '[MATH]', '1006.0407-1-14-0': 'then, with probability at least [MATH], [EQUATION].', '1006.0407-1-15-0': 'It is worth noting that the sampling algorithm implied by Corollary [REF] can not be implemented in one pass, since we would need a priory knowledge of the spectral norm of [MATH] in order to implement Step [MATH] of Algorithm [MATH].', '1006.0407-1-16-0': '# Background', '1006.0407-1-17-0': '## Notation', '1006.0407-1-18-0': 'We let [MATH] denote the set [MATH].', '1006.0407-1-18-1': 'We will use the notation [MATH] to denote the probability of the event in the parentheses and [MATH] to denote the expectation of a random variable [MATH].', '1006.0407-1-18-2': 'When [MATH] is a matrix, [MATH] denotes the element-wise expectation of each entry of [MATH].', '1006.0407-1-18-3': 'For a matrix [MATH], [MATH] will denote the [MATH]-th column of [MATH] as a column vector and, similarly, [MATH] will denote the [MATH]-th row of [MATH] as a row vector (for any [MATH] or [MATH] in [MATH]).', '1006.0407-1-18-4': 'The Frobenius norm [MATH] of the matrix [MATH] is defined as', '1006.0407-1-19-0': '[MATH]', '1006.0407-1-20-0': 'and the spectral norm [MATH] of the matrix [MATH] is defined as', '1006.0407-1-21-0': '[MATH].', '1006.0407-1-22-0': 'Finally, [MATH] denotes the identity matrix of size [MATH] and [MATH] denotes the natural logarithm of [MATH].', '1006.0407-1-23-0': '## Implementing the sampling in one pass over the input matrix', '1006.0407-1-24-0': 'We now discuss the implementation of Algorithm 1 in one pass over the input matrix [MATH].', '1006.0407-1-24-1': 'Towards that end, we will leverage (a slightly modified version of) Algorithm Select (p. 137 of [CITATION]).', '1006.0407-1-25-0': 'One-pass Select algorithm [1] Input: [MATH] for all [MATH], arbitrarily ordered.', '1006.0407-1-25-1': '[MATH].', '1006.0407-1-25-2': 'For all [MATH] such that [MATH]', '1006.0407-1-26-0': 'Output: Return [MATH], [MATH] and [MATH].', '1006.0407-1-27-0': 'We note that Step [MATH] essentially operates on [MATH].', '1006.0407-1-27-1': 'Clearly, in a single pass over the data we can run in parallel [MATH] copies of the Select Algorithm (using a total of [MATH] memory) to effectively return [MATH] independent samples from [MATH].', '1006.0407-1-27-2': "Lemma [MATH] (page [MATH] of [CITATION], note that the sequence of the [MATH]'s is all-positive) guarantees that each of the [MATH] copies of Select returns a sample satisfying:", '1006.0407-1-28-0': '[EQUATION].', '1006.0407-1-29-0': 'Finally, in the parlance of Step [MATH] of Algorithm [MATH], [MATH] is set to [MATH] and [MATH] is set to [MATH] for all [MATH].', '1006.0407-1-30-0': '# Proving Theorem [REF]', '1006.0407-1-31-0': 'Our proof of Theorem [REF] will combine Lemmas [REF] and [REF] in order to bound [MATH] as follows: [EQUATION].', '1006.0407-1-32-0': 'The failure probability of Theorem [REF] emerges from Lemma [REF], which fails with probability [MATH] for the choice of [MATH] in Eqn. ([REF]).', '1006.0407-1-32-1': 'The proof of Lemma [REF] will involve an elegant matrix-valued Chernoff bound proven by Recht in [CITATION].', '1006.0407-1-33-0': '## Bounding [MATH]', '1006.0407-1-34-0': 'Using the notation of Algorithm 1, [MATH].', '1006.0407-1-35-0': 'Recall that the entries of [MATH] are either equal to the corresponding entries of [MATH] or they are set to zero if the corresponding entry of [MATH] is (in absolute value) smaller than [MATH].', '1006.0407-1-35-1': 'Thus, [EQUATION]', '1006.0407-1-36-0': '## Bounding [MATH]', '1006.0407-1-37-0': 'In order to prove our main result in this section (Lemma [REF]) we will leverage a powerful matrix-valued Chernoff bound originally proven in [CITATION] (Theorem 3.2).', '1006.0407-1-37-1': 'We restate this theorem, slightly rephrased to better suit our notation.', '1006.0407-1-38-0': 'Let [MATH] be independent, zero-mean random matrices in [MATH].', '1006.0407-1-38-1': 'Suppose [MATH] and [MATH] for all [MATH].', '1006.0407-1-38-2': 'Then, for any [MATH], [EQUATION] holds, subject to a failure probability of at most [EQUATION].', '1006.0407-1-39-0': 'In order to apply the above theorem, first observe that, using the notation of Algorithm 1, [EQUATION] where [EQUATION] for all [MATH].', '1006.0407-1-39-1': 'It is easy to argue that [MATH] for all [MATH].', '1006.0407-1-39-2': 'Indeed,', '1006.0407-1-40-0': '[EQUATION].', '1006.0407-1-41-0': 'In the above we used [MATH] and [MATH].', '1006.0407-1-41-1': 'Our next lemma bounds [MATH] for all [MATH].', '1006.0407-1-42-0': 'Using our notation, [MATH] for all [MATH].', '1006.0407-1-43-0': 'First, using the definition of [MATH] and the fact that [MATH], [EQUATION].', '1006.0407-1-44-0': 'The last inequality follows since all entries of [MATH] are at least [MATH] and the fact that [MATH].', '1006.0407-1-44-1': 'We can now assume that [EQUATION] to conclude the proof of the lemma.', '1006.0407-1-44-2': 'To justify our assumption in Eqn. ([REF]), we note that if it is violated, then it must be the case that [MATH].', '1006.0407-1-44-3': 'If that were true, then all entries of [MATH] would be equal to zero.', '1006.0407-1-44-4': '(Recall that all entries of [MATH] are either zero or, in absolute value, larger than [MATH].)', '1006.0407-1-44-5': 'Also, if [MATH] were identically zero, then (i) [MATH] would also be identically zero and, (ii) all entries of [MATH] would be at most [MATH].', '1006.0407-1-44-6': 'Thus, [EQUATION].', '1006.0407-1-45-0': 'Thus, if the assumption of Eqn. ([REF]) is not satisfied, the resulting all-zeros [MATH] still satisfies Theorem [REF].', '1006.0407-1-46-0': 'Our next step towards applying Theorem [REF] involves bounding the spectral norm of the expectation of [MATH].', '1006.0407-1-46-1': '(The spectral norm of the expectation of [MATH] admits a similar analysis and the same bound and is omitted.)', '1006.0407-1-47-0': 'Using our notation, [MATH] for any [MATH].', '1006.0407-1-48-0': 'We start by evaluating [MATH]; recall that [MATH]: [EQUATION]', '1006.0407-1-48-1': 'We now simplify the above result using a few simple observations: [MATH], [MATH], [MATH], [MATH], and [MATH].', '1006.0407-1-48-2': 'Thus, we get [EQUATION]', '1006.0407-1-48-3': 'To conclude the proof of the lemma note that (using [MATH]) [EQUATION]', '1006.0407-1-48-4': 'We can now apply Theorem [REF] on Eqn. ([REF]) with [MATH], [MATH] (Lemma [REF]), and [MATH] (Lemma [REF]) .', '1006.0407-1-48-5': 'Thus, we get that [MATH] holds, subject to a failure probability of at most [EQUATION].', '1006.0407-1-49-0': 'Bounding the failure probability by [MATH] and solving for [MATH], we get that [MATH] must be at least [EQUATION]', '1006.0407-1-49-1': 'Assuming that [MATH] we get [MATH].', '1006.0407-1-49-2': 'Using [MATH] (by construction) concludes the proof of the following lemma, which is the main result of this section.', '1006.0407-1-50-0': 'Using the notation of Algorithm 1, if [MATH] and', '1006.0407-1-51-0': '[MATH]', '1006.0407-1-52-0': 'then, with probability at least [MATH], [EQUATION]'}

{'1006.0407-2-0-0': 'Given a matrix [MATH], we present a simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements of [MATH] and then retains some of the remaining elements with probabilities proportional to the square of their magnitudes.', '1006.0407-2-0-1': 'We analyze the approximation accuracy of the proposed algorithm using a recent, elegant non-commutative Bernstein inequality, and compare our bounds with all existing (to the best of our knowledge) element-wise matrix sparsification algorithms.', '1006.0407-2-1-0': '# Introduction', '1006.0407-2-2-0': 'Element-wise matrix sparsification was pioneered by Achlioptas and McSherry [CITATION], who described sampling-based algorithms to select a small number of elements from an input matrix [MATH] in order to construct a sparse sketch [MATH], which is close to [MATH] in the operator norm.', '1006.0407-2-2-1': 'Such sketches were used in approximate eigenvector computations [CITATION], semi-definite programming solvers [CITATION], and matrix completion problems [CITATION].', '1006.0407-2-2-2': 'Motivated by their work, we present a simple matrix sparsification algorithm that achieves the best known upper bounds for element-wise matrix sparsification.', '1006.0407-2-3-0': 'Our main algorithm (Algorithm 1) zeroes out "small" elements of [MATH] and randomly samples the remaining elements of [MATH] with respect to a probability distribution that favors "larger" entries.', '1006.0407-2-4-0': 'Matrix Sparsification Algorithm [1]', '1006.0407-2-5-0': 'Input: [MATH], accuracy parameter [MATH].', '1006.0407-2-6-0': 'Let [MATH] and zero-out all entries of [MATH] that are smaller (in absolute value) than [MATH].', '1006.0407-2-7-0': 'Set [MATH] as in Eqn. [REF].', '1006.0407-2-8-0': 'For [MATH] (i.i.d. trials with replacement) randomly sample indices [MATH] (entries of [MATH]), with', '1006.0407-2-9-0': '[EQUATION]', '1006.0407-2-9-1': 'Output: [EQUATION].', '1006.0407-2-10-0': 'In Algorithm [MATH], we let [MATH] denote the standard basis vectors for [MATH] (see Section [REF] for more notation).', '1006.0407-2-10-1': 'Our sampling procedure selects [MATH] entries from [MATH] (note that [MATH] from the description of Algorithm [MATH] is simply [MATH], but with elements less than or equal to [MATH] zeroed out) in [MATH] independent, identically distributed (i.i.d.) trials with replacement.', '1006.0407-2-10-2': 'In each trial, elements of [MATH] are retained with probability proportional to their squared magnitude.', '1006.0407-2-10-3': 'Note that the same element of [MATH] could be selected multiple times and that [MATH] contains at most [MATH] non-zero entries.', '1006.0407-2-10-4': 'Theorem [REF] is our main quality-of-approximation result for Algorithm [MATH] and achieves sparsity bounds proportional to [MATH].', '1006.0407-2-11-0': 'Let [MATH] be any matrix, let [MATH] be an accuracy parameter, and let [MATH] be the sparse sketch of [MATH] constructed via Algorithm 1.', '1006.0407-2-11-1': 'If [EQUATION] then, with probability at least [MATH], [EQUATION] [MATH] has at most [MATH] non-zero entries and the construction of [MATH] can be implemented in one pass over the input matrix [MATH] (see Section [REF]).', '1006.0407-2-12-0': 'We conclude this section with Corollary [REF], which is a re-statement of Theorem [REF] involving the stable rank of [MATH], denoted by [MATH] (recall that the stable rank of any matrix [MATH] is defined as the ratio [MATH], which is upper bounded by the rank of [MATH]).', '1006.0407-2-12-1': 'The corollary guarantees relative error approximations for matrices of - say - constant stable rank, such as the ones that arise in [CITATION].', '1006.0407-2-13-0': 'Let [MATH] be any matrix and let [MATH] be an accuracy parameter.', '1006.0407-2-13-1': 'Let [MATH] be the sparse sketch of [MATH] constructed via Algorithm 1 (with [MATH]).', '1006.0407-2-13-2': 'If', '1006.0407-2-14-0': '[MATH]', '1006.0407-2-15-0': 'then, with probability at least [MATH], [EQUATION].', '1006.0407-2-16-0': 'It is worth noting that the sampling algorithm implied by Corollary [REF] can not be implemented in one pass, since we would need a priori knowledge of the spectral norm of [MATH] in order to implement Step [MATH] of Algorithm [MATH].', '1006.0407-2-17-0': '# Related Work', '1006.0407-2-18-0': 'In this section (as well as in Table [REF]), we present a head-to-head comparison of our result with all existing (to the best of our knowledge) bounds on matrix sparsification.', '1006.0407-2-18-1': 'In [CITATION] the authors presented a sampling method that requires in expectation [MATH] non-zero entries in [MATH] in order to achieve an accuracy guarantee [MATH] with a failure probability of at most [MATH].', '1006.0407-2-18-2': 'Compared with our result, their bound holds only when [MATH] and, in this range, our bounds are superior when [MATH].', '1006.0407-2-18-3': 'It is worth mentioning that the constant involved in [CITATION] is two orders of magnitude larger than ours and, more importantly, that the results of [CITATION] hold only when [MATH].', '1006.0407-2-19-0': 'In [CITATION], the authors study the [MATH] and [MATH] norms in the matrix sparsification context and they also present a sampling scheme analogous to ours.', '1006.0407-2-19-1': 'They achieve (in expectation) a sparsity bound of [MATH] when [MATH]; here [MATH].', '1006.0407-2-19-2': 'Thus, our results are superior (in the above range of [MATH]) when [MATH].', '1006.0407-2-20-0': 'It is harder to compare our method to the work of [CITATION], which depends on the [MATH].', '1006.0407-2-20-1': 'The latter quantity is, in general, upper bounded only by [MATH], in which case the sampling complexity of [CITATION] is much worse, namely [MATH].', '1006.0407-2-21-0': "Finally, the recent bounds on matrix sparsification via the non-commutative Khintchine's inequality in [CITATION] are inferior compared to ours in terms of sparsity guarantees by at least [MATH].", '1006.0407-2-21-1': 'However, we should mention that the bounds of [CITATION] can be extended to multi-dimensional matrices (tensors), whereas our result does not generalize to this setting; see [CITATION] for details.', '1006.0407-2-22-0': '# Background', '1006.0407-2-23-0': '## Notation', '1006.0407-2-24-0': 'We let [MATH] denote the set [MATH].', '1006.0407-2-24-1': 'We will use the notation [MATH] to denote the probability of the event in the parentheses and [MATH] to denote the expectation of a random variable [MATH].', '1006.0407-2-24-2': 'When [MATH] is a matrix, [MATH] denotes the element-wise expectation of each entry of [MATH].', '1006.0407-2-24-3': 'For a matrix [MATH], [MATH] will denote the [MATH]-th column of [MATH] as a column vector and, similarly, [MATH] will denote the [MATH]-th row of [MATH] as a row vector (for any [MATH] or [MATH] in [MATH]).', '1006.0407-2-24-4': 'The Frobenius norm [MATH] of the matrix [MATH] is defined as', '1006.0407-2-25-0': '[MATH]', '1006.0407-2-26-0': 'and the spectral norm [MATH] of the matrix [MATH] is defined as', '1006.0407-2-27-0': '[MATH].', '1006.0407-2-28-0': 'For two symmetric matrices [MATH] we say that [MATH] if and only if [MATH] is a positive semi-definite matrix.', '1006.0407-2-28-1': 'Finally, [MATH] denotes the identity matrix of size [MATH] and [MATH] denotes the natural logarithm of [MATH].', '1006.0407-2-29-0': '## Implementing the Sampling in one Pass over the Input Matrix', '1006.0407-2-30-0': 'We now discuss the implementation of Algorithm 1 in one pass over the input matrix [MATH].', '1006.0407-2-30-1': 'Towards that end, we will leverage (a slightly modified version of) Algorithm Select (p. 137 of [CITATION]).', '1006.0407-2-31-0': 'One-pass Select algorithm [1] Input: [MATH] for all [MATH], arbitrarily ordered and [MATH].', '1006.0407-2-31-1': '[MATH].', '1006.0407-2-31-2': 'For all [MATH] such that [MATH]', '1006.0407-2-32-0': 'Output: Return [MATH], [MATH] and [MATH].', '1006.0407-2-33-0': 'We note that Step [MATH] essentially operates on [MATH].', '1006.0407-2-33-1': 'Clearly, in a single pass over the data we can run in parallel [MATH] copies of the Select Algorithm (using a total of [MATH] memory) to effectively return [MATH] independent samples from [MATH].', '1006.0407-2-33-2': "Lemma [MATH] (page [MATH] of [CITATION], note that the sequence of the [MATH]'s is all-positive) guarantees that each of the [MATH] copies of Select returns a sample satisfying:", '1006.0407-2-34-0': '[EQUATION].', '1006.0407-2-35-0': 'Finally, in the parlance of Step [MATH] of Algorithm [MATH], [MATH] is set to [MATH] and [MATH] is set to [MATH] for all [MATH].', '1006.0407-2-36-0': '# Proof of Theorem [REF]', '1006.0407-2-37-0': 'The proof of Theorem [REF] will combine Lemmas [REF] and [REF] in order to bound [MATH] as follows: [EQUATION]', '1006.0407-2-37-1': 'The failure probability of Theorem [REF] emerges from Lemma [REF], which fails with probability at most [MATH] for the choice of [MATH] in Eqn. ([REF]).', '1006.0407-2-37-2': 'The proof of Lemma [REF] will involve an elegant matrix-valued Bernstein bound proven in [CITATION].', '1006.0407-2-37-3': 'See also [CITATION] or [CITATION] for similar bounds.', '1006.0407-2-38-0': '## Bounding [MATH]', '1006.0407-2-39-0': 'Using the notation of Algorithm 1, [MATH].', '1006.0407-2-40-0': 'Recall that the entries of [MATH] are either equal to the corresponding entries of [MATH] or they are set to zero if the corresponding entry of [MATH] is (in absolute value) smaller than [MATH].', '1006.0407-2-40-1': 'Thus, [EQUATION]', '1006.0407-2-41-0': '## Bounding [MATH]', '1006.0407-2-42-0': 'In order to prove our main result in this section (Lemma [REF]) we will leverage a powerful matrix-valued Bernstein bound originally proven in [CITATION] (Theorem 3.2).', '1006.0407-2-42-1': 'We restate this theorem, slightly rephrased to better suit our notation.', '1006.0407-2-43-0': 'Let [MATH] be independent, zero-mean random matrices in [MATH].', '1006.0407-2-43-1': 'Suppose [MATH] and [MATH] for all [MATH].', '1006.0407-2-43-2': 'Then, for any [MATH], [EQUATION] holds, subject to a failure probability of at most [EQUATION].', '1006.0407-2-44-0': 'In order to apply the above theorem, using the notation of Algorithm [MATH], we set [MATH] for all [MATH] to obtain [EQUATION]', '1006.0407-2-44-1': 'Let [MATH] denote the all-zeros matrix of size [MATH].', '1006.0407-2-44-2': 'It is easy to argue that [MATH] for all [MATH].', '1006.0407-2-44-3': 'Indeed, if we consider that [MATH] and [MATH] we obtain', '1006.0407-2-45-0': '[EQUATION].', '1006.0407-2-46-0': 'Our next lemma bounds [MATH] for all [MATH].', '1006.0407-2-47-0': 'Using our notation, [MATH] for all [MATH].', '1006.0407-2-48-0': 'First, using the definition of [MATH] and the fact that [MATH], [EQUATION].', '1006.0407-2-49-0': 'The last inequality follows since all entries of [MATH] are at least [MATH] and the fact that [MATH].', '1006.0407-2-49-1': 'We can now assume that [EQUATION] to conclude the proof of the lemma.', '1006.0407-2-49-2': 'To justify our assumption in Eqn. ([REF]), we note that if it is violated, then it must be the case that [MATH].', '1006.0407-2-49-3': 'If that were true, then all entries of [MATH] would be equal to zero.', '1006.0407-2-49-4': '(Recall that all entries of [MATH] are either zero or, in absolute value, larger than [MATH].)', '1006.0407-2-49-5': 'Also, if [MATH] were identically zero, then (i) [MATH] would also be identically zero and, (ii) all entries of [MATH] would be at most [MATH].', '1006.0407-2-49-6': 'Thus, [EQUATION].', '1006.0407-2-50-0': 'Thus, if the assumption of Eqn. ([REF]) is not satisfied, the resulting all-zeros [MATH] still satisfies Theorem [REF].', '1006.0407-2-51-0': 'Our next step towards applying Theorem [REF] involves bounding the spectral norm of the expectation of [MATH].', '1006.0407-2-51-1': 'The spectral norm of the expectation of [MATH] admits a similar analysis and the same bound and is omitted.', '1006.0407-2-52-0': 'Using our notation, [MATH] for any [MATH].', '1006.0407-2-53-0': 'We start by evaluating [MATH]; recall that [MATH]: [EQUATION] where [MATH] is the number of non-zeroes of the [MATH]-th row of [MATH].', '1006.0407-2-54-0': 'We now simplify the above result using a few simple observations: [MATH], [MATH], [MATH], and [MATH].', '1006.0407-2-54-1': 'Thus, we get [EQUATION]', '1006.0407-2-54-2': "Since [MATH] and using Weyl's inequality (Theorem [MATH] of [CITATION]), which states that by adding a positive semi-definite matrix to a symmetric matrix all its eigenvalues will increase, we get that [EQUATION].", '1006.0407-2-55-0': 'Consequently [MATH].', '1006.0407-2-56-0': 'We can now apply Theorem [REF] on Eqn. ([REF]) with [MATH], [MATH] (Lemma [REF]), and [MATH] (Lemma [REF]) .', '1006.0407-2-56-1': 'Thus, we get that [MATH] holds, subject to a failure probability of at most [EQUATION].', '1006.0407-2-57-0': 'Bounding the failure probability by [MATH] and solving for [MATH], we get that [EQUATION]', '1006.0407-2-57-1': 'Using [MATH] (by construction) concludes the proof of the following lemma, which is the main result of this section.', '1006.0407-2-58-0': 'Using the notation of Algorithm [MATH], if', '1006.0407-2-59-0': '[MATH]', '1006.0407-2-60-0': 'then, with probability at least [MATH], [EQUATION]'}

[['1006.0407-1-0-0', '1006.0407-2-0-0'], ['1006.0407-1-44-0', '1006.0407-2-49-0'], ['1006.0407-1-44-1', '1006.0407-2-49-1'], ['1006.0407-1-44-2', '1006.0407-2-49-2'], ['1006.0407-1-44-3', '1006.0407-2-49-3'], ['1006.0407-1-44-4', '1006.0407-2-49-4'], ['1006.0407-1-44-5', '1006.0407-2-49-5'], ['1006.0407-1-12-1', '1006.0407-2-13-1'], ['1006.0407-1-43-0', '1006.0407-2-48-0'], ['1006.0407-1-20-0', '1006.0407-2-26-0'], ['1006.0407-1-29-0', '1006.0407-2-35-0'], ['1006.0407-1-9-1', '1006.0407-2-11-1'], ['1006.0407-1-10-2', '1006.0407-2-20-0'], ['1006.0407-1-10-3', '1006.0407-2-20-1'], ['1006.0407-1-18-0', '1006.0407-2-24-0'], ['1006.0407-1-18-1', '1006.0407-2-24-1'], ['1006.0407-1-18-2', '1006.0407-2-24-2'], ['1006.0407-1-18-3', '1006.0407-2-24-3'], ['1006.0407-1-18-4', '1006.0407-2-24-4'], ['1006.0407-1-39-1', '1006.0407-2-44-2'], ['1006.0407-1-35-0', '1006.0407-2-40-0'], ['1006.0407-1-46-0', '1006.0407-2-51-0'], ['1006.0407-1-22-0', '1006.0407-2-28-1'], ['1006.0407-1-24-0', '1006.0407-2-30-0'], ['1006.0407-1-24-1', '1006.0407-2-30-1'], ['1006.0407-1-11-2', '1006.0407-2-12-1'], ['1006.0407-1-45-0', '1006.0407-2-50-0'], ['1006.0407-1-25-2', '1006.0407-2-31-2'], ['1006.0407-1-27-0', '1006.0407-2-33-0'], ['1006.0407-1-27-1', '1006.0407-2-33-1'], ['1006.0407-1-4-0', '1006.0407-2-6-0'], ['1006.0407-1-6-0', '1006.0407-2-8-0'], ['1006.0407-1-8-3', '1006.0407-2-10-3'], ['1006.0407-1-49-2', '1006.0407-2-57-1'], ['1006.0407-1-37-1', '1006.0407-2-42-1'], ['1006.0407-1-2-1', '1006.0407-2-2-1'], ['1006.0407-1-2-3', '1006.0407-2-3-0'], ['1006.0407-1-48-2', '1006.0407-2-54-1'], ['1006.0407-1-48-4', '1006.0407-2-56-0'], ['1006.0407-1-48-5', '1006.0407-2-56-1'], ['1006.0407-1-38-0', '1006.0407-2-43-0'], ['1006.0407-1-38-1', '1006.0407-2-43-1'], ['1006.0407-1-38-2', '1006.0407-2-43-2'], ['1006.0407-1-15-0', '1006.0407-2-16-0'], ['1006.0407-1-46-1', '1006.0407-2-51-1'], ['1006.0407-1-25-0', '1006.0407-2-31-0'], ['1006.0407-1-8-0', '1006.0407-2-10-0'], ['1006.0407-1-8-1', '1006.0407-2-10-1'], ['1006.0407-1-8-2', '1006.0407-2-10-2'], ['1006.0407-1-49-0', '1006.0407-2-57-0'], ['1006.0407-1-37-0', '1006.0407-2-42-0'], ['1006.0407-1-2-0', '1006.0407-2-2-0'], ['1006.0407-1-2-2', '1006.0407-2-2-2'], ['1006.0407-1-48-1', '1006.0407-2-54-0'], ['1006.0407-1-31-0', '1006.0407-2-37-0'], ['1006.0407-1-32-0', '1006.0407-2-37-1'], ['1006.0407-1-32-1', '1006.0407-2-37-2'], ['1006.0407-1-0-1', '1006.0407-2-0-1'], ['1006.0407-1-12-0', '1006.0407-2-13-0'], ['1006.0407-1-9-0', '1006.0407-2-11-0'], ['1006.0407-1-39-0', '1006.0407-2-44-0'], ['1006.0407-1-11-0', '1006.0407-2-12-0'], ['1006.0407-1-11-1', '1006.0407-2-12-0'], ['1006.0407-1-8-4', '1006.0407-2-10-4'], ['1006.0407-1-48-0', '1006.0407-2-53-0']]

[['1006.0407-1-0-0', '1006.0407-2-0-0'], ['1006.0407-1-44-0', '1006.0407-2-49-0'], ['1006.0407-1-44-1', '1006.0407-2-49-1'], ['1006.0407-1-44-2', '1006.0407-2-49-2'], ['1006.0407-1-44-3', '1006.0407-2-49-3'], ['1006.0407-1-44-4', '1006.0407-2-49-4'], ['1006.0407-1-44-5', '1006.0407-2-49-5'], ['1006.0407-1-12-1', '1006.0407-2-13-1'], ['1006.0407-1-43-0', '1006.0407-2-48-0'], ['1006.0407-1-20-0', '1006.0407-2-26-0'], ['1006.0407-1-29-0', '1006.0407-2-35-0'], ['1006.0407-1-9-1', '1006.0407-2-11-1'], ['1006.0407-1-10-2', '1006.0407-2-20-0'], ['1006.0407-1-10-3', '1006.0407-2-20-1'], ['1006.0407-1-18-0', '1006.0407-2-24-0'], ['1006.0407-1-18-1', '1006.0407-2-24-1'], ['1006.0407-1-18-2', '1006.0407-2-24-2'], ['1006.0407-1-18-3', '1006.0407-2-24-3'], ['1006.0407-1-18-4', '1006.0407-2-24-4'], ['1006.0407-1-39-1', '1006.0407-2-44-2'], ['1006.0407-1-35-0', '1006.0407-2-40-0'], ['1006.0407-1-46-0', '1006.0407-2-51-0'], ['1006.0407-1-22-0', '1006.0407-2-28-1'], ['1006.0407-1-24-0', '1006.0407-2-30-0'], ['1006.0407-1-24-1', '1006.0407-2-30-1'], ['1006.0407-1-11-2', '1006.0407-2-12-1'], ['1006.0407-1-45-0', '1006.0407-2-50-0'], ['1006.0407-1-25-2', '1006.0407-2-31-2'], ['1006.0407-1-27-0', '1006.0407-2-33-0'], ['1006.0407-1-27-1', '1006.0407-2-33-1'], ['1006.0407-1-4-0', '1006.0407-2-6-0'], ['1006.0407-1-6-0', '1006.0407-2-8-0'], ['1006.0407-1-8-3', '1006.0407-2-10-3'], ['1006.0407-1-49-2', '1006.0407-2-57-1'], ['1006.0407-1-37-1', '1006.0407-2-42-1'], ['1006.0407-1-2-1', '1006.0407-2-2-1'], ['1006.0407-1-2-3', '1006.0407-2-3-0'], ['1006.0407-1-48-2', '1006.0407-2-54-1'], ['1006.0407-1-48-4', '1006.0407-2-56-0'], ['1006.0407-1-48-5', '1006.0407-2-56-1'], ['1006.0407-1-38-0', '1006.0407-2-43-0'], ['1006.0407-1-38-1', '1006.0407-2-43-1'], ['1006.0407-1-38-2', '1006.0407-2-43-2']]

[['1006.0407-1-15-0', '1006.0407-2-16-0'], ['1006.0407-1-46-1', '1006.0407-2-51-1'], ['1006.0407-1-25-0', '1006.0407-2-31-0'], ['1006.0407-1-8-0', '1006.0407-2-10-0'], ['1006.0407-1-8-1', '1006.0407-2-10-1'], ['1006.0407-1-8-2', '1006.0407-2-10-2'], ['1006.0407-1-49-0', '1006.0407-2-57-0'], ['1006.0407-1-37-0', '1006.0407-2-42-0'], ['1006.0407-1-2-0', '1006.0407-2-2-0'], ['1006.0407-1-2-2', '1006.0407-2-2-2'], ['1006.0407-1-48-1', '1006.0407-2-54-0'], ['1006.0407-1-31-0', '1006.0407-2-37-0'], ['1006.0407-1-32-0', '1006.0407-2-37-1'], ['1006.0407-1-32-1', '1006.0407-2-37-2']]

[]

[['1006.0407-1-0-1', '1006.0407-2-0-1'], ['1006.0407-1-12-0', '1006.0407-2-13-0'], ['1006.0407-1-9-0', '1006.0407-2-11-0'], ['1006.0407-1-39-0', '1006.0407-2-44-0'], ['1006.0407-1-11-0', '1006.0407-2-12-0'], ['1006.0407-1-11-1', '1006.0407-2-12-0'], ['1006.0407-1-8-4', '1006.0407-2-10-4'], ['1006.0407-1-48-0', '1006.0407-2-53-0']]

[]

['1006.0407-1-3-0', '1006.0407-1-5-0', '1006.0407-1-7-0', '1006.0407-1-7-1', '1006.0407-1-12-2', '1006.0407-1-13-0', '1006.0407-1-14-0', '1006.0407-1-19-0', '1006.0407-1-21-0', '1006.0407-1-25-1', '1006.0407-1-26-0', '1006.0407-1-27-2', '1006.0407-1-28-0', '1006.0407-1-34-0', '1006.0407-1-35-1', '1006.0407-1-39-2', '1006.0407-1-40-0', '1006.0407-1-42-0', '1006.0407-1-44-6', '1006.0407-1-47-0', '1006.0407-1-50-0', '1006.0407-1-51-0', '1006.0407-1-52-0', '1006.0407-2-4-0', '1006.0407-2-5-0', '1006.0407-2-7-0', '1006.0407-2-9-0', '1006.0407-2-9-1', '1006.0407-2-13-2', '1006.0407-2-14-0', '1006.0407-2-15-0', '1006.0407-2-25-0', '1006.0407-2-27-0', '1006.0407-2-31-1', '1006.0407-2-32-0', '1006.0407-2-33-2', '1006.0407-2-34-0', '1006.0407-2-39-0', '1006.0407-2-40-1', '1006.0407-2-45-0', '1006.0407-2-46-0', '1006.0407-2-47-0', '1006.0407-2-49-6', '1006.0407-2-52-0', '1006.0407-2-55-0', '1006.0407-2-58-0', '1006.0407-2-59-0', '1006.0407-2-60-0']

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

https://arxiv.org/abs/1006.0407

1509.05652

{'1509.05652-1-0-0': '# Introduction', '1509.05652-1-1-0': 'After more than 100 years of cosmic-ray measurements, the composition and the energy spectrum is relatively well known only in the energy range [MATH]eV due to direct measurements.', '1509.05652-1-1-1': 'At higher energies the flux of the cosmic-ray is too low to perform statistically significant direct measurements.', '1509.05652-1-1-2': 'Consequently, the knowledge is poorer and relies on the measurement of secondary-particle cascades called air-showers.', '1509.05652-1-1-3': 'These indirect cosmic-ray measurements use the atmosphere as calorimeter.', '1509.05652-1-1-4': 'The energy of the primary particle can be estimated from the energy contained in the shower, the type of the primary particle, for example, from the longitudinal shower development.', '1509.05652-1-1-5': 'Heavy primary particles, like iron nuclei, on average interact earlier in the atmosphere than light particles, like protons.', '1509.05652-1-1-6': 'Thus, the composition can be statistically estimated from measurements of the atmospheric depth of the shower maximum, [MATH].', '1509.05652-1-2-0': 'Established air-shower techniques are the measurement of secondary particles on ground, the measurement of fluorescence or air-Cherenkov light by dedicated telescopes or non-imaging detectors on ground.', '1509.05652-1-2-1': 'The latter two methods have a relatively high accuracy for the energy and for [MATH], but are limited to dark and clear nights.', '1509.05652-1-2-2': 'Detection of the radio emission by air showers is an additional technique, which does not suffer from this intrinsic limitation of the exposure.', '1509.05652-1-2-3': 'The radio signal is sensitive to the shower energy [CITATION] and [MATH] [CITATION], too, but it has not yet been demonstrated experimentally that the accuracy for both variables is competitive with the other techniques, as indicated by simulation-based studies, e.g., [CITATION].', '1509.05652-1-3-0': 'The main origin of the radio emission is the geomagnetic deflection of relativistic electrons and positrons in the shower inducing a time-variable current [CITATION].', '1509.05652-1-3-1': 'The amplitude of this emission is proportional to the shower energy and to [MATH], with [MATH] the angle between the geomagnetic field and the shower direction.', '1509.05652-1-3-2': 'On a weaker level the Askaryan effect contributes, i.e., radio emission due to the time variation of the net charge excess in the shower front [CITATION].', '1509.05652-1-3-3': 'The interference of both effects leads to an azimuthal asymmetry of the radio footprint [CITATION].', '1509.05652-1-3-4': 'Finally, the refractive index of air causes Cherenkov-like effects for both emission mechanisms [CITATION].', '1509.05652-1-4-0': 'Tunka-Rex is the radio extension of the Tunka observatory for cosmic-ray air showers [CITATION].', '1509.05652-1-4-1': 'Its main detector, Tunka-133, is an array of non-imaging photomultipliers detecting the Cherenkov light emitted by the air-showers in the atmosphere in the energy range of [MATH]eV.', '1509.05652-1-4-2': 'For Tunka-133 measurements, the energy of the primary particle [MATH] is reconstructed from the flux of the Cherenkov light at [MATH]m distance to the shower axis, and [MATH] is reconstructed from the steepness of the amplitude-distance function.', '1509.05652-1-4-3': 'Using these methods, Tunka-133 features an energy resolution of about [MATH], and an [MATH] resolution of about [MATH]g/cm2 [CITATION].', '1509.05652-1-4-4': 'For comparison, the average difference between the extreme cases of a pure proton and a pure iron composition is in the order of [MATH]g/cm2.', '1509.05652-1-4-5': 'Since Tunka-133 triggers Tunka-Rex, the same air showers are measured simultaneously with the air-Cherenkov and the radio detector.', '1509.05652-1-5-0': 'These combined measurements are used for cross-check of both methods.', '1509.05652-1-5-1': 'In particular the precision of Tunka-Rex is estimated by a comparison of the energy and [MATH] reconstructions to Tunka-133.', '1509.05652-1-5-2': 'Reconstruction methods for Tunka-Rex have been developed with CoREAS simulations [CITATION] (the radio extension of CORSIKA [CITATION]) taking into account previous theoretical predictions [CITATION], and experience by other experiments, in particular LOPES [CITATION], AERA [CITATION] and LOFAR [CITATION].', '1509.05652-1-6-0': 'The parameters in the reconstruction methods have been determined from the simulations and not tuned against the Tunka-133 measurements.', '1509.05652-1-6-1': 'If at all, there might be an implicit tuning on the Tunka-133 results: during development of the reconstruction methods and definition of quality cuts we compared the energy and [MATH] values resulting for Tunka-Rex with the values of Tunka-133.', '1509.05652-1-6-2': 'These have been available for us for the data recorded from October 2012 to April 2013 (effectively 280 hours of measurements).', '1509.05652-1-6-3': 'Results of this tuning season are available in reference [CITATION].', '1509.05652-1-6-4': 'They showed that energy and [MATH] of Tunka-Rex and Tunka-133 agree well within uncertainties on an absolute level, when using all parameters for the Tunka-Rex reconstruction directly as they come out of the CoREAS simulation study.', '1509.05652-1-6-5': 'For this reason, we did not follow our original idea of a cross-calibration, i.e., an explicit tuning of the radio reconstruction against the air-Cherenkov reconstruction.', '1509.05652-1-6-6': 'Instead, we use the methods and the absolute scale of Tunka-Rex as resulting from the CoREAS simulation study.', '1509.05652-1-7-0': 'We applied these methods also to the second season of Tunka-Rex (October 2013 - April 2014, effectively 260 hours of measurements), for which the Tunka-133 reconstruction of energy and shower maximum was blinded.', '1509.05652-1-7-1': 'These Tunka-Rex predictions of the energy and shower maximum have been uploaded to a public webpage before unblinding (see appendix [REF] for details).', '1509.05652-1-7-2': 'After unblinding we now present the comparison of Tunka-Rex and Tunka-133 for both seasons.', '1509.05652-1-7-3': 'Doing so, this paper is focused on the experimental results demonstrating the feasibility of the reconstruction methods, which are described in more detail in reference [CITATION].', '1509.05652-1-8-0': '# Experimental setup and data selection', '1509.05652-1-9-0': 'Tunka-Rex started in autumn 2012 with 21 antenna stations (but 3 of the outer stations were not yet operated), and in summer 2013, 4 further antennas were added giving a total number of 25 antenna stations used for the present analysis.', '1509.05652-1-9-1': 'In the central area covering about [MATH]km2 the spacing is approximately [MATH]m.', '1509.05652-1-9-2': 'The outer antennas contribute only to a few events (see figure [REF]).', '1509.05652-1-9-3': 'The air-Cherenkov array Tunka-133 covers the inner area with 133 photomultipliers, and features additional photomultipliers in the outer area.', '1509.05652-1-9-4': 'Upon a coincidence trigger of the photomultipliers, both the radio and the air-Cherenkov detector are read out in parallel.', '1509.05652-1-9-5': 'The electric field of the radio signal measured by Tunka-Rex is reconstructed in an effective bandwidth of [MATH]MHz.', '1509.05652-1-9-6': 'Details on the detector setup and its calibration can be read in reference [CITATION].', '1509.05652-1-10-0': 'The reconstruction of Tunka-133 is reliable for events with zenith angles [MATH], which are used here.', '1509.05652-1-10-1': 'For these events the Tunka-133 reconstruction of the shower provides: energy of the primary particle [MATH], atmospheric depth of the shower maximum [MATH], and axis, i.e., the direction and point of incidence on the ground (= shower core).', '1509.05652-1-10-2': '[MATH] and [MATH] were blinded for the second season of data.', '1509.05652-1-10-3': 'For all Tunka-Rex events we used the shower core of Tunka-133 as input, because of the denser spacing of the photomultipliers compared to the antennas.', '1509.05652-1-10-4': 'The direction was only used as cross-check: events are considered false positive, i.e., contaminated by background pulses and excluded from the analysis, if the directions reconstructed from the radio and air-Cherenkov arrival times deviate by more than [MATH] [CITATION].', '1509.05652-1-11-0': 'For selection of Tunka-Rex events we applied a quality cut based on the signal-to-noise ratio in individual antenna stations.', '1509.05652-1-11-1': 'The signal has to exceed a signal-to-noise ratio, [MATH] in at least three antenna stations, where the signal [MATH] is the maximum of an envelope on the electric-field strength of the pulse, and the noise [MATH] the root-mean square of the electric field in a time window before the signal.', '1509.05652-1-11-2': 'This cut is set such that pure background has a chance of about [MATH] to pass for an individual antenna, which balances efficiency and purity.', '1509.05652-1-11-3': 'Given 25 antenna stations, this leads to a non-negligible probability that in a true event antenna stations with false-positive signals are contained.', '1509.05652-1-11-4': 'To exclude most of these false-positive antenna stations, we sort the antennas by their distance of the shower axis.', '1509.05652-1-11-5': 'After two antenna stations failing the SNR cut, any further antenna stations are excluded from the analysis for the particular event.', '1509.05652-1-11-6': 'Finally, for [MATH] reconstruction we require at least one antenna above threshold at a distance [MATH]m to the shower axis, since [MATH] is reconstructed from the slope of the lateral distribution, which requires sufficient lever arm.', '1509.05652-1-12-0': "Reconstruction of the electric-field strengths (= amplitude) in individual antenna stations is done with a modified version of the 'Auger Offline' software developed by the Pierre Auger Collaboration [CITATION].", '1509.05652-1-12-1': 'This software applies all known experimental characteristics to the recorded signals, in particular the frequency-dependent gain and pulse distortion of the various components in the signal chain as determined by calibration measurements [CITATION].', '1509.05652-1-12-2': 'For the reconstruction of the electric-field vector the simulated direction dependence of the antenna is used, which has been validated for a few selected directions by the calibration measurements.', '1509.05652-1-13-0': 'For the suppression of narrow-band radio interferences (RFI) we apply rectangular band-stop filters at selected frequencies often contaminated by RFI background .', '1509.05652-1-13-1': 'Since the radio emission by the air shower is broadband, it is significantly less affected by the filter than the narrow-band disturbances.', '1509.05652-1-13-2': 'The background remaining after filtering still on average increases the measured pulse height [MATH] compared to the true pulse height [MATH], as already seen at the LOPES experiment [CITATION] and assumed for early experiments [CITATION].', '1509.05652-1-13-3': 'We studied this effect by adding measured background to CoREAS simulations (figure [REF]), and found that the parametrization given in reference [CITATION] has to be slightly modified introducing a normalization factor [MATH] adapted for Tunka-Rex.', '1509.05652-1-13-4': 'Thus, we use the following formula to correct for the average effect of noise: [EQUATION] with the signal-to-noise ratio [MATH] defined above.', '1509.05652-1-14-0': 'An overview of the available events statistics is provided in table [REF], and figure [REF] shows the distribution of the selected events over energy and [MATH].', '1509.05652-1-14-1': 'Most of the events used for energy reconstruction are at energies above [MATH]eV.', '1509.05652-1-14-2': 'For the high-quality events used for [MATH] reconstruction, the threshold is slightly higher: above an energy of about [MATH]eV, all events used for energy reconstruction pass also the high-quality cuts required for [MATH] reconstruction (see figure [REF]).', '1509.05652-1-14-3': 'The total number of Tunka-Rex events passing the quality cuts is larger [CITATION], since in contrast to the air-Cherenkov detection, radio detection is more efficient for more inclined events.', '1509.05652-1-14-4': 'Nevertheless, for the present analysis only events with [MATH] have been selected, since Tunka-133 is designed for this zenith angle range.', '1509.05652-1-14-5': 'Thus, the selection ensures that a valid reconstruction is available for both detectors.', '1509.05652-1-15-0': '# Reconstruction of energy and shower maximum', '1509.05652-1-16-0': 'In the present analysis, the reconstruction of energy and shower maximum is based on the lateral distribution, i.e., the dependence of the radio amplitude on the distance to the shower axis.', '1509.05652-1-16-1': 'This approach has been used earlier.', '1509.05652-1-16-2': 'Different experiments have shown that the amplitude at a certain distance is correlated with the primary energy, after correcting for the geomagnetic angle [CITATION].', '1509.05652-1-16-3': 'Moreover, it has been shown that the shape of the lateral distribution depends on the distance to the shower maximum [CITATION].', '1509.05652-1-17-0': 'Because of the different polarization of the radio emission generated by the Askaryan and the geomagnetic effects [CITATION], the radio amplitude on ground depends not only on the geomagnetic angle [MATH] and distance to the shower axis [MATH], but also on the azimuth [MATH] relative to the shower axis.', '1509.05652-1-17-1': 'Fitting the asymmetry for individual events by a two-dimensional lateral distribution function (LDF) introduces additional parameters [CITATION].', '1509.05652-1-17-2': 'To maintain events with low station multiplicity, we developed an alternative approach described in reference [CITATION], which is well-suited for sparse arrays like Tunka-Rex.', '1509.05652-1-18-0': 'Our approach makes use of the fact that the asymmetry depends in first order only on the shower geometry, which can be reconstructed independent from the LDF by arrival time measurements.', '1509.05652-1-18-1': 'Based on CoREAS simulations made for the Tunka-Rex setup, the azimuthal asymmetry of the footprint is parametrized as function of the antenna position using a constant typical value of [MATH] for the relative strength of the Askaryan effect.', '1509.05652-1-18-2': 'Using this parametrization, we correct all measured amplitudes in individual antennas for the azimuthal asymmetry, and obtain a one-dimensional lateral distribution in which the amplitude depends only on the distance to the shower axis.', '1509.05652-1-18-3': 'Simultaneously, we correct for the relative strength of the geomagnetic Lorentz force, since this depends only on the shower geometry, too, namely on the geomagnetic angle [MATH]: [EQUATION] with [MATH] the measured amplitude after noise correction by equation [REF], [MATH] the distance to the shower axis, [MATH] the azimuth angle relative to the shower axis (with [MATH], when a station is in the direction of the geomagnetic Lorentz force), [MATH] the geomagnetic angle, [MATH] the size of the asymmetry, and [MATH] the amplitude after correction for the asymmetry and the geomagnetic angle.', '1509.05652-1-18-4': 'After this correction of the amplitude measured at each antenna station, the lateral distribution can be fitted with a one dimensional function whose parameters depend on energy and distance to the shower maximum.', '1509.05652-1-19-0': 'The amplitude (= electric field strength) scale of this one-dimensional lateral distribution is proportional to the shower energy, because the radio emission in the shower is coherent.', '1509.05652-1-19-1': 'Moreover, the shape of the lateral distribution depends mainly on the distance to the shower maximum and has two main features: a bump caused by Cherenkov-like effects [CITATION], whose existence, size and position depend on the distance to the shower maximum, and an exponential tail [CITATION].', '1509.05652-1-19-2': 'This means that even a one-dimensional LDF should contain at least three free parameters: one for the energy scale, and two for the shape, namely the width of the bump, and the slope of exponential fall-off.', '1509.05652-1-19-3': 'For this purpose we use the following lateral distribution function: [EQUATION]', '1509.05652-1-19-4': 'The shape parameter [MATH] determines the slope of the exponential fall-off, and the shape parameter [MATH] is related to the width of the Cherenkov bump.', '1509.05652-1-19-5': 'The distance parameter [MATH] does not determine the function, but it affects the values for the fitted free parameters, their uncertainties and correlations.', '1509.05652-1-19-6': 'We set this distance parameter [MATH] to [MATH]m for energy reconstruction, and to [MATH]m for [MATH] reconstruction, as this turned out to be the optimum values for energy and [MATH] reconstruction in the simulation study (see figure [REF] for an example).', '1509.05652-1-20-0': 'In the CoREAS simulations made for Tunka-Rex we observe that both shape parameters, [MATH] and [MATH], are correlated, which complicates the reconstruction of [MATH].', '1509.05652-1-20-1': 'Since [MATH] depends significantly stronger on the distance to the shower maximum than the shape parameter [MATH], we decided to express [MATH] as function of the zenith angle [MATH] and the primary energy [MATH].', '1509.05652-1-20-2': 'The applied parametrization [MATH] has been defined by studying the dependence in the CoREAS simulations and is given in appendix [REF].', '1509.05652-1-20-3': 'After fixing [MATH], the LDF is fit to the data points, and the remaining free shape parameter [MATH] carries the whole remaining [MATH] sensitivity of the LDF.', '1509.05652-1-20-4': 'Furthermore, reducing the number of free parameters in the LDF from three to two brings the additional advantage that we can use events with only three antenna stations above threshold (about half of the Tunka-Rex events in the present analysis).', '1509.05652-1-21-0': 'The energy [MATH] is determined using a simplified form of the equation presented in reference [CITATION].', '1509.05652-1-21-1': 'The simplification is to assume that the energy is exactly proportional to the amplitude, as expected for [MATH] coherent emission, since we find no evidence in the data justifying a different assumption (for this reason we here use a different value for the constant [MATH]).', '1509.05652-1-21-2': 'For [MATH] we use the equation of reference [CITATION] without any simplifications: [EQUATION] where [MATH]g/cm[MATH] is the atmospheric depth of the detector (typical value for the site), and the parameters are determined with CoREAS simulations: [MATH], [MATH]g/cm[MATH], [MATH]g/cm[MATH], [MATH] given in units of meters, and [MATH].', '1509.05652-1-22-0': 'To summarize the reconstruction procedure:', '1509.05652-1-23-0': 'We correct the lateral distribution for the azimuthal asymmetry and the geomagnetic angle [MATH].', '1509.05652-1-23-1': 'We fix the shape parameter [MATH] of the Gaussian LDF describing the Cherenkov bump of the lateral distribution.', '1509.05652-1-23-2': 'We fit the Gaussian LDF with the two remaining free parameters [MATH] and [MATH] describing the amplitude scale and the slope of the exponential tail, respectively.', '1509.05652-1-23-3': 'For each event, this fit is performed twice: once setting the reference distance [MATH]m, and once setting [MATH]m.', '1509.05652-1-23-4': 'We use the fit result with [MATH]m to reconstruct the shower energy [MATH] from the amplitude parameter [MATH], and the fit result with [MATH]m to reconstruct the atmospheric depth of the shower maximum [MATH] from the shape parameter [MATH].', '1509.05652-1-24-0': '# Results', '1509.05652-1-25-0': 'In reference [CITATION] it was already shown with simulations that the reconstruction methods for energy and [MATH] perform well.', '1509.05652-1-25-1': 'This section presents the experimental result of Tunka-Rex measurements and the comparison to the Tunka-133 air-Cherenkov measurements.', '1509.05652-1-26-0': 'There is a clear correlation between the energy reconstructed from the radio amplitude measured by Tunka-Rex and the energy reconstructed from the air-Cherenkov light measured by Tunka-133 (figure [REF]).', '1509.05652-1-26-1': 'The average deviation between Tunka-Rex and Tunka-133 is almost equal for the tuning and prediction seasons (figure [REF]).', '1509.05652-1-26-2': 'This indicates that energy reconstruction works reliable with the presented method.', '1509.05652-1-26-3': 'When combining both seasons, the average deviation between the Tunka-Rex and Tunka-133 energy values is [MATH], which is only slightly larger than the [MATH] energy resolution reported by Tunka-133 [CITATION].', '1509.05652-1-27-0': 'Also for the position of the shower maximum there is a clear correlation between the radio and air-Cherenkov reconstructions (figure [REF]).', '1509.05652-1-27-1': 'Due to the stricter quality cuts the event statistics is lower than for the energy correlation.', '1509.05652-1-27-2': 'Still, the correlation is significant for both seasons.', '1509.05652-1-27-3': 'This again indicates that also the reconstruction of [MATH] works reliable for the selected, high-quality radio events.', '1509.05652-1-27-4': 'Because of the lower statistics, the [MATH] reconstruction precision of Tunka-Rex cannot be estimated very accurately.', '1509.05652-1-27-5': 'The average difference between the Tunka-Rex and Tunka-133 reconstructions is consistent within statistical uncertainties for both seasons (figure [REF]).', '1509.05652-1-27-6': 'Combining both seasons, the average [MATH] difference between both detectors is ([MATH]g/cm2, which reflects the combination of the unknown Tunka-Rex uncertainty and the Tunka-133 uncertainty of [MATH]g/cm2 [CITATION].', '1509.05652-1-27-7': 'This is consistent with the applied quality cut on the Tunka-Rex fit uncertainty [MATH]g/cm2.', '1509.05652-1-28-0': 'A posteriori we discovered that the average [MATH] resolution is degraded by five events with shower core outside of the central area (dashed circle in figure [REF]).', '1509.05652-1-28-1': 'The average [MATH] difference between both detectors is ([MATH]g/cm2 for the 42 events inside the central area.', '1509.05652-1-28-2': 'This corresponds to a radio-only precision of better than [MATH]g/cm2, when assuming that the resolutions of Tunka-133 and Tunka-Rex add in squares to the total deviation.', '1509.05652-1-28-3': 'Finally, changing the quality cut on [MATH] can increase the combined resolution or alternatively the event statistics, on the cost of the other, which provides the possibility for optimization on the quality and quantity needs of specific science goals.', '1509.05652-1-29-0': 'For both, the energy and [MATH] reconstruction, the observed correlation is statistically compatible with a 1:1 correlation of both experiments: the mean of all Gaussians fitted in figures [REF] and [REF] is compatible with 0, where 0 corresponds to no bias (= systematic scale offset).', '1509.05652-1-29-1': 'This means that the energy and [MATH] scales of both detectors agree on an absolute level within the measurement uncertainty.', '1509.05652-1-29-2': 'This is remarkable, since the Tunka-Rex scale is defined by CoREAS simulations and has not been tuned against Tunka-133.', '1509.05652-1-29-3': 'Thus, it will be worthwhile to study in future, whether the agreement is coincidence, or whether some deeper, not-obvious relation causes equal absolute scales for Tunka-133 and Tunka-Rex.', '1509.05652-1-29-4': 'Independent of the reason, with already consistent scales there is no need for a true cross-calibration, i.e., en explicit calibration of both detectors against each other.', '1509.05652-1-30-0': '# Discussion', '1509.05652-1-31-0': '## Validity of the results', '1509.05652-1-32-0': 'The present analysis is the first event-by-event comparison of radio and air-Cherenkov measurements of the same air showers.', '1509.05652-1-32-1': 'Since both detection methods are sensitive to the energy and the position of the maximum of the electromagnetic shower component, a strong correlation is expected between radio and air-Cherenkov observables.', '1509.05652-1-32-2': 'This correlation is indeed seen for the reconstructed energy and the reconstructed distance to the shower maximum in two independent data sets of the tuning and the prediction season.', '1509.05652-1-33-0': 'This is especially noteworthy, since the Tunka-Rex reconstruction is based on simulations, but not tuned against the Tunka-133 measurements we compared to.', '1509.05652-1-33-1': 'Even for the tuning season, only details of the method have been decided after looking to the data, e.g., the filter used to remove narrow-band interferences or the exact quality cuts.', '1509.05652-1-33-2': 'Neither of these small decisions made the [MATH] and energy correlations appear or vanish.', '1509.05652-1-33-3': 'With the blind analysis approach on the independent data set of the second season, we definitely confirmed that the correlations are real.', '1509.05652-1-34-0': '## Precision and accuracy', '1509.05652-1-35-0': 'The average deviation between the Tunka-Rex and Tunka-133 reconstruction values for the same events can be used to estimate the reconstruction precision of Tunka-Rex.', '1509.05652-1-35-1': 'Assuming independent Gaussian uncertainties of both reconstructions, the average deviation between Tunka-Rex and Tunka-133 is the quadratic sum of both uncertainties.', '1509.05652-1-35-2': 'If Tunka-Rex would feature the same precision as Tunka-133, thus, the average deviation should be a factor of [MATH] larger than the Tunka-133 precision alone, i.e.: [MATH] for the energy, and [MATH]g/cm[MATH]g/cm[MATH] for [MATH].', '1509.05652-1-36-0': 'Consequently, the energy precision of Tunka-Rex seems to be at least equal to Tunka-133.', '1509.05652-1-36-1': 'For [MATH], the Tunka-Rex precision is slightly worse than for Tunka-133.', '1509.05652-1-36-2': 'Nevertheless, by rejecting events outside of the central area and by using even stronger quality cuts on the radio reconstruction, the resulting deviation to Tunka-133 can be further reduced.', '1509.05652-1-36-3': 'Moreover, the [MATH] resolution depends on the quality cuts.', '1509.05652-1-36-4': 'With the chosen quality cuts the total statistics of events usable for [MATH] reconstruction is significantly lower than the statistics for energy reconstruction.', '1509.05652-1-36-5': 'This is basically a threshold effect (cf. figure [REF]): Due to the cosmic-ray flux falling steeply falling with energy, even a slight increase in energy threshold leads to a significant decrease in statistics.', '1509.05652-1-36-6': 'Nevertheless, at the highest energies ([MATH]eV) the statistics of events for energy and [MATH] reconstruction is equal and similar to the Tunka-133 statistics.', '1509.05652-1-36-7': 'This means, that at the highest energies almost all events have sufficient quality for an accurate [MATH] reconstruction from the radio signal.', '1509.05652-1-37-0': 'Hence, also the finally achievable [MATH] uncertainty of the radio technique might become equal to the one of the air-Cherenkov technique, when high-quality events are used.', '1509.05652-1-37-1': 'In the next year, the number of antennas will be further increased: at each Tunka cluster already now we have two antenna stations instead of one, and will deploy a third one next year.', '1509.05652-1-37-2': 'This enables an experimental test how the antenna density and the core resolution affect the [MATH] resolution.', '1509.05652-1-37-3': 'With more statistics and independent measurements of the electromagnetic and muonic shower components by the recently deployed array of particle detectors named Tunka-Grande [CITATION], also possible correlations and systematic uncertainties can be checked.', '1509.05652-1-38-0': 'In addition to the precision also the total accuracy is of importance.', '1509.05652-1-38-1': 'This is dominated by the [MATH] scale uncertainty of the amplitude calibration [CITATION], but at least two further systematic scale uncertainties have to be considered: First, if the fraction of the total primary energy going in the electromagnetic shower component would be simulated wrong by CORSIKA, the energy scale would be wrong by roughly the same value.', '1509.05652-1-38-2': 'However, the electromagnetic shower component is relatively well understood, and as estimated by other experiments [CITATION], this uncertainty on the invisible energy is only a few percent.', '1509.05652-1-38-3': 'Second, the parameters in the method slightly depend on the assumed mass composition of the primary particles, in particular also the parameter [MATH] determining the energy scale.', '1509.05652-1-38-4': 'For the present analysis, we determined [MATH] as average of proton and iron simulations with the hadronic interaction model QGSJET-II.04 [CITATION].', '1509.05652-1-38-5': 'If we would have taken a pure proton or a pure iron composition to determine [MATH] of equation [REF], the reconstructed energy would be [MATH] smaller or larger, respectively.', '1509.05652-1-38-6': 'Consequently, both uncertainties (invisible energy and composition dependence) are small compared to the scale uncertainty of the amplitude calibration of [MATH].', '1509.05652-1-38-7': 'Thus, the total accuracy of the Tunka-Rex energy scale should be slightly better than [MATH].', '1509.05652-1-39-0': 'The Tunka-Rex energy scale defined by CoREAS simulations agrees within the uncertainties to the Tunka-133 scale.', '1509.05652-1-39-1': 'This means that the measurements of Tunka-Rex and Tunka-133 are not only correlated, but in addition agree on an absolute level, which is another confirmation, that CoREAS seems to predict the radio signal correctly.', '1509.05652-1-39-2': 'It opens prospects to cross-calibrate the energy scales of different air-shower experiments via radio measurements, which can be a very economic add-on to existing detectors as shown by Tunka-Rex.', '1509.05652-1-40-0': '## Comparison with other experiments', '1509.05652-1-41-0': 'The energy precision of Tunka-Rex is of the same order or slightly better as that of other radio arrays, like AERA [CITATION], LOPES [CITATION], CODALEMA [CITATION], and LOFAR [CITATION].', '1509.05652-1-41-1': 'Also the scale uncertainty roughly corresponds to that of AERA [CITATION] and LOPES [CITATION], which compared the radio measurements to air-fluorescence and particle measurements on ground.', '1509.05652-1-41-2': 'For [MATH] reconstruction, the resolution of LOPES had been significantly worse, using a simpler method in a more radio-loud environment.', '1509.05652-1-41-3': 'The much denser array LOFAR is located in a more radio-quiet environment.', '1509.05652-1-41-4': 'Its [MATH] reconstruction is also based on CoREAS simulations using two different methods.', '1509.05652-1-41-5': 'With an [MATH] reconstruction based on a fitted LDF (like in our approach), the precision of [MATH] at LOFAR is only slightly better than our resolution [CITATION].', '1509.05652-1-41-6': 'However, with a more computing-intensive method using many simulations for each individual event [CITATION], the precision claimed by LOFAR is at least twice as good as ours.', '1509.05652-1-41-7': 'This indicates that in addition to the planned deployment of additional antennas, also further improvements in the reconstruction methods could increase the [MATH] resolution of Tunka-Rex.', '1509.05652-1-42-0': 'The real scientific value of the present analysis is the experimental comparision of radio measurements against another, established technique.', '1509.05652-1-42-1': 'For the first time, energy and [MATH] reconstructions based on absolutely-calibrated radio measurements have been compared to air-Cherenkov measurements.', '1509.05652-1-42-2': 'While the principle sensitivity of the radio signal on the longitudinal shower development was already demonstrated experimentally [CITATION], this is the first time that the radio reconstruction of [MATH] is cross-checked with an independent technique.', '1509.05652-1-42-3': 'This also gives more confidence in the results of other radio arrays whose reconstruction procedures are developed with the same CORSIKA + CoREAS Monte Carlo codes (or other codes yielding consistent results, see, e.g., reference [CITATION] for a comparison of different codes.)', '1509.05652-1-43-0': '# Conclusion', '1509.05652-1-44-0': 'We compared the reconstructed energy and [MATH] of Tunka-Rex and Tunka-133 measurements of the same air showers in the energy range [MATH]eV.', '1509.05652-1-44-1': 'The reconstruction methods have been developed and tuned using CORSIKA simulations and its radio extension CoREAS.', '1509.05652-1-44-2': 'For both parameters we find a strong correlation between Tunka-Rex and Tunka-133 for two independent data sets, namely two seasons of data taking.', '1509.05652-1-44-3': 'Since for the second season the Tunka-133 energy and [MATH] values were blinded, this confirms experimentally that the radio measurements are indeed sensitive to energy and shower maximum.', '1509.05652-1-44-4': 'For [MATH] this is the first confirmation based on a cross-check with a different, established experimental technique, namely air-Cherenkov measurements.', '1509.05652-1-45-0': 'The Tunka-Rex energy precision seems to be at least as good as the published Tunka-133 resolution of [MATH].', '1509.05652-1-45-1': 'The total scale uncertainty of Tunka-Rex is dominated by the uncertainty of the amplitude calibration, and in total is in the order of [MATH].', '1509.05652-1-45-2': 'This is comparable to the scale uncertainty of particle detector arrays, like KASCADE-Grande [CITATION].', '1509.05652-1-45-3': 'By further improving the absolute calibration of the radio measurements, it seams feasible to further improve the accuracy to the same level as the currently leading fluorescence and air-Cherenkov techniques.', '1509.05652-1-45-4': 'At the Pierre Auger Observatory, the scale accuracy of fluorescence measurements is [MATH] [CITATION].', '1509.05652-1-45-5': 'This now is in reasonable reach for future radio measurements, which unlike fluorescence measurements are available around-the-clock.', '1509.05652-1-45-6': 'Hence, radio measurements could be used to determine the absolute energy scale of air-shower measurements.', '1509.05652-1-46-0': 'The [MATH] precision of Tunka-Rex is roughly [MATH]g/cm2, and can be slightly increased by setting stricter quality cuts on the cost of statistics.', '1509.05652-1-46-1': 'This resolution is sufficient to statistically distinguish light from heavy primary particles.', '1509.05652-1-46-2': 'Still, the resolution is worse than that currently achieved by non-imaging air-Cherenkov measurements ([MATH]g/cm2 for Tunka-133 [CITATION]), by air-fluorescence measurements ([MATH]g/cm2 for the Pierre Auger Observatory [CITATION]), and by very dense radio arrays ([MATH]g/cm2 claimed by LOFAR [CITATION]).', '1509.05652-1-46-3': 'Nevertheless, the resolution of Tunka-Rex is not yet at its limit and will likely be improved in future: First, additional antennas will be deployed at each Tunka cluster increasing the quality of the data points in the LDF (each data point will be complemented by two additional near-by points corresponding to the other two antenna stations per Tunka cluster).', '1509.05652-1-46-4': 'Second, other methods for [MATH] reconstruction based on other quantities of the radio signal can be used to improve the total accuracy, in particular the shape of the wavefront [CITATION] and the slope of the frequency spectrum [CITATION].', '1509.05652-1-47-0': 'For astrophysical applications the total accuracy for the mass composition as function of energy counts, e.g., to better study the transition from galactic to yet unknown extra-galactic cosmic-ray sources assumed in the energy range of Tunka-Rex [CITATION].', '1509.05652-1-47-1': 'While unimportant for the present analysis focused on the feasibility of the reconstruction methods, for reconstruction of the cosmic-ray composition also further systematic uncertainties like selection biases have to be investigated.', '1509.05652-1-47-2': 'Independent muon measurements on ground with the newly deployed particle-detector array Tunka-Grande [CITATION] can enhance the total accuracy, since the electron-muon ratio provides complementary mass information to [MATH] [CITATION].', '1509.05652-1-47-3': 'In addition, the Tunka-Grande measurement accuracy will be enhanced by a cross-calibration on hybrid measurements with Tunka-Rex.', '1509.05652-1-47-4': 'Finally, the day-time trigger provided by Tunka-Grande, will drastically enhance the statistics of Tunka-Rex, which in turn will increase the total event rate of the Tunka cosmic-ray facility around [MATH]eV by an order of magnitude.', '1509.05652-1-47-5': 'Hence, Tunka-Rex will provide additional statistics exactly in the energy range where current Tunka-133 analyses are limited by statistics.', '1509.05652-1-47-6': 'Consequently, Tunka-Rex will contribute to cosmic-ray physics in at least two ways: indirectly, by improving the calibration of Tunka-Grande in the full energy range of [MATH]eV to a few times [MATH]eV, and directly at [MATH]eV by increasing the accuracy for energy and mass composition.', '1509.05652-1-48-0': 'Tunka-Rex has been funded by the German Helmholtz association and the Russian Foundation for Basic Research (grant HRJRG-303).', '1509.05652-1-48-1': 'Moreover, this work was supported by the Helmholtz Alliance for Astroparticle Physics (HAP), by the Russian Federation Ministry of Education and Science (agreement 14.', '1509.05652-1-48-2': 'B25.31.0010, zadanie 3.889.2014/K), the Russian Foundation for Basic Research (Grants 12-02-91323, 13-02-00214, 13-02-12095, 14002-10002) and the President of the Russian Federation (grant MK-1170.2013.2).', '1509.05652-1-49-0': '# Parametrization of LDF parameter [MATH]', '1509.05652-1-50-0': 'In the Gaussian lateral distribution function (LDF, equation [REF]), the parameter [MATH] defining the bump has been set by the following parameterization determined with CoREAS simulations made for the situation of Tunka-Rex [CITATION]: [EQUATION]', '1509.05652-1-50-1': 'As shown in reference [CITATION], a simple exponential LDF without parameter [MATH] can be used to reconstruct the primary energy of Tunka-Rex events with a precision only slightly worse than in our approach here.', '1509.05652-1-50-2': 'Thus, [MATH] in the parametrization above is estimated by fitting the following simple LDF: [EQUATION] and then using equation [REF] for energy reconstruction.', '1509.05652-1-51-0': 'As alternative we tried to fix [MATH] to a typical, constant value instead of using the parametrization.', '1509.05652-1-51-1': 'This simplifies the reconstruction procedure, but in turn slightly degrades the precision for the energy and the shower maximum for the tuning season.', '1509.05652-1-51-2': 'Therefore, before unblinding of the second season, we decided for the more complicated reconstruction method using this parametrization.', '1509.05652-1-51-3': 'For denser arrays featuring many antennas with signal in a typical event, fitting [MATH] to the measured data might be a suitable option, too.', '1509.05652-1-52-0': '# Used events', '1509.05652-1-53-0': 'As supplementary material we will upload the list of the events used for the present analysis.', '1509.05652-1-53-1': 'This list contains the reconstructed energy and [MATH] values of the events.', '1509.05652-1-53-2': 'However, the values should not be used for reconstruction of the cosmic-ray energy spectrum or the mass composition.', '1509.05652-1-53-3': 'While unimportant for the present study comparing the Tunka-Rex to the Tunka-133 reconstruction, for such an analyses selection biases have to be taken into account, which cannot be derived from the event list alone.', '1509.05652-1-54-0': 'The original event lists for the tuning and prediction seasons as created before unblinding are available at http://www.ikp.kit.edu/tunka-rex/ and can be encrypted using the following Linux command:', '1509.05652-1-55-0': 'openssl aes-256-cbc -d -in encryptedFile -out decryptedFile', '1509.05652-1-56-0': 'The passwords are TunkaRex1H9AoxFywAt for the tuning season,', '1509.05652-1-57-0': 'and TunkaRex2z3DumFNfsq for the prediction season.', '1509.05652-1-58-0': 'All these events are also contained in the above mentioned list submitted as supplementary material.'}

{'1509.05652-2-0-0': '# Introduction', '1509.05652-2-1-0': 'After more than 100 years of cosmic-ray measurements, the mass composition and the energy spectrum of the primary particles is relatively well-known only in the energy range accessible by direct measurements, [MATH]eV.', '1509.05652-2-1-1': 'At higher energies the flux of cosmic-rays is too low to perform direct measurements with statistical significance.', '1509.05652-2-1-2': 'Consequently, the knowledge is poorer and relies on the measurement of secondary-particle cascades called air showers.', '1509.05652-2-1-3': 'These indirect cosmic-ray measurements use the atmosphere as a calorimeter.', '1509.05652-2-1-4': 'The energy of the primary particle can be estimated from the energy contained in the shower.', '1509.05652-2-1-5': 'The type of the primary particle can, for example, be deduced from the longitudinal shower development.', '1509.05652-2-1-6': 'Heavy primary particles such as iron nuclei on average interact earlier in the atmosphere than light particles such as protons.', '1509.05652-2-1-7': 'Thus, the mass composition can be statistically estimated from measurements of the atmospheric depth of the shower maximum, [MATH].', '1509.05652-2-2-0': 'Established air-shower techniques are the measurement of secondary particles on ground, the measurement of fluorescence or air-Cherenkov light by dedicated telescopes or non-imaging detectors on ground.', '1509.05652-2-2-1': 'The latter two methods have a relatively high accuracy for the energy and for [MATH], but are limited to dark and clear nights.', '1509.05652-2-2-2': 'Detection of the radio emission by air showers is an additional technique, which does not suffer from this intrinsic limitation of the exposure.', '1509.05652-2-2-3': 'The radio signal is sensitive to the shower energy [CITATION] and [MATH] [CITATION], too, but it has not yet been demonstrated experimentally that the accuracy for both variables is competitive with those of the other techniques, as indicated by simulation-based studies, e.g., [CITATION].', '1509.05652-2-3-0': 'The main origin of the radio emission is the geomagnetic deflection of relativistic electrons and positrons in the shower, which induces a time-variable current [CITATION].', '1509.05652-2-3-1': 'The amplitude of this emission is proportional to the shower energy and to [MATH], with [MATH] the angle between the geomagnetic field and the shower direction.', '1509.05652-2-3-2': 'On a weaker level the Askaryan effect contributes, i.e., radio emission due to the time variation of the net charge excess in the shower front [CITATION].', '1509.05652-2-3-3': 'The interference of both effects leads to an azimuthal asymmetry of the radio footprint [CITATION].', '1509.05652-2-3-4': 'Finally, the refractive index of air causes Cherenkov-like effects for both emission mechanisms [CITATION].', '1509.05652-2-4-0': 'Tunka-Rex [CITATION] is the radio extension of the Tunka observatory for cosmic-ray air showers.', '1509.05652-2-4-1': 'Its main detector, Tunka-133, is an array of non-imaging photomultipliers detecting the Cherenkov light emitted by air-showers in the atmosphere in the energy range of [MATH]eV.', '1509.05652-2-4-2': 'Tunka-133 is fully efficient for all zenith angles [MATH] at energies above [MATH]eV [CITATION], which covers the full energy range of Tunka-Rex.', '1509.05652-2-4-3': 'For Tunka-133 measurements, the energy of the primary particle [MATH] is reconstructed from the flux of the Cherenkov light at [MATH]m distance to the shower axis, and [MATH] is reconstructed from the steepness of the amplitude-distance function.', '1509.05652-2-4-4': 'Using these methods, Tunka-133 features an energy resolution of about [MATH], and an [MATH] resolution of about [MATH]g/cm2 [CITATION].', '1509.05652-2-4-5': 'For comparison, the average difference between the extreme cases of a pure proton and a pure iron composition is in the order of [MATH]g/cm2.', '1509.05652-2-4-6': 'Since Tunka-133 triggers Tunka-Rex, the same air showers are measured simultaneously with the air-Cherenkov and the radio detector.', '1509.05652-2-5-0': 'These combined measurements are used for a cross-check of both methods.', '1509.05652-2-5-1': 'In particular the precision of Tunka-Rex is estimated by a comparison of the energy and [MATH] reconstructions to Tunka-133.', '1509.05652-2-5-2': 'Reconstruction methods for Tunka-Rex have been developed with CoREAS simulations [CITATION] (the radio extension of CORSIKA [CITATION]) taking into account previous theoretical predictions [CITATION], and experience by other experiments, in particular LOPES [CITATION], AERA [CITATION] and LOFAR [CITATION].', '1509.05652-2-6-0': 'The parameters in the reconstruction methods have been determined from the simulations (cf. reference [CITATION] for details), and not tuned against the Tunka-133 measurements.', '1509.05652-2-6-1': 'First results on energy and [MATH] reconstruction have already been presented in reference [CITATION] using the first season of Tunka-Rex measurements from October 2012 to April 2013 (effectively 280 hours of measurements).', '1509.05652-2-6-2': 'The same reconstruction method is now applied also to the second season of data from October 2013 to April 2014 (effectively 260 hours of measurements), and the combined results and the cross-check between both measurement seasons are presented in this article.', '1509.05652-2-6-3': 'In the experimental data we observe a direct correlation between the Tunka-Rex and Tunka-133 reconstructions, which is used to estimate precision and accuracy of Tunka-Rex for energy and [MATH].', '1509.05652-2-7-0': '# Experimental setup and data selection', '1509.05652-2-8-0': 'Tunka-Rex started in autumn 2012 with 19 antenna stations, and in summer 2013, 6 further antennas went into operation giving a total number of 25 antenna stations used for the present analysis.', '1509.05652-2-8-1': 'In the central area covering about [MATH]km2 the spacing is approximately [MATH]m.', '1509.05652-2-8-2': 'The outer antennas contribute only to a few events (see figure [REF]).', '1509.05652-2-8-3': 'The air-Cherenkov array Tunka-133 covers the inner area with 133 photomultipliers, and features additional photomultipliers in the outer area.', '1509.05652-2-8-4': 'Upon a coincidence trigger of the photomultipliers, both the radio and the air-Cherenkov detector are read out in parallel.', '1509.05652-2-8-5': 'The electric field of the radio signal measured by Tunka-Rex is reconstructed in an effective bandwidth of [MATH]MHz.', '1509.05652-2-8-6': 'Details on the detector setup and its calibration can be read in reference [CITATION].', '1509.05652-2-9-0': 'The reconstruction of Tunka-133 is fully efficient and has a reliable reconstruction for events with zenith angles [MATH].', '1509.05652-2-9-1': 'Therefore, only events with [MATH] have been selected for the present analysis (see figure [REF]).', '1509.05652-2-9-2': 'For these events the Tunka-133 reconstruction of the shower provides: energy of the primary particle [MATH], atmospheric depth of the shower maximum [MATH], and axis, i.e., the direction and point of incidence on the ground (= shower core).', '1509.05652-2-9-3': 'For all Tunka-Rex events we used the shower core of Tunka-133 as input, because of the denser spacing of the photomultipliers compared to that of the antennas.', '1509.05652-2-9-4': 'The direction was only used as cross-check: events are considered false-positive, i.e., contaminated by background pulses and excluded from the analysis, if the directions reconstructed from the radio and air-Cherenkov arrival times deviate by more than [MATH] [CITATION].', '1509.05652-2-10-0': 'For the selection of Tunka-Rex events we applied a quality cut based on the signal-to-noise ratio in individual antenna stations.', '1509.05652-2-10-1': 'The signal has to exceed a signal-to-noise ratio, [MATH] in at least three antenna stations, where the signal [MATH] is the maximum of an envelope on the electric-field strength of the pulse, and the noise [MATH] the root-mean square of the electric field in a time window before the signal.', '1509.05652-2-10-2': 'This cut is set such that pure background has a chance of about [MATH] to pass in an individual antenna, which balances efficiency and purity.', '1509.05652-2-10-3': 'Given 25 antenna stations, this leads to a non-negligible probability that in a true event antenna stations with false-positive signals are contained.', '1509.05652-2-10-4': 'To exclude most of these false-positive signals, we sort the antennas by their distance to the shower axis.', '1509.05652-2-10-5': 'After two antenna stations failing the SNR cut, any further antenna stations are excluded from the analysis for the particular event.', '1509.05652-2-10-6': 'Thus, the risk that an event is contaminated by a station with false-positive signal is reduced to approximately [MATH].', '1509.05652-2-10-7': 'If surviving false-positive signals significantly impact the event reconstruction, they are removed by the cut requiring the [MATH] agreement between the direction reconstructed by Tunka-Rex and Tunka-133.', '1509.05652-2-10-8': 'Thus, the remaining data set is assumed to be practically free of false-positive signals.', '1509.05652-2-10-9': 'Finally, for the [MATH] reconstruction we apply additional quality cuts: At least one antenna above threshold has to be at a distance [MATH]m to the shower axis, since [MATH] is reconstructed from the slope of the lateral distribution, which requires a sufficient lever arm.', '1509.05652-2-10-10': 'Moreover, the shower core has to be inside the central area of the array (dashed circle in figure [REF]), and the estimator for the statistical reconstruction uncertainty of Tunka-Rex must not exceed a certain value: [MATH]g/cm2.', '1509.05652-2-11-0': "Reconstruction of the electric-field strengths (= amplitude) in individual antenna stations is performed with a modified version of the 'Auger Offline' software developed by the Pierre Auger Collaboration [CITATION].", '1509.05652-2-11-1': 'This software applies all known experimental characteristics to the recorded signals, in particular the frequency-dependent gain and pulse distortion of the various components in the signal chain as determined by calibration measurements [CITATION].', '1509.05652-2-11-2': 'For the reconstruction of the electric-field vector the simulated direction dependence of the antenna gain is used, which has been validated for a few selected directions by calibration measurements.', '1509.05652-2-12-0': 'For the suppression of narrow-band radio-frequency interferences (RFI) we apply rectangular band-stop filters of [MATH]MHz width (corresponding to 3 bins) at each integer [MATH]MHz ([MATH], [MATH], ..., [MATH]MHz), since these frequencies are often contaminated by RFI background .', '1509.05652-2-12-1': 'Since the radio emission from the air shower is broadband, it is significantly less affected by the filter than the narrow-band disturbances.', '1509.05652-2-12-2': 'The background remaining after filtering still on average increases the measured pulse height [MATH] compared to the true pulse height [MATH], as already seen at the LOPES experiment [CITATION] and assumed for early experiments [CITATION].', '1509.05652-2-12-3': 'We studied this effect by adding measured background to about 300 CoREAS simulations (figure [REF]), and found that the parametrization given in reference [CITATION] has to be slightly modified introducing a normalization factor [MATH] adapted for Tunka-Rex.', '1509.05652-2-12-4': 'This normalization factor reflects the different ways how the noise level is measured in LOPES (mean of local maxima in the instantaneous amplitude) and Tunka-Rex (root mean square of the electric field, which corresponds to the mean power of noise).', '1509.05652-2-12-5': 'Thus, we use the following formula to correct for the average effect of noise: [EQUATION] with the signal-to-noise ratio [MATH] defined above.', '1509.05652-2-13-0': 'An overview of the available event statistics is provided in table [REF], and figure [REF] shows the distribution of the selected events over energy and [MATH].', '1509.05652-2-13-1': 'Most of the events used for the energy reconstruction are at energies above [MATH]eV.', '1509.05652-2-13-2': 'The fraction does not reach [MATH] because some Tunka-133 events have small geomagnetic angles.', '1509.05652-2-13-3': 'Also, there were occasional technical issues in single antenna stations.', '1509.05652-2-13-4': 'For the high-quality events used for the [MATH] reconstruction, the threshold is slightly higher: above an energy of about [MATH]eV, all events used for energy reconstruction pass also the high-quality cuts required for [MATH] reconstruction (see figure [REF]).', '1509.05652-2-13-5': 'Apart from technical issues (e.g., missing or malfunctioning antennas), the fraction shown in figure [REF] is the efficiency of Tunka-Rex for zenith angles [MATH], because the reference is the fully efficient Tunka-133 trigger.', '1509.05652-2-13-6': 'The total number of Tunka-Rex events passing the quality cuts is larger [CITATION], since in contrast to the air-Cherenkov detection, radio detection is more efficient for inclined air showers.', '1509.05652-2-13-7': 'Nevertheless, for the present analysis only events with [MATH] have been selected, since Tunka-133 is fully efficient for this zenith angle range.', '1509.05652-2-14-0': '# Reconstruction of energy and shower maximum', '1509.05652-2-15-0': 'In the present analysis, the reconstruction of energy and shower maximum is based on the lateral distribution, i.e., the dependence of the radio amplitude on the distance to the shower axis.', '1509.05652-2-15-1': 'This approach has been used earlier.', '1509.05652-2-15-2': 'Different experiments have shown that the amplitude at a certain distance is correlated with the primary energy, after correcting for the geomagnetic angle [CITATION].', '1509.05652-2-15-3': 'Moreover, it has been shown that the shape of the lateral distribution depends on the distance to the shower maximum [CITATION].', '1509.05652-2-16-0': 'Our reconstruction method in brief is:', '1509.05652-2-17-0': 'We correct the lateral distribution for the azimuthal asymmetry caused by the interference of the geomagnetic and the Askaryan effects, and for the geomagnetic angle [MATH].', '1509.05652-2-17-1': 'We fit a Gaussian LDF, the shape of which depends on three parameters: One scale parameter, [MATH], which is the amplitude at [MATH]m, and two shape parameters [MATH] and [MATH], which describe the slope and the width of the Gaussian LDF, respectively.', '1509.05652-2-17-2': '[MATH] and [MATH] are fit to the data points, [MATH] is fixed using a parametrization depending on energy and zenith angle.', '1509.05652-2-17-3': 'We use the fit result [MATH] to reconstruct the primary energy [MATH], and the fit result for [MATH] at [MATH]m axis distance, representing the slope of the lateral distribution, to reconstruct the atmospheric depth of the shower maximum [MATH].', '1509.05652-2-18-0': '## Asymmetry correction', '1509.05652-2-19-0': 'Because of the different polarizations of the radio emission generated by the Askaryan and the geomagnetic effects [CITATION], the radio amplitude on ground depends not only on the geomagnetic angle [MATH] and distance to the shower axis [MATH], but also on the azimuth angle [MATH] of each antenna relative to the shower axis.', '1509.05652-2-19-1': 'Fitting the asymmetry for individual events by a two-dimensional lateral distribution function (LDF) introduces additional parameters [CITATION].', '1509.05652-2-19-2': 'To maintain events with low station multiplicity, we have developed an alternative approach described in reference [CITATION], which is well-suited for sparse arrays like Tunka-Rex.', '1509.05652-2-20-0': 'Our approach makes use of the fact that the asymmetry to first order only depends on the shower geometry, which can be reconstructed independently from the LDF by arrival time measurements.', '1509.05652-2-20-1': 'Based on CoREAS simulations made for the Tunka-Rex setup, the azimuthal asymmetry of the footprint is parametrized as function of the antenna position using a constant typical value of [MATH] for the relative strength of the Askaryan effect.', '1509.05652-2-20-2': 'Using this parametrization, we correct all measured amplitudes in individual antennas for the azimuthal asymmetry, and obtain a one-dimensional lateral distribution in which the amplitude depends only on the distance to the shower axis.', '1509.05652-2-20-3': 'Simultaneously, we correct for the relative strength of the geomagnetic Lorentz force, since this depends only on the shower geometry, too, namely on the geomagnetic angle [MATH].', '1509.05652-2-20-4': 'Thus, we use the following equation to obtain the corrected amplitude [MATH], which is later used to fit a radially symmetric lateral-distribution function: [EQUATION] with [MATH] the measured amplitude after noise correction ([MATH] of equation [REF]), [MATH] the distance to the shower axis, [MATH] the azimuth angle relative to the shower axis (with [MATH], when a station is in the direction of the geomagnetic Lorentz force), [MATH] the geomagnetic angle, [MATH] the size of the asymmetry, and [MATH] the amplitude after correction for the asymmetry and the geomagnetic angle.', '1509.05652-2-20-5': 'After this correction of the amplitude measured at each antenna station, the lateral distribution is approximately azimuthally symmetric around the shower axis.', '1509.05652-2-20-6': 'Thus, to first order the amplitude depends only on the distance to the shower axis [MATH], and the lateral distribution can be fitted with a one-dimensional function the parameters of which depend on energy and distance to the shower maximum.', '1509.05652-2-21-0': 'Figure [REF] shows an example event with many stations, where the asymmetry correction leaves the energy reconstruction almost unchanged, since the LDF fit averages the asymmetry out.', '1509.05652-2-21-1': 'However, the [MATH] reconstruction is slightly improved for this event (Tunka-Rex [MATH] is [MATH]g/cm2 and [MATH]g/cm2 before and after correction, respectively, and the Tunka-133 [MATH] is [MATH]g/cm2).', '1509.05652-2-21-2': 'Especially for events with few stations, the asymmetry correction is important, and can have an effect on the energy reconstruction in the order of [MATH].', '1509.05652-2-22-0': '## Lateral distribution function (LDF)', '1509.05652-2-23-0': 'The amplitude (= electric field strength) of this one-dimensional lateral distribution is proportional to the shower energy, because the radio emission in the shower is coherent.', '1509.05652-2-23-1': 'Moreover, the shape of the lateral distribution depends mainly on the distance to the shower maximum and has two main features: first, a bump caused by Cherenkov-like effects [CITATION], the existence, size and position of which depend on the distance to the shower maximum; second, an exponential tail [CITATION], the slope of which depends on the distance to the shower maximum, too.', '1509.05652-2-23-2': 'This means that even a one-dimensional LDF should contain at least three free parameters: one for the energy scale, and two for the shape, namely the width of the bump, and the slope of the exponential fall-off.', '1509.05652-2-23-3': 'For this purpose we use the following lateral distribution function: [EQUATION] with the amplitude [MATH] after correction for the azimuthal asymmetry and for the geomagnetic angle as function of axis distance [MATH], and the fitted scale parameter [MATH], which is the amplitude at the axis distance [MATH].', '1509.05652-2-23-4': 'The shape parameter [MATH] determines the slope of the exponential fall-off, and the shape parameter [MATH] is related to the width of the Cherenkov bump.', '1509.05652-2-23-5': 'The distance parameter [MATH] is a purely technical parameter.', '1509.05652-2-23-6': 'Using a minimizing chi-square fit, the shape of the function does not depend on the choice of [MATH], i.e, for different [MATH] the resulting function is exactly the same, but just described with a different set of values for the parameters [MATH], [MATH] and [MATH].', '1509.05652-2-23-7': 'However, the choice of [MATH] affects the uncertainties and correlations of the fit parameters, which is why a specific choice of [MATH] can simplify the reconstruction of [MATH], [MATH], and their statistical uncertainties.', '1509.05652-2-23-8': 'We set this distance parameter [MATH] to [MATH]m for the reconstruction of the primary energy [MATH] based on [MATH], and to [MATH]m for the [MATH] reconstruction based on [MATH], as these turned out to be the optimum values in the CoREAS simulation study.', '1509.05652-2-24-0': 'In the CoREAS simulations made for Tunka-Rex we observe that the scale parameter [MATH] is strongly correlated with the primary energy [MATH].', '1509.05652-2-24-1': 'The shape parameter [MATH], i.e., the slope at [MATH]m axis distance is strongly correlated with the distance to the shower maximum.', '1509.05652-2-24-2': 'The second shape parameter [MATH] is correlated with [MATH], and weakly depends on energy and zenith angle.', '1509.05652-2-24-3': 'Thus, fitting [MATH] as a free parameter would complicate the reconstruction of [MATH] significantly.', '1509.05652-2-24-4': 'For this reason we decided to perform the fit with only two free parameters, and to fix the parameter [MATH] using a parametrization depending on zenith angle [MATH] and the primary energy [MATH] (details in appendix [REF]).', '1509.05652-2-24-5': 'After fixing [MATH], the LDF is fit to the data points.', '1509.05652-2-24-6': '[MATH] carries the whole sensitivity to the primary energy [MATH], and the remaining free shape parameter [MATH] carries the whole [MATH] sensitivity of the LDF.', '1509.05652-2-24-7': 'Furthermore, reducing the number of free parameters in the LDF from three to two brings the additional advantage that we can use events with only three antenna stations above threshold (about half of the Tunka-Rex events in the present analysis).', '1509.05652-2-25-0': '## Primary energy [MATH], and atmospheric depth of shower maximum [MATH]', '1509.05652-2-26-0': 'Energy [MATH] and [MATH] are determined using the equations presented in reference [CITATION].', '1509.05652-2-26-1': 'While for [MATH] we use exactly the same equation as in the reference, i.e., the same values for all parameters in the equation, for [MATH] we use a simplification.', '1509.05652-2-26-2': 'The simplification is to assume that the energy is exactly proportional to the amplitude, as expected for [MATH] coherent emission.', '1509.05652-2-26-3': 'This means that the exponent [MATH] of reference [CITATION] is set to [MATH] instead and, thus, is not present in the equation used here.', '1509.05652-2-26-4': 'The reason for applying this simplification is that with the present statistics and quality of the measured data, we have found no evidence for [MATH].', '1509.05652-2-26-5': 'Moreover, there is no significant difference in the resulting energy precision whether [MATH] is set to [MATH] or [MATH].', '1509.05652-2-26-6': 'Consequently, at the moment we lack experimental justification for introducing [MATH] as an additional parameter and decided for the simpler solution.', '1509.05652-2-26-7': 'The simplification requires a different proportionality coefficient [MATH], which has been determined with the same set of CoREAS simulations.', '1509.05652-2-26-8': 'Summarizing, we use the following equations for the reconstruction of [MATH] and [MATH] with Tunka-Rex: [EQUATION] where [MATH]g/cm[MATH] is the atmospheric depth of the detector (typical value for the site used in the simulations), and the parameters are determined with CoREAS simulations: [MATH], [MATH]g/cm[MATH], [MATH]g/cm[MATH], [MATH] given in units of meters, and [MATH].', '1509.05652-2-26-9': 'As explained above, the technical parameter of the LDF, [MATH], is set to [MATH]m for the determination of the amplitude at [MATH]m, [MATH], and to [MATH]m for the determination of the slope parameter [MATH].', '1509.05652-2-27-0': 'The energy and [MATH] reconstructions based on simulations were already presented in reference [CITATION].', '1509.05652-2-27-1': 'The following section presents the experimental result of Tunka-Rex measurements and the comparison to the Tunka-133 air-Cherenkov measurements.', '1509.05652-2-28-0': '# Results', '1509.05652-2-29-0': 'There is a clear correlation between the energy reconstructed from the radio amplitude measured by Tunka-Rex and the energy reconstructed from the air-Cherenkov light measured by Tunka-133 (figure [REF]).', '1509.05652-2-29-1': 'This indicates that the energy reconstruction works reliably with the presented method.', '1509.05652-2-29-2': 'When combining both seasons, the average deviation between the Tunka-Rex and Tunka-133 energy values is [MATH], which is only slightly larger than the [MATH] energy resolution reported by Tunka-133 [CITATION].', '1509.05652-2-30-0': 'Also for the position of the shower maximum there is a clear correlation between the radio and air-Cherenkov reconstructions (figure [REF]).', '1509.05652-2-30-1': 'Due to the stricter quality cuts the event statistics is lower than for the energy correlation.', '1509.05652-2-30-2': 'Still, the correlation is significant.', '1509.05652-2-30-3': 'This again indicates that also the reconstruction of [MATH] works reliably for the selected, high-quality radio events.', '1509.05652-2-30-4': 'Because of the lower statistics, the [MATH] reconstruction precision of Tunka-Rex cannot be estimated very accurately.', '1509.05652-2-30-5': 'The standard deviation of the [MATH] difference between both detectors is ([MATH]g/cm2, which reflects the combination of the unknown Tunka-Rex uncertainty and the Tunka-133 uncertainty of [MATH]g/cm2 [CITATION].', '1509.05652-2-30-6': 'This corresponds to a radio-only precision of better than [MATH]g/cm2, when assuming that the resolutions of Tunka-133 and Tunka-Rex add in squares to the total deviation.', '1509.05652-2-31-0': 'For both the energy and [MATH] reconstructions the observed correlation is statistically compatible with a 1:1 correlation of Tunka-133 and Tunka-Rex: the means of the Gaussians fitted in figures [REF] and [REF] are compatible with 0, where 0 corresponds to no bias (= systematic scale offset).', '1509.05652-2-31-1': 'This means that the energy and [MATH] scales of both detectors agree on an absolute level within the measurement uncertainty.', '1509.05652-2-31-2': 'This is remarkable, since the Tunka-Rex scale is defined by CoREAS simulations and has not been tuned against Tunka-133.', '1509.05652-2-31-3': 'With already consistent scales there is no need for a true cross-calibration, i.e., en explicit calibration of both detectors against each other.', '1509.05652-2-32-0': '# Discussion', '1509.05652-2-33-0': '## Validity of the results', '1509.05652-2-34-0': 'The present analysis is the first event-by-event comparison of radio and air-Cherenkov measurements of the same air showers.', '1509.05652-2-34-1': 'Since both detection methods are sensitive to the energy and the position of the maximum of the electromagnetic shower component, a strong correlation is expected between radio and air-Cherenkov observables.', '1509.05652-2-34-2': 'This is especially noteworthy since the Tunka-Rex reconstruction is based on simulations and not tuned against the Tunka-133 measurements we compared with.', '1509.05652-2-34-3': 'Only details of the reconstruction method have been decided after looking at the data of the first season, e.g., the filter used to remove narrow-band interferences or the exact quality cuts.', '1509.05652-2-34-4': 'Neither of these small decisions made the [MATH] and energy correlations appear or vanish.', '1509.05652-2-35-0': 'As additional cross-check of the validity of the results, we decided to blind the energy and [MATH] reconstruction of Tunka-133 for the second season.', '1509.05652-2-35-1': 'These values were revealed to us only after we froze the reconstruction method, including digital filters and quality cuts, and after we decided to use the simplified equation [REF] for energy reconstruction.', '1509.05652-2-35-2': 'In appendix [REF] the results of both seasons are compared, and found to be compatible within statistical uncertainties.', '1509.05652-2-35-3': 'Thus, we conclude that our results are not corrupted by any kind of implicit tuning on the data set of the first year.', '1509.05652-2-35-4': 'Moreover, the quality cuts are general enough to cover the slightly different array configurations in both seasons, i.e., the missing antennas in the first season do not have significant impact on any of the present results.', '1509.05652-2-36-0': 'Potential biases not excluded by the cross-check concern possible special events.', '1509.05652-2-36-1': 'For example, events with their shower maximum very close to the detector will have a very steep lateral distribution and a small footprint.', '1509.05652-2-36-2': 'Thus, they will likely not be detected or will be rejected by the quality cuts.', '1509.05652-2-36-3': 'Still, since Tunka-133 is fully efficient and has a reliable reconstruction in the full zenith and energy range used in this study, the non-detection of certain events is not assumed to cause any significant bias in the observed correlations.', '1509.05652-2-36-4': 'Furthermore, at high energies less than a third of the events are rejected (cf. figure [REF]), and this due to technical reasons (missing antenna stations) or due to small geomagnetic angles leading to small radio amplitudes.', '1509.05652-2-36-5': 'Thus, the potential problem of a bias introduced by the rejection of any kind of special event can concern only a small fraction of events.', '1509.05652-2-36-6': 'Nevertheless, this has to be studied in more detail when larger statistics will be available.', '1509.05652-2-37-0': 'Consequently, the cross-check of both seasons cannot totally exclude all imaginable systematic biases implicit in the experimental setup or in the simulations simulations, but it definitely confirms that the observed correlations are real.', '1509.05652-2-38-0': '## Precision and accuracy', '1509.05652-2-39-0': 'The spread of the deviation between the Tunka-Rex and Tunka-133 reconstruction values for the same events can be used to estimate the reconstruction precision of Tunka-Rex.', '1509.05652-2-39-1': 'Assuming independent Gaussian uncertainties of both reconstructions, the standard deviation of the difference or ratio between Tunka-Rex and Tunka-133 is the quadratic sum of the absolute or relative uncertainties, receptively.', '1509.05652-2-39-2': 'If Tunka-Rex would feature the same precision as Tunka-133, thus, this standard deviation should be a factor of [MATH] larger than the Tunka-133 precision alone, i.e.: [MATH] for the energy, which is slightly larger than the observed standard deviation of [MATH].', '1509.05652-2-39-3': 'Consequently, the energy precision of Tunka-Rex seems to be at least equal to that of Tunka-133.', '1509.05652-2-40-0': 'For [MATH], we would expect a standard deviation of the difference of [MATH]g/cm[MATH]g/cm[MATH] if Tunka-Rex had the same precision as Tunka-133 ([MATH]g/cm[MATH]), but observe a standard deviation of the difference of [MATH]g/cm[MATH].', '1509.05652-2-40-1': 'The derived Tunka-Rex precision is slightly better than [MATH]g/cm[MATH], as expected from CoREAS simulations when including realistic background [CITATION].', '1509.05652-2-40-2': 'Since the average spacing of the Tunka-133 photomultipliers is half of the average antenna spacing, this result does not necessarily imply that the radio [MATH] precision is intrinsically worse than the air-Cherenkov precision.', '1509.05652-2-40-3': 'Instead, that later studies will show how the Tunka-133 and Tunka-Rex resolutions compare for equal detector spacing, and by how much the Tunka-Rex precision will improve due to additionally deployed antennas.', '1509.05652-2-41-0': 'Using CoREAS simulations we searched for systematic uncertainties potentially degrading the resolution further, in particular biases of the [MATH] reconstruction over energy or zenith angle, but we did not find any significant effect.', '1509.05652-2-41-1': 'The finally achievable [MATH] uncertainty of the radio technique might become equal to the one of the air-Cherenkov technique, when events of even higher quality are used.', '1509.05652-2-41-2': 'The spread between the Tunka-133 and Tunka-Rex [MATH] values approximately corresponds to the cut placed on the estimator [MATH] for the statistical uncertainty of Tunka-Rex.', '1509.05652-2-41-3': 'This means that this estimator is reliable and can be used to set stronger quality cuts when higher statistics are available.', '1509.05652-2-42-0': 'In addition to the precision also the total accuracy is of importance.', '1509.05652-2-42-1': 'This is dominated by the [MATH] scale uncertainty of the amplitude calibration [CITATION], but at least three further systematic scale uncertainties have to be considered:', '1509.05652-2-43-0': 'First, if the fraction of the total primary energy going in the electromagnetic shower component was simulated wrongly in CORSIKA, the energy scale would be wrong by roughly the same value.', '1509.05652-2-43-1': 'However, the electromagnetic shower component is relatively well understood, and as estimated by other experiments [CITATION], this uncertainty on the invisible energy is only a few percent.', '1509.05652-2-43-2': 'Moreover, we assume that the CoREAS radio extension of CORSIKA simulates the absolute radio amplitude emitted by the electromagnetic shower component correctly, i.e., with errors which are small against the calibration scale uncertainty of [MATH], which is supported by recent experimental tests [CITATION].', '1509.05652-2-43-3': 'Still, the fraction of energy going in the electromagnetic shower component depends on the mass of the primary particle.', '1509.05652-2-43-4': 'Thus, the parameters in the method slightly depend on the assumed mass composition of the primary particles, in particular also the parameter [MATH] determining the energy scale.', '1509.05652-2-43-5': 'For the present analysis, we determined [MATH] as average of proton and iron simulations with the hadronic interaction model QGSJET-II.04 [CITATION].', '1509.05652-2-43-6': 'If we had taken a pure proton or a pure iron composition to determine [MATH] of equation [REF], the reconstructed energy would be [MATH] smaller or larger, respectively.', '1509.05652-2-44-0': 'Second, there is some freedom in the equation used for energy reconstruction.', '1509.05652-2-44-1': 'For example, if we had used exactly the energy equation of reference [CITATION] instead of equation [REF], then the reconstructed primary energy would be about [MATH] lower, although both equations have been tuned against the same set of CoREAS simulations.', '1509.05652-2-44-2': 'This difference is well within the [MATH] scale uncertainty of the amplitude calibration, but indicates that a more detailed study of the reconstruction method will become necessary once a more accurate calibration becomes available.', '1509.05652-2-45-0': 'Third, using CoREAS simulations we found a systematic bias of the energy reconstruction depending on the distance to the shower maximum.', '1509.05652-2-45-1': 'The mean effect is smaller than the difference between proton- and iron-initiated showers, and thus negligible.', '1509.05652-2-45-2': 'However, for very close or far distances to the shower maximum ([MATH]g/cm[MATH] or [MATH]g/cm[MATH]) the bias becomes larger than [MATH], and will have to be corrected for in such rare events.', '1509.05652-2-45-3': 'Consequently, all known systematic uncertainties are small compared to the scale uncertainty of the amplitude calibration of [MATH].', '1509.05652-2-45-4': 'Thus, the total accuracy of the Tunka-Rex energy scale is approximately [MATH].', '1509.05652-2-46-0': 'The Tunka-Rex energy scale defined by CoREAS simulations agrees within the uncertainties to the Tunka-133 scale.', '1509.05652-2-46-1': 'This means that the measurements of Tunka-Rex and Tunka-133 are not only correlated, but in addition agree on an absolute level, which is another confirmation, that CoREAS seems to predict the radio signal correctly.', '1509.05652-2-47-0': '## Comparison with other experiments', '1509.05652-2-48-0': 'The [MATH] energy precision of Tunka-Rex is at least as good as that of other radio arrays, like AERA [CITATION], LOPES [CITATION], and CODALEMA [CITATION], which all reported energy precisions around [MATH].', '1509.05652-2-48-1': 'The Tunka-Rex scale uncertainty of [MATH] roughly corresponds to that of AERA [CITATION] and LOPES [CITATION], which compared the radio measurements to air-fluorescence and particle measurements on ground.', '1509.05652-2-48-2': 'LOFAR as well features an absolute calibration of the radio amplitude with similar accuracy [CITATION], but experimental checks of the energy accuracy are limited by the poor energy resolution of the LORA particle detector array [CITATION].', '1509.05652-2-48-3': 'For [MATH] reconstruction, the resolution of LOPES has been significantly worse, using a simpler method in a more radio-loud environment than Tunka-Rex.', '1509.05652-2-48-4': 'The much denser array LOFAR is located in a more radio-quiet environment similar to the situation of Tunka-Rex.', '1509.05652-2-48-5': 'Its [MATH] reconstruction is also based on CoREAS simulations using two different methods.', '1509.05652-2-48-6': 'With an [MATH] reconstruction based on a fitted LDF (like in our approach), the precision of [MATH] at LOFAR is only slightly better than our resolution [CITATION].', '1509.05652-2-48-7': 'However, with a more computing-intensive method using many simulations for each individual event [CITATION], the precision by LOFAR is at least twice as good as ours.', '1509.05652-2-48-8': 'This indicates that in addition to the planned deployment of additional antennas, also further improvements in the reconstruction methods could increase the [MATH] resolution of Tunka-Rex.', '1509.05652-2-49-0': 'The most important scientific value of the present analysis is the experimental comparision of radio measurements against another, established technique.', '1509.05652-2-49-1': 'For the first time, energy and [MATH] reconstructions based on absolutely-calibrated radio measurements have been compared to air-Cherenkov measurements.', '1509.05652-2-49-2': 'While the principle sensitivity of the radio signal on the longitudinal shower development was already demonstrated experimentally by comparing LOPES lateral distributions to measurements of the KASCADE muon-tracking detector [CITATION], KASCADE did not feature direct [MATH] measurements.', '1509.05652-2-49-3': 'Thus, with the present comparison of Tunka-Rex and Tunka-133, for the first time the radio reconstruction of [MATH] is directly cross-checked with an independent [MATH] measurement.', '1509.05652-2-49-4': 'This also gives more confidence in the results of other radio arrays whose reconstruction procedures are developed with the same CORSIKA + CoREAS Monte Carlo codes.', '1509.05652-2-50-0': '# Outlook', '1509.05652-2-51-0': 'The presently achieved accuracy for energy and [MATH] reconstruction by Tunka-Rex is not yet at its principle limits.', '1509.05652-2-51-1': 'In particular for the [MATH] reconstruction, there is room for improvement.', '1509.05652-2-51-2': 'As shown by LOFAR [CITATION], the precision can be as good as [MATH]g/cm2, when using a denser radio array and a simulation-driven method.', '1509.05652-2-51-3': 'Thus, we plan to deploy additional antennas and to further improve the reconstruction method.', '1509.05652-2-51-4': 'With an increased antenna density, the average event will contain more antenna stations with signal, which allows fitting of lateral distributions with more free parameters.', '1509.05652-2-51-5': 'For example, the weak energy and [MATH] sensitivity of the LDF parameter [MATH] could be exploited when more antenna stations contribute to the fit.', '1509.05652-2-52-0': 'Compared to this study the antenna density will be tripled.', '1509.05652-2-52-1': 'At each of the 19 Tunka clusters in the inner area, already now a second antenna station is deployed, and a third one will be deployed in summer 2016.', '1509.05652-2-52-2': 'This enables an experimental test how the antenna density and the core resolution affect the [MATH] resolution.', '1509.05652-2-52-3': 'Moreover, additional methods for [MATH] reconstruction based on other quantities of the radio signal can be used to improve the total accuracy.', '1509.05652-2-52-4': 'The shape of the wavefront [CITATION] can be reconstructed by arrival time measurements, and the slope of the frequency spectrum [CITATION] by sampling the pulse shape.', '1509.05652-2-52-5': 'Future studies will show by how much these methods can increase the total [MATH] accuracy under practical conditions.', '1509.05652-2-53-0': 'The statistics of Tunka-Rex will be dramatically increased by joint measurements with the newly deployed particle-detector array Tunka-Grande [CITATION], which provides a day-time trigger for Tunka-Rex.', '1509.05652-2-53-1': 'Tunka-Grande consists of 19 stations, one at each inner cluster, where each station features scintillator detectors on ground and under ground, for measurement of secondary air-shower electrons and muons, respectively.', '1509.05652-2-53-2': 'While unimportant for the present analysis focused on the feasibility of the reconstruction methods, for reconstruction of the cosmic-ray composition possible systematic uncertainties like selection biases are relevant.', '1509.05652-2-53-3': 'Using the additional statistics and independent measurements provided by Tunka-Grande such systematic uncertainties can be checked experimentally.', '1509.05652-2-53-4': 'Moreover, the additional muon measurements can enhance the total accuracy for the mass composition, since the electron-muon ratio provides complementary mass information to [MATH] [CITATION].', '1509.05652-2-53-5': 'In addition, the Tunka-Grande measurement accuracy will be enhanced by a cross-calibration on hybrid measurements with Tunka-Rex.', '1509.05652-2-54-0': 'Finally, the absolute energy measurement of air-showers with a radio array opens prospects to cross-calibrate the energy scales of different air-shower experiments via radio measurements.', '1509.05652-2-54-1': 'This aspect is worth to be investigated more deeply, in particular since antenna arrays can be a very economic add-on to existing detectors as shown by Tunka-Rex.', '1509.05652-2-55-0': '# Conclusion', '1509.05652-2-56-0': 'We have compared energy and [MATH] reconstructed from Tunka-Rex and Tunka-133 measurements of the same air showers in the energy range [MATH]eV.', '1509.05652-2-56-1': 'The reconstruction methods have been developed and tuned using simulations with CORSIKA and its radio extension CoREAS.', '1509.05652-2-56-2': 'For both parameters we find a strong correlation between Tunka-Rex and Tunka-133 consistently in two seasons of data taking.', '1509.05652-2-56-3': 'This confirms experimentally that the radio measurements are indeed sensitive to energy and shower maximum.', '1509.05652-2-56-4': 'For [MATH] this is the first direct confirmation based on a cross-check with a different, established experimental technique, namely air-Cherenkov measurements.', '1509.05652-2-57-0': 'The Tunka-Rex energy precision seems to be at least as good as the published Tunka-133 resolution of [MATH].', '1509.05652-2-57-1': 'The total scale uncertainty of Tunka-Rex is dominated by the uncertainty of the amplitude calibration, and in total is in the order of [MATH].', '1509.05652-2-57-2': 'This is comparable to the scale uncertainty of particle detector arrays, like KASCADE-Grande [CITATION].', '1509.05652-2-57-3': 'Since the calibration uncertainty dominates the total scale uncertainty of Tunka-Rex, any improvement in the calibration accuracy will directly propagate to the energy-scale accuracy.', '1509.05652-2-57-4': 'Consequently, further efforts on the calibration will be necessary to increase the accuracy to the same level as the currently leading fluorescence and air-Cherenkov techniques.', '1509.05652-2-57-5': 'At the Pierre Auger Observatory, the scale accuracy of fluorescence measurements is [MATH] [CITATION].', '1509.05652-2-57-6': 'This now is in reasonable reach for future radio measurements, which unlike fluorescence measurements are available around-the-clock.', '1509.05652-2-57-7': 'Hence, radio measurements can be used to determine the absolute energy scale of air-shower measurements.', '1509.05652-2-58-0': 'The [MATH] precision of Tunka-Rex is roughly [MATH]g/cm2, and can be slightly increased by setting stricter quality cuts at the cost of statistics.', '1509.05652-2-58-1': 'This resolution is sufficient to statistically distinguish light from heavy primary particles.', '1509.05652-2-58-2': 'The resolution is worse than the one currently achieved by non-imaging air-Cherenkov measurements ([MATH]g/cm2 for Tunka-133 [CITATION]) and by air-fluorescence measurements ([MATH]g/cm2 for the Pierre Auger Observatory [CITATION]).', '1509.05652-2-58-3': 'Nevertheless, the resolution of Tunka-Rex is not yet at its limit and can likely be increased by further improvements of the reconstruction method, and by deploying additional antennas.', '1509.05652-2-59-0': 'For astrophysical applications the total accuracy for the reconstruction of the mass composition as a function of energy counts, e.g., to better study the transition from Galactic to yet-unknown extra-galactic cosmic-ray sources assumed in the energy range of Tunka-Rex [CITATION].', '1509.05652-2-59-1': 'Tunka-Rex will provide additional statistics exactly in the energy range where current Tunka-133 analyses are limited by statistics.', '1509.05652-2-59-2': 'Moreover, hybrid measurements of air-showers by Tunka-Rex and Tunka-Grande started in autumn 2015.', '1509.05652-2-59-3': 'These hybrid measurements can be used to enhance the total accuracy for the mass composition by combining radio and muon measurements.', '1509.05652-2-60-0': 'Tunka-Rex has been funded by the German Helmholtz association and the Russian Foundation for Basic Research (grant HRJRG-303).', '1509.05652-2-60-1': 'Moreover, this work was supported by the Helmholtz Alliance for Astroparticle Physics (HAP), and by the Russian grant RSF 15-12-20022.', '1509.05652-2-61-0': '# Parametrization of LDF parameter [MATH]', '1509.05652-2-62-0': 'In the Gaussian lateral distribution function (LDF, equation [REF]), the parameter [MATH] defining the bump has been set by the following parameterization determined with CoREAS simulations made for the situation of Tunka-Rex [CITATION]: [EQUATION]', '1509.05652-2-62-1': 'Since the [MATH] parameter is a function of the primary energy [MATH], the shape of the Gaussian LDF used to reconstruct the primary energy depends implicitly on the energy itself.', '1509.05652-2-62-2': 'This reciprocal dependence on [MATH] can either be solved by an iterative approach, or by using a pre-estimate for [MATH].', '1509.05652-2-62-3': 'We decided for the latter solution, using the energy estimator based on a simpler exponential LDF presented in reference [CITATION], since its precision is only slightly worse than than in our more advanced approach here.', '1509.05652-2-63-0': '# Comparison of both seasons', '1509.05652-2-64-0': 'The data set of Tunka-Rex is split in two seasons of about equal size.', '1509.05652-2-64-1': 'As additional cross-check of the validity of our methods, the Tunka-133 reconstruction of energy and [MATH] had been blinded for the second season.', '1509.05652-2-64-2': 'Only for the first season the Tunka-133 energy and [MATH] values had been known, and were used for occasional cross-checks during the development of the Tunka-Rex methods.', '1509.05652-2-64-3': 'After we frozen the Tunka-Rex reconstruction methods, we predicted the energy and [MATH] for the second season based on the radio measurements.', '1509.05652-2-64-4': 'Only afterwards, Tunka-133 unblinded their reconstruction of the second season to us.', '1509.05652-2-65-0': 'The plots in this appendix show the comparison of the Tunka-Rex and Tunka-133 energy and [MATH] reconstructions (figures [REF] and [REF]) separately for both seasons.', '1509.05652-2-65-1': 'The results of both seasons are consistent within statistical uncertainties.', '1509.05652-2-65-2': 'The quality cut on events inside of the inner area ([MATH]m, dashed circle in figure [REF]) was discovered a posteriori by us, i.e., after we had frozen the reconstruction methods for the unblinding procedure.', '1509.05652-2-65-3': 'Thus, in figure [REF] 5 additional events are present (2 in the first season, 3 in the second season), which are outside of the inner area and slightly degrade the [MATH] resolution.', '1509.05652-2-66-0': 'As supplementary material we will upload the list of the events used for the present analysis.', '1509.05652-2-66-1': 'This list contains the reconstructed energy and [MATH] values of the events.', '1509.05652-2-66-2': 'However, the values should not be used for reconstruction of the cosmic-ray energy spectrum or the mass composition.', '1509.05652-2-66-3': 'While unimportant for the present study comparing the Tunka-Rex to the Tunka-133 reconstruction, for such analyses selection biases have to be taken into account, which cannot be derived from the event list alone.', '1509.05652-2-67-0': 'The original event lists for the first tuning and second prediction seasons, as created after freezing the reconstruction methods, but before unblinding, are available at:', '1509.05652-2-68-0': 'http://www.ikp.kit.edu/tunka-rex/', '1509.05652-2-69-0': 'They can be decrypted using the following Linux command:', '1509.05652-2-70-0': 'openssl aes-256-cbc -d -in encryptedFile -out decryptedFile', '1509.05652-2-71-0': 'The passwords are TunkaRex1H9AoxFywAt for the tuning season,', '1509.05652-2-72-0': 'and TunkaRex2z3DumFNfsq for the prediction season.', '1509.05652-2-73-0': 'All these events are also contained in the above-mentioned list submitted as supplementary material.'}

[['1509.05652-1-46-1', '1509.05652-2-58-1'], ['1509.05652-1-13-2', '1509.05652-2-12-2'], ['1509.05652-1-13-4', '1509.05652-2-12-5'], ['1509.05652-1-4-2', '1509.05652-2-4-3'], ['1509.05652-1-4-3', '1509.05652-2-4-4'], ['1509.05652-1-4-4', '1509.05652-2-4-5'], ['1509.05652-1-4-5', '1509.05652-2-4-6'], ['1509.05652-1-53-0', '1509.05652-2-66-0'], ['1509.05652-1-53-1', '1509.05652-2-66-1'], ['1509.05652-1-53-2', '1509.05652-2-66-2'], ['1509.05652-1-26-0', '1509.05652-2-29-0'], ['1509.05652-1-26-3', '1509.05652-2-29-2'], ['1509.05652-1-45-0', '1509.05652-2-57-0'], ['1509.05652-1-45-1', '1509.05652-2-57-1'], ['1509.05652-1-45-2', '1509.05652-2-57-2'], ['1509.05652-1-45-4', '1509.05652-2-57-5'], ['1509.05652-1-45-5', '1509.05652-2-57-6'], ['1509.05652-1-42-1', '1509.05652-2-49-1'], ['1509.05652-1-16-0', '1509.05652-2-15-0'], ['1509.05652-1-16-1', '1509.05652-2-15-1'], ['1509.05652-1-16-2', '1509.05652-2-15-2'], ['1509.05652-1-16-3', '1509.05652-2-15-3'], ['1509.05652-1-19-4', '1509.05652-2-23-4'], ['1509.05652-1-9-1', '1509.05652-2-8-1'], ['1509.05652-1-9-2', '1509.05652-2-8-2'], ['1509.05652-1-9-3', '1509.05652-2-8-3'], ['1509.05652-1-9-4', '1509.05652-2-8-4'], ['1509.05652-1-9-5', '1509.05652-2-8-5'], ['1509.05652-1-9-6', '1509.05652-2-8-6'], ['1509.05652-1-50-0', '1509.05652-2-62-0'], ['1509.05652-1-20-4', '1509.05652-2-24-7'], ['1509.05652-1-10-1', '1509.05652-2-9-2'], ['1509.05652-1-17-1', '1509.05652-2-19-1'], ['1509.05652-1-11-1', '1509.05652-2-10-1'], ['1509.05652-1-11-3', '1509.05652-2-10-3'], ['1509.05652-1-11-5', '1509.05652-2-10-5'], ['1509.05652-1-37-2', '1509.05652-2-52-2'], ['1509.05652-1-2-0', '1509.05652-2-2-0'], ['1509.05652-1-2-1', '1509.05652-2-2-1'], ['1509.05652-1-2-2', '1509.05652-2-2-2'], ['1509.05652-1-18-1', '1509.05652-2-20-1'], ['1509.05652-1-18-2', '1509.05652-2-20-2'], ['1509.05652-1-5-1', '1509.05652-2-5-1'], ['1509.05652-1-5-2', '1509.05652-2-5-2'], ['1509.05652-1-12-1', '1509.05652-2-11-1'], ['1509.05652-1-48-0', '1509.05652-2-60-0'], ['1509.05652-1-3-1', '1509.05652-2-3-1'], ['1509.05652-1-3-2', '1509.05652-2-3-2'], ['1509.05652-1-3-3', '1509.05652-2-3-3'], ['1509.05652-1-3-4', '1509.05652-2-3-4'], ['1509.05652-1-41-4', '1509.05652-2-48-5'], ['1509.05652-1-41-5', '1509.05652-2-48-6'], ['1509.05652-1-41-7', '1509.05652-2-48-8'], ['1509.05652-1-29-1', '1509.05652-2-31-1'], ['1509.05652-1-29-2', '1509.05652-2-31-2'], ['1509.05652-1-39-0', '1509.05652-2-46-0'], ['1509.05652-1-39-1', '1509.05652-2-46-1'], ['1509.05652-1-38-0', '1509.05652-2-42-0'], ['1509.05652-1-38-2', '1509.05652-2-43-1'], ['1509.05652-1-38-4', '1509.05652-2-43-5'], ['1509.05652-1-47-3', '1509.05652-2-53-5'], ['1509.05652-1-27-0', '1509.05652-2-30-0'], ['1509.05652-1-27-1', '1509.05652-2-30-1'], ['1509.05652-1-27-4', '1509.05652-2-30-4'], ['1509.05652-1-28-2', '1509.05652-2-30-6'], ['1509.05652-1-32-0', '1509.05652-2-34-0'], ['1509.05652-1-32-1', '1509.05652-2-34-1'], ['1509.05652-1-33-2', '1509.05652-2-34-4'], ['1509.05652-1-46-0', '1509.05652-2-58-0'], ['1509.05652-1-13-1', '1509.05652-2-12-1'], ['1509.05652-1-13-3', '1509.05652-2-12-3'], ['1509.05652-1-4-0', '1509.05652-2-4-0'], ['1509.05652-1-4-1', '1509.05652-2-4-1'], ['1509.05652-1-53-3', '1509.05652-2-66-3'], ['1509.05652-1-26-2', '1509.05652-2-29-1'], ['1509.05652-1-45-6', '1509.05652-2-57-7'], ['1509.05652-1-42-0', '1509.05652-2-49-0'], ['1509.05652-1-19-0', '1509.05652-2-23-0'], ['1509.05652-1-19-1', '1509.05652-2-23-1'], ['1509.05652-1-19-2', '1509.05652-2-23-2'], ['1509.05652-1-58-0', '1509.05652-2-73-0'], ['1509.05652-1-10-3', '1509.05652-2-9-3'], ['1509.05652-1-10-4', '1509.05652-2-9-4'], ['1509.05652-1-1-2', '1509.05652-2-1-2'], ['1509.05652-1-1-3', '1509.05652-2-1-3'], ['1509.05652-1-1-6', '1509.05652-2-1-7'], ['1509.05652-1-17-0', '1509.05652-2-19-0'], ['1509.05652-1-17-2', '1509.05652-2-19-2'], ['1509.05652-1-11-0', '1509.05652-2-10-0'], ['1509.05652-1-11-2', '1509.05652-2-10-2'], ['1509.05652-1-11-4', '1509.05652-2-10-4'], ['1509.05652-1-11-6', '1509.05652-2-10-9'], ['1509.05652-1-2-3', '1509.05652-2-2-3'], ['1509.05652-1-18-0', '1509.05652-2-20-0'], ['1509.05652-1-5-0', '1509.05652-2-5-0'], ['1509.05652-1-14-0', '1509.05652-2-13-0'], ['1509.05652-1-14-1', '1509.05652-2-13-1'], ['1509.05652-1-14-2', '1509.05652-2-13-4'], ['1509.05652-1-14-3', '1509.05652-2-13-6'], ['1509.05652-1-14-4', '1509.05652-2-13-7'], ['1509.05652-1-12-0', '1509.05652-2-11-0'], ['1509.05652-1-12-2', '1509.05652-2-11-2'], ['1509.05652-1-6-0', '1509.05652-2-6-0'], ['1509.05652-1-3-0', '1509.05652-2-3-0'], ['1509.05652-1-41-1', '1509.05652-2-48-1'], ['1509.05652-1-41-2', '1509.05652-2-48-3'], ['1509.05652-1-41-6', '1509.05652-2-48-7'], ['1509.05652-1-44-0', '1509.05652-2-56-0'], ['1509.05652-1-44-1', '1509.05652-2-56-1'], ['1509.05652-1-44-2', '1509.05652-2-56-2'], ['1509.05652-1-44-4', '1509.05652-2-56-4'], ['1509.05652-1-25-1', '1509.05652-2-27-1'], ['1509.05652-1-29-0', '1509.05652-2-31-0'], ['1509.05652-1-29-4', '1509.05652-2-31-3'], ['1509.05652-1-38-3', '1509.05652-2-43-4'], ['1509.05652-1-38-5', '1509.05652-2-43-6'], ['1509.05652-1-47-0', '1509.05652-2-59-0'], ['1509.05652-1-47-1', '1509.05652-2-53-2'], ['1509.05652-1-47-5', '1509.05652-2-59-1'], ['1509.05652-1-27-3', '1509.05652-2-30-3'], ['1509.05652-1-27-6', '1509.05652-2-30-5'], ['1509.05652-1-35-0', '1509.05652-2-39-0'], ['1509.05652-1-35-2', '1509.05652-2-39-2'], ['1509.05652-1-36-0', '1509.05652-2-39-3'], ['1509.05652-1-33-1', '1509.05652-2-34-3'], ['1509.05652-1-46-2', '1509.05652-2-58-2'], ['1509.05652-1-46-3', '1509.05652-2-58-3'], ['1509.05652-1-13-0', '1509.05652-2-12-0'], ['1509.05652-1-45-3', '1509.05652-2-57-4'], ['1509.05652-1-21-0', '1509.05652-2-26-0'], ['1509.05652-1-21-1', '1509.05652-2-26-2'], ['1509.05652-1-21-2', '1509.05652-2-26-8'], ['1509.05652-1-42-2', '1509.05652-2-49-2'], ['1509.05652-1-42-3', '1509.05652-2-49-4'], ['1509.05652-1-19-3', '1509.05652-2-23-3'], ['1509.05652-1-19-6', '1509.05652-2-23-8'], ['1509.05652-1-9-0', '1509.05652-2-8-0'], ['1509.05652-1-50-1', '1509.05652-2-62-1'], ['1509.05652-1-50-1', '1509.05652-2-62-3'], ['1509.05652-1-20-0', '1509.05652-2-24-0'], ['1509.05652-1-20-1', '1509.05652-2-24-2'], ['1509.05652-1-20-3', '1509.05652-2-24-5'], ['1509.05652-1-20-3', '1509.05652-2-24-6'], ['1509.05652-1-10-0', '1509.05652-2-9-0'], ['1509.05652-1-1-0', '1509.05652-2-1-0'], ['1509.05652-1-1-1', '1509.05652-2-1-1'], ['1509.05652-1-1-4', '1509.05652-2-1-4'], ['1509.05652-1-1-4', '1509.05652-2-1-5'], ['1509.05652-1-1-5', '1509.05652-2-1-6'], ['1509.05652-1-18-3', '1509.05652-2-20-3'], ['1509.05652-1-18-3', '1509.05652-2-20-4'], ['1509.05652-1-18-4', '1509.05652-2-20-5'], ['1509.05652-1-18-4', '1509.05652-2-20-6'], ['1509.05652-1-6-1', '1509.05652-2-6-3'], ['1509.05652-1-6-2', '1509.05652-2-6-1'], ['1509.05652-1-6-4', '1509.05652-2-6-3'], ['1509.05652-1-48-1', '1509.05652-2-60-1'], ['1509.05652-1-23-0', '1509.05652-2-17-0'], ['1509.05652-1-23-2', '1509.05652-2-17-1'], ['1509.05652-1-23-4', '1509.05652-2-17-3'], ['1509.05652-1-41-0', '1509.05652-2-48-0'], ['1509.05652-1-41-3', '1509.05652-2-48-4'], ['1509.05652-1-44-3', '1509.05652-2-56-3'], ['1509.05652-1-39-2', '1509.05652-2-54-0'], ['1509.05652-1-39-2', '1509.05652-2-54-1'], ['1509.05652-1-38-6', '1509.05652-2-45-3'], ['1509.05652-1-38-7', '1509.05652-2-45-4'], ['1509.05652-1-47-2', '1509.05652-2-53-0'], ['1509.05652-1-47-2', '1509.05652-2-53-4'], ['1509.05652-1-47-4', '1509.05652-2-53-0'], ['1509.05652-1-27-2', '1509.05652-2-30-2'], ['1509.05652-1-33-0', '1509.05652-2-34-2']]

[['1509.05652-1-46-1', '1509.05652-2-58-1'], ['1509.05652-1-13-2', '1509.05652-2-12-2'], ['1509.05652-1-13-4', '1509.05652-2-12-5'], ['1509.05652-1-4-2', '1509.05652-2-4-3'], ['1509.05652-1-4-3', '1509.05652-2-4-4'], ['1509.05652-1-4-4', '1509.05652-2-4-5'], ['1509.05652-1-4-5', '1509.05652-2-4-6'], ['1509.05652-1-53-0', '1509.05652-2-66-0'], ['1509.05652-1-53-1', '1509.05652-2-66-1'], ['1509.05652-1-53-2', '1509.05652-2-66-2'], ['1509.05652-1-26-0', '1509.05652-2-29-0'], ['1509.05652-1-26-3', '1509.05652-2-29-2'], ['1509.05652-1-45-0', '1509.05652-2-57-0'], ['1509.05652-1-45-1', '1509.05652-2-57-1'], ['1509.05652-1-45-2', '1509.05652-2-57-2'], ['1509.05652-1-45-4', '1509.05652-2-57-5'], ['1509.05652-1-45-5', '1509.05652-2-57-6'], ['1509.05652-1-42-1', '1509.05652-2-49-1'], ['1509.05652-1-16-0', '1509.05652-2-15-0'], ['1509.05652-1-16-1', '1509.05652-2-15-1'], ['1509.05652-1-16-2', '1509.05652-2-15-2'], ['1509.05652-1-16-3', '1509.05652-2-15-3'], ['1509.05652-1-19-4', '1509.05652-2-23-4'], ['1509.05652-1-9-1', '1509.05652-2-8-1'], ['1509.05652-1-9-2', '1509.05652-2-8-2'], ['1509.05652-1-9-3', '1509.05652-2-8-3'], ['1509.05652-1-9-4', '1509.05652-2-8-4'], ['1509.05652-1-9-5', '1509.05652-2-8-5'], ['1509.05652-1-9-6', '1509.05652-2-8-6'], ['1509.05652-1-50-0', '1509.05652-2-62-0'], ['1509.05652-1-20-4', '1509.05652-2-24-7'], ['1509.05652-1-10-1', '1509.05652-2-9-2'], ['1509.05652-1-17-1', '1509.05652-2-19-1'], ['1509.05652-1-11-1', '1509.05652-2-10-1'], ['1509.05652-1-11-3', '1509.05652-2-10-3'], ['1509.05652-1-11-5', '1509.05652-2-10-5'], ['1509.05652-1-37-2', '1509.05652-2-52-2'], ['1509.05652-1-2-0', '1509.05652-2-2-0'], ['1509.05652-1-2-1', '1509.05652-2-2-1'], ['1509.05652-1-2-2', '1509.05652-2-2-2'], ['1509.05652-1-18-1', '1509.05652-2-20-1'], ['1509.05652-1-18-2', '1509.05652-2-20-2'], ['1509.05652-1-5-1', '1509.05652-2-5-1'], ['1509.05652-1-5-2', '1509.05652-2-5-2'], ['1509.05652-1-12-1', '1509.05652-2-11-1'], ['1509.05652-1-48-0', '1509.05652-2-60-0'], ['1509.05652-1-3-1', '1509.05652-2-3-1'], ['1509.05652-1-3-2', '1509.05652-2-3-2'], ['1509.05652-1-3-3', '1509.05652-2-3-3'], ['1509.05652-1-3-4', '1509.05652-2-3-4'], ['1509.05652-1-41-4', '1509.05652-2-48-5'], ['1509.05652-1-41-5', '1509.05652-2-48-6'], ['1509.05652-1-41-7', '1509.05652-2-48-8'], ['1509.05652-1-29-1', '1509.05652-2-31-1'], ['1509.05652-1-29-2', '1509.05652-2-31-2'], ['1509.05652-1-39-0', '1509.05652-2-46-0'], ['1509.05652-1-39-1', '1509.05652-2-46-1'], ['1509.05652-1-38-0', '1509.05652-2-42-0'], ['1509.05652-1-38-2', '1509.05652-2-43-1'], ['1509.05652-1-38-4', '1509.05652-2-43-5'], ['1509.05652-1-47-3', '1509.05652-2-53-5'], ['1509.05652-1-27-0', '1509.05652-2-30-0'], ['1509.05652-1-27-1', '1509.05652-2-30-1'], ['1509.05652-1-27-4', '1509.05652-2-30-4'], ['1509.05652-1-28-2', '1509.05652-2-30-6'], ['1509.05652-1-32-0', '1509.05652-2-34-0'], ['1509.05652-1-32-1', '1509.05652-2-34-1'], ['1509.05652-1-33-2', '1509.05652-2-34-4']]

[['1509.05652-1-46-0', '1509.05652-2-58-0'], ['1509.05652-1-13-1', '1509.05652-2-12-1'], ['1509.05652-1-13-3', '1509.05652-2-12-3'], ['1509.05652-1-4-0', '1509.05652-2-4-0'], ['1509.05652-1-4-1', '1509.05652-2-4-1'], ['1509.05652-1-53-3', '1509.05652-2-66-3'], ['1509.05652-1-26-2', '1509.05652-2-29-1'], ['1509.05652-1-45-6', '1509.05652-2-57-7'], ['1509.05652-1-42-0', '1509.05652-2-49-0'], ['1509.05652-1-19-0', '1509.05652-2-23-0'], ['1509.05652-1-19-1', '1509.05652-2-23-1'], ['1509.05652-1-19-2', '1509.05652-2-23-2'], ['1509.05652-1-58-0', '1509.05652-2-73-0'], ['1509.05652-1-10-3', '1509.05652-2-9-3'], ['1509.05652-1-10-4', '1509.05652-2-9-4'], ['1509.05652-1-1-2', '1509.05652-2-1-2'], ['1509.05652-1-1-3', '1509.05652-2-1-3'], ['1509.05652-1-1-6', '1509.05652-2-1-7'], ['1509.05652-1-17-0', '1509.05652-2-19-0'], ['1509.05652-1-17-2', '1509.05652-2-19-2'], ['1509.05652-1-11-0', '1509.05652-2-10-0'], ['1509.05652-1-11-2', '1509.05652-2-10-2'], ['1509.05652-1-11-4', '1509.05652-2-10-4'], ['1509.05652-1-11-6', '1509.05652-2-10-9'], ['1509.05652-1-2-3', '1509.05652-2-2-3'], ['1509.05652-1-18-0', '1509.05652-2-20-0'], ['1509.05652-1-5-0', '1509.05652-2-5-0'], ['1509.05652-1-14-0', '1509.05652-2-13-0'], ['1509.05652-1-14-1', '1509.05652-2-13-1'], ['1509.05652-1-14-2', '1509.05652-2-13-4'], ['1509.05652-1-14-3', '1509.05652-2-13-6'], ['1509.05652-1-14-4', '1509.05652-2-13-7'], ['1509.05652-1-12-0', '1509.05652-2-11-0'], ['1509.05652-1-12-2', '1509.05652-2-11-2'], ['1509.05652-1-6-0', '1509.05652-2-6-0'], ['1509.05652-1-3-0', '1509.05652-2-3-0'], ['1509.05652-1-41-1', '1509.05652-2-48-1'], ['1509.05652-1-41-2', '1509.05652-2-48-3'], ['1509.05652-1-41-6', '1509.05652-2-48-7'], ['1509.05652-1-44-0', '1509.05652-2-56-0'], ['1509.05652-1-44-1', '1509.05652-2-56-1'], ['1509.05652-1-44-2', '1509.05652-2-56-2'], ['1509.05652-1-44-4', '1509.05652-2-56-4'], ['1509.05652-1-25-1', '1509.05652-2-27-1'], ['1509.05652-1-29-0', '1509.05652-2-31-0'], ['1509.05652-1-29-4', '1509.05652-2-31-3'], ['1509.05652-1-38-3', '1509.05652-2-43-4'], ['1509.05652-1-38-5', '1509.05652-2-43-6'], ['1509.05652-1-47-0', '1509.05652-2-59-0'], ['1509.05652-1-47-1', '1509.05652-2-53-2'], ['1509.05652-1-47-5', '1509.05652-2-59-1'], ['1509.05652-1-27-3', '1509.05652-2-30-3'], ['1509.05652-1-27-6', '1509.05652-2-30-5'], ['1509.05652-1-35-0', '1509.05652-2-39-0'], ['1509.05652-1-35-2', '1509.05652-2-39-2'], ['1509.05652-1-36-0', '1509.05652-2-39-3'], ['1509.05652-1-33-1', '1509.05652-2-34-3']]

[]

[['1509.05652-1-46-2', '1509.05652-2-58-2'], ['1509.05652-1-46-3', '1509.05652-2-58-3'], ['1509.05652-1-13-0', '1509.05652-2-12-0'], ['1509.05652-1-45-3', '1509.05652-2-57-4'], ['1509.05652-1-21-0', '1509.05652-2-26-0'], ['1509.05652-1-21-1', '1509.05652-2-26-2'], ['1509.05652-1-21-2', '1509.05652-2-26-8'], ['1509.05652-1-42-2', '1509.05652-2-49-2'], ['1509.05652-1-42-3', '1509.05652-2-49-4'], ['1509.05652-1-19-3', '1509.05652-2-23-3'], ['1509.05652-1-19-6', '1509.05652-2-23-8'], ['1509.05652-1-9-0', '1509.05652-2-8-0'], ['1509.05652-1-50-1', '1509.05652-2-62-1'], ['1509.05652-1-50-1', '1509.05652-2-62-3'], ['1509.05652-1-20-0', '1509.05652-2-24-0'], ['1509.05652-1-20-1', '1509.05652-2-24-2'], ['1509.05652-1-20-3', '1509.05652-2-24-5'], ['1509.05652-1-20-3', '1509.05652-2-24-6'], ['1509.05652-1-10-0', '1509.05652-2-9-0'], ['1509.05652-1-1-0', '1509.05652-2-1-0'], ['1509.05652-1-1-1', '1509.05652-2-1-1'], ['1509.05652-1-1-4', '1509.05652-2-1-4'], ['1509.05652-1-1-4', '1509.05652-2-1-5'], ['1509.05652-1-1-5', '1509.05652-2-1-6'], ['1509.05652-1-18-3', '1509.05652-2-20-3'], ['1509.05652-1-18-3', '1509.05652-2-20-4'], ['1509.05652-1-18-4', '1509.05652-2-20-5'], ['1509.05652-1-18-4', '1509.05652-2-20-6'], ['1509.05652-1-6-1', '1509.05652-2-6-3'], ['1509.05652-1-6-2', '1509.05652-2-6-1'], ['1509.05652-1-6-4', '1509.05652-2-6-3'], ['1509.05652-1-48-1', '1509.05652-2-60-1'], ['1509.05652-1-23-0', '1509.05652-2-17-0'], ['1509.05652-1-23-2', '1509.05652-2-17-1'], ['1509.05652-1-23-4', '1509.05652-2-17-3'], ['1509.05652-1-41-0', '1509.05652-2-48-0'], ['1509.05652-1-41-3', '1509.05652-2-48-4'], ['1509.05652-1-44-3', '1509.05652-2-56-3'], ['1509.05652-1-39-2', '1509.05652-2-54-0'], ['1509.05652-1-39-2', '1509.05652-2-54-1'], ['1509.05652-1-38-6', '1509.05652-2-45-3'], ['1509.05652-1-38-7', '1509.05652-2-45-4'], ['1509.05652-1-47-2', '1509.05652-2-53-0'], ['1509.05652-1-47-2', '1509.05652-2-53-4'], ['1509.05652-1-47-4', '1509.05652-2-53-0'], ['1509.05652-1-27-2', '1509.05652-2-30-2'], ['1509.05652-1-33-0', '1509.05652-2-34-2']]

[]

['1509.05652-1-22-0', '1509.05652-1-48-2', '1509.05652-1-54-0', '1509.05652-1-55-0', '1509.05652-1-56-0', '1509.05652-1-57-0', '1509.05652-2-16-0', '1509.05652-2-42-1', '1509.05652-2-67-0', '1509.05652-2-68-0', '1509.05652-2-69-0', '1509.05652-2-70-0', '1509.05652-2-71-0', '1509.05652-2-72-0']

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

https://arxiv.org/abs/1509.05652

1412.8298

{'1412.8298-1-0-0': 'The magnetoresistance under pressure has been measured for WTe[MATH] single crystal, in which extremely large magnetoresistance was discovered recently.', '1412.8298-1-0-1': 'At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under 23.6 kbar.', '1412.8298-1-0-2': 'By analyzing the Shubnikov-de Haas oscillations, two Fermi surfaces are found to persist to high pressure.', '1412.8298-1-0-3': 'The sizes of these two pockets are comparable, but show increasing difference with pressure, which may cause the strong suppression of the magnetoresistance.', '1412.8298-1-0-4': 'These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe[MATH].', '1412.8298-1-1-0': 'The magnetoresistance (MR) is an important transport property of condensed matters.', '1412.8298-1-1-1': 'For simple metals, the MR is usually very small, showing quadratic field dependence in low field and saturating in high field [CITATION].', '1412.8298-1-1-2': 'In contrast, giant magnetoresistance (GMR) was discovered in magnetic multilayers [CITATION], and colossal magnetoresistance (CMR) was found in manganites [CITATION].', '1412.8298-1-1-3': 'Interestingly, in recent few years extremely large ([MATH]%) and non-saturating positive MR (XMR) was discovered in some nonmagnetic compounds including PtSn[MATH] [CITATION], Cd[MATH]As[MATH] [CITATION], WTe[MATH] [CITATION], and NbSb[MATH] [CITATION].', '1412.8298-1-2-0': 'The very recent discovery of XMR in WTe[MATH] is of particular interest [CITATION].', '1412.8298-1-2-1': 'WTe[MATH] is a layered transition-metal dichalcogenide, with the crystal structure shown in Fig. 1(a).', '1412.8298-1-2-2': 'In the dichalcogenide layers, W chains are formed along the [MATH] axis.', '1412.8298-1-2-3': 'An XMR of 4.5 [MATH] in 14.7 T at 4.5 K was found when the current is along the [MATH] axis and magnetic field is applied along the [MATH] axis [CITATION].', '1412.8298-1-2-4': 'More remarkably, it increases to as high as 1.3 [MATH] in 60 T at 0.53 K, without any sign of saturation [CITATION].', '1412.8298-1-2-5': 'Such an XMR makes WTe[MATH] useful in devices such as highly sensitive low-temperature magnetic-field sensors or high-field temperature sensors in cryogenics, and from the material perspective, WTe[MATH] is also very suitable for the development of advanced devices.', '1412.8298-1-3-0': 'While the XMR in Cd[MATH]As[MATH] and NbSb[MATH] may relate to the high mobility of Dirac carriers [CITATION], the XMR in PtSn[MATH] and WTe[MATH] was attributed to the compensation between the electron and hole populations [CITATION].', '1412.8298-1-3-1': 'Actually, such a two-band model for charge transport in semimetals was previously used to explain the very large MR observed in high-purity graphite and bismuth [CITATION].', '1412.8298-1-3-2': 'However, in both graphite and Bi, the MR saturates beyond a few Tesla, due to the slight deviation from perfect compensation [CITATION].', '1412.8298-1-3-3': 'The non-saturating XMR in WTe[MATH] may provide the first example of perfectly balanced electron-hole populations [CITATION].', '1412.8298-1-3-4': 'The subsequent angle-resolved photoemission spectroscopy (ARPES) experiments indeed observed hole and electron pockets of approximately the same size at low temperature [CITATION].', '1412.8298-1-3-5': 'Since the perfect balance between electron and hole populations should be very sensitive to some tuning parameters such as doping and pressure, investigating the evolution of the XMR in WTe[MATH] with these parameters may further clarify its origin.', '1412.8298-1-4-0': 'In this Letter, we present the magnetoresistance measurements on WTe[MATH] single crystals under various pressures.', '1412.8298-1-4-1': 'A strong suppression of the XMR with increasing pressure is observed.', '1412.8298-1-4-2': 'By analyzing the Shubnikov-de Haas oscillations of magnetoresistance at low temperature, the evolution of Fermi surfaces with pressure is revealed.', '1412.8298-1-4-3': 'The correlation between the suppression of XMR and the change of Fermi surfaces provides strong support for the mechanism that the XMR of WTe[MATH] originates from the electron-hole compensation.', '1412.8298-1-5-0': 'The WTe[MATH] single crystals were grown by using a chemical vapor transport method similar to that described in Ref. [CITATION].', '1412.8298-1-5-1': 'A mixture of stoichiometric W and Te powder were sealed into an evacuated quartz tube with transport agent Br[MATH].', '1412.8298-1-5-2': 'The quartz tube was then placed in a double zone furnace with temperature gradient of 100 [MATH]C between 750 [MATH]C and 650 [MATH]C. Large single crystals of centimeter size were obtained after one week.', '1412.8298-1-5-3': 'The samples were cut and cleaved to a typical size of 2.0 [MATH] 0.7 [MATH] 0.02 mm[MATH], in which the longest edge is along [MATH] axis and the thinnest dimension is along the [MATH] axis.', '1412.8298-1-5-4': 'Figure 1(b) shows the photo of one sample.', '1412.8298-1-5-5': 'Standard four-probe method was used for resistivity measurements, with current along the [MATH] axis.', '1412.8298-1-5-6': 'The contacts were made with silver epoxy.', '1412.8298-1-5-7': 'The resistivity was measured in a [MATH]He cryostat from 300 K to 2 K, and in a [MATH]He cryostat down to 0.3 K. For measurements under pressure, samples were pressurized in a piston-cylinder clamp cell made of Be-Cu alloy, with Daphne oil as the pressure medium.', '1412.8298-1-5-8': 'The pressure inside the cell was determined from the [MATH] of a tin wire.', '1412.8298-1-5-9': 'Magnetic field was applied along the [MATH]-axis direction up to 14.5 T.', '1412.8298-1-6-0': 'The inset of Fig. 1(c) shows the temperature dependence of resistivity in zero field for WTe[MATH] single crystal, with current along the [MATH] axis.', '1412.8298-1-6-1': 'It has [MATH](295 K) = 355 [MATH] cm, with the residual resistivity ratio RRR = [MATH](295 K)/[MATH](2 K) = 184.', '1412.8298-1-6-2': 'This RRR value is about half of that in Ref. [CITATION].', '1412.8298-1-6-3': 'The main panel of Fig. 1(c) presents the magnetoresistance up to 14.5 T at various temperatures.', '1412.8298-1-6-4': 'The MR is defined by MR = [[MATH]) - [MATH](0 T)]/[MATH](0 T) [MATH].', '1412.8298-1-6-5': 'At 0.3 K and in 14.5 T, the MR reaches as high as 1.25 [MATH]% (taken from the smooth background of the curve).', '1412.8298-1-6-6': 'This value is lower than that in Ref. [CITATION], which is attributed to the relatively smaller RRR of our sample.', '1412.8298-1-6-7': 'With increasing temperature, the MR decreases rapidly and the oscillations disappear above 15 K, which are consistent with Ref. [CITATION].', '1412.8298-1-7-0': 'Figure 2(a) shows the MR of WTe[MATH] single crystal under various pressures up to 23.6 kbar, measured at [MATH] = 0.3 K. With increasing pressure, the MR is strongly suppressed and the oscillations also gradually disappear.', '1412.8298-1-7-1': 'The pressure dependence of the MR in the highest field [MATH] = 14.5 T is plotted in Fig. 2(b).', '1412.8298-1-7-2': 'It decreases from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under [MATH] = 23.6 kbar.', '1412.8298-1-8-0': 'To find out the cause of this strong suppression of MR in WTe[MATH], we check the evolution of its Fermi surface with pressure by analyzing the Shubnikov-de Haas (SdH) oscillations.', '1412.8298-1-8-1': 'In Fig. 3(a), the oscillatory MR under ambient pressure is firstly analyzed by employing Fast Fourier Transform (FFT).', '1412.8298-1-8-2': 'The FFT spectrum shows four major peaks at 94.7, 132, 148, and 166 T oscillation frequency, labeled as [MATH] and [MATH].', '1412.8298-1-8-3': 'The higher harmonics of the [MATH], [MATH], and [MATH] peaks are also observed.', '1412.8298-1-8-4': 'In metal or semiconductor, the SdH oscillation is a useful technique to detect the Fermi surface topology [CITATION].', '1412.8298-1-8-5': 'The SdH oscillation frequency [MATH] is proportional to the extremal cross-sectional area of the Fermi surface normal to the field, according to the Onsager relation [MATH].', '1412.8298-1-8-6': 'The [MATH] and [MATH] peaks seen in Fig. 3(a) clearly indicate that there exist four Fermi pockets normal to the field.', '1412.8298-1-8-7': 'It is possible that two pockets belong to the same three-dimensional (3D) Fermi surface, as we will show below.', '1412.8298-1-9-0': 'The SdH oscillations of MR can be described by the Lifshitz-Kosevich (LK) formula [CITATION].', '1412.8298-1-9-1': 'The temperature dependence of the oscillation amplitude is determined by the thermal damping factor [MATH] in the LK formula, defined as [MATH], where the effective mass [MATH] and [MATH] 14.69 T/K.', '1412.8298-1-9-2': '[MATH] is the Boltzmann constant and [MATH] is the bare electron mass.', '1412.8298-1-9-3': 'Figure 3(b) shows the fits of the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor [MATH] from 0.3 to 6 K.', '1412.8298-1-9-4': 'The effective masses [MATH] = 0.387, 0.414, and 0.462[MATH] are obtained for [MATH], and [MATH] pockets, respectively.', '1412.8298-1-9-5': 'Since the amplitude of the [MATH] peak can not be reliably extracted above 3 K, we did not do the fitting for the [MATH] pocket.', '1412.8298-1-10-0': 'The Fermi surface of WTe[MATH], calculated including spin orbital coupling, shows electron and hole pockets displaced from the [MATH] point and aligned along the chain direction [CITATION].', '1412.8298-1-10-1': 'Such a Fermi surface topology was confirmed by the ARPES experiments at 20 K, which found a pair of electron and hole pockets with comparable size, as sketched in Fig. 4(a) [CITATION].', '1412.8298-1-10-2': 'Interestingly, according to the calculated electronic structure [CITATION], a potential second set of electron and hole pockets forming along [MATH] (parallel to the [MATH] direction, but shifted along [MATH] into the perpendicular face of the Brillouin zone) represents a potential second crossing that would change the pockets into tubes in the Fermi surface.', '1412.8298-1-10-3': 'We can not identify the four major peaks with only the FFT spectrum under ambient pressure, therefore we further analyze the SdH oscillations under high pressure.', '1412.8298-1-11-0': 'Figure 4(b)-(f) show the FFT spectrums from ambient pressure to 18.1 kbar.', '1412.8298-1-11-1': 'Above 18.1 kbar, the oscillations are too weak to give a reliable FFT spectrum.', '1412.8298-1-11-2': 'One can see that both [MATH] and [MATH] peaks persist all the way to the highest pressure.', '1412.8298-1-11-3': 'According to the scenario in Ref. [CITATION], the nearly perfect balance of electron and hole populations is responsible for the XMR in WTe[MATH].', '1412.8298-1-11-4': 'Since under 12.2 kbar, the MR is still as large as 8.28 [MATH]% and only the [MATH] and [MATH] peaks remain, we tentatively identify the [MATH] and [MATH] peaks as the pair of electron and hole pockets along the [MATH] direction.', '1412.8298-1-11-5': 'Then we examine the sizes of these two pockets.', '1412.8298-1-11-6': 'According to the Onsager relation [MATH], [MATH] = 0.0125 [MATH] and [MATH] = 0.0141 [MATH] are obtained for ambient pressure.', '1412.8298-1-11-7': 'By assuming a circular pocket, the Fermi momentums [MATH] 0.063 [MATH] and [MATH] 0.067 [MATH] are estimated.', '1412.8298-1-11-8': 'These values are roughly consistent with [MATH] 0.08 [MATH] for the electron and hole pockets observed by ARPES experiments [CITATION].', '1412.8298-1-11-9': 'Note that the two values of [MATH] and [MATH] are comparable but slightly different.', '1412.8298-1-11-10': 'To contain the same electron and hole populations, it simply requires the two corresponding 3D Fermi surfaces to have slightly different shapes in the 3D Brillouin zone, as sketched in Ref. [CITATION].', '1412.8298-1-12-0': 'From Fig. 4, the [MATH] peak with higher frequency disappears first as the pressure [MATH] = 8.1 kbar is applied, then the [MATH] peak with lower frequency also disappears above [MATH] = 12.2 kbar.', '1412.8298-1-12-1': 'We identify the [MATH] and [MATH] peaks as the second set of electron and hole pockets forming along [MATH], as suggested in Ref. [CITATION].', '1412.8298-1-12-2': 'Indeed, the ARPES experiments with different photon energies found a tube-like Fermi surface along [MATH] [CITATION].', '1412.8298-1-12-3': 'To get a clear 3D Fermi surface topology of WTe[MATH], more high-resolution ARPES or angle-dependent quantum oscillation measurements are highly desired.', '1412.8298-1-12-4': 'Nevertheless, our pressure study shows that the [MATH] and [MATH] pockets, likely forming along [MATH], is very sensitive to the lattice parameters, while the [MATH] and [MATH] pockets along the [MATH] are very robust.', '1412.8298-1-13-0': 'With increasing pressure, the absolute sizes of [MATH] and [MATH] pockets increase, because of the shrink of lattice.', '1412.8298-1-13-1': 'The relative size of [MATH] and [MATH] pockets, defined as [MATH]/[MATH], is plotted in Fig. 2(b) together with the pressure dependence of MR. The two curves are clearly correlated, suggesting that the increasing difference between the sizes of [MATH] and [MATH] pockets is the cause of the strong suppression of MR with pressure in WTe[MATH].', '1412.8298-1-13-2': 'Although our observation is only the relative size change of the two two-dimensional pockets, it may reflect the increasingly different populations of electrons and holes contained in the two corresponding 3D Fermi surfaces.', '1412.8298-1-13-3': 'In this sense, our result supports the scenario that the perfect balance between the electron and hole populations is the origin of the XMR in WTe[MATH].', '1412.8298-1-14-0': 'In summary, we use pressure as the tuning parameter to study its effect on the XMR of WTe[MATH].', '1412.8298-1-14-1': 'A strong suppression of the XMR under pressure is observed.', '1412.8298-1-14-2': 'Four Fermi pockets are revealed by analyzing the SdH oscillations.', '1412.8298-1-14-3': 'Two of them have comparable size and persist under high pressure, which are identified as the pair of electron and hole pockets along the [MATH] direction.', '1412.8298-1-14-4': 'The other two pockets disappear with increasing pressure, which are identified as the second set of electron and hole pockets forming along [MATH].', '1412.8298-1-14-5': 'The relative size of the pair of electron and hole pockets along the [MATH] direction decreases with increasing pressure, which may cause the strong suppression of the XMR.', '1412.8298-1-14-6': 'Our results confirm the importance of the perfect balance between the electron and hole populations to the XMR in W[MATH].', '1412.8298-1-15-0': 'We thank J. K. Dong, P. S. Wang, and Z. J. Xiang for helpful discussions.', '1412.8298-1-15-1': 'This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.', '1412.8298-1-15-2': 'The work at Tulane is supported by NSF under Grant No. DMR-1205469.'}

{'1412.8298-2-0-0': 'The quantum oscillations of the magnetoresistance under ambient and high pressure have been studied for WTe[MATH] single crystals, in which extremely large magnetoresistance was discovered recently.', '1412.8298-2-0-1': 'By analyzing the Shubnikov-de Haas oscillations, four Fermi surfaces are identified for the first time, and two of them are found to persist to high pressure.', '1412.8298-2-0-2': 'The sizes of these two pockets are comparable, but show increasing difference with pressure.', '1412.8298-2-0-3': 'At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under 23.6 kbar, which is likely caused by the relative change of Fermi surfaces.', '1412.8298-2-0-4': 'These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe[MATH].', '1412.8298-2-1-0': 'The magnetoresistance (MR) is an important transport property of condensed matters.', '1412.8298-2-1-1': 'For simple metals, the MR is usually very small, showing quadratic field dependence in low field and saturating in high field [CITATION].', '1412.8298-2-1-2': 'In contrast, giant magnetoresistance (GMR) was discovered in magnetic multilayers [CITATION], and colossal magnetoresistance (CMR) was found in manganites [CITATION].', '1412.8298-2-1-3': 'Interestingly, in recent few years extremely large ([MATH]%) and non-saturating positive MR (XMR) was discovered in some nonmagnetic compounds including PtSn[MATH] [CITATION], Cd[MATH]As[MATH] [CITATION], WTe[MATH] [CITATION], and NbSb[MATH] [CITATION].', '1412.8298-2-2-0': 'The very recent discovery of XMR in WTe[MATH] is of particular interest [CITATION].', '1412.8298-2-2-1': 'WTe[MATH] is a layered transition-metal dichalcogenide, with the crystal structure shown in Fig. 1(a).', '1412.8298-2-2-2': 'In the dichalcogenide layers, W chains are formed along the [MATH] axis.', '1412.8298-2-2-3': 'An XMR of 4.5 [MATH] in 14.7 T at 4.5 K was found when the current is along the [MATH] axis and magnetic field is applied along the [MATH] axis [CITATION].', '1412.8298-2-2-4': 'More remarkably, it increases to as high as 1.3 [MATH] in 60 T at 0.53 K, without any sign of saturation [CITATION].', '1412.8298-2-2-5': 'Such an XMR makes WTe[MATH] useful in devices such as highly sensitive low-temperature magnetic-field sensors or high-field temperature sensors in cryogenics.', '1412.8298-2-2-6': 'From the material perspective, WTe[MATH] is also quite suitable for the development of advanced devices.', '1412.8298-2-3-0': 'While the XMR in Cd[MATH]As[MATH] and NbSb[MATH] may relate to the high mobility of Dirac carriers [CITATION], the XMR in PtSn[MATH] and WTe[MATH] was attributed to the compensation between the electron and hole populations [CITATION].', '1412.8298-2-3-1': 'Actually, such a two-band model for charge transport in semimetals was previously used to explain the very large MR observed in high-purity graphite and bismuth [CITATION].', '1412.8298-2-3-2': 'However, in both graphite and Bi, the MR saturates beyond a few Tesla, due to the slight deviation from perfect compensation [CITATION].', '1412.8298-2-3-3': 'The non-saturating XMR in WTe[MATH] may provide the first example of perfectly balanced electron-hole populations [CITATION].', '1412.8298-2-3-4': 'While the subsequent angle-resolved photoemission spectroscopy (ARPES) experiments observed one pair of hole and electron pockets of approximately the same size at low temperature [CITATION], a very recent ARPES work showed a complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], therefore more experiments like quantum oscillation measurement are highly desired to clarify it.', '1412.8298-2-3-5': 'On the other hand, since the perfect balance between electron and hole populations should be very sensitive to some tuning parameters such as doping and pressure, investigating the evolution of the XMR in WTe[MATH] with these parameters may further clarify its origin.', '1412.8298-2-4-0': 'In this Letter, we present the quantum oscillation study of the magnetoresistance for WTe[MATH] single crystals under ambient and high pressure.', '1412.8298-2-4-1': 'By analyzing the Shubnikov-de Haas oscillations of magnetoresistance at low temperature, four Fermi surfaces are revealed, which should be the two pairs of hole and electron pockets later observed by ARPES experiments [CITATION].', '1412.8298-2-4-2': 'A drastic suppression of the XMR with increasing pressure is observed, which is accompanied by a change of the Fermi surface topology.', '1412.8298-2-4-3': 'The correlation between them provides support for the mechanism that the XMR of WTe[MATH] originates from the electron-hole compensation.', '1412.8298-2-5-0': 'The WTe[MATH] single crystals were grown by using a chemical vapor transport method similar to that described in Ref. [CITATION].', '1412.8298-2-5-1': 'A mixture of stoichiometric W and Te powder were sealed into an evacuated quartz tube with transport agent Br[MATH].', '1412.8298-2-5-2': 'The quartz tube was then placed in a double zone furnace with temperature gradient of 100 [MATH]C between 750 [MATH]C and 650 [MATH]C. Large single crystals of centimeter size were obtained after one week.', '1412.8298-2-5-3': 'The samples were cut and cleaved to a typical size of 2.0 [MATH] 0.7 [MATH] 0.02 mm[MATH], in which the longest edge is along [MATH] axis and the thinnest dimension is along the [MATH] axis.', '1412.8298-2-5-4': 'Figure 1(b) shows the photo of one sample.', '1412.8298-2-5-5': 'X-ray diffraction (XRD) measurement was performed by using an X-ray diffractometer (D8 Advance, Bruker).', '1412.8298-2-5-6': 'Standard four-probe method was used for resistivity measurements, with current along the [MATH] axis.', '1412.8298-2-5-7': 'The contacts were made with silver epoxy.', '1412.8298-2-5-8': 'The resistivity was measured in a [MATH]He cryostat from 300 K to 2 K, and in a [MATH]He cryostat down to 0.3 K. For measurements under pressure, samples were pressurized in a piston-cylinder clamp cell made of Be-Cu alloy, with Daphne oil as the pressure medium.', '1412.8298-2-5-9': 'The pressure inside the cell was determined from the [MATH] of a tin wire.', '1412.8298-2-5-10': 'Magnetic field was applied along the [MATH]-axis direction up to 14.5 T.', '1412.8298-2-6-0': 'Figure 1(c) presents the XRD result of WTe[MATH] single crystal.', '1412.8298-2-6-1': 'Only reflections of (0 0 2[MATH]) show up, suggesting good orientation along the [MATH] axis.', '1412.8298-2-6-2': 'The lattice parameter [MATH] 14.054 is estimated from the XRD data, which agrees well with previous reports [CITATION].', '1412.8298-2-7-0': 'Figure 2(a) shows the temperature dependence of resistivity in zero field for WTe[MATH] single crystal, with current along the [MATH] axis.', '1412.8298-2-7-1': 'It has [MATH](295 K) = 355 [MATH] cm, with the residual resistivity ratio RRR = [MATH](295 K)/[MATH](2 K) = 184.', '1412.8298-2-7-2': 'This RRR value is about half of that in Ref. [CITATION].', '1412.8298-2-7-3': 'Figure 2(b) presents the magnetoresistance up to 14.5 T at various temperatures.', '1412.8298-2-7-4': 'The MR is defined by MR = [[MATH]) - [MATH](0 T)]/[MATH](0 T) [MATH].', '1412.8298-2-7-5': 'At 0.3 K and in 14.5 T, the MR reaches as high as 1.25 [MATH]% (taken from the smooth background of the curve).', '1412.8298-2-7-6': 'This value is lower than that in Ref. [CITATION], which is attributed to the slightly lower quality (smaller RRR) of our sample.', '1412.8298-2-7-7': 'With increasing temperature, the MR decreases rapidly and the oscillations disappear above 15 K, which are consistent with Ref. [CITATION].', '1412.8298-2-8-0': 'The SdH oscillation is a useful technique to detect the Fermi surface topology [CITATION].', '1412.8298-2-8-1': 'In Fig. 3(a), the oscillatory MR is analyzed by employing Fast Fourier Transform (FFT) for various temperatures from 0.3 to 6 K.', '1412.8298-2-8-2': 'The FFT spectrum shows four major peaks at 94.7, 132, 148, and 166 T oscillation frequency, labeled as [MATH] and [MATH].', '1412.8298-2-8-3': 'The higher harmonics of the [MATH] and [MATH] peaks are also observed.', '1412.8298-2-8-4': 'The SdH oscillation frequency [MATH] is proportional to the extremal cross-sectional area of the Fermi surface normal to the field, according to the Onsager relation [MATH].', '1412.8298-2-8-5': 'The [MATH] and [MATH] peaks seen in Fig. 3(a) clearly indicate that there exist four Fermi pockets normal to the field.', '1412.8298-2-9-0': 'The SdH oscillations of MR can be described by the Lifshitz-Kosevich (LK) formula [CITATION].', '1412.8298-2-9-1': 'The temperature dependence of the oscillation amplitude is determined by the thermal damping factor [MATH] in the LK formula, defined as [MATH], where [MATH] is the effective mass, [MATH], and [MATH] is the Boltzmann constant.', '1412.8298-2-9-2': 'Figure 3(b) shows the fits of the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor [MATH] from 0.3 to 6 K.', '1412.8298-2-9-3': 'The effective masses [MATH] = 0.387[MATH], 0.414[MATH], and 0.462[MATH] are obtained for [MATH], and [MATH] pockets, respectively, where [MATH] is the bare electron mass.', '1412.8298-2-9-4': 'Since the amplitude of the [MATH] peak can not be reliably extracted above 3 K, we did not do the fitting for the [MATH] pocket.', '1412.8298-2-10-0': 'Next we investigate the pressure effect on the XMR of WTe[MATH].', '1412.8298-2-10-1': 'Figure 4(a) shows the MR of WTe[MATH] single crystal under various pressures up to 23.6 kbar, measured at [MATH] = 0.3 K. With increasing pressure, the MR is strongly suppressed and the oscillations also gradually disappear.', '1412.8298-2-10-2': 'The pressure dependence of the MR in the highest field [MATH] = 14.5 T is plotted in Fig. 4(b).', '1412.8298-2-10-3': 'It decreases from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under [MATH] = 23.6 kbar.', '1412.8298-2-11-0': 'To find out the cause of this strong suppression of MR in WTe[MATH], we check the evolution of its Fermi surfaces with pressure.', '1412.8298-2-11-1': 'First let us identify the four Fermi pockets obtained at ambient pressure, by comparing with the two ARPES results [CITATION].', '1412.8298-2-11-2': 'One ARPES group found a pair of hole and electron pockets with comparable size along the [MATH] direction [CITATION], however, another ARPES group revealed a more complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], as sketched in Fig. 5(a).', '1412.8298-2-11-3': 'The sizes of the pair of hole and electron pockets in Ref. [CITATION] are comparable with [MATH] 0.08 [MATH], while the sizes of the two pairs of hole and electron pockets in Ref. [CITATION] are smaller and show slight difference.', '1412.8298-2-11-4': 'We examine the sizes of the four Fermi pockets obtained from our SdH oscillations of MR.', '1412.8298-2-11-5': 'According to the Onsager relation [MATH], [MATH] = 0.0090, 0.0125, 0.0141, and 0.0158 [MATH] are obtained for [MATH], [MATH], [MATH], and [MATH] pockets, respectively.', '1412.8298-2-11-6': 'By assuming a circular pocket, the Fermi momentums [MATH] 0.054 [MATH], [MATH] 0.063 [MATH], [MATH] 0.067 [MATH], and [MATH] 0.071 [MATH] are estimated.', '1412.8298-2-11-7': 'Therefore, both the number and sizes of the Fermi pockets obtained from our measurements support the Fermi surface topology of WTe[MATH] revealed in Ref. [CITATION].', '1412.8298-2-11-8': 'Note that an additional hole pocket around [MATH] was observed in some samples but absent in some other samples [CITATION], therefore we do not sketch it in Fig. 5(a).', '1412.8298-2-11-9': 'It is likely absent in our samples.', '1412.8298-2-12-0': 'Figure 5(b)-(f) show the FFT spectrums from ambient pressure to 18.1 kbar.', '1412.8298-2-12-1': 'Above 18.1 kbar, the oscillations are too weak to give a reliable FFT spectrum.', '1412.8298-2-12-2': 'One can see that both [MATH] and [MATH] peaks persist all the way to the highest pressure.', '1412.8298-2-12-3': 'According to the scenario in Ref. [CITATION], the nearly perfect balance of electron and hole populations is responsible for the XMR in WTe[MATH].', '1412.8298-2-12-4': 'Since under 12.2 kbar, the MR is still as large as 8.28 [MATH]% and only the [MATH] and [MATH] peaks remain, we identify [MATH] and [MATH] as one of the two pairs of hole and electron pockets.', '1412.8298-2-12-5': 'It will be very interesting to do electronic structure calculation under pressure, to show how another pair of hole and electron pockets, [MATH] and [MATH], disappears with increasing pressure.', '1412.8298-2-12-6': "We note a recent theoretical calculation shows that the electronic structure of monolayer 1T'-WTe[MATH] is sensitive to the tensile strain, which may be crucial for realizing the quantum spin Hall effect in this two-dimensional transition metal dichalcogenide [CITATION].", '1412.8298-2-13-0': 'With increasing pressure, the absolute sizes of [MATH] and [MATH] pockets increase, because of the shrink of lattice.', '1412.8298-2-13-1': 'The relative size of [MATH] and [MATH] pockets, defined as [MATH]/[MATH], is plotted in Fig. 4(b) together with the pressure dependence of MR. The two curves are clearly correlated, suggesting that the increasing difference between the sizes of [MATH] and [MATH] pockets is the cause of the strong suppression of MR with pressure in WTe[MATH].', '1412.8298-2-13-2': 'In this sense, our result confirms the importance of the perfect balance between the electron and hole populations to the XMR in WTe[MATH].', '1412.8298-2-13-3': 'The high sensitivity of MR to pressure (strain) provides additional way to use WTe[MATH] in devices, for example, as a pressure sensor.', '1412.8298-2-14-0': 'In summary, we study the quantum oscillations of magnetoresistance under ambient and high pressure for WTe[MATH] single crystals.', '1412.8298-2-14-1': 'Under ambient pressure, four Fermi surfaces are identified by analyzing the SdH oscillations, which are likely two pairs of hole and electron pockets along the [MATH] direction.', '1412.8298-2-14-2': 'With increasing pressure, drastic change of Fermi surface topology and strong suppression of the XMR are observed.', '1412.8298-2-14-3': 'While one pair of hole and electron pockets ([MATH] and [MATH]) persists to high pressure, the other pair of hole and electron pockets ([MATH] and [MATH]) disappears with increasing pressure.', '1412.8298-2-14-4': 'The relative size of the [MATH] and [MATH] pockets decreases with increasing pressure, which may cause the strong suppression of the XMR.', '1412.8298-2-14-5': 'Our results support the scenario that the perfect balance between the electron and hole populations is the origin of the XMR in WTe[MATH].', '1412.8298-2-15-0': 'We thank J. K. Dong, D. L. Feng, J. Jiang, P. S. Wang, and Z. J. Xiang for helpful discussions.', '1412.8298-2-15-1': 'This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200).', '1412.8298-2-15-2': 'The work at Tulane is supported by NSF under Grant No. DMR-1205469.', '1412.8298-2-16-0': 'Note added: After we put this work in arXiv (1412.8298), superconductivity was discovered in WTe[MATH] by applying higher pressure than ours [CITATION], which coincides with the suppression of the XMR.'}

[['1412.8298-1-5-0', '1412.8298-2-5-0'], ['1412.8298-1-5-1', '1412.8298-2-5-1'], ['1412.8298-1-5-2', '1412.8298-2-5-2'], ['1412.8298-1-5-3', '1412.8298-2-5-3'], ['1412.8298-1-5-4', '1412.8298-2-5-4'], ['1412.8298-1-5-5', '1412.8298-2-5-6'], ['1412.8298-1-5-6', '1412.8298-2-5-7'], ['1412.8298-1-5-7', '1412.8298-2-5-8'], ['1412.8298-1-5-8', '1412.8298-2-5-9'], ['1412.8298-1-5-9', '1412.8298-2-5-10'], ['1412.8298-1-9-0', '1412.8298-2-9-0'], ['1412.8298-1-9-3', '1412.8298-2-9-2'], ['1412.8298-1-9-5', '1412.8298-2-9-4'], ['1412.8298-1-6-2', '1412.8298-2-7-2'], ['1412.8298-1-6-5', '1412.8298-2-7-5'], ['1412.8298-1-6-7', '1412.8298-2-7-7'], ['1412.8298-1-3-0', '1412.8298-2-3-0'], ['1412.8298-1-3-1', '1412.8298-2-3-1'], ['1412.8298-1-3-2', '1412.8298-2-3-2'], ['1412.8298-1-3-3', '1412.8298-2-3-3'], ['1412.8298-1-13-0', '1412.8298-2-13-0'], ['1412.8298-1-1-0', '1412.8298-2-1-0'], ['1412.8298-1-1-1', '1412.8298-2-1-1'], ['1412.8298-1-1-2', '1412.8298-2-1-2'], ['1412.8298-1-1-3', '1412.8298-2-1-3'], ['1412.8298-1-2-0', '1412.8298-2-2-0'], ['1412.8298-1-2-1', '1412.8298-2-2-1'], ['1412.8298-1-2-2', '1412.8298-2-2-2'], ['1412.8298-1-2-3', '1412.8298-2-2-3'], ['1412.8298-1-2-4', '1412.8298-2-2-4'], ['1412.8298-1-11-1', '1412.8298-2-12-1'], ['1412.8298-1-11-2', '1412.8298-2-12-2'], ['1412.8298-1-11-3', '1412.8298-2-12-3'], ['1412.8298-1-0-4', '1412.8298-2-0-4'], ['1412.8298-1-8-2', '1412.8298-2-8-2'], ['1412.8298-1-8-5', '1412.8298-2-8-4'], ['1412.8298-1-8-6', '1412.8298-2-8-5'], ['1412.8298-1-7-2', '1412.8298-2-10-3'], ['1412.8298-2-1-0', '1412.8298-3-1-0'], ['1412.8298-2-1-1', '1412.8298-3-1-1'], ['1412.8298-2-1-2', '1412.8298-3-1-2'], ['1412.8298-2-1-3', '1412.8298-3-1-3'], ['1412.8298-2-11-0', '1412.8298-3-11-0'], ['1412.8298-2-11-1', '1412.8298-3-11-1'], ['1412.8298-2-11-2', '1412.8298-3-11-2'], ['1412.8298-2-11-3', '1412.8298-3-11-3'], ['1412.8298-2-11-4', '1412.8298-3-11-4'], ['1412.8298-2-11-7', '1412.8298-3-11-7'], ['1412.8298-2-11-8', '1412.8298-3-11-8'], ['1412.8298-2-11-9', '1412.8298-3-11-9'], ['1412.8298-2-9-0', '1412.8298-3-9-0'], ['1412.8298-2-9-1', '1412.8298-3-9-1'], ['1412.8298-2-9-2', '1412.8298-3-9-2'], ['1412.8298-2-9-3', '1412.8298-3-9-3'], ['1412.8298-2-9-4', '1412.8298-3-9-4'], ['1412.8298-2-16-0', '1412.8298-3-17-0'], ['1412.8298-2-14-0', '1412.8298-3-15-0'], ['1412.8298-2-14-1', '1412.8298-3-15-1'], ['1412.8298-2-14-2', '1412.8298-3-15-2'], ['1412.8298-2-14-3', '1412.8298-3-15-3'], ['1412.8298-2-14-4', '1412.8298-3-15-4'], ['1412.8298-2-14-5', '1412.8298-3-15-5'], ['1412.8298-2-0-0', '1412.8298-3-0-0'], ['1412.8298-2-0-1', '1412.8298-3-0-1'], ['1412.8298-2-0-2', '1412.8298-3-0-2'], ['1412.8298-2-0-3', '1412.8298-3-0-3'], ['1412.8298-2-0-4', '1412.8298-3-0-4'], ['1412.8298-2-15-0', '1412.8298-3-16-0'], ['1412.8298-2-15-1', '1412.8298-3-16-1'], ['1412.8298-2-5-0', '1412.8298-3-5-0'], ['1412.8298-2-5-1', '1412.8298-3-5-1'], ['1412.8298-2-5-2', '1412.8298-3-5-2'], ['1412.8298-2-5-3', '1412.8298-3-5-3'], ['1412.8298-2-5-4', '1412.8298-3-5-4'], ['1412.8298-2-5-5', '1412.8298-3-5-5'], ['1412.8298-2-5-6', '1412.8298-3-5-6'], ['1412.8298-2-5-7', '1412.8298-3-5-7'], ['1412.8298-2-5-8', '1412.8298-3-5-8'], ['1412.8298-2-5-9', '1412.8298-3-5-9'], ['1412.8298-2-5-10', '1412.8298-3-5-10'], ['1412.8298-2-10-0', '1412.8298-3-10-0'], ['1412.8298-2-10-1', '1412.8298-3-10-1'], ['1412.8298-2-10-2', '1412.8298-3-10-2'], ['1412.8298-2-10-3', '1412.8298-3-10-3'], ['1412.8298-2-2-0', '1412.8298-3-2-0'], ['1412.8298-2-2-1', '1412.8298-3-2-1'], ['1412.8298-2-2-2', '1412.8298-3-2-2'], ['1412.8298-2-2-3', '1412.8298-3-2-3'], ['1412.8298-2-2-4', '1412.8298-3-2-4'], ['1412.8298-2-3-1', '1412.8298-3-3-1'], ['1412.8298-2-3-2', '1412.8298-3-3-2'], ['1412.8298-2-3-3', '1412.8298-3-3-3'], ['1412.8298-2-3-5', '1412.8298-3-3-5'], ['1412.8298-2-6-0', '1412.8298-3-6-0'], ['1412.8298-2-6-1', '1412.8298-3-6-1'], ['1412.8298-2-6-2', '1412.8298-3-6-2'], ['1412.8298-2-7-0', '1412.8298-3-7-0'], ['1412.8298-2-7-2', '1412.8298-3-7-2'], ['1412.8298-2-7-3', '1412.8298-3-7-3'], ['1412.8298-2-7-5', '1412.8298-3-7-5'], ['1412.8298-2-7-6', '1412.8298-3-7-6'], ['1412.8298-2-7-7', '1412.8298-3-7-7'], ['1412.8298-2-12-0', '1412.8298-3-12-0'], ['1412.8298-2-12-1', '1412.8298-3-12-1'], ['1412.8298-2-12-2', '1412.8298-3-12-2'], ['1412.8298-2-12-3', '1412.8298-3-12-3'], ['1412.8298-2-12-4', '1412.8298-3-12-4'], ['1412.8298-2-12-5', '1412.8298-3-12-5'], ['1412.8298-2-12-6', '1412.8298-3-12-6'], ['1412.8298-2-13-0', '1412.8298-3-13-0'], ['1412.8298-2-13-1', '1412.8298-3-13-1'], ['1412.8298-2-13-2', '1412.8298-3-13-2'], ['1412.8298-2-8-0', '1412.8298-3-8-0'], ['1412.8298-2-8-1', '1412.8298-3-8-1'], ['1412.8298-2-8-2', '1412.8298-3-8-2'], ['1412.8298-2-8-4', '1412.8298-3-8-4'], ['1412.8298-2-8-5', '1412.8298-3-8-5'], ['1412.8298-2-4-0', '1412.8298-3-4-0'], ['1412.8298-2-4-1', '1412.8298-3-4-1'], ['1412.8298-2-4-2', '1412.8298-3-4-2'], ['1412.8298-2-4-3', '1412.8298-3-4-3'], ['1412.8298-3-4-0', '1412.8298-4-4-0'], ['1412.8298-3-4-2', '1412.8298-4-4-2'], ['1412.8298-3-4-3', '1412.8298-4-4-3'], ['1412.8298-3-11-1', '1412.8298-4-11-1'], ['1412.8298-3-11-2', '1412.8298-4-11-2'], ['1412.8298-3-11-3', '1412.8298-4-11-3'], ['1412.8298-3-11-4', '1412.8298-4-11-4'], ['1412.8298-3-11-8', '1412.8298-4-11-8'], ['1412.8298-3-11-9', '1412.8298-4-11-9'], ['1412.8298-3-2-0', '1412.8298-4-2-0'], ['1412.8298-3-2-1', '1412.8298-4-2-1'], ['1412.8298-3-2-2', '1412.8298-4-2-2'], ['1412.8298-3-2-3', '1412.8298-4-2-3'], ['1412.8298-3-2-4', '1412.8298-4-2-4'], ['1412.8298-3-2-5', '1412.8298-4-2-5'], ['1412.8298-3-13-0', '1412.8298-4-13-0'], ['1412.8298-3-13-1', '1412.8298-4-13-1'], ['1412.8298-3-13-2', '1412.8298-4-13-2'], ['1412.8298-3-7-2', '1412.8298-4-7-2'], ['1412.8298-3-7-3', '1412.8298-4-7-3'], ['1412.8298-3-7-5', '1412.8298-4-7-5'], ['1412.8298-3-7-6', '1412.8298-4-7-6'], ['1412.8298-3-7-7', '1412.8298-4-7-7'], ['1412.8298-3-8-0', '1412.8298-4-8-0'], ['1412.8298-3-8-1', '1412.8298-4-8-1'], ['1412.8298-3-8-2', '1412.8298-4-8-2'], ['1412.8298-3-8-3', '1412.8298-4-8-3'], ['1412.8298-3-8-4', '1412.8298-4-8-4'], ['1412.8298-3-8-5', '1412.8298-4-8-5'], ['1412.8298-3-12-0', '1412.8298-4-12-0'], ['1412.8298-3-12-1', '1412.8298-4-12-1'], ['1412.8298-3-12-2', '1412.8298-4-12-2'], ['1412.8298-3-12-3', '1412.8298-4-12-3'], ['1412.8298-3-12-4', '1412.8298-4-12-4'], ['1412.8298-3-12-5', '1412.8298-4-12-5'], ['1412.8298-3-12-6', '1412.8298-4-12-6'], ['1412.8298-3-15-0', '1412.8298-4-15-0'], ['1412.8298-3-15-1', '1412.8298-4-15-1'], ['1412.8298-3-15-2', '1412.8298-4-15-2'], ['1412.8298-3-15-3', '1412.8298-4-15-3'], ['1412.8298-3-15-4', '1412.8298-4-15-4'], ['1412.8298-3-15-5', '1412.8298-4-15-5'], ['1412.8298-3-5-0', '1412.8298-4-5-0'], ['1412.8298-3-5-1', '1412.8298-4-5-1'], ['1412.8298-3-5-2', '1412.8298-4-5-2'], ['1412.8298-3-5-3', '1412.8298-4-5-3'], ['1412.8298-3-5-4', '1412.8298-4-5-4'], ['1412.8298-3-5-6', '1412.8298-4-5-6'], ['1412.8298-3-5-7', '1412.8298-4-5-7'], ['1412.8298-3-5-8', '1412.8298-4-5-8'], ['1412.8298-3-5-9', '1412.8298-4-5-9'], ['1412.8298-3-5-10', '1412.8298-4-5-10'], ['1412.8298-3-3-1', '1412.8298-4-3-1'], ['1412.8298-3-3-2', '1412.8298-4-3-2'], ['1412.8298-3-3-4', '1412.8298-4-3-4'], ['1412.8298-3-3-5', '1412.8298-4-3-5'], ['1412.8298-3-1-0', '1412.8298-4-1-0'], ['1412.8298-3-1-1', '1412.8298-4-1-1'], ['1412.8298-3-1-2', '1412.8298-4-1-2'], ['1412.8298-3-1-3', '1412.8298-4-1-3'], ['1412.8298-3-9-0', '1412.8298-4-9-0'], ['1412.8298-3-9-2', '1412.8298-4-9-2'], ['1412.8298-3-9-3', '1412.8298-4-9-3'], ['1412.8298-3-9-4', '1412.8298-4-9-4'], ['1412.8298-3-10-0', '1412.8298-4-10-0'], ['1412.8298-3-10-1', '1412.8298-4-10-1'], ['1412.8298-3-10-2', '1412.8298-4-10-2'], ['1412.8298-3-10-3', '1412.8298-4-10-3'], ['1412.8298-3-6-0', '1412.8298-4-6-0'], ['1412.8298-3-16-0', '1412.8298-4-16-0'], ['1412.8298-3-16-1', '1412.8298-4-16-1'], ['1412.8298-3-17-0', '1412.8298-4-17-0'], ['1412.8298-3-0-0', '1412.8298-4-0-0'], ['1412.8298-3-0-2', '1412.8298-4-0-2'], ['1412.8298-3-0-3', '1412.8298-4-0-3'], ['1412.8298-3-0-4', '1412.8298-4-0-4'], ['1412.8298-4-17-0', '1412.8298-5-17-0'], ['1412.8298-4-8-0', '1412.8298-5-8-0'], ['1412.8298-4-8-1', '1412.8298-5-8-1'], ['1412.8298-4-8-2', '1412.8298-5-8-2'], ['1412.8298-4-8-3', '1412.8298-5-8-3'], ['1412.8298-4-8-4', '1412.8298-5-8-4'], ['1412.8298-4-8-5', '1412.8298-5-8-5'], ['1412.8298-4-14-0', '1412.8298-5-14-0'], ['1412.8298-4-14-1', '1412.8298-5-14-1'], ['1412.8298-4-14-3', '1412.8298-5-14-3'], ['1412.8298-4-14-6', '1412.8298-5-14-5'], ['1412.8298-4-14-9', '1412.8298-5-14-7'], ['1412.8298-4-10-0', '1412.8298-5-10-0'], ['1412.8298-4-10-1', '1412.8298-5-10-1'], ['1412.8298-4-10-2', '1412.8298-5-10-2'], ['1412.8298-4-10-3', '1412.8298-5-10-3'], ['1412.8298-4-16-0', '1412.8298-5-16-0'], ['1412.8298-4-16-1', '1412.8298-5-16-1'], ['1412.8298-4-13-0', '1412.8298-5-13-0'], ['1412.8298-4-13-1', '1412.8298-5-13-1'], ['1412.8298-4-13-2', '1412.8298-5-13-2'], ['1412.8298-4-0-0', '1412.8298-5-0-0'], ['1412.8298-4-0-1', '1412.8298-5-0-1'], ['1412.8298-4-0-2', '1412.8298-5-0-2'], ['1412.8298-4-0-3', '1412.8298-5-0-3'], ['1412.8298-4-0-4', '1412.8298-5-0-4'], ['1412.8298-4-2-0', '1412.8298-5-2-0'], ['1412.8298-4-2-1', '1412.8298-5-2-1'], ['1412.8298-4-2-2', '1412.8298-5-2-2'], ['1412.8298-4-2-3', '1412.8298-5-2-3'], ['1412.8298-4-2-4', '1412.8298-5-2-4'], ['1412.8298-4-2-5', '1412.8298-5-2-5'], ['1412.8298-4-5-0', '1412.8298-5-5-0'], ['1412.8298-4-5-1', '1412.8298-5-5-1'], ['1412.8298-4-5-2', '1412.8298-5-5-2'], ['1412.8298-4-5-3', '1412.8298-5-5-3'], ['1412.8298-4-5-4', '1412.8298-5-5-4'], ['1412.8298-4-5-5', '1412.8298-5-5-5'], ['1412.8298-4-5-6', '1412.8298-5-5-6'], ['1412.8298-4-5-7', '1412.8298-5-5-7'], ['1412.8298-4-5-8', '1412.8298-5-5-8'], ['1412.8298-4-5-9', '1412.8298-5-5-9'], ['1412.8298-4-5-10', '1412.8298-5-5-10'], ['1412.8298-4-6-0', '1412.8298-5-6-0'], ['1412.8298-4-6-1', '1412.8298-5-6-1'], ['1412.8298-4-6-2', '1412.8298-5-6-2'], ['1412.8298-4-4-0', '1412.8298-5-4-0'], ['1412.8298-4-4-1', '1412.8298-5-4-1'], ['1412.8298-4-4-2', '1412.8298-5-4-2'], ['1412.8298-4-4-3', '1412.8298-5-4-3'], ['1412.8298-4-11-0', '1412.8298-5-11-0'], ['1412.8298-4-11-1', '1412.8298-5-11-1'], ['1412.8298-4-11-2', '1412.8298-5-11-2'], ['1412.8298-4-11-3', '1412.8298-5-11-3'], ['1412.8298-4-11-4', '1412.8298-5-11-4'], ['1412.8298-4-11-7', '1412.8298-5-11-7'], ['1412.8298-4-11-8', '1412.8298-5-11-8'], ['1412.8298-4-11-9', '1412.8298-5-11-9'], ['1412.8298-4-15-0', '1412.8298-5-15-0'], ['1412.8298-4-15-1', '1412.8298-5-15-1'], ['1412.8298-4-15-2', '1412.8298-5-15-2'], ['1412.8298-4-15-3', '1412.8298-5-15-3'], ['1412.8298-4-15-4', '1412.8298-5-15-4'], ['1412.8298-4-15-5', '1412.8298-5-15-5'], ['1412.8298-4-3-0', '1412.8298-5-3-0'], ['1412.8298-4-3-1', '1412.8298-5-3-1'], ['1412.8298-4-3-2', '1412.8298-5-3-2'], ['1412.8298-4-3-3', '1412.8298-5-3-3'], ['1412.8298-4-3-4', '1412.8298-5-3-4'], ['1412.8298-4-3-5', '1412.8298-5-3-5'], ['1412.8298-4-9-0', '1412.8298-5-9-0'], ['1412.8298-4-9-1', '1412.8298-5-9-1'], ['1412.8298-4-9-2', '1412.8298-5-9-2'], ['1412.8298-4-9-3', '1412.8298-5-9-3'], ['1412.8298-4-9-4', '1412.8298-5-9-4'], ['1412.8298-4-7-0', '1412.8298-5-7-0'], ['1412.8298-4-7-2', '1412.8298-5-7-2'], ['1412.8298-4-7-3', '1412.8298-5-7-3'], ['1412.8298-4-7-5', '1412.8298-5-7-5'], ['1412.8298-4-7-6', '1412.8298-5-7-6'], ['1412.8298-4-7-7', '1412.8298-5-7-7'], ['1412.8298-4-1-0', '1412.8298-5-1-0'], ['1412.8298-4-1-1', '1412.8298-5-1-1'], ['1412.8298-4-1-2', '1412.8298-5-1-2'], ['1412.8298-4-1-3', '1412.8298-5-1-3'], ['1412.8298-4-12-0', '1412.8298-5-12-0'], ['1412.8298-4-12-1', '1412.8298-5-12-1'], ['1412.8298-4-12-2', '1412.8298-5-12-2'], ['1412.8298-4-12-3', '1412.8298-5-12-3'], ['1412.8298-4-12-4', '1412.8298-5-12-4'], ['1412.8298-4-12-5', '1412.8298-5-12-5'], ['1412.8298-4-12-6', '1412.8298-5-12-6'], ['1412.8298-1-9-1', '1412.8298-2-9-1'], ['1412.8298-1-6-0', '1412.8298-2-7-0'], ['1412.8298-1-3-5', '1412.8298-2-3-5'], ['1412.8298-1-13-1', '1412.8298-2-13-1'], ['1412.8298-1-15-0', '1412.8298-2-15-0'], ['1412.8298-1-15-1', '1412.8298-2-15-1'], ['1412.8298-1-11-0', '1412.8298-2-12-0'], ['1412.8298-1-11-4', '1412.8298-2-12-4'], ['1412.8298-1-8-3', '1412.8298-2-8-3'], ['1412.8298-1-8-4', '1412.8298-2-8-0'], ['1412.8298-1-14-5', '1412.8298-2-14-4'], ['1412.8298-1-7-0', '1412.8298-2-10-1'], ['1412.8298-1-7-1', '1412.8298-2-10-2'], ['1412.8298-2-3-4', '1412.8298-3-3-4'], ['1412.8298-3-4-1', '1412.8298-4-4-1'], ['1412.8298-3-11-0', '1412.8298-4-11-0'], ['1412.8298-3-11-7', '1412.8298-4-11-7'], ['1412.8298-3-7-0', '1412.8298-4-7-0'], ['1412.8298-3-5-5', '1412.8298-4-5-5'], ['1412.8298-3-3-0', '1412.8298-4-3-0'], ['1412.8298-3-3-3', '1412.8298-4-3-3'], ['1412.8298-3-9-1', '1412.8298-4-9-1'], ['1412.8298-3-6-2', '1412.8298-4-6-2'], ['1412.8298-3-0-1', '1412.8298-4-0-1'], ['1412.8298-4-14-2', '1412.8298-5-14-2'], ['1412.8298-4-14-4', '1412.8298-5-14-4'], ['1412.8298-4-14-8', '1412.8298-5-14-6'], ['1412.8298-1-6-6', '1412.8298-2-7-6'], ['1412.8298-1-3-4', '1412.8298-2-3-4'], ['1412.8298-1-13-3', '1412.8298-2-13-2'], ['1412.8298-1-4-0', '1412.8298-2-4-0'], ['1412.8298-1-4-1', '1412.8298-2-4-2'], ['1412.8298-1-4-2', '1412.8298-2-4-1'], ['1412.8298-1-4-2', '1412.8298-2-4-2'], ['1412.8298-1-4-3', '1412.8298-2-4-2'], ['1412.8298-1-4-3', '1412.8298-2-4-3'], ['1412.8298-1-2-5', '1412.8298-2-2-5'], ['1412.8298-1-2-5', '1412.8298-2-2-6'], ['1412.8298-1-0-0', '1412.8298-2-0-0'], ['1412.8298-1-0-1', '1412.8298-2-0-3'], ['1412.8298-1-0-2', '1412.8298-2-0-1'], ['1412.8298-1-0-3', '1412.8298-2-0-2'], ['1412.8298-1-8-1', '1412.8298-2-8-1'], ['1412.8298-1-14-0', '1412.8298-2-14-0'], ['1412.8298-1-14-1', '1412.8298-2-14-2'], ['1412.8298-1-14-2', '1412.8298-2-14-1'], ['1412.8298-1-14-6', '1412.8298-2-14-5'], ['1412.8298-2-3-0', '1412.8298-3-3-0'], ['1412.8298-3-6-1', '1412.8298-4-6-1']]

[['1412.8298-1-5-0', '1412.8298-2-5-0'], ['1412.8298-1-5-1', '1412.8298-2-5-1'], ['1412.8298-1-5-2', '1412.8298-2-5-2'], ['1412.8298-1-5-3', '1412.8298-2-5-3'], ['1412.8298-1-5-4', '1412.8298-2-5-4'], ['1412.8298-1-5-5', '1412.8298-2-5-6'], ['1412.8298-1-5-6', '1412.8298-2-5-7'], ['1412.8298-1-5-7', '1412.8298-2-5-8'], ['1412.8298-1-5-8', '1412.8298-2-5-9'], ['1412.8298-1-5-9', '1412.8298-2-5-10'], ['1412.8298-1-9-0', '1412.8298-2-9-0'], ['1412.8298-1-9-3', '1412.8298-2-9-2'], ['1412.8298-1-9-5', '1412.8298-2-9-4'], ['1412.8298-1-6-2', '1412.8298-2-7-2'], ['1412.8298-1-6-5', '1412.8298-2-7-5'], ['1412.8298-1-6-7', '1412.8298-2-7-7'], ['1412.8298-1-3-0', '1412.8298-2-3-0'], ['1412.8298-1-3-1', '1412.8298-2-3-1'], ['1412.8298-1-3-2', '1412.8298-2-3-2'], ['1412.8298-1-3-3', '1412.8298-2-3-3'], ['1412.8298-1-13-0', '1412.8298-2-13-0'], ['1412.8298-1-1-0', '1412.8298-2-1-0'], ['1412.8298-1-1-1', '1412.8298-2-1-1'], ['1412.8298-1-1-2', '1412.8298-2-1-2'], ['1412.8298-1-1-3', '1412.8298-2-1-3'], ['1412.8298-1-2-0', '1412.8298-2-2-0'], ['1412.8298-1-2-1', '1412.8298-2-2-1'], ['1412.8298-1-2-2', '1412.8298-2-2-2'], ['1412.8298-1-2-3', '1412.8298-2-2-3'], ['1412.8298-1-2-4', '1412.8298-2-2-4'], ['1412.8298-1-11-1', '1412.8298-2-12-1'], ['1412.8298-1-11-2', '1412.8298-2-12-2'], ['1412.8298-1-11-3', '1412.8298-2-12-3'], ['1412.8298-1-0-4', '1412.8298-2-0-4'], ['1412.8298-1-8-2', '1412.8298-2-8-2'], ['1412.8298-1-8-5', '1412.8298-2-8-4'], ['1412.8298-1-8-6', '1412.8298-2-8-5'], ['1412.8298-1-7-2', '1412.8298-2-10-3'], ['1412.8298-2-1-0', '1412.8298-3-1-0'], ['1412.8298-2-1-1', '1412.8298-3-1-1'], ['1412.8298-2-1-2', '1412.8298-3-1-2'], ['1412.8298-2-1-3', '1412.8298-3-1-3'], ['1412.8298-2-11-0', '1412.8298-3-11-0'], ['1412.8298-2-11-1', '1412.8298-3-11-1'], ['1412.8298-2-11-2', '1412.8298-3-11-2'], ['1412.8298-2-11-3', '1412.8298-3-11-3'], ['1412.8298-2-11-4', '1412.8298-3-11-4'], ['1412.8298-2-11-7', '1412.8298-3-11-7'], ['1412.8298-2-11-8', '1412.8298-3-11-8'], ['1412.8298-2-11-9', '1412.8298-3-11-9'], ['1412.8298-2-9-0', '1412.8298-3-9-0'], ['1412.8298-2-9-1', '1412.8298-3-9-1'], ['1412.8298-2-9-2', '1412.8298-3-9-2'], ['1412.8298-2-9-3', '1412.8298-3-9-3'], ['1412.8298-2-9-4', '1412.8298-3-9-4'], ['1412.8298-2-16-0', '1412.8298-3-17-0'], ['1412.8298-2-14-0', '1412.8298-3-15-0'], ['1412.8298-2-14-1', '1412.8298-3-15-1'], ['1412.8298-2-14-2', '1412.8298-3-15-2'], ['1412.8298-2-14-3', '1412.8298-3-15-3'], ['1412.8298-2-14-4', '1412.8298-3-15-4'], ['1412.8298-2-14-5', '1412.8298-3-15-5'], ['1412.8298-2-0-0', '1412.8298-3-0-0'], ['1412.8298-2-0-1', '1412.8298-3-0-1'], ['1412.8298-2-0-2', '1412.8298-3-0-2'], ['1412.8298-2-0-3', '1412.8298-3-0-3'], ['1412.8298-2-0-4', '1412.8298-3-0-4'], ['1412.8298-2-15-0', '1412.8298-3-16-0'], ['1412.8298-2-15-1', '1412.8298-3-16-1'], ['1412.8298-2-5-0', '1412.8298-3-5-0'], ['1412.8298-2-5-1', '1412.8298-3-5-1'], ['1412.8298-2-5-2', '1412.8298-3-5-2'], ['1412.8298-2-5-3', '1412.8298-3-5-3'], ['1412.8298-2-5-4', '1412.8298-3-5-4'], ['1412.8298-2-5-5', '1412.8298-3-5-5'], ['1412.8298-2-5-6', '1412.8298-3-5-6'], ['1412.8298-2-5-7', '1412.8298-3-5-7'], ['1412.8298-2-5-8', '1412.8298-3-5-8'], ['1412.8298-2-5-9', '1412.8298-3-5-9'], ['1412.8298-2-5-10', '1412.8298-3-5-10'], ['1412.8298-2-10-0', '1412.8298-3-10-0'], ['1412.8298-2-10-1', '1412.8298-3-10-1'], ['1412.8298-2-10-2', '1412.8298-3-10-2'], ['1412.8298-2-10-3', '1412.8298-3-10-3'], ['1412.8298-2-2-0', '1412.8298-3-2-0'], ['1412.8298-2-2-1', '1412.8298-3-2-1'], ['1412.8298-2-2-2', '1412.8298-3-2-2'], ['1412.8298-2-2-3', '1412.8298-3-2-3'], ['1412.8298-2-2-4', '1412.8298-3-2-4'], ['1412.8298-2-3-1', '1412.8298-3-3-1'], ['1412.8298-2-3-2', '1412.8298-3-3-2'], ['1412.8298-2-3-3', '1412.8298-3-3-3'], ['1412.8298-2-3-5', '1412.8298-3-3-5'], ['1412.8298-2-6-0', '1412.8298-3-6-0'], ['1412.8298-2-6-1', '1412.8298-3-6-1'], ['1412.8298-2-6-2', '1412.8298-3-6-2'], ['1412.8298-2-7-0', '1412.8298-3-7-0'], ['1412.8298-2-7-2', '1412.8298-3-7-2'], ['1412.8298-2-7-3', '1412.8298-3-7-3'], ['1412.8298-2-7-5', '1412.8298-3-7-5'], ['1412.8298-2-7-6', '1412.8298-3-7-6'], ['1412.8298-2-7-7', '1412.8298-3-7-7'], ['1412.8298-2-12-0', '1412.8298-3-12-0'], ['1412.8298-2-12-1', '1412.8298-3-12-1'], ['1412.8298-2-12-2', '1412.8298-3-12-2'], ['1412.8298-2-12-3', '1412.8298-3-12-3'], ['1412.8298-2-12-4', '1412.8298-3-12-4'], ['1412.8298-2-12-5', '1412.8298-3-12-5'], ['1412.8298-2-12-6', '1412.8298-3-12-6'], ['1412.8298-2-13-0', '1412.8298-3-13-0'], ['1412.8298-2-13-1', '1412.8298-3-13-1'], ['1412.8298-2-13-2', '1412.8298-3-13-2'], ['1412.8298-2-8-0', '1412.8298-3-8-0'], ['1412.8298-2-8-1', '1412.8298-3-8-1'], ['1412.8298-2-8-2', '1412.8298-3-8-2'], ['1412.8298-2-8-4', '1412.8298-3-8-4'], ['1412.8298-2-8-5', '1412.8298-3-8-5'], ['1412.8298-2-4-0', '1412.8298-3-4-0'], ['1412.8298-2-4-1', '1412.8298-3-4-1'], ['1412.8298-2-4-2', '1412.8298-3-4-2'], ['1412.8298-2-4-3', '1412.8298-3-4-3'], ['1412.8298-3-4-0', '1412.8298-4-4-0'], ['1412.8298-3-4-2', '1412.8298-4-4-2'], ['1412.8298-3-4-3', '1412.8298-4-4-3'], ['1412.8298-3-11-1', '1412.8298-4-11-1'], ['1412.8298-3-11-2', '1412.8298-4-11-2'], ['1412.8298-3-11-3', '1412.8298-4-11-3'], ['1412.8298-3-11-4', '1412.8298-4-11-4'], ['1412.8298-3-11-8', '1412.8298-4-11-8'], ['1412.8298-3-11-9', '1412.8298-4-11-9'], ['1412.8298-3-2-0', '1412.8298-4-2-0'], ['1412.8298-3-2-1', '1412.8298-4-2-1'], ['1412.8298-3-2-2', '1412.8298-4-2-2'], ['1412.8298-3-2-3', '1412.8298-4-2-3'], ['1412.8298-3-2-4', '1412.8298-4-2-4'], ['1412.8298-3-2-5', '1412.8298-4-2-5'], ['1412.8298-3-13-0', '1412.8298-4-13-0'], ['1412.8298-3-13-1', '1412.8298-4-13-1'], ['1412.8298-3-13-2', '1412.8298-4-13-2'], ['1412.8298-3-7-2', '1412.8298-4-7-2'], ['1412.8298-3-7-3', '1412.8298-4-7-3'], ['1412.8298-3-7-5', '1412.8298-4-7-5'], ['1412.8298-3-7-6', '1412.8298-4-7-6'], ['1412.8298-3-7-7', '1412.8298-4-7-7'], ['1412.8298-3-8-0', '1412.8298-4-8-0'], ['1412.8298-3-8-1', '1412.8298-4-8-1'], ['1412.8298-3-8-2', '1412.8298-4-8-2'], ['1412.8298-3-8-3', '1412.8298-4-8-3'], ['1412.8298-3-8-4', '1412.8298-4-8-4'], ['1412.8298-3-8-5', '1412.8298-4-8-5'], ['1412.8298-3-12-0', '1412.8298-4-12-0'], ['1412.8298-3-12-1', '1412.8298-4-12-1'], ['1412.8298-3-12-2', '1412.8298-4-12-2'], ['1412.8298-3-12-3', '1412.8298-4-12-3'], ['1412.8298-3-12-4', '1412.8298-4-12-4'], ['1412.8298-3-12-5', '1412.8298-4-12-5'], ['1412.8298-3-12-6', '1412.8298-4-12-6'], ['1412.8298-3-15-0', '1412.8298-4-15-0'], ['1412.8298-3-15-1', '1412.8298-4-15-1'], ['1412.8298-3-15-2', '1412.8298-4-15-2'], ['1412.8298-3-15-3', '1412.8298-4-15-3'], ['1412.8298-3-15-4', '1412.8298-4-15-4'], ['1412.8298-3-15-5', '1412.8298-4-15-5'], ['1412.8298-3-5-0', '1412.8298-4-5-0'], ['1412.8298-3-5-1', '1412.8298-4-5-1'], ['1412.8298-3-5-2', '1412.8298-4-5-2'], ['1412.8298-3-5-3', '1412.8298-4-5-3'], ['1412.8298-3-5-4', '1412.8298-4-5-4'], ['1412.8298-3-5-6', '1412.8298-4-5-6'], ['1412.8298-3-5-7', '1412.8298-4-5-7'], ['1412.8298-3-5-8', '1412.8298-4-5-8'], ['1412.8298-3-5-9', '1412.8298-4-5-9'], ['1412.8298-3-5-10', '1412.8298-4-5-10'], ['1412.8298-3-3-1', '1412.8298-4-3-1'], ['1412.8298-3-3-2', '1412.8298-4-3-2'], ['1412.8298-3-3-4', '1412.8298-4-3-4'], ['1412.8298-3-3-5', '1412.8298-4-3-5'], ['1412.8298-3-1-0', '1412.8298-4-1-0'], ['1412.8298-3-1-1', '1412.8298-4-1-1'], ['1412.8298-3-1-2', '1412.8298-4-1-2'], ['1412.8298-3-1-3', '1412.8298-4-1-3'], ['1412.8298-3-9-0', '1412.8298-4-9-0'], ['1412.8298-3-9-2', '1412.8298-4-9-2'], ['1412.8298-3-9-3', '1412.8298-4-9-3'], ['1412.8298-3-9-4', '1412.8298-4-9-4'], ['1412.8298-3-10-0', '1412.8298-4-10-0'], ['1412.8298-3-10-1', '1412.8298-4-10-1'], ['1412.8298-3-10-2', '1412.8298-4-10-2'], ['1412.8298-3-10-3', '1412.8298-4-10-3'], ['1412.8298-3-6-0', '1412.8298-4-6-0'], ['1412.8298-3-16-0', '1412.8298-4-16-0'], ['1412.8298-3-16-1', '1412.8298-4-16-1'], ['1412.8298-3-17-0', '1412.8298-4-17-0'], ['1412.8298-3-0-0', '1412.8298-4-0-0'], ['1412.8298-3-0-2', '1412.8298-4-0-2'], ['1412.8298-3-0-3', '1412.8298-4-0-3'], ['1412.8298-3-0-4', '1412.8298-4-0-4'], ['1412.8298-4-17-0', '1412.8298-5-17-0'], ['1412.8298-4-8-0', '1412.8298-5-8-0'], ['1412.8298-4-8-1', '1412.8298-5-8-1'], ['1412.8298-4-8-2', '1412.8298-5-8-2'], ['1412.8298-4-8-3', '1412.8298-5-8-3'], ['1412.8298-4-8-4', '1412.8298-5-8-4'], ['1412.8298-4-8-5', '1412.8298-5-8-5'], ['1412.8298-4-14-0', '1412.8298-5-14-0'], ['1412.8298-4-14-1', '1412.8298-5-14-1'], ['1412.8298-4-14-3', '1412.8298-5-14-3'], ['1412.8298-4-14-6', '1412.8298-5-14-5'], ['1412.8298-4-14-9', '1412.8298-5-14-7'], ['1412.8298-4-10-0', '1412.8298-5-10-0'], ['1412.8298-4-10-1', '1412.8298-5-10-1'], ['1412.8298-4-10-2', '1412.8298-5-10-2'], ['1412.8298-4-10-3', '1412.8298-5-10-3'], ['1412.8298-4-16-0', '1412.8298-5-16-0'], ['1412.8298-4-16-1', '1412.8298-5-16-1'], ['1412.8298-4-13-0', '1412.8298-5-13-0'], ['1412.8298-4-13-1', '1412.8298-5-13-1'], ['1412.8298-4-13-2', '1412.8298-5-13-2'], ['1412.8298-4-0-0', '1412.8298-5-0-0'], ['1412.8298-4-0-1', '1412.8298-5-0-1'], ['1412.8298-4-0-2', '1412.8298-5-0-2'], ['1412.8298-4-0-3', '1412.8298-5-0-3'], ['1412.8298-4-0-4', '1412.8298-5-0-4'], ['1412.8298-4-2-0', '1412.8298-5-2-0'], ['1412.8298-4-2-1', '1412.8298-5-2-1'], ['1412.8298-4-2-2', '1412.8298-5-2-2'], ['1412.8298-4-2-3', '1412.8298-5-2-3'], ['1412.8298-4-2-4', '1412.8298-5-2-4'], ['1412.8298-4-2-5', '1412.8298-5-2-5'], ['1412.8298-4-5-0', '1412.8298-5-5-0'], ['1412.8298-4-5-1', '1412.8298-5-5-1'], ['1412.8298-4-5-2', '1412.8298-5-5-2'], ['1412.8298-4-5-3', '1412.8298-5-5-3'], ['1412.8298-4-5-4', '1412.8298-5-5-4'], ['1412.8298-4-5-5', '1412.8298-5-5-5'], ['1412.8298-4-5-6', '1412.8298-5-5-6'], ['1412.8298-4-5-7', '1412.8298-5-5-7'], ['1412.8298-4-5-8', '1412.8298-5-5-8'], ['1412.8298-4-5-9', '1412.8298-5-5-9'], ['1412.8298-4-5-10', '1412.8298-5-5-10'], ['1412.8298-4-6-0', '1412.8298-5-6-0'], ['1412.8298-4-6-1', '1412.8298-5-6-1'], ['1412.8298-4-6-2', '1412.8298-5-6-2'], ['1412.8298-4-4-0', '1412.8298-5-4-0'], ['1412.8298-4-4-1', '1412.8298-5-4-1'], ['1412.8298-4-4-2', '1412.8298-5-4-2'], ['1412.8298-4-4-3', '1412.8298-5-4-3'], ['1412.8298-4-11-0', '1412.8298-5-11-0'], ['1412.8298-4-11-1', '1412.8298-5-11-1'], ['1412.8298-4-11-2', '1412.8298-5-11-2'], ['1412.8298-4-11-3', '1412.8298-5-11-3'], ['1412.8298-4-11-4', '1412.8298-5-11-4'], ['1412.8298-4-11-7', '1412.8298-5-11-7'], ['1412.8298-4-11-8', '1412.8298-5-11-8'], ['1412.8298-4-11-9', '1412.8298-5-11-9'], ['1412.8298-4-15-0', '1412.8298-5-15-0'], ['1412.8298-4-15-1', '1412.8298-5-15-1'], ['1412.8298-4-15-2', '1412.8298-5-15-2'], ['1412.8298-4-15-3', '1412.8298-5-15-3'], ['1412.8298-4-15-4', '1412.8298-5-15-4'], ['1412.8298-4-15-5', '1412.8298-5-15-5'], ['1412.8298-4-3-0', '1412.8298-5-3-0'], ['1412.8298-4-3-1', '1412.8298-5-3-1'], ['1412.8298-4-3-2', '1412.8298-5-3-2'], ['1412.8298-4-3-3', '1412.8298-5-3-3'], ['1412.8298-4-3-4', '1412.8298-5-3-4'], ['1412.8298-4-3-5', '1412.8298-5-3-5'], ['1412.8298-4-9-0', '1412.8298-5-9-0'], ['1412.8298-4-9-1', '1412.8298-5-9-1'], ['1412.8298-4-9-2', '1412.8298-5-9-2'], ['1412.8298-4-9-3', '1412.8298-5-9-3'], ['1412.8298-4-9-4', '1412.8298-5-9-4'], ['1412.8298-4-7-0', '1412.8298-5-7-0'], ['1412.8298-4-7-2', '1412.8298-5-7-2'], ['1412.8298-4-7-3', '1412.8298-5-7-3'], ['1412.8298-4-7-5', '1412.8298-5-7-5'], ['1412.8298-4-7-6', '1412.8298-5-7-6'], ['1412.8298-4-7-7', '1412.8298-5-7-7'], ['1412.8298-4-1-0', '1412.8298-5-1-0'], ['1412.8298-4-1-1', '1412.8298-5-1-1'], ['1412.8298-4-1-2', '1412.8298-5-1-2'], ['1412.8298-4-1-3', '1412.8298-5-1-3'], ['1412.8298-4-12-0', '1412.8298-5-12-0'], ['1412.8298-4-12-1', '1412.8298-5-12-1'], ['1412.8298-4-12-2', '1412.8298-5-12-2'], ['1412.8298-4-12-3', '1412.8298-5-12-3'], ['1412.8298-4-12-4', '1412.8298-5-12-4'], ['1412.8298-4-12-5', '1412.8298-5-12-5'], ['1412.8298-4-12-6', '1412.8298-5-12-6']]

[['1412.8298-1-9-1', '1412.8298-2-9-1'], ['1412.8298-1-6-0', '1412.8298-2-7-0'], ['1412.8298-1-3-5', '1412.8298-2-3-5'], ['1412.8298-1-13-1', '1412.8298-2-13-1'], ['1412.8298-1-15-0', '1412.8298-2-15-0'], ['1412.8298-1-15-1', '1412.8298-2-15-1'], ['1412.8298-1-11-0', '1412.8298-2-12-0'], ['1412.8298-1-11-4', '1412.8298-2-12-4'], ['1412.8298-1-8-3', '1412.8298-2-8-3'], ['1412.8298-1-8-4', '1412.8298-2-8-0'], ['1412.8298-1-14-5', '1412.8298-2-14-4'], ['1412.8298-1-7-0', '1412.8298-2-10-1'], ['1412.8298-1-7-1', '1412.8298-2-10-2'], ['1412.8298-2-3-4', '1412.8298-3-3-4'], ['1412.8298-3-4-1', '1412.8298-4-4-1'], ['1412.8298-3-11-0', '1412.8298-4-11-0'], ['1412.8298-3-11-7', '1412.8298-4-11-7'], ['1412.8298-3-7-0', '1412.8298-4-7-0'], ['1412.8298-3-5-5', '1412.8298-4-5-5'], ['1412.8298-3-3-0', '1412.8298-4-3-0'], ['1412.8298-3-3-3', '1412.8298-4-3-3'], ['1412.8298-3-9-1', '1412.8298-4-9-1'], ['1412.8298-3-6-2', '1412.8298-4-6-2'], ['1412.8298-3-0-1', '1412.8298-4-0-1'], ['1412.8298-4-14-2', '1412.8298-5-14-2'], ['1412.8298-4-14-4', '1412.8298-5-14-4'], ['1412.8298-4-14-8', '1412.8298-5-14-6']]

[]

[['1412.8298-1-6-6', '1412.8298-2-7-6'], ['1412.8298-1-3-4', '1412.8298-2-3-4'], ['1412.8298-1-13-3', '1412.8298-2-13-2'], ['1412.8298-1-4-0', '1412.8298-2-4-0'], ['1412.8298-1-4-1', '1412.8298-2-4-2'], ['1412.8298-1-4-2', '1412.8298-2-4-1'], ['1412.8298-1-4-2', '1412.8298-2-4-2'], ['1412.8298-1-4-3', '1412.8298-2-4-2'], ['1412.8298-1-4-3', '1412.8298-2-4-3'], ['1412.8298-1-2-5', '1412.8298-2-2-5'], ['1412.8298-1-2-5', '1412.8298-2-2-6'], ['1412.8298-1-0-0', '1412.8298-2-0-0'], ['1412.8298-1-0-1', '1412.8298-2-0-3'], ['1412.8298-1-0-2', '1412.8298-2-0-1'], ['1412.8298-1-0-3', '1412.8298-2-0-2'], ['1412.8298-1-8-1', '1412.8298-2-8-1'], ['1412.8298-1-14-0', '1412.8298-2-14-0'], ['1412.8298-1-14-1', '1412.8298-2-14-2'], ['1412.8298-1-14-2', '1412.8298-2-14-1'], ['1412.8298-1-14-6', '1412.8298-2-14-5'], ['1412.8298-2-3-0', '1412.8298-3-3-0'], ['1412.8298-3-6-1', '1412.8298-4-6-1']]

[]

['1412.8298-1-6-1', '1412.8298-1-6-4', '1412.8298-1-9-4', '1412.8298-1-15-2', '1412.8298-2-7-1', '1412.8298-2-7-4', '1412.8298-2-11-5', '1412.8298-2-11-6', '1412.8298-2-15-2', '1412.8298-3-7-1', '1412.8298-3-7-4', '1412.8298-3-11-5', '1412.8298-3-11-6', '1412.8298-3-16-2', '1412.8298-4-7-1', '1412.8298-4-7-4', '1412.8298-4-11-5', '1412.8298-4-11-6', '1412.8298-4-16-2', '1412.8298-5-7-1', '1412.8298-5-7-4', '1412.8298-5-11-5', '1412.8298-5-11-6', '1412.8298-5-16-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1412.8298

{'1412.8298-3-0-0': 'The quantum oscillations of the magnetoresistance under ambient and high pressure have been studied for WTe[MATH] single crystals, in which extremely large magnetoresistance was discovered recently.', '1412.8298-3-0-1': 'By analyzing the Shubnikov-de Haas oscillations, four Fermi surfaces are identified for the first time, and two of them are found to persist to high pressure.', '1412.8298-3-0-2': 'The sizes of these two pockets are comparable, but show increasing difference with pressure.', '1412.8298-3-0-3': 'At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under 23.6 kbar, which is likely caused by the relative change of Fermi surfaces.', '1412.8298-3-0-4': 'These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe[MATH].', '1412.8298-3-1-0': 'The magnetoresistance (MR) is an important transport property of condensed matters.', '1412.8298-3-1-1': 'For simple metals, the MR is usually very small, showing quadratic field dependence in low field and saturating in high field [CITATION].', '1412.8298-3-1-2': 'In contrast, giant magnetoresistance (GMR) was discovered in magnetic multilayers [CITATION], and colossal magnetoresistance (CMR) was found in manganites [CITATION].', '1412.8298-3-1-3': 'Interestingly, in recent few years extremely large ([MATH]%) and non-saturating positive MR (XMR) was discovered in some nonmagnetic compounds including PtSn[MATH] [CITATION], Cd[MATH]As[MATH] [CITATION], WTe[MATH] [CITATION], and NbSb[MATH] [CITATION].', '1412.8298-3-2-0': 'The very recent discovery of XMR in WTe[MATH] is of particular interest [CITATION].', '1412.8298-3-2-1': 'WTe[MATH] is a layered transition-metal dichalcogenide, with the crystal structure shown in Fig. 1(a).', '1412.8298-3-2-2': 'In the dichalcogenide layers, W chains are formed along the [MATH] axis.', '1412.8298-3-2-3': 'An XMR of 4.5 [MATH] in 14.7 T at 4.5 K was found when the current is along the [MATH] axis and magnetic field is applied along the [MATH] axis [CITATION].', '1412.8298-3-2-4': 'More remarkably, it increases to as high as 1.3 [MATH] in 60 T at 0.53 K, without any sign of saturation [CITATION].', '1412.8298-3-2-5': 'Such an XMR makes WTe[MATH] outstanding among transition-metal dichalcogenides, in which various interesting physics such as charge density wave and superconductivity have been extensively studied [CITATION].', '1412.8298-3-3-0': 'The XMR in WTe[MATH] was attributed to the compensation between the electron and hole populations [CITATION].', '1412.8298-3-3-1': 'Actually, such a two-band model for charge transport in semimetals was previously used to explain the very large MR observed in high-purity graphite and bismuth [CITATION].', '1412.8298-3-3-2': 'However, in both graphite and Bi, the MR saturates beyond a few Tesla, due to the slight deviation from perfect compensation [CITATION].', '1412.8298-3-3-3': 'The non-saturating XMR in WTe[MATH] may provide the first example of perfectly balanced electron-hole populations [CITATION].', '1412.8298-3-3-4': 'While the subsequent angle-resolved photoemission spectroscopy (ARPES) experiments observed one pair of hole and electron pockets of approximately the same size at low temperature [CITATION], a very recent ARPES work showed a complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], therefore more experiments like quantum oscillation measurement are highly desired to clarify this important issue.', '1412.8298-3-3-5': 'On the other hand, since the perfect balance between electron and hole populations should be very sensitive to some tuning parameters such as doping and pressure, investigating the evolution of the XMR in WTe[MATH] with these parameters may further clarify its origin.', '1412.8298-3-4-0': 'In this Letter, we present the quantum oscillation study of the magnetoresistance for WTe[MATH] single crystals under ambient and high pressure.', '1412.8298-3-4-1': 'By analyzing the Shubnikov-de Haas oscillations of magnetoresistance at low temperature, four Fermi surfaces are revealed, which should be the two pairs of hole and electron pockets later observed by ARPES experiments [CITATION].', '1412.8298-3-4-2': 'A drastic suppression of the XMR with increasing pressure is observed, which is accompanied by a change of the Fermi surface topology.', '1412.8298-3-4-3': 'The correlation between them provides support for the mechanism that the XMR of WTe[MATH] originates from the electron-hole compensation.', '1412.8298-3-5-0': 'The WTe[MATH] single crystals were grown by using a chemical vapor transport method similar to that described in Ref. [CITATION].', '1412.8298-3-5-1': 'A mixture of stoichiometric W and Te powder were sealed into an evacuated quartz tube with transport agent Br[MATH].', '1412.8298-3-5-2': 'The quartz tube was then placed in a double zone furnace with temperature gradient of 100 [MATH]C between 750 [MATH]C and 650 [MATH]C. Large single crystals of centimeter size were obtained after one week.', '1412.8298-3-5-3': 'The samples were cut and cleaved to a typical size of 2.0 [MATH] 0.7 [MATH] 0.02 mm[MATH], in which the longest edge is along [MATH] axis and the thinnest dimension is along the [MATH] axis.', '1412.8298-3-5-4': 'Figure 1(b) shows the photo of one sample.', '1412.8298-3-5-5': 'X-ray diffraction (XRD) measurement was performed by using an X-ray diffractometer (D8 Advance, Bruker).', '1412.8298-3-5-6': 'Standard four-probe method was used for resistivity measurements, with current along the [MATH] axis.', '1412.8298-3-5-7': 'The contacts were made with silver epoxy.', '1412.8298-3-5-8': 'The resistivity was measured in a [MATH]He cryostat from 300 K to 2 K, and in a [MATH]He cryostat down to 0.3 K. For measurements under pressure, samples were pressurized in a piston-cylinder clamp cell made of Be-Cu alloy, with Daphne oil as the pressure medium.', '1412.8298-3-5-9': 'The pressure inside the cell was determined from the [MATH] of a tin wire.', '1412.8298-3-5-10': 'Magnetic field was applied along the [MATH]-axis direction up to 14.5 T.', '1412.8298-3-6-0': 'Figure 1(c) presents the XRD result of WTe[MATH] single crystal.', '1412.8298-3-6-1': 'Only reflections of (0 0 2[MATH]) show up, suggesting good orientation along the [MATH] axis.', '1412.8298-3-6-2': 'The lattice parameter [MATH] 14.054 is estimated from the XRD data, which agrees well with previous reports [CITATION].', '1412.8298-3-7-0': 'Figure 2(a) shows the temperature dependence of resistivity in zero field for WTe[MATH] single crystal, with current along the [MATH] axis.', '1412.8298-3-7-1': 'It has [MATH](295 K) = 355 [MATH] cm, with the residual resistivity ratio RRR = [MATH](295 K)/[MATH](2 K) = 184.', '1412.8298-3-7-2': 'This RRR value is about half of that in Ref. [CITATION].', '1412.8298-3-7-3': 'Figure 2(b) presents the magnetoresistance up to 14.5 T at various temperatures.', '1412.8298-3-7-4': 'The MR is defined by MR = [[MATH]) - [MATH](0 T)]/[MATH](0 T) [MATH].', '1412.8298-3-7-5': 'At 0.3 K and in 14.5 T, the MR reaches as high as 1.25 [MATH]% (taken from the smooth background of the curve).', '1412.8298-3-7-6': 'This value is lower than that in Ref. [CITATION], which is attributed to the slightly lower quality (smaller RRR) of our sample.', '1412.8298-3-7-7': 'With increasing temperature, the MR decreases rapidly and the oscillations disappear above 15 K, which are consistent with Ref. [CITATION].', '1412.8298-3-8-0': 'The SdH oscillation is a useful technique to detect the Fermi surface topology [CITATION].', '1412.8298-3-8-1': 'In Fig. 3(a), the oscillatory MR is analyzed by employing Fast Fourier Transform (FFT) for various temperatures from 0.3 to 6 K.', '1412.8298-3-8-2': 'The FFT spectrum shows four major peaks at 94.7, 132, 148, and 166 T oscillation frequency, labeled as [MATH] and [MATH].', '1412.8298-3-8-3': 'The second harmonics 2[MATH] and 2[MATH], likely due to spin-splitting, and the sum of [MATH] and [MATH] due to magnetic breakdown are also observed [CITATION].', '1412.8298-3-8-4': 'The SdH oscillation frequency [MATH] is proportional to the extremal cross-sectional area of the Fermi surface normal to the field, according to the Onsager relation [MATH].', '1412.8298-3-8-5': 'The [MATH] and [MATH] peaks seen in Fig. 3(a) clearly indicate that there exist four Fermi pockets normal to the field.', '1412.8298-3-9-0': 'The SdH oscillations of MR can be described by the Lifshitz-Kosevich (LK) formula [CITATION].', '1412.8298-3-9-1': 'The temperature dependence of the oscillation amplitude is determined by the thermal damping factor [MATH] in the LK formula, defined as [MATH], where [MATH] is the effective mass, [MATH], and [MATH] is the Boltzmann constant.', '1412.8298-3-9-2': 'Figure 3(b) shows the fits of the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor [MATH] from 0.3 to 6 K.', '1412.8298-3-9-3': 'The effective masses [MATH] = 0.387[MATH], 0.414[MATH], and 0.462[MATH] are obtained for [MATH], and [MATH] pockets, respectively, where [MATH] is the bare electron mass.', '1412.8298-3-9-4': 'Since the amplitude of the [MATH] peak can not be reliably extracted above 3 K, we did not do the fitting for the [MATH] pocket.', '1412.8298-3-10-0': 'Next we investigate the pressure effect on the XMR of WTe[MATH].', '1412.8298-3-10-1': 'Figure 4(a) shows the MR of WTe[MATH] single crystal under various pressures up to 23.6 kbar, measured at [MATH] = 0.3 K. With increasing pressure, the MR is strongly suppressed and the oscillations also gradually disappear.', '1412.8298-3-10-2': 'The pressure dependence of the MR in the highest field [MATH] = 14.5 T is plotted in Fig. 4(b).', '1412.8298-3-10-3': 'It decreases from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under [MATH] = 23.6 kbar.', '1412.8298-3-11-0': 'To find out the cause of this strong suppression of MR in WTe[MATH], we check the evolution of its Fermi surfaces with pressure.', '1412.8298-3-11-1': 'First let us identify the four Fermi pockets obtained at ambient pressure, by comparing with the two ARPES results [CITATION].', '1412.8298-3-11-2': 'One ARPES group found a pair of hole and electron pockets with comparable size along the [MATH] direction [CITATION], however, another ARPES group revealed a more complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], as sketched in Fig. 5(a).', '1412.8298-3-11-3': 'The sizes of the pair of hole and electron pockets in Ref. [CITATION] are comparable with [MATH] 0.08 [MATH], while the sizes of the two pairs of hole and electron pockets in Ref. [CITATION] are smaller and show slight difference.', '1412.8298-3-11-4': 'We examine the sizes of the four Fermi pockets obtained from our SdH oscillations of MR.', '1412.8298-3-11-5': 'According to the Onsager relation [MATH], [MATH] = 0.0090, 0.0125, 0.0141, and 0.0158 [MATH] are obtained for [MATH], [MATH], [MATH], and [MATH] pockets, respectively.', '1412.8298-3-11-6': 'By assuming a circular pocket, the Fermi momentums [MATH] 0.054 [MATH], [MATH] 0.063 [MATH], [MATH] 0.067 [MATH], and [MATH] 0.071 [MATH] are estimated.', '1412.8298-3-11-7': 'Therefore, both the number and sizes of the Fermi pockets obtained from our measurements support the Fermi surface topology of WTe[MATH] revealed in Ref. [CITATION].', '1412.8298-3-11-8': 'Note that an additional hole pocket around [MATH] was observed in some samples but absent in some other samples [CITATION], therefore we do not sketch it in Fig. 5(a).', '1412.8298-3-11-9': 'It is likely absent in our samples.', '1412.8298-3-12-0': 'Figure 5(b)-(f) show the FFT spectrums from ambient pressure to 18.1 kbar.', '1412.8298-3-12-1': 'Above 18.1 kbar, the oscillations are too weak to give a reliable FFT spectrum.', '1412.8298-3-12-2': 'One can see that both [MATH] and [MATH] peaks persist all the way to the highest pressure.', '1412.8298-3-12-3': 'According to the scenario in Ref. [CITATION], the nearly perfect balance of electron and hole populations is responsible for the XMR in WTe[MATH].', '1412.8298-3-12-4': 'Since under 12.2 kbar, the MR is still as large as 8.28 [MATH]% and only the [MATH] and [MATH] peaks remain, we identify [MATH] and [MATH] as one of the two pairs of hole and electron pockets.', '1412.8298-3-12-5': 'It will be very interesting to do electronic structure calculation under pressure, to show how another pair of hole and electron pockets, [MATH] and [MATH], disappears with increasing pressure.', '1412.8298-3-12-6': "We note a recent theoretical calculation shows that the electronic structure of monolayer 1T'-WTe[MATH] is sensitive to the tensile strain, which may be crucial for realizing the quantum spin Hall effect in this two-dimensional transition metal dichalcogenide [CITATION].", '1412.8298-3-13-0': 'With increasing pressure, the absolute sizes of [MATH] and [MATH] pockets increase, because of the shrink of lattice.', '1412.8298-3-13-1': 'The relative size of [MATH] and [MATH] pockets, defined as [MATH]/[MATH], is plotted in Fig. 4(b) together with the pressure dependence of MR. The two curves are clearly correlated, suggesting that the increasing difference between the sizes of [MATH] and [MATH] pockets is the cause of the strong suppression of MR with pressure in WTe[MATH].', '1412.8298-3-13-2': 'In this sense, our result confirms the importance of the perfect balance between the electron and hole populations to the XMR in WTe[MATH].', '1412.8298-3-14-0': 'From the perspective of application, the XMR makes WTe[MATH] useful in devices such as highly sensitive low-temperature magnetic-field sensors or high-field temperature sensors in cryogenics.', '1412.8298-3-14-1': 'The high sensitivity of MR to pressure we observe here provides additional way to use WTe[MATH] in devices, for example, as a pressure sensor.', '1412.8298-3-14-2': 'WTe[MATH] itself is also quite suitable for the development of advanced devices.', '1412.8298-3-15-0': 'In summary, we study the quantum oscillations of magnetoresistance under ambient and high pressure for WTe[MATH] single crystals.', '1412.8298-3-15-1': 'Under ambient pressure, four Fermi surfaces are identified by analyzing the SdH oscillations, which are likely two pairs of hole and electron pockets along the [MATH] direction.', '1412.8298-3-15-2': 'With increasing pressure, drastic change of Fermi surface topology and strong suppression of the XMR are observed.', '1412.8298-3-15-3': 'While one pair of hole and electron pockets ([MATH] and [MATH]) persists to high pressure, the other pair of hole and electron pockets ([MATH] and [MATH]) disappears with increasing pressure.', '1412.8298-3-15-4': 'The relative size of the [MATH] and [MATH] pockets decreases with increasing pressure, which may cause the strong suppression of the XMR.', '1412.8298-3-15-5': 'Our results support the scenario that the perfect balance between the electron and hole populations is the origin of the XMR in WTe[MATH].', '1412.8298-3-16-0': 'We thank J. K. Dong, D. L. Feng, J. Jiang, P. S. Wang, and Z. J. Xiang for helpful discussions.', '1412.8298-3-16-1': 'This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200).', '1412.8298-3-16-2': 'The work at Tulane is supported by NSF under Grant No. DMR-1205469.', '1412.8298-3-17-0': 'Note added: After we put this work in arXiv (1412.8298), superconductivity was discovered in WTe[MATH] by applying higher pressure than ours [CITATION], which coincides with the suppression of the XMR.'}

{'1412.8298-4-0-0': 'The quantum oscillations of the magnetoresistance under ambient and high pressure have been studied for WTe[MATH] single crystals, in which extremely large magnetoresistance was discovered recently.', '1412.8298-4-0-1': 'By analyzing the Shubnikov-de Haas oscillations, four Fermi surfaces are identified, and two of them are found to persist to high pressure.', '1412.8298-4-0-2': 'The sizes of these two pockets are comparable, but show increasing difference with pressure.', '1412.8298-4-0-3': 'At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under 23.6 kbar, which is likely caused by the relative change of Fermi surfaces.', '1412.8298-4-0-4': 'These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe[MATH].', '1412.8298-4-1-0': 'The magnetoresistance (MR) is an important transport property of condensed matters.', '1412.8298-4-1-1': 'For simple metals, the MR is usually very small, showing quadratic field dependence in low field and saturating in high field [CITATION].', '1412.8298-4-1-2': 'In contrast, giant magnetoresistance (GMR) was discovered in magnetic multilayers [CITATION], and colossal magnetoresistance (CMR) was found in manganites [CITATION].', '1412.8298-4-1-3': 'Interestingly, in recent few years extremely large ([MATH]%) and non-saturating positive MR (XMR) was discovered in some nonmagnetic compounds including PtSn[MATH] [CITATION], Cd[MATH]As[MATH] [CITATION], WTe[MATH] [CITATION], and NbSb[MATH] [CITATION].', '1412.8298-4-2-0': 'The very recent discovery of XMR in WTe[MATH] is of particular interest [CITATION].', '1412.8298-4-2-1': 'WTe[MATH] is a layered transition-metal dichalcogenide, with the crystal structure shown in Fig. 1(a).', '1412.8298-4-2-2': 'In the dichalcogenide layers, W chains are formed along the [MATH] axis.', '1412.8298-4-2-3': 'An XMR of 4.5 [MATH] in 14.7 T at 4.5 K was found when the current is along the [MATH] axis and magnetic field is applied along the [MATH] axis [CITATION].', '1412.8298-4-2-4': 'More remarkably, it increases to as high as 1.3 [MATH] in 60 T at 0.53 K, without any sign of saturation [CITATION].', '1412.8298-4-2-5': 'Such an XMR makes WTe[MATH] outstanding among transition-metal dichalcogenides, in which various interesting physics such as charge density wave and superconductivity have been extensively studied [CITATION].', '1412.8298-4-3-0': 'Based on electronic structure calculations, the XMR in WTe[MATH] was attributed to the compensation between the electron and hole populations [CITATION].', '1412.8298-4-3-1': 'Actually, such a two-band model for charge transport in semimetals was previously used to explain the very large MR observed in high-purity graphite and bismuth [CITATION].', '1412.8298-4-3-2': 'However, in both graphite and Bi, the MR saturates beyond a few Tesla, due to the slight deviation from perfect compensation [CITATION].', '1412.8298-4-3-3': 'In this sense, the non-saturating XMR in WTe[MATH] may provide the first example of perfectly balanced electron-hole populations [CITATION].', '1412.8298-4-3-4': 'While the subsequent angle-resolved photoemission spectroscopy (ARPES) experiments observed one pair of hole and electron pockets of approximately the same size at low temperature [CITATION], a very recent ARPES work showed a complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], therefore more experiments like quantum oscillation measurement are highly desired to clarify this important issue.', '1412.8298-4-3-5': 'On the other hand, since the perfect balance between electron and hole populations should be very sensitive to some tuning parameters such as doping and pressure, investigating the evolution of the XMR in WTe[MATH] with these parameters may further clarify its origin.', '1412.8298-4-4-0': 'In this Letter, we present the quantum oscillation study of the magnetoresistance for WTe[MATH] single crystals under ambient and high pressure.', '1412.8298-4-4-1': 'By analyzing the Shubnikov-de Haas oscillations of magnetoresistance at low temperature, four Fermi surfaces are revealed, which should correspond to the two pairs of hole and electron pockets later probed by ARPES experiments [CITATION].', '1412.8298-4-4-2': 'A drastic suppression of the XMR with increasing pressure is observed, which is accompanied by a change of the Fermi surface topology.', '1412.8298-4-4-3': 'The correlation between them provides support for the mechanism that the XMR of WTe[MATH] originates from the electron-hole compensation.', '1412.8298-4-5-0': 'The WTe[MATH] single crystals were grown by using a chemical vapor transport method similar to that described in Ref. [CITATION].', '1412.8298-4-5-1': 'A mixture of stoichiometric W and Te powder were sealed into an evacuated quartz tube with transport agent Br[MATH].', '1412.8298-4-5-2': 'The quartz tube was then placed in a double zone furnace with temperature gradient of 100 [MATH]C between 750 [MATH]C and 650 [MATH]C. Large single crystals of centimeter size were obtained after one week.', '1412.8298-4-5-3': 'The samples were cut and cleaved to a typical size of 2.0 [MATH] 0.7 [MATH] 0.02 mm[MATH], in which the longest edge is along [MATH] axis and the thinnest dimension is along the [MATH] axis.', '1412.8298-4-5-4': 'Figure 1(b) shows the photo of one sample.', '1412.8298-4-5-5': 'X-ray diffraction (XRD) measurement was performed using an X-ray diffractometer (D8 Advance, Bruker).', '1412.8298-4-5-6': 'Standard four-probe method was used for resistivity measurements, with current along the [MATH] axis.', '1412.8298-4-5-7': 'The contacts were made with silver epoxy.', '1412.8298-4-5-8': 'The resistivity was measured in a [MATH]He cryostat from 300 K to 2 K, and in a [MATH]He cryostat down to 0.3 K. For measurements under pressure, samples were pressurized in a piston-cylinder clamp cell made of Be-Cu alloy, with Daphne oil as the pressure medium.', '1412.8298-4-5-9': 'The pressure inside the cell was determined from the [MATH] of a tin wire.', '1412.8298-4-5-10': 'Magnetic field was applied along the [MATH]-axis direction up to 14.5 T.', '1412.8298-4-6-0': 'Figure 1(c) presents the XRD result of WTe[MATH] single crystal.', '1412.8298-4-6-1': 'Only reflections of (0 0 2[MATH]) show up, indicating the [MATH]-axis orientation.', '1412.8298-4-6-2': 'The lattice parameter [MATH] 14.054 is determined from the XRD data, which agrees well with previous reports [CITATION].', '1412.8298-4-7-0': 'Figure 2(a) shows the temperature dependence of resistivity in zero field for a WTe[MATH] single crystal, with current along the [MATH] axis.', '1412.8298-4-7-1': 'It has [MATH](295 K) = 355 [MATH] cm, with the residual resistivity ratio RRR = [MATH](295 K)/[MATH](2 K) = 184.', '1412.8298-4-7-2': 'This RRR value is about half of that in Ref. [CITATION].', '1412.8298-4-7-3': 'Figure 2(b) presents the magnetoresistance up to 14.5 T at various temperatures.', '1412.8298-4-7-4': 'The MR is defined by MR = [[MATH]) - [MATH](0 T)]/[MATH](0 T) [MATH].', '1412.8298-4-7-5': 'At 0.3 K and in 14.5 T, the MR reaches as high as 1.25 [MATH]% (taken from the smooth background of the curve).', '1412.8298-4-7-6': 'This value is lower than that in Ref. [CITATION], which is attributed to the slightly lower quality (smaller RRR) of our sample.', '1412.8298-4-7-7': 'With increasing temperature, the MR decreases rapidly and the oscillations disappear above 15 K, which are consistent with Ref. [CITATION].', '1412.8298-4-8-0': 'The SdH oscillation is a useful technique to detect the Fermi surface topology [CITATION].', '1412.8298-4-8-1': 'In Fig. 3(a), the oscillatory MR is analyzed by employing Fast Fourier Transform (FFT) for various temperatures from 0.3 to 6 K.', '1412.8298-4-8-2': 'The FFT spectrum shows four major peaks at 94.7, 132, 148, and 166 T oscillation frequency, labeled as [MATH] and [MATH].', '1412.8298-4-8-3': 'The second harmonics 2[MATH] and 2[MATH], likely due to spin-splitting, and the sum of [MATH] and [MATH] due to magnetic breakdown are also observed [CITATION].', '1412.8298-4-8-4': 'The SdH oscillation frequency [MATH] is proportional to the extremal cross-sectional area of the Fermi surface normal to the field, according to the Onsager relation [MATH].', '1412.8298-4-8-5': 'The [MATH] and [MATH] peaks seen in Fig. 3(a) clearly indicate that there exist four Fermi pockets normal to the field.', '1412.8298-4-9-0': 'The SdH oscillations of MR can be described by the Lifshitz-Kosevich (LK) formula [CITATION].', '1412.8298-4-9-1': 'The temperature dependence of the oscillation amplitude is determined by the thermal damping factor [MATH] in the LK formula, defined as [MATH], where [MATH] is the temperature, [MATH] is the effective mass, 1/[MATH] is the average inverse field of the Fourier window, [MATH], and [MATH] is the Boltzmann constant.', '1412.8298-4-9-2': 'Figure 3(b) shows the fits of the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor [MATH] from 0.3 to 6 K.', '1412.8298-4-9-3': 'The effective masses [MATH] = 0.387[MATH], 0.414[MATH], and 0.462[MATH] are obtained for [MATH], and [MATH] pockets, respectively, where [MATH] is the bare electron mass.', '1412.8298-4-9-4': 'Since the amplitude of the [MATH] peak can not be reliably extracted above 3 K, we did not do the fitting for the [MATH] pocket.', '1412.8298-4-10-0': 'Next we investigate the pressure effect on the XMR of WTe[MATH].', '1412.8298-4-10-1': 'Figure 4(a) shows the MR of WTe[MATH] single crystal under various pressures up to 23.6 kbar, measured at [MATH] = 0.3 K. With increasing pressure, the MR is strongly suppressed and the oscillations also gradually disappear.', '1412.8298-4-10-2': 'The pressure dependence of the MR in the highest field [MATH] = 14.5 T is plotted in Fig. 4(b).', '1412.8298-4-10-3': 'It decreases from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under [MATH] = 23.6 kbar.', '1412.8298-4-11-0': 'To find out the cause of this strong suppression of MR in WTe[MATH], we examine the evolution of its Fermi surfaces with pressure.', '1412.8298-4-11-1': 'First let us identify the four Fermi pockets obtained at ambient pressure, by comparing with the two ARPES results [CITATION].', '1412.8298-4-11-2': 'One ARPES group found a pair of hole and electron pockets with comparable size along the [MATH] direction [CITATION], however, another ARPES group revealed a more complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], as sketched in Fig. 5(a).', '1412.8298-4-11-3': 'The sizes of the pair of hole and electron pockets in Ref. [CITATION] are comparable with [MATH] 0.08 [MATH], while the sizes of the two pairs of hole and electron pockets in Ref. [CITATION] are smaller and show slight difference.', '1412.8298-4-11-4': 'We examine the sizes of the four Fermi pockets obtained from our SdH oscillations of MR.', '1412.8298-4-11-5': 'According to the Onsager relation [MATH], [MATH] = 0.0090, 0.0125, 0.0141, and 0.0158 [MATH] are obtained for [MATH], [MATH], [MATH], and [MATH] pockets, respectively.', '1412.8298-4-11-6': 'By assuming a circular pocket, the Fermi momentums [MATH] 0.054 [MATH], [MATH] 0.063 [MATH], [MATH] 0.067 [MATH], and [MATH] 0.071 [MATH] are estimated.', '1412.8298-4-11-7': 'Therefore, both the number and sizes of the Fermi pockets obtained from our measurements are consistent with the Fermi surface topology of WTe[MATH] revealed in Ref. [CITATION].', '1412.8298-4-11-8': 'Note that an additional hole pocket around [MATH] was observed in some samples but absent in some other samples [CITATION], therefore we do not sketch it in Fig. 5(a).', '1412.8298-4-11-9': 'It is likely absent in our samples.', '1412.8298-4-12-0': 'Figure 5(b)-(f) show the FFT spectrums from ambient pressure to 18.1 kbar.', '1412.8298-4-12-1': 'Above 18.1 kbar, the oscillations are too weak to give a reliable FFT spectrum.', '1412.8298-4-12-2': 'One can see that both [MATH] and [MATH] peaks persist all the way to the highest pressure.', '1412.8298-4-12-3': 'According to the scenario in Ref. [CITATION], the nearly perfect balance of electron and hole populations is responsible for the XMR in WTe[MATH].', '1412.8298-4-12-4': 'Since under 12.2 kbar, the MR is still as large as 8.28 [MATH]% and only the [MATH] and [MATH] peaks remain, we identify [MATH] and [MATH] as one of the two pairs of hole and electron pockets.', '1412.8298-4-12-5': 'It will be very interesting to do electronic structure calculation under pressure, to show how another pair of hole and electron pockets, [MATH] and [MATH], disappears with increasing pressure.', '1412.8298-4-12-6': "We note a recent theoretical calculation shows that the electronic structure of monolayer 1T'-WTe[MATH] is sensitive to the tensile strain, which may be crucial for realizing the quantum spin Hall effect in this two-dimensional transition metal dichalcogenide [CITATION].", '1412.8298-4-13-0': 'With increasing pressure, the absolute sizes of [MATH] and [MATH] pockets increase, because of the shrink of lattice.', '1412.8298-4-13-1': 'The relative size of [MATH] and [MATH] pockets, defined as [MATH]/[MATH], is plotted in Fig. 4(b) together with the pressure dependence of MR. The two curves are clearly correlated, suggesting that the increasing difference between the sizes of [MATH] and [MATH] pockets is the cause of the strong suppression of MR with pressure in WTe[MATH].', '1412.8298-4-13-2': 'In this sense, our result confirms the importance of the perfect balance between the electron and hole populations to the XMR in WTe[MATH].', '1412.8298-4-14-0': 'Note that the four major frequencies we observed ([MATH] = 94.7 T, [MATH] = 132 T, [MATH] = 148 T, and [MATH] = 166 T) were confirmed by a later quantum oscillation study of WTe[MATH] [CITATION].', '1412.8298-4-14-1': 'According to the FFT analysis of the quantum oscillations of the Seebeck coefficient with field applied along the [MATH] axis, they found four major frequencies [MATH] = 92 T, [MATH] = 125 T, [MATH] = 142 T, and [MATH] = 162 T [CITATION], which are nearly the same as ours.', '1412.8298-4-14-2': 'However, they wrongly identified [MATH] and [MATH] and [MATH] in our Fig. 5) as two electron pockets, based on their band calculations [CITATION].', '1412.8298-4-14-3': 'In fact, the band calculations of WTe[MATH] are very subtle, in the number and size of electron and hole pockets [CITATION].', '1412.8298-4-14-4': 'The calculated electronic structure only partially reproduces the experimental bands and Fermi surface [CITATION], thus it is risky to identify the four major frequencies only based on band calculations.', '1412.8298-4-14-5': 'Quantitatively, it is not appropriate to identify the well-separated [MATH] and [MATH] peaks to the two nearly degenerated electron pockets [CITATION].', '1412.8298-4-14-6': 'This situation highlights the importance of our pressure study, in which the large MR = 8.28 [MATH]% and the remaining [MATH] and [MATH] peaks under 12.2 kbar enable us to identify [MATH] and [MATH] and [MATH] in Ref. [CITATION]) as one pair of electron and hole pockets.', '1412.8298-4-14-7': 'Therefore, the wrong Fermi surface topology ("Russian doll" structure) of WTe[MATH] given in Ref. [CITATION] is quite misleading.', '1412.8298-4-14-8': 'The correct Fermi surface topology is very likely that revealed by the ARPES experiments in Ref. [CITATION], which is consistent with our results.', '1412.8298-4-14-9': 'Such a Fermi surface topology is the base to understand the XMR in WTe[MATH].', '1412.8298-4-15-0': 'In summary, we study the quantum oscillations of magnetoresistance under ambient and high pressure for WTe[MATH] single crystals.', '1412.8298-4-15-1': 'Under ambient pressure, four Fermi surfaces are identified by analyzing the SdH oscillations, which are likely two pairs of hole and electron pockets along the [MATH] direction.', '1412.8298-4-15-2': 'With increasing pressure, drastic change of Fermi surface topology and strong suppression of the XMR are observed.', '1412.8298-4-15-3': 'While one pair of hole and electron pockets ([MATH] and [MATH]) persists to high pressure, the other pair of hole and electron pockets ([MATH] and [MATH]) disappears with increasing pressure.', '1412.8298-4-15-4': 'The relative size of the [MATH] and [MATH] pockets decreases with increasing pressure, which may cause the strong suppression of the XMR.', '1412.8298-4-15-5': 'Our results support the scenario that the perfect balance between the electron and hole populations is the origin of the XMR in WTe[MATH].', '1412.8298-4-16-0': 'We thank J. K. Dong, D. L. Feng, J. Jiang, P. S. Wang, and Z. J. Xiang for helpful discussions.', '1412.8298-4-16-1': 'This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200).', '1412.8298-4-16-2': 'The work at Tulane is supported by NSF under Grant No. DMR-1205469.', '1412.8298-4-17-0': 'Note added: After we put this work in arXiv (1412.8298), superconductivity was discovered in WTe[MATH] by applying higher pressure than ours [CITATION], which coincides with the suppression of the XMR.'}

{'1412.8298-5-0-0': 'The quantum oscillations of the magnetoresistance under ambient and high pressure have been studied for WTe[MATH] single crystals, in which extremely large magnetoresistance was discovered recently.', '1412.8298-5-0-1': 'By analyzing the Shubnikov-de Haas oscillations, four Fermi surfaces are identified, and two of them are found to persist to high pressure.', '1412.8298-5-0-2': 'The sizes of these two pockets are comparable, but show increasing difference with pressure.', '1412.8298-5-0-3': 'At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under 23.6 kbar, which is likely caused by the relative change of Fermi surfaces.', '1412.8298-5-0-4': 'These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe[MATH].', '1412.8298-5-1-0': 'The magnetoresistance (MR) is an important transport property of condensed matters.', '1412.8298-5-1-1': 'For simple metals, the MR is usually very small, showing quadratic field dependence in low field and saturating in high field [CITATION].', '1412.8298-5-1-2': 'In contrast, giant magnetoresistance (GMR) was discovered in magnetic multilayers [CITATION], and colossal magnetoresistance (CMR) was found in manganites [CITATION].', '1412.8298-5-1-3': 'Interestingly, in recent few years extremely large ([MATH]%) and non-saturating positive MR (XMR) was discovered in some nonmagnetic compounds including PtSn[MATH] [CITATION], Cd[MATH]As[MATH] [CITATION], WTe[MATH] [CITATION], and NbSb[MATH] [CITATION].', '1412.8298-5-2-0': 'The very recent discovery of XMR in WTe[MATH] is of particular interest [CITATION].', '1412.8298-5-2-1': 'WTe[MATH] is a layered transition-metal dichalcogenide, with the crystal structure shown in Fig. 1(a).', '1412.8298-5-2-2': 'In the dichalcogenide layers, W chains are formed along the [MATH] axis.', '1412.8298-5-2-3': 'An XMR of 4.5 [MATH] in 14.7 T at 4.5 K was found when the current is along the [MATH] axis and magnetic field is applied along the [MATH] axis [CITATION].', '1412.8298-5-2-4': 'More remarkably, it increases to as high as 1.3 [MATH] in 60 T at 0.53 K, without any sign of saturation [CITATION].', '1412.8298-5-2-5': 'Such an XMR makes WTe[MATH] outstanding among transition-metal dichalcogenides, in which various interesting physics such as charge density wave and superconductivity have been extensively studied [CITATION].', '1412.8298-5-3-0': 'Based on electronic structure calculations, the XMR in WTe[MATH] was attributed to the compensation between the electron and hole populations [CITATION].', '1412.8298-5-3-1': 'Actually, such a two-band model for charge transport in semimetals was previously used to explain the very large MR observed in high-purity graphite and bismuth [CITATION].', '1412.8298-5-3-2': 'However, in both graphite and Bi, the MR saturates beyond a few Tesla, due to the slight deviation from perfect compensation [CITATION].', '1412.8298-5-3-3': 'In this sense, the non-saturating XMR in WTe[MATH] may provide the first example of perfectly balanced electron-hole populations [CITATION].', '1412.8298-5-3-4': 'While the subsequent angle-resolved photoemission spectroscopy (ARPES) experiments observed one pair of hole and electron pockets of approximately the same size at low temperature [CITATION], a very recent ARPES work showed a complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], therefore more experiments like quantum oscillation measurement are highly desired to clarify this important issue.', '1412.8298-5-3-5': 'On the other hand, since the perfect balance between electron and hole populations should be very sensitive to some tuning parameters such as doping and pressure, investigating the evolution of the XMR in WTe[MATH] with these parameters may further clarify its origin.', '1412.8298-5-4-0': 'In this Letter, we present the quantum oscillation study of the magnetoresistance for WTe[MATH] single crystals under ambient and high pressure.', '1412.8298-5-4-1': 'By analyzing the Shubnikov-de Haas oscillations of magnetoresistance at low temperature, four Fermi surfaces are revealed, which should correspond to the two pairs of hole and electron pockets later probed by ARPES experiments [CITATION].', '1412.8298-5-4-2': 'A drastic suppression of the XMR with increasing pressure is observed, which is accompanied by a change of the Fermi surface topology.', '1412.8298-5-4-3': 'The correlation between them provides support for the mechanism that the XMR of WTe[MATH] originates from the electron-hole compensation.', '1412.8298-5-5-0': 'The WTe[MATH] single crystals were grown by using a chemical vapor transport method similar to that described in Ref. [CITATION].', '1412.8298-5-5-1': 'A mixture of stoichiometric W and Te powder were sealed into an evacuated quartz tube with transport agent Br[MATH].', '1412.8298-5-5-2': 'The quartz tube was then placed in a double zone furnace with temperature gradient of 100 [MATH]C between 750 [MATH]C and 650 [MATH]C. Large single crystals of centimeter size were obtained after one week.', '1412.8298-5-5-3': 'The samples were cut and cleaved to a typical size of 2.0 [MATH] 0.7 [MATH] 0.02 mm[MATH], in which the longest edge is along [MATH] axis and the thinnest dimension is along the [MATH] axis.', '1412.8298-5-5-4': 'Figure 1(b) shows the photo of one sample.', '1412.8298-5-5-5': 'X-ray diffraction (XRD) measurement was performed using an X-ray diffractometer (D8 Advance, Bruker).', '1412.8298-5-5-6': 'Standard four-probe method was used for resistivity measurements, with current along the [MATH] axis.', '1412.8298-5-5-7': 'The contacts were made with silver epoxy.', '1412.8298-5-5-8': 'The resistivity was measured in a [MATH]He cryostat from 300 K to 2 K, and in a [MATH]He cryostat down to 0.3 K. For measurements under pressure, samples were pressurized in a piston-cylinder clamp cell made of Be-Cu alloy, with Daphne oil as the pressure medium.', '1412.8298-5-5-9': 'The pressure inside the cell was determined from the [MATH] of a tin wire.', '1412.8298-5-5-10': 'Magnetic field was applied along the [MATH]-axis direction up to 14.5 T.', '1412.8298-5-6-0': 'Figure 1(c) presents the XRD result of WTe[MATH] single crystal.', '1412.8298-5-6-1': 'Only reflections of (0 0 2[MATH]) show up, indicating the [MATH]-axis orientation.', '1412.8298-5-6-2': 'The lattice parameter [MATH] 14.054 is determined from the XRD data, which agrees well with previous reports [CITATION].', '1412.8298-5-7-0': 'Figure 2(a) shows the temperature dependence of resistivity in zero field for a WTe[MATH] single crystal, with current along the [MATH] axis.', '1412.8298-5-7-1': 'It has [MATH](295 K) = 355 [MATH] cm, with the residual resistivity ratio RRR = [MATH](295 K)/[MATH](2 K) = 184.', '1412.8298-5-7-2': 'This RRR value is about half of that in Ref. [CITATION].', '1412.8298-5-7-3': 'Figure 2(b) presents the magnetoresistance up to 14.5 T at various temperatures.', '1412.8298-5-7-4': 'The MR is defined by MR = [[MATH]) - [MATH](0 T)]/[MATH](0 T) [MATH].', '1412.8298-5-7-5': 'At 0.3 K and in 14.5 T, the MR reaches as high as 1.25 [MATH]% (taken from the smooth background of the curve).', '1412.8298-5-7-6': 'This value is lower than that in Ref. [CITATION], which is attributed to the slightly lower quality (smaller RRR) of our sample.', '1412.8298-5-7-7': 'With increasing temperature, the MR decreases rapidly and the oscillations disappear above 15 K, which are consistent with Ref. [CITATION].', '1412.8298-5-8-0': 'The SdH oscillation is a useful technique to detect the Fermi surface topology [CITATION].', '1412.8298-5-8-1': 'In Fig. 3(a), the oscillatory MR is analyzed by employing Fast Fourier Transform (FFT) for various temperatures from 0.3 to 6 K.', '1412.8298-5-8-2': 'The FFT spectrum shows four major peaks at 94.7, 132, 148, and 166 T oscillation frequency, labeled as [MATH] and [MATH].', '1412.8298-5-8-3': 'The second harmonics 2[MATH] and 2[MATH], likely due to spin-splitting, and the sum of [MATH] and [MATH] due to magnetic breakdown are also observed [CITATION].', '1412.8298-5-8-4': 'The SdH oscillation frequency [MATH] is proportional to the extremal cross-sectional area of the Fermi surface normal to the field, according to the Onsager relation [MATH].', '1412.8298-5-8-5': 'The [MATH] and [MATH] peaks seen in Fig. 3(a) clearly indicate that there exist four Fermi pockets normal to the field.', '1412.8298-5-9-0': 'The SdH oscillations of MR can be described by the Lifshitz-Kosevich (LK) formula [CITATION].', '1412.8298-5-9-1': 'The temperature dependence of the oscillation amplitude is determined by the thermal damping factor [MATH] in the LK formula, defined as [MATH], where [MATH] is the temperature, [MATH] is the effective mass, 1/[MATH] is the average inverse field of the Fourier window, [MATH], and [MATH] is the Boltzmann constant.', '1412.8298-5-9-2': 'Figure 3(b) shows the fits of the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor [MATH] from 0.3 to 6 K.', '1412.8298-5-9-3': 'The effective masses [MATH] = 0.387[MATH], 0.414[MATH], and 0.462[MATH] are obtained for [MATH], and [MATH] pockets, respectively, where [MATH] is the bare electron mass.', '1412.8298-5-9-4': 'Since the amplitude of the [MATH] peak can not be reliably extracted above 3 K, we did not do the fitting for the [MATH] pocket.', '1412.8298-5-10-0': 'Next we investigate the pressure effect on the XMR of WTe[MATH].', '1412.8298-5-10-1': 'Figure 4(a) shows the MR of WTe[MATH] single crystal under various pressures up to 23.6 kbar, measured at [MATH] = 0.3 K. With increasing pressure, the MR is strongly suppressed and the oscillations also gradually disappear.', '1412.8298-5-10-2': 'The pressure dependence of the MR in the highest field [MATH] = 14.5 T is plotted in Fig. 4(b).', '1412.8298-5-10-3': 'It decreases from 1.25 [MATH]% under ambient pressure to 7.47 [MATH]% under [MATH] = 23.6 kbar.', '1412.8298-5-11-0': 'To find out the cause of this strong suppression of MR in WTe[MATH], we examine the evolution of its Fermi surfaces with pressure.', '1412.8298-5-11-1': 'First let us identify the four Fermi pockets obtained at ambient pressure, by comparing with the two ARPES results [CITATION].', '1412.8298-5-11-2': 'One ARPES group found a pair of hole and electron pockets with comparable size along the [MATH] direction [CITATION], however, another ARPES group revealed a more complex Fermi surface topology with two pairs of hole and electron pockets [CITATION], as sketched in Fig. 5(a).', '1412.8298-5-11-3': 'The sizes of the pair of hole and electron pockets in Ref. [CITATION] are comparable with [MATH] 0.08 [MATH], while the sizes of the two pairs of hole and electron pockets in Ref. [CITATION] are smaller and show slight difference.', '1412.8298-5-11-4': 'We examine the sizes of the four Fermi pockets obtained from our SdH oscillations of MR.', '1412.8298-5-11-5': 'According to the Onsager relation [MATH], [MATH] = 0.0090, 0.0125, 0.0141, and 0.0158 [MATH] are obtained for [MATH], [MATH], [MATH], and [MATH] pockets, respectively.', '1412.8298-5-11-6': 'By assuming a circular pocket, the Fermi momentums [MATH] 0.054 [MATH], [MATH] 0.063 [MATH], [MATH] 0.067 [MATH], and [MATH] 0.071 [MATH] are estimated.', '1412.8298-5-11-7': 'Therefore, both the number and sizes of the Fermi pockets obtained from our measurements are consistent with the Fermi surface topology of WTe[MATH] revealed in Ref. [CITATION].', '1412.8298-5-11-8': 'Note that an additional hole pocket around [MATH] was observed in some samples but absent in some other samples [CITATION], therefore we do not sketch it in Fig. 5(a).', '1412.8298-5-11-9': 'It is likely absent in our samples.', '1412.8298-5-12-0': 'Figure 5(b)-(f) show the FFT spectrums from ambient pressure to 18.1 kbar.', '1412.8298-5-12-1': 'Above 18.1 kbar, the oscillations are too weak to give a reliable FFT spectrum.', '1412.8298-5-12-2': 'One can see that both [MATH] and [MATH] peaks persist all the way to the highest pressure.', '1412.8298-5-12-3': 'According to the scenario in Ref. [CITATION], the nearly perfect balance of electron and hole populations is responsible for the XMR in WTe[MATH].', '1412.8298-5-12-4': 'Since under 12.2 kbar, the MR is still as large as 8.28 [MATH]% and only the [MATH] and [MATH] peaks remain, we identify [MATH] and [MATH] as one of the two pairs of hole and electron pockets.', '1412.8298-5-12-5': 'It will be very interesting to do electronic structure calculation under pressure, to show how another pair of hole and electron pockets, [MATH] and [MATH], disappears with increasing pressure.', '1412.8298-5-12-6': "We note a recent theoretical calculation shows that the electronic structure of monolayer 1T'-WTe[MATH] is sensitive to the tensile strain, which may be crucial for realizing the quantum spin Hall effect in this two-dimensional transition metal dichalcogenide [CITATION].", '1412.8298-5-13-0': 'With increasing pressure, the absolute sizes of [MATH] and [MATH] pockets increase, because of the shrink of lattice.', '1412.8298-5-13-1': 'The relative size of [MATH] and [MATH] pockets, defined as [MATH]/[MATH], is plotted in Fig. 4(b) together with the pressure dependence of MR. The two curves are clearly correlated, suggesting that the increasing difference between the sizes of [MATH] and [MATH] pockets is the cause of the strong suppression of MR with pressure in WTe[MATH].', '1412.8298-5-13-2': 'In this sense, our result confirms the importance of the perfect balance between the electron and hole populations to the XMR in WTe[MATH].', '1412.8298-5-14-0': 'Note that the four major frequencies we observed ([MATH] = 94.7 T, [MATH] = 132 T, [MATH] = 148 T, and [MATH] = 166 T) were confirmed by a later quantum oscillation study of WTe[MATH] [CITATION].', '1412.8298-5-14-1': 'According to the FFT analysis of the quantum oscillations of the Seebeck coefficient with field applied along the [MATH] axis, they found four major frequencies [MATH] = 92 T, [MATH] = 125 T, [MATH] = 142 T, and [MATH] = 162 T [CITATION], which are nearly the same as ours.', '1412.8298-5-14-2': 'However, they identified [MATH] and [MATH] and [MATH] in our Fig. 5) as two electron pockets ("Russian doll" structure), based on their band calculations [CITATION].', '1412.8298-5-14-3': 'In fact, the band calculations of WTe[MATH] are very subtle, in the number and size of electron and hole pockets [CITATION].', '1412.8298-5-14-4': 'The calculated electronic structure only partially reproduces the experimental bands and Fermi surface [CITATION], thus it is not appropriate to identify the four major frequencies only based on band calculations [CITATION].', '1412.8298-5-14-5': 'This situation highlights the importance of our pressure study, in which the large MR = 8.28 [MATH]% and the remaining [MATH] and [MATH] peaks under 12.2 kbar enable us to identify [MATH] and [MATH] and [MATH] in Ref. [CITATION]) as one pair of electron and hole pockets.', '1412.8298-5-14-6': 'Therefore, the correct Fermi surface topology is very likely that revealed by the ARPES experiments in Ref. [CITATION], which is consistent with our results.', '1412.8298-5-14-7': 'Such a Fermi surface topology is the base to understand the XMR in WTe[MATH].', '1412.8298-5-15-0': 'In summary, we study the quantum oscillations of magnetoresistance under ambient and high pressure for WTe[MATH] single crystals.', '1412.8298-5-15-1': 'Under ambient pressure, four Fermi surfaces are identified by analyzing the SdH oscillations, which are likely two pairs of hole and electron pockets along the [MATH] direction.', '1412.8298-5-15-2': 'With increasing pressure, drastic change of Fermi surface topology and strong suppression of the XMR are observed.', '1412.8298-5-15-3': 'While one pair of hole and electron pockets ([MATH] and [MATH]) persists to high pressure, the other pair of hole and electron pockets ([MATH] and [MATH]) disappears with increasing pressure.', '1412.8298-5-15-4': 'The relative size of the [MATH] and [MATH] pockets decreases with increasing pressure, which may cause the strong suppression of the XMR.', '1412.8298-5-15-5': 'Our results support the scenario that the perfect balance between the electron and hole populations is the origin of the XMR in WTe[MATH].', '1412.8298-5-16-0': 'We thank J. K. Dong, D. L. Feng, J. Jiang, P. S. Wang, and Z. J. Xiang for helpful discussions.', '1412.8298-5-16-1': 'This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200).', '1412.8298-5-16-2': 'The work at Tulane is supported by NSF under Grant No. DMR-1205469.', '1412.8298-5-17-0': 'Note added: After we put this work in arXiv (1412.8298), superconductivity was discovered in WTe[MATH] by applying higher pressure than ours [CITATION], which coincides with the suppression of the XMR.'}

cond-mat-0111501

{'cond-mat-0111501-1-0-0': 'We report on the formation of ultra-cold Na[MATH] molecules using single-photon photoassociation of a Bose-Einstein condensate.', 'cond-mat-0111501-1-0-1': 'The photoassociation rate, linewidth and light shift of the [MATH], [MATH] vibrational level of the [MATH] molecular bound state have been measured.', 'cond-mat-0111501-1-0-2': 'We find that the photoassociation rate constant increases linearly with intensity, even where it is predicted that many-body effects might limit the rate.', 'cond-mat-0111501-1-0-3': 'Our observations are everywhere in good agreement with a two-body theory having no free parameters.', 'cond-mat-0111501-1-1-0': 'Bose-Einstein condensates (BECs) of atomic gases [CITATION] are versatile systems for the experimental study of quantum behavior.', 'cond-mat-0111501-1-1-1': 'Of particular interest are the suggestions for the coherent coupling of a BEC of atoms with a BEC of molecules [CITATION] and the possibility of creating entangled pairs of atoms in the BEC via coupling with molecular levels [CITATION].', 'cond-mat-0111501-1-1-2': 'Two-photon photoassociation processes using stimulated Raman transitions have formed ground state molecules from ground state atoms in a BEC [CITATION] but at very low rates.', 'cond-mat-0111501-1-1-3': 'Here we explore the fundamental upper limits of molecule formation by making them at high rates using the elementary process of single-photon photoassociation.', 'cond-mat-0111501-1-2-0': 'In single-photon photoassociation two atoms collide in the presence of a light field and form an excited state molecule.', 'cond-mat-0111501-1-2-1': 'Photoassociative spectroscopy is used extensively to study collisions between laser cooled atoms [CITATION].', 'cond-mat-0111501-1-2-2': 'Photoassociation in a BEC presents quite a different regime: the collision energies are orders of magnitude lower than in a laser cooled sample (the de Broglie wavelength is as big as the sample) and the densities are higher.', 'cond-mat-0111501-1-2-3': 'This puts us in a regime where many-body effects may be important.', 'cond-mat-0111501-1-3-0': 'We concentrate on a particular photoassociation transition and measure the photoassociation spectra for various intensities and durations of the light pulse.', 'cond-mat-0111501-1-3-1': 'From these we determine the photoassociation rate, lineshape and the shift of the resonance.', 'cond-mat-0111501-1-3-2': 'Finally we examine various limits on the photoassociation rate.', 'cond-mat-0111501-1-4-0': 'Figure [REF] shows the photoassociation process.', 'cond-mat-0111501-1-4-1': 'The molecular level chosen for study is the [MATH], [MATH], rotational-vibrational level of the [MATH] Na[MATH] molecular state, excited from free atoms by a laser frequency of 16913.37(2) cm[MATH] .', 'cond-mat-0111501-1-4-2': 'We chose this level because its detuning from the D[MATH] resonance is far enough (43 cm[MATH]) for atomic absorption to be negligible and because earlier experiments using laser cooled atoms indicated a high photoassociation rate.', 'cond-mat-0111501-1-4-3': 'The lifetime of our excited molecules is about 8.6 ns.', 'cond-mat-0111501-1-4-4': 'Decay of excited molecules into hot atoms or ground state molecules constitutes loss from the condensate.', 'cond-mat-0111501-1-4-5': 'This loss is how we detect photoassociation.', 'cond-mat-0111501-1-5-0': 'We prepare an almost pure condensate of [MATH] sodium atoms in the [MATH] ground state with a peak density of [MATH] cm[MATH].', 'cond-mat-0111501-1-5-1': 'The condensate is held in an anisotropic magnetic TOP [CITATION] trap with oscillation frequencies of [MATH] Hz and corresponding Thomas-Fermi radii of [MATH].', 'cond-mat-0111501-1-6-0': 'To induce photoassociation we illuminate the condensate with a Gaussian laser beam focused to 120 [MATH] FWHM at the condensate.', 'cond-mat-0111501-1-6-1': 'The peak intensity is varied from 50 to 1200 W[MATH]cm[MATH].', 'cond-mat-0111501-1-6-2': 'The polarization is linear and parallel with the rotation ([MATH]) axis of the TOP trap bias field.', 'cond-mat-0111501-1-6-3': 'The light is applied as a square pulse for between 10 and 400 [MATH] with rise and fall times of less than 0.5 [MATH].', 'cond-mat-0111501-1-7-0': 'The condensate number is measured using phase contrast imaging [CITATION], taking two images before and two images after the photoassociation pulse to determine loss.', 'cond-mat-0111501-1-7-1': 'The imaging occurs at 40 ms intervals using a 100 [MATH]s probe pulse from a laser tuned 1.78 GHz to the red of the 3S[MATH]P[MATH] transition.', 'cond-mat-0111501-1-7-2': 'The imaging rate is limited by the readout time of the CCD camera.', 'cond-mat-0111501-1-7-3': 'The photoassociation pulse begins halfway between the second and third images.', 'cond-mat-0111501-1-7-4': 'We use multiple imaging pulses to improve statistics and to partially correct for small losses other than those due to the photoassociation pulse.', 'cond-mat-0111501-1-7-5': 'These losses are typically 4% between images.', 'cond-mat-0111501-1-7-6': 'Once the number of atoms is extracted from the images we calculate [MATH], the fraction of atoms remaining after the photoassociation pulse.', 'cond-mat-0111501-1-8-0': 'Figure [REF] shows a typical photoassociation spectrum.', 'cond-mat-0111501-1-8-1': 'Each point represents a freshly prepared condensate.', 'cond-mat-0111501-1-8-2': 'We use a Fabry-Perot etalon and a reference laser locked to an atomic Na line to measure differences in the photoassociation frequency with a precision of 5 MHz.', 'cond-mat-0111501-1-8-3': 'The laser linewidth is [MATH] MHz.', 'cond-mat-0111501-1-8-4': 'All detunings quoted are relative to the center of the photoassociation line in the low intensity limit.', 'cond-mat-0111501-1-8-5': 'For small trap loss we expect the line to be Lorentzian (in contrast to photoassociation lines in an uncondensed thermal sample where the kinetic energy distribution distorts the line shape [CITATION]).', 'cond-mat-0111501-1-8-6': 'For significant trap loss, as in figure [REF], one must account for the change of the density profile during the photoassociation pulse.', 'cond-mat-0111501-1-9-0': 'A two-body collisional loss process changes the local atomic density as: [MATH], where [MATH] is the intensity and frequency dependent photoassociation rate constant.', 'cond-mat-0111501-1-9-1': 'Because the characteristic time for the motion of the atoms, the trap oscillation period, is long compared to the photoassociation pulse, we assume that the local density only changes due to photoassociation.', 'cond-mat-0111501-1-9-2': 'We then obtain: [EQUATION]', 'cond-mat-0111501-1-9-3': 'The density distribution flattens with time.', 'cond-mat-0111501-1-9-4': 'Spatially integrating equation ([REF]), assuming an initial, parabolic (Thomas-Fermi), density distribution and a uniform intensity [MATH], leads to an expression for the fraction of atoms remaining in the condensate [EQUATION] where [MATH].', 'cond-mat-0111501-1-10-0': 'We use a three parameter fit of equation ([REF]) to the spectra to extract the on-resonance rate constant [MATH], effective line width [MATH] and central frequency [MATH] (for example, the dotted line in figure [REF]).', 'cond-mat-0111501-1-10-1': 'The fit is good.', 'cond-mat-0111501-1-10-2': 'To further verify equation ([REF]) we plot the measured [MATH] as a function of pulse length for [MATH] W[MATH]cm[MATH] and [MATH], along with a one parameter ([MATH]) fit to the data (figure [REF]).', 'cond-mat-0111501-1-10-3': 'The error bars are the fitting uncertainty.', 'cond-mat-0111501-1-11-0': 'By fitting spectra obtained at various intensities we measure [MATH], [MATH], and [MATH].', 'cond-mat-0111501-1-11-1': 'We calculate the unbroadened molecular linewidth of the chosen state to be [MATH] MHz (nearly twice the atomic linewidth).', 'cond-mat-0111501-1-11-2': 'This is in good agreement with the measured linewidth of 19.5(25) MHz in the low intensity limit where it is independent of intensity.', 'cond-mat-0111501-1-11-3': 'At higher intensities we observe broadening with a maximum linewidth of 60 MHz at 1 kW[MATH]cm[MATH].', 'cond-mat-0111501-1-11-4': 'Homogeneous power broadening is calculated to be three orders of magnitude too low to explain this width.', 'cond-mat-0111501-1-11-5': 'It is, however, partially explained by differential light shifts of the unresolved molecular hyperfine states.', 'cond-mat-0111501-1-11-6': 'These states are calculated to be split by less than 1 MHz at low intensities and about 30 MHz at our maximum intensity.', 'cond-mat-0111501-1-11-7': 'Another possible contribution is the inhomogeneity of the photoassociation beam intensity combined with the large light shift (discussed below).', 'cond-mat-0111501-1-11-8': 'Variations due to either local spatial inhomogeneity (e.g. interference fringes) or displacement of the cloud from the center of the Gaussian beam could account for the extra width.', 'cond-mat-0111501-1-11-9': 'Assuming these inhomogeneous broadening mechanisms do not change the area of the line (verified by a simulation), we take the on-resonance photoassociation rate constant to be [MATH].', 'cond-mat-0111501-1-12-0': 'Figure [REF] shows corrected and uncorrected [MATH] as a function of intensity (for various pulse lengths).', 'cond-mat-0111501-1-12-1': 'The error bars are the fitting uncertainties.', 'cond-mat-0111501-1-12-2': 'Once we correct for the inhomogeneous broadening we get a linear dependence on intensity with a slope of [MATH].', 'cond-mat-0111501-1-12-3': 'The uncertainties are due, respectively, to fitting and to the combined uncertainties in the measurement of the intensity and density.', 'cond-mat-0111501-1-12-4': 'For intensities above 1.2 kW[MATH]cm[MATH], which we could only achieve by more tightly focusing the photoassociation laser, atomic dipole forces significantly perturb the condensate, thwarting meaningful measurements.', 'cond-mat-0111501-1-12-5': 'A coupled-channels, two-body scattering calculation with no adjustable parameters [CITATION] yields a photoassociation rate constant of [MATH] for our range of intensities.', 'cond-mat-0111501-1-12-6': 'This includes a factor of 2 decrease relative to [MATH] for a non-condensed gas and agrees well with our experimental result.', 'cond-mat-0111501-1-13-0': 'We study the photoassociation light shift, previously observed in a non-condensed gas [CITATION], in a set of experiments where the total fluence (intensity [MATH] pulse length) of the pulse was kept constant, to maintain the depth of the photoassociation dip in an easily observable regime.', 'cond-mat-0111501-1-13-1': 'The results are shown in figure [REF].', 'cond-mat-0111501-1-13-2': 'The measured light shift is [MATH], which leads at high intensity to a shift large compared to the linewidth.', 'cond-mat-0111501-1-13-3': 'The principal contribution to the uncertainty is the intensity calibration.', 'cond-mat-0111501-1-13-4': 'During preparation of this work we became aware of similar results in lithium [CITATION].', 'cond-mat-0111501-1-14-0': 'While the strength of the photoassociation resonance is dominated by s-wave scattering, the dominant contribution to the light shift is due to a d-wave shape resonance.', 'cond-mat-0111501-1-14-1': 'A theoretical calculation of the light shifts using equation (3.7) of reference [CITATION], including the effect of the d-wave shape resonance embedded in the continuum, gives an average value of [MATH] with a spread of [MATH] due to the hyperfine structure.', 'cond-mat-0111501-1-15-0': 'We now consider the upper limit to the photoassociation rate constant K [this implies a lower limit on the photoassociation time [MATH]].', 'cond-mat-0111501-1-15-1': 'If one uses a semi-classical theory that is commonly applied to collisions of laser-cooled atoms [CITATION] then [MATH], where [MATH] is the Condon radius (see figure [REF]) and [MATH] is the probability of a photoassociative transition with a maximum value of 1.', 'cond-mat-0111501-1-15-2': 'If we take the relative velocity [MATH] of the atoms to be [MATH] mm[MATH]s[MATH], where [MATH] is the atomic mass, then the maximum photoassociation rate constant is four orders of magnitude lower than our highest measured value.', 'cond-mat-0111501-1-15-3': 'This reveals the inadequacy of a semi-classical approach, which fails to take into account threshold laws [CITATION].', 'cond-mat-0111501-1-16-0': 'Quantum theories for the photoassociation rate constant can be compared by expressing [MATH] as [MATH], where [MATH] is a characteristic length.', 'cond-mat-0111501-1-16-1': 'Two-body s-wave scattering theory for a BEC gives [MATH] where [MATH] is the relative collision momentum and [MATH] is the [MATH]-matrix element for atom loss via photoassociation.', 'cond-mat-0111501-1-16-2': 'References [CITATION] show that, on resonance, [EQUATION] where [MATH] is the Fermi-golden-rule stimulated-decay width of the excited molecular state [MATH] due to the optical coupling [MATH] with the colliding atoms.', 'cond-mat-0111501-1-16-3': 'Since [MATH] as [MATH], and [MATH] in our range of power and collision energy, [MATH] is independent of [MATH].', 'cond-mat-0111501-1-16-4': 'We calculate [MATH] MHz.', 'cond-mat-0111501-1-16-5': '[MATH] is linear in [MATH] for our experimental conditions and [MATH] is calculated to be 24 nm[MATH].', 'cond-mat-0111501-1-16-6': 'This gives the above-quoted rate constant in good agreement with the experiment.', 'cond-mat-0111501-1-16-7': 'In our power range, [MATH] can be significantly larger than the Condon point for the transition, 2.0 nm.', 'cond-mat-0111501-1-16-8': 'Note that equation ([REF]) shows that [MATH] will saturate with increasing [MATH] and decrease for sufficiently large [MATH].', 'cond-mat-0111501-1-17-0': 'The upper limit to the two-body quantum [MATH] is the unitarity limit where [MATH] so [MATH], where [MATH] is the de Broglie wavelength.', 'cond-mat-0111501-1-17-1': 'Since [MATH] is on the order of the BEC size [MATH] so that our experiment is well below the unitarity constraint.', 'cond-mat-0111501-1-18-0': 'Recent many-body theoretical work [CITATION] has suggested an upper limit to [MATH] in a BEC of [MATH], where [MATH] and [MATH] is the mean distance between particles.', 'cond-mat-0111501-1-18-1': 'One might question if two-body scattering methods are applicable at densities where [MATH] becomes larger than [MATH].', 'cond-mat-0111501-1-18-2': 'At our maximum density [MATH] nm, so [MATH] at our highest intensities.', 'cond-mat-0111501-1-18-3': 'This is the regime where one might expect two-body theory to start to fail.', 'cond-mat-0111501-1-18-4': 'Nevertheless the linearity of [MATH] at our highest intensities (figure [REF]) suggests that two-body theory continues to be valid.', 'cond-mat-0111501-1-19-0': 'Larger values of [MATH] might be accessible by a modification to our experimental design.', 'cond-mat-0111501-1-19-1': 'We can use the atomic dipole force (which currently limits our ability to use high intensities) to our advantage by trapping the atoms with the photoassociation laser.', 'cond-mat-0111501-1-19-2': 'Without changing the atomic dipole forces, the laser can be suddenly brought from far off molecular resonance to on molecular resonance to induce photoassociation.', 'cond-mat-0111501-1-19-3': 'Difficulties due to the molecular light shift might be reduced by finding a transition with a smaller light shift.', 'cond-mat-0111501-1-20-0': 'In conclusion, we have measured the single-photon photoassociation in a BEC, in good agreement with two-body theory.', 'cond-mat-0111501-1-20-1': 'This agreement represents a confirmation of the factor-of-two reduction for a two-body inelastic process in a BEC.', 'cond-mat-0111501-1-20-2': 'The characteristic time for photoassociation is as short as 5 [MATH]s, much shorter than the 100 [MATH]s to traverse the mean distance between atoms, another demonstration of the extreme quantum nature of the collisions.', 'cond-mat-0111501-1-20-3': 'Our largest rate is still much smaller than the unitarity limit, but is about equal to a limit suggested on the basis of many-body effects; we do not see evidence that the rate saturates at this limit.', 'cond-mat-0111501-1-21-0': 'We acknowledge funding support from the US Office of Naval Research and NASA.', 'cond-mat-0111501-1-21-1': 'J.H.D. and H.H. acknowledge funding from the Alexander von Humboldt foundation.', 'cond-mat-0111501-1-21-2': 'A.B. was partially supported by DGA (France).'}

{'cond-mat-0111501-2-0-0': 'We report on the formation of ultra-cold Na[MATH] molecules using single-photon photoassociation of a Bose-Einstein condensate.', 'cond-mat-0111501-2-0-1': 'The photoassociation rate, linewidth and light shift of the [MATH], [MATH] vibrational level of the [MATH] molecular bound state have been measured.', 'cond-mat-0111501-2-0-2': 'We find that the photoassociation rate constant increases linearly with intensity, even where it is predicted that many-body effects might limit the rate.', 'cond-mat-0111501-2-0-3': 'Our observations are everywhere in good agreement with a two-body theory having no free parameters.', 'cond-mat-0111501-2-1-0': 'Bose-Einstein condensates (BECs) of atomic gases [CITATION] are versatile systems for the experimental study of quantum behavior.', 'cond-mat-0111501-2-1-1': 'Of particular interest are the suggestions for the coherent coupling of a BEC of atoms with a BEC of molecules [CITATION] and the possibility of creating entangled pairs of atoms in the BEC via coupling with molecular levels [CITATION].', 'cond-mat-0111501-2-1-2': 'Two-photon photoassociation processes using stimulated Raman transitions have formed ground state molecules from ground state atoms in a BEC [CITATION] but at very low rates.', 'cond-mat-0111501-2-1-3': 'Here we explore the fundamental upper limits of molecule formation by making them at high rates using the elementary process of single-photon photoassociation.', 'cond-mat-0111501-2-2-0': 'In single-photon photoassociation two atoms collide in the presence of a light field and form an excited state molecule.', 'cond-mat-0111501-2-2-1': 'Photoassociative spectroscopy is used extensively to study collisions between laser cooled atoms [CITATION].', 'cond-mat-0111501-2-2-2': 'Photoassociation in a BEC presents quite a different regime: the collision energies are orders of magnitude lower than in a laser cooled sample (the de Broglie wavelength is as big as the sample) and the densities are higher.', 'cond-mat-0111501-2-2-3': 'This puts us in a regime where many-body effects may be important.', 'cond-mat-0111501-2-3-0': 'We concentrate on a particular photoassociation transition and measure the photoassociation spectra for various intensities and durations of the light pulse.', 'cond-mat-0111501-2-3-1': 'From these we determine the photoassociation rate, lineshape and the shift of the resonance.', 'cond-mat-0111501-2-3-2': 'Finally we examine various limits on the photoassociation rate.', 'cond-mat-0111501-2-4-0': 'Figure [REF] shows the photoassociation process.', 'cond-mat-0111501-2-4-1': 'The molecular level chosen for study is the [MATH], [MATH], rotational-vibrational level of the [MATH] Na[MATH] molecular state, excited from free atoms by a laser frequency of 16913.37(2) cm[MATH] .', 'cond-mat-0111501-2-4-2': 'We chose this level because its detuning from the D[MATH] resonance is far enough (43 cm[MATH]) for atomic absorption to be negligible and because earlier experiments using laser cooled atoms indicated a high photoassociation rate.', 'cond-mat-0111501-2-4-3': 'The lifetime of our excited molecules is about 8.6 ns.', 'cond-mat-0111501-2-4-4': 'Decay of excited molecules into hot atoms or ground state molecules constitutes loss from the condensate.', 'cond-mat-0111501-2-4-5': 'This loss is how we detect photoassociation.', 'cond-mat-0111501-2-5-0': 'We prepare an almost pure condensate of [MATH] sodium atoms in the [MATH] ground state with a peak density of [MATH] cm[MATH].', 'cond-mat-0111501-2-5-1': 'The condensate is held in an anisotropic magnetic TOP [CITATION] trap with oscillation frequencies of [MATH] Hz and corresponding Thomas-Fermi radii of [MATH].', 'cond-mat-0111501-2-6-0': 'To induce photoassociation we illuminate the condensate with a Gaussian laser beam focused to 120 [MATH] FWHM at the condensate.', 'cond-mat-0111501-2-6-1': 'The peak intensity is varied from 50 to 1200 W[MATH]cm[MATH].', 'cond-mat-0111501-2-6-2': 'The polarization is linear and parallel with the rotation ([MATH]) axis of the TOP trap bias field.', 'cond-mat-0111501-2-6-3': 'The light is applied as a square pulse for between 10 and 400 [MATH] with rise and fall times of less than 0.5 [MATH].', 'cond-mat-0111501-2-7-0': 'The condensate number is measured using phase contrast imaging [CITATION], taking two images before and two images after the photoassociation pulse to determine loss.', 'cond-mat-0111501-2-7-1': 'The imaging occurs at 40 ms intervals using a 100 [MATH]s probe pulse from a laser tuned 1.78 GHz to the red of the 3S[MATH]P[MATH] transition.', 'cond-mat-0111501-2-7-2': 'The imaging rate is limited by the readout time of the CCD camera.', 'cond-mat-0111501-2-7-3': 'The photoassociation pulse begins halfway between the second and third images.', 'cond-mat-0111501-2-7-4': 'We use multiple imaging pulses to improve statistics and to partially correct for small losses other than those due to the photoassociation pulse.', 'cond-mat-0111501-2-7-5': 'These losses are typically 4% between images.', 'cond-mat-0111501-2-7-6': 'Once the number of atoms is extracted from the images we calculate [MATH], the fraction of atoms remaining after the photoassociation pulse.', 'cond-mat-0111501-2-8-0': 'Figure [REF] shows a typical photoassociation spectrum.', 'cond-mat-0111501-2-8-1': 'Each point represents a freshly prepared condensate.', 'cond-mat-0111501-2-8-2': 'We use a Fabry-Perot etalon and a reference laser locked to an atomic Na line to measure differences in the photoassociation frequency with a precision of 5 MHz.', 'cond-mat-0111501-2-8-3': 'The laser linewidth is [MATH] MHz.', 'cond-mat-0111501-2-8-4': 'All detunings quoted are relative to the center of the photoassociation line in the low intensity limit.', 'cond-mat-0111501-2-8-5': 'For small trap loss we expect the line to be Lorentzian (in contrast to photoassociation lines in an uncondensed thermal sample where the kinetic energy distribution distorts the line shape [CITATION]).', 'cond-mat-0111501-2-8-6': 'For significant trap loss, as in figure [REF], one must account for the change of the density profile during the photoassociation pulse.', 'cond-mat-0111501-2-9-0': 'A two-body collisional loss process changes the local atomic density as: [MATH], where [MATH] is the intensity and frequency dependent photoassociation rate constant.', 'cond-mat-0111501-2-9-1': 'Because the characteristic time for the motion of the atoms, the trap oscillation period, is long compared to the photoassociation pulse, we assume that the local density only changes due to photoassociation.', 'cond-mat-0111501-2-9-2': 'We then obtain: [EQUATION]', 'cond-mat-0111501-2-9-3': 'The density distribution flattens with time.', 'cond-mat-0111501-2-9-4': 'Spatially integrating equation ([REF]), assuming an initial, parabolic (Thomas-Fermi), density distribution and a uniform intensity [MATH], leads to an expression for the fraction of atoms remaining in the condensate [EQUATION] where [MATH].', 'cond-mat-0111501-2-10-0': 'We use a three parameter fit of equation ([REF]) to the spectra to extract the on-resonance rate constant [MATH], effective line width [MATH] and central frequency [MATH] (for example, the dotted line in figure [REF]).', 'cond-mat-0111501-2-10-1': 'The fit is good.', 'cond-mat-0111501-2-10-2': 'To further verify equation ([REF]) we plot the measured [MATH] as a function of pulse length for [MATH] W[MATH]cm[MATH] and [MATH], along with a one parameter ([MATH]) fit to the data (figure [REF]).', 'cond-mat-0111501-2-10-3': 'The error bars are the fitting uncertainty.', 'cond-mat-0111501-2-11-0': 'By fitting spectra obtained at various intensities we measure [MATH], [MATH], and [MATH].', 'cond-mat-0111501-2-11-1': 'We calculate the unbroadened molecular linewidth of the chosen state to be [MATH] MHz (nearly twice the atomic linewidth).', 'cond-mat-0111501-2-11-2': 'This is in good agreement with the measured linewidth of 19.5(25) MHz in the low intensity limit where it is independent of intensity.', 'cond-mat-0111501-2-11-3': 'At higher intensities we observe broadening with a maximum linewidth of 60 MHz at 1 kW[MATH]cm[MATH].', 'cond-mat-0111501-2-11-4': 'Homogeneous power broadening is calculated to be three orders of magnitude too low to explain this width.', 'cond-mat-0111501-2-11-5': 'It is, however, partially explained by differential light shifts of the unresolved molecular hyperfine states.', 'cond-mat-0111501-2-11-6': 'These states are calculated to be split by less than 1 MHz at low intensities and about 30 MHz at our maximum intensity.', 'cond-mat-0111501-2-11-7': 'Another possible contribution is the inhomogeneity of the photoassociation beam intensity combined with the large light shift (discussed below).', 'cond-mat-0111501-2-11-8': 'Variations due to either local spatial inhomogeneity (e.g. interference fringes) or displacement of the cloud from the center of the Gaussian beam could account for the extra width.', 'cond-mat-0111501-2-11-9': 'Assuming these inhomogeneous broadening mechanisms do not change the area of the line (verified by a simulation), we take the on-resonance photoassociation rate constant to be [MATH].', 'cond-mat-0111501-2-12-0': 'Figure [REF] shows corrected and uncorrected [MATH] as a function of intensity (for various pulse lengths).', 'cond-mat-0111501-2-12-1': 'The error bars are the fitting uncertainties.', 'cond-mat-0111501-2-12-2': 'Once we correct for the inhomogeneous broadening we get a linear dependence on intensity with a slope of [MATH].', 'cond-mat-0111501-2-12-3': 'The uncertainties are due, respectively, to fitting and to the combined uncertainties in the measurement of the intensity and density.', 'cond-mat-0111501-2-12-4': 'For intensities above 1.2 kW[MATH]cm[MATH], which we could only achieve by more tightly focusing the photoassociation laser, atomic dipole forces significantly perturb the condensate, thwarting meaningful measurements.', 'cond-mat-0111501-2-12-5': 'A coupled-channels, two-body scattering calculation with no adjustable parameters [CITATION] yields a photoassociation rate constant of [MATH] for our range of intensities.', 'cond-mat-0111501-2-12-6': 'This includes a factor of 2 decrease relative to [MATH] for a non-condensed gas and agrees well with our experimental result.', 'cond-mat-0111501-2-13-0': 'We study the photoassociation light shift, previously observed in a non-condensed gas [CITATION], in a set of experiments where the total fluence (intensity [MATH] pulse length) of the pulse was kept constant, to maintain the depth of the photoassociation dip in an easily observable regime.', 'cond-mat-0111501-2-13-1': 'The results are shown in figure [REF].', 'cond-mat-0111501-2-13-2': 'The measured light shift is [MATH], which leads at high intensity to a shift large compared to the linewidth.', 'cond-mat-0111501-2-13-3': 'The principal contribution to the uncertainty is the intensity calibration.', 'cond-mat-0111501-2-13-4': 'During preparation of this work we became aware of similar results in lithium [CITATION].', 'cond-mat-0111501-2-14-0': 'While the strength of the photoassociation resonance is dominated by s-wave scattering, the dominant contribution to the light shift is due to a d-wave shape resonance.', 'cond-mat-0111501-2-14-1': 'A theoretical calculation of the light shifts using equation (3.7) of reference [CITATION], including the effect of the d-wave shape resonance embedded in the continuum, gives an average value of [MATH] with a spread of [MATH] due to the hyperfine structure.', 'cond-mat-0111501-2-15-0': 'We now consider the upper limit to the photoassociation rate constant K [this implies a lower limit on the photoassociation time [MATH]].', 'cond-mat-0111501-2-15-1': 'If one uses a semi-classical theory that is commonly applied to collisions of laser-cooled atoms [CITATION] then [MATH], where [MATH] is the Condon radius (see figure [REF]) and [MATH] is the probability of a photoassociative transition with a maximum value of 1.', 'cond-mat-0111501-2-15-2': 'If we take the relative velocity [MATH] of the atoms to be [MATH] mm[MATH]s[MATH], where [MATH] is the atomic mass, then the maximum photoassociation rate constant is four orders of magnitude lower than our highest measured value.', 'cond-mat-0111501-2-15-3': 'This reveals the inadequacy of a semi-classical approach, which fails to take into account threshold laws [CITATION].', 'cond-mat-0111501-2-16-0': 'Quantum theories for the photoassociation rate constant can be compared by expressing [MATH] as [MATH], where [MATH] is a characteristic length.', 'cond-mat-0111501-2-16-1': 'Two-body s-wave scattering theory for a BEC gives [MATH] where [MATH] is the relative collision momentum and [MATH] is the [MATH]-matrix element for atom loss via photoassociation.', 'cond-mat-0111501-2-16-2': 'References [CITATION] show that, on resonance, [EQUATION] where [MATH] is the Fermi-golden-rule stimulated-decay width of the excited molecular state [MATH] due to the optical coupling [MATH] with the colliding atoms.', 'cond-mat-0111501-2-16-3': 'Since [MATH] as [MATH], and [MATH] in our range of power and collision energy, [MATH] is independent of [MATH].', 'cond-mat-0111501-2-16-4': 'We calculate [MATH] MHz.', 'cond-mat-0111501-2-16-5': '[MATH] is linear in [MATH] for our experimental conditions and [MATH] is calculated to be 24 nm[MATH].', 'cond-mat-0111501-2-16-6': 'This gives the above-quoted rate constant in good agreement with the experiment.', 'cond-mat-0111501-2-16-7': 'In our power range, [MATH] can be significantly larger than the Condon point for the transition, 2.0 nm.', 'cond-mat-0111501-2-16-8': 'Note that equation ([REF]) shows that [MATH] will saturate with increasing [MATH] and decrease for sufficiently large [MATH].', 'cond-mat-0111501-2-17-0': 'The upper limit to the two-body quantum [MATH] is the unitarity limit where [MATH] so [MATH], where [MATH] is the de Broglie wavelength.', 'cond-mat-0111501-2-17-1': 'Since [MATH] is on the order of the BEC size [MATH] so that our experiment is well below the unitarity constraint.', 'cond-mat-0111501-2-18-0': 'Recent many-body theoretical work [CITATION] has suggested an upper limit to [MATH] in a BEC of [MATH], where [MATH] and [MATH] is the mean distance between particles.', 'cond-mat-0111501-2-18-1': 'One might question if two-body scattering methods are applicable at densities where [MATH] becomes larger than [MATH].', 'cond-mat-0111501-2-18-2': 'At our maximum density [MATH] nm, so [MATH] at our highest intensities.', 'cond-mat-0111501-2-18-3': 'This is the regime where one might expect two-body theory to start to fail.', 'cond-mat-0111501-2-18-4': 'Nevertheless the linearity of [MATH] at our highest intensities (figure [REF]) suggests that two-body theory continues to be valid.', 'cond-mat-0111501-2-19-0': 'Larger values of [MATH] might be accessible by a modification to our experimental design.', 'cond-mat-0111501-2-19-1': 'We can use the atomic dipole force (which currently limits our ability to use high intensities) to our advantage by trapping the atoms with the photoassociation laser.', 'cond-mat-0111501-2-19-2': 'Without changing the atomic dipole forces, the laser can be suddenly brought from far off molecular resonance to on molecular resonance to induce photoassociation.', 'cond-mat-0111501-2-19-3': 'Difficulties due to the molecular light shift might be reduced by finding a transition with a smaller light shift.', 'cond-mat-0111501-2-20-0': 'In conclusion, we have measured the single-photon photoassociation in a BEC, in good agreement with two-body theory.', 'cond-mat-0111501-2-20-1': 'This agreement represents a confirmation of the factor-of-two reduction for a two-body inelastic process in a BEC.', 'cond-mat-0111501-2-20-2': 'The characteristic time for photoassociation is as short as 5 [MATH]s, much shorter than the 100 [MATH]s to traverse the mean distance between atoms, another demonstration of the extreme quantum nature of the collisions.', 'cond-mat-0111501-2-20-3': 'Our largest rate is still much smaller than the unitarity limit, but is about equal to a limit suggested on the basis of many-body effects; we do not see evidence that the rate saturates at this limit.', 'cond-mat-0111501-2-21-0': 'We acknowledge funding support from the US Office of Naval Research and NASA.', 'cond-mat-0111501-2-21-1': 'J.H.D. and H.H. acknowledge funding from the Alexander von Humboldt foundation.', 'cond-mat-0111501-2-21-2': 'A.B. was partially supported by DGA (France).'}

[['cond-mat-0111501-1-17-0', 'cond-mat-0111501-2-17-0'], ['cond-mat-0111501-1-17-1', 'cond-mat-0111501-2-17-1'], ['cond-mat-0111501-1-3-0', 'cond-mat-0111501-2-3-0'], ['cond-mat-0111501-1-3-1', 'cond-mat-0111501-2-3-1'], ['cond-mat-0111501-1-3-2', 'cond-mat-0111501-2-3-2'], ['cond-mat-0111501-1-20-0', 'cond-mat-0111501-2-20-0'], ['cond-mat-0111501-1-20-1', 'cond-mat-0111501-2-20-1'], ['cond-mat-0111501-1-20-2', 'cond-mat-0111501-2-20-2'], ['cond-mat-0111501-1-20-3', 'cond-mat-0111501-2-20-3'], ['cond-mat-0111501-1-4-0', 'cond-mat-0111501-2-4-0'], ['cond-mat-0111501-1-4-1', 'cond-mat-0111501-2-4-1'], ['cond-mat-0111501-1-4-2', 'cond-mat-0111501-2-4-2'], ['cond-mat-0111501-1-4-3', 'cond-mat-0111501-2-4-3'], ['cond-mat-0111501-1-4-4', 'cond-mat-0111501-2-4-4'], ['cond-mat-0111501-1-4-5', 'cond-mat-0111501-2-4-5'], ['cond-mat-0111501-1-6-0', 'cond-mat-0111501-2-6-0'], ['cond-mat-0111501-1-6-1', 'cond-mat-0111501-2-6-1'], ['cond-mat-0111501-1-6-2', 'cond-mat-0111501-2-6-2'], ['cond-mat-0111501-1-6-3', 'cond-mat-0111501-2-6-3'], ['cond-mat-0111501-1-0-0', 'cond-mat-0111501-2-0-0'], ['cond-mat-0111501-1-0-1', 'cond-mat-0111501-2-0-1'], ['cond-mat-0111501-1-0-2', 'cond-mat-0111501-2-0-2'], ['cond-mat-0111501-1-0-3', 'cond-mat-0111501-2-0-3'], ['cond-mat-0111501-1-12-0', 'cond-mat-0111501-2-12-0'], ['cond-mat-0111501-1-12-1', 'cond-mat-0111501-2-12-1'], ['cond-mat-0111501-1-12-2', 'cond-mat-0111501-2-12-2'], ['cond-mat-0111501-1-12-3', 'cond-mat-0111501-2-12-3'], ['cond-mat-0111501-1-12-4', 'cond-mat-0111501-2-12-4'], ['cond-mat-0111501-1-12-5', 'cond-mat-0111501-2-12-5'], ['cond-mat-0111501-1-12-6', 'cond-mat-0111501-2-12-6'], ['cond-mat-0111501-1-14-0', 'cond-mat-0111501-2-14-0'], ['cond-mat-0111501-1-14-1', 'cond-mat-0111501-2-14-1'], ['cond-mat-0111501-1-18-0', 'cond-mat-0111501-2-18-0'], ['cond-mat-0111501-1-18-1', 'cond-mat-0111501-2-18-1'], ['cond-mat-0111501-1-18-2', 'cond-mat-0111501-2-18-2'], ['cond-mat-0111501-1-18-3', 'cond-mat-0111501-2-18-3'], ['cond-mat-0111501-1-18-4', 'cond-mat-0111501-2-18-4'], ['cond-mat-0111501-1-7-0', 'cond-mat-0111501-2-7-0'], ['cond-mat-0111501-1-7-1', 'cond-mat-0111501-2-7-1'], ['cond-mat-0111501-1-7-2', 'cond-mat-0111501-2-7-2'], ['cond-mat-0111501-1-7-3', 'cond-mat-0111501-2-7-3'], ['cond-mat-0111501-1-7-4', 'cond-mat-0111501-2-7-4'], ['cond-mat-0111501-1-7-5', 'cond-mat-0111501-2-7-5'], ['cond-mat-0111501-1-7-6', 'cond-mat-0111501-2-7-6'], ['cond-mat-0111501-1-10-0', 'cond-mat-0111501-2-10-0'], ['cond-mat-0111501-1-10-1', 'cond-mat-0111501-2-10-1'], ['cond-mat-0111501-1-10-2', 'cond-mat-0111501-2-10-2'], ['cond-mat-0111501-1-10-3', 'cond-mat-0111501-2-10-3'], ['cond-mat-0111501-1-9-0', 'cond-mat-0111501-2-9-0'], ['cond-mat-0111501-1-9-1', 'cond-mat-0111501-2-9-1'], ['cond-mat-0111501-1-9-2', 'cond-mat-0111501-2-9-2'], ['cond-mat-0111501-1-9-3', 'cond-mat-0111501-2-9-3'], ['cond-mat-0111501-1-9-4', 'cond-mat-0111501-2-9-4'], ['cond-mat-0111501-1-2-0', 'cond-mat-0111501-2-2-0'], ['cond-mat-0111501-1-2-1', 'cond-mat-0111501-2-2-1'], ['cond-mat-0111501-1-2-2', 'cond-mat-0111501-2-2-2'], ['cond-mat-0111501-1-2-3', 'cond-mat-0111501-2-2-3'], ['cond-mat-0111501-1-11-0', 'cond-mat-0111501-2-11-0'], ['cond-mat-0111501-1-11-1', 'cond-mat-0111501-2-11-1'], ['cond-mat-0111501-1-11-2', 'cond-mat-0111501-2-11-2'], ['cond-mat-0111501-1-11-3', 'cond-mat-0111501-2-11-3'], ['cond-mat-0111501-1-11-4', 'cond-mat-0111501-2-11-4'], ['cond-mat-0111501-1-11-5', 'cond-mat-0111501-2-11-5'], ['cond-mat-0111501-1-11-6', 'cond-mat-0111501-2-11-6'], ['cond-mat-0111501-1-11-7', 'cond-mat-0111501-2-11-7'], ['cond-mat-0111501-1-11-8', 'cond-mat-0111501-2-11-8'], ['cond-mat-0111501-1-11-9', 'cond-mat-0111501-2-11-9'], ['cond-mat-0111501-1-15-0', 'cond-mat-0111501-2-15-0'], ['cond-mat-0111501-1-15-1', 'cond-mat-0111501-2-15-1'], ['cond-mat-0111501-1-15-2', 'cond-mat-0111501-2-15-2'], ['cond-mat-0111501-1-15-3', 'cond-mat-0111501-2-15-3'], ['cond-mat-0111501-1-8-0', 'cond-mat-0111501-2-8-0'], ['cond-mat-0111501-1-8-1', 'cond-mat-0111501-2-8-1'], ['cond-mat-0111501-1-8-2', 'cond-mat-0111501-2-8-2'], ['cond-mat-0111501-1-8-3', 'cond-mat-0111501-2-8-3'], ['cond-mat-0111501-1-8-4', 'cond-mat-0111501-2-8-4'], ['cond-mat-0111501-1-8-5', 'cond-mat-0111501-2-8-5'], ['cond-mat-0111501-1-8-6', 'cond-mat-0111501-2-8-6'], ['cond-mat-0111501-1-21-0', 'cond-mat-0111501-2-21-0'], ['cond-mat-0111501-1-21-1', 'cond-mat-0111501-2-21-1'], ['cond-mat-0111501-1-21-2', 'cond-mat-0111501-2-21-2'], ['cond-mat-0111501-1-16-0', 'cond-mat-0111501-2-16-0'], ['cond-mat-0111501-1-16-1', 'cond-mat-0111501-2-16-1'], ['cond-mat-0111501-1-16-2', 'cond-mat-0111501-2-16-2'], ['cond-mat-0111501-1-16-3', 'cond-mat-0111501-2-16-3'], ['cond-mat-0111501-1-16-4', 'cond-mat-0111501-2-16-4'], ['cond-mat-0111501-1-16-5', 'cond-mat-0111501-2-16-5'], ['cond-mat-0111501-1-16-6', 'cond-mat-0111501-2-16-6'], ['cond-mat-0111501-1-16-7', 'cond-mat-0111501-2-16-7'], ['cond-mat-0111501-1-16-8', 'cond-mat-0111501-2-16-8'], ['cond-mat-0111501-1-1-0', 'cond-mat-0111501-2-1-0'], ['cond-mat-0111501-1-1-1', 'cond-mat-0111501-2-1-1'], ['cond-mat-0111501-1-1-2', 'cond-mat-0111501-2-1-2'], ['cond-mat-0111501-1-1-3', 'cond-mat-0111501-2-1-3'], ['cond-mat-0111501-1-13-0', 'cond-mat-0111501-2-13-0'], ['cond-mat-0111501-1-13-1', 'cond-mat-0111501-2-13-1'], ['cond-mat-0111501-1-13-2', 'cond-mat-0111501-2-13-2'], ['cond-mat-0111501-1-13-3', 'cond-mat-0111501-2-13-3'], ['cond-mat-0111501-1-13-4', 'cond-mat-0111501-2-13-4'], ['cond-mat-0111501-1-19-0', 'cond-mat-0111501-2-19-0'], ['cond-mat-0111501-1-19-1', 'cond-mat-0111501-2-19-1'], ['cond-mat-0111501-1-19-2', 'cond-mat-0111501-2-19-2'], ['cond-mat-0111501-1-19-3', 'cond-mat-0111501-2-19-3'], ['cond-mat-0111501-1-5-0', 'cond-mat-0111501-2-5-0'], ['cond-mat-0111501-1-5-1', 'cond-mat-0111501-2-5-1']]

[['cond-mat-0111501-1-17-0', 'cond-mat-0111501-2-17-0'], ['cond-mat-0111501-1-17-1', 'cond-mat-0111501-2-17-1'], ['cond-mat-0111501-1-3-0', 'cond-mat-0111501-2-3-0'], ['cond-mat-0111501-1-3-1', 'cond-mat-0111501-2-3-1'], ['cond-mat-0111501-1-3-2', 'cond-mat-0111501-2-3-2'], ['cond-mat-0111501-1-20-0', 'cond-mat-0111501-2-20-0'], ['cond-mat-0111501-1-20-1', 'cond-mat-0111501-2-20-1'], ['cond-mat-0111501-1-20-2', 'cond-mat-0111501-2-20-2'], ['cond-mat-0111501-1-20-3', 'cond-mat-0111501-2-20-3'], ['cond-mat-0111501-1-4-0', 'cond-mat-0111501-2-4-0'], ['cond-mat-0111501-1-4-1', 'cond-mat-0111501-2-4-1'], ['cond-mat-0111501-1-4-2', 'cond-mat-0111501-2-4-2'], ['cond-mat-0111501-1-4-3', 'cond-mat-0111501-2-4-3'], ['cond-mat-0111501-1-4-4', 'cond-mat-0111501-2-4-4'], ['cond-mat-0111501-1-4-5', 'cond-mat-0111501-2-4-5'], ['cond-mat-0111501-1-6-0', 'cond-mat-0111501-2-6-0'], ['cond-mat-0111501-1-6-1', 'cond-mat-0111501-2-6-1'], ['cond-mat-0111501-1-6-2', 'cond-mat-0111501-2-6-2'], ['cond-mat-0111501-1-6-3', 'cond-mat-0111501-2-6-3'], ['cond-mat-0111501-1-0-0', 'cond-mat-0111501-2-0-0'], ['cond-mat-0111501-1-0-1', 'cond-mat-0111501-2-0-1'], ['cond-mat-0111501-1-0-2', 'cond-mat-0111501-2-0-2'], ['cond-mat-0111501-1-0-3', 'cond-mat-0111501-2-0-3'], ['cond-mat-0111501-1-12-0', 'cond-mat-0111501-2-12-0'], ['cond-mat-0111501-1-12-1', 'cond-mat-0111501-2-12-1'], ['cond-mat-0111501-1-12-2', 'cond-mat-0111501-2-12-2'], ['cond-mat-0111501-1-12-3', 'cond-mat-0111501-2-12-3'], ['cond-mat-0111501-1-12-4', 'cond-mat-0111501-2-12-4'], ['cond-mat-0111501-1-12-5', 'cond-mat-0111501-2-12-5'], ['cond-mat-0111501-1-12-6', 'cond-mat-0111501-2-12-6'], ['cond-mat-0111501-1-14-0', 'cond-mat-0111501-2-14-0'], ['cond-mat-0111501-1-14-1', 'cond-mat-0111501-2-14-1'], ['cond-mat-0111501-1-18-0', 'cond-mat-0111501-2-18-0'], ['cond-mat-0111501-1-18-1', 'cond-mat-0111501-2-18-1'], ['cond-mat-0111501-1-18-2', 'cond-mat-0111501-2-18-2'], ['cond-mat-0111501-1-18-3', 'cond-mat-0111501-2-18-3'], ['cond-mat-0111501-1-18-4', 'cond-mat-0111501-2-18-4'], ['cond-mat-0111501-1-7-0', 'cond-mat-0111501-2-7-0'], ['cond-mat-0111501-1-7-1', 'cond-mat-0111501-2-7-1'], ['cond-mat-0111501-1-7-2', 'cond-mat-0111501-2-7-2'], ['cond-mat-0111501-1-7-3', 'cond-mat-0111501-2-7-3'], ['cond-mat-0111501-1-7-4', 'cond-mat-0111501-2-7-4'], ['cond-mat-0111501-1-7-5', 'cond-mat-0111501-2-7-5'], ['cond-mat-0111501-1-7-6', 'cond-mat-0111501-2-7-6'], ['cond-mat-0111501-1-10-0', 'cond-mat-0111501-2-10-0'], ['cond-mat-0111501-1-10-1', 'cond-mat-0111501-2-10-1'], ['cond-mat-0111501-1-10-2', 'cond-mat-0111501-2-10-2'], ['cond-mat-0111501-1-10-3', 'cond-mat-0111501-2-10-3'], ['cond-mat-0111501-1-9-0', 'cond-mat-0111501-2-9-0'], ['cond-mat-0111501-1-9-1', 'cond-mat-0111501-2-9-1'], ['cond-mat-0111501-1-9-2', 'cond-mat-0111501-2-9-2'], ['cond-mat-0111501-1-9-3', 'cond-mat-0111501-2-9-3'], ['cond-mat-0111501-1-9-4', 'cond-mat-0111501-2-9-4'], ['cond-mat-0111501-1-2-0', 'cond-mat-0111501-2-2-0'], ['cond-mat-0111501-1-2-1', 'cond-mat-0111501-2-2-1'], ['cond-mat-0111501-1-2-2', 'cond-mat-0111501-2-2-2'], ['cond-mat-0111501-1-2-3', 'cond-mat-0111501-2-2-3'], ['cond-mat-0111501-1-11-0', 'cond-mat-0111501-2-11-0'], ['cond-mat-0111501-1-11-1', 'cond-mat-0111501-2-11-1'], ['cond-mat-0111501-1-11-2', 'cond-mat-0111501-2-11-2'], ['cond-mat-0111501-1-11-3', 'cond-mat-0111501-2-11-3'], ['cond-mat-0111501-1-11-4', 'cond-mat-0111501-2-11-4'], ['cond-mat-0111501-1-11-5', 'cond-mat-0111501-2-11-5'], ['cond-mat-0111501-1-11-6', 'cond-mat-0111501-2-11-6'], ['cond-mat-0111501-1-11-7', 'cond-mat-0111501-2-11-7'], ['cond-mat-0111501-1-11-8', 'cond-mat-0111501-2-11-8'], ['cond-mat-0111501-1-11-9', 'cond-mat-0111501-2-11-9'], ['cond-mat-0111501-1-15-0', 'cond-mat-0111501-2-15-0'], ['cond-mat-0111501-1-15-1', 'cond-mat-0111501-2-15-1'], ['cond-mat-0111501-1-15-2', 'cond-mat-0111501-2-15-2'], ['cond-mat-0111501-1-15-3', 'cond-mat-0111501-2-15-3'], ['cond-mat-0111501-1-8-0', 'cond-mat-0111501-2-8-0'], ['cond-mat-0111501-1-8-1', 'cond-mat-0111501-2-8-1'], ['cond-mat-0111501-1-8-2', 'cond-mat-0111501-2-8-2'], ['cond-mat-0111501-1-8-3', 'cond-mat-0111501-2-8-3'], ['cond-mat-0111501-1-8-4', 'cond-mat-0111501-2-8-4'], ['cond-mat-0111501-1-8-5', 'cond-mat-0111501-2-8-5'], ['cond-mat-0111501-1-8-6', 'cond-mat-0111501-2-8-6'], ['cond-mat-0111501-1-21-0', 'cond-mat-0111501-2-21-0'], ['cond-mat-0111501-1-21-1', 'cond-mat-0111501-2-21-1'], ['cond-mat-0111501-1-21-2', 'cond-mat-0111501-2-21-2'], ['cond-mat-0111501-1-16-0', 'cond-mat-0111501-2-16-0'], ['cond-mat-0111501-1-16-1', 'cond-mat-0111501-2-16-1'], ['cond-mat-0111501-1-16-2', 'cond-mat-0111501-2-16-2'], ['cond-mat-0111501-1-16-3', 'cond-mat-0111501-2-16-3'], ['cond-mat-0111501-1-16-4', 'cond-mat-0111501-2-16-4'], ['cond-mat-0111501-1-16-5', 'cond-mat-0111501-2-16-5'], ['cond-mat-0111501-1-16-6', 'cond-mat-0111501-2-16-6'], ['cond-mat-0111501-1-16-7', 'cond-mat-0111501-2-16-7'], ['cond-mat-0111501-1-16-8', 'cond-mat-0111501-2-16-8'], ['cond-mat-0111501-1-1-0', 'cond-mat-0111501-2-1-0'], ['cond-mat-0111501-1-1-1', 'cond-mat-0111501-2-1-1'], ['cond-mat-0111501-1-1-2', 'cond-mat-0111501-2-1-2'], ['cond-mat-0111501-1-1-3', 'cond-mat-0111501-2-1-3'], ['cond-mat-0111501-1-13-0', 'cond-mat-0111501-2-13-0'], ['cond-mat-0111501-1-13-1', 'cond-mat-0111501-2-13-1'], ['cond-mat-0111501-1-13-2', 'cond-mat-0111501-2-13-2'], ['cond-mat-0111501-1-13-3', 'cond-mat-0111501-2-13-3'], ['cond-mat-0111501-1-13-4', 'cond-mat-0111501-2-13-4'], ['cond-mat-0111501-1-19-0', 'cond-mat-0111501-2-19-0'], ['cond-mat-0111501-1-19-1', 'cond-mat-0111501-2-19-1'], ['cond-mat-0111501-1-19-2', 'cond-mat-0111501-2-19-2'], ['cond-mat-0111501-1-19-3', 'cond-mat-0111501-2-19-3'], ['cond-mat-0111501-1-5-0', 'cond-mat-0111501-2-5-0'], ['cond-mat-0111501-1-5-1', 'cond-mat-0111501-2-5-1']]

[]

[]

[]

[]

[]

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

https://arxiv.org/abs/cond-mat/0111501

1405.7650

{'1405.7650-1-0-0': 'We consider the question of how well points in a manifold [MATH] can be approximated by rational points of [MATH].', '1405.7650-1-0-1': 'This contrasts with the more common setup of approximating points in [MATH] by all rational points in [MATH].', '1405.7650-1-0-2': 'Our theorems are of two classes: theorems which apply to all nondegenerate manifolds - which by their nature must be of the form "many or most points of [MATH] are not very well approximable by rationals" - and theorems concerning intrinsic approximation for rational quadratic hypersurfaces.', '1405.7650-1-0-3': 'In the latter case, we provide a complete answer to the four major questions of Diophantine approximation in this context.', '1405.7650-1-0-4': "Extending results of Dani ('86) and Kleinbock-Margulis ('99), our methods utilize a correspondence between the intrinsic Diophantine approximation theory on a rational quadratic hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface.", '1405.7650-1-1-0': '# Introduction and motivation', '1405.7650-1-2-0': 'Classical theorems in Diophantine approximation theory address questions regarding the way points [MATH] are approximated by rational points, considering the trade-off between the height of the rational point - the size of its denominator - and its distance to [MATH]; see [CITATION] for a general introduction.', '1405.7650-1-2-1': 'Often [MATH] is assumed to lie on a certain subset of [MATH], for example a smooth manifold [MATH], leading to Diophantine approximation on manifolds.', '1405.7650-1-2-2': 'This area of research has experienced rapid progress during the last two decades, owing much of it to methods coming from flows on homogeneous spaces; see e.g. [CITATION] for a proof of long-standing conjectures of A. Baker and V. G. Sprindzuk.', '1405.7650-1-2-3': 'See also [CITATION] for more recent developments.', '1405.7650-1-3-0': 'It was observed in [CITATION] that all sufficiently good rational approximants to points on certain rational varieties must in fact be intrinsic - that is, they are rational points lying on the variety itself.', '1405.7650-1-3-1': 'These results, in part, have motivated a new field of intrinsic approximation, which examines the quality to which points on a manifold or variety are approximated by rationals lying on that same subset.', '1405.7650-1-3-2': 'Questions about the quality of these approximations were raised already by S. Lang [CITATION] and K. Mahler [CITATION].', '1405.7650-1-3-3': 'Nevertheless, for the most part these questions have gone unnoticed, with the exceptions of [CITATION], where E. Schmutz uses elementary methods to obtain a crude bound on the quality of rational approximations, and [CITATION], where more general [MATH]-adic approximations on varieties are considered (see [CITATION] for a detailed discussion).', '1405.7650-1-3-4': 'This paper seeks to fully explore intrinsic approximation on manifolds, obtaining definitive results when the manifolds are quadratic hypersurfaces.', '1405.7650-1-3-5': 'In the latter case, our approach has some similarities to that of [CITATION], but significant new ideas are needed to extend the results from the case [MATH] to the case of an arbitrary rational quadratic hypersurface.', '1405.7650-1-3-6': 'The important parameters to consider are the [MATH]-rank and [MATH]-rank of the hypersurface (see Definition [REF]), which we will denote by [MATH] and [MATH] respectively.', '1405.7650-1-3-7': 'It turns out that there are qualitative differences between the intrinsic Diophantine approximation theories of forms with different rank pairs; obviously, such subtleties will not be noticed when studying only the case [MATH] (the case considered in [CITATION]).', '1405.7650-1-4-0': 'Convention 1.', '1405.7650-1-4-1': 'The symbols [MATH], [MATH], and [MATH] will denote asymptotics; a subscript of [MATH] indicates that the asymptotic is additive, and a subscript of [MATH] indicates that it is multiplicative.', '1405.7650-1-4-2': 'For example, [MATH] means that there exists a constant [MATH] (the implied constant), depending only on [MATH], such that [MATH].', '1405.7650-1-4-3': '[MATH] means that there exist constants [MATH] so that [MATH].', '1405.7650-1-4-4': 'In general, dependence of the implied constant(s) on universal objects such as the manifold [MATH] will be omitted from the notation.', '1405.7650-1-5-0': 'Convention 2.', '1405.7650-1-5-1': 'For any [MATH] we let [EQUATION]'}

{'1405.7650-2-0-0': 'We consider the question of how well points in a manifold [MATH] can be approximated by rational points of [MATH].', '1405.7650-2-0-1': 'This contrasts with the more common setup of approximating points in [MATH] by all rational points in [MATH].', '1405.7650-2-0-2': 'Our theorems are of two classes: theorems which apply to all nondegenerate manifolds - which by their nature must be of the form "many or most points of [MATH] are not very well approximable by rationals" - and theorems concerning intrinsic approximation for rational quadratic hypersurfaces.', '1405.7650-2-0-3': 'In the latter case, we provide a complete answer to the four major questions of Diophantine approximation in this context.', '1405.7650-2-0-4': "Extending results of Dani ('86) and Kleinbock-Margulis ('99), our methods utilize a correspondence between the intrinsic Diophantine approximation theory on a rational quadratic hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface.", '1405.7650-2-1-0': '# Introduction and motivation', '1405.7650-2-2-0': 'Classical theorems in Diophantine approximation theory address questions regarding the way points [MATH] are approximated by rational points, considering the trade-off between the height of the rational point - the size of its denominator - and its distance to [MATH]; see [CITATION] for a general introduction.', '1405.7650-2-2-1': 'Often [MATH] is assumed to lie on a certain subset of [MATH], for example a smooth manifold [MATH], leading to Diophantine approximation on manifolds.', '1405.7650-2-2-2': 'This area of research has experienced rapid progress during the last two decades, owing much of it to methods coming from flows on homogeneous spaces; see e.g. [CITATION] for a proof of long-standing conjectures of A. Baker and V. G. Sprindzuk.', '1405.7650-2-2-3': 'See also [CITATION] for more recent developments.', '1405.7650-2-3-0': 'It was observed in [CITATION] that all sufficiently good rational approximants to points on certain rational varieties must in fact be intrinsic - that is, they are rational points lying on the variety itself.', '1405.7650-2-3-1': 'These results, in part, have motivated a new field of intrinsic approximation, which examines the quality to which points on a manifold or variety are approximated by rationals lying on that same subset.', '1405.7650-2-3-2': 'Questions about the quality of these approximations were raised already by S. Lang [CITATION] and K. Mahler [CITATION].', '1405.7650-2-3-3': 'Nevertheless, for the most part these questions have gone unnoticed, with the exceptions of [CITATION], where E. Schmutz uses elementary methods to obtain a crude bound on the quality of rational approximations on spheres, and [CITATION], where more general [MATH]-adic approximations on varieties are considered (see [CITATION] for a detailed discussion).', '1405.7650-2-3-4': 'This paper seeks to fully explore intrinsic approximation on manifolds, obtaining definitive results when the manifolds are quadratic hypersurfaces.', '1405.7650-2-3-5': 'In the latter case, our approach has some similarities to that of [CITATION], but significant new ideas are needed to extend the results from the case [MATH] to the case of an arbitrary rational quadratic hypersurface.', '1405.7650-2-3-6': 'The important parameters to consider are the [MATH]-rank and [MATH]-rank of the hypersurface (see Definition [REF]), which we will denote by [MATH] and [MATH] respectively.', '1405.7650-2-3-7': 'It turns out that there are qualitative differences between the intrinsic Diophantine approximation theories of forms with different rank pairs; obviously, such subtleties will not be noticed when studying only the case [MATH] (the case considered in [CITATION]).', '1405.7650-2-4-0': 'Convention 1.', '1405.7650-2-4-1': 'The symbols [MATH], [MATH], and [MATH] will denote asymptotics; a subscript of [MATH] indicates that the asymptotic is additive, and a subscript of [MATH] indicates that it is multiplicative.', '1405.7650-2-4-2': 'For example, [MATH] means that there exists a constant [MATH] (the implied constant), depending only on [MATH], such that [MATH].', '1405.7650-2-4-3': '[MATH] means that there exist constants [MATH] so that [MATH].', '1405.7650-2-4-4': 'In general, dependence of the implied constant(s) on universal objects such as the manifold [MATH] will be omitted from the notation.', '1405.7650-2-5-0': 'Convention 2.', '1405.7650-2-5-1': 'For any [MATH] we let [EQUATION]'}

[['1405.7650-1-4-1', '1405.7650-2-4-1'], ['1405.7650-1-4-2', '1405.7650-2-4-2'], ['1405.7650-1-4-3', '1405.7650-2-4-3'], ['1405.7650-1-4-4', '1405.7650-2-4-4'], ['1405.7650-1-0-0', '1405.7650-2-0-0'], ['1405.7650-1-0-1', '1405.7650-2-0-1'], ['1405.7650-1-0-2', '1405.7650-2-0-2'], ['1405.7650-1-0-3', '1405.7650-2-0-3'], ['1405.7650-1-0-4', '1405.7650-2-0-4'], ['1405.7650-1-3-0', '1405.7650-2-3-0'], ['1405.7650-1-3-1', '1405.7650-2-3-1'], ['1405.7650-1-3-2', '1405.7650-2-3-2'], ['1405.7650-1-3-4', '1405.7650-2-3-4'], ['1405.7650-1-3-5', '1405.7650-2-3-5'], ['1405.7650-1-3-6', '1405.7650-2-3-6'], ['1405.7650-1-3-7', '1405.7650-2-3-7'], ['1405.7650-1-2-0', '1405.7650-2-2-0'], ['1405.7650-1-2-1', '1405.7650-2-2-1'], ['1405.7650-1-2-2', '1405.7650-2-2-2'], ['1405.7650-1-2-3', '1405.7650-2-2-3'], ['1405.7650-2-2-0', '1405.7650-3-2-0'], ['1405.7650-2-2-1', '1405.7650-3-2-1'], ['1405.7650-2-4-1', '1405.7650-3-4-0'], ['1405.7650-2-4-2', '1405.7650-3-4-1'], ['1405.7650-2-4-3', '1405.7650-3-4-2'], ['1405.7650-2-4-4', '1405.7650-3-4-3'], ['1405.7650-2-3-0', '1405.7650-3-3-0'], ['1405.7650-3-2-0', '1405.7650-4-2-0'], ['1405.7650-3-2-1', '1405.7650-4-2-1'], ['1405.7650-3-2-2', '1405.7650-4-2-2'], ['1405.7650-3-5-0', '1405.7650-4-5-0'], ['1405.7650-3-5-1', '1405.7650-4-5-1'], ['1405.7650-3-4-0', '1405.7650-4-4-0'], ['1405.7650-3-4-1', '1405.7650-4-4-1'], ['1405.7650-3-4-2', '1405.7650-4-4-2'], ['1405.7650-3-4-3', '1405.7650-4-4-3'], ['1405.7650-3-0-0', '1405.7650-4-0-0'], ['1405.7650-3-0-1', '1405.7650-4-0-1'], ['1405.7650-3-0-2', '1405.7650-4-0-2'], ['1405.7650-3-0-3', '1405.7650-4-0-3'], ['1405.7650-3-0-4', '1405.7650-4-0-4'], ['1405.7650-3-3-0', '1405.7650-4-3-0'], ['1405.7650-3-3-1', '1405.7650-4-3-1'], ['1405.7650-3-3-2', '1405.7650-4-3-2'], ['1405.7650-3-3-3', '1405.7650-4-3-3'], ['1405.7650-3-3-4', '1405.7650-4-3-4'], ['1405.7650-3-3-5', '1405.7650-4-3-5'], ['1405.7650-3-3-6', '1405.7650-4-3-6'], ['1405.7650-3-3-7', '1405.7650-4-3-7'], ['1405.7650-4-3-0', '1405.7650-5-3-0'], ['1405.7650-4-3-1', '1405.7650-5-3-1'], ['1405.7650-4-3-2', '1405.7650-5-3-2'], ['1405.7650-4-3-3', '1405.7650-5-3-3'], ['1405.7650-4-3-4', '1405.7650-5-3-4'], ['1405.7650-4-3-5', '1405.7650-5-3-5'], ['1405.7650-4-3-6', '1405.7650-5-3-6'], ['1405.7650-4-3-7', '1405.7650-5-3-7'], ['1405.7650-4-2-0', '1405.7650-5-2-0'], ['1405.7650-4-2-1', '1405.7650-5-2-1'], ['1405.7650-4-2-2', '1405.7650-5-2-2'], ['1405.7650-4-5-0', '1405.7650-5-5-0'], ['1405.7650-4-5-1', '1405.7650-5-5-1'], ['1405.7650-4-4-0', '1405.7650-5-4-0'], ['1405.7650-4-4-1', '1405.7650-5-4-1'], ['1405.7650-4-4-2', '1405.7650-5-4-2'], ['1405.7650-4-4-3', '1405.7650-5-4-3'], ['1405.7650-4-0-0', '1405.7650-5-0-0'], ['1405.7650-4-0-1', '1405.7650-5-0-1'], ['1405.7650-4-0-2', '1405.7650-5-0-2'], ['1405.7650-4-0-3', '1405.7650-5-0-3'], ['1405.7650-4-0-4', '1405.7650-5-0-4'], ['1405.7650-1-3-3', '1405.7650-2-3-3'], ['1405.7650-2-0-0', '1405.7650-3-0-0'], ['1405.7650-2-0-1', '1405.7650-3-0-1'], ['1405.7650-2-0-3', '1405.7650-3-0-2'], ['1405.7650-2-3-1', '1405.7650-3-3-1'], ['1405.7650-2-3-2', '1405.7650-3-3-2'], ['1405.7650-2-0-4', '1405.7650-3-0-4'], ['1405.7650-2-2-2', '1405.7650-3-2-2'], ['1405.7650-2-3-4', '1405.7650-3-3-3'], ['1405.7650-2-3-7', '1405.7650-3-3-5']]

[['1405.7650-1-4-1', '1405.7650-2-4-1'], ['1405.7650-1-4-2', '1405.7650-2-4-2'], ['1405.7650-1-4-3', '1405.7650-2-4-3'], ['1405.7650-1-4-4', '1405.7650-2-4-4'], ['1405.7650-1-0-0', '1405.7650-2-0-0'], ['1405.7650-1-0-1', '1405.7650-2-0-1'], ['1405.7650-1-0-2', '1405.7650-2-0-2'], ['1405.7650-1-0-3', '1405.7650-2-0-3'], ['1405.7650-1-0-4', '1405.7650-2-0-4'], ['1405.7650-1-3-0', '1405.7650-2-3-0'], ['1405.7650-1-3-1', '1405.7650-2-3-1'], ['1405.7650-1-3-2', '1405.7650-2-3-2'], ['1405.7650-1-3-4', '1405.7650-2-3-4'], ['1405.7650-1-3-5', '1405.7650-2-3-5'], ['1405.7650-1-3-6', '1405.7650-2-3-6'], ['1405.7650-1-3-7', '1405.7650-2-3-7'], ['1405.7650-1-2-0', '1405.7650-2-2-0'], ['1405.7650-1-2-1', '1405.7650-2-2-1'], ['1405.7650-1-2-2', '1405.7650-2-2-2'], ['1405.7650-1-2-3', '1405.7650-2-2-3'], ['1405.7650-2-2-0', '1405.7650-3-2-0'], ['1405.7650-2-2-1', '1405.7650-3-2-1'], ['1405.7650-2-4-1', '1405.7650-3-4-0'], ['1405.7650-2-4-2', '1405.7650-3-4-1'], ['1405.7650-2-4-3', '1405.7650-3-4-2'], ['1405.7650-2-4-4', '1405.7650-3-4-3'], ['1405.7650-2-3-0', '1405.7650-3-3-0'], ['1405.7650-3-2-0', '1405.7650-4-2-0'], ['1405.7650-3-2-1', '1405.7650-4-2-1'], ['1405.7650-3-2-2', '1405.7650-4-2-2'], ['1405.7650-3-5-0', '1405.7650-4-5-0'], ['1405.7650-3-5-1', '1405.7650-4-5-1'], ['1405.7650-3-4-0', '1405.7650-4-4-0'], ['1405.7650-3-4-1', '1405.7650-4-4-1'], ['1405.7650-3-4-2', '1405.7650-4-4-2'], ['1405.7650-3-4-3', '1405.7650-4-4-3'], ['1405.7650-3-0-0', '1405.7650-4-0-0'], ['1405.7650-3-0-1', '1405.7650-4-0-1'], ['1405.7650-3-0-2', '1405.7650-4-0-2'], ['1405.7650-3-0-3', '1405.7650-4-0-3'], ['1405.7650-3-0-4', '1405.7650-4-0-4'], ['1405.7650-3-3-0', '1405.7650-4-3-0'], ['1405.7650-3-3-1', '1405.7650-4-3-1'], ['1405.7650-3-3-2', '1405.7650-4-3-2'], ['1405.7650-3-3-3', '1405.7650-4-3-3'], ['1405.7650-3-3-4', '1405.7650-4-3-4'], ['1405.7650-3-3-5', '1405.7650-4-3-5'], ['1405.7650-3-3-6', '1405.7650-4-3-6'], ['1405.7650-3-3-7', '1405.7650-4-3-7'], ['1405.7650-4-3-0', '1405.7650-5-3-0'], ['1405.7650-4-3-1', '1405.7650-5-3-1'], ['1405.7650-4-3-2', '1405.7650-5-3-2'], ['1405.7650-4-3-3', '1405.7650-5-3-3'], ['1405.7650-4-3-4', '1405.7650-5-3-4'], ['1405.7650-4-3-5', '1405.7650-5-3-5'], ['1405.7650-4-3-6', '1405.7650-5-3-6'], ['1405.7650-4-3-7', '1405.7650-5-3-7'], ['1405.7650-4-2-0', '1405.7650-5-2-0'], ['1405.7650-4-2-1', '1405.7650-5-2-1'], ['1405.7650-4-2-2', '1405.7650-5-2-2'], ['1405.7650-4-5-0', '1405.7650-5-5-0'], ['1405.7650-4-5-1', '1405.7650-5-5-1'], ['1405.7650-4-4-0', '1405.7650-5-4-0'], ['1405.7650-4-4-1', '1405.7650-5-4-1'], ['1405.7650-4-4-2', '1405.7650-5-4-2'], ['1405.7650-4-4-3', '1405.7650-5-4-3'], ['1405.7650-4-0-0', '1405.7650-5-0-0'], ['1405.7650-4-0-1', '1405.7650-5-0-1'], ['1405.7650-4-0-2', '1405.7650-5-0-2'], ['1405.7650-4-0-3', '1405.7650-5-0-3'], ['1405.7650-4-0-4', '1405.7650-5-0-4']]

[['1405.7650-1-3-3', '1405.7650-2-3-3'], ['1405.7650-2-0-0', '1405.7650-3-0-0'], ['1405.7650-2-0-1', '1405.7650-3-0-1'], ['1405.7650-2-0-3', '1405.7650-3-0-2'], ['1405.7650-2-3-1', '1405.7650-3-3-1'], ['1405.7650-2-3-2', '1405.7650-3-3-2']]

[]

[['1405.7650-2-0-4', '1405.7650-3-0-4'], ['1405.7650-2-2-2', '1405.7650-3-2-2'], ['1405.7650-2-3-4', '1405.7650-3-3-3'], ['1405.7650-2-3-7', '1405.7650-3-3-5']]

[]

['1405.7650-1-4-0', '1405.7650-1-5-0', '1405.7650-1-5-1', '1405.7650-2-4-0', '1405.7650-2-5-0', '1405.7650-2-5-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1405.7650

{'1405.7650-3-0-0': 'We consider the question of how well points in a quadric hypersurface [MATH] can be approximated by rational points of [MATH].', '1405.7650-3-0-1': 'This contrasts with the more common setup of approximating points in a manifold by all rational points in [MATH].', '1405.7650-3-0-2': 'We provide complete answers to major questions of Diophantine approximation in this context.', '1405.7650-3-0-3': 'Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated.', '1405.7650-3-0-4': "Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock-Margulis ('99).", '1405.7650-3-1-0': '# Introduction and motivation', '1405.7650-3-2-0': 'Classical theorems in Diophantine approximation theory address questions regarding the way points [MATH] are approximated by rational points, considering the trade-off between the height of the rational point - the size of its denominator - and its distance to [MATH]; see [CITATION] for a general introduction.', '1405.7650-3-2-1': 'Often [MATH] is assumed to lie on a certain subset of [MATH], for example a smooth manifold [MATH], leading to Diophantine approximation on manifolds.', '1405.7650-3-2-2': 'This area of research has experienced rapid progress during the last two decades, owing much of it to methods coming from flows on homogeneous spaces.', '1405.7650-3-3-0': 'It was observed in [CITATION] that all sufficiently good rational approximants to points on certain rational varieties must in fact be intrinsic - that is, they are rational points lying on the variety itself.', '1405.7650-3-3-1': 'These results, in part, have motivated a new field of intrinsic approximation, which examines the degree to which points on a manifold or variety can be approximated by rationals lying on that same subset.', '1405.7650-3-3-2': 'Questions about the quality of these approximations were raised already by Lang [CITATION] and Mahler [CITATION].', '1405.7650-3-3-3': 'Following some recent results on quadric hypersurfaces [CITATION] and a comprehensive treatment of Diophantine approximation on spheres [CITATION], this paper seeks to fully explore the topic of intrinsic approximation on quadrics.', '1405.7650-3-3-4': 'One of the most novel and important aspects of our work is an elucidation of the role of the [MATH]-rank and the [MATH]-rank of the defining quadratic form (see Definition [REF]).', '1405.7650-3-3-5': 'It turns out there are qualitative differences between the intrinsic approximation theories of forms with different rank pairs, highlighting the importance of rank, rather than the dimension of the hypersurface.', '1405.7650-3-3-6': 'In particular, we will see below that our Dirichlet-type theorem, Theorem [REF], is independent of the dimension [MATH], but changes depending on whether the [MATH]-rank and [MATH]-rank are equal or different.', '1405.7650-3-3-7': 'We remark that [CITATION] considers only the case where both ranks equal 1, therefore the dependence on the ranks is not explored there, and significant new ideas have had to be developed in the present paper.', '1405.7650-3-4-0': 'The symbols [MATH], [MATH], and [MATH] will denote asymptotics; a subscript of [MATH] indicates that the asymptotic is additive, and a subscript of [MATH] indicates that it is multiplicative.', '1405.7650-3-4-1': 'For example, [MATH] means that there exists a constant [MATH] (the implied constant), depending only on [MATH], such that [MATH].', '1405.7650-3-4-2': '[MATH] means that there exist constants [MATH] so that [MATH].', '1405.7650-3-4-3': 'In general, dependence of the implied constant(s) on universal objects such as the manifold [MATH] will be omitted from the notation.', '1405.7650-3-5-0': 'For any [MATH] we let [EQUATION]', '1405.7650-3-5-1': 'The symbol [MATH] will be used to indicate the end of a nested proof.'}

{'1405.7650-4-0-0': 'We consider the question of how well points in a quadric hypersurface [MATH] can be approximated by rational points of [MATH].', '1405.7650-4-0-1': 'This contrasts with the more common setup of approximating points in a manifold by all rational points in [MATH].', '1405.7650-4-0-2': 'We provide complete answers to major questions of Diophantine approximation in this context.', '1405.7650-4-0-3': 'Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated.', '1405.7650-4-0-4': "Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock-Margulis ('99).", '1405.7650-4-1-0': '# Introduction and motivation', '1405.7650-4-2-0': 'Classical theorems in Diophantine approximation theory address questions regarding the way points [MATH] are approximated by rational points, considering the trade-off between the height of the rational point - the size of its denominator - and its distance to [MATH]; see [CITATION] for a general introduction.', '1405.7650-4-2-1': 'Often [MATH] is assumed to lie on a certain subset of [MATH], for example a smooth manifold [MATH], leading to Diophantine approximation on manifolds.', '1405.7650-4-2-2': 'This area of research has experienced rapid progress during the last two decades, owing much of it to methods coming from flows on homogeneous spaces.', '1405.7650-4-3-0': 'It was observed in [CITATION] that all sufficiently good rational approximants to points on certain rational varieties must in fact be intrinsic - that is, they are rational points lying on the variety itself.', '1405.7650-4-3-1': 'These results, in part, have motivated a new field of intrinsic approximation, which examines the degree to which points on a manifold or variety can be approximated by rationals lying on that same subset.', '1405.7650-4-3-2': 'Questions about the quality of these approximations were raised already by Lang [CITATION] and Mahler [CITATION].', '1405.7650-4-3-3': 'Following some recent results on quadric hypersurfaces [CITATION] and a comprehensive treatment of Diophantine approximation on spheres [CITATION], this paper seeks to fully explore the topic of intrinsic approximation on quadrics.', '1405.7650-4-3-4': 'One of the most novel and important aspects of our work is an elucidation of the role of the [MATH]-rank and the [MATH]-rank of the defining quadratic form (see Definition [REF]).', '1405.7650-4-3-5': 'It turns out there are qualitative differences between the intrinsic approximation theories of forms with different rank pairs, highlighting the importance of rank, rather than the dimension of the hypersurface.', '1405.7650-4-3-6': 'In particular, we will see below that our Dirichlet-type theorem, Theorem [REF], is independent of the dimension [MATH], but changes depending on whether the [MATH]-rank and [MATH]-rank are equal or different.', '1405.7650-4-3-7': 'We remark that [CITATION] considers only the case where both ranks equal 1, therefore the dependence on the ranks is not explored there, and significant new ideas have had to be developed in the present paper.', '1405.7650-4-4-0': 'The symbols [MATH], [MATH], and [MATH] will denote asymptotics; a subscript of [MATH] indicates that the asymptotic is additive, and a subscript of [MATH] indicates that it is multiplicative.', '1405.7650-4-4-1': 'For example, [MATH] means that there exists a constant [MATH] (the implied constant), depending only on [MATH], such that [MATH].', '1405.7650-4-4-2': '[MATH] means that there exist constants [MATH] so that [MATH].', '1405.7650-4-4-3': 'In general, dependence of the implied constant(s) on universal objects such as the manifold [MATH] will be omitted from the notation.', '1405.7650-4-5-0': 'For any [MATH] we let [EQUATION]', '1405.7650-4-5-1': 'The symbol [MATH] will be used to indicate the end of a nested proof.'}

{'1405.7650-5-0-0': 'We consider the question of how well points in a quadric hypersurface [MATH] can be approximated by rational points of [MATH].', '1405.7650-5-0-1': 'This contrasts with the more common setup of approximating points in a manifold by all rational points in [MATH].', '1405.7650-5-0-2': 'We provide complete answers to major questions of Diophantine approximation in this context.', '1405.7650-5-0-3': 'Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated.', '1405.7650-5-0-4': "Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock-Margulis ('99).", '1405.7650-5-1-0': '# Introduction and motivation', '1405.7650-5-2-0': 'Classical theorems in Diophantine approximation theory address questions regarding the way points [MATH] are approximated by rational points, considering the trade-off between the height of the rational point - the size of its denominator - and its distance to [MATH]; see [CITATION] for a general introduction.', '1405.7650-5-2-1': 'Often [MATH] is assumed to lie on a certain subset of [MATH], for example a smooth manifold [MATH], leading to Diophantine approximation on manifolds.', '1405.7650-5-2-2': 'This area of research has experienced rapid progress during the last two decades, owing much of it to methods coming from flows on homogeneous spaces.', '1405.7650-5-3-0': 'It was observed in [CITATION] that all sufficiently good rational approximants to points on certain rational varieties must in fact be intrinsic - that is, they are rational points lying on the variety itself.', '1405.7650-5-3-1': 'These results, in part, have motivated a new field of intrinsic approximation, which examines the degree to which points on a manifold or variety can be approximated by rationals lying on that same subset.', '1405.7650-5-3-2': 'Questions about the quality of these approximations were raised already by Lang [CITATION] and Mahler [CITATION].', '1405.7650-5-3-3': 'Following some recent results on quadric hypersurfaces [CITATION] and a comprehensive treatment of Diophantine approximation on spheres [CITATION], this paper seeks to fully explore the topic of intrinsic approximation on quadrics.', '1405.7650-5-3-4': 'One of the most novel and important aspects of our work is an elucidation of the role of the [MATH]-rank and the [MATH]-rank of the defining quadratic form (see Definition [REF]).', '1405.7650-5-3-5': 'It turns out there are qualitative differences between the intrinsic approximation theories of forms with different rank pairs, highlighting the importance of rank, rather than the dimension of the hypersurface.', '1405.7650-5-3-6': 'In particular, we will see below that our Dirichlet-type theorem, Theorem [REF], is independent of the dimension [MATH], but changes depending on whether the [MATH]-rank and [MATH]-rank are equal or different.', '1405.7650-5-3-7': 'We remark that [CITATION] considers only the case where both ranks equal 1, therefore the dependence on the ranks is not explored there, and significant new ideas have had to be developed in the present paper.', '1405.7650-5-4-0': 'The symbols [MATH], [MATH], and [MATH] will denote asymptotics; a subscript of [MATH] indicates that the asymptotic is additive, and a subscript of [MATH] indicates that it is multiplicative.', '1405.7650-5-4-1': 'For example, [MATH] means that there exists a constant [MATH] (the implied constant), depending only on [MATH], such that [MATH].', '1405.7650-5-4-2': '[MATH] means that there exist constants [MATH] so that [MATH].', '1405.7650-5-4-3': 'In general, dependence of the implied constant(s) on universal objects such as the manifold [MATH] will be omitted from the notation.', '1405.7650-5-5-0': 'For any [MATH] we let [EQUATION]', '1405.7650-5-5-1': 'The symbol [MATH] will be used to indicate the end of a nested proof.'}

0806.2805

{'0806.2805-1-0-0': '*.25A relatively simple gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling constant [MATH], and a standard matter action.', '0806.2805-1-0-1': 'This particular theory provides a concrete realization of the generic vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics, [MATH].', '0806.2805-1-0-2': 'The theory gives a flat Friedmann-Robertson-Walker (FRW) universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-1-0-3': 'Extrapolating to the present age of the universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical density of the present universe.', '0806.2805-1-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter (CDM) in the standard FRW framework.', '0806.2805-1-1-0': '# Introduction', '0806.2805-1-2-0': 'In a previous article [CITATION] we have described a Lorentz-invariant vacuum characterized by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-1-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability, compressibility, and response to perturbations.', '0806.2805-1-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-1-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-1-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-1-3-2': 'The macroscopic vacuum energy density, which contributes to the effective gravitational fields equations at low energies, is determined by a thermodynamic quantity, [MATH].', '0806.2805-1-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, this effective (coarse-grained) vacuum energy density is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-1-3-4': 'This implies that the effective cosmological constant of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-1-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value which shifts the effective vacuum energy density [MATH] away from zero.', '0806.2805-1-4-1': 'The same happens with other types of perturbations which violate Lorentz invariance, such as the existence of a spacetime boundary or interface.', '0806.2805-1-4-2': 'According to this approach, the present value of [MATH] is small because the universe is close to equilibrium and all perturbations of the quantum vacuum are small (compared to the ultraviolet scale).', '0806.2805-1-5-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-1-5-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION] which is automatically conserved.', '0806.2805-1-5-2': 'Using this explicit realization with a four-form field [MATH], we here extended our effective theory to the study of the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective "cosmological constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-1-5-3': 'In short, the present cosmological constant is small because the universe is old.', '0806.2805-1-6-0': 'The decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-1-6-1': 'This suggests that part of the inferred CDM may come from the vacuum energy density and may also give a clue on the coincidence problem, namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-1-7-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-1-8-0': 'Here and in the following, the vacuum variable [MATH] is represented by a four-form [MATH] field.', '0806.2805-1-8-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-1-8-2': 'Such a quadratic function corresponds to a gas-like vacuum.', '0806.2805-1-8-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-1-8-4': 'A self-sustained vacuum requires a more general function [MATH], so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-1-8-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-1-9-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-1-9-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-1-9-2': 'Moreover, a [MATH] dependence allows for the cosmological constant to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-1-10-0': 'Specifically, the action considered in the present article takes the following form ([MATH]): [EQUATION] where [MATH] denotes a generic low-energy matter field with a scalar Lagrange density [MATH] without [MATH]-field dependence, [MATH] the covariant derivative, and square brackets around indices complete anti-symmetrization.', '0806.2805-1-10-1': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-1-11-0': "The variation of the action ([REF]) over the 3-form field [MATH] gives the generalized Maxwell equations, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equations, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH] in [REF], and [MATH] the effective vacuum energy density [CITATION] [EQUATION]", '0806.2805-1-11-1': 'Remark that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a covariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-1-11-2': 'This last observation explains the origin of the second term on the left-hand side of [REF].', '0806.2805-1-12-0': 'Using ([REF]) for [MATH], we obtain the Maxwell equations ([REF]) in the form [EQUATION]', '0806.2805-1-12-1': 'The solution is [EQUATION] where [MATH] is an integration constant.', '0806.2805-1-12-2': 'Here, the constant [MATH] is seen to emerge dynamically.', '0806.2805-1-12-3': 'In the thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-1-12-4': 'In the equilibrium state with vanishing Ricci scalar [MATH], the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH].', '0806.2805-1-13-0': 'Excluding [MATH] from ([REF]) and ([REF]), the generalized Einstein equations become [EQUATION]', '0806.2805-1-13-1': 'Equations [REF] and [REF] are also obtained if we use, instead of the original action, [EQUATION]', '0806.2805-1-13-2': 'If the four-form field is considered to be given in terms of a three-form potential, the [MATH] term in [REF] does not contribute to the equations of motion [REF], because it is a total derivative: [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH].', '0806.2805-1-13-3': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-1-13-4': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-1-14-0': 'Equations ([REF]) and ([REF]) are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-1-14-1': 'The [MATH] field can be replaced by any conserved variable [MATH] (cf. Ref. [CITATION]).', '0806.2805-1-14-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity which is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-1-14-3': 'The functions [MATH] and [MATH] can be considered as phenomenological parameters of the effective theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-1-15-0': '# de-Sitter expansion', '0806.2805-1-16-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations.', '0806.2805-1-16-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-1-16-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-1-17-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH], which allows us to consider this universe as a thermodynamic equilibrium system.', '0806.2805-1-17-1': 'Using [EQUATION] we get from [REF]-[REF] two equations for the constants [MATH] and [MATH]: [EQUATION] where [MATH] denotes the [MATH] value corresponding to a given positive value of the Hubble constant [MATH].', '0806.2805-1-18-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-1-19-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-1-19-1': 'The corresponding equilibrium values of [MATH] and [MATH] in the perfect quantum vacuum are determined from the equations [EQUATION]', '0806.2805-1-19-2': 'If [MATH] is small compared to the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-1-19-3': 'Then, the correction [MATH] due to the [MATH] term is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-1-19-4': 'The chemical potential is also modified by the expansion: [EQUATION]', '0806.2805-1-19-5': 'Instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-1-19-6': 'From ([REF]), we then obtain the other parameters as functions of [MATH]: [MATH], [MATH] and the cosmological constant [MATH].', '0806.2805-1-19-7': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion [[MATH]].', '0806.2805-1-20-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included, as will be discussed in the next section.', '0806.2805-1-21-0': '# Dynamics of a flat FRW universe', '0806.2805-1-22-0': '## General equations', '0806.2805-1-23-0': 'In this section and the next, we consider only spatially flat universes, because of two reasons.', '0806.2805-1-23-1': 'The first reason is that flatness is suggested by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-1-23-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-1-24-0': 'For a spatially flat ([MATH]) FRW universe with time-dependent Hubble parameter [MATH], we have from the reduced Maxwell equation ([REF]): [EQUATION] and from the Einstein equations [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-1-24-1': 'With definition [REF], the reduced Maxwell equation [REF] can be written as [EQUATION]', '0806.2805-1-24-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-1-25-0': '## Model for [MATH]', '0806.2805-1-26-0': 'The above equations allow us to study the development of the universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-1-26-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-1-26-2': 'The only conditions are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum compressibility [REF] is positive [[MATH]].', '0806.2805-1-26-3': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] is the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-1-26-4': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-1-26-5': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale: [MATH] and [MATH].', '0806.2805-1-26-6': 'From [REF], we then see that [MATH].', '0806.2805-1-26-7': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION]', '0806.2805-1-26-8': 'The corresponding normalized vacuum and matter energy densities are defined as follows [EQUATION]', '0806.2805-1-26-9': 'From the Maxwell equation [REF], the Einstein equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH] and [MATH]: [EQUATION] with the matter equation-of-state (EOS) parameter [MATH].', '0806.2805-1-27-0': '## Model for [MATH]', '0806.2805-1-28-0': 'Now we need an appropriate Ansatz for the function [MATH] or the function [MATH] in dimensionless units.', '0806.2805-1-28-1': 'There are several possible types of behavior for [MATH], but we adopt the following reasoning.', '0806.2805-1-28-2': 'It is possible to argue that for [MATH] (i.e., in the gas-like vacuum), the role of the Planck scale is played by [MATH].', '0806.2805-1-28-3': 'The gravitational coupling constant would then be given by [EQUATION]', '0806.2805-1-28-4': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH].', '0806.2805-1-28-5': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single dimensionless parameter [MATH] also taken positive.', '0806.2805-1-29-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-1-29-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-1-29-2': 'Anyway, this parameter can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-1-29-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for parameter [MATH] of the matter EOS.', '0806.2805-1-30-0': '# Equilibrium approach', '0806.2805-1-31-0': '## Equations at the equilibrium point [MATH]', '0806.2805-1-32-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat universe towards the stationary state, if the initial universe was far away from equilibrium.', '0806.2805-1-32-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum with [MATH] and [MATH].', '0806.2805-1-32-2': 'Here, we consider the situation where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for the Ansatz vacuum energy density [REF].', '0806.2805-1-33-0': 'Such an equilibrium vacuum state can be reached only if the normalized chemical potential [MATH] corresponds to full equilibrium: [MATH] in [REF] or [MATH] in dimensionless units.', '0806.2805-1-33-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-1-33-2': 'Indeed, the starting non-equilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-1-33-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-1-33-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-1-34-0': 'In order to avoid having to consider quantum corrections to the Einstein equations, which typically appear near the Planck time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-1-34-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium, [MATH].', '0806.2805-1-35-0': 'Taking the time derivative of [REF] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-1-35-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-1-35-2': 'The normalized vacuum energy density [REF] for the normalized equilibrium chemical potential [MATH] is given by [EQUATION]', '0806.2805-1-36-0': '## Vacuum oscillations', '0806.2805-1-37-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-1-37-1': 'The solution of these equations describes fast oscillations near the equilibrium point: [EQUATION]', '0806.2805-1-37-2': 'The oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-1-37-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-1-38-0': '## Vacuum energy decay', '0806.2805-1-39-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the parameters in [REF], namely, the amplitude of oscillations [MATH], the Hubble parameter [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-1-39-1': 'To find the explicit behavior, we expand in powers of [MATH] and keep terms up to [MATH]: [EQUATION]', '0806.2805-1-39-2': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear equations [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-1-39-3': 'The solution of these equations is given by [REF]: [EQUATION] where [MATH] and [MATH] are numerical coefficients which determine the decay of [MATH] and [MATH].', '0806.2805-1-40-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-1-40-1': 'This leads to inhomogeneous linear equations for functions [MATH] and [MATH].', '0806.2805-1-40-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-1-40-3': 'To find these coefficients it suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-1-40-4': 'As a result we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-1-40-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-1-40-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which gives [MATH].', '0806.2805-1-41-0': 'Combining these results for the coefficients, we have the following asymptotic behavior of [MATH] and [MATH] for [MATH]: [EQUATION]', '0806.2805-1-41-1': 'This behavior has some remarkable properties.', '0806.2805-1-41-2': 'First, it does not depend on the parameter [MATH] of the matter EOS.', '0806.2805-1-41-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], whereas the average scale parameter increases as [MATH].', '0806.2805-1-41-4': 'In fact, the average vacuum energy density behaves as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed further below.', '0806.2805-1-42-0': '## Numerical results', '0806.2805-1-43-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled differential equations [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-1-43-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equations can be expected (cf. Sec. [REF]).', '0806.2805-1-43-2': 'Still, the results show clearly that', '0806.2805-1-44-0': 'the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); the asymptotic FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; the same [MATH] behavior occurs if there is initially nonrelativistic matter; see Figs. [REF] and [REF] (which, purely for illustrative purpose, have an initial energy density ten times smaller than the one of ultrarelativistic matter in the first two figures); for an integration constant (chemical potential) [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is shown in Fig. [REF].', '0806.2805-1-45-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results, whereas the last item shows that, after an initial oscillating stage, the universe approaches a de-Sitter stage.', '0806.2805-1-46-0': '## Effective CDM-like behavior', '0806.2805-1-47-0': 'The main result found in the previous two subsections is quite interesting: the oscillating vacuum energy density and the corresponding oscillating gravitational coupling constant behave in the same way as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-1-47-1': 'Recall the standard behavior [MATH], which matches the behavior found in [REF].', '0806.2805-1-48-0': 'The explanation is as follows.', '0806.2805-1-48-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure serve as a source of the slowly varying gravitational field.', '0806.2805-1-48-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] form a dynamic field with Lagrangian density [MATH].', '0806.2805-1-48-3': 'Originally, the [MATH] (or [MATH]) field has no kinetic term in [REF], but it emerges due to the interaction of the [MATH] field with the gravity field ([MATH] in the present context).', '0806.2805-1-48-4': 'The effective Lagrange density [MATH] is, therefore, induced by gravity.', '0806.2805-1-48-5': 'The pressure of this rapidly oscillating field [MATH] is given by [MATH].', '0806.2805-1-48-6': 'Hence, the rapidly oscillating vacuum pressure is zero on average and the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-1-49-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-1-49-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales which determine the microscopic dynamics of the vacuum energy density.', '0806.2805-1-49-2': 'But surprises are, of course, not excluded.', '0806.2805-1-50-0': '## Extrapolation to large times', '0806.2805-1-51-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-1-51-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-row panels of Figs. [REF] and [REF].', '0806.2805-1-52-0': 'Extrapolating this evolution to the present age of the universe ([MATH]), the numerical value of the average vacuum energy density is given by [EQUATION]', '0806.2805-1-52-1': 'The order of magnitude of the above estimate is in agreement with the observed energy density of the present universe which is close to the critical density (cf. Refs. [CITATION] and references therein).', '0806.2805-1-52-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-1-52-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-1-53-0': 'Even though the order of magnitude [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-1-53-1': 'Of course, there may be many other processes which intervene between the very early (Planckian) phase of the universe and the present epoch, including particle production (for example, by parametric resonance [CITATION]) which can be expected to be effective because of the very rapid but small oscillations.', '0806.2805-1-53-2': 'Still, there is possibility that these other processes are secondary effects and that the main mechanism of dark-energy dynamics is the decay of vacuum energy density by oscillations.', '0806.2805-1-54-0': '# Conclusion', '0806.2805-1-55-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide some clues.', '0806.2805-1-55-1': 'Specifically, the new results are', '0806.2805-1-56-0': 'Expanding on the last point, another important consequence of oscillations is that they naturally lead to the creation of ultrarelativistic matter from the vacuum.', '0806.2805-1-56-1': 'This effective mechanism of the energy exchange between vacuum and matter will be considered in a forthcoming publication.', '0806.2805-1-57-0': '# ACKNOWLEDGMENTS GEV is supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (grant 4899.2008.2)'}

{'0806.2805-2-0-0': '*.25A particular gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-2-0-1': 'This theory provides a concrete realization for the generic vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics, [MATH].', '0806.2805-2-0-2': 'The theory gives a flat Friedmann-Robertson-Walker (FRW) universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-2-0-3': 'Extrapolating to the present age of the universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-2-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter (CDM) in the standard FRW framework.', '0806.2805-2-1-0': '# Introduction', '0806.2805-2-2-0': 'In a previous article [CITATION], we have proposed to characterize a Lorentz-invariant vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-2-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability, compressibility, and response to perturbations.', '0806.2805-2-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-2-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-2-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-2-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density which contributes to the effective gravitational field equations at low energies.', '0806.2805-2-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-2-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-2-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-2-4-1': 'The same happens with other types of perturbations which violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-2-4-2': 'According to this approach, the present value of [MATH] is small because the universe is close to equilibrium and perturbations of the quantum vacuum are small (compared to the ultraviolet scale which sets [MATH]).', '0806.2805-2-5-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-2-5-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-2-5-2': 'Using this explicit realization with a four-form field [MATH], we here extend our effective theory to the study of the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective "cosmological constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-2-5-3': 'In short, the present cosmological constant is small because the universe is old.', '0806.2805-2-6-0': 'The decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-2-6-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-2-7-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-2-8-0': 'Here and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-2-8-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-2-8-2': 'Such a quadratic function corresponds to a gas-like vacuum [CITATION].', '0806.2805-2-8-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-2-8-4': 'A self-sustained vacuum requires a more general function [MATH], so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-2-8-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-2-9-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-2-9-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-2-9-2': 'Moreover, a [MATH] dependence allows for the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-2-10-0': 'Specifically, the action considered in the present article takes the following form ([MATH]): [EQUATION] where [MATH] denotes the covariant derivative, [MATH] the Levi-Civita symbol, and square brackets around spacetime indices complete anti-symmetrization.', '0806.2805-2-10-1': 'The field [MATH] in [REF] stands for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants to [MATH]-dependent parameters).', '0806.2805-2-10-2': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-2-11-0': "The variation of the action [REF] over the 3-form field [MATH] gives the generalized Maxwell equations, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equations, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [CITATION] [EQUATION]", '0806.2805-2-11-1': 'Remark that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-2-11-2': 'This last observation explains the origin of the second term on the left-hand side of [REF].', '0806.2805-2-12-0': 'Using [REF] for [MATH], we obtain the Maxwell equations [REF] in the form [EQUATION]', '0806.2805-2-12-1': 'The solution is [EQUATION] with integration constant [MATH].', '0806.2805-2-12-2': 'Here, the constant [MATH] is seen to emerge dynamically.', '0806.2805-2-12-3': 'In the thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-2-12-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-2-13-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equations become [EQUATION]', '0806.2805-2-13-1': 'Equations [REF] and [REF] are also obtained if we use, instead of the original action, [EQUATION]', '0806.2805-2-13-2': 'If the four-form field is considered to be given in terms of a three-form potential, the [MATH] term in [REF] does not contribute to the equations of motion [REF], because it is a total derivative: [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH].', '0806.2805-2-13-3': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-2-13-4': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-2-14-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-2-14-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-2-14-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity which is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-2-14-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-2-15-0': '# de-Sitter expansion', '0806.2805-2-16-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations.', '0806.2805-2-16-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-2-16-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-2-17-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH], which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-2-17-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] where [MATH] denotes the [MATH] value corresponding to a given positive value of the Hubble constant [MATH].', '0806.2805-2-18-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-2-19-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-2-19-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION]', '0806.2805-2-19-2': 'If [MATH] is small compared to the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-2-19-3': 'Then, the correction [MATH] due to the [MATH] term is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-2-19-4': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-2-19-5': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-2-19-6': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and the cosmological constant [MATH].', '0806.2805-2-19-7': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion, [MATH].', '0806.2805-2-20-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-2-21-0': '# Dynamics of a flat FRW universe', '0806.2805-2-22-0': '## General equations', '0806.2805-2-23-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-2-23-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-2-23-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-2-23-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and pressure [MATH].', '0806.2805-2-24-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with time-dependent Hubble parameter [MATH], we have from the reduced Maxwell equation [REF]: [EQUATION] and from the Einstein equations [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-2-24-1': 'With definition [REF], the reduced Maxwell equation [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to [MATH].', '0806.2805-2-24-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-2-25-0': '## Model for [MATH]', '0806.2805-2-26-0': 'The above equations allow us to study the development of the universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-2-26-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-2-26-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum compressibility [REF] is positive [[MATH]].', '0806.2805-2-26-3': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-2-26-4': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-2-26-5': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale: [MATH] and [MATH].', '0806.2805-2-26-6': 'From [REF], we then see that [MATH].', '0806.2805-2-26-7': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION]', '0806.2805-2-26-8': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] corresponds to [EQUATION]', '0806.2805-2-26-9': 'From the Maxwell equation [REF], the Einstein equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH] and [MATH]: [EQUATION] with the matter equation-of-state (EOS) parameter [MATH].', '0806.2805-2-27-0': '## Model for [MATH]', '0806.2805-2-28-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the function [MATH] in dimensionless units.', '0806.2805-2-28-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-2-29-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-2-29-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-2-29-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given on the lines below [REF].', '0806.2805-2-29-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single dimensionless parameter [MATH] also taken positive.', '0806.2805-2-30-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-2-30-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-2-30-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-2-30-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (later taken to be time independent).', '0806.2805-2-31-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-2-32-0': '## Equations at the equilibrium point [MATH]', '0806.2805-2-33-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-2-33-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-2-33-2': 'Here, we consider the latter situation where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for the Ansatz vacuum energy density [REF].', '0806.2805-2-34-0': 'Such an equilibrium vacuum state can be reached only if the normalized chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in dimensionless units.', '0806.2805-2-34-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-2-34-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-2-34-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-2-34-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-2-35-0': 'In order to avoid having to consider quantum corrections to the Einstein equations, which typically appear near the Planck time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-2-35-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium, [MATH].', '0806.2805-2-36-0': 'Taking the time derivative of [REF] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-2-36-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-2-36-2': 'The normalized vacuum energy density [REF] for the normalized equilibrium chemical potential [MATH] is given by [EQUATION] which vanishes in the equilibrium state [MATH].', '0806.2805-2-37-0': '## Vacuum oscillations', '0806.2805-2-38-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-2-38-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-2-38-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-2-38-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-2-38-4': 'Both oscillation periods will be apparent in the numerical results of Sec. [REF].', '0806.2805-2-39-0': '## Vacuum energy decay', '0806.2805-2-40-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the parameters in [REF], namely, the amplitude of oscillations [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-2-40-1': 'To find the explicit behavior, we expand in powers of [MATH] and keep terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-2-40-2': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear equations [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-2-40-3': 'The solution of these equations is given by [REF]: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-2-41-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-2-41-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-2-41-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-2-41-3': 'To find these coefficients it suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-2-41-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-2-41-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-2-41-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which gives [MATH].', '0806.2805-2-42-0': 'Combining these results for the coefficients [MATH] and [MATH], we have the following asymptotic behavior of [MATH] and [MATH] for [MATH]: [EQUATION] with dimensionless frequency [MATH].', '0806.2805-2-42-1': 'This asymptotic solution has some remarkable properties.', '0806.2805-2-42-2': 'First, it does not depend on the parameter [MATH] of the matter EOS (that is, the asymptotic solution appears to act as an "attractor," which is also confirmed by the numerical results in the next subsection).', '0806.2805-2-42-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-2-42-4': 'In fact, the average vacuum energy density behaves as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed in Sec. [REF].', '0806.2805-2-43-0': '## Numerical results', '0806.2805-2-44-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled differential equations [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-2-44-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equations can be expected (cf. Sec. [REF]).', '0806.2805-2-44-2': 'Still, the numerical results show clearly that', '0806.2805-2-45-0': 'the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] (which, purely for illustrative purpose, have an initial energy density ten times smaller than the one of ultrarelativistic matter in Figs. [REF] and [REF]); for a chemical potential (integration constant) [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-2-46-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the universe approaches a de-Sitter stage.', '0806.2805-2-47-0': '## Effective CDM-like behavior', '0806.2805-2-48-0': 'The main result found in the previous two subsections is quite interesting: the oscillating vacuum energy density and the corresponding oscillating gravitational coupling parameter behave in the same way as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-2-48-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-2-49-0': 'The explanation is as follows.', '0806.2805-2-49-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-2-49-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-2-49-3': 'Originally, the [MATH] (or [MATH]) field has no kinetic term in [REF], but it emerges due to the interaction of the [MATH] field with the gravity field ([MATH] in the present context).', '0806.2805-2-49-4': 'The effective Lagrange density [MATH] is, therefore, induced by gravity.', '0806.2805-2-49-5': 'The pressure of this rapidly oscillating field [MATH] is given by [MATH].', '0806.2805-2-49-6': 'Hence, the rapidly oscillating vacuum pressure is zero on average and the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-2-50-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-2-50-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales which determine the microscopic dynamics of the vacuum energy density.', '0806.2805-2-50-2': 'But surprises are, of course, not excluded.', '0806.2805-2-51-0': '## Extrapolation to large times', '0806.2805-2-52-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-2-52-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF] and [REF].', '0806.2805-2-53-0': 'Extrapolating this evolution to the present age of the universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] with [MATH].', '0806.2805-2-53-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-2-53-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-2-53-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-2-54-0': 'Let us expand on the previous remarks.', '0806.2805-2-54-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-2-54-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-2-54-3': 'This also explains the result [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-2-55-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-2-55-1': 'Clearly, there may be many other processes which intervene between the very early (Planckian) phase of the universe and the present epoch, including particle production (for example, by parametric resonance [CITATION]) which can be expected to be effective because of the very rapid but small oscillations.', '0806.2805-2-55-2': 'Still, there is possibility that these other processes are secondary effects and that the main mechanism of dark-energy dynamics is the decay of vacuum energy density by oscillations.', '0806.2805-2-56-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-2-56-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-2-56-2': "The behavior [REF] is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-2-57-0': '# Conclusion', '0806.2805-2-58-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-2-58-1': 'Specifically, the new results are', '0806.2805-2-59-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-2-59-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-2-60-0': '# ACKNOWLEDGMENTS GEV is supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (grant 4899.2008.2)'}

[['0806.2805-1-8-1', '0806.2805-2-8-1'], ['0806.2805-1-8-3', '0806.2805-2-8-3'], ['0806.2805-1-8-4', '0806.2805-2-8-4'], ['0806.2805-1-8-5', '0806.2805-2-8-5'], ['0806.2805-1-19-0', '0806.2805-2-19-0'], ['0806.2805-1-19-2', '0806.2805-2-19-2'], ['0806.2805-1-19-3', '0806.2805-2-19-3'], ['0806.2805-1-11-2', '0806.2805-2-11-2'], ['0806.2805-1-3-0', '0806.2805-2-3-0'], ['0806.2805-1-3-1', '0806.2805-2-3-1'], ['0806.2805-1-2-1', '0806.2805-2-2-1'], ['0806.2805-1-2-2', '0806.2805-2-2-2'], ['0806.2805-1-6-0', '0806.2805-2-6-0'], ['0806.2805-1-9-1', '0806.2805-2-9-1'], ['0806.2805-1-14-2', '0806.2805-2-14-2'], ['0806.2805-1-55-1', '0806.2805-2-58-1'], ['0806.2805-1-52-2', '0806.2805-2-53-2'], ['0806.2805-1-52-3', '0806.2805-2-53-3'], ['0806.2805-1-39-2', '0806.2805-2-40-2'], ['0806.2805-1-16-0', '0806.2805-2-16-0'], ['0806.2805-1-16-1', '0806.2805-2-16-1'], ['0806.2805-1-16-2', '0806.2805-2-16-2'], ['0806.2805-1-53-2', '0806.2805-2-55-2'], ['0806.2805-1-37-0', '0806.2805-2-38-0'], ['0806.2805-1-37-3', '0806.2805-2-38-3'], ['0806.2805-1-0-2', '0806.2805-2-0-2'], ['0806.2805-1-0-4', '0806.2805-2-0-4'], ['0806.2805-1-40-0', '0806.2805-2-41-0'], ['0806.2805-1-40-2', '0806.2805-2-41-2'], ['0806.2805-1-40-3', '0806.2805-2-41-3'], ['0806.2805-1-40-5', '0806.2805-2-41-5'], ['0806.2805-1-40-6', '0806.2805-2-41-6'], ['0806.2805-1-13-1', '0806.2805-2-13-1'], ['0806.2805-1-13-2', '0806.2805-2-13-2'], ['0806.2805-1-13-3', '0806.2805-2-13-3'], ['0806.2805-1-13-4', '0806.2805-2-13-4'], ['0806.2805-1-34-0', '0806.2805-2-35-0'], ['0806.2805-1-34-1', '0806.2805-2-35-1'], ['0806.2805-1-35-0', '0806.2805-2-36-0'], ['0806.2805-1-35-1', '0806.2805-2-36-1'], ['0806.2805-1-23-2', '0806.2805-2-23-2'], ['0806.2805-1-5-0', '0806.2805-2-5-0'], ['0806.2805-1-5-3', '0806.2805-2-5-3'], ['0806.2805-1-10-1', '0806.2805-2-10-2'], ['0806.2805-1-51-0', '0806.2805-2-52-0'], ['0806.2805-1-43-0', '0806.2805-2-44-0'], ['0806.2805-1-43-1', '0806.2805-2-44-1'], ['0806.2805-1-18-0', '0806.2805-2-18-0'], ['0806.2805-1-26-0', '0806.2805-2-26-0'], ['0806.2805-1-26-1', '0806.2805-2-26-1'], ['0806.2805-1-26-4', '0806.2805-2-26-4'], ['0806.2805-1-26-5', '0806.2805-2-26-5'], ['0806.2805-1-26-6', '0806.2805-2-26-6'], ['0806.2805-1-26-7', '0806.2805-2-26-7'], ['0806.2805-1-26-9', '0806.2805-2-26-9'], ['0806.2805-1-12-2', '0806.2805-2-12-2'], ['0806.2805-1-12-3', '0806.2805-2-12-3'], ['0806.2805-1-49-0', '0806.2805-2-50-0'], ['0806.2805-1-49-1', '0806.2805-2-50-1'], ['0806.2805-1-49-2', '0806.2805-2-50-2'], ['0806.2805-1-29-0', '0806.2805-2-30-0'], ['0806.2805-1-29-1', '0806.2805-2-30-1'], ['0806.2805-1-24-2', '0806.2805-2-24-2'], ['0806.2805-1-48-0', '0806.2805-2-49-0'], ['0806.2805-1-48-3', '0806.2805-2-49-3'], ['0806.2805-1-48-4', '0806.2805-2-49-4'], ['0806.2805-1-48-5', '0806.2805-2-49-5'], ['0806.2805-1-48-6', '0806.2805-2-49-6'], ['0806.2805-1-33-1', '0806.2805-2-34-1'], ['0806.2805-1-33-3', '0806.2805-2-34-3'], ['0806.2805-1-33-4', '0806.2805-2-34-4'], ['0806.2805-2-40-0', '0806.2805-3-46-0'], ['0806.2805-2-40-2', '0806.2805-3-46-2'], ['0806.2805-2-41-0', '0806.2805-3-47-0'], ['0806.2805-2-41-1', '0806.2805-3-47-1'], ['0806.2805-2-41-2', '0806.2805-3-47-2'], ['0806.2805-2-41-4', '0806.2805-3-47-4'], ['0806.2805-2-41-5', '0806.2805-3-47-5'], ['0806.2805-2-4-0', '0806.2805-3-4-0'], ['0806.2805-2-4-1', '0806.2805-3-4-1'], ['0806.2805-2-53-0', '0806.2805-3-60-0'], ['0806.2805-2-53-1', '0806.2805-3-60-1'], ['0806.2805-2-53-2', '0806.2805-3-60-2'], ['0806.2805-2-53-3', '0806.2805-3-60-3'], ['0806.2805-2-44-0', '0806.2805-3-51-0'], ['0806.2805-2-44-1', '0806.2805-3-51-1'], ['0806.2805-2-44-2', '0806.2805-3-51-2'], ['0806.2805-2-12-0', '0806.2805-3-14-0'], ['0806.2805-2-12-1', '0806.2805-3-14-1'], ['0806.2805-2-12-2', '0806.2805-3-14-2'], ['0806.2805-2-12-4', '0806.2805-3-14-4'], ['0806.2805-2-0-2', '0806.2805-3-0-2'], ['0806.2805-2-0-3', '0806.2805-3-0-3'], ['0806.2805-2-0-4', '0806.2805-3-0-4'], ['0806.2805-2-33-0', '0806.2805-3-38-0'], ['0806.2805-2-33-1', '0806.2805-3-38-1'], ['0806.2805-2-52-0', '0806.2805-3-59-0'], ['0806.2805-2-2-1', '0806.2805-3-2-1'], ['0806.2805-2-2-2', '0806.2805-3-2-2'], ['0806.2805-2-59-0', '0806.2805-3-66-0'], ['0806.2805-2-59-1', '0806.2805-3-66-1'], ['0806.2805-2-10-1', '0806.2805-3-12-1'], ['0806.2805-2-10-2', '0806.2805-3-12-2'], ['0806.2805-2-19-0', '0806.2805-3-23-0'], ['0806.2805-2-19-2', '0806.2805-3-23-2'], ['0806.2805-2-19-4', '0806.2805-3-23-4'], ['0806.2805-2-19-5', '0806.2805-3-23-5'], ['0806.2805-2-19-6', '0806.2805-3-23-6'], ['0806.2805-2-19-7', '0806.2805-3-23-7'], ['0806.2805-2-28-1', '0806.2805-3-33-1'], ['0806.2805-2-55-0', '0806.2805-3-62-0'], ['0806.2805-2-42-3', '0806.2805-3-49-3'], ['0806.2805-2-20-0', '0806.2805-3-24-0'], ['0806.2805-2-34-1', '0806.2805-3-39-1'], ['0806.2805-2-34-2', '0806.2805-3-39-2'], ['0806.2805-2-34-3', '0806.2805-3-39-3'], ['0806.2805-2-34-4', '0806.2805-3-39-4'], ['0806.2805-2-3-0', '0806.2805-3-3-0'], ['0806.2805-2-3-1', '0806.2805-3-3-1'], ['0806.2805-2-3-2', '0806.2805-3-3-2'], ['0806.2805-2-3-3', '0806.2805-3-3-3'], ['0806.2805-2-3-4', '0806.2805-3-3-4'], ['0806.2805-2-38-0', '0806.2805-3-44-0'], ['0806.2805-2-38-1', '0806.2805-3-44-1'], ['0806.2805-2-38-2', '0806.2805-3-44-2'], ['0806.2805-2-38-3', '0806.2805-3-44-3'], ['0806.2805-2-50-0', '0806.2805-3-57-0'], ['0806.2805-2-50-1', '0806.2805-3-57-1'], ['0806.2805-2-50-2', '0806.2805-3-57-2'], ['0806.2805-2-58-0', '0806.2805-3-65-0'], ['0806.2805-2-58-1', '0806.2805-3-65-1'], ['0806.2805-2-6-1', '0806.2805-3-7-1'], ['0806.2805-2-36-0', '0806.2805-3-41-0'], ['0806.2805-2-36-1', '0806.2805-3-41-1'], ['0806.2805-2-36-2', '0806.2805-3-41-2'], ['0806.2805-2-18-0', '0806.2805-3-22-0'], ['0806.2805-2-14-0', '0806.2805-3-17-0'], ['0806.2805-2-14-1', '0806.2805-3-17-1'], ['0806.2805-2-14-2', '0806.2805-3-17-2'], ['0806.2805-2-14-3', '0806.2805-3-17-3'], ['0806.2805-2-23-0', '0806.2805-3-27-0'], ['0806.2805-2-23-1', '0806.2805-3-27-1'], ['0806.2805-2-23-2', '0806.2805-3-27-2'], ['0806.2805-2-35-0', '0806.2805-3-40-0'], ['0806.2805-2-35-1', '0806.2805-3-40-1'], ['0806.2805-2-16-1', '0806.2805-3-20-1'], ['0806.2805-2-16-2', '0806.2805-3-20-2'], ['0806.2805-2-8-0', '0806.2805-3-10-0'], ['0806.2805-2-8-1', '0806.2805-3-10-1'], ['0806.2805-2-8-2', '0806.2805-3-10-2'], ['0806.2805-2-8-3', '0806.2805-3-10-3'], ['0806.2805-2-8-5', '0806.2805-3-10-5'], ['0806.2805-2-26-0', '0806.2805-3-31-0'], ['0806.2805-2-26-1', '0806.2805-3-31-1'], ['0806.2805-2-26-3', '0806.2805-3-31-3'], ['0806.2805-2-26-4', '0806.2805-3-31-4'], ['0806.2805-2-26-5', '0806.2805-3-31-5'], ['0806.2805-2-26-6', '0806.2805-3-31-6'], ['0806.2805-2-26-7', '0806.2805-3-31-7'], ['0806.2805-2-26-9', '0806.2805-3-31-9'], ['0806.2805-2-46-0', '0806.2805-3-53-0'], ['0806.2805-2-17-0', '0806.2805-3-21-0'], ['0806.2805-2-17-1', '0806.2805-3-21-1'], ['0806.2805-2-29-0', '0806.2805-3-34-0'], ['0806.2805-2-29-1', '0806.2805-3-34-1'], ['0806.2805-2-29-2', '0806.2805-3-34-2'], ['0806.2805-2-30-0', '0806.2805-3-35-0'], ['0806.2805-2-30-1', '0806.2805-3-35-1'], ['0806.2805-2-30-2', '0806.2805-3-35-2'], ['0806.2805-2-30-3', '0806.2805-3-35-3'], ['0806.2805-2-9-0', '0806.2805-3-11-0'], ['0806.2805-2-9-1', '0806.2805-3-11-1'], ['0806.2805-2-9-2', '0806.2805-3-11-2'], ['0806.2805-2-54-0', '0806.2805-3-61-0'], ['0806.2805-2-54-1', '0806.2805-3-61-1'], ['0806.2805-2-54-2', '0806.2805-3-61-2'], ['0806.2805-2-56-0', '0806.2805-3-63-0'], ['0806.2805-2-56-1', '0806.2805-3-63-1'], ['0806.2805-2-5-0', '0806.2805-3-6-0'], ['0806.2805-2-5-1', '0806.2805-3-6-1'], ['0806.2805-2-5-3', '0806.2805-3-6-3'], ['0806.2805-2-48-0', '0806.2805-3-55-0'], ['0806.2805-2-48-1', '0806.2805-3-55-1'], ['0806.2805-2-49-0', '0806.2805-3-56-0'], ['0806.2805-2-49-1', '0806.2805-3-56-1'], ['0806.2805-2-49-2', '0806.2805-3-56-2'], ['0806.2805-2-49-6', '0806.2805-3-56-6'], ['0806.2805-3-7-0', '0806.2805-4-7-0'], ['0806.2805-3-7-1', '0806.2805-4-7-1'], ['0806.2805-3-24-0', '0806.2805-4-26-0'], ['0806.2805-3-21-0', '0806.2805-4-23-0'], ['0806.2805-3-11-0', '0806.2805-4-11-0'], ['0806.2805-3-11-1', '0806.2805-4-11-1'], ['0806.2805-3-61-0', '0806.2805-4-65-0'], ['0806.2805-3-61-1', '0806.2805-4-65-1'], ['0806.2805-3-61-2', '0806.2805-4-65-2'], ['0806.2805-3-61-4', '0806.2805-4-65-4'], ['0806.2805-3-20-0', '0806.2805-4-22-0'], ['0806.2805-3-20-1', '0806.2805-4-22-1'], ['0806.2805-3-20-2', '0806.2805-4-22-2'], ['0806.2805-3-33-0', '0806.2805-4-36-0'], ['0806.2805-3-33-1', '0806.2805-4-36-1'], ['0806.2805-3-51-0', '0806.2805-4-55-0'], ['0806.2805-3-51-1', '0806.2805-4-55-1'], ['0806.2805-3-51-2', '0806.2805-4-55-2'], ['0806.2805-3-14-0', '0806.2805-4-15-0'], ['0806.2805-3-14-3', '0806.2805-4-15-3'], ['0806.2805-3-14-4', '0806.2805-4-15-4'], ['0806.2805-3-22-0', '0806.2805-4-24-0'], ['0806.2805-3-41-0', '0806.2805-4-44-0'], ['0806.2805-3-41-1', '0806.2805-4-44-1'], ['0806.2805-3-60-1', '0806.2805-4-64-1'], ['0806.2805-3-60-2', '0806.2805-4-64-2'], ['0806.2805-3-60-3', '0806.2805-4-64-3'], ['0806.2805-3-65-0', '0806.2805-4-69-0'], ['0806.2805-3-65-1', '0806.2805-4-69-1'], ['0806.2805-3-2-0', '0806.2805-4-2-0'], ['0806.2805-3-2-2', '0806.2805-4-2-2'], ['0806.2805-3-3-0', '0806.2805-4-3-0'], ['0806.2805-3-3-1', '0806.2805-4-3-1'], ['0806.2805-3-3-2', '0806.2805-4-3-2'], ['0806.2805-3-3-3', '0806.2805-4-3-3'], ['0806.2805-3-3-4', '0806.2805-4-3-4'], ['0806.2805-3-34-0', '0806.2805-4-37-0'], ['0806.2805-3-34-1', '0806.2805-4-37-1'], ['0806.2805-3-34-3', '0806.2805-4-37-3'], ['0806.2805-3-15-0', '0806.2805-4-16-0'], ['0806.2805-3-66-0', '0806.2805-4-70-0'], ['0806.2805-3-66-1', '0806.2805-4-70-1'], ['0806.2805-3-27-0', '0806.2805-4-29-0'], ['0806.2805-3-27-1', '0806.2805-4-29-1'], ['0806.2805-3-27-2', '0806.2805-4-29-2'], ['0806.2805-3-27-3', '0806.2805-4-29-3'], ['0806.2805-3-6-0', '0806.2805-4-6-0'], ['0806.2805-3-6-1', '0806.2805-4-6-1'], ['0806.2805-3-6-2', '0806.2805-4-6-2'], ['0806.2805-3-23-0', '0806.2805-4-25-0'], ['0806.2805-3-23-1', '0806.2805-4-25-1'], ['0806.2805-3-23-2', '0806.2805-4-25-2'], ['0806.2805-3-23-3', '0806.2805-4-25-3'], ['0806.2805-3-23-4', '0806.2805-4-25-4'], ['0806.2805-3-23-5', '0806.2805-4-25-5'], ['0806.2805-3-23-7', '0806.2805-4-25-7'], ['0806.2805-3-5-0', '0806.2805-4-5-0'], ['0806.2805-3-42-0', '0806.2805-4-45-0'], ['0806.2805-3-44-0', '0806.2805-4-47-0'], ['0806.2805-3-44-1', '0806.2805-4-47-1'], ['0806.2805-3-44-2', '0806.2805-4-47-2'], ['0806.2805-3-44-3', '0806.2805-4-47-3'], ['0806.2805-3-38-0', '0806.2805-4-41-0'], ['0806.2805-3-38-1', '0806.2805-4-41-1'], ['0806.2805-3-35-0', '0806.2805-4-38-0'], ['0806.2805-3-35-1', '0806.2805-4-38-1'], ['0806.2805-3-35-2', '0806.2805-4-38-2'], ['0806.2805-3-12-2', '0806.2805-4-12-3'], ['0806.2805-3-57-0', '0806.2805-4-61-0'], ['0806.2805-3-57-1', '0806.2805-4-61-1'], ['0806.2805-3-57-2', '0806.2805-4-61-2'], ['0806.2805-3-10-1', '0806.2805-4-10-1'], ['0806.2805-3-10-2', '0806.2805-4-10-2'], ['0806.2805-3-10-3', '0806.2805-4-10-3'], ['0806.2805-3-10-5', '0806.2805-4-10-5'], ['0806.2805-3-62-0', '0806.2805-4-66-0'], ['0806.2805-3-62-3', '0806.2805-4-66-4'], ['0806.2805-3-62-4', '0806.2805-4-66-5'], ['0806.2805-3-63-0', '0806.2805-4-67-0'], ['0806.2805-3-63-1', '0806.2805-4-67-1'], ['0806.2805-3-4-0', '0806.2805-4-4-0'], ['0806.2805-3-4-1', '0806.2805-4-4-1'], ['0806.2805-3-4-2', '0806.2805-4-4-2'], ['0806.2805-3-0-0', '0806.2805-4-0-0'], ['0806.2805-3-0-2', '0806.2805-4-0-2'], ['0806.2805-3-0-3', '0806.2805-4-0-3'], ['0806.2805-3-0-4', '0806.2805-4-0-4'], ['0806.2805-3-47-0', '0806.2805-4-51-0'], ['0806.2805-3-47-1', '0806.2805-4-51-1'], ['0806.2805-3-47-2', '0806.2805-4-51-2'], ['0806.2805-3-47-3', '0806.2805-4-51-3'], ['0806.2805-3-47-4', '0806.2805-4-51-4'], ['0806.2805-3-47-5', '0806.2805-4-51-5'], ['0806.2805-3-47-6', '0806.2805-4-51-6'], ['0806.2805-3-49-1', '0806.2805-4-53-1'], ['0806.2805-3-49-3', '0806.2805-4-53-3'], ['0806.2805-3-49-4', '0806.2805-4-53-4'], ['0806.2805-3-67-1', '0806.2805-4-71-1'], ['0806.2805-3-39-0', '0806.2805-4-42-0'], ['0806.2805-3-39-1', '0806.2805-4-42-1'], ['0806.2805-3-39-2', '0806.2805-4-42-2'], ['0806.2805-3-39-3', '0806.2805-4-42-3'], ['0806.2805-3-39-4', '0806.2805-4-42-4'], ['0806.2805-3-17-0', '0806.2805-4-19-0'], ['0806.2805-3-17-1', '0806.2805-4-19-1'], ['0806.2805-3-17-2', '0806.2805-4-19-2'], ['0806.2805-3-17-3', '0806.2805-4-19-3'], ['0806.2805-3-16-2', '0806.2805-4-18-1'], ['0806.2805-3-16-3', '0806.2805-4-18-2'], ['0806.2805-3-18-2', '0806.2805-4-20-2'], ['0806.2805-3-18-3', '0806.2805-4-20-3'], ['0806.2805-3-55-1', '0806.2805-4-59-1'], ['0806.2805-3-29-2', '0806.2805-4-31-2'], ['0806.2805-3-59-0', '0806.2805-4-63-0'], ['0806.2805-3-59-1', '0806.2805-4-63-1'], ['0806.2805-3-48-0', '0806.2805-4-52-0'], ['0806.2805-3-28-0', '0806.2805-4-30-0'], ['0806.2805-3-28-1', '0806.2805-4-30-1'], ['0806.2805-3-8-2', '0806.2805-4-8-2'], ['0806.2805-3-8-4', '0806.2805-4-8-4'], ['0806.2805-3-56-0', '0806.2805-4-60-0'], ['0806.2805-3-56-1', '0806.2805-4-60-1'], ['0806.2805-3-56-2', '0806.2805-4-60-2'], ['0806.2805-3-56-3', '0806.2805-4-60-3'], ['0806.2805-3-56-4', '0806.2805-4-60-4'], ['0806.2805-3-56-5', '0806.2805-4-60-5'], ['0806.2805-4-67-0', '0806.2805-5-67-0'], ['0806.2805-4-67-1', '0806.2805-5-67-1'], ['0806.2805-4-67-2', '0806.2805-5-67-2'], ['0806.2805-4-67-3', '0806.2805-5-67-3'], ['0806.2805-4-67-4', '0806.2805-5-67-4'], ['0806.2805-4-53-0', '0806.2805-5-53-0'], ['0806.2805-4-53-1', '0806.2805-5-53-1'], ['0806.2805-4-53-2', '0806.2805-5-53-2'], ['0806.2805-4-53-3', '0806.2805-5-53-3'], ['0806.2805-4-53-4', '0806.2805-5-53-4'], ['0806.2805-4-24-0', '0806.2805-5-24-0'], ['0806.2805-4-0-0', '0806.2805-5-0-0'], ['0806.2805-4-0-2', '0806.2805-5-0-2'], ['0806.2805-4-0-3', '0806.2805-5-0-3'], ['0806.2805-4-0-4', '0806.2805-5-0-4'], ['0806.2805-4-65-0', '0806.2805-5-65-0'], ['0806.2805-4-65-1', '0806.2805-5-65-1'], ['0806.2805-4-65-2', '0806.2805-5-65-2'], ['0806.2805-4-65-3', '0806.2805-5-65-3'], ['0806.2805-4-65-4', '0806.2805-5-65-4'], ['0806.2805-4-29-0', '0806.2805-5-29-0'], ['0806.2805-4-29-1', '0806.2805-5-29-1'], ['0806.2805-4-29-2', '0806.2805-5-29-2'], ['0806.2805-4-29-3', '0806.2805-5-29-3'], ['0806.2805-4-29-4', '0806.2805-5-29-4'], ['0806.2805-4-56-0', '0806.2805-5-56-0'], ['0806.2805-4-66-0', '0806.2805-5-66-0'], ['0806.2805-4-66-1', '0806.2805-5-66-1'], ['0806.2805-4-66-5', '0806.2805-5-66-5'], ['0806.2805-4-14-0', '0806.2805-5-14-0'], ['0806.2805-4-14-1', '0806.2805-5-14-1'], ['0806.2805-4-14-2', '0806.2805-5-14-2'], ['0806.2805-4-14-3', '0806.2805-5-14-3'], ['0806.2805-4-14-4', '0806.2805-5-14-4'], ['0806.2805-4-42-0', '0806.2805-5-42-0'], ['0806.2805-4-42-1', '0806.2805-5-42-1'], ['0806.2805-4-42-2', '0806.2805-5-42-2'], ['0806.2805-4-42-3', '0806.2805-5-42-3'], ['0806.2805-4-42-4', '0806.2805-5-42-4'], ['0806.2805-4-36-0', '0806.2805-5-36-0'], ['0806.2805-4-36-1', '0806.2805-5-36-1'], ['0806.2805-4-50-0', '0806.2805-5-50-0'], ['0806.2805-4-50-1', '0806.2805-5-50-1'], ['0806.2805-4-50-2', '0806.2805-5-50-2'], ['0806.2805-4-52-0', '0806.2805-5-52-0'], ['0806.2805-4-52-1', '0806.2805-5-52-1'], ['0806.2805-4-71-1', '0806.2805-5-72-1'], ['0806.2805-4-43-0', '0806.2805-5-43-0'], ['0806.2805-4-43-1', '0806.2805-5-43-1'], ['0806.2805-4-70-0', '0806.2805-5-70-0'], ['0806.2805-4-70-1', '0806.2805-5-70-1'], ['0806.2805-4-55-0', '0806.2805-5-55-0'], ['0806.2805-4-55-1', '0806.2805-5-55-1'], ['0806.2805-4-55-2', '0806.2805-5-55-2'], ['0806.2805-4-38-0', '0806.2805-5-38-0'], ['0806.2805-4-38-1', '0806.2805-5-38-1'], ['0806.2805-4-38-2', '0806.2805-5-38-2'], ['0806.2805-4-38-3', '0806.2805-5-38-3'], ['0806.2805-4-59-0', '0806.2805-5-59-0'], ['0806.2805-4-59-1', '0806.2805-5-59-1'], ['0806.2805-4-12-0', '0806.2805-5-12-0'], ['0806.2805-4-12-1', '0806.2805-5-12-1'], ['0806.2805-4-12-2', '0806.2805-5-12-3'], ['0806.2805-4-12-3', '0806.2805-5-12-4'], ['0806.2805-4-8-0', '0806.2805-5-8-0'], ['0806.2805-4-8-1', '0806.2805-5-8-1'], ['0806.2805-4-8-2', '0806.2805-5-8-2'], ['0806.2805-4-8-3', '0806.2805-5-8-3'], ['0806.2805-4-8-4', '0806.2805-5-8-4'], ['0806.2805-4-47-0', '0806.2805-5-47-0'], ['0806.2805-4-47-1', '0806.2805-5-47-1'], ['0806.2805-4-47-2', '0806.2805-5-47-2'], ['0806.2805-4-47-3', '0806.2805-5-47-3'], ['0806.2805-4-47-4', '0806.2805-5-47-4'], ['0806.2805-4-61-0', '0806.2805-5-61-0'], ['0806.2805-4-61-1', '0806.2805-5-61-1'], ['0806.2805-4-61-2', '0806.2805-5-61-2'], ['0806.2805-4-45-0', '0806.2805-5-45-0'], ['0806.2805-4-10-0', '0806.2805-5-10-0'], ['0806.2805-4-10-1', '0806.2805-5-10-1'], ['0806.2805-4-10-2', '0806.2805-5-10-2'], ['0806.2805-4-10-3', '0806.2805-5-10-3'], ['0806.2805-4-10-4', '0806.2805-5-10-4'], ['0806.2805-4-10-5', '0806.2805-5-10-5'], ['0806.2805-4-22-0', '0806.2805-5-22-0'], ['0806.2805-4-22-1', '0806.2805-5-22-1'], ['0806.2805-4-22-2', '0806.2805-5-22-2'], ['0806.2805-4-16-0', '0806.2805-5-16-0'], ['0806.2805-4-57-0', '0806.2805-5-57-0'], ['0806.2805-4-20-0', '0806.2805-5-20-0'], ['0806.2805-4-20-1', '0806.2805-5-20-1'], ['0806.2805-4-20-2', '0806.2805-5-20-2'], ['0806.2805-4-20-3', '0806.2805-5-20-3'], ['0806.2805-4-20-4', '0806.2805-5-20-4'], ['0806.2805-4-20-5', '0806.2805-5-20-5'], ['0806.2805-4-20-6', '0806.2805-5-20-6'], ['0806.2805-4-33-0', '0806.2805-5-33-0'], ['0806.2805-4-33-1', '0806.2805-5-33-1'], ['0806.2805-4-33-2', '0806.2805-5-33-2'], ['0806.2805-4-44-0', '0806.2805-5-44-0'], ['0806.2805-4-44-1', '0806.2805-5-44-1'], ['0806.2805-4-44-2', '0806.2805-5-44-2'], ['0806.2805-4-19-0', '0806.2805-5-19-0'], ['0806.2805-4-19-1', '0806.2805-5-19-1'], ['0806.2805-4-19-2', '0806.2805-5-19-2'], ['0806.2805-4-19-3', '0806.2805-5-19-3'], ['0806.2805-4-69-0', '0806.2805-5-69-0'], ['0806.2805-4-69-1', '0806.2805-5-69-1'], ['0806.2805-4-13-0', '0806.2805-5-13-0'], ['0806.2805-4-26-0', '0806.2805-5-26-0'], ['0806.2805-4-37-0', '0806.2805-5-37-0'], ['0806.2805-4-37-1', '0806.2805-5-37-1'], ['0806.2805-4-37-2', '0806.2805-5-37-2'], ['0806.2805-4-37-3', '0806.2805-5-37-3'], ['0806.2805-4-2-0', '0806.2805-5-2-0'], ['0806.2805-4-2-1', '0806.2805-5-2-1'], ['0806.2805-4-2-2', '0806.2805-5-2-2'], ['0806.2805-4-17-0', '0806.2805-5-17-0'], ['0806.2805-4-17-1', '0806.2805-5-17-1'], ['0806.2805-4-17-2', '0806.2805-5-17-2'], ['0806.2805-4-17-3', '0806.2805-5-17-3'], ['0806.2805-4-49-0', '0806.2805-5-49-0'], ['0806.2805-4-15-0', '0806.2805-5-15-0'], ['0806.2805-4-15-1', '0806.2805-5-15-1'], ['0806.2805-4-15-2', '0806.2805-5-15-2'], ['0806.2805-4-15-3', '0806.2805-5-15-3'], ['0806.2805-4-15-4', '0806.2805-5-15-4'], ['0806.2805-4-23-0', '0806.2805-5-23-0'], ['0806.2805-4-23-1', '0806.2805-5-23-1'], ['0806.2805-4-31-0', '0806.2805-5-31-0'], ['0806.2805-4-31-1', '0806.2805-5-31-1'], ['0806.2805-4-31-2', '0806.2805-5-31-2'], ['0806.2805-4-4-0', '0806.2805-5-4-0'], ['0806.2805-4-4-1', '0806.2805-5-4-1'], ['0806.2805-4-4-2', '0806.2805-5-4-2'], ['0806.2805-4-25-0', '0806.2805-5-25-0'], ['0806.2805-4-25-1', '0806.2805-5-25-1'], ['0806.2805-4-25-2', '0806.2805-5-25-2'], ['0806.2805-4-25-3', '0806.2805-5-25-3'], ['0806.2805-4-25-4', '0806.2805-5-25-4'], ['0806.2805-4-25-5', '0806.2805-5-25-5'], ['0806.2805-4-25-6', '0806.2805-5-25-6'], ['0806.2805-4-25-7', '0806.2805-5-25-7'], ['0806.2805-4-25-8', '0806.2805-5-25-8'], ['0806.2805-4-5-0', '0806.2805-5-5-0'], ['0806.2805-4-5-1', '0806.2805-5-5-1'], ['0806.2805-4-11-0', '0806.2805-5-11-0'], ['0806.2805-4-11-1', '0806.2805-5-11-1'], ['0806.2805-4-11-2', '0806.2805-5-11-2'], ['0806.2805-4-63-0', '0806.2805-5-63-0'], ['0806.2805-4-63-1', '0806.2805-5-63-1'], ['0806.2805-4-34-0', '0806.2805-5-34-0'], ['0806.2805-4-34-1', '0806.2805-5-34-1'], ['0806.2805-4-34-2', '0806.2805-5-34-2'], ['0806.2805-4-34-3', '0806.2805-5-34-3'], ['0806.2805-4-34-4', '0806.2805-5-34-4'], ['0806.2805-4-34-5', '0806.2805-5-34-5'], ['0806.2805-4-34-6', '0806.2805-5-34-6'], ['0806.2805-4-18-0', '0806.2805-5-18-0'], ['0806.2805-4-18-1', '0806.2805-5-18-1'], ['0806.2805-4-18-2', '0806.2805-5-18-2'], ['0806.2805-4-41-0', '0806.2805-5-41-0'], ['0806.2805-4-41-1', '0806.2805-5-41-1'], ['0806.2805-4-41-2', '0806.2805-5-41-2'], ['0806.2805-4-7-0', '0806.2805-5-7-0'], ['0806.2805-4-7-1', '0806.2805-5-7-1'], ['0806.2805-4-51-0', '0806.2805-5-51-0'], ['0806.2805-4-51-1', '0806.2805-5-51-1'], ['0806.2805-4-51-2', '0806.2805-5-51-2'], ['0806.2805-4-51-3', '0806.2805-5-51-3'], ['0806.2805-4-51-4', '0806.2805-5-51-4'], ['0806.2805-4-51-5', '0806.2805-5-51-5'], ['0806.2805-4-51-6', '0806.2805-5-51-6'], ['0806.2805-4-64-0', '0806.2805-5-64-0'], ['0806.2805-4-64-1', '0806.2805-5-64-1'], ['0806.2805-4-64-2', '0806.2805-5-64-2'], ['0806.2805-4-64-3', '0806.2805-5-64-3'], ['0806.2805-4-30-0', '0806.2805-5-30-0'], ['0806.2805-4-30-1', '0806.2805-5-30-1'], ['0806.2805-4-6-0', '0806.2805-5-6-0'], ['0806.2805-4-6-1', '0806.2805-5-6-1'], ['0806.2805-4-6-2', '0806.2805-5-6-2'], ['0806.2805-4-6-3', '0806.2805-5-6-3'], ['0806.2805-4-3-0', '0806.2805-5-3-0'], ['0806.2805-4-3-1', '0806.2805-5-3-1'], ['0806.2805-4-3-2', '0806.2805-5-3-2'], ['0806.2805-4-3-3', '0806.2805-5-3-3'], ['0806.2805-4-3-4', '0806.2805-5-3-4'], ['0806.2805-4-60-0', '0806.2805-5-60-0'], ['0806.2805-4-60-1', '0806.2805-5-60-1'], ['0806.2805-4-60-2', '0806.2805-5-60-2'], ['0806.2805-4-60-3', '0806.2805-5-60-3'], ['0806.2805-4-60-4', '0806.2805-5-60-4'], ['0806.2805-4-60-5', '0806.2805-5-60-5'], ['0806.2805-4-60-6', '0806.2805-5-60-6'], ['0806.2805-5-2-1', '0806.2805-6-2-1'], ['0806.2805-5-2-2', '0806.2805-6-2-2'], ['0806.2805-5-30-0', '0806.2805-6-32-0'], ['0806.2805-5-30-1', '0806.2805-6-32-1'], ['0806.2805-5-61-0', '0806.2805-6-65-0'], ['0806.2805-5-61-2', '0806.2805-6-65-2'], ['0806.2805-5-67-0', '0806.2805-6-71-0'], ['0806.2805-5-67-1', '0806.2805-6-71-1'], ['0806.2805-5-67-2', '0806.2805-6-71-2'], ['0806.2805-5-67-3', '0806.2805-6-71-3'], ['0806.2805-5-67-4', '0806.2805-6-71-4'], ['0806.2805-5-7-0', '0806.2805-6-7-0'], ['0806.2805-5-7-1', '0806.2805-6-7-1'], ['0806.2805-5-37-0', '0806.2805-6-40-0'], ['0806.2805-5-37-1', '0806.2805-6-40-1'], ['0806.2805-5-37-2', '0806.2805-6-40-2'], ['0806.2805-5-37-3', '0806.2805-6-40-3'], ['0806.2805-5-14-0', '0806.2805-6-14-0'], ['0806.2805-5-14-1', '0806.2805-6-14-1'], ['0806.2805-5-14-2', '0806.2805-6-14-2'], ['0806.2805-5-14-3', '0806.2805-6-14-3'], ['0806.2805-5-14-4', '0806.2805-6-14-4'], ['0806.2805-5-17-0', '0806.2805-6-17-0'], ['0806.2805-5-17-1', '0806.2805-6-17-1'], ['0806.2805-5-17-2', '0806.2805-6-17-2'], ['0806.2805-5-29-0', '0806.2805-6-31-0'], ['0806.2805-5-29-1', '0806.2805-6-31-1'], ['0806.2805-5-29-2', '0806.2805-6-31-2'], ['0806.2805-5-29-3', '0806.2805-6-31-3'], ['0806.2805-5-29-4', '0806.2805-6-31-4'], ['0806.2805-5-41-0', '0806.2805-6-44-0'], ['0806.2805-5-41-1', '0806.2805-6-44-1'], ['0806.2805-5-41-2', '0806.2805-6-44-2'], ['0806.2805-5-66-0', '0806.2805-6-70-0'], ['0806.2805-5-64-2', '0806.2805-6-68-2'], ['0806.2805-5-64-3', '0806.2805-6-68-3'], ['0806.2805-5-65-0', '0806.2805-6-69-0'], ['0806.2805-5-65-1', '0806.2805-6-69-1'], ['0806.2805-5-65-2', '0806.2805-6-69-2'], ['0806.2805-5-65-3', '0806.2805-6-69-3'], ['0806.2805-5-65-4', '0806.2805-6-69-4'], ['0806.2805-5-50-0', '0806.2805-6-53-0'], ['0806.2805-5-50-2', '0806.2805-6-53-2'], ['0806.2805-5-53-1', '0806.2805-6-56-1'], ['0806.2805-5-53-3', '0806.2805-6-56-3'], ['0806.2805-5-3-0', '0806.2805-6-3-0'], ['0806.2805-5-3-1', '0806.2805-6-3-1'], ['0806.2805-5-3-3', '0806.2805-6-3-3'], ['0806.2805-5-3-4', '0806.2805-6-3-4'], ['0806.2805-5-38-0', '0806.2805-6-41-0'], ['0806.2805-5-38-1', '0806.2805-6-41-1'], ['0806.2805-5-38-2', '0806.2805-6-41-2'], ['0806.2805-5-38-3', '0806.2805-6-41-3'], ['0806.2805-5-33-1', '0806.2805-6-35-1'], ['0806.2805-5-33-2', '0806.2805-6-35-2'], ['0806.2805-5-49-0', '0806.2805-6-52-0'], ['0806.2805-5-51-0', '0806.2805-6-54-0'], ['0806.2805-5-51-1', '0806.2805-6-54-1'], ['0806.2805-5-51-2', '0806.2805-6-54-2'], ['0806.2805-5-51-3', '0806.2805-6-54-3'], ['0806.2805-5-51-4', '0806.2805-6-54-4'], ['0806.2805-5-51-5', '0806.2805-6-54-5'], ['0806.2805-5-51-6', '0806.2805-6-54-6'], ['0806.2805-5-15-1', '0806.2805-6-15-1'], ['0806.2805-5-15-2', '0806.2805-6-15-2'], ['0806.2805-5-15-3', '0806.2805-6-15-3'], ['0806.2805-5-15-4', '0806.2805-6-15-4'], ['0806.2805-5-31-2', '0806.2805-6-33-2'], ['0806.2805-5-69-0', '0806.2805-6-73-0'], ['0806.2805-5-69-1', '0806.2805-6-73-1'], ['0806.2805-5-11-0', '0806.2805-6-11-0'], ['0806.2805-5-11-1', '0806.2805-6-11-1'], ['0806.2805-5-11-2', '0806.2805-6-11-2'], ['0806.2805-5-5-0', '0806.2805-6-5-0'], ['0806.2805-5-20-0', '0806.2805-6-21-0'], ['0806.2805-5-20-1', '0806.2805-6-21-1'], ['0806.2805-5-20-2', '0806.2805-6-21-2'], ['0806.2805-5-20-3', '0806.2805-6-21-3'], ['0806.2805-5-20-5', '0806.2805-6-21-5'], ['0806.2805-5-20-6', '0806.2805-6-21-6'], ['0806.2805-5-36-0', '0806.2805-6-39-0'], ['0806.2805-5-36-1', '0806.2805-6-39-1'], ['0806.2805-5-44-1', '0806.2805-6-47-1'], ['0806.2805-5-22-0', '0806.2805-6-23-0'], ['0806.2805-5-22-1', '0806.2805-6-23-1'], ['0806.2805-5-22-2', '0806.2805-6-23-2'], ['0806.2805-5-26-0', '0806.2805-6-28-0'], ['0806.2805-5-59-0', '0806.2805-6-62-0'], ['0806.2805-5-59-1', '0806.2805-6-62-1'], ['0806.2805-5-8-0', '0806.2805-6-8-0'], ['0806.2805-5-8-1', '0806.2805-6-8-1'], ['0806.2805-5-8-2', '0806.2805-6-8-2'], ['0806.2805-5-8-3', '0806.2805-6-8-3'], ['0806.2805-5-8-4', '0806.2805-6-8-4'], ['0806.2805-5-63-0', '0806.2805-6-67-0'], ['0806.2805-5-63-1', '0806.2805-6-67-1'], ['0806.2805-5-0-0', '0806.2805-6-0-0'], ['0806.2805-5-0-1', '0806.2805-6-0-1'], ['0806.2805-5-42-1', '0806.2805-6-45-1'], ['0806.2805-5-42-2', '0806.2805-6-45-2'], ['0806.2805-5-42-3', '0806.2805-6-45-3'], ['0806.2805-5-42-4', '0806.2805-6-45-4'], ['0806.2805-5-12-2', '0806.2805-6-12-3'], ['0806.2805-5-12-4', '0806.2805-6-12-5'], ['0806.2805-5-52-0', '0806.2805-6-55-0'], ['0806.2805-5-52-1', '0806.2805-6-55-1'], ['0806.2805-5-23-1', '0806.2805-6-24-1'], ['0806.2805-5-19-0', '0806.2805-6-20-0'], ['0806.2805-5-19-1', '0806.2805-6-20-1'], ['0806.2805-5-19-3', '0806.2805-6-20-3'], ['0806.2805-5-72-0', '0806.2805-6-76-0'], ['0806.2805-5-43-1', '0806.2805-6-46-1'], ['0806.2805-5-4-0', '0806.2805-6-4-0'], ['0806.2805-5-24-0', '0806.2805-6-25-0'], ['0806.2805-5-47-0', '0806.2805-6-50-0'], ['0806.2805-5-47-1', '0806.2805-6-50-1'], ['0806.2805-5-47-2', '0806.2805-6-50-2'], ['0806.2805-5-47-3', '0806.2805-6-50-3'], ['0806.2805-5-47-4', '0806.2805-6-50-4'], ['0806.2805-5-55-2', '0806.2805-6-58-2'], ['0806.2805-5-70-0', '0806.2805-6-74-0'], ['0806.2805-5-70-1', '0806.2805-6-74-1'], ['0806.2805-5-10-0', '0806.2805-6-10-0'], ['0806.2805-5-10-1', '0806.2805-6-10-1'], ['0806.2805-5-10-3', '0806.2805-6-10-3'], ['0806.2805-5-10-4', '0806.2805-6-10-4'], ['0806.2805-5-10-5', '0806.2805-6-10-5'], ['0806.2805-5-6-0', '0806.2805-6-6-0'], ['0806.2805-5-6-1', '0806.2805-6-6-1'], ['0806.2805-5-6-2', '0806.2805-6-6-2'], ['0806.2805-6-45-0', '0806.2805-7-45-0'], ['0806.2805-6-45-1', '0806.2805-7-45-1'], ['0806.2805-6-45-2', '0806.2805-7-45-2'], ['0806.2805-6-45-3', '0806.2805-7-45-3'], ['0806.2805-6-45-4', '0806.2805-7-45-4'], ['0806.2805-6-67-0', '0806.2805-7-67-0'], ['0806.2805-6-67-1', '0806.2805-7-67-1'], ['0806.2805-6-12-0', '0806.2805-7-12-0'], ['0806.2805-6-12-1', '0806.2805-7-12-1'], ['0806.2805-6-12-2', '0806.2805-7-12-2'], ['0806.2805-6-12-3', '0806.2805-7-12-3'], ['0806.2805-6-12-4', '0806.2805-7-12-4'], ['0806.2805-6-12-5', '0806.2805-7-12-5'], ['0806.2805-6-37-0', '0806.2805-7-37-0'], ['0806.2805-6-39-0', '0806.2805-7-39-0'], ['0806.2805-6-39-1', '0806.2805-7-39-1'], ['0806.2805-6-41-0', '0806.2805-7-41-0'], ['0806.2805-6-41-1', '0806.2805-7-41-1'], ['0806.2805-6-41-2', '0806.2805-7-41-2'], ['0806.2805-6-41-3', '0806.2805-7-41-3'], ['0806.2805-6-53-0', '0806.2805-7-53-0'], ['0806.2805-6-53-1', '0806.2805-7-53-1'], ['0806.2805-6-53-2', '0806.2805-7-53-2'], ['0806.2805-6-47-0', '0806.2805-7-47-0'], ['0806.2805-6-47-1', '0806.2805-7-47-1'], ['0806.2805-6-47-2', '0806.2805-7-47-2'], ['0806.2805-6-59-0', '0806.2805-7-59-0'], ['0806.2805-6-5-0', '0806.2805-7-5-0'], ['0806.2805-6-5-1', '0806.2805-7-5-1'], ['0806.2805-6-54-0', '0806.2805-7-54-0'], ['0806.2805-6-54-1', '0806.2805-7-54-1'], ['0806.2805-6-54-2', '0806.2805-7-54-2'], ['0806.2805-6-54-3', '0806.2805-7-54-3'], ['0806.2805-6-54-4', '0806.2805-7-54-4'], ['0806.2805-6-54-5', '0806.2805-7-54-5'], ['0806.2805-6-54-6', '0806.2805-7-54-6'], ['0806.2805-6-13-0', '0806.2805-7-13-0'], ['0806.2805-6-19-0', '0806.2805-7-19-0'], ['0806.2805-6-19-1', '0806.2805-7-19-1'], ['0806.2805-6-27-0', '0806.2805-7-27-0'], ['0806.2805-6-27-1', '0806.2805-7-27-1'], ['0806.2805-6-27-2', '0806.2805-7-27-2'], ['0806.2805-6-27-3', '0806.2805-7-27-3'], ['0806.2805-6-27-4', '0806.2805-7-27-4'], ['0806.2805-6-27-5', '0806.2805-7-27-5'], ['0806.2805-6-27-6', '0806.2805-7-27-6'], ['0806.2805-6-26-0', '0806.2805-7-26-0'], ['0806.2805-6-26-1', '0806.2805-7-26-1'], ['0806.2805-6-76-0', '0806.2805-7-76-0'], ['0806.2805-6-76-1', '0806.2805-7-76-1'], ['0806.2805-6-70-0', '0806.2805-7-70-0'], ['0806.2805-6-70-1', '0806.2805-7-70-1'], ['0806.2805-6-70-2', '0806.2805-7-70-2'], ['0806.2805-6-70-3', '0806.2805-7-70-3'], ['0806.2805-6-70-4', '0806.2805-7-70-4'], ['0806.2805-6-70-5', '0806.2805-7-70-5'], ['0806.2805-6-21-0', '0806.2805-7-21-0'], ['0806.2805-6-21-1', '0806.2805-7-21-1'], ['0806.2805-6-21-2', '0806.2805-7-21-2'], ['0806.2805-6-21-3', '0806.2805-7-21-3'], ['0806.2805-6-21-4', '0806.2805-7-21-4'], ['0806.2805-6-21-5', '0806.2805-7-21-5'], ['0806.2805-6-21-6', '0806.2805-7-21-6'], ['0806.2805-6-20-0', '0806.2805-7-20-0'], ['0806.2805-6-20-1', '0806.2805-7-20-1'], ['0806.2805-6-20-2', '0806.2805-7-20-2'], ['0806.2805-6-20-3', '0806.2805-7-20-3'], ['0806.2805-6-16-0', '0806.2805-7-16-0'], ['0806.2805-6-17-0', '0806.2805-7-17-0'], ['0806.2805-6-17-1', '0806.2805-7-17-1'], ['0806.2805-6-17-2', '0806.2805-7-17-2'], ['0806.2805-6-17-3', '0806.2805-7-17-3'], ['0806.2805-6-64-0', '0806.2805-7-64-0'], ['0806.2805-6-64-1', '0806.2805-7-64-1'], ['0806.2805-6-71-0', '0806.2805-7-71-0'], ['0806.2805-6-71-1', '0806.2805-7-71-1'], ['0806.2805-6-71-2', '0806.2805-7-71-2'], ['0806.2805-6-71-3', '0806.2805-7-71-3'], ['0806.2805-6-71-4', '0806.2805-7-71-4'], ['0806.2805-6-15-0', '0806.2805-7-15-0'], ['0806.2805-6-15-1', '0806.2805-7-15-1'], ['0806.2805-6-15-2', '0806.2805-7-15-2'], ['0806.2805-6-15-3', '0806.2805-7-15-3'], ['0806.2805-6-15-4', '0806.2805-7-15-4'], ['0806.2805-6-73-0', '0806.2805-7-73-0'], ['0806.2805-6-73-1', '0806.2805-7-73-1'], ['0806.2805-6-63-0', '0806.2805-7-63-0'], ['0806.2805-6-63-1', '0806.2805-7-63-1'], ['0806.2805-6-63-2', '0806.2805-7-63-2'], ['0806.2805-6-63-3', '0806.2805-7-63-3'], ['0806.2805-6-63-4', '0806.2805-7-63-4'], ['0806.2805-6-63-5', '0806.2805-7-63-5'], ['0806.2805-6-63-6', '0806.2805-7-63-6'], ['0806.2805-6-58-0', '0806.2805-7-58-0'], ['0806.2805-6-58-1', '0806.2805-7-58-1'], ['0806.2805-6-58-2', '0806.2805-7-58-2'], ['0806.2805-6-56-0', '0806.2805-7-56-0'], ['0806.2805-6-56-1', '0806.2805-7-56-1'], ['0806.2805-6-56-2', '0806.2805-7-56-2'], ['0806.2805-6-56-3', '0806.2805-7-56-3'], ['0806.2805-6-56-4', '0806.2805-7-56-4'], ['0806.2805-6-2-0', '0806.2805-7-2-0'], ['0806.2805-6-2-1', '0806.2805-7-2-1'], ['0806.2805-6-2-2', '0806.2805-7-2-2'], ['0806.2805-6-60-0', '0806.2805-7-60-0'], ['0806.2805-6-52-0', '0806.2805-7-52-0'], ['0806.2805-6-36-0', '0806.2805-7-36-0'], ['0806.2805-6-36-1', '0806.2805-7-36-1'], ['0806.2805-6-36-2', '0806.2805-7-36-2'], ['0806.2805-6-36-3', '0806.2805-7-36-3'], ['0806.2805-6-36-4', '0806.2805-7-36-4'], ['0806.2805-6-36-5', '0806.2805-7-36-5'], ['0806.2805-6-69-0', '0806.2805-7-69-0'], ['0806.2805-6-69-1', '0806.2805-7-69-1'], ['0806.2805-6-69-2', '0806.2805-7-69-2'], ['0806.2805-6-69-3', '0806.2805-7-69-3'], ['0806.2805-6-69-4', '0806.2805-7-69-4'], ['0806.2805-6-50-0', '0806.2805-7-50-0'], ['0806.2805-6-50-1', '0806.2805-7-50-1'], ['0806.2805-6-50-2', '0806.2805-7-50-2'], ['0806.2805-6-50-3', '0806.2805-7-50-3'], ['0806.2805-6-50-4', '0806.2805-7-50-4'], ['0806.2805-6-78-0', '0806.2805-7-78-0'], ['0806.2805-6-78-1', '0806.2805-7-78-1'], ['0806.2805-6-10-0', '0806.2805-7-10-0'], ['0806.2805-6-10-1', '0806.2805-7-10-1'], ['0806.2805-6-10-2', '0806.2805-7-10-2'], ['0806.2805-6-10-3', '0806.2805-7-10-3'], ['0806.2805-6-10-4', '0806.2805-7-10-4'], ['0806.2805-6-10-5', '0806.2805-7-10-5'], ['0806.2805-6-11-0', '0806.2805-7-11-0'], ['0806.2805-6-11-1', '0806.2805-7-11-1'], ['0806.2805-6-11-2', '0806.2805-7-11-2'], ['0806.2805-6-68-0', '0806.2805-7-68-0'], ['0806.2805-6-68-1', '0806.2805-7-68-1'], ['0806.2805-6-68-2', '0806.2805-7-68-2'], ['0806.2805-6-68-3', '0806.2805-7-68-3'], ['0806.2805-6-14-0', '0806.2805-7-14-0'], ['0806.2805-6-14-1', '0806.2805-7-14-1'], ['0806.2805-6-14-2', '0806.2805-7-14-2'], ['0806.2805-6-14-3', '0806.2805-7-14-3'], ['0806.2805-6-14-4', '0806.2805-7-14-4'], ['0806.2805-6-25-0', '0806.2805-7-25-0'], ['0806.2805-6-4-0', '0806.2805-7-4-0'], ['0806.2805-6-4-1', '0806.2805-7-4-1'], ['0806.2805-6-4-2', '0806.2805-7-4-2'], ['0806.2805-6-33-0', '0806.2805-7-33-0'], ['0806.2805-6-33-1', '0806.2805-7-33-1'], ['0806.2805-6-33-2', '0806.2805-7-33-2'], ['0806.2805-6-32-0', '0806.2805-7-32-0'], ['0806.2805-6-32-1', '0806.2805-7-32-1'], ['0806.2805-6-0-0', '0806.2805-7-0-0'], ['0806.2805-6-0-1', '0806.2805-7-0-1'], ['0806.2805-6-0-2', '0806.2805-7-0-2'], ['0806.2805-6-0-3', '0806.2805-7-0-3'], ['0806.2805-6-0-4', '0806.2805-7-0-4'], ['0806.2805-6-6-0', '0806.2805-7-6-0'], ['0806.2805-6-6-1', '0806.2805-7-6-1'], ['0806.2805-6-6-2', '0806.2805-7-6-2'], ['0806.2805-6-6-3', '0806.2805-7-6-3'], ['0806.2805-6-44-0', '0806.2805-7-44-0'], ['0806.2805-6-44-1', '0806.2805-7-44-1'], ['0806.2805-6-44-2', '0806.2805-7-44-2'], ['0806.2805-6-62-0', '0806.2805-7-62-0'], ['0806.2805-6-62-1', '0806.2805-7-62-1'], ['0806.2805-6-3-0', '0806.2805-7-3-0'], ['0806.2805-6-3-1', '0806.2805-7-3-1'], ['0806.2805-6-3-2', '0806.2805-7-3-2'], ['0806.2805-6-3-3', '0806.2805-7-3-3'], ['0806.2805-6-3-4', '0806.2805-7-3-4'], ['0806.2805-6-23-0', '0806.2805-7-23-0'], ['0806.2805-6-23-1', '0806.2805-7-23-1'], ['0806.2805-6-23-2', '0806.2805-7-23-2'], ['0806.2805-6-28-0', '0806.2805-7-28-0'], ['0806.2805-6-31-0', '0806.2805-7-31-0'], ['0806.2805-6-31-1', '0806.2805-7-31-1'], ['0806.2805-6-31-2', '0806.2805-7-31-2'], ['0806.2805-6-31-3', '0806.2805-7-31-3'], ['0806.2805-6-31-4', '0806.2805-7-31-4'], ['0806.2805-6-24-0', '0806.2805-7-24-0'], ['0806.2805-6-24-1', '0806.2805-7-24-1'], ['0806.2805-6-40-0', '0806.2805-7-40-0'], ['0806.2805-6-40-1', '0806.2805-7-40-1'], ['0806.2805-6-40-2', '0806.2805-7-40-2'], ['0806.2805-6-40-3', '0806.2805-7-40-3'], ['0806.2805-6-46-0', '0806.2805-7-46-0'], ['0806.2805-6-46-1', '0806.2805-7-46-1'], ['0806.2805-6-48-0', '0806.2805-7-48-0'], ['0806.2805-6-8-0', '0806.2805-7-8-0'], ['0806.2805-6-8-1', '0806.2805-7-8-1'], ['0806.2805-6-8-2', '0806.2805-7-8-2'], ['0806.2805-6-8-3', '0806.2805-7-8-3'], ['0806.2805-6-8-4', '0806.2805-7-8-4'], ['0806.2805-6-55-0', '0806.2805-7-55-0'], ['0806.2805-6-55-1', '0806.2805-7-55-1'], ['0806.2805-6-18-0', '0806.2805-7-18-0'], ['0806.2805-6-65-0', '0806.2805-7-65-0'], ['0806.2805-6-65-1', '0806.2805-7-65-1'], ['0806.2805-6-65-2', '0806.2805-7-65-2'], ['0806.2805-6-35-0', '0806.2805-7-35-0'], ['0806.2805-6-35-1', '0806.2805-7-35-1'], ['0806.2805-6-35-2', '0806.2805-7-35-2'], ['0806.2805-6-74-0', '0806.2805-7-74-0'], ['0806.2805-6-74-1', '0806.2805-7-74-1'], ['0806.2805-6-7-0', '0806.2805-7-7-0'], ['0806.2805-6-7-1', '0806.2805-7-7-1'], ['0806.2805-1-28-5', '0806.2805-2-29-3'], ['0806.2805-2-13-3', '0806.2805-3-16-2'], ['0806.2805-2-13-4', '0806.2805-3-16-3'], ['0806.2805-2-24-1', '0806.2805-3-29-1'], ['0806.2805-2-24-2', '0806.2805-3-29-2'], ['0806.2805-3-46-2', '0806.2805-4-50-1'], ['0806.2805-3-46-3', '0806.2805-4-50-2'], ['0806.2805-3-13-1', '0806.2805-4-14-0'], ['0806.2805-3-13-2', '0806.2805-4-14-1'], ['0806.2805-3-13-3', '0806.2805-4-14-2'], ['0806.2805-3-13-4', '0806.2805-4-14-3'], ['0806.2805-3-31-1', '0806.2805-4-33-1'], ['0806.2805-3-31-2', '0806.2805-4-33-2'], ['0806.2805-3-31-3', '0806.2805-4-34-0'], ['0806.2805-3-31-4', '0806.2805-4-34-1'], ['0806.2805-3-31-5', '0806.2805-4-34-2'], ['0806.2805-3-31-6', '0806.2805-4-34-3'], ['0806.2805-3-31-8', '0806.2805-4-34-5'], ['0806.2805-5-18-1', '0806.2805-6-19-0'], ['0806.2805-5-18-2', '0806.2805-6-19-1'], ['0806.2805-5-60-0', '0806.2805-6-63-0'], ['0806.2805-5-60-1', '0806.2805-6-63-1'], ['0806.2805-5-60-2', '0806.2805-6-63-2'], ['0806.2805-5-60-4', '0806.2805-6-63-4'], ['0806.2805-5-60-5', '0806.2805-6-63-5'], ['0806.2805-5-60-6', '0806.2805-6-63-6'], ['0806.2805-5-60-7', '0806.2805-6-64-0'], ['0806.2805-5-60-8', '0806.2805-6-64-1'], ['0806.2805-5-34-0', '0806.2805-6-36-0'], ['0806.2805-5-34-1', '0806.2805-6-36-1'], ['0806.2805-5-34-3', '0806.2805-6-36-3'], ['0806.2805-5-34-4', '0806.2805-6-36-4'], ['0806.2805-5-25-0', '0806.2805-6-26-0'], ['0806.2805-5-25-3', '0806.2805-6-27-1'], ['0806.2805-5-25-4', '0806.2805-6-27-2'], ['0806.2805-5-25-5', '0806.2805-6-27-3'], ['0806.2805-5-25-6', '0806.2805-6-27-4'], ['0806.2805-5-25-8', '0806.2805-6-27-6'], ['0806.2805-1-8-0', '0806.2805-2-8-0'], ['0806.2805-1-8-2', '0806.2805-2-8-2'], ['0806.2805-1-19-1', '0806.2805-2-19-1'], ['0806.2805-1-19-5', '0806.2805-2-19-5'], ['0806.2805-1-19-6', '0806.2805-2-19-6'], ['0806.2805-1-19-7', '0806.2805-2-19-7'], ['0806.2805-1-11-0', '0806.2805-2-11-0'], ['0806.2805-1-11-1', '0806.2805-2-11-1'], ['0806.2805-1-3-3', '0806.2805-2-3-3'], ['0806.2805-1-3-4', '0806.2805-2-3-4'], ['0806.2805-1-2-0', '0806.2805-2-2-0'], ['0806.2805-1-6-1', '0806.2805-2-6-1'], ['0806.2805-1-9-0', '0806.2805-2-9-0'], ['0806.2805-1-9-2', '0806.2805-2-9-2'], ['0806.2805-1-17-0', '0806.2805-2-17-0'], ['0806.2805-1-17-1', '0806.2805-2-17-1'], ['0806.2805-1-14-0', '0806.2805-2-14-0'], ['0806.2805-1-14-3', '0806.2805-2-14-3'], ['0806.2805-1-55-0', '0806.2805-2-58-0'], ['0806.2805-1-52-1', '0806.2805-2-53-1'], ['0806.2805-1-39-0', '0806.2805-2-40-0'], ['0806.2805-1-39-3', '0806.2805-2-40-3'], ['0806.2805-1-47-0', '0806.2805-2-48-0'], ['0806.2805-1-53-0', '0806.2805-2-55-0'], ['0806.2805-1-53-1', '0806.2805-2-55-1'], ['0806.2805-1-37-1', '0806.2805-2-38-1'], ['0806.2805-1-37-2', '0806.2805-2-38-2'], ['0806.2805-1-44-0', '0806.2805-2-45-0'], ['0806.2805-1-0-0', '0806.2805-2-0-0'], ['0806.2805-1-0-1', '0806.2805-2-0-1'], ['0806.2805-1-0-3', '0806.2805-2-0-3'], ['0806.2805-1-40-1', '0806.2805-2-41-1'], ['0806.2805-1-40-4', '0806.2805-2-41-4'], ['0806.2805-1-35-2', '0806.2805-2-36-2'], ['0806.2805-1-4-0', '0806.2805-2-4-0'], ['0806.2805-1-4-1', '0806.2805-2-4-1'], ['0806.2805-1-4-2', '0806.2805-2-4-2'], ['0806.2805-1-32-0', '0806.2805-2-33-0'], ['0806.2805-1-32-1', '0806.2805-2-33-1'], ['0806.2805-1-32-2', '0806.2805-2-33-2'], ['0806.2805-1-23-1', '0806.2805-2-23-1'], ['0806.2805-1-5-1', '0806.2805-2-5-1'], ['0806.2805-1-5-2', '0806.2805-2-5-2'], ['0806.2805-1-51-1', '0806.2805-2-52-1'], ['0806.2805-1-43-2', '0806.2805-2-44-2'], ['0806.2805-1-26-2', '0806.2805-2-26-2'], ['0806.2805-1-26-3', '0806.2805-2-26-3'], ['0806.2805-1-26-8', '0806.2805-2-26-8'], ['0806.2805-1-12-0', '0806.2805-2-12-0'], ['0806.2805-1-12-4', '0806.2805-2-12-4'], ['0806.2805-1-29-2', '0806.2805-2-30-2'], ['0806.2805-1-24-0', '0806.2805-2-24-0'], ['0806.2805-1-56-0', '0806.2805-2-59-0'], ['0806.2805-1-48-1', '0806.2805-2-49-1'], ['0806.2805-1-33-0', '0806.2805-2-34-0'], ['0806.2805-1-33-2', '0806.2805-2-34-2'], ['0806.2805-1-41-3', '0806.2805-2-42-3'], ['0806.2805-1-41-4', '0806.2805-2-42-4'], ['0806.2805-2-40-1', '0806.2805-3-46-1'], ['0806.2805-2-40-3', '0806.2805-3-46-3'], ['0806.2805-2-41-3', '0806.2805-3-47-3'], ['0806.2805-2-41-6', '0806.2805-3-47-6'], ['0806.2805-2-4-2', '0806.2805-3-4-2'], ['0806.2805-2-12-3', '0806.2805-3-14-3'], ['0806.2805-2-0-0', '0806.2805-3-0-0'], ['0806.2805-2-0-1', '0806.2805-3-0-1'], ['0806.2805-2-33-2', '0806.2805-3-38-2'], ['0806.2805-2-52-1', '0806.2805-3-59-1'], ['0806.2805-2-2-0', '0806.2805-3-2-0'], ['0806.2805-2-10-0', '0806.2805-3-12-0'], ['0806.2805-2-19-3', '0806.2805-3-23-3'], ['0806.2805-2-28-0', '0806.2805-3-33-0'], ['0806.2805-2-55-2', '0806.2805-3-62-4'], ['0806.2805-2-42-4', '0806.2805-3-49-4'], ['0806.2805-2-34-0', '0806.2805-3-39-0'], ['0806.2805-2-38-4', '0806.2805-3-44-4'], ['0806.2805-2-6-0', '0806.2805-3-7-0'], ['0806.2805-2-23-3', '0806.2805-3-27-3'], ['0806.2805-2-8-4', '0806.2805-3-10-4'], ['0806.2805-2-26-2', '0806.2805-3-31-2'], ['0806.2805-2-26-8', '0806.2805-3-31-8'], ['0806.2805-2-29-3', '0806.2805-3-34-3'], ['0806.2805-2-45-0', '0806.2805-3-52-0'], ['0806.2805-2-11-0', '0806.2805-3-13-0'], ['0806.2805-2-11-1', '0806.2805-3-13-2'], ['0806.2805-2-56-2', '0806.2805-3-63-2'], ['0806.2805-2-5-2', '0806.2805-3-6-2'], ['0806.2805-2-49-5', '0806.2805-3-56-5'], ['0806.2805-3-11-2', '0806.2805-4-11-2'], ['0806.2805-3-61-3', '0806.2805-4-65-3'], ['0806.2805-3-14-1', '0806.2805-4-15-1'], ['0806.2805-3-14-2', '0806.2805-4-15-2'], ['0806.2805-3-52-0', '0806.2805-4-56-0'], ['0806.2805-3-41-2', '0806.2805-4-44-2'], ['0806.2805-3-60-0', '0806.2805-4-64-0'], ['0806.2805-3-2-1', '0806.2805-4-2-1'], ['0806.2805-3-40-0', '0806.2805-4-43-0'], ['0806.2805-3-34-2', '0806.2805-4-37-2'], ['0806.2805-3-53-0', '0806.2805-4-57-0'], ['0806.2805-3-6-3', '0806.2805-4-6-3'], ['0806.2805-3-23-6', '0806.2805-4-25-6'], ['0806.2805-3-5-1', '0806.2805-4-5-1'], ['0806.2805-3-44-4', '0806.2805-4-47-4'], ['0806.2805-3-38-2', '0806.2805-4-41-2'], ['0806.2805-3-12-0', '0806.2805-4-12-0'], ['0806.2805-3-12-1', '0806.2805-4-12-1'], ['0806.2805-3-10-0', '0806.2805-4-10-0'], ['0806.2805-3-10-4', '0806.2805-4-10-4'], ['0806.2805-3-62-1', '0806.2805-4-66-2'], ['0806.2805-3-62-2', '0806.2805-4-66-3'], ['0806.2805-3-63-2', '0806.2805-4-67-2'], ['0806.2805-3-0-1', '0806.2805-4-0-1'], ['0806.2805-3-49-0', '0806.2805-4-53-0'], ['0806.2805-3-49-2', '0806.2805-4-53-2'], ['0806.2805-3-18-0', '0806.2805-4-20-0'], ['0806.2805-3-18-1', '0806.2805-4-20-1'], ['0806.2805-3-18-4', '0806.2805-4-20-4'], ['0806.2805-3-18-5', '0806.2805-4-20-6'], ['0806.2805-3-29-0', '0806.2805-4-31-0'], ['0806.2805-3-29-1', '0806.2805-4-31-1'], ['0806.2805-3-48-1', '0806.2805-4-52-1'], ['0806.2805-3-8-0', '0806.2805-4-8-0'], ['0806.2805-3-8-1', '0806.2805-4-8-1'], ['0806.2805-3-8-3', '0806.2805-4-8-3'], ['0806.2805-3-56-6', '0806.2805-4-60-6'], ['0806.2805-4-0-1', '0806.2805-5-0-1'], ['0806.2805-4-66-2', '0806.2805-5-66-2'], ['0806.2805-4-66-3', '0806.2805-5-66-3'], ['0806.2805-4-66-4', '0806.2805-5-66-4'], ['0806.2805-5-2-0', '0806.2805-6-2-0'], ['0806.2805-5-61-1', '0806.2805-6-65-1'], ['0806.2805-5-57-0', '0806.2805-6-60-0'], ['0806.2805-5-17-3', '0806.2805-6-17-3'], ['0806.2805-5-66-1', '0806.2805-6-70-1'], ['0806.2805-5-66-2', '0806.2805-6-70-2'], ['0806.2805-5-66-3', '0806.2805-6-70-3'], ['0806.2805-5-66-4', '0806.2805-6-70-4'], ['0806.2805-5-66-5', '0806.2805-6-70-5'], ['0806.2805-5-64-0', '0806.2805-6-68-0'], ['0806.2805-5-64-1', '0806.2805-6-68-1'], ['0806.2805-5-50-1', '0806.2805-6-53-1'], ['0806.2805-5-53-0', '0806.2805-6-56-0'], ['0806.2805-5-53-2', '0806.2805-6-56-2'], ['0806.2805-5-53-4', '0806.2805-6-56-4'], ['0806.2805-5-3-2', '0806.2805-6-3-2'], ['0806.2805-5-56-0', '0806.2805-6-59-0'], ['0806.2805-5-33-0', '0806.2805-6-35-0'], ['0806.2805-5-15-0', '0806.2805-6-15-0'], ['0806.2805-5-16-0', '0806.2805-6-16-0'], ['0806.2805-5-31-0', '0806.2805-6-33-0'], ['0806.2805-5-31-1', '0806.2805-6-33-1'], ['0806.2805-5-5-1', '0806.2805-6-5-1'], ['0806.2805-5-20-4', '0806.2805-6-21-4'], ['0806.2805-5-44-0', '0806.2805-6-47-0'], ['0806.2805-5-44-2', '0806.2805-6-47-2'], ['0806.2805-5-13-0', '0806.2805-6-13-0'], ['0806.2805-5-0-2', '0806.2805-6-0-2'], ['0806.2805-5-0-3', '0806.2805-6-0-3'], ['0806.2805-5-0-4', '0806.2805-6-0-4'], ['0806.2805-5-42-0', '0806.2805-6-45-0'], ['0806.2805-5-12-0', '0806.2805-6-12-0'], ['0806.2805-5-12-1', '0806.2805-6-12-2'], ['0806.2805-5-12-3', '0806.2805-6-12-4'], ['0806.2805-5-23-0', '0806.2805-6-24-0'], ['0806.2805-5-19-2', '0806.2805-6-20-2'], ['0806.2805-5-72-1', '0806.2805-6-76-1'], ['0806.2805-5-43-0', '0806.2805-6-46-0'], ['0806.2805-5-4-1', '0806.2805-6-4-1'], ['0806.2805-5-4-2', '0806.2805-6-4-2'], ['0806.2805-5-45-0', '0806.2805-6-48-0'], ['0806.2805-5-55-0', '0806.2805-6-58-0'], ['0806.2805-5-55-1', '0806.2805-6-58-1'], ['0806.2805-5-10-2', '0806.2805-6-10-2'], ['0806.2805-5-6-3', '0806.2805-6-6-3'], ['0806.2805-1-28-0', '0806.2805-2-28-0'], ['0806.2805-1-28-2', '0806.2805-2-29-0'], ['0806.2805-1-28-3', '0806.2805-2-29-1'], ['0806.2805-2-13-1', '0806.2805-3-16-0'], ['0806.2805-2-13-2', '0806.2805-3-16-1'], ['0806.2805-2-24-0', '0806.2805-3-29-0'], ['0806.2805-3-46-0', '0806.2805-4-49-0'], ['0806.2805-3-46-1', '0806.2805-4-50-0'], ['0806.2805-3-13-0', '0806.2805-4-13-0'], ['0806.2805-3-13-5', '0806.2805-4-14-4'], ['0806.2805-3-31-0', '0806.2805-4-33-0'], ['0806.2805-3-31-9', '0806.2805-4-34-6'], ['0806.2805-5-18-0', '0806.2805-6-18-0'], ['0806.2805-5-60-3', '0806.2805-6-63-3'], ['0806.2805-5-34-2', '0806.2805-6-36-2'], ['0806.2805-5-34-5', '0806.2805-6-36-5'], ['0806.2805-5-34-6', '0806.2805-6-37-0'], ['0806.2805-5-25-1', '0806.2805-6-26-1'], ['0806.2805-5-25-2', '0806.2805-6-27-0'], ['0806.2805-5-25-7', '0806.2805-6-27-5'], ['0806.2805-1-19-4', '0806.2805-2-19-4'], ['0806.2805-1-3-2', '0806.2805-2-3-2'], ['0806.2805-1-14-1', '0806.2805-2-14-1'], ['0806.2805-1-52-0', '0806.2805-2-53-0'], ['0806.2805-1-39-1', '0806.2805-2-40-1'], ['0806.2805-1-47-1', '0806.2805-2-48-1'], ['0806.2805-1-45-0', '0806.2805-2-46-0'], ['0806.2805-1-20-0', '0806.2805-2-20-0'], ['0806.2805-1-13-0', '0806.2805-2-13-0'], ['0806.2805-1-23-0', '0806.2805-2-23-0'], ['0806.2805-1-10-0', '0806.2805-2-10-0'], ['0806.2805-1-10-0', '0806.2805-2-10-1'], ['0806.2805-1-12-1', '0806.2805-2-12-1'], ['0806.2805-1-29-3', '0806.2805-2-30-3'], ['0806.2805-1-24-1', '0806.2805-2-24-1'], ['0806.2805-1-56-1', '0806.2805-2-59-1'], ['0806.2805-1-48-2', '0806.2805-2-49-2'], ['0806.2805-1-41-0', '0806.2805-2-42-0'], ['0806.2805-1-41-1', '0806.2805-2-42-1'], ['0806.2805-1-41-2', '0806.2805-2-42-2'], ['0806.2805-2-19-1', '0806.2805-3-23-1'], ['0806.2805-2-55-1', '0806.2805-3-62-1'], ['0806.2805-2-55-1', '0806.2805-3-62-2'], ['0806.2805-2-42-0', '0806.2805-3-49-0'], ['0806.2805-2-42-2', '0806.2805-3-49-2'], ['0806.2805-2-16-0', '0806.2805-3-20-0'], ['0806.2805-2-11-2', '0806.2805-3-13-3'], ['0806.2805-2-54-3', '0806.2805-3-61-3'], ['0806.2805-2-49-3', '0806.2805-3-56-3'], ['0806.2805-2-49-4', '0806.2805-3-56-4'], ['0806.2805-3-21-1', '0806.2805-4-23-1'], ['0806.2805-3-40-1', '0806.2805-4-43-1'], ['0806.2805-3-35-3', '0806.2805-4-38-3'], ['0806.2805-3-16-1', '0806.2805-4-18-0'], ['0806.2805-3-55-0', '0806.2805-4-59-0'], ['0806.2805-1-28-1', '0806.2805-2-28-1'], ['0806.2805-1-28-4', '0806.2805-2-29-2'], ['0806.2805-2-13-0', '0806.2805-3-15-0'], ['0806.2805-3-31-7', '0806.2805-4-34-4']]

[['0806.2805-1-8-1', '0806.2805-2-8-1'], ['0806.2805-1-8-3', '0806.2805-2-8-3'], ['0806.2805-1-8-4', '0806.2805-2-8-4'], ['0806.2805-1-8-5', '0806.2805-2-8-5'], ['0806.2805-1-19-0', '0806.2805-2-19-0'], ['0806.2805-1-19-2', '0806.2805-2-19-2'], ['0806.2805-1-19-3', '0806.2805-2-19-3'], ['0806.2805-1-11-2', '0806.2805-2-11-2'], ['0806.2805-1-3-0', '0806.2805-2-3-0'], ['0806.2805-1-3-1', '0806.2805-2-3-1'], ['0806.2805-1-2-1', '0806.2805-2-2-1'], ['0806.2805-1-2-2', '0806.2805-2-2-2'], ['0806.2805-1-6-0', '0806.2805-2-6-0'], ['0806.2805-1-9-1', '0806.2805-2-9-1'], ['0806.2805-1-14-2', '0806.2805-2-14-2'], ['0806.2805-1-55-1', '0806.2805-2-58-1'], ['0806.2805-1-52-2', '0806.2805-2-53-2'], ['0806.2805-1-52-3', '0806.2805-2-53-3'], ['0806.2805-1-39-2', '0806.2805-2-40-2'], ['0806.2805-1-16-0', '0806.2805-2-16-0'], ['0806.2805-1-16-1', '0806.2805-2-16-1'], ['0806.2805-1-16-2', '0806.2805-2-16-2'], ['0806.2805-1-53-2', '0806.2805-2-55-2'], ['0806.2805-1-37-0', '0806.2805-2-38-0'], ['0806.2805-1-37-3', '0806.2805-2-38-3'], ['0806.2805-1-0-2', '0806.2805-2-0-2'], ['0806.2805-1-0-4', '0806.2805-2-0-4'], ['0806.2805-1-40-0', '0806.2805-2-41-0'], ['0806.2805-1-40-2', '0806.2805-2-41-2'], ['0806.2805-1-40-3', '0806.2805-2-41-3'], ['0806.2805-1-40-5', '0806.2805-2-41-5'], ['0806.2805-1-40-6', '0806.2805-2-41-6'], ['0806.2805-1-13-1', '0806.2805-2-13-1'], ['0806.2805-1-13-2', '0806.2805-2-13-2'], ['0806.2805-1-13-3', '0806.2805-2-13-3'], ['0806.2805-1-13-4', '0806.2805-2-13-4'], ['0806.2805-1-34-0', '0806.2805-2-35-0'], ['0806.2805-1-34-1', '0806.2805-2-35-1'], ['0806.2805-1-35-0', '0806.2805-2-36-0'], ['0806.2805-1-35-1', '0806.2805-2-36-1'], ['0806.2805-1-23-2', '0806.2805-2-23-2'], ['0806.2805-1-5-0', '0806.2805-2-5-0'], ['0806.2805-1-5-3', '0806.2805-2-5-3'], ['0806.2805-1-10-1', '0806.2805-2-10-2'], ['0806.2805-1-51-0', '0806.2805-2-52-0'], ['0806.2805-1-43-0', '0806.2805-2-44-0'], ['0806.2805-1-43-1', '0806.2805-2-44-1'], ['0806.2805-1-18-0', '0806.2805-2-18-0'], ['0806.2805-1-26-0', '0806.2805-2-26-0'], ['0806.2805-1-26-1', '0806.2805-2-26-1'], ['0806.2805-1-26-4', '0806.2805-2-26-4'], ['0806.2805-1-26-5', '0806.2805-2-26-5'], ['0806.2805-1-26-6', '0806.2805-2-26-6'], ['0806.2805-1-26-7', '0806.2805-2-26-7'], ['0806.2805-1-26-9', '0806.2805-2-26-9'], ['0806.2805-1-12-2', '0806.2805-2-12-2'], ['0806.2805-1-12-3', '0806.2805-2-12-3'], ['0806.2805-1-49-0', '0806.2805-2-50-0'], ['0806.2805-1-49-1', '0806.2805-2-50-1'], ['0806.2805-1-49-2', '0806.2805-2-50-2'], ['0806.2805-1-29-0', '0806.2805-2-30-0'], ['0806.2805-1-29-1', '0806.2805-2-30-1'], ['0806.2805-1-24-2', '0806.2805-2-24-2'], ['0806.2805-1-48-0', '0806.2805-2-49-0'], ['0806.2805-1-48-3', '0806.2805-2-49-3'], ['0806.2805-1-48-4', '0806.2805-2-49-4'], ['0806.2805-1-48-5', '0806.2805-2-49-5'], ['0806.2805-1-48-6', '0806.2805-2-49-6'], ['0806.2805-1-33-1', '0806.2805-2-34-1'], ['0806.2805-1-33-3', '0806.2805-2-34-3'], ['0806.2805-1-33-4', '0806.2805-2-34-4'], ['0806.2805-2-40-0', '0806.2805-3-46-0'], ['0806.2805-2-40-2', '0806.2805-3-46-2'], ['0806.2805-2-41-0', '0806.2805-3-47-0'], ['0806.2805-2-41-1', '0806.2805-3-47-1'], ['0806.2805-2-41-2', '0806.2805-3-47-2'], ['0806.2805-2-41-4', '0806.2805-3-47-4'], ['0806.2805-2-41-5', '0806.2805-3-47-5'], ['0806.2805-2-4-0', '0806.2805-3-4-0'], ['0806.2805-2-4-1', '0806.2805-3-4-1'], ['0806.2805-2-53-0', '0806.2805-3-60-0'], ['0806.2805-2-53-1', '0806.2805-3-60-1'], ['0806.2805-2-53-2', '0806.2805-3-60-2'], ['0806.2805-2-53-3', '0806.2805-3-60-3'], ['0806.2805-2-44-0', '0806.2805-3-51-0'], ['0806.2805-2-44-1', '0806.2805-3-51-1'], ['0806.2805-2-44-2', '0806.2805-3-51-2'], ['0806.2805-2-12-0', '0806.2805-3-14-0'], ['0806.2805-2-12-1', '0806.2805-3-14-1'], ['0806.2805-2-12-2', '0806.2805-3-14-2'], ['0806.2805-2-12-4', '0806.2805-3-14-4'], ['0806.2805-2-0-2', '0806.2805-3-0-2'], ['0806.2805-2-0-3', '0806.2805-3-0-3'], ['0806.2805-2-0-4', '0806.2805-3-0-4'], ['0806.2805-2-33-0', '0806.2805-3-38-0'], ['0806.2805-2-33-1', '0806.2805-3-38-1'], ['0806.2805-2-52-0', '0806.2805-3-59-0'], ['0806.2805-2-2-1', '0806.2805-3-2-1'], ['0806.2805-2-2-2', '0806.2805-3-2-2'], ['0806.2805-2-59-0', '0806.2805-3-66-0'], ['0806.2805-2-59-1', '0806.2805-3-66-1'], ['0806.2805-2-10-1', '0806.2805-3-12-1'], ['0806.2805-2-10-2', '0806.2805-3-12-2'], ['0806.2805-2-19-0', '0806.2805-3-23-0'], ['0806.2805-2-19-2', '0806.2805-3-23-2'], ['0806.2805-2-19-4', '0806.2805-3-23-4'], ['0806.2805-2-19-5', '0806.2805-3-23-5'], ['0806.2805-2-19-6', '0806.2805-3-23-6'], ['0806.2805-2-19-7', '0806.2805-3-23-7'], ['0806.2805-2-28-1', '0806.2805-3-33-1'], ['0806.2805-2-55-0', '0806.2805-3-62-0'], ['0806.2805-2-42-3', '0806.2805-3-49-3'], ['0806.2805-2-20-0', '0806.2805-3-24-0'], ['0806.2805-2-34-1', '0806.2805-3-39-1'], ['0806.2805-2-34-2', '0806.2805-3-39-2'], ['0806.2805-2-34-3', '0806.2805-3-39-3'], ['0806.2805-2-34-4', '0806.2805-3-39-4'], ['0806.2805-2-3-0', '0806.2805-3-3-0'], ['0806.2805-2-3-1', '0806.2805-3-3-1'], ['0806.2805-2-3-2', '0806.2805-3-3-2'], ['0806.2805-2-3-3', '0806.2805-3-3-3'], ['0806.2805-2-3-4', '0806.2805-3-3-4'], ['0806.2805-2-38-0', '0806.2805-3-44-0'], ['0806.2805-2-38-1', '0806.2805-3-44-1'], ['0806.2805-2-38-2', '0806.2805-3-44-2'], ['0806.2805-2-38-3', '0806.2805-3-44-3'], ['0806.2805-2-50-0', '0806.2805-3-57-0'], ['0806.2805-2-50-1', '0806.2805-3-57-1'], ['0806.2805-2-50-2', '0806.2805-3-57-2'], ['0806.2805-2-58-0', '0806.2805-3-65-0'], ['0806.2805-2-58-1', '0806.2805-3-65-1'], ['0806.2805-2-6-1', '0806.2805-3-7-1'], ['0806.2805-2-36-0', '0806.2805-3-41-0'], ['0806.2805-2-36-1', '0806.2805-3-41-1'], ['0806.2805-2-36-2', '0806.2805-3-41-2'], ['0806.2805-2-18-0', '0806.2805-3-22-0'], ['0806.2805-2-14-0', '0806.2805-3-17-0'], ['0806.2805-2-14-1', '0806.2805-3-17-1'], ['0806.2805-2-14-2', '0806.2805-3-17-2'], ['0806.2805-2-14-3', '0806.2805-3-17-3'], ['0806.2805-2-23-0', '0806.2805-3-27-0'], ['0806.2805-2-23-1', '0806.2805-3-27-1'], ['0806.2805-2-23-2', '0806.2805-3-27-2'], ['0806.2805-2-35-0', '0806.2805-3-40-0'], ['0806.2805-2-35-1', '0806.2805-3-40-1'], ['0806.2805-2-16-1', '0806.2805-3-20-1'], ['0806.2805-2-16-2', '0806.2805-3-20-2'], ['0806.2805-2-8-0', '0806.2805-3-10-0'], ['0806.2805-2-8-1', '0806.2805-3-10-1'], ['0806.2805-2-8-2', '0806.2805-3-10-2'], ['0806.2805-2-8-3', '0806.2805-3-10-3'], ['0806.2805-2-8-5', '0806.2805-3-10-5'], ['0806.2805-2-26-0', '0806.2805-3-31-0'], ['0806.2805-2-26-1', '0806.2805-3-31-1'], ['0806.2805-2-26-3', '0806.2805-3-31-3'], ['0806.2805-2-26-4', '0806.2805-3-31-4'], ['0806.2805-2-26-5', '0806.2805-3-31-5'], ['0806.2805-2-26-6', '0806.2805-3-31-6'], ['0806.2805-2-26-7', '0806.2805-3-31-7'], ['0806.2805-2-26-9', '0806.2805-3-31-9'], ['0806.2805-2-46-0', '0806.2805-3-53-0'], ['0806.2805-2-17-0', '0806.2805-3-21-0'], ['0806.2805-2-17-1', '0806.2805-3-21-1'], ['0806.2805-2-29-0', '0806.2805-3-34-0'], ['0806.2805-2-29-1', '0806.2805-3-34-1'], ['0806.2805-2-29-2', '0806.2805-3-34-2'], ['0806.2805-2-30-0', '0806.2805-3-35-0'], ['0806.2805-2-30-1', '0806.2805-3-35-1'], ['0806.2805-2-30-2', '0806.2805-3-35-2'], ['0806.2805-2-30-3', '0806.2805-3-35-3'], ['0806.2805-2-9-0', '0806.2805-3-11-0'], ['0806.2805-2-9-1', '0806.2805-3-11-1'], ['0806.2805-2-9-2', '0806.2805-3-11-2'], ['0806.2805-2-54-0', '0806.2805-3-61-0'], ['0806.2805-2-54-1', '0806.2805-3-61-1'], ['0806.2805-2-54-2', '0806.2805-3-61-2'], ['0806.2805-2-56-0', '0806.2805-3-63-0'], ['0806.2805-2-56-1', '0806.2805-3-63-1'], ['0806.2805-2-5-0', '0806.2805-3-6-0'], ['0806.2805-2-5-1', '0806.2805-3-6-1'], ['0806.2805-2-5-3', '0806.2805-3-6-3'], ['0806.2805-2-48-0', '0806.2805-3-55-0'], ['0806.2805-2-48-1', '0806.2805-3-55-1'], ['0806.2805-2-49-0', '0806.2805-3-56-0'], ['0806.2805-2-49-1', '0806.2805-3-56-1'], ['0806.2805-2-49-2', '0806.2805-3-56-2'], ['0806.2805-2-49-6', '0806.2805-3-56-6'], ['0806.2805-3-7-0', '0806.2805-4-7-0'], ['0806.2805-3-7-1', '0806.2805-4-7-1'], ['0806.2805-3-24-0', '0806.2805-4-26-0'], ['0806.2805-3-21-0', '0806.2805-4-23-0'], ['0806.2805-3-11-0', '0806.2805-4-11-0'], ['0806.2805-3-11-1', '0806.2805-4-11-1'], ['0806.2805-3-61-0', '0806.2805-4-65-0'], ['0806.2805-3-61-1', '0806.2805-4-65-1'], ['0806.2805-3-61-2', '0806.2805-4-65-2'], ['0806.2805-3-61-4', '0806.2805-4-65-4'], ['0806.2805-3-20-0', '0806.2805-4-22-0'], ['0806.2805-3-20-1', '0806.2805-4-22-1'], ['0806.2805-3-20-2', '0806.2805-4-22-2'], ['0806.2805-3-33-0', '0806.2805-4-36-0'], ['0806.2805-3-33-1', '0806.2805-4-36-1'], ['0806.2805-3-51-0', '0806.2805-4-55-0'], ['0806.2805-3-51-1', '0806.2805-4-55-1'], ['0806.2805-3-51-2', '0806.2805-4-55-2'], ['0806.2805-3-14-0', '0806.2805-4-15-0'], ['0806.2805-3-14-3', '0806.2805-4-15-3'], ['0806.2805-3-14-4', '0806.2805-4-15-4'], ['0806.2805-3-22-0', '0806.2805-4-24-0'], ['0806.2805-3-41-0', '0806.2805-4-44-0'], ['0806.2805-3-41-1', '0806.2805-4-44-1'], ['0806.2805-3-60-1', '0806.2805-4-64-1'], ['0806.2805-3-60-2', '0806.2805-4-64-2'], ['0806.2805-3-60-3', '0806.2805-4-64-3'], ['0806.2805-3-65-0', '0806.2805-4-69-0'], ['0806.2805-3-65-1', '0806.2805-4-69-1'], ['0806.2805-3-2-0', '0806.2805-4-2-0'], ['0806.2805-3-2-2', '0806.2805-4-2-2'], ['0806.2805-3-3-0', '0806.2805-4-3-0'], ['0806.2805-3-3-1', '0806.2805-4-3-1'], ['0806.2805-3-3-2', '0806.2805-4-3-2'], ['0806.2805-3-3-3', '0806.2805-4-3-3'], ['0806.2805-3-3-4', '0806.2805-4-3-4'], ['0806.2805-3-34-0', '0806.2805-4-37-0'], ['0806.2805-3-34-1', '0806.2805-4-37-1'], ['0806.2805-3-34-3', '0806.2805-4-37-3'], ['0806.2805-3-15-0', '0806.2805-4-16-0'], ['0806.2805-3-66-0', '0806.2805-4-70-0'], ['0806.2805-3-66-1', '0806.2805-4-70-1'], ['0806.2805-3-27-0', '0806.2805-4-29-0'], ['0806.2805-3-27-1', '0806.2805-4-29-1'], ['0806.2805-3-27-2', '0806.2805-4-29-2'], ['0806.2805-3-27-3', '0806.2805-4-29-3'], ['0806.2805-3-6-0', '0806.2805-4-6-0'], ['0806.2805-3-6-1', '0806.2805-4-6-1'], ['0806.2805-3-6-2', '0806.2805-4-6-2'], ['0806.2805-3-23-0', '0806.2805-4-25-0'], ['0806.2805-3-23-1', '0806.2805-4-25-1'], ['0806.2805-3-23-2', '0806.2805-4-25-2'], ['0806.2805-3-23-3', '0806.2805-4-25-3'], ['0806.2805-3-23-4', '0806.2805-4-25-4'], ['0806.2805-3-23-5', '0806.2805-4-25-5'], ['0806.2805-3-23-7', '0806.2805-4-25-7'], ['0806.2805-3-5-0', '0806.2805-4-5-0'], ['0806.2805-3-42-0', '0806.2805-4-45-0'], ['0806.2805-3-44-0', '0806.2805-4-47-0'], ['0806.2805-3-44-1', '0806.2805-4-47-1'], ['0806.2805-3-44-2', '0806.2805-4-47-2'], ['0806.2805-3-44-3', '0806.2805-4-47-3'], ['0806.2805-3-38-0', '0806.2805-4-41-0'], ['0806.2805-3-38-1', '0806.2805-4-41-1'], ['0806.2805-3-35-0', '0806.2805-4-38-0'], ['0806.2805-3-35-1', '0806.2805-4-38-1'], ['0806.2805-3-35-2', '0806.2805-4-38-2'], ['0806.2805-3-12-2', '0806.2805-4-12-3'], ['0806.2805-3-57-0', '0806.2805-4-61-0'], ['0806.2805-3-57-1', '0806.2805-4-61-1'], ['0806.2805-3-57-2', '0806.2805-4-61-2'], ['0806.2805-3-10-1', '0806.2805-4-10-1'], ['0806.2805-3-10-2', '0806.2805-4-10-2'], ['0806.2805-3-10-3', '0806.2805-4-10-3'], ['0806.2805-3-10-5', '0806.2805-4-10-5'], ['0806.2805-3-62-0', '0806.2805-4-66-0'], ['0806.2805-3-62-3', '0806.2805-4-66-4'], ['0806.2805-3-62-4', '0806.2805-4-66-5'], ['0806.2805-3-63-0', '0806.2805-4-67-0'], ['0806.2805-3-63-1', '0806.2805-4-67-1'], ['0806.2805-3-4-0', '0806.2805-4-4-0'], ['0806.2805-3-4-1', '0806.2805-4-4-1'], ['0806.2805-3-4-2', '0806.2805-4-4-2'], ['0806.2805-3-0-0', '0806.2805-4-0-0'], ['0806.2805-3-0-2', '0806.2805-4-0-2'], ['0806.2805-3-0-3', '0806.2805-4-0-3'], ['0806.2805-3-0-4', '0806.2805-4-0-4'], ['0806.2805-3-47-0', '0806.2805-4-51-0'], ['0806.2805-3-47-1', '0806.2805-4-51-1'], ['0806.2805-3-47-2', '0806.2805-4-51-2'], ['0806.2805-3-47-3', '0806.2805-4-51-3'], ['0806.2805-3-47-4', '0806.2805-4-51-4'], ['0806.2805-3-47-5', '0806.2805-4-51-5'], ['0806.2805-3-47-6', '0806.2805-4-51-6'], ['0806.2805-3-49-1', '0806.2805-4-53-1'], ['0806.2805-3-49-3', '0806.2805-4-53-3'], ['0806.2805-3-49-4', '0806.2805-4-53-4'], ['0806.2805-3-67-1', '0806.2805-4-71-1'], ['0806.2805-3-39-0', '0806.2805-4-42-0'], ['0806.2805-3-39-1', '0806.2805-4-42-1'], ['0806.2805-3-39-2', '0806.2805-4-42-2'], ['0806.2805-3-39-3', '0806.2805-4-42-3'], ['0806.2805-3-39-4', '0806.2805-4-42-4'], ['0806.2805-3-17-0', '0806.2805-4-19-0'], ['0806.2805-3-17-1', '0806.2805-4-19-1'], ['0806.2805-3-17-2', '0806.2805-4-19-2'], ['0806.2805-3-17-3', '0806.2805-4-19-3'], ['0806.2805-3-16-2', '0806.2805-4-18-1'], ['0806.2805-3-16-3', '0806.2805-4-18-2'], ['0806.2805-3-18-2', '0806.2805-4-20-2'], ['0806.2805-3-18-3', '0806.2805-4-20-3'], ['0806.2805-3-55-1', '0806.2805-4-59-1'], ['0806.2805-3-29-2', '0806.2805-4-31-2'], ['0806.2805-3-59-0', '0806.2805-4-63-0'], ['0806.2805-3-59-1', '0806.2805-4-63-1'], ['0806.2805-3-48-0', '0806.2805-4-52-0'], ['0806.2805-3-28-0', '0806.2805-4-30-0'], ['0806.2805-3-28-1', '0806.2805-4-30-1'], ['0806.2805-3-8-2', '0806.2805-4-8-2'], ['0806.2805-3-8-4', '0806.2805-4-8-4'], ['0806.2805-3-56-0', '0806.2805-4-60-0'], ['0806.2805-3-56-1', '0806.2805-4-60-1'], ['0806.2805-3-56-2', '0806.2805-4-60-2'], ['0806.2805-3-56-3', '0806.2805-4-60-3'], ['0806.2805-3-56-4', '0806.2805-4-60-4'], ['0806.2805-3-56-5', '0806.2805-4-60-5'], ['0806.2805-4-67-0', '0806.2805-5-67-0'], ['0806.2805-4-67-1', '0806.2805-5-67-1'], ['0806.2805-4-67-2', '0806.2805-5-67-2'], ['0806.2805-4-67-3', '0806.2805-5-67-3'], ['0806.2805-4-67-4', '0806.2805-5-67-4'], ['0806.2805-4-53-0', '0806.2805-5-53-0'], ['0806.2805-4-53-1', '0806.2805-5-53-1'], ['0806.2805-4-53-2', '0806.2805-5-53-2'], ['0806.2805-4-53-3', '0806.2805-5-53-3'], ['0806.2805-4-53-4', '0806.2805-5-53-4'], ['0806.2805-4-24-0', '0806.2805-5-24-0'], ['0806.2805-4-0-0', '0806.2805-5-0-0'], ['0806.2805-4-0-2', '0806.2805-5-0-2'], ['0806.2805-4-0-3', '0806.2805-5-0-3'], ['0806.2805-4-0-4', '0806.2805-5-0-4'], ['0806.2805-4-65-0', '0806.2805-5-65-0'], ['0806.2805-4-65-1', '0806.2805-5-65-1'], ['0806.2805-4-65-2', '0806.2805-5-65-2'], ['0806.2805-4-65-3', '0806.2805-5-65-3'], ['0806.2805-4-65-4', '0806.2805-5-65-4'], ['0806.2805-4-29-0', '0806.2805-5-29-0'], ['0806.2805-4-29-1', '0806.2805-5-29-1'], ['0806.2805-4-29-2', '0806.2805-5-29-2'], ['0806.2805-4-29-3', '0806.2805-5-29-3'], ['0806.2805-4-29-4', '0806.2805-5-29-4'], ['0806.2805-4-56-0', '0806.2805-5-56-0'], ['0806.2805-4-66-0', '0806.2805-5-66-0'], ['0806.2805-4-66-1', '0806.2805-5-66-1'], ['0806.2805-4-66-5', '0806.2805-5-66-5'], ['0806.2805-4-14-0', '0806.2805-5-14-0'], ['0806.2805-4-14-1', '0806.2805-5-14-1'], ['0806.2805-4-14-2', '0806.2805-5-14-2'], ['0806.2805-4-14-3', '0806.2805-5-14-3'], ['0806.2805-4-14-4', '0806.2805-5-14-4'], ['0806.2805-4-42-0', '0806.2805-5-42-0'], ['0806.2805-4-42-1', '0806.2805-5-42-1'], ['0806.2805-4-42-2', '0806.2805-5-42-2'], ['0806.2805-4-42-3', '0806.2805-5-42-3'], ['0806.2805-4-42-4', '0806.2805-5-42-4'], ['0806.2805-4-36-0', '0806.2805-5-36-0'], ['0806.2805-4-36-1', '0806.2805-5-36-1'], ['0806.2805-4-50-0', '0806.2805-5-50-0'], ['0806.2805-4-50-1', '0806.2805-5-50-1'], ['0806.2805-4-50-2', '0806.2805-5-50-2'], ['0806.2805-4-52-0', '0806.2805-5-52-0'], ['0806.2805-4-52-1', '0806.2805-5-52-1'], ['0806.2805-4-71-1', '0806.2805-5-72-1'], ['0806.2805-4-43-0', '0806.2805-5-43-0'], ['0806.2805-4-43-1', '0806.2805-5-43-1'], ['0806.2805-4-70-0', '0806.2805-5-70-0'], ['0806.2805-4-70-1', '0806.2805-5-70-1'], ['0806.2805-4-55-0', '0806.2805-5-55-0'], ['0806.2805-4-55-1', '0806.2805-5-55-1'], ['0806.2805-4-55-2', '0806.2805-5-55-2'], ['0806.2805-4-38-0', '0806.2805-5-38-0'], ['0806.2805-4-38-1', '0806.2805-5-38-1'], ['0806.2805-4-38-2', '0806.2805-5-38-2'], ['0806.2805-4-38-3', '0806.2805-5-38-3'], ['0806.2805-4-59-0', '0806.2805-5-59-0'], ['0806.2805-4-59-1', '0806.2805-5-59-1'], ['0806.2805-4-12-0', '0806.2805-5-12-0'], ['0806.2805-4-12-1', '0806.2805-5-12-1'], ['0806.2805-4-12-2', '0806.2805-5-12-3'], ['0806.2805-4-12-3', '0806.2805-5-12-4'], ['0806.2805-4-8-0', '0806.2805-5-8-0'], ['0806.2805-4-8-1', '0806.2805-5-8-1'], ['0806.2805-4-8-2', '0806.2805-5-8-2'], ['0806.2805-4-8-3', '0806.2805-5-8-3'], ['0806.2805-4-8-4', '0806.2805-5-8-4'], ['0806.2805-4-47-0', '0806.2805-5-47-0'], ['0806.2805-4-47-1', '0806.2805-5-47-1'], ['0806.2805-4-47-2', '0806.2805-5-47-2'], ['0806.2805-4-47-3', '0806.2805-5-47-3'], ['0806.2805-4-47-4', '0806.2805-5-47-4'], ['0806.2805-4-61-0', '0806.2805-5-61-0'], ['0806.2805-4-61-1', '0806.2805-5-61-1'], ['0806.2805-4-61-2', '0806.2805-5-61-2'], ['0806.2805-4-45-0', '0806.2805-5-45-0'], ['0806.2805-4-10-0', '0806.2805-5-10-0'], ['0806.2805-4-10-1', '0806.2805-5-10-1'], ['0806.2805-4-10-2', '0806.2805-5-10-2'], ['0806.2805-4-10-3', '0806.2805-5-10-3'], ['0806.2805-4-10-4', '0806.2805-5-10-4'], ['0806.2805-4-10-5', '0806.2805-5-10-5'], ['0806.2805-4-22-0', '0806.2805-5-22-0'], ['0806.2805-4-22-1', '0806.2805-5-22-1'], ['0806.2805-4-22-2', '0806.2805-5-22-2'], ['0806.2805-4-16-0', '0806.2805-5-16-0'], ['0806.2805-4-57-0', '0806.2805-5-57-0'], ['0806.2805-4-20-0', '0806.2805-5-20-0'], ['0806.2805-4-20-1', '0806.2805-5-20-1'], ['0806.2805-4-20-2', '0806.2805-5-20-2'], ['0806.2805-4-20-3', '0806.2805-5-20-3'], ['0806.2805-4-20-4', '0806.2805-5-20-4'], ['0806.2805-4-20-5', '0806.2805-5-20-5'], ['0806.2805-4-20-6', '0806.2805-5-20-6'], ['0806.2805-4-33-0', '0806.2805-5-33-0'], ['0806.2805-4-33-1', '0806.2805-5-33-1'], ['0806.2805-4-33-2', '0806.2805-5-33-2'], ['0806.2805-4-44-0', '0806.2805-5-44-0'], ['0806.2805-4-44-1', '0806.2805-5-44-1'], ['0806.2805-4-44-2', '0806.2805-5-44-2'], ['0806.2805-4-19-0', '0806.2805-5-19-0'], ['0806.2805-4-19-1', '0806.2805-5-19-1'], ['0806.2805-4-19-2', '0806.2805-5-19-2'], ['0806.2805-4-19-3', '0806.2805-5-19-3'], ['0806.2805-4-69-0', '0806.2805-5-69-0'], ['0806.2805-4-69-1', '0806.2805-5-69-1'], ['0806.2805-4-13-0', '0806.2805-5-13-0'], ['0806.2805-4-26-0', '0806.2805-5-26-0'], ['0806.2805-4-37-0', '0806.2805-5-37-0'], ['0806.2805-4-37-1', '0806.2805-5-37-1'], ['0806.2805-4-37-2', '0806.2805-5-37-2'], ['0806.2805-4-37-3', '0806.2805-5-37-3'], ['0806.2805-4-2-0', '0806.2805-5-2-0'], ['0806.2805-4-2-1', '0806.2805-5-2-1'], ['0806.2805-4-2-2', '0806.2805-5-2-2'], ['0806.2805-4-17-0', '0806.2805-5-17-0'], ['0806.2805-4-17-1', '0806.2805-5-17-1'], ['0806.2805-4-17-2', '0806.2805-5-17-2'], ['0806.2805-4-17-3', '0806.2805-5-17-3'], ['0806.2805-4-49-0', '0806.2805-5-49-0'], ['0806.2805-4-15-0', '0806.2805-5-15-0'], ['0806.2805-4-15-1', '0806.2805-5-15-1'], ['0806.2805-4-15-2', '0806.2805-5-15-2'], ['0806.2805-4-15-3', '0806.2805-5-15-3'], ['0806.2805-4-15-4', '0806.2805-5-15-4'], ['0806.2805-4-23-0', '0806.2805-5-23-0'], ['0806.2805-4-23-1', '0806.2805-5-23-1'], ['0806.2805-4-31-0', '0806.2805-5-31-0'], ['0806.2805-4-31-1', '0806.2805-5-31-1'], ['0806.2805-4-31-2', '0806.2805-5-31-2'], ['0806.2805-4-4-0', '0806.2805-5-4-0'], ['0806.2805-4-4-1', '0806.2805-5-4-1'], ['0806.2805-4-4-2', '0806.2805-5-4-2'], ['0806.2805-4-25-0', '0806.2805-5-25-0'], ['0806.2805-4-25-1', '0806.2805-5-25-1'], ['0806.2805-4-25-2', '0806.2805-5-25-2'], ['0806.2805-4-25-3', '0806.2805-5-25-3'], ['0806.2805-4-25-4', '0806.2805-5-25-4'], ['0806.2805-4-25-5', '0806.2805-5-25-5'], ['0806.2805-4-25-6', '0806.2805-5-25-6'], ['0806.2805-4-25-7', '0806.2805-5-25-7'], ['0806.2805-4-25-8', '0806.2805-5-25-8'], ['0806.2805-4-5-0', '0806.2805-5-5-0'], ['0806.2805-4-5-1', '0806.2805-5-5-1'], ['0806.2805-4-11-0', '0806.2805-5-11-0'], ['0806.2805-4-11-1', '0806.2805-5-11-1'], ['0806.2805-4-11-2', '0806.2805-5-11-2'], ['0806.2805-4-63-0', '0806.2805-5-63-0'], ['0806.2805-4-63-1', '0806.2805-5-63-1'], ['0806.2805-4-34-0', '0806.2805-5-34-0'], ['0806.2805-4-34-1', '0806.2805-5-34-1'], ['0806.2805-4-34-2', '0806.2805-5-34-2'], ['0806.2805-4-34-3', '0806.2805-5-34-3'], ['0806.2805-4-34-4', '0806.2805-5-34-4'], ['0806.2805-4-34-5', '0806.2805-5-34-5'], ['0806.2805-4-34-6', '0806.2805-5-34-6'], ['0806.2805-4-18-0', '0806.2805-5-18-0'], ['0806.2805-4-18-1', '0806.2805-5-18-1'], ['0806.2805-4-18-2', '0806.2805-5-18-2'], ['0806.2805-4-41-0', '0806.2805-5-41-0'], ['0806.2805-4-41-1', '0806.2805-5-41-1'], ['0806.2805-4-41-2', '0806.2805-5-41-2'], ['0806.2805-4-7-0', '0806.2805-5-7-0'], ['0806.2805-4-7-1', '0806.2805-5-7-1'], ['0806.2805-4-51-0', '0806.2805-5-51-0'], ['0806.2805-4-51-1', '0806.2805-5-51-1'], ['0806.2805-4-51-2', '0806.2805-5-51-2'], ['0806.2805-4-51-3', '0806.2805-5-51-3'], ['0806.2805-4-51-4', '0806.2805-5-51-4'], ['0806.2805-4-51-5', '0806.2805-5-51-5'], ['0806.2805-4-51-6', '0806.2805-5-51-6'], ['0806.2805-4-64-0', '0806.2805-5-64-0'], ['0806.2805-4-64-1', '0806.2805-5-64-1'], ['0806.2805-4-64-2', '0806.2805-5-64-2'], ['0806.2805-4-64-3', '0806.2805-5-64-3'], ['0806.2805-4-30-0', '0806.2805-5-30-0'], ['0806.2805-4-30-1', '0806.2805-5-30-1'], ['0806.2805-4-6-0', '0806.2805-5-6-0'], ['0806.2805-4-6-1', '0806.2805-5-6-1'], ['0806.2805-4-6-2', '0806.2805-5-6-2'], ['0806.2805-4-6-3', '0806.2805-5-6-3'], ['0806.2805-4-3-0', '0806.2805-5-3-0'], ['0806.2805-4-3-1', '0806.2805-5-3-1'], ['0806.2805-4-3-2', '0806.2805-5-3-2'], ['0806.2805-4-3-3', '0806.2805-5-3-3'], ['0806.2805-4-3-4', '0806.2805-5-3-4'], ['0806.2805-4-60-0', '0806.2805-5-60-0'], ['0806.2805-4-60-1', '0806.2805-5-60-1'], ['0806.2805-4-60-2', '0806.2805-5-60-2'], ['0806.2805-4-60-3', '0806.2805-5-60-3'], ['0806.2805-4-60-4', '0806.2805-5-60-4'], ['0806.2805-4-60-5', '0806.2805-5-60-5'], ['0806.2805-4-60-6', '0806.2805-5-60-6'], ['0806.2805-5-2-1', '0806.2805-6-2-1'], ['0806.2805-5-2-2', '0806.2805-6-2-2'], ['0806.2805-5-30-0', '0806.2805-6-32-0'], ['0806.2805-5-30-1', '0806.2805-6-32-1'], ['0806.2805-5-61-0', '0806.2805-6-65-0'], ['0806.2805-5-61-2', '0806.2805-6-65-2'], ['0806.2805-5-67-0', '0806.2805-6-71-0'], ['0806.2805-5-67-1', '0806.2805-6-71-1'], ['0806.2805-5-67-2', '0806.2805-6-71-2'], ['0806.2805-5-67-3', '0806.2805-6-71-3'], ['0806.2805-5-67-4', '0806.2805-6-71-4'], ['0806.2805-5-7-0', '0806.2805-6-7-0'], ['0806.2805-5-7-1', '0806.2805-6-7-1'], ['0806.2805-5-37-0', '0806.2805-6-40-0'], ['0806.2805-5-37-1', '0806.2805-6-40-1'], ['0806.2805-5-37-2', '0806.2805-6-40-2'], ['0806.2805-5-37-3', '0806.2805-6-40-3'], ['0806.2805-5-14-0', '0806.2805-6-14-0'], ['0806.2805-5-14-1', '0806.2805-6-14-1'], ['0806.2805-5-14-2', '0806.2805-6-14-2'], ['0806.2805-5-14-3', '0806.2805-6-14-3'], ['0806.2805-5-14-4', '0806.2805-6-14-4'], ['0806.2805-5-17-0', '0806.2805-6-17-0'], ['0806.2805-5-17-1', '0806.2805-6-17-1'], ['0806.2805-5-17-2', '0806.2805-6-17-2'], ['0806.2805-5-29-0', '0806.2805-6-31-0'], ['0806.2805-5-29-1', '0806.2805-6-31-1'], ['0806.2805-5-29-2', '0806.2805-6-31-2'], ['0806.2805-5-29-3', '0806.2805-6-31-3'], ['0806.2805-5-29-4', '0806.2805-6-31-4'], ['0806.2805-5-41-0', '0806.2805-6-44-0'], ['0806.2805-5-41-1', '0806.2805-6-44-1'], ['0806.2805-5-41-2', '0806.2805-6-44-2'], ['0806.2805-5-66-0', '0806.2805-6-70-0'], ['0806.2805-5-64-2', '0806.2805-6-68-2'], ['0806.2805-5-64-3', '0806.2805-6-68-3'], ['0806.2805-5-65-0', '0806.2805-6-69-0'], ['0806.2805-5-65-1', '0806.2805-6-69-1'], ['0806.2805-5-65-2', '0806.2805-6-69-2'], ['0806.2805-5-65-3', '0806.2805-6-69-3'], ['0806.2805-5-65-4', '0806.2805-6-69-4'], ['0806.2805-5-50-0', '0806.2805-6-53-0'], ['0806.2805-5-50-2', '0806.2805-6-53-2'], ['0806.2805-5-53-1', '0806.2805-6-56-1'], ['0806.2805-5-53-3', '0806.2805-6-56-3'], ['0806.2805-5-3-0', '0806.2805-6-3-0'], ['0806.2805-5-3-1', '0806.2805-6-3-1'], ['0806.2805-5-3-3', '0806.2805-6-3-3'], ['0806.2805-5-3-4', '0806.2805-6-3-4'], ['0806.2805-5-38-0', '0806.2805-6-41-0'], ['0806.2805-5-38-1', '0806.2805-6-41-1'], ['0806.2805-5-38-2', '0806.2805-6-41-2'], ['0806.2805-5-38-3', '0806.2805-6-41-3'], ['0806.2805-5-33-1', '0806.2805-6-35-1'], ['0806.2805-5-33-2', '0806.2805-6-35-2'], ['0806.2805-5-49-0', '0806.2805-6-52-0'], ['0806.2805-5-51-0', '0806.2805-6-54-0'], ['0806.2805-5-51-1', '0806.2805-6-54-1'], ['0806.2805-5-51-2', '0806.2805-6-54-2'], ['0806.2805-5-51-3', '0806.2805-6-54-3'], ['0806.2805-5-51-4', '0806.2805-6-54-4'], ['0806.2805-5-51-5', '0806.2805-6-54-5'], ['0806.2805-5-51-6', '0806.2805-6-54-6'], ['0806.2805-5-15-1', '0806.2805-6-15-1'], ['0806.2805-5-15-2', '0806.2805-6-15-2'], ['0806.2805-5-15-3', '0806.2805-6-15-3'], ['0806.2805-5-15-4', '0806.2805-6-15-4'], ['0806.2805-5-31-2', '0806.2805-6-33-2'], ['0806.2805-5-69-0', '0806.2805-6-73-0'], ['0806.2805-5-69-1', '0806.2805-6-73-1'], ['0806.2805-5-11-0', '0806.2805-6-11-0'], ['0806.2805-5-11-1', '0806.2805-6-11-1'], ['0806.2805-5-11-2', '0806.2805-6-11-2'], ['0806.2805-5-5-0', '0806.2805-6-5-0'], ['0806.2805-5-20-0', '0806.2805-6-21-0'], ['0806.2805-5-20-1', '0806.2805-6-21-1'], ['0806.2805-5-20-2', '0806.2805-6-21-2'], ['0806.2805-5-20-3', '0806.2805-6-21-3'], ['0806.2805-5-20-5', '0806.2805-6-21-5'], ['0806.2805-5-20-6', '0806.2805-6-21-6'], ['0806.2805-5-36-0', '0806.2805-6-39-0'], ['0806.2805-5-36-1', '0806.2805-6-39-1'], ['0806.2805-5-44-1', '0806.2805-6-47-1'], ['0806.2805-5-22-0', '0806.2805-6-23-0'], ['0806.2805-5-22-1', '0806.2805-6-23-1'], ['0806.2805-5-22-2', '0806.2805-6-23-2'], ['0806.2805-5-26-0', '0806.2805-6-28-0'], ['0806.2805-5-59-0', '0806.2805-6-62-0'], ['0806.2805-5-59-1', '0806.2805-6-62-1'], ['0806.2805-5-8-0', '0806.2805-6-8-0'], ['0806.2805-5-8-1', '0806.2805-6-8-1'], ['0806.2805-5-8-2', '0806.2805-6-8-2'], ['0806.2805-5-8-3', '0806.2805-6-8-3'], ['0806.2805-5-8-4', '0806.2805-6-8-4'], ['0806.2805-5-63-0', '0806.2805-6-67-0'], ['0806.2805-5-63-1', '0806.2805-6-67-1'], ['0806.2805-5-0-0', '0806.2805-6-0-0'], ['0806.2805-5-0-1', '0806.2805-6-0-1'], ['0806.2805-5-42-1', '0806.2805-6-45-1'], ['0806.2805-5-42-2', '0806.2805-6-45-2'], ['0806.2805-5-42-3', '0806.2805-6-45-3'], ['0806.2805-5-42-4', '0806.2805-6-45-4'], ['0806.2805-5-12-2', '0806.2805-6-12-3'], ['0806.2805-5-12-4', '0806.2805-6-12-5'], ['0806.2805-5-52-0', '0806.2805-6-55-0'], ['0806.2805-5-52-1', '0806.2805-6-55-1'], ['0806.2805-5-23-1', '0806.2805-6-24-1'], ['0806.2805-5-19-0', '0806.2805-6-20-0'], ['0806.2805-5-19-1', '0806.2805-6-20-1'], ['0806.2805-5-19-3', '0806.2805-6-20-3'], ['0806.2805-5-72-0', '0806.2805-6-76-0'], ['0806.2805-5-43-1', '0806.2805-6-46-1'], ['0806.2805-5-4-0', '0806.2805-6-4-0'], ['0806.2805-5-24-0', '0806.2805-6-25-0'], ['0806.2805-5-47-0', '0806.2805-6-50-0'], ['0806.2805-5-47-1', '0806.2805-6-50-1'], ['0806.2805-5-47-2', '0806.2805-6-50-2'], ['0806.2805-5-47-3', '0806.2805-6-50-3'], ['0806.2805-5-47-4', '0806.2805-6-50-4'], ['0806.2805-5-55-2', '0806.2805-6-58-2'], ['0806.2805-5-70-0', '0806.2805-6-74-0'], ['0806.2805-5-70-1', '0806.2805-6-74-1'], ['0806.2805-5-10-0', '0806.2805-6-10-0'], ['0806.2805-5-10-1', '0806.2805-6-10-1'], ['0806.2805-5-10-3', '0806.2805-6-10-3'], ['0806.2805-5-10-4', '0806.2805-6-10-4'], ['0806.2805-5-10-5', '0806.2805-6-10-5'], ['0806.2805-5-6-0', '0806.2805-6-6-0'], ['0806.2805-5-6-1', '0806.2805-6-6-1'], ['0806.2805-5-6-2', '0806.2805-6-6-2'], ['0806.2805-6-45-0', '0806.2805-7-45-0'], ['0806.2805-6-45-1', '0806.2805-7-45-1'], ['0806.2805-6-45-2', '0806.2805-7-45-2'], ['0806.2805-6-45-3', '0806.2805-7-45-3'], ['0806.2805-6-45-4', '0806.2805-7-45-4'], ['0806.2805-6-67-0', '0806.2805-7-67-0'], ['0806.2805-6-67-1', '0806.2805-7-67-1'], ['0806.2805-6-12-0', '0806.2805-7-12-0'], ['0806.2805-6-12-1', '0806.2805-7-12-1'], ['0806.2805-6-12-2', '0806.2805-7-12-2'], ['0806.2805-6-12-3', '0806.2805-7-12-3'], ['0806.2805-6-12-4', '0806.2805-7-12-4'], ['0806.2805-6-12-5', '0806.2805-7-12-5'], ['0806.2805-6-37-0', '0806.2805-7-37-0'], ['0806.2805-6-39-0', '0806.2805-7-39-0'], ['0806.2805-6-39-1', '0806.2805-7-39-1'], ['0806.2805-6-41-0', '0806.2805-7-41-0'], ['0806.2805-6-41-1', '0806.2805-7-41-1'], ['0806.2805-6-41-2', '0806.2805-7-41-2'], ['0806.2805-6-41-3', '0806.2805-7-41-3'], ['0806.2805-6-53-0', '0806.2805-7-53-0'], ['0806.2805-6-53-1', '0806.2805-7-53-1'], ['0806.2805-6-53-2', '0806.2805-7-53-2'], ['0806.2805-6-47-0', '0806.2805-7-47-0'], ['0806.2805-6-47-1', '0806.2805-7-47-1'], ['0806.2805-6-47-2', '0806.2805-7-47-2'], ['0806.2805-6-59-0', '0806.2805-7-59-0'], ['0806.2805-6-5-0', '0806.2805-7-5-0'], ['0806.2805-6-5-1', '0806.2805-7-5-1'], ['0806.2805-6-54-0', '0806.2805-7-54-0'], ['0806.2805-6-54-1', '0806.2805-7-54-1'], ['0806.2805-6-54-2', '0806.2805-7-54-2'], ['0806.2805-6-54-3', '0806.2805-7-54-3'], ['0806.2805-6-54-4', '0806.2805-7-54-4'], ['0806.2805-6-54-5', '0806.2805-7-54-5'], ['0806.2805-6-54-6', '0806.2805-7-54-6'], ['0806.2805-6-13-0', '0806.2805-7-13-0'], ['0806.2805-6-19-0', '0806.2805-7-19-0'], ['0806.2805-6-19-1', '0806.2805-7-19-1'], ['0806.2805-6-27-0', '0806.2805-7-27-0'], ['0806.2805-6-27-1', '0806.2805-7-27-1'], ['0806.2805-6-27-2', '0806.2805-7-27-2'], ['0806.2805-6-27-3', '0806.2805-7-27-3'], ['0806.2805-6-27-4', '0806.2805-7-27-4'], ['0806.2805-6-27-5', '0806.2805-7-27-5'], ['0806.2805-6-27-6', '0806.2805-7-27-6'], ['0806.2805-6-26-0', '0806.2805-7-26-0'], ['0806.2805-6-26-1', '0806.2805-7-26-1'], ['0806.2805-6-76-0', '0806.2805-7-76-0'], ['0806.2805-6-76-1', '0806.2805-7-76-1'], ['0806.2805-6-70-0', '0806.2805-7-70-0'], ['0806.2805-6-70-1', '0806.2805-7-70-1'], ['0806.2805-6-70-2', '0806.2805-7-70-2'], ['0806.2805-6-70-3', '0806.2805-7-70-3'], ['0806.2805-6-70-4', '0806.2805-7-70-4'], ['0806.2805-6-70-5', '0806.2805-7-70-5'], ['0806.2805-6-21-0', '0806.2805-7-21-0'], ['0806.2805-6-21-1', '0806.2805-7-21-1'], ['0806.2805-6-21-2', '0806.2805-7-21-2'], ['0806.2805-6-21-3', '0806.2805-7-21-3'], ['0806.2805-6-21-4', '0806.2805-7-21-4'], ['0806.2805-6-21-5', '0806.2805-7-21-5'], ['0806.2805-6-21-6', '0806.2805-7-21-6'], ['0806.2805-6-20-0', '0806.2805-7-20-0'], ['0806.2805-6-20-1', '0806.2805-7-20-1'], ['0806.2805-6-20-2', '0806.2805-7-20-2'], ['0806.2805-6-20-3', '0806.2805-7-20-3'], ['0806.2805-6-16-0', '0806.2805-7-16-0'], ['0806.2805-6-17-0', '0806.2805-7-17-0'], ['0806.2805-6-17-1', '0806.2805-7-17-1'], ['0806.2805-6-17-2', '0806.2805-7-17-2'], ['0806.2805-6-17-3', '0806.2805-7-17-3'], ['0806.2805-6-64-0', '0806.2805-7-64-0'], ['0806.2805-6-64-1', '0806.2805-7-64-1'], ['0806.2805-6-71-0', '0806.2805-7-71-0'], ['0806.2805-6-71-1', '0806.2805-7-71-1'], ['0806.2805-6-71-2', '0806.2805-7-71-2'], ['0806.2805-6-71-3', '0806.2805-7-71-3'], ['0806.2805-6-71-4', '0806.2805-7-71-4'], ['0806.2805-6-15-0', '0806.2805-7-15-0'], ['0806.2805-6-15-1', '0806.2805-7-15-1'], ['0806.2805-6-15-2', '0806.2805-7-15-2'], ['0806.2805-6-15-3', '0806.2805-7-15-3'], ['0806.2805-6-15-4', '0806.2805-7-15-4'], ['0806.2805-6-73-0', '0806.2805-7-73-0'], ['0806.2805-6-73-1', '0806.2805-7-73-1'], ['0806.2805-6-63-0', '0806.2805-7-63-0'], ['0806.2805-6-63-1', '0806.2805-7-63-1'], ['0806.2805-6-63-2', '0806.2805-7-63-2'], ['0806.2805-6-63-3', '0806.2805-7-63-3'], ['0806.2805-6-63-4', '0806.2805-7-63-4'], ['0806.2805-6-63-5', '0806.2805-7-63-5'], ['0806.2805-6-63-6', '0806.2805-7-63-6'], ['0806.2805-6-58-0', '0806.2805-7-58-0'], ['0806.2805-6-58-1', '0806.2805-7-58-1'], ['0806.2805-6-58-2', '0806.2805-7-58-2'], ['0806.2805-6-56-0', '0806.2805-7-56-0'], ['0806.2805-6-56-1', '0806.2805-7-56-1'], ['0806.2805-6-56-2', '0806.2805-7-56-2'], ['0806.2805-6-56-3', '0806.2805-7-56-3'], ['0806.2805-6-56-4', '0806.2805-7-56-4'], ['0806.2805-6-2-0', '0806.2805-7-2-0'], ['0806.2805-6-2-1', '0806.2805-7-2-1'], ['0806.2805-6-2-2', '0806.2805-7-2-2'], ['0806.2805-6-60-0', '0806.2805-7-60-0'], ['0806.2805-6-52-0', '0806.2805-7-52-0'], ['0806.2805-6-36-0', '0806.2805-7-36-0'], ['0806.2805-6-36-1', '0806.2805-7-36-1'], ['0806.2805-6-36-2', '0806.2805-7-36-2'], ['0806.2805-6-36-3', '0806.2805-7-36-3'], ['0806.2805-6-36-4', '0806.2805-7-36-4'], ['0806.2805-6-36-5', '0806.2805-7-36-5'], ['0806.2805-6-69-0', '0806.2805-7-69-0'], ['0806.2805-6-69-1', '0806.2805-7-69-1'], ['0806.2805-6-69-2', '0806.2805-7-69-2'], ['0806.2805-6-69-3', '0806.2805-7-69-3'], ['0806.2805-6-69-4', '0806.2805-7-69-4'], ['0806.2805-6-50-0', '0806.2805-7-50-0'], ['0806.2805-6-50-1', '0806.2805-7-50-1'], ['0806.2805-6-50-2', '0806.2805-7-50-2'], ['0806.2805-6-50-3', '0806.2805-7-50-3'], ['0806.2805-6-50-4', '0806.2805-7-50-4'], ['0806.2805-6-78-0', '0806.2805-7-78-0'], ['0806.2805-6-78-1', '0806.2805-7-78-1'], ['0806.2805-6-10-0', '0806.2805-7-10-0'], ['0806.2805-6-10-1', '0806.2805-7-10-1'], ['0806.2805-6-10-2', '0806.2805-7-10-2'], ['0806.2805-6-10-3', '0806.2805-7-10-3'], ['0806.2805-6-10-4', '0806.2805-7-10-4'], ['0806.2805-6-10-5', '0806.2805-7-10-5'], ['0806.2805-6-11-0', '0806.2805-7-11-0'], ['0806.2805-6-11-1', '0806.2805-7-11-1'], ['0806.2805-6-11-2', '0806.2805-7-11-2'], ['0806.2805-6-68-0', '0806.2805-7-68-0'], ['0806.2805-6-68-1', '0806.2805-7-68-1'], ['0806.2805-6-68-2', '0806.2805-7-68-2'], ['0806.2805-6-68-3', '0806.2805-7-68-3'], ['0806.2805-6-14-0', '0806.2805-7-14-0'], ['0806.2805-6-14-1', '0806.2805-7-14-1'], ['0806.2805-6-14-2', '0806.2805-7-14-2'], ['0806.2805-6-14-3', '0806.2805-7-14-3'], ['0806.2805-6-14-4', '0806.2805-7-14-4'], ['0806.2805-6-25-0', '0806.2805-7-25-0'], ['0806.2805-6-4-0', '0806.2805-7-4-0'], ['0806.2805-6-4-1', '0806.2805-7-4-1'], ['0806.2805-6-4-2', '0806.2805-7-4-2'], ['0806.2805-6-33-0', '0806.2805-7-33-0'], ['0806.2805-6-33-1', '0806.2805-7-33-1'], ['0806.2805-6-33-2', '0806.2805-7-33-2'], ['0806.2805-6-32-0', '0806.2805-7-32-0'], ['0806.2805-6-32-1', '0806.2805-7-32-1'], ['0806.2805-6-0-0', '0806.2805-7-0-0'], ['0806.2805-6-0-1', '0806.2805-7-0-1'], ['0806.2805-6-0-2', '0806.2805-7-0-2'], ['0806.2805-6-0-3', '0806.2805-7-0-3'], ['0806.2805-6-0-4', '0806.2805-7-0-4'], ['0806.2805-6-6-0', '0806.2805-7-6-0'], ['0806.2805-6-6-1', '0806.2805-7-6-1'], ['0806.2805-6-6-2', '0806.2805-7-6-2'], ['0806.2805-6-6-3', '0806.2805-7-6-3'], ['0806.2805-6-44-0', '0806.2805-7-44-0'], ['0806.2805-6-44-1', '0806.2805-7-44-1'], ['0806.2805-6-44-2', '0806.2805-7-44-2'], ['0806.2805-6-62-0', '0806.2805-7-62-0'], ['0806.2805-6-62-1', '0806.2805-7-62-1'], ['0806.2805-6-3-0', '0806.2805-7-3-0'], ['0806.2805-6-3-1', '0806.2805-7-3-1'], ['0806.2805-6-3-2', '0806.2805-7-3-2'], ['0806.2805-6-3-3', '0806.2805-7-3-3'], ['0806.2805-6-3-4', '0806.2805-7-3-4'], ['0806.2805-6-23-0', '0806.2805-7-23-0'], ['0806.2805-6-23-1', '0806.2805-7-23-1'], ['0806.2805-6-23-2', '0806.2805-7-23-2'], ['0806.2805-6-28-0', '0806.2805-7-28-0'], ['0806.2805-6-31-0', '0806.2805-7-31-0'], ['0806.2805-6-31-1', '0806.2805-7-31-1'], ['0806.2805-6-31-2', '0806.2805-7-31-2'], ['0806.2805-6-31-3', '0806.2805-7-31-3'], ['0806.2805-6-31-4', '0806.2805-7-31-4'], ['0806.2805-6-24-0', '0806.2805-7-24-0'], ['0806.2805-6-24-1', '0806.2805-7-24-1'], ['0806.2805-6-40-0', '0806.2805-7-40-0'], ['0806.2805-6-40-1', '0806.2805-7-40-1'], ['0806.2805-6-40-2', '0806.2805-7-40-2'], ['0806.2805-6-40-3', '0806.2805-7-40-3'], ['0806.2805-6-46-0', '0806.2805-7-46-0'], ['0806.2805-6-46-1', '0806.2805-7-46-1'], ['0806.2805-6-48-0', '0806.2805-7-48-0'], ['0806.2805-6-8-0', '0806.2805-7-8-0'], ['0806.2805-6-8-1', '0806.2805-7-8-1'], ['0806.2805-6-8-2', '0806.2805-7-8-2'], ['0806.2805-6-8-3', '0806.2805-7-8-3'], ['0806.2805-6-8-4', '0806.2805-7-8-4'], ['0806.2805-6-55-0', '0806.2805-7-55-0'], ['0806.2805-6-55-1', '0806.2805-7-55-1'], ['0806.2805-6-18-0', '0806.2805-7-18-0'], ['0806.2805-6-65-0', '0806.2805-7-65-0'], ['0806.2805-6-65-1', '0806.2805-7-65-1'], ['0806.2805-6-65-2', '0806.2805-7-65-2'], ['0806.2805-6-35-0', '0806.2805-7-35-0'], ['0806.2805-6-35-1', '0806.2805-7-35-1'], ['0806.2805-6-35-2', '0806.2805-7-35-2'], ['0806.2805-6-74-0', '0806.2805-7-74-0'], ['0806.2805-6-74-1', '0806.2805-7-74-1'], ['0806.2805-6-7-0', '0806.2805-7-7-0'], ['0806.2805-6-7-1', '0806.2805-7-7-1'], ['0806.2805-1-28-5', '0806.2805-2-29-3'], ['0806.2805-2-13-3', '0806.2805-3-16-2'], ['0806.2805-2-13-4', '0806.2805-3-16-3'], ['0806.2805-2-24-1', '0806.2805-3-29-1'], ['0806.2805-2-24-2', '0806.2805-3-29-2'], ['0806.2805-3-46-2', '0806.2805-4-50-1'], ['0806.2805-3-46-3', '0806.2805-4-50-2'], ['0806.2805-3-13-1', '0806.2805-4-14-0'], ['0806.2805-3-13-2', '0806.2805-4-14-1'], ['0806.2805-3-13-3', '0806.2805-4-14-2'], ['0806.2805-3-13-4', '0806.2805-4-14-3'], ['0806.2805-3-31-1', '0806.2805-4-33-1'], ['0806.2805-3-31-2', '0806.2805-4-33-2'], ['0806.2805-3-31-3', '0806.2805-4-34-0'], ['0806.2805-3-31-4', '0806.2805-4-34-1'], ['0806.2805-3-31-5', '0806.2805-4-34-2'], ['0806.2805-3-31-6', '0806.2805-4-34-3'], ['0806.2805-3-31-8', '0806.2805-4-34-5'], ['0806.2805-5-18-1', '0806.2805-6-19-0'], ['0806.2805-5-18-2', '0806.2805-6-19-1'], ['0806.2805-5-60-0', '0806.2805-6-63-0'], ['0806.2805-5-60-1', '0806.2805-6-63-1'], ['0806.2805-5-60-2', '0806.2805-6-63-2'], ['0806.2805-5-60-4', '0806.2805-6-63-4'], ['0806.2805-5-60-5', '0806.2805-6-63-5'], ['0806.2805-5-60-6', '0806.2805-6-63-6'], ['0806.2805-5-60-7', '0806.2805-6-64-0'], ['0806.2805-5-60-8', '0806.2805-6-64-1'], ['0806.2805-5-34-0', '0806.2805-6-36-0'], ['0806.2805-5-34-1', '0806.2805-6-36-1'], ['0806.2805-5-34-3', '0806.2805-6-36-3'], ['0806.2805-5-34-4', '0806.2805-6-36-4'], ['0806.2805-5-25-0', '0806.2805-6-26-0'], ['0806.2805-5-25-3', '0806.2805-6-27-1'], ['0806.2805-5-25-4', '0806.2805-6-27-2'], ['0806.2805-5-25-5', '0806.2805-6-27-3'], ['0806.2805-5-25-6', '0806.2805-6-27-4'], ['0806.2805-5-25-8', '0806.2805-6-27-6']]

[['0806.2805-1-8-0', '0806.2805-2-8-0'], ['0806.2805-1-8-2', '0806.2805-2-8-2'], ['0806.2805-1-19-1', '0806.2805-2-19-1'], ['0806.2805-1-19-5', '0806.2805-2-19-5'], ['0806.2805-1-19-6', '0806.2805-2-19-6'], ['0806.2805-1-19-7', '0806.2805-2-19-7'], ['0806.2805-1-11-0', '0806.2805-2-11-0'], ['0806.2805-1-11-1', '0806.2805-2-11-1'], ['0806.2805-1-3-3', '0806.2805-2-3-3'], ['0806.2805-1-3-4', '0806.2805-2-3-4'], ['0806.2805-1-2-0', '0806.2805-2-2-0'], ['0806.2805-1-6-1', '0806.2805-2-6-1'], ['0806.2805-1-9-0', '0806.2805-2-9-0'], ['0806.2805-1-9-2', '0806.2805-2-9-2'], ['0806.2805-1-17-0', '0806.2805-2-17-0'], ['0806.2805-1-17-1', '0806.2805-2-17-1'], ['0806.2805-1-14-0', '0806.2805-2-14-0'], ['0806.2805-1-14-3', '0806.2805-2-14-3'], ['0806.2805-1-55-0', '0806.2805-2-58-0'], ['0806.2805-1-52-1', '0806.2805-2-53-1'], ['0806.2805-1-39-0', '0806.2805-2-40-0'], ['0806.2805-1-39-3', '0806.2805-2-40-3'], ['0806.2805-1-47-0', '0806.2805-2-48-0'], ['0806.2805-1-53-0', '0806.2805-2-55-0'], ['0806.2805-1-53-1', '0806.2805-2-55-1'], ['0806.2805-1-37-1', '0806.2805-2-38-1'], ['0806.2805-1-37-2', '0806.2805-2-38-2'], ['0806.2805-1-44-0', '0806.2805-2-45-0'], ['0806.2805-1-0-0', '0806.2805-2-0-0'], ['0806.2805-1-0-1', '0806.2805-2-0-1'], ['0806.2805-1-0-3', '0806.2805-2-0-3'], ['0806.2805-1-40-1', '0806.2805-2-41-1'], ['0806.2805-1-40-4', '0806.2805-2-41-4'], ['0806.2805-1-35-2', '0806.2805-2-36-2'], ['0806.2805-1-4-0', '0806.2805-2-4-0'], ['0806.2805-1-4-1', '0806.2805-2-4-1'], ['0806.2805-1-4-2', '0806.2805-2-4-2'], ['0806.2805-1-32-0', '0806.2805-2-33-0'], ['0806.2805-1-32-1', '0806.2805-2-33-1'], ['0806.2805-1-32-2', '0806.2805-2-33-2'], ['0806.2805-1-23-1', '0806.2805-2-23-1'], ['0806.2805-1-5-1', '0806.2805-2-5-1'], ['0806.2805-1-5-2', '0806.2805-2-5-2'], ['0806.2805-1-51-1', '0806.2805-2-52-1'], ['0806.2805-1-43-2', '0806.2805-2-44-2'], ['0806.2805-1-26-2', '0806.2805-2-26-2'], ['0806.2805-1-26-3', '0806.2805-2-26-3'], ['0806.2805-1-26-8', '0806.2805-2-26-8'], ['0806.2805-1-12-0', '0806.2805-2-12-0'], ['0806.2805-1-12-4', '0806.2805-2-12-4'], ['0806.2805-1-29-2', '0806.2805-2-30-2'], ['0806.2805-1-24-0', '0806.2805-2-24-0'], ['0806.2805-1-56-0', '0806.2805-2-59-0'], ['0806.2805-1-48-1', '0806.2805-2-49-1'], ['0806.2805-1-33-0', '0806.2805-2-34-0'], ['0806.2805-1-33-2', '0806.2805-2-34-2'], ['0806.2805-1-41-3', '0806.2805-2-42-3'], ['0806.2805-1-41-4', '0806.2805-2-42-4'], ['0806.2805-2-40-1', '0806.2805-3-46-1'], ['0806.2805-2-40-3', '0806.2805-3-46-3'], ['0806.2805-2-41-3', '0806.2805-3-47-3'], ['0806.2805-2-41-6', '0806.2805-3-47-6'], ['0806.2805-2-4-2', '0806.2805-3-4-2'], ['0806.2805-2-12-3', '0806.2805-3-14-3'], ['0806.2805-2-0-0', '0806.2805-3-0-0'], ['0806.2805-2-0-1', '0806.2805-3-0-1'], ['0806.2805-2-33-2', '0806.2805-3-38-2'], ['0806.2805-2-52-1', '0806.2805-3-59-1'], ['0806.2805-2-2-0', '0806.2805-3-2-0'], ['0806.2805-2-10-0', '0806.2805-3-12-0'], ['0806.2805-2-19-3', '0806.2805-3-23-3'], ['0806.2805-2-28-0', '0806.2805-3-33-0'], ['0806.2805-2-55-2', '0806.2805-3-62-4'], ['0806.2805-2-42-4', '0806.2805-3-49-4'], ['0806.2805-2-34-0', '0806.2805-3-39-0'], ['0806.2805-2-38-4', '0806.2805-3-44-4'], ['0806.2805-2-6-0', '0806.2805-3-7-0'], ['0806.2805-2-23-3', '0806.2805-3-27-3'], ['0806.2805-2-8-4', '0806.2805-3-10-4'], ['0806.2805-2-26-2', '0806.2805-3-31-2'], ['0806.2805-2-26-8', '0806.2805-3-31-8'], ['0806.2805-2-29-3', '0806.2805-3-34-3'], ['0806.2805-2-45-0', '0806.2805-3-52-0'], ['0806.2805-2-11-0', '0806.2805-3-13-0'], ['0806.2805-2-11-1', '0806.2805-3-13-2'], ['0806.2805-2-56-2', '0806.2805-3-63-2'], ['0806.2805-2-5-2', '0806.2805-3-6-2'], ['0806.2805-2-49-5', '0806.2805-3-56-5'], ['0806.2805-3-11-2', '0806.2805-4-11-2'], ['0806.2805-3-61-3', '0806.2805-4-65-3'], ['0806.2805-3-14-1', '0806.2805-4-15-1'], ['0806.2805-3-14-2', '0806.2805-4-15-2'], ['0806.2805-3-52-0', '0806.2805-4-56-0'], ['0806.2805-3-41-2', '0806.2805-4-44-2'], ['0806.2805-3-60-0', '0806.2805-4-64-0'], ['0806.2805-3-2-1', '0806.2805-4-2-1'], ['0806.2805-3-40-0', '0806.2805-4-43-0'], ['0806.2805-3-34-2', '0806.2805-4-37-2'], ['0806.2805-3-53-0', '0806.2805-4-57-0'], ['0806.2805-3-6-3', '0806.2805-4-6-3'], ['0806.2805-3-23-6', '0806.2805-4-25-6'], ['0806.2805-3-5-1', '0806.2805-4-5-1'], ['0806.2805-3-44-4', '0806.2805-4-47-4'], ['0806.2805-3-38-2', '0806.2805-4-41-2'], ['0806.2805-3-12-0', '0806.2805-4-12-0'], ['0806.2805-3-12-1', '0806.2805-4-12-1'], ['0806.2805-3-10-0', '0806.2805-4-10-0'], ['0806.2805-3-10-4', '0806.2805-4-10-4'], ['0806.2805-3-62-1', '0806.2805-4-66-2'], ['0806.2805-3-62-2', '0806.2805-4-66-3'], ['0806.2805-3-63-2', '0806.2805-4-67-2'], ['0806.2805-3-0-1', '0806.2805-4-0-1'], ['0806.2805-3-49-0', '0806.2805-4-53-0'], ['0806.2805-3-49-2', '0806.2805-4-53-2'], ['0806.2805-3-18-0', '0806.2805-4-20-0'], ['0806.2805-3-18-1', '0806.2805-4-20-1'], ['0806.2805-3-18-4', '0806.2805-4-20-4'], ['0806.2805-3-18-5', '0806.2805-4-20-6'], ['0806.2805-3-29-0', '0806.2805-4-31-0'], ['0806.2805-3-29-1', '0806.2805-4-31-1'], ['0806.2805-3-48-1', '0806.2805-4-52-1'], ['0806.2805-3-8-0', '0806.2805-4-8-0'], ['0806.2805-3-8-1', '0806.2805-4-8-1'], ['0806.2805-3-8-3', '0806.2805-4-8-3'], ['0806.2805-3-56-6', '0806.2805-4-60-6'], ['0806.2805-4-0-1', '0806.2805-5-0-1'], ['0806.2805-4-66-2', '0806.2805-5-66-2'], ['0806.2805-4-66-3', '0806.2805-5-66-3'], ['0806.2805-4-66-4', '0806.2805-5-66-4'], ['0806.2805-5-2-0', '0806.2805-6-2-0'], ['0806.2805-5-61-1', '0806.2805-6-65-1'], ['0806.2805-5-57-0', '0806.2805-6-60-0'], ['0806.2805-5-17-3', '0806.2805-6-17-3'], ['0806.2805-5-66-1', '0806.2805-6-70-1'], ['0806.2805-5-66-2', '0806.2805-6-70-2'], ['0806.2805-5-66-3', '0806.2805-6-70-3'], ['0806.2805-5-66-4', '0806.2805-6-70-4'], ['0806.2805-5-66-5', '0806.2805-6-70-5'], ['0806.2805-5-64-0', '0806.2805-6-68-0'], ['0806.2805-5-64-1', '0806.2805-6-68-1'], ['0806.2805-5-50-1', '0806.2805-6-53-1'], ['0806.2805-5-53-0', '0806.2805-6-56-0'], ['0806.2805-5-53-2', '0806.2805-6-56-2'], ['0806.2805-5-53-4', '0806.2805-6-56-4'], ['0806.2805-5-3-2', '0806.2805-6-3-2'], ['0806.2805-5-56-0', '0806.2805-6-59-0'], ['0806.2805-5-33-0', '0806.2805-6-35-0'], ['0806.2805-5-15-0', '0806.2805-6-15-0'], ['0806.2805-5-16-0', '0806.2805-6-16-0'], ['0806.2805-5-31-0', '0806.2805-6-33-0'], ['0806.2805-5-31-1', '0806.2805-6-33-1'], ['0806.2805-5-5-1', '0806.2805-6-5-1'], ['0806.2805-5-20-4', '0806.2805-6-21-4'], ['0806.2805-5-44-0', '0806.2805-6-47-0'], ['0806.2805-5-44-2', '0806.2805-6-47-2'], ['0806.2805-5-13-0', '0806.2805-6-13-0'], ['0806.2805-5-0-2', '0806.2805-6-0-2'], ['0806.2805-5-0-3', '0806.2805-6-0-3'], ['0806.2805-5-0-4', '0806.2805-6-0-4'], ['0806.2805-5-42-0', '0806.2805-6-45-0'], ['0806.2805-5-12-0', '0806.2805-6-12-0'], ['0806.2805-5-12-1', '0806.2805-6-12-2'], ['0806.2805-5-12-3', '0806.2805-6-12-4'], ['0806.2805-5-23-0', '0806.2805-6-24-0'], ['0806.2805-5-19-2', '0806.2805-6-20-2'], ['0806.2805-5-72-1', '0806.2805-6-76-1'], ['0806.2805-5-43-0', '0806.2805-6-46-0'], ['0806.2805-5-4-1', '0806.2805-6-4-1'], ['0806.2805-5-4-2', '0806.2805-6-4-2'], ['0806.2805-5-45-0', '0806.2805-6-48-0'], ['0806.2805-5-55-0', '0806.2805-6-58-0'], ['0806.2805-5-55-1', '0806.2805-6-58-1'], ['0806.2805-5-10-2', '0806.2805-6-10-2'], ['0806.2805-5-6-3', '0806.2805-6-6-3'], ['0806.2805-1-28-0', '0806.2805-2-28-0'], ['0806.2805-1-28-2', '0806.2805-2-29-0'], ['0806.2805-1-28-3', '0806.2805-2-29-1'], ['0806.2805-2-13-1', '0806.2805-3-16-0'], ['0806.2805-2-13-2', '0806.2805-3-16-1'], ['0806.2805-2-24-0', '0806.2805-3-29-0'], ['0806.2805-3-46-0', '0806.2805-4-49-0'], ['0806.2805-3-46-1', '0806.2805-4-50-0'], ['0806.2805-3-13-0', '0806.2805-4-13-0'], ['0806.2805-3-13-5', '0806.2805-4-14-4'], ['0806.2805-3-31-0', '0806.2805-4-33-0'], ['0806.2805-3-31-9', '0806.2805-4-34-6'], ['0806.2805-5-18-0', '0806.2805-6-18-0'], ['0806.2805-5-60-3', '0806.2805-6-63-3'], ['0806.2805-5-34-2', '0806.2805-6-36-2'], ['0806.2805-5-34-5', '0806.2805-6-36-5'], ['0806.2805-5-34-6', '0806.2805-6-37-0'], ['0806.2805-5-25-1', '0806.2805-6-26-1'], ['0806.2805-5-25-2', '0806.2805-6-27-0'], ['0806.2805-5-25-7', '0806.2805-6-27-5']]

[]

[['0806.2805-1-19-4', '0806.2805-2-19-4'], ['0806.2805-1-3-2', '0806.2805-2-3-2'], ['0806.2805-1-14-1', '0806.2805-2-14-1'], ['0806.2805-1-52-0', '0806.2805-2-53-0'], ['0806.2805-1-39-1', '0806.2805-2-40-1'], ['0806.2805-1-47-1', '0806.2805-2-48-1'], ['0806.2805-1-45-0', '0806.2805-2-46-0'], ['0806.2805-1-20-0', '0806.2805-2-20-0'], ['0806.2805-1-13-0', '0806.2805-2-13-0'], ['0806.2805-1-23-0', '0806.2805-2-23-0'], ['0806.2805-1-10-0', '0806.2805-2-10-0'], ['0806.2805-1-10-0', '0806.2805-2-10-1'], ['0806.2805-1-12-1', '0806.2805-2-12-1'], ['0806.2805-1-29-3', '0806.2805-2-30-3'], ['0806.2805-1-24-1', '0806.2805-2-24-1'], ['0806.2805-1-56-1', '0806.2805-2-59-1'], ['0806.2805-1-48-2', '0806.2805-2-49-2'], ['0806.2805-1-41-0', '0806.2805-2-42-0'], ['0806.2805-1-41-1', '0806.2805-2-42-1'], ['0806.2805-1-41-2', '0806.2805-2-42-2'], ['0806.2805-2-19-1', '0806.2805-3-23-1'], ['0806.2805-2-55-1', '0806.2805-3-62-1'], ['0806.2805-2-55-1', '0806.2805-3-62-2'], ['0806.2805-2-42-0', '0806.2805-3-49-0'], ['0806.2805-2-42-2', '0806.2805-3-49-2'], ['0806.2805-2-16-0', '0806.2805-3-20-0'], ['0806.2805-2-11-2', '0806.2805-3-13-3'], ['0806.2805-2-54-3', '0806.2805-3-61-3'], ['0806.2805-2-49-3', '0806.2805-3-56-3'], ['0806.2805-2-49-4', '0806.2805-3-56-4'], ['0806.2805-3-21-1', '0806.2805-4-23-1'], ['0806.2805-3-40-1', '0806.2805-4-43-1'], ['0806.2805-3-35-3', '0806.2805-4-38-3'], ['0806.2805-3-16-1', '0806.2805-4-18-0'], ['0806.2805-3-55-0', '0806.2805-4-59-0'], ['0806.2805-1-28-1', '0806.2805-2-28-1'], ['0806.2805-1-28-4', '0806.2805-2-29-2'], ['0806.2805-2-13-0', '0806.2805-3-15-0'], ['0806.2805-3-31-7', '0806.2805-4-34-4']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '6': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '7': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0806.2805

{'0806.2805-3-0-0': 'A modified-gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-3-0-1': 'This theory provides a concrete realization for the general vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics, [MATH].', '0806.2805-3-0-2': 'The theory gives a flat Friedmann-Robertson-Walker (FRW) universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-3-0-3': 'Extrapolating to the present age of the universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-3-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter (CDM) in the standard FRW framework.', '0806.2805-3-1-0': '# Introduction', '0806.2805-3-2-0': 'In a previous article [CITATION], we have proposed to characterize a Lorentz-invariant quantum vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-3-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability, compressibility, and response to perturbations.', '0806.2805-3-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-3-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-3-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-3-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density which contributes to the effective gravitational field equations at low energies.', '0806.2805-3-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-3-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-3-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-3-4-1': 'The same happens with other types of perturbations which violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-3-4-2': 'According to this approach, the present value of [MATH] is nonzero but small because the universe is close to equilibrium and Lorentz-noninvariant perturbations of the quantum vacuum are small (compared to the ultraviolet scale which sets the microscopic energy density [MATH]).', '0806.2805-3-5-0': 'The situation is different for Lorentz-invariant perturbations of the vacuum, such as the formation of scalar condensates as discussed in Ref. [CITATION] or quark/gluon condensates derived from quantum chromodynamics (cf. Ref. [CITATION]).', '0806.2805-3-5-1': 'In this case, the variable [MATH] shifts in such a way that it completely compensates the energy density of the perturbation and the cosmological constant remains zero in the new Lorentz-invariant equilibrium vacuum.', '0806.2805-3-6-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-3-6-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-3-6-2': 'Here, we use this explicit realization with a four-form field [MATH] to study the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective cosmological "constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-3-6-3': 'In short, the present cosmological constant is small because the universe is old.', '0806.2805-3-7-0': 'The results of the present article show that, for the type of theory considered, the decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and that the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-3-7-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-3-8-0': 'These interesting results are obtained by the following steps.', '0806.2805-3-8-1': 'In Sec. [REF], the gravity theory with a four-form field [MATH] is defined in terms of general functions for the microscopic energy density [MATH] and variable gravitational coupling parameter [MATH].', '0806.2805-3-8-2': 'In Sec. [REF], the dynamics of the corresponding de-Sitter universe without matter is discussed and, in Sec. [REF], the dynamics of a flat FRW universe with matter, using simple Ansatze for the functions [MATH] and [MATH].', '0806.2805-3-8-3': 'In Sec. [REF], the approach to equilibrium in such a FRW universe is studied in detail and the above mention vacuum oscillations are established.', '0806.2805-3-8-4': 'In Sec. [REF], the main results are summarized.', '0806.2805-3-9-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-3-10-0': 'Here and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-3-10-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-3-10-2': 'Such a quadratic function corresponds to a gas-like vacuum [CITATION].', '0806.2805-3-10-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-3-10-4': 'A self-sustained vacuum requires a more complicated function [MATH], so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-3-10-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-3-11-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-3-11-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-3-11-2': 'Moreover, a [MATH] dependence allows for the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-3-12-0': 'Specifically, the action considered takes the following form ([MATH]): [EQUATION] where [MATH] denotes the covariant derivative, [MATH] the Levi-Civita symbol, and square brackets around spacetime indices complete anti-symmetrization.', '0806.2805-3-12-1': 'The field [MATH] in [REF] stands for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants to [MATH]-dependent parameters).', '0806.2805-3-12-2': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-3-13-0': "The variation of the action [REF] over the three-form gauge field [MATH] gives the generalized Maxwell equations, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equations, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [CITATION] [EQUATION]", '0806.2805-3-13-1': 'At this point two remarks may be helpful.', '0806.2805-3-13-2': 'First, observe that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-3-13-3': 'This implicit metric dependence of [MATH] explains the origin of the second term on the left-hand side of [REF].', '0806.2805-3-13-4': 'Second, observe that the three-form gauge field [MATH] does not propagate physical degrees of freedom in flat spacetime [CITATION].', '0806.2805-3-13-5': 'Still, [MATH] has gravitational effects, both classically in the modified-gravity theory with [MATH] as discussed in the present article (see, in particular, Sec. [REF]) and quantum-mechanically already in the standard gravity theory with [MATH] (having, for example, a nonvanishing gravitational trace anomaly [CITATION]).', '0806.2805-3-14-0': 'Using [REF] for [MATH], we obtain the Maxwell equations [REF] in the form [EQUATION]', '0806.2805-3-14-1': 'The solution is [EQUATION] with integration constant [MATH].', '0806.2805-3-14-2': 'Here, the constant [MATH] is seen to emerge dynamically.', '0806.2805-3-14-3': 'In a thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-3-14-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-3-15-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equations become [EQUATION] which will be used in the rest of this article, together with [REF].', '0806.2805-3-16-0': 'Equations [REF] and [REF] are also obtained if we use, instead of the original action, the effective action [EQUATION]', '0806.2805-3-16-1': 'If the four-form field is considered to be given in terms of a three-form potential, the [MATH] term in [REF] does not contribute to the equations of motion [REF], because it is a total derivative: [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH] (see also the related discussion in Refs. [CITATION]).', '0806.2805-3-16-2': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-3-16-3': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-3-17-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-3-17-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-3-17-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity which is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-3-17-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-3-18-0': 'Before we turn to the cosmological solutions of our particular [MATH] theory, as defined by [REF], it may be useful to mention the connection with so-called [MATH] theories which have recently received much attention in cosmology (see, e.g., Refs. [CITATION] and references therein).', '0806.2805-3-18-1': 'The latter are purely phenomenological theories, in which the linear function of the Ricci scalar [MATH] from the Einstein-Hilbert action term is replaced by a more general function [MATH].', '0806.2805-3-18-2': 'This function [MATH] can, in principle, be adjusted to fit the astronomical observations and to produce a viable cosmological model.', '0806.2805-3-18-3': 'Returning to our [MATH] theory, we can express [MATH] in terms of [MATH] by use of [REF] and substitute the resulting expression [MATH] into [REF].', '0806.2805-3-18-4': 'This gives equations for the metric field which are similar to those of [MATH] cosmology [CITATION].', '0806.2805-3-18-5': 'In this way, the [MATH] theory introduced in this section (or, more generally, [MATH] theory as mentioned in the previous paragraph) may give a microscopic justification for the phenomenological [MATH] theories used in cosmology and may allow for a choice between different classes of model functions [MATH] based on fundamental physics.', '0806.2805-3-19-0': '# de-Sitter expansion', '0806.2805-3-20-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations from the effective action [REF].', '0806.2805-3-20-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-3-20-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-3-21-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH], which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-3-21-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] where [MATH] denotes the [MATH] value corresponding to a given positive value of the Hubble constant [MATH].', '0806.2805-3-22-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-3-23-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-3-23-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION] which are obtained from [REF] and [REF] by recalling that the perfect quantum vacuum is the equilibrium vacuum in the absence of matter and gravity fields, that is, for [MATH].', '0806.2805-3-23-2': 'If [MATH] is small compared to the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-3-23-3': 'Then, the correction [MATH] due to the expansion is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-3-23-4': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-3-23-5': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-3-23-6': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and the cosmological constant [MATH].', '0806.2805-3-23-7': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion, [MATH].', '0806.2805-3-24-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-3-25-0': '# Dynamics of a flat FRW universe', '0806.2805-3-26-0': '## General equations', '0806.2805-3-27-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-3-27-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-3-27-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-3-27-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and isotropic pressure [MATH].', '0806.2805-3-28-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with expansion factor [MATH], the homogenous matter has, in general, a time-dependent energy density [MATH] and pressure [MATH].', '0806.2805-3-28-1': 'Equally, the scalar field entering the four-form field-strength tensor [REF] is taken to be homogenous and time dependent, [MATH].', '0806.2805-3-29-0': 'With a time-dependent Hubble parameter [MATH], we then have from the reduced Maxwell equation [REF]: [EQUATION] and from the Einstein equations [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-3-29-1': 'With definition [REF], the reduced Maxwell equation [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to [MATH].', '0806.2805-3-29-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-3-30-0': '## Model for [MATH]', '0806.2805-3-31-0': 'The above equations allow us to study the development of the universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-3-31-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-3-31-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum is stable [i.e., the vacuum compressibility [REF] is positive, [MATH]].', '0806.2805-3-31-3': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-3-31-4': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-3-31-5': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale: [MATH] and [MATH].', '0806.2805-3-31-6': 'From [REF], we then see that [MATH].', '0806.2805-3-31-7': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION]', '0806.2805-3-31-8': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] becomes [EQUATION]', '0806.2805-3-31-9': 'From the Maxwell equation [REF], the Einstein equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH] and [MATH]: [EQUATION] with the matter equation-of-state (EOS) parameter [MATH].', '0806.2805-3-32-0': '## Model for [MATH]', '0806.2805-3-33-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the dimensionless function [MATH] in microscopic units.', '0806.2805-3-33-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-3-34-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-3-34-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-3-34-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given on the lines below [REF].', '0806.2805-3-34-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single time-independent dimensionless parameter [MATH] also taken positive.', '0806.2805-3-35-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-3-35-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-3-35-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-3-35-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (later taken to be time independent).', '0806.2805-3-36-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-3-37-0': '## Equations at the equilibrium point [MATH]', '0806.2805-3-38-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-3-38-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-3-38-2': 'Here, we consider the latter situation where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for the vacuum energy density [REF].', '0806.2805-3-39-0': 'Such an equilibrium vacuum state can be reached only if the chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in microscopic units.', '0806.2805-3-39-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-3-39-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-3-39-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-3-39-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-3-40-0': 'In order to avoid having to consider quantum corrections to the Einstein equations, which typically appear near the Planck time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-3-40-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium, [MATH].', '0806.2805-3-41-0': 'Taking the time derivative of [REF] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-3-41-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-3-41-2': 'The normalized vacuum energy density [REF] for the normalized equilibrium chemical potential [MATH] is given by [EQUATION] which vanishes in the equilibrium state [MATH].', '0806.2805-3-42-0': 'In order to simplify the analysis, we, first, consider matter with a nonzero time-independent EOS parameter, [EQUATION] so that the matter energy density from [REF] can be neglected asymptotically.', '0806.2805-3-43-0': '## Vacuum oscillations', '0806.2805-3-44-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-3-44-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-3-44-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-3-44-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-3-44-4': 'Both oscillation periods will be observed in the numerical results of Sec. [REF].', '0806.2805-3-45-0': '## Vacuum energy decay', '0806.2805-3-46-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the parameters in [REF], namely, the amplitude of oscillations [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-3-46-1': 'To find the explicit behavior, we expand the functions [MATH] and [MATH] in powers of [MATH] and keep terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-3-46-2': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear equations [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-3-46-3': 'The solution of these equations is given by [REF] with the same replacements: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-3-47-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-3-47-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-3-47-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-3-47-3': 'It suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-3-47-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-3-47-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-3-47-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which, for [MATH], gives [MATH].', '0806.2805-3-48-0': 'The above results for the coefficients [MATH] and [MATH] hold for the generic case [MATH], as stated in [REF].', '0806.2805-3-48-1': 'For the special case [MATH], inspection of [REF] shows that the same Ansatze for [MATH] and [MATH] can be used, but with the following coefficients: [EQUATION] where a Kronecker delta has been employed in the expression for the damping factor [MATH] and the coefficient [MATH] of the [MATH] asymptotic energy density [MATH] is assumed to be less than [MATH].', '0806.2805-3-49-0': 'Altogether, we have the following behavior of [MATH], [MATH], and [MATH] for [MATH]:[EQUATION] with dimensionless frequency [MATH].', '0806.2805-3-49-1': 'This asymptotic solution has some remarkable properties (in a different context, the same oscillatory behavior of [MATH] has been found in Ref. [CITATION]; see also the discussion in the last paragraph of Sec. [REF]).', '0806.2805-3-49-2': 'First, it depends rather weakly on the parameter [MATH] of the matter EOS (this is also confirmed by the numerical results of the next subsection).', '0806.2805-3-49-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-3-49-4': 'Combined, the average vacuum energy density is found to behave as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed in Sec. [REF].', '0806.2805-3-50-0': '## Numerical results', '0806.2805-3-51-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled differential equations [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-3-51-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equations can be expected (cf. Sec. [REF]).', '0806.2805-3-51-2': 'Still, the numerical results show clearly that', '0806.2805-3-52-0': 'the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] for a relatively small initial energy density and by Fig. [REF] for a relatively large initial energy density; for a chemical potential (integration constant) [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-3-53-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the universe approaches a de-Sitter stage.', '0806.2805-3-54-0': '## Effective CDM-like behavior', '0806.2805-3-55-0': 'The main result found in the previous two subsections is quite interesting: the oscillating vacuum energy density and the corresponding oscillating gravitational coupling parameter behave in the same way as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-3-55-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-3-56-0': 'The explanation is as follows.', '0806.2805-3-56-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-3-56-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-3-56-3': 'The [MATH] (or [MATH]) field has no explicit kinetic term in the action [REF], but derivatives of [MATH] appear in the field equations [REF] via terms with covariant derivatives of [MATH], which trace back to the Einstein-Hilbert-like term [MATH] in [REF].', '0806.2805-3-56-4': 'In a way, the effective Lagrange density [MATH] can be said to be induced by gravity.', '0806.2805-3-56-5': 'The pressure of this rapidly oscillating field [MATH] is now given by [MATH].', '0806.2805-3-56-6': 'Hence, the rapidly oscillating vacuum pressure is zero on average and the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-3-57-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-3-57-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales which determine the microscopic dynamics of the vacuum energy density.', '0806.2805-3-57-2': 'But surprises are, of course, not excluded.', '0806.2805-3-58-0': '## Extrapolation to large times', '0806.2805-3-59-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-3-59-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF], [REF], and [REF].', '0806.2805-3-60-0': 'Extrapolating this evolution to the present age of the universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] with [MATH].', '0806.2805-3-60-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-3-60-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-3-60-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-3-61-0': 'Let us expand on the previous remarks.', '0806.2805-3-61-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-3-61-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-3-61-3': 'This explains the following result for the case of a nonzero matter EOS parameter [MATH]: [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-3-61-4': 'For nonrelativistic matter ([MATH]), the right-hand side of [REF] is multiplied by a further reduction factor [MATH], according to the results of Sec. [REF].', '0806.2805-3-62-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-3-62-1': 'Clearly, there are many other processes which intervene between the very early (Planckian) phase of the universe and the present epoch.', '0806.2805-3-62-2': 'An example of such a process is particle production (for example, by parametric resonance [CITATION]), which can be expected to be effective because of the very rapid but small oscillations.', '0806.2805-3-62-3': 'A further possible source for a modified vacuum energy behavior would be the change of EOS parameter [MATH] to [MATH], which occurs when the expanding universe leaves the radiation dominated epoch.', '0806.2805-3-62-4': 'Still, there is possibility that these and other processes are only secondary effects and that the main mechanism of dark-energy dynamics at the early stage is the decay of vacuum energy density by oscillations.', '0806.2805-3-63-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-3-63-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-3-63-2': "The behavior [REF] (shown qualitatively by the [MATH] panels in Figs. [REF]-[REF]) is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-3-64-0': '# Conclusion', '0806.2805-3-65-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-3-65-1': 'Specifically, the new results are', '0806.2805-3-66-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-3-66-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-3-67-0': '# ACKNOWLEDGMENTS It is a pleasure to thank A.A. Starobinsky for informative discussions.', '0806.2805-3-67-1': 'GEV is supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (grant 4899.2008.2)'}

{'0806.2805-4-0-0': 'A modified-gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-4-0-1': 'This theory provides a concrete realization of the general vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics, [MATH].', '0806.2805-4-0-2': 'The theory gives a flat Friedmann-Robertson-Walker (FRW) universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-4-0-3': 'Extrapolating to the present age of the universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-4-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter (CDM) in the standard FRW framework.', '0806.2805-4-1-0': '# Introduction', '0806.2805-4-2-0': 'In a previous article [CITATION], we have proposed to characterize a Lorentz-invariant quantum vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-4-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability and compressibility.', '0806.2805-4-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-4-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-4-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-4-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density which contributes to the effective gravitational field equations at low energies.', '0806.2805-4-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-4-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-4-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-4-4-1': 'The same happens with other types of perturbations which violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-4-4-2': 'According to this approach, the present value of [MATH] is nonzero but small because the universe is close to equilibrium and Lorentz-noninvariant perturbations of the quantum vacuum are small (compared to the ultraviolet scale which sets the microscopic energy density [MATH]).', '0806.2805-4-5-0': 'The situation is different for Lorentz-invariant perturbations of the vacuum, such as the formation of scalar condensates as discussed in Ref. [CITATION] or quark/gluon condensates derived from quantum chromodynamics (cf. Ref. [CITATION]).', '0806.2805-4-5-1': 'In this case, the variable [MATH] shifts in such a way that it completely compensates the energy density of the perturbation and the effective cosmological constant remains zero in the new Lorentz-invariant equilibrium vacuum.', '0806.2805-4-6-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-4-6-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-4-6-2': 'Here, we use this explicit realization with a four-form field [MATH] to study the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective cosmological "constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-4-6-3': 'In short, the present cosmological constant is small because the universe happens to be old.', '0806.2805-4-7-0': 'The results of the present article show that, for the type of theory considered, the decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and that the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-4-7-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-4-8-0': 'These results are obtained by the following steps.', '0806.2805-4-8-1': 'In Sec. [REF], a modified-gravity theory with a four-form field [MATH] is defined in terms of general functions for the microscopic energy density [MATH] and variable gravitational coupling parameter [MATH].', '0806.2805-4-8-2': 'In Sec. [REF], the dynamics of the corresponding de-Sitter universe without matter is discussed and, in Sec. [REF], the dynamics of a flat FRW universe with matter, using simple Ansatze for the functions [MATH] and [MATH].', '0806.2805-4-8-3': 'In Sec. [REF], the approach to equilibrium in such a FRW universe is studied in detail and the above mentioned vacuum oscillations are established.', '0806.2805-4-8-4': 'In Sec. [REF], the main results are summarized.', '0806.2805-4-9-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-4-10-0': 'Here, and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-4-10-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-4-10-2': 'Such a quadratic function corresponds to a gas-like vacuum [CITATION].', '0806.2805-4-10-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-4-10-4': 'A self-sustained vacuum requires a more complicated function [MATH] in the action, so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-4-10-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-4-11-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-4-11-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-4-11-2': 'Moreover, a [MATH] dependence allows the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-4-12-0': 'Specifically, the action considered takes the following form ([MATH]): [EQUATION] where [MATH] denotes a covariant derivative and a square bracket around spacetime indices complete anti-symmetrization.', '0806.2805-4-12-1': 'The field [MATH] in [REF] stands for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants in [MATH] to [MATH]-dependent parameters).', '0806.2805-4-12-2': 'In this section, the low-energy fields are indicated by lower case letters, namely, [MATH] and [MATH], whereas the fields originating from the microscopic theory are indicated by upper case letters, namely, [MATH] and [MATH] [later also [MATH]].', '0806.2805-4-12-3': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-4-13-0': "The variation of the action [REF] over the three-form gauge field [MATH] gives the generalized Maxwell equations, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equations, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [EQUATION] whose precise form has been argued on thermodynamic grounds in Ref. [CITATION].", '0806.2805-4-14-0': 'At this point two remarks may be helpful.', '0806.2805-4-14-1': 'First, observe that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-4-14-2': 'This implicit metric dependence of [MATH] explains the origin of the second term on the left-hand side of [REF].', '0806.2805-4-14-3': 'Second, observe that the three-form gauge field [MATH] does not propagate physical degrees of freedom in flat spacetime [CITATION].', '0806.2805-4-14-4': 'Still, [MATH] has gravitational effects, both classically in the modified-gravity theory with [MATH] as discussed in the present article (see, in particular, Sec. [REF]) and quantum-mechanically already in the standard gravity theory with [MATH] (giving, for example, a nonvanishing gravitational trace anomaly [CITATION]).', '0806.2805-4-15-0': 'Using [REF] for [MATH], we obtain the Maxwell equations [REF] in the form [EQUATION]', '0806.2805-4-15-1': 'The solution is simply [EQUATION] with an integration constant [MATH].', '0806.2805-4-15-2': 'Hence, the constant [MATH] is seen to emerge dynamically.', '0806.2805-4-15-3': 'In a thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-4-15-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-4-16-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equations become [EQUATION] which will be used in the rest of this article, together with [REF].', '0806.2805-4-17-0': 'Equations [REF] and [REF] can also be obtained if we use, instead of the original action, an effective action in terms of a Brans-Dicke-type scalar field [MATH] with mass dimension 2, setting [MATH] afterwards.', '0806.2805-4-17-1': 'Specifically, this effective action is given by [EQUATION]', '0806.2805-4-17-2': 'The potential term in [REF] contains, different from a conventional Brans-Dicke potential [MATH], a linear term, [MATH], for a constant [MATH] of mass dimension 2.', '0806.2805-4-17-3': 'This linear term reflects the fact that our effective scalar field [MATH] is not an arbitrary field but should be a conserved quantity, for which the constant parameter [MATH] plays the role of a chemical potential which is thermodynamically conjugate to [MATH].', '0806.2805-4-18-0': 'Indeed, if [MATH] in [REF] is replaced by a four-form field [MATH] given in terms of the three-form potential [MATH], the resulting [MATH] term in the effective action does not contribute to the equations of motion [REF], because it is a total derivative, [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH] (see also the related discussion in Refs. [CITATION], where [MATH] is compared to the [MATH] parameter of quantum chromodynamics).', '0806.2805-4-18-1': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-4-18-2': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-4-19-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-4-19-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-4-19-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity which is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-4-19-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-4-20-0': 'Before we turn to the cosmological solutions of our particular [MATH] theory [REF], it may be useful to mention the connection with so-called [MATH] models which have recently received considerable attention (see, e.g., Refs. [CITATION] and references therein).', '0806.2805-4-20-1': 'The latter are purely phenomenological models, in which the linear function of the Ricci scalar [MATH] from the Einstein-Hilbert action term is replaced by a more general function [MATH].', '0806.2805-4-20-2': 'This function [MATH] can, in principle, be adjusted to fit the astronomical observations and to produce a viable cosmological model.', '0806.2805-4-20-3': 'Returning to our [MATH] theory, we can express [MATH] in terms of [MATH] by use of [REF] and substitute the resulting expression [MATH] into [REF].', '0806.2805-4-20-4': 'This gives equations for the metric field which are identical to those of [MATH] cosmology.', '0806.2805-4-20-5': '(The latter result is not altogether surprising as the metric [MATH] model is known to be equivalent to a Brans-Dicke model without kinetic term [CITATION] and the same holds for our effective action [REF] at the classical level.)', '0806.2805-4-20-6': 'In this way, the [MATH] theory introduced in this section (or, more generally, [MATH] theory as mentioned in the previous paragraph) may give a microscopic justification for the phenomenological [MATH] models used in theoretical cosmology and may allow for a choice between different classes of model functions [MATH] based on fundamental physics.', '0806.2805-4-21-0': '# de-Sitter expansion', '0806.2805-4-22-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations from the effective action [REF].', '0806.2805-4-22-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-4-22-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-4-23-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH], which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-4-23-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] with [MATH] considered given.', '0806.2805-4-24-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-4-25-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-4-25-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION] which are obtained from [REF] and [REF] by recalling that the perfect quantum vacuum is the equilibrium vacuum in the absence of matter and gravity fields, that is, for [MATH].', '0806.2805-4-25-2': 'If [MATH] is small compared to the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-4-25-3': 'Then, the correction [MATH] due to the expansion is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-4-25-4': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-4-25-5': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-4-25-6': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and [MATH].', '0806.2805-4-25-7': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion, [MATH].', '0806.2805-4-25-8': 'From now on, the physical situation considered will be the one determined by having a fixed chemical potential [MATH].', '0806.2805-4-26-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-4-27-0': '# Dynamics of a flat FRW universe', '0806.2805-4-28-0': '## General equations', '0806.2805-4-29-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-4-29-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-4-29-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-4-29-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and isotropic pressure [MATH].', '0806.2805-4-29-4': 'As mentioned in the previous section, the physics of the [MATH] field is considered to be specified by a fixed chemical potential [MATH].', '0806.2805-4-30-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with expansion factor [MATH], the homogenous matter has, in general, a time-dependent energy density [MATH] and pressure [MATH].', '0806.2805-4-30-1': 'Equally, the scalar field entering the four-form field-strength tensor [REF] is taken to be homogenous and time dependent, [MATH].', '0806.2805-4-31-0': 'With a time-dependent Hubble parameter [MATH], we then have from the reduced Maxwell equations [REF]: [EQUATION] and from the Einstein equations [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-4-31-1': 'With definition [REF], the reduced Maxwell equations [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to [MATH].', '0806.2805-4-31-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-4-32-0': '## Model for [MATH]', '0806.2805-4-33-0': 'The equations of Sec. [REF] allow us to study the development of the universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-4-33-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-4-33-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum is stable [i.e., the vacuum compressibility [REF] is positive, [MATH]].', '0806.2805-4-34-0': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-4-34-1': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-4-34-2': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale: [MATH] and [MATH].', '0806.2805-4-34-3': 'From [REF], we then see that [MATH].', '0806.2805-4-34-4': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION] where the variable [MATH] has been introduced in anticipation of the calculations of Sec. [REF].', '0806.2805-4-34-5': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] becomes [EQUATION]', '0806.2805-4-34-6': 'From the Maxwell equation [REF], the Einstein equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH] and [MATH]: [EQUATION] with matter equation-of-state (EOS) parameter [MATH].', '0806.2805-4-35-0': '## Model for [MATH]', '0806.2805-4-36-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the dimensionless function [MATH] in microscopic units.', '0806.2805-4-36-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-4-37-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-4-37-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-4-37-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given a few lines below [REF].', '0806.2805-4-37-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single time-independent dimensionless parameter [MATH] also taken positive.', '0806.2805-4-38-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-4-38-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-4-38-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-4-38-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (taken to be time independent in the analysis of the next section).', '0806.2805-4-39-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-4-40-0': '## Equations at the equilibrium point [MATH]', '0806.2805-4-41-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-4-41-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-4-41-2': 'Here, we consider the latter possibility where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for vacuum energy density [REF].', '0806.2805-4-42-0': 'Such an equilibrium vacuum state can be reached only if the chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in microscopic units.', '0806.2805-4-42-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-4-42-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-4-42-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-4-42-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-4-43-0': 'In order to avoid having to consider quantum corrections to the Einstein equations, which typically appear near the time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-4-43-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium as given by the variable [MATH] defined in [REF].', '0806.2805-4-44-0': 'Taking the time derivative of [REF] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-4-44-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-4-44-2': 'The dimensionless vacuum energy density [REF] for the dimensionless equilibrium chemical potential [MATH] is given by [EQUATION] which vanishes in the equilibrium state [MATH].', '0806.2805-4-45-0': 'In order to simplify the analysis, we, first, consider matter with a nonzero time-independent EOS parameter, [EQUATION] so that the matter energy density from [REF] can be neglected asymptotically.', '0806.2805-4-46-0': '## Vacuum oscillations', '0806.2805-4-47-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-4-47-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-4-47-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-4-47-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-4-47-4': 'Both oscillation periods will be manifest in the numerical results of Sec. [REF].', '0806.2805-4-48-0': '## Vacuum energy decay', '0806.2805-4-49-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the amplitudes in [REF], namely, the [MATH]-field oscillation amplitude [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-4-50-0': 'The explicit behavior is found by expanding the functions [MATH] and [MATH] in powers of [MATH] and keeping terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-4-50-1': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear equations [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-4-50-2': 'The solution of these equations is given by [REF] with the same replacements: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-4-51-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-4-51-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-4-51-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-4-51-3': 'It suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-4-51-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-4-51-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-4-51-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which, for [MATH], gives [MATH].', '0806.2805-4-52-0': 'The above results for the coefficients [MATH] and [MATH] hold for the generic case [MATH], as stated in [REF].', '0806.2805-4-52-1': 'For the special case [MATH], inspection of [REF] shows that the same Ansatze for [MATH] and [MATH] can be used, but with the following coefficients: [EQUATION] where a Kronecker delta has been employed in the expression for the damping factor [MATH] and the coefficient [MATH] of the [MATH] asymptotic energy density [MATH] has been assumed to be less than [MATH].', '0806.2805-4-53-0': 'Altogether, we have the following behavior of [MATH], [MATH], and [MATH] for [MATH]:[EQUATION] with dimensionless frequency [MATH] and damping factor [MATH] given by [REF].', '0806.2805-4-53-1': 'This asymptotic solution has some remarkable properties (in a different context, the same oscillatory behavior of [MATH] has been found in Ref. [CITATION]; see also the discussion in the last paragraph of Sec. [REF]).', '0806.2805-4-53-2': 'First, it depends rather weakly on the parameter [MATH] of the matter EOS, which is confirmed by the numerical results of the next subsection.', '0806.2805-4-53-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-4-53-4': 'Combined, the average vacuum energy density is found to behave as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed in Sec. [REF].', '0806.2805-4-54-0': '## Numerical results', '0806.2805-4-55-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled differential equations [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-4-55-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equations can be expected (cf. Sec. [REF]).', '0806.2805-4-55-2': 'Still, the numerical results show clearly that', '0806.2805-4-56-0': 'the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] for a relatively small initial energy density and by Fig. [REF] for a relatively large initial energy density; for a chemical potential [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-4-57-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict, in fact, oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the universe approaches a de-Sitter stage (see, in particular, the middle panel of the second row of Fig. [REF]).', '0806.2805-4-58-0': '## Effective CDM-like behavior', '0806.2805-4-59-0': 'The main result of the previous two subsections can be summarized as follows: the oscillating vacuum energy density [MATH] and the corresponding oscillating gravitational coupling parameter [MATH] conspire to give the same Hubble expansion as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-4-59-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-4-60-0': 'The explanation is as follows.', '0806.2805-4-60-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-4-60-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-4-60-3': 'The [MATH] (or [MATH]) field has no explicit kinetic term in the action [REF], but derivatives of [MATH] appear in the field equations [REF] via terms with covariant derivatives of [MATH], which trace back to the Einstein-Hilbert-like term [MATH] in [REF].', '0806.2805-4-60-4': 'In a way, the effective Lagrange density [MATH] can be said to be induced by gravity.', '0806.2805-4-60-5': 'The pressure of this rapidly oscillating field [MATH] is now given by [MATH].', '0806.2805-4-60-6': 'In turn, this implies that the rapidly oscillating vacuum pressure is zero on average and that the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-4-61-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-4-61-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales which determine the microscopic dynamics of the vacuum energy density.', '0806.2805-4-61-2': 'But surprises are, of course, not excluded.', '0806.2805-4-62-0': '## Extrapolation to large times', '0806.2805-4-63-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-4-63-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF], [REF], and [REF].', '0806.2805-4-64-0': 'Extrapolating this evolution to the present age of the universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] for [MATH].', '0806.2805-4-64-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-4-64-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-4-64-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-4-65-0': 'Let us expand on the previous remarks.', '0806.2805-4-65-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-4-65-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-4-65-3': 'This explains the following result for the case of a nonzero matter EOS parameter ([MATH]): [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-4-65-4': 'For nonrelativistic matter ([MATH]), the right-hand side of [REF] is multiplied by a further reduction factor [MATH], according to the results of Sec. [REF].', '0806.2805-4-66-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-4-66-1': 'A related problem is the CDM-like expansion of the universe, [MATH], whereas big bang nucleosynthesis (BBN) requires radiation-like expansion, [MATH], at least for the relevant temperature range.', '0806.2805-4-66-2': 'Clearly, there are many other processes which intervene between the very early (Planckian) phase of the universe and later phases, including the BBN era and the present epoch.', '0806.2805-4-66-3': 'An example of such a process is particle production (e.g., by parametric resonance [CITATION]), which can be expected to be effective because of the very rapid but small oscillations.', '0806.2805-4-66-4': 'A further possible source for a modified vacuum energy behavior would be the change of EOS parameter [MATH] to [MATH], which occurs when the expanding universe leaves the radiation dominated epoch.', '0806.2805-4-66-5': 'Still, there is possibility that these and other processes are only secondary effects and that the main mechanism of dark-energy dynamics at the early stage is the decay of vacuum energy density by oscillations.', '0806.2805-4-67-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-4-67-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-4-67-2': "The behavior [REF], shown qualitatively by the [MATH] panels in Figs. [REF]-[REF], is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-4-67-3': 'For the present universe and the solar system in it, the gravitational coupling parameter [REF] would have minuscule oscillations.', '0806.2805-4-67-4': 'Combined with the Planck-scale mass of the [MATH] degree of freedom (cf. the discussion in Sec. [REF]), this would suggest that all solar-system experimental bounds are satisfied, but, again, surprises are not excluded.', '0806.2805-4-68-0': '# Conclusion', '0806.2805-4-69-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-4-69-1': 'Specifically, the new results are', '0806.2805-4-70-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-4-70-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-4-71-0': '# ACKNOWLEDGMENTS It is a pleasure to thank A.A. Starobinsky for informative discussions.', '0806.2805-4-71-1': 'GEV is supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (grant 4899.2008.2)'}

{'0806.2805-5-0-0': 'A modified-gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-5-0-1': 'This theory provides a concrete realization of the general vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics.', '0806.2805-5-0-2': 'The theory gives a flat Friedmann-Robertson-Walker (FRW) universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-5-0-3': 'Extrapolating to the present age of the universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-5-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter (CDM) in the standard FRW framework.', '0806.2805-5-1-0': '# Introduction', '0806.2805-5-2-0': 'In a previous article [CITATION], we have proposed to characterize a Lorentz-invariant quantum vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-5-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability and compressibility.', '0806.2805-5-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-5-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-5-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-5-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density which contributes to the effective gravitational field equations at low energies.', '0806.2805-5-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-5-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-5-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-5-4-1': 'The same happens with other types of perturbations which violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-5-4-2': 'According to this approach, the present value of [MATH] is nonzero but small because the universe is close to equilibrium and Lorentz-noninvariant perturbations of the quantum vacuum are small (compared to the ultraviolet scale which sets the microscopic energy density [MATH]).', '0806.2805-5-5-0': 'The situation is different for Lorentz-invariant perturbations of the vacuum, such as the formation of scalar condensates as discussed in Ref. [CITATION] or quark/gluon condensates derived from quantum chromodynamics (cf. Ref. [CITATION]).', '0806.2805-5-5-1': 'In this case, the variable [MATH] shifts in such a way that it completely compensates the energy density of the perturbation and the effective cosmological constant remains zero in the new Lorentz-invariant equilibrium vacuum.', '0806.2805-5-6-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-5-6-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-5-6-2': 'Here, we use this explicit realization with a four-form field [MATH] to study the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective cosmological "constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-5-6-3': 'In short, the present cosmological constant is small because the universe happens to be old.', '0806.2805-5-7-0': 'The results of the present article show that, for the type of theory considered, the decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and that the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-5-7-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-5-8-0': 'These results are obtained by the following steps.', '0806.2805-5-8-1': 'In Sec. [REF], a modified-gravity theory with a four-form field [MATH] is defined in terms of general functions for the microscopic energy density [MATH] and variable gravitational coupling parameter [MATH].', '0806.2805-5-8-2': 'In Sec. [REF], the dynamics of the corresponding de-Sitter universe without matter is discussed and, in Sec. [REF], the dynamics of a flat FRW universe with matter, using simple Ansatze for the functions [MATH] and [MATH].', '0806.2805-5-8-3': 'In Sec. [REF], the approach to equilibrium in such a FRW universe is studied in detail and the above mentioned vacuum oscillations are established.', '0806.2805-5-8-4': 'In Sec. [REF], the main results are summarized.', '0806.2805-5-9-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-5-10-0': 'Here, and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-5-10-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-5-10-2': 'Such a quadratic function corresponds to a gas-like vacuum [CITATION].', '0806.2805-5-10-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-5-10-4': 'A self-sustained vacuum requires a more complicated function [MATH] in the action, so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-5-10-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-5-11-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-5-11-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-5-11-2': 'Moreover, a [MATH] dependence allows the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-5-12-0': 'Specifically, the action considered takes the following form ([MATH]): [EQUATION] where [MATH] denotes a covariant derivative and a square bracket around spacetime indices complete anti-symmetrization.', '0806.2805-5-12-1': 'The field [MATH] in [REF] stands for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants in [MATH] to [MATH]-dependent parameters).', '0806.2805-5-12-2': 'It is also assumed that a possible constant term [MATH] in [MATH] has been absorbed in [MATH], so that, in the end, [MATH] contains only [MATH]-dependent terms.', '0806.2805-5-12-3': 'In this section, the low-energy fields are indicated by lower case letters, namely, [MATH] and [MATH], whereas the fields originating from the microscopic theory are indicated by upper case letters, namely, [MATH] and [MATH] [later also [MATH]].', '0806.2805-5-12-4': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-5-13-0': "The variation of the action [REF] over the three-form gauge field [MATH] gives the generalized Maxwell equations, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equations, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [EQUATION] whose precise form has been argued on thermodynamic grounds in Ref. [CITATION].", '0806.2805-5-14-0': 'At this point two remarks may be helpful.', '0806.2805-5-14-1': 'First, observe that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-5-14-2': 'This implicit metric dependence of [MATH] explains the origin of the second term on the left-hand side of [REF].', '0806.2805-5-14-3': 'Second, observe that the three-form gauge field [MATH] does not propagate physical degrees of freedom in flat spacetime [CITATION].', '0806.2805-5-14-4': 'Still, [MATH] has gravitational effects, both classically in the modified-gravity theory with [MATH] as discussed in the present article (see, in particular, Sec. [REF]) and quantum-mechanically already in the standard gravity theory with [MATH] (giving, for example, a nonvanishing gravitational trace anomaly [CITATION]).', '0806.2805-5-15-0': 'Using [REF] for [MATH], we obtain the Maxwell equations [REF] in the form [EQUATION]', '0806.2805-5-15-1': 'The solution is simply [EQUATION] with an integration constant [MATH].', '0806.2805-5-15-2': 'Hence, the constant [MATH] is seen to emerge dynamically.', '0806.2805-5-15-3': 'In a thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-5-15-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-5-16-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equations become [EQUATION] which will be used in the rest of this article, together with [REF].', '0806.2805-5-17-0': 'Equations [REF] and [REF] can also be obtained if we use, instead of the original action, an effective action in terms of a Brans-Dicke-type scalar field [MATH] with mass dimension 2, setting [MATH] afterwards.', '0806.2805-5-17-1': 'Specifically, this effective action is given by [EQUATION]', '0806.2805-5-17-2': 'The potential term in [REF] contains, different from a conventional Brans-Dicke potential [MATH], a linear term, [MATH], for a constant [MATH] of mass dimension 2.', '0806.2805-5-17-3': 'This linear term reflects the fact that our effective scalar field [MATH] is not an arbitrary field but should be a conserved quantity, for which the constant parameter [MATH] plays the role of a chemical potential which is thermodynamically conjugate to [MATH].', '0806.2805-5-18-0': 'Indeed, if [MATH] in [REF] is replaced by a four-form field [MATH] given in terms of the three-form potential [MATH], the resulting [MATH] term in the effective action does not contribute to the equations of motion [REF], because it is a total derivative, [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH] (see also the related discussion in Refs. [CITATION], where [MATH] is compared to the [MATH] parameter of quantum chromodynamics).', '0806.2805-5-18-1': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-5-18-2': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-5-19-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-5-19-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-5-19-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity which is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-5-19-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-5-20-0': 'Before we turn to the cosmological solutions of our particular [MATH] theory [REF], it may be useful to mention the connection with so-called [MATH] models which have recently received considerable attention (see, e.g., Refs. [CITATION] and references therein).', '0806.2805-5-20-1': 'The latter are purely phenomenological models, in which the linear function of the Ricci scalar [MATH] from the Einstein-Hilbert action term is replaced by a more general function [MATH].', '0806.2805-5-20-2': 'This function [MATH] can, in principle, be adjusted to fit the astronomical observations and to produce a viable cosmological model.', '0806.2805-5-20-3': 'Returning to our [MATH] theory, we can express [MATH] in terms of [MATH] by use of [REF] and substitute the resulting expression [MATH] into [REF].', '0806.2805-5-20-4': 'This gives equations for the metric field which are identical to those of [MATH] cosmology.', '0806.2805-5-20-5': '(The latter result is not altogether surprising as the metric [MATH] model is known to be equivalent to a Brans-Dicke model without kinetic term [CITATION] and the same holds for our effective action [REF] at the classical level.)', '0806.2805-5-20-6': 'In this way, the [MATH] theory introduced in this section (or, more generally, [MATH] theory as mentioned in the previous paragraph) may give a microscopic justification for the phenomenological [MATH] models used in theoretical cosmology and may allow for a choice between different classes of model functions [MATH] based on fundamental physics.', '0806.2805-5-21-0': '# de-Sitter expansion', '0806.2805-5-22-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations from the effective action [REF].', '0806.2805-5-22-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-5-22-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-5-23-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH], which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-5-23-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] with [MATH] considered given.', '0806.2805-5-24-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-5-25-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-5-25-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION] which are obtained from [REF] and [REF] by recalling that the perfect quantum vacuum is the equilibrium vacuum in the absence of matter and gravity fields, that is, for [MATH].', '0806.2805-5-25-2': 'If [MATH] is small compared to the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-5-25-3': 'Then, the correction [MATH] due to the expansion is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-5-25-4': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-5-25-5': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-5-25-6': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and [MATH].', '0806.2805-5-25-7': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion, [MATH].', '0806.2805-5-25-8': 'From now on, the physical situation considered will be the one determined by having a fixed chemical potential [MATH].', '0806.2805-5-26-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-5-27-0': '# Dynamics of a flat FRW universe', '0806.2805-5-28-0': '## General equations', '0806.2805-5-29-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-5-29-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-5-29-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-5-29-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and isotropic pressure [MATH].', '0806.2805-5-29-4': 'As mentioned in the previous section, the physics of the [MATH] field is considered to be specified by a fixed chemical potential [MATH].', '0806.2805-5-30-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with expansion factor [MATH], the homogenous matter has, in general, a time-dependent energy density [MATH] and pressure [MATH].', '0806.2805-5-30-1': 'Equally, the scalar field entering the four-form field-strength tensor [REF] is taken to be homogenous and time dependent, [MATH].', '0806.2805-5-31-0': 'With a time-dependent Hubble parameter [MATH], we then have from the reduced Maxwell equations [REF]: [EQUATION] and from the Einstein equations [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-5-31-1': 'With definition [REF], the reduced Maxwell equations [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to [MATH].', '0806.2805-5-31-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-5-32-0': '## Model for [MATH]', '0806.2805-5-33-0': 'The equations of Sec. [REF] allow us to study the development of the universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-5-33-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-5-33-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum is stable [i.e., the vacuum compressibility [REF] is positive, [MATH]].', '0806.2805-5-34-0': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-5-34-1': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-5-34-2': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale: [MATH] and [MATH].', '0806.2805-5-34-3': 'From [REF], we then see that [MATH].', '0806.2805-5-34-4': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION] where the variable [MATH] has been introduced in anticipation of the calculations of Sec. [REF].', '0806.2805-5-34-5': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] becomes [EQUATION]', '0806.2805-5-34-6': 'From the Maxwell equation [REF], the Einstein equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH] and [MATH]: [EQUATION] with matter equation-of-state (EOS) parameter [MATH].', '0806.2805-5-35-0': '## Model for [MATH]', '0806.2805-5-36-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the dimensionless function [MATH] in microscopic units.', '0806.2805-5-36-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-5-37-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-5-37-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-5-37-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given a few lines below [REF].', '0806.2805-5-37-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single time-independent dimensionless parameter [MATH] also taken positive.', '0806.2805-5-38-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-5-38-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-5-38-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-5-38-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (taken to be time independent in the analysis of the next section).', '0806.2805-5-39-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-5-40-0': '## Equations at the equilibrium point [MATH]', '0806.2805-5-41-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-5-41-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-5-41-2': 'Here, we consider the latter possibility where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for vacuum energy density [REF].', '0806.2805-5-42-0': 'Such an equilibrium vacuum state can be reached only if the chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in microscopic units.', '0806.2805-5-42-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-5-42-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-5-42-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-5-42-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-5-43-0': 'In order to avoid having to consider quantum corrections to the Einstein equations, which typically appear near the time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-5-43-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium as given by the variable [MATH] defined in [REF].', '0806.2805-5-44-0': 'Taking the time derivative of [REF] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-5-44-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-5-44-2': 'The dimensionless vacuum energy density [REF] for the dimensionless equilibrium chemical potential [MATH] is given by [EQUATION] which vanishes in the equilibrium state [MATH].', '0806.2805-5-45-0': 'In order to simplify the analysis, we, first, consider matter with a nonzero time-independent EOS parameter, [EQUATION] so that the matter energy density from [REF] can be neglected asymptotically.', '0806.2805-5-46-0': '## Vacuum oscillations', '0806.2805-5-47-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-5-47-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-5-47-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-5-47-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-5-47-4': 'Both oscillation periods will be manifest in the numerical results of Sec. [REF].', '0806.2805-5-48-0': '## Vacuum energy decay', '0806.2805-5-49-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the amplitudes in [REF], namely, the [MATH]-field oscillation amplitude [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-5-50-0': 'The explicit behavior is found by expanding the functions [MATH] and [MATH] in powers of [MATH] and keeping terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-5-50-1': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear equations [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-5-50-2': 'The solution of these equations is given by [REF] with the same replacements: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-5-51-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-5-51-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-5-51-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-5-51-3': 'It suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-5-51-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-5-51-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-5-51-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which, for [MATH], gives [MATH].', '0806.2805-5-52-0': 'The above results for the coefficients [MATH] and [MATH] hold for the generic case [MATH], as stated in [REF].', '0806.2805-5-52-1': 'For the special case [MATH], inspection of [REF] shows that the same Ansatze for [MATH] and [MATH] can be used, but with the following coefficients: [EQUATION] where a Kronecker delta has been employed in the expression for the damping factor [MATH] and the coefficient [MATH] of the [MATH] asymptotic energy density [MATH] has been assumed to be less than [MATH].', '0806.2805-5-53-0': 'Altogether, we have the following behavior of [MATH], [MATH], and [MATH] for [MATH]:[EQUATION] with dimensionless frequency [MATH] and damping factor [MATH] given by [REF].', '0806.2805-5-53-1': 'This asymptotic solution has some remarkable properties (in a different context, the same oscillatory behavior of [MATH] has been found in Ref. [CITATION]; see also the discussion in the last paragraph of Sec. [REF]).', '0806.2805-5-53-2': 'First, it depends rather weakly on the parameter [MATH] of the matter EOS, which is confirmed by the numerical results of the next subsection.', '0806.2805-5-53-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-5-53-4': 'Combined, the average vacuum energy density is found to behave as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed in Sec. [REF].', '0806.2805-5-54-0': '## Numerical results', '0806.2805-5-55-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled differential equations [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-5-55-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equations can be expected (cf. Sec. [REF]).', '0806.2805-5-55-2': 'Still, the numerical results show clearly that', '0806.2805-5-56-0': 'the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] for a relatively small initial energy density and by Fig. [REF] for a relatively large initial energy density; for a chemical potential [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-5-57-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict, in fact, oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the universe approaches a de-Sitter stage (see, in particular, the middle panel of the second row of Fig. [REF]).', '0806.2805-5-58-0': '## Effective CDM-like behavior', '0806.2805-5-59-0': 'The main result of the previous two subsections can be summarized as follows: the oscillating vacuum energy density [MATH] and the corresponding oscillating gravitational coupling parameter [MATH] conspire to give the same Hubble expansion as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-5-59-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-5-60-0': 'The explanation is as follows.', '0806.2805-5-60-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-5-60-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-5-60-3': 'The [MATH] (or [MATH]) field has no explicit kinetic term in the action [REF], but derivatives of [MATH] appear in the field equations [REF] via terms with covariant derivatives of [MATH], which trace back to the Einstein-Hilbert-like term [MATH] in [REF].', '0806.2805-5-60-4': 'In a way, the effective Lagrange density [MATH] can be said to be induced by gravity.', '0806.2805-5-60-5': 'The pressure of this rapidly oscillating field [MATH] is now given by [MATH].', '0806.2805-5-60-6': 'In turn, this implies that the rapidly oscillating vacuum pressure is zero on average and that the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-5-60-7': 'Observe that, while the [MATH]-field itself has an EOS parameter [MATH] corresponding to vacuum energy density, the net effect of the dampened [MATH] oscillations is to mimic the evolution of cold dark matter with [MATH] in a standard flat FRW universe.', '0806.2805-5-60-8': 'As mentioned before, this effective EOS parameter [MATH] is induced by the interaction of the [MATH] and gravity fields.', '0806.2805-5-61-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-5-61-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales which determine the microscopic dynamics of the vacuum energy density.', '0806.2805-5-61-2': 'But surprises are, of course, not excluded.', '0806.2805-5-62-0': '## Extrapolation to large times', '0806.2805-5-63-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-5-63-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF], [REF], and [REF].', '0806.2805-5-64-0': 'Extrapolating this evolution to the present age of the universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] for [MATH].', '0806.2805-5-64-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-5-64-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-5-64-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-5-65-0': 'Let us expand on the previous remarks.', '0806.2805-5-65-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-5-65-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-5-65-3': 'This explains the following result for the case of a nonzero matter EOS parameter ([MATH]): [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-5-65-4': 'For nonrelativistic matter ([MATH]), the right-hand side of [REF] is multiplied by a further reduction factor [MATH], according to the results of Sec. [REF].', '0806.2805-5-66-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-5-66-1': 'A related problem is the CDM-like expansion of the universe, [MATH], whereas big bang nucleosynthesis (BBN) requires radiation-like expansion, [MATH], at least for the relevant temperature range.', '0806.2805-5-66-2': 'Clearly, there are many other processes which intervene between the very early (Planckian) phase of the universe and later phases such as the BBN era and the present epoch.', '0806.2805-5-66-3': 'An example of a relevant process may be particle production (e.g., by parametric resonance [CITATION]), which can be expected to be effective because of the very rapid but small oscillations.', '0806.2805-5-66-4': 'A further possible source of modified vacuum energy behavior may be the change of EOS parameter [MATH] to [MATH], which occurs when the expanding universe leaves the radiation dominated epoch.', '0806.2805-5-66-5': 'Still, there is possibility that these and other processes are only secondary effects and that the main mechanism of dark-energy dynamics at the early stage is the decay of vacuum energy density by oscillations.', '0806.2805-5-67-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-5-67-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-5-67-2': "The behavior [REF], shown qualitatively by the [MATH] panels in Figs. [REF]-[REF], is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-5-67-3': 'For the present universe and the solar system in it, the gravitational coupling parameter [REF] would have minuscule oscillations.', '0806.2805-5-67-4': 'Combined with the Planck-scale mass of the [MATH] degree of freedom (cf. the discussion in Sec. [REF]), this would suggest that all solar-system experimental bounds are satisfied, but, again, surprises are not excluded.', '0806.2805-5-68-0': '# Conclusion', '0806.2805-5-69-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-5-69-1': 'Specifically, the new results are', '0806.2805-5-70-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-5-70-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-5-71-0': '# ACKNOWLEDGMENTS', '0806.2805-5-72-0': 'It is a pleasure to thank A.A. Starobinsky for informative discussions.', '0806.2805-5-72-1': 'GEV is supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (grant 4899.2008.2)'}

{'0806.2805-6-0-0': 'A modified-gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-6-0-1': 'This theory provides a concrete realization of the general vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics.', '0806.2805-6-0-2': 'The theory gives a flat Friedmann-Robertson-Walker universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-6-0-3': 'Extrapolating to the present age of the Universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-6-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter in the standard Friedmann-Robertson-Walker framework.', '0806.2805-6-1-0': '# Introduction', '0806.2805-6-2-0': 'In a previous article [CITATION], we proposed to characterize a Lorentz-invariant quantum vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-6-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability and compressibility.', '0806.2805-6-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-6-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-6-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-6-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density that contributes to the effective gravitational field equations at low energies.', '0806.2805-6-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-6-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-6-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-6-4-1': 'The same happens with other types of perturbations that violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-6-4-2': 'According to this approach, the present value of [MATH] is nonzero but small because the universe is close to equilibrium and Lorentz-noninvariant perturbations of the quantum vacuum are small (compared with the ultraviolet scale which sets the microscopic energy density [MATH]).', '0806.2805-6-5-0': 'The situation is different for Lorentz-invariant perturbations of the vacuum, such as the formation of scalar condensates as discussed in Ref. [CITATION] or quark/gluon condensates derived from quantum chromodynamics (cf. Ref. [CITATION]).', '0806.2805-6-5-1': 'In this case, the variable [MATH] shifts in such a way that it completely compensates the energy density of the perturbation and the effective cosmological constant is again zero in the new Lorentz-invariant equilibrium vacuum.', '0806.2805-6-6-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-6-6-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-6-6-2': 'Here, we use this explicit realization with a four-form field [MATH] to study the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective cosmological "constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-6-6-3': 'In short, the present cosmological constant is small because the Universe happens to be old.', '0806.2805-6-7-0': 'The results of the present article show that, for the type of theory considered, the decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and that the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-6-7-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-6-8-0': 'These results are obtained by the following steps.', '0806.2805-6-8-1': 'In Sec. [REF], a modified-gravity theory with a four-form field [MATH] is defined in terms of general functions for the microscopic energy density [MATH] and variable gravitational coupling parameter [MATH].', '0806.2805-6-8-2': 'In Sec. [REF], the dynamics of the corresponding de-Sitter universe without matter is discussed and, in Sec. [REF], the dynamics of a flat FRW universe with matter, using simple Ansatze for the functions [MATH] and [MATH].', '0806.2805-6-8-3': 'In Sec. [REF], the approach to equilibrium in such a FRW universe is studied in detail and the above mentioned vacuum oscillations are established.', '0806.2805-6-8-4': 'In Sec. [REF], the main results are summarized.', '0806.2805-6-9-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-6-10-0': 'Here, and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-6-10-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-6-10-2': 'Such a quadratic function gives rise to a gas-like vacuum [CITATION].', '0806.2805-6-10-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-6-10-4': 'A self-sustained vacuum requires a more complicated function [MATH] in the action, so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-6-10-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-6-11-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-6-11-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-6-11-2': 'Moreover, a [MATH] dependence allows the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-6-12-0': 'Specifically, the action considered takes the following form ([MATH]): [EQUATION] where [MATH] denotes a covariant derivative and a square bracket around spacetime indices complete antisymmetrization.', '0806.2805-6-12-1': 'The functional dependence on [MATH] has been kept implicit on the right-hand side of [REF], showing only the dependence on [MATH] and [MATH].', '0806.2805-6-12-2': 'The field [MATH] in [REF] stands, in fact, for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants in [MATH] to [MATH]-dependent parameters).', '0806.2805-6-12-3': 'It is also assumed that a possible constant term [MATH] in [MATH] has been absorbed in [MATH], so that, in the end, [MATH] contains only [MATH]-dependent terms.', '0806.2805-6-12-4': 'In this section, the low-energy fields are indicated by lower-case letters, namely, [MATH] and [MATH], whereas the fields originating from the microscopic theory are indicated by upper-case letters, namely, [MATH] and [MATH] [later also [MATH]].', '0806.2805-6-12-5': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-6-13-0': "The variation of the action [REF] over the three-form gauge field [MATH] gives the generalized Maxwell equation, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equation, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [EQUATION] whose precise form has been argued on thermodynamic grounds in Ref. [CITATION].", '0806.2805-6-14-0': 'At this point two remarks may be helpful.', '0806.2805-6-14-1': 'First, observe that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-6-14-2': 'This implicit metric dependence of [MATH] explains the origin of the second term on the left-hand side of [REF].', '0806.2805-6-14-3': 'Second, observe that the three-form gauge field [MATH] does not propagate physical degrees of freedom in flat spacetime [CITATION].', '0806.2805-6-14-4': 'Still, [MATH] has gravitational effects, both classically in the modified-gravity theory with [MATH] as discussed in the present article (see, in particular, Sec. [REF]) and quantum-mechanically already in the standard gravity theory with [MATH] (giving, for example, a nonvanishing gravitational trace anomaly [CITATION]).', '0806.2805-6-15-0': 'Using [REF] for [MATH], we obtain the Maxwell equation [REF] in the form [EQUATION]', '0806.2805-6-15-1': 'The solution is simply [EQUATION] with an integration constant [MATH].', '0806.2805-6-15-2': 'Hence, the constant [MATH] is seen to emerge dynamically.', '0806.2805-6-15-3': 'In a thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-6-15-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-6-16-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equation becomes [EQUATION] which will be used in the rest of this article, together with [REF].', '0806.2805-6-17-0': 'Equations [REF] and [REF] can also be obtained if we use, instead of the original action, an effective action in terms of a Brans-Dicke-type scalar field [MATH] with mass dimension 2, setting [MATH] afterwards.', '0806.2805-6-17-1': 'Specifically, this effective action is given by [EQUATION]', '0806.2805-6-17-2': 'The potential term in [REF] contains, different from a conventional Brans-Dicke potential [MATH], a linear term, [MATH], for a constant [MATH] of mass dimension 2.', '0806.2805-6-17-3': 'This linear term reflects the fact that our effective scalar field [MATH] is not an arbitrary field but should be a conserved quantity, for which the constant parameter [MATH] plays the role of a chemical potential that is thermodynamically conjugate to [MATH].', '0806.2805-6-18-0': 'Indeed, if [MATH] in [REF] is replaced by a four-form field [MATH] given in terms of the three-form potential [MATH], the resulting [MATH] term in the effective action does not contribute to the equations of motion [REF], because it is a total derivative, [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH] (see also the discussion in Refs. [CITATION], where [MATH] is compared with the [MATH] parameter of quantum chromodynamics).', '0806.2805-6-19-0': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-6-19-1': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-6-20-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-6-20-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-6-20-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity that is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-6-20-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-6-21-0': 'Before we turn to the cosmological solutions of our particular [MATH] theory [REF], it may be useful to mention the connection with so-called [MATH] models which have recently received considerable attention (see, e.g., Refs. [CITATION] and references therein).', '0806.2805-6-21-1': 'The latter are purely phenomenological models, in which the linear function of the Ricci scalar [MATH] from the Einstein-Hilbert action term is replaced by a more general function [MATH].', '0806.2805-6-21-2': 'This function [MATH] can, in principle, be adjusted to fit the astronomical observations and to produce a viable cosmological model.', '0806.2805-6-21-3': 'Returning to our [MATH] theory, we can express [MATH] in terms of [MATH] by use of [REF] and substitute the resulting expression [MATH] into [REF].', '0806.2805-6-21-4': 'This gives an equation for the metric field, which is identical to the one of [MATH] cosmology.', '0806.2805-6-21-5': '(The latter result is not altogether surprising as the metric [MATH] model is known to be equivalent to a Brans-Dicke model without kinetic term [CITATION] and the same holds for our effective action [REF] at the classical level.)', '0806.2805-6-21-6': 'In this way, the [MATH] theory introduced in this section (or, more generally, [MATH] theory as mentioned in the previous paragraph) may give a microscopic justification for the phenomenological [MATH] models used in theoretical cosmology and may allow for a choice between different classes of model functions [MATH] based on fundamental physics.', '0806.2805-6-22-0': '# de-Sitter expansion', '0806.2805-6-23-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations from the effective action [REF].', '0806.2805-6-23-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-6-23-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-6-24-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH] (that is, a genuine Hubble constant [MATH]), which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-6-24-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] with [MATH] considered given.', '0806.2805-6-25-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-6-26-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-6-26-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION] which are obtained from [REF] and [REF] by recalling that the perfect quantum vacuum is the equilibrium vacuum in the absence of matter and gravity fields ([MATH]).', '0806.2805-6-27-0': 'If [MATH] is nonzero but small compared with the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-6-27-1': 'Then, the correction [MATH] due to the expansion is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-6-27-2': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-6-27-3': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-6-27-4': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and [MATH].', '0806.2805-6-27-5': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion [[MATH]].', '0806.2805-6-27-6': 'From now on, the physical situation considered will be the one determined by having a fixed chemical potential [MATH].', '0806.2805-6-28-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-6-29-0': '# Dynamics of a flat FRW universe', '0806.2805-6-30-0': '## General equations', '0806.2805-6-31-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-6-31-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-6-31-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-6-31-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and isotropic pressure [MATH].', '0806.2805-6-31-4': 'As mentioned in the previous section, the physics of the [MATH] field is considered to be specified by a fixed chemical potential [MATH].', '0806.2805-6-32-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with expansion factor [MATH], the homogenous matter has, in general, a time-dependent energy density [MATH] and pressure [MATH].', '0806.2805-6-32-1': 'Equally, the scalar field entering the four-form field-strength tensor [REF] is taken to be homogenous and time dependent, [MATH].', '0806.2805-6-33-0': 'With a time-dependent Hubble parameter [MATH], we then have from the reduced Maxwell equation [REF]: [EQUATION] and from the Einstein equation [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-6-33-1': 'With definition [REF], the reduced Maxwell equation [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to cosmic time [MATH].', '0806.2805-6-33-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-6-34-0': '## Model for [MATH]', '0806.2805-6-35-0': 'The equations of Sec. [REF] allow us to study the development of the Universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-6-35-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-6-35-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum is stable [i.e., the vacuum compressibility [REF] is positive, [MATH]].', '0806.2805-6-36-0': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-6-36-1': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-6-36-2': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale, [MATH] and [MATH].', '0806.2805-6-36-3': 'From [REF], we then see that [MATH].', '0806.2805-6-36-4': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION] where the variable [MATH] has been introduced in anticipation of the calculations of Sec. [REF].', '0806.2805-6-36-5': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] gives [EQUATION] with [MATH] from [REF].', '0806.2805-6-37-0': 'From the Maxwell equation [REF], the Friedmann equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH], and [MATH]: [EQUATION] with matter equation-of-state (EOS) parameter [MATH].', '0806.2805-6-38-0': '## Model for [MATH]', '0806.2805-6-39-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the dimensionless function [MATH] in microscopic units.', '0806.2805-6-39-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-6-40-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-6-40-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-6-40-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given a few lines below [REF].', '0806.2805-6-40-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single time-independent dimensionless parameter [MATH] also taken positive.', '0806.2805-6-41-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-6-41-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-6-41-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-6-41-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (taken to be time independent in the analysis of the next section).', '0806.2805-6-42-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-6-43-0': '## Equations at the equilibrium point [MATH]', '0806.2805-6-44-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-6-44-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-6-44-2': 'Here, we consider the latter possibility where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for vacuum energy density [REF].', '0806.2805-6-45-0': 'Such an equilibrium vacuum state can be reached only if the chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in microscopic units [REF].', '0806.2805-6-45-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-6-45-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-6-45-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-6-45-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-6-46-0': 'In order to avoid having to consider quantum corrections to the Einstein equation, which typically appear near the time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-6-46-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium as given by the variable [MATH] defined in [REF].', '0806.2805-6-47-0': 'Taking the time derivative of [REF] for [MATH] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-6-47-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-6-47-2': 'The dimensionless vacuum energy density [REF] for the dimensionless equilibrium chemical potential [MATH] is given by [EQUATION] which obviously vanishes in the equilibrium state [MATH].', '0806.2805-6-48-0': 'In order to simplify the analysis, we, first, consider matter with a nonzero time-independent EOS parameter, [EQUATION] so that the matter energy density from [REF] can be neglected asymptotically, as will become clear later on.', '0806.2805-6-49-0': '## Vacuum oscillations', '0806.2805-6-50-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-6-50-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-6-50-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-6-50-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-6-50-4': 'Both oscillation periods will be manifest in the numerical results of Sec. [REF].', '0806.2805-6-51-0': '## Vacuum energy decay', '0806.2805-6-52-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the amplitudes in [REF], namely, the [MATH]-field oscillation amplitude [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-6-53-0': 'The explicit behavior is found by expanding the functions [MATH] and [MATH] in powers of [MATH] and keeping terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-6-53-1': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear ODEs [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-6-53-2': 'The solution of these equations is given by [REF] with the same replacements: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-6-54-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-6-54-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-6-54-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-6-54-3': 'It suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-6-54-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-6-54-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-6-54-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which, for [MATH], gives [MATH].', '0806.2805-6-55-0': 'The above results for the coefficients [MATH] and [MATH] hold for the generic case [MATH], as stated in [REF].', '0806.2805-6-55-1': 'For the special case [MATH], inspection of [REF] shows that the same Ansatze for [MATH] and [MATH] can be used, but with the following coefficients: [EQUATION] where a Kronecker delta has been employed in the expression for the damping factor [MATH] and the coefficient [MATH] of the [MATH] asymptotic energy density [MATH] has been assumed to be less than [MATH].', '0806.2805-6-56-0': 'Altogether, we have the following behavior of [MATH], [MATH], and [MATH] for [MATH]: [EQUATION] with dimensionless frequency [MATH] and damping factor [MATH] given by [REF].', '0806.2805-6-56-1': 'This asymptotic solution has some remarkable properties (in a different context, the same oscillatory behavior of [MATH] has been found in Ref. [CITATION]; see also the discussion in the last paragraph of Sec. [REF]).', '0806.2805-6-56-2': 'First, the solution depends rather weakly on the parameter [MATH] of the matter EOS, which is confirmed by the numerical results of the next subsection.', '0806.2805-6-56-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-6-56-4': 'Combined, the average vacuum energy density is found to behave as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed further in Sec. [REF].', '0806.2805-6-57-0': '## Numerical results', '0806.2805-6-58-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled ODEs [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-6-58-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equation can be expected (cf. Sec. [REF]).', '0806.2805-6-58-2': 'Still, the numerical results show clearly that', '0806.2805-6-59-0': '[(i)] the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); [(ii)] the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; [(iii)] the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] for a relatively small initial energy density and by Fig. [REF] for a relatively large initial energy density; [(iv)] for a chemical potential [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-6-60-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict, in fact, oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the model universe approaches a de-Sitter stage (see, in particular, the middle panel of the second row of Fig. [REF]).', '0806.2805-6-61-0': '## Effective CDM-like behavior', '0806.2805-6-62-0': 'The main result of the previous two subsections can be summarized as follows: the oscillating vacuum energy density [MATH] and the corresponding oscillating gravitational coupling parameter [MATH] conspire to give the same Hubble expansion as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-6-62-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-6-63-0': 'The explanation is as follows.', '0806.2805-6-63-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-6-63-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-6-63-3': 'The [MATH] (or [MATH]) field has no explicit kinetic term in the action [REF], but derivatives of [MATH] appear in the generalized Einstein Eq. [REF] via terms with covariant derivatives of [MATH], which trace back to the Einstein-Hilbert-like term [MATH] in [REF].', '0806.2805-6-63-4': 'In a way, the effective Lagrange density [MATH] can be said to be induced by gravity.', '0806.2805-6-63-5': 'The pressure of this rapidly oscillating field [MATH] is now given by [MATH].', '0806.2805-6-63-6': 'In turn, this implies that the rapidly oscillating vacuum pressure is zero on average and that the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-6-64-0': 'Observe that, while the [MATH]-field itself has an EOS parameter [MATH] corresponding to vacuum energy density, the net effect of the dampened [MATH] oscillations is to mimic the evolution of cold dark matter with [MATH] in a standard flat FRW universe.', '0806.2805-6-64-1': 'As mentioned before, this effective EOS parameter [MATH] is induced by the interaction of the [MATH] and gravity fields.', '0806.2805-6-65-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-6-65-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales that determine the microscopic dynamics of the vacuum energy density.', '0806.2805-6-65-2': 'But surprises are, of course, not excluded.', '0806.2805-6-66-0': '## Extrapolation to large times', '0806.2805-6-67-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-6-67-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF], [REF], and [REF].', '0806.2805-6-68-0': 'Extrapolating this evolution to the present age of the Universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] for [MATH].', '0806.2805-6-68-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe, which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-6-68-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-6-68-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-6-69-0': 'Let us expand on the previous remarks.', '0806.2805-6-69-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-6-69-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-6-69-3': 'This explains the following result for the case of a nonzero matter EOS parameter ([MATH]): [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-6-69-4': 'For nonrelativistic matter ([MATH]), the right-hand side of [REF] is multiplied by a further reduction factor [MATH], according to the results of Sec. [REF].', '0806.2805-6-70-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-6-70-1': 'A related problem is the CDM-like expansion of the model universe, [MATH], whereas big bang nucleosynthesis requires radiation-like expansion, [MATH], at least for the relevant temperature range.', '0806.2805-6-70-2': 'Clearly, there are many other processes that intervene between the very early (Planckian) phase of the Universe and later phases such as the nucleosynthesis era and the present epoch.', '0806.2805-6-70-3': 'An example of a relevant process may be particle production (e.g., by parametric resonance [CITATION]), which can be expected to be effective because of the very rapid (but small-amplitude) oscillations.', '0806.2805-6-70-4': 'A further possible source of modified vacuum energy behavior may be the change of EOS parameter [MATH] to [MATH], which occurs when the expanding universe leaves the radiation-dominated epoch.', '0806.2805-6-70-5': 'Still, there is a possibility that these and other processes are only secondary effects and that the main mechanism of dark-energy dynamics at the early stage is the decay of vacuum energy density by oscillations.', '0806.2805-6-71-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-6-71-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-6-71-2': "The behavior [REF], shown qualitatively by the [MATH] panels in Figs. [REF]-[REF], is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-6-71-3': 'For the present universe and the solar system in it, the gravitational coupling parameter [REF] would have minuscule oscillations.', '0806.2805-6-71-4': 'Combined with the Planck-scale mass of the [MATH] degree of freedom (cf. the discussion in Sec. [REF]), this would suggest that all solar-system experimental bounds are satisfied, but, again, surprises are not excluded.', '0806.2805-6-72-0': '# Conclusion', '0806.2805-6-73-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-6-73-1': 'Specifically, the new results are', '0806.2805-6-74-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-6-74-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-6-75-0': '# ACKNOWLEDGMENTS', '0806.2805-6-76-0': 'It is a pleasure to thank A.A. Starobinsky for informative discussions.', '0806.2805-6-76-1': 'GEV is supported in part by the Russian Foundation for Basic Research (Grant No. 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (Grant No. 4899.2008.2).', '0806.2805-6-77-0': '# NOTE ADDED IN PROOF', '0806.2805-6-78-0': 'Following up on the remarks in the last paragraph of Sec. [REF], we have recently shown [CITATION] that, close to equilibrium, the [MATH]-theory of the quantum vacuum gives rise to an effective [MATH]-model which belongs to the [MATH] class of models with a Planck-scale mass [MATH].', '0806.2805-6-78-1': 'We have also extended our analysis to a quantum vacuum containing several conserved [MATH]-fields, which allows for the coexistence of different vacua.'}

{'0806.2805-7-0-0': 'A modified-gravity theory is considered with a four-form field strength [MATH], a variable gravitational coupling parameter [MATH], and a standard matter action.', '0806.2805-7-0-1': 'This theory provides a concrete realization of the general vacuum variable [MATH] as the four-form amplitude [MATH] and allows for a study of its dynamics.', '0806.2805-7-0-2': 'The theory gives a flat Friedmann-Robertson-Walker universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically.', '0806.2805-7-0-3': 'Extrapolating to the present age of the Universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe.', '0806.2805-7-0-4': 'It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter in the standard Friedmann-Robertson-Walker framework.', '0806.2805-7-1-0': '# Introduction', '0806.2805-7-2-0': 'In a previous article [CITATION], we proposed to characterize a Lorentz-invariant quantum vacuum by a nonzero conserved relativistic "charge" [MATH].', '0806.2805-7-2-1': 'This approach allowed us to discuss the thermodynamics of the quantum vacuum, in particular, thermodynamic properties as stability and compressibility.', '0806.2805-7-2-2': 'We found that the vacuum energy density appears in two guises.', '0806.2805-7-3-0': 'The microscopic vacuum energy density is characterized by an ultraviolet energy scale, [MATH].', '0806.2805-7-3-1': 'For definiteness, we will take this energy scale [MATH] to be close to the Planck energy scale [MATH].', '0806.2805-7-3-2': 'The macroscopic vacuum energy density is, however, determined by a particular thermodynamic quantity, [MATH], and it is this energy density that contributes to the effective gravitational field equations at low energies.', '0806.2805-7-3-3': 'For a self-sustained vacuum in full thermodynamic equilibrium and in the absence of matter, the effective (coarse-grained) vacuum energy density [MATH] is automatically nullified (without fine tuning) by the spontaneous adjustment of the vacuum variable [MATH] to its equilibrium value [MATH], so that [MATH].', '0806.2805-7-3-4': 'This implies that the effective cosmological constant [MATH] of a perfect quantum vacuum is strictly zero, which is consistent with the requirement of Lorentz invariance.', '0806.2805-7-4-0': 'The presence of thermal matter makes the vacuum state Lorentz noninvariant and leads to a readjustment of the variable [MATH] to a new equilibrium value, [MATH], which shifts the effective vacuum energy density away from zero, [MATH].', '0806.2805-7-4-1': 'The same happens with other types of perturbations that violate Lorentz invariance, such as the existence of a spacetime boundary or an interface.', '0806.2805-7-4-2': 'According to this approach, the present value of [MATH] is nonzero but small because the universe is close to equilibrium and Lorentz-noninvariant perturbations of the quantum vacuum are small (compared with the ultraviolet scale which sets the microscopic energy density [MATH]).', '0806.2805-7-5-0': 'The situation is different for Lorentz-invariant perturbations of the vacuum, such as the formation of scalar condensates as discussed in Ref. [CITATION] or quark/gluon condensates derived from quantum chromodynamics (cf. Ref. [CITATION]).', '0806.2805-7-5-1': 'In this case, the variable [MATH] shifts in such a way that it completely compensates the energy density of the perturbation and the effective cosmological constant is again zero in the new Lorentz-invariant equilibrium vacuum.', '0806.2805-7-6-0': 'The possible origin of the conserved vacuum charge [MATH] in the perfect Lorentz-invariant quantum vacuum was discussed in Ref. [CITATION] in general terms.', '0806.2805-7-6-1': 'But a specific example was also given in terms of a four-form field strength [MATH] [CITATION].', '0806.2805-7-6-2': 'Here, we use this explicit realization with a four-form field [MATH] to study the dynamics of the vacuum energy, which describes the relaxation of the vacuum energy density [MATH] (effective cosmological "constant") from its natural Planck-scale value at early times to a naturally small value at late times.', '0806.2805-7-6-3': 'In short, the present cosmological constant is small because the Universe happens to be old.', '0806.2805-7-7-0': 'The results of the present article show that, for the type of theory considered, the decay of [MATH] is accompanied by rapid oscillations of the vacuum variable [MATH] and that the relaxation of [MATH] mimics the behavior of cold dark matter (CDM) in a standard Friedmann-Robertson-Walker (FRW) universe.', '0806.2805-7-7-1': 'This suggests that part of the inferred CDM may come from dynamic vacuum energy density and may also give a clue to the solution of the so-called coincidence problem [CITATION], namely, why the approximately constant vacuum energy density is precisely now of the same order as the time-dependent CDM energy density.', '0806.2805-7-8-0': 'These results are obtained by the following steps.', '0806.2805-7-8-1': 'In Sec. [REF], a modified-gravity theory with a four-form field [MATH] is defined in terms of general functions for the microscopic energy density [MATH] and variable gravitational coupling parameter [MATH].', '0806.2805-7-8-2': 'In Sec. [REF], the dynamics of the corresponding de-Sitter universe without matter is discussed and, in Sec. [REF], the dynamics of a flat FRW universe with matter, using simple Ansatze for the functions [MATH] and [MATH].', '0806.2805-7-8-3': 'In Sec. [REF], the approach to equilibrium in such a FRW universe is studied in detail and the above mentioned vacuum oscillations are established.', '0806.2805-7-8-4': 'In Sec. [REF], the main results are summarized.', '0806.2805-7-9-0': '# Gravity with [MATH] field and variable gravitational coupling', '0806.2805-7-10-0': 'Here, and in the following, the vacuum variable [MATH] is represented by a four-form field [MATH].', '0806.2805-7-10-1': 'The corresponding action is given by a generalization of the action in which only a quadratic function of [MATH] is used (see, e.g., Refs. [CITATION]).', '0806.2805-7-10-2': 'Such a quadratic function gives rise to a gas-like vacuum [CITATION].', '0806.2805-7-10-3': 'But a gas-like vacuum cannot exist in equilibrium without external pressure, as the equilibrium vacuum charge vanishes, [MATH].', '0806.2805-7-10-4': 'A self-sustained vacuum requires a more complicated function [MATH] in the action, so that the equilibrium at zero external pressure occurs for [MATH].', '0806.2805-7-10-5': 'An example of an appropriate function [MATH] will be given in Sec. [REF].', '0806.2805-7-11-0': "The action is chosen as in Ref. [CITATION] but with one important modification: Newton's constant [MATH] is replaced by a gravitational coupling parameter [MATH] which is taken to depend on the state of the vacuum and thus on the vacuum variable [MATH].", '0806.2805-7-11-1': 'Such a [MATH] dependence is natural and must, in principle, occur in the quantum vacuum.', '0806.2805-7-11-2': 'Moreover, a [MATH] dependence allows the cosmological "constant" to change with time, which is otherwise prohibited by the Bianchi identities and energy-momentum conservation [CITATION].', '0806.2805-7-12-0': 'Specifically, the action considered takes the following form ([MATH]): [EQUATION] where [MATH] denotes a covariant derivative and a square bracket around spacetime indices complete antisymmetrization.', '0806.2805-7-12-1': 'The functional dependence on [MATH] has been kept implicit on the right-hand side of [REF], showing only the dependence on [MATH] and [MATH].', '0806.2805-7-12-2': 'The field [MATH] in [REF] stands, in fact, for a generic low-energy matter field with a scalar Lagrange density, [MATH], which is assumed to be without [MATH]-field dependence (this assumption can be relaxed later by changing the low-energy constants in [MATH] to [MATH]-dependent parameters).', '0806.2805-7-12-3': 'It is also assumed that a possible constant term [MATH] in [MATH] has been absorbed in [MATH], so that, in the end, [MATH] contains only [MATH]-dependent terms.', '0806.2805-7-12-4': 'In this section, the low-energy fields are indicated by lower-case letters, namely, [MATH] and [MATH], whereas the fields originating from the microscopic theory are indicated by upper-case letters, namely, [MATH] and [MATH] [later also [MATH]].', '0806.2805-7-12-5': 'Throughout, we use the conventions of Ref. [CITATION], in particular, those for the Riemann tensor and the metric signature [MATH].', '0806.2805-7-13-0': "The variation of the action [REF] over the three-form gauge field [MATH] gives the generalized Maxwell equation, [EQUATION] and the variation over the metric [MATH] gives the generalized Einstein equation, [EQUATION] where [MATH] is the invariant d'Alembertian, [MATH] the energy-momentum tensor of the matter field [MATH], and [MATH] the effective vacuum energy density [EQUATION] whose precise form has been argued on thermodynamic grounds in Ref. [CITATION].", '0806.2805-7-14-0': 'At this point two remarks may be helpful.', '0806.2805-7-14-1': 'First, observe that the action [REF] is not quite the one of Brans-Dicke theory [CITATION], as the argument of [MATH] is not a fundamental scalar field but involves the inverse metric [needed to change the covariant tensor [MATH] into a contravariant tensor [MATH] for the definition of [MATH] according to [REF]].', '0806.2805-7-14-2': 'This implicit metric dependence of [MATH] explains the origin of the second term on the left-hand side of [REF].', '0806.2805-7-14-3': 'Second, observe that the three-form gauge field [MATH] does not propagate physical degrees of freedom in flat spacetime [CITATION].', '0806.2805-7-14-4': 'Still, [MATH] has gravitational effects, both classically in the modified-gravity theory with [MATH] as discussed in the present article (see, in particular, Sec. [REF]) and quantum-mechanically already in the standard gravity theory with [MATH] (giving, for example, a nonvanishing gravitational trace anomaly [CITATION]).', '0806.2805-7-15-0': 'Using [REF] for [MATH], we obtain the Maxwell equation [REF] in the form [EQUATION]', '0806.2805-7-15-1': 'The solution is simply [EQUATION] with an integration constant [MATH].', '0806.2805-7-15-2': 'Hence, the constant [MATH] is seen to emerge dynamically.', '0806.2805-7-15-3': 'In a thermodynamic equilibrium state, this constant becomes a genuine chemical potential corresponding to the conservation law obeyed by the vacuum "charge" [MATH].', '0806.2805-7-15-4': 'Indeed, the integration constant [MATH] is, according to [REF], thermodynamically conjugate to [MATH] in an equilibrium state with vanishing Ricci scalar [MATH].', '0806.2805-7-16-0': 'Eliminating [MATH] from [REF] by use of [REF], the generalized Einstein equation becomes [EQUATION] which will be used in the rest of this article, together with [REF].', '0806.2805-7-17-0': 'Equations [REF] and [REF] can also be obtained if we use, instead of the original action, an effective action in terms of a Brans-Dicke-type scalar field [MATH] with mass dimension 2, setting [MATH] afterwards.', '0806.2805-7-17-1': 'Specifically, this effective action is given by [EQUATION]', '0806.2805-7-17-2': 'The potential term in [REF] contains, different from a conventional Brans-Dicke potential [MATH], a linear term, [MATH], for a constant [MATH] of mass dimension 2.', '0806.2805-7-17-3': 'This linear term reflects the fact that our effective scalar field [MATH] is not an arbitrary field but should be a conserved quantity, for which the constant parameter [MATH] plays the role of a chemical potential that is thermodynamically conjugate to [MATH].', '0806.2805-7-18-0': 'Indeed, if [MATH] in [REF] is replaced by a four-form field [MATH] given in terms of the three-form potential [MATH], the resulting [MATH] term in the effective action does not contribute to the equations of motion [REF], because it is a total derivative, [EQUATION] where the constant [MATH] plays the role of a Lagrange multiplier related to the conservation of vacuum "charge" [MATH] (see also the discussion in Refs. [CITATION], where [MATH] is compared with the [MATH] parameter of quantum chromodynamics).', '0806.2805-7-19-0': 'Instead of the large microscopic energy density [MATH] in the original action [REF], the potentially smaller macroscopic vacuum energy density [MATH] enters the effective action [REF].', '0806.2805-7-19-1': 'Precisely this macroscopic vacuum energy density gravitates and determines the cosmological term in the gravitational field equations [REF].', '0806.2805-7-20-0': 'Equations [REF] and [REF] are universal: they do not depend on the particular origin of the vacuum field [MATH].', '0806.2805-7-20-1': 'The [MATH] field can be replaced by any conserved variable [MATH], as discussed in Ref. [CITATION].', '0806.2805-7-20-2': 'Observe that, for thermodynamics, the parameter [MATH] is the quantity that is thermodynamically conjugate to [MATH] and that, for dynamics, [MATH] plays the role of a Lagrange multiplier.', '0806.2805-7-20-3': 'The functions [MATH] and [MATH] can be considered to be phenomenological parameters in an effective low-energy theory (see also the general discussion in the Appendix of Ref. [CITATION]).', '0806.2805-7-21-0': 'Before we turn to the cosmological solutions of our particular [MATH] theory [REF], it may be useful to mention the connection with so-called [MATH] models which have recently received considerable attention (see, e.g., Refs. [CITATION] and references therein).', '0806.2805-7-21-1': 'The latter are purely phenomenological models, in which the linear function of the Ricci scalar [MATH] from the Einstein-Hilbert action term is replaced by a more general function [MATH].', '0806.2805-7-21-2': 'This function [MATH] can, in principle, be adjusted to fit the astronomical observations and to produce a viable cosmological model.', '0806.2805-7-21-3': 'Returning to our [MATH] theory, we can express [MATH] in terms of [MATH] by use of [REF] and substitute the resulting expression [MATH] into [REF].', '0806.2805-7-21-4': 'This gives an equation for the metric field, which is identical to the one of [MATH] cosmology.', '0806.2805-7-21-5': '(The latter result is not altogether surprising as the metric [MATH] model is known to be equivalent to a Brans-Dicke model without kinetic term [CITATION] and the same holds for our effective action [REF] at the classical level.)', '0806.2805-7-21-6': 'In this way, the [MATH] theory introduced in this section (or, more generally, [MATH] theory as mentioned in the previous paragraph) may give a microscopic justification for the phenomenological [MATH] models used in theoretical cosmology and may allow for a choice between different classes of model functions [MATH] based on fundamental physics.', '0806.2805-7-22-0': '# de-Sitter expansion', '0806.2805-7-23-0': 'Let us, first, consider stationary solutions of the generalized Maxwell-Einstein equations from the effective action [REF].', '0806.2805-7-23-1': 'At this moment, we are primarily interested in the class of spatially flat, homogeneous, and isotropic universes.', '0806.2805-7-23-2': 'In this class, only the matter-free de-Sitter universe is stationary.', '0806.2805-7-24-0': 'The de-Sitter universe is characterized by a time-independent Hubble parameter [MATH] (that is, a genuine Hubble constant [MATH]), which allows us to regard this universe as a thermodynamic equilibrium system.', '0806.2805-7-24-1': 'Using [EQUATION] we get from [REF] and [REF] two equations for the constants [MATH] and [MATH]: [EQUATION] with [MATH] considered given.', '0806.2805-7-25-0': 'Eliminating the chemical potential [MATH] from the above equations, we find the following equation for [MATH]: [EQUATION] where the functions [MATH] and [MATH] are assumed to be known.', '0806.2805-7-26-0': 'The perfect quantum vacuum corresponds to [MATH] and describes Minkowski spacetime.', '0806.2805-7-26-1': 'The corresponding equilibrium values [MATH] and [MATH] in the perfect quantum vacuum are determined from the following equations: [EQUATION] which are obtained from [REF] and [REF] by recalling that the perfect quantum vacuum is the equilibrium vacuum in the absence of matter and gravity fields ([MATH]).', '0806.2805-7-27-0': 'If [MATH] is nonzero but small compared with the Planck energy scale, the [MATH] term on the right-hand side of [REF] can be considered as a perturbation.', '0806.2805-7-27-1': 'Then, the correction [MATH] due to the expansion is given by [EQUATION] where [MATH] is the vacuum compressibility introduced in Ref. [CITATION], [EQUATION]', '0806.2805-7-27-2': 'Equally, the chemical potential is modified by the expansion ([MATH]): [EQUATION]', '0806.2805-7-27-3': 'But, instead of fixing [MATH], it is also possible to fix the integration constant [MATH].', '0806.2805-7-27-4': 'From [REF], we then obtain the other parameters as functions of [MATH]: [MATH], [MATH], and [MATH].', '0806.2805-7-27-5': 'The cosmological constant [MATH] is zero for [MATH], which corresponds to thermodynamic equilibrium in the absence of external pressure and expansion [[MATH]].', '0806.2805-7-27-6': 'From now on, the physical situation considered will be the one determined by having a fixed chemical potential [MATH].', '0806.2805-7-28-0': 'The de-Sitter universe is of interest because it is an equilibrium system and, therefore, may serve as the final state of a dynamic universe with matter included (see Sec. [REF]).', '0806.2805-7-29-0': '# Dynamics of a flat FRW universe', '0806.2805-7-30-0': '## General equations', '0806.2805-7-31-0': 'The discussion of this section and the next is restricted to a spatially flat FRW universe, because of two reasons.', '0806.2805-7-31-1': 'The first reason is that flatness is indicated by the data from observational cosmology (cf. Refs. [CITATION] and references therein).', '0806.2805-7-31-2': 'The second reason is that flatness is a natural property of the quantum vacuum in an emergent gravity theory (cf. Ref. [CITATION] and references therein).', '0806.2805-7-31-3': 'In addition, the matter energy-momentum tensor for the model universe is taken as that of a perfect fluid characterized by the energy density [MATH] and isotropic pressure [MATH].', '0806.2805-7-31-4': 'As mentioned in the previous section, the physics of the [MATH] field is considered to be specified by a fixed chemical potential [MATH].', '0806.2805-7-32-0': 'For a spatially flat ([MATH]) FRW universe [CITATION] with expansion factor [MATH], the homogenous matter has, in general, a time-dependent energy density [MATH] and pressure [MATH].', '0806.2805-7-32-1': 'Equally, the scalar field entering the four-form field-strength tensor [REF] is taken to be homogenous and time dependent, [MATH].', '0806.2805-7-33-0': 'With a time-dependent Hubble parameter [MATH], we then have from the reduced Maxwell equation [REF]: [EQUATION] and from the Einstein equation [REF]: [EQUATION] with total energy density and pressure [EQUATION] for the effective vacuum energy density [EQUATION]', '0806.2805-7-33-1': 'With definition [REF], the reduced Maxwell equation [REF] can be written as [EQUATION] where the overdot stands for differentiation with respect to cosmic time [MATH].', '0806.2805-7-33-2': 'The above equations give automatically energy-conservation of matter, [EQUATION] as should be the case for a standard matter field [MATH] (recall that [MATH] follows from the invariance of [MATH] under general coordinate transformations; cf. Appendix E of Ref. [CITATION]).', '0806.2805-7-34-0': '## Model for [MATH]', '0806.2805-7-35-0': 'The equations of Sec. [REF] allow us to study the development of the Universe from very small (near-Planckian) time scales to macroscopic time scales.', '0806.2805-7-35-1': 'Because the results do not depend very much on the details of the functions [MATH] and [MATH], it is possible to choose the simplest functions for an exploratory investigation.', '0806.2805-7-35-2': 'The only requirements are that the vacuum is self-sustained [i.e., [REF] has a solution with nonzero [MATH]] and that the vacuum is stable [i.e., the vacuum compressibility [REF] is positive, [MATH]].', '0806.2805-7-36-0': 'A simple choice for the function [MATH] is [EQUATION] where [MATH] is a constant parameter (vacuum compressibility) and [MATH] the value of [MATH] in a particular equilibrium vacuum satisfying [REF].', '0806.2805-7-36-1': 'The equilibrium value of the chemical potential [MATH] in the perfect vacuum is then given by [EQUATION]', '0806.2805-7-36-2': 'The microscopic parameters [MATH] and [MATH] are presumably determined by the Planck energy scale, [MATH] and [MATH].', '0806.2805-7-36-3': 'From [REF], we then see that [MATH].', '0806.2805-7-36-4': 'Let us now rewrite our equations in microscopic (Planckian) units by introducing appropriate dimensionless variables [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH]: [EQUATION] where the variable [MATH] has been introduced in anticipation of the calculations of Sec. [REF].', '0806.2805-7-36-5': 'The corresponding normalized vacuum and matter energy densities are defined as follows: [EQUATION] and Ansatz [REF] gives [EQUATION] with [MATH] from [REF].', '0806.2805-7-37-0': 'From the Maxwell equation [REF], the Friedmann equation [REF], and the matter conservation equation [REF], we finally obtain a closed system of three ordinary differential equations (ODEs) for the three dimensionless variables [MATH], [MATH], and [MATH]: [EQUATION] with matter equation-of-state (EOS) parameter [MATH].', '0806.2805-7-38-0': '## Model for [MATH]', '0806.2805-7-39-0': 'Next, we need an appropriate Ansatz for the function [MATH] or the dimensionless function [MATH] in microscopic units.', '0806.2805-7-39-1': 'There are several possible types of behavior for [MATH], but we may reason as follows.', '0806.2805-7-40-0': 'It is possible that for [MATH] (i.e., in the gas-like vacuum) the role of the Planck scale is played by [MATH].', '0806.2805-7-40-1': 'The gravitational coupling parameter would then be given by [EQUATION]', '0806.2805-7-40-2': 'This equation also gives the correct estimate for [MATH] in the equilibrium vacuum: [MATH], according to the estimates given a few lines below [REF].', '0806.2805-7-40-3': 'Thus, a simple choice for the function [MATH] is [EQUATION] with [MATH] taken positive (in fact, [MATH] for [MATH]) and a single time-independent dimensionless parameter [MATH] also taken positive.', '0806.2805-7-41-0': 'Assuming [REF], the three ODEs [REF] become [EQUATION] with [MATH] given by [REF] and a single free parameter [MATH].', '0806.2805-7-41-1': 'This dimensionless parameter [MATH] is of order [MATH] if the physics of [MATH] field is solely determined by the Planck energy scale (i.e., for [MATH]).', '0806.2805-7-41-2': 'Anyway, the parameter [MATH] can be absorbed in [MATH] and [MATH] by the redefinition [MATH] and [MATH].', '0806.2805-7-41-3': 'Henceforth, we set [MATH] in [REF], so that there are no more free parameters except for the EOS parameter [MATH] (taken to be time independent in the analysis of the next section).', '0806.2805-7-42-0': '# Equilibrium approach in a flat FRW universe', '0806.2805-7-43-0': '## Equations at the equilibrium point [MATH]', '0806.2805-7-44-0': 'Equations [REF]-[REF] allow us to study the evolution of the flat FRW universe towards a stationary state, if the initial universe was far away from equilibrium.', '0806.2805-7-44-1': 'The final state can be either the de-Sitter universe of Sec. [REF] with [MATH] and [MATH] or the perfect quantum vacuum (Minkowski spacetime) with [MATH] and [MATH].', '0806.2805-7-44-2': 'Here, we consider the latter possibility where the system approaches one of the two perfect quantum vacuum states with [MATH], which correspond to either [MATH] or [MATH] for vacuum energy density [REF].', '0806.2805-7-45-0': 'Such an equilibrium vacuum state can be reached only if the chemical potential [MATH] corresponds to full equilibrium: [MATH] as given by [REF] or [MATH] in microscopic units [REF].', '0806.2805-7-45-1': 'Since [MATH] is an integration constant, there may be a physical reason for the special value [MATH].', '0806.2805-7-45-2': 'Indeed, the starting nonequilibrium state could, in turn, be obtained by a large perturbation of an initial equilibrium vacuum.', '0806.2805-7-45-3': 'In this case, the integration constant would remember the original perfect equilibrium.', '0806.2805-7-45-4': '(The evolution towards a de-Sitter universe for [MATH] will be only briefly discussed in Sec. [REF].)', '0806.2805-7-46-0': 'In order to avoid having to consider quantum corrections to the Einstein equation, which typically appear near the time [MATH] (or [MATH]), we consider times [MATH], where the quantum corrections can be expected to be small.', '0806.2805-7-46-1': 'For these relatively large times, [MATH] is close to unity and we may focus on the deviation from equilibrium as given by the variable [MATH] defined in [REF].', '0806.2805-7-47-0': 'Taking the time derivative of [REF] for [MATH] and using [REF] and [REF], we obtain [EQUATION] where, from now on, the overdot stands for differentiation with respect to [MATH].', '0806.2805-7-47-1': 'Next, eliminate the matter density [MATH] from equations [REF] and [REF], in order to obtain a system of two equations for the two variables [MATH] and [MATH]: [EQUATION] where the last equation corresponds to [REF] for [MATH].', '0806.2805-7-47-2': 'The dimensionless vacuum energy density [REF] for the dimensionless equilibrium chemical potential [MATH] is given by [EQUATION] which obviously vanishes in the equilibrium state [MATH].', '0806.2805-7-48-0': 'In order to simplify the analysis, we, first, consider matter with a nonzero time-independent EOS parameter, [EQUATION] so that the matter energy density from [REF] can be neglected asymptotically, as will become clear later on.', '0806.2805-7-49-0': '## Vacuum oscillations', '0806.2805-7-50-0': 'Close to equilibrium, equations [REF] and [REF] can be linearized: [EQUATION]', '0806.2805-7-50-1': 'The solution of these equations describes rapid oscillations near the equilibrium point: [EQUATION]', '0806.2805-7-50-2': 'The (dimensionless) oscillation period of [MATH] and [MATH] is given by [EQUATION]', '0806.2805-7-50-3': 'The corresponding oscillation period of the vacuum energy density [MATH] is smaller by a factor [MATH], so that numerically this period is given by [MATH].', '0806.2805-7-50-4': 'Both oscillation periods will be manifest in the numerical results of Sec. [REF].', '0806.2805-7-51-0': '## Vacuum energy decay', '0806.2805-7-52-0': 'The neglected quadratic terms in equations [REF] and [REF] provide the slow decay of the amplitudes in [REF], namely, the [MATH]-field oscillation amplitude [MATH], the Hubble term [MATH], and the vacuum energy density averaged over fast oscillations [MATH].', '0806.2805-7-53-0': 'The explicit behavior is found by expanding the functions [MATH] and [MATH] in powers of [MATH] and keeping terms up to [MATH]: [EQUATION] where the equality sign has been used rather freely.', '0806.2805-7-53-1': 'Collecting the [MATH] terms, we get homogeneous linear equations for [MATH] and [MATH], which are actually the same as the linear ODEs [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '0806.2805-7-53-2': 'The solution of these equations is given by [REF] with the same replacements: [EQUATION] where [MATH] and [MATH] are numerical coefficients which ultimately determine the decay of [MATH] and [MATH].', '0806.2805-7-54-0': 'In order to obtain these coefficients, we must collect the [MATH] terms.', '0806.2805-7-54-1': 'This leads to inhomogeneous linear equations for the functions [MATH] and [MATH].', '0806.2805-7-54-2': 'The consistency of these equations determines the coefficients [MATH] and [MATH].', '0806.2805-7-54-3': 'It suffices to keep only the zeroth and first harmonics in the functions [MATH] and [MATH]: [EQUATION]', '0806.2805-7-54-4': 'As a result, we obtain the following equations for [MATH] and [MATH]: [EQUATION]', '0806.2805-7-54-5': 'From the consistency of these equations for the first harmonics of [MATH] and [MATH], we obtain [EQUATION] which gives [MATH].', '0806.2805-7-54-6': 'Similarly, we find from the zeroth harmonic of [REF] [EQUATION] which, for [MATH], gives [MATH].', '0806.2805-7-55-0': 'The above results for the coefficients [MATH] and [MATH] hold for the generic case [MATH], as stated in [REF].', '0806.2805-7-55-1': 'For the special case [MATH], inspection of [REF] shows that the same Ansatze for [MATH] and [MATH] can be used, but with the following coefficients: [EQUATION] where a Kronecker delta has been employed in the expression for the damping factor [MATH] and the coefficient [MATH] of the [MATH] asymptotic energy density [MATH] has been assumed to be less than [MATH].', '0806.2805-7-56-0': 'Altogether, we have the following behavior of [MATH], [MATH], and [MATH] for [MATH]: [EQUATION] with dimensionless frequency [MATH] and damping factor [MATH] given by [REF].', '0806.2805-7-56-1': 'This asymptotic solution has some remarkable properties (in a different context, the same oscillatory behavior of [MATH] has been found in Ref. [CITATION]; see also the discussion in the last paragraph of Sec. [REF]).', '0806.2805-7-56-2': 'First, the solution depends rather weakly on the parameter [MATH] of the matter EOS, which is confirmed by the numerical results of the next subsection.', '0806.2805-7-56-3': 'Second, the average value of the vacuum energy density decays as [MATH] and the average value of the Hubble parameter as [MATH], while the average scale parameter increases as [MATH].', '0806.2805-7-56-4': 'Combined, the average vacuum energy density is found to behave as [MATH], which is the same behavior as that of CDM in a standard FRW universe, as will be discussed further in Sec. [REF].', '0806.2805-7-57-0': '## Numerical results', '0806.2805-7-58-0': 'For ultrarelativistic matter ([MATH]), chemical potential [MATH], and parameter [MATH], the numerical solution of the coupled ODEs [REF]-[REF] is given in Figs. [REF] and [REF].', '0806.2805-7-58-1': 'The behavior near [MATH] is only indicative, as significant quantum corrections to the classical Einstein equation can be expected (cf. Sec. [REF]).', '0806.2805-7-58-2': 'Still, the numerical results show clearly that', '0806.2805-7-59-0': '[(i)] the equilibrium vacuum is approached asymptotically ([MATH] for [MATH]); [(ii)] the FRW universe (averaged over time intervals larger than the Planck-scale oscillation period) does not have the expected behavior [MATH] for ultrarelativistic matter but rather [MATH]; [(iii)] the same [MATH] behavior occurs if there is initially nonrelativistic matter, as demonstrated by Figs. [REF] and [REF] for a relatively small initial energy density and by Fig. [REF] for a relatively large initial energy density; [(iv)] for a chemical potential [MATH] slightly different from the equilibrium value [MATH], the vacuum decay is displayed in Fig. [REF].', '0806.2805-7-60-0': 'The first three items of the above list of numerical results confirm the previous asymptotic analytic results of Sec. [REF] (these asymptotic results predict, in fact, oscillations between [MATH] for the particular combinations shown on the bottom-row panels of Figs. [REF]-[REF]), while the last item shows that, after an initial oscillating stage, the model universe approaches a de-Sitter stage (see, in particular, the middle panel of the second row of Fig. [REF]).', '0806.2805-7-61-0': '## Effective CDM-like behavior', '0806.2805-7-62-0': 'The main result of the previous two subsections can be summarized as follows: the oscillating vacuum energy density [MATH] and the corresponding oscillating gravitational coupling parameter [MATH] conspire to give the same Hubble expansion as pressureless matter (e.g., CDM) in a standard FRW universe with fixed gravitational coupling constant [MATH].', '0806.2805-7-62-1': 'Recall that the standard behavior of the CDM energy density is given by [MATH], which matches the average behavior found in [REF].', '0806.2805-7-63-0': 'The explanation is as follows.', '0806.2805-7-63-1': 'The average values of the rapidly oscillating vacuum energy density and vacuum pressure act as a source for the slowly varying gravitational field.', '0806.2805-7-63-2': 'The rapidly oscillating parts of [MATH] and [MATH] in the linearized equation [REF] correspond to a dynamic system with Lagrangian density [MATH] for a time-dependent homogenous field [MATH].', '0806.2805-7-63-3': 'The [MATH] (or [MATH]) field has no explicit kinetic term in the action [REF], but derivatives of [MATH] appear in the generalized Einstein Eq. [REF] via terms with covariant derivatives of [MATH], which trace back to the Einstein-Hilbert-like term [MATH] in [REF].', '0806.2805-7-63-4': 'In a way, the effective Lagrange density [MATH] can be said to be induced by gravity.', '0806.2805-7-63-5': 'The pressure of this rapidly oscillating field [MATH] is now given by [MATH].', '0806.2805-7-63-6': 'In turn, this implies that the rapidly oscillating vacuum pressure is zero on average and that the main contribution of the oscillating vacuum energy density behaves effectively as cold dark matter.', '0806.2805-7-64-0': 'Observe that, while the [MATH]-field itself has an EOS parameter [MATH] corresponding to vacuum energy density, the net effect of the dampened [MATH] oscillations is to mimic the evolution of cold dark matter with [MATH] in a standard flat FRW universe.', '0806.2805-7-64-1': 'As mentioned before, this effective EOS parameter [MATH] is induced by the interaction of the [MATH] and gravity fields.', '0806.2805-7-65-0': 'An outstanding task is to establish the clustering properties of this type of oscillating vacuum energy density.', '0806.2805-7-65-1': 'A priori, we may expect the same properties as CDM, because the relevant astronomical length scales are very much larger than the ultraviolet length scales that determine the microscopic dynamics of the vacuum energy density.', '0806.2805-7-65-2': 'But surprises are, of course, not excluded.', '0806.2805-7-66-0': '## Extrapolation to large times', '0806.2805-7-67-0': 'In Secs. [REF] and [REF], we have established that the average vacuum energy density decreases quadratically with cosmic time.', '0806.2805-7-67-1': 'This behavior follows, analytically, from [REF] and, numerically, from the bottom-right panels of Figs. [REF], [REF], and [REF].', '0806.2805-7-68-0': 'Extrapolating this evolution to the present age of the Universe ([MATH]) and using [MATH] for [MATH], the numerical value of the average vacuum energy density is given by [EQUATION] for [MATH].', '0806.2805-7-68-1': 'The order of magnitude of the above estimate is in agreement with the observed vacuum energy density of the present universe, which is close to the critical density of a standard FRW universe (cf. Refs. [CITATION] and references therein).', '0806.2805-7-68-2': 'If the behavior found had been [MATH] for an integer [MATH], this agreement would be lost altogether.', '0806.2805-7-68-3': 'In other words, the dynamic behavior established in [REF] is quite nontrivial.', '0806.2805-7-69-0': 'Let us expand on the previous remarks.', '0806.2805-7-69-1': 'For a standard flat FRW universe, the total energy density is, of course, always equal to the critical density [MATH].', '0806.2805-7-69-2': 'But, here, the gravitational coupling parameter is variable, [MATH], and there are rapid oscillations, so that, for example, [MATH].', '0806.2805-7-69-3': 'This explains the following result for the case of a nonzero matter EOS parameter ([MATH]): [EQUATION] which is of order [MATH] but not exactly equal to [MATH].', '0806.2805-7-69-4': 'For nonrelativistic matter ([MATH]), the right-hand side of [REF] is multiplied by a further reduction factor [MATH], according to the results of Sec. [REF].', '0806.2805-7-70-0': 'Even though the order of magnitude of [REF] or [REF] appears to be relevant to the observed universe, the [MATH] behavior of [MATH] contradicts the current astronomical data on "cosmic acceleration" [CITATION].', '0806.2805-7-70-1': 'A related problem is the CDM-like expansion of the model universe, [MATH], whereas big bang nucleosynthesis requires radiation-like expansion, [MATH], at least for the relevant temperature range.', '0806.2805-7-70-2': 'Clearly, there are many other processes that intervene between the very early (Planckian) phase of the Universe and later phases such as the nucleosynthesis era and the present epoch.', '0806.2805-7-70-3': 'An example of a relevant process may be particle production (e.g., by parametric resonance [CITATION]), which can be expected to be effective because of the very rapid (but small-amplitude) oscillations.', '0806.2805-7-70-4': 'A further possible source of modified vacuum energy behavior may be the change of EOS parameter [MATH] to [MATH], which occurs when the expanding universe leaves the radiation-dominated epoch.', '0806.2805-7-70-5': 'Still, there is a possibility that these and other processes are only secondary effects and that the main mechanism of dark-energy dynamics at the early stage is the decay of vacuum energy density by oscillations.', '0806.2805-7-71-0': 'Another aspect of the large-time extrapolation concerns the variation of Newton\'s "constant."', '0806.2805-7-71-1': "For the theory [REF] and the particular Ansatz [REF], the gravitational coupling parameter [MATH] is found to relax to an equilibrium value in the following way: [EQUATION] with [MATH] a constant of order unity, [MATH] a gravitational constant presumably very close to the Cavendish-type value for Newton's constant [MATH], and [MATH] an ultraviolet timescale of the order of the corresponding Planckian time scale [MATH].", '0806.2805-7-71-2': "The behavior [REF], shown qualitatively by the [MATH] panels in Figs. [REF]-[REF], is very different from previous suggestions for the dynamics of [MATH], including Dirac's original suggestion [MATH] (cf. Sec. 16.4 of Ref. [CITATION]).", '0806.2805-7-71-3': 'For the present universe and the solar system in it, the gravitational coupling parameter [REF] would have minuscule oscillations.', '0806.2805-7-71-4': 'Combined with the Planck-scale mass of the [MATH] degree of freedom (cf. the discussion in Sec. [REF]), this would suggest that all solar-system experimental bounds are satisfied, but, again, surprises are not excluded.', '0806.2805-7-72-0': '# Conclusion', '0806.2805-7-73-0': 'The considerations of the present article and its predecessor [CITATION] by no means solve the cosmological constant problems, but may provide hints.', '0806.2805-7-73-1': 'Specifically, the new results are', '0806.2805-7-74-0': 'Expanding on the last point, another consequence of [MATH] oscillations is that they naturally lead to the creation of hot (ultrarelativistic) matter from the vacuum.', '0806.2805-7-74-1': 'This effective mechanism of energy exchange between vacuum and matter deserves further study.', '0806.2805-7-75-0': '# ACKNOWLEDGMENTS', '0806.2805-7-76-0': 'It is a pleasure to thank A.A. Starobinsky for informative discussions.', '0806.2805-7-76-1': 'GEV is supported in part by the Russian Foundation for Basic Research (Grant No. 06-02-16002-a) and the Khalatnikov-Starobinsky leading scientific school (Grant No. 4899.2008.2).', '0806.2805-7-77-0': '# NOTE ADDED IN PROOF', '0806.2805-7-78-0': 'Following up on the remarks in the last paragraph of Sec. [REF], we have recently shown [CITATION] that, close to equilibrium, the [MATH]-theory of the quantum vacuum gives rise to an effective [MATH]-model which belongs to the [MATH] class of models with a Planck-scale mass [MATH].', '0806.2805-7-78-1': 'We have also extended our analysis to a quantum vacuum containing several conserved [MATH]-fields, which allows for the coexistence of different vacua.'}

0802.0971

{'0802.0971-1-0-0': 'The pion-polarizability functions are structure functions of pion-Compton scattering.', '0802.0971-1-0-1': 'They can be assessed in high-energy pion-nucleus bremsstrahlung reactions, [MATH].', '0802.0971-1-0-2': 'We present numerical expectations for pion-nucleus bremsstrahlung cross sections in the Coulomb region, i.e. the small-angle region where the nuclear scattering is dominated by the Coulomb interaction.', '0802.0971-1-0-3': 'We investigate the prospects of measuring the polarizability functions for pion-Compton c.m. energies from threshold up to 1 GeV.', '0802.0971-1-0-4': 'A meson-exchange model is used for the pion-Compton amplitude.', '0802.0971-1-1-0': '# Introduction', '0802.0971-1-2-0': 'The cross-section distribution for high-energy pionic bremsstrahlung [EQUATION] is at small-momentum transfers to the nucleus dominated by the one-photon-exchange contribution.', '0802.0971-1-2-1': 'Hence, by measuring cross-section distributions for pion-nucleus bremsstrahlung at small momentum transfers one gains at the same time information about the cross-section distribution for pion-Compton scattering [EQUATION] the photon of the initial state being the virtual photon exchanged between pion and nucleus.', '0802.0971-1-2-2': 'In particular, it becomes possible to determine pion polarizabilities.', '0802.0971-1-3-0': "The theoretical underpinnings of the pion-bremsstrahlung reaction were investigated by Gal'perin et al. [CITATION], and an experiment was susequently performed by Antipov et al. [CITATION], which showed that pion polarizabilities could indeed be determined.", '0802.0971-1-3-1': 'The COMPASS experiment at CERN [CITATION] is a refinement of the Antipov experiment.', '0802.0971-1-4-0': 'In ref. [CITATION] a mathematical expression for the bremsstrahlung cross section was derived assuming a pion-Compton amplitude that in additon to the Born contributions contained also contributions from the [MATH], [MATH], and [MATH], exchanges.', '0802.0971-1-4-1': 'The kinematics of the nuclear reaction was defined through [EQUATION] and the kinematics of the related pion-Compton reaction through [EQUATION] with [MATH].', '0802.0971-1-4-2': 'In terms of these kinematic variables the cross-section distribution can be written as [CITATION] [EQUATION]', '0802.0971-1-4-3': 'This expression is valid when the transverse momenta of the emerging pion and photon are much smaller than their longitudinal momenta.', '0802.0971-1-4-4': 'The parameter [MATH] is the fine-structure constant, [MATH] the pion mass, [MATH] the ratio [EQUATION] [MATH], and [MATH] the minimum-longitudinal-momentum transfer [EQUATION]', '0802.0971-1-4-5': 'At high energies [MATH] is extremely small.', '0802.0971-1-5-0': 'The functions [MATH] and [MATH] characterize the pion-Compton amplitude.', '0802.0971-1-5-1': 'In the Born approximation [EQUATION] and generally [EQUATION] with [MATH] and [MATH] generalized-polarizability functions.', '0802.0971-1-5-2': 'Their mathematical expressions, in the meson-exchange model, are given in [CITATION].', '0802.0971-1-6-0': 'The structure of the cross-section distribution is mainly determined by the inelastic-Coulomb factor [EQUATION] which vanishes when the transverse-momentum transfer [MATH]q[MATH] to the nucleus vanishes.', '0802.0971-1-6-1': 'This is the classic Primakoff factor, which exhibits a maximum at [MATH].', '0802.0971-1-6-2': 'When the momentum transfer [MATH] becomes much larger than [MATH] the Coulomb contribution to the bremsstrahlung cross section becomes small and the hadronic contribution dominates [CITATION].', '0802.0971-1-6-3': 'This contribution is not considered in the present paper.', '0802.0971-1-7-0': 'The Mandelstam kinematic variables of the pion-Compton scattering ([REF]) can in our application be approximated as follows; [EQUATION]', '0802.0971-1-8-0': '# Kinematical considerations', '0802.0971-1-9-0': 'We are interested in studying the pion-Compton-scattering process.', '0802.0971-1-9-1': 'In order to do so in a transparent way we need to make specific cuts in the pion-nucleus-bremsstrahlung phase space.', '0802.0971-1-9-2': 'To this end a few kinematical observations might be helpful.', '0802.0971-1-10-0': 'Let us first introduce, instead of the pion-Compton c.m. energy and the photon-transverse momentum, the corresponding dimensionless variables [EQUATION] with [MATH] and [MATH].', '0802.0971-1-10-1': 'The first entry of Eq. ([REF]), defining the pion-Compton c.m. energy squared, can then be reformulated as [EQUATION]', '0802.0971-1-10-2': 'Since [MATH] is positive we conclude that for a given fixed value of [MATH] the variable [MATH] must be in the interval [EQUATION]', '0802.0971-1-10-3': 'The maximal value of [MATH] occurs at the midpoint of this interval, [EQUATION]', '0802.0971-1-10-4': 'We may also rewrite Eq.([REF]) as [EQUATION] showing directly that the photon-transverse momentum vanishes at the kinematic end-points [MATH] and [MATH].', '0802.0971-1-11-0': 'For a fixed value of [MATH], on the other hand, it follows from the positivity of [MATH] of Eq.( [REF]) that the pion-Compton c.m. energy is bounded from below [EQUATION]', '0802.0971-1-11-1': 'The boundary value obtains in the forward direction, where the photon-transverse momentum vanishes.', '0802.0971-1-12-0': 'In Fig.1 the photon-transverse momentum [MATH] is plotted for a few values of the pion-Compton c.m. energy [MATH].', '0802.0971-1-12-1': 'As an illustration consider [MATH], a mass at which the threshold approximation to the polarizability functions should be acceptable.', '0802.0971-1-12-2': 'In this case the maximal value of [MATH] is [MATH] and the maximal value of the photon-transverse momentum [MATH] is [MATH].', '0802.0971-1-13-0': '# Low-mass-Compton region', '0802.0971-1-14-0': 'Hard-pion bremsstrahlung offers a unique possibility of determining pion-threshold polarizabilities.', '0802.0971-1-14-1': 'However, this will succeed only if we restrain the c.m. mass in the pion-photon system to masses well below the [MATH]-meson mass, so as to avoid the [MATH]-channel [MATH]-exchange contribution to the Compton amplitude.', '0802.0971-1-15-0': 'Hence, we first concentrate our efforts on the region [MATH], where we can confidently approximate the structure functions [MATH] and [MATH] of Eqs. ([REF]) and ([REF]) by their near-threshold values, with [EQUATION] and [MATH] and [MATH] the standard pion-electric and pion-magnetic polarizabilities.', '0802.0971-1-15-1': 'These values given are obtained in the meson-exchange model when fixing the [MATH]-meson coupling so that the chiral Lagrangian value [CITATION] of [MATH] is reproduced.', '0802.0971-1-15-2': 'Our present knowledge of the pion polarizabilities is reviewed in [CITATION].', '0802.0971-1-16-0': 'The cross-section distribution ([REF]) contains angular dependent terms, through [MATH].', '0802.0971-1-16-1': 'Measured variables are however [MATH] and [MATH], but since, [MATH], with [MATH], the angular variations of [MATH] and [MATH] are independent.', '0802.0971-1-16-2': 'Thus, the integrations over the angles of [MATH] and [MATH] in Eq. ([REF]) are unrestricted and we may replace [MATH] by its angular average, which is a half, and at the same make the replacement [EQUATION]', '0802.0971-1-16-3': 'After integration over angles the cross-section distribution factorizes into factors that depend either on [MATH] and [MATH], or on [MATH] and [MATH].', '0802.0971-1-16-4': 'The integration over the inelastic-Coulomb factor ([REF]) is an integration over [MATH], but there is an [MATH] dependence in [MATH], [EQUATION]', '0802.0971-1-16-5': 'It is tempting to integrate over all possible momentum transfers [MATH], but that is not possible since the integral diverges.', '0802.0971-1-16-6': 'The reason is that we have neglected pion and nuclear form factors.', '0802.0971-1-16-7': 'Also, for large values of [MATH] the hadronic-bremsstrahlung contribution dominates the Coulomb contribution.', '0802.0971-1-16-8': 'Consequently, we must make a cut so as to avoid the hadronic contribution.', '0802.0971-1-16-9': 'The best approach is to make a cut in the same way for all [MATH] values.', '0802.0971-1-16-10': 'This is achieved by cutting off the integral at [MATH] on the low-momentum side and at [MATH] on the high-momentum side of the Coulomb peak.', '0802.0971-1-16-11': 'From ([REF]) we derive [EQUATION]', '0802.0971-1-16-12': 'This integral is obviously independent of [MATH].', '0802.0971-1-16-13': 'The precise values of [MATH] and [MATH] to be chosen can only be decided after inspection of the experimental distribution.', '0802.0971-1-17-0': 'Next we integrate the cross-section distribution Eq.([REF]) over transverse-photon momenta in the domain [MATH].', '0802.0971-1-17-1': 'Since the polarizability functions [MATH] and [MATH] entering the structure functions [MATH] and [MATH] of Eqs.([REF]) and ([REF]) are now constants the integration is straightforward, and we get [EQUATION] with [EQUATION] and the distribution function [EQUATION]', '0802.0971-1-17-2': 'The [MATH]-dependent prefactor in Eq.([REF]) is simply the ratio of the final state momenta, [MATH], and the singularity at [MATH] is the well-known soft-photon-radiation singularity.', '0802.0971-1-18-0': 'The functions [MATH], [MATH], and [MATH], of Eq.([REF]) are all elementary and given in the Appendix.', '0802.0971-1-18-1': 'They are graphed in Fig.2a.', '0802.0971-1-18-2': 'For small values of the argument [MATH], or transverse momenta such that [MATH], we have [EQUATION]', '0802.0971-1-18-3': 'For large values of the argument [MATH], or transverse momenta such that [MATH], we have instead [EQUATION]', '0802.0971-1-18-4': 'In particular we note for large values of [MATH] the ratio [MATH].', '0802.0971-1-19-0': 'An alternative to integrating over transverse-photon momenta is to integrate over Compton masses [MATH] in the domain [MATH].', '0802.0971-1-19-1': 'This is simply achieved by making the integration limit [MATH] depend on [MATH].', '0802.0971-1-19-2': 'According to Eq.([REF]), or ([REF]), the relation between [MATH] and [MATH] allows us to write [EQUATION] with [EQUATION]', '0802.0971-1-19-3': 'At the same time as being the definition of the variable [MATH], Eq.([REF]) displays the functional relation between [MATH] and [MATH].', '0802.0971-1-20-0': 'We now specialize to [MATH] which is the cut for the threshold region employed by the COMPASS collaboration.', '0802.0971-1-20-1': 'This implies, via Eq. ([REF]), a maximal [MATH]-value of [MATH].', '0802.0971-1-20-2': 'In Fig.2b the three distribution functions [MATH], [MATH], and [MATH] are plotted as functions of [MATH] for this value of [MATH].', '0802.0971-1-20-3': 'From this figure we draw the following conclusions.', '0802.0971-1-21-0': 'Near the upper end of the [MATH]-distribution the three functions [MATH], [MATH], and [MATH], may be approximated by their threshold values, Eq.([REF]).', '0802.0971-1-21-1': 'This is a consequence of Eq.([REF]), which demonstrates that even though the factor [MATH] is large, when [MATH] is sufficiently close to [MATH] the variable [MATH] will be small.', '0802.0971-1-21-2': 'This observation leads to a simplification of the cross-section-distribution function [MATH] of Eq.([REF]), [EQUATION]', '0802.0971-1-21-3': 'Numerically, we expect [MATH] and small, but enters multiplied by a large factor, [MATH], and therefore makes an important contribution to the cross-section distribution.', '0802.0971-1-22-0': 'In the mid-range [MATH]-region the value of [MATH] is large, due to the large value of [MATH], and we may here approximate the functions [MATH] and [MATH] by their asymptotic expressions, Eq. ([REF]).', '0802.0971-1-22-1': 'This allows us to simplify the expression for the function [MATH], yielding the approximate formula [EQUATION]', '0802.0971-1-22-2': 'The function [MATH], representing the Born approximation, must be evaluated exactly as we are looking for small deviations from it.', '0802.0971-1-22-3': 'The numerical value of the coefficient multiplying the middle [MATH] term in Eq.( [REF]) is large.', '0802.0971-1-22-4': 'Nevertheless, this term has essentially no effect on the cross section since according to Eq.([REF]) [MATH] is proportional to the sum [MATH] which vanishes, or is very small compared with [MATH].', '0802.0971-1-22-5': 'The important term is the [MATH] term.', '0802.0971-1-22-6': 'Over most of the [MATH]-range its strength is fixed by Eq. ([REF]).', '0802.0971-1-22-7': 'But this approximation cannot be used near the upper end-point since it would there overestimate the polarizability contribution by a factor of 3/2, as a comparison with expression ([REF]) would demonstrate.', '0802.0971-1-23-0': 'In Fig.3a the cross-section-distribution function [MATH] of Eq. ([REF]) is graphed in the full calculation and in the Born approximation, i.e., when polarizability parameters [MATH].', '0802.0971-1-23-1': 'In Fig.3b the ratio of the two distributions is plotted.', '0802.0971-1-23-2': 'Clearly, the polarizabilities are important only in a region near the end-point of the [MATH]- distibution, and therefore the experimental efforts are concentrated on this region [CITATION].', '0802.0971-1-24-0': 'In the COMPASS experiment the apparatus is blind in a small region around the forward direction.', '0802.0971-1-24-1': 'The transverse momenta of the final state pions and photons must be larger than [MATH] to be detected.', '0802.0971-1-24-2': 'As a result the distribution function [MATH] of Eq. ([REF]) must be replaced by [EQUATION] where from Eq. ([REF]) [EQUATION]', '0802.0971-1-24-3': 'In the COMPASS experiment [MATH] MeV/[MATH] and [MATH] is small except when [MATH] is near [MATH].', '0802.0971-1-24-4': 'The subtraction procedure of Eq. ([REF]) works unless there are momentum transfers [MATH] for which [MATH].', '0802.0971-1-24-5': 'This will not happen as long as [MATH] remains in the domain [EQUATION] which is the case in the COMPASS application.', '0802.0971-1-24-6': 'The effect of the cut on the cross section is illustrated in Fig.4.', '0802.0971-1-25-0': '# Rho-mass-Compton region', '0802.0971-1-26-0': 'At large Compton masses the polarizability functions are complex and depend in a complicated manner on the varables [MATH] and [MATH].', '0802.0971-1-26-1': 'As before, after angular integration, the cross-section distribution factorizes into one factor dependening on [MATH] and [MATH], and another one depending on [MATH] and [MATH].', '0802.0971-1-27-0': 'The integration over the Coulomb-peak factor is unchanged and yields the factor [MATH] of Eq.([REF]).', '0802.0971-1-27-1': 'As the Compton energy [MATH] increases and we move closer to the end-point of the [MATH] distribution, the minimum-momentum transfer, Eq.([REF]), increases.', '0802.0971-1-27-2': 'The maximal value is obtained at the end-point itself, which is located at [MATH], and where [MATH].', '0802.0971-1-27-3': 'At a Compton mass equals the rho mass and an energy [MATH] GeV this means [MATH] MeV[MATH].', '0802.0971-1-27-4': 'Away from [MATH] the value is much smaller.', '0802.0971-1-28-0': 'The angular integration of the right hand side of Eq.([REF]) is also unchanged.', '0802.0971-1-28-1': 'However, it is convenient to replace the variable [MATH] by [MATH], variables which are related by Eq.([REF]).', '0802.0971-1-28-2': 'The cross-section distribution obtained is [EQUATION]', '0802.0971-1-28-3': 'The distribution function [MATH] is given by the expresssion [EQUATION] with [MATH] related to [MATH] via Eq. ([REF]) and [EQUATION] with the obvious restriction [MATH].', '0802.0971-1-29-0': 'It is instructive to look at the structure functions as functions of [MATH] and [MATH].', '0802.0971-1-29-1': 'From Eqs ([REF]) and ([REF]) we derive the expressions [EQUATION]', '0802.0971-1-29-2': 'In the Born approximation [MATH] and [MATH], as stated in Eqs. ([REF]) and ([REF]).', '0802.0971-1-29-3': 'The Born contribution, which is by far the dominant contribution, is in the cross-section distribution ([REF]) multiplied by the polynomial factor [EQUATION] with [MATH].', '0802.0971-1-29-4': 'Since [MATH] it follows that [EQUATION]', '0802.0971-1-29-5': 'For a fixed value of [MATH], the maximal value of the pre-factor, [MATH], is obtained at the boundary points [MATH] and [MATH].', '0802.0971-1-29-6': 'The minimum value, [MATH], is obtained at [MATH].', '0802.0971-1-29-7': 'Thus, for large values of [MATH] there is rapid variation in the kinematical factor [MATH] near the upper end-point of the [MATH] region.', '0802.0971-1-29-8': 'In the remaining part of phase-space [MATH] is smoothly varying and near to one, since there the variable [MATH] itself will be quite small.', '0802.0971-1-29-9': 'This is all illustrated in Fig.5.', '0802.0971-1-30-0': 'The pion structure functions [MATH] and [MATH] depend on the Mandelstam variables [MATH], [MATH], and [MATH], of the pion-Compton scattering, or equivalently [MATH] and [MATH].', '0802.0971-1-30-1': 'In the region we are considering the Born terms dominate and the polarizability terms mainly come in through their interference with the Born terms.', '0802.0971-1-30-2': 'In Fig.6a we plot the quantity [MATH].', '0802.0971-1-30-3': 'From Eq. ([REF]) it is clear that this is a measure of the strength of the [MATH] term.', '0802.0971-1-30-4': 'The graph shows that the [MATH] term is small, except for [MATH]-values near [MATH], that is grows with [MATH] and exhibits a structure related to the [MATH]- and [MATH]-meson exchanges.', '0802.0971-1-31-0': 'In Fig.6b we plot the quantity [MATH] which measures the strength of the [MATH] term.', '0802.0971-1-31-1': 'The graph demonstrates that [MATH] is important in a region of phase space, essentially disjoint from that of [MATH], namely the region near the upper boundary line of [MATH]-values.', '0802.0971-1-31-2': 'Its strength grows with increasing value of [MATH].', '0802.0971-1-31-3': 'The structure related to the [MATH]- and [MATH]-meson exchanges is visible but less pronounced.', '0802.0971-1-31-4': 'It is [MATH] that contains the [MATH]-meson exchange term.', '0802.0971-1-31-5': 'According to Eq. ([REF]) the interference between [MATH] and the Born term has an extra factor [MATH].', '0802.0971-1-31-6': 'This factor results in a sharper structure in the interference term than seen in [MATH] alone.', '0802.0971-1-32-0': 'Fig.7 illustrates the behaviour of the dynamic factor [MATH] of the integrand ([REF]).', '0802.0971-1-32-1': 'The plotted quantity is the ratio [MATH], i.e. the ratio of [MATH] to its Born approximation [MATH].', '0802.0971-1-32-2': 'We notice that in the overwhelming part of phase space the Born approximation is reasonably accurate.', '0802.0971-1-32-3': 'The exceptions are at small [MATH]-values where the polarizability function [MATH] causes substantial deviations from unity, and at [MATH]-values near the upper-boundary line where the polarizability function [MATH] contributes to the structure.', '0802.0971-1-33-0': 'Unfortunately, in the latter region precise measurements are complicated by the sharp structure in the distribution function [MATH].', '0802.0971-1-34-0': 'In Fig.8a the distribution function [MATH] is graphed as a function of [MATH] for fixed [MATH].', '0802.0971-1-34-1': 'The dependence on the pion-polarizability functions is strong, with a first dip at the [MATH]-meson mass.', '0802.0971-1-34-2': 'As pointed out above this structure is essentially caused by the [MATH] contribution.', '0802.0971-1-34-3': 'This is seen already in Eqs. ([REF]) and ([REF]) where the [MATH] contribution is multiplied by the factor [MATH] and the [MATH] contribution by the factor [MATH].', '0802.0971-1-35-0': 'In Fig.8b the distribution function [MATH] is instead graphed as a function of [MATH] for fixed [MATH].', '0802.0971-1-35-1': 'This Compton mass is right on top of the [MATH]-meson mass.', '0802.0971-1-35-2': 'Again we observe that the deviation from the Born contribution is larger for small [MATH]-values than for large [MATH]-values.', '0802.0971-1-35-3': 'Near the upper end-point there is a narrow dip, a structure mainly caused be the [MATH]-dependence.', '0802.0971-1-36-0': '# Summary', '0802.0971-1-37-0': 'Hard bremsstrahlung in high-energy pion-nucleus collisions in the Coulomb region is induced by single photon excitation.', '0802.0971-1-37-1': 'Therefore, this reaction can be used to extract information about pion-Compton scattering.', '0802.0971-1-38-0': 'The pion-Compton amplitude is dominated by Born terms that give the exact amplitude for point-like pions.', '0802.0971-1-38-1': 'The non-pointlike structure is described by two independent polarizability terms.', '0802.0971-1-38-2': 'We describe them in a meson-exchange model with [MATH], [MATH], and [MATH], mesons.', '0802.0971-1-38-3': 'The polarizability contributions are small but are important to study.', '0802.0971-1-38-4': 'At threshold in the pion-Compton system [MATH] is predicted to vanish and [MATH] predicted to be small.', '0802.0971-1-38-5': 'We investigate, through the pion-bremsstrahlung mechanism, the polarizability functions for pion-Compton c.m. energies [MATH] below 1 GeV.', '0802.0971-1-39-0': 'In that part of phase space where most of the final-state energy is carried by the photon ([MATH]), the contribution from [MATH] is negligible.', '0802.0971-1-39-1': 'The contribution from [MATH] is small but its strength increases with increasing [MATH].', '0802.0971-1-39-2': 'Some structure caused by the [MATH]-and [MATH]-meson exchanges can be seen.', '0802.0971-1-39-3': 'At the pion-Compton threshold the contribution from the [MATH]-exchange term dominates.', '0802.0971-1-39-4': 'Experimental efforts to measure the polarizability contributions are at the moment concentrated to this region.', '0802.0971-1-40-0': 'In that part of phase space where most of the final-state energy is carried by the pion ([MATH]), the contribution from [MATH] is negligible.', '0802.0971-1-40-1': 'The contribution from [MATH] is clearly seen, and exhibits a strong variation due to the [MATH]-and [MATH]-meson exchanges.', '0802.0971-1-40-2': 'There is no contribution from [MATH] exchange in [MATH].', '0802.0971-1-41-0': 'Our analysis is valid at high energies when the transverse momenta are small compared with the longitudinal momenta.', '0802.0971-1-41-1': 'In addition the momentum transfer to the nucleus must be in the Coulomb region, i.e. very small.'}

{'0802.0971-2-0-0': 'The pion-polarizability functions are structure functions of pion-Compton scattering.', '0802.0971-2-0-1': 'They can be assessed in high-energy pion-nucleus bremsstrahlung reactions, [MATH].', '0802.0971-2-0-2': 'We present numerical expectations for pion-nucleus bremsstrahlung cross sections in the Coulomb region, i.e. the small-angle region where the nuclear scattering is dominated by the Coulomb interaction.', '0802.0971-2-0-3': 'We investigate the prospects of measuring the polarizability functions for pion-Compton c.m. energies from threshold up to 1 GeV.', '0802.0971-2-0-4': 'A meson-exchange model is used for the pion-Compton amplitude.', '0802.0971-2-1-0': '# Introduction', '0802.0971-2-2-0': 'The cross-section distribution for high-energy pionic bremsstrahlung [EQUATION] is at small-momentum transfers to the nucleus dominated by the one-photon-exchange contribution.', '0802.0971-2-2-1': 'Hence, by measuring cross-section distributions for pion-nucleus bremsstrahlung at small momentum transfers one gains at the same time information about the cross-section distribution for pion-Compton scattering [EQUATION] the photon of the initial state being the virtual photon exchanged between pion and nucleus.', '0802.0971-2-2-2': 'In particular, it becomes possible to determine pion polarizabilities.', '0802.0971-2-3-0': "The theoretical underpinnings of the pion-bremsstrahlung reaction were investigated by Gal'perin et al. [CITATION], and an experiment was susequently performed by Antipov et al. [CITATION], which showed that pion polarizabilities could indeed be determined.", '0802.0971-2-3-1': 'The COMPASS experiment at CERN [CITATION] is a refinement of the Antipov experiment.', '0802.0971-2-4-0': 'In ref. [CITATION] a mathematical expression for the bremsstrahlung cross section was derived assuming a pion-Compton amplitude that in additon to the Born contributions contained also contributions from the [MATH], [MATH], and [MATH], exchanges.', '0802.0971-2-4-1': 'The kinematics of the nuclear reaction was defined through [EQUATION] and the kinematics of the related pion-Compton reaction through [EQUATION] with [MATH].', '0802.0971-2-4-2': 'In terms of these kinematic variables the cross-section distribution can be written as [CITATION] [EQUATION]', '0802.0971-2-4-3': 'This expression is valid when the transverse momenta of the emerging pion and photon are much smaller than their longitudinal momenta.', '0802.0971-2-4-4': 'The parameter [MATH] is the fine-structure constant, [MATH] the pion mass, [MATH] the ratio [EQUATION] [MATH], and [MATH] the minimum-longitudinal-momentum transfer [EQUATION]', '0802.0971-2-4-5': 'At high energies [MATH] is extremely small.', '0802.0971-2-5-0': 'The functions [MATH] and [MATH] characterize the pion-Compton amplitude.', '0802.0971-2-5-1': 'In the Born approximation [EQUATION] and generally [EQUATION] with [MATH] and [MATH] generalized-polarizability functions.', '0802.0971-2-5-2': 'Their mathematical expressions, in the meson-exchange model, are given in [CITATION].', '0802.0971-2-6-0': 'The structure of the cross-section distribution is mainly determined by the inelastic-Coulomb factor [EQUATION] which vanishes when the transverse-momentum transfer [MATH]q[MATH] to the nucleus vanishes.', '0802.0971-2-6-1': 'This is the classic Primakoff factor, which exhibits a maximum at [MATH].', '0802.0971-2-6-2': 'When the momentum transfer [MATH] becomes much larger than [MATH] the Coulomb contribution to the bremsstrahlung cross section becomes small and the hadronic contribution dominates [CITATION].', '0802.0971-2-6-3': 'This contribution is not considered in the present paper.', '0802.0971-2-7-0': 'The Mandelstam kinematic variables of the pion-Compton scattering ([REF]) can in our application be approximated as follows; [EQUATION]', '0802.0971-2-8-0': '# Kinematical considerations', '0802.0971-2-9-0': 'We are interested in studying the pion-Compton-scattering process.', '0802.0971-2-9-1': 'In order to do so in a transparent way we need to make specific cuts in the pion-nucleus-bremsstrahlung phase space.', '0802.0971-2-9-2': 'To this end a few kinematical observations might be helpful.', '0802.0971-2-10-0': 'Let us first introduce, instead of the pion-Compton c.m. energy and the photon-transverse momentum, the corresponding dimensionless variables [EQUATION] with [MATH] and [MATH].', '0802.0971-2-10-1': 'The first entry of Eq. ([REF]), defining the pion-Compton c.m. energy squared, can then be reformulated as [EQUATION]', '0802.0971-2-10-2': 'Since [MATH] is positive we conclude that for a given fixed value of [MATH] the variable [MATH] must be in the interval [EQUATION]', '0802.0971-2-10-3': 'The maximal value of [MATH] occurs at the midpoint of this interval, [EQUATION]', '0802.0971-2-10-4': 'We may also rewrite Eq.([REF]) as [EQUATION] showing directly that the photon-transverse momentum vanishes at the kinematic end-points [MATH] and [MATH].', '0802.0971-2-11-0': 'For a fixed value of [MATH], on the other hand, it follows from the positivity of [MATH] of Eq.( [REF]) that the pion-Compton c.m. energy is bounded from below [EQUATION]', '0802.0971-2-11-1': 'The boundary value obtains in the forward direction, where the photon-transverse momentum vanishes.', '0802.0971-2-12-0': 'In Fig.1 the photon-transverse momentum [MATH] is plotted for a few values of the pion-Compton c.m. energy [MATH].', '0802.0971-2-12-1': 'As an illustration consider [MATH], a mass at which the threshold approximation to the polarizability functions should be acceptable.', '0802.0971-2-12-2': 'In this case the maximal value of [MATH] is [MATH] and the maximal value of the photon-transverse momentum [MATH] is [MATH].', '0802.0971-2-13-0': '# Low-mass-Compton region', '0802.0971-2-14-0': 'Hard-pion bremsstrahlung offers a unique possibility of determining pion-threshold polarizabilities.', '0802.0971-2-14-1': 'However, this will succeed only if we restrain the c.m. mass in the pion-photon system to masses well below the [MATH]-meson mass, so as to avoid the [MATH]-channel [MATH]-exchange contribution to the Compton amplitude.', '0802.0971-2-15-0': 'Hence, we first concentrate our efforts on the region [MATH], where we can confidently approximate the structure functions [MATH] and [MATH] of Eqs. ([REF]) and ([REF]) by their near-threshold values, with [EQUATION] and [MATH] and [MATH] the standard pion-electric and pion-magnetic polarizabilities.', '0802.0971-2-15-1': 'These values given are obtained in the meson-exchange model when fixing the [MATH]-meson coupling so that the chiral Lagrangian value [CITATION] of [MATH] is reproduced.', '0802.0971-2-15-2': 'Our present knowledge of the pion polarizabilities is reviewed in [CITATION].', '0802.0971-2-16-0': 'The cross-section distribution ([REF]) contains angular dependent terms, through [MATH].', '0802.0971-2-16-1': 'Measured variables are however [MATH] and [MATH], but since, [MATH], with [MATH], the angular variations of [MATH] and [MATH] are independent.', '0802.0971-2-16-2': 'Thus, the integrations over the angles of [MATH] and [MATH] in Eq. ([REF]) are unrestricted and we may replace [MATH] by its angular average, which is a half, and at the same make the replacement [EQUATION]', '0802.0971-2-16-3': 'After integration over angles the cross-section distribution factorizes into factors that depend either on [MATH] and [MATH], or on [MATH] and [MATH].', '0802.0971-2-16-4': 'The integration over the inelastic-Coulomb factor ([REF]) is an integration over [MATH], but there is an [MATH] dependence in [MATH], [EQUATION]', '0802.0971-2-16-5': 'It is tempting to integrate over all possible momentum transfers [MATH], but that is not possible since the integral diverges.', '0802.0971-2-16-6': 'The reason is that we have neglected pion and nuclear form factors.', '0802.0971-2-16-7': 'Also, for large values of [MATH] the hadronic-bremsstrahlung contribution dominates the Coulomb contribution.', '0802.0971-2-16-8': 'Consequently, we must make a cut so as to avoid the hadronic contribution.', '0802.0971-2-16-9': 'The best approach is to make a cut in the same way for all [MATH] values.', '0802.0971-2-16-10': 'This is achieved by cutting off the integral at [MATH] on the low-momentum side and at [MATH] on the high-momentum side of the Coulomb peak.', '0802.0971-2-16-11': 'From ([REF]) we derive [EQUATION]', '0802.0971-2-16-12': 'This integral is obviously independent of [MATH].', '0802.0971-2-16-13': 'The precise values of [MATH] and [MATH] to be chosen can only be decided after inspection of the experimental distribution.', '0802.0971-2-17-0': 'Next we integrate the cross-section distribution Eq.([REF]) over transverse-photon momenta in the domain [MATH].', '0802.0971-2-17-1': 'Since the polarizability functions [MATH] and [MATH] entering the structure functions [MATH] and [MATH] of Eqs.([REF]) and ([REF]) are now constants the integration is straightforward, and we get [EQUATION] with [EQUATION] and the distribution function [EQUATION]', '0802.0971-2-17-2': 'The [MATH]-dependent prefactor in Eq.([REF]) is simply the ratio of the final state momenta, [MATH], and the singularity at [MATH] is the well-known soft-photon-radiation singularity.', '0802.0971-2-18-0': 'The functions [MATH], [MATH], and [MATH], of Eq.([REF]) are all elementary and given in the Appendix.', '0802.0971-2-18-1': 'They are graphed in Fig.2a.', '0802.0971-2-18-2': 'For small values of the argument [MATH], or transverse momenta such that [MATH], we have [EQUATION]', '0802.0971-2-18-3': 'For large values of the argument [MATH], or transverse momenta such that [MATH], we have instead [EQUATION]', '0802.0971-2-18-4': 'In particular we note for large values of [MATH] the ratio [MATH].', '0802.0971-2-19-0': 'An alternative to integrating over transverse-photon momenta is to integrate over Compton masses [MATH] in the domain [MATH].', '0802.0971-2-19-1': 'This is simply achieved by making the integration limit [MATH] depend on [MATH].', '0802.0971-2-19-2': 'According to Eq.([REF]), or ([REF]), the relation between [MATH] and [MATH] allows us to write [EQUATION] with [EQUATION]', '0802.0971-2-19-3': 'At the same time as being the definition of the variable [MATH], Eq.([REF]) displays the functional relation between [MATH] and [MATH].', '0802.0971-2-20-0': 'We now specialize to [MATH] which is the cut for the threshold region employed by the COMPASS collaboration.', '0802.0971-2-20-1': 'This implies, via Eq. ([REF]), a maximal [MATH]-value of [MATH].', '0802.0971-2-20-2': 'In Fig.2b the three distribution functions [MATH], [MATH], and [MATH] are plotted as functions of [MATH] for this value of [MATH].', '0802.0971-2-20-3': 'From this figure we draw the following conclusions.', '0802.0971-2-21-0': 'Near the upper end of the [MATH]-distribution the three functions [MATH], [MATH], and [MATH], may be approximated by their threshold values, Eq.([REF]).', '0802.0971-2-21-1': 'This is a consequence of Eq.([REF]), which demonstrates that even though the factor [MATH] is large, when [MATH] is sufficiently close to [MATH] the variable [MATH] will be small.', '0802.0971-2-21-2': 'This observation leads to a simplification of the cross-section-distribution function [MATH] of Eq.([REF]), [EQUATION]', '0802.0971-2-21-3': 'Numerically, we expect [MATH] and small, but enters multiplied by a large factor, [MATH], and therefore makes an important contribution to the cross-section distribution.', '0802.0971-2-22-0': 'In the mid-range [MATH]-region the value of [MATH] is large, due to the large value of [MATH], and we may here approximate the functions [MATH] and [MATH] by their asymptotic expressions, Eq. ([REF]).', '0802.0971-2-22-1': 'This allows us to simplify the expression for the function [MATH], yielding the approximate formula [EQUATION]', '0802.0971-2-22-2': 'The function [MATH], representing the Born approximation, must be evaluated exactly as we are looking for small deviations from it.', '0802.0971-2-22-3': 'The numerical value of the coefficient multiplying the middle [MATH] term in Eq.( [REF]) is large.', '0802.0971-2-22-4': 'Nevertheless, this term has essentially no effect on the cross section since according to Eq.([REF]) [MATH] is proportional to the sum [MATH] which vanishes, or is very small compared with [MATH].', '0802.0971-2-22-5': 'The important term is the [MATH] term.', '0802.0971-2-22-6': 'Over most of the [MATH]-range its strength is fixed by Eq. ([REF]).', '0802.0971-2-22-7': 'But this approximation cannot be used near the upper end-point since it would there overestimate the polarizability contribution by a factor of 3/2, as a comparison with expression ([REF]) would demonstrate.', '0802.0971-2-23-0': 'In Fig.3a the cross-section-distribution function [MATH] of Eq. ([REF]) is graphed in the full calculation and in the Born approximation, i.e., when polarizability parameters [MATH].', '0802.0971-2-23-1': 'In Fig.3b the ratio of the two distributions is plotted.', '0802.0971-2-23-2': 'Clearly, the polarizabilities are important only in a region near the end-point of the [MATH]- distibution, and therefore the experimental efforts are concentrated on this region [CITATION].', '0802.0971-2-24-0': 'In the COMPASS experiment the apparatus is blind in a small region around the forward direction.', '0802.0971-2-24-1': 'The transverse momenta of the final state pions and photons must be larger than [MATH] to be detected.', '0802.0971-2-24-2': 'As a result the distribution function [MATH] of Eq. ([REF]) must be replaced by [EQUATION] where from Eq. ([REF]) [EQUATION]', '0802.0971-2-24-3': 'In the COMPASS experiment [MATH] MeV/[MATH] and [MATH] is small except when [MATH] is near [MATH].', '0802.0971-2-24-4': 'The subtraction procedure of Eq. ([REF]) works unless there are momentum transfers [MATH] for which [MATH].', '0802.0971-2-24-5': 'This will not happen as long as [MATH] remains in the domain [EQUATION] which is the case in the COMPASS application.', '0802.0971-2-24-6': 'The effect of the cut on the cross section is illustrated in Fig.4.', '0802.0971-2-25-0': '# Rho-mass-Compton region', '0802.0971-2-26-0': 'At large Compton masses the polarizability functions are complex and depend in a complicated manner on the varables [MATH] and [MATH].', '0802.0971-2-26-1': 'As before, after angular integration, the cross-section distribution factorizes into one factor dependening on [MATH] and [MATH], and another one depending on [MATH] and [MATH].', '0802.0971-2-27-0': 'The integration over the Coulomb-peak factor is unchanged and yields the factor [MATH] of Eq.([REF]).', '0802.0971-2-27-1': 'As the Compton energy [MATH] increases and we move closer to the end-point of the [MATH] distribution, the minimum-momentum transfer, Eq.([REF]), increases.', '0802.0971-2-27-2': 'The maximal value is obtained at the end-point itself, which is located at [MATH], and where [MATH].', '0802.0971-2-27-3': 'At a Compton mass equals the rho mass and an energy [MATH] GeV this means [MATH] MeV[MATH].', '0802.0971-2-27-4': 'Away from [MATH] the value is much smaller.', '0802.0971-2-28-0': 'The angular integration of the right hand side of Eq.([REF]) is also unchanged.', '0802.0971-2-28-1': 'However, it is convenient to replace the variable [MATH] by [MATH], variables which are related by Eq.([REF]).', '0802.0971-2-28-2': 'The cross-section distribution obtained is [EQUATION]', '0802.0971-2-28-3': 'The distribution function [MATH] is given by the expresssion [EQUATION] with [MATH] related to [MATH] via Eq. ([REF]) and [EQUATION] with the obvious restriction [MATH].', '0802.0971-2-29-0': 'It is instructive to look at the structure functions as functions of [MATH] and [MATH].', '0802.0971-2-29-1': 'From Eqs ([REF]) and ([REF]) we derive the expressions [EQUATION]', '0802.0971-2-29-2': 'In the Born approximation [MATH] and [MATH], as stated in Eqs. ([REF]) and ([REF]).', '0802.0971-2-29-3': 'The Born contribution, which is by far the dominant contribution, is in the cross-section distribution ([REF]) multiplied by the polynomial factor [EQUATION] with [MATH].', '0802.0971-2-29-4': 'Since [MATH] it follows that [EQUATION]', '0802.0971-2-29-5': 'For a fixed value of [MATH], the maximal value of the pre-factor, [MATH], is obtained at the boundary points [MATH] and [MATH].', '0802.0971-2-29-6': 'The minimum value, [MATH], is obtained at [MATH].', '0802.0971-2-29-7': 'Thus, for large values of [MATH] there is rapid variation in the kinematical factor [MATH] near the upper end-point of the [MATH] region.', '0802.0971-2-29-8': 'In the remaining part of phase-space [MATH] is smoothly varying and near to one, since there the variable [MATH] itself will be quite small.', '0802.0971-2-29-9': 'This is all illustrated in Fig.5.', '0802.0971-2-30-0': 'The pion structure functions [MATH] and [MATH] depend on the Mandelstam variables [MATH], [MATH], and [MATH], of the pion-Compton scattering, or equivalently [MATH] and [MATH].', '0802.0971-2-30-1': 'In the region we are considering the Born terms dominate and the polarizability terms mainly come in through their interference with the Born terms.', '0802.0971-2-30-2': 'In Fig.6a we plot the quantity [MATH].', '0802.0971-2-30-3': 'From Eq. ([REF]) it is clear that this is a measure of the strength of the [MATH] term.', '0802.0971-2-30-4': 'The graph shows that the [MATH] term is small, except for [MATH]-values near [MATH], that is grows with [MATH] and exhibits a structure related to the [MATH]- and [MATH]-meson exchanges.', '0802.0971-2-31-0': 'In Fig.6b we plot the quantity [MATH] which measures the strength of the [MATH] term.', '0802.0971-2-31-1': 'The graph demonstrates that [MATH] is important in a region of phase space, essentially disjoint from that of [MATH], namely the region near the upper boundary line of [MATH]-values.', '0802.0971-2-31-2': 'Its strength grows with increasing value of [MATH].', '0802.0971-2-31-3': 'The structure related to the [MATH]- and [MATH]-meson exchanges is visible but less pronounced.', '0802.0971-2-31-4': 'It is [MATH] that contains the [MATH]-meson exchange term.', '0802.0971-2-31-5': 'According to Eq. ([REF]) the interference between [MATH] and the Born term has an extra factor [MATH].', '0802.0971-2-31-6': 'This factor results in a sharper structure in the interference term than seen in [MATH] alone.', '0802.0971-2-32-0': 'Fig.7 illustrates the behaviour of the dynamic factor [MATH] of the integrand ([REF]).', '0802.0971-2-32-1': 'The plotted quantity is the ratio [MATH], i.e. the ratio of [MATH] to its Born approximation [MATH].', '0802.0971-2-32-2': 'We notice that in the overwhelming part of phase space the Born approximation is reasonably accurate.', '0802.0971-2-32-3': 'The exceptions are at small [MATH]-values where the polarizability function [MATH] causes substantial deviations from unity, and at [MATH]-values near the upper-boundary line where the polarizability function [MATH] contributes to the structure.', '0802.0971-2-33-0': 'Unfortunately, in the latter region precise measurements are complicated by the sharp structure in the distribution function [MATH].', '0802.0971-2-34-0': 'In Fig.8a the distribution function [MATH] is graphed as a function of [MATH] for fixed [MATH].', '0802.0971-2-34-1': 'The dependence on the pion-polarizability functions is strong, with a first dip at the [MATH]-meson mass.', '0802.0971-2-34-2': 'As pointed out above this structure is essentially caused by the [MATH] contribution.', '0802.0971-2-34-3': 'This is seen already in Eqs. ([REF]) and ([REF]) where the [MATH] contribution is multiplied by the factor [MATH] and the [MATH] contribution by the factor [MATH].', '0802.0971-2-35-0': 'In Fig.8b the distribution function [MATH] is instead graphed as a function of [MATH] for fixed [MATH].', '0802.0971-2-35-1': 'This Compton mass is right on top of the [MATH]-meson mass.', '0802.0971-2-35-2': 'Again we observe that the deviation from the Born contribution is larger for small [MATH]-values than for large [MATH]-values.', '0802.0971-2-35-3': 'Near the upper end-point there is a narrow dip, a structure mainly caused be the [MATH]-dependence.', '0802.0971-2-36-0': '# Summary', '0802.0971-2-37-0': 'Hard bremsstrahlung in high-energy pion-nucleus collisions in the Coulomb region is induced by single photon excitation.', '0802.0971-2-37-1': 'Therefore, this reaction can be used to extract information about pion-Compton scattering.', '0802.0971-2-38-0': 'The pion-Compton amplitude is dominated by Born terms that give the exact amplitude for point-like pions.', '0802.0971-2-38-1': 'The non-pointlike structure is described by two independent polarizability terms.', '0802.0971-2-38-2': 'We describe them in a meson-exchange model with [MATH], [MATH], and [MATH], mesons.', '0802.0971-2-38-3': 'The polarizability contributions are small but are important to study.', '0802.0971-2-38-4': 'At threshold in the pion-Compton system [MATH] is predicted to vanish and [MATH] predicted to be small.', '0802.0971-2-38-5': 'We investigate, through the pion-bremsstrahlung mechanism, the polarizability functions for pion-Compton c.m. energies [MATH] below 1 GeV.', '0802.0971-2-39-0': 'In that part of phase space where most of the final-state energy is carried by the photon ([MATH]), the contribution from [MATH] is negligible.', '0802.0971-2-39-1': 'The contribution from [MATH] is small but its strength increases with increasing [MATH].', '0802.0971-2-39-2': 'Some structure caused by the [MATH]-and [MATH]-meson exchanges can be seen.', '0802.0971-2-39-3': 'At the pion-Compton threshold the contribution from the [MATH]-exchange term dominates.', '0802.0971-2-39-4': 'Experimental efforts to measure the polarizability contributions are at the moment concentrated to this region.', '0802.0971-2-40-0': 'In that part of phase space where most of the final-state energy is carried by the pion ([MATH]), the contribution from [MATH] is negligible.', '0802.0971-2-40-1': 'The contribution from [MATH] is clearly seen, and exhibits a strong variation due to the [MATH]-and [MATH]-meson exchanges.', '0802.0971-2-40-2': 'There is no contribution from [MATH] exchange in [MATH].', '0802.0971-2-41-0': 'Our analysis is valid at high energies when the transverse momenta are small compared with the longitudinal momenta.', '0802.0971-2-41-1': 'In addition the momentum transfer to the nucleus must be in the Coulomb region, i.e. very small.'}

[['0802.0971-1-29-0', '0802.0971-2-29-0'], ['0802.0971-1-29-1', '0802.0971-2-29-1'], ['0802.0971-1-29-2', '0802.0971-2-29-2'], ['0802.0971-1-29-3', '0802.0971-2-29-3'], ['0802.0971-1-29-4', '0802.0971-2-29-4'], ['0802.0971-1-29-5', '0802.0971-2-29-5'], ['0802.0971-1-29-6', '0802.0971-2-29-6'], ['0802.0971-1-29-7', '0802.0971-2-29-7'], ['0802.0971-1-29-8', '0802.0971-2-29-8'], ['0802.0971-1-29-9', '0802.0971-2-29-9'], ['0802.0971-1-5-0', '0802.0971-2-5-0'], ['0802.0971-1-5-1', '0802.0971-2-5-1'], ['0802.0971-1-5-2', '0802.0971-2-5-2'], ['0802.0971-1-28-0', '0802.0971-2-28-0'], ['0802.0971-1-28-1', '0802.0971-2-28-1'], ['0802.0971-1-28-2', '0802.0971-2-28-2'], ['0802.0971-1-28-3', '0802.0971-2-28-3'], ['0802.0971-1-39-0', '0802.0971-2-39-0'], ['0802.0971-1-39-1', '0802.0971-2-39-1'], ['0802.0971-1-39-2', '0802.0971-2-39-2'], ['0802.0971-1-39-3', '0802.0971-2-39-3'], ['0802.0971-1-39-4', '0802.0971-2-39-4'], ['0802.0971-1-6-0', '0802.0971-2-6-0'], ['0802.0971-1-6-1', '0802.0971-2-6-1'], ['0802.0971-1-6-2', '0802.0971-2-6-2'], ['0802.0971-1-6-3', '0802.0971-2-6-3'], ['0802.0971-1-22-0', '0802.0971-2-22-0'], ['0802.0971-1-22-1', '0802.0971-2-22-1'], ['0802.0971-1-22-2', '0802.0971-2-22-2'], ['0802.0971-1-22-3', '0802.0971-2-22-3'], ['0802.0971-1-22-4', '0802.0971-2-22-4'], ['0802.0971-1-22-5', '0802.0971-2-22-5'], ['0802.0971-1-22-6', '0802.0971-2-22-6'], ['0802.0971-1-22-7', '0802.0971-2-22-7'], ['0802.0971-1-40-0', '0802.0971-2-40-0'], ['0802.0971-1-40-1', '0802.0971-2-40-1'], ['0802.0971-1-40-2', '0802.0971-2-40-2'], ['0802.0971-1-3-0', '0802.0971-2-3-0'], ['0802.0971-1-3-1', '0802.0971-2-3-1'], ['0802.0971-1-35-0', '0802.0971-2-35-0'], ['0802.0971-1-35-1', '0802.0971-2-35-1'], ['0802.0971-1-35-2', '0802.0971-2-35-2'], ['0802.0971-1-35-3', '0802.0971-2-35-3'], ['0802.0971-1-31-0', '0802.0971-2-31-0'], ['0802.0971-1-31-1', '0802.0971-2-31-1'], ['0802.0971-1-31-2', '0802.0971-2-31-2'], ['0802.0971-1-31-3', '0802.0971-2-31-3'], ['0802.0971-1-31-4', '0802.0971-2-31-4'], ['0802.0971-1-31-5', '0802.0971-2-31-5'], ['0802.0971-1-31-6', '0802.0971-2-31-6'], ['0802.0971-1-26-0', '0802.0971-2-26-0'], ['0802.0971-1-26-1', '0802.0971-2-26-1'], ['0802.0971-1-27-0', '0802.0971-2-27-0'], ['0802.0971-1-27-1', '0802.0971-2-27-1'], ['0802.0971-1-27-2', '0802.0971-2-27-2'], ['0802.0971-1-27-3', '0802.0971-2-27-3'], ['0802.0971-1-27-4', '0802.0971-2-27-4'], ['0802.0971-1-2-0', '0802.0971-2-2-0'], ['0802.0971-1-2-1', '0802.0971-2-2-1'], ['0802.0971-1-2-2', '0802.0971-2-2-2'], ['0802.0971-1-20-0', '0802.0971-2-20-0'], ['0802.0971-1-20-1', '0802.0971-2-20-1'], ['0802.0971-1-20-2', '0802.0971-2-20-2'], ['0802.0971-1-20-3', '0802.0971-2-20-3'], ['0802.0971-1-18-0', '0802.0971-2-18-0'], ['0802.0971-1-18-1', '0802.0971-2-18-1'], ['0802.0971-1-18-2', '0802.0971-2-18-2'], ['0802.0971-1-18-3', '0802.0971-2-18-3'], ['0802.0971-1-18-4', '0802.0971-2-18-4'], ['0802.0971-1-21-0', '0802.0971-2-21-0'], ['0802.0971-1-21-1', '0802.0971-2-21-1'], ['0802.0971-1-21-2', '0802.0971-2-21-2'], ['0802.0971-1-21-3', '0802.0971-2-21-3'], ['0802.0971-1-24-0', '0802.0971-2-24-0'], ['0802.0971-1-24-1', '0802.0971-2-24-1'], ['0802.0971-1-24-2', '0802.0971-2-24-2'], ['0802.0971-1-24-3', '0802.0971-2-24-3'], ['0802.0971-1-24-4', '0802.0971-2-24-4'], ['0802.0971-1-24-5', '0802.0971-2-24-5'], ['0802.0971-1-24-6', '0802.0971-2-24-6'], ['0802.0971-1-9-0', '0802.0971-2-9-0'], ['0802.0971-1-9-1', '0802.0971-2-9-1'], ['0802.0971-1-9-2', '0802.0971-2-9-2'], ['0802.0971-1-11-0', '0802.0971-2-11-0'], ['0802.0971-1-11-1', '0802.0971-2-11-1'], ['0802.0971-1-30-0', '0802.0971-2-30-0'], ['0802.0971-1-30-1', '0802.0971-2-30-1'], ['0802.0971-1-30-2', '0802.0971-2-30-2'], ['0802.0971-1-30-3', '0802.0971-2-30-3'], ['0802.0971-1-30-4', '0802.0971-2-30-4'], ['0802.0971-1-0-0', '0802.0971-2-0-0'], ['0802.0971-1-0-1', '0802.0971-2-0-1'], ['0802.0971-1-0-2', '0802.0971-2-0-2'], ['0802.0971-1-0-3', '0802.0971-2-0-3'], ['0802.0971-1-0-4', '0802.0971-2-0-4'], ['0802.0971-1-19-0', '0802.0971-2-19-0'], ['0802.0971-1-19-1', '0802.0971-2-19-1'], ['0802.0971-1-19-2', '0802.0971-2-19-2'], ['0802.0971-1-19-3', '0802.0971-2-19-3'], ['0802.0971-1-41-0', '0802.0971-2-41-0'], ['0802.0971-1-41-1', '0802.0971-2-41-1'], ['0802.0971-1-34-0', '0802.0971-2-34-0'], ['0802.0971-1-34-1', '0802.0971-2-34-1'], ['0802.0971-1-34-2', '0802.0971-2-34-2'], ['0802.0971-1-34-3', '0802.0971-2-34-3'], ['0802.0971-1-12-0', '0802.0971-2-12-0'], ['0802.0971-1-12-1', '0802.0971-2-12-1'], ['0802.0971-1-12-2', '0802.0971-2-12-2'], ['0802.0971-1-7-0', '0802.0971-2-7-0'], ['0802.0971-1-15-0', '0802.0971-2-15-0'], ['0802.0971-1-15-1', '0802.0971-2-15-1'], ['0802.0971-1-15-2', '0802.0971-2-15-2'], ['0802.0971-1-37-0', '0802.0971-2-37-0'], ['0802.0971-1-37-1', '0802.0971-2-37-1'], ['0802.0971-1-17-0', '0802.0971-2-17-0'], ['0802.0971-1-17-1', '0802.0971-2-17-1'], ['0802.0971-1-17-2', '0802.0971-2-17-2'], ['0802.0971-1-4-0', '0802.0971-2-4-0'], ['0802.0971-1-4-1', '0802.0971-2-4-1'], ['0802.0971-1-4-2', '0802.0971-2-4-2'], ['0802.0971-1-4-3', '0802.0971-2-4-3'], ['0802.0971-1-4-4', '0802.0971-2-4-4'], ['0802.0971-1-4-5', '0802.0971-2-4-5'], ['0802.0971-1-32-0', '0802.0971-2-32-0'], ['0802.0971-1-32-1', '0802.0971-2-32-1'], ['0802.0971-1-32-2', '0802.0971-2-32-2'], ['0802.0971-1-32-3', '0802.0971-2-32-3'], ['0802.0971-1-14-0', '0802.0971-2-14-0'], ['0802.0971-1-14-1', '0802.0971-2-14-1'], ['0802.0971-1-10-0', '0802.0971-2-10-0'], ['0802.0971-1-10-1', '0802.0971-2-10-1'], ['0802.0971-1-10-2', '0802.0971-2-10-2'], ['0802.0971-1-10-3', '0802.0971-2-10-3'], ['0802.0971-1-10-4', '0802.0971-2-10-4'], ['0802.0971-1-23-0', '0802.0971-2-23-0'], ['0802.0971-1-23-1', '0802.0971-2-23-1'], ['0802.0971-1-23-2', '0802.0971-2-23-2'], ['0802.0971-1-38-0', '0802.0971-2-38-0'], ['0802.0971-1-38-1', '0802.0971-2-38-1'], ['0802.0971-1-38-2', '0802.0971-2-38-2'], ['0802.0971-1-38-3', '0802.0971-2-38-3'], ['0802.0971-1-38-4', '0802.0971-2-38-4'], ['0802.0971-1-38-5', '0802.0971-2-38-5'], ['0802.0971-1-33-0', '0802.0971-2-33-0'], ['0802.0971-1-16-0', '0802.0971-2-16-0'], ['0802.0971-1-16-1', '0802.0971-2-16-1'], ['0802.0971-1-16-2', '0802.0971-2-16-2'], ['0802.0971-1-16-3', '0802.0971-2-16-3'], ['0802.0971-1-16-4', '0802.0971-2-16-4'], ['0802.0971-1-16-5', '0802.0971-2-16-5'], ['0802.0971-1-16-6', '0802.0971-2-16-6'], ['0802.0971-1-16-7', '0802.0971-2-16-7'], ['0802.0971-1-16-8', '0802.0971-2-16-8'], ['0802.0971-1-16-9', '0802.0971-2-16-9'], ['0802.0971-1-16-10', '0802.0971-2-16-10'], ['0802.0971-1-16-11', '0802.0971-2-16-11'], ['0802.0971-1-16-12', '0802.0971-2-16-12'], ['0802.0971-1-16-13', '0802.0971-2-16-13']]

[['0802.0971-1-29-0', '0802.0971-2-29-0'], ['0802.0971-1-29-1', '0802.0971-2-29-1'], ['0802.0971-1-29-2', '0802.0971-2-29-2'], ['0802.0971-1-29-3', '0802.0971-2-29-3'], ['0802.0971-1-29-4', '0802.0971-2-29-4'], ['0802.0971-1-29-5', '0802.0971-2-29-5'], ['0802.0971-1-29-6', '0802.0971-2-29-6'], ['0802.0971-1-29-7', '0802.0971-2-29-7'], ['0802.0971-1-29-8', '0802.0971-2-29-8'], ['0802.0971-1-29-9', '0802.0971-2-29-9'], ['0802.0971-1-5-0', '0802.0971-2-5-0'], ['0802.0971-1-5-1', '0802.0971-2-5-1'], ['0802.0971-1-5-2', '0802.0971-2-5-2'], ['0802.0971-1-28-0', '0802.0971-2-28-0'], ['0802.0971-1-28-1', '0802.0971-2-28-1'], ['0802.0971-1-28-2', '0802.0971-2-28-2'], ['0802.0971-1-28-3', '0802.0971-2-28-3'], ['0802.0971-1-39-0', '0802.0971-2-39-0'], ['0802.0971-1-39-1', '0802.0971-2-39-1'], ['0802.0971-1-39-2', '0802.0971-2-39-2'], ['0802.0971-1-39-3', '0802.0971-2-39-3'], ['0802.0971-1-39-4', '0802.0971-2-39-4'], ['0802.0971-1-6-0', '0802.0971-2-6-0'], ['0802.0971-1-6-1', '0802.0971-2-6-1'], ['0802.0971-1-6-2', '0802.0971-2-6-2'], ['0802.0971-1-6-3', '0802.0971-2-6-3'], ['0802.0971-1-22-0', '0802.0971-2-22-0'], ['0802.0971-1-22-1', '0802.0971-2-22-1'], ['0802.0971-1-22-2', '0802.0971-2-22-2'], ['0802.0971-1-22-3', '0802.0971-2-22-3'], ['0802.0971-1-22-4', '0802.0971-2-22-4'], ['0802.0971-1-22-5', '0802.0971-2-22-5'], ['0802.0971-1-22-6', '0802.0971-2-22-6'], ['0802.0971-1-22-7', '0802.0971-2-22-7'], ['0802.0971-1-40-0', '0802.0971-2-40-0'], ['0802.0971-1-40-1', '0802.0971-2-40-1'], ['0802.0971-1-40-2', '0802.0971-2-40-2'], ['0802.0971-1-3-0', '0802.0971-2-3-0'], ['0802.0971-1-3-1', '0802.0971-2-3-1'], ['0802.0971-1-35-0', '0802.0971-2-35-0'], ['0802.0971-1-35-1', '0802.0971-2-35-1'], ['0802.0971-1-35-2', '0802.0971-2-35-2'], ['0802.0971-1-35-3', '0802.0971-2-35-3'], ['0802.0971-1-31-0', '0802.0971-2-31-0'], ['0802.0971-1-31-1', '0802.0971-2-31-1'], ['0802.0971-1-31-2', '0802.0971-2-31-2'], ['0802.0971-1-31-3', '0802.0971-2-31-3'], ['0802.0971-1-31-4', '0802.0971-2-31-4'], ['0802.0971-1-31-5', '0802.0971-2-31-5'], ['0802.0971-1-31-6', '0802.0971-2-31-6'], ['0802.0971-1-26-0', '0802.0971-2-26-0'], ['0802.0971-1-26-1', '0802.0971-2-26-1'], ['0802.0971-1-27-0', '0802.0971-2-27-0'], ['0802.0971-1-27-1', '0802.0971-2-27-1'], ['0802.0971-1-27-2', '0802.0971-2-27-2'], ['0802.0971-1-27-3', '0802.0971-2-27-3'], ['0802.0971-1-27-4', '0802.0971-2-27-4'], ['0802.0971-1-2-0', '0802.0971-2-2-0'], ['0802.0971-1-2-1', '0802.0971-2-2-1'], ['0802.0971-1-2-2', '0802.0971-2-2-2'], ['0802.0971-1-20-0', '0802.0971-2-20-0'], ['0802.0971-1-20-1', '0802.0971-2-20-1'], ['0802.0971-1-20-2', '0802.0971-2-20-2'], ['0802.0971-1-20-3', '0802.0971-2-20-3'], ['0802.0971-1-18-0', '0802.0971-2-18-0'], ['0802.0971-1-18-1', '0802.0971-2-18-1'], ['0802.0971-1-18-2', '0802.0971-2-18-2'], ['0802.0971-1-18-3', '0802.0971-2-18-3'], ['0802.0971-1-18-4', '0802.0971-2-18-4'], ['0802.0971-1-21-0', '0802.0971-2-21-0'], ['0802.0971-1-21-1', '0802.0971-2-21-1'], ['0802.0971-1-21-2', '0802.0971-2-21-2'], ['0802.0971-1-21-3', '0802.0971-2-21-3'], ['0802.0971-1-24-0', '0802.0971-2-24-0'], ['0802.0971-1-24-1', '0802.0971-2-24-1'], ['0802.0971-1-24-2', '0802.0971-2-24-2'], ['0802.0971-1-24-3', '0802.0971-2-24-3'], ['0802.0971-1-24-4', '0802.0971-2-24-4'], ['0802.0971-1-24-5', '0802.0971-2-24-5'], ['0802.0971-1-24-6', '0802.0971-2-24-6'], ['0802.0971-1-9-0', '0802.0971-2-9-0'], ['0802.0971-1-9-1', '0802.0971-2-9-1'], ['0802.0971-1-9-2', '0802.0971-2-9-2'], ['0802.0971-1-11-0', '0802.0971-2-11-0'], ['0802.0971-1-11-1', '0802.0971-2-11-1'], ['0802.0971-1-30-0', '0802.0971-2-30-0'], ['0802.0971-1-30-1', '0802.0971-2-30-1'], ['0802.0971-1-30-2', '0802.0971-2-30-2'], ['0802.0971-1-30-3', '0802.0971-2-30-3'], ['0802.0971-1-30-4', '0802.0971-2-30-4'], ['0802.0971-1-0-0', '0802.0971-2-0-0'], ['0802.0971-1-0-1', '0802.0971-2-0-1'], ['0802.0971-1-0-2', '0802.0971-2-0-2'], ['0802.0971-1-0-3', '0802.0971-2-0-3'], ['0802.0971-1-0-4', '0802.0971-2-0-4'], ['0802.0971-1-19-0', '0802.0971-2-19-0'], ['0802.0971-1-19-1', '0802.0971-2-19-1'], ['0802.0971-1-19-2', '0802.0971-2-19-2'], ['0802.0971-1-19-3', '0802.0971-2-19-3'], ['0802.0971-1-41-0', '0802.0971-2-41-0'], ['0802.0971-1-41-1', '0802.0971-2-41-1'], ['0802.0971-1-34-0', '0802.0971-2-34-0'], ['0802.0971-1-34-1', '0802.0971-2-34-1'], ['0802.0971-1-34-2', '0802.0971-2-34-2'], ['0802.0971-1-34-3', '0802.0971-2-34-3'], ['0802.0971-1-12-0', '0802.0971-2-12-0'], ['0802.0971-1-12-1', '0802.0971-2-12-1'], ['0802.0971-1-12-2', '0802.0971-2-12-2'], ['0802.0971-1-7-0', '0802.0971-2-7-0'], ['0802.0971-1-15-0', '0802.0971-2-15-0'], ['0802.0971-1-15-1', '0802.0971-2-15-1'], ['0802.0971-1-15-2', '0802.0971-2-15-2'], ['0802.0971-1-37-0', '0802.0971-2-37-0'], ['0802.0971-1-37-1', '0802.0971-2-37-1'], ['0802.0971-1-17-0', '0802.0971-2-17-0'], ['0802.0971-1-17-1', '0802.0971-2-17-1'], ['0802.0971-1-17-2', '0802.0971-2-17-2'], ['0802.0971-1-4-0', '0802.0971-2-4-0'], ['0802.0971-1-4-1', '0802.0971-2-4-1'], ['0802.0971-1-4-2', '0802.0971-2-4-2'], ['0802.0971-1-4-3', '0802.0971-2-4-3'], ['0802.0971-1-4-4', '0802.0971-2-4-4'], ['0802.0971-1-4-5', '0802.0971-2-4-5'], ['0802.0971-1-32-0', '0802.0971-2-32-0'], ['0802.0971-1-32-1', '0802.0971-2-32-1'], ['0802.0971-1-32-2', '0802.0971-2-32-2'], ['0802.0971-1-32-3', '0802.0971-2-32-3'], ['0802.0971-1-14-0', '0802.0971-2-14-0'], ['0802.0971-1-14-1', '0802.0971-2-14-1'], ['0802.0971-1-10-0', '0802.0971-2-10-0'], ['0802.0971-1-10-1', '0802.0971-2-10-1'], ['0802.0971-1-10-2', '0802.0971-2-10-2'], ['0802.0971-1-10-3', '0802.0971-2-10-3'], ['0802.0971-1-10-4', '0802.0971-2-10-4'], ['0802.0971-1-23-0', '0802.0971-2-23-0'], ['0802.0971-1-23-1', '0802.0971-2-23-1'], ['0802.0971-1-23-2', '0802.0971-2-23-2'], ['0802.0971-1-38-0', '0802.0971-2-38-0'], ['0802.0971-1-38-1', '0802.0971-2-38-1'], ['0802.0971-1-38-2', '0802.0971-2-38-2'], ['0802.0971-1-38-3', '0802.0971-2-38-3'], ['0802.0971-1-38-4', '0802.0971-2-38-4'], ['0802.0971-1-38-5', '0802.0971-2-38-5'], ['0802.0971-1-33-0', '0802.0971-2-33-0'], ['0802.0971-1-16-0', '0802.0971-2-16-0'], ['0802.0971-1-16-1', '0802.0971-2-16-1'], ['0802.0971-1-16-2', '0802.0971-2-16-2'], ['0802.0971-1-16-3', '0802.0971-2-16-3'], ['0802.0971-1-16-4', '0802.0971-2-16-4'], ['0802.0971-1-16-5', '0802.0971-2-16-5'], ['0802.0971-1-16-6', '0802.0971-2-16-6'], ['0802.0971-1-16-7', '0802.0971-2-16-7'], ['0802.0971-1-16-8', '0802.0971-2-16-8'], ['0802.0971-1-16-9', '0802.0971-2-16-9'], ['0802.0971-1-16-10', '0802.0971-2-16-10'], ['0802.0971-1-16-11', '0802.0971-2-16-11'], ['0802.0971-1-16-12', '0802.0971-2-16-12'], ['0802.0971-1-16-13', '0802.0971-2-16-13']]

[]

[]

[]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0802.0971

1301.4457

{'1301.4457-1-0-0': 'We use the holographic hardwall model to calculate the masses of light glueball states with odd spin and [MATH] associated with Odderons.', '1301.4457-1-0-1': 'Using Dirichlet and Neumann boundary conditions we obtain expressions for the Odderon Regge trajectories consistent with those calculated by other approaches.', '1301.4457-1-1-0': '# Introduction', '1301.4457-1-2-0': 'Mesons and baryons have their total angular momenta [MATH] related to the square of their masses [MATH] through approximate linear functions known as Regge trajectories: [EQUATION] where [MATH] and [MATH] are constants characteristic of each hadronic branch.', '1301.4457-1-2-1': 'Analogously one can find Regge trajectories for odd spin glueballs with [MATH] which are related to the Odderon.', '1301.4457-1-3-0': 'The Regge trajectories for the Odderon were obtained by Llanes-Estrada, Bicudo and Cotanch [CITATION] using two different methods.', '1301.4457-1-3-1': 'The first one is based on a relativistic many-body (RMB) formulation which gives (masses are expressed in GeV throughout this paper): [EQUATION]', '1301.4457-1-3-2': 'The second method is based on a nonrelativistic constituent model (NRCM) resulting in: [EQUATION]', '1301.4457-1-3-3': 'A very interesting and deep study of the Odderon in gauge/string duality was presented in ref [CITATION] but without discussing its Regge trajectories.', '1301.4457-1-4-0': 'In this work we obtain the masses of odd spin glueballs from the holographic hardwall model and derive the corresponding Regge trajectories for the Odderon compatible with the above results.', '1301.4457-1-5-0': 'Since its conception quantum chromodynamics (QCD) has been used as the standard theory to explain the phenomenology of strong interactions.', '1301.4457-1-5-1': 'Due to asymptotic freedom, the coupling of strong interactions decreases when the energy of the process increases.', '1301.4457-1-5-2': 'This result is obtained using perturbation theory and is valid only for small couplings [MATH].', '1301.4457-1-5-3': 'Extrapolating this result to low energies one obtains strong coupling [MATH], outside the perturbative regime.', '1301.4457-1-5-4': 'Regge trajectories are an exemple of non-pertubative behavior of strong interactions and so are difficult to explain using QCD.', '1301.4457-1-6-0': 'The AdS/CFT correspondence [CITATION] brought new perspectives for string and quantum field theories since it relates [MATH] supersymmetric and conformal Yang-Mills field theory for [MATH], in flat Minkowski spacetime with [MATH] dimensions, with a string theory in a curved 10 dimensional spacetime, the [MATH] space.', '1301.4457-1-6-1': 'In the supergravity approximation of string theory in this space one can relate both theories through [CITATION]: [EQUATION] where [MATH] is a non-normalizable supergravity field, [MATH] is the corresponding on shell supergravity action, [MATH] is the value of [MATH] at the boundary [MATH] and [MATH] is the associated operator of the conformal field theory (CFT).', '1301.4457-1-6-2': 'From this equation, one can obtain 4 dimensional correlation functions, for instance: [EQUATION]', '1301.4457-1-6-3': 'In particular, the scalar glueball [MATH] is represented by the operator [MATH] associated with a dilaton in the [MATH] space.', '1301.4457-1-7-0': '# Odd spin glueballs masses in the Hardwall model', '1301.4457-1-8-0': 'Glueballs are characterized by [MATH], where [MATH] represents the total angular momentum, [MATH] defines how a state behaves under spatial inversion ([MATH]-parity) and [MATH] shows the behavior of a state under charge conjugation ([MATH]-parity).', '1301.4457-1-9-0': 'In this paper we are interested in glueballs in the [MATH] and [MATH] sector with odd spins [MATH] which are associated with a particle called the Odderon.', '1301.4457-1-9-1': "The concept of the Odderon emerged in the 70's [CITATION], within the context of asymptotic theorems, reappearing later in perturbative QCD [CITATION].", '1301.4457-1-9-2': 'The Odderon have also been linked, for instance, to the color glass condensate [CITATION].', '1301.4457-1-9-3': 'Although the Odderon has not been detected so far, there is some experimental evidence of its existence and it could be regarded as a test of QCD [CITATION].', '1301.4457-1-9-4': 'The Odderon is a bound state of three gluons, without color, which represents a singularity in the complex plane [MATH], close to [MATH], in the odd-under-crossing amplitude [MATH] [CITATION].', '1301.4457-1-9-5': 'For a review see for instance [CITATION].', '1301.4457-1-10-0': 'The AdS/CFT correspondence can not be used directly as a tool for the study of hadrons, because the dual theory is a supersymmetric conformal theory which is very different from QCD.', '1301.4457-1-10-1': 'However, it was noticed that the energy [MATH] of a process in the [MATH] theory is related to the radial coordinate [MATH] in [MATH] space as [EQUATION]', '1301.4457-1-10-2': 'This motivated the holographic hardwall model proposed by Polchinski and Strassler [CITATION] to calculate the scattering of glueballs in [MATH]-dimensions using a dilaton field in [MATH] space.', '1301.4457-1-10-3': 'The works [CITATION] introduced a cut-off at a certain value [MATH] of the [MATH] coordinate and considered an [MATH] slice in the region [MATH].', '1301.4457-1-10-4': 'An immediate consequence of introducing a cut-off is the breaking of conformal invariance, so that particles on the 4-dimensional boundary acquire mass.', '1301.4457-1-10-5': 'Futhermore, one can associate the size of the [MATH] slice with the energy scale of QCD: [EQUATION]', '1301.4457-1-10-6': 'The hardwall model assumes an approximate duality between a string theory in an [MATH] space with metric defined by: [EQUATION] where [MATH] is the [MATH] radius, and a pure Yang-Mills theory in four dimensions with symmetry group [MATH] in the large [MATH] limit.', '1301.4457-1-10-7': 'In this model it is assumed that the AdS/CFT dictionary between supergravity fields in [MATH] space and operators on the 4d boundary, as given by eqs. ([REF]) and ([REF]), still holds after breaking the conformal invariance.', '1301.4457-1-10-8': 'This implies that the conformal dimension [MATH] of an operator [MATH] related to a [MATH]-form [MATH] field with mass [MATH] is given by [CITATION] (here and in the following we are disregarding excitations on de [MATH] subspace): [EQUATION]', '1301.4457-1-10-9': 'In particular, the operator that describes the glueball [MATH] is [EQUATION] with conformal dimension [MATH].', '1301.4457-1-10-10': 'This operator is associated with the Ramond-Ramond tensor [MATH] described in a single [MATH]-brane, by the action [CITATION]: [EQUATION]', '1301.4457-1-10-11': 'From this action one can obtain the equations of motion for the Ramond-Ramond field.', '1301.4457-1-10-12': 'With a suitable polarization choice [MATH] where [MATH] is a constant polarization tensor and [MATH] is a scalar field, it can be shown that these equations can be reduced to [CITATION]: [EQUATION] where [MATH] is the [MATH]-dimensional Minkowski metric.', '1301.4457-1-11-0': 'We use a plane wave ansatz in the 4-dimensional space for the [MATH]-form field [MATH] [EQUATION] where [MATH] is a normalization constant, [MATH] and the discrete modes [MATH] will be calculated by imposing appropriate boundary conditions.', '1301.4457-1-12-0': 'It has been proposed in the literature [CITATION] that the glueball operator with spin [MATH], could be obtained by the insertion of symmetrized covariant derivatives in the operator [MATH], such that [MATH] with conformal dimension [MATH].', '1301.4457-1-12-1': 'This approach was used in ref. [CITATION] to calculate the masses of glueball states [MATH], [MATH], [MATH], [MATH], etc and to obtain the corresponding Pomeron Regge trajectory.', '1301.4457-1-13-0': 'Here we are going to follow a similar approach for the glueball states [MATH], [MATH], [MATH], [MATH], ... .', '1301.4457-1-13-1': 'The state [MATH] is described by the operator [MATH].', '1301.4457-1-13-2': 'Inserting covariant derivatives as described above, one obtains [MATH] with [MATH] satisfying equations similar to ([REF]) and ([REF]) with a shift in the index of the Bessel function [MATH], where [MATH] is the spin of each state [MATH], [MATH], [MATH], etc.', '1301.4457-1-14-0': 'Following the approach of ref. [CITATION], we impose Dirichlet and Neumann boundary conditions to calculate glueball masses within the hardwall model.', '1301.4457-1-14-1': 'For the Dirichlet boundary condition: [EQUATION] one obtains from ([REF]), the following relation: [EQUATION]', '1301.4457-1-14-2': 'On the other hand, for the Neumann boundary condition: [EQUATION] one gets: [EQUATION] where [EQUATION]', '1301.4457-1-14-3': 'Using these boundary conditions we obtain glueball masses in the sector [MATH].', '1301.4457-1-14-4': 'Our results are shown in Table I.', '1301.4457-1-14-5': 'We also show for comparison the values for these masses found in the literature [CITATION] using other methods.', '1301.4457-1-15-0': '# Odderon Regge trajectories in the Hardwall model', '1301.4457-1-16-0': 'Using the data for odd spin glueballs obtained in the previous section we are going to built up the Regge trajectories for the Odderon.', '1301.4457-1-17-0': 'Using Dirichlet boundary condition and the set of states, [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-1-17-1': 'The errors for the slope and linear coefficient come from the linear fit.', '1301.4457-1-17-2': 'The plot relative to this trajectory can be seen in Figure [MATH].', '1301.4457-1-17-3': 'This result is in agreement with that found in [CITATION], with the relativistic many-body Hamiltonian formulation, described by equation ([REF]).', '1301.4457-1-18-0': 'It has been argued in ref. [CITATION] that the state [MATH] might not be part of the spectrum of the Odderon.', '1301.4457-1-18-1': 'To test this possibility we also consider another set of states, [MATH], [MATH], [MATH], [MATH], for which we obtain the following Regge trajectory: [EQUATION]', '1301.4457-1-18-2': 'This result is also consistent with the Regge trajectory for Odderon, equation ([REF]).', '1301.4457-1-19-0': 'Now using Neumann boundary condition and the set of states, [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-1-19-1': 'The plot relative to this trajectory can be seen in Figure 2.', '1301.4457-1-20-0': 'We also consider here the possibility of excluding the state [MATH] from the spectrum of the Odderon.', '1301.4457-1-20-1': 'For the set of states [MATH], we find the following Regge trajectory [EQUATION]', '1301.4457-1-20-2': 'This result is in agreement with that found in [CITATION], with the nonrelativistic constituent model, equation ([REF]).', '1301.4457-1-21-0': '# conclusions', '1301.4457-1-22-0': 'In this work we obtained odd spin glueball masses in the sector [MATH] using the holographic hardwall model with Dirichlet and Neumann boundary conditions.', '1301.4457-1-22-1': 'These glueball masses lie in approximate linear Regge trajectories compatible with results for the Odderon, both in the relativistic many-body as well in the non-relativistic constituent model presented in ref. [CITATION].', '1301.4457-1-22-2': 'The present analysis gives support to the conclusion of ref. [CITATION] about the general properties of the Odderon Regge trajectories, i.e., a low intercept and a slope similar to that of the Pomeron.', '1301.4457-1-23-0': 'Some aspects of the holographic approach for the Odderon Regge trajectories remain open.', '1301.4457-1-23-1': 'In our approach, we used Dirichlet and Neumann boundary conditions in the hardwall model obtaining results compatible with those of ref. [CITATION].', '1301.4457-1-23-2': 'The hardwall model was used before to obtain the Regge trajectory for the Pomeron in ref. [CITATION].', '1301.4457-1-23-3': 'In that work it was possible to conclude that Neumann boundary condition was more appropriate than the Dirichlet boundary condition by comparison with experimental data.', '1301.4457-1-23-4': 'Here in this work, it is not possible to reach a similar conclusion about boundary conditions because there is no clear experimental data for the Odderon Regge trajectories.', '1301.4457-1-24-0': 'Another open question in the Odderon Regge trajectories regards the state [MATH].', '1301.4457-1-24-1': 'It was argued in ref. [CITATION] that the glueball state [MATH] does not belong to the Odderon Regge trajectory.', '1301.4457-1-24-2': 'However, from our analysis it is not possible to reach this conclusion since we have found trajectories compatible with Odderon including the state [MATH] (eqs. ([REF]) and ([REF])) as well excluding it (eqs. ([REF]) and ([REF])).', '1301.4457-1-25-0': 'As a final remark, let us comment on our choice for the holographic model to obtain glueball masses and the Odderon Regge trajectories.', '1301.4457-1-25-1': 'This model is very interesting since masses can be obtained from the zeros of the corresponding Bessel functions.', '1301.4457-1-25-2': 'However, it is well known that holographic hardwall model leads to asymptotic non linear Regge trajectories for very high states.', '1301.4457-1-25-3': 'Nevertheless, for light states, as discussed in this work, approximate linear Regge trajectories were found.', '1301.4457-1-25-4': 'In this regard, it wil be interesting to investigate the glueball masses in the [MATH] sector within other holographic approaches, as the softwall model [CITATION] which is known to provide linear Regge trajectories.', '1301.4457-1-25-5': 'We leave this study for future work.', '1301.4457-1-26-0': 'We would like to thank Felipe J. Llanes-Estrada for interesting discussions.', '1301.4457-1-26-1': 'The autors are partially supported by CAPES, CNPq and FAPERJ, Brazilian agencies.'}

{'1301.4457-2-0-0': 'We use the holographic hardwall model to calculate the masses of light glueball states with odd spin and [MATH] associated with Odderons.', '1301.4457-2-0-1': 'Using Dirichlet and Neumann boundary conditions we obtain expressions for the Odderon Regge trajectories consistent with those calculated by other approaches.', '1301.4457-2-1-0': '# Introduction', '1301.4457-2-2-0': 'Mesons and baryons have their total angular momenta [MATH] related to the square of their masses [MATH] through approximate linear functions known as Regge trajectories: [EQUATION] where [MATH] and [MATH] are constants characteristic of each hadronic branch.', '1301.4457-2-2-1': 'Analogously one can find Regge trajectories for odd spin glueballs with [MATH] which are related to the Odderon.', '1301.4457-2-3-0': 'The Regge trajectories for the Odderon were obtained by Llanes-Estrada, Bicudo and Cotanch [CITATION] using two different methods.', '1301.4457-2-3-1': 'The first one is based on a relativistic many-body (RMB) formulation which gives (masses are expressed in GeV throughout this paper): [EQUATION]', '1301.4457-2-3-2': 'The second method is based on a nonrelativistic constituent model (NRCM) resulting in: [EQUATION]', '1301.4457-2-3-3': 'Interesting studies of the Odderon in gauge/string dualities were presented in refs. [CITATION].', '1301.4457-2-4-0': 'In this work we obtain the masses of odd spin glueballs from the holographic hardwall model and derive the corresponding Regge trajectories for the Odderon compatible with the above results.', '1301.4457-2-5-0': 'Since its conception quantum chromodynamics (QCD) has been used as the standard theory to explain the phenomenology of strong interactions.', '1301.4457-2-5-1': 'Due to asymptotic freedom, the coupling of strong interactions decreases when the energy of the process increases.', '1301.4457-2-5-2': 'This result is obtained using perturbation theory and is valid only for small couplings [MATH].', '1301.4457-2-5-3': 'Extrapolating this result to low energies one obtains strong coupling [MATH], outside the perturbative regime.', '1301.4457-2-5-4': 'Regge trajectories are an exemple of non-pertubative behavior of strong interactions and so are difficult to explain using QCD.', '1301.4457-2-6-0': 'The AdS/CFT correspondence [CITATION] brought new perspectives for string and quantum field theories since it relates [MATH] supersymmetric and conformal Yang-Mills field theory for [MATH], in flat Minkowski spacetime with [MATH] dimensions, with a string theory in a curved 10 dimensional spacetime, the [MATH] space.', '1301.4457-2-6-1': 'In the supergravity approximation of string theory in this space one can relate both theories through [CITATION]: [EQUATION] where [MATH] is a non-normalizable supergravity field, [MATH] is the corresponding on shell supergravity action, [MATH] is the value of [MATH] at the boundary [MATH] and [MATH] is the associated operator of the conformal field theory (CFT).', '1301.4457-2-6-2': 'From this equation, one can obtain 4 dimensional correlation functions, for instance: [EQUATION]', '1301.4457-2-6-3': 'In particular, the scalar glueball [MATH] is represented by the operator [MATH] associated with a dilaton in the [MATH] space.', '1301.4457-2-7-0': '# Odd spin glueballs masses in the Hardwall model', '1301.4457-2-8-0': 'Glueballs are characterized by [MATH], where [MATH] represents the total angular momentum, [MATH] defines how a state behaves under spatial inversion ([MATH]-parity) and [MATH] shows the behavior of a state under charge conjugation ([MATH]-parity).', '1301.4457-2-9-0': 'In this paper we are interested in glueballs in the [MATH] and [MATH] sector with odd spins [MATH] which are associated with a particle called the Odderon.', '1301.4457-2-9-1': "The concept of the Odderon emerged in the 70's [CITATION], within the context of asymptotic theorems, reappearing later in perturbative QCD [CITATION].", '1301.4457-2-9-2': 'The Odderon have also been linked, for instance, to the color glass condensate [CITATION].', '1301.4457-2-9-3': 'Although the Odderon has not been detected so far, there is some experimental evidence of its existence and it could be regarded as a test of QCD [CITATION].', '1301.4457-2-9-4': 'The Odderon is a bound state of three gluons, without color, which represents a singularity in the complex plane [MATH], close to [MATH], in the odd-under-crossing amplitude [MATH] [CITATION].', '1301.4457-2-9-5': 'For a review see for instance [CITATION].', '1301.4457-2-10-0': 'The AdS/CFT correspondence can not be used directly as a tool for the study of hadrons, because the dual theory is a supersymmetric conformal theory which is very different from QCD.', '1301.4457-2-10-1': 'However, it was noticed that the energy [MATH] of a process in the [MATH] theory is related to the radial coordinate [MATH] in [MATH] space as [EQUATION]', '1301.4457-2-10-2': 'This motivated the holographic hardwall model proposed by Polchinski and Strassler [CITATION] to calculate the scattering of glueballs in [MATH]-dimensions using a dilaton field in [MATH] space.', '1301.4457-2-10-3': 'The works [CITATION] introduced a cut-off at a certain value [MATH] of the [MATH] coordinate and considered an [MATH] slice in the region [MATH].', '1301.4457-2-10-4': 'An immediate consequence of introducing a cut-off is the breaking of conformal invariance, so that particles on the 4-dimensional boundary acquire mass.', '1301.4457-2-10-5': 'Futhermore, one can associate the size of the [MATH] slice with the energy scale of QCD: [EQUATION]', '1301.4457-2-10-6': 'The hardwall model assumes an approximate duality between a string theory in an [MATH] space with metric defined by: [EQUATION] where [MATH] is the [MATH] radius, and a pure Yang-Mills theory in four dimensions with symmetry group [MATH] in the large [MATH] limit.', '1301.4457-2-10-7': 'In this model it is assumed that the AdS/CFT dictionary between supergravity fields in [MATH] space and operators on the 4d boundary, as given by eqs. ([REF]) and ([REF]), still holds after breaking the conformal invariance.', '1301.4457-2-10-8': 'This implies that the conformal dimension [MATH] of an operator [MATH] related to a [MATH]-form [MATH] field with mass [MATH] is given by [CITATION] (here and in the following we are disregarding excitations on de [MATH] subspace): [EQUATION]', '1301.4457-2-10-9': 'In particular, the operator that describes the glueball [MATH] is [EQUATION] with conformal dimension [MATH].', '1301.4457-2-10-10': 'This operator is associated with the Ramond-Ramond tensor [MATH] described in a single [MATH]-brane, by the action [CITATION]: [EQUATION]', '1301.4457-2-10-11': 'From this action one can obtain the equations of motion for the Ramond-Ramond field.', '1301.4457-2-10-12': 'With a suitable polarization choice [MATH] where [MATH] is a constant polarization tensor and [MATH] is a scalar field, it can be shown that these equations can be reduced to [CITATION]: [EQUATION] where [MATH] is the [MATH]-dimensional Minkowski metric.', '1301.4457-2-11-0': 'We use a plane wave ansatz in the 4-dimensional space for the [MATH]-form field [MATH] [EQUATION] where [MATH] is a normalization constant, [MATH] and the discrete modes [MATH] will be calculated by imposing appropriate boundary conditions.', '1301.4457-2-12-0': 'It has been proposed in the literature [CITATION] that the glueball operator with spin [MATH], could be obtained by the insertion of symmetrized covariant derivatives in the operator [MATH], such that [MATH] with conformal dimension [MATH].', '1301.4457-2-12-1': 'This approach was used in ref. [CITATION] to calculate the masses of glueball states [MATH], [MATH], [MATH], [MATH], etc and to obtain the corresponding Pomeron Regge trajectory.', '1301.4457-2-13-0': 'Here we are going to follow a similar approach for the glueball states [MATH], [MATH], [MATH], [MATH], ... .', '1301.4457-2-13-1': 'The state [MATH] is described by the operator [MATH].', '1301.4457-2-13-2': 'Inserting covariant derivatives as described above, one obtains [MATH] with [MATH] satisfying equations similar to ([REF]) and ([REF]) with a shift in the index of the Bessel function [MATH], where [MATH] is the spin of each state [MATH], [MATH], [MATH], etc.', '1301.4457-2-14-0': 'Following the approach of ref. [CITATION], we impose Dirichlet and Neumann boundary conditions to calculate glueball masses within the hardwall model.', '1301.4457-2-14-1': 'For the Dirichlet boundary condition: [EQUATION] one obtains from ([REF]), the following relation: [EQUATION]', '1301.4457-2-14-2': 'On the other hand, for the Neumann boundary condition: [EQUATION] one gets: [EQUATION] where [EQUATION]', '1301.4457-2-14-3': 'Using these boundary conditions we obtain glueball masses in the sector [MATH].', '1301.4457-2-14-4': 'Our results are shown in Table I.', '1301.4457-2-14-5': 'We also show for comparison the values for these masses found in the literature [CITATION] using other methods.', '1301.4457-2-15-0': '# Odderon Regge trajectories in the Hardwall model', '1301.4457-2-16-0': 'Using the data for odd spin glueballs obtained in the previous section we are going to built up the Regge trajectories for the Odderon.', '1301.4457-2-17-0': 'Using Dirichlet boundary condition and the set of states, [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-2-17-1': 'The errors for the slope and linear coefficient come from the linear fit.', '1301.4457-2-17-2': 'The plot relative to this trajectory can be seen in Figure [MATH].', '1301.4457-2-17-3': 'This result is in agreement with that found in [CITATION], with the relativistic many-body Hamiltonian formulation, described by equation ([REF]).', '1301.4457-2-18-0': 'It has been argued in ref. [CITATION] that the state [MATH] might not be part of the spectrum of the Odderon.', '1301.4457-2-18-1': 'To test this possibility we also consider another set of states, [MATH], [MATH], [MATH], [MATH], for which we obtain the following Regge trajectory: [EQUATION]', '1301.4457-2-18-2': 'This result is also consistent with the Regge trajectory for Odderon, equation ([REF]).', '1301.4457-2-19-0': 'Now using Neumann boundary condition and the set of states, [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-2-19-1': 'The plot relative to this trajectory can be seen in Figure 2.', '1301.4457-2-20-0': 'We also consider here the possibility of excluding the state [MATH] from the spectrum of the Odderon.', '1301.4457-2-20-1': 'For the set of states [MATH], we find the following Regge trajectory [EQUATION]', '1301.4457-2-20-2': 'This result is in agreement with that found in [CITATION], with the nonrelativistic constituent model, equation ([REF]).', '1301.4457-2-21-0': '# conclusions', '1301.4457-2-22-0': 'In this work we obtained odd spin glueball masses in the sector [MATH] using the holographic hardwall model with Dirichlet and Neumann boundary conditions.', '1301.4457-2-22-1': 'These glueball masses lie in approximate linear Regge trajectories compatible with results for the Odderon, both in the relativistic many-body as well in the non-relativistic constituent model presented in ref. [CITATION].', '1301.4457-2-22-2': 'The present analysis gives support to the conclusion of ref. [CITATION] about the general properties of the Odderon Regge trajectories, i.e., a low intercept and a slope similar to that of the Pomeron.', '1301.4457-2-23-0': 'Some aspects of the holographic approach for the Odderon Regge trajectories remain open.', '1301.4457-2-23-1': 'In our approach, we used Dirichlet and Neumann boundary conditions in the hardwall model obtaining results compatible with those of ref. [CITATION].', '1301.4457-2-23-2': 'The hardwall model was used before to obtain the Regge trajectory for the Pomeron in ref. [CITATION].', '1301.4457-2-23-3': 'In that work it was possible to conclude that Neumann boundary condition was more appropriate than the Dirichlet boundary condition by comparison with experimental data.', '1301.4457-2-23-4': 'Here in this work, it is not possible to reach a similar conclusion about boundary conditions because there is no clear experimental data for the Odderon Regge trajectories.', '1301.4457-2-24-0': 'Another open question in the Odderon Regge trajectories regards the state [MATH].', '1301.4457-2-24-1': 'It was argued in ref. [CITATION] that the glueball state [MATH] does not belong to the Odderon Regge trajectory.', '1301.4457-2-24-2': 'However, from our analysis it is not possible to reach this conclusion since we have found trajectories compatible with Odderon including the state [MATH] (eqs. ([REF]) and ([REF])) as well excluding it (eqs. ([REF]) and ([REF])).', '1301.4457-2-25-0': 'As a final remark, let us comment on our choice for the holographic model to obtain glueball masses and the Odderon Regge trajectories.', '1301.4457-2-25-1': 'This model is very interesting since masses can be obtained from the zeros of the corresponding Bessel functions.', '1301.4457-2-25-2': 'However, it is well known that holographic hardwall model leads to asymptotic non linear Regge trajectories for very high states.', '1301.4457-2-25-3': 'Nevertheless, for light states, as discussed in this work, approximate linear Regge trajectories were found.', '1301.4457-2-25-4': 'In this regard, it wil be interesting to investigate the glueball masses in the [MATH] sector within other holographic approaches, as the softwall model [CITATION] which is known to provide linear Regge trajectories.', '1301.4457-2-25-5': 'We leave this study for future work.', '1301.4457-2-26-0': 'We would like to thank Felipe J. Llanes-Estrada for interesting discussions.', '1301.4457-2-26-1': 'The autors are partially supported by CAPES, CNPq and FAPERJ, Brazilian agencies.'}

[['1301.4457-1-24-0', '1301.4457-2-24-0'], ['1301.4457-1-24-1', '1301.4457-2-24-1'], ['1301.4457-1-24-2', '1301.4457-2-24-2'], ['1301.4457-1-8-0', '1301.4457-2-8-0'], ['1301.4457-1-14-0', '1301.4457-2-14-0'], ['1301.4457-1-14-1', '1301.4457-2-14-1'], ['1301.4457-1-14-2', '1301.4457-2-14-2'], ['1301.4457-1-14-3', '1301.4457-2-14-3'], ['1301.4457-1-14-4', '1301.4457-2-14-4'], ['1301.4457-1-14-5', '1301.4457-2-14-5'], ['1301.4457-1-23-0', '1301.4457-2-23-0'], ['1301.4457-1-23-1', '1301.4457-2-23-1'], ['1301.4457-1-23-2', '1301.4457-2-23-2'], ['1301.4457-1-23-3', '1301.4457-2-23-3'], ['1301.4457-1-23-4', '1301.4457-2-23-4'], ['1301.4457-1-22-0', '1301.4457-2-22-0'], ['1301.4457-1-22-1', '1301.4457-2-22-1'], ['1301.4457-1-22-2', '1301.4457-2-22-2'], ['1301.4457-1-5-0', '1301.4457-2-5-0'], ['1301.4457-1-5-1', '1301.4457-2-5-1'], ['1301.4457-1-5-2', '1301.4457-2-5-2'], ['1301.4457-1-5-3', '1301.4457-2-5-3'], ['1301.4457-1-5-4', '1301.4457-2-5-4'], ['1301.4457-1-3-0', '1301.4457-2-3-0'], ['1301.4457-1-3-1', '1301.4457-2-3-1'], ['1301.4457-1-3-2', '1301.4457-2-3-2'], ['1301.4457-1-17-0', '1301.4457-2-17-0'], ['1301.4457-1-17-1', '1301.4457-2-17-1'], ['1301.4457-1-17-2', '1301.4457-2-17-2'], ['1301.4457-1-17-3', '1301.4457-2-17-3'], ['1301.4457-1-0-0', '1301.4457-2-0-0'], ['1301.4457-1-0-1', '1301.4457-2-0-1'], ['1301.4457-1-13-0', '1301.4457-2-13-0'], ['1301.4457-1-13-1', '1301.4457-2-13-1'], ['1301.4457-1-13-2', '1301.4457-2-13-2'], ['1301.4457-1-26-0', '1301.4457-2-26-0'], ['1301.4457-1-26-1', '1301.4457-2-26-1'], ['1301.4457-1-4-0', '1301.4457-2-4-0'], ['1301.4457-1-2-0', '1301.4457-2-2-0'], ['1301.4457-1-2-1', '1301.4457-2-2-1'], ['1301.4457-1-10-0', '1301.4457-2-10-0'], ['1301.4457-1-10-1', '1301.4457-2-10-1'], ['1301.4457-1-10-2', '1301.4457-2-10-2'], ['1301.4457-1-10-3', '1301.4457-2-10-3'], ['1301.4457-1-10-4', '1301.4457-2-10-4'], ['1301.4457-1-10-5', '1301.4457-2-10-5'], ['1301.4457-1-10-6', '1301.4457-2-10-6'], ['1301.4457-1-10-7', '1301.4457-2-10-7'], ['1301.4457-1-10-8', '1301.4457-2-10-8'], ['1301.4457-1-10-9', '1301.4457-2-10-9'], ['1301.4457-1-10-10', '1301.4457-2-10-10'], ['1301.4457-1-10-11', '1301.4457-2-10-11'], ['1301.4457-1-10-12', '1301.4457-2-10-12'], ['1301.4457-1-18-0', '1301.4457-2-18-0'], ['1301.4457-1-18-1', '1301.4457-2-18-1'], ['1301.4457-1-18-2', '1301.4457-2-18-2'], ['1301.4457-1-6-0', '1301.4457-2-6-0'], ['1301.4457-1-6-1', '1301.4457-2-6-1'], ['1301.4457-1-6-2', '1301.4457-2-6-2'], ['1301.4457-1-6-3', '1301.4457-2-6-3'], ['1301.4457-1-20-0', '1301.4457-2-20-0'], ['1301.4457-1-20-1', '1301.4457-2-20-1'], ['1301.4457-1-20-2', '1301.4457-2-20-2'], ['1301.4457-1-9-0', '1301.4457-2-9-0'], ['1301.4457-1-9-1', '1301.4457-2-9-1'], ['1301.4457-1-9-2', '1301.4457-2-9-2'], ['1301.4457-1-9-3', '1301.4457-2-9-3'], ['1301.4457-1-9-4', '1301.4457-2-9-4'], ['1301.4457-1-9-5', '1301.4457-2-9-5'], ['1301.4457-1-11-0', '1301.4457-2-11-0'], ['1301.4457-1-19-0', '1301.4457-2-19-0'], ['1301.4457-1-19-1', '1301.4457-2-19-1'], ['1301.4457-1-16-0', '1301.4457-2-16-0'], ['1301.4457-1-12-0', '1301.4457-2-12-0'], ['1301.4457-1-12-1', '1301.4457-2-12-1'], ['1301.4457-1-25-0', '1301.4457-2-25-0'], ['1301.4457-1-25-1', '1301.4457-2-25-1'], ['1301.4457-1-25-2', '1301.4457-2-25-2'], ['1301.4457-1-25-3', '1301.4457-2-25-3'], ['1301.4457-1-25-4', '1301.4457-2-25-4'], ['1301.4457-1-25-5', '1301.4457-2-25-5'], ['1301.4457-2-5-2', '1301.4457-3-5-2'], ['1301.4457-2-14-3', '1301.4457-3-16-3'], ['1301.4457-2-14-4', '1301.4457-3-16-7'], ['1301.4457-2-14-5', '1301.4457-3-16-8'], ['1301.4457-2-22-0', '1301.4457-3-24-0'], ['1301.4457-2-13-1', '1301.4457-3-15-1'], ['1301.4457-2-13-2', '1301.4457-3-15-2'], ['1301.4457-2-18-1', '1301.4457-3-20-1'], ['1301.4457-2-25-1', '1301.4457-3-27-1'], ['1301.4457-2-25-3', '1301.4457-3-27-3'], ['1301.4457-2-25-5', '1301.4457-3-27-5'], ['1301.4457-2-17-1', '1301.4457-3-19-1'], ['1301.4457-2-10-9', '1301.4457-3-12-3'], ['1301.4457-2-10-11', '1301.4457-3-12-5'], ['1301.4457-2-8-0', '1301.4457-3-8-0'], ['1301.4457-2-5-0', '1301.4457-3-5-0'], ['1301.4457-2-5-1', '1301.4457-3-5-1'], ['1301.4457-2-5-3', '1301.4457-3-5-3'], ['1301.4457-2-14-0', '1301.4457-3-16-0'], ['1301.4457-2-14-1', '1301.4457-3-16-1'], ['1301.4457-2-22-1', '1301.4457-3-24-1'], ['1301.4457-2-22-2', '1301.4457-3-24-2'], ['1301.4457-2-13-0', '1301.4457-3-15-0'], ['1301.4457-2-12-0', '1301.4457-3-14-0'], ['1301.4457-2-12-1', '1301.4457-3-14-1'], ['1301.4457-2-2-1', '1301.4457-3-2-2'], ['1301.4457-2-18-0', '1301.4457-3-20-0'], ['1301.4457-2-18-2', '1301.4457-3-20-2'], ['1301.4457-2-23-0', '1301.4457-3-25-0'], ['1301.4457-2-23-1', '1301.4457-3-25-1'], ['1301.4457-2-23-2', '1301.4457-3-25-2'], ['1301.4457-2-23-3', '1301.4457-3-25-3'], ['1301.4457-2-23-4', '1301.4457-3-25-4'], ['1301.4457-2-4-0', '1301.4457-3-4-0'], ['1301.4457-2-19-0', '1301.4457-3-21-0'], ['1301.4457-2-19-1', '1301.4457-3-21-1'], ['1301.4457-2-0-0', '1301.4457-3-0-0'], ['1301.4457-2-0-1', '1301.4457-3-0-1'], ['1301.4457-2-3-1', '1301.4457-3-3-2'], ['1301.4457-2-3-2', '1301.4457-3-3-3'], ['1301.4457-2-3-3', '1301.4457-3-3-4'], ['1301.4457-2-24-0', '1301.4457-3-26-0'], ['1301.4457-2-24-1', '1301.4457-3-26-1'], ['1301.4457-2-26-1', '1301.4457-3-28-1'], ['1301.4457-2-20-0', '1301.4457-3-22-0'], ['1301.4457-2-20-1', '1301.4457-3-22-1'], ['1301.4457-2-20-2', '1301.4457-3-22-2'], ['1301.4457-2-6-0', '1301.4457-3-6-0'], ['1301.4457-2-6-1', '1301.4457-3-6-1'], ['1301.4457-2-6-2', '1301.4457-3-6-2'], ['1301.4457-2-6-3', '1301.4457-3-6-3'], ['1301.4457-2-9-0', '1301.4457-3-9-0'], ['1301.4457-2-9-1', '1301.4457-3-9-1'], ['1301.4457-2-9-2', '1301.4457-3-9-2'], ['1301.4457-2-9-4', '1301.4457-3-9-4'], ['1301.4457-2-11-0', '1301.4457-3-13-0'], ['1301.4457-2-25-0', '1301.4457-3-27-0'], ['1301.4457-2-25-2', '1301.4457-3-27-2'], ['1301.4457-2-25-4', '1301.4457-3-27-4'], ['1301.4457-2-17-0', '1301.4457-3-19-0'], ['1301.4457-2-17-2', '1301.4457-3-19-2'], ['1301.4457-2-17-3', '1301.4457-3-19-3'], ['1301.4457-2-10-0', '1301.4457-3-11-0'], ['1301.4457-2-10-1', '1301.4457-3-11-1'], ['1301.4457-2-10-2', '1301.4457-3-11-2'], ['1301.4457-2-10-3', '1301.4457-3-11-3'], ['1301.4457-2-10-4', '1301.4457-3-11-4'], ['1301.4457-2-10-6', '1301.4457-3-12-0'], ['1301.4457-2-10-7', '1301.4457-3-12-1'], ['1301.4457-2-10-8', '1301.4457-3-12-2'], ['1301.4457-2-10-10', '1301.4457-3-12-4'], ['1301.4457-2-10-12', '1301.4457-3-12-6'], ['1301.4457-1-3-3', '1301.4457-2-3-3'], ['1301.4457-2-5-4', '1301.4457-3-5-4'], ['1301.4457-2-14-2', '1301.4457-3-16-2'], ['1301.4457-2-14-2', '1301.4457-3-16-6'], ['1301.4457-2-2-0', '1301.4457-3-2-0'], ['1301.4457-2-2-0', '1301.4457-3-2-1'], ['1301.4457-2-3-0', '1301.4457-3-3-0'], ['1301.4457-2-3-0', '1301.4457-3-3-1'], ['1301.4457-2-24-2', '1301.4457-3-26-2'], ['1301.4457-2-26-0', '1301.4457-3-28-0'], ['1301.4457-2-9-3', '1301.4457-3-9-3'], ['1301.4457-2-16-0', '1301.4457-3-18-0'], ['1301.4457-2-10-5', '1301.4457-3-11-5']]

[['1301.4457-1-24-0', '1301.4457-2-24-0'], ['1301.4457-1-24-1', '1301.4457-2-24-1'], ['1301.4457-1-24-2', '1301.4457-2-24-2'], ['1301.4457-1-8-0', '1301.4457-2-8-0'], ['1301.4457-1-14-0', '1301.4457-2-14-0'], ['1301.4457-1-14-1', '1301.4457-2-14-1'], ['1301.4457-1-14-2', '1301.4457-2-14-2'], ['1301.4457-1-14-3', '1301.4457-2-14-3'], ['1301.4457-1-14-4', '1301.4457-2-14-4'], ['1301.4457-1-14-5', '1301.4457-2-14-5'], ['1301.4457-1-23-0', '1301.4457-2-23-0'], ['1301.4457-1-23-1', '1301.4457-2-23-1'], ['1301.4457-1-23-2', '1301.4457-2-23-2'], ['1301.4457-1-23-3', '1301.4457-2-23-3'], ['1301.4457-1-23-4', '1301.4457-2-23-4'], ['1301.4457-1-22-0', '1301.4457-2-22-0'], ['1301.4457-1-22-1', '1301.4457-2-22-1'], ['1301.4457-1-22-2', '1301.4457-2-22-2'], ['1301.4457-1-5-0', '1301.4457-2-5-0'], ['1301.4457-1-5-1', '1301.4457-2-5-1'], ['1301.4457-1-5-2', '1301.4457-2-5-2'], ['1301.4457-1-5-3', '1301.4457-2-5-3'], ['1301.4457-1-5-4', '1301.4457-2-5-4'], ['1301.4457-1-3-0', '1301.4457-2-3-0'], ['1301.4457-1-3-1', '1301.4457-2-3-1'], ['1301.4457-1-3-2', '1301.4457-2-3-2'], ['1301.4457-1-17-0', '1301.4457-2-17-0'], ['1301.4457-1-17-1', '1301.4457-2-17-1'], ['1301.4457-1-17-2', '1301.4457-2-17-2'], ['1301.4457-1-17-3', '1301.4457-2-17-3'], ['1301.4457-1-0-0', '1301.4457-2-0-0'], ['1301.4457-1-0-1', '1301.4457-2-0-1'], ['1301.4457-1-13-0', '1301.4457-2-13-0'], ['1301.4457-1-13-1', '1301.4457-2-13-1'], ['1301.4457-1-13-2', '1301.4457-2-13-2'], ['1301.4457-1-26-0', '1301.4457-2-26-0'], ['1301.4457-1-26-1', '1301.4457-2-26-1'], ['1301.4457-1-4-0', '1301.4457-2-4-0'], ['1301.4457-1-2-0', '1301.4457-2-2-0'], ['1301.4457-1-2-1', '1301.4457-2-2-1'], ['1301.4457-1-10-0', '1301.4457-2-10-0'], ['1301.4457-1-10-1', '1301.4457-2-10-1'], ['1301.4457-1-10-2', '1301.4457-2-10-2'], ['1301.4457-1-10-3', '1301.4457-2-10-3'], ['1301.4457-1-10-4', '1301.4457-2-10-4'], ['1301.4457-1-10-5', '1301.4457-2-10-5'], ['1301.4457-1-10-6', '1301.4457-2-10-6'], ['1301.4457-1-10-7', '1301.4457-2-10-7'], ['1301.4457-1-10-8', '1301.4457-2-10-8'], ['1301.4457-1-10-9', '1301.4457-2-10-9'], ['1301.4457-1-10-10', '1301.4457-2-10-10'], ['1301.4457-1-10-11', '1301.4457-2-10-11'], ['1301.4457-1-10-12', '1301.4457-2-10-12'], ['1301.4457-1-18-0', '1301.4457-2-18-0'], ['1301.4457-1-18-1', '1301.4457-2-18-1'], ['1301.4457-1-18-2', '1301.4457-2-18-2'], ['1301.4457-1-6-0', '1301.4457-2-6-0'], ['1301.4457-1-6-1', '1301.4457-2-6-1'], ['1301.4457-1-6-2', '1301.4457-2-6-2'], ['1301.4457-1-6-3', '1301.4457-2-6-3'], ['1301.4457-1-20-0', '1301.4457-2-20-0'], ['1301.4457-1-20-1', '1301.4457-2-20-1'], ['1301.4457-1-20-2', '1301.4457-2-20-2'], ['1301.4457-1-9-0', '1301.4457-2-9-0'], ['1301.4457-1-9-1', '1301.4457-2-9-1'], ['1301.4457-1-9-2', '1301.4457-2-9-2'], ['1301.4457-1-9-3', '1301.4457-2-9-3'], ['1301.4457-1-9-4', '1301.4457-2-9-4'], ['1301.4457-1-9-5', '1301.4457-2-9-5'], ['1301.4457-1-11-0', '1301.4457-2-11-0'], ['1301.4457-1-19-0', '1301.4457-2-19-0'], ['1301.4457-1-19-1', '1301.4457-2-19-1'], ['1301.4457-1-16-0', '1301.4457-2-16-0'], ['1301.4457-1-12-0', '1301.4457-2-12-0'], ['1301.4457-1-12-1', '1301.4457-2-12-1'], ['1301.4457-1-25-0', '1301.4457-2-25-0'], ['1301.4457-1-25-1', '1301.4457-2-25-1'], ['1301.4457-1-25-2', '1301.4457-2-25-2'], ['1301.4457-1-25-3', '1301.4457-2-25-3'], ['1301.4457-1-25-4', '1301.4457-2-25-4'], ['1301.4457-1-25-5', '1301.4457-2-25-5'], ['1301.4457-2-5-2', '1301.4457-3-5-2'], ['1301.4457-2-14-3', '1301.4457-3-16-3'], ['1301.4457-2-14-4', '1301.4457-3-16-7'], ['1301.4457-2-14-5', '1301.4457-3-16-8'], ['1301.4457-2-22-0', '1301.4457-3-24-0'], ['1301.4457-2-13-1', '1301.4457-3-15-1'], ['1301.4457-2-13-2', '1301.4457-3-15-2'], ['1301.4457-2-18-1', '1301.4457-3-20-1'], ['1301.4457-2-25-1', '1301.4457-3-27-1'], ['1301.4457-2-25-3', '1301.4457-3-27-3'], ['1301.4457-2-25-5', '1301.4457-3-27-5'], ['1301.4457-2-17-1', '1301.4457-3-19-1'], ['1301.4457-2-10-9', '1301.4457-3-12-3'], ['1301.4457-2-10-11', '1301.4457-3-12-5']]

[['1301.4457-2-8-0', '1301.4457-3-8-0'], ['1301.4457-2-5-0', '1301.4457-3-5-0'], ['1301.4457-2-5-1', '1301.4457-3-5-1'], ['1301.4457-2-5-3', '1301.4457-3-5-3'], ['1301.4457-2-14-0', '1301.4457-3-16-0'], ['1301.4457-2-14-1', '1301.4457-3-16-1'], ['1301.4457-2-22-1', '1301.4457-3-24-1'], ['1301.4457-2-22-2', '1301.4457-3-24-2'], ['1301.4457-2-13-0', '1301.4457-3-15-0'], ['1301.4457-2-12-0', '1301.4457-3-14-0'], ['1301.4457-2-12-1', '1301.4457-3-14-1'], ['1301.4457-2-2-1', '1301.4457-3-2-2'], ['1301.4457-2-18-0', '1301.4457-3-20-0'], ['1301.4457-2-18-2', '1301.4457-3-20-2'], ['1301.4457-2-23-0', '1301.4457-3-25-0'], ['1301.4457-2-23-1', '1301.4457-3-25-1'], ['1301.4457-2-23-2', '1301.4457-3-25-2'], ['1301.4457-2-23-3', '1301.4457-3-25-3'], ['1301.4457-2-23-4', '1301.4457-3-25-4'], ['1301.4457-2-4-0', '1301.4457-3-4-0'], ['1301.4457-2-19-0', '1301.4457-3-21-0'], ['1301.4457-2-19-1', '1301.4457-3-21-1'], ['1301.4457-2-0-0', '1301.4457-3-0-0'], ['1301.4457-2-0-1', '1301.4457-3-0-1'], ['1301.4457-2-3-1', '1301.4457-3-3-2'], ['1301.4457-2-3-2', '1301.4457-3-3-3'], ['1301.4457-2-3-3', '1301.4457-3-3-4'], ['1301.4457-2-24-0', '1301.4457-3-26-0'], ['1301.4457-2-24-1', '1301.4457-3-26-1'], ['1301.4457-2-26-1', '1301.4457-3-28-1'], ['1301.4457-2-20-0', '1301.4457-3-22-0'], ['1301.4457-2-20-1', '1301.4457-3-22-1'], ['1301.4457-2-20-2', '1301.4457-3-22-2'], ['1301.4457-2-6-0', '1301.4457-3-6-0'], ['1301.4457-2-6-1', '1301.4457-3-6-1'], ['1301.4457-2-6-2', '1301.4457-3-6-2'], ['1301.4457-2-6-3', '1301.4457-3-6-3'], ['1301.4457-2-9-0', '1301.4457-3-9-0'], ['1301.4457-2-9-1', '1301.4457-3-9-1'], ['1301.4457-2-9-2', '1301.4457-3-9-2'], ['1301.4457-2-9-4', '1301.4457-3-9-4'], ['1301.4457-2-11-0', '1301.4457-3-13-0'], ['1301.4457-2-25-0', '1301.4457-3-27-0'], ['1301.4457-2-25-2', '1301.4457-3-27-2'], ['1301.4457-2-25-4', '1301.4457-3-27-4'], ['1301.4457-2-17-0', '1301.4457-3-19-0'], ['1301.4457-2-17-2', '1301.4457-3-19-2'], ['1301.4457-2-17-3', '1301.4457-3-19-3'], ['1301.4457-2-10-0', '1301.4457-3-11-0'], ['1301.4457-2-10-1', '1301.4457-3-11-1'], ['1301.4457-2-10-2', '1301.4457-3-11-2'], ['1301.4457-2-10-3', '1301.4457-3-11-3'], ['1301.4457-2-10-4', '1301.4457-3-11-4'], ['1301.4457-2-10-6', '1301.4457-3-12-0'], ['1301.4457-2-10-7', '1301.4457-3-12-1'], ['1301.4457-2-10-8', '1301.4457-3-12-2'], ['1301.4457-2-10-10', '1301.4457-3-12-4'], ['1301.4457-2-10-12', '1301.4457-3-12-6']]

[]

[['1301.4457-1-3-3', '1301.4457-2-3-3'], ['1301.4457-2-5-4', '1301.4457-3-5-4'], ['1301.4457-2-14-2', '1301.4457-3-16-2'], ['1301.4457-2-14-2', '1301.4457-3-16-6'], ['1301.4457-2-2-0', '1301.4457-3-2-0'], ['1301.4457-2-2-0', '1301.4457-3-2-1'], ['1301.4457-2-3-0', '1301.4457-3-3-0'], ['1301.4457-2-3-0', '1301.4457-3-3-1'], ['1301.4457-2-24-2', '1301.4457-3-26-2'], ['1301.4457-2-26-0', '1301.4457-3-28-0'], ['1301.4457-2-9-3', '1301.4457-3-9-3'], ['1301.4457-2-16-0', '1301.4457-3-18-0'], ['1301.4457-2-10-5', '1301.4457-3-11-5']]

[]

[]

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

https://arxiv.org/abs/1301.4457

{'1301.4457-3-0-0': 'We use the holographic hardwall model to calculate the masses of light glueball states with odd spin and [MATH] associated with odderons.', '1301.4457-3-0-1': 'Considering Dirichlet and Neumann boundary conditions we obtain expressions for the odderon Regge trajectories consistent with those calculated using other approaches.', '1301.4457-3-1-0': '# Introduction', '1301.4457-3-2-0': 'For mesons and baryons, there is an approximate relation between the total angular momenta [MATH] and the square of the masses [MATH].', '1301.4457-3-2-1': 'This relation, known as Regge trajectories has the form [EQUATION] where [MATH] and [MATH] are constants.', '1301.4457-3-2-2': 'Analogously one can find Regge trajectories for odd spin glueballs with [MATH], which are related to the odderon.', '1301.4457-3-3-0': 'The Regge trajectories for the odderon were obtained by Llanes-Estrada et.', '1301.4457-3-3-1': 'al. [CITATION] using two different methods.', '1301.4457-3-3-2': 'The first one is based on a relativistic many-body (RMB) formulation that gives [EQUATION] where the masses are expressed in GeV throughout this article.', '1301.4457-3-3-3': 'The second method is based on a nonrelativistic constituent model (NRCM) resulting in [EQUATION]', '1301.4457-3-3-4': 'Interesting studies of the odderon in gauge/string dualities were presented in Refs. [CITATION].', '1301.4457-3-4-0': 'In this work we obtain the masses of odd spin glueballs from the holographic hardwall model and derive the corresponding Regge trajectories for the odderon.', '1301.4457-3-4-1': 'We find results compatible with those above, given by Eqs. ([REF]) and ([REF]).', '1301.4457-3-5-0': 'Since its conception, quantum chromodynamics (QCD) has been used as the standard theory to explain the phenomenology of strong interactions.', '1301.4457-3-5-1': 'As a consequence of asymptotic freedom, the coupling of strong interactions decreases when the energy of the process increases.', '1301.4457-3-5-2': 'This result is obtained using perturbation theory and is valid only for small couplings [MATH].', '1301.4457-3-5-3': 'Extrapolating this result to low energies one obtains strong coupling [MATH] outside the perturbative regime.', '1301.4457-3-5-4': 'Regge trajectories are an example of nonpertubative behavior of strong interactions difficult to model using QCD.', '1301.4457-3-6-0': 'The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [CITATION] brought new perspectives for string and quantum field theories since it relates [MATH] supersymmetric and conformal Yang-Mills field theory for [MATH], in flat Minkowski spacetime with [MATH] dimensions, with a string theory in a curved 10-dimensional spacetime, the AdS[MATH] space.', '1301.4457-3-6-1': 'In the supergravity approximation of string theory in this space one can relate both theories through [CITATION] [EQUATION] where [MATH] is a non-normalizable supergravity field, [MATH] is the corresponding on shell supergravity action, [MATH] is the value of [MATH] at the boundary [MATH], and [MATH] is the associated operator of the conformal field theory (CFT).', '1301.4457-3-6-2': 'From this equation, one can obtain four-dimensional correlation functions, for instance [EQUATION]', '1301.4457-3-6-3': 'In particular, the scalar glueball [MATH] is represented by the operator [MATH] associated with a dilaton in the AdS[MATH] space.', '1301.4457-3-7-0': '# Odd spin glueball masses in the Hardwall model', '1301.4457-3-8-0': 'Glueballs are characterized by [MATH], where [MATH] represents the total angular momentum, [MATH] defines how a state behaves under spatial inversion ([MATH]-parity), and [MATH] shows the behavior of a state under charge conjugation ([MATH]-parity).', '1301.4457-3-9-0': 'In this paper we are interested in glueballs in the [MATH] and [MATH] sector with odd spins [MATH], which are associated with a particle called the odderon.', '1301.4457-3-9-1': 'The concept of the odderon emerged in the 1970s [CITATION] within the context of asymptotic theorems, reappearing later in perturbative QCD [CITATION].', '1301.4457-3-9-2': 'The odderon has also been linked, for instance, to the color glass condensate [CITATION].', '1301.4457-3-9-3': 'Although the odderon has not been detected so far, it is regarded as a crucial test of QCD [CITATION].', '1301.4457-3-9-4': 'The odderon is a bound state of three gluons, without color, which represents a singularity in the complex plane [MATH], close to [MATH], in the odd-under-crossing amplitude [MATH] [CITATION].', '1301.4457-3-10-0': 'The best experimental evidence for the odderon occurred in 1985 at ISR CERN.', '1301.4457-3-10-1': 'A difference between differential cross sections for [MATH] and [MATH] in the dip-shoulder region [MATH] GeV[MATH] at [MATH] GeV was measured, but these results were not confirmed [CITATION].', '1301.4457-3-10-2': 'There are two more evidences related to the nonperturbative odderon, that is, the change of shape in the polarization in [MATH] from [MATH] GeV/[MATH] [CITATION] to [MATH] GeV/[MATH] [CITATION] and a strange structure seen in the [MATH] data for [MATH] scattering at [MATH] GeV, namely a bump centered at [MATH] GeV[MATH] [CITATION].', '1301.4457-3-10-3': 'Some other experiments to detect the odderon were proposed for the HERA [CITATION] and recently for the LHC CERN through the study of coherent hadron-hadron interactions [CITATION].', '1301.4457-3-10-4': 'In this article, we are dealing with the nonperturbative odderon that is related with glueball states via its Regge trajectory, although the Regge trajectory of the odderon is not yet well understood.', '1301.4457-3-10-5': 'For reviews see, for instance, [CITATION].', '1301.4457-3-11-0': 'The AdS/CFT correspondence cannot be used directly as a tool for the study of hadrons because the dual theory is a supersymmetric conformal theory that is very different from QCD.', '1301.4457-3-11-1': 'However, it was noticed that the energy [MATH] of a process in the [MATH] theory is related to the radial coordinate [MATH] in AdS space as [EQUATION]', '1301.4457-3-11-2': 'This motivated the holographic hardwall model proposed by Polchinski and Strassler [CITATION] to calculate the scattering of glueballs in four dimensions using a dilaton field in AdS[MATH] space.', '1301.4457-3-11-3': 'The works [CITATION] introduced a cutoff at a certain value [MATH] of the [MATH] coordinate and considered an AdS slice in the region [MATH].', '1301.4457-3-11-4': 'An immediate consequence of introducing a cutoff is the breaking of conformal invariance, so that particles on the four-dimensional boundary acquire mass.', '1301.4457-3-11-5': 'Furthermore, one can associate the size of the AdS slice with the energy scale of QCD [EQUATION]', '1301.4457-3-11-6': 'Hadron masses can be determined using the hardwall model with a given mass scale (infrared cutoff).', '1301.4457-3-11-7': 'This can be used to build up Regge trajectories for the hadrons, as it was done in Refs. [CITATION].', '1301.4457-3-11-8': 'One can note that the asymptotic behavior of these Regge trajectories is not linear.', '1301.4457-3-11-9': 'Despite this problem one can find approximate linear Regge trajectories for the first few light states of each hadronic branch.', '1301.4457-3-11-10': 'It should be noticed that there is another holographic model that presents exact linear Regge trajectories: the softwall model [CITATION].', '1301.4457-3-11-11': 'The analysis presented in [CITATION] for vector mesons can be extended to glueballs as was done in [CITATION].', '1301.4457-3-11-12': 'However, despite the fact that the Regge trajectories are linear in this case, the glueball masses are too low compared with lattice data.', '1301.4457-3-11-13': 'So, in the following, we are going to use the hardwall model to study the odd spin glueball masses and to obtain the odderon Regge trajectories.', '1301.4457-3-12-0': 'The hardwall model assumes an approximate duality between a string theory in an AdS[MATH] space with metric defined by [EQUATION] where [MATH] is the AdS radius, and a pure Yang-Mills theory in four dimensions with symmetry group [MATH] is in the large [MATH] limit.', '1301.4457-3-12-1': 'In this model it is assumed that the AdS/CFT dictionary between supergravity fields in AdS[MATH] space and operators on the [MATH] boundary, as given by Eqs. ([REF]) and ([REF]), still holds after breaking the conformal invariance.', '1301.4457-3-12-2': 'This implies that the conformal dimension [MATH] of an operator [MATH] related to a [MATH]-form AdS[MATH] field with mass [MATH] is given by [CITATION] (here and in the following we are disregarding excitations on the [MATH] subspace) [EQUATION]', '1301.4457-3-12-3': 'In particular, the operator that describes the glueball [MATH] is [EQUATION] with conformal dimension [MATH].', '1301.4457-3-12-4': 'This operator is associated with the Ramond-Ramond tensor [MATH] described in a single [MATH]-brane, by the action [CITATION] [EQUATION]', '1301.4457-3-12-5': 'From this action one can obtain the equations of motion for the Ramond-Ramond field.', '1301.4457-3-12-6': 'With a suitable polarization choice [MATH] where [MATH] is a constant polarization tensor and [MATH] is a scalar field, it can be shown that these equations can be reduced to [CITATION] [EQUATION] where [MATH] is the four-dimensional Minkowski metric.', '1301.4457-3-13-0': 'We use a plane wave ansatz in the four-dimensional space for the [MATH]-form field [MATH] [EQUATION] where [MATH] is a normalization constant, [MATH] is the Bessel function of order [MATH] with [MATH], and the discrete modes [MATH] corresponding to the glueball masses will be calculated by imposing appropriate boundary conditions.', '1301.4457-3-13-1': 'Note that [MATH] represent radial excitations of glueballs, but we will only consider in this paper the case [MATH].', '1301.4457-3-14-0': 'It has been proposed in the literature [CITATION] that the glueball operator with spin [MATH] could be obtained by the insertion of symmetrized covariant derivatives in the operator [MATH], such that [MATH] with conformal dimension [MATH].', '1301.4457-3-14-1': 'This approach was used in Ref. [CITATION] to calculate the masses of glueball states [MATH], [MATH], [MATH], [MATH], etc and to obtain the corresponding Pomeron Regge trajectory.', '1301.4457-3-15-0': 'Here we are going to follow a similar approach for the glueball states [MATH], [MATH], [MATH], [MATH], [MATH].', '1301.4457-3-15-1': 'The state [MATH] is described by the operator [MATH].', '1301.4457-3-15-2': 'Inserting covariant derivatives as described above, one obtains [MATH] with [MATH] satisfying equations similar to ([REF]) and ([REF]) with a shift in the index of the Bessel function [MATH], where [MATH] is the spin of each state [MATH], [MATH], [MATH], etc.', '1301.4457-3-16-0': 'Following the approach of Ref. [CITATION], we impose Dirichlet and Neumann boundary conditions to calculate glueball masses within the hardwall model.', '1301.4457-3-16-1': 'For the Dirichlet boundary condition [EQUATION] one obtains from ([REF]), the following masses: [EQUATION]', '1301.4457-3-16-2': 'On the other hand, for the Neumann boundary condition [EQUATION] one gets [EQUATION] and the masses are now [EQUATION]', '1301.4457-3-16-3': 'Using these boundary conditions we obtain glueball masses in the sector [MATH].', '1301.4457-3-16-4': 'We take the mass [MATH] of the state [MATH] from the isotropic lattice ([MATH] GeV) found in Refs. [CITATION] to fix [MATH] (and [MATH], and then calculate the other odd spin glueball masses [MATH] for the states [MATH], [MATH], [MATH], using Eqs. ([REF]) and ([REF]), respectively, for the Dirichlet and Neumann boundary conditions.', '1301.4457-3-16-5': 'For instance, for the state [MATH] with the Dirichlet boundary condition we have [EQUATION] so we get 4.09 GeV for the mass [MATH] of the [MATH] state, etc.', '1301.4457-3-16-6': 'A similar calculation is done for the Neumann boundary condition.', '1301.4457-3-16-7': 'Our results are shown in Table I.', '1301.4457-3-16-8': 'We also show for comparison the values for these masses found in the literature [CITATION] using other methods.', '1301.4457-3-16-9': 'Then from our results we obtain different Regge trajectories for the odderon as discussed in the next section.', '1301.4457-3-17-0': '# Odderon Regge trajectories in the Hardwall model', '1301.4457-3-18-0': 'Taking the data for odd spin glueball masses obtained in the previous section we are going to build up the Regge trajectories for the odderon, using linear regression.', '1301.4457-3-19-0': 'For the Dirichlet boundary condition and the set of states, [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-3-19-1': 'The errors for the slope and linear coefficient come from the linear fit.', '1301.4457-3-19-2': 'The plot relative to this trajectory can be seen in Fig. [MATH].', '1301.4457-3-19-3': 'This result is in agreement with that found in [CITATION], with the relativistic many-body Hamiltonian formulation, described by Eq. ([REF]).', '1301.4457-3-20-0': 'It has been argued in Ref. [CITATION] that the state [MATH] might not be part of the spectrum of the odderon.', '1301.4457-3-20-1': 'To test this possibility we also consider another set of states, [MATH], [MATH], [MATH], [MATH], for which we obtain the following Regge trajectory: [EQUATION]', '1301.4457-3-20-2': 'This result is also consistent with the Regge trajectory for odderon, Eq. ([REF]).', '1301.4457-3-21-0': 'Now using the Neumann boundary condition and the set of states, [MATH], [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-3-21-1': 'The plot relative to this trajectory can be seen in Fig. 2.', '1301.4457-3-22-0': 'We also consider here the possibility of excluding the state [MATH] from the spectrum of the odderon.', '1301.4457-3-22-1': 'For the set of states [MATH], we find the following Regge trajectory: [EQUATION]', '1301.4457-3-22-2': 'This result is in agreement with that found in [CITATION], with the nonrelativistic constituent model, Eq. ([REF]).', '1301.4457-3-23-0': '# conclusions', '1301.4457-3-24-0': 'In this work we obtained odd spin glueball masses in the sector [MATH] using the holographic hardwall model with Dirichlet and Neumann boundary conditions.', '1301.4457-3-24-1': 'These glueball masses lie in approximate linear Regge trajectories compatible with results for the odderon, both in the relativistic many-body as well in the nonrelativistic constituent models presented in Ref. [CITATION].', '1301.4457-3-24-2': 'The present analysis gives support to the conclusion of Ref. [CITATION] about the general properties of the odderon Regge trajectories, i.e., a low intercept and a slope similar to that of the Pomeron.', '1301.4457-3-25-0': 'Some aspects of the holographic approach for the odderon Regge trajectories remain open.', '1301.4457-3-25-1': 'In our approach, we used Dirichlet and Neumann boundary conditions in the hardwall model obtaining results compatible with those of Ref. [CITATION].', '1301.4457-3-25-2': 'The hardwall model was used before to obtain the Regge trajectory for the Pomeron in Ref. [CITATION].', '1301.4457-3-25-3': 'In that work it was possible to conclude that the Neumann boundary condition is more appropriate than the Dirichlet boundary condition by comparison with experimental data.', '1301.4457-3-25-4': 'Here in this work, it is not possible to reach a similar conclusion about boundary conditions because there is no clear experimental data for the odderon Regge trajectories.', '1301.4457-3-26-0': 'Another open question in the odderon Regge trajectories regards the state [MATH].', '1301.4457-3-26-1': 'It was argued in Ref. [CITATION] that the glueball state [MATH] does not belong to the odderon Regge trajectory.', '1301.4457-3-26-2': 'However, our analysis is not conclusive regarding this point since we have found trajectories compatible with odderon including the state [MATH] [Eqs. ([REF]) and ([REF])] as well as excluding it [Eqs. ([REF]) and ([REF])].', '1301.4457-3-27-0': 'As a final remark, let us comment on our choice for the holographic model to obtain glueball masses and the odderon Regge trajectories.', '1301.4457-3-27-1': 'This model is very interesting since masses can be obtained from the zeros of the corresponding Bessel functions.', '1301.4457-3-27-2': 'However, it is well known that a holographic hardwall model leads to asymptotic nonlinear Regge trajectories for very high states.', '1301.4457-3-27-3': 'Nevertheless, for light states, as discussed in this work, approximate linear Regge trajectories were found.', '1301.4457-3-27-4': 'In this regard, it will be interesting to investigate the glueball masses in the [MATH] sector within other holographic approaches, such as the softwall model [CITATION], which is known to provide linear Regge trajectories.', '1301.4457-3-27-5': 'We leave this study for future work.', '1301.4457-3-28-0': 'We would like to thank Felipe J. Llanes-Estrada for interesting discussions and Nelson R. F. Braga for a careful reading of the manuscript.', '1301.4457-3-28-1': 'The authors are partially supported by CAPES, CNPq and FAPERJ, Brazilian agencies.'}

1709.05614

{'1709.05614-1-0-0': 'Let us concern the quasi-periodic Schrodinger operator in the continuous case, [EQUATION] where [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable.', '1709.05614-1-0-1': 'Let [MATH] be the Lyapunov exponent of [MATH].', '1709.05614-1-0-2': 'Define [MATH] as [EQUATION]', '1709.05614-1-0-3': 'We prove that [MATH] admits no eigenvalue in regime [MATH].', '1709.05614-1-1-0': '# Introduction', '1709.05614-1-2-0': 'In this note, we study the continuous quasi-periodic Schrodinger operator, which is given by [EQUATION] where [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable and [MATH] is frequency.', '1709.05614-1-2-1': 'Here [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable means [MATH] is measurable and there exists [MATH] and [MATH]) such that [EQUATION]', '1709.05614-1-2-2': 'Such class of function contains a special case [MATH], where [MATH] is measurable and [MATH] is piecewisely [MATH]-Holder continuous.', '1709.05614-1-3-0': 'Recently, there has been a remarkable development of arithmetically spectral transition for discrete quasiperiodic operator [CITATION], in particular for explicit models: almost Mathieu operator [CITATION], Maryland model [CITATION] and extended Harper model [CITATION](Jacobi operator).', '1709.05614-1-3-1': 'One important part of their quantitative arguments is to show the absence of eigenvalues if the frequency can be approximated by a rational number well (Gordon type potential in one dimension) , which improved the previous results [CITATION], obtaining sharp thresholds for the smallness of small denominators in terms of the Lyapunov exponents.', '1709.05614-1-3-2': "For various other recent developments on the Gordon-type potentials, see Damanik-Stolz [CITATION], Damanik's survey paper [CITATION] and references therein, and Jitomirskaya-Zhang [CITATION].", '1709.05614-1-3-3': 'The purpose of this note is to obtain a similar sharp result for the continuous case,', '1709.05614-1-4-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable.', '1709.05614-1-4-1': 'Let [MATH] be the Lyapunov exponent of [MATH].', '1709.05614-1-4-2': 'Define [MATH] as [EQUATION]', '1709.05614-1-4-3': 'Then [MATH] does not have any eigenvalue in the regime [MATH].', '1709.05614-1-5-0': 'As a direct corollary, we obtain', '1709.05614-1-6-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is a Holder continuous with respect to the second variable.', '1709.05614-1-6-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-1-6-2': 'Then [MATH] does not have any eigenvalue.', '1709.05614-1-7-0': 'If the analytical potential [MATH] is small or the energy [MATH] is large, the Lyapunov exponent is zero in the spectrum [CITATION].', '1709.05614-1-7-1': 'Thus we have the following result.', '1709.05614-1-8-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is analytic.', '1709.05614-1-8-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-1-8-2': 'Then there exists [MATH] such that [MATH] does not have any eigenvalue for [MATH].', '1709.05614-1-9-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is analytic.', '1709.05614-1-9-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-1-9-2': 'Then there exists [MATH] such that [MATH] does not have any eigenvalue in regime [MATH].', '1709.05614-1-10-0': '# Proof of Theorem [REF]', '1709.05614-1-11-0': 'Let [MATH] be a solution of [MATH].', '1709.05614-1-11-1': 'Define the transfer matrix [MATH] as [EQUATION] [CITATION] Let [MATH] and [MATH] be a unit vector in [MATH], then [EQUATION].', '1709.05614-1-12-0': 'Let [MATH] be the continued fraction expansion to [MATH].', '1709.05614-1-12-1': 'By [REF], one has [EQUATION]', '1709.05614-1-12-2': 'Below, [MATH] is arbitrarily small, and [MATH]) is small (large) constant depending on [MATH], [MATH] and the potential [MATH].', '1709.05614-1-13-0': 'Since [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable, there exists [MATH] and [MATH]) such that [REF] holds.', '1709.05614-1-14-0': 'Write down the eigen-equation [MATH] in first order, [EQUATION]', '1709.05614-1-14-1': 'By the definition of transfer matrix and Lyapunov exponent, we have [EQUATION]', '1709.05614-1-14-2': 'For simplicity, sometimes we ignore the dependence of [MATH] in [MATH] and [MATH].', '1709.05614-1-15-0': 'By [REF], there exists a sequence [MATH] such that [EQUATION]', '1709.05614-1-15-1': 'By the property of continued fraction expansion, one has [EQUATION] where [MATH].', '1709.05614-1-16-0': 'We start the proof with some basic Lemmata.', '1709.05614-1-17-0': 'The following estimate holds, [EQUATION]', '1709.05614-1-17-1': 'Let [EQUATION]', '1709.05614-1-17-2': 'By [REF] and the assumption on [MATH], one has [EQUATION]', '1709.05614-1-17-3': 'We also have [EQUATION] where [MATH].', '1709.05614-1-17-4': 'Thus we obtain [EQUATION]', '1709.05614-1-17-5': 'The following estimates hold [EQUATION] and [EQUATION]', '1709.05614-1-17-6': 'Consider the two differential equations [MATH] on [MATH] with the same initial condition at [MATH], [EQUATION]', '1709.05614-1-17-7': 'Let [MATH].', '1709.05614-1-18-0': 'Denote by [EQUATION] and [EQUATION]', '1709.05614-1-18-1': 'Thus [EQUATION] and [MATH], since [MATH].', '1709.05614-1-18-2': 'By the constant variation method, we obtain that [EQUATION]', '1709.05614-1-18-3': 'By Lemma [REF],', '1709.05614-1-19-0': 'we have for [MATH], [EQUATION] where the second inequality holds by [REF].', '1709.05614-1-20-0': 'For [MATH], by [REF] again, one has [EQUATION]', '1709.05614-1-20-1': 'By [REF], [REF] and [REF], we obtain [EQUATION] which implies [REF] by the arbitrary choice of [MATH].', '1709.05614-1-21-0': 'By the fact that [MATH] and following the similar arguments of proof of [REF], we can prove [REF].', '1709.05614-1-22-0': 'The following lemma is well known for Schrodinger operator.', '1709.05614-1-22-1': 'We give the proof for completeness.', '1709.05614-1-23-0': 'Suppose [MATH] is an eigensolution, that is [MATH] and [MATH] for some [MATH], then [EQUATION]', '1709.05614-1-23-1': 'Suppose [EQUATION]', '1709.05614-1-23-2': 'By equation [REF], one has [EQUATION]', '1709.05614-1-23-3': 'By a result of [CITATION], we have [EQUATION]', '1709.05614-1-23-4': 'Thus [EQUATION] which leads to [REF] by the fact that [MATH].', '1709.05614-1-24-0': 'Proof of Theorem [REF]', '1709.05614-1-25-0': 'Let [MATH] be such that [MATH].', '1709.05614-1-25-1': 'Suppose [MATH] and [MATH].', '1709.05614-1-25-2': 'Setting [MATH] and applying Lemma [REF], we have [EQUATION] where [MATH] is unit.', '1709.05614-1-25-3': 'We claim that [EQUATION]', '1709.05614-1-25-4': 'If [MATH], there is nothing to prove.', '1709.05614-1-25-5': 'If [MATH], by [REF], [EQUATION]', '1709.05614-1-25-6': 'If [MATH] and [MATH], by [REF], [EQUATION]', '1709.05614-1-25-7': 'This implies [EQUATION]', '1709.05614-1-25-8': 'We finish the proof of the claim.', '1709.05614-1-25-9': 'By Lemma [REF], this is impossible.'}

{'1709.05614-2-0-0': 'Let us concern the quasi-periodic Schrodinger operator in the continuous case, [EQUATION] where [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable.', '1709.05614-2-0-1': 'Let [MATH] be the Lyapunov exponent of [MATH].', '1709.05614-2-0-2': 'Define [MATH] as [EQUATION]', '1709.05614-2-0-3': 'We prove that [MATH] admits no eigenvalue in regime [MATH].', '1709.05614-2-1-0': '# Introduction', '1709.05614-2-2-0': 'In this note, we study the continuous quasi-periodic Schrodinger operator, which is given by [EQUATION] where [MATH] is the potential and [MATH] is frequency.', '1709.05614-2-3-0': 'We are interested in a particular class of functions [MATH].', '1709.05614-2-3-1': '[MATH] is called piecewisely [MATH]-Holder continuous with respect to the second variable, denoted by [MATH], if [MATH] is measurable and there exist [MATH] and [MATH]) such that [EQUATION]', '1709.05614-2-3-2': 'We emphasize that [MATH] implies [MATH] is bounded and continuous with respect to [MATH].', '1709.05614-2-3-3': '[MATH] contains a special case [MATH], where [MATH] is bounded measurable function and [MATH] is piecewisely [MATH]-Holder continuous.', '1709.05614-2-3-4': 'We always assume that potentials [MATH] in this paper.', '1709.05614-2-4-0': 'Recently, there has been a remarkable development of arithmetically spectral transition (singular spectrum and Anderson localization) for discrete quasiperiodic operator [CITATION], in particular for explicit models: almost Mathieu operator [CITATION], Maryland model [CITATION] and extended Harper model [CITATION](Jacobi operator).', '1709.05614-2-4-1': 'For the continuous, arithmetic phase transitions are currently very far from being established.', '1709.05614-2-4-2': 'The quantitative arguments for discrete case is to show the absence of eigenvalues if the frequency can be approximated by a rational number well (Gordon type potential in one dimension) , which improved the previous results [CITATION], obtaining sharp thresholds for the smallness of small denominators in terms of the Lyapunov exponents.', '1709.05614-2-4-3': "For various other recent developments on the Gordon-type potentials, see Damanik-Stolz [CITATION], Damanik's survey paper [CITATION] and references therein, and Jitomirskaya-Zhang [CITATION].", '1709.05614-2-4-4': 'The purpose of this note is to obtain a similar sharp result for the continuous case.', '1709.05614-2-4-5': 'We should mention that the localization part for continuous case is only known for a full Lebesgue measure subset of Diophantine frequencies [CITATION].', '1709.05614-2-5-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is piecewisely [MATH]-Holder continuous with respect to the second variable.', '1709.05614-2-5-1': 'Let [MATH] be the Lyapunov exponent of [MATH].', '1709.05614-2-5-2': 'Define [MATH] as [EQUATION]', '1709.05614-2-5-3': 'Then [MATH] does not have any eigenvalue in the regime [MATH].', '1709.05614-2-6-0': 'As a direct corollary, we obtain', '1709.05614-2-7-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is a Holder continuous with respect to the second variable.', '1709.05614-2-7-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-2-7-2': 'Then [MATH] does not have any eigenvalue.', '1709.05614-2-8-0': 'If the analytical potential [MATH] is small or the energy [MATH] is large, the Lyapunov exponent is zero in the spectrum [CITATION].', '1709.05614-2-8-1': 'Thus we have the following result.', '1709.05614-2-9-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is analytic.', '1709.05614-2-9-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-2-9-2': 'Then there exists [MATH] such that [MATH] does not have any eigenvalue for [MATH].', '1709.05614-2-10-0': 'Let [MATH] be a quasiperiodic Schrodinger operator, [EQUATION] where [MATH] is analytic.', '1709.05614-2-10-1': 'Suppose the frequency [MATH] satisfies [EQUATION]', '1709.05614-2-10-2': 'Then there exists [MATH] such that [MATH] does not have any eigenvalue in regime [MATH].', '1709.05614-2-11-0': '# Proof of Theorem [REF]', '1709.05614-2-12-0': 'Let [MATH] be a solution of [MATH].', '1709.05614-2-12-1': 'Define the transfer matrix [MATH] as [EQUATION] [CITATION] Let [MATH] and [MATH] be a unit vector in [MATH], then [EQUATION].', '1709.05614-2-13-0': 'Let [MATH] be the continued fraction expansion to [MATH].', '1709.05614-2-13-1': 'By [REF], one has [EQUATION]', '1709.05614-2-13-2': 'Below, [MATH] is arbitrarily small, and [MATH]) is small (large) constant depending on [MATH], [MATH] and the potential [MATH].', '1709.05614-2-14-0': 'Since [MATH], there exist [MATH] and [MATH]) such that [REF] holds.', '1709.05614-2-15-0': 'Write down the eigen-equation [MATH] in first order, [EQUATION]', '1709.05614-2-15-1': 'By the definition of transfer matrix and Lyapunov exponent (see [CITATION]), we have [EQUATION]', '1709.05614-2-15-2': 'For simplicity, sometimes we ignore the dependence of [MATH] in [MATH] and [MATH].', '1709.05614-2-16-0': 'By [REF], there exists a sequence [MATH] such that [EQUATION]', '1709.05614-2-16-1': 'By the property of continued fraction expansion, one has [EQUATION] where [MATH].', '1709.05614-2-17-0': 'We start the proof with some basic Lemmas.', '1709.05614-2-18-0': 'The following estimate holds, [EQUATION]', '1709.05614-2-18-1': 'Let [EQUATION]', '1709.05614-2-18-2': 'By [REF] and the assumption on [MATH], one has [EQUATION]', '1709.05614-2-18-3': 'We also have [EQUATION] where [MATH].', '1709.05614-2-18-4': 'Thus we obtain [EQUATION]', '1709.05614-2-18-5': 'The following estimates hold [EQUATION] and [EQUATION]', '1709.05614-2-18-6': 'Consider the two differential equations [MATH] on [MATH] with the same initial condition at [MATH], [EQUATION] and [EQUATION]', '1709.05614-2-18-7': 'Let [MATH].', '1709.05614-2-19-0': 'Denote by [EQUATION] and [EQUATION]', '1709.05614-2-19-1': 'Thus [EQUATION] and [MATH], since [MATH].', '1709.05614-2-19-2': 'By the constant variation method, we obtain that [EQUATION]', '1709.05614-2-19-3': 'We give the proof of [REF] in the Appendix.', '1709.05614-2-20-0': 'By Lemma [REF],', '1709.05614-2-21-0': 'we have for [MATH], [EQUATION] where the second inequality holds by [REF].', '1709.05614-2-22-0': 'For [MATH], by [REF] again, one has [EQUATION]', '1709.05614-2-22-1': 'By [REF], [REF] and [REF], we obtain [EQUATION] which implies [REF] by the arbitrary choice of [MATH].', '1709.05614-2-23-0': 'By the fact that [MATH] and following the similar arguments of proof of [REF], we can prove [REF].', '1709.05614-2-24-0': 'The following lemma is well known for Schrodinger operator.', '1709.05614-2-24-1': 'We give the proof for completeness.', '1709.05614-2-25-0': 'Suppose [MATH] is an eigensolution, that is [MATH] and [MATH] for some [MATH], then [EQUATION]', '1709.05614-2-25-1': 'Suppose [EQUATION]', '1709.05614-2-25-2': 'By equation [REF], one has [EQUATION]', '1709.05614-2-25-3': 'By a result of [CITATION], we have [EQUATION]', '1709.05614-2-25-4': 'Thus [EQUATION] which leads to [REF] by the fact that [MATH].', '1709.05614-2-26-0': 'Proof of Theorem [REF]', '1709.05614-2-27-0': 'Let [MATH] be such that [MATH].', '1709.05614-2-27-1': 'Suppose [MATH] and [MATH].', '1709.05614-2-27-2': 'Setting [MATH] and applying Lemma [REF], we have [EQUATION] where [MATH] is unit.', '1709.05614-2-27-3': 'We claim that [EQUATION]', '1709.05614-2-27-4': 'If [MATH], there is nothing to prove.', '1709.05614-2-27-5': 'If [MATH], by [REF], [EQUATION]', '1709.05614-2-27-6': 'If [MATH] and [MATH], by [REF], [EQUATION]', '1709.05614-2-27-7': 'This implies [EQUATION]', '1709.05614-2-27-8': 'We finish the proof of the claim.', '1709.05614-2-27-9': 'By Lemma [REF], this is impossible.', '1709.05614-2-28-0': '# Appendix.', '1709.05614-2-28-1': 'Proof of [REF]', '1709.05614-2-29-0': 'Let [MATH] be [EQUATION]', '1709.05614-2-29-1': 'Obviously, [EQUATION]', '1709.05614-2-29-2': 'By the definition of [MATH] and [REF], we have [EQUATION]', '1709.05614-2-29-3': 'By [REF] and [REF], we have [EQUATION]', '1709.05614-2-29-4': 'Thus [MATH] and [MATH] satisfy the same differential equation and initial condition.', '1709.05614-2-29-5': 'This implies [MATH].'}

[['1709.05614-1-19-0', '1709.05614-2-21-0'], ['1709.05614-1-6-0', '1709.05614-2-7-0'], ['1709.05614-1-6-1', '1709.05614-2-7-1'], ['1709.05614-1-6-2', '1709.05614-2-7-2'], ['1709.05614-1-25-0', '1709.05614-2-27-0'], ['1709.05614-1-25-1', '1709.05614-2-27-1'], ['1709.05614-1-25-2', '1709.05614-2-27-2'], ['1709.05614-1-25-3', '1709.05614-2-27-3'], ['1709.05614-1-25-4', '1709.05614-2-27-4'], ['1709.05614-1-25-8', '1709.05614-2-27-8'], ['1709.05614-1-25-9', '1709.05614-2-27-9'], ['1709.05614-1-0-0', '1709.05614-2-0-0'], ['1709.05614-1-0-1', '1709.05614-2-0-1'], ['1709.05614-1-0-2', '1709.05614-2-0-2'], ['1709.05614-1-0-3', '1709.05614-2-0-3'], ['1709.05614-1-3-2', '1709.05614-2-4-3'], ['1709.05614-1-15-0', '1709.05614-2-16-0'], ['1709.05614-1-15-1', '1709.05614-2-16-1'], ['1709.05614-1-17-0', '1709.05614-2-18-0'], ['1709.05614-1-17-2', '1709.05614-2-18-2'], ['1709.05614-1-17-3', '1709.05614-2-18-3'], ['1709.05614-1-17-4', '1709.05614-2-18-4'], ['1709.05614-1-17-5', '1709.05614-2-18-5'], ['1709.05614-1-18-0', '1709.05614-2-19-0'], ['1709.05614-1-18-1', '1709.05614-2-19-1'], ['1709.05614-1-18-2', '1709.05614-2-19-2'], ['1709.05614-1-23-0', '1709.05614-2-25-0'], ['1709.05614-1-23-2', '1709.05614-2-25-2'], ['1709.05614-1-23-3', '1709.05614-2-25-3'], ['1709.05614-1-23-4', '1709.05614-2-25-4'], ['1709.05614-1-9-0', '1709.05614-2-10-0'], ['1709.05614-1-9-1', '1709.05614-2-10-1'], ['1709.05614-1-9-2', '1709.05614-2-10-2'], ['1709.05614-1-7-0', '1709.05614-2-8-0'], ['1709.05614-1-7-1', '1709.05614-2-8-1'], ['1709.05614-1-4-0', '1709.05614-2-5-0'], ['1709.05614-1-4-1', '1709.05614-2-5-1'], ['1709.05614-1-4-2', '1709.05614-2-5-2'], ['1709.05614-1-4-3', '1709.05614-2-5-3'], ['1709.05614-1-14-0', '1709.05614-2-15-0'], ['1709.05614-1-14-2', '1709.05614-2-15-2'], ['1709.05614-1-8-0', '1709.05614-2-9-0'], ['1709.05614-1-8-1', '1709.05614-2-9-1'], ['1709.05614-1-8-2', '1709.05614-2-9-2'], ['1709.05614-1-12-0', '1709.05614-2-13-0'], ['1709.05614-1-12-1', '1709.05614-2-13-1'], ['1709.05614-1-12-2', '1709.05614-2-13-2'], ['1709.05614-1-21-0', '1709.05614-2-23-0'], ['1709.05614-1-22-0', '1709.05614-2-24-0'], ['1709.05614-1-22-1', '1709.05614-2-24-1'], ['1709.05614-1-3-0', '1709.05614-2-4-0'], ['1709.05614-1-3-1', '1709.05614-2-4-2'], ['1709.05614-1-17-6', '1709.05614-2-18-6'], ['1709.05614-1-14-1', '1709.05614-2-15-1'], ['1709.05614-1-2-0', '1709.05614-2-2-0'], ['1709.05614-1-2-2', '1709.05614-2-3-3'], ['1709.05614-1-2-1', '1709.05614-2-3-1']]

[['1709.05614-1-19-0', '1709.05614-2-21-0'], ['1709.05614-1-6-0', '1709.05614-2-7-0'], ['1709.05614-1-6-1', '1709.05614-2-7-1'], ['1709.05614-1-6-2', '1709.05614-2-7-2'], ['1709.05614-1-25-0', '1709.05614-2-27-0'], ['1709.05614-1-25-1', '1709.05614-2-27-1'], ['1709.05614-1-25-2', '1709.05614-2-27-2'], ['1709.05614-1-25-3', '1709.05614-2-27-3'], ['1709.05614-1-25-4', '1709.05614-2-27-4'], ['1709.05614-1-25-8', '1709.05614-2-27-8'], ['1709.05614-1-25-9', '1709.05614-2-27-9'], ['1709.05614-1-0-0', '1709.05614-2-0-0'], ['1709.05614-1-0-1', '1709.05614-2-0-1'], ['1709.05614-1-0-2', '1709.05614-2-0-2'], ['1709.05614-1-0-3', '1709.05614-2-0-3'], ['1709.05614-1-3-2', '1709.05614-2-4-3'], ['1709.05614-1-15-0', '1709.05614-2-16-0'], ['1709.05614-1-15-1', '1709.05614-2-16-1'], ['1709.05614-1-17-0', '1709.05614-2-18-0'], ['1709.05614-1-17-2', '1709.05614-2-18-2'], ['1709.05614-1-17-3', '1709.05614-2-18-3'], ['1709.05614-1-17-4', '1709.05614-2-18-4'], ['1709.05614-1-17-5', '1709.05614-2-18-5'], ['1709.05614-1-18-0', '1709.05614-2-19-0'], ['1709.05614-1-18-1', '1709.05614-2-19-1'], ['1709.05614-1-18-2', '1709.05614-2-19-2'], ['1709.05614-1-23-0', '1709.05614-2-25-0'], ['1709.05614-1-23-2', '1709.05614-2-25-2'], ['1709.05614-1-23-3', '1709.05614-2-25-3'], ['1709.05614-1-23-4', '1709.05614-2-25-4'], ['1709.05614-1-9-0', '1709.05614-2-10-0'], ['1709.05614-1-9-1', '1709.05614-2-10-1'], ['1709.05614-1-9-2', '1709.05614-2-10-2'], ['1709.05614-1-7-0', '1709.05614-2-8-0'], ['1709.05614-1-7-1', '1709.05614-2-8-1'], ['1709.05614-1-4-0', '1709.05614-2-5-0'], ['1709.05614-1-4-1', '1709.05614-2-5-1'], ['1709.05614-1-4-2', '1709.05614-2-5-2'], ['1709.05614-1-4-3', '1709.05614-2-5-3'], ['1709.05614-1-14-0', '1709.05614-2-15-0'], ['1709.05614-1-14-2', '1709.05614-2-15-2'], ['1709.05614-1-8-0', '1709.05614-2-9-0'], ['1709.05614-1-8-1', '1709.05614-2-9-1'], ['1709.05614-1-8-2', '1709.05614-2-9-2'], ['1709.05614-1-12-0', '1709.05614-2-13-0'], ['1709.05614-1-12-1', '1709.05614-2-13-1'], ['1709.05614-1-12-2', '1709.05614-2-13-2'], ['1709.05614-1-21-0', '1709.05614-2-23-0'], ['1709.05614-1-22-0', '1709.05614-2-24-0'], ['1709.05614-1-22-1', '1709.05614-2-24-1']]

[['1709.05614-1-3-0', '1709.05614-2-4-0'], ['1709.05614-1-3-1', '1709.05614-2-4-2'], ['1709.05614-1-17-6', '1709.05614-2-18-6'], ['1709.05614-1-14-1', '1709.05614-2-15-1'], ['1709.05614-1-2-0', '1709.05614-2-2-0'], ['1709.05614-1-2-2', '1709.05614-2-3-3']]

[]

[['1709.05614-1-2-1', '1709.05614-2-3-1']]

[]

['1709.05614-1-3-3', '1709.05614-1-5-0', '1709.05614-1-11-0', '1709.05614-1-11-1', '1709.05614-1-16-0', '1709.05614-1-17-1', '1709.05614-1-17-7', '1709.05614-1-18-3', '1709.05614-1-20-0', '1709.05614-1-20-1', '1709.05614-1-23-1', '1709.05614-1-24-0', '1709.05614-1-25-5', '1709.05614-1-25-6', '1709.05614-1-25-7', '1709.05614-2-6-0', '1709.05614-2-12-0', '1709.05614-2-12-1', '1709.05614-2-14-0', '1709.05614-2-17-0', '1709.05614-2-18-1', '1709.05614-2-18-7', '1709.05614-2-20-0', '1709.05614-2-22-0', '1709.05614-2-22-1', '1709.05614-2-25-1', '1709.05614-2-26-0', '1709.05614-2-27-5', '1709.05614-2-27-6', '1709.05614-2-27-7', '1709.05614-2-28-1', '1709.05614-2-29-0', '1709.05614-2-29-1', '1709.05614-2-29-2', '1709.05614-2-29-3', '1709.05614-2-29-4', '1709.05614-2-29-5']

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

https://arxiv.org/abs/1709.05614

hep-ph-0406253

{'hep-ph-0406253-1-0-0': 'We consider the production of neutralino, chargino, and charged Higgs boson pairs in the MSSM framework at future [MATH] colliders.', 'hep-ph-0406253-1-0-1': 'We show that, for c.m. energies in the one TeV range and in a "moderately" light SUSY scenario, a combined analysis of the slopes of these production cross sections could lead to a strong consistency test involving the soft supersymmetric breaking parameters [MATH], [MATH], the Higgsino parameter [MATH], and [MATH].', 'hep-ph-0406253-1-1-0': 'It is a widespread hope within the members of the high energy elementary particle physics community that the upcoming experiments at Tevatron [CITATION] and at LHC [CITATION] will finally reveal the existence of supersymmetric particles, via direct production of sparticle-antisparticle pairs.', 'hep-ph-0406253-1-1-1': 'In such an exciting hypothesis, the simplest available theoretical supersymmetric model, the MSSM, will acquire the same role that belonged to the SM after the W, Z discoveries, and a long and crucial period of precision tests will start, aiming to confirm, or to disprove, the main theoretical assumptions that were used in the practical construction of the model.', 'hep-ph-0406253-1-2-0': 'For the MSSM case, this program requires the construction of a linear [MATH] collider [CITATION] that explores the c.m. TeV energy region.', 'hep-ph-0406253-1-2-1': 'This machine should be sufficiently accurate to provide the same kind of consistency tests of the model that were achieved in the hundred GeV region at LEP for the Standard Model, from detailed analyses of several independent one-loop virtual effects.', 'hep-ph-0406253-1-3-0': 'If such an ambitious project is to be carried on, a clean investigation strategy will undoubtedly be welcome, given the fact that the involved theoretical model has a not simple structure in which several independent parameters are implied.', 'hep-ph-0406253-1-3-1': 'As a consequence of this feature, it might be appreciable to identify clean experimental measurements that, possibly, isolate in a relatively simple way a reduced subset of parameters and of theoretical assumptions behind them.', 'hep-ph-0406253-1-4-0': 'The aim of this short paper is to show that a dedicated combined analysis of the process of production of neutralino, chargino and charged Higgs boson pairs might lead to a relevant precision test of the theoretical assumptions that were used to fix, not only the basic supersymmetric sector (i.e. the chiral superpotential and the gauge multiplet), but also the gaugino component of the soft supersymmetry breaking of the model.', 'hep-ph-0406253-1-4-1': 'This will be done in the paper by assuming a scenario of "moderately" light SUSY particles (i.e. the various masses are all below, roughly, 350 GeV) and a c.m. energy [MATH] in the one TeV region.', 'hep-ph-0406253-1-4-2': 'Under these two assumptions, a simple asymptotic expansion of so called "Sudakov type" is effective [CITATION].', 'hep-ph-0406253-1-4-3': 'But the approach would not change for larger SUSY masses [MATH] GeV since a related c.m. energy rescaling would still be possible, from our previous experience, up to [MATH] values of about 2 TeV, where the resummation to higher orders is still not necessary [CITATION].', 'hep-ph-0406253-1-5-0': 'After this brief description of the strategy of our approach we are now ready to illustrate the practical details of our work.', 'hep-ph-0406253-1-5-1': "With this aim, we must recall, for the reader's convenience, that the relevant details of an analogous procedure, limited to the combination of chargino and charged Higgs pairs production, have already appeared in a previous paper[CITATION].", 'hep-ph-0406253-1-5-2': 'It was shown there that from the combined analyses of the slopes of the total cross sections in a c.m. energy range of about 1 TeV, assuming a "light" SUSY scenario where all the relevant sparticle masses lie below [MATH] 350-400 GeV , a strong constraint on [MATH] (mostly produced by the Higgs cross section) and a strip in the [MATH] plane (only produced by the chargino cross sections) were derivable.', 'hep-ph-0406253-1-5-3': 'Here [MATH] is the soft SUSY breaking wino mass, [MATH] is the Higgsino mass parameter, and [MATH] is the ratio of the two Higgs doublets vevs.', 'hep-ph-0406253-1-5-4': 'The novel process that we consider in this paper is neutralino pair production, and we shall show that its addition to the other processes will lead to highly improved constraints on the previous parameters, with the additional presence of [MATH], the soft SUSY breaking bino mass.', 'hep-ph-0406253-1-5-5': 'With this aim, we now briefly list the relevant theoretical formulae that we need for the analysis.', 'hep-ph-0406253-1-6-0': 'At the Born level, the neutralino pair production process [MATH] is described by three s, t, u channel components of the scattering amplitude, using the following notations: [EQUATION] with the tree level values for the s channel [EQUATION] and for the t, u channels [EQUATION]', 'hep-ph-0406253-1-6-1': 'The indices [MATH], [MATH] label the electron and the [MATH]-th neutralino chiralities.', 'hep-ph-0406253-1-6-2': 'The quantities [MATH] are the elements of the 4x4 mixing matrix, defined in a conventional way [CITATION].', 'hep-ph-0406253-1-6-3': 'For simplicity, we have neglected in the considered asymptotic region the selectron mass in Eq. (1).', 'hep-ph-0406253-1-7-0': 'For the purposes of a precision test, it becomes mandatory to compute the following perturbative one-loop expansion of the scattering amplitude.', 'hep-ph-0406253-1-7-1': 'This requires the calculation of a large number of Feynman diagrams.', 'hep-ph-0406253-1-7-2': 'The resulting expressions are valid for any value of the c.m. energy [MATH].', 'hep-ph-0406253-1-7-3': 'The full one-loop result has been calculated recently [CITATION].', 'hep-ph-0406253-1-7-4': 'In this work we propose an alternative approach, valid for values of [MATH] "much" larger than all the SUSY sparticle masses involved in the process.', 'hep-ph-0406253-1-7-5': 'In fact, our strategy should be now made very clear.', 'hep-ph-0406253-1-7-6': 'We assume a previous production of the charginos, of the charged Higgs bosons and of at least two neutralinos with mass [MATH].', 'hep-ph-0406253-1-7-7': 'Calling [MATH] the heaviest of the real and virtual SUSY particles that appear in the processes, and assuming a "reasonable" limit [MATH] GeV, we shall choose the value [MATH] TeV value to proceed with our approach.', 'hep-ph-0406253-1-7-8': 'This is due to the fact that, in a previous paper only concerned with the charged Higgs pair production [CITATION], we proved that in such a configuration an asymptotic energy logarithmic expansion of so called "Sudakov type" was reliable, with the only addition of a constant term to the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-1-7-9': 'This conclusion allows to propose a determination of the SUSY parameters entering the Sudakov logarithms, based on measurements of the slope of the total cross section, in which the (complicated) constant terms cancel.', 'hep-ph-0406253-1-8-0': 'The approach that we shall follow in this paper, that was already used in the combined chargino-charged Higgs analysis [CITATION], will assume that a similar expansion is valid, with only logarithmic and constant terms.', 'hep-ph-0406253-1-8-1': 'To prove this statement would require a detailed comparison of the complete existing calculation [CITATION] with the assumed asymptotic expansion.', 'hep-ph-0406253-1-8-2': 'This will be the goal of a forthcoming rigorous analysis.', 'hep-ph-0406253-1-8-3': 'For the moment, we shall assume its validity as a working Ansatz, and we shall show the main relevant consequences that it will be able to produce.', 'hep-ph-0406253-1-9-0': 'After this preliminary discussion, we are now ready to write the relevant asymptotic expansion of the scattering amplitude at one loop, in which we shall only retain the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-1-9-1': 'For this purpose, we shall use the following formal decomposition (valid also for the separate [MATH], [MATH] and [MATH] subamplitudes) [EQUATION]', 'hep-ph-0406253-1-9-2': 'In the asymptotic expansion, two different kinds of logarithmic terms appear.', 'hep-ph-0406253-1-9-3': 'The first ones are the standard linear ones of RG origin.', 'hep-ph-0406253-1-9-4': 'They are well known and can be derived in a straightforward way replacing in the Born quantities the various bare couplings with running ones.', 'hep-ph-0406253-1-9-5': 'For sake of completeness, we write the related formulae, noticing that they are only requested for the s-channel (purely Higgsino) component, as: [EQUATION] where [MATH] and [MATH].', 'hep-ph-0406253-1-9-6': 'The second type of logarithms which arise asymptotically is that of the "genuine weak" Sudakov terms.', 'hep-ph-0406253-1-9-7': 'These are usually classified [CITATION] as logarithms of gauge non universal, gauge universal and Yukawa origin.', 'hep-ph-0406253-1-9-8': 'The gauge non universal, scattering angle dependent ones, stem from box diagrams and t,u channel vertices and have the following expressions: [EQUATION]', 'hep-ph-0406253-1-9-9': 'The gauge universal ones are due both to the initial and to the final vertices.', 'hep-ph-0406253-1-9-10': 'The initial contribution is fixed by the coefficients [EQUATION]', 'hep-ph-0406253-1-9-11': 'The contribution from final neutralino legs is different for the s or u, t channels.', 'hep-ph-0406253-1-9-12': 'For the s-channel we have higgsino components [EQUATION] where [EQUATION]', 'hep-ph-0406253-1-9-13': 'The logarithms of Yukawa origin are only due to final s-channel vertices with virtual heavy (b,t) quark-squark pairs.', 'hep-ph-0406253-1-9-14': 'These are the only logarithmic terms that contain (also) the vevs ratio [MATH] and their expression is [EQUATION]', 'hep-ph-0406253-1-9-15': 'To conclude, the final contribution to the u and t channel amplitudes is due to gaugino components for which there is no Yukawa contribution: [EQUATION] with [EQUATION]', 'hep-ph-0406253-1-9-16': 'Eqs. ([REF]-[REF]) represent the complete logarithmic contributions at one loop to the considered process, and are the main original result of this paper.', 'hep-ph-0406253-1-9-17': 'We expect from our previous discussion their reliability for what concerns the calculation of the slope of the total cross section in the 1 TeV range.', 'hep-ph-0406253-1-9-18': 'To perform the latter, one must first compute the expression of the differential cross section that is readily obtained from the above amplitude.', 'hep-ph-0406253-1-9-19': 'After angular integration, one obtains the approximate (to next-to-leading logarithmic order) asymptotic expression of the total cross section.', 'hep-ph-0406253-1-9-20': 'This will be sufficient to determine the effective asymptotic expression of the slope of the cross section, where by definition possible extra "next-to-next-to-leading" (i.e. constant) terms disappear.', 'hep-ph-0406253-1-9-21': 'This expression will only depend on [MATH] and on the mixing parameters [MATH].', 'hep-ph-0406253-1-9-22': 'The latter ones, in turn, can be expressed as functions of the supersymmetric parameters [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0406253-1-9-23': 'Thus, the final form of the slope will depend on these four parameters.', 'hep-ph-0406253-1-10-0': 'In order to achieve the maximal theoretical information using our approach, we have combined the theoretical expressions of the slope of the neutralino pair total cross section, given in this paper, with those of the chargino and charged Higgs pairs, given in [CITATION].', 'hep-ph-0406253-1-10-1': 'In practice, we have limited our analysis to the production of the two light observable final states ([MATH], [MATH]) for neutralinos, combined with the production of all the three chargino pairs and of the charged Higgs pair.', 'hep-ph-0406253-1-10-2': 'To work in a consistent way, we have chosen as relevant examples for the masses of the produced pairs three sets of values denoted by [MATH], [MATH], and [MATH].', 'hep-ph-0406253-1-10-3': 'The first, [MATH], is the Tesla benchmark point RR2[CITATION] with the two lightest neutralinos being respectively 95 bino and 82 wino.', 'hep-ph-0406253-1-10-4': 'The set [MATH] is a mixed scenario with neutralinos having non negligible gaugino and Higgsino components; [MATH] is 86 bino and 13 higgsino, [MATH] is 11 bino, 48 wino and 41 higgsino.', 'hep-ph-0406253-1-10-5': 'Finally, [MATH] is a purely Higgsino one with the two lightest neutralinos being 92 and 98 higgsino like.', 'hep-ph-0406253-1-10-6': 'The values of the input parameters as well as the masses of the two charginos and of the two lightest neutralinos are summarized in Tab.', 'hep-ph-0406253-1-10-7': 'We have performed a standard [MATH] analysis assuming in the various scenarios 10-12 experimental points at energies ranging from 700-850 GeV (depending on the scenario) up to 1200 GeV and assuming an experimental accuracy of 1 for the cross sections of gauginos and 2 for that of Higgs bosons.', 'hep-ph-0406253-1-10-8': 'For each scenario we have checked the condition [MATH].', 'hep-ph-0406253-1-10-9': 'The results of our analysis are shown in Figs. (1-3).', 'hep-ph-0406253-1-10-10': 'From inspection of the figures one can draw the following conclusions.', 'hep-ph-0406253-1-11-0': 'In Fig. (1) we consider the [MATH] benchmark point.', 'hep-ph-0406253-1-11-1': 'One notices that a combined analysis is able to generate closed contours in the planes of all possible six couples of parameters.', 'hep-ph-0406253-1-11-2': 'This possibility is obviously existing for the first 3 cases i.e. [MATH], [MATH], [MATH] under the condition that the neutralino information is added to the remaining ones, that do not depend on [MATH].', 'hep-ph-0406253-1-11-3': 'Less obvious and more illustrative is the closure of the undetermined strip in the [MATH] plane (drawn in the absence of neutralino data) when neutralino data are added to the remaining ones.', 'hep-ph-0406253-1-11-4': 'This addition does not practically improve, on the contrary, the bounds on [MATH] and [MATH] obtained from chargino-Higgs data.', 'hep-ph-0406253-1-11-5': 'One sees typical errors, under the assumed experimental conditions, of about 10 GeV for [MATH], [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-1-12-0': 'In Fig.(2) we consider the [MATH] scenario.', 'hep-ph-0406253-1-12-1': 'One finds, approximately, the same features for the set of projections.', 'hep-ph-0406253-1-12-2': 'The size of errors is now of approximately [MATH] GeV for [MATH], [MATH], and [MATH], a relative 40-50 for [MATH].', 'hep-ph-0406253-1-13-0': 'Fig.(3) describes [MATH], a typical Higgsino scenario.', 'hep-ph-0406253-1-13-1': 'Here the new (negative) feature concerns the lack of boundary for [MATH], as seen from the first three plots.', 'hep-ph-0406253-1-13-2': 'This can be understood since, at the considered [MATH], [MATH] and [MATH] values, the mixing parameters [MATH] do not vary appreciably when [MATH] increases.', 'hep-ph-0406253-1-13-3': 'On the contrary, the three plots in the bottom line retain the typical features of the previous cases, with particular relevance of the neutralino role for the determination of a boundary for [MATH].', 'hep-ph-0406253-1-13-4': 'The errors are now , typically, of about 20 GeV for [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-1-14-0': 'The previous results should illustrate the aimed possible outcomes of the testing strategy that we are proposing in this preliminary qualitative paper.', 'hep-ph-0406253-1-14-1': 'Our assumed physical inputs will necessarily be the masses of the (supposedly produced) (light) neutralinos, charginos and charged Higgs.', 'hep-ph-0406253-1-14-2': 'Their values will depend on the four parameters [MATH], [MATH], [MATH], [MATH] but, in general there will not be a 1-1 correspondence, and different sets of parameters might reproduce "essentially" the same masses [CITATION].', 'hep-ph-0406253-1-14-3': 'Within the errors that we have illustrated in our examples, these different sets can be discriminated via the [MATH] analysis that we have proposed, and the "correct" set is selected by the true experimental data, if the latter are actually described by the model.', 'hep-ph-0406253-1-14-4': 'If none of the candidate sets were compatible with the analysis, an indication would arise that some of the details of the model might need a suitable modification.'}

{'hep-ph-0406253-2-0-0': 'We consider the production of neutralino, chargino, and charged Higgs boson pairs in the MSSM framework at future [MATH] colliders.', 'hep-ph-0406253-2-0-1': 'We show that, for c.m. energies in the one TeV range and in a "moderately" light SUSY scenario, a combined analysis of the slopes of these production cross sections could lead to a strong consistency test involving the soft supersymmetric breaking parameters [MATH], [MATH], the Higgsino parameter [MATH], and [MATH].', 'hep-ph-0406253-2-1-0': 'It is a widespread hope within the members of the high energy elementary particle physics community that the upcoming experiments at Tevatron [CITATION] and at LHC [CITATION] will finally reveal the existence of supersymmetric particles, via direct production of sparticle-antisparticle pairs.', 'hep-ph-0406253-2-1-1': 'In such an exciting hypothesis, the simplest available theoretical supersymmetric model, the MSSM, will acquire the same role that belonged to the SM after the W, Z discoveries, and a long and crucial period of precision tests will start, aiming to confirm, or to disprove, the main theoretical assumptions that were used in the practical construction of the model.', 'hep-ph-0406253-2-2-0': 'For the MSSM case, this program requires the construction of a linear [MATH] collider [CITATION] that explores the c.m. TeV energy region.', 'hep-ph-0406253-2-2-1': 'This machine should be sufficiently accurate to provide the same kind of consistency tests of the model that were achieved in the hundred GeV region at LEP for the Standard Model, from detailed analyses of several independent one-loop virtual effects.', 'hep-ph-0406253-2-3-0': 'If such an ambitious project is to be carried on, a clean investigation strategy will undoubtedly be welcome, given the fact that the involved theoretical model has a not simple structure in which several independent parameters are implied.', 'hep-ph-0406253-2-3-1': 'As a consequence of this feature, it might be appreciable to identify clean experimental measurements that, possibly, isolate in a relatively simple way a reduced subset of parameters and of theoretical assumptions behind them.', 'hep-ph-0406253-2-4-0': 'The aim of this short paper is to show that a dedicated combined analysis of the process of production of neutralino, chargino and charged Higgs boson pairs might lead to a relevant precision test of the theoretical assumptions that were used to fix, not only the basic supersymmetric sector (i.e. the chiral superpotential and the gauge multiplet), but also the gaugino component of the soft supersymmetry breaking of the model.', 'hep-ph-0406253-2-4-1': 'This will be done in the paper by assuming a scenario of "moderately" light SUSY particles (i.e. the various masses are all below, roughly, 350 GeV) and a c.m. energy [MATH] in the one TeV region.', 'hep-ph-0406253-2-4-2': 'Under these two assumptions, a simple asymptotic expansion of so called "Sudakov type" is effective [CITATION].', 'hep-ph-0406253-2-4-3': 'But the approach would not change for larger SUSY masses [MATH] GeV since a related c.m. energy rescaling would still be possible, from our previous experience, up to [MATH] values of about 2 TeV, where the resummation to higher orders is still not necessary [CITATION].', 'hep-ph-0406253-2-5-0': 'After this brief description of the strategy of our approach we are now ready to illustrate the practical details of our work.', 'hep-ph-0406253-2-5-1': "With this aim, we must recall, for the reader's convenience, that the relevant details of an analogous procedure, limited to the combination of chargino and charged Higgs pairs production, have already appeared in a previous paper[CITATION].", 'hep-ph-0406253-2-5-2': 'It was shown there that from the combined analyses of the slopes of the total cross sections in a c.m. energy range of about 1 TeV, assuming a "light" SUSY scenario where all the relevant sparticle masses lie below [MATH] 350-400 GeV , a strong constraint on [MATH] (mostly produced by the Higgs cross section) and a strip in the [MATH] plane (only produced by the chargino cross sections) were derivable.', 'hep-ph-0406253-2-5-3': 'Here [MATH] is the soft SUSY breaking wino mass, [MATH] is the Higgsino mass parameter, and [MATH] is the ratio of the two Higgs doublets vevs.', 'hep-ph-0406253-2-5-4': 'The novel process that we consider in this paper is neutralino pair production, and we shall show that its addition to the other processes will lead to highly improved constraints on the previous parameters, with the additional presence of [MATH], the soft SUSY breaking bino mass.', 'hep-ph-0406253-2-5-5': 'With this aim, we now briefly list the relevant theoretical formulae that we need for the analysis.', 'hep-ph-0406253-2-6-0': 'At the Born level, the neutralino pair production process [MATH] is described by three s, t, u channel components of the scattering amplitude, using the following notations (the indices [MATH], [MATH] label the electron and the [MATH]-th neutralino chiralities): [EQUATION] with the tree level values for the s channel [EQUATION] and for the t, u channels [EQUATION]', 'hep-ph-0406253-2-6-1': 'The quantities [MATH] are the elements of the 4x4 mixing matrix, defined in a conventional way [CITATION].', 'hep-ph-0406253-2-6-2': 'For simplicity, we have neglected in the considered asymptotic region the selectron mass in Eq. (1).', 'hep-ph-0406253-2-7-0': 'For the purposes of a precision test, it becomes mandatory to compute the following perturbative one-loop expansion of the scattering amplitude.', 'hep-ph-0406253-2-7-1': 'This requires the calculation of a large number of Feynman diagrams.', 'hep-ph-0406253-2-7-2': 'The resulting expressions are valid for any value of the c.m. energy [MATH].', 'hep-ph-0406253-2-7-3': 'The full one-loop result has been calculated recently [CITATION].', 'hep-ph-0406253-2-7-4': 'In this work we propose an alternative approach, valid for values of [MATH] "much" larger than all the SUSY sparticle masses involved in the process.', 'hep-ph-0406253-2-7-5': 'In fact, our strategy should be now made very clear.', 'hep-ph-0406253-2-7-6': 'We assume a previous production of the charginos, of the charged Higgs bosons and of at least two neutralinos with mass [MATH].', 'hep-ph-0406253-2-7-7': 'Calling [MATH] the heaviest of the real and virtual SUSY particles that appear in the processes, and assuming a "reasonable" limit [MATH] GeV, we shall choose the value [MATH] TeV value to proceed with our approach.', 'hep-ph-0406253-2-7-8': 'This is due to the fact that, in a previous paper only concerned with the charged Higgs pair production [CITATION], we proved that in such a configuration an asymptotic energy logarithmic expansion of so called "Sudakov type" was reliable, with the only addition of a constant term to the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-2-7-9': 'This conclusion allows to propose a determination of the SUSY parameters entering the Sudakov logarithms, based on measurements of the slope of the total cross section, in which the (complicated) constant terms cancel.', 'hep-ph-0406253-2-8-0': 'The approach that we shall follow in this paper, that was already used in the combined chargino-charged Higgs analysis [CITATION], will assume that a similar expansion is valid, with only logarithmic and constant terms.', 'hep-ph-0406253-2-8-1': 'To prove this statement would require a detailed comparison of the complete existing calculation [CITATION] with the assumed asymptotic expansion.', 'hep-ph-0406253-2-8-2': 'This will be the goal of a forthcoming rigorous analysis.', 'hep-ph-0406253-2-8-3': 'For the moment, we shall assume its validity as a working Ansatz, and we shall show the main relevant consequences that it will be able to produce.', 'hep-ph-0406253-2-9-0': 'After this preliminary discussion, we are now ready to write the relevant asymptotic expansion of the scattering amplitude at one loop, in which we shall only retain the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-2-9-1': 'For this purpose, we shall use the following formal decomposition the separate [MATH], [MATH] or [MATH] subamplitudes (the Born values are those listed in Eqs. ([REF]-[REF])): [EQUATION]', 'hep-ph-0406253-2-9-2': 'In the asymptotic expansion, two different kinds of logarithmic terms appear.', 'hep-ph-0406253-2-9-3': 'The first ones are the standard linear ones of RG origin.', 'hep-ph-0406253-2-9-4': 'They are well known and can be derived in a straightforward way replacing in the Born quantities the various bare couplings with running ones.', 'hep-ph-0406253-2-9-5': 'For sake of completeness, we write the related formulae, noticing that they are only requested for the s-channel (purely Higgsino) component, as: [EQUATION] where [MATH] and [MATH].', 'hep-ph-0406253-2-9-6': 'The second type of logarithms which arise asymptotically is that of the "genuine weak" Sudakov terms.', 'hep-ph-0406253-2-9-7': 'These are usually classified [CITATION] as logarithms of gauge non universal, gauge universal and Yukawa origin.', 'hep-ph-0406253-2-9-8': 'The gauge non universal, scattering angle dependent ones, stem from box diagrams and t,u channel vertices and have the following expressions: [EQUATION]', 'hep-ph-0406253-2-9-9': 'The gauge universal ones are due both to the initial and to the final vertices.', 'hep-ph-0406253-2-9-10': 'The initial contribution is fixed by the coefficients [EQUATION]', 'hep-ph-0406253-2-9-11': 'The contribution from final neutralino legs is different for the s or u, t channels.', 'hep-ph-0406253-2-9-12': 'For the s-channel we have higgsino components [EQUATION] where [EQUATION]', 'hep-ph-0406253-2-9-13': 'The logarithms of Yukawa origin are only due to final s-channel vertices with virtual heavy (b,t) quark-squark pairs.', 'hep-ph-0406253-2-9-14': 'These are the only logarithmic terms that contain (also) the vevs ratio [MATH] and their expression is [EQUATION]', 'hep-ph-0406253-2-9-15': 'To conclude, the final contribution to the u and t channel amplitudes is due to gaugino components for which there is no Yukawa contribution: [EQUATION] with [EQUATION]', 'hep-ph-0406253-2-9-16': 'Eqs. ([REF]-[REF]) represent the complete logarithmic contributions at one loop to the considered process, and are the main original result of this paper.', 'hep-ph-0406253-2-9-17': 'We expect from our previous discussion their reliability for what concerns the calculation of the slope of the total cross section in the 1 TeV range.', 'hep-ph-0406253-2-9-18': 'To perform the latter, one must first compute the expression of the differential cross section that is readily obtained from the above amplitude.', 'hep-ph-0406253-2-9-19': 'After angular integration, one obtains the approximate (to next-to-leading logarithmic order) asymptotic expression of the total cross section.', 'hep-ph-0406253-2-9-20': 'This will be sufficient to determine the effective asymptotic expression of the slope of the cross section, where by definition possible extra "next-to-next-to-leading" (i.e. constant) terms disappear.', 'hep-ph-0406253-2-9-21': 'This expression will only depend on [MATH] and on the mixing parameters [MATH].', 'hep-ph-0406253-2-9-22': 'The latter ones, in turn, can be expressed as functions of the supersymmetric parameters [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0406253-2-9-23': 'Thus, the final form of the slope will depend on these four parameters.', 'hep-ph-0406253-2-10-0': 'In order to achieve the maximal theoretical information using our approach, we have combined the theoretical expressions of the slope of the neutralino pair total cross section, given in this paper, with those of the chargino and charged Higgs pairs, given in [CITATION].', 'hep-ph-0406253-2-10-1': 'In practice, we have limited our analysis to the production of the two light observable final states ([MATH], [MATH]) for neutralinos, combined with the production of all the three chargino pairs and of the charged Higgs pair.', 'hep-ph-0406253-2-10-2': 'To work in a consistent way, we have chosen as relevant examples for the masses of the produced pairs three sets of values denoted by [MATH], [MATH], and [MATH].', 'hep-ph-0406253-2-10-3': 'The first, [MATH], is the Tesla benchmark point RR2[CITATION] with the two lightest neutralinos being respectively 95 bino and 82 wino.', 'hep-ph-0406253-2-10-4': 'The set [MATH] is a mixed scenario with neutralinos having non negligible gaugino and Higgsino components; [MATH] is 86 bino and 13 higgsino, [MATH] is 11 bino, 48 wino and 41 higgsino.', 'hep-ph-0406253-2-10-5': 'Finally, [MATH] is a purely Higgsino one with the two lightest neutralinos being 92 and 98 higgsino like.', 'hep-ph-0406253-2-10-6': 'The values of the input parameters as well as the masses of the two charginos and of the two lightest neutralinos are summarized in Tab.', 'hep-ph-0406253-2-10-7': 'We have performed a standard [MATH] analysis assuming in the various scenarios 10-12 experimental points at energies ranging from 700-850 GeV (depending on the scenario) up to 1200 GeV and assuming an experimental accuracy of 1 for the cross sections of gauginos and 2 for that of Higgs bosons.', 'hep-ph-0406253-2-10-8': 'For each scenario we have checked the condition [MATH].', 'hep-ph-0406253-2-10-9': 'The results of our analysis are shown in Figs. (1-3).', 'hep-ph-0406253-2-10-10': 'From inspection of the figures one can draw the following conclusions.', 'hep-ph-0406253-2-11-0': 'In Fig. (1) we consider the [MATH] benchmark point.', 'hep-ph-0406253-2-11-1': 'One notices that a combined analysis is able to generate closed contours in the planes of all possible six couples of parameters.', 'hep-ph-0406253-2-11-2': 'This possibility is obviously existing for the first 3 cases i.e. [MATH], [MATH], [MATH] under the condition that the neutralino information is added to the remaining ones, that do not depend on [MATH].', 'hep-ph-0406253-2-11-3': 'Less obvious and more illustrative is the closure of the undetermined strip in the [MATH] plane (drawn in the absence of neutralino data) when neutralino data are added to the remaining ones.', 'hep-ph-0406253-2-11-4': 'This addition does not practically improve, on the contrary, the bounds on [MATH] and [MATH] obtained from chargino-Higgs data.', 'hep-ph-0406253-2-11-5': 'One sees typical errors, under the assumed experimental conditions, of about 10 GeV for [MATH], [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-2-12-0': 'In Fig.(2) we consider the [MATH] scenario.', 'hep-ph-0406253-2-12-1': 'One finds, approximately, the same features for the set of projections.', 'hep-ph-0406253-2-12-2': 'The size of errors is now of approximately [MATH] GeV for [MATH], [MATH], and [MATH], a relative 40-50 for [MATH].', 'hep-ph-0406253-2-13-0': 'Fig.(3) describes [MATH], a typical Higgsino scenario.', 'hep-ph-0406253-2-13-1': 'Here the new (negative) feature concerns the lack of boundary for [MATH], as seen from the first three plots.', 'hep-ph-0406253-2-13-2': 'This can be understood since, at the considered [MATH], [MATH] and [MATH] values, the mixing parameters [MATH] do not vary appreciably when [MATH] increases.', 'hep-ph-0406253-2-13-3': 'On the contrary, the three plots in the bottom line retain the typical features of the previous cases, with particular relevance of the neutralino role for the determination of a boundary for [MATH].', 'hep-ph-0406253-2-13-4': 'The errors are now , typically, of about 20 GeV for [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-2-14-0': 'The previous results should illustrate the aimed possible outcomes of the testing strategy that we are proposing in this preliminary qualitative paper.', 'hep-ph-0406253-2-14-1': 'Our assumed physical inputs will necessarily be the masses of the (supposedly produced) (light) neutralinos, charginos and charged Higgs.', 'hep-ph-0406253-2-14-2': 'Their values will depend on the four parameters [MATH], [MATH], [MATH], [MATH] but, in general there will not be a 1-1 correspondence, and different sets of parameters might reproduce "essentially" the same masses [CITATION].', 'hep-ph-0406253-2-14-3': 'Within the errors that we have illustrated in our examples, these different sets can be discriminated via the [MATH] analysis that we have proposed, and the "correct" set is selected by the true experimental data, if the latter are actually described by the model.', 'hep-ph-0406253-2-14-4': 'If none of the candidate sets were compatible with the analysis, an indication would arise that some of the details of the model might need a suitable modification.'}

[['hep-ph-0406253-1-3-0', 'hep-ph-0406253-2-3-0'], ['hep-ph-0406253-1-3-1', 'hep-ph-0406253-2-3-1'], ['hep-ph-0406253-1-0-0', 'hep-ph-0406253-2-0-0'], ['hep-ph-0406253-1-0-1', 'hep-ph-0406253-2-0-1'], ['hep-ph-0406253-1-4-0', 'hep-ph-0406253-2-4-0'], ['hep-ph-0406253-1-4-1', 'hep-ph-0406253-2-4-1'], ['hep-ph-0406253-1-4-2', 'hep-ph-0406253-2-4-2'], ['hep-ph-0406253-1-4-3', 'hep-ph-0406253-2-4-3'], ['hep-ph-0406253-1-5-0', 'hep-ph-0406253-2-5-0'], ['hep-ph-0406253-1-5-1', 'hep-ph-0406253-2-5-1'], ['hep-ph-0406253-1-5-2', 'hep-ph-0406253-2-5-2'], ['hep-ph-0406253-1-5-3', 'hep-ph-0406253-2-5-3'], ['hep-ph-0406253-1-5-4', 'hep-ph-0406253-2-5-4'], ['hep-ph-0406253-1-5-5', 'hep-ph-0406253-2-5-5'], ['hep-ph-0406253-1-11-0', 'hep-ph-0406253-2-11-0'], ['hep-ph-0406253-1-11-1', 'hep-ph-0406253-2-11-1'], ['hep-ph-0406253-1-11-2', 'hep-ph-0406253-2-11-2'], ['hep-ph-0406253-1-11-3', 'hep-ph-0406253-2-11-3'], ['hep-ph-0406253-1-11-4', 'hep-ph-0406253-2-11-4'], ['hep-ph-0406253-1-11-5', 'hep-ph-0406253-2-11-5'], ['hep-ph-0406253-1-12-0', 'hep-ph-0406253-2-12-0'], ['hep-ph-0406253-1-12-1', 'hep-ph-0406253-2-12-1'], ['hep-ph-0406253-1-12-2', 'hep-ph-0406253-2-12-2'], ['hep-ph-0406253-1-10-0', 'hep-ph-0406253-2-10-0'], ['hep-ph-0406253-1-10-1', 'hep-ph-0406253-2-10-1'], ['hep-ph-0406253-1-10-2', 'hep-ph-0406253-2-10-2'], ['hep-ph-0406253-1-10-3', 'hep-ph-0406253-2-10-3'], ['hep-ph-0406253-1-10-4', 'hep-ph-0406253-2-10-4'], ['hep-ph-0406253-1-10-5', 'hep-ph-0406253-2-10-5'], ['hep-ph-0406253-1-10-6', 'hep-ph-0406253-2-10-6'], ['hep-ph-0406253-1-10-7', 'hep-ph-0406253-2-10-7'], ['hep-ph-0406253-1-10-8', 'hep-ph-0406253-2-10-8'], ['hep-ph-0406253-1-10-9', 'hep-ph-0406253-2-10-9'], ['hep-ph-0406253-1-10-10', 'hep-ph-0406253-2-10-10'], ['hep-ph-0406253-1-14-0', 'hep-ph-0406253-2-14-0'], ['hep-ph-0406253-1-14-1', 'hep-ph-0406253-2-14-1'], ['hep-ph-0406253-1-14-2', 'hep-ph-0406253-2-14-2'], ['hep-ph-0406253-1-14-3', 'hep-ph-0406253-2-14-3'], ['hep-ph-0406253-1-14-4', 'hep-ph-0406253-2-14-4'], ['hep-ph-0406253-1-6-2', 'hep-ph-0406253-2-6-1'], ['hep-ph-0406253-1-6-3', 'hep-ph-0406253-2-6-2'], ['hep-ph-0406253-1-2-0', 'hep-ph-0406253-2-2-0'], ['hep-ph-0406253-1-2-1', 'hep-ph-0406253-2-2-1'], ['hep-ph-0406253-1-1-0', 'hep-ph-0406253-2-1-0'], ['hep-ph-0406253-1-1-1', 'hep-ph-0406253-2-1-1'], ['hep-ph-0406253-1-7-0', 'hep-ph-0406253-2-7-0'], ['hep-ph-0406253-1-7-1', 'hep-ph-0406253-2-7-1'], ['hep-ph-0406253-1-7-2', 'hep-ph-0406253-2-7-2'], ['hep-ph-0406253-1-7-3', 'hep-ph-0406253-2-7-3'], ['hep-ph-0406253-1-7-4', 'hep-ph-0406253-2-7-4'], ['hep-ph-0406253-1-7-5', 'hep-ph-0406253-2-7-5'], ['hep-ph-0406253-1-7-6', 'hep-ph-0406253-2-7-6'], ['hep-ph-0406253-1-7-7', 'hep-ph-0406253-2-7-7'], ['hep-ph-0406253-1-7-8', 'hep-ph-0406253-2-7-8'], ['hep-ph-0406253-1-7-9', 'hep-ph-0406253-2-7-9'], ['hep-ph-0406253-1-8-0', 'hep-ph-0406253-2-8-0'], ['hep-ph-0406253-1-8-1', 'hep-ph-0406253-2-8-1'], ['hep-ph-0406253-1-8-2', 'hep-ph-0406253-2-8-2'], ['hep-ph-0406253-1-8-3', 'hep-ph-0406253-2-8-3'], ['hep-ph-0406253-1-9-0', 'hep-ph-0406253-2-9-0'], ['hep-ph-0406253-1-9-2', 'hep-ph-0406253-2-9-2'], ['hep-ph-0406253-1-9-3', 'hep-ph-0406253-2-9-3'], ['hep-ph-0406253-1-9-4', 'hep-ph-0406253-2-9-4'], ['hep-ph-0406253-1-9-5', 'hep-ph-0406253-2-9-5'], ['hep-ph-0406253-1-9-6', 'hep-ph-0406253-2-9-6'], ['hep-ph-0406253-1-9-7', 'hep-ph-0406253-2-9-7'], ['hep-ph-0406253-1-9-8', 'hep-ph-0406253-2-9-8'], ['hep-ph-0406253-1-9-9', 'hep-ph-0406253-2-9-9'], ['hep-ph-0406253-1-9-10', 'hep-ph-0406253-2-9-10'], ['hep-ph-0406253-1-9-11', 'hep-ph-0406253-2-9-11'], ['hep-ph-0406253-1-9-12', 'hep-ph-0406253-2-9-12'], ['hep-ph-0406253-1-9-13', 'hep-ph-0406253-2-9-13'], ['hep-ph-0406253-1-9-14', 'hep-ph-0406253-2-9-14'], ['hep-ph-0406253-1-9-15', 'hep-ph-0406253-2-9-15'], ['hep-ph-0406253-1-9-16', 'hep-ph-0406253-2-9-16'], ['hep-ph-0406253-1-9-17', 'hep-ph-0406253-2-9-17'], ['hep-ph-0406253-1-9-18', 'hep-ph-0406253-2-9-18'], ['hep-ph-0406253-1-9-19', 'hep-ph-0406253-2-9-19'], ['hep-ph-0406253-1-9-20', 'hep-ph-0406253-2-9-20'], ['hep-ph-0406253-1-9-21', 'hep-ph-0406253-2-9-21'], ['hep-ph-0406253-1-9-22', 'hep-ph-0406253-2-9-22'], ['hep-ph-0406253-1-9-23', 'hep-ph-0406253-2-9-23'], ['hep-ph-0406253-1-13-0', 'hep-ph-0406253-2-13-0'], ['hep-ph-0406253-1-13-1', 'hep-ph-0406253-2-13-1'], ['hep-ph-0406253-1-13-2', 'hep-ph-0406253-2-13-2'], ['hep-ph-0406253-1-13-3', 'hep-ph-0406253-2-13-3'], ['hep-ph-0406253-1-13-4', 'hep-ph-0406253-2-13-4'], ['hep-ph-0406253-2-6-0', 'hep-ph-0406253-3-6-0'], ['hep-ph-0406253-2-6-1', 'hep-ph-0406253-3-6-1'], ['hep-ph-0406253-2-6-2', 'hep-ph-0406253-3-6-2'], ['hep-ph-0406253-2-7-0', 'hep-ph-0406253-3-7-0'], ['hep-ph-0406253-2-7-1', 'hep-ph-0406253-3-7-1'], ['hep-ph-0406253-2-7-2', 'hep-ph-0406253-3-7-2'], ['hep-ph-0406253-2-7-3', 'hep-ph-0406253-3-7-3'], ['hep-ph-0406253-2-7-4', 'hep-ph-0406253-3-7-4'], ['hep-ph-0406253-2-7-5', 'hep-ph-0406253-3-7-5'], ['hep-ph-0406253-2-7-6', 'hep-ph-0406253-3-7-6'], ['hep-ph-0406253-2-7-7', 'hep-ph-0406253-3-7-7'], ['hep-ph-0406253-2-7-8', 'hep-ph-0406253-3-7-8'], ['hep-ph-0406253-2-7-9', 'hep-ph-0406253-3-7-9'], ['hep-ph-0406253-2-9-0', 'hep-ph-0406253-3-9-0'], ['hep-ph-0406253-2-9-1', 'hep-ph-0406253-3-9-1'], ['hep-ph-0406253-2-9-2', 'hep-ph-0406253-3-9-2'], ['hep-ph-0406253-2-9-3', 'hep-ph-0406253-3-9-3'], ['hep-ph-0406253-2-9-4', 'hep-ph-0406253-3-9-4'], ['hep-ph-0406253-2-9-5', 'hep-ph-0406253-3-9-5'], ['hep-ph-0406253-2-9-6', 'hep-ph-0406253-3-9-6'], ['hep-ph-0406253-2-9-7', 'hep-ph-0406253-3-9-7'], ['hep-ph-0406253-2-9-8', 'hep-ph-0406253-3-9-8'], ['hep-ph-0406253-2-9-9', 'hep-ph-0406253-3-9-9'], ['hep-ph-0406253-2-9-10', 'hep-ph-0406253-3-9-10'], ['hep-ph-0406253-2-9-11', 'hep-ph-0406253-3-9-11'], ['hep-ph-0406253-2-9-12', 'hep-ph-0406253-3-9-12'], ['hep-ph-0406253-2-9-13', 'hep-ph-0406253-3-9-13'], ['hep-ph-0406253-2-9-14', 'hep-ph-0406253-3-9-14'], ['hep-ph-0406253-2-9-15', 'hep-ph-0406253-3-9-15'], ['hep-ph-0406253-2-9-16', 'hep-ph-0406253-3-9-16'], ['hep-ph-0406253-2-9-17', 'hep-ph-0406253-3-9-17'], ['hep-ph-0406253-2-9-18', 'hep-ph-0406253-3-9-18'], ['hep-ph-0406253-2-9-19', 'hep-ph-0406253-3-9-19'], ['hep-ph-0406253-2-9-20', 'hep-ph-0406253-3-9-20'], ['hep-ph-0406253-2-9-21', 'hep-ph-0406253-3-9-21'], ['hep-ph-0406253-2-9-22', 'hep-ph-0406253-3-9-22'], ['hep-ph-0406253-2-9-23', 'hep-ph-0406253-3-9-23'], ['hep-ph-0406253-2-12-0', 'hep-ph-0406253-3-12-0'], ['hep-ph-0406253-2-12-1', 'hep-ph-0406253-3-12-1'], ['hep-ph-0406253-2-12-2', 'hep-ph-0406253-3-12-2'], ['hep-ph-0406253-2-3-0', 'hep-ph-0406253-3-3-0'], ['hep-ph-0406253-2-3-1', 'hep-ph-0406253-3-3-1'], ['hep-ph-0406253-2-13-0', 'hep-ph-0406253-3-13-0'], ['hep-ph-0406253-2-13-1', 'hep-ph-0406253-3-13-1'], ['hep-ph-0406253-2-13-2', 'hep-ph-0406253-3-13-2'], ['hep-ph-0406253-2-13-4', 'hep-ph-0406253-3-13-4'], ['hep-ph-0406253-2-10-0', 'hep-ph-0406253-3-10-0'], ['hep-ph-0406253-2-10-1', 'hep-ph-0406253-3-10-1'], ['hep-ph-0406253-2-10-2', 'hep-ph-0406253-3-10-2'], ['hep-ph-0406253-2-10-3', 'hep-ph-0406253-3-10-3'], ['hep-ph-0406253-2-10-4', 'hep-ph-0406253-3-10-4'], ['hep-ph-0406253-2-10-5', 'hep-ph-0406253-3-10-5'], ['hep-ph-0406253-2-10-6', 'hep-ph-0406253-3-10-6'], ['hep-ph-0406253-2-10-7', 'hep-ph-0406253-3-10-7'], ['hep-ph-0406253-2-10-8', 'hep-ph-0406253-3-10-8'], ['hep-ph-0406253-2-10-9', 'hep-ph-0406253-3-10-9'], ['hep-ph-0406253-2-10-10', 'hep-ph-0406253-3-10-10'], ['hep-ph-0406253-2-11-0', 'hep-ph-0406253-3-11-0'], ['hep-ph-0406253-2-11-1', 'hep-ph-0406253-3-11-1'], ['hep-ph-0406253-2-11-2', 'hep-ph-0406253-3-11-2'], ['hep-ph-0406253-2-11-3', 'hep-ph-0406253-3-11-3'], ['hep-ph-0406253-2-11-4', 'hep-ph-0406253-3-11-4'], ['hep-ph-0406253-2-11-5', 'hep-ph-0406253-3-11-5'], ['hep-ph-0406253-2-5-0', 'hep-ph-0406253-3-5-0'], ['hep-ph-0406253-2-5-1', 'hep-ph-0406253-3-5-1'], ['hep-ph-0406253-2-5-2', 'hep-ph-0406253-3-5-2'], ['hep-ph-0406253-2-5-3', 'hep-ph-0406253-3-5-3'], ['hep-ph-0406253-2-5-4', 'hep-ph-0406253-3-5-4'], ['hep-ph-0406253-2-5-5', 'hep-ph-0406253-3-5-5'], ['hep-ph-0406253-2-1-0', 'hep-ph-0406253-3-1-0'], ['hep-ph-0406253-2-1-1', 'hep-ph-0406253-3-1-1'], ['hep-ph-0406253-2-8-0', 'hep-ph-0406253-3-8-0'], ['hep-ph-0406253-2-8-1', 'hep-ph-0406253-3-8-1'], ['hep-ph-0406253-2-8-2', 'hep-ph-0406253-3-8-2'], ['hep-ph-0406253-2-8-3', 'hep-ph-0406253-3-8-3'], ['hep-ph-0406253-2-0-0', 'hep-ph-0406253-3-0-0'], ['hep-ph-0406253-2-0-1', 'hep-ph-0406253-3-0-1'], ['hep-ph-0406253-2-14-0', 'hep-ph-0406253-3-14-0'], ['hep-ph-0406253-2-14-1', 'hep-ph-0406253-3-14-1'], ['hep-ph-0406253-2-14-2', 'hep-ph-0406253-3-14-2'], ['hep-ph-0406253-2-14-3', 'hep-ph-0406253-3-14-3'], ['hep-ph-0406253-2-14-4', 'hep-ph-0406253-3-14-4'], ['hep-ph-0406253-2-4-0', 'hep-ph-0406253-3-4-0'], ['hep-ph-0406253-2-4-1', 'hep-ph-0406253-3-4-1'], ['hep-ph-0406253-2-4-2', 'hep-ph-0406253-3-4-2'], ['hep-ph-0406253-2-4-3', 'hep-ph-0406253-3-4-3'], ['hep-ph-0406253-2-2-0', 'hep-ph-0406253-3-2-0'], ['hep-ph-0406253-2-2-1', 'hep-ph-0406253-3-2-1'], ['hep-ph-0406253-1-6-0', 'hep-ph-0406253-2-6-0'], ['hep-ph-0406253-2-13-3', 'hep-ph-0406253-3-13-3'], ['hep-ph-0406253-1-9-1', 'hep-ph-0406253-2-9-1']]

[['hep-ph-0406253-1-3-0', 'hep-ph-0406253-2-3-0'], ['hep-ph-0406253-1-3-1', 'hep-ph-0406253-2-3-1'], ['hep-ph-0406253-1-0-0', 'hep-ph-0406253-2-0-0'], ['hep-ph-0406253-1-0-1', 'hep-ph-0406253-2-0-1'], ['hep-ph-0406253-1-4-0', 'hep-ph-0406253-2-4-0'], ['hep-ph-0406253-1-4-1', 'hep-ph-0406253-2-4-1'], ['hep-ph-0406253-1-4-2', 'hep-ph-0406253-2-4-2'], ['hep-ph-0406253-1-4-3', 'hep-ph-0406253-2-4-3'], ['hep-ph-0406253-1-5-0', 'hep-ph-0406253-2-5-0'], ['hep-ph-0406253-1-5-1', 'hep-ph-0406253-2-5-1'], ['hep-ph-0406253-1-5-2', 'hep-ph-0406253-2-5-2'], ['hep-ph-0406253-1-5-3', 'hep-ph-0406253-2-5-3'], ['hep-ph-0406253-1-5-4', 'hep-ph-0406253-2-5-4'], ['hep-ph-0406253-1-5-5', 'hep-ph-0406253-2-5-5'], ['hep-ph-0406253-1-11-0', 'hep-ph-0406253-2-11-0'], ['hep-ph-0406253-1-11-1', 'hep-ph-0406253-2-11-1'], ['hep-ph-0406253-1-11-2', 'hep-ph-0406253-2-11-2'], ['hep-ph-0406253-1-11-3', 'hep-ph-0406253-2-11-3'], ['hep-ph-0406253-1-11-4', 'hep-ph-0406253-2-11-4'], ['hep-ph-0406253-1-11-5', 'hep-ph-0406253-2-11-5'], ['hep-ph-0406253-1-12-0', 'hep-ph-0406253-2-12-0'], ['hep-ph-0406253-1-12-1', 'hep-ph-0406253-2-12-1'], ['hep-ph-0406253-1-12-2', 'hep-ph-0406253-2-12-2'], ['hep-ph-0406253-1-10-0', 'hep-ph-0406253-2-10-0'], ['hep-ph-0406253-1-10-1', 'hep-ph-0406253-2-10-1'], ['hep-ph-0406253-1-10-2', 'hep-ph-0406253-2-10-2'], ['hep-ph-0406253-1-10-3', 'hep-ph-0406253-2-10-3'], ['hep-ph-0406253-1-10-4', 'hep-ph-0406253-2-10-4'], ['hep-ph-0406253-1-10-5', 'hep-ph-0406253-2-10-5'], ['hep-ph-0406253-1-10-6', 'hep-ph-0406253-2-10-6'], ['hep-ph-0406253-1-10-7', 'hep-ph-0406253-2-10-7'], ['hep-ph-0406253-1-10-8', 'hep-ph-0406253-2-10-8'], ['hep-ph-0406253-1-10-9', 'hep-ph-0406253-2-10-9'], ['hep-ph-0406253-1-10-10', 'hep-ph-0406253-2-10-10'], ['hep-ph-0406253-1-14-0', 'hep-ph-0406253-2-14-0'], ['hep-ph-0406253-1-14-1', 'hep-ph-0406253-2-14-1'], ['hep-ph-0406253-1-14-2', 'hep-ph-0406253-2-14-2'], ['hep-ph-0406253-1-14-3', 'hep-ph-0406253-2-14-3'], ['hep-ph-0406253-1-14-4', 'hep-ph-0406253-2-14-4'], ['hep-ph-0406253-1-6-2', 'hep-ph-0406253-2-6-1'], ['hep-ph-0406253-1-6-3', 'hep-ph-0406253-2-6-2'], ['hep-ph-0406253-1-2-0', 'hep-ph-0406253-2-2-0'], ['hep-ph-0406253-1-2-1', 'hep-ph-0406253-2-2-1'], ['hep-ph-0406253-1-1-0', 'hep-ph-0406253-2-1-0'], ['hep-ph-0406253-1-1-1', 'hep-ph-0406253-2-1-1'], ['hep-ph-0406253-1-7-0', 'hep-ph-0406253-2-7-0'], ['hep-ph-0406253-1-7-1', 'hep-ph-0406253-2-7-1'], ['hep-ph-0406253-1-7-2', 'hep-ph-0406253-2-7-2'], ['hep-ph-0406253-1-7-3', 'hep-ph-0406253-2-7-3'], ['hep-ph-0406253-1-7-4', 'hep-ph-0406253-2-7-4'], ['hep-ph-0406253-1-7-5', 'hep-ph-0406253-2-7-5'], ['hep-ph-0406253-1-7-6', 'hep-ph-0406253-2-7-6'], ['hep-ph-0406253-1-7-7', 'hep-ph-0406253-2-7-7'], ['hep-ph-0406253-1-7-8', 'hep-ph-0406253-2-7-8'], ['hep-ph-0406253-1-7-9', 'hep-ph-0406253-2-7-9'], ['hep-ph-0406253-1-8-0', 'hep-ph-0406253-2-8-0'], ['hep-ph-0406253-1-8-1', 'hep-ph-0406253-2-8-1'], ['hep-ph-0406253-1-8-2', 'hep-ph-0406253-2-8-2'], ['hep-ph-0406253-1-8-3', 'hep-ph-0406253-2-8-3'], ['hep-ph-0406253-1-9-0', 'hep-ph-0406253-2-9-0'], ['hep-ph-0406253-1-9-2', 'hep-ph-0406253-2-9-2'], ['hep-ph-0406253-1-9-3', 'hep-ph-0406253-2-9-3'], ['hep-ph-0406253-1-9-4', 'hep-ph-0406253-2-9-4'], ['hep-ph-0406253-1-9-5', 'hep-ph-0406253-2-9-5'], ['hep-ph-0406253-1-9-6', 'hep-ph-0406253-2-9-6'], ['hep-ph-0406253-1-9-7', 'hep-ph-0406253-2-9-7'], ['hep-ph-0406253-1-9-8', 'hep-ph-0406253-2-9-8'], ['hep-ph-0406253-1-9-9', 'hep-ph-0406253-2-9-9'], ['hep-ph-0406253-1-9-10', 'hep-ph-0406253-2-9-10'], ['hep-ph-0406253-1-9-11', 'hep-ph-0406253-2-9-11'], ['hep-ph-0406253-1-9-12', 'hep-ph-0406253-2-9-12'], ['hep-ph-0406253-1-9-13', 'hep-ph-0406253-2-9-13'], ['hep-ph-0406253-1-9-14', 'hep-ph-0406253-2-9-14'], ['hep-ph-0406253-1-9-15', 'hep-ph-0406253-2-9-15'], ['hep-ph-0406253-1-9-16', 'hep-ph-0406253-2-9-16'], ['hep-ph-0406253-1-9-17', 'hep-ph-0406253-2-9-17'], ['hep-ph-0406253-1-9-18', 'hep-ph-0406253-2-9-18'], ['hep-ph-0406253-1-9-19', 'hep-ph-0406253-2-9-19'], ['hep-ph-0406253-1-9-20', 'hep-ph-0406253-2-9-20'], ['hep-ph-0406253-1-9-21', 'hep-ph-0406253-2-9-21'], ['hep-ph-0406253-1-9-22', 'hep-ph-0406253-2-9-22'], ['hep-ph-0406253-1-9-23', 'hep-ph-0406253-2-9-23'], ['hep-ph-0406253-1-13-0', 'hep-ph-0406253-2-13-0'], ['hep-ph-0406253-1-13-1', 'hep-ph-0406253-2-13-1'], ['hep-ph-0406253-1-13-2', 'hep-ph-0406253-2-13-2'], ['hep-ph-0406253-1-13-3', 'hep-ph-0406253-2-13-3'], ['hep-ph-0406253-1-13-4', 'hep-ph-0406253-2-13-4'], ['hep-ph-0406253-2-6-0', 'hep-ph-0406253-3-6-0'], ['hep-ph-0406253-2-6-1', 'hep-ph-0406253-3-6-1'], ['hep-ph-0406253-2-6-2', 'hep-ph-0406253-3-6-2'], ['hep-ph-0406253-2-7-0', 'hep-ph-0406253-3-7-0'], ['hep-ph-0406253-2-7-1', 'hep-ph-0406253-3-7-1'], ['hep-ph-0406253-2-7-2', 'hep-ph-0406253-3-7-2'], ['hep-ph-0406253-2-7-3', 'hep-ph-0406253-3-7-3'], ['hep-ph-0406253-2-7-4', 'hep-ph-0406253-3-7-4'], ['hep-ph-0406253-2-7-5', 'hep-ph-0406253-3-7-5'], ['hep-ph-0406253-2-7-6', 'hep-ph-0406253-3-7-6'], ['hep-ph-0406253-2-7-7', 'hep-ph-0406253-3-7-7'], ['hep-ph-0406253-2-7-8', 'hep-ph-0406253-3-7-8'], ['hep-ph-0406253-2-7-9', 'hep-ph-0406253-3-7-9'], ['hep-ph-0406253-2-9-0', 'hep-ph-0406253-3-9-0'], ['hep-ph-0406253-2-9-1', 'hep-ph-0406253-3-9-1'], ['hep-ph-0406253-2-9-2', 'hep-ph-0406253-3-9-2'], ['hep-ph-0406253-2-9-3', 'hep-ph-0406253-3-9-3'], ['hep-ph-0406253-2-9-4', 'hep-ph-0406253-3-9-4'], ['hep-ph-0406253-2-9-5', 'hep-ph-0406253-3-9-5'], ['hep-ph-0406253-2-9-6', 'hep-ph-0406253-3-9-6'], ['hep-ph-0406253-2-9-7', 'hep-ph-0406253-3-9-7'], ['hep-ph-0406253-2-9-8', 'hep-ph-0406253-3-9-8'], ['hep-ph-0406253-2-9-9', 'hep-ph-0406253-3-9-9'], ['hep-ph-0406253-2-9-10', 'hep-ph-0406253-3-9-10'], ['hep-ph-0406253-2-9-11', 'hep-ph-0406253-3-9-11'], ['hep-ph-0406253-2-9-12', 'hep-ph-0406253-3-9-12'], ['hep-ph-0406253-2-9-13', 'hep-ph-0406253-3-9-13'], ['hep-ph-0406253-2-9-14', 'hep-ph-0406253-3-9-14'], ['hep-ph-0406253-2-9-15', 'hep-ph-0406253-3-9-15'], ['hep-ph-0406253-2-9-16', 'hep-ph-0406253-3-9-16'], ['hep-ph-0406253-2-9-17', 'hep-ph-0406253-3-9-17'], ['hep-ph-0406253-2-9-18', 'hep-ph-0406253-3-9-18'], ['hep-ph-0406253-2-9-19', 'hep-ph-0406253-3-9-19'], ['hep-ph-0406253-2-9-20', 'hep-ph-0406253-3-9-20'], ['hep-ph-0406253-2-9-21', 'hep-ph-0406253-3-9-21'], ['hep-ph-0406253-2-9-22', 'hep-ph-0406253-3-9-22'], ['hep-ph-0406253-2-9-23', 'hep-ph-0406253-3-9-23'], ['hep-ph-0406253-2-12-0', 'hep-ph-0406253-3-12-0'], ['hep-ph-0406253-2-12-1', 'hep-ph-0406253-3-12-1'], ['hep-ph-0406253-2-12-2', 'hep-ph-0406253-3-12-2'], ['hep-ph-0406253-2-3-0', 'hep-ph-0406253-3-3-0'], ['hep-ph-0406253-2-3-1', 'hep-ph-0406253-3-3-1'], ['hep-ph-0406253-2-13-0', 'hep-ph-0406253-3-13-0'], ['hep-ph-0406253-2-13-1', 'hep-ph-0406253-3-13-1'], ['hep-ph-0406253-2-13-2', 'hep-ph-0406253-3-13-2'], ['hep-ph-0406253-2-13-4', 'hep-ph-0406253-3-13-4'], ['hep-ph-0406253-2-10-0', 'hep-ph-0406253-3-10-0'], ['hep-ph-0406253-2-10-1', 'hep-ph-0406253-3-10-1'], ['hep-ph-0406253-2-10-2', 'hep-ph-0406253-3-10-2'], ['hep-ph-0406253-2-10-3', 'hep-ph-0406253-3-10-3'], ['hep-ph-0406253-2-10-4', 'hep-ph-0406253-3-10-4'], ['hep-ph-0406253-2-10-5', 'hep-ph-0406253-3-10-5'], ['hep-ph-0406253-2-10-6', 'hep-ph-0406253-3-10-6'], ['hep-ph-0406253-2-10-7', 'hep-ph-0406253-3-10-7'], ['hep-ph-0406253-2-10-8', 'hep-ph-0406253-3-10-8'], ['hep-ph-0406253-2-10-9', 'hep-ph-0406253-3-10-9'], ['hep-ph-0406253-2-10-10', 'hep-ph-0406253-3-10-10'], ['hep-ph-0406253-2-11-0', 'hep-ph-0406253-3-11-0'], ['hep-ph-0406253-2-11-1', 'hep-ph-0406253-3-11-1'], ['hep-ph-0406253-2-11-2', 'hep-ph-0406253-3-11-2'], ['hep-ph-0406253-2-11-3', 'hep-ph-0406253-3-11-3'], ['hep-ph-0406253-2-11-4', 'hep-ph-0406253-3-11-4'], ['hep-ph-0406253-2-11-5', 'hep-ph-0406253-3-11-5'], ['hep-ph-0406253-2-5-0', 'hep-ph-0406253-3-5-0'], ['hep-ph-0406253-2-5-1', 'hep-ph-0406253-3-5-1'], ['hep-ph-0406253-2-5-2', 'hep-ph-0406253-3-5-2'], ['hep-ph-0406253-2-5-3', 'hep-ph-0406253-3-5-3'], ['hep-ph-0406253-2-5-4', 'hep-ph-0406253-3-5-4'], ['hep-ph-0406253-2-5-5', 'hep-ph-0406253-3-5-5'], ['hep-ph-0406253-2-1-0', 'hep-ph-0406253-3-1-0'], ['hep-ph-0406253-2-1-1', 'hep-ph-0406253-3-1-1'], ['hep-ph-0406253-2-8-0', 'hep-ph-0406253-3-8-0'], ['hep-ph-0406253-2-8-1', 'hep-ph-0406253-3-8-1'], ['hep-ph-0406253-2-8-2', 'hep-ph-0406253-3-8-2'], ['hep-ph-0406253-2-8-3', 'hep-ph-0406253-3-8-3'], ['hep-ph-0406253-2-0-0', 'hep-ph-0406253-3-0-0'], ['hep-ph-0406253-2-0-1', 'hep-ph-0406253-3-0-1'], ['hep-ph-0406253-2-14-0', 'hep-ph-0406253-3-14-0'], ['hep-ph-0406253-2-14-1', 'hep-ph-0406253-3-14-1'], ['hep-ph-0406253-2-14-2', 'hep-ph-0406253-3-14-2'], ['hep-ph-0406253-2-14-3', 'hep-ph-0406253-3-14-3'], ['hep-ph-0406253-2-14-4', 'hep-ph-0406253-3-14-4'], ['hep-ph-0406253-2-4-0', 'hep-ph-0406253-3-4-0'], ['hep-ph-0406253-2-4-1', 'hep-ph-0406253-3-4-1'], ['hep-ph-0406253-2-4-2', 'hep-ph-0406253-3-4-2'], ['hep-ph-0406253-2-4-3', 'hep-ph-0406253-3-4-3'], ['hep-ph-0406253-2-2-0', 'hep-ph-0406253-3-2-0'], ['hep-ph-0406253-2-2-1', 'hep-ph-0406253-3-2-1']]

[['hep-ph-0406253-1-6-0', 'hep-ph-0406253-2-6-0'], ['hep-ph-0406253-2-13-3', 'hep-ph-0406253-3-13-3']]

[]

[['hep-ph-0406253-1-9-1', 'hep-ph-0406253-2-9-1']]

[]

[]

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

https://arxiv.org/abs/hep-ph/0406253

{'hep-ph-0406253-3-0-0': 'We consider the production of neutralino, chargino, and charged Higgs boson pairs in the MSSM framework at future [MATH] colliders.', 'hep-ph-0406253-3-0-1': 'We show that, for c.m. energies in the one TeV range and in a "moderately" light SUSY scenario, a combined analysis of the slopes of these production cross sections could lead to a strong consistency test involving the soft supersymmetric breaking parameters [MATH], [MATH], the Higgsino parameter [MATH], and [MATH].', 'hep-ph-0406253-3-1-0': 'It is a widespread hope within the members of the high energy elementary particle physics community that the upcoming experiments at Tevatron [CITATION] and at LHC [CITATION] will finally reveal the existence of supersymmetric particles, via direct production of sparticle-antisparticle pairs.', 'hep-ph-0406253-3-1-1': 'In such an exciting hypothesis, the simplest available theoretical supersymmetric model, the MSSM, will acquire the same role that belonged to the SM after the W, Z discoveries, and a long and crucial period of precision tests will start, aiming to confirm, or to disprove, the main theoretical assumptions that were used in the practical construction of the model.', 'hep-ph-0406253-3-2-0': 'For the MSSM case, this program requires the construction of a linear [MATH] collider [CITATION] that explores the c.m. TeV energy region.', 'hep-ph-0406253-3-2-1': 'This machine should be sufficiently accurate to provide the same kind of consistency tests of the model that were achieved in the hundred GeV region at LEP for the Standard Model, from detailed analyses of several independent one-loop virtual effects.', 'hep-ph-0406253-3-3-0': 'If such an ambitious project is to be carried on, a clean investigation strategy will undoubtedly be welcome, given the fact that the involved theoretical model has a not simple structure in which several independent parameters are implied.', 'hep-ph-0406253-3-3-1': 'As a consequence of this feature, it might be appreciable to identify clean experimental measurements that, possibly, isolate in a relatively simple way a reduced subset of parameters and of theoretical assumptions behind them.', 'hep-ph-0406253-3-4-0': 'The aim of this short paper is to show that a dedicated combined analysis of the process of production of neutralino, chargino and charged Higgs boson pairs might lead to a relevant precision test of the theoretical assumptions that were used to fix, not only the basic supersymmetric sector (i.e. the chiral superpotential and the gauge multiplet), but also the gaugino component of the soft supersymmetry breaking of the model.', 'hep-ph-0406253-3-4-1': 'This will be done in the paper by assuming a scenario of "moderately" light SUSY particles (i.e. the various masses are all below, roughly, 350 GeV) and a c.m. energy [MATH] in the one TeV region.', 'hep-ph-0406253-3-4-2': 'Under these two assumptions, a simple asymptotic expansion of so called "Sudakov type" is effective [CITATION].', 'hep-ph-0406253-3-4-3': 'But the approach would not change for larger SUSY masses [MATH] GeV since a related c.m. energy rescaling would still be possible, from our previous experience, up to [MATH] values of about 2 TeV, where the resummation to higher orders is still not necessary [CITATION].', 'hep-ph-0406253-3-5-0': 'After this brief description of the strategy of our approach we are now ready to illustrate the practical details of our work.', 'hep-ph-0406253-3-5-1': "With this aim, we must recall, for the reader's convenience, that the relevant details of an analogous procedure, limited to the combination of chargino and charged Higgs pairs production, have already appeared in a previous paper[CITATION].", 'hep-ph-0406253-3-5-2': 'It was shown there that from the combined analyses of the slopes of the total cross sections in a c.m. energy range of about 1 TeV, assuming a "light" SUSY scenario where all the relevant sparticle masses lie below [MATH] 350-400 GeV , a strong constraint on [MATH] (mostly produced by the Higgs cross section) and a strip in the [MATH] plane (only produced by the chargino cross sections) were derivable.', 'hep-ph-0406253-3-5-3': 'Here [MATH] is the soft SUSY breaking wino mass, [MATH] is the Higgsino mass parameter, and [MATH] is the ratio of the two Higgs doublets vevs.', 'hep-ph-0406253-3-5-4': 'The novel process that we consider in this paper is neutralino pair production, and we shall show that its addition to the other processes will lead to highly improved constraints on the previous parameters, with the additional presence of [MATH], the soft SUSY breaking bino mass.', 'hep-ph-0406253-3-5-5': 'With this aim, we now briefly list the relevant theoretical formulae that we need for the analysis.', 'hep-ph-0406253-3-6-0': 'At the Born level, the neutralino pair production process [MATH] is described by three s, t, u channel components of the scattering amplitude, using the following notations (the indices [MATH], [MATH] label the electron and the [MATH]-th neutralino chiralities): [EQUATION] with the tree level values for the s channel [EQUATION] and for the t, u channels [EQUATION]', 'hep-ph-0406253-3-6-1': 'The quantities [MATH] are the elements of the 4x4 mixing matrix, defined in a conventional way [CITATION].', 'hep-ph-0406253-3-6-2': 'For simplicity, we have neglected in the considered asymptotic region the selectron mass in Eq. (1).', 'hep-ph-0406253-3-7-0': 'For the purposes of a precision test, it becomes mandatory to compute the following perturbative one-loop expansion of the scattering amplitude.', 'hep-ph-0406253-3-7-1': 'This requires the calculation of a large number of Feynman diagrams.', 'hep-ph-0406253-3-7-2': 'The resulting expressions are valid for any value of the c.m. energy [MATH].', 'hep-ph-0406253-3-7-3': 'The full one-loop result has been calculated recently [CITATION].', 'hep-ph-0406253-3-7-4': 'In this work we propose an alternative approach, valid for values of [MATH] "much" larger than all the SUSY sparticle masses involved in the process.', 'hep-ph-0406253-3-7-5': 'In fact, our strategy should be now made very clear.', 'hep-ph-0406253-3-7-6': 'We assume a previous production of the charginos, of the charged Higgs bosons and of at least two neutralinos with mass [MATH].', 'hep-ph-0406253-3-7-7': 'Calling [MATH] the heaviest of the real and virtual SUSY particles that appear in the processes, and assuming a "reasonable" limit [MATH] GeV, we shall choose the value [MATH] TeV value to proceed with our approach.', 'hep-ph-0406253-3-7-8': 'This is due to the fact that, in a previous paper only concerned with the charged Higgs pair production [CITATION], we proved that in such a configuration an asymptotic energy logarithmic expansion of so called "Sudakov type" was reliable, with the only addition of a constant term to the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-3-7-9': 'This conclusion allows to propose a determination of the SUSY parameters entering the Sudakov logarithms, based on measurements of the slope of the total cross section, in which the (complicated) constant terms cancel.', 'hep-ph-0406253-3-8-0': 'The approach that we shall follow in this paper, that was already used in the combined chargino-charged Higgs analysis [CITATION], will assume that a similar expansion is valid, with only logarithmic and constant terms.', 'hep-ph-0406253-3-8-1': 'To prove this statement would require a detailed comparison of the complete existing calculation [CITATION] with the assumed asymptotic expansion.', 'hep-ph-0406253-3-8-2': 'This will be the goal of a forthcoming rigorous analysis.', 'hep-ph-0406253-3-8-3': 'For the moment, we shall assume its validity as a working Ansatz, and we shall show the main relevant consequences that it will be able to produce.', 'hep-ph-0406253-3-9-0': 'After this preliminary discussion, we are now ready to write the relevant asymptotic expansion of the scattering amplitude at one loop, in which we shall only retain the leading quadratic and next-to-leading linear logarithmic terms.', 'hep-ph-0406253-3-9-1': 'For this purpose, we shall use the following formal decomposition the separate [MATH], [MATH] or [MATH] subamplitudes (the Born values are those listed in Eqs. ([REF]-[REF])): [EQUATION]', 'hep-ph-0406253-3-9-2': 'In the asymptotic expansion, two different kinds of logarithmic terms appear.', 'hep-ph-0406253-3-9-3': 'The first ones are the standard linear ones of RG origin.', 'hep-ph-0406253-3-9-4': 'They are well known and can be derived in a straightforward way replacing in the Born quantities the various bare couplings with running ones.', 'hep-ph-0406253-3-9-5': 'For sake of completeness, we write the related formulae, noticing that they are only requested for the s-channel (purely Higgsino) component, as: [EQUATION] where [MATH] and [MATH].', 'hep-ph-0406253-3-9-6': 'The second type of logarithms which arise asymptotically is that of the "genuine weak" Sudakov terms.', 'hep-ph-0406253-3-9-7': 'These are usually classified [CITATION] as logarithms of gauge non universal, gauge universal and Yukawa origin.', 'hep-ph-0406253-3-9-8': 'The gauge non universal, scattering angle dependent ones, stem from box diagrams and t,u channel vertices and have the following expressions: [EQUATION]', 'hep-ph-0406253-3-9-9': 'The gauge universal ones are due both to the initial and to the final vertices.', 'hep-ph-0406253-3-9-10': 'The initial contribution is fixed by the coefficients [EQUATION]', 'hep-ph-0406253-3-9-11': 'The contribution from final neutralino legs is different for the s or u, t channels.', 'hep-ph-0406253-3-9-12': 'For the s-channel we have higgsino components [EQUATION] where [EQUATION]', 'hep-ph-0406253-3-9-13': 'The logarithms of Yukawa origin are only due to final s-channel vertices with virtual heavy (b,t) quark-squark pairs.', 'hep-ph-0406253-3-9-14': 'These are the only logarithmic terms that contain (also) the vevs ratio [MATH] and their expression is [EQUATION]', 'hep-ph-0406253-3-9-15': 'To conclude, the final contribution to the u and t channel amplitudes is due to gaugino components for which there is no Yukawa contribution: [EQUATION] with [EQUATION]', 'hep-ph-0406253-3-9-16': 'Eqs. ([REF]-[REF]) represent the complete logarithmic contributions at one loop to the considered process, and are the main original result of this paper.', 'hep-ph-0406253-3-9-17': 'We expect from our previous discussion their reliability for what concerns the calculation of the slope of the total cross section in the 1 TeV range.', 'hep-ph-0406253-3-9-18': 'To perform the latter, one must first compute the expression of the differential cross section that is readily obtained from the above amplitude.', 'hep-ph-0406253-3-9-19': 'After angular integration, one obtains the approximate (to next-to-leading logarithmic order) asymptotic expression of the total cross section.', 'hep-ph-0406253-3-9-20': 'This will be sufficient to determine the effective asymptotic expression of the slope of the cross section, where by definition possible extra "next-to-next-to-leading" (i.e. constant) terms disappear.', 'hep-ph-0406253-3-9-21': 'This expression will only depend on [MATH] and on the mixing parameters [MATH].', 'hep-ph-0406253-3-9-22': 'The latter ones, in turn, can be expressed as functions of the supersymmetric parameters [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0406253-3-9-23': 'Thus, the final form of the slope will depend on these four parameters.', 'hep-ph-0406253-3-10-0': 'In order to achieve the maximal theoretical information using our approach, we have combined the theoretical expressions of the slope of the neutralino pair total cross section, given in this paper, with those of the chargino and charged Higgs pairs, given in [CITATION].', 'hep-ph-0406253-3-10-1': 'In practice, we have limited our analysis to the production of the two light observable final states ([MATH], [MATH]) for neutralinos, combined with the production of all the three chargino pairs and of the charged Higgs pair.', 'hep-ph-0406253-3-10-2': 'To work in a consistent way, we have chosen as relevant examples for the masses of the produced pairs three sets of values denoted by [MATH], [MATH], and [MATH].', 'hep-ph-0406253-3-10-3': 'The first, [MATH], is the Tesla benchmark point RR2[CITATION] with the two lightest neutralinos being respectively 95 bino and 82 wino.', 'hep-ph-0406253-3-10-4': 'The set [MATH] is a mixed scenario with neutralinos having non negligible gaugino and Higgsino components; [MATH] is 86 bino and 13 higgsino, [MATH] is 11 bino, 48 wino and 41 higgsino.', 'hep-ph-0406253-3-10-5': 'Finally, [MATH] is a purely Higgsino one with the two lightest neutralinos being 92 and 98 higgsino like.', 'hep-ph-0406253-3-10-6': 'The values of the input parameters as well as the masses of the two charginos and of the two lightest neutralinos are summarized in Tab.', 'hep-ph-0406253-3-10-7': 'We have performed a standard [MATH] analysis assuming in the various scenarios 10-12 experimental points at energies ranging from 700-850 GeV (depending on the scenario) up to 1200 GeV and assuming an experimental accuracy of 1 for the cross sections of gauginos and 2 for that of Higgs bosons.', 'hep-ph-0406253-3-10-8': 'For each scenario we have checked the condition [MATH].', 'hep-ph-0406253-3-10-9': 'The results of our analysis are shown in Figs. (1-3).', 'hep-ph-0406253-3-10-10': 'From inspection of the figures one can draw the following conclusions.', 'hep-ph-0406253-3-11-0': 'In Fig. (1) we consider the [MATH] benchmark point.', 'hep-ph-0406253-3-11-1': 'One notices that a combined analysis is able to generate closed contours in the planes of all possible six couples of parameters.', 'hep-ph-0406253-3-11-2': 'This possibility is obviously existing for the first 3 cases i.e. [MATH], [MATH], [MATH] under the condition that the neutralino information is added to the remaining ones, that do not depend on [MATH].', 'hep-ph-0406253-3-11-3': 'Less obvious and more illustrative is the closure of the undetermined strip in the [MATH] plane (drawn in the absence of neutralino data) when neutralino data are added to the remaining ones.', 'hep-ph-0406253-3-11-4': 'This addition does not practically improve, on the contrary, the bounds on [MATH] and [MATH] obtained from chargino-Higgs data.', 'hep-ph-0406253-3-11-5': 'One sees typical errors, under the assumed experimental conditions, of about 10 GeV for [MATH], [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-3-12-0': 'In Fig.(2) we consider the [MATH] scenario.', 'hep-ph-0406253-3-12-1': 'One finds, approximately, the same features for the set of projections.', 'hep-ph-0406253-3-12-2': 'The size of errors is now of approximately [MATH] GeV for [MATH], [MATH], and [MATH], a relative 40-50 for [MATH].', 'hep-ph-0406253-3-13-0': 'Fig.(3) describes [MATH], a typical Higgsino scenario.', 'hep-ph-0406253-3-13-1': 'Here the new (negative) feature concerns the lack of boundary for [MATH], as seen from the first three plots.', 'hep-ph-0406253-3-13-2': 'This can be understood since, at the considered [MATH], [MATH] and [MATH] values, the mixing parameters [MATH] do not vary appreciably when [MATH] increases.', 'hep-ph-0406253-3-13-3': 'On the contrary, the three plots in the bottom line retain the typical features of the previous cases, with particular relevance of the neutralino role for the determination of lower limits for [MATH] and [MATH].', 'hep-ph-0406253-3-13-4': 'The errors are now , typically, of about 20 GeV for [MATH], [MATH] and of a relative [MATH] for [MATH].', 'hep-ph-0406253-3-14-0': 'The previous results should illustrate the aimed possible outcomes of the testing strategy that we are proposing in this preliminary qualitative paper.', 'hep-ph-0406253-3-14-1': 'Our assumed physical inputs will necessarily be the masses of the (supposedly produced) (light) neutralinos, charginos and charged Higgs.', 'hep-ph-0406253-3-14-2': 'Their values will depend on the four parameters [MATH], [MATH], [MATH], [MATH] but, in general there will not be a 1-1 correspondence, and different sets of parameters might reproduce "essentially" the same masses [CITATION].', 'hep-ph-0406253-3-14-3': 'Within the errors that we have illustrated in our examples, these different sets can be discriminated via the [MATH] analysis that we have proposed, and the "correct" set is selected by the true experimental data, if the latter are actually described by the model.', 'hep-ph-0406253-3-14-4': 'If none of the candidate sets were compatible with the analysis, an indication would arise that some of the details of the model might need a suitable modification.'}

1907.06611

{'1907.06611-1-0-0': 'Next-generation optical imaging surveys will revolutionise the observations of weak gravitational lensing by galaxy clusters and provide stringent constraints on growth of structure and cosmic acceleration.', '1907.06611-1-0-1': 'In these experiments, accurate modelling of covariance matrices of cluster weak lensing plays the key role in obtaining robust measurements of the mean mass of clusters and cosmological parameters.', '1907.06611-1-0-2': 'We use a combination of analytical calculations and high-resolution N-body simulations to derive accurate covariance matrices that span from the virial regime to linear scales of the cluster-matter cross-correlation.', '1907.06611-1-0-3': 'We validate this calculation using a public ray-tracing lensing simulation and provide a software package for calculating covariance matrices for a wide range of cluster and source sample choices.', '1907.06611-1-0-4': 'We discuss the relative importance of shape noise and density fluctuations, the impact of radial bin size, and the impact of off-diagonal elements.', '1907.06611-1-0-5': 'For a weak lensing source density [MATH] arcmin[MATH], shape noise typically dominates the variance on comoving scales [MATH].', '1907.06611-1-0-6': 'However, for [MATH] arcmin[MATH], potentially achievable with future weak lensing experiments, density fluctuations typically dominate the variance at [MATH] and remain comparable to shape noise on smaller scales.', '1907.06611-1-1-0': '# Introduction', '1907.06611-1-2-0': 'Understanding the origin of cosmic acceleration requires measurements of both the expansion rate of the Universe and the growth rate of large-scale structure .', '1907.06611-1-2-1': 'The number counts of galaxy clusters as a function of mass and redshift are sensitive to both expansion rate and the growth of structure ; also see [CITATION] for a review.', '1907.06611-1-2-2': 'With optical surveys, one can identify clusters to lower mass thresholds at high redshifts relative to X-ray or submillimeter surveys.', '1907.06611-1-2-3': 'Regardless of identification method, calibrating the mass scale associated with observed cluster properties plays a key role in extracting cosmological information from galaxy clusters.', '1907.06611-1-2-4': 'In optical imaging surveys, one can use weak gravitational lensing to constrain mean cluster mass profiles .', '1907.06611-1-2-5': 'These profiles can often be measured with high precision far beyond the cluster virial radius, to scales of tens of Mpc, where they probe the linear regime of the cluster-matter cross-correlation function .', '1907.06611-1-2-6': 'The combination of cluster space densities and weak lensing profiles is in many ways analogous to the combination of galaxy clustering and galaxy-galaxy lensing, but applied to the high mass end of the dark matter halo population.', '1907.06611-1-2-7': 'In these experiments, accurate modelling of the signals and the covariance matrices is essential for obtaining robust constraints on cluster masses and cosmological parameters.', '1907.06611-1-3-0': 'To date, most of the weak gravitational lensing measurements of galaxy clusters are dominated by shape noise; that is, because of the modest number density of source galaxies, the noise of lensing is dominated by the intrinsic ellipticities of source galaxies.', '1907.06611-1-3-1': 'However, for the next-generation optical surveys like the Large Synoptic Survey Telescope (LSST, ), Euclid , and the Wide Field Infrared Survey Telescope (WFIRST, ), the number density of source galaxies will be significantly higher and the shape noise will no longer dominate.', '1907.06611-1-3-2': 'Instead, the density fluctuations related to the intrinsic variation of halo density profiles will dominate the uncertainties at small scales, and the contribution of uncorrelated LSS will dominate the uncertainties at large scales.', '1907.06611-1-3-3': 'In this paper, we detail how to calculate covariance matrices for cluster lensing at all scales with and without shape noise, and we provide a software package that can calculate the covariance matrix for a wide range of survey assumption.', '1907.06611-1-4-0': 'The covariance matrices of cluster lensing can be calculated directly from data using jackknife methods , analytical formulae , or simulations .', '1907.06611-1-4-1': 'In this paper, we focus on the latter two approaches.', '1907.06611-1-4-2': 'The literature for galaxy cluster lensing covariance matrices is relatively small.', '1907.06611-1-4-3': '[CITATION] quantified the impact of intrinsic variation of halo density profiles on the covariance matrices of cluster weak lensing by combining analytical calculations with numerical simulations.', '1907.06611-1-4-4': 'However, as we will explain in Section [REF], they quantified halo-to-halo covariance rather than patch-to-patch covariance, and the latter is relevant for stacked cluster lensing.', '1907.06611-1-5-0': 'The covariance matrices of galaxy-galaxy lensing are more widely studied.', '1907.06611-1-5-1': 'Galaxy-galaxy lensing is analogous to cluster lensing and refers to using galaxies (lower-mass haloes) instead of galaxy clusters as lenses.', '1907.06611-1-5-2': '[CITATION] provided analytical formulae for galaxy-galaxy lensing covariance matrices.', '1907.06611-1-5-3': 'Using observed and simulated catalogues, they compared various sources of errors and demonstrated the importance of subtracting lensing signals around random points.', '1907.06611-1-5-4': 'Their analysis at small scales is dominated by shape noise and is therefore not directly applicable for the next-generation cluster lensing surveys.', '1907.06611-1-5-5': '[CITATION] used simulated full-sky weak lensing maps to compare various ways for calculating covariance matrices and demonstrated the accuracy of jackknife covariance.', '1907.06611-1-5-6': '[CITATION] developed large-volume simulations for calculating the cross-probe covariance for cosmic shear, galaxy-galaxy lensing, and galaxy clustering.', '1907.06611-1-5-7': 'Other earlier work includes [CITATION].', '1907.06611-1-5-8': 'These papers focus on lower-mass haloes and large scales; therefore, they do not provide the small-scale variances we need for cluster lensing mass calibration.', '1907.06611-1-6-0': 'Because of the reasons stated above, for the next-generation cluster lensing measurements by LSST, Euclid, and WFIRST, we need to come up with a new treatment of covariance matrices.', '1907.06611-1-6-1': 'In this paper, we first compare the analytical covariance assuming Gaussian random fields with the covariance from simulated lensing maps.', '1907.06611-1-6-2': 'We then identify the scales where the Gaussian-field covariance is insufficient, and we add corrections based on high-resolution N-body simulations.', '1907.06611-1-6-3': 'We provide readers with a set of user-friendly equations with short and heuristic derivations, a software package, and tabulated results from simulations.', '1907.06611-1-7-0': 'Readers might ask why we do not use simulated gravitational lensing ray-tracing maps to calculate covariance matrices at all scales.', '1907.06611-1-7-1': 'For example, [CITATION] and [CITATION] produced full-sky lensing maps and used them to study the covariance matrices of galaxy-galaxy lensing, and we use these maps to cross-check our calculations.', '1907.06611-1-7-2': 'However, because of the substantial computational resources required, these maps are limited to one cosmology and are unable to resolve the inner profiles of clusters.', '1907.06611-1-7-3': 'We need the contribution from very small scales (to account for the intrinsic variation of density profiles) to very large scales (to account for LSS contribution).', '1907.06611-1-7-4': 'For the former, we use high-resolution N-body simulation boxes to characterise the non-linear evolution of dark matter haloes.', '1907.06611-1-7-5': 'For the latter, it is impractical to use N-body simulation boxes because we would need to include dark matter particles extending to a few Gpc, while analytical calculations are much more efficient in this regime.', '1907.06611-1-7-6': 'We will show that this approach agrees with the ray-tracing simulations of [CITATION] at intermediate and large scales.', '1907.06611-1-8-0': 'We use two sets of public simulations: the ray-tracing lensing maps from [CITATION] and the N-body simulations from Abacus Cosmos .', '1907.06611-1-8-1': 'Table [REF] summarises the specifications of the two simulations.', '1907.06611-1-8-2': 'The difference in these two sets of cosmological parameters leads to negligible difference in covariance matrices.', '1907.06611-1-9-0': 'Throughout this work, we use [MATH], the mass defined by the radius within which the overdensity is 200 times the mean density of the Universe.', '1907.06611-1-9-1': 'All distances are in comoving [MATH]; [MATH] denotes the line-of-sight distances, and [MATH] denotes projected (transverse) distances.', '1907.06611-1-9-2': 'Since cluster lensing covariance matrices are inversely proportional to the survey area, we normalise all of the covariance values to those of a volume of 1 [MATH].', '1907.06611-1-9-3': 'Table [REF] summarises the notations used in this work.', '1907.06611-1-9-4': 'We use "variance" to indicate the diagonal elements of a covariance matrix.', '1907.06611-1-10-0': 'This paper is organised as follows.', '1907.06611-1-10-1': 'Section [REF] provides the basic equations of cluster weak lensing.', '1907.06611-1-10-2': 'Section [REF] describes how we measure covariance matrices from lensing simulations and possible pitfalls in this procedure.', '1907.06611-1-10-3': 'In Section [REF] we focus on large-scale covariance matrices for tangential shear assuming Gaussian random fields, and in Section [REF] we present the analogous calculations for excess surface density.', '1907.06611-1-10-4': 'In Section [REF] we use high-resolution N-body simulations to calculate small-scale covariance matrices and combine simulations with analytical calculations.', '1907.06611-1-10-5': 'We discuss the correlation between radial bins in Section [REF].', '1907.06611-1-10-6': 'In Section [REF] we discuss shape noise, the dependence of covariance matrices on halo mass and redshift, and cross-mass covariance matrices.', '1907.06611-1-10-7': 'Section [REF] summarises our results.', '1907.06611-1-11-0': '# Background of cluster weak lensing', '1907.06611-1-12-0': 'Below we briefly describe the basic equations for cluster weak lensing.', '1907.06611-1-12-1': 'We refer interested readers to, e.g., [CITATION] for comprehensive reviews.', '1907.06611-1-13-0': 'The gravitational lensing signal is quantified by the Jacobian matrix for the coordinate transformation from the source plane to the lens plane, [EQUATION] where [MATH] corresponds to convergence, and ([MATH], [MATH]) corresponds to the two components of shear.', '1907.06611-1-13-1': 'We ignore higher-order effects in this work.', '1907.06611-1-13-2': 'If we assume that the line-of-sight dimension of a dark matter halo is much smaller than the distance between the observer and the source galaxy (the thin-lens approximation), the azimuthally averaged convergence is related to the surface density ([MATH]) of a halo via [EQUATION] where [MATH] is the projected distance on the lens plane in comoving units, and [MATH] is the critical surface density defined as [EQUATION] where [MATH] denotes comoving distances; we use subscripts h and s to denote the redshifts or distances of haloes (galaxy clusters) and sources (background galaxies).', '1907.06611-1-13-3': 'Note that we have [MATH] in the denominator because we use comoving units; if one uses physical units, [MATH] would differ by [MATH] and would usually be written in terms of angular diameter distances.', '1907.06611-1-14-0': 'The two shear components depend on the choice of coordinate system, and a physical quantity is the tangential shear, [EQUATION] where [MATH] is the position angle of the source galaxy with respect to the cluster centre.', '1907.06611-1-14-1': 'The azimuthally averaged tangential shear is related to the excess surface density ([MATH]) via [EQUATION]', '1907.06611-1-14-2': 'The observables are the reduced shears of individual source galaxies, [MATH], [MATH].', '1907.06611-1-14-3': 'In the weak lensing regime, [MATH], and [MATH] is the observable for practical purposes.', '1907.06611-1-14-4': 'In the context of cluster lensing, one often converts [MATH] to [MATH] because the latter can be interpreted physically as the excess surface density profiles of clusters.', '1907.06611-1-15-0': 'The two quantities, [MATH] and [MATH], have different advantages and disadvantages.', '1907.06611-1-15-1': 'For a given lens, [MATH] is independent of source redshift, while [MATH] is higher for a higher source redshift.', '1907.06611-1-15-2': 'On the other hand, to calculate [MATH] from the observed [MATH] one needs to assume a cosmology and know the redshifts of sources and lenses.', '1907.06611-1-15-3': 'In this work, we will first focus on the covariance matrices of [MATH] in Section [REF] because the contribution from LSS to the noise of [MATH] is easier to understand.', '1907.06611-1-15-4': 'We will then discuss the analogous covariance matrices of [MATH] in Section [REF].', '1907.06611-1-16-0': 'For an ensemble of galaxy clusters at a given redshift, the mean [MATH] profile can be computed from the halo-matter correlation function [MATH] using [EQUATION]', '1907.06611-1-16-1': 'This real-space expression is useful for calculating mean profiles of [MATH] or [MATH], while the equivalent Fourier-space expression (equation [REF] below) is more useful for calculating covariance matrices.', '1907.06611-1-17-0': 'To calculate [MATH], we use the approach in [CITATION].', '1907.06611-1-17-1': 'On scales smaller than the virial radius, [MATH] represents the average density profile.', '1907.06611-1-17-2': 'We use an Navarro-Frenk-White (NFW) profile , [MATH], with the concentration-mass relation from [CITATION].', '1907.06611-1-17-3': 'We assume no scatter between concentration and mass.', '1907.06611-1-17-4': 'At scales much greater than the virial radius, [MATH] is the linear matter correlation function [MATH] multiplied by the halo bias factor [MATH].', '1907.06611-1-17-5': 'For intermediate scales, we use the larger of the two; that is, [EQUATION] where [EQUATION]', '1907.06611-1-18-0': '# Covariance matrices from weak lensing maps', '1907.06611-1-19-0': 'We start by measuring the covariance matrices from the weak lensing maps of [CITATION] and discussing pitfalls associated with these measurements.', '1907.06611-1-19-1': 'We will later use these measurements to cross-check our calculations combining analytical formulae and N-body simulations.', '1907.06611-1-20-0': '## Takahashi ray-tracing simulations', '1907.06611-1-21-0': 'We use the publicly available lensing maps produced by [CITATION].', '1907.06611-1-21-1': 'These authors built full-sky lightcones using N-body simulations based on Gadget2 .', '1907.06611-1-21-2': 'The N-body simulations are based on a flat [MATH]CDM cosmology consistent with WMAP9 , [MATH] = 0.279; [MATH] = 0.721; [MATH] = 0.046; [MATH] = 0.7; [MATH] = 0.82; [MATH] = 0.97.', '1907.06611-1-21-3': 'The dark matter haloes identified using Rockstar .', '1907.06611-1-21-4': 'For the lightcones, starting from [MATH], for every 450 [MATH] the authors used a different N-body simulation volume with progressively lower resolution.', '1907.06611-1-21-5': 'For the lensing simulation, they set a source plane every 150 [MATH] and generated full-sky maps of convergence, shear, and rotation using ray-tracing calculations for all the dark matter in front of the source plane.', '1907.06611-1-21-6': 'They provided maps with resolution with Healpix [MATH] = 4096, 8192, and 16384.', '1907.06611-1-21-7': 'In our calculations, we use the maps with [MATH] = 4096 (corresponding to [MATH] pixels in the full sky), and the angular resolution of each pixel is [MATH] radian or 0.86 arcmin.', '1907.06611-1-21-8': 'We have checked that the [MATH]=8192 maps give nearly identical results.', '1907.06611-1-22-0': 'Since the lightcone exhibits discontinuities every 450 [MATH], we choose lens redshift bins that avoid these discontinuities; that is, a halo sample we choose comes from one original N-body simulation box and thus has continuous LSS.', '1907.06611-1-22-1': 'For our fiducial calculation, we use haloes from lens plane 10, which corresponds to a comoving distance range (1350, 1500) [MATH] and a redshift range (0.508, 0.574).', '1907.06611-1-22-2': 'The haloes in this redshift range are generated from an N-body simulation of box size 1800 [MATH] with 2048[MATH] particles; the mass resolution is [MATH], and the softening length is 32 [MATH].', '1907.06611-1-22-3': 'We use source plane 18, which corresponds to a redshift range (1.218, 1.318); see Tables 1 and 2 in [CITATION].', '1907.06611-1-23-0': 'We divide the full-sky catalogues into 48 equal-area samples.', '1907.06611-1-23-1': 'We use 10 of the 108 realisations provided by the authors; that is, we have 480 realisations of a survey with [MATH] (860 deg[MATH]).', '1907.06611-1-23-2': 'To calculate the averaged [MATH] profiles for each sample, we take all the haloes with [MATH] in this sample (on average 785 haloes per sample) and cross-correlate with all pixels in the shear map using the publicly available code TreeCorr.', '1907.06611-1-23-3': 'In addition, we generate random points in the same area (20 times the number of haloes) and calculate the [MATH] profile.', '1907.06611-1-23-4': 'We subtract the shear signal around random points from the cluster lensing shear signal.', '1907.06611-1-23-5': 'In this way, we obtain the mean shear profiles of [MATH] for each patch of [MATH], and we calculate the covariance matrix of these 480 realisations.', '1907.06611-1-23-6': 'We use 5 logarithmically-spaced bins per decade of angular separation (15 bins for [MATH]).', '1907.06611-1-23-7': 'The number of bins (15) is much smaller than the number of realisations (480).', '1907.06611-1-23-8': 'As the number of realisations is much greater than the number of bins, we do not need to correct the inverse covariance as in [CITATION].', '1907.06611-1-24-0': '## Pitfalls of calculating covariance matrices', '1907.06611-1-25-0': 'Before going into the details of covariance matrix calculations, we discuss two possible pitfalls in calculating and interpreting the cluster lensing covariance matrices from simulations.', '1907.06611-1-25-1': 'Fig. [REF] shows an example for calculating the covariance matrix from the Takahashi simulations (for the fiducial calculation described in Section [REF]).', '1907.06611-1-26-0': 'The first pitfall is calculating covariance matrices without subtracting the lensing signal around random locations.', '1907.06611-1-26-1': '[CITATION] demonstrated that when measuring the shear signal around galaxies, subtracting the shear signal around random lenses can remove shear systematics .', '1907.06611-1-26-2': 'We find that this subtraction is necessary not only for observations but also for simulated shear maps.', '1907.06611-1-26-3': 'Fig. [REF] shows that, even when using simulations free of systematics, the patch-to-patch variance at large scales is erroneously high if one does not subtract the mean random signal (orange).', '1907.06611-1-26-4': 'A similar excess at large-scale has also been shown in [CITATION] using real sources from SDSS.', '1907.06611-1-26-5': 'Subtracting the shear at random points removes the impact of over- or under-densities on the scales of the sample and is roughly analogous to using a [CITATION] estimator for the correlation function.', '1907.06611-1-26-6': 'For small scales, the shear signal centred on random points is small compared to that centred on clusters, and this subtraction makes negligible difference.', '1907.06611-1-27-0': 'The second pitfall is using halo-to-halo covariance instead of patch-to-patch covariance.', '1907.06611-1-27-1': 'One obvious way to estimate the covariance matrix for a sample of clusters, either observed or simulated, is to measure the [MATH] or [MATH] profile cluster by cluster, compute the covariance matrix of these profiles, and divide by the cluster number to estimate the covariance matrix of the mean profile for the full sample .', '1907.06611-1-27-2': 'However, this halo-to-halo covariance generally underestimates the true uncertainty of the mean profile from the sample.', '1907.06611-1-27-3': 'Instead, the covariance matrix should be computed from the mean profiles derived from patches of sky that are substantially larger than the largest scales being measured.', '1907.06611-1-27-4': 'Fig. [REF] demonstrates this point using the Takahashi simulations: at scales [MATH] radian ([MATH]), the variance of [MATH] from halo to halo divided by the number of haloes in a patch (green) is smaller than the variance among the 860 deg[MATH] patches (blue).', '1907.06611-1-27-5': 'It is this latter variance that provides an estimate of the uncertainty on the mean [MATH] profile for an 860 deg[MATH] survey.', '1907.06611-1-28-0': 'At small scales, the halo-to-halo and patch-to-patch estimates of the covariance are similar.', '1907.06611-1-28-1': 'The difference occurs at large scales, where the dominant contribution to the covariance is from the uncorrelated LSS.', '1907.06611-1-28-2': 'The grey dash curve in Fig. [REF] shows the analytical prediction for this LSS contribution (equation [REF] below), which agrees with the patch-to-patch simulation results.', '1907.06611-1-28-3': 'The grey dotted curve corresponds to equation ([REF]) without the halo clustering term [MATH] and is close to the halo-to-halo variance.', '1907.06611-1-28-4': 'At large scale, in addition to shot noise, the halo-to-halo covariance includes the three-point function [MATH], which is negligible, while the patch-to-patch covariance includes the four-point function [MATH], which is non-negligible and includes the effect of halo clustering.', '1907.06611-1-28-5': 'Therefore, the halo clustering affects the latter but not the former.', '1907.06611-1-28-6': 'Physically, the excess variance arises because clustered haloes do not independently sample the foreground/background cosmic shear signal.', '1907.06611-1-29-0': 'For samples in which individual massive clusters are widely separated in the sky, the halo-to-halo and patch-to-patch variances are equivalent by definition.', '1907.06611-1-29-1': 'Thus, the effect illustrated in Fig. [REF] is not important for individual cluster measurements like Weighing the Giants .', '1907.06611-1-29-2': 'However, it will be relevant for surveys that measure stacked weak lensing profiles to large scales for large cluster samples over contiguous areas of the sky, such as DES, LSST, Euclid, and WFIRST.', '1907.06611-1-30-0': '# Analytical Gaussian-field covariance for tangential shear ([MATH])', '1907.06611-1-31-0': 'For analytical calculations, we start by calculating the covariance of [MATH] derived from linear theory, with the assumption that the underlying matter density and halo number density follow Gaussian random fields.', '1907.06611-1-31-1': 'Such an assumption is only valid at large scales.', '1907.06611-1-31-2': 'We call such covariance "Gaussian-field" covariance.', '1907.06611-1-31-3': 'In this section, we focus on the covariance of [MATH], and we will discuss the covariance of [MATH] in Section [REF].', '1907.06611-1-31-4': 'The Gaussian-field covariance of [MATH] is easier to derive and interpret because the contribution from LSS can be intuitively understood as additional noise of [MATH].', '1907.06611-1-32-0': 'To ensure readability, we write the equations in their simplest forms, i.e., assuming no weighting for lenses or sources (see e.g., for equations incorporating weighting).', '1907.06611-1-32-1': 'Our notations are similar to those in [CITATION] and [CITATION], and we use angular power spectrum [MATH] (based on the spherical harmonics transform of a map on a sphere) for the spatial distributions of lenses and sources.', '1907.06611-1-32-2': 'Some authors use the two-dimensional power spectrum, [MATH], which is equivalent to the angular power spectrum on scales where sky curvature is negligible ([MATH]).', '1907.06611-1-33-0': '## Covariance matrices from Fourier space', '1907.06611-1-34-0': 'The lensing covariance is more easily derived in Fourier space than in real space because in Fourier space, [MATH] and [MATH] only differ by a phase and thus have the same power spectrum (see, e.g. ).', '1907.06611-1-34-1': 'In contrast, in real space, they are related via equation ([REF]).', '1907.06611-1-35-0': 'We can analytically calculate the mean tangential shear of clusters using a Hankel transform of the halo-lensing power spectrum: [EQUATION] where [MATH] is the cross angular power spectrum between haloes and shear (Section [REF]), and [MATH] is the Bessel function of order 2.', '1907.06611-1-35-1': 'This [MATH] arises from the fact that [MATH] is a spin-2 field; for [MATH] (scalar), we simply replace [MATH] by [MATH].', '1907.06611-1-35-2': 'Equation ([REF]) is equivalent to equation ([REF]) based on the real-space correlation function [MATH].', '1907.06611-1-35-3': 'While the Fourier-space expression is more useful for understanding the covariance matrix derivation, the real-space expression is easier to use for calculating the mean lensing signal.', '1907.06611-1-36-0': 'If we assume that both the halo number overdensity and the matter overdensity follow Gaussian random fields, then the covariance of [MATH] is given by (see, e.g., and Appendix [REF]) [EQUATION]', '1907.06611-1-36-1': 'Here [MATH] is the sky fraction of the survey; [MATH] and [MATH] are the surface number densities of haloes and sources in the unit of sr[MATH]; [MATH], [MATH], [MATH] are the angular power spectra of halo-halo, lensing-lensing, and halo-lensing, respectively (see Section [REF]).', '1907.06611-1-36-2': 'We use the bin-averaged [MATH] to take into account the finite radial bin size (e.g., equation 26 in ): [EQUATION]', '1907.06611-1-36-3': 'Appendix [REF] presents the properties of the bin-averaged Bessel function and the impact of bin size.', '1907.06611-1-36-4': 'We give the analytical expressions for [MATH], [MATH], [MATH] in terms of 3D power spectra using the Limber approximation in Section [REF], the derivations in Appendix [REF].', '1907.06611-1-36-5': 'The halo model we use is described in Appendix [REF].', '1907.06611-1-36-6': 'Appendix [REF] demonstrates the relative importance of [MATH], [MATH], [MATH], and [MATH].', '1907.06611-1-37-0': 'We use equation ([REF]) for modelling the Gaussian-field variance.', '1907.06611-1-37-1': 'This equation first appeared in [CITATION] and was used to calculate the covariance matrix at several hundred Mpc scales and to assess the sensitivity of galaxy-galaxy lensing to primordial non-Gaussianity.', '1907.06611-1-37-2': 'In this work, we use it to calculate cluster lensing at the scales of tens to 100 Mpc.', '1907.06611-1-38-0': '## Interpreting the three components of the covariance matrix', '1907.06611-1-39-0': 'Equation ([REF]) can be interpreted as the contribution from three components: shape noise (involving [MATH]), LSS (involving [MATH]), and the intrinsic variation of halo density profiles (involving [MATH]).', '1907.06611-1-39-1': '[EQUATION]', '1907.06611-1-39-2': 'We will show, based on comparisons to simulations, that the LSS term is sufficiently accurate, while the intrinsic term must be replaced by non-Gaussian calculations (e.g., numerical simulations).', '1907.06611-1-40-0': 'The contribution from shape noise is given by [EQUATION]', '1907.06611-1-40-1': 'Here [MATH] and [MATH] are associated with halo clustering and shot noise, respectively.', '1907.06611-1-40-2': 'When shot noise dominates the variance of halo number counts [MATH], the off-diagonal elements approach zero because of the orthogonality of [MATH], and the shape noise reduces to the more intuitive form [EQUATION]', '1907.06611-1-40-3': 'In Appendix [REF] we provide the derivation.', '1907.06611-1-40-4': 'We assume that each source galaxy contributes to only one halo and one radial bin, and violation of this assumption would lead to extra contributions to off-diagonal elements.', '1907.06611-1-41-0': 'For [MATH], halo clustering is non-negligible compared with shot noise (see the left-hand panel of Fig. [REF] and Fig. [REF]).', '1907.06611-1-41-1': 'For a higher mass threshold, both terms increase, and shot noise dominates for [MATH] at [MATH], and [MATH] at [MATH].', '1907.06611-1-41-2': 'The exact mass of transition slightly depends on the width of the redshift bin.', '1907.06611-1-42-0': 'The contribution from the LSS is given by [EQUATION]', '1907.06611-1-42-1': 'This contribution is basically the convergence power spectrum ([MATH], see for a review) multiplied by the noise of halo number counts, which include both the shot noise ([MATH]) and the clustering of haloes ([MATH]).', '1907.06611-1-42-2': 'Compared with shape noise, this LSS term is less sensitive to the radial bin size because both [MATH] and [MATH] decrease at high [MATH] and only the integration over the first peak of [MATH] has significant contribution (see Fig. [REF]).', '1907.06611-1-42-3': 'At large scales, [MATH] is usually higher than [MATH] (see the middle panel of Fig. [REF]), and thus [MATH].', '1907.06611-1-43-0': 'The contribution from intrinsic variation of halo density profiles is given by [EQUATION]', '1907.06611-1-43-1': 'This equation has limited use because it assumes that matter inside haloes follows a Gaussian random field, which is not true.', '1907.06611-1-43-2': 'We will replace this term with N-body calculations.', '1907.06611-1-44-0': 'Fig. [REF] displays the mean and variance of lensing calculated from simulations and analytical formulae.', '1907.06611-1-44-1': 'We use the fiducial calculations specified in Section [REF] ([MATH], [MATH]), with no shape noise.', '1907.06611-1-44-2': 'We convert from [MATH] to [MATH] by calculating the [MATH] corresponding to the source and lens redshifts; we show [MATH] and [MATH] at left and bottom axes, and the equivalent [MATH] and [MATH] on the top and right axes.', '1907.06611-1-45-0': 'The left panel shows the mean value of [MATH] calculated analytically from equation ([REF]).', '1907.06611-1-45-1': 'The right panel shows the contribution from LSS ([MATH], green) and from the intrinsic variation of halo density profiles ([MATH], grey dotted).', '1907.06611-1-45-2': 'The LSS contribution fully accounts for the variance above [MATH] radian ([MATH]).', '1907.06611-1-45-3': 'The intrinsic variation (grey dotted) significantly under-predicts the small-scale variance and will be replaced by N-body simulations (Section [REF]).', '1907.06611-1-46-0': 'We do not include shape noise in this figure.', '1907.06611-1-46-1': 'When halo clustering is negligible, the shape noise is inversely proportional to the area of the radial bin and the surface density of sources (equation [REF]).', '1907.06611-1-46-2': 'At small scales, the shape noise is usually higher than the intrinsic variation of the halo density profile.', '1907.06611-1-46-3': 'At large scales, the shape noise becomes subdominant to the contribution from LSS, and the transition scale depends on the source density.', '1907.06611-1-46-4': 'We will discuss the relative importance of shape noise in detail in Section [REF].', '1907.06611-1-46-5': 'For the high source densities of LSST, Euclid, and WFIRST, the small-scale intrinsic variation of halo density profiles will become non-negligible, and it is imperative to accurately characterise the covariance at small scales.', '1907.06611-1-47-0': '## Angular power spectra', '1907.06611-1-48-0': 'We use the Limber approximation to calculate the angular power spectra in equation ([REF]).', '1907.06611-1-48-1': 'In Appendix [REF] we provide the derivations.', '1907.06611-1-49-0': 'The auto angular power spectrum of halo number density is given by [EQUATION] the [MATH] integration is over the comoving distance range ([MATH], [MATH]) of the cluster redshift bin, and [EQUATION]', '1907.06611-1-49-1': 'The left panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations, which include the contribution from shot noise [MATH].', '1907.06611-1-49-2': 'We use the linear matter power spectrum multiplied by the halo bias from [CITATION].', '1907.06611-1-49-3': 'Wider redshift bins correspond to lower [MATH] and [MATH], while the relative importance of the two remains unchanged (see Appendix [REF]).', '1907.06611-1-50-0': 'For the convergence power spectrum, we assume that the source galaxies follow a redshift distribution [MATH] normalised such that [MATH].', '1907.06611-1-50-1': 'The auto angular power spectrum of convergence is given by [EQUATION] where we integrate all the LSS along the line of sight, and [EQUATION]', '1907.06611-1-50-2': 'Equation ([REF]) corresponds to the contribution of lensing from all the LSS in front of the sources, integrated from zero distance to the farthest source galaxy.', '1907.06611-1-50-3': 'Equation ([REF]) shows that the intervening LSS is weighted by the lensing kernel.', '1907.06611-1-50-4': 'Additional weights on sources can be absorbed in [MATH].', '1907.06611-1-50-5': 'This equation is equivalent to the equation 29 in [CITATION].', '1907.06611-1-51-0': 'The middle panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations.', '1907.06611-1-51-1': 'In the analytical calculation, the linear matter power spectrum underestimates the small-scale (high-[MATH]) power, while the non-linear matter power spectrum from Halofit agrees with simulations out to [MATH].', '1907.06611-1-51-2': 'We also show the shape noise term [MATH] for [MATH] and [MATH]; the shape noise dominates at small scale.', '1907.06611-1-52-0': 'The cross angular power spectrum of halo and lensing is given by [EQUATION]', '1907.06611-1-52-1': 'Similar to [MATH], this integration is over the line-of-sight distance range of the lens sample.', '1907.06611-1-52-2': 'The right panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations.', '1907.06611-1-52-3': 'We use the 3D halo-matter cross power spectrum [MATH] from the halo model; i.e., the sum of the two-halo term (linear power spectrum multiplied by halo bias) and the one-halo term computed for an NFW density profile (see Appendix [REF]).', '1907.06611-1-53-0': '# Analytical Gaussian-field covariance for excess surface density ([MATH])', '1907.06611-1-54-0': 'In this section, we focus on the covariance of the excess surface density ([MATH]).', '1907.06611-1-54-1': 'Since [MATH] and [MATH] are proportional to each other, their fractional errors are the same.', '1907.06611-1-54-2': 'When we consider a single source redshift (that is, if [MATH] is close to a delta function), the covariance matrices of [MATH] and [MATH] simply differ by a constant [MATH].', '1907.06611-1-54-3': 'However, when we consider a broad range of source redshifts, we need to integrate [MATH] and change the order of integration; see equation ([REF]) below.', '1907.06611-1-54-4': 'Therefore, we find it necessary to detail the analytical expressions for the covariance of [MATH].', '1907.06611-1-54-5': 'The equations in this section are equivalent to those in [CITATION], who used the 2D spectra [MATH] instead of the angular power spectra [MATH].', '1907.06611-1-55-0': 'The contribution of LSS to the covariance of [MATH] is less intuitive than that of [MATH], because it is the projected mass density weighted by the lensing kernel and is equivalent to taking the [MATH] coming from large scale structure between redshift 0 and [MATH] and treating it as if it is at [MATH].', '1907.06611-1-55-1': 'This component is not the projected mass density of LSS because the thin-lens approximation does not hold for LSS.', '1907.06611-1-56-0': 'The covariance matrix analogous to equation ([REF]) is given by [EQUATION] where [EQUATION]', '1907.06611-1-56-1': 'To derive this equation, we use the covariance of [MATH] (equation [REF]) for a single source redshift and a single lens redshift, and then integrate over [MATH] twice.', '1907.06611-1-56-2': 'To convert from ([MATH], [MATH]) to ([MATH], [MATH]), we assume [MATH] and [MATH]; therefore, this expression only applies to a thin lens redshift bin.', '1907.06611-1-57-0': 'Here we introduce two extra angular power spectra: [MATH] corresponds to the auto spectrum for projected matter (analogous to [MATH]), and [MATH] corresponds to the cross spectrum between halo and projected matter (analogous to [MATH]).', '1907.06611-1-57-1': 'Using the Limber approximation (see Appendix [REF]), the auto power spectrum for projected matter is given by [EQUATION]', '1907.06611-1-57-2': 'Here we integrate the LSS along the line-of-sight from zero to infinity and weight the LSS by the window function [EQUATION]', '1907.06611-1-57-3': "The [MATH] in the denominator comes from the lensing kernel (the same as in equation [REF]), while the [MATH] in the numerator comes from the fact that we interpret all the line-of-sight structure as the noise to halo profiles at the haloes' redshift [MATH].", '1907.06611-1-58-0': 'The halo-matter cross power spectrum is given by [EQUATION]', '1907.06611-1-58-1': 'Here we integrate over the redshift range of the halo sample (where the two fields [MATH] and [MATH] overlap), and [MATH] is given by equation ([REF]).', '1907.06611-1-58-2': 'Here [MATH] has the same dimension as [MATH] and [MATH] has the same dimension of as [MATH].', '1907.06611-1-59-0': '# Small-scale covariance from N-body simulations', '1907.06611-1-60-0': 'In cases where shape noise does not dominate, Fig. [REF] shows that the analytical formula ([REF]) underestimates the covariance at small scales, a consequence of treating matter in haloes as a Gaussian field.', '1907.06611-1-60-1': 'The actual covariance in this regime will include the effects of variation in halo density profiles, sub-structures, and orientations, none of which are captured by the Gaussian field approximation.', '1907.06611-1-60-2': 'Similar to [CITATION], we calculate the contribution of halo-matter correlation using fully non-linear N-body simulations.', '1907.06611-1-61-0': '## Abacus N-body simulations', '1907.06611-1-62-0': 'We use the publicly available Abacus Cosmos simulations based on the Abacus N-body code .', '1907.06611-1-62-1': 'We use the 20 realisations of the simulation named AbacusCosmos720boxplanck.', '1907.06611-1-62-2': 'This suite of boxes are based on a [CITATION] cosmology ([MATH]=0.314, [MATH], [MATH]=0.67) with different phases in the initial condition.', '1907.06611-1-62-3': 'Each realisation has 1440[MATH] particles in a box of side length 720 [MATH], a mass resolution of [MATH], and a spline softening length of 41 [MATH].', '1907.06611-1-62-4': 'Dark matter haloes are identified using Rockstar .', '1907.06611-1-63-0': 'For computing the covariance matrices, we divide each box in the [MATH]-[MATH] plane into 9 equal prism-shaped subvolumes, each of which has a dimension [MATH].', '1907.06611-1-63-1': 'Each subvolume includes approximately 580 haloes with [MATH].', '1907.06611-1-63-2': 'We calculate the covariance matrix from these 20 [MATH] 9 = 180 subvolumes.', '1907.06611-1-63-3': 'Since the covariance is inversely proportional to the simulation volume, we multiply the covariance by the volume in the unit of [MATH] so that the resulting covariance corresponds to a 1 [MATH] volume.', '1907.06611-1-64-0': 'We use the 10% down-sampled particles to measure the azimuthally averaged [MATH] profiles around haloes.', '1907.06611-1-64-1': 'We use an integration depth of [MATH] along the [MATH]-direction of the box; this integration depth is sufficient for a convergent [MATH] profile but insufficient to include the contribution of uncorrelated LSS to the covariance matrix.', '1907.06611-1-64-2': 'We will use the analytical calculations to capture the contribution from the LSS outside this [MATH] integration depth.', '1907.06611-1-65-0': 'To compute [MATH], we cross-correlate the haloes in a subvolume with the particles in the full-volume, applying periodic boundary condition.', '1907.06611-1-65-1': 'For haloes near the boundary of each subvolume, we use particles outside the boundary to measure their [MATH].', '1907.06611-1-65-2': 'For counting halo-particle pairs, we use the public code Corrfunc .', '1907.06611-1-65-3': 'Similar to the [MATH] calculation, we calculate [MATH] around random points (30 times the number of haloes) in each subvolume and subtract it from the [MATH] around haloes.', '1907.06611-1-66-0': 'Fig. [REF] compares the results from the Abacus simulations with the Takahashi simulations and with analytical calculations.', '1907.06611-1-66-1': 'The left-hand panel shows the mean [MATH].', '1907.06611-1-66-2': 'For Abacus, we use the [MATH] output.', '1907.06611-1-66-3': 'For Takahashi, we use the lens redshift [MATH] and source redshift [MATH], converting from [MATH] to [MATH] using the corresponding [MATH].', '1907.06611-1-66-4': 'The slight difference of the two simulations is due to the slightly different redshift and cosmology ([MATH]=0.314 for Abacus and 0.279 for Takahashi).', '1907.06611-1-66-5': 'Because of the relatively low resolution of Takahashi, the density profile is underestimated below 1 [MATH].', '1907.06611-1-66-6': 'We calculate the mean profile analytically using equation ([REF]) using [MATH] from the halo model (assuming the same cosmology as the Abacus boxes we use).', '1907.06611-1-67-0': 'The right-hand panel of Fig. [REF] shows the variance of [MATH].', '1907.06611-1-67-1': 'The Takahashi result is significantly higher than the Abacus result at large scales because the former includes the lensing effects from all the LSS, while the latter only takes into account [MATH].', '1907.06611-1-67-2': 'At intermediate scales, the Takahashi result approaches the Abacus result because intrinsic variation of halo density profiles starts to dominate.', '1907.06611-1-68-0': '## Combining analytical and numerical treatments', '1907.06611-1-69-0': 'Our approach to a full calculation of covariance matrices is to combine the analytical expressions for shape noise and LSS contributions (equations [REF] and [REF]) with Abacus calculations, which model the intrinsic contribution and replace the inaccurate Gaussian-field model at small scales (equation [REF]).', '1907.06611-1-69-1': 'However, our Abacus calculations also include the LSS contribution from a [MATH] slice.', '1907.06611-1-69-2': 'We calculate the contribution from this slice by integrating [MATH] or [MATH] from [MATH] to [MATH], with [MATH] corresponding to the halo redshift and [MATH].', '1907.06611-1-69-3': 'We then subtract this slice from the full LSS contribution.', '1907.06611-1-70-0': 'Fig. [REF] shows how we combine small-scale simulation results with large-scale analytical results.', '1907.06611-1-70-1': 'At small scales, the Abacus calculation dominates; at large scale, the Abacus calculation is similar to but slightly higher than the Gaussian-field LSS in a slice of [MATH] (grey dotted).', '1907.06611-1-70-2': 'This difference could be related to the correlated structure within the slice.', '1907.06611-1-70-3': 'To graft with the large-scale analytical calculation, we subtract this slice of LSS from the full LSS and add the Abacus variance.', '1907.06611-1-70-4': 'In practice, subtracting this slice has negligible effect.', '1907.06611-1-70-5': 'The combined result is the heavy black curve.', '1907.06611-1-70-6': 'We also show the results from the Takahashi shear maps; we use the [MATH] measurements from Takahashi in a narrow lens range ([MATH]) and a narrow source range ([MATH]), calculate the corresponding [MATH], and scale to a 1 [MATH] volume.', '1907.06611-1-70-7': 'The Takahashi results agree well with our Abacus+analytical approach at intermediate and large scales.', '1907.06611-1-70-8': 'At small scales, the Takahashi simulation cannot resolve the inner profile of clusters, which has also been shown in Fig. [REF].', '1907.06611-1-71-0': 'We have also attempted to calculate the non-Gaussian small-scale covariance analytically.', '1907.06611-1-71-1': 'We find that the analytical non-Gaussian results are higher than simulations (see Appendix [REF]).', '1907.06611-1-71-2': 'This discrepancy does not affect any of the calculations in the main text.', '1907.06611-1-72-0': '# Correlation between radial bins', '1907.06611-1-73-0': 'In this section, we discuss the properties of full covariance matrices, focusing on the importance of off-diagonal elements.', '1907.06611-1-73-1': 'Fig. [REF] shows the full covariance matrices corresponds to Fig. [REF]: [MATH], [MATH], [MATH], with no shape noise.', '1907.06611-1-73-2': 'We combine Abacus simulations and analytical calculations, and we use 15 or 30 logarithmically-spaced [MATH] bins between 0.1 and 100 [MATH].', '1907.06611-1-74-0': 'The top panels show the diagonal elements and the off-diagonal elements parallel to the diagonal.', '1907.06611-1-74-1': 'The top curves correspond to the diagonal elements, the next curve corresponds to the elements next to the diagonal (offset from the diagonal by 1 element), and so on.', '1907.06611-1-74-2': 'In the top left panel, we show both the Abacus+analytical (blue) and Takahashi (orange crosses) results for 15 radial bins.', '1907.06611-1-74-3': 'The values of the off-diagonal elements drop by approximately an order of magnitude at the fourth curve (offset=3), which corresponds to a 0.6 dex difference between [MATH] and [MATH].', '1907.06611-1-74-4': 'The off-diagonal elements drop more rapidly at large scales than at small scales because small-scale structures are more correlated.', '1907.06611-1-75-0': 'The top right panel compares the elements for 15 radial bins (blue solid) and 30 radial bins (orange dash).', '1907.06611-1-75-1': 'For the diagonal elements, the dependence on bin size is rather weak.', '1907.06611-1-75-2': 'This is explained by the property of the bin-averaged Bessel function [MATH] (see Appendix [REF] and Fig. [REF]).', '1907.06611-1-75-3': 'The peak of [MATH] is insensitive to the radial bin size; given the steep negative slope of [MATH] and [MATH] at large [MATH], only the first peak of [MATH] contributes to the integration in equation ([REF]), and thus the dependence on the bin size is weak.', '1907.06611-1-75-4': 'This is contrary to the case of shape noise, which is inversely proportional to the bin area and only contributes to the diagonal elements.', '1907.06611-1-75-5': 'The insensitivity to the radial bin size at small scales in the N-body results also indicates that the particle shot noise is negligible; if we have a low number of dark matter particles in N-body simulations, the small-scale variance would be inversely proportional to the number of particles in a radial bin.', '1907.06611-1-76-0': 'For the 30-bin case (orange dash), the first, third, and fifth curves agree with the first, second, and third curves of the 15-bin case (blue solid).', '1907.06611-1-76-1': 'That is, every other curve of the 30-bin case agrees with the 15-bin case, and each of the even-numbered orange dash curves is approximately the geometric mean of the two neighbouring curves.', '1907.06611-1-76-2': 'In other words, for a given ([MATH], [MATH]), the covariance is almost independent of the bin size.', '1907.06611-1-76-3': 'By halving the bin size, we increase the number of correlated bins, while the covariance between the two scales remains the same.', '1907.06611-1-77-0': 'The two bottom panels show the correlation matrices (with the diagonal elements normalised to 1), [EQUATION] for 15 and 30 radial bins, respectively.', '1907.06611-1-77-1': 'The two matrices have similar structures apart from the different bin sizes.', '1907.06611-1-78-0': 'To compare the information content of these matrices, we assume a parameter [MATH] multiplying the [MATH] data vector.', '1907.06611-1-78-1': 'The Fisher information for [MATH] is given by [EQUATION] where the superscript [MATH] denotes the transpose.', '1907.06611-1-78-2': 'This expression is equivalent to the square of the signal-to-noise ratio.', '1907.06611-1-78-3': 'The constraint on [MATH] is given by [EQUATION]', '1907.06611-1-78-4': 'The [MATH] values from the two matrices are almost identical (0.0032 and 0.0031), confirming that the information content is independent of the binning.', '1907.06611-1-79-0': 'Fig. [REF] further demonstrates the impact of bin size, shape noise, and off-diagonal elements on [MATH].', '1907.06611-1-79-1': 'We assume a survey area of 5000 deg[MATH] and a redshift range [MATH], which corresponds to a comoving volume 1.2 [MATH].', '1907.06611-1-79-2': 'We perform the calculation for two mass thresholds: [MATH] and [MATH].', '1907.06611-1-79-3': 'The left and right panels correspond to 15 and 30 radial bins, respectively.', '1907.06611-1-79-4': 'The [MATH]-axis corresponds to different levels of shape noise indicated by the source density.', '1907.06611-1-79-5': 'The solid curves correspond to [MATH] calculated using the full covariance matrix.', '1907.06611-1-79-6': 'The two panels show identical results, indicating that the information content, or the total signal-to-noise ratio, does not depend on the radial bin size.', '1907.06611-1-79-7': 'Comparing the two mass ranges, we can see that the low-mass threshold is more sensitive to the shape noise than the high-mass threshold because of its larger statistical power.', '1907.06611-1-79-8': 'In addition, the dash curves demonstrate that ignoring off-diagonal elements leads to underestimation of [MATH].', '1907.06611-1-79-9': 'The underestimation is worse for narrower radial bins, lower shape noise (high source density), and higher halo mass.', '1907.06611-1-80-0': 'Since the diagonal elements drop rapidly, the elements far from the diagonal should be negligible.', '1907.06611-1-80-1': 'Fig. [REF] demonstrates how many off-diagonal elements we need to avoid underestimation of [MATH].', '1907.06611-1-80-2': 'We compare the [MATH] using the full covariance matrix vs. part of the covariance matrix (ignoring elements far from the diagonal).', '1907.06611-1-80-3': 'The two panels correspond to 15 and 30 radial bins.', '1907.06611-1-80-4': 'The [MATH]-axis corresponds to the number of the off-diagonal lines used; [MATH] corresponds to using only the diagonal elements (marked by crosses), [MATH] corresponds to adding the first off-diagonal line, and so on.', '1907.06611-1-80-5': 'The [MATH]-axis corresponds to the ratio of the [MATH] calculated with the partial covariance matrix specified by the [MATH]-axis and the [MATH] obtained from the full matrix.', '1907.06611-1-80-6': 'Various curves correspond to source densities 10 arcmin[MATH] (similar to DES), 20, 60 (similar to WFIRST), and [MATH] (no shape noise).', '1907.06611-1-81-0': 'The left panel shows that for 15 radial bins, for the case with no shape noise (black curves), using only the diagonal elements ([MATH]) underestimates the error bar by a factor of 2, while including 3 off-diagonal lines ([MATH]) is almost equivalent to using the full covariance matrix.', '1907.06611-1-81-1': 'Comparing the two panels we can see that when we halve the bin size, we need to double the number of off-diagonal lines we use.', '1907.06611-1-81-2': 'When the source density is low (shape noise is high), the effect of off-diagonal elements is weaker but still non-negligible.', '1907.06611-1-81-3': 'Even with 10 arcmin[MATH], we cannot completely ignore the off-diagonal elements because the shape noise is subdominant at large scales.', '1907.06611-1-81-4': 'The results in Fig. [REF] are similar for other redshifts and mass thresholds.', '1907.06611-1-82-0': '# Discussion', '1907.06611-1-83-0': 'In this section, we discuss the importance of shape noise ([REF]), the mass and redshift dependence of covariance matrices ([REF]), the cross-covariance of two different mass bins ([REF]), and potential systematic uncertainties that can affect our calculations ([REF]).', '1907.06611-1-84-0': '## Importance of shape noise', '1907.06611-1-85-0': 'Fig. [REF] shows the relative importance of shape noise (various colour curves) and the shape noise-free part (black curves, we call it density variance hereafter) for the variance of [MATH].', '1907.06611-1-85-1': 'We use lenses at [MATH] and 15 [MATH] bins between 0.1 and 100 [MATH], and the two panels correspond to mass thresholds of [MATH] and [MATH].', '1907.06611-1-85-2': 'For lenses at [MATH] and [MATH], the results are nearly identical to the [MATH] case shown here.', '1907.06611-1-85-3': 'Various line styles correspond to various source redshifts.', '1907.06611-1-85-4': 'For simplicity, we assume that all source galaxies are at the same redshift; in reality, the source galaxies are distributed in a wide redshift range.', '1907.06611-1-86-0': 'For the density variance (black), the LSS contribution at large scales depends on the source redshift because we integrate all the line-of-sight structure in front of the source redshift.', '1907.06611-1-86-1': 'This redshift dependence is weak but non-monotonic.', '1907.06611-1-86-2': 'The LSS contribution involves the ratio [MATH]; see equation ([REF]).', '1907.06611-1-86-3': 'As [MATH] increases, both the numerator and the denominator decrease.', '1907.06611-1-86-4': 'When [MATH] is low and close to [MATH] (e.g., 0.75), as [MATH] increases the numerator decreases rapidly with [MATH], leading to a smaller covariance.', '1907.06611-1-86-5': 'When [MATH] is high, as [MATH] increase, the numerator is almost constant; since we integrate over a wider range of line-of-sight structure, the total covariance increases.', '1907.06611-1-87-0': 'On the other hand, at a fixed source density, the shape noise is sensitive to the source redshift.', '1907.06611-1-87-1': 'At a fixed [MATH], [MATH] decreases with [MATH], and a higher [MATH] corresponds to a lower shape noise due the [MATH] factor in equation ([REF]).', '1907.06611-1-87-2': 'If we plot the variance of [MATH] instead, the shape noise-free part would depend on the source redshift (because [MATH] is higher for a higher source redshift), while the shape noise would be independent of the source redshift (equation [REF]).', '1907.06611-1-87-3': 'In both cases, the relative importance of shape noise and density variance is the same.', '1907.06611-1-88-0': 'The blue and orange curves correspond to source densities of 10 arcmin[MATH] (DES-like) and 60 arcmin[MATH] (WFIRST-like).', '1907.06611-1-88-1': 'For 10 arcmin[MATH], the small-scale variance is dominated by shape noise.', '1907.06611-1-88-2': 'For 60 arcmin[MATH], small-scale density variance becomes more important.', '1907.06611-1-88-3': 'Comparing the two panels, we can see that the density variance is more important for high-mass haloes.', '1907.06611-1-89-0': 'When halo clustering is negligible, the shape noise is inversely proportional to the radial bin area.', '1907.06611-1-89-1': 'For example, if we use 30 radial bins instead of 15, the shape noise would double, while the density variance would remain approximately the same.', '1907.06611-1-89-2': 'With 30 radial bins, although the diagonal elements have larger shape noise, the off-diagonal elements play more important roles than the 15-bin case.', '1907.06611-1-90-0': '## Mass and redshift dependence', '1907.06611-1-91-0': 'In this section, we show that the fractional errors of [MATH], in the case of negligible shape noise, are almost independent of redshift and only weakly depend on mass.', '1907.06611-1-91-1': 'Fig. [REF] compares the fractional errors for [MATH] from Abacus+analytical, for halo redshifts 0.3, 0.5, and 0.7, and for mass bins [MATH], [MATH], and [MATH].', '1907.06611-1-91-2': 'We assume source redshifts [MATH].', '1907.06611-1-91-3': 'We multiply the covariance matrix of [MATH] by the number of haloes in each bin.', '1907.06611-1-91-4': 'For the [MATH]-axis, we take the square root of this product and divide it by the mean of [MATH].', '1907.06611-1-91-5': 'For the [MATH]-axis, we divide the projected distance by the mean [MATH] in each bin.', '1907.06611-1-91-6': 'The left panel corresponds to the diagonal elements, and the right panel corresponds to the elements next to the diagonal.', '1907.06611-1-92-0': 'We use different colours to indicate different redshift bins, and different line styles to indicate different mass bins.', '1907.06611-1-92-1': 'For a given mass bin (same line style), the fractional error profiles are almost independent of redshift.', '1907.06611-1-92-2': 'Comparing different mass bins, we see that higher mass haloes have slightly lower fractional error (higher signal-to-noise ratio).', '1907.06611-1-92-3': 'Both panels show very similar trends.', '1907.06611-1-92-4': 'Our numerical calculations choose specific mass and redshift bins, and the weak dependence demonstrated in Fig. [REF] can be useful for scaling our results to other redshift or mass bin choices, after which shape noise can be added using our analytical formulae.', '1907.06611-1-93-0': '## Cross-mass covariance', '1907.06611-1-94-0': 'If we use multiple mass bins to perform stacked weak lensing measurements and to jointly constrain cosmological parameters, we need to take into account the covariance between mass bins.', '1907.06611-1-94-1': 'To calculate the cross-mass covariance, we again combine Abacus simulations at small scales and analytical calculations at large scales.', '1907.06611-1-95-0': 'To calculate the Gaussian-field covariance analytically for two mass bins at the same redshift (denoted as h1 and h2), we can write the covariance matrix analogous to equation ([REF]): [EQUATION]', '1907.06611-1-95-1': 'The cross-mass covariance has no shot noise associated with halo number counts, and the halo clustering is described by the cross-mass power spectrum [EQUATION]', '1907.06611-1-95-2': 'In the expression above, the 3D cross power spectrum of halo samples is given by [EQUATION] where [MATH] and [MATH] are the halo bias values of the two samples.', '1907.06611-1-95-3': 'Since we consider two samples at the same redshift, they have the same window function [MATH], as given by equation ([REF]).', '1907.06611-1-95-4': 'In this calculation, the large-scale cross-mass covariance matrix is symmetric.', '1907.06611-1-95-5': 'The last term ([MATH]) is replaced by the N-body calculations from Abacus.', '1907.06611-1-96-0': 'Fig. [REF] shows the cross-mass covariance matrices for two mass bins: [MATH] and [MATH], for Abacus+analytical and Takahashi.', '1907.06611-1-96-1': 'The left panel shows the diagonal elements of each block.', '1907.06611-1-96-2': 'The blue, orange, and green curves correspond to the first mass bin, the second mass bin, and the cross-bin covariance.', '1907.06611-1-96-3': 'The solid curves correspond to the Abacus+analytical results, while the dash curves correspond to the Takahashi results.', '1907.06611-1-96-4': 'The second mass bin has a higher variance than the first mass bin due to its higher mass (and thus higher shot noise).', '1907.06611-1-96-5': 'At large scale, the cross-mass covariance is closer to the low mass bin because both are dominated by halo clustering.', '1907.06611-1-96-6': 'For the cross-mass variance, shot noise has no contribution; for the low mass bin, halo clustering is higher than shot noise; for the high mass bin, shot noise dominates.', '1907.06611-1-96-7': 'At small scales, the cross-mass covariance is approximately an order of magnitude smaller than the first mass bin.', '1907.06611-1-97-0': 'The right panel shows the full correlation matrix.', '1907.06611-1-97-1': 'The upper left block corresponds to the first mass bin (15 radial bins between 0.1 and 100 [MATH]), and the lower right block corresponds to the second mass bin calculated with the same radial binning.', '1907.06611-1-97-2': 'The upper right and lower left blocks correspond to the cross-mass covariance of the two mass bins and are the transpose of each other.', '1907.06611-1-98-0': 'To understand the contribution of the cross-mass block to the error budget, we again calculate the constraints on a parameter multiplying the [MATH] for both mass bins.', '1907.06611-1-98-1': 'We combine the two data vectors to form [EQUATION] and we again use equation ([REF]).', '1907.06611-1-98-2': 'We calculate [MATH] using the full matrix vs. a block-diagonal matrix (ignoring the cross-mass covariance).', '1907.06611-1-98-3': 'Both calculations give [MATH] for a 1 [MATH] volume, indicating that the cross-mass covariance is negligible in the absence of shot noise.', '1907.06611-1-98-4': 'However, if source galaxy shape noise dominates, we need to add diagonal shape noise to the cross-mass covariance block.', '1907.06611-1-99-0': '## Potential systematic uncertainties', '1907.06611-1-100-0': 'In this section we discuss the impact of baryons, cluster miscentering, and mass-observable relation on the covariance calculations.', '1907.06611-1-101-0': 'In this work, the small-scale cluster lensing signal is derived from dark matter-only simulations.', '1907.06611-1-101-1': 'Baryonic effects can change the inner density profiles of clusters and thus change the lensing signal.', '1907.06611-1-101-2': 'Since the effect is mainly at very small scales (less than 10 kpc), which will be dominated by shape noise for most of the surveys, its impact is likely to be small for covariance matrices.', '1907.06611-1-102-0': 'We calculate the cluster lensing signals with respect to the centres of dark matter haloes.', '1907.06611-1-102-1': 'In optically-selected cluster samples, the locations of the brightest cluster galaxies may not coincide with the centres of dark matter haloes (see, e.g., for comparisons between optically defined centres and X-ray centres).', '1907.06611-1-102-2': 'This miscentering would lead to shallower lensing profiles at small scales.', '1907.06611-1-102-3': 'For current surveys, the scales affected by miscentering are dominated by shape noise, and the covariance matrices are unaffected.', '1907.06611-1-102-4': 'For future surveys, when shape noise no longer dominates small scales, the miscentering effects would need to be taken into account in modelling the small-scale covariance matrices.', '1907.06611-1-103-0': 'In this work we select clusters based on their masses; in real surveys, clusters would be selected by some observed property (e.g., optical richness), which has a scaling relation and a scatter around the true mass.', '1907.06611-1-103-1': 'If this mass-observable relation is not biased by some property that can also bias lensing, we can simply incorporate this effect by convolving the halo mass function with the mass-observable relation.', '1907.06611-1-103-2': 'However, if the mass-observable relation is biased by some property that can also bias the lensing signal, for example, the orientation of a triaxial halo, we will need to take into account this extra property in order to obtain unbiased lensing measurements and robust covariance matrices.', '1907.06611-1-103-3': 'We will explore this in future work.', '1907.06611-1-104-0': '# Summary', '1907.06611-1-105-0': 'We calculate accurate covariance matrices required for the cosmological interpretation of weak gravitational lensing by galaxy clusters that will be observed by LSST, Euclid, and WFIRST.', '1907.06611-1-105-1': 'We combine analytical calculations at large scales with the Abacus N-body simulations at small scales, and we validate our approach with the Takahashi full-sky ray-tracing simulations at intermediate and large scales.', '1907.06611-1-105-2': 'Our main results are summarised as follows.', '1907.06611-1-106-0': 'The covariance matrices provided in this paper can be used for robust forecast and survey optimisation for future surveys like LSST, Euclid, and WFIRST.', '1907.06611-1-106-1': 'Given the high signal-to-noise ratio of these surveys, cluster lensing will have unprecedented constraining power on the growth of structure.', '1907.06611-1-106-2': 'In companion papers ( and Wu et al. in prep), we present forecasts on combining cluster counts and lensing, as well as how we can effectively combine the cluster lensing and cross correlation functions of clusters and galaxies to constrain cosmological parameters.'}

{'1907.06611-2-0-0': 'Next-generation optical imaging surveys will revolutionise the observations of weak gravitational lensing by galaxy clusters and provide stringent constraints on growth of structure and cosmic acceleration.', '1907.06611-2-0-1': 'In these experiments, accurate modelling of covariance matrices of cluster weak lensing plays the key role in obtaining robust measurements of the mean mass of clusters and cosmological parameters.', '1907.06611-2-0-2': 'We use a combination of analytical calculations and high-resolution N-body simulations to derive accurate covariance matrices that span from the virial regime to linear scales of the cluster-matter cross-correlation.', '1907.06611-2-0-3': 'We validate this calculation using a public ray-tracing lensing simulation and provide a software package for calculating covariance matrices for a wide range of cluster and source sample choices.', '1907.06611-2-0-4': 'We discuss the relative importance of shape noise and density fluctuations, the impact of radial bin size, and the impact of off-diagonal elements.', '1907.06611-2-0-5': 'For a weak lensing source density [MATH] arcmin[MATH], shape noise typically dominates the variance on comoving scales [MATH].', '1907.06611-2-0-6': 'However, for [MATH] arcmin[MATH], potentially achievable with future weak lensing experiments, density fluctuations typically dominate the variance at [MATH] and remain comparable to shape noise on smaller scales.', '1907.06611-2-1-0': '# Introduction', '1907.06611-2-2-0': 'Understanding the origin of cosmic acceleration requires measurements of both the expansion rate of the Universe and the growth rate of large-scale structure .', '1907.06611-2-2-1': 'The number counts of galaxy clusters as a function of mass and redshift are sensitive to both expansion rate and the growth of structure ; also see [CITATION] for a review.', '1907.06611-2-2-2': 'With optical surveys, one can identify clusters to lower mass thresholds at high redshifts relative to X-ray or submillimeter surveys.', '1907.06611-2-2-3': 'Regardless of identification method, calibrating the mass scale associated with observed cluster properties plays a key role in extracting cosmological information from galaxy clusters.', '1907.06611-2-2-4': 'In optical imaging surveys, one can use weak gravitational lensing to constrain mean cluster mass density profiles .', '1907.06611-2-2-5': 'These profiles can often be measured with high precision far beyond the cluster virial radius, to scales of tens of Mpc, where they probe the linear regime of the cluster-matter cross-correlation function .', '1907.06611-2-2-6': 'The combination of cluster space densities and weak lensing profiles is in many ways analogous to the combination of galaxy clustering and galaxy-galaxy lensing, but applied to the high mass end of the dark matter halo population.', '1907.06611-2-2-7': 'In these experiments, accurate modelling of the signals and the covariance matrices is essential for obtaining robust constraints on cluster masses and cosmological parameters.', '1907.06611-2-3-0': 'To date, most of the weak gravitational lensing measurements of galaxy clusters are dominated by shape noise; that is, because of the modest number density of source galaxies, the noise of lensing is dominated by the intrinsic ellipticities of source galaxies.', '1907.06611-2-3-1': 'However, for the next-generation optical surveys like the Large Synoptic Survey Telescope (LSST, ), Euclid , and the Wide Field Infrared Survey Telescope (WFIRST, ), the number density of source galaxies will be significantly higher and shape noise will no longer dominate.', '1907.06611-2-3-2': 'Instead, the density fluctuations related to the intrinsic variation of halo density profiles will dominate the uncertainties at small scales, and the contribution of uncorrelated LSS will dominate the uncertainties at large scales.', '1907.06611-2-3-3': 'In this paper, we detail how to calculate covariance matrices for cluster lensing at all scales with and without shape noise, and we provide a software package that can calculate covariance matrices for a wide range of survey assumptions.', '1907.06611-2-4-0': 'The covariance matrices of cluster lensing or galaxy-galaxy lensing can be calculated directly from data using jackknife methods , analytical formulae , or simulations .', '1907.06611-2-4-1': 'In this paper, we focus on the latter two approaches.', '1907.06611-2-4-2': 'The literature for cluster lensing covariance matrices is relatively small.', '1907.06611-2-4-3': '[CITATION] quantified the impact of intrinsic variation of halo density profiles on the covariance matrices of cluster weak lensing by combining analytical calculations with numerical simulations.', '1907.06611-2-4-4': 'However, as we will explain in Section [REF], they quantified the halo-to-halo covariance rather than the patch-to-patch covariance, and the latter is relevant for stacked cluster lensing.', '1907.06611-2-5-0': 'The covariance matrices of galaxy-galaxy lensing are more widely studied.', '1907.06611-2-5-1': 'Galaxy-galaxy lensing is analogous to cluster lensing and refers to using galaxies (lower-mass haloes) instead of galaxy clusters as lenses.', '1907.06611-2-5-2': '[CITATION] provided analytical formulae for galaxy-galaxy lensing covariance matrices.', '1907.06611-2-5-3': 'Using observed and simulated catalogues, they compared various sources of errors and demonstrated the importance of subtracting lensing signals around random points.', '1907.06611-2-5-4': 'Their analysis at small scales is dominated by shape noise and is therefore not directly applicable for the next-generation cluster lensing surveys, where the non-Gaussian covariance will be significant compared with shape noise.', '1907.06611-2-5-5': '[CITATION] used simulated full-sky weak lensing maps to compare various methods for calculating covariance matrices and demonstrated the accuracy of jackknife covariance.', '1907.06611-2-5-6': '[CITATION] developed large-volume simulations for calculating the cross-probe covariance for cosmic shear, galaxy-galaxy lensing, and galaxy clustering.', '1907.06611-2-5-7': 'Other earlier work includes [CITATION].', '1907.06611-2-5-8': 'These papers focus on lower-mass haloes and large scales; therefore, they do not provide the small-scale covariance matrices we need for cluster lensing mass calibration.', '1907.06611-2-6-0': 'Because of the reasons stated above, for the next-generation cluster lensing measurements by LSST, Euclid, and WFIRST, we need to come up with a new treatment for covariance matrices.', '1907.06611-2-6-1': 'In this paper, we first compare the analytical covariance assuming Gaussian random fields with the covariance from simulated lensing maps.', '1907.06611-2-6-2': 'We then identify the scales where the Gaussian-field covariance is insufficient, and we add corrections based on high-resolution N-body simulations.', '1907.06611-2-6-3': 'We provide readers with a set of user-friendly equations with short and heuristic derivations, a software package, and tabulated results from simulations.', '1907.06611-2-7-0': 'Readers might ask why we do not use simulated gravitational lensing ray-tracing maps to calculate covariance matrices at all scales.', '1907.06611-2-7-1': 'For example, [CITATION] and [CITATION] produced full-sky lensing maps and used them to study the covariance matrices of galaxy-galaxy lensing, and we use these maps to cross-check our calculations.', '1907.06611-2-7-2': 'However, because of the substantial computational resources required, these maps are limited to one cosmology and are unable to resolve the inner profiles of clusters.', '1907.06611-2-7-3': 'We need the contribution from very small scales (to account for the intrinsic variation of density profiles) to very large scales (to account for the LSS contribution).', '1907.06611-2-7-4': 'For the former, we use high-resolution N-body simulation boxes to characterise the non-linear evolution of dark matter haloes.', '1907.06611-2-7-5': 'For the latter, it is impractical to use N-body simulation boxes because we would need to include dark matter particles extending to a few Gpc, while analytical calculations are much more efficient in this regime.', '1907.06611-2-7-6': 'We will show that this approach agrees with the ray-tracing simulations of [CITATION] at intermediate and large scales.', '1907.06611-2-8-0': 'We use two sets of public simulations: the ray-tracing lensing maps from [CITATION] and the N-body simulations from Abacus Cosmos .', '1907.06611-2-8-1': 'Table [REF] summarises the specifications of the two simulations.', '1907.06611-2-8-2': 'The difference in their cosmological parameters leads to negligible difference in covariance matrices.', '1907.06611-2-9-0': 'Throughout this work, we use [MATH], the mass defined by the radius within which the overdensity is 200 times the mean density of the Universe.', '1907.06611-2-9-1': 'All distances are in comoving [MATH]; [MATH] denotes the line-of-sight distances, and [MATH] denotes projected (transverse) distances.', '1907.06611-2-9-2': 'Since cluster lensing covariance matrices are inversely proportional to the survey area, we normalise all of the covariance values to those of a volume of 1 [MATH].', '1907.06611-2-9-3': 'Table [REF] summarises the notations used in this work.', '1907.06611-2-9-4': 'We use "variance" to indicate the diagonal elements of a covariance matrix.', '1907.06611-2-10-0': 'This paper is organised as follows.', '1907.06611-2-10-1': 'Section [REF] provides the basic equations of cluster weak lensing.', '1907.06611-2-10-2': 'Section [REF] describes how we measure covariance matrices from lensing simulations and possible pitfalls in this procedure.', '1907.06611-2-10-3': 'In Section [REF] we focus on large-scale covariance matrices for tangential shear assuming Gaussian random fields, and in Section [REF] we present the analogous calculations for the excess surface density.', '1907.06611-2-10-4': 'In Section [REF] we use high-resolution N-body simulations to calculate small-scale covariance matrices and combine simulations with analytical calculations.', '1907.06611-2-10-5': 'We discuss the correlation between radial bins in Section [REF].', '1907.06611-2-10-6': 'In Section [REF] we discuss shape noise, the dependence of covariance matrices on halo mass and redshift, and cross-mass covariance matrices.', '1907.06611-2-10-7': 'Section [REF] summarises our results.', '1907.06611-2-11-0': '# Background of cluster weak lensing', '1907.06611-2-12-0': 'Below we briefly describe the basic equations for cluster weak lensing.', '1907.06611-2-12-1': 'We refer interested readers to e.g. [CITATION] for comprehensive reviews.', '1907.06611-2-13-0': 'The gravitational lensing signal is quantified by the Jacobian matrix for the coordinate transformation from the source plane to the lens plane, [EQUATION] where [MATH] corresponds to convergence, and ([MATH], [MATH]) corresponds to the two components of shear.', '1907.06611-2-13-1': 'We ignore higher-order effects in this work.', '1907.06611-2-13-2': 'If we assume that the line-of-sight dimension of a dark matter halo is much smaller than the distance between the observer and the source galaxy (the thin-lens approximation), the azimuthally averaged convergence is related to the surface density ([MATH]) of a halo via [EQUATION] where [MATH] is the projected distance on the lens plane in comoving units, and [MATH] is the critical surface density defined as [EQUATION] where [MATH] denotes comoving distances; we use subscripts h and s to denote the redshifts or distances of haloes (galaxy clusters) and sources (background galaxies).', '1907.06611-2-13-3': 'Note that we have [MATH] in the denominator because we use comoving units; if one uses physical units, [MATH] would differ by [MATH] and would usually be written in terms of angular diameter distances.', '1907.06611-2-14-0': 'The two shear components depend on the choice of coordinate system, and a physical quantity is the tangential shear, [EQUATION] where [MATH] is the position angle of the source galaxy with respect to the cluster centre.', '1907.06611-2-14-1': 'The azimuthally averaged tangential shear is related to the excess surface density ([MATH]) via [EQUATION]', '1907.06611-2-14-2': 'The observable is the reduced shear of individual source galaxies, [MATH], [MATH].', '1907.06611-2-14-3': 'In the weak lensing regime, [MATH], and [MATH] is the observable for practical purposes.', '1907.06611-2-14-4': 'In the context of cluster lensing, one often converts [MATH] to [MATH] because the latter can be interpreted physically as the excess surface density profiles of clusters.', '1907.06611-2-15-0': 'The two quantities, [MATH] and [MATH], have different advantages and disadvantages.', '1907.06611-2-15-1': 'For a given lens, [MATH] is independent of source redshift, while [MATH] is higher for a higher source redshift.', '1907.06611-2-15-2': 'On the other hand, to calculate [MATH] from the observed [MATH] one needs to assume a cosmology and know the redshifts of sources and lenses.', '1907.06611-2-15-3': 'In this work, we will first focus on the covariance matrices of [MATH] in Section [REF] because the contribution from LSS to the noise of [MATH] is easier to understand.', '1907.06611-2-15-4': 'We will then discuss the analogous covariance matrices of [MATH] in Section [REF].', '1907.06611-2-16-0': 'For an ensemble of galaxy clusters at a given redshift, the mean [MATH] profile can be computed from the halo-matter correlation function [MATH] using [EQUATION]', '1907.06611-2-16-1': 'This real-space expression is useful for calculating mean profiles of [MATH] or [MATH], while the equivalent Fourier-space expression (equation [REF] below) is more useful for calculating covariance matrices.', '1907.06611-2-17-0': 'To calculate [MATH], we use the approach in [CITATION].', '1907.06611-2-17-1': 'On scales smaller than the virial radius, [MATH] represents the average density profile.', '1907.06611-2-17-2': 'We use an Navarro-Frenk-White (NFW) profile , [MATH], with the concentration-mass relation from [CITATION].', '1907.06611-2-17-3': 'We assume no scatter between concentration and mass.', '1907.06611-2-17-4': 'On scales much greater than the virial radius, [MATH] is the linear matter correlation function [MATH] multiplied by the halo bias factor [MATH].', '1907.06611-2-17-5': 'For intermediate scales, we use the larger of the two; that is, [EQUATION] where [EQUATION]', '1907.06611-2-18-0': '# Covariance matrices from weak lensing maps', '1907.06611-2-19-0': 'We start by measuring the covariance matrices from the weak lensing maps of [CITATION] and discussing pitfalls associated with these measurements.', '1907.06611-2-19-1': 'We will later use these measurements to cross-check our calculations combining analytical formulae and N-body simulations.', '1907.06611-2-20-0': '## Takahashi ray-tracing simulations', '1907.06611-2-21-0': 'We use the publicly available lensing maps produced by [CITATION].', '1907.06611-2-21-1': 'These authors built full-sky lightcones using N-body simulations based on Gadget2 .', '1907.06611-2-21-2': 'The N-body simulations are based on a flat [MATH]CDM cosmology consistent with WMAP9 , [MATH] = 0.279; [MATH] = 0.721; [MATH] = 0.046; [MATH] = 0.7; [MATH] = 0.82; [MATH] = 0.97.', '1907.06611-2-21-3': 'The dark matter haloes are identified using Rockstar .', '1907.06611-2-21-4': 'For the lightcones, starting from [MATH], for every 450 [MATH] the authors used a different N-body simulation volume with progressively lower resolution.', '1907.06611-2-21-5': 'For the lensing simulations, they set a source plane every 150 [MATH] and generated full-sky maps of convergence, shear, and rotation using ray-tracing calculations for all the dark matter in front of the source plane.', '1907.06611-2-21-6': 'They provided maps in the Healpix format with resolution [MATH] = 4096, 8192, and 16384.', '1907.06611-2-21-7': 'In our calculations, we use the maps with [MATH] = 4096 (corresponding to [MATH] pixels in the full sky), and the angular resolution of each pixel is [MATH] radian or 0.86 arcmin.', '1907.06611-2-21-8': 'We have checked that the [MATH]=8192 maps give nearly identical results.', '1907.06611-2-22-0': 'Since the lightcone exhibits discontinuities every 450 [MATH], we choose lens redshift bins that avoid these discontinuities; that is, a halo sample we choose comes from one original N-body simulation box and thus has continuous LSS.', '1907.06611-2-22-1': 'For our fiducial calculation, we use haloes from lens plane 10, which corresponds to a comoving distance range (1350, 1500) [MATH] and a redshift range (0.508, 0.574).', '1907.06611-2-22-2': 'The haloes in this redshift range are generated from an N-body simulation of box size 1800 [MATH] with 2048[MATH] particles; the mass resolution is [MATH], and the softening length is 32 [MATH].', '1907.06611-2-22-3': 'We use source plane 18, which corresponds to a redshift range (1.218, 1.318); see Tables 1 and 2 in [CITATION].', '1907.06611-2-23-0': 'We divide the full-sky catalogues into 48 equal-area samples.', '1907.06611-2-23-1': 'We use 10 of the 108 realisations provided by the authors; that is, we have 480 realisations of a survey with [MATH] (860 deg[MATH]).', '1907.06611-2-23-2': 'To calculate the averaged [MATH] profiles, we take all the haloes with [MATH] in each sample (on average 785 haloes per sample) and cross-correlate them with all pixels in the shear map using the publicly available code TreeCorr.', '1907.06611-2-23-3': 'In addition, we generate random points in the same area (20 times the number of haloes) and calculate the [MATH] profile.', '1907.06611-2-23-4': 'We subtract the shear signal around random points from the cluster lensing shear signal.', '1907.06611-2-23-5': 'In this way, we obtain the mean shear profiles of [MATH] for each patch of [MATH], and we calculate the covariance matrix of these 480 realisations.', '1907.06611-2-23-6': 'We use 5 logarithmically-spaced bins per decade of angular separation (15 bins for [MATH]).', '1907.06611-2-23-7': 'As the number of realisations (480) is much greater than the number of bins (15), we do not need to correct the inverse covariance as in [CITATION].', '1907.06611-2-24-0': '## Pitfalls of calculating covariance matrices', '1907.06611-2-25-0': 'Before going into the details of covariance matrix calculations, we discuss two possible pitfalls of calculating and interpreting cluster lensing covariance matrices from simulations.', '1907.06611-2-25-1': 'Fig. [REF] shows an example for calculating the covariance matrix from the Takahashi simulations (for the fiducial calculation described in Section [REF]).', '1907.06611-2-26-0': 'The first pitfall is calculating covariance matrices without subtracting the lensing signal around random locations.', '1907.06611-2-26-1': '[CITATION] demonstrated that when measuring the shear signal around galaxies, subtracting the shear signal around random lenses can remove shear systematics .', '1907.06611-2-26-2': 'We find that this subtraction is not only necessary for observations but also necessary for simulated shear maps.', '1907.06611-2-26-3': 'Fig. [REF] shows that, even when using simulations free of systematics, the patch-to-patch variance at large scales is erroneously high if one does not subtract the mean random signal (orange).', '1907.06611-2-26-4': 'A similar excess at large scales has also been shown in [CITATION] using real sources from the Sloan Digital Sky Survey.', '1907.06611-2-26-5': 'Subtracting the shear at random points removes the impact of over- or under-densities on the scales of the survey and is roughly analogous to using a [CITATION] estimator for the correlation function.', '1907.06611-2-26-6': 'For small scales, the shear signal centred on random points is small compared to that centred on clusters, and this subtraction makes negligible difference.', '1907.06611-2-27-0': 'The second pitfall is using the halo-to-halo covariance instead of the patch-to-patch covariance.', '1907.06611-2-27-1': 'One obvious way to estimate the covariance matrix for a sample of clusters, either observed or simulated, is to measure the [MATH] or [MATH] profile cluster by cluster, compute the covariance matrix of these profiles, and divide by the cluster number .', '1907.06611-2-27-2': 'However, this halo-to-halo covariance generally underestimates the true uncertainty of the mean profile of a stacked lens sample.', '1907.06611-2-27-3': 'Instead, the covariance matrix should be computed from the mean profiles derived from patches of sky that are substantially larger than the largest scales being measured.', '1907.06611-2-27-4': 'Fig. [REF] demonstrates this point using the Takahashi simulations: at scales [MATH] radian ([MATH]), the variance of [MATH] from halo to halo divided by the number of haloes in a patch (green) is smaller than the variance among the multiple patches of 860 deg[MATH] (blue).', '1907.06611-2-27-5': 'It is this latter variance that provides an estimate of the uncertainty on the mean [MATH] profile for an 860 deg[MATH] survey.', '1907.06611-2-28-0': 'At small scales, the halo-to-halo and the patch-to-patch estimates of the covariance are similar.', '1907.06611-2-28-1': 'The difference occurs at large scales, where the dominant contribution to the covariance is from foreground and background LSS that is not correlated with clusters themselves.', '1907.06611-2-28-2': 'The grey dash curve in Fig. [REF] shows the analytical prediction for this LSS contribution (equation [REF] below), which agrees with the patch-to-patch simulation results.', '1907.06611-2-28-3': 'The grey dotted curve corresponds to equation ([REF]) without the halo clustering term [MATH] and is close to the halo-to-halo variance.', '1907.06611-2-28-4': 'Physically, the excess variance due to [MATH] arises because clustered haloes do not independently sample the foreground and background cosmic shear signal.', '1907.06611-2-28-5': 'Mathematically, at large scale, in addition to halo shot noise, the halo-to-halo covariance includes the three-point function [MATH], which is negligible, while the patch-to-patch covariance includes the four-point function [MATH], which is non-negligible and describes the effect of halo clustering.', '1907.06611-2-28-6': 'Therefore, the halo clustering affects the latter but not the former.', '1907.06611-2-29-0': 'For a sample of individual massive clusters widely separated in the sky, the halo-to-halo and the patch-to-patch variances are equivalent by definition.', '1907.06611-2-29-1': 'In this case, halo clustering is much smaller than shot noise.', '1907.06611-2-29-2': 'Thus, the effect illustrated in Fig. [REF] is not important for individual cluster measurements like Weighing the Giants .', '1907.06611-2-29-3': 'However, it will be relevant for surveys that measure stacked weak lensing profiles to large scales for large cluster samples over contiguous areas of the sky, such as DES, LSST, Euclid, and WFIRST.', '1907.06611-2-30-0': '# Analytical Gaussian-field covariance for tangential shear ([MATH])', '1907.06611-2-31-0': 'We start by calculating the covariance of [MATH] derived from linear theory, with the assumption that the underlying matter density and halo number density follow Gaussian random fields.', '1907.06611-2-31-1': 'Such an assumption is only valid at large scales.', '1907.06611-2-31-2': 'We call such covariance "Gaussian-field" covariance.', '1907.06611-2-31-3': 'In this section we focus on the covariance of [MATH], and in Section [REF] we will discuss the covariance of [MATH].', '1907.06611-2-31-4': 'The Gaussian-field covariance of [MATH] is easier to derive and interpret because the contribution from LSS can be intuitively understood as an additional noise of [MATH].', '1907.06611-2-32-0': 'To ensure readability, we write equations in their simplest forms, i.e., assuming no weighting for lenses or sources (see e.g. for equations incorporating weighting).', '1907.06611-2-32-1': 'Our notations are similar to those in [CITATION] and [CITATION], and we use the angular power spectrum [MATH] to describe the spatial distributions of lenses and sources.', '1907.06611-2-32-2': 'Some authors use the two-dimensional power spectrum, [MATH], which is equivalent to [MATH] on scales where sky curvature is negligible ([MATH]).', '1907.06611-2-32-3': 'While [MATH] arises from the spherical harmonics transform of a field on a sphere and is usually dimensionless, [MATH] describes the perpendicular (transverse) modes in a 3D field assuming a flat geometry and usually has the dimension of distance squared.', '1907.06611-2-33-0': '## Covariance matrices from Fourier space', '1907.06611-2-34-0': 'Weak lensing covariance matrices are more easily derived in Fourier space than in real space because in Fourier space, [MATH] and [MATH] only differ by a phase and thus have the same power spectrum for large [MATH] .', '1907.06611-2-34-1': 'In contrast, in real space, they are related via equation ([REF]).', '1907.06611-2-35-0': 'We can analytically calculate the mean tangential shear of clusters using a Hankel transform of the halo-lensing power spectrum: [EQUATION] where [MATH] is the cross angular power spectrum between haloes and shear (Section [REF]), and [MATH] is the Bessel function of order 2.', '1907.06611-2-35-1': 'This [MATH] arises from the fact that [MATH] is a spin-2 field; for [MATH] (scalar), we simply replace [MATH] by [MATH].', '1907.06611-2-35-2': 'Equation ([REF]) is equivalent to equation ([REF]) based on the real-space correlation function [MATH].', '1907.06611-2-35-3': 'While the Fourier-space expression is more useful for understanding the covariance matrix derivation, the real-space expression is easier to use for calculating the mean lensing signal.', '1907.06611-2-36-0': 'If we assume that both the halo number overdensity and the matter overdensity follow Gaussian random fields, then the covariance of [MATH] is given by (see e.g. and Appendix [REF]) [EQUATION]', '1907.06611-2-36-1': 'Here [MATH] is the sky fraction of the survey; [MATH] and [MATH] are the surface number densities of haloes and sources in the unit of sr[MATH]; [MATH], [MATH], [MATH] are the angular power spectra of halo-halo, lensing-lensing, and halo-lensing, respectively (see Section [REF]).', '1907.06611-2-36-2': 'We use the bin-averaged [MATH] to take into account the finite radial bin size (e.g. equation 26 in ): [EQUATION]', '1907.06611-2-36-3': 'Appendix [REF] presents the properties of the bin-averaged Bessel function and the impact of bin size.', '1907.06611-2-36-4': 'We give the analytical expressions for [MATH], [MATH], [MATH] in terms of 3D power spectra using the Limber approximation in Section [REF], and we provide the derivations in Appendix [REF].', '1907.06611-2-36-5': 'The halo model we use is described in Appendix [REF].', '1907.06611-2-36-6': 'Appendix [REF] demonstrates the relative importance of [MATH], [MATH], [MATH], and [MATH].', '1907.06611-2-37-0': 'We use equation ([REF]) for modelling the Gaussian-field variance.', '1907.06611-2-37-1': 'This equation first appeared in [CITATION] and was used to calculate the galaxy-galaxy lensing covariance matrix at several hundred Mpc scales and to assess the sensitivity of galaxy-galaxy lensing to primordial non-Gaussianity.', '1907.06611-2-37-2': 'In this work, we use it to calculate cluster lensing at the scales of tens to 100 Mpc.', '1907.06611-2-38-0': '## Interpreting the three components of the covariance matrix', '1907.06611-2-39-0': 'Equation ([REF]) can be interpreted as the contribution from three components: shape noise (involving [MATH]), LSS (involving [MATH]), and the intrinsic variation of halo density profiles (involving [MATH]).', '1907.06611-2-39-1': '[EQUATION]', '1907.06611-2-39-2': 'We will show, based on comparisons to simulations, that the LSS term is sufficiently accurate, while the intrinsic term must be supplemented by non-Gaussian calculations (e.g. numerical simulations).', '1907.06611-2-40-0': 'The contribution from shape noise is given by [EQUATION]', '1907.06611-2-40-1': 'Here [MATH] and [MATH] are associated with halo clustering and shot noise, respectively.', '1907.06611-2-40-2': 'When shot noise dominates the variance of halo number counts [MATH], the off-diagonal elements approach zero because of the orthogonality of [MATH], and the shape noise reduces to the more intuitive form [EQUATION]', '1907.06611-2-40-3': 'In Appendix [REF] we provide the derivation.', '1907.06611-2-41-0': 'For [MATH], halo clustering is comparable to shot noise (see the left-hand panel of Fig. [REF] and Fig. [REF]).', '1907.06611-2-41-1': 'For a higher mass threshold, both terms increase, and shot noise dominates for [MATH] at [MATH], and for [MATH] at [MATH].', '1907.06611-2-41-2': 'The exact mass of transition slightly depends on the width of the redshift bin.', '1907.06611-2-42-0': 'The contribution from the LSS is given by [EQUATION]', '1907.06611-2-42-1': 'This contribution is basically the convergence power spectrum ([MATH], see for a review) multiplied by the noise of halo number counts, which include both the shot noise ([MATH]) and the clustering of haloes ([MATH]).', '1907.06611-2-42-2': 'Compared with shape noise, this LSS term is less sensitive to the radial bin size because both [MATH] and [MATH] decrease at high [MATH] and only the integration over the first peak of [MATH] has significant contribution (see Fig. [REF]).', '1907.06611-2-42-3': 'At large scales, [MATH] is usually higher than [MATH] (see the middle panel of Fig. [REF]), and thus [MATH].', '1907.06611-2-43-0': 'The contribution from the intrinsic variation of halo density profiles is given by [EQUATION]', '1907.06611-2-43-1': 'This equation assumes that matter inside haloes follows a Gaussian random field; however, the intrinsic variation of halo density profiles includes significant non-Gaussian contributions.', '1907.06611-2-43-2': 'In this work we use N-body simulations to account for both Gaussian and non-Gaussian contributions.', '1907.06611-2-44-0': 'Fig. [REF] displays the mean and variance of lensing calculated from simulations and from analytical formulae.', '1907.06611-2-44-1': 'We use the fiducial calculations specified in Section [REF] ([MATH]; [MATH] and [MATH] for Takahashi; [MATH] for Abacus), with no shape noise.', '1907.06611-2-44-2': 'We convert from [MATH] to [MATH] by calculating the [MATH] corresponding to the source and lens redshifts; we show [MATH] and [MATH] at left and bottom axes, and the equivalent [MATH] and [MATH] on the top and right axes.', '1907.06611-2-45-0': 'The left-hand panel shows the mean value of [MATH] calculated analytically from equation ([REF]).', '1907.06611-2-45-1': 'The right-hand panel shows the contributions from LSS ([MATH], green) and from the Gaussian intrinsic term ([MATH], grey dotted).', '1907.06611-2-45-2': 'The LSS contribution fully accounts for the variance above [MATH] radian ([MATH]).', '1907.06611-2-45-3': 'The Gaussian intrinsic term significantly under-predicts the small-scale variance and will be replaced by N-body simulations (Section [REF]).', '1907.06611-2-46-0': 'We do not include shape noise in this figure.', '1907.06611-2-46-1': 'When halo clustering is negligible, the shape noise is inversely proportional to the area of the radial bin and the surface density of sources (equation [REF]).', '1907.06611-2-46-2': 'At small scales, the shape noise is usually higher than the intrinsic variation of the halo density profile.', '1907.06611-2-46-3': 'At large scales, the shape noise becomes subdominant to the contribution from LSS, and the transition scale depends on the source density.', '1907.06611-2-46-4': 'We will discuss the relative importance of shape noise in detail in Section [REF].', '1907.06611-2-46-5': 'For the high source densities of LSST, Euclid, and WFIRST, the small-scale intrinsic variation of halo density profiles will become non-negligible, and it is imperative to accurately characterise the covariance at small scales.', '1907.06611-2-47-0': '## Angular power spectra', '1907.06611-2-48-0': 'We use the Limber approximation to calculate the angular power spectra in equation ([REF]).', '1907.06611-2-48-1': 'In Appendix [REF] we provide the derivations (also see ).', '1907.06611-2-49-0': 'The auto angular power spectrum of the halo number density is given by [EQUATION] the [MATH] integration is over the comoving distance range ([MATH], [MATH]) of the cluster redshift bin, and [EQUATION]', '1907.06611-2-49-1': 'For [MATH], we use the linear matter power spectrum multiplied by the halo bias from [CITATION].', '1907.06611-2-49-2': 'The left-hand panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations, which agree with the sum of [MATH] and the shot noise [MATH].', '1907.06611-2-49-3': 'In Appendix [REF] we show that wider redshift bins correspond to lower [MATH] and [MATH], while the relative importance of the two remains unchanged.', '1907.06611-2-50-0': 'For the convergence power spectrum, we assume that the source galaxies follow a redshift distribution [MATH] normalised such that [MATH].', '1907.06611-2-50-1': 'The auto angular power spectrum of convergence is given by [EQUATION] where we integrate all the LSS along the line of sight, and [EQUATION]', '1907.06611-2-50-2': 'Equation ([REF]) corresponds to the contribution of lensing from all the LSS in front of the sources, integrated from zero distance to the farthest source galaxy.', '1907.06611-2-50-3': 'Equation ([REF]) shows that the intervening LSS is weighted by the lensing kernel.', '1907.06611-2-50-4': 'Additional weights on sources can be absorbed in [MATH].', '1907.06611-2-50-5': 'This equation is equivalent to the equation 29 in [CITATION].', '1907.06611-2-51-0': 'The middle panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations.', '1907.06611-2-51-1': 'In the analytical calculation, the linear matter power spectrum underestimates the small-scale (high-[MATH]) power, while the non-linear matter power spectrum from Halofit agrees with simulations out to [MATH].', '1907.06611-2-51-2': 'We also show the shape noise term [MATH] for [MATH] and [MATH]; the shape noise dominates at small scale.', '1907.06611-2-52-0': 'The cross angular power spectrum of halo and lensing is given by [EQUATION]', '1907.06611-2-52-1': 'Similar to [MATH], this integration is over the line-of-sight distance range of the lens sample.', '1907.06611-2-52-2': 'The right-hand panel of Fig. [REF] shows the [MATH] calculated from equation ([REF]) and from the Takahashi simulations.', '1907.06611-2-52-3': 'We use the 3D halo-matter cross power spectrum [MATH] from the halo model; i.e., the sum of the two-halo term (linear power spectrum multiplied by halo bias) and the one-halo term computed for an NFW density profile (see Appendix [REF]).', '1907.06611-2-53-0': '# Analytical Gaussian-field covariance for excess surface density ([MATH])', '1907.06611-2-54-0': 'In this section, we focus on the covariance of the excess surface density ([MATH]).', '1907.06611-2-54-1': 'Since [MATH] and [MATH] are proportional to each other, their fractional errors are the same.', '1907.06611-2-54-2': 'When we consider a single source redshift (that is, if [MATH] is close to a delta function), the covariance matrices of [MATH] and [MATH] simply differ by a constant [MATH].', '1907.06611-2-54-3': 'However, when we consider a broad range of source redshifts, we need to integrate [MATH] and change the order of integration; see equation ([REF]) below.', '1907.06611-2-54-4': 'Therefore, we find it necessary to detail the analytical expressions for the covariance of [MATH].', '1907.06611-2-55-0': 'The contribution of LSS to the covariance of [MATH] is less intuitive than that of [MATH], because it is the projected mass density weighted by the lensing kernel and is equivalent to taking the [MATH] coming from large scale structure between redshift 0 and [MATH] and treating it as if it is at [MATH].', '1907.06611-2-55-1': 'This component is not the projected mass density of LSS because the thin-lens approximation does not hold for LSS.', '1907.06611-2-56-0': 'The covariance matrix analogous to equation ([REF]) is given by [EQUATION] where [EQUATION]', '1907.06611-2-56-1': 'To derive this equation, we use the covariance of [MATH] (equation [REF]) for a single source redshift and a single lens redshift, and then integrate over [MATH] twice.', '1907.06611-2-56-2': 'To convert from ([MATH], [MATH]) to ([MATH], [MATH]), we assume [MATH] and [MATH]; therefore, this expression only applies to a thin lens redshift bin.', '1907.06611-2-57-0': 'Here we introduce two extra angular power spectra: [MATH] corresponds to the auto spectrum for projected matter (analogous to [MATH]), and [MATH] corresponds to the cross spectrum between halo and projected matter (analogous to [MATH]).', '1907.06611-2-57-1': 'Using the Limber approximation (see Appendix [REF]), the auto power spectrum for projected matter is given by [EQUATION]', '1907.06611-2-57-2': 'Here we integrate the LSS along the line-of-sight from zero to infinity and weight the LSS by the window function [EQUATION]', '1907.06611-2-57-3': "The [MATH] in the denominator comes from the lensing kernel (the same as in equation [REF]), while the [MATH] in the numerator comes from the fact that we interpret all the line-of-sight structure as the noise to halo profiles at the haloes' redshift [MATH].", '1907.06611-2-58-0': 'The halo-matter cross power spectrum is given by [EQUATION]', '1907.06611-2-58-1': 'Here we integrate over the redshift range of the halo sample (where the two fields [MATH] and [MATH] overlap), and [MATH] is given by equation ([REF]).', '1907.06611-2-58-2': 'Here [MATH] has the same dimension as [MATH] and [MATH] has the same dimension as [MATH].', '1907.06611-2-59-0': '# Small-scale covariance from N-body simulations', '1907.06611-2-60-0': 'When shape noise is subdominant, Fig. [REF] shows that equation ([REF]) underestimates the covariance at small scales, a consequence of treating matter in haloes as a Gaussian random field.', '1907.06611-2-60-1': 'The actual covariance in this regime will include the effects of variation in halo concentration, sub-structures, and orientations, none of which are captured by the Gaussian field approximation.', '1907.06611-2-60-2': 'Similar to [CITATION], we calculate cluster lensing using N-body simulations.', '1907.06611-2-61-0': '## Abacus N-body simulations', '1907.06611-2-62-0': 'We use the publicly available Abacus Cosmos simulations based on the Abacus N-body code .', '1907.06611-2-62-1': 'We use the 20 realisations of AbacusCosmos720boxplanck.', '1907.06611-2-62-2': 'This suite of boxes are based on a [CITATION] cosmology ([MATH]=0.314, [MATH], [MATH]=0.67) with different phases in the initial condition.', '1907.06611-2-62-3': 'Each realisation has 1440[MATH] particles in a box of side length 720 [MATH], a mass resolution of [MATH], and a spline softening length of 41 [MATH].', '1907.06611-2-62-4': 'Dark matter haloes are identified using Rockstar .', '1907.06611-2-63-0': 'For computing the covariance matrices, we divide each box in the [MATH]-[MATH] plane into 9 equal prism-shaped subvolumes, each of which has a dimension of [MATH].', '1907.06611-2-63-1': 'Each subvolume includes approximately 580 haloes with [MATH].', '1907.06611-2-63-2': 'We calculate the covariance matrix from these 20 [MATH] 9 = 180 subvolumes.', '1907.06611-2-63-3': 'Since the covariance is inversely proportional to the simulation volume, we multiply the covariance by the volume in the unit of [MATH] so that the resulting covariance matrix corresponds to a 1 [MATH] volume.', '1907.06611-2-64-0': 'We use the 10% down-sampled particles to measure the azimuthally averaged [MATH] profiles around haloes.', '1907.06611-2-64-1': 'We use an integration depth of [MATH] along the [MATH]-direction of the box; this integration depth is sufficient for a convergent [MATH] profile but insufficient to include the contribution of uncorrelated LSS to the covariance matrix.', '1907.06611-2-64-2': 'We will use analytical calculations to capture the contribution from the LSS outside this [MATH] integration depth.', '1907.06611-2-65-0': 'To compute [MATH], we cross-correlate the haloes in a subvolume with the particles in the full-volume, applying the periodic boundary condition.', '1907.06611-2-65-1': 'For haloes near the boundary of each subvolume, we use particles outside the boundary to measure their [MATH].', '1907.06611-2-65-2': 'For counting halo-particle pairs, we use the public code Corrfunc .', '1907.06611-2-65-3': 'Similar to the [MATH] calculation, we calculate [MATH] around random points (30 times the number of haloes) in each subvolume and subtract it from the [MATH] around haloes.', '1907.06611-2-66-0': 'Fig. [REF] compares the results from the Abacus simulations with the Takahashi simulations and with analytical calculations.', '1907.06611-2-66-1': 'The left-hand panel shows the mean [MATH].', '1907.06611-2-66-2': 'For Abacus, we use the [MATH] output.', '1907.06611-2-66-3': 'For Takahashi, we use the lens redshift [MATH] and source redshift [MATH], converting from [MATH] to [MATH] using the corresponding [MATH].', '1907.06611-2-66-4': 'The slight difference of the two simulations is due to the slightly different redshift and cosmology ([MATH]=0.314 for Abacus and 0.279 for Takahashi).', '1907.06611-2-66-5': 'Because of the relatively low resolution of Takahashi, the density profile is underestimated below 1 [MATH].', '1907.06611-2-66-6': 'We calculate the mean profile analytically using equation ([REF]) using [MATH] from the halo model (assuming the same cosmology as the Abacus boxes we use).', '1907.06611-2-67-0': 'The right-hand panel of Fig. [REF] shows the variance of [MATH].', '1907.06611-2-67-1': 'The Takahashi result is significantly higher than the Abacus result at large scales because the former includes the lensing effects from all the LSS, while the latter only takes into account [MATH].', '1907.06611-2-67-2': 'At intermediate scales, the Takahashi result approaches the Abacus result because the intrinsic variation of halo density profiles starts to dominate.', '1907.06611-2-68-0': '## Combining analytical and numerical treatments', '1907.06611-2-69-0': 'Our approach to a full calculation of covariance matrices is to combine the analytical expressions for shape noise and LSS contributions (equations [REF] and [REF]) with Abacus calculations, which model the intrinsic halo profile contribution and replace the inaccurate Gaussian-field model at small scales (equation [REF]).', '1907.06611-2-69-1': 'However, our Abacus calculations also include the LSS contribution from a [MATH] slice.', '1907.06611-2-69-2': 'We calculate the contribution from this slice by integrating [MATH] or [MATH] from [MATH] to [MATH], with [MATH] corresponding to the halo redshift and [MATH].', '1907.06611-2-69-3': 'We then subtract this slice from the full LSS contribution.', '1907.06611-2-70-0': 'Fig. [REF] shows how we combine small-scale simulation results with large-scale analytical results.', '1907.06611-2-70-1': 'At small scales, the Abacus calculation dominates; at large scale, the Abacus calculation is similar to but slightly higher than the Gaussian-field LSS in a slice of [MATH] (grey dotted).', '1907.06611-2-70-2': 'This difference could be related to the correlated structure within the slice.', '1907.06611-2-70-3': 'We subtract this slice of LSS from the full LSS and add the Abacus variance.', '1907.06611-2-70-4': 'In practice, subtracting this slice has negligible effect.', '1907.06611-2-70-5': 'The combined result is the heavy black curve.', '1907.06611-2-70-6': 'We also show the results from the Takahashi shear maps; we use the [MATH] measurements from Takahashi in a narrow lens range ([MATH]) and a narrow source range ([MATH]), calculate the corresponding [MATH], and scale to a 1 [MATH] volume.', '1907.06611-2-70-7': 'The Takahashi results agree well with our Abacus+analytical approach at intermediate and large scales.', '1907.06611-2-70-8': 'At small scales, the Takahashi simulation cannot resolve the inner profile of clusters, which has been shown in Fig. [REF].', '1907.06611-2-71-0': 'We have also attempted to calculate the non-Gaussian small-scale covariance analytically.', '1907.06611-2-71-1': 'We find that the analytical non-Gaussian results are higher than simulations (see Appendix [REF]).', '1907.06611-2-71-2': 'This discrepancy does not affect any of the calculations in the main text.', '1907.06611-2-72-0': '# Correlation between radial bins', '1907.06611-2-73-0': 'In this section, we discuss the properties of full covariance matrices, focusing on the importance of off-diagonal elements.', '1907.06611-2-73-1': 'Fig. [REF] shows the full covariance matrices corresponds to Fig. [REF]: [MATH], [MATH], [MATH], with no shape noise.', '1907.06611-2-73-2': 'We combine Abacus simulations and analytical calculations, and we use 15 or 30 logarithmically-spaced [MATH] bins between 0.1 and 100 [MATH].', '1907.06611-2-74-0': 'The top panels show the diagonal elements and the off-diagonal elements parallel to the diagonal.', '1907.06611-2-74-1': 'The top curves correspond to the diagonal elements, the second curves correspond to the elements next to the diagonal (offset from the diagonal by 1 element), and so on.', '1907.06611-2-74-2': 'In the top left panel, we show both the Abacus+analytical (blue) and Takahashi (orange crosses) results for 15 radial bins.', '1907.06611-2-74-3': 'The values of the off-diagonal elements drop by approximately an order of magnitude at the fourth curve (offset=3), which corresponds to a 0.6 dex difference between [MATH] and [MATH].', '1907.06611-2-74-4': 'The off-diagonal elements drop more rapidly at large scales than at small scales because small-scale structures are more correlated.', '1907.06611-2-75-0': 'The top right panel compares the elements for 15 radial bins (blue solid) and 30 radial bins (orange dash).', '1907.06611-2-75-1': 'For the diagonal elements, the dependence on bin size is rather weak.', '1907.06611-2-75-2': 'This is explained by the property of the bin-averaged Bessel function [MATH] (see Appendix [REF] and Fig. [REF]).', '1907.06611-2-75-3': 'The peak of [MATH] is insensitive to the radial bin size; given the steep negative slope of [MATH] and [MATH] at large [MATH], only the first peak of [MATH] contributes to the integration in equation ([REF]), and thus the dependence on the bin size is weak.', '1907.06611-2-75-4': 'This is contrary to the case of shape noise, which is inversely proportional to the bin area and only contributes to the diagonal elements when halo shot noise dominates.', '1907.06611-2-75-5': 'The insensitivity to the radial bin size at small scales in the N-body results also indicates that the particle shot noise is negligible; if we have a low number of dark matter particles in N-body simulations, the small-scale variance would be inversely proportional to the number of particles in a radial bin.', '1907.06611-2-76-0': 'For the 30-bin case (orange dash), the first, third, and fifth curves agree with the first, second, and third curves of the 15-bin case (blue solid).', '1907.06611-2-76-1': 'That is, every other curve of the 30-bin case agrees with the 15-bin case, and each of the even-numbered orange dash curves is approximately the geometric mean of the two neighbouring curves.', '1907.06611-2-76-2': 'In other words, for a given ([MATH], [MATH]), the covariance is almost independent of the bin size.', '1907.06611-2-76-3': 'By halving the bin size, we increase the number of correlated bins, while the covariance between the two scales remains the same.', '1907.06611-2-77-0': 'The two bottom panels show the correlation matrices (with the diagonal elements normalised to 1), [EQUATION] for 15 and 30 radial bins, respectively.', '1907.06611-2-77-1': 'The two matrices have similar structures apart from the different bin sizes.', '1907.06611-2-78-0': 'To compare the information content of these matrices, we assume a parameter [MATH] multiplying the [MATH] data vector.', '1907.06611-2-78-1': 'The Fisher information for [MATH] is given by [EQUATION] where the superscript [MATH] denotes the transpose.', '1907.06611-2-78-2': 'This expression is equivalent to the square of the signal-to-noise ratio.', '1907.06611-2-78-3': 'The constraint on [MATH] is given by [EQUATION] and corresponds to the constraint on the lensing amplitude.', '1907.06611-2-78-4': 'The [MATH] values from the two matrices are almost identical (0.0032 and 0.0031), confirming that the information content is independent of the radial bin size.', '1907.06611-2-79-0': 'Fig. [REF] further demonstrates the impact of bin size, shape noise, and off-diagonal elements on [MATH].', '1907.06611-2-79-1': 'We assume a survey area of 5000 deg[MATH] and a redshift range [MATH], which corresponds to a comoving volume of 1.2 [MATH].', '1907.06611-2-79-2': 'We perform the calculation for two mass thresholds: [MATH] and [MATH].', '1907.06611-2-79-3': 'The left- and right-hand panels correspond to 15 and 30 radial bins, respectively.', '1907.06611-2-79-4': 'The [MATH]-axis corresponds to different levels of shape noise indicated by the source density.', '1907.06611-2-79-5': 'The solid curves correspond to [MATH] calculated using full covariance matrices, while the dash curves use only the diagonal elements and underestimate [MATH].', '1907.06611-2-79-6': 'For the full-covariance cases (solid curves), the two panels show identical results, indicating that the total signal-to-noise ratio does not depend on the radial bin size.', '1907.06611-2-79-7': 'Comparing the two mass ranges, we can see that the low-mass threshold is more sensitive to the shape noise than the high-mass threshold because of its larger statistical power.', '1907.06611-2-80-0': 'For the diagonal-only cases (dash curves), the two panels have almost the same results when shape noise dominates (low source density) because most of the off-diagonal elements are negligible.', '1907.06611-2-80-1': 'When the shape noise is low (high source density), the two panels show slightly different results, which originate from the slightly different diagonal elements for different bin sizes (see the top right panel of Fig. [REF], two top curves).', '1907.06611-2-80-2': 'In addition, the underestimation of [MATH] is worse for narrower radial bins, lower shape noises (high source densities), and higher halo mass.', '1907.06611-2-81-0': 'Since the diagonal elements drop rapidly, the elements far from the diagonal should be negligible.', '1907.06611-2-81-1': 'Fig. [REF] demonstrates how many off-diagonal elements are needed to avoid underestimation of [MATH].', '1907.06611-2-81-2': 'We compare the [MATH] using the full covariance matrix vs. part of the covariance matrix (ignoring elements far from the diagonal).', '1907.06611-2-81-3': 'The two panels correspond to 15 and 30 radial bins.', '1907.06611-2-81-4': 'The [MATH]-axis corresponds to the number of the off-diagonal lines used; [MATH] corresponds to using only the diagonal elements (marked by crosses), [MATH] corresponds to adding the first off-diagonal line, and so on.', '1907.06611-2-81-5': 'The [MATH]-axis corresponds to the ratio of the [MATH] calculated with the partial covariance matrix specified by the [MATH]-axis and the [MATH] obtained from the full matrix.', '1907.06611-2-81-6': 'Various curves correspond to source densities 10 arcmin[MATH] (similar to DES), 20, 60 (similar to WFIRST), and [MATH] (no shape noise).', '1907.06611-2-82-0': 'The left-hand panel shows that for 15 radial bins, for the case with no shape noise (black curves), using only the diagonal elements ([MATH]) underestimates the error bar by a factor of 2, while including 3 off-diagonal lines ([MATH]) is almost equivalent to using the full covariance matrix.', '1907.06611-2-82-1': 'Comparing the two panels we can see that when we halve the bin size, we need to double the number of off-diagonal lines we use.', '1907.06611-2-82-2': 'When the source density is low (shape noise is high), the effect of off-diagonal elements is weaker but still non-negligible.', '1907.06611-2-82-3': 'Even with 10 arcmin[MATH], we cannot completely ignore the off-diagonal elements because the shape noise is subdominant at large scales.', '1907.06611-2-82-4': 'Other redshifts and mass thresholds give very similar results.', '1907.06611-2-83-0': '# Discussion', '1907.06611-2-84-0': 'In this section, we discuss the importance of shape noise ([REF]), the mass and redshift dependence of covariance matrices ([REF]), the cross-covariance of two different mass bins ([REF]), and potential systematic uncertainties that can affect our calculations ([REF]).', '1907.06611-2-85-0': '## Importance of shape noise', '1907.06611-2-86-0': 'Fig. [REF] shows the relative importance of shape noise (various colour curves) and the shape noise-free part (black curves, we call it density variance hereafter) for the variance of [MATH].', '1907.06611-2-86-1': 'We use lenses at [MATH] and 15 [MATH] bins between 0.1 and 100 [MATH], and the two panels correspond to mass thresholds of [MATH] and [MATH].', '1907.06611-2-86-2': 'For lenses at [MATH] and [MATH], the results are nearly identical to the [MATH] case shown here.', '1907.06611-2-86-3': 'Various line styles correspond to various source redshifts.', '1907.06611-2-86-4': 'For simplicity, we assume that all source galaxies are at the same redshift; in reality, the source galaxies are distributed in a redshift range.', '1907.06611-2-87-0': 'For the density variance (black), the LSS contribution at large scales depends on the source redshift because we integrate all the line-of-sight structure in front of the source redshift.', '1907.06611-2-87-1': 'This redshift dependence is weak but non-monotonic.', '1907.06611-2-87-2': 'The LSS contribution involves the ratio [MATH]; see equation ([REF]).', '1907.06611-2-87-3': 'As [MATH] increases, both the numerator and the denominator decrease.', '1907.06611-2-87-4': 'When [MATH] is low and close to [MATH] (e.g. 0.75), as [MATH] increases the numerator decreases rapidly, leading to a smaller covariance.', '1907.06611-2-87-5': 'When [MATH] is high, as [MATH] increases, the numerator is almost constant; since we integrate over a wider range of line-of-sight structure, the total covariance increases.', '1907.06611-2-88-0': 'On the other hand, at a fixed source density, the shape noise is sensitive to the source redshift.', '1907.06611-2-88-1': 'At a fixed [MATH], [MATH] decreases with [MATH], and a higher [MATH] corresponds to a lower shape noise due to the [MATH] factor in equation ([REF]).', '1907.06611-2-88-2': 'If we plot the variance of [MATH] instead, the density variance part would depend on the source redshift (because [MATH] is higher for a higher source redshift), while the shape noise would be independent of the source redshift (equation [REF]).', '1907.06611-2-88-3': 'In both cases, the relative importance of shape noise and density variance is the same.', '1907.06611-2-89-0': 'The blue and orange curves correspond to source densities of 10 arcmin[MATH] (DES-like) and 60 arcmin[MATH] (WFIRST-like).', '1907.06611-2-89-1': 'For 10 arcmin[MATH], the small-scale variance is dominated by shape noise.', '1907.06611-2-89-2': 'For 60 arcmin[MATH], small-scale density variance becomes more important.', '1907.06611-2-89-3': 'Comparing the two panels, we can see that the density variance is more important for high-mass haloes.', '1907.06611-2-90-0': 'When halo clustering is negligible, the shape noise is inversely proportional to the radial bin area.', '1907.06611-2-90-1': 'For example, if we use 30 radial bins instead of 15, the shape noise would double, while the density variance would remain approximately the same.', '1907.06611-2-90-2': 'With 30 radial bins, although the diagonal elements have larger shape noise, the off-diagonal elements play more important roles than the 15-bin case.', '1907.06611-2-91-0': '## Mass and redshift dependence', '1907.06611-2-92-0': 'In this section, we show that the fractional errors of [MATH], in the case of negligible shape noise, are almost independent of redshift and only weakly depend on mass.', '1907.06611-2-92-1': 'Fig. [REF] compares the fractional errors for [MATH] from Abacus+analytical, for halo redshifts [MATH] 0.3, 0.5, and 0.7, and mass ranges [MATH], [MATH], and [MATH].', '1907.06611-2-92-2': 'We assume source redshifts [MATH].', '1907.06611-2-92-3': 'We multiply the covariance matrix of [MATH] by the number of haloes in each bin.', '1907.06611-2-92-4': 'For the [MATH]-axis, we take the square root of this product and divide it by the mean of [MATH].', '1907.06611-2-92-5': 'For the [MATH]-axis, we divide the projected distance by the mean [MATH] in each bin.', '1907.06611-2-92-6': 'The left-hand panel corresponds to the diagonal elements, and the right-hand panel corresponds to the elements next to the diagonal.', '1907.06611-2-93-0': 'We use different colours to indicate different redshift bins, and different line styles to indicate different mass bins.', '1907.06611-2-93-1': 'For a given mass bin (same line style), the fractional error profiles are almost independent of redshift.', '1907.06611-2-93-2': 'Comparing different mass bins, we see that higher mass haloes have slightly lower fractional errors (higher signal-to-noise ratio).', '1907.06611-2-93-3': 'Both panels show very similar trends.', '1907.06611-2-93-4': 'Our numerical calculations focus on specific mass and redshift bins, and the weak dependence demonstrated in Fig. [REF] can be useful for scaling our results to other redshift or mass bin choices, after which shape noise can be added using our analytical formulae.', '1907.06611-2-94-0': '## Cross-mass covariance', '1907.06611-2-95-0': 'If we use multiple mass bins to perform stacked weak lensing measurements and to jointly constrain cosmological parameters, we need to take into account the covariance between mass bins.', '1907.06611-2-95-1': 'To calculate the cross-mass covariance, we again combine Abacus simulations at small scales and analytical calculations at large scales.', '1907.06611-2-96-0': 'To calculate the Gaussian-field covariance analytically for two mass bins at the same redshift (denoted as h1 and h2), we can write the covariance matrix analogous to equation ([REF]): [EQUATION]', '1907.06611-2-96-1': 'The cross-mass covariance has no shot noise associated with halo number counts, and the halo clustering is described by the cross-mass power spectrum [EQUATION]', '1907.06611-2-96-2': 'In the expression above, the 3D cross power spectrum of halo samples is given by [EQUATION] where [MATH] and [MATH] are the halo bias values of the two samples.', '1907.06611-2-96-3': 'Since we consider two samples at the same redshift, they have the same window function [MATH], which is given by equation ([REF]).', '1907.06611-2-96-4': 'In this calculation, the large-scale cross-mass covariance matrix is symmetric.', '1907.06611-2-96-5': 'The last term ([MATH]) and the non-Gaussian part is calculated using Abacus simulations.', '1907.06611-2-97-0': 'Fig. [REF] shows the cross-mass covariance matrices for two mass bins: [MATH] and [MATH], for Abacus+analytical and Takahashi.', '1907.06611-2-97-1': 'The left-hand panel shows the diagonal elements of each block.', '1907.06611-2-97-2': 'The blue, orange, and green curves correspond to the low-mass bin, the high-mass bin, and the cross-mass covariance.', '1907.06611-2-97-3': 'The solid curves correspond to the Abacus+analytical results, while the dash curves correspond to the Takahashi results.', '1907.06611-2-97-4': 'The high-mass bin has a higher variance than the low-mass bin due to its higher shot noise.', '1907.06611-2-97-5': 'At large scale, the cross-mass covariance is closer to the low-mass bin because both are dominated by halo clustering.', '1907.06611-2-97-6': 'For the cross-mass variance, shot noise has no contribution; for the low-mass bin, halo clustering is higher than shot noise; for the high-mass bin, shot noise dominates.', '1907.06611-2-97-7': 'At small scales, the cross-mass covariance is approximately an order of magnitude smaller than the covariance of the low-mass bin.', '1907.06611-2-98-0': 'The right-hand panel shows the full correlation matrix.', '1907.06611-2-98-1': 'The upper left block corresponds to the low-mass bin (15 radial bins between 0.1 and 100 [MATH]), and the lower right block corresponds to the high-mass bin calculated with the same radial binning.', '1907.06611-2-98-2': 'The upper right and lower left blocks correspond to the cross-mass covariance of the two mass bins and are the transpose of each other.', '1907.06611-2-99-0': 'To understand the contribution of the cross-mass block to the error budget, we again calculate the constraints on a parameter multiplying the [MATH] for both mass bins.', '1907.06611-2-99-1': 'We combine the two data vectors to form [EQUATION] and we again use equation ([REF]).', '1907.06611-2-99-2': 'We calculate [MATH] using the full matrix vs. a block-diagonal matrix (ignoring the cross-mass covariance).', '1907.06611-2-99-3': 'Both calculations give [MATH] for a 1 [MATH] volume, indicating that the cross-mass covariance is negligible in the absence of shape noise.', '1907.06611-2-99-4': 'However, if shape noise dominates, we need to add diagonal shape noise to the cross-mass covariance block.', '1907.06611-2-100-0': '## Potential systematic uncertainties', '1907.06611-2-101-0': 'In this section we discuss the impact of baryons, cluster miscentering, and mass-observable relation on the covariance calculations.', '1907.06611-2-102-0': 'In this work, the small-scale cluster lensing signal is derived from dark matter-only simulations.', '1907.06611-2-102-1': 'Baryonic effects can change the inner density profiles of clusters and thus change the lensing signal.', '1907.06611-2-102-2': 'Since the effect is mainly at very small scales (less than 10 kpc), which will be dominated by shape noise for most of the surveys, its impact is likely to be small for covariance matrices.', '1907.06611-2-103-0': 'We calculate the cluster lensing signals with respect to the centres of dark matter haloes.', '1907.06611-2-103-1': 'In optically-selected cluster samples, the locations of the brightest cluster galaxies may not coincide with the centres of dark matter haloes (see e.g. for comparisons between optically defined centres and X-ray centres).', '1907.06611-2-103-2': 'This miscentering would lead to shallower lensing profiles at small scales.', '1907.06611-2-103-3': 'For current surveys, the scales affected by miscentering are dominated by shape noise, and the covariance matrices are unaffected.', '1907.06611-2-103-4': 'For future surveys, when shape noise no longer dominates at small scales, the miscentering effects would need to be taken into account in modelling the small-scale covariance matrices.', '1907.06611-2-104-0': 'In this work we select clusters based on their masses; in real surveys, clusters would be selected by some observed property (e.g. optical richness), which has a scaling relation and a scatter around the true mass.', '1907.06611-2-104-1': 'If this mass-observable relation is not biased by some property that can also bias lensing, we can simply incorporate this effect by convolving the halo mass function with the mass-observable relation.', '1907.06611-2-104-2': 'However, if the mass-observable relation is biased by some property that can also bias the lensing signal, for example, the orientation of a triaxial halo, we will need to take into account this extra property in order to obtain unbiased lensing measurements and robust covariance matrices.', '1907.06611-2-104-3': 'We will explore this in future work.', '1907.06611-2-105-0': '# Summary', '1907.06611-2-106-0': 'We calculate accurate covariance matrices required for the cosmological interpretation of weak gravitational lensing by galaxy clusters that will be observed by LSST, Euclid, and WFIRST.', '1907.06611-2-106-1': 'We combine analytical calculations at large scales with the Abacus N-body simulations at small scales, and we validate our approach with the Takahashi full-sky ray-tracing simulations at intermediate and large scales.', '1907.06611-2-106-2': 'Our main results are summarised as follows.', '1907.06611-2-107-0': 'The covariance matrices provided in this paper can be used for robust forecast and survey optimisation for future surveys like LSST, Euclid, and WFIRST.', '1907.06611-2-107-1': 'Given the high signal-to-noise ratio of these surveys, cluster lensing will have unprecedented constraining power on the growth of structure.', '1907.06611-2-107-2': 'In Wu et al. (in preparation) and [CITATION], we present forecasts on combining cluster counts and lensing, as well as how we can effectively combine the cluster lensing and cross correlation functions of clusters and galaxies to constrain cosmological parameters.'}

[['1907.06611-1-9-0', '1907.06611-2-9-0'], ['1907.06611-1-9-1', '1907.06611-2-9-1'], ['1907.06611-1-9-2', '1907.06611-2-9-2'], ['1907.06611-1-9-3', '1907.06611-2-9-3'], ['1907.06611-1-9-4', '1907.06611-2-9-4'], ['1907.06611-1-44-2', '1907.06611-2-44-2'], ['1907.06611-1-91-0', '1907.06611-2-92-0'], ['1907.06611-1-91-2', '1907.06611-2-92-2'], ['1907.06611-1-91-3', '1907.06611-2-92-3'], ['1907.06611-1-91-4', '1907.06611-2-92-4'], ['1907.06611-1-91-5', '1907.06611-2-92-5'], ['1907.06611-1-16-0', '1907.06611-2-16-0'], ['1907.06611-1-16-1', '1907.06611-2-16-1'], ['1907.06611-1-26-0', '1907.06611-2-26-0'], ['1907.06611-1-26-1', '1907.06611-2-26-1'], ['1907.06611-1-26-3', '1907.06611-2-26-3'], ['1907.06611-1-26-6', '1907.06611-2-26-6'], ['1907.06611-1-80-0', '1907.06611-2-81-0'], ['1907.06611-1-80-2', '1907.06611-2-81-2'], ['1907.06611-1-80-3', '1907.06611-2-81-3'], ['1907.06611-1-80-4', '1907.06611-2-81-4'], ['1907.06611-1-80-5', '1907.06611-2-81-5'], ['1907.06611-1-80-6', '1907.06611-2-81-6'], ['1907.06611-1-62-0', '1907.06611-2-62-0'], ['1907.06611-1-62-2', '1907.06611-2-62-2'], ['1907.06611-1-62-3', '1907.06611-2-62-3'], ['1907.06611-1-62-4', '1907.06611-2-62-4'], ['1907.06611-1-78-0', '1907.06611-2-78-0'], ['1907.06611-1-78-1', '1907.06611-2-78-1'], ['1907.06611-1-78-2', '1907.06611-2-78-2'], ['1907.06611-1-86-0', '1907.06611-2-87-0'], ['1907.06611-1-86-1', '1907.06611-2-87-1'], ['1907.06611-1-86-2', '1907.06611-2-87-2'], ['1907.06611-1-86-3', '1907.06611-2-87-3'], ['1907.06611-1-40-0', '1907.06611-2-40-0'], ['1907.06611-1-40-1', '1907.06611-2-40-1'], ['1907.06611-1-40-2', '1907.06611-2-40-2'], ['1907.06611-1-40-3', '1907.06611-2-40-3'], ['1907.06611-1-22-0', '1907.06611-2-22-0'], ['1907.06611-1-22-1', '1907.06611-2-22-1'], ['1907.06611-1-22-2', '1907.06611-2-22-2'], ['1907.06611-1-22-3', '1907.06611-2-22-3'], ['1907.06611-1-19-0', '1907.06611-2-19-0'], ['1907.06611-1-19-1', '1907.06611-2-19-1'], ['1907.06611-1-29-1', '1907.06611-2-29-2'], ['1907.06611-1-29-2', '1907.06611-2-29-3'], ['1907.06611-1-28-2', '1907.06611-2-28-2'], ['1907.06611-1-28-3', '1907.06611-2-28-3'], ['1907.06611-1-28-5', '1907.06611-2-28-6'], ['1907.06611-1-37-0', '1907.06611-2-37-0'], ['1907.06611-1-37-2', '1907.06611-2-37-2'], ['1907.06611-1-98-0', '1907.06611-2-99-0'], ['1907.06611-1-98-1', '1907.06611-2-99-1'], ['1907.06611-1-98-2', '1907.06611-2-99-2'], ['1907.06611-1-27-3', '1907.06611-2-27-3'], ['1907.06611-1-27-5', '1907.06611-2-27-5'], ['1907.06611-1-92-0', '1907.06611-2-93-0'], ['1907.06611-1-92-1', '1907.06611-2-93-1'], ['1907.06611-1-92-3', '1907.06611-2-93-3'], ['1907.06611-1-34-1', '1907.06611-2-34-1'], ['1907.06611-1-41-2', '1907.06611-2-41-2'], ['1907.06611-1-106-0', '1907.06611-2-107-0'], ['1907.06611-1-106-1', '1907.06611-2-107-1'], ['1907.06611-1-55-0', '1907.06611-2-55-0'], ['1907.06611-1-55-1', '1907.06611-2-55-1'], ['1907.06611-1-42-0', '1907.06611-2-42-0'], ['1907.06611-1-42-1', '1907.06611-2-42-1'], ['1907.06611-1-42-2', '1907.06611-2-42-2'], ['1907.06611-1-42-3', '1907.06611-2-42-3'], ['1907.06611-1-51-0', '1907.06611-2-51-0'], ['1907.06611-1-51-1', '1907.06611-2-51-1'], ['1907.06611-1-51-2', '1907.06611-2-51-2'], ['1907.06611-1-87-0', '1907.06611-2-88-0'], ['1907.06611-1-87-3', '1907.06611-2-88-3'], ['1907.06611-1-56-0', '1907.06611-2-56-0'], ['1907.06611-1-56-1', '1907.06611-2-56-1'], ['1907.06611-1-56-2', '1907.06611-2-56-2'], ['1907.06611-1-64-0', '1907.06611-2-64-0'], ['1907.06611-1-64-1', '1907.06611-2-64-1'], ['1907.06611-1-6-1', '1907.06611-2-6-1'], ['1907.06611-1-6-2', '1907.06611-2-6-2'], ['1907.06611-1-6-3', '1907.06611-2-6-3'], ['1907.06611-1-103-1', '1907.06611-2-104-1'], ['1907.06611-1-103-2', '1907.06611-2-104-2'], ['1907.06611-1-103-3', '1907.06611-2-104-3'], ['1907.06611-1-83-0', '1907.06611-2-84-0'], ['1907.06611-1-71-0', '1907.06611-2-71-0'], ['1907.06611-1-71-1', '1907.06611-2-71-1'], ['1907.06611-1-71-2', '1907.06611-2-71-2'], ['1907.06611-1-21-0', '1907.06611-2-21-0'], ['1907.06611-1-21-1', '1907.06611-2-21-1'], ['1907.06611-1-21-4', '1907.06611-2-21-4'], ['1907.06611-1-21-7', '1907.06611-2-21-7'], ['1907.06611-1-21-8', '1907.06611-2-21-8'], ['1907.06611-1-105-0', '1907.06611-2-106-0'], ['1907.06611-1-105-1', '1907.06611-2-106-1'], ['1907.06611-1-105-2', '1907.06611-2-106-2'], ['1907.06611-1-8-0', '1907.06611-2-8-0'], ['1907.06611-1-8-1', '1907.06611-2-8-1'], ['1907.06611-1-57-0', '1907.06611-2-57-0'], ['1907.06611-1-57-1', '1907.06611-2-57-1'], ['1907.06611-1-57-2', '1907.06611-2-57-2'], ['1907.06611-1-57-3', '1907.06611-2-57-3'], ['1907.06611-1-5-0', '1907.06611-2-5-0'], ['1907.06611-1-5-1', '1907.06611-2-5-1'], ['1907.06611-1-5-2', '1907.06611-2-5-2'], ['1907.06611-1-5-3', '1907.06611-2-5-3'], ['1907.06611-1-5-6', '1907.06611-2-5-6'], ['1907.06611-1-5-7', '1907.06611-2-5-7'], ['1907.06611-1-13-0', '1907.06611-2-13-0'], ['1907.06611-1-13-1', '1907.06611-2-13-1'], ['1907.06611-1-13-2', '1907.06611-2-13-2'], ['1907.06611-1-13-3', '1907.06611-2-13-3'], ['1907.06611-1-39-0', '1907.06611-2-39-0'], ['1907.06611-1-25-1', '1907.06611-2-25-1'], ['1907.06611-1-3-0', '1907.06611-2-3-0'], ['1907.06611-1-3-2', '1907.06611-2-3-2'], ['1907.06611-1-67-0', '1907.06611-2-67-0'], ['1907.06611-1-67-1', '1907.06611-2-67-1'], ['1907.06611-1-89-0', '1907.06611-2-90-0'], ['1907.06611-1-89-1', '1907.06611-2-90-1'], ['1907.06611-1-89-2', '1907.06611-2-90-2'], ['1907.06611-1-12-0', '1907.06611-2-12-0'], ['1907.06611-1-14-0', '1907.06611-2-14-0'], ['1907.06611-1-14-1', '1907.06611-2-14-1'], ['1907.06611-1-14-3', '1907.06611-2-14-3'], ['1907.06611-1-14-4', '1907.06611-2-14-4'], ['1907.06611-1-36-1', '1907.06611-2-36-1'], ['1907.06611-1-36-3', '1907.06611-2-36-3'], ['1907.06611-1-36-5', '1907.06611-2-36-5'], ['1907.06611-1-36-6', '1907.06611-2-36-6'], ['1907.06611-1-69-1', '1907.06611-2-69-1'], ['1907.06611-1-69-2', '1907.06611-2-69-2'], ['1907.06611-1-69-3', '1907.06611-2-69-3'], ['1907.06611-1-95-0', '1907.06611-2-96-0'], ['1907.06611-1-95-1', '1907.06611-2-96-1'], ['1907.06611-1-95-2', '1907.06611-2-96-2'], ['1907.06611-1-95-4', '1907.06611-2-96-4'], ['1907.06611-1-45-2', '1907.06611-2-45-2'], ['1907.06611-1-52-0', '1907.06611-2-52-0'], ['1907.06611-1-52-1', '1907.06611-2-52-1'], ['1907.06611-1-52-3', '1907.06611-2-52-3'], ['1907.06611-1-97-2', '1907.06611-2-98-2'], ['1907.06611-1-88-0', '1907.06611-2-89-0'], ['1907.06611-1-88-1', '1907.06611-2-89-1'], ['1907.06611-1-88-2', '1907.06611-2-89-2'], ['1907.06611-1-88-3', '1907.06611-2-89-3'], ['1907.06611-1-4-1', '1907.06611-2-4-1'], ['1907.06611-1-4-3', '1907.06611-2-4-3'], ['1907.06611-1-100-0', '1907.06611-2-101-0'], ['1907.06611-1-48-0', '1907.06611-2-48-0'], ['1907.06611-1-15-0', '1907.06611-2-15-0'], ['1907.06611-1-15-1', '1907.06611-2-15-1'], ['1907.06611-1-15-2', '1907.06611-2-15-2'], ['1907.06611-1-15-3', '1907.06611-2-15-3'], ['1907.06611-1-15-4', '1907.06611-2-15-4'], ['1907.06611-1-77-0', '1907.06611-2-77-0'], ['1907.06611-1-77-1', '1907.06611-2-77-1'], ['1907.06611-1-46-0', '1907.06611-2-46-0'], ['1907.06611-1-46-1', '1907.06611-2-46-1'], ['1907.06611-1-46-2', '1907.06611-2-46-2'], ['1907.06611-1-46-3', '1907.06611-2-46-3'], ['1907.06611-1-46-4', '1907.06611-2-46-4'], ['1907.06611-1-46-5', '1907.06611-2-46-5'], ['1907.06611-1-76-0', '1907.06611-2-76-0'], ['1907.06611-1-76-1', '1907.06611-2-76-1'], ['1907.06611-1-76-2', '1907.06611-2-76-2'], ['1907.06611-1-76-3', '1907.06611-2-76-3'], ['1907.06611-1-81-1', '1907.06611-2-82-1'], ['1907.06611-1-81-2', '1907.06611-2-82-2'], ['1907.06611-1-81-3', '1907.06611-2-82-3'], ['1907.06611-1-23-0', '1907.06611-2-23-0'], ['1907.06611-1-23-1', '1907.06611-2-23-1'], ['1907.06611-1-23-3', '1907.06611-2-23-3'], ['1907.06611-1-23-4', '1907.06611-2-23-4'], ['1907.06611-1-23-5', '1907.06611-2-23-5'], ['1907.06611-1-23-6', '1907.06611-2-23-6'], ['1907.06611-1-63-1', '1907.06611-2-63-1'], ['1907.06611-1-63-2', '1907.06611-2-63-2'], ['1907.06611-1-50-0', '1907.06611-2-50-0'], ['1907.06611-1-50-1', '1907.06611-2-50-1'], ['1907.06611-1-50-2', '1907.06611-2-50-2'], ['1907.06611-1-50-3', '1907.06611-2-50-3'], ['1907.06611-1-50-4', '1907.06611-2-50-4'], ['1907.06611-1-50-5', '1907.06611-2-50-5'], ['1907.06611-1-10-0', '1907.06611-2-10-0'], ['1907.06611-1-10-1', '1907.06611-2-10-1'], ['1907.06611-1-10-2', '1907.06611-2-10-2'], ['1907.06611-1-10-4', '1907.06611-2-10-4'], ['1907.06611-1-10-5', '1907.06611-2-10-5'], ['1907.06611-1-10-6', '1907.06611-2-10-6'], ['1907.06611-1-10-7', '1907.06611-2-10-7'], ['1907.06611-1-35-0', '1907.06611-2-35-0'], ['1907.06611-1-35-1', '1907.06611-2-35-1'], ['1907.06611-1-35-2', '1907.06611-2-35-2'], ['1907.06611-1-35-3', '1907.06611-2-35-3'], ['1907.06611-1-2-0', '1907.06611-2-2-0'], ['1907.06611-1-2-1', '1907.06611-2-2-1'], ['1907.06611-1-2-2', '1907.06611-2-2-2'], ['1907.06611-1-2-3', '1907.06611-2-2-3'], ['1907.06611-1-2-5', '1907.06611-2-2-5'], ['1907.06611-1-2-6', '1907.06611-2-2-6'], ['1907.06611-1-2-7', '1907.06611-2-2-7'], ['1907.06611-1-0-0', '1907.06611-2-0-0'], ['1907.06611-1-0-1', '1907.06611-2-0-1'], ['1907.06611-1-0-2', '1907.06611-2-0-2'], ['1907.06611-1-0-3', '1907.06611-2-0-3'], ['1907.06611-1-0-4', '1907.06611-2-0-4'], ['1907.06611-1-0-5', '1907.06611-2-0-5'], ['1907.06611-1-0-6', '1907.06611-2-0-6'], ['1907.06611-1-75-0', '1907.06611-2-75-0'], ['1907.06611-1-75-1', '1907.06611-2-75-1'], ['1907.06611-1-75-2', '1907.06611-2-75-2'], ['1907.06611-1-75-3', '1907.06611-2-75-3'], ['1907.06611-1-75-5', '1907.06611-2-75-5'], ['1907.06611-1-54-0', '1907.06611-2-54-0'], ['1907.06611-1-54-1', '1907.06611-2-54-1'], ['1907.06611-1-54-2', '1907.06611-2-54-2'], ['1907.06611-1-54-3', '1907.06611-2-54-3'], ['1907.06611-1-54-4', '1907.06611-2-54-4'], ['1907.06611-1-7-0', '1907.06611-2-7-0'], ['1907.06611-1-7-1', '1907.06611-2-7-1'], ['1907.06611-1-7-2', '1907.06611-2-7-2'], ['1907.06611-1-7-4', '1907.06611-2-7-4'], ['1907.06611-1-7-5', '1907.06611-2-7-5'], ['1907.06611-1-7-6', '1907.06611-2-7-6'], ['1907.06611-1-66-0', '1907.06611-2-66-0'], ['1907.06611-1-66-1', '1907.06611-2-66-1'], ['1907.06611-1-66-2', '1907.06611-2-66-2'], ['1907.06611-1-66-3', '1907.06611-2-66-3'], ['1907.06611-1-66-4', '1907.06611-2-66-4'], ['1907.06611-1-66-5', '1907.06611-2-66-5'], ['1907.06611-1-66-6', '1907.06611-2-66-6'], ['1907.06611-1-94-0', '1907.06611-2-95-0'], ['1907.06611-1-94-1', '1907.06611-2-95-1'], ['1907.06611-1-58-0', '1907.06611-2-58-0'], ['1907.06611-1-58-1', '1907.06611-2-58-1'], ['1907.06611-1-31-1', '1907.06611-2-31-1'], ['1907.06611-1-31-2', '1907.06611-2-31-2'], ['1907.06611-1-102-0', '1907.06611-2-103-0'], ['1907.06611-1-102-2', '1907.06611-2-103-2'], ['1907.06611-1-102-3', '1907.06611-2-103-3'], ['1907.06611-1-74-0', '1907.06611-2-74-0'], ['1907.06611-1-74-2', '1907.06611-2-74-2'], ['1907.06611-1-74-3', '1907.06611-2-74-3'], ['1907.06611-1-74-4', '1907.06611-2-74-4'], ['1907.06611-1-65-1', '1907.06611-2-65-1'], ['1907.06611-1-65-2', '1907.06611-2-65-2'], ['1907.06611-1-65-3', '1907.06611-2-65-3'], ['1907.06611-1-101-0', '1907.06611-2-102-0'], ['1907.06611-1-101-1', '1907.06611-2-102-1'], ['1907.06611-1-101-2', '1907.06611-2-102-2'], ['1907.06611-1-70-0', '1907.06611-2-70-0'], ['1907.06611-1-70-1', '1907.06611-2-70-1'], ['1907.06611-1-70-2', '1907.06611-2-70-2'], ['1907.06611-1-70-4', '1907.06611-2-70-4'], ['1907.06611-1-70-5', '1907.06611-2-70-5'], ['1907.06611-1-70-6', '1907.06611-2-70-6'], ['1907.06611-1-70-7', '1907.06611-2-70-7'], ['1907.06611-1-17-0', '1907.06611-2-17-0'], ['1907.06611-1-17-1', '1907.06611-2-17-1'], ['1907.06611-1-17-2', '1907.06611-2-17-2'], ['1907.06611-1-17-3', '1907.06611-2-17-3'], ['1907.06611-1-17-5', '1907.06611-2-17-5'], ['1907.06611-1-73-0', '1907.06611-2-73-0'], ['1907.06611-1-73-1', '1907.06611-2-73-1'], ['1907.06611-1-73-2', '1907.06611-2-73-2'], ['1907.06611-1-85-0', '1907.06611-2-86-0'], ['1907.06611-1-85-1', '1907.06611-2-86-1'], ['1907.06611-1-85-2', '1907.06611-2-86-2'], ['1907.06611-1-85-3', '1907.06611-2-86-3'], ['1907.06611-1-96-0', '1907.06611-2-97-0'], ['1907.06611-1-96-3', '1907.06611-2-97-3'], ['1907.06611-1-79-0', '1907.06611-2-79-0'], ['1907.06611-1-79-2', '1907.06611-2-79-2'], ['1907.06611-1-79-4', '1907.06611-2-79-4'], ['1907.06611-1-79-7', '1907.06611-2-79-7'], ['1907.06611-1-44-0', '1907.06611-2-44-0'], ['1907.06611-1-91-1', '1907.06611-2-92-1'], ['1907.06611-1-91-6', '1907.06611-2-92-6'], ['1907.06611-1-26-2', '1907.06611-2-26-2'], ['1907.06611-1-26-4', '1907.06611-2-26-4'], ['1907.06611-1-26-5', '1907.06611-2-26-5'], ['1907.06611-1-80-1', '1907.06611-2-81-1'], ['1907.06611-1-62-1', '1907.06611-2-62-1'], ['1907.06611-1-78-4', '1907.06611-2-78-4'], ['1907.06611-1-86-4', '1907.06611-2-87-4'], ['1907.06611-1-86-5', '1907.06611-2-87-5'], ['1907.06611-1-29-0', '1907.06611-2-29-0'], ['1907.06611-1-28-0', '1907.06611-2-28-0'], ['1907.06611-1-28-4', '1907.06611-2-28-5'], ['1907.06611-1-37-1', '1907.06611-2-37-1'], ['1907.06611-1-98-3', '1907.06611-2-99-3'], ['1907.06611-1-98-4', '1907.06611-2-99-4'], ['1907.06611-1-27-0', '1907.06611-2-27-0'], ['1907.06611-1-27-1', '1907.06611-2-27-1'], ['1907.06611-1-27-2', '1907.06611-2-27-2'], ['1907.06611-1-27-4', '1907.06611-2-27-4'], ['1907.06611-1-92-2', '1907.06611-2-93-2'], ['1907.06611-1-92-4', '1907.06611-2-93-4'], ['1907.06611-1-34-0', '1907.06611-2-34-0'], ['1907.06611-1-41-0', '1907.06611-2-41-0'], ['1907.06611-1-41-1', '1907.06611-2-41-1'], ['1907.06611-1-106-2', '1907.06611-2-107-2'], ['1907.06611-1-87-1', '1907.06611-2-88-1'], ['1907.06611-1-87-2', '1907.06611-2-88-2'], ['1907.06611-1-64-2', '1907.06611-2-64-2'], ['1907.06611-1-6-0', '1907.06611-2-6-0'], ['1907.06611-1-103-0', '1907.06611-2-104-0'], ['1907.06611-1-21-3', '1907.06611-2-21-3'], ['1907.06611-1-21-5', '1907.06611-2-21-5'], ['1907.06611-1-21-6', '1907.06611-2-21-6'], ['1907.06611-1-5-5', '1907.06611-2-5-5'], ['1907.06611-1-5-8', '1907.06611-2-5-8'], ['1907.06611-1-39-2', '1907.06611-2-39-2'], ['1907.06611-1-25-0', '1907.06611-2-25-0'], ['1907.06611-1-3-1', '1907.06611-2-3-1'], ['1907.06611-1-3-3', '1907.06611-2-3-3'], ['1907.06611-1-67-2', '1907.06611-2-67-2'], ['1907.06611-1-14-2', '1907.06611-2-14-2'], ['1907.06611-1-36-0', '1907.06611-2-36-0'], ['1907.06611-1-36-2', '1907.06611-2-36-2'], ['1907.06611-1-36-4', '1907.06611-2-36-4'], ['1907.06611-1-69-0', '1907.06611-2-69-0'], ['1907.06611-1-95-3', '1907.06611-2-96-3'], ['1907.06611-1-45-0', '1907.06611-2-45-0'], ['1907.06611-1-45-3', '1907.06611-2-45-3'], ['1907.06611-1-52-2', '1907.06611-2-52-2'], ['1907.06611-1-97-0', '1907.06611-2-98-0'], ['1907.06611-1-97-1', '1907.06611-2-98-1'], ['1907.06611-1-4-0', '1907.06611-2-4-0'], ['1907.06611-1-4-2', '1907.06611-2-4-2'], ['1907.06611-1-4-4', '1907.06611-2-4-4'], ['1907.06611-1-60-1', '1907.06611-2-60-1'], ['1907.06611-1-43-0', '1907.06611-2-43-0'], ['1907.06611-1-81-0', '1907.06611-2-82-0'], ['1907.06611-1-49-0', '1907.06611-2-49-0'], ['1907.06611-1-49-1', '1907.06611-2-49-2'], ['1907.06611-1-49-2', '1907.06611-2-49-1'], ['1907.06611-1-23-2', '1907.06611-2-23-2'], ['1907.06611-1-23-8', '1907.06611-2-23-7'], ['1907.06611-1-63-0', '1907.06611-2-63-0'], ['1907.06611-1-63-3', '1907.06611-2-63-3'], ['1907.06611-1-10-3', '1907.06611-2-10-3'], ['1907.06611-1-2-4', '1907.06611-2-2-4'], ['1907.06611-1-75-4', '1907.06611-2-75-4'], ['1907.06611-1-7-3', '1907.06611-2-7-3'], ['1907.06611-1-58-2', '1907.06611-2-58-2'], ['1907.06611-1-31-0', '1907.06611-2-31-0'], ['1907.06611-1-31-3', '1907.06611-2-31-3'], ['1907.06611-1-31-4', '1907.06611-2-31-4'], ['1907.06611-1-102-1', '1907.06611-2-103-1'], ['1907.06611-1-102-4', '1907.06611-2-103-4'], ['1907.06611-1-74-1', '1907.06611-2-74-1'], ['1907.06611-1-65-0', '1907.06611-2-65-0'], ['1907.06611-1-70-8', '1907.06611-2-70-8'], ['1907.06611-1-32-0', '1907.06611-2-32-0'], ['1907.06611-1-32-2', '1907.06611-2-32-2'], ['1907.06611-1-17-4', '1907.06611-2-17-4'], ['1907.06611-1-85-4', '1907.06611-2-86-4'], ['1907.06611-1-96-1', '1907.06611-2-97-1'], ['1907.06611-1-96-2', '1907.06611-2-97-2'], ['1907.06611-1-96-5', '1907.06611-2-97-5'], ['1907.06611-1-96-6', '1907.06611-2-97-6'], ['1907.06611-1-96-7', '1907.06611-2-97-7'], ['1907.06611-1-79-1', '1907.06611-2-79-1'], ['1907.06611-1-79-3', '1907.06611-2-79-3'], ['1907.06611-1-79-9', '1907.06611-2-80-2'], ['1907.06611-1-44-1', '1907.06611-2-44-1'], ['1907.06611-1-78-3', '1907.06611-2-78-3'], ['1907.06611-1-28-1', '1907.06611-2-28-1'], ['1907.06611-1-28-6', '1907.06611-2-28-4'], ['1907.06611-1-8-2', '1907.06611-2-8-2'], ['1907.06611-1-5-4', '1907.06611-2-5-4'], ['1907.06611-1-12-1', '1907.06611-2-12-1'], ['1907.06611-1-95-5', '1907.06611-2-96-5'], ['1907.06611-1-45-1', '1907.06611-2-45-1'], ['1907.06611-1-48-1', '1907.06611-2-48-1'], ['1907.06611-1-60-0', '1907.06611-2-60-0'], ['1907.06611-1-60-2', '1907.06611-2-60-2'], ['1907.06611-1-43-1', '1907.06611-2-43-1'], ['1907.06611-1-81-4', '1907.06611-2-82-4'], ['1907.06611-1-49-3', '1907.06611-2-49-3'], ['1907.06611-1-70-3', '1907.06611-2-70-3'], ['1907.06611-1-32-1', '1907.06611-2-32-1'], ['1907.06611-1-96-4', '1907.06611-2-97-4'], ['1907.06611-1-79-5', '1907.06611-2-79-5'], ['1907.06611-1-79-8', '1907.06611-2-79-5']]

[['1907.06611-1-9-0', '1907.06611-2-9-0'], ['1907.06611-1-9-1', '1907.06611-2-9-1'], ['1907.06611-1-9-2', '1907.06611-2-9-2'], ['1907.06611-1-9-3', '1907.06611-2-9-3'], ['1907.06611-1-9-4', '1907.06611-2-9-4'], ['1907.06611-1-44-2', '1907.06611-2-44-2'], ['1907.06611-1-91-0', '1907.06611-2-92-0'], ['1907.06611-1-91-2', '1907.06611-2-92-2'], ['1907.06611-1-91-3', '1907.06611-2-92-3'], ['1907.06611-1-91-4', '1907.06611-2-92-4'], ['1907.06611-1-91-5', '1907.06611-2-92-5'], ['1907.06611-1-16-0', '1907.06611-2-16-0'], ['1907.06611-1-16-1', '1907.06611-2-16-1'], ['1907.06611-1-26-0', '1907.06611-2-26-0'], ['1907.06611-1-26-1', '1907.06611-2-26-1'], ['1907.06611-1-26-3', '1907.06611-2-26-3'], ['1907.06611-1-26-6', '1907.06611-2-26-6'], ['1907.06611-1-80-0', '1907.06611-2-81-0'], ['1907.06611-1-80-2', '1907.06611-2-81-2'], ['1907.06611-1-80-3', '1907.06611-2-81-3'], ['1907.06611-1-80-4', '1907.06611-2-81-4'], ['1907.06611-1-80-5', '1907.06611-2-81-5'], ['1907.06611-1-80-6', '1907.06611-2-81-6'], ['1907.06611-1-62-0', '1907.06611-2-62-0'], ['1907.06611-1-62-2', '1907.06611-2-62-2'], ['1907.06611-1-62-3', '1907.06611-2-62-3'], ['1907.06611-1-62-4', '1907.06611-2-62-4'], ['1907.06611-1-78-0', '1907.06611-2-78-0'], ['1907.06611-1-78-1', '1907.06611-2-78-1'], ['1907.06611-1-78-2', '1907.06611-2-78-2'], ['1907.06611-1-86-0', '1907.06611-2-87-0'], ['1907.06611-1-86-1', '1907.06611-2-87-1'], ['1907.06611-1-86-2', '1907.06611-2-87-2'], ['1907.06611-1-86-3', '1907.06611-2-87-3'], ['1907.06611-1-40-0', '1907.06611-2-40-0'], ['1907.06611-1-40-1', '1907.06611-2-40-1'], ['1907.06611-1-40-2', '1907.06611-2-40-2'], ['1907.06611-1-40-3', '1907.06611-2-40-3'], ['1907.06611-1-22-0', '1907.06611-2-22-0'], ['1907.06611-1-22-1', '1907.06611-2-22-1'], ['1907.06611-1-22-2', '1907.06611-2-22-2'], ['1907.06611-1-22-3', '1907.06611-2-22-3'], ['1907.06611-1-19-0', '1907.06611-2-19-0'], ['1907.06611-1-19-1', '1907.06611-2-19-1'], ['1907.06611-1-29-1', '1907.06611-2-29-2'], ['1907.06611-1-29-2', '1907.06611-2-29-3'], ['1907.06611-1-28-2', '1907.06611-2-28-2'], ['1907.06611-1-28-3', '1907.06611-2-28-3'], ['1907.06611-1-28-5', '1907.06611-2-28-6'], ['1907.06611-1-37-0', '1907.06611-2-37-0'], ['1907.06611-1-37-2', '1907.06611-2-37-2'], ['1907.06611-1-98-0', '1907.06611-2-99-0'], ['1907.06611-1-98-1', '1907.06611-2-99-1'], ['1907.06611-1-98-2', '1907.06611-2-99-2'], ['1907.06611-1-27-3', '1907.06611-2-27-3'], ['1907.06611-1-27-5', '1907.06611-2-27-5'], ['1907.06611-1-92-0', '1907.06611-2-93-0'], ['1907.06611-1-92-1', '1907.06611-2-93-1'], ['1907.06611-1-92-3', '1907.06611-2-93-3'], ['1907.06611-1-34-1', '1907.06611-2-34-1'], ['1907.06611-1-41-2', '1907.06611-2-41-2'], ['1907.06611-1-106-0', '1907.06611-2-107-0'], ['1907.06611-1-106-1', '1907.06611-2-107-1'], ['1907.06611-1-55-0', '1907.06611-2-55-0'], ['1907.06611-1-55-1', '1907.06611-2-55-1'], ['1907.06611-1-42-0', '1907.06611-2-42-0'], ['1907.06611-1-42-1', '1907.06611-2-42-1'], ['1907.06611-1-42-2', '1907.06611-2-42-2'], ['1907.06611-1-42-3', '1907.06611-2-42-3'], ['1907.06611-1-51-0', '1907.06611-2-51-0'], ['1907.06611-1-51-1', '1907.06611-2-51-1'], ['1907.06611-1-51-2', '1907.06611-2-51-2'], ['1907.06611-1-87-0', '1907.06611-2-88-0'], ['1907.06611-1-87-3', '1907.06611-2-88-3'], ['1907.06611-1-56-0', '1907.06611-2-56-0'], ['1907.06611-1-56-1', '1907.06611-2-56-1'], ['1907.06611-1-56-2', '1907.06611-2-56-2'], ['1907.06611-1-64-0', '1907.06611-2-64-0'], ['1907.06611-1-64-1', '1907.06611-2-64-1'], ['1907.06611-1-6-1', '1907.06611-2-6-1'], ['1907.06611-1-6-2', '1907.06611-2-6-2'], ['1907.06611-1-6-3', '1907.06611-2-6-3'], ['1907.06611-1-103-1', '1907.06611-2-104-1'], ['1907.06611-1-103-2', '1907.06611-2-104-2'], ['1907.06611-1-103-3', '1907.06611-2-104-3'], ['1907.06611-1-83-0', '1907.06611-2-84-0'], ['1907.06611-1-71-0', '1907.06611-2-71-0'], ['1907.06611-1-71-1', '1907.06611-2-71-1'], ['1907.06611-1-71-2', '1907.06611-2-71-2'], ['1907.06611-1-21-0', '1907.06611-2-21-0'], ['1907.06611-1-21-1', '1907.06611-2-21-1'], ['1907.06611-1-21-4', '1907.06611-2-21-4'], ['1907.06611-1-21-7', '1907.06611-2-21-7'], ['1907.06611-1-21-8', '1907.06611-2-21-8'], ['1907.06611-1-105-0', '1907.06611-2-106-0'], ['1907.06611-1-105-1', '1907.06611-2-106-1'], ['1907.06611-1-105-2', '1907.06611-2-106-2'], ['1907.06611-1-8-0', '1907.06611-2-8-0'], ['1907.06611-1-8-1', '1907.06611-2-8-1'], ['1907.06611-1-57-0', '1907.06611-2-57-0'], ['1907.06611-1-57-1', '1907.06611-2-57-1'], ['1907.06611-1-57-2', '1907.06611-2-57-2'], ['1907.06611-1-57-3', '1907.06611-2-57-3'], ['1907.06611-1-5-0', '1907.06611-2-5-0'], ['1907.06611-1-5-1', '1907.06611-2-5-1'], ['1907.06611-1-5-2', '1907.06611-2-5-2'], ['1907.06611-1-5-3', '1907.06611-2-5-3'], ['1907.06611-1-5-6', '1907.06611-2-5-6'], ['1907.06611-1-5-7', '1907.06611-2-5-7'], ['1907.06611-1-13-0', '1907.06611-2-13-0'], ['1907.06611-1-13-1', '1907.06611-2-13-1'], ['1907.06611-1-13-2', '1907.06611-2-13-2'], ['1907.06611-1-13-3', '1907.06611-2-13-3'], ['1907.06611-1-39-0', '1907.06611-2-39-0'], ['1907.06611-1-25-1', '1907.06611-2-25-1'], ['1907.06611-1-3-0', '1907.06611-2-3-0'], ['1907.06611-1-3-2', '1907.06611-2-3-2'], ['1907.06611-1-67-0', '1907.06611-2-67-0'], ['1907.06611-1-67-1', '1907.06611-2-67-1'], ['1907.06611-1-89-0', '1907.06611-2-90-0'], ['1907.06611-1-89-1', '1907.06611-2-90-1'], ['1907.06611-1-89-2', '1907.06611-2-90-2'], ['1907.06611-1-12-0', '1907.06611-2-12-0'], ['1907.06611-1-14-0', '1907.06611-2-14-0'], ['1907.06611-1-14-1', '1907.06611-2-14-1'], ['1907.06611-1-14-3', '1907.06611-2-14-3'], ['1907.06611-1-14-4', '1907.06611-2-14-4'], ['1907.06611-1-36-1', '1907.06611-2-36-1'], ['1907.06611-1-36-3', '1907.06611-2-36-3'], ['1907.06611-1-36-5', '1907.06611-2-36-5'], ['1907.06611-1-36-6', '1907.06611-2-36-6'], ['1907.06611-1-69-1', '1907.06611-2-69-1'], ['1907.06611-1-69-2', '1907.06611-2-69-2'], ['1907.06611-1-69-3', '1907.06611-2-69-3'], ['1907.06611-1-95-0', '1907.06611-2-96-0'], ['1907.06611-1-95-1', '1907.06611-2-96-1'], ['1907.06611-1-95-2', '1907.06611-2-96-2'], ['1907.06611-1-95-4', '1907.06611-2-96-4'], ['1907.06611-1-45-2', '1907.06611-2-45-2'], ['1907.06611-1-52-0', '1907.06611-2-52-0'], ['1907.06611-1-52-1', '1907.06611-2-52-1'], ['1907.06611-1-52-3', '1907.06611-2-52-3'], ['1907.06611-1-97-2', '1907.06611-2-98-2'], ['1907.06611-1-88-0', '1907.06611-2-89-0'], ['1907.06611-1-88-1', '1907.06611-2-89-1'], ['1907.06611-1-88-2', '1907.06611-2-89-2'], ['1907.06611-1-88-3', '1907.06611-2-89-3'], ['1907.06611-1-4-1', '1907.06611-2-4-1'], ['1907.06611-1-4-3', '1907.06611-2-4-3'], ['1907.06611-1-100-0', '1907.06611-2-101-0'], ['1907.06611-1-48-0', '1907.06611-2-48-0'], ['1907.06611-1-15-0', '1907.06611-2-15-0'], ['1907.06611-1-15-1', '1907.06611-2-15-1'], ['1907.06611-1-15-2', '1907.06611-2-15-2'], ['1907.06611-1-15-3', '1907.06611-2-15-3'], ['1907.06611-1-15-4', '1907.06611-2-15-4'], ['1907.06611-1-77-0', '1907.06611-2-77-0'], ['1907.06611-1-77-1', '1907.06611-2-77-1'], ['1907.06611-1-46-0', '1907.06611-2-46-0'], ['1907.06611-1-46-1', '1907.06611-2-46-1'], ['1907.06611-1-46-2', '1907.06611-2-46-2'], ['1907.06611-1-46-3', '1907.06611-2-46-3'], ['1907.06611-1-46-4', '1907.06611-2-46-4'], ['1907.06611-1-46-5', '1907.06611-2-46-5'], ['1907.06611-1-76-0', '1907.06611-2-76-0'], ['1907.06611-1-76-1', '1907.06611-2-76-1'], ['1907.06611-1-76-2', '1907.06611-2-76-2'], ['1907.06611-1-76-3', '1907.06611-2-76-3'], ['1907.06611-1-81-1', '1907.06611-2-82-1'], ['1907.06611-1-81-2', '1907.06611-2-82-2'], ['1907.06611-1-81-3', '1907.06611-2-82-3'], ['1907.06611-1-23-0', '1907.06611-2-23-0'], ['1907.06611-1-23-1', '1907.06611-2-23-1'], ['1907.06611-1-23-3', '1907.06611-2-23-3'], ['1907.06611-1-23-4', '1907.06611-2-23-4'], ['1907.06611-1-23-5', '1907.06611-2-23-5'], ['1907.06611-1-23-6', '1907.06611-2-23-6'], ['1907.06611-1-63-1', '1907.06611-2-63-1'], ['1907.06611-1-63-2', '1907.06611-2-63-2'], ['1907.06611-1-50-0', '1907.06611-2-50-0'], ['1907.06611-1-50-1', '1907.06611-2-50-1'], ['1907.06611-1-50-2', '1907.06611-2-50-2'], ['1907.06611-1-50-3', '1907.06611-2-50-3'], ['1907.06611-1-50-4', '1907.06611-2-50-4'], ['1907.06611-1-50-5', '1907.06611-2-50-5'], ['1907.06611-1-10-0', '1907.06611-2-10-0'], ['1907.06611-1-10-1', '1907.06611-2-10-1'], ['1907.06611-1-10-2', '1907.06611-2-10-2'], ['1907.06611-1-10-4', '1907.06611-2-10-4'], ['1907.06611-1-10-5', '1907.06611-2-10-5'], ['1907.06611-1-10-6', '1907.06611-2-10-6'], ['1907.06611-1-10-7', '1907.06611-2-10-7'], ['1907.06611-1-35-0', '1907.06611-2-35-0'], ['1907.06611-1-35-1', '1907.06611-2-35-1'], ['1907.06611-1-35-2', '1907.06611-2-35-2'], ['1907.06611-1-35-3', '1907.06611-2-35-3'], ['1907.06611-1-2-0', '1907.06611-2-2-0'], ['1907.06611-1-2-1', '1907.06611-2-2-1'], ['1907.06611-1-2-2', '1907.06611-2-2-2'], ['1907.06611-1-2-3', '1907.06611-2-2-3'], ['1907.06611-1-2-5', '1907.06611-2-2-5'], ['1907.06611-1-2-6', '1907.06611-2-2-6'], ['1907.06611-1-2-7', '1907.06611-2-2-7'], ['1907.06611-1-0-0', '1907.06611-2-0-0'], ['1907.06611-1-0-1', '1907.06611-2-0-1'], ['1907.06611-1-0-2', '1907.06611-2-0-2'], ['1907.06611-1-0-3', '1907.06611-2-0-3'], ['1907.06611-1-0-4', '1907.06611-2-0-4'], ['1907.06611-1-0-5', '1907.06611-2-0-5'], ['1907.06611-1-0-6', '1907.06611-2-0-6'], ['1907.06611-1-75-0', '1907.06611-2-75-0'], ['1907.06611-1-75-1', '1907.06611-2-75-1'], ['1907.06611-1-75-2', '1907.06611-2-75-2'], ['1907.06611-1-75-3', '1907.06611-2-75-3'], ['1907.06611-1-75-5', '1907.06611-2-75-5'], ['1907.06611-1-54-0', '1907.06611-2-54-0'], ['1907.06611-1-54-1', '1907.06611-2-54-1'], ['1907.06611-1-54-2', '1907.06611-2-54-2'], ['1907.06611-1-54-3', '1907.06611-2-54-3'], ['1907.06611-1-54-4', '1907.06611-2-54-4'], ['1907.06611-1-7-0', '1907.06611-2-7-0'], ['1907.06611-1-7-1', '1907.06611-2-7-1'], ['1907.06611-1-7-2', '1907.06611-2-7-2'], ['1907.06611-1-7-4', '1907.06611-2-7-4'], ['1907.06611-1-7-5', '1907.06611-2-7-5'], ['1907.06611-1-7-6', '1907.06611-2-7-6'], ['1907.06611-1-66-0', '1907.06611-2-66-0'], ['1907.06611-1-66-1', '1907.06611-2-66-1'], ['1907.06611-1-66-2', '1907.06611-2-66-2'], ['1907.06611-1-66-3', '1907.06611-2-66-3'], ['1907.06611-1-66-4', '1907.06611-2-66-4'], ['1907.06611-1-66-5', '1907.06611-2-66-5'], ['1907.06611-1-66-6', '1907.06611-2-66-6'], ['1907.06611-1-94-0', '1907.06611-2-95-0'], ['1907.06611-1-94-1', '1907.06611-2-95-1'], ['1907.06611-1-58-0', '1907.06611-2-58-0'], ['1907.06611-1-58-1', '1907.06611-2-58-1'], ['1907.06611-1-31-1', '1907.06611-2-31-1'], ['1907.06611-1-31-2', '1907.06611-2-31-2'], ['1907.06611-1-102-0', '1907.06611-2-103-0'], ['1907.06611-1-102-2', '1907.06611-2-103-2'], ['1907.06611-1-102-3', '1907.06611-2-103-3'], ['1907.06611-1-74-0', '1907.06611-2-74-0'], ['1907.06611-1-74-2', '1907.06611-2-74-2'], ['1907.06611-1-74-3', '1907.06611-2-74-3'], ['1907.06611-1-74-4', '1907.06611-2-74-4'], ['1907.06611-1-65-1', '1907.06611-2-65-1'], ['1907.06611-1-65-2', '1907.06611-2-65-2'], ['1907.06611-1-65-3', '1907.06611-2-65-3'], ['1907.06611-1-101-0', '1907.06611-2-102-0'], ['1907.06611-1-101-1', '1907.06611-2-102-1'], ['1907.06611-1-101-2', '1907.06611-2-102-2'], ['1907.06611-1-70-0', '1907.06611-2-70-0'], ['1907.06611-1-70-1', '1907.06611-2-70-1'], ['1907.06611-1-70-2', '1907.06611-2-70-2'], ['1907.06611-1-70-4', '1907.06611-2-70-4'], ['1907.06611-1-70-5', '1907.06611-2-70-5'], ['1907.06611-1-70-6', '1907.06611-2-70-6'], ['1907.06611-1-70-7', '1907.06611-2-70-7'], ['1907.06611-1-17-0', '1907.06611-2-17-0'], ['1907.06611-1-17-1', '1907.06611-2-17-1'], ['1907.06611-1-17-2', '1907.06611-2-17-2'], ['1907.06611-1-17-3', '1907.06611-2-17-3'], ['1907.06611-1-17-5', '1907.06611-2-17-5'], ['1907.06611-1-73-0', '1907.06611-2-73-0'], ['1907.06611-1-73-1', '1907.06611-2-73-1'], ['1907.06611-1-73-2', '1907.06611-2-73-2'], ['1907.06611-1-85-0', '1907.06611-2-86-0'], ['1907.06611-1-85-1', '1907.06611-2-86-1'], ['1907.06611-1-85-2', '1907.06611-2-86-2'], ['1907.06611-1-85-3', '1907.06611-2-86-3'], ['1907.06611-1-96-0', '1907.06611-2-97-0'], ['1907.06611-1-96-3', '1907.06611-2-97-3'], ['1907.06611-1-79-0', '1907.06611-2-79-0'], ['1907.06611-1-79-2', '1907.06611-2-79-2'], ['1907.06611-1-79-4', '1907.06611-2-79-4'], ['1907.06611-1-79-7', '1907.06611-2-79-7']]

[['1907.06611-1-44-0', '1907.06611-2-44-0'], ['1907.06611-1-91-1', '1907.06611-2-92-1'], ['1907.06611-1-91-6', '1907.06611-2-92-6'], ['1907.06611-1-26-2', '1907.06611-2-26-2'], ['1907.06611-1-26-4', '1907.06611-2-26-4'], ['1907.06611-1-26-5', '1907.06611-2-26-5'], ['1907.06611-1-80-1', '1907.06611-2-81-1'], ['1907.06611-1-62-1', '1907.06611-2-62-1'], ['1907.06611-1-78-4', '1907.06611-2-78-4'], ['1907.06611-1-86-4', '1907.06611-2-87-4'], ['1907.06611-1-86-5', '1907.06611-2-87-5'], ['1907.06611-1-29-0', '1907.06611-2-29-0'], ['1907.06611-1-28-0', '1907.06611-2-28-0'], ['1907.06611-1-28-4', '1907.06611-2-28-5'], ['1907.06611-1-37-1', '1907.06611-2-37-1'], ['1907.06611-1-98-3', '1907.06611-2-99-3'], ['1907.06611-1-98-4', '1907.06611-2-99-4'], ['1907.06611-1-27-0', '1907.06611-2-27-0'], ['1907.06611-1-27-1', '1907.06611-2-27-1'], ['1907.06611-1-27-2', '1907.06611-2-27-2'], ['1907.06611-1-27-4', '1907.06611-2-27-4'], ['1907.06611-1-92-2', '1907.06611-2-93-2'], ['1907.06611-1-92-4', '1907.06611-2-93-4'], ['1907.06611-1-34-0', '1907.06611-2-34-0'], ['1907.06611-1-41-0', '1907.06611-2-41-0'], ['1907.06611-1-41-1', '1907.06611-2-41-1'], ['1907.06611-1-106-2', '1907.06611-2-107-2'], ['1907.06611-1-87-1', '1907.06611-2-88-1'], ['1907.06611-1-87-2', '1907.06611-2-88-2'], ['1907.06611-1-64-2', '1907.06611-2-64-2'], ['1907.06611-1-6-0', '1907.06611-2-6-0'], ['1907.06611-1-103-0', '1907.06611-2-104-0'], ['1907.06611-1-21-3', '1907.06611-2-21-3'], ['1907.06611-1-21-5', '1907.06611-2-21-5'], ['1907.06611-1-21-6', '1907.06611-2-21-6'], ['1907.06611-1-5-5', '1907.06611-2-5-5'], ['1907.06611-1-5-8', '1907.06611-2-5-8'], ['1907.06611-1-39-2', '1907.06611-2-39-2'], ['1907.06611-1-25-0', '1907.06611-2-25-0'], ['1907.06611-1-3-1', '1907.06611-2-3-1'], ['1907.06611-1-3-3', '1907.06611-2-3-3'], ['1907.06611-1-67-2', '1907.06611-2-67-2'], ['1907.06611-1-14-2', '1907.06611-2-14-2'], ['1907.06611-1-36-0', '1907.06611-2-36-0'], ['1907.06611-1-36-2', '1907.06611-2-36-2'], ['1907.06611-1-36-4', '1907.06611-2-36-4'], ['1907.06611-1-69-0', '1907.06611-2-69-0'], ['1907.06611-1-95-3', '1907.06611-2-96-3'], ['1907.06611-1-45-0', '1907.06611-2-45-0'], ['1907.06611-1-45-3', '1907.06611-2-45-3'], ['1907.06611-1-52-2', '1907.06611-2-52-2'], ['1907.06611-1-97-0', '1907.06611-2-98-0'], ['1907.06611-1-97-1', '1907.06611-2-98-1'], ['1907.06611-1-4-0', '1907.06611-2-4-0'], ['1907.06611-1-4-2', '1907.06611-2-4-2'], ['1907.06611-1-4-4', '1907.06611-2-4-4'], ['1907.06611-1-60-1', '1907.06611-2-60-1'], ['1907.06611-1-43-0', '1907.06611-2-43-0'], ['1907.06611-1-81-0', '1907.06611-2-82-0'], ['1907.06611-1-49-0', '1907.06611-2-49-0'], ['1907.06611-1-49-1', '1907.06611-2-49-2'], ['1907.06611-1-49-2', '1907.06611-2-49-1'], ['1907.06611-1-23-2', '1907.06611-2-23-2'], ['1907.06611-1-23-8', '1907.06611-2-23-7'], ['1907.06611-1-63-0', '1907.06611-2-63-0'], ['1907.06611-1-63-3', '1907.06611-2-63-3'], ['1907.06611-1-10-3', '1907.06611-2-10-3'], ['1907.06611-1-2-4', '1907.06611-2-2-4'], ['1907.06611-1-75-4', '1907.06611-2-75-4'], ['1907.06611-1-7-3', '1907.06611-2-7-3'], ['1907.06611-1-58-2', '1907.06611-2-58-2'], ['1907.06611-1-31-0', '1907.06611-2-31-0'], ['1907.06611-1-31-3', '1907.06611-2-31-3'], ['1907.06611-1-31-4', '1907.06611-2-31-4'], ['1907.06611-1-102-1', '1907.06611-2-103-1'], ['1907.06611-1-102-4', '1907.06611-2-103-4'], ['1907.06611-1-74-1', '1907.06611-2-74-1'], ['1907.06611-1-65-0', '1907.06611-2-65-0'], ['1907.06611-1-70-8', '1907.06611-2-70-8'], ['1907.06611-1-32-0', '1907.06611-2-32-0'], ['1907.06611-1-32-2', '1907.06611-2-32-2'], ['1907.06611-1-17-4', '1907.06611-2-17-4'], ['1907.06611-1-85-4', '1907.06611-2-86-4'], ['1907.06611-1-96-1', '1907.06611-2-97-1'], ['1907.06611-1-96-2', '1907.06611-2-97-2'], ['1907.06611-1-96-5', '1907.06611-2-97-5'], ['1907.06611-1-96-6', '1907.06611-2-97-6'], ['1907.06611-1-96-7', '1907.06611-2-97-7'], ['1907.06611-1-79-1', '1907.06611-2-79-1'], ['1907.06611-1-79-3', '1907.06611-2-79-3'], ['1907.06611-1-79-9', '1907.06611-2-80-2']]

[]

[['1907.06611-1-44-1', '1907.06611-2-44-1'], ['1907.06611-1-78-3', '1907.06611-2-78-3'], ['1907.06611-1-28-1', '1907.06611-2-28-1'], ['1907.06611-1-28-6', '1907.06611-2-28-4'], ['1907.06611-1-8-2', '1907.06611-2-8-2'], ['1907.06611-1-5-4', '1907.06611-2-5-4'], ['1907.06611-1-12-1', '1907.06611-2-12-1'], ['1907.06611-1-95-5', '1907.06611-2-96-5'], ['1907.06611-1-45-1', '1907.06611-2-45-1'], ['1907.06611-1-48-1', '1907.06611-2-48-1'], ['1907.06611-1-60-0', '1907.06611-2-60-0'], ['1907.06611-1-60-2', '1907.06611-2-60-2'], ['1907.06611-1-43-1', '1907.06611-2-43-1'], ['1907.06611-1-81-4', '1907.06611-2-82-4'], ['1907.06611-1-49-3', '1907.06611-2-49-3'], ['1907.06611-1-70-3', '1907.06611-2-70-3'], ['1907.06611-1-32-1', '1907.06611-2-32-1'], ['1907.06611-1-96-4', '1907.06611-2-97-4'], ['1907.06611-1-79-5', '1907.06611-2-79-5'], ['1907.06611-1-79-8', '1907.06611-2-79-5']]

[]

['1907.06611-1-21-2', '1907.06611-1-39-1', '1907.06611-2-21-2', '1907.06611-2-39-1']

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

https://arxiv.org/abs/1907.06611

0705.0163

{'0705.0163-1-0-0': 'Several dark energy experiments are available from a single large-area imaging survey, and may be combined to improve cosmological parameter constraints and/or test inherent systematics.', '0705.0163-1-0-1': 'Two promising experiments are cosmic shear power spectra and counts of galaxy clusters.', '0705.0163-1-0-2': 'However the two experiments probe the same cosmic mass density field in large-scale structure, therefore the combination may be less powerful than first thought.', '0705.0163-1-1-0': 'In this paper we develop a formulation to compute the cross-covariance between the lensing power spectra and the cluster counts based on the halo model approach.', '0705.0163-1-1-1': 'The cross-covariance arises from the three-point correlations of the underlying mass density field for which a cold dark matter structure formation model can provide specific predictions via a plausible modeling of non-linear gravitation clustering.', '0705.0163-1-1-2': 'Fully taking into account the cross-covariance as well as non-Gaussian errors on the lensing power spectrum covariance, we find a significant cross-correlation between the lensing power spectrum signals at multipoles [MATH] and the cluster counts containing halos with masses [MATH] > [MATH].', '0705.0163-1-1-3': 'Including the cross-covariance for the combined measurement degrades and in some cases improves the total signal-to-noise ratios up to [MATH] relative to when the two are independent.', '0705.0163-1-1-4': 'For cosmological parameter determination, the cross-covariance has a smaller effect as a result of working in a multi-dimensional parameter space, implying that the two observables can be considered independent to a good approximation.', '0705.0163-1-1-5': 'We also found that cluster count experiments using lensing selected mass peaks are more complementary to lensing tomography than mass-selected cluster counts.', '0705.0163-1-1-6': 'Using lensing selected clusters with a realistic usable detection threshold ([MATH] for a ground-based survey), the uncertainty on each dark energy parameter is roughly halved by combining cluster counts and lensing power spectra, relative to using power spectra alone.', '0705.0163-1-2-0': '# Introduction', '0705.0163-1-3-0': 'In recent years great observational progress has been made in measuring the constituents of the universe (e.g. [CITATION]).', '0705.0163-1-3-1': 'It appears that the universe is currently dominated by an unexpected component that is causing the universe to accelerate in its expansion.', '0705.0163-1-3-2': 'This component is dubbed "dark energy".', '0705.0163-1-3-3': 'Understanding the nature of dark energy is one of most fundamental questions that remain unresolved with the current cosmological data sets (e.g. [CITATION]).', '0705.0163-1-3-4': 'This is now the focus of several planned future surveys [CITATION].', '0705.0163-1-4-0': "Whether the accelerating expansion is as a consequence of the cosmological constant, a new fluid or a modification to Einstein's gravity, these future surveys will provide key information.", '0705.0163-1-4-1': 'In addition they will provide a wealth of further cosmological information, such as constraints on the neutrino mass and the spectrum of primordial perturbations generated in the early universe (e.g. [CITATION]).', '0705.0163-1-5-0': 'Combining several techniques accessible from different cosmological observables is often a powerful way to improve constraints on cosmology.', '0705.0163-1-5-1': 'However, care must be taken if the observables are not completely independent.', '0705.0163-1-5-2': 'Two of the most promising methods for constraining the dark energy are galaxy cluster counts and cosmic shear (e.g. [CITATION]).', '0705.0163-1-6-0': 'Clusters of galaxies contain galaxies, hot gas and dark matter in ratio approximately 1:10:100.', '0705.0163-1-6-1': 'They are the largest gravitationally bound objects in the universe and the number of clusters of galaxies has long been recognized as a powerful probe of cosmology [CITATION].', '0705.0163-1-6-2': 'Counting clusters of galaxies as a function of redshift allows a combination of structure growth and geometrical information to be extracted, thus potentially allowing constraints on the nature of dark energy [CITATION].', '0705.0163-1-6-3': 'If cluster masses can be measured accurately then the shape of the mass function also helps to break degeneracies [CITATION].', '0705.0163-1-6-4': 'The distribution of clusters on the sky (e.g. two-point correlation function) carries additional information on dark energy [CITATION].', '0705.0163-1-7-0': 'The bending of light by mass, gravitational lensing, causes images of distant galaxies to be distorted.', '0705.0163-1-7-1': 'These sheared source galaxies are mostly too weakly distorted for us to measure the effect in single galaxies, but require surveys containing millions of galaxies to detect the signal.', '0705.0163-1-7-2': 'This cosmic shear signal has been observed [CITATION] and used to constrain cosmology (most recently [CITATION]).', '0705.0163-1-7-3': 'By using redshift information of source galaxies the evolution of the dark matter distribution with redshift can be inferred.', '0705.0163-1-7-4': 'Hence, measuring the cosmic shear two-point function as a function of redshift and separation between pairs of galaxies can be used to constrain the geometry of the universe as well as the growth of mass clustering.', '0705.0163-1-7-5': 'This method has emerged as one of the most promising to obtain precise constraints on the nature of dark energy if systematics are well under control [CITATION].', '0705.0163-1-8-0': 'Future optical imaging surveys suitable for cosmic shear analysis will also allow the identification of clusters of galaxies.', '0705.0163-1-8-1': 'This could be done either using the colors of the cluster members (e.g. [CITATION]) or using peaks in the gravitational lensing shear field (e.g. [CITATION]).', '0705.0163-1-8-2': "In addition cluster surveys in other wavebands will overlap with the cosmic shear surveys allowing detection using X-rays and the Sunyaev-Zel'dovich effect.", '0705.0163-1-9-0': 'Clusters of galaxies produce a large gravitational lensing effect on distant galaxies, therefore cluster counts and cosmic shear will not be strictly statistically independent.', '0705.0163-1-9-1': 'The volume surveyed is finite and therefore the number of clusters observed will not be exactly equal to the average over all universe realizations.', '0705.0163-1-9-2': 'If the number of clusters happens to be higher for a given survey region, then the cosmic shear signal is also likely to be higher.', '0705.0163-1-9-3': 'Although the volumes will be large, and thus the deviation is small, this may amount to a significant uncertainty in the dark energy parameters as obtained by cluster counts, and dominates the non-Gaussian errors on the cosmic shear [CITATION].', '0705.0163-1-10-0': 'One aspect of this cross-correlation was discussed in [CITATION] and found to be negligible.', '0705.0163-1-10-1': 'However, here we make a full treatment of this effect using the halo model for non-linear structure formation, and quantify the resulting change in joint constraints on the dark energy parameters.', '0705.0163-1-11-0': 'The structure of our paper is as follows.', '0705.0163-1-11-1': 'In [REF] we describe how our observables, cluster number counts and lensing power spectra, can be expressed in terms of the background cosmological model and the density perturbations.', '0705.0163-1-11-2': 'In [REF], we describe a methodology to compute covariances of the cluster counts and the lensing power spectra, and the cross-covariance between the two observables.', '0705.0163-1-11-3': 'The detailed derivations of the covariances are presented in Appendix.', '0705.0163-1-11-4': 'In [REF] we first study the total signal-to-noise ratios expected for a joint experiment of the cluster counts and the lensing power spectrum fully including the cross-covariance predicted from the [MATH]CDM cosmologies.', '0705.0163-1-11-5': 'We then present forecasts for cosmological parameter determination for the joint experiment, with particular focus on forecasts for the dark energy parameter constraints.', '0705.0163-1-11-6': 'Finally, we present conclusions and discussion in [REF].', '0705.0163-1-12-0': '# Preliminaries', '0705.0163-1-13-0': '## A CDM model', '0705.0163-1-14-0': 'We work in the context of spatially flat cold dark matter models for structure formation.', '0705.0163-1-14-1': 'The expansion history of the universe is given by the scale factor [MATH] in a homogeneous and isotropic universe (e.g., see [CITATION]).', '0705.0163-1-14-2': 'We describe the Universe in terms of the matter density [MATH] (the cold dark matter plus the baryons) and dark energy density [MATH] at present (in units of the critical density [MATH], where [MATH] is the Hubble parameter at present).', '0705.0163-1-14-3': 'In general the expansion rate, the Hubble parameter, is given by [EQUATION] where we have employed the normalization [MATH] today and [MATH] specifies the equation of state for dark energy as [MATH].', '0705.0163-1-14-4': 'Note that [MATH] corresponds to a cosmological constant.', '0705.0163-1-14-5': 'The comoving distance [MATH] from an observer at [MATH] to a source at [MATH] is expressed in terms of the Hubble parameter as [EQUATION]', '0705.0163-1-14-6': 'This gives the distance-redshift relation [MATH] via the relation [MATH].', '0705.0163-1-15-0': 'Next we need the redshift growth of density perturbations.', '0705.0163-1-15-1': 'In linear theory after matter-radiation equality, all Fourier modes of the mass density perturbation, [MATH]x[MATH]x[MATH], grow at the same rate, the growth rate (e.g. see Eq. 10 in [CITATION] for details).', '0705.0163-1-16-0': '## Number counts of galaxy clusters', '0705.0163-1-17-0': 'The galaxy cluster observables we will consider in this paper are the number counts drawn from a given survey region.', '0705.0163-1-17-1': "Clusters can be found via their notable observational properties such as gravitational lensing, member galaxies, [MATH]-ray emission and the Sunyaev-Zel'dovich effect.", '0705.0163-1-17-2': 'For number counts we simply treat clusters as points; in other words, we do not care about the distribution of mass within a cluster.', '0705.0163-1-17-3': 'Hence, the number density field of clusters at redshift [MATH] can be expressed as [EQUATION] where [MATH]x[MATH] is the three-dimensional Dirac delta function.', '0705.0163-1-17-4': 'The summation runs over halos (the subscript [MATH] stands for the [MATH]-th halo), and [MATH] denotes the selection function that discriminates the halos used for the cluster number counts statistic from other halos.', '0705.0163-1-18-0': 'In this paper, we will consider the following two toy models for the selection function, to develop intuition for the importance of cross-correlation between cluster counts and the lensing power spectrum and to make a comparison between cosmological parameter estimations derived from different cluster samples.', '0705.0163-1-19-0': 'A mass-limited cluster sample - The first toy model we will consider is a mass-limited cluster sample.', '0705.0163-1-19-1': 'For this model, we include all halos with masses above a given mass threshold: [EQUATION]', '0705.0163-1-19-2': "To a zero-th order approximation, the mass-limited selection may mimic a cluster sample derived from a flux-limited survey of clusters via the Sunyaev-Zel'dovich effect, as this effect is free of the surface brightness dimming effect (e.g. see ).", '0705.0163-1-20-0': 'A lensing-based cluster sample - A lensing measurement allows one to make a reconstruction of the two-dimensional mass distribution projected along the line of sight [CITATION].', '0705.0163-1-20-1': 'A high peak in the mass map provides a strong candidate for a massive cluster (see [CITATION] for an implementation of this method to actual data).', '0705.0163-1-20-2': 'To be more explicit, one can define height or significance for each peak in the reconstructed mass map using the effective signal-to-noise ratio (see [CITATION] for details): [EQUATION]', '0705.0163-1-20-3': 'Here [MATH] is the convergence amplitude due to a given cluster at redshift [MATH] and with mass [MATH], and [MATH] is the rms intrinsic ellipticity noise of the background galaxies.', '0705.0163-1-20-4': 'Note that we assume an NFW profile [CITATION] with profile parameters modeled in [CITATION], and consider the convergence field smoothed with a Gaussian smoothing function of angular scale [MATH], which determines the noise [MATH] by the number of background galaxies contained within the smoothing aperture.', '0705.0163-1-20-5': 'To compute the [MATH] for a cluster at redshift [MATH], we take into account the remaining fraction of background galaxies behind the cluster for a given redshift distribution of whole galaxy population (see [REF]).', '0705.0163-1-20-6': 'This accounts for the variation of mean redshift and number density of the background galaxies with cluster redshift, which also changes the signal and the intrinsic noise.', '0705.0163-1-21-0': 'From the reconstructed mass map, a cluster sample may be constructed by counting mass peaks with heights above a given threshold, [MATH]: the selection function is given by [EQUATION]', '0705.0163-1-21-1': 'As carefully investigated in [CITATION], the minimum mass of clusters detectable with a given threshold varies with cluster redshift; clusters at medium redshift between observer and a typical source redshift are most easily detectable, while only more massive clusters can be detected at redshifts smaller and greater than the medium redshift, as discussed below.', '0705.0163-1-22-0': 'We will employ the halo model to quantify the statistical properties of cluster observables.', '0705.0163-1-22-1': 'In the halo model approach, we assume that all the matter is in halos.', '0705.0163-1-22-2': 'Following the formulation developed in [CITATION] (also see [CITATION] and Appendix [REF]), the ensemble average of Eq. ([REF]) can be computed as [EQUATION] where [MATH] is the halo mass function corresponding to the redshift considered and we have used the ensemble average [MATH].', '0705.0163-1-22-3': 'Thus, as expected, the ensemble average of the cluster number density field is given by the integral of the halo mass function, which does not depend on the cluster distribution.', '0705.0163-1-22-4': 'For the halo mass function, we employ the fitting formula proposed by [CITATION], modified from the original Press-Schechter function [CITATION], so that the formula better matches the cosmological simulation results.', '0705.0163-1-22-5': 'Note that we use parameter values [MATH] and [MATH] in the formula following the discussion in [CITATION].', '0705.0163-1-22-6': 'We assume that the mass function can be applied to dark energy cosmologies by replacing the growth rate appearing in the formula with that for a dark energy model [CITATION].', '0705.0163-1-23-0': 'A more useful quantity often considered in the literature is the total number counts of clusters available from a given survey, which is obtained by integrating the three-dimensional number density field over a range of redshifts surveyed.', '0705.0163-1-23-1': 'Cluster redshifts are rather easily available even from a multicolor imaging survey alone because their central bright galaxies, or red sequence galaxies, have secure photometric redshift estimates.', '0705.0163-1-23-2': 'Having these facts in mind we will use as our observable the angular number density averaged over a survey area and divided into redshift bins: [EQUATION] where [MATH] is the window function of the survey defined so that it is normalized as [MATH], and the comoving volume per unit comoving distance and unit steradian is given by [MATH] for a flat universe.', '0705.0163-1-23-3': 'The subscript in the round bracket, [MATH], stands for the [MATH]-th redshift bin for the cluster number counts.', '0705.0163-1-23-4': 'In the following, we will simply consider the sharp redshift selection function [EQUATION]', '0705.0163-1-23-5': 'Note that the redshift [MATH] appearing in the argument of [MATH] is related to the comoving distance [MATH] via the relation [MATH].', '0705.0163-1-24-0': 'Using the halo model, the expectation value of the angular number density can be computed from the ensemble average of Eq. ([REF]) as [EQUATION]', '0705.0163-1-24-1': 'Thus, the expectation value again does not depend on the cluster distribution.', '0705.0163-1-24-2': 'The sensitivity of the number density to dark energy arises from the comoving volume and the mass function [MATH] [CITATION] (and, more generically, comes through the conversion relations of observables to halo masses as discussed in [CITATION]).', '0705.0163-1-25-0': 'Fig. [REF] shows the average number density of halos with masses greater than a given threshold, per unit square arcminute and per unit redshift interval assuming the fiducial model defined in [REF].', '0705.0163-1-25-1': 'Increasing the dark energy equation of state from our fiducial model [MATH] to [MATH] decreases the number density, because the change decreases both the comoving volume [MATH] and the number density of cluster-scale halos, for a given CMB normalization of density perturbations.', '0705.0163-1-25-2': 'Comparing the results for mass thresholds [MATH] and [MATH] clarifies that a factor 5 increase in the mass threshold leads to a significant decrease in the number density, reflecting the mass sensitivity of the halo mass function in its exponential tail.', '0705.0163-1-26-0': 'In Fig. [REF] we present the number density for the lensing-based cluster sample in which clusters having a lensing signal greater than a given threshold are included in the sample as discussed around Eq. ([REF]).', '0705.0163-1-26-1': 'Note that to compute the results shown in this plot we considered the redshift distribution of galaxies described in [REF] and assumed an NFW mass profile to model the cluster lensing (see [CITATION]).', '0705.0163-1-26-2': 'Note that high detection thresholds such as [MATH] > [MATH] are necessary in order to make robust estimates for cluster counts, because contamination of false peaks due to intrinsic ellipticities or the projection effect are expected to be low for such high thresholds (see [CITATION] for the details).', '0705.0163-1-26-3': 'Comparing with the number density for a mass-selected sample shown by the dot-dashed curves (also see Fig. [REF]), one can roughly find which mass and redshift ranges of clusters are probed by the lensing-based cluster sample (see [CITATION] for more details).', '0705.0163-1-26-4': 'For example, the cluster sample with lensing signal [MATH] contains massive clusters with masses [MATH] > [MATH] over redshift ranges [MATH] < [MATH], while only even more massive clusters are included in the sample at the higher redshifts.', '0705.0163-1-26-5': 'This cluster sample has a narrower redshift coverage than the simple mass threshold; all the curves peak at a redshift [MATH].', '0705.0163-1-26-6': 'The peak redshift is mainly attributed to redshift dependence of the lensing efficiency for source galaxies of [MATH] in our redshift distribution.', '0705.0163-1-26-7': 'A change of [MATH] from [MATH] to [MATH] leads to a decrease in the number density, as seen in Fig. [REF].', '0705.0163-1-26-8': 'The effect comes partially arises as before from the decrease in comoving volume and the change in the halo mass function.', '0705.0163-1-26-9': 'Unlike the simple mass threshold case, there is now an additional contribution to the decrease in number density caused by the lower lensing efficiency and thus lower [MATH] for a cluster of a given mass and redshift.', '0705.0163-1-27-0': '## Lensing power spectrum with tomography', '0705.0163-1-28-0': 'Gravitational shear can be simply related to the lensing convergence: the weighted mass distribution integrated along the line of sight.', '0705.0163-1-28-1': 'Photometric redshift information on source galaxies allows us to subdivide galaxies into redshift bins.', '0705.0163-1-28-2': 'This allows more cosmological information to be extracted, which is referred to as lensing tomography (e.g., see [CITATION] for a thorough review, and see [CITATION] for the details of lensing tomography).', '0705.0163-1-29-0': 'In the context of cosmological gravitational lensing the convergence field with tomographic information is expressed as a weighted projection of the three-dimensional mass density fluctuation field: [EQUATION] where [MATH][MATH] is the angular position on the sky, [MATH] is the distance to the Hubble horizon, and [MATH] is the gravitational lensing weight function for source galaxies sitting into the [MATH]-th redshift bin (see Eq. (10) in [CITATION] for the definition).', '0705.0163-1-29-1': 'Note that, hereafter, quantities with subscripts in the round bracket such as [MATH] stands for those for the [MATH]-th redshift bin.', '0705.0163-1-29-2': 'To avoid confusion, throughout this paper we use [MATH] or [MATH] for the lensing power spectrum redshift bins, and [MATH] for the cluster count redshift bins.', '0705.0163-1-30-0': 'The lensing tomographic information allows us to extract redshift evolution of the lensing weight function as well as the growth rate of mass clustering.', '0705.0163-1-30-1': 'These are both sensitive to dark energy.', '0705.0163-1-30-2': 'For example, increasing the equation of state parameter [MATH] from [MATH] lowers [MATH] as well as suppressing the growth rate at lower redshifts.', '0705.0163-1-30-3': 'Therefore when the CMB normalization of density perturbations is employed, an increase in [MATH] decreases the lensing power spectrum due to both the lower [MATH] and the lower matter power spectrum amplitude.', '0705.0163-1-30-4': 'The sensitivity of lensing observables to the dark energy equation of state roughly arises equally from the two effects (e.g., see [CITATION]).', '0705.0163-1-31-0': 'The cosmic shear fields are measurable only in a statistical sense.', '0705.0163-1-31-1': 'The most conventional methods used in the literature are the shear two-point correlation function.', '0705.0163-1-31-2': 'The Fourier transformed counterpart is the shear power spectrum.', '0705.0163-1-31-3': 'The convergence power spectrum is identical to the shear power spectrum but is easier to work with as it is a scalar.', '0705.0163-1-31-4': 'Using the flat-sky approximation [CITATION], the angular power spectrum between the convergence fields of redshift bins [MATH] and [MATH] is found to be [EQUATION] where [MATH] is the three-dimensional mass power spectrum.', '0705.0163-1-31-5': 'For [MATH] > [MATH] the major contribution to [MATH] comes from non-linear clustering (e.g., see Fig. 2 in [CITATION]).', '0705.0163-1-31-6': 'We employ the fitting formula for the non-linear [MATH] proposed in Smith et al. [CITATION], assuming that it can be applied to dark energy cosmologies by replacing the growth rate used in the formula with that for a given dark energy model.', '0705.0163-1-31-7': 'We note in passing that the issue of accurate power spectra for general dark energy cosmologies still needs to be addressed carefully (see [CITATION] for related discussion).', '0705.0163-1-32-0': 'Fig. [REF] demonstrates how lensing of background galaxies by clusters contributes to the lensing power spectrum.', '0705.0163-1-32-1': 'Note here that we have employed the halo model developed in [CITATION] to compute the mass power spectrum, although we will use the Smith et al. fitting formula in most parts of this paper.', '0705.0163-1-32-2': 'Briefly, to compute the spectra based on the halo model approach, we need to model three ingredients: (i) the halo mass function (see also the description below Eq. [[REF]]); (ii) the profile for the mass distribution around a halo; and (iii) the halo bias parameter.', '0705.0163-1-32-3': 'All of these can be fully specified by halo masses within the framework of a CDM structure formation.', '0705.0163-1-33-0': 'It is clear that the convergence on scales [MATH] > [MATH] is significantly boosted by the existence of non-linear structures, halos.', '0705.0163-1-33-1': 'In this paper we are especially interested in using the lensing information inherent in angular scales [MATH] < [MATH] to constrain dark energy, and most of the power at scales [MATH] arises from massive halos with [MATH] > [MATH].', '0705.0163-1-33-2': 'The 1-halo term contribution is given by redshift-space integral of the halo mass function and halo profiles weighted with the lensing efficiency.', '0705.0163-1-33-3': 'The results imply that, if massive clusters with [MATH] > [MATH] happen to be less or more populated in a survey region, amplitudes of the observed lensing power spectrum from the survey are very likely to be smaller or greater than expected, respectively.', '0705.0163-1-33-4': 'Therefore, a cross-correlation between the lensing power spectrum and the cluster counts are intuitively expected, if both of the observables are measured from the same survey region.', '0705.0163-1-33-5': 'This is the main focus of this paper which we will carefully address in the following.', '0705.0163-1-34-0': 'In reality, the observed power spectrum is contaminated by the intrinsic ellipticity noise.', '0705.0163-1-34-1': 'Assuming that the intrinsic ellipticity distribution is uncorrelated between different galaxies, the observed power spectrum between redshift bins [MATH] and [MATH] can be expressed as [EQUATION] where [MATH] is the rms of intrinsic ellipticities per component, and [MATH] denotes the average number density of galaxies in the [MATH]-th redshift bin.', '0705.0163-1-34-2': 'The Kronecker delta function, [MATH], accounts for the fact that the cross-spectra of different redshift bins ([MATH]) are not affected by the shot noise contamination.', '0705.0163-1-34-3': "We will omit the superscript 'obs' when referring to [MATH] in the following for notational simplicity.", '0705.0163-1-35-0': '# Covariances of lensing power spectrum and cluster observables', '0705.0163-1-36-0': 'To estimate a realistic forecast for cosmological parameter constraints for a given survey we have to quantify sources of statistical error on observables of interest, the cluster number counts and the lensing power spectrum, and then propagate the errors into the parameter forecasts.', '0705.0163-1-36-1': 'In this section, we will present the covariance matrices of the observables.', '0705.0163-1-37-0': '## Covariances of the cluster number counts', '0705.0163-1-38-0': 'The cluster observables can be naturally incorporated in the halo model approach, allowing us to compute the statistical properties in a straightforward way.', '0705.0163-1-38-1': 'In this paper we focus on the average number density of clusters drawn from a survey, also subdivided into redshift bins as described in [REF].', '0705.0163-1-38-2': 'The covariance between the average number densities in redshift bins [MATH] and [MATH], given by Eq. ([REF]), is defined as [EQUATION]', '0705.0163-1-38-3': 'Based on the halo model the covariances of the angular number density can be straightforwardly derived in Appendix [REF] as [EQUATION] where [MATH] is the halo bias parameter ([CITATION]; we use the model derived in [CITATION]), [MATH] is the linear mass power spectrum, and [MATH] is the Fourier transform of the survey window function; for this we simply employ [MATH] is the 1-st order Bessel function) assuming a circular geometry of the survey region, [MATH] (see [CITATION]).', '0705.0163-1-38-4': 'In the following, the tilde symbol is used to denote the Fourier components of quantities.', '0705.0163-1-38-5': 'To derive the covariance ([REF]), we have ignored correlations between the number densities between different redshift bins, which would be a good approximation for a redshift bin thicker than the correlation length of the cluster distribution.', '0705.0163-1-39-0': 'The first and second terms in Eq. ([REF]) arise from the 1- and 2-halo terms in the halo model calculation; the former gives the shot noise due to the imperfect sampling of fluctuations by a finite number of clusters, while the latter represents the sampling variance arising from fluctuations of the cluster distribution due to a finite survey volume.', '0705.0163-1-39-1': 'It should be noted that our formulation allows us to derive the shot noise term without ad hoc introducing as often done in the literature (e.g., see [CITATION]).', '0705.0163-1-39-2': 'The two terms in Eq. ([REF]) depend on sky coverage in slightly different ways, and the relative importance depends on the survey area; for a larger survey, the sampling variance could be more important than the shot noise [CITATION].', '0705.0163-1-40-0': '## Covariances of lensing power spectra', '0705.0163-1-41-0': 'In reality the lensing power spectrum has to be estimated from the Fourier or spherical harmonic coefficients of the observed lensing fields constructed for a finite survey.', '0705.0163-1-41-1': 'In this paper we assume a flat sky approximation and thus use Fourier wavenumbers [MATH], which are equivalent to spherical harmonic multipoles [MATH] in the limit [MATH] [CITATION].', '0705.0163-1-41-2': 'Because the survey is finite, an infinite number of Fourier modes are not available, and rather the discrete Fourier decomposition has to be constructed in terms of the fundamental mode that is limited by the survey size; [MATH], where [MATH] is the survey area.', '0705.0163-1-41-3': 'In this paper we assume a homogeneous survey geometry for simplicity and do not consider any complex boundary and/or masking effects.', '0705.0163-1-41-4': 'The lensing power spectrum of a multipole [MATH] is observationally estimated by averaging over wavenumber direction in an annulus of width [MATH] [EQUATION] where the integration range is confined to the Fourier modes of [MATH] satisfying the bin condition [MATH] and [MATH] denotes the integration area in the Fourier space approximately given by [MATH].', '0705.0163-1-41-5': 'This is discussed in more detail in Appendix [REF].', '0705.0163-1-42-0': 'Once an estimator of the lensing power spectrum is defined, it is straightforward to compute the covariance [CITATION] (also see [CITATION] for the detailed derivation).', '0705.0163-1-42-1': 'From Eq. ([REF]), the covariance to describe the correlation between the lensing power spectra of different multipoles and redshift bins is given by [EQUATION] where [MATH] is the sky coverage ([MATH]) and the lensing trispectrum [MATH] is defined in terms of the 3D mass trispectrum [MATH] as [EQUATION] with [MATH].', '0705.0163-1-42-2': 'Note that the power spectra [MATH] appearing on the r.h.s. of Eq. ([REF]) are the observed spectra given in Eq. ([REF]), and therefore include the intrinsic ellipticity contamination.', '0705.0163-1-42-3': 'The indices [MATH] denote elements in the lensing power spectrum covariance and run over the multipole bins and redshift bins.', '0705.0163-1-42-4': 'For tomography with [MATH] redshift bins, there are [MATH] different spectra available at each multipole.', '0705.0163-1-42-5': 'Hence, if assuming [MATH] multipole bins, the indices [MATH] run as [MATH].', '0705.0163-1-42-6': 'In most of this paper we adopt [MATH] multipole bins logarithmically spaced, which are sufficient to well capture all the relevant features in the lensing power spectrum.', '0705.0163-1-42-7': 'For example, for tomography with 3 redshift bins, the covariance matrix [MATH] has dimension of [MATH] for [MATH].', '0705.0163-1-43-0': 'The first term of the covariance matrix (second line of Eq. [[REF]]) represents the Gaussian error contribution ensuring that the two power spectra of different multipoles are uncorrelated via [MATH], while the second term gives the non-Gaussian errors to describe correlation between the different power spectra.', '0705.0163-1-43-1': 'The two terms both scale with sky coverage as [MATH].', '0705.0163-1-43-2': 'Note that the non-Gaussian term does not depend on the multipole bin width [MATH] because of [MATH], and taking a wider bin only reduces the Gaussian contribution or equivalently enhances the relative importance of the non-Gaussian contribution.', '0705.0163-1-43-3': 'Although this sounds somewhat odd, the signal-to-noise ratio and parameter forecasts we will show below do not depend on the multipole bin width if the bin width is fine enough (see [CITATION] for the details).', '0705.0163-1-44-0': 'We employ a further simplification to make quick computations of the lensing covariance matrices.', '0705.0163-1-44-1': 'We use the halo model approach to compute the lensing covariance matrices.', '0705.0163-1-44-2': 'We know that most of the signal in the power spectrum comes from small angular scales at [MATH] to which the 1-halo term provides dominant contribution as shown in Fig. [REF].', '0705.0163-1-44-3': 'In addition, the non-Gaussian errors are important only at small angular scales.', '0705.0163-1-44-4': 'For these reasons, we only include the 1-halo term contribution to the lensing trispectrum to compute the non-Gaussian errors.', '0705.0163-1-44-5': 'Although the trispectrum generally depends on four vectors in the Fourier space such as [MATH] and [MATH], the 1-halo term does not depend on any angle between the vectors, but rather depends only on the length of each vector; [MATH] (see [CITATION]), reflecting spherical mass distribution around a halo in a statistical average sense, which does not have any preferred direction in the Fourier space.', '0705.0163-1-44-6': 'Therefore, the non-Gaussian term in Eq. ([REF]) can be further simplified as [EQUATION] where we have assumed that the lensing trispectrum does not change significantly within the multipole bin, which is a good approximation for the lensing fields of interest.', '0705.0163-1-45-0': '## Cross-covariances of the cluster number counts and lensing power spectra', '0705.0163-1-46-0': 'The cluster observables and the weak lensing power spectra probe the same density fluctuation fields in large-scale structure, if the two observables are drawn from the same survey region.', '0705.0163-1-46-1': 'As implied in Fig. [REF], a somewhat significant correlation between the two observables is expected if the small-scale lensing power spectrum is considered.', '0705.0163-1-46-2': 'We again use the halo model to compute the cross-covariance.', '0705.0163-1-46-3': 'The detailed derivation is described in Appendix [REF], and the cross-covariances can be expressed as [EQUATION]', '0705.0163-1-46-4': 'Here [MATH] is the 3D bispectrum corresponding to the three-point function of the cluster distribution and the two mass density fluctuation fields.', '0705.0163-1-46-5': 'The cross-covariance arises from two contributions of the 3D bispectrum, the 1- and 2-halo terms: [EQUATION] where [MATH] is the Fourier transform of a halo profile for which we assume an NFW profile [CITATION] as explicitly defined in Eq. ([REF]).', '0705.0163-1-46-6': 'The cross-covariance arising from the 1-halo term represents correlation between one cluster, treated as a point, and two lensing effects on background galaxies due to the same cluster.', '0705.0163-1-46-7': 'The 2-halo term contribution shows the correlation between one cluster, the lensing field on a background galaxies around the cluster, and the lensing field due to another cluster.', '0705.0163-1-47-0': 'Fig. [REF] shows the cross-covariance between the mass-selected cluster counts and the weak lensing power spectrum as a function of angular multipole [MATH], for a concordance [MATH]CDM model.', '0705.0163-1-47-1': 'For this Figure we use a single redshift bin for both of the cluster counts and the lensing power spectrum.', '0705.0163-1-47-2': 'The dashed, solid and dotted curves are the results obtained when minimum halo masses of [MATH] and [MATH] are assumed for the cluster counts, respectively.', '0705.0163-1-47-3': 'The two thin, solid curves show the 1- and 2-halo term contribution to the total power (bold solid curve) for the [MATH] mass cut.', '0705.0163-1-47-4': 'It is apparent that the cross-covariance at small angular scales [MATH] > [MATH] arises mainly from the 1-halo term contribution.', '0705.0163-1-47-5': 'Comparing the dashed, solid and dotted curves clarifies that the covariance amplitude gets greater with decreasing minimum halo mass, as the weak lensing and the cluster counts probe more similar density fields in the large-scale structure.', '0705.0163-1-48-0': "A more useful quantity is the cross-correlation coefficients defined as [EQUATION] where the subscript '[MATH]' denotes the first redshift bin for the cluster number density as we considered a single redshift bin.", '0705.0163-1-48-1': 'The coefficients quantify the relative importance of the cross-covariance to the auto-covariances at a given [MATH].', '0705.0163-1-48-2': 'The upper panel of Fig. [REF] shows the correlation coefficients for model parameters assumed in Fig. [REF].', '0705.0163-1-48-3': 'The coefficients depend on the multipole bin width taken in the lensing power spectrum covariance calculation as well as on a survey sky coverage; we here assumed [MATH] and [MATH] deg[MATH], except for the thin solid curve where a full-sky survey [MATH] is considered.', '0705.0163-1-48-4': 'Note, however, that the [MATH] ratios and the parameter forecasts we will investigate below are independent of the bin width assumed.', '0705.0163-1-49-0': 'The coefficients peak around [MATH], and decrease at smaller scales.', '0705.0163-1-49-1': 'On the intermediate scales there is a significant cross-correlation since the 1-halo term in the lensing power spectrum depends so strongly on the number of clusters (Fig. [REF]).', '0705.0163-1-49-2': 'However on smaller scales the lensing covariance is dominated by shot noise in the intrinsic galaxy shapes (e.g., see Fig. 1 in [CITATION]), which do not correlate with the cluster counts.', '0705.0163-1-49-3': 'Comparing the thin and bold solid curves shows that the coefficients have only weak dependence on the sky coverage, reflecting that the sampling variance of the cluster count covariance ([REF]) roughly scales as [MATH] for a large area survey of our interest, which is same dependence as the other elements in the covariances.', '0705.0163-1-50-0': 'The lower panel of Fig. [REF] shows the correlation coefficients with varying mass thresholds in the cluster counts, for a fixed multipole [MATH] of the lensing power spectrum.', '0705.0163-1-50-1': 'The lensing power spectrum at [MATH] is found to be most correlated with the cluster counts for the [MATH] mass cut.', '0705.0163-1-50-2': 'The correlation decreases at high mass thresholds when the number of clusters is very small and therefore not representative of the lensing field.', '0705.0163-1-50-3': 'The correlation also decreases at smaller masses since the contribution to the lensing power spectrum is small for light halos (Fig [REF]).', '0705.0163-1-50-4': 'As can be seen from comparison of the bold and thin solid curves, an inclusion of the intrinsic ellipticity noise suppresses the correlation coefficients.', '0705.0163-1-51-0': 'We now consider multiple redshift bins in the cluster catalogue.', '0705.0163-1-51-1': 'Fig. [REF] shows the relative contribution of each cluster count redshift bin to the cross-covariance at a given multipole, where we assume the same minimum halo masses as in Fig. [REF].', '0705.0163-1-51-2': 'It is clear that clusters at [MATH] contribute most to the cross-covariance for an angular scale of [MATH].', '0705.0163-1-51-3': 'Since the number density of mass-selected cluster counts has a weak redshift dependence as shown in Fig. [REF], one can notice that the peak redshift reflects redshift dependence of the lensing efficiency function for a source redshift [MATH] that corresponds to the survey depth we assumed; structures at [MATH] most efficiently cause the lensing effect on source galaxies at [MATH].', '0705.0163-1-51-4': 'It is also worth noting that clusters at higher redshifts have a smaller angular size (smaller virial radii) than [MATH] (e.g. see the right panel of Fig. 2 in [CITATION]).', '0705.0163-1-51-5': 'In other words, clusters at [MATH] > [MATH] carry complementary information to the lensing power spectrum; we shall come back to this issue below.', '0705.0163-1-51-6': 'On the other hand, for an angular scale of [MATH], clusters at lower redshifts [MATH] contribute most to the covariance, because the cluster virial radius matches such a large angular scale only if the cluster is located at lower redshift.', '0705.0163-1-52-0': '# Results: Signal-to-Noise and Parameter Forecasts', '0705.0163-1-53-0': '## A CDM model and survey parameters', '0705.0163-1-54-0': 'To compute the observables of interest we need to specify cosmological model and we assume survey parameters similar to those of future surveys in order to estimate realistic measurement errors.', '0705.0163-1-55-0': 'We include the key parameters that may affect the observables within an adiabatic CDM dominated model with dark energy component: the density parameters are [MATH], [MATH], and [MATH] (note that we assume a flat universe); the primordial power spectrum parameters are the spectral tilt, [MATH], the running index, [MATH], and the normalization parameter of primordial curvature perturbation, [MATH] (the values in the parentheses denote the fiducial model).', '0705.0163-1-55-1': 'We employ the transfer function of matter perturbations, [MATH], with baryon oscillations smoothed out [CITATION].', '0705.0163-1-55-2': 'The dark energy equation of state parameters are [MATH], with fiducial values [MATH] and [MATH].', '0705.0163-1-56-0': 'We specify survey parameters that well resemble a future ground-based survey (e.g., see [CITATION]).', '0705.0163-1-56-1': 'We model the redshift distribution of galaxies by using a toy model given by Eq. (4) in [CITATION]; we employ the parameter value [MATH] leading the redshift distribution to peak at [MATH] and have a mean redshift of [MATH].', '0705.0163-1-56-2': 'The intrinsic ellipticities dilute the lensing shear measurements according to Eq. ([REF]); we simply assume that the shot noise contamination is modeled by the rms ellipticity per component, [MATH], and the total number density of galaxies, [MATH] arcmin[MATH].', '0705.0163-1-56-3': 'We assume a survey area of [MATH] degree[MATH] for our fiducial choice, but will vary the survey area parameter [MATH] to see how our results change with [MATH].', '0705.0163-1-56-4': 'Note that throughout this paper we will assume that the two observables we are interested in, the cluster number counts and the lensing power spectrum, are taken from the same survey region, to study how the cross-correlation affects the parameter constraints as the two methods probe the same cosmic structure.', '0705.0163-1-57-0': 'The redshift information inherent in the cluster number counts and the lensing power spectrum can be very powerful to significantly tighten cosmological parameter errors, especially dark energy parameters (e.g., see [CITATION]).', '0705.0163-1-57-1': 'For our results on the dark energy parameters, we will assume [MATH] redshift bins for lensing tomography and [MATH] redshift bins for the cluster number counts for our fiducial choice.', '0705.0163-1-58-0': '## A signal-to-noise ratio', '0705.0163-1-59-0': 'It is instructive to investigate the expected signal-to-noise ([MATH]) ratio for a combined measurement of the cluster counts and the lensing power spectrum, in order to highlight how the cross-correlation between the two observables affects the measurement accuracies.', '0705.0163-1-59-1': 'The [MATH] can be estimated using the covariance matrix derived in [REF] as [EQUATION]', '0705.0163-1-59-2': 'Here the data vector of our observables, [MATH], constructed from the lensing power spectrum tomography with [MATH] redshift bins and the cluster number counts with [MATH]-redshift bins is defined as [EQUATION]', '0705.0163-1-59-3': 'Note that the dimension of [MATH] is [MATH] when the lensing tomography with [MATH] multipole bins and [MATH] redshift bins and the cluster counts with [MATH] redshift bins are considered (also see below Eq. [[REF]]).', '0705.0163-1-59-4': 'For a case of [MATH], [MATH] and [MATH], the dimension of [MATH] is 610.', '0705.0163-1-59-5': 'The full covariance matrix for the joint measurement, [MATH], can be constructed from Eqs. ([REF]), ([REF]), and ([REF]) as [EQUATION]', '0705.0163-1-59-6': 'Note that [MATH] appearing in Eq. ([REF]) is the inverse matrix of [MATH].', '0705.0163-1-60-0': 'For comparison we consider the [MATH] from each of cluster counts and weak lensing alone by using the relevant part of the data vector in Eq. ([REF]) and the covariance matrices.', '0705.0163-1-60-1': 'We also compare with the [MATH] if the cross-correlation is not taken into account, i.e. a matrix of zeros is used instead of the matrix [MATH] in Eq. ([REF]).', '0705.0163-1-60-2': 'In this case that the two are independent, e.g. measured from non-overlapping two survey regions, the [MATH] values from each of the cluster counts and the lensing power spectrum alone therefore add in quadrature to form the joint [MATH].', '0705.0163-1-61-0': 'When computing [MATH] in Eq. ([REF]) care must be taken with numerical accuracy of the matrix inversion.', '0705.0163-1-61-1': 'The observables of interest, the angular number density of clusters and the lensing power spectrum, have different units and their amplitudes could therefore differ from each other by many orders of magnitudes.', '0705.0163-1-61-2': 'To avoid numerical inaccuracies caused by this fact, we have used the dimension-less covariance matrix [MATH] normalized by the data vector as [EQUATION]', '0705.0163-1-61-3': 'In terms of the re-defined covariance matrix, the total [MATH] can be computed simply as [MATH].', '0705.0163-1-62-0': 'Fig. [REF] shows the [MATH] ratios expected from a ground-based survey with area [MATH] deg[MATH], as a function of minimum halo mass used in the mass-selected cluster counts.', '0705.0163-1-62-1': 'Here we include all the clusters with masses greater than a given mass threshold over a range of [MATH] as in Fig. [REF], and include the lensing power spectrum at multipoles [MATH] assuming the redshift distribution of galaxies described in [REF].', '0705.0163-1-62-2': 'For this Figure we considered a single redshift bin for both of the two for simplicity.', '0705.0163-1-62-3': 'Note that the signal-to-noise ratios only slightly increase by adding redshift binned information (e.g., see Fig. 5 in [CITATION]).', '0705.0163-1-62-4': 'First of all, we should notice that the lensing power spectrum and the cluster number counts have similar [MATH] ratios, when the mass threshold [MATH] is used.', '0705.0163-1-62-5': 'At mass thresholds smaller than [MATH], the cluster counts (dotted curve) have a greater [MATH] than the lensing power spectrum (dot-dashed curve) due to an increase in the number of sampled clusters, while the lensing power spectrum has a greater [MATH] at the greater mass threshold.', '0705.0163-1-63-0': 'The solid curve shows the total [MATH] for a combined measurement of the cluster counts and the lensing power spectrum, when the cross-correlation between the two observables is correctly taken into account for the full covariance matrix (see Eq. [[REF]]).', '0705.0163-1-63-1': 'We compare this to the standard approach in which the two probes are considered to be independent (dashed curve).', '0705.0163-1-63-2': 'The lower panel explicitly shows the percentage difference in [MATH] with and without the cross-covariance.', '0705.0163-1-64-0': 'At small mass thresholds [MATH] < [MATH], the total [MATH] taking into account the cross-covariance is degraded compared to when the probes are considered independent.', '0705.0163-1-64-1': 'This is because the cosmic density field probed is only partially shared by the two observables and therefore an inclusion of the cross-covariance reduces independence of the two observables.', '0705.0163-1-64-2': 'However, the degradation ceases at a critical mass scale where the total [MATH] (including the covariance) is equal to the [MATH] for weak lensing alone.', '0705.0163-1-64-3': 'In other words, the total [MATH] is never smaller than the [MATH] obtained from either alone of the lensing power spectrum or the cluster counts.', '0705.0163-1-65-0': 'Then, an intriguing result is found: the total [MATH] is slightly improved by including the cross-covariance as the mass threshold is increased up to [MATH], where the improvement is up to [MATH] as shown in the lower panel.', '0705.0163-1-65-1': 'This occurs even though the [MATH] ratio from the cluster counts alone is much less than that for the lensing power spectrum alone.', '0705.0163-1-65-2': 'The peak mass scale of the total [MATH] corresponds to the mass scale at which the correlation coefficient of the covariance peaks as can be found in the lower panel of Fig. [REF].', '0705.0163-1-65-3': 'That is, the improvement in [MATH] could happen when the two observables are highly correlated.', '0705.0163-1-65-4': 'Since the cross-covariance describes how the two observables are correlated with each other, it appears that a knowledge of the number of such massive clusters with [MATH] > [MATH] for a given survey region helps to improve the amount of information that can be extracted from the weak lensing measurement.', '0705.0163-1-65-5': 'In simpler words, if a smaller or greater number of massive clusters than the ensemble average value was observed from a given survey region, the observed lensing power spectrum will most likely have smaller or greater amplitudes at [MATH], respectively.', '0705.0163-1-66-0': 'We reproduce this qualitative behavior using a simple toy model described in the Appendix [REF], where the lensing power spectrum is modeled to be given solely by the number of halos, ignoring the halo mass profile and the clustering of different halos.', '0705.0163-1-66-1': 'Based on this toy model we attribute the increase in the total [MATH] for high cluster mass thresholds to the fact that the lensing power spectrum amplitude is sensitive to the number of such massive clusters as demonstrated in Fig. [REF].', '0705.0163-1-66-2': 'The fact that the lensing power spectrum is sensitive to the number counts weighted by the mass squared, means that it adds complementary information to the unweighted sum from the cluster counts.', '0705.0163-1-67-0': 'It should, however, be noted that the improvement in [MATH] is achievable only if the cross-covariance is a priori known from the theoretical prediction e.g. based on CDM structure formation scenarios.', '0705.0163-1-67-1': 'Alternatively it could be obtained from a measurement of the cross-correlation from the survey region.', '0705.0163-1-68-0': 'In Fig. [REF] we study which model ingredient in the full-covariance calculation mainly derives the results in Fig. [REF].', '0705.0163-1-68-1': 'The dashed curve shows the percentage difference in [MATH] when only the 1-halo terms are included in each element of the full covariance matrix ([REF]), which corresponds to a simplified case that there is no clustering between halos.', '0705.0163-1-68-2': 'Compared to the solid line, or Fig. [REF], the results are little different.', '0705.0163-1-68-3': 'The dot-dashed curve shows the result obtained when we ignore the intrinsic ellipticity noise that contributes only to the diagonal elements of the weak lensing power spectrum covariance.', '0705.0163-1-68-4': 'Again only a small difference is found.', '0705.0163-1-68-5': 'On the other hand, the thin dashed curve shows the results when only the 2-halo term contribution to the cross-covariance is included, which attempts to reproduce the results in the previous work [CITATION].', '0705.0163-1-68-6': 'For this case, the impact of the cross-covariance on the [MATH] is negligible as concluded in [CITATION].', '0705.0163-1-68-7': 'Rather, it turns out that the most important effect comes from the lensing trispectrum contribution to the lensing power spectrum covariance.', '0705.0163-1-68-8': 'If we switch off the non-Gaussian contribution, the percentage difference in [MATH] is significantly changed.', '0705.0163-1-68-9': 'In particular, there is a significant improvement in [MATH] by adding the cluster counts with [MATH] > [MATH], because ignoring the lensing trispectrum decreases the diagonal elements in the full covariance matrix and thus enhances the relative importance of the cross-covariance.', '0705.0163-1-68-10': 'This also implies that the weak lensing fields are highly non-Gaussian as carefully investigated in [CITATION].', '0705.0163-1-69-0': 'Fig. [REF] demonstrates how this percentage difference in [MATH] depends on the sky coverage ([MATH]) and the maximum multipole ([MATH]) of the lensing power spectrum.', '0705.0163-1-69-1': 'All the curves in Fig. [REF] are very similar, showing a weak dependence on [MATH] and [MATH].', '0705.0163-1-69-2': '(Note however that the absolute [MATH] itself has a strong dependence.)', '0705.0163-1-69-3': 'Nevertheless, there are several points to note when interpreting the results.', '0705.0163-1-69-4': 'Comparing the dotted, solid and dot-dashed curves clarify that, with increasing [MATH], the mass threshold corresponding to the dip in the percentage difference in [MATH] increases.', '0705.0163-1-69-5': 'This is because the lensing power spectrum at higher multipoles is more sensitive to the cosmic density fields down to smaller length scales, and therefore the cluster counts including less massive halos are more correlated with the lensing power spectra.', '0705.0163-1-69-6': 'Also the impact of the cross-covariance is reduced when assuming [MATH], compared to the fiducial case of [MATH], because the intrinsic ellipticity noise (shot noise) is dominant in the lensing power spectrum covariance at such small scales.', '0705.0163-1-70-0': 'Similarly to in Fig. [REF], Fig. [REF] shows the total [MATH] ratios for a combined measurement of the lensing-based cluster counts and the lensing power spectrum, as a function of the cluster lensing signal thresholds (see Eq. [[REF]] and Fig. [REF]).', '0705.0163-1-70-1': 'Notice that the plotting range of [MATH]-axis is roughly half of that in Fig. [REF].', '0705.0163-1-70-2': 'Because the number densities for the lensing signal thresholds of interest are less than that for the mass-selected cluster sample (as shown in Fig. [REF]) the lensing-based cluster counts do not contribute much to the total [MATH] ratios, compared to Fig. [REF].', '0705.0163-1-70-3': 'Other than this difference, the behavior for the [MATH] curves found in Fig. [REF] are similar to those in Fig. [REF].', '0705.0163-1-71-0': '## Fisher analysis for cosmological parameter constraints', '0705.0163-1-72-0': 'We have so far computed the observables of interest: the cluster counts and the lensing power spectra, and their covariances for a given survey.', '0705.0163-1-72-1': 'Therefore we can now estimate accuracies of cosmological parameter determination, given the measurement accuracies of the observables, using the Fisher matrix formalism [CITATION].', '0705.0163-1-72-2': 'This formalism assesses how well given observables can distinguish the true ("fiducial") cosmological model from other models.', '0705.0163-1-72-3': 'The parameter forecasts we obtain depend on the fiducial model and are also sensitive to the choice of free parameters.', '0705.0163-1-72-4': 'Furthermore, the Fisher matrix gives only a lower limit to the parameter uncertainties, being exact if the likelihood surface around the local minimum is Gaussian in multi-dimensional parameter space.', '0705.0163-1-72-5': 'Ideally a more quantitative method would be used to explore the global structure to realize more accurate parameter forecasts.', '0705.0163-1-72-6': 'As described in [REF], we include all the key parameters that can describe varieties in the observables within [MATH]CDM model cosmologies.', '0705.0163-1-73-0': 'The Fisher information matrix available from the lensing power spectrum tomography is given by [EQUATION] where the partial derivative with respect to the [MATH]-th cosmological parameter [MATH] is evaluated around the fiducial model, with other parameters [MATH]) being fixed to their fiducial values.', '0705.0163-1-73-1': 'As noted above, the Fisher matrix quantifies the best statistical errors achievable on parameter determination with a given data set: the error on a parameter [MATH], marginalized over other parameter uncertainties, is given by [MATH], where [MATH] is the inverse of the Fisher matrix.', '0705.0163-1-74-0': 'Similarly, the Fisher matrix for the cluster counts is given by [EQUATION]', '0705.0163-1-74-1': 'For a combined measurement of the lensing power spectrum and the cluster counts, the Fisher matrix is calculated using the full covariance matrix defined by Eq. ([REF]) (also see Eq. [[REF]]) as [EQUATION] where the summation indices [MATH] run over the redshift and multipole bins of the tomographic lensing power spectra as well as the redshift bins of the cluster counts.', '0705.0163-1-75-0': 'A single probe of structure formation is not powerful enough to constrain all the cosmological parameters simultaneously and well.', '0705.0163-1-75-1': 'Rather, combining the large-scale structure measurements with constraints from CMB temperature and polarization anisotropies significantly helps to lift parameter degeneracies and, in particular, the dark energy parameters (e.g. [CITATION]).', '0705.0163-1-75-2': 'When computing the Fisher matrix for a given CMB data set, we employ 9 parameters: the 8 parameters described in [REF] plus the Thomson scattering optical depth to the last scattering surface, [MATH].', '0705.0163-1-75-3': 'Note that we ignore the contribution to the CMB spectra from the primordial gravitational waves for simplicity.', '0705.0163-1-75-4': 'We use the publicly-available CMBFAST code [CITATION] to compute the angular power spectra of temperature anisotropy, [MATH], [MATH]-mode polarization, [MATH], and their cross-correlation, [MATH].', '0705.0163-1-75-5': 'To model the measurement accuracies we assume the noise per pixel and the angular resolution of the Planck experiment that were assumed in [CITATION].', '0705.0163-1-75-6': 'To be conservative, however, we do not include the CMB information on the dark energy equation of state parameters, [MATH] and [MATH], based on the following steps.', '0705.0163-1-75-7': 'We first compute the inverse of the CMB Fisher matrix, [MATH], for the 9 parameters in order to obtain marginalized errors on the parameters, and then re-invert a sub-matrix of the inverse Fisher matrix that includes only the rows and columns for the parameters besides [MATH] and [MATH].', '0705.0163-1-75-8': 'The sub-matrix of the CMB Fisher matrix derived in this way describes accuracies of the 7 parameter determination, having marginalized over degeneracies of the dark energy parameters [MATH] and [MATH] with other parameters for the hypothetical Planck data sets.', '0705.0163-1-75-9': 'In addition, we use only the CMB information in the range of multipoles [MATH], and therefore we do not include the ISW effect at low multipoles [MATH] < [MATH] which might be sensitive to gravitational waves as well as dark energy perturbations we ignore in this paper.', '0705.0163-1-76-0': 'To obtain the Fisher matrix for a joint experiment combining the lensing and/or cluster experiments with the CMB information, we simply sum the two Fisher matrices as, e.g. [MATH] because the CMB information can be safely considered as an independent probe to the low-[MATH] universe probes.', '0705.0163-1-76-1': 'Note that the final Fisher matrix such as [MATH] has [MATH] dimensions.', '0705.0163-1-77-0': '## Forecasts for parameter constraints', '0705.0163-1-78-0': 'Now we present forecasts for cosmological parameter determination for a combined measurement of cluster counts and lensing power spectrum tomography.', '0705.0163-1-78-1': 'We first consider mass-selected cluster counts, and Fig. [REF] shows the expected [MATH] C.L. errors on each of the parameters [MATH], [MATH] (primordial curvature perturbation), [MATH] and [MATH], as a function of mass thresholds in the cluster counts.', '0705.0163-1-78-2': 'In each case we have marginalised over the remaining eight cosmological parameters (see [REF] and [REF] for the cosmological parameters used).', '0705.0163-1-78-3': 'We assume 3 redshift bins for the lensing power spectrum tomography, and 10 redshift bins for the cluster counts over redshifts [MATH], for a survey with [MATH] deg[MATH] area.', '0705.0163-1-78-4': 'Note that the number of redshift bins used for the cluster counts analysis is not important; a small change in the number of bins has little effect on the results shown below.', '0705.0163-1-78-5': 'It should be also noted that the errors for the combined measurement are enlarged only by [MATH] without the CMB priors.', '0705.0163-1-79-0': 'In the upper panel of each plot, the solid curve shows the error on a given parameter when the cross-covariance between the two observables is correctly taken into account, while the dot-dashed curve shows the error obtained from the lensing tomography alone.', '0705.0163-1-79-1': 'Comparing the solid and dot-dashed curves demonstrates that adding the cluster counts for the smaller mass cuts into the lensing tomography can more improve the errors on dark energy parameters, [MATH], [MATH] and [MATH], because the two observables depend on cosmological parameters in different ways and combining the two can lift the parameter degeneracies (also see [CITATION]).', '0705.0163-1-79-2': 'To be more explicit, the errors are improved by [MATH] for mass threshold [MATH], while the errors are improved only slightly by [MATH] for [MATH].', '0705.0163-1-79-3': 'On the other hand, there is a complex behavior in the error on the primordial curvature perturbation amplitude, [MATH].', '0705.0163-1-79-4': 'This is explained as follows.', '0705.0163-1-79-5': 'The cluster counts are very sensitive to the normalization parameter of the linear mass power spectrum ([MATH] for our case or [MATH] often used in the literature) through the sharp exponential cut-off of the halo mass function at high mass end.', '0705.0163-1-79-6': 'As we reduce the minimum mass threshold we are losing information about the number of very high mass clusters, which are diluted in the total count by the large number of low mass clusters.', '0705.0163-1-79-7': 'Therefore we lose information from the sharp exponential cut-off in cluster counts at high mass.', '0705.0163-1-80-0': 'The impact of the cross-covariance between the two observables on the parameter forecasts can be found from comparison of the solid and dashed curves: the dashed curve shows the error obtained when the two observables are considered to be independent.', '0705.0163-1-80-1': 'Further, the lower panel of each plot explicitly presents the percentage difference in the errors.', '0705.0163-1-80-2': 'The impact of the cross-covariance on the parameter errors is generally very small, at only a few per cent.', '0705.0163-1-81-0': 'Nevertheless, interestingly, in some cases inclusion of the cross-covariance leads to an improvement in the parameter errors; for example, the errors on [MATH] or [MATH] are improved over a range of the mass thresholds we have considered.', '0705.0163-1-81-1': 'This counter-intuitive result is in part due to working in 9 dimensional parameter space, and could also be explained as follows (also see [CITATION] for related discussion).', '0705.0163-1-81-2': 'As we have carefully investigated, the cross-covariance predicted from a CDM model quantifies how the cluster counts and the lensing power spectrum amplitude are correlated with each other in redshift and multipole space.', '0705.0163-1-81-3': 'The positive cross-correlation shown in Fig. [REF] implies that for a given survey region, if the number of clusters probed happens to be higher or lower than the ensemble average, the lensing power spectrum will be expected to have larger or smaller amplitudes, respectively.', '0705.0163-1-81-4': 'Therefore, such a correlated offset in the two observables makes it difficult to determine their true amplitudes compared to the case in which the two observables are independent, thereby degrading the errors of parameters that are primarily sensitive to the amplitudes of the two observables.', '0705.0163-1-81-5': 'This explains the degradation in the errors on [MATH] and [MATH] for some range of mass thresholds.', '0705.0163-1-81-6': 'On the other hand, the correlated offset rather preserves functional relations between the cluster counts and the lensing spectrum amplitudes in redshift and multipole space.', '0705.0163-1-81-7': 'That is, a priori knowledge on the cross-covariance leads to an improvement in the errors on parameters that are primarily sensitive to the redshift and multipole dependence of the cluster counts and the lensing power spectrum.', '0705.0163-1-81-8': 'This is the case for the parameters [MATH] and [MATH].', '0705.0163-1-82-0': 'Fig. [REF] shows the results for the lensing-based cluster counts, as a function of the cluster lensing signal, where clusters having a lensing signal greater than a given threshold, [MATH], are included in the counts.', '0705.0163-1-82-1': 'As in the previous plot, we consider [MATH] redshift bins for the cluster counts over redshifts [MATH] and [MATH] redshift bins for the lensing tomography.', '0705.0163-1-82-2': 'Note that the plotting range of [MATH]-axis in the upper panel of each plot is same as that in Fig. [REF], while the plotting range in the lower panel is different.', '0705.0163-1-83-0': 'First of all, it should be noted that adding the lensing-based cluster counts into the lensing tomography does tighten the errors on [MATH], [MATH] and [MATH] significantly even though the cluster counts include fewer clusters than the mass-selected counts (see Fig. [REF]).', '0705.0163-1-83-1': 'For the threshold [MATH], which includes clusters with masses [MATH] > [MATH] and mainly covers a narrow redshift range of [MATH] < [MATH] < [MATH], the cluster counts still improve the dark energy parameters by [MATH], in contrast with only [MATH] improvement for the mass-selected cluster counts with [MATH] > [MATH] in Fig. [REF].', '0705.0163-1-83-2': 'We find the same percentage improvement when the CMB priors are not included.', '0705.0163-1-83-3': 'With the reasonable value of [MATH] the uncertainties are halved by adding cluster counts to lensing power spectra.', '0705.0163-1-83-4': 'As a result, [MATH] and [MATH] may be attainable for a [MATH] square degree survey.', '0705.0163-1-83-5': 'The relatively amplified sensitivity to dark energy is attributed to the fact that the cluster lensing signal itself depends on the dark energy parameters via the lensing efficiency, even for a fixed halo mass (see Eq. [[REF]]).', '0705.0163-1-84-0': 'For the primordial curvature perturbation [MATH], adding the cluster counts does not improve the error much, compared to the result in Fig. [REF].', '0705.0163-1-84-1': 'The parameter [MATH] does not affect the amount of lensing for a given cluster so the poorer accuracy just arises from larger statistical errors due to the smaller number of clusters, compared to the mass-selected counts.', '0705.0163-1-85-0': 'As shown in the lower panel of each plot, the cross-covariance has more influence on the parameter forecasts, compared to Fig. [REF].', '0705.0163-1-85-1': 'This is because lensing-based cluster counts and lensing tomography both pick up halos over a very similar range in redshifts.', '0705.0163-1-85-2': 'Therefore there are more significant cross-correlations between the two observables.', '0705.0163-1-85-3': 'However, for the lensing signal thresholds of our interest such as [MATH] > [MATH], where contamination to the mass peaks due to false peaks is expected to be very small [CITATION], the effect of the cross-covariances on the parameter errors is small, by less than [MATH].', '0705.0163-1-86-0': '# Conclusion and Discussion', '0705.0163-1-87-0': 'In this paper we have estimated accuracies on cosmological parameters derivable from a joint experiment of cluster counts and lensing power spectrum tomography when the two are drawn from the same survey region.', '0705.0163-1-87-1': 'For the first time we have properly taken into account the cross-covariance between the two observables, which describes how the two observables are correlated in redshift and multipole space.', '0705.0163-1-87-2': 'This is necessary because the two experiments probe the same cosmic density fields.', '0705.0163-1-87-3': 'We will below summarize our findings, and then will discuss the applications and the remaining issues.', '0705.0163-1-88-0': 'We have developed a formulation to compute the cross-covariance between the cluster counts and the lensing power spectrum based on the dark matter halo approach within the framework of a CDM structure formation model (see Appendix).', '0705.0163-1-88-1': 'The cross-covariance arises from the three-point correlation function between the cluster distribution and two points of the mass density fields.', '0705.0163-1-88-2': 'This is because the lensing power spectrum measures the 3D mass power spectrum weighted with the lensing efficiency along the line-of-sight direction.', '0705.0163-1-88-3': 'It is found that there is a significant positive cross-correlation between the cluster counts probing clusters with masses [MATH] > [MATH] and the lensing power spectrum amplitudes at multipoles [MATH] > [MATH].', '0705.0163-1-88-4': "Here the term 'positive' is used to mean that if fewer or more massive clusters are found from a given survey region than the ensemble average, the lensing power spectra will most likely have smaller or larger amplitudes, respectively.", '0705.0163-1-88-5': 'The cross-correlation on angular and mass scales of interest arises mainly from the 1-halo term contribution of the three-point correlations: the correlation between one point within a given cluster and the two lensing fields of background galaxies due to the same cluster.', '0705.0163-1-88-6': 'Our results are more accurate than the earlier work presented in [CITATION], because their work ignored the 1-halo term contribution to the cross-covariance and only included the 2-halo term contribution, which is dominant only on large angular scales where the useful cosmological information can not be extracted.', '0705.0163-1-89-0': 'To quantify the impact of the cross-covariance, we first investigated the total signal-to-noise ([MATH]) ratios for a joint experiment of the cluster counts and the lensing power spectrum, fully taking into account the cross-covariance contribution to the full covariance matrix.', '0705.0163-1-89-1': 'It was shown that inclusion of the cross-covariance leads to degradation and, also interestingly, improvement in the [MATH] ratios up to [MATH] compared to the case that the two observable are considered to be independent.', '0705.0163-1-89-2': 'The improvement occurs when the cluster counts including massive halos [MATH] > [MATH] are combined with the lensing power spectrum measurement.', '0705.0163-1-89-3': 'This occurs even though the [MATH] ratio for the cluster counts alone is much less than that for the lensing power spectrum alone (see Figs. [REF] and [REF]).', '0705.0163-1-89-4': 'That is, knowledge of the number of such massive clusters for a given survey region helps improve accuracies of the joint measurement.', '0705.0163-1-89-5': 'This improvement is achievable only if the cross-covariance is a priori known by using the theoretical predictions or by directly estimating the cross-correlation from the survey region.', '0705.0163-1-89-6': 'We also note that the results change greatly if we ignore the non-Gaussian error contribution to the lensing power spectrum covariance, which arises from the lensing trispectrum (see Fig. [REF]).', '0705.0163-1-89-7': 'This implies that the lensing fields are highly non-Gaussian (see [CITATION] for an extensive discussion).', '0705.0163-1-90-0': 'We then presented forecasts for accuracies of the cosmological parameter determination for the joint experiment.', '0705.0163-1-90-1': 'To do this we included redshift binning for both the cluster counts and the lensing power spectrum, motivated by the fact that the additional redshift information is very useful to tighten the cosmological parameter constraints, especially the dark energy parameters.', '0705.0163-1-90-2': 'In this paper we considered two simplified cluster selection criteria: one is a mass-selected cluster sample, and the other is the lensing-based cluster sample, where the latter contains clusters having the lensing signal greater than a given threshold in the sample.', '0705.0163-1-90-3': 'For the mass-selected cluster counts, it was found that combining the cluster counts and the lensing tomography leads to significant improvement in the errors on the dark energy parameters by [MATH] only if the cluster counts including less massive halos such as [MATH] > [MATH] are considered.', '0705.0163-1-90-4': 'The improvement is due to different dependence of the two observables on the cosmological parameters.', '0705.0163-1-91-0': 'On the other hand, for the lensing-based cluster counts, adding the cluster counts to the lensing power spectrum tomography is more powerful to tighten the errors on the dark energy parameters than the mass-selected cluster counts (see Figs. [REF] and [REF]).', '0705.0163-1-91-1': 'For example, adding the counts of clusters with the high lensing signals [MATH] > [MATH] still improves the dark energy errors by [MATH], even though the counts contain many fewer clusters contain massive clusters with [MATH] > [MATH] and probe a narrower redshift range [MATH] < [MATH] < [MATH] than the mass-selected clusters (see Fig. [REF]).', '0705.0163-1-91-2': 'This result is encouraging because such massive halos are rare and therefore it seems relatively easy to make follow-up observations in order to obtain well-calibrated relations between cluster mass and observables (see also the discussion below).', '0705.0163-1-91-3': 'The reason lensing-based cluster counts are more powerful is ascribed to the fact that the cluster lensing signal itself depends on the cosmological parameters via the lensing efficiency and the dependence amplifies the sensitivity of the cluster counts to the dark energy parameters.', '0705.0163-1-92-0': 'For the impact of the cross-covariance on the parameter determination, the effect is generally small for both the mass-selected and lensing-based cluster counts.', '0705.0163-1-92-1': 'This is partly because the lensing power spectra are sensitive to the total number of clusters roughly weighted by the cluster mass squared whereas for the cluster counts we simply added up the number of clusters.', '0705.0163-1-92-2': 'This means that the two probes are not measuring such a similar quantity and the cross-covariance is smaller than if they both measured the unweighted total number of clusters.', '0705.0163-1-92-3': 'Further, the redshift weighting is different for the lensing power spectra and the cluster counts, so not all the halos are in common.', '0705.0163-1-92-4': 'It is also partly a result of working in multi-dimensional parameter space (9 parameters for our case).', '0705.0163-1-93-0': 'Inclusion of the cross-covariance leads to degradation/improvement in the parameter errors in a complex way, again due to the complicated parameter dependence that the two observables possess.', '0705.0163-1-93-1': 'Yet, it is intriguing to note that the dark energy parameters are in most cases improved by including the cross-covariance (see the lower panels of each plot in Figs. [REF] and [REF]; also see [CITATION] for the related discussion).', '0705.0163-1-93-2': 'In summary, a joint experiment of cluster counts and lensing power spectrum tomography will be worth exploring in order to exploit full information on the cosmological parameters from future massive surveys, and including the cross-covariance will be needed in order to correctly estimate the error bars.', '0705.0163-1-93-3': 'In this work we have assumed that cluster counts measures the total number of clusters above some threshold, in a number of redshift bins.', '0705.0163-1-93-4': 'In principle the mass or lensing signal from each cluster could also be included to improve cosmological parameter constraints .', '0705.0163-1-93-5': 'This would make the improvement on including cluster counts to lensing power spectra even more impressive, however the covariance may be more important than we find in this paper.', '0705.0163-1-94-0': 'Throughout this paper we have ignored possible systematics involved in both the cluster counts and the lensing power spectrum measurement for simplicity.', '0705.0163-1-94-1': 'Understanding the systematics is currently one of the most important issues to be carefully addressed.', '0705.0163-1-94-2': 'For the lensing power spectrum, problematic systematics are caused by biases in photometric redshift estimates (hereafter, simply photo-[MATH]), in the shape measurements of source galaxies and in contamination by intrinsic alignments; for future surveys, the biases in terms of the cosmic shear amplitudes need to be calibrated with better than a few [MATH] accuracy at each multipole bin as carefully investigated in [CITATION].', '0705.0163-1-94-3': 'For cluster counts we need reliable methods for estimating cluster masses, taking into account where necessary gas and AGN physics and projection effects.', '0705.0163-1-94-4': 'The resulting scatter on the mass-observable relations also needs to be carefully incorporated.', '0705.0163-1-95-0': 'A promising way to tackle these obstacles would be to combine different observables, because the different methods depend on the cosmological parameters in different ways and also are sensitive to different systematics (e.g. see [CITATION]).', '0705.0163-1-95-1': 'If properties of the systematics are fairly well modeled, one may reach to the self-calibration regime, which is the possibility to constrain the cosmological parameters as well as the model systematics simultaneously, from the combined measurements.', '0705.0163-1-95-2': 'Further, in this direction, the cross-covariances or cross-correlations between the different observables could play an important role, because (1) the cross-correlations are cosmological signals arising from the cosmic mass density field in large-scale structures or, in other words, there would be generally no cross-correlation between the systematics involved in different observables, and (2) a CDM structure formation model provides specific predictions for the cross-covariances between the given observables as we have shown.', '0705.0163-1-95-3': 'Thus including the cross-covariances in the parameter estimations for the combined measurements could be another viable monitor of the systematics.', '0705.0163-1-96-0': 'One interesting possibility to be further addressed is as follows.', '0705.0163-1-96-1': 'We have shown that the counts of massive clusters are complementary to the lensing tomography for the parameter determination.', '0705.0163-1-96-2': 'It is known that member galaxies in such massive clusters provide much more secure photo-[MATH] estimates than those for other majority of source galaxies used for the lensing tomography.', '0705.0163-1-96-3': 'Even follow-up spectroscopic observation of member galaxies, most efficiently central bright galaxies, would be relatively feasible because such massive clusters are rare objects.', '0705.0163-1-96-4': 'These high quality redshifts would allow much finer redshift binning of the cluster distribution than redshift bins of the lensing tomography.', '0705.0163-1-96-5': 'Then, taking the cross-correlation between the clusters with known redshifts and a fair sub-sample of the galaxies used for the lensing tomography may be used to self-calibrate the photo-[MATH] errors, because the cross-correlation is non-vanishing only if the source galaxies are physically associated with the clusters.', '0705.0163-1-96-6': 'This is a very interesting issue, and will be presented elsewhere.', '0705.0163-1-97-0': 'Focusing now on the systematics that affect cluster counts: a dominant systematic is uncertainty in relating cluster mass to observables.', '0705.0163-1-97-1': 'There are some promising ways to tackle this obstacle.', '0705.0163-1-97-2': 'It may be possible to develop a well-calibrated model for the mass-observable relation using hydrodynamical simulations of cluster formation including the associated physical processes of intracluster medium.', '0705.0163-1-97-3': 'Or one may rather employ an empirical approach for this issue, which is to investigate properties of known massive clusters in great detail combining different various techniques (radio, optical, cluster lensing, [MATH]-ray and so on).', '0705.0163-1-98-0': 'Alternatively, a promising way to self-calibrate the mass-observable relation is to use a measurement of the two-point correlation function of clusters as proposed in [CITATION].', '0705.0163-1-98-1': 'Since amplitudes of the cluster two-point correlation function are very sensitive to halo bias parameters that are fairly well specified by halo masses within a CDM model, the measured amplitude of the two-point function can be used to better know masses of the clusters probed in a statistical sense.', '0705.0163-1-98-2': 'Therefore, by combining the cluster counts, the cluster correlation functions and the lensing tomography, we can further improve the parameter constraints as well as protect the constraints against the possible systematics in the mass-observable relation, fully taking into account the covariances between the three observables.', '0705.0163-1-98-3': 'For this issue, the formulation developed in this paper would be straightforwardly extended to compute the full covariances.', '0705.0163-1-99-0': 'We have, in this paper, made several assumptions in the model calculations for simplicity.', '0705.0163-1-99-1': 'Firstly we ignored variations in the mass density profile to compute the cluster lensing signal which is used for the lensing based cluster counts.', '0705.0163-1-99-2': 'In reality the cluster mass distribution displays rich variants: asphericity, scatters in the relation between mass and halo concentration, substructures and so on.', '0705.0163-1-99-3': 'In addition, we did not include possible variations in the redshift distribution of background galaxies behind a cluster: on angular scales relevant for cluster lensing, 10 arcminute scales at most, the redshift distribution of background galaxies could be quite different from the averaged distribution over the whole sample of imaging galaxies, changing from cluster to cluster.', '0705.0163-1-99-4': 'In turn, these variations cause mean redshift and number density of the background galaxies to be changed, which are sensitive to the cluster lensing signal and the noise due to the intrinsic galaxy shapes.', '0705.0163-1-99-5': 'These effects are very likely to introduce statistical scatters and, also generally, biases in the cluster lensing signals, thereby changing the halo model predictions for the cluster counts as well as the covariances.', '0705.0163-1-99-6': 'These issues have to be further carefully investigated.', '0705.0163-1-100-0': 'Finally, we comment on a possibility for ultimate experiments combining all observables available from one survey region.', '0705.0163-1-100-1': 'As we have extensively shown, one can combine different observables to improve accuracies of the cosmological parameter determination, even though the observables probe the same cosmic density fields.', '0705.0163-1-100-2': 'Besides the cluster counts and the lensing power spectrum considered in this paper, there will be other various observables available: the lensing bispectrum or more generally [MATH]-point correlation functions of the lensing shear fields [CITATION], [MATH]-point correlation functions of cluster and galaxy distributions [CITATION], small-scale cluster lensing signals [CITATION], flexion correlation functions [CITATION] and so on.', '0705.0163-1-100-3': 'Then, one natural question raises: Can we combine all the observables in order to improve the parameter constraints as much as possible?', '0705.0163-1-100-4': 'Or, in the presence of the systematics, is there an optimal combination of the observables to maximize the parameter constraints as well as protect the constraints against the systematics in a most secure way.', '0705.0163-1-100-5': 'However, to address this interesting issue quantitatively, all the covariances between the observables used have to be correctly taken into account.', '0705.0163-1-100-6': 'Since a useful cosmological information inherent in large-scale structures quite often comes from the cosmic density fields on small, non-linear scales, there are inevitably cross-correlations between the different observables because the relevant fields are very likely to display non-Gaussian features in the non-linear scales.', '0705.0163-1-100-7': 'We believe that the formulation developed in this paper would be useful to develop a method for computing the covariances for any observables and the combinations.', '0705.0163-1-100-8': 'This kind of study will be worthwhile exploring in order to exploit the full potential of future expensive surveys for constraining the nature of mysterious dark energy components and possible modifications of gravity.'}

{'0705.0163-2-0-0': 'Several dark energy experiments are available from a single large-area imaging survey, and may be combined to improve cosmological parameter constraints and/or test inherent systematics.', '0705.0163-2-0-1': 'Two promising experiments are cosmic shear power spectra and counts of galaxy clusters.', '0705.0163-2-0-2': 'However the two experiments probe the same cosmic mass density field in large-scale structure, therefore the combination may be less powerful than first thought.', '0705.0163-2-1-0': 'We investigate the cross-covariance between the cosmic shear power spectra and the cluster counts based on the halo model approach, where the cross-covariance arises from the three-point correlations of the underlying mass density field.', '0705.0163-2-1-1': 'Fully taking into account the cross-covariance as well as non-Gaussian errors on the lensing power spectrum covariance, we find a significant cross-correlation between the lensing power spectrum signals at multipoles [MATH] and the cluster counts containing halos with masses [MATH] > [MATH].', '0705.0163-2-1-2': 'Including the cross-covariance for the combined measurement degrades and in some cases improves the total signal-to-noise ratios up to [MATH] relative to when the two are independent.', '0705.0163-2-1-3': 'For cosmological parameter determination, the cross-covariance has a smaller effect as a result of working in a multi-dimensional parameter space, implying that the two observables can be considered independent to a good approximation.', '0705.0163-2-1-4': 'We also discuss that cluster count experiments using lensing-selected mass peaks could be more complementary to cosmic shear tomography than mass-selected cluster counts of the corresponding mass threshold.', '0705.0163-2-1-5': 'Using lensing selected clusters with a realistic usable detection threshold ([MATH] for a ground-based survey), the uncertainty on each dark energy parameter may be roughly halved by the combined experiments, relative to using the power spectra alone.', '0705.0163-2-2-0': '# Introduction', '0705.0163-2-3-0': 'In recent years great observational progress has been made in measuring the constituents of the universe (e.g. [CITATION]).', '0705.0163-2-3-1': 'It appears that the universe is currently dominated by an unexpected component that is causing the universe to accelerate in its expansion.', '0705.0163-2-3-2': 'This component is dubbed "dark energy".', '0705.0163-2-3-3': 'Understanding the nature of dark energy is one of most fundamental questions that remain unresolved with the current cosmological data sets (e.g. [CITATION]).', '0705.0163-2-3-4': 'This is now the focus of several planned future surveys [CITATION].', '0705.0163-2-4-0': "Whether the accelerating expansion is as a consequence of the cosmological constant, a new fluid or a modification to Einstein's gravity, these future surveys will provide key information.", '0705.0163-2-4-1': 'In addition they will provide a wealth of further cosmological information, such as constraints on the neutrino mass and the spectrum of primordial perturbations generated in the early universe (e.g. [CITATION]).', '0705.0163-2-5-0': 'Combining several techniques accessible from different cosmological observables is often a powerful way to improve constraints on cosmology.', '0705.0163-2-5-1': 'However, care must be taken if the observables are not completely independent.', '0705.0163-2-5-2': 'Two of the most promising methods for constraining the dark energy are galaxy cluster counts and cosmic shear (e.g. [CITATION]).', '0705.0163-2-6-0': 'Clusters of galaxies contain galaxies, hot gas and dark matter in ratio approximately 1:10:100 [CITATION].', '0705.0163-2-6-1': 'They are the largest gravitationally bound objects in the universe and the number of clusters of galaxies has long been recognized as a powerful probe of cosmology [CITATION].', '0705.0163-2-6-2': 'Counting clusters of galaxies as a function of redshift allows a combination of structure growth and geometrical information to be extracted, thus potentially allowing constraints on the nature of dark energy [CITATION].', '0705.0163-2-6-3': 'If cluster masses can be measured accurately then the shape of the mass function also helps to break degeneracies [CITATION].', '0705.0163-2-6-4': 'The distribution of clusters on the sky (e.g. two-point correlation function) carries additional information on dark energy [CITATION].', '0705.0163-2-7-0': 'The bending of light by mass, gravitational lensing, causes images of distant galaxies to be distorted.', '0705.0163-2-7-1': 'These sheared source galaxies are mostly too weakly distorted for us to measure the effect in single galaxies, but require surveys containing a few million galaxies to detect the signal in a statistical way.', '0705.0163-2-7-2': 'This cosmic shear signal has been observed [CITATION] and used to constrain cosmology (most recently [CITATION]).', '0705.0163-2-7-3': 'By using redshift information of source galaxies the evolution of the dark matter distribution with redshift can be inferred.', '0705.0163-2-7-4': 'Hence, measuring the cosmic shear two-point function as a function of redshift and separation between pairs of galaxies can be used to constrain the geometry of the universe as well as the growth of mass clustering.', '0705.0163-2-7-5': 'This method has emerged as one of the most promising to obtain precise constraints on the nature of dark energy if systematics are well under control [CITATION].', '0705.0163-2-8-0': 'Future optical imaging surveys suitable for cosmic shear analysis will also allow the identification of clusters of galaxies.', '0705.0163-2-8-1': 'This could be done either using the colors of the cluster members (e.g. [CITATION]) or using peaks in the gravitational lensing shear field (e.g. [CITATION]).', '0705.0163-2-8-2': "In addition cluster surveys in other wavebands will overlap with the cosmic shear surveys allowing detection using X-rays and the thermal Sunyaev-Zel'dovich (SZ) effect.", '0705.0163-2-9-0': 'Clusters of galaxies produce a large gravitational lensing effect on distant galaxies, therefore cluster counts and cosmic shear will not be strictly statistically independent.', '0705.0163-2-9-1': 'The volume surveyed is finite and therefore the number of clusters observed will not be exactly equal to the average over all universe realizations.', '0705.0163-2-9-2': 'If the number of clusters happens to be higher for a given survey region, then the cosmic shear signal is also likely to be higher.', '0705.0163-2-9-3': 'Although the volumes will be large, and thus the deviation is small, this may amount to a significant uncertainty in the dark energy parameters as obtained by cluster counts, and dominates the non-Gaussian errors on the cosmic shear [CITATION].', '0705.0163-2-10-0': 'One aspect of this cross-correlation was discussed in [CITATION] and found to be negligible.', '0705.0163-2-10-1': 'However, here we make a full treatment of this effect using the halo model for non-linear structure formation, and quantify the resulting change in joint constraints on the dark energy parameters.', '0705.0163-2-11-0': 'The structure of our paper is as follows.', '0705.0163-2-11-1': 'In [REF] we describe how our observables, cluster number counts and lensing power spectra, can be expressed in terms of the background cosmological model and the density perturbations.', '0705.0163-2-11-2': 'In [REF], we describe a methodology to compute covariances of the cluster counts and the lensing power spectra, and the cross-covariance between the two observables.', '0705.0163-2-11-3': 'The detailed derivations of the covariances are presented in Appendix.', '0705.0163-2-11-4': 'In [REF] we first study the total signal-to-noise ratios expected for a joint experiment of the cluster counts and the lensing power spectrum fully including the cross-covariance predicted from the [MATH]CDM cosmologies.', '0705.0163-2-11-5': 'We then present forecasts for cosmological parameter determination for the joint experiment, with particular focus on forecasts for the dark energy parameter constraints.', '0705.0163-2-11-6': 'Finally, we present conclusions and discussion in [REF].', '0705.0163-2-12-0': '# Preliminaries', '0705.0163-2-13-0': '## A CDM model', '0705.0163-2-14-0': 'We work in the context of spatially flat cold dark matter models for structure formation.', '0705.0163-2-14-1': 'The expansion history of the universe is given by the scale factor [MATH] in a homogeneous and isotropic universe (e.g., see [CITATION]).', '0705.0163-2-14-2': 'We describe the Universe in terms of the matter density [MATH] (the cold dark matter plus the baryons) and dark energy density [MATH] at present (in units of the critical density [MATH], where [MATH] is the Hubble parameter at present).', '0705.0163-2-14-3': 'In general the expansion rate, the Hubble parameter, is given by [EQUATION] where we have employed the normalization [MATH] today and [MATH] specifies the equation of state for dark energy as [MATH].', '0705.0163-2-14-4': 'Note that [MATH] and [MATH] corresponds to a cosmological constant.', '0705.0163-2-14-5': 'The comoving distance [MATH] from an observer at [MATH] to a source at [MATH] is expressed in terms of the Hubble parameter as [EQUATION]', '0705.0163-2-14-6': 'This gives the distance-redshift relation [MATH] via the relation [MATH].', '0705.0163-2-15-0': 'Next we need the redshift growth of density perturbations.', '0705.0163-2-15-1': 'In linear theory after matter-radiation equality, all Fourier modes of the mass density perturbation, [MATH]x[MATH]x[MATH], grow at the same rate, the growth rate (e.g. see Eq. 10 in [CITATION] for details).', '0705.0163-2-16-0': '## Number counts of galaxy clusters', '0705.0163-2-17-0': 'The galaxy cluster observables we will consider in this paper are the number counts drawn from a given survey region.', '0705.0163-2-17-1': 'Clusters can be found via their notable observational properties such as gravitational lensing, member galaxies, X-ray emission and the SZ effect.', '0705.0163-2-17-2': 'For number counts we simply treat clusters as points; in other words, we do not care about the distribution of mass within a cluster.', '0705.0163-2-17-3': 'Hence, the number density field of clusters at redshift [MATH] can be expressed as [EQUATION] where [MATH]x[MATH] is the three-dimensional Dirac delta function.', '0705.0163-2-17-4': 'The summation runs over halos (the subscript [MATH] stands for the [MATH]-th halo), and [MATH] denotes the selection function that discriminates the halos used for the cluster number counts statistic from other halos.', '0705.0163-2-18-0': 'In this paper, we will consider the following two toy models for the selection function, to develop intuition for the importance of cross-correlation between cluster counts and the lensing power spectrum and to make a comparison between cosmological parameter estimations derived from different cluster samples.', '0705.0163-2-18-1': 'Note that throughout this paper we will ignore uncertainties associated with cluster mass-observable relation, which could significantly degrade the ability of cluster counts for constraining cosmological parameters (e.g. [CITATION]).', '0705.0163-2-18-2': 'We shall discuss this issue in [REF].', '0705.0163-2-19-0': 'A mass-limited cluster sample - The first toy model we will consider is a mass-limited cluster sample.', '0705.0163-2-19-1': 'For this model, we include all halos with masses above a given mass threshold: [EQUATION]', '0705.0163-2-19-2': 'To a zero-th order approximation, the mass-limited selection may mimic a cluster sample derived from a flux-limited survey of clusters via the SZ effect, as this effect is free of the surface brightness dimming effect (e.g. see ).', '0705.0163-2-20-0': 'A lensing-based cluster sample - A lensing measurement allows one to make a reconstruction of the two-dimensional mass distribution projected along the line of sight [CITATION].', '0705.0163-2-20-1': 'A high peak in the mass map provides a strong candidate for a massive cluster (see [CITATION] for an implementation of this method to actual data).', '0705.0163-2-20-2': 'To be more explicit, one can define height or significance for each peak in the reconstructed mass map using the effective signal-to-noise ratio (see [CITATION] for details): [EQUATION]', '0705.0163-2-20-3': 'Here [MATH] is the convergence amplitude due to a given cluster at redshift [MATH] and with mass [MATH], and [MATH] is the rms fluctuations in [MATH] due to the intrinsic ellipticity noise arising from a finite number of the background galaxies.', '0705.0163-2-20-4': 'Note that we assume an NFW profile [CITATION] with profile parameters modeled in [CITATION], and consider the convergence field smoothed with a Gaussian filter of angular scale [MATH].', '0705.0163-2-20-5': 'To compute the [MATH] for a cluster at redshift [MATH], we take into account the remaining fraction of background galaxies behind the cluster for a given redshift distribution of whole galaxy population (see [REF]).', '0705.0163-2-20-6': 'This accounts for the variation of mean redshift and number density of the background galaxies with cluster redshift, which changes both the signal and the intrinsic noise in Eq. ([REF]).', '0705.0163-2-21-0': 'From the reconstructed mass map, a cluster sample may be constructed by counting mass peaks with heights above a given threshold, [MATH]: the selection function is given by [EQUATION]', '0705.0163-2-21-1': 'As carefully investigated in [CITATION], the minimum mass of clusters detectable with a given threshold varies with cluster redshift; clusters at medium redshift between observer and a typical source redshift are most easily detectable, while only more massive clusters can be detected at redshifts smaller and greater than the medium redshift, as discussed below.', '0705.0163-2-22-0': 'We will employ the halo model to quantify the statistical properties of cluster observables.', '0705.0163-2-22-1': 'In the halo model approach, we assume that all the matter is in halos.', '0705.0163-2-22-2': 'Following the formulation developed in [CITATION] (also see Appendix [REF] and [CITATION] for a thorough review), the ensemble average of Eq. ([REF]) can be computed as [EQUATION] where [MATH] is the halo mass function corresponding to the redshift considered and we have used the ensemble average [MATH].', '0705.0163-2-22-3': 'Thus, as expected, the ensemble average of the cluster number density field is given by the integral of the halo mass function, which does not depend on the cluster distribution and spatial position.', '0705.0163-2-22-4': 'For the halo mass function, we employ the Sheth-Tormen fitting formula [CITATION], modified from the original Press-Schechter function [CITATION].', '0705.0163-2-22-5': 'Note that we use parameter values [MATH] and [MATH] in the formula following the discussion in [CITATION].', '0705.0163-2-22-6': 'We assume that the mass function can be applied to dark energy cosmologies by replacing the growth rate appearing in the formula with that for a dark energy model [CITATION].', '0705.0163-2-23-0': 'A more useful quantity often considered in the literature is the total number counts of clusters available from a given survey, which is obtained by integrating the three-dimensional number density field over a range of redshifts surveyed.', '0705.0163-2-23-1': 'Cluster redshifts are rather easily available even from a multicolor imaging survey alone because their central bright galaxies, or red sequence galaxies, have secure photometric redshift estimates.', '0705.0163-2-23-2': 'Having these facts in mind we will use as our observable the angular number density averaged over a survey area and divided into redshift bins: [EQUATION] where [MATH] is the window function of the survey defined so that it is normalized as [MATH], [MATH] is the distance to the Hubble horizon, and the comoving volume per unit comoving distance and unit steradian is given by [MATH] for a flat universe.', '0705.0163-2-23-3': 'The subscript in the round bracket, [MATH], stands for the [MATH]-th redshift bin for the cluster number counts.', '0705.0163-2-23-4': 'In the following, we will simply consider the sharp redshift selection function [EQUATION]', '0705.0163-2-23-5': 'Note that the redshift [MATH] appearing in the argument of [MATH] is related to the comoving distance [MATH] via the relation [MATH].', '0705.0163-2-24-0': 'Using the halo model, the expectation value of the angular number density can be computed from the ensemble average of Eq. ([REF]) as [EQUATION]', '0705.0163-2-24-1': 'Thus, the expectation value again does not depend on the cluster distribution.', '0705.0163-2-24-2': 'The sensitivity of the number density to dark energy arises from the comoving volume and the mass function [MATH] [CITATION].', '0705.0163-2-25-0': 'Fig. [REF] shows the average angular number density of halos with masses greater than a given threshold, per unit square arcminute and per unit redshift interval assuming the fiducial model defined in [REF].', '0705.0163-2-25-1': 'Increasing the dark energy equation of state from our fiducial model [MATH] to [MATH] decreases the number density, because the change decreases both the comoving volume [MATH] and the number density of cluster-scale halos, for a given CMB normalization of density perturbations.', '0705.0163-2-25-2': 'Comparing the results for mass thresholds [MATH] and [MATH] clarifies that a factor 5 increase in the mass threshold leads to a significant decrease in the number density, reflecting the mass sensitivity of the halo mass function in its exponential tail.', '0705.0163-2-26-0': 'In Fig. [REF] we present the number density for the lensing-based cluster sample in which clusters having a lensing signal greater than a given detection threshold are included in the sample as discussed around Eq. ([REF]).', '0705.0163-2-26-1': 'Note that to compute the results shown in this plot we assumed the redshift distribution of galaxies described in [REF] and the NFW mass profile to model the cluster lensing.', '0705.0163-2-26-2': 'In practice high detection thresholds such as [MATH] > [MATH] are necessary in order to make robust estimates for cluster counts, because contamination of false peaks due to intrinsic ellipticities or the projection effect are expected to be low for such high thresholds (see [CITATION] for the details).', '0705.0163-2-26-3': 'Comparing with the number density for a mass-selected sample shown by the dot-dashed curves, one can roughly find which mass and redshift ranges of clusters are probed by the lensing-based cluster sample.', '0705.0163-2-26-4': 'For example, the cluster sample with lensing signal [MATH] contains massive clusters with masses [MATH] > [MATH] over redshift ranges [MATH] < [MATH], while only even more massive clusters are included in the sample at the higher redshifts.', '0705.0163-2-26-5': 'This cluster sample has a narrower redshift coverage than the simple mass threshold; all the curves peak at a redshift [MATH].', '0705.0163-2-26-6': 'The peak redshift is mainly attributed to redshift dependence of the lensing efficiency for source galaxies of [MATH] in our redshift distribution.', '0705.0163-2-26-7': 'A change of [MATH] from [MATH] to [MATH] leads to a decrease in the number density, as seen in Fig. [REF].', '0705.0163-2-26-8': 'As before the effect comes partially arises from the decrease in comoving volume and the change in the halo mass function.', '0705.0163-2-26-9': 'Unlike the simple mass threshold case, there is now an additional contribution to the decrease in number density caused by the lower lensing efficiency and thus lower [MATH] for a cluster of a given mass and redshift.', '0705.0163-2-27-0': '## Lensing power spectrum with tomography', '0705.0163-2-28-0': 'Gravitational shear can be simply related to the lensing convergence: the weighted mass distribution integrated along the line of sight.', '0705.0163-2-28-1': 'Photometric redshift information on source galaxies allows us to subdivide galaxies into redshift bins (we will discuss possible effects of photometric redshift errors on our results in [REF]).', '0705.0163-2-28-2': 'This allows more cosmological information to be extracted, which is referred to as lensing tomography (e.g., see [CITATION] for a thorough review, and see [CITATION] for the details of lensing tomography).', '0705.0163-2-29-0': 'In the context of cosmological gravitational lensing the convergence field with tomographic information is expressed as a weighted projection of the three-dimensional mass density fluctuation field: [EQUATION] where [MATH][MATH] is the angular position on the sky, and [MATH] is the gravitational lensing weight function for source galaxies sitting in the [MATH]-th redshift bin (see Eq. (10) in [CITATION] for the definition).', '0705.0163-2-29-1': 'Note that, hereafter, quantities with subscripts in the round bracket such as [MATH] stands for those for the [MATH]-th redshift bin.', '0705.0163-2-29-2': 'To avoid confusion, throughout this paper we use [MATH] or [MATH] for the lensing power spectrum redshift bins, and [MATH] for the cluster count redshift bins.', '0705.0163-2-30-0': 'The lensing tomographic information allows us to extract redshift evolution of the lensing weight function as well as the growth rate of mass clustering.', '0705.0163-2-30-1': 'These are both sensitive to dark energy.', '0705.0163-2-30-2': 'For example, increasing the equation of state parameter [MATH] from [MATH] lowers [MATH] as well as suppressing the growth rate at lower redshifts.', '0705.0163-2-30-3': 'Therefore when the CMB normalization of density perturbations is employed, an increase in [MATH] decreases the lensing power spectrum due to both the lower [MATH] and the lower matter power spectrum amplitude.', '0705.0163-2-30-4': 'The sensitivity of lensing observables to the dark energy equation of state roughly arises equally from the two effects (e.g., see [CITATION]).', '0705.0163-2-31-0': 'The cosmic shear fields are measurable only in a statistical way.', '0705.0163-2-31-1': 'The most conventional methods used in the literature are the shear two-point correlation function.', '0705.0163-2-31-2': 'The Fourier transformed counterpart is the shear power spectrum.', '0705.0163-2-31-3': 'The convergence power spectrum is identical to the shear power spectrum but is easier to work with as it is a scalar.', '0705.0163-2-31-4': 'Using the flat-sky approximation [CITATION], the angular power spectrum between the convergence fields of redshift bins [MATH] and [MATH] is found to be [EQUATION] where [MATH] is the three-dimensional mass power spectrum.', '0705.0163-2-31-5': 'We can safely employ the flat-sky approximation for our purpose, because a most accurate measurement for the lensing power spectrum is available around multipoles [MATH] for a ground-based survey of our interest (e.g. see Fig. 1 in [CITATION]), and the flat-sky approximation serves as a very good approximation on these small scales [CITATION].', '0705.0163-2-32-0': 'For [MATH] > [MATH] the major contribution to [MATH] comes from non-linear clustering (e.g., see Fig. 2 in [CITATION]).', '0705.0163-2-32-1': 'We employ the fitting formula for the non-linear [MATH] proposed in Smith et al. [CITATION], assuming that it can be applied to dark energy cosmologies by replacing the growth rate used in the formula with that for a given dark energy model.', '0705.0163-2-32-2': 'We note in passing that the issue of accurate power spectra for general dark energy cosmologies still needs to be addressed carefully (see [CITATION] for related discussion).', '0705.0163-2-33-0': 'Fig. [REF] demonstrates how lensing of background galaxies by clusters contributes to the lensing power spectrum.', '0705.0163-2-33-1': 'Note here that we have employed the halo model developed in Takada Jain [CITATION] to compute the mass power spectrum, although we will use the Smith et al. fitting formula to compute the lensing power spectrum in most parts of this paper instead.', '0705.0163-2-33-2': 'Briefly, to compute the spectra based on the halo model approach, we need to model three ingredients: (i) the halo mass function (see also the description below Eq. [[REF]]); (ii) the profile for the mass distribution around a halo; and (iii) the halo bias parameter.', '0705.0163-2-34-0': 'It is clear that the convergence on scales [MATH] > [MATH] is significantly boosted by the existence of non-linear structures, halos.', '0705.0163-2-34-1': 'In this paper we are especially interested in using the lensing information inherent in angular scales [MATH] < [MATH] to constrain dark energy, and a fair fraction of the power at scales [MATH], up to [MATH] of the total power, arises from massive halos with [MATH] > [MATH].', '0705.0163-2-34-2': 'The 1-halo term contribution is given by redshift-space integral of the halo mass function and halo profiles weighted with the lensing efficiency.', '0705.0163-2-34-3': 'The results imply that, if massive clusters with [MATH] > [MATH] happen to be less or more populated in a survey region, amplitudes of the observed lensing power spectrum from the survey are very likely to be smaller or greater than expected, respectively.', '0705.0163-2-34-4': 'Therefore, a cross-correlation between the lensing power spectrum and the cluster counts are intuitively expected, if both of the observables are measured from the same survey region.', '0705.0163-2-35-0': 'In reality, the observed power spectrum is contaminated by the intrinsic ellipticity noise.', '0705.0163-2-35-1': 'Assuming that the intrinsic ellipticity distribution is uncorrelated between different galaxies, the observed power spectrum between redshift bins [MATH] and [MATH] can be expressed as [EQUATION] where [MATH] is the rms of intrinsic ellipticities per component, and [MATH] denotes the average number density of galaxies in the [MATH]-th redshift bin.', '0705.0163-2-35-2': 'The Kronecker delta function, [MATH], accounts for the fact that the cross-spectra of different redshift bins ([MATH]) are not affected by the shot noise contamination.', '0705.0163-2-35-3': "We will omit the superscript 'obs' when referring to [MATH] in the following for notational simplicity.", '0705.0163-2-36-0': '# Covariances of lensing power spectrum and cluster observables', '0705.0163-2-37-0': 'To estimate a realistic forecast for cosmological parameter constraints for a given survey we have to quantify sources of statistical error on observables of interest, the cluster number counts and the lensing power spectrum, and then propagate the errors into the parameter forecasts.', '0705.0163-2-37-1': 'In this section, we will present the covariance matrices of the observables.', '0705.0163-2-38-0': '## Covariances of the cluster number counts', '0705.0163-2-39-0': 'The cluster observables can be naturally incorporated in the halo model approach, allowing us to compute the statistical properties in a straightforward way.', '0705.0163-2-39-1': 'In this paper we focus on the average angular number density of clusters drawn from a survey, also subdivided into redshift bins as described in [REF].', '0705.0163-2-39-2': 'The covariance between the average number densities in redshift bins [MATH] and [MATH], given by Eq. ([REF]), is defined as [EQUATION]', '0705.0163-2-39-3': 'Based on the halo model the covariances of the angular number density can be derived in Appendix [REF] (also see [CITATION] for the original derivation) as [EQUATION] where [MATH] is the halo bias parameter ([CITATION]; we use the model derived in [CITATION]), [MATH] is the linear mass power spectrum, and [MATH] is the Fourier transform of the survey window function; for this we simply employ [MATH] is the 1-st order Bessel function) assuming a circular geometry of the survey region, [MATH].', '0705.0163-2-39-4': 'In the following, the tilde symbol is used to denote the Fourier components of quantities.', '0705.0163-2-39-5': 'To derive the covariance ([REF]), we have ignored correlations between the number densities between different redshift bins, which would be a good approximation for a redshift bin thicker than the correlation length of the cluster distribution.', '0705.0163-2-40-0': 'The first and second terms in Eq. ([REF]) arise from the 1- and 2-halo terms in the halo model calculation; the former gives the shot noise due to the imperfect sampling of fluctuations by a finite number of clusters, while the latter represents the sampling variance arising from fluctuations of the cluster distribution due to a finite survey volume.', '0705.0163-2-40-1': 'It should be noted that our formulation allows us to derive the shot noise term without ad hoc introducing as often done in the literature (e.g., see [CITATION]).', '0705.0163-2-40-2': 'The two terms in Eq. ([REF]) depend on sky coverage in slightly different ways, and the relative importance depends on the survey area; for a larger survey, the sampling variance could be more important than the shot noise [CITATION].', '0705.0163-2-41-0': '## Covariances of lensing power spectra', '0705.0163-2-42-0': 'In reality the lensing power spectrum has to be estimated from the Fourier or spherical harmonic coefficients of the observed lensing fields constructed for a finite survey.', '0705.0163-2-42-1': 'In this paper we assume the flat-sky approximation and thus use Fourier wavenumbers [MATH], which are equivalent to spherical harmonic multipoles [MATH] in the limit [MATH] [CITATION].', '0705.0163-2-42-2': 'Because the survey is finite, an infinite number of Fourier modes are not available, and rather the discrete Fourier decomposition has to be constructed in terms of the fundamental mode that is limited by the survey size; [MATH], where [MATH] is the survey area.', '0705.0163-2-42-3': 'We assume a homogeneous survey geometry for simplicity and do not consider any complex boundary and/or masking effects.', '0705.0163-2-42-4': 'The lensing power spectrum of a multipole [MATH] is observationally estimated by averaging over wavenumber direction in an annulus of width [MATH] [EQUATION] where the integration range is confined to the Fourier modes of [MATH] satisfying the bin condition [MATH] and [MATH] denotes the integration area in the Fourier space approximately given by [MATH].', '0705.0163-2-42-5': 'This is discussed in more detail in Appendix [REF].', '0705.0163-2-43-0': 'Once an estimator of the lensing power spectrum is defined, it is straightforward to compute the covariance [CITATION] (also see [CITATION] for the detailed derivation).', '0705.0163-2-43-1': 'From Eq. ([REF]), the covariance to describe the correlation between the lensing power spectra of different multipoles and redshift bins is given by [EQUATION] where [MATH] is the sky coverage ([MATH]) and the lensing trispectrum [MATH] is defined in terms of the 3D mass trispectrum [MATH] as [EQUATION] with [MATH].', '0705.0163-2-43-2': 'Note that the power spectra [MATH] appearing on the r.h.s. of Eq. ([REF]) are the observed spectra given in Eq. ([REF]), and therefore include the intrinsic ellipticity noise.', '0705.0163-2-43-3': 'The indices [MATH] denote elements in the lensing power spectrum covariance and run over the multipole bins and redshift bins.', '0705.0163-2-43-4': 'For tomography with [MATH] redshift bins, there are [MATH] different spectra available at each multipole.', '0705.0163-2-43-5': 'Hence, if assuming [MATH] multipole bins, the indices [MATH] run as [MATH].', '0705.0163-2-43-6': 'In most parts of this paper we adopt [MATH] multipole bins logarithmically spaced, which are sufficient to capture all the relevant features in the lensing power spectrum.', '0705.0163-2-43-7': 'For example, for tomography with 3 redshift bins, the covariance matrix [MATH] has dimension of [MATH] for [MATH].', '0705.0163-2-44-0': 'The first term of the covariance matrix (second line of Eq. [[REF]]) represents the Gaussian error contribution ensuring that the two power spectra of different multipoles are uncorrelated via [MATH], while the second term gives the non-Gaussian errors to describe correlation between the different power spectra.', '0705.0163-2-44-1': 'The two terms both scale with sky coverage as [MATH].', '0705.0163-2-44-2': 'Note that the non-Gaussian term does not depend on the multipole bin width [MATH] because of [MATH], and taking a wider bin only reduces the Gaussian contribution or equivalently enhances the relative importance of the non-Gaussian contribution.', '0705.0163-2-44-3': 'Naturally, however, the signal-to-noise ratio and parameter forecasts we will show below do not depend on the multipole bin width if the bin width is not very coarse (see [CITATION] for the details).', '0705.0163-2-45-0': 'We employ a further simplification to make quick computations of the lensing covariance matrices.', '0705.0163-2-45-1': 'We use the halo model approach to compute the lensing covariance matrices.', '0705.0163-2-45-2': 'We know that most of the signal in the power spectrum comes from small angular scales at [MATH] to which the 1-halo term provides dominant contribution as shown in Fig. [REF].', '0705.0163-2-45-3': 'In addition, the non-Gaussian errors are important only at small angular scales.', '0705.0163-2-45-4': 'For these reasons, we only include the 1-halo term contribution to the lensing trispectrum to compute the non-Gaussian errors.', '0705.0163-2-45-5': 'Although the trispectrum generally depends on four vectors in the Fourier space such as [MATH] and [MATH], the 1-halo term does not depend on any angle between the vectors, but rather depends only on the length of each vector; [MATH] (see [CITATION]), reflecting spherical mass distribution around a halo in a statistical average sense, which does not have any preferred direction in the Fourier space.', '0705.0163-2-45-6': 'Therefore, the non-Gaussian term in Eq. ([REF]) can be further simplified as [EQUATION] where we have assumed that the lensing trispectrum does not change significantly within the multipole bin, which is a good approximation for the lensing fields.', '0705.0163-2-46-0': '## Cross-covariances of the cluster number counts and lensing power spectra', '0705.0163-2-47-0': 'The cluster observables and the weak lensing power spectra probe the same density fluctuation fields in large-scale structure, if the two observables are drawn from the same survey region.', '0705.0163-2-47-1': 'As implied in Fig. [REF], a somewhat significant correlation between the two observables is expected if the small-scale lensing power spectrum is considered.', '0705.0163-2-47-2': 'We again use the halo model to compute the cross-covariance.', '0705.0163-2-47-3': 'The detailed derivation is described in Appendix [REF], and the cross-covariances can be expressed as [EQUATION]', '0705.0163-2-47-4': 'Here [MATH] is the 3D bispectrum corresponding to the three-point function of the cluster distribution and the two mass density fluctuation fields.', '0705.0163-2-47-5': 'The cross-covariance arises from two contributions of the 3D bispectrum, the 1- and 2-halo terms: [EQUATION] where [MATH] is the Fourier transform of a halo profile for which we assume an NFW profile [CITATION] as explicitly defined in Eq. ([REF]).', '0705.0163-2-47-6': 'The cross-covariance arising from the 1-halo term represents correlation between one cluster, treated as a point, and the lensing effects on two different background galaxies due to the same cluster.', '0705.0163-2-47-7': 'The 2-halo term contribution shows the correlation between one cluster, the lensing field on a background galaxies around the cluster, and the lensing field due to another cluster.', '0705.0163-2-47-8': 'Note that the cross-covariance ([REF]) is derived assuming the flat-sky approximation as we focus mainly on small angular scale information, but the full-sky expression can be derived combining the methods developed in this paper and in [CITATION].', '0705.0163-2-48-0': 'Fig. [REF] shows the cross-covariance between the mass-selected cluster counts and the weak lensing power spectrum as a function of angular multipole [MATH], for a concordance [MATH]CDM model.', '0705.0163-2-48-1': 'For illustrative clarity we use a single redshift bin for both of the cluster counts and the lensing power spectrum.', '0705.0163-2-48-2': 'The dashed, solid and dotted curves are the results obtained when minimum halo masses of [MATH] and [MATH] are assumed for the cluster counts, respectively.', '0705.0163-2-48-3': 'The two thin, solid curves show the 1- and 2-halo term contribution to the total power (bold solid curve) for the [MATH] mass cut.', '0705.0163-2-48-4': 'It is apparent that the cross-covariance at small angular scales [MATH] > [MATH] arises mainly from the 1-halo term contribution.', '0705.0163-2-48-5': 'Comparing the dashed, solid and dotted curves clarifies that the covariance amplitude gets greater with decreasing minimum halo mass, as the weak lensing and the cluster counts probe more similar density fields in the large-scale structure as implied in Fig. [REF].', '0705.0163-2-49-0': "A more useful quantity is the cross-correlation coefficients defined as [EQUATION] where the subscript '[MATH]' denotes the first redshift bin because for this calculation we are putting all the clusters into a single redshift bin, for illustration (the cluster redshift bin index [MATH] for this case).", '0705.0163-2-49-1': 'The coefficients quantify the relative importance of the cross-covariance to the auto-covariances at a given [MATH].', '0705.0163-2-49-2': 'The upper panel of Fig. [REF] shows the correlation coefficients for model parameters assumed in Fig. [REF].', '0705.0163-2-49-3': 'The coefficients depend on the multipole bin width taken in the lensing power spectrum covariance calculation as well as on a survey sky coverage; we here assumed [MATH] and [MATH] deg[MATH], except for the thin solid curve where a full-sky survey [MATH] is considered.', '0705.0163-2-50-0': 'The upper panel of Fig. [REF] shows that the coefficients peak around [MATH], and decrease at smaller scales.', '0705.0163-2-50-1': 'On the intermediate scales there is a significant cross-correlation since the 1-halo term in the lensing power spectrum depends so strongly on the number of clusters (Fig. [REF]).', '0705.0163-2-50-2': 'However on smaller scales the lensing covariance is dominated by shot noise in the intrinsic galaxy shapes (e.g., see Fig. 1 in [CITATION]), which do not correlate with the cluster counts.', '0705.0163-2-50-3': 'Comparing the thin and bold solid curves shows that the coefficients have only weak dependence on the sky coverage, reflecting that the sampling variance of the cluster count covariance ([REF]) roughly scales as [MATH] for a large area survey of our interest, which is same dependence as the other elements in the covariances.', '0705.0163-2-51-0': 'The lower panel of Fig. [REF] shows the correlation coefficients with varying mass thresholds in the cluster counts, for a fixed multipole [MATH] of the lensing power spectrum.', '0705.0163-2-51-1': 'The lensing power spectrum at [MATH] is found to be most correlated with the cluster counts for the [MATH] mass cut.', '0705.0163-2-51-2': 'The correlation decreases at high mass thresholds when the number of clusters is very small and therefore not representative of the lensing field.', '0705.0163-2-51-3': 'The correlation also decreases at smaller masses since the contribution to the lensing power spectrum is small for light halos (Fig [REF]).', '0705.0163-2-51-4': 'As can be seen from comparison of the bold and thin solid curves, an inclusion of the intrinsic ellipticity noise suppresses the correlation coefficients.', '0705.0163-2-52-0': 'We now consider multiple redshift bins in the cluster catalog.', '0705.0163-2-52-1': 'Fig. [REF] shows the relative contribution of each cluster count redshift bin to the cross-covariance at a given multipole.', '0705.0163-2-52-2': 'It is clear that clusters at [MATH] contribute most to the cross-covariance for an angular scale of [MATH].', '0705.0163-2-52-3': 'Since the number density of mass-selected cluster counts has a weak redshift dependence as shown in Fig. [REF], one can notice that the redshift peak reflects redshift dependence of the lensing efficiency function for a source redshift [MATH] that corresponds to the survey depth we assumed; structures at [MATH] most efficiently cause the lensing effect on source galaxies at [MATH].', '0705.0163-2-52-4': 'It is also worth noting that clusters at higher redshifts have a smaller angular size (smaller virial radii) than [MATH] (e.g. see the right panel of Fig. 2 in [CITATION]).', '0705.0163-2-52-5': 'In other words, clusters at [MATH] > [MATH] may carry complementary information to the lensing power spectrum.', '0705.0163-2-52-6': 'On the other hand, for an angular scale of [MATH], clusters at lower redshifts [MATH] contribute most to the covariance, because the cluster virial radius matches such a large angular scale only if the cluster is located at lower redshift.', '0705.0163-2-53-0': '# Results: Signal-to-Noise and Parameter Forecasts', '0705.0163-2-54-0': '## A CDM model and survey parameters', '0705.0163-2-55-0': 'To compute the observables of interest we need to specify cosmological model and we assume survey parameters similar to those of future surveys in order to estimate realistic measurement errors.', '0705.0163-2-56-0': 'We include the key parameters that may affect the observables within an adiabatic CDM dominated model with dark energy component: the density parameters are [MATH], [MATH], and [MATH] (note that we assume a flat universe); the primordial power spectrum parameters are the spectral tilt, [MATH], the running index, [MATH], and the normalization parameter of primordial curvature perturbation, [MATH] (the values in the parentheses denote the fiducial model).', '0705.0163-2-56-1': 'We employ the transfer function of matter perturbations, [MATH], with baryon oscillations smoothed out [CITATION].', '0705.0163-2-56-2': 'We employ the dark energy model [CITATION] parametrized as [MATH], with fiducial values [MATH] and [MATH].', '0705.0163-2-57-0': 'We specify survey parameters that well resemble a future ground-based survey (e.g., see [CITATION]).', '0705.0163-2-57-1': 'We model the redshift distribution of galaxies by using a toy model given by Eq. (4) in [CITATION]; we employ the parameter value [MATH] leading the redshift distribution to peak at [MATH] and have a mean redshift of [MATH].', '0705.0163-2-57-2': 'The intrinsic ellipticities dilute the lensing shear measurements according to Eq. ([REF]); we simply assume that the shot noise contamination is modeled by the rms ellipticity per component, [MATH], and the total number density of galaxies, [MATH] arcmin[MATH].', '0705.0163-2-57-3': 'The survey area is taken to be [MATH] degree[MATH] for our fiducial choice.', '0705.0163-2-57-4': 'Note that throughout this paper we will assume that the two observables we are interested in, the cluster number counts and the lensing power spectrum, are taken from the same survey region, to study how the cross-correlation affects the parameter constraints.', '0705.0163-2-57-5': 'as the two methods probe the same cosmic structure.', '0705.0163-2-58-0': '## A signal-to-noise ratio', '0705.0163-2-59-0': 'It is instructive to investigate the expected signal-to-noise ([MATH]) ratio for a combined measurement of the cluster counts and the lensing power spectrum, in order to highlight how the cross-correlation between the two observables affects the measurement accuracies.', '0705.0163-2-59-1': 'The [MATH] can be estimated using the covariance matrix derived in [REF] as [EQUATION]', '0705.0163-2-59-2': 'Here the data vector of our observables, [MATH], constructed from the lensing power spectrum tomography with [MATH] redshift bins and the cluster number counts with [MATH]-redshift bins is defined as [EQUATION]', '0705.0163-2-59-3': 'Note that the dimension of [MATH] is [MATH] when the lensing tomography with [MATH] multipole bins and [MATH] redshift bins and the cluster counts with [MATH] redshift bins are considered (also see below Eq. [[REF]]).', '0705.0163-2-59-4': 'For a case of [MATH], [MATH] and [MATH], the dimension of [MATH] is 610.', '0705.0163-2-59-5': 'The full covariance matrix for the joint measurement, [MATH], can be constructed from Eqs. ([REF]), ([REF]), and ([REF]) as [EQUATION]', '0705.0163-2-59-6': 'Note that [MATH] appearing in Eq. ([REF]) is the inverse matrix of [MATH].', '0705.0163-2-60-0': 'For comparison we consider the [MATH] from each of cluster counts and weak lensing alone by using the relevant part of the data vector in Eq. ([REF]) and the covariance matrices.', '0705.0163-2-60-1': 'We also compare with the [MATH] if the cross-correlation is not taken into account, i.e. a matrix of zeros is used instead of the matrix [MATH] in Eq. ([REF]).', '0705.0163-2-60-2': 'In this case that the two are independent, e.g. measured from non-overlapping two survey regions, the [MATH] values from each of the cluster counts and the lensing power spectrum alone therefore add in quadrature to form the joint [MATH].', '0705.0163-2-61-0': 'When computing [MATH] in Eq. ([REF]) care must be taken with numerical accuracy of the matrix inversion.', '0705.0163-2-61-1': 'The observables of interest, the angular number density of clusters and the lensing power spectrum, have different units and their amplitudes could therefore differ from each other by many orders of magnitudes.', '0705.0163-2-61-2': 'To avoid numerical inaccuracies caused by this fact, we have used the dimension-less covariance matrix [MATH] normalized by the data vector as [EQUATION]', '0705.0163-2-61-3': 'In terms of the re-defined covariance matrix, the total [MATH] can be computed simply as [MATH].', '0705.0163-2-62-0': 'Fig. [REF] shows the [MATH] ratios expected for a ground-based survey with area [MATH] deg[MATH] and our fiducial [MATH]CDM model, as a function of minimum halo mass used in the mass-selected cluster counts.', '0705.0163-2-62-1': 'Here we include all the clusters with masses greater than a given mass threshold over a range of [MATH], and include the lensing power spectrum at multipoles [MATH] assuming the redshift distribution of galaxies described in [REF].', '0705.0163-2-62-2': 'Note that we here consider a single redshift bin for both the cluster counts and cosmic shear power spectrum for simplicity, and the signal-to-noise ratios only slightly increase by adding redshift binned information (e.g., see Fig. 5 in [CITATION]).', '0705.0163-2-62-3': 'First of all, we should notice that the lensing power spectrum and the cluster number counts have similar [MATH] ratios, when the mass threshold [MATH] is used.', '0705.0163-2-62-4': 'At mass thresholds smaller than [MATH], the cluster counts (dotted curve) have a greater [MATH] than the lensing power spectrum (dot-dashed curve) due to an increase in the number of sampled clusters, while the lensing power spectrum has a greater [MATH] at the greater mass threshold.', '0705.0163-2-63-0': 'The solid curve shows the total [MATH] for a combined measurement of the cluster counts and the lensing power spectrum, when the cross-correlation between the two observables is correctly taken into account for the full covariance matrix (see Eq. [[REF]]).', '0705.0163-2-63-1': 'We compare this to the standard approach in which the two probes are considered to be independent (dashed curve).', '0705.0163-2-63-2': 'The lower panel explicitly shows the percentage difference in [MATH] with and without the cross-covariance.', '0705.0163-2-64-0': 'At small mass thresholds [MATH] < [MATH], the total [MATH] taking into account the cross-covariance is degraded compared to when the probes are considered independent.', '0705.0163-2-64-1': 'This is because the cosmic density field probed is shared by the two observables and therefore an inclusion of the cross-covariance reduces independence of the two observables.', '0705.0163-2-64-2': 'However, the degradation ceases at a critical mass scale where the total [MATH] (including the covariance) is equal to the [MATH] for the lensing power spectrum alone.', '0705.0163-2-64-3': 'In other words, the total [MATH] is never smaller than the [MATH] obtained from either alone of the lensing power spectrum or the cluster counts.', '0705.0163-2-65-0': 'Then, an intriguing result is found: the total [MATH] is slightly improved by including the cross-covariance as the mass threshold is increased up to [MATH], where the improvement is up to [MATH] as shown in the lower panel.', '0705.0163-2-65-1': 'This occurs even though the [MATH] ratio from the cluster counts alone is much less than that for the lensing power spectrum alone.', '0705.0163-2-65-2': 'The peak mass scale of the total [MATH] corresponds to the mass scale at which the correlation coefficient of the covariance peaks as can be found in the lower panel of Fig. [REF].', '0705.0163-2-65-3': 'That is, the improvement in [MATH] could happen when the two observables are highly correlated.', '0705.0163-2-65-4': 'Since the cross-covariance describes how the two observables are correlated with each other, it appears that a knowledge of the number of such massive clusters with [MATH] > [MATH] for a given survey region helps to improve the amount of information that can be extracted from the weak lensing measurement (also see [CITATION] for the related discussion).', '0705.0163-2-65-5': 'In simpler words, if a smaller or greater number of massive clusters than the ensemble average value was observed from a given survey region, the observed lensing power spectrum will most likely have smaller or greater amplitudes at [MATH], respectively.', '0705.0163-2-66-0': 'We reproduce this qualitative behavior using a simple toy model described in the Appendix [REF], where the lensing power spectrum is modeled to be given solely by the number of halos, ignoring the halo mass profile and the clustering of different halos.', '0705.0163-2-66-1': 'Based on this toy model we attribute the increase in the total [MATH] for high cluster mass thresholds to the fact that the lensing power spectrum amplitude is sensitive to the number of such massive clusters as demonstrated in Fig. [REF].', '0705.0163-2-66-2': 'The fact that the lensing power spectrum is sensitive to the number counts weighted by the mass squared, means that it adds complementary information to the unweighted sum from the cluster counts.', '0705.0163-2-67-0': 'It should, however, be noted that the improvement in [MATH] is achievable only if the cross-covariance is a priori known from the theoretical prediction e.g. based on CDM structure formation scenarios.', '0705.0163-2-67-1': 'Alternatively it could be obtained from a measurement of the cross-correlation from the survey region.', '0705.0163-2-68-0': 'In Fig. [REF] we study which model ingredient in the full-covariance calculation mainly leads to the results in Fig. [REF].', '0705.0163-2-68-1': 'The dashed curve shows the percentage difference in [MATH] when only the 1-halo terms are included in each element of the full covariance matrix ([REF]), which corresponds to a simplified case that there is no clustering between halos.', '0705.0163-2-68-2': 'Compared to the solid line, or Fig. [REF], the results are little different.', '0705.0163-2-68-3': 'The dot-dashed curve shows the result obtained when we ignore the intrinsic ellipticity noise that contributes only to the diagonal elements of the weak lensing power spectrum covariance.', '0705.0163-2-68-4': 'Again only a small difference is found.', '0705.0163-2-68-5': 'On the other hand, the thin dashed curve shows the results when only the 2-halo term contribution to the cross-covariance is included, which attempts to reproduce the results in the previous work [CITATION].', '0705.0163-2-68-6': 'For this case, the impact of the cross-covariance on the [MATH] is negligible as concluded in [CITATION].', '0705.0163-2-68-7': 'Rather, it turns out that the most important effect comes from the lensing trispectrum contribution to the lensing power spectrum covariance.', '0705.0163-2-68-8': 'If we switch off the non-Gaussian contribution, the percentage difference in [MATH] is significantly changed.', '0705.0163-2-68-9': 'In particular, there is a significant improvement in [MATH] by adding the cluster counts with [MATH] > [MATH], because ignoring the lensing trispectrum decreases the diagonal elements in the full covariance matrix and thus enhances the relative importance of the cross-covariance.', '0705.0163-2-68-10': 'This also implies that the cosmic shear fields are highly non-Gaussian as carefully investigated in [CITATION].', '0705.0163-2-69-0': 'Fig. [REF] demonstrates how this percentage difference in [MATH] depends on the sky coverage ([MATH]) and the maximum multipole ([MATH]) of the lensing power spectrum.', '0705.0163-2-69-1': 'All the curves in Fig. [REF] are very similar, showing a weak dependence on [MATH] and [MATH].', '0705.0163-2-69-2': '(Note however that the absolute [MATH] itself has a strong dependence.)', '0705.0163-2-69-3': 'Nevertheless, there are several points to note when interpreting the results.', '0705.0163-2-69-4': 'Comparing the dotted, solid and dot-dashed curves clarify that, with increasing [MATH], the mass threshold corresponding to the dip in the percentage difference in [MATH] increases.', '0705.0163-2-69-5': 'This is because the lensing power spectrum at higher multipoles is more sensitive to the cosmic density fields down to smaller length scales, and therefore the cluster counts including less massive halos are more correlated with the lensing power spectra.', '0705.0163-2-69-6': 'Also the impact of the cross-covariance is reduced when assuming [MATH], compared to the fiducial case of [MATH], because the intrinsic ellipticity noise (shot noise) is dominant in the lensing power spectrum covariance at such small scales.', '0705.0163-2-70-0': 'Similar to Fig. [REF], Fig. [REF] shows the total [MATH] ratios for a combined measurement of the lensing-based cluster counts and the lensing power spectrum, as a function of the cluster lensing-signal thresholds (see Eq. [[REF]] and Fig. [REF]).', '0705.0163-2-70-1': 'Notice that the plotting range of [MATH]-axis is roughly half of that in Fig. [REF].', '0705.0163-2-70-2': 'Because the number densities for the lensing signal thresholds of interest are less than that for the mass-selected cluster sample (as shown in Fig. [REF]) the lensing-based cluster counts do not contribute much to the total [MATH] ratios, compared to Fig. [REF].', '0705.0163-2-70-3': 'Other than this difference, the behavior for the [MATH] curves found in Fig. [REF] are similar to those in Fig. [REF].', '0705.0163-2-71-0': '## Fisher analysis for cosmological parameter constraints', '0705.0163-2-72-0': 'We now estimate accuracies of cosmological parameter determination, given the measurement accuracies of the observables, using the Fisher matrix formalism [CITATION].', '0705.0163-2-72-1': 'This formalism assesses how well given observables can constrain cosmological parameters around a fiducial cosmological model.', '0705.0163-2-72-2': 'The parameter forecasts we obtain depend on the fiducial model and are also sensitive to the choice of free parameters.', '0705.0163-2-72-3': 'Furthermore, the Fisher matrix gives only a lower limit to the parameter uncertainties, being exact if the likelihood surface around the local minimum is Gaussian in multi-dimensional parameter space.', '0705.0163-2-72-4': 'Ideally a more quantitative method would be used to explore the global structure to realize more accurate parameter forecasts.', '0705.0163-2-72-5': 'As described in [REF], we include all the key parameters that can describe varieties in the observables within [MATH]CDM model cosmologies.', '0705.0163-2-73-0': 'The Fisher information matrix available from the lensing power spectrum tomography is given by [EQUATION] where the partial derivative with respect to the [MATH]-th cosmological parameter [MATH] is evaluated around the fiducial model, with other parameters [MATH]) being fixed to their fiducial values.', '0705.0163-2-73-1': 'The error on a parameter [MATH], marginalized over other parameter uncertainties, is given by [MATH], where [MATH] is the inverse of the Fisher matrix.', '0705.0163-2-74-0': 'Similarly, the Fisher matrix for the cluster counts is given by [EQUATION]', '0705.0163-2-74-1': 'For a combined measurement of the lensing power spectrum and the cluster counts, the Fisher matrix is calculated using the full covariance matrix defined by Eq. ([REF]) (also see Eq. [[REF]]) as [EQUATION] where the summation indices [MATH] run over the redshift and multipole bins of the tomographic lensing power spectra as well as the redshift bins of the cluster counts.', '0705.0163-2-75-0': 'Using probes of structure formation alone is not powerful enough to constrain all the cosmological parameters simultaneously and well.', '0705.0163-2-75-1': 'Rather, combining the large-scale structure probes with constraints from CMB temperature and polarization anisotropies significantly helps to lift parameter degeneracies and, in particular, the dark energy parameters (e.g. [CITATION]).', '0705.0163-2-75-2': 'When computing the Fisher matrix for a given CMB data set, we employ 9 parameters: the 8 parameters described in [REF] plus the Thomson scattering optical depth to the last scattering surface, [MATH].', '0705.0163-2-75-3': 'Note that we ignore the contribution to the CMB spectra from the primordial gravitational waves for simplicity.', '0705.0163-2-75-4': 'We use the publicly-available CMBFAST code [CITATION] to compute the angular power spectra of temperature anisotropy, [MATH], [MATH]-mode polarization, [MATH], and their cross-correlation, [MATH].', '0705.0163-2-75-5': 'To model the measurement accuracies we assume the noise per pixel and the angular resolution of the Planck experiment that were assumed in [CITATION].', '0705.0163-2-76-0': 'To be conservative, however, we do not include the CMB information on the dark energy equation of state parameters, [MATH] and [MATH].', '0705.0163-2-76-1': 'We do this because essentially angular positions of the CMB acoustic peaks constrain a degenerate combination of the curvature of the universe and the dark energy parameters, through their dependences on the angular diameter distance to the last scattering surface.', '0705.0163-2-76-2': 'We are assuming a flat universe and therefore wish to remove the artificially good constraint on dark energy that we would get from the CMB.', '0705.0163-2-76-3': 'Note, however, that our parameter forecasts shown below would not change significantly even for a non-flat universe, because we focus on large-scale structure probes in combination with the CMB constraints, as carefully shown in [CITATION].', '0705.0163-2-77-0': 'We remove the CMB information on the dark energy parameters using the following steps.', '0705.0163-2-77-1': 'We first compute the inverse of the CMB Fisher matrix, [MATH], for the 9 parameters in order to obtain marginalized errors on the parameters, and then re-invert a sub-matrix of the inverse Fisher matrix that includes only the rows and columns for the parameters besides [MATH] and [MATH].', '0705.0163-2-77-2': 'The sub-matrix of the CMB Fisher matrix derived in this way describes accuracies of the 7 parameter determination, having marginalized over degeneracies of the dark energy parameters [MATH] and [MATH] with other parameters for the hypothetical Planck data sets.', '0705.0163-2-77-3': 'In addition, we use only the CMB information in the range of multipoles [MATH], and therefore we do not include the integrated Sachs-Wolfe (ISW) effect that contribute to the CMB spectra mainly at low multipoles [MATH] < [MATH], because the ISW effect is very likely correlated with the cosmic shear power spectrum and cluster counts, and we will ignore the correlations in this paper.', '0705.0163-2-78-0': 'To obtain the Fisher matrix for a joint experiment combining the lensing and/or cluster experiments with the CMB information, we simply sum the two Fisher matrices as, e.g. [MATH] because the CMB information can be safely considered as an independent probe to the low-[MATH] universe probes in our setting.', '0705.0163-2-78-1': 'Note that the final Fisher matrix such as [MATH] has [MATH] dimensions.', '0705.0163-2-79-0': '## Forecasts for parameter constraints', '0705.0163-2-80-0': 'When forecasting cosmological parameter determination, we should notice that the redshift information inherent in the cluster number counts and the lensing power spectrum can be very powerful to significantly tighten cosmological parameter errors, especially dark energy parameters (e.g., see [CITATION]).', '0705.0163-2-80-1': 'In the following, we will assume [MATH] redshift bins for lensing tomography and [MATH] redshift bins for the cluster number counts over [MATH], for a survey with 5000 deg[MATH] area.', '0705.0163-2-80-2': "The 'three' redshift bins for lensing tomography is a minimal choice to obtain non-degenerate constraints on the 'three' dark energy parameters, [MATH], [MATH] and [MATH] as implied from Fig. 3 in [CITATION], although four or more redshift bins lead to further improvement, albeit not so much, in the parameter errors.", '0705.0163-2-80-3': 'Since we ignore various systematic errors for both the cluster counts and the lensing tomography, we adopt a rather conservative setting for the lensing tomographic binning.', '0705.0163-2-80-4': 'However note that we have checked a different redshift binning for cluster counts in combination with the lensing tomography does not change the main results below significantly.', '0705.0163-2-80-5': 'An investigation into survey optimization for survey parameters (area and depth) and redshift binning will be presented elsewhere in a more practical manner taking into account possible effects of the systematic errors.', '0705.0163-2-81-0': 'We first consider mass-selected cluster counts, and Fig. [REF] shows the expected [MATH] limits on each of the parameters [MATH], [MATH], [MATH] and [MATH], as a function of mass thresholds in the cluster counts.', '0705.0163-2-81-1': 'In each case we have marginalized over the remaining 8 cosmological parameters (see [REF] and [REF] for the cosmological parameters used).', '0705.0163-2-81-2': 'It should be also noted that the errors on these 4 parameters are enlarged only by [MATH] without the CMB priors.', '0705.0163-2-81-3': 'In the upper panel of each plot, the solid curve shows the error on a given parameter when the cross-covariance between the two observables is correctly taken into account, while the dot-dashed curve shows the error obtained from the lensing tomography alone.', '0705.0163-2-81-4': 'Comparing the solid and dot-dashed curves demonstrates that adding the cluster counts for the smaller mass cuts into the lensing tomography can more improve the errors on dark energy parameters, [MATH], [MATH] and [MATH], because the two observables depend on cosmological parameters in different ways and combining the two can lift the parameter degeneracies (also see [CITATION]).', '0705.0163-2-81-5': 'To be more explicit, the errors are improved by [MATH] for mass threshold [MATH], while the errors are improved only slightly by [MATH] for [MATH].', '0705.0163-2-82-0': 'On the other hand, there is a complex behavior in the error on the primordial curvature perturbation, [MATH].', '0705.0163-2-82-1': 'This is explained as follows.', '0705.0163-2-82-2': 'The cluster counts are very sensitive to the normalization parameter of the linear mass power spectrum ([MATH] for our case or [MATH] often used in the literature) through the sharp exponential cut-off of the halo mass function at high mass end.', '0705.0163-2-82-3': 'As we reduce the minimum mass threshold down to the range [MATH] < [MATH] < [MATH] we are beginning to lose information about the number of very high mass clusters, which are diluted in the total count by the large number of low mass clusters.', '0705.0163-2-82-4': 'At much lower mass cuts [MATH] < [MATH], the cluster counts come back to yield a tighter constraint on [MATH] than the lensing tomography through the better measurement accuracy due to the larger number of very low mass halos.', '0705.0163-2-82-5': 'It would be also worth pointing out that knowing the number of clusters can effectively allow the lensing power spectrum to yield more information on the linear theory part of the power spectrum (e.g. one could subtract off the contribution from the clusters to get at the two halo term).', '0705.0163-2-82-6': 'Therefore the constraint on the power spectrum amplitude can be partly improved from the joint constraint from the mass function and the linear power spectrum.', '0705.0163-2-82-7': 'Note that in this paper we consider a simple mass threshold for the cluster counts, however if clusters can be binned by mass then the information would be restored and we may also use the shape of the mass function to constrain cosmological parameters (e.g. [CITATION]).', '0705.0163-2-83-0': 'The impact of the cross-covariance between the two observables on the parameter forecasts can be found from comparison of the solid and dashed curves: the dashed curve shows the error obtained when the two observables are considered to be independent.', '0705.0163-2-83-1': 'Further, the lower panel of each plot explicitly presents the percentage difference in the errors.', '0705.0163-2-83-2': 'The impact of the cross-covariance on the parameter errors is generally very small, at only a few per cent.', '0705.0163-2-84-0': 'Nevertheless, interestingly, in some cases an inclusion of the cross-covariance leads to an improvement in the parameter errors; for example, the errors on [MATH] or [MATH] are improved over a range of the mass thresholds we have considered.', '0705.0163-2-84-1': 'This perhaps counter-intuitive result is in part due to working in 9 dimensional parameter space, and could also be explained as follows (also see [CITATION] for related discussion).', '0705.0163-2-84-2': 'As we have carefully investigated, the cross-covariance predicted from a CDM model quantifies how the cluster counts and the lensing power spectrum amplitude are correlated with each other in redshift and multipole space.', '0705.0163-2-84-3': 'The positive cross-correlation shown in Fig. [REF] implies that for a given survey region, if the number of clusters probed happens to be higher or lower than the ensemble average, the lensing power spectrum will be expected to have larger or smaller amplitudes, respectively.', '0705.0163-2-84-4': 'Therefore, such a correlated offset in the two observables makes it difficult to determine their true amplitudes compared to the case in which the two observables are independent, thereby degrading the errors of parameters that are primarily sensitive to the amplitudes of the two observables.', '0705.0163-2-84-5': 'This explains the degradation in the errors on [MATH] and [MATH] for some range of mass thresholds.', '0705.0163-2-84-6': 'On the other hand, the correlated offset rather preserves relative values between the cluster counts and the lensing spectrum amplitudes in redshift and multipole space.', '0705.0163-2-84-7': 'That is, a priori knowledge on the cross-covariance leads to an improvement in the errors on parameters that imprint characteristic redshift and multipole dependences onto the cluster counts and the lensing power spectrum.', '0705.0163-2-84-8': 'This is the case for the parameters [MATH] and [MATH].', '0705.0163-2-85-0': 'In Fig. [REF] we show the projected 68% C.L. error ellipses in the dark energy parameter space, for one particular example mass threshold, [MATH].', '0705.0163-2-85-1': 'The error ellipses in a two-parameter subspace highlight how the two parameters considered are degenerate for a given observable and the degeneracies are broken by combining different observables.', '0705.0163-2-85-2': 'It can be seen that the lensing power spectrum tomography and the cluster counts have similar degeneracy directions in constraining the dark energy parameters.', '0705.0163-2-85-3': 'Adding the cluster counts only slightly improves the parameter errors compared to the errors from the lensing tomography alone.', '0705.0163-2-85-4': 'The plot also shows that the cross-covariance has a negligible effect on the error ellipses.', '0705.0163-2-86-0': 'Fig. [REF] shows the results for the lensing-based cluster counts, as a function of the cluster lensing signal, where clusters having a lensing signal greater than a given threshold, [MATH], are included in the counts.', '0705.0163-2-86-1': 'As in Fig. [REF], we consider [MATH] redshift bins for the cluster counts over redshifts [MATH] and [MATH] redshift bins for the lensing tomography.', '0705.0163-2-86-2': 'Note that the plotting range of [MATH]-axis in the upper panel of each plot is same as that in Fig. [REF], while the plotting range in the lower panel is different.', '0705.0163-2-87-0': 'First of all, it should be noted that adding the lensing-based cluster counts into the lensing tomography does tighten the errors on [MATH], [MATH] and [MATH] significantly even though the cluster counts include fewer clusters than the mass-selected counts (see Fig. [REF]).', '0705.0163-2-87-1': 'For the threshold [MATH], which includes clusters with masses [MATH] > [MATH] and mainly covers a narrow redshift range of [MATH] < [MATH] < [MATH], the cluster counts still improve the dark energy parameters by [MATH], in contrast with only [MATH] improvement for the mass-selected cluster counts with [MATH] > [MATH] in Fig. [REF].', '0705.0163-2-87-2': 'We find the same percentage improvement when the CMB priors are not included.', '0705.0163-2-87-3': 'With the reasonable value of [MATH] the uncertainties are halved by adding cluster counts to lensing power spectra.', '0705.0163-2-87-4': 'The relatively amplified sensitivity to dark energy is attributed to the fact that the cluster lensing signal itself depends on the dark energy parameters via the lensing efficiency, even for a fixed halo mass (see Eq. [[REF]]).', '0705.0163-2-88-0': 'For the primordial curvature perturbation [MATH], adding the cluster counts does not improve the error much, compared to the result in Fig. [REF].', '0705.0163-2-88-1': 'The parameter [MATH] does not affect the amount of lensing for a given cluster so the poorer accuracy just arises from larger statistical errors due to the smaller number of clusters, compared to the mass-selected counts.', '0705.0163-2-89-0': 'As shown in the lower panel of each plot, the cross-covariance has more influence on the parameter forecasts, compared to Fig. [REF].', '0705.0163-2-89-1': 'This is because lensing-based cluster counts and lensing tomography both pick up halos over a very similar range in redshifts.', '0705.0163-2-89-2': 'Therefore there are more significant cross-correlations between the two observables.', '0705.0163-2-89-3': 'However, the effect of the cross-covariances on the parameter errors is small, by less than [MATH], for [MATH] > [MATH].', '0705.0163-2-90-0': 'Fig. [REF] shows the marginalized error ellipses, for the lensing-signal detection threshold [MATH].', '0705.0163-2-90-1': 'The degeneracy directions in dark energy parameter constraints are very similar between the cluster counts and the lensing tomography.', '0705.0163-2-90-2': 'Compared to Fig. [REF], the lensing-based cluster counts have a better accuracy of constraining the dark energy parameters, thereby leading the error ellipses to more shrink when the cluster counts and the lensing tomography combined.', '0705.0163-2-90-3': 'This can be explained because the contours in the dark energy equation of state parameter space is just a projection of the full 9d space.', '0705.0163-2-90-4': 'We have investigated eigendirections in cosmological parameter space and identified differences between cluster counts and lensing for [MATH] and [MATH] (effectively [MATH] given that [MATH] is a parameter in the Fisher matrix).', '0705.0163-2-90-5': 'For example, we find that if we plot [MATH] against [MATH] then we see that the cluster counts plus CMB contours are more aligned with the [MATH] axis whereas the lensing plus CMB contours are more aligned with the [MATH] axis.', '0705.0163-2-90-6': 'When combined together, both degeneracies are broken and the error bar on [MATH] is reduced.', '0705.0163-2-90-7': 'Similarly for [MATH] and [MATH] and [MATH] with each dark energy parameter.', '0705.0163-2-90-8': 'This explains why the combined contours (cluster counts plus lensing plus CMB) are smaller than either separate contour (cluster counts plus CMB or lensing plus CMB), even though the separate constraints have the same degeneracy directions when projected down onto [MATH] versus [MATH] space.', '0705.0163-2-91-0': 'Table [REF] summarizes the results shown in Figs. [REF] and [REF] showing the marginalized uncertainties for determination of the 4 parameters [MATH] and [MATH], where [MATH] and [MATH] are employed for the mass-selected cluster counts and the lensing-based cluster counts, respectively.', '0705.0163-2-91-1': 'The error on [MATH], [MATH], shows the error in the dark energy equation of state at the best constrained redshift for given observables, or equivalently the error on the constant equation of state parameter [MATH] obtained by fixing [MATH].', '0705.0163-2-91-2': 'Note that the pivot redshift [MATH] is similar for all the cases: [MATH].', '0705.0163-2-91-3': 'The numerical value [MATH] is proportional to the area of error ellipses in the right panels of Figs. [REF] and [REF].', '0705.0163-2-91-4': 'For the mass selected clusters there is a mild improvement in both the error on [MATH] and [MATH] when adding cluster counts to lensing tomography.', '0705.0163-2-91-5': 'For the lensing selected clusters case, the improvement is mostly in [MATH].', '0705.0163-2-92-0': '## Discussion of systematic errors', '0705.0163-2-93-0': 'We have considered idealized cases: we have ignored possible systematic errors involved in both the cluster counts and the lensing power spectrum measurement for simplicity.', '0705.0163-2-93-1': 'In this subsection, we present some discussion of possible effects of the ignored systematic errors on our results.', '0705.0163-2-94-0': 'An imperfect knowledge of galaxy redshifts inferred from multi-band imaging data (photometric redshifts hereafter simply photo-[MATH]) could affect both measurements of cosmic shear and cluster counts.', '0705.0163-2-94-1': 'For cosmic shear, statistical errors in photo-[MATH]s are unlikely to be a dominant source of the error budget of cosmic shear measurement, if they are well characterized [CITATION].', '0705.0163-2-94-2': 'This is because gravitational lensing has a broad redshift sensitivity function.', '0705.0163-2-94-3': 'As carefully investigated in [CITATION], the dominant source of the systematic error is rather caused by mean bias in photo-[MATH]s, causing mean redshifts of the tomographic bins to be shifted relative to the true mean redshifts.', '0705.0163-2-94-4': 'For planned future surveys, the mean redshifts need to be known to better than a few tenths of a percent accuracy in redshift in order to avoid much degradation in cosmological parameter errors.', '0705.0163-2-94-5': 'To achieve this requirement, a large representative spectroscopic redshift sub-sample of the full set of galaxies used for lensing may be needed to calibrate/correct photo-[MATH] errors.', '0705.0163-2-95-0': 'For cluster counts, photo-[MATH] errors cause uncertainties in redshift estimates of individual clusters and in addition cause uncertainties in the lensing signal of individual clusters, if a lensing-based cluster catalog is used.', '0705.0163-2-95-1': 'For the lensing signal, the requirements on photo-[MATH] accuracy would be similar to that for cosmic shear, as discussed in the previous paragraph.', '0705.0163-2-95-2': 'To estimate the redshift of a cluster using photo-[MATH]s, we often have old red-sequence galaxies for which good photo-[MATH]s are easier to obtain due to a strong 4,000[MATH] break (e.g., [CITATION]).', '0705.0163-2-95-3': 'Further, the redshift of the cluster is an average over the redshifts of the cluster members, thus reducing the uncertainty yet further.', '0705.0163-2-95-4': 'In addition, since clusters are relatively rare objects it would not be very expensive to perform follow-up spectroscopy on a central bright galaxy or some member galaxies.', '0705.0163-2-95-5': 'These high-quality redshifts would allow much finer redshift binning of the cluster distribution than redshift bins of the cosmic shear tomography.', '0705.0163-2-95-6': 'Then, taking the cross-correlation between the clusters with known redshifts and a fair sub-sample of the galaxies used for the cosmic shear tomography may be used to calibrate the photo-[MATH] errors, because the cross-correlation is non-vanishing only if the source galaxies are physically associated with the clusters (see [CITATION] for the related discussion).', '0705.0163-2-95-7': 'This issue would be worth exploring further, and will be presented elsewhere.', '0705.0163-2-96-0': 'Intrinsic alignments of galaxy ellipticities are another potential source of systematic errors for cosmic shear measurement (see [CITATION] for the detail and references therein).', '0705.0163-2-96-1': 'There are two kinds of the contamination.', '0705.0163-2-96-2': 'The first is intrinsic-intrinsic galaxy alignment (II) that may arise from neighboring galaxies residing in a similar tidal field of large-scale structure [CITATION].', '0705.0163-2-96-3': 'The second effect is a cross-correlation between intrinsic ellipticity of a foreground galaxy and lensing distortion of background galaxy shape (GI) because the foreground tidal field affecting the intrinsic ellipticity of a foreground galaxy may also cause lensing shear of a distant, background galaxy ([CITATION]).', '0705.0163-2-96-4': 'In general member galaxies of a cluster tend to be more elliptical and therefore the width of the ellipticity histogram (intrinsic ellipticity dispersion) will be smaller for cluster members due to the absence of e.g. edge-on spirals.', '0705.0163-2-96-5': 'Therefore if there are by chance more clusters in a surveyed area then the noise on the shear power spectrum will be slightly smaller.', '0705.0163-2-96-6': 'This is likely a tiny effect and can be safely neglected.', '0705.0163-2-96-7': 'Another interesting possibility is that if there are many clusters in a surveyed area then both the II and GI contamination to the cosmic shear power spectra would be larger, since member galaxies of a cluster are more aligned with each other.', '0705.0163-2-96-8': 'Also the stronger tidal field due to the cluster may also cause the cluster members to be more anti-aligned with background galaxy shapes due to lensing distortion by the cluster.', '0705.0163-2-96-9': 'However, identifying clusters within a given survey region could be a useful way to remove/correct this II and GI contamination, which is another interesting possibility of the combined cluster counts and cosmic shear to use for future surveys.', '0705.0163-2-97-0': 'For cluster counts, a most problematic source of the systematic errors is the uncertainty in relating cluster observables to halo mass.', '0705.0163-2-97-1': 'One traditional way to tackle this obstacle is to investigate properties of known massive clusters in great detail combing various techniques (radio, optical, cluster lensing, X-ray and the SZ effect).', '0705.0163-2-97-2': 'Or one might develop a reliable model for the mass-observable relation using hydrodynamical simulations of cluster formation fully taking into account the associated physical processes in the intracluster medium.', '0705.0163-2-97-3': 'Then the mass-observable relation obtained in these ways could be used for cluster counting statistics if the derived relation is a fair representation of the mass-observable distribution of clusters in the sample.', '0705.0163-2-98-0': 'For lensing-based cluster counts, projection effects on a cluster lensing signal due to mass along line-of-sight that is not associated with the cluster introduce additional statistical errors in the mass estimates of individual clusters [CITATION].', '0705.0163-2-98-1': 'In addition, the scatter will be correlated with the cosmic shear power spectrum, which we have also not taken into account.', '0705.0163-2-98-2': 'To estimate the mass estimate uncertainty and the effect of the ignored correlation in a quantitative way, ray-tracing simulations of cosmic shear including cluster lensing contributions will be needed.', '0705.0163-2-98-3': 'Also in practice traditional methods (optical, X-ray, the SZ effect) will need to be combined to exclude false clusters from the sample.', '0705.0163-2-98-4': 'These issues are beyond the scope of this paper.', '0705.0163-2-99-0': 'One may develop a model to describe the mass-observable relation in terms of nuisance parameters.', '0705.0163-2-99-1': "Then we could use cluster observables available from a given survey to 'self-calibrate', i.e. determine both the cosmological parameters and the nuisance parameters concurrently.", '0705.0163-2-99-2': 'In particular, it was shown in [CITATION] that adding the two-point correlation function of clusters to cluster counts, both of which are drawn from the same survey region, can be a useful way to self-calibrate the model systematic errors in the mass-observable relation because the amplitude of the cluster two-point function is very sensitive to halo bias that is fairly well specified by halo masses.', '0705.0163-2-100-0': 'Having the discussion above in mind, it would be interesting to address whether the self-calibration regime could be attained for the combined measurements of cluster counts and cosmic shear tomography, taking into account the effects of systematic errors involved in each observable.', '0705.0163-2-100-1': 'The cluster counts and cosmic shear depend on the cosmological parameters in different ways and are sensitive to different systematic errors.', '0705.0163-2-100-2': 'Hence one can use the combined measurement to constrain simultaneously the cosmological parameters as well as the nuisance parameters of systematic errors, mitigating degradation in the cosmological parameter determination due to the systematic errors.', '0705.0163-2-100-3': 'Also importantly one could realize, for a given survey, the requirements on the control of the systematic errors (photo-[MATH], mass-observable relation etc) to attain the desired accuracy of constraining dark energy parameters.', '0705.0163-2-100-4': 'In this direction, the cross-covariances between the cluster counts and cosmic shear tomography may play an intriguing role, because (1) the cross-correlations are cosmological signals arising from the cosmic mass density field in large-scale structures or, in other words, there is in general little cross-correlation between the systematic errors in the two observables, and (2) a CDM structure formation model provides accurate predictions for the cosmological cross-covariances.', '0705.0163-2-100-5': 'Hence including the cross-covariances in the parameter estimations may be used as another viable monitor of the systematic errors.', '0705.0163-2-100-6': 'This interesting issue is beyond the scope of this paper and will be presented elsewhere.', '0705.0163-2-101-0': '# Conclusion and Discussion', '0705.0163-2-102-0': 'In this paper we have estimated accuracies on cosmological parameters derivable from a joint experiment of cluster counts and cosmic shear power spectrum tomography when the two are drawn from the same survey region.', '0705.0163-2-102-1': 'In doing this we have properly taken into account the cross-covariance between the two observables, which describes how the two observables are correlated in redshift and multipole space.', '0705.0163-2-102-2': 'This is necessary because the two experiments probe the same cosmic density fields.', '0705.0163-2-102-3': 'However note that, since we have ignored possible systematic errors, all the results shown in this paper demonstrate pure cosmological powers for the combined method.', '0705.0163-2-102-4': 'We will below summarize our findings, and then will discuss the remaining issues.', '0705.0163-2-103-0': 'We have developed a formulation to compute the cross-covariance between the cluster counts and the cosmic shear power spectra based on the dark matter halo approach within the framework of a CDM structure formation model (see Appendix).', '0705.0163-2-103-1': 'The cross-covariance arises from the three-point correlation function between the cluster distribution and two points of the mass density fields.', '0705.0163-2-103-2': 'It is found that there is a significant positive cross-correlation between the cluster counts probing clusters with masses [MATH] > [MATH] and the lensing power spectrum amplitudes at multipoles [MATH] > [MATH].', '0705.0163-2-103-3': "Here the term 'positive' is used to mean that if fewer or more massive clusters are found from a given survey region than the ensemble average, the lensing power spectra will most likely have smaller or larger amplitudes, respectively.", '0705.0163-2-103-4': 'The cross-correlation on angular and mass scales of interest arises mainly from the 1-halo term contribution of the three-point correlations: the correlation between one point within a given cluster and the shearing effects on two different background galaxies due to the same cluster.', '0705.0163-2-103-5': 'Our results are more accurate than the earlier work presented in [CITATION], because their work ignored the 1-halo term contribution to the cross-covariance and only included the 2-halo term contribution, which is dominant only on large angular scales where the useful cosmological information can not be extracted.', '0705.0163-2-104-0': 'To quantify the impact of the cross-covariance, we first investigated the total signal-to-noise ([MATH]) ratios for a joint measurement of the cluster counts and the lensing power spectrum.', '0705.0163-2-104-1': 'It was shown that an inclusion of the cross-covariance leads to degradation and, depending on the mass thresholds or the lensing detection thresholds, improvement in the [MATH] ratios up to [MATH] compared to the case that the two observable are considered to be independent (see Figs. [REF] and [REF]).', '0705.0163-2-104-2': 'The improvement occurs when the cluster counts including massive halos [MATH] > [MATH] are combined with the lensing power spectrum measurement (also see [CITATION] for the related discussion).', '0705.0163-2-104-3': 'This occurs even though the [MATH] ratio for the cluster counts alone is much less than that for the lensing power spectrum alone.', '0705.0163-2-104-4': 'That is, a knowledge of the number of such massive clusters for a given survey region helps improve accuracies of the joint measurement.', '0705.0163-2-104-5': 'This improvement is achievable only if the cross-covariance is a priori known by using the theoretical predictions or by directly estimating the cross-correlation from the survey region.', '0705.0163-2-104-6': 'We also note that the results change greatly if we ignore the non-Gaussian error contribution to the lensing power spectrum covariance, which arises from the lensing trispectrum (see Fig. [REF]).', '0705.0163-2-104-7': 'This implies that the lensing fields are highly non-Gaussian (see [CITATION] for an extensive discussion).', '0705.0163-2-105-0': 'We then presented forecasts for accuracies of the cosmological parameter determination for the joint experiment.', '0705.0163-2-105-1': 'To do this we included redshift binning for both the cluster counts and the lensing power spectrum, motivated by the fact that the additional redshift information is very useful to tighten the cosmological parameter constraints, especially the dark energy parameters.', '0705.0163-2-105-2': 'In this paper we considered two simplified cluster selection criteria: one is a mass-selected cluster sample, and the other is the lensing-based cluster sample, where the latter contains clusters having the lensing signal greater than a given threshold in the sample.', '0705.0163-2-105-3': 'For the mass-selected cluster counts, it was found that combining the cluster counts and the lensing tomography leads to significant improvement in the errors on the dark energy parameters by [MATH] only if the cluster counts including down to less massive halos such as [MATH] > [MATH] are considered (see Fig. [REF]).', '0705.0163-2-105-4': 'The improvement is due to different dependence of the two observables on the cosmological parameters.', '0705.0163-2-106-0': 'On the other hand, for the lensing-based cluster counts, adding the cluster counts to the lensing power spectrum tomography is more complementary to tighten the errors on the dark energy parameters than the mass-selected cluster counts (see Fig. [REF]).', '0705.0163-2-106-1': 'For example, adding the counts of clusters with the high lensing signals [MATH] > [MATH] improves the dark energy errors by a factor of 2, even though the counts contain many fewer clusters and probe a narrower redshift range than the mass-selected clusters of [MATH] > [MATH] (see Fig. [REF]).', '0705.0163-2-106-2': 'This result is encouraging because such massive halos are rare and therefore it seems relatively easy to make follow-up observations, e.g., in order to obtain well-calibrated relations between cluster mass and observables (also see [REF] for the discussion).', '0705.0163-2-106-3': 'The reason lensing-based cluster counts are more powerful is ascribed to the fact that the cluster lensing signal itself depends on the cosmological parameters via the lensing efficiency and the dependence amplifies the sensitivity of the cluster counts to the dark energy parameters.', '0705.0163-2-106-4': 'However, with low detection thresholds such as [MATH] < [MATH] the lensing-based counts begin to suffer too much from projection effects due to large-scale structures that are not associated with the cluster.', '0705.0163-2-106-5': 'Hence, if traditional mass-selected cluster counts can go to lower masses then they might catch up with, or overtake, the lensing-based counts in their constraining power.', '0705.0163-2-107-0': 'For the impact of the cross-covariance on the parameter determination, the effect is generally small for both the mass-selected and lensing-based cluster counts.', '0705.0163-2-107-1': 'This is partly because the lensing power spectra are sensitive to the total number of clusters roughly weighted by the cluster mass squared whereas for the cluster counts we simply added up the number of clusters (see Appendix [REF]).', '0705.0163-2-107-2': 'This means that the two probes are not measuring such a similar quantity and the cross-covariance is smaller than if they both measured the unweighted total number of clusters.', '0705.0163-2-107-3': 'Further, the redshift weighting is different for the lensing power spectra and the cluster counts, so not all the halos are in common.', '0705.0163-2-107-4': 'It is also partly a result of working in multi-dimensional parameter space (9 parameters for our case).', '0705.0163-2-107-5': 'Yet, it is intriguing to note that the dark energy parameters are in most cases improved by including the cross-covariance (see the lower panels of each plot in Figs. [REF] and [REF]; also see [CITATION] for the related discussion).', '0705.0163-2-107-6': 'In summary, a joint experiment of cluster counts and lensing power spectrum tomography will be worth exploring in order to exploit full information on the cosmological parameters from future massive surveys, and including the cross-covariance will be needed in order to correctly estimate the error bars.', '0705.0163-2-108-0': 'In this work we have assumed that cluster counts measure the total number of clusters above some threshold, in a number of redshift bins.', '0705.0163-2-108-1': 'In principle subdividing cluster counts in mass or lensing signal bins could also improve cosmological parameter constraints .', '0705.0163-2-108-2': 'This would make the improvement on including cluster counts to lensing power spectra even more impressive, however the covariance may be more important than we find in this paper.', '0705.0163-2-109-0': 'Finally, we comment on a possibility for ultimate experiments combining all observables available from one survey region.', '0705.0163-2-109-1': 'As we have shown, one can combine different observables to improve accuracies of the cosmological parameter determination, even though the observables probe the same cosmic density fields.', '0705.0163-2-109-2': 'Besides the cluster counts and the lensing power spectrum considered in this paper, there will be other various observables available: cosmic shear bispectra or more generally [MATH]-point correlation functions of the cosmic shear fields [CITATION], [MATH]-point correlation functions of cluster and galaxy distributions [CITATION], small-scale cluster lensing signals [CITATION], cosmic flexion correlation functions [CITATION] and so on.', '0705.0163-2-109-3': 'Then, one natural question raises: Can we combine all the observables in order to improve the parameter constraints as much as possible?', '0705.0163-2-109-4': 'Or, in the presence of the systematic errors, is there an optimal combination of the observables to maximize the parameter constraints as well as most mitigate degradation in the parameter constraints due the systematic errors.', '0705.0163-2-109-5': 'However, to address this interesting issue quantitatively, all the covariances between the observables used have to be correctly taken into account.', '0705.0163-2-109-6': 'We believe that the formulation developed in this paper would be useful to compute the covariances for any observables and the combinations.', '0705.0163-2-109-7': 'This kinds of study will be worthwhile exploring in order to exploit the full potential of future expensive surveys for constraining the nature of mysterious dark energy components and possible modifications of gravity.'}

[['0705.0163-1-42-0', '0705.0163-2-43-0'], ['0705.0163-1-42-1', '0705.0163-2-43-1'], ['0705.0163-1-42-3', '0705.0163-2-43-3'], ['0705.0163-1-42-4', '0705.0163-2-43-4'], ['0705.0163-1-42-5', '0705.0163-2-43-5'], ['0705.0163-1-42-7', '0705.0163-2-43-7'], ['0705.0163-1-0-0', '0705.0163-2-0-0'], ['0705.0163-1-0-1', '0705.0163-2-0-1'], ['0705.0163-1-0-2', '0705.0163-2-0-2'], ['0705.0163-1-1-2', '0705.0163-2-1-1'], ['0705.0163-1-1-3', '0705.0163-2-1-2'], ['0705.0163-1-1-4', '0705.0163-2-1-3'], ['0705.0163-1-81-2', '0705.0163-2-84-2'], ['0705.0163-1-81-3', '0705.0163-2-84-3'], ['0705.0163-1-81-4', '0705.0163-2-84-4'], ['0705.0163-1-81-5', '0705.0163-2-84-5'], ['0705.0163-1-81-8', '0705.0163-2-84-8'], ['0705.0163-1-51-2', '0705.0163-2-52-2'], ['0705.0163-1-51-4', '0705.0163-2-52-4'], ['0705.0163-1-51-6', '0705.0163-2-52-6'], ['0705.0163-1-38-0', '0705.0163-2-39-0'], ['0705.0163-1-38-2', '0705.0163-2-39-2'], ['0705.0163-1-38-4', '0705.0163-2-39-4'], ['0705.0163-1-38-5', '0705.0163-2-39-5'], ['0705.0163-1-92-0', '0705.0163-2-107-0'], ['0705.0163-1-92-2', '0705.0163-2-107-2'], ['0705.0163-1-92-3', '0705.0163-2-107-3'], ['0705.0163-1-92-4', '0705.0163-2-107-4'], ['0705.0163-1-8-0', '0705.0163-2-8-0'], ['0705.0163-1-8-1', '0705.0163-2-8-1'], ['0705.0163-1-91-3', '0705.0163-2-106-3'], ['0705.0163-1-36-0', '0705.0163-2-37-0'], ['0705.0163-1-36-1', '0705.0163-2-37-1'], ['0705.0163-1-41-0', '0705.0163-2-42-0'], ['0705.0163-1-41-2', '0705.0163-2-42-2'], ['0705.0163-1-41-4', '0705.0163-2-42-4'], ['0705.0163-1-41-5', '0705.0163-2-42-5'], ['0705.0163-1-65-0', '0705.0163-2-65-0'], ['0705.0163-1-65-1', '0705.0163-2-65-1'], ['0705.0163-1-65-2', '0705.0163-2-65-2'], ['0705.0163-1-65-3', '0705.0163-2-65-3'], ['0705.0163-1-65-5', '0705.0163-2-65-5'], ['0705.0163-1-6-1', '0705.0163-2-6-1'], ['0705.0163-1-6-2', '0705.0163-2-6-2'], ['0705.0163-1-6-3', '0705.0163-2-6-3'], ['0705.0163-1-6-4', '0705.0163-2-6-4'], ['0705.0163-1-44-0', '0705.0163-2-45-0'], ['0705.0163-1-44-1', '0705.0163-2-45-1'], ['0705.0163-1-44-2', '0705.0163-2-45-2'], ['0705.0163-1-44-3', '0705.0163-2-45-3'], ['0705.0163-1-44-4', '0705.0163-2-45-4'], ['0705.0163-1-44-5', '0705.0163-2-45-5'], ['0705.0163-1-9-0', '0705.0163-2-9-0'], ['0705.0163-1-9-1', '0705.0163-2-9-1'], ['0705.0163-1-9-2', '0705.0163-2-9-2'], ['0705.0163-1-9-3', '0705.0163-2-9-3'], ['0705.0163-1-4-0', '0705.0163-2-4-0'], ['0705.0163-1-4-1', '0705.0163-2-4-1'], ['0705.0163-1-61-0', '0705.0163-2-61-0'], ['0705.0163-1-61-1', '0705.0163-2-61-1'], ['0705.0163-1-61-2', '0705.0163-2-61-2'], ['0705.0163-1-61-3', '0705.0163-2-61-3'], ['0705.0163-1-32-0', '0705.0163-2-33-0'], ['0705.0163-1-32-2', '0705.0163-2-33-2'], ['0705.0163-1-28-0', '0705.0163-2-28-0'], ['0705.0163-1-28-2', '0705.0163-2-28-2'], ['0705.0163-1-60-0', '0705.0163-2-60-0'], ['0705.0163-1-60-1', '0705.0163-2-60-1'], ['0705.0163-1-60-2', '0705.0163-2-60-2'], ['0705.0163-1-67-0', '0705.0163-2-67-0'], ['0705.0163-1-67-1', '0705.0163-2-67-1'], ['0705.0163-1-10-0', '0705.0163-2-10-0'], ['0705.0163-1-10-1', '0705.0163-2-10-1'], ['0705.0163-1-34-0', '0705.0163-2-35-0'], ['0705.0163-1-34-1', '0705.0163-2-35-1'], ['0705.0163-1-34-2', '0705.0163-2-35-2'], ['0705.0163-1-34-3', '0705.0163-2-35-3'], ['0705.0163-1-49-1', '0705.0163-2-50-1'], ['0705.0163-1-49-2', '0705.0163-2-50-2'], ['0705.0163-1-49-3', '0705.0163-2-50-3'], ['0705.0163-1-50-0', '0705.0163-2-51-0'], ['0705.0163-1-50-1', '0705.0163-2-51-1'], ['0705.0163-1-50-2', '0705.0163-2-51-2'], ['0705.0163-1-50-3', '0705.0163-2-51-3'], ['0705.0163-1-50-4', '0705.0163-2-51-4'], ['0705.0163-1-54-0', '0705.0163-2-55-0'], ['0705.0163-1-55-0', '0705.0163-2-56-0'], ['0705.0163-1-55-1', '0705.0163-2-56-1'], ['0705.0163-1-82-0', '0705.0163-2-86-0'], ['0705.0163-1-82-2', '0705.0163-2-86-2'], ['0705.0163-1-14-0', '0705.0163-2-14-0'], ['0705.0163-1-14-1', '0705.0163-2-14-1'], ['0705.0163-1-14-2', '0705.0163-2-14-2'], ['0705.0163-1-14-3', '0705.0163-2-14-3'], ['0705.0163-1-14-5', '0705.0163-2-14-5'], ['0705.0163-1-14-6', '0705.0163-2-14-6'], ['0705.0163-1-85-0', '0705.0163-2-89-0'], ['0705.0163-1-85-1', '0705.0163-2-89-1'], ['0705.0163-1-85-2', '0705.0163-2-89-2'], ['0705.0163-1-17-0', '0705.0163-2-17-0'], ['0705.0163-1-17-2', '0705.0163-2-17-2'], ['0705.0163-1-17-3', '0705.0163-2-17-3'], ['0705.0163-1-17-4', '0705.0163-2-17-4'], ['0705.0163-1-87-2', '0705.0163-2-102-2'], ['0705.0163-1-29-1', '0705.0163-2-29-1'], ['0705.0163-1-29-2', '0705.0163-2-29-2'], ['0705.0163-1-25-1', '0705.0163-2-25-1'], ['0705.0163-1-25-2', '0705.0163-2-25-2'], ['0705.0163-1-73-0', '0705.0163-2-73-0'], ['0705.0163-1-74-0', '0705.0163-2-74-0'], ['0705.0163-1-74-1', '0705.0163-2-74-1'], ['0705.0163-1-56-0', '0705.0163-2-57-0'], ['0705.0163-1-56-1', '0705.0163-2-57-1'], ['0705.0163-1-56-2', '0705.0163-2-57-2'], ['0705.0163-1-76-1', '0705.0163-2-78-1'], ['0705.0163-1-3-0', '0705.0163-2-3-0'], ['0705.0163-1-3-1', '0705.0163-2-3-1'], ['0705.0163-1-3-2', '0705.0163-2-3-2'], ['0705.0163-1-3-3', '0705.0163-2-3-3'], ['0705.0163-1-3-4', '0705.0163-2-3-4'], ['0705.0163-1-48-1', '0705.0163-2-49-1'], ['0705.0163-1-48-2', '0705.0163-2-49-2'], ['0705.0163-1-48-3', '0705.0163-2-49-3'], ['0705.0163-1-66-0', '0705.0163-2-66-0'], ['0705.0163-1-66-1', '0705.0163-2-66-1'], ['0705.0163-1-66-2', '0705.0163-2-66-2'], ['0705.0163-1-30-0', '0705.0163-2-30-0'], ['0705.0163-1-30-1', '0705.0163-2-30-1'], ['0705.0163-1-30-2', '0705.0163-2-30-2'], ['0705.0163-1-30-3', '0705.0163-2-30-3'], ['0705.0163-1-30-4', '0705.0163-2-30-4'], ['0705.0163-1-84-0', '0705.0163-2-88-0'], ['0705.0163-1-84-1', '0705.0163-2-88-1'], ['0705.0163-1-11-0', '0705.0163-2-11-0'], ['0705.0163-1-11-1', '0705.0163-2-11-1'], ['0705.0163-1-11-2', '0705.0163-2-11-2'], ['0705.0163-1-11-3', '0705.0163-2-11-3'], ['0705.0163-1-11-4', '0705.0163-2-11-4'], ['0705.0163-1-11-5', '0705.0163-2-11-5'], ['0705.0163-1-11-6', '0705.0163-2-11-6'], ['0705.0163-1-46-0', '0705.0163-2-47-0'], ['0705.0163-1-46-1', '0705.0163-2-47-1'], ['0705.0163-1-46-2', '0705.0163-2-47-2'], ['0705.0163-1-46-3', '0705.0163-2-47-3'], ['0705.0163-1-46-4', '0705.0163-2-47-4'], ['0705.0163-1-46-5', '0705.0163-2-47-5'], ['0705.0163-1-46-7', '0705.0163-2-47-7'], ['0705.0163-1-88-1', '0705.0163-2-103-1'], ['0705.0163-1-88-3', '0705.0163-2-103-2'], ['0705.0163-1-88-4', '0705.0163-2-103-3'], ['0705.0163-1-88-6', '0705.0163-2-103-5'], ['0705.0163-1-89-5', '0705.0163-2-104-5'], ['0705.0163-1-89-6', '0705.0163-2-104-6'], ['0705.0163-1-89-7', '0705.0163-2-104-7'], ['0705.0163-1-18-0', '0705.0163-2-18-0'], ['0705.0163-1-22-0', '0705.0163-2-22-0'], ['0705.0163-1-22-1', '0705.0163-2-22-1'], ['0705.0163-1-22-5', '0705.0163-2-22-5'], ['0705.0163-1-22-6', '0705.0163-2-22-6'], ['0705.0163-1-43-0', '0705.0163-2-44-0'], ['0705.0163-1-43-1', '0705.0163-2-44-1'], ['0705.0163-1-43-2', '0705.0163-2-44-2'], ['0705.0163-1-83-0', '0705.0163-2-87-0'], ['0705.0163-1-83-1', '0705.0163-2-87-1'], ['0705.0163-1-83-2', '0705.0163-2-87-2'], ['0705.0163-1-83-3', '0705.0163-2-87-3'], ['0705.0163-1-83-5', '0705.0163-2-87-4'], ['0705.0163-1-24-0', '0705.0163-2-24-0'], ['0705.0163-1-24-1', '0705.0163-2-24-1'], ['0705.0163-1-64-0', '0705.0163-2-64-0'], ['0705.0163-1-64-3', '0705.0163-2-64-3'], ['0705.0163-1-39-0', '0705.0163-2-40-0'], ['0705.0163-1-39-1', '0705.0163-2-40-1'], ['0705.0163-1-39-2', '0705.0163-2-40-2'], ['0705.0163-1-100-0', '0705.0163-2-109-0'], ['0705.0163-1-100-3', '0705.0163-2-109-3'], ['0705.0163-1-100-5', '0705.0163-2-109-5'], ['0705.0163-1-59-0', '0705.0163-2-59-0'], ['0705.0163-1-59-1', '0705.0163-2-59-1'], ['0705.0163-1-59-2', '0705.0163-2-59-2'], ['0705.0163-1-59-3', '0705.0163-2-59-3'], ['0705.0163-1-59-4', '0705.0163-2-59-4'], ['0705.0163-1-59-5', '0705.0163-2-59-5'], ['0705.0163-1-59-6', '0705.0163-2-59-6'], ['0705.0163-1-15-0', '0705.0163-2-15-0'], ['0705.0163-1-15-1', '0705.0163-2-15-1'], ['0705.0163-1-7-0', '0705.0163-2-7-0'], ['0705.0163-1-7-2', '0705.0163-2-7-2'], ['0705.0163-1-7-3', '0705.0163-2-7-3'], ['0705.0163-1-7-4', '0705.0163-2-7-4'], ['0705.0163-1-7-5', '0705.0163-2-7-5'], ['0705.0163-1-5-0', '0705.0163-2-5-0'], ['0705.0163-1-5-1', '0705.0163-2-5-1'], ['0705.0163-1-5-2', '0705.0163-2-5-2'], ['0705.0163-1-21-0', '0705.0163-2-21-0'], ['0705.0163-1-21-1', '0705.0163-2-21-1'], ['0705.0163-1-62-4', '0705.0163-2-62-3'], ['0705.0163-1-62-5', '0705.0163-2-62-4'], ['0705.0163-1-47-0', '0705.0163-2-48-0'], ['0705.0163-1-47-2', '0705.0163-2-48-2'], ['0705.0163-1-47-3', '0705.0163-2-48-3'], ['0705.0163-1-47-4', '0705.0163-2-48-4'], ['0705.0163-1-69-0', '0705.0163-2-69-0'], ['0705.0163-1-69-1', '0705.0163-2-69-1'], ['0705.0163-1-69-2', '0705.0163-2-69-2'], ['0705.0163-1-69-3', '0705.0163-2-69-3'], ['0705.0163-1-69-4', '0705.0163-2-69-4'], ['0705.0163-1-69-5', '0705.0163-2-69-5'], ['0705.0163-1-69-6', '0705.0163-2-69-6'], ['0705.0163-1-70-1', '0705.0163-2-70-1'], ['0705.0163-1-70-2', '0705.0163-2-70-2'], ['0705.0163-1-70-3', '0705.0163-2-70-3'], ['0705.0163-1-20-0', '0705.0163-2-20-0'], ['0705.0163-1-20-1', '0705.0163-2-20-1'], ['0705.0163-1-20-2', '0705.0163-2-20-2'], ['0705.0163-1-20-5', '0705.0163-2-20-5'], ['0705.0163-1-72-3', '0705.0163-2-72-2'], ['0705.0163-1-72-4', '0705.0163-2-72-3'], ['0705.0163-1-72-5', '0705.0163-2-72-4'], ['0705.0163-1-72-6', '0705.0163-2-72-5'], ['0705.0163-1-90-0', '0705.0163-2-105-0'], ['0705.0163-1-90-1', '0705.0163-2-105-1'], ['0705.0163-1-90-2', '0705.0163-2-105-2'], ['0705.0163-1-90-4', '0705.0163-2-105-4'], ['0705.0163-1-68-1', '0705.0163-2-68-1'], ['0705.0163-1-68-2', '0705.0163-2-68-2'], ['0705.0163-1-68-3', '0705.0163-2-68-3'], ['0705.0163-1-68-4', '0705.0163-2-68-4'], ['0705.0163-1-68-5', '0705.0163-2-68-5'], ['0705.0163-1-68-6', '0705.0163-2-68-6'], ['0705.0163-1-68-7', '0705.0163-2-68-7'], ['0705.0163-1-68-8', '0705.0163-2-68-8'], ['0705.0163-1-68-9', '0705.0163-2-68-9'], ['0705.0163-1-80-0', '0705.0163-2-83-0'], ['0705.0163-1-80-1', '0705.0163-2-83-1'], ['0705.0163-1-80-2', '0705.0163-2-83-2'], ['0705.0163-1-19-0', '0705.0163-2-19-0'], ['0705.0163-1-19-1', '0705.0163-2-19-1'], ['0705.0163-1-93-5', '0705.0163-2-108-2'], ['0705.0163-1-23-0', '0705.0163-2-23-0'], ['0705.0163-1-23-1', '0705.0163-2-23-1'], ['0705.0163-1-23-3', '0705.0163-2-23-3'], ['0705.0163-1-23-4', '0705.0163-2-23-4'], ['0705.0163-1-23-5', '0705.0163-2-23-5'], ['0705.0163-1-63-0', '0705.0163-2-63-0'], ['0705.0163-1-63-1', '0705.0163-2-63-1'], ['0705.0163-1-63-2', '0705.0163-2-63-2'], ['0705.0163-1-26-4', '0705.0163-2-26-4'], ['0705.0163-1-26-5', '0705.0163-2-26-5'], ['0705.0163-1-26-6', '0705.0163-2-26-6'], ['0705.0163-1-26-7', '0705.0163-2-26-7'], ['0705.0163-1-26-9', '0705.0163-2-26-9'], ['0705.0163-1-33-0', '0705.0163-2-34-0'], ['0705.0163-1-33-2', '0705.0163-2-34-2'], ['0705.0163-1-33-3', '0705.0163-2-34-3'], ['0705.0163-1-33-4', '0705.0163-2-34-4'], ['0705.0163-1-75-2', '0705.0163-2-75-2'], ['0705.0163-1-75-3', '0705.0163-2-75-3'], ['0705.0163-1-75-4', '0705.0163-2-75-4'], ['0705.0163-1-75-5', '0705.0163-2-75-5'], ['0705.0163-1-75-7', '0705.0163-2-77-1'], ['0705.0163-1-75-8', '0705.0163-2-77-2'], ['0705.0163-1-31-1', '0705.0163-2-31-1'], ['0705.0163-1-31-2', '0705.0163-2-31-2'], ['0705.0163-1-31-3', '0705.0163-2-31-3'], ['0705.0163-1-31-4', '0705.0163-2-31-4'], ['0705.0163-1-31-5', '0705.0163-2-32-0'], ['0705.0163-1-31-6', '0705.0163-2-32-1'], ['0705.0163-1-31-7', '0705.0163-2-32-2'], ['0705.0163-1-79-4', '0705.0163-2-82-1'], ['0705.0163-1-79-5', '0705.0163-2-82-2'], ['0705.0163-1-79-0', '0705.0163-2-81-3'], ['0705.0163-1-79-1', '0705.0163-2-81-4'], ['0705.0163-1-79-2', '0705.0163-2-81-5'], ['0705.0163-1-42-2', '0705.0163-2-43-2'], ['0705.0163-1-42-6', '0705.0163-2-43-6'], ['0705.0163-1-1-6', '0705.0163-2-1-5'], ['0705.0163-1-81-0', '0705.0163-2-84-0'], ['0705.0163-1-81-1', '0705.0163-2-84-1'], ['0705.0163-1-81-6', '0705.0163-2-84-6'], ['0705.0163-1-81-7', '0705.0163-2-84-7'], ['0705.0163-1-51-0', '0705.0163-2-52-0'], ['0705.0163-1-51-3', '0705.0163-2-52-3'], ['0705.0163-1-38-1', '0705.0163-2-39-1'], ['0705.0163-1-38-3', '0705.0163-2-39-3'], ['0705.0163-1-92-1', '0705.0163-2-107-1'], ['0705.0163-1-8-2', '0705.0163-2-8-2'], ['0705.0163-1-91-0', '0705.0163-2-106-0'], ['0705.0163-1-91-1', '0705.0163-2-106-1'], ['0705.0163-1-91-2', '0705.0163-2-106-2'], ['0705.0163-1-41-1', '0705.0163-2-42-1'], ['0705.0163-1-41-3', '0705.0163-2-42-3'], ['0705.0163-1-65-4', '0705.0163-2-65-4'], ['0705.0163-1-6-0', '0705.0163-2-6-0'], ['0705.0163-1-44-6', '0705.0163-2-45-6'], ['0705.0163-1-32-1', '0705.0163-2-33-1'], ['0705.0163-1-82-1', '0705.0163-2-86-1'], ['0705.0163-1-14-4', '0705.0163-2-14-4'], ['0705.0163-1-17-1', '0705.0163-2-17-1'], ['0705.0163-1-87-0', '0705.0163-2-102-0'], ['0705.0163-1-87-1', '0705.0163-2-102-1'], ['0705.0163-1-87-3', '0705.0163-2-102-4'], ['0705.0163-1-29-0', '0705.0163-2-29-0'], ['0705.0163-1-25-0', '0705.0163-2-25-0'], ['0705.0163-1-56-4', '0705.0163-2-57-4'], ['0705.0163-1-76-0', '0705.0163-2-78-0'], ['0705.0163-1-57-0', '0705.0163-2-80-0'], ['0705.0163-1-46-6', '0705.0163-2-47-6'], ['0705.0163-1-88-0', '0705.0163-2-103-0'], ['0705.0163-1-88-5', '0705.0163-2-103-4'], ['0705.0163-1-89-3', '0705.0163-2-104-3'], ['0705.0163-1-89-4', '0705.0163-2-104-4'], ['0705.0163-1-96-4', '0705.0163-2-95-5'], ['0705.0163-1-96-5', '0705.0163-2-95-6'], ['0705.0163-1-22-2', '0705.0163-2-22-2'], ['0705.0163-1-22-3', '0705.0163-2-22-3'], ['0705.0163-1-64-1', '0705.0163-2-64-1'], ['0705.0163-1-64-2', '0705.0163-2-64-2'], ['0705.0163-1-100-1', '0705.0163-2-109-1'], ['0705.0163-1-100-2', '0705.0163-2-109-2'], ['0705.0163-1-100-7', '0705.0163-2-109-6'], ['0705.0163-1-100-8', '0705.0163-2-109-7'], ['0705.0163-1-7-1', '0705.0163-2-7-1'], ['0705.0163-1-62-0', '0705.0163-2-62-0'], ['0705.0163-1-62-1', '0705.0163-2-62-1'], ['0705.0163-1-47-1', '0705.0163-2-48-1'], ['0705.0163-1-47-5', '0705.0163-2-48-5'], ['0705.0163-1-70-0', '0705.0163-2-70-0'], ['0705.0163-1-20-3', '0705.0163-2-20-3'], ['0705.0163-1-20-6', '0705.0163-2-20-6'], ['0705.0163-1-72-1', '0705.0163-2-72-0'], ['0705.0163-1-90-3', '0705.0163-2-105-3'], ['0705.0163-1-68-0', '0705.0163-2-68-0'], ['0705.0163-1-68-10', '0705.0163-2-68-10'], ['0705.0163-1-19-2', '0705.0163-2-19-2'], ['0705.0163-1-93-3', '0705.0163-2-108-0'], ['0705.0163-1-23-2', '0705.0163-2-23-2'], ['0705.0163-1-26-0', '0705.0163-2-26-0'], ['0705.0163-1-26-1', '0705.0163-2-26-1'], ['0705.0163-1-26-2', '0705.0163-2-26-2'], ['0705.0163-1-26-3', '0705.0163-2-26-3'], ['0705.0163-1-26-8', '0705.0163-2-26-8'], ['0705.0163-1-33-1', '0705.0163-2-34-1'], ['0705.0163-1-75-0', '0705.0163-2-75-0'], ['0705.0163-1-75-1', '0705.0163-2-75-1'], ['0705.0163-1-75-6', '0705.0163-2-76-0'], ['0705.0163-1-31-0', '0705.0163-2-31-0'], ['0705.0163-1-79-3', '0705.0163-2-82-0'], ['0705.0163-1-79-6', '0705.0163-2-82-3'], ['0705.0163-1-78-1', '0705.0163-2-81-0'], ['0705.0163-1-78-2', '0705.0163-2-81-1'], ['0705.0163-1-1-5', '0705.0163-2-1-4'], ['0705.0163-1-51-1', '0705.0163-2-52-1'], ['0705.0163-1-51-5', '0705.0163-2-52-5'], ['0705.0163-1-28-1', '0705.0163-2-28-1'], ['0705.0163-1-49-0', '0705.0163-2-50-0'], ['0705.0163-1-95-0', '0705.0163-2-100-1'], ['0705.0163-1-95-1', '0705.0163-2-100-2'], ['0705.0163-1-95-2', '0705.0163-2-100-4'], ['0705.0163-1-95-3', '0705.0163-2-100-5'], ['0705.0163-1-94-0', '0705.0163-2-93-0'], ['0705.0163-1-55-2', '0705.0163-2-56-2'], ['0705.0163-1-73-1', '0705.0163-2-73-1'], ['0705.0163-1-48-0', '0705.0163-2-49-0'], ['0705.0163-1-57-1', '0705.0163-2-80-1'], ['0705.0163-1-89-0', '0705.0163-2-104-0'], ['0705.0163-1-89-1', '0705.0163-2-104-1'], ['0705.0163-1-89-2', '0705.0163-2-104-2'], ['0705.0163-1-96-2', '0705.0163-2-95-0'], ['0705.0163-1-22-4', '0705.0163-2-22-4'], ['0705.0163-1-43-3', '0705.0163-2-44-3'], ['0705.0163-1-24-2', '0705.0163-2-24-2'], ['0705.0163-1-100-4', '0705.0163-2-109-4'], ['0705.0163-1-97-2', '0705.0163-2-97-2'], ['0705.0163-1-97-3', '0705.0163-2-97-1'], ['0705.0163-1-62-3', '0705.0163-2-62-2'], ['0705.0163-1-20-4', '0705.0163-2-20-4'], ['0705.0163-1-72-2', '0705.0163-2-72-1'], ['0705.0163-1-93-4', '0705.0163-2-108-1'], ['0705.0163-1-75-9', '0705.0163-2-77-3'], ['0705.0163-1-78-5', '0705.0163-2-81-2']]

[['0705.0163-1-42-0', '0705.0163-2-43-0'], ['0705.0163-1-42-1', '0705.0163-2-43-1'], ['0705.0163-1-42-3', '0705.0163-2-43-3'], ['0705.0163-1-42-4', '0705.0163-2-43-4'], ['0705.0163-1-42-5', '0705.0163-2-43-5'], ['0705.0163-1-42-7', '0705.0163-2-43-7'], ['0705.0163-1-0-0', '0705.0163-2-0-0'], ['0705.0163-1-0-1', '0705.0163-2-0-1'], ['0705.0163-1-0-2', '0705.0163-2-0-2'], ['0705.0163-1-1-2', '0705.0163-2-1-1'], ['0705.0163-1-1-3', '0705.0163-2-1-2'], ['0705.0163-1-1-4', '0705.0163-2-1-3'], ['0705.0163-1-81-2', '0705.0163-2-84-2'], ['0705.0163-1-81-3', '0705.0163-2-84-3'], ['0705.0163-1-81-4', '0705.0163-2-84-4'], ['0705.0163-1-81-5', '0705.0163-2-84-5'], ['0705.0163-1-81-8', '0705.0163-2-84-8'], ['0705.0163-1-51-2', '0705.0163-2-52-2'], ['0705.0163-1-51-4', '0705.0163-2-52-4'], ['0705.0163-1-51-6', '0705.0163-2-52-6'], ['0705.0163-1-38-0', '0705.0163-2-39-0'], ['0705.0163-1-38-2', '0705.0163-2-39-2'], ['0705.0163-1-38-4', '0705.0163-2-39-4'], ['0705.0163-1-38-5', '0705.0163-2-39-5'], ['0705.0163-1-92-0', '0705.0163-2-107-0'], ['0705.0163-1-92-2', '0705.0163-2-107-2'], ['0705.0163-1-92-3', '0705.0163-2-107-3'], ['0705.0163-1-92-4', '0705.0163-2-107-4'], ['0705.0163-1-8-0', '0705.0163-2-8-0'], ['0705.0163-1-8-1', '0705.0163-2-8-1'], ['0705.0163-1-91-3', '0705.0163-2-106-3'], ['0705.0163-1-36-0', '0705.0163-2-37-0'], ['0705.0163-1-36-1', '0705.0163-2-37-1'], ['0705.0163-1-41-0', '0705.0163-2-42-0'], ['0705.0163-1-41-2', '0705.0163-2-42-2'], ['0705.0163-1-41-4', '0705.0163-2-42-4'], ['0705.0163-1-41-5', '0705.0163-2-42-5'], ['0705.0163-1-65-0', '0705.0163-2-65-0'], ['0705.0163-1-65-1', '0705.0163-2-65-1'], ['0705.0163-1-65-2', '0705.0163-2-65-2'], ['0705.0163-1-65-3', '0705.0163-2-65-3'], ['0705.0163-1-65-5', '0705.0163-2-65-5'], ['0705.0163-1-6-1', '0705.0163-2-6-1'], ['0705.0163-1-6-2', '0705.0163-2-6-2'], ['0705.0163-1-6-3', '0705.0163-2-6-3'], ['0705.0163-1-6-4', '0705.0163-2-6-4'], ['0705.0163-1-44-0', '0705.0163-2-45-0'], ['0705.0163-1-44-1', '0705.0163-2-45-1'], ['0705.0163-1-44-2', '0705.0163-2-45-2'], ['0705.0163-1-44-3', '0705.0163-2-45-3'], ['0705.0163-1-44-4', '0705.0163-2-45-4'], ['0705.0163-1-44-5', '0705.0163-2-45-5'], ['0705.0163-1-9-0', '0705.0163-2-9-0'], ['0705.0163-1-9-1', '0705.0163-2-9-1'], ['0705.0163-1-9-2', '0705.0163-2-9-2'], ['0705.0163-1-9-3', '0705.0163-2-9-3'], ['0705.0163-1-4-0', '0705.0163-2-4-0'], ['0705.0163-1-4-1', '0705.0163-2-4-1'], ['0705.0163-1-61-0', '0705.0163-2-61-0'], ['0705.0163-1-61-1', '0705.0163-2-61-1'], ['0705.0163-1-61-2', '0705.0163-2-61-2'], ['0705.0163-1-61-3', '0705.0163-2-61-3'], ['0705.0163-1-32-0', '0705.0163-2-33-0'], ['0705.0163-1-32-2', '0705.0163-2-33-2'], ['0705.0163-1-28-0', '0705.0163-2-28-0'], ['0705.0163-1-28-2', '0705.0163-2-28-2'], ['0705.0163-1-60-0', '0705.0163-2-60-0'], ['0705.0163-1-60-1', '0705.0163-2-60-1'], ['0705.0163-1-60-2', '0705.0163-2-60-2'], ['0705.0163-1-67-0', '0705.0163-2-67-0'], ['0705.0163-1-67-1', '0705.0163-2-67-1'], ['0705.0163-1-10-0', '0705.0163-2-10-0'], ['0705.0163-1-10-1', '0705.0163-2-10-1'], ['0705.0163-1-34-0', '0705.0163-2-35-0'], ['0705.0163-1-34-1', '0705.0163-2-35-1'], ['0705.0163-1-34-2', '0705.0163-2-35-2'], ['0705.0163-1-34-3', '0705.0163-2-35-3'], ['0705.0163-1-49-1', '0705.0163-2-50-1'], ['0705.0163-1-49-2', '0705.0163-2-50-2'], ['0705.0163-1-49-3', '0705.0163-2-50-3'], ['0705.0163-1-50-0', '0705.0163-2-51-0'], ['0705.0163-1-50-1', '0705.0163-2-51-1'], ['0705.0163-1-50-2', '0705.0163-2-51-2'], ['0705.0163-1-50-3', '0705.0163-2-51-3'], ['0705.0163-1-50-4', '0705.0163-2-51-4'], ['0705.0163-1-54-0', '0705.0163-2-55-0'], ['0705.0163-1-55-0', '0705.0163-2-56-0'], ['0705.0163-1-55-1', '0705.0163-2-56-1'], ['0705.0163-1-82-0', '0705.0163-2-86-0'], ['0705.0163-1-82-2', '0705.0163-2-86-2'], ['0705.0163-1-14-0', '0705.0163-2-14-0'], ['0705.0163-1-14-1', '0705.0163-2-14-1'], ['0705.0163-1-14-2', '0705.0163-2-14-2'], ['0705.0163-1-14-3', '0705.0163-2-14-3'], ['0705.0163-1-14-5', '0705.0163-2-14-5'], ['0705.0163-1-14-6', '0705.0163-2-14-6'], ['0705.0163-1-85-0', '0705.0163-2-89-0'], ['0705.0163-1-85-1', '0705.0163-2-89-1'], ['0705.0163-1-85-2', '0705.0163-2-89-2'], ['0705.0163-1-17-0', '0705.0163-2-17-0'], ['0705.0163-1-17-2', '0705.0163-2-17-2'], ['0705.0163-1-17-3', '0705.0163-2-17-3'], ['0705.0163-1-17-4', '0705.0163-2-17-4'], ['0705.0163-1-87-2', '0705.0163-2-102-2'], ['0705.0163-1-29-1', '0705.0163-2-29-1'], ['0705.0163-1-29-2', '0705.0163-2-29-2'], ['0705.0163-1-25-1', '0705.0163-2-25-1'], ['0705.0163-1-25-2', '0705.0163-2-25-2'], ['0705.0163-1-73-0', '0705.0163-2-73-0'], ['0705.0163-1-74-0', '0705.0163-2-74-0'], ['0705.0163-1-74-1', '0705.0163-2-74-1'], ['0705.0163-1-56-0', '0705.0163-2-57-0'], ['0705.0163-1-56-1', '0705.0163-2-57-1'], ['0705.0163-1-56-2', '0705.0163-2-57-2'], ['0705.0163-1-76-1', '0705.0163-2-78-1'], ['0705.0163-1-3-0', '0705.0163-2-3-0'], ['0705.0163-1-3-1', '0705.0163-2-3-1'], ['0705.0163-1-3-2', '0705.0163-2-3-2'], ['0705.0163-1-3-3', '0705.0163-2-3-3'], ['0705.0163-1-3-4', '0705.0163-2-3-4'], ['0705.0163-1-48-1', '0705.0163-2-49-1'], ['0705.0163-1-48-2', '0705.0163-2-49-2'], ['0705.0163-1-48-3', '0705.0163-2-49-3'], ['0705.0163-1-66-0', '0705.0163-2-66-0'], ['0705.0163-1-66-1', '0705.0163-2-66-1'], ['0705.0163-1-66-2', '0705.0163-2-66-2'], ['0705.0163-1-30-0', '0705.0163-2-30-0'], ['0705.0163-1-30-1', '0705.0163-2-30-1'], ['0705.0163-1-30-2', '0705.0163-2-30-2'], ['0705.0163-1-30-3', '0705.0163-2-30-3'], ['0705.0163-1-30-4', '0705.0163-2-30-4'], ['0705.0163-1-84-0', '0705.0163-2-88-0'], ['0705.0163-1-84-1', '0705.0163-2-88-1'], ['0705.0163-1-11-0', '0705.0163-2-11-0'], ['0705.0163-1-11-1', '0705.0163-2-11-1'], ['0705.0163-1-11-2', '0705.0163-2-11-2'], ['0705.0163-1-11-3', '0705.0163-2-11-3'], ['0705.0163-1-11-4', '0705.0163-2-11-4'], ['0705.0163-1-11-5', '0705.0163-2-11-5'], ['0705.0163-1-11-6', '0705.0163-2-11-6'], ['0705.0163-1-46-0', '0705.0163-2-47-0'], ['0705.0163-1-46-1', '0705.0163-2-47-1'], ['0705.0163-1-46-2', '0705.0163-2-47-2'], ['0705.0163-1-46-3', '0705.0163-2-47-3'], ['0705.0163-1-46-4', '0705.0163-2-47-4'], ['0705.0163-1-46-5', '0705.0163-2-47-5'], ['0705.0163-1-46-7', '0705.0163-2-47-7'], ['0705.0163-1-88-1', '0705.0163-2-103-1'], ['0705.0163-1-88-3', '0705.0163-2-103-2'], ['0705.0163-1-88-4', '0705.0163-2-103-3'], ['0705.0163-1-88-6', '0705.0163-2-103-5'], ['0705.0163-1-89-5', '0705.0163-2-104-5'], ['0705.0163-1-89-6', '0705.0163-2-104-6'], ['0705.0163-1-89-7', '0705.0163-2-104-7'], ['0705.0163-1-18-0', '0705.0163-2-18-0'], ['0705.0163-1-22-0', '0705.0163-2-22-0'], ['0705.0163-1-22-1', '0705.0163-2-22-1'], ['0705.0163-1-22-5', '0705.0163-2-22-5'], ['0705.0163-1-22-6', '0705.0163-2-22-6'], ['0705.0163-1-43-0', '0705.0163-2-44-0'], ['0705.0163-1-43-1', '0705.0163-2-44-1'], ['0705.0163-1-43-2', '0705.0163-2-44-2'], ['0705.0163-1-83-0', '0705.0163-2-87-0'], ['0705.0163-1-83-1', '0705.0163-2-87-1'], ['0705.0163-1-83-2', '0705.0163-2-87-2'], ['0705.0163-1-83-3', '0705.0163-2-87-3'], ['0705.0163-1-83-5', '0705.0163-2-87-4'], ['0705.0163-1-24-0', '0705.0163-2-24-0'], ['0705.0163-1-24-1', '0705.0163-2-24-1'], ['0705.0163-1-64-0', '0705.0163-2-64-0'], ['0705.0163-1-64-3', '0705.0163-2-64-3'], ['0705.0163-1-39-0', '0705.0163-2-40-0'], ['0705.0163-1-39-1', '0705.0163-2-40-1'], ['0705.0163-1-39-2', '0705.0163-2-40-2'], ['0705.0163-1-100-0', '0705.0163-2-109-0'], ['0705.0163-1-100-3', '0705.0163-2-109-3'], ['0705.0163-1-100-5', '0705.0163-2-109-5'], ['0705.0163-1-59-0', '0705.0163-2-59-0'], ['0705.0163-1-59-1', '0705.0163-2-59-1'], ['0705.0163-1-59-2', '0705.0163-2-59-2'], ['0705.0163-1-59-3', '0705.0163-2-59-3'], ['0705.0163-1-59-4', '0705.0163-2-59-4'], ['0705.0163-1-59-5', '0705.0163-2-59-5'], ['0705.0163-1-59-6', '0705.0163-2-59-6'], ['0705.0163-1-15-0', '0705.0163-2-15-0'], ['0705.0163-1-15-1', '0705.0163-2-15-1'], ['0705.0163-1-7-0', '0705.0163-2-7-0'], ['0705.0163-1-7-2', '0705.0163-2-7-2'], ['0705.0163-1-7-3', '0705.0163-2-7-3'], ['0705.0163-1-7-4', '0705.0163-2-7-4'], ['0705.0163-1-7-5', '0705.0163-2-7-5'], ['0705.0163-1-5-0', '0705.0163-2-5-0'], ['0705.0163-1-5-1', '0705.0163-2-5-1'], ['0705.0163-1-5-2', '0705.0163-2-5-2'], ['0705.0163-1-21-0', '0705.0163-2-21-0'], ['0705.0163-1-21-1', '0705.0163-2-21-1'], ['0705.0163-1-62-4', '0705.0163-2-62-3'], ['0705.0163-1-62-5', '0705.0163-2-62-4'], ['0705.0163-1-47-0', '0705.0163-2-48-0'], ['0705.0163-1-47-2', '0705.0163-2-48-2'], ['0705.0163-1-47-3', '0705.0163-2-48-3'], ['0705.0163-1-47-4', '0705.0163-2-48-4'], ['0705.0163-1-69-0', '0705.0163-2-69-0'], ['0705.0163-1-69-1', '0705.0163-2-69-1'], ['0705.0163-1-69-2', '0705.0163-2-69-2'], ['0705.0163-1-69-3', '0705.0163-2-69-3'], ['0705.0163-1-69-4', '0705.0163-2-69-4'], ['0705.0163-1-69-5', '0705.0163-2-69-5'], ['0705.0163-1-69-6', '0705.0163-2-69-6'], ['0705.0163-1-70-1', '0705.0163-2-70-1'], ['0705.0163-1-70-2', '0705.0163-2-70-2'], ['0705.0163-1-70-3', '0705.0163-2-70-3'], ['0705.0163-1-20-0', '0705.0163-2-20-0'], ['0705.0163-1-20-1', '0705.0163-2-20-1'], ['0705.0163-1-20-2', '0705.0163-2-20-2'], ['0705.0163-1-20-5', '0705.0163-2-20-5'], ['0705.0163-1-72-3', '0705.0163-2-72-2'], ['0705.0163-1-72-4', '0705.0163-2-72-3'], ['0705.0163-1-72-5', '0705.0163-2-72-4'], ['0705.0163-1-72-6', '0705.0163-2-72-5'], ['0705.0163-1-90-0', '0705.0163-2-105-0'], ['0705.0163-1-90-1', '0705.0163-2-105-1'], ['0705.0163-1-90-2', '0705.0163-2-105-2'], ['0705.0163-1-90-4', '0705.0163-2-105-4'], ['0705.0163-1-68-1', '0705.0163-2-68-1'], ['0705.0163-1-68-2', '0705.0163-2-68-2'], ['0705.0163-1-68-3', '0705.0163-2-68-3'], ['0705.0163-1-68-4', '0705.0163-2-68-4'], ['0705.0163-1-68-5', '0705.0163-2-68-5'], ['0705.0163-1-68-6', '0705.0163-2-68-6'], ['0705.0163-1-68-7', '0705.0163-2-68-7'], ['0705.0163-1-68-8', '0705.0163-2-68-8'], ['0705.0163-1-68-9', '0705.0163-2-68-9'], ['0705.0163-1-80-0', '0705.0163-2-83-0'], ['0705.0163-1-80-1', '0705.0163-2-83-1'], ['0705.0163-1-80-2', '0705.0163-2-83-2'], ['0705.0163-1-19-0', '0705.0163-2-19-0'], ['0705.0163-1-19-1', '0705.0163-2-19-1'], ['0705.0163-1-93-5', '0705.0163-2-108-2'], ['0705.0163-1-23-0', '0705.0163-2-23-0'], ['0705.0163-1-23-1', '0705.0163-2-23-1'], ['0705.0163-1-23-3', '0705.0163-2-23-3'], ['0705.0163-1-23-4', '0705.0163-2-23-4'], ['0705.0163-1-23-5', '0705.0163-2-23-5'], ['0705.0163-1-63-0', '0705.0163-2-63-0'], ['0705.0163-1-63-1', '0705.0163-2-63-1'], ['0705.0163-1-63-2', '0705.0163-2-63-2'], ['0705.0163-1-26-4', '0705.0163-2-26-4'], ['0705.0163-1-26-5', '0705.0163-2-26-5'], ['0705.0163-1-26-6', '0705.0163-2-26-6'], ['0705.0163-1-26-7', '0705.0163-2-26-7'], ['0705.0163-1-26-9', '0705.0163-2-26-9'], ['0705.0163-1-33-0', '0705.0163-2-34-0'], ['0705.0163-1-33-2', '0705.0163-2-34-2'], ['0705.0163-1-33-3', '0705.0163-2-34-3'], ['0705.0163-1-33-4', '0705.0163-2-34-4'], ['0705.0163-1-75-2', '0705.0163-2-75-2'], ['0705.0163-1-75-3', '0705.0163-2-75-3'], ['0705.0163-1-75-4', '0705.0163-2-75-4'], ['0705.0163-1-75-5', '0705.0163-2-75-5'], ['0705.0163-1-75-7', '0705.0163-2-77-1'], ['0705.0163-1-75-8', '0705.0163-2-77-2'], ['0705.0163-1-31-1', '0705.0163-2-31-1'], ['0705.0163-1-31-2', '0705.0163-2-31-2'], ['0705.0163-1-31-3', '0705.0163-2-31-3'], ['0705.0163-1-31-4', '0705.0163-2-31-4'], ['0705.0163-1-31-5', '0705.0163-2-32-0'], ['0705.0163-1-31-6', '0705.0163-2-32-1'], ['0705.0163-1-31-7', '0705.0163-2-32-2'], ['0705.0163-1-79-4', '0705.0163-2-82-1'], ['0705.0163-1-79-5', '0705.0163-2-82-2'], ['0705.0163-1-79-0', '0705.0163-2-81-3'], ['0705.0163-1-79-1', '0705.0163-2-81-4'], ['0705.0163-1-79-2', '0705.0163-2-81-5']]

[['0705.0163-1-42-2', '0705.0163-2-43-2'], ['0705.0163-1-42-6', '0705.0163-2-43-6'], ['0705.0163-1-1-6', '0705.0163-2-1-5'], ['0705.0163-1-81-0', '0705.0163-2-84-0'], ['0705.0163-1-81-1', '0705.0163-2-84-1'], ['0705.0163-1-81-6', '0705.0163-2-84-6'], ['0705.0163-1-81-7', '0705.0163-2-84-7'], ['0705.0163-1-51-0', '0705.0163-2-52-0'], ['0705.0163-1-51-3', '0705.0163-2-52-3'], ['0705.0163-1-38-1', '0705.0163-2-39-1'], ['0705.0163-1-38-3', '0705.0163-2-39-3'], ['0705.0163-1-92-1', '0705.0163-2-107-1'], ['0705.0163-1-8-2', '0705.0163-2-8-2'], ['0705.0163-1-91-0', '0705.0163-2-106-0'], ['0705.0163-1-91-1', '0705.0163-2-106-1'], ['0705.0163-1-91-2', '0705.0163-2-106-2'], ['0705.0163-1-41-1', '0705.0163-2-42-1'], ['0705.0163-1-41-3', '0705.0163-2-42-3'], ['0705.0163-1-65-4', '0705.0163-2-65-4'], ['0705.0163-1-6-0', '0705.0163-2-6-0'], ['0705.0163-1-44-6', '0705.0163-2-45-6'], ['0705.0163-1-32-1', '0705.0163-2-33-1'], ['0705.0163-1-82-1', '0705.0163-2-86-1'], ['0705.0163-1-14-4', '0705.0163-2-14-4'], ['0705.0163-1-17-1', '0705.0163-2-17-1'], ['0705.0163-1-87-0', '0705.0163-2-102-0'], ['0705.0163-1-87-1', '0705.0163-2-102-1'], ['0705.0163-1-87-3', '0705.0163-2-102-4'], ['0705.0163-1-29-0', '0705.0163-2-29-0'], ['0705.0163-1-25-0', '0705.0163-2-25-0'], ['0705.0163-1-56-4', '0705.0163-2-57-4'], ['0705.0163-1-76-0', '0705.0163-2-78-0'], ['0705.0163-1-57-0', '0705.0163-2-80-0'], ['0705.0163-1-46-6', '0705.0163-2-47-6'], ['0705.0163-1-88-0', '0705.0163-2-103-0'], ['0705.0163-1-88-5', '0705.0163-2-103-4'], ['0705.0163-1-89-3', '0705.0163-2-104-3'], ['0705.0163-1-89-4', '0705.0163-2-104-4'], ['0705.0163-1-96-4', '0705.0163-2-95-5'], ['0705.0163-1-96-5', '0705.0163-2-95-6'], ['0705.0163-1-22-2', '0705.0163-2-22-2'], ['0705.0163-1-22-3', '0705.0163-2-22-3'], ['0705.0163-1-64-1', '0705.0163-2-64-1'], ['0705.0163-1-64-2', '0705.0163-2-64-2'], ['0705.0163-1-100-1', '0705.0163-2-109-1'], ['0705.0163-1-100-2', '0705.0163-2-109-2'], ['0705.0163-1-100-7', '0705.0163-2-109-6'], ['0705.0163-1-100-8', '0705.0163-2-109-7'], ['0705.0163-1-7-1', '0705.0163-2-7-1'], ['0705.0163-1-62-0', '0705.0163-2-62-0'], ['0705.0163-1-62-1', '0705.0163-2-62-1'], ['0705.0163-1-47-1', '0705.0163-2-48-1'], ['0705.0163-1-47-5', '0705.0163-2-48-5'], ['0705.0163-1-70-0', '0705.0163-2-70-0'], ['0705.0163-1-20-3', '0705.0163-2-20-3'], ['0705.0163-1-20-6', '0705.0163-2-20-6'], ['0705.0163-1-72-1', '0705.0163-2-72-0'], ['0705.0163-1-90-3', '0705.0163-2-105-3'], ['0705.0163-1-68-0', '0705.0163-2-68-0'], ['0705.0163-1-68-10', '0705.0163-2-68-10'], ['0705.0163-1-19-2', '0705.0163-2-19-2'], ['0705.0163-1-93-3', '0705.0163-2-108-0'], ['0705.0163-1-23-2', '0705.0163-2-23-2'], ['0705.0163-1-26-0', '0705.0163-2-26-0'], ['0705.0163-1-26-1', '0705.0163-2-26-1'], ['0705.0163-1-26-2', '0705.0163-2-26-2'], ['0705.0163-1-26-3', '0705.0163-2-26-3'], ['0705.0163-1-26-8', '0705.0163-2-26-8'], ['0705.0163-1-33-1', '0705.0163-2-34-1'], ['0705.0163-1-75-0', '0705.0163-2-75-0'], ['0705.0163-1-75-1', '0705.0163-2-75-1'], ['0705.0163-1-75-6', '0705.0163-2-76-0'], ['0705.0163-1-31-0', '0705.0163-2-31-0'], ['0705.0163-1-79-3', '0705.0163-2-82-0'], ['0705.0163-1-79-6', '0705.0163-2-82-3'], ['0705.0163-1-78-1', '0705.0163-2-81-0'], ['0705.0163-1-78-2', '0705.0163-2-81-1']]

[]

[['0705.0163-1-1-5', '0705.0163-2-1-4'], ['0705.0163-1-51-1', '0705.0163-2-52-1'], ['0705.0163-1-51-5', '0705.0163-2-52-5'], ['0705.0163-1-28-1', '0705.0163-2-28-1'], ['0705.0163-1-49-0', '0705.0163-2-50-0'], ['0705.0163-1-95-0', '0705.0163-2-100-1'], ['0705.0163-1-95-1', '0705.0163-2-100-2'], ['0705.0163-1-95-2', '0705.0163-2-100-4'], ['0705.0163-1-95-3', '0705.0163-2-100-5'], ['0705.0163-1-94-0', '0705.0163-2-93-0'], ['0705.0163-1-55-2', '0705.0163-2-56-2'], ['0705.0163-1-73-1', '0705.0163-2-73-1'], ['0705.0163-1-48-0', '0705.0163-2-49-0'], ['0705.0163-1-57-1', '0705.0163-2-80-1'], ['0705.0163-1-89-0', '0705.0163-2-104-0'], ['0705.0163-1-89-1', '0705.0163-2-104-1'], ['0705.0163-1-89-2', '0705.0163-2-104-2'], ['0705.0163-1-96-2', '0705.0163-2-95-0'], ['0705.0163-1-22-4', '0705.0163-2-22-4'], ['0705.0163-1-43-3', '0705.0163-2-44-3'], ['0705.0163-1-24-2', '0705.0163-2-24-2'], ['0705.0163-1-100-4', '0705.0163-2-109-4'], ['0705.0163-1-97-2', '0705.0163-2-97-2'], ['0705.0163-1-97-3', '0705.0163-2-97-1'], ['0705.0163-1-62-3', '0705.0163-2-62-2'], ['0705.0163-1-20-4', '0705.0163-2-20-4'], ['0705.0163-1-72-2', '0705.0163-2-72-1'], ['0705.0163-1-93-4', '0705.0163-2-108-1'], ['0705.0163-1-75-9', '0705.0163-2-77-3'], ['0705.0163-1-78-5', '0705.0163-2-81-2']]

[]

[]

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

https://arxiv.org/abs/0705.0163

1712.01561

{'1712.01561-1-0-0': 'A way to measure the community structure of a network is the clustering coefficient.', '1712.01561-1-0-1': 'Such a quantity is based on the number of existing triangles around the nodes over the theoretical ones.', '1712.01561-1-0-2': 'To the best of our knowledge, scarce attention has been paid to the fictitious triangles due to the presence of indirect connections among the nodes of the network.', '1712.01561-1-0-3': 'This paper fills this gap by providing a new definition of the clustering coefficient for weighted networks when missing links might be also considered.', '1712.01561-1-0-4': 'Specifically, a novel concept of triangles is here introduced by assuming that a strong enough aggregate weight of two arcs sharing a node induces a link between the not common nodes.', '1712.01561-1-0-5': 'Beyond the intuitive meaning of such social triangles, we also explore the usefulness of them for gaining insights on the topological structure of the underline network.', '1712.01561-1-0-6': 'Empirical experiments on the standard networks of 500 commercial US airports and on the nervous system of the Caenorhabditis elegans support the theoretical framework.', '1712.01561-1-1-0': 'Keywords: networks; clustering coefficient; weighted arcs; social triangles.', '1712.01561-1-2-0': '# Introduction', '1712.01561-1-3-0': 'The network approach to complex systems has revealed several general and unexpected findings applicable to a large number of systems, such as the ubiquity of scale freeness, the frequent appearance of high clustering, and the relationship between functionality and the presence of specific motifs ([CITATION], [CITATION]).', '1712.01561-1-4-0': 'In recent years, it has become clear that it is relevant to consider the heterogeneity of the interactions and their correlations with the network structure in order to understand the characteristics of the system.', '1712.01561-1-4-1': 'In particular, a survey of measures of complex networks is reported in [CITATION] while an extended approach related to some centrality measures is presented in [CITATION].', '1712.01561-1-4-2': 'Such quantities are applied in different real networks, for example, in [CITATION] are considered measures of importance and power in terrorist networks, in [CITATION] are analyzed corporate systems while in [CITATION] are examined social networks.', '1712.01561-1-5-0': 'We consider, in this work, the tendency for clustering, i.e. the link formation between neighboring vertices [CITATION] that reveals the clustering of edges in tightly connected neighborhood and identifies the local group cohesiveness.', '1712.01561-1-6-0': 'The measure for assessing the tendency of vertices to cluster is the local cluster coefficient [CITATION].', '1712.01561-1-6-1': 'Such quantity has been extensively studied by several authors and applied in different networks ([CITATION], [CITATION], [CITATION], [CITATION]).', '1712.01561-1-6-2': 'It captures the degree of social embeddedness of the nodes in a network and is based on local density [CITATION].', '1712.01561-1-6-3': 'Indeed, especially in social networks, vertices tend to create tightly knit groups that are characterized by a relatively high density of links [CITATION].', '1712.01561-1-7-0': "The clustering coefficient assesses the connectivity in a node's neighborhood; a node has a high value of clustering coefficient of its neighbors tends to be directly connected with each other [CITATION].", '1712.01561-1-7-1': 'This quantity is relevant to determine the small-world property of a network [CITATION] and can be considered as an index of the redundancy of a node ([CITATION], [CITATION]).', '1712.01561-1-7-2': 'In the contribution of Benati et al. [CITATION], it is proposed a new combinatorial model to detect clusters that takes into consideration the standard individual data pertinent to a single population unit and the data describing the connections among units.', '1712.01561-1-7-3': 'More in general, a review of the dominant set clustering framework is presented in [CITATION].', '1712.01561-1-7-4': 'Regarding the clustering coefficient in weighted networks, it has been analyzed in ([CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION]) as reported in Section 3.', '1712.01561-1-7-5': 'Related to the clustering coefficient in terms of link prediction, the estimation of the likelihood of new link creation is presented in ([CITATION], [CITATION]).', '1712.01561-1-7-6': 'The connections between nodes using suitable definitions of the matrix governing the connections has been considered by [CITATION].', '1712.01561-1-8-0': 'The analysis of the weights along the edges and their correlations is able to change the view of the hierarchical and structural organization of the systems.', '1712.01561-1-8-1': 'This is evident if we consider, as an example, a network in which the weights of all links forming triangles of interconnected vertices are extremely small.', '1712.01561-1-8-2': 'In this case, even for a large clustering coefficient, these triangles play a minimal role in the network dynamics and organization, and the clustering features are certainly overestimated by a simple structural analysis [CITATION].', '1712.01561-1-8-3': 'Also, vertices with high degree can be attached to a majority of low-degree nodes whilst concentrating the largest portion of their strength only on the vertices with high degree.', '1712.01561-1-8-4': 'In this situation, the topology reveals a disassortative characteristic of the network, whereas the system could be considered assortative since the more relevant edges in terms of weights are linked to the high-degree vertices.', '1712.01561-1-9-0': 'In this paper, we aim at elaborating on the presence of large weights on the arcs of the network, which are able to induce indirect links among the nodes.', '1712.01561-1-10-0': 'Specifically, we provide a new definition of the concept of triangles of a network of social type, by allowing the presence of indirect links.', '1712.01561-1-10-1': 'One has triangles even in the case of one missing side, but under the constraint that the weights of the remaining sides - suitably aggregated - are large enough.', '1712.01561-1-10-2': 'These new triangles admit the introduction of social intensity as a potential mechanism for links formation.', '1712.01561-1-11-0': 'In particular, we define two different new kinds of triangle around a node [MATH]: the first one is composed by two not linked nodes which are adjacent to [MATH]; the second one is composed by a trajectory of length two from [MATH] such that [MATH] is not linked with the extreme node of the trajectory.', '1712.01561-1-12-0': 'The constraints on the weights of the arcs are driven by two thresholds, one for each type of triangle.', '1712.01561-1-12-1': 'As we will see below in the formalization of our setting, null thresholds mean no constraints - and all the two-sided figures are triangles - while a large value of the thresholds is associated to very restrictive constraints - and a small number of two-sided figures will be accepted as triangles.', '1712.01561-1-13-0': 'It is very important to notice that the case of zero thresholds gives also insights on the topological structure of the unweighted graph associated to the network.', '1712.01561-1-13-1': 'We address the reader to the empirical analysis section for an intuitive explanation of this point.', '1712.01561-1-14-0': 'Starting from the considered definition of triangles, a definition of a new generalized weighted clustering coefficient is provided.', '1712.01561-1-14-1': 'This measure is able to give useful hints on the community structure of the network, when weights play a relevant role in inducing missing links among the node.', '1712.01561-1-14-2': 'Moreover, this measure is also useful for predicting the fictitious links that may appear in the future of evolving networks.', '1712.01561-1-14-3': 'Link prediction is a relevant problem that attempt to estimate the likelihood of the existence of a link between two vertices based on observed links and the attributes of nodes [CITATION].', '1712.01561-1-14-4': 'Such prediction can be used to analyze a network to suggest promising interactions or collaborations that have not yet identified or is related to the problem of inferring missing or additional links that, while not directly visible, are likely to exist ([CITATION], [CITATION]).', '1712.01561-1-14-5': 'Our generalized clustering coefficient has a further very relevant property: it assumes unitary value in several situations and not only when the graph is a clique.', '1712.01561-1-14-6': 'Specifically, the community structure of the network is intended to include also the realistic cases of the presence of indirect connections among two agents induced by their strong links with a third node.', '1712.01561-1-15-0': 'To gain more information, we have also implemented a sensitivity analysis of the clustering coefficient with respect to the exogenous thresholds defining the triangles.', '1712.01561-1-15-1': 'Moreover, a discussion on the way to aggregate the weights for identifying the triangles has been provided, in order to specify different concepts of sociality.', '1712.01561-1-15-2': 'The proposed clustering coefficient is tested on two well-established empirical settings.', '1712.01561-1-16-0': 'The paper is structured as follows; after preliminaries and notations about the graph theory (Section 2), we review the literature on the clustering coefficient in weighted networks (Section 3), we propose a generalized clustering coefficient and generalized triangles with the relative interpretation (Section 4), the computational experience of two empirical networks: the nervous system of the nematode Caenorhabditis elegans and the network among the 500 commercial airports in the United States is presented in Section 5.', '1712.01561-1-16-1': 'The paper ends with some conclusions and remarks on future research directions.', '1712.01561-1-17-0': '# Preliminaries and notations about graph theory', '1712.01561-1-18-0': 'Networks can be considered as a valuable representation of many complex systems found in the real world.', '1712.01561-1-18-1': 'The abstract representation of a network is a graph.', '1712.01561-1-18-2': 'It consists of a set of vertices (or nodes) and a set of edges (or links).', '1712.01561-1-18-3': 'The presence of an edge between two vertices signifies the presence of some kind of interactions or connections between the vertices.', '1712.01561-1-18-4': 'The classical mathematical abstraction of a network is a graph [MATH], where [MATH] is the set of [MATH] nodes (or vertices) and [MATH] is the set of [MATH] links (or edges) stating the relationships among the nodes.', '1712.01561-1-18-5': 'We refer to a node by an index [MATH], meaning that we allow a one-to-one correspondence between an index in [MATH] and a node in [MATH].', '1712.01561-1-18-6': 'Generally, the existence of a link between two given nodes is captured by a binary variable, whose value is 1 if the link exists and 0 otherwise.', '1712.01561-1-18-7': 'In so doing, the set [MATH] can be conceptualized through a squared matrix of order [MATH] - the so-called adjacent matrix [MATH] - which is filled by 0s and 1s and whose general element [MATH] is 1 if the link between [MATH] and [MATH] exists and 0 otherwise.', '1712.01561-1-18-8': 'The graph is symmetric when [MATH], for each [MATH], and asymmetric otherwise.', '1712.01561-1-18-9': 'In this latter case, links are (and the graph is) oriented.', '1712.01561-1-18-10': 'In this paper, we examine weighted networks that can be denoted mathematically by an adjacency matrix [MATH] with elements [MATH] which represent the weights on the link connecting nodes [MATH] and [MATH], with [MATH].', '1712.01561-1-18-11': "Thus, [MATH] denotes the intensity of the interactions between two nodes [MATH] and [MATH] and allows for modeling the ties' strength of the observed system.", '1712.01561-1-19-0': 'The degree [MATH] of the node [MATH] is a nonnegative integer representing the number of links incident upon [MATH].', '1712.01561-1-19-1': 'The degree gives a measure of how well a node is connected to the other elements of the graph, and provides also information on which nodes tend to cluster together.', '1712.01561-1-20-0': '# Literature review on the clustering coefficient for unweighted and weighted networks', '1712.01561-1-21-0': '## Unweighted networks', '1712.01561-1-22-0': 'The local clustering coefficient is defined for any vertex [MATH] as the fraction of its connected neighbors and captures the capacity of edge creations among neighbors, i.e. the tendency in the network to create stable groups [CITATION].', '1712.01561-1-22-1': 'Thus, the clustering around a vertex [MATH] is quantified by the (unweighted) local clustering coefficient [MATH] defined as the number of triangles in which vertex [MATH] participates normalized by the maximum possible number of such triangles: [EQUATION] where [MATH] represents the number of triangles around [MATH].', '1712.01561-1-22-2': 'The local clustering coefficient quantifies how well a node is connected to the other elements of the graph together with how nodes tend to cluster together.', '1712.01561-1-22-3': 'Therefore, [MATH] if none of the neighbors of a node are connected and [MATH] if all of the neighbors are linked.', '1712.01561-1-23-0': 'The value of the local clustering coefficient is influenced by the nodes degrees, a node with several neighbours is likely to be embedded in rather fewer closed triangles.', '1712.01561-1-23-1': 'Hence it has a smaller local clustering coefficient respect to a node linked to fewer neighbors where they are more likely to be clustered in triangles [CITATION].', '1712.01561-1-24-0': 'The clustering coefficient for a given graph is computed in two classical modes [CITATION].', '1712.01561-1-24-1': 'The first one is the averaged clustering coefficient [MATH] given as the average of all the local clustering coefficients, while the second, called the global clustering coefficient and denoted by [MATH], is defined as the ratio among three times the number of closed triangles in the graph and the number of its triplets, i.e. the number of 2-paths among three nodes.', '1712.01561-1-25-0': 'Note that both [MATH] and [MATH] assume values from [MATH] to [MATH] and are equal to [MATH] in case of a clique, i.e. a fully coupled network.', '1712.01561-1-25-1': 'In real networks, the evidence shows that nodes are inclined to cluster into densely connected groups ([CITATION], [CITATION]) and the difficulty to compare the values of clustering of nodes with different degrees makes the average value of local clustering sensitive with respect to how degrees are distributed across the whole network.', '1712.01561-1-26-0': 'The quantities [MATH] and [MATH] are specifically talored to unweighted networks, and they cannot be satisfactorily employed for describing the community structure of the network in presence of weights on the links and when arcs are of direct type.', '1712.01561-1-27-0': 'Next section is devoted to the analysis of the more general weighted case.', '1712.01561-1-28-0': '## Weighted networks', '1712.01561-1-29-0': 'In many real networks, connections are relevant not only in terms of the classical binary state - whenever they exist or do not exist - but also with regards to their strength which, for any node [MATH], is defined as: [EQUATION]', '1712.01561-1-29-1': 'The introduction of weights and strengths extends the study of the macroscopic properties of the network, by adding to the mere interactions some forms of entity of connections and capability.', '1712.01561-1-29-2': 'In particular, the strength integrates information about the vertex connectivity and the weights of its links [CITATION].', '1712.01561-1-29-3': 'It is considered a natural measure of the importance or centrality of a vertex [MATH].', '1712.01561-1-29-4': 'Indeed, the identification of the most central nodes represents a major issue in network characterization [CITATION].', '1712.01561-1-30-0': 'Barrat et al. [CITATION] combine the topological information of the network with the distribution of weights along links and define the weighted clustering coefficient for a node [MATH] as follows: [EQUATION]', '1712.01561-1-30-1': 'This coefficient is a quantity of the local cohesiveness that considers the importance of the clustered structure taking into account the intensity of the interactions found on the local triangles.', '1712.01561-1-30-2': 'This measure counts for each triangle created in the neighborhood of the node [MATH] the weight of the two related edges.', '1712.01561-1-30-3': 'The authors refer not just to the number of the closed triangles in the neighborhood of a node but also to their total relative weight with respect to the strength of the nodes.', '1712.01561-1-31-0': 'The normalization factor [MATH] accounts for the weights of each edge times the maximum possible number of triangles in which it may participate, and it ensures that [MATH].', '1712.01561-1-31-1': 'The definition of [MATH] recovers the topological clustering coefficient in the case where [MATH] is constant, for each [MATH].', '1712.01561-1-32-0': 'Therefore, the authors introduce the weighted clustering coefficient averaged over all nodes of the network, say [MATH], and over all nodes with degree [MATH], say [MATH].', '1712.01561-1-32-1': 'These measures offer global information on the correlation between weights and topology by comparing them with their topological analogs.', '1712.01561-1-33-0': 'Note that [MATH], so [MATH] can be written as: [EQUATION] where [MATH].', '1712.01561-1-33-1': 'In such equation the contribution of each triangle is weighted by a ratio of the average weight of the two adjacent links of the triangle to the average weight [MATH].', '1712.01561-1-34-0': 'Thus, [MATH] compares the weights related with triangles to the average weight of edges connected to the local node.', '1712.01561-1-35-0': 'Zhang and Horvath [CITATION] describe the weighted clustering coefficient in the context of gene co-expression networks.', '1712.01561-1-35-1': 'Unlike the unweighted clustering coefficient, the weighted clustering coefficient is not inversely related to the connectivity.', '1712.01561-1-35-2': 'Authors show a model that reveals how an inverse relationship between clustering coefficient and connectivity occurs from hard thresholding.', '1712.01561-1-35-3': 'In formula: [EQUATION] where the weights have been normalized by [MATH].', '1712.01561-1-35-4': 'The number of triangles around the node [MATH] can be written in terms of the adjacency matrix elements as [MATH] and the numerator of the above equation is a weighted generalization of the formula.', '1712.01561-1-35-5': 'The denominator has been selected by considering the upper bound of the numerator, ensuring [MATH] .', '1712.01561-1-35-6': 'The equation can be written as: [EQUATION]', '1712.01561-1-35-7': 'In [CITATION] a similar definition has been shown, indeed the edge weights are considered as probabilities such that in an ensemble of networks, [MATH] and [MATH] are linked with probability [MATH].', '1712.01561-1-35-8': 'Finally, Holme et al [CITATION] discuss the definition of weights and express a redefined weighted clustering coefficient as: [EQUATION] where [MATH] indicates a matrix where each entry equals [MATH].', '1712.01561-1-35-9': 'This equation seems similar to those of [CITATION] though, [MATH] is not required in the denominator sum.', '1712.01561-1-36-0': 'Onnela et al. ([CITATION], [CITATION]) refer to the notion of motif defining it as a set (ensemble) of topologically equivalent subgraphs of a network.', '1712.01561-1-36-1': 'In case of weighted systems, it is relevant to deal with intensities instead of numbers of occurrence.', '1712.01561-1-36-2': 'Moreover, the latter concept is considered as a special case of the former one.', '1712.01561-1-36-3': 'For the authors the triangles are among the simplest nontrivial motifs and have a crucial role as one of the classic quantities of network characterization in defining the clustering coefficient of a node [MATH].', '1712.01561-1-36-4': 'They propose a weighted clustering coefficient taking into consideration the subgraph intensity that is defined as the geometric average of subgraph edge weights.', '1712.01561-1-36-5': 'In formula: [EQUATION] where [MATH] are the edge weights normalized by the maximum weight in the network of the edges linking [MATH] to the other nodes of [MATH].', '1712.01561-1-37-0': 'Formula ([REF]) shows that triangles contribute to the creation of [MATH] according to the weights associated to their three edges.', '1712.01561-1-37-1': 'More specifically, [MATH] disregards the strength of the local node and measure triangle weights only in relation to the maximum edge weight.', '1712.01561-1-38-0': 'Moreover, [MATH] collapses to [MATH] when, for each [MATH], one has [MATH], hence in the unweighted case.', '1712.01561-1-39-0': '# The generalized clustering coefficient', '1712.01561-1-40-0': 'This section contains our proposal for a new definition of the clustering coefficient of weighted networks.', '1712.01561-1-40-1': 'The ground of the definition is a novel concept of triangles, to include also the presence of real indirect connections among individuals.', '1712.01561-1-40-2': 'To our purpose, we first provide the definition of the triangles and discuss it, and then we introduce the clustering coefficient.', '1712.01561-1-41-0': '## Generalized triangles', '1712.01561-1-42-0': 'Here, we propose a generalization of the concept of triangle, and rewrite accordingly the indices [MATH] in ([REF]) for the case of weighted networks.', '1712.01561-1-43-0': 'Let us consider a weighted non-oriented graph [MATH] with vertices [MATH], symmetric adjacent matrix [MATH] and weight matrix [MATH], with nonnegative weights.', '1712.01561-1-43-1': 'Moreover, let us take [MATH] and a function [MATH] which is not decreasing in its arguments.', '1712.01561-1-44-0': 'For each triple of distinct vertices [MATH], a subgraph [MATH] is a triangle around [MATH] if one of the following conditions are satisfied:', '1712.01561-1-45-0': 'We denote the set of triangles associated to case [MATH] as [MATH], for [MATH].', '1712.01561-1-45-1': 'By definition, [MATH].', '1712.01561-1-45-2': 'We denote the set collecting all the triangles by [MATH].', '1712.01561-1-46-0': 'Figure [REF] reports the three different type of triangles, respectively [MATH], [MATH] and [MATH].', '1712.01561-1-46-1': 'Clearly, in case [MATH], the concept of triangle given in Definition [REF] coincides with the standard one.', '1712.01561-1-47-0': 'Note that with [MATH] nodes, the maximum number of possible triangles is [MATH].', '1712.01561-1-47-1': 'This is the case of a clique with [MATH], for each [MATH].', '1712.01561-1-48-0': 'When considering the maximum number of candidates triangles for a node [MATH] to belong to [MATH], it is [MATH].', '1712.01561-1-48-1': 'Then, in this case for the node [MATH] the number of triangles is [MATH]', '1712.01561-1-49-0': 'Triangles in [MATH] for node [MATH] are path of length [MATH] (i.e. triplets), that can be computed considering the square of the adjacency matrix.', '1712.01561-1-49-1': 'Indeed.', '1712.01561-1-49-2': 'the number of different path of length [MATH] from [MATH] to [MATH] equals the entry [MATH] of [MATH] [CITATION].', '1712.01561-1-49-3': 'For a given row [MATH] of [MATH], the sum of the element (excluding the element [MATH]) equals the maximum potential number of pseudo-triangles of type [MATH].', '1712.01561-1-50-0': 'Herein we define the elements in [MATH] and [MATH] as pseudo-triangles since they do not are contained in [MATH] but can be seen as triangles under conditions on the weights of the two edges.', '1712.01561-1-51-0': 'Figure [REF] shows the types of triangles, without emphasis on the conditions on the weights.', '1712.01561-1-52-0': '### Interpretation of the triangles and equivalent graphs', '1712.01561-1-53-0': 'The pseudo-triangles [MATH] and [MATH] have an interpretation whose ground is the theory of social networks.', '1712.01561-1-54-0': 'The former case describes a situation in which agent [MATH] has a direct relationship with agents [MATH] and [MATH].', '1712.01561-1-54-1': 'One can say that there exists a triangle among the three if the strength of the connections of [MATH] with the others is sufficiently high - in the sense described by function [MATH].', '1712.01561-1-54-2': 'The idea is that the cooperation between [MATH] and the other agents is so effective and fruitful that the presence of a direct link between [MATH] and [MATH] is not required.', '1712.01561-1-55-0': 'The latter case is associated to the presence of a strong link between [MATH] and [MATH] and between [MATH] and [MATH], always in terms of the entities of the weights - in the sense described by function [MATH].', '1712.01561-1-55-1': 'In this peculiar situation, the node [MATH] represents the intermediate agent letting also the (indirect) collaboration between [MATH] and [MATH] be possible.', '1712.01561-1-56-0': 'In our empirical experiments (see below), we have considered four cases of function [MATH]:', '1712.01561-1-57-0': 'The selection of the specific function [MATH] - to be implemented among [MATH] defined above - provides further insights on the interpretation of the triangles of type [MATH] and [MATH].', '1712.01561-1-57-1': 'Indeed, once [MATH] and [MATH] are kept fixed, then [MATH] and [MATH] state that both weights of the considered edges should be taken into account in an identical way by considering their mere aggregation in the former case or their mean in the latter one.', '1712.01561-1-57-2': 'When considering functions [MATH] and [MATH], only one of the weights is relevant for the measurement of the strength of the connections - the minimum weight and the maximum one, respectively.', '1712.01561-1-57-3': 'Naturally, the former case is more restrictive than the latter one, since it implicitly assumes that both weights should be greater than [MATH] or [MATH] for having a triangle of type [MATH] or [MATH].', '1712.01561-1-58-0': "Social sciences can suggest other functions [MATH]'s to be considered in Definition ([REF]) in order to capture some peculiarities of the system under observation.", '1712.01561-1-59-0': 'Notice also that [MATH] and [MATH] are decreasing functions of [MATH] and [MATH], respectively, as Definition [REF] immediately gives.', '1712.01561-1-60-0': 'Triangles [MATH], [MATH] and [MATH] serve also for deriving topological information on the graph.', '1712.01561-1-60-1': 'In particular, assume that [MATH], so that the number of [MATH] and [MATH] around each node does not depend on the specific selection of function [MATH].', '1712.01561-1-60-2': 'In this case, we know that [MATH], so that we are able to infer the degree of the node [MATH] by the knowledge of the number of triangles of type [MATH] around it.', '1712.01561-1-60-3': 'Differently, [MATH] represents the number of existing paths of length two having [MATH] as one of the extreme nodes.', '1712.01561-1-60-4': 'By collecting the number of the triangles [MATH], [MATH] and [MATH] for each node of the graph, we are able to identify a class of graphs.', '1712.01561-1-61-0': 'Formally, consider a [MATH] matrix collecting [MATH], [MATH] and [MATH], for each node [MATH].', '1712.01561-1-61-1': 'Denote by [MATH] the set of all the matrices with dimension [MATH] and filled by integer nonnegative numbers.', '1712.01561-1-62-0': 'Thus, each matrix [MATH] identifies a not unique graph having [MATH] nodes and with edges described by [MATH].', '1712.01561-1-62-1': 'We call such matrix as the triangles matrix.', '1712.01561-1-62-2': 'In this sense, [MATH] can be viewed as an equivalent class in the set of the graph with [MATH] nodes, where two graphs [MATH] and [MATH] are said to be equivalent when they share the same matrix [MATH].', '1712.01561-1-63-0': 'Figure [REF] and matrix in ([REF]) provides an example of two equivalent class along with their common triangles matrix [MATH].', '1712.01561-1-63-1': 'In particular, notice that matrix [MATH] is symmetric, hence suggesting that the equivalent class identified by [MATH] contains more than one graph.', '1712.01561-1-64-0': '## Conceptualization of the generalized clustering coefficient', '1712.01561-1-65-0': 'Under Definition ([REF]), we can introduce a generalization of the clustering coefficients presented in Formula ([REF]) for weighted networks.', '1712.01561-1-66-0': 'Given a graph [MATH] and a node [MATH], the generalized unweighted clustering coefficient of [MATH] is [EQUATION] where [MATH], where [MATH] is the minimum distance between the nodes [MATH] and [MATH].', '1712.01561-1-67-0': "The term unweighted in Definition [REF] points to absence of [MATH]'s in the coefficient in ([REF]).", '1712.01561-1-67-1': 'However, weights intervene in the identification of the triangles, according to Definition [REF].', '1712.01561-1-68-0': 'In particular, formula ([REF]) extend ([REF]).', '1712.01561-1-68-1': 'Indeed, notice that [MATH] when [MATH].', '1712.01561-1-68-2': 'This happens in the case of [MATH] and [MATH] large enough, so that there are no pseudo-triangles around a given node of the network.', '1712.01561-1-69-0': 'Importantly, [MATH] assumes unitary value not only in the clique case, but also when any missing link is compensated by the high weights of the other two links.', '1712.01561-1-69-1': 'This property of the generalized clustering coefficient is very relevant, since it allows to extend the sense of community given by the clustering coefficient to the case of presence of indirect links, according to the definition of pseudo-triangles [MATH] and [MATH].', '1712.01561-1-70-0': '# Test on real instances', '1712.01561-1-71-0': 'Herein we considered the analysis of the generalized clustering coefficient on two empirical networks: the nervous system of the nematode Caenorhabditis elegans ([CITATION]; [CITATION]) and the network among the [MATH] busiest US commercial airports [CITATION].', '1712.01561-1-71-1': 'The data processing, the network analysis and all simulations are conducted using the software R ([CITATION]) with the igraph package ([CITATION]).', '1712.01561-1-71-2': 'The dataset are obtained from the R packege tnet authored by Tore Opsahl (http://toreopsahl.com).', '1712.01561-1-72-0': 'The network of nematode Caenorhabditis elegans (C.elegans) has [MATH] nodes representing neurons and [MATH] edges occurring when two neurons are connected by either a synapse or a gap junction; for each edge, weights are equal to the number of junctions between node [MATH] and [MATH].', '1712.01561-1-72-1': 'The network has a scale-free organization with [MATH] ([CITATION]; [CITATION]).', '1712.01561-1-73-0': 'The US commercial airport network has [MATH] nodes denoting airports and [MATH] edges representing flight connections.', '1712.01561-1-73-1': 'In this network, weights are the number of seats available on that connections in 2010.', '1712.01561-1-73-2': 'The network has both small-world and scale-free organization with [MATH] [CITATION].', '1712.01561-1-74-0': 'In Figure [REF] we show the network visualization while in Table [REF] are reported some basic measures as the density [MATH], the averaged clustering coefficient [MATH], the global clustering coefficient [MATH] and the minimum, the maximum and the average degree, weight and strength.', '1712.01561-1-74-1': 'Note that for the C.elegans network we considered the giant component of [MATH] nodes while the complete network is composed of [MATH] nodes.', '1712.01561-1-75-0': 'In Figure [REF] we report the strength distributions for the two networks, with the strength ([MATH]) as the sum of the weights of the links incident on [MATH].', '1712.01561-1-75-1': 'The two networks are very different, especially in the distribution of low and high values of strength.', '1712.01561-1-75-2': 'The weight profiles in Figure [REF] confirm such differences, mostly caused by a difference of scale in the values.', '1712.01561-1-76-0': 'According to Definition ([REF]), [MATH] and [MATH], i.e. the pseudo-triangles for every nodes in a network, can be computed considering [MATH] and [MATH].', '1712.01561-1-76-1': 'Since in this case a generic function [MATH] is always satisfied we obtain every pseudo-triangles in [MATH] and [MATH].', '1712.01561-1-76-2': 'Concerning the sets [MATH], such triangles can be easily computed by a built-in function in [MATH].', '1712.01561-1-77-0': 'As example, in Figure [REF] we show the arcs composing the pseudo-triangles in [MATH] and in [MATH] for the neighborhood of order [MATH] of node [MATH] in the US airport network.', '1712.01561-1-77-1': 'Such node has a degree [MATH], a second order neighborhood of cardinality [MATH] and a local clustering coefficient [MATH], since it close [MATH] triangles up to the theoretical [MATH].', '1712.01561-1-77-2': 'Thus, [MATH] while triangles in [MATH] are computed obtaining [MATH].', '1712.01561-1-77-3': 'Note that the blue arcs in the right panel of Figure [REF] are [MATH] because some arcs can be mentioned twice in the set, since arc [MATH] can derive from [MATH] as well as from [MATH].', '1712.01561-1-78-0': 'The generalized clustering coefficieng has value [MATH], much lower than [MATH] since the proportion of closed triangles and pseudo-triangles when [MATH] and [MATH] is smaller than the original network.', '1712.01561-1-79-0': 'Figures [REF] for C.elegans network and [REF] for US airports network report three curves, respectively and for each node, the total number of triangles [MATH], the number of potential pseudo-triangles of type [MATH] and the number of potential pseudo-triangles of type [MATH].', '1712.01561-1-79-1': 'Figures [REF] and [REF] compare the degree [MATH] and the local clustering coefficient [MATH] for each node [MATH].', '1712.01561-1-79-2': 'Note that in these benchmark instances nodes in C.elegans network are enumerated without a particular rule, while the nodes in US airport network are enumerated in non-increasing order of their degree.', '1712.01561-1-80-0': 'The nodes in C.elegans network have a relatively small values of local clustering coefficient for nodes with high degree while nodes with small degree have in general, higher values of local clustering coefficient.', '1712.01561-1-80-1': 'This means that small degree nodes tends to form dense local neighborhoods, while the neighborhood of hubs is much sparser.', '1712.01561-1-80-2': 'Such observations motivate the limited number of pseudo-triangles in [MATH] because, for each node [MATH], they are in number of [MATH], thus to dense neighborhood corresponds a small number of possible pseudo-triangles.', '1712.01561-1-81-0': 'Note in Figure [REF] that node [MATH] has a peak because [MATH] when the thresholds [MATH] and [MATH] are null (this is the case of potential pseudo-triangles).', '1712.01561-1-81-1': 'This is motivated by its particular neighborhood composed of a limited number of triangles in which it is embedded ([MATH] and [MATH]) despite its degree ([MATH].', '1712.01561-1-81-2': 'Choosing two edges on [MATH] lead to [MATH] potential pseudo triangle of type [MATH] and when subtract [MATH] it results [MATH].', '1712.01561-1-81-3': 'Such a remarkable presence of triangles of type [MATH] for a single node for the case of [MATH] suggests that the C.elegans network is star-shaped.', '1712.01561-1-82-0': 'Similar arguments can be considering for [MATH], indeed, we have a small number of potential pseudo-triangles for both small degree nodes and hubs since low values of degree allow for a small number of transitive closure.', '1712.01561-1-83-0': 'The analysis of Figures [REF] and [REF] depict a very different picture for the US airport network.', '1712.01561-1-83-1': 'The nodes with indices until [MATH] have values of degree and clustering coefficient which allow for a large number of pseudo-triangles [MATH] and a significant number of pseudo-triangles [MATH].', '1712.01561-1-83-2': 'Then, when the degree decreases and the local clustering coefficient increases the local neighborhoods preclude the formation of pseudo-triangles.', '1712.01561-1-84-0': 'Figure [REF] shows the global values of the generalized clustering coefficient [MATH] for the C.elegans network when considering the four different functions [MATH] and [MATH].', '1712.01561-1-84-1': 'In each figure, the values are presented for every combination of [MATH] and [MATH] while the horizontal axis report the values of [MATH] as averaged over every nodes in the network.', '1712.01561-1-84-2': 'For the simulations, the value of [MATH] and [MATH] are [MATH] for the C.elegans network and [MATH] for the US airport network.', '1712.01561-1-84-3': 'Max values are chosen considering that the function [MATH] could possibly be true also when considering arcs with the higher weights and the step is set as to have [MATH] runs for each coefficient.', '1712.01561-1-84-4': 'Thus, we performed [MATH] computations for every network.', '1712.01561-1-85-0': 'As expected, higher values of [MATH] are obtained for lower values of [MATH] and [MATH] and, globally, we have a non increasing trend with a higher slope for functions [MATH] and [MATH] since the average function smooths the values and as well as for the min function, they are true only for small values of weights.', '1712.01561-1-85-1': 'Regarding [MATH] and [MATH], they are more prone to be true for higher values of arcs weight and the slope declines slower.', '1712.01561-1-86-0': 'A common behavior for all four cases is that the magnitude of [MATH] is more dependent from pseudo-triangles [MATH] than those in [MATH].', '1712.01561-1-86-1': 'This is due to the tendency of high degree nodes to have also high strength.', '1712.01561-1-86-2': 'Therefore, the functions are more prone to be true for pseudo-triangles [MATH] than for pseudo-triangles in [MATH] since the adjacent links could possibly lie in a low degree node with a low value of strength.', '1712.01561-1-87-0': 'Figure [REF] reports the same plots for the US airport network and same comments on the general behavior can be repeated as for the C.Elegans network.', '1712.01561-1-87-1': 'The main differences are in the steepest slope since the profile of weights distribution is more concentrated on lower values (see Figure [REF]).', '1712.01561-1-88-0': 'In order to study the evolution of the generalized clustering coefficient [MATH] when varying [MATH] and [MATH], we provide a series of diagrams in which, for the two networks under examination, the density of the [MATH] values are reported when considering fixed values of [MATH] or [MATH] and varying the other coefficient.', '1712.01561-1-89-0': 'In particular, for the C.Elegans network Figure [REF] shows different density values for each [MATH] when [MATH] and Figure [REF] for each [MATH] when [MATH].', '1712.01561-1-89-1': 'Same setting is proposed in Figure [REF] and Figure [REF] for the US Airport network.', '1712.01561-1-89-2': 'In all the figures are also reported the density values of the local clustering coefficient [MATH].', '1712.01561-1-90-0': 'When [MATH] (see Figures [REF] and [REF]) we can observe the contribution of pseudo-triangles in [MATH] to [MATH].', '1712.01561-1-90-1': 'For both network we observe the density of [MATH] more concentrated around the max value [MATH] when [MATH], while when [MATH] starts to growth the values shift to be close to [MATH].', '1712.01561-1-90-2': 'As in the previous figure, the effect is more evident for the C.elegans network.', '1712.01561-1-91-0': 'For [MATH], Figures [REF] and [REF] highlight that [MATH] receives a small contribute from pseudo-triangles in [MATH] and the values lay around [MATH] as soon [MATH] grows.', '1712.01561-1-92-0': 'The different figures confirm that for the two networks under observation, the main contribution to [MATH] is provided by the pseudo-triangles in [MATH], i.e. their structures and weight profiles cause the networks to be more prone to close pseudo-triangles in [MATH] rather than pseudo-triangles in [MATH].', '1712.01561-1-93-0': '# Conclusions', '1712.01561-1-94-0': 'This paper deals with a novel definition of the clustering coefficient for weighted networks when triangles are viewed under a social perspective, even when one of the sides is missing.', '1712.01561-1-94-1': 'The thresholds [MATH] and [MATH] provides a key information on the strength of the links able to induce missing arcs.', '1712.01561-1-95-0': 'Triangles [MATH] and [MATH] serve for two scopes: by one side, they model the evidence that transitive relations among the nodes appear when the direct links are strong enough; by the other side, the knowledge of number and types of the triangles around the nodes when [MATH] identify equivalent classes of networks on the basis of their topological structures.', '1712.01561-1-96-0': 'Interestingly, our setting leaves some unsolved problems.', '1712.01561-1-97-0': 'First, one can deal with the definition of more complex ways to connect indirectly nodes, beyond the mere triangles.', '1712.01561-1-97-1': 'In particular, the concept of social polygon with more than three sides can be introduced and explored.', '1712.01561-1-97-2': 'In so doing, a wider concept of community structure of the network can be effectively provided, with a novel definition of clustering coefficient where triangles are replaced by polygons.', '1712.01561-1-97-3': 'Of course, this generalization offers a high degree of complexity in implementing the empirical experiments.', '1712.01561-1-98-0': 'Second, the analysis of the topological structure of the network can be discussed in more details.', '1712.01561-1-98-1': 'In this respect, notice that one can introduce a novel formulation of the concepts of hubs and centrality measures on the basis of the social connections among the nodes, according to our definition of induced indirect links.', '1712.01561-1-98-2': 'In this context, one is able to generalize the exploration to the case of [MATH] and [MATH] not necessarily null.'}

{'1712.01561-2-0-0': 'In the context of expert systems approach, the problem of community detection can be afford as a clustering model for networks.', '1712.01561-2-0-1': 'In this respect, a way to measure the community structure is the clustering coefficient.', '1712.01561-2-0-2': 'Such a quantity is based on the number of existing triangles around the nodes over the theoretical ones.', '1712.01561-2-0-3': 'To the best of our knowledge, scarce attention has been paid to the fictitious triangles due to the presence of indirect connections among the nodes of the network.', '1712.01561-2-0-4': 'This paper fills this gap by providing a new definition of the clustering coefficient for weighted networks when missing links might be also considered.', '1712.01561-2-0-5': 'Specifically, a novel concept of triangles is here introduced by assuming that a strong enough aggregate weight of two arcs sharing a node induces a link between the not common nodes.', '1712.01561-2-0-6': 'Beyond the intuitive meaning of such social triangles, we also explore the usefulness of them for gaining insights on the topological structure of the underline network.', '1712.01561-2-0-7': 'Empirical experiments on the standard networks of 500 commercial US airports and on the nervous system of the Caenorhabditis elegans support the theoretical framework.', '1712.01561-2-1-0': 'Keywords: complex networks; local cohesiveness, clustering coefficient; social triangles.', '1712.01561-2-2-0': '# Introduction', '1712.01561-2-3-0': 'The network approach to complex systems has revealed several general and unexpected findings applicable to a large number of systems, such as the ubiquity of scale freeness, the frequent appearance of high clustering, and the relationship between functionality and the presence of specific motifs ([CITATION], [CITATION]).', '1712.01561-2-4-0': 'In recent years, it has become clear that it is relevant to consider the heterogeneity of the interactions and their correlations with the network structure in order to understand the characteristics of the system.', '1712.01561-2-4-1': 'In particular, a survey of measures of complex networks is reported in [CITATION] while an extended approach related to some centrality measures is presented in [CITATION].', '1712.01561-2-5-0': 'We consider, in this work, the tendency for clustering, i.e. the link formation between neighboring vertices [CITATION] leading to the identification of the local groups cohesiveness.', '1712.01561-2-5-1': 'Such a theme is of paramount relevance in that it allows to assess the community structure of a group of interconnected units.', '1712.01561-2-6-0': 'In this respect, we agree with [CITATION] who states that, in agreement with the expert systems perspective, the problem of community detection can be faced as a clustering model for networks.', '1712.01561-2-6-1': 'This explains also why community detection is nowadays at the core of the debate on social networks (see e.g. [CITATION], [CITATION], [CITATION]).', '1712.01561-2-7-0': 'The measure for assessing the tendency of vertices to cluster is the local cluster coefficient [CITATION].', '1712.01561-2-7-1': 'Such quantity has been extensively studied by several authors and applied in different networks ([CITATION], [CITATION], [CITATION], [CITATION]).', '1712.01561-2-7-2': 'It captures the degree of social embeddedness of the nodes in a network and is based on local density [CITATION].', '1712.01561-2-7-3': 'Indeed, especially in social networks, vertices tend to create tightly knit groups that are characterized by a relatively high density of links [CITATION].', '1712.01561-2-8-0': "The clustering coefficient assesses the connectivity in a node's neighborhood; a node has a high value of clustering coefficient of its neighbors tends to be directly connected with each other [CITATION], [CITATION].", '1712.01561-2-8-1': 'This quantity is relevant to determine the small-world property of a network [CITATION] and can be considered as an index of the redundancy of a node ([CITATION], [CITATION]).', '1712.01561-2-8-2': 'In the contribution of Benati et al. [CITATION], a new combinatorial model is proposed to detect clusters.', '1712.01561-2-8-3': 'Such a model takes into consideration the standard individual data pertinent to a single population unit and data describing the connections among units.', '1712.01561-2-8-4': 'Regarding the clustering coefficient in weighted networks, it has been analyzed in ([CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION]) as reported in Section 3.', '1712.01561-2-8-5': 'Related to the clustering coefficient in terms of link prediction, the estimation of the likelihood of new link creation is presented in ([CITATION], [CITATION]).', '1712.01561-2-8-6': 'The connections between nodes using suitable definitions of the matrix governing the connections has been considered by [CITATION].', '1712.01561-2-9-0': 'The analysis of the weights along the edges and their correlations is able to change the view of the hierarchical and structural organization of the systems.', '1712.01561-2-9-1': 'This is evident if we consider, as an example, a network in which the weights of all links forming triangles of interconnected vertices are extremely small.', '1712.01561-2-9-2': 'In this case, even for a large clustering coefficient, these triangles play a minimal role in the network dynamics and organization, and the clustering features are certainly overestimated by a simple structural analysis [CITATION].', '1712.01561-2-9-3': 'Also, vertices with high degree can be attached to a majority of low-degree nodes whilst concentrating the largest portion of their strength only on the vertices with high degree.', '1712.01561-2-9-4': 'In this situation, the topology reveals a disassortative characteristic of the network, whereas the system could be considered assortative since the more relevant edges in terms of weights are linked to the high-degree vertices.', '1712.01561-2-10-0': 'In this paper, we aim at elaborating on the presence of large weights on the arcs of the network, which are able to induce indirect links among the nodes.', '1712.01561-2-11-0': 'Specifically, we provide a new definition of the concept of triangles of a network of social type, by allowing the presence of indirect links.', '1712.01561-2-11-1': 'One has triangles even in the case of one missing side, but under the constraint that the weights of the remaining sides - suitably aggregated - are large enough.', '1712.01561-2-11-2': 'These new triangles admit the introduction of social intensity as a potential mechanism for links formation.', '1712.01561-2-12-0': 'In particular, we define two different new kinds of triangle around a node [MATH]: the first one is composed by two not linked nodes which are adjacent to [MATH]; the second one is composed by a path of length two from [MATH] such that [MATH] is not linked with the extreme node of the trajectory.', '1712.01561-2-13-0': 'The constraints on the weights of the arcs are driven by two thresholds, one for each type of triangle.', '1712.01561-2-13-1': 'As we will see below in the formalization of our setting, null thresholds mean no constraints - and all the two-sided figures are triangles - while a large value of the thresholds is associated to very restrictive constraints - and a small number of two-sided figures will be accepted as triangles.', '1712.01561-2-14-0': 'It is very important to notice that the case of zero thresholds gives also insights on the topological structure of the unweighted graph associated to the network.', '1712.01561-2-14-1': 'We address the reader to the empirical analysis section for an intuitive explanation of this point.', '1712.01561-2-15-0': 'Starting from the considered definition of triangles, a definition of a new generalized weighted clustering coefficient is provided.', '1712.01561-2-15-1': 'This measure is able to give useful hints on the community structure of the network, when weights play a relevant role in inducing missing links among the node.', '1712.01561-2-15-2': 'Moreover, this measure is also useful for predicting the fictitious links that may appear in the future of evolving networks.', '1712.01561-2-15-3': 'Link prediction is a relevant problem that attempt to estimate the likelihood of the existence of a link between two vertices based on observed links and the attributes of nodes [CITATION].', '1712.01561-2-15-4': 'Such prediction can be used to analyze a network to suggest promising interactions or collaborations that have not yet identified or is related to the problem of inferring missing or additional links that, while not directly visible, are likely to exist ([CITATION], [CITATION]).', '1712.01561-2-15-5': 'Our generalized clustering coefficient has a further very relevant property: it assumes unitary value in several situations and not only when the graph is a clique.', '1712.01561-2-15-6': 'Specifically, the community structure of the network is intended to include also the realistic cases of the presence of indirect connections among two agents induced by their strong links with a third node.', '1712.01561-2-16-0': 'To gain more information, we have also implemented a sensitivity analysis of the clustering coefficient with respect to the exogenous thresholds defining the triangles.', '1712.01561-2-16-1': 'Moreover, a discussion on the way to aggregate the weights for identifying the triangles has been provided, in order to specify different concepts of sociality.', '1712.01561-2-16-2': 'The proposed clustering coefficient is tested on two well-established empirical settings.', '1712.01561-2-17-0': 'The paper is structured as follows; after preliminaries and notations about the graph theory (Section 2), we review the literature on the clustering coefficient in weighted networks (Section 3), we propose a generalized clustering coefficient and generalized triangles with the relative interpretation (Section 4), the computational experience of two empirical networks: the nervous system of the nematode Caenorhabditis elegans and the network among the 500 commercial airports in the United States is presented in Section 5.', '1712.01561-2-17-1': 'The paper ends with some conclusions and remarks on future research directions.', '1712.01561-2-18-0': '# Preliminaries and notations about graph theory', '1712.01561-2-19-0': 'Networks can be considered as a valuable representation of many complex systems found in the real world.', '1712.01561-2-19-1': 'The abstract representation of a network is a graph.', '1712.01561-2-19-2': 'It consists of a set of vertices (or nodes) and a set of edges (or links).', '1712.01561-2-19-3': 'The presence of an edge between two vertices signifies the presence of some kind of interactions or connections between the vertices.', '1712.01561-2-19-4': 'The classical mathematical abstraction of a network is a graph [MATH], where [MATH] is the set of [MATH] nodes (or vertices) and [MATH] is the set of [MATH] links (or edges) stating the relationships among the nodes.', '1712.01561-2-19-5': 'We refer to a node by an index [MATH], meaning that we allow a one-to-one correspondence between an index in [MATH] and a node in [MATH].', '1712.01561-2-19-6': 'Generally, the existence or not of a link between two given nodes is captured by a binary variable, whose value is 1 if the link exists and 0 otherwise.', '1712.01561-2-19-7': 'In so doing, the set [MATH] can be conceptualized through a squared matrix of order [MATH] - the so-called adjacency matrix [MATH] - whose generic element [MATH] is 1 if the link between [MATH] and [MATH] exists and 0 otherwise.', '1712.01561-2-19-8': 'The graph is undirected when [MATH], for each [MATH], and directed otherwise.', '1712.01561-2-19-9': 'The degree [MATH] of the node [MATH] is a nonnegative integer representing the number of links incident upon [MATH].', '1712.01561-2-19-10': 'The degree gives a measure of how well a node is connected to the other elements of the graph, and provides also information on which nodes tend to cluster together.', '1712.01561-2-20-0': 'In this paper, we examine weighted networks that can be denoted mathematically by a weighted adjacency matrix [MATH] whose elements [MATH] represent the weights on the link connecting nodes [MATH] and [MATH], with [MATH].', '1712.01561-2-20-1': 'Clearly, [MATH] stands for absence of a link between [MATH] and [MATH].', '1712.01561-2-20-2': "Thus, [MATH] denotes the intensity of the interactions between two nodes [MATH] and [MATH] and allows for modeling the ties' strength of the observed system.", '1712.01561-2-21-0': '# Literature review on the clustering coefficient for unweighted and weighted networks', '1712.01561-2-22-0': '## Unweighted networks', '1712.01561-2-23-0': 'The local clustering coefficient can be defined for any vertex [MATH] and captures the capacity of edge creations among neighbors, i.e. the tendency in the network to create stable groups [CITATION].', '1712.01561-2-23-1': 'Thus, the cohesion around a vertex [MATH] is quantified by the local clustering coefficient [MATH] defined as the number of triangles [MATH] in which vertex [MATH] participates normalized by the maximum possible number of such triangles: [EQUATION]', '1712.01561-2-23-2': 'The local clustering coefficient quantifies how a node takes part of a cohesive group.', '1712.01561-2-23-3': 'Therefore, [MATH] if none of the neighbors of a node are connected and [MATH] if all of the neighbors are linked.', '1712.01561-2-24-0': 'The value of the local clustering coefficient is influenced by the nodes degrees, a node with several neighbours is likely to be embedded in rather fewer closed triangles.', '1712.01561-2-24-1': 'Hence it has a smaller local clustering coefficient respect to a node linked to fewer neighbors where they are more likely to be clustered in triangles [CITATION].', '1712.01561-2-25-0': 'The clustering coefficient for a given graph is computed in two classical modes [CITATION].', '1712.01561-2-25-1': 'The first one is the averaged clustering coefficient [MATH] given as the average of all the local clustering coefficients, while the second, called the global clustering coefficient and denoted by [MATH], is defined as the ratio among three times the number of closed triangles in the graph and the number of its triplets, i.e. the number of 2-paths among three nodes.', '1712.01561-2-26-0': 'Note that both [MATH] and [MATH] assume values from [MATH] to [MATH] and are equal to [MATH] in case of a clique, i.e. a fully coupled network.', '1712.01561-2-26-1': 'In real networks, the evidence shows that nodes are inclined to cluster into densely connected groups ([CITATION], [CITATION]) and the difficulty to compare the values of clustering of nodes with different degrees makes the average value of local clustering sensitive with respect to how degrees are distributed across the whole network.', '1712.01561-2-27-0': 'The quantities [MATH] and [MATH] are specifically tailored to unweighted networks, and they cannot be satisfactorily employed for describing the community structure of the network in presence of weights on the links and when arcs are of direct type.', '1712.01561-2-28-0': 'Next section is devoted to the analysis of the more general weighted case.', '1712.01561-2-29-0': '## Weighted networks', '1712.01561-2-30-0': 'In many real networks, connections are relevant not only in terms of the classical binary state - whenever they exist or do not exist - but also with regards to their strength which, for any node [MATH], is defined as: [EQUATION]', '1712.01561-2-30-1': 'The introduction of weights and strengths extends the study of the macroscopic properties of the network, by adding to the mere interactions some forms of entity of connections and capability.', '1712.01561-2-30-2': 'In particular, the strength integrates information about the vertex connectivity and the weights of its links [CITATION].', '1712.01561-2-30-3': 'It is considered a natural measure of the importance or centrality of a vertex [MATH].', '1712.01561-2-30-4': 'Indeed, the identification of the most central nodes represents a major issue in network characterization [CITATION].', '1712.01561-2-31-0': 'Barrat et al. [CITATION] combine the topological information of the network with the distribution of weights along links and define the weighted clustering coefficient for a node [MATH] as follows: [EQUATION]', '1712.01561-2-31-1': 'This coefficient is a quantity of the local cohesiveness that considers the importance of the clustered structure taking into account the intensity of the interactions found on the local triangles.', '1712.01561-2-31-2': 'This measure counts, for each triangle created in the neighborhood of the node [MATH], the weight of the two related edges.', '1712.01561-2-31-3': 'The authors refer not just to the number of the closed triangles in the neighborhood of a node but also to their total relative weight with respect to the strength of the nodes.', '1712.01561-2-32-0': 'The normalization factor [MATH] accounts for the strength [MATH] and the maximum possible number of triangles in which the node [MATH] may participate, and it ensures that [MATH].', '1712.01561-2-32-1': 'The definition of [MATH] recovers the topological clustering coefficient in the case where [MATH] is constant, for each [MATH].', '1712.01561-2-33-0': 'Therefore, the authors introduce the weighted clustering coefficient averaged over all nodes of the network, say [MATH], and over all nodes with degree [MATH], say [MATH].', '1712.01561-2-33-1': 'These measures offer global information on the correlation between weights and topology by comparing them with their topological analogs.', '1712.01561-2-34-0': 'Note that [MATH], so [MATH] can be written as: [EQUATION] where [MATH].', '1712.01561-2-34-1': 'In such equation the contribution of each triangle is weighted by a ratio of the average weight of the two adjacent links of the triangle to the average weight [MATH].', '1712.01561-2-35-0': 'Thus, [MATH] compares the weights related with triangles to the average weight of edges connected to the local node.', '1712.01561-2-36-0': 'Zhang and Horvath [CITATION] describe the weighted clustering coefficient in the context of gene co-expression networks.', '1712.01561-2-36-1': 'Unlike the unweighted clustering coefficient, the weighted clustering coefficient is not inversely related to the connectivity.', '1712.01561-2-36-2': 'Authors show a model that reveals how an inverse relationship between clustering coefficient and connectivity occurs from hard thresholding.', '1712.01561-2-36-3': 'In formula: [EQUATION] where the weights have been normalized by [MATH].', '1712.01561-2-36-4': 'The number of triangles around the node [MATH] can be written in terms of the adjacency matrix elements as [MATH] and the numerator of the above equation is a weighted generalization of the formula.', '1712.01561-2-36-5': 'The denominator has been selected by considering the upper bound of the numerator, ensuring [MATH] .', '1712.01561-2-36-6': 'The equation ([REF]) can be written as: [EQUATION]', '1712.01561-2-36-7': 'In [CITATION] a similar definition has been shown, indeed the edge weights are considered as probabilities such that in an ensemble of networks, [MATH] and [MATH] are linked with probability [MATH].', '1712.01561-2-36-8': 'Finally, Holme et al [CITATION] discuss the definition of weights and express a redefined weighted clustering coefficient as: [EQUATION] where [MATH] indicates a matrix where each entry equals [MATH].', '1712.01561-2-36-9': 'This equation seems similar to those of [CITATION] though, [MATH] is not required in the denominator sum.', '1712.01561-2-37-0': 'Onnela et al. ([CITATION], [CITATION]) refer to the notion of motif defining it as a set (ensemble) of topologically equivalent subgraphs of a network.', '1712.01561-2-37-1': 'In case of weighted systems, it is relevant to deal with intensities instead of numbers of occurrence.', '1712.01561-2-37-2': 'Moreover, the latter concept is considered as a special case of the former one.', '1712.01561-2-37-3': 'For the authors the triangles are among the simplest nontrivial motifs and have a crucial role as one of the classic quantities of network characterization in defining the clustering coefficient of a node [MATH].', '1712.01561-2-37-4': 'They propose a weighted clustering coefficient taking into consideration the subgraph intensity that is defined as the geometric average of subgraph edge weights.', '1712.01561-2-37-5': 'In formula: [EQUATION] where [MATH] are the edge weights normalized by the maximum weight in the network of the edges linking [MATH] to the other nodes of [MATH].', '1712.01561-2-38-0': 'Formula ([REF]) shows that triangles contribute to the creation of [MATH] according to the weights associated to their three edges.', '1712.01561-2-38-1': 'More specifically, [MATH] disregards the strength of the local node and measure triangle weights only in relation to the maximum edge weight.', '1712.01561-2-39-0': 'Moreover, [MATH] collapses to [MATH] when, for each [MATH], one has [MATH], hence in the unweighted case.', '1712.01561-2-40-0': '# The generalized clustering coefficient', '1712.01561-2-41-0': 'This section contains our proposal for a new definition of the clustering coefficient of weighted networks.', '1712.01561-2-41-1': 'The ground of the definition is a novel concept of triangles, to include also the presence of real indirect connections among individuals.', '1712.01561-2-41-2': 'To our purpose, we first provide the definition of the triangles and discuss it, and then we introduce the clustering coefficient.', '1712.01561-2-42-0': '## Generalized triangles', '1712.01561-2-43-0': 'Here, we propose a generalization of the concept of triangle, and rewrite accordingly the coefficient [MATH] in ([REF]) for the case of weighted networks.', '1712.01561-2-44-0': 'Let us consider a weighted non-oriented graph [MATH] with vertices [MATH], symmetric adjacent matrix [MATH] and weight matrix [MATH], with nonnegative weights.', '1712.01561-2-44-1': 'Moreover, let us take [MATH] and a function [MATH] which is not decreasing in its arguments.', '1712.01561-2-45-0': 'For each triple of distinct vertices [MATH], a subgraph [MATH] is a triangle around [MATH] if one of the following conditions are satisfied:', '1712.01561-2-46-0': 'We denote the set of triangles associated to case [MATH] as [MATH], for [MATH].', '1712.01561-2-46-1': 'By definition, [MATH].', '1712.01561-2-46-2': 'We denote the set collecting all the triangles by [MATH].', '1712.01561-2-47-0': 'Figure [REF] reports the three different type of triangles, respectively [MATH], [MATH] and [MATH].', '1712.01561-2-47-1': 'Clearly, in case [MATH], the concept of triangle given in Definition [REF] coincides with the standard one.', '1712.01561-2-48-0': 'Note that with [MATH] nodes, the maximum number of possible triangles is [MATH].', '1712.01561-2-48-1': 'This is the case of a clique with [MATH], for each [MATH].', '1712.01561-2-49-0': 'When considering the maximum number of candidates triangles for a node [MATH] to belong to [MATH], it is [MATH].', '1712.01561-2-49-1': 'Then, in this case for the node [MATH] the number of triangles is [MATH]', '1712.01561-2-50-0': 'Triangles in [MATH] for node [MATH] are path of length [MATH] (i.e. triplets), that can be computed considering the square of the adjacency matrix.', '1712.01561-2-50-1': 'Indeed.', '1712.01561-2-50-2': 'the number of different path of length [MATH] from [MATH] to [MATH] equals the entry [MATH] of [MATH] [CITATION].', '1712.01561-2-50-3': 'For a given row [MATH] of [MATH], the sum of the element (excluding the element [MATH]) equals the maximum potential number of pseudo-triangles of type [MATH].', '1712.01561-2-51-0': 'Herein we define the elements in [MATH] and [MATH] as pseudo-triangles since they do not are contained in [MATH] but can be seen as triangles under conditions on the weights of the two edges.', '1712.01561-2-52-0': 'Figure [REF] shows the types of triangles, without emphasis on the conditions on the weights.', '1712.01561-2-53-0': '### Interpretation of the triangles and equivalent graphs', '1712.01561-2-54-0': 'The pseudo-triangles [MATH] and [MATH] have an interpretation whose ground is the theory of social networks.', '1712.01561-2-55-0': 'The former case describes a situation in which agent [MATH] has a direct relationship with agents [MATH] and [MATH].', '1712.01561-2-55-1': 'One can say that there exists a triangle among the three if the strength of the connections of [MATH] with the others is sufficiently high - in the sense described by function [MATH].', '1712.01561-2-55-2': 'The idea is that the cooperation between [MATH] and the other agents is so effective and fruitful that the presence of a direct link between [MATH] and [MATH] is not required.', '1712.01561-2-56-0': 'The latter case is associated to the presence of a strong link between [MATH] and [MATH] and between [MATH] and [MATH], always in terms of the entities of the weights - in the sense described by function [MATH].', '1712.01561-2-56-1': 'In this peculiar situation, the node [MATH] represents the intermediate agent letting also the (indirect) collaboration between [MATH] and [MATH] be possible.', '1712.01561-2-57-0': 'In our empirical experiments (see below), we have considered four cases of function [MATH]:', '1712.01561-2-58-0': 'The selection of the specific function [MATH] - to be implemented among [MATH] defined above - provides further insights on the interpretation of the triangles of type [MATH] and [MATH].', '1712.01561-2-58-1': 'Indeed, once [MATH] and [MATH] are kept fixed, then [MATH] and [MATH] state that both weights of the considered edges should be taken into account in an identical way by considering their mere aggregation in the former case or their mean in the latter one.', '1712.01561-2-58-2': 'When considering functions [MATH] and [MATH], only one of the weights is relevant for the measurement of the strength of the connections - the minimum weight and the maximum one, respectively.', '1712.01561-2-58-3': 'Naturally, the former case is more restrictive than the latter one, since it implicitly assumes that both weights should be greater than [MATH] or [MATH] for having a triangle of type [MATH] or [MATH].', '1712.01561-2-59-0': "Social sciences can suggest other functions [MATH]'s to be considered in Definition ([REF]) in order to capture some peculiarities of the system under observation.", '1712.01561-2-60-0': 'Notice also that [MATH] and [MATH] are decreasing functions of [MATH] and [MATH], respectively, as Definition [REF] immediately gives.', '1712.01561-2-61-0': 'Triangles [MATH], [MATH] and [MATH] serve also for deriving topological information on the graph.', '1712.01561-2-61-1': 'In particular, assume that [MATH], so that the number of [MATH] and [MATH] around each node does not depend on the specific selection of function [MATH].', '1712.01561-2-61-2': 'In this case, we know that [MATH], so that we are able to infer the degree of the node [MATH] by the knowledge of the number of triangles of type [MATH] around it.', '1712.01561-2-61-3': 'Differently, [MATH] represents the number of existing paths of length two having [MATH] as one of the extreme nodes.', '1712.01561-2-61-4': 'By collecting the number of the triangles [MATH], [MATH] and [MATH] for each node of the graph, we are able to identify a class of graphs.', '1712.01561-2-62-0': 'Formally, consider a [MATH] matrix collecting [MATH], [MATH] and [MATH], for each node [MATH].', '1712.01561-2-62-1': 'Denote by [MATH] the set of all the matrices with dimension [MATH] and filled by integer nonnegative numbers.', '1712.01561-2-63-0': 'Thus, each matrix [MATH] identifies a not unique graph having [MATH] nodes and with edges described by [MATH].', '1712.01561-2-63-1': 'We call such matrix as the triangles matrix.', '1712.01561-2-63-2': 'In this sense, [MATH] can be viewed as an equivalent class in the set of the graph with [MATH] nodes, where two graphs [MATH] and [MATH] are said to be equivalent when they share the same matrix [MATH].', '1712.01561-2-64-0': 'Figure [REF] and matrix in ([REF]) provides an example of two equivalent class along with their common triangles matrix [MATH].', '1712.01561-2-64-1': 'In particular, notice matrix [MATH] is the same for the two considered graphs, hence suggesting that the equivalent class identified by [MATH] contains more than one graph.', '1712.01561-2-65-0': '## Conceptualization of the generalized clustering coefficient', '1712.01561-2-66-0': 'Under Definition ([REF]), we can introduce a generalization of the clustering coefficients presented in Formula ([REF]) for weighted networks.', '1712.01561-2-67-0': 'Given a graph [MATH] and a node [MATH], the generalized unweighted clustering coefficient of [MATH] is [EQUATION] where [MATH], where [MATH] is the minimum distance between the nodes [MATH] and [MATH].', '1712.01561-2-68-0': "The term unweighted in Definition [REF] points to absence of [MATH]'s in the coefficient in ([REF]).", '1712.01561-2-68-1': 'However, weights intervene in the identification of the triangles, according to Definition [REF].', '1712.01561-2-69-0': 'In particular, formula ([REF]) extends ([REF]).', '1712.01561-2-69-1': 'As an example, notice that [MATH] in the clique case.', '1712.01561-2-70-0': 'Importantly, [MATH] assumes unitary value not only in the clique case, but also when any missing link is compensated by the high weights of the other two links.', '1712.01561-2-70-1': 'This property of the generalized clustering coefficient is very relevant, since it allows to extend the sense of community given by the clustering coefficient to the case of presence of indirect links, according to the definition of pseudo-triangles [MATH] and [MATH].', '1712.01561-2-71-0': '# Test on real instances', '1712.01561-2-72-0': 'Herein we considered the analysis of the generalized clustering coefficient on two empirical networks: the nervous system of the nematode Caenorhabditis elegans ([CITATION]; [CITATION]) and the network among the [MATH] busiest US commercial airports [CITATION].', '1712.01561-2-72-1': 'The data processing, the network analysis and all simulations are conducted using the software R ([CITATION]) with the igraph package ([CITATION]).', '1712.01561-2-72-2': 'The dataset are obtained from the R packege tnet authored by Tore Opsahl (http://toreopsahl.com).', '1712.01561-2-73-0': 'The network of nematode Caenorhabditis elegans (C.elegans) has [MATH] nodes representing neurons and [MATH] edges occurring when two neurons are connected by either a synapse or a gap junction; for each edge, weights are equal to the number of junctions between node [MATH] and [MATH].', '1712.01561-2-73-1': 'The network has a scale-free organization with [MATH] ([CITATION]; [CITATION]).', '1712.01561-2-74-0': 'The US commercial airport network has [MATH] nodes denoting airports and [MATH] edges representing flight connections.', '1712.01561-2-74-1': 'In this network, weights are the number of seats available on that connections in 2010.', '1712.01561-2-74-2': 'The network has both small-world and scale-free organization with [MATH] [CITATION].', '1712.01561-2-75-0': 'In Figure [REF] we show the network visualization while in Table [REF] are reported some basic measures as the density [MATH], the averaged clustering coefficient [MATH], the global clustering coefficient [MATH] and the minimum, the maximum and the average degree, weight and strength.', '1712.01561-2-75-1': 'Note that for the C.elegans network we considered the giant component of [MATH] nodes while the complete network is composed of [MATH] nodes.', '1712.01561-2-76-0': 'In Figure [REF] we report the strength distributions for the two networks, with the strength ([MATH]) as the sum of the weights of the links incident on [MATH].', '1712.01561-2-76-1': 'The two networks are very different, especially in the distribution of low and high values of strength.', '1712.01561-2-76-2': 'The weight profiles in Figure [REF] confirm such differences, mostly caused by a difference of scale in the values.', '1712.01561-2-77-0': 'According to Definition ([REF]), [MATH] and [MATH], i.e. the pseudo-triangles for every nodes in a network, can be computed considering [MATH] and [MATH].', '1712.01561-2-77-1': 'Since in this case a generic function [MATH] is always satisfied we obtain every pseudo-triangles in [MATH] and [MATH].', '1712.01561-2-77-2': 'Concerning the sets [MATH], such triangles can be easily computed by a built-in function in [MATH].', '1712.01561-2-78-0': 'As example, in Figure [REF] we show the arcs composing the pseudo-triangles in [MATH] and in [MATH] for the neighborhood of order [MATH] of node [MATH] in the US airport network.', '1712.01561-2-78-1': 'Such node has a degree [MATH], a second order neighborhood of cardinality [MATH] and a local clustering coefficient [MATH], since it close [MATH] triangles up to the theoretical [MATH].', '1712.01561-2-78-2': 'Thus, [MATH] while triangles in [MATH] are computed obtaining [MATH].', '1712.01561-2-78-3': 'Note that the blue arcs in the right panel of Figure [REF] are [MATH] because some arcs can be mentioned twice in the set, since arc [MATH] can derive from [MATH] as well as from [MATH].', '1712.01561-2-79-0': 'The generalized clustering coefficieng has value [MATH], much lower than [MATH] since the proportion of closed triangles and pseudo-triangles when [MATH] and [MATH] is smaller than the original network.', '1712.01561-2-80-0': 'Figures [REF] for C.elegans network and [REF] for US airports network report three curves, respectively and for each node, the total number of triangles [MATH], the number of potential pseudo-triangles of type [MATH] and the number of potential pseudo-triangles of type [MATH].', '1712.01561-2-80-1': 'Figures [REF] and [REF] compare the degree [MATH] and the local clustering coefficient [MATH] for each node [MATH].', '1712.01561-2-80-2': 'Note that in these benchmark instances nodes in C.elegans network are enumerated without a particular rule, while the nodes in US airport network are enumerated in non-increasing order of their degree.', '1712.01561-2-81-0': 'The nodes in C.elegans network have a relatively small values of local clustering coefficient for nodes with high degree while nodes with small degree have in general, higher values of local clustering coefficient.', '1712.01561-2-81-1': 'This means that small degree nodes tends to form dense local neighborhoods, while the neighborhood of hubs is much sparser.', '1712.01561-2-81-2': 'Such observations motivate the limited number of pseudo-triangles in [MATH] because, for each node [MATH], they are in number of [MATH], thus to dense neighborhood corresponds a small number of possible pseudo-triangles.', '1712.01561-2-82-0': 'Note in Figure [REF] that node [MATH] has a peak because [MATH] when the thresholds [MATH] and [MATH] are null (this is the case of potential pseudo-triangles).', '1712.01561-2-82-1': 'This is motivated by its particular neighborhood composed of a limited number of triangles in which it is embedded ([MATH] and [MATH]) despite its degree ([MATH].', '1712.01561-2-82-2': 'Choosing two edges on [MATH] lead to [MATH] potential pseudo triangle of type [MATH] and when subtract [MATH] it results [MATH].', '1712.01561-2-82-3': 'Such a remarkable presence of triangles of type [MATH] for a single node for the case of [MATH] suggests that the C.elegans network is star-shaped.', '1712.01561-2-83-0': 'Similar arguments can be considering for [MATH], indeed, we have a small number of potential pseudo-triangles for both small degree nodes and hubs since low values of degree allow for a small number of transitive closure.', '1712.01561-2-84-0': 'The analysis of Figures [REF] and [REF] depict a very different picture for the US airport network.', '1712.01561-2-84-1': 'The nodes with indices until [MATH] have values of degree and clustering coefficient which allow for a large number of pseudo-triangles [MATH] and a significant number of pseudo-triangles [MATH].', '1712.01561-2-84-2': 'Then, when the degree decreases and the local clustering coefficient increases the local neighborhoods preclude the formation of pseudo-triangles.', '1712.01561-2-85-0': 'Figure [REF] shows the global values of the generalized clustering coefficient [MATH] for the C.elegans network when considering the four different functions [MATH] and [MATH].', '1712.01561-2-85-1': 'In each figure, the values are presented for every combination of [MATH] and [MATH] while the horizontal axis report the values of [MATH] as averaged over every nodes in the network.', '1712.01561-2-85-2': 'For the simulations, the value of [MATH] and [MATH] are [MATH] for the C.elegans network and [MATH] for the US airport network.', '1712.01561-2-85-3': 'Max values are chosen considering that the function [MATH] could possibly be true also when considering arcs with the higher weights and the step is set as to have [MATH] runs for each coefficient.', '1712.01561-2-85-4': 'Thus, we performed [MATH] computations for every network.', '1712.01561-2-86-0': 'As expected, higher values of [MATH] are obtained for lower values of [MATH] and [MATH] and, globally, we have a non increasing trend with a higher slope for functions [MATH] and [MATH] since the average function smooths the values and as well as for the min function, they are true only for small values of weights.', '1712.01561-2-86-1': 'Regarding [MATH] and [MATH], they are more prone to be true for higher values of arcs weight and the slope declines slower.', '1712.01561-2-87-0': 'A common behavior for all four cases is that the magnitude of [MATH] is more dependent from pseudo-triangles [MATH] than those in [MATH].', '1712.01561-2-87-1': 'This is due to the tendency of high degree nodes to have also high strength.', '1712.01561-2-87-2': 'Therefore, the functions are more prone to be true for pseudo-triangles [MATH] than for pseudo-triangles in [MATH] since the adjacent links could possibly lie in a low degree node with a low value of strength.', '1712.01561-2-88-0': 'Figure [REF] reports the same plots for the US airport network and same comments on the general behavior can be repeated as for the C.Elegans network.', '1712.01561-2-88-1': 'The main differences are in the steepest slope since the profile of weights distribution is more concentrated on lower values (see Figure [REF]).', '1712.01561-2-89-0': 'In order to study the evolution of the generalized clustering coefficient [MATH] when varying [MATH] and [MATH], we provide a series of diagrams in which, for the two networks under examination, the density of the [MATH] values are reported when considering fixed values of [MATH] or [MATH] and varying the other coefficient.', '1712.01561-2-90-0': 'In particular, for the C.Elegans network Figure [REF] shows different density values for each [MATH] when [MATH] and Figure [REF] for each [MATH] when [MATH].', '1712.01561-2-90-1': 'Same setting is proposed in Figure [REF] and Figure [REF] for the US Airport network.', '1712.01561-2-90-2': 'In all the figures are also reported the density values of the local clustering coefficient [MATH].', '1712.01561-2-91-0': 'When [MATH] (see Figures [REF] and [REF]) we can observe the contribution of pseudo-triangles in [MATH] to [MATH].', '1712.01561-2-91-1': 'For both network we observe the density of [MATH] more concentrated around the max value [MATH] when [MATH], while when [MATH] starts to growth the values shift to be close to [MATH].', '1712.01561-2-91-2': 'As in the previous figure, the effect is more evident for the C.elegans network.', '1712.01561-2-92-0': 'For [MATH], Figures [REF] and [REF] highlight that [MATH] receives a small contribute from pseudo-triangles in [MATH] and the values lay around [MATH] as soon [MATH] grows.', '1712.01561-2-93-0': 'The different figures confirm that for the two networks under observation, the main contribution to [MATH] is provided by the pseudo-triangles in [MATH], i.e. their structures and weight profiles cause the networks to be more prone to close pseudo-triangles in [MATH] rather than pseudo-triangles in [MATH].', '1712.01561-2-94-0': '# Conclusions and future research lines', '1712.01561-2-95-0': 'This paper deals with a novel definition of the clustering coefficient for weighted networks when triangles are viewed under a social perspective, even when one of the sides is missing.', '1712.01561-2-95-1': 'The thresholds [MATH] and [MATH] provides a key information on the strength of the links able to induce missing arcs.', '1712.01561-2-96-0': 'Triangles [MATH] and [MATH] serve for two scopes: by one side, they model the evidence that transitive relations among the nodes appear when the direct links are strong enough; by the other side, the knowledge of number and types of the triangles around the nodes when [MATH] identify equivalent classes of networks on the basis of their topological structures.', '1712.01561-2-97-0': 'Interestingly, our setting leaves some unsolved problems.', '1712.01561-2-98-0': 'First, one can deal with the definition of more complex ways to connect indirectly nodes, beyond the mere triangles.', '1712.01561-2-98-1': 'In particular, the concept of social polygon with more than three sides can be introduced and explored.', '1712.01561-2-98-2': 'In so doing, a wider concept of community structure of the network can be effectively provided, with a novel definition of clustering coefficient where triangles are replaced by polygons.', '1712.01561-2-98-3': 'Of course, this generalization offers a high degree of complexity in implementing the empirical experiments.', '1712.01561-2-99-0': 'Second, the analysis of the topological structure of the network can be discussed in more details.', '1712.01561-2-99-1': 'In this respect, notice that one can introduce a novel formulation of the concepts of hubs and centrality measures on the basis of the social connections among the nodes, according to our definition of induced indirect links.', '1712.01561-2-99-2': 'In this context, one is able to generalize the exploration to the case of [MATH] and [MATH] not necessarily null.'}

[['1712.01561-1-8-0', '1712.01561-2-9-0'], ['1712.01561-1-8-1', '1712.01561-2-9-1'], ['1712.01561-1-8-2', '1712.01561-2-9-2'], ['1712.01561-1-8-3', '1712.01561-2-9-3'], ['1712.01561-1-8-4', '1712.01561-2-9-4'], ['1712.01561-1-54-0', '1712.01561-2-55-0'], ['1712.01561-1-54-1', '1712.01561-2-55-1'], ['1712.01561-1-54-2', '1712.01561-2-55-2'], ['1712.01561-1-30-0', '1712.01561-2-31-0'], ['1712.01561-1-30-1', '1712.01561-2-31-1'], ['1712.01561-1-30-3', '1712.01561-2-31-3'], ['1712.01561-1-79-0', '1712.01561-2-80-0'], ['1712.01561-1-79-1', '1712.01561-2-80-1'], ['1712.01561-1-79-2', '1712.01561-2-80-2'], ['1712.01561-1-47-0', '1712.01561-2-48-0'], ['1712.01561-1-47-1', '1712.01561-2-48-1'], ['1712.01561-1-97-0', '1712.01561-2-98-0'], ['1712.01561-1-97-1', '1712.01561-2-98-1'], ['1712.01561-1-97-2', '1712.01561-2-98-2'], ['1712.01561-1-97-3', '1712.01561-2-98-3'], ['1712.01561-1-72-0', '1712.01561-2-73-0'], ['1712.01561-1-72-1', '1712.01561-2-73-1'], ['1712.01561-1-82-0', '1712.01561-2-83-0'], ['1712.01561-1-84-0', '1712.01561-2-85-0'], ['1712.01561-1-84-1', '1712.01561-2-85-1'], ['1712.01561-1-84-2', '1712.01561-2-85-2'], ['1712.01561-1-84-3', '1712.01561-2-85-3'], ['1712.01561-1-84-4', '1712.01561-2-85-4'], ['1712.01561-1-63-0', '1712.01561-2-64-0'], ['1712.01561-1-24-0', '1712.01561-2-25-0'], ['1712.01561-1-24-1', '1712.01561-2-25-1'], ['1712.01561-1-55-0', '1712.01561-2-56-0'], ['1712.01561-1-55-1', '1712.01561-2-56-1'], ['1712.01561-1-66-0', '1712.01561-2-67-0'], ['1712.01561-1-53-0', '1712.01561-2-54-0'], ['1712.01561-1-37-0', '1712.01561-2-38-0'], ['1712.01561-1-37-1', '1712.01561-2-38-1'], ['1712.01561-1-32-0', '1712.01561-2-33-0'], ['1712.01561-1-32-1', '1712.01561-2-33-1'], ['1712.01561-1-83-0', '1712.01561-2-84-0'], ['1712.01561-1-83-1', '1712.01561-2-84-1'], ['1712.01561-1-83-2', '1712.01561-2-84-2'], ['1712.01561-1-29-0', '1712.01561-2-30-0'], ['1712.01561-1-29-1', '1712.01561-2-30-1'], ['1712.01561-1-29-2', '1712.01561-2-30-2'], ['1712.01561-1-29-3', '1712.01561-2-30-3'], ['1712.01561-1-29-4', '1712.01561-2-30-4'], ['1712.01561-1-88-0', '1712.01561-2-89-0'], ['1712.01561-1-7-1', '1712.01561-2-8-1'], ['1712.01561-1-7-4', '1712.01561-2-8-4'], ['1712.01561-1-7-5', '1712.01561-2-8-5'], ['1712.01561-1-7-6', '1712.01561-2-8-6'], ['1712.01561-1-71-0', '1712.01561-2-72-0'], ['1712.01561-1-71-1', '1712.01561-2-72-1'], ['1712.01561-1-71-2', '1712.01561-2-72-2'], ['1712.01561-1-4-0', '1712.01561-2-4-0'], ['1712.01561-1-4-1', '1712.01561-2-4-1'], ['1712.01561-1-43-0', '1712.01561-2-44-0'], ['1712.01561-1-43-1', '1712.01561-2-44-1'], ['1712.01561-1-76-0', '1712.01561-2-77-0'], ['1712.01561-1-76-1', '1712.01561-2-77-1'], ['1712.01561-1-76-2', '1712.01561-2-77-2'], ['1712.01561-1-65-0', '1712.01561-2-66-0'], ['1712.01561-1-89-0', '1712.01561-2-90-0'], ['1712.01561-1-89-1', '1712.01561-2-90-1'], ['1712.01561-1-89-2', '1712.01561-2-90-2'], ['1712.01561-1-67-0', '1712.01561-2-68-0'], ['1712.01561-1-67-1', '1712.01561-2-68-1'], ['1712.01561-1-51-0', '1712.01561-2-52-0'], ['1712.01561-1-57-0', '1712.01561-2-58-0'], ['1712.01561-1-57-1', '1712.01561-2-58-1'], ['1712.01561-1-57-2', '1712.01561-2-58-2'], ['1712.01561-1-57-3', '1712.01561-2-58-3'], ['1712.01561-1-40-0', '1712.01561-2-41-0'], ['1712.01561-1-40-1', '1712.01561-2-41-1'], ['1712.01561-1-40-2', '1712.01561-2-41-2'], ['1712.01561-1-34-0', '1712.01561-2-35-0'], ['1712.01561-1-3-0', '1712.01561-2-3-0'], ['1712.01561-1-45-0', '1712.01561-2-46-0'], ['1712.01561-1-45-2', '1712.01561-2-46-2'], ['1712.01561-1-69-0', '1712.01561-2-70-0'], ['1712.01561-1-69-1', '1712.01561-2-70-1'], ['1712.01561-1-60-0', '1712.01561-2-61-0'], ['1712.01561-1-60-1', '1712.01561-2-61-1'], ['1712.01561-1-60-2', '1712.01561-2-61-2'], ['1712.01561-1-60-3', '1712.01561-2-61-3'], ['1712.01561-1-60-4', '1712.01561-2-61-4'], ['1712.01561-1-22-3', '1712.01561-2-23-3'], ['1712.01561-1-9-0', '1712.01561-2-10-0'], ['1712.01561-1-6-0', '1712.01561-2-7-0'], ['1712.01561-1-6-1', '1712.01561-2-7-1'], ['1712.01561-1-6-2', '1712.01561-2-7-2'], ['1712.01561-1-6-3', '1712.01561-2-7-3'], ['1712.01561-1-46-0', '1712.01561-2-47-0'], ['1712.01561-1-46-1', '1712.01561-2-47-1'], ['1712.01561-1-23-0', '1712.01561-2-24-0'], ['1712.01561-1-23-1', '1712.01561-2-24-1'], ['1712.01561-1-95-0', '1712.01561-2-96-0'], ['1712.01561-1-75-0', '1712.01561-2-76-0'], ['1712.01561-1-75-1', '1712.01561-2-76-1'], ['1712.01561-1-75-2', '1712.01561-2-76-2'], ['1712.01561-1-59-0', '1712.01561-2-60-0'], ['1712.01561-1-77-0', '1712.01561-2-78-0'], ['1712.01561-1-77-1', '1712.01561-2-78-1'], ['1712.01561-1-77-2', '1712.01561-2-78-2'], ['1712.01561-1-77-3', '1712.01561-2-78-3'], ['1712.01561-1-78-0', '1712.01561-2-79-0'], ['1712.01561-1-13-0', '1712.01561-2-14-0'], ['1712.01561-1-13-1', '1712.01561-2-14-1'], ['1712.01561-1-90-0', '1712.01561-2-91-0'], ['1712.01561-1-90-1', '1712.01561-2-91-1'], ['1712.01561-1-90-2', '1712.01561-2-91-2'], ['1712.01561-1-62-0', '1712.01561-2-63-0'], ['1712.01561-1-62-1', '1712.01561-2-63-1'], ['1712.01561-1-62-2', '1712.01561-2-63-2'], ['1712.01561-1-61-0', '1712.01561-2-62-0'], ['1712.01561-1-61-1', '1712.01561-2-62-1'], ['1712.01561-1-94-0', '1712.01561-2-95-0'], ['1712.01561-1-94-1', '1712.01561-2-95-1'], ['1712.01561-1-98-0', '1712.01561-2-99-0'], ['1712.01561-1-98-1', '1712.01561-2-99-1'], ['1712.01561-1-98-2', '1712.01561-2-99-2'], ['1712.01561-1-48-0', '1712.01561-2-49-0'], ['1712.01561-1-48-1', '1712.01561-2-49-1'], ['1712.01561-1-38-0', '1712.01561-2-39-0'], ['1712.01561-1-14-0', '1712.01561-2-15-0'], ['1712.01561-1-14-1', '1712.01561-2-15-1'], ['1712.01561-1-14-2', '1712.01561-2-15-2'], ['1712.01561-1-14-3', '1712.01561-2-15-3'], ['1712.01561-1-14-4', '1712.01561-2-15-4'], ['1712.01561-1-14-5', '1712.01561-2-15-5'], ['1712.01561-1-14-6', '1712.01561-2-15-6'], ['1712.01561-1-12-0', '1712.01561-2-13-0'], ['1712.01561-1-12-1', '1712.01561-2-13-1'], ['1712.01561-1-86-0', '1712.01561-2-87-0'], ['1712.01561-1-86-1', '1712.01561-2-87-1'], ['1712.01561-1-86-2', '1712.01561-2-87-2'], ['1712.01561-1-16-0', '1712.01561-2-17-0'], ['1712.01561-1-16-1', '1712.01561-2-17-1'], ['1712.01561-1-27-0', '1712.01561-2-28-0'], ['1712.01561-1-36-0', '1712.01561-2-37-0'], ['1712.01561-1-36-1', '1712.01561-2-37-1'], ['1712.01561-1-36-2', '1712.01561-2-37-2'], ['1712.01561-1-36-3', '1712.01561-2-37-3'], ['1712.01561-1-36-4', '1712.01561-2-37-4'], ['1712.01561-1-36-5', '1712.01561-2-37-5'], ['1712.01561-1-10-0', '1712.01561-2-11-0'], ['1712.01561-1-10-1', '1712.01561-2-11-1'], ['1712.01561-1-10-2', '1712.01561-2-11-2'], ['1712.01561-1-15-0', '1712.01561-2-16-0'], ['1712.01561-1-15-1', '1712.01561-2-16-1'], ['1712.01561-1-15-2', '1712.01561-2-16-2'], ['1712.01561-1-31-1', '1712.01561-2-32-1'], ['1712.01561-1-85-0', '1712.01561-2-86-0'], ['1712.01561-1-85-1', '1712.01561-2-86-1'], ['1712.01561-1-33-0', '1712.01561-2-34-0'], ['1712.01561-1-33-1', '1712.01561-2-34-1'], ['1712.01561-1-35-0', '1712.01561-2-36-0'], ['1712.01561-1-35-1', '1712.01561-2-36-1'], ['1712.01561-1-35-2', '1712.01561-2-36-2'], ['1712.01561-1-35-3', '1712.01561-2-36-3'], ['1712.01561-1-35-4', '1712.01561-2-36-4'], ['1712.01561-1-35-5', '1712.01561-2-36-5'], ['1712.01561-1-35-7', '1712.01561-2-36-7'], ['1712.01561-1-35-8', '1712.01561-2-36-8'], ['1712.01561-1-35-9', '1712.01561-2-36-9'], ['1712.01561-1-87-0', '1712.01561-2-88-0'], ['1712.01561-1-87-1', '1712.01561-2-88-1'], ['1712.01561-1-0-1', '1712.01561-2-0-2'], ['1712.01561-1-0-2', '1712.01561-2-0-3'], ['1712.01561-1-0-3', '1712.01561-2-0-4'], ['1712.01561-1-0-4', '1712.01561-2-0-5'], ['1712.01561-1-0-5', '1712.01561-2-0-6'], ['1712.01561-1-0-6', '1712.01561-2-0-7'], ['1712.01561-1-73-0', '1712.01561-2-74-0'], ['1712.01561-1-73-1', '1712.01561-2-74-1'], ['1712.01561-1-73-2', '1712.01561-2-74-2'], ['1712.01561-1-81-0', '1712.01561-2-82-0'], ['1712.01561-1-81-1', '1712.01561-2-82-1'], ['1712.01561-1-81-2', '1712.01561-2-82-2'], ['1712.01561-1-81-3', '1712.01561-2-82-3'], ['1712.01561-1-80-0', '1712.01561-2-81-0'], ['1712.01561-1-80-1', '1712.01561-2-81-1'], ['1712.01561-1-80-2', '1712.01561-2-81-2'], ['1712.01561-1-74-0', '1712.01561-2-75-0'], ['1712.01561-1-74-1', '1712.01561-2-75-1'], ['1712.01561-1-49-0', '1712.01561-2-50-0'], ['1712.01561-1-49-2', '1712.01561-2-50-2'], ['1712.01561-1-49-3', '1712.01561-2-50-3'], ['1712.01561-1-92-0', '1712.01561-2-93-0'], ['1712.01561-1-25-0', '1712.01561-2-26-0'], ['1712.01561-1-25-1', '1712.01561-2-26-1'], ['1712.01561-1-91-0', '1712.01561-2-92-0'], ['1712.01561-1-50-0', '1712.01561-2-51-0'], ['1712.01561-1-58-0', '1712.01561-2-59-0'], ['1712.01561-2-23-0', '1712.01561-3-31-0'], ['1712.01561-2-23-1', '1712.01561-3-31-1'], ['1712.01561-2-23-3', '1712.01561-3-31-3'], ['1712.01561-2-62-0', '1712.01561-3-76-0'], ['1712.01561-2-62-1', '1712.01561-3-76-1'], ['1712.01561-2-52-0', '1712.01561-3-60-0'], ['1712.01561-2-30-2', '1712.01561-3-38-2'], ['1712.01561-2-30-3', '1712.01561-3-38-3'], ['1712.01561-2-30-4', '1712.01561-3-38-4'], ['1712.01561-2-26-0', '1712.01561-3-34-0'], ['1712.01561-2-44-0', '1712.01561-3-52-0'], ['1712.01561-2-44-1', '1712.01561-3-52-1'], ['1712.01561-2-77-2', '1712.01561-3-87-1'], ['1712.01561-2-47-0', '1712.01561-3-56-0'], ['1712.01561-2-35-0', '1712.01561-3-43-0'], ['1712.01561-2-38-0', '1712.01561-3-46-0'], ['1712.01561-2-49-0', '1712.01561-3-58-0'], ['1712.01561-2-61-1', '1712.01561-3-75-1'], ['1712.01561-2-61-4', '1712.01561-3-75-4'], ['1712.01561-2-56-0', '1712.01561-3-71-0'], ['1712.01561-2-63-2', '1712.01561-3-77-2'], ['1712.01561-2-33-0', '1712.01561-3-41-0'], ['1712.01561-2-33-1', '1712.01561-3-41-1'], ['1712.01561-2-58-1', '1712.01561-3-84-1'], ['1712.01561-2-58-2', '1712.01561-3-84-2'], ['1712.01561-2-37-2', '1712.01561-3-45-2'], ['1712.01561-2-37-5', '1712.01561-3-45-5'], ['1712.01561-2-31-2', '1712.01561-3-39-2'], ['1712.01561-2-46-2', '1712.01561-3-55-2'], ['1712.01561-2-55-1', '1712.01561-3-70-2'], ['1712.01561-2-19-4', '1712.01561-3-27-1'], ['1712.01561-2-19-5', '1712.01561-3-27-2'], ['1712.01561-2-19-8', '1712.01561-3-27-4'], ['1712.01561-2-19-9', '1712.01561-3-27-5'], ['1712.01561-2-32-0', '1712.01561-3-40-0'], ['1712.01561-2-32-1', '1712.01561-3-40-1'], ['1712.01561-2-36-0', '1712.01561-3-44-0'], ['1712.01561-2-36-1', '1712.01561-3-44-1'], ['1712.01561-2-36-3', '1712.01561-3-44-3'], ['1712.01561-2-36-4', '1712.01561-3-44-4'], ['1712.01561-2-36-5', '1712.01561-3-44-5'], ['1712.01561-2-36-6', '1712.01561-3-44-6'], ['1712.01561-2-7-2', '1712.01561-3-4-2'], ['1712.01561-2-7-3', '1712.01561-3-4-3'], ['1712.01561-2-74-1', '1712.01561-3-89-1'], ['1712.01561-2-74-2', '1712.01561-3-89-2'], ['1712.01561-2-41-0', '1712.01561-3-49-0'], ['1712.01561-2-9-1', '1712.01561-3-6-1'], ['1712.01561-2-9-2', '1712.01561-3-6-2'], ['1712.01561-2-9-3', '1712.01561-3-6-3'], ['1712.01561-2-20-1', '1712.01561-3-28-1'], ['1712.01561-2-48-0', '1712.01561-3-57-0'], ['1712.01561-2-48-1', '1712.01561-3-57-1'], ['1712.01561-2-43-0', '1712.01561-3-51-0'], ['1712.01561-2-25-0', '1712.01561-3-33-0'], ['1712.01561-2-34-0', '1712.01561-3-42-0'], ['1712.01561-2-34-1', '1712.01561-3-42-1'], ['1712.01561-2-78-2', '1712.01561-3-93-0'], ['1712.01561-2-78-3', '1712.01561-3-93-1'], ['1712.01561-2-67-0', '1712.01561-3-63-0'], ['1712.01561-2-15-2', '1712.01561-3-11-2'], ['1712.01561-2-15-6', '1712.01561-3-12-1'], ['1712.01561-2-68-1', '1712.01561-3-64-1'], ['1712.01561-2-69-0', '1712.01561-3-64-2'], ['1712.01561-2-69-1', '1712.01561-3-64-3'], ['1712.01561-1-18-0', '1712.01561-2-19-0'], ['1712.01561-1-18-1', '1712.01561-2-19-1'], ['1712.01561-1-18-2', '1712.01561-2-19-2'], ['1712.01561-1-18-3', '1712.01561-2-19-3'], ['1712.01561-1-18-4', '1712.01561-2-19-4'], ['1712.01561-1-18-5', '1712.01561-2-19-5'], ['1712.01561-1-18-11', '1712.01561-2-20-2'], ['1712.01561-1-19-0', '1712.01561-2-19-9'], ['1712.01561-1-19-1', '1712.01561-2-19-10'], ['1712.01561-1-30-2', '1712.01561-2-31-2'], ['1712.01561-1-63-1', '1712.01561-2-64-1'], ['1712.01561-1-26-0', '1712.01561-2-27-0'], ['1712.01561-1-7-0', '1712.01561-2-8-0'], ['1712.01561-1-22-0', '1712.01561-2-23-0'], ['1712.01561-1-22-1', '1712.01561-2-23-1'], ['1712.01561-1-11-0', '1712.01561-2-12-0'], ['1712.01561-1-68-0', '1712.01561-2-69-0'], ['1712.01561-1-31-0', '1712.01561-2-32-0'], ['1712.01561-1-35-6', '1712.01561-2-36-6'], ['1712.01561-1-42-0', '1712.01561-2-43-0'], ['1712.01561-2-23-2', '1712.01561-3-31-2'], ['1712.01561-2-27-0', '1712.01561-3-35-0'], ['1712.01561-2-30-0', '1712.01561-3-38-0'], ['1712.01561-2-30-1', '1712.01561-3-38-1'], ['1712.01561-2-14-0', '1712.01561-3-15-0'], ['1712.01561-2-14-1', '1712.01561-3-15-1'], ['1712.01561-2-93-0', '1712.01561-3-117-0'], ['1712.01561-2-26-1', '1712.01561-3-34-1'], ['1712.01561-2-77-0', '1712.01561-3-87-0'], ['1712.01561-2-47-1', '1712.01561-3-56-1'], ['1712.01561-2-38-1', '1712.01561-3-46-1'], ['1712.01561-2-49-1', '1712.01561-3-58-1'], ['1712.01561-2-61-0', '1712.01561-3-75-0'], ['1712.01561-2-61-2', '1712.01561-3-75-2'], ['1712.01561-2-61-3', '1712.01561-3-75-3'], ['1712.01561-2-99-0', '1712.01561-3-125-1'], ['1712.01561-2-99-1', '1712.01561-3-125-2'], ['1712.01561-2-99-2', '1712.01561-3-125-3'], ['1712.01561-2-56-1', '1712.01561-3-71-1'], ['1712.01561-2-28-0', '1712.01561-3-36-0'], ['1712.01561-2-58-0', '1712.01561-3-84-0'], ['1712.01561-2-58-3', '1712.01561-3-84-3'], ['1712.01561-2-87-0', '1712.01561-3-97-0'], ['1712.01561-2-87-1', '1712.01561-3-97-1'], ['1712.01561-2-87-2', '1712.01561-3-97-2'], ['1712.01561-2-37-0', '1712.01561-3-45-0'], ['1712.01561-2-37-1', '1712.01561-3-45-1'], ['1712.01561-2-37-3', '1712.01561-3-45-3'], ['1712.01561-2-37-4', '1712.01561-3-45-4'], ['1712.01561-2-50-2', '1712.01561-3-59-1'], ['1712.01561-2-50-3', '1712.01561-3-59-2'], ['1712.01561-2-31-0', '1712.01561-3-39-0'], ['1712.01561-2-31-1', '1712.01561-3-39-1'], ['1712.01561-2-31-3', '1712.01561-3-39-3'], ['1712.01561-2-46-0', '1712.01561-3-55-0'], ['1712.01561-2-55-0', '1712.01561-3-70-1'], ['1712.01561-2-55-2', '1712.01561-3-70-3'], ['1712.01561-2-64-0', '1712.01561-3-78-0'], ['1712.01561-2-64-1', '1712.01561-3-78-1'], ['1712.01561-2-36-2', '1712.01561-3-44-2'], ['1712.01561-2-36-7', '1712.01561-3-44-7'], ['1712.01561-2-36-8', '1712.01561-3-44-8'], ['1712.01561-2-36-9', '1712.01561-3-44-9'], ['1712.01561-2-81-1', '1712.01561-3-108-1'], ['1712.01561-2-7-0', '1712.01561-3-4-0'], ['1712.01561-2-7-1', '1712.01561-3-4-1'], ['1712.01561-2-79-0', '1712.01561-3-94-0'], ['1712.01561-2-72-0', '1712.01561-3-80-0'], ['1712.01561-2-72-1', '1712.01561-3-80-1'], ['1712.01561-2-74-0', '1712.01561-3-89-0'], ['1712.01561-2-70-0', '1712.01561-3-69-0'], ['1712.01561-2-41-2', '1712.01561-3-49-2'], ['1712.01561-2-9-4', '1712.01561-3-6-4'], ['1712.01561-2-82-0', '1712.01561-3-109-0'], ['1712.01561-2-82-1', '1712.01561-3-109-1'], ['1712.01561-2-82-3', '1712.01561-3-109-3'], ['1712.01561-2-88-0', '1712.01561-3-111-0'], ['1712.01561-2-20-2', '1712.01561-3-28-2'], ['1712.01561-2-73-0', '1712.01561-3-104-0'], ['1712.01561-2-73-1', '1712.01561-3-104-1'], ['1712.01561-2-66-0', '1712.01561-3-62-0'], ['1712.01561-2-83-0', '1712.01561-3-110-0'], ['1712.01561-2-39-0', '1712.01561-3-47-0'], ['1712.01561-2-13-1', '1712.01561-3-14-0'], ['1712.01561-2-25-1', '1712.01561-3-33-1'], ['1712.01561-2-78-1', '1712.01561-3-92-1'], ['1712.01561-2-92-0', '1712.01561-3-101-0'], ['1712.01561-2-75-0', '1712.01561-3-90-0'], ['1712.01561-2-75-1', '1712.01561-3-105-1'], ['1712.01561-2-90-0', '1712.01561-3-99-0'], ['1712.01561-2-91-0', '1712.01561-3-100-0'], ['1712.01561-2-89-0', '1712.01561-3-98-0'], ['1712.01561-2-76-1', '1712.01561-3-106-1'], ['1712.01561-2-76-2', '1712.01561-3-106-2'], ['1712.01561-2-15-3', '1712.01561-3-10-1'], ['1712.01561-2-15-4', '1712.01561-3-10-2'], ['1712.01561-2-68-0', '1712.01561-3-64-0'], ['1712.01561-2-5-1', '1712.01561-3-3-2'], ['1712.01561-2-6-0', '1712.01561-3-3-3'], ['1712.01561-2-6-1', '1712.01561-3-3-4'], ['1712.01561-2-59-0', '1712.01561-3-85-0'], ['1712.01561-1-18-6', '1712.01561-2-19-6'], ['1712.01561-1-18-7', '1712.01561-2-19-7'], ['1712.01561-1-18-10', '1712.01561-2-20-0'], ['1712.01561-2-85-2', '1712.01561-3-86-0'], ['1712.01561-2-86-0', '1712.01561-3-96-2'], ['1712.01561-2-86-1', '1712.01561-3-96-3'], ['1712.01561-2-85-0', '1712.01561-3-96-0'], ['1712.01561-2-85-1', '1712.01561-3-96-1'], ['1712.01561-2-84-2', '1712.01561-3-95-3'], ['1712.01561-2-80-1', '1712.01561-3-107-2'], ['1712.01561-2-80-0', '1712.01561-3-95-0'], ['1712.01561-2-80-1', '1712.01561-3-95-1'], ['1712.01561-2-17-0', '1712.01561-3-17-0'], ['1712.01561-1-7-2', '1712.01561-2-8-2'], ['1712.01561-1-7-2', '1712.01561-2-8-3'], ['1712.01561-1-22-2', '1712.01561-2-23-2'], ['1712.01561-1-0-0', '1712.01561-2-0-1'], ['1712.01561-1-5-0', '1712.01561-2-5-0'], ['1712.01561-2-96-0', '1712.01561-3-123-0'], ['1712.01561-2-17-1', '1712.01561-3-17-8'], ['1712.01561-2-24-0', '1712.01561-3-32-0'], ['1712.01561-2-24-0', '1712.01561-3-32-1'], ['1712.01561-2-24-1', '1712.01561-3-32-1'], ['1712.01561-2-63-0', '1712.01561-3-77-0'], ['1712.01561-2-63-1', '1712.01561-3-77-1'], ['1712.01561-2-51-0', '1712.01561-3-54-0'], ['1712.01561-2-50-0', '1712.01561-3-59-0'], ['1712.01561-2-19-7', '1712.01561-3-27-3'], ['1712.01561-2-81-0', '1712.01561-3-108-0'], ['1712.01561-2-81-2', '1712.01561-3-108-2'], ['1712.01561-2-72-2', '1712.01561-3-80-2'], ['1712.01561-2-70-1', '1712.01561-3-69-1'], ['1712.01561-2-41-1', '1712.01561-3-49-1'], ['1712.01561-2-9-0', '1712.01561-3-6-0'], ['1712.01561-2-82-2', '1712.01561-3-109-2'], ['1712.01561-2-88-1', '1712.01561-3-111-1'], ['1712.01561-2-20-0', '1712.01561-3-28-0'], ['1712.01561-2-8-0', '1712.01561-3-5-0'], ['1712.01561-2-8-1', '1712.01561-3-5-1'], ['1712.01561-2-8-4', '1712.01561-3-5-2'], ['1712.01561-2-0-0', '1712.01561-3-0-0'], ['1712.01561-2-0-1', '1712.01561-3-0-1'], ['1712.01561-2-0-4', '1712.01561-3-0-2'], ['1712.01561-2-0-5', '1712.01561-3-0-3'], ['1712.01561-2-0-6', '1712.01561-3-0-4'], ['1712.01561-2-0-7', '1712.01561-3-0-5'], ['1712.01561-2-95-0', '1712.01561-3-122-0'], ['1712.01561-2-95-1', '1712.01561-3-122-1'], ['1712.01561-2-78-0', '1712.01561-3-92-0'], ['1712.01561-2-90-2', '1712.01561-3-99-1'], ['1712.01561-2-90-2', '1712.01561-3-113-1'], ['1712.01561-2-91-1', '1712.01561-3-100-1'], ['1712.01561-2-91-1', '1712.01561-3-114-0'], ['1712.01561-2-91-2', '1712.01561-3-114-1'], ['1712.01561-2-76-0', '1712.01561-3-91-0'], ['1712.01561-2-76-0', '1712.01561-3-106-0'], ['1712.01561-2-15-0', '1712.01561-3-11-0'], ['1712.01561-2-15-5', '1712.01561-3-12-0'], ['1712.01561-2-5-0', '1712.01561-3-3-1'], ['1712.01561-2-60-0', '1712.01561-3-85-1'], ['1712.01561-1-18-8', '1712.01561-2-19-8'], ['1712.01561-2-85-3', '1712.01561-3-86-1'], ['1712.01561-2-85-3', '1712.01561-3-86-2'], ['1712.01561-2-85-4', '1712.01561-3-86-3'], ['1712.01561-2-84-0', '1712.01561-3-95-0'], ['1712.01561-2-84-1', '1712.01561-3-95-2'], ['1712.01561-2-80-0', '1712.01561-3-107-1'], ['1712.01561-2-80-2', '1712.01561-3-107-3'], ['1712.01561-2-80-2', '1712.01561-3-95-2']]

[['1712.01561-1-8-0', '1712.01561-2-9-0'], ['1712.01561-1-8-1', '1712.01561-2-9-1'], ['1712.01561-1-8-2', '1712.01561-2-9-2'], ['1712.01561-1-8-3', '1712.01561-2-9-3'], ['1712.01561-1-8-4', '1712.01561-2-9-4'], ['1712.01561-1-54-0', '1712.01561-2-55-0'], ['1712.01561-1-54-1', '1712.01561-2-55-1'], ['1712.01561-1-54-2', '1712.01561-2-55-2'], ['1712.01561-1-30-0', '1712.01561-2-31-0'], ['1712.01561-1-30-1', '1712.01561-2-31-1'], ['1712.01561-1-30-3', '1712.01561-2-31-3'], ['1712.01561-1-79-0', '1712.01561-2-80-0'], ['1712.01561-1-79-1', '1712.01561-2-80-1'], ['1712.01561-1-79-2', '1712.01561-2-80-2'], ['1712.01561-1-47-0', '1712.01561-2-48-0'], ['1712.01561-1-47-1', '1712.01561-2-48-1'], ['1712.01561-1-97-0', '1712.01561-2-98-0'], ['1712.01561-1-97-1', '1712.01561-2-98-1'], ['1712.01561-1-97-2', '1712.01561-2-98-2'], ['1712.01561-1-97-3', '1712.01561-2-98-3'], ['1712.01561-1-72-0', '1712.01561-2-73-0'], ['1712.01561-1-72-1', '1712.01561-2-73-1'], ['1712.01561-1-82-0', '1712.01561-2-83-0'], ['1712.01561-1-84-0', '1712.01561-2-85-0'], ['1712.01561-1-84-1', '1712.01561-2-85-1'], ['1712.01561-1-84-2', '1712.01561-2-85-2'], ['1712.01561-1-84-3', '1712.01561-2-85-3'], ['1712.01561-1-84-4', '1712.01561-2-85-4'], ['1712.01561-1-63-0', '1712.01561-2-64-0'], ['1712.01561-1-24-0', '1712.01561-2-25-0'], ['1712.01561-1-24-1', '1712.01561-2-25-1'], ['1712.01561-1-55-0', '1712.01561-2-56-0'], ['1712.01561-1-55-1', '1712.01561-2-56-1'], ['1712.01561-1-66-0', '1712.01561-2-67-0'], ['1712.01561-1-53-0', '1712.01561-2-54-0'], ['1712.01561-1-37-0', '1712.01561-2-38-0'], ['1712.01561-1-37-1', '1712.01561-2-38-1'], ['1712.01561-1-32-0', '1712.01561-2-33-0'], ['1712.01561-1-32-1', '1712.01561-2-33-1'], ['1712.01561-1-83-0', '1712.01561-2-84-0'], ['1712.01561-1-83-1', '1712.01561-2-84-1'], ['1712.01561-1-83-2', '1712.01561-2-84-2'], ['1712.01561-1-29-0', '1712.01561-2-30-0'], ['1712.01561-1-29-1', '1712.01561-2-30-1'], ['1712.01561-1-29-2', '1712.01561-2-30-2'], ['1712.01561-1-29-3', '1712.01561-2-30-3'], ['1712.01561-1-29-4', '1712.01561-2-30-4'], ['1712.01561-1-88-0', '1712.01561-2-89-0'], ['1712.01561-1-7-1', '1712.01561-2-8-1'], ['1712.01561-1-7-4', '1712.01561-2-8-4'], ['1712.01561-1-7-5', '1712.01561-2-8-5'], ['1712.01561-1-7-6', '1712.01561-2-8-6'], ['1712.01561-1-71-0', '1712.01561-2-72-0'], ['1712.01561-1-71-1', '1712.01561-2-72-1'], ['1712.01561-1-71-2', '1712.01561-2-72-2'], ['1712.01561-1-4-0', '1712.01561-2-4-0'], ['1712.01561-1-4-1', '1712.01561-2-4-1'], ['1712.01561-1-43-0', '1712.01561-2-44-0'], ['1712.01561-1-43-1', '1712.01561-2-44-1'], ['1712.01561-1-76-0', '1712.01561-2-77-0'], ['1712.01561-1-76-1', '1712.01561-2-77-1'], ['1712.01561-1-76-2', '1712.01561-2-77-2'], ['1712.01561-1-65-0', '1712.01561-2-66-0'], ['1712.01561-1-89-0', '1712.01561-2-90-0'], ['1712.01561-1-89-1', '1712.01561-2-90-1'], ['1712.01561-1-89-2', '1712.01561-2-90-2'], ['1712.01561-1-67-0', '1712.01561-2-68-0'], ['1712.01561-1-67-1', '1712.01561-2-68-1'], ['1712.01561-1-51-0', '1712.01561-2-52-0'], ['1712.01561-1-57-0', '1712.01561-2-58-0'], ['1712.01561-1-57-1', '1712.01561-2-58-1'], ['1712.01561-1-57-2', '1712.01561-2-58-2'], ['1712.01561-1-57-3', '1712.01561-2-58-3'], ['1712.01561-1-40-0', '1712.01561-2-41-0'], ['1712.01561-1-40-1', '1712.01561-2-41-1'], ['1712.01561-1-40-2', '1712.01561-2-41-2'], ['1712.01561-1-34-0', '1712.01561-2-35-0'], ['1712.01561-1-3-0', '1712.01561-2-3-0'], ['1712.01561-1-45-0', '1712.01561-2-46-0'], ['1712.01561-1-45-2', '1712.01561-2-46-2'], ['1712.01561-1-69-0', '1712.01561-2-70-0'], ['1712.01561-1-69-1', '1712.01561-2-70-1'], ['1712.01561-1-60-0', '1712.01561-2-61-0'], ['1712.01561-1-60-1', '1712.01561-2-61-1'], ['1712.01561-1-60-2', '1712.01561-2-61-2'], ['1712.01561-1-60-3', '1712.01561-2-61-3'], ['1712.01561-1-60-4', '1712.01561-2-61-4'], ['1712.01561-1-22-3', '1712.01561-2-23-3'], ['1712.01561-1-9-0', '1712.01561-2-10-0'], ['1712.01561-1-6-0', '1712.01561-2-7-0'], ['1712.01561-1-6-1', '1712.01561-2-7-1'], ['1712.01561-1-6-2', '1712.01561-2-7-2'], ['1712.01561-1-6-3', '1712.01561-2-7-3'], ['1712.01561-1-46-0', '1712.01561-2-47-0'], ['1712.01561-1-46-1', '1712.01561-2-47-1'], ['1712.01561-1-23-0', '1712.01561-2-24-0'], ['1712.01561-1-23-1', '1712.01561-2-24-1'], ['1712.01561-1-95-0', '1712.01561-2-96-0'], ['1712.01561-1-75-0', '1712.01561-2-76-0'], ['1712.01561-1-75-1', '1712.01561-2-76-1'], ['1712.01561-1-75-2', '1712.01561-2-76-2'], ['1712.01561-1-59-0', '1712.01561-2-60-0'], ['1712.01561-1-77-0', '1712.01561-2-78-0'], ['1712.01561-1-77-1', '1712.01561-2-78-1'], ['1712.01561-1-77-2', '1712.01561-2-78-2'], ['1712.01561-1-77-3', '1712.01561-2-78-3'], ['1712.01561-1-78-0', '1712.01561-2-79-0'], ['1712.01561-1-13-0', '1712.01561-2-14-0'], ['1712.01561-1-13-1', '1712.01561-2-14-1'], ['1712.01561-1-90-0', '1712.01561-2-91-0'], ['1712.01561-1-90-1', '1712.01561-2-91-1'], ['1712.01561-1-90-2', '1712.01561-2-91-2'], ['1712.01561-1-62-0', '1712.01561-2-63-0'], ['1712.01561-1-62-1', '1712.01561-2-63-1'], ['1712.01561-1-62-2', '1712.01561-2-63-2'], ['1712.01561-1-61-0', '1712.01561-2-62-0'], ['1712.01561-1-61-1', '1712.01561-2-62-1'], ['1712.01561-1-94-0', '1712.01561-2-95-0'], ['1712.01561-1-94-1', '1712.01561-2-95-1'], ['1712.01561-1-98-0', '1712.01561-2-99-0'], ['1712.01561-1-98-1', '1712.01561-2-99-1'], ['1712.01561-1-98-2', '1712.01561-2-99-2'], ['1712.01561-1-48-0', '1712.01561-2-49-0'], ['1712.01561-1-48-1', '1712.01561-2-49-1'], ['1712.01561-1-38-0', '1712.01561-2-39-0'], ['1712.01561-1-14-0', '1712.01561-2-15-0'], ['1712.01561-1-14-1', '1712.01561-2-15-1'], ['1712.01561-1-14-2', '1712.01561-2-15-2'], ['1712.01561-1-14-3', '1712.01561-2-15-3'], ['1712.01561-1-14-4', '1712.01561-2-15-4'], ['1712.01561-1-14-5', '1712.01561-2-15-5'], ['1712.01561-1-14-6', '1712.01561-2-15-6'], ['1712.01561-1-12-0', '1712.01561-2-13-0'], ['1712.01561-1-12-1', '1712.01561-2-13-1'], ['1712.01561-1-86-0', '1712.01561-2-87-0'], ['1712.01561-1-86-1', '1712.01561-2-87-1'], ['1712.01561-1-86-2', '1712.01561-2-87-2'], ['1712.01561-1-16-0', '1712.01561-2-17-0'], ['1712.01561-1-16-1', '1712.01561-2-17-1'], ['1712.01561-1-27-0', '1712.01561-2-28-0'], ['1712.01561-1-36-0', '1712.01561-2-37-0'], ['1712.01561-1-36-1', '1712.01561-2-37-1'], ['1712.01561-1-36-2', '1712.01561-2-37-2'], ['1712.01561-1-36-3', '1712.01561-2-37-3'], ['1712.01561-1-36-4', '1712.01561-2-37-4'], ['1712.01561-1-36-5', '1712.01561-2-37-5'], ['1712.01561-1-10-0', '1712.01561-2-11-0'], ['1712.01561-1-10-1', '1712.01561-2-11-1'], ['1712.01561-1-10-2', '1712.01561-2-11-2'], ['1712.01561-1-15-0', '1712.01561-2-16-0'], ['1712.01561-1-15-1', '1712.01561-2-16-1'], ['1712.01561-1-15-2', '1712.01561-2-16-2'], ['1712.01561-1-31-1', '1712.01561-2-32-1'], ['1712.01561-1-85-0', '1712.01561-2-86-0'], ['1712.01561-1-85-1', '1712.01561-2-86-1'], ['1712.01561-1-33-0', '1712.01561-2-34-0'], ['1712.01561-1-33-1', '1712.01561-2-34-1'], ['1712.01561-1-35-0', '1712.01561-2-36-0'], ['1712.01561-1-35-1', '1712.01561-2-36-1'], ['1712.01561-1-35-2', '1712.01561-2-36-2'], ['1712.01561-1-35-3', '1712.01561-2-36-3'], ['1712.01561-1-35-4', '1712.01561-2-36-4'], ['1712.01561-1-35-5', '1712.01561-2-36-5'], ['1712.01561-1-35-7', '1712.01561-2-36-7'], ['1712.01561-1-35-8', '1712.01561-2-36-8'], ['1712.01561-1-35-9', '1712.01561-2-36-9'], ['1712.01561-1-87-0', '1712.01561-2-88-0'], ['1712.01561-1-87-1', '1712.01561-2-88-1'], ['1712.01561-1-0-1', '1712.01561-2-0-2'], ['1712.01561-1-0-2', '1712.01561-2-0-3'], ['1712.01561-1-0-3', '1712.01561-2-0-4'], ['1712.01561-1-0-4', '1712.01561-2-0-5'], ['1712.01561-1-0-5', '1712.01561-2-0-6'], ['1712.01561-1-0-6', '1712.01561-2-0-7'], ['1712.01561-1-73-0', '1712.01561-2-74-0'], ['1712.01561-1-73-1', '1712.01561-2-74-1'], ['1712.01561-1-73-2', '1712.01561-2-74-2'], ['1712.01561-1-81-0', '1712.01561-2-82-0'], ['1712.01561-1-81-1', '1712.01561-2-82-1'], ['1712.01561-1-81-2', '1712.01561-2-82-2'], ['1712.01561-1-81-3', '1712.01561-2-82-3'], ['1712.01561-1-80-0', '1712.01561-2-81-0'], ['1712.01561-1-80-1', '1712.01561-2-81-1'], ['1712.01561-1-80-2', '1712.01561-2-81-2'], ['1712.01561-1-74-0', '1712.01561-2-75-0'], ['1712.01561-1-74-1', '1712.01561-2-75-1'], ['1712.01561-1-49-0', '1712.01561-2-50-0'], ['1712.01561-1-49-2', '1712.01561-2-50-2'], ['1712.01561-1-49-3', '1712.01561-2-50-3'], ['1712.01561-1-92-0', '1712.01561-2-93-0'], ['1712.01561-1-25-0', '1712.01561-2-26-0'], ['1712.01561-1-25-1', '1712.01561-2-26-1'], ['1712.01561-1-91-0', '1712.01561-2-92-0'], ['1712.01561-1-50-0', '1712.01561-2-51-0'], ['1712.01561-1-58-0', '1712.01561-2-59-0'], ['1712.01561-2-23-0', '1712.01561-3-31-0'], ['1712.01561-2-23-1', '1712.01561-3-31-1'], ['1712.01561-2-23-3', '1712.01561-3-31-3'], ['1712.01561-2-62-0', '1712.01561-3-76-0'], ['1712.01561-2-62-1', '1712.01561-3-76-1'], ['1712.01561-2-52-0', '1712.01561-3-60-0'], ['1712.01561-2-30-2', '1712.01561-3-38-2'], ['1712.01561-2-30-3', '1712.01561-3-38-3'], ['1712.01561-2-30-4', '1712.01561-3-38-4'], ['1712.01561-2-26-0', '1712.01561-3-34-0'], ['1712.01561-2-44-0', '1712.01561-3-52-0'], ['1712.01561-2-44-1', '1712.01561-3-52-1'], ['1712.01561-2-77-2', '1712.01561-3-87-1'], ['1712.01561-2-47-0', '1712.01561-3-56-0'], ['1712.01561-2-35-0', '1712.01561-3-43-0'], ['1712.01561-2-38-0', '1712.01561-3-46-0'], ['1712.01561-2-49-0', '1712.01561-3-58-0'], ['1712.01561-2-61-1', '1712.01561-3-75-1'], ['1712.01561-2-61-4', '1712.01561-3-75-4'], ['1712.01561-2-56-0', '1712.01561-3-71-0'], ['1712.01561-2-63-2', '1712.01561-3-77-2'], ['1712.01561-2-33-0', '1712.01561-3-41-0'], ['1712.01561-2-33-1', '1712.01561-3-41-1'], ['1712.01561-2-58-1', '1712.01561-3-84-1'], ['1712.01561-2-58-2', '1712.01561-3-84-2'], ['1712.01561-2-37-2', '1712.01561-3-45-2'], ['1712.01561-2-37-5', '1712.01561-3-45-5'], ['1712.01561-2-31-2', '1712.01561-3-39-2'], ['1712.01561-2-46-2', '1712.01561-3-55-2'], ['1712.01561-2-55-1', '1712.01561-3-70-2'], ['1712.01561-2-19-4', '1712.01561-3-27-1'], ['1712.01561-2-19-5', '1712.01561-3-27-2'], ['1712.01561-2-19-8', '1712.01561-3-27-4'], ['1712.01561-2-19-9', '1712.01561-3-27-5'], ['1712.01561-2-32-0', '1712.01561-3-40-0'], ['1712.01561-2-32-1', '1712.01561-3-40-1'], ['1712.01561-2-36-0', '1712.01561-3-44-0'], ['1712.01561-2-36-1', '1712.01561-3-44-1'], ['1712.01561-2-36-3', '1712.01561-3-44-3'], ['1712.01561-2-36-4', '1712.01561-3-44-4'], ['1712.01561-2-36-5', '1712.01561-3-44-5'], ['1712.01561-2-36-6', '1712.01561-3-44-6'], ['1712.01561-2-7-2', '1712.01561-3-4-2'], ['1712.01561-2-7-3', '1712.01561-3-4-3'], ['1712.01561-2-74-1', '1712.01561-3-89-1'], ['1712.01561-2-74-2', '1712.01561-3-89-2'], ['1712.01561-2-41-0', '1712.01561-3-49-0'], ['1712.01561-2-9-1', '1712.01561-3-6-1'], ['1712.01561-2-9-2', '1712.01561-3-6-2'], ['1712.01561-2-9-3', '1712.01561-3-6-3'], ['1712.01561-2-20-1', '1712.01561-3-28-1'], ['1712.01561-2-48-0', '1712.01561-3-57-0'], ['1712.01561-2-48-1', '1712.01561-3-57-1'], ['1712.01561-2-43-0', '1712.01561-3-51-0'], ['1712.01561-2-25-0', '1712.01561-3-33-0'], ['1712.01561-2-34-0', '1712.01561-3-42-0'], ['1712.01561-2-34-1', '1712.01561-3-42-1'], ['1712.01561-2-78-2', '1712.01561-3-93-0'], ['1712.01561-2-78-3', '1712.01561-3-93-1'], ['1712.01561-2-67-0', '1712.01561-3-63-0'], ['1712.01561-2-15-2', '1712.01561-3-11-2'], ['1712.01561-2-15-6', '1712.01561-3-12-1'], ['1712.01561-2-68-1', '1712.01561-3-64-1'], ['1712.01561-2-69-0', '1712.01561-3-64-2'], ['1712.01561-2-69-1', '1712.01561-3-64-3'], ['1712.01561-1-18-0', '1712.01561-2-19-0'], ['1712.01561-1-18-1', '1712.01561-2-19-1'], ['1712.01561-1-18-2', '1712.01561-2-19-2'], ['1712.01561-1-18-3', '1712.01561-2-19-3'], ['1712.01561-1-18-4', '1712.01561-2-19-4'], ['1712.01561-1-18-5', '1712.01561-2-19-5'], ['1712.01561-1-18-11', '1712.01561-2-20-2'], ['1712.01561-1-19-0', '1712.01561-2-19-9'], ['1712.01561-1-19-1', '1712.01561-2-19-10']]

[['1712.01561-1-30-2', '1712.01561-2-31-2'], ['1712.01561-1-63-1', '1712.01561-2-64-1'], ['1712.01561-1-26-0', '1712.01561-2-27-0'], ['1712.01561-1-7-0', '1712.01561-2-8-0'], ['1712.01561-1-22-0', '1712.01561-2-23-0'], ['1712.01561-1-22-1', '1712.01561-2-23-1'], ['1712.01561-1-11-0', '1712.01561-2-12-0'], ['1712.01561-1-68-0', '1712.01561-2-69-0'], ['1712.01561-1-31-0', '1712.01561-2-32-0'], ['1712.01561-1-35-6', '1712.01561-2-36-6'], ['1712.01561-1-42-0', '1712.01561-2-43-0'], ['1712.01561-2-23-2', '1712.01561-3-31-2'], ['1712.01561-2-27-0', '1712.01561-3-35-0'], ['1712.01561-2-30-0', '1712.01561-3-38-0'], ['1712.01561-2-30-1', '1712.01561-3-38-1'], ['1712.01561-2-14-0', '1712.01561-3-15-0'], ['1712.01561-2-14-1', '1712.01561-3-15-1'], ['1712.01561-2-93-0', '1712.01561-3-117-0'], ['1712.01561-2-26-1', '1712.01561-3-34-1'], ['1712.01561-2-77-0', '1712.01561-3-87-0'], ['1712.01561-2-47-1', '1712.01561-3-56-1'], ['1712.01561-2-38-1', '1712.01561-3-46-1'], ['1712.01561-2-49-1', '1712.01561-3-58-1'], ['1712.01561-2-61-0', '1712.01561-3-75-0'], ['1712.01561-2-61-2', '1712.01561-3-75-2'], ['1712.01561-2-61-3', '1712.01561-3-75-3'], ['1712.01561-2-99-0', '1712.01561-3-125-1'], ['1712.01561-2-99-1', '1712.01561-3-125-2'], ['1712.01561-2-99-2', '1712.01561-3-125-3'], ['1712.01561-2-56-1', '1712.01561-3-71-1'], ['1712.01561-2-28-0', '1712.01561-3-36-0'], ['1712.01561-2-58-0', '1712.01561-3-84-0'], ['1712.01561-2-58-3', '1712.01561-3-84-3'], ['1712.01561-2-87-0', '1712.01561-3-97-0'], ['1712.01561-2-87-1', '1712.01561-3-97-1'], ['1712.01561-2-87-2', '1712.01561-3-97-2'], ['1712.01561-2-37-0', '1712.01561-3-45-0'], ['1712.01561-2-37-1', '1712.01561-3-45-1'], ['1712.01561-2-37-3', '1712.01561-3-45-3'], ['1712.01561-2-37-4', '1712.01561-3-45-4'], ['1712.01561-2-50-2', '1712.01561-3-59-1'], ['1712.01561-2-50-3', '1712.01561-3-59-2'], ['1712.01561-2-31-0', '1712.01561-3-39-0'], ['1712.01561-2-31-1', '1712.01561-3-39-1'], ['1712.01561-2-31-3', '1712.01561-3-39-3'], ['1712.01561-2-46-0', '1712.01561-3-55-0'], ['1712.01561-2-55-0', '1712.01561-3-70-1'], ['1712.01561-2-55-2', '1712.01561-3-70-3'], ['1712.01561-2-64-0', '1712.01561-3-78-0'], ['1712.01561-2-64-1', '1712.01561-3-78-1'], ['1712.01561-2-36-2', '1712.01561-3-44-2'], ['1712.01561-2-36-7', '1712.01561-3-44-7'], ['1712.01561-2-36-8', '1712.01561-3-44-8'], ['1712.01561-2-36-9', '1712.01561-3-44-9'], ['1712.01561-2-81-1', '1712.01561-3-108-1'], ['1712.01561-2-7-0', '1712.01561-3-4-0'], ['1712.01561-2-7-1', '1712.01561-3-4-1'], ['1712.01561-2-79-0', '1712.01561-3-94-0'], ['1712.01561-2-72-0', '1712.01561-3-80-0'], ['1712.01561-2-72-1', '1712.01561-3-80-1'], ['1712.01561-2-74-0', '1712.01561-3-89-0'], ['1712.01561-2-70-0', '1712.01561-3-69-0'], ['1712.01561-2-41-2', '1712.01561-3-49-2'], ['1712.01561-2-9-4', '1712.01561-3-6-4'], ['1712.01561-2-82-0', '1712.01561-3-109-0'], ['1712.01561-2-82-1', '1712.01561-3-109-1'], ['1712.01561-2-82-3', '1712.01561-3-109-3'], ['1712.01561-2-88-0', '1712.01561-3-111-0'], ['1712.01561-2-20-2', '1712.01561-3-28-2'], ['1712.01561-2-73-0', '1712.01561-3-104-0'], ['1712.01561-2-73-1', '1712.01561-3-104-1'], ['1712.01561-2-66-0', '1712.01561-3-62-0'], ['1712.01561-2-83-0', '1712.01561-3-110-0'], ['1712.01561-2-39-0', '1712.01561-3-47-0'], ['1712.01561-2-13-1', '1712.01561-3-14-0'], ['1712.01561-2-25-1', '1712.01561-3-33-1'], ['1712.01561-2-78-1', '1712.01561-3-92-1'], ['1712.01561-2-92-0', '1712.01561-3-101-0'], ['1712.01561-2-75-0', '1712.01561-3-90-0'], ['1712.01561-2-75-1', '1712.01561-3-105-1'], ['1712.01561-2-90-0', '1712.01561-3-99-0'], ['1712.01561-2-91-0', '1712.01561-3-100-0'], ['1712.01561-2-89-0', '1712.01561-3-98-0'], ['1712.01561-2-76-1', '1712.01561-3-106-1'], ['1712.01561-2-76-2', '1712.01561-3-106-2'], ['1712.01561-2-15-3', '1712.01561-3-10-1'], ['1712.01561-2-15-4', '1712.01561-3-10-2'], ['1712.01561-2-68-0', '1712.01561-3-64-0'], ['1712.01561-2-5-1', '1712.01561-3-3-2'], ['1712.01561-2-6-0', '1712.01561-3-3-3'], ['1712.01561-2-6-1', '1712.01561-3-3-4'], ['1712.01561-2-59-0', '1712.01561-3-85-0'], ['1712.01561-1-18-6', '1712.01561-2-19-6'], ['1712.01561-1-18-7', '1712.01561-2-19-7'], ['1712.01561-1-18-10', '1712.01561-2-20-0'], ['1712.01561-2-85-2', '1712.01561-3-86-0'], ['1712.01561-2-86-0', '1712.01561-3-96-2'], ['1712.01561-2-86-1', '1712.01561-3-96-3'], ['1712.01561-2-85-0', '1712.01561-3-96-0'], ['1712.01561-2-85-1', '1712.01561-3-96-1'], ['1712.01561-2-84-2', '1712.01561-3-95-3'], ['1712.01561-2-80-1', '1712.01561-3-107-2'], ['1712.01561-2-80-0', '1712.01561-3-95-0'], ['1712.01561-2-80-1', '1712.01561-3-95-1']]

[['1712.01561-2-17-0', '1712.01561-3-17-0']]

[['1712.01561-1-7-2', '1712.01561-2-8-2'], ['1712.01561-1-7-2', '1712.01561-2-8-3'], ['1712.01561-1-22-2', '1712.01561-2-23-2'], ['1712.01561-1-0-0', '1712.01561-2-0-1'], ['1712.01561-1-5-0', '1712.01561-2-5-0'], ['1712.01561-2-96-0', '1712.01561-3-123-0'], ['1712.01561-2-17-1', '1712.01561-3-17-8'], ['1712.01561-2-24-0', '1712.01561-3-32-0'], ['1712.01561-2-24-0', '1712.01561-3-32-1'], ['1712.01561-2-24-1', '1712.01561-3-32-1'], ['1712.01561-2-63-0', '1712.01561-3-77-0'], ['1712.01561-2-63-1', '1712.01561-3-77-1'], ['1712.01561-2-51-0', '1712.01561-3-54-0'], ['1712.01561-2-50-0', '1712.01561-3-59-0'], ['1712.01561-2-19-7', '1712.01561-3-27-3'], ['1712.01561-2-81-0', '1712.01561-3-108-0'], ['1712.01561-2-81-2', '1712.01561-3-108-2'], ['1712.01561-2-72-2', '1712.01561-3-80-2'], ['1712.01561-2-70-1', '1712.01561-3-69-1'], ['1712.01561-2-41-1', '1712.01561-3-49-1'], ['1712.01561-2-9-0', '1712.01561-3-6-0'], ['1712.01561-2-82-2', '1712.01561-3-109-2'], ['1712.01561-2-88-1', '1712.01561-3-111-1'], ['1712.01561-2-20-0', '1712.01561-3-28-0'], ['1712.01561-2-8-0', '1712.01561-3-5-0'], ['1712.01561-2-8-1', '1712.01561-3-5-1'], ['1712.01561-2-8-4', '1712.01561-3-5-2'], ['1712.01561-2-0-0', '1712.01561-3-0-0'], ['1712.01561-2-0-1', '1712.01561-3-0-1'], ['1712.01561-2-0-4', '1712.01561-3-0-2'], ['1712.01561-2-0-5', '1712.01561-3-0-3'], ['1712.01561-2-0-6', '1712.01561-3-0-4'], ['1712.01561-2-0-7', '1712.01561-3-0-5'], ['1712.01561-2-95-0', '1712.01561-3-122-0'], ['1712.01561-2-95-1', '1712.01561-3-122-1'], ['1712.01561-2-78-0', '1712.01561-3-92-0'], ['1712.01561-2-90-2', '1712.01561-3-99-1'], ['1712.01561-2-90-2', '1712.01561-3-113-1'], ['1712.01561-2-91-1', '1712.01561-3-100-1'], ['1712.01561-2-91-1', '1712.01561-3-114-0'], ['1712.01561-2-91-2', '1712.01561-3-114-1'], ['1712.01561-2-76-0', '1712.01561-3-91-0'], ['1712.01561-2-76-0', '1712.01561-3-106-0'], ['1712.01561-2-15-0', '1712.01561-3-11-0'], ['1712.01561-2-15-5', '1712.01561-3-12-0'], ['1712.01561-2-5-0', '1712.01561-3-3-1'], ['1712.01561-2-60-0', '1712.01561-3-85-1'], ['1712.01561-1-18-8', '1712.01561-2-19-8'], ['1712.01561-2-85-3', '1712.01561-3-86-1'], ['1712.01561-2-85-3', '1712.01561-3-86-2'], ['1712.01561-2-85-4', '1712.01561-3-86-3'], ['1712.01561-2-84-0', '1712.01561-3-95-0'], ['1712.01561-2-84-1', '1712.01561-3-95-2'], ['1712.01561-2-80-0', '1712.01561-3-107-1'], ['1712.01561-2-80-2', '1712.01561-3-107-3'], ['1712.01561-2-80-2', '1712.01561-3-95-2']]

[]

['1712.01561-1-1-0', '1712.01561-1-44-0', '1712.01561-1-45-1', '1712.01561-1-49-1', '1712.01561-1-56-0', '1712.01561-1-96-0', '1712.01561-2-1-0', '1712.01561-2-45-0', '1712.01561-2-46-1', '1712.01561-2-50-1', '1712.01561-2-57-0', '1712.01561-2-97-0', '1712.01561-3-1-0', '1712.01561-3-53-0', '1712.01561-3-55-1', '1712.01561-3-83-0']

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

https://arxiv.org/abs/1712.01561

{'1712.01561-3-0-0': 'Based on an expert systems approach, the issue of community detection can be conceptualized as a clustering model for networks.', '1712.01561-3-0-1': 'Building upon this further, community structure can be measured through a clustering coefficient, which is generated from the number of existing triangles around the nodes over the number of triangles that can be hypothetically constructed.', '1712.01561-3-0-2': 'This paper provides a new definition of the clustering coefficient for weighted networks under a generalized definition of triangles.', '1712.01561-3-0-3': 'Specifically, a novel concept of triangles is introduced, based on the assumption that, should the aggregate weight of two arcs be strong enough, a link between the uncommon nodes can be induced.', '1712.01561-3-0-4': 'Beyond the intuitive meaning of such generalized triangles in the social context, we also explore the usefulness of them for gaining insights into the topological structure of the underlying network.', '1712.01561-3-0-5': 'Empirical experiments on the standard networks of 500 commercial US airports and on the nervous system of the Caenorhabditis elegans support the theoretical framework and allow a comparison between our proposal and the standard definition of clustering coefficient.', '1712.01561-3-1-0': 'Keywords: complex networks; local cohesiveness, clustering coefficient; generalized triangles.', '1712.01561-3-2-0': '# Introduction', '1712.01561-3-3-0': 'Networks represent an effective methodological device for modeling the main features of several complex systems [CITATION].', '1712.01561-3-3-1': 'This paper builds on such a premise by focusing on the tendency of nodes in a network to cluster, i.e. the link formation between neighboring vertices [CITATION] leading to the identification of the local groups cohesiveness.', '1712.01561-3-3-2': 'Such a theme is of paramount relevance in that it allows one to assess the community structure of a group of interconnected units [CITATION].', '1712.01561-3-3-3': 'In this respect, we are in accord with Liu and Juan Ban [CITATION], who state that, in agreement with the expert systems perspective, the problem of community detection can be dealt with as a clustering model for networks.', '1712.01561-3-3-4': 'This explains also why community detection is nowadays at the core of most discourse surrounding social networks (see e.g. [CITATION]).', '1712.01561-3-4-0': 'One of the most acknowledged and employed measures for assessing the tendency of vertices to cluster is the local cluster coefficient [CITATION].', '1712.01561-3-4-1': 'Such a quantity has been extensively studied by several authors and applied in different networks [CITATION].', '1712.01561-3-4-2': 'It captures the degree of social embeddedness of the nodes in a network and is based on local density [CITATION].', '1712.01561-3-4-3': 'Indeed, especially in social networks, vertices tend to create tightly knit groups that are characterized by a relatively high density of links [CITATION].', '1712.01561-3-5-0': 'The clustering coefficient assesses the connectivity of node neighborhoods; a node having a high value of clustering coefficient tends to be directly connected with well-established communities of nodes [CITATION].', '1712.01561-3-5-1': 'Clustering coefficient is relevant when determining the small-world property of a network [CITATION] and can be considered as an index of the redundancy of a node [CITATION].', '1712.01561-3-5-2': 'In the context of weighted networks, the clustering coefficient has been analyzed in Grinrod [CITATION], Onnela et al. [CITATION], Barrat et al. [CITATION], Zhang and Horvath [CITATION] and Opsahl and Panzarasa [CITATION], as reported in Section [REF].', '1712.01561-3-6-0': 'The weighted framework is of paramount relevance, in that the analysis of the weights along the edges and their correlations is able to provide a description of the hierarchical and structural organization of the systems.', '1712.01561-3-6-1': 'This is evident if we consider, as an example, a network in which the weights of all links forming triangles of interconnected vertices are extremely small.', '1712.01561-3-6-2': 'In this case, even for a large clustering coefficient, these triangles play a minimal role in the network dynamics and organization, and the clustering features are certainly overestimated by a simple structural analysis [CITATION].', '1712.01561-3-6-3': 'Also, vertices with high degree can be attached to a majority of low-degree nodes whilst concentrating the largest portion of their strength only on the vertices with high degree.', '1712.01561-3-6-4': 'In this situation, the topology reveals a disassortative characteristic of the network, whereas the system could be considered assortative since the more relevant edges in terms of weights are linked to the high-degree vertices [CITATION].', '1712.01561-3-7-0': 'Despite several measures being proposed for the local and global clustering coefficients, they are all only able to capture the clustering of ego networks or the overall statistics regarding the network [CITATION].', '1712.01561-3-7-1': 'An ego is a focal individual and the ego network is composed of the nodes directly connected to him (also called alters) and the links among him and others (see e.g. [CITATION]).', '1712.01561-3-8-0': 'Thus, in this paper we are interested in two relevant cases.', '1712.01561-3-8-1': 'In the first, the ego is connected to two alters not mutually connected and we aim to understand if the strength of the connections with the ego is strong enough to induce a certain level of interaction as can be found when they are connected.', '1712.01561-3-9-0': 'In the second, the alter of an alter is not directly connected to the ego.', '1712.01561-3-9-1': 'Also in this case, we advance the proposal that the strength of the existing connections induces interactions between the ego and the alter of the alter.', '1712.01561-3-10-0': 'It is worth noting that all the considered aspects can be interpreted in the context of link formation as reasonable premises.', '1712.01561-3-10-1': 'Link prediction is relevant in that it attempts to estimate the likelihood of a link existing between two vertices based on observed links and the attributes of nodes [CITATION].', '1712.01561-3-10-2': 'Such a prediction can be used to analyze a network to suggest promising interactions or collaborations that have not yet been identified, or is related to the problem of inferring missing or additional links that, while not directly visible, are likely to exist [CITATION].', '1712.01561-3-11-0': 'The specific aim of this paper is to introduce a novel definition of a generalized clustering coefficient by including also the triples of the two cases presented above.', '1712.01561-3-11-1': "In so doing, our concept of community captures the weighted network's propensity for close triples.", '1712.01561-3-11-2': 'Moreover, this measure is also useful for predicting the fictitious links that may appear in the future of evolving networks.', '1712.01561-3-12-0': 'Our generalized clustering coefficient has a further relevant property: it assumes unitary value not only when the graph is a clique, but in a number of different situations.', '1712.01561-3-12-1': 'Specifically, the community structure of the network is intended to include also the realistic cases of the presence of indirect connections among two agents induced by their strong links with a third node.', '1712.01561-3-13-0': 'The ground of our study is quite intuitive.', '1712.01561-3-13-1': 'Indeed, in the context of community structure of weighted networks, there is evidence that strong enough connections among two individuals are prone to creating triangles among their neighborhood.', '1712.01561-3-13-2': 'Formally, this means that it is possible to introduce a threshold for stating when the weight of a link can be defined as strong enough.', '1712.01561-3-13-3': 'We reasonably take that the larger the threshold, the stronger the link.', '1712.01561-3-14-0': 'In this respect, as we will see below in the formalization of our setting, null thresholds mean no constraints - and all the two-sided figures can be viewed as triples - while a large value of the thresholds is associated to very restrictive constraints - and a small number of two-sided figures will be accepted as triples.', '1712.01561-3-15-0': 'It is very important to note that the case of zero thresholds gives further insights into the topological structure of the unweighted graph associated to the network.', '1712.01561-3-15-1': 'We direct the reader to the empirical analysis section for an intuitive explanation of this point.', '1712.01561-3-15-2': 'In this regard, we have also implemented a comparison between our definition and the standard clustering coefficient for weighted networks.', '1712.01561-3-16-0': 'Based on such a perspective, this paper also implements a wide computational analysis to explore the reaction of the proposed clustering coefficient to threshold variations.', '1712.01561-3-17-0': 'The paper is structured as follows.', '1712.01561-3-17-1': 'Section 2 outlines the motivations - based also on real-world applications - behind the present study and the novel definition of clustering coefficient.', '1712.01561-3-17-2': 'Such a motivating discussion is proposed before the formal definition to immediately convince the reader of the usefulness of the presented scientific proposal.', '1712.01561-3-17-3': 'For some more formal insights on the generalized clustering coefficient and on the generalized triangles, please refer to Section 5, where a detailed discussion of definitions and concepts is carried out.', '1712.01561-3-17-4': 'Section 3 is devoted to the outline of certain relevant preliminaries and the employed notations about the graph theory.', '1712.01561-3-17-5': 'Section 4 contains a review of the literature on the clustering coefficient in both cases of weighted and unweighted networks.', '1712.01561-3-17-6': 'Section 5 introduces and discusses the proposed definition of generalized clustering coefficient and generalized triples, along with the related interpretation.', '1712.01561-3-17-7': 'Section 6 focuses on the computational experience of two empirical networks: the network among the 500 commercial airports in the United States and the nervous system of the nematode Caenorhabditis elegans.', '1712.01561-3-17-8': 'The final section offers some conclusive remarks and outlines directions for future research.', '1712.01561-3-18-0': '# Motivation for the generalized clustering coefficient and real-world applications', '1712.01561-3-19-0': 'One of the major fields of study in the empirical investigation of networks is the uncovering of subgroups of nodes according to a given criteria.', '1712.01561-3-19-1': 'Such subgroups, or communities, are interesting since they can help to understand a wide variety of possible group organizations, and they occur in networks in biology, computer science, economics, politics and more [CITATION].', '1712.01561-3-19-2': "Recently, community discovery has been used in social media, such as in [CITATION], where authors propose a community-aware approach to constructing resource profiles via social filtering, in [CITATION], where communities are discovered from social media by low-rank matrix recovery, and in [CITATION], where communities are studied by means of the network's internal structural properties.", '1712.01561-3-20-0': 'The behavior of nodes is often highly influenced by the behavior of their neighbors or community members [CITATION].', '1712.01561-3-20-1': 'From this point of view, the clustering coefficient is one of the main measures used to understand the level of cohesion around a node.', '1712.01561-3-21-0': 'The generalized concept of clustering coefficient presented here describes community structures which are not established, but are indirectly induced by strong cooperations among the formally linked nodes.', '1712.01561-3-21-1': 'More specifically, the existence of a powerful link between two nodes is assumed to be able to form the connection between nodes that are disconnected but adjacent to the considered ones.', '1712.01561-3-22-0': 'In this respect, social motivations from the perspective of a single node (ego) support the study of the proposed generalized clustering coefficient.', '1712.01561-3-22-1': 'In particular, the knowledge for two alters of a common ego increases the opportunities for them to meet since they could be engaged in similar interests, which would ultimately provide a basis for them to trust one another [CITATION].', '1712.01561-3-22-2': 'Moreover, social psychology suggests that an ego has an incentive to connect to its alters in order to reduce its isolation [CITATION].', '1712.01561-3-22-3': 'An ego might also be interested in connecting to its neighbors for proximity reasons, taking into consideration the shortest path distance between them [CITATION].', '1712.01561-3-23-0': 'Here, we also consider the possibility of having an indirect connection of an ego with an alter of one of its alters.', '1712.01561-3-23-1': 'Therefore, we consider two forms of engagements between an ego and various alters, as shown in Figure [REF].', '1712.01561-3-23-2': 'More details on the figure will be given in Section [REF].', '1712.01561-3-24-0': 'The perspective adopted here represents the basis for several real-world cases.', '1712.01561-3-24-1': 'For example, in [CITATION], affiliation networks allow one to observe the connections among individuals indirectly, i.e. not through directly observed social interactions, while in [CITATION], a new measure of the clustering coefficient is proposed, with applications in the study of segregation and homophily.', '1712.01561-3-24-2': 'Finally, in biology, gene expression data can be studied with the weighted clustering coefficient in order to reveal differences between normal and tumour networks [CITATION].', '1712.01561-3-25-0': 'It is also worth mentioning how the concepts of triples introduced here could relate to the issue of link formation.', '1712.01561-3-25-1': 'Indeed, the presence of a strong connection between two units would probably induce cooperation also among the units connected with those considered in the near future.', '1712.01561-3-25-2': 'In this regard, link creation was studied from the perspective of the clustering coefficient.', '1712.01561-3-25-3': 'We mention [CITATION], where authors investigate the origins of homophily and tie formation by means of triadic closures and proximity, and [CITATION], where a new method for triple estimation is presented.', '1712.01561-3-26-0': '# Preliminaries and notations about graph theory', '1712.01561-3-27-0': 'For the convenience of the reader, we shall now provide some preliminaries and notations.', '1712.01561-3-27-1': 'The classical mathematical abstraction of a network is a graph [MATH], where [MATH] is the set of [MATH] nodes (or vertices) and [MATH] is the set of [MATH] links (or edges) stating the relationships among the nodes.', '1712.01561-3-27-2': 'We refer to a node by an index [MATH], meaning that we allow a one-to-one correspondence between an index in [MATH] and a node in [MATH].', '1712.01561-3-27-3': 'The set [MATH] can be conceptualized through the adjacency matrix [MATH], whose generic element [MATH] is 1 if the link between [MATH] and [MATH] exists and 0 otherwise.', '1712.01561-3-27-4': 'The graph is undirected when [MATH], for each [MATH], and directed otherwise.', '1712.01561-3-27-5': 'The degree [MATH] of the node [MATH] is a nonnegative integer representing the number of links incident upon [MATH].', '1712.01561-3-28-0': 'In this paper, we examine weighted networks, and we refer to a weighted adjacency matrix [MATH] whose elements [MATH] represent the weights on the link connecting nodes [MATH] and [MATH], with [MATH].', '1712.01561-3-28-1': 'Clearly, [MATH] stands for absence of a link between [MATH] and [MATH].', '1712.01561-3-28-2': "Thus, [MATH] denotes the intensity of the interactions between two nodes [MATH] and [MATH] and allows for the modeling of the ties' strength of the observed system.", '1712.01561-3-29-0': '# Literature review on the clustering coefficient for unweighted and weighted networks', '1712.01561-3-30-0': '## Unweighted networks', '1712.01561-3-31-0': 'The local clustering coefficient can be defined for any vertex [MATH] and captures the capacity of edge creations among neighbors, i.e. the tendency in the network to create stable groups [CITATION].', '1712.01561-3-31-1': 'Thus, the cohesion around a vertex [MATH] is quantified by the local clustering coefficient [MATH] defined as the number of triangles [MATH] in which vertex [MATH] participates normalized by the maximum possible number of such triangles: [EQUATION]', '1712.01561-3-31-2': 'The local clustering coefficient quantifies how a node takes part in a cohesive group.', '1712.01561-3-31-3': 'Therefore, [MATH] if none of the neighbors of a node are connected and [MATH] if all of the neighbors are linked.', '1712.01561-3-32-0': 'The value of the local clustering coefficient is influenced by the nodes degrees.', '1712.01561-3-32-1': 'A node with several neighbors is likely to be embedded in fewer closed triangles; hence, it has a smaller local clustering coefficient when compared to a node linked to fewer neighbors, where they are more likely to be clustered in triangles [CITATION].', '1712.01561-3-33-0': 'The clustering coefficient for a given graph is computed in two classical modes [CITATION].', '1712.01561-3-33-1': 'The first is the averaged clustering coefficient [MATH], given as the average of all the local clustering coefficients, while the second, called the global clustering coefficient and denoted by [MATH], is defined as the ratio among three times the number of closed triangles in the graph and the number of its triples, i.e. the number of 2-paths among three nodes.', '1712.01561-3-34-0': 'Note that both [MATH] and [MATH] assume values from [MATH] to [MATH] and are equal to [MATH] in case of a clique, i.e. a fully coupled network.', '1712.01561-3-34-1': 'In real networks, the evidence shows that nodes are inclined to cluster into densely connected groups [CITATION] and the difficulty of comparing the values of clustering nodes with different degrees makes the average value of local clustering sensitive to the way in which degrees are distributed across the whole network.', '1712.01561-3-35-0': 'The quantities [MATH] and [MATH] are specifically tailored to unweighted networks, and they cannot be satisfactorily employed to describe the community structure of the network in the presence of weights on links and when arcs are of the direct type.', '1712.01561-3-36-0': 'The next section is devoted to the analysis of the more general weighted cases.', '1712.01561-3-37-0': '## Weighted networks', '1712.01561-3-38-0': 'In many real networks, connections are relevant not only in terms of the classical binary state - whether they exist or do not exist - but also with regards to their strength which, for any node [MATH], is defined as: [EQUATION]', '1712.01561-3-38-1': 'The introduction of weights and strengths extends the study of the macroscopic properties of the network by adding some forms of entity of connections and capability to the mere interactions.', '1712.01561-3-38-2': 'In particular, the strength integrates information about the vertex connectivity and the weights of its links [CITATION].', '1712.01561-3-38-3': 'It is considered a natural measure of the importance or centrality of a vertex [MATH].', '1712.01561-3-38-4': 'Indeed, the identification of the most central nodes represents a major issue in network characterization [CITATION].', '1712.01561-3-39-0': 'In [CITATION], Barrat et al. combine the topological information of the network with the distribution of weights along links, and define the weighted clustering coefficient for a node [MATH] as follows: [EQUATION]', '1712.01561-3-39-1': 'This coefficient is a quantity of the local cohesiveness, which considers the importance of the clustered structure by taking into account the intensity of the interactions found on the local triangles.', '1712.01561-3-39-2': 'This measure counts, for each triangle created in the neighborhood of the node [MATH], the weight of the two related edges.', '1712.01561-3-39-3': 'The authors refer not to the mere number of the triangles in the neighborhood of a node but also to their total relative weight with respect to the strength of the nodes.', '1712.01561-3-40-0': 'The normalization factor [MATH] accounts for the strength [MATH] and the maximum possible number of triangles in which the node [MATH] may participate, and it ensures that [MATH].', '1712.01561-3-40-1': 'The definition of [MATH] recovers the topological clustering coefficient in the case where [MATH] is constant, for each [MATH].', '1712.01561-3-41-0': 'Therefore, the authors introduce the weighted clustering coefficient averaged over all nodes of the network, say [MATH], and over all nodes with degree [MATH], say [MATH].', '1712.01561-3-41-1': 'These measures offer global information on the correlation between weights and topology by comparing them with their topological analogs.', '1712.01561-3-42-0': 'Note that [MATH], so [MATH] can be written as: [EQUATION] where [MATH].', '1712.01561-3-42-1': 'In such equation the contribution of each triangle is weighted by a ratio of the average weight of the two adjacent links of the triangle to the average weight [MATH].', '1712.01561-3-43-0': 'Thus, [MATH] compares the weights related with triangles to the average weight of edges connected to the local node.', '1712.01561-3-44-0': 'Zhang and Horvath [CITATION] describe the weighted clustering coefficient in the context of gene co-expression networks.', '1712.01561-3-44-1': 'Unlike the unweighted clustering coefficient, the weighted clustering coefficient is not inversely related to the connectivity.', '1712.01561-3-44-2': 'Authors show a model that reveals how an inverse relationship between the clustering coefficient and connectivity occurs from hard thresholding.', '1712.01561-3-44-3': 'In formula: [EQUATION] where the weights have been normalized by [MATH].', '1712.01561-3-44-4': 'The number of triangles around the node [MATH] can be written in terms of the adjacency matrix elements as [MATH] and the numerator of the above equation is a weighted generalization of the formula.', '1712.01561-3-44-5': 'The denominator has been selected by considering the upper bound of the numerator, ensuring [MATH] .', '1712.01561-3-44-6': 'The equation ([REF]) can be written as: [EQUATION]', '1712.01561-3-44-7': 'In Grindrod [CITATION], a similar definition has been shown; indeed, the edge weights are considered as probabilities such that in an ensemble of networks, [MATH] and [MATH] are linked with probability [MATH].', '1712.01561-3-44-8': 'Finally, Holme et al. [CITATION] discuss the definition of weights and express a redefined weighted clustering coefficient as: [EQUATION] where [MATH] indicates a matrix where each entry equals [MATH].', '1712.01561-3-44-9': 'This equation seems similar to those in [CITATION], though, [MATH] is not required in the denominator sum.', '1712.01561-3-45-0': 'Onnela et al. [CITATION] refer to the notion of motif, defining it as a set (ensemble) of topologically equivalent subgraphs of a network.', '1712.01561-3-45-1': 'In cases of weighted systems, it is necessary to deal with intensities rather than numbers of occurrence.', '1712.01561-3-45-2': 'Moreover, the latter concept is considered as a special case of the former one.', '1712.01561-3-45-3': 'For the authors, the triangles are among the simplest nontrivial motifs and have a crucial role as one of the classic quantities of network characterization in defining the clustering coefficient of a node [MATH].', '1712.01561-3-45-4': 'They propose a weighted clustering coefficient taking into consideration the subgraph intensity, which is defined as the geometric average of subgraph edge weights.', '1712.01561-3-45-5': 'In formula: [EQUATION] where [MATH] are the edge weights normalized by the maximum weight in the network of the edges linking [MATH] to the other nodes of [MATH].', '1712.01561-3-46-0': 'Formula ([REF]) shows that triangles contribute to the creation of [MATH] according to the weights associated to their three edges.', '1712.01561-3-46-1': 'More specifically, [MATH] disregards the strength of the local node and measures triangle weights only in relation to the maximum edge weight.', '1712.01561-3-47-0': 'Moreover, [MATH] collapses to [MATH] when, for each [MATH], one has [MATH], and is thus in the unweighted case.', '1712.01561-3-48-0': '# The generalized clustering coefficient', '1712.01561-3-49-0': 'This section contains our proposal for a new definition of the clustering coefficient of weighted networks.', '1712.01561-3-49-1': 'Based on a novel concept of triangles, this definition includes the presence of real indirect connections among individuals.', '1712.01561-3-49-2': 'For our purpose, we first provide and discuss the definition of the triangles, and then we introduce the clustering coefficient.', '1712.01561-3-50-0': '## Generalized triples', '1712.01561-3-51-0': 'Here, we propose a generalization of the concept of triangle, and rewrite accordingly the coefficient [MATH] in ([REF]) for the case of weighted networks.', '1712.01561-3-52-0': 'Let us consider a weighted non-oriented graph [MATH] with vertices [MATH], symmetric adjacent matrix [MATH] and weight matrix [MATH], with nonnegative weights.', '1712.01561-3-52-1': 'Moreover, let us take [MATH] and a function [MATH] which is not decreasing in its arguments.', '1712.01561-3-53-0': 'For each triple of distinct vertices [MATH], a subgraph [MATH] is a generalized triangle (or, simply, a triangle) around [MATH] if one of the following conditions are satisfied:', '1712.01561-3-54-0': 'Herein we denote the elements of types [MATH] and [MATH] as triples since they are not really triangles since they are not contained in [MATH].', '1712.01561-3-54-1': 'They can be seen as a generalization of triangles by including the missing side, which is induced by conditions on the weights of the two existing edges.', '1712.01561-3-55-0': 'We denote the set of generalized triangles associated to case [MATH] as [MATH], for [MATH].', '1712.01561-3-55-1': 'By definition, [MATH].', '1712.01561-3-55-2': 'We denote the set collecting all the triangles by [MATH].', '1712.01561-3-56-0': 'Figure [REF] reports the three different type of triangles, respectively [MATH], [MATH] and [MATH].', '1712.01561-3-56-1': 'Clearly, in the case of [MATH], the concept of triangle given in Definition [REF] coincides with the standard one.', '1712.01561-3-57-0': 'Note that with [MATH] nodes, the maximum number of possible triangles is [MATH].', '1712.01561-3-57-1': 'This is the case of a clique with [MATH], for each [MATH].', '1712.01561-3-58-0': 'When considering the maximum number of candidates triangles for a node [MATH] to belong to [MATH], it is [MATH].', '1712.01561-3-58-1': 'Then, in this case for the node [MATH] the number of triples is [MATH].', '1712.01561-3-59-0': 'Triples in [MATH] for node [MATH] are the paths of length [MATH], which can be computed by considering the square of the adjacency matrix.', '1712.01561-3-59-1': 'Indeed, the number of different paths of length [MATH] from [MATH] to [MATH] equals the entry [MATH] of [MATH] [CITATION].', '1712.01561-3-59-2': 'For a given row [MATH] of [MATH], the sum of the element (excluding the element [MATH]) equals the maximum potential number of triples of type [MATH].', '1712.01561-3-60-0': 'Figure [REF] shows the types of triangles, without emphasis on the conditions on the weights.', '1712.01561-3-61-0': '## Conceptualization of the generalized clustering coefficient', '1712.01561-3-62-0': 'Under Definition [REF], we can introduce a generalization of the clustering coefficients presented in Formula ([REF]) for weighted networks.', '1712.01561-3-63-0': 'Given a graph [MATH] and a node [MATH], the generalized unweighted clustering coefficient of [MATH] is [EQUATION] where [MATH], where [MATH] is the minimum distance between the nodes [MATH] and [MATH].', '1712.01561-3-64-0': "The term unweighted in Definition [REF] points to the absence of [MATH]'s in the coefficient in ([REF]).", '1712.01561-3-64-1': 'However, weights intervene in the identification of the triangles, according to Definition [REF].', '1712.01561-3-64-2': 'In particular, formula ([REF]) extends ([REF]).', '1712.01561-3-64-3': 'As an example, notice that [MATH] in the clique case.', '1712.01561-3-65-0': '## Implications of the generalized clustering coefficient and equivalent graphs', '1712.01561-3-66-0': 'The classical local clustering coefficient [MATH] not only captures the proportion of closed triples on all possible triples depending on the degree of the ego/node [MATH], but it also identifies its level of cohesion.', '1712.01561-3-66-1': 'While the averaged clustering coefficient [MATH] captures the whole level of network cohesion.', '1712.01561-3-67-0': 'The proposed generalized clustering coefficient extends the same setting also to the triples in [MATH] and [MATH], i.e. it is the proportion of triangles of type [MATH], [MATH] and [MATH] on all possible triangles.', '1712.01561-3-67-1': 'This process depends not only on the degree of the ego but also on the two thresholds [MATH] and [MATH], which take into account the strength profile around the ego, and thus have the possibility of creating triangles [MATH] and [MATH].', '1712.01561-3-68-0': 'The values of the generalized clustering coefficient are [MATH] as well as for the averaged measure [MATH] and differ from the usual measure because they depend on the thresholds [MATH] and [MATH].', '1712.01561-3-69-0': 'Importantly, [MATH] assumes unitary value not only in the clique case, but also when any missing link is compensated by the high weights of the other two links, i.e. when simultaneously [MATH] and [MATH].', '1712.01561-3-69-1': 'This property of the generalized clustering coefficient is very relevant, since it allows one to extend the sense of community given by the classical clustering coefficient to the case of indirect links being present, as seen in the definition of triples [MATH] and [MATH].', '1712.01561-3-70-0': 'The triples [MATH] and [MATH] can be described as follows (see Figure [REF]).', '1712.01561-3-70-1': 'The former describes a situation in which an ego [MATH] has a direct relationship with alters [MATH] and [MATH].', '1712.01561-3-70-2': 'One can say that there exists a triangle among the three if the strength of the connections of [MATH] with the others is sufficiently high - in the sense described by function [MATH].', '1712.01561-3-70-3': 'The idea is that the cooperation and/or the common interests between [MATH] and the alters is so effective and fruitful that the presence of a direct link between [MATH] and [MATH] is not required.', '1712.01561-3-71-0': 'The latter case is associated to the presence of a strong link between [MATH] and [MATH] and between [MATH] and [MATH], always in terms of the entities of the weights - in the sense described by function [MATH].', '1712.01561-3-71-1': 'In this peculiar situation, the node [MATH] represents the intermediate alter letting also the (indirect) collaboration between [MATH] and [MATH] be possible.', '1712.01561-3-72-0': 'Finally, the thresholds have a double meaning.', '1712.01561-3-72-1': 'In fact, if we consider a network in which interactions between alters could be facilitated, an external decision-maker could implement policies aiming to define the correspondent values of [MATH] and [MATH] low.', '1712.01561-3-72-2': 'For example, in the case of inter-organizational innovation networks, the presence of triangles is positively related to the establishment of stable groups, as well as to the amount of produced efforts, the straightening of transitive relationships and the innovation capacity [CITATION].', '1712.01561-3-73-0': 'On the other hand, if a decision-maker were to prevent interactions among alters, the policies with correspondent values [MATH] and [MATH] could be deemed sufficiently large.', '1712.01561-3-73-1': 'Such an instance can be found in the prevention of community formation in criminal organizations [CITATION].', '1712.01561-3-74-0': '### Equivalent graphs', '1712.01561-3-75-0': 'Triangles [MATH], [MATH] and [MATH] also serve in deriving topological information from the graph.', '1712.01561-3-75-1': 'In particular, assume that [MATH], so that the number of [MATH] and [MATH] around each node does not depend on the specific selection of function [MATH].', '1712.01561-3-75-2': 'In this case, we know that [MATH], meaning we are able to infer the degree of the node [MATH] by the knowledge of the number of triangles of type [MATH] around it.', '1712.01561-3-75-3': 'Conversely, [MATH] represents the number of existing paths of length 2 having [MATH] as one of the extreme nodes.', '1712.01561-3-75-4': 'By collecting the number of the triangles [MATH], [MATH] and [MATH] for each node of the graph, we are able to identify a class of graphs.', '1712.01561-3-76-0': 'Formally, consider a [MATH] matrix collecting [MATH], [MATH] and [MATH], for each node [MATH].', '1712.01561-3-76-1': 'Denote by [MATH] the set of all the matrices with dimension [MATH] and filled by integer nonnegative numbers.', '1712.01561-3-77-0': 'Thus, each matrix [MATH] identifies a non-unique graph that has [MATH] nodes and edges described by [MATH].', '1712.01561-3-77-1': 'We refer to such a matrix as the triangles matrix.', '1712.01561-3-77-2': 'In this sense, [MATH] can be viewed as an equivalent class in the set of the graph with [MATH] nodes, where two graphs [MATH] and [MATH] are said to be equivalent when they share the same matrix [MATH].', '1712.01561-3-78-0': 'Figure [REF] and the matrix in ([REF]) provide an example of two equivalent classes, along with their common triangles matrix [MATH].', '1712.01561-3-78-1': 'In particular, notice that matrix [MATH] is the same for the two considered graphs, thus suggesting that the equivalent class identified by [MATH] contains more than one graph.', '1712.01561-3-79-0': '# Applications', '1712.01561-3-80-0': 'Herein we considered the analysis of the generalized clustering coefficient on two empirical networks: the network among the [MATH] busiest US commercial airports [CITATION] and the nervous system of the nematode Caenorhabditis elegans [CITATION].', '1712.01561-3-80-1': 'The data processing, the network analysis and all simulations were conducted using the software R [CITATION] with the igraph package [CITATION].', '1712.01561-3-80-2': 'The datasets were obtained from the [MATH] packege tnet, authored by Tore Opsahl [CITATION].', '1712.01561-3-80-3': 'Code in the [MATH] programming language is available upon request.', '1712.01561-3-81-0': 'For the sake of readability we report in Table [REF] the notations used hereafter.', '1712.01561-3-82-0': '## General settings', '1712.01561-3-83-0': 'In the empirical experiments, we consider four cases of function [MATH]:', '1712.01561-3-84-0': 'The selection of the specific function [MATH] - to be implemented among [MATH] defined above - provides further insights into the interpretation of the triples of type [MATH] and [MATH].', '1712.01561-3-84-1': 'Indeed, once [MATH] and [MATH] are kept fixed, then [MATH] and [MATH] state that both weights of the considered edges should be taken into account in an identical way by considering their mere aggregation in the former case or their mean in the latter one.', '1712.01561-3-84-2': 'When considering functions [MATH] and [MATH], only one of the weights is relevant for the measurement of the strength of the connections - the minimum weight and the maximum one, respectively.', '1712.01561-3-84-3': 'Naturally, the former case is more restrictive than the latter one, since it implicitly assumes that both weights should be greater than [MATH] or [MATH] for having a triples of type [MATH] or [MATH].', '1712.01561-3-85-0': "Social sciences suggest other functions [MATH]'s to be considered in Definition ([REF]) to capture certain peculiarities of the system under observation.", '1712.01561-3-85-1': 'Notice also that [MATH] and [MATH] are not increasing functions of [MATH] and [MATH], respectively, as Definition ([REF]) implies.', '1712.01561-3-86-0': 'For the simulations, the value of [MATH] and [MATH] are [MATH] for the US airports network and [MATH] for the C.elegans network.', '1712.01561-3-86-1': 'The max values were chosen on the ground that function [MATH] could possibly be true also when considering arcs with the higher weights.', '1712.01561-3-86-2': 'As such, [MATH] runs were implemented for each considered value.', '1712.01561-3-86-3': 'Thus, we performed [MATH] computations for the US airports network and [MATH] computations for the C.elegans network.', '1712.01561-3-87-0': 'According to Definition ([REF]), [MATH] and [MATH], i.e. the triangles for every node in a network, can be computed considering [MATH] and [MATH].', '1712.01561-3-87-1': 'Concerning the sets [MATH], such triangles can be easily computed by a built-in function in [MATH].', '1712.01561-3-88-0': '## Analysis of the US commercial airports network', '1712.01561-3-89-0': 'The US commercial airports network has [MATH] nodes denoting airports and [MATH] edges representing flight connections.', '1712.01561-3-89-1': 'In this network, weights are the number of seats available on that connections in 2010.', '1712.01561-3-89-2': 'The network has both small-world and scale-free organization with [MATH] [CITATION].', '1712.01561-3-90-0': 'In Figure [REF] (left) we show the network visualization, while Table [REF] reports some basic measures: the density [MATH], the averaged clustering coefficient [MATH], the global clustering coefficient [MATH] and the minimum, maximum and average degree, weight and strength.', '1712.01561-3-91-0': 'In Figure [REF] (left) we report the strength distribution for this network, with the strength [MATH] as the sum of the weights of the links incident on [MATH], while Figure [REF] (left) uses a histogram to display the weights.', '1712.01561-3-92-0': 'Functioning as an example, Figure [REF] shows the arcs composing the triples in [MATH] and in [MATH] for the neighborhood of order [MATH] of node [MATH], i.e. its 2-step ego network.', '1712.01561-3-92-1': 'Such a node has a degree [MATH], a second order neighborhood of cardinality [MATH] and a local clustering coefficient [MATH], because [MATH] triangles are closed out of a theoretical [MATH].', '1712.01561-3-93-0': 'Thus, [MATH] while triangles in [MATH] are computed obtaining [MATH].', '1712.01561-3-93-1': 'Note that the blue arcs in the right panel of Figure [REF] are [MATH] because some arcs can be mentioned twice in the set, since arc [MATH] can derive from [MATH] as well as from [MATH].', '1712.01561-3-94-0': 'The generalized clustering coefficient has value [MATH], which is much lower than [MATH] since the proportion of closed triangles when [MATH] and [MATH] is smaller than the basic setting.', '1712.01561-3-95-0': 'Figure [REF] for the US airports network reports three curves for each node: the total number of triangles [MATH], the number of potential triples of type [MATH] and the number of potential triples of type [MATH].', '1712.01561-3-95-1': 'Figure [REF] compares the degree [MATH] and the local clustering coefficient [MATH] for each node [MATH].', '1712.01561-3-95-2': 'Note that nodes in the US airports network are enumerated in non-increasing order of their degree and the nodes with indices until [MATH] have values of degree and clustering coefficient, which allow for a large number of triples [MATH] and a significant number of triples [MATH].', '1712.01561-3-95-3': 'Then, when the degree decreases and the local clustering coefficient increases, the local neighborhoods preclude the formation of triangles.', '1712.01561-3-96-0': 'Figure [REF] shows the averaged values of the generalized clustering coefficient [MATH] for the US airports network when considering the four different functions [MATH] and [MATH].', '1712.01561-3-96-1': 'In each figure, the values are presented for every combination of [MATH] and [MATH] while the horizontal axis reports the values of [MATH] as averaged over every node in the network.', '1712.01561-3-96-2': 'As expected, higher values of [MATH] are obtained for lower values of [MATH] and [MATH] and, globally, we have a non-increasing trend with a higher slope for functions [MATH] and [MATH] since the average and the min functions smooth the values, thus indicating that the functions are true only for small values of weights.', '1712.01561-3-96-3': 'Regarding [MATH] and [MATH], they are more prone to being true for higher values of arc weight, meaning the slope declines at slower rate.', '1712.01561-3-97-0': 'A common behavior for all four cases is that the magnitude of [MATH] is more dependent on triples [MATH] than those in [MATH].', '1712.01561-3-97-1': 'This is due to the tendency of high-degree nodes to have a higher strength.', '1712.01561-3-97-2': 'Therefore, the functions are more prone to being true for triples [MATH] than for triples in [MATH] since the adjacent links could possibly lie in a low-degree node with a low value of strength.', '1712.01561-3-98-0': 'In order to study the evolution of the generalized clustering coefficient [MATH] when varying [MATH] and [MATH], we provide a series of diagrams in which, for the network under examination, the density of the [MATH] values are reported when considering fixed values of [MATH] or [MATH] and when the other thresholds varing.', '1712.01561-3-99-0': 'In particular, for the network under observation, Figure [REF] shows different density values for each [MATH] when [MATH], and Figure [REF] shows each [MATH] when [MATH].', '1712.01561-3-99-1': 'All the figures also report the density values of the local clustering coefficient [MATH] (colored in light green).', '1712.01561-3-100-0': 'When [MATH] (see Figure [REF]) we can observe the contribution of triples in [MATH] to [MATH].', '1712.01561-3-100-1': 'The density of [MATH] is more concentrated around the max value [MATH] when [MATH]; however, when [MATH] starts to grow the values shift closer to [MATH].', '1712.01561-3-101-0': 'For [MATH], Figure [REF] highlights that [MATH] receives a small contribution from triples in [MATH] and the values lay around [MATH] as soon [MATH] grows.', '1712.01561-3-102-0': 'The density of [MATH] shows that values are concentrated mainly around [MATH] and [MATH], meaning that many airports have a single connection with another airport or have a strong cohesive structure.', '1712.01561-3-102-1': 'When studying [MATH] it is possible to infer that some airports with a single connection with a common node have weight profiles that involve a certain level of interaction for a given threshold.', '1712.01561-3-102-2': 'For example, for low values of [MATH] passengers from or to airports [MATH] and [MATH] often use connection [MATH], thus suggesting the establishment of a direct and intended connection between the two airports.', '1712.01561-3-102-3': 'This is not true when considering triples in [MATH]; here the analysis suggests that a direct connection among [MATH] and [MATH] is less useful and passengers still prefer to fly by [MATH].', '1712.01561-3-103-0': '## Analysis of the C.elegans network', '1712.01561-3-104-0': 'The network of nematode Caenorhabditis elegans (C.elegans) has [MATH] nodes representing neurons and [MATH] edges occurring when two neurons are connected by either a synapse or a gap junction; for each edge, weights are equal to the number of junctions between nodes [MATH] and [MATH].', '1712.01561-3-104-1': 'The network has a scale-free organization with [MATH] [CITATION].', '1712.01561-3-105-0': 'In Figure [REF] (right) we show the network visualization, while Table [REF] reports the basic measures.', '1712.01561-3-105-1': 'Note that for this network we considered the giant component of [MATH] nodes while the complete network is composed of [MATH] nodes.', '1712.01561-3-106-0': 'In Figure [REF] (right) we report the strength distributions for the C.elegans network.', '1712.01561-3-106-1': 'Note that the two networks under observation are very different, especially in the distribution of low and high values of strength.', '1712.01561-3-106-2': 'The weight profiles in Figure [REF] confirm such differences, which are mostly caused by a difference of scale in the values.', '1712.01561-3-107-0': 'The analysis of Figures [REF] and [REF] depicts a very different picture for the C.elegans network when compared to the US airports network.', '1712.01561-3-107-1': 'Again, Figure [REF] reports the three curves representing the total number of triangles [MATH], the number of potential triples of type [MATH] and the number of potential triples of type [MATH].', '1712.01561-3-107-2': 'Figure [REF] compares the degree [MATH] and the local clustering coefficient [MATH] for each node [MATH].', '1712.01561-3-107-3': 'Note that in these benchmark instances, nodes are enumerated without a particular rule.', '1712.01561-3-108-0': 'In the C.elegans networks, nodes with a higher degree have relatively small values of local clustering coefficient, whilst nodes with a smaller degree have, in general, higher values of local clustering coefficient.', '1712.01561-3-108-1': 'This means that small-degree nodes tends to form dense local neighborhoods, while the neighborhood of hubs is much sparser.', '1712.01561-3-108-2': 'Such observations motivate the limited number of triples in [MATH] because, for each node [MATH], they are in number of [MATH], thus implying that denser neighborhoods have a smaller number of possible triples.', '1712.01561-3-109-0': 'Note in Figure [REF] that node [MATH] has a peak because [MATH] when the thresholds [MATH] and [MATH] are null (this is the case of all potential triples).', '1712.01561-3-109-1': 'This is motivated by the fact that its particular neighborhood is composed of a limited number of triangles in which it is embedded ([MATH] and [MATH]) despite its degree ([MATH].', '1712.01561-3-109-2': 'Choosing two edges on [MATH] leads to [MATH] potential triples of type [MATH] and subtracting [MATH] results in [MATH].', '1712.01561-3-109-3': 'Such a remarkable presence of triples of type [MATH] for a single node for the case of [MATH] suggests that the C.elegans network is star-shaped.', '1712.01561-3-110-0': 'Similar arguments can be considered for [MATH]; indeed, we have a small number of potential triples for both small-degree nodes and hubs, since low values of degree allow for a smaller amount of transitive closure.', '1712.01561-3-111-0': 'Figure [REF] reports the same plots for the C.elegans network and same comments on the general behavior can be repeated as for the previous network.', '1712.01561-3-111-1': 'The main difference is the gentler slope, which occurs due to the profile of weight distribution being less concentrated on lower values when compared to the US airports network (see also Figure [REF]).', '1712.01561-3-112-0': 'The analysis of the C.elegans network is completed with the series of diagrams in which the density of the [MATH] values are reported when considering fixed values of [MATH] or [MATH] and varying the other threshold.', '1712.01561-3-113-0': 'Even for this network, Figure [REF] shows different density values for each [MATH] when [MATH] and Figure [REF] for each [MATH] when [MATH].', '1712.01561-3-113-1': 'Note that all the figures report the density values of the local clustering coefficient [MATH] (colored in light green).', '1712.01561-3-114-0': 'When [MATH] (see Figure [REF]) the contribution of triples in [MATH] makes the density of [MATH] more concentrated around the max value [MATH] when [MATH]; when [MATH] starts to grow the values shift closer to [MATH].', '1712.01561-3-114-1': 'Such an effect is present in both the networks under observation but it is more evident for the C.elegans.', '1712.01561-3-115-0': 'Similarly, at the US airports network, for [MATH], Figure [REF] highlights that [MATH] receives a small contribution from triples in [MATH] and the values lay around [MATH] as soon as [MATH] grows.', '1712.01561-3-116-0': 'The observation of the way in which [MATH] density lays seems to affirm that the network has a small cohesive structure, since the values are mostly around [MATH] or on low values ([MATH]).', '1712.01561-3-116-1': 'The study of [MATH] values highlights that for small values of [MATH] there exists an intense interaction among alters, i.e. many neurons undergo a certain level of mutual influence when connected to a common neuron.', '1712.01561-3-116-2': 'When considering triples in [MATH] and in particular for [MATH], the density remains away from [MATH], i.e. transitive influence is always present even for growing values of [MATH], mostly because of the very high strength of node [MATH].', '1712.01561-3-117-0': 'The different figures confirm that, for the two networks under observation, the main contribution to [MATH] is provided by the triangles in [MATH], i.e. their structures and weight profiles cause the networks to be more prone to close triples in [MATH] rather than in [MATH].', '1712.01561-3-118-0': '# Conclusions and future research lines', '1712.01561-3-119-0': 'In complex systems, the way in which members behave is influenced by their interactions with one another, as well as by other, not always explicit, phenomena.', '1712.01561-3-119-1': 'Networks are a special case in which interactions can be studied in more formal ways.', '1712.01561-3-119-2': 'In this regards, certain aspects of the network structure, for instance, the neighborhood around a node or different ways of clustering, allow one to study important characteristics as local or global cohesive groups.', '1712.01561-3-120-0': 'A classical measure used to study the local cohesiveness is the cluster coefficient, which has been used in almost every network analysis.', '1712.01561-3-120-1': 'However, when a weighted network is considered, such a measure starts to become ambiguous since all the introduced measures are sensitive to the degree, as well as the strength profiles, of a node.', '1712.01561-3-121-0': 'Despite the classical clustering coefficient being defined as a measure of the combinatoric structure of the network, it does not have any ability to provide information when links, rather than being established, are indirectly induced by strong cooperations among the formally linked nodes.', '1712.01561-3-121-1': 'This occurs when two alters of a common ego have an increased likelihood of meeting due to the fact that the social motivations are strong enough or that the weight between an alter of its alters has such an intensity that a connection with the ego is admissible.', '1712.01561-3-122-0': 'This paper deals with a novel definition of the clustering coefficient for weighted networks in that triangles are viewed under such social perspectives, thus allowing consideration of cases whereby one of the edges is missing between three nodes.', '1712.01561-3-122-1': "The propensity to induce missing edges is studied by means of two thresholds [MATH] and [MATH], which capture key information on the strength profile of a node's neighborhood.", '1712.01561-3-123-0': 'The definition of two types of triangles, [MATH] and [MATH], serves two purposes: on the one hand, they model the evidence that transitive relations among the nodes appear when the existing links are strong enough; on the other hand, an understanding of the number and types of the triangles around the nodes when [MATH] identify equivalent classes of networks on the basis of their topological structures.', '1712.01561-3-124-0': 'The experiments on two real networks, with many different peculiar characteristics, highlight the ability of the proposed measure to express the hidden influences between nodes according to the weight profiles.', '1712.01561-3-124-1': 'A thorough computational exercise has also shown the sensitivity of the networks to the thresholds values, thus allowing us to obtain further information.', '1712.01561-3-125-0': 'Future research should be devoted in order to extend this approach to more complicated problems.', '1712.01561-3-125-1': 'For instance, the topological structure of the network can be discussed in more details.', '1712.01561-3-125-2': 'In this respect, note that one can introduce a novel formulation of the concepts of hubs and centrality measures on the basis of the social connections among the nodes, according to our definition of induced indirect links.', '1712.01561-3-125-3': 'In this context, one is able to generalize the exploration to the cases when [MATH] and [MATH] are not necessarily equal to zero.'}

1111.2698

{'1111.2698-1-0-0': 'The nonlinear propagation of large amplitude dust ion-acoustic (DIA) solitary waves (SWs) in an ion-beam plasma with stationary charged dusts is investigated.', '1111.2698-1-0-1': 'For typical plasma parameters relevant for experiments [J. Plasma Phys.', '1111.2698-1-0-2': '60, 69 (1998)], when the beam speed is larger than the DIA speed ([MATH]), three stable waves, namely the "fast" and "slow" ion-beam modes and the plasma DIA wave are shown to exist.', '1111.2698-1-0-3': 'These modes can propagate as SWs in the beam plasmas.', '1111.2698-1-0-4': 'However, in the other regime ([MATH]), one of the beam modes when coupled to the DIA mode may become unstable.', '1111.2698-1-0-5': 'The SWs with positive (negative) potential may exist when the difference of the nonlinear wave speed ([MATH]) and the beam speed is such that [MATH]).', '1111.2698-1-0-6': 'Furthermore, for real density perturbations, the wave potential [MATH] is found to be limited by a critical value which typically depends on [MATH], [MATH] as well as the ion/beam temperature.', '1111.2698-1-0-7': 'The conditions for the existence of DIA solitons are obtained and their properties are analyzed numerically in terms of the system parameters.', '1111.2698-1-0-8': 'While the system supports both the compressive and rarefactive large amplitude SWs, the small amplitude solitons exist only of the compressive type.', '1111.2698-1-0-9': 'The theoretical results may be useful for observation of soliton excitations in laboratory ion-beam driven plasmas as well as in space plasmas where the charged dusts play as impurities.', '1111.2698-1-1-0': '# Introduction', '1111.2698-1-2-0': "Large amplitude solitary waves (SWs) in plasmas with a high energy ion beam have been observed at various space plasma environments, e.g., the Earth's magnetopause, in the Van Allen radiation belts as well as in the auroral zone [CITATION].", '1111.2698-1-2-1': 'These nonlinear waves may be driven by the momentum exchange between the electrons and protons, or between different ion populations in multi-ion plasmas.', '1111.2698-1-2-2': 'Propagation of such ion-acoustic (IA) SWs (IASWs) in magnetized or unmagnetized collisionless plasmas under different physical situations has been of considerable interest in recent years [CITATION].', '1111.2698-1-2-3': 'Using different methods, many authors have studied the behavior and characteristics of IA solitons both theoretically [CITATION] and experimentally [CITATION] in ion-beam plasmas.', '1111.2698-1-3-0': 'It has been found that the presence of energetic charged particles like ion beam in plasmas can significantly modify the propagation behaviors of SWs [CITATION].', '1111.2698-1-3-1': 'The latter with negative potentials have been found in the vicinity of ion beam regions of the auroral zone in the upper atmosphere [CITATION].', '1111.2698-1-3-2': "The spacecraft observations in the Earth's plasma sheet boundary indicate that the electron and ion beams can also drive the broadband electrostatic waves [CITATION].", '1111.2698-1-3-3': 'Furthermore, the low-temperature plasmas containing charged dusts are ubiquitous in various space plasmas [CITATION], laboratory devices [CITATION] as well as in industrial processes [CITATION].', '1111.2698-1-3-4': 'The presence of such negatively charged dusts can significantly influence the collective behaviors of plasmas [CITATION].', '1111.2698-1-4-0': 'On the other hand, it has been shown that when an ion beam is injected into an unmagnetized plasma, three stable normal modes, namely the "fast" and "slow" ion-beam modes and the plasma IA wave exist in plasmas [CITATION].', '1111.2698-1-4-1': 'Furthermore, depending on whether the beam speed is of the order of or larger (e.g. 2 times) than the IA speed, the IA waves when coupled to slow beam mode may become unstable.', '1111.2698-1-4-2': 'Such ion-ion instability has been confirmed experimentally by Gresillon et al [CITATION].', '1111.2698-1-4-3': 'Thus, there may exist either three stable modes or one unstable and two stable modes.', '1111.2698-1-4-4': 'The nonlinear wave evolution of such three modes was investigated by Yajima et al [CITATION] in an ion-beam plasma.', '1111.2698-1-4-5': 'They showed that each of these modes can propagate as small amplitude Korteweg-de Vries (KdV) solitons when the beam density is smaller than the electron density and the amplitude is smaller than a critical value.', '1111.2698-1-4-6': 'In an another work, Nakamura et al [CITATION] had experimentally shown that the propagation of large amplitude IASWs (compressive type) is possible for beam speed larger than the IA speed in an ion-beam plasma with two groups (high and low-temperature) of electrons.', '1111.2698-1-4-7': 'However, it seems that the effects of adiabatic positive ions and ion beam as well as their finite temperatures on the propagation of arbitrary amplitude dust IASWs (DIASWs) in a plasma with charged dust impurities, have not yet been considered in detail.', '1111.2698-1-5-0': 'In this paper, we study the properties of three linear eigenmodes, which may propagate as stable DIASWs.', '1111.2698-1-5-1': 'For typical plasma parameters relevant for experimental conditions [CITATION], one of the modes may become unstable when the beam speed to dust ion-acoustic (DIA) speed ratio lies in [MATH].', '1111.2698-1-5-2': 'The conditions for the existence of large amplitude DIASWs are analyzed by pseudopotential approach and their properties are studied numerically for fast modes.', '1111.2698-1-5-3': 'Depending on the parameters, we show the existence of compressive as well rarefactive DIASWs.', '1111.2698-1-5-4': 'The latter exist when the ion to electron temperature ratio is of the order [MATH] or more.', '1111.2698-1-5-5': 'On the other hand, the DIASWs with small amplitudes are shown to propagate only of the compressive type.', '1111.2698-1-6-0': '# Basic equations and the dispersion relation', '1111.2698-1-7-0': 'We consider an unmagnetized collisionless plasma composed of adiabatic positive ions, positive beam ions, Boltzmann distributed electrons and negatively charged (stationary) dust grains in the background plasma.', '1111.2698-1-7-1': 'The finite temperatures of both the beam and plasma ions together with an equilibrium flow of the beam are considered.', '1111.2698-1-7-2': 'The normalized equations read [EQUATION] where [MATH], [MATH] and [MATH] respectively denote the number density, speed and the pressure of singly charged beam [MATH] and plasma ions [MATH], normalized by the unperturbed number density [MATH], the ion-acoustic speed [MATH] and [MATH].', '1111.2698-1-7-3': 'Here [MATH] is the Boltzmann constant, [MATH] is the temperature for [MATH]-th species ([MATH] for electrons) and [MATH] is the particle mass with [MATH].', '1111.2698-1-7-4': 'Furthermore, [MATH] is the DIA wave potential normalized by [MATH], where [MATH] is the elementary charge.', '1111.2698-1-7-5': 'The space and time variables are respectively normalized by the Debye length [MATH] and the ion plasma period [MATH].', '1111.2698-1-7-6': 'The overall charge neutrality condition in the background plasma reads [EQUATION] where [MATH] and [MATH] in which [MATH] is the dust number density and [MATH] is the number of electrons on the dust grain.', '1111.2698-1-7-7': 'In Eq. ([REF]), [MATH] is the temperature ratio for [MATH], [MATH].', '1111.2698-1-7-8': 'We have neglected the electron inertia, since the electron thermal speed is much larger than the ion/beam speed.', '1111.2698-1-8-0': 'It has been theoretically and experimentally shown that when beam ions are injected into an unmagnetized plasma, three longitudinal modes involving ion motions, namely an ion-acoustic wave and the fast and the slow space charge waves in the beam, can propagate [CITATION].', '1111.2698-1-8-1': 'In order to identify those modes in our dusty beam plasma system we linearize the basic equations and assume that the perturbations vary as [MATH], where [MATH] is the angular frequency normalized by [MATH] and [MATH] is the wave number normalized by [MATH].', '1111.2698-1-8-2': 'Thus, we obtain the following dispersion relation.', '1111.2698-1-8-3': '[EQUATION] where [MATH] and [MATH] is the speed (drift) of beam ions in equilibrium.', '1111.2698-1-8-4': 'In absence of the ion beam, DIA mode propagates with the phase speed given by [EQUATION]', '1111.2698-1-8-5': 'However, in absence of plasma ions the fast (F) and slow (S) modes propagate in the beam with the phase speeds, [EQUATION]', '1111.2698-1-8-6': 'When the beam speed is much larger than the ion-acoustic speed, the beam and background ions are not strongly coupled.', '1111.2698-1-8-7': 'The background ions support slightly modified ion-acoustic waves and the beam ions support ion-acoustic waves (F and S modes) whose phase speeds are shifted by the beam speed [MATH].', '1111.2698-1-8-8': 'Equation [REF] shows that the phase speed of the fast (slow) modes is larger (smaller) than the equilibrium beam speed [MATH].', '1111.2698-1-8-9': 'Now, the dispersion equation ([REF]) can be expressed as a polynomial equation in [MATH] of degree [MATH], and when the phase speed is much larger than the DIA speed, the dispersion relation gives real roots of the wave frequency.', '1111.2698-1-8-10': 'However, in the opposite case, the coupled wave modes can be stable for [MATH] where [EQUATION] and unstable when [EQUATION]', '1111.2698-1-8-11': 'For typical laboratory plasma parameters [CITATION] [MATH], [MATH], [MATH] and [MATH], the critical value of [MATH] above which the modes are stable is [MATH], and for unstable modes we have [MATH].', '1111.2698-1-8-12': 'Thus, it is reasonable to consider higher values of the beam speed (i.e. roughly greater than 2 times the DIA speed) in order to avoid ion-ion instability [CITATION].', '1111.2698-1-8-13': 'The latter has been confirmed in experiments [CITATION].', '1111.2698-1-9-0': 'Next, we numerically examine the dispersion relation ([REF]) for the three wave modes which propagate along the beam direction.', '1111.2698-1-9-1': 'The results are shown in Fig. 1 where a quartic equation in [MATH] is contour plotted against the wave number.', '1111.2698-1-9-2': 'The behaviors of the modes are found similar as experimentally observed modes but in different beam plasmas [CITATION].', '1111.2698-1-9-3': 'It clearly shows how the frequencies of the DIA wave as well as the F and S modes get modified by the increase of the beam or ion density, the beam speed, the ion temperature as well as the percentage of impurity in the background plasma.', '1111.2698-1-9-4': 'We find that as [MATH] increases [Fig. 1(a)] the phase speed of the F-modes (S-modes) decreases (increases) and that for the IA mode increases.', '1111.2698-1-9-5': 'However, the ion-temperature only modifies the DIA mode as in Fig. 1(b).', '1111.2698-1-9-6': 'Figure 1(c) shows that the effect of the beam speed is to enhance the phase speeds of both the F and S modes.', '1111.2698-1-9-7': 'In contrast to Fig. 1(a), the enhancement of the charged dust concentration [MATH] results into the increase (decrease) of the phase speed of F (S) modes, whereas the DIA modes remain almost unchanged by the effect of [MATH].', '1111.2698-1-9-8': 'In the following section we determine the conditions for which these stable modes would indeed propagate as SWs, and analyze their properties with different plasma parameters using the pseudopotential approach.', '1111.2698-1-10-0': '# Large amplitude solitons: Pseudopotential approach', '1111.2698-1-11-0': 'Assuming that the perturbations vary in the moving frame of reference [MATH], where [MATH] is the nonlinear wave speed normalized by [MATH] (If [MATH] be normalized by the phase speed of the ion-acoustic waves, it would then be called as the Mach number), we obtain from Eqs. ([REF])-([REF]) the following relation for the densities.', '1111.2698-1-11-1': '[EQUATION] where [MATH], [MATH], [MATH], [MATH] and [MATH].', '1111.2698-1-11-2': 'In our nonlinear theory we will consider [MATH] to be larger than the beam speed in order to examine whether the fast modes propagate as solitary waves [CITATION].', '1111.2698-1-11-3': 'Similar analysis can also be done for the slow modes with [MATH].', '1111.2698-1-11-4': 'In obtaining Eq. ([REF]) we have used the boundary conditions, namely [MATH], [MATH], [MATH] and [MATH] as [MATH].', '1111.2698-1-11-5': 'Furthermore, inspecting on Eq. ([REF]), we find that the number densities are real when there exists a critical value [MATH] of [MATH] such that [MATH] where [EQUATION]', '1111.2698-1-11-6': 'Note that the above restriction is valid only for large amplitude waves with positive potential.', '1111.2698-1-11-7': 'However, for waves with negative potential, the values of [MATH] are not limited by the ratios, namely [MATH], [MATH] or [MATH], [MATH].', '1111.2698-1-11-8': 'Typically, for [MATH] and [MATH], [MATH] assumes the second term in the curly brackets in Eq. ([REF]).', '1111.2698-1-11-9': 'However, this value must be considered together with the conditions for the existence of SWs.', '1111.2698-1-11-10': 'Introducing now the relation ([REF]) into Eq. ([REF]), and integrating it we obtain the following energy balance equation for an oscillating particle of unit mass at the pseudoposition [MATH] and pseudotime [MATH].', '1111.2698-1-11-11': '[EQUATION] where the pseudopotential [MATH] is given by [EQUATION]', '1111.2698-1-11-12': 'Here [MATH].', '1111.2698-1-11-13': 'Equations ([REF]) and ([REF]) are valid for arbitrary amplitude stationary perturbations like SWs and/or double layers.', '1111.2698-1-11-14': 'The conditions for the existence of such perturbations can be obtained as follows:', '1111.2698-1-12-0': '(i) [MATH].', '1111.2698-1-12-1': 'This has already been satisfied in obtaining Eq. ([REF]), and by using the charge neutrality condition ([REF]), one can easily verify that [MATH] at [MATH];', '1111.2698-1-13-0': '(ii) [MATH] at [MATH].', '1111.2698-1-13-1': 'This is satisfied when the following inequality holds.', '1111.2698-1-13-2': '[EQUATION] (iii) [MATH] and [MATH] according to whether the SWs are compressive (with positive potential, i.e. [MATH]) or rarefactive (with negative potential, i.e. [MATH]).', '1111.2698-1-13-3': 'Here [MATH] represents the amplitude of the solitary waves or double layers, if exist.', '1111.2698-1-14-0': 'In order that the three modes propagate as SWs the conditions (i)-(iii) must be satisfied.', '1111.2698-1-14-1': 'However, for the double layers to exist there must be an additional condition, i.e., [MATH] to be satisfied along with (i)-(iii).', '1111.2698-1-14-2': 'We numerically analyze the conditions (ii) and (iii) for some typical plasma parameters as relevant for experiments [CITATION].', '1111.2698-1-14-3': 'The condition (ii), which results to Eq. ([REF]) is presented in the [MATH] plane as shown in Fig. 2 for different sets of parameters.', '1111.2698-1-14-4': 'The white (gray or shaded) regions show the parameter space where the inequality ([REF]) is satisfied (not satisfied).', '1111.2698-1-14-5': 'The upper boundary curves that separate the white and gray regions represent the minimum values of [MATH] for the existence of solitary waves or double layers.', '1111.2698-1-14-6': 'The upper limits of [MATH] (with [MATH] and [MATH]) for the existence of positive SWs can be obtained by substituting [MATH] into [MATH] [cf. Eq. ([REF])].', '1111.2698-1-14-7': 'These limiting values of [MATH] are represented by the thick (blue) curves in Fig. 2.', '1111.2698-1-14-8': 'Typically, for [MATH], Fig. 2 shows that the lower and upper limits of [MATH] increase with the beam speed as well as the charged dust concentration.', '1111.2698-1-14-9': 'However, the case of ion temperature effect [c.f. Figs. 2(a) and (c)] is different.', '1111.2698-1-14-10': 'Here the range of [MATH]) remains almost the same with the increase of [MATH].', '1111.2698-1-14-11': 'Thus, for [MATH], SWs may exist in the upper parts of the white regions.', '1111.2698-1-15-0': 'Figure 3 shows the contour lines of [MATH] [thin (red) curves] and [MATH] [thick (blue) curves] for different values of [MATH].', '1111.2698-1-15-1': 'There are, in general, wide ranges of values of [MATH] and [MATH], for which [MATH] and [MATH] can be satisfied.', '1111.2698-1-15-2': 'However, all the numerical values of [MATH] are not admissible as values of [MATH] is limited by [MATH] such that [MATH] [cf. Eq. [REF]].', '1111.2698-1-15-3': 'Such [MATH] changes with [MATH] or some other parameters [MATH], [MATH].', '1111.2698-1-15-4': 'Thus, for each set of parameters we can obtain a value of [MATH] below (for positive [MATH]) which SWs may exist.', '1111.2698-1-15-5': 'Figures 3(a) and (d) explain the variation of the charged dust impurity and Figs. 3(b) and (c) that due to ion temperature in different (higher and lower) regimes of [MATH].', '1111.2698-1-15-6': 'We find that the enhancement of the ion temperature ([MATH]) can lead to the existence of negative SWs [Figs. 3(b) and (c)].', '1111.2698-1-15-7': 'The common parameter regimes (left to or above the curves) in Figures 3(a) and (d) where both the conditions [MATH] and [MATH] are satisfied gives the region of the existence of positive SWs, whereas the conditions [MATH] and [MATH] are satisfied only to the left to or down the curves in Figs. 3(b) and (c), implying the existence of negative SWs.', '1111.2698-1-15-8': 'Note, however, that with similar parameter values as in Fig. 3(b) or (c), the formation of negative SWs was not reported in beam-plasma experiments [CITATION].', '1111.2698-1-15-9': 'Since there is no common point of intersection of [MATH] except at [MATH], we conclude that the double layer formation may not be possible in our system.', '1111.2698-1-16-0': 'Next, the pseudopotential [MATH] can be plotted against [MATH] for a given set of physical parameters.', '1111.2698-1-16-1': 'We see from Fig. 4 that for both positive and negative values of [MATH], [MATH] crosses the [MATH]-axis for the fast modes, implying the existence of both positive and negative SWs.', '1111.2698-1-16-2': 'These different crossing points for certain values of the parameters give the height or amplitude [MATH] of the SWs.', '1111.2698-1-16-3': 'However, the width of the SWs with speed [MATH] can be obtained from the shape of [MATH] by the numerical solution of Eq. ([REF]).', '1111.2698-1-16-4': 'We can also verify the wave amplitude and width [[MATH], where [MATH] is the minimum value of [MATH]] obtained from the plots of [MATH] with those from the numerical solution of Eq. ([REF]).', '1111.2698-1-16-5': 'Since the values of [MATH] are limited by [MATH], i.e., [MATH] , and [MATH] changes with [MATH] and other parameters, [MATH] can cross the [MATH]-axis only for a set of parameters.', '1111.2698-1-16-6': 'We find that the amplitude of the wave increases with the enhancement of the ion concentration until [MATH] [Fig. 4(a)] as well as the wave speed [MATH] [Fig. 4(b)], and it decreases with the increase of the dust impurity, [MATH] [Fig. 4(d)].', '1111.2698-1-16-7': 'On the other hand, Fig. 4(c) shows the existence of negative SWs with large negative values of [MATH].', '1111.2698-1-16-8': 'The amplitudes (in absolute value) of these SWs increase with the nonlinear wave speed [MATH].', '1111.2698-1-16-9': 'For the parameters as in Fig. 4(c) we find that in contrast to positive SWs in which upper limits of [MATH] exist, there exist a lower limit of [MATH] above which the SWs with negative potential exist.', '1111.2698-1-17-0': 'To verify the results as in Fig. 4 (especially the amplitude and width of the SWs) and to obtain the shape of [MATH], we numerically integrate Eq. ([REF]).', '1111.2698-1-17-1': 'The results are presented in Figs. 5 and 6 for the positive and negative SWs respectively.', '1111.2698-1-17-2': 'We find that for compressive SWs (Fig. 5), the amplitude increases and the width decreases with increasing values of [MATH] and [MATH], whereas both the amplitude and the width are enhanced by the effect of charged dust impurity [MATH].', '1111.2698-1-17-3': 'Figure 6 shows that the effects of [MATH] and [MATH] respectively are to increase and decrease the amplitude as well as the width of the negative SWs.', '1111.2698-1-18-0': '# Summary and Conclusion', '1111.2698-1-19-0': 'The nonlinear evolution of large amplitude SWs in a dust contaminated ion-beam driven plasma is investigated by using a pseudopotential approach.', '1111.2698-1-19-1': 'The dispersion properties of three linear modes, namely the fast and slow ion-beam modes and the DIA wave are analyzed numerically.', '1111.2698-1-19-2': 'The conditions for which these three modes propagate as (large amplitude) SWs and their (soliton) properties are studied numerically in terms of the system parameters.', '1111.2698-1-19-3': 'While the system supports both compressive (positive) and rarefactive (negative) large amplitude DIASWs, the small amplitude DIA solitons exist only of the compressive type (see Fig. 5).', '1111.2698-1-19-4': 'The presence of charged dusts significantly alters the existence regions as well as the properties of solitons.', '1111.2698-1-19-5': 'We also show that for typical plasma parameters as in experiments [CITATION] the formation of double layers is not possible.', '1111.2698-1-19-6': 'Our results can be summarized as follows:', '1111.2698-1-20-0': '(i) When the beam speed is larger than ([MATH] times) the DIA speed, each of the stable modes, namely the fast and slow beam mode as well as the DIA mode can propagate as SWs in ion-beam plasmas.', '1111.2698-1-20-1': 'Otherwise, there may exist one unstable beam mode coupled to the DIA wave and two stable modes in the plasma.', '1111.2698-1-20-2': 'With the enhancement of [MATH], while the phase speed of the F-modes (S-modes) decreases (increases) that for the DIA mode increases.', '1111.2698-1-20-3': 'These behaviors are in contrast to the effect of charged dusts [MATH] in which the DIA modes remain almost unchanged.', '1111.2698-1-20-4': 'The effect of the beam speed is to enhance the phase speed of both the F and S-modes.', '1111.2698-1-20-5': 'In the limit of the phase speed larger than the DIA speed, the modes propagate as stable waves.', '1111.2698-1-20-6': 'However, in the opposite limit, the DIA mode, coupled to a beam mode, may become unstable in [MATH] for typical plasma parameters [MATH], [MATH], [MATH] at [MATH], as relevant for experiments [CITATION].', '1111.2698-1-20-7': 'Thus, it is reasonable to consider higher values of [MATH]) for the DIASWs to exist.', '1111.2698-1-21-0': '(ii) For the perturbations to be real, the wave potential [MATH] is to be less than a critical value, which typically depends on the nonlinear wave speed, the beam speed as well as the beam or the ion temperature.', '1111.2698-1-22-0': '(iii) Both the compressive and rarefactive large amplitude DIASWs may coexist, whereas the small amplitude soliton exists only of the compressive type.', '1111.2698-1-22-1': 'In order that the the SWs with [MATH]) may exist, the regime of the nonlinear wave speed ([MATH]) and the beam speed ([MATH]) is such that [MATH].', '1111.2698-1-23-0': '(iv) For large amplitude positive SWs, the effects of [MATH] and [MATH] are to enhance the wave amplitude and to reduce the width.', '1111.2698-1-23-1': 'However, both the amplitude and the width may be increased (in magnitude) by the charged dust concentration [MATH].', '1111.2698-1-23-2': 'These behaviors are similar to the case of large amplitude negative SWs by the effect of [MATH].', '1111.2698-1-24-0': 'The theoretical results could be useful for soliton excitation in laboratory ion-beam driven plasmas as well as in space plasmas where negatively charged dusts are considered as impurities.', '1111.2698-1-25-0': 'Authors sincerely thank the Referee for his useful comments which improved the manuscript in the present form.', '1111.2698-1-25-1': 'APM was supported by the Kempe Foundations, Sweden through Grant No. SMK-2647.'}

{'1111.2698-2-0-0': 'The nonlinear propagation of large amplitude dust ion-acoustic (DIA) solitary waves (SWs) in an ion-beam plasma with stationary charged dusts is investigated.', '1111.2698-2-0-1': 'For typical plasma parameters relevant for experiments [J. Plasma Phys.', '1111.2698-2-0-2': '60, 69 (1998)], when the beam speed is larger than the DIA speed ([MATH]), three stable waves, namely the "fast" and "slow" ion-beam modes and the plasma DIA wave are shown to exist.', '1111.2698-2-0-3': 'These modes can propagate as SWs in the beam plasmas.', '1111.2698-2-0-4': 'However, in the other regime ([MATH]), one of the beam modes when coupled to the DIA mode may become unstable.', '1111.2698-2-0-5': 'The SWs with positive (negative) potential may exist when the difference of the nonlinear wave speed ([MATH]) and the beam speed is such that [MATH]).', '1111.2698-2-0-6': 'Furthermore, for real density perturbations, the wave potential [MATH] is found to be limited by a critical value which typically depends on [MATH], [MATH] as well as the ion/beam temperature.', '1111.2698-2-0-7': 'The conditions for the existence of DIA solitons are obtained and their properties are analyzed numerically in terms of the system parameters.', '1111.2698-2-0-8': 'While the system supports both the compressive and rarefactive large amplitude SWs, the small amplitude solitons exist only of the compressive type.', '1111.2698-2-0-9': 'The theoretical results may be useful for observation of soliton excitations in laboratory ion-beam driven plasmas as well as in space plasmas where the charged dusts play as impurities.', '1111.2698-2-1-0': '# Introduction', '1111.2698-2-2-0': "Large amplitude solitary waves (SWs) in plasmas with a high energy ion beam have been observed at various space plasma environments, e.g., the Earth's magnetopause, in the Van Allen radiation belts as well as in the auroral zone [CITATION].", '1111.2698-2-2-1': 'These nonlinear waves may be driven by the momentum exchange between the electrons and protons, or between different ion populations in multi-ion plasmas.', '1111.2698-2-2-2': 'Propagation of such ion-acoustic (IA) SWs (IASWs) in magnetized or unmagnetized collisionless plasmas under different physical situations has been of considerable interest in recent years [CITATION].', '1111.2698-2-2-3': 'Using different methods, many authors have studied the behavior and characteristics of IA solitons both theoretically [CITATION] and experimentally [CITATION] in ion-beam plasmas.', '1111.2698-2-3-0': 'It has been found that the presence of energetic charged particles like ion beam in plasmas can significantly modify the propagation behaviors of SWs [CITATION].', '1111.2698-2-3-1': 'The latter with negative potentials have been found in the vicinity of ion beam regions of the auroral zone in the upper atmosphere [CITATION].', '1111.2698-2-3-2': "The spacecraft observations in the Earth's plasma sheet boundary indicate that the electron and ion beams can also drive the broadband electrostatic waves [CITATION].", '1111.2698-2-3-3': 'Furthermore, the low-temperature plasmas containing charged dusts are ubiquitous in various space plasmas [CITATION], laboratory devices [CITATION] as well as in industrial processes [CITATION].', '1111.2698-2-3-4': 'The presence of such negatively charged dusts can significantly influence the collective behaviors of plasmas [CITATION].', '1111.2698-2-4-0': 'On the other hand, it has been shown that when an ion beam is injected into an unmagnetized plasma, three stable normal modes, namely the "fast" and "slow" ion-beam modes and the plasma IA wave exist in plasmas [CITATION].', '1111.2698-2-4-1': 'Furthermore, depending on whether the beam speed is of the order of or larger (e.g. 2 times) than the IA speed, the IA waves when coupled to slow beam mode may become unstable.', '1111.2698-2-4-2': 'Such ion-ion instability has been confirmed experimentally by Gresillon et al [CITATION].', '1111.2698-2-4-3': 'Thus, there may exist either three stable modes or one unstable and two stable modes.', '1111.2698-2-4-4': 'The nonlinear wave evolution of such three modes was investigated by Yajima et al [CITATION] in an ion-beam plasma.', '1111.2698-2-4-5': 'They showed that each of these modes can propagate as small amplitude Korteweg-de Vries (KdV) solitons when the beam density is smaller than the electron density and the amplitude is smaller than a critical value.', '1111.2698-2-4-6': 'In an another work, Nakamura et al [CITATION] had experimentally shown that the propagation of large amplitude IASWs (compressive type) is possible for beam speed larger than the IA speed in an ion-beam plasma with two groups (high and low-temperature) of electrons.', '1111.2698-2-4-7': 'However, it seems that the effects of adiabatic positive ions and ion beam as well as their finite temperatures on the propagation of arbitrary amplitude dust IASWs (DIASWs) in a plasma with charged dust impurities, have not yet been considered in detail.', '1111.2698-2-5-0': 'In this paper, we study the properties of three linear eigenmodes, which may propagate as stable DIASWs.', '1111.2698-2-5-1': 'For typical plasma parameters relevant for experimental conditions [CITATION], one of the modes may become unstable when the beam speed to dust ion-acoustic (DIA) speed ratio lies in [MATH].', '1111.2698-2-5-2': 'The conditions for the existence of large amplitude DIASWs are analyzed by pseudopotential approach and their properties are studied numerically for fast modes.', '1111.2698-2-5-3': 'Depending on the parameters, we show the existence of compressive as well rarefactive DIASWs.', '1111.2698-2-5-4': 'The latter exist when the ion to electron temperature ratio is of the order [MATH] or more.', '1111.2698-2-5-5': 'On the other hand, the DIASWs with small amplitudes are shown to propagate only of the compressive type.', '1111.2698-2-6-0': '# Basic equations and the dispersion relation', '1111.2698-2-7-0': 'We consider an unmagnetized collisionless plasma composed of adiabatic positive ions, positive beam ions, Boltzmann distributed electrons and negatively charged (stationary) dust grains in the background plasma.', '1111.2698-2-7-1': 'The finite temperatures of both the beam and plasma ions together with an equilibrium flow of the beam are considered.', '1111.2698-2-7-2': 'The normalized equations read [EQUATION] where [MATH], [MATH] and [MATH] respectively denote the number density, speed and the pressure of singly charged beam [MATH] and plasma ions [MATH], normalized by the unperturbed number density [MATH], the ion-acoustic speed [MATH] and [MATH].', '1111.2698-2-7-3': 'Here [MATH] is the Boltzmann constant, [MATH] is the temperature for [MATH]-th species ([MATH] for electrons) and [MATH] is the particle mass with [MATH].', '1111.2698-2-7-4': 'Furthermore, [MATH] is the DIA wave potential normalized by [MATH], where [MATH] is the elementary charge.', '1111.2698-2-7-5': 'The space and time variables are respectively normalized by the Debye length [MATH] and the ion plasma period [MATH].', '1111.2698-2-7-6': 'The overall charge neutrality condition in the background plasma reads [EQUATION] where [MATH] and [MATH] in which [MATH] is the dust number density and [MATH] is the number of electrons on the dust grain.', '1111.2698-2-7-7': 'In Eq. ([REF]), [MATH] is the temperature ratio for [MATH], [MATH].', '1111.2698-2-7-8': 'We have neglected the electron inertia, since the electron thermal speed is much larger than the ion/beam speed.', '1111.2698-2-8-0': 'It has been theoretically and experimentally shown that when beam ions are injected into an unmagnetized plasma, three longitudinal modes involving ion motions, namely an ion-acoustic wave and the fast and the slow space charge waves in the beam, can propagate [CITATION].', '1111.2698-2-8-1': 'In order to identify those modes in our dusty beam plasma system we linearize the basic equations and assume that the perturbations vary as [MATH], where [MATH] is the angular frequency normalized by [MATH] and [MATH] is the wave number normalized by [MATH].', '1111.2698-2-8-2': 'Thus, we obtain the following dispersion relation.', '1111.2698-2-8-3': '[EQUATION] where [MATH] and [MATH] is the speed (drift) of beam ions in equilibrium.', '1111.2698-2-8-4': 'In absence of the ion beam, DIA mode propagates with the phase speed given by [EQUATION]', '1111.2698-2-8-5': 'However, in absence of plasma ions the fast (F) and slow (S) modes propagate in the beam with the phase speeds, [EQUATION]', '1111.2698-2-8-6': 'When the beam speed is much larger than the ion-acoustic speed, the beam and background ions are not strongly coupled.', '1111.2698-2-8-7': 'The background ions support slightly modified ion-acoustic waves and the beam ions support ion-acoustic waves (F and S modes) whose phase speeds are shifted by the beam speed [MATH].', '1111.2698-2-8-8': 'Equation [REF] shows that the phase speed of the fast (slow) modes is larger (smaller) than the equilibrium beam speed [MATH].', '1111.2698-2-8-9': 'Now, the dispersion equation ([REF]) can be expressed as a polynomial equation in [MATH] of degree [MATH], and when the phase speed is much larger than the DIA speed, the dispersion relation gives real roots of the wave frequency.', '1111.2698-2-8-10': 'However, in the opposite case, the coupled wave modes can be stable for [MATH] where [EQUATION] and unstable when [EQUATION]', '1111.2698-2-8-11': 'For typical laboratory plasma parameters [CITATION] [MATH], [MATH], [MATH] and [MATH], the critical value of [MATH] above which the modes are stable is [MATH], and for unstable modes we have [MATH].', '1111.2698-2-8-12': 'Thus, it is reasonable to consider higher values of the beam speed (i.e. roughly greater than 2 times the DIA speed) in order to avoid ion-ion instability [CITATION].', '1111.2698-2-8-13': 'The latter has been confirmed in experiments [CITATION].', '1111.2698-2-9-0': 'Next, we numerically examine the dispersion relation ([REF]) for the three wave modes which propagate along the beam direction.', '1111.2698-2-9-1': 'The results are shown in Fig. 1 where a quartic equation in [MATH] is contour plotted against the wave number.', '1111.2698-2-9-2': 'The behaviors of the modes are found similar as experimentally observed modes but in different beam plasmas [CITATION].', '1111.2698-2-9-3': 'It clearly shows how the frequencies of the DIA wave as well as the F and S modes get modified by the increase of the beam or ion density, the beam speed, the ion temperature as well as the percentage of impurity in the background plasma.', '1111.2698-2-9-4': 'We find that as [MATH] increases [Fig. 1(a)] the phase speed of the F-modes (S-modes) decreases (increases) and that for the IA mode increases.', '1111.2698-2-9-5': 'However, the ion-temperature only modifies the DIA mode as in Fig. 1(b).', '1111.2698-2-9-6': 'Figure 1(c) shows that the effect of the beam speed is to enhance the phase speeds of both the F and S modes.', '1111.2698-2-9-7': 'In contrast to Fig. 1(a), the enhancement of the charged dust concentration [MATH] results into the increase (decrease) of the phase speed of F (S) modes, whereas the DIA modes remain almost unchanged by the effect of [MATH].', '1111.2698-2-9-8': 'In the following section we determine the conditions for which these stable modes would indeed propagate as SWs, and analyze their properties with different plasma parameters using the pseudopotential approach.', '1111.2698-2-10-0': '# Large amplitude solitons: Pseudopotential approach', '1111.2698-2-11-0': 'Assuming that the perturbations vary in the moving frame of reference [MATH], where [MATH] is the nonlinear wave speed normalized by [MATH] (If [MATH] be normalized by the phase speed of the ion-acoustic waves, it would then be called as the Mach number), we obtain from Eqs. ([REF])-([REF]) the following relation for the densities.', '1111.2698-2-11-1': '[EQUATION] where [MATH], [MATH], [MATH], [MATH] and [MATH].', '1111.2698-2-11-2': 'In our nonlinear theory we will consider [MATH] to be larger than the beam speed in order to examine whether the fast modes propagate as solitary waves [CITATION].', '1111.2698-2-11-3': 'Similar analysis can also be done for the slow modes with [MATH].', '1111.2698-2-11-4': 'In obtaining Eq. ([REF]) we have used the boundary conditions, namely [MATH], [MATH], [MATH] and [MATH] as [MATH].', '1111.2698-2-11-5': 'Furthermore, inspecting on Eq. ([REF]), we find that the number densities are real when there exists a critical value [MATH] of [MATH] such that [MATH] where [EQUATION]', '1111.2698-2-11-6': 'Note that the above restriction is valid only for large amplitude waves with positive potential.', '1111.2698-2-11-7': 'However, for waves with negative potential, the values of [MATH] are not limited by the ratios, namely [MATH], [MATH] or [MATH], [MATH].', '1111.2698-2-11-8': 'Typically, for [MATH] and [MATH], [MATH] assumes the second term in the curly brackets in Eq. ([REF]).', '1111.2698-2-11-9': 'However, this value must be considered together with the conditions for the existence of SWs.', '1111.2698-2-11-10': 'Introducing now the relation ([REF]) into Eq. ([REF]), and integrating it we obtain the following energy balance equation for an oscillating particle of unit mass at the pseudoposition [MATH] and pseudotime [MATH].', '1111.2698-2-11-11': '[EQUATION] where the pseudopotential [MATH] is given by [EQUATION]', '1111.2698-2-11-12': 'Here [MATH].', '1111.2698-2-11-13': 'Equations ([REF]) and ([REF]) are valid for arbitrary amplitude stationary perturbations like SWs and/or double layers.', '1111.2698-2-11-14': 'The conditions for the existence of such perturbations can be obtained as follows:', '1111.2698-2-12-0': '(i) [MATH].', '1111.2698-2-12-1': 'This has already been satisfied in obtaining Eq. ([REF]), and by using the charge neutrality condition ([REF]), one can easily verify that [MATH] at [MATH];', '1111.2698-2-13-0': '(ii) [MATH] at [MATH].', '1111.2698-2-13-1': 'This is satisfied when the following inequality holds.', '1111.2698-2-13-2': '[EQUATION] (iii) [MATH] and [MATH] according to whether the SWs are compressive (with positive potential, i.e. [MATH]) or rarefactive (with negative potential, i.e. [MATH]).', '1111.2698-2-13-3': 'Here [MATH] represents the amplitude of the solitary waves or double layers, if exist.', '1111.2698-2-14-0': 'In order that the three modes propagate as SWs the conditions (i)-(iii) must be satisfied.', '1111.2698-2-14-1': 'However, for the double layers to exist there must be an additional condition, i.e., [MATH] to be satisfied along with (i)-(iii).', '1111.2698-2-14-2': 'We numerically analyze the conditions (ii) and (iii) for some typical plasma parameters as relevant for experiments [CITATION].', '1111.2698-2-14-3': 'The condition (ii), which results to Eq. ([REF]) is presented in the [MATH] plane as shown in Fig. 2 for different sets of parameters.', '1111.2698-2-14-4': 'The white (gray or shaded) regions show the parameter space where the inequality ([REF]) is satisfied (not satisfied).', '1111.2698-2-14-5': 'The upper boundary curves that separate the white and gray regions represent the minimum values of [MATH] for the existence of solitary waves or double layers.', '1111.2698-2-14-6': 'The upper limits of [MATH] (with [MATH] and [MATH]) for the existence of positive SWs can be obtained by substituting [MATH] into [MATH] [cf. Eq. ([REF])].', '1111.2698-2-14-7': 'These limiting values of [MATH] are represented by the thick (blue) curves in Fig. 2.', '1111.2698-2-14-8': 'Typically, for [MATH], Fig. 2 shows that the lower and upper limits of [MATH] increase with the beam speed as well as the charged dust concentration.', '1111.2698-2-14-9': 'However, the case of ion temperature effect [c.f. Figs. 2(a) and (c)] is different.', '1111.2698-2-14-10': 'Here the range of [MATH]) remains almost the same with the increase of [MATH].', '1111.2698-2-14-11': 'Thus, for [MATH], SWs may exist in the upper parts of the white regions.', '1111.2698-2-15-0': 'Figure 3 shows the contour lines of [MATH] [thin (red) curves] and [MATH] [thick (blue) curves] for different values of [MATH].', '1111.2698-2-15-1': 'There are, in general, wide ranges of values of [MATH] and [MATH], for which [MATH] and [MATH] can be satisfied.', '1111.2698-2-15-2': 'However, all the numerical values of [MATH] are not admissible as values of [MATH] is limited by [MATH] such that [MATH] [cf. Eq. [REF]].', '1111.2698-2-15-3': 'Such [MATH] changes with [MATH] or some other parameters [MATH], [MATH].', '1111.2698-2-15-4': 'Thus, for each set of parameters we can obtain a value of [MATH] below (for positive [MATH]) which SWs may exist.', '1111.2698-2-15-5': 'Figures 3(a) and (d) explain the variation of the charged dust impurity and Figs. 3(b) and (c) that due to ion temperature in different (higher and lower) regimes of [MATH].', '1111.2698-2-15-6': 'We find that the enhancement of the ion temperature ([MATH]) can lead to the existence of negative SWs [Figs. 3(b) and (c)].', '1111.2698-2-15-7': 'The common parameter regimes (left to or above the curves) in Figures 3(a) and (d) where both the conditions [MATH] and [MATH] are satisfied gives the region of the existence of positive SWs, whereas the conditions [MATH] and [MATH] are satisfied only to the left to or down the curves in Figs. 3(b) and (c), implying the existence of negative SWs.', '1111.2698-2-15-8': 'Note, however, that with similar parameter values as in Fig. 3(b) or (c), the formation of negative SWs was not reported in beam-plasma experiments [CITATION].', '1111.2698-2-15-9': 'Since there is no common point of intersection of [MATH] except at [MATH], we conclude that the double layer formation may not be possible in our system.', '1111.2698-2-16-0': 'Next, the pseudopotential [MATH] can be plotted against [MATH] for a given set of physical parameters.', '1111.2698-2-16-1': 'We see from Fig. 4 that for both positive and negative values of [MATH], [MATH] crosses the [MATH]-axis for the fast modes, implying the existence of both positive and negative SWs.', '1111.2698-2-16-2': 'These different crossing points for certain values of the parameters give the height or amplitude [MATH] of the SWs.', '1111.2698-2-16-3': 'However, the width of the SWs with speed [MATH] can be obtained from the shape of [MATH] by the numerical solution of Eq. ([REF]).', '1111.2698-2-16-4': 'We can also verify the wave amplitude and width [[MATH], where [MATH] is the minimum value of [MATH]] obtained from the plots of [MATH] with those from the numerical solution of Eq. ([REF]).', '1111.2698-2-16-5': 'Since the values of [MATH] are limited by [MATH], i.e., [MATH] , and [MATH] changes with [MATH] and other parameters, [MATH] can cross the [MATH]-axis only for a set of parameters.', '1111.2698-2-16-6': 'We find that the amplitude of the wave increases with the enhancement of the ion concentration until [MATH] [Fig. 4(a)] as well as the wave speed [MATH] [Fig. 4(b)], and it decreases with the increase of the dust impurity, [MATH] [Fig. 4(d)].', '1111.2698-2-16-7': 'On the other hand, Fig. 4(c) shows the existence of negative SWs with large negative values of [MATH].', '1111.2698-2-16-8': 'The amplitudes (in absolute value) of these SWs increase with the nonlinear wave speed [MATH].', '1111.2698-2-16-9': 'For the parameters as in Fig. 4(c) we find that in contrast to positive SWs in which upper limits of [MATH] exist, there exist a lower limit of [MATH] above which the SWs with negative potential exist.', '1111.2698-2-17-0': 'To verify the results as in Fig. 4 (especially the amplitude and width of the SWs) and to obtain the shape of [MATH], we numerically integrate Eq. ([REF]).', '1111.2698-2-17-1': 'The results are presented in Figs. 5 and 6 for the positive and negative SWs respectively.', '1111.2698-2-17-2': 'We find that for compressive SWs (Fig. 5), the amplitude increases and the width decreases with increasing values of [MATH] and [MATH], whereas both the amplitude and the width are enhanced by the effect of charged dust impurity [MATH].', '1111.2698-2-17-3': 'Figure 6 shows that the effects of [MATH] and [MATH] respectively are to increase and decrease the amplitude as well as the width of the negative SWs.', '1111.2698-2-18-0': '# Summary and Conclusion', '1111.2698-2-19-0': 'The nonlinear evolution of large amplitude SWs in a dust contaminated ion-beam driven plasma is investigated by using a pseudopotential approach.', '1111.2698-2-19-1': 'The dispersion properties of three linear modes, namely the fast and slow ion-beam modes and the DIA wave are analyzed numerically.', '1111.2698-2-19-2': 'The conditions for which these three modes propagate as (large amplitude) SWs and their (soliton) properties are studied numerically in terms of the system parameters.', '1111.2698-2-19-3': 'While the system supports both compressive (positive) and rarefactive (negative) large amplitude DIASWs, the small amplitude DIA solitons exist only of the compressive type (see Fig. 5).', '1111.2698-2-19-4': 'The presence of charged dusts significantly alters the existence regions as well as the properties of solitons.', '1111.2698-2-19-5': 'We also show that for typical plasma parameters as in experiments [CITATION] the formation of double layers is not possible.', '1111.2698-2-19-6': 'Our results can be summarized as follows:', '1111.2698-2-20-0': '(i) When the beam speed is larger than ([MATH] times) the DIA speed, each of the stable modes, namely the fast and slow beam mode as well as the DIA mode can propagate as SWs in ion-beam plasmas.', '1111.2698-2-20-1': 'Otherwise, there may exist one unstable beam mode coupled to the DIA wave and two stable modes in the plasma.', '1111.2698-2-20-2': 'With the enhancement of [MATH], while the phase speed of the F-modes (S-modes) decreases (increases) that for the DIA mode increases.', '1111.2698-2-20-3': 'These behaviors are in contrast to the effect of charged dusts [MATH] in which the DIA modes remain almost unchanged.', '1111.2698-2-20-4': 'The effect of the beam speed is to enhance the phase speed of both the F and S-modes.', '1111.2698-2-20-5': 'In the limit of the phase speed larger than the DIA speed, the modes propagate as stable waves.', '1111.2698-2-20-6': 'However, in the opposite limit, the DIA mode, coupled to a beam mode, may become unstable in [MATH] for typical plasma parameters [MATH], [MATH], [MATH] at [MATH], as relevant for experiments [CITATION].', '1111.2698-2-20-7': 'Thus, it is reasonable to consider higher values of [MATH]) for the DIASWs to exist.', '1111.2698-2-21-0': '(ii) For the perturbations to be real, the wave potential [MATH] is to be less than a critical value, which typically depends on the nonlinear wave speed, the beam speed as well as the beam or the ion temperature.', '1111.2698-2-22-0': '(iii) Both the compressive and rarefactive large amplitude DIASWs may coexist, whereas the small amplitude soliton exists only of the compressive type.', '1111.2698-2-22-1': 'In order that the the SWs with [MATH]) may exist, the regime of the nonlinear wave speed ([MATH]) and the beam speed ([MATH]) is such that [MATH].', '1111.2698-2-23-0': '(iv) For large amplitude positive SWs, the effects of [MATH] and [MATH] are to enhance the wave amplitude and to reduce the width.', '1111.2698-2-23-1': 'However, both the amplitude and the width may be increased (in magnitude) by the charged dust concentration [MATH].', '1111.2698-2-23-2': 'These behaviors are similar to the case of large amplitude negative SWs by the effect of [MATH].', '1111.2698-2-24-0': 'The theoretical results could be useful for soliton excitation in laboratory ion-beam driven plasmas as well as in space plasmas where negatively charged dusts are considered as impurities.', '1111.2698-2-25-0': 'Authors sincerely thank the Referee for his useful comments which improved the manuscript in the present form.', '1111.2698-2-25-1': 'APM was supported by the Kempe Foundations, Sweden through Grant No. SMK-2647.'}

[['1111.2698-1-20-0', '1111.2698-2-20-0'], ['1111.2698-1-20-1', '1111.2698-2-20-1'], ['1111.2698-1-20-2', '1111.2698-2-20-2'], ['1111.2698-1-20-3', '1111.2698-2-20-3'], ['1111.2698-1-20-4', '1111.2698-2-20-4'], ['1111.2698-1-20-5', '1111.2698-2-20-5'], ['1111.2698-1-20-6', '1111.2698-2-20-6'], ['1111.2698-1-20-7', '1111.2698-2-20-7'], ['1111.2698-1-24-0', '1111.2698-2-24-0'], ['1111.2698-1-8-0', '1111.2698-2-8-0'], ['1111.2698-1-8-1', '1111.2698-2-8-1'], ['1111.2698-1-8-2', '1111.2698-2-8-2'], ['1111.2698-1-8-3', '1111.2698-2-8-3'], ['1111.2698-1-8-4', '1111.2698-2-8-4'], ['1111.2698-1-8-5', '1111.2698-2-8-5'], ['1111.2698-1-8-6', '1111.2698-2-8-6'], ['1111.2698-1-8-7', '1111.2698-2-8-7'], ['1111.2698-1-8-8', '1111.2698-2-8-8'], ['1111.2698-1-8-9', '1111.2698-2-8-9'], ['1111.2698-1-8-10', '1111.2698-2-8-10'], ['1111.2698-1-8-11', '1111.2698-2-8-11'], ['1111.2698-1-8-12', '1111.2698-2-8-12'], ['1111.2698-1-8-13', '1111.2698-2-8-13'], ['1111.2698-1-3-0', '1111.2698-2-3-0'], ['1111.2698-1-3-1', '1111.2698-2-3-1'], ['1111.2698-1-3-2', '1111.2698-2-3-2'], ['1111.2698-1-3-3', '1111.2698-2-3-3'], ['1111.2698-1-3-4', '1111.2698-2-3-4'], ['1111.2698-1-21-0', '1111.2698-2-21-0'], ['1111.2698-1-16-0', '1111.2698-2-16-0'], ['1111.2698-1-16-1', '1111.2698-2-16-1'], ['1111.2698-1-16-2', '1111.2698-2-16-2'], ['1111.2698-1-16-3', '1111.2698-2-16-3'], ['1111.2698-1-16-4', '1111.2698-2-16-4'], ['1111.2698-1-16-5', '1111.2698-2-16-5'], ['1111.2698-1-16-6', '1111.2698-2-16-6'], ['1111.2698-1-16-7', '1111.2698-2-16-7'], ['1111.2698-1-16-8', '1111.2698-2-16-8'], ['1111.2698-1-16-9', '1111.2698-2-16-9'], ['1111.2698-1-17-0', '1111.2698-2-17-0'], ['1111.2698-1-17-1', '1111.2698-2-17-1'], ['1111.2698-1-17-2', '1111.2698-2-17-2'], ['1111.2698-1-17-3', '1111.2698-2-17-3'], ['1111.2698-1-15-0', '1111.2698-2-15-0'], ['1111.2698-1-15-1', '1111.2698-2-15-1'], ['1111.2698-1-15-2', '1111.2698-2-15-2'], ['1111.2698-1-15-3', '1111.2698-2-15-3'], ['1111.2698-1-15-4', '1111.2698-2-15-4'], ['1111.2698-1-15-5', '1111.2698-2-15-5'], ['1111.2698-1-15-6', '1111.2698-2-15-6'], ['1111.2698-1-15-7', '1111.2698-2-15-7'], ['1111.2698-1-15-8', '1111.2698-2-15-8'], ['1111.2698-1-15-9', '1111.2698-2-15-9'], ['1111.2698-1-23-0', '1111.2698-2-23-0'], ['1111.2698-1-23-1', '1111.2698-2-23-1'], ['1111.2698-1-23-2', '1111.2698-2-23-2'], ['1111.2698-1-13-1', '1111.2698-2-13-1'], ['1111.2698-1-13-2', '1111.2698-2-13-2'], ['1111.2698-1-13-3', '1111.2698-2-13-3'], ['1111.2698-1-5-0', '1111.2698-2-5-0'], ['1111.2698-1-5-1', '1111.2698-2-5-1'], ['1111.2698-1-5-2', '1111.2698-2-5-2'], ['1111.2698-1-5-3', '1111.2698-2-5-3'], ['1111.2698-1-5-4', '1111.2698-2-5-4'], ['1111.2698-1-5-5', '1111.2698-2-5-5'], ['1111.2698-1-19-0', '1111.2698-2-19-0'], ['1111.2698-1-19-1', '1111.2698-2-19-1'], ['1111.2698-1-19-2', '1111.2698-2-19-2'], ['1111.2698-1-19-3', '1111.2698-2-19-3'], ['1111.2698-1-19-4', '1111.2698-2-19-4'], ['1111.2698-1-19-5', '1111.2698-2-19-5'], ['1111.2698-1-2-0', '1111.2698-2-2-0'], ['1111.2698-1-2-1', '1111.2698-2-2-1'], ['1111.2698-1-2-2', '1111.2698-2-2-2'], ['1111.2698-1-2-3', '1111.2698-2-2-3'], ['1111.2698-1-11-0', '1111.2698-2-11-0'], ['1111.2698-1-11-2', '1111.2698-2-11-2'], ['1111.2698-1-11-3', '1111.2698-2-11-3'], ['1111.2698-1-11-4', '1111.2698-2-11-4'], ['1111.2698-1-11-5', '1111.2698-2-11-5'], ['1111.2698-1-11-6', '1111.2698-2-11-6'], ['1111.2698-1-11-7', '1111.2698-2-11-7'], ['1111.2698-1-11-8', '1111.2698-2-11-8'], ['1111.2698-1-11-9', '1111.2698-2-11-9'], ['1111.2698-1-11-10', '1111.2698-2-11-10'], ['1111.2698-1-11-11', '1111.2698-2-11-11'], ['1111.2698-1-11-13', '1111.2698-2-11-13'], ['1111.2698-1-7-0', '1111.2698-2-7-0'], ['1111.2698-1-7-1', '1111.2698-2-7-1'], ['1111.2698-1-7-2', '1111.2698-2-7-2'], ['1111.2698-1-7-3', '1111.2698-2-7-3'], ['1111.2698-1-7-4', '1111.2698-2-7-4'], ['1111.2698-1-7-5', '1111.2698-2-7-5'], ['1111.2698-1-7-6', '1111.2698-2-7-6'], ['1111.2698-1-7-7', '1111.2698-2-7-7'], ['1111.2698-1-7-8', '1111.2698-2-7-8'], ['1111.2698-1-14-0', '1111.2698-2-14-0'], ['1111.2698-1-14-1', '1111.2698-2-14-1'], ['1111.2698-1-14-2', '1111.2698-2-14-2'], ['1111.2698-1-14-3', '1111.2698-2-14-3'], ['1111.2698-1-14-4', '1111.2698-2-14-4'], ['1111.2698-1-14-5', '1111.2698-2-14-5'], ['1111.2698-1-14-6', '1111.2698-2-14-6'], ['1111.2698-1-14-7', '1111.2698-2-14-7'], ['1111.2698-1-14-8', '1111.2698-2-14-8'], ['1111.2698-1-14-9', '1111.2698-2-14-9'], ['1111.2698-1-14-10', '1111.2698-2-14-10'], ['1111.2698-1-14-11', '1111.2698-2-14-11'], ['1111.2698-1-0-0', '1111.2698-2-0-0'], ['1111.2698-1-0-1', '1111.2698-2-0-1'], ['1111.2698-1-0-2', '1111.2698-2-0-2'], ['1111.2698-1-0-3', '1111.2698-2-0-3'], ['1111.2698-1-0-4', '1111.2698-2-0-4'], ['1111.2698-1-0-5', '1111.2698-2-0-5'], ['1111.2698-1-0-6', '1111.2698-2-0-6'], ['1111.2698-1-0-7', '1111.2698-2-0-7'], ['1111.2698-1-0-8', '1111.2698-2-0-8'], ['1111.2698-1-0-9', '1111.2698-2-0-9'], ['1111.2698-1-9-0', '1111.2698-2-9-0'], ['1111.2698-1-9-1', '1111.2698-2-9-1'], ['1111.2698-1-9-2', '1111.2698-2-9-2'], ['1111.2698-1-9-3', '1111.2698-2-9-3'], ['1111.2698-1-9-4', '1111.2698-2-9-4'], ['1111.2698-1-9-5', '1111.2698-2-9-5'], ['1111.2698-1-9-6', '1111.2698-2-9-6'], ['1111.2698-1-9-7', '1111.2698-2-9-7'], ['1111.2698-1-9-8', '1111.2698-2-9-8'], ['1111.2698-1-22-0', '1111.2698-2-22-0'], ['1111.2698-1-22-1', '1111.2698-2-22-1'], ['1111.2698-1-4-0', '1111.2698-2-4-0'], ['1111.2698-1-4-1', '1111.2698-2-4-1'], ['1111.2698-1-4-2', '1111.2698-2-4-2'], ['1111.2698-1-4-3', '1111.2698-2-4-3'], ['1111.2698-1-4-4', '1111.2698-2-4-4'], ['1111.2698-1-4-5', '1111.2698-2-4-5'], ['1111.2698-1-4-6', '1111.2698-2-4-6'], ['1111.2698-1-4-7', '1111.2698-2-4-7'], ['1111.2698-1-25-0', '1111.2698-2-25-0'], ['1111.2698-1-12-1', '1111.2698-2-12-1']]

[['1111.2698-1-20-0', '1111.2698-2-20-0'], ['1111.2698-1-20-1', '1111.2698-2-20-1'], ['1111.2698-1-20-2', '1111.2698-2-20-2'], ['1111.2698-1-20-3', '1111.2698-2-20-3'], ['1111.2698-1-20-4', '1111.2698-2-20-4'], ['1111.2698-1-20-5', '1111.2698-2-20-5'], ['1111.2698-1-20-6', '1111.2698-2-20-6'], ['1111.2698-1-20-7', '1111.2698-2-20-7'], ['1111.2698-1-24-0', '1111.2698-2-24-0'], ['1111.2698-1-8-0', '1111.2698-2-8-0'], ['1111.2698-1-8-1', '1111.2698-2-8-1'], ['1111.2698-1-8-2', '1111.2698-2-8-2'], ['1111.2698-1-8-3', '1111.2698-2-8-3'], ['1111.2698-1-8-4', '1111.2698-2-8-4'], ['1111.2698-1-8-5', '1111.2698-2-8-5'], ['1111.2698-1-8-6', '1111.2698-2-8-6'], ['1111.2698-1-8-7', '1111.2698-2-8-7'], ['1111.2698-1-8-8', '1111.2698-2-8-8'], ['1111.2698-1-8-9', '1111.2698-2-8-9'], ['1111.2698-1-8-10', '1111.2698-2-8-10'], ['1111.2698-1-8-11', '1111.2698-2-8-11'], ['1111.2698-1-8-12', '1111.2698-2-8-12'], ['1111.2698-1-8-13', '1111.2698-2-8-13'], ['1111.2698-1-3-0', '1111.2698-2-3-0'], ['1111.2698-1-3-1', '1111.2698-2-3-1'], ['1111.2698-1-3-2', '1111.2698-2-3-2'], ['1111.2698-1-3-3', '1111.2698-2-3-3'], ['1111.2698-1-3-4', '1111.2698-2-3-4'], ['1111.2698-1-21-0', '1111.2698-2-21-0'], ['1111.2698-1-16-0', '1111.2698-2-16-0'], ['1111.2698-1-16-1', '1111.2698-2-16-1'], ['1111.2698-1-16-2', '1111.2698-2-16-2'], ['1111.2698-1-16-3', '1111.2698-2-16-3'], ['1111.2698-1-16-4', '1111.2698-2-16-4'], ['1111.2698-1-16-5', '1111.2698-2-16-5'], ['1111.2698-1-16-6', '1111.2698-2-16-6'], ['1111.2698-1-16-7', '1111.2698-2-16-7'], ['1111.2698-1-16-8', '1111.2698-2-16-8'], ['1111.2698-1-16-9', '1111.2698-2-16-9'], ['1111.2698-1-17-0', '1111.2698-2-17-0'], ['1111.2698-1-17-1', '1111.2698-2-17-1'], ['1111.2698-1-17-2', '1111.2698-2-17-2'], ['1111.2698-1-17-3', '1111.2698-2-17-3'], ['1111.2698-1-15-0', '1111.2698-2-15-0'], ['1111.2698-1-15-1', '1111.2698-2-15-1'], ['1111.2698-1-15-2', '1111.2698-2-15-2'], ['1111.2698-1-15-3', '1111.2698-2-15-3'], ['1111.2698-1-15-4', '1111.2698-2-15-4'], ['1111.2698-1-15-5', '1111.2698-2-15-5'], ['1111.2698-1-15-6', '1111.2698-2-15-6'], ['1111.2698-1-15-7', '1111.2698-2-15-7'], ['1111.2698-1-15-8', '1111.2698-2-15-8'], ['1111.2698-1-15-9', '1111.2698-2-15-9'], ['1111.2698-1-23-0', '1111.2698-2-23-0'], ['1111.2698-1-23-1', '1111.2698-2-23-1'], ['1111.2698-1-23-2', '1111.2698-2-23-2'], ['1111.2698-1-13-1', '1111.2698-2-13-1'], ['1111.2698-1-13-2', '1111.2698-2-13-2'], ['1111.2698-1-13-3', '1111.2698-2-13-3'], ['1111.2698-1-5-0', '1111.2698-2-5-0'], ['1111.2698-1-5-1', '1111.2698-2-5-1'], ['1111.2698-1-5-2', '1111.2698-2-5-2'], ['1111.2698-1-5-3', '1111.2698-2-5-3'], ['1111.2698-1-5-4', '1111.2698-2-5-4'], ['1111.2698-1-5-5', '1111.2698-2-5-5'], ['1111.2698-1-19-0', '1111.2698-2-19-0'], ['1111.2698-1-19-1', '1111.2698-2-19-1'], ['1111.2698-1-19-2', '1111.2698-2-19-2'], ['1111.2698-1-19-3', '1111.2698-2-19-3'], ['1111.2698-1-19-4', '1111.2698-2-19-4'], ['1111.2698-1-19-5', '1111.2698-2-19-5'], ['1111.2698-1-2-0', '1111.2698-2-2-0'], ['1111.2698-1-2-1', '1111.2698-2-2-1'], ['1111.2698-1-2-2', '1111.2698-2-2-2'], ['1111.2698-1-2-3', '1111.2698-2-2-3'], ['1111.2698-1-11-0', '1111.2698-2-11-0'], ['1111.2698-1-11-2', '1111.2698-2-11-2'], ['1111.2698-1-11-3', '1111.2698-2-11-3'], ['1111.2698-1-11-4', '1111.2698-2-11-4'], ['1111.2698-1-11-5', '1111.2698-2-11-5'], ['1111.2698-1-11-6', '1111.2698-2-11-6'], ['1111.2698-1-11-7', '1111.2698-2-11-7'], ['1111.2698-1-11-8', '1111.2698-2-11-8'], ['1111.2698-1-11-9', '1111.2698-2-11-9'], ['1111.2698-1-11-10', '1111.2698-2-11-10'], ['1111.2698-1-11-11', '1111.2698-2-11-11'], ['1111.2698-1-11-13', '1111.2698-2-11-13'], ['1111.2698-1-7-0', '1111.2698-2-7-0'], ['1111.2698-1-7-1', '1111.2698-2-7-1'], ['1111.2698-1-7-2', '1111.2698-2-7-2'], ['1111.2698-1-7-3', '1111.2698-2-7-3'], ['1111.2698-1-7-4', '1111.2698-2-7-4'], ['1111.2698-1-7-5', '1111.2698-2-7-5'], ['1111.2698-1-7-6', '1111.2698-2-7-6'], ['1111.2698-1-7-7', '1111.2698-2-7-7'], ['1111.2698-1-7-8', '1111.2698-2-7-8'], ['1111.2698-1-14-0', '1111.2698-2-14-0'], ['1111.2698-1-14-1', '1111.2698-2-14-1'], ['1111.2698-1-14-2', '1111.2698-2-14-2'], ['1111.2698-1-14-3', '1111.2698-2-14-3'], ['1111.2698-1-14-4', '1111.2698-2-14-4'], ['1111.2698-1-14-5', '1111.2698-2-14-5'], ['1111.2698-1-14-6', '1111.2698-2-14-6'], ['1111.2698-1-14-7', '1111.2698-2-14-7'], ['1111.2698-1-14-8', '1111.2698-2-14-8'], ['1111.2698-1-14-9', '1111.2698-2-14-9'], ['1111.2698-1-14-10', '1111.2698-2-14-10'], ['1111.2698-1-14-11', '1111.2698-2-14-11'], ['1111.2698-1-0-0', '1111.2698-2-0-0'], ['1111.2698-1-0-1', '1111.2698-2-0-1'], ['1111.2698-1-0-2', '1111.2698-2-0-2'], ['1111.2698-1-0-3', '1111.2698-2-0-3'], ['1111.2698-1-0-4', '1111.2698-2-0-4'], ['1111.2698-1-0-5', '1111.2698-2-0-5'], ['1111.2698-1-0-6', '1111.2698-2-0-6'], ['1111.2698-1-0-7', '1111.2698-2-0-7'], ['1111.2698-1-0-8', '1111.2698-2-0-8'], ['1111.2698-1-0-9', '1111.2698-2-0-9'], ['1111.2698-1-9-0', '1111.2698-2-9-0'], ['1111.2698-1-9-1', '1111.2698-2-9-1'], ['1111.2698-1-9-2', '1111.2698-2-9-2'], ['1111.2698-1-9-3', '1111.2698-2-9-3'], ['1111.2698-1-9-4', '1111.2698-2-9-4'], ['1111.2698-1-9-5', '1111.2698-2-9-5'], ['1111.2698-1-9-6', '1111.2698-2-9-6'], ['1111.2698-1-9-7', '1111.2698-2-9-7'], ['1111.2698-1-9-8', '1111.2698-2-9-8'], ['1111.2698-1-22-0', '1111.2698-2-22-0'], ['1111.2698-1-22-1', '1111.2698-2-22-1'], ['1111.2698-1-4-0', '1111.2698-2-4-0'], ['1111.2698-1-4-1', '1111.2698-2-4-1'], ['1111.2698-1-4-2', '1111.2698-2-4-2'], ['1111.2698-1-4-3', '1111.2698-2-4-3'], ['1111.2698-1-4-4', '1111.2698-2-4-4'], ['1111.2698-1-4-5', '1111.2698-2-4-5'], ['1111.2698-1-4-6', '1111.2698-2-4-6'], ['1111.2698-1-4-7', '1111.2698-2-4-7'], ['1111.2698-1-25-0', '1111.2698-2-25-0'], ['1111.2698-1-12-1', '1111.2698-2-12-1']]

[]

[]

[]

[]

['1111.2698-1-11-1', '1111.2698-1-11-12', '1111.2698-1-11-14', '1111.2698-1-12-0', '1111.2698-1-13-0', '1111.2698-1-19-6', '1111.2698-1-25-1', '1111.2698-2-11-1', '1111.2698-2-11-12', '1111.2698-2-11-14', '1111.2698-2-12-0', '1111.2698-2-13-0', '1111.2698-2-19-6', '1111.2698-2-25-1']

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

https://arxiv.org/abs/1111.2698

0803.0912

{'0803.0912-1-0-0': 'We investigate a generalized two-dimensional Weyl Hamiltonian, which may describe the low-energy properties of mechanically deformed graphene and of the organic compound [MATH](BEDT-TTF)[MATH]I[MATH] under pressure.', '0803.0912-1-0-1': 'The associated dispersion has generically the form of tilted anisotropic Dirac cones.', '0803.0912-1-0-2': 'The tilt arises due to next-nearest-neighbor hopping when the Dirac points, where the valence band touches the conduction band, do not coincide with crystallographic high-symmetry points within the first Brillouin zone.', '0803.0912-1-0-3': 'Within a semiclassical treatment, we describe the formation of Landau levels in a strong magnetic field, the relativistic form of which is reminiscent to that of graphene, with a renormalized Fermi velocity due to the tilt of the Dirac cones.', '0803.0912-1-0-4': 'These relativistic Landau levels, experimentally accessible via spectroscopy or even a quantum Hall effect measurement, may be used as a direct experimental verification of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-1-0': '# Introduction', '0803.0912-1-2-0': 'The discovery of a particular quantum Hall effect in graphene[CITATION] has shown that the low-energy electronic properties in this two-dimensional (2D) carbon crystal are described not in terms of a Schrodinger-type wave equation but by a relativistic Dirac equation.', '0803.0912-1-2-1': '[CITATION] Due to a [MATH]-band, which shrinks at half-filling to two inequivalent points at the corners of the first Brillouin zone (BZ), the electronic energy dispersion is almost linear resulting in Dirac cones.', '0803.0912-1-2-2': 'This is reminiscent of the case of massless relativistic particles, where the speed of light [MATH] is replaced by a Fermi velocity [MATH], which is roughly 300 times smaller than [MATH].', '0803.0912-1-3-0': 'Another material where Dirac cones are expected to occur is the organic 2D compound [MATH](BEDT-TTF)[MATH]I[MATH] under pressure.[', '0803.0912-1-3-1': '[CITATION] The relativistic behavior of the carriers may be at the origin[CITATION] of an experimentally observed [MATH] dependence of the carrier density.[', '0803.0912-1-3-2': '[CITATION] Whereas in graphene, the Dirac cones at the corners of the first BZ are isotropic, they are situated within the first BZ in [MATH](BEDT-TTF)[MATH]I[MATH], strongly anisotropic, and tilted in the wave-vector energy space ([MATH]).[', '0803.0912-1-3-3': '[CITATION] The electronic properties are described by a generalized Weyl Hamiltonian with terms linear in the 2D wave vector [MATH].', '0803.0912-1-3-4': 'However, in contrast to graphene, there is yet no direct experimental evidence for the presence of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH] or whether the system is simply a narrow-gap semiconductor.', '0803.0912-1-4-0': 'In the present paper, we study the structure of the generalized Weyl Hamiltonian, which yields energy dispersions in form of tilted anisotropic Dirac cones.', '0803.0912-1-4-1': 'In the presence of a strong magnetic field, the dispersion is quantized in relativistic Landau levels (LLs), with the characteristic [MATH] behavior known from graphene.', '0803.0912-1-4-2': 'The tilt and the anisotropy of the Dirac cones give rise to a renormalization of the effective Fermi velocity and therefore of the typical LL spacing.', '0803.0912-1-5-0': 'One example of a 2D system described by such generalized Weyl equation may be the above-mentioned organic material [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-5-1': 'We show, within an effective tight-binding model on an anisotropic triangular lattice with two atoms per unit cell,[CITATION] that the tilting of the Dirac cones is due to next-nearest-neighbor (nnn) hopping, which may be in [MATH](BEDT-TTF)[MATH]I[MATH] on the same order of magnitude as nearest-neighbor (nn) hopping.', '0803.0912-1-5-2': '[CITATION] A necessary condition for nnn hopping to cause a tilt of the Dirac cones is that they are situated at points in the first BZ different from those of high crystallographic symmetry, such as its corners.', '0803.0912-1-5-3': 'Furthermore, we show that it may equally apply to graphene when the Dirac points, [MATH] and [MATH] move away from the high-symmetry points [MATH] and [MATH] at the corners of the first BZ.', '0803.0912-1-5-4': 'In this case the wave-vector expansion of the nnn term yields a linear contribution, whereas it is quadratic when the Dirac points coincide with the BZ corners [MATH] and [MATH].', '0803.0912-1-5-5': 'Such motion of the Dirac points may indeed be induced by a quinoid-type lattice distortion[CITATION] of the graphene sheet.', '0803.0912-1-5-6': 'However, we show that the tilt of the Dirac cones is much less pronounced than in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-5-7': 'Alternatively, this motion of Dirac points may be studied in cold atoms in an optical lattice where one may deform the honeycomb lattice and fine-tune the nn and nnn hopping parameters with the help of the laser intensities, wavelengths, and relative orientation.', '0803.0912-1-5-8': '[CITATION]', '0803.0912-1-6-0': 'The paper is organized as follows.', '0803.0912-1-6-1': 'We start with a theoretical discussion of the generalized Weyl Hamiltonian in Sec. II.', '0803.0912-1-6-2': 'Sec. III is devoted to the LL formation in a strong magnetic field, for the case of tilted Dirac cones.', '0803.0912-1-6-3': 'Possible experimental realizations in distorted graphene and [MATH](BEDT-TTF)[MATH]I[MATH] are discussed in Sec. IV, which we conclude with an analysis of a possible quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-7-0': '# Generalized Weyl Hamiltonian', '0803.0912-1-8-0': 'We consider a model of two-spinor fermions restricted to a 2D space.', '0803.0912-1-8-1': 'Whereas the two-spinor form is in general dictated by relativistic invariance in two space dimensions, it naturally arises in the condensed matter situation of a lattice with two inequivalent sites.', '0803.0912-1-8-2': 'The most general Hamiltonian linear in the 2D wave vector [MATH], is given by the "generalized Weyl Hamiltonian", [EQUATION] in terms of the velocities [MATH], and the [MATH] Pauli matrices [MATH].', '0803.0912-1-8-3': 'Here and in the following parts, we choose a unit system with [MATH].', '0803.0912-1-8-4': 'Both 2D space components of the velocities, [MATH] and [MATH], are in themselves vectors in the 4D spin space [the space of SU(2) matrices] spanned by the Pauli matrices.', '0803.0912-1-8-5': 'The usual 2D Weyl Hamiltonian, which describes for instance low-energy massless electrons in graphene,[CITATION] is included in ([REF]) if one considers [MATH], [MATH], and [MATH], in terms of the Fermi velocity [MATH].', '0803.0912-1-9-0': 'Although, at first sight, the Weyl Hamiltonian is described by eight different parameters, given by the four two-component velocities [MATH], it is indeed overspecified.', '0803.0912-1-9-1': 'In order to illustrate this point, we rewrite the Hamiltonian ([REF]) in a different manner, [EQUATION]', '0803.0912-1-9-2': 'One may get rid of two parameters ([MATH]) by choosing the 3-quantization axis in the SU(2) space perpendicular to the vectors [MATH] and [MATH].', '0803.0912-1-10-0': 'This point is indeed remarkable and needs to be discussed in the light of graphene physics.', '0803.0912-1-10-1': 'In this case, a constant [MATH] term breaks the inversion symmetry of the honeycomb lattice, e.g. due to a different on-site energy of the two triangular sublattices.', '0803.0912-1-10-2': 'Usually, this gives rise to a mass term and breaks the particle-hole symmetry.', '0803.0912-1-10-3': 'In the generalized Weyl Hamiltonian, this is not the case because the [MATH] term is linear in the wave vector and therefore does not affect the zero-energy state at [MATH].', '0803.0912-1-11-0': 'Furthermore, one may rotate the 2D reference system in the physical space, [EQUATION] accompanied by a unitary transformation in the SU(2) space, [EQUATION] which leaves the [MATH]-quantization axis invariant and describes a rotation in the [MATH]-plane in the SU([MATH]) spin space, [EQUATION]', '0803.0912-1-11-1': 'If one chooses [EQUATION] and [EQUATION] one obtains the "minimal" Weyl Hamiltonian [EQUATION] in terms of the four effective velocities [MATH], [MATH] and [MATH].', '0803.0912-1-11-2': 'In terms of the original velocities, they read [EQUATION]', '0803.0912-1-11-3': 'The diagonalization of the minimal Weyl Hamiltonian yields the energy dispersions [EQUATION] where [MATH] plays the role of the band index.', '0803.0912-1-12-0': 'For [MATH] and [MATH], one obtains the isotropic model, which applies e.g. to the low-energy electronic properties in graphene: the Fermi velocities are the same in the [MATH]- and [MATH]-direction.', '0803.0912-1-12-1': 'The rotational symmetry is broken if [MATH] (anisotropic model).', '0803.0912-1-12-2': 'Such case may be obtained e.g. if the graphene sheet is constrained by a uniaxial pressure, as is discussed in Sec. IV A. For [MATH], the Dirac cones are tilted away from the [MATH]-axis, as is shown in Fig. [REF].', '0803.0912-1-13-0': 'Notice that not all values of the tilt parameter [MATH] are indeed physical.', '0803.0912-1-13-1': 'In order to be able to associate [MATH] to a positive and [MATH] to a negative energy state, one obtains the condition [EQUATION]', '0803.0912-1-13-2': 'Unless this condition is satisfied, the iso-energetic lines are no longer ellipses but hyperbolas.', '0803.0912-1-13-3': 'Notice that, here, we aim to use the generalized Weyl Hamiltonian ([REF]) and its resulting energy dispersion ([REF]) to describe the electronic properties of particular 2D materials.', '0803.0912-1-13-4': 'Although it may be interesting to speculate about the resulting properties of a model that violates the condition ([REF]), we are not aware of any physical example which might correspond to such a case.', '0803.0912-1-14-0': 'In order to discuss the symmetry properties of the generalized Weyl Hamiltonian ([REF]), it is convenient to introduce the unitary and Hermitian chirality operator [EQUATION] which commutes naturally with the Hamiltonian.', '0803.0912-1-14-1': 'The associated eigenvalues are [MATH] and coincide with the band indices [MATH].', '0803.0912-1-14-2': 'As exemplified in Sec. IV, this is generally not the case in a physical condensed-matter situation - the Weyl Hamiltonian corresponds to the effective model at Dirac points, where the conduction band touches the valence band; these Dirac points occur in pairs, at inequivalent points in the first BZ, which yields a twofold valley degeneracy.', '0803.0912-1-14-3': 'In this case, the effective model is rather given by [MATH], where [MATH] denotes the two valleys, and the relation between band index, chirality, and valley index is given by [EQUATION]', '0803.0912-1-14-4': 'In the present discussion, we may however identify the band index with the chirality, for simplicity.', '0803.0912-1-15-0': 'The eigenstates of the chirality operator are [EQUATION] where [MATH].', '0803.0912-1-15-1': 'These eigenstates are also the natural eigenstates for the generalized Weyl Hamiltonian.', '0803.0912-1-16-0': '# Tilted Dirac Cones in a Magnetic Field', '0803.0912-1-17-0': 'We use the Peierls substitution to obtain the generalized Weyl Hamiltonian in a magnetic field [EQUATION] where [MATH] is the magnetic length, and [MATH] is the usual lowering (rising) operator, which satisfies the commutation relation [MATH].', '0803.0912-1-17-1': 'The Peierls substitution yields the quantum Hamiltonian [EQUATION] where we have defined an average Fermi velocity [MATH], [MATH], and [MATH].', '0803.0912-1-18-0': 'Instead of the full solution of the quantum Hamiltonian ([REF]), we consider the effect of the magnetic field in a semiclassical treatment.', '0803.0912-1-18-1': 'The Onsager relation[CITATION] states that the surface [MATH] enclosed by a trajectory of constant energy [MATH] in reciprocal space is quantized, [EQUATION] where [MATH] is an integer denoting the energy level which coincides with the Landau level in the full quantum treatment.', '0803.0912-1-18-2': 'The additional contribution [MATH] is related to a Berry phase.', '0803.0912-1-18-3': 'Usually, one has [MATH] except if there is an extra Berry phase of [MATH], which yields [MATH] as in the case of graphene with no tilt.', '0803.0912-1-18-4': '[CITATION] If one considers a density of states which scales as [MATH], the energy levels thus scale as [EQUATION] in the large-[MATH] limit.', '0803.0912-1-18-5': 'In usual (non-relativistic) 2D electron systems, one finds a constant density of states, i.e. [MATH], and [MATH].', '0803.0912-1-18-6': 'The scaling of the conventional Landau levels is therefore [MATH].', '0803.0912-1-18-7': 'In the relativistic case of electrons in graphene, the density of states vanishes linearly at the Dirac points, and one therefore obtains [MATH] because [MATH] and [MATH].', '0803.0912-1-18-8': 'The relation ([REF]) has been generalized to the case of a spatially anisotropic density of states by Dietl et al. [CITATION]', '0803.0912-1-19-0': 'From the scaling argument ([REF]), one may notice that the [MATH]-field scaling of the levels must be the same as the [MATH] scaling.', '0803.0912-1-19-1': 'Furthermore, one sees from the quantum Hamiltonian ([REF]) that the energy must scale as [MATH].', '0803.0912-1-19-2': 'Therefore, the energy levels must obey, in the large-[MATH] limit, the equation [EQUATION] as in the case of the Weyl equation for massless charged particles, such as in graphene, apart from a renormalization of the Fermi velocity.', '0803.0912-1-20-0': 'The renormalization of the Fermi velocity may be obtained from the calculation of the density of states.', '0803.0912-1-20-1': 'The total number of states below a given energy [MATH] within the positive energy cone is given by [EQUATION] where we have defined [MATH], and the renormalized Fermi velocity is written in integral form, [EQUATION] in terms of the effective tilt parameter [EQUATION]', '0803.0912-1-20-2': 'The integer [MATH] takes into account a degeneracy due to possible internal degrees of freedom, such as e.g. the fourfold spin-valley degeneracy in graphene.', '0803.0912-1-20-3': 'One notices from Eq. ([REF]) that if the condition ([REF]), [MATH], is not satisfied, the expression under the integral diverges because the denominator may become zero.', '0803.0912-1-20-4': 'This result is not suprising because the Onsager quantization relation, which yields the energy levels ([REF]) is only valid for closed orbits, given e.g. by the elliptic isoenergetic lines.', '0803.0912-1-20-5': 'As already mentioned, the orbits for [MATH] are open hyperbolas, and the expression ([REF]) is no longer valid.', '0803.0912-1-21-0': 'The density of states is obtained by differentiation of the number of states, [EQUATION] which is the concise expression for both the positive and negative parts of the tilted Dirac cones.', '0803.0912-1-21-1': 'In agreement with the above scaling arguments, we have [MATH].', '0803.0912-1-22-0': 'The [MATH] behavior of Eq. ([REF]) is, strictly speaking, valid only in the large-[MATH] limit.', '0803.0912-1-22-1': 'However, empirically it yields extremely good estimates for the levels down to values as small as [MATH].', '0803.0912-1-22-2': 'Special care is needed for the discussion of the [MATH] level, which requires a quantum treatment of the Hamiltonian ([REF]) and which may only exist if [MATH].', '0803.0912-1-22-3': 'The behavior of this level may be understood with the help of the quantum treatment of the Hamiltonian for [MATH].', '0803.0912-1-22-4': 'It may be easily diagonalized via the introduction of the auxiliary ladder operators [EQUATION] with [MATH], in order to satisfy the commutation relation [MATH].', '0803.0912-1-22-5': 'In this case, the expression ([REF]) is exact with [MATH], which is also the [MATH]-limit of the expression ([REF]).', '0803.0912-1-22-6': 'There exists thus a zero-energy level for [MATH], which has the same degeneracy, [MATH] as all other levels [MATH], in terms of the abovementioned internal degeneracy [MATH] and the number of flux quanta [MATH] threading the total surface [MATH].', '0803.0912-1-23-0': 'For non-zero values of [MATH], one may investigate the fate of the zero-energy level within a perturbative treatment, where the perturbation term is given by [EQUATION].', '0803.0912-1-23-1': 'We have checked that the zero-energy level persists up to fourth order in perturbation theory, as one expects for an electron-hole symmetric system.', '0803.0912-1-23-2': 'The zero-energy level may indeed split into sublevels if the internal degeneracy [MATH] is lifted, e.g. in the case of a Zeeman field which lifts the spin degeneracy.', '0803.0912-1-23-3': 'However, if one considers only orbital effects, one may exclude a splitting of the zero-energy level into two sublevels, [MATH] and [MATH], for finite values of [MATH].', '0803.0912-1-23-4': 'This would indeed lead to an unphysical doubling of the number of quantum states because each level, [MATH] and [MATH], would have to be [MATH] times degenerate.', '0803.0912-1-23-5': 'Therefore the number of states in the [MATH] level, which one obtains when [MATH] and [MATH] merge at [MATH], would be [MATH], in contrast to the result obtained from the above quantum treatment for [MATH].', '0803.0912-1-24-0': 'Furthermore, we do not exclude a parity anomaly which consists of a different behavior of the [MATH] level at two inequivalent Dirac points in a lattice model.', '0803.0912-1-24-1': 'This parity anomaly is, however, expected to play no physical role in the continuum limit with [MATH], where [MATH] is the lattice spacing.', '0803.0912-1-25-0': '# Physical Examples of Tilted Dirac Cones', '0803.0912-1-26-0': 'After this rather technical discussion of the generalized Weyl Hamiltonian and tilted Dirac cones, we discuss, here, two physical systems which may display these properties.', '0803.0912-1-26-1': 'We find that whereas the tilt of the Dirac cones is well pronounced and thus strongly affects the Landau level quantization in [MATH](BEDT-TTF)[MATH]I[MATH], it is much more difficult to induce a tilt in graphene via a lattice deformation.', '0803.0912-1-26-2': 'However, a quinoid-type lattice deformation is also discussed for pedagogical reasons because the general physical origin of tilted Dirac cones becomes transparent.', '0803.0912-1-27-0': '## Quinoid-type graphene under uniaxial strain', '0803.0912-1-28-0': 'As a first example, we consider a graphene sheet which is deformed in one of its principle symmetry axes.', '0803.0912-1-28-1': 'This particular deformation results in a quinoid variety of the honeycomb lattice.', '0803.0912-1-28-2': '[CITATION] We treat its electronic properties within the tight-binding approximation.', '0803.0912-1-28-3': 'Starting from the graphene honeycomb lattice, with equal bond length [MATH] nm and equal nn hopping energy [MATH] eV, the bond length and hopping energy are modified in the deformation axis (see Fig. [REF]), [EQUATION] and kept unchanged otherwise.', '0803.0912-1-28-4': 'We call [MATH] the relative strain.', '0803.0912-1-28-5': 'Here, we consider a moderate deformation, [MATH], such that one may linearize the hopping energy around its nondeformed value [MATH], and [MATH] eV/[MATH].', '0803.0912-1-28-6': "[CITATION] This value agrees with an evaluation based on Harrison's law[CITATION] according to which [MATH], where [MATH] is a numerical prefactor of order one.", '0803.0912-1-28-7': 'Derivation with respect to [MATH] yields [EQUATION]', '0803.0912-1-28-8': 'For simplicity and as a first approximation, one may keep the bond angles fixed at [MATH].', '0803.0912-1-28-9': 'The underlying Bravais lattice is no longer triangular but oblique with the basis vectors [EQUATION] and the reciprocal lattice is spanned by the vectors [EQUATION].', '0803.0912-1-28-10': 'Furthermore, we take into account nnn hopping, with a characteristic energy of[CITATION] [MATH] in the undeformed horizontal axes.', '0803.0912-1-28-11': 'The deformation yields, in the same manner as for the nn hopping energies, different hopping energies for the other directions (see Fig. [REF]), [EQUATION].', '0803.0912-1-29-0': 'The tight-binding model may be described by the Hamiltonian [EQUATION] in reciprocal space, where [MATH] and [MATH] are the Fourier components of the annihilation (creation) operators on the A and B sublattices, respectively.', '0803.0912-1-29-1': 'The Hamiltonian [MATH] matrix [EQUATION] is given in terms of the elements [EQUATION] and [EQUATION]', '0803.0912-1-29-2': 'The energy dispersion is obtained from the eigenvalues of [MATH], [EQUATION] and is plotted in Fig. [REF] for a deformation of [MATH].', '0803.0912-1-29-3': 'The two bands, [MATH] and [MATH], touch each other at the Dirac points [MATH], which are obtained from the condition [MATH],[CITATION] [EQUATION] where [MATH] denotes the two inequivalent Dirac points [MATH] and [MATH], respectively.', '0803.0912-1-29-4': 'In the absence of any distortion, the Dirac points [MATH] and [MATH] coincide with the crystallographic points [MATH] and [MATH], respectively, at the corners of the first BZ.', '0803.0912-1-29-5': 'The distortion makes both pairs of points move in the same direction due to the negative value of [MATH].', '0803.0912-1-29-6': 'However, unless the parameters are fine-tuned, this motion is different, and the two pairs of points no longer coincide.', '0803.0912-1-29-7': '[CITATION]', '0803.0912-1-30-0': 'The low-energy properties of electrons in a quinoid-type distorted graphene sheet are described by the linearized model around the Dirac points, which is exactly of the form ([REF]) of the Weyl Hamiltonian, [EQUATION] with the effective velocities [EQUATION] where we have defined [MATH].', '0803.0912-1-30-1': 'The corresponding energy dispersion is independent of [MATH], which is at the origin of the twofold valley degeneracy.', '0803.0912-1-30-2': 'In order to obtain the concise form of Eq. ([REF]), we have chosen the spinor representation [MATH] at the [MATH] Dirac point and [MATH] for [MATH], i.e. interchanged the sublattice components at [MATH].', '0803.0912-1-30-3': 'As mentioned in Sed.', '0803.0912-1-30-4': 'II, the relation between the band index [MATH], chirality [MATH], and valley index [MATH] is given by Eq. ([REF]), [MATH], due to the global sign [MATH] in the Hamiltonian ([REF]).', '0803.0912-1-30-5': 'The constant term [MATH] has been absorbed in a renormalization of the chemical potential, the position of which is determined by the electronic half-filling of the graphene sheet.', '0803.0912-1-31-0': 'One notices from the Eqs. ([REF]) that the quinoid-type distortion yields an anisotropy in the Fermi velocities, [MATH], and that the Dirac cones are tilted due to [MATH].', '0803.0912-1-31-1': 'The isotropic graphene model is retrieved at [MATH] - one has then [MATH] eV and [MATH] because [MATH], [MATH], and [MATH], in the undeformed case.', '0803.0912-1-31-2': 'Without deformation, nnn hopping therefore does not affect the energy dispersion at linear order, but only at second order.', '0803.0912-1-31-3': 'This is due to the fact that the Dirac points are then situated at the high-symmetry crystallographic points [MATH] and [MATH].', '0803.0912-1-31-4': 'Indeed, this yields a parabolic correction, which breaks the original electron-hole symmetry.', '0803.0912-1-31-5': '[CITATION]', '0803.0912-1-32-0': 'To summarize, in order to obtain tilted Dirac cones in graphene, two ingredients are required: (i) nnn hopping, which generates the diagonal components [MATH] in the Hamiltonian ([REF]); and (ii) for a linear contribution arising from this term, the Dirac points [MATH] and [MATH] need to be shifted away from the high-symmetry points [MATH] and [MATH].', '0803.0912-1-32-1': 'This shift may be obtained by constraining the graphene sheet into such a quinoid type.', '0803.0912-1-33-0': 'In the presence of a magnetic field, the LL spacing is affected by the deformation because the Fermi velocity is renormalized according to Eq. ([REF]), [EQUATION] for small values of the effective tilt parameter [MATH].', '0803.0912-1-33-1': 'It may be evaluated from the model parameters, [EQUATION]', '0803.0912-1-33-2': 'In order to estimate [MATH], we use the "atomic orbitals overlap law" familiar in the context of the extended Huckel model,[CITATION] [EQUATION] where [MATH] is the nn distance, [MATH] is the nnn distance, and [MATH] AA is a caracteristic distance related to the overlap of atomic orbitals.', '0803.0912-1-33-3': 'In the undeformed graphene [MATH], whereas in the quinoid type graphene [MATH] and [MATH].', '0803.0912-1-33-4': 'This gives [MATH] and [MATH].', '0803.0912-1-33-5': 'Therefore, the effective tilt parameter is given by [EQUATION].', '0803.0912-1-33-6': 'As the correction to the Fermi velocity appears as [MATH] [see Eq. ([REF])], this effect remains extremely small, and the tilt affects the LL spacing in a negligible manner.', '0803.0912-1-34-0': 'The main contribution to the renormalized Fermi velocity therefore arises not from the tilt of the Dirac cones (effect of order [MATH]), but from the anisotropy in the Fermi velocities (effect of order [MATH]), and one finds [EQUATION] which may yield an experimentally observable effect in the percent range for a strain of [MATH].', '0803.0912-1-35-0': 'From an experimental point of view, such quinoid-type deformation may be realized if one uses a piezoelectric substrate, on which the graphene sheet is posed, instead of the most commonly used SiO[MATH].', '0803.0912-1-35-1': 'Another possibility would be to use a mechanical deformation of the underlying substrate.', '0803.0912-1-35-2': 'Such bending has been exploited e.g. to investigate carbon nanotubes under strain.[', '0803.0912-1-35-3': '[CITATION] More recently, graphene on polydimethylsiloxane (PDMS) has been put under uniaxial strain by bending of the PDMS.[', '0803.0912-1-35-4': '[CITATION] The elastic regime in graphene requires that the strain is smaller than [MATH] and the rupture occurs around [MATH].', '0803.0912-1-35-5': 'Therefore an upper bound for [MATH] is certainly [MATH].', '0803.0912-1-36-0': '## Organic 2D compounds', '0803.0912-1-37-0': 'Another example of a 2D metal, where tilted Dirac cones may occur, is the layered organic compound [MATH](BEDT-TTF)[MATH]I[MATH] under (uniaxial) pressure.', '0803.0912-1-37-1': '[CITATION] Each layer may be described by an oblique lattice with four sites per unit cell, and the electronic filling is [MATH].', '0803.0912-1-37-2': 'In the vicinity of the Fermi energy, only two out of the four bands are relevant for the low-energy electronic properties.', '0803.0912-1-37-3': 'It has indeed been shown that the band structure may be modeled with great precision within a tight-binding model on a half-filled anisotropic triangular lattice with nn and nnn hopping, where each site corresponds to a dimer.', '0803.0912-1-37-4': '[CITATION] This is a natural assumption for [MATH]- and [MATH]-(BEDT-TTF)[MATH]I[MATH], where there exists one hopping energy which is largely enhanced with respect to the others.', '0803.0912-1-37-5': 'In contrast to these compounds, the assumption may seem hasardous at first sight in the case of [MATH](BEDT-TTF)[MATH]I[MATH], where there is no such clearly enhanced hopping energy, such that the dimerization is expected to be rather weak.', '0803.0912-1-37-6': 'Furthermore, these organic materials exhibit strong electronic correlations, and a tight-binding calculation for quasi-free electrons sweeps a lot of interesting physics under the carpet.', '0803.0912-1-37-7': 'However, the high-pressure limit corresponds to a regime where the electrons are less strongly correlated and where interaction effects may be taken into account via renormalized effective hopping parameters.', '0803.0912-1-37-8': '[CITATION]', '0803.0912-1-38-0': 'The tight-binding model on the anisotropic triangular lattice is depicted in Fig. [REF].', '0803.0912-1-38-1': 'The nn are situated at the vectors [MATH] and [MATH], with [EQUATION] which connect sites on the different sublattices, [MATH] and [MATH] and the vectors [EQUATION] span the underlying Bravais lattice, which is chosen to be a square lattice, for simplicity.', '0803.0912-1-38-2': 'Notice that the lattice may also be viewed as an anisotropic 2D NaCl lattice (two inequivalent interpenetrating square lattices).', '0803.0912-1-38-3': 'The bond length is set to unity, [MATH].', '0803.0912-1-38-4': 'The nn hopping energies are [MATH] and [MATH] in the directions [MATH], and [MATH] and [MATH] in the directions [MATH], respectively.', '0803.0912-1-38-5': 'The nnn hopping energy is [MATH].', '0803.0912-1-39-0': 'The effective tight-binding model may be written in the same manner ([REF]) as for the case of quinoid-type graphene, with the matrix elements [EQUATION] and [EQUATION].', '0803.0912-1-39-1': 'The energy dispersion is obtained from Eq. ([REF]), and the position of the Dirac points is calculated from [EQUATION]', '0803.0912-1-39-2': 'One may directly see that the r.h.s of both equations must be positive in order to have a pair of Dirac points ([MATH] and [MATH]) within the first BZ, [MATH].', '0803.0912-1-40-0': 'An expansion around the Dirac points yields the generalized Weyl Hamiltonian ([REF]), [EQUATION] in terms of the velocities [EQUATION]', '0803.0912-1-40-1': 'Here, we have used the same spinor representation as for quinoid-type graphene, i.e. we have interchanged the sublattice components when changing the valley.', '0803.0912-1-40-2': 'One notices that the Dirac cones are tilted only if the Dirac points are not situated at the border of the first BZ, [MATH].', '0803.0912-1-40-3': 'This corresponds to the high-symmetry crystallographic points in graphene, and nnn hopping affects the effective model again only at second order in the expansion around the Dirac points.', '0803.0912-1-41-0': '## Possible quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH]', '0803.0912-1-42-0': 'Although it is a delicate issue to yield energy values for the hopping parameters [MATH] and [MATH] from the overlap integrals in [MATH](BEDT-TTF)[MATH]I[MATH],[CITATION] we expect that the good agreement between band-structure calculations in the full model with four sites per unit cell and the anisotropic triangular lattice model[CITATION] yields the correct orders of magnitude for the effective velocities ([REF]).', '0803.0912-1-42-1': 'Using the prescription proposed by Hotta[CITATION] and the overlap integrals calculated by Mori et al.,[CITATION] we may estimate [MATH] meV, [MATH] meV, [MATH] meV, [MATH] meV, and [MATH] meV.', '0803.0912-1-42-2': 'These values yield a pair of Dirac points at [MATH] and [MATH], with [MATH], in units of the inverse lattice constant, which is on the order of [MATH] .', '0803.0912-1-42-3': '[CITATION] With the help of Eqs. ([REF]), one thus obtains the effective velocities [MATH] eV, [MATH] eV, [MATH] eV, [MATH] eV, [MATH], and [MATH] eV.', '0803.0912-1-42-4': 'One notices a variation by almost two orders of magnitude, and one may therefore expect rather large anisotropies.', '0803.0912-1-43-0': 'The effective velocities in the minimal model are calculated with the help of Eqs. ([REF]), and one finds a rotation angle of [MATH] and the velocities [MATH] eV, [MATH] eV, [MATH] eV, and [MATH] eV.', '0803.0912-1-43-1': 'The average Fermi velocity is therefore [MATH] eV, which is roughly one order of magnitude smaller than that in graphene.', '0803.0912-1-43-2': 'The tilt parameter ([REF]) is [EQUATION] and thus much larger than in the case of a quinoid-type deformation of a graphene sheet.', '0803.0912-1-43-3': 'The tilt therefore leads to a reduction of the average Fermi velocity, and one finds form Eq. ([REF]) a renormalized velocity of [EQUATION].', '0803.0912-1-44-0': 'The renormalized Fermi velocity allows one to extract the typical energy scale for the Landau levels in [MATH](BEDT-TTF)[MATH]I[MATH], and one finds from Eq. ([REF]) [MATH], with a characteristic "cyclotron" frequency of [EQUATION] which is, due to the smaller Fermi velocity, roughly one order of magnitude smaller than that in graphene.', '0803.0912-1-44-1': 'However, this energy scale is comparable to the cyclotron frequency in GaAs heterostructures ([MATH] meV), which are most commonly used in the study of quantum Hall physics.', '0803.0912-1-44-2': '[CITATION] One may therefore expect that a relativistic quantum Hall effect[CITATION] could principally also occur in [MATH](BEDT-TTF)[MATH]I[MATH] if disorder does not prevent LL formation.', '0803.0912-1-45-0': 'Experimentally, thin (BEDT-TTF)[MATH]I[MATH] films have already been synthesized.', '0803.0912-1-45-1': '[CITATION] Alternatively, one may hope that the exfoliation technique,[CITATION] which has proven to be particularly successful in the fabrication of single-layer graphene sheets, also yields reasonably thin [MATH](BEDT-TTF)[MATH]I[MATH] samples.', '0803.0912-1-45-2': 'However, (BEDT-TTF)[MATH]I[MATH] crystals are generally of lower mechanical stability than carbon crystals, due to the relatively large lattice constants and the reduced binding energies.', '0803.0912-1-46-0': 'Apart from a direct measurement of a quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH] compounds, one may probe the system via transmission spectroscopy in a magnetic field.', '0803.0912-1-46-1': 'This would allow for a direct measurement of the cyclotron frequency and for a check of the relativistic character of electrons in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-46-2': 'Transmission spectroscopy has indeed been successfully applied to epitaxial[CITATION] and exfoliated[CITATION] graphene and yields a [MATH] scaling of the transmission lines, as expected for the relativistic quantum Hall effect in graphene.', '0803.0912-1-47-0': '# Conclusions', '0803.0912-1-48-0': 'In conclusion, we have investigated tilted Dirac cones in deformed graphene and the organic 2D material [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-1-48-1': 'The low-energy electronic properties are described by a generalized Weyl Hamiltonian, which may in both physical systems be derived from a tight-binding model on a lattice with two inequivalent sites.', '0803.0912-1-48-2': 'Whereas the presence of pairs of Dirac points is due to nn hopping, which couples neighboring sites on inequivalent sublattices, the tilt of the Dirac cones arises from nnn hopping if the Dirac points are shifted away from the points of high crystallographic symmetry in the first Brillouin zone.', '0803.0912-1-49-0': 'In the presence of a strong magnetic field, a semiclassical analysis yields the same structure of relativistic LLs as in non-deformed graphene, but with a renormalized effective Fermi velocity due to the tilt of the Dirac cones.', '0803.0912-1-49-1': 'Whereas this effect is expected to be small in a quinoid-type deformation of the graphene, our estimates for the effective velocities for [MATH](BEDT-TTF)[MATH]I[MATH] indicate that the tilt yields a significant reduction of the effective Fermi velocity, which determines the LL spacing.', '0803.0912-1-49-2': 'The largest spacing of the [MATH] and [MATH] LL transitions is on the order of [MATH] meV, which is on the order of the (equidistant) LL spacing in GaAs heterostructures most commonly used in quantum Hall effect measurements.', '0803.0912-1-49-3': 'Such measurements in [MATH](BEDT-TTF)[MATH]I[MATH], as well as LL spectroscopy, may be a possible experimental verification of the yet weakly corroborated presence of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH].'}

{'0803.0912-2-0-0': 'We investigate a generalized two-dimensional Weyl Hamiltonian, which may describe the low-energy properties of mechanically deformed graphene and of the organic compound [MATH](BEDT-TTF)[MATH]I[MATH] under pressure.', '0803.0912-2-0-1': 'The associated dispersion has generically the form of tilted anisotropic Dirac cones.', '0803.0912-2-0-2': 'The tilt arises due to next-nearest-neighbor hopping when the Dirac points, where the valence band touches the conduction band, do not coincide with crystallographic high-symmetry points within the first Brillouin zone.', '0803.0912-2-0-3': 'Within a semiclassical treatment, we describe the formation of Landau levels in a strong magnetic field, the relativistic form of which is reminiscent to that of graphene, with a renormalized Fermi velocity due to the tilt of the Dirac cones.', '0803.0912-2-0-4': 'These relativistic Landau levels, experimentally accessible via spectroscopy or even a quantum Hall effect measurement, may be used as a direct experimental verification of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-1-0': '# Introduction', '0803.0912-2-2-0': 'The discovery of a particular quantum Hall effect in graphene[CITATION] has shown that the low-energy electronic properties in this two-dimensional (2D) carbon crystal are described not in terms of a Schrodinger-type wave equation but by a relativistic Dirac equation.', '0803.0912-2-2-1': '[CITATION] Due to a [MATH]-band, which shrinks at half-filling to two inequivalent points at the corners of the first Brillouin zone (BZ), the electronic energy dispersion is almost linear resulting in Dirac cones.', '0803.0912-2-2-2': 'This is reminiscent of the case of massless relativistic particles, where the speed of light [MATH] is replaced by a Fermi velocity [MATH], which is roughly 300 times smaller than [MATH].', '0803.0912-2-3-0': 'Another material where Dirac cones are expected to occur is the organic 2D compound [MATH](BEDT-TTF)[MATH]I[MATH] under pressure.[', '0803.0912-2-3-1': '[CITATION] The relativistic behavior of the carriers may be at the origin[CITATION] of an experimentally observed [MATH] dependence of the carrier density.[', '0803.0912-2-3-2': '[CITATION] Whereas in graphene, the Dirac cones at the corners of the first BZ are isotropic, they are situated within the first BZ in [MATH](BEDT-TTF)[MATH]I[MATH], strongly anisotropic, and tilted in the wave-vector energy space ([MATH]).[', '0803.0912-2-3-3': '[CITATION] The electronic properties are described by a generalized Weyl Hamiltonian with terms linear in the 2D wave vector [MATH].', '0803.0912-2-3-4': 'However, in contrast to graphene, there is yet no direct experimental evidence for the presence of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH] or whether the system is simply a narrow-gap semiconductor.', '0803.0912-2-4-0': 'In the present paper, we study the structure of the generalized Weyl Hamiltonian, which yields energy dispersions in form of tilted anisotropic Dirac cones.', '0803.0912-2-4-1': 'In the presence of a strong magnetic field, the dispersion is quantized in relativistic Landau levels (LLs), with the characteristic [MATH] behavior known from graphene.', '0803.0912-2-4-2': 'The tilt and the anisotropy of the Dirac cones give rise to a renormalization of the effective Fermi velocity and therefore of the typical LL spacing.', '0803.0912-2-5-0': 'One example of a 2D system described by such generalized Weyl equation may be the above-mentioned organic material [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-5-1': 'We show, within an effective tight-binding model on an anisotropic triangular lattice with two atoms per unit cell,[CITATION] that the tilting of the Dirac cones is due to next-nearest-neighbor (nnn) hopping, which may be in [MATH](BEDT-TTF)[MATH]I[MATH] on the same order of magnitude as nearest-neighbor (nn) hopping.', '0803.0912-2-5-2': '[CITATION] A necessary condition for nnn hopping to cause a tilt of the Dirac cones is that they are situated at points in the first BZ different from those of high crystallographic symmetry, such as its corners.', '0803.0912-2-5-3': 'Furthermore, we show that it may equally apply to graphene when the Dirac points, [MATH] and [MATH] move away from the high-symmetry points [MATH] and [MATH] at the corners of the first BZ.', '0803.0912-2-5-4': 'In this case the wave-vector expansion of the nnn term yields a linear contribution, whereas it is quadratic when the Dirac points coincide with the BZ corners [MATH] and [MATH].', '0803.0912-2-5-5': 'Such motion of the Dirac points may indeed be induced by a quinoid-type lattice distortion[CITATION] of the graphene sheet.', '0803.0912-2-5-6': 'However, we show that the tilt of the Dirac cones is much less pronounced than in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-5-7': 'Alternatively, this motion of Dirac points may be studied in cold atoms in an optical lattice where one may deform the honeycomb lattice and fine-tune the nn and nnn hopping parameters with the help of the laser intensities, wavelengths, and relative orientation.', '0803.0912-2-5-8': '[CITATION]', '0803.0912-2-6-0': 'The paper is organized as follows.', '0803.0912-2-6-1': 'We start with a theoretical discussion of the generalized Weyl Hamiltonian in Sec. II.', '0803.0912-2-6-2': 'Sec. III is devoted to the LL formation in a strong magnetic field, for the case of tilted Dirac cones.', '0803.0912-2-6-3': 'Possible experimental realizations in distorted graphene and [MATH](BEDT-TTF)[MATH]I[MATH] are discussed in Sec. IV, which we conclude with an analysis of a possible quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-7-0': '# Generalized Weyl Hamiltonian', '0803.0912-2-8-0': 'We consider a model of two-spinor fermions restricted to a 2D space.', '0803.0912-2-8-1': 'Whereas the two-spinor form is in general dictated by relativistic invariance in two space dimensions, it naturally arises in the condensed matter situation of a lattice with two inequivalent sites.', '0803.0912-2-8-2': 'The most general Hamiltonian linear in the 2D wave vector [MATH], is given by the "generalized Weyl Hamiltonian", [EQUATION] in terms of the velocities [MATH], and the [MATH] Pauli matrices [MATH].', '0803.0912-2-8-3': 'Here and in the following parts, we choose a unit system with [MATH].', '0803.0912-2-8-4': 'Both 2D space components of the velocities, [MATH] and [MATH], are in themselves vectors in the 4D spin space [the space of SU(2) matrices] spanned by the Pauli matrices.', '0803.0912-2-8-5': 'The usual 2D Weyl Hamiltonian, which describes for instance low-energy massless electrons in graphene,[CITATION] is included in ([REF]) if one considers [MATH], [MATH], and [MATH], in terms of the Fermi velocity [MATH].', '0803.0912-2-9-0': 'Although, at first sight, the Weyl Hamiltonian is described by eight different parameters, given by the four two-component velocities [MATH], it is indeed overspecified.', '0803.0912-2-9-1': 'In order to illustrate this point, we rewrite the Hamiltonian ([REF]) in a different manner, [EQUATION]', '0803.0912-2-9-2': 'One may get rid of two parameters ([MATH]) by choosing the 3-quantization axis in the SU(2) space perpendicular to the vectors [MATH] and [MATH].', '0803.0912-2-10-0': 'This point is indeed remarkable and needs to be discussed in the light of graphene physics.', '0803.0912-2-10-1': 'In this case, a constant [MATH] term breaks the inversion symmetry of the honeycomb lattice, e.g. due to a different on-site energy of the two triangular sublattices.', '0803.0912-2-10-2': 'Usually, this gives rise to a mass term and breaks the particle-hole symmetry.', '0803.0912-2-10-3': 'In the generalized Weyl Hamiltonian, this is not the case because the [MATH] term is linear in the wave vector and therefore does not affect the zero-energy state at [MATH].', '0803.0912-2-11-0': 'One may furthermore reduce the number of relevant model parameters by a simple rotation of the 2D frame of reference, accompanied by a unitary transformation in the SU(2) space, which leaves the 3-quantization axis invariant.', '0803.0912-2-11-1': 'One, thus, obtains the "minimal" Weyl Hamiltonian [EQUATION] in terms of the four effective velocities [MATH], [MATH] and [MATH].', '0803.0912-2-11-2': 'A detailed discussion of the involved transformations and a derivation of the exact expressions for the effective velocities may be found in the Appendix [REF].', '0803.0912-2-12-0': 'The diagonalization of the minimal Weyl Hamiltonian yields the energy dispersions [EQUATION] where [MATH] plays the role of the band index.', '0803.0912-2-13-0': 'For [MATH] and [MATH], one obtains the isotropic model, which applies e.g. to the low-energy electronic properties in graphene: the Fermi velocities are the same in the [MATH]- and [MATH]-direction.', '0803.0912-2-13-1': 'The rotational symmetry is broken if [MATH] (anisotropic model).', '0803.0912-2-13-2': 'Such case may be obtained e.g. if the graphene sheet is constrained by a uniaxial pressure, as is discussed in Sec. IV A. For [MATH], the Dirac cones are tilted away from the [MATH]-axis, as is shown in Fig. [REF].', '0803.0912-2-14-0': 'Notice that not all values of the tilt parameter [MATH] are indeed physical.', '0803.0912-2-14-1': 'In order to be able to associate [MATH] to a positive and [MATH] to a negative energy state, one obtains the condition [EQUATION]', '0803.0912-2-14-2': 'Unless this condition is satisfied, the iso-energetic lines are no longer ellipses but hyperbolas.', '0803.0912-2-14-3': 'Notice that, here, we aim to use the generalized Weyl Hamiltonian ([REF]) and its resulting energy dispersion ([REF]) to describe the electronic properties of particular 2D materials.', '0803.0912-2-14-4': 'Although it may be interesting to speculate about the resulting properties of a model that violates the condition ([REF]), we are not aware of any physical example which might correspond to such a case.', '0803.0912-2-15-0': 'In a 2D lattice system with valley degeneracy, a generalized Weyl Hamiltonian may describe the low-energy excitations in different valleys separately.', '0803.0912-2-15-1': 'In the remainder of this paper, we will in general only consider a single valley (and explicitly mention the inclusion of the twofold valley degeneracy when needed).', '0803.0912-2-15-2': 'Note also that we do not consider the true electron spin and do not include the corresponding twofold spin degeneracy.', '0803.0912-2-16-0': 'In order to discuss the symmetry properties of the generalized Weyl Hamiltonian ([REF]), it is convenient to introduce the unitary and Hermitian chirality operator [EQUATION] which commutes naturally with the Hamiltonian.', '0803.0912-2-16-1': 'The associated eigenvalues are [MATH] and coincide with the band indices [MATH].', '0803.0912-2-16-2': 'As exemplified in Sec. IV, this is generally not the case in a physical condensed-matter situation - the Weyl Hamiltonian corresponds to the effective model at Dirac points, where the conduction band touches the valence band; these Dirac points occur in pairs, at inequivalent points in the first BZ, which yields a twofold valley degeneracy.', '0803.0912-2-16-3': 'In this case, the effective model is rather given by [MATH], where [MATH] denotes the two valleys, and the relation between band index, chirality, and valley index is given by [EQUATION]', '0803.0912-2-16-4': 'In the present discussion, we may however identify the band index with the chirality, for simplicity.', '0803.0912-2-17-0': 'The eigenstates of the chirality operator are [EQUATION] where [MATH].', '0803.0912-2-17-1': 'These eigenstates are also the natural eigenstates for the generalized Weyl Hamiltonian.', '0803.0912-2-18-0': '# Tilted Dirac Cones in a Magnetic Field', '0803.0912-2-19-0': 'We use the Peierls substitution to obtain the generalized Weyl Hamiltonian in a magnetic field [EQUATION] where [MATH] is the vector potential that generates the (uniform) magnetic field [MATH] perpendicular to the 2D plane.', '0803.0912-2-19-1': 'With the help of the ladder operators [EQUATION] in terms of the magnetic length [MATH], one obtains the Hamiltonian [EQUATION] where we have defined [EQUATION] in terms of the effective tilt parameter [EQUATION]', '0803.0912-2-19-2': 'Instead of the full solution of the Hamiltonian ([REF]), we consider the effect of the magnetic field in a semiclassical treatment.', '0803.0912-2-19-3': 'The Onsager relation[CITATION] states that the surface [MATH] enclosed by a trajectory of constant energy [MATH] in reciprocal space is quantized as [EQUATION] where [MATH] is an integer denoting the energy level which coincides with the Landau level in the full quantum treatment.', '0803.0912-2-19-4': 'The additional contribution [MATH] is related to a Berry phase acquired by an electron during its cyclotron orbit.', '0803.0912-2-19-5': 'Usually, one has [MATH] except if there is an extra Berry phase of [MATH], which in our case yields [MATH], as in the case of graphene with no tilt.', '0803.0912-2-19-6': '[CITATION] If one considers a density of states which scales as [MATH], the energy levels thus scale as [EQUATION] in the large-[MATH] limit.', '0803.0912-2-19-7': 'In usual (non-relativistic) 2D electron systems, one finds a constant density of states, i.e. [MATH], and [MATH].', '0803.0912-2-19-8': 'The scaling of the conventional Landau levels is therefore [MATH].', '0803.0912-2-19-9': 'In the relativistic case of electrons in graphene, the density of states vanishes linearly at the Dirac points, and one therefore obtains [MATH] because [MATH] and [MATH].', '0803.0912-2-19-10': 'The relation ([REF]) has been generalized to the case of a spatially anisotropic density of states by Dietl et al. [CITATION]', '0803.0912-2-20-0': 'From the scaling argument ([REF]) in the large-[MATH] limit, one may notice that the [MATH]-field scaling of the levels must be the same as the [MATH] scaling.', '0803.0912-2-20-1': 'Furthermore, one sees from the quantum Hamiltonian ([REF]) that the energy must scale as [MATH].', '0803.0912-2-20-2': 'Therefore, the energy levels must obey, in the large-[MATH] limit, the equation [EQUATION] as in the case of the Weyl equation for massless charged particles, such as in graphene, apart from a renormalization of the Fermi velocity.', '0803.0912-2-21-0': 'The renormalization of the Fermi velocity may be obtained from the calculation of the density of states.', '0803.0912-2-21-1': 'The total number of states below a given energy [MATH] within the positive energy cone is given by [EQUATION] where we have defined [MATH], and the renormalized Fermi velocity is written in integral form, [EQUATION] in terms of the effective tilt parameter ([REF]).', '0803.0912-2-21-2': 'One notices from Eq. ([REF]) that if the condition ([REF]), [MATH], is not satisfied, the expression under the integral diverges because the denominator may become zero.', '0803.0912-2-21-3': 'This result is not suprising because the Onsager quantization relation, which yields the energy levels ([REF]) is only valid for closed orbits, given e.g. by the elliptic isoenergetic lines.', '0803.0912-2-21-4': 'As already mentioned, the orbits for [MATH] are open hyperbolas, and the expression ([REF]) is no longer valid.', '0803.0912-2-22-0': 'The density of states is obtained by differentiation of the number of states, [EQUATION] which is the concise expression for both the positive and negative parts of the tilted Dirac cones.', '0803.0912-2-23-0': 'The [MATH] behavior of Eq. ([REF]) is, strictly speaking, valid only in the large-[MATH] limit.', '0803.0912-2-23-1': 'However, usually it yields extremely good estimates for the levels down to values as small as [MATH].', '0803.0912-2-23-2': 'Special care is needed for the discussion of the [MATH] level, which requires a quantum treatment of the Hamiltonian ([REF]).', '0803.0912-2-23-3': 'In the following, we discuss the fate of the zero-energy Landau level.', '0803.0912-2-24-0': 'The behavior of this level may be understood with the help of the quantum treatment of the Hamiltonian for [MATH].', '0803.0912-2-24-1': 'In this case, the expression ([REF]) is exact with [MATH], which is also the [MATH]-limit of the expression ([REF]).', '0803.0912-2-24-2': 'There exists thus a zero-energy level for [MATH], which has the same degeneracy, [MATH] as all other levels [MATH], in terms of the number of flux quanta [MATH] threading the total surface [MATH].', '0803.0912-2-25-0': 'For non-zero values of [MATH], the Hamiltonian ([REF]) may not be diagonalized by a simple canonical transformation.', '0803.0912-2-25-1': 'However, the Hamiltonian ([REF]) is transformed as [MATH] under space inversion, [MATH], as shown in the Appendix [REF].', '0803.0912-2-25-2': 'This implies that the energy spectrum is symmetric around zero energy (see Appendix [REF]).', '0803.0912-2-25-3': 'Therefore, starting from [MATH] and adiabatically turning on [MATH], there are only two possibilities for the evolution of the zero-energy level: (i) either it remains at zero energy or (ii) it splits into (at least) two sublevels [MATH] and [MATH], which are symmetric around zero energy.', '0803.0912-2-25-4': 'However, splitting of the zero-energy level into sublevels can be excluded on account of the degeneracy of this level.', '0803.0912-2-25-5': 'Indeed, when [MATH], the exact degeneracy of the zero energy ([MATH]) Landau level is given by [MATH] (remember that we only consider a single valley here).', '0803.0912-2-25-6': 'When [MATH], it can, therefore, not split since this would indeed lead to an unphysical doubling of the number of quantum states because each level, [MATH] and [MATH], would have to be [MATH] times degenerate.', '0803.0912-2-25-7': 'Therefore, for all magnetic field strength, there exist a zero-energy Landau level.', '0803.0912-2-25-8': 'The explicit expressions for the zero-energy wave functions may be found in the Appendix [REF].', '0803.0912-2-25-9': 'Notice, that this is consistent with the semiclassical spectrum with [MATH].', '0803.0912-2-26-0': 'In the above treatment, we only considered a single valley.', '0803.0912-2-26-1': 'We note however that the magnetic field might introduce a coupling between the two valleys.', '0803.0912-2-26-2': 'In such a case, we do not exclude a parity anomaly which consists of a different behavior of the [MATH] level at the two inequivalent Dirac points at non-zero wave vectors in a lattice model.', '0803.0912-2-26-3': 'In this case, space inversion would involve the low-energy Hamiltonians at both Dirac points, and the spectrum is only symmetric around zero energy if one accounts for both valleys.', '0803.0912-2-26-4': 'The parity anomaly is, however, expected to play no physical role in the continuum limit with [MATH], where [MATH] is the lattice spacing.', '0803.0912-2-27-0': 'In conclusion, we have obtained the semiclassical spectrum of Landau levels [see Eqs ([REF])-([REF]], valid when [MATH]) and checked that the zero energy level ([MATH]) indeed exists in a full quantum treatment.', '0803.0912-2-27-1': 'Based on this two calculations, we expect the semiclassical spectrum to be a very good approximation to the true quantum spectrum of Landau levels, for all [MATH].', '0803.0912-2-27-2': 'This is one of the main result of the present paper.', '0803.0912-2-28-0': '# Physical Examples of Tilted Dirac Cones', '0803.0912-2-29-0': 'After this rather technical discussion of the generalized Weyl Hamiltonian and tilted Dirac cones, we discuss, here, two physical systems which may display these properties.', '0803.0912-2-29-1': 'We find that whereas the tilt of the Dirac cones is well pronounced and thus strongly affects the Landau level quantization in [MATH](BEDT-TTF)[MATH]I[MATH], it is much more difficult to induce a tilt in graphene via a lattice deformation.', '0803.0912-2-29-2': 'However, a quinoid-type lattice deformation is also discussed for pedagogical reasons because the general physical origin of tilted Dirac cones becomes transparent.', '0803.0912-2-30-0': '## Quinoid-type graphene under uniaxial strain', '0803.0912-2-31-0': 'As a first example, we consider a graphene sheet which is deformed in one of its principle symmetry axes.', '0803.0912-2-31-1': 'This particular deformation results in a quinoid variety of the honeycomb lattice.', '0803.0912-2-31-2': '[CITATION] We treat its electronic properties within the tight-binding approximation.', '0803.0912-2-31-3': 'Starting from the graphene honeycomb lattice, with equal bond length [MATH] nm and equal nn hopping energy [MATH] eV, the bond length and hopping energy are modified in the deformation axis (see Fig. [REF]), [EQUATION] and kept unchanged otherwise.', '0803.0912-2-31-4': 'We call [MATH] the relative strain.', '0803.0912-2-31-5': 'Here, we consider a moderate deformation, [MATH], such that one may linearize the hopping energy around its nondeformed value [MATH], and [MATH] eV/[MATH].', '0803.0912-2-31-6': "[CITATION] This value agrees with an evaluation based on Harrison's law[CITATION] according to which [MATH], where [MATH] is a numerical prefactor of order one.", '0803.0912-2-31-7': 'Derivation with respect to [MATH] yields [EQUATION]', '0803.0912-2-31-8': 'For simplicity and as a first approximation, one may keep the bond angles fixed at [MATH].', '0803.0912-2-31-9': 'The underlying Bravais lattice is no longer triangular but oblique with the basis vectors [EQUATION] and the reciprocal lattice is spanned by the vectors [EQUATION].', '0803.0912-2-31-10': 'Furthermore, we take into account nnn hopping, with a characteristic energy of[CITATION] [MATH] in the undeformed horizontal axes.', '0803.0912-2-31-11': 'The deformation yields, in the same manner as for the nn hopping energies, different hopping energies for the other directions (see Fig. [REF]), [EQUATION].', '0803.0912-2-32-0': 'The tight-binding model may be described by the Hamiltonian [EQUATION] in reciprocal space, where [MATH] and [MATH] are the Fourier components of the annihilation (creation) operators on the A and B sublattices, respectively.', '0803.0912-2-32-1': 'The Hamiltonian [MATH] matrix [EQUATION] is given in terms of the elements [EQUATION] and [EQUATION]', '0803.0912-2-32-2': 'The energy dispersion is obtained from the eigenvalues of [MATH], [EQUATION] and is plotted in Fig. [REF] for a deformation of [MATH].', '0803.0912-2-32-3': 'The two bands, [MATH] and [MATH], touch each other at the Dirac points [MATH], which are obtained from the condition [MATH],[CITATION] [EQUATION] where [MATH] denotes the two inequivalent Dirac points [MATH] and [MATH], respectively.', '0803.0912-2-32-4': 'In the absence of any distortion, the Dirac points [MATH] and [MATH] coincide with the crystallographic points [MATH] and [MATH], respectively, at the corners of the first BZ.', '0803.0912-2-32-5': 'The distortion makes both pairs of points move in the same direction due to the negative value of [MATH].', '0803.0912-2-32-6': 'However, unless the parameters are fine-tuned, this motion is different, and the two pairs of points no longer coincide.', '0803.0912-2-32-7': '[CITATION]', '0803.0912-2-33-0': 'The low-energy properties of electrons in a quinoid-type distorted graphene sheet are described by the linearized model around the Dirac points, which is exactly of the form ([REF]) of the Weyl Hamiltonian, [EQUATION] with the effective velocities [EQUATION] where we have defined [MATH].', '0803.0912-2-33-1': 'The corresponding energy dispersion is independent of [MATH], which is at the origin of the twofold valley degeneracy.', '0803.0912-2-33-2': 'In order to obtain the concise form of Eq. ([REF]), we have chosen the spinor representation [MATH] at the [MATH] Dirac point and [MATH] for [MATH], i.e. interchanged the sublattice components at [MATH].', '0803.0912-2-33-3': 'As mentioned in Sed.', '0803.0912-2-33-4': 'II, the relation between the band index [MATH], chirality [MATH], and valley index [MATH] is given by Eq. ([REF]), [MATH], due to the global sign [MATH] in the Hamiltonian ([REF]).', '0803.0912-2-33-5': 'The constant term [MATH] has been absorbed in a renormalization of the chemical potential, the position of which is determined by the electronic half-filling of the graphene sheet.', '0803.0912-2-34-0': 'One notices from the Eqs. ([REF]) that the quinoid-type distortion yields an anisotropy in the Fermi velocities, [MATH], and that the Dirac cones are tilted due to [MATH].', '0803.0912-2-34-1': 'The isotropic graphene model is retrieved at [MATH] - one has then [MATH] eV and [MATH] because [MATH], [MATH], and [MATH], in the undeformed case.', '0803.0912-2-34-2': 'Without deformation, nnn hopping therefore does not affect the energy dispersion at linear order, but only at second order.', '0803.0912-2-34-3': 'This is due to the fact that the Dirac points are then situated at the high-symmetry crystallographic points [MATH] and [MATH].', '0803.0912-2-34-4': 'Indeed, this yields a parabolic correction, which breaks the original electron-hole symmetry.', '0803.0912-2-34-5': '[CITATION]', '0803.0912-2-35-0': 'To summarize, in order to obtain tilted Dirac cones in graphene, two ingredients are required: (i) nnn hopping, which generates the diagonal components [MATH] in the Hamiltonian ([REF]); and (ii) for a linear contribution arising from this term, the Dirac points [MATH] and [MATH] need to be shifted away from the high-symmetry points [MATH] and [MATH].', '0803.0912-2-35-1': 'This shift may be obtained by constraining the graphene sheet into such a quinoid type.', '0803.0912-2-36-0': 'In the presence of a magnetic field, the LL spacing is affected by the deformation because the Fermi velocity is renormalized according to Eq. ([REF]), [EQUATION] for small values of the effective tilt parameter [MATH].', '0803.0912-2-36-1': 'It may be evaluated from the model parameters, [EQUATION]', '0803.0912-2-36-2': 'In order to estimate [MATH], we use the "atomic orbitals overlap law" familiar in the context of the extended Huckel model,[CITATION] [EQUATION] where [MATH] is the nn distance, [MATH] is the nnn distance, and [MATH] AA is a caracteristic distance related to the overlap of atomic orbitals.', '0803.0912-2-36-3': 'In the undeformed graphene [MATH], whereas in the quinoid type graphene [MATH] and [MATH].', '0803.0912-2-36-4': 'This gives [MATH] and [MATH].', '0803.0912-2-36-5': 'Therefore, the effective tilt parameter is given by [EQUATION].', '0803.0912-2-36-6': 'As the correction to the Fermi velocity appears as [MATH] [see Eq. ([REF])], this effect remains extremely small, and the tilt affects the LL spacing in a negligible manner.', '0803.0912-2-37-0': 'The main contribution to the renormalized Fermi velocity therefore arises not from the tilt of the Dirac cones (effect of order [MATH]), but from the anisotropy in the Fermi velocities (effect of order [MATH]), and one finds [EQUATION] which may yield an experimentally observable effect in the percent range for a strain of [MATH].', '0803.0912-2-38-0': '>From an experimental point of view, such quinoid-type deformation may be realized if one uses a piezoelectric substrate, on which the graphene sheet is posed, instead of the most commonly used SiO[MATH].', '0803.0912-2-38-1': 'Another possibility would be to use a mechanical deformation of the underlying substrate.', '0803.0912-2-38-2': 'Such bending has been exploited e.g. to investigate carbon nanotubes under strain.[', '0803.0912-2-38-3': '[CITATION] More recently, graphene on polydimethylsiloxane (PDMS) has been put under uniaxial strain by bending of the PDMS.[', '0803.0912-2-38-4': '[CITATION] The elastic regime in graphene requires that the strain is smaller than [MATH] and the rupture occurs around [MATH].', '0803.0912-2-38-5': 'Therefore an upper bound for [MATH] is certainly [MATH].', '0803.0912-2-39-0': '## Organic 2D compounds', '0803.0912-2-40-0': 'Another example of a 2D metal, where tilted Dirac cones may occur, is the layered organic compound [MATH](BEDT-TTF)[MATH]I[MATH] under (uniaxial) pressure.', '0803.0912-2-40-1': '[CITATION] Each layer may be described by an oblique lattice with four sites per unit cell, and the electronic filling is [MATH].', '0803.0912-2-40-2': 'In the vicinity of the Fermi energy, only two out of the four bands are relevant for the low-energy electronic properties.', '0803.0912-2-40-3': 'It has indeed been shown that the band structure may be modeled with great precision within a tight-binding model on a half-filled anisotropic triangular lattice with nn and nnn hopping, where each site corresponds to a dimer.', '0803.0912-2-40-4': '[CITATION] This is a natural assumption for [MATH]- and [MATH]-(BEDT-TTF)[MATH]I[MATH], where there exists one hopping energy which is largely enhanced with respect to the others.', '0803.0912-2-40-5': 'In contrast to these compounds, the assumption may seem hasardous at first sight in the case of [MATH](BEDT-TTF)[MATH]I[MATH], where there is no such clearly enhanced hopping energy, such that the dimerization is expected to be rather weak.', '0803.0912-2-40-6': 'Furthermore, these organic materials exhibit strong electronic correlations, and a tight-binding calculation for quasi-free electrons sweeps a lot of interesting physics under the carpet.', '0803.0912-2-40-7': 'However, the high-pressure limit corresponds to a regime where the electrons are less strongly correlated and where interaction effects may be taken into account via renormalized effective hopping parameters.', '0803.0912-2-40-8': '[CITATION]', '0803.0912-2-41-0': 'The tight-binding model on the anisotropic triangular lattice is depicted in Fig. [REF].', '0803.0912-2-41-1': 'The nn are situated at the vectors [MATH] and [MATH], with [EQUATION] which connect sites on the different sublattices, [MATH] and [MATH] and the vectors [EQUATION] span the underlying Bravais lattice, which is chosen to be a square lattice, for simplicity.', '0803.0912-2-41-2': 'Notice that the lattice may also be viewed as an anisotropic 2D NaCl lattice (two inequivalent interpenetrating square lattices).', '0803.0912-2-41-3': 'The bond length is set to unity, [MATH].', '0803.0912-2-41-4': 'The nn hopping energies are [MATH] and [MATH] in the directions [MATH], and [MATH] and [MATH] in the directions [MATH], respectively.', '0803.0912-2-41-5': 'The nnn hopping energy is [MATH].', '0803.0912-2-42-0': 'The effective tight-binding model may be written in the same manner ([REF]) as for the case of quinoid-type graphene, with the matrix elements [EQUATION] and [EQUATION].', '0803.0912-2-42-1': 'The energy dispersion is obtained from Eq. ([REF]), and the position of the Dirac points is calculated from [EQUATION]', '0803.0912-2-42-2': 'One may directly see that the r.h.s of both equations must be positive in order to have a pair of Dirac points ([MATH] and [MATH]) within the first BZ, [MATH].', '0803.0912-2-43-0': 'An expansion around the Dirac points yields the generalized Weyl Hamiltonian ([REF]), [EQUATION] in terms of the velocities [EQUATION]', '0803.0912-2-43-1': 'Here, we have used the same spinor representation as for quinoid-type graphene, i.e. we have interchanged the sublattice components when changing the valley.', '0803.0912-2-43-2': 'One notices that the Dirac cones are tilted only if the Dirac points are not situated at the border of the first BZ, [MATH].', '0803.0912-2-43-3': 'This corresponds to the high-symmetry crystallographic points in graphene, and nnn hopping affects the effective model again only at second order in the expansion around the Dirac points.', '0803.0912-2-44-0': 'The experimental evidence for (tilted) Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH] compounds under pressure is yet rather weak.', '0803.0912-2-44-1': 'Whereas at ambiant pressure, the material is an insulator due to charge ordering, temperature-dependent transport measurements under high hydrostatic pressure have revealed a [MATH] dependence of the carrier density below [MATH] K,[CITATION] as one would expect for relativistic electrons with a linear dispersion relation.', '0803.0912-2-44-2': '[CITATION] It is, however, not clear whether the compound has, under these circumstances, a truely vanishing gap as for massless relativistic electrons or whether a tiny gap persists.', '0803.0912-2-44-3': 'Furthermore the [MATH] dependence of the carrier density is accompanied by a temperature-dependent mobility, which results in an essentially constant conductivity over a large temperature range.[', '0803.0912-2-45-0': 'A more direct evidence for the relevance of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH] would be a measurement of the characteristic properties of relativistic quantum Hall physics, as in the case of graphene.[', '0803.0912-2-45-1': '[CITATION] The following final part of this paper is devoted to the discussion of possible quantum Hall physics in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-46-0': '## Possible quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH]', '0803.0912-2-47-0': 'Although it is a delicate issue to yield energy values for the hopping parameters [MATH] and [MATH] from the overlap integrals in [MATH](BEDT-TTF)[MATH]I[MATH],[CITATION] we expect that the good agreement between band-structure calculations in the full model with four sites per unit cell and the anisotropic triangular lattice model[CITATION] yields the correct orders of magnitude for the effective velocities ([REF]).', '0803.0912-2-47-1': 'Using the prescription proposed by Hotta[CITATION] and the overlap integrals calculated by Mori et al.,[CITATION] we may estimate [MATH] meV, [MATH] meV, [MATH] meV, [MATH] meV, and [MATH] meV.', '0803.0912-2-47-2': 'These values yield a pair of Dirac points at [MATH] and [MATH], with [MATH], in units of the inverse lattice constant, which is on the order of [MATH] .', '0803.0912-2-47-3': '[CITATION] With the help of Eqs. ([REF]), one thus obtains the effective velocities [MATH] eV, [MATH] eV, [MATH] eV, [MATH] eV, [MATH], and [MATH] eV.', '0803.0912-2-47-4': 'One notices a variation by almost two orders of magnitude, and one may therefore expect rather large anisotropies.', '0803.0912-2-48-0': 'The effective velocities in the minimal model are calculated with the help of Eqs. ([REF]), and one finds a rotation angle of [MATH] and the velocities [MATH] eV, [MATH] eV, [MATH] eV, and [MATH] eV.', '0803.0912-2-48-1': 'The average Fermi velocity is therefore [MATH] eV, which is roughly one order of magnitude smaller than that in graphene.', '0803.0912-2-48-2': 'The tilt parameter ([REF]) is [EQUATION] and thus much larger than in the case of a quinoid-type deformation of a graphene sheet.', '0803.0912-2-48-3': 'The tilt therefore leads to a reduction of the average Fermi velocity, and one finds from Eq. ([REF]) a renormalized velocity of [EQUATION].', '0803.0912-2-49-0': 'The renormalized Fermi velocity allows one to extract the typical energy scale for the Landau levels in [MATH](BEDT-TTF)[MATH]I[MATH], and one finds from Eq. ([REF]) [MATH], with a characteristic "cyclotron" frequency of [EQUATION] which is, due to the smaller Fermi velocity, roughly one order of magnitude smaller than that in graphene.', '0803.0912-2-49-1': 'However, this energy scale is comparable to the cyclotron frequency in GaAs heterostructures ([MATH] meV), which are most commonly used in the study of quantum Hall physics.', '0803.0912-2-49-2': '[CITATION] One may therefore expect that a relativistic quantum Hall effect[CITATION] could principally also occur in [MATH](BEDT-TTF)[MATH]I[MATH] if disorder does not prevent LL formation.', '0803.0912-2-50-0': 'Experimentally, thin (BEDT-TTF)[MATH]I[MATH] films have already been synthesized.', '0803.0912-2-50-1': '[CITATION] Alternatively, one may hope that the exfoliation technique,[CITATION] which has proven to be particularly successful in the fabrication of single-layer graphene sheets, also yields reasonably thin [MATH](BEDT-TTF)[MATH]I[MATH] samples.', '0803.0912-2-50-2': 'However, (BEDT-TTF)[MATH]I[MATH] crystals are generally of lower mechanical stability than carbon crystals, due to the relatively large lattice constants and the reduced binding energies.', '0803.0912-2-51-0': 'Apart from a direct measurement of a quantum Hall effect in [MATH](BEDT-TTF)[MATH]I[MATH] compounds, one may probe the system via transmission spectroscopy in a magnetic field.', '0803.0912-2-51-1': 'This would allow for a direct measurement of the cyclotron frequency and for a check of the relativistic character of electrons in [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-51-2': 'Transmission spectroscopy has indeed been successfully applied to epitaxial[CITATION] and exfoliated[CITATION] graphene and yields a [MATH] scaling of the transmission lines, as expected for the relativistic quantum Hall effect in graphene.', '0803.0912-2-52-0': '# Conclusions', '0803.0912-2-53-0': 'In conclusion, we have investigated tilted Dirac cones in deformed graphene and the organic 2D material [MATH](BEDT-TTF)[MATH]I[MATH].', '0803.0912-2-53-1': 'The low-energy electronic properties are described by a generalized Weyl Hamiltonian, which may in both physical systems be derived from a tight-binding model on a lattice with two inequivalent sites.', '0803.0912-2-53-2': 'Whereas the presence of pairs of Dirac points is due to nn hopping, which couples neighboring sites on inequivalent sublattices, the tilt of the Dirac cones arises from nnn hopping if the Dirac points are shifted away from the points of high crystallographic symmetry in the first Brillouin zone.', '0803.0912-2-54-0': 'In the presence of a strong magnetic field, a semiclassical analysis yields the same structure of relativistic LLs as in non-deformed graphene, but with a renormalized effective Fermi velocity due to the tilt and the anisotropy of the Dirac cones.', '0803.0912-2-54-1': 'Whereas this effect is expected to be small in a quinoid-type deformation of the graphene, our estimates for the effective velocities for [MATH](BEDT-TTF)[MATH]I[MATH] indicate that the tilt yields a significant reduction of the effective Fermi velocity, which determines the LL spacing.', '0803.0912-2-54-2': 'The largest spacing of the [MATH] and [MATH] LL transitions is on the order of [MATH] meV, which is on the order of the (equidistant) LL spacing in GaAs heterostructures most commonly used in quantum Hall effect measurements.', '0803.0912-2-54-3': 'Such measurements in [MATH](BEDT-TTF)[MATH]I[MATH], as well as LL spectroscopy, may be a possible experimental verification of the yet weakly corroborated presence of Dirac cones in [MATH](BEDT-TTF)[MATH]I[MATH].'}

[['0803.0912-1-3-0', '0803.0912-2-3-0'], ['0803.0912-1-3-1', '0803.0912-2-3-1'], ['0803.0912-1-3-2', '0803.0912-2-3-2'], ['0803.0912-1-3-3', '0803.0912-2-3-3'], ['0803.0912-1-3-4', '0803.0912-2-3-4'], ['0803.0912-1-39-0', '0803.0912-2-42-0'], ['0803.0912-1-39-1', '0803.0912-2-42-1'], ['0803.0912-1-39-2', '0803.0912-2-42-2'], ['0803.0912-1-31-0', '0803.0912-2-34-0'], ['0803.0912-1-31-1', '0803.0912-2-34-1'], ['0803.0912-1-31-2', '0803.0912-2-34-2'], ['0803.0912-1-31-3', '0803.0912-2-34-3'], ['0803.0912-1-31-4', '0803.0912-2-34-4'], ['0803.0912-1-48-0', '0803.0912-2-53-0'], ['0803.0912-1-48-1', '0803.0912-2-53-1'], ['0803.0912-1-48-2', '0803.0912-2-53-2'], ['0803.0912-1-4-0', '0803.0912-2-4-0'], ['0803.0912-1-4-1', '0803.0912-2-4-1'], ['0803.0912-1-4-2', '0803.0912-2-4-2'], ['0803.0912-1-9-0', '0803.0912-2-9-0'], ['0803.0912-1-9-1', '0803.0912-2-9-1'], ['0803.0912-1-9-2', '0803.0912-2-9-2'], ['0803.0912-1-10-0', '0803.0912-2-10-0'], ['0803.0912-1-10-1', '0803.0912-2-10-1'], ['0803.0912-1-10-2', '0803.0912-2-10-2'], ['0803.0912-1-10-3', '0803.0912-2-10-3'], ['0803.0912-1-46-0', '0803.0912-2-51-0'], ['0803.0912-1-46-1', '0803.0912-2-51-1'], ['0803.0912-1-46-2', '0803.0912-2-51-2'], ['0803.0912-1-30-0', '0803.0912-2-33-0'], ['0803.0912-1-30-1', '0803.0912-2-33-1'], ['0803.0912-1-30-2', '0803.0912-2-33-2'], ['0803.0912-1-30-3', '0803.0912-2-33-3'], ['0803.0912-1-30-4', '0803.0912-2-33-4'], ['0803.0912-1-30-5', '0803.0912-2-33-5'], ['0803.0912-1-29-0', '0803.0912-2-32-0'], ['0803.0912-1-29-1', '0803.0912-2-32-1'], ['0803.0912-1-29-2', '0803.0912-2-32-2'], ['0803.0912-1-29-3', '0803.0912-2-32-3'], ['0803.0912-1-29-4', '0803.0912-2-32-4'], ['0803.0912-1-29-5', '0803.0912-2-32-5'], ['0803.0912-1-29-6', '0803.0912-2-32-6'], ['0803.0912-1-33-0', '0803.0912-2-36-0'], ['0803.0912-1-33-1', '0803.0912-2-36-1'], ['0803.0912-1-33-2', '0803.0912-2-36-2'], ['0803.0912-1-33-3', '0803.0912-2-36-3'], ['0803.0912-1-33-4', '0803.0912-2-36-4'], ['0803.0912-1-33-5', '0803.0912-2-36-5'], ['0803.0912-1-33-6', '0803.0912-2-36-6'], ['0803.0912-1-43-0', '0803.0912-2-48-0'], ['0803.0912-1-43-1', '0803.0912-2-48-1'], ['0803.0912-1-43-2', '0803.0912-2-48-2'], ['0803.0912-1-2-0', '0803.0912-2-2-0'], ['0803.0912-1-2-1', '0803.0912-2-2-1'], ['0803.0912-1-2-2', '0803.0912-2-2-2'], ['0803.0912-1-15-0', '0803.0912-2-17-0'], ['0803.0912-1-15-1', '0803.0912-2-17-1'], ['0803.0912-1-5-0', '0803.0912-2-5-0'], ['0803.0912-1-5-1', '0803.0912-2-5-1'], ['0803.0912-1-5-2', '0803.0912-2-5-2'], ['0803.0912-1-5-3', '0803.0912-2-5-3'], ['0803.0912-1-5-4', '0803.0912-2-5-4'], ['0803.0912-1-5-5', '0803.0912-2-5-5'], ['0803.0912-1-5-6', '0803.0912-2-5-6'], ['0803.0912-1-5-7', '0803.0912-2-5-7'], ['0803.0912-1-28-0', '0803.0912-2-31-0'], ['0803.0912-1-28-1', '0803.0912-2-31-1'], ['0803.0912-1-28-2', '0803.0912-2-31-2'], ['0803.0912-1-28-3', '0803.0912-2-31-3'], ['0803.0912-1-28-4', '0803.0912-2-31-4'], ['0803.0912-1-28-5', '0803.0912-2-31-5'], ['0803.0912-1-28-6', '0803.0912-2-31-6'], ['0803.0912-1-28-7', '0803.0912-2-31-7'], ['0803.0912-1-28-8', '0803.0912-2-31-8'], ['0803.0912-1-28-9', '0803.0912-2-31-9'], ['0803.0912-1-28-10', '0803.0912-2-31-10'], ['0803.0912-1-28-11', '0803.0912-2-31-11'], ['0803.0912-1-44-0', '0803.0912-2-49-0'], ['0803.0912-1-44-1', '0803.0912-2-49-1'], ['0803.0912-1-44-2', '0803.0912-2-49-2'], ['0803.0912-1-32-0', '0803.0912-2-35-0'], ['0803.0912-1-32-1', '0803.0912-2-35-1'], ['0803.0912-1-6-0', '0803.0912-2-6-0'], ['0803.0912-1-6-1', '0803.0912-2-6-1'], ['0803.0912-1-6-2', '0803.0912-2-6-2'], ['0803.0912-1-6-3', '0803.0912-2-6-3'], ['0803.0912-1-20-0', '0803.0912-2-21-0'], ['0803.0912-1-20-3', '0803.0912-2-21-2'], ['0803.0912-1-20-4', '0803.0912-2-21-3'], ['0803.0912-1-20-5', '0803.0912-2-21-4'], ['0803.0912-1-0-0', '0803.0912-2-0-0'], ['0803.0912-1-0-1', '0803.0912-2-0-1'], ['0803.0912-1-0-2', '0803.0912-2-0-2'], ['0803.0912-1-0-3', '0803.0912-2-0-3'], ['0803.0912-1-0-4', '0803.0912-2-0-4'], ['0803.0912-1-38-0', '0803.0912-2-41-0'], ['0803.0912-1-38-1', '0803.0912-2-41-1'], ['0803.0912-1-38-2', '0803.0912-2-41-2'], ['0803.0912-1-38-3', '0803.0912-2-41-3'], ['0803.0912-1-38-4', '0803.0912-2-41-4'], ['0803.0912-1-38-5', '0803.0912-2-41-5'], ['0803.0912-1-19-1', '0803.0912-2-20-1'], ['0803.0912-1-19-2', '0803.0912-2-20-2'], ['0803.0912-1-45-0', '0803.0912-2-50-0'], ['0803.0912-1-45-1', '0803.0912-2-50-1'], ['0803.0912-1-45-2', '0803.0912-2-50-2'], ['0803.0912-1-21-0', '0803.0912-2-22-0'], ['0803.0912-1-12-0', '0803.0912-2-13-0'], ['0803.0912-1-12-1', '0803.0912-2-13-1'], ['0803.0912-1-12-2', '0803.0912-2-13-2'], ['0803.0912-1-26-0', '0803.0912-2-29-0'], ['0803.0912-1-26-1', '0803.0912-2-29-1'], ['0803.0912-1-26-2', '0803.0912-2-29-2'], ['0803.0912-1-42-0', '0803.0912-2-47-0'], ['0803.0912-1-42-1', '0803.0912-2-47-1'], ['0803.0912-1-42-2', '0803.0912-2-47-2'], ['0803.0912-1-42-3', '0803.0912-2-47-3'], ['0803.0912-1-42-4', '0803.0912-2-47-4'], ['0803.0912-1-49-1', '0803.0912-2-54-1'], ['0803.0912-1-49-2', '0803.0912-2-54-2'], ['0803.0912-1-49-3', '0803.0912-2-54-3'], ['0803.0912-1-13-0', '0803.0912-2-14-0'], ['0803.0912-1-13-1', '0803.0912-2-14-1'], ['0803.0912-1-13-2', '0803.0912-2-14-2'], ['0803.0912-1-13-3', '0803.0912-2-14-3'], ['0803.0912-1-13-4', '0803.0912-2-14-4'], ['0803.0912-1-37-0', '0803.0912-2-40-0'], ['0803.0912-1-37-1', '0803.0912-2-40-1'], ['0803.0912-1-37-2', '0803.0912-2-40-2'], ['0803.0912-1-37-3', '0803.0912-2-40-3'], ['0803.0912-1-37-4', '0803.0912-2-40-4'], ['0803.0912-1-37-5', '0803.0912-2-40-5'], ['0803.0912-1-37-6', '0803.0912-2-40-6'], ['0803.0912-1-37-7', '0803.0912-2-40-7'], ['0803.0912-1-14-0', '0803.0912-2-16-0'], ['0803.0912-1-14-1', '0803.0912-2-16-1'], ['0803.0912-1-14-2', '0803.0912-2-16-2'], ['0803.0912-1-14-3', '0803.0912-2-16-3'], ['0803.0912-1-14-4', '0803.0912-2-16-4'], ['0803.0912-1-40-0', '0803.0912-2-43-0'], ['0803.0912-1-40-1', '0803.0912-2-43-1'], ['0803.0912-1-40-2', '0803.0912-2-43-2'], ['0803.0912-1-40-3', '0803.0912-2-43-3'], ['0803.0912-1-8-0', '0803.0912-2-8-0'], ['0803.0912-1-8-1', '0803.0912-2-8-1'], ['0803.0912-1-8-2', '0803.0912-2-8-2'], ['0803.0912-1-8-3', '0803.0912-2-8-3'], ['0803.0912-1-8-4', '0803.0912-2-8-4'], ['0803.0912-1-8-5', '0803.0912-2-8-5'], ['0803.0912-1-34-0', '0803.0912-2-37-0'], ['0803.0912-1-35-1', '0803.0912-2-38-1'], ['0803.0912-1-35-2', '0803.0912-2-38-2'], ['0803.0912-1-35-3', '0803.0912-2-38-3'], ['0803.0912-1-35-4', '0803.0912-2-38-4'], ['0803.0912-1-35-5', '0803.0912-2-38-5'], ['0803.0912-1-22-0', '0803.0912-2-23-0'], ['0803.0912-1-22-3', '0803.0912-2-24-0'], ['0803.0912-1-22-5', '0803.0912-2-24-1'], ['0803.0912-1-11-3', '0803.0912-2-12-0'], ['0803.0912-1-18-4', '0803.0912-2-19-6'], ['0803.0912-1-18-5', '0803.0912-2-19-7'], ['0803.0912-1-18-6', '0803.0912-2-19-8'], ['0803.0912-1-18-7', '0803.0912-2-19-9'], ['0803.0912-1-18-8', '0803.0912-2-19-10'], ['0803.0912-1-24-0', '0803.0912-2-26-2'], ['0803.0912-1-24-1', '0803.0912-2-26-4'], ['0803.0912-1-43-3', '0803.0912-2-48-3'], ['0803.0912-1-20-1', '0803.0912-2-21-1'], ['0803.0912-1-19-0', '0803.0912-2-20-0'], ['0803.0912-1-49-0', '0803.0912-2-54-0'], ['0803.0912-1-23-4', '0803.0912-2-25-6'], ['0803.0912-1-35-0', '0803.0912-2-38-0'], ['0803.0912-1-22-1', '0803.0912-2-23-1'], ['0803.0912-1-22-6', '0803.0912-2-24-2'], ['0803.0912-1-11-1', '0803.0912-2-11-1'], ['0803.0912-1-18-0', '0803.0912-2-19-2'], ['0803.0912-1-18-1', '0803.0912-2-19-3'], ['0803.0912-1-18-3', '0803.0912-2-19-5'], ['0803.0912-1-23-3', '0803.0912-2-25-5'], ['0803.0912-1-22-2', '0803.0912-2-23-2'], ['0803.0912-1-11-0', '0803.0912-2-11-0'], ['0803.0912-1-17-0', '0803.0912-2-19-0'], ['0803.0912-1-18-2', '0803.0912-2-19-4']]

[['0803.0912-1-3-0', '0803.0912-2-3-0'], ['0803.0912-1-3-1', '0803.0912-2-3-1'], ['0803.0912-1-3-2', '0803.0912-2-3-2'], ['0803.0912-1-3-3', '0803.0912-2-3-3'], ['0803.0912-1-3-4', '0803.0912-2-3-4'], ['0803.0912-1-39-0', '0803.0912-2-42-0'], ['0803.0912-1-39-1', '0803.0912-2-42-1'], ['0803.0912-1-39-2', '0803.0912-2-42-2'], ['0803.0912-1-31-0', '0803.0912-2-34-0'], ['0803.0912-1-31-1', '0803.0912-2-34-1'], ['0803.0912-1-31-2', '0803.0912-2-34-2'], ['0803.0912-1-31-3', '0803.0912-2-34-3'], ['0803.0912-1-31-4', '0803.0912-2-34-4'], ['0803.0912-1-48-0', '0803.0912-2-53-0'], ['0803.0912-1-48-1', '0803.0912-2-53-1'], ['0803.0912-1-48-2', '0803.0912-2-53-2'], ['0803.0912-1-4-0', '0803.0912-2-4-0'], ['0803.0912-1-4-1', '0803.0912-2-4-1'], ['0803.0912-1-4-2', '0803.0912-2-4-2'], ['0803.0912-1-9-0', '0803.0912-2-9-0'], ['0803.0912-1-9-1', '0803.0912-2-9-1'], ['0803.0912-1-9-2', '0803.0912-2-9-2'], ['0803.0912-1-10-0', '0803.0912-2-10-0'], ['0803.0912-1-10-1', '0803.0912-2-10-1'], ['0803.0912-1-10-2', '0803.0912-2-10-2'], ['0803.0912-1-10-3', '0803.0912-2-10-3'], ['0803.0912-1-46-0', '0803.0912-2-51-0'], ['0803.0912-1-46-1', '0803.0912-2-51-1'], ['0803.0912-1-46-2', '0803.0912-2-51-2'], ['0803.0912-1-30-0', '0803.0912-2-33-0'], ['0803.0912-1-30-1', '0803.0912-2-33-1'], ['0803.0912-1-30-2', '0803.0912-2-33-2'], ['0803.0912-1-30-3', '0803.0912-2-33-3'], ['0803.0912-1-30-4', '0803.0912-2-33-4'], ['0803.0912-1-30-5', '0803.0912-2-33-5'], ['0803.0912-1-29-0', '0803.0912-2-32-0'], ['0803.0912-1-29-1', '0803.0912-2-32-1'], ['0803.0912-1-29-2', '0803.0912-2-32-2'], ['0803.0912-1-29-3', '0803.0912-2-32-3'], ['0803.0912-1-29-4', '0803.0912-2-32-4'], ['0803.0912-1-29-5', '0803.0912-2-32-5'], ['0803.0912-1-29-6', '0803.0912-2-32-6'], ['0803.0912-1-33-0', '0803.0912-2-36-0'], ['0803.0912-1-33-1', '0803.0912-2-36-1'], ['0803.0912-1-33-2', '0803.0912-2-36-2'], ['0803.0912-1-33-3', '0803.0912-2-36-3'], ['0803.0912-1-33-4', '0803.0912-2-36-4'], ['0803.0912-1-33-5', '0803.0912-2-36-5'], ['0803.0912-1-33-6', '0803.0912-2-36-6'], ['0803.0912-1-43-0', '0803.0912-2-48-0'], ['0803.0912-1-43-1', '0803.0912-2-48-1'], ['0803.0912-1-43-2', '0803.0912-2-48-2'], ['0803.0912-1-2-0', '0803.0912-2-2-0'], ['0803.0912-1-2-1', '0803.0912-2-2-1'], ['0803.0912-1-2-2', '0803.0912-2-2-2'], ['0803.0912-1-15-0', '0803.0912-2-17-0'], ['0803.0912-1-15-1', '0803.0912-2-17-1'], ['0803.0912-1-5-0', '0803.0912-2-5-0'], ['0803.0912-1-5-1', '0803.0912-2-5-1'], ['0803.0912-1-5-2', '0803.0912-2-5-2'], ['0803.0912-1-5-3', '0803.0912-2-5-3'], ['0803.0912-1-5-4', '0803.0912-2-5-4'], ['0803.0912-1-5-5', '0803.0912-2-5-5'], ['0803.0912-1-5-6', '0803.0912-2-5-6'], ['0803.0912-1-5-7', '0803.0912-2-5-7'], ['0803.0912-1-28-0', '0803.0912-2-31-0'], ['0803.0912-1-28-1', '0803.0912-2-31-1'], ['0803.0912-1-28-2', '0803.0912-2-31-2'], ['0803.0912-1-28-3', '0803.0912-2-31-3'], ['0803.0912-1-28-4', '0803.0912-2-31-4'], ['0803.0912-1-28-5', '0803.0912-2-31-5'], ['0803.0912-1-28-6', '0803.0912-2-31-6'], ['0803.0912-1-28-7', '0803.0912-2-31-7'], ['0803.0912-1-28-8', '0803.0912-2-31-8'], ['0803.0912-1-28-9', '0803.0912-2-31-9'], ['0803.0912-1-28-10', '0803.0912-2-31-10'], ['0803.0912-1-28-11', '0803.0912-2-31-11'], ['0803.0912-1-44-0', '0803.0912-2-49-0'], ['0803.0912-1-44-1', '0803.0912-2-49-1'], ['0803.0912-1-44-2', '0803.0912-2-49-2'], ['0803.0912-1-32-0', '0803.0912-2-35-0'], ['0803.0912-1-32-1', '0803.0912-2-35-1'], ['0803.0912-1-6-0', '0803.0912-2-6-0'], ['0803.0912-1-6-1', '0803.0912-2-6-1'], ['0803.0912-1-6-2', '0803.0912-2-6-2'], ['0803.0912-1-6-3', '0803.0912-2-6-3'], ['0803.0912-1-20-0', '0803.0912-2-21-0'], ['0803.0912-1-20-3', '0803.0912-2-21-2'], ['0803.0912-1-20-4', '0803.0912-2-21-3'], ['0803.0912-1-20-5', '0803.0912-2-21-4'], ['0803.0912-1-0-0', '0803.0912-2-0-0'], ['0803.0912-1-0-1', '0803.0912-2-0-1'], ['0803.0912-1-0-2', '0803.0912-2-0-2'], ['0803.0912-1-0-3', '0803.0912-2-0-3'], ['0803.0912-1-0-4', '0803.0912-2-0-4'], ['0803.0912-1-38-0', '0803.0912-2-41-0'], ['0803.0912-1-38-1', '0803.0912-2-41-1'], ['0803.0912-1-38-2', '0803.0912-2-41-2'], ['0803.0912-1-38-3', '0803.0912-2-41-3'], ['0803.0912-1-38-4', '0803.0912-2-41-4'], ['0803.0912-1-38-5', '0803.0912-2-41-5'], ['0803.0912-1-19-1', '0803.0912-2-20-1'], ['0803.0912-1-19-2', '0803.0912-2-20-2'], ['0803.0912-1-45-0', '0803.0912-2-50-0'], ['0803.0912-1-45-1', '0803.0912-2-50-1'], ['0803.0912-1-45-2', '0803.0912-2-50-2'], ['0803.0912-1-21-0', '0803.0912-2-22-0'], ['0803.0912-1-12-0', '0803.0912-2-13-0'], ['0803.0912-1-12-1', '0803.0912-2-13-1'], ['0803.0912-1-12-2', '0803.0912-2-13-2'], ['0803.0912-1-26-0', '0803.0912-2-29-0'], ['0803.0912-1-26-1', '0803.0912-2-29-1'], ['0803.0912-1-26-2', '0803.0912-2-29-2'], ['0803.0912-1-42-0', '0803.0912-2-47-0'], ['0803.0912-1-42-1', '0803.0912-2-47-1'], ['0803.0912-1-42-2', '0803.0912-2-47-2'], ['0803.0912-1-42-3', '0803.0912-2-47-3'], ['0803.0912-1-42-4', '0803.0912-2-47-4'], ['0803.0912-1-49-1', '0803.0912-2-54-1'], ['0803.0912-1-49-2', '0803.0912-2-54-2'], ['0803.0912-1-49-3', '0803.0912-2-54-3'], ['0803.0912-1-13-0', '0803.0912-2-14-0'], ['0803.0912-1-13-1', '0803.0912-2-14-1'], ['0803.0912-1-13-2', '0803.0912-2-14-2'], ['0803.0912-1-13-3', '0803.0912-2-14-3'], ['0803.0912-1-13-4', '0803.0912-2-14-4'], ['0803.0912-1-37-0', '0803.0912-2-40-0'], ['0803.0912-1-37-1', '0803.0912-2-40-1'], ['0803.0912-1-37-2', '0803.0912-2-40-2'], ['0803.0912-1-37-3', '0803.0912-2-40-3'], ['0803.0912-1-37-4', '0803.0912-2-40-4'], ['0803.0912-1-37-5', '0803.0912-2-40-5'], ['0803.0912-1-37-6', '0803.0912-2-40-6'], ['0803.0912-1-37-7', '0803.0912-2-40-7'], ['0803.0912-1-14-0', '0803.0912-2-16-0'], ['0803.0912-1-14-1', '0803.0912-2-16-1'], ['0803.0912-1-14-2', '0803.0912-2-16-2'], ['0803.0912-1-14-3', '0803.0912-2-16-3'], ['0803.0912-1-14-4', '0803.0912-2-16-4'], ['0803.0912-1-40-0', '0803.0912-2-43-0'], ['0803.0912-1-40-1', '0803.0912-2-43-1'], ['0803.0912-1-40-2', '0803.0912-2-43-2'], ['0803.0912-1-40-3', '0803.0912-2-43-3'], ['0803.0912-1-8-0', '0803.0912-2-8-0'], ['0803.0912-1-8-1', '0803.0912-2-8-1'], ['0803.0912-1-8-2', '0803.0912-2-8-2'], ['0803.0912-1-8-3', '0803.0912-2-8-3'], ['0803.0912-1-8-4', '0803.0912-2-8-4'], ['0803.0912-1-8-5', '0803.0912-2-8-5'], ['0803.0912-1-34-0', '0803.0912-2-37-0'], ['0803.0912-1-35-1', '0803.0912-2-38-1'], ['0803.0912-1-35-2', '0803.0912-2-38-2'], ['0803.0912-1-35-3', '0803.0912-2-38-3'], ['0803.0912-1-35-4', '0803.0912-2-38-4'], ['0803.0912-1-35-5', '0803.0912-2-38-5'], ['0803.0912-1-22-0', '0803.0912-2-23-0'], ['0803.0912-1-22-3', '0803.0912-2-24-0'], ['0803.0912-1-22-5', '0803.0912-2-24-1'], ['0803.0912-1-11-3', '0803.0912-2-12-0'], ['0803.0912-1-18-4', '0803.0912-2-19-6'], ['0803.0912-1-18-5', '0803.0912-2-19-7'], ['0803.0912-1-18-6', '0803.0912-2-19-8'], ['0803.0912-1-18-7', '0803.0912-2-19-9'], ['0803.0912-1-18-8', '0803.0912-2-19-10']]

[['0803.0912-1-24-0', '0803.0912-2-26-2'], ['0803.0912-1-24-1', '0803.0912-2-26-4'], ['0803.0912-1-43-3', '0803.0912-2-48-3'], ['0803.0912-1-20-1', '0803.0912-2-21-1'], ['0803.0912-1-19-0', '0803.0912-2-20-0'], ['0803.0912-1-49-0', '0803.0912-2-54-0'], ['0803.0912-1-23-4', '0803.0912-2-25-6'], ['0803.0912-1-35-0', '0803.0912-2-38-0'], ['0803.0912-1-22-1', '0803.0912-2-23-1'], ['0803.0912-1-22-6', '0803.0912-2-24-2'], ['0803.0912-1-11-1', '0803.0912-2-11-1'], ['0803.0912-1-18-0', '0803.0912-2-19-2'], ['0803.0912-1-18-1', '0803.0912-2-19-3'], ['0803.0912-1-18-3', '0803.0912-2-19-5']]

[]

[['0803.0912-1-23-3', '0803.0912-2-25-5'], ['0803.0912-1-22-2', '0803.0912-2-23-2'], ['0803.0912-1-11-0', '0803.0912-2-11-0'], ['0803.0912-1-17-0', '0803.0912-2-19-0'], ['0803.0912-1-18-2', '0803.0912-2-19-4']]

[]

['0803.0912-1-5-8', '0803.0912-1-29-7', '0803.0912-1-31-5', '0803.0912-1-37-8', '0803.0912-2-5-8', '0803.0912-2-32-7', '0803.0912-2-34-5', '0803.0912-2-40-8']

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

https://arxiv.org/abs/0803.0912

astro-ph-0107382

{'astro-ph-0107382-1-0-0': 'The recent identification of the perpendicular mode of radio polarization as the primary one in the Vela pulsar by Lai et al (2001) is interpreted in terms of the maser mechanism proposed by Luo Melrose (1995).', 'astro-ph-0107382-1-0-1': 'We suggest that such a mechanism may also be operative for the parallel mode which opens up the possibility of accounting for all types of polarization observed in pulsars.', 'astro-ph-0107382-1-0-2': 'We propose an alternative interpretation of the arcs in the nebular X-radiation observed by Pavlov et al (2000) Helfand et al (2001) with the Chandra Observatory, and interpreted by the latter as an equatorial wind.', 'astro-ph-0107382-1-0-3': 'We interpret the arcs as traces of the particle beams from the two magnetic poles at the shock front.', 'astro-ph-0107382-1-0-4': 'We also propose that the alignment with the rotation axis of the jet-like feature bisecting the arcs is an effect of projection on the sky plane and that there is no physical jet along the axis of rotation.', 'astro-ph-0107382-1-1-0': 'Vela, X-ray nebula pulsar polarization', 'astro-ph-0107382-1-2-0': '# Introduction', 'astro-ph-0107382-1-3-0': 'Unlike in the case of most other non-thermal radio sources, the polarization of the radiation from pulsars played an early and fundamental role in attempts to understand and model the operative emission mechanism.', 'astro-ph-0107382-1-3-1': 'The high percentage of linear polarization, well over the maximum theoretical limit for synchrotron radiation, together with a special type of systematic sweep of the P.A. observed in the Vela Pulsar led to the "magnetic pole model" (Radhakrishnan Cooke 1969).', 'astro-ph-0107382-1-3-2': "The sweep of the P.A. across the pulse was interpreted in terms of the line of sight tangentially encountering different field lines close to the magnetic pole as the pulsar rotated; and the parameters of the so called 'S' curve of the P.A. sweep have ever since been interpreted in terms of [MATH] and [MATH], the angles made by the magnetic axis to the rotational axis and to the line of sight (at minimum impact angle) respectively.", 'astro-ph-0107382-1-3-3': "An important point is that while the geometry of the 'S' curve is intimately related, through [MATH] and [MATH], to the locus of the sight line, the actual angle between the plane of polarization and the operative magnetic field line can have any value, as long as it remains fixed.", 'astro-ph-0107382-1-3-4': 'In the case of synchrotron radiation, the most widespread emission mechanism invoked for non-thermal sources before the discovery of pulsars, the electric vector of the radiation would be perpendicular to the projected magnetic field, as the acceleration of the charged particles was due to their gyration around the field lines.', 'astro-ph-0107382-1-4-0': 'In the case of pulsars, the systematics of the polarization sweep, and its independence of observing frequency, indicated clearly that the radiation emanated from close to the polar cap in a region that had no internal Faraday rotation.', 'astro-ph-0107382-1-4-1': 'The strength of the fields associated with these regions was so high that any transverse momentum and energy would be radiated away "instantly", and the charged particles would be in their lowest Landau levels and constrained to move along the magnetic field lines, like beads on a string.', 'astro-ph-0107382-1-4-2': 'An appreciation of this constraint led to the suggestion (Radhakrishnan 1969) that the radiation could be due to the acceleration in the plane of the curved field lines, and has been known since then as "curvature radiation".', 'astro-ph-0107382-1-4-3': 'As the motion of the particles, whether electrons or positrons, could be only along the field lines, the polarization of the emitted radiation should have the electric vector parallel to the projected field lines.', 'astro-ph-0107382-1-4-4': 'A consequence of this was the identification of the intrinsic plane of polarization at the centre of the pulse (or more correctly the inflexion point of the S curve), with the projection of the rotation axis of the pulsar on the sky.', 'astro-ph-0107382-1-4-5': 'This has had important implications for a variety of studies over the years relating to the space velocities of pulsars.', 'astro-ph-0107382-1-5-0': 'According to the above picture, the PA of the polarization can have one and only one value at any pulse longitude since the angle of the projected field line is fixed.', 'astro-ph-0107382-1-5-1': 'But as early as 1975 (Manchester, Taylor Huguenin 1975; Backer, Rankin Campbell 1976) it was discovered that the PA could have more than one value at a given longitude!', 'astro-ph-0107382-1-5-2': 'Closer investigation revealed that the PA switched between two modes, taking any one of two values which were orthogonal to each other (Backer Rankin, 1980).', 'astro-ph-0107382-1-5-3': 'The polarization sweep pattern in any one mode appeared identical to that in the other, barring the 90[MATH] shift in P.A.', 'astro-ph-0107382-1-5-4': 'There has been no shortage of attempted models for the radiation mechanism, but in the absence of any other that could be meaningfully compared with observations, the simple picture of the magnetic pole model, with its rules for deriving [MATH] and [MATH], has survived for over three decades, despite the blatant sweeping under the rug of the observed freedom of the polarization vector to take one of two orthogonal values, neither of which was ever shown to have a definite orientation with respect to the field direction!!', 'astro-ph-0107382-1-6-0': '# THE X-RAY VELA STORY', 'astro-ph-0107382-1-7-0': 'We turn now to a discussion of some observations which appear to offer for the first time the possibility of establishing a clear relationship between the directions of polarization and the magnetic field of the pulsar.', 'astro-ph-0107382-1-8-0': 'Recent observations of the Vela pulsar, and its immediate surroundings, with the Chandra X-ray Observatory show a two-sided jet at a position angle coinciding with that of the proper motion of the pulsar (Pavlov et al 2000; Helfand, Gotthelf Halpern 2001).', 'astro-ph-0107382-1-8-1': 'Lai, Chernoff and Cordes (2001) have argued that the symmetric morphology of the X-ray emission about the jet direction suggests strongly that the jet is along the spin axis of the pulsar.', 'astro-ph-0107382-1-8-2': 'As corroborating this interpretation, they cite polarization observations using data from Deshpande, Ramachandran and Radhakrishnan (1999) which have been corrected for Faraday rotation both in the interstellar medium and in the ionosphere.', 'astro-ph-0107382-1-8-3': 'As the radio P.A. is at right angles to the direction of the X-ray jet, Lai et al (2001) conclude that the polarization mode dominant in the Vela pulsar is one where the electric field in radio emission is orthogonal to the magnetospheric field.', 'astro-ph-0107382-1-9-0': 'Further support for this picture comes from recent radio observations of the region that have revealed a double lobe structure, with well separated lobes of comparable intensity (Dodson et al 2001; private communication).', 'astro-ph-0107382-1-9-1': "These radio lobes are symmetrically placed on either side of the X-ray jet with their diffuse inner edges very close to the boundaries of the X-ray nebula, and independently suggest that the projection of the star's spin axis must match the observed direction of the jet in the sky plane.", 'astro-ph-0107382-1-10-0': "From what has been discussed above, it must be concluded that the electric vector of the Vela pulsar's radiation is perpendicular to the plane containing the magnetic field-line, and not parallel to it as one has generally assumed.", 'astro-ph-0107382-1-10-1': 'This was also noted by Helfand et al (2001) although their major concern was with the morphology of the X-ray nebula and its interpretation.', 'astro-ph-0107382-1-10-2': 'We shall also discuss the X-ray nebula shortly, but first the implications of the polarization P.A.', 'astro-ph-0107382-1-11-0': '# IMPLICATIONS FOR THE EMISSION/AMPLIFICATION MECHANISM', 'astro-ph-0107382-1-12-0': 'There is no question that the charged particles in the polar cap regions of any pulsar will be constrained to move essentially along the field lines as already mentioned earlier.', 'astro-ph-0107382-1-12-1': 'There is also no question that there will be radiation from these relativistic particles due to the acceleration associated with the curvature of the field lines, and that the polarization of this radiation will be linear and parallel to the projection of the field lines, assumed planar for the moment.', 'astro-ph-0107382-1-12-2': 'But the brightness temperature of such radiation cannot exceed the kinetic temperature of the electrons (and positrons), which for even extreme values of the magnetic field and spin period are unlikely to be within orders of magnitude of the inferred brightness temperatures, as has been well known since the earliest observations.', 'astro-ph-0107382-1-12-3': 'The absolute need for maser-like amplification, whatever the mechanism of the input radiation, has thus always been recognised, and has motivated numerous attempts over the decades to propose models for the radiation mechanism of pulsars.', 'astro-ph-0107382-1-13-0': 'As noted at the beginning, one of the striking characteristics of pulsar radiation, which must be accounted for in any theoretical model, is its polarization behaviour.', 'astro-ph-0107382-1-13-1': 'Another, as just seen, is the extremely high brightness temperature.', 'astro-ph-0107382-1-13-2': 'We find it remarkable that both these characteristics seem to be well accounted for in the model put forward by Luo and Melrose (1995).', 'astro-ph-0107382-1-13-3': 'They assume that the input signal is curvature radiation, which as discussed in detail above, seems eminently reasonable to us.', 'astro-ph-0107382-1-13-4': 'The amplification process requires a certain non-planarity of the field lines, for which there has long been evidence from many different lines of investigation on pulsars (Radhakrishnan 1992) The surprising aspect is that while the spontaneous curvature radiation is polarized parallel to the field lines, the amplified output is perpendicular to it, in agreement with the observations discussed in the last section.', 'astro-ph-0107382-1-13-5': 'It would appear therefore that the "normal" polarization mode is really "orthogonal" (to the field lines) for reasons associated with the physics of the amplification process, without which pulsars would not be detectable.', 'astro-ph-0107382-1-14-0': 'It should be pointed out that the above is not a violation of the fundamental requirement that in any amplifier the stimulated emission should be indistinguishable from the stimulating input signal.', 'astro-ph-0107382-1-14-1': 'Because of the torsion in the field geometry, the spontaneous emission has a small component in the direction perpendicular to the field in the amplifying region, and it is this component that, according to Luo Melrose (1995), is preferentially amplified and dominates the output.', 'astro-ph-0107382-1-15-0': 'The pattern of switching to orthogonal modes of polarization varies from pulsar to pulsar and can happen in different parts of the pulse profile for different pulsars.', 'astro-ph-0107382-1-15-1': 'For Vela, the existance of both modes was noticed as early as 1983 by Krishnamohan Downs.', 'astro-ph-0107382-1-15-2': 'Although the radiation in the other mode is on the average much weaker, it is important to appreciate that detectable radiation in any mode, in any pulsar, always corresponds to brightness temperatures that require enormous amplification.', 'astro-ph-0107382-1-15-3': 'We are thus forced to conclude that there must be more than one mode of amplification if the input signal is the spontaneous "curvature radiation", as is manifest by the "S" curve outputs of the amplifier, in whichever mode it is operating.', 'astro-ph-0107382-1-15-4': 'This implies that there must be conditions when the parallel mode develops more negative absorption than the perpendicular one favoured by Luo Melrose, and results in a polarization flip.', 'astro-ph-0107382-1-15-5': 'The probability of this happening could be influenced by the particular field distortions present over the longitude range in question.', 'astro-ph-0107382-1-15-6': 'But the fact that either mode can occur is very reminiscent of maser processes in general, where different allowable modes typically compete with each other resulting in one of them rapidly taking over all of the available power.', 'astro-ph-0107382-1-16-0': 'The observation in many pulsars, of elliptically polarized radiation of detectable strength, is further evidence of the existence of amplification in both modes, but now simultaneously with some phase difference and different gains.', 'astro-ph-0107382-1-16-1': 'A major obstacle in the understanding of pulsar polarization until now has been the difficulty of seeing how electric fields could be generated perpendicular to the ultrastrong magnetic field lines, only along which the charges were constrained to move.', 'astro-ph-0107382-1-16-2': 'The mechanism of Luo Melrose (1995) - thanks to torsion - allows and predicts radiation perpendicular to the field lines, and now reduces the explanation of any type of polarization in pulsars to a matter of detail, as opposed to a difficulty of principle.', 'astro-ph-0107382-1-17-0': '# THE X-RAY ARCS: TRACERS OF THE RADIATION BEAMS?', 'astro-ph-0107382-1-18-0': 'We turn now to the spectacular X-ray image provided by Chandra with two remarkably symmetrical arcs bisected by the jet like feature mentioned earlier.', 'astro-ph-0107382-1-18-1': 'Helfand et al (2001) have put forward a detailed model where they "assume that the two arc-like features lie along circular rings highlighting shocks in which the energy of an outflowing equatorial wind is dissipated to become the source of synchrotron emission for the compact nebula" and attribute the incompleteness of the rings to preferential Doppler boosting of the emission in the forward direction.', 'astro-ph-0107382-1-18-2': 'They also "assume that the two rings straddle the equator symmetrically and suppose that the deficit of emission exactly in the equatorial plane is related to the fact that this is where the direction of a toroidally wrapped magnetic field changes sign i.e. the field may vanish there".', 'astro-ph-0107382-1-18-3': 'They go on to derive the half opening angle of the wind [MATH] as 23[MATH].3, and the radius of the shock [MATH] as a/dCos[MATH] cm for d = 250 pc.', 'astro-ph-0107382-1-19-0': 'We would like to propose a somewhat different model for the X-ray arcs, starting from the magnetic pole model for pulsar radiation discussed at length earlier, and that is invariably miscalled the "rotating vector model".', 'astro-ph-0107382-1-19-1': 'In that model, the radiation (and also its amplification as just seen) are produced by highly relativistic particles streaming out along the open field lines from both magnetic poles.', 'astro-ph-0107382-1-19-2': 'We now examine the X-ray data for the possibility that the two arcs reflect the traces of these two particle beams as they encounter the "walls" surrounding the central cavity created by the pulsar.', 'astro-ph-0107382-1-19-3': 'Such a cavity was elaborated in the classic paper by Rees and Gunn (1974) for the Crab, and has since formed a part of most, if not all, subsequent discussions and models of pulsar created nebulae.', 'astro-ph-0107382-1-20-0': 'We assume that the particles leave the weakening field lines at some point well before sweep-back effects set in close to the light cylinder, and proceed "ballistically" outwards.', 'astro-ph-0107382-1-20-1': 'If this picture is valid, one of the two arcs should pass close to our sight line to the pulsar, as indeed it does.', 'astro-ph-0107382-1-20-2': 'To assess this further, we have modelled the arcs as the near-side portions of two rings (seen in projection) traced by sweeps of the (magnetic) polar cones.', 'astro-ph-0107382-1-20-3': 'The rotating magnetic-axis vector is as described in Deshpande et al (1999).', 'astro-ph-0107382-1-20-4': 'The model parameters are the pair of radial distances ([MATH] as measured from the star location and expressed in arcseconds) associated with the two ring traces, the inclination ([MATH]) of the magnetic axis of the star with its rotation axis, the angle of closest approach (the impact angle [MATH]) of the magnetic axis to our sight-line, and the position angle (PAo) of the rotation axis projection on the sky-plane.', 'astro-ph-0107382-1-20-5': 'The angle ([MATH]) between the rotation axis and our sight-line is simply [MATH].', 'astro-ph-0107382-1-20-6': 'Desired consistency with radio polarization observations would allow only certain combinations of the viewing geometry; that is [MATH] should be equal to the steepest sweep rate ([MATH]) of the polarization position angle with respect to the rotational longitude.', 'astro-ph-0107382-1-20-7': 'When we constrain the [MATH] combinations using [MATH] of -9 degree/degree (as listed by Lyne Manchester 1988), the best fit PA[MATH] is found to be 129 degrees (measured from North through East), the radial distances [MATH] are unequal (about 22 29 arcsec for the near and the opposite polar cones respectively) and [MATH] 71 degrees ( [MATH]-6 degrees correspondingly).', 'astro-ph-0107382-1-20-8': 'Note that this implies a value of [MATH] of about 65 degrees, significantly larger than the 53 degrees estimated by Helfand et al (2001) based on their model of an equatorial torus.', 'astro-ph-0107382-1-20-9': 'The above model is illustrated in Fig. 1.', 'astro-ph-0107382-1-21-0': 'The changing direction of the magnetic axis viewed in projection on the sky plane as the star rotates is described in exactly the same way as that for the position angle of the radio polarization (Deshpande et al 1999) whatever their relative difference.', 'astro-ph-0107382-1-21-1': 'The proposed association of the arcs with the traces of the polar emission beams thus provides a new and independent means to probe the viewing geometry.', 'astro-ph-0107382-1-21-2': 'They can sample (as they do in the present case) a much larger fraction of the rotation cycle and can provide additional constraints on the viewing geometry.', 'astro-ph-0107382-1-21-3': 'One crucial such constraint becomes available via the trace of the polar emission from the other pole that is generally not available unless interpulses are observable.', 'astro-ph-0107382-1-21-4': 'Also, interestingly, the "sign" of the impact angle [MATH] would become readily apparent from the beam traces even if they do not sample large fractions of the rotation cycle, and without needing to know the sense of rotation of the star.', 'astro-ph-0107382-1-22-0': 'A simple calculation shows that the visible extent of the arcs is consistent with an X-radiation spread confined to about 70 degrees around the respective directions to which the magnetic axis points as the star rotates.', 'astro-ph-0107382-1-22-1': 'We consider this as explained simply in terms of the initial dispersion of the motions as the particle beam ploughs into the region of compressed toroidal field - the shocked region as usually described - and the consequent spread of the pitch angle distribution.', 'astro-ph-0107382-1-22-2': 'By just assuming that we will see radiation as long as there is some component of the motion towards our line of sight (i.e. [MATH]) our simulation reproduces remarkably well the Chandra observations of the arcs.', 'astro-ph-0107382-1-23-0': '# THE "AXIAL" JET !', 'astro-ph-0107382-1-24-0': 'We now turn to the jet like feature whose symmetric bisection of the arcs discussed above has prompted immediate identification with its rotation axis.', 'astro-ph-0107382-1-24-1': 'The fact that the projection on the sky-plane of the rotational axis as derived from the polarization data (with a 90[MATH] shift) is also in agreement has been interpreted as further and strong evidence of the above supposition.', 'astro-ph-0107382-1-24-2': 'If the jet originates from the pulsar magnetosphere, as seems likely, it is most natural to associate the jet axis with the pulsar spin axis."', 'astro-ph-0107382-1-24-3': 'The alignment of the jet with the observed proper motion for Vela seems also to have prompted an association with, and raised hopes of an explanation for the space velocity of the neutron star.', 'astro-ph-0107382-1-24-4': 'The fact that a similar jet was already seen in the Crab, also aligned with the derived P.A. of the spin axis, and its proper motion, heralds an incipient industry as more X-ray images of radio pulsar nebulae become available.', 'astro-ph-0107382-1-25-0': 'A radio pulsar is not an accreting object like an x-ray pulsar, a black hole binary, or an AGN.', 'astro-ph-0107382-1-25-1': 'In our view, it is not only unlikely, but unphysical to expect an actual jet along the rotation axis of a radio pulsar.', 'astro-ph-0107382-1-25-2': "There is an immense amount we don't know about the magnetospheres of radio pulsars, but one thing we have known for over three decades is that the particle emission is along the magnetic axis (Goldreich Julian 1969), that is often at large angles to the rotation axis.", 'astro-ph-0107382-1-25-3': 'The only connection that there can possibly be between the X-ray jet and the rotation axis is one of projection on the sky-plane as we shall argue below.', 'astro-ph-0107382-1-26-0': 'In our picture of the relativistic particle beams leaving the magnetic poles and proceeding ballistically outwards to the cavity walls, we can ask what if any deviations they are likely to suffer.', 'astro-ph-0107382-1-26-1': 'The only field along their trajectory through the cavity is the toroidal field (that they are carrying out) and that later gets compressed and strengthened at the wall.', 'astro-ph-0107382-1-26-2': 'This toroidal field will, over most or all of the trajectory, be perpendicular to the path of the particles, and, of course, the rotation axis.', 'astro-ph-0107382-1-26-3': 'Depending on the spread of their energies, it should not be surprising if a small fraction of the particles acquired a spread of velocities in the latitudinal direction making their synchrotron radiation visible to an observer when their motion is tangential to the sight line.', 'astro-ph-0107382-1-27-0': 'One should, as a consequence, expect to receive radiation from these particles when the projection of the magnetic axis coincides with that of the rotation axis, precisely as in the case of the radio pulse, but now over a larger range of angles in latitude.', 'astro-ph-0107382-1-27-1': 'Wherever the observer, the apparent jet will appear along the minor axis of the projected ellipses of the arcs but the extent over which it is visible will depend on the spread of particle velocities, and the angles the beams from the two poles make to the line of sight.', 'astro-ph-0107382-1-27-2': 'These particles must also reach the cavity walls and will then create a diffuse glow around the arc regions but with a greater spread, exactly as seen in the Chandra image.', 'astro-ph-0107382-1-27-3': 'We have assessed the spread of this weak fan beam from the size of the diffuse glow, and from the poor or non-visibility of the corresponding radiation from the beam of the other magnetic pole.', 'astro-ph-0107382-1-27-4': 'We estimate a spread of about [MATH] from the observed extent of the jet and have used this value in our simulation (fig. 2) of the jet and the associated diffuse components.', 'astro-ph-0107382-1-28-0': 'If this interpretation of the "jet" is correct, it has a major implication for the picture of particle flow from the pulsar to the nebula.', 'astro-ph-0107382-1-28-1': 'In section 4 when attempting to explain the formation of the X-ray arcs, we made the assumption that the particle beam separated from the polar field lines well within the light cylinder and before the effect of any sweep-back.', 'astro-ph-0107382-1-28-2': 'The precise agreement of the apparent jet with the direction of the pulsar can be explained in no other way and justifies this assumption.', 'astro-ph-0107382-1-29-0': 'The total picture that emerges is as follows.', 'astro-ph-0107382-1-29-1': 'There is a cavity of radius of order [MATH] cms, which we can presume was created by the dipole radiation originally of much higher frequency than the present 11 Hz.', 'astro-ph-0107382-1-29-2': 'Inside is a double cone of half angle [MATH], along which there is a relativistic particle flow in straight lines with a drift time of [MATH]0.1 year before they encounter the "shock".', 'astro-ph-0107382-1-29-3': 'In addition, there is a low density of particles separating from the cones but whose trajectories are in planes containing the rotation axis.', 'astro-ph-0107382-1-29-4': 'Any observer will see radiation only from those particles in the plane containing the observer and the rotation axis and we predict that this radiation should be highly linearly polarized with the electric vector parallel to the rotation axis.', 'astro-ph-0107382-1-30-0': '# DISCUSSION', 'astro-ph-0107382-1-31-0': 'In the long march towards the elucidation of the mysterious ways of pulsars, now numbering around a thousand, a few special ones have taught us more than most of the rest put together.', 'astro-ph-0107382-1-31-1': 'The Vela Pulsar is one such, and played an important role in several ways within months of its discovery in 1968.', 'astro-ph-0107382-1-31-2': 'The superb capabilities of the Chandra telescope have put this pulsar in the limelight again by providing a spectacular image in X-rays of the surrounding nebula.', 'astro-ph-0107382-1-31-3': 'Its form and proportions, reminiscent of pre-Columbian pectoral ornaments are loaded richly with information about many aspects.', 'astro-ph-0107382-1-31-4': 'To begin with, the symmetry of the nebula.', 'astro-ph-0107382-1-31-5': 'has provided compelling evidence for the identification of the P.A. of the rotational axis of the pulsar.', 'astro-ph-0107382-1-31-6': 'The near precise orthogonality of this P.A. to that inferred with certain assumptions from polarization measurements, has had important implications.', 'astro-ph-0107382-1-31-7': 'We support the finding of Lai et al (2001) and Helfand et al (2001) that the dominant polarization mode in the Vela Pulsar has the electric field orthogonal to the magnetospheric field.', 'astro-ph-0107382-1-31-8': 'We find this is in accordance with the mechanism of Luo Melrose (1995) that appears to explain three important characteristics of the radio radiation.', 'astro-ph-0107382-1-31-9': 'They are the high brightness temperatures, the dominant polarization mode, and the observed sweep of the P.A. across the pulse, the last clearly identifying the input signal to the maser as curvature radiation from the field lines in the polar neighbourhood.', 'astro-ph-0107382-1-31-10': 'Since the first and third characteristics also apply to the parallel polarization mode to which the radiation occasionally flips, as seen in many pulsars, we propose that the same amplification mechanism must also be operative at times in the other mode.', 'astro-ph-0107382-1-31-11': 'If the Luo Melrose mechanism can in fact operate in both modes, the explanation of any type of elliptic polarization becomes a matter of detail, a topic which we shall address later.', 'astro-ph-0107382-1-32-0': 'Noting the clear separation of the X-ray nebular emission into two elliptic arcs symmetrically located with respect to the inferred rotation axis, we propose that they are the traces of the two particle-beams from the magnetic poles on the walls of the cavity [MATH] the shocked region.', 'astro-ph-0107382-1-32-1': 'We explain the visible extent of the arcs as arising simply from the spread of the pitch angles at the shock front.', 'astro-ph-0107382-1-32-2': 'We also explain the alignment with the rotational axis of the jet like feature bisecting the arcs as simply a projection effect on the sky plane.', 'astro-ph-0107382-1-32-3': 'We attribute its visibility to a latitudinal spread of the velocities of a small fraction of the particles in the beam, an explanation strongly supported by the presence of diffuse emission around the arcs.', 'astro-ph-0107382-1-32-4': 'We present a simulation of the expected X-radiation around the pulsar based on the above model for the arcs, the jet, and the diffuse glow around both (see fig. 3), and find gratifying agreement with the Chandra observations.', 'astro-ph-0107382-1-32-5': 'We predict that the polarization of the jet feature will be linear and parallel to it, and claim that there can be no physical jet along the rotational axis of the pulsar.', 'astro-ph-0107382-1-33-0': "Note that the apparent 'jet', as radiation from the fan beam, is visible only at certain phases of the rotational cycle of the pulsar, namely when in line with the rotation axis as viewed by a given distant observer.", 'astro-ph-0107382-1-33-1': 'This means, that in principle, the entire jet is pulsed, and looking at a small portion of it should show pulses.', 'astro-ph-0107382-1-33-2': 'But their phase would differ from that of the radio pulse produced close to the star surface because of the extra distance of the two legs over the direct path from the polar cap.', 'astro-ph-0107382-1-34-0': 'As there are about 3[MATH] bunches of particles along any line from the pulsar to the cavity wall, even the slightest dispersion in velocity would smooth this out to a uniform stream at some short distance from the pulsar.', 'astro-ph-0107382-1-34-1': 'But the fact that the radiation comes from a distributed source not perfectly aligned with the line of sight immediately allows for the possibility of both shifting and spreading of the phases.', 'astro-ph-0107382-1-34-2': 'The resultant smearing would wash out its pulsed character, making the associated radiation appear continuous and contributing to a part of the observed un-pulsed fraction in X-rays.', 'astro-ph-0107382-1-34-3': 'This fraction will then depend on the size of the region around the pulsar from which radiation is collected.', 'astro-ph-0107382-1-35-0': 'In our modelling of the pair of arcs as traces of the two polar beams on a cavity wall (e.g. as illustrated in Fig 1), we find that an assumption of equatorial symmetry does not fit the observations well.', 'astro-ph-0107382-1-35-1': 'As already mentioned in an earlier section, the best fit [MATH] is significantly different from [MATH] (i.e. [MATH]1.35).', 'astro-ph-0107382-1-35-2': 'Note that, in our model, [MATH] represent the implied distances to the "wall" from the star along the \'rotating vectors\' associated with the near and the opposite poles respectively.', 'astro-ph-0107382-1-35-3': "The arcs provide us important information including about the otherwise 'unseen' pole.", 'astro-ph-0107382-1-35-4': 'Significantly improved fits are obtained when unequal values of [MATH] (the half angles of the polar cones associated with the two poles) are allowed in the model.', 'astro-ph-0107382-1-35-5': 'The corresponding [MATH],[MATH] are unequal again.', 'astro-ph-0107382-1-35-6': 'Even better fits are obtained if [MATH] is not constrained by the radio observations.', 'astro-ph-0107382-1-35-7': 'But interestingly, the implied value of [MATH]), the angle between the rotation axis and our line of sight, is about the same as in the other cases.', 'astro-ph-0107382-1-35-8': "Further detailed modelling, than has been possible presently, may provide better estimates of the above parameters and clues about the size as well as the shape of the 'cavity'.", 'astro-ph-0107382-1-36-0': 'Inequality between [MATH] has implications, particularly for certain acceleration mechanisms for the origin of pulsar velocities.', 'astro-ph-0107382-1-36-1': "For example, the 'rocket' mechanism of Harrison Tademaru (1975) does necessarily require such an inequality amounting to a tilted and offset dipole.", 'astro-ph-0107382-1-37-0': 'Other possible/plausible implications are the following.', 'astro-ph-0107382-1-38-0': "i) If the dominant radio polarization mode is indeed the 'orthogonal' mode, then any analysis based on the assumption that the observed central polarization P.A. is the same as the P.A. of the rotation axis needs to be reviewed.", 'astro-ph-0107382-1-38-1': 'This applies, for example, to the comparisons of the proper motion directions of pulsars with the orientations of their rotation axis.', 'astro-ph-0107382-1-38-2': 'In the work of Deshpande et al (1999), they allowed for mode ambiguity in cases where emission of both modes is observed, and have assessed the distribution of the proper motion direction (relative to the rotation axis P.A.) as shown in their Fig 1b.', 'astro-ph-0107382-1-38-3': 'We argue that the required revision will amount to simply replacing the relative angle by its complement.', 'astro-ph-0107382-1-38-4': 'Since no particular preference was found for any relative angle value, the conclusions of Deshpande et al remain unaltered.', 'astro-ph-0107382-1-39-0': "ii) Considering the generality of the situation giving rise to the 'jet-like' feature in X-rays, it should not be surprising if a similar feature exists in radio too at some level.", 'astro-ph-0107382-1-39-1': "As discussed above, the major part of such a 'jet' would contribute to a continuous (i.e. un-pulsed) component.", 'astro-ph-0107382-1-39-2': 'But a tiny part of it across a sub-milliarcsec scale close to the pulsar direction may be pulsed and at about the phase of the direct pulse.', 'astro-ph-0107382-1-39-3': 'If this turns out to be indeed so, then it has an important bearing on certain fine-structure details seen in dynamic spectra of pulsars (e.g. see Stinebring et al, 2001) as well as to VLBI investigations towards pulsars.', 'astro-ph-0107382-1-40-0': 'iii) Given the similarities observed between the morphologies of the surrounding nebulae as well as other properties of the Vela Crab pulsars, it would not surprise us if a similar arc structure is revealed around the Crab pulsar by observations with improved spatial resolution.', 'astro-ph-0107382-1-41-0': 'We would like to thank Don Melrose, Qinghuan Lou and Mark Walker for many useful discussions, and Richard Dodson for kindly providing information prior to publication.'}

{'astro-ph-0107382-2-0-0': 'The recent identification of the perpendicular mode of radio polarization as the primary one in the Vela pulsar by Lai et al. (2001) is interpreted in terms of the maser mechanism proposed by Luo Melrose (1995).', 'astro-ph-0107382-2-0-1': 'We suggest that such a mechanism may also be operative for the parallel mode which opens up the possibility of accounting for all types of polarization observed in pulsars.', 'astro-ph-0107382-2-0-2': 'We propose an alternative interpretation of the arcs in the nebular X-radiation observed by Pavlov et al. (2000) Helfand et al. (2001) with the Chandra Observatory, and interpreted by the latter as an equatorial wind.', 'astro-ph-0107382-2-0-3': 'We interpret the arcs as traces of the particle beams from the two magnetic poles at the shock front.', 'astro-ph-0107382-2-0-4': 'We also propose that the alignment with the rotation axis of the jet-like feature bisecting the arcs is an effect of projection on the sky plane and that there is no physical jet along the axis of rotation.', 'astro-ph-0107382-2-1-0': 'Vela, X-ray nebula pulsar polarization', 'astro-ph-0107382-2-2-0': '# Introduction', 'astro-ph-0107382-2-3-0': 'Unlike in the case of most other non-thermal radio sources, the polarization of the radiation from pulsars played an early and fundamental role in attempts to understand and model the operative emission mechanism.', 'astro-ph-0107382-2-3-1': 'The high percentage of linear polarization, well over the maximum theoretical limit for synchrotron radiation, together with a special type of systematic sweep of the P.A. observed in the Vela Pulsar led to the "magnetic pole model" (Radhakrishnan Cooke 1969).', 'astro-ph-0107382-2-3-2': "The sweep of the P.A. across the pulse was interpreted in terms of the line of sight tangentially encountering different field lines close to the magnetic pole as the pulsar rotated; and the parameters of the so called 'S' curve of the P.A. sweep have ever since been interpreted in terms of [MATH] and [MATH], the angles made by the magnetic axis to the rotational axis and to the line of sight (at minimum impact angle) respectively.", 'astro-ph-0107382-2-3-3': "An important point is that while the geometry of the 'S' curve is intimately related, through [MATH] and [MATH], to the locus of the sight line, the actual angle between the plane of polarization and the operative magnetic field line can have any value, as long as it remains fixed.", 'astro-ph-0107382-2-3-4': 'In the case of synchrotron radiation, the most widespread emission mechanism invoked for non-thermal sources before the discovery of pulsars, the electric vector of the radiation would be perpendicular to the projected magnetic field, as the acceleration of the charged particles was due to their gyration around the field lines.', 'astro-ph-0107382-2-4-0': 'In the case of pulsars, the systematics of the polarization sweep, and its independence of observing frequency, indicated clearly that the radiation emanated from close to the polar cap in a region that had no internal Faraday rotation.', 'astro-ph-0107382-2-4-1': 'The strength of the fields associated with these regions was so high that any transverse momentum and energy would be radiated away "instantly", and the charged particles would be in their lowest Landau levels and constrained to move along the magnetic field lines, like beads on a string.', 'astro-ph-0107382-2-4-2': 'An appreciation of this constraint led to the suggestion (Radhakrishnan 1969) that the radiation could be due to the acceleration in the plane of the curved field lines, and has been known since then as "curvature radiation".', 'astro-ph-0107382-2-4-3': 'As the motion of the particles, whether electrons or positrons, could be only along the field lines, the polarization of the emitted radiation should have the electric vector parallel to the projected field lines.', 'astro-ph-0107382-2-4-4': 'A consequence of this was the identification of the intrinsic plane of polarization at the centre of the pulse (or more correctly the inflexion point of the S curve), with the projection of the rotation axis of the pulsar on the sky.', 'astro-ph-0107382-2-4-5': 'This has had important implications for a variety of studies over the years relating to the space velocities of pulsars.', 'astro-ph-0107382-2-5-0': 'According to the above picture, the PA of the polarization can have one and only one value at any pulse longitude since the angle of the projected field line is fixed.', 'astro-ph-0107382-2-5-1': 'But as early as 1975 (Manchester et al. 1975; Backer et al. 1976) it was discovered that the PA could have more than one value at a given longitude!', 'astro-ph-0107382-2-5-2': 'Closer investigation revealed that the PA switched between two modes, taking any one of two values which were orthogonal to each other (Backer Rankin 1980).', 'astro-ph-0107382-2-5-3': 'The polarization sweep pattern in any one mode appeared identical to that in the other, barring the 90[MATH] shift in P.A.', 'astro-ph-0107382-2-5-4': 'There has been no shortage of attempted models for the radiation mechanism, but in the absence of any other that could be meaningfully compared with observations, the simple picture of the magnetic pole model, with its rules for deriving [MATH] and [MATH], has survived for over three decades, despite the blatant sweeping under the rug of the observed freedom of the polarization vector to take one of two orthogonal values, neither of which was ever shown to have a definite orientation with respect to the field direction!!', 'astro-ph-0107382-2-6-0': '# THE X-RAY VELA STORY', 'astro-ph-0107382-2-7-0': 'We turn now to a discussion of some observations which appear to offer for the first time the possibility of establishing a clear relationship between the directions of polarization and the magnetic field of the pulsar.', 'astro-ph-0107382-2-8-0': 'Recent observations of the Vela pulsar, and its immediate surroundings, with the Chandra X-ray Observatory show a two-sided jet at a position angle coinciding with that of the proper motion of the pulsar (Pavlov et al. 2000; Helfand et al. 2001).', 'astro-ph-0107382-2-8-1': 'Lai et al. (2001) have argued that the symmetric morphology of the X-ray emission about the jet direction suggests strongly that the jet is along the spin axis of the pulsar.', 'astro-ph-0107382-2-8-2': 'As corroborating this interpretation, they cite polarization observations using data from Deshpande et al. (1999) which have been corrected for Faraday rotation both in the interstellar medium and in the ionosphere.', 'astro-ph-0107382-2-8-3': 'As the radio P.A. is at right angles to the direction of the X-ray jet, Lai et al. (2001) conclude that the polarization mode dominant in the Vela pulsar is one where the electric field in radio emission is orthogonal to the magnetospheric field.', 'astro-ph-0107382-2-9-0': 'Further support for this picture comes from recent radio observations of the region that have revealed a double lobe structure, with well separated lobes of comparable intensity (Dodson et al. 2001; private communication).', 'astro-ph-0107382-2-9-1': "These radio lobes are symmetrically placed on either side of the X-ray jet with their diffuse inner edges very close to the boundaries of the X-ray nebula, and independently suggest that the projection of the star's spin axis must match the observed direction of the jet in the sky plane.", 'astro-ph-0107382-2-10-0': "From what has been discussed above, it must be concluded that the electric vector of the Vela pulsar's radiation is perpendicular to the plane containing the magnetic field-line, and not parallel to it as one has generally assumed.", 'astro-ph-0107382-2-10-1': 'This was also noted by Helfand et al. (2001) although their major concern was with the morphology of the X-ray nebula and its interpretation.', 'astro-ph-0107382-2-10-2': 'We shall also discuss the X-ray nebula shortly, but first the implications of the polarization P.A.', 'astro-ph-0107382-2-11-0': '# IMPLICATIONS FOR THE EMISSION/AMPLIFICATION MECHANISM', 'astro-ph-0107382-2-12-0': 'There is no question that the charged particles in the polar cap regions of any pulsar will be constrained to move essentially along the field lines as already mentioned earlier.', 'astro-ph-0107382-2-12-1': 'There is also no question that there will be radiation from these relativistic particles due to the acceleration associated with the curvature of the field lines, and that the polarization of this radiation will be linear and parallel to the projection of the field lines, assumed planar for the moment.', 'astro-ph-0107382-2-12-2': 'But the brightness temperature of such radiation cannot exceed the kinetic temperature of the electrons (and positrons), which for even extreme values of the magnetic field and spin period are unlikely to be within orders of magnitude of the inferred brightness temperatures, as has been well known since the earliest observations.', 'astro-ph-0107382-2-12-3': 'The absolute need for maser-like amplification, whatever the mechanism of the input radiation, has thus always been recognised, and has motivated numerous attempts over the decades to propose models for the radiation mechanism of pulsars.', 'astro-ph-0107382-2-13-0': 'As noted at the beginning, one of the striking characteristics of pulsar radiation, which must be accounted for in any theoretical model, is its polarization behaviour.', 'astro-ph-0107382-2-13-1': 'Another, as just seen, is the extremely high brightness temperature.', 'astro-ph-0107382-2-13-2': 'We find it remarkable that both these characteristics seem to be well accounted for in the model put forward by Luo Melrose (1995).', 'astro-ph-0107382-2-13-3': 'They assume that the input signal is curvature radiation, which as discussed in detail above, seems eminently reasonable to us.', 'astro-ph-0107382-2-13-4': 'The amplification process requires a certain non-planarity of the field lines, for which there has long been evidence from many different lines of investigation on pulsars (Radhakrishnan 1992) The surprising aspect is that while the spontaneous curvature radiation is polarized parallel to the field lines, the amplified output is perpendicular to it, in agreement with the observations discussed in the last section.', 'astro-ph-0107382-2-13-5': 'It would appear therefore that the "normal" polarization mode is really "orthogonal" (to the field lines) for reasons associated with the physics of the amplification process, without which pulsars would not be detectable.', 'astro-ph-0107382-2-14-0': 'It should be pointed out that the above is not a violation of the fundamental requirement that in any amplifier the stimulated emission should be indistinguishable from the stimulating input signal.', 'astro-ph-0107382-2-14-1': 'Because of the torsion in the field geometry, the spontaneous emission has a small component in the direction perpendicular to the field in the amplifying region, and it is this component that, according to Luo Melrose (1995), is preferentially amplified and dominates the output.', 'astro-ph-0107382-2-15-0': 'The pattern of switching to orthogonal modes of polarization varies from pulsar to pulsar and can happen in different parts of the pulse profile for different pulsars.', 'astro-ph-0107382-2-15-1': 'For Vela, the existance of both modes was noticed as early as 1983 by Krishnamohan Downs.', 'astro-ph-0107382-2-15-2': 'Although the radiation in the other mode is on the average much weaker, it is important to appreciate that detectable radiation in any mode, in any pulsar, always corresponds to brightness temperatures that require enormous amplification.', 'astro-ph-0107382-2-15-3': 'We are thus forced to conclude that there must be more than one mode of amplification if the input signal is the spontaneous "curvature radiation", as is manifest by the "S" curve outputs of the amplifier, in whichever mode it is operating.', 'astro-ph-0107382-2-15-4': 'This implies that there must be conditions when the parallel mode develops more negative absorption than the perpendicular one favoured by Luo Melrose, and results in a polarization flip.', 'astro-ph-0107382-2-15-5': 'The probability of this happening could be influenced by the particular field distortions present over the longitude range in question.', 'astro-ph-0107382-2-15-6': 'But the fact that either mode can occur is very reminiscent of maser processes in general, where different allowable modes typically compete with each other resulting in one of them rapidly taking over all of the available power.', 'astro-ph-0107382-2-16-0': 'The observation in many pulsars, of elliptically polarized radiation of detectable strength, is further evidence of the existence of amplification in both modes, but now simultaneously with some phase difference and different gains.', 'astro-ph-0107382-2-16-1': 'A major obstacle in the understanding of pulsar polarization until now has been the difficulty of seeing how electric fields could be generated perpendicular to the ultrastrong magnetic field lines, only along which the charges were constrained to move.', 'astro-ph-0107382-2-16-2': 'The mechanism of Luo Melrose (1995) - thanks to torsion - allows and predicts radiation perpendicular to the field lines, and now reduces the explanation of any type of polarization in pulsars to a matter of detail, as opposed to a difficulty of principle.', 'astro-ph-0107382-2-17-0': '# THE X-RAY ARCS: TRACERS OF THE RADIATION BEAMS?', 'astro-ph-0107382-2-18-0': 'We turn now to the spectacular X-ray image provided by Chandra with two remarkably symmetrical arcs bisected by the jet like feature mentioned earlier.', 'astro-ph-0107382-2-18-1': 'Helfand et al. (2001) have put forward a detailed model where they "assume that the two arc-like features lie along circular rings highlighting shocks in which the energy of an outflowing equatorial wind is dissipated to become the source of synchrotron emission for the compact nebula" and attribute the incompleteness of the rings to preferential Doppler boosting of the emission in the forward direction.', 'astro-ph-0107382-2-18-2': 'They also "assume that the two rings straddle the equator symmetrically and suppose that the deficit of emission exactly in the equatorial plane is related to the fact that this is where the direction of a toroidally wrapped magnetic field changes sign i.e. the field may vanish there".', 'astro-ph-0107382-2-18-3': 'They go on to derive the half opening angle of the wind [MATH] as 23[MATH].3, and the radius of the shock [MATH] as a/dCos[MATH] cm for d = 250 pc.', 'astro-ph-0107382-2-19-0': 'We would like to propose a somewhat different model for the X-ray arcs, starting from the magnetic pole model for pulsar radiation discussed at length earlier, and that is invariably miscalled the "rotating vector model".', 'astro-ph-0107382-2-19-1': 'In that model, the radiation (and also its amplification as just seen) are produced by highly relativistic particles streaming out along the open field lines from both magnetic poles.', 'astro-ph-0107382-2-19-2': 'We now examine the X-ray data for the possibility that the two arcs reflect the traces of these two particle beams as they encounter the "walls" surrounding the central cavity created by the pulsar.', 'astro-ph-0107382-2-19-3': 'Such a cavity was elaborated in the classic paper by Rees Gunn (1974) for the Crab, and has since formed a part of most, if not all, subsequent discussions and models of pulsar created nebulae.', 'astro-ph-0107382-2-20-0': 'We assume that the particles leave the weakening field lines at some point well before sweep-back effects set in close to the light cylinder, and proceed "ballistically" outwards.', 'astro-ph-0107382-2-20-1': 'If this picture is valid, one of the two arcs should pass close to our sight line to the pulsar, as indeed it does.', 'astro-ph-0107382-2-20-2': 'To assess this further, we have modelled the arcs as the near-side portions of two rings (seen in projection) traced by sweeps of the (magnetic) polar cones.', 'astro-ph-0107382-2-20-3': 'The rotating magnetic-axis vector is as described in Deshpande et al. (1999).', 'astro-ph-0107382-2-20-4': 'The model parameters are the pair of radial distances ([MATH] as measured from the star location and expressed in arcseconds) associated with the two ring traces, the inclination ([MATH]) of the magnetic axis of the star to its rotation axis, the angle of closest approach (the impact angle [MATH]) of the magnetic axis to our sight-line, and the position angle (PAo) of the rotation axis projection on the sky-plane.', 'astro-ph-0107382-2-20-5': 'The angle ([MATH]) between the rotation axis and our sight-line is simply [MATH].', 'astro-ph-0107382-2-20-6': 'Desired consistency with radio polarization observations would allow only certain combinations of the viewing geometry; that is [MATH] should be equal to the steepest sweep rate ([MATH]) of the polarization position angle with respect to the rotational longitude.', 'astro-ph-0107382-2-20-7': 'When we constrain the [MATH] combinations using [MATH] of -9 degree/degree (as listed by Lyne Manchester 1988), the best fit PA[MATH] is found to be 129 degrees (measured from North through East), the radial distances [MATH] are unequal (about 22 29 arcsec for the near and the opposite polar cones respectively) and [MATH] 71 degrees ( [MATH]-6 degrees correspondingly).', 'astro-ph-0107382-2-20-8': 'Note that this implies a value of [MATH] of about 65 degrees, significantly larger than the 53 degrees estimated by Helfand et al. (2001) based on their model of an equatorial torus.', 'astro-ph-0107382-2-20-9': 'The above model is illustrated in Fig. 1.', 'astro-ph-0107382-2-21-0': 'The changing direction of the magnetic axis viewed in projection on the sky plane as the star rotates is described in exactly the same way as that for the position angle of the radio polarization (Deshpande et al. 1999) whatever their relative difference.', 'astro-ph-0107382-2-21-1': 'The proposed association of the arcs with the traces of the polar emission beams thus provides a new and independent means to probe the viewing geometry.', 'astro-ph-0107382-2-21-2': 'They can sample (as they do in the present case) a much larger fraction of the rotation cycle and can provide additional constraints on the viewing geometry.', 'astro-ph-0107382-2-21-3': 'One crucial such constraint becomes available via the trace of the polar emission from the other pole that is generally not available unless interpulses are observable.', 'astro-ph-0107382-2-21-4': 'Also, interestingly, the "sign" of the impact angle [MATH] would become readily apparent from the beam traces even if they do not sample large fractions of the rotation cycle, and without needing to know the sense of rotation of the star.', 'astro-ph-0107382-2-22-0': 'A simple calculation shows that the visible extent of the arcs is consistent with an X-radiation spread confined to about 70 degrees around the respective directions to which the magnetic axis points as the star rotates.', 'astro-ph-0107382-2-22-1': 'We consider this as explained simply in terms of the initial dispersion of the motions as the particle beam ploughs into the region of compressed toroidal field - the shocked region as usually described - and the consequent spread of the pitch angle distribution.', 'astro-ph-0107382-2-22-2': 'By just assuming that we will see radiation as long as there is some component of the motion towards our line of sight (i.e. [MATH]) our simulation reproduces remarkably well the Chandra observations of the arcs.', 'astro-ph-0107382-2-23-0': '# THE "AXIAL" JET !', 'astro-ph-0107382-2-24-0': 'We now turn to the jet like feature whose symmetric bisection of the arcs discussed above has prompted immediate identification with its rotation axis.', 'astro-ph-0107382-2-24-1': 'The fact that the projection on the sky-plane of the rotational axis as derived from the polarization data (with a 90[MATH] shift) is also in agreement has been interpreted as further and strong evidence of the above supposition.', 'astro-ph-0107382-2-24-2': 'If the jet originates from the pulsar magnetosphere, as seems likely, it is most natural to associate the jet axis with the pulsar spin axis."', 'astro-ph-0107382-2-24-3': 'The alignment of the jet with the observed proper motion for Vela seems also to have prompted an association with, and raised hopes of an explanation for the space velocity of the neutron star.', 'astro-ph-0107382-2-24-4': 'The fact that a similar jet was already seen in the Crab, also aligned with the derived P.A. of the spin axis, and its proper motion, heralds an incipient industry as more X-ray images of radio pulsar nebulae become available.', 'astro-ph-0107382-2-25-0': 'A radio pulsar is not an accreting object like an x-ray pulsar, a black hole binary, or an AGN.', 'astro-ph-0107382-2-25-1': 'In our view, it is not only unlikely, but unphysical to expect an actual jet along the rotation axis of a radio pulsar.', 'astro-ph-0107382-2-25-2': "There is an immense amount we don't know about the magnetospheres of radio pulsars, but one thing we have known for over three decades is that the particle emission is along the magnetic axis (Goldreich Julian 1969), that is often at large angles to the rotation axis.", 'astro-ph-0107382-2-25-3': 'The only connection that there can possibly be between the X-ray jet and the rotation axis is one of projection on the sky-plane as we shall argue below.', 'astro-ph-0107382-2-26-0': 'In our picture of the relativistic particle beams leaving the magnetic poles and proceeding ballistically outwards to the cavity walls, we can ask what if any deviations they are likely to suffer.', 'astro-ph-0107382-2-26-1': 'The only field along their trajectory through the cavity is the toroidal field (that they are carrying out) and that later gets compressed and strengthened at the wall.', 'astro-ph-0107382-2-26-2': 'This toroidal field will, over most or all of the trajectory, be perpendicular to the path of the particles, and, of course, the rotation axis.', 'astro-ph-0107382-2-26-3': 'Depending on the spread of their energies, it should not be surprising if a small fraction of the particles acquired a spread of velocities in the latitudinal direction making their synchrotron radiation visible to an observer when their motion is tangential to the sight line.', 'astro-ph-0107382-2-27-0': 'One should, as a consequence, expect to receive radiation from these particles when the projection of the magnetic axis coincides with that of the rotation axis, precisely as in the case of the radio pulse, but now over a larger range of angles in latitude.', 'astro-ph-0107382-2-27-1': 'Wherever the observer, the apparent jet will appear along the minor axis of the projected ellipses of the arcs but the extent over which it is visible will depend on the spread of particle velocities, and the angles the beams from the two poles make to the line of sight.', 'astro-ph-0107382-2-27-2': 'These particles must also reach the cavity walls and will then create a diffuse glow around the arc regions but with a greater spread, exactly as seen in the Chandra image.', 'astro-ph-0107382-2-27-3': 'We have assessed the spread of this weak fan beam from the size of the diffuse glow, and from the poor or non-visibility of the corresponding radiation from the beam of the other magnetic pole.', 'astro-ph-0107382-2-27-4': 'We estimate a spread of about [MATH] from the observed extent of the jet and have used this value in our simulation (fig. 2) of the jet and the associated diffuse components.', 'astro-ph-0107382-2-28-0': 'If this interpretation of the "jet" is correct, it has a major implication for the picture of particle flow from the pulsar to the nebula.', 'astro-ph-0107382-2-28-1': 'In section 4 when attempting to explain the formation of the X-ray arcs, we made the assumption that the particle beam separated from the polar field lines well within the light cylinder and before the effect of any sweep-back.', 'astro-ph-0107382-2-28-2': 'The justification for this assumption is now seen as the absence of any misalignment between the apparent jet (defined by the trajectory of particles radiating towards us) and the projected rotation axis, the very observation that prompted the physical misinterpretation of the jet referred to earlier.', 'astro-ph-0107382-2-29-0': 'The total picture that emerges is as follows.', 'astro-ph-0107382-2-29-1': 'There is a cavity of radius of order [MATH] cms, which we can presume was created by the dipole radiation originally of much higher frequency than the present 11 Hz.', 'astro-ph-0107382-2-29-2': 'Inside is a double cone of half angle [MATH], along which there is a relativistic particle flow in straight lines with a drift time of [MATH]0.1 year before they encounter the "shock".', 'astro-ph-0107382-2-29-3': 'In addition, there is a low density of particles separating from the cones but whose trajectories are in planes containing the rotation axis.', 'astro-ph-0107382-2-29-4': 'Any observer will see radiation only from those particles in the plane containing the observer and the rotation axis and we predict that this radiation should be highly linearly polarized with the electric vector parallel to the rotation axis.', 'astro-ph-0107382-2-30-0': '# DISCUSSION', 'astro-ph-0107382-2-31-0': 'In the long march towards the elucidation of the mysterious ways of pulsars, now numbering around a thousand, a few special ones have taught us more than most of the rest put together.', 'astro-ph-0107382-2-31-1': 'The Vela Pulsar is one such, and played an important role in several ways within months of its discovery in 1968.', 'astro-ph-0107382-2-31-2': 'The superb capabilities of the Chandra telescope have put this pulsar in the limelight again by providing a spectacular image in X-rays of the surrounding nebula.', 'astro-ph-0107382-2-31-3': 'Its form and proportions, reminiscent of pre-Columbian pectoral ornaments are loaded richly with information about many aspects.', 'astro-ph-0107382-2-31-4': 'To begin with, the symmetry of the nebula has provided compelling evidence for the identification of the P.A. of the rotational axis of the pulsar.', 'astro-ph-0107382-2-31-5': 'The near precise orthogonality of this P.A. to that inferred with certain assumptions from polarization measurements, has had important implications.', 'astro-ph-0107382-2-31-6': 'We support the finding of Lai et al. (2001) and Helfand et al. (2001) that the dominant polarization mode in the Vela Pulsar has the electric field orthogonal to the magnetospheric field.', 'astro-ph-0107382-2-31-7': 'We find this is in accordance with the mechanism of Luo Melrose (1995) that appears to explain three important characteristics of the radio radiation.', 'astro-ph-0107382-2-31-8': 'They are the high brightness temperatures, the dominant polarization mode, and the observed sweep of the P.A. across the pulse, the last clearly identifying the input signal to the maser as curvature radiation from the field lines in the polar neighbourhood.', 'astro-ph-0107382-2-31-9': 'Since the first and third characteristics also apply to the parallel polarization mode to which the radiation occasionally flips, as seen in many pulsars, we propose that the same amplification mechanism must also be operative at times in the other mode.', 'astro-ph-0107382-2-31-10': 'If the Luo Melrose mechanism can in fact operate in both modes, the explanation of any type of elliptic polarization becomes a matter of detail, a topic which we shall address later.', 'astro-ph-0107382-2-32-0': 'Noting the clear separation of the X-ray nebular emission into two elliptic arcs symmetrically located with respect to the inferred rotation axis, we propose that they are the traces of the two particle-beams from the magnetic poles on the walls of the cavity [MATH] the shocked region.', 'astro-ph-0107382-2-32-1': 'We explain the visible extent of the arcs as arising simply from the spread of the pitch angles at the shock front.', 'astro-ph-0107382-2-32-2': 'We also explain the alignment with the rotational axis of the jet like feature bisecting the arcs as simply a projection effect on the sky plane.', 'astro-ph-0107382-2-32-3': 'We attribute its visibility to a latitudinal spread of the velocities of a small fraction of the particles in the beam, an explanation strongly supported by the presence of diffuse emission around the arcs.', 'astro-ph-0107382-2-32-4': 'We present a simulation of the expected X-radiation around the pulsar based on the above model for the arcs, the jet, and the diffuse glow around both (see fig. 3), and find gratifying agreement with the Chandra observations.', 'astro-ph-0107382-2-32-5': 'We predict that the polarization of the jet feature will be linear and parallel to it, and claim that there can be no physical jet along the rotational axis of the pulsar.', 'astro-ph-0107382-2-33-0': "Note that the 'jet' is really an apparent one.", 'astro-ph-0107382-2-33-1': 'There are at least 3[MATH] input bunches of particles along any line from the pulsar to the cavity wall, and even the slightest dispersion in velocity would smooth this out to a uniform flow at a very short distance from the pulsar.', 'astro-ph-0107382-2-33-2': 'Even in the absence of velocity dispersion, the finite gamma of the bunch combined with the distance over which it is radiating will result in smearing.', 'astro-ph-0107382-2-33-3': 'The radiation will thus appear continuous and be a part of the unpulsed fraction in X-rays, the fraction depending on the extent of the jet/nebula from which radiation is collected.', 'astro-ph-0107382-2-34-0': 'In our modelling of the pair of arcs as traces of the two polar beams on a cavity wall (e.g. as illustrated in Fig 1), we find that an assumption of equatorial symmetry does not fit the observations well.', 'astro-ph-0107382-2-34-1': 'As already mentioned in an earlier section, the best fit [MATH] is significantly different from [MATH] (i.e. [MATH]1.35).', 'astro-ph-0107382-2-34-2': 'Note that, in our model, [MATH] represent the implied distances to the "wall" from the star along the \'rotating vectors\' associated with the near and the opposite poles respectively.', 'astro-ph-0107382-2-34-3': "The arcs provide us important information including about the otherwise 'unseen' pole.", 'astro-ph-0107382-2-34-4': 'Significantly improved fits are obtained when unequal values of [MATH] (the half angles of the polar cones associated with the two poles) are allowed in the model.', 'astro-ph-0107382-2-34-5': 'The corresponding [MATH],[MATH] are unequal again.', 'astro-ph-0107382-2-34-6': 'Even better fits are obtained if [MATH] is not constrained by the radio observations.', 'astro-ph-0107382-2-34-7': 'But interestingly, the implied value of [MATH]), the angle between the rotation axis and our line of sight, is about the same as in the other cases.', 'astro-ph-0107382-2-34-8': "Further detailed modelling, than has been possible presently, may provide better estimates of the above parameters and clues about the size as well as the shape of the 'cavity'.", 'astro-ph-0107382-2-35-0': 'Inequality between [MATH] has implications, particularly for certain acceleration mechanisms for the origin of pulsar velocities.', 'astro-ph-0107382-2-35-1': "For example, the 'rocket' mechanism of Harrison Tademaru (1975) does necessarily require such an inequality amounting to a tilted and offset dipole.", 'astro-ph-0107382-2-36-0': 'Other possible/plausible implications are the following.', 'astro-ph-0107382-2-37-0': "i) If the dominant radio polarization mode is indeed the 'orthogonal' mode, then any analysis based on the assumption that the observed central polarization P.A. is the same as the P.A. of the rotation axis needs to be reviewed.", 'astro-ph-0107382-2-37-1': 'This applies, for example, to the comparisons of the proper motion directions of pulsars with the orientations of their rotation axis.', 'astro-ph-0107382-2-37-2': 'In the work of Deshpande et al. (1999), they allowed for mode ambiguity in cases where emission of both modes is observed, and have assessed the distribution of the proper motion direction (relative to the rotation axis P.A.) as shown in their Fig 1b.', 'astro-ph-0107382-2-37-3': 'We argue that the required revision will amount to simply replacing the relative angle by its complement.', 'astro-ph-0107382-2-37-4': 'Since no particular preference was found for any relative angle value, the conclusions of Deshpande et al. remain unaltered.', 'astro-ph-0107382-2-38-0': 'ii) Given the similarities observed between the morphologies of the surrounding nebulae as well as other properties of the Vela Crab pulsars, it would not surprise us if a similar arc structure is revealed around the Crab pulsar by observations with improved spatial resolution.', 'astro-ph-0107382-2-39-0': 'We would like to thank Don Melrose, Qinghuan Luo and Mark Walker for many useful discussions, and Richard Dodson for kindly providing information prior to publication.'}

[['astro-ph-0107382-1-13-0', 'astro-ph-0107382-2-13-0'], ['astro-ph-0107382-1-13-1', 'astro-ph-0107382-2-13-1'], ['astro-ph-0107382-1-13-3', 'astro-ph-0107382-2-13-3'], ['astro-ph-0107382-1-13-4', 'astro-ph-0107382-2-13-4'], ['astro-ph-0107382-1-13-5', 'astro-ph-0107382-2-13-5'], ['astro-ph-0107382-1-20-0', 'astro-ph-0107382-2-20-0'], ['astro-ph-0107382-1-20-1', 'astro-ph-0107382-2-20-1'], ['astro-ph-0107382-1-20-2', 'astro-ph-0107382-2-20-2'], ['astro-ph-0107382-1-20-5', 'astro-ph-0107382-2-20-5'], ['astro-ph-0107382-1-20-6', 'astro-ph-0107382-2-20-6'], ['astro-ph-0107382-1-20-7', 'astro-ph-0107382-2-20-7'], ['astro-ph-0107382-1-20-9', 'astro-ph-0107382-2-20-9'], ['astro-ph-0107382-1-38-0', 'astro-ph-0107382-2-37-0'], ['astro-ph-0107382-1-38-1', 'astro-ph-0107382-2-37-1'], ['astro-ph-0107382-1-38-3', 'astro-ph-0107382-2-37-3'], ['astro-ph-0107382-1-7-0', 'astro-ph-0107382-2-7-0'], ['astro-ph-0107382-1-9-1', 'astro-ph-0107382-2-9-1'], ['astro-ph-0107382-1-31-0', 'astro-ph-0107382-2-31-0'], ['astro-ph-0107382-1-31-1', 'astro-ph-0107382-2-31-1'], ['astro-ph-0107382-1-31-2', 'astro-ph-0107382-2-31-2'], ['astro-ph-0107382-1-31-3', 'astro-ph-0107382-2-31-3'], ['astro-ph-0107382-1-31-6', 'astro-ph-0107382-2-31-5'], ['astro-ph-0107382-1-31-8', 'astro-ph-0107382-2-31-7'], ['astro-ph-0107382-1-31-9', 'astro-ph-0107382-2-31-8'], ['astro-ph-0107382-1-31-10', 'astro-ph-0107382-2-31-9'], ['astro-ph-0107382-1-31-11', 'astro-ph-0107382-2-31-10'], ['astro-ph-0107382-1-22-0', 'astro-ph-0107382-2-22-0'], ['astro-ph-0107382-1-22-1', 'astro-ph-0107382-2-22-1'], ['astro-ph-0107382-1-22-2', 'astro-ph-0107382-2-22-2'], ['astro-ph-0107382-1-29-0', 'astro-ph-0107382-2-29-0'], ['astro-ph-0107382-1-29-1', 'astro-ph-0107382-2-29-1'], ['astro-ph-0107382-1-29-2', 'astro-ph-0107382-2-29-2'], ['astro-ph-0107382-1-29-3', 'astro-ph-0107382-2-29-3'], ['astro-ph-0107382-1-29-4', 'astro-ph-0107382-2-29-4'], ['astro-ph-0107382-1-35-0', 'astro-ph-0107382-2-34-0'], ['astro-ph-0107382-1-35-1', 'astro-ph-0107382-2-34-1'], ['astro-ph-0107382-1-35-2', 'astro-ph-0107382-2-34-2'], ['astro-ph-0107382-1-35-3', 'astro-ph-0107382-2-34-3'], ['astro-ph-0107382-1-35-4', 'astro-ph-0107382-2-34-4'], ['astro-ph-0107382-1-35-5', 'astro-ph-0107382-2-34-5'], ['astro-ph-0107382-1-35-6', 'astro-ph-0107382-2-34-6'], ['astro-ph-0107382-1-35-7', 'astro-ph-0107382-2-34-7'], ['astro-ph-0107382-1-35-8', 'astro-ph-0107382-2-34-8'], ['astro-ph-0107382-1-4-0', 'astro-ph-0107382-2-4-0'], ['astro-ph-0107382-1-4-1', 'astro-ph-0107382-2-4-1'], ['astro-ph-0107382-1-4-2', 'astro-ph-0107382-2-4-2'], ['astro-ph-0107382-1-4-3', 'astro-ph-0107382-2-4-3'], ['astro-ph-0107382-1-4-4', 'astro-ph-0107382-2-4-4'], ['astro-ph-0107382-1-4-5', 'astro-ph-0107382-2-4-5'], ['astro-ph-0107382-1-14-0', 'astro-ph-0107382-2-14-0'], ['astro-ph-0107382-1-14-1', 'astro-ph-0107382-2-14-1'], ['astro-ph-0107382-1-36-0', 'astro-ph-0107382-2-35-0'], ['astro-ph-0107382-1-36-1', 'astro-ph-0107382-2-35-1'], ['astro-ph-0107382-1-5-0', 'astro-ph-0107382-2-5-0'], ['astro-ph-0107382-1-5-3', 'astro-ph-0107382-2-5-3'], ['astro-ph-0107382-1-5-4', 'astro-ph-0107382-2-5-4'], ['astro-ph-0107382-1-24-0', 'astro-ph-0107382-2-24-0'], ['astro-ph-0107382-1-24-1', 'astro-ph-0107382-2-24-1'], ['astro-ph-0107382-1-24-2', 'astro-ph-0107382-2-24-2'], ['astro-ph-0107382-1-24-3', 'astro-ph-0107382-2-24-3'], ['astro-ph-0107382-1-24-4', 'astro-ph-0107382-2-24-4'], ['astro-ph-0107382-1-25-0', 'astro-ph-0107382-2-25-0'], ['astro-ph-0107382-1-25-1', 'astro-ph-0107382-2-25-1'], ['astro-ph-0107382-1-25-2', 'astro-ph-0107382-2-25-2'], ['astro-ph-0107382-1-25-3', 'astro-ph-0107382-2-25-3'], ['astro-ph-0107382-1-18-0', 'astro-ph-0107382-2-18-0'], ['astro-ph-0107382-1-18-2', 'astro-ph-0107382-2-18-2'], ['astro-ph-0107382-1-18-3', 'astro-ph-0107382-2-18-3'], ['astro-ph-0107382-1-3-0', 'astro-ph-0107382-2-3-0'], ['astro-ph-0107382-1-3-1', 'astro-ph-0107382-2-3-1'], ['astro-ph-0107382-1-3-2', 'astro-ph-0107382-2-3-2'], ['astro-ph-0107382-1-3-3', 'astro-ph-0107382-2-3-3'], ['astro-ph-0107382-1-3-4', 'astro-ph-0107382-2-3-4'], ['astro-ph-0107382-1-15-0', 'astro-ph-0107382-2-15-0'], ['astro-ph-0107382-1-15-1', 'astro-ph-0107382-2-15-1'], ['astro-ph-0107382-1-15-2', 'astro-ph-0107382-2-15-2'], ['astro-ph-0107382-1-15-3', 'astro-ph-0107382-2-15-3'], ['astro-ph-0107382-1-15-4', 'astro-ph-0107382-2-15-4'], ['astro-ph-0107382-1-15-5', 'astro-ph-0107382-2-15-5'], ['astro-ph-0107382-1-15-6', 'astro-ph-0107382-2-15-6'], ['astro-ph-0107382-1-26-0', 'astro-ph-0107382-2-26-0'], ['astro-ph-0107382-1-26-1', 'astro-ph-0107382-2-26-1'], ['astro-ph-0107382-1-26-2', 'astro-ph-0107382-2-26-2'], ['astro-ph-0107382-1-26-3', 'astro-ph-0107382-2-26-3'], ['astro-ph-0107382-1-28-0', 'astro-ph-0107382-2-28-0'], ['astro-ph-0107382-1-28-1', 'astro-ph-0107382-2-28-1'], ['astro-ph-0107382-1-27-0', 'astro-ph-0107382-2-27-0'], ['astro-ph-0107382-1-27-1', 'astro-ph-0107382-2-27-1'], ['astro-ph-0107382-1-27-2', 'astro-ph-0107382-2-27-2'], ['astro-ph-0107382-1-27-3', 'astro-ph-0107382-2-27-3'], ['astro-ph-0107382-1-27-4', 'astro-ph-0107382-2-27-4'], ['astro-ph-0107382-1-12-0', 'astro-ph-0107382-2-12-0'], ['astro-ph-0107382-1-12-1', 'astro-ph-0107382-2-12-1'], ['astro-ph-0107382-1-12-2', 'astro-ph-0107382-2-12-2'], ['astro-ph-0107382-1-12-3', 'astro-ph-0107382-2-12-3'], ['astro-ph-0107382-1-21-1', 'astro-ph-0107382-2-21-1'], ['astro-ph-0107382-1-21-2', 'astro-ph-0107382-2-21-2'], ['astro-ph-0107382-1-21-3', 'astro-ph-0107382-2-21-3'], ['astro-ph-0107382-1-21-4', 'astro-ph-0107382-2-21-4'], ['astro-ph-0107382-1-32-0', 'astro-ph-0107382-2-32-0'], ['astro-ph-0107382-1-32-1', 'astro-ph-0107382-2-32-1'], ['astro-ph-0107382-1-32-2', 'astro-ph-0107382-2-32-2'], ['astro-ph-0107382-1-32-3', 'astro-ph-0107382-2-32-3'], ['astro-ph-0107382-1-32-4', 'astro-ph-0107382-2-32-4'], ['astro-ph-0107382-1-32-5', 'astro-ph-0107382-2-32-5'], ['astro-ph-0107382-1-0-1', 'astro-ph-0107382-2-0-1'], ['astro-ph-0107382-1-0-3', 'astro-ph-0107382-2-0-3'], ['astro-ph-0107382-1-0-4', 'astro-ph-0107382-2-0-4'], ['astro-ph-0107382-1-19-0', 'astro-ph-0107382-2-19-0'], ['astro-ph-0107382-1-19-1', 'astro-ph-0107382-2-19-1'], ['astro-ph-0107382-1-19-2', 'astro-ph-0107382-2-19-2'], ['astro-ph-0107382-1-16-0', 'astro-ph-0107382-2-16-0'], ['astro-ph-0107382-1-16-1', 'astro-ph-0107382-2-16-1'], ['astro-ph-0107382-1-16-2', 'astro-ph-0107382-2-16-2'], ['astro-ph-0107382-1-10-0', 'astro-ph-0107382-2-10-0'], ['astro-ph-0107382-1-10-2', 'astro-ph-0107382-2-10-2'], ['astro-ph-0107382-1-13-2', 'astro-ph-0107382-2-13-2'], ['astro-ph-0107382-1-20-3', 'astro-ph-0107382-2-20-3'], ['astro-ph-0107382-1-20-4', 'astro-ph-0107382-2-20-4'], ['astro-ph-0107382-1-20-8', 'astro-ph-0107382-2-20-8'], ['astro-ph-0107382-1-40-0', 'astro-ph-0107382-2-38-0'], ['astro-ph-0107382-1-38-2', 'astro-ph-0107382-2-37-2'], ['astro-ph-0107382-1-38-4', 'astro-ph-0107382-2-37-4'], ['astro-ph-0107382-1-9-0', 'astro-ph-0107382-2-9-0'], ['astro-ph-0107382-1-31-5', 'astro-ph-0107382-2-31-4'], ['astro-ph-0107382-1-31-7', 'astro-ph-0107382-2-31-6'], ['astro-ph-0107382-1-8-0', 'astro-ph-0107382-2-8-0'], ['astro-ph-0107382-1-8-1', 'astro-ph-0107382-2-8-1'], ['astro-ph-0107382-1-8-2', 'astro-ph-0107382-2-8-2'], ['astro-ph-0107382-1-8-3', 'astro-ph-0107382-2-8-3'], ['astro-ph-0107382-1-5-2', 'astro-ph-0107382-2-5-2'], ['astro-ph-0107382-1-18-1', 'astro-ph-0107382-2-18-1'], ['astro-ph-0107382-1-34-0', 'astro-ph-0107382-2-33-1'], ['astro-ph-0107382-1-21-0', 'astro-ph-0107382-2-21-0'], ['astro-ph-0107382-1-0-0', 'astro-ph-0107382-2-0-0'], ['astro-ph-0107382-1-0-2', 'astro-ph-0107382-2-0-2'], ['astro-ph-0107382-1-19-3', 'astro-ph-0107382-2-19-3'], ['astro-ph-0107382-1-41-0', 'astro-ph-0107382-2-39-0'], ['astro-ph-0107382-1-10-1', 'astro-ph-0107382-2-10-1'], ['astro-ph-0107382-1-5-1', 'astro-ph-0107382-2-5-1'], ['astro-ph-0107382-1-34-3', 'astro-ph-0107382-2-33-3']]

[['astro-ph-0107382-1-13-0', 'astro-ph-0107382-2-13-0'], ['astro-ph-0107382-1-13-1', 'astro-ph-0107382-2-13-1'], ['astro-ph-0107382-1-13-3', 'astro-ph-0107382-2-13-3'], ['astro-ph-0107382-1-13-4', 'astro-ph-0107382-2-13-4'], ['astro-ph-0107382-1-13-5', 'astro-ph-0107382-2-13-5'], ['astro-ph-0107382-1-20-0', 'astro-ph-0107382-2-20-0'], ['astro-ph-0107382-1-20-1', 'astro-ph-0107382-2-20-1'], ['astro-ph-0107382-1-20-2', 'astro-ph-0107382-2-20-2'], ['astro-ph-0107382-1-20-5', 'astro-ph-0107382-2-20-5'], ['astro-ph-0107382-1-20-6', 'astro-ph-0107382-2-20-6'], ['astro-ph-0107382-1-20-7', 'astro-ph-0107382-2-20-7'], ['astro-ph-0107382-1-20-9', 'astro-ph-0107382-2-20-9'], ['astro-ph-0107382-1-38-0', 'astro-ph-0107382-2-37-0'], ['astro-ph-0107382-1-38-1', 'astro-ph-0107382-2-37-1'], ['astro-ph-0107382-1-38-3', 'astro-ph-0107382-2-37-3'], ['astro-ph-0107382-1-7-0', 'astro-ph-0107382-2-7-0'], ['astro-ph-0107382-1-9-1', 'astro-ph-0107382-2-9-1'], ['astro-ph-0107382-1-31-0', 'astro-ph-0107382-2-31-0'], ['astro-ph-0107382-1-31-1', 'astro-ph-0107382-2-31-1'], ['astro-ph-0107382-1-31-2', 'astro-ph-0107382-2-31-2'], ['astro-ph-0107382-1-31-3', 'astro-ph-0107382-2-31-3'], ['astro-ph-0107382-1-31-6', 'astro-ph-0107382-2-31-5'], ['astro-ph-0107382-1-31-8', 'astro-ph-0107382-2-31-7'], ['astro-ph-0107382-1-31-9', 'astro-ph-0107382-2-31-8'], ['astro-ph-0107382-1-31-10', 'astro-ph-0107382-2-31-9'], ['astro-ph-0107382-1-31-11', 'astro-ph-0107382-2-31-10'], ['astro-ph-0107382-1-22-0', 'astro-ph-0107382-2-22-0'], ['astro-ph-0107382-1-22-1', 'astro-ph-0107382-2-22-1'], ['astro-ph-0107382-1-22-2', 'astro-ph-0107382-2-22-2'], ['astro-ph-0107382-1-29-0', 'astro-ph-0107382-2-29-0'], ['astro-ph-0107382-1-29-1', 'astro-ph-0107382-2-29-1'], ['astro-ph-0107382-1-29-2', 'astro-ph-0107382-2-29-2'], ['astro-ph-0107382-1-29-3', 'astro-ph-0107382-2-29-3'], ['astro-ph-0107382-1-29-4', 'astro-ph-0107382-2-29-4'], ['astro-ph-0107382-1-35-0', 'astro-ph-0107382-2-34-0'], ['astro-ph-0107382-1-35-1', 'astro-ph-0107382-2-34-1'], ['astro-ph-0107382-1-35-2', 'astro-ph-0107382-2-34-2'], ['astro-ph-0107382-1-35-3', 'astro-ph-0107382-2-34-3'], ['astro-ph-0107382-1-35-4', 'astro-ph-0107382-2-34-4'], ['astro-ph-0107382-1-35-5', 'astro-ph-0107382-2-34-5'], ['astro-ph-0107382-1-35-6', 'astro-ph-0107382-2-34-6'], ['astro-ph-0107382-1-35-7', 'astro-ph-0107382-2-34-7'], ['astro-ph-0107382-1-35-8', 'astro-ph-0107382-2-34-8'], ['astro-ph-0107382-1-4-0', 'astro-ph-0107382-2-4-0'], ['astro-ph-0107382-1-4-1', 'astro-ph-0107382-2-4-1'], ['astro-ph-0107382-1-4-2', 'astro-ph-0107382-2-4-2'], ['astro-ph-0107382-1-4-3', 'astro-ph-0107382-2-4-3'], ['astro-ph-0107382-1-4-4', 'astro-ph-0107382-2-4-4'], ['astro-ph-0107382-1-4-5', 'astro-ph-0107382-2-4-5'], ['astro-ph-0107382-1-14-0', 'astro-ph-0107382-2-14-0'], ['astro-ph-0107382-1-14-1', 'astro-ph-0107382-2-14-1'], ['astro-ph-0107382-1-36-0', 'astro-ph-0107382-2-35-0'], ['astro-ph-0107382-1-36-1', 'astro-ph-0107382-2-35-1'], ['astro-ph-0107382-1-5-0', 'astro-ph-0107382-2-5-0'], ['astro-ph-0107382-1-5-3', 'astro-ph-0107382-2-5-3'], ['astro-ph-0107382-1-5-4', 'astro-ph-0107382-2-5-4'], ['astro-ph-0107382-1-24-0', 'astro-ph-0107382-2-24-0'], ['astro-ph-0107382-1-24-1', 'astro-ph-0107382-2-24-1'], ['astro-ph-0107382-1-24-2', 'astro-ph-0107382-2-24-2'], ['astro-ph-0107382-1-24-3', 'astro-ph-0107382-2-24-3'], ['astro-ph-0107382-1-24-4', 'astro-ph-0107382-2-24-4'], ['astro-ph-0107382-1-25-0', 'astro-ph-0107382-2-25-0'], ['astro-ph-0107382-1-25-1', 'astro-ph-0107382-2-25-1'], ['astro-ph-0107382-1-25-2', 'astro-ph-0107382-2-25-2'], ['astro-ph-0107382-1-25-3', 'astro-ph-0107382-2-25-3'], ['astro-ph-0107382-1-18-0', 'astro-ph-0107382-2-18-0'], ['astro-ph-0107382-1-18-2', 'astro-ph-0107382-2-18-2'], ['astro-ph-0107382-1-18-3', 'astro-ph-0107382-2-18-3'], ['astro-ph-0107382-1-3-0', 'astro-ph-0107382-2-3-0'], ['astro-ph-0107382-1-3-1', 'astro-ph-0107382-2-3-1'], ['astro-ph-0107382-1-3-2', 'astro-ph-0107382-2-3-2'], ['astro-ph-0107382-1-3-3', 'astro-ph-0107382-2-3-3'], ['astro-ph-0107382-1-3-4', 'astro-ph-0107382-2-3-4'], ['astro-ph-0107382-1-15-0', 'astro-ph-0107382-2-15-0'], ['astro-ph-0107382-1-15-1', 'astro-ph-0107382-2-15-1'], ['astro-ph-0107382-1-15-2', 'astro-ph-0107382-2-15-2'], ['astro-ph-0107382-1-15-3', 'astro-ph-0107382-2-15-3'], ['astro-ph-0107382-1-15-4', 'astro-ph-0107382-2-15-4'], ['astro-ph-0107382-1-15-5', 'astro-ph-0107382-2-15-5'], ['astro-ph-0107382-1-15-6', 'astro-ph-0107382-2-15-6'], ['astro-ph-0107382-1-26-0', 'astro-ph-0107382-2-26-0'], ['astro-ph-0107382-1-26-1', 'astro-ph-0107382-2-26-1'], ['astro-ph-0107382-1-26-2', 'astro-ph-0107382-2-26-2'], ['astro-ph-0107382-1-26-3', 'astro-ph-0107382-2-26-3'], ['astro-ph-0107382-1-28-0', 'astro-ph-0107382-2-28-0'], ['astro-ph-0107382-1-28-1', 'astro-ph-0107382-2-28-1'], ['astro-ph-0107382-1-27-0', 'astro-ph-0107382-2-27-0'], ['astro-ph-0107382-1-27-1', 'astro-ph-0107382-2-27-1'], ['astro-ph-0107382-1-27-2', 'astro-ph-0107382-2-27-2'], ['astro-ph-0107382-1-27-3', 'astro-ph-0107382-2-27-3'], ['astro-ph-0107382-1-27-4', 'astro-ph-0107382-2-27-4'], ['astro-ph-0107382-1-12-0', 'astro-ph-0107382-2-12-0'], ['astro-ph-0107382-1-12-1', 'astro-ph-0107382-2-12-1'], ['astro-ph-0107382-1-12-2', 'astro-ph-0107382-2-12-2'], ['astro-ph-0107382-1-12-3', 'astro-ph-0107382-2-12-3'], ['astro-ph-0107382-1-21-1', 'astro-ph-0107382-2-21-1'], ['astro-ph-0107382-1-21-2', 'astro-ph-0107382-2-21-2'], ['astro-ph-0107382-1-21-3', 'astro-ph-0107382-2-21-3'], ['astro-ph-0107382-1-21-4', 'astro-ph-0107382-2-21-4'], ['astro-ph-0107382-1-32-0', 'astro-ph-0107382-2-32-0'], ['astro-ph-0107382-1-32-1', 'astro-ph-0107382-2-32-1'], ['astro-ph-0107382-1-32-2', 'astro-ph-0107382-2-32-2'], ['astro-ph-0107382-1-32-3', 'astro-ph-0107382-2-32-3'], ['astro-ph-0107382-1-32-4', 'astro-ph-0107382-2-32-4'], ['astro-ph-0107382-1-32-5', 'astro-ph-0107382-2-32-5'], ['astro-ph-0107382-1-0-1', 'astro-ph-0107382-2-0-1'], ['astro-ph-0107382-1-0-3', 'astro-ph-0107382-2-0-3'], ['astro-ph-0107382-1-0-4', 'astro-ph-0107382-2-0-4'], ['astro-ph-0107382-1-19-0', 'astro-ph-0107382-2-19-0'], ['astro-ph-0107382-1-19-1', 'astro-ph-0107382-2-19-1'], ['astro-ph-0107382-1-19-2', 'astro-ph-0107382-2-19-2'], ['astro-ph-0107382-1-16-0', 'astro-ph-0107382-2-16-0'], ['astro-ph-0107382-1-16-1', 'astro-ph-0107382-2-16-1'], ['astro-ph-0107382-1-16-2', 'astro-ph-0107382-2-16-2'], ['astro-ph-0107382-1-10-0', 'astro-ph-0107382-2-10-0'], ['astro-ph-0107382-1-10-2', 'astro-ph-0107382-2-10-2']]

[['astro-ph-0107382-1-13-2', 'astro-ph-0107382-2-13-2'], ['astro-ph-0107382-1-20-3', 'astro-ph-0107382-2-20-3'], ['astro-ph-0107382-1-20-4', 'astro-ph-0107382-2-20-4'], ['astro-ph-0107382-1-20-8', 'astro-ph-0107382-2-20-8'], ['astro-ph-0107382-1-40-0', 'astro-ph-0107382-2-38-0'], ['astro-ph-0107382-1-38-2', 'astro-ph-0107382-2-37-2'], ['astro-ph-0107382-1-38-4', 'astro-ph-0107382-2-37-4'], ['astro-ph-0107382-1-9-0', 'astro-ph-0107382-2-9-0'], ['astro-ph-0107382-1-31-5', 'astro-ph-0107382-2-31-4'], ['astro-ph-0107382-1-31-7', 'astro-ph-0107382-2-31-6'], ['astro-ph-0107382-1-8-0', 'astro-ph-0107382-2-8-0'], ['astro-ph-0107382-1-8-1', 'astro-ph-0107382-2-8-1'], ['astro-ph-0107382-1-8-2', 'astro-ph-0107382-2-8-2'], ['astro-ph-0107382-1-8-3', 'astro-ph-0107382-2-8-3'], ['astro-ph-0107382-1-5-2', 'astro-ph-0107382-2-5-2'], ['astro-ph-0107382-1-18-1', 'astro-ph-0107382-2-18-1'], ['astro-ph-0107382-1-34-0', 'astro-ph-0107382-2-33-1'], ['astro-ph-0107382-1-21-0', 'astro-ph-0107382-2-21-0'], ['astro-ph-0107382-1-0-0', 'astro-ph-0107382-2-0-0'], ['astro-ph-0107382-1-0-2', 'astro-ph-0107382-2-0-2'], ['astro-ph-0107382-1-19-3', 'astro-ph-0107382-2-19-3'], ['astro-ph-0107382-1-41-0', 'astro-ph-0107382-2-39-0'], ['astro-ph-0107382-1-10-1', 'astro-ph-0107382-2-10-1']]

[]

[['astro-ph-0107382-1-5-1', 'astro-ph-0107382-2-5-1'], ['astro-ph-0107382-1-34-3', 'astro-ph-0107382-2-33-3']]

[]

['astro-ph-0107382-1-1-0', 'astro-ph-0107382-1-37-0', 'astro-ph-0107382-2-1-0', 'astro-ph-0107382-2-36-0']

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

https://arxiv.org/abs/astro-ph/0107382

1502.04201

{'1502.04201-1-0-0': 'Given positive numbers [MATH] and [MATH] , the function [MATH] is exponentially convex function of [MATH] on the whole real axis.', '1502.04201-1-0-1': 'Three proofs of this result are presented.', '1502.04201-1-1-0': 'Keywords: Exponentially convex functions, BMV conjecture.', '1502.04201-1-2-0': '# The exponential convexity result', '1502.04201-1-3-0': 'A function [MATH] on [MATH] , [MATH] , is said to be exponentially convex if', '1502.04201-1-4-0': '[1.]', '1502.04201-1-4-1': 'For every nonnegative integer [MATH] , for every choice of real numbers [MATH] , [MATH] , and complex numbers [MATH] , [MATH] , the inequality holds [EQUATION] [2.]', '1502.04201-1-4-2': 'The function [MATH] is continuous on [MATH] .', '1502.04201-1-5-0': 'The class of exponentially convex functions was introduced by S.N.Bernstein, [CITATION], see 15 there.', '1502.04201-1-6-0': 'From [REF] it follows that the inequality [MATH] holds for every [MATH] .', '1502.04201-1-6-1': 'Thus the alternative takes place: If [MATH] is an exponentially convex function then either [MATH] , or [MATH] for every [MATH] .', '1502.04201-1-7-0': 'Properties of the class of exponentially convex functions.', '1502.04201-1-8-0': '[P1.]', '1502.04201-1-8-1': 'If [MATH] if an exponentially convex function and [MATH] is a nonnegative constant, then the function [MATH] is exponentially convex.', '1502.04201-1-8-2': '[P2.]', '1502.04201-1-8-3': 'If [MATH] and [MATH] are exponentially convex functions, then their sum [MATH] is exponentially convex.', '1502.04201-1-8-4': '[P3.]', '1502.04201-1-8-5': 'If [MATH] and [MATH] are exponentially convex functions, then their product [MATH] is exponentially convex.', '1502.04201-1-8-6': '[P4.]', '1502.04201-1-8-7': 'Let [MATH] be a sequence of exponentially convex functions.', '1502.04201-1-8-8': 'We assume that for each [MATH] there exists the limit [MATH] , and that [MATH] .', '1502.04201-1-8-9': 'Then the limiting function [MATH] is exponentially convex.', '1502.04201-1-9-0': 'From the functional equation for the exponential function it follows that for each real number [MATH] , for every choice of real numbers [MATH] , [MATH] and complex numbers [MATH] , [MATH] , the equality holds [EQUATION]', '1502.04201-1-9-1': 'The relation [REF] can be formulated as', '1502.04201-1-10-0': 'For each real [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-1-11-0': 'For [MATH] , the function [MATH] , which is called the hyperbolic cosine of [MATH] , is defined as [EQUATION]', '1502.04201-1-11-1': 'From Lemma [REF] and property P2 we obtain', '1502.04201-1-12-0': 'For each real [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-1-13-0': 'The following result is well known.', '1502.04201-1-13-1': '[The representation theorem]', '1502.04201-1-14-0': '[1.]', '1502.04201-1-14-1': 'Let [MATH] be a nonnegative measure on the real axis, and let the function [MATH] be a two-sided Laplace transform of the measure [MATH] : [EQUATION] for any [MATH] .', '1502.04201-1-14-2': 'Then the function [MATH] is exponentially convex.', '1502.04201-1-14-3': '[2. ]', '1502.04201-1-14-4': 'Let [MATH] be an exponentially convex function.', '1502.04201-1-14-5': 'Then this function [MATH] can be represented on [MATH] as a two-sided Laplace transform [REF] of a nonnegative measure [MATH] .', '1502.04201-1-14-6': '(In particular, the integral in the right hand side of [REF] is finite for any [MATH] .)', '1502.04201-1-14-7': 'The representing measure [MATH] is unique.', '1502.04201-1-15-0': 'The assertion 1 of the representation theorem is an evident consequence of Lemma [REF], of the properties P1,P2, P4, and of the definition of the integration.', '1502.04201-1-16-0': 'The proof of the assertion 2 can be found in [CITATION],Theorem 5.5.4, and in [CITATION],Theorem 21.', '1502.04201-1-17-0': "Of course, Lemma [REF] is a special case of the representation theorem which corresponds to the representing measure [MATH] , where [MATH] are Dirak's [MATH] -functions supported at the points [MATH] .", '1502.04201-1-18-0': 'The expression [EQUATION] is well defined for every complex numbers [MATH] .', '1502.04201-1-18-1': 'The function [MATH] is an entire function of complex variables [MATH] .', '1502.04201-1-18-2': 'For each fixed [MATH] and [MATH] , the function [MATH] , considered as a function of [MATH] , is an entire function of exponential type [MATH] .', '1502.04201-1-19-0': 'Proof.', '1502.04201-1-19-1': 'The function [MATH] is a superposition of the entire function [MATH] of variable [MATH] and the quadratic polynomial [MATH] .', '1502.04201-1-19-2': 'The assertion concerning the growth of this function is evident.', '1502.04201-1-19-3': '*[MATH]', '1502.04201-1-20-0': 'For each fixed [MATH] and [MATH] , the function [MATH] defined by [REF] is an exponentially convex function of variable [MATH] .', '1502.04201-1-21-0': 'In what follow we present three different proofs of Theorem [REF].', '1502.04201-1-21-1': 'The first and the second proofs are based on the representation theorem.', '1502.04201-1-21-2': 'We prove that the function [MATH] in [REF] takes positive values for [MATH] .', '1502.04201-1-21-3': 'In the first proof we calculate the function [MATH] explicitly expressing this function in terms of the modified Bessel function [MATH] .', '1502.04201-1-21-4': 'In the second proof, we prove the positivity of the function [MATH] using the reasoning by Herbert Stahl in [CITATION].', '1502.04201-1-21-5': '(We use a very simple special case of this reasoning.)', '1502.04201-1-21-6': 'The third proof is based on the Taylor expansion of the function [MATH] , [REF], with respect to parameter [MATH] .', '1502.04201-1-21-7': 'This proof does not use any integration in the complex plane.', '1502.04201-1-21-8': 'It based only on Lemma [REF] and on the properties P1 - P4 of the class of exponentially convex functions.', '1502.04201-1-21-9': 'As a by-product of this proof we obtain that all coefficient of this Taylor expansion are exponentially convex functions.', '1502.04201-1-21-10': 'However we can not conclude directly from this proof that the restriction of the representing measure on the open interval [MATH] is an absolutely continuous measure.', '1502.04201-1-22-0': 'For each fixed [MATH] , the function [MATH] defined by [REF] is representable in the form [EQUATION] where the function [MATH] possesses the properties', '1502.04201-1-23-0': '[1.]', '1502.04201-1-24-0': '[EQUATION] [2.]', '1502.04201-1-24-1': 'The function [MATH] is continuous with respect to [MATH] on the closed interval [MATH] , takes real values there, and is even.', '1502.04201-1-24-2': '[3.]', '1502.04201-1-24-3': 'The values of the function [MATH] at the end points [MATH] of the interval [MATH] are: [EQUATION]', '1502.04201-1-24-4': 'Proof.', '1502.04201-1-24-5': 'We introduce the function [EQUATION] of variables [MATH] .', '1502.04201-1-24-6': 'Considered as a function of [MATH] for fixed positive [MATH] and [MATH] , [MATH] is entire function of exponential type [MATH] .', '1502.04201-1-24-7': 'On the imaginary axis [MATH] takes the form [EQUATION]', '1502.04201-1-24-8': 'From [REF] it follows that the function [MATH] is a bounded and decaying on the imaginary axis: [MATH] , [MATH] .', '1502.04201-1-24-9': 'By the Wiener-Paley theorem, the function [MATH] is representable in the form [EQUATION] where the function [MATH] satisfies the condition [REF].', '1502.04201-1-24-10': 'The equality [REF] serves as a definition of the function [MATH] .', '1502.04201-1-24-11': 'So, this function is defined only for [MATH] .', '1502.04201-1-25-0': 'Since the function [MATH] is even with respect to [MATH] and real valued, its inverse Fourier transform [MATH] is even with respect to [MATH] and real valued.', '1502.04201-1-26-0': 'From [REF] we obtain that [EQUATION]', '1502.04201-1-26-1': 'Hence the function in the left hand side of [REF] is a Fourier transform of some function [MATH] which is square summable and continuous at every [MATH] .', '1502.04201-1-26-2': 'We remark that [EQUATION] where [MATH] is a constant function.', '1502.04201-1-26-3': 'Hence [EQUATION]', '1502.04201-1-26-4': 'Since [MATH] for [MATH] , also [MATH] .', '1502.04201-1-26-5': 'Thus, [REF] holds.', '1502.04201-1-26-6': '[MATH]', '1502.04201-1-27-0': '# Representation the function [MATH] by a contour integral.', '1502.04201-1-28-0': 'Let [MATH] be a segment of the imaginary axis: [EQUATION]', '1502.04201-1-28-1': 'The function [MATH] is a single value function of [MATH] in the complex plane slitted along the vertical segment [MATH] .', '1502.04201-1-28-2': 'We choose the branch of this function which takes positive values for large real [MATH] .', '1502.04201-1-29-0': 'The function [MATH] , which was defined by [REF], admits the integral representation [EQUATION] where [MATH] is an arbitrary counterclockwise oriented closed Jordan curve which contains the slit [MATH] inside.', '1502.04201-1-30-0': 'Proof.', '1502.04201-1-30-1': 'According the inversion formula for the Fourier transform, [EQUATION]', '1502.04201-1-30-2': 'We interpret the integral in the right hand side of [REF] as the integral along the the vertical straight line [MATH] : [EQUATION]', '1502.04201-1-30-3': 'Since the function [MATH] is bounded in each vertical strip [MATH] and tends to zero as [MATH] within this strip, the value of the integral in [REF] does not change if we integrate along any vertical line [MATH] , where [MATH] is an arbitrary real number: [EQUATION]', '1502.04201-1-30-4': 'Choosing [MATH] , we split the integral in [REF] into the sum [EQUATION] where [EQUATION]', '1502.04201-1-30-5': 'The function [MATH] is holomorphic in the halfplane [MATH] and admits the estimate [EQUATION] there, where [MATH] is a constant.', '1502.04201-1-30-6': 'Therefore [EQUATION]', '1502.04201-1-30-7': 'The function [MATH] is holomorphic in the slitted half plane [MATH] and admits the estimate [EQUATION] there.', '1502.04201-1-30-8': 'Therefore [EQUATION] where [MATH] is an arbitrary closed Jourdan curve which is oriented clockwise and contains the slit [MATH] in its interior.', '1502.04201-1-30-9': 'Since the function [MATH] is entire, [MATH] .', '1502.04201-1-30-10': 'So [EQUATION] where the integral in the right hand side is taken over the curve [MATH] which is oriented counterclockwise.', '1502.04201-1-30-11': 'Comparing [REF], [REF], [REF], and [REF], we obtain [REF].', '1502.04201-1-30-12': '[MATH]', '1502.04201-1-31-0': '# The first proof of Theorem [REF].', '1502.04201-1-32-0': 'The function [MATH] which was defined by [REF] admits the integral representation [EQUATION]', '1502.04201-1-32-1': 'Proof.', '1502.04201-1-32-2': 'We derive Lemma [REF] from Lemma [REF] showing that [EQUATION]', '1502.04201-1-33-0': 'The function [MATH] is holomorphic in the domain [MATH] and continuous up to boundary [MATH] of this domain.', '1502.04201-1-33-1': 'Therefore the integral of this function over [MATH] does not change if we shrink the original contour [MATH] to the boundary [MATH] : [EQUATION]', '1502.04201-1-33-2': 'To one "geometric" point [MATH] there corresponds two topologically different "boundary" points [MATH] and [MATH] lying on the right edge [MATH] and the left edge [MATH] of the slit [MATH] respectively.', '1502.04201-1-33-3': 'The chosen branch of the function [MATH] takes the following values on the boundary of the domain [MATH] : [EQUATION]', '1502.04201-1-33-4': 'If the point [MATH] runs over [MATH] counterclockwise, the [MATH] increases from [MATH] to [MATH] if [MATH] and [MATH] decreases from [MATH] to [MATH] if [MATH] .', '1502.04201-1-34-0': 'The function [MATH] which was defined by [REF] can be expressed explicitly in terms of the modified Bessel function [MATH] : [EQUATION] [2.]T', '1502.04201-1-34-1': 'The function [MATH] is representable as the sum of the series', '1502.04201-1-35-0': 'The expression in the right hand sides of [REF] is an entire function of three variables [MATH] .', '1502.04201-1-35-1': 'However the equalities [REF], [REF], [REF] hold only for [MATH] .', '1502.04201-1-35-2': '(We recall that the function [MATH] was defined by [REF] only for [MATH] .)', '1502.04201-1-36-0': 'Proof of Lemma [REF].', '1502.04201-1-36-1': 'We start from the formula [REF].', '1502.04201-1-36-2': 'Using the Taylor expansion of the hyperbolic [MATH] function, we obtain [EQUATION]', '1502.04201-1-37-0': 'Substituting the last equality into [REF], we obtain the equality [EQUATION]', '1502.04201-1-37-1': 'Taking into account the duplication formula for the Gamma-function, [CITATION]]AS: [EQUATION] we transform the last equality to the form [EQUATION]', '1502.04201-1-37-2': 'Now we would like to reduce the equality [REF] to the form which occurs in the so called Multiplication Theorem, see [CITATION]]AS: [EQUATION]', '1502.04201-1-37-3': 'Let us introduce [MATH] , i.e. [EQUATION]', '1502.04201-1-37-4': 'Then the equality [REF] takes the form [EQUATION]', '1502.04201-1-37-5': 'According to the Multiplication Theorem, [EQUATION]', '1502.04201-1-37-6': 'Taking into account that [MATH] , we reduce the equality [REF] to the form [REF].', '1502.04201-1-38-0': 'Using the Taylor expansion of the modified Bessel function [MATH] , [CITATION]]AS, we represent the function [MATH] as the sum of the series [REF].', '1502.04201-1-38-1': '*[MATH]', '1502.04201-1-39-0': 'The first proof of Theorem [REF].', '1502.04201-1-39-1': 'From the equality [REF] is evident that [EQUATION]', '1502.04201-1-39-2': 'Theorem [REF] follows from [REF] and [REF].', '1502.04201-1-39-3': '*[MATH]', '1502.04201-1-40-0': 'For each [MATH] , the function [MATH] which was introduced in [REF] admits the Taylor expansion with respect to [MATH] : [EQUATION]', '1502.04201-1-40-1': 'Proof.', '1502.04201-1-40-2': 'The expansion [REF] can be presented as a Taylor expansion with respect to [MATH] : [EQUATION]', '1502.04201-1-40-3': 'Substituting the expansion [REF] into the integrand in [REF], we obtain the expansion [REF].', '1502.04201-1-40-4': 'It is evident that [MATH] for [MATH] .', '1502.04201-1-40-5': 'The exponential convexity of the function [MATH] follows from the representation [REF].', '1502.04201-1-40-6': '*[MATH]', '1502.04201-1-41-0': 'The function [MATH] , [REF], can be expressed in terms of the modified Bessel function [MATH] : [EQUATION]', '1502.04201-1-41-1': 'See [CITATION]]AS.', '1502.04201-1-42-0': 'The formula [REF] appeared in [CITATION], see formulas (7.22) and (7.23) there.', '1502.04201-1-42-1': 'In [CITATION], the derivation of the expansion [REF] was done by a direct calculation, without any reference to the multiplication theorem for Bessel function.', '1502.04201-1-42-2': 'It should be mention that the series in the right hand side of [REF] appeared in [CITATION] as a perturbation series related to the BMV conjection for [MATH] matrices.', '1502.04201-1-43-0': '# The second proof of Theorem [REF].', '1502.04201-1-44-0': 'The starting point of the first as well as of the second is the representation of the value [MATH] by the contour integral [REF].', '1502.04201-1-44-1': 'See Lemma [REF].', '1502.04201-1-45-0': 'In the first proof, we shrank the contour of integration over the slit [MATH] , so the contour of integration was the same for every [MATH] .', '1502.04201-1-46-0': 'In contrast to this, in the second proof we choose the contour [MATH] in such a way that the exponent [MATH] of the integrand [MATH] in [REF] takes real values on [MATH] .', '1502.04201-1-46-1': '(So the contour [MATH] depends on [MATH] !)', '1502.04201-1-46-2': 'We denote this contour by [MATH]', '1502.04201-1-47-0': 'The function [MATH] is even with respect to [MATH] .', '1502.04201-1-47-1': 'Therefore to prove the exponential convexity of the function [MATH] , it is enough to prove that the value [MATH] is positive for each [EQUATION]', '1502.04201-1-47-2': 'We choose an arbitrary [MATH] satisfying the condition [REF] and fix this choice in the course of the proof.', '1502.04201-1-48-0': 'Let us introduce the functions [EQUATION] where [MATH] is the vertical slit [REF] and the branch of the function [MATH] in [MATH] is chosen which takes positive values for large real [MATH] .', '1502.04201-1-49-0': 'Let us assume that [MATH] and [MATH] satisfies the condition [REF].', '1502.04201-1-49-1': 'Then there exist [MATH] , [MATH] , such that [EQUATION]', '1502.04201-1-49-2': 'Proof.', '1502.04201-1-49-3': 'From the identity [EQUATION] we derive that [EQUATION] where [MATH] .', '1502.04201-1-49-4': 'Thus [EQUATION] and [EQUATION]', '1502.04201-1-49-5': 'It is clear that [MATH] as [MATH] , [MATH] as [MATH] .', '1502.04201-1-49-6': 'Since [MATH] , the inequality [REF] holds if if [MATH] is large enough.', '1502.04201-1-49-7': 'Since [MATH] , the inequality [REF] holds if [MATH] is small enough.', '1502.04201-1-49-8': '*[MATH]', '1502.04201-1-50-0': 'Let [MATH] be the set [EQUATION]', '1502.04201-1-51-0': '[1.]', '1502.04201-1-51-1': 'The set [MATH] is the union of the real axis and an ellipse [MATH] : [EQUATION] where the ellipse [MATH] is described by the equation: [EQUATION] with [EQUATION] [2.]', '1502.04201-1-51-2': 'The slit [MATH] is contained in the interior of the ellipse [MATH] .', '1502.04201-1-52-0': 'Proof.', '1502.04201-1-53-0': '1.', '1502.04201-1-53-1': 'Let [MATH] , where [MATH] are real numbers.', '1502.04201-1-53-2': 'The equality [EQUATION] is equivalent to the system of equalities [EQUATION]', '1502.04201-1-53-3': 'Here [MATH].', '1502.04201-1-53-4': 'Clearly [MATH] .', '1502.04201-1-54-0': 'Let [MATH] .', '1502.04201-1-54-1': 'This means that [MATH] , i.e. [EQUATION]', '1502.04201-1-54-2': 'Substituting the equalities [REF] into the first equality of the system [REF], we obtain that the equality [REF] holds for [MATH] .', '1502.04201-1-54-3': 'Thus we proved that [EQUATION]', '1502.04201-1-54-4': 'Let [EQUATION] be the upper and the lower half-plane respectively.', '1502.04201-1-55-0': 'According to Lemma [REF], there exist points [MATH] where [MATH] and points [MATH] where [MATH] .', '1502.04201-1-55-1': 'This means that the set [MATH] , [REF], separates the domain [MATH] .', '1502.04201-1-55-2': 'In other words, the open set [MATH] is disconnected.', '1502.04201-1-55-3': 'Since [MATH] , the set [MATH] is symmetric with respect to the real axis.', '1502.04201-1-55-4': 'The set [MATH] also is symmetric with respect to the real axis.', '1502.04201-1-55-5': 'Since [REF], the set [MATH] can not separate the domain [MATH] if [MATH] .', '1502.04201-1-56-0': '2.', '1502.04201-1-56-1': 'In view of [REF], the inequality [MATH] hold.', '1502.04201-1-56-2': 'So [MATH] is the minor semiaxis of the ellips [MATH] and [MATH] is its major semiaxis.', '1502.04201-1-56-3': 'Moreover, the inequality [MATH] holds.', '1502.04201-1-56-4': 'This means that the slit [MATH] is contained inside the ellipse [MATH] .', '1502.04201-1-56-5': '*[MATH]', '1502.04201-1-57-0': '[1.]', '1502.04201-1-57-1': 'The functions [MATH] and [MATH] are conjugate harmonic function of [MATH] in the domain [MATH] .', '1502.04201-1-57-2': '[2.]', '1502.04201-1-57-3': 'The only critical points of the the functions [MATH] and [MATH] in the domain [MATH] are the points [EQUATION] that is the points where the ellipse [MATH] and the real axis [MATH] intersect. [3.]', '1502.04201-1-57-4': 'If [MATH] lies outside the contour [MATH] , then [MATH] .', '1502.04201-1-57-5': 'If [MATH] lies inside the contour [MATH] , then [MATH] .', '1502.04201-1-58-0': 'Proof.', '1502.04201-1-58-1': 'The functions [MATH] and [MATH] are the real and the imaginary parts of the holomorphic function [MATH] .', '1502.04201-1-58-2': 'From the Cauchy-Riemann equation it follows that the functions [MATH] and [MATH] have the same critical points.', '1502.04201-1-58-3': 'Moreover the point [MATH] is critical for [MATH] if and only if [MATH] is a root of the derivative [MATH] of the function [MATH] .', '1502.04201-1-58-4': 'An explicit calculation shows that this derivative has only two roots [MATH] and [MATH] , [REF].', '1502.04201-1-59-0': 'Let [MATH] and [MATH] be the exterior and the exterior of the contour [MATH] respectively.', '1502.04201-1-59-1': 'Each of the sets [MATH] and [MATH] , [EQUATION] is a connected open set.', '1502.04201-1-59-2': 'According to [REF] and [REF], the continuous real valued function [MATH] does not vanish on any of these two sets.', '1502.04201-1-59-3': 'Hence the values [MATH] have the same sign, say [MATH] , at all points [MATH] of the set [MATH] , and the same same sign, say [MATH] , at all points [MATH] of the set [MATH] .', '1502.04201-1-59-4': 'Now the assertion 3 of Lemma [REF] is a consequence of Lemma [REF].', '1502.04201-1-59-5': '*[MATH]', '1502.04201-1-60-0': 'Completion of the proof of Theorem [REF].', '1502.04201-1-60-1': 'Let us chose the ellipse [MATH] as the contour of integration [MATH] in the integral in the right hand side of [REF].', '1502.04201-1-60-2': 'Since the imaginary part [MATH] of the exponent of the integrand vanishes on [MATH] , the integral representation [REF] takes the form [EQUATION]', '1502.04201-1-60-3': 'Since [MATH] , we can split the integral in [REF]: [EQUATION]', '1502.04201-1-60-4': 'Since the values [MATH] , [MATH] , [MATH] , and [MATH] are real, the first integral in the right hand side of [REF] vanishes.', '1502.04201-1-60-5': 'So the equality [REF] takes the form [EQUATION]', '1502.04201-1-60-6': 'Since the contour [MATH] is symmetric with respect to the real axis [MATH] and the function [MATH] also is symmetric: [MATH] , the equality [REF] can be reduced to the form [EQUATION] where [MATH] is the upper half of the contour [MATH] .', '1502.04201-1-60-7': 'Integrating by parts in [REF], we obtain [EQUATION] (The values [MATH] at the end points [MATH] , [REF], of the integration path [MATH] vanish.)', '1502.04201-1-61-0': 'The differential [MATH] in [REF] can be represented as [EQUATION] where [MATH] is a natural parameter on [MATH] .', '1502.04201-1-61-1': 'In other words, the differential [MATH] can be represented as [EQUATION] where [MATH] is the tangent vector to the curve [MATH] at the point [MATH] .', '1502.04201-1-61-2': 'The direction of the vector [MATH] corresponds to the motion of the point [MATH] along the path [MATH] from its left end point [MATH] to the right end point [MATH] .', '1502.04201-1-61-3': 'If [MATH] is the vector of the exterior normal to [MATH] at the point [MATH] , then the orientation of the frame [MATH] coincides with the orientation of the natural frame of [MATH] .', '1502.04201-1-61-4': 'According the Cauchy-Riemann equations, [EQUATION]', '1502.04201-1-61-5': 'Thus the representation [REF] can be reduced to the form [EQUATION]', '1502.04201-1-61-6': 'According the assertion 3 of Lemma [REF], [EQUATION]', '1502.04201-1-61-7': 'The inequality in [REF] is strict because [MATH] and the gradient [MATH] of the function [MATH] vanishes only at the critical points [MATH] of the function [MATH] , which are the end points of the integration path [MATH] .', '1502.04201-1-61-8': 'Evidently [MATH] and [MATH] at every point [MATH] .', '1502.04201-1-61-9': 'Thus the integrand in [REF] is strictly positive at every point [MATH] .', '1502.04201-1-61-10': 'So the inequality [MATH] holds.', '1502.04201-1-61-11': '*[MATH]', '1502.04201-1-62-0': 'The method which we use in the second proof of Theorem [REF] is the lite version of the method which Herbert Stahl, [CITATION], used in his proof of the BMV conjecture.', '1502.04201-1-63-0': '# The third proof of Theorem [REF].', '1502.04201-1-64-0': 'For each fixed [MATH] , the function [MATH] is an entire function of the variables [MATH] .', '1502.04201-1-64-1': 'Therefore, the Taylor expansion holds [EQUATION] where [EQUATION]', '1502.04201-1-64-2': 'It turns out that for every fixed real [MATH] and for every [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-1-64-3': 'We prove this by induction in [MATH] .', '1502.04201-1-64-4': 'Therefore for [MATH] , the sum of the series in [REF] is an exponentially convex function of [MATH] .', '1502.04201-1-64-5': 'To obtain Theorem [REF], we put [MATH] , [MATH] in [REF].', '1502.04201-1-64-6': '(For [MATH] , the statement of Theorem [REF] is trivially true.)', '1502.04201-1-65-0': 'Our proof of the exponential convexity of the functions [MATH] is based on the identity [EQUATION] which holds for every [MATH] .', '1502.04201-1-65-1': 'Substituting the expression [EQUATION] into this identity, we obtain the equality [EQUATION]', '1502.04201-1-65-2': 'Using the equality [EQUATION] which holds for every [MATH] , we obtain the equality [EQUATION]', '1502.04201-1-65-3': 'By successive differentiation the equality [REF] with respect to [MATH] , we obtain the equality [EQUATION] where [MATH] .', '1502.04201-1-65-4': 'In [REF], the summation is extended over all sequences [MATH] of non-negative integers for which [MATH].', '1502.04201-1-66-0': 'The equality [REF] holds for every [MATH] .', '1502.04201-1-66-1': 'Restricting this equality to the value [MATH] , we obtain the equality [EQUATION] which holds for every [MATH] , [MATH] , and [MATH] .', '1502.04201-1-66-2': 'In [REF], the summation is extended over all sequences [MATH] of non-negative integers for which [MATH].', '1502.04201-1-67-0': 'Let [MATH] be an arbitrary real number.', '1502.04201-1-67-1': 'By Lemma [REF], the function [EQUATION] of [MATH] is exponentially convex.', '1502.04201-1-67-2': 'Moreover, the function [MATH] is exponentially convex for every [MATH] (The number [MATH] here plays the same role as the number [MATH] in [REF]: it is an arbitrary real number.)', '1502.04201-1-68-0': 'Given [MATH] , assume that all functions [MATH] with [MATH] are exponentially convex functions of [MATH] .', '1502.04201-1-68-1': 'Then for each sequence [MATH] with [MATH] , the inequalities [MATH] hold.', '1502.04201-1-68-2': 'Thus, all the factors [MATH] which appears in the product [MATH] are exponentially convex functions of [MATH] .', '1502.04201-1-68-3': 'Hence the product itself is an exponentially convex function.', '1502.04201-1-68-4': 'Finally, the function [MATH] , [REF], which is essentially equal to the sum of all such products with [MATH] , is exponentially convex.', '1502.04201-1-68-5': 'This finishes the proof.', '1502.04201-1-68-6': '[MATH]', '1502.04201-1-69-0': 'Comparing the expansions [REF] and [REF], we see that [EQUATION]', '1502.04201-1-69-1': 'As a byproduct of the third proof of Theorem [REF], we proved that each of the functions [MATH] is exponentially convex.', '1502.04201-1-69-2': 'Thus we have given a second proof of Theorem [REF].', '1502.04201-1-70-0': 'Actually we proved more then we formulated in Theorem [REF].', '1502.04201-1-70-1': 'Namely we proved that for any sequence [MATH] of non-negative numbers the sum of series [EQUATION] is an exponentially convex function if this series converges for every real [MATH] .', '1502.04201-1-71-0': 'If [MATH] is a positive integer and [MATH] , then the Taylor expansion [EQUATION] is of the form [REF] with [MATH] for [MATH] , [MATH] for [MATH].', '1502.04201-1-72-0': 'In particular, for [MATH] the following result holds:', '1502.04201-1-73-0': 'For any [MATH] and [MATH] , the function [MATH] is an exponentially convex function of the variable [MATH] .'}

{'1502.04201-2-0-0': 'Given positive numbers [MATH] and [MATH] , the function [MATH] is exponentially convex function of [MATH] on the whole real axis.', '1502.04201-2-0-1': 'Three proofs of this result are presented.', '1502.04201-2-1-0': '# The exponential convexity result', '1502.04201-2-2-0': 'A function [MATH] on [MATH] , [MATH] , is said to be exponentially convex if', '1502.04201-2-3-0': '[1.]', '1502.04201-2-3-1': 'For every positive integer [MATH] , for every choice of real numbers [MATH] , [MATH] , and complex numbers [MATH] , [MATH] , the inequality holds [EQUATION] [2.]', '1502.04201-2-3-2': 'The function [MATH] is continuous on [MATH] .', '1502.04201-2-4-0': 'The class of exponentially convex functions was introduced by S.N.Bernstein, [CITATION], see 15 there.', '1502.04201-2-4-1': 'Russian translation of the paper [CITATION] can be found in [CITATION].', '1502.04201-2-5-0': 'From [REF] it follows that the inequality [MATH] holds for every [MATH] .', '1502.04201-2-5-1': 'Thus the alternative takes place: If [MATH] is an exponentially convex function then either [MATH] , or [MATH] for every [MATH] .', '1502.04201-2-6-0': 'Properties of the class of exponentially convex functions.', '1502.04201-2-7-0': '[P1.]', '1502.04201-2-7-1': 'If [MATH] if an exponentially convex function and [MATH] is a nonnegative constant, then the function [MATH] is exponentially convex.', '1502.04201-2-7-2': '[P2.]', '1502.04201-2-7-3': 'If [MATH] and [MATH] are exponentially convex functions, then their sum [MATH] is exponentially convex.', '1502.04201-2-7-4': '[P3.]', '1502.04201-2-7-5': 'If [MATH] and [MATH] are exponentially convex functions, then their product [MATH] is exponentially convex.', '1502.04201-2-7-6': '[P4.]', '1502.04201-2-7-7': 'Let [MATH] be a sequence of exponentially convex functions.', '1502.04201-2-7-8': 'We assume that for each [MATH] there exists the limit [MATH] , and that [MATH] .', '1502.04201-2-7-9': 'Then the limiting function [MATH] is exponentially convex.', '1502.04201-2-8-0': 'From the functional equation for the exponential function it follows that for each real number [MATH] , for every choice of real numbers [MATH] , [MATH] and complex numbers [MATH] , [MATH] , the equality holds [EQUATION]', '1502.04201-2-8-1': 'The relation [REF] can be formulated as', '1502.04201-2-9-0': 'For each real [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-2-10-0': 'For [MATH] , the function [MATH] , which is called the hyperbolic cosine of [MATH] , is defined as [EQUATION]', '1502.04201-2-10-1': 'From Lemma [REF] and property P2 we obtain', '1502.04201-2-11-0': 'For each real [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-2-12-0': 'The following result is well known.', '1502.04201-2-12-1': '[The representation theorem]', '1502.04201-2-13-0': '[1.]', '1502.04201-2-13-1': 'Let [MATH] be a nonnegative measure on the real axis, and let the function [MATH] be a two-sided Laplace transform of the measure [MATH] : [EQUATION] for any [MATH] .', '1502.04201-2-13-2': 'Then the function [MATH] is exponentially convex.', '1502.04201-2-13-3': '[2. ]', '1502.04201-2-13-4': 'Let [MATH] be an exponentially convex function.', '1502.04201-2-13-5': 'Then this function [MATH] can be represented on [MATH] as a two-sided Laplace transform [REF] of a nonnegative measure [MATH] .', '1502.04201-2-13-6': '(In particular, the integral in the right hand side of [REF] is finite for any [MATH] .)', '1502.04201-2-13-7': 'The representing measure [MATH] is unique.', '1502.04201-2-14-0': 'The assertion 1 of the representation theorem is an evident consequence of Lemma [REF], of the properties P1,P2, P4, and of the definition of the integration.', '1502.04201-2-15-0': 'The proof of the assertion 2 can be found in [CITATION],Theorem 5.5.4, and in [CITATION],Theorem 21.', '1502.04201-2-16-0': "Of course, Lemma [REF] is a special case of the representation theorem which corresponds to the representing measure [MATH] , where [MATH] are Dirak's [MATH] -functions supported at the points [MATH] .", '1502.04201-2-17-0': 'The expression [EQUATION] is well defined for every complex numbers [MATH] .', '1502.04201-2-17-1': 'The function [MATH] is an entire function of complex variables [MATH] .', '1502.04201-2-17-2': 'For each fixed [MATH] and [MATH] , the function [MATH] , considered as a function of [MATH] , is an entire function of exponential type [MATH] .', '1502.04201-2-18-0': 'Proof.', '1502.04201-2-18-1': 'The function [MATH] is a superposition of the entire function [MATH] of variable [MATH] and the quadratic polynomial [MATH] .', '1502.04201-2-18-2': 'The assertion concerning the growth of this function is evident.', '1502.04201-2-18-3': '*[MATH]', '1502.04201-2-19-0': 'In the paper [CITATION] a conjecture was formulated which now is commonly known as the BMV conjecture:', '1502.04201-2-20-0': 'The BMV Conjecture.', '1502.04201-2-20-1': 'Let [MATH] and [MATH] be Hermitian matrices of size [MATH] .', '1502.04201-2-20-2': 'Then the function [EQUATION] of the variable [MATH] is representable as a bilateral Laplace transform of a non-negative measure [MATH] compactly supported on the real axis: [EQUATION]', '1502.04201-2-20-3': 'In general case, if the matrices [MATH] and [MATH] do not commute, the BMV conjecture remained an open question for longer than 35 years.', '1502.04201-2-20-4': 'In 2011, Herbert Stahl, [CITATION], gave an affirmative answer to the BMV conjecture.', '1502.04201-2-21-0': 'Theorem(H.Stahl) Let [MATH] and [MATH] be [MATH] Hermitian matrices.', '1502.04201-2-21-1': 'Then the function [MATH] defined by [REF] is representable as the bilateral Laplace transform [REF] of a non-negative measure [MATH] supported on the closed interval [MATH] .', '1502.04201-2-22-0': 'The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions.', '1502.04201-2-23-0': 'In the case [MATH] matrices [MATH] and [MATH] , the BMV conjecture is equivalent to the exponential convexity with respect to [MATH] for each [MATH] and [MATH] of the function [MATH] which was introduced in [REF] .', '1502.04201-2-24-0': 'For each fixed [MATH] and [MATH] , the function [MATH] defined by [REF] is an exponentially convex function of variable [MATH] .', '1502.04201-2-25-0': 'In what follow we present three different proofs of Theorem [REF].', '1502.04201-2-25-1': 'The first and the second proofs are based on the representation theorem.', '1502.04201-2-25-2': 'We prove that the function [MATH] which is defined by [REF] below takes positive values for [MATH] .', '1502.04201-2-25-3': 'In the first proof we calculate the function [MATH] explicitly expressing this function in terms of the modified Bessel function [MATH] .', '1502.04201-2-25-4': 'In the second proof, we prove the positivity of the function [MATH] using the reasoning by Herbert Stahl in [CITATION].', '1502.04201-2-25-5': '(We use a very simple special case of this reasoning.)', '1502.04201-2-25-6': 'The third proof is based on the Taylor expansion of the function [MATH] , [REF], with respect to parameter [MATH] .', '1502.04201-2-25-7': 'This proof does not use any integration in the complex plane.', '1502.04201-2-25-8': 'It based only on Lemma [REF] and on the properties P1 - P4 of the class of exponentially convex functions.', '1502.04201-2-25-9': 'As a by-product of this proof we obtain that all coefficients of this Taylor expansion are exponentially convex functions.', '1502.04201-2-25-10': 'However we can not conclude directly from this proof that the restriction of the representing measure on the open interval [MATH] is an absolutely continuous measure.', '1502.04201-2-26-0': 'For each fixed [MATH] , the function [MATH] defined by [REF] is representable in the form [EQUATION] where the function [MATH] possesses the properties', '1502.04201-2-27-0': '[1.]', '1502.04201-2-28-0': '[EQUATION] [2.]', '1502.04201-2-28-1': 'The function [MATH] is continuous with respect to [MATH] on the closed interval [MATH] , takes real values there, and is even.', '1502.04201-2-28-2': '[3.]', '1502.04201-2-28-3': 'The values of the function [MATH] at the end points [MATH] of the interval [MATH] are: [EQUATION]', '1502.04201-2-29-0': 'Proof.', '1502.04201-2-29-1': 'We introduce the function [EQUATION] of variables [MATH] .', '1502.04201-2-29-2': 'Considered as a function of [MATH] for fixed positive [MATH] and [MATH] , [MATH] is entire function of exponential type [MATH] .', '1502.04201-2-29-3': 'On the imaginary axis [MATH] takes the form [EQUATION]', '1502.04201-2-29-4': 'From [REF] it follows that the function [MATH] is a bounded and decaying on the imaginary axis: [MATH] , [MATH] .', '1502.04201-2-29-5': 'By the Wiener-Paley theorem, the function [MATH] is representable in the form [EQUATION] where the function [MATH] satisfies the condition [REF].', '1502.04201-2-29-6': 'The equality [REF] serves as a definition of the function [MATH] .', '1502.04201-2-29-7': 'So, this function is defined only for [MATH] .', '1502.04201-2-30-0': 'Since the function [MATH] is even with respect to [MATH] and real valued, its inverse Fourier transform [MATH] is even with respect to [MATH] and real valued.', '1502.04201-2-31-0': 'From [REF] we obtain that [EQUATION]', '1502.04201-2-31-1': 'Hence the function in the left hand side of [REF] is a Fourier transform of some function [MATH] which is square summable and continuous at every [MATH] .', '1502.04201-2-31-2': 'We remark that [EQUATION] where [MATH] is a constant function.', '1502.04201-2-31-3': 'Hence [EQUATION]', '1502.04201-2-31-4': 'Since [MATH] for [MATH] , also [MATH] .', '1502.04201-2-31-5': 'Thus, [REF] holds.', '1502.04201-2-31-6': '[MATH]', '1502.04201-2-32-0': '# Representation the function [MATH] by a contour integral.', '1502.04201-2-33-0': 'Let [MATH] be a segment of the imaginary axis: [EQUATION]', '1502.04201-2-33-1': 'The function [MATH] is a single value function of [MATH] in the complex plane slitted along the vertical segment [MATH] .', '1502.04201-2-33-2': 'We choose the branch of this function which takes positive values for large real [MATH] .', '1502.04201-2-34-0': 'The function [MATH] , which was defined by [REF], admits the integral representation [EQUATION] where [MATH] is an arbitrary counterclockwise oriented closed Jordan curve which contains the slit [MATH] inside.', '1502.04201-2-35-0': 'Proof.', '1502.04201-2-35-1': 'According the inversion formula for the Fourier transform, [EQUATION]', '1502.04201-2-35-2': 'We interpret the integral in the right hand side of [REF] as the integral along the the vertical straight line [MATH] : [EQUATION]', '1502.04201-2-35-3': 'Since the function [MATH] is bounded in each vertical strip [MATH] and tends to zero as [MATH] within this strip, the value of the integral in [REF] does not change if we integrate along any vertical line [MATH] , where [MATH] is an arbitrary real number: [EQUATION]', '1502.04201-2-35-4': 'Choosing [MATH] , we split the integral in [REF] into the sum [EQUATION] where [EQUATION]', '1502.04201-2-35-5': 'The function [MATH] is holomorphic in the halfplane [MATH] and admits the estimate [EQUATION] there, where [MATH] is a constant.', '1502.04201-2-35-6': 'Therefore [EQUATION]', '1502.04201-2-35-7': 'The function [MATH] is holomorphic in the slitted half plane [MATH] and admits the estimate [EQUATION] there.', '1502.04201-2-35-8': 'Therefore [EQUATION] where [MATH] is an arbitrary closed Jourdan curve which is oriented clockwise and contains the slit [MATH] in its interior.', '1502.04201-2-35-9': 'Since the function [MATH] is entire, [MATH] .', '1502.04201-2-35-10': 'So [EQUATION] where the integral in the right hand side is taken over the curve [MATH] which is oriented counterclockwise.', '1502.04201-2-35-11': 'Comparing [REF], [REF], [REF], and [REF], we obtain [REF].', '1502.04201-2-35-12': '[MATH]', '1502.04201-2-36-0': '# Explicite calculation of the function [MATH] .', '1502.04201-2-37-0': 'The function [MATH] which was defined by [REF] admits the integral representation [EQUATION]', '1502.04201-2-37-1': 'Proof.', '1502.04201-2-37-2': 'We derive Lemma [REF] from Lemma [REF] showing that [EQUATION]', '1502.04201-2-38-0': 'The function [MATH] is holomorphic in the domain [MATH] and continuous up to boundary [MATH] of this domain.', '1502.04201-2-38-1': 'Therefore the integral of this function over [MATH] does not change if we shrink the original contour [MATH] to the boundary [MATH] : [EQUATION]', '1502.04201-2-38-2': 'To one "geometric" point [MATH] there corresponds two topologically different "boundary" points [MATH] and [MATH] lying on the right edge [MATH] and the left edge [MATH] of the slit [MATH] respectively.', '1502.04201-2-38-3': 'The chosen branch of the function [MATH] takes the following values on the boundary of the domain [MATH] : [EQUATION]', '1502.04201-2-38-4': 'If the point [MATH] runs over [MATH] counterclockwise, the [MATH] increases from [MATH] to [MATH] if [MATH] and [MATH] decreases from [MATH] to [MATH] if [MATH] .', '1502.04201-2-39-0': 'The function [MATH] which was defined by [REF] can be expressed explicitly in terms of the modified Bessel function [MATH] : [EQUATION] [2.]T', '1502.04201-2-39-1': 'The function [MATH] is representable as the sum of the series', '1502.04201-2-40-0': 'The expression in the right hand sides of [REF] is an entire function of three variables [MATH] .', '1502.04201-2-40-1': 'However the equalities [REF], [REF], [REF] hold only for [MATH] .', '1502.04201-2-40-2': '(We recall that the function [MATH] was defined by [REF] only for [MATH] .)', '1502.04201-2-41-0': 'Proof of Lemma [REF].', '1502.04201-2-41-1': 'We start from the formula [REF].', '1502.04201-2-41-2': 'Using the Taylor expansion of the hyperbolic [MATH] function, we obtain [EQUATION]', '1502.04201-2-42-0': 'Substituting the last equality into [REF], we obtain the equality [EQUATION]', '1502.04201-2-42-1': 'Taking into account the duplication formula for the Gamma-function, [CITATION]]AS: [EQUATION] we transform the last equality to the form [EQUATION]', '1502.04201-2-42-2': 'Now we would like to reduce the equality [REF] to the form which occurs in the so called Multiplication Theorem, see [CITATION]]AS: [EQUATION]', '1502.04201-2-42-3': 'Let us introduce [MATH] , i.e. [EQUATION]', '1502.04201-2-42-4': 'Then the equality [REF] takes the form [EQUATION]', '1502.04201-2-42-5': 'According to the Multiplication Theorem, [EQUATION]', '1502.04201-2-42-6': 'Taking into account that [MATH] , we reduce the equality [REF] to the form [REF].', '1502.04201-2-43-0': 'Using the Taylor expansion of the modified Bessel function [MATH] , [CITATION]]AS, we represent the function [MATH] as the sum of the series [REF].', '1502.04201-2-43-1': '*[MATH]', '1502.04201-2-44-0': '# The first proof of Theorem [REF].', '1502.04201-2-44-1': 'From the equality [REF] is evident that [EQUATION]', '1502.04201-2-44-2': 'Theorem [REF] follows from [REF] and [REF].', '1502.04201-2-44-3': '*[MATH]', '1502.04201-2-45-0': 'For each [MATH] , the function [MATH] which was introduced in [REF] admits the Taylor expansion with respect to [MATH] : [EQUATION]', '1502.04201-2-45-1': 'Proof.', '1502.04201-2-45-2': 'The expansion [REF] can be presented as a Taylor expansion with respect to [MATH] : [EQUATION]', '1502.04201-2-45-3': 'Substituting the expansion [REF] into the integrand in [REF], we obtain the expansion [REF].', '1502.04201-2-45-4': 'It is evident that [MATH] for [MATH] .', '1502.04201-2-45-5': 'The exponential convexity of the function [MATH] follows from the representation [REF].', '1502.04201-2-45-6': '*[MATH]', '1502.04201-2-46-0': 'The function [MATH] , [REF], can be expressed in terms of the modified Bessel function [MATH] : [EQUATION]', '1502.04201-2-46-1': 'See [CITATION]]AS.', '1502.04201-2-47-0': 'The formula [REF] appeared in [CITATION], see formulas (7.22) and (7.23) there.', '1502.04201-2-47-1': 'In [CITATION], the derivation of the expansion [REF] was done by a direct calculation, without any reference to the multiplication theorem for Bessel function.', '1502.04201-2-47-2': 'It should be mention that the series in the right hand side of [REF] appeared in [CITATION] as a perturbation series related to the BMV conjection for [MATH] matrices.', '1502.04201-2-48-0': '# The second proof of Theorem [REF].', '1502.04201-2-49-0': 'The starting point of the first as well as of the second is the representation of the value [MATH] by the contour integral [REF].', '1502.04201-2-49-1': 'See Lemma [REF].', '1502.04201-2-50-0': 'In the first proof, we shrank the contour of integration over the slit [MATH] , so the contour of integration was the same for every [MATH] .', '1502.04201-2-51-0': 'In contrast to this, in the second proof we choose the contour [MATH] in such a way that the exponent [MATH] of the integrand [MATH] in [REF] takes real values on [MATH] .', '1502.04201-2-51-1': '(So the contour [MATH] depends on [MATH] !)', '1502.04201-2-51-2': 'We denote this contour by [MATH]', '1502.04201-2-52-0': 'The function [MATH] is even with respect to [MATH] .', '1502.04201-2-52-1': 'Therefore to prove the exponential convexity of the function [MATH] , it is enough to prove that the value [MATH] is positive for each [EQUATION]', '1502.04201-2-52-2': 'We choose an arbitrary [MATH] satisfying the condition [REF] and fix this choice in the course of the proof.', '1502.04201-2-53-0': 'Let us introduce the functions [EQUATION] where [MATH] is the vertical slit [REF] and the branch of the function [MATH] in [MATH] is chosen which takes positive values for large real [MATH] .', '1502.04201-2-54-0': 'Let us assume that [MATH] and [MATH] satisfies the condition [REF].', '1502.04201-2-54-1': 'Then there exist [MATH] , [MATH] , such that [EQUATION]', '1502.04201-2-54-2': 'Proof.', '1502.04201-2-54-3': 'From the identity [EQUATION] we derive that [EQUATION] where [MATH] .', '1502.04201-2-54-4': 'Thus [EQUATION] and [EQUATION]', '1502.04201-2-54-5': 'It is clear that [MATH] as [MATH] , [MATH] as [MATH] .', '1502.04201-2-54-6': 'Since [MATH] , the inequality [REF] holds if if [MATH] is large enough.', '1502.04201-2-54-7': 'Since [MATH] , the inequality [REF] holds if [MATH] is small enough.', '1502.04201-2-54-8': '*[MATH]', '1502.04201-2-55-0': 'Let [MATH] be the set [EQUATION]', '1502.04201-2-56-0': '[1.]', '1502.04201-2-56-1': 'The set [MATH] is the union of the real axis and an ellipse [MATH] : [EQUATION] where the ellipse [MATH] is described by the equation: [EQUATION] with [EQUATION] [2.]', '1502.04201-2-56-2': 'The slit [MATH] is contained in the interior of the ellipse [MATH] .', '1502.04201-2-57-0': 'Proof.', '1502.04201-2-58-0': '1.', '1502.04201-2-58-1': 'Let [MATH] , where [MATH] are real numbers.', '1502.04201-2-58-2': 'The equality [EQUATION] is equivalent to the system of equalities [EQUATION]', '1502.04201-2-58-3': 'Here [MATH].', '1502.04201-2-58-4': 'Clearly [MATH] .', '1502.04201-2-59-0': 'Let [MATH] .', '1502.04201-2-59-1': 'This means that [MATH] , i.e. [EQUATION]', '1502.04201-2-59-2': 'Substituting the equalities [REF] into the first equality of the system [REF], we obtain that the equality [REF] holds for [MATH] .', '1502.04201-2-59-3': 'Thus we proved that [EQUATION]', '1502.04201-2-59-4': 'Let [EQUATION] be the upper and the lower half-plane respectively.', '1502.04201-2-60-0': 'According to Lemma [REF], there exist points [MATH] where [MATH] and points [MATH] where [MATH] .', '1502.04201-2-60-1': 'This means that the set [MATH] , [REF], separates the domain [MATH] .', '1502.04201-2-60-2': 'In other words, the open set [MATH] is disconnected.', '1502.04201-2-60-3': 'Since [MATH] , the set [MATH] is symmetric with respect to the real axis.', '1502.04201-2-60-4': 'The set [MATH] also is symmetric with respect to the real axis.', '1502.04201-2-60-5': 'Since [REF], the set [MATH] can not separate the domain [MATH] if [MATH] .', '1502.04201-2-61-0': '2.', '1502.04201-2-61-1': 'In view of [REF], the inequality [MATH] hold.', '1502.04201-2-61-2': 'So [MATH] is the minor semiaxis of the ellips [MATH] and [MATH] is its major semiaxis.', '1502.04201-2-61-3': 'Moreover, the inequality [MATH] holds.', '1502.04201-2-61-4': 'This means that the slit [MATH] is contained inside the ellipse [MATH] .', '1502.04201-2-61-5': '*[MATH]', '1502.04201-2-62-0': '[1.]', '1502.04201-2-62-1': 'The functions [MATH] and [MATH] are conjugate harmonic function of [MATH] in the domain [MATH] .', '1502.04201-2-62-2': '[2.]', '1502.04201-2-62-3': 'The only critical points of the the functions [MATH] and [MATH] in the domain [MATH] are the points [EQUATION] that is the points where the ellipse [MATH] and the real axis [MATH] intersect. [3.]', '1502.04201-2-62-4': 'If [MATH] lies outside the contour [MATH] , then [MATH] .', '1502.04201-2-62-5': 'If [MATH] lies inside the contour [MATH] , then [MATH] .', '1502.04201-2-63-0': 'Proof.', '1502.04201-2-63-1': 'The functions [MATH] and [MATH] are the real and the imaginary parts of the holomorphic function [MATH] .', '1502.04201-2-63-2': 'From the Cauchy-Riemann equation it follows that the functions [MATH] and [MATH] have the same critical points.', '1502.04201-2-63-3': 'Moreover the point [MATH] is critical for [MATH] if and only if [MATH] is a root of the derivative [MATH] of the function [MATH] .', '1502.04201-2-63-4': 'An explicit calculation shows that this derivative has only two roots [MATH] and [MATH] , [REF].', '1502.04201-2-64-0': 'Let [MATH] and [MATH] be the exterior and the exterior of the contour [MATH] respectively.', '1502.04201-2-64-1': 'Each of the sets [MATH] and [MATH] , [EQUATION] is a connected open set.', '1502.04201-2-64-2': 'According to [REF] and [REF], the continuous real valued function [MATH] does not vanish on any of these two sets.', '1502.04201-2-64-3': 'Hence the values [MATH] have the same sign, say [MATH] , at all points [MATH] of the set [MATH] , and the same same sign, say [MATH] , at all points [MATH] of the set [MATH] .', '1502.04201-2-64-4': 'Now the assertion 3 of Lemma [REF] is a consequence of Lemma [REF].', '1502.04201-2-64-5': '*[MATH]', '1502.04201-2-65-0': 'Completion of the proof of Theorem [REF].', '1502.04201-2-65-1': 'Let us chose the ellipse [MATH] as the contour of integration [MATH] in the integral in the right hand side of [REF].', '1502.04201-2-65-2': 'Since the imaginary part [MATH] of the exponent of the integrand vanishes on [MATH] , the integral representation [REF] takes the form [EQUATION]', '1502.04201-2-65-3': 'Since [MATH] , we can split the integral in [REF]: [EQUATION]', '1502.04201-2-65-4': 'Since the values [MATH] , [MATH] , [MATH] , and [MATH] are real, the first integral in the right hand side of [REF] vanishes.', '1502.04201-2-65-5': 'So the equality [REF] takes the form [EQUATION]', '1502.04201-2-65-6': 'Since the contour [MATH] is symmetric with respect to the real axis [MATH] and the function [MATH] also is symmetric: [MATH] , the equality [REF] can be reduced to the form [EQUATION] where [MATH] is the upper half of the contour [MATH] .', '1502.04201-2-65-7': 'Integrating by parts in [REF], we obtain [EQUATION] (The values [MATH] at the end points [MATH] , [REF], of the integration path [MATH] vanish.)', '1502.04201-2-66-0': 'The differential [MATH] in [REF] can be represented as [EQUATION] where [MATH] is a natural parameter on [MATH] .', '1502.04201-2-66-1': 'In other words, the differential [MATH] can be represented as [EQUATION] where [MATH] is the tangent vector to the curve [MATH] at the point [MATH] .', '1502.04201-2-66-2': 'The direction of the vector [MATH] corresponds to the motion of the point [MATH] along the path [MATH] from its left end point [MATH] to the right end point [MATH] .', '1502.04201-2-66-3': 'If [MATH] is the vector of the exterior normal to [MATH] at the point [MATH] , then the orientation of the frame [MATH] coincides with the orientation of the natural frame of [MATH] .', '1502.04201-2-66-4': 'According the Cauchy-Riemann equations, [EQUATION]', '1502.04201-2-66-5': 'Thus the representation [REF] can be reduced to the form [EQUATION]', '1502.04201-2-66-6': 'According the assertion 3 of Lemma [REF], [EQUATION]', '1502.04201-2-66-7': 'The inequality in [REF] is strict because [MATH] and the gradient [MATH] of the function [MATH] vanishes only at the critical points [MATH] of the function [MATH] , which are the end points of the integration path [MATH] .', '1502.04201-2-66-8': 'Evidently [MATH] and [MATH] at every point [MATH] .', '1502.04201-2-66-9': 'Thus the integrand in [REF] is strictly positive at every point [MATH] .', '1502.04201-2-66-10': 'So the inequality [MATH] holds.', '1502.04201-2-66-11': '*[MATH]', '1502.04201-2-67-0': 'The method which we use in the second proof of Theorem [REF] is the lite version of the method which Herbert Stahl, [CITATION], used in his proof of the BMV conjecture.', '1502.04201-2-68-0': '# The third proof of Theorem [REF].', '1502.04201-2-69-0': 'For each fixed [MATH] , the function [MATH] is an entire function of the variables [MATH] .', '1502.04201-2-69-1': 'Therefore, the Taylor expansion holds [EQUATION] where [EQUATION]', '1502.04201-2-69-2': 'It turns out that for every fixed real [MATH] and for every [MATH] , the function [MATH] of the variable [MATH] is exponentially convex.', '1502.04201-2-69-3': 'We prove this by induction in [MATH] .', '1502.04201-2-69-4': 'Therefore for [MATH] , the sum of the series in [REF] is an exponentially convex function of [MATH] .', '1502.04201-2-69-5': 'To obtain Theorem [REF], we put [MATH] , [MATH] in [REF].', '1502.04201-2-69-6': '(For [MATH] , the statement of Theorem [REF] is trivially true.)', '1502.04201-2-70-0': 'Our proof of the exponential convexity of the functions [MATH] is based on the identity [EQUATION] which holds for every [MATH] .', '1502.04201-2-70-1': 'Substituting the expression [EQUATION] into this identity, we obtain the equality [EQUATION]', '1502.04201-2-70-2': 'Using the equality [EQUATION] which holds for every [MATH] , we obtain the equality [EQUATION]', '1502.04201-2-70-3': 'By successive differentiation the equality [REF] with respect to [MATH] , we obtain the equality [EQUATION] where [MATH] .', '1502.04201-2-70-4': 'In [REF], the summation is extended over all sequences [MATH] of non-negative integers for which [MATH].', '1502.04201-2-71-0': 'The equality [REF] holds for every [MATH] .', '1502.04201-2-71-1': 'Restricting this equality to the value [MATH] , we obtain the equality [EQUATION] which holds for every [MATH] , [MATH] , and [MATH] .', '1502.04201-2-71-2': 'In [REF], the summation is extended over all sequences [MATH] of non-negative integers for which [MATH].', '1502.04201-2-72-0': 'Let [MATH] be an arbitrary real number.', '1502.04201-2-72-1': 'By Lemma [REF], the function [EQUATION] of [MATH] is exponentially convex.', '1502.04201-2-72-2': 'Moreover, the function [MATH] is exponentially convex for every [MATH] (The number [MATH] here plays the same role as the number [MATH] in [REF]: it is an arbitrary real number.)', '1502.04201-2-73-0': 'Given [MATH] , assume that all functions [MATH] with [MATH] are exponentially convex functions of [MATH] .', '1502.04201-2-73-1': 'Then for each sequence [MATH] with [MATH] , the inequalities [MATH] hold.', '1502.04201-2-73-2': 'Thus, all the factors [MATH] which appears in the product [MATH] are exponentially convex functions of [MATH] .', '1502.04201-2-73-3': 'Hence the product itself is an exponentially convex function.', '1502.04201-2-73-4': 'Finally, the function [MATH] , [REF], which is essentially equal to the sum of all such products with [MATH] , is exponentially convex.', '1502.04201-2-73-5': 'This finishes the proof.', '1502.04201-2-73-6': '[MATH]', '1502.04201-2-74-0': 'Comparing the expansions [REF] and [REF], we see that [EQUATION]', '1502.04201-2-74-1': 'As a byproduct of the third proof of Theorem [REF], we proved that each of the functions [MATH] is exponentially convex.', '1502.04201-2-74-2': 'Thus we have given a second proof of Theorem [REF].', '1502.04201-2-75-0': 'Actually we proved more then we formulated in Theorem [REF].', '1502.04201-2-75-1': 'Namely we proved that for any sequence [MATH] of non-negative numbers the sum of series [EQUATION] is an exponentially convex function if this series converges for every real [MATH] .', '1502.04201-2-76-0': 'If [MATH] is a positive integer and [MATH] , then the Taylor expansion [EQUATION] is of the form [REF] with [MATH] for [MATH] , [MATH] for [MATH].', '1502.04201-2-77-0': 'In particular, for [MATH] the following result holds:', '1502.04201-2-78-0': 'For any [MATH] and [MATH] , the function [MATH] is an exponentially convex function of the variable [MATH] .'}

[['1502.04201-1-6-0', '1502.04201-2-5-0'], ['1502.04201-1-6-1', '1502.04201-2-5-1'], ['1502.04201-1-16-0', '1502.04201-2-15-0'], ['1502.04201-1-55-0', '1502.04201-2-60-0'], ['1502.04201-1-55-1', '1502.04201-2-60-1'], ['1502.04201-1-55-2', '1502.04201-2-60-2'], ['1502.04201-1-55-3', '1502.04201-2-60-3'], ['1502.04201-1-55-4', '1502.04201-2-60-4'], ['1502.04201-1-55-5', '1502.04201-2-60-5'], ['1502.04201-1-67-0', '1502.04201-2-72-0'], ['1502.04201-1-67-1', '1502.04201-2-72-1'], ['1502.04201-1-67-2', '1502.04201-2-72-2'], ['1502.04201-1-71-0', '1502.04201-2-76-0'], ['1502.04201-1-58-1', '1502.04201-2-63-1'], ['1502.04201-1-58-2', '1502.04201-2-63-2'], ['1502.04201-1-58-3', '1502.04201-2-63-3'], ['1502.04201-1-58-4', '1502.04201-2-63-4'], ['1502.04201-1-45-0', '1502.04201-2-50-0'], ['1502.04201-1-8-1', '1502.04201-2-7-1'], ['1502.04201-1-8-3', '1502.04201-2-7-3'], ['1502.04201-1-8-5', '1502.04201-2-7-5'], ['1502.04201-1-8-7', '1502.04201-2-7-7'], ['1502.04201-1-8-8', '1502.04201-2-7-8'], ['1502.04201-1-8-9', '1502.04201-2-7-9'], ['1502.04201-1-42-0', '1502.04201-2-47-0'], ['1502.04201-1-42-1', '1502.04201-2-47-1'], ['1502.04201-1-42-2', '1502.04201-2-47-2'], ['1502.04201-1-26-0', '1502.04201-2-31-0'], ['1502.04201-1-26-1', '1502.04201-2-31-1'], ['1502.04201-1-26-2', '1502.04201-2-31-2'], ['1502.04201-1-26-4', '1502.04201-2-31-4'], ['1502.04201-1-59-0', '1502.04201-2-64-0'], ['1502.04201-1-59-1', '1502.04201-2-64-1'], ['1502.04201-1-59-2', '1502.04201-2-64-2'], ['1502.04201-1-59-3', '1502.04201-2-64-3'], ['1502.04201-1-59-4', '1502.04201-2-64-4'], ['1502.04201-1-60-0', '1502.04201-2-65-0'], ['1502.04201-1-60-1', '1502.04201-2-65-1'], ['1502.04201-1-60-2', '1502.04201-2-65-2'], ['1502.04201-1-60-3', '1502.04201-2-65-3'], ['1502.04201-1-60-4', '1502.04201-2-65-4'], ['1502.04201-1-60-5', '1502.04201-2-65-5'], ['1502.04201-1-60-6', '1502.04201-2-65-6'], ['1502.04201-1-60-7', '1502.04201-2-65-7'], ['1502.04201-1-39-1', '1502.04201-2-44-1'], ['1502.04201-1-39-2', '1502.04201-2-44-2'], ['1502.04201-1-70-0', '1502.04201-2-75-0'], ['1502.04201-1-70-1', '1502.04201-2-75-1'], ['1502.04201-1-29-0', '1502.04201-2-34-0'], ['1502.04201-1-36-0', '1502.04201-2-41-0'], ['1502.04201-1-36-1', '1502.04201-2-41-1'], ['1502.04201-1-36-2', '1502.04201-2-41-2'], ['1502.04201-1-61-0', '1502.04201-2-66-0'], ['1502.04201-1-61-1', '1502.04201-2-66-1'], ['1502.04201-1-61-2', '1502.04201-2-66-2'], ['1502.04201-1-61-3', '1502.04201-2-66-3'], ['1502.04201-1-61-4', '1502.04201-2-66-4'], ['1502.04201-1-61-5', '1502.04201-2-66-5'], ['1502.04201-1-61-6', '1502.04201-2-66-6'], ['1502.04201-1-61-7', '1502.04201-2-66-7'], ['1502.04201-1-61-8', '1502.04201-2-66-8'], ['1502.04201-1-61-9', '1502.04201-2-66-9'], ['1502.04201-1-61-10', '1502.04201-2-66-10'], ['1502.04201-1-17-0', '1502.04201-2-16-0'], ['1502.04201-1-28-0', '1502.04201-2-33-0'], ['1502.04201-1-28-1', '1502.04201-2-33-1'], ['1502.04201-1-28-2', '1502.04201-2-33-2'], ['1502.04201-1-30-1', '1502.04201-2-35-1'], ['1502.04201-1-30-2', '1502.04201-2-35-2'], ['1502.04201-1-30-3', '1502.04201-2-35-3'], ['1502.04201-1-30-4', '1502.04201-2-35-4'], ['1502.04201-1-30-5', '1502.04201-2-35-5'], ['1502.04201-1-30-7', '1502.04201-2-35-7'], ['1502.04201-1-30-8', '1502.04201-2-35-8'], ['1502.04201-1-30-9', '1502.04201-2-35-9'], ['1502.04201-1-30-10', '1502.04201-2-35-10'], ['1502.04201-1-20-0', '1502.04201-2-24-0'], ['1502.04201-1-53-1', '1502.04201-2-58-1'], ['1502.04201-1-53-2', '1502.04201-2-58-2'], ['1502.04201-1-49-0', '1502.04201-2-54-0'], ['1502.04201-1-49-1', '1502.04201-2-54-1'], ['1502.04201-1-49-3', '1502.04201-2-54-3'], ['1502.04201-1-49-4', '1502.04201-2-54-4'], ['1502.04201-1-49-5', '1502.04201-2-54-5'], ['1502.04201-1-49-6', '1502.04201-2-54-6'], ['1502.04201-1-49-7', '1502.04201-2-54-7'], ['1502.04201-1-46-0', '1502.04201-2-51-0'], ['1502.04201-1-46-1', '1502.04201-2-51-1'], ['1502.04201-1-46-2', '1502.04201-2-51-2'], ['1502.04201-1-47-0', '1502.04201-2-52-0'], ['1502.04201-1-47-1', '1502.04201-2-52-1'], ['1502.04201-1-47-2', '1502.04201-2-52-2'], ['1502.04201-1-73-0', '1502.04201-2-78-0'], ['1502.04201-1-40-0', '1502.04201-2-45-0'], ['1502.04201-1-40-2', '1502.04201-2-45-2'], ['1502.04201-1-40-3', '1502.04201-2-45-3'], ['1502.04201-1-40-4', '1502.04201-2-45-4'], ['1502.04201-1-40-5', '1502.04201-2-45-5'], ['1502.04201-1-56-1', '1502.04201-2-61-1'], ['1502.04201-1-56-2', '1502.04201-2-61-2'], ['1502.04201-1-56-3', '1502.04201-2-61-3'], ['1502.04201-1-56-4', '1502.04201-2-61-4'], ['1502.04201-1-32-0', '1502.04201-2-37-0'], ['1502.04201-1-32-2', '1502.04201-2-37-2'], ['1502.04201-1-21-0', '1502.04201-2-25-0'], ['1502.04201-1-21-1', '1502.04201-2-25-1'], ['1502.04201-1-21-3', '1502.04201-2-25-3'], ['1502.04201-1-21-4', '1502.04201-2-25-4'], ['1502.04201-1-21-5', '1502.04201-2-25-5'], ['1502.04201-1-21-6', '1502.04201-2-25-6'], ['1502.04201-1-21-7', '1502.04201-2-25-7'], ['1502.04201-1-21-8', '1502.04201-2-25-8'], ['1502.04201-1-21-10', '1502.04201-2-25-10'], ['1502.04201-1-19-1', '1502.04201-2-18-1'], ['1502.04201-1-19-2', '1502.04201-2-18-2'], ['1502.04201-1-15-0', '1502.04201-2-14-0'], ['1502.04201-1-33-0', '1502.04201-2-38-0'], ['1502.04201-1-33-1', '1502.04201-2-38-1'], ['1502.04201-1-33-2', '1502.04201-2-38-2'], ['1502.04201-1-33-3', '1502.04201-2-38-3'], ['1502.04201-1-33-4', '1502.04201-2-38-4'], ['1502.04201-1-41-0', '1502.04201-2-46-0'], ['1502.04201-1-68-0', '1502.04201-2-73-0'], ['1502.04201-1-68-1', '1502.04201-2-73-1'], ['1502.04201-1-68-2', '1502.04201-2-73-2'], ['1502.04201-1-68-3', '1502.04201-2-73-3'], ['1502.04201-1-68-4', '1502.04201-2-73-4'], ['1502.04201-1-68-5', '1502.04201-2-73-5'], ['1502.04201-1-9-0', '1502.04201-2-8-0'], ['1502.04201-1-9-1', '1502.04201-2-8-1'], ['1502.04201-1-34-0', '1502.04201-2-39-0'], ['1502.04201-1-34-1', '1502.04201-2-39-1'], ['1502.04201-1-37-0', '1502.04201-2-42-0'], ['1502.04201-1-37-1', '1502.04201-2-42-1'], ['1502.04201-1-37-2', '1502.04201-2-42-2'], ['1502.04201-1-37-3', '1502.04201-2-42-3'], ['1502.04201-1-37-4', '1502.04201-2-42-4'], ['1502.04201-1-37-5', '1502.04201-2-42-5'], ['1502.04201-1-37-6', '1502.04201-2-42-6'], ['1502.04201-1-4-2', '1502.04201-2-3-2'], ['1502.04201-1-62-0', '1502.04201-2-67-0'], ['1502.04201-1-54-1', '1502.04201-2-59-1'], ['1502.04201-1-54-2', '1502.04201-2-59-2'], ['1502.04201-1-54-3', '1502.04201-2-59-3'], ['1502.04201-1-54-4', '1502.04201-2-59-4'], ['1502.04201-1-0-0', '1502.04201-2-0-0'], ['1502.04201-1-0-1', '1502.04201-2-0-1'], ['1502.04201-1-12-0', '1502.04201-2-11-0'], ['1502.04201-1-51-1', '1502.04201-2-56-1'], ['1502.04201-1-51-2', '1502.04201-2-56-2'], ['1502.04201-1-38-0', '1502.04201-2-43-0'], ['1502.04201-1-11-0', '1502.04201-2-10-0'], ['1502.04201-1-11-1', '1502.04201-2-10-1'], ['1502.04201-1-66-0', '1502.04201-2-71-0'], ['1502.04201-1-66-1', '1502.04201-2-71-1'], ['1502.04201-1-66-2', '1502.04201-2-71-2'], ['1502.04201-1-65-0', '1502.04201-2-70-0'], ['1502.04201-1-65-1', '1502.04201-2-70-1'], ['1502.04201-1-65-2', '1502.04201-2-70-2'], ['1502.04201-1-65-3', '1502.04201-2-70-3'], ['1502.04201-1-65-4', '1502.04201-2-70-4'], ['1502.04201-1-44-0', '1502.04201-2-49-0'], ['1502.04201-1-25-0', '1502.04201-2-30-0'], ['1502.04201-1-10-0', '1502.04201-2-9-0'], ['1502.04201-1-22-0', '1502.04201-2-26-0'], ['1502.04201-1-35-0', '1502.04201-2-40-0'], ['1502.04201-1-35-1', '1502.04201-2-40-1'], ['1502.04201-1-35-2', '1502.04201-2-40-2'], ['1502.04201-1-5-0', '1502.04201-2-4-0'], ['1502.04201-1-18-0', '1502.04201-2-17-0'], ['1502.04201-1-18-1', '1502.04201-2-17-1'], ['1502.04201-1-18-2', '1502.04201-2-17-2'], ['1502.04201-1-69-0', '1502.04201-2-74-0'], ['1502.04201-1-69-1', '1502.04201-2-74-1'], ['1502.04201-1-69-2', '1502.04201-2-74-2'], ['1502.04201-1-57-1', '1502.04201-2-62-1'], ['1502.04201-1-57-3', '1502.04201-2-62-3'], ['1502.04201-1-57-4', '1502.04201-2-62-4'], ['1502.04201-1-57-5', '1502.04201-2-62-5'], ['1502.04201-1-64-0', '1502.04201-2-69-0'], ['1502.04201-1-64-1', '1502.04201-2-69-1'], ['1502.04201-1-64-2', '1502.04201-2-69-2'], ['1502.04201-1-64-3', '1502.04201-2-69-3'], ['1502.04201-1-64-4', '1502.04201-2-69-4'], ['1502.04201-1-64-5', '1502.04201-2-69-5'], ['1502.04201-1-64-6', '1502.04201-2-69-6'], ['1502.04201-1-48-0', '1502.04201-2-53-0'], ['1502.04201-1-3-0', '1502.04201-2-2-0'], ['1502.04201-1-14-1', '1502.04201-2-13-1'], ['1502.04201-1-14-2', '1502.04201-2-13-2'], ['1502.04201-1-14-4', '1502.04201-2-13-4'], ['1502.04201-1-14-5', '1502.04201-2-13-5'], ['1502.04201-1-14-6', '1502.04201-2-13-6'], ['1502.04201-1-14-7', '1502.04201-2-13-7'], ['1502.04201-1-24-1', '1502.04201-2-28-1'], ['1502.04201-1-24-3', '1502.04201-2-28-3'], ['1502.04201-1-24-5', '1502.04201-2-29-1'], ['1502.04201-1-24-6', '1502.04201-2-29-2'], ['1502.04201-1-24-7', '1502.04201-2-29-3'], ['1502.04201-1-24-8', '1502.04201-2-29-4'], ['1502.04201-1-24-9', '1502.04201-2-29-5'], ['1502.04201-1-24-10', '1502.04201-2-29-6'], ['1502.04201-1-24-11', '1502.04201-2-29-7'], ['1502.04201-1-21-9', '1502.04201-2-25-9'], ['1502.04201-1-4-1', '1502.04201-2-3-1'], ['1502.04201-1-21-2', '1502.04201-2-25-2']]

[['1502.04201-1-6-0', '1502.04201-2-5-0'], ['1502.04201-1-6-1', '1502.04201-2-5-1'], ['1502.04201-1-16-0', '1502.04201-2-15-0'], ['1502.04201-1-55-0', '1502.04201-2-60-0'], ['1502.04201-1-55-1', '1502.04201-2-60-1'], ['1502.04201-1-55-2', '1502.04201-2-60-2'], ['1502.04201-1-55-3', '1502.04201-2-60-3'], ['1502.04201-1-55-4', '1502.04201-2-60-4'], ['1502.04201-1-55-5', '1502.04201-2-60-5'], ['1502.04201-1-67-0', '1502.04201-2-72-0'], ['1502.04201-1-67-1', '1502.04201-2-72-1'], ['1502.04201-1-67-2', '1502.04201-2-72-2'], ['1502.04201-1-71-0', '1502.04201-2-76-0'], ['1502.04201-1-58-1', '1502.04201-2-63-1'], ['1502.04201-1-58-2', '1502.04201-2-63-2'], ['1502.04201-1-58-3', '1502.04201-2-63-3'], ['1502.04201-1-58-4', '1502.04201-2-63-4'], ['1502.04201-1-45-0', '1502.04201-2-50-0'], ['1502.04201-1-8-1', '1502.04201-2-7-1'], ['1502.04201-1-8-3', '1502.04201-2-7-3'], ['1502.04201-1-8-5', '1502.04201-2-7-5'], ['1502.04201-1-8-7', '1502.04201-2-7-7'], ['1502.04201-1-8-8', '1502.04201-2-7-8'], ['1502.04201-1-8-9', '1502.04201-2-7-9'], ['1502.04201-1-42-0', '1502.04201-2-47-0'], ['1502.04201-1-42-1', '1502.04201-2-47-1'], ['1502.04201-1-42-2', '1502.04201-2-47-2'], ['1502.04201-1-26-0', '1502.04201-2-31-0'], ['1502.04201-1-26-1', '1502.04201-2-31-1'], ['1502.04201-1-26-2', '1502.04201-2-31-2'], ['1502.04201-1-26-4', '1502.04201-2-31-4'], ['1502.04201-1-59-0', '1502.04201-2-64-0'], ['1502.04201-1-59-1', '1502.04201-2-64-1'], ['1502.04201-1-59-2', '1502.04201-2-64-2'], ['1502.04201-1-59-3', '1502.04201-2-64-3'], ['1502.04201-1-59-4', '1502.04201-2-64-4'], ['1502.04201-1-60-0', '1502.04201-2-65-0'], ['1502.04201-1-60-1', '1502.04201-2-65-1'], ['1502.04201-1-60-2', '1502.04201-2-65-2'], ['1502.04201-1-60-3', '1502.04201-2-65-3'], ['1502.04201-1-60-4', '1502.04201-2-65-4'], ['1502.04201-1-60-5', '1502.04201-2-65-5'], ['1502.04201-1-60-6', '1502.04201-2-65-6'], ['1502.04201-1-60-7', '1502.04201-2-65-7'], ['1502.04201-1-39-1', '1502.04201-2-44-1'], ['1502.04201-1-39-2', '1502.04201-2-44-2'], ['1502.04201-1-70-0', '1502.04201-2-75-0'], ['1502.04201-1-70-1', '1502.04201-2-75-1'], ['1502.04201-1-29-0', '1502.04201-2-34-0'], ['1502.04201-1-36-0', '1502.04201-2-41-0'], ['1502.04201-1-36-1', '1502.04201-2-41-1'], ['1502.04201-1-36-2', '1502.04201-2-41-2'], ['1502.04201-1-61-0', '1502.04201-2-66-0'], ['1502.04201-1-61-1', '1502.04201-2-66-1'], ['1502.04201-1-61-2', '1502.04201-2-66-2'], ['1502.04201-1-61-3', '1502.04201-2-66-3'], ['1502.04201-1-61-4', '1502.04201-2-66-4'], ['1502.04201-1-61-5', '1502.04201-2-66-5'], ['1502.04201-1-61-6', '1502.04201-2-66-6'], ['1502.04201-1-61-7', '1502.04201-2-66-7'], ['1502.04201-1-61-8', '1502.04201-2-66-8'], ['1502.04201-1-61-9', '1502.04201-2-66-9'], ['1502.04201-1-61-10', '1502.04201-2-66-10'], ['1502.04201-1-17-0', '1502.04201-2-16-0'], ['1502.04201-1-28-0', '1502.04201-2-33-0'], ['1502.04201-1-28-1', '1502.04201-2-33-1'], ['1502.04201-1-28-2', '1502.04201-2-33-2'], ['1502.04201-1-30-1', '1502.04201-2-35-1'], ['1502.04201-1-30-2', '1502.04201-2-35-2'], ['1502.04201-1-30-3', '1502.04201-2-35-3'], ['1502.04201-1-30-4', '1502.04201-2-35-4'], ['1502.04201-1-30-5', '1502.04201-2-35-5'], ['1502.04201-1-30-7', '1502.04201-2-35-7'], ['1502.04201-1-30-8', '1502.04201-2-35-8'], ['1502.04201-1-30-9', '1502.04201-2-35-9'], ['1502.04201-1-30-10', '1502.04201-2-35-10'], ['1502.04201-1-20-0', '1502.04201-2-24-0'], ['1502.04201-1-53-1', '1502.04201-2-58-1'], ['1502.04201-1-53-2', '1502.04201-2-58-2'], ['1502.04201-1-49-0', '1502.04201-2-54-0'], ['1502.04201-1-49-1', '1502.04201-2-54-1'], ['1502.04201-1-49-3', '1502.04201-2-54-3'], ['1502.04201-1-49-4', '1502.04201-2-54-4'], ['1502.04201-1-49-5', '1502.04201-2-54-5'], ['1502.04201-1-49-6', '1502.04201-2-54-6'], ['1502.04201-1-49-7', '1502.04201-2-54-7'], ['1502.04201-1-46-0', '1502.04201-2-51-0'], ['1502.04201-1-46-1', '1502.04201-2-51-1'], ['1502.04201-1-46-2', '1502.04201-2-51-2'], ['1502.04201-1-47-0', '1502.04201-2-52-0'], ['1502.04201-1-47-1', '1502.04201-2-52-1'], ['1502.04201-1-47-2', '1502.04201-2-52-2'], ['1502.04201-1-73-0', '1502.04201-2-78-0'], ['1502.04201-1-40-0', '1502.04201-2-45-0'], ['1502.04201-1-40-2', '1502.04201-2-45-2'], ['1502.04201-1-40-3', '1502.04201-2-45-3'], ['1502.04201-1-40-4', '1502.04201-2-45-4'], ['1502.04201-1-40-5', '1502.04201-2-45-5'], ['1502.04201-1-56-1', '1502.04201-2-61-1'], ['1502.04201-1-56-2', '1502.04201-2-61-2'], ['1502.04201-1-56-3', '1502.04201-2-61-3'], ['1502.04201-1-56-4', '1502.04201-2-61-4'], ['1502.04201-1-32-0', '1502.04201-2-37-0'], ['1502.04201-1-32-2', '1502.04201-2-37-2'], ['1502.04201-1-21-0', '1502.04201-2-25-0'], ['1502.04201-1-21-1', '1502.04201-2-25-1'], ['1502.04201-1-21-3', '1502.04201-2-25-3'], ['1502.04201-1-21-4', '1502.04201-2-25-4'], ['1502.04201-1-21-5', '1502.04201-2-25-5'], ['1502.04201-1-21-6', '1502.04201-2-25-6'], ['1502.04201-1-21-7', '1502.04201-2-25-7'], ['1502.04201-1-21-8', '1502.04201-2-25-8'], ['1502.04201-1-21-10', '1502.04201-2-25-10'], ['1502.04201-1-19-1', '1502.04201-2-18-1'], ['1502.04201-1-19-2', '1502.04201-2-18-2'], ['1502.04201-1-15-0', '1502.04201-2-14-0'], ['1502.04201-1-33-0', '1502.04201-2-38-0'], ['1502.04201-1-33-1', '1502.04201-2-38-1'], ['1502.04201-1-33-2', '1502.04201-2-38-2'], ['1502.04201-1-33-3', '1502.04201-2-38-3'], ['1502.04201-1-33-4', '1502.04201-2-38-4'], ['1502.04201-1-41-0', '1502.04201-2-46-0'], ['1502.04201-1-68-0', '1502.04201-2-73-0'], ['1502.04201-1-68-1', '1502.04201-2-73-1'], ['1502.04201-1-68-2', '1502.04201-2-73-2'], ['1502.04201-1-68-3', '1502.04201-2-73-3'], ['1502.04201-1-68-4', '1502.04201-2-73-4'], ['1502.04201-1-68-5', '1502.04201-2-73-5'], ['1502.04201-1-9-0', '1502.04201-2-8-0'], ['1502.04201-1-9-1', '1502.04201-2-8-1'], ['1502.04201-1-34-0', '1502.04201-2-39-0'], ['1502.04201-1-34-1', '1502.04201-2-39-1'], ['1502.04201-1-37-0', '1502.04201-2-42-0'], ['1502.04201-1-37-1', '1502.04201-2-42-1'], ['1502.04201-1-37-2', '1502.04201-2-42-2'], ['1502.04201-1-37-3', '1502.04201-2-42-3'], ['1502.04201-1-37-4', '1502.04201-2-42-4'], ['1502.04201-1-37-5', '1502.04201-2-42-5'], ['1502.04201-1-37-6', '1502.04201-2-42-6'], ['1502.04201-1-4-2', '1502.04201-2-3-2'], ['1502.04201-1-62-0', '1502.04201-2-67-0'], ['1502.04201-1-54-1', '1502.04201-2-59-1'], ['1502.04201-1-54-2', '1502.04201-2-59-2'], ['1502.04201-1-54-3', '1502.04201-2-59-3'], ['1502.04201-1-54-4', '1502.04201-2-59-4'], ['1502.04201-1-0-0', '1502.04201-2-0-0'], ['1502.04201-1-0-1', '1502.04201-2-0-1'], ['1502.04201-1-12-0', '1502.04201-2-11-0'], ['1502.04201-1-51-1', '1502.04201-2-56-1'], ['1502.04201-1-51-2', '1502.04201-2-56-2'], ['1502.04201-1-38-0', '1502.04201-2-43-0'], ['1502.04201-1-11-0', '1502.04201-2-10-0'], ['1502.04201-1-11-1', '1502.04201-2-10-1'], ['1502.04201-1-66-0', '1502.04201-2-71-0'], ['1502.04201-1-66-1', '1502.04201-2-71-1'], ['1502.04201-1-66-2', '1502.04201-2-71-2'], ['1502.04201-1-65-0', '1502.04201-2-70-0'], ['1502.04201-1-65-1', '1502.04201-2-70-1'], ['1502.04201-1-65-2', '1502.04201-2-70-2'], ['1502.04201-1-65-3', '1502.04201-2-70-3'], ['1502.04201-1-65-4', '1502.04201-2-70-4'], ['1502.04201-1-44-0', '1502.04201-2-49-0'], ['1502.04201-1-25-0', '1502.04201-2-30-0'], ['1502.04201-1-10-0', '1502.04201-2-9-0'], ['1502.04201-1-22-0', '1502.04201-2-26-0'], ['1502.04201-1-35-0', '1502.04201-2-40-0'], ['1502.04201-1-35-1', '1502.04201-2-40-1'], ['1502.04201-1-35-2', '1502.04201-2-40-2'], ['1502.04201-1-5-0', '1502.04201-2-4-0'], ['1502.04201-1-18-0', '1502.04201-2-17-0'], ['1502.04201-1-18-1', '1502.04201-2-17-1'], ['1502.04201-1-18-2', '1502.04201-2-17-2'], ['1502.04201-1-69-0', '1502.04201-2-74-0'], ['1502.04201-1-69-1', '1502.04201-2-74-1'], ['1502.04201-1-69-2', '1502.04201-2-74-2'], ['1502.04201-1-57-1', '1502.04201-2-62-1'], ['1502.04201-1-57-3', '1502.04201-2-62-3'], ['1502.04201-1-57-4', '1502.04201-2-62-4'], ['1502.04201-1-57-5', '1502.04201-2-62-5'], ['1502.04201-1-64-0', '1502.04201-2-69-0'], ['1502.04201-1-64-1', '1502.04201-2-69-1'], ['1502.04201-1-64-2', '1502.04201-2-69-2'], ['1502.04201-1-64-3', '1502.04201-2-69-3'], ['1502.04201-1-64-4', '1502.04201-2-69-4'], ['1502.04201-1-64-5', '1502.04201-2-69-5'], ['1502.04201-1-64-6', '1502.04201-2-69-6'], ['1502.04201-1-48-0', '1502.04201-2-53-0'], ['1502.04201-1-3-0', '1502.04201-2-2-0'], ['1502.04201-1-14-1', '1502.04201-2-13-1'], ['1502.04201-1-14-2', '1502.04201-2-13-2'], ['1502.04201-1-14-4', '1502.04201-2-13-4'], ['1502.04201-1-14-5', '1502.04201-2-13-5'], ['1502.04201-1-14-6', '1502.04201-2-13-6'], ['1502.04201-1-14-7', '1502.04201-2-13-7'], ['1502.04201-1-24-1', '1502.04201-2-28-1'], ['1502.04201-1-24-3', '1502.04201-2-28-3'], ['1502.04201-1-24-5', '1502.04201-2-29-1'], ['1502.04201-1-24-6', '1502.04201-2-29-2'], ['1502.04201-1-24-7', '1502.04201-2-29-3'], ['1502.04201-1-24-8', '1502.04201-2-29-4'], ['1502.04201-1-24-9', '1502.04201-2-29-5'], ['1502.04201-1-24-10', '1502.04201-2-29-6'], ['1502.04201-1-24-11', '1502.04201-2-29-7']]

[['1502.04201-1-21-9', '1502.04201-2-25-9'], ['1502.04201-1-4-1', '1502.04201-2-3-1']]

[]

[['1502.04201-1-21-2', '1502.04201-2-25-2']]

[]

['1502.04201-1-1-0', '1502.04201-1-4-0', '1502.04201-1-7-0', '1502.04201-1-8-0', '1502.04201-1-8-2', '1502.04201-1-8-4', '1502.04201-1-8-6', '1502.04201-1-13-0', '1502.04201-1-13-1', '1502.04201-1-14-0', '1502.04201-1-14-3', '1502.04201-1-19-0', '1502.04201-1-19-3', '1502.04201-1-23-0', '1502.04201-1-24-0', '1502.04201-1-24-2', '1502.04201-1-24-4', '1502.04201-1-26-3', '1502.04201-1-26-5', '1502.04201-1-26-6', '1502.04201-1-30-0', '1502.04201-1-30-6', '1502.04201-1-30-11', '1502.04201-1-30-12', '1502.04201-1-32-1', '1502.04201-1-38-1', '1502.04201-1-39-3', '1502.04201-1-40-1', '1502.04201-1-40-6', '1502.04201-1-41-1', '1502.04201-1-44-1', '1502.04201-1-49-2', '1502.04201-1-49-8', '1502.04201-1-50-0', '1502.04201-1-51-0', '1502.04201-1-52-0', '1502.04201-1-53-0', '1502.04201-1-53-3', '1502.04201-1-53-4', '1502.04201-1-54-0', '1502.04201-1-56-0', '1502.04201-1-56-5', '1502.04201-1-57-0', '1502.04201-1-57-2', '1502.04201-1-58-0', '1502.04201-1-59-5', '1502.04201-1-61-11', '1502.04201-1-68-6', '1502.04201-1-72-0', '1502.04201-2-3-0', '1502.04201-2-6-0', '1502.04201-2-7-0', '1502.04201-2-7-2', '1502.04201-2-7-4', '1502.04201-2-7-6', '1502.04201-2-12-0', '1502.04201-2-12-1', '1502.04201-2-13-0', '1502.04201-2-13-3', '1502.04201-2-18-0', '1502.04201-2-18-3', '1502.04201-2-19-0', '1502.04201-2-20-0', '1502.04201-2-27-0', '1502.04201-2-28-0', '1502.04201-2-28-2', '1502.04201-2-29-0', '1502.04201-2-31-3', '1502.04201-2-31-5', '1502.04201-2-31-6', '1502.04201-2-35-0', '1502.04201-2-35-6', '1502.04201-2-35-11', '1502.04201-2-35-12', '1502.04201-2-37-1', '1502.04201-2-43-1', '1502.04201-2-44-3', '1502.04201-2-45-1', '1502.04201-2-45-6', '1502.04201-2-46-1', '1502.04201-2-49-1', '1502.04201-2-54-2', '1502.04201-2-54-8', '1502.04201-2-55-0', '1502.04201-2-56-0', '1502.04201-2-57-0', '1502.04201-2-58-0', '1502.04201-2-58-3', '1502.04201-2-58-4', '1502.04201-2-59-0', '1502.04201-2-61-0', '1502.04201-2-61-5', '1502.04201-2-62-0', '1502.04201-2-62-2', '1502.04201-2-63-0', '1502.04201-2-64-5', '1502.04201-2-66-11', '1502.04201-2-73-6', '1502.04201-2-77-0']

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

https://arxiv.org/abs/1502.04201

1402.0074

{'1402.0074-1-0-0': 'A simple construction of indecomposable higher Chow cycles in a product of an elliptic curve', '1402.0074-1-1-0': 'Dedicated to the 60th birthday of Professor James D. Lewis', '1402.0074-1-2-0': '# Introduction', '1402.0074-1-3-0': "Let [MATH] be Bloch's higher Chow groups of a projective smooth variety [MATH] over [MATH].", '1402.0074-1-3-1': 'A higher Chow cycle [MATH] is called indecomposable if it does not belong to the image of the map of the product [EQUATION].', '1402.0074-1-3-2': 'In particular, [MATH] is indecomposable if and only if it does not belong to the image of [MATH].', '1402.0074-1-4-0': 'Quite a lot of examples of indecomposable higher Chow cycles are obtained by many people.', '1402.0074-1-4-1': 'One of them is the following theorem due to Lewis and Gordon .', '1402.0074-1-4-2': '[Lewis - Gordon, [CITATION] Thm.1] Let [MATH] be a product of elliptic curves over [MATH].', '1402.0074-1-4-3': 'If [MATH] is sufficiently general, then there exists a [MATH]-regulator indecomposable cycle [MATH] .', '1402.0074-1-5-0': 'Here we mean by [MATH]-regulator indecomposable for [MATH] or [MATH] that the regulator class [MATH] in the Deligne-Beilinson cohomology group with coefficients in [MATH] does not belong to the image of [MATH].', '1402.0074-1-5-1': 'In other words, [MATH] is [MATH]-regulator indecomposable if and only if [EQUATION].', '1402.0074-1-5-2': 'Obviously [MATH]', '1402.0074-1-6-0': 'In this note, we give a new proof of Theorem [REF] based on the technique of the Kummer surface with an elliptic fibration.', '1402.0074-1-6-1': 'The higher Chow cycle which we will discuss is different from the cycles studied in [CITATION] or [CITATION], and the proof is simpler.', '1402.0074-1-7-0': '# Alternative proof of the theorem of Lewis-Gordon', '1402.0074-1-8-0': '## Kummer surface', '1402.0074-1-9-0': 'Let [EQUATION] be an elliptic curve over [MATH].', '1402.0074-1-9-1': 'Let [MATH] be the Kummer surface associated to the product [MATH].', '1402.0074-1-9-2': 'The birational model of [MATH] is given by the double sextic [EQUATION].', '1402.0074-1-9-3': 'Changing the variables [MATH], [MATH] and [MATH], we get [EQUATION]', '1402.0074-1-9-4': 'This gives an elliptic fibration [EQUATION] where [MATH] is the affine parameter of [MATH].', '1402.0074-1-9-5': 'We fix a section [MATH] of [MATH].', '1402.0074-1-9-6': 'The singular fibers are located at [MATH].', '1402.0074-1-9-7': 'Two additive fibers appear over [MATH] and both are of Kodaira type [MATH].', '1402.0074-1-9-8': 'If [MATH], the singular fibers over [MATH] are multiplicative of type [MATH], [MATH], [MATH] respectively, and if [MATH], two fibers over [MATH] are multiplicative of type [MATH].', '1402.0074-1-9-9': 'Let [MATH] be the subgroup generated by irreducible components of singular fibers and the section [MATH].', '1402.0074-1-9-10': 'The rank of [MATH] is [MATH] if [MATH], and [MATH] if [MATH].', '1402.0074-1-9-11': 'As is well-known, [EQUATION]', '1402.0074-1-9-12': 'If [MATH], there is the isomorphism [EQUATION]', '1402.0074-1-9-13': 'We will construct a [MATH]-regulator indecomposable cycle in [MATH] for a general [MATH].', '1402.0074-1-9-14': 'This is equivalent to do so for [MATH] by [REF].', '1402.0074-1-10-0': '## A higher Chow cycle arising from a multiplicative fiber', '1402.0074-1-11-0': 'Let [MATH]) be a higher Chow cycle associated to [MATH] which is vertical to [MATH] (see Lemma [REF] below).', '1402.0074-1-11-1': "The goal is to show nonvanishing [EQUATION] for uncountably many [MATH]'s.", '1402.0074-1-11-2': 'Indeed, since [MATH] is vertical to [MATH], [REF] implies nonvanishing [EQUATION]', '1402.0074-1-11-3': 'Furthermore, if [MATH] has no CM, then [REF] means that [MATH] is [MATH]-regulator indecomposable by [REF].', '1402.0074-1-11-4': "Since there are at most countably many [MATH]'s such that [MATH] has a CM, Theorem [REF] follows.", '1402.0074-1-12-0': 'Let [MATH] and [MATH] be the family of [MATH].', '1402.0074-1-12-1': '[EQUATION]', '1402.0074-1-12-2': 'Put [MATH].', '1402.0074-1-12-3': 'Let [MATH] be the higher cycle such that [MATH].', '1402.0074-1-13-0': 'Let [MATH] be the boundary map arising from the localization exact sequence.', '1402.0074-1-13-1': 'Then [MATH] is non-torsion.', '1402.0074-1-14-0': 'We look at [MATH] over a neighborhood of [MATH].', '1402.0074-1-14-1': 'Let [MATH].', '1402.0074-1-14-2': 'We take a blow-up [MATH] along [MATH] and [MATH] and let [MATH] be the exceptional surfaces respectively.', '1402.0074-1-14-3': 'Let [MATH] be the generic fiber of [MATH].', '1402.0074-1-14-4': 'Then [MATH] and [MATH] where [MATH] and [MATH] are the proper transforms of fibers over [MATH] in [MATH], and [MATH] and [MATH] are the generic fibers of [MATH].', '1402.0074-1-14-5': 'If [MATH], the fiber [MATH] is the Neron 4-gon, say [MATH] where [MATH] and [MATH] are the special fiber in [MATH] and [MATH] and [MATH] is the proper transform of the fiber over [MATH] in [MATH].', '1402.0074-1-15-0': '0.1in ( 55.3000, 38.1000)( 10.2000,-42.3000)', '1402.0074-1-16-0': 'pn 8 pa 4840 1260 pa 4840 2840 fp', '1402.0074-1-17-0': 'pn 8 pa 4570 2620 pa 6550 2630 fp', '1402.0074-1-18-0': 'pn 8 ar 2120 1340 650 900 1.5378593 4.7123890', '1402.0074-1-19-0': 'pn 8 ar 1350 1320 650 900 4.7123890 1.6037333', '1402.0074-1-20-0': 'pn 8 ar 2092 3330 650 900 1.5378593 4.7123890', '1402.0074-1-21-0': 'pn 8 ar 1322 3310 650 900 4.7123890 1.6037333', '1402.0074-1-22-0': 'pn 8 pa 3060 1500 pa 4010 1850 fp sh 1 pa 4010 1850 pa 3954 1808 pa 3960 1832 pa 3942 1846 pa 4010 1850 fp', '1402.0074-1-23-0': 'pn 8 pa 3080 3020 pa 3980 2620 fp sh 1 pa 3980 2620 pa 3912 2630 pa 3932 2642 pa 3928 2666 pa 3980 2620 fp', '1402.0074-1-24-0': 'pn 8 pa 4510 1490 pa 6460 1490 fp', '1402.0074-1-25-0': 'pn 8 pa 6190 1230 pa 6190 2820 fp', '1402.0074-1-26-0': '(53.8000,-14.0000)(0,0)[lb][MATH]', '1402.0074-1-27-0': '(53.9000,-28.8000)(0,0)[lb][MATH]', '1402.0074-1-28-0': '(45.1000,-20.1000)(0,0)[lb][MATH]', '1402.0074-1-29-0': '(63.8000,-19.8000)(0,0)[lb][MATH]', '1402.0074-1-30-0': '(10.2000,-13.1000)(0,0)[lb][MATH]', '1402.0074-1-31-0': '(21.4000,-13.1000)(0,0)[lb][MATH]', '1402.0074-1-32-0': '(21.5000,-33.5000)(0,0)[lb][MATH]', '1402.0074-1-33-0': '(10.4000,-33.5000)(0,0)[lb][MATH]', '1402.0074-1-34-0': '(32.7000,-36.6000)(0,0)[lb][MATH]', '1402.0074-1-35-0': 'Let [MATH] the specialization map.', '1402.0074-1-35-1': 'Now it is not hard to see [EQUATION] and [EQUATION]', '1402.0074-1-35-2': 'This completes the proof.', '1402.0074-1-36-0': '## Proof of Theorem [REF]', '1402.0074-1-37-0': 'Let [EQUATION] be the Deligne-Beilinson cohomology class.', '1402.0074-1-37-1': 'Lemma [REF] together with the commutative diagram [EQUATION] yields nonvanishing [MATH].', '1402.0074-1-37-2': 'There is the isomorphism [EQUATION] with the extension group of admissible variations of mixed Hodge structures on [MATH].', '1402.0074-1-37-3': 'Let [MATH] be a polarized variation of Hodge structures whose fiber [MATH] at [MATH] is as follows [EQUATION]', '1402.0074-1-37-4': "Since [MATH] is vertical to [MATH] by the construction, it turns out that the class [MATH] belongs to [MATH], and hence it defines a non-trivial extension [EQUATION] of admissible VMHS's on [MATH].", '1402.0074-1-37-5': "Tensoring with [MATH], one has an extension [EQUATION] of real VMHS's.", '1402.0074-1-37-6': 'One can show that this is also non-trivial in the same way as above.', '1402.0074-1-38-0': "To finish the proof of Theorem [REF] it is enough to show that for uncountably many [MATH], [EQUATION] is a nontrivial extension of real MHS's.", '1402.0074-1-39-0': 'Let [MATH] and [MATH] be the Hodge bundles.', '1402.0074-1-39-1': 'For [MATH] or [MATH], we put [MATH] and [EQUATION] the sheaf of horizontal sections where [MATH] denotes the Gauss-Manin connection.', '1402.0074-1-39-2': 'As is well-known, there is the injective map [EQUATION]', '1402.0074-1-39-3': 'Let us wite [MATH] the normal function associated to [MATH].', '1402.0074-1-39-4': 'If [MATH], the zero locus of the normal function is at most a countable set.', '1402.0074-1-39-5': 'Thus it immediately follows that [MATH] is [MATH]-regulator indecomposable for a general [MATH].', '1402.0074-1-39-6': 'However to show the "[MATH]-regulator indecomposability", we need a little more.', '1402.0074-1-40-0': 'Let [MATH] be a unit disk, and [MATH] a polarized VHS over [MATH] with Hodge numbers [MATH].', '1402.0074-1-40-1': 'Let [MATH] and let [MATH] be the sheaf of horizontal sections.', '1402.0074-1-40-2': 'Then the restriction map [EQUATION] is injective.', '1402.0074-1-41-0': 'Lemma [REF] finishes the proof of Theorem [REF].', '1402.0074-1-41-1': 'Indeed, there is [MATH] such that [MATH] in [MATH].', '1402.0074-1-41-2': 'Then obviously [MATH] for all [MATH], which is equivalent to say that [REF] is non-trivial.', '1402.0074-1-41-3': 'This is the desired assertion.', '1402.0074-1-42-0': 'The proof below shows that the injectivity of [REF] fails if we replace [MATH] with [MATH].', '1402.0074-1-42-1': "I don't know whether the injectivity on [MATH] is true for a general [MATH].", '1402.0074-1-43-0': 'Proof of Lemma [REF].', '1402.0074-1-43-1': 'Note [MATH] is a constant sheaf.', '1402.0074-1-43-2': 'We may assume that the polarization matrix on [MATH] is as follows.', '1402.0074-1-43-3': '[EQUATION].', '1402.0074-1-43-4': 'Then the period domain [MATH] for the type [MATH] and [MATH] is given as follows [EQUATION].', '1402.0074-1-43-5': 'There is a holomorphic map [MATH] such that [MATH] where [MATH] denotes the universal polarized VHS on [MATH].', '1402.0074-1-44-0': 'Let [MATH] be a normal function, and fix a lifting [EQUATION].', '1402.0074-1-44-1': 'Assume [MATH] for all [MATH].', '1402.0074-1-44-2': 'We then want to show [MATH].', '1402.0074-1-44-3': 'Note [EQUATION].', '1402.0074-1-44-4': 'Therefore there are [MATH]-valued functions [MATH] such that [EQUATION]', '1402.0074-1-44-5': 'This implies [EQUATION]', '1402.0074-1-44-6': 'Moreover since [MATH], [MATH] is uniquely determined, and hence so are [MATH] and [MATH].', '1402.0074-1-44-7': 'In particular they are [MATH]-valued [MATH]-functions.', '1402.0074-1-45-0': 'Assume first that [MATH] is not a constant.', '1402.0074-1-45-1': 'Let us write [MATH] and [MATH] with [MATH].', '1402.0074-1-45-2': 'Since the LHS of [REF] is [MATH]-harmonic, one has [EQUATION] by using the Cauchy-Riemann equations [MATH], [MATH] etc.', '1402.0074-1-45-3': 'This implies [EQUATION] with some [MATH]-valued function [MATH].', '1402.0074-1-45-4': 'Since [MATH] is [MATH]-valued, it must be a real constant [MATH].', '1402.0074-1-45-5': 'So that we have [MATH] for [MATH].', '1402.0074-1-45-6': 'By [REF], one has [MATH].', '1402.0074-1-45-7': 'Since [MATH] is holomorphic, it must be a real constant [MATH].', '1402.0074-1-45-8': 'Summing up this, we have [EQUATION]', '1402.0074-1-45-9': 'Since [MATH] is a horizontal section, [MATH], namely [EQUATION] so that we have [MATH].', '1402.0074-1-45-10': 'Hence [MATH], the desired assertion.', '1402.0074-1-46-0': 'There remains the case that [MATH] is a constant.', '1402.0074-1-46-1': 'Then since [MATH] is a [MATH]-valued holomorphic function by [REF], it must be a real constant.', '1402.0074-1-46-2': 'Hence [EQUATION].', '1402.0074-1-46-3': 'Since [MATH] is a horizontal section, it satisfies [EQUATION].', '1402.0074-1-46-4': 'Since [MATH], [MATH] namely [MATH] is a constant.', '1402.0074-1-46-5': 'Hence [MATH] and all [MATH] are constants by [REF].', '1402.0074-1-46-6': 'Now it is immediate to see [MATH].', '1402.0074-1-47-0': 'This completes the proof of Lemma [REF].', '1402.0074-1-48-0': '## Complement', '1402.0074-1-49-0': 'In [REF], we used the following result to construct the higher Chow cycle [MATH] or [MATH].', '1402.0074-1-49-1': "Since I don't find a literature where the proof is written, I put it here for the sake of completeness.", '1402.0074-1-50-0': 'Let [MATH] be an elliptic fibration over a field [MATH] of characteristic[MATH], with a (fixed) section [MATH].', '1402.0074-1-50-1': 'Let [MATH] be a split multiplicative fiber, and let [MATH].', '1402.0074-1-50-2': 'Then there is an exact sequence [EQUATION] where [MATH] is the normalization.', '1402.0074-1-50-3': 'There exists a higher Chow cycle [MATH] such that [MATH] and [MATH] is vertical to the section [MATH] and fibral divisors (i.e. divisors in fibers).', '1402.0074-1-50-4': 'Here "vertical to [MATH]" means that it lies in the kernel of the composition [MATH].', '1402.0074-1-51-0': 'We omit to show the exact sequence [REF] (easy exercise).', '1402.0074-1-51-1': 'We show the existence of [MATH].', '1402.0074-1-51-2': 'In this proof, we use [MATH]-groups rather than higher Chow groups.', '1402.0074-1-51-3': 'Let [MATH] be the irreducible decomposition.', '1402.0074-1-51-4': 'It is enough to construct [MATH] such that [MATH] and [MATH] is vertical to each [MATH] and [MATH] because it is obviously vertical to the other fibral divisors.', '1402.0074-1-51-5': 'Since [MATH] is split multiplicative, each [MATH] is geometrically irreducible and the singularities of [MATH] are [MATH]-rational.', '1402.0074-1-51-6': 'Therefore we may assume [MATH] by the standard norm argument.', '1402.0074-1-52-0': 'It is enough to show that the image of the composition [EQUATION] coincides with that of [EQUATION].', '1402.0074-1-52-1': 'Indeed, the above implies that there is a cycle [MATH] such that [MATH] and [MATH] is vertical to each [MATH].', '1402.0074-1-52-2': 'Let [MATH].', '1402.0074-1-52-3': 'Then [MATH] for suitable [MATH] can be vertical to the section [MATH] and [MATH].', '1402.0074-1-52-4': 'Moreover [MATH].', '1402.0074-1-53-0': 'To do the above we may replace [MATH] with [MATH] the formal neighborhood around [MATH].', '1402.0074-1-53-1': 'Then it is enough to show that there is [MATH] such that [MATH] and [MATH] in [MATH].', '1402.0074-1-53-2': 'To do this, we may further replace [MATH] with [MATH] the minimal desingularization of [MATH] for some [MATH] due to a commutative diagram [EQUATION] where [MATH].', '1402.0074-1-53-3': 'Thus we can assume [MATH] is defined by a Weierstrass equation [EQUATION].', '1402.0074-1-53-4': 'Then letting [MATH] be the boundary map, we put [EQUATION].', '1402.0074-1-53-5': 'This satisfies [MATH] and [MATH] in [MATH].', '1402.0074-1-54-0': 'Department of Mathematics, Hokkaido University, Sapporo 060-0810, JAPAN', '1402.0074-1-55-0': 'asakura@math.sci.hokudai.ac.jp'}

{'1402.0074-2-0-0': 'A simple construction of indecomposable higher Chow cycles in elliptic surfaces', '1402.0074-2-1-0': 'Dedicated to the 60th birthday of Professor James D. Lewis', '1402.0074-2-2-0': '# Introduction', '1402.0074-2-3-0': "Let [MATH] be Bloch's higher Chow groups of a projective smooth variety [MATH] over [MATH].", '1402.0074-2-3-1': 'A higher Chow cycle [MATH] is called indecomposable if it does not belong to the image of the map of the product [EQUATION].', '1402.0074-2-3-2': 'Of particular interest to us is [MATH].', '1402.0074-2-3-3': 'For [MATH] or [MATH], we say [MATH]-regulator indecomposable if the regulator class [MATH] in the Deligne-Beilinson cohomology group with coefficients in [MATH] does not belong to the image of [MATH].', '1402.0074-2-3-4': 'In other words, [MATH] is [MATH]-regulator indecomposable if and only if [EQUATION].', '1402.0074-2-3-5': 'Obviously [MATH]', '1402.0074-2-4-0': 'Quite a lot of examples of [MATH] or [MATH]-regulator indecomposable cycles are obtained by many people ([CITATION], [CITATION], [CITATION], [CITATION], [CITATION] and more).', '1402.0074-2-5-0': 'In this note we construct [MATH]-regulator indecomposable cycles for [MATH] an elliptic surface which satisfies certain conditions.', '1402.0074-2-5-1': 'The main theorem is the following.', '1402.0074-2-5-2': '[EQUATION]', '1402.0074-2-5-3': 'The key assumption is a "degeneration of [MATH]", which often appears in a family of elliptic surfaces.', '1402.0074-2-6-0': 'Let [MATH] be a smooth irreducible curve over [MATH].', '1402.0074-2-6-1': 'Let [EQUATION] be an elliptic fibration over [MATH] with a section [MATH].', '1402.0074-2-6-2': 'This means that [MATH] and [MATH] are projective smooth morphisms of relative dimension [MATH] and [MATH] respectively, and the general fiber of [MATH] is an elliptic curve.', '1402.0074-2-6-3': 'For a point [MATH] we denote [MATH] or [MATH] the fibers over [MATH].', '1402.0074-2-6-4': 'Assume that the following conditions hold.', '1402.0074-2-7-0': '[(1)] Let [MATH] be the generic point of [MATH].', '1402.0074-2-7-1': 'Then there is a split multiplicative fiber [MATH] of Kodaira type [MATH], [MATH].', '1402.0074-2-7-2': 'Then there is a closed point [MATH] such that the specialization [MATH] is multiplicative of type [MATH] with [MATH].', '1402.0074-2-8-0': 'Then the composition [EQUATION] is non-zero for a general [MATH].', '1402.0074-2-8-1': 'Here [MATH] denotes the subgroup generated by components of singular fibers and the section [MATH].', '1402.0074-2-8-2': 'In particular, if [MATH], then there is a [MATH]-regulator indecomposable higher Chow cycle.', '1402.0074-2-9-0': 'The key assumption is a "degeneration of [MATH]", which often appears in a family of elliptic surfaces.', '1402.0074-2-10-0': 'In [REF], we will apply Theorem [REF] to construct a [MATH]-reg.', '1402.0074-2-10-1': 'indecomp.', '1402.0074-2-10-2': 'cycle in a self-product of elliptic curves.', '1402.0074-2-10-3': 'However, to do it in more general situation, the computation of the Picard number might be an obstacle.', '1402.0074-2-10-4': 'Indeed it is easy to compute the rank of [MATH], whereas there is no general method to do it for [MATH], and usually it is done by case-by-case analysis.'}

[['1402.0074-1-3-0', '1402.0074-2-3-0'], ['1402.0074-1-3-1', '1402.0074-2-3-1'], ['1402.0074-1-5-1', '1402.0074-2-3-4'], ['1402.0074-1-0-0', '1402.0074-2-0-0'], ['1402.0074-1-5-0', '1402.0074-2-3-3'], ['1402.0074-1-14-3', '1402.0074-2-7-0'], ['1402.0074-1-9-13', '1402.0074-2-5-0'], ['1402.0074-1-50-0', '1402.0074-2-6-1'], ['1402.0074-1-9-0', '1402.0074-2-6-1'], ['1402.0074-1-9-9', '1402.0074-2-8-1'], ['1402.0074-1-4-3', '1402.0074-2-8-2']]

[['1402.0074-1-3-0', '1402.0074-2-3-0'], ['1402.0074-1-3-1', '1402.0074-2-3-1'], ['1402.0074-1-5-1', '1402.0074-2-3-4']]

[['1402.0074-1-0-0', '1402.0074-2-0-0'], ['1402.0074-1-5-0', '1402.0074-2-3-3']]

[]

[['1402.0074-1-14-3', '1402.0074-2-7-0'], ['1402.0074-1-9-13', '1402.0074-2-5-0']]

[['1402.0074-1-50-0', '1402.0074-2-6-1'], ['1402.0074-1-9-0', '1402.0074-2-6-1'], ['1402.0074-1-9-9', '1402.0074-2-8-1'], ['1402.0074-1-4-3', '1402.0074-2-8-2']]

['1402.0074-1-1-0', '1402.0074-1-5-2', '1402.0074-1-12-1', '1402.0074-1-12-2', '1402.0074-1-14-1', '1402.0074-1-15-0', '1402.0074-1-16-0', '1402.0074-1-17-0', '1402.0074-1-18-0', '1402.0074-1-19-0', '1402.0074-1-20-0', '1402.0074-1-21-0', '1402.0074-1-22-0', '1402.0074-1-23-0', '1402.0074-1-24-0', '1402.0074-1-25-0', '1402.0074-1-26-0', '1402.0074-1-27-0', '1402.0074-1-28-0', '1402.0074-1-29-0', '1402.0074-1-30-0', '1402.0074-1-31-0', '1402.0074-1-32-0', '1402.0074-1-33-0', '1402.0074-1-34-0', '1402.0074-1-43-3', '1402.0074-1-44-3', '1402.0074-1-44-5', '1402.0074-1-45-6', '1402.0074-1-46-2', '1402.0074-1-47-0', '1402.0074-1-52-2', '1402.0074-1-52-4', '1402.0074-1-54-0', '1402.0074-1-55-0', '1402.0074-2-1-0', '1402.0074-2-3-5', '1402.0074-2-4-0', '1402.0074-2-5-2', '1402.0074-2-5-3', '1402.0074-2-10-1']

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

https://arxiv.org/abs/1402.0074

1203.5879

{'1203.5879-1-0-0': 'Using a generalized genetic algorithm, we propose four new sp[MATH] carbon allotropes with 5-6-7 (5-6-7-type Z-ACA and Z-CACB) or 4-6-8 (4-6-8-type Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH]) carbon rings.', '1203.5879-1-0-1': 'Their stability, mechanical and electronic properties are systematically studied using first-principles method.', '1203.5879-1-0-2': 'We find that all these four carbon allotropes show amazing stability in comparison with recently proposed carbon phases.', '1203.5879-1-0-3': 'Both ZACA and Z-CACB are direct-band-gap semiconductors with band gaps of 2.261 eV and 4.196 eV, respectively.', '1203.5879-1-0-4': 'Whereas Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] are indirect-band-gap semiconductors with band gaps of 3.105 eV and 3.271 eV, respectively.', '1203.5879-1-0-5': 'Their mechanical properties reveal that all these four carbon allotropes are superhard materials comparable to diamond.', '1203.5879-1-1-0': 'Carbon is considered as the most active element in the periodic table due to its broad sp, sp[MATH] and sp[MATH] hybridizing ability.', '1203.5879-1-1-1': 'Besides the four best-known carbon allotropes, graphite, cubic diamond (C-diamond), hexagonal diamond (H-diamond) and amorphous carbon, an unknown superhard phase of carbon has been reported in experiment [CITATION] along with the structural phase transition in cold compressing graphite.', '1203.5879-1-1-2': 'Several structures have been proposed as the candidates for this superhard phase, such as the monoclinic M-carbon [CITATION], cubic body center C4 carbon (bct-C4) [CITATION] and the orthorhombic W-carbon [CITATION].', '1203.5879-1-1-3': 'Very recently, another new carbon allotrope, named as Z-carbon, was proposed and investigated at almost the same time by three independent research groups [CITATION] (this structure was also named as oC16II in reference [8] and Cco-C8 in reference [10]).', '1203.5879-1-1-4': 'Z-carbon is more stable (its cohesive energy is about 129 meV per atom above diamond) and harder than bct-C4, M-carbon and W-carbon.', '1203.5879-1-1-5': 'Moreover, its transition pressure is around 10 Gpa which is lower than those of bct-C4, M-carbon and W-carbon.', '1203.5879-1-1-6': 'Thus, it is believed that cold compressing graphite prefers forming Z-carbon.', '1203.5879-1-1-7': 'Although none of them can solely fit the experimental results satisfactorily, these theoretically proposed intermediate phases are significant in understanding the experimental process of cold compressing graphite and the cold compressing carbon nanotubes [CITATION].', '1203.5879-1-1-8': 'These theoretical studies also arise many interests on superhard carbon materials [CITATION] and analogical BN phases [CITATION].', '1203.5879-1-2-0': 'All above new carbon phases can be discovered from the recently developed particle-swarm optimization method [CITATION] or graph theoretical methods [CITATION].', '1203.5879-1-2-1': 'They can be divided into two groups: 5-6-7-type (M-carbon [CITATION] and W-carbon [CITATION] contain 5-, 6-, 7-carbon rings in corresponding structures) and 4-6-8-type (bct-C4 [CITATION] and Z-carbon [CITATION] contain 4-, 6-, 8-carbon rings).', '1203.5879-1-2-2': 'We notice that, they all can be constructed through mutating H-diamonds or combing the segments of H-diamond and C-diamond.', '1203.5879-1-2-3': 'H-diamond and C-diamond are the most favorable sp[MATH] carbon allotropes in nature that could be used as excellent parents for finding new carbon allotropes through hybridizing their stable segments.', '1203.5879-1-2-4': 'For example, bct-C4 and Z-carbon can be looked upon as hybridizations of H-diamond and mutated H-diamond.', '1203.5879-1-2-5': 'M-carbon and W-carbon are hybridizations of distorted H-diamond and C-diamond segments.', '1203.5879-1-2-6': 'The 4H [CITATION], 6H [CITATION] and 12R [CITATION] carbon allotropes can be regarded as the superlattice of stacked C-diamond (along 111 direction) and H-diamond (along 001direction).', '1203.5879-1-2-7': 'Hybridizing C-diamond and H-diamond with different manners can obtain almost all previously proposed carbon structures.', '1203.5879-1-2-8': 'Such structural construction process is compatible with the essence of genetic algorithm (GA).', '1203.5879-1-2-9': 'In this paper, using this generalized GA, we choose H-diamond and C-diamond as parents to hybridize new carbon allotropes.', '1203.5879-1-2-10': 'Four new carbon allotropes with 5-6-7 (Z-ACA and Z-CACB) or 4-6-8 (Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH]) carbon rings are proposed in our present work and their stability, electronic and mechanical properties are systematically studied using first-principles calculations based on the density functional theory.', '1203.5879-1-2-11': 'Such four carbon allotropes are more favorable than bct-C4.', '1203.5879-1-2-12': 'Our results indicate that all of them are superhard insulators with direct or indirect band gaps.', '1203.5879-1-2-13': 'Because the four new allotropes are more stable than graphite under certain pressures, all of them are expected to be obtained from cold compressing graphite.', '1203.5879-1-3-0': '# Models and Methods', '1203.5879-1-4-0': '## Models', '1203.5879-1-5-0': 'To compare with previously proposed carbon allotropes, we restrict the atoms in the primitive unit cell no larger than 16 carbon atoms.', '1203.5879-1-5-1': 'The primitive cells and the side and top views of Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are shown in Fig. [REF].', '1203.5879-1-5-2': 'Here Z and A denote that the framework of the systems are constructed with zigzag and armchair carbon chains along [100] direction, respectively.', '1203.5879-1-5-3': 'From the side views, we can see that Z-carbon (Z4-A[MATH]B[MATH], as shown in Fig. [REF] (a)) is composed of two A gene segments derived from H-diamond and two B gene segments derived from mutated H-diamond (the mutation is a c/2 translation along the [001] orientation).', '1203.5879-1-5-4': 'At the AA-BB interface, a series of 4 and 8 carbon rings appear.', '1203.5879-1-5-5': 'According to such denomination, the previously proposed bct-C4 is Z2-A[MATH]B[MATH] (Z4-A[MATH]B[MATH]A[MATH]B[MATH]) due to the fact that there is one A gene segment and one B gene segment in its crystal cell.', '1203.5879-1-5-6': 'Z4-A[MATH]B[MATH] as shown in Fig. [REF](b) is hybridized by a triple A gene segment derived from H-diamond and a B segment derived from mutated H-diamond.', '1203.5879-1-5-7': 'The primitive cell of A4-A[MATH]B[MATH] contains 16 carbon atoms and its crystal cell contains a double A gene segment and a double B segment of H-diamond, as shown in Fig. [REF](c).', '1203.5879-1-5-8': 'Z3- systems are restricted according to the topological requirement.', '1203.5879-1-5-9': 'The systems of A3-, A4-A[MATH]B[MATH], Zn- and An- (n is larger than 4) containing more than 16 atoms in the primitive cell are not included in this letter.', '1203.5879-1-5-10': 'The systems of A2-A[MATH]B[MATH] and A4-A[MATH]B[MATH]A[MATH]B[MATH] are energy unstable in compare with bct-C4.', '1203.5879-1-5-11': 'In Fig. [REF] (d) and (e), we show the situations of hybridizing H-diamond and C-diamond in the zigzag direction with 5-7 carbon rings at the C/A or C/B interface.', '1203.5879-1-5-12': 'Hybridizing H-diamond and C-diamond in the armchair direction is abandoned because it creates non-four-connected carbon atoms forming unstable systems.', '1203.5879-1-5-13': 'In fact, hybridizing H-diamond and C-diamond segments can also construct many new allotropes.', '1203.5879-1-5-14': 'In this letter, two low energy allotropes named as Z-ACA and Z-CACB containing 16 and 12 atoms in their primitive cell, respectively, are taken as examples.', '1203.5879-1-5-15': 'Here, A, B and C denote the H-diamond, mutated H-diamond and C-diamond gene segment, respectively.', '1203.5879-1-6-0': '## Methods', '1203.5879-1-7-0': 'To investigate the stability, electronic and mechanical properties of these new carbon allotropes, first-principles calculations based on the density functional theory are employed.', '1203.5879-1-7-1': 'All calculations are performed within general gradient approximation (GGA) [CITATION] as implemented in Vienna ab initio simulation package (VASP) [CITATION].', '1203.5879-1-7-2': 'The interactions between nucleus and the 2s[MATH]2p[MATH] valence electrons of carbon are described by the projector augmented wave (PAW) method [CITATION].', '1203.5879-1-7-3': 'A plane-wave basis with a cutoff energy of 500 eV is used to expand the wave functions.', '1203.5879-1-7-4': 'The Brillouin Zone (BZ) sample meshes are set to be dense enough (less than 0.21/AA) to ensure the accuracy of our calculations.', '1203.5879-1-7-5': 'Crystal lattices and atom positions of graphite, diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are fully optimized up to the residual force on every atom less than 0.005 eV/AA through the conjugate-gradient algorithm.', '1203.5879-1-7-6': 'Vibration properties are calculated by using the phonon package [CITATION] with the forces calculated from VASP.', '1203.5879-1-7-7': 'To evaluate the transition pressure from graphite to each superhard phase, the exchange-correlation functional is describe by LDA [CITATION] for the consideration that LDA can give reasonable interlayer distances, mechanical properties of graphite sheets due to a delicate error cancelation of the exchange and correlation interactions in comparison with that of semi-local generalized gradient approximation (GGA).', '1203.5879-1-8-0': '# Results and Discussion', '1203.5879-1-9-0': '## Structures', '1203.5879-1-10-0': 'Similar to bct-C4 and Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] belong to 4-6-8-type.', '1203.5879-1-10-1': 'The crystal structure of Z4-A[MATH]B[MATH] belongs to PMMN space group.', '1203.5879-1-10-2': 'At zero pressure, its GGA calculated equilibrium lattice constants are a=8.762 AA, b=4.263 AA and c=2.514 AA.', '1203.5879-1-10-3': 'Four inequivalent atoms (16 atoms per primitive cell) in this crystal occupy the positions at (0.041, 0.312, 0.5), (0.208, 0.185, 0.5), (0.285, 0.316, 1.0) and (0.464, 0.318, 1.0), respectively.', '1203.5879-1-10-4': 'A4-A[MATH]B[MATH] belongs to CMCA space group and its equilibrium lattice constants obtained from GGA calculations are a=4.257 AA, b=10.114 AA and c=4.363 AA.', '1203.5879-1-10-5': 'There have only two inequivalent atoms (16 atoms per primitive cell) in A4-A[MATH]B[MATH] locating at positions of (0.317, 0.067, 0.088) and (0.314, 0.188, 0.584).', '1203.5879-1-10-6': 'Recently proposed Z-carbon holds CMMM symmetry with equilibrium lattice parameters a=8.772 AA, b=4.256 AA and c=2.514 AA.', '1203.5879-1-10-7': 'There are only two inequivalent atoms in Z-carbon locating at (0.089, 0.316, 0.5), and (0.167, 0.185, 1.0).', '1203.5879-1-10-8': 'Z-ACA and Z-CACB belong to 5-6-7-type similar to M-carbon and W-carbon containing 5-, 6-, and 7-rings.', '1203.5879-1-10-9': 'Z-ACA belongs to PMMN space group and contains five inequivalent atoms (16 atoms per primitive cell) in its orthorhombic lattice with constants of a=4.760 AA, b=2.521 AA and c=7.93 AA.', '1203.5879-1-10-10': 'The five inequivalent atom positions are listed in Tab.', '1203.5879-1-10-11': 'The structure information of Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB derived from GGA calculations is listed in Tab.', '1203.5879-1-10-12': 'All these new carbon allotropes can be constructed from hybridizing C-diamond and H-diamond as sketched in fig. [REF].(', '1203.5879-1-10-13': 'Moreover, similar to the previously proposed bct-C4, Z-carbon, M-carbon and W-carbon, all of them can also be considered as potential products in the process of cold compressing graphite.', '1203.5879-1-11-0': '## Stability', '1203.5879-1-12-0': 'The relative stability of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB is evaluated through comparing their cohesive energy per atom.', '1203.5879-1-12-1': 'All these four new allotropes are more energy stable than bct-C4.', '1203.5879-1-12-2': 'At zero pressure, the cohesive energy of Z-ACA is about 30 meV lower than bct-C4 and only 5 meV higher than that of M-carbon.', '1203.5879-1-12-3': 'Z-CACB is more favorable than both M-carbon and W-carbon.', '1203.5879-1-12-4': 'Its cohesive energy is -7.556 eV per atom that is about 17 meV lower than that of W-carbon (-7.539 eV per atom) and 8 meV above than Z-carbon (7.564 eV per atom).', '1203.5879-1-12-5': 'The most stable two allotropes are Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] and their cohesive energies are -7.568 eV and -7.565 eV per atom, respectively.', '1203.5879-1-12-6': 'Both of them are more stable than Z-carbon and they are the stablest new carbon phases theoretically predicated so far.', '1203.5879-1-12-7': 'The enthalpy per atom for diamond, H-diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB as a function of pressure relative to graphite derived from LDA calculations is shown in Fig. [REF].', '1203.5879-1-12-8': 'The results indicate that Z-ACA is more stable than M-carbon under external pressure and becomes more stable than W-carbon when the pressure is above 35 Gpa.', '1203.5879-1-12-9': 'The transition pressures for M-carbon, W-carbon and Z-ACA are very close to each other (located at around 12.1-13.3 Gpa).', '1203.5879-1-12-10': 'Z-CACB is always more favorable than bct-C4, M-carbon, W-carbon and Z-ACA and is more stable than graphite when the external pressure is larger than 10 GPa.', '1203.5879-1-12-11': 'Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] have almost the same relative stability and transition pressure.', '1203.5879-1-12-12': 'They are more stable than graphite when the external pressure is larger than 9.16 Gpa.', '1203.5879-1-12-13': 'To further confirm the dynamic stability of Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB, their phonon band structures and phonon density of states are calculated.', '1203.5879-1-12-14': 'The phonon band structures and phonon density of states at zero pressure are show in Fig. [REF].', '1203.5879-1-12-15': 'There is no negative frequency for all these new structures up to 40 Gpa, confirming that these allotropes are dynamic stable phases of carbon.', '1203.5879-1-13-0': '## Mechanical and electronic properties', '1203.5879-1-14-0': 'Mass density, band gaps, cohesive energies, bulk modulus and hardness of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are summarized in Tab. [REF].', '1203.5879-1-14-1': 'The results indicate that all these allotropes are superhard intermediate phases between graphite and diamond due to their considerable bulk modulus and hardness.', '1203.5879-1-14-2': 'Bulk modulus (B0) is obtained by fitting the total energy as a function of volume to the third-order Birch-Murnaghan equation of state.', '1203.5879-1-14-3': 'Further hardness evaluation is considered according to the recently introduced empirical scheme[CITATION] which correlates the Vickers hardness to the bulk modulus (B[MATH]) and shear modulus (G) through the formula: H[MATH]=2(G[MATH]/B[MATH]-3.', '1203.5879-1-14-4': 'From Tab. [REF] we can see that Z-carobn, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] hold almost the same stability, band gap, mass density, bulk modulus and hardness.', '1203.5879-1-14-5': 'The values of bulk modulus are 415.83 Gpa, 415.49 Gpa and 413.62GP, respectively.', '1203.5879-1-14-6': 'The values of Vickers hardness are 81.09 Gpa, 80.54 Gpa and 83.18 Gpa, respectively, which are comparable to that of diamond (88.31 Gpa).', '1203.5879-1-14-7': 'M-carbon, W-carbon, Z-ACA and Z-CACB hold similar stability, density, bulk modulus and Vickers hardness.', '1203.5879-1-14-8': 'Their calculated values of Vickers hardness (79.24 Gpa, 79.08 Gpa, 78.74 Gpa and 82.01 Gpa, respectively) are also comparable to the value for diamond.', '1203.5879-1-14-9': 'These results indicate that all these allotropes are superhard materials comparable to diamond.', '1203.5879-1-15-0': 'Electronic properties of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA, Z-CACB are investigated and the band structures of Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are shown in Fig. [REF].', '1203.5879-1-15-1': 'The results indicate that all these superhard carbon allotropes are indirect wide band gap semiconductors except for Z-ACA and Z-CACB.', '1203.5879-1-15-2': 'From the results of Fig. [REF] we can see that Z4-A[MATH]B[MATH] is an indirect-wide-gap semiconductor and its band gap (3.105 eV ) is larger than that of bct-C4 (2.491 eV) and close to that of Z-carbon (3.273 eV).', '1203.5879-1-15-3': 'A4-A[MATH]B[MATH] is also an indirect-band-gap semiconductor and its band gap (3.271 eV) is close to those of Z-carbon and Z4-A[MATH]B[MATH].', '1203.5879-1-15-4': 'Different from diamond, bct-C4, M-carbon, W-cabon, Z-carbon, Z4-A[MATH]B[MATH], and A4-A[MATH]B[MATH], both Z-ACA and Z-CACB are direct-band-gap semiconductors with gaps of 2.261 eV and 4.196 eV, respectively.', '1203.5879-1-15-5': 'The wide band gaps of all these four new carbon allotropes indicates that all of them are transparent carbon phases.', '1203.5879-1-16-0': '# Conclusion', '1203.5879-1-17-0': 'In summary, using a generalized genetic-algorithm, we propose four new carbon allotropes.', '1203.5879-1-17-1': 'The stability, electronic and mechanical properties of the four new carbon allotropes are investigated using first-principle method.', '1203.5879-1-17-2': 'The dynamic stability of all these new carbon phases are confirmed from the phonon band calculation.', '1203.5879-1-17-3': 'Under proper external pressure, these four new allotropes of carbon are expected to be obtained from cold compressing graphite.', '1203.5879-1-17-4': 'Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] are the stablest new carbon phases theoretically predicted so far.', '1203.5879-1-17-5': 'All these four new carbon allotropes are transparent superhard carbon phases with values of bulk modulus and hardness comparable to that of diamond.'}

{'1203.5879-2-0-0': 'Using a generalized genetic algorithm, we propose four new sp[MATH] carbon allotropes with 5-6-7 (5-6-7-type Z-ACA and Z-CACB) or 4-6-8 (4-6-8-type Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH]) carbon rings.', '1203.5879-2-0-1': 'Their stability, mechanical and electronic properties are systematically studied using first-principles method.', '1203.5879-2-0-2': 'We find that the four new carbon allotropes show amazing stability in comparison with the carbon phases proposed recently.', '1203.5879-2-0-3': 'Both 5-6-7-type ZACA and Z-CACB are direct-band-gap semiconductors with band gaps of 2.261 eV and 4.196 eV, respectively.', '1203.5879-2-0-4': 'However, the 4-6-8-type Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] are indirect-band-gap semiconductors with band gaps of 3.105 eV and 3.271 eV, respectively.', '1203.5879-2-0-5': 'Their mechanical properties reveal that all the four carbon allotropes proposed in present work are superhard materials comparable to diamond.', '1203.5879-2-1-0': 'Carbon is considered as the most active element in the periodic table due to its broad sp, sp[MATH] and sp[MATH] hybridizing ability.', '1203.5879-2-1-1': 'Besides the four best-known carbon allotropes, graphite, cubic diamond (C-diamond), hexagonal diamond (H-diamond) and amorphous carbon, an unknown superhard phase of carbon has been reported in experiment [CITATION] along with the structural phase transition in cold compressing graphite.', '1203.5879-2-1-2': 'Several structures have been proposed theoretically as the candidate for this superhard phase, such as the monoclinic M-carbon[CITATION], cubic body center C4 carbon (bct-C4) [CITATION] and the orthorhombic W-carbon [CITATION].', '1203.5879-2-1-3': 'Although the monoclinic one, namely M-carbon, has been preliminarily identified by a following experimental process of compressing graphite[CITATION], bct-C4 and orthorhombic W-carbon can also fit the experimental XRD-data to some extent.', '1203.5879-2-1-4': 'These discoveries attract great interest in theoretical predication [CITATION] and experimental research [CITATION] on such superhard carbon allotropes.', '1203.5879-2-1-5': 'Very recently, another new carbon allotrope, named as Z-carbon, was proposed and investigated at almost the same time by three independent research groups [CITATION] (this structure was also named as oC16II in reference [8] and Cco-C8 in reference [10]).', '1203.5879-2-1-6': 'Z-carbon is more stable (its cohesive energy is about 129 meV per atom above diamond) and harder than bct-C4, M-carbon and W-carbon.', '1203.5879-2-1-7': 'Moreover, its transition pressure is around 10 Gpa which is lower than those of bct-C4, M-carbon and W-carbon.', '1203.5879-2-1-8': 'Thus, it is believed that cold compressing graphite prefers forming Z-carbon.', '1203.5879-2-1-9': 'Although none of them can solely fit the experimental results satisfactorily, these theoretically proposed intermediate phases are significant in understanding the experimental process of cold compressing graphite and the cold compressing carbon nanotubes [CITATION].', '1203.5879-2-1-10': 'Moreover, these theoretical studies also arouse great interest on the superhard carbon materials [CITATION] and analogical superhard BN phases [CITATION].', '1203.5879-2-2-0': 'All above new carbon phases can be designed by using the recently developed particle-swarm optimization method [CITATION], graph theoretical methods [CITATION] and the evolutionary algorithm USPEX developed by Oganov [CITATION].', '1203.5879-2-2-1': 'All of them can be divided into two groups: 5-6-7-type (M-carbon [CITATION] and W-carbon [CITATION] containing 5-, 6-, 7-carbon rings) and 4-6-8-type (bct-C4 [CITATION] and Z-carbon [CITATION] containing 4-, 6-, 8-carbon rings).', '1203.5879-2-2-2': 'We notice that, from the point of view of structure, they all can be constructed through mutating H-diamonds or combing the segments of H-diamond and C-diamond.', '1203.5879-2-2-3': 'H-diamond and C-diamond are the most favorable sp[MATH] carbon allotropes in nature that could be used as excellent parents for finding new carbon allotropes through hybridizing their stable segments.', '1203.5879-2-2-4': 'For example, bct-C4 and Z-carbon can be looked upon as the hybridization of H-diamond and mutated H-diamond.', '1203.5879-2-2-5': 'M-carbon and W-carbon can be taken as the hybridization of distorted H-diamond and C-diamond segments.', '1203.5879-2-2-6': 'The 4H [CITATION], 6H [CITATION] and 12R [CITATION] carbon allotropes can be regarded as the superlattice of C-diamond (along [111] direction) and H-diamond (along 001 direction).', '1203.5879-2-2-7': 'By hybridizing C-diamond and H-diamond in different manners, almost all previously proposed carbon structures can be obtained.', '1203.5879-2-2-8': 'Such a structural construction process is compatible with the essence of genetic algorithm (GA).', '1203.5879-2-2-9': 'The genetic algorithm is widely used in searching for zero-dimensional (0D) element clusters such as the carbon fullerenes [CITATION].', '1203.5879-2-2-10': 'Moreover, it is an effective method in prediction three-dimensional (3D) superhard carbon phases [CITATION].', '1203.5879-2-2-11': 'In this paper, using a generalized GA, we choose H-diamond and C-diamond as parents to hybridize new carbon allotropes.', '1203.5879-2-2-12': 'Four new carbon allotropes with 5-6-7 (Z-ACA and Z-CACB) or 4-6-8 (Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH]) carbon rings are proposed in our present work and their stability, electronic and mechanical properties are systematically studied using first-principles calculations based on the density functional theory.', '1203.5879-2-2-13': 'All the four carbon allotropes in our present work are more favorable than bct-C4.', '1203.5879-2-2-14': 'Our results indicate that all of them are superhard insulators with direct or indirect band gaps.', '1203.5879-2-2-15': 'Because the four new allotropes are more stable than graphite under certain pressures, all of them are expected to be obtained from cold compressing graphite.', '1203.5879-2-3-0': '# Models and Methods', '1203.5879-2-4-0': '## Models', '1203.5879-2-5-0': 'To compare with previously proposed carbon allotropes, we restrict the atoms in the primitive unit cell no larger than 16 carbon atoms.', '1203.5879-2-5-1': 'The primitive cells and the side and top views of Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are shown in Fig. [REF].', '1203.5879-2-5-2': 'Here Z and A denote that the framework of the systems are constructed with zigzag and armchair carbon chains along [100] direction, respectively.', '1203.5879-2-5-3': 'From the side view, we can see that Z-carbon (Z4-A[MATH]B[MATH], as shown in Fig. [REF] (a)) is composed of two A gene segments derived from H-diamond and two B gene segments derived from mutated H-diamond (the mutation is a c/2 translation along the [001] orientation).', '1203.5879-2-5-4': 'At the AA-BB interface, a series of 4 and 8 carbon rings appear.', '1203.5879-2-5-5': 'According to such denomination, the previously proposed bct-C4 is Z2-A[MATH]B[MATH] (Z4-A[MATH]B[MATH]A[MATH]B[MATH]) due to the fact that there are one A gene segment and one B gene segment in its crystal cell.', '1203.5879-2-5-6': 'Z4-A[MATH]B[MATH] as shown in Fig. [REF](b) is hybridized by a triple A gene segment derived from H-diamond and one B segment derived from mutated H-diamond.', '1203.5879-2-5-7': 'The primitive cell of A4-A[MATH]B[MATH] contains 16 carbon atoms and its crystal cell contains one double A gene segment and one double B segment of H-diamond, as shown in Fig. [REF](c).', '1203.5879-2-5-8': 'Z3- systems are restricted according to the topological requirement.', '1203.5879-2-5-9': 'The systems of A3-, A4-A[MATH]B[MATH], Zn- and An- (n is larger than 4) containing more than 16 atoms in the primitive cell are not included in present work.', '1203.5879-2-5-10': 'The systems of A2-A[MATH]B[MATH] and A4-A[MATH]B[MATH]A[MATH]B[MATH] are energetically unstable in comparison with bct-C4.', '1203.5879-2-5-11': 'In Fig. [REF] (d) and (e), we show the situations of hybridizing H-diamond and C-diamond in the zigzag direction with 5-7 carbon rings at the C/A or C/B interface.', '1203.5879-2-5-12': 'The system by hybridizing H-diamond and C-diamond in the armchair direction is unstable because it is contained non-four-connected carbon atoms.', '1203.5879-2-5-13': 'In fact, by hybridizing H-diamond and C-diamond segments, many new allotropes can also be constructed.', '1203.5879-2-5-14': 'In present work, two low energy allotropes named as Z-ACA and Z-CACB containing 16 and 12 atoms in their primitive cell, respectively, are taken as examples.', '1203.5879-2-5-15': 'Here, A, B and C denote the H-diamond, mutated H-diamond and C-diamond gene segment, respectively.', '1203.5879-2-6-0': '## Methods', '1203.5879-2-7-0': 'To investigate the stability, electronic and mechanical properties of these new carbon allotropes, first-principles calculations based on the density functional theory are employed.', '1203.5879-2-7-1': 'All calculations are performed within general gradient approximation (GGA) [CITATION] as implemented in Vienna ab initio simulation package (VASP) [CITATION].', '1203.5879-2-7-2': 'The interactions between nucleus and the 2s[MATH]2p[MATH] valence electrons of carbon are described by the projector augmented wave (PAW) method [CITATION].', '1203.5879-2-7-3': 'A plane-wave basis with a cutoff energy of 500 eV is used to expand the wave functions.', '1203.5879-2-7-4': 'The Brillouin Zone (BZ) sample meshes are set to be dense enough (less than 0.21/AA) to ensure the accuracy of our calculations.', '1203.5879-2-7-5': 'Crystal lattices and atom positions of graphite, diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are fully optimized up to the residual force on every atom less than 0.005 eV/AA through the conjugate-gradient algorithm.', '1203.5879-2-7-6': 'Vibration properties are calculated by using the phonon package [CITATION] with the forces calculated from VASP.', '1203.5879-2-7-7': 'To evaluate the transition pressure from graphite to superhard phase, the exchange-correlation functional is describe by LDA [CITATION] for the consideration that LDA can give reasonable interlayer distances, mechanical properties of graphite sheets due to a delicate error cancelation of the exchange and correlation interactions in comparison with that of semi-local generalized gradient approximation (GGA).', '1203.5879-2-8-0': '# Results and Discussion', '1203.5879-2-9-0': '## Structures', '1203.5879-2-10-0': 'Similar to bct-C4 and Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] belong to 4-6-8-type.', '1203.5879-2-10-1': 'The crystal structure of Z4-A[MATH]B[MATH] belongs to Pmmn space group.', '1203.5879-2-10-2': 'At zero pressure, its equilibrium lattice constants calculated from GGA are a=8.762 AA, b=4.263 AA and c=2.514 AA.', '1203.5879-2-10-3': 'Four inequivalent atoms (16 atoms per primitive cell) in its unit cell occupy the positions at (0.041, 0.312, 0.5), (0.208, 0.185, 0.5), (0.285, 0.316, 1.0) and (0.464, 0.318, 1.0), respectively.', '1203.5879-2-10-4': 'A4-A[MATH]B[MATH] belongs to Cmca space group and its equilibrium lattice constants obtained from GGA are a=4.257 AA, b=10.114 AA and c=4.363 AA.', '1203.5879-2-10-5': 'There are only two inequivalent atoms (16 atoms per primitive cell) in A4-A[MATH]B[MATH] locating at positions of (0.317, 0.067, 0.088) and (0.314, 0.188, 0.584).', '1203.5879-2-10-6': 'Recently proposed Z-carbon holds Cmmm symmetry with equilibrium lattice parameters a=8.772 AA, b=4.256 AA and c=2.514 AA.', '1203.5879-2-10-7': 'There are only two inequivalent atoms in Z-carbon unit cell locating at (0.089, 0.316, 0.5), and (0.167, 0.185, 1.0).', '1203.5879-2-10-8': 'Z-ACA and Z-CACB belong to 5-6-7-type similar to M-carbon and W-carbon containing 5-, 6-, and 7-carbon rings.', '1203.5879-2-10-9': 'Z-ACA belongs to Pmmn space group and contains five inequivalent atoms (16 atoms per primitive cell) in its orthorhombic lattice with constants of a=4.760 AA, b=2.521 AA and c=7.93 AA.', '1203.5879-2-10-10': 'The five inequivalent atom positions are listed in Tab.', '1203.5879-2-10-11': 'The structure information of Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB derived from GGA calculations is listed in Tab.', '1203.5879-2-10-12': 'All these new carbon allotropes can be constructed by hybridizing C-diamond and H-diamond as sketched in Fig. [REF].(', '1203.5879-2-10-13': 'Moreover, similar to the previously proposed bct-C4, Z-carbon, M-carbon and W-carbon, all of them can be considered as potential products in the process of cold compressing graphite.', '1203.5879-2-11-0': '## Stability', '1203.5879-2-12-0': 'The relative stability of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB is evaluated through comparing their cohesive energy per atom.', '1203.5879-2-12-1': 'All these four new allotropes are more energically stable than bct-C4.', '1203.5879-2-12-2': 'At zero pressure, the cohesive energy of Z-ACA is about 30 meV lower than bct-C4 and only 5 meV higher than that of M-carbon.', '1203.5879-2-12-3': 'Z-CACB is more favorable than both M-carbon and W-carbon.', '1203.5879-2-12-4': 'Its cohesive energy is -7.556 eV per atom that is about 17 meV lower than that of W-carbon (-7.539 eV per atom) and 8 meV larger than that of Z-carbon (7.564 eV per atom).', '1203.5879-2-12-5': 'The most stable two allotropes are Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] and their cohesive energies are -7.568 eV and -7.565 eV per atom, respectively.', '1203.5879-2-12-6': 'Both of them are more stable than Z-carbon and they are the most stable new carbon phases theoretically predicated so far.', '1203.5879-2-12-7': 'The enthalpy per atom for diamond, H-diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB as a function of pressure relative to graphite derived from LDA calculations is shown in Fig. [REF].', '1203.5879-2-12-8': 'The results indicate that Z-ACA is more stable than M-carbon under external pressure and becomes more stable than W-carbon when the pressure is above 35 Gpa.', '1203.5879-2-12-9': 'The transition pressures for M-carbon, W-carbon and Z-ACA are very close to each other (located at around 12.1-13.3 Gpa).', '1203.5879-2-12-10': 'Z-CACB is always more favorable than bct-C4, M-carbon, W-carbon and Z-ACA and is more stable than graphite when the external pressure is larger than 10 GPa.', '1203.5879-2-12-11': 'Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] have almost the same relative stability and transition pressure.', '1203.5879-2-12-12': 'They are more stable than graphite when the external pressure is larger than 9.16 Gpa.', '1203.5879-2-12-13': 'To further confirm the dynamic stability of Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB, their phonon band structures and phonon density of states are calculated.', '1203.5879-2-12-14': 'The phonon band structures and phonon density of states at zero pressure are show in Fig. [REF].', '1203.5879-2-12-15': 'There is no negative frequency for all of the four new carbon allotropes up to 40 Gpa, confirming that these allotropes are dynamic stable phases of carbon.', '1203.5879-2-13-0': '## Mechanical and electronic properties', '1203.5879-2-14-0': 'Mass density, band gaps, cohesive energies, bulk modulus and hardness of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are summarized in Tab. [REF].', '1203.5879-2-14-1': 'The results indicate that all above carbon allotropes are superhard intermediate phases between graphite and diamond due to their considerable bulk modulus and hardness.', '1203.5879-2-14-2': 'Bulk modulus (B0) is obtained by fitting the total energy as a function of volume to the third-order Birch-Murnaghan equation of state.', '1203.5879-2-14-3': 'Further hardness evaluation is considered according to the recently introduced empirical scheme[CITATION] which correlates the Vickers hardness to the bulk modulus (B[MATH]) and shear modulus (G) through the formula: H[MATH]=2(G[MATH]/B[MATH]-3.', '1203.5879-2-14-4': 'From Tab. [REF] we can see that Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] hold almost the same stability, band gap, mass density, bulk modulus and hardness.', '1203.5879-2-14-5': 'The values of bulk modulus are 415.83 Gpa, 415.49 Gpa and 413.62GP for Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH], respectively.', '1203.5879-2-14-6': 'The values of Vickers hardness are 81.09 Gpa, 80.54 Gpa and 83.18 Gpa for Z-carbon, Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH], respectively, which are comparable to that of diamond (88.31 Gpa).', '1203.5879-2-14-7': 'M-carbon, W-carbon, Z-ACA and Z-CACB hold similar stability, density, bulk modulus and Vickers hardness.', '1203.5879-2-14-8': 'Their calculated values of Vickers hardness (79.24 Gpa, 79.08 Gpa, 78.74 Gpa and 82.01 Gpa, respectively) are also comparable to the value for diamond.', '1203.5879-2-14-9': 'The results indicate that the four new carbon allotropes proposed in present work are superhard materials comparable to diamond.', '1203.5879-2-15-0': 'Electronic properties of diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA, Z-CACB are investigated and the band structures of Z4-A[MATH]B[MATH], A4-A[MATH]B[MATH], Z-ACA and Z-CACB are shown in Fig. [REF].', '1203.5879-2-15-1': 'The results indicate that all these superhard carbon allotropes are indirect wide band gap semiconductors except for Z-ACA and Z-CACB.', '1203.5879-2-15-2': 'From the results of Fig. [REF] we can see that Z4-A[MATH]B[MATH] is an indirect-wide-gap semiconductor and its band gap (3.105 eV ) is larger than that of bct-C4 (2.491 eV) and close to that of Z-carbon (3.273 eV).', '1203.5879-2-15-3': 'A4-A[MATH]B[MATH] is also an indirect-band-gap semiconductor and its band gap (3.271 eV) is close to those of Z-carbon and Z4-A[MATH]B[MATH].', '1203.5879-2-15-4': 'Different from diamond, bct-C4, M-carbon, W-carbon, Z-carbon, Z4-A[MATH]B[MATH], and A4-A[MATH]B[MATH], both Z-ACA and Z-CACB are direct-band-gap semiconductors with gaps of 2.261 eV and 4.196 eV, respectively.', '1203.5879-2-15-5': 'The wide band gaps of all these four new carbon allotropes indicate that all of them are transparent carbon phases.', '1203.5879-2-16-0': '# Conclusion', '1203.5879-2-17-0': 'In summary, using a generalized genetic-algorithm, we proposed four new carbon allotropes.', '1203.5879-2-17-1': 'The stability, electronic and mechanical properties of the four new carbon allotropes are investigated using first-principles method.', '1203.5879-2-17-2': 'The dynamic stability of all these new carbon phases is confirmed from the phonon band calculations.', '1203.5879-2-17-3': 'Under proper external pressure, these four new allotropes of carbon are expected to be obtained from cold compressing graphite.', '1203.5879-2-17-4': 'Z4-A[MATH]B[MATH] and A4-A[MATH]B[MATH] are the most stable new carbon phases theoretically predicted so far.', '1203.5879-2-17-5': 'All these four new carbon allotropes are transparent superhard carbon phases with values of bulk modulus and hardness comparable to that of diamond.'}

[['1203.5879-1-7-0', '1203.5879-2-7-0'], ['1203.5879-1-7-1', '1203.5879-2-7-1'], ['1203.5879-1-7-2', '1203.5879-2-7-2'], ['1203.5879-1-7-3', '1203.5879-2-7-3'], ['1203.5879-1-7-4', '1203.5879-2-7-4'], ['1203.5879-1-7-5', '1203.5879-2-7-5'], ['1203.5879-1-7-6', '1203.5879-2-7-6'], ['1203.5879-1-15-0', '1203.5879-2-15-0'], ['1203.5879-1-15-1', '1203.5879-2-15-1'], ['1203.5879-1-15-2', '1203.5879-2-15-2'], ['1203.5879-1-15-3', '1203.5879-2-15-3'], ['1203.5879-1-5-0', '1203.5879-2-5-0'], ['1203.5879-1-5-1', '1203.5879-2-5-1'], ['1203.5879-1-5-2', '1203.5879-2-5-2'], ['1203.5879-1-5-4', '1203.5879-2-5-4'], ['1203.5879-1-5-8', '1203.5879-2-5-8'], ['1203.5879-1-5-11', '1203.5879-2-5-11'], ['1203.5879-1-5-15', '1203.5879-2-5-15'], ['1203.5879-1-1-0', '1203.5879-2-1-0'], ['1203.5879-1-1-1', '1203.5879-2-1-1'], ['1203.5879-1-1-3', '1203.5879-2-1-5'], ['1203.5879-1-1-4', '1203.5879-2-1-6'], ['1203.5879-1-1-5', '1203.5879-2-1-7'], ['1203.5879-1-1-6', '1203.5879-2-1-8'], ['1203.5879-1-1-7', '1203.5879-2-1-9'], ['1203.5879-1-0-0', '1203.5879-2-0-0'], ['1203.5879-1-0-1', '1203.5879-2-0-1'], ['1203.5879-1-14-0', '1203.5879-2-14-0'], ['1203.5879-1-14-2', '1203.5879-2-14-2'], ['1203.5879-1-14-3', '1203.5879-2-14-3'], ['1203.5879-1-14-7', '1203.5879-2-14-7'], ['1203.5879-1-14-8', '1203.5879-2-14-8'], ['1203.5879-1-17-3', '1203.5879-2-17-3'], ['1203.5879-1-17-5', '1203.5879-2-17-5'], ['1203.5879-1-10-10', '1203.5879-2-10-10'], ['1203.5879-1-10-11', '1203.5879-2-10-11'], ['1203.5879-1-12-0', '1203.5879-2-12-0'], ['1203.5879-1-12-2', '1203.5879-2-12-2'], ['1203.5879-1-12-3', '1203.5879-2-12-3'], ['1203.5879-1-12-5', '1203.5879-2-12-5'], ['1203.5879-1-12-7', '1203.5879-2-12-7'], ['1203.5879-1-12-8', '1203.5879-2-12-8'], ['1203.5879-1-12-9', '1203.5879-2-12-9'], ['1203.5879-1-12-10', '1203.5879-2-12-10'], ['1203.5879-1-12-11', '1203.5879-2-12-11'], ['1203.5879-1-12-12', '1203.5879-2-12-12'], ['1203.5879-1-12-13', '1203.5879-2-12-13'], ['1203.5879-1-12-14', '1203.5879-2-12-14'], ['1203.5879-1-2-3', '1203.5879-2-2-3'], ['1203.5879-1-2-10', '1203.5879-2-2-12'], ['1203.5879-1-2-12', '1203.5879-2-2-14'], ['1203.5879-1-2-13', '1203.5879-2-2-15'], ['1203.5879-1-7-7', '1203.5879-2-7-7'], ['1203.5879-1-15-4', '1203.5879-2-15-4'], ['1203.5879-1-15-5', '1203.5879-2-15-5'], ['1203.5879-1-5-3', '1203.5879-2-5-3'], ['1203.5879-1-5-5', '1203.5879-2-5-5'], ['1203.5879-1-5-6', '1203.5879-2-5-6'], ['1203.5879-1-5-7', '1203.5879-2-5-7'], ['1203.5879-1-5-9', '1203.5879-2-5-9'], ['1203.5879-1-5-10', '1203.5879-2-5-10'], ['1203.5879-1-5-14', '1203.5879-2-5-14'], ['1203.5879-1-1-2', '1203.5879-2-1-2'], ['1203.5879-1-1-8', '1203.5879-2-1-10'], ['1203.5879-1-0-3', '1203.5879-2-0-3'], ['1203.5879-1-0-5', '1203.5879-2-0-5'], ['1203.5879-1-14-1', '1203.5879-2-14-1'], ['1203.5879-1-14-4', '1203.5879-2-14-4'], ['1203.5879-1-14-6', '1203.5879-2-14-6'], ['1203.5879-1-17-0', '1203.5879-2-17-0'], ['1203.5879-1-17-1', '1203.5879-2-17-1'], ['1203.5879-1-17-2', '1203.5879-2-17-2'], ['1203.5879-1-17-4', '1203.5879-2-17-4'], ['1203.5879-1-10-1', '1203.5879-2-10-1'], ['1203.5879-1-10-2', '1203.5879-2-10-2'], ['1203.5879-1-10-4', '1203.5879-2-10-4'], ['1203.5879-1-10-6', '1203.5879-2-10-6'], ['1203.5879-1-10-8', '1203.5879-2-10-8'], ['1203.5879-1-10-9', '1203.5879-2-10-9'], ['1203.5879-1-10-12', '1203.5879-2-10-12'], ['1203.5879-1-10-13', '1203.5879-2-10-13'], ['1203.5879-1-12-1', '1203.5879-2-12-1'], ['1203.5879-1-12-4', '1203.5879-2-12-4'], ['1203.5879-1-12-6', '1203.5879-2-12-6'], ['1203.5879-1-12-15', '1203.5879-2-12-15'], ['1203.5879-1-2-1', '1203.5879-2-2-1'], ['1203.5879-1-2-2', '1203.5879-2-2-2'], ['1203.5879-1-2-4', '1203.5879-2-2-4'], ['1203.5879-1-2-5', '1203.5879-2-2-5'], ['1203.5879-1-2-6', '1203.5879-2-2-6'], ['1203.5879-1-2-8', '1203.5879-2-2-8'], ['1203.5879-1-2-9', '1203.5879-2-2-11'], ['1203.5879-1-5-12', '1203.5879-2-5-12'], ['1203.5879-1-5-13', '1203.5879-2-5-12'], ['1203.5879-1-5-13', '1203.5879-2-5-13'], ['1203.5879-1-0-2', '1203.5879-2-0-2'], ['1203.5879-1-0-4', '1203.5879-2-0-4'], ['1203.5879-1-14-9', '1203.5879-2-14-9'], ['1203.5879-1-2-0', '1203.5879-2-2-0'], ['1203.5879-1-2-7', '1203.5879-2-2-7'], ['1203.5879-1-2-11', '1203.5879-2-2-13']]

[['1203.5879-1-7-0', '1203.5879-2-7-0'], ['1203.5879-1-7-1', '1203.5879-2-7-1'], ['1203.5879-1-7-2', '1203.5879-2-7-2'], ['1203.5879-1-7-3', '1203.5879-2-7-3'], ['1203.5879-1-7-4', '1203.5879-2-7-4'], ['1203.5879-1-7-5', '1203.5879-2-7-5'], ['1203.5879-1-7-6', '1203.5879-2-7-6'], ['1203.5879-1-15-0', '1203.5879-2-15-0'], ['1203.5879-1-15-1', '1203.5879-2-15-1'], ['1203.5879-1-15-2', '1203.5879-2-15-2'], ['1203.5879-1-15-3', '1203.5879-2-15-3'], ['1203.5879-1-5-0', '1203.5879-2-5-0'], ['1203.5879-1-5-1', '1203.5879-2-5-1'], ['1203.5879-1-5-2', '1203.5879-2-5-2'], ['1203.5879-1-5-4', '1203.5879-2-5-4'], ['1203.5879-1-5-8', '1203.5879-2-5-8'], ['1203.5879-1-5-11', '1203.5879-2-5-11'], ['1203.5879-1-5-15', '1203.5879-2-5-15'], ['1203.5879-1-1-0', '1203.5879-2-1-0'], ['1203.5879-1-1-1', '1203.5879-2-1-1'], ['1203.5879-1-1-3', '1203.5879-2-1-5'], ['1203.5879-1-1-4', '1203.5879-2-1-6'], ['1203.5879-1-1-5', '1203.5879-2-1-7'], ['1203.5879-1-1-6', '1203.5879-2-1-8'], ['1203.5879-1-1-7', '1203.5879-2-1-9'], ['1203.5879-1-0-0', '1203.5879-2-0-0'], ['1203.5879-1-0-1', '1203.5879-2-0-1'], ['1203.5879-1-14-0', '1203.5879-2-14-0'], ['1203.5879-1-14-2', '1203.5879-2-14-2'], ['1203.5879-1-14-3', '1203.5879-2-14-3'], ['1203.5879-1-14-7', '1203.5879-2-14-7'], ['1203.5879-1-14-8', '1203.5879-2-14-8'], ['1203.5879-1-17-3', '1203.5879-2-17-3'], ['1203.5879-1-17-5', '1203.5879-2-17-5'], ['1203.5879-1-10-10', '1203.5879-2-10-10'], ['1203.5879-1-10-11', '1203.5879-2-10-11'], ['1203.5879-1-12-0', '1203.5879-2-12-0'], ['1203.5879-1-12-2', '1203.5879-2-12-2'], ['1203.5879-1-12-3', '1203.5879-2-12-3'], ['1203.5879-1-12-5', '1203.5879-2-12-5'], ['1203.5879-1-12-7', '1203.5879-2-12-7'], ['1203.5879-1-12-8', '1203.5879-2-12-8'], ['1203.5879-1-12-9', '1203.5879-2-12-9'], ['1203.5879-1-12-10', '1203.5879-2-12-10'], ['1203.5879-1-12-11', '1203.5879-2-12-11'], ['1203.5879-1-12-12', '1203.5879-2-12-12'], ['1203.5879-1-12-13', '1203.5879-2-12-13'], ['1203.5879-1-12-14', '1203.5879-2-12-14'], ['1203.5879-1-2-3', '1203.5879-2-2-3'], ['1203.5879-1-2-10', '1203.5879-2-2-12'], ['1203.5879-1-2-12', '1203.5879-2-2-14'], ['1203.5879-1-2-13', '1203.5879-2-2-15']]

[['1203.5879-1-7-7', '1203.5879-2-7-7'], ['1203.5879-1-15-4', '1203.5879-2-15-4'], ['1203.5879-1-15-5', '1203.5879-2-15-5'], ['1203.5879-1-5-3', '1203.5879-2-5-3'], ['1203.5879-1-5-5', '1203.5879-2-5-5'], ['1203.5879-1-5-6', '1203.5879-2-5-6'], ['1203.5879-1-5-7', '1203.5879-2-5-7'], ['1203.5879-1-5-9', '1203.5879-2-5-9'], ['1203.5879-1-5-10', '1203.5879-2-5-10'], ['1203.5879-1-5-14', '1203.5879-2-5-14'], ['1203.5879-1-1-2', '1203.5879-2-1-2'], ['1203.5879-1-1-8', '1203.5879-2-1-10'], ['1203.5879-1-0-3', '1203.5879-2-0-3'], ['1203.5879-1-0-5', '1203.5879-2-0-5'], ['1203.5879-1-14-1', '1203.5879-2-14-1'], ['1203.5879-1-14-4', '1203.5879-2-14-4'], ['1203.5879-1-14-6', '1203.5879-2-14-6'], ['1203.5879-1-17-0', '1203.5879-2-17-0'], ['1203.5879-1-17-1', '1203.5879-2-17-1'], ['1203.5879-1-17-2', '1203.5879-2-17-2'], ['1203.5879-1-17-4', '1203.5879-2-17-4'], ['1203.5879-1-10-1', '1203.5879-2-10-1'], ['1203.5879-1-10-2', '1203.5879-2-10-2'], ['1203.5879-1-10-4', '1203.5879-2-10-4'], ['1203.5879-1-10-6', '1203.5879-2-10-6'], ['1203.5879-1-10-8', '1203.5879-2-10-8'], ['1203.5879-1-10-9', '1203.5879-2-10-9'], ['1203.5879-1-10-12', '1203.5879-2-10-12'], ['1203.5879-1-10-13', '1203.5879-2-10-13'], ['1203.5879-1-12-1', '1203.5879-2-12-1'], ['1203.5879-1-12-4', '1203.5879-2-12-4'], ['1203.5879-1-12-6', '1203.5879-2-12-6'], ['1203.5879-1-12-15', '1203.5879-2-12-15'], ['1203.5879-1-2-1', '1203.5879-2-2-1'], ['1203.5879-1-2-2', '1203.5879-2-2-2'], ['1203.5879-1-2-4', '1203.5879-2-2-4'], ['1203.5879-1-2-5', '1203.5879-2-2-5'], ['1203.5879-1-2-6', '1203.5879-2-2-6'], ['1203.5879-1-2-8', '1203.5879-2-2-8'], ['1203.5879-1-2-9', '1203.5879-2-2-11']]

[]

[['1203.5879-1-5-12', '1203.5879-2-5-12'], ['1203.5879-1-5-13', '1203.5879-2-5-12'], ['1203.5879-1-5-13', '1203.5879-2-5-13'], ['1203.5879-1-0-2', '1203.5879-2-0-2'], ['1203.5879-1-0-4', '1203.5879-2-0-4'], ['1203.5879-1-14-9', '1203.5879-2-14-9'], ['1203.5879-1-2-0', '1203.5879-2-2-0'], ['1203.5879-1-2-7', '1203.5879-2-2-7'], ['1203.5879-1-2-11', '1203.5879-2-2-13']]

[]

['1203.5879-1-10-0', '1203.5879-1-10-3', '1203.5879-1-10-5', '1203.5879-1-10-7', '1203.5879-1-14-5', '1203.5879-2-10-0', '1203.5879-2-10-3', '1203.5879-2-10-5', '1203.5879-2-10-7', '1203.5879-2-14-5']

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

https://arxiv.org/abs/1203.5879

1306.5766

{'1306.5766-1-0-0': 'Using cosmological galaxy formation simulations from the MaGICC project, spanning more than three magnitudes in stellar mass ([MATH]M[MATH] to 3[MATH]M[MATH]), we trace the baryonic cycle of infalling gas from the virial radius through to its eventual participation in the star formation process.', '1306.5766-1-0-1': 'An emphasis is placed upon the temporal history of chemical enrichment during its passage through the corona and circumgalactic medium.', '1306.5766-1-0-2': 'We derive the distributions of time between gas crossing the virial radius and being accreted to the star forming region (which allows for mixing within the corona), as well as the time between gas being accreted to the star forming region and then ultimately forming stars (which allows for mixing within the disc).', '1306.5766-1-0-3': 'We show that significant numbers of stars are formed from gas that cycles back through the hot halo after first accreting to the star forming region.', '1306.5766-1-0-4': 'Gas entering high mass galaxies is pre-enriched in low mass proto-galaxies prior to entering the virial radius of the central progenitor, with only small amounts of primordial gas accreted, even at high redshift ([MATH]M[MATH], with M[MATH]99 to [MATH]M[MATH], local collapsing over-densities outside the central galaxy result in subhalo progenitors that form stars and enrich the IGM during the hierarchical build-up of the galaxies.', '1306.5766-1-1-0': 'Inflow rates peak between [MATH]12 Gyr ago (3[MATH]2), in all cases except SG4, which has a late merger which brings in significant amounts of gas [MATH]8 Gyr ago ([MATH]5 is taken as primordial and is shown in the plots at [Z][MATH]M[MATH]1 to [MATH]1.7/Gyr, with a mean of [MATH]5 is shown in the plots at [Z][MATH] of the universal baryon fraction accreting to the virial radius of a galaxy of virial mass [MATH]% for galaxies with virial mass [MATH]1.2/Gyr.', '1306.5766-1-1-1': 'Enrichment in the star forming region: Low star formation efficiency in disc galaxies results in a broad distribution in the time that baryons spend between first being accreted to the star forming region, and then subsequently forming stars.', '1306.5766-1-1-2': 'This gas may remain in the star forming region, or cycle through the hot halo and re-accrete to the star forming region to form stars (see ).', '1306.5766-1-1-3': 'The result is that gas accreted to the star forming region is significantly more enriched than gas crossing the virial radius.', '1306.5766-1-1-4': 'The distribution of timescales between gas passing to the star forming region and finally forming a star is reasonably approximated by an exponential with exponent [MATH], with intermediate mass halos (M[MATH]M[MATH]) having the most such stars: gas in lower mass galaxies has an increasing tendency to be expelled without being re-accreted, while at the more massive en the larger potential well in higher mass galaxies means that gas is not blown as far into the hot halo.', '1306.5766-1-1-5': 'The distribution of timescales for which gas remains within a galactic fountain is similar to that of the time taken from crossing the virial radius to arriving at the star forming region.', '1306.5766-1-1-6': 'As we showed in [CITATION], the distance that fountain gas reaches from the centre of the galaxy drops off exponentially.', '1306.5766-1-1-7': 'Hence, significantly more star forming gas could be involved in smaller-scale galactic fountains, particularly perpendicular to the disc, in the manner of e.g. [CITATION].'}

{'1306.5766-2-0-0': 'Using cosmological galaxy formation simulations from the MaGICC project, spanning stellar mass from [MATH]M[MATH] to 3[MATH]M[MATH], we trace the baryonic cycle of infalling gas from the virial radius through to its eventual participation in the star formation process.', '1306.5766-2-0-1': 'An emphasis is placed upon the temporal history of chemical enrichment during its passage through the corona and circum-galactic medium.', '1306.5766-2-0-2': 'We derive the distributions of time between gas crossing the virial radius and being accreted to the star forming region (which allows for mixing within the corona), as well as the time between gas being accreted to the star forming region and then ultimately forming stars (which allows for mixing within the disc).', '1306.5766-2-0-3': 'Significant numbers of stars are formed from gas that cycles back through the hot halo after first accreting to the star forming region.', '1306.5766-2-0-4': 'Gas entering high mass galaxies is pre-enriched in low mass proto-galaxies prior to entering the virial radius of the central progenitor, with only small amounts of primordial gas accreted, even at high redshift ([MATH]M[MATH], with M[MATH]99 to [MATH]M[MATH], local collapsing over-densities outside the central galaxy result in subhalo progenitors that form stars and enrich the IGM during the hierarchical build-up of the galaxies.', '1306.5766-2-1-0': 'Inflow rates peak between [MATH]12 Gyr ago (3[MATH]2), in all cases except SG4, which has a late merger which brings in significant amounts of gas [MATH]8 Gyr ago ([MATH]5 is taken as primordial and is shown in the plots at [Z][MATH]M[MATH]1 to [MATH]1.7/Gyr, with a mean of [MATH]1.', '1306.5766-2-2-0': '## Effective Yields', '1306.5766-2-3-0': 'The effective yield measures how a galaxys metallicity deviates from what would be expected for a closed box model of galaxy formation, i.e. a galaxy with the same gas mass fraction that had evolved without inflow or outflow of gas.', '1306.5766-2-3-1': 'A closed box galaxy evolution obeys a simple relationship between gas metallicity and the gas mass fraction.', '1306.5766-2-3-2': 'Thus, effective yields place constrains on the baryon cycle, which must have a combination of inflows and outflows of gas and metals that result in matching observed values of effective yields.', '1306.5766-2-3-3': 'The effective yield is defined as:', '1306.5766-2-4-0': '[EQUATION]', '1306.5766-2-4-1': 'Clearly, our simulations are far from closed box models as our earlier analysis has shown, and we measure here how such deviations are reflected in the effective yields of our simulations.', '1306.5766-2-4-2': 'Figure [REF] shows the effective yields as a function of rotation velocity (V[MATH]) and as a function of gas fraction (f[MATH]).', '1306.5766-2-4-3': 'A flattening of the relation for galaxies with [MATH]5 is shown in the plots at [Z][MATH] of the universal baryon fraction accreting to the virial radius of a galaxy of virial mass [MATH]% for galaxies with virial mass [MATH]1.2/Gyr.', '1306.5766-2-4-4': 'Enrichment in the star forming region: Low star formation efficiency in disc galaxies results in a broad distribution in the time that baryons spend between first being accreted to the star forming region, and then subsequently forming stars.', '1306.5766-2-4-5': 'This gas may remain in the star forming region, or cycle through the hot halo and re-accrete to the star forming region to form stars (see ).', '1306.5766-2-4-6': 'The result is that gas accreted to the star forming region is significantly more enriched than gas crossing the virial radius.', '1306.5766-2-4-7': 'The distribution of timescales between gas passing to the star forming region and finally forming a star is reasonably approximated by an exponential with exponent [MATH], with intermediate mass halos (M[MATH]M[MATH]) having the most such stars: gas in lower mass galaxies has an increasing tendency to be expelled without being re-accreted, while at the more massive en the larger potential well in higher mass galaxies means that gas is not blown as far into the hot halo.', '1306.5766-2-4-8': 'The distribution of timescales for which gas remains within a galactic fountain is similar to that of the time taken from crossing the virial radius to arriving at the star forming region.', '1306.5766-2-4-9': 'As we showed in [CITATION], the distance that fountain gas reaches from the centre of the galaxy drops off exponentially.', '1306.5766-2-4-10': 'Hence, significantly more star forming gas could be involved in smaller-scale galactic fountains, particularly perpendicular to the disc, in the manner of e.g. [CITATION].'}

[['1306.5766-1-0-2', '1306.5766-2-0-2'], ['1306.5766-1-0-4', '1306.5766-2-0-4'], ['1306.5766-1-1-1', '1306.5766-2-4-4'], ['1306.5766-1-1-2', '1306.5766-2-4-5'], ['1306.5766-1-1-3', '1306.5766-2-4-6'], ['1306.5766-1-1-4', '1306.5766-2-4-7'], ['1306.5766-1-1-5', '1306.5766-2-4-8'], ['1306.5766-1-1-6', '1306.5766-2-4-9'], ['1306.5766-1-1-7', '1306.5766-2-4-10'], ['1306.5766-1-0-0', '1306.5766-2-0-0'], ['1306.5766-1-0-1', '1306.5766-2-0-1'], ['1306.5766-1-0-3', '1306.5766-2-0-3']]

[['1306.5766-1-0-2', '1306.5766-2-0-2'], ['1306.5766-1-0-4', '1306.5766-2-0-4'], ['1306.5766-1-1-1', '1306.5766-2-4-4'], ['1306.5766-1-1-2', '1306.5766-2-4-5'], ['1306.5766-1-1-3', '1306.5766-2-4-6'], ['1306.5766-1-1-4', '1306.5766-2-4-7'], ['1306.5766-1-1-5', '1306.5766-2-4-8'], ['1306.5766-1-1-6', '1306.5766-2-4-9'], ['1306.5766-1-1-7', '1306.5766-2-4-10']]

[['1306.5766-1-0-0', '1306.5766-2-0-0'], ['1306.5766-1-0-1', '1306.5766-2-0-1'], ['1306.5766-1-0-3', '1306.5766-2-0-3']]

[]

[]

[]

['1306.5766-2-3-3', '1306.5766-2-4-0']

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

https://arxiv.org/abs/1306.5766

astro-ph-9903306

{'astro-ph-9903306-1-0-0': 'We present an analysis of the RXTE observations of recently discovered Galactic microquasar XTE J1748-288 during its 1998 outburst.', 'astro-ph-9903306-1-0-1': 'General spectral and temporal properties of the source and their evolution were very typical for the Galactic black hole candidates and, in particular, black hole X-ray Novae.', 'astro-ph-9903306-1-1-0': 'The spectral evolution of the source during the outburst can be considered as a sequence of three qualitatively distinct states.', 'astro-ph-9903306-1-1-1': 'During the first observations corresponding to the maximum of X-ray flux the spectrum of the source was formed by the sum of the dominating hard power law component and soft thermal component, which can be described by the model of multicolor disk emission.', 'astro-ph-9903306-1-1-2': 'Hard component contributed [MATH]80% to the X-ray luminosity in 3-25 keV energy band.', 'astro-ph-9903306-1-1-3': 'This spectral shape is quite different from spectra of high and low spectral states typical for Galactic black hole systems.', 'astro-ph-9903306-1-1-4': 'Later on, as the X-ray source faded, its energy spectrum qualitatively changed, and first high, then low standard states have been observed.', 'astro-ph-9903306-1-2-0': 'In correlation with the spectral changes the character of fast variability have changed dramatically also.', 'astro-ph-9903306-1-2-1': 'Initially the power density spectrum was formed by the dominating band-limited noise component, QPO features at 20-30 Hz and at [MATH]0.5 Hz, and very low frequency noise component.', 'astro-ph-9903306-1-2-2': 'After transition to high state the amplitude of the fractional variability decreased by order of magnitude and PDS spectrum became of power-law shape with broad QPO peak around 0.03 Hz.', 'astro-ph-9903306-1-2-3': 'When the systems switched to low state, PDS shape changed again and QPOs have not been detected anymore.', 'astro-ph-9903306-1-3-0': 'Clear correlation between QPO parameters and X-ray flux have been found, when the source have been in its initial bright hard state.', 'astro-ph-9903306-1-3-1': 'Such a correlation is of special interest for understanding of the accretion process in X-ray black hole binaries.', 'astro-ph-9903306-1-4-0': '# Introduction', 'astro-ph-9903306-1-5-0': 'The XTE J1748-288 was discovered as a new transient source on June 4, 1998 by All Sky Monitor (ASM) aboard Rossi X-ray Timing Explorer (RXTE) observatory (Smith, Levine Wood 1998).', 'astro-ph-9903306-1-5-1': 'X-ray source was localized with an accuracy of 1 arcmin in multiple scans of the region by the PCA/RXTE experiment (Strohmayer Marshall 1998).', 'astro-ph-9903306-1-5-2': 'Observations in radio band by VLA revealed the presence of unresolved radio source with the position consistent with one found by PCA: R.A.=17[MATH].06, Dec=-28[MATH].8 (equinox 2000.0, position uncertainty 0.', 'astro-ph-9903306-1-5-3': '[MATH]6, Hjellming, Rupen Miodiszewski 1998).', 'astro-ph-9903306-1-5-4': 'Significant variability of the new radio source (Hjellming et al. 1998a, Fender Stappers 1998) strongly supported its association with the X-ray transient.', 'astro-ph-9903306-1-5-5': 'In June 14.31 radio source became extended, with proper motions of 20-40 mas/day (Rupen, Hjellming Miodiszewski 1998).', 'astro-ph-9903306-1-5-6': 'The intrinsic velocity of the moving jet was higher than [MATH]c, for the distance [MATH]8 kpc derived from an 21 cm HI absorption measurement (Hjellming et al. 1998b).', 'astro-ph-9903306-1-6-0': 'Two quasi periodical oscillations (hereafter QPO) with central frequencies [MATH]0.5 Hz and [MATH]32 Hz have been found in the power density spectrum (PDS) of the source in the PCA observation of June 6, 1998 (Fox Lewin 1998).', 'astro-ph-9903306-1-7-0': 'The flux from the source had been above ASM/RXTE detection limit until August of 1998.', 'astro-ph-9903306-1-7-1': 'Pointed instruments of RXTE - PCA and HEXTE - observed the source periodically in June-September, obtaining a good coverage of the whole outburst.', 'astro-ph-9903306-1-7-2': 'Here we present results of spectral and timing analysis for these RXTE observations.', 'astro-ph-9903306-1-8-0': '# Observations and analysis', 'astro-ph-9903306-1-9-0': 'In our analysis we used all publicly available data obtained from RXTE TOO (Target Of Opportunity) archive including 21 pointed observations and slew parts of two PI restricted observations.', 'astro-ph-9903306-1-9-1': 'The 23 observations quasi evenly cover the 1998 outburst of the source with total exposure [MATH] ksec. The list of observations is presented in the Table [REF].', 'astro-ph-9903306-1-10-0': 'For data reduction we used standard FTOOLS package.', 'astro-ph-9903306-1-10-1': 'PCA data collected in 3-25 keV energy range and response matrix v3.5 have been used in spectral analysis.', 'astro-ph-9903306-1-10-2': 'For PCA background estimations we applied Very Large Events (VLE) model.', 'astro-ph-9903306-1-10-3': 'For some observations, where standard VLE-based background subtraction was not good enough for our purposes, we used activation component for background models separately (see details in the RXTE Cook Book, "Using the Latest PCABACKEST").', 'astro-ph-9903306-1-10-4': 'All spectra were deadtime corrected.', 'astro-ph-9903306-1-10-5': 'The analysis of the Crab nebula spectra confirmed that the uncertainties attributed to the response matrix do not exceed 1.0-1.5%.', 'astro-ph-9903306-1-10-6': 'To account for the uncertainty in the knowledge of the spectral response, a 1 per cent systematic error was added to the statistical error in each PCA channel.', 'astro-ph-9903306-1-11-0': 'For our timing analysis we used PCA timing modes [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9903306-1-11-1': 'Background subtracted light curves from [MATH] mode data were used to generate the power density spectra for very low frequencies (below [MATH] Hz).', 'astro-ph-9903306-1-12-0': 'Unfortunately, an analysis of the HEXTE data was strongly complicated by the difficulties in the background subtraction.', 'astro-ph-9903306-1-12-1': 'The source/background beamswitching of the HEXTE clusters (they "rock" periodically to [MATH]3.0[MATH] from the source position), is not very efficient for precise background subtraction when the source is located in densely populated Galactic Center region.', 'astro-ph-9903306-1-12-2': 'There is a number of X-ray sources around the XTE J1748-288, namely GX 5-1, GRS 1758-258, GX 3+1, 1E1740.7-2942 to mention few, each of them luminous enough to affect the result of the background measurement.', 'astro-ph-9903306-1-12-3': 'We have performed a check of the quality of the background subtraction in any given observation by comparing the m-background spectra (the spectra obtained during the -3.0[MATH] offset from the source) with the p-background one (+3.0[MATH] offset).', 'astro-ph-9903306-1-12-4': 'Only observations with adequate quality of the subtraction were accepted for the analysis.', 'astro-ph-9903306-1-12-5': 'Because the rocking planes for clusters A and B are perpendicular to each other, the different sky regions are within field of view of these clusters, when performing the background measurements.', 'astro-ph-9903306-1-12-6': 'So A clusters provide more accurate background measurements for some observations, and B clusters - for others.', 'astro-ph-9903306-1-12-7': 'After the 8-th observation, when the source flux in the 20-100 keV energy band dropped below [MATH]10 cnts/s/cluster, the accuracy of our knowledge of the background becomes to be comparable to the source flux, so we excluded these HEXTE data from our spectral analysis.', 'astro-ph-9903306-1-12-8': 'However, we presented the HEXTE data for observations of the source in low state in the Fig. [REF].', 'astro-ph-9903306-1-12-9': 'The spectrum presented there has been obtained from carefully selected data, but still should be treated with care.', 'astro-ph-9903306-1-13-0': '# Results', 'astro-ph-9903306-1-14-0': '## Evolution of the source during the outburst', 'astro-ph-9903306-1-15-0': 'X-ray flux histories of the XTE J1748-288 outburst based on the [MATH]/PCA and ASM data in the 3-15 and 15-30 keV energy ranges are shown in Fig. [REF].', 'astro-ph-9903306-1-15-1': 'The evolution of the source flux in the soft X-ray band (3-15 keV) was characterized by the fast initial rise to the level of [MATH]600-700 mCrab on a time scale of [MATH]2-3 days followed by the [MATH] day-long maximum and a nearly exponential decay to the quiescent level with e-folding time [MATH] days, while the evolution of the flux in the hard band (15-30 keV) has fast initial rise to the [MATH]300-400 mCrab level followed by the maximum phase of a [MATH]-day duration and abrupt decay, which is similar to the behavior of other Galactic X-ray Novae, like Nova Muscae GS/GRS 1124-68 [CITATION] and KS/GRS 1730-312 [CITATION].', 'astro-ph-9903306-1-15-2': 'Based on the results of the spectral and timing analysis, the evolution of the source could be subdivided into three distinct parts, corresponding to three phenomenological states: bright hard state (BHS), high state (HS) and low state (LS) described below and marked in Fig. [REF].', 'astro-ph-9903306-1-16-0': '[Bright hard state(BHS)] During the first eight RXTE pointed observations (June 4 - 11, 1998) the source was found in an extremely bright state.', 'astro-ph-9903306-1-16-1': 'The broad-band [MATH] keV energy spectrum of XTE J1748-288 in this state can be satisfactorily described by the sum of two components: relatively weak soft thermal component (contributing only [MATH] per cent of the total X-ray flux in the [MATH] keV energy band) with a characteristic temperature [MATH] keV and a strong hard component, which has approximately a power law form and does not show the evidence of the high energy cut-off up to [MATH] keV .', 'astro-ph-9903306-1-16-2': 'The corresponding source luminosity in the [MATH] keV band not corrected for the interstellar absorption was at the level [MATH] ergs/s (assuming 8 kpc distance) during this period.', 'astro-ph-9903306-1-17-0': 'The broad-band power density spectrum (PDS) of the source is formed by the dominating band-limited noise component (corresponding rms amplitude [MATH]), strong QPO feature with centroid frequency [MATH] Hz and a very low frequency noise (VLFN) component with a slope [MATH] and rms amplitude [MATH] (Fig. [REF], upper panels).', 'astro-ph-9903306-1-18-0': 'The correlated spectral and timing evolution of the source during this period is of special interest.', 'astro-ph-9903306-1-18-1': 'Local rise of the contribution of the soft spectral component to the total X-ray flux in the observations [MATH] (June 6 - 7, 1998) was accompanied by steepening of the hard spectral component and significant changes in the power density spectrum (Table 2, 3.', 'astro-ph-9903306-1-18-2': 'Fig. [REF], left upper panel):', 'astro-ph-9903306-1-19-0': '- increase of the BLN characteristic break frequency and decrease of its rms amplitude;', 'astro-ph-9903306-1-20-0': '- increase of the QPO centroid frequency from [MATH] Hz to [MATH] Hz;', 'astro-ph-9903306-1-21-0': '- appearance of the additional broad QPO feature at [MATH] Hz (Fox Lewin 1998);', 'astro-ph-9903306-1-22-0': '- notable rise of the VLFN rms amplitude.', 'astro-ph-9903306-1-22-1': 'Detected correlations between spectral and timing properties of the source in this state discussed in more detail below (see 3.4) holds on a wide range of the timescales from seconds to several days.', 'astro-ph-9903306-1-23-0': 'Two-component spectrum, consisting of the soft thermal component and a hard power law component is a signature of the high spectral state of the Galactic black hole candidates (see Tanaka Shibazaki 1996 and references herein).', 'astro-ph-9903306-1-23-1': 'Although the general spectral properties of XTE J1748-288 during the period of maximum luminosity (which we call bright hard state - BHS) satisfy these criteria, we would like to distinguish this state from the canonical high state.', 'astro-ph-9903306-1-23-2': 'The main reason for such a distinction is an exceptional role of the hard spectral component which dominates the energy spectrum in the X-ray band (the soft spectral component contribution to the total flux is [MATH]10-20 per cent instead of [MATH]60-90 per cent as typical for the HS) and crucially change the properties of the fast variability in the source flux (the BLN and QPO components of the power density spectrum are commonly related with the presence of the hard power-law component in the energy spectrum).', 'astro-ph-9903306-1-24-0': '[High State (HS)] As the flux of XTE J1748-288 dropped below [MATH]4200 cnts/s/PCA (observations [MATH]), the spectral and timing properties of the source changed drastically.', 'astro-ph-9903306-1-24-1': 'The strength of the hard component in energy spectrum decreased rapidly, while the contribution of the soft thermal component to the total luminosity in the [MATH] keV energy range became higher than 60 per cent as typical for HS (Fig. [REF], [REF]; Table 2).', 'astro-ph-9903306-1-25-0': 'Simultaneously the shape of power density spectrum changed qualitatively.', 'astro-ph-9903306-1-25-1': 'The band-limited noise (BLN) component and high frequency [MATH] Hz QPOs became undetectable, and the amplitude of the fractional variability decreased from [MATH]10 to [MATH]1 per cent.', 'astro-ph-9903306-1-25-2': 'Power density spectrum during this period was composed by power law component with a slope 1.0-1.5 and a broad QPO peak at [MATH]0.03 Hz (see Fig. [REF],[REF]; Table 3).', 'astro-ph-9903306-1-26-0': '[Low State(LS)] Subsequent decline in the intensity of XTE J1748-288 X-ray flux was accompanied by gradual decrease of the strength of the soft spectral component (Table [REF]) and in 16th observation (July 18, 1998) the source was found already in the standard low state.', 'astro-ph-9903306-1-26-1': 'It should be noted that in this state the source flux was extremely weak and substantial part of the observed emission in the PCA energy band can be attributed to the galactic diffuse and point sources, in particular to Sgr B2 (the source offset [MATH]).', 'astro-ph-9903306-1-27-0': 'The low state X-ray spectrum in the 3-25 keV energy band was composed by a power law component with photon index [MATH]2, complex of strong emission lines (6.4, 6.7, 6.9 and 8.0 keV) and low energy absorption.', 'astro-ph-9903306-1-27-1': 'Absolute intensity of the lines appeared to be very stable against the decrease of X-ray continuum, which strongly supports the interpretation that they originate from some of the sources within field of view, but not from the XTE J1748-288.', 'astro-ph-9903306-1-27-2': 'We have performed an additional study to get an estimation for the intensity of the emission line attributed to the XTE J1748-288 itself assuming centroid energy of the line to be equal to 6.4 keV.', 'astro-ph-9903306-1-27-3': 'Using the fact that quadruple decline in the continuum flux did not result to the statistically significant change in the observed line flux, we have obtained an 2[MATH] upper limit [MATH]80 eV on the equivalent width of the intrinsic emission line.', 'astro-ph-9903306-1-28-0': '## Energy spectrum fitting', 'astro-ph-9903306-1-29-0': 'We generated the energy spectra of XTE J1748-288 averaging the data over the whole single observations.', 'astro-ph-9903306-1-29-1': 'To trace the evolution of the spectral parameters during the individual observations, an additional analysis of the spectra accumulated in 256 sec time intervals was performed.', 'astro-ph-9903306-1-29-2': 'This procedure allowed us to study the evolution of the spectral parameters in a wide range of time scales from hundreds of seconds to weeks.', 'astro-ph-9903306-1-30-0': 'For the spectral analysis of the first 15 observations we used model consisted of the multicolor disk blackbody emission component (Shakura Sunyaev 1973, Mitsuda et al. 1984), power law component and low energy absorption.', 'astro-ph-9903306-1-31-0': 'To fit low state (LS) spectra we used a combination of power law and four gaussian lines with the fixed centroid energies (6.4 keV, 6.7 keV, 6.9 keV and 8.0 keV ) and the widths frozen on the 0.1 keV.', 'astro-ph-9903306-1-31-1': 'The inferred line fluxes were constant within the accuracy of measurements for all low state observations, namely, [MATH] phot/s/cm[MATH] for 6.4 keV line, [MATH] phot/s/cm[MATH] for combined 6.7 and 6.9 keV lines ) and [MATH] phot/s/cm[MATH] for 8.0 keV complex.', 'astro-ph-9903306-1-31-2': 'The corresponding equivalent width of the iron complex is [MATH]1.3 keV.', 'astro-ph-9903306-1-32-0': 'The results of fitting of the PCA data with analytical models described above are presented on Table 2.', 'astro-ph-9903306-1-32-1': 'Fig. [REF] shows an evolution of fitting parameters with time.', 'astro-ph-9903306-1-32-2': 'Mention evident correlation between changes in soft fraction, disk temperature and power-law slope during bright hard and high states.', 'astro-ph-9903306-1-33-0': '## Power density spectrum', 'astro-ph-9903306-1-34-0': 'To perform the analysis of the XTE J1748-288 timing properties we generated power density spectra (PDS) in the 0.01-250 Hz frequency range (2-13 keV energy band) using a short stretches of data.', 'astro-ph-9903306-1-34-1': 'The resulting spectra were logarithmically rebinned when necessary to reduce scatter at high frequencies and normalized to square root of fractional variability rms.', 'astro-ph-9903306-1-34-2': 'White-noise level due to the Poissonian statistics corrected for the dead-time effects was subtracted (Vikhlinin, Gilfanov Churazov 1994, Zhang et al. 1995).', 'astro-ph-9903306-1-34-3': 'To obtain PDS in the lower frequency band ([MATH] Hz - [MATH]0.01 Hz) we used the 16-sec integrated data of Standard 2 mode, because it allowed us to take into account the influence of the background variation on the PDS, which might be of importance at such frequencies.', 'astro-ph-9903306-1-35-0': 'To study the evolution of the basic timing and spectral parameters of XTE J1748-288 within the individual observations with a relatively high level of variability we generated power density spectra of the source averaged over 256 s time intervals according to the procedure described above.', 'astro-ph-9903306-1-36-0': 'We fitted the power density spectra of the source in the 0.02-150 Hz range to analytic models using [MATH] minimization technique to quantify their characteristics.', 'astro-ph-9903306-1-36-1': 'For the approximation of the PDS obtained during the first 8 observations corresponding to the bright hard state, we used the sum of the flat-topped band-limited noise (BLN) component: [MATH], where [MATH] - is a characteristic break frequency of the BLN continuum, a power law (PL) component - very low frequency noise (VLFN) and up to three harmonically related Lorentzian groups to express the QPO features (see explanation below).', 'astro-ph-9903306-1-37-0': 'Fig. [REF] presents the data and model fit for one observation (June 10).', 'astro-ph-9903306-1-37-1': 'As it is clearly seen from the inset on this figure, the single Lorentzian approximation of the 20-30 Hz QPO features does not provide an acceptable fit.', 'astro-ph-9903306-1-37-2': 'This fact could be explained in terms of the existence of the additional high frequency shoulder, similar to one observed in Nova Muscae 1991 and GX 339-4 (Belloni et al. 1997).', 'astro-ph-9903306-1-37-3': 'As we demonstrate below, the QPO centroid frequency derived from the simple Lorentzian fit moves during the individual observations (Fig. [REF]).', 'astro-ph-9903306-1-37-4': 'Since the strength of the QPO peak is anticorrelated with the QPO centroid frequency, averaging over the whole observation should lead to the steepening of the resulting QPO profile in its low frequency wing and flattening in its high frequency wing forming a shoulder-like structure.', 'astro-ph-9903306-1-37-5': 'Thus, the appearance of observed QPO shoulder could be a result of such motion of QPO centroid frequency, then it seems natural to approximate the QPO feature by a sum of several Lorentzians.', 'astro-ph-9903306-1-37-6': 'To obtain a satisfactory fit we used sum of the two Lorentzians for every harmonic.', 'astro-ph-9903306-1-37-7': 'The fit shows that the frequencies of these additional Lorentzians are also harmonically related that supports our interpretation for the shoulders origin.', 'astro-ph-9903306-1-38-0': 'High state (observations [MATH]) power density spectra were approximated by the sum of the power law very low frequency noise (VLFN) component and the Lorentzian profile representing the low frequency QPO.', 'astro-ph-9903306-1-39-0': 'The best fit parameters of band limited noise and QPO components are presented on the Table [REF].', 'astro-ph-9903306-1-39-1': 'QPO rms amplitude is calculated as a quadratic sum of the corresponding rms amplitudes for the Lorentzians used for the QPO approximation.', 'astro-ph-9903306-1-39-2': 'Parameter errors correspond to [MATH] confidence level.', 'astro-ph-9903306-1-39-3': 'These models approximate the data reasonably well, as indicated by the values of reduced [MATH] for the fits.', 'astro-ph-9903306-1-40-0': '## Correlations between spectral and variability parameters in the bright hard state', 'astro-ph-9903306-1-41-0': 'Significant evolution of the spectral and temporal properties of XTE J1748-288 in the bright hard state provides an excellent opportunity to study their mutual relationship in a wide range of timescales from seconds to several days.', 'astro-ph-9903306-1-41-1': 'In the course of our extended timing analysis the general correlation between the main parameters describing the power density spectrum of the source (QPO centroid frequency, characteristic break frequency of the BLN component, BLN rms, QPO rms) was observed.', 'astro-ph-9903306-1-41-2': 'As an example the dependence of the rms amplitude of main QPO peak on its centroid frequency is shown in Fig. [REF].', 'astro-ph-9903306-1-42-0': 'The most striking is the established close relation between the evolution of the spectral and timing parameters of the source.', 'astro-ph-9903306-1-42-1': 'The change of the QPO centroid frequency is correlated with change of the spectral parameters, derived from the energy spectra fits.', 'astro-ph-9903306-1-42-2': 'In particular, there is a clear trend of rising the QPO centroid frequency with rise of the soft component flux (Fig. [REF]).', 'astro-ph-9903306-1-42-3': 'It should be noted that this type of correlation holds on the wide range of time scales (from seconds to several days).', 'astro-ph-9903306-1-43-0': 'Similar type of the correlation between the QPO parameters and X-ray flux have been reported also for other Galactic superluminal jet source - GRS 1915+105 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999), X-ray Nova XTE J1550-564 (Cui et al. 1999) and XTE J1806-246 suspected to be a system containing a neutron star (Revnivtsev, Borozdin Emelyanov 1999).', 'astro-ph-9903306-1-43-1': 'As in the cases of GRS 1915+105 and XTE J1550-564 the QPO peak in the power density spectrum of XTE J1748-288 was detected only in the observations characterized by the domination of the hard spectral component.', 'astro-ph-9903306-1-43-2': 'This fact establishes the direct link between the processes of QPO and hard spectral component formation.', 'astro-ph-9903306-1-44-0': '# Discussion', 'astro-ph-9903306-1-45-0': 'Many bright X-ray transients are reputed black hole candidates and demonstrate similar X-ray spectra and fast variability.', 'astro-ph-9903306-1-45-1': 'During the outburst these systems are typically found in one of two qualitatively distinguishable spectral states: in high state composed by bright thermal component and extended hard power-law, or in low state with hard power-law spectrum with photon index [MATH]1.5 and an exponential high energy cutoff.', 'astro-ph-9903306-1-45-2': 'More detailed description of these states could be found elsewhere [CITATION].', 'astro-ph-9903306-1-45-3': 'Spectral evolution of X-ray transients is usually in correlation with X-ray flux variations, so it is widely believed that state transitions are driven by the variable accretion rate.', 'astro-ph-9903306-1-45-4': 'Such transitions were observed also in the persistent black hole systems, namely, Cyg X-1 and GX 339-4, but dynamics of these systems is much slower, so that they are more often observed in one of these states and typically switch to another one once per several years [CITATION].', 'astro-ph-9903306-1-46-0': 'Transient X-ray source XTE J1748-288 resembles other black hole X-ray transients in many ways.', 'astro-ph-9903306-1-46-1': 'General properties of the source light curves are similar to the properties of light curves of black hole X-ray Novae such as A 0620-00, GS 2000+25, GS/GRS 1124-68, GRS 1009-45, GRS1739-278 and others (see Chen, Shrader Livio 1997 for review, and references herein).', 'astro-ph-9903306-1-46-2': 'During 1998 outburst the source was observed by RXTE in high and low spectral states, which are very typical for black hole systems.', 'astro-ph-9903306-1-46-3': 'However, during several first observations corresponding to the peak X-ray flux the spectrum of the source was qualitatively different from any of these states.', 'astro-ph-9903306-1-46-4': 'Similar spectra were detected at the beginning of outbursts of X-ray black hole Novae GS/GRS1124-68 (Ebisawa et al. 1994, Miyamoto et al. 1994), KS/GRS 1730-312 [CITATION] and GRS 1739-278 [CITATION].', 'astro-ph-9903306-1-46-5': 'This time we studied this spectral state, which we labeled bright hard state (BHS), in more detail and using timing analysis together with spectral one.', 'astro-ph-9903306-1-47-0': 'The shape of broad-band power density spectrum of XTE J1748-288 in the BHS was similar to the PDS observed for the Galactic black hole candidates GX 339-4, Nova Muscae 1991 and GRS 1915+105 (Belloni et al. 1997; Trudolyubov, Churazov Gilfanov 1999): it was dominated by strong band-limited noise with QPO components.', 'astro-ph-9903306-1-47-1': 'We note, however, that energy spectra were not the same in those other cases.', 'astro-ph-9903306-1-48-0': 'The correlated spectral and timing evolution of the source during the BHS and, in particular, the observed trend of rising the QPO centroid frequency with rise of the soft component flux on the wide range of time scales are of special importance.', 'astro-ph-9903306-1-48-1': 'Similar type of correlation between the QPO parameters and X-ray flux have been reported also for other Galactic black hole candidates: GRS 1915+105 and XTE J1550-564 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999; Cui et al. 1999), which hints on the general mechanism of the X-ray emission generation for this class of objects.', 'astro-ph-9903306-1-49-0': 'One of the possible and probably the most promising way to explain these observational results for black hole systems is to use the models based on the idea of a shock front formation between two distinct parts of the accretion flow producing the bulk of X-rays: the central region responsible for the generation of the hard spectral component and optically thick accretion disk (Chakrabarti Titarchuk, 1995).', 'astro-ph-9903306-1-49-1': 'The oscillations in shock geometrical parameters (i.e. its height, width) may modulate the flux of the "seed" photons from the inner boundary of the optically thick accretion disk and change the state of matter in the optically thin region - and as a result, QPOs of X-ray flux are to be detected (Titarchuk, Lapidus Muslimov 1998; Molteni, Sponholz Chakrabarti, 1996).', 'astro-ph-9903306-1-49-2': 'In this case the time scale of the oscillations is determined by position of the boundary between the optically thick and thin regions, which in turn is directly related with the luminosities of the soft and hard spectral components.', 'astro-ph-9903306-1-49-3': 'Hence the observed correlation between the QPO frequency and X-ray flux could be naturally explained in the framework of this model.', 'astro-ph-9903306-1-50-0': 'Transition of XTE J1748-288 from the bright hard state to the high state coincides with the maximum of the luminosity of the soft spectral component and radical change of the properties of short-term flux variability, similar to the case of Nova Muscae 1991 (Miyamoto et al. 1994).', 'astro-ph-9903306-1-50-1': 'Power density spectrum (PDS) of the source in the high state has a power law shape of the continuum and shows the presence of the low-frequency QPO peak at [MATH] Hz, resembling the properties of the high state PDS of the well-known black hole candidate LMC X-1 (its form was described by the sum of the power law component and a QPO peak with centroid frequency [MATH] Hz) (Ebisawa et al. 1989).', 'astro-ph-9903306-1-51-0': 'Following decrease of the luminosity of XTE J1748-288 switched the source to the low state typical for the black hole X-ray Novae at this stage of the outburst evolution.', 'astro-ph-9903306-1-52-0': 'The observed complex of general spectral and timing properties of XTE J1748-288 is similar to that of the dynamically proven black hole X-ray Novae and Galactic superluminal jet sources.', 'astro-ph-9903306-1-52-1': 'This fact could hint on the universal character of evolution of the accretion flow in the black hole systems and its direct link to the mechanism of the formation of relativistic ejections of matter.'}

{'astro-ph-9903306-2-0-0': 'We present an analysis of the RXTE observations of the recently discovered Galactic microquasar XTE J1748-288 during its 1998 outburst.', 'astro-ph-9903306-2-0-1': 'General spectral and temporal properties of the source and their evolution were very typical for the Galactic black hole candidates (BHC) and, in particular, black hole X-ray Novae.', 'astro-ph-9903306-2-1-0': 'The spectral evolution of the source during the outburst can be considered a sequence of qualitatively distinct states.', 'astro-ph-9903306-2-1-1': 'During the first observations, corresponding to the maximum of X-ray flux, the spectrum of the source consisted of a dominating hard power law component and a soft thermal component, which can be described by the model of multicolor disk emission.', 'astro-ph-9903306-2-1-2': 'The hard component contributed [MATH]80% to the X-ray luminosity in the 3-25 keV energy band.', 'astro-ph-9903306-2-1-3': 'Overall two-component spectral shape is an attribute of very high state (VHS) observed previously in BHC, but the domination of hard component is unusual.', 'astro-ph-9903306-2-1-4': 'Later on, as the X-ray source faded, its energy spectrum qualitatively changed, showing high (HS) and then low (LS) states, both typical for black hole binaries.', 'astro-ph-9903306-2-2-0': 'As the energy spectrum changed, the fast variability also evolved dramatically.', 'astro-ph-9903306-2-2-1': 'Initially the power density spectrum was formed by a dominating band-limited noise component, QPO features at 20-30 Hz and at [MATH]0.5 Hz, and a very low frequency noise component.', 'astro-ph-9903306-2-2-2': 'After a significant decrease of the contribution of the hard spectral component the amplitude of the fractional variability decreased by an order of magnitude and the PDS spectrum adopted a power-law shape with a broad QPO peak around 0.03 Hz.', 'astro-ph-9903306-2-2-3': 'When the system switched to the LS, the PDS shape changed again and the QPOs have not been detected since.', 'astro-ph-9903306-2-3-0': 'When the source was observed in VHS, a clear correlation between QPO parameters and X-ray flux was seen.', 'astro-ph-9903306-2-3-1': 'Such a correlation gives an insight into our understanding of the accretion process in X-ray black hole binaries.', 'astro-ph-9903306-2-4-0': '# Introduction', 'astro-ph-9903306-2-5-0': 'The XTE J1748-288 was discovered as a new transient source on June 4, 1998 by All Sky Monitor (ASM) aboard the Rossi X-ray Timing Explorer (RXTE) observatory (Smith, Levine Wood 1998).', 'astro-ph-9903306-2-5-1': 'The X-ray source was localized with an accuracy of 1 arcmin in multiple scans of the region by the PCA/RXTE experiment (Strohmayer Marshall 1998).', 'astro-ph-9903306-2-5-2': 'Observations in the radio band by VLA revealed the presence of an unresolved radio source with a position consistent with one found by PCA: R.A.=17[MATH].06, Dec=-28[MATH].8 (equinox 2000.0, position uncertainty 0.', 'astro-ph-9903306-2-5-3': '[MATH]6, Hjellming, Rupen Mioduszewski 1998).', 'astro-ph-9903306-2-5-4': 'Significant variability of the new radio source (Hjellming et al. 1998a; Fender Stappers 1998) strongly supports its association with the X-ray transient.', 'astro-ph-9903306-2-5-5': 'In June 14.31 radio source became extended, with a proper motion of 20-40 mas/day (Rupen, Hjellming Mioduszewski 1998).', 'astro-ph-9903306-2-5-6': 'The intrinsic velocity of the moving jet was higher than [MATH]c for the distance [MATH]8 kpc derived from a 21 cm HI absorption measurement (Hjellming et al. 1998b).', 'astro-ph-9903306-2-6-0': 'Two quasi periodical oscillations (hereafter QPO) with the central frequencies [MATH]0.5 Hz and [MATH]32 Hz were found in the power density spectrum (PDS) of the source in the PCA observation of June 6, 1998 (Fox Lewin 1998).', 'astro-ph-9903306-2-7-0': 'The flux from the source was above ASM/RXTE detection limit until August of 1998.', 'astro-ph-9903306-2-7-1': 'Pointed instruments of RXTE - PCA and HEXTE - observed the source quasi evenly from June till September, obtaining a good coverage of the whole outburst.', 'astro-ph-9903306-2-7-2': 'Here we present the results of a spectral and timing analysis of these RXTE observations.', 'astro-ph-9903306-2-8-0': '# Observations and analysis', 'astro-ph-9903306-2-9-0': 'In our analysis we used all the publicly available data obtained from RXTE TOO (Target Of Opportunity) archive including 21 pointed observations and "slew" parts of two PI restricted observations.', 'astro-ph-9903306-2-9-1': 'The 23 observations quasi evenly cover the 1998 outburst of the source with a total exposure of [MATH] ksec. The list of observations is presented in Table [REF].', 'astro-ph-9903306-2-10-0': 'For data reduction we used the standard FTOOLS package.', 'astro-ph-9903306-2-10-1': 'For the spectral analysis we used PCA data collected in the 3-25 keV energy range.', 'astro-ph-9903306-2-10-2': 'The response matrix was constructed for every single observation using PCARMF v3.5.', 'astro-ph-9903306-2-10-3': 'For the PCA background estimations we applied a Very Large Events (VLE)-based model.', 'astro-ph-9903306-2-10-4': 'For some observations, where the standard VLE-based background subtraction was not good enough for our purposes, we used the activation component for background models separately (see details in the RXTE Cook Book, "Using the Latest PCABACKEST").', 'astro-ph-9903306-2-10-5': 'All spectra were dead time corrected.', 'astro-ph-9903306-2-10-6': 'The analysis of the spectra of the Crab nebula confirmed that the uncertainties attributed to the response matrix do not exceed 0.5-1.0%.', 'astro-ph-9903306-2-10-7': 'To account for the uncertainty in the knowledge of the spectral response, a 1 per cent systematic error was added to the statistical error in each PCA channel.', 'astro-ph-9903306-2-11-0': 'For our timing analysis we used the PCA timing modes [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9903306-2-11-1': 'Background subtracted light curves from [MATH] mode data were used to generate the power density spectra for very low frequencies (below [MATH] Hz).', 'astro-ph-9903306-2-12-0': 'Unfortunately, an analysis of the HEXTE data was strongly complicated by the difficulties in the background subtraction.', 'astro-ph-9903306-2-12-1': 'The source/background beam-switching of the HEXTE clusters (they "rock" periodically to [MATH]3.0[MATH] from the source position), is not very efficient for precise background subtraction when the source is located in the densely populated Galactic Center region.', 'astro-ph-9903306-2-12-2': 'There are a number of X-ray sources around XTE J1748-288, namely GX 5-1, GRS 1758-258, GX 3+1 and 1E1740.7-2942 to mention few, each of them luminous enough to affect the results of the background measurement.', 'astro-ph-9903306-2-12-3': 'We have performed a check of the quality of the background subtraction in any given observation by comparing the m-background spectra (the spectra obtained during the -3.0[MATH] offset from the source) with the p-background (+3.0[MATH] offset).', 'astro-ph-9903306-2-12-4': 'Only observations with an adequate quality of subtraction were accepted for the analysis.', 'astro-ph-9903306-2-12-5': 'Because the rocking planes for clusters A and B are perpendicular to each other, different sky regions are within the field of view of these clusters during the background measurements.', 'astro-ph-9903306-2-12-6': 'Therefore A clusters can provide more accurate background measurements for some observations, and B clusters - for others.', 'astro-ph-9903306-2-12-7': 'After the 8-th observation, when the source flux in the 20-100 keV energy band had dropped below [MATH]10 cnts/s/cluster, the accuracy of our knowledge of the background became comparable to the source flux, and we excluded these later HEXTE data from our spectral analysis.', 'astro-ph-9903306-2-12-8': 'However, we present the HEXTE observations of the source in its LS in Fig. [REF].', 'astro-ph-9903306-2-12-9': 'The spectrum plotted there was obtained from carefully selected data, but still should be treated with care.', 'astro-ph-9903306-2-13-0': '# Results', 'astro-ph-9903306-2-14-0': '## Energy spectra approximation', 'astro-ph-9903306-2-15-0': 'We generated the energy spectra of XTE J1748-288 averaging the data over the whole single observations.', 'astro-ph-9903306-2-15-1': 'To trace the evolution of the spectral parameters within some observations, an additional analysis of the spectra, accumulated in 256 sec time intervals, was performed.', 'astro-ph-9903306-2-15-2': 'These spectra were then used to study the correlation of the spectral and variability parameters (see section 3.4).', 'astro-ph-9903306-2-16-0': 'For the spectral approximation of the first 15 observations we used the model consisted of a multicolor disk blackbody emission component (Shakura Sunyaev 1973; Mitsuda et al. 1984), a power law component and low energy absorption.', 'astro-ph-9903306-2-17-0': 'Overall spectral shape for the remaining observations (16-23) can be approximated by power law with low energy absorption.', 'astro-ph-9903306-2-17-1': 'However, strong emission line-like feature around [MATH]6.5 keV cannot be ignored.', 'astro-ph-9903306-2-17-2': 'This feature may be approximated reasonably well by a Gaussian with centroid energy [MATH] keV and width [MATH] keV.', 'astro-ph-9903306-2-17-3': 'Its absolute intensity appeared to be stable against a decrease of X-ray continuum by a factor of [MATH]4, thus we believe that this line emission originates from the diffuse X-ray source near the Center of the Galaxy rather than from XTE J1748-288 itself.', 'astro-ph-9903306-2-17-4': 'The measured intensity of this feature, [MATH] phot/s/cm[MATH], is in a reasonable agreement with previous measurements of the diffuse line emission from the Galactic Center (e.g. Yamauchi Koyama 1993).', 'astro-ph-9903306-2-17-5': 'The inclusion of the most prominent Galactic diffuse lines - 6.4, 6.7, and 6.9 keV - also improves the spectral approximation significantly (the widths of the lines were frozen at 0.1 keV - value comparable with the PCA energy resolution).', 'astro-ph-9903306-2-17-6': 'The best fit line intensities ([MATH] phot/s/cm[MATH] for 6.4 keV line and [MATH] phot/s/cm[MATH] for 6.7 keV and 6.9 keV lines together ) were approximately constant within the accuracy of the measurements for all LS observations.', 'astro-ph-9903306-2-17-7': 'The equivalent width of the whole [MATH] feature increased from [MATH]300 eV in observation 16 to [MATH]1.1 keV in observation 23.', 'astro-ph-9903306-2-18-0': 'Spectral fits revealed the presence of an additional line in the spectrum, with the central energy around 8 keV and an intensity [MATH] phot/s/cm[MATH].', 'astro-ph-9903306-2-18-1': 'Even given the moderate energy resolution of the PCA, the presence of this line is evident both when the Fe [MATH] complex is fitted by by three narrow lines as described above, and when one fits it by one broader line with central energy [MATH]6.5 keV instead.', 'astro-ph-9903306-2-18-2': 'The 8 keV line feature is likely a complex of the Fe [MATH] or/and Ni lines of the hard ([MATH] keV) optically thin diffuse plasma emission of the Galactic Center region [CITATION].', 'astro-ph-9903306-2-18-3': 'Note, that the similar spectral feature was observed in the spectrum of XTE J0421+560/CI Cam [CITATION], that was likely caused by optically thin plasma thermal emission.', 'astro-ph-9903306-2-19-0': 'The results of the spectral approximation of the PCA data with the analytical models described above are presented in Table 2.', 'astro-ph-9903306-2-19-1': 'Fig. [REF] shows an evolution of some best fit parameters with time.', 'astro-ph-9903306-2-19-2': 'A correlation between changes in the soft fraction, the disk temperature and the power-law slope during the first 15 observations (VHS) is clearly seen.', 'astro-ph-9903306-2-20-0': '## Power density spectrum', 'astro-ph-9903306-2-21-0': 'In order to analyze the timing properties of XTE J1748-288, we generated power density spectra (PDS) in the 0.01-250 Hz frequency range (for 2-13 keV energy band) using short stretches of data.', 'astro-ph-9903306-2-21-1': 'The resulting spectra were logarithmically rebinned when necessary to reduce the scatter at high frequencies and normalized to the square of the fractional variability.', 'astro-ph-9903306-2-21-2': 'The white-noise level due to the Poissonian statistics corrected for the dead-time effects was subtracted (Vikhlinin, Gilfanov Churazov 1994; Zhang et al. 1995).', 'astro-ph-9903306-2-21-3': 'To obtain PDS in the lower frequency band ([MATH] Hz - [MATH]0.01 Hz) we used the 16-sec integrated data of Standard 2 mode, because it allowed us to take into account the influence of the background variation on the PDS, which might be of importance at these frequencies.', 'astro-ph-9903306-2-22-0': 'To study the evolution of the basic timing and spectral parameters of XTE J1748-288 within the individual observations with a relatively high level of variability we generated power density spectra of the source averaged over 256 s time intervals according to the procedure described above (for the results of this study see section 3.4).', 'astro-ph-9903306-2-23-0': 'We fitted the power density spectra of the source in the 0.02-150 Hz range to analytic models using the [MATH] minimization technique to quantify their characteristics.', 'astro-ph-9903306-2-23-1': 'For the approximation of the PDS obtained during the first 8 observations, we used the sum of a flat-topped band-limited noise (BLN) component: [MATH], where [MATH] - is a characteristic break frequency of the BLN continuum, a power law (PL) component - very low frequency noise (VLFN) and up to three harmonically related Lorentzian groups to describe the QPO features (see explanation below).', 'astro-ph-9903306-2-24-0': 'Fig. [REF] presents the data and model fit for one observation (June 10).', 'astro-ph-9903306-2-24-1': 'As is clearly seen from the inset of this figure, an approximation of the 20-30 Hz QPO features by a single Lorentzian component does not provide an acceptable fit.', 'astro-ph-9903306-2-24-2': 'This fact could be explained in terms of the existence of an additional high frequency shoulder, similar to those observed in Nova Muscae 1991 and GX 339-4 (Belloni et al. 1997).', 'astro-ph-9903306-2-24-3': 'As we demonstrate below (section 3.4), the QPO centroid frequency derived from the simple Lorentzian fit shifts during the individual observations (Fig. [REF]).', 'astro-ph-9903306-2-24-4': 'Since the strength of the QPO peak is anti-correlated with the QPO centroid frequency, averaging over the whole observation should lead to a steepening of the resulting QPO profile in its low frequency wing and a flattening in its high frequency wing forming a shoulder-like structure.', 'astro-ph-9903306-2-24-5': 'Thus, the appearance of the observed QPO shoulder could be a result of such a displacement of the QPO centroid frequency, for this reason it seems natural to approximate the QPO feature by a sum of several Lorentzians.', 'astro-ph-9903306-2-24-6': 'To obtain a satisfactory fit we used the sum of two Lorentzians for every harmonic.', 'astro-ph-9903306-2-24-7': 'The approximation shows that the frequencies of these additional Lorentzians are also harmonically related which supports our interpretation of the origin of the shoulders.', 'astro-ph-9903306-2-25-0': 'In observations [MATH] the power density spectra were approximated by a power law component, the very low frequency noise (VLFN) component and a Lorentzian profile representing the low frequency QPO.', 'astro-ph-9903306-2-26-0': 'The best fit parameters of the band limited noise and QPO components are presented in Table [REF].', 'astro-ph-9903306-2-26-1': 'QPO rms amplitude is calculated as the quadratic sum of the corresponding rms amplitudes for the Lorentzians used for the QPO approximation.', 'astro-ph-9903306-2-26-2': 'Parameter errors correspond to the [MATH] confidence level.', 'astro-ph-9903306-2-26-3': 'These models approximate the data reasonably well, as indicated by the values of reduced [MATH] for the fits.', 'astro-ph-9903306-2-27-0': '## Evolution of the source during the outburst', 'astro-ph-9903306-2-28-0': 'The X-ray flux histories of the XTE J1748-288 outburst based on the [MATH]/PCA and ASM data in the 3-15 and 15-30 keV energy ranges are shown in Fig. [REF].', 'astro-ph-9903306-2-28-1': 'The evolution of the source flux in the soft X-ray band (3-15 keV) was characterized by the fast initial rise to a level of [MATH]600-700 mCrab on a time scale of [MATH]2-3 days followed by the [MATH] day-long maximum and a nearly exponential decay to the quiescent level with e-folding time [MATH] days.', 'astro-ph-9903306-2-28-2': 'The flux in the hard band (15-30 keV) undergone a fast initial rise to the [MATH]300-400 mCrab level followed by the maximum phase of a [MATH]-day duration and an abrupt decay, which is similar to the behavior of Nova Muscae GS/GRS 1124-683 [CITATION] during its outburst of 1991.', 'astro-ph-9903306-2-28-3': 'Based on the results of the spectral and timing analysis, the evolution of XTE J1748-288, in analogy to Nova Muscae 1991, could be divided into three distinct parts, corresponding to three main phenomenological states: very high (VHS), high (HS) and low (LS).', 'astro-ph-9903306-2-28-4': 'At the same time we would like to note, that in the first 8 observations the spectrum of XTE J1748-288 have an extremely bright hard component (the power law component contribution to the total 3-25 keV flux [MATH]80%) which is somewhat different from the usual VHS spectrum.', 'astro-ph-9903306-2-28-5': 'Very similar spectra were observed only with Ginga at the very beginning of the outburst of GS 1124-683 (Kitamoto et al. 1992; Ebisawa et al. 1994), and with Mir-Kvant at the beginning of the outbursts of X-ray Novae KS1730-312 [CITATION] and GRS 1739-278 [CITATION], but observations of XTE J1748-288 outburst provided an unique opportunity to study this unusually hard VHS spectrum in detail.', 'astro-ph-9903306-2-28-6': 'It is also worth mentioning that contrary to many previous observations of the VHS, which could be distinguished from the HS according to the difference in the power spectrum only, in our case both the energy spectrum and power spectrum in VHS were distinctly different from those in the HS.', 'astro-ph-9903306-2-29-0': '[Very High State] During the first eight RXTE pointed observations (June 4 - 11, 1998) the source was found in an extremely bright state.', 'astro-ph-9903306-2-29-1': 'The broad-band 3-150 keV energy spectrum of XTE J1748-288 in this state can be satisfactorily described by the sum of two components: a relatively weak soft thermal component (contributing only [MATH] 10-20 per cent to the total X-ray flux in the 3-25 keV energy band) with a characteristic temperature of [MATH] 0.8-1.4 keV and a strong hard component, which is approximately a power law shape and does not show the evidence of a high energy cut-off up to [MATH] keV .', 'astro-ph-9903306-2-29-2': 'The corresponding luminosity in the 3-25 keV band uncorrected for the interstellar absorption was at a level of 1-1.5[MATH] ergs/s (assuming a 8 kpc distance) during this period.', 'astro-ph-9903306-2-30-0': 'The broad-band power density spectrum (PDS) of the source is formed by the dominating band-limited noise component (corresponding rms amplitude [MATH] 5-9 %), the strong QPO feature with centroid frequency [MATH] 20-30 Hz and a very low frequency noise (VLFN) component with a slope [MATH] and rms amplitude [MATH] 1.0-1.5 % (Fig. [REF], upper panels).', 'astro-ph-9903306-2-31-0': 'The correlated spectral and timing evolution of the source during this period is of particular interest.', 'astro-ph-9903306-2-31-1': 'The local increase of the contribution of the soft spectral component to the total X-ray flux in observations 3-4 (June 6 - 7, 1998) was accompanied by a steepening of the hard spectral component and significant changes in the power density spectrum (Table 2, 3.', 'astro-ph-9903306-2-31-2': 'Fig. [REF], left upper panel), i.e.:', 'astro-ph-9903306-2-32-0': '- the increase of the BLN characteristic break frequency and the decrease of its rms amplitude;', 'astro-ph-9903306-2-33-0': '- the increase of the QPO centroid frequency from [MATH] Hz to [MATH] Hz;', 'astro-ph-9903306-2-34-0': '- the appearance of the additional broad QPO feature at [MATH] Hz (Fox Lewin 1998);', 'astro-ph-9903306-2-35-0': '- the notable rise of the VLFN rms amplitude.', 'astro-ph-9903306-2-35-1': 'Detected correlations between the spectral and timing properties of the source in this state, discussed in more detail below (see 3.4), holds a wide range of the time scales from seconds to several days.', 'astro-ph-9903306-2-36-0': 'The two-component spectrum, consisting of the soft thermal component and a hard power law component is a signature of the high spectral state of the Galactic black hole candidates (see Tanaka Shibazaki 1996, and references therein).', 'astro-ph-9903306-2-36-1': 'The very high state differs from high state mostly by the much stronger fast variability, while the differences in energy spectra are not significant and no good criterium was suggested to distinguish between these two states on spectral ground.', 'astro-ph-9903306-2-36-2': 'In case of XTE J1748-288 the spectrum was sufficiently different from the typical spectra observed previously in the HS and the VHS, because power law component dominated in the 3-25 keV energy band.', 'astro-ph-9903306-2-36-3': 'Fast variability and shape of power spectrum was typical for the VHS of Galactic black hole binaries (e.g. Belloni et al. 1997).', 'astro-ph-9903306-2-37-0': '[High State].', 'astro-ph-9903306-2-37-1': 'As the flux of XTE J1748-288 dropped below [MATH]4200 cnts/s/PCA (observations 9-15), the spectral and timing properties of the source changed drastically.', 'astro-ph-9903306-2-37-2': 'The strength of the hard component in the energy spectrum decreased rapidly and the contribution of the soft thermal component to the total luminosity in the 3-25 keV energy range increased to more than 60 per cent as typical for HS (Fig. [REF], [REF]; Table 2).', 'astro-ph-9903306-2-38-0': 'Simultaneously the shape of the power density spectrum changed qualitatively.', 'astro-ph-9903306-2-38-1': 'The band-limited noise (BLN) component and the high frequency 20-30 Hz QPOs became undetectable, and the amplitude of the fractional variability decreased from [MATH]10 to [MATH]1 per cent.', 'astro-ph-9903306-2-38-2': 'The power density spectrum during this period was composed of the power law component with a slope of -1.0-1.5 and a broad QPO peak at [MATH]0.03 Hz (see Fig. [REF],[REF]; Table 3).', 'astro-ph-9903306-2-39-0': '[Low State].', 'astro-ph-9903306-2-39-1': 'The subsequent decline in the intensity of the XTE J1748-288 X-ray flux was accompanied by a gradual decrease of the strength of the soft spectral component (Table [REF]) and in 16th observation (July 18, 1998) the source was found already in the standard LS.', 'astro-ph-9903306-2-39-2': 'It should be noted that in this state the source flux was extremely weak and a substantial part of the observed emission in the PCA energy band can be attributed to the galactic diffuse and point sources, in particular to Sgr B2 (the offset from XTE J1748-288 [MATH]).', 'astro-ph-9903306-2-40-0': 'The low state X-ray spectrum in the 3-25 keV energy band was composed by a power law component with a photon index of [MATH]2, a complex of strong emission lines (around 6.5 keV and at [MATH]8 keV) and low energy absorption.', 'astro-ph-9903306-2-40-1': 'The absolute intensity of the lines appeared to be very stable against the decrease of X-ray continuum, which strongly supports the interpretation that they originate from some of the sources within the field of view, but not from the XTE J1748-288.', 'astro-ph-9903306-2-40-2': 'We have performed an additional study to get an estimation for the intensity of the emission line attributed to the XTE J1748-288 itself, assuming the centroid energy of the intrinsic source line to be equal to 6.4 keV.', 'astro-ph-9903306-2-40-3': 'Using the fact that the strong decline in the continuum flux did not result in the statistically significant change in the observed 6.4 keV line flux, we obtained 2[MATH] upper limit [MATH]80 eV on the equivalent width of the intrinsic emission line.', 'astro-ph-9903306-2-41-0': '## Correlations between spectral and variability parameters in the very high state', 'astro-ph-9903306-2-42-0': 'The significant evolution of the spectral and temporal properties of XTE J1748-288 in the VHS provides an excellent opportunity to study their mutual relationship in a wide range of time scales from seconds to several days.', 'astro-ph-9903306-2-42-1': 'In the course of our extended timing analysis the general correlation between the main parameters describing the power density spectrum of the source (QPO centroid frequency, characteristic break frequency of the BLN component, BLN rms, QPO rms) was observed.', 'astro-ph-9903306-2-42-2': 'The dependence of the rms amplitude of main QPO peak on its centroid frequency is shown in Fig. [REF] .', 'astro-ph-9903306-2-42-3': 'The plot demonstrates evident anticorrelation between these two parameters.', 'astro-ph-9903306-2-42-4': 'Other PDS parameters also appeared to be strongly inter-correlated.', 'astro-ph-9903306-2-43-0': 'Most striking is the established close relation between the evolution of the spectral and timing parameters of the source.', 'astro-ph-9903306-2-43-1': 'The change of the QPO centroid frequency is correlated with change of the spectral parameters, derived from the energy spectra fits.', 'astro-ph-9903306-2-43-2': 'In particular, there is a clear trend of increasing the QPO centroid frequency with rise of the soft component flux (Fig. [REF]).', 'astro-ph-9903306-2-43-3': 'It should be noted that this type of correlation holds on the wide range of time scales (from seconds to several days).', 'astro-ph-9903306-2-44-0': 'A similar type of the correlation between the QPO parameters and X-ray flux has also been reported for other Galactic superluminal jet source GRS 1915+105 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999), X-ray Nova XTE J1550-564 (Cui et al. 1999), and XTE J1806-246, that is suspected to be a system containing a neutron star (Revnivtsev, Borozdin Emelyanov 1999).', 'astro-ph-9903306-2-44-1': 'As in the cases of GRS 1915+105 and XTE J1550-564 the QPO peak in the power density spectrum of XTE J1748-288 was detected only in observations characterized by the domination of the hard spectral component.', 'astro-ph-9903306-2-44-2': 'This fact hints on the direct link between the processes of QPO and hard spectral component formation.', 'astro-ph-9903306-2-45-0': '# Discussion', 'astro-ph-9903306-2-46-0': 'Many bright X-ray transients are reputed black hole candidates and demonstrate similar X-ray spectra and fast variability.', 'astro-ph-9903306-2-46-1': 'During the outburst these systems are typically found in one of two qualitatively distinguishable spectral states: in the high state composed of the bright thermal component and the extended hard power-law, or in the low state with the hard power-law spectrum with a photon index of [MATH]1.5 and an exponential high energy cutoff.', 'astro-ph-9903306-2-46-2': 'A more detailed description of these states can be found elsewhere [CITATION].', 'astro-ph-9903306-2-46-3': 'A third, very high, state has been recognized also, with two component spectrum similar to the HS, but with somewhat stronger power-law component and with much stronger fast variability (Miyamoto et al. 1991, 1994; Takizawa et al. 1997).', 'astro-ph-9903306-2-46-4': 'The spectral evolution of X-ray transients is usually in correlation with X-ray flux changes, so it is widely believed that state transitions are driven by the variable accretion rate.', 'astro-ph-9903306-2-46-5': 'Such transitions were also observed in the persistent black hole systems, namely, Cyg X-1 and GX 339-4, but the dynamics of these systems is much slower, so they are more often observed in one of these states and typically switch to another state once every several years [CITATION].', 'astro-ph-9903306-2-47-0': 'The transient X-ray source XTE J1748-288 resembles other black hole X-ray transients in many ways.', 'astro-ph-9903306-2-47-1': 'The general properties of the source light curves are similar to the properties of the light curves of black hole X-ray Novae such as A 0620-00, GS 2000+25, GS/GRS 1124-68, GRS 1009-45, GRS1739-278 and others (see Chen, Shrader Livio 1997 for review, and references therein).', 'astro-ph-9903306-2-47-2': 'During the 1998 outburst the source was observed by RXTE in very high, high and low spectral states, which are very typical for black hole systems.', 'astro-ph-9903306-2-47-3': 'During several observations corresponding to the peak X-ray flux, the spectrum of the source was in an unusual VHS, a very bright hard component was observed.', 'astro-ph-9903306-2-47-4': 'Similar spectra were detected at the beginning of the outbursts of X-ray black hole Novae GS/GRS1124-68 [CITATION], KS/GRS 1730-312 [CITATION] and GRS 1739-278 [CITATION].', 'astro-ph-9903306-2-47-5': 'Here we studied this spectrum in more detail.', 'astro-ph-9903306-2-48-0': 'The transition of XTE J1748-288 from the very high state to the high state changed both the spectral and fast variability properties of the X-ray flux of XTE J1748-288.', 'astro-ph-9903306-2-48-1': 'Unlike the case of Nova Muscae [CITATION], hard component dominated spectrum during all VHS observations.', 'astro-ph-9903306-2-48-2': 'The soft component in the spectrum of the XTE J1748-288 increased rapidly by a factor of [MATH]4 after the transition, while the total flux in the energy band 3-25 keV continued to decrease smoothly.', 'astro-ph-9903306-2-48-3': 'The amplitude of the fast variability dropped down during the transition and PDS shape changed qualitatively (see Fig. [REF]).', 'astro-ph-9903306-2-48-4': 'PDS of the source X-ray flux in the HS has a power law shape of the continuum and shows the presence of the low-frequency QPO peak at [MATH] Hz, resembling the properties of the high state PDS of the well-known black hole candidate LMC X-1 (its form was described by the sum of the power law component and a QPO peak with a centroid frequency of [MATH] Hz) (Ebisawa et al. 1989).', 'astro-ph-9903306-2-49-0': 'After the subsequent decrease of the luminosity, the XTE J1748-288 switched the source to the low state typical for the black hole X-ray Novae at this stage of the outburst evolution.', 'astro-ph-9903306-2-50-0': 'The shape of the broad-band power density spectrum of XTE J1748-288 in the VHS was similar to the PDS observed for the Galactic black hole candidates GX 339-4, Nova Muscae 1991 and GRS 1915+105 (Miyamoto et al. 1994; Belloni et al. 1997; Trudolyubov, Churazov Gilfanov 1999): it was dominated by a strong band-limited noise with QPO components.', 'astro-ph-9903306-2-51-0': 'The correlated spectral and timing evolution of the source during the VHS and, in particular, the observed trend of increasing the QPO centroid frequency with the rise of the soft component flux on the wide range of time scales is of special importance.', 'astro-ph-9903306-2-51-1': 'A similar type of correlation between the QPO parameters and X-ray flux have also been reported for other Galactic black hole candidates: GRS 1915+105 and XTE J1550-564 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999; Cui et al. 1999), which hints at the general mechanism of the X-ray emission generation for this class of objects.', 'astro-ph-9903306-2-52-0': 'One of the possible and probably the most promising way to explain these observational results for black hole systems is to use the models based on the idea of a shock front formation between two distinct parts of the accretion flow producing the bulk of X-rays: the central region responsible for the generation of the hard spectral component and optically thick accretion disk (Chakrabarti Titarchuk 1995).', 'astro-ph-9903306-2-52-1': 'The oscillations in shock geometrical parameters (i.e. height, width) may modulate the flux of the "seed" photons from the inner boundary of the optically thick accretion disk and change the state of matter in the optically thin region - and as a result, QPOs of X-ray flux are to be detected (Titarchuk, Lapidus Muslimov 1998; Molteni, Sponholz Chakrabarti, 1996).', 'astro-ph-9903306-2-52-2': 'In this case the time scale of the oscillations is determined by the position of the boundary between the optically thick and thin regions, which in turn is directly related to the luminosities of the soft and hard spectral components.', 'astro-ph-9903306-2-52-3': 'Hence the observed correlation between the QPO frequency and X-ray flux could be naturally explained in the framework of this model.', 'astro-ph-9903306-2-53-0': 'The observed complex of the general spectral and timing properties of XTE J1748-288 is similar to that of the dynamically proven black hole X-ray Novae and Galactic superluminal jet sources.', 'astro-ph-9903306-2-53-1': 'This fact could hint at the universal character of evolution of the accretion flow in the black hole systems and its direct link to the mechanism of the formation of relativistic ejections of matter.'}

[['astro-ph-9903306-1-45-0', 'astro-ph-9903306-2-46-0'], ['astro-ph-9903306-1-49-3', 'astro-ph-9903306-2-52-3'], ['astro-ph-9903306-1-37-0', 'astro-ph-9903306-2-24-0'], ['astro-ph-9903306-1-11-1', 'astro-ph-9903306-2-11-1'], ['astro-ph-9903306-1-41-1', 'astro-ph-9903306-2-42-1'], ['astro-ph-9903306-1-10-6', 'astro-ph-9903306-2-10-7'], ['astro-ph-9903306-1-42-1', 'astro-ph-9903306-2-43-1'], ['astro-ph-9903306-1-42-3', 'astro-ph-9903306-2-43-3'], ['astro-ph-9903306-1-39-3', 'astro-ph-9903306-2-26-3'], ['astro-ph-9903306-1-12-0', 'astro-ph-9903306-2-12-0'], ['astro-ph-9903306-2-38-0', 'astro-ph-9903306-3-38-0'], ['astro-ph-9903306-2-38-1', 'astro-ph-9903306-3-38-1'], ['astro-ph-9903306-2-38-2', 'astro-ph-9903306-3-38-2'], ['astro-ph-9903306-2-1-0', 'astro-ph-9903306-3-1-0'], ['astro-ph-9903306-2-1-1', 'astro-ph-9903306-3-1-1'], ['astro-ph-9903306-2-1-2', 'astro-ph-9903306-3-1-2'], ['astro-ph-9903306-2-1-3', 'astro-ph-9903306-3-1-3'], ['astro-ph-9903306-2-1-4', 'astro-ph-9903306-3-1-4'], ['astro-ph-9903306-2-42-0', 'astro-ph-9903306-3-42-0'], ['astro-ph-9903306-2-42-1', 'astro-ph-9903306-3-42-1'], ['astro-ph-9903306-2-42-2', 'astro-ph-9903306-3-42-2'], ['astro-ph-9903306-2-42-3', 'astro-ph-9903306-3-42-3'], ['astro-ph-9903306-2-42-4', 'astro-ph-9903306-3-42-4'], ['astro-ph-9903306-2-53-0', 'astro-ph-9903306-3-53-0'], ['astro-ph-9903306-2-53-1', 'astro-ph-9903306-3-53-1'], ['astro-ph-9903306-2-7-0', 'astro-ph-9903306-3-7-0'], ['astro-ph-9903306-2-7-1', 'astro-ph-9903306-3-7-1'], ['astro-ph-9903306-2-7-2', 'astro-ph-9903306-3-7-2'], ['astro-ph-9903306-2-43-0', 'astro-ph-9903306-3-43-0'], ['astro-ph-9903306-2-43-1', 'astro-ph-9903306-3-43-1'], ['astro-ph-9903306-2-43-2', 'astro-ph-9903306-3-43-2'], ['astro-ph-9903306-2-43-3', 'astro-ph-9903306-3-43-3'], ['astro-ph-9903306-2-31-0', 'astro-ph-9903306-3-31-0'], ['astro-ph-9903306-2-31-1', 'astro-ph-9903306-3-31-1'], ['astro-ph-9903306-2-37-1', 'astro-ph-9903306-3-37-1'], ['astro-ph-9903306-2-37-2', 'astro-ph-9903306-3-37-2'], ['astro-ph-9903306-2-39-1', 'astro-ph-9903306-3-39-1'], ['astro-ph-9903306-2-39-2', 'astro-ph-9903306-3-39-2'], ['astro-ph-9903306-2-17-0', 'astro-ph-9903306-3-17-0'], ['astro-ph-9903306-2-17-1', 'astro-ph-9903306-3-17-1'], ['astro-ph-9903306-2-17-2', 'astro-ph-9903306-3-17-2'], ['astro-ph-9903306-2-17-3', 'astro-ph-9903306-3-17-3'], ['astro-ph-9903306-2-17-4', 'astro-ph-9903306-3-17-4'], ['astro-ph-9903306-2-17-5', 'astro-ph-9903306-3-17-5'], ['astro-ph-9903306-2-17-6', 'astro-ph-9903306-3-17-6'], ['astro-ph-9903306-2-17-7', 'astro-ph-9903306-3-17-7'], ['astro-ph-9903306-2-25-0', 'astro-ph-9903306-3-25-0'], ['astro-ph-9903306-2-47-0', 'astro-ph-9903306-3-47-0'], ['astro-ph-9903306-2-47-1', 'astro-ph-9903306-3-47-1'], ['astro-ph-9903306-2-47-2', 'astro-ph-9903306-3-47-2'], ['astro-ph-9903306-2-47-3', 'astro-ph-9903306-3-47-3'], ['astro-ph-9903306-2-47-4', 'astro-ph-9903306-3-47-4'], ['astro-ph-9903306-2-47-5', 'astro-ph-9903306-3-47-5'], ['astro-ph-9903306-2-49-0', 'astro-ph-9903306-3-49-0'], ['astro-ph-9903306-2-9-0', 'astro-ph-9903306-3-9-0'], ['astro-ph-9903306-2-9-1', 'astro-ph-9903306-3-9-1'], ['astro-ph-9903306-2-24-0', 'astro-ph-9903306-3-24-0'], ['astro-ph-9903306-2-24-1', 'astro-ph-9903306-3-24-1'], ['astro-ph-9903306-2-24-2', 'astro-ph-9903306-3-24-2'], ['astro-ph-9903306-2-24-3', 'astro-ph-9903306-3-24-3'], ['astro-ph-9903306-2-24-4', 'astro-ph-9903306-3-24-4'], ['astro-ph-9903306-2-24-5', 'astro-ph-9903306-3-24-5'], ['astro-ph-9903306-2-24-6', 'astro-ph-9903306-3-24-6'], ['astro-ph-9903306-2-24-7', 'astro-ph-9903306-3-24-7'], ['astro-ph-9903306-2-46-0', 'astro-ph-9903306-3-46-0'], ['astro-ph-9903306-2-46-1', 'astro-ph-9903306-3-46-1'], ['astro-ph-9903306-2-46-2', 'astro-ph-9903306-3-46-2'], ['astro-ph-9903306-2-46-3', 'astro-ph-9903306-3-46-3'], ['astro-ph-9903306-2-46-4', 'astro-ph-9903306-3-46-4'], ['astro-ph-9903306-2-46-5', 'astro-ph-9903306-3-46-5'], ['astro-ph-9903306-2-26-0', 'astro-ph-9903306-3-26-0'], ['astro-ph-9903306-2-26-1', 'astro-ph-9903306-3-26-1'], ['astro-ph-9903306-2-26-2', 'astro-ph-9903306-3-26-2'], ['astro-ph-9903306-2-26-3', 'astro-ph-9903306-3-26-3'], ['astro-ph-9903306-2-16-0', 'astro-ph-9903306-3-16-0'], ['astro-ph-9903306-2-44-0', 'astro-ph-9903306-3-44-0'], ['astro-ph-9903306-2-44-1', 'astro-ph-9903306-3-44-1'], ['astro-ph-9903306-2-44-2', 'astro-ph-9903306-3-44-2'], ['astro-ph-9903306-2-48-0', 'astro-ph-9903306-3-48-0'], ['astro-ph-9903306-2-48-1', 'astro-ph-9903306-3-48-1'], ['astro-ph-9903306-2-48-2', 'astro-ph-9903306-3-48-2'], ['astro-ph-9903306-2-48-3', 'astro-ph-9903306-3-48-3'], ['astro-ph-9903306-2-48-4', 'astro-ph-9903306-3-48-4'], ['astro-ph-9903306-2-6-0', 'astro-ph-9903306-3-6-0'], ['astro-ph-9903306-2-34-0', 'astro-ph-9903306-3-34-0'], ['astro-ph-9903306-2-35-0', 'astro-ph-9903306-3-35-0'], ['astro-ph-9903306-2-35-1', 'astro-ph-9903306-3-35-1'], ['astro-ph-9903306-2-36-0', 'astro-ph-9903306-3-36-0'], ['astro-ph-9903306-2-36-1', 'astro-ph-9903306-3-36-1'], ['astro-ph-9903306-2-36-2', 'astro-ph-9903306-3-36-2'], ['astro-ph-9903306-2-36-3', 'astro-ph-9903306-3-36-3'], ['astro-ph-9903306-2-0-0', 'astro-ph-9903306-3-0-0'], ['astro-ph-9903306-2-0-1', 'astro-ph-9903306-3-0-1'], ['astro-ph-9903306-2-2-0', 'astro-ph-9903306-3-2-0'], ['astro-ph-9903306-2-2-1', 'astro-ph-9903306-3-2-1'], ['astro-ph-9903306-2-2-2', 'astro-ph-9903306-3-2-2'], ['astro-ph-9903306-2-2-3', 'astro-ph-9903306-3-2-3'], ['astro-ph-9903306-2-52-0', 'astro-ph-9903306-3-52-0'], ['astro-ph-9903306-2-52-1', 'astro-ph-9903306-3-52-1'], ['astro-ph-9903306-2-52-2', 'astro-ph-9903306-3-52-2'], ['astro-ph-9903306-2-52-3', 'astro-ph-9903306-3-52-3'], ['astro-ph-9903306-2-11-0', 'astro-ph-9903306-3-11-0'], ['astro-ph-9903306-2-11-1', 'astro-ph-9903306-3-11-1'], ['astro-ph-9903306-2-32-0', 'astro-ph-9903306-3-32-0'], ['astro-ph-9903306-2-15-0', 'astro-ph-9903306-3-15-0'], ['astro-ph-9903306-2-15-1', 'astro-ph-9903306-3-15-1'], ['astro-ph-9903306-2-15-2', 'astro-ph-9903306-3-15-2'], ['astro-ph-9903306-2-3-0', 'astro-ph-9903306-3-3-0'], ['astro-ph-9903306-2-3-1', 'astro-ph-9903306-3-3-1'], ['astro-ph-9903306-2-29-0', 'astro-ph-9903306-3-29-0'], ['astro-ph-9903306-2-29-1', 'astro-ph-9903306-3-29-1'], ['astro-ph-9903306-2-29-2', 'astro-ph-9903306-3-29-2'], ['astro-ph-9903306-2-12-0', 'astro-ph-9903306-3-12-0'], ['astro-ph-9903306-2-12-1', 'astro-ph-9903306-3-12-1'], ['astro-ph-9903306-2-12-2', 'astro-ph-9903306-3-12-2'], ['astro-ph-9903306-2-12-3', 'astro-ph-9903306-3-12-3'], ['astro-ph-9903306-2-12-4', 'astro-ph-9903306-3-12-4'], ['astro-ph-9903306-2-12-5', 'astro-ph-9903306-3-12-5'], ['astro-ph-9903306-2-12-6', 'astro-ph-9903306-3-12-6'], ['astro-ph-9903306-2-12-7', 'astro-ph-9903306-3-12-7'], ['astro-ph-9903306-2-12-8', 'astro-ph-9903306-3-12-8'], ['astro-ph-9903306-2-12-9', 'astro-ph-9903306-3-12-9'], ['astro-ph-9903306-2-18-0', 'astro-ph-9903306-3-18-0'], ['astro-ph-9903306-2-18-1', 'astro-ph-9903306-3-18-1'], ['astro-ph-9903306-2-18-2', 'astro-ph-9903306-3-18-2'], ['astro-ph-9903306-2-18-3', 'astro-ph-9903306-3-18-3'], ['astro-ph-9903306-2-50-0', 'astro-ph-9903306-3-50-0'], ['astro-ph-9903306-2-28-0', 'astro-ph-9903306-3-28-0'], ['astro-ph-9903306-2-28-1', 'astro-ph-9903306-3-28-1'], ['astro-ph-9903306-2-28-2', 'astro-ph-9903306-3-28-2'], ['astro-ph-9903306-2-28-3', 'astro-ph-9903306-3-28-3'], ['astro-ph-9903306-2-28-4', 'astro-ph-9903306-3-28-4'], ['astro-ph-9903306-2-28-5', 'astro-ph-9903306-3-28-5'], ['astro-ph-9903306-2-28-6', 'astro-ph-9903306-3-28-6'], ['astro-ph-9903306-2-30-0', 'astro-ph-9903306-3-30-0'], ['astro-ph-9903306-2-22-0', 'astro-ph-9903306-3-22-0'], ['astro-ph-9903306-2-23-0', 'astro-ph-9903306-3-23-0'], ['astro-ph-9903306-2-23-1', 'astro-ph-9903306-3-23-1'], ['astro-ph-9903306-2-5-0', 'astro-ph-9903306-3-5-0'], ['astro-ph-9903306-2-5-1', 'astro-ph-9903306-3-5-1'], ['astro-ph-9903306-2-5-2', 'astro-ph-9903306-3-5-2'], ['astro-ph-9903306-2-5-3', 'astro-ph-9903306-3-5-3'], ['astro-ph-9903306-2-5-4', 'astro-ph-9903306-3-5-4'], ['astro-ph-9903306-2-5-5', 'astro-ph-9903306-3-5-5'], ['astro-ph-9903306-2-5-6', 'astro-ph-9903306-3-5-6'], ['astro-ph-9903306-2-10-0', 'astro-ph-9903306-3-10-0'], ['astro-ph-9903306-2-10-1', 'astro-ph-9903306-3-10-1'], ['astro-ph-9903306-2-10-2', 'astro-ph-9903306-3-10-2'], ['astro-ph-9903306-2-10-3', 'astro-ph-9903306-3-10-3'], ['astro-ph-9903306-2-10-4', 'astro-ph-9903306-3-10-4'], ['astro-ph-9903306-2-10-5', 'astro-ph-9903306-3-10-5'], ['astro-ph-9903306-2-10-6', 'astro-ph-9903306-3-10-6'], ['astro-ph-9903306-2-10-7', 'astro-ph-9903306-3-10-7'], ['astro-ph-9903306-2-40-0', 'astro-ph-9903306-3-40-0'], ['astro-ph-9903306-2-40-1', 'astro-ph-9903306-3-40-1'], ['astro-ph-9903306-2-40-2', 'astro-ph-9903306-3-40-2'], ['astro-ph-9903306-2-40-3', 'astro-ph-9903306-3-40-3'], ['astro-ph-9903306-2-33-0', 'astro-ph-9903306-3-33-0'], ['astro-ph-9903306-2-51-0', 'astro-ph-9903306-3-51-0'], ['astro-ph-9903306-2-51-1', 'astro-ph-9903306-3-51-1'], ['astro-ph-9903306-2-21-0', 'astro-ph-9903306-3-21-0'], ['astro-ph-9903306-2-21-1', 'astro-ph-9903306-3-21-1'], ['astro-ph-9903306-2-21-2', 'astro-ph-9903306-3-21-2'], ['astro-ph-9903306-2-21-3', 'astro-ph-9903306-3-21-3'], ['astro-ph-9903306-2-19-0', 'astro-ph-9903306-3-19-0'], ['astro-ph-9903306-2-19-1', 'astro-ph-9903306-3-19-1'], ['astro-ph-9903306-2-19-2', 'astro-ph-9903306-3-19-2'], ['astro-ph-9903306-1-48-0', 'astro-ph-9903306-2-51-0'], ['astro-ph-9903306-1-48-1', 'astro-ph-9903306-2-51-1'], ['astro-ph-9903306-1-0-0', 'astro-ph-9903306-2-0-0'], ['astro-ph-9903306-1-0-1', 'astro-ph-9903306-2-0-1'], ['astro-ph-9903306-1-45-1', 'astro-ph-9903306-2-46-1'], ['astro-ph-9903306-1-45-2', 'astro-ph-9903306-2-46-2'], ['astro-ph-9903306-1-45-3', 'astro-ph-9903306-2-46-4'], ['astro-ph-9903306-1-45-4', 'astro-ph-9903306-2-46-5'], ['astro-ph-9903306-1-47-0', 'astro-ph-9903306-2-50-0'], ['astro-ph-9903306-1-49-0', 'astro-ph-9903306-2-52-0'], ['astro-ph-9903306-1-49-1', 'astro-ph-9903306-2-52-1'], ['astro-ph-9903306-1-49-2', 'astro-ph-9903306-2-52-2'], ['astro-ph-9903306-1-37-1', 'astro-ph-9903306-2-24-1'], ['astro-ph-9903306-1-37-2', 'astro-ph-9903306-2-24-2'], ['astro-ph-9903306-1-37-3', 'astro-ph-9903306-2-24-3'], ['astro-ph-9903306-1-37-4', 'astro-ph-9903306-2-24-4'], ['astro-ph-9903306-1-37-5', 'astro-ph-9903306-2-24-5'], ['astro-ph-9903306-1-37-6', 'astro-ph-9903306-2-24-6'], ['astro-ph-9903306-1-11-0', 'astro-ph-9903306-2-11-0'], ['astro-ph-9903306-1-20-0', 'astro-ph-9903306-2-33-0'], ['astro-ph-9903306-1-50-1', 'astro-ph-9903306-2-48-4'], ['astro-ph-9903306-1-19-0', 'astro-ph-9903306-2-32-0'], ['astro-ph-9903306-1-41-0', 'astro-ph-9903306-2-42-0'], ['astro-ph-9903306-1-41-2', 'astro-ph-9903306-2-42-2'], ['astro-ph-9903306-1-10-0', 'astro-ph-9903306-2-10-0'], ['astro-ph-9903306-1-10-2', 'astro-ph-9903306-2-10-3'], ['astro-ph-9903306-1-10-3', 'astro-ph-9903306-2-10-4'], ['astro-ph-9903306-1-10-5', 'astro-ph-9903306-2-10-6'], ['astro-ph-9903306-1-51-0', 'astro-ph-9903306-2-49-0'], ['astro-ph-9903306-1-42-0', 'astro-ph-9903306-2-43-0'], ['astro-ph-9903306-1-42-2', 'astro-ph-9903306-2-43-2'], ['astro-ph-9903306-1-21-0', 'astro-ph-9903306-2-34-0'], ['astro-ph-9903306-1-26-0', 'astro-ph-9903306-2-39-1'], ['astro-ph-9903306-1-26-1', 'astro-ph-9903306-2-39-2'], ['astro-ph-9903306-1-5-0', 'astro-ph-9903306-2-5-0'], ['astro-ph-9903306-1-5-1', 'astro-ph-9903306-2-5-1'], ['astro-ph-9903306-1-5-2', 'astro-ph-9903306-2-5-2'], ['astro-ph-9903306-1-5-4', 'astro-ph-9903306-2-5-4'], ['astro-ph-9903306-1-5-5', 'astro-ph-9903306-2-5-5'], ['astro-ph-9903306-1-5-6', 'astro-ph-9903306-2-5-6'], ['astro-ph-9903306-1-2-1', 'astro-ph-9903306-2-2-1'], ['astro-ph-9903306-1-2-3', 'astro-ph-9903306-2-2-3'], ['astro-ph-9903306-1-39-0', 'astro-ph-9903306-2-26-0'], ['astro-ph-9903306-1-39-1', 'astro-ph-9903306-2-26-1'], ['astro-ph-9903306-1-39-2', 'astro-ph-9903306-2-26-2'], ['astro-ph-9903306-1-6-0', 'astro-ph-9903306-2-6-0'], ['astro-ph-9903306-1-27-0', 'astro-ph-9903306-2-40-0'], ['astro-ph-9903306-1-27-1', 'astro-ph-9903306-2-40-1'], ['astro-ph-9903306-1-27-2', 'astro-ph-9903306-2-40-2'], ['astro-ph-9903306-1-27-3', 'astro-ph-9903306-2-40-3'], ['astro-ph-9903306-1-9-0', 'astro-ph-9903306-2-9-0'], ['astro-ph-9903306-1-9-1', 'astro-ph-9903306-2-9-1'], ['astro-ph-9903306-1-36-0', 'astro-ph-9903306-2-23-0'], ['astro-ph-9903306-1-36-1', 'astro-ph-9903306-2-23-1'], ['astro-ph-9903306-1-22-0', 'astro-ph-9903306-2-35-0'], ['astro-ph-9903306-1-22-1', 'astro-ph-9903306-2-35-1'], ['astro-ph-9903306-1-1-0', 'astro-ph-9903306-2-1-0'], ['astro-ph-9903306-1-1-1', 'astro-ph-9903306-2-1-1'], ['astro-ph-9903306-1-1-2', 'astro-ph-9903306-2-1-2'], ['astro-ph-9903306-1-34-0', 'astro-ph-9903306-2-21-0'], ['astro-ph-9903306-1-34-1', 'astro-ph-9903306-2-21-1'], ['astro-ph-9903306-1-34-2', 'astro-ph-9903306-2-21-2'], ['astro-ph-9903306-1-34-3', 'astro-ph-9903306-2-21-3'], ['astro-ph-9903306-1-43-0', 'astro-ph-9903306-2-44-0'], ['astro-ph-9903306-1-43-1', 'astro-ph-9903306-2-44-1'], ['astro-ph-9903306-1-43-2', 'astro-ph-9903306-2-44-2'], ['astro-ph-9903306-1-52-0', 'astro-ph-9903306-2-53-0'], ['astro-ph-9903306-1-52-1', 'astro-ph-9903306-2-53-1'], ['astro-ph-9903306-1-46-0', 'astro-ph-9903306-2-47-0'], ['astro-ph-9903306-1-46-1', 'astro-ph-9903306-2-47-1'], ['astro-ph-9903306-1-46-2', 'astro-ph-9903306-2-47-2'], ['astro-ph-9903306-1-46-4', 'astro-ph-9903306-2-47-4'], ['astro-ph-9903306-1-25-0', 'astro-ph-9903306-2-38-0'], ['astro-ph-9903306-1-25-1', 'astro-ph-9903306-2-38-1'], ['astro-ph-9903306-1-25-2', 'astro-ph-9903306-2-38-2'], ['astro-ph-9903306-1-12-1', 'astro-ph-9903306-2-12-1'], ['astro-ph-9903306-1-12-2', 'astro-ph-9903306-2-12-2'], ['astro-ph-9903306-1-12-3', 'astro-ph-9903306-2-12-3'], ['astro-ph-9903306-1-12-4', 'astro-ph-9903306-2-12-4'], ['astro-ph-9903306-1-12-5', 'astro-ph-9903306-2-12-5'], ['astro-ph-9903306-1-12-6', 'astro-ph-9903306-2-12-6'], ['astro-ph-9903306-1-12-7', 'astro-ph-9903306-2-12-7'], ['astro-ph-9903306-1-12-9', 'astro-ph-9903306-2-12-9'], ['astro-ph-9903306-1-7-0', 'astro-ph-9903306-2-7-0'], ['astro-ph-9903306-1-7-1', 'astro-ph-9903306-2-7-1'], ['astro-ph-9903306-1-7-2', 'astro-ph-9903306-2-7-2'], ['astro-ph-9903306-1-18-0', 'astro-ph-9903306-2-31-0'], ['astro-ph-9903306-1-18-1', 'astro-ph-9903306-2-31-1'], ['astro-ph-9903306-1-37-7', 'astro-ph-9903306-2-24-7'], ['astro-ph-9903306-1-3-0', 'astro-ph-9903306-2-3-0'], ['astro-ph-9903306-1-3-1', 'astro-ph-9903306-2-3-1'], ['astro-ph-9903306-1-50-0', 'astro-ph-9903306-2-48-0'], ['astro-ph-9903306-1-10-1', 'astro-ph-9903306-2-10-1'], ['astro-ph-9903306-1-10-4', 'astro-ph-9903306-2-10-5'], ['astro-ph-9903306-1-5-3', 'astro-ph-9903306-2-5-3'], ['astro-ph-9903306-1-2-2', 'astro-ph-9903306-2-2-2'], ['astro-ph-9903306-1-1-4', 'astro-ph-9903306-2-1-4'], ['astro-ph-9903306-1-46-3', 'astro-ph-9903306-2-47-3'], ['astro-ph-9903306-1-12-8', 'astro-ph-9903306-2-12-8'], ['astro-ph-9903306-1-15-0', 'astro-ph-9903306-2-28-0'], ['astro-ph-9903306-1-15-1', 'astro-ph-9903306-2-28-1'], ['astro-ph-9903306-1-15-1', 'astro-ph-9903306-2-28-2'], ['astro-ph-9903306-1-15-2', 'astro-ph-9903306-2-28-3'], ['astro-ph-9903306-1-16-0', 'astro-ph-9903306-2-29-0'], ['astro-ph-9903306-1-16-1', 'astro-ph-9903306-2-29-1'], ['astro-ph-9903306-1-16-2', 'astro-ph-9903306-2-29-2'], ['astro-ph-9903306-1-17-0', 'astro-ph-9903306-2-30-0'], ['astro-ph-9903306-1-23-0', 'astro-ph-9903306-2-36-0'], ['astro-ph-9903306-1-24-0', 'astro-ph-9903306-2-37-1'], ['astro-ph-9903306-1-24-1', 'astro-ph-9903306-2-37-2'], ['astro-ph-9903306-1-29-0', 'astro-ph-9903306-2-15-0'], ['astro-ph-9903306-1-29-1', 'astro-ph-9903306-2-15-1'], ['astro-ph-9903306-1-30-0', 'astro-ph-9903306-2-16-0'], ['astro-ph-9903306-1-31-1', 'astro-ph-9903306-2-17-6'], ['astro-ph-9903306-1-32-0', 'astro-ph-9903306-2-19-0'], ['astro-ph-9903306-1-32-1', 'astro-ph-9903306-2-19-1'], ['astro-ph-9903306-1-32-2', 'astro-ph-9903306-2-19-2'], ['astro-ph-9903306-1-35-0', 'astro-ph-9903306-2-22-0'], ['astro-ph-9903306-1-38-0', 'astro-ph-9903306-2-25-0']]

[['astro-ph-9903306-1-45-0', 'astro-ph-9903306-2-46-0'], ['astro-ph-9903306-1-49-3', 'astro-ph-9903306-2-52-3'], ['astro-ph-9903306-1-37-0', 'astro-ph-9903306-2-24-0'], ['astro-ph-9903306-1-11-1', 'astro-ph-9903306-2-11-1'], ['astro-ph-9903306-1-41-1', 'astro-ph-9903306-2-42-1'], ['astro-ph-9903306-1-10-6', 'astro-ph-9903306-2-10-7'], ['astro-ph-9903306-1-42-1', 'astro-ph-9903306-2-43-1'], ['astro-ph-9903306-1-42-3', 'astro-ph-9903306-2-43-3'], ['astro-ph-9903306-1-39-3', 'astro-ph-9903306-2-26-3'], ['astro-ph-9903306-1-12-0', 'astro-ph-9903306-2-12-0'], ['astro-ph-9903306-2-38-0', 'astro-ph-9903306-3-38-0'], ['astro-ph-9903306-2-38-1', 'astro-ph-9903306-3-38-1'], ['astro-ph-9903306-2-38-2', 'astro-ph-9903306-3-38-2'], ['astro-ph-9903306-2-1-0', 'astro-ph-9903306-3-1-0'], ['astro-ph-9903306-2-1-1', 'astro-ph-9903306-3-1-1'], ['astro-ph-9903306-2-1-2', 'astro-ph-9903306-3-1-2'], ['astro-ph-9903306-2-1-3', 'astro-ph-9903306-3-1-3'], ['astro-ph-9903306-2-1-4', 'astro-ph-9903306-3-1-4'], ['astro-ph-9903306-2-42-0', 'astro-ph-9903306-3-42-0'], ['astro-ph-9903306-2-42-1', 'astro-ph-9903306-3-42-1'], ['astro-ph-9903306-2-42-2', 'astro-ph-9903306-3-42-2'], ['astro-ph-9903306-2-42-3', 'astro-ph-9903306-3-42-3'], ['astro-ph-9903306-2-42-4', 'astro-ph-9903306-3-42-4'], ['astro-ph-9903306-2-53-0', 'astro-ph-9903306-3-53-0'], ['astro-ph-9903306-2-53-1', 'astro-ph-9903306-3-53-1'], ['astro-ph-9903306-2-7-0', 'astro-ph-9903306-3-7-0'], ['astro-ph-9903306-2-7-1', 'astro-ph-9903306-3-7-1'], ['astro-ph-9903306-2-7-2', 'astro-ph-9903306-3-7-2'], ['astro-ph-9903306-2-43-0', 'astro-ph-9903306-3-43-0'], ['astro-ph-9903306-2-43-1', 'astro-ph-9903306-3-43-1'], ['astro-ph-9903306-2-43-2', 'astro-ph-9903306-3-43-2'], ['astro-ph-9903306-2-43-3', 'astro-ph-9903306-3-43-3'], ['astro-ph-9903306-2-31-0', 'astro-ph-9903306-3-31-0'], ['astro-ph-9903306-2-31-1', 'astro-ph-9903306-3-31-1'], ['astro-ph-9903306-2-37-1', 'astro-ph-9903306-3-37-1'], ['astro-ph-9903306-2-37-2', 'astro-ph-9903306-3-37-2'], ['astro-ph-9903306-2-39-1', 'astro-ph-9903306-3-39-1'], ['astro-ph-9903306-2-39-2', 'astro-ph-9903306-3-39-2'], ['astro-ph-9903306-2-17-0', 'astro-ph-9903306-3-17-0'], ['astro-ph-9903306-2-17-1', 'astro-ph-9903306-3-17-1'], ['astro-ph-9903306-2-17-2', 'astro-ph-9903306-3-17-2'], ['astro-ph-9903306-2-17-3', 'astro-ph-9903306-3-17-3'], ['astro-ph-9903306-2-17-4', 'astro-ph-9903306-3-17-4'], ['astro-ph-9903306-2-17-5', 'astro-ph-9903306-3-17-5'], ['astro-ph-9903306-2-17-6', 'astro-ph-9903306-3-17-6'], ['astro-ph-9903306-2-17-7', 'astro-ph-9903306-3-17-7'], ['astro-ph-9903306-2-25-0', 'astro-ph-9903306-3-25-0'], ['astro-ph-9903306-2-47-0', 'astro-ph-9903306-3-47-0'], ['astro-ph-9903306-2-47-1', 'astro-ph-9903306-3-47-1'], ['astro-ph-9903306-2-47-2', 'astro-ph-9903306-3-47-2'], ['astro-ph-9903306-2-47-3', 'astro-ph-9903306-3-47-3'], ['astro-ph-9903306-2-47-4', 'astro-ph-9903306-3-47-4'], ['astro-ph-9903306-2-47-5', 'astro-ph-9903306-3-47-5'], ['astro-ph-9903306-2-49-0', 'astro-ph-9903306-3-49-0'], ['astro-ph-9903306-2-9-0', 'astro-ph-9903306-3-9-0'], ['astro-ph-9903306-2-9-1', 'astro-ph-9903306-3-9-1'], ['astro-ph-9903306-2-24-0', 'astro-ph-9903306-3-24-0'], ['astro-ph-9903306-2-24-1', 'astro-ph-9903306-3-24-1'], ['astro-ph-9903306-2-24-2', 'astro-ph-9903306-3-24-2'], ['astro-ph-9903306-2-24-3', 'astro-ph-9903306-3-24-3'], ['astro-ph-9903306-2-24-4', 'astro-ph-9903306-3-24-4'], ['astro-ph-9903306-2-24-5', 'astro-ph-9903306-3-24-5'], ['astro-ph-9903306-2-24-6', 'astro-ph-9903306-3-24-6'], ['astro-ph-9903306-2-24-7', 'astro-ph-9903306-3-24-7'], ['astro-ph-9903306-2-46-0', 'astro-ph-9903306-3-46-0'], ['astro-ph-9903306-2-46-1', 'astro-ph-9903306-3-46-1'], ['astro-ph-9903306-2-46-2', 'astro-ph-9903306-3-46-2'], ['astro-ph-9903306-2-46-3', 'astro-ph-9903306-3-46-3'], ['astro-ph-9903306-2-46-4', 'astro-ph-9903306-3-46-4'], ['astro-ph-9903306-2-46-5', 'astro-ph-9903306-3-46-5'], ['astro-ph-9903306-2-26-0', 'astro-ph-9903306-3-26-0'], ['astro-ph-9903306-2-26-1', 'astro-ph-9903306-3-26-1'], ['astro-ph-9903306-2-26-2', 'astro-ph-9903306-3-26-2'], ['astro-ph-9903306-2-26-3', 'astro-ph-9903306-3-26-3'], ['astro-ph-9903306-2-16-0', 'astro-ph-9903306-3-16-0'], ['astro-ph-9903306-2-44-0', 'astro-ph-9903306-3-44-0'], ['astro-ph-9903306-2-44-1', 'astro-ph-9903306-3-44-1'], ['astro-ph-9903306-2-44-2', 'astro-ph-9903306-3-44-2'], ['astro-ph-9903306-2-48-0', 'astro-ph-9903306-3-48-0'], ['astro-ph-9903306-2-48-1', 'astro-ph-9903306-3-48-1'], ['astro-ph-9903306-2-48-2', 'astro-ph-9903306-3-48-2'], ['astro-ph-9903306-2-48-3', 'astro-ph-9903306-3-48-3'], ['astro-ph-9903306-2-48-4', 'astro-ph-9903306-3-48-4'], ['astro-ph-9903306-2-6-0', 'astro-ph-9903306-3-6-0'], ['astro-ph-9903306-2-34-0', 'astro-ph-9903306-3-34-0'], ['astro-ph-9903306-2-35-0', 'astro-ph-9903306-3-35-0'], ['astro-ph-9903306-2-35-1', 'astro-ph-9903306-3-35-1'], ['astro-ph-9903306-2-36-0', 'astro-ph-9903306-3-36-0'], ['astro-ph-9903306-2-36-1', 'astro-ph-9903306-3-36-1'], ['astro-ph-9903306-2-36-2', 'astro-ph-9903306-3-36-2'], ['astro-ph-9903306-2-36-3', 'astro-ph-9903306-3-36-3'], ['astro-ph-9903306-2-0-0', 'astro-ph-9903306-3-0-0'], ['astro-ph-9903306-2-0-1', 'astro-ph-9903306-3-0-1'], ['astro-ph-9903306-2-2-0', 'astro-ph-9903306-3-2-0'], ['astro-ph-9903306-2-2-1', 'astro-ph-9903306-3-2-1'], ['astro-ph-9903306-2-2-2', 'astro-ph-9903306-3-2-2'], ['astro-ph-9903306-2-2-3', 'astro-ph-9903306-3-2-3'], ['astro-ph-9903306-2-52-0', 'astro-ph-9903306-3-52-0'], ['astro-ph-9903306-2-52-1', 'astro-ph-9903306-3-52-1'], ['astro-ph-9903306-2-52-2', 'astro-ph-9903306-3-52-2'], ['astro-ph-9903306-2-52-3', 'astro-ph-9903306-3-52-3'], ['astro-ph-9903306-2-11-0', 'astro-ph-9903306-3-11-0'], ['astro-ph-9903306-2-11-1', 'astro-ph-9903306-3-11-1'], ['astro-ph-9903306-2-32-0', 'astro-ph-9903306-3-32-0'], ['astro-ph-9903306-2-15-0', 'astro-ph-9903306-3-15-0'], ['astro-ph-9903306-2-15-1', 'astro-ph-9903306-3-15-1'], ['astro-ph-9903306-2-15-2', 'astro-ph-9903306-3-15-2'], ['astro-ph-9903306-2-3-0', 'astro-ph-9903306-3-3-0'], ['astro-ph-9903306-2-3-1', 'astro-ph-9903306-3-3-1'], ['astro-ph-9903306-2-29-0', 'astro-ph-9903306-3-29-0'], ['astro-ph-9903306-2-29-1', 'astro-ph-9903306-3-29-1'], ['astro-ph-9903306-2-29-2', 'astro-ph-9903306-3-29-2'], ['astro-ph-9903306-2-12-0', 'astro-ph-9903306-3-12-0'], ['astro-ph-9903306-2-12-1', 'astro-ph-9903306-3-12-1'], ['astro-ph-9903306-2-12-2', 'astro-ph-9903306-3-12-2'], ['astro-ph-9903306-2-12-3', 'astro-ph-9903306-3-12-3'], ['astro-ph-9903306-2-12-4', 'astro-ph-9903306-3-12-4'], ['astro-ph-9903306-2-12-5', 'astro-ph-9903306-3-12-5'], ['astro-ph-9903306-2-12-6', 'astro-ph-9903306-3-12-6'], ['astro-ph-9903306-2-12-7', 'astro-ph-9903306-3-12-7'], ['astro-ph-9903306-2-12-8', 'astro-ph-9903306-3-12-8'], ['astro-ph-9903306-2-12-9', 'astro-ph-9903306-3-12-9'], ['astro-ph-9903306-2-18-0', 'astro-ph-9903306-3-18-0'], ['astro-ph-9903306-2-18-1', 'astro-ph-9903306-3-18-1'], ['astro-ph-9903306-2-18-2', 'astro-ph-9903306-3-18-2'], ['astro-ph-9903306-2-18-3', 'astro-ph-9903306-3-18-3'], ['astro-ph-9903306-2-50-0', 'astro-ph-9903306-3-50-0'], ['astro-ph-9903306-2-28-0', 'astro-ph-9903306-3-28-0'], ['astro-ph-9903306-2-28-1', 'astro-ph-9903306-3-28-1'], ['astro-ph-9903306-2-28-2', 'astro-ph-9903306-3-28-2'], ['astro-ph-9903306-2-28-3', 'astro-ph-9903306-3-28-3'], ['astro-ph-9903306-2-28-4', 'astro-ph-9903306-3-28-4'], ['astro-ph-9903306-2-28-5', 'astro-ph-9903306-3-28-5'], ['astro-ph-9903306-2-28-6', 'astro-ph-9903306-3-28-6'], ['astro-ph-9903306-2-30-0', 'astro-ph-9903306-3-30-0'], ['astro-ph-9903306-2-22-0', 'astro-ph-9903306-3-22-0'], ['astro-ph-9903306-2-23-0', 'astro-ph-9903306-3-23-0'], ['astro-ph-9903306-2-23-1', 'astro-ph-9903306-3-23-1'], ['astro-ph-9903306-2-5-0', 'astro-ph-9903306-3-5-0'], ['astro-ph-9903306-2-5-1', 'astro-ph-9903306-3-5-1'], ['astro-ph-9903306-2-5-2', 'astro-ph-9903306-3-5-2'], ['astro-ph-9903306-2-5-3', 'astro-ph-9903306-3-5-3'], ['astro-ph-9903306-2-5-4', 'astro-ph-9903306-3-5-4'], ['astro-ph-9903306-2-5-5', 'astro-ph-9903306-3-5-5'], ['astro-ph-9903306-2-5-6', 'astro-ph-9903306-3-5-6'], ['astro-ph-9903306-2-10-0', 'astro-ph-9903306-3-10-0'], ['astro-ph-9903306-2-10-1', 'astro-ph-9903306-3-10-1'], ['astro-ph-9903306-2-10-2', 'astro-ph-9903306-3-10-2'], ['astro-ph-9903306-2-10-3', 'astro-ph-9903306-3-10-3'], ['astro-ph-9903306-2-10-4', 'astro-ph-9903306-3-10-4'], ['astro-ph-9903306-2-10-5', 'astro-ph-9903306-3-10-5'], ['astro-ph-9903306-2-10-6', 'astro-ph-9903306-3-10-6'], ['astro-ph-9903306-2-10-7', 'astro-ph-9903306-3-10-7'], ['astro-ph-9903306-2-40-0', 'astro-ph-9903306-3-40-0'], ['astro-ph-9903306-2-40-1', 'astro-ph-9903306-3-40-1'], ['astro-ph-9903306-2-40-2', 'astro-ph-9903306-3-40-2'], ['astro-ph-9903306-2-40-3', 'astro-ph-9903306-3-40-3'], ['astro-ph-9903306-2-33-0', 'astro-ph-9903306-3-33-0'], ['astro-ph-9903306-2-51-0', 'astro-ph-9903306-3-51-0'], ['astro-ph-9903306-2-51-1', 'astro-ph-9903306-3-51-1'], ['astro-ph-9903306-2-21-0', 'astro-ph-9903306-3-21-0'], ['astro-ph-9903306-2-21-1', 'astro-ph-9903306-3-21-1'], ['astro-ph-9903306-2-21-2', 'astro-ph-9903306-3-21-2'], ['astro-ph-9903306-2-21-3', 'astro-ph-9903306-3-21-3'], ['astro-ph-9903306-2-19-0', 'astro-ph-9903306-3-19-0'], ['astro-ph-9903306-2-19-1', 'astro-ph-9903306-3-19-1'], ['astro-ph-9903306-2-19-2', 'astro-ph-9903306-3-19-2']]

[['astro-ph-9903306-1-48-0', 'astro-ph-9903306-2-51-0'], ['astro-ph-9903306-1-48-1', 'astro-ph-9903306-2-51-1'], ['astro-ph-9903306-1-0-0', 'astro-ph-9903306-2-0-0'], ['astro-ph-9903306-1-0-1', 'astro-ph-9903306-2-0-1'], ['astro-ph-9903306-1-45-1', 'astro-ph-9903306-2-46-1'], ['astro-ph-9903306-1-45-2', 'astro-ph-9903306-2-46-2'], ['astro-ph-9903306-1-45-3', 'astro-ph-9903306-2-46-4'], ['astro-ph-9903306-1-45-4', 'astro-ph-9903306-2-46-5'], ['astro-ph-9903306-1-47-0', 'astro-ph-9903306-2-50-0'], ['astro-ph-9903306-1-49-0', 'astro-ph-9903306-2-52-0'], ['astro-ph-9903306-1-49-1', 'astro-ph-9903306-2-52-1'], ['astro-ph-9903306-1-49-2', 'astro-ph-9903306-2-52-2'], ['astro-ph-9903306-1-37-1', 'astro-ph-9903306-2-24-1'], ['astro-ph-9903306-1-37-2', 'astro-ph-9903306-2-24-2'], ['astro-ph-9903306-1-37-3', 'astro-ph-9903306-2-24-3'], ['astro-ph-9903306-1-37-4', 'astro-ph-9903306-2-24-4'], ['astro-ph-9903306-1-37-5', 'astro-ph-9903306-2-24-5'], ['astro-ph-9903306-1-37-6', 'astro-ph-9903306-2-24-6'], ['astro-ph-9903306-1-11-0', 'astro-ph-9903306-2-11-0'], ['astro-ph-9903306-1-20-0', 'astro-ph-9903306-2-33-0'], ['astro-ph-9903306-1-50-1', 'astro-ph-9903306-2-48-4'], ['astro-ph-9903306-1-19-0', 'astro-ph-9903306-2-32-0'], ['astro-ph-9903306-1-41-0', 'astro-ph-9903306-2-42-0'], ['astro-ph-9903306-1-41-2', 'astro-ph-9903306-2-42-2'], ['astro-ph-9903306-1-10-0', 'astro-ph-9903306-2-10-0'], ['astro-ph-9903306-1-10-2', 'astro-ph-9903306-2-10-3'], ['astro-ph-9903306-1-10-3', 'astro-ph-9903306-2-10-4'], ['astro-ph-9903306-1-10-5', 'astro-ph-9903306-2-10-6'], ['astro-ph-9903306-1-51-0', 'astro-ph-9903306-2-49-0'], ['astro-ph-9903306-1-42-0', 'astro-ph-9903306-2-43-0'], ['astro-ph-9903306-1-42-2', 'astro-ph-9903306-2-43-2'], ['astro-ph-9903306-1-21-0', 'astro-ph-9903306-2-34-0'], ['astro-ph-9903306-1-26-0', 'astro-ph-9903306-2-39-1'], ['astro-ph-9903306-1-26-1', 'astro-ph-9903306-2-39-2'], ['astro-ph-9903306-1-5-0', 'astro-ph-9903306-2-5-0'], ['astro-ph-9903306-1-5-1', 'astro-ph-9903306-2-5-1'], ['astro-ph-9903306-1-5-2', 'astro-ph-9903306-2-5-2'], ['astro-ph-9903306-1-5-4', 'astro-ph-9903306-2-5-4'], ['astro-ph-9903306-1-5-5', 'astro-ph-9903306-2-5-5'], ['astro-ph-9903306-1-5-6', 'astro-ph-9903306-2-5-6'], ['astro-ph-9903306-1-2-1', 'astro-ph-9903306-2-2-1'], ['astro-ph-9903306-1-2-3', 'astro-ph-9903306-2-2-3'], ['astro-ph-9903306-1-39-0', 'astro-ph-9903306-2-26-0'], ['astro-ph-9903306-1-39-1', 'astro-ph-9903306-2-26-1'], ['astro-ph-9903306-1-39-2', 'astro-ph-9903306-2-26-2'], ['astro-ph-9903306-1-6-0', 'astro-ph-9903306-2-6-0'], ['astro-ph-9903306-1-27-0', 'astro-ph-9903306-2-40-0'], ['astro-ph-9903306-1-27-1', 'astro-ph-9903306-2-40-1'], ['astro-ph-9903306-1-27-2', 'astro-ph-9903306-2-40-2'], ['astro-ph-9903306-1-27-3', 'astro-ph-9903306-2-40-3'], ['astro-ph-9903306-1-9-0', 'astro-ph-9903306-2-9-0'], ['astro-ph-9903306-1-9-1', 'astro-ph-9903306-2-9-1'], ['astro-ph-9903306-1-36-0', 'astro-ph-9903306-2-23-0'], ['astro-ph-9903306-1-36-1', 'astro-ph-9903306-2-23-1'], ['astro-ph-9903306-1-22-0', 'astro-ph-9903306-2-35-0'], ['astro-ph-9903306-1-22-1', 'astro-ph-9903306-2-35-1'], ['astro-ph-9903306-1-1-0', 'astro-ph-9903306-2-1-0'], ['astro-ph-9903306-1-1-1', 'astro-ph-9903306-2-1-1'], ['astro-ph-9903306-1-1-2', 'astro-ph-9903306-2-1-2'], ['astro-ph-9903306-1-34-0', 'astro-ph-9903306-2-21-0'], ['astro-ph-9903306-1-34-1', 'astro-ph-9903306-2-21-1'], ['astro-ph-9903306-1-34-2', 'astro-ph-9903306-2-21-2'], ['astro-ph-9903306-1-34-3', 'astro-ph-9903306-2-21-3'], ['astro-ph-9903306-1-43-0', 'astro-ph-9903306-2-44-0'], ['astro-ph-9903306-1-43-1', 'astro-ph-9903306-2-44-1'], ['astro-ph-9903306-1-43-2', 'astro-ph-9903306-2-44-2'], ['astro-ph-9903306-1-52-0', 'astro-ph-9903306-2-53-0'], ['astro-ph-9903306-1-52-1', 'astro-ph-9903306-2-53-1'], ['astro-ph-9903306-1-46-0', 'astro-ph-9903306-2-47-0'], ['astro-ph-9903306-1-46-1', 'astro-ph-9903306-2-47-1'], ['astro-ph-9903306-1-46-2', 'astro-ph-9903306-2-47-2'], ['astro-ph-9903306-1-46-4', 'astro-ph-9903306-2-47-4'], ['astro-ph-9903306-1-25-0', 'astro-ph-9903306-2-38-0'], ['astro-ph-9903306-1-25-1', 'astro-ph-9903306-2-38-1'], ['astro-ph-9903306-1-25-2', 'astro-ph-9903306-2-38-2'], ['astro-ph-9903306-1-12-1', 'astro-ph-9903306-2-12-1'], ['astro-ph-9903306-1-12-2', 'astro-ph-9903306-2-12-2'], ['astro-ph-9903306-1-12-3', 'astro-ph-9903306-2-12-3'], ['astro-ph-9903306-1-12-4', 'astro-ph-9903306-2-12-4'], ['astro-ph-9903306-1-12-5', 'astro-ph-9903306-2-12-5'], ['astro-ph-9903306-1-12-6', 'astro-ph-9903306-2-12-6'], ['astro-ph-9903306-1-12-7', 'astro-ph-9903306-2-12-7'], ['astro-ph-9903306-1-12-9', 'astro-ph-9903306-2-12-9'], ['astro-ph-9903306-1-7-0', 'astro-ph-9903306-2-7-0'], ['astro-ph-9903306-1-7-1', 'astro-ph-9903306-2-7-1'], ['astro-ph-9903306-1-7-2', 'astro-ph-9903306-2-7-2'], ['astro-ph-9903306-1-18-0', 'astro-ph-9903306-2-31-0'], ['astro-ph-9903306-1-18-1', 'astro-ph-9903306-2-31-1']]

[]

[['astro-ph-9903306-1-37-7', 'astro-ph-9903306-2-24-7'], ['astro-ph-9903306-1-3-0', 'astro-ph-9903306-2-3-0'], ['astro-ph-9903306-1-3-1', 'astro-ph-9903306-2-3-1'], ['astro-ph-9903306-1-50-0', 'astro-ph-9903306-2-48-0'], ['astro-ph-9903306-1-10-1', 'astro-ph-9903306-2-10-1'], ['astro-ph-9903306-1-10-4', 'astro-ph-9903306-2-10-5'], ['astro-ph-9903306-1-5-3', 'astro-ph-9903306-2-5-3'], ['astro-ph-9903306-1-2-2', 'astro-ph-9903306-2-2-2'], ['astro-ph-9903306-1-1-4', 'astro-ph-9903306-2-1-4'], ['astro-ph-9903306-1-46-3', 'astro-ph-9903306-2-47-3'], ['astro-ph-9903306-1-12-8', 'astro-ph-9903306-2-12-8']]

[['astro-ph-9903306-1-15-0', 'astro-ph-9903306-2-28-0'], ['astro-ph-9903306-1-15-1', 'astro-ph-9903306-2-28-1'], ['astro-ph-9903306-1-15-1', 'astro-ph-9903306-2-28-2'], ['astro-ph-9903306-1-15-2', 'astro-ph-9903306-2-28-3'], ['astro-ph-9903306-1-16-0', 'astro-ph-9903306-2-29-0'], ['astro-ph-9903306-1-16-1', 'astro-ph-9903306-2-29-1'], ['astro-ph-9903306-1-16-2', 'astro-ph-9903306-2-29-2'], ['astro-ph-9903306-1-17-0', 'astro-ph-9903306-2-30-0'], ['astro-ph-9903306-1-23-0', 'astro-ph-9903306-2-36-0'], ['astro-ph-9903306-1-24-0', 'astro-ph-9903306-2-37-1'], ['astro-ph-9903306-1-24-1', 'astro-ph-9903306-2-37-2'], ['astro-ph-9903306-1-29-0', 'astro-ph-9903306-2-15-0'], ['astro-ph-9903306-1-29-1', 'astro-ph-9903306-2-15-1'], ['astro-ph-9903306-1-30-0', 'astro-ph-9903306-2-16-0'], ['astro-ph-9903306-1-31-1', 'astro-ph-9903306-2-17-6'], ['astro-ph-9903306-1-32-0', 'astro-ph-9903306-2-19-0'], ['astro-ph-9903306-1-32-1', 'astro-ph-9903306-2-19-1'], ['astro-ph-9903306-1-32-2', 'astro-ph-9903306-2-19-2'], ['astro-ph-9903306-1-35-0', 'astro-ph-9903306-2-22-0'], ['astro-ph-9903306-1-38-0', 'astro-ph-9903306-2-25-0']]

['astro-ph-9903306-1-18-2', 'astro-ph-9903306-2-31-2', 'astro-ph-9903306-2-37-0', 'astro-ph-9903306-2-39-0', 'astro-ph-9903306-3-31-2', 'astro-ph-9903306-3-37-0', 'astro-ph-9903306-3-39-0']

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

https://arxiv.org/abs/astro-ph/9903306

{'astro-ph-9903306-3-0-0': 'We present an analysis of the RXTE observations of the recently discovered Galactic microquasar XTE J1748-288 during its 1998 outburst.', 'astro-ph-9903306-3-0-1': 'General spectral and temporal properties of the source and their evolution were very typical for the Galactic black hole candidates (BHC) and, in particular, black hole X-ray Novae.', 'astro-ph-9903306-3-1-0': 'The spectral evolution of the source during the outburst can be considered a sequence of qualitatively distinct states.', 'astro-ph-9903306-3-1-1': 'During the first observations, corresponding to the maximum of X-ray flux, the spectrum of the source consisted of a dominating hard power law component and a soft thermal component, which can be described by the model of multicolor disk emission.', 'astro-ph-9903306-3-1-2': 'The hard component contributed [MATH]80% to the X-ray luminosity in the 3-25 keV energy band.', 'astro-ph-9903306-3-1-3': 'Overall two-component spectral shape is an attribute of very high state (VHS) observed previously in BHC, but the domination of hard component is unusual.', 'astro-ph-9903306-3-1-4': 'Later on, as the X-ray source faded, its energy spectrum qualitatively changed, showing high (HS) and then low (LS) states, both typical for black hole binaries.', 'astro-ph-9903306-3-2-0': 'As the energy spectrum changed, the fast variability also evolved dramatically.', 'astro-ph-9903306-3-2-1': 'Initially the power density spectrum was formed by a dominating band-limited noise component, QPO features at 20-30 Hz and at [MATH]0.5 Hz, and a very low frequency noise component.', 'astro-ph-9903306-3-2-2': 'After a significant decrease of the contribution of the hard spectral component the amplitude of the fractional variability decreased by an order of magnitude and the PDS spectrum adopted a power-law shape with a broad QPO peak around 0.03 Hz.', 'astro-ph-9903306-3-2-3': 'When the system switched to the LS, the PDS shape changed again and the QPOs have not been detected since.', 'astro-ph-9903306-3-3-0': 'When the source was observed in VHS, a clear correlation between QPO parameters and X-ray flux was seen.', 'astro-ph-9903306-3-3-1': 'Such a correlation gives an insight into our understanding of the accretion process in X-ray black hole binaries.', 'astro-ph-9903306-3-4-0': '# Introduction', 'astro-ph-9903306-3-5-0': 'The XTE J1748-288 was discovered as a new transient source on June 4, 1998 by All Sky Monitor (ASM) aboard the Rossi X-ray Timing Explorer (RXTE) observatory (Smith, Levine Wood 1998).', 'astro-ph-9903306-3-5-1': 'The X-ray source was localized with an accuracy of 1 arcmin in multiple scans of the region by the PCA/RXTE experiment (Strohmayer Marshall 1998).', 'astro-ph-9903306-3-5-2': 'Observations in the radio band by VLA revealed the presence of an unresolved radio source with a position consistent with one found by PCA: R.A.=17[MATH].06, Dec=-28[MATH].8 (equinox 2000.0, position uncertainty 0.', 'astro-ph-9903306-3-5-3': '[MATH]6, Hjellming, Rupen Mioduszewski 1998).', 'astro-ph-9903306-3-5-4': 'Significant variability of the new radio source (Hjellming et al. 1998a; Fender Stappers 1998) strongly supports its association with the X-ray transient.', 'astro-ph-9903306-3-5-5': 'In June 14.31 radio source became extended, with a proper motion of 20-40 mas/day (Rupen, Hjellming Mioduszewski 1998).', 'astro-ph-9903306-3-5-6': 'The intrinsic velocity of the moving jet was higher than [MATH]c for the distance [MATH]8 kpc derived from a 21 cm HI absorption measurement (Hjellming et al. 1998b).', 'astro-ph-9903306-3-6-0': 'Two quasi periodical oscillations (hereafter QPO) with the central frequencies [MATH]0.5 Hz and [MATH]32 Hz were found in the power density spectrum (PDS) of the source in the PCA observation of June 6, 1998 (Fox Lewin 1998).', 'astro-ph-9903306-3-7-0': 'The flux from the source was above ASM/RXTE detection limit until August of 1998.', 'astro-ph-9903306-3-7-1': 'Pointed instruments of RXTE - PCA and HEXTE - observed the source quasi evenly from June till September, obtaining a good coverage of the whole outburst.', 'astro-ph-9903306-3-7-2': 'Here we present the results of a spectral and timing analysis of these RXTE observations.', 'astro-ph-9903306-3-8-0': '# Observations and analysis', 'astro-ph-9903306-3-9-0': 'In our analysis we used all the publicly available data obtained from RXTE TOO (Target Of Opportunity) archive including 21 pointed observations and "slew" parts of two PI restricted observations.', 'astro-ph-9903306-3-9-1': 'The 23 observations quasi evenly cover the 1998 outburst of the source with a total exposure of [MATH] ksec. The list of observations is presented in Table [REF].', 'astro-ph-9903306-3-10-0': 'For data reduction we used the standard FTOOLS package.', 'astro-ph-9903306-3-10-1': 'For the spectral analysis we used PCA data collected in the 3-25 keV energy range.', 'astro-ph-9903306-3-10-2': 'The response matrix was constructed for every single observation using PCARMF v3.5.', 'astro-ph-9903306-3-10-3': 'For the PCA background estimations we applied a Very Large Events (VLE)-based model.', 'astro-ph-9903306-3-10-4': 'For some observations, where the standard VLE-based background subtraction was not good enough for our purposes, we used the activation component for background models separately (see details in the RXTE Cook Book, "Using the Latest PCABACKEST").', 'astro-ph-9903306-3-10-5': 'All spectra were dead time corrected.', 'astro-ph-9903306-3-10-6': 'The analysis of the spectra of the Crab nebula confirmed that the uncertainties attributed to the response matrix do not exceed 0.5-1.0%.', 'astro-ph-9903306-3-10-7': 'To account for the uncertainty in the knowledge of the spectral response, a 1 per cent systematic error was added to the statistical error in each PCA channel.', 'astro-ph-9903306-3-11-0': 'For our timing analysis we used the PCA timing modes [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9903306-3-11-1': 'Background subtracted light curves from [MATH] mode data were used to generate the power density spectra for very low frequencies (below [MATH] Hz).', 'astro-ph-9903306-3-12-0': 'Unfortunately, an analysis of the HEXTE data was strongly complicated by the difficulties in the background subtraction.', 'astro-ph-9903306-3-12-1': 'The source/background beam-switching of the HEXTE clusters (they "rock" periodically to [MATH]3.0[MATH] from the source position), is not very efficient for precise background subtraction when the source is located in the densely populated Galactic Center region.', 'astro-ph-9903306-3-12-2': 'There are a number of X-ray sources around XTE J1748-288, namely GX 5-1, GRS 1758-258, GX 3+1 and 1E1740.7-2942 to mention few, each of them luminous enough to affect the results of the background measurement.', 'astro-ph-9903306-3-12-3': 'We have performed a check of the quality of the background subtraction in any given observation by comparing the m-background spectra (the spectra obtained during the -3.0[MATH] offset from the source) with the p-background (+3.0[MATH] offset).', 'astro-ph-9903306-3-12-4': 'Only observations with an adequate quality of subtraction were accepted for the analysis.', 'astro-ph-9903306-3-12-5': 'Because the rocking planes for clusters A and B are perpendicular to each other, different sky regions are within the field of view of these clusters during the background measurements.', 'astro-ph-9903306-3-12-6': 'Therefore A clusters can provide more accurate background measurements for some observations, and B clusters - for others.', 'astro-ph-9903306-3-12-7': 'After the 8-th observation, when the source flux in the 20-100 keV energy band had dropped below [MATH]10 cnts/s/cluster, the accuracy of our knowledge of the background became comparable to the source flux, and we excluded these later HEXTE data from our spectral analysis.', 'astro-ph-9903306-3-12-8': 'However, we present the HEXTE observations of the source in its LS in Fig. [REF].', 'astro-ph-9903306-3-12-9': 'The spectrum plotted there was obtained from carefully selected data, but still should be treated with care.', 'astro-ph-9903306-3-13-0': '# Results', 'astro-ph-9903306-3-14-0': '## Energy spectra approximation', 'astro-ph-9903306-3-15-0': 'We generated the energy spectra of XTE J1748-288 averaging the data over the whole single observations.', 'astro-ph-9903306-3-15-1': 'To trace the evolution of the spectral parameters within some observations, an additional analysis of the spectra, accumulated in 256 sec time intervals, was performed.', 'astro-ph-9903306-3-15-2': 'These spectra were then used to study the correlation of the spectral and variability parameters (see section 3.4).', 'astro-ph-9903306-3-16-0': 'For the spectral approximation of the first 15 observations we used the model consisted of a multicolor disk blackbody emission component (Shakura Sunyaev 1973; Mitsuda et al. 1984), a power law component and low energy absorption.', 'astro-ph-9903306-3-17-0': 'Overall spectral shape for the remaining observations (16-23) can be approximated by power law with low energy absorption.', 'astro-ph-9903306-3-17-1': 'However, strong emission line-like feature around [MATH]6.5 keV cannot be ignored.', 'astro-ph-9903306-3-17-2': 'This feature may be approximated reasonably well by a Gaussian with centroid energy [MATH] keV and width [MATH] keV.', 'astro-ph-9903306-3-17-3': 'Its absolute intensity appeared to be stable against a decrease of X-ray continuum by a factor of [MATH]4, thus we believe that this line emission originates from the diffuse X-ray source near the Center of the Galaxy rather than from XTE J1748-288 itself.', 'astro-ph-9903306-3-17-4': 'The measured intensity of this feature, [MATH] phot/s/cm[MATH], is in a reasonable agreement with previous measurements of the diffuse line emission from the Galactic Center (e.g. Yamauchi Koyama 1993).', 'astro-ph-9903306-3-17-5': 'The inclusion of the most prominent Galactic diffuse lines - 6.4, 6.7, and 6.9 keV - also improves the spectral approximation significantly (the widths of the lines were frozen at 0.1 keV - value comparable with the PCA energy resolution).', 'astro-ph-9903306-3-17-6': 'The best fit line intensities ([MATH] phot/s/cm[MATH] for 6.4 keV line and [MATH] phot/s/cm[MATH] for 6.7 keV and 6.9 keV lines together ) were approximately constant within the accuracy of the measurements for all LS observations.', 'astro-ph-9903306-3-17-7': 'The equivalent width of the whole [MATH] feature increased from [MATH]300 eV in observation 16 to [MATH]1.1 keV in observation 23.', 'astro-ph-9903306-3-18-0': 'Spectral fits revealed the presence of an additional line in the spectrum, with the central energy around 8 keV and an intensity [MATH] phot/s/cm[MATH].', 'astro-ph-9903306-3-18-1': 'Even given the moderate energy resolution of the PCA, the presence of this line is evident both when the Fe [MATH] complex is fitted by by three narrow lines as described above, and when one fits it by one broader line with central energy [MATH]6.5 keV instead.', 'astro-ph-9903306-3-18-2': 'The 8 keV line feature is likely a complex of the Fe [MATH] or/and Ni lines of the hard ([MATH] keV) optically thin diffuse plasma emission of the Galactic Center region [CITATION].', 'astro-ph-9903306-3-18-3': 'Note, that the similar spectral feature was observed in the spectrum of XTE J0421+560/CI Cam [CITATION], that was likely caused by optically thin plasma thermal emission.', 'astro-ph-9903306-3-19-0': 'The results of the spectral approximation of the PCA data with the analytical models described above are presented in Table 2.', 'astro-ph-9903306-3-19-1': 'Fig. [REF] shows an evolution of some best fit parameters with time.', 'astro-ph-9903306-3-19-2': 'A correlation between changes in the soft fraction, the disk temperature and the power-law slope during the first 15 observations (VHS) is clearly seen.', 'astro-ph-9903306-3-20-0': '## Power density spectrum', 'astro-ph-9903306-3-21-0': 'In order to analyze the timing properties of XTE J1748-288, we generated power density spectra (PDS) in the 0.01-250 Hz frequency range (for 2-13 keV energy band) using short stretches of data.', 'astro-ph-9903306-3-21-1': 'The resulting spectra were logarithmically rebinned when necessary to reduce the scatter at high frequencies and normalized to the square of the fractional variability.', 'astro-ph-9903306-3-21-2': 'The white-noise level due to the Poissonian statistics corrected for the dead-time effects was subtracted (Vikhlinin, Gilfanov Churazov 1994; Zhang et al. 1995).', 'astro-ph-9903306-3-21-3': 'To obtain PDS in the lower frequency band ([MATH] Hz - [MATH]0.01 Hz) we used the 16-sec integrated data of Standard 2 mode, because it allowed us to take into account the influence of the background variation on the PDS, which might be of importance at these frequencies.', 'astro-ph-9903306-3-22-0': 'To study the evolution of the basic timing and spectral parameters of XTE J1748-288 within the individual observations with a relatively high level of variability we generated power density spectra of the source averaged over 256 s time intervals according to the procedure described above (for the results of this study see section 3.4).', 'astro-ph-9903306-3-23-0': 'We fitted the power density spectra of the source in the 0.02-150 Hz range to analytic models using the [MATH] minimization technique to quantify their characteristics.', 'astro-ph-9903306-3-23-1': 'For the approximation of the PDS obtained during the first 8 observations, we used the sum of a flat-topped band-limited noise (BLN) component: [MATH], where [MATH] - is a characteristic break frequency of the BLN continuum, a power law (PL) component - very low frequency noise (VLFN) and up to three harmonically related Lorentzian groups to describe the QPO features (see explanation below).', 'astro-ph-9903306-3-24-0': 'Fig. [REF] presents the data and model fit for one observation (June 10).', 'astro-ph-9903306-3-24-1': 'As is clearly seen from the inset of this figure, an approximation of the 20-30 Hz QPO features by a single Lorentzian component does not provide an acceptable fit.', 'astro-ph-9903306-3-24-2': 'This fact could be explained in terms of the existence of an additional high frequency shoulder, similar to those observed in Nova Muscae 1991 and GX 339-4 (Belloni et al. 1997).', 'astro-ph-9903306-3-24-3': 'As we demonstrate below (section 3.4), the QPO centroid frequency derived from the simple Lorentzian fit shifts during the individual observations (Fig. [REF]).', 'astro-ph-9903306-3-24-4': 'Since the strength of the QPO peak is anti-correlated with the QPO centroid frequency, averaging over the whole observation should lead to a steepening of the resulting QPO profile in its low frequency wing and a flattening in its high frequency wing forming a shoulder-like structure.', 'astro-ph-9903306-3-24-5': 'Thus, the appearance of the observed QPO shoulder could be a result of such a displacement of the QPO centroid frequency, for this reason it seems natural to approximate the QPO feature by a sum of several Lorentzians.', 'astro-ph-9903306-3-24-6': 'To obtain a satisfactory fit we used the sum of two Lorentzians for every harmonic.', 'astro-ph-9903306-3-24-7': 'The approximation shows that the frequencies of these additional Lorentzians are also harmonically related which supports our interpretation of the origin of the shoulders.', 'astro-ph-9903306-3-25-0': 'In observations [MATH] the power density spectra were approximated by a power law component, the very low frequency noise (VLFN) component and a Lorentzian profile representing the low frequency QPO.', 'astro-ph-9903306-3-26-0': 'The best fit parameters of the band limited noise and QPO components are presented in Table [REF].', 'astro-ph-9903306-3-26-1': 'QPO rms amplitude is calculated as the quadratic sum of the corresponding rms amplitudes for the Lorentzians used for the QPO approximation.', 'astro-ph-9903306-3-26-2': 'Parameter errors correspond to the [MATH] confidence level.', 'astro-ph-9903306-3-26-3': 'These models approximate the data reasonably well, as indicated by the values of reduced [MATH] for the fits.', 'astro-ph-9903306-3-27-0': '## Evolution of the source during the outburst', 'astro-ph-9903306-3-28-0': 'The X-ray flux histories of the XTE J1748-288 outburst based on the [MATH]/PCA and ASM data in the 3-15 and 15-30 keV energy ranges are shown in Fig. [REF].', 'astro-ph-9903306-3-28-1': 'The evolution of the source flux in the soft X-ray band (3-15 keV) was characterized by the fast initial rise to a level of [MATH]600-700 mCrab on a time scale of [MATH]2-3 days followed by the [MATH] day-long maximum and a nearly exponential decay to the quiescent level with e-folding time [MATH] days.', 'astro-ph-9903306-3-28-2': 'The flux in the hard band (15-30 keV) undergone a fast initial rise to the [MATH]300-400 mCrab level followed by the maximum phase of a [MATH]-day duration and an abrupt decay, which is similar to the behavior of Nova Muscae GS/GRS 1124-683 [CITATION] during its outburst of 1991.', 'astro-ph-9903306-3-28-3': 'Based on the results of the spectral and timing analysis, the evolution of XTE J1748-288, in analogy to Nova Muscae 1991, could be divided into three distinct parts, corresponding to three main phenomenological states: very high (VHS), high (HS) and low (LS).', 'astro-ph-9903306-3-28-4': 'At the same time we would like to note, that in the first 8 observations the spectrum of XTE J1748-288 have an extremely bright hard component (the power law component contribution to the total 3-25 keV flux [MATH]80%) which is somewhat different from the usual VHS spectrum.', 'astro-ph-9903306-3-28-5': 'Very similar spectra were observed only with Ginga at the very beginning of the outburst of GS 1124-683 (Kitamoto et al. 1992; Ebisawa et al. 1994), and with Mir-Kvant at the beginning of the outbursts of X-ray Novae KS1730-312 [CITATION] and GRS 1739-278 [CITATION], but observations of XTE J1748-288 outburst provided an unique opportunity to study this unusually hard VHS spectrum in detail.', 'astro-ph-9903306-3-28-6': 'It is also worth mentioning that contrary to many previous observations of the VHS, which could be distinguished from the HS according to the difference in the power spectrum only, in our case both the energy spectrum and power spectrum in VHS were distinctly different from those in the HS.', 'astro-ph-9903306-3-29-0': '[Very High State] During the first eight RXTE pointed observations (June 4 - 11, 1998) the source was found in an extremely bright state.', 'astro-ph-9903306-3-29-1': 'The broad-band 3-150 keV energy spectrum of XTE J1748-288 in this state can be satisfactorily described by the sum of two components: a relatively weak soft thermal component (contributing only [MATH] 10-20 per cent to the total X-ray flux in the 3-25 keV energy band) with a characteristic temperature of [MATH] 0.8-1.4 keV and a strong hard component, which is approximately a power law shape and does not show the evidence of a high energy cut-off up to [MATH] keV .', 'astro-ph-9903306-3-29-2': 'The corresponding luminosity in the 3-25 keV band uncorrected for the interstellar absorption was at a level of 1-1.5[MATH] ergs/s (assuming a 8 kpc distance) during this period.', 'astro-ph-9903306-3-30-0': 'The broad-band power density spectrum (PDS) of the source is formed by the dominating band-limited noise component (corresponding rms amplitude [MATH] 5-9 %), the strong QPO feature with centroid frequency [MATH] 20-30 Hz and a very low frequency noise (VLFN) component with a slope [MATH] and rms amplitude [MATH] 1.0-1.5 % (Fig. [REF], upper panels).', 'astro-ph-9903306-3-31-0': 'The correlated spectral and timing evolution of the source during this period is of particular interest.', 'astro-ph-9903306-3-31-1': 'The local increase of the contribution of the soft spectral component to the total X-ray flux in observations 3-4 (June 6 - 7, 1998) was accompanied by a steepening of the hard spectral component and significant changes in the power density spectrum (Table 2, 3.', 'astro-ph-9903306-3-31-2': 'Fig. [REF], left upper panel), i.e.:', 'astro-ph-9903306-3-32-0': '- the increase of the BLN characteristic break frequency and the decrease of its rms amplitude;', 'astro-ph-9903306-3-33-0': '- the increase of the QPO centroid frequency from [MATH] Hz to [MATH] Hz;', 'astro-ph-9903306-3-34-0': '- the appearance of the additional broad QPO feature at [MATH] Hz (Fox Lewin 1998);', 'astro-ph-9903306-3-35-0': '- the notable rise of the VLFN rms amplitude.', 'astro-ph-9903306-3-35-1': 'Detected correlations between the spectral and timing properties of the source in this state, discussed in more detail below (see 3.4), holds a wide range of the time scales from seconds to several days.', 'astro-ph-9903306-3-36-0': 'The two-component spectrum, consisting of the soft thermal component and a hard power law component is a signature of the high spectral state of the Galactic black hole candidates (see Tanaka Shibazaki 1996, and references therein).', 'astro-ph-9903306-3-36-1': 'The very high state differs from high state mostly by the much stronger fast variability, while the differences in energy spectra are not significant and no good criterium was suggested to distinguish between these two states on spectral ground.', 'astro-ph-9903306-3-36-2': 'In case of XTE J1748-288 the spectrum was sufficiently different from the typical spectra observed previously in the HS and the VHS, because power law component dominated in the 3-25 keV energy band.', 'astro-ph-9903306-3-36-3': 'Fast variability and shape of power spectrum was typical for the VHS of Galactic black hole binaries (e.g. Belloni et al. 1997).', 'astro-ph-9903306-3-37-0': '[High State].', 'astro-ph-9903306-3-37-1': 'As the flux of XTE J1748-288 dropped below [MATH]4200 cnts/s/PCA (observations 9-15), the spectral and timing properties of the source changed drastically.', 'astro-ph-9903306-3-37-2': 'The strength of the hard component in the energy spectrum decreased rapidly and the contribution of the soft thermal component to the total luminosity in the 3-25 keV energy range increased to more than 60 per cent as typical for HS (Fig. [REF], [REF]; Table 2).', 'astro-ph-9903306-3-38-0': 'Simultaneously the shape of the power density spectrum changed qualitatively.', 'astro-ph-9903306-3-38-1': 'The band-limited noise (BLN) component and the high frequency 20-30 Hz QPOs became undetectable, and the amplitude of the fractional variability decreased from [MATH]10 to [MATH]1 per cent.', 'astro-ph-9903306-3-38-2': 'The power density spectrum during this period was composed of the power law component with a slope of -1.0-1.5 and a broad QPO peak at [MATH]0.03 Hz (see Fig. [REF],[REF]; Table 3).', 'astro-ph-9903306-3-39-0': '[Low State].', 'astro-ph-9903306-3-39-1': 'The subsequent decline in the intensity of the XTE J1748-288 X-ray flux was accompanied by a gradual decrease of the strength of the soft spectral component (Table [REF]) and in 16th observation (July 18, 1998) the source was found already in the standard LS.', 'astro-ph-9903306-3-39-2': 'It should be noted that in this state the source flux was extremely weak and a substantial part of the observed emission in the PCA energy band can be attributed to the galactic diffuse and point sources, in particular to Sgr B2 (the offset from XTE J1748-288 [MATH]).', 'astro-ph-9903306-3-40-0': 'The low state X-ray spectrum in the 3-25 keV energy band was composed by a power law component with a photon index of [MATH]2, a complex of strong emission lines (around 6.5 keV and at [MATH]8 keV) and low energy absorption.', 'astro-ph-9903306-3-40-1': 'The absolute intensity of the lines appeared to be very stable against the decrease of X-ray continuum, which strongly supports the interpretation that they originate from some of the sources within the field of view, but not from the XTE J1748-288.', 'astro-ph-9903306-3-40-2': 'We have performed an additional study to get an estimation for the intensity of the emission line attributed to the XTE J1748-288 itself, assuming the centroid energy of the intrinsic source line to be equal to 6.4 keV.', 'astro-ph-9903306-3-40-3': 'Using the fact that the strong decline in the continuum flux did not result in the statistically significant change in the observed 6.4 keV line flux, we obtained 2[MATH] upper limit [MATH]80 eV on the equivalent width of the intrinsic emission line.', 'astro-ph-9903306-3-41-0': '## Correlations between spectral and variability parameters in the very high state', 'astro-ph-9903306-3-42-0': 'The significant evolution of the spectral and temporal properties of XTE J1748-288 in the VHS provides an excellent opportunity to study their mutual relationship in a wide range of time scales from seconds to several days.', 'astro-ph-9903306-3-42-1': 'In the course of our extended timing analysis the general correlation between the main parameters describing the power density spectrum of the source (QPO centroid frequency, characteristic break frequency of the BLN component, BLN rms, QPO rms) was observed.', 'astro-ph-9903306-3-42-2': 'The dependence of the rms amplitude of main QPO peak on its centroid frequency is shown in Fig. [REF] .', 'astro-ph-9903306-3-42-3': 'The plot demonstrates evident anticorrelation between these two parameters.', 'astro-ph-9903306-3-42-4': 'Other PDS parameters also appeared to be strongly inter-correlated.', 'astro-ph-9903306-3-43-0': 'Most striking is the established close relation between the evolution of the spectral and timing parameters of the source.', 'astro-ph-9903306-3-43-1': 'The change of the QPO centroid frequency is correlated with change of the spectral parameters, derived from the energy spectra fits.', 'astro-ph-9903306-3-43-2': 'In particular, there is a clear trend of increasing the QPO centroid frequency with rise of the soft component flux (Fig. [REF]).', 'astro-ph-9903306-3-43-3': 'It should be noted that this type of correlation holds on the wide range of time scales (from seconds to several days).', 'astro-ph-9903306-3-44-0': 'A similar type of the correlation between the QPO parameters and X-ray flux has also been reported for other Galactic superluminal jet source GRS 1915+105 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999), X-ray Nova XTE J1550-564 (Cui et al. 1999), and XTE J1806-246, that is suspected to be a system containing a neutron star (Revnivtsev, Borozdin Emelyanov 1999).', 'astro-ph-9903306-3-44-1': 'As in the cases of GRS 1915+105 and XTE J1550-564 the QPO peak in the power density spectrum of XTE J1748-288 was detected only in observations characterized by the domination of the hard spectral component.', 'astro-ph-9903306-3-44-2': 'This fact hints on the direct link between the processes of QPO and hard spectral component formation.', 'astro-ph-9903306-3-45-0': '# Discussion', 'astro-ph-9903306-3-46-0': 'Many bright X-ray transients are reputed black hole candidates and demonstrate similar X-ray spectra and fast variability.', 'astro-ph-9903306-3-46-1': 'During the outburst these systems are typically found in one of two qualitatively distinguishable spectral states: in the high state composed of the bright thermal component and the extended hard power-law, or in the low state with the hard power-law spectrum with a photon index of [MATH]1.5 and an exponential high energy cutoff.', 'astro-ph-9903306-3-46-2': 'A more detailed description of these states can be found elsewhere [CITATION].', 'astro-ph-9903306-3-46-3': 'A third, very high, state has been recognized also, with two component spectrum similar to the HS, but with somewhat stronger power-law component and with much stronger fast variability (Miyamoto et al. 1991, 1994; Takizawa et al. 1997).', 'astro-ph-9903306-3-46-4': 'The spectral evolution of X-ray transients is usually in correlation with X-ray flux changes, so it is widely believed that state transitions are driven by the variable accretion rate.', 'astro-ph-9903306-3-46-5': 'Such transitions were also observed in the persistent black hole systems, namely, Cyg X-1 and GX 339-4, but the dynamics of these systems is much slower, so they are more often observed in one of these states and typically switch to another state once every several years [CITATION].', 'astro-ph-9903306-3-47-0': 'The transient X-ray source XTE J1748-288 resembles other black hole X-ray transients in many ways.', 'astro-ph-9903306-3-47-1': 'The general properties of the source light curves are similar to the properties of the light curves of black hole X-ray Novae such as A 0620-00, GS 2000+25, GS/GRS 1124-68, GRS 1009-45, GRS1739-278 and others (see Chen, Shrader Livio 1997 for review, and references therein).', 'astro-ph-9903306-3-47-2': 'During the 1998 outburst the source was observed by RXTE in very high, high and low spectral states, which are very typical for black hole systems.', 'astro-ph-9903306-3-47-3': 'During several observations corresponding to the peak X-ray flux, the spectrum of the source was in an unusual VHS, a very bright hard component was observed.', 'astro-ph-9903306-3-47-4': 'Similar spectra were detected at the beginning of the outbursts of X-ray black hole Novae GS/GRS1124-68 [CITATION], KS/GRS 1730-312 [CITATION] and GRS 1739-278 [CITATION].', 'astro-ph-9903306-3-47-5': 'Here we studied this spectrum in more detail.', 'astro-ph-9903306-3-48-0': 'The transition of XTE J1748-288 from the very high state to the high state changed both the spectral and fast variability properties of the X-ray flux of XTE J1748-288.', 'astro-ph-9903306-3-48-1': 'Unlike the case of Nova Muscae [CITATION], hard component dominated spectrum during all VHS observations.', 'astro-ph-9903306-3-48-2': 'The soft component in the spectrum of the XTE J1748-288 increased rapidly by a factor of [MATH]4 after the transition, while the total flux in the energy band 3-25 keV continued to decrease smoothly.', 'astro-ph-9903306-3-48-3': 'The amplitude of the fast variability dropped down during the transition and PDS shape changed qualitatively (see Fig. [REF]).', 'astro-ph-9903306-3-48-4': 'PDS of the source X-ray flux in the HS has a power law shape of the continuum and shows the presence of the low-frequency QPO peak at [MATH] Hz, resembling the properties of the high state PDS of the well-known black hole candidate LMC X-1 (its form was described by the sum of the power law component and a QPO peak with a centroid frequency of [MATH] Hz) (Ebisawa et al. 1989).', 'astro-ph-9903306-3-49-0': 'After the subsequent decrease of the luminosity, the XTE J1748-288 switched the source to the low state typical for the black hole X-ray Novae at this stage of the outburst evolution.', 'astro-ph-9903306-3-50-0': 'The shape of the broad-band power density spectrum of XTE J1748-288 in the VHS was similar to the PDS observed for the Galactic black hole candidates GX 339-4, Nova Muscae 1991 and GRS 1915+105 (Miyamoto et al. 1994; Belloni et al. 1997; Trudolyubov, Churazov Gilfanov 1999): it was dominated by a strong band-limited noise with QPO components.', 'astro-ph-9903306-3-51-0': 'The correlated spectral and timing evolution of the source during the VHS and, in particular, the observed trend of increasing the QPO centroid frequency with the rise of the soft component flux on the wide range of time scales is of special importance.', 'astro-ph-9903306-3-51-1': 'A similar type of correlation between the QPO parameters and X-ray flux have also been reported for other Galactic black hole candidates: GRS 1915+105 and XTE J1550-564 (Trudolyubov, Churazov Gilfanov 1999; Markwardt, Swank Taam 1999; Cui et al. 1999), which hints at the general mechanism of the X-ray emission generation for this class of objects.', 'astro-ph-9903306-3-52-0': 'One of the possible and probably the most promising way to explain these observational results for black hole systems is to use the models based on the idea of a shock front formation between two distinct parts of the accretion flow producing the bulk of X-rays: the central region responsible for the generation of the hard spectral component and optically thick accretion disk (Chakrabarti Titarchuk 1995).', 'astro-ph-9903306-3-52-1': 'The oscillations in shock geometrical parameters (i.e. height, width) may modulate the flux of the "seed" photons from the inner boundary of the optically thick accretion disk and change the state of matter in the optically thin region - and as a result, QPOs of X-ray flux are to be detected (Titarchuk, Lapidus Muslimov 1998; Molteni, Sponholz Chakrabarti, 1996).', 'astro-ph-9903306-3-52-2': 'In this case the time scale of the oscillations is determined by the position of the boundary between the optically thick and thin regions, which in turn is directly related to the luminosities of the soft and hard spectral components.', 'astro-ph-9903306-3-52-3': 'Hence the observed correlation between the QPO frequency and X-ray flux could be naturally explained in the framework of this model.', 'astro-ph-9903306-3-53-0': 'The observed complex of the general spectral and timing properties of XTE J1748-288 is similar to that of the dynamically proven black hole X-ray Novae and Galactic superluminal jet sources.', 'astro-ph-9903306-3-53-1': 'This fact could hint at the universal character of evolution of the accretion flow in the black hole systems and its direct link to the mechanism of the formation of relativistic ejections of matter.'}

cond-mat-9905173

{'cond-mat-9905173-1-0-0': 'Velocity distributions in a vibrated granular monolayer are investigated experimentally.', 'cond-mat-9905173-1-0-1': 'Non-Gaussian velocity distributions are observed at low vibration amplitudes but cross over smoothly to Gaussian distributions as the amplitude is increased.', 'cond-mat-9905173-1-0-2': 'Cross-correlations between fluctuations in density and temperature are present only when the velocity distributions are strongly non-Gaussian.', 'cond-mat-9905173-1-0-3': 'Confining the expansion of the granular layer results in non-Gaussian velocity distributions that persist to high vibration amplitudes.', 'cond-mat-9905173-1-1-0': 'The effects of inelasticity on the statistical properties of a granular gas has been a topic of recent intense theoretical and experimental interest.', 'cond-mat-9905173-1-1-1': 'In freely cooling granular media, analytic results and simulations show that dissipative inter-particle collisions result in clustering [CITATION], non-Gaussian velocity distributions [CITATION], and eventually to the breakdown of hydrodynamics [CITATION].', 'cond-mat-9905173-1-1-2': 'In driven granular gases, where the energy lost through collisions is balanced by energy input from external forcing, the effects of inelasticity can be observed in the steady state statistical properties of the gas.', 'cond-mat-9905173-1-1-3': 'Experimental studies have shown clustering [CITATION] and non-Gaussian velocity distributions [CITATION], but the dynamical origin of the velocity distributions remains unclear.', 'cond-mat-9905173-1-1-4': 'A model of a granular gas coupled to a thermal reservoir shows long-range correlations in density and velocity, and velocity distributions that fall off with [MATH] in the tails [CITATION].', 'cond-mat-9905173-1-1-5': 'A model with a discrete random forcing produces strong clustering, and a cross-correlation between the fluctuations in density and granular temperature (average kinetic energy) [CITATION].', 'cond-mat-9905173-1-1-6': 'The cross-correlation is due to the same mechanism as the clustering instability in a freely cooling granular gas: fluctuations of increased density result in more frequent inter-particle collisions, producing increased dissipation and a reduced local granular temperature.', 'cond-mat-9905173-1-2-0': 'Recent work has demonstrated various collective phenomena for a large number ([MATH]) of identical, uniform ball bearings constituting less than one layer coverage on a vertically shaken, horizontal plate [CITATION].', 'cond-mat-9905173-1-2-1': 'At peak plate accelerations above 1 g and for most densities, the particles behave as a rapidly fluctuating gas.', 'cond-mat-9905173-1-2-2': 'Reducing the acceleration amplitude decreases the mean square velocity, or granular temperature, T[MATH], and effectively "cools" the gas, leading to an observed increase in clustering.', 'cond-mat-9905173-1-2-3': 'Continued cooling eventually leads to the formation of a collapse, a condensate of motionless particles that remain in contact with the plate and each other.', 'cond-mat-9905173-1-2-4': 'In the range of 0.8 - 1.0 g, all of the velocity distributions of the gas appear to scale with the second moment of the distribution to an universal curve [CITATION].', 'cond-mat-9905173-1-2-5': 'The velocity distributions demonstrate strong deviations from a Gaussian distribution in both the high velocity tails and at low velocities.', 'cond-mat-9905173-1-3-0': 'This report presents the results of a further investigation into the nature of the non-Gaussian velocity distributions and their relation to the observed density fluctuations that stem from the inelastic collisions.', 'cond-mat-9905173-1-3-1': 'In order to understand the non-Gaussian velocity distributions seen previously in this system [CITATION], measurements were made over a larger range of [MATH], the dimensionless acceleration of the plate, where [MATH] is the frequency in Hz and g is the acceleration due to gravity.', 'cond-mat-9905173-1-3-2': 'Our results demonstrate both non-Gaussian velocity distributions and a cross-correlation between density and temperature, similar to that seen in the model system of Puglisi et al. [CITATION].', 'cond-mat-9905173-1-3-3': 'However, the data reported here clearly shows that the two effects are independent: there is a large range over which the velocity distributions are non-Gaussian but for which the granular temperature is not measurably dependent on the density.', 'cond-mat-9905173-1-3-4': 'Only at low [MATH], where the clustering is the strongest, is the cross-correlation between density and temperature observed in our experiment.', 'cond-mat-9905173-1-4-0': 'For these experiments, the plate vibration is sinusoidal, [MATH] 70 Hz and unless otherwise stated, the particle density is [MATH].', 'cond-mat-9905173-1-4-1': 'For accelerations below [MATH], the system was initially fluidized by shaking at [MATH].', 'cond-mat-9905173-1-4-2': 'Two different particle species were used: The small spheres were 302 stainless steel with an average diameter of 0.1191 cm [MATH] 0.00024 cm and the large spheres were 316 stainless with an average diameter of 0.1588 cm [MATH] 0.00032 cm.', 'cond-mat-9905173-1-4-3': 'The coefficient of restitution for both particle species is approximately 0.9 [CITATION].', 'cond-mat-9905173-1-5-0': 'The measured velocity distributions at [MATH], [MATH] and [MATH] are shown in Fig. [REF].', 'cond-mat-9905173-1-5-1': 'The distribution crosses over from one with approximately exponential tails as reported in [CITATION] to a Gaussian distribution.', 'cond-mat-9905173-1-5-2': 'This remarkable evolution is superficially similar to what is observed in freely cooling granular media, where an initial Gaussian velocity distribution becomes non-Gaussian as the system cools.', 'cond-mat-9905173-1-5-3': 'In that case, the evolution is determined by the strength of the inelasticity and the integrated number of collisions per particle, which does not have an obvious analog in the driven system.', 'cond-mat-9905173-1-6-0': 'As previously reported, the non-Gaussian velocity distributions observed at low accelerations are accompanied by clustering, as demonstrated by a dramatic increase in the structure of the pair correlation function.', 'cond-mat-9905173-1-6-1': 'Figure [REF](a) shows the pair correlation function determined via analysis of images taken from above the granular layer for the same accelerations as the velocity distributions in Fig. [REF].', 'cond-mat-9905173-1-6-2': 'The results of the measured correlation function approach that of an uncorrelated dilute hard sphere gas (solid line).', 'cond-mat-9905173-1-6-3': 'Thus, the crossover to Gaussian velocity distributions is accompanied by the disappearance of spatial correlations, consistent with the suggestions that the non-Gaussian velocity distributions arise from a coupling between density and temperature fluctuations [CITATION].', 'cond-mat-9905173-1-7-0': 'Increasing the steady state kinetic energy of the granular gas by increasing the amplitude of the acceleration at constant frequency causes the gas to change from primarily two-dimensional, where the particles never hop over one another, to essentially three-dimensional [CITATION].', 'cond-mat-9905173-1-7-1': 'This transition can be observed in the pair correlation function, [MATH], by the increase in its value for [MATH].', 'cond-mat-9905173-1-7-2': '(The correlation function includes only particle separations in the hortizontal plane.)', 'cond-mat-9905173-1-7-3': 'This transition can affect the dynamics in several ways: the effective density is decreased, so that excluded volume effects are less important; the inter-particle collisions can occur at angles closer to vertical, affecting transfer of energy and momentum from the vertical direction to the horizontal; and the change in the dimensionality itself can have important consequences.', 'cond-mat-9905173-1-7-4': 'In order to separate these effects from the direct consequences of increasing the kinetic energy of the gas, a plexiglass lid was added to the system at a height of 0.254 cm, or 1.6 ball diameters for the larger particles.', 'cond-mat-9905173-1-7-5': 'For this plate-to-lid separation, the larger particles cannot pass over top of one another, although enough room remains for collisions between particles at sufficiently different heights to transfer momentum from the vertical to the horizontal direction.', 'cond-mat-9905173-1-8-0': 'Figure [REF] (b) demonstrates the persistence of the particle-particle correlations when the system is constrained to 2D.', 'cond-mat-9905173-1-8-1': 'The particle-particle correlation function decreases slightly from [MATH] to [MATH], and then remains essentially constant up to [MATH].', 'cond-mat-9905173-1-8-2': 'The small value of [MATH] for distances less than one ball diameter indicates that the system remains 2D as [MATH] is increased.', 'cond-mat-9905173-1-8-3': 'The structure observed in the correlation function is essentially the same as that of an equilibrium elastic hard sphere gas at the same density, indicating that the correlations that exist are due to excluded volume effects.', 'cond-mat-9905173-1-9-0': 'Figure [REF](a) shows that the presence of the lid adds an energy sink to the system at high [MATH].', 'cond-mat-9905173-1-9-1': 'At low acceleration ([MATH]), very few, if any, particles strike the lid and the lid has no significant effect.', 'cond-mat-9905173-1-9-2': 'At larger [MATH] it is clear that the horizontal granular temperature is reduced as a significant number of particles strike the lid, dissipating energy.', 'cond-mat-9905173-1-9-3': 'It is interesting to note that T[MATH] approaches zero lineraly at finite [MATH], indicating that the relationship between the driving and the horizontal granular temperature is of the form T[MATH].', 'cond-mat-9905173-1-10-0': 'In order to demonstrate the effect of the lid on the velocity distributions, we use a simple quantitative measure of the non-Gaussian nature, the flatness (or kurtosis) of the distribution: [EQUATION]', 'cond-mat-9905173-1-10-1': 'For a Gaussian distribution, the flatness is 3 and for the broader exponential distribution, the flatness is 6.', 'cond-mat-9905173-1-10-2': 'In the absence of a lid, the flatness demonstrates a smooth transition from non-Gaussian to Gaussian behavior as the granular temperature is raised (Fig. [REF](b)) whether the smaller (circles) or larger (stars) particles are used.', 'cond-mat-9905173-1-10-3': 'With the lid on, the velocity distributions remain more non-Gaussian than in the free system for identical granular temperatures and density (diamonds).', 'cond-mat-9905173-1-10-4': 'The crossover from Gaussian to non-Gaussian behavior observed without the lid is therefore not simply an effect of increasing the vertical kinetic energy of the particles, but rather related to the transfer of energy from the vertical to horizontal motion in the system via collisions, the change in the density, or the change in dimensionality of the gas.', 'cond-mat-9905173-1-11-0': 'To determine the relative contribution of density changes to the non-Gaussian velocity distributions in the gas, the number of particles on the plate was increased by [MATH] and decreased by [MATH] from the value of [MATH] and the lid was kept on.', 'cond-mat-9905173-1-11-1': 'For all accelerations, the flatness decreased with increased density.', 'cond-mat-9905173-1-11-2': 'This surprising result may be related to the fact that strongly non-Gaussian distributions observed at low [MATH] are accompanied by strong clustering [CITATION].', 'cond-mat-9905173-1-11-3': 'If the average density is increased, the larger excluded volume means that less phase space remains for fluctuations to persist.', 'cond-mat-9905173-1-11-4': 'The fact that increasing the density with the lid on makes the velocity distribution more Gaussian suggests that the crossover to Gaussian observed without the lid is not due to the decrease in density of the gas.', 'cond-mat-9905173-1-12-0': 'Puglisi et al. [CITATION] have proposed a model which relates strong clustering to non-Gaussian velocity distributions in a driven granular medium.', 'cond-mat-9905173-1-12-1': 'In their framework, at each local density the velocity distributions are Gaussian, and the non-Gaussian behavior arises from the relative weighting of the temperature by local density in the following manner: [EQUATION] where [MATH] is the second moment of the distribution for the number of boxes, n, that contain N particles.', 'cond-mat-9905173-1-12-2': 'In this model, the local temperature is a decreasing function of the local density, and the velocity distributions conditioned on the local density are Gaussian.', 'cond-mat-9905173-1-13-0': 'In our experiment, this feature can be examined by conditioning the local horizontal granular temperature on the local density.', 'cond-mat-9905173-1-13-1': 'That is, examining the distribution of velocities for data at a constant number of particles in the frame of the camera in the strongly clustering regime at [MATH] [CITATION].', 'cond-mat-9905173-1-13-2': 'In the strongly clustering regime, we do observe a direct correlation between local density and temperature.', 'cond-mat-9905173-1-13-3': 'Figure [REF] is a plot of the local temperature as a function of particle number in the camera frame normalized by the granular temperature, including data from the open system for both the smaller and larger particles as well as for the confined system using the larger particles.', 'cond-mat-9905173-1-14-0': 'The result is similar to the model of Puglisi et al. [CITATION]: At low T[MATH], when the particle-particle correlations are strongest (and larger than those of an equilibrium hard sphere gas [CITATION]), there is a density dependence to the granular temperature (filled circles).', 'cond-mat-9905173-1-14-1': 'At [MATH], where all of the particles are essentially uncorrelated in a 3D volume in the absence of a lid, there is no density dependence (open circles, stars).', 'cond-mat-9905173-1-14-2': 'However, even in the confined system at [MATH], where the distribution is still not Gaussian, no appreciable density dependence is observed (diamonds), suggesting that the non-Gaussian velocity distributions and density-dependent temperature are not as simply dependent upon one another as they are in the model of Puglisi et al. [CITATION].', 'cond-mat-9905173-1-14-3': 'In fact, while there is a clear density dependence on the local temperature at low [MATH], the measured velocity distribution conditioned on the local temperature is not Gaussian.', 'cond-mat-9905173-1-14-4': 'At each density, the velocity distribution function is almost identical to that of the whole: when the entire distribution is non-Gaussian, the distribution at a single density is non-Gaussian, and when the whole is Gaussian, each conditional velocity distribution is Gaussian.', 'cond-mat-9905173-1-15-0': 'A more general form of Eq. [REF] represents the total velocity distribution as a product of local Gaussian velocity distributions with a distribution of local temperatures:', 'cond-mat-9905173-1-16-0': '[EQUATION] where [MATH] is the local temperature that is varying in space and time.', 'cond-mat-9905173-1-16-1': 'Conditioning on the local temperature would then recover the underlying Maxwell statistics in the fluctuations [CITATION].', 'cond-mat-9905173-1-17-0': 'Performing this analysis on our data does not succeed in producing Gaussian statistics.', 'cond-mat-9905173-1-17-1': 'Within small windows of local temperature, the distributions remain non-Gaussian.', 'cond-mat-9905173-1-17-2': 'Indeed, the analysis can be extended to condition on both the local temperature and density in the system, but with similarly limited success except for the slowest of particles in the most dilute regions of the system, although all of the conditioned distributions are closer to Gaussian than the full distribution.', 'cond-mat-9905173-1-17-3': 'Curiously, if the local velocities are normalized by the magnitude of the local granular temperature, then the total distribution is Gaussian [CITATION].', 'cond-mat-9905173-1-18-0': 'These results demonstrate several important new characteristics of a 2D granular gas.', 'cond-mat-9905173-1-18-1': 'Extreme non-Gaussian velocity distributions observed previously [CITATION] involve strong cross-correlations between density and velocity fluctuations.', 'cond-mat-9905173-1-18-2': 'This is not the whole picture, however, as a 2D constrained granular gas also demonstrates strongly non-Gaussian behavior without a significant cross-correlation between density and velocity distributions.', 'cond-mat-9905173-1-18-3': 'Finally, the crossover to Gaussian distributions observed when the gas is fully 3D is probably associated with the change in dimensionality or momentum transfer, rather than the increase in kinetic energy or the decrease in density.', 'cond-mat-9905173-1-19-0': 'This work was supported by an award from the Research Corporation, a grant from the Petroleum Research Fund and grant DMR-9875529 from the NSF.', 'cond-mat-9905173-1-19-1': 'One of us (JSU) was supported by a fellowship from the Sloan Foundation.'}

{'cond-mat-9905173-2-0-0': 'Velocity distributions in a vibrated granular monolayer are investigated experimentally.', 'cond-mat-9905173-2-0-1': 'Non-Gaussian velocity distributions are observed at low vibration amplitudes but cross over smoothly to Gaussian distributions as the amplitude is increased.', 'cond-mat-9905173-2-0-2': 'Cross-correlations between fluctuations in density and temperature are present only when the velocity distributions are strongly non-Gaussian.', 'cond-mat-9905173-2-0-3': 'Confining the expansion of the granular layer results in non-Gaussian velocity distributions that persist to high vibration amplitudes.', 'cond-mat-9905173-2-1-0': 'The effects of inelasticity on the statistical properties of a granular gas has been a topic of recent intense theoretical and experimental interest.', 'cond-mat-9905173-2-1-1': 'In freely cooling granular media, analytic results and simulations show that dissipative inter-particle collisions result in clustering [CITATION], non-Gaussian velocity distributions [CITATION], and eventually to the breakdown of hydrodynamics [CITATION].', 'cond-mat-9905173-2-1-2': 'In driven granular gases, where the energy lost through collisions is balanced by energy input from external forcing, the effects of inelasticity can be observed in the steady state statistical properties of the gas.', 'cond-mat-9905173-2-1-3': 'Experimental studies have shown clustering [CITATION] and non-Gaussian velocity distributions [CITATION], but the dynamical origin of the velocity distributions remains unclear.', 'cond-mat-9905173-2-1-4': 'A model of a granular gas coupled to a thermal reservoir shows long-range correlations in density and velocity, and velocity distributions that fall off with [MATH] in the tails [CITATION].', 'cond-mat-9905173-2-1-5': 'A model with a discrete random forcing produces strong clustering, and a cross-correlation between the fluctuations in density and granular temperature (average kinetic energy) [CITATION].', 'cond-mat-9905173-2-1-6': 'The cross-correlation is due to the same mechanism as the clustering instability in a freely cooling granular gas: fluctuations of increased density result in more frequent inter-particle collisions, producing increased dissipation and a reduced local granular temperature.', 'cond-mat-9905173-2-2-0': 'Recent work has demonstrated various collective phenomena for a large number ([MATH]) of identical, uniform ball bearings constituting less than one layer coverage on a vertically shaken, horizontal plate [CITATION].', 'cond-mat-9905173-2-2-1': 'At peak plate accelerations above 1 g and for most densities, the particles behave as a rapidly fluctuating gas.', 'cond-mat-9905173-2-2-2': 'Reducing the acceleration amplitude decreases the mean square velocity, or granular temperature, T[MATH], and effectively "cools" the gas, leading to an observed increase in clustering.', 'cond-mat-9905173-2-2-3': 'Continued cooling eventually leads to the formation of a collapse, a condensate of motionless particles that remain in contact with the plate and each other.', 'cond-mat-9905173-2-2-4': 'In the range of 0.8 - 1.0 g, all of the velocity distributions of the gas appear to scale with the second moment of the distribution to an universal curve [CITATION].', 'cond-mat-9905173-2-2-5': 'The velocity distributions demonstrate strong deviations from a Gaussian distribution in both the high velocity tails and at low velocities.', 'cond-mat-9905173-2-3-0': 'This report presents the results of a further investigation into the nature of the non-Gaussian velocity distributions and their relation to the observed density fluctuations that stem from the inelastic collisions.', 'cond-mat-9905173-2-3-1': 'In order to understand the non-Gaussian velocity distributions seen previously in this system [CITATION], measurements were made over a larger range of [MATH], the dimensionless acceleration of the plate, where [MATH] is the frequency in Hz and g is the acceleration due to gravity.', 'cond-mat-9905173-2-3-2': 'Our results demonstrate both non-Gaussian velocity distributions and a cross-correlation between density and temperature, similar to that seen in the model system of Puglisi et al. [CITATION].', 'cond-mat-9905173-2-3-3': 'However, the data reported here clearly shows that the two effects are independent: there is a large range over which the velocity distributions are non-Gaussian but for which the granular temperature is not measurably dependent on the density.', 'cond-mat-9905173-2-3-4': 'Only at low [MATH], where the clustering is the strongest, is the cross-correlation between density and temperature observed in our experiment.', 'cond-mat-9905173-2-4-0': 'For these experiments, the plate vibration is sinusoidal, [MATH] 70 Hz and unless otherwise stated, the particle density is [MATH].', 'cond-mat-9905173-2-4-1': 'For accelerations below [MATH], the system was initially fluidized by shaking at [MATH].', 'cond-mat-9905173-2-4-2': 'Two different particle species were used: The small spheres were 302 stainless steel with an average diameter of 0.1191 cm [MATH] 0.00024 cm and the large spheres were 316 stainless with an average diameter of 0.1588 cm [MATH] 0.00032 cm.', 'cond-mat-9905173-2-4-3': 'The coefficient of restitution for both particle species is approximately 0.9 [CITATION].', 'cond-mat-9905173-2-5-0': 'The measured velocity distributions at [MATH], [MATH] and [MATH] are shown in Fig. [REF].', 'cond-mat-9905173-2-5-1': 'The distribution crosses over from one with approximately exponential tails as reported in [CITATION] to a Gaussian distribution.', 'cond-mat-9905173-2-5-2': 'This remarkable evolution is superficially similar to what is observed in freely cooling granular media, where an initial Gaussian velocity distribution becomes non-Gaussian as the system cools.', 'cond-mat-9905173-2-5-3': 'In that case, the evolution is determined by the strength of the inelasticity and the integrated number of collisions per particle, which does not have an obvious analog in the driven system.', 'cond-mat-9905173-2-6-0': 'As previously reported, the non-Gaussian velocity distributions observed at low accelerations are accompanied by clustering, as demonstrated by a dramatic increase in the structure of the pair correlation function.', 'cond-mat-9905173-2-6-1': 'Figure [REF](a) shows the pair correlation function determined via analysis of images taken from above the granular layer for the same accelerations as the velocity distributions in Fig. [REF].', 'cond-mat-9905173-2-6-2': 'The results of the measured correlation function approach that of an uncorrelated dilute hard sphere gas (solid line).', 'cond-mat-9905173-2-6-3': 'Thus, the crossover to Gaussian velocity distributions is accompanied by the disappearance of spatial correlations, consistent with the suggestions that the non-Gaussian velocity distributions arise from a coupling between density and temperature fluctuations [CITATION].', 'cond-mat-9905173-2-7-0': 'Increasing the steady state kinetic energy of the granular gas by increasing the amplitude of the acceleration at constant frequency causes the gas to change from primarily two-dimensional, where the particles never hop over one another, to essentially three-dimensional [CITATION].', 'cond-mat-9905173-2-7-1': 'This transition can be observed in the pair correlation function, [MATH], by the increase in its value for [MATH].', 'cond-mat-9905173-2-7-2': '(The correlation function includes only particle separations in the hortizontal plane.)', 'cond-mat-9905173-2-7-3': 'This transition can affect the dynamics in several ways: the effective density is decreased, so that excluded volume effects are less important; the inter-particle collisions can occur at angles closer to vertical, affecting transfer of energy and momentum from the vertical direction to the horizontal; and the change in the dimensionality itself can have important consequences.', 'cond-mat-9905173-2-7-4': 'In order to separate these effects from the direct consequences of increasing the kinetic energy of the gas, a plexiglass lid was added to the system at a height of 0.254 cm, or 1.6 ball diameters for the larger particles.', 'cond-mat-9905173-2-7-5': 'For this plate-to-lid separation, the larger particles cannot pass over top of one another, although enough room remains for collisions between particles at sufficiently different heights to transfer momentum from the vertical to the horizontal direction.', 'cond-mat-9905173-2-8-0': 'Figure [REF] (b) demonstrates the persistence of the particle-particle correlations when the system is constrained to 2D.', 'cond-mat-9905173-2-8-1': 'The particle-particle correlation function decreases slightly from [MATH] to [MATH], and then remains essentially constant up to [MATH].', 'cond-mat-9905173-2-8-2': 'The small value of [MATH] for distances less than one ball diameter indicates that the system remains 2D as [MATH] is increased.', 'cond-mat-9905173-2-8-3': 'The structure observed in the correlation function is essentially the same as that of an equilibrium elastic hard sphere gas at the same density, indicating that the correlations that exist are due to excluded volume effects.', 'cond-mat-9905173-2-9-0': 'Figure [REF](a) shows that the presence of the lid adds an energy sink to the system at high [MATH].', 'cond-mat-9905173-2-9-1': 'At low acceleration ([MATH]), very few, if any, particles strike the lid and the lid has no significant effect.', 'cond-mat-9905173-2-9-2': 'At larger [MATH] it is clear that the horizontal granular temperature is reduced as a significant number of particles strike the lid, dissipating energy.', 'cond-mat-9905173-2-9-3': 'It is interesting to note that T[MATH] approaches zero lineraly at finite [MATH], indicating that the relationship between the driving and the horizontal granular temperature is of the form T[MATH].', 'cond-mat-9905173-2-10-0': 'In order to demonstrate the effect of the lid on the velocity distributions, we use a simple quantitative measure of the non-Gaussian nature, the flatness (or kurtosis) of the distribution: [EQUATION]', 'cond-mat-9905173-2-10-1': 'For a Gaussian distribution, the flatness is 3 and for the broader exponential distribution, the flatness is 6.', 'cond-mat-9905173-2-10-2': 'In the absence of a lid, the flatness demonstrates a smooth transition from non-Gaussian to Gaussian behavior as the granular temperature is raised (Fig. [REF](b)) whether the smaller (circles) or larger (stars) particles are used.', 'cond-mat-9905173-2-10-3': 'With the lid on, the velocity distributions remain more non-Gaussian than in the free system for identical granular temperatures and density (diamonds).', 'cond-mat-9905173-2-10-4': 'The crossover from Gaussian to non-Gaussian behavior observed without the lid is therefore not simply an effect of increasing the vertical kinetic energy of the particles, but rather related to the transfer of energy from the vertical to horizontal motion in the system via collisions, the change in the density, or the change in dimensionality of the gas.', 'cond-mat-9905173-2-11-0': 'To determine the relative contribution of density changes to the non-Gaussian velocity distributions in the gas, the number of particles on the plate was increased by [MATH] and decreased by [MATH] from the value of [MATH] and the lid was kept on.', 'cond-mat-9905173-2-11-1': 'For all accelerations, the flatness decreased with increased density.', 'cond-mat-9905173-2-11-2': 'This surprising result may be related to the fact that strongly non-Gaussian distributions observed at low [MATH] are accompanied by strong clustering [CITATION].', 'cond-mat-9905173-2-11-3': 'If the average density is increased, the larger excluded volume means that less phase space remains for fluctuations to persist.', 'cond-mat-9905173-2-11-4': 'The fact that increasing the density with the lid on makes the velocity distribution more Gaussian suggests that the crossover to Gaussian observed without the lid is not due to the decrease in density of the gas.', 'cond-mat-9905173-2-12-0': 'Puglisi et al. [CITATION] have proposed a model which relates strong clustering to non-Gaussian velocity distributions in a driven granular medium.', 'cond-mat-9905173-2-12-1': 'In their framework, at each local density the velocity distributions are Gaussian, and the non-Gaussian behavior arises from the relative weighting of the temperature by local density in the following manner: [EQUATION] where [MATH] is the second moment of the distribution for the number of boxes, n, that contain N particles.', 'cond-mat-9905173-2-12-2': 'In this model, the local temperature is a decreasing function of the local density, and the velocity distributions conditioned on the local density are Gaussian.', 'cond-mat-9905173-2-13-0': 'In our experiment, this feature can be examined by conditioning the local horizontal granular temperature on the local density.', 'cond-mat-9905173-2-13-1': 'That is, examining the distribution of velocities for data at a constant number of particles in the frame of the camera in the strongly clustering regime at [MATH] [CITATION].', 'cond-mat-9905173-2-13-2': 'In the strongly clustering regime, we do observe a direct correlation between local density and temperature.', 'cond-mat-9905173-2-13-3': 'Figure [REF] is a plot of the local temperature as a function of particle number in the camera frame normalized by the granular temperature, including data from the open system for both the smaller and larger particles as well as for the confined system using the larger particles.', 'cond-mat-9905173-2-14-0': 'The result is similar to the model of Puglisi et al. [CITATION]: At low T[MATH], when the particle-particle correlations are strongest (and larger than those of an equilibrium hard sphere gas [CITATION]), there is a density dependence to the granular temperature (filled circles).', 'cond-mat-9905173-2-14-1': 'At [MATH], where all of the particles are essentially uncorrelated in a 3D volume in the absence of a lid, there is no density dependence (open circles, stars).', 'cond-mat-9905173-2-14-2': 'However, even in the confined system at [MATH], where the distribution is still not Gaussian, no appreciable density dependence is observed (diamonds), suggesting that the non-Gaussian velocity distributions and density-dependent temperature are not as simply dependent upon one another as they are in the model of Puglisi et al. [CITATION].', 'cond-mat-9905173-2-14-3': 'In fact, while there is a clear density dependence on the local temperature at low [MATH], the measured velocity distribution conditioned on the local temperature is not Gaussian.', 'cond-mat-9905173-2-14-4': 'At each density, the velocity distribution function is almost identical to that of the whole: when the entire distribution is non-Gaussian, the distribution at a single density is non-Gaussian, and when the whole is Gaussian, each conditional velocity distribution is Gaussian.', 'cond-mat-9905173-2-15-0': 'A more general form of Eq. [REF] represents the total velocity distribution as a product of local Gaussian velocity distributions with a distribution of local temperatures:', 'cond-mat-9905173-2-16-0': '[EQUATION] where [MATH] is the local temperature that is varying in space and time.', 'cond-mat-9905173-2-16-1': 'Conditioning on the local temperature would then recover the underlying Maxwell statistics in the fluctuations [CITATION].', 'cond-mat-9905173-2-17-0': 'Performing this analysis on our data does not succeed in producing Gaussian statistics.', 'cond-mat-9905173-2-17-1': 'Within small windows of local temperature, the distributions remain non-Gaussian.', 'cond-mat-9905173-2-17-2': 'Indeed, the analysis can be extended to condition on both the local temperature and density in the system, but with similarly limited success except for the slowest of particles in the most dilute regions of the system, although all of the conditioned distributions are closer to Gaussian than the full distribution.', 'cond-mat-9905173-2-17-3': 'Curiously, if the local velocities are normalized by the magnitude of the local granular temperature, then the total distribution is Gaussian [CITATION].', 'cond-mat-9905173-2-18-0': 'These results demonstrate several important new characteristics of a 2D granular gas.', 'cond-mat-9905173-2-18-1': 'Extreme non-Gaussian velocity distributions observed previously [CITATION] involve strong cross-correlations between density and velocity fluctuations.', 'cond-mat-9905173-2-18-2': 'This is not the whole picture, however, as a 2D constrained granular gas also demonstrates strongly non-Gaussian behavior without a significant cross-correlation between density and velocity distributions.', 'cond-mat-9905173-2-18-3': 'Finally, the crossover to Gaussian distributions observed when the gas is fully 3D is probably associated with the change in dimensionality or momentum transfer, rather than the increase in kinetic energy or the decrease in density.', 'cond-mat-9905173-2-19-0': 'This work was supported by an award from the Research Corporation, a grant from the Petroleum Research Fund and grant DMR-9875529 from the NSF.', 'cond-mat-9905173-2-19-1': 'One of us (JSU) was supported by a fellowship from the Sloan Foundation.'}

[['cond-mat-9905173-1-11-0', 'cond-mat-9905173-2-11-0'], ['cond-mat-9905173-1-11-1', 'cond-mat-9905173-2-11-1'], ['cond-mat-9905173-1-11-2', 'cond-mat-9905173-2-11-2'], ['cond-mat-9905173-1-11-3', 'cond-mat-9905173-2-11-3'], ['cond-mat-9905173-1-11-4', 'cond-mat-9905173-2-11-4'], ['cond-mat-9905173-1-14-0', 'cond-mat-9905173-2-14-0'], ['cond-mat-9905173-1-14-1', 'cond-mat-9905173-2-14-1'], ['cond-mat-9905173-1-14-2', 'cond-mat-9905173-2-14-2'], ['cond-mat-9905173-1-14-3', 'cond-mat-9905173-2-14-3'], ['cond-mat-9905173-1-14-4', 'cond-mat-9905173-2-14-4'], ['cond-mat-9905173-1-12-0', 'cond-mat-9905173-2-12-0'], ['cond-mat-9905173-1-12-1', 'cond-mat-9905173-2-12-1'], ['cond-mat-9905173-1-12-2', 'cond-mat-9905173-2-12-2'], ['cond-mat-9905173-1-6-0', 'cond-mat-9905173-2-6-0'], ['cond-mat-9905173-1-6-1', 'cond-mat-9905173-2-6-1'], ['cond-mat-9905173-1-6-2', 'cond-mat-9905173-2-6-2'], ['cond-mat-9905173-1-6-3', 'cond-mat-9905173-2-6-3'], ['cond-mat-9905173-1-2-0', 'cond-mat-9905173-2-2-0'], ['cond-mat-9905173-1-2-1', 'cond-mat-9905173-2-2-1'], ['cond-mat-9905173-1-2-2', 'cond-mat-9905173-2-2-2'], ['cond-mat-9905173-1-2-3', 'cond-mat-9905173-2-2-3'], ['cond-mat-9905173-1-2-4', 'cond-mat-9905173-2-2-4'], ['cond-mat-9905173-1-2-5', 'cond-mat-9905173-2-2-5'], ['cond-mat-9905173-1-0-0', 'cond-mat-9905173-2-0-0'], ['cond-mat-9905173-1-0-1', 'cond-mat-9905173-2-0-1'], ['cond-mat-9905173-1-0-2', 'cond-mat-9905173-2-0-2'], ['cond-mat-9905173-1-0-3', 'cond-mat-9905173-2-0-3'], ['cond-mat-9905173-1-18-0', 'cond-mat-9905173-2-18-0'], ['cond-mat-9905173-1-18-1', 'cond-mat-9905173-2-18-1'], ['cond-mat-9905173-1-18-2', 'cond-mat-9905173-2-18-2'], ['cond-mat-9905173-1-18-3', 'cond-mat-9905173-2-18-3'], ['cond-mat-9905173-1-13-0', 'cond-mat-9905173-2-13-0'], ['cond-mat-9905173-1-13-1', 'cond-mat-9905173-2-13-1'], ['cond-mat-9905173-1-13-2', 'cond-mat-9905173-2-13-2'], ['cond-mat-9905173-1-13-3', 'cond-mat-9905173-2-13-3'], ['cond-mat-9905173-1-4-0', 'cond-mat-9905173-2-4-0'], ['cond-mat-9905173-1-4-1', 'cond-mat-9905173-2-4-1'], ['cond-mat-9905173-1-4-2', 'cond-mat-9905173-2-4-2'], ['cond-mat-9905173-1-4-3', 'cond-mat-9905173-2-4-3'], ['cond-mat-9905173-1-5-0', 'cond-mat-9905173-2-5-0'], ['cond-mat-9905173-1-5-1', 'cond-mat-9905173-2-5-1'], ['cond-mat-9905173-1-5-2', 'cond-mat-9905173-2-5-2'], ['cond-mat-9905173-1-5-3', 'cond-mat-9905173-2-5-3'], ['cond-mat-9905173-1-16-0', 'cond-mat-9905173-2-16-0'], ['cond-mat-9905173-1-16-1', 'cond-mat-9905173-2-16-1'], ['cond-mat-9905173-1-7-0', 'cond-mat-9905173-2-7-0'], ['cond-mat-9905173-1-7-1', 'cond-mat-9905173-2-7-1'], ['cond-mat-9905173-1-7-2', 'cond-mat-9905173-2-7-2'], ['cond-mat-9905173-1-7-3', 'cond-mat-9905173-2-7-3'], ['cond-mat-9905173-1-7-4', 'cond-mat-9905173-2-7-4'], ['cond-mat-9905173-1-7-5', 'cond-mat-9905173-2-7-5'], ['cond-mat-9905173-1-3-0', 'cond-mat-9905173-2-3-0'], ['cond-mat-9905173-1-3-1', 'cond-mat-9905173-2-3-1'], ['cond-mat-9905173-1-3-2', 'cond-mat-9905173-2-3-2'], ['cond-mat-9905173-1-3-3', 'cond-mat-9905173-2-3-3'], ['cond-mat-9905173-1-3-4', 'cond-mat-9905173-2-3-4'], ['cond-mat-9905173-1-9-0', 'cond-mat-9905173-2-9-0'], ['cond-mat-9905173-1-9-1', 'cond-mat-9905173-2-9-1'], ['cond-mat-9905173-1-9-2', 'cond-mat-9905173-2-9-2'], ['cond-mat-9905173-1-9-3', 'cond-mat-9905173-2-9-3'], ['cond-mat-9905173-1-10-0', 'cond-mat-9905173-2-10-0'], ['cond-mat-9905173-1-10-1', 'cond-mat-9905173-2-10-1'], ['cond-mat-9905173-1-10-2', 'cond-mat-9905173-2-10-2'], ['cond-mat-9905173-1-10-3', 'cond-mat-9905173-2-10-3'], ['cond-mat-9905173-1-10-4', 'cond-mat-9905173-2-10-4'], ['cond-mat-9905173-1-17-0', 'cond-mat-9905173-2-17-0'], ['cond-mat-9905173-1-17-1', 'cond-mat-9905173-2-17-1'], ['cond-mat-9905173-1-17-2', 'cond-mat-9905173-2-17-2'], ['cond-mat-9905173-1-17-3', 'cond-mat-9905173-2-17-3'], ['cond-mat-9905173-1-19-0', 'cond-mat-9905173-2-19-0'], ['cond-mat-9905173-1-19-1', 'cond-mat-9905173-2-19-1'], ['cond-mat-9905173-1-1-0', 'cond-mat-9905173-2-1-0'], ['cond-mat-9905173-1-1-1', 'cond-mat-9905173-2-1-1'], ['cond-mat-9905173-1-1-2', 'cond-mat-9905173-2-1-2'], ['cond-mat-9905173-1-1-3', 'cond-mat-9905173-2-1-3'], ['cond-mat-9905173-1-1-4', 'cond-mat-9905173-2-1-4'], ['cond-mat-9905173-1-1-5', 'cond-mat-9905173-2-1-5'], ['cond-mat-9905173-1-1-6', 'cond-mat-9905173-2-1-6'], ['cond-mat-9905173-1-8-0', 'cond-mat-9905173-2-8-0'], ['cond-mat-9905173-1-8-1', 'cond-mat-9905173-2-8-1'], ['cond-mat-9905173-1-8-2', 'cond-mat-9905173-2-8-2'], ['cond-mat-9905173-1-8-3', 'cond-mat-9905173-2-8-3']]

[['cond-mat-9905173-1-11-0', 'cond-mat-9905173-2-11-0'], ['cond-mat-9905173-1-11-1', 'cond-mat-9905173-2-11-1'], ['cond-mat-9905173-1-11-2', 'cond-mat-9905173-2-11-2'], ['cond-mat-9905173-1-11-3', 'cond-mat-9905173-2-11-3'], ['cond-mat-9905173-1-11-4', 'cond-mat-9905173-2-11-4'], ['cond-mat-9905173-1-14-0', 'cond-mat-9905173-2-14-0'], ['cond-mat-9905173-1-14-1', 'cond-mat-9905173-2-14-1'], ['cond-mat-9905173-1-14-2', 'cond-mat-9905173-2-14-2'], ['cond-mat-9905173-1-14-3', 'cond-mat-9905173-2-14-3'], ['cond-mat-9905173-1-14-4', 'cond-mat-9905173-2-14-4'], ['cond-mat-9905173-1-12-0', 'cond-mat-9905173-2-12-0'], ['cond-mat-9905173-1-12-1', 'cond-mat-9905173-2-12-1'], ['cond-mat-9905173-1-12-2', 'cond-mat-9905173-2-12-2'], ['cond-mat-9905173-1-6-0', 'cond-mat-9905173-2-6-0'], ['cond-mat-9905173-1-6-1', 'cond-mat-9905173-2-6-1'], ['cond-mat-9905173-1-6-2', 'cond-mat-9905173-2-6-2'], ['cond-mat-9905173-1-6-3', 'cond-mat-9905173-2-6-3'], ['cond-mat-9905173-1-2-0', 'cond-mat-9905173-2-2-0'], ['cond-mat-9905173-1-2-1', 'cond-mat-9905173-2-2-1'], ['cond-mat-9905173-1-2-2', 'cond-mat-9905173-2-2-2'], ['cond-mat-9905173-1-2-3', 'cond-mat-9905173-2-2-3'], ['cond-mat-9905173-1-2-4', 'cond-mat-9905173-2-2-4'], ['cond-mat-9905173-1-2-5', 'cond-mat-9905173-2-2-5'], ['cond-mat-9905173-1-0-0', 'cond-mat-9905173-2-0-0'], ['cond-mat-9905173-1-0-1', 'cond-mat-9905173-2-0-1'], ['cond-mat-9905173-1-0-2', 'cond-mat-9905173-2-0-2'], ['cond-mat-9905173-1-0-3', 'cond-mat-9905173-2-0-3'], ['cond-mat-9905173-1-18-0', 'cond-mat-9905173-2-18-0'], ['cond-mat-9905173-1-18-1', 'cond-mat-9905173-2-18-1'], ['cond-mat-9905173-1-18-2', 'cond-mat-9905173-2-18-2'], ['cond-mat-9905173-1-18-3', 'cond-mat-9905173-2-18-3'], ['cond-mat-9905173-1-13-0', 'cond-mat-9905173-2-13-0'], ['cond-mat-9905173-1-13-1', 'cond-mat-9905173-2-13-1'], ['cond-mat-9905173-1-13-2', 'cond-mat-9905173-2-13-2'], ['cond-mat-9905173-1-13-3', 'cond-mat-9905173-2-13-3'], ['cond-mat-9905173-1-4-0', 'cond-mat-9905173-2-4-0'], ['cond-mat-9905173-1-4-1', 'cond-mat-9905173-2-4-1'], ['cond-mat-9905173-1-4-2', 'cond-mat-9905173-2-4-2'], ['cond-mat-9905173-1-4-3', 'cond-mat-9905173-2-4-3'], ['cond-mat-9905173-1-5-0', 'cond-mat-9905173-2-5-0'], ['cond-mat-9905173-1-5-1', 'cond-mat-9905173-2-5-1'], ['cond-mat-9905173-1-5-2', 'cond-mat-9905173-2-5-2'], ['cond-mat-9905173-1-5-3', 'cond-mat-9905173-2-5-3'], ['cond-mat-9905173-1-16-0', 'cond-mat-9905173-2-16-0'], ['cond-mat-9905173-1-16-1', 'cond-mat-9905173-2-16-1'], ['cond-mat-9905173-1-7-0', 'cond-mat-9905173-2-7-0'], ['cond-mat-9905173-1-7-1', 'cond-mat-9905173-2-7-1'], ['cond-mat-9905173-1-7-2', 'cond-mat-9905173-2-7-2'], ['cond-mat-9905173-1-7-3', 'cond-mat-9905173-2-7-3'], ['cond-mat-9905173-1-7-4', 'cond-mat-9905173-2-7-4'], ['cond-mat-9905173-1-7-5', 'cond-mat-9905173-2-7-5'], ['cond-mat-9905173-1-3-0', 'cond-mat-9905173-2-3-0'], ['cond-mat-9905173-1-3-1', 'cond-mat-9905173-2-3-1'], ['cond-mat-9905173-1-3-2', 'cond-mat-9905173-2-3-2'], ['cond-mat-9905173-1-3-3', 'cond-mat-9905173-2-3-3'], ['cond-mat-9905173-1-3-4', 'cond-mat-9905173-2-3-4'], ['cond-mat-9905173-1-9-0', 'cond-mat-9905173-2-9-0'], ['cond-mat-9905173-1-9-1', 'cond-mat-9905173-2-9-1'], ['cond-mat-9905173-1-9-2', 'cond-mat-9905173-2-9-2'], ['cond-mat-9905173-1-9-3', 'cond-mat-9905173-2-9-3'], ['cond-mat-9905173-1-10-0', 'cond-mat-9905173-2-10-0'], ['cond-mat-9905173-1-10-1', 'cond-mat-9905173-2-10-1'], ['cond-mat-9905173-1-10-2', 'cond-mat-9905173-2-10-2'], ['cond-mat-9905173-1-10-3', 'cond-mat-9905173-2-10-3'], ['cond-mat-9905173-1-10-4', 'cond-mat-9905173-2-10-4'], ['cond-mat-9905173-1-17-0', 'cond-mat-9905173-2-17-0'], ['cond-mat-9905173-1-17-1', 'cond-mat-9905173-2-17-1'], ['cond-mat-9905173-1-17-2', 'cond-mat-9905173-2-17-2'], ['cond-mat-9905173-1-17-3', 'cond-mat-9905173-2-17-3'], ['cond-mat-9905173-1-19-0', 'cond-mat-9905173-2-19-0'], ['cond-mat-9905173-1-19-1', 'cond-mat-9905173-2-19-1'], ['cond-mat-9905173-1-1-0', 'cond-mat-9905173-2-1-0'], ['cond-mat-9905173-1-1-1', 'cond-mat-9905173-2-1-1'], ['cond-mat-9905173-1-1-2', 'cond-mat-9905173-2-1-2'], ['cond-mat-9905173-1-1-3', 'cond-mat-9905173-2-1-3'], ['cond-mat-9905173-1-1-4', 'cond-mat-9905173-2-1-4'], ['cond-mat-9905173-1-1-5', 'cond-mat-9905173-2-1-5'], ['cond-mat-9905173-1-1-6', 'cond-mat-9905173-2-1-6'], ['cond-mat-9905173-1-8-0', 'cond-mat-9905173-2-8-0'], ['cond-mat-9905173-1-8-1', 'cond-mat-9905173-2-8-1'], ['cond-mat-9905173-1-8-2', 'cond-mat-9905173-2-8-2'], ['cond-mat-9905173-1-8-3', 'cond-mat-9905173-2-8-3']]

[]

[]

[]

[]

['cond-mat-9905173-1-15-0', 'cond-mat-9905173-2-15-0']

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

https://arxiv.org/abs/cond-mat/9905173

hep-ex-0412078

{'hep-ex-0412078-1-0-0': 'Polarized antiprotons produced by spin filtering with an internal polarized gas target provide access to a wealth of single- and double-spin observables, thereby opening a window to physics uniquely accessible with the HESR at FAIR.', 'hep-ex-0412078-1-0-1': 'This includes a first measurement of the transversity distribution of the valence quarks in the proton, a test of the predicted opposite sign of the Sivers-function, related to the quark distribution inside a transversely polarized nucleon, in Drell-Yan (DY) as compared to semi-inclusive DIS, and a first measurement of the moduli and the relative phase of the time-like electric and magnetic form factors [MATH] of the proton.', 'hep-ex-0412078-1-0-2': 'In polarized and unpolarized [MATH] elastic scattering open questions like the contribution from the odd charge-symmetry Landshoff-mechanism at large [MATH] and spin-effects in the extraction of the forward scattering amplitude at low [MATH] can be addressed.', 'hep-ex-0412078-1-1-0': '# Physics Case', 'hep-ex-0412078-1-2-0': 'The polarized antiproton-proton interactions at the High Energy Storage Ring (HESR) at the future Facility for Antiproton and Ion Research (FAIR) will provide unique access to a number of new fundamental physics observables, which can be studied neither at other facilities nor at HESR without transverse polarization of protons and antiprotons.', 'hep-ex-0412078-1-3-0': '## The transversity distribution is the last leading-twist missing piece of the QCD description of the partonic structure of the nucleon.', 'hep-ex-0412078-1-3-1': 'It describes the quark transverse polarization inside a transversely polarized proton [CITATION].', 'hep-ex-0412078-1-3-2': 'Unlike the more conventional unpolarized quark distribution [MATH] and the helicity distribution [MATH], the transversity [MATH] can neither be accessed in deep-inelastic scattering of leptons off nucleons nor can it be reconstructed from the knowledge of [MATH] and [MATH].', 'hep-ex-0412078-1-3-3': 'It may contribute to some single-spin observables, but always coupled to other unknown functions.', 'hep-ex-0412078-1-3-4': 'The transversity distribution is directly accessible uniquely via the double transverse spin asymmetry [MATH] in the Drell-Yan production of lepton pairs.', 'hep-ex-0412078-1-3-5': 'The theoretical expectations for [MATH] in the Drell-Yan process with transversely polarized antiprotons interacting with a transversely polarized proton target at HESR are in the 0.3-0.4 range [CITATION]; with the expected beam polarization achieved using a dedicated low-energy antiproton polarizer ring (AP) of [MATH] and the luminosity of HESR, the PAX experiment is uniquely suited for the definitive observation of [MATH] of the proton for the valence quarks.', 'hep-ex-0412078-1-3-6': 'The determination of [MATH] will open new pathways to the QCD interpretation of single-spin asymmetry (SSA) measurements.', 'hep-ex-0412078-1-3-7': 'In conjunction with the data on SSA from the HERMES collaboration [CITATION], the PAX measurements of the SSA in Drell-Yan production on polarized protons can for the first time provide a test of the theoretical prediction [CITATION] of the reversal of the sign of the Sivers function [CITATION] from semi-inclusive DIS to Drell-Yan production.', 'hep-ex-0412078-1-4-0': '## Magnetic and electric form factors The origin of the unexpected [MATH]-dependence of the ratio of the magnetic and electric form factors of the proton as observed at the Jefferson laboratory [CITATION] can be clarified by a measurement of their relative phase in the time-like region, which discriminates strongly between the models for the form factor.', 'hep-ex-0412078-1-4-1': 'This phase can only be measured via SSA in the annihilation [MATH] on a transversely polarized target [CITATION].', 'hep-ex-0412078-1-4-2': 'The first ever measurement of this phase at PAX will also contribute to the understanding of the onset of the pQCD asymptotics in the time-like region and will serve as a stringent test of dispersion theory approaches to the relationship between the space-like and time-like form factors [CITATION].', 'hep-ex-0412078-1-4-3': 'The double-spin asymmetry will allow independently the [MATH] separation and serve as a check of the Rosenbluth separation in the time-like region which has not been carried out so far.', 'hep-ex-0412078-1-5-0': '## Hard scattering Arguably, in [MATH] elastic scattering the hard scattering mechanism can be checked beyond [MATH] accessible in the [MATH]-[MATH]-symmetric [MATH] scattering, because in the [MATH] case the [MATH]-channel exchange contribution can only originate from the strongly suppressed exotic dibaryon exchange.', 'hep-ex-0412078-1-5-1': 'Consequently, in the [MATH] case the hard mechanisms [CITATION] can be tested at [MATH] almost twice as large as in [MATH] scattering.', 'hep-ex-0412078-1-5-2': 'Even unpolarized large angle [MATH] scattering data can shed light on the origin of the intriguing oscillations around the [MATH] behavior of the [MATH] scattering cross section in the [MATH] channel and put stringent constraints on the much disputed odd-charge conjugation Landshoff mechanism [CITATION].', 'hep-ex-0412078-1-5-3': 'If the Landshoff mechanism is suppressed then the double transverse asymmetry in [MATH] scattering is expected to be as large as the one observed in the [MATH] case.', 'hep-ex-0412078-1-6-0': '# Towards an asymmetric polarized antiproton-proton collider at FAIR', 'hep-ex-0412078-1-7-0': 'The possibility to test the nucleon structure via double spin asymmetries in polarized proton-antiproton reactions at the HESR ring of FAIR at GSI has been suggested by the PAX collaboration in 2004 [CITATION].', 'hep-ex-0412078-1-7-1': 'Since then, there has been much progress, both in understanding the physics potential of such an experiment [CITATION] and in studying the feasibility of efficiently producing polarized antiprotons [CITATION].', 'hep-ex-0412078-1-7-2': 'The physics program of such a facility would extend to a new domain the exceptionally fruitful studies of the nucleon structure performed in unpolarized and polarized deep inelastic scattering (DIS), which have been at the center of high energy physics during the past four decades.', 'hep-ex-0412078-1-7-3': 'As mentioned earlier, a direct measurement of the transversity distribution function [MATH], one of the last missing fundamental pieces in the QCD description of the nucleon, is unique.', 'hep-ex-0412078-1-7-4': 'In the available kinematic domain of the proposed experiment, which covers the valence region, the Drell-Yan double transverse spin asymmetry was recently predicted to be as large as 0.3 [CITATION].', 'hep-ex-0412078-1-7-5': 'Other novel tests of QCD at such a facility include the polarized elastic hard scattering of antiprotons on protons and the measurement of the phases of the time-like form factors of the proton (see Ref. [CITATION]).', 'hep-ex-0412078-1-7-6': 'A viable practical scheme which allows us to reach a polarization of the stored antiprotons at HESR-FAIR of [MATH] has been worked out and published in Ref. [CITATION].', 'hep-ex-0412078-1-8-0': 'The PAX Letter-of-Intent was submitted on January 15, 2004.', 'hep-ex-0412078-1-8-1': 'The physics program of PAX has been positively reviewed by the QCD Program Advisory Committee (PAC) on May 14-16, 2004 [CITATION].', 'hep-ex-0412078-1-8-2': 'The proposal by the ASSIA collaboration [CITATION] to utilize a polarized solid target and bombard it with a 45 GeV unpolarized antiproton beam extracted from the synchrotron SIS100 has been rejected by the GSI management.', 'hep-ex-0412078-1-8-3': 'Such measurements would not allow one to determine [MATH], because in single spin measurements [MATH] appears always coupled to another unknown fragmentation function.', 'hep-ex-0412078-1-8-4': 'Following the QCD-PAC report and the recommendation of the Chairman of the committee on Scientific and Technological Issues (STI) [CITATION] and the FAIR project coordinator, the PAX collaboration has optimized the technique to achieve a sizable antiproton polarization and the proposal for experiments at GSI with polarized antiprotons [CITATION].', 'hep-ex-0412078-1-8-5': 'From various working group meetings of the PAX collaboration, presented in part in 2004 at several workshops and conferences [CITATION], we conclude:', 'hep-ex-0412078-1-9-0': 'The PAX collaboration proposes an approach that is composed of two phases.', 'hep-ex-0412078-1-9-1': 'During these the major milestones of the project can be tested and optimized before the final goal is approached: An asymmetric proton-antiproton collider, in which polarized protons with momenta of about 3.5 GeV/c collide with polarized antiprotons with momenta up to 15 GeV/c.', 'hep-ex-0412078-1-9-2': 'These circulate in the HESR, which has already been approved and will serve the PANDA experiment.', 'hep-ex-0412078-1-9-3': 'In the following, we will briefly describe the overall machine setup of the HESR complex, schematically depicted in Fig. [REF].', 'hep-ex-0412078-1-10-0': 'Let us summarize the main features of the accelerator setup:', 'hep-ex-0412078-1-11-0': 'It is worthwhile to stress that, through the employment of the CSR, effectively a second interaction point is formed with minimum interference with PANDA.', 'hep-ex-0412078-1-11-1': 'The proposed solution opens the possibility to run two different experiments at the same time.', 'hep-ex-0412078-1-12-0': 'In the following sections, we discuss the physics program, which should be pursued in two different phases.', 'hep-ex-0412078-1-13-0': '## Phase I', 'hep-ex-0412078-1-14-0': 'A beam of unpolarized or polarized antiprotons with momentum up to 3.5 GeV/c in the CSR ring, colliding on a polarized hydrogen target in the PAX detector.', 'hep-ex-0412078-1-14-1': 'This phase is independent of the HESR performance.', 'hep-ex-0412078-1-15-0': 'This first phase, at moderately high energy, will allow for the first time the measurement of the time-like proton form factors in single and double polarized [MATH] interactions in a wide kinematical range, from close to threshold up to [MATH] GeV[MATH].', 'hep-ex-0412078-1-15-1': 'It would enable to determine several double spin asymmetries in elastic [MATH] scattering.', 'hep-ex-0412078-1-15-2': 'By detecting back scattered antiprotons one can also explore hard scattering regions of large [MATH]: In proton-proton scattering the same region of [MATH] requires twice the energy.', 'hep-ex-0412078-1-15-3': 'There are no competing facilities at which these topical issues can be addressed.', 'hep-ex-0412078-1-15-4': 'For the theoretical background, see the PAX LoI [CITATION] and the recent review paper [CITATION].', 'hep-ex-0412078-1-16-0': '## Phase II', 'hep-ex-0412078-1-17-0': 'This phase will allow the first ever direct measurement of the quark transversity distribution [MATH], by measuring the double transverse spin asymmetry [MATH] in Drell-Yan processes [MATH] as a function of Bjorken [MATH] and [MATH]) [EQUATION] where [MATH], [MATH] is the invariant mass of the lepton pair and [MATH], of the order of one, is the calculable double-spin asymmetry of the QED elementary process [MATH].', 'hep-ex-0412078-1-17-1': 'Two possible scenarios might be foreseen to perform the measurement, which are discussed below.', 'hep-ex-0412078-1-18-0': '### Asymmetric collider', 'hep-ex-0412078-1-19-0': 'A beam of polarized antiprotons from 1.5 GeV/c up to 15 GeV/c circulating in the HESR, colliding on a beam of polarized protons with momenta up to 3.5 GeV/c circulating in the CSR.', 'hep-ex-0412078-1-19-1': 'This scenario however requires to demonstrate that a suitable luminosity is reachable.', 'hep-ex-0412078-1-19-2': 'Deflection of the HESR beam to the PAX detector in the CSR is necessary (see Fig. [REF]).', 'hep-ex-0412078-1-20-0': 'By proper variation of the energy of the two colliding beams, this setup would allow a measurement of the transversity distribution [MATH] in the valence region of [MATH], with corresponding [MATH] (see Fig. [REF]).', 'hep-ex-0412078-1-20-1': '[MATH] is predicted to be larger than 0.3 over the full kinematic range, up to the highest reachable center-of-mass energy of [MATH].', 'hep-ex-0412078-1-20-2': 'The cross section is large as well: With a luminosity of [MATH] cm[MATH] about [MATH] events per day can be expected.', 'hep-ex-0412078-1-20-3': 'For the transversity distribution [MATH], such an experiment can be considered as the analogue of polarized DIS for the determination of the helicity structure function [MATH], i.e. of the helicity distribution [MATH]; the kinematical coverage [MATH] will be similar to that of the HERMES experiment.', 'hep-ex-0412078-1-21-0': '### High luminosity fixed target experiment', 'hep-ex-0412078-1-22-0': 'If the required luminosity in the collider mode is not achievable, a fixed target experiment can be conducted.', 'hep-ex-0412078-1-22-1': 'A beam of 22 GeV/c (15 GeV/c) polarized antiprotons circulating in the HESR is used to collide with a polarized internal hydrogen target.', 'hep-ex-0412078-1-22-2': 'Also this scenario requires the deflection of the HESR beam to the PAX detector in the CSR (see Fig. [REF]).', 'hep-ex-0412078-1-23-0': 'A theoretical discussion of the significance of the measurement of [MATH] for a 22 GeV/c (15 GeV/c) beam impinging on a fixed target is given in Refs. [CITATION] and the recent review paper [CITATION].', 'hep-ex-0412078-1-23-1': 'The theoretical work on the [MATH]-factors for the transversity determination is in progress [CITATION].', 'hep-ex-0412078-1-23-2': 'This measurement will explore the valence region of [MATH], with corresponding [MATH] (see Fig. [REF]).', 'hep-ex-0412078-1-23-3': 'In this region [MATH] is predicted to be large (of the order of 0.3, or more) and the expected number of events can be of the order of 2000 per day.', 'hep-ex-0412078-1-24-0': 'We would like to mention, that we are also investigating whether the PANDA detector, properly modified, is compatible with the transversity measurements in the collider mode, where an efficient identification of the Drell-Yan pairs is required.', 'hep-ex-0412078-1-24-1': 'At the interaction point, the spins of the colliding protons and antiprotons should be vertical, with no significant component along the beam direction.', 'hep-ex-0412078-1-25-0': '# Conclusion', 'hep-ex-0412078-1-26-0': 'To summarize, we note that the storage of polarized antiprotons at HESR will open unique possibilities to test QCD in hitherto unexplored domains.', 'hep-ex-0412078-1-26-1': 'This will provide another cornerstone to the antiproton program at FAIR.'}

{'hep-ex-0412078-2-0-0': 'Polarized antiprotons produced by spin filtering with an internal polarized gas target provide access to a wealth of single- and double-spin observables, thereby opening a window to physics uniquely accessible with the HESR at FAIR.', 'hep-ex-0412078-2-0-1': 'This includes a first measurement of the transversity distribution of the valence quarks in the proton, a test of the predicted opposite sign of the Sivers-function, related to the quark distribution inside a transversely polarized nucleon, in Drell-Yan (DY) as compared to semi-inclusive DIS, and a first measurement of the moduli and the relative phase of the time-like electric and magnetic form factors [MATH] of the proton.', 'hep-ex-0412078-2-0-2': 'In polarized and unpolarized [MATH] elastic scattering open questions like the contribution from the odd charge-symmetry Landshoff-mechanism at large [MATH] and spin-effects in the extraction of the forward scattering amplitude at low [MATH] can be addressed.', 'hep-ex-0412078-2-1-0': '# Physics Case', 'hep-ex-0412078-2-2-0': 'The polarized antiproton-proton interactions at the High Energy Storage Ring (HESR) at the future Facility for Antiproton and Ion Research (FAIR) will provide unique access to a number of new fundamental physics observables, which can be studied neither at other facilities nor at HESR without transverse polarization of protons and antiprotons.', 'hep-ex-0412078-2-3-0': '## The transversity distribution is the last leading-twist missing piece of the QCD description of the partonic structure of the nucleon.', 'hep-ex-0412078-2-3-1': 'It describes the quark transverse polarization inside a transversely polarized proton [CITATION].', 'hep-ex-0412078-2-3-2': 'Unlike the more conventional unpolarized quark distribution [MATH] and the helicity distribution [MATH], the transversity [MATH] can neither be accessed in inclusive deep-inelastic scattering of leptons off nucleons nor can it be reconstructed from the knowledge of [MATH] and [MATH].', 'hep-ex-0412078-2-3-3': 'It may contribute to some single-spin observables, but always coupled to other unknown functions.', 'hep-ex-0412078-2-3-4': 'The transversity distribution is directly accessible uniquely via the double transverse spin asymmetry [MATH] in the Drell-Yan production of lepton pairs.', 'hep-ex-0412078-2-3-5': 'The theoretical expectations for [MATH] in the Drell-Yan process with transversely polarized antiprotons interacting with a transversely polarized proton target at HESR are in the 0.3-0.4 range [CITATION]; with the expected beam polarization achieved using a dedicated low-energy antiproton polarizer ring (AP) of [MATH] and the luminosity of HESR, the PAX experiment is uniquely suited for the definitive observation of [MATH] of the proton for the valence quarks.', 'hep-ex-0412078-2-3-6': 'The determination of [MATH] will open new pathways to the QCD interpretation of single-spin asymmetry (SSA) measurements.', 'hep-ex-0412078-2-3-7': 'In conjunction with the data on SSA from the HERMES collaboration [CITATION], the PAX measurements of the SSA in Drell-Yan production on polarized protons can for the first time provide a test of the theoretical prediction [CITATION] of the reversal of the sign of the Sivers function [CITATION] from semi-inclusive DIS to Drell-Yan production.', 'hep-ex-0412078-2-4-0': '## Magnetic and electric form factors The origin of the unexpected [MATH]-dependence of the ratio of the magnetic and electric form factors of the proton as observed at the Jefferson laboratory [CITATION] can be clarified by a measurement of their relative phase in the time-like region, which discriminates strongly between the models for the form factor.', 'hep-ex-0412078-2-4-1': 'This phase can be measured via SSA in the annihilation [MATH] on a transversely polarized target [CITATION].', 'hep-ex-0412078-2-4-2': 'The first ever measurement of this phase at PAX will also contribute to the understanding of the onset of the pQCD asymptotics in the time-like region and will serve as a stringent test of dispersion theory approaches to the relationship between the space-like and time-like form factors [CITATION].', 'hep-ex-0412078-2-4-3': 'The double-spin asymmetry will allow independently the [MATH] separation and serve as a check of the Rosenbluth separation in the time-like region which has not been carried out so far.', 'hep-ex-0412078-2-5-0': '## Hard scattering Arguably, in [MATH] elastic scattering the hard scattering mechanism can be checked beyond [MATH] accessible in the [MATH]-[MATH]-symmetric [MATH] scattering, because in the [MATH] case the [MATH]-channel exchange contribution can only originate from the strongly suppressed exotic dibaryon exchange.', 'hep-ex-0412078-2-5-1': 'Consequently, in the [MATH] case the hard mechanisms [CITATION] can be tested at [MATH] almost twice as large as in [MATH] scattering.', 'hep-ex-0412078-2-5-2': 'Even unpolarized large angle [MATH] scattering data can shed light on the origin of the intriguing oscillations around the [MATH] behavior of the [MATH] scattering cross section in the [MATH] channel and put stringent constraints on the much disputed odd-charge conjugation Landshoff mechanism [CITATION].', 'hep-ex-0412078-2-5-3': 'If the Landshoff mechanism is suppressed then the double transverse asymmetry in [MATH] scattering is expected to be as large as the one observed in the [MATH] case.', 'hep-ex-0412078-2-6-0': '# Towards an asymmetric polarized antiproton-proton collider at FAIR', 'hep-ex-0412078-2-7-0': 'The possibility to test the nucleon structure via double spin asymmetries in polarized proton-antiproton reactions at the HESR ring of FAIR at GSI has been suggested by the PAX collaboration in 2004 [CITATION].', 'hep-ex-0412078-2-7-1': 'Since then, there has been much progress, both in understanding the physics potential of such an experiment [CITATION] and in studying the feasibility of efficiently producing polarized antiprotons [CITATION].', 'hep-ex-0412078-2-7-2': 'The physics program of such a facility would extend to a new domain the exceptionally fruitful studies of the nucleon structure performed in unpolarized and polarized deep inelastic scattering (DIS), which have been at the center of high energy physics during the past four decades.', 'hep-ex-0412078-2-7-3': 'As mentioned earlier, a direct measurement of the transversity distribution function [MATH], one of the last missing fundamental pieces in the QCD description of the nucleon, is unique.', 'hep-ex-0412078-2-7-4': 'In the available kinematic domain of the proposed experiment, which covers the valence region, the Drell-Yan double transverse spin asymmetry was recently predicted to be as large as 0.3 [CITATION].', 'hep-ex-0412078-2-7-5': 'Other novel tests of QCD at such a facility include the polarized elastic hard scattering of antiprotons on protons and the measurement of the phases of the time-like form factors of the proton (see Ref. [CITATION]).', 'hep-ex-0412078-2-7-6': 'A viable practical scheme which allows us to reach a polarization of the stored antiprotons at HESR-FAIR of [MATH] has been worked out and published in Ref. [CITATION].', 'hep-ex-0412078-2-8-0': 'The PAX Letter-of-Intent was submitted on January 15, 2004.', 'hep-ex-0412078-2-8-1': 'The physics program of PAX has been positively reviewed by the QCD Program Advisory Committee (PAC) on May 14-16, 2004 [CITATION].', 'hep-ex-0412078-2-8-2': 'The proposal by the ASSIA collaboration [CITATION] to utilize a polarized solid target and bombard it with a 45 GeV unpolarized antiproton beam extracted from the synchrotron SIS100 has been rejected by the GSI management.', 'hep-ex-0412078-2-8-3': 'Such measurements would not allow one to determine [MATH], because in single spin measurements [MATH] appears always coupled to another unknown fragmentation function.', 'hep-ex-0412078-2-8-4': 'Following the QCD-PAC report and the recommendation of the Chairman of the committee on Scientific and Technological Issues (STI) [CITATION] and the FAIR project coordinator, the PAX collaboration has optimized the technique to achieve a sizable antiproton polarization and the proposal for experiments at GSI with polarized antiprotons [CITATION].', 'hep-ex-0412078-2-8-5': 'From various working group meetings of the PAX collaboration, presented in part in 2004 at several workshops and conferences [CITATION], we conclude:', 'hep-ex-0412078-2-9-0': 'The PAX collaboration proposes an approach that is composed of two phases.', 'hep-ex-0412078-2-9-1': 'During these the major milestones of the project can be tested and optimized before the final goal is approached: An asymmetric proton-antiproton collider, in which polarized protons with momenta of about 3.5 GeV/c collide with polarized antiprotons with momenta up to 15 GeV/c.', 'hep-ex-0412078-2-9-2': 'These circulate in the HESR, which has already been approved and will serve the PANDA experiment.', 'hep-ex-0412078-2-9-3': 'In the following, we will briefly describe the overall machine setup of the HESR complex, schematically depicted in Fig. [REF].', 'hep-ex-0412078-2-10-0': 'Let us summarize the main features of the accelerator setup:', 'hep-ex-0412078-2-11-0': 'It is worthwhile to stress that, through the employment of the CSR, effectively a second interaction point is formed with minimum interference with PANDA.', 'hep-ex-0412078-2-11-1': 'The proposed solution opens the possibility to run two different experiments at the same time.', 'hep-ex-0412078-2-12-0': 'In the following sections, we discuss the physics program, which should be pursued in two different phases.', 'hep-ex-0412078-2-13-0': '## Phase I', 'hep-ex-0412078-2-14-0': 'A beam of unpolarized or polarized antiprotons with momentum up to 3.5 GeV/c in the CSR ring, colliding on a polarized hydrogen target in the PAX detector.', 'hep-ex-0412078-2-14-1': 'This phase is independent of the HESR performance.', 'hep-ex-0412078-2-15-0': 'This first phase, at moderately high energy, will allow for the first time the measurement of the time-like proton form factors in single and double polarized [MATH] interactions in a wide kinematical range, from close to threshold up to [MATH] GeV[MATH].', 'hep-ex-0412078-2-15-1': 'It would enable to determine several double spin asymmetries in elastic [MATH] scattering.', 'hep-ex-0412078-2-15-2': 'By detecting back scattered antiprotons one can also explore hard scattering regions of large [MATH]: In proton-proton scattering the same region of [MATH] requires twice the energy.', 'hep-ex-0412078-2-15-3': 'There are no competing facilities at which these topical issues can be addressed.', 'hep-ex-0412078-2-15-4': 'For the theoretical background, see the PAX LoI [CITATION] and the recent review paper [CITATION].', 'hep-ex-0412078-2-16-0': '## Phase II', 'hep-ex-0412078-2-17-0': 'This phase will allow the first ever direct measurement of the quark transversity distribution [MATH], by measuring the double transverse spin asymmetry [MATH] in Drell-Yan processes [MATH] as a function of Bjorken [MATH] and [MATH]) [EQUATION] where [MATH], [MATH] is the invariant mass of the lepton pair and [MATH], of the order of one, is the calculable double-spin asymmetry of the QED elementary process [MATH].', 'hep-ex-0412078-2-17-1': 'Two possible scenarios might be foreseen to perform the measurement, which are discussed below.', 'hep-ex-0412078-2-18-0': '### Asymmetric collider', 'hep-ex-0412078-2-19-0': 'A beam of polarized antiprotons from 1.5 GeV/c up to 15 GeV/c circulating in the HESR, colliding on a beam of polarized protons with momenta up to 3.5 GeV/c circulating in the CSR.', 'hep-ex-0412078-2-19-1': 'This scenario however requires to demonstrate that a suitable luminosity is reachable.', 'hep-ex-0412078-2-19-2': 'Deflection of the HESR beam to the PAX detector in the CSR is necessary (see Fig. [REF]).', 'hep-ex-0412078-2-20-0': 'By proper variation of the energy of the two colliding beams, this setup would allow a measurement of the transversity distribution [MATH] in the valence region of [MATH], with corresponding [MATH] (see Fig. [REF]).', 'hep-ex-0412078-2-20-1': '[MATH] is predicted to be larger than 0.3 over the full kinematic range, up to the highest reachable center-of-mass energy of [MATH].', 'hep-ex-0412078-2-20-2': 'The cross section is large as well: With a luminosity of [MATH] cm[MATH] about [MATH] events per day can be expected.', 'hep-ex-0412078-2-20-3': 'For the transversity distribution [MATH], such an experiment can be considered as the analogue of polarized DIS for the determination of the helicity structure function [MATH], i.e. of the helicity distribution [MATH]; the kinematical coverage [MATH] will be similar to that of the HERMES experiment.', 'hep-ex-0412078-2-21-0': '### High luminosity fixed target experiment', 'hep-ex-0412078-2-22-0': 'If the required luminosity in the collider mode is not achievable, a fixed target experiment can be conducted.', 'hep-ex-0412078-2-22-1': 'A beam of 22 GeV/c (15 GeV/c) polarized antiprotons circulating in the HESR is used to collide with a polarized internal hydrogen target.', 'hep-ex-0412078-2-22-2': 'Also this scenario requires the deflection of the HESR beam to the PAX detector in the CSR (see Fig. [REF]).', 'hep-ex-0412078-2-23-0': 'A theoretical discussion of the significance of the measurement of [MATH] for a 22 GeV/c (15 GeV/c) beam impinging on a fixed target is given in Refs. [CITATION] and the recent review paper [CITATION].', 'hep-ex-0412078-2-23-1': 'The theoretical work on the [MATH]-factors for the transversity determination is in progress [CITATION].', 'hep-ex-0412078-2-23-2': 'This measurement will explore the valence region of [MATH], with corresponding [MATH] (see Fig. [REF]).', 'hep-ex-0412078-2-23-3': 'In this region [MATH] is predicted to be large (of the order of 0.3, or more) and the expected number of events can be of the order of 2000 per day.', 'hep-ex-0412078-2-24-0': 'We would like to mention, that we are also investigating whether the PANDA detector, properly modified, is compatible with the transversity measurements in the collider mode, where an efficient identification of the Drell-Yan pairs is required.', 'hep-ex-0412078-2-24-1': 'At the interaction point, the spins of the colliding protons and antiprotons should be vertical, with no significant component along the beam direction.', 'hep-ex-0412078-2-25-0': '# Conclusion', 'hep-ex-0412078-2-26-0': 'To summarize, we note that the storage of polarized antiprotons at HESR will open unique possibilities to test QCD in hitherto unexplored domains.', 'hep-ex-0412078-2-26-1': 'This will provide another cornerstone to the antiproton program at FAIR.'}

[['hep-ex-0412078-1-17-0', 'hep-ex-0412078-2-17-0'], ['hep-ex-0412078-1-17-1', 'hep-ex-0412078-2-17-1'], ['hep-ex-0412078-1-0-0', 'hep-ex-0412078-2-0-0'], ['hep-ex-0412078-1-0-1', 'hep-ex-0412078-2-0-1'], ['hep-ex-0412078-1-0-2', 'hep-ex-0412078-2-0-2'], ['hep-ex-0412078-1-14-0', 'hep-ex-0412078-2-14-0'], ['hep-ex-0412078-1-14-1', 'hep-ex-0412078-2-14-1'], ['hep-ex-0412078-1-9-0', 'hep-ex-0412078-2-9-0'], ['hep-ex-0412078-1-9-1', 'hep-ex-0412078-2-9-1'], ['hep-ex-0412078-1-9-2', 'hep-ex-0412078-2-9-2'], ['hep-ex-0412078-1-9-3', 'hep-ex-0412078-2-9-3'], ['hep-ex-0412078-1-19-0', 'hep-ex-0412078-2-19-0'], ['hep-ex-0412078-1-19-1', 'hep-ex-0412078-2-19-1'], ['hep-ex-0412078-1-19-2', 'hep-ex-0412078-2-19-2'], ['hep-ex-0412078-1-7-0', 'hep-ex-0412078-2-7-0'], ['hep-ex-0412078-1-7-1', 'hep-ex-0412078-2-7-1'], ['hep-ex-0412078-1-7-2', 'hep-ex-0412078-2-7-2'], ['hep-ex-0412078-1-7-3', 'hep-ex-0412078-2-7-3'], ['hep-ex-0412078-1-7-4', 'hep-ex-0412078-2-7-4'], ['hep-ex-0412078-1-7-5', 'hep-ex-0412078-2-7-5'], ['hep-ex-0412078-1-7-6', 'hep-ex-0412078-2-7-6'], ['hep-ex-0412078-1-23-0', 'hep-ex-0412078-2-23-0'], ['hep-ex-0412078-1-23-1', 'hep-ex-0412078-2-23-1'], ['hep-ex-0412078-1-23-2', 'hep-ex-0412078-2-23-2'], ['hep-ex-0412078-1-23-3', 'hep-ex-0412078-2-23-3'], ['hep-ex-0412078-1-11-0', 'hep-ex-0412078-2-11-0'], ['hep-ex-0412078-1-11-1', 'hep-ex-0412078-2-11-1'], ['hep-ex-0412078-1-24-0', 'hep-ex-0412078-2-24-0'], ['hep-ex-0412078-1-24-1', 'hep-ex-0412078-2-24-1'], ['hep-ex-0412078-1-20-0', 'hep-ex-0412078-2-20-0'], ['hep-ex-0412078-1-20-1', 'hep-ex-0412078-2-20-1'], ['hep-ex-0412078-1-20-2', 'hep-ex-0412078-2-20-2'], ['hep-ex-0412078-1-20-3', 'hep-ex-0412078-2-20-3'], ['hep-ex-0412078-1-12-0', 'hep-ex-0412078-2-12-0'], ['hep-ex-0412078-1-4-2', 'hep-ex-0412078-2-4-2'], ['hep-ex-0412078-1-4-3', 'hep-ex-0412078-2-4-3'], ['hep-ex-0412078-1-15-0', 'hep-ex-0412078-2-15-0'], ['hep-ex-0412078-1-15-1', 'hep-ex-0412078-2-15-1'], ['hep-ex-0412078-1-15-2', 'hep-ex-0412078-2-15-2'], ['hep-ex-0412078-1-15-3', 'hep-ex-0412078-2-15-3'], ['hep-ex-0412078-1-15-4', 'hep-ex-0412078-2-15-4'], ['hep-ex-0412078-1-22-0', 'hep-ex-0412078-2-22-0'], ['hep-ex-0412078-1-22-1', 'hep-ex-0412078-2-22-1'], ['hep-ex-0412078-1-22-2', 'hep-ex-0412078-2-22-2'], ['hep-ex-0412078-1-2-0', 'hep-ex-0412078-2-2-0'], ['hep-ex-0412078-1-5-1', 'hep-ex-0412078-2-5-1'], ['hep-ex-0412078-1-5-2', 'hep-ex-0412078-2-5-2'], ['hep-ex-0412078-1-5-3', 'hep-ex-0412078-2-5-3'], ['hep-ex-0412078-1-8-0', 'hep-ex-0412078-2-8-0'], ['hep-ex-0412078-1-8-1', 'hep-ex-0412078-2-8-1'], ['hep-ex-0412078-1-8-2', 'hep-ex-0412078-2-8-2'], ['hep-ex-0412078-1-8-3', 'hep-ex-0412078-2-8-3'], ['hep-ex-0412078-1-8-4', 'hep-ex-0412078-2-8-4'], ['hep-ex-0412078-1-26-0', 'hep-ex-0412078-2-26-0'], ['hep-ex-0412078-1-26-1', 'hep-ex-0412078-2-26-1'], ['hep-ex-0412078-1-3-1', 'hep-ex-0412078-2-3-1'], ['hep-ex-0412078-1-3-3', 'hep-ex-0412078-2-3-3'], ['hep-ex-0412078-1-3-4', 'hep-ex-0412078-2-3-4'], ['hep-ex-0412078-1-3-5', 'hep-ex-0412078-2-3-5'], ['hep-ex-0412078-1-3-6', 'hep-ex-0412078-2-3-6'], ['hep-ex-0412078-1-3-7', 'hep-ex-0412078-2-3-7'], ['hep-ex-0412078-1-4-1', 'hep-ex-0412078-2-4-1'], ['hep-ex-0412078-1-3-2', 'hep-ex-0412078-2-3-2']]

[['hep-ex-0412078-1-17-0', 'hep-ex-0412078-2-17-0'], ['hep-ex-0412078-1-17-1', 'hep-ex-0412078-2-17-1'], ['hep-ex-0412078-1-0-0', 'hep-ex-0412078-2-0-0'], ['hep-ex-0412078-1-0-1', 'hep-ex-0412078-2-0-1'], ['hep-ex-0412078-1-0-2', 'hep-ex-0412078-2-0-2'], ['hep-ex-0412078-1-14-0', 'hep-ex-0412078-2-14-0'], ['hep-ex-0412078-1-14-1', 'hep-ex-0412078-2-14-1'], ['hep-ex-0412078-1-9-0', 'hep-ex-0412078-2-9-0'], ['hep-ex-0412078-1-9-1', 'hep-ex-0412078-2-9-1'], ['hep-ex-0412078-1-9-2', 'hep-ex-0412078-2-9-2'], ['hep-ex-0412078-1-9-3', 'hep-ex-0412078-2-9-3'], ['hep-ex-0412078-1-19-0', 'hep-ex-0412078-2-19-0'], ['hep-ex-0412078-1-19-1', 'hep-ex-0412078-2-19-1'], ['hep-ex-0412078-1-19-2', 'hep-ex-0412078-2-19-2'], ['hep-ex-0412078-1-7-0', 'hep-ex-0412078-2-7-0'], ['hep-ex-0412078-1-7-1', 'hep-ex-0412078-2-7-1'], ['hep-ex-0412078-1-7-2', 'hep-ex-0412078-2-7-2'], ['hep-ex-0412078-1-7-3', 'hep-ex-0412078-2-7-3'], ['hep-ex-0412078-1-7-4', 'hep-ex-0412078-2-7-4'], ['hep-ex-0412078-1-7-5', 'hep-ex-0412078-2-7-5'], ['hep-ex-0412078-1-7-6', 'hep-ex-0412078-2-7-6'], ['hep-ex-0412078-1-23-0', 'hep-ex-0412078-2-23-0'], ['hep-ex-0412078-1-23-1', 'hep-ex-0412078-2-23-1'], ['hep-ex-0412078-1-23-2', 'hep-ex-0412078-2-23-2'], ['hep-ex-0412078-1-23-3', 'hep-ex-0412078-2-23-3'], ['hep-ex-0412078-1-11-0', 'hep-ex-0412078-2-11-0'], ['hep-ex-0412078-1-11-1', 'hep-ex-0412078-2-11-1'], ['hep-ex-0412078-1-24-0', 'hep-ex-0412078-2-24-0'], ['hep-ex-0412078-1-24-1', 'hep-ex-0412078-2-24-1'], ['hep-ex-0412078-1-20-0', 'hep-ex-0412078-2-20-0'], ['hep-ex-0412078-1-20-1', 'hep-ex-0412078-2-20-1'], ['hep-ex-0412078-1-20-2', 'hep-ex-0412078-2-20-2'], ['hep-ex-0412078-1-20-3', 'hep-ex-0412078-2-20-3'], ['hep-ex-0412078-1-12-0', 'hep-ex-0412078-2-12-0'], ['hep-ex-0412078-1-4-2', 'hep-ex-0412078-2-4-2'], ['hep-ex-0412078-1-4-3', 'hep-ex-0412078-2-4-3'], ['hep-ex-0412078-1-15-0', 'hep-ex-0412078-2-15-0'], ['hep-ex-0412078-1-15-1', 'hep-ex-0412078-2-15-1'], ['hep-ex-0412078-1-15-2', 'hep-ex-0412078-2-15-2'], ['hep-ex-0412078-1-15-3', 'hep-ex-0412078-2-15-3'], ['hep-ex-0412078-1-15-4', 'hep-ex-0412078-2-15-4'], ['hep-ex-0412078-1-22-0', 'hep-ex-0412078-2-22-0'], ['hep-ex-0412078-1-22-1', 'hep-ex-0412078-2-22-1'], ['hep-ex-0412078-1-22-2', 'hep-ex-0412078-2-22-2'], ['hep-ex-0412078-1-2-0', 'hep-ex-0412078-2-2-0'], ['hep-ex-0412078-1-5-1', 'hep-ex-0412078-2-5-1'], ['hep-ex-0412078-1-5-2', 'hep-ex-0412078-2-5-2'], ['hep-ex-0412078-1-5-3', 'hep-ex-0412078-2-5-3'], ['hep-ex-0412078-1-8-0', 'hep-ex-0412078-2-8-0'], ['hep-ex-0412078-1-8-1', 'hep-ex-0412078-2-8-1'], ['hep-ex-0412078-1-8-2', 'hep-ex-0412078-2-8-2'], ['hep-ex-0412078-1-8-3', 'hep-ex-0412078-2-8-3'], ['hep-ex-0412078-1-8-4', 'hep-ex-0412078-2-8-4'], ['hep-ex-0412078-1-26-0', 'hep-ex-0412078-2-26-0'], ['hep-ex-0412078-1-26-1', 'hep-ex-0412078-2-26-1'], ['hep-ex-0412078-1-3-1', 'hep-ex-0412078-2-3-1'], ['hep-ex-0412078-1-3-3', 'hep-ex-0412078-2-3-3'], ['hep-ex-0412078-1-3-4', 'hep-ex-0412078-2-3-4'], ['hep-ex-0412078-1-3-5', 'hep-ex-0412078-2-3-5'], ['hep-ex-0412078-1-3-6', 'hep-ex-0412078-2-3-6'], ['hep-ex-0412078-1-3-7', 'hep-ex-0412078-2-3-7']]

[['hep-ex-0412078-1-4-1', 'hep-ex-0412078-2-4-1'], ['hep-ex-0412078-1-3-2', 'hep-ex-0412078-2-3-2']]

[]

[]

[]

['hep-ex-0412078-1-8-5', 'hep-ex-0412078-1-10-0', 'hep-ex-0412078-2-8-5', 'hep-ex-0412078-2-10-0']

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

https://arxiv.org/abs/hep-ex/0412078

1007.1599

{'1007.1599-1-0-0': 'We address the question of whether the [MATH]CDM model produces enough substructure in galaxy scale dark matter halos to cause the observed image flux anomalies in lensed QSOs observed in the radio and mid-infrared.', '1007.1599-1-0-1': 'We create a very large number of simulated lenses with finite source sizes to compare with the data.', '1007.1599-1-0-2': 'After analysing these simulations, our conclusions are: 1) The finite size of the source is important.', '1007.1599-1-0-3': 'The point source approximation commonly used can cause large and biased results.', '1007.1599-1-0-4': '2) When we randomly select lens models that are distorted from a Singular Isothermal Ellipsoid in reasonable ways, but do not contain substructure, the flux anomalies are not reproduced.', '1007.1599-1-0-5': '3) We find new upper bounds on the amount of substructure from the constraint that no simple single-galaxy lenses have been observed with a single source having more than four well separated images.', '1007.1599-1-0-6': '4) The lower bound on the amount of substructure is set by the frequency of image flux anomalies and is largely a bound on the surface mass density in substructure and not on the number density normalization and average mass separately.', '1007.1599-1-0-7': '5) Substructure models with the same size-mass relation produce similar numbers of flux anomalies even when their internal mass profiles are different.', '1007.1599-1-0-8': '6) Substructures with shallower mass profiles produce less extra images.', '1007.1599-1-0-9': '7) Assuming no constribution form extragalactic small-scale structure, we find a lower bound of 0.3% on the fraction of mass in substructures within a projected distance of [MATH] from the center of a galaxy with velocity dispersion of 200 km/s.', '1007.1599-1-0-10': 'This bound is consistent with the results of [MATH]CDM Nbody simulations.', '1007.1599-1-1-0': '# introduction', '1007.1599-1-2-0': 'The Cold Dark Matter (CDM) model with a cosmological constant ([MATH]CDM) has become the standard model of cosmology.', '1007.1599-1-2-1': 'This model is in good agreement with a variety of observational probes of the large scales distribution of matter and galaxies in the Universe and is in general agreement with probes of the distribution of mass in galaxy clusters and in large galaxies.', '1007.1599-1-2-2': 'In the [MATH]CDM model, dark matter clumps into halos and galaxies form in the halos.', '1007.1599-1-2-3': 'On small scales, [MATH]CDM predicts that dark matter halos exist down to very small masses; the exact lower limit depending on the properties of the CDM particle and its thermal history.', '1007.1599-1-2-4': 'It has long been recognized that the number of observed dwarf galaxies in the local group of galaxies falls well short of the number of predicted halos .', '1007.1599-1-2-5': 'This is referred to as the substructure problem.', '1007.1599-1-2-6': 'Either galaxy formation is highly suppressed in small mass halos or [MATH]CDM needs to be modified in some way by, for example, changing the properties of the dark matter particle or the initial conditions for the density fluctuation in the Universe.', '1007.1599-1-2-7': 'Warm Dark Matter (WDM) is a popular alternative.', '1007.1599-1-2-8': 'Whether or not these small mass halos exist has been one of the most pressing unanswered question in cosmology for a decade.', '1007.1599-1-3-0': '[CITATION] demonstrated that if small-scale structure exists in the distribution of dark matter it will have a strong effect on the magnifications of quasar images in strong gravitational lenses.', '1007.1599-1-3-1': 'This effect causes the flux ratio between images to disagree with any lens model with a smooth distribution of matter.', '1007.1599-1-3-2': 'These cases are call anomalous flux ratios.', '1007.1599-1-3-3': 'A particular case had been studied by [CITATION] and subsequently it was shown that anomalies are common in quasar lensed .', '1007.1599-1-3-4': 'This work and a number of subsequent studies [CITATION] relied on fitting lens models to individual lens systems.', '1007.1599-1-3-5': 'It has not yet been shown clearly what can be causing these anomalies and what cannot be causing them.', '1007.1599-1-4-0': 'In a parallel approach, we and others have tried to simulate the lenses directly from cosmological Nbody simulations to determine if they are consistent with the observed frequency of flux anomalies .', '1007.1599-1-4-1': 'The first study predicted a large number of anomalies, but it may have been strongly affected by shot noise.', '1007.1599-1-4-2': 'The two more recent and higher resolution studies found that the substructure in the Nbody simulations is not sufficient to cause the observed flux anomalies (also the conclusion of [CITATION]).', '1007.1599-1-4-3': 'This is largely because of the small number density of substructures near the radii where images form (typically around 10 kpc in projection).', '1007.1599-1-4-4': 'These studies relied on only a few projections of a single high resolution halo each.', '1007.1599-1-4-5': 'It is possible that these results are a statistical fluke or that the observed anomalies are largely caused by dark matter objects along the line of sight but not inside the halo of the primary lens [CITATION].', '1007.1599-1-4-6': 'Answering the question of whether the Nbody simulations have enough small-scale structure in them to account for the flux ratio anomalies is one of the primary goals of this paper.', '1007.1599-1-5-0': 'It is very difficult to realistically simulate strong QSO lenses from an Nbody simulation.', '1007.1599-1-5-1': 'The first, and most important, problem is that shot noise from the discrete particles has a strong effect on the image magnifications.', '1007.1599-1-5-2': 'Roughly, the error in the magnification goes as [MATH] where [MATH] is the magnification and [MATH] is the number of particles over which the smoothing is done.', '1007.1599-1-5-3': 'Since [MATH] can be large, 100 or larger in the best cases for detecting substructure, the amount of smoothing needed to obtain an accuracy of even 10% is very large.', '1007.1599-1-5-4': 'So much smoothing can even smooth out the very substructures one wants to detect.', '1007.1599-1-5-5': 'Because of this [CITATION] replace an Nbody simulation with a simple analytic model fit to an Nbody simulation.', '1007.1599-1-5-6': 'A second problem is that the highest resolution simulations do not contain baryons.', '1007.1599-1-5-7': 'Baryons have a strong effect on the profile of the lens and in some cases dominate the mass within one Einstein radius.', '1007.1599-1-5-8': 'The baryons need to be put in "by hand".', '1007.1599-1-5-9': 'A third problem is that the extremely high resolution simulations required provide one, or at best a few, dark matter halos.', '1007.1599-1-5-10': 'Variations between halos make their lensing properties and their tendency to produce anomalies very different.', '1007.1599-1-5-11': 'It will be demonstrated in this paper that only very limited conclustions about the CDM model can be drawn from a single simultated lens.', '1007.1599-1-6-0': 'To avoid these problems, we take a different approach in this paper.', '1007.1599-1-6-1': 'We produce a large number of analytic lens models that are meant to reproduce the population of lenses expected in the [MATH]CDM model.', '1007.1599-1-6-2': 'We then determine the frequency of flux ratio anomalies in these lenses and compare it to the observed frequency.', '1007.1599-1-6-3': 'We adjust the properties and abundance of the substructures to see what kind of substructure is consistent with observations.', '1007.1599-1-6-4': 'The allowed statistical properties of the substructures are compared with the properties of Nbody halos.', '1007.1599-1-7-0': 'All previous studies, except [CITATION], have also suffered from the problem that the sources are treated as infinitely small points.', '1007.1599-1-7-1': 'The magnification of individual images are calculated by taking derivatives of the gravitational force at the position of the image.', '1007.1599-1-7-2': 'It will be shown in this paper, that since the physical size of the quasar radio or mid-infrared emission regions are similar to the sizes of the substructures of interest the point source magnifications are not accurate approximations.', '1007.1599-1-7-3': 'We use a new, high speed lensing code called GLAMER (Gravitational Lensing with Adaptive MEsh Refinement) that is the first one capable of producing a very large number of simulated lenses with finite sources in a reasonable amount of time.', '1007.1599-1-7-4': 'It does this through an adaptive mesh refinement algorithm that will be briefly described in section [REF].', '1007.1599-1-8-0': 'In section [REF], the models and techniques used to create simulated lenses are described.', '1007.1599-1-8-1': 'In section [REF], the results of those simulations are discussed.', '1007.1599-1-8-2': 'Ways of comparing the results to the available lensing data are presented in section [REF].', '1007.1599-1-8-3': 'The results are compared with the predictions of cosmological Nbody simulations in section [REF].', '1007.1599-1-8-4': 'A summery and discussion are given in section [REF].', '1007.1599-1-9-0': '# lens simulations', '1007.1599-1-10-0': 'Our approach in this paper is to produce a large population of realistic simulated lenses and then compare their statistical properties to the observed population of lenses.', '1007.1599-1-10-1': 'To do this, we must develop a model for the population of gravitational lens that includes the host, galaxy + dark matter halo, and the substructures within the host.', '1007.1599-1-10-2': 'We will not consider the effects of companion galaxies with masses roughly equivalent to the primary lens in this paper.', '1007.1599-1-11-0': '## Host lens model', '1007.1599-1-12-0': 'There is significant evidence from lensing and X-ray observations that early-type galaxies have a [MATH] mass profiles .', '1007.1599-1-12-1': 'In accordance with this finding, we model the host lenses as Distorted Singular Isothermal Ellipsoids (DSIE).', '1007.1599-1-12-2': 'The surface density for this model is [EQUATION] where the Einstein radius is [EQUATION] and the critical surface density is [EQUATION] where [MATH], [MATH] and [MATH] are the angular size distance to the lens, to the source and between the lens and the source respectively.', '1007.1599-1-12-3': 'The first part ([REF]) is a Singular Isothermal Ellipsoids whose lensing properties have been extensively studied (see [CITATION] for example).', '1007.1599-1-12-4': 'The deflection angle and shear caused by the series in ([REF]) have been worked out by [CITATION], although with different notation.', '1007.1599-1-13-0': 'The perturbations [MATH] are assumed to be of the same order as the observed perturbations in the surface brightness profile of of early-type galaxies.', '1007.1599-1-13-1': 'Typical values for [MATH] and [MATH] are two or three percent, but accurate statistics are not available .', '1007.1599-1-13-2': 'We draw random values from a Gaussian distribution with variance 0.005 for [MATH] and [MATH] and 0.01 for [MATH].', '1007.1599-1-13-3': 'We take [MATH] and [MATH] terms to be zero.', '1007.1599-1-13-4': 'In the observations, [MATH] is usually defined with the orientation of this mode fixed to the same axis as the axis of the elliptical component to define the "diskyness" or "boxyness" of the galaxy.', '1007.1599-1-13-5': 'Since the alignment has important effects on the lensing properties, we relax this requirement somewhat and allow [MATH] to vary from the position angle of the elliptical component.', '1007.1599-1-13-6': 'The misalignment is normally distributed with variance 3 degrees.', '1007.1599-1-14-0': 'We also include background shear and convergence in the model.', '1007.1599-1-14-1': '[CITATION] calculated the expected distribution of [MATH] and [MATH] in an Nbody simulation at potential lenses.', '1007.1599-1-14-2': 'They found that [MATH] and [MATH] are both roughly lognormally distributed with a variance of [MATH].', '1007.1599-1-14-3': 'We assume this distribution in our model.', '1007.1599-1-14-4': 'Analytic estimates by [CITATION] are in agreement with this result, as are observations .', '1007.1599-1-15-0': 'The model described above is what will be called the "standard" host model.', '1007.1599-1-15-1': 'To test how sensitive magnification anomalies are to the host model, we perform a series of tests where the distortions to the lens are increased.', '1007.1599-1-15-2': 'For the "extra distorted model", we triple the variance in the distortion modes and decouple their orientation from the orientation of the elliptical component.', '1007.1599-1-15-3': 'For the "extra shear model", we triple the variance in the background shear and convergence.', '1007.1599-1-16-0': '### Distributions of host properties', '1007.1599-1-17-0': "Calculating the expected distribution of the lenses' redshifts, velocity dispersions and ellipticities requires knowing not only the source luminosity and redshift distributions of lenses and sources, but also the many selection effects that might be important.", '1007.1599-1-17-1': 'The sample of lenses we wish to compare our results with were discovered in many different ways and do not have a uniform, well defined selection criterion.', '1007.1599-1-17-2': 'Instead of trying to model these biases, we use the distributions of already known lenses when possible.', '1007.1599-1-18-0': 'For the lens and sources redshifts, we use the observed values for the Castles lenses.', '1007.1599-1-18-1': 'There are 60 lenses with measured source and lens redshift pairs.', '1007.1599-1-18-2': 'We draw randomly from these sets of redshifts.', '1007.1599-1-18-3': 'The lenses discussed in section [REF] are a subsample of these.', '1007.1599-1-19-0': 'To get a sample of host velocity dispersions, [MATH], we use the velocity dispersions from the SLACS lenses .', '1007.1599-1-19-1': 'This sample of 61 lenses is used to make a cumulative distribution of [MATH].', '1007.1599-1-19-2': 'The discrete distribution is linearly interpolated to get a continuous cumulative distribution and then this is randomly sampled from.', '1007.1599-1-19-3': 'In the SLACS sample, the measured velocity dispersion of stars and the velocity dispersion of the best-fit SIE models have statistically indistinguishable distributions.', '1007.1599-1-19-4': 'We choose to use the best-fit SIE velocity dispersions.', '1007.1599-1-19-5': 'These values range from 160 to [MATH]^-1[MATH].', '1007.1599-1-20-0': 'The axis ratios, [MATH], are sampled independently from the SLACS lenses in the same way as the velocity dispersions.', '1007.1599-1-20-1': 'No possible correlations between the internal structure of the lenses and their redshift are reproduced in this sampling.', '1007.1599-1-20-2': 'The SLACS lenses are at relatively low redshift because of their selection criterion, but observations indicate that the internal structure of early-type galaxies do not evolve significantly between [MATH] and 0 .', '1007.1599-1-21-0': 'We consider only four image quasar lenses in this paper, while the SLACS lenses include two image lenses.', '1007.1599-1-21-1': 'The asymmetry of the lens changes the area enclosed in the tangential caustic and thus a sample of four image lenses will tend to have more asymmetric lenses than a sample that includes all multiple image cases.', '1007.1599-1-21-2': 'To correct for this bias, we calculate the ratio of the area within the tangential caustic to the area within the radial caustic (or "cut" in the case of a DSIE).', '1007.1599-1-21-3': 'The number of sources used for the lens is then proportional to this ratio.', '1007.1599-1-21-4': 'More circular galaxies will have less lenses in the final sample.', '1007.1599-1-21-5': 'This corrects for the bias in the SLACS lenses relative to the four image quasars.', '1007.1599-1-21-6': 'From 0 to [MATH], source positions are used for each lens model.', '1007.1599-1-22-0': '## Substructure model', '1007.1599-1-23-0': 'We wish to construct a substructure model that reflects the expectations we have from Nbody simulation, but is relatively simple and has a small number of parameters that can be varied to measure the agreement or disagreement with [MATH]CDM.', '1007.1599-1-24-0': 'Simulations show that the mass fraction in substructure within a projected radius increases roughly linearly with projected radius .', '1007.1599-1-24-1': 'With a SIE mass model, this implies that the surface density of substructure is constant at least near the Einstein radius and interior to it.', '1007.1599-1-24-2': 'This can be understood as a consequence of the substructures being mostly at larger 3D radii than the Einstein radius.', '1007.1599-1-24-3': 'This will be assumed in all cases.', '1007.1599-1-25-0': 'The mass function of the substructure will be [EQUATION] where the number density of substructures is [MATH].', '1007.1599-1-25-1': 'Nbody simulations indicate that [MATH] up to about [MATH] of the halo mass without any resolved lower mass limit.', '1007.1599-1-25-2': 'It is assumed that this parameter is not a function of host mass.', '1007.1599-1-26-0': 'The maximum mass in the mass function must be a function of host halo size.', '1007.1599-1-26-1': 'A mass scale for the host can be defined as the mass within a fixed radius ([MATH]) or the mass within a radius where the average density reaches a fixed threshold ([MATH]).', '1007.1599-1-26-2': 'The latter is the one commonly used to define the mass of a halo in cosmology although the virial radius is generally larger than the radii over which one would expect the SIE model to hold.', '1007.1599-1-26-3': 'However, if the concentration of the halos does not vary greatly within the range of host lenses then the same scaling would be expected in the inner regions.', '1007.1599-1-26-4': 'Making the maximum substructure mass a fixed fraction of the host halo mass results in [EQUATION]', '1007.1599-1-26-5': 'The same scaling is assumed for the minimum mass.', '1007.1599-1-26-6': "[MATH] is used as an adjustable parameter to change the mass scale and test the data's consistency with a mass cutoff as would be expected in many alternatives theories to CDM.", '1007.1599-1-26-7': 'The normalizing halo is fixes to [MATH]^-1[MATH].', '1007.1599-1-27-0': 'The normalization of the mass function ([REF]) needs to be set.', '1007.1599-1-27-1': 'To agree with Nbody simulations, the fraction of mass in substructure at a fixed fraction of the virial radius should be the same in all halos.', '1007.1599-1-27-2': 'Since [MATH] and ([REF]) makes average mass scale like [MATH] the normalization must scale like [MATH].', '1007.1599-1-27-3': 'Explicitly the result is [EQUATION]', '1007.1599-1-27-4': 'Although the mass fraction in substructure at a fixed fraction of the halo radius is the same for all lenses, the same is not true at the Einstein radius.', '1007.1599-1-27-5': 'Since [MATH], the total surface density at [MATH] is independent of [MATH] for lenses and sources at the same redshift, which makes the mass fraction scale as [MATH] at this radius.', '1007.1599-1-27-6': 'As a result, we might expect substructure to be more important for larger lenses.', '1007.1599-1-28-0': 'The internal structure of the substructures is, for simplicity, a simple power-law with a cutoff radius [EQUATION]', '1007.1599-1-28-1': "In the classical analytic treatment, the average density within the tidal radius is proportional to the average density of the host within the substructure's orbit .", '1007.1599-1-28-2': 'This implies [MATH] if all the substructures are at the same distance from the center of the host, which we assume.', '1007.1599-1-28-3': 'Since the density at a fixed fraction of the host halo radius is independent of the host size, it is expected that this relation is independent of the host size: [EQUATION]', '1007.1599-1-28-4': 'In a more realistic model, there would be a significant scatter in the [MATH]-[MATH]-[MATH] relation, but for our purposes this relation is sufficient.', '1007.1599-1-29-0': 'In summery, the substructure model has the free parameters [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and the normalization host velocity dispersion [MATH] which we fix at [MATH]^-1[MATH].', '1007.1599-1-30-0': '## Ray-shooting', '1007.1599-1-31-0': 'The sources that we wish to use in our simulation have sizes of [MATH] and the substructures can have similar sizes.', '1007.1599-1-31-1': 'Therefore, it is essential that we be able to calculate the magnification of finite size sources.', '1007.1599-1-31-2': 'This requirement has been widely ignored in the literature because it is difficult to map the image of a finite source in a short enough amount of time to make it possible to create the large number of simulated lenses required for this problem.', '1007.1599-1-31-3': 'A new code, GLAMER, has been developed for this and other applications.', '1007.1599-1-31-4': 'This code employs a highly optimized adaptive mesh refinement scheme which allows the shapes of the images and their area to be calculated rapidly.', '1007.1599-1-31-5': 'This allows us to make millions of mock lenses with a finite size source in a relatively short amount of time.', '1007.1599-1-31-6': 'Figure [REF] illustrates how the grid is refined to find all the images and their areas.', '1007.1599-1-31-7': 'Figure [REF] shows the critical curve and caustic structure for one example lens.', '1007.1599-1-31-8': 'For more details on this code, see [CITATION].', '1007.1599-1-32-0': 'The range of positions in which a substructure will make a significant change to the magnification of an image depends on the mass of the substructure.', '1007.1599-1-32-1': 'To optimize calculations, small-mass substructures that are far away from the lens are omitted from the calculation while more massive substructures further from the lens are included.', '1007.1599-1-32-2': 'To accomplish this, a mass dependent cutoff radius from the center of the lens is used: [EQUATION]', '1007.1599-1-32-3': 'The first two terms ensure that all substructures within two Einstein radii plus the radius of the substructure are included.', '1007.1599-1-32-4': 'The third term ensures that any substructure close enough to cause a perturbation to the lens that is not well approximated as a pure shear will be included.', '1007.1599-1-32-5': 'The parameter [MATH] controls how large the variation in the shear across the Einstein radius are allowed to be.', '1007.1599-1-32-6': 'We set this parameter to [MATH].', '1007.1599-1-32-7': 'The contribution from substructures or companions outside this range is considered to be part of the background shear discussed in section [REF] as part of the host lens model.', '1007.1599-1-33-0': 'For each lens model (host and substructure), the critical curves and caustics are found first.', '1007.1599-1-33-1': 'There are sometimes multiple, disconnected critical curves.', '1007.1599-1-33-2': 'The main tangential caustic is found by requiring its critical curve to be the one that encompasses the most area while also surrounding the center of the lens.', '1007.1599-1-33-3': 'The area within the tangential caustic is calculated and the number of source positions that will be used for that lens is calculated as described in section [REF].', '1007.1599-1-33-4': 'The sources are required to have their centers inside the tangential caustic, but they are otherwise randomly distributed.', '1007.1599-1-33-5': 'Because of the finite source size, some images will be merged and this results in less than four images.', '1007.1599-1-34-0': 'Some lenses have more than the four images that the undistorted host model alone would predict.', '1007.1599-1-34-1': 'Some of these additional images are very small and/or so close to another image that they would not be observed as separate images.', '1007.1599-1-34-2': 'We do a rough initial cut in all cases by merging together any images with centroids that are less than 0.1 arcsec apart.', '1007.1599-1-34-3': 'Further discussion of additional images is given in the next section.', '1007.1599-1-35-0': 'Table [REF] lists the simulation runs that were performed.', '1007.1599-1-35-1': 'They are in batches of 100,000 lenses with fixed substructure parameters.', '1007.1599-1-36-0': '# results', '1007.1599-1-37-0': 'We create several million simulated lenses and save the image positions and magnifications.', '1007.1599-1-37-1': 'We also store the point source magnifications at the centroid of each image and the point source magnification for the point in the image that is closest to the center of the source.', '1007.1599-1-37-2': 'Some of the host lens parameters are also stored.', '1007.1599-1-37-3': 'In this paper, for ease of comparison, we classify the observed and simulated lenses and reduce the position and magnification information to two parameters.', '1007.1599-1-37-4': 'The parameter [MATH] is defined in figure [REF].', '1007.1599-1-37-5': 'A small value of [MATH] indicates the source is near a cusp in the caustic.', '1007.1599-1-37-6': 'Figure [REF] also describes what a long- and short-axis lenses are.', '1007.1599-1-37-7': 'We have found that a good observational way of sorting the lenses into these categories is by comparing the angular distance between the center of the lens and the singlet image to the distance between the center of the lens and the central image of the triplet.', '1007.1599-1-37-8': 'If the former is greater, then the lens is a short-axis lens.', '1007.1599-1-37-9': 'Otherwise, it is a long-axis lense.', '1007.1599-1-38-0': 'The second parameter used to characterize each lens is [EQUATION] where "[MATH]" is for long-axis lenses and "[MATH]" for short-axis lenses.', '1007.1599-1-38-1': 'The magnifications for the images in the triplet are [MATH], [MATH] and [MATH], with [MATH] being for the central image.', '1007.1599-1-38-2': 'The motivation for this parameter is that [MATH] asymptotically as a point source approaches a cusp in the caustic .', '1007.1599-1-38-3': 'The [MATH] parameter has been widely used because of this model independent prediction.', '1007.1599-1-38-4': 'In practice, [MATH] is not constrained to a very small region around zero because of finite source effects and the invalidity of the lowest-order expansion of the lensing equation around the cusp.', '1007.1599-1-38-5': 'Although this property motivates the parameter [MATH], it is not used anywhere in this analysis.', '1007.1599-1-39-0': 'Figures [REF] shows the distribution of [MATH] and [MATH] for a sample of simulations.', '1007.1599-1-39-1': "An additional 10% error on each image's flux is added to conservatively account for typical obervational uncertainties.", '1007.1599-1-39-2': 'It can be seen that the simulated lenses occupy a well localized regions in these diagrams when no substructure is present.', '1007.1599-1-39-3': 'Even when substructure is present at the levels investigated, the majority of lenses occupy the same regions with a smaller number of cases spread out in tails to the distribution.', '1007.1599-1-40-0': 'Figure [REF] shows the fractional error made in the magnifications when the point source magnification is used.', '1007.1599-1-40-1': 'It can be seen there that the fractional error is small for magnifications less than around 5.', '1007.1599-1-40-2': 'This is confirmation that the numerical errors made by the ray-tracing code are small.', '1007.1599-1-40-3': 'At higher magnifications, larger errors are made when the source is 10 pc.', '1007.1599-1-40-4': 'This is not a numerical effect.', '1007.1599-1-40-5': 'It can also be seen in figure [REF] that substructure causes the errors made by using the point magnification to increase when the source is 10 pc, but less so when the source is 1 pc.', '1007.1599-1-40-6': 'This is in agreement with expectations because the source size of 10 pc is closer to the characteristic scale of the substructures.', '1007.1599-1-41-0': 'Figure [REF] shows the ratio between the point source magnifications and the finite source magnifications.', '1007.1599-1-41-1': 'Again, it can be seen that numerical errors are clearly not playing a large part.', '1007.1599-1-41-2': 'It is evident that the point source magnifications are not evenly distributed around the finite source magnifications.', '1007.1599-1-41-3': 'Centroid point source magnifications tend to overestimate the real magnification; in some cases by a large factor.', '1007.1599-1-41-4': 'This is the magnification that would be calculated when fitting a lens model to an observed lens.', '1007.1599-1-41-5': 'The nearest point magnification is much less biased and in the opposite direction; the magnification is underestimated.', '1007.1599-1-41-6': 'Both effects are much smaller for a smaller source size, as they should be.', '1007.1599-1-42-0': 'Some of the images were merged because their centroids were within 0.1 arcsec. It makes no sense to take the closest point magnification in these cases since the closest point is not unique.', '1007.1599-1-42-1': 'Unsurprisingly, the magnification at the centroid point is an even worse approximation in these cases, as can be seen in figures [REF] and [REF].', '1007.1599-1-42-2': 'In exceptional cases, the centroid might not even be in one of the images that are merged.', '1007.1599-1-42-3': 'As expected, these cases only arise when substructure is present.', '1007.1599-1-43-0': 'Figures [REF] and [REF] should give one pause before using the point source approximation for the magnification in any substructure lensing study or when interpreting the results of any studies that use this approximation.', '1007.1599-1-44-0': '# comparison with data', '1007.1599-1-45-0': 'To avoid contamination from microlensing by stars in the lens galaxy, differential extinction and variability of the source on time-scales smaller than the image delay times, we compare our simulations only to quad lenses measured in the radio and the mid-infrared.', '1007.1599-1-45-1': 'Since we have not included companion galaxies to the primary lens in our simulations, we also remove lenses with nearby galaxies that appear to have similar masses to the primary.', '1007.1599-1-45-2': 'This removes 1608+656 and 1004+4112 from the list.', '1007.1599-1-45-3': 'There is a very faint dwarf galaxy within the Einstein radius of 2045+265 , but we will consider this to be a substructure and not a companion galaxy because it is small.', '1007.1599-1-45-4': 'Lens models show that this substructure would need to be unnaturally elongated to cause the flux anomaly in this system, so there is probably another substructure present.', '1007.1599-1-45-5': 'The lenses must also have a detected lens galaxy which eliminates 0134-0931 and 0128+437.', '1007.1599-1-45-6': 'Table [REF] lists the lenses used and their [MATH] and [MATH] values are plotted in figure [REF].', '1007.1599-1-46-0': 'The most striking thing in figure [REF] is that two of the lenses have significantly higher [MATH] than is expected in the absence of substructure.', '1007.1599-1-46-1': 'We would like to quantify how likely it is to get such outliers for each substructure model.', '1007.1599-1-46-2': 'In doing this, one must confront the fact that there are selection effects that will strongly change the distribution of [MATH] in the observed lens sample.', '1007.1599-1-46-3': 'Cuspy lenses with small [MATH] have higher magnifications and thus are more likely to be found.', '1007.1599-1-46-4': 'On the other hand, small [MATH] lenses have small image separations and may not be as easily recognized as lenses.', '1007.1599-1-46-5': 'The heterogeneous nature of the lens sample makes quantifying these selection effects difficult and unreliable.', '1007.1599-1-46-6': 'Without modeling these selection effects, a two dimensional ([MATH] and [MATH]) comparison is not possible.', '1007.1599-1-46-7': 'Instead, we compare the distributions in [MATH] only.', '1007.1599-1-46-8': 'There remains the problem that the [MATH] distribution is clearly dependent on [MATH].', '1007.1599-1-46-9': 'We avoid this by binning simulations into intervals in [MATH] and using rank statistics which do not depend on any assumptions about the underlying distribution.', '1007.1599-1-46-10': 'In this way, information from data points at different [MATH] can be combined.', '1007.1599-1-47-0': 'Within a bin in [MATH], if the simulations and data came from the same distribution the rank of the data point (the number of simulations with a larger value of [MATH]) would be evenly distributed between 0 and the total number of simulations in that bin [MATH].', '1007.1599-1-47-1': 'The probability of getting a rank smaller than [MATH] is [MATH].', '1007.1599-1-47-2': 'The probability of getting [MATH] or more observed cases with ranks below [MATH] out of a sample of [MATH] lenses is [EQUATION]', '1007.1599-1-47-3': 'If [MATH] is set to the highest rank of the two outlier lenses, then [MATH] gives the probability of getting at least 2 outliers if the observed lenses came from the same distribution as the simulations.', '1007.1599-1-47-4': 'We use this probability for hypothesis testing.', '1007.1599-1-47-5': 'If it is low, the model is unlikely.', '1007.1599-1-47-6': 'If it is high, the model is consistent.', '1007.1599-1-48-0': 'We set the [MATH] bin sizes by finding the 3000 simulations with [MATH] closest to but less than the observed [MATH] and the 3000 closest simulations with larger [MATH].', '1007.1599-1-48-1': 'This is done for each lens.', '1007.1599-1-48-2': 'The rank used in ([REF]) is the second smallest rank found.', '1007.1599-1-49-0': 'The calculated [MATH] for simulation sets 1,2 and 3 are shown in figure [REF] as a function of the substructure number density [MATH].', '1007.1599-1-49-1': 'Also shown in figure [REF] are estimates of two sources of noise.', '1007.1599-1-49-2': 'Random, Gaussian distributed noise with a standard deviation of 10% is added to each simulated image flux to represent observational errors.', '1007.1599-1-49-3': 'In the [MATH] case, five realizations of this noise are shown to gauge how much of an influence it has on the results.', '1007.1599-1-49-4': 'There is also an uncertainty that comes from the fact that we have only a finite number of simulations.', '1007.1599-1-49-5': 'To estimate this sampling error we do a bootstrap resampling of the simulations for each [MATH] set.', '1007.1599-1-49-6': 'The 2-[MATH] range in [MATH] are shown as the shaded regions in figures [REF] through [REF].', '1007.1599-1-49-7': 'Both of these sources of error are small when [MATH] is small.', '1007.1599-1-49-8': 'This allows us to make conclusions about where [MATH] crosses the 5% line, which we will take as the threshhold for an acceptable substructure model.', '1007.1599-1-50-0': 'In the top panel of figure [REF], we have shifted the x-axis from figure [REF] so that it is the total surface mass density of substructures.', '1007.1599-1-50-1': 'The curves nearly lie on top of one another in this plot.', '1007.1599-1-50-2': 'This implies that, for the mass function assumed here ([MATH]), the frequency of anomalies is largely dependent on the total mass in substructure and not so much on the lower mass cutoff.', '1007.1599-1-50-3': 'This degeneracy between mass function normalization and lower mass cutoff will limit how well monochromatic QSO lensing can disentangle these parameters.', '1007.1599-1-50-4': 'The assumed mass function implies that the fraction of mass in objects with masses above [MATH] are 41% for [MATH] and 56% for [MATH].', '1007.1599-1-51-0': 'We introduce another constraint in the bottom panel of figure [REF] based on the fraction of simulations with more than four images.', '1007.1599-1-51-1': '(This does not include the central demagnified image that forms near the center of the lens for nonsingular lens mass profiles.', '1007.1599-1-51-2': 'In our case, the density in the center of the lens diverges like [MATH], and this image never appears; it is infinitely demagnified.)', '1007.1599-1-51-3': 'Even after merging images with centroids less than 0.1 arcsec apart ,there are cases where the substructures cause further splitting of the images.', '1007.1599-1-51-4': 'Of the 32 QSO lenses in the Castles list of lenses with more than four images and simple lenses, none have more than 4 images of a single source separated by more than 0.1 arcsec. This puts a strong constraint on the allowed fraction of lenses that have more than four images, [MATH].', '1007.1599-1-51-5': 'There would be less than a 5% chance of this happening in the observed sample if [MATH] is greater than 0.089 and less than 10% chance if [MATH].', '1007.1599-1-51-6': 'These are the dotted lines in the bottom panels of figures [REF] and [REF].', '1007.1599-1-51-7': 'For equal surface density, more massive substructures cause more high image multiplicity lenses.', '1007.1599-1-52-0': 'For the models of set 2, the constraints discussed above are shown as dashed vertical lines in figures [REF] and [REF].', '1007.1599-1-52-1': 'They are, in terms of surface density, [MATH].', '1007.1599-1-52-2': 'The upper limit is reduced to [MATH] for set 1 ([MATH]).', '1007.1599-1-52-3': 'For set 3 (the [MATH] case), the upper limit for [MATH] is not reached in the simulation, but appears to be somewhere greater than [MATH].', '1007.1599-1-52-4': 'The lower limit on [MATH] is effectively the same in sets 1 through 3.', '1007.1599-1-53-0': 'Figure [REF] shows the importance of the internal structure of the subclumps on this constraint.', '1007.1599-1-53-1': 'If the size of the subsclumps is reduced by reducing [MATH] to 0.25 kpc, the frequency of flux anomalies is not strongly affected, but the frequency of high multiplicity lenses increases which reduces the acceptable range of [MATH].', '1007.1599-1-53-2': 'Increasing the sizes of the subclumps by increasing [MATH] to 1.0 kpc has a stronger effect on the frequency of anomalies and substantially reduces the frequency of high multiplicity lenses.', '1007.1599-1-53-3': 'The [MATH] limit is no longer of relevance in this case.', '1007.1599-1-53-4': 'Flattening the internal mass profile to [MATH] has little effect on the anomalies, but it does weaken the [MATH] limit to [MATH].', '1007.1599-1-54-0': 'To test if the [MATH] distribution is strongly dependent on the host lens distribution we chose, [MATH] is calculated for the "extra distortion" and "extra shear" models.', '1007.1599-1-54-1': 'The results are [MATH] and [MATH], respectively.', '1007.1599-1-54-2': 'The anomalies are very unlikely to be the result of a large scale distortion of the host lens.', '1007.1599-1-55-0': '# expectations for small-scale structure within the CDM model', '1007.1599-1-56-0': 'A good point of comparison is the fraction of mass in substructure within a projected radius of 10 kpc.', '1007.1599-1-56-1': 'This is easily measured in the simulations and since the Einstein radius is typically around 10 kpc, it is close to what is actually constrained by the lensing data.', '1007.1599-1-56-2': 'In our model, this quantity is given by [EQUATION] where the fiducial value [MATH]^-1[MATH] has been used.', '1007.1599-1-56-3': 'Note that this fraction scales with host mass in our model and in the simulations.', '1007.1599-1-56-4': 'Our limits for set 1 translate to [EQUATION]', '1007.1599-1-56-5': 'The upper limit will be larger if the substructure is less concentrated.', '1007.1599-1-57-0': '[CITATION] give [MATH] for the Via Lactea simulation, and [CITATION] give [MATH] with a large scatter in the Aquarius simulations.', '1007.1599-1-57-1': 'These simulations should be resolving substructure to below [MATH].', '1007.1599-1-57-2': 'Although there is some tension between our lower limit and these simulations, we consider them to be consistent considering the uncertainties in the predictions.', '1007.1599-1-57-3': 'For example, the baryons need to be inserted by hand into these Nbody simulations.', '1007.1599-1-57-4': 'The mass fraction decreases with the inclusion of baryons.', '1007.1599-1-57-5': 'The CDM clumps are likely to have a shallower slope then [MATH], so the upper limit on [MATH] from image multiplicities is probably as large as our [MATH] case.', '1007.1599-1-57-6': 'Taking these into consideration, it does not appear that there is any strong contradiction between our limits and the Nbody predictions.', '1007.1599-1-58-0': '[CITATION], [CITATION] and [CITATION] come to the conclusion that the substructure present in the simulations is not enough to cause the observed frequency of flux anomalies.', '1007.1599-1-58-1': 'It is possible that this is a result of too few simulated lenses to be statistically significant or it could be caused by over smoothing.', '1007.1599-1-58-2': 'It can be seen from the bootstrap resampling in figures [REF] and [REF] that the variations between lenses are substantial and even with 100,000 simulations they have a non-negligible effect on the calculated frequency of flux anomalies.', '1007.1599-1-58-3': 'The simulations also differ from the Nbody simulations in that a much larger ranges of lens mass, lens redshift, source redshift and lens ellipticities are used.', '1007.1599-1-58-4': '[CITATION] may have used too low a substructure mass range ([MATH]) to cause enough anomalies.', '1007.1599-1-58-5': 'The point source approximation to the magnifications might also play a role.', '1007.1599-1-59-0': 'Because we appear to be consistent with the simulations on the frequency of [MATH] anomalies does not mean that some other test, such as fitting each simulated lens to a smooth lens model, would not show some inconsistency.', '1007.1599-1-60-0': '# conclusions discussion', '1007.1599-1-61-0': 'We have preformed the largest number of lens simulations ever done with finite size sources.', '1007.1599-1-61-1': 'This was made possible by the new adaptive ray-tracing code GLAMER.', '1007.1599-1-61-2': 'We find that accounting for the finite size of the source is necessary for drawing accurate conclusions from the lensed QSO data.', '1007.1599-1-62-0': 'We find rough consistency between the [MATH]CDM predictions and observations.', '1007.1599-1-62-1': 'There is some tension between them in that our lower limit on the mass fraction in substructure is near the [MATH]CDM predictions from pure dark matter simulations.', '1007.1599-1-62-2': 'We do not think this is a serious problem yet, but with more data and improvements in the simulations these may come into conflict.', '1007.1599-1-62-3': 'In addition to the substructure within the primary lens there should be some contribution from intergalactic small-scale structure so one should expect the limits derived from the data to be somewhat higher than the limits derived from Nbody simulations of individual dark matter halos.', '1007.1599-1-62-4': 'The baryons also clearly play a role in shaping the lensing properties and they are not fully taken into account in the simulations at the necessary resolution.', '1007.1599-1-63-0': 'We have limited our study here to a substructure mass function of the form [MATH] with [MATH].', '1007.1599-1-63-1': 'This seems well motivated by the simulations and our goal here has been to test for consistency between the simulations and the data.', '1007.1599-1-63-2': 'With this mass function, the small mass substructure plays a smaller part in causing flux anomalies because most of the mass resides in larger mass objects.', '1007.1599-1-63-3': 'This will make it difficult to measure any possible lower mass cutoff using monochromatic QSO lensing alone.', '1007.1599-1-63-4': 'Fortunately there are some other prospects for probing the mass function in the future such as spectroscopic gravitation and Einstein rings .', '1007.1599-1-63-5': 'If the slope of the mass function is steeper than [MATH], the smaller structures will play a larger role in the lensing.', '1007.1599-1-64-0': 'It is clear that what is really required to make a more conclusive measurement of the amount of substructure in dark matter halos is more data.', '1007.1599-1-64-1': 'With 7 lenses, only limited conclusions can be made from a statistical point of view.', '1007.1599-1-64-2': 'We are also vulnerable to systematic errors.', '1007.1599-1-64-3': 'For the kind of study done here, more strong lenses measured in the radio and/or mid-infrared are needed.', '1007.1599-1-64-4': 'Planned large scale imaging surveys expect to increase the number of lensed QSOs in the visible by an order of magnitude so we look forward to great improvements in this field.'}

{'1007.1599-2-0-0': 'We investigate the statistics of flux anomalies in gravitationally lensed QSOs as a function of dark matter halo properties such as substructure content and halo ellipticity.', '1007.1599-2-0-1': 'We do this by creating a very large number of simulated lenses with finite source sizes to compare with the data.', '1007.1599-2-0-2': 'After analysing these simulations, our conclusions are: 1) The finite size of the source is important.', '1007.1599-2-0-3': 'The point source approximation commonly used can cause biased results.', '1007.1599-2-0-4': "2) The widely used [MATH] statistic is sensitive to halo ellipticity as well as the lens' substructure content.", '1007.1599-2-0-5': '3) For compact substructure, we find new upper bounds on the amount of substructure from the the fact that no simple single-galaxy lenses have been observed with a single source having more than four well separated images.', '1007.1599-2-0-6': '4) The frequency of image flux anomalies is largely dependent on the total surface mass density in substructures and the size-mass relation for the substructures, and not on the range of substructure masses.', '1007.1599-2-0-7': '5) Substructure models with the same size-mass relation produce similar numbers of flux anomalies even when their internal mass profiles are different.', '1007.1599-2-0-8': "6) The lack of high image multiplicity lenses puts a limit on a combination of the substructures' size-mass relation, surface density and mass.", '1007.1599-2-0-9': '7) Substructures with shallower mass profiles and/or larger sizes produce less extra images.', '1007.1599-2-0-10': '8) The constraints that we are able to measure here with current data are roughly consistent with [MATH]CDM Nbody simulations.', '1007.1599-2-1-0': '# introduction', '1007.1599-2-2-0': 'The Cold Dark Matter (CDM) model with a cosmological constant ([MATH]CDM) has become the standard model of cosmology.', '1007.1599-2-2-1': 'This model is in good agreement with a variety of observational probes of the large scales distribution of matter and galaxies in the Universe and is in general agreement with probes of the distribution of mass in galaxy clusters and in large galaxies.', '1007.1599-2-2-2': 'In the [MATH]CDM model, dark matter clumps into halos and galaxies form in the halos.', '1007.1599-2-2-3': 'On small scales, [MATH]CDM predicts that dark matter halos exist down to very small masses; the exact lower limit depending on the properties of the CDM particle and its thermal history.', '1007.1599-2-2-4': 'It has long been recognized that the number of observed dwarf galaxies in the local group of galaxies falls well short of the number of predicted halos .', '1007.1599-2-2-5': 'This is referred to as the substructure problem.', '1007.1599-2-2-6': 'Either galaxy formation is highly suppressed in small mass halos or [MATH]CDM needs to be modified in some way by, for example, changing the properties of the dark matter particle or the initial conditions for the density fluctuation in the Universe.', '1007.1599-2-2-7': 'Warm Dark Matter (WDM) is a popular alternative.', '1007.1599-2-2-8': 'Whether or not these small mass halos exist has been one of the most pressing unanswered question in cosmology for a decade.', '1007.1599-2-3-0': '[CITATION] demonstrated that if small-scale structure exists in the distribution of dark matter it will have a strong effect on the magnifications of quasar images in strong gravitational lenses.', '1007.1599-2-3-1': 'This effect causes the flux ratio between images to disagree with any lens model with a smooth distribution of matter.', '1007.1599-2-3-2': 'These cases are call anomalous flux ratios.', '1007.1599-2-3-3': 'A particular case had been studied by [CITATION] and subsequently it was shown that anomalies are common in quasar lenses .', '1007.1599-2-3-4': 'This work and a number of subsequent studies [CITATION] relied on fitting lens models to individual lens systems.', '1007.1599-2-3-5': 'It has not yet been shown clearly what can be causing these anomalies and what cannot be causing them.', '1007.1599-2-4-0': 'In a parallel approach, we and others have tried to simulate the lenses directly from cosmological Nbody simulations to determine if they are consistent with the observed frequency of flux anomalies .', '1007.1599-2-4-1': 'The first study predicted a large number of anomalies, but it may have been strongly affected by shot noise.', '1007.1599-2-4-2': 'The two more recent and higher resolution studies found that the substructure in the Nbody simulations is not sufficient to cause the observed flux anomalies (also the conclusion of [CITATION]).', '1007.1599-2-4-3': 'This is largely because of the small number density of substructures near the radii where images form (typically around 10 kpc in projection).', '1007.1599-2-4-4': 'These studies relied on only a few projections of a small number of high resolution halos.', '1007.1599-2-4-5': 'It is possible that these results are a statistical fluke or that the observed anomalies are largely caused by dark matter objects along the line of sight but not inside the halo of the primary lens [CITATION].', '1007.1599-2-4-6': 'Answering the question of whether the Nbody simulations have enough small-scale structure in them to account for the flux ratio anomalies is one of the primary goals of this paper.', '1007.1599-2-5-0': 'It is very difficult to realistically simulate strong QSO lenses from an Nbody simulation.', '1007.1599-2-5-1': 'The first, and most important, problem is that shot noise from the discrete particles has a strong effect on the image magnifications.', '1007.1599-2-5-2': 'Roughly, the error in the magnification goes as [MATH] where [MATH] is the magnification and [MATH] is the number of particles over which the smoothing is done.', '1007.1599-2-5-3': 'Since [MATH] can be large, 100 or larger in the best cases for detecting substructure, the amount of smoothing needed to obtain an accuracy of even 10% is very large.', '1007.1599-2-5-4': 'So much smoothing can even smooth out the very substructures one wants to detect.', '1007.1599-2-5-5': 'Because of this [CITATION] replace an Nbody simulation with a simple analytic model fit to an Nbody simulation.', '1007.1599-2-5-6': 'A second problem is that the highest resolution simulations do not contain baryons.', '1007.1599-2-5-7': 'Baryons have a strong effect on the profile of the lens and in some cases dominate the mass within one Einstein radius.', '1007.1599-2-5-8': 'The baryons need to be put in "by hand".', '1007.1599-2-5-9': 'A third problem is that the extremely high resolution simulations required provide one, or at best a few, dark matter halos.', '1007.1599-2-5-10': 'Variations between halos make their lensing properties and their tendency to produce anomalies very different.', '1007.1599-2-5-11': 'It will be demonstrated in this paper that only very limited conclustions about the CDM model can be drawn from a single simultated lens.', '1007.1599-2-6-0': 'To avoid these problems, we take a different approach in this paper.', '1007.1599-2-6-1': 'We produce a large number of analytic lens models that are meant to reproduce the population of lenses expected in the [MATH]CDM model.', '1007.1599-2-6-2': 'We then determine the frequency of flux ratio anomalies in these lenses and compare it to the observed frequency.', '1007.1599-2-6-3': 'We adjust the properties and abundance of the substructures to see what kind of substructure is consistent with observations.', '1007.1599-2-6-4': 'The allowed statistical properties of the substructures are compared with the properties of Nbody halos.', '1007.1599-2-7-0': 'All previous studies, except [CITATION], have also suffered from the problem that the sources are treated as infinitely small points.', '1007.1599-2-7-1': 'The magnification of individual images are calculated by taking derivatives of the gravitational force at the position of the image.', '1007.1599-2-7-2': 'It will be shown in this paper, that since the physical size of the quasar radio or mid-infrared emission regions are similar to the sizes of the substructures of interest the point source magnifications are not accurate approximations.', '1007.1599-2-7-3': 'We use a new, high speed lensing code called GLAMER (Gravitational Lensing with Adaptive MEsh Refinement) that is the first one capable of producing a very large number of simulated lenses with finite sources in a reasonable amount of time.', '1007.1599-2-7-4': 'It does this through an adaptive mesh refinement algorithm that will be briefly described in section [REF].', '1007.1599-2-8-0': 'In section [REF], the models and techniques used to create simulated lenses are described.', '1007.1599-2-8-1': 'In section [REF], the results of those simulations are discussed.', '1007.1599-2-8-2': 'Ways of comparing the results to the available lensing data are presented in section [REF].', '1007.1599-2-8-3': 'The results are compared with the predictions of cosmological Nbody simulations in section [REF].', '1007.1599-2-8-4': 'A summery and discussion are given in section [REF].', '1007.1599-2-9-0': '# lens simulations', '1007.1599-2-10-0': 'Our approach in this paper is to produce a large population of realistic simulated lenses and then compare their statistical properties to the observed population of lenses.', '1007.1599-2-10-1': 'To do this, we must develop a model for the population of gravitational lens that includes the host, galaxy + dark matter halo, and the substructures within the host.', '1007.1599-2-10-2': 'We will not consider the effects of companion galaxies with masses roughly equivalent to the primary lens in this paper.', '1007.1599-2-11-0': '## Host lens model', '1007.1599-2-12-0': 'There is significant evidence from lensing and X-ray observations that early-type galaxies have a [MATH] mass profiles .', '1007.1599-2-12-1': 'In accordance with this finding, we model the host lenses as Distorted Singular Isothermal Ellipsoids (DSIE).', '1007.1599-2-12-2': 'The surface mass density for this model is [EQUATION] where the Einstein radius is [EQUATION] and the critical surface density is [EQUATION] where [MATH], [MATH] and [MATH] are the angular size distance to the lens, to the source and between the lens and the source respectively.', '1007.1599-2-12-3': 'The first part ([REF]) is a Singular Isothermal Ellipsoids whose lensing properties have been extensively studied (see [CITATION] for example).', '1007.1599-2-12-4': 'The deflection angle and shear caused by the series in ([REF]) have been worked out by [CITATION], although with different notation.', '1007.1599-2-13-0': 'The perturbations [MATH] are assumed to be of the same order as the observed perturbations in the surface brightness profile of of early-type galaxies.', '1007.1599-2-13-1': 'Typical values for [MATH] and [MATH] are two or three percent, but accurate statistics are not available .', '1007.1599-2-13-2': 'We draw random values from a Gaussian distribution with variance 0.005 for [MATH] and [MATH] and 0.01 for [MATH].', '1007.1599-2-13-3': 'We take [MATH] terms to be zero.', '1007.1599-2-13-4': 'In the observations, [MATH] is usually defined with the orientation of this mode fixed to the same axis as the axis of the elliptical component to define the "diskyness" or "boxyness" of the galaxy.', '1007.1599-2-13-5': 'Since the alignment has important effects on the lensing properties, we relax this requirement somewhat and allow [MATH] to vary from the position angle of the elliptical component.', '1007.1599-2-13-6': 'The misalignment is normally distributed with variance 3 degrees.', '1007.1599-2-14-0': 'We also include background shear and convergence in the model.', '1007.1599-2-14-1': '[CITATION] calculated the expected distribution of [MATH] and [MATH] in an Nbody simulation at potential lenses.', '1007.1599-2-14-2': 'They found that [MATH] and [MATH] are both roughly lognormally distributed with a variance of [MATH].', '1007.1599-2-14-3': 'We assume this distribution in our model.', '1007.1599-2-14-4': 'Analytic estimates by [CITATION] are in agreement with this result, as are observations .', '1007.1599-2-15-0': 'The model described above is what will be called the "standard" host model.', '1007.1599-2-15-1': 'To test how sensitive magnification anomalies are to the host model, we perform a series of tests where the distortions to the lens are increased.', '1007.1599-2-15-2': 'For the "extra distorted model", we triple the variance in the distortion modes and decouple their orientation from the orientation of the elliptical component.', '1007.1599-2-15-3': 'For the "extra shear model", we triple the variance in the background shear and convergence.', '1007.1599-2-16-0': '### Distributions of host properties', '1007.1599-2-17-0': "Calculating the expected distribution of the lenses' redshifts, velocity dispersions and ellipticities requires knowing not only the source luminosity and redshift distributions of lenses and sources, but also the many selection effects that might be important.", '1007.1599-2-17-1': 'The sample of lenses we wish to compare our results with were discovered in many different ways and do not have a uniform, well defined selection criterion.', '1007.1599-2-17-2': 'Instead of trying to model these biases, we use the distributions of already known lenses when possible.', '1007.1599-2-18-0': 'For the lens and sources redshifts, we use the observed values for the Castles lenses.', '1007.1599-2-18-1': 'There are 60 lenses with measured source and lens redshift pairs.', '1007.1599-2-18-2': 'We draw randomly from these sets of redshifts.', '1007.1599-2-18-3': 'The lenses discussed in section [REF] are a subsample of these.', '1007.1599-2-19-0': 'To get a sample of host velocity dispersions, [MATH], we use the velocity dispersions from the SLACS lenses .', '1007.1599-2-19-1': 'This sample of 61 lenses is used to make a cumulative distribution of [MATH].', '1007.1599-2-19-2': 'The discrete distribution is linearly interpolated to get a continuous cumulative distribution and then this is randomly sampled from.', '1007.1599-2-19-3': 'In the SLACS sample, the measured velocity dispersion of stars and the velocity dispersion of the best-fit SIE models have statistically indistinguishable distributions.', '1007.1599-2-19-4': 'We choose to use the best-fit SIE velocity dispersions.', '1007.1599-2-19-5': 'These values range from 160 to [MATH]^-1[MATH].', '1007.1599-2-20-0': 'The axis ratios, [MATH], are sampled independently from the SLACS lenses in the same way as the velocity dispersions.', '1007.1599-2-20-1': 'No possible correlations between the internal structure of the lenses and their redshift are reproduced in this sampling.', '1007.1599-2-20-2': 'The average of this distribution is [MATH], the standard deviation 0.14 and the range is [MATH].', '1007.1599-2-20-3': 'The SLACS lenses are at relatively low redshift because of their selection criterion, but observations indicate that the internal structure of early-type galaxies do not evolve significantly between [MATH] and 0 .', '1007.1599-2-21-0': 'We consider only four image quasar lenses in this paper, while the SLACS lenses include two image lenses.', '1007.1599-2-21-1': 'The asymmetry of the lens changes the area enclosed in the tangential caustic and thus a sample of four image lenses will tend to have more asymmetric lenses than a sample that includes all multiple image cases.', '1007.1599-2-21-2': 'To correct for this bias, we calculate the ratio of the area within the tangential caustic to the area within the radial caustic (or "cut" in the case of a DSIE).', '1007.1599-2-21-3': 'The number of sources used for the lens is then proportional to this ratio.', '1007.1599-2-21-4': 'More circular galaxies will have less lenses in the final sample.', '1007.1599-2-21-5': 'This corrects for the bias in the SLACS lenses relative to the four image quasars.', '1007.1599-2-21-6': 'From 0 to [MATH], source positions are used for each lens model.', '1007.1599-2-21-7': 'This method of using a variable number of sources per lens is something of a compromise; ideally one would have a population of lenses that reflected the biases and one source per lens, but to do this the caustic structure of each lens would need to be calculated and then many of those with a small cross-sections for producing four images would be discarded.', '1007.1599-2-21-8': 'This would be computationally inefficient.', '1007.1599-2-21-9': 'A small number of sources per lens means that the population of high cross-section lenses will be better sampled, but if the average number of sources per lens is set too low all the lenses with small cross-sections will have zero sources.', '1007.1599-2-21-10': 'We have set the number of sources per lens so that lenses with zero sources are rare ([MATH] 1%).', '1007.1599-2-22-0': 'For each lens model the source centers are chosen to randomly cover a region that encloses the region within the tangential caustic.', '1007.1599-2-22-1': 'Some of these source positions give rise to less than four images (when the source intersects the caustic or is completely outside the caustic) and some give rise to more than four images (when caustics structure is more complicated).', '1007.1599-2-22-2': 'The cases with less than four images are discarded in the analysis that follow.', '1007.1599-2-23-0': '## Substructure model', '1007.1599-2-24-0': 'We wish to construct a substructure model that reflects the expectations we have from Nbody simulation, but is relatively simple and has a small number of parameters that can be varied to measure the agreement or disagreement with [MATH]CDM.', '1007.1599-2-25-0': 'Simulations show that the mass fraction in substructure within a projected radius increases roughly linearly with projected radius .', '1007.1599-2-25-1': 'With a SIE mass model, this implies that the surface mass density of substructure is constant at least near the Einstein radius and interior to it.', '1007.1599-2-25-2': 'This will be assumed in all cases.', '1007.1599-2-26-0': 'The mass function of subhalos in Nbody simulations is found to be a power-law [EQUATION] where [MATH] is the number of substructures in a halo.', '1007.1599-2-26-1': '[CITATION] found that [MATH] up to about [MATH] of the halo mass without any resolved lower mass limit in halos as a whole.', '1007.1599-2-26-2': 'Transforming mass function into a projected mass function in 2 dimensions is not straightforward because of mass segregation in the host halo.', '1007.1599-2-26-3': 'The projected substructure number density will be denoted [MATH] and the projected mass function will be [MATH].', '1007.1599-2-27-0': 'It found that the substructures of different masses are distributed within host halos in remarkably similar ways except that at each radius the mass function has an upper mass cutoff .', '1007.1599-2-27-1': 'If it were not for this mass cutoff the projected mass function (surface number density) would have the same slope as the the total mass function.', '1007.1599-2-27-2': 'Instead, the projected mass function will become steeper than [MATH] above some mass scale.', '1007.1599-2-27-3': 'We represent this effect in our model crudely with an upper mass cutoff that is smaller than the one found for the complete mass function [EQUATION] with [MATH].', '1007.1599-2-27-4': 'This is a crude model that could be improved on in the future.', '1007.1599-2-28-0': 'The maximum mass in the mass function must be a function of host halo size.', '1007.1599-2-28-1': 'A mass scale for the host can be defined as the mass within a fixed radius ([MATH]) or the mass within a radius where the average density reaches a fixed threshold ([MATH]).', '1007.1599-2-28-2': 'The latter is the one commonly used to define the mass of a halo in cosmology although the virial radius is generally larger than the radii over which one would expect the SIE model to hold.', '1007.1599-2-28-3': 'However, if the concentration of the halos does not vary greatly within the range of host lenses then the same scaling would be expected in the inner regions.', '1007.1599-2-28-4': 'Making the maximum substructure mass a fixed fraction of the host halo mass results in [EQUATION]', '1007.1599-2-28-5': 'The same scaling is assumed for the minimum mass.', '1007.1599-2-28-6': "[MATH] is used as an adjustable parameter to change the mass scale and test the data's consistency with a mass cutoff as would be expected in many alternatives theories to CDM.", '1007.1599-2-28-7': 'The normalizing halo is fixes to [MATH]^-1[MATH].', '1007.1599-2-29-0': 'The normalization of the mass function ([REF]) needs to be set.', '1007.1599-2-29-1': 'To agree with Nbody simulations, the fraction of mass in substructure at a fixed fraction of the virial radius should be the same in all halos.', '1007.1599-2-29-2': 'Since [MATH] and ([REF]) makes average mass scale like [MATH] the normalization must scale like [MATH].', '1007.1599-2-29-3': 'Explicitly the result is [EQUATION]', '1007.1599-2-29-4': 'The parameter [MATH] is then the total surface number density of substructures is a host with [MATH] and is not a function of projected radius.', '1007.1599-2-30-0': 'Although the mass fraction in substructure at a fixed fraction of the halo radius is the same for all lenses, the same is not true at the Einstein radius.', '1007.1599-2-30-1': 'Since [MATH], the total surface density at [MATH] is independent of [MATH] for lenses and sources at the same redshift, which makes the mass fraction scale as [MATH] at this radius.', '1007.1599-2-30-2': 'As a result, we might expect substructure to be more important for larger lenses.', '1007.1599-2-31-0': 'The internal structure of the substructures is, for simplicity, a simple power-law with a cutoff radius [EQUATION]', '1007.1599-2-31-1': "In the classical analytic treatment, the average mass density within the tidal radius is proportional to the average mass density of the host within the substructure's orbit .", '1007.1599-2-31-2': 'This implies [MATH] if all the substructures are at the same distance from the center of the host, which we assume.', '1007.1599-2-31-3': 'Since the mass density at a fixed fraction of the host halo radius is independent of the host size, it is expected that this relation is independent of the host size: [EQUATION]', '1007.1599-2-31-4': 'Here [MATH] is a free parameter describing the size of the most massive substructures.', '1007.1599-2-31-5': 'In a more realistic model, there would be a significant scatter in the [MATH]-[MATH]-[MATH] relation, but for our purposes this relation is sufficient.', '1007.1599-2-31-6': 'Using the classical tidal radius, the three dimensional distance from the center of the lens that this cutoff radius corresponds to is [EQUATION]', '1007.1599-2-31-7': 'Our fiducial model will have [MATH] and [MATH] so [MATH] is a representative distance which is, perhaps, optimistically compact.', '1007.1599-2-31-8': 'We will vary [MATH] from [MATH] to [MATH].', '1007.1599-2-32-0': 'It should be noted that the appropriate [MATH] for lensing would be significantly smaller than the average [MATH] for subhalos in general.', '1007.1599-2-32-1': 'Most subhalos are at large radii ([MATH] > [MATH]) because there is so much volume at large radii to make up for the lower weighted number density.', '1007.1599-2-32-2': 'Projecting along the line-of-sight weights the inner regions of the halo more.', '1007.1599-2-32-3': 'The difference is an order of magnitude or more.', '1007.1599-2-32-4': 'This means that the substructure that are important for lensing will tend to be denser than the overall population.', '1007.1599-2-33-0': 'In summery, the substructure model has the free parameters [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and the normalization host velocity dispersion [MATH] which we fix at [MATH]^-1[MATH].', '1007.1599-2-33-1': 'However, in the simulations described in the following [MATH] and [MATH] are fixed and the remaining parameters are varied.', '1007.1599-2-34-0': '## Ray-shooting', '1007.1599-2-35-0': 'The sources that we wish to use in our simulation have sizes of [MATH] and the substructures can have similar sizes.', '1007.1599-2-35-1': 'Therefore, it is essential that we be able to calculate the magnification of finite size sources.', '1007.1599-2-35-2': 'This requirement has been widely ignored in the literature because it is difficult to map the image of a finite source in a short enough amount of time to make it possible to create the large number of simulated lenses required for this problem.', '1007.1599-2-35-3': 'A new code, GLAMER, has been developed for this and other applications.', '1007.1599-2-35-4': 'This code employs a highly optimized adaptive mesh refinement scheme which allows the shapes of the images and their area to be calculated rapidly.', '1007.1599-2-35-5': '(Because of surface brightness conservation, the area of a uniform brightness image is proportional to its magnification.)', '1007.1599-2-35-6': 'This allows us to make millions of mock lenses with a finite size source in a relatively short amount of time.', '1007.1599-2-35-7': 'Figure [REF] illustrates how the grid is refined to find all the images and their areas.', '1007.1599-2-35-8': 'Figure [REF] shows the critical curve and caustic structure for one example lens.', '1007.1599-2-35-9': 'For more details on this code, see [CITATION].', '1007.1599-2-36-0': 'The range of positions in which a substructure will make a significant change to the magnification of an image depends on the mass of the substructure.', '1007.1599-2-36-1': 'To optimize calculations, small-mass substructures that are far away from the lens are omitted from the calculation while more massive substructures further from the lens are included.', '1007.1599-2-36-2': 'To accomplish this, a mass dependent cutoff radius from the center of the lens is used: [EQUATION]', '1007.1599-2-36-3': 'The first two terms ensure that all substructures within two Einstein radii plus the radius of the substructure are included.', '1007.1599-2-36-4': 'The third term ensures that any substructure close enough to cause a perturbation to the lens that is not well approximated as a pure shear will be included.', '1007.1599-2-36-5': 'The parameter [MATH] controls how large the variation in the shear across the Einstein radius are allowed to be.', '1007.1599-2-36-6': 'We set this parameter to [MATH].', '1007.1599-2-36-7': 'The contribution from substructures or companions outside this range is considered to be part of the background shear discussed in section [REF] as part of the host lens model.', '1007.1599-2-37-0': 'For each lens model (host and substructure), the critical curves and caustics are found first.', '1007.1599-2-37-1': 'There are sometimes multiple, disconnected critical curves.', '1007.1599-2-37-2': 'The main tangential caustic is found by requiring its critical curve to be the one that encompasses the most area while also surrounding the center of the lens.', '1007.1599-2-37-3': 'The area within the tangential caustic is calculated and the number of source positions that will be used for that lens is calculated as described in section [REF].', '1007.1599-2-37-4': 'The sources are required to have their centers inside the tangential caustic, but they are otherwise randomly distributed.', '1007.1599-2-37-5': 'Because of the finite source size, some images will be merged and this results in less than four images.', '1007.1599-2-38-0': 'Some lenses have more than the four images that the undistorted host model alone would predict.', '1007.1599-2-38-1': 'Some of these additional images are very small and/or so close to another image that they would not be observed as separate images.', '1007.1599-2-38-2': 'We do a rough initial cut in all cases by merging together any images with centroids that are less than 0.1 arcsec apart, roughly the resolution of the Hubble Space Telescope (HST).', '1007.1599-2-38-3': 'Further discussion of additional images is given in the next section.', '1007.1599-2-39-0': 'Table [REF] lists the simulation runs that were performed.', '1007.1599-2-39-1': 'They are in batches of 100,000 lenses with fixed substructure parameters.', '1007.1599-2-39-2': 'The first five sets of simulations have no substructure in them and are used to evaluate the importance of distortions to the host lens model and establish a baseline from which to measure the importance of substructure.', '1007.1599-2-39-3': 'The the parameters for the remaining twelve simulation were chosen to explore the importance of particular substructure properties for lensing.', '1007.1599-2-39-4': 'Set 2 is taken to be a fiducial model.', '1007.1599-2-39-5': 'This is a somewhat arbitrary choice, but we do believe that it is similar to the predictions of Nbody simulations except for the internal profile of the substructures which, as will be shown, has relatively little effect on the lensing properties.', '1007.1599-2-39-6': 'Relative to simulation set 2, set 1 has a higher minimum mass (and average mass), set 3 has a lower minimum mass, set 4 has a smaller source size, set 5 has more compact substructure (a smaller [MATH]), sets 6 and 8 have less compact substructure and set 7 has a shallower internal mass profile for the substructures.', '1007.1599-2-39-7': "In sets 9 and 10, the upper mass cutoff is increased to [MATH] which is about 10% of the host's virial mass.", '1007.1599-2-39-8': 'Set 9 has more compact substructures than set 10.', '1007.1599-2-39-9': 'The [MATH] values are set here so that the size-mass relation is the same as in sets 7 and 8.', '1007.1599-2-39-10': 'For eample, a [MATH] substructure has the same size in sets 9 and 7.', '1007.1599-2-39-11': 'The rescaling is nessisary because the size-mass relation is normalized at the maximum mass in each model which changes between these models.', '1007.1599-2-39-12': 'In sets 11 and 12, the upper mass cutoff is decreased to [MATH].', '1007.1599-2-39-13': 'Set 11 has more compact substructures than set 12.', '1007.1599-2-39-14': 'Again the [MATH] values are set to preserve the mass-size relation between sets 7 and 11, and between sets 8 and 12.', '1007.1599-2-40-0': 'The range in surface number density in the simulation sets is meant to span the credible range within a CDM-like model .', '1007.1599-2-40-1': 'In set 3 the number density of substructures is much higher for the same mass density so because of computer time constraints the mass density range for this set does not go as high as in the others although the number density goes higher.', '1007.1599-2-40-2': 'The ranges in [MATH] are were chosen to cover the realistic range in a CDM-like model.', '1007.1599-2-41-0': '# results', '1007.1599-2-42-0': 'We create several million simulated lenses and save the image positions and magnifications.', '1007.1599-2-42-1': 'We also store the point source magnifications at the centroid of each image and the point source magnification for the point in the image that is closest to the center of the source.', '1007.1599-2-42-2': 'Some of the host lens parameters are also stored.', '1007.1599-2-42-3': 'In this paper, for ease of comparison, we classify the observed and simulated lenses and reduce the position and magnification information to two parameters.', '1007.1599-2-42-4': 'The parameter [MATH] is defined in figure [REF].', '1007.1599-2-42-5': 'A small value of [MATH] indicates the source is near a cusp in the caustic.', '1007.1599-2-42-6': 'Figure [REF] also describes what a long- and short-axis lenses are.', '1007.1599-2-42-7': 'We have found that a good observational way of sorting the lenses into these categories is by comparing the angular distance between the center of the lens and the singlet image to the distance between the center of the lens and the central image of the triplet.', '1007.1599-2-42-8': 'If the former is greater, then the lens is a short-axis lens.', '1007.1599-2-42-9': 'Otherwise, it is a long-axis lenses.', '1007.1599-2-43-0': 'The second parameter used to characterize each lens is [EQUATION] where "[MATH]" is for long-axis lenses and "[MATH]" for short-axis lenses.', '1007.1599-2-43-1': 'The magnifications for the images in the triplet are [MATH], [MATH] and [MATH], with [MATH] being for the central image.', '1007.1599-2-43-2': 'The original motivation for this parameter was that [MATH] asymptotically as a point source approaches a cusp in the caustic .', '1007.1599-2-43-3': 'The [MATH] parameter has been widely used because of this model independent prediction.', '1007.1599-2-43-4': 'In practice, [MATH] is not constrained to a very small region around zero because of finite source effects and the invalidity of the lowest-order expansion of the lensing equation around the cusp.', '1007.1599-2-43-5': 'And, as will be shown, the distribution of [MATH] is not very model independent.', '1007.1599-2-44-0': 'Figures [REF] and [REF] shows the distribution of [MATH] and [MATH] for the sample of simulations listed in the captions.', '1007.1599-2-44-1': 'It can be seen that the simulated lenses occupy a well localized regions in these diagrams when no substructure is present.', '1007.1599-2-44-2': 'Even when substructure is present at the levels investigated, the majority of lenses occupy the same regions with a smaller number of cases spread out in tails to the distribution.', '1007.1599-2-45-0': 'Figure [REF] shows how important the ellipticity of the host lens is to the distribution of [MATH] values.', '1007.1599-2-45-1': 'Distortions to the SIE model and background shear do broaden the distribution, but ellipticity has a particularly strong effect.', '1007.1599-2-45-2': 'If only low ellipticity lenses are considered the [MATH] values are restricted to a much narrower band.', '1007.1599-2-45-3': 'The sample of lenses is biased toward high ellipticities relative to the general population of lenses because the cross-section for producing four images (the area within the tangential caustic) is increases with increasing ellipticity.', '1007.1599-2-45-4': 'At the same time, Nbody simulations might be biased toward low ellipticity since generally only dynamically well relaxed systems are chosen for very high resolution simulations.', '1007.1599-2-45-5': 'This can explain some of the discrepancies between simulations and observations that have been reported .', '1007.1599-2-45-6': 'This will be further discussed in section [REF].', '1007.1599-2-46-0': 'Figure [REF] is similar to figure [REF], but the effect of substructure on the [MATH]-[MATH] distribution is illustrated.', '1007.1599-2-46-1': "An additional 10% error on each image's flux is added to conservatively account for typical obervational uncertainties.", '1007.1599-2-46-2': 'Substructure has the effect of producing a papulation of extreme outliers in this distribution.', '1007.1599-2-47-0': 'Figure [REF] shows the fractional error made in the magnifications when the point source magnification is used.', '1007.1599-2-47-1': 'It can be seen there that the fractional error is small for magnifications less than around 5.', '1007.1599-2-47-2': 'This is confirmation that the numerical errors made by the ray-tracing code are small.', '1007.1599-2-47-3': 'At higher magnifications, larger errors are made when the source is 10 pc.', '1007.1599-2-47-4': 'This is not a numerical effect.', '1007.1599-2-47-5': 'It can also be seen in figure [REF] that substructure causes the errors made by using the point magnification to increase when the source size is 10 pc, but less so when the source size is 1 pc.', '1007.1599-2-47-6': 'This is in agreement with expectations because the source size of 10 pc is closer to the characteristic scale of the substructures.', '1007.1599-2-48-0': 'Figure [REF] shows the ratio between the point source magnifications and the finite source magnifications.', '1007.1599-2-48-1': 'Again, it can be seen that numerical errors are not playing a large part.', '1007.1599-2-48-2': 'It is evident that the point source magnifications are not evenly distributed around the finite source magnifications.', '1007.1599-2-48-3': 'Centroid point source magnifications tend to overestimate the real magnification; in some cases by a large factor.', '1007.1599-2-48-4': 'This is the magnification that would be calculated when fitting a lens model to an observed lens.', '1007.1599-2-48-5': 'In the simulation, the centroid is calculated by doing a flux weighted average over the pixels on the simulation grid.', '1007.1599-2-48-6': 'The nearest point magnification is much less biased and in the opposite direction; the magnification is underestimated.', '1007.1599-2-48-7': 'In other lensing simulations, the source position is often fixed and the images are found by an iterative minimization algorithm.', '1007.1599-2-48-8': 'This would give essentially the same result as our nearest point magnification.', '1007.1599-2-48-9': 'Both effects are much smaller for a smaller source size, as they should be.', '1007.1599-2-49-0': 'Many images were merged because their centroids were within 0.1 arcsec. In these cases, it makes no sense to take the closest point magnification since the closest point is not unique.', '1007.1599-2-49-1': 'Unsurprisingly, the magnification at the centroid point is an even worse approximation in these cases, as can be seen in figures [REF] and [REF].', '1007.1599-2-49-2': 'In exceptional cases, the centroid might not even be in one of the images that are merged.', '1007.1599-2-49-3': 'As expected, these cases only arise when substructure is present.', '1007.1599-2-50-0': 'Figures [REF] and [REF] should give one pause before using the point source approximation for the magnification in any substructure lensing study or when interpreting the results of any studies that use this approximation.', '1007.1599-2-51-0': '## frequency of [MATH] - [MATH] outliers', '1007.1599-2-52-0': 'To determine how often it would be expected for a lens to have [MATH] and [MATH] values that are inconsistent with a smooth lens model we define a region around the distribution in the case where no substructure is present and find how many simulated lenses lie outside this region when substructure is added.', '1007.1599-2-52-1': 'We define this region by taking bins in [MATH] that contain 2000 simulations taking the long-axis and short-axis cases separately.', '1007.1599-2-52-2': 'Upper and lower boundaries within each bin are set such that 2.5% of the simulations in the bin are greater than the upper bound and an equal number are less than the lower bound.', '1007.1599-2-52-3': 'The bins completely cover the full possible range of [MATH].', '1007.1599-2-52-4': 'Without substructure, 5% of a lens lie outside of this region.', '1007.1599-2-52-5': 'The fraction of simulated lenses outside this region when substructure is added will be called the fraction of outliers.', '1007.1599-2-53-0': 'Figure [REF] shows the fraction of outliers as a function of the substructure surface number density, [MATH], for different substructure minimum masses (simulation sets 1, 2 and 3).', '1007.1599-2-53-1': 'A significant fraction of the lenses are found to be outliers.', '1007.1599-2-53-2': 'The top panels of figures [REF] through [REF] show the same outlier fraction, but as a function of surface mass density.', '1007.1599-2-54-0': 'It is surprising that in figure [REF] the outlier fraction appears dependent only on the total surface mass density and not on the lower mass cutoff.', '1007.1599-2-54-1': 'One might think that all the lensing is being done by the most massive substructures and this is why the lower mass cutoff is not important in these cases.', '1007.1599-2-54-2': 'This does not seem to be the case; from set 1 ([MATH]) to set 3 ([MATH]) the mass density in the highest decade of mass ([MATH] to [MATH]) drops by 60% for the same total surface mass density and yet the number of outliers is unchanged.', '1007.1599-2-55-0': 'Figure [REF] shows the importance of compactness and internal structure on the number of outliers.', '1007.1599-2-55-1': 'The substructure mass function is the same for all the models in this figure.', '1007.1599-2-55-2': 'The slope of the internal density profile, [MATH], seems to have very little effect on the outlier fraction.', '1007.1599-2-55-3': 'On the other hand, the size of the substructures, or their compactness, does have a strong influence of the outlier fraction.', '1007.1599-2-55-4': 'Between [MATH] and [MATH] the fraction decreases significantly.', '1007.1599-2-55-5': 'Since the size-mass relation of the substructures is related to their galactocentric distance through tidal stripping, this sensitivity would provide information on where the substructures are within the lens halo or outside of it.', '1007.1599-2-56-0': 'In figure [REF], the upper substructure mass limit is changed to investigate further the insensitivity to mass range.', '1007.1599-2-56-1': 'It is seen again that for the same mass-size relation the fraction of outliers is dependent on the total surface mass density and relatively insensitive to the upper mass cutoff.', '1007.1599-2-56-2': 'The sensitivity to substructure compactness is again clearly present.', '1007.1599-2-56-3': 'Set 9 with [MATH] appears to produce slightly less outliers than set 2 with [MATH].', '1007.1599-2-56-4': 'This could be because large substructures will sometimes displace the image positions and magnifications significantly while preserving a low [MATH] value; the cusp in the caustic is moved, but its shape remains relatively intact.', '1007.1599-2-57-0': 'From the upper panels of figures [REF] through [REF], it can be seen that if the size-mass relation is held fixed the outlier fraction is largely a function of the total surface mass density in substructures and not the range of substructure masses.', '1007.1599-2-57-1': 'This conclusion may depend on the function used here ([MATH]).', '1007.1599-2-57-2': 'Further simulations will be needed to investigate this.', '1007.1599-2-57-3': 'Changing the size-mass relation so that the substructures are less dense does reduce the fraction of anomalies (sets 6 and 8).', '1007.1599-2-58-0': '# comparison with data', '1007.1599-2-59-0': 'To avoid contamination from microlensing by stars in the lens galaxy, differential extinction and variability of the source on time-scales smaller than the image delay times, we compare our simulations only to quad lenses measured in the radio and the mid-infrared.', '1007.1599-2-59-1': 'Since we have not included companion galaxies to the primary lens in our simulations, we also remove lenses with nearby galaxies that appear to have similar masses to the primary.', '1007.1599-2-59-2': 'This removes 1608+656 and 1004+4112 from the list.', '1007.1599-2-59-3': 'There is a very faint dwarf galaxy within the Einstein radius of 2045+265 , but we will consider this to be a substructure and not a companion galaxy because it is small.', '1007.1599-2-59-4': 'Lens models show that this substructure would need to be unnaturally elongated to cause the flux anomaly in this system, so there is probably another substructure present.', '1007.1599-2-59-5': 'The lenses must also have a detected lens galaxy which eliminates 0134-0931 and 0128+437.', '1007.1599-2-59-6': 'Table [REF] lists the lenses used and their [MATH] and [MATH] values are plotted in figures [REF] and [REF].', '1007.1599-2-60-0': 'The most striking thing in figures [REF] and [REF] is that one of the lenses, 2045+265, has significantly higher [MATH] than is expected in the absence of substructure, but that all the other lenses have [MATH]-[MATH] values that are not particularly anomalous.', '1007.1599-2-60-1': 'In the bottom two rows of figure [REF] it can be seen that if only the low ellipticity lenses (axis ratio [MATH]) were considered three or four of the observed lenses would have anomalous [MATH]-[MATH] values.', '1007.1599-2-60-2': 'Since the authors that have compared Nbody simulations to the data using [MATH] values in the past have used very few simulated lenses and all with axis ratios [MATH] it is now not surprising that they concluded that the simulations did not produce enough anomalies.', '1007.1599-2-61-0': "It should be emphasized that just because the lenses' [MATH]-[MATH] values are not anomalous does not mean that they do not have anomalous flux ratios.", '1007.1599-2-61-1': 'Some of these cases clearly cannot be fit by reasonable models without substructure when all the image positions and fluxes are taken into account .', '1007.1599-2-61-2': 'With so few observed lenses and only one clear anomaly in [MATH]-[MATH] space, it is impossible to make any strong conclusion about the aloud properties for substructure using only the [MATH]-[MATH] distribution.', '1007.1599-2-61-3': 'About one anomalies out of the seven lenses is about what one would expect from studying the top panels of figures [REF] and [REF] for a substructure surface density of [MATH].', '1007.1599-2-61-4': 'Other flux-based constraints are possible and will be investigated in future papers.', '1007.1599-2-62-0': 'We introduce another constraint in the bottom panels of figure [REF] through [REF] based on the fraction of simulations with more than four images.', '1007.1599-2-62-1': '(This does not include the central demagnified image that forms near the center of the lens for nonsingular lens mass profiles.', '1007.1599-2-62-2': 'In our case, the mass density in the center of the lens diverges like [MATH], and this image never appears; it is infinitely demagnified.)', '1007.1599-2-62-3': 'Even after merging images with centroids less than 0.1 arcsec apart, there are cases where the substructures cause further splitting of the images.', '1007.1599-2-62-4': 'Of the 32 QSO lenses in the Castles list of lenses with more than four images and simple lenses, none have more than 4 images of a single source separated by more than 0.1 arcsec. This puts a strong constraint on the allowed fraction of lenses that have more than four images, [MATH].', '1007.1599-2-62-5': 'The probability of getting zero cases of [MATH] images in 33, given that the probability of getting such a case is [MATH], is a binomial distribution.', '1007.1599-2-62-6': 'There would be less than a 5% chance of this happening in the observed sample if [MATH] is greater than 0.089 and less than 10% chance if [MATH].', '1007.1599-2-62-7': 'These are the dotted lines in the bottom panels of figures [REF] through [REF].', '1007.1599-2-62-8': 'For equal surface mass density, more massive substructures cause more high image multiplicity lenses.', '1007.1599-2-63-0': 'The multiplicity constraint does change significantly if the resolution cutoff of [MATH] arcsec is changed.', '1007.1599-2-63-1': 'There are a large number of lenses where the images are merged in some cases (up to [MATH]).', '1007.1599-2-63-2': 'With improved resolution or a more careful anaylisis of the data, we believe this constaint could be made significantly stronger.', '1007.1599-2-64-0': 'Within the ranges of [MATH] studied here, the only models that are limited by this image multiplicity constraint are set 1 (high lower mass cutoff and compact), set 5 (super-compact) and set 9 (high upper mass cutoff and compact).', '1007.1599-2-64-1': 'The constraints are [MATH], [MATH] and [MATH] respectively.', '1007.1599-2-64-2': 'The more compact and massive the substructures are the more high multiplicity cases are created.', '1007.1599-2-64-3': 'This constraint is in contrast to the [MATH] constraint which depends only on the mass density and compactness.', '1007.1599-2-64-4': 'With more lenses this constraint could become significantly stronger in the future.', '1007.1599-2-65-0': '# expectations for small-scale structure within the CDM model', '1007.1599-2-66-0': 'A good point of comparison between lens simulations and Nbody simulations is the fraction of mass in substructure within a projected radius of 10 kpc.', '1007.1599-2-66-1': 'This is easily measured in the simulations and since the Einstein radius is typically around 10 kpc, it is close to what is actually constrained by the lensing data.', '1007.1599-2-66-2': 'In our model, this quantity is given by [EQUATION] where the fiducial value [MATH]^-1[MATH] has been used.', '1007.1599-2-66-3': 'Note that this fraction scales with host mass in our model and in the simulations.', '1007.1599-2-67-0': '[CITATION] give [MATH] for the Via Lactea simulation, and [CITATION] give [MATH] with a large scatter in the Aquarius simulations.', '1007.1599-2-67-1': 'These simulations should be resolving substructure to below [MATH].', '1007.1599-2-67-2': 'These translate to [MATH] and [MATH] respectively.', '1007.1599-2-67-3': 'Accounting for the extra mass below there resolution and judging from figures [REF] through [REF] we would expect about a 10% chance of a clear outlier in the [MATH]-[MATH] distribution for the high compactness cases which is consistant with the one out of seven observed.', '1007.1599-2-67-4': 'For the larger size-mass relation (sets 8, 10 and 12) the expected fraction is increased by only a few percent from the no substructure case, but with only one observed outlier, we do not consider this a significant contradition.', '1007.1599-2-68-0': '[CITATION], [CITATION] and [CITATION] come to the conclusion that the substructure present in the simulations is not enough to cause the observed frequency of [MATH] anomalies.', '1007.1599-2-68-1': 'In light of the findings in this paper we believe that these conclusions were flawed because the full range of host lens ellipticiites was not represented in the simulations.', '1007.1599-2-68-2': '[CITATION] may have used too low a substructure mass range ([MATH]) to cause enough anomalies.', '1007.1599-2-69-0': 'There are a number other complicating factors that make comparing observations to the true predictions of CDM difficult.', '1007.1599-2-69-1': 'For example, the baryons are not accounted for in the Nbody simulations.', '1007.1599-2-69-2': 'This impacts the predictions in several ways.', '1007.1599-2-69-3': 'First, the host galaxy needs to be inserted by hand into these Nbody simulations for them to be realistic lenses.', '1007.1599-2-69-4': 'The mass fraction decreases with the inclusion of baryons.', '1007.1599-2-69-5': 'Second, the baryons are expected to have some effect on the internal structure of the substructures, either expanding or contracting them, which will affect their tidal stripping and disruption in the host halo.', '1007.1599-2-69-6': 'The resident galaxy might also have a significant effect on the survival of substructures.', '1007.1599-2-69-7': 'As discussed in section [REF], the typical galactocentric distance for substructures that are important for lensing is significantly smaller than the typical distance of substructures in general.', '1007.1599-2-69-8': 'The substructure population probed by lensing is likely to be more compact and have a steeper mass function, at least above [MATH], than the general population.', '1007.1599-2-69-9': 'This steepening of the mass function at high masses has been only crudely accounted for in our model by the [MATH] cutoff parameter.', '1007.1599-2-70-0': 'Because we appear to be consistent with the simulations on the frequency of [MATH] anomalies does not mean that some other test, such as fitting each simulated lens to a smooth lens model, would not show some inconsistency.', '1007.1599-2-70-1': 'Modeling the lens puts constraints on the ellipticity.', '1007.1599-2-70-2': 'Our argument is that [MATH] is not a good test for the existence of substructure without further constraints.', '1007.1599-2-71-0': '# conclusions discussion', '1007.1599-2-72-0': 'We have preformed the largest number of lens simulations ever done with finite size sources.', '1007.1599-2-72-1': 'This was made possible by the new adaptive ray-tracing code GLAMER.', '1007.1599-2-72-2': 'We find that accounting for the finite size of the source is necessary for drawing accurate conclusions from the lensed QSO data.', '1007.1599-2-73-0': 'We find rough consistency between the [MATH]CDM predictions and observations.', '1007.1599-2-73-1': '[MATH] is found to be a poor discriminator between lenses with substructure and without because of its sensitivity to the ellipticity of the lens.', '1007.1599-2-73-2': 'The distribution of ellipticities used in our lens models is based on the ellipticities of observed lenses so we do no think the ellipticities required to explain the observed [MATH] distribution (accepting lens 2045+265) are atypical.', '1007.1599-2-73-3': 'Other methods for comparing observations to models are likely to be more fruitful.', '1007.1599-2-73-4': 'And as the data improves more precise comparisons will be possible.', '1007.1599-2-73-5': 'In addition to the substructure within the primary lens, there should be some contribution from intergalactic small-scale structure so one should expect the limits derived from the data to be somewhat higher than the limits derived from Nbody simulations of individual dark matter halos.', '1007.1599-2-73-6': 'The baryons also clearly play a role in shaping the lensing properties and they are not fully taken into account in the simulations at the necessary resolution.', '1007.1599-2-74-0': 'We have limited our study here to a substructure mass function of the form [MATH] with [MATH].', '1007.1599-2-74-1': 'This seems well motivated by the simulations on small mass-scales, but could be steeper on larger mass-scales because of tidal stripping and disruption in the central regions of the lens.', '1007.1599-2-74-2': 'With the [MATH] mass function, the smaller mass substructures plays a smaller part in causing flux anomalies because most of the mass resides in larger mass objects.', '1007.1599-2-74-3': 'This will make it difficult to measure any possible lower mass cutoff using monochromatic QSO lensing alone.', '1007.1599-2-74-4': 'Fortunately there are some other prospects for probing the mass function in the future such as spectroscopic gravitation and Einstein rings .', '1007.1599-2-74-5': 'If the slope of the mass function is steeper than [MATH], the smaller structures will play a larger role in the lensing.', '1007.1599-2-75-0': 'It is clear that what is really required to make a more conclusive measurement of the amount of substructure in dark matter halos is more data.', '1007.1599-2-75-1': 'With 7 lenses, only limited conclusions can be made from a statistical point of view.', '1007.1599-2-75-2': 'We are also vulnerable to systematic errors.', '1007.1599-2-75-3': 'For the kind of study done here, more strong lenses measured in the radio and/or mid-infrared are needed.', '1007.1599-2-75-4': 'Planned large scale imaging surveys expect to increase the number of lensed QSOs in the visible by an order of magnitude so we look forward to great improvements in this field.'}

[['1007.1599-1-63-0', '1007.1599-2-74-0'], ['1007.1599-1-63-3', '1007.1599-2-74-3'], ['1007.1599-1-63-4', '1007.1599-2-74-4'], ['1007.1599-1-63-5', '1007.1599-2-74-5'], ['1007.1599-1-64-0', '1007.1599-2-75-0'], ['1007.1599-1-64-1', '1007.1599-2-75-1'], ['1007.1599-1-64-2', '1007.1599-2-75-2'], ['1007.1599-1-64-3', '1007.1599-2-75-3'], ['1007.1599-1-64-4', '1007.1599-2-75-4'], ['1007.1599-1-18-0', '1007.1599-2-18-0'], ['1007.1599-1-18-1', '1007.1599-2-18-1'], ['1007.1599-1-18-2', '1007.1599-2-18-2'], ['1007.1599-1-18-3', '1007.1599-2-18-3'], ['1007.1599-1-40-0', '1007.1599-2-47-0'], ['1007.1599-1-40-1', '1007.1599-2-47-1'], ['1007.1599-1-40-2', '1007.1599-2-47-2'], ['1007.1599-1-40-3', '1007.1599-2-47-3'], ['1007.1599-1-40-4', '1007.1599-2-47-4'], ['1007.1599-1-40-6', '1007.1599-2-47-6'], ['1007.1599-1-20-0', '1007.1599-2-20-0'], ['1007.1599-1-20-1', '1007.1599-2-20-1'], ['1007.1599-1-20-2', '1007.1599-2-20-3'], ['1007.1599-1-41-0', '1007.1599-2-48-0'], ['1007.1599-1-41-2', '1007.1599-2-48-2'], ['1007.1599-1-41-3', '1007.1599-2-48-3'], ['1007.1599-1-41-4', '1007.1599-2-48-4'], ['1007.1599-1-41-5', '1007.1599-2-48-6'], ['1007.1599-1-41-6', '1007.1599-2-48-9'], ['1007.1599-1-12-0', '1007.1599-2-12-0'], ['1007.1599-1-12-1', '1007.1599-2-12-1'], ['1007.1599-1-12-3', '1007.1599-2-12-3'], ['1007.1599-1-12-4', '1007.1599-2-12-4'], ['1007.1599-1-34-0', '1007.1599-2-38-0'], ['1007.1599-1-34-1', '1007.1599-2-38-1'], ['1007.1599-1-34-3', '1007.1599-2-38-3'], ['1007.1599-1-58-4', '1007.1599-2-68-2'], ['1007.1599-1-2-0', '1007.1599-2-2-0'], ['1007.1599-1-2-1', '1007.1599-2-2-1'], ['1007.1599-1-2-2', '1007.1599-2-2-2'], ['1007.1599-1-2-3', '1007.1599-2-2-3'], ['1007.1599-1-2-4', '1007.1599-2-2-4'], ['1007.1599-1-2-5', '1007.1599-2-2-5'], ['1007.1599-1-2-6', '1007.1599-2-2-6'], ['1007.1599-1-2-7', '1007.1599-2-2-7'], ['1007.1599-1-2-8', '1007.1599-2-2-8'], ['1007.1599-1-0-2', '1007.1599-2-0-2'], ['1007.1599-1-0-7', '1007.1599-2-0-7'], ['1007.1599-1-15-0', '1007.1599-2-15-0'], ['1007.1599-1-15-1', '1007.1599-2-15-1'], ['1007.1599-1-15-2', '1007.1599-2-15-2'], ['1007.1599-1-15-3', '1007.1599-2-15-3'], ['1007.1599-1-24-0', '1007.1599-2-25-0'], ['1007.1599-1-24-3', '1007.1599-2-25-2'], ['1007.1599-1-8-0', '1007.1599-2-8-0'], ['1007.1599-1-8-1', '1007.1599-2-8-1'], ['1007.1599-1-8-2', '1007.1599-2-8-2'], ['1007.1599-1-8-3', '1007.1599-2-8-3'], ['1007.1599-1-8-4', '1007.1599-2-8-4'], ['1007.1599-1-21-0', '1007.1599-2-21-0'], ['1007.1599-1-21-1', '1007.1599-2-21-1'], ['1007.1599-1-21-2', '1007.1599-2-21-2'], ['1007.1599-1-21-3', '1007.1599-2-21-3'], ['1007.1599-1-21-4', '1007.1599-2-21-4'], ['1007.1599-1-21-5', '1007.1599-2-21-5'], ['1007.1599-1-21-6', '1007.1599-2-21-6'], ['1007.1599-1-23-0', '1007.1599-2-24-0'], ['1007.1599-1-62-0', '1007.1599-2-73-0'], ['1007.1599-1-62-4', '1007.1599-2-73-6'], ['1007.1599-1-42-1', '1007.1599-2-49-1'], ['1007.1599-1-42-2', '1007.1599-2-49-2'], ['1007.1599-1-42-3', '1007.1599-2-49-3'], ['1007.1599-1-28-0', '1007.1599-2-31-0'], ['1007.1599-1-28-2', '1007.1599-2-31-2'], ['1007.1599-1-28-4', '1007.1599-2-31-5'], ['1007.1599-1-13-0', '1007.1599-2-13-0'], ['1007.1599-1-13-1', '1007.1599-2-13-1'], ['1007.1599-1-13-2', '1007.1599-2-13-2'], ['1007.1599-1-13-4', '1007.1599-2-13-4'], ['1007.1599-1-13-5', '1007.1599-2-13-5'], ['1007.1599-1-13-6', '1007.1599-2-13-6'], ['1007.1599-1-43-0', '1007.1599-2-50-0'], ['1007.1599-1-32-0', '1007.1599-2-36-0'], ['1007.1599-1-32-1', '1007.1599-2-36-1'], ['1007.1599-1-32-2', '1007.1599-2-36-2'], ['1007.1599-1-32-3', '1007.1599-2-36-3'], ['1007.1599-1-32-4', '1007.1599-2-36-4'], ['1007.1599-1-32-5', '1007.1599-2-36-5'], ['1007.1599-1-32-6', '1007.1599-2-36-6'], ['1007.1599-1-32-7', '1007.1599-2-36-7'], ['1007.1599-1-14-0', '1007.1599-2-14-0'], ['1007.1599-1-14-1', '1007.1599-2-14-1'], ['1007.1599-1-14-2', '1007.1599-2-14-2'], ['1007.1599-1-14-3', '1007.1599-2-14-3'], ['1007.1599-1-14-4', '1007.1599-2-14-4'], ['1007.1599-1-19-0', '1007.1599-2-19-0'], ['1007.1599-1-19-1', '1007.1599-2-19-1'], ['1007.1599-1-19-2', '1007.1599-2-19-2'], ['1007.1599-1-19-3', '1007.1599-2-19-3'], ['1007.1599-1-19-4', '1007.1599-2-19-4'], ['1007.1599-1-19-5', '1007.1599-2-19-5'], ['1007.1599-1-61-0', '1007.1599-2-72-0'], ['1007.1599-1-61-1', '1007.1599-2-72-1'], ['1007.1599-1-61-2', '1007.1599-2-72-2'], ['1007.1599-1-6-0', '1007.1599-2-6-0'], ['1007.1599-1-6-1', '1007.1599-2-6-1'], ['1007.1599-1-6-2', '1007.1599-2-6-2'], ['1007.1599-1-6-3', '1007.1599-2-6-3'], ['1007.1599-1-6-4', '1007.1599-2-6-4'], ['1007.1599-1-10-0', '1007.1599-2-10-0'], ['1007.1599-1-10-1', '1007.1599-2-10-1'], ['1007.1599-1-10-2', '1007.1599-2-10-2'], ['1007.1599-1-59-0', '1007.1599-2-70-0'], ['1007.1599-1-38-0', '1007.1599-2-43-0'], ['1007.1599-1-38-1', '1007.1599-2-43-1'], ['1007.1599-1-38-3', '1007.1599-2-43-3'], ['1007.1599-1-38-4', '1007.1599-2-43-4'], ['1007.1599-1-37-0', '1007.1599-2-42-0'], ['1007.1599-1-37-1', '1007.1599-2-42-1'], ['1007.1599-1-37-2', '1007.1599-2-42-2'], ['1007.1599-1-37-3', '1007.1599-2-42-3'], ['1007.1599-1-37-4', '1007.1599-2-42-4'], ['1007.1599-1-37-5', '1007.1599-2-42-5'], ['1007.1599-1-37-6', '1007.1599-2-42-6'], ['1007.1599-1-37-7', '1007.1599-2-42-7'], ['1007.1599-1-37-8', '1007.1599-2-42-8'], ['1007.1599-1-35-0', '1007.1599-2-39-0'], ['1007.1599-1-35-1', '1007.1599-2-39-1'], ['1007.1599-1-26-0', '1007.1599-2-28-0'], ['1007.1599-1-26-1', '1007.1599-2-28-1'], ['1007.1599-1-26-2', '1007.1599-2-28-2'], ['1007.1599-1-26-3', '1007.1599-2-28-3'], ['1007.1599-1-26-4', '1007.1599-2-28-4'], ['1007.1599-1-26-5', '1007.1599-2-28-5'], ['1007.1599-1-26-6', '1007.1599-2-28-6'], ['1007.1599-1-26-7', '1007.1599-2-28-7'], ['1007.1599-1-51-1', '1007.1599-2-62-1'], ['1007.1599-1-51-4', '1007.1599-2-62-4'], ['1007.1599-1-51-5', '1007.1599-2-62-6'], ['1007.1599-1-17-0', '1007.1599-2-17-0'], ['1007.1599-1-17-1', '1007.1599-2-17-1'], ['1007.1599-1-17-2', '1007.1599-2-17-2'], ['1007.1599-1-4-0', '1007.1599-2-4-0'], ['1007.1599-1-4-1', '1007.1599-2-4-1'], ['1007.1599-1-4-2', '1007.1599-2-4-2'], ['1007.1599-1-4-3', '1007.1599-2-4-3'], ['1007.1599-1-4-5', '1007.1599-2-4-5'], ['1007.1599-1-4-6', '1007.1599-2-4-6'], ['1007.1599-1-31-0', '1007.1599-2-35-0'], ['1007.1599-1-31-1', '1007.1599-2-35-1'], ['1007.1599-1-31-2', '1007.1599-2-35-2'], ['1007.1599-1-31-3', '1007.1599-2-35-3'], ['1007.1599-1-31-4', '1007.1599-2-35-4'], ['1007.1599-1-31-5', '1007.1599-2-35-6'], ['1007.1599-1-31-6', '1007.1599-2-35-7'], ['1007.1599-1-31-7', '1007.1599-2-35-8'], ['1007.1599-1-31-8', '1007.1599-2-35-9'], ['1007.1599-1-5-0', '1007.1599-2-5-0'], ['1007.1599-1-5-1', '1007.1599-2-5-1'], ['1007.1599-1-5-2', '1007.1599-2-5-2'], ['1007.1599-1-5-3', '1007.1599-2-5-3'], ['1007.1599-1-5-4', '1007.1599-2-5-4'], ['1007.1599-1-5-5', '1007.1599-2-5-5'], ['1007.1599-1-5-6', '1007.1599-2-5-6'], ['1007.1599-1-5-7', '1007.1599-2-5-7'], ['1007.1599-1-5-8', '1007.1599-2-5-8'], ['1007.1599-1-5-9', '1007.1599-2-5-9'], ['1007.1599-1-5-10', '1007.1599-2-5-10'], ['1007.1599-1-5-11', '1007.1599-2-5-11'], ['1007.1599-1-33-0', '1007.1599-2-37-0'], ['1007.1599-1-33-1', '1007.1599-2-37-1'], ['1007.1599-1-33-2', '1007.1599-2-37-2'], ['1007.1599-1-33-3', '1007.1599-2-37-3'], ['1007.1599-1-33-4', '1007.1599-2-37-4'], ['1007.1599-1-33-5', '1007.1599-2-37-5'], ['1007.1599-1-45-0', '1007.1599-2-59-0'], ['1007.1599-1-45-1', '1007.1599-2-59-1'], ['1007.1599-1-45-3', '1007.1599-2-59-3'], ['1007.1599-1-45-4', '1007.1599-2-59-4'], ['1007.1599-1-45-5', '1007.1599-2-59-5'], ['1007.1599-1-7-0', '1007.1599-2-7-0'], ['1007.1599-1-7-1', '1007.1599-2-7-1'], ['1007.1599-1-7-2', '1007.1599-2-7-2'], ['1007.1599-1-7-3', '1007.1599-2-7-3'], ['1007.1599-1-7-4', '1007.1599-2-7-4'], ['1007.1599-1-56-1', '1007.1599-2-66-1'], ['1007.1599-1-56-2', '1007.1599-2-66-2'], ['1007.1599-1-56-3', '1007.1599-2-66-3'], ['1007.1599-1-3-0', '1007.1599-2-3-0'], ['1007.1599-1-3-1', '1007.1599-2-3-1'], ['1007.1599-1-3-2', '1007.1599-2-3-2'], ['1007.1599-1-3-4', '1007.1599-2-3-4'], ['1007.1599-1-3-5', '1007.1599-2-3-5'], ['1007.1599-1-57-0', '1007.1599-2-67-0'], ['1007.1599-1-57-1', '1007.1599-2-67-1'], ['1007.1599-1-57-4', '1007.1599-2-69-4'], ['1007.1599-1-39-1', '1007.1599-2-46-1'], ['1007.1599-1-39-2', '1007.1599-2-44-1'], ['1007.1599-1-39-3', '1007.1599-2-44-2'], ['1007.1599-1-27-0', '1007.1599-2-29-0'], ['1007.1599-1-27-1', '1007.1599-2-29-1'], ['1007.1599-1-27-2', '1007.1599-2-29-2'], ['1007.1599-1-27-3', '1007.1599-2-29-3'], ['1007.1599-1-27-4', '1007.1599-2-30-0'], ['1007.1599-1-27-5', '1007.1599-2-30-1'], ['1007.1599-1-27-6', '1007.1599-2-30-2'], ['1007.1599-1-63-2', '1007.1599-2-74-2'], ['1007.1599-1-40-5', '1007.1599-2-47-5'], ['1007.1599-1-41-1', '1007.1599-2-48-1'], ['1007.1599-1-12-2', '1007.1599-2-12-2'], ['1007.1599-1-58-0', '1007.1599-2-68-0'], ['1007.1599-1-0-1', '1007.1599-2-0-1'], ['1007.1599-1-0-3', '1007.1599-2-0-3'], ['1007.1599-1-0-5', '1007.1599-2-0-5'], ['1007.1599-1-24-1', '1007.1599-2-25-1'], ['1007.1599-1-62-3', '1007.1599-2-73-5'], ['1007.1599-1-42-0', '1007.1599-2-49-0'], ['1007.1599-1-28-1', '1007.1599-2-31-1'], ['1007.1599-1-28-3', '1007.1599-2-31-3'], ['1007.1599-1-13-3', '1007.1599-2-13-3'], ['1007.1599-1-38-2', '1007.1599-2-43-2'], ['1007.1599-1-37-9', '1007.1599-2-42-9'], ['1007.1599-1-54-1', '1007.1599-2-64-1'], ['1007.1599-1-51-0', '1007.1599-2-62-0'], ['1007.1599-1-51-2', '1007.1599-2-62-2'], ['1007.1599-1-51-3', '1007.1599-2-62-3'], ['1007.1599-1-51-6', '1007.1599-2-62-7'], ['1007.1599-1-51-7', '1007.1599-2-62-8'], ['1007.1599-1-45-6', '1007.1599-2-59-6'], ['1007.1599-1-56-0', '1007.1599-2-66-0'], ['1007.1599-1-3-3', '1007.1599-2-3-3'], ['1007.1599-1-63-1', '1007.1599-2-74-1'], ['1007.1599-1-46-0', '1007.1599-2-60-0'], ['1007.1599-1-34-2', '1007.1599-2-38-2'], ['1007.1599-1-0-0', '1007.1599-2-0-0'], ['1007.1599-1-0-6', '1007.1599-2-0-6'], ['1007.1599-1-0-8', '1007.1599-2-0-9'], ['1007.1599-1-25-0', '1007.1599-2-26-0'], ['1007.1599-1-25-0', '1007.1599-2-26-3'], ['1007.1599-1-25-1', '1007.1599-2-26-1'], ['1007.1599-1-4-4', '1007.1599-2-4-4'], ['1007.1599-1-57-3', '1007.1599-2-69-1'], ['1007.1599-1-57-3', '1007.1599-2-69-3'], ['1007.1599-1-39-0', '1007.1599-2-44-0'], ['1007.1599-1-50-2', '1007.1599-2-0-6'], ['1007.1599-1-53-0', '1007.1599-2-55-0']]

[['1007.1599-1-63-0', '1007.1599-2-74-0'], ['1007.1599-1-63-3', '1007.1599-2-74-3'], ['1007.1599-1-63-4', '1007.1599-2-74-4'], ['1007.1599-1-63-5', '1007.1599-2-74-5'], ['1007.1599-1-64-0', '1007.1599-2-75-0'], ['1007.1599-1-64-1', '1007.1599-2-75-1'], ['1007.1599-1-64-2', '1007.1599-2-75-2'], ['1007.1599-1-64-3', '1007.1599-2-75-3'], ['1007.1599-1-64-4', '1007.1599-2-75-4'], ['1007.1599-1-18-0', '1007.1599-2-18-0'], ['1007.1599-1-18-1', '1007.1599-2-18-1'], ['1007.1599-1-18-2', '1007.1599-2-18-2'], ['1007.1599-1-18-3', '1007.1599-2-18-3'], ['1007.1599-1-40-0', '1007.1599-2-47-0'], ['1007.1599-1-40-1', '1007.1599-2-47-1'], ['1007.1599-1-40-2', '1007.1599-2-47-2'], ['1007.1599-1-40-3', '1007.1599-2-47-3'], ['1007.1599-1-40-4', '1007.1599-2-47-4'], ['1007.1599-1-40-6', '1007.1599-2-47-6'], ['1007.1599-1-20-0', '1007.1599-2-20-0'], ['1007.1599-1-20-1', '1007.1599-2-20-1'], ['1007.1599-1-20-2', '1007.1599-2-20-3'], ['1007.1599-1-41-0', '1007.1599-2-48-0'], ['1007.1599-1-41-2', '1007.1599-2-48-2'], ['1007.1599-1-41-3', '1007.1599-2-48-3'], ['1007.1599-1-41-4', '1007.1599-2-48-4'], ['1007.1599-1-41-5', '1007.1599-2-48-6'], ['1007.1599-1-41-6', '1007.1599-2-48-9'], ['1007.1599-1-12-0', '1007.1599-2-12-0'], ['1007.1599-1-12-1', '1007.1599-2-12-1'], ['1007.1599-1-12-3', '1007.1599-2-12-3'], ['1007.1599-1-12-4', '1007.1599-2-12-4'], ['1007.1599-1-34-0', '1007.1599-2-38-0'], ['1007.1599-1-34-1', '1007.1599-2-38-1'], ['1007.1599-1-34-3', '1007.1599-2-38-3'], ['1007.1599-1-58-4', '1007.1599-2-68-2'], ['1007.1599-1-2-0', '1007.1599-2-2-0'], ['1007.1599-1-2-1', '1007.1599-2-2-1'], ['1007.1599-1-2-2', '1007.1599-2-2-2'], ['1007.1599-1-2-3', '1007.1599-2-2-3'], ['1007.1599-1-2-4', '1007.1599-2-2-4'], ['1007.1599-1-2-5', '1007.1599-2-2-5'], ['1007.1599-1-2-6', '1007.1599-2-2-6'], ['1007.1599-1-2-7', '1007.1599-2-2-7'], ['1007.1599-1-2-8', '1007.1599-2-2-8'], ['1007.1599-1-0-2', '1007.1599-2-0-2'], ['1007.1599-1-0-7', '1007.1599-2-0-7'], ['1007.1599-1-15-0', '1007.1599-2-15-0'], ['1007.1599-1-15-1', '1007.1599-2-15-1'], ['1007.1599-1-15-2', '1007.1599-2-15-2'], ['1007.1599-1-15-3', '1007.1599-2-15-3'], ['1007.1599-1-24-0', '1007.1599-2-25-0'], ['1007.1599-1-24-3', '1007.1599-2-25-2'], ['1007.1599-1-8-0', '1007.1599-2-8-0'], ['1007.1599-1-8-1', '1007.1599-2-8-1'], ['1007.1599-1-8-2', '1007.1599-2-8-2'], ['1007.1599-1-8-3', '1007.1599-2-8-3'], ['1007.1599-1-8-4', '1007.1599-2-8-4'], ['1007.1599-1-21-0', '1007.1599-2-21-0'], ['1007.1599-1-21-1', '1007.1599-2-21-1'], ['1007.1599-1-21-2', '1007.1599-2-21-2'], ['1007.1599-1-21-3', '1007.1599-2-21-3'], ['1007.1599-1-21-4', '1007.1599-2-21-4'], ['1007.1599-1-21-5', '1007.1599-2-21-5'], ['1007.1599-1-21-6', '1007.1599-2-21-6'], ['1007.1599-1-23-0', '1007.1599-2-24-0'], ['1007.1599-1-62-0', '1007.1599-2-73-0'], ['1007.1599-1-62-4', '1007.1599-2-73-6'], ['1007.1599-1-42-1', '1007.1599-2-49-1'], ['1007.1599-1-42-2', '1007.1599-2-49-2'], ['1007.1599-1-42-3', '1007.1599-2-49-3'], ['1007.1599-1-28-0', '1007.1599-2-31-0'], ['1007.1599-1-28-2', '1007.1599-2-31-2'], ['1007.1599-1-28-4', '1007.1599-2-31-5'], ['1007.1599-1-13-0', '1007.1599-2-13-0'], ['1007.1599-1-13-1', '1007.1599-2-13-1'], ['1007.1599-1-13-2', '1007.1599-2-13-2'], ['1007.1599-1-13-4', '1007.1599-2-13-4'], ['1007.1599-1-13-5', '1007.1599-2-13-5'], ['1007.1599-1-13-6', '1007.1599-2-13-6'], ['1007.1599-1-43-0', '1007.1599-2-50-0'], ['1007.1599-1-32-0', '1007.1599-2-36-0'], ['1007.1599-1-32-1', '1007.1599-2-36-1'], ['1007.1599-1-32-2', '1007.1599-2-36-2'], ['1007.1599-1-32-3', '1007.1599-2-36-3'], ['1007.1599-1-32-4', '1007.1599-2-36-4'], ['1007.1599-1-32-5', '1007.1599-2-36-5'], ['1007.1599-1-32-6', '1007.1599-2-36-6'], ['1007.1599-1-32-7', '1007.1599-2-36-7'], ['1007.1599-1-14-0', '1007.1599-2-14-0'], ['1007.1599-1-14-1', '1007.1599-2-14-1'], ['1007.1599-1-14-2', '1007.1599-2-14-2'], ['1007.1599-1-14-3', '1007.1599-2-14-3'], ['1007.1599-1-14-4', '1007.1599-2-14-4'], ['1007.1599-1-19-0', '1007.1599-2-19-0'], ['1007.1599-1-19-1', '1007.1599-2-19-1'], ['1007.1599-1-19-2', '1007.1599-2-19-2'], ['1007.1599-1-19-3', '1007.1599-2-19-3'], ['1007.1599-1-19-4', '1007.1599-2-19-4'], ['1007.1599-1-19-5', '1007.1599-2-19-5'], ['1007.1599-1-61-0', '1007.1599-2-72-0'], ['1007.1599-1-61-1', '1007.1599-2-72-1'], ['1007.1599-1-61-2', '1007.1599-2-72-2'], ['1007.1599-1-6-0', '1007.1599-2-6-0'], ['1007.1599-1-6-1', '1007.1599-2-6-1'], ['1007.1599-1-6-2', '1007.1599-2-6-2'], ['1007.1599-1-6-3', '1007.1599-2-6-3'], ['1007.1599-1-6-4', '1007.1599-2-6-4'], ['1007.1599-1-10-0', '1007.1599-2-10-0'], ['1007.1599-1-10-1', '1007.1599-2-10-1'], ['1007.1599-1-10-2', '1007.1599-2-10-2'], ['1007.1599-1-59-0', '1007.1599-2-70-0'], ['1007.1599-1-38-0', '1007.1599-2-43-0'], ['1007.1599-1-38-1', '1007.1599-2-43-1'], ['1007.1599-1-38-3', '1007.1599-2-43-3'], ['1007.1599-1-38-4', '1007.1599-2-43-4'], ['1007.1599-1-37-0', '1007.1599-2-42-0'], ['1007.1599-1-37-1', '1007.1599-2-42-1'], ['1007.1599-1-37-2', '1007.1599-2-42-2'], ['1007.1599-1-37-3', '1007.1599-2-42-3'], ['1007.1599-1-37-4', '1007.1599-2-42-4'], ['1007.1599-1-37-5', '1007.1599-2-42-5'], ['1007.1599-1-37-6', '1007.1599-2-42-6'], ['1007.1599-1-37-7', '1007.1599-2-42-7'], ['1007.1599-1-37-8', '1007.1599-2-42-8'], ['1007.1599-1-35-0', '1007.1599-2-39-0'], ['1007.1599-1-35-1', '1007.1599-2-39-1'], ['1007.1599-1-26-0', '1007.1599-2-28-0'], ['1007.1599-1-26-1', '1007.1599-2-28-1'], ['1007.1599-1-26-2', '1007.1599-2-28-2'], ['1007.1599-1-26-3', '1007.1599-2-28-3'], ['1007.1599-1-26-4', '1007.1599-2-28-4'], ['1007.1599-1-26-5', '1007.1599-2-28-5'], ['1007.1599-1-26-6', '1007.1599-2-28-6'], ['1007.1599-1-26-7', '1007.1599-2-28-7'], ['1007.1599-1-51-1', '1007.1599-2-62-1'], ['1007.1599-1-51-4', '1007.1599-2-62-4'], ['1007.1599-1-51-5', '1007.1599-2-62-6'], ['1007.1599-1-17-0', '1007.1599-2-17-0'], ['1007.1599-1-17-1', '1007.1599-2-17-1'], ['1007.1599-1-17-2', '1007.1599-2-17-2'], ['1007.1599-1-4-0', '1007.1599-2-4-0'], ['1007.1599-1-4-1', '1007.1599-2-4-1'], ['1007.1599-1-4-2', '1007.1599-2-4-2'], ['1007.1599-1-4-3', '1007.1599-2-4-3'], ['1007.1599-1-4-5', '1007.1599-2-4-5'], ['1007.1599-1-4-6', '1007.1599-2-4-6'], ['1007.1599-1-31-0', '1007.1599-2-35-0'], ['1007.1599-1-31-1', '1007.1599-2-35-1'], ['1007.1599-1-31-2', '1007.1599-2-35-2'], ['1007.1599-1-31-3', '1007.1599-2-35-3'], ['1007.1599-1-31-4', '1007.1599-2-35-4'], ['1007.1599-1-31-5', '1007.1599-2-35-6'], ['1007.1599-1-31-6', '1007.1599-2-35-7'], ['1007.1599-1-31-7', '1007.1599-2-35-8'], ['1007.1599-1-31-8', '1007.1599-2-35-9'], ['1007.1599-1-5-0', '1007.1599-2-5-0'], ['1007.1599-1-5-1', '1007.1599-2-5-1'], ['1007.1599-1-5-2', '1007.1599-2-5-2'], ['1007.1599-1-5-3', '1007.1599-2-5-3'], ['1007.1599-1-5-4', '1007.1599-2-5-4'], ['1007.1599-1-5-5', '1007.1599-2-5-5'], ['1007.1599-1-5-6', '1007.1599-2-5-6'], ['1007.1599-1-5-7', '1007.1599-2-5-7'], ['1007.1599-1-5-8', '1007.1599-2-5-8'], ['1007.1599-1-5-9', '1007.1599-2-5-9'], ['1007.1599-1-5-10', '1007.1599-2-5-10'], ['1007.1599-1-5-11', '1007.1599-2-5-11'], ['1007.1599-1-33-0', '1007.1599-2-37-0'], ['1007.1599-1-33-1', '1007.1599-2-37-1'], ['1007.1599-1-33-2', '1007.1599-2-37-2'], ['1007.1599-1-33-3', '1007.1599-2-37-3'], ['1007.1599-1-33-4', '1007.1599-2-37-4'], ['1007.1599-1-33-5', '1007.1599-2-37-5'], ['1007.1599-1-45-0', '1007.1599-2-59-0'], ['1007.1599-1-45-1', '1007.1599-2-59-1'], ['1007.1599-1-45-3', '1007.1599-2-59-3'], ['1007.1599-1-45-4', '1007.1599-2-59-4'], ['1007.1599-1-45-5', '1007.1599-2-59-5'], ['1007.1599-1-7-0', '1007.1599-2-7-0'], ['1007.1599-1-7-1', '1007.1599-2-7-1'], ['1007.1599-1-7-2', '1007.1599-2-7-2'], ['1007.1599-1-7-3', '1007.1599-2-7-3'], ['1007.1599-1-7-4', '1007.1599-2-7-4'], ['1007.1599-1-56-1', '1007.1599-2-66-1'], ['1007.1599-1-56-2', '1007.1599-2-66-2'], ['1007.1599-1-56-3', '1007.1599-2-66-3'], ['1007.1599-1-3-0', '1007.1599-2-3-0'], ['1007.1599-1-3-1', '1007.1599-2-3-1'], ['1007.1599-1-3-2', '1007.1599-2-3-2'], ['1007.1599-1-3-4', '1007.1599-2-3-4'], ['1007.1599-1-3-5', '1007.1599-2-3-5'], ['1007.1599-1-57-0', '1007.1599-2-67-0'], ['1007.1599-1-57-1', '1007.1599-2-67-1'], ['1007.1599-1-57-4', '1007.1599-2-69-4'], ['1007.1599-1-39-1', '1007.1599-2-46-1'], ['1007.1599-1-39-2', '1007.1599-2-44-1'], ['1007.1599-1-39-3', '1007.1599-2-44-2'], ['1007.1599-1-27-0', '1007.1599-2-29-0'], ['1007.1599-1-27-1', '1007.1599-2-29-1'], ['1007.1599-1-27-2', '1007.1599-2-29-2'], ['1007.1599-1-27-3', '1007.1599-2-29-3'], ['1007.1599-1-27-4', '1007.1599-2-30-0'], ['1007.1599-1-27-5', '1007.1599-2-30-1'], ['1007.1599-1-27-6', '1007.1599-2-30-2']]

[['1007.1599-1-63-2', '1007.1599-2-74-2'], ['1007.1599-1-40-5', '1007.1599-2-47-5'], ['1007.1599-1-41-1', '1007.1599-2-48-1'], ['1007.1599-1-12-2', '1007.1599-2-12-2'], ['1007.1599-1-58-0', '1007.1599-2-68-0'], ['1007.1599-1-0-1', '1007.1599-2-0-1'], ['1007.1599-1-0-3', '1007.1599-2-0-3'], ['1007.1599-1-0-5', '1007.1599-2-0-5'], ['1007.1599-1-24-1', '1007.1599-2-25-1'], ['1007.1599-1-62-3', '1007.1599-2-73-5'], ['1007.1599-1-42-0', '1007.1599-2-49-0'], ['1007.1599-1-28-1', '1007.1599-2-31-1'], ['1007.1599-1-28-3', '1007.1599-2-31-3'], ['1007.1599-1-13-3', '1007.1599-2-13-3'], ['1007.1599-1-38-2', '1007.1599-2-43-2'], ['1007.1599-1-37-9', '1007.1599-2-42-9'], ['1007.1599-1-54-1', '1007.1599-2-64-1'], ['1007.1599-1-51-0', '1007.1599-2-62-0'], ['1007.1599-1-51-2', '1007.1599-2-62-2'], ['1007.1599-1-51-3', '1007.1599-2-62-3'], ['1007.1599-1-51-6', '1007.1599-2-62-7'], ['1007.1599-1-51-7', '1007.1599-2-62-8'], ['1007.1599-1-45-6', '1007.1599-2-59-6'], ['1007.1599-1-56-0', '1007.1599-2-66-0'], ['1007.1599-1-3-3', '1007.1599-2-3-3']]

[]

[['1007.1599-1-63-1', '1007.1599-2-74-1'], ['1007.1599-1-46-0', '1007.1599-2-60-0'], ['1007.1599-1-34-2', '1007.1599-2-38-2'], ['1007.1599-1-0-0', '1007.1599-2-0-0'], ['1007.1599-1-0-6', '1007.1599-2-0-6'], ['1007.1599-1-0-8', '1007.1599-2-0-9'], ['1007.1599-1-25-0', '1007.1599-2-26-0'], ['1007.1599-1-25-0', '1007.1599-2-26-3'], ['1007.1599-1-25-1', '1007.1599-2-26-1'], ['1007.1599-1-4-4', '1007.1599-2-4-4'], ['1007.1599-1-57-3', '1007.1599-2-69-1'], ['1007.1599-1-57-3', '1007.1599-2-69-3'], ['1007.1599-1-39-0', '1007.1599-2-44-0']]

[['1007.1599-1-50-2', '1007.1599-2-0-6'], ['1007.1599-1-53-0', '1007.1599-2-55-0']]

['1007.1599-1-29-0', '1007.1599-1-45-2', '1007.1599-2-59-2']

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

https://arxiv.org/abs/1007.1599

0911.4449

{'0911.4449-1-0-0': 'We report two complementary measurements of the diboson ([MATH]) cross section in the final state consisting of an electron or muon, missing transverse energy, and jets, performed using [MATH] collision data at [MATH]1.96 TeV collected by the Collider Detector at Fermilab.', '0911.4449-1-0-1': 'The first method uses the dijet invariant mass distribution while the second method uses more of the kinematic information in the event through matrix-element calculations of the signal and background processes and has a higher sensitivity.', '0911.4449-1-0-2': 'The result from the second method has a signal significance of 5.4[MATH] and is the first observation of [MATH] production using this signature.', '0911.4449-1-0-3': 'Combining the results from both methods gives [MATH], in agreement with the standard model prediction.', '0911.4449-1-1-0': 'Measurements involving heavy vector boson pairs ([MATH], [MATH], and [MATH]) are important tests of the electroweak sector of the standard model (SM).', '0911.4449-1-1-1': 'Deviations of the production cross section from predictions could arise from anomalous triple gauge boson interactions [CITATION] or from new resonances decaying to vector bosons.', '0911.4449-1-1-2': 'Furthermore, the topology of diboson events is similar to that of events in which a Higgs boson is produced in association with a [MATH] or a [MATH], allowing diboson measurements to provide an important step towards future measurements of Higgs boson production.', '0911.4449-1-2-0': 'Diboson production has been observed at the Tevatron in channels in which both bosons decay leptonically [CITATION].', '0911.4449-1-2-1': 'Extraction of the diboson signal in hadronic channels is more challenging because of significantly larger backgrounds.', '0911.4449-1-2-2': 'In addition, due to limited detector resolution, it is difficult to distinguish hadronically decaying [MATH] bosons from [MATH] bosons.', '0911.4449-1-2-3': 'We report on two measurements of the cross section, [MATH], that use different techniques applied to the leptonic decay of one [MATH] and the hadronic decay of the associated [MATH] or [MATH], where [MATH] represents an electron or muon).', '0911.4449-1-2-4': 'Our result represents the first observation of this signal in the lepton + jets channel.', '0911.4449-1-2-5': 'Evidence has previously been reported by the D0 collaboration [CITATION], and the CDF collaboration set a limit on its cross section times branching ratio [CITATION].', '0911.4449-1-2-6': 'In addition the CDF collaboration has reported observation of [MATH] in a different hadronic channel with large missing transverse energy and jets [CITATION].', '0911.4449-1-3-0': 'The first method uses the invariant mass of the two-jet system ([MATH]) to extract a signal peak from data corresponding to 3.9 fb[MATH] of [MATH] collisions at [MATH] TeV.', '0911.4449-1-3-1': 'The second method takes advantage of more kinematic information in the event by constructing a discriminant based on calculations of the differential cross sections of the signal and background processes.', '0911.4449-1-3-2': 'This so-called matrix-element (ME) method has been employed in a search for a low-mass Higgs produced in association with a [MATH] boson [CITATION] and in a measurement of single top production [CITATION].', '0911.4449-1-3-3': 'It is expected to achieve greater discriminating power and here uses data corresponding to an integrated luminosity of 2.7 fb[MATH].', '0911.4449-1-4-0': 'The aspects of the CDF II detector [CITATION] relevant to these analyses are briefly described here.', '0911.4449-1-4-1': 'The tracking system is composed of silicon microstrip detectors and an open-cell drift chamber inside a 1.4 T solenoid.', '0911.4449-1-4-2': 'Electromagnetic lead-scintillator and hadronic iron-scintillator sampling calorimeters segmented in a projective geometry surround the tracking detectors.', '0911.4449-1-4-3': 'A central calorimeter covers a pseudorapidity range of [MATH] while plug calorimeters extend the acceptance into the region [MATH].', '0911.4449-1-4-4': 'Outside of the calorimeters are muon detectors composed of scintillators and drift chambers.', '0911.4449-1-4-5': 'Cherenkov counters around the beam pipe and in the plug calorimeters count the inelastic collisions per bunch crossing and provide the luminosity measurement.', '0911.4449-1-5-0': 'Data samples common to both analyses use trigger selections requiring a central electron (muon) with [MATH]E_T[MATH]p_T[MATH]18 GeV.', '0911.4449-1-5-1': 'The ME method utilizes an additional sample derived from a trigger requiring two jets and large missing transverse energy ([MATH][MATH]) [CITATION].', '0911.4449-1-6-0': 'Offline we select events with electron (muon) candidates with [MATH]E_T[MATH]p_T[MATH] 20 GeV, and with [MATH][MATH], jet, and other kinematic requirements chosen differently for the two methods.', '0911.4449-1-6-1': 'Jets are clustered using a fixed-cone algorithm with radius [MATH] and their energies are corrected for detector effects [CITATION].', '0911.4449-1-6-2': 'Cosmic ray and photon conversion candidates are identified and removed.', '0911.4449-1-7-0': 'Further event selection requirements are made to reduce backgrounds and the sensitivity to systematic uncertainties.', '0911.4449-1-7-1': 'In the [MATH] method, we require events to have [MATH].3ex[MATH]E_T[MATH] GeV, at least two jets with [MATH]E_T[MATH] GeV and [MATH], and the dijet vector boson candidate to have [MATH]p_T[MATH] GeV/[MATH].', '0911.4449-1-7-2': 'As a result of these selection criteria, the [MATH] distribution for background is smoothly falling in the region where the signal is expected to peak.', '0911.4449-1-7-3': 'The invariant mass of the dijet vector boson candidate, [MATH], is evaluated from the two most energetic jets.', '0911.4449-1-7-4': 'Additional requirements are made to reduce backgrounds and improve the Monte Carlo modeling of event kinematics: the transverse mass of the lepton and [MATH][MATH] system ([MATH] [CITATION]) must be greater than 30 GeV/[MATH], and the two most energetic jets must be separated by [MATH].', '0911.4449-1-8-0': 'In the ME method, we require events to have [MATH].3ex[MATH]E_T[MATH] 20 GeV and exactly two jets with [MATH]E_T[MATH] GeV and [MATH].', '0911.4449-1-8-1': 'Additional selection criteria to reduce backgrounds and achieve good modeling of the quantities used in the matrix element calculation include the rejection of events with either an additional jet of [MATH]E_T[MATH] GeV or a second charged lepton.', '0911.4449-1-8-2': 'The latter reduces [MATH]+jets, [MATH], and leptonic diboson backgrounds.', '0911.4449-1-8-3': 'For events with an electron candidate, there is a significant background from production of multiple jets (multi-jet in the following) by quantum chromodynamical (QCD) processes, where the electron is faked by a hadronic jet.', '0911.4449-1-8-4': 'The ME method deals with this background by applying stringent selection criteria, while the [MATH] method assigns a systematic uncertainty to the background shape.', '0911.4449-1-8-5': 'The reduction of the multi-jet QCD background in the ME analysis is achieved by raising the [MATH].3ex[MATH]E_T[MATH] cut to 40 GeV, requiring [MATH] GeV/[MATH], and imposing additional cuts on the angles between the jets, the lepton, and the [MATH]0.3ex[MATH][MATH] [CITATION].', '0911.4449-1-8-6': 'There is a less stringent requirement of [MATH] GeV/[MATH] imposed on muon events to reduce the QCD background in that channel.', '0911.4449-1-9-0': 'After these selections for both methods, the dominant background to the diboson signal is a [MATH] boson produced with accompanying jets ([MATH]+jets), where the [MATH] decays leptonically.', '0911.4449-1-9-1': 'Smaller but non-negligible backgrounds come from QCD multi-jet (where one jet mimics a lepton signature), [MATH]+jets, [MATH], and single top production.', '0911.4449-1-9-2': 'QCD multi-jet events are modeled using data with loosened lepton selection criteria.', '0911.4449-1-9-3': 'All other signal and background processes are modeled using event generators and a geant-based CDF II detector simulation.', '0911.4449-1-9-4': 'The diboson signals as well as the [MATH] and single top backgrounds are simulated using the pythia event generator [CITATION].', '0911.4449-1-9-5': 'The [MATH]+jets and [MATH]+jets backgrounds are simulated using the tree-level event generator alpgen [CITATION], with an interface to pythia providing parton showering and hadronization.', '0911.4449-1-10-0': 'The normalization of the [MATH]+jets background is based on the measured cross section while for [MATH] and single top backgrounds the NLO predicted cross section is used [CITATION].', '0911.4449-1-10-1': 'The efficiencies for the [MATH]+jets, [MATH], and single top backgrounds are estimated from simulation.', '0911.4449-1-10-2': 'The normalization of the QCD background is estimated by fitting the [MATH].3ex[MATH]E_T[MATH] spectrum in data to the sum of all contributing processes, where the QCD and [MATH]+jets normalizations float in the fit.', '0911.4449-1-10-3': 'In the final signal extractions from both methods, the multi-jet QCD background is Gaussian constrained to the result of this [MATH].3ex[MATH]E_T[MATH] fit and the [MATH]+jets background is left unconstrained.', '0911.4449-1-11-0': 'We now describe the methodology and results from each technique.', '0911.4449-1-11-1': 'In the [MATH] method we extract the signal fraction from the data by performing a [MATH] fit to the dijet invariant mass spectrum, separately for electron and muon events.', '0911.4449-1-11-2': 'Templates of [MATH] distributions are constructed with the multi-jet QCD background, the signal [MATH] processes, and the sum of the electroweak backgrounds ([MATH], [MATH], and [MATH] production).', '0911.4449-1-12-0': 'Figure [REF] shows the fit results superimposed on data after the electron and muon samples are combined.', '0911.4449-1-12-1': 'Also shown is the data [MATH] distribution after having subtracted the estimated background, superimposed on the signal Monte Carlo shape extracted from the fit.', '0911.4449-1-12-2': 'Combining the two [MATH] fit results we get a total of [MATH] events, of which about 60% are muon events and 40% are electron events.', '0911.4449-1-12-3': 'The observed significance is 4.6[MATH] where 4.9[MATH] is expected, which is obtained by combining the separate results from the electron and muon channels.', '0911.4449-1-12-4': 'The resultant [MATH] production cross section measurement is [MATH].', '0911.4449-1-12-5': 'The sources of systematic uncertainty in this measurement are discussed together with those from the ME method below.', '0911.4449-1-13-0': 'In the ME method a probability density [MATH] that an event was produced by a given process is determined using the standard model differential cross section for that process.', '0911.4449-1-13-1': 'For an event with measured quantities [MATH], we integrate the appropriate differential cross section [MATH] over the partonic quantities [MATH] convolved with the parton distribution functions (PDFs), [MATH], and over transfer functions describing detector resolution effects, [MATH]: [EQUATION]', '0911.4449-1-13-2': 'We use the CTEQ5L PDF parameterization [CITATION].', '0911.4449-1-13-3': '[MATH] is a mapping of measured jet energy to partonic energy derived using the full detector simulation, while the lepton momenta and jet angles are assumed to be measured exactly.', '0911.4449-1-13-4': 'The integration is performed over the energy of the partons and the longitudinal momentum of the neutrino.', '0911.4449-1-13-5': 'The matrix element is calculated with tree-level diagrams from madgraph [CITATION].', '0911.4449-1-13-6': 'Event probability densities [MATH] are calculated for the signal processes [MATH] and [MATH], as well as for [MATH] plus two parton and single top background processes.', '0911.4449-1-13-7': 'The event probabilities are combined into an event probability discriminant: [MATH], where [MATH] and [MATH].', '0911.4449-1-13-8': 'We make templates of the [MATH] for all signal and background processes and ultimately extract the signal using a fit of the observed [MATH] distribution to a sum of the signal and background templates.', '0911.4449-1-13-9': "The expected event yields are as shown in Table [REF] for the ME method's event selection.", '0911.4449-1-14-0': 'Figure [REF] shows the dijet mass in bins of [MATH].', '0911.4449-1-14-1': 'Most of the background events have low [MATH].', '0911.4449-1-14-2': 'Events with [MATH] have a dijet mass peak close to the expected [MATH] resonance, and the signal-to-background ratio improves with increasing [MATH].', '0911.4449-1-15-0': 'Before comparing the observed [MATH] to the prediction, we validate the Monte Carlo modeling of the quantities that enter the matrix element calculation.', '0911.4449-1-15-1': 'We compare the observed distributions to the predicted ones in control regions with very little signal and also in the signal-rich region.', '0911.4449-1-15-2': 'The different regions are chosen according to the invariant mass of the two-jet system ([MATH]): the signal-rich region has [MATH] GeV and the control regions cover the rest of the [MATH] range.', '0911.4449-1-15-3': 'We also check the modeling of the properties (mass, [MATH], and [MATH]) of the leptonic [MATH] boson and the hadronic [MATH] or [MATH] boson candidate.', '0911.4449-1-15-4': 'All of these quantities are well described by the simulation for our event selection.', '0911.4449-1-15-5': 'There is a small discrepancy in the description of [MATH] in the control regions, as is visible in the low-[MATH] region of Figure [REF].', '0911.4449-1-15-6': 'Associated with this discrepancy we assign a systematic mismodeling uncertainty which is derived in the control regions and extrapolated through the signal region.', '0911.4449-1-15-7': 'This uncertainty has a negligible effect on the results, because most background events lie in the first few bins of the [MATH] distribution.', '0911.4449-1-15-8': 'Small changes in modeling of those background events do not change the shape of the [MATH].', '0911.4449-1-16-0': 'The observed and predicted [MATH]s are shown in Figure [REF].', '0911.4449-1-16-1': 'We use a binned-likelihood fit of the observed [MATH] to a sum of templates, testing both a background-only hypothesis and a signal-plus-background ([MATH]) hypothesis.', '0911.4449-1-16-2': 'Systematic uncertainties, discussed further below, are included in the fit as constrained parameters.', '0911.4449-1-16-3': 'We perform pseudo experiments to calculate the probability ([MATH]-value) that the background-only discriminant fluctuates up to the observed result (observed [MATH]-value) and up to the median expected [MATH] result (expected [MATH]-value).', '0911.4449-1-16-4': 'We observe a [MATH]-value of [MATH], corresponding to a signal significance of 5.4[MATH], where [MATH] is expected.', '0911.4449-1-16-5': 'The observed [MATH] cross section is [MATH].', '0911.4449-1-17-0': 'We consider several sources of systematic uncertainty in both methods, taking into account their effect on both the signal acceptance and the shape of the background and signal templates.', '0911.4449-1-17-1': 'The uncertainty on the normalization of the backgrounds is taken as part of the statistical uncertainty.', '0911.4449-1-17-2': 'In the [MATH] method the largest systematic uncertainties are due to the modeling of the electroweak and QCD shapes, about 8 and 6 respectively.', '0911.4449-1-17-3': 'In the ME method the uncertainty in the jet energy scale is the largest systematic uncertainty, at about 10, which includes contributions both from the signal acceptance and from the shapes of the signal templates.', '0911.4449-1-17-4': 'In the [MATH] method this uncertainty is about 6.', '0911.4449-1-17-5': 'Both methods include an uncertainty of about 5 due to initial and final state radiation and a 6 uncertainty on the integrated luminosity.', '0911.4449-1-17-6': 'Smaller contributions arise from PDFs, jet energy resolution, the factorization and renormalization scales used in the [MATH]+jets simulation, and trigger and lepton identification efficiencies.', '0911.4449-1-18-0': 'One measure of how the two methods are correlated is the expected overlap of [MATH] signal.', '0911.4449-1-18-1': 'Accounting for the different integrated luminosities used, 15 of the signal in the [MATH] analysis is common to that in the [MATH] analysis.', '0911.4449-1-18-2': 'Conversely, 29 of the signal in the [MATH] analysis is common to that in the [MATH] analysis.', '0911.4449-1-18-3': 'This corresponds to a statistical correlation of about 21.', '0911.4449-1-18-4': 'If we assume the systematic uncertainties are 100 correlated, then the total correlation between the two analyses is 49, leading to a combined [CITATION] result of [MATH].', '0911.4449-1-18-5': 'Because the total uncertainties on the two input measurements are so similar, the combined central value does not depend significantly on the correlation assumed.', '0911.4449-1-18-6': 'The total uncertainty in the combined result increases with increasing correlation and we quote the value assuming maximum possible correlation.', '0911.4449-1-18-7': 'The signal overlap with the CDF [MATH] observation in the [MATH]0.3ex[MATH][MATH]+jets channel [CITATION] is also studied.', '0911.4449-1-18-8': 'While that analysis requires much larger [MATH]0.3ex[MATH][MATH], it does not veto events with identified leptons.', '0911.4449-1-18-9': 'We found that about 15% of the [MATH] signal from the [MATH]0.3ex[MATH][MATH]+jets analysis appears in the analyses presented here.', '0911.4449-1-19-0': 'In summary, we observe [MATH] production in the lepton plus jets plus [MATH].3ex[MATH]E_T[MATH] final state.', '0911.4449-1-19-1': 'We perform two searches: one seeking a resonance on top of a smoothly falling dijet mass distribution, and another building a discriminant using a matrix element technique.', '0911.4449-1-19-2': 'The combined [MATH] cross section from these two methods is measured to be [MATH], in good agreement with the SM prediction of [MATH] pb [CITATION].', '0911.4449-1-19-3': 'Measurements of these diboson processes are a necessary step toward validating Higgs boson search techniques at the Tevatron.', '0911.4449-1-20-0': 'We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions.', '0911.4449-1-20-1': 'This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium fur Bildung und Forschung, Germany; the World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovacion, and Programa Consolider-Ingenio 2010, Spain; the Slovak RD Agency; and the Academy of Finland.'}

{'0911.4449-2-0-0': 'We report two complementary measurements of the [MATH] cross section in the final state consisting of an electron or muon, missing transverse energy, and jets, performed using [MATH] collision data at [MATH]1.96 TeV collected by the CDF II detector.', '0911.4449-2-0-1': 'The first method uses the dijet invariant mass distribution while the second more sensitive method uses matrix-element calculations.', '0911.4449-2-0-2': 'The result from the second method has a signal significance of 5.4[MATH] and is the first observation of [MATH] production using this signature.', '0911.4449-2-0-3': 'Combining the results gives [MATH], in agreement with the standard model prediction.', '0911.4449-2-1-0': 'Measurements involving heavy vector boson pairs ([MATH], [MATH], and [MATH]) are important tests of the electroweak sector of the standard model (SM).', '0911.4449-2-1-1': 'Deviations of the production cross section from predictions could arise from anomalous triple gauge boson interactions [CITATION] or from new resonances decaying to vector bosons.', '0911.4449-2-1-2': 'Furthermore, the topology of diboson events is similar to that of events in which a Higgs boson is produced in association with a [MATH] or a [MATH], allowing diboson measurements to provide an important step towards future measurements of Higgs boson production.', '0911.4449-2-2-0': 'Diboson production has been observed at the Tevatron in channels in which both bosons decay leptonically [CITATION].', '0911.4449-2-2-1': 'Extraction of the diboson signal in hadronic channels is more challenging because of significantly larger backgrounds.', '0911.4449-2-2-2': 'In addition, due to limited detector resolution, it is difficult to distinguish hadronically decaying [MATH] bosons from [MATH] bosons.', '0911.4449-2-2-3': 'We report on two measurements of the cross section, [MATH], with the CDF II detector [CITATION] that use different techniques applied to the leptonic decay of one [MATH] and the hadronic decay of the associated [MATH] or [MATH], where [MATH] represents a high-[MATH]p_T[MATH] electron or muon).', '0911.4449-2-2-4': 'Our result represents the first observation of this signal in the lepton + jets channel.', '0911.4449-2-2-5': 'Evidence has previously been reported by the D0 collaboration [CITATION], and the CDF collaboration set a limit on its cross section times branching ratio [CITATION].', '0911.4449-2-2-6': 'In addition the CDF collaboration has reported observation of [MATH] in a different hadronic channel with large missing transverse energy and jets [CITATION].', '0911.4449-2-3-0': 'The first method uses the invariant mass of the two-jet system ([MATH]) to extract a signal peak from data corresponding to 3.9 fb[MATH] of [MATH] collisions at [MATH] TeV.', '0911.4449-2-3-1': 'The second method takes advantage of more kinematic information in the event by constructing a discriminant based on calculations of the differential cross sections of the signal and background processes.', '0911.4449-2-3-2': 'This so-called matrix-element (ME) method has been employed in a search for a low-mass Higgs produced in association with a [MATH] boson [CITATION] and in a measurement of single top production [CITATION].', '0911.4449-2-3-3': 'It is expected to achieve greater discriminating power and here uses data corresponding to an integrated luminosity of 2.7 fb[MATH].', '0911.4449-2-4-0': 'Data samples common to both analyses use trigger selections requiring a central electron (muon) with [MATH]E_T[MATH]p_T[MATH]18 GeV.', '0911.4449-2-4-1': 'The ME method utilizes an additional sample derived from a trigger requiring two jets and large missing transverse energy ([MATH][MATH]) [CITATION].', '0911.4449-2-5-0': 'Offline we select events with electron (muon) candidates with [MATH]E_T[MATH]p_T[MATH] 20 GeV, and with [MATH][MATH], jet, and other kinematic requirements chosen differently for the two methods.', '0911.4449-2-5-1': 'Jets are clustered using a fixed-cone algorithm with radius [MATH] and their energies are corrected for detector effects [CITATION].', '0911.4449-2-5-2': 'Cosmic ray and photon conversion candidates are identified and removed.', '0911.4449-2-6-0': 'Further event selection requirements are made to reduce backgrounds and the sensitivity to systematic uncertainties.', '0911.4449-2-6-1': 'In the [MATH] method, we require events to have [MATH].3ex[MATH]E_T[MATH] GeV, at least two jets with [MATH]E_T[MATH] GeV and [MATH], and the dijet vector boson candidate to have [MATH]p_T[MATH] GeV/[MATH].', '0911.4449-2-6-2': 'As a result of these selection criteria, the [MATH] distribution for background is smoothly falling in the region where the signal is expected to peak.', '0911.4449-2-6-3': 'The invariant mass of the dijet vector boson candidate, [MATH], is evaluated from the two most energetic jets.', '0911.4449-2-6-4': 'Additional requirements are made to reduce backgrounds and improve the Monte Carlo modeling of event kinematics: the transverse mass of the lepton and [MATH][MATH] system ([MATH] [CITATION]) must be greater than 30 GeV/[MATH], and the two most energetic jets must be separated by [MATH].', '0911.4449-2-7-0': 'In the ME method, we require events to have [MATH].3ex[MATH]E_T[MATH] 20 GeV and exactly two jets with [MATH]E_T[MATH] GeV and [MATH].', '0911.4449-2-7-1': 'Additional selection criteria to reduce backgrounds and achieve good modeling of the quantities used in the matrix element calculation include the rejection of events with either an additional jet of [MATH]E_T[MATH] GeV or a second high-[MATH]p_T[MATH] charged lepton.', '0911.4449-2-7-2': 'The latter reduces [MATH]+jets, [MATH], and leptonic diboson backgrounds.', '0911.4449-2-7-3': 'For events with an electron candidate, there is a significant background from production of multiple jets (multi-jet in the following) by quantum chromodynamical (QCD) processes, where the electron is faked by a hadronic jet.', '0911.4449-2-7-4': 'The ME method deals with this background by applying stringent selection criteria, while the [MATH] method assigns a systematic uncertainty to the background shape.', '0911.4449-2-7-5': 'The reduction of the multi-jet QCD background in the ME analysis is achieved by raising the [MATH].3ex[MATH]E_T[MATH] cut to 40 GeV, requiring [MATH] GeV/[MATH], and imposing additional cuts on the angles between the jets, the lepton, and the [MATH]0.3ex[MATH][MATH] [CITATION].', '0911.4449-2-7-6': 'There is a less stringent requirement of [MATH] GeV/[MATH] imposed on muon events to reduce the QCD background in that channel.', '0911.4449-2-8-0': 'After these selections for both methods, the dominant background to the diboson signal is a [MATH] boson produced with accompanying jets ([MATH]+jets), where the [MATH] decays leptonically.', '0911.4449-2-8-1': 'Smaller but non-negligible backgrounds come from QCD multi-jet (where one jet mimics a lepton signature), [MATH]+jets, [MATH], and single top production.', '0911.4449-2-8-2': 'QCD multi-jet events are modeled using data with loosened lepton selection criteria.', '0911.4449-2-8-3': 'Signal and other background processes are modeled using event generators and a geant-based CDF II detector simulation.', '0911.4449-2-8-4': 'The diboson signals and the [MATH] and single top backgrounds are simulated using the pythia event generator [CITATION].', '0911.4449-2-8-5': 'The [MATH]+jets and [MATH]+jets backgrounds are simulated using the tree-level event generator alpgen [CITATION], with an interface to pythia providing parton showering and hadronization.', '0911.4449-2-9-0': 'The normalization of the [MATH]+jets background is based on the measured cross section while for [MATH] and single top backgrounds the NLO predicted cross section is used [CITATION].', '0911.4449-2-9-1': 'The efficiencies for the [MATH]+jets, [MATH], and single top backgrounds are estimated from simulation.', '0911.4449-2-9-2': 'The normalization of the QCD background is estimated by fitting the [MATH].3ex[MATH]E_T[MATH] spectrum in data to the sum of all contributing processes, where the QCD and [MATH]+jets normalizations float in the fit.', '0911.4449-2-9-3': 'In the final signal extractions from both methods, the multi-jet QCD background is Gaussian constrained to the result of this [MATH].3ex[MATH]E_T[MATH] fit and the [MATH]+jets background is left unconstrained.', '0911.4449-2-10-0': 'We now describe the methodology and results from each technique.', '0911.4449-2-10-1': 'In the [MATH] method we extract the signal fraction from the data by performing a [MATH] fit to the dijet invariant mass spectrum separately for electron and muon events.', '0911.4449-2-10-2': 'Templates of [MATH] distributions are constructed with the multi-jet QCD background, the signal [MATH] processes, and the sum of the electroweak backgrounds ([MATH], [MATH], and [MATH] production).', '0911.4449-2-11-0': 'Figure [REF] shows the fit results superimposed on data after the electron and muon samples are combined.', '0911.4449-2-11-1': 'Also shown is the data [MATH] distribution after having subtracted the estimated background, superimposed on the signal Monte Carlo normalized to the fit result.', '0911.4449-2-11-2': 'Combining the two [MATH] fit results we get a total of [MATH] events, of which about 60% are muon events and 40% are electron events.', '0911.4449-2-11-3': 'The observed significance is 4.6[MATH] where 4.9[MATH] is expected.', '0911.4449-2-11-4': 'The resultant [MATH] production cross section measurement is [MATH].', '0911.4449-2-11-5': 'The sources of systematic uncertainty in this measurement are discussed together with those from the ME method below.', '0911.4449-2-12-0': 'In the ME method a probability density [MATH] that an event was produced by a given process is determined using the standard model differential cross section for that process.', '0911.4449-2-12-1': 'For an event with measured quantities [MATH], we integrate the appropriate differential cross section [MATH] over the partonic quantities [MATH] convolved with the parton distribution functions (PDFs) and a transfer function: [EQUATION]', '0911.4449-2-12-2': 'The PDFs ([MATH] and [MATH]) are evaluated according to the CTEQ5L parameterization [CITATION].', '0911.4449-2-12-3': 'The transfer function [MATH] relates [MATH] to [MATH], encoding the effects of the detector resolution.', '0911.4449-2-12-4': 'The momenta of electrons, muons, and the angles of jets are assumed to be measured exactly and a mapping of measured jet energy to partonic energy is derived using the full detector simulation.', '0911.4449-2-12-5': 'The integration is performed over the energy of the partons and the longitudinal momentum of the neutrino.', '0911.4449-2-12-6': 'The matrix element is calculated with tree-level diagrams from madgraph [CITATION].', '0911.4449-2-12-7': 'Event probability densities are calculated for the signal processes as well as for [MATH]+jets and single top background processes.', '0911.4449-2-12-8': 'The event probabilities are combined into an event probability discriminant: [MATH], where [MATH] and [MATH].', '0911.4449-2-12-9': 'We make templates of the [MATH] for all signal and background processes and ultimately extract the signal using a fit of the observed [MATH] distribution to a sum of the templates.', '0911.4449-2-12-10': "The expected event yields are as shown in Table [REF] for the ME method's event selection.", '0911.4449-2-13-0': 'Figure [REF] shows the dijet mass in bins of [MATH].', '0911.4449-2-13-1': 'Most of the background events have low [MATH].', '0911.4449-2-13-2': 'Events with [MATH] have a dijet mass peak close to the expected [MATH] resonance, and the signal-to-background ratio improves with increasing [MATH].', '0911.4449-2-14-0': 'Before comparing the observed [MATH] to the prediction, we validate the Monte Carlo modeling of the quantities that enter the matrix element calculation.', '0911.4449-2-14-1': 'We compare the observed distributions to the predicted ones in control regions with very little signal and also in the signal-rich region.', '0911.4449-2-14-2': 'The different regions are chosen according to the invariant mass of the two-jet system ([MATH]): the signal-rich region has [MATH] GeV and the control regions cover the rest of the [MATH] range.', '0911.4449-2-14-3': 'We also check the modeling of the properties (mass, [MATH], and [MATH]) of the leptonic [MATH] boson and the hadronic [MATH] or [MATH] boson candidate.', '0911.4449-2-14-4': 'All of these quantities are well described by the simulation for our event selection.', '0911.4449-2-14-5': 'There is a small discrepancy in the description of [MATH] in the control regions, as is visible in the low-[MATH] region of Figure [REF].', '0911.4449-2-14-6': 'Associated with this discrepancy we assign a systematic mismodeling uncertainty which is derived in the control regions and extrapolated through the signal region.', '0911.4449-2-14-7': 'This uncertainty has a negligible effect on the results, because most background events lie in the first few bins of the [MATH] distribution.', '0911.4449-2-14-8': 'Small changes in modeling of those background events do not change the shape of the [MATH].', '0911.4449-2-15-0': 'The observed and predicted [MATH]s are shown in Figure [REF].', '0911.4449-2-15-1': 'We use a binned-likelihood fit of the observed [MATH] to a sum of templates, testing both a background-only hypothesis and a signal-plus-background ([MATH]) hypothesis.', '0911.4449-2-15-2': 'Systematic uncertainties, discussed further below, are included in the fit as constrained parameters.', '0911.4449-2-15-3': 'We perform pseudo experiments to calculate the probability ([MATH]-value) that the background-only discriminant fluctuates up to the observed result (observed [MATH]-value) and up to the median expected [MATH] result (expected [MATH]-value).', '0911.4449-2-15-4': 'We observe a [MATH]-value of [MATH], corresponding to a signal significance of 5.4[MATH], where [MATH] is expected.', '0911.4449-2-15-5': 'The observed [MATH] cross section is [MATH].', '0911.4449-2-16-0': 'We consider several sources of systematic uncertainty in both methods, taking into account their effect on both the signal acceptance and the shape of the background and signal templates.', '0911.4449-2-16-1': 'The uncertainty on the normalization of the backgrounds is taken as part of the statistical uncertainty.', '0911.4449-2-16-2': 'In the [MATH] method the largest systematic uncertainties are due to the modeling of the electroweak and QCD shapes, about 8 and 6 respectively.', '0911.4449-2-16-3': 'In the ME method the uncertainty in the jet energy scale is the largest systematic uncertainty, at about 10, which includes contributions both from the signal acceptance and from the shapes of the signal templates.', '0911.4449-2-16-4': 'In the [MATH] method this uncertainty is about 6.', '0911.4449-2-16-5': 'Both methods include an uncertainty of about 5 due to initial and final state radiation and a 6 uncertainty on the integrated luminosity.', '0911.4449-2-16-6': 'Smaller contributions arise from PDFs, jet energy resolution, the factorization and renormalization scales used in the [MATH]+jets simulation, and trigger and lepton identification efficiencies.', '0911.4449-2-17-0': 'One measure of how the two methods are correlated is the expected overlap of [MATH] signal.', '0911.4449-2-17-1': 'Accounting for the different integrated luminosities used, 15 of the signal in the [MATH] analysis is common to that in the [MATH] analysis.', '0911.4449-2-17-2': 'Conversely, 29 of the signal in the [MATH] analysis is common to that in the [MATH] analysis.', '0911.4449-2-17-3': 'This corresponds to a statistical correlation of about 21.', '0911.4449-2-17-4': 'If we assume the systematic uncertainties are 100 correlated, then the total correlation between the two analyses is 49, leading to a combined [CITATION] result of [MATH].', '0911.4449-2-17-5': 'Because the total uncertainties on the two input measurements are similar, the combined central value does not depend significantly on the correlation assumed.', '0911.4449-2-17-6': 'The total uncertainty in the combined result increases with increasing correlation and we quote the value assuming maximum possible correlation.', '0911.4449-2-17-7': 'The signal overlap with the CDF [MATH] observation in the [MATH]0.3ex[MATH][MATH]+jets channel [CITATION] is also studied.', '0911.4449-2-17-8': 'While that analysis requires much larger [MATH]0.3ex[MATH][MATH], it does not veto events with identified leptons.', '0911.4449-2-17-9': 'We found that about 15% of the [MATH] signal from the [MATH]0.3ex[MATH][MATH]+jets analysis appears in the analyses presented here.', '0911.4449-2-18-0': 'In summary, we observe [MATH] production in the lepton plus jets plus [MATH].3ex[MATH]E_T[MATH] final state.', '0911.4449-2-18-1': 'We perform two searches: one seeking a resonance on top of a smoothly falling dijet mass distribution, and another building a discriminant using a matrix element technique.', '0911.4449-2-18-2': 'The combined [MATH] cross section from these two methods is measured to be [MATH], in good agreement with the prediction of [MATH] pb [CITATION].', '0911.4449-2-18-3': 'Measurements of these diboson processes are tests of electroweak theory and a necessary step toward validating Higgs boson search techniques at the Tevatron.', '0911.4449-2-19-0': 'We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions.', '0911.4449-2-19-1': 'This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium fur Bildung und Forschung, Germany; the World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovacion, and Programa Consolider-Ingenio 2010, Spain; the Slovak RD Agency; and the Academy of Finland.'}

[['0911.4449-1-9-0', '0911.4449-2-8-0'], ['0911.4449-1-9-1', '0911.4449-2-8-1'], ['0911.4449-1-9-2', '0911.4449-2-8-2'], ['0911.4449-1-9-5', '0911.4449-2-8-5'], ['0911.4449-1-5-0', '0911.4449-2-4-0'], ['0911.4449-1-5-1', '0911.4449-2-4-1'], ['0911.4449-1-6-0', '0911.4449-2-5-0'], ['0911.4449-1-6-1', '0911.4449-2-5-1'], ['0911.4449-1-6-2', '0911.4449-2-5-2'], ['0911.4449-1-19-0', '0911.4449-2-18-0'], ['0911.4449-1-19-1', '0911.4449-2-18-1'], ['0911.4449-1-0-2', '0911.4449-2-0-2'], ['0911.4449-1-18-0', '0911.4449-2-17-0'], ['0911.4449-1-18-1', '0911.4449-2-17-1'], ['0911.4449-1-18-2', '0911.4449-2-17-2'], ['0911.4449-1-18-3', '0911.4449-2-17-3'], ['0911.4449-1-18-4', '0911.4449-2-17-4'], ['0911.4449-1-18-6', '0911.4449-2-17-6'], ['0911.4449-1-18-7', '0911.4449-2-17-7'], ['0911.4449-1-18-8', '0911.4449-2-17-8'], ['0911.4449-1-18-9', '0911.4449-2-17-9'], ['0911.4449-1-1-0', '0911.4449-2-1-0'], ['0911.4449-1-1-1', '0911.4449-2-1-1'], ['0911.4449-1-1-2', '0911.4449-2-1-2'], ['0911.4449-1-14-0', '0911.4449-2-13-0'], ['0911.4449-1-14-1', '0911.4449-2-13-1'], ['0911.4449-1-14-2', '0911.4449-2-13-2'], ['0911.4449-1-7-0', '0911.4449-2-6-0'], ['0911.4449-1-7-1', '0911.4449-2-6-1'], ['0911.4449-1-7-2', '0911.4449-2-6-2'], ['0911.4449-1-7-3', '0911.4449-2-6-3'], ['0911.4449-1-7-4', '0911.4449-2-6-4'], ['0911.4449-1-20-0', '0911.4449-2-19-0'], ['0911.4449-1-20-1', '0911.4449-2-19-1'], ['0911.4449-1-13-0', '0911.4449-2-12-0'], ['0911.4449-1-13-4', '0911.4449-2-12-5'], ['0911.4449-1-13-5', '0911.4449-2-12-6'], ['0911.4449-1-13-7', '0911.4449-2-12-8'], ['0911.4449-1-13-9', '0911.4449-2-12-10'], ['0911.4449-1-17-0', '0911.4449-2-16-0'], ['0911.4449-1-17-1', '0911.4449-2-16-1'], ['0911.4449-1-17-2', '0911.4449-2-16-2'], ['0911.4449-1-17-3', '0911.4449-2-16-3'], ['0911.4449-1-17-4', '0911.4449-2-16-4'], ['0911.4449-1-17-5', '0911.4449-2-16-5'], ['0911.4449-1-17-6', '0911.4449-2-16-6'], ['0911.4449-1-16-0', '0911.4449-2-15-0'], ['0911.4449-1-16-1', '0911.4449-2-15-1'], ['0911.4449-1-16-2', '0911.4449-2-15-2'], ['0911.4449-1-16-3', '0911.4449-2-15-3'], ['0911.4449-1-16-4', '0911.4449-2-15-4'], ['0911.4449-1-16-5', '0911.4449-2-15-5'], ['0911.4449-1-10-0', '0911.4449-2-9-0'], ['0911.4449-1-10-1', '0911.4449-2-9-1'], ['0911.4449-1-10-2', '0911.4449-2-9-2'], ['0911.4449-1-10-3', '0911.4449-2-9-3'], ['0911.4449-1-11-0', '0911.4449-2-10-0'], ['0911.4449-1-11-2', '0911.4449-2-10-2'], ['0911.4449-1-3-0', '0911.4449-2-3-0'], ['0911.4449-1-3-1', '0911.4449-2-3-1'], ['0911.4449-1-3-2', '0911.4449-2-3-2'], ['0911.4449-1-3-3', '0911.4449-2-3-3'], ['0911.4449-1-2-0', '0911.4449-2-2-0'], ['0911.4449-1-2-1', '0911.4449-2-2-1'], ['0911.4449-1-2-2', '0911.4449-2-2-2'], ['0911.4449-1-2-4', '0911.4449-2-2-4'], ['0911.4449-1-2-5', '0911.4449-2-2-5'], ['0911.4449-1-2-6', '0911.4449-2-2-6'], ['0911.4449-1-12-0', '0911.4449-2-11-0'], ['0911.4449-1-12-2', '0911.4449-2-11-2'], ['0911.4449-1-12-4', '0911.4449-2-11-4'], ['0911.4449-1-12-5', '0911.4449-2-11-5'], ['0911.4449-1-15-0', '0911.4449-2-14-0'], ['0911.4449-1-15-1', '0911.4449-2-14-1'], ['0911.4449-1-15-2', '0911.4449-2-14-2'], ['0911.4449-1-15-3', '0911.4449-2-14-3'], ['0911.4449-1-15-4', '0911.4449-2-14-4'], ['0911.4449-1-15-5', '0911.4449-2-14-5'], ['0911.4449-1-15-6', '0911.4449-2-14-6'], ['0911.4449-1-15-7', '0911.4449-2-14-7'], ['0911.4449-1-15-8', '0911.4449-2-14-8'], ['0911.4449-1-8-0', '0911.4449-2-7-0'], ['0911.4449-1-8-2', '0911.4449-2-7-2'], ['0911.4449-1-8-3', '0911.4449-2-7-3'], ['0911.4449-1-8-4', '0911.4449-2-7-4'], ['0911.4449-1-8-5', '0911.4449-2-7-5'], ['0911.4449-1-8-6', '0911.4449-2-7-6'], ['0911.4449-1-9-3', '0911.4449-2-8-3'], ['0911.4449-1-9-4', '0911.4449-2-8-4'], ['0911.4449-1-19-2', '0911.4449-2-18-2'], ['0911.4449-1-19-3', '0911.4449-2-18-3'], ['0911.4449-1-0-0', '0911.4449-2-0-0'], ['0911.4449-1-0-3', '0911.4449-2-0-3'], ['0911.4449-1-18-5', '0911.4449-2-17-5'], ['0911.4449-1-13-1', '0911.4449-2-12-1'], ['0911.4449-1-13-6', '0911.4449-2-12-7'], ['0911.4449-1-13-8', '0911.4449-2-12-9'], ['0911.4449-1-11-1', '0911.4449-2-10-1'], ['0911.4449-1-2-3', '0911.4449-2-2-3'], ['0911.4449-1-8-1', '0911.4449-2-7-1'], ['0911.4449-1-0-1', '0911.4449-2-0-1'], ['0911.4449-1-13-2', '0911.4449-2-12-2'], ['0911.4449-1-13-3', '0911.4449-2-12-4'], ['0911.4449-1-12-1', '0911.4449-2-11-1'], ['0911.4449-1-12-3', '0911.4449-2-11-3']]

[['0911.4449-1-9-0', '0911.4449-2-8-0'], ['0911.4449-1-9-1', '0911.4449-2-8-1'], ['0911.4449-1-9-2', '0911.4449-2-8-2'], ['0911.4449-1-9-5', '0911.4449-2-8-5'], ['0911.4449-1-5-0', '0911.4449-2-4-0'], ['0911.4449-1-5-1', '0911.4449-2-4-1'], ['0911.4449-1-6-0', '0911.4449-2-5-0'], ['0911.4449-1-6-1', '0911.4449-2-5-1'], ['0911.4449-1-6-2', '0911.4449-2-5-2'], ['0911.4449-1-19-0', '0911.4449-2-18-0'], ['0911.4449-1-19-1', '0911.4449-2-18-1'], ['0911.4449-1-0-2', '0911.4449-2-0-2'], ['0911.4449-1-18-0', '0911.4449-2-17-0'], ['0911.4449-1-18-1', '0911.4449-2-17-1'], ['0911.4449-1-18-2', '0911.4449-2-17-2'], ['0911.4449-1-18-3', '0911.4449-2-17-3'], ['0911.4449-1-18-4', '0911.4449-2-17-4'], ['0911.4449-1-18-6', '0911.4449-2-17-6'], ['0911.4449-1-18-7', '0911.4449-2-17-7'], ['0911.4449-1-18-8', '0911.4449-2-17-8'], ['0911.4449-1-18-9', '0911.4449-2-17-9'], ['0911.4449-1-1-0', '0911.4449-2-1-0'], ['0911.4449-1-1-1', '0911.4449-2-1-1'], ['0911.4449-1-1-2', '0911.4449-2-1-2'], ['0911.4449-1-14-0', '0911.4449-2-13-0'], ['0911.4449-1-14-1', '0911.4449-2-13-1'], ['0911.4449-1-14-2', '0911.4449-2-13-2'], ['0911.4449-1-7-0', '0911.4449-2-6-0'], ['0911.4449-1-7-1', '0911.4449-2-6-1'], ['0911.4449-1-7-2', '0911.4449-2-6-2'], ['0911.4449-1-7-3', '0911.4449-2-6-3'], ['0911.4449-1-7-4', '0911.4449-2-6-4'], ['0911.4449-1-20-0', '0911.4449-2-19-0'], ['0911.4449-1-20-1', '0911.4449-2-19-1'], ['0911.4449-1-13-0', '0911.4449-2-12-0'], ['0911.4449-1-13-4', '0911.4449-2-12-5'], ['0911.4449-1-13-5', '0911.4449-2-12-6'], ['0911.4449-1-13-7', '0911.4449-2-12-8'], ['0911.4449-1-13-9', '0911.4449-2-12-10'], ['0911.4449-1-17-0', '0911.4449-2-16-0'], ['0911.4449-1-17-1', '0911.4449-2-16-1'], ['0911.4449-1-17-2', '0911.4449-2-16-2'], ['0911.4449-1-17-3', '0911.4449-2-16-3'], ['0911.4449-1-17-4', '0911.4449-2-16-4'], ['0911.4449-1-17-5', '0911.4449-2-16-5'], ['0911.4449-1-17-6', '0911.4449-2-16-6'], ['0911.4449-1-16-0', '0911.4449-2-15-0'], ['0911.4449-1-16-1', '0911.4449-2-15-1'], ['0911.4449-1-16-2', '0911.4449-2-15-2'], ['0911.4449-1-16-3', '0911.4449-2-15-3'], ['0911.4449-1-16-4', '0911.4449-2-15-4'], ['0911.4449-1-16-5', '0911.4449-2-15-5'], ['0911.4449-1-10-0', '0911.4449-2-9-0'], ['0911.4449-1-10-1', '0911.4449-2-9-1'], ['0911.4449-1-10-2', '0911.4449-2-9-2'], ['0911.4449-1-10-3', '0911.4449-2-9-3'], ['0911.4449-1-11-0', '0911.4449-2-10-0'], ['0911.4449-1-11-2', '0911.4449-2-10-2'], ['0911.4449-1-3-0', '0911.4449-2-3-0'], ['0911.4449-1-3-1', '0911.4449-2-3-1'], ['0911.4449-1-3-2', '0911.4449-2-3-2'], ['0911.4449-1-3-3', '0911.4449-2-3-3'], ['0911.4449-1-2-0', '0911.4449-2-2-0'], ['0911.4449-1-2-1', '0911.4449-2-2-1'], ['0911.4449-1-2-2', '0911.4449-2-2-2'], ['0911.4449-1-2-4', '0911.4449-2-2-4'], ['0911.4449-1-2-5', '0911.4449-2-2-5'], ['0911.4449-1-2-6', '0911.4449-2-2-6'], ['0911.4449-1-12-0', '0911.4449-2-11-0'], ['0911.4449-1-12-2', '0911.4449-2-11-2'], ['0911.4449-1-12-4', '0911.4449-2-11-4'], ['0911.4449-1-12-5', '0911.4449-2-11-5'], ['0911.4449-1-15-0', '0911.4449-2-14-0'], ['0911.4449-1-15-1', '0911.4449-2-14-1'], ['0911.4449-1-15-2', '0911.4449-2-14-2'], ['0911.4449-1-15-3', '0911.4449-2-14-3'], ['0911.4449-1-15-4', '0911.4449-2-14-4'], ['0911.4449-1-15-5', '0911.4449-2-14-5'], ['0911.4449-1-15-6', '0911.4449-2-14-6'], ['0911.4449-1-15-7', '0911.4449-2-14-7'], ['0911.4449-1-15-8', '0911.4449-2-14-8'], ['0911.4449-1-8-0', '0911.4449-2-7-0'], ['0911.4449-1-8-2', '0911.4449-2-7-2'], ['0911.4449-1-8-3', '0911.4449-2-7-3'], ['0911.4449-1-8-4', '0911.4449-2-7-4'], ['0911.4449-1-8-5', '0911.4449-2-7-5'], ['0911.4449-1-8-6', '0911.4449-2-7-6']]

[['0911.4449-1-9-3', '0911.4449-2-8-3'], ['0911.4449-1-9-4', '0911.4449-2-8-4'], ['0911.4449-1-19-2', '0911.4449-2-18-2'], ['0911.4449-1-19-3', '0911.4449-2-18-3'], ['0911.4449-1-0-0', '0911.4449-2-0-0'], ['0911.4449-1-0-3', '0911.4449-2-0-3'], ['0911.4449-1-18-5', '0911.4449-2-17-5'], ['0911.4449-1-13-1', '0911.4449-2-12-1'], ['0911.4449-1-13-6', '0911.4449-2-12-7'], ['0911.4449-1-13-8', '0911.4449-2-12-9'], ['0911.4449-1-11-1', '0911.4449-2-10-1'], ['0911.4449-1-2-3', '0911.4449-2-2-3'], ['0911.4449-1-8-1', '0911.4449-2-7-1']]

[]

[['0911.4449-1-0-1', '0911.4449-2-0-1'], ['0911.4449-1-13-2', '0911.4449-2-12-2'], ['0911.4449-1-13-3', '0911.4449-2-12-4'], ['0911.4449-1-12-1', '0911.4449-2-11-1'], ['0911.4449-1-12-3', '0911.4449-2-11-3']]

[]

[]

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

https://arxiv.org/abs/0911.4449

1503.04159

{'1503.04159-1-0-0': 'We address the impact of sterile fermions on the lepton flavor violating decays of quarkonia as well as of the [MATH] boson.', '1503.04159-1-0-1': 'We compute the relevant Wilson coefficients and show that the [MATH], where [MATH], [MATH] can be significantly enhanced in the case of large sterile fermion masses and a non-negligible active-sterile mixing.', '1503.04159-1-0-2': 'We illustrate that feature in a specific minimal realization of the inverse seesaw mechanism, known as [MATH]-ISS, and in an effective model in which the presence of non-standard sterile fermions is parameterized by means of one heavy sterile (Majorana) neutrino.', '1503.04159-1-1-0': 'PACS: 14.60.', '1503.04159-1-1-1': 'St, 14.60.', '1503.04159-1-1-2': 'Pq, 13.20.', '1503.04159-1-1-3': 'Gd, 13.38.', '1503.04159-1-1-4': 'Dg.', '1503.04159-1-2-0': '# Introduction', '1503.04159-1-3-0': 'So far no signal of new physics has been observed but its search is important in order to understand how to enlarge the Standard Model (SM) to solve both the hierarchy and flavor problems.', '1503.04159-1-3-1': 'One of the most significant observations requiring to go beyond the Standard Model (BSM) is the assessment that neutrinos are massive and that they mix [CITATION].', '1503.04159-1-3-2': 'Possible SM extensions aiming at incorporating massive neutrinos give rise to interesting collider signatures and open the door to new phenomena such as lepton flavor violating (LFV) decays.', '1503.04159-1-4-0': 'Currently, the search for manifestations of LFV constitutes a goal of several experimental facilities dedicated to rare lepton decays, such as [MATH] and [MATH], and to the neutrinoless [MATH] conversion in muonic atoms.', '1503.04159-1-4-1': 'One of the most stringent bounds from these searches is the one derived by the MEG collaboration, [MATH] [CITATION], which is expected to be improved to a planned sensitivity of [MATH] [CITATION].', '1503.04159-1-4-2': 'Moreover, the bound [MATH], set by the SINDRUM experiment [CITATION], is expected to be improved by the Mu3e experiment where a sensitivity [MATH] is planed [CITATION].', '1503.04159-1-4-3': 'Limits on the [MATH] radiative decays [CITATION] and the [MATH]-body decays of [MATH] [CITATION] appear to be less stringent right now, but are likely to be improved at Belle II [CITATION], where the search of LFV decays of [MATH]-meson will be made too [CITATION].', '1503.04159-1-4-4': 'The most promising developments regarding LFV are those related to the [MATH] conversion in nuclei.', '1503.04159-1-4-5': 'The present bound for [MATH] conversion rate is [MATH] [CITATION], and the planned sensitivity is [MATH] [CITATION].', '1503.04159-1-4-6': 'Similar is the case for gold and aluminum [CITATION].', '1503.04159-1-5-0': 'Searches for LFV are also conducted in high-energy experiments and a first bound on the Higgs boson LFV decay [MATH] has been reported by the CMS collaboration [CITATION].', '1503.04159-1-5-1': 'The LHCb collaboration, instead, reported the bound [MATH] [CITATION], which is likely to be improved in the near future [CITATION].', '1503.04159-1-5-2': 'Notice also that they already improved the bounds on [MATH] by an order of magnitude [CITATION].', '1503.04159-1-6-0': 'In this work we will focus on the indirect probes of New Physics (NP) through the LFV processes of neutral vector bosons, namely [MATH], with [MATH], and [MATH], where [MATH] stands for [MATH] and its radial excitations, and similarly for [MATH].', '1503.04159-1-6-1': 'Most of the research in this direction reported so far is related to the [MATH] decay modes.', '1503.04159-1-6-2': 'More specifically, the experimental bounds, obtained at LEP are found to be [MATH] [CITATION], [MATH] [CITATION], and [MATH] [CITATION].', '1503.04159-1-6-3': 'One of these bounds has been improved at LHC, namely [MATH] [CITATION].', '1503.04159-1-6-4': 'On the theory side, the [MATH] decays have been analyzed in the extensions of the SM involving additional massive and sterile neutrinos that could mix with the standard (active) ones and thus give rise to the LFV decay rates [CITATION].', '1503.04159-1-6-5': 'A similar approach has been also adopted in ref. [CITATION], in the perspective of a Tera-[MATH] factory FCC-ee [CITATION] for which a targeted sensitivity is expected to be [MATH] [CITATION].', '1503.04159-1-7-0': 'Lepton flavor conserving decays of quarkonia have been measured to a high accuracy which can be actually used to fix the hadronic parameters (decay constants).', '1503.04159-1-7-1': 'Otherwise, one can use the results of numerical simulations of QCD on the lattice which are nowadays accurate as well [CITATION].', '1503.04159-1-7-2': 'The experimentally established bounds for the simplest LFV decays of quarkonia are [CITATION]: [EQUATION] where each mode is to be understood as [MATH].', '1503.04159-1-8-0': 'Despite the appreciable experimental work on the latter observables, only a few theoretical studies have been carried out so far.', '1503.04159-1-8-1': 'The authors of ref. [CITATION] applied a vector meson dominance approximation to [MATH] and expressed the width of the latter process, [MATH].', '1503.04159-1-8-2': 'Since the values of [MATH] are very well known experimentally [CITATION], the experimental bound on [MATH] is then used to obtain an upper bound on the phenomenological coupling [MATH], which is then converted to an upper bound on [MATH].', '1503.04159-1-8-3': 'Similar approach has been used in ref. [CITATION] where instead of [MATH], the authors considered the [MATH] conversion in nuclei ([MATH]), which they described in terms of a product of couplings [MATH] and [MATH].', '1503.04159-1-8-4': 'The latter could be extracted from the experimentally measured [MATH], and with that knowledge the experimental upper bound on [MATH] results in an upper bound on [MATH].', '1503.04159-1-8-5': 'A more dynamical approach in modeling the [MATH] processes has been made in a supersymmetric extension of the SM with type-I seesaw [CITATION].', '1503.04159-1-9-0': 'Sterile fermions were proposed in various neutrino mass generation mechanisms, but the interest in their properties was further motivated by the reactor/accelerator anomalies [CITATION], a possibility to offer a warm dark matter candidate [CITATION], and by indications from the large scale structure formation [CITATION].', '1503.04159-1-10-0': 'Incorporating neutrino oscillations (masses and mixing [CITATION]) into the SM implies that the charged current is modified to [EQUATION] [MATH] being the leptonic mixing matrix, [MATH] the flavor of a charged lepton, and [MATH] denotes a physical neutrino state.', '1503.04159-1-10-1': 'If one assumes that only three massive neutrinos are present, the matrix [MATH] corresponds to the unitary PMNS matrix.', '1503.04159-1-10-2': 'In that situation the GIM mechanism makes the decay rates B([MATH]) completely negligible, [MATH].', '1503.04159-1-10-3': 'That feature, however, can be drastically changed in the presence of a non-negligible mixing with heavy sterile fermions.', '1503.04159-1-10-4': 'In what follows we will consider such situations, derive analytical expressions for B([MATH]), and discuss a specific realization of the inverse seesaw mechanism, known as (2,3)-ISS [CITATION].', '1503.04159-1-10-5': 'We will also discuss a simplified model in which the effect of the heavy sterile neutrinos is described by one effective sterile neutrino state with non-negligible mixing with active neutrinos.', '1503.04159-1-10-6': 'Despite several differences, our approach is similar to the one discussed in ref. [CITATION], where the SM has been extended by new, heavy, Dirac neutrinos, singlets under [MATH], and applied to a number of low energy decay processes.', '1503.04159-1-10-7': 'Our sterile neutrinos are Majorana and we apply the approach to the leptonic decays of quarkonia for the first time.', '1503.04159-1-11-0': 'The remainder of this paper is organized as follows: In Sec. [REF] we formulate the problem in terms of a low energy effective theory of a larger theory which contains heavy sterile neutrinos, we derive expression for B([MATH]) and compute the Wilson coefficients.', '1503.04159-1-11-1': 'In Sec. [REF] we briefly describe the specific models with sterile neutrinos which are used in this paper to produce our results presented in Sec. [REF].', '1503.04159-1-11-2': 'We finally conclude in Sec. [REF].', '1503.04159-1-12-0': '# LFV decay of Quarkonia - Effective Theory', '1503.04159-1-13-0': 'In this section we formulate a low energy effective theory of the LFV decays of quarkonia of type [MATH], and express the decay amplitude in terms of the quarkonium decay constants and the corresponding Wilson coefficients.', '1503.04159-1-13-1': 'The latter are then computed in the extensions of the SM which include the heavy sterile neutrinos.', '1503.04159-1-13-2': 'We also derive the expression relevant to [MATH].', '1503.04159-1-14-0': '## Effective Hamiltonian', '1503.04159-1-15-0': 'Keeping in mind the fact that we are extending the SM by adding sterile fermions, without touching the gauge sector of the theory, the decays of vector quarkonia, [MATH], can only occur through the photon and the [MATH]-boson exchange at tree level.', '1503.04159-1-15-1': 'In the lepton flavor conserving processes the [MATH]-exchange terms are very small with respect to those arising from the electromagnetic interaction and are usually neglected.', '1503.04159-1-15-2': 'The generic effective Hamiltonian can be written as [EQUATION] where [MATH] is the electric charge of the quark [MATH], [MATH] is the mass of quarkonium [MATH] which is dominated by the valence quark configuration [MATH], [MATH] are the Wilson coefficients, [MATH] is the momentum of one of the outgoing leptons, and [MATH].', '1503.04159-1-15-3': 'Contributions to the scalar (left and right) terms are suppressed by [MATH], where [MATH] are the charged lepton masses.', '1503.04159-1-15-4': 'In this section we will keep such terms so that our expressions can be useful to approaches in which the scalar bosons are taken in consideration.', '1503.04159-1-15-5': 'For our phenomenological discussion, however, it is worth emphasizing that [MATH].', '1503.04159-1-16-0': 'Without entering the details of calculation it is easy to verify that the only relevant diagrams are those shown in Fig. [REF], and therefore the structure of the Wilson coefficients [MATH] reads, [EQUATION] where [MATH] are the contributions arising from the photon and the [MATH]-boson exchange, while [MATH] comes from the box diagram that involves the Cabibbo-Kobayashi-Maskawa coupling [MATH].', '1503.04159-1-16-1': 'In the above expressions [MATH].', '1503.04159-1-16-2': 'The blob in the diagram shown in Fig. [REF] stands for the lepton loop diagrams that may contain one or two neutrino states and which, in the extensions of the SM involving a heavy neutrino sector, will give rise to the LFV decay due to the effect of mixing which is parametrized by the matrix [MATH] [see eq. ([REF])].', '1503.04159-1-16-3': 'Separate contributions coming from different diagrams can be further reduced by factoring out the neutrino mixing matrix elements, namely [EQUATION] where we see that the term involving two neutrino eigenstates appears only in the [MATH] coefficient because it is related to the vertex [MATH].', '1503.04159-1-17-0': 'Using the effective Hamiltonian ([REF]) and parameterizing the hadronic matrix as [EQUATION] where [MATH] is the decay constant of a quarkonium [MATH] with momentum [MATH] and in a polarization state [MATH], we can write the decay rate as, [EQUATION] with [EQUATION] and [EQUATION] which gives, [EQUATION]', '1503.04159-1-17-1': 'As we mentioned above, we consider in our framework [MATH], and therefore we can write, [EQUATION] where [MATH] is given in eq. ([REF]).', '1503.04159-1-17-2': 'In this last expression we also used [MATH].', '1503.04159-1-18-0': 'Besides quarkonia we will also revisit the issue of adding extra species of sterile neutrinos to the decay of [MATH].', '1503.04159-1-18-1': 'In that case the effective Hamiltonian can be written as, [EQUATION] where the Wilson coefficients are now denoted by [MATH] and take the form, [EQUATION]', '1503.04159-1-18-2': 'The decay rate in the similar limit, [MATH], reads, [EQUATION]', '1503.04159-1-19-0': '## Wilson coefficients', '1503.04159-1-20-0': 'Concerning the computation of the Wilson coefficients we stress again that our results are obtained in a theory in which the Standard Model is extended to include extra species of sterile fermions, without changing the gauge sector.', '1503.04159-1-20-1': 'The origin of the leptonic mixing matrix [MATH] is model dependent and in order to be able to do a phenomenological analysis, we will have to adopt a specific model which will be discussed in the next Section.', '1503.04159-1-21-0': 'The blob in the diagram shown in Fig. [REF] stands for a series of diagrams such as those displayed in Fig. [REF].', '1503.04159-1-21-1': 'All of them, including the box diagram in Fig. [REF], have been computed in the Feynman gauge and the results are collected in appendix A.', '1503.04159-1-22-0': 'Here we focus on the most important contributions in the case of large masses of sterile (Majorana) neutrinos.', '1503.04159-1-22-1': 'Contributions to the Wilson coefficients coming from vertex diagrams can be divided into two pieces: those involving only one neutrino in the loop, [MATH], where [MATH], and those with two neutrinos in the loop, [MATH].', '1503.04159-1-22-2': 'In the limit of large values of [MATH], we find the following behavior [EQUATION]', '1503.04159-1-22-3': 'To illustrate the relative contribution of the different diagrams we fix the values of the coefficients [MATH], and plot [MATH] and [MATH] for the case of [MATH], cf. Fig. [REF].', '1503.04159-1-22-4': 'We see that only for very large masses the diagrams with two neutrinos in the loop become more important than those with one neutrino state.', '1503.04159-1-22-5': 'We should stress that each contribution to [MATH], i.e. [MATH] and [MATH], scales as [MATH] for large values of [MATH], except for [MATH] which goes to a constant in the same limit.', '1503.04159-1-22-6': 'That can also be seen in Fig. [REF] where in the left plot we show the dependence of the total [MATH] on [MATH] and in the right plot we show [MATH] and its dependence on the mass of the initial decaying meson, [MATH], [MATH] and [MATH].', '1503.04159-1-22-7': 'The contribution of sterile neutrinos to the LFV decay of [MATH] is larger than the one to lighter mesons, since the Wilson coefficients are also proportional the mass of the initial particle.', '1503.04159-1-23-0': '# SM in presence of sterile fermions', '1503.04159-1-24-0': 'With the expressions derived above, we now have to specify a model for lepton mixing (couplings) [MATH] in the presence of heavy sterile neutrinos propagating in the loops.', '1503.04159-1-24-1': 'We opt for a minimal realization of the inverse seesaw mechanism for generation of neutrino masses, which is nowadays rather well constrained by the available experimental data.', '1503.04159-1-24-2': 'Furthermore, we will use a parametric model containing one effective sterile neutrino, which essentially mimics the behavior at low energy scales of mechanisms involving heavy sterile fermions.', '1503.04159-1-25-0': '## The (2,3)-inverse seesaw realization', '1503.04159-1-26-0': 'Among many possible realizations to account for massive neutrinos, the inverse seesaw mechanism (ISS) [CITATION] offers a possibility to accommodate the smallness of the active neutrino masses for a comparatively low seesaw scale, but still with natural [MATH] Yukawa couplings, which renders this scenario phenomenologically appealing.', '1503.04159-1-26-1': 'Indeed, depending on their masses and mixing with active neutrinos, the new states can be produced in collider and/or low-energy experiments, and their contribution to physical processes can be sizable.', '1503.04159-1-26-2': 'ISS, embedded in the SM, results in a mass term for neutrinos of the form, [EQUATION] where [MATH] is the charge conjugation matrix and [MATH].', '1503.04159-1-26-3': 'Here [MATH], [MATH], denotes the active (left-handed) neutrino states of the SM, while [MATH]) and [MATH]) are right-handed neutrino fields and additional fermionic gauge singlets, respectively.', '1503.04159-1-26-4': 'The neutrino mass matrix [MATH] then has the form, [EQUATION] where [MATH] are complex matrices.', '1503.04159-1-27-0': 'The Dirac mass matrix [MATH] arises from the Yukawa couplings to the SM Higgs boson, [MATH], [EQUATION] while the matrix [MATH], instead, contains the Majorana mass terms for the sterile fermions [MATH].', '1503.04159-1-27-1': 'By assigning a leptonic charge [MATH] to both [MATH] and [MATH], one makes sure that the off diagonal terms are lepton number conserving, while [MATH] violates the lepton number by two units.', '1503.04159-1-27-2': 'Furthermore, the interesting feature of this seesaw realization is that the entries of [MATH] can be made small in order to accommodate for [MATH]) masses of active neutrinos, with large Yukawa couplings.', '1503.04159-1-27-3': 'This is not in conflict with naturalness since the lepton number is restored in the limit of [MATH].', '1503.04159-1-28-0': 'Concerning the additional sterile states [MATH] and [MATH], since up to now there is no direct evidence for their existence and because they do not contribute to anomalies, their number is unknown.', '1503.04159-1-28-1': 'In ref. [CITATION] it was shown that it is possible to construct several minimal distinct realizations of ISS, each reproducing the correct neutrino mass spectrum and satisfying all phenomenological constraints.', '1503.04159-1-28-2': 'More specifically, it was shown that, depending on the number of additional fields, the neutrino mass spectrum obtained for each ISS realization is characterized by either 2 or 3 mass scales, one corresponding to [MATH] (light neutrino masses), one corresponding to the heavy mass eigenstates [mass scale of the matrix [MATH] of eq. ([REF])], and finally an intermediate scale [MATH], only present if [MATH].', '1503.04159-1-28-3': 'This allows to identify two truly minimal ISS realizations that comply with all experimental bounds, namely the (2,2)-ISS model, which corresponds to the SM extended by two RH neutrinos and two additional sterile states, leading to a three-flavor mixing scheme, and the (2,3)-ISS realization, where the SM is extended by two RH neutrinos and three sterile states leading to a 3+1-mixing scheme.', '1503.04159-1-28-4': 'Interestingly, the lightest sterile neutrino with a mass around eV in the (2,3)-ISS can be used to explain the short baseline (reactor/accelerator) anomaly [CITATION] if its mass lies around eV, or to provide a dark matter candidate if the lightest sterile state were in the keV range [CITATION].', '1503.04159-1-29-0': '## A model with one effective sterile fermion', '1503.04159-1-30-0': 'Since the generic idea of obtaining a significant contribution to our observables applies to any model in which the active neutrinos have sizable mixing with some additional singlet states (sterile fermions), we can use an effective model with [MATH] light active neutrinos plus [MATH] extra sterile neutrino.', '1503.04159-1-31-0': 'The introduction of this extra state implies three new active-sterile mixing angles ([MATH]), two extra Dirac CP violating phases ([MATH]) and one additional Majorana phase ([MATH]).', '1503.04159-1-31-1': 'The lepton mixing matrix is then a product of [MATH] rotations times the Majorana phases, namely, [EQUATION] where the rotation matrices [MATH] can be defined as: [EQUATION]', '1503.04159-1-31-2': 'In the framework of the SM extended by sterile fermion states, which have a non-vanishing mixing with active neutrinos, the Lagrangian describing the leptonic charged currents becomes [EQUATION] where [MATH] denotes the physical neutrino states, and [MATH] are the flavors of charged leptons.', '1503.04159-1-31-3': 'In the case of the SM with three neutrino generations, [MATH] is the PMNS matrix, while in the case of [MATH], the [MATH] submatrix ([MATH]), is not unitary anymore and one can parameterize it as [EQUATION] where [MATH] is a matrix that accounts for the deviation of [MATH] from unitarity [CITATION], due to the presence of extra fermion states.', '1503.04159-1-31-4': 'Many observables are sensitive to the active-sterile mixing and their current experimental values can be used to constrain the [MATH]-matrix [CITATION].', '1503.04159-1-32-0': 'In order to express the deviation from unitarity in terms of a single parameter, we define [EQUATION] which, in the case of the extension of the SM by only one sterile fermion and in terms of the mixing angles defined above, reads [EQUATION]', '1503.04159-1-33-0': '# Results and discussion', '1503.04159-1-34-0': 'In this section we present and discuss our results.', '1503.04159-1-35-0': 'Since the Wilson coefficients of the processes discussed here are proportional to the mass of the decaying particle, it is quite obvious that the most significant enhancement of B([MATH]) will occur for [MATH] and its radial excitations.', '1503.04159-1-35-1': 'For this reason we will present plots of our results for this decay channel.', '1503.04159-1-35-2': 'Plots for other channels are completely similar which is why we do not display them.', '1503.04159-1-35-3': 'Before we further discuss the impact of the active-sterile neutrino mixing on the LFV decay rates, we first specify the constraints on parameters of both of our models.', '1503.04159-1-36-0': 'In Fig. [REF] (left panel), we plot the dependence of [MATH] with respect to the mass of the effective sterile neutrino [MATH].', '1503.04159-1-37-0': 'Gray points in that plot are obtained by varying the mass of the lightest neutrino, [MATH] eV, and by imposing the following constraints: (i) Neutrino data (masses and mixing angles) respect the normal hierarchy, with [MATH] eV, and [MATH] eV [CITATION].', '1503.04159-1-37-1': 'We checked that our final results do not change in any significant manner if the inverse hierarchy is adopted.', '1503.04159-1-37-2': 'Furthermore, we vary the three mixing angles with the fourth neutrino by assuming [MATH], while keeping the other three mixing angles to their best-fit values, namely [MATH], [MATH], [MATH] [CITATION].', '1503.04159-1-37-3': '(ii) The selected points satisfy the upper bound [MATH] [CITATION].', '1503.04159-1-37-4': '(iii) The results for [MATH], [MATH], [MATH] and for [MATH], remain consistent with experimental findings.', '1503.04159-1-37-5': 'We see that for all (heavy) sterile neutrino masses the unitarity breaking parameter is [MATH].', '1503.04159-1-37-6': 'That parameter space is not compatible with the perturbative unitarity requirement, which for [MATH] translates into [CITATION], [EQUATION]', '1503.04159-1-37-7': 'The resulting region, i.e. the one that satisfies constraints (i,ii,iii) and eq. ([REF]), is depicted by blue points (dark region) in Fig. [REF], where we see that the parameter [MATH] is indeed diminishing with the increase of the heavy sterile mass [MATH].', '1503.04159-1-37-8': 'In other words, the decoupling of a very heavy sterile neutrino entails the unitarity of the [MATH] submatrix [MATH].', '1503.04159-1-37-9': 'Decoupling from active neutrinos for very large masses was also explicitly emphasized in ref. [CITATION].', '1503.04159-1-37-10': 'We should mention that, besides the above constraints, we also implemented the constraint coming from [MATH] [CITATION], but it turns out that the present experimental bound does not bring any additional improvement.', '1503.04159-1-38-0': 'By imposing the constraints (i) and eq. ([REF]) on the (2,3)-ISS model, we get a similar region of allowed (blue) points in the right plot of Fig. [REF].', '1503.04159-1-38-1': 'A notable difference with respect to the situation with one effective sterile neutrino is that the region of very small mixing angles is excluded due to relations between the active neutrino masses and the active-sterile neutrino mixing, cf. ref. [CITATION].', '1503.04159-1-38-2': "For very heavy [MATH], on the other hand, the range of allowed [MATH]'s shrinks and eventually vanish with [MATH].", '1503.04159-1-38-3': 'Furthermore, we use the results of ref. [CITATION] which are derived in the Minimal Unitarity Violation scheme in which the heavy sterile neutrino fields are integrated out, and therefore the observables computed in that scheme are functions of the deviation of PMNS matrix from unitarity only [CITATION].', '1503.04159-1-38-4': 'We adapt and apply them to our (2,3)-ISS model and get a region of the bright blue points, as shown in the right panel of Fig. [REF].', '1503.04159-1-38-5': 'To further constrain the parameter space we find it useful to account for the experimental bound on [MATH], as discussed in refs. [CITATION].', '1503.04159-1-38-6': 'This latter constraint appears to be superfluous in most of the parameter space, once the constraints of eq. ([REF]) and ref. [CITATION] are taken into account, except in the range 10 TeV [MATH] 100 TeV, where the bound [MATH] restricts the parameter space relevant to B([MATH]).', '1503.04159-1-39-0': 'We also mention that we attempted implementing the constraints coming from various laboratory experiments, summarized in ref. [CITATION], but since those results only impact the region of relatively small sterile neutrino masses ([MATH] GeV), they are of not relevance for the present study.', '1503.04159-1-40-0': 'After having completed the discussion on several constraints, we present our results for branching fractions B([MATH]) depending on the mass of heavy sterile neutrino(s).', '1503.04159-1-40-1': 'In Fig. [REF] we plot our results for [MATH] and [MATH], for which the enhancement is more pronounced.', '1503.04159-1-40-2': 'Other cases of [MATH] result in similar shapes but the upper bound becomes lower.', '1503.04159-1-40-3': 'In Tab. [REF] we collect our results for three values of the heavy sterile neutrino(s) mass.', '1503.04159-1-41-0': 'To better appreciate the enhancement of the LFV decay rates shown in Fig. [REF], we emphasize that both of them are [MATH] in the absence of heavy sterile neutrinos.', '1503.04159-1-41-1': 'Current experimental bounds in both cases are shown by dashed lines.', '1503.04159-1-41-2': 'Since those bounds are expected to improve in near future, a possibility of seeing the LFV modes discussed in this paper might become realistic.', '1503.04159-1-41-3': 'Conversely, an observation of the LFV modes [MATH], with branching fractions significantly larger than the bounds presented in Tab. [REF] would be a way to disfavor many of the models containing heavy sterile neutrinos as being the unique source of lepton flavor violation.', '1503.04159-1-41-4': 'In obtaining the bounds presented in Tab. [REF] we used masses and decay constants listed in Appendix B.', '1503.04159-1-42-0': 'Finally, we compare in Tab. [REF] our upper bounds for the modes for which we could find predictions in the literature.', '1503.04159-1-43-0': '# Conclusions', '1503.04159-1-44-0': 'In this paper we discussed the enhancement of the LFV decays of flavorless vector bosons, [MATH], with [MATH], induced by mixing between the active and sterile neutrinos.', '1503.04159-1-44-1': 'The enhancement grows with mass of the heavy sterile neutrino(s), as can be seen from the mass dependence of Wilson coefficients that we explicitly calculated.', '1503.04159-1-44-2': 'We find that the most significant diagram that gives rise to the LFV decay amplitudes is the one coming from the [MATH]-vertex, which suggests a steady growth of the decay rate with the mass of sterile neutrino(s).', '1503.04159-1-44-3': 'In the physical amplitude, however, the region of very large mass of the sterile neutrino(s) is suppressed as the decoupling takes place, i.e. mixing between the active and sterile neutrinos rapidly falls.', '1503.04159-1-45-0': 'We illustrated the enhancement of [MATH] in two scenarios: a model with one effective sterile neutrino that mimics the effect of a generic extensions of the SM including heavy sterile fermions, and in a minimal realization of the inverse seesaw scenario compatible with current observations.', '1503.04159-1-45-1': 'Our results for upper bounds on [MATH] [[MATH]] are still considerably smaller than the current experimental bounds (when available), but that situation might change in the future as more experimental research will be conducted at Belle II, BESIII, LHC, and hopefully at FCC-ee (TLEP).', '1503.04159-1-45-2': 'If one of the decays studied here is observed and turns out to have a branching fraction larger than the upper bounds reported here, then sources of LFV other than those coming from mixing with heavy sterile neutrinos must be accounted for.'}

{'1503.04159-2-0-0': 'We address the impact of sterile fermions on the lepton flavor violating decays of quarkonia as well as of the [MATH] boson.', '1503.04159-2-0-1': 'We compute the relevant Wilson coefficients and show that the [MATH], where [MATH], [MATH] can be significantly enhanced in the case of large sterile fermion masses and a non-negligible active-sterile mixing.', '1503.04159-2-0-2': 'We illustrate that feature in a specific minimal realization of the inverse seesaw mechanism, known as [MATH]-ISS, and in an effective model in which the presence of nonstandard sterile fermions is parameterized by means of one heavy sterile (Majorana) neutrino.', '1503.04159-2-1-0': 'PACS: 14.60.', '1503.04159-2-1-1': 'St, 14.60.', '1503.04159-2-1-2': 'Pq, 13.20.', '1503.04159-2-1-3': 'Gd, 13.38.', '1503.04159-2-1-4': 'Dg.', '1503.04159-2-2-0': '# Introduction', '1503.04159-2-3-0': 'So far no signal of new physics has been observed but its search is important in order to understand how to enlarge the Standard Model (SM) to solve both the hierarchy and the flavor problems.', '1503.04159-2-3-1': 'One of the most significant observations requiring us to go beyond the Standard Model is the assessment that neutrinos are massive and that they mix [CITATION].', '1503.04159-2-3-2': 'Possible SM extensions aiming at incorporating massive neutrinos give rise to interesting collider signatures and open the door to new phenomena such as lepton flavor violating (LFV) decays.', '1503.04159-2-4-0': 'Currently, the search for manifestations of LFV constitutes a goal of several experimental facilities dedicated to rare lepton decays, such as [MATH] and [MATH], and to the neutrinoless [MATH] conversion in muonic atoms.', '1503.04159-2-4-1': 'One of the most stringent bounds from these searches is the one derived by the MEG Collaboration, [MATH] [CITATION], which is expected to be improved to a planned sensitivity of [MATH] [CITATION].', '1503.04159-2-4-2': 'Moreover, the bound [MATH], set by the SINDRUM experiment [CITATION], is expected to be improved by the Mu3e experiment where a sensitivity [MATH] is planned [CITATION].', '1503.04159-2-4-3': 'Limits on the [MATH] radiative decays [CITATION] and the three-body decays of [MATH] [CITATION] appear to be less stringent right now, but are likely to be improved at Belle II [CITATION], where the search for LFV decays of the [MATH]-meson will be made too [CITATION].', '1503.04159-2-4-4': 'The most promising developments regarding LFV are those related to the [MATH] conversion in nuclei.', '1503.04159-2-4-5': 'The present bound for the [MATH] conversion rate is [MATH] [CITATION], and the planned sensitivity is [MATH] [CITATION].', '1503.04159-2-4-6': 'Similar is the case for gold and aluminum [CITATION].', '1503.04159-2-5-0': 'Searches for LFV are also conducted in high-energy experiments and a first bound on the Higgs boson LFV decay [MATH] has been reported by the CMS Collaboration [CITATION].', '1503.04159-2-5-1': 'The LHCb Collaboration, instead, reported the bound [MATH] [CITATION], which is likely to be improved in the near future [CITATION].', '1503.04159-2-5-2': 'Notice also that they already improved the bounds on [MATH] by an order of magnitude [CITATION].', '1503.04159-2-6-0': 'In this work we will focus on the indirect probes of new physics through the LFV processes of neutral vector bosons, namely [MATH], with [MATH], and [MATH], where [MATH] stands for [MATH] and its radial excitations, and similarly for [MATH].', '1503.04159-2-6-1': 'Most of the research in this direction reported so far is related to the [MATH] decay modes.', '1503.04159-2-6-2': 'More specifically, the experimental bounds, obtained at LEP are found to be [MATH] [CITATION], [MATH] [CITATION], and [MATH] [CITATION].', '1503.04159-2-6-3': 'One of these bounds has been improved at LHC, namely [MATH] [CITATION].', '1503.04159-2-6-4': 'On the theory side, the [MATH] decays have been analyzed in the extensions of the SM involving additional massive and sterile neutrinos that could mix with the standard (active) ones and thus give rise to the LFV decay rates [CITATION].', '1503.04159-2-6-5': 'A similar approach has been also adopted in Ref. [CITATION], in the perspective of a Tera-[MATH] factory FCC-ee [CITATION] for which a targeted sensitivity is expected to be [MATH] [CITATION].', '1503.04159-2-7-0': 'Lepton flavor conserving decays of quarkonia have been measured to a high accuracy which can actually be used to fix the hadronic parameters (decay constants).', '1503.04159-2-7-1': 'Otherwise, one can use the results of numerical simulations of QCD on the lattice, which are nowadays accurate as well [CITATION].', '1503.04159-2-7-2': 'The experimentally established bounds for the simplest LFV decays of quarkonia are [CITATION]: [EQUATION] where each mode is to be understood as [MATH].', '1503.04159-2-8-0': 'Despite the appreciable experimental work on the latter observables, only a few theoretical studies have been carried out so far.', '1503.04159-2-8-1': 'The authors of Ref. [CITATION] applied a vector meson dominance approximation to [MATH] and expressed the width of the latter process, [MATH].', '1503.04159-2-8-2': 'Since the values of [MATH] are very well known experimentally [CITATION], the experimental bound on [MATH] is then used to obtain an upper bound on the phenomenological coupling [MATH], which is then converted to an upper bound on [MATH].', '1503.04159-2-8-3': 'A similar approach has been used in Ref. [CITATION] where instead of [MATH], the authors considered the [MATH] conversion in nuclei ([MATH]), which they described in terms of a product of couplings [MATH] and [MATH].', '1503.04159-2-8-4': 'The latter could be extracted from the experimentally measured [MATH], and with that knowledge the experimental upper bound on [MATH] results in an upper bound on [MATH].', '1503.04159-2-8-5': 'A more dynamical approach in modeling the [MATH] processes has been made in a supersymmetric extension of the SM with type I seesaw [CITATION].', '1503.04159-2-9-0': 'Sterile fermions were proposed in various neutrino mass generation mechanisms, but the interest in their properties was further motivated by the reactor/accelerator anomalies [CITATION], a possibility to offer a warm dark matter candidate [CITATION], and by indications from the large scale structure formation [CITATION].', '1503.04159-2-10-0': 'Incorporating neutrino oscillations (masses and mixing [CITATION]) into the SM implies that the charged current is modified to [EQUATION] [MATH] being the leptonic mixing matrix, [MATH] the flavor of a charged lepton, and [MATH] denotes a physical neutrino state.', '1503.04159-2-10-1': 'If one assumes that only three massive neutrinos are present, the matrix [MATH] corresponds to the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.', '1503.04159-2-10-2': 'In that situation the GIM mechanism makes the decay rates B([MATH]) completely negligible, [MATH].', '1503.04159-2-10-3': 'That feature, however, can be drastically changed in the presence of a non-negligible mixing with heavy sterile fermions.', '1503.04159-2-10-4': 'In what follows we will consider such situations, derive analytical expressions for B([MATH]), and discuss a specific realization of the inverse seesaw mechanism, known as (2,3)-ISS [CITATION].', '1503.04159-2-10-5': 'We will also discuss a simplified model in which the effect of the heavy sterile neutrinos is described by one effective sterile neutrino state with non-negligible mixing with active neutrinos.', '1503.04159-2-10-6': 'Despite several differences, our approach is similar to the one discussed in Ref. [CITATION], where the SM has been extended by new, heavy, Dirac neutrinos, singlets under [MATH], and applied to a number of low energy decay processes.', '1503.04159-2-10-7': 'Our sterile neutrinos are Majorana and we apply the approach to the leptonic decays of quarkonia for the first time.', '1503.04159-2-11-0': 'The remainder of this paper is organized as follows: In Sec. [REF] we formulate the problem in terms of a low energy effective theory of a larger theory which contains heavy sterile neutrinos, we derive expression for B([MATH]) and compute the Wilson coefficients.', '1503.04159-2-11-1': 'In Sec. [REF] we briefly describe the specific models with sterile neutrinos which are used in this paper to produce our results presented in Sec. [REF].', '1503.04159-2-11-2': 'We finally conclude in Sec. [REF].', '1503.04159-2-12-0': '# LFV decay of Quarkonia - Effective Theory', '1503.04159-2-13-0': 'In this section we formulate a low energy effective theory of the LFV decays of quarkonia of type [MATH], and express the decay amplitude in terms of the quarkonium decay constants and the corresponding Wilson coefficients.', '1503.04159-2-13-1': 'The latter are then computed in the extensions of the SM which include the heavy sterile neutrinos.', '1503.04159-2-13-2': 'We also derive the expression relevant to [MATH].', '1503.04159-2-14-0': '## Effective Hamiltonian', '1503.04159-2-15-0': 'Keeping in mind the fact that we are extending the SM by adding sterile fermions, without touching the gauge sector of the theory, the decays of vector quarkonia, [MATH], can only occur through the photon and the [MATH]-boson exchange at tree level.', '1503.04159-2-15-1': 'In the lepton flavor conserving processes the [MATH]-exchange terms are very small with respect to those arising from the electromagnetic interaction and are usually neglected.', '1503.04159-2-15-2': 'The generic effective Hamiltonian can be written as [EQUATION] where [MATH] is the electric charge of the quark [MATH], [MATH] is the mass of quarkonium [MATH] which is dominated by the valence quark configuration [MATH], [MATH] are the Wilson coefficients, [MATH] is the momentum of one of the outgoing leptons, and [MATH].', '1503.04159-2-15-3': 'Contributions to the scalar (left and right) terms are suppressed by [MATH], where [MATH] are the charged lepton masses.', '1503.04159-2-15-4': 'In this section we will keep such terms so that our expressions can be useful to approaches in which the scalar bosons are taken in consideration.', '1503.04159-2-15-5': 'For our phenomenological discussion, however, it is worth emphasizing that [MATH].', '1503.04159-2-16-0': 'Without entering the details of calculation it is easy to verify that the only relevant diagrams are those shown in Fig. [REF], and therefore the structure of the Wilson coefficients [MATH] reads, [EQUATION] where [MATH] are the contributions arising from the photon and the [MATH]-boson exchange, while [MATH] comes from the box diagram that involves the Cabibbo-Kobayashi-Maskawa coupling [MATH].', '1503.04159-2-16-1': 'In the above expressions [MATH].', '1503.04159-2-16-2': 'The blob in the diagram shown in Fig. [REF] stands for the lepton loop diagrams that may contain one or two neutrino states and which, in the extensions of the SM involving a heavy neutrino sector, will give rise to the LFV decay due to the effect of mixing which is parametrized by the matrix [MATH] [see Eq. ([REF])].', '1503.04159-2-16-3': 'Separate contributions coming from different diagrams can be further reduced by factoring out the neutrino mixing matrix elements, namely [EQUATION] where we see that the term involving two neutrino eigenstates appears only in the [MATH] coefficient because it is related to the vertex [MATH].', '1503.04159-2-16-4': 'It is worth emphasizing that the tensor structure in Eq. ([REF]) can be easily obtained from the coefficients [MATH] by applying the Gordon identity.', '1503.04159-2-16-5': 'Such contributions are [MATH] suppressed, and thus completely negligible, which is why we do not give explicit expressions for these coefficients.', '1503.04159-2-17-0': 'Using the effective Hamiltonian ([REF]) and parameterizing the hadronic matrix as [EQUATION] where [MATH] is the decay constant of a quarkonium [MATH] with momentum [MATH] and in a polarization state [MATH], we can write the decay rate as, [EQUATION] with [EQUATION] and [EQUATION] which gives [EQUATION]', '1503.04159-2-17-1': 'As we mentioned above, we consider in our framework [MATH], and therefore we can write [EQUATION] where [MATH] is given in Eq. ([REF]).', '1503.04159-2-17-2': 'In this last expression we also used [MATH].', '1503.04159-2-18-0': 'Besides quarkonia we will also revisit the issue of adding extra species of sterile neutrinos to the decay of [MATH].', '1503.04159-2-18-1': 'In that case the effective Hamiltonian can be written as [EQUATION] where the Wilson coefficients are now denoted by [MATH] and take the form [EQUATION]', '1503.04159-2-18-2': 'The decay rate in the similar limit, [MATH], reads [EQUATION]', '1503.04159-2-19-0': '## Wilson coefficients', '1503.04159-2-20-0': 'Concerning the computation of the Wilson coefficients we stress again that our results are obtained in a theory in which the Standard Model is extended to include extra species of sterile fermions, without changing the gauge sector.', '1503.04159-2-20-1': 'The origin of the leptonic mixing matrix [MATH] is model dependent and in order to be able to do a phenomenological analysis, we will have to adopt a specific model which will be discussed in the next section.', '1503.04159-2-21-0': 'The blob in the diagram shown in Fig. [REF] stands for a series of diagrams such as those displayed in Fig. [REF].', '1503.04159-2-21-1': 'All of them, including the box diagram in Fig. [REF], have been computed in the Feynman gauge and the results are collected in Appendix A.', '1503.04159-2-22-0': 'Here we focus on the most important contributions in the case of large masses of sterile (Majorana) neutrinos.', '1503.04159-2-22-1': 'Contributions to the Wilson coefficients coming from vertex diagrams can be divided into two pieces: those involving only one neutrino in the loop, [MATH], where [MATH], and those with two neutrinos in the loop, [MATH].', '1503.04159-2-22-2': 'In the limit of large values of [MATH], we find the following behavior [EQUATION]', '1503.04159-2-22-3': 'To illustrate the relative contribution of the different diagrams we fix the values of the coefficients [MATH], and plot [MATH] and [MATH] for the case of [MATH], cf. Fig. [REF].', '1503.04159-2-22-4': 'We see that only for very large masses the diagrams with two neutrinos in the loop become more important than those with one neutrino state.', '1503.04159-2-22-5': 'We should stress that each contribution to [MATH], i.e. [MATH] and [MATH], scales as [MATH] for large values of [MATH], except for [MATH] which goes to a constant in the same limit.', '1503.04159-2-22-6': 'That can also be seen in Fig. [REF] where in the left panel we show the dependence of the total [MATH] on [MATH] and in the right panel we show [MATH] and its dependence on the mass of the initial decaying meson, [MATH], [MATH], and [MATH].', '1503.04159-2-22-7': 'The contribution of sterile neutrinos to the LFV decay of [MATH] is larger than the one to lighter mesons, since the Wilson coefficients are also proportional to the mass of the initial particle.', '1503.04159-2-23-0': 'Before closing this section we should reiterate that our Wilson coefficients have been computed in the Feynman gauge.', '1503.04159-2-23-1': 'Since all divergencies cancel out, our results are finite and gauge invariant, as was already observed in Refs. [CITATION].', '1503.04159-2-24-0': '# SM in the presence of sterile fermions', '1503.04159-2-25-0': 'With the expressions derived above, we now have to specify a model for lepton mixing (couplings) [MATH] in the presence of heavy sterile neutrinos propagating in the loops.', '1503.04159-2-25-1': 'We opt for a minimal realization of the inverse seesaw mechanism for the generation of neutrino masses, which is nowadays rather well constrained by the available experimental data.', '1503.04159-2-25-2': 'Furthermore, we will use a parametric model containing one effective sterile neutrino, which essentially mimics the behavior at low energy scales of mechanisms involving heavy sterile fermions.', '1503.04159-2-26-0': '## The (2,3)-inverse seesaw realization', '1503.04159-2-27-0': 'Among many possible realizations of accounting for massive neutrinos, the inverse seesaw mechanism (ISS) [CITATION] offers the possibility of accommodating the smallness of the active neutrino masses for a comparatively low seesaw scale, but still with natural [MATH] Yukawa couplings, which renders this scenario phenomenologically appealing.', '1503.04159-2-27-1': 'Indeed, depending on their masses and mixing with active neutrinos, the new states can be produced in collider and/or low energy experiments, and their contribution to physical processes can be sizable.', '1503.04159-2-27-2': 'ISS, embedded in the SM, results in a mass term for neutrinos of the form [EQUATION] where [MATH] is the charge conjugation matrix and [MATH].', '1503.04159-2-27-3': 'Here [MATH], [MATH] denotes the active (left-handed) neutrino states of the SM, while [MATH]) and [MATH]) are right-handed neutrino fields and additional fermionic gauge singlets, respectively.', '1503.04159-2-27-4': 'The neutrino mass matrix [MATH] then has the form [EQUATION] where [MATH] are complex matrices.', '1503.04159-2-28-0': 'The Dirac mass matrix [MATH] arises from the Yukawa couplings to the SM Higgs boson, [MATH], [EQUATION] while the matrix [MATH], instead, contains the Majorana mass terms for the sterile fermions [MATH].', '1503.04159-2-28-1': 'By assigning a leptonic charge [MATH] to both [MATH] and [MATH], one makes sure that the off diagonal terms are lepton number conserving, while [MATH] violates the lepton number by two units.', '1503.04159-2-28-2': 'Furthermore, the interesting feature of this seesaw realization is that the entries of [MATH] can be made small in order to accommodate for the [MATH]) masses of active neutrinos, with large Yukawa couplings.', '1503.04159-2-28-3': 'This is not in conflict with naturalness since the lepton number is restored in the limit of [MATH].', '1503.04159-2-29-0': 'Concerning the additional sterile states [MATH] and [MATH], since up to now there is no direct evidence for their existence and because they do not contribute to anomalies, their number is unknown.', '1503.04159-2-29-1': 'In Ref. [CITATION] it was shown that it is possible to construct several minimal distinct realizations of ISS, each reproducing the correct neutrino mass spectrum and satisfying all phenomenological constraints.', '1503.04159-2-29-2': 'More specifically, it was shown that, depending on the number of additional fields, the neutrino mass spectrum obtained for each ISS realization is characterized by either two or three mass scales, one corresponding to [MATH] (light neutrino masses), one corresponding to the heavy mass eigenstates [the mass scale of the matrix [MATH] of Eq. ([REF])], and finally an intermediate scale [MATH], only present if [MATH].', '1503.04159-2-29-3': 'This allows us to identify two truly minimal ISS realizations that comply with all experimental bounds, namely the (2,2)-ISS model, which corresponds to the SM extended by two right-handed (RH) neutrinos and two additional sterile states, leading to a three-flavor mixing scheme, and the (2,3)-ISS realization, where the SM is extended by two RH neutrinos and three sterile states leading to a 3+1-mixing scheme.', '1503.04159-2-29-4': 'Interestingly, the lightest sterile neutrino with a mass around eV in the (2,3)-ISS can be used to explain the short baseline (reactor/accelerator) anomaly [CITATION] if its mass lies around eV, or to provide a dark matter candidate if the lightest sterile state were in the keV range [CITATION].', '1503.04159-2-30-0': '## A model with one effective sterile fermion', '1503.04159-2-31-0': 'Since the generic idea of obtaining a significant contribution to our observables applies to any model in which the active neutrinos have sizable mixing with some additional singlet states (sterile fermions), we can use an effective model with three light active neutrinos plus one extra sterile neutrino.', '1503.04159-2-32-0': 'The introduction of this extra state implies three new active-sterile mixing angles ([MATH]), two extra Dirac [MATH] violating phases ([MATH]) and one additional Majorana phase ([MATH]).', '1503.04159-2-32-1': 'The lepton mixing matrix is then a product of six rotations times the Majorana phases, namely [EQUATION] where the rotation matrices [MATH] can be defined as: [EQUATION]', '1503.04159-2-32-2': 'In the framework of the SM extended by sterile fermion states, which have a nonvanishing mixing with active neutrinos, the Lagrangian describing the leptonic charged currents becomes [EQUATION] where [MATH] denotes the physical neutrino states, and [MATH] are the flavors of the charged leptons.', '1503.04159-2-32-3': 'In the case of the SM with three neutrino generations, [MATH] is the PMNS matrix, while in the case of [MATH], the [MATH] submatrix ([MATH]) is not unitary anymore and one can parameterize it as [EQUATION] where [MATH] is a matrix that accounts for the deviation of [MATH] from unitarity [CITATION], due to the presence of extra fermion states.', '1503.04159-2-32-4': 'Many observables are sensitive to the active-sterile mixing and their current experimental values can be used to constrain the [MATH] matrix [CITATION].', '1503.04159-2-33-0': 'In order to express the deviation from unitarity in terms of a single parameter, we define [EQUATION] which, in the case of the extension of the SM by only one sterile fermion and in terms of the mixing angles defined above, reads [EQUATION]', '1503.04159-2-34-0': '# Results and discussion', '1503.04159-2-35-0': 'In this section we present and discuss our results.', '1503.04159-2-36-0': 'Since the Wilson coefficients of the processes discussed here are proportional to the mass of the decaying particle, it is quite obvious that the most significant enhancement of B([MATH]) will occur for [MATH] and its radial excitations.', '1503.04159-2-36-1': 'For this reason we will present plots of our results for this decay channel.', '1503.04159-2-36-2': 'Plots for other channels are completely similar which is why we do not display them.', '1503.04159-2-36-3': 'Before we discuss the impact of the active-sterile neutrino mixing on the LFV decay rates further, we first specify the constraints on parameters of both of our models.', '1503.04159-2-37-0': 'In Fig. [REF] (left panel), we plot the dependence of [MATH] with respect to the mass of the effective sterile neutrino [MATH].', '1503.04159-2-38-0': 'Gray points in that plot are obtained by varying the mass of the lightest neutrino, [MATH] eV, and by imposing the following constraints: (i) Neutrino data (masses and mixing angles) respect the normal hierarchy, with [MATH] eV, and [MATH] eV [CITATION].', '1503.04159-2-38-1': 'We checked to see that our final results do not change in any significant manner if the inverse hierarchy is adopted.', '1503.04159-2-38-2': 'Furthermore, we vary the three mixing angles with the fourth neutrino by assuming [MATH], while keeping the other three mixing angles to their best-fit values, namely [MATH], [MATH], [MATH] [CITATION].', '1503.04159-2-38-3': '(ii) The selected points satisfy the upper bound [MATH] [CITATION].', '1503.04159-2-38-4': '(iii) The results for [MATH], [MATH], [MATH], and [MATH], remain consistent with experimental findings.', '1503.04159-2-38-5': 'We see that for all (heavy) sterile neutrino masses the unitarity breaking parameter is [MATH].', '1503.04159-2-38-6': 'That parameter space is not compatible with the perturbative unitarity requirement, which for [MATH] translates into [CITATION], [EQUATION]', '1503.04159-2-38-7': 'The resulting region, i.e. the one that satisfies constraints (i), (ii), (iii) and Eq. ([REF]), is depicted by blue points (the dark region) in Fig. [REF], where we see that the parameter [MATH] is indeed diminishing with the increase of the heavy sterile mass [MATH].', '1503.04159-2-38-8': 'In other words, the decoupling of a very heavy sterile neutrino entails the unitarity of the [MATH] submatrix [MATH].', '1503.04159-2-38-9': 'Decoupling from active neutrinos for very large masses was also explicitly emphasized in Ref. [CITATION].', '1503.04159-2-38-10': 'We should mention that, besides the above constraints, we also implemented the constraint coming from [MATH] [CITATION], but it turns out that the present experimental bound does not bring any additional improvement.', '1503.04159-2-39-0': 'By imposing the constraints (i) and Eq. ([REF]) on the (2,3)-ISS model, we get a similar region of allowed (blue) points in the right panel of Fig. [REF].', '1503.04159-2-39-1': 'A notable difference with respect to the situation with one effective sterile neutrino is that the region of very small mixing angles is excluded due to relations between the active neutrino masses and the active-sterile neutrino mixing, cf. Ref. [CITATION].', '1503.04159-2-39-2': "For very heavy [MATH], on the other hand, the range of allowed [MATH]'s shrinks and eventually vanishes with [MATH].", '1503.04159-2-39-3': 'Furthermore, we use the results of Ref. [CITATION] which are derived in the minimal unitarity violation scheme in which the heavy sterile neutrino fields are integrated out, and therefore the observables computed in that scheme are functions of the deviation of PMNS matrix from unitarity only [CITATION].', '1503.04159-2-39-4': 'We adapt and apply them to our (2,3)-ISS model and get a region of the bright-blue points, as shown in the right panel of Fig. [REF].', '1503.04159-2-39-5': 'To further constrain the parameter space we find it useful to account for the experimental bound on [MATH], as is discussed in Refs. [CITATION].', '1503.04159-2-39-6': 'This latter constraint appears to be superfluous in most of the parameter space, once the constraints of Eq. ([REF]) and Ref. [CITATION] are taken into account, except in the range [MATH], where the bound [MATH] restricts the parameter space relevant to B([MATH]).', '1503.04159-2-40-0': 'We also mention that we attempted implementing the constraints coming from various laboratory experiments, summarized in Ref. [CITATION], but since those results only impact the region of relatively small sterile neutrino masses ([MATH] GeV), they are of no relevance to the present study.', '1503.04159-2-41-0': 'After having completed the discussion on several constraints, we present our results for branching fractions B([MATH]) depending on the mass of heavy sterile neutrino(s).', '1503.04159-2-41-1': 'In Fig. [REF] we plot our results for [MATH] and [MATH], for which the enhancement is more pronounced.', '1503.04159-2-41-2': 'Other cases of [MATH] result in similar shapes but the upper bound becomes lower.', '1503.04159-2-41-3': 'In Table [REF] we collect our results for three values of the heavy sterile neutrino(s) mass.', '1503.04159-2-42-0': 'To better appreciate the enhancement of the LFV decay rates shown in Fig. [REF], we emphasize that both of them are [MATH] in the absence of heavy sterile neutrinos.', '1503.04159-2-42-1': 'Current experimental bounds in both cases are shown by dashed lines.', '1503.04159-2-42-2': 'Since those bounds are expected to improve in the near future, a possibility of seeing the LFV modes discussed in this paper might become realistic.', '1503.04159-2-42-3': 'Conversely, an observation of the LFV modes [MATH], with branching fractions significantly larger than the bounds presented in Table [REF] would be a way to disfavor many of the models containing heavy sterile neutrinos as being the unique source of lepton flavor violation.', '1503.04159-2-42-4': 'In obtaining the bounds presented in Table [REF] we used masses and decay constants listed in Appendix B.', '1503.04159-2-42-5': 'In presenting our results (the upper bounds) for lepton flavor violating modes, we used the parameters from Ref. [CITATION] which were determined at 90 C.L. For that reason, we treated all other input data to 2 [MATH] as well.', '1503.04159-2-42-6': 'Therefore, our final results in Table [REF] are also obtained at 2 [MATH] level.', '1503.04159-2-43-0': 'Finally, we compare in Table [REF] our upper bounds for the modes for which we could find predictions in the literature.', '1503.04159-2-44-0': '# Conclusions', '1503.04159-2-45-0': 'In this paper we discussed the enhancement of the LFV decays of flavorless vector bosons, [MATH], with [MATH], induced by a mixing between the active and sterile neutrinos.', '1503.04159-2-45-1': 'The enhancement grows with the mass of the heavy sterile neutrino(s), as can be seen from the mass dependence of the Wilson coefficients that we explicitly calculated.', '1503.04159-2-45-2': 'We find that the most significant diagram that gives rise to the LFV decay amplitudes is the one coming from the [MATH] vertex, which suggests a steady growth of the decay rate with the mass of the sterile neutrino(s).', '1503.04159-2-45-3': 'In the physical amplitude, however, the region of very large mass of the sterile neutrino(s) is suppressed as the decoupling takes place, i.e. mixing between the active and sterile neutrinos rapidly falls.', '1503.04159-2-46-0': 'We illustrated the enhancement of [MATH] in two scenarios: a model with one effective sterile neutrino that mimics the effect of a generic extensions of the SM including heavy sterile fermions, and in a minimal realization of the inverse seesaw scenario compatible with current observations.', '1503.04159-2-46-1': 'Our results for upper bounds on [MATH] [[MATH]] are still considerably smaller than the current experimental bounds (when available), but that situation might change in the future as more experimental research will be conducted at Belle II, BESIII, LHC, and hopefully at FCC-ee (TLEP).', '1503.04159-2-46-2': 'If one of the decays studied here is observed and turns out to have a branching fraction larger than the upper bounds reported here, then sources of LFV other than those coming from mixing with heavy sterile neutrinos must be accounted for.'}

[['1503.04159-1-17-2', '1503.04159-2-17-2'], ['1503.04159-1-11-0', '1503.04159-2-11-0'], ['1503.04159-1-11-1', '1503.04159-2-11-1'], ['1503.04159-1-11-2', '1503.04159-2-11-2'], ['1503.04159-1-18-0', '1503.04159-2-18-0'], ['1503.04159-1-4-0', '1503.04159-2-4-0'], ['1503.04159-1-4-4', '1503.04159-2-4-4'], ['1503.04159-1-4-6', '1503.04159-2-4-6'], ['1503.04159-1-13-0', '1503.04159-2-13-0'], ['1503.04159-1-13-1', '1503.04159-2-13-1'], ['1503.04159-1-13-2', '1503.04159-2-13-2'], ['1503.04159-1-37-0', '1503.04159-2-38-0'], ['1503.04159-1-37-2', '1503.04159-2-38-2'], ['1503.04159-1-37-3', '1503.04159-2-38-3'], ['1503.04159-1-37-5', '1503.04159-2-38-5'], ['1503.04159-1-37-6', '1503.04159-2-38-6'], ['1503.04159-1-37-8', '1503.04159-2-38-8'], ['1503.04159-1-37-10', '1503.04159-2-38-10'], ['1503.04159-1-40-0', '1503.04159-2-41-0'], ['1503.04159-1-40-1', '1503.04159-2-41-1'], ['1503.04159-1-40-2', '1503.04159-2-41-2'], ['1503.04159-1-27-0', '1503.04159-2-28-0'], ['1503.04159-1-27-1', '1503.04159-2-28-1'], ['1503.04159-1-27-3', '1503.04159-2-28-3'], ['1503.04159-1-10-0', '1503.04159-2-10-0'], ['1503.04159-1-10-2', '1503.04159-2-10-2'], ['1503.04159-1-10-3', '1503.04159-2-10-3'], ['1503.04159-1-10-4', '1503.04159-2-10-4'], ['1503.04159-1-10-5', '1503.04159-2-10-5'], ['1503.04159-1-10-7', '1503.04159-2-10-7'], ['1503.04159-1-3-2', '1503.04159-2-3-2'], ['1503.04159-1-5-2', '1503.04159-2-5-2'], ['1503.04159-1-32-0', '1503.04159-2-33-0'], ['1503.04159-1-21-0', '1503.04159-2-21-0'], ['1503.04159-1-20-0', '1503.04159-2-20-0'], ['1503.04159-1-22-0', '1503.04159-2-22-0'], ['1503.04159-1-22-1', '1503.04159-2-22-1'], ['1503.04159-1-22-2', '1503.04159-2-22-2'], ['1503.04159-1-22-3', '1503.04159-2-22-3'], ['1503.04159-1-22-4', '1503.04159-2-22-4'], ['1503.04159-1-22-5', '1503.04159-2-22-5'], ['1503.04159-1-45-0', '1503.04159-2-46-0'], ['1503.04159-1-45-1', '1503.04159-2-46-1'], ['1503.04159-1-45-2', '1503.04159-2-46-2'], ['1503.04159-1-41-0', '1503.04159-2-42-0'], ['1503.04159-1-41-1', '1503.04159-2-42-1'], ['1503.04159-1-0-0', '1503.04159-2-0-0'], ['1503.04159-1-0-1', '1503.04159-2-0-1'], ['1503.04159-1-15-0', '1503.04159-2-15-0'], ['1503.04159-1-15-1', '1503.04159-2-15-1'], ['1503.04159-1-15-2', '1503.04159-2-15-2'], ['1503.04159-1-15-3', '1503.04159-2-15-3'], ['1503.04159-1-15-4', '1503.04159-2-15-4'], ['1503.04159-1-15-5', '1503.04159-2-15-5'], ['1503.04159-1-8-0', '1503.04159-2-8-0'], ['1503.04159-1-8-2', '1503.04159-2-8-2'], ['1503.04159-1-8-4', '1503.04159-2-8-4'], ['1503.04159-1-7-2', '1503.04159-2-7-2'], ['1503.04159-1-36-0', '1503.04159-2-37-0'], ['1503.04159-1-6-1', '1503.04159-2-6-1'], ['1503.04159-1-6-2', '1503.04159-2-6-2'], ['1503.04159-1-6-3', '1503.04159-2-6-3'], ['1503.04159-1-6-4', '1503.04159-2-6-4'], ['1503.04159-1-44-3', '1503.04159-2-45-3'], ['1503.04159-1-9-0', '1503.04159-2-9-0'], ['1503.04159-1-35-0', '1503.04159-2-36-0'], ['1503.04159-1-35-1', '1503.04159-2-36-1'], ['1503.04159-1-35-2', '1503.04159-2-36-2'], ['1503.04159-1-28-0', '1503.04159-2-29-0'], ['1503.04159-1-28-4', '1503.04159-2-29-4'], ['1503.04159-1-24-0', '1503.04159-2-25-0'], ['1503.04159-1-24-2', '1503.04159-2-25-2'], ['1503.04159-1-16-0', '1503.04159-2-16-0'], ['1503.04159-1-16-1', '1503.04159-2-16-1'], ['1503.04159-1-16-3', '1503.04159-2-16-3'], ['1503.04159-1-39-0', '1503.04159-2-40-0'], ['1503.04159-1-17-0', '1503.04159-2-17-0'], ['1503.04159-1-17-1', '1503.04159-2-17-1'], ['1503.04159-1-18-1', '1503.04159-2-18-1'], ['1503.04159-1-18-2', '1503.04159-2-18-2'], ['1503.04159-1-4-1', '1503.04159-2-4-1'], ['1503.04159-1-4-2', '1503.04159-2-4-2'], ['1503.04159-1-4-3', '1503.04159-2-4-3'], ['1503.04159-1-4-5', '1503.04159-2-4-5'], ['1503.04159-1-37-1', '1503.04159-2-38-1'], ['1503.04159-1-37-4', '1503.04159-2-38-4'], ['1503.04159-1-37-7', '1503.04159-2-38-7'], ['1503.04159-1-37-9', '1503.04159-2-38-9'], ['1503.04159-1-40-3', '1503.04159-2-41-3'], ['1503.04159-1-27-2', '1503.04159-2-28-2'], ['1503.04159-1-10-1', '1503.04159-2-10-1'], ['1503.04159-1-10-6', '1503.04159-2-10-6'], ['1503.04159-1-38-0', '1503.04159-2-39-0'], ['1503.04159-1-38-1', '1503.04159-2-39-1'], ['1503.04159-1-38-2', '1503.04159-2-39-2'], ['1503.04159-1-38-3', '1503.04159-2-39-3'], ['1503.04159-1-38-4', '1503.04159-2-39-4'], ['1503.04159-1-38-5', '1503.04159-2-39-5'], ['1503.04159-1-38-6', '1503.04159-2-39-6'], ['1503.04159-1-31-0', '1503.04159-2-32-0'], ['1503.04159-1-31-1', '1503.04159-2-32-1'], ['1503.04159-1-31-2', '1503.04159-2-32-2'], ['1503.04159-1-31-3', '1503.04159-2-32-3'], ['1503.04159-1-31-4', '1503.04159-2-32-4'], ['1503.04159-1-3-0', '1503.04159-2-3-0'], ['1503.04159-1-3-1', '1503.04159-2-3-1'], ['1503.04159-1-5-0', '1503.04159-2-5-0'], ['1503.04159-1-5-1', '1503.04159-2-5-1'], ['1503.04159-1-42-0', '1503.04159-2-43-0'], ['1503.04159-1-21-1', '1503.04159-2-21-1'], ['1503.04159-1-20-1', '1503.04159-2-20-1'], ['1503.04159-1-22-6', '1503.04159-2-22-6'], ['1503.04159-1-22-7', '1503.04159-2-22-7'], ['1503.04159-1-26-0', '1503.04159-2-27-0'], ['1503.04159-1-26-1', '1503.04159-2-27-1'], ['1503.04159-1-26-2', '1503.04159-2-27-2'], ['1503.04159-1-26-3', '1503.04159-2-27-3'], ['1503.04159-1-26-4', '1503.04159-2-27-4'], ['1503.04159-1-41-2', '1503.04159-2-42-2'], ['1503.04159-1-0-2', '1503.04159-2-0-2'], ['1503.04159-1-8-1', '1503.04159-2-8-1'], ['1503.04159-1-8-3', '1503.04159-2-8-3'], ['1503.04159-1-8-5', '1503.04159-2-8-5'], ['1503.04159-1-7-0', '1503.04159-2-7-0'], ['1503.04159-1-7-1', '1503.04159-2-7-1'], ['1503.04159-1-6-0', '1503.04159-2-6-0'], ['1503.04159-1-6-5', '1503.04159-2-6-5'], ['1503.04159-1-44-0', '1503.04159-2-45-0'], ['1503.04159-1-44-1', '1503.04159-2-45-1'], ['1503.04159-1-44-2', '1503.04159-2-45-2'], ['1503.04159-1-35-3', '1503.04159-2-36-3'], ['1503.04159-1-28-1', '1503.04159-2-29-1'], ['1503.04159-1-28-2', '1503.04159-2-29-2'], ['1503.04159-1-28-3', '1503.04159-2-29-3'], ['1503.04159-1-24-1', '1503.04159-2-25-1'], ['1503.04159-1-16-2', '1503.04159-2-16-2'], ['1503.04159-1-30-0', '1503.04159-2-31-0'], ['1503.04159-1-41-3', '1503.04159-2-42-3'], ['1503.04159-1-41-4', '1503.04159-2-42-4']]

[['1503.04159-1-17-2', '1503.04159-2-17-2'], ['1503.04159-1-11-0', '1503.04159-2-11-0'], ['1503.04159-1-11-1', '1503.04159-2-11-1'], ['1503.04159-1-11-2', '1503.04159-2-11-2'], ['1503.04159-1-18-0', '1503.04159-2-18-0'], ['1503.04159-1-4-0', '1503.04159-2-4-0'], ['1503.04159-1-4-4', '1503.04159-2-4-4'], ['1503.04159-1-4-6', '1503.04159-2-4-6'], ['1503.04159-1-13-0', '1503.04159-2-13-0'], ['1503.04159-1-13-1', '1503.04159-2-13-1'], ['1503.04159-1-13-2', '1503.04159-2-13-2'], ['1503.04159-1-37-0', '1503.04159-2-38-0'], ['1503.04159-1-37-2', '1503.04159-2-38-2'], ['1503.04159-1-37-3', '1503.04159-2-38-3'], ['1503.04159-1-37-5', '1503.04159-2-38-5'], ['1503.04159-1-37-6', '1503.04159-2-38-6'], ['1503.04159-1-37-8', '1503.04159-2-38-8'], ['1503.04159-1-37-10', '1503.04159-2-38-10'], ['1503.04159-1-40-0', '1503.04159-2-41-0'], ['1503.04159-1-40-1', '1503.04159-2-41-1'], ['1503.04159-1-40-2', '1503.04159-2-41-2'], ['1503.04159-1-27-0', '1503.04159-2-28-0'], ['1503.04159-1-27-1', '1503.04159-2-28-1'], ['1503.04159-1-27-3', '1503.04159-2-28-3'], ['1503.04159-1-10-0', '1503.04159-2-10-0'], ['1503.04159-1-10-2', '1503.04159-2-10-2'], ['1503.04159-1-10-3', '1503.04159-2-10-3'], ['1503.04159-1-10-4', '1503.04159-2-10-4'], ['1503.04159-1-10-5', '1503.04159-2-10-5'], ['1503.04159-1-10-7', '1503.04159-2-10-7'], ['1503.04159-1-3-2', '1503.04159-2-3-2'], ['1503.04159-1-5-2', '1503.04159-2-5-2'], ['1503.04159-1-32-0', '1503.04159-2-33-0'], ['1503.04159-1-21-0', '1503.04159-2-21-0'], ['1503.04159-1-20-0', '1503.04159-2-20-0'], ['1503.04159-1-22-0', '1503.04159-2-22-0'], ['1503.04159-1-22-1', '1503.04159-2-22-1'], ['1503.04159-1-22-2', '1503.04159-2-22-2'], ['1503.04159-1-22-3', '1503.04159-2-22-3'], ['1503.04159-1-22-4', '1503.04159-2-22-4'], ['1503.04159-1-22-5', '1503.04159-2-22-5'], ['1503.04159-1-45-0', '1503.04159-2-46-0'], ['1503.04159-1-45-1', '1503.04159-2-46-1'], ['1503.04159-1-45-2', '1503.04159-2-46-2'], ['1503.04159-1-41-0', '1503.04159-2-42-0'], ['1503.04159-1-41-1', '1503.04159-2-42-1'], ['1503.04159-1-0-0', '1503.04159-2-0-0'], ['1503.04159-1-0-1', '1503.04159-2-0-1'], ['1503.04159-1-15-0', '1503.04159-2-15-0'], ['1503.04159-1-15-1', '1503.04159-2-15-1'], ['1503.04159-1-15-2', '1503.04159-2-15-2'], ['1503.04159-1-15-3', '1503.04159-2-15-3'], ['1503.04159-1-15-4', '1503.04159-2-15-4'], ['1503.04159-1-15-5', '1503.04159-2-15-5'], ['1503.04159-1-8-0', '1503.04159-2-8-0'], ['1503.04159-1-8-2', '1503.04159-2-8-2'], ['1503.04159-1-8-4', '1503.04159-2-8-4'], ['1503.04159-1-7-2', '1503.04159-2-7-2'], ['1503.04159-1-36-0', '1503.04159-2-37-0'], ['1503.04159-1-6-1', '1503.04159-2-6-1'], ['1503.04159-1-6-2', '1503.04159-2-6-2'], ['1503.04159-1-6-3', '1503.04159-2-6-3'], ['1503.04159-1-6-4', '1503.04159-2-6-4'], ['1503.04159-1-44-3', '1503.04159-2-45-3'], ['1503.04159-1-9-0', '1503.04159-2-9-0'], ['1503.04159-1-35-0', '1503.04159-2-36-0'], ['1503.04159-1-35-1', '1503.04159-2-36-1'], ['1503.04159-1-35-2', '1503.04159-2-36-2'], ['1503.04159-1-28-0', '1503.04159-2-29-0'], ['1503.04159-1-28-4', '1503.04159-2-29-4'], ['1503.04159-1-24-0', '1503.04159-2-25-0'], ['1503.04159-1-24-2', '1503.04159-2-25-2'], ['1503.04159-1-16-0', '1503.04159-2-16-0'], ['1503.04159-1-16-1', '1503.04159-2-16-1'], ['1503.04159-1-16-3', '1503.04159-2-16-3']]

[['1503.04159-1-39-0', '1503.04159-2-40-0'], ['1503.04159-1-17-0', '1503.04159-2-17-0'], ['1503.04159-1-17-1', '1503.04159-2-17-1'], ['1503.04159-1-18-1', '1503.04159-2-18-1'], ['1503.04159-1-18-2', '1503.04159-2-18-2'], ['1503.04159-1-4-1', '1503.04159-2-4-1'], ['1503.04159-1-4-2', '1503.04159-2-4-2'], ['1503.04159-1-4-3', '1503.04159-2-4-3'], ['1503.04159-1-4-5', '1503.04159-2-4-5'], ['1503.04159-1-37-1', '1503.04159-2-38-1'], ['1503.04159-1-37-4', '1503.04159-2-38-4'], ['1503.04159-1-37-7', '1503.04159-2-38-7'], ['1503.04159-1-37-9', '1503.04159-2-38-9'], ['1503.04159-1-40-3', '1503.04159-2-41-3'], ['1503.04159-1-27-2', '1503.04159-2-28-2'], ['1503.04159-1-10-1', '1503.04159-2-10-1'], ['1503.04159-1-10-6', '1503.04159-2-10-6'], ['1503.04159-1-38-0', '1503.04159-2-39-0'], ['1503.04159-1-38-1', '1503.04159-2-39-1'], ['1503.04159-1-38-2', '1503.04159-2-39-2'], ['1503.04159-1-38-3', '1503.04159-2-39-3'], ['1503.04159-1-38-4', '1503.04159-2-39-4'], ['1503.04159-1-38-5', '1503.04159-2-39-5'], ['1503.04159-1-38-6', '1503.04159-2-39-6'], ['1503.04159-1-31-0', '1503.04159-2-32-0'], ['1503.04159-1-31-1', '1503.04159-2-32-1'], ['1503.04159-1-31-2', '1503.04159-2-32-2'], ['1503.04159-1-31-3', '1503.04159-2-32-3'], ['1503.04159-1-31-4', '1503.04159-2-32-4'], ['1503.04159-1-3-0', '1503.04159-2-3-0'], ['1503.04159-1-3-1', '1503.04159-2-3-1'], ['1503.04159-1-5-0', '1503.04159-2-5-0'], ['1503.04159-1-5-1', '1503.04159-2-5-1'], ['1503.04159-1-42-0', '1503.04159-2-43-0'], ['1503.04159-1-21-1', '1503.04159-2-21-1'], ['1503.04159-1-20-1', '1503.04159-2-20-1'], ['1503.04159-1-22-6', '1503.04159-2-22-6'], ['1503.04159-1-22-7', '1503.04159-2-22-7'], ['1503.04159-1-26-0', '1503.04159-2-27-0'], ['1503.04159-1-26-1', '1503.04159-2-27-1'], ['1503.04159-1-26-2', '1503.04159-2-27-2'], ['1503.04159-1-26-3', '1503.04159-2-27-3'], ['1503.04159-1-26-4', '1503.04159-2-27-4'], ['1503.04159-1-41-2', '1503.04159-2-42-2'], ['1503.04159-1-0-2', '1503.04159-2-0-2'], ['1503.04159-1-8-1', '1503.04159-2-8-1'], ['1503.04159-1-8-3', '1503.04159-2-8-3'], ['1503.04159-1-8-5', '1503.04159-2-8-5'], ['1503.04159-1-7-0', '1503.04159-2-7-0'], ['1503.04159-1-7-1', '1503.04159-2-7-1'], ['1503.04159-1-6-0', '1503.04159-2-6-0'], ['1503.04159-1-6-5', '1503.04159-2-6-5'], ['1503.04159-1-44-0', '1503.04159-2-45-0'], ['1503.04159-1-44-1', '1503.04159-2-45-1'], ['1503.04159-1-44-2', '1503.04159-2-45-2'], ['1503.04159-1-35-3', '1503.04159-2-36-3'], ['1503.04159-1-28-1', '1503.04159-2-29-1'], ['1503.04159-1-28-2', '1503.04159-2-29-2'], ['1503.04159-1-28-3', '1503.04159-2-29-3'], ['1503.04159-1-24-1', '1503.04159-2-25-1'], ['1503.04159-1-16-2', '1503.04159-2-16-2'], ['1503.04159-1-30-0', '1503.04159-2-31-0'], ['1503.04159-1-41-3', '1503.04159-2-42-3'], ['1503.04159-1-41-4', '1503.04159-2-42-4']]

[]

[]

[]

['1503.04159-1-1-0', '1503.04159-1-1-1', '1503.04159-1-1-2', '1503.04159-1-1-3', '1503.04159-1-1-4', '1503.04159-1-34-0', '1503.04159-2-1-0', '1503.04159-2-1-1', '1503.04159-2-1-2', '1503.04159-2-1-3', '1503.04159-2-1-4', '1503.04159-2-35-0']

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

https://arxiv.org/abs/1503.04159

hep-th-0307178

{'hep-th-0307178-1-0-0': 'We solve the topological Poisson Sigma model for a Poisson-Lie group [MATH] and its dual [MATH].', 'hep-th-0307178-1-0-1': 'We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target.', 'hep-th-0307178-1-0-2': 'The resolution of both models in the open geometry reveals that there exists a map from the reduced phase of each model ([MATH] and [MATH]) to the main symplectic leaf of the Heisenberg double ([MATH]) such that the symplectic forms on [MATH], [MATH] are obtained as the pull-back by those maps of the symplectic structure on [MATH].', 'hep-th-0307178-1-0-3': 'This uncovers a duality between [MATH] and [MATH] under the exchange of bulk degrees of freedom of one model with boundary degrees of freedom of the other one.', 'hep-th-0307178-1-0-4': 'We finally solve the Poisson Sigma model for the Poisson structure on [MATH] given by a pair of [MATH]-matrices that generalizes the Poisson-Lie case.', 'hep-th-0307178-1-0-5': 'The Hamiltonian analysis of the theory requires the introduction of a deformation of the Heisenberg double.', 'hep-th-0307178-1-1-0': '# Introduction', 'hep-th-0307178-1-2-0': 'Since their appearance in [CITATION],[CITATION] the Poisson Sigma models have played a central role in the study of two dimensional gauge theories.', 'hep-th-0307178-1-2-1': 'They are related to pure gravity, to Wess-Zumino-Witten models and Yang-Mills in two dimensions.', 'hep-th-0307178-1-2-2': 'Generalizations of the model that include supergravity have been recently proposed in [CITATION],[CITATION].', 'hep-th-0307178-1-3-0': 'The Poisson Sigma models are topological and besides the two dimensional (space-time) manifold [MATH] all we require is a Poisson structure on the target [MATH] (supplemented by boundary conditions if [MATH]).', 'hep-th-0307178-1-3-1': 'The theory seems then suitable for the study of the underlying Poisson structure and in fact in [CITATION] it was shown that the semiclassical expansion of these models (with [MATH] the unit disk and insertions at the boundary) reproduces the [MATH]-product introduced by Kontsevich in [CITATION] that quantizes by deformation the Poisson bracket.', 'hep-th-0307178-1-4-0': 'In this paper we shall be interested in Poisson Sigma models in which [MATH] is a Lie group.', 'hep-th-0307178-1-4-1': 'In this case it is natural to ask for a consistency relation between its Poisson and group structures and this leads to the concept of Poisson-Lie group ([CITATION],[CITATION]).', 'hep-th-0307178-1-4-2': 'Poisson-Lie groups are the classical counterparts of quantum groups and they lead to the so called hidden symmetries of integrable systems that induce dressing transformations in the phase space.', 'hep-th-0307178-1-4-3': 'We shall give later a more detailed description of these objects, but for the moment we would like to stress that Poisson-Lie groups come in dual pairs under the exchange of the roles of Poisson and Lie brackets.', 'hep-th-0307178-1-4-4': 'Examples of Poisson-Lie Sigma models have been studied in [CITATION],[CITATION] in connection to [MATH] Wess-Zumino-Witten theories.', 'hep-th-0307178-1-4-5': 'Here we pursue a systematic study of the matter and besides we shall also consider some generalizations of the Poisson structure for which the product with the corresponding Poisson-Lie group acts as a Poisson action.', 'hep-th-0307178-1-5-0': 'One of the simplest examples of Poisson-Lie Sigma models is the linear one.', 'hep-th-0307178-1-5-1': 'Here [MATH] is a vector space (abelian group) and its Poisson structure is linear.', 'hep-th-0307178-1-5-2': 'This model is related to [MATH] and Yang-Mills and can be considered as dual of the trivial Poisson-Lie Sigma model in an (in general) non-abelian group with vanishing Poisson bracket.', 'hep-th-0307178-1-5-3': 'Our models can be regarded as the simplest generalizations of the linear ones with which they share some properties that will be stressed in the sequel.', 'hep-th-0307178-1-6-0': 'Another aspect that will deserve our attention is that of duality, i.e. we shall try to relate the two dual models, which is not obvious at the Lagrangian level.', 'hep-th-0307178-1-6-1': 'However, in the Hamiltonian approach, in the open geometry, the duality becomes evident: it consists of the exchange of bulk and boundary degrees of freedom.', 'hep-th-0307178-1-6-2': 'We would like to relate this fact with the non-abelian T-duality for WZW Sigma models ([CITATION]).', 'hep-th-0307178-1-7-0': 'We begin with a brief review of the essential results about Poisson Sigma models and Poisson-Lie groups in sections [REF] and [REF].', 'hep-th-0307178-1-7-1': 'Sections [REF] and [REF] are devoted to the study of the Poisson-Lie Sigma model of a simple Poisson-Lie group and its dual one, respectively.', 'hep-th-0307178-1-7-2': 'By using the thechniques of the two previous sections we are able to solve more general Poisson Sigma models on Lie groups and we introduce a generalization of the Heisenberg double in section [REF].', 'hep-th-0307178-1-7-3': 'The conclusions and open questions are presented in section [REF].', 'hep-th-0307178-1-8-0': '# Poisson Sigma models [MATH] PSmodels The Poisson Sigma model is a two-dimensional topological Sigma model with the target manifold [MATH] equipped with a Poisson structure [MATH], i.e. a bivector field on [MATH].', 'hep-th-0307178-1-8-1': 'The Poisson bracket of two functions on [MATH] is given by the contraction of [MATH]: [MATH].', 'hep-th-0307178-1-9-0': 'The fields of the model are [MATH] and a 1-form [MATH] on [MATH] with values in the pullback by [MATH] of the cotangent bundle of [MATH].', 'hep-th-0307178-1-9-1': 'The action functional has the form [EQUATION] where [MATH] denotes the pairing between the tangent and the cotangent vectors to [MATH].', 'hep-th-0307178-1-10-0': 'If [MATH] are local coordinates on [MATH], [MATH] local coordinates on [MATH], [MATH] the components of the Poisson structure in these coordinates and [MATH], [MATH] the action reads [EQUATION]', 'hep-th-0307178-1-10-1': 'It is straightforward to work out the equations of motion: [EQUATION]', 'hep-th-0307178-1-10-2': 'One can show ([CITATION]) that for solutions of ([REF],a) the image of [MATH] lies within one of the symplectic leaves of the foliation of [MATH].', 'hep-th-0307178-1-11-0': 'Under the infinitesimal transformation [EQUATION] where [MATH] is a section of [MATH], the action ([REF]) transforms by a boundary term [EQUATION]', 'hep-th-0307178-1-11-1': 'One of the special properties of these models is that the commutator of two consecutive gauge tansformation of ([REF]) is not of the same form i.e. [EQUATION] where for a vector field [MATH], [MATH].', 'hep-th-0307178-1-11-2': 'Later, we shall make extensive use of this commutator in the context of Poisson-Lie groups.', 'hep-th-0307178-1-11-3': 'Note that the term in parenthesis in ([REF],b) is the equation of motion ([REF],a), and then it vanishes on-shell.', 'hep-th-0307178-1-11-4': 'However, as remarked in [CITATION], due to the [MATH] dependence of [MATH] even on-shell one can not talk properly about a Lie algebra structure in the space of parameters, unless we enlarge its definition to account for field dependent gauge transformations.', 'hep-th-0307178-1-11-5': 'The situation will become simpler in the linear case that we describe below.', 'hep-th-0307178-1-12-0': 'One of the simplest examples of Poisson Sigma models is the linear one.', 'hep-th-0307178-1-12-1': 'We take a finite dimensional vector space [MATH] as the target space and define a linear Poisson bracket on it, i.e. for any [MATH], their Poisson bracket [MATH].', 'hep-th-0307178-1-12-2': 'The linear Poisson bracket on [MATH] defines then a Lie bracket [MATH] on [MATH].', 'hep-th-0307178-1-12-3': 'Assuming that [MATH] is semisimple we can use the Cartan-Killing form [MATH] to identify [MATH] and [MATH].', 'hep-th-0307178-1-12-4': 'The fields are then [MATH] and [MATH] and the action reads: [EQUATION]', 'hep-th-0307178-1-12-5': 'The equations of motion are: [EQUATION]', 'hep-th-0307178-1-12-6': 'For [MATH] the gauge transformation [EQUATION] induces the change of the action ([REF]) by a boundary term [EQUATION]', 'hep-th-0307178-1-12-7': 'Note that in this case the gauge transformations close even off-shell [EQUATION] and induce the Lie algebra structure of [MATH] in the space of parameters.', 'hep-th-0307178-1-13-0': '# Poisson-Lie groups [MATH] PoissonLiegroups In this section we shall review some basic features of the theory of Poisson-Lie groups and fix our notation.', 'hep-th-0307178-1-13-1': 'See [CITATION] for details.', 'hep-th-0307178-1-13-2': 'A Poisson-Lie group is a Lie group equipped with a Poisson structure which makes the product [MATH] a Poisson map if [MATH] is considered with the product Poisson structure.', 'hep-th-0307178-1-13-3': 'Linearization of the Poisson structure at the unit [MATH] of [MATH] provides a Lie algebra structure on [MATH] by the formula [EQUATION]', 'hep-th-0307178-1-13-4': 'The Poisson-Lie structure of [MATH] yields the compatibility condition [EQUATION] which allows to define a Lie bracket in [MATH] by the formula[EQUATION]', 'hep-th-0307178-1-13-5': 'If [MATH] is connected and simply connected, ([REF]) is enough to integrate [MATH] to a Poisson structure on [MATH] that makes it Poisson-Lie and the Poisson structure is unique.', 'hep-th-0307178-1-13-6': 'The symmetry between [MATH] and [MATH] in ([REF]), implies that one has also a Poisson-Lie group [MATH] with Lie algebra [MATH] and a Poisson structure whose linearization at [MATH] gives the bracket [MATH].', 'hep-th-0307178-1-13-7': '[MATH] is the dual Poisson-Lie group of [MATH].', 'hep-th-0307178-1-14-0': 'Let us take [MATH] a complex, simple, connected, simply connected Lie group and give the above construction explicitly.', 'hep-th-0307178-1-14-1': 'The (essentially unique) nondegenerate, invariant, bilinear form tr on [MATH] establishes an isomorphism between [MATH] and [MATH].', 'hep-th-0307178-1-14-2': 'The Poisson structure [MATH] contracted with the right-invariant forms [MATH], will be denoted [EQUATION]', 'hep-th-0307178-1-14-3': 'For a general Poisson-Lie structure on [MATH] ([CITATION]), [EQUATION] where [MATH] is an antisymmetric linear operator such that [EQUATION]', 'hep-th-0307178-1-14-4': 'Using the operator [MATH] it is possible to define a new Lie bracket in g, [EQUATION] [MATH] is isomorphic to [MATH] via tr.', 'hep-th-0307178-1-15-0': 'If [MATH] we can rescale the bilinear form in [MATH] and [MATH] so that we leave the Poisson bracket unchanged but [MATH].', 'hep-th-0307178-1-15-1': 'In the following we shall consider this case ([MATH]) that corresponds to the factorizable Lie bialgebras of ref. [CITATION].', 'hep-th-0307178-1-15-2': 'Take [MATH], [MATH] and [MATH].', 'hep-th-0307178-1-15-3': 'Then, [EQUATION] and the embedding [MATH] defines an homomorphism from [MATH] to [MATH].', 'hep-th-0307178-1-15-4': 'We give an invariant, nondegenerate bilinear form on [MATH] by [EQUATION]', 'hep-th-0307178-1-15-5': 'This allows to identify [MATH] with the subalgebra [MATH] and [MATH] as the subgroup [MATH] in [MATH] corresponding to the Lie subalgebra [MATH].', 'hep-th-0307178-1-15-6': 'We denote by [MATH] the elements of [MATH].', 'hep-th-0307178-1-15-7': 'In a natural way, [MATH] and [MATH].', 'hep-th-0307178-1-15-8': 'Notice that [MATH] is a discrete subgroup of [MATH].', 'hep-th-0307178-1-16-0': 'The induced Poisson-Lie structure on [MATH] contracted with the right-invariant forms on [MATH], [MATH] for [MATH], takes the form [EQUATION]', 'hep-th-0307178-1-16-1': 'The [MATH]-matrices are used not only to give explicit realizations of the dual group and to construct Poisson-Lie structures but to construct more general Poisson structures on Lie groups.', 'hep-th-0307178-1-16-2': 'Following Semenov-Tian-Shansky ([CITATION]), if [MATH] and [MATH] are antisymmetric and satisfy ([REF]) with the same value for [MATH], we can define the Poisson structure (contracted with the right-invariant forms), [EQUATION] [MATH] equipped with ([REF]) is denoted by [MATH].', 'hep-th-0307178-1-16-3': 'One can verify that [MATH] is a Poisson map.', 'hep-th-0307178-1-16-4': 'When [MATH] and [MATH] we get ([REF]) and the Poisson structure is, indeed, Poisson-Lie.', 'hep-th-0307178-1-17-0': 'Given a [MATH]-matrix in [MATH] it is possible to define a [MATH]-matrix in [MATH] (which we shall denote by [MATH]) in a natural way: [MATH] where [MATH] and [MATH] are the projectors on [MATH] and [MATH] respectively, parallel to the complementary subalgebra.', 'hep-th-0307178-1-17-1': 'Hence, [MATH].', 'hep-th-0307178-1-18-0': 'We shall need the description of the Heisenberg double (which is not a Poisson-Lie group; in fact, with the notation introduced before, it corresponds to [MATH]).', 'hep-th-0307178-1-18-1': 'Its main symplectic leaf is [MATH] (that contains a neighborhood of the unit of [MATH]).', 'hep-th-0307178-1-18-2': 'The symplectic structure obtained by inverting the Poisson structure on it reads [EQUATION] where [MATH].', 'hep-th-0307178-1-18-3': 'Note that although these decompositions are not unique, the ambiguity (the product by an element of [MATH]) does not affect the values of [MATH] which is well defined in [MATH].', 'hep-th-0307178-1-19-0': '# Poisson-Lie Sigma model on [MATH] G Recall the action ([REF]) and take the Poisson-Lie group of the previous section as the target.', 'hep-th-0307178-1-19-1': 'The action for [MATH] and [MATH] reads: [EQUATION] which is what we call the Poisson-Lie Sigma model for [MATH].', 'hep-th-0307178-1-20-0': 'We proceed now to a detailed study of this model, from which we shall learn techniques applicable to the resolution of more general Poisson Sigma models on Lie groups.', 'hep-th-0307178-1-21-0': 'The equations of motion are: [EQUATION]', 'hep-th-0307178-1-21-1': 'From the previous two equations one can derive also a zero curvature equation for [MATH] itself [EQUATION] which can be deduced directly from the stationary condition for the action under variations of the fields that keep [MATH] invariant.', 'hep-th-0307178-1-22-0': 'The gauge symmetry of the action, for [MATH], is given in its infinitesimal form by[EQUATION]', 'hep-th-0307178-1-22-1': 'Under this transformation the action changes by a boundary term, namely [MATH].', 'hep-th-0307178-1-22-2': 'Note that ([REF],a) corresponds to the right dressing vector fields of ref. [CITATION] translated to the origin by right multiplication in [MATH].', 'hep-th-0307178-1-22-3': 'Its integration (local as in general the vector field is not complete) gives rise to the dressing transformation of [MATH].', 'hep-th-0307178-1-22-4': 'Also in ([REF],b) the third term vanishes on-shell; it corresponds to a trivial transformation (see refs. [CITATION],[CITATION]), proportional to the equations of motion.', 'hep-th-0307178-1-22-5': 'If we forget about that term (or alternatively, working on-shell) [EQUATION].', 'hep-th-0307178-1-22-6': 'Note that unlike the general case this relation defines a Lie algebra structure (that of the gauge group corresponding to [MATH]) in the space of gauge transformations.', 'hep-th-0307178-1-22-7': 'This is also what happens in the linear case discussed in Sect. 2.', 'hep-th-0307178-1-22-8': 'However, in this case the relation holds uniquely on-shell while in the linear case the Lie algebra structure persists off-shell.', 'hep-th-0307178-1-22-9': 'Another difference is that the vector fields (on-shell) induced by the gauge transformation are not complete in the general case.', 'hep-th-0307178-1-22-10': 'This is a second obstruction to properly talk about a gauge group, so we shall restrict ourselves to infinitesimal gauge transformations.', 'hep-th-0307178-1-23-0': 'Now we proceed to integrate the equations of motion.', 'hep-th-0307178-1-23-1': 'We shall consider [MATH] in eq. ([REF]) although as remarked before the case [MATH] can be treated similarly.', 'hep-th-0307178-1-23-2': 'As the equation of motion for [MATH] is independent of [MATH], like in the linear model, we first solve the eqs. ([REF],b) and ([REF]).', 'hep-th-0307178-1-23-3': 'This is in contrast with the general case of ref. [CITATION].', 'hep-th-0307178-1-23-4': 'Locally [EQUATION]', 'hep-th-0307178-1-23-5': 'In order to fix the ambiguity in the new fields we choose a point [MATH] and take [MATH].', 'hep-th-0307178-1-24-0': 'We need to work a bit more to get explicit solutions for [MATH].', 'hep-th-0307178-1-24-1': 'The equation of motion ([REF],a) can be equivalently written as [EQUATION] and recalling that [MATH], [EQUATION] [MATH] projects on the [MATH] component of the elements of g and yields [EQUATION] which is equivalent to [EQUATION]', 'hep-th-0307178-1-24-2': 'The same procedure can be carried out by using [MATH], [EQUATION]', 'hep-th-0307178-1-24-3': 'The solutions for the equations of motion can then be expressed by the single relation in [MATH]: [EQUATION] which in particular implies that [MATH], and [MATH] takes values in the connected component of orbits of [MATH] by dressing transformations.', 'hep-th-0307178-1-24-4': 'These orbits are the symplectic leaves of the Poisson-Lie group [MATH].', 'hep-th-0307178-1-25-0': 'Now we proceed to the Hamiltonian analysis of the model.', 'hep-th-0307178-1-25-1': 'To that end we fix the open geometry for [MATH] and take free boundary conditions for [MATH] and [MATH] vanishing on vectors tangent to [MATH].', 'hep-th-0307178-1-25-2': 'The boundary conditions enforce that [MATH] are constant along the connected components of the boundary, [MATH], [MATH], [MATH], [MATH].', 'hep-th-0307178-1-25-3': 'The presymplectic form is [EQUATION] where [MATH] and [MATH] varies in the space of solutions of the equations of motion and [MATH] is any curve joining the two components of the boundary.', 'hep-th-0307178-1-25-4': 'The result is, of course, independent of the particular choice of [MATH].', 'hep-th-0307178-1-26-0': 'If we parametrize the solutions in terms of [MATH] with [MATH] we obtain [EQUATION] where [MATH] acts on the space-time variables in [MATH].', 'hep-th-0307178-1-26-1': 'Then [MATH] depends only on the values at the boundaries, namely [EQUATION]', 'hep-th-0307178-1-26-2': 'Or if we take [MATH], i.e. [MATH] [EQUATION]', 'hep-th-0307178-1-26-3': 'From the analysis of the degeneracy of [MATH] we conclude that the vector fields induced by infinitesimal gauge transformations that are the identity at the boundary form its kernel and we reduce the space of solutions by this transformations.', 'hep-th-0307178-1-26-4': 'The reduced phase space [MATH] is then the set of pairs [MATH] with [MATH] a homotopy class of maps from [MATH] to [MATH] which are the identity at [MATH] and have fixed value at [MATH] and such that [MATH].', 'hep-th-0307178-1-26-5': 'The symplectic form on [MATH] can be viewed as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-1-27-0': 'The set of solutions may be endowed with the structure of a symplectic groupoid ([CITATION]).', 'hep-th-0307178-1-27-1': 'The product of two such solutions [MATH] and [MATH] is defined if [MATH] and is given by [MATH] with [EQUATION]', 'hep-th-0307178-1-27-2': 'The results obtained in this section have their dual counterpart if we consider the dual group [MATH], which is accomplished in the next section.', 'hep-th-0307178-1-28-0': '# Poisson-Lie Sigma model on [MATH] Gdual In order to unravel the role of the duality between Poisson-Lie groups in the context of Poisson-Lie Sigma models we proceed now to the study of the theory for [MATH].', 'hep-th-0307178-1-28-1': 'Using the Poisson structure given in ([REF]), the action of the model reads, [EQUATION]', 'hep-th-0307178-1-28-2': 'Note that, similarly to the previous case for the group [MATH], the relation between [MATH] and [MATH] is nothing but the coadjoint action of [MATH] for the group [MATH].', 'hep-th-0307178-1-28-3': 'Like in the former case it is posible to deduce, from the previous equations, a zero curvature condition for [MATH] itself [EQUATION]', 'hep-th-0307178-1-28-4': 'The gauge transformations, for [MATH], read: [EQUATION] where [MATH].', 'hep-th-0307178-1-28-5': 'The change in the action is [MATH].', 'hep-th-0307178-1-28-6': 'As before, if we forget about the third term on the right hand side of ([REF],b), that vanishes on-shell, the transformations close.', 'hep-th-0307178-1-28-7': 'Namely [MATH] which corresponds to the gauge group [MATH].', 'hep-th-0307178-1-29-0': 'The solutions of the equations of motion can be obtained along the same lines as before.', 'hep-th-0307178-1-29-1': 'From ([REF]) [EQUATION] in any contractible, open set.', 'hep-th-0307178-1-29-2': 'Without loss of generality we fix again a point [MATH] such that [MATH].', 'hep-th-0307178-1-29-3': 'Now we write together the equations for [MATH] using the solution for [MATH].', 'hep-th-0307178-1-29-4': '[EQUATION] or equivalently [EQUATION] where [MATH].', 'hep-th-0307178-1-29-5': 'Hence [EQUATION] and once more, we can obtain the solutions from an equation in [MATH] [EQUATION] which means that [MATH] is the dressing-transformed of [MATH] by [MATH].', 'hep-th-0307178-1-29-6': 'Orbits of the dressing transformation are the symplectic leaves of [MATH].', 'hep-th-0307178-1-29-7': 'Note, at this point, the complete symmetry with the previous situation under the exchange of the roles of [MATH] and [MATH].', 'hep-th-0307178-1-30-0': 'We consider again the open geometry [MATH] with boundary conditions for [MATH] vanishing on vectors tangent to the boundary and [MATH] free.', 'hep-th-0307178-1-30-1': 'The boundary conditions impose that [MATH] and [MATH] are constant in every connected component of [MATH] i.e. [MATH], [MATH], [MATH], [MATH].', 'hep-th-0307178-1-31-0': 'The presymplectic form of the model [MATH] can be written [EQUATION] where again [MATH] acts on the [MATH] variables.', 'hep-th-0307178-1-31-1': 'Therefore [EQUATION] and taking [MATH], or [MATH], [EQUATION]', 'hep-th-0307178-1-31-2': 'Points of the phase space reduced by gauge symmetries [MATH], can be described by pairs [MATH] where by [MATH] we denote a homotopy class of maps [MATH] with fixed boundary values [MATH], [MATH] and such that [MATH].', 'hep-th-0307178-1-31-3': 'The symplectic form on [MATH] can be described as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-1-32-0': 'Note the duality between [MATH] and [MATH].', 'hep-th-0307178-1-32-1': 'The symplectic forms of the two models coincide upon the exchange of [MATH] with [MATH] and [MATH] with [MATH].', 'hep-th-0307178-1-32-2': 'Variables [MATH] in the [MATH] model or [MATH] in the [MATH] model are associated to boundary values of their corresponding fields (this can be more explicitly seen working in a manifold [MATH] whose boundary has more than two connected components) while [MATH] and [MATH] are associated to the bulk.', 'hep-th-0307178-1-32-3': 'In this sense one can say that the duality previously described between the Poisson-Lie Sigma models for [MATH] and [MATH] can be stated as a bulk-boundary duality.', 'hep-th-0307178-1-33-0': '# More general Poisson Sigma models on [MATH] generalization In this section we solve the model with Poisson structure ([REF]) following the lines of Section 4.', 'hep-th-0307178-1-33-1': 'As remarked in Section 3 this structure does not make [MATH] a Poisson Lie group, but indeed the left and right product by the corresponding Poisson-Lie groups are Poisson actions.', 'hep-th-0307178-1-33-2': 'In the resolution of the model we shall introduce a generalization of the dressing transformation and of the Heissenberg double and we shall be able to identify the symplectic leaves for the Poissson structure defined on [MATH].', 'hep-th-0307178-1-34-0': 'The action for the model is, [EQUATION] where as stressed in Section 3 [MATH] and [MATH] are two solutions of the modified Yang-Baxter equation ([REF]) with the same value for [MATH].', 'hep-th-0307178-1-34-1': 'The equations of motion are [EQUATION]', 'hep-th-0307178-1-34-2': 'From the previous equations, or performing in the action variations of the fields that keep [MATH] unchanged, we obtain [EQUATION]', 'hep-th-0307178-1-34-3': 'The gauge symmetry in its infinitesimal form is [EQUATION]', 'hep-th-0307178-1-34-4': 'The transformation for [MATH] corresponds to the right dressing vector field that comes, like in the Poisson-Lie case of Section 4, from the contraction of the Poisson structure with the right-invariant forms.', 'hep-th-0307178-1-34-5': 'On-shell, [MATH].', 'hep-th-0307178-1-34-6': 'Hence, the symmetry group is the one corresponding to the matrix [MATH], i.e. [MATH].', 'hep-th-0307178-1-34-7': 'The reason for the preferred role of [MATH] against [MATH] in the gauge symmetry is simply the choice of right-invariant one forms to express the Poisson structure.', 'hep-th-0307178-1-34-8': 'Had we chosen left-invariant forms (i.e. changing the variable [MATH] by [MATH] in the action) the symmetry algebra would have been that of [MATH].', 'hep-th-0307178-1-35-0': 'As before we shall take [MATH].', 'hep-th-0307178-1-35-1': 'Then we can consider [MATH] or [MATH] as different subgroups of the same double group [MATH].', 'hep-th-0307178-1-35-2': 'We shall denote by [MATH] (resp. [MATH]) the elements of [MATH] (resp. [MATH]).', 'hep-th-0307178-1-36-0': 'Using the methods of Section 4, we can easily solve the model.', 'hep-th-0307178-1-36-1': 'As before, we first write locally the solutions for [MATH] and [MATH], [EQUATION] with [MATH]', 'hep-th-0307178-1-37-0': 'The equation of motion for [MATH] eq. ([REF],a) can be transformed into [EQUATION] and inserting the solutions for [MATH], [MATH] [EQUATION]', 'hep-th-0307178-1-37-1': 'Or equivalently [EQUATION] with [MATH].', 'hep-th-0307178-1-38-0': 'We can write the general solution as an equation in [MATH] for [MATH], [EQUATION]', 'hep-th-0307178-1-38-1': 'If we now define [MATH], we see that for solutions of the equations of motion [MATH].', 'hep-th-0307178-1-38-2': 'The symplectic leaves of [MATH] are connected components of the orbits of the generalized dressing transformation of [MATH] by [MATH], that comes from solving eq. ([REF]) in [MATH].', 'hep-th-0307178-1-39-0': 'In order to describe the presymplectic structure in the space of solutions we need to introduce a new Poisson bracket in [MATH].', 'hep-th-0307178-1-39-1': 'Recall that the Heisenberg double was defined in Section 3 as [MATH] whith [MATH].', 'hep-th-0307178-1-39-2': 'We can generalize this construction by introducing another [MATH]-matrix [MATH] that give rise to [MATH].', 'hep-th-0307178-1-39-3': 'The Poisson structure in the double we are interested in is [MATH] that is again non-degenerate around the unit.', 'hep-th-0307178-1-39-4': 'Its main symplectic leaf is [MATH].', 'hep-th-0307178-1-39-5': 'If we parametrize the points in [MATH] by [MATH], the symplectic structure in [MATH] obtained by inverting the Poisson bracket is [EQUATION].', 'hep-th-0307178-1-40-0': 'Note again that although for a given point of [MATH] factors [MATH] in general are not uniquely determined (different choices differ by elements of the discrete groups [MATH] or [MATH]) the form [MATH] is not affected by the ambiguity and is indeed well defined in [MATH].', 'hep-th-0307178-1-41-0': 'The presymplectic structure of the [MATH] Poisson-Sigma model [MATH] in the open geometry ([MATH]) with the boundary conditions of Sect. 4 can then be written in terms of [MATH].', 'hep-th-0307178-1-41-1': 'It reads [EQUATION] with [MATH].', 'hep-th-0307178-1-41-2': 'And if we take [MATH], i.e. [MATH], [EQUATION].', 'hep-th-0307178-1-42-0': 'The discussion of the gauge transformations and the reduced phase space goes parallel to the previous models.', 'hep-th-0307178-1-42-1': 'The points of the reduced phase space [MATH] are pairs [MATH] with [MATH] the homotopy class of maps [MATH] with fixed boundary values and such that [MATH].', 'hep-th-0307178-1-42-2': 'The symplectic form in [MATH] can be obtained as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-1-43-0': '# Conclusions [MATH] conclusions We have solved the Poisson Sigma model for a Poisson-Lie group [MATH] and for its dual group [MATH].', 'hep-th-0307178-1-43-1': 'This models have non-regular foliations.', 'hep-th-0307178-1-43-2': 'For example, as both are Poisson-Lie groups, the unit of the respective groups forms itself a symplectic leaf.', 'hep-th-0307178-1-43-3': 'Due to this fact they fall out of the general construction of solutions of [CITATION] and, as we have the explicit solutions at hand, they are an interesting example in which we can test the methods of [CITATION] for the case of non regular leaves.', 'hep-th-0307178-1-43-4': 'Actually in many aspects our models can be taken as a deformation of the linear one.', 'hep-th-0307178-1-44-0': 'Another remarkable aspect of the models under consideration is their duality.', 'hep-th-0307178-1-44-1': 'We can take [MATH] and [MATH] as subgroups of the same double group [MATH], [MATH] and [MATH] respectively.', 'hep-th-0307178-1-44-2': 'In the resolution of the two models on the strip [MATH], we find a close relation between their reduced phase spaces (obtained by taking quotient by the gauge symmetry).', 'hep-th-0307178-1-44-3': 'Actually we see that there exist maps from these spaces to [MATH] such that their symplectic forms are obtained as the pullback by these maps of the symplectic form on the main symplectic leaf of the Heisenberg double.', 'hep-th-0307178-1-44-4': 'This study reveals a bulk-boundary duality, such that the degrees of freedom naturally associated to the bulk in one model are mapped into those corresponding to the boundary in the other one.', 'hep-th-0307178-1-44-5': 'This suggests a connection with the Poisson-Lie non-abelian T-duality of [CITATION] for the case of WZW models.', 'hep-th-0307178-1-44-6': 'The concrete form of this correspondence is not clear to us.', 'hep-th-0307178-1-45-0': 'We have also solved the model on [MATH] for the more general Poisson structure associated to a pair of [MATH] matrices introduced in [CITATION].', 'hep-th-0307178-1-45-1': 'The solution is obtained along similar lines to those of the Poisson-Lie case.', 'hep-th-0307178-1-45-2': 'Now the symplectic form in the reduced phase space is obtained from a new Poisson structure in the double that generalizes the Heisenberg double of [CITATION][CITATION][CITATION].', 'hep-th-0307178-1-46-0': 'Throughout the paper we have considered [MATH]- matrices that are solutions of the modified Yang-Baxter equation ([REF]) with [MATH], i.e. factorizable Lie bialgebras of ref. [CITATION].', 'hep-th-0307178-1-46-1': 'In this case we can take the same double group [MATH] independently of the choice of [MATH] because only [MATH], the embedding of [MATH] in [MATH], depends on it.', 'hep-th-0307178-1-46-2': 'This allows us to carry out the study of the general case for a pair of [MATH]-matrices in Section 6.', 'hep-th-0307178-1-46-3': 'The construction can be generalized without difficulties if [MATH], but for the case [MATH] our methods have to be modified.', 'hep-th-0307178-1-46-4': 'In particular the double group [MATH] will depend on the actual choice of the [MATH]-matrix and the explicit form of the solutions requires further work.', 'hep-th-0307178-1-46-5': 'Despite this fact we would expect that the main results about duality remain essentially unchanged.', 'hep-th-0307178-1-47-0': 'For instance, for the trivial model of eq. ([REF]) with [MATH] and its dual, the linear one of eq. ([REF]), the reduced phase space is [MATH].', 'hep-th-0307178-1-47-1': 'In the trivial model the bulk degrees of freedom are in [MATH] and those of the boundary in [MATH], whereas for the linear model the situation is the converse.', 'hep-th-0307178-1-47-2': 'The symplectic form in both cases is the canonical form in [MATH].', 'hep-th-0307178-1-48-0': 'As extensions of the work carried out in the paper one might consider more general geometries and boundary conditions.', 'hep-th-0307178-1-48-1': 'Another point that is worth mentioning is that of possible non-topological deformations of our models.', 'hep-th-0307178-1-48-2': 'In order to keep the gauge symmetry of the model one usually considers the addition of Cassimir functions on the Poisson manifold.', 'hep-th-0307178-1-48-3': 'One of these deformations, for the linear model, gives rise to the two dimensional Yang-Mills theory.', 'hep-th-0307178-1-48-4': 'It would be interesting to consider a generalizations of the latter in the context of Poisson-Lie groups.', 'hep-th-0307178-1-49-0': 'An aspect that we do not cover in this paper is that of quantization of the models.', 'hep-th-0307178-1-49-1': 'The cases for Poisson-Lie groups have been extensively studied in the literature; see for instance [CITATION], [CITATION], [CITATION].', 'hep-th-0307178-1-49-2': 'However, quantization of the generalized Heisenberg double [MATH] of Section 6 is much less known.', 'hep-th-0307178-1-49-3': 'It will be the subject of further research.'}

{'hep-th-0307178-2-0-0': '# Introduction', 'hep-th-0307178-2-1-0': 'Since their appearance in [CITATION] the Poisson Sigma models have played a central role in the study of two dimensional gauge theories.', 'hep-th-0307178-2-1-1': 'They are related to pure gravity, to Wess-Zumino-Witten models and Yang-Mills in two dimensions.', 'hep-th-0307178-2-1-2': 'Generalizations of the model that include supergravity have been recently proposed in [CITATION].', 'hep-th-0307178-2-2-0': 'The Poisson Sigma models are topological and besides the two dimensional (space-time) manifold [MATH] all we require is a Poisson structure on the target [MATH] (supplemented by boundary conditions if [MATH]).', 'hep-th-0307178-2-2-1': 'The theory seems then suitable for the study of the underlying Poisson structure and in fact in [CITATION] it was shown that the semiclassical expansion of these models (with [MATH] the unit disk and insertions at the boundary) reproduces the [MATH]-product introduced by Kontsevich in [CITATION] that quantizes by deformation the Poisson bracket.', 'hep-th-0307178-2-3-0': 'In this paper we shall be interested in Poisson Sigma models in which [MATH] is a Lie group.', 'hep-th-0307178-2-3-1': 'In this case it is natural to ask for a consistency relation between its Poisson and group structures and this leads to the concept of Poisson-Lie group ([CITATION]).', 'hep-th-0307178-2-3-2': 'Poisson-Lie groups are the classical counterparts of quantum groups and they lead to the so called hidden symmetries of integrable systems that induce dressing transformations in the phase space.', 'hep-th-0307178-2-3-3': 'We shall give later a more detailed description of these objects, but for the moment we would like to stress that Poisson-Lie groups come in dual pairs under the exchange of the roles of Poisson and Lie brackets.', 'hep-th-0307178-2-3-4': 'Examples of Poisson-Lie Sigma models have been studied in [CITATION] in connection to [MATH] Wess-Zumino-Witten theories.', 'hep-th-0307178-2-3-5': 'Here we pursue a systematic study of the matter and besides we shall also consider some generalizations of the Poisson structure for which the product with the corresponding Poisson-Lie group acts as a Poisson action.', 'hep-th-0307178-2-4-0': '[3]', 'hep-th-0307178-2-5-0': 'One of the simplest examples of Poisson-Lie Sigma models is the linear one.', 'hep-th-0307178-2-5-1': 'Here [MATH] is a vector space (abelian group) and its Poisson structure is linear.', 'hep-th-0307178-2-5-2': 'This model is related to [MATH] and Yang-Mills and can be considered as dual of the trivial Poisson-Lie Sigma model in an (in general) non-abelian group with vanishing Poisson bracket.', 'hep-th-0307178-2-5-3': 'Our models can be regarded as the simplest generalizations of the linear ones with which they share some properties that will be stressed in the sequel.', 'hep-th-0307178-2-6-0': 'Another aspect that will deserve our attention is that of duality, i.e. we shall try to relate the two dual models, which is not obvious at the lagrangian level.', 'hep-th-0307178-2-6-1': 'However, in the hamiltonian approach, in the open geometry, the duality becomes evident: it consists of the exchange of bulk and boundary degrees of freedom.', 'hep-th-0307178-2-6-2': 'We would like to relate this fact with the non-abelian T-duality for WZW Sigma models ([CITATION]).', 'hep-th-0307178-2-7-0': 'We begin with a brief review of the essential results about Poisson Sigma models and Poisson-Lie groups in sections [REF] and [REF].', 'hep-th-0307178-2-7-1': 'Sections [REF] and [REF] are devoted to the study of the Poisson-Lie Sigma model of a simple Poisson-Lie group and its dual one, respectively.', 'hep-th-0307178-2-7-2': 'By using the techniques of the two previous sections we are able to solve more general Poisson Sigma models on Lie groups and we introduce a generalization of the Heisenberg double in section [REF].', 'hep-th-0307178-2-7-3': 'The conclusions and open questions are presented in section [REF].', 'hep-th-0307178-2-8-0': '# Poisson Sigma models', 'hep-th-0307178-2-9-0': 'The Poisson Sigma model is a two-dimensional topological Sigma model with the target manifold [MATH] equipped with a Poisson structure [MATH], i.e. a bivector field on [MATH].', 'hep-th-0307178-2-9-1': 'The Poisson bracket of two functions on [MATH] is given by the contraction of [MATH]: [MATH].', 'hep-th-0307178-2-10-0': 'The fields of the model are [MATH] and a 1-form [MATH] on [MATH] with values in the pullback by [MATH] of the cotangent bundle of [MATH].', 'hep-th-0307178-2-10-1': 'The action functional has the form [EQUATION] where [MATH] denotes the pairing between the tangent and the cotangent vectors to [MATH].', 'hep-th-0307178-2-11-0': 'If [MATH] are local coordinates on [MATH], [MATH] local coordinates on [MATH], [MATH] the components of the Poisson structure in these coordinates and [MATH], [MATH], [MATH] the action reads [EQUATION]', 'hep-th-0307178-2-11-1': 'It is straightforward to work out the equations of motion: [EQUATION]', 'hep-th-0307178-2-11-2': 'One can show ([CITATION]) that for solutions of ([REF]) the image of [MATH] lies within one of the symplectic leaves of the foliation of [MATH].', 'hep-th-0307178-2-12-0': 'Under the infinitesimal transformation [EQUATION] where [MATH] is a section of [MATH], the action ([REF]) transforms by a boundary term [EQUATION]', 'hep-th-0307178-2-12-1': 'One of the special properties of these models is that the commutator of two consecutive gauge tansformation of ([REF]) is not of the same form i.e. [EQUATION] where for a vector field [MATH], [MATH].', 'hep-th-0307178-2-12-2': 'Later, we shall make extensive use of this commutator in the context of Poisson-Lie groups.', 'hep-th-0307178-2-12-3': 'Note that the term in parenthesis in ([REF]) is the equation of motion ([REF]), and then it vanishes on-shell.', 'hep-th-0307178-2-12-4': 'However, as remarked in [CITATION], due to the [MATH] dependence of [MATH] even on-shell one can not talk properly about a Lie algebra structure in the space of parameters, unless we enlarge its definition to account for field dependent gauge transformations.', 'hep-th-0307178-2-12-5': 'The situation will become simpler in the linear case that we describe below.', 'hep-th-0307178-2-13-0': 'One of the simplest examples of Poisson Sigma models is the linear one (2D BF theory).', 'hep-th-0307178-2-13-1': 'We take a finite dimensional vector space [MATH] as the target space and define a linear Poisson bracket on it, i.e. for any [MATH], their Poisson bracket [MATH].', 'hep-th-0307178-2-13-2': 'The linear Poisson bracket on [MATH] defines then a Lie bracket [MATH] on [MATH].', 'hep-th-0307178-2-13-3': 'Assuming that [MATH] is semisimple we can use the Cartan-Killing form [MATH] to identify [MATH] and [MATH].', 'hep-th-0307178-2-13-4': 'The fields are then [MATH] and [MATH] and the action reads: [EQUATION]', 'hep-th-0307178-2-13-5': 'The equations of motion are: [EQUATION]', 'hep-th-0307178-2-13-6': 'For [MATH] the gauge transformation [EQUATION] induces the change of the action ([REF]) by a boundary term [EQUATION]', 'hep-th-0307178-2-13-7': 'Note that in this case the gauge transformations close even off-shell [EQUATION] and induce the Lie algebra structure of [MATH] in the space of parameters.', 'hep-th-0307178-2-14-0': '# Poisson-Lie groups', 'hep-th-0307178-2-15-0': 'In this section we shall review some basic features of the theory of Poisson-Lie groups and fix our notation.', 'hep-th-0307178-2-15-1': 'See [CITATION] for details.', 'hep-th-0307178-2-15-2': 'A Poisson-Lie group is a Lie group equipped with a Poisson structure which makes the product [MATH] a Poisson map if [MATH] is considered with the product Poisson structure.', 'hep-th-0307178-2-15-3': 'Linearization of the Poisson structure at the unit [MATH] of [MATH] provides a Lie algebra structure on [MATH] by the formula [EQUATION]', 'hep-th-0307178-2-15-4': 'The Poisson-Lie structure of [MATH] yields the compatibility condition [EQUATION] which allows to define a Lie bracket in [MATH] by the formula [EQUATION]', 'hep-th-0307178-2-15-5': 'If [MATH] is connected and simply connected, ([REF]) is enough to integrate [MATH] to a Poisson structure on [MATH] that makes it Poisson-Lie and the Poisson structure is unique.', 'hep-th-0307178-2-15-6': 'The symmetry between [MATH] and [MATH] in ([REF]), implies that one has also a Poisson-Lie group [MATH] with Lie algebra [MATH] and a Poisson structure whose linearization at [MATH] gives the bracket [MATH].', 'hep-th-0307178-2-15-7': '[MATH] is the dual Poisson-Lie group of [MATH].', 'hep-th-0307178-2-16-0': 'Let us take [MATH] a complex, simple, connected, simply connected Lie group and give the above construction explicitly.', 'hep-th-0307178-2-16-1': 'The (essentially unique) nondegenerate, invariant, bilinear form tr on [MATH] establishes an isomorphism between [MATH] and [MATH].', 'hep-th-0307178-2-16-2': 'The Poisson structure [MATH] contracted with the right-invariant forms [MATH], will be denoted [EQUATION]', 'hep-th-0307178-2-16-3': 'For a general Poisson-Lie structure on [MATH] ([CITATION]), [EQUATION] where [MATH] is an antisymmetric linear operator such that [EQUATION]', 'hep-th-0307178-2-16-4': 'Using the operator [MATH] it is possible to define a new Lie bracket in [MATH], [EQUATION] [MATH] is isomorphic to [MATH] via tr.', 'hep-th-0307178-2-17-0': 'If [MATH] we can rescale the bilinear form in [MATH] and [MATH] so that we leave the Poisson bracket unchanged but [MATH].', 'hep-th-0307178-2-17-1': 'In the following we shall consider this case ([MATH]) that corresponds to the factorizable Lie bialgebras of ref. [CITATION].', 'hep-th-0307178-2-17-2': 'Take [MATH], [MATH] and [MATH].', 'hep-th-0307178-2-17-3': 'Then, [EQUATION] and the embedding [MATH] defines an homomorphism from [MATH] to [MATH].', 'hep-th-0307178-2-17-4': 'We give an invariant, nondegenerate bilinear form on [MATH] by [EQUATION]', 'hep-th-0307178-2-17-5': 'This allows to identify [MATH] with the subalgebra [MATH] and [MATH] as the subgroup [MATH] in [MATH] corresponding to the Lie subalgebra [MATH].', 'hep-th-0307178-2-17-6': 'We denote by [MATH] the elements of [MATH].', 'hep-th-0307178-2-17-7': 'In a natural way, [MATH] and [MATH].', 'hep-th-0307178-2-17-8': 'Notice that [MATH] is a discrete subgroup of [MATH].', 'hep-th-0307178-2-18-0': 'The induced Poisson-Lie structure on [MATH] contracted with the right-invariant forms on [MATH], [MATH] for [MATH], takes the form [EQUATION]', 'hep-th-0307178-2-18-1': 'The [MATH]-matrices are used not only to give explicit realizations of the dual group and to construct Poisson-Lie structures but to construct more general Poisson structures on Lie groups.', 'hep-th-0307178-2-18-2': 'Following Semenov-Tian-Shansky ([CITATION]), if [MATH] and [MATH] are antisymmetric and satisfy ([REF]) with the same value for [MATH], we can define the Poisson structure (contracted with the right-invariant forms), [EQUATION] [MATH] equipped with ([REF]) is denoted by [MATH].', 'hep-th-0307178-2-18-3': 'One can verify that [MATH] is a Poisson map.', 'hep-th-0307178-2-18-4': 'When [MATH] and [MATH] then we get ([REF]) and the Poisson structure is, indeed, Poisson-Lie.', 'hep-th-0307178-2-19-0': 'Given a [MATH]-matrix in [MATH] it is possible to define a [MATH]-matrix in [MATH] (which we shall denote by [MATH]) in a natural way: [MATH] where [MATH] and [MATH] are the projectors on [MATH] and [MATH] respectively, parallel to the complementary subalgebra.', 'hep-th-0307178-2-19-1': 'Hence, [MATH].', 'hep-th-0307178-2-20-0': 'We shall need the description of the Heisenberg double (which is not a Poisson-Lie group; in fact, with the notation introduced before, it corresponds to [MATH]).', 'hep-th-0307178-2-20-1': 'Its main symplectic leaf is [MATH] (that contains a neighborhood of the unit of [MATH]).', 'hep-th-0307178-2-20-2': 'The symplectic structure obtained by inverting the Poisson structure on it reads [EQUATION] where [MATH].', 'hep-th-0307178-2-20-3': 'Note that although these decompositions are not unique, the ambiguity (the product by an element of [MATH]) does not affect the values of [MATH] which is well defined in [MATH].', 'hep-th-0307178-2-21-0': '# Poisson-Lie Sigma model on [MATH]', 'hep-th-0307178-2-22-0': 'Recall the action ([REF]) and take the Poisson-Lie group of the previous section as the target.', 'hep-th-0307178-2-22-1': 'The action for [MATH] and [MATH] reads: [EQUATION] which is what we call the Poisson-Lie Sigma model for [MATH].', 'hep-th-0307178-2-23-0': '[3]', 'hep-th-0307178-2-24-0': 'We proceed now to a detailed study of this model, from which we shall learn techniques applicable to the resolution of more general Poisson Sigma models on Lie groups.', 'hep-th-0307178-2-25-0': 'The equations of motion are: [EQUATION]', 'hep-th-0307178-2-25-1': 'From the previous two equations one can derive also a zero curvature equation for [MATH] itself [EQUATION] which can be deduced directly from the stationary condition for the action under variations of the fields that keep [MATH] invariant.', 'hep-th-0307178-2-26-0': 'The gauge symmetry of the action, for [MATH], is given in its infinitesimal form by[EQUATION]', 'hep-th-0307178-2-26-1': 'Under this transformation the action changes by a boundary term, namely [MATH].', 'hep-th-0307178-2-26-2': 'Note that ([REF]) corresponds to the right dressing vector fields of ref. [CITATION] translated to the origin by right multiplication in [MATH].', 'hep-th-0307178-2-26-3': 'Its integration (local as in general the vector field is not complete) gives rise to the dressing transformation of [MATH].', 'hep-th-0307178-2-26-4': 'Also in ([REF]) the third term vanishes on-shell; it corresponds to a trivial transformation (see refs. [CITATION]), proportional to the equations of motion.', 'hep-th-0307178-2-26-5': 'If we forget about that term (or alternatively, working on-shell) [EQUATION].', 'hep-th-0307178-2-26-6': 'Note that unlike the general case this relation defines a Lie algebra structure (that of the gauge group corresponding to [MATH]) in the space of gauge transformations.', 'hep-th-0307178-2-26-7': 'This is also what happens in the linear case discussed in section [REF].', 'hep-th-0307178-2-26-8': 'However, in this case the relation holds uniquely on-shell while in the linear case the Lie algebra structure persists off-shell.', 'hep-th-0307178-2-26-9': 'Another difference is that the vector fields (on-shell) induced by the gauge transformation are not complete in the general case.', 'hep-th-0307178-2-26-10': 'This is a second obstruction to properly talk about a gauge group, so we shall restrict ourselves to infinitesimal gauge transformations.', 'hep-th-0307178-2-27-0': 'Now we proceed to integrate the equations of motion.', 'hep-th-0307178-2-27-1': 'We shall consider [MATH] in eq. ([REF]) although as remarked before the case [MATH] can be treated similarly.', 'hep-th-0307178-2-27-2': 'As the equation of motion for [MATH] is independent of [MATH], like in the linear model, we first solve the eqs. ([REF]) and ([REF]).', 'hep-th-0307178-2-27-3': 'This is in contrast with the general case of ref. [CITATION].', 'hep-th-0307178-2-27-4': 'Locally [EQUATION]', 'hep-th-0307178-2-27-5': 'In order to fix the ambiguity in the new fields we choose a point [MATH] and take [MATH].', 'hep-th-0307178-2-28-0': 'We need to work a bit more to get explicit solutions for [MATH].', 'hep-th-0307178-2-28-1': 'The equation of motion ([REF]) can be equivalently written as [EQUATION] and recalling that [MATH], [EQUATION] [MATH] projects on the [MATH] component of the elements of [MATH] and yields [EQUATION] which is equivalent to [EQUATION]', 'hep-th-0307178-2-28-2': 'The same procedure can be carried out by using [MATH], [EQUATION]', 'hep-th-0307178-2-28-3': 'The solutions for the equations of motion can then be expressed by the single relation in [MATH]: [EQUATION] which in particular implies that [MATH], and [MATH] takes values in the connected component of orbits of [MATH] by dressing transformations.', 'hep-th-0307178-2-28-4': 'These orbits are the symplectic leaves of the Poisson-Lie group [MATH].', 'hep-th-0307178-2-29-0': 'Now we proceed to the hamiltonian analysis of the model.', 'hep-th-0307178-2-29-1': 'To that end we fix the open geometry for [MATH] and take free boundary conditions for [MATH] and [MATH] vanishing on vectors tangent to [MATH].', 'hep-th-0307178-2-29-2': 'The boundary conditions enforce that [MATH] are constant along the connected components of the boundary, [MATH], [MATH], [MATH], [MATH].', 'hep-th-0307178-2-29-3': 'The presymplectic form is [EQUATION] where [MATH] and [MATH] varies in the space of solutions of the equations of motion and [MATH] is any curve joining the two components of the boundary.', 'hep-th-0307178-2-29-4': 'The result is, of course, independent of the particular choice of [MATH].', 'hep-th-0307178-2-30-0': 'If we parametrize the solutions in terms of [MATH] with [MATH] we obtain [EQUATION] where [MATH] acts on the space-time variables in [MATH].', 'hep-th-0307178-2-30-1': 'Then [MATH] depends only on the values at the boundaries, namely [EQUATION]', 'hep-th-0307178-2-30-2': 'Or if we take [MATH], i.e. [MATH] [EQUATION].', 'hep-th-0307178-2-31-0': 'From the analysis of the degeneracy of [MATH] we conclude that the vector fields induced by infinitesimal gauge transformations that are the identity at the boundary form its kernel and we reduce the space of solutions by this transformations.', 'hep-th-0307178-2-31-1': 'The reduced phase space [MATH] is then the set of pairs [MATH] with [MATH] a homotopy class of maps from [MATH] to [MATH] which are the identity at [MATH] and have fixed value at [MATH] and such that [MATH].', 'hep-th-0307178-2-31-2': 'The symplectic form on [MATH] can be viewed as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-2-32-0': 'The set of solutions may be endowed with the structure of a symplectic groupoid ([CITATION]).', 'hep-th-0307178-2-32-1': 'The product of two such solutions [MATH] and [MATH] is defined if [MATH] and is given by [MATH] with [EQUATION]', 'hep-th-0307178-2-32-2': 'The results obtained in this section have their dual counterpart if we consider the dual group [MATH], which is accomplished in the next section.', 'hep-th-0307178-2-33-0': '# Poisson-Lie Sigma model on [MATH]', 'hep-th-0307178-2-34-0': 'In order to unravel the role of the duality between Poisson-Lie groups in the context of Poisson-Lie Sigma models we proceed now to the study of the theory for [MATH].', 'hep-th-0307178-2-34-1': 'Using the Poisson structure given in ([REF]), the action of the model reads, [EQUATION]', 'hep-th-0307178-2-34-2': 'As shown in [CITATION], this Poisson-Lie Sigma model with target [MATH] and fields [MATH] and [MATH] is closely related to the [MATH] gauged WZW model with fields [MATH] and [MATH].', 'hep-th-0307178-2-34-3': 'We do not pursue further this relation in the present work.', 'hep-th-0307178-2-34-4': 'We rather shall make a study analogous to that of the previous section, stressing its similarities and differences with respect to the Poisson-Lie Sigma model for [MATH].', 'hep-th-0307178-2-35-0': 'The equations of motion of the model are [EQUATION] [3]', 'hep-th-0307178-2-36-0': 'Note that, similarly to the previous case for the group [MATH], the relation between [MATH] and [MATH] is nothing but the coadjoint action of [MATH] for the group [MATH].', 'hep-th-0307178-2-36-1': 'Like in the former case it is posible to deduce, from the previous equations, a zero curvature condition for [MATH] itself [EQUATION]', 'hep-th-0307178-2-36-2': 'The gauge transformations, for [MATH], read: [EQUATION] where [MATH].', 'hep-th-0307178-2-36-3': 'The change in the action is [MATH].', 'hep-th-0307178-2-36-4': 'As before, if we forget about the third term on the right hand side of ([REF]), that vanishes on-shell, the transformations close.', 'hep-th-0307178-2-36-5': 'Namely [MATH] which corresponds to the gauge group [MATH].', 'hep-th-0307178-2-37-0': 'The solutions of the equations of motion can be obtained along the same lines as before.', 'hep-th-0307178-2-37-1': 'From ([REF]) [EQUATION] in any contractible, open set.', 'hep-th-0307178-2-37-2': 'Without loss of generality we fix again a point [MATH] such that [MATH].', 'hep-th-0307178-2-37-3': 'Now we write together the equations for [MATH] using the solution for [MATH].', 'hep-th-0307178-2-37-4': '[EQUATION] or equivalently [EQUATION] where [MATH].', 'hep-th-0307178-2-37-5': 'Hence [EQUATION] and once more, we can obtain the solutions from an equation in [MATH] [EQUATION] which means that [MATH] is the dressing-transformed of [MATH] by [MATH].', 'hep-th-0307178-2-37-6': 'Orbits of the dressing transformation are the symplectic leaves of [MATH].', 'hep-th-0307178-2-37-7': 'Note, at this point, the complete symmetry with the previous situation under the exchange of the roles of [MATH] and [MATH].', 'hep-th-0307178-2-38-0': 'We consider again the open geometry [MATH] with boundary conditions for [MATH] vanishing on vectors tangent to the boundary and [MATH] free.', 'hep-th-0307178-2-38-1': 'The boundary conditions impose that [MATH] and [MATH] are constant in every connected component of [MATH] i.e. [MATH], [MATH], [MATH], [MATH].', 'hep-th-0307178-2-39-0': 'The presymplectic form of the model [MATH] can be written [EQUATION] where again [MATH] acts on the [MATH] variables.', 'hep-th-0307178-2-39-1': 'Therefore [EQUATION] and taking [MATH], or [MATH], [EQUATION]', 'hep-th-0307178-2-39-2': 'Points of the phase space reduced by gauge symmetries [MATH], can be described by pairs [MATH] where by [MATH] we denote a homotopy class of maps [MATH] with fixed boundary values [MATH], [MATH] and such that [MATH].', 'hep-th-0307178-2-39-3': 'The symplectic form on [MATH] can be described as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-2-40-0': 'Note the duality between [MATH] and [MATH].', 'hep-th-0307178-2-40-1': 'The symplectic forms of the two models coincide upon the exchange of [MATH] with [MATH] and [MATH] with [MATH].', 'hep-th-0307178-2-40-2': 'Variables [MATH] in the [MATH] model or [MATH] in the [MATH] model are associated to boundary values of their corresponding fields (this can be more explicitly seen working in a manifold [MATH] whose boundary has more than two connected components) while [MATH] and [MATH] are associated to the bulk.', 'hep-th-0307178-2-40-3': 'In this sense one can say that the duality previously described between the Poisson-Lie Sigma models for [MATH] and [MATH] can be stated as a bulk-boundary duality.', 'hep-th-0307178-2-41-0': '# More general Poisson Sigma models on [MATH]', 'hep-th-0307178-2-42-0': 'In this section we solve the model with Poisson structure ([REF]) following the lines of section [REF].', 'hep-th-0307178-2-42-1': 'As remarked in section [REF] this structure does not make [MATH] a Poisson Lie group, but indeed the left and right product by the corresponding Poisson-Lie groups are Poisson actions.', 'hep-th-0307178-2-42-2': 'In the resolution of the model we shall introduce a generalization of the dressing transformation and of the Heissenberg double and we shall be able to identify the symplectic leaves for the Poissson structure defined on [MATH].', 'hep-th-0307178-2-43-0': 'The action for the model is, [EQUATION] where as stressed in section [REF] [MATH] and [MATH] are two solutions of the modified Yang-Baxter equation ([REF]) with the same value for [MATH].', 'hep-th-0307178-2-43-1': 'The equations of motion are [EQUATION]', 'hep-th-0307178-2-43-2': 'From the previous equations, or performing in the action variations of the fields that keep [MATH] unchanged, we obtain [EQUATION]', 'hep-th-0307178-2-43-3': 'The gauge symmetry in its infinitesimal form is [EQUATION]', 'hep-th-0307178-2-43-4': 'The transformation for [MATH] corresponds to the right dressing vector field that comes, like in the Poisson-Lie case of section [REF], from the contraction of the Poisson structure with the right-invariant forms.', 'hep-th-0307178-2-43-5': 'On-shell, [MATH].', 'hep-th-0307178-2-43-6': 'Hence, the symmetry group is the one corresponding to the matrix [MATH], i.e. [MATH].', 'hep-th-0307178-2-43-7': 'The reason for the preferred role of [MATH] against [MATH] in the gauge symmetry is simply the choice of right-invariant one forms to express the Poisson structure.', 'hep-th-0307178-2-43-8': 'Had we chosen left-invariant forms (i.e. changing the variable [MATH] by [MATH] in the action) the symmetry algebra would have been that of [MATH].', 'hep-th-0307178-2-44-0': 'As before we shall take [MATH].', 'hep-th-0307178-2-44-1': 'Then we can consider [MATH] or [MATH] as different subgroups of the same double group [MATH].', 'hep-th-0307178-2-44-2': 'We shall denote by [MATH] (resp. [MATH]) the elements of [MATH] (resp. [MATH]).', 'hep-th-0307178-2-45-0': 'Using the methods of section [REF], we can easily solve the model.', 'hep-th-0307178-2-45-1': 'As before, we first write locally the solutions for [MATH] and [MATH], [EQUATION] with [MATH]', 'hep-th-0307178-2-46-0': 'The equation of motion for [MATH] eq. ([REF]) can be transformed into [EQUATION] and inserting the solutions for [MATH], [MATH] [EQUATION]', 'hep-th-0307178-2-46-1': 'Or equivalently [EQUATION] with [MATH].', 'hep-th-0307178-2-47-0': 'We can write the general solution as an equation in [MATH] for [MATH], [EQUATION]', 'hep-th-0307178-2-47-1': 'If we now define [MATH], we see that for solutions of the equations of motion [MATH].', 'hep-th-0307178-2-47-2': 'The symplectic leaves of [MATH] are connected components of the orbits of the generalized dressing transformation of [MATH] by [MATH], that comes from solving eq. ([REF]) in [MATH].', 'hep-th-0307178-2-48-0': 'In order to describe the presymplectic structure in the space of solutions we need to introduce a new Poisson bracket in [MATH].', 'hep-th-0307178-2-48-1': 'Recall that the Heisenberg double was defined in section [REF] as [MATH] with [MATH].', 'hep-th-0307178-2-48-2': 'We can generalize this construction by introducing another [MATH]-matrix [MATH] that give rise to [MATH].', 'hep-th-0307178-2-48-3': 'The Poisson structure in the double we are interested in is [MATH] that is again non-degenerate around the unit.', 'hep-th-0307178-2-48-4': 'Its main symplectic leaf is [MATH].', 'hep-th-0307178-2-48-5': 'If we parametrize the points in [MATH] by [MATH], the symplectic structure in [MATH] obtained by inverting the Poisson bracket is [EQUATION].', 'hep-th-0307178-2-49-0': 'Note again that although for a given point of [MATH] factors [MATH] in general are not uniquely determined (different choices differ by elements of the discrete groups [MATH] or [MATH]) the form [MATH] is not affected by the ambiguity and is indeed well defined in [MATH].', 'hep-th-0307178-2-50-0': 'The presymplectic structure of the [MATH] Poisson-Sigma model [MATH] in the open geometry ([MATH]) with the boundary conditions of section [REF] can then be written in terms of [MATH].', 'hep-th-0307178-2-50-1': 'It reads [EQUATION] with [MATH].', 'hep-th-0307178-2-50-2': 'And if we take [MATH], i.e. [MATH], [EQUATION].', 'hep-th-0307178-2-51-0': 'The discussion of the gauge transformations and the reduced phase space goes parallel to the previous models.', 'hep-th-0307178-2-51-1': 'The points of the reduced phase space [MATH] are pairs [MATH] with [MATH] the homotopy class of maps [MATH] with fixed boundary values and such that [MATH].', 'hep-th-0307178-2-51-2': 'The symplectic form in [MATH] can be obtained as the pullback of [MATH] by the map [MATH].', 'hep-th-0307178-2-52-0': '# Conclusions', 'hep-th-0307178-2-53-0': 'We have solved the Poisson Sigma model for a Poisson-Lie group [MATH] and for its dual group [MATH].', 'hep-th-0307178-2-53-1': 'This models have non-regular foliations.', 'hep-th-0307178-2-53-2': 'For example, as both are Poisson-Lie groups, the unit of the respective groups forms itself a symplectic leaf.', 'hep-th-0307178-2-53-3': 'Due to this fact they fall out of the general construction of solutions of [CITATION] and, as we have the explicit solutions at hand, they are an interesting example in which we can test the methods of [CITATION] for the case of non regular leaves.', 'hep-th-0307178-2-53-4': 'Actually in many aspects our models can be taken as a deformation of the linear one.', 'hep-th-0307178-2-54-0': 'Another remarkable aspect of the models under consideration is their duality.', 'hep-th-0307178-2-54-1': 'We can take [MATH] and [MATH] as subgroups of the same double group [MATH], [MATH] and [MATH] respectively.', 'hep-th-0307178-2-54-2': 'In the resolution of the two models on the strip [MATH], we find a close relation between their reduced phase spaces (obtained by taking quotient by the gauge symmetry).', 'hep-th-0307178-2-54-3': 'Actually we see that there exist maps from these spaces to [MATH] such that their symplectic forms are obtained as the pullback by these maps of the symplectic form on the main symplectic leaf of the Heisenberg double.', 'hep-th-0307178-2-54-4': 'This study reveals a bulk-boundary duality, such that the degrees of freedom naturally associated to the bulk in one model are mapped into those corresponding to the boundary in the other one.', 'hep-th-0307178-2-54-5': 'This suggests a connection with the Poisson-Lie non-abelian T-duality of [CITATION] for the case of WZW models.', 'hep-th-0307178-2-54-6': 'The concrete form of this correspondence is not clear to us.', 'hep-th-0307178-2-55-0': 'We have also solved the model on [MATH] for the more general Poisson structure associated to a pair of [MATH] matrices introduced in [CITATION].', 'hep-th-0307178-2-55-1': 'The solution is obtained along similar lines to those of the Poisson-Lie case.', 'hep-th-0307178-2-55-2': 'Now the symplectic form in the reduced phase space is obtained from a new Poisson structure in the double that generalizes the Heisenberg double of [CITATION].', 'hep-th-0307178-2-56-0': 'Throughout the paper we have considered [MATH]- matrices that are solutions of the modified Yang-Baxter equation ([REF]) with [MATH], i.e. factorizable Lie bialgebras of ref. [CITATION].', 'hep-th-0307178-2-56-1': 'In this case we can take the same double group [MATH] independently of the choice of [MATH] because only [MATH], the embedding of [MATH] in [MATH], depends on it.', 'hep-th-0307178-2-56-2': 'This allows us to carry out the study of the general case for a pair of [MATH]-matrices in section [REF].', 'hep-th-0307178-2-56-3': 'The construction can be generalized without difficulties if [MATH], but for the case [MATH] our methods have to be modified and the explicit form of the solutions requires further work.', 'hep-th-0307178-2-56-4': 'Despite this fact we would expect that the main results about duality remain essentially unchanged.', 'hep-th-0307178-2-57-0': 'For instance, for the trivial model of eq. ([REF]) with [MATH] and its dual, the linear one of eq. ([REF]), the reduced phase space is [MATH].', 'hep-th-0307178-2-57-1': 'In the trivial model the bulk degrees of freedom are in [MATH] and those of the boundary in [MATH], whereas for the linear model the situation is the converse.', 'hep-th-0307178-2-57-2': 'The symplectic form in both cases is the canonical form in [MATH].', 'hep-th-0307178-2-58-0': 'As extensions of the work carried out in the paper one might consider more general geometries and boundary conditions.', 'hep-th-0307178-2-58-1': 'Another point that is worth mentioning is that of possible non-topological deformations of our models.', 'hep-th-0307178-2-58-2': 'In order to keep the gauge symmetry of the model one usually considers the addition of Cassimir functions on the Poisson manifold.', 'hep-th-0307178-2-58-3': 'One of these deformations, for the linear model, gives rise to the two dimensional Yang-Mills theory.', 'hep-th-0307178-2-58-4': 'It would be interesting to consider a generalizations of the latter in the context of Poisson-Lie groups.', 'hep-th-0307178-2-59-0': 'An aspect that we do not cover in this paper is that of quantization of the models.', 'hep-th-0307178-2-59-1': 'The cases for Poisson-Lie groups have been extensively studied in the literature; see for instance [CITATION].', 'hep-th-0307178-2-59-2': 'However, quantization of the generalized Heisenberg double [MATH] of section [REF] is much less known.', 'hep-th-0307178-2-59-3': 'It will be the subject of further research.'}

[['hep-th-0307178-1-36-1', 'hep-th-0307178-2-45-1'], ['hep-th-0307178-1-43-1', 'hep-th-0307178-2-53-1'], ['hep-th-0307178-1-43-2', 'hep-th-0307178-2-53-2'], ['hep-th-0307178-1-43-3', 'hep-th-0307178-2-53-3'], ['hep-th-0307178-1-43-4', 'hep-th-0307178-2-53-4'], ['hep-th-0307178-1-7-0', 'hep-th-0307178-2-7-0'], ['hep-th-0307178-1-7-1', 'hep-th-0307178-2-7-1'], ['hep-th-0307178-1-7-3', 'hep-th-0307178-2-7-3'], ['hep-th-0307178-1-34-1', 'hep-th-0307178-2-43-1'], ['hep-th-0307178-1-34-2', 'hep-th-0307178-2-43-2'], ['hep-th-0307178-1-34-3', 'hep-th-0307178-2-43-3'], ['hep-th-0307178-1-34-6', 'hep-th-0307178-2-43-6'], ['hep-th-0307178-1-34-7', 'hep-th-0307178-2-43-7'], ['hep-th-0307178-1-34-8', 'hep-th-0307178-2-43-8'], ['hep-th-0307178-1-15-0', 'hep-th-0307178-2-17-0'], ['hep-th-0307178-1-15-1', 'hep-th-0307178-2-17-1'], ['hep-th-0307178-1-15-3', 'hep-th-0307178-2-17-3'], ['hep-th-0307178-1-15-4', 'hep-th-0307178-2-17-4'], ['hep-th-0307178-1-15-5', 'hep-th-0307178-2-17-5'], ['hep-th-0307178-1-15-6', 'hep-th-0307178-2-17-6'], ['hep-th-0307178-1-15-7', 'hep-th-0307178-2-17-7'], ['hep-th-0307178-1-15-8', 'hep-th-0307178-2-17-8'], ['hep-th-0307178-1-20-0', 'hep-th-0307178-2-24-0'], ['hep-th-0307178-1-23-0', 'hep-th-0307178-2-27-0'], ['hep-th-0307178-1-23-1', 'hep-th-0307178-2-27-1'], ['hep-th-0307178-1-23-3', 'hep-th-0307178-2-27-3'], ['hep-th-0307178-1-23-5', 'hep-th-0307178-2-27-5'], ['hep-th-0307178-1-5-0', 'hep-th-0307178-2-5-0'], ['hep-th-0307178-1-5-1', 'hep-th-0307178-2-5-1'], ['hep-th-0307178-1-5-2', 'hep-th-0307178-2-5-2'], ['hep-th-0307178-1-5-3', 'hep-th-0307178-2-5-3'], ['hep-th-0307178-1-6-2', 'hep-th-0307178-2-6-2'], ['hep-th-0307178-1-48-0', 'hep-th-0307178-2-58-0'], ['hep-th-0307178-1-48-1', 'hep-th-0307178-2-58-1'], ['hep-th-0307178-1-48-2', 'hep-th-0307178-2-58-2'], ['hep-th-0307178-1-48-3', 'hep-th-0307178-2-58-3'], ['hep-th-0307178-1-48-4', 'hep-th-0307178-2-58-4'], ['hep-th-0307178-1-42-0', 'hep-th-0307178-2-51-0'], ['hep-th-0307178-1-42-1', 'hep-th-0307178-2-51-1'], ['hep-th-0307178-1-42-2', 'hep-th-0307178-2-51-2'], ['hep-th-0307178-1-21-0', 'hep-th-0307178-2-25-0'], ['hep-th-0307178-1-21-1', 'hep-th-0307178-2-25-1'], ['hep-th-0307178-1-12-1', 'hep-th-0307178-2-13-1'], ['hep-th-0307178-1-12-2', 'hep-th-0307178-2-13-2'], ['hep-th-0307178-1-12-3', 'hep-th-0307178-2-13-3'], ['hep-th-0307178-1-12-4', 'hep-th-0307178-2-13-4'], ['hep-th-0307178-1-12-5', 'hep-th-0307178-2-13-5'], ['hep-th-0307178-1-12-6', 'hep-th-0307178-2-13-6'], ['hep-th-0307178-1-12-7', 'hep-th-0307178-2-13-7'], ['hep-th-0307178-1-47-0', 'hep-th-0307178-2-57-0'], ['hep-th-0307178-1-47-1', 'hep-th-0307178-2-57-1'], ['hep-th-0307178-1-47-2', 'hep-th-0307178-2-57-2'], ['hep-th-0307178-1-32-0', 'hep-th-0307178-2-40-0'], ['hep-th-0307178-1-32-1', 'hep-th-0307178-2-40-1'], ['hep-th-0307178-1-32-2', 'hep-th-0307178-2-40-2'], ['hep-th-0307178-1-32-3', 'hep-th-0307178-2-40-3'], ['hep-th-0307178-1-49-0', 'hep-th-0307178-2-59-0'], ['hep-th-0307178-1-49-3', 'hep-th-0307178-2-59-3'], ['hep-th-0307178-1-46-0', 'hep-th-0307178-2-56-0'], ['hep-th-0307178-1-46-1', 'hep-th-0307178-2-56-1'], ['hep-th-0307178-1-46-5', 'hep-th-0307178-2-56-4'], ['hep-th-0307178-1-16-0', 'hep-th-0307178-2-18-0'], ['hep-th-0307178-1-16-1', 'hep-th-0307178-2-18-1'], ['hep-th-0307178-1-16-2', 'hep-th-0307178-2-18-2'], ['hep-th-0307178-1-16-3', 'hep-th-0307178-2-18-3'], ['hep-th-0307178-1-22-0', 'hep-th-0307178-2-26-0'], ['hep-th-0307178-1-22-1', 'hep-th-0307178-2-26-1'], ['hep-th-0307178-1-22-3', 'hep-th-0307178-2-26-3'], ['hep-th-0307178-1-22-5', 'hep-th-0307178-2-26-5'], ['hep-th-0307178-1-22-6', 'hep-th-0307178-2-26-6'], ['hep-th-0307178-1-22-8', 'hep-th-0307178-2-26-8'], ['hep-th-0307178-1-22-9', 'hep-th-0307178-2-26-9'], ['hep-th-0307178-1-22-10', 'hep-th-0307178-2-26-10'], ['hep-th-0307178-1-30-0', 'hep-th-0307178-2-38-0'], ['hep-th-0307178-1-30-1', 'hep-th-0307178-2-38-1'], ['hep-th-0307178-1-45-0', 'hep-th-0307178-2-55-0'], ['hep-th-0307178-1-45-1', 'hep-th-0307178-2-55-1'], ['hep-th-0307178-1-8-1', 'hep-th-0307178-2-9-1'], ['hep-th-0307178-1-25-1', 'hep-th-0307178-2-29-1'], ['hep-th-0307178-1-25-2', 'hep-th-0307178-2-29-2'], ['hep-th-0307178-1-25-3', 'hep-th-0307178-2-29-3'], ['hep-th-0307178-1-25-4', 'hep-th-0307178-2-29-4'], ['hep-th-0307178-1-14-0', 'hep-th-0307178-2-16-0'], ['hep-th-0307178-1-14-1', 'hep-th-0307178-2-16-1'], ['hep-th-0307178-1-14-2', 'hep-th-0307178-2-16-2'], ['hep-th-0307178-1-14-3', 'hep-th-0307178-2-16-3'], ['hep-th-0307178-1-40-0', 'hep-th-0307178-2-49-0'], ['hep-th-0307178-1-27-0', 'hep-th-0307178-2-32-0'], ['hep-th-0307178-1-27-1', 'hep-th-0307178-2-32-1'], ['hep-th-0307178-1-27-2', 'hep-th-0307178-2-32-2'], ['hep-th-0307178-1-19-1', 'hep-th-0307178-2-22-1'], ['hep-th-0307178-1-2-1', 'hep-th-0307178-2-1-1'], ['hep-th-0307178-1-38-0', 'hep-th-0307178-2-47-0'], ['hep-th-0307178-1-38-1', 'hep-th-0307178-2-47-1'], ['hep-th-0307178-1-38-2', 'hep-th-0307178-2-47-2'], ['hep-th-0307178-1-9-0', 'hep-th-0307178-2-10-0'], ['hep-th-0307178-1-9-1', 'hep-th-0307178-2-10-1'], ['hep-th-0307178-1-33-2', 'hep-th-0307178-2-42-2'], ['hep-th-0307178-1-29-0', 'hep-th-0307178-2-37-0'], ['hep-th-0307178-1-29-1', 'hep-th-0307178-2-37-1'], ['hep-th-0307178-1-29-2', 'hep-th-0307178-2-37-2'], ['hep-th-0307178-1-29-3', 'hep-th-0307178-2-37-3'], ['hep-th-0307178-1-29-4', 'hep-th-0307178-2-37-4'], ['hep-th-0307178-1-29-5', 'hep-th-0307178-2-37-5'], ['hep-th-0307178-1-29-6', 'hep-th-0307178-2-37-6'], ['hep-th-0307178-1-29-7', 'hep-th-0307178-2-37-7'], ['hep-th-0307178-1-13-1', 'hep-th-0307178-2-15-1'], ['hep-th-0307178-1-13-2', 'hep-th-0307178-2-15-2'], ['hep-th-0307178-1-13-3', 'hep-th-0307178-2-15-3'], ['hep-th-0307178-1-13-5', 'hep-th-0307178-2-15-5'], ['hep-th-0307178-1-13-6', 'hep-th-0307178-2-15-6'], ['hep-th-0307178-1-13-7', 'hep-th-0307178-2-15-7'], ['hep-th-0307178-1-17-0', 'hep-th-0307178-2-19-0'], ['hep-th-0307178-1-11-0', 'hep-th-0307178-2-12-0'], ['hep-th-0307178-1-11-1', 'hep-th-0307178-2-12-1'], ['hep-th-0307178-1-11-2', 'hep-th-0307178-2-12-2'], ['hep-th-0307178-1-11-4', 'hep-th-0307178-2-12-4'], ['hep-th-0307178-1-11-5', 'hep-th-0307178-2-12-5'], ['hep-th-0307178-1-24-0', 'hep-th-0307178-2-28-0'], ['hep-th-0307178-1-24-2', 'hep-th-0307178-2-28-2'], ['hep-th-0307178-1-24-3', 'hep-th-0307178-2-28-3'], ['hep-th-0307178-1-24-4', 'hep-th-0307178-2-28-4'], ['hep-th-0307178-1-41-1', 'hep-th-0307178-2-50-1'], ['hep-th-0307178-1-35-0', 'hep-th-0307178-2-44-0'], ['hep-th-0307178-1-35-1', 'hep-th-0307178-2-44-1'], ['hep-th-0307178-1-35-2', 'hep-th-0307178-2-44-2'], ['hep-th-0307178-1-31-0', 'hep-th-0307178-2-39-0'], ['hep-th-0307178-1-31-1', 'hep-th-0307178-2-39-1'], ['hep-th-0307178-1-31-2', 'hep-th-0307178-2-39-2'], ['hep-th-0307178-1-31-3', 'hep-th-0307178-2-39-3'], ['hep-th-0307178-1-39-0', 'hep-th-0307178-2-48-0'], ['hep-th-0307178-1-39-2', 'hep-th-0307178-2-48-2'], ['hep-th-0307178-1-39-3', 'hep-th-0307178-2-48-3'], ['hep-th-0307178-1-39-4', 'hep-th-0307178-2-48-4'], ['hep-th-0307178-1-39-5', 'hep-th-0307178-2-48-5'], ['hep-th-0307178-1-18-0', 'hep-th-0307178-2-20-0'], ['hep-th-0307178-1-18-1', 'hep-th-0307178-2-20-1'], ['hep-th-0307178-1-18-2', 'hep-th-0307178-2-20-2'], ['hep-th-0307178-1-18-3', 'hep-th-0307178-2-20-3'], ['hep-th-0307178-1-4-0', 'hep-th-0307178-2-3-0'], ['hep-th-0307178-1-4-2', 'hep-th-0307178-2-3-2'], ['hep-th-0307178-1-4-3', 'hep-th-0307178-2-3-3'], ['hep-th-0307178-1-4-5', 'hep-th-0307178-2-3-5'], ['hep-th-0307178-1-10-1', 'hep-th-0307178-2-11-1'], ['hep-th-0307178-1-3-0', 'hep-th-0307178-2-2-0'], ['hep-th-0307178-1-3-1', 'hep-th-0307178-2-2-1'], ['hep-th-0307178-1-44-0', 'hep-th-0307178-2-54-0'], ['hep-th-0307178-1-44-1', 'hep-th-0307178-2-54-1'], ['hep-th-0307178-1-44-2', 'hep-th-0307178-2-54-2'], ['hep-th-0307178-1-44-3', 'hep-th-0307178-2-54-3'], ['hep-th-0307178-1-44-4', 'hep-th-0307178-2-54-4'], ['hep-th-0307178-1-44-5', 'hep-th-0307178-2-54-5'], ['hep-th-0307178-1-44-6', 'hep-th-0307178-2-54-6'], ['hep-th-0307178-1-26-0', 'hep-th-0307178-2-30-0'], ['hep-th-0307178-1-26-1', 'hep-th-0307178-2-30-1'], ['hep-th-0307178-1-26-3', 'hep-th-0307178-2-31-0'], ['hep-th-0307178-1-26-4', 'hep-th-0307178-2-31-1'], ['hep-th-0307178-1-26-5', 'hep-th-0307178-2-31-2'], ['hep-th-0307178-1-28-1', 'hep-th-0307178-2-34-1'], ['hep-th-0307178-1-28-2', 'hep-th-0307178-2-36-0'], ['hep-th-0307178-1-28-3', 'hep-th-0307178-2-36-1'], ['hep-th-0307178-1-28-4', 'hep-th-0307178-2-36-2'], ['hep-th-0307178-1-28-5', 'hep-th-0307178-2-36-3'], ['hep-th-0307178-1-28-7', 'hep-th-0307178-2-36-5'], ['hep-th-0307178-1-36-0', 'hep-th-0307178-2-45-0'], ['hep-th-0307178-1-7-2', 'hep-th-0307178-2-7-2'], ['hep-th-0307178-1-34-0', 'hep-th-0307178-2-43-0'], ['hep-th-0307178-1-34-4', 'hep-th-0307178-2-43-4'], ['hep-th-0307178-1-23-2', 'hep-th-0307178-2-27-2'], ['hep-th-0307178-1-6-0', 'hep-th-0307178-2-6-0'], ['hep-th-0307178-1-6-1', 'hep-th-0307178-2-6-1'], ['hep-th-0307178-1-12-0', 'hep-th-0307178-2-13-0'], ['hep-th-0307178-1-49-1', 'hep-th-0307178-2-59-1'], ['hep-th-0307178-1-49-2', 'hep-th-0307178-2-59-2'], ['hep-th-0307178-1-46-2', 'hep-th-0307178-2-56-2'], ['hep-th-0307178-1-16-4', 'hep-th-0307178-2-18-4'], ['hep-th-0307178-1-22-2', 'hep-th-0307178-2-26-2'], ['hep-th-0307178-1-22-4', 'hep-th-0307178-2-26-4'], ['hep-th-0307178-1-22-7', 'hep-th-0307178-2-26-7'], ['hep-th-0307178-1-45-2', 'hep-th-0307178-2-55-2'], ['hep-th-0307178-1-25-0', 'hep-th-0307178-2-29-0'], ['hep-th-0307178-1-14-4', 'hep-th-0307178-2-16-4'], ['hep-th-0307178-1-2-0', 'hep-th-0307178-2-1-0'], ['hep-th-0307178-1-2-2', 'hep-th-0307178-2-1-2'], ['hep-th-0307178-1-33-1', 'hep-th-0307178-2-42-1'], ['hep-th-0307178-1-13-4', 'hep-th-0307178-2-15-4'], ['hep-th-0307178-1-11-3', 'hep-th-0307178-2-12-3'], ['hep-th-0307178-1-24-1', 'hep-th-0307178-2-28-1'], ['hep-th-0307178-1-41-0', 'hep-th-0307178-2-50-0'], ['hep-th-0307178-1-39-1', 'hep-th-0307178-2-48-1'], ['hep-th-0307178-1-4-1', 'hep-th-0307178-2-3-1'], ['hep-th-0307178-1-4-4', 'hep-th-0307178-2-3-4'], ['hep-th-0307178-1-10-0', 'hep-th-0307178-2-11-0'], ['hep-th-0307178-1-10-2', 'hep-th-0307178-2-11-2'], ['hep-th-0307178-1-28-6', 'hep-th-0307178-2-36-4'], ['hep-th-0307178-1-46-3', 'hep-th-0307178-2-56-3']]

[['hep-th-0307178-1-36-1', 'hep-th-0307178-2-45-1'], ['hep-th-0307178-1-43-1', 'hep-th-0307178-2-53-1'], ['hep-th-0307178-1-43-2', 'hep-th-0307178-2-53-2'], ['hep-th-0307178-1-43-3', 'hep-th-0307178-2-53-3'], ['hep-th-0307178-1-43-4', 'hep-th-0307178-2-53-4'], ['hep-th-0307178-1-7-0', 'hep-th-0307178-2-7-0'], ['hep-th-0307178-1-7-1', 'hep-th-0307178-2-7-1'], ['hep-th-0307178-1-7-3', 'hep-th-0307178-2-7-3'], ['hep-th-0307178-1-34-1', 'hep-th-0307178-2-43-1'], ['hep-th-0307178-1-34-2', 'hep-th-0307178-2-43-2'], ['hep-th-0307178-1-34-3', 'hep-th-0307178-2-43-3'], ['hep-th-0307178-1-34-6', 'hep-th-0307178-2-43-6'], ['hep-th-0307178-1-34-7', 'hep-th-0307178-2-43-7'], ['hep-th-0307178-1-34-8', 'hep-th-0307178-2-43-8'], ['hep-th-0307178-1-15-0', 'hep-th-0307178-2-17-0'], ['hep-th-0307178-1-15-1', 'hep-th-0307178-2-17-1'], ['hep-th-0307178-1-15-3', 'hep-th-0307178-2-17-3'], ['hep-th-0307178-1-15-4', 'hep-th-0307178-2-17-4'], ['hep-th-0307178-1-15-5', 'hep-th-0307178-2-17-5'], ['hep-th-0307178-1-15-6', 'hep-th-0307178-2-17-6'], ['hep-th-0307178-1-15-7', 'hep-th-0307178-2-17-7'], ['hep-th-0307178-1-15-8', 'hep-th-0307178-2-17-8'], ['hep-th-0307178-1-20-0', 'hep-th-0307178-2-24-0'], ['hep-th-0307178-1-23-0', 'hep-th-0307178-2-27-0'], ['hep-th-0307178-1-23-1', 'hep-th-0307178-2-27-1'], ['hep-th-0307178-1-23-3', 'hep-th-0307178-2-27-3'], ['hep-th-0307178-1-23-5', 'hep-th-0307178-2-27-5'], ['hep-th-0307178-1-5-0', 'hep-th-0307178-2-5-0'], ['hep-th-0307178-1-5-1', 'hep-th-0307178-2-5-1'], ['hep-th-0307178-1-5-2', 'hep-th-0307178-2-5-2'], ['hep-th-0307178-1-5-3', 'hep-th-0307178-2-5-3'], ['hep-th-0307178-1-6-2', 'hep-th-0307178-2-6-2'], ['hep-th-0307178-1-48-0', 'hep-th-0307178-2-58-0'], ['hep-th-0307178-1-48-1', 'hep-th-0307178-2-58-1'], ['hep-th-0307178-1-48-2', 'hep-th-0307178-2-58-2'], ['hep-th-0307178-1-48-3', 'hep-th-0307178-2-58-3'], ['hep-th-0307178-1-48-4', 'hep-th-0307178-2-58-4'], ['hep-th-0307178-1-42-0', 'hep-th-0307178-2-51-0'], ['hep-th-0307178-1-42-1', 'hep-th-0307178-2-51-1'], ['hep-th-0307178-1-42-2', 'hep-th-0307178-2-51-2'], ['hep-th-0307178-1-21-0', 'hep-th-0307178-2-25-0'], ['hep-th-0307178-1-21-1', 'hep-th-0307178-2-25-1'], ['hep-th-0307178-1-12-1', 'hep-th-0307178-2-13-1'], ['hep-th-0307178-1-12-2', 'hep-th-0307178-2-13-2'], ['hep-th-0307178-1-12-3', 'hep-th-0307178-2-13-3'], ['hep-th-0307178-1-12-4', 'hep-th-0307178-2-13-4'], ['hep-th-0307178-1-12-5', 'hep-th-0307178-2-13-5'], ['hep-th-0307178-1-12-6', 'hep-th-0307178-2-13-6'], ['hep-th-0307178-1-12-7', 'hep-th-0307178-2-13-7'], ['hep-th-0307178-1-47-0', 'hep-th-0307178-2-57-0'], ['hep-th-0307178-1-47-1', 'hep-th-0307178-2-57-1'], ['hep-th-0307178-1-47-2', 'hep-th-0307178-2-57-2'], ['hep-th-0307178-1-32-0', 'hep-th-0307178-2-40-0'], ['hep-th-0307178-1-32-1', 'hep-th-0307178-2-40-1'], ['hep-th-0307178-1-32-2', 'hep-th-0307178-2-40-2'], ['hep-th-0307178-1-32-3', 'hep-th-0307178-2-40-3'], ['hep-th-0307178-1-49-0', 'hep-th-0307178-2-59-0'], ['hep-th-0307178-1-49-3', 'hep-th-0307178-2-59-3'], ['hep-th-0307178-1-46-0', 'hep-th-0307178-2-56-0'], ['hep-th-0307178-1-46-1', 'hep-th-0307178-2-56-1'], ['hep-th-0307178-1-46-5', 'hep-th-0307178-2-56-4'], ['hep-th-0307178-1-16-0', 'hep-th-0307178-2-18-0'], ['hep-th-0307178-1-16-1', 'hep-th-0307178-2-18-1'], ['hep-th-0307178-1-16-2', 'hep-th-0307178-2-18-2'], ['hep-th-0307178-1-16-3', 'hep-th-0307178-2-18-3'], ['hep-th-0307178-1-22-0', 'hep-th-0307178-2-26-0'], ['hep-th-0307178-1-22-1', 'hep-th-0307178-2-26-1'], ['hep-th-0307178-1-22-3', 'hep-th-0307178-2-26-3'], ['hep-th-0307178-1-22-5', 'hep-th-0307178-2-26-5'], ['hep-th-0307178-1-22-6', 'hep-th-0307178-2-26-6'], ['hep-th-0307178-1-22-8', 'hep-th-0307178-2-26-8'], ['hep-th-0307178-1-22-9', 'hep-th-0307178-2-26-9'], ['hep-th-0307178-1-22-10', 'hep-th-0307178-2-26-10'], ['hep-th-0307178-1-30-0', 'hep-th-0307178-2-38-0'], ['hep-th-0307178-1-30-1', 'hep-th-0307178-2-38-1'], ['hep-th-0307178-1-45-0', 'hep-th-0307178-2-55-0'], ['hep-th-0307178-1-45-1', 'hep-th-0307178-2-55-1'], ['hep-th-0307178-1-8-1', 'hep-th-0307178-2-9-1'], ['hep-th-0307178-1-25-1', 'hep-th-0307178-2-29-1'], ['hep-th-0307178-1-25-2', 'hep-th-0307178-2-29-2'], ['hep-th-0307178-1-25-3', 'hep-th-0307178-2-29-3'], ['hep-th-0307178-1-25-4', 'hep-th-0307178-2-29-4'], ['hep-th-0307178-1-14-0', 'hep-th-0307178-2-16-0'], ['hep-th-0307178-1-14-1', 'hep-th-0307178-2-16-1'], ['hep-th-0307178-1-14-2', 'hep-th-0307178-2-16-2'], ['hep-th-0307178-1-14-3', 'hep-th-0307178-2-16-3'], ['hep-th-0307178-1-40-0', 'hep-th-0307178-2-49-0'], ['hep-th-0307178-1-27-0', 'hep-th-0307178-2-32-0'], ['hep-th-0307178-1-27-1', 'hep-th-0307178-2-32-1'], ['hep-th-0307178-1-27-2', 'hep-th-0307178-2-32-2'], ['hep-th-0307178-1-19-1', 'hep-th-0307178-2-22-1'], ['hep-th-0307178-1-2-1', 'hep-th-0307178-2-1-1'], ['hep-th-0307178-1-38-0', 'hep-th-0307178-2-47-0'], ['hep-th-0307178-1-38-1', 'hep-th-0307178-2-47-1'], ['hep-th-0307178-1-38-2', 'hep-th-0307178-2-47-2'], ['hep-th-0307178-1-9-0', 'hep-th-0307178-2-10-0'], ['hep-th-0307178-1-9-1', 'hep-th-0307178-2-10-1'], ['hep-th-0307178-1-33-2', 'hep-th-0307178-2-42-2'], ['hep-th-0307178-1-29-0', 'hep-th-0307178-2-37-0'], ['hep-th-0307178-1-29-1', 'hep-th-0307178-2-37-1'], ['hep-th-0307178-1-29-2', 'hep-th-0307178-2-37-2'], ['hep-th-0307178-1-29-3', 'hep-th-0307178-2-37-3'], ['hep-th-0307178-1-29-4', 'hep-th-0307178-2-37-4'], ['hep-th-0307178-1-29-5', 'hep-th-0307178-2-37-5'], ['hep-th-0307178-1-29-6', 'hep-th-0307178-2-37-6'], ['hep-th-0307178-1-29-7', 'hep-th-0307178-2-37-7'], ['hep-th-0307178-1-13-1', 'hep-th-0307178-2-15-1'], ['hep-th-0307178-1-13-2', 'hep-th-0307178-2-15-2'], ['hep-th-0307178-1-13-3', 'hep-th-0307178-2-15-3'], ['hep-th-0307178-1-13-5', 'hep-th-0307178-2-15-5'], ['hep-th-0307178-1-13-6', 'hep-th-0307178-2-15-6'], ['hep-th-0307178-1-13-7', 'hep-th-0307178-2-15-7'], ['hep-th-0307178-1-17-0', 'hep-th-0307178-2-19-0'], ['hep-th-0307178-1-11-0', 'hep-th-0307178-2-12-0'], ['hep-th-0307178-1-11-1', 'hep-th-0307178-2-12-1'], ['hep-th-0307178-1-11-2', 'hep-th-0307178-2-12-2'], ['hep-th-0307178-1-11-4', 'hep-th-0307178-2-12-4'], ['hep-th-0307178-1-11-5', 'hep-th-0307178-2-12-5'], ['hep-th-0307178-1-24-0', 'hep-th-0307178-2-28-0'], ['hep-th-0307178-1-24-2', 'hep-th-0307178-2-28-2'], ['hep-th-0307178-1-24-3', 'hep-th-0307178-2-28-3'], ['hep-th-0307178-1-24-4', 'hep-th-0307178-2-28-4'], ['hep-th-0307178-1-41-1', 'hep-th-0307178-2-50-1'], ['hep-th-0307178-1-35-0', 'hep-th-0307178-2-44-0'], ['hep-th-0307178-1-35-1', 'hep-th-0307178-2-44-1'], ['hep-th-0307178-1-35-2', 'hep-th-0307178-2-44-2'], ['hep-th-0307178-1-31-0', 'hep-th-0307178-2-39-0'], ['hep-th-0307178-1-31-1', 'hep-th-0307178-2-39-1'], ['hep-th-0307178-1-31-2', 'hep-th-0307178-2-39-2'], ['hep-th-0307178-1-31-3', 'hep-th-0307178-2-39-3'], ['hep-th-0307178-1-39-0', 'hep-th-0307178-2-48-0'], ['hep-th-0307178-1-39-2', 'hep-th-0307178-2-48-2'], ['hep-th-0307178-1-39-3', 'hep-th-0307178-2-48-3'], ['hep-th-0307178-1-39-4', 'hep-th-0307178-2-48-4'], ['hep-th-0307178-1-39-5', 'hep-th-0307178-2-48-5'], ['hep-th-0307178-1-18-0', 'hep-th-0307178-2-20-0'], ['hep-th-0307178-1-18-1', 'hep-th-0307178-2-20-1'], ['hep-th-0307178-1-18-2', 'hep-th-0307178-2-20-2'], ['hep-th-0307178-1-18-3', 'hep-th-0307178-2-20-3'], ['hep-th-0307178-1-4-0', 'hep-th-0307178-2-3-0'], ['hep-th-0307178-1-4-2', 'hep-th-0307178-2-3-2'], ['hep-th-0307178-1-4-3', 'hep-th-0307178-2-3-3'], ['hep-th-0307178-1-4-5', 'hep-th-0307178-2-3-5'], ['hep-th-0307178-1-10-1', 'hep-th-0307178-2-11-1'], ['hep-th-0307178-1-3-0', 'hep-th-0307178-2-2-0'], ['hep-th-0307178-1-3-1', 'hep-th-0307178-2-2-1'], ['hep-th-0307178-1-44-0', 'hep-th-0307178-2-54-0'], ['hep-th-0307178-1-44-1', 'hep-th-0307178-2-54-1'], ['hep-th-0307178-1-44-2', 'hep-th-0307178-2-54-2'], ['hep-th-0307178-1-44-3', 'hep-th-0307178-2-54-3'], ['hep-th-0307178-1-44-4', 'hep-th-0307178-2-54-4'], ['hep-th-0307178-1-44-5', 'hep-th-0307178-2-54-5'], ['hep-th-0307178-1-44-6', 'hep-th-0307178-2-54-6'], ['hep-th-0307178-1-26-0', 'hep-th-0307178-2-30-0'], ['hep-th-0307178-1-26-1', 'hep-th-0307178-2-30-1'], ['hep-th-0307178-1-26-3', 'hep-th-0307178-2-31-0'], ['hep-th-0307178-1-26-4', 'hep-th-0307178-2-31-1'], ['hep-th-0307178-1-26-5', 'hep-th-0307178-2-31-2'], ['hep-th-0307178-1-28-1', 'hep-th-0307178-2-34-1'], ['hep-th-0307178-1-28-2', 'hep-th-0307178-2-36-0'], ['hep-th-0307178-1-28-3', 'hep-th-0307178-2-36-1'], ['hep-th-0307178-1-28-4', 'hep-th-0307178-2-36-2'], ['hep-th-0307178-1-28-5', 'hep-th-0307178-2-36-3'], ['hep-th-0307178-1-28-7', 'hep-th-0307178-2-36-5']]

[['hep-th-0307178-1-36-0', 'hep-th-0307178-2-45-0'], ['hep-th-0307178-1-7-2', 'hep-th-0307178-2-7-2'], ['hep-th-0307178-1-34-0', 'hep-th-0307178-2-43-0'], ['hep-th-0307178-1-34-4', 'hep-th-0307178-2-43-4'], ['hep-th-0307178-1-23-2', 'hep-th-0307178-2-27-2'], ['hep-th-0307178-1-6-0', 'hep-th-0307178-2-6-0'], ['hep-th-0307178-1-6-1', 'hep-th-0307178-2-6-1'], ['hep-th-0307178-1-12-0', 'hep-th-0307178-2-13-0'], ['hep-th-0307178-1-49-1', 'hep-th-0307178-2-59-1'], ['hep-th-0307178-1-49-2', 'hep-th-0307178-2-59-2'], ['hep-th-0307178-1-46-2', 'hep-th-0307178-2-56-2'], ['hep-th-0307178-1-16-4', 'hep-th-0307178-2-18-4'], ['hep-th-0307178-1-22-2', 'hep-th-0307178-2-26-2'], ['hep-th-0307178-1-22-4', 'hep-th-0307178-2-26-4'], ['hep-th-0307178-1-22-7', 'hep-th-0307178-2-26-7'], ['hep-th-0307178-1-45-2', 'hep-th-0307178-2-55-2'], ['hep-th-0307178-1-25-0', 'hep-th-0307178-2-29-0'], ['hep-th-0307178-1-14-4', 'hep-th-0307178-2-16-4'], ['hep-th-0307178-1-2-0', 'hep-th-0307178-2-1-0'], ['hep-th-0307178-1-2-2', 'hep-th-0307178-2-1-2'], ['hep-th-0307178-1-33-1', 'hep-th-0307178-2-42-1'], ['hep-th-0307178-1-13-4', 'hep-th-0307178-2-15-4'], ['hep-th-0307178-1-11-3', 'hep-th-0307178-2-12-3'], ['hep-th-0307178-1-24-1', 'hep-th-0307178-2-28-1'], ['hep-th-0307178-1-41-0', 'hep-th-0307178-2-50-0'], ['hep-th-0307178-1-39-1', 'hep-th-0307178-2-48-1'], ['hep-th-0307178-1-4-1', 'hep-th-0307178-2-3-1'], ['hep-th-0307178-1-4-4', 'hep-th-0307178-2-3-4'], ['hep-th-0307178-1-10-0', 'hep-th-0307178-2-11-0'], ['hep-th-0307178-1-10-2', 'hep-th-0307178-2-11-2'], ['hep-th-0307178-1-28-6', 'hep-th-0307178-2-36-4']]

[]

[['hep-th-0307178-1-46-3', 'hep-th-0307178-2-56-3']]

[]

['hep-th-0307178-1-15-2', 'hep-th-0307178-1-17-1', 'hep-th-0307178-1-23-4', 'hep-th-0307178-1-26-2', 'hep-th-0307178-1-34-5', 'hep-th-0307178-1-37-0', 'hep-th-0307178-1-37-1', 'hep-th-0307178-1-41-2', 'hep-th-0307178-2-4-0', 'hep-th-0307178-2-17-2', 'hep-th-0307178-2-19-1', 'hep-th-0307178-2-23-0', 'hep-th-0307178-2-27-4', 'hep-th-0307178-2-30-2', 'hep-th-0307178-2-35-0', 'hep-th-0307178-2-43-5', 'hep-th-0307178-2-46-0', 'hep-th-0307178-2-46-1', 'hep-th-0307178-2-50-2']

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

https://arxiv.org/abs/hep-th/0307178

1805.08659

{'1805.08659-1-0-0': 'We explore Two Higgs Doublet Models with non-standard flavor structures.', '1805.08659-1-0-1': 'In analogy to the four, well studied, models with natural flavor conservation (type 1, type 2, lepton-specific, flipped), we identify four models that preserve an approximate [MATH] flavor symmetry acting on the first two generations.', '1805.08659-1-0-2': 'In all four models, the couplings of the 125 GeV Higgs are modified in characteristic flavor non-universal ways.', '1805.08659-1-0-3': 'The heavy neutral and charged Higgs bosons show an interesting non-standard phenomenology.', '1805.08659-1-0-4': 'We discuss their production and decay modes and identify the most sensitive search channels at the LHC.', '1805.08659-1-0-5': 'We study the effects on low energy flavor violating processes finding relevant constraints from [MATH] and [MATH] meson oscillations and from the rare decay [MATH].', '1805.08659-1-0-6': 'We also find that lepton flavor violating [MATH] meson decays like [MATH] and [MATH] might have branching ratios at an observable level.', '1805.08659-1-1-0': '# Introduction', '1805.08659-1-2-0': 'Measurements of Higgs rates at the LHC show that the Standard Model (SM) Higgs mechanism provides the bulk of the masses of the third generation fermions.', '1805.08659-1-2-1': 'The decay [MATH] has been observed at a rate compatible with the SM prediction [CITATION].', '1805.08659-1-2-2': 'Similarly, evidence exists for a SM-like [MATH] decay [CITATION].', '1805.08659-1-2-3': 'Recently, production of the Higgs in association with top quarks has been observed in agreement with the SM [CITATION].', '1805.08659-1-3-0': 'Much less is known about the origin of the first and second generation masses.', '1805.08659-1-3-1': 'With the exception of the muon, direct measurements of Higgs couplings to the light fermions are extremely challenging.', '1805.08659-1-3-2': 'It is therefore unknown if the light fermions obtain their mass from the Higgs boson.', '1805.08659-1-3-3': 'A complementary approach to probe the origin of light fermion masses is to search for signatures of alternatives to the SM Higgs mechanism in which the light fermion masses originate from a new source of electroweak symmetry breaking.', '1805.08659-1-3-4': 'The simplest realization of such a setup is the Two Higgs Doublet Model (2HDM).', '1805.08659-1-4-0': 'In [CITATION] a 2HDM setup was proposed in which one Higgs doublet couples only to the third generation fermions, and a second Higgs doublet couples mainly to the first and second generation (see also [CITATION]).', '1805.08659-1-4-1': 'A dynamical generation of such a coupling structure can be achieved using the flavor-locking mechanism [CITATION].', '1805.08659-1-4-2': 'The collider phenomenology of this "flavorful" 2HDM scenario was discussed in [CITATION].', '1805.08659-1-5-0': 'The proposed 2HDM goes beyond the principle of natural flavor conservation (NFC) [CITATION] and introduces flavor changing neutral currents (FCNCs) at tree level.', '1805.08659-1-5-1': 'However, the Yukawa couplings of the first Higgs doublet to the third generation preserve a [MATH] flavor symmetry, which is only broken by the small couplings of the second Higgs doublet.', '1805.08659-1-5-2': 'The approximate [MATH] symmetry protects the most sensitive flavor violating transitions between the second and first generation.', '1805.08659-1-6-0': 'In this work we explore additional flavor structures for 2HDMs that approximately preserve a [MATH] flavor symmetry for the first two generations.', '1805.08659-1-6-1': 'Starting from the flavorful 2HDM scenario of [CITATION] we "twist" the Yukawa couplings of the down-type quarks and/or leptons by exchanging the Higgs doublets these fermions couple to.', '1805.08659-1-6-2': 'In analogy to the four well studied 2HDMs with natural flavor conservation (type 1, type 2, lepton-specific, flipped) we obtain four flavorful 2HDMs in which the third and first two generations of each fermion type (up-type quarks, down-type quarks, leptons) obtain the bulk of their mass from a different source.', '1805.08659-1-6-3': 'The non-standard flavor structures of these four 2HDMs lead to (i) distinct, flavor non-universal modifications of all Higgs couplings with respect to the models with NFC, (ii) potentially sizable flavor violating Higgs couplings involving the third generation fermions.', '1805.08659-1-6-4': 'This implies an interesting characteristic collider and flavor phenomenology.', '1805.08659-1-6-5': '(For recent work on 2HDMs with other non-standard flavor structures see [CITATION].)', '1805.08659-1-7-0': 'The paper is structured as follows: In section [REF] we introduce the four flavorful 2HDMs, discuss the Yukawa textures and the couplings of the fermions to the various Higgs boson mass eigenstates.', '1805.08659-1-7-1': 'In section [REF] we consider the phenomenology of the 125 GeV Higgs boson, comparing the predicted production and decay rates in our models to measurements at the LHC.', '1805.08659-1-7-2': 'In sections [REF] and [REF] we evaluate the production cross sections and decay branching ratios of the heavy neutral and charged Higgs bosons.', '1805.08659-1-7-3': 'We then compare the model predictions to the limits from current searches for extra Higgs bosons that are being performed at the LHC and identify the most sensitive collider probes of the models.', '1805.08659-1-7-4': 'In section [REF] we investigate the characteristic effects of the new sources of flavor violation on low energy flavor violating processes such as meson mixing and rare [MATH] meson decays.', '1805.08659-1-7-5': 'We conclude in section [REF].', '1805.08659-1-8-0': '# Flavorful Two Higgs Doublet Models', '1805.08659-1-9-0': 'One of the simplest realizations of a viable alternative framework of mass generation are 2HDMs with one doublet coupling only to the third generation, and a second doublet coupling mainly to the first and second generation.', '1805.08659-1-9-1': 'Such a setup was proposed in [CITATION] (see also [CITATION]).', '1805.08659-1-9-2': 'The masses of the SM fermions arise from two sources, the vacuum expectation values of two Higgs doublets [MATH] and [MATH].', '1805.08659-1-9-3': 'The relevant part of the 2HDM Lagrangian is [EQUATION] where [MATH].', '1805.08659-1-9-4': 'The three generations of quark and lepton doublets are denoted by [MATH], [MATH], and [MATH], [MATH], [MATH] are the up quark, down quark, and charged lepton singlets.', '1805.08659-1-9-5': 'The [MATH] and [MATH] matrices are the Yukawa couplings.', '1805.08659-1-10-0': 'The above setup for the Higgs couplings violates the principle of natural flavor conservation.', '1805.08659-1-10-1': 'Both of the Higgs doublets couple to the leptons, the up-type quarks, and the down-type quarks, leading to FCNCs at tree level.', '1805.08659-1-11-0': '## Yukawa textures', '1805.08659-1-12-0': 'We are interested in Yukawa couplings beyond NFC that do not introduce an unacceptably large amount of flavor violation.', '1805.08659-1-12-1': 'This can be achieved by demanding that one set of the Yukawa couplings preserves a [MATH] flavor symmetry, acting on the first two generations.', '1805.08659-1-12-2': 'In this case, flavor transitions between the first and second generation are protected.', '1805.08659-1-12-3': 'Such transitions are absent at first order in flavor symmetry breaking and arise only at second order as an effective [MATH] transition.', '1805.08659-1-12-4': 'As we will discuss in section [REF], effects in neutral Kaon and [MATH] meson oscillations are indeed typically well below present constraints.', '1805.08659-1-13-0': 'We consider the following set of Yukawa matrices in the flavor basis [EQUATION]', '1805.08659-1-13-1': 'Due to the rank 1 nature of the [MATH], [MATH], [MATH] Yukawa couplings, the [MATH] flavor symmetry acting on the first two generations is only broken by the small [MATH], [MATH], [MATH] Yukawa couplings.', '1805.08659-1-13-2': 'Such a pattern of textures can be obtained using for example the flavor locking mechanism [CITATION].', '1805.08659-1-13-3': 'The entries in the Yukawa couplings of the first and second generations are chosen such that the mass eigenvalues reproduce the observed values without any tuning.', '1805.08659-1-13-4': 'The structure in the down sector leads naturally to the observed pattern in the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.', '1805.08659-1-13-5': 'Alternatively, the CKM matrix could also be generated in the up sector, but we will not consider this option here.', '1805.08659-1-13-6': 'The entries in the above matrices are given up to [MATH] factors that, in all generality, can be complex.', '1805.08659-1-14-0': 'The vacuum expectation values [MATH] in Eqs. ([REF]) - ([REF]) correspond to either [MATH] or [MATH], depending on the model under consideration.', '1805.08659-1-14-1': 'The Yukawa couplings for the third or first two generations are identified with the [MATH] and [MATH] couplings introduced in Eq. ([REF]), accordingly.', '1805.08659-1-14-2': 'Without loss of generality, we denote the Higgs doublet that couples to the top quark with [MATH] [CITATION], i.e. [MATH], [MATH] and [MATH], [MATH].', '1805.08659-1-14-3': 'This leaves us with four distinct "flavorful" possibilities to assign the two Higgs doublets to the down-quarks and leptons.', '1805.08659-1-14-4': 'In analogy to the four well known 2HDMs with natural flavor conservation (that we refer to as type 1A, type 2A, lepton-specific A, and flipped A, in the following) we denote our four flavorful models as type 1B, type 2B, lepton-specific B, and flipped B.', '1805.08659-1-14-5': 'The type 1B model was studied in some detail in [CITATION].', '1805.08659-1-14-6': 'The coupling structure of all four flavorful models is summarized in table [REF].', '1805.08659-1-15-0': 'Rotating the fermions into mass eigenstates, we define the following mass parameters [EQUATION] with quark mass eigenstates [MATH].', '1805.08659-1-15-1': 'These mass parameters obey [MATH], where [MATH] are the observed up-type quark masses.', '1805.08659-1-15-2': 'Analogous definitions and identities hold for the down-type quarks and the charged leptons.', '1805.08659-1-15-3': 'We find the following values for the up mass parameters in all four types of flavorful models [EQUATION]', '1805.08659-1-15-4': 'For leptons we find analogous expressions for the off-diagonal mass parameters in all four types [EQUATION]', '1805.08659-1-15-5': 'Finally, for the down quarks we find for all four types [EQUATION]', '1805.08659-1-15-6': 'As we assume that the CKM matrix is generated in the down sector, the CKM elements [MATH] and [MATH] appear in several of the down-type mass parameters.', '1805.08659-1-16-0': 'The [MATH] terms in the above expressions are free parameters that in general can be complex.', '1805.08659-1-16-1': 'It is worth noting that due to those [MATH] terms, the off-diagonal mass parameters [MATH] and [MATH] need not be the same for any type of fermion.', '1805.08659-1-16-2': 'It is also important to note that in all cases the mass parameters that are responsible for flavor mixing between the first and second generation are suppressed by small mass ratios and not independent from the mass entries that parameterize mixing with the third generation.', '1805.08659-1-16-3': 'All [MATH] mixing is given by an effective [MATH] mixing.', '1805.08659-1-16-4': 'This is a consequence of the breaking of the [MATH] symmetry by only one set of Yukawa couplings.', '1805.08659-1-17-0': '## Couplings of the Higgs bosons', '1805.08659-1-18-0': 'Next, we discuss the couplings of the physical Higgs bosons in the four different models.', '1805.08659-1-18-1': 'We largely follow the notation and conventions in [CITATION] and state only the relevant results.', '1805.08659-1-19-0': 'The part of the Lagrangian that parametrizes the couplings to the three neutral scalars, [MATH], [MATH], and [MATH] (we identify [MATH] with the 125 GeV Higgs), as well as the charged Higgs [MATH] to mass eigenstate fermions is written as [EQUATION]', '1805.08659-1-19-1': 'For the flavor diagonal and off-diagonal couplings of the neutral Higgs bosons to leptons one finds [EQUATION] where we introduced the coupling modifiers [MATH] with respect to the SM Higgs couplings.', '1805.08659-1-19-2': 'We use the notation [MATH], [MATH], and [MATH].', '1805.08659-1-19-3': 'The angle [MATH] parametrizes the mixing between the neutral CP-even components of the two Higgs doublets and [MATH] is the ratio of Higgs vacuum expectation values.', '1805.08659-1-19-4': 'Completely analogous expressions hold for the neutral Higgs couplings to the up-type and down-type quarks.', '1805.08659-1-20-0': 'Ignoring neutrino mixing (which is of no relevance for our study) one finds for the charged Higgs couplings to leptons [EQUATION]', '1805.08659-1-20-1': 'In the expressions for the charged Higgs couplings to quarks, the CKM matrix [MATH] enters.', '1805.08659-1-20-2': 'We find [EQUATION]', '1805.08659-1-20-3': 'All of these expressions for the couplings are completely generic and can be applied to any of our flavorful models.', '1805.08659-1-20-4': 'The only terms that change in the different models are the [MATH] mass parameters, as given in Eqs. ([REF]), ([REF]), and ([REF]).', '1805.08659-1-21-0': 'In tables [REF], [REF], and [REF], we show the leading order coupling modifiers for the flavor diagonal couplings of the Higgs bosons [MATH] as an expansion in [MATH], where [MATH].', '1805.08659-1-21-1': 'We compare the coupling modifiers of all four flavorful 2HDM types to those of the four 2HDM types with natural flavor conservation.', '1805.08659-1-21-2': 'As is well known, the coupling modifiers are flavor universal in the models with natural flavor conservation.', '1805.08659-1-21-3': 'In the flavorful models the modifiers are flavor dependent and differentiate between the third generation and the first two generations.', '1805.08659-1-22-0': '# Light Higgs phenomenology', '1805.08659-1-23-0': '## Constraints from Higgs signal strength measurements', '1805.08659-1-24-0': 'The introduction of a second doublet alters the couplings to the 125 GeV Higgs boson [MATH] as shown in table [REF] as well as Eq. ([REF]).', '1805.08659-1-24-1': 'We can compare the Higgs production and decay rates predicted by our models to those measured by ATLAS and CMS in order to constrain the new physics parameter space.', '1805.08659-1-25-0': 'To determine the constraints from the measured Higgs signals we construct a [MATH] function [EQUATION] where [MATH], [MATH], and [MATH] are the experimental measurements, the Standard Model predictions, and flavorful 2HDM predictions for the production cross sections times branching ratio of the various measured channels.', '1805.08659-1-26-0': 'The ratios of experimental measurements and SM predictions that enter Eq. ([REF]) are given by the signal strength modifiers that are reported by ATLAS and CMS.', '1805.08659-1-26-1': 'The SM predictions for the production cross sections and branching ratios are taken from [CITATION].', '1805.08659-1-26-2': 'The ratios of BSM and SM predictions for individual channels can be obtained in a straight-forward way as functions of the coupling modifiers.', '1805.08659-1-26-3': 'For the gluon-gluon fusion production (ggf), vector boson fusion production (VBF), production in association with [MATH] and [MATH] bosons (Wh, Zh), and production in association with top quarks (tth), we have [EQUATION] where for the loop induced gluon-gluon fusion we take into account top and bottom contributions at 1-loop.', '1805.08659-1-26-4': 'For tree level decays, the partial widths simply scale with the appropriate coupling modifiers.', '1805.08659-1-26-5': 'In the case of the loop induced [MATH] decay width we take into account top and bottom contributions at 1-loop (We explicitly checked that loops with lighter quarks do not lead to any appreciable effects).', '1805.08659-1-26-6': 'For [MATH] we consider [MATH], top, and bottom loops.', '1805.08659-1-26-7': 'We neglect charged Higgs loops, that are typically tiny [CITATION] [EQUATION]', '1805.08659-1-26-8': 'The covariance matrix in Eq. ([REF]) contains the experimental uncertainties and (where available) the correlations among the uncertainties.', '1805.08659-1-26-9': 'We assume that theory uncertainties in the ratio of BSM and SM predictions are negligible compared to current experimental uncertainties.', '1805.08659-1-26-10': 'We take into account the Higgs signal strengths from the LHC run 1 combination [CITATION], as well as several individual run 2 results, in particular measurements of [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], and [MATH] [CITATION].', '1805.08659-1-26-11': 'We also include results on Higgs production is association with top quarks [CITATION].', '1805.08659-1-26-12': '(See [CITATION] for recent Higgs signal strength studies of 2HDMs with natural flavor conservation.)', '1805.08659-1-27-0': 'The couplings of [MATH] are largely determined by the parameters [MATH] and [MATH].', '1805.08659-1-27-1': 'Subleading corrections enter through the [MATH] mass parameters, see Eq. ([REF]).', '1805.08659-1-27-2': 'We use the [MATH] function to put constraints on the [MATH] and [MATH] parameters allowing the [MATH] coefficients in the subleading corrections to vary in the range [MATH].', '1805.08659-1-27-3': 'The allowed regions in the [MATH] vs. [MATH] plane that we obtain in this way are shown in Fig. [REF].', '1805.08659-1-27-4': 'The dark (light) green regions correspond to the [MATH] and [MATH] allowed regions (that we define as [MATH]) in the four flavorful models.', '1805.08659-1-27-5': 'We also compare these regions to the 2[MATH] constraint in the corresponding models with natural flavor conservation (dashed contours).', '1805.08659-1-27-6': 'The plot for the type 1B model updates the corresponding result in [CITATION].', '1805.08659-1-28-0': 'The couplings of the 125 GeV Higgs to the third generation fermions are already constrained by current data to be SM-like at the level of 10% - 20%.', '1805.08659-1-28-1': 'By coupling the [MATH] and/or [MATH] to the second doublet (as in the type 2B, lepton-specific B, and flipped B models), we therefore find the parameter space to be more strongly constrained than in the type 1B model.', '1805.08659-1-28-2': 'Note that in those models there are two distinct regions of parameter space: one region close to the alignment limit [MATH], where the mixing between the 125 GeV Higgs and heavy Higgs is tiny and all [MATH] couplings become SM-like, and a second narrow strip where the bottom and/or the tau coupling have opposite sign with respect to the SM prediction.', '1805.08659-1-28-3': 'The constraints for the type 2B and flipped B models are very similar, implying that the bottom coupling (which largely determines the total width of [MATH]) is the most important factor in determining the parameter space of these models.', '1805.08659-1-28-4': 'Generally, as [MATH] gets very large or very small the [MATH] values can deviate substantially from 1, resulting in strong constraints.', '1805.08659-1-28-5': 'Moderate values for [MATH] are the least constrained.', '1805.08659-1-29-0': 'Currently, the only decay of the Higgs into a non-third generation fermion which has been constrained in a relevant way at the LHC is the decay to [MATH] [CITATION].', '1805.08659-1-29-1': 'However, the current sensitivities to the [MATH] decay are not sufficient to impose strong constraints on our parameter space, yet.', '1805.08659-1-29-2': 'Future precise measurements of [MATH] can potentially constrain large parts of the open parameter space of the type 1B model.', '1805.08659-1-29-3': 'The type 2B and flipped B models will be mainly constrained by improved measurements of [MATH].', '1805.08659-1-29-4': 'For the lepton-specific B model, future precision measurements of [MATH] will give the most relevant constraints.', '1805.08659-1-30-0': '## Flavor violating decays', '1805.08659-1-31-0': 'Along with altering the flavor diagonal couplings of the light Higgs, the introduction of the second doublet also introduces flavor violating couplings of [MATH] to the fermions.', '1805.08659-1-31-1': 'We expect in our models a number of FCNC decays that are extremely suppressed in the SM, most notably rare top decays [MATH] and [MATH] as well as lepton flavor violating Higgs decays [MATH], [MATH], and [MATH].', '1805.08659-1-32-0': 'In all four flavorful models the branching ratio of [MATH] is given by [EQUATION]', '1805.08659-1-32-1': 'From our study of Higgs signal strength measurements described in section [REF] we find in all four flavorful models the constraint [MATH].', '1805.08659-1-32-2': 'Combined with the generic expectation [MATH], this implies that BR[MATH] is typically not larger than [MATH].', '1805.08659-1-32-3': 'While this is much larger than the SM prediction of [MATH] [CITATION], it is below the current and expected sensitivities at the LHC [CITATION].', '1805.08659-1-32-4': 'The decay [MATH] is further suppressed by the up quark mass and generically not larger than [MATH], i.e. far below any foreseeable experimental sensitivity.', '1805.08659-1-32-5': 'Rare top decays could have much larger branching ratios if the CKM matrix is generated in the up-sector.', '1805.08659-1-33-0': 'The branching ratio for the rare Higgs decay [MATH] is given by [EQUATION] where [MATH] MeV is the total Higgs width.', '1805.08659-1-33-1': 'This expression holds in all four flavorful models, and we generically expect branching ratios up to [MATH].', '1805.08659-1-33-2': 'This has to be compared to the current bounds on this branching ratio from CMS [CITATION] and ATLAS [CITATION] [EQUATION]', '1805.08659-1-33-3': 'Future searches for [MATH] will start to probe interesting new physics parameter space.', '1805.08659-1-34-0': 'In all our models, the branching ratio of [MATH] is suppressed by a factor of [MATH] compared to [MATH] and therefore outside the reach of foreseeable experiments.', '1805.08659-1-34-1': 'The branching ratio of [MATH] is further suppressed and generically not larger than [MATH].', '1805.08659-1-35-0': '# Heavy neutral Higgs production and decays', '1805.08659-1-36-0': 'We expect a distinct collider phenomenology for the heavy Higgs bosons in each of our models.', '1805.08659-1-36-1': 'In contrast to models with natural flavor conservation, flavor alignment, or minimal flavor violation [CITATION], the coupling modifiers of the heavy Higgs bosons to fermions are not flavor universal.', '1805.08659-1-36-2': 'The difference is particularly striking for moderate and large [MATH].', '1805.08659-1-36-3': 'As shown in table [REF], for [MATH] and [MATH], whenever the coupling to a third generation fermion is suppressed by a factor [MATH], the couplings to the corresponding first and second generation fermions are enhanced by a factor [MATH], and vice versa.', '1805.08659-1-36-4': 'Depending on the type of flavorful model, a specific set of fermions can dominate the decay of the heavy Higgs bosons and cause different types of production modes to be more or less relevant.', '1805.08659-1-36-5': 'In the following we will focus on the type 2B, lepton-specific B, and flipped B models.', '1805.08659-1-36-6': 'The collider phenomenology of the type 1B model has been discussed previously in [CITATION].', '1805.08659-1-37-0': 'For the numerical results that will be presented in this section as well as in the subsequent charged Higgs section we will consider a fixed set of [MATH] mass parameters.', '1805.08659-1-37-1': 'To choose [MATH] parameters in the up and lepton sectors, we start with the Yukawa textures from Eqs. ([REF]) and ([REF]) setting all free [MATH] parameters to [MATH].', '1805.08659-1-37-2': 'The precise values for [MATH] and [MATH] in Eqs. ([REF]) and ([REF]) are then fully determined by demanding that the mass eigenvalues reproduce the known fermion masses (we use [MATH] masses at a scale of 500 GeV).', '1805.08659-1-37-3': 'In the down sector, the entries in Eq. ([REF]) of [MATH], [MATH], and [MATH] are chosen to reproduce the CKM matrix.', '1805.08659-1-37-4': 'The [MATH] parameters in Eq. ([REF]) are determined by the known down quark masses, setting the remaining free [MATH] parameters to [MATH].', '1805.08659-1-37-5': 'Generically, choosing different [MATH] parameters does not lead to a qualitative change of the heavy neutral and charged Higgs phenomenology.', '1805.08659-1-37-6': 'We will discuss the quantitative impact of varying the [MATH] parameters where appropriate.', '1805.08659-1-38-0': '## Production cross sections', '1805.08659-1-39-0': 'As we have seen in section [REF], the type 2B, lepton-specific B, and flipped B models are strongly constrained by Higgs signal strength measurements.', '1805.08659-1-39-1': 'In order to have maximal freedom in choosing a value for [MATH], we will limit our discussion to the decoupling limit and thus set [MATH].', '1805.08659-1-39-2': 'In this limit the couplings of the heavy scalar and pseudoscalar Higgs to fermions are identical and, furthermore, their couplings to gauge bosons vanish.', '1805.08659-1-39-3': 'The main production modes of the heavy neutral Higgs bosons are therefore: gluon-gluon fusion, production in association with tops or bottoms, and direct production from a [MATH] initial state.', '1805.08659-1-39-4': 'Vector boson fusion and production in association with gauge bosons is absent.', '1805.08659-1-40-0': 'We compute the cross section of the [MATH] processes by convoluting the leading order parton level cross section with the appropriate MMHT 2014 quark parton distribution functions (PDF) [CITATION] [EQUATION] where [MATH] is the center of mass energy of the protons.', '1805.08659-1-40-1': 'We take into account [MATH], [MATH], [MATH], and [MATH] initial states.', '1805.08659-1-40-2': "Given the small couplings to the lighter quark generations, we find that the remaining possible quark combinations are always sub-dominant (despite the larger PDF's).", '1805.08659-1-40-3': 'We do not include higher order corrections where one or two [MATH] quarks appear in the final state, keeping in mind that such processes might modify our [MATH] and [MATH] results by an [MATH] amount [CITATION].', '1805.08659-1-40-4': 'In the type 2B and flipped B models, we expect that the [MATH] production is the most relevant for moderate and large [MATH], thanks to the enhanced couplings to bottom quarks.', '1805.08659-1-40-5': "Production from initial state charm benefits from slightly larger PDF's but is suppressed by the significantly smaller charm mass.", '1805.08659-1-40-6': 'In the lepton-specific B model instead, we expect [MATH] to dominate for moderate and large [MATH], as the couplings to bottom are suppressed.', '1805.08659-1-41-0': 'We estimate the [MATH] production cross section by scaling the corresponding cross section of a heavy Higgs with SM-like couplings from [CITATION] by the ratio of the leading order [MATH] partial width in our model and a heavy Higgs with SM-like couplings.', '1805.08659-1-41-1': 'We take the expression for the partial width from [CITATION].', '1805.08659-1-42-0': 'Top associated production arises from diagrams like those shown in Fig. [REF].', '1805.08659-1-42-1': 'The corresponding cross section is identical for all four flavorful models.', '1805.08659-1-42-2': "We use the cross section from [CITATION], which was obtained by summing over the initial state quarks [MATH] and [MATH] and convoluting the parton cross section with the appropriate PDF's.", '1805.08659-1-43-0': 'The plots on the left hand side of Fig. [REF] show the various production cross sections for the three considered types of models as function of [MATH], for fixed Higgs mass of [MATH] GeV, and [MATH].', '1805.08659-1-43-1': 'In the type 2B and flipped B models, production involving bottom quarks is typically most relevant, while in the lepton-specific B model either production from [MATH] or gluon-gluon fusion dominates.', '1805.08659-1-43-2': 'For large [MATH], the gluon-gluon fusion production is sub-dominant in all cases due to the suppressed Higgs coupling to tops.', '1805.08659-1-43-3': 'Gluon-gluon fusion is minimal for intermediate values of [MATH], where the heavy Higgs coupling to tops accidentally vanishes.', '1805.08659-1-43-4': 'The precise location of the minimum depends on the choice of [MATH] parameters and can shift by an [MATH] factor.', '1805.08659-1-43-5': 'For large and small [MATH] the shown production cross sections are robust with respect to [MATH] changes in the [MATH] parameters.', '1805.08659-1-44-0': 'Overall, the total production cross section of a heavy Higgs of mass 500 GeV ranges from several hundred [MATH] to several [MATH] in the type 2B and flipped B models, and from tens of [MATH] to several [MATH] in the lepton-specific B model.', '1805.08659-1-44-1': 'The results for the type 2B and flipped B models are very similar to the corresponding models with natural flavor conservation.', '1805.08659-1-44-2': 'The reason is that the dominant production modes are governed by the top and bottom couplings that behave very similarly in those type A and B models.', '1805.08659-1-44-3': 'The results for the lepton-specific B model, however, differ markedly from the corresponding results of the lepton-specific A model.', '1805.08659-1-44-4': 'In the type A model, all couplings to quarks are universally suppressed by [MATH] leading to tiny production cross sections.', '1805.08659-1-44-5': 'In the type B model the couplings to charm are enhanced, leading to an appreciable amount of heavy Higgs production.', '1805.08659-1-45-0': '## Branching ratios', '1805.08659-1-46-0': 'The heavy Higgs bosons can in principle decay to SM fermions, to the SM gauge bosons, and to other Higgs bosons.', '1805.08659-1-46-1': 'In the decoupling limit [MATH], the decays of [MATH] and [MATH] to final states with massive vector bosons vanish.', '1805.08659-1-46-2': 'Decays into photons and gluons are loop suppressed and typically tiny.', '1805.08659-1-46-3': 'We assume that the heavy Higgs bosons are sufficiently degenerate, such that decays into each other are kinematically forbidden.', '1805.08659-1-46-4': 'The decay into two light Higgs bosons is in principle possible.', '1805.08659-1-46-5': 'The corresponding trilinear couplings depend on the couplings in the Higgs potential and can be made arbitrarily small.', '1805.08659-1-46-6': 'In the following, we will only consider decays into fermions.', '1805.08659-1-46-7': 'Generically, the decay widths of the heavy scalar [MATH] to two fermions are [EQUATION] where we assumed that the mass of the fermions is negligible [MATH].', '1805.08659-1-46-8': 'The color factor is [MATH] for leptons and [MATH] for quarks.', '1805.08659-1-46-9': 'This expression is sufficiently generic to describe both flavor conserving and flavor violating decays.', '1805.08659-1-46-10': 'In the case where one or both of the fermions is a top quark, top mass effects have to be included [EQUATION]', '1805.08659-1-46-11': 'We show the branching ratios of the heavy Higgs as function of [MATH] in the plots on the right hand side of Fig. [REF].', '1805.08659-1-46-12': 'The heavy Higgs mass is set to [MATH] GeV and [MATH].', '1805.08659-1-46-13': 'The main decay modes of the heavy Higgs to the fermions are easily understood from table [REF], that shows to which fermions the [MATH] doublet couples.', '1805.08659-1-46-14': 'In the type 2B and flipped B models we expect the [MATH] decay to dominate at large [MATH].', '1805.08659-1-46-15': 'For the lepton-specific setup we expect the [MATH] decay to be the primary branching ratio.', '1805.08659-1-46-16': 'In the flipped B model, the [MATH] decay is instead strongly suppressed.', '1805.08659-1-46-17': 'For low [MATH], decays into [MATH] dominate (if kinematically allowed).', '1805.08659-1-46-18': 'These are the same patterns as in the models with natural flavor conservation.', '1805.08659-1-47-0': 'In contrast to the models with natural flavor conservation, decays involving charm quarks ([MATH] and [MATH]) can have branching ratios of [MATH] in all three flavorful models.', '1805.08659-1-47-1': 'Also the decay into [MATH] has branching ratios of several [MATH] for large [MATH], due to terms in the coupling of the heavy Higgs to tops that are proprotional to [MATH].', '1805.08659-1-47-2': 'For [MATH] there can be a cancellation between the leading [MATH] suppressed term and the [MATH] correction, leading to an accidental vanishing of the [MATH] branching ratio.', '1805.08659-1-48-0': 'Also lepton flavor violating decays can arise.', '1805.08659-1-48-1': 'In the lepton-specific B model, we find the decay [MATH] can have branching ratios of up to [MATH].', '1805.08659-1-48-2': 'In the type 2B and flipped B model, the branching ratio of this decay mode is smaller by a factor of few, as it has to compete with the dominant decay into [MATH].', '1805.08659-1-49-0': 'The branching ratios of flavor diagonal decay modes like [MATH], [MATH], and [MATH] are fairly robust against changes in the [MATH] mass parameters.', '1805.08659-1-49-1': 'The branching ratios of flavor violating decays can change by a factor of few if the relevant [MATH] parameters are modified by an [MATH] amount.', '1805.08659-1-50-0': 'In the decoupling limit, the scalar and pseudoscalar Higgs couplings are identical.', '1805.08659-1-50-1': 'Consequently, the production cross sections and branching ratios of the pseudoscalar Higgs are very similar to the scalar Higgs and we do not show the plots for the pseudoscalar.', '1805.08659-1-51-0': '## Constraints from direct searches', '1805.08659-1-52-0': 'Having examined the main production and decay modes of the heavy neutral Higgs bosons of the flavorful models we now compare results from current heavy Higgs searches at the LHC with the model predictions.', '1805.08659-1-52-1': 'We find the most relevant constraints come from', '1805.08659-1-53-0': 'In Fig. [REF] we show the ratio of the experimentally excluded rate [MATH] to the rate predicted in our flavorful 2HDMs [MATH] as function of the heavy Higgs mass for a benchmark scenario with [MATH] and [MATH].', '1805.08659-1-53-1': 'If this ratio is below 1, the model is excluded for the given set of parameters.', '1805.08659-1-54-0': 'Concerning, the experimental searches that target Higgs production in association with bottom quarks, we estimate the theoretical production cross section from [MATH], keeping in mind that higher order corrections might change the result by an [MATH] amount.', '1805.08659-1-54-1': 'The corresponding constraints in the plots of Fig. [REF] are labeled with the subscript "bbF".', '1805.08659-1-54-2': 'If experimental constraints assume gluon-gluon fusion production, we take into account both gluon-gluon fusion and also production from [MATH], which should lead to the same experimental signature.', '1805.08659-1-54-3': 'The corresponding constraints are labeled "ggF".', '1805.08659-1-54-4': 'If no particular production mode is singled out by the experimental search, we add up all the production mechanisms.', '1805.08659-1-54-5': 'For each individual channel we show the strongest constraint among the considered experimental analyses.', '1805.08659-1-55-0': 'We observe that for [MATH] the type 2B and the lepton-specific B models are strongly constrained by searches for heavy Higgs decaying to a [MATH] final state.', '1805.08659-1-55-1': 'Heavy Higgs masses up to [MATH] TeV (type 2B) and up to [MATH] GeV (lepton-specific B) are already excluded in this case.', '1805.08659-1-55-2': 'The constraints are much weaker in the flipped B model.', '1805.08659-1-55-3': 'Searches for di-muon, [MATH] and di-jet resonance searches have sensitivities that start to approach the model predictions, but currently do not exclude parameter space with [MATH].', '1805.08659-1-56-0': 'Note that the excluded mass ranges are extremely sensitive to the values of [MATH].', '1805.08659-1-56-1': 'For large [MATH] the production cross sections in all models are approximately proportional to [MATH].', '1805.08659-1-56-2': 'So, the cross section ratios quickly go below the exclusion line.', '1805.08659-1-56-3': 'However, as [MATH] becomes small the constraints generically get weaker and the constraints in the type 2B and lepton-specific B case can be easily avoided.', '1805.08659-1-57-0': '# Charged Higgs production and decays', '1805.08659-1-58-0': 'The collider phenomenology of the charged Higgs in the type 1B model has been discussed previously in [CITATION].', '1805.08659-1-58-1': 'Here we discuss the phenomenology of the charged Higgs in the type 2B, lepton-specific B, and flipped B models.', '1805.08659-1-59-0': '## Production cross sections', '1805.08659-1-60-0': 'As for the neutral Higgses, the main production mode is again primarily from [MATH] fusion.', '1805.08659-1-60-1': 'We estimate these cross sections using an expression analogous to Eq. ([REF]) along with the MMHT 2014 PDFs [CITATION].', '1805.08659-1-60-2': 'Also production in association with a top quark (see diagrams in Fig. [REF]) can become important.', '1805.08659-1-60-3': 'The corresponding production cross section is taken from [CITATION].', '1805.08659-1-61-0': 'We show the production cross sections as function of [MATH] in Fig. [REF].', '1805.08659-1-61-1': 'As an example, we use the charged Higgs mass [MATH] GeV and set [MATH].', '1805.08659-1-62-0': 'At low [MATH], the production in association with a top quark dominates in all three flavorful models.', '1805.08659-1-62-1': 'In the type 2B and flipped B models production in association with a top quark remains dominant also for large [MATH] due to the enhanced couplings to bottom in this region of parameter space.', '1805.08659-1-62-2': 'In the lepton-specific B model, however, large [MATH] implies suppression of both top and bottom couplings and the top associated charged Higgs production is suppressed.', '1805.08659-1-63-0': 'We find that the charged Higgs production from [MATH] fusion is dominated by initial states containing charm quarks.', '1805.08659-1-63-1': 'All three combinations [MATH], [MATH], and [MATH] have production cross sections of the same order of magnitude.', '1805.08659-1-63-2': 'While the coupling to [MATH] is suppressed by a factor of [MATH] compared to the [MATH] and [MATH] couplings, this suppression is partially compensated by the larger down PDF.', '1805.08659-1-63-3': 'Furthermore, the [MATH] production cross sections are mainly determined by couplings of the charged Higgs involving right handed charm quarks.', '1805.08659-1-63-4': 'Those couplings have the same scaling with [MATH] for all three flavorful models and we indeed observe that also the corresponding cross sections are approximately equal in the three models.', '1805.08659-1-64-0': 'This is particularly interesting for the lepton-specific B case.', '1805.08659-1-64-1': 'In the lepton-specific A model, all couplings to quarks are suppressed at large [MATH], and charged Higgs production is tiny.', '1805.08659-1-64-2': 'In the "B-type" of the model, however, the enhanced couplings to charm open up the possibility to directly probe this region of parameter space at the LHC.', '1805.08659-1-65-0': '## Branching Ratios', '1805.08659-1-66-0': 'In the considered scenario with [MATH], the charged Higgs decays either to quarks or leptons.', '1805.08659-1-66-1': 'The decay to [MATH] is absent.', '1805.08659-1-66-2': 'The decay rate to fermions is given analogous to the neutral Higgs, Eq. ([REF]).', '1805.08659-1-67-0': 'In the type 2B and flipped B models we expect the dominant branching ratio to be [MATH] both for small [MATH] (where the coupling to top is large) and at large [MATH] (where the coupling to bottom is enhanced).', '1805.08659-1-67-1': 'This can be clearly seen in the plots of Fig. [REF] that show the most relevant branching ratios as function of [MATH] for [MATH] GeV and [MATH].', '1805.08659-1-68-0': 'In the type 2B model, the [MATH] decay mode has the second largest branching ratio at large [MATH].', '1805.08659-1-68-1': 'This is very similar to the type 2A model with natural flavor conservation.', '1805.08659-1-68-2': 'In contrast to the type 2A, decay modes including charm quarks, like [MATH] and [MATH], can have branching ratios of several [MATH] in the flavorful type 2B model.', '1805.08659-1-68-3': 'Also in the flipped B model, [MATH] and [MATH] can have branching ratios of several [MATH].', '1805.08659-1-68-4': 'The decay to [MATH] on the other hand is strongly suppressed.', '1805.08659-1-68-5': 'The rather clean [MATH] final state can reach branching ratios of [MATH], which is orders of magnitude larger than in the flipped A model.', '1805.08659-1-69-0': 'In the lepton-specific B model, the branching ratio to [MATH] dominates at large [MATH] and is typically around [MATH].', '1805.08659-1-69-1': 'Decay modes involving charm ([MATH] and [MATH]) as well as top ([MATH] and [MATH]) have typical branching ratios of [MATH].', '1805.08659-1-70-0': 'For [MATH] above [MATH] most branching ratios stay approximately constant.', '1805.08659-1-70-1': 'One exception is the [MATH] branching ratio in the lepton-specific B model which changes considerably with [MATH].', '1805.08659-1-70-2': 'For [MATH] the relevant coupling of the charged Higgs to [MATH] vanishes, due to an accidental cancellation between the [MATH] term and the term of [MATH] in Eq. ([REF]).', '1805.08659-1-70-3': 'The same cancellation is also responsible for the dip in the top associated production in the lepton-specific B model shown on the left-hand side of Fig. [REF].', '1805.08659-1-70-4': 'The precise value of [MATH] where this cancellation happens depends on the sign and exact size of the free [MATH] parameters in the [MATH] mass parameters, see Eq. ([REF]).', '1805.08659-1-70-5': 'In general, variation of the [MATH] mass parameters can change the branching ratios of flavor violating decays by a factor of few.', '1805.08659-1-71-0': '## Constraints from direct searches', '1805.08659-1-72-0': 'The constraints in this section are implemented with the same process we used in section [REF].', '1805.08659-1-72-1': 'The strongest constraints come from', '1805.08659-1-73-0': 'For low mass charged Higgs at [MATH], the type 2B and flipped B models are ruled out due to [MATH] decays.', '1805.08659-1-73-1': 'However, in the lepton-specific B case the parameter space for charged Higgs bosons lighter than the top quark is still open, motivating continued search for charged Higgs bosons in top decays [MATH].', '1805.08659-1-73-2': 'For [MATH] the high mass region is still largely unconstrained.', '1805.08659-1-73-3': 'For the flipped B and type 2B models, searches for [MATH] need to improve by approximately an order of magnitude to begin to probe the high mass region.', '1805.08659-1-73-4': 'The type 2B and lepton-specific B models can also be probed by [MATH] searches if their sensitivities improve one order or magnitude in the future.', '1805.08659-1-74-0': '# Effects on flavor violating processes', '1805.08659-1-75-0': 'The flavor violating couplings of the neutral Higgs bosons also affect low energy flavor observables like meson mixing and rare meson decays.', '1805.08659-1-75-1': 'In the following we consider neutral [MATH] meson, Kaon, and [MATH] meson mixing as well as the branching ratios of several rare meson decays [MATH], [MATH], [MATH], [MATH], and [MATH].', '1805.08659-1-76-0': '## Meson oscillations', '1805.08659-1-77-0': 'The SM Higgs, as well as the heavy scalar and pseudoscalar Higgs add contributions to neutral [MATH] meson mixing at tree level.', '1805.08659-1-77-1': 'For the new physics contribution to the [MATH] mixing amplitude normalized to the SM amplitude we have [CITATION] [EQUATION] where [MATH] is a SM loop function.', '1805.08659-1-77-2': 'The corresponding expression for the [MATH] mixing amplitude is analogous.', '1805.08659-1-77-3': 'Note that this expression holds for all four flavorful 2HDMs.', '1805.08659-1-77-4': 'The [MATH] factors in Eq. ([REF]) contain leading order QCD running corrections and ratios of hadronic matrix elements [MATH], [MATH], [MATH], see [CITATION].', '1805.08659-1-77-5': 'The first value listed corresponds to [MATH] and the second to [MATH].', '1805.08659-1-77-6': 'From the above new physics contribution we can find values for the meson oscillation frequencies as well as the mixing phases [EQUATION]', '1805.08659-1-77-7': 'We confront our models with experimental constraints by constructing a [MATH] function that includes the mass differences and mixing phases in [MATH] and [MATH] mixing.', '1805.08659-1-77-8': 'The SM predictions and experimental results are taken from [CITATION] (see also [CITATION] for a recent discussion of [MATH] mixing constraints).', '1805.08659-1-77-9': 'Note that in our models the [MATH] and [MATH] mass parameters are largely fixed by the CKM matrix, see Eqs. ([REF]) and ([REF]).', '1805.08659-1-77-10': 'Thus we use the [MATH] mixing observables to constrain the free [MATH] and [MATH] mass parameters, setting [MATH] and [MATH] (with the signs depending on the type of flavorful model).', '1805.08659-1-78-0': 'In Fig. [REF] we show constraints on the absolute values and phases of [MATH] (left) and [MATH] (right) for a benchmark scenario with [MATH] (as favored by the Higgs signal strengths measurements, see section [REF]), [MATH], and [MATH] GeV.', '1805.08659-1-78-1': 'The constraints on the [MATH] parameter scale approximately as [MATH], i.e. they become weaker for larger Higgs masses and stronger for larger [MATH].', '1805.08659-1-78-2': 'The shown constraints hold in the type 1B and lepton-specific B models.', '1805.08659-1-78-3': 'In the type 2B and flipped B models, the [MATH] and [MATH] mass parameters have the opposite sign.', '1805.08659-1-78-4': 'This results in constraints that are shifted in phase by Arg[MATH] Arg[MATH].', '1805.08659-1-79-0': 'We observe that both [MATH] and [MATH] are strongly constrained by [MATH] and [MATH] mixing for large [MATH] and for heavy Higgs bosons below [MATH] TeV.', '1805.08659-1-79-1': 'The fact that these mass parameters have to be much smaller than the generic prediction of our flavor textures, [MATH] and [MATH] might call for an underlying flavor model.', '1805.08659-1-80-0': 'Similarly to [MATH] meson mixing, also the Kaon mixing amplitude obtains additional contributions.', '1805.08659-1-80-1': 'The new physics amplitude is [EQUATION] with the Kaon decay constant [MATH] MeV [CITATION].', '1805.08659-1-80-2': 'The bag parameters [MATH], [MATH], [MATH] are taken from [CITATION] (see also [CITATION]).', '1805.08659-1-80-3': 'The parameters [MATH], [MATH], and [MATH] (see [CITATION]) encode one loop renormalization group effects.', '1805.08659-1-81-0': 'The relevant observables in Kaon mixing are the mass difference [MATH] and the CP violating parameter [MATH].', '1805.08659-1-81-1': 'They can be calculated via [EQUATION] with [MATH] [CITATION].', '1805.08659-1-81-2': 'In Eqs. ([REF]) and ([REF]) we saw that the [MATH] parameters that are responsible for Kaon mixing are not independent parameters but given in terms of the parameters that govern [MATH] and [MATH] mixing.', '1805.08659-1-81-3': 'Given the constraints from [MATH] and [MATH] mixing, we find that new physics effects in Kaon mixing are generically below the current bounds.', '1805.08659-1-81-4': 'In particular, we find that new physics effects in [MATH] are at most at the permille level, while effects in [MATH] are [MATH].', '1805.08659-1-82-0': 'Analogously to Kaon mixing, the new physics contributions to neutral [MATH] meson mixing are given by [EQUATION]', '1805.08659-1-82-1': 'According to Eqs. ([REF]) and ([REF]), the [MATH] and [MATH] parameters are strongly suppressed, generically of the order of [MATH].', '1805.08659-1-82-2': 'We find that the resulting new physics contributions to the mixing amplitude are many orders of magnitude below the current sensitivities [CITATION] in all the models we consider.', '1805.08659-1-83-0': '## The rare [MATH] Bs -> mu+ mu- decay', '1805.08659-1-84-0': 'The rare FCNC decay [MATH] is known to be a highly sensitive probe of new physics (see e.g. [CITATION]).', '1805.08659-1-84-1': 'The decay has been observed at the LHC [CITATION] and the latest experimental result for the time integrated branching ratio from LHCb [CITATION] [EQUATION] agrees well with the SM prediction [CITATION] [EQUATION]', '1805.08659-1-84-2': 'A generic expression for the branching ratio in presence of NP reads [CITATION] [EQUATION] where [MATH] is the life-time difference of the [MATH] mesons, [MATH] [CITATION].', '1805.08659-1-84-3': 'In the above expression we do not consider corrections due to a possible non-standard [MATH] mixing phase [MATH] [CITATION].', '1805.08659-1-84-4': 'Given the existing constraint on [MATH] [CITATION], such corrections to the branching ratio are negligible.', '1805.08659-1-85-0': 'In the SM, the coefficients [MATH] and [MATH].', '1805.08659-1-85-1': 'Corrections due to tree level exchange of the neutral Higgs bosons are collected in the appendix [REF].', '1805.08659-1-85-2': 'As [MATH] meson mixing puts strong constraints on [MATH] we will set it to zero in the following discussion.', '1805.08659-1-85-3': 'In the alignment limit and for [MATH], as well as neglecting the life time difference, the expression for BR[MATH] simplifies to [EQUATION] with the SM Wilson coefficient [MATH].', '1805.08659-1-85-4': 'The plus (minus) sign in the first term holds in the type 1B and the lepton-specific B models (type 2B and flipped B models).', '1805.08659-1-85-5': 'Note that the [MATH] parameter is approximately given by [MATH] in the type 1B and flipped B models.', '1805.08659-1-85-6': 'In the type 2B and lepton-specific B models, [MATH] is a free parameter of [MATH].', '1805.08659-1-85-7': 'Consequently, we expect much more stringent constraints in the type 1B and flipped B models as compared to the type 2B and lepton-specific B models.', '1805.08659-1-86-0': 'In Fig. [REF] we show constraints in the plane of heavy Higgs mass [MATH] vs. [MATH] from [MATH] in the four flavorful models.', '1805.08659-1-86-1': 'In all four models we set [MATH] and [MATH].', '1805.08659-1-86-2': 'In the type 1B and flipped B models we set the (small) higher order corrections to [MATH] to zero, i.e. [MATH].', '1805.08659-1-86-3': 'In the type 2B and lepton-specific B models we set [MATH].', '1805.08659-1-87-0': 'The constraints in the type 2B and lepton-specific B models depend strongly on the choice of [MATH].', '1805.08659-1-87-1': 'If [MATH] accidentally vanishes, the [MATH] constraint can be even completely avoided in these models.', '1805.08659-1-87-2': 'The bounds in the type 1B and flipped B models, however, are robust.', '1805.08659-1-87-3': 'The higher order corrections to [MATH] modify them typically by [MATH] or less.', '1805.08659-1-87-4': 'In these models, the shown bounds from [MATH] can only be avoided by postulating that the CKM matrix is generated in the up-sector.', '1805.08659-1-88-0': 'In comparison to the constraints from direct searches we observe that [MATH] gives stronger bounds in the type 1B and flipped B models.', '1805.08659-1-88-1': 'In the type 2B and lepton-specific models, the direct searches in the [MATH] channel tend to be more constraining, instead.', '1805.08659-1-89-0': '## Lepton flavor violating B meson decays', '1805.08659-1-90-0': 'In the SM, the lepton flavor violating decays based on the [MATH] transition are suppressed by the tiny neutrino masses and are far below any imaginable experimental sensitivities.', '1805.08659-1-90-1': 'Observation of these decays would be clear sign of new physics.', '1805.08659-1-90-2': 'In our setup, tree level exchange of neutral Higgs bosons can induce these decays at levels that might become experimentally accessible.', '1805.08659-1-91-0': 'Similarly to the lepton flavor conserving decay [MATH] we express the branching ratio of the two body decay [MATH] as [EQUATION] where the last line takes into account the effect of a non-zero life time difference in the [MATH] system.', '1805.08659-1-91-1': 'An analogous expression holds for the decay [MATH].', '1805.08659-1-91-2': 'We will use the notation [MATH].', '1805.08659-1-91-3': 'The expressions for the coefficients [MATH] and [MATH] are collected in the appendix [REF].', '1805.08659-1-92-0': 'As in our discussion of the [MATH] decay, we set [MATH], [MATH], [MATH], and neglect the life time difference.', '1805.08659-1-92-1': 'In this case we find [EQUATION]', '1805.08659-1-92-2': 'This expression holds in all four flavorful 2HDMs.', '1805.08659-1-92-3': 'For all types we have [MATH].', '1805.08659-1-92-4': 'In the type 1B and the flipped B models, the possible values for BR[MATH] are bounded by the measured BR[MATH].', '1805.08659-1-92-5': 'Considering [MATH] and [MATH] GeV [MATH] TeV, we find the following upper bounds [EQUATION]', '1805.08659-1-92-6': 'In the type 2B and lepton-specific B models, the constraint from [MATH] is much weaker.', '1805.08659-1-92-7': 'In those models the strongest constraint comes from direct searches for the heavy Higgs bosons in the [MATH] channel (see Fig. [REF]).', '1805.08659-1-92-8': 'Values of BR[MATH] are possible in those models.', '1805.08659-1-93-0': 'Lepton flavor changing decays involving electrons on the other hand are tiny.', '1805.08659-1-93-1': 'Generically we expect in all models [EQUATION]', '1805.08659-1-93-2': 'In addition to the [MATH] decay, tree level exchange of flavor violating Higgs bosons also leads to three body semi-leptonic [MATH] meson decays like [MATH], [MATH], and [MATH].', '1805.08659-1-94-0': 'We find that the [MATH] and [MATH] branching ratios are directly correlated to the [MATH] branching ratio.', '1805.08659-1-94-1': 'Ignoring the life-time difference in the [MATH] system and using the results from [CITATION] we obtain for the differential branching ratio [EQUATION] where [MATH].', '1805.08659-1-94-2': 'An analogous expression holds for [MATH].', '1805.08659-1-94-3': 'For the [MATH] meson decay constant we use [MATH] MeV [CITATION].', '1805.08659-1-94-4': 'The [MATH] and [MATH] form factors [MATH] are taken from [CITATION].', '1805.08659-1-94-5': 'Integrating over [MATH], we find [EQUATION]', '1805.08659-1-94-6': 'Using the bounds and generic expectations for [MATH] in the different flavorful models discussed above, we find that [MATH] and [MATH] can be at most few [MATH] in the type 1B model and [MATH] in the flipped B model, respectively.', '1805.08659-1-94-7': 'In the type 2B and lepton-specific B models, however, these branching ratios can be as large as [MATH].', '1805.08659-1-95-0': 'We find similar results also for the [MATH] decay.', '1805.08659-1-95-1': 'The fact that [MATH] is a pseudoscalar to pseudoscalar transition, while [MATH] and [MATH] are pseudoscalar to vector transitions has little impact numerically.', '1805.08659-1-95-2': 'We find that [MATH] can be as large as few [MATH] in the type 1B model, [MATH] in the flipped B model, and [MATH] in the type 2B and lepton-specific B models.', '1805.08659-1-96-0': '# Conclusions', '1805.08659-1-97-0': 'Little is known experimentally about the tiny couplings of the Higgs boson to the light flavors of quarks and leptons.', '1805.08659-1-97-1': 'It is thus interesting to study possible alternative origins of mass for the light flavors beyond the 125 GeV Higgs boson.', '1805.08659-1-97-2': 'As an example, we analyzed a particular class of 2HDMs with non-trivial flavor structure.', '1805.08659-1-97-3': 'In analogy to the four, well studied 2HDMs with natural flavor conservation (NFC), we identified four models that preserve an approximate [MATH] flavor symmetry acting on the first two generations.', '1805.08659-1-97-4': 'We refer to them as type 1B, type 2B, lepton-specific B, and flipped B.', '1805.08659-1-97-5': 'In these flavorful 2HDMs, interesting flavor violating phenomena involving the third generation of fermions can be expected, while the [MATH] flavor symmetry still protects flavor violating transitions between the first and second generations.', '1805.08659-1-98-0': 'We studied the production and decay modes of the neutral and charged Higgs bosons of the models, as well as various low energy flavor violating observables, and identified the signatures of the flavorful models that are qualitatively different from the models with NFC.', '1805.08659-1-99-0': 'With regards to the collider phenomenology we find:', '1805.08659-1-100-0': 'The most interesting features in the flavor phenomenology are:'}

{'1805.08659-2-0-0': 'We explore Two Higgs Doublet Models with non-standard flavor structures.', '1805.08659-2-0-1': 'In analogy to the four, well studied, models with natural flavor conservation (type 1, type 2, lepton-specific, flipped), we identify four models that preserve an approximate [MATH] flavor symmetry acting on the first two generations.', '1805.08659-2-0-2': 'In all four models, the couplings of the 125 GeV Higgs are modified in characteristic flavor non-universal ways.', '1805.08659-2-0-3': 'The heavy neutral and charged Higgs bosons show an interesting non-standard phenomenology.', '1805.08659-2-0-4': 'We discuss their production and decay modes and identify the most sensitive search channels at the LHC.', '1805.08659-2-0-5': 'We study the effects on low energy flavor violating processes finding relevant constraints from [MATH] and [MATH] meson oscillations and from the rare decay [MATH].', '1805.08659-2-0-6': 'We also find that lepton flavor violating [MATH] meson decays like [MATH] and [MATH] might have branching ratios at an observable level.', '1805.08659-2-1-0': '# Introduction', '1805.08659-2-2-0': 'Measurements of Higgs rates at the LHC show that the Standard Model (SM) Higgs mechanism provides the bulk of the masses of the third generation fermions.', '1805.08659-2-2-1': 'The decay [MATH] has been observed at a rate compatible with the SM prediction [CITATION].', '1805.08659-2-2-2': 'Similarly, evidence exists for a SM-like [MATH] decay [CITATION].', '1805.08659-2-2-3': 'Recently, production of the Higgs in association with top quarks has been observed in agreement with the SM [CITATION].', '1805.08659-2-3-0': 'Much less is known about the origin of the first and second generation masses.', '1805.08659-2-3-1': 'With the exception of the muon, direct measurements of Higgs couplings to the light fermions are extremely challenging.', '1805.08659-2-3-2': 'It is therefore unknown if the light fermions obtain their mass from the Higgs boson.', '1805.08659-2-3-3': 'A complementary approach to probe the origin of light fermion masses is to search for signatures of alternatives to the SM Higgs mechanism in which the light fermion masses originate from a new source of electroweak symmetry breaking.', '1805.08659-2-3-4': 'The simplest realization of such a setup is the Two Higgs Doublet Model (2HDM).', '1805.08659-2-4-0': 'In [CITATION] a 2HDM setup was proposed in which one Higgs doublet couples only to the third generation fermions, and a second Higgs doublet couples mainly to the first and second generation (see also [CITATION]).', '1805.08659-2-4-1': 'A dynamical generation of such a coupling structure can be achieved using the flavor-locking mechanism [CITATION].', '1805.08659-2-4-2': 'The collider phenomenology of this "flavorful" 2HDM scenario was discussed in [CITATION].', '1805.08659-2-5-0': 'The proposed 2HDM goes beyond the principle of natural flavor conservation (NFC) [CITATION] and introduces flavor changing neutral currents (FCNCs) at tree level.', '1805.08659-2-5-1': 'However, the Yukawa couplings of the first Higgs doublet to the third generation preserve a [MATH] flavor symmetry, which is only broken by the small couplings of the second Higgs doublet.', '1805.08659-2-5-2': 'The approximate [MATH] symmetry protects the most sensitive flavor violating transitions between the second and first generation.', '1805.08659-2-6-0': 'In this work we explore additional flavor structures for 2HDMs that approximately preserve a [MATH] flavor symmetry for the first two generations.', '1805.08659-2-6-1': 'Starting from the flavorful 2HDM scenario of [CITATION] we "twist" the Yukawa couplings of the down-type quarks and/or leptons by exchanging the Higgs doublets these fermions couple to.', '1805.08659-2-6-2': 'In analogy to the four well studied 2HDMs with natural flavor conservation (type 1, type 2, lepton-specific, flipped) we obtain four flavorful 2HDMs in which the third and first two generations of each fermion type (up-type quarks, down-type quarks, leptons) obtain the bulk of their mass from a different source.', '1805.08659-2-6-3': 'The non-standard flavor structures of these four 2HDMs lead to (i) distinct, flavor non-universal modifications of all Higgs couplings with respect to the models with NFC, (ii) potentially sizable flavor violating Higgs couplings involving the third generation fermions.', '1805.08659-2-6-4': 'This implies an interesting characteristic collider and flavor phenomenology.', '1805.08659-2-6-5': '(For recent work on 2HDMs with other non-standard flavor structures see [CITATION].)', '1805.08659-2-7-0': 'The paper is structured as follows: In section [REF] we introduce the four flavorful 2HDMs, discuss the Yukawa textures and the couplings of the fermions to the various Higgs boson mass eigenstates.', '1805.08659-2-7-1': 'In section [REF] we consider the phenomenology of the 125 GeV Higgs boson, comparing the predicted production and decay rates in our models to measurements at the LHC.', '1805.08659-2-7-2': 'In sections [REF] and [REF] we evaluate the production cross sections and decay branching ratios of the heavy neutral and charged Higgs bosons.', '1805.08659-2-7-3': 'We then compare the model predictions to the limits from current searches for extra Higgs bosons that are being performed at the LHC and identify the most sensitive collider probes of the models.', '1805.08659-2-7-4': 'In section [REF] we investigate the characteristic effects of the new sources of flavor violation on low energy flavor violating processes such as meson mixing and rare [MATH] meson decays.', '1805.08659-2-7-5': 'We conclude in section [REF].', '1805.08659-2-8-0': '# Flavorful Two Higgs Doublet Models', '1805.08659-2-9-0': 'One of the simplest realizations of a viable alternative framework of mass generation are 2HDMs with one doublet coupling only to the third generation, and a second doublet coupling mainly to the first and second generation.', '1805.08659-2-9-1': 'Such a setup was proposed in [CITATION] (see also [CITATION]).', '1805.08659-2-9-2': 'The masses of the SM fermions arise from two sources, the vacuum expectation values of two Higgs doublets [MATH] and [MATH].', '1805.08659-2-9-3': 'The relevant part of the 2HDM Lagrangian is [EQUATION] where [MATH].', '1805.08659-2-9-4': 'The three generations of quark and lepton doublets are denoted by [MATH], [MATH], and [MATH], [MATH], [MATH] are the up quark, down quark, and charged lepton singlets.', '1805.08659-2-9-5': 'The [MATH] and [MATH] matrices are the Yukawa couplings.', '1805.08659-2-10-0': 'The above setup for the Higgs couplings violates the principle of natural flavor conservation.', '1805.08659-2-10-1': 'Both of the Higgs doublets couple to the leptons, the up-type quarks, and the down-type quarks, leading to FCNCs at tree level.', '1805.08659-2-11-0': '## Yukawa textures', '1805.08659-2-12-0': 'We are interested in Yukawa couplings beyond NFC that do not introduce an unacceptably large amount of flavor violation.', '1805.08659-2-12-1': 'This can be achieved by demanding that one set of the Yukawa couplings preserves a [MATH] flavor symmetry, acting on the first two generations.', '1805.08659-2-12-2': 'In this case, flavor transitions between the first and second generation are protected.', '1805.08659-2-12-3': 'Such transitions are absent at first order in flavor symmetry breaking and arise only at second order as an effective [MATH] transition.', '1805.08659-2-12-4': 'As we will discuss in section [REF], effects in neutral Kaon and [MATH] meson oscillations are indeed typically well below present constraints.', '1805.08659-2-13-0': 'We consider the following set of Yukawa matrices in the flavor basis [EQUATION]', '1805.08659-2-13-1': 'Due to the rank 1 nature of the [MATH], [MATH], [MATH] Yukawa couplings, the [MATH] flavor symmetry acting on the first two generations is only broken by the small [MATH], [MATH], [MATH] Yukawa couplings.', '1805.08659-2-13-2': 'Such a pattern of textures can be obtained using for example the flavor locking mechanism [CITATION].', '1805.08659-2-13-3': 'Note that the above Yukawa couplings contain additional structure that is not dictated by the approximate [MATH] flavor symmetry.', '1805.08659-2-13-4': 'Our choice is motivated on the one hand by simplicity (the Yukawa matrices do not contain any unnecessary hierarchies) and on the other hand by robustness: The entries in the Yukawa couplings of the first and second generations are chosen such that the mass eigenvalues reproduce the observed values without any tuning.', '1805.08659-2-13-5': 'The structure in the down sector leads naturally to the observed pattern in the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.', '1805.08659-2-13-6': 'Alternatively, the CKM matrix could also be generated in the up sector, but we will not consider this option here as it requires additional hierarchies in the up Yukawa coupling.', '1805.08659-2-13-7': 'The entries in the above matrices are given up to [MATH] factors that, in all generality, can be complex.', '1805.08659-2-14-0': 'The vacuum expectation values [MATH] in Eqs. ([REF]) - ([REF]) correspond to either [MATH] or [MATH], depending on the model under consideration.', '1805.08659-2-14-1': 'The Yukawa couplings for the third or first two generations are identified with the [MATH] and [MATH] couplings introduced in Eq. ([REF]), accordingly.', '1805.08659-2-14-2': 'Without loss of generality, we denote the Higgs doublet that couples to the top quark with [MATH] [CITATION], i.e. [MATH], [MATH] and [MATH], [MATH].', '1805.08659-2-14-3': 'This leaves us with four distinct "flavorful" possibilities to assign the two Higgs doublets to the down-quarks and leptons.', '1805.08659-2-14-4': 'In analogy to the four well known 2HDMs with natural flavor conservation (that we refer to as type 1A, type 2A, lepton-specific A, and flipped A, in the following) we denote our four flavorful models as type 1B, type 2B, lepton-specific B, and flipped B.', '1805.08659-2-14-5': 'The type 1B model was studied in some detail in [CITATION].', '1805.08659-2-14-6': 'The coupling structure of all four flavorful models is summarized in table [REF].', '1805.08659-2-15-0': 'Rotating the fermions into mass eigenstates, we define the following mass parameters [EQUATION] with quark mass eigenstates [MATH].', '1805.08659-2-15-1': 'These mass parameters obey [MATH], where [MATH] are the observed up-type quark masses.', '1805.08659-2-15-2': 'Analogous definitions and identities hold for the down-type quarks and the charged leptons.', '1805.08659-2-15-3': 'We derive expressions for the [MATH] mass parameters in the mass eigenstate basis that automatically reproduce the observed fermion masses and CKM matrix elements.', '1805.08659-2-15-4': 'We find the following values for the up mass parameters in all four types of flavorful models [EQUATION]', '1805.08659-2-15-5': 'For leptons we find analogous expressions for the off-diagonal mass parameters in all four types [EQUATION]', '1805.08659-2-15-6': 'Finally, for the down quarks we find for all four types [EQUATION]', '1805.08659-2-15-7': 'As we assume that the CKM matrix is generated in the down sector, the CKM elements [MATH] and [MATH] appear in several of the down-type mass parameters.', '1805.08659-2-16-0': 'The [MATH] terms in the above expressions are free parameters that in general can be complex.', '1805.08659-2-16-1': 'It is worth noting that due to those [MATH] terms, the off-diagonal mass parameters [MATH] and [MATH] need not be the same for any type of fermion.', '1805.08659-2-16-2': 'It is also important to note that in all cases the mass parameters that are responsible for flavor mixing between the first and second generation are suppressed by small mass ratios and not independent from the mass entries that parameterize mixing with the third generation.', '1805.08659-2-16-3': 'All [MATH] mixing is given by an effective [MATH] mixing.', '1805.08659-2-16-4': 'This is a consequence of the breaking of the [MATH] symmetry by only one set of Yukawa couplings.', '1805.08659-2-17-0': '## Couplings of the Higgs bosons', '1805.08659-2-18-0': 'Next, we discuss the couplings of the physical Higgs bosons in the four different models.', '1805.08659-2-18-1': 'We largely follow the notation and conventions in [CITATION] and state only the relevant results.', '1805.08659-2-19-0': 'The part of the Lagrangian that parametrizes the couplings to the three neutral scalars, [MATH], [MATH], and [MATH] (we identify [MATH] with the 125 GeV Higgs), as well as the charged Higgs [MATH] to mass eigenstate fermions is written as [EQUATION]', '1805.08659-2-19-1': 'For the flavor diagonal and off-diagonal couplings of the neutral Higgs bosons to leptons one finds [EQUATION] where we introduced the coupling modifiers [MATH] with respect to the SM Higgs couplings.', '1805.08659-2-19-2': 'We use the notation [MATH], [MATH], and [MATH].', '1805.08659-2-19-3': 'The angle [MATH] parametrizes the mixing between the neutral CP-even components of the two Higgs doublets and [MATH] is the ratio of Higgs vacuum expectation values.', '1805.08659-2-19-4': 'Completely analogous expressions hold for the neutral Higgs couplings to the up-type and down-type quarks.', '1805.08659-2-20-0': 'Ignoring neutrino mixing (which is of no relevance for our study) one finds for the charged Higgs couplings to leptons [EQUATION]', '1805.08659-2-20-1': 'In the expressions for the charged Higgs couplings to quarks, the CKM matrix [MATH] enters.', '1805.08659-2-20-2': 'We find [EQUATION]', '1805.08659-2-20-3': 'All of these expressions for the couplings are completely generic and can be applied to any of our flavorful models.', '1805.08659-2-20-4': 'The only terms that change in the different models are the [MATH] mass parameters, as given in Eqs. ([REF]), ([REF]), and ([REF]).', '1805.08659-2-21-0': 'In tables [REF], [REF], and [REF], we show the leading order coupling modifiers for the flavor diagonal couplings of the Higgs bosons [MATH] as an expansion in [MATH], where [MATH].', '1805.08659-2-21-1': 'We compare the coupling modifiers of all four flavorful 2HDM types to those of the four 2HDM types with natural flavor conservation.', '1805.08659-2-21-2': 'As is well known, the coupling modifiers are flavor universal in the models with natural flavor conservation.', '1805.08659-2-21-3': 'In the flavorful models the modifiers are flavor dependent and differentiate between the third generation and the first two generations.', '1805.08659-2-22-0': '# Light Higgs phenomenology', '1805.08659-2-23-0': '## Constraints from Higgs signal strength measurements', '1805.08659-2-24-0': 'The introduction of a second doublet alters the couplings to the 125 GeV Higgs boson [MATH] as shown in table [REF] as well as Eq. ([REF]).', '1805.08659-2-24-1': 'We can compare the Higgs production and decay rates predicted by our models to those measured by ATLAS and CMS in order to constrain the new physics parameter space.', '1805.08659-2-25-0': 'To determine the constraints from the measured Higgs signals we construct a [MATH] function [EQUATION] where [MATH], [MATH], and [MATH] are the experimental measurements, the Standard Model predictions, and flavorful 2HDM predictions for the production cross sections times branching ratio of the various measured channels.', '1805.08659-2-26-0': 'The ratios of experimental measurements and SM predictions that enter Eq. ([REF]) are given by the signal strength modifiers that are reported by ATLAS and CMS.', '1805.08659-2-26-1': 'The SM predictions for the production cross sections and branching ratios are taken from [CITATION].', '1805.08659-2-26-2': 'The ratios of BSM and SM predictions for individual channels can be obtained in a straight-forward way as functions of the coupling modifiers.', '1805.08659-2-26-3': 'For the gluon-gluon fusion production (ggf), vector boson fusion production (VBF), production in association with [MATH] and [MATH] bosons (Wh, Zh), and production in association with top quarks (tth), we have [EQUATION] where for the loop induced gluon-gluon fusion we take into account top and bottom contributions at 1-loop.', '1805.08659-2-26-4': 'For tree level decays, the partial widths simply scale with the appropriate coupling modifiers.', '1805.08659-2-26-5': 'In the case of the loop induced [MATH] decay width we take into account top and bottom contributions at 1-loop (We explicitly checked that loops with lighter quarks do not lead to any appreciable effects).', '1805.08659-2-26-6': 'For [MATH] we consider [MATH], top, and bottom loops.', '1805.08659-2-26-7': 'We neglect charged Higgs loops, that are typically tiny [CITATION] [EQUATION]', '1805.08659-2-26-8': 'The covariance matrix in Eq. ([REF]) contains the experimental uncertainties and (where available) the correlations among the uncertainties.', '1805.08659-2-26-9': 'We assume that theory uncertainties in the ratio of BSM and SM predictions are negligible compared to current experimental uncertainties.', '1805.08659-2-26-10': 'We take into account the Higgs signal strengths from the LHC run 1 combination [CITATION], as well as several individual run 2 results, in particular measurements of [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], and [MATH] [CITATION].', '1805.08659-2-26-11': 'We also include results on Higgs production is association with top quarks [CITATION].', '1805.08659-2-26-12': '(See [CITATION] for recent Higgs signal strength studies of 2HDMs with natural flavor conservation.)', '1805.08659-2-27-0': 'The couplings of [MATH] are largely determined by the parameters [MATH] and [MATH].', '1805.08659-2-27-1': 'Subleading corrections enter through the [MATH] mass parameters, see Eq. ([REF]).', '1805.08659-2-27-2': 'We use the [MATH] function to put constraints on the [MATH] and [MATH] parameters allowing the [MATH] coefficients in the subleading corrections to vary in the range [MATH].', '1805.08659-2-27-3': 'The allowed regions in the [MATH] vs. [MATH] plane that we obtain in this way are shown in Fig. [REF].', '1805.08659-2-27-4': 'The dark (light) green regions correspond to the [MATH] and [MATH] allowed regions (that we define as [MATH]) in the four flavorful models.', '1805.08659-2-27-5': 'We also compare these regions to the 2[MATH] constraint in the corresponding models with natural flavor conservation (dashed contours).', '1805.08659-2-27-6': 'The plot for the type 1B model updates the corresponding result in [CITATION].', '1805.08659-2-28-0': 'The couplings of the 125 GeV Higgs to the third generation fermions are already constrained by current data to be SM-like at the level of 10% - 20%.', '1805.08659-2-28-1': 'By coupling the [MATH] and/or [MATH] to the second doublet (as in the type 2B, lepton-specific B, and flipped B models), we therefore find the parameter space to be more strongly constrained than in the type 1B model.', '1805.08659-2-28-2': 'Note that in those models there are two distinct regions of parameter space: one region close to the alignment limit [MATH], where the mixing between the 125 GeV Higgs and heavy Higgs is tiny and all [MATH] couplings become SM-like, and a second narrow strip where the bottom and/or the tau coupling have opposite sign with respect to the SM prediction.', '1805.08659-2-28-3': 'The constraints for the type 2B and flipped B models are very similar, implying that the bottom coupling (which largely determines the total width of [MATH]) is the most important factor in determining the parameter space of these models.', '1805.08659-2-28-4': 'Generally, as [MATH] gets very large or very small the [MATH] values can deviate substantially from 1, resulting in strong constraints.', '1805.08659-2-28-5': 'Moderate values for [MATH] are the least constrained.', '1805.08659-2-29-0': 'Currently, the only decay of the Higgs into a non-third generation fermion which has been constrained in a relevant way at the LHC is the decay to [MATH] [CITATION].', '1805.08659-2-29-1': 'However, the current sensitivities to the [MATH] decay are not sufficient to impose strong constraints on our parameter space, yet.', '1805.08659-2-29-2': 'Future precise measurements of [MATH] can potentially constrain large parts of the open parameter space of the type 1B model.', '1805.08659-2-29-3': 'The type 2B and flipped B models will be mainly constrained by improved measurements of [MATH].', '1805.08659-2-29-4': 'For the lepton-specific B model, future precision measurements of [MATH] will give the most relevant constraints.', '1805.08659-2-30-0': '## Flavor violating decays', '1805.08659-2-31-0': 'Along with altering the flavor diagonal couplings of the light Higgs, the introduction of the second doublet also introduces flavor violating couplings of [MATH] to the fermions.', '1805.08659-2-31-1': 'We expect in our models a number of FCNC decays that are extremely suppressed in the SM, most notably rare top decays [MATH] and [MATH] as well as lepton flavor violating Higgs decays [MATH], [MATH], and [MATH].', '1805.08659-2-32-0': 'In all four flavorful models the branching ratio of [MATH] is given by [EQUATION]', '1805.08659-2-32-1': 'From our study of Higgs signal strength measurements described in section [REF] we find in all four flavorful models the constraint [MATH].', '1805.08659-2-32-2': 'Combined with the generic expectation [MATH], this implies that BR[MATH] is typically not larger than [MATH].', '1805.08659-2-32-3': 'While this is much larger than the SM prediction of [MATH] [CITATION], it is below the current and expected sensitivities at the LHC [CITATION].', '1805.08659-2-32-4': 'The decay [MATH] is further suppressed by the up quark mass and generically not larger than [MATH], i.e. far below any foreseeable experimental sensitivity.', '1805.08659-2-32-5': 'Rare top decays could have much larger branching ratios if the CKM matrix is generated in the up-sector.', '1805.08659-2-33-0': 'The branching ratio for the rare Higgs decay [MATH] is given by [EQUATION] where [MATH] MeV is the total Higgs width.', '1805.08659-2-33-1': 'This expression holds in all four flavorful models, and we generically expect branching ratios up to [MATH].', '1805.08659-2-33-2': 'This has to be compared to the current bounds on this branching ratio from CMS [CITATION] and ATLAS [CITATION] [EQUATION]', '1805.08659-2-33-3': 'Future searches for [MATH] will start to probe interesting new physics parameter space.', '1805.08659-2-34-0': 'In all our models, the branching ratio of [MATH] is suppressed by a factor of [MATH] compared to [MATH] and therefore outside the reach of foreseeable experiments.', '1805.08659-2-34-1': 'The branching ratio of [MATH] is further suppressed and generically not larger than [MATH].', '1805.08659-2-35-0': '# Heavy neutral Higgs production and decays', '1805.08659-2-36-0': 'We expect a distinct collider phenomenology for the heavy Higgs bosons in each of our models.', '1805.08659-2-36-1': 'In contrast to models with natural flavor conservation, flavor alignment, or minimal flavor violation [CITATION], the coupling modifiers of the heavy Higgs bosons to fermions are not flavor universal.', '1805.08659-2-36-2': 'The difference is particularly striking for moderate and large [MATH].', '1805.08659-2-36-3': 'As shown in table [REF], for [MATH] and [MATH], whenever the coupling to a third generation fermion is suppressed by a factor [MATH], the couplings to the corresponding first and second generation fermions are enhanced by a factor [MATH], and vice versa.', '1805.08659-2-36-4': 'Depending on the type of flavorful model, a specific set of fermions can dominate the decay of the heavy Higgs bosons and cause different types of production modes to be more or less relevant.', '1805.08659-2-36-5': 'In the following we will focus on the type 2B, lepton-specific B, and flipped B models.', '1805.08659-2-36-6': 'The collider phenomenology of the type 1B model has been discussed previously in [CITATION].', '1805.08659-2-37-0': 'For the numerical results that will be presented in this section as well as in the subsequent charged Higgs section we will consider a fixed set of [MATH] mass parameters.', '1805.08659-2-37-1': 'To choose [MATH] parameters in the up and lepton sectors, we start with the Yukawa textures from Eqs. ([REF]) and ([REF]) setting all free [MATH] parameters to [MATH].', '1805.08659-2-37-2': 'The precise values for [MATH] and [MATH] in Eqs. ([REF]) and ([REF]) are then fully determined by demanding that the mass eigenvalues reproduce the known fermion masses (we use [MATH] masses at a scale of 500 GeV).', '1805.08659-2-37-3': 'In the down sector, the entries in Eq. ([REF]) of [MATH], [MATH], and [MATH] are chosen to reproduce the CKM matrix.', '1805.08659-2-37-4': 'The [MATH] parameters in Eq. ([REF]) are determined by the known down quark masses, setting the remaining free [MATH] parameters to [MATH].', '1805.08659-2-37-5': 'Generically, choosing different [MATH] parameters does not lead to a qualitative change of the heavy neutral and charged Higgs phenomenology.', '1805.08659-2-37-6': 'We will discuss the quantitative impact of varying the [MATH] parameters where appropriate.', '1805.08659-2-38-0': '## Production cross sections', '1805.08659-2-39-0': 'As we have seen in section [REF], the type 2B, lepton-specific B, and flipped B models are strongly constrained by Higgs signal strength measurements.', '1805.08659-2-39-1': 'In order to have maximal freedom in choosing a value for [MATH], we will limit our discussion to the decoupling limit and thus set [MATH].', '1805.08659-2-39-2': 'In this limit the couplings of the heavy scalar and pseudoscalar Higgs to fermions are identical and, furthermore, their couplings to gauge bosons vanish.', '1805.08659-2-39-3': 'The main production modes of the heavy neutral Higgs bosons are therefore: gluon-gluon fusion, production in association with tops or bottoms, and direct production from a [MATH] initial state.', '1805.08659-2-39-4': 'Vector boson fusion and production in association with gauge bosons is absent.', '1805.08659-2-40-0': 'We compute the cross section of the [MATH] processes by convoluting the leading order parton level cross section with the appropriate MMHT 2014 quark parton distribution functions (PDF) [CITATION] [EQUATION] where [MATH] is the center of mass energy of the protons.', '1805.08659-2-40-1': 'We take into account [MATH], [MATH], [MATH], and [MATH] initial states.', '1805.08659-2-40-2': "Given the small couplings to the lighter quark generations, we find that the remaining possible quark combinations are always sub-dominant (despite the larger PDF's).", '1805.08659-2-40-3': 'We do not include higher order corrections where one or two [MATH] quarks appear in the final state, keeping in mind that such processes might modify our [MATH] and [MATH] results by an [MATH] amount [CITATION].', '1805.08659-2-40-4': 'In the type 2B and flipped B models, we expect that the [MATH] production is the most relevant for moderate and large [MATH], thanks to the enhanced couplings to bottom quarks.', '1805.08659-2-40-5': "Production from initial state charm benefits from slightly larger PDF's but is suppressed by the significantly smaller charm mass.", '1805.08659-2-40-6': 'In the lepton-specific B model instead, we expect [MATH] to dominate for moderate and large [MATH], as the couplings to bottom are suppressed.', '1805.08659-2-41-0': 'We estimate the [MATH] production cross section by scaling the corresponding cross section of a heavy Higgs with SM-like couplings from [CITATION] by the ratio of the leading order [MATH] partial width in our model and a heavy Higgs with SM-like couplings.', '1805.08659-2-41-1': 'We take the expression for the partial width from [CITATION].', '1805.08659-2-42-0': 'Top associated production arises from diagrams like those shown in Fig. [REF].', '1805.08659-2-42-1': 'The corresponding cross section is identical for all four flavorful models.', '1805.08659-2-42-2': "We use the cross section from [CITATION], which was obtained by summing over the initial state quarks [MATH] and [MATH] and convoluting the parton cross section with the appropriate PDF's.", '1805.08659-2-43-0': 'The plots on the left hand side of Fig. [REF] show the various production cross sections for the three considered types of models as function of [MATH], for fixed Higgs mass of [MATH] GeV, and [MATH].', '1805.08659-2-43-1': 'In the type 2B and flipped B models, production involving bottom quarks is typically most relevant, while in the lepton-specific B model either production from [MATH] or gluon-gluon fusion dominates.', '1805.08659-2-43-2': 'For large [MATH], the gluon-gluon fusion production is sub-dominant in all cases due to the suppressed Higgs coupling to tops.', '1805.08659-2-43-3': 'Gluon-gluon fusion is minimal for intermediate values of [MATH], where the heavy Higgs coupling to tops accidentally vanishes.', '1805.08659-2-43-4': 'The precise location of the minimum depends on the choice of [MATH] parameters and can shift by an [MATH] factor.', '1805.08659-2-43-5': 'For large and small [MATH] the shown production cross sections are robust with respect to [MATH] changes in the [MATH] parameters.', '1805.08659-2-44-0': 'Overall, the total production cross section of a heavy Higgs of mass 500 GeV ranges from several hundred [MATH] to several [MATH] in the type 2B and flipped B models, and from tens of [MATH] to several [MATH] in the lepton-specific B model.', '1805.08659-2-44-1': 'The results for the type 2B and flipped B models are very similar to the corresponding models with natural flavor conservation.', '1805.08659-2-44-2': 'The reason is that the dominant production modes are governed by the top and bottom couplings that behave very similarly in those type A and B models.', '1805.08659-2-44-3': 'The results for the lepton-specific B model, however, differ markedly from the corresponding results of the lepton-specific A model.', '1805.08659-2-44-4': 'In the type A model, all couplings to quarks are universally suppressed by [MATH] leading to tiny production cross sections.', '1805.08659-2-44-5': 'In the type B model the couplings to charm are enhanced, leading to an appreciable amount of heavy Higgs production.', '1805.08659-2-45-0': '## Branching ratios', '1805.08659-2-46-0': 'The heavy Higgs bosons can in principle decay to SM fermions, to the SM gauge bosons, and to other Higgs bosons.', '1805.08659-2-46-1': 'In the decoupling limit [MATH], the decays of [MATH] and [MATH] to final states with massive vector bosons vanish.', '1805.08659-2-46-2': 'Decays into photons and gluons are loop suppressed and typically tiny.', '1805.08659-2-46-3': 'We assume that the heavy Higgs bosons are sufficiently degenerate, such that decays into each other are kinematically forbidden.', '1805.08659-2-46-4': 'The decay into two light Higgs bosons is in principle possible.', '1805.08659-2-46-5': 'The corresponding trilinear couplings depend on the couplings in the Higgs potential and can be made arbitrarily small.', '1805.08659-2-46-6': 'In the following, we will only consider decays into fermions.', '1805.08659-2-46-7': 'Generically, the decay widths of the heavy scalar [MATH] to two fermions are [EQUATION] where we assumed that the mass of the fermions is negligible [MATH].', '1805.08659-2-46-8': 'The color factor is [MATH] for leptons and [MATH] for quarks.', '1805.08659-2-46-9': 'This expression is sufficiently generic to describe both flavor conserving and flavor violating decays.', '1805.08659-2-46-10': 'In the case where one or both of the fermions is a top quark, top mass effects have to be included [EQUATION]', '1805.08659-2-46-11': 'We show the branching ratios of the heavy Higgs as function of [MATH] in the plots on the right hand side of Fig. [REF].', '1805.08659-2-46-12': 'The heavy Higgs mass is set to [MATH] GeV and [MATH].', '1805.08659-2-46-13': 'The main decay modes of the heavy Higgs to the fermions are easily understood from table [REF], that shows to which fermions the [MATH] doublet couples.', '1805.08659-2-46-14': 'In the type 2B and flipped B models we expect the [MATH] decay to dominate at large [MATH].', '1805.08659-2-46-15': 'For the lepton-specific setup we expect the [MATH] decay to be the primary branching ratio.', '1805.08659-2-46-16': 'In the flipped B model, the [MATH] decay is instead strongly suppressed.', '1805.08659-2-46-17': 'For low [MATH], decays into [MATH] dominate (if kinematically allowed).', '1805.08659-2-46-18': 'These are the same patterns as in the models with natural flavor conservation.', '1805.08659-2-47-0': 'In contrast to the models with natural flavor conservation, decays involving charm quarks ([MATH] and [MATH]) can have branching ratios of [MATH] in all three flavorful models.', '1805.08659-2-47-1': 'Also the decay into [MATH] has branching ratios of several [MATH] for large [MATH], due to terms in the coupling of the heavy Higgs to tops that are proprotional to [MATH].', '1805.08659-2-47-2': 'For [MATH] there can be a cancellation between the leading [MATH] suppressed term and the [MATH] correction, leading to an accidental vanishing of the [MATH] branching ratio.', '1805.08659-2-48-0': 'Also lepton flavor violating decays can arise.', '1805.08659-2-48-1': 'In the lepton-specific B model, we find the decay [MATH] can have branching ratios of up to [MATH].', '1805.08659-2-48-2': 'In the type 2B and flipped B model, the branching ratio of this decay mode is smaller by a factor of few, as it has to compete with the dominant decay into [MATH].', '1805.08659-2-49-0': 'The branching ratios of flavor diagonal decay modes like [MATH], [MATH], and [MATH] are fairly robust against changes in the [MATH] mass parameters.', '1805.08659-2-49-1': 'The branching ratios of flavor violating decays can change by a factor of few if the relevant [MATH] parameters are modified by an [MATH] amount.', '1805.08659-2-50-0': 'In the decoupling limit, the scalar and pseudoscalar Higgs couplings are identical.', '1805.08659-2-50-1': 'Consequently, the production cross sections and branching ratios of the pseudoscalar Higgs are very similar to the scalar Higgs and we do not show the plots for the pseudoscalar.', '1805.08659-2-51-0': '## Constraints from direct searches', '1805.08659-2-52-0': 'Having examined the main production and decay modes of the heavy neutral Higgs bosons of the flavorful models we now compare results from current heavy Higgs searches at the LHC with the model predictions.', '1805.08659-2-52-1': 'We find the most relevant constraints come from', '1805.08659-2-53-0': 'In Fig. [REF] we show the ratio of the experimentally excluded rate [MATH] to the rate predicted in our flavorful 2HDMs [MATH] as function of the heavy Higgs mass for a benchmark scenario with [MATH] and [MATH].', '1805.08659-2-53-1': 'If this ratio is below 1, the model is excluded for the given set of parameters.', '1805.08659-2-54-0': 'Concerning, the experimental searches that target Higgs production in association with bottom quarks, we estimate the theoretical production cross section from [MATH], keeping in mind that higher order corrections might change the result by an [MATH] amount.', '1805.08659-2-54-1': 'The corresponding constraints in the plots of Fig. [REF] are labeled with the subscript "bbF".', '1805.08659-2-54-2': 'If experimental constraints assume gluon-gluon fusion production, we take into account both gluon-gluon fusion and also production from [MATH], which should lead to the same experimental signature.', '1805.08659-2-54-3': 'The corresponding constraints are labeled "ggF".', '1805.08659-2-54-4': 'If no particular production mode is singled out by the experimental search, we add up all the production mechanisms.', '1805.08659-2-54-5': 'For each individual channel we show the strongest constraint among the considered experimental analyses.', '1805.08659-2-55-0': 'We observe that for [MATH] the type 2B and the lepton-specific B models are strongly constrained by searches for heavy Higgs decaying to a [MATH] final state.', '1805.08659-2-55-1': 'Heavy Higgs masses up to [MATH] TeV (type 2B) and up to [MATH] GeV (lepton-specific B) are already excluded in this case.', '1805.08659-2-55-2': 'The constraints are much weaker in the flipped B model.', '1805.08659-2-55-3': 'Searches for di-muon, [MATH] and di-jet resonance searches have sensitivities that start to approach the model predictions, but currently do not exclude parameter space with [MATH].', '1805.08659-2-56-0': 'Note that the excluded mass ranges are extremely sensitive to the values of [MATH].', '1805.08659-2-56-1': 'For large [MATH] the production cross sections in all models are approximately proportional to [MATH].', '1805.08659-2-56-2': 'So, the cross section ratios quickly go below the exclusion line.', '1805.08659-2-56-3': 'However, as [MATH] becomes small the constraints generically get weaker and the constraints in the type 2B and lepton-specific B case can be easily avoided.', '1805.08659-2-57-0': '# Charged Higgs production and decays', '1805.08659-2-58-0': 'The collider phenomenology of the charged Higgs in the type 1B model has been discussed previously in [CITATION].', '1805.08659-2-58-1': 'Here we discuss the phenomenology of the charged Higgs in the type 2B, lepton-specific B, and flipped B models.', '1805.08659-2-59-0': '## Production cross sections', '1805.08659-2-60-0': 'As for the neutral Higgses, the main production mode is again primarily from [MATH] fusion.', '1805.08659-2-60-1': 'We estimate these cross sections using an expression analogous to Eq. ([REF]) along with the MMHT 2014 PDFs [CITATION].', '1805.08659-2-60-2': 'Also production in association with a top quark (see diagrams in Fig. [REF]) can become important.', '1805.08659-2-60-3': 'The corresponding production cross section is taken from [CITATION].', '1805.08659-2-61-0': 'We show the production cross sections as function of [MATH] in Fig. [REF].', '1805.08659-2-61-1': 'As an example, we use the charged Higgs mass [MATH] GeV and set [MATH].', '1805.08659-2-62-0': 'At low [MATH], the production in association with a top quark dominates in all three flavorful models.', '1805.08659-2-62-1': 'In the type 2B and flipped B models production in association with a top quark remains dominant also for large [MATH] due to the enhanced couplings to bottom in this region of parameter space.', '1805.08659-2-62-2': 'In the lepton-specific B model, however, large [MATH] implies suppression of both top and bottom couplings and the top associated charged Higgs production is suppressed.', '1805.08659-2-63-0': 'We find that the charged Higgs production from [MATH] fusion is dominated by initial states containing charm quarks.', '1805.08659-2-63-1': 'All three combinations [MATH], [MATH], and [MATH] have production cross sections of the same order of magnitude.', '1805.08659-2-63-2': 'While the coupling to [MATH] is suppressed by a factor of [MATH] compared to the [MATH] and [MATH] couplings, this suppression is partially compensated by the larger down PDF.', '1805.08659-2-63-3': 'Furthermore, the [MATH] production cross sections are mainly determined by couplings of the charged Higgs involving right handed charm quarks.', '1805.08659-2-63-4': 'Those couplings have the same scaling with [MATH] for all three flavorful models and we indeed observe that also the corresponding cross sections are approximately equal in the three models.', '1805.08659-2-64-0': 'This is particularly interesting for the lepton-specific B case.', '1805.08659-2-64-1': 'In the lepton-specific A model, all couplings to quarks are suppressed at large [MATH], and charged Higgs production is tiny.', '1805.08659-2-64-2': 'In the "B-type" of the model, however, the enhanced couplings to charm open up the possibility to directly probe this region of parameter space at the LHC.', '1805.08659-2-65-0': '## Branching Ratios', '1805.08659-2-66-0': 'In the considered scenario with [MATH], the charged Higgs decays either to quarks or leptons.', '1805.08659-2-66-1': 'The decay to [MATH] is absent.', '1805.08659-2-66-2': 'The decay rate to fermions is given analogous to the neutral Higgs, Eq. ([REF]).', '1805.08659-2-67-0': 'In the type 2B and flipped B models we expect the dominant branching ratio to be [MATH] both for small [MATH] (where the coupling to top is large) and at large [MATH] (where the coupling to bottom is enhanced).', '1805.08659-2-67-1': 'This can be clearly seen in the plots of Fig. [REF] that show the most relevant branching ratios as function of [MATH] for [MATH] GeV and [MATH].', '1805.08659-2-68-0': 'In the type 2B model, the [MATH] decay mode has the second largest branching ratio at large [MATH].', '1805.08659-2-68-1': 'This is very similar to the type 2A model with natural flavor conservation.', '1805.08659-2-68-2': 'In contrast to the type 2A, decay modes including charm quarks, like [MATH] and [MATH], can have branching ratios of several [MATH] in the flavorful type 2B model.', '1805.08659-2-68-3': 'Also in the flipped B model, [MATH] and [MATH] can have branching ratios of several [MATH].', '1805.08659-2-68-4': 'The decay to [MATH] on the other hand is strongly suppressed.', '1805.08659-2-68-5': 'The rather clean [MATH] final state can reach branching ratios of [MATH], which is orders of magnitude larger than in the flipped A model.', '1805.08659-2-69-0': 'In the lepton-specific B model, the branching ratio to [MATH] dominates at large [MATH] and is typically around [MATH].', '1805.08659-2-69-1': 'Decay modes involving charm ([MATH] and [MATH]) as well as top ([MATH] and [MATH]) have typical branching ratios of [MATH].', '1805.08659-2-70-0': 'For [MATH] above [MATH] most branching ratios stay approximately constant.', '1805.08659-2-70-1': 'One exception is the [MATH] branching ratio in the lepton-specific B model which changes considerably with [MATH].', '1805.08659-2-70-2': 'For [MATH] the relevant coupling of the charged Higgs to [MATH] vanishes, due to an accidental cancellation between the [MATH] term and the term of [MATH] in Eq. ([REF]).', '1805.08659-2-70-3': 'The same cancellation is also responsible for the dip in the top associated production in the lepton-specific B model shown on the left-hand side of Fig. [REF].', '1805.08659-2-70-4': 'The precise value of [MATH] where this cancellation happens depends on the sign and exact size of the free [MATH] parameters in the [MATH] mass parameters, see Eq. ([REF]).', '1805.08659-2-70-5': 'In general, variation of the [MATH] mass parameters can change the branching ratios of flavor violating decays by a factor of few.', '1805.08659-2-71-0': '## Constraints from direct searches', '1805.08659-2-72-0': 'The constraints in this section are implemented with the same process we used in section [REF].', '1805.08659-2-72-1': 'The strongest constraints come from', '1805.08659-2-73-0': 'For low mass charged Higgs at [MATH], the type 2B and flipped B models are ruled out due to [MATH] decays.', '1805.08659-2-73-1': 'However, in the lepton-specific B case the parameter space for charged Higgs bosons lighter than the top quark is still open, motivating continued search for charged Higgs bosons in top decays [MATH].', '1805.08659-2-73-2': 'For [MATH] the high mass region is still largely unconstrained.', '1805.08659-2-73-3': 'For the flipped B and type 2B models, searches for [MATH] need to improve by approximately an order of magnitude to begin to probe the high mass region.', '1805.08659-2-73-4': 'The type 2B and lepton-specific B models can also be probed by [MATH] searches if their sensitivities improve one order or magnitude in the future.', '1805.08659-2-74-0': '# Effects on flavor violating processes', '1805.08659-2-75-0': 'The flavor violating couplings of the neutral Higgs bosons also affect low energy flavor observables like meson mixing and rare meson decays.', '1805.08659-2-75-1': 'In the following we consider neutral [MATH] meson, Kaon, and [MATH] meson mixing as well as the branching ratios of several rare meson decays [MATH], [MATH], [MATH], [MATH], and [MATH].', '1805.08659-2-76-0': '## Meson oscillations', '1805.08659-2-77-0': 'The SM Higgs, as well as the heavy scalar and pseudoscalar Higgs add contributions to neutral [MATH] meson mixing at tree level.', '1805.08659-2-77-1': 'For the new physics contribution to the [MATH] mixing amplitude normalized to the SM amplitude we have [CITATION] [EQUATION] where [MATH] is a SM loop function.', '1805.08659-2-77-2': 'The corresponding expression for the [MATH] mixing amplitude is analogous.', '1805.08659-2-77-3': 'Note that this expression holds for all four flavorful 2HDMs.', '1805.08659-2-77-4': 'The [MATH] factors in Eq. ([REF]) contain leading order QCD running corrections and ratios of hadronic matrix elements [MATH], [MATH], [MATH], see [CITATION].', '1805.08659-2-77-5': 'The first value listed corresponds to [MATH] and the second to [MATH].', '1805.08659-2-77-6': 'From the above new physics contribution we can find values for the meson oscillation frequencies as well as the mixing phases [EQUATION]', '1805.08659-2-77-7': 'We confront our models with experimental constraints by constructing a [MATH] function that includes the mass differences and mixing phases in [MATH] and [MATH] mixing.', '1805.08659-2-77-8': 'The SM predictions and experimental results are taken from [CITATION] (see also [CITATION] for a recent discussion of [MATH] mixing constraints).', '1805.08659-2-77-9': 'Note that in our models the [MATH] and [MATH] mass parameters are largely fixed by the CKM matrix, see Eqs. ([REF]) and ([REF]).', '1805.08659-2-77-10': 'Thus we use the [MATH] mixing observables to constrain the free [MATH] and [MATH] mass parameters, setting [MATH] and [MATH] (with the signs depending on the type of flavorful model).', '1805.08659-2-78-0': 'In Fig. [REF] we show constraints on the absolute values and phases of [MATH] (left) and [MATH] (right) for a benchmark scenario with [MATH] (as favored by the Higgs signal strengths measurements, see section [REF]), [MATH], and [MATH] GeV.', '1805.08659-2-78-1': 'The constraints on the [MATH] parameter scale approximately as [MATH], i.e. they become weaker for larger Higgs masses and stronger for larger [MATH].', '1805.08659-2-78-2': 'The shown constraints hold in the type 1B and lepton-specific B models.', '1805.08659-2-78-3': 'In the type 2B and flipped B models, the [MATH] and [MATH] mass parameters have the opposite sign.', '1805.08659-2-78-4': 'This results in constraints that are shifted in phase by Arg[MATH] Arg[MATH].', '1805.08659-2-79-0': 'We observe that both [MATH] and [MATH] are strongly constrained by [MATH] and [MATH] mixing for large [MATH] and for heavy Higgs bosons below [MATH] TeV.', '1805.08659-2-79-1': 'The fact that these mass parameters have to be much smaller than the generic prediction of our flavor textures, [MATH] and [MATH] might call for an underlying flavor model.', '1805.08659-2-80-0': 'Similarly to [MATH] meson mixing, also the Kaon mixing amplitude obtains additional contributions.', '1805.08659-2-80-1': 'The new physics amplitude is [EQUATION] with the Kaon decay constant [MATH] MeV [CITATION].', '1805.08659-2-80-2': 'The bag parameters [MATH], [MATH], [MATH] are taken from [CITATION] (see also [CITATION]).', '1805.08659-2-80-3': 'The parameters [MATH], [MATH], and [MATH] (see [CITATION]) encode one loop renormalization group effects.', '1805.08659-2-81-0': 'The relevant observables in Kaon mixing are the mass difference [MATH] and the CP violating parameter [MATH].', '1805.08659-2-81-1': 'They can be calculated via [EQUATION] with [MATH] [CITATION].', '1805.08659-2-81-2': 'In Eqs. ([REF]) and ([REF]) we saw that the [MATH] parameters that are responsible for Kaon mixing are not independent parameters but given in terms of the parameters that govern [MATH] and [MATH] mixing.', '1805.08659-2-81-3': 'Given the constraints from [MATH] and [MATH] mixing, we find that new physics effects in Kaon mixing are generically below the current bounds.', '1805.08659-2-81-4': 'In particular, we find that new physics effects in [MATH] are at most at the permille level, while effects in [MATH] are [MATH].', '1805.08659-2-82-0': 'Analogously to Kaon mixing, the new physics contributions to neutral [MATH] meson mixing are given by [EQUATION]', '1805.08659-2-82-1': 'According to Eqs. ([REF]) and ([REF]), the [MATH] and [MATH] parameters are strongly suppressed, generically of the order of [MATH].', '1805.08659-2-82-2': 'We find that the resulting new physics contributions to the mixing amplitude are many orders of magnitude below the current sensitivities [CITATION] in all the models we consider.', '1805.08659-2-83-0': '## The rare [MATH] Bs -> mu+ mu- decay', '1805.08659-2-84-0': 'The rare FCNC decay [MATH] is known to be a highly sensitive probe of new physics (see e.g. [CITATION]).', '1805.08659-2-84-1': 'The decay has been observed at the LHC [CITATION] and the latest experimental result for the time integrated branching ratio from LHCb [CITATION] [EQUATION] agrees well with the SM prediction [CITATION] [EQUATION]', '1805.08659-2-84-2': 'A generic expression for the branching ratio in presence of NP reads [CITATION] [EQUATION] where [MATH] is the life-time difference of the [MATH] mesons, [MATH] [CITATION].', '1805.08659-2-84-3': 'In the above expression we do not consider corrections due to a possible non-standard [MATH] mixing phase [MATH] [CITATION].', '1805.08659-2-84-4': 'Given the existing constraint on [MATH] [CITATION], such corrections to the branching ratio are negligible.', '1805.08659-2-85-0': 'In the SM, the coefficients [MATH] and [MATH].', '1805.08659-2-85-1': 'Corrections due to tree level exchange of the neutral Higgs bosons are collected in the appendix [REF].', '1805.08659-2-85-2': 'As [MATH] meson mixing puts strong constraints on [MATH] we will set it to zero in the following discussion.', '1805.08659-2-85-3': 'In the alignment limit and for [MATH], as well as neglecting the life time difference, the expression for BR[MATH] simplifies to [EQUATION] with the SM Wilson coefficient [MATH].', '1805.08659-2-85-4': 'The plus (minus) sign in the first term holds in the type 1B and the lepton-specific B models (type 2B and flipped B models).', '1805.08659-2-85-5': 'Note that the [MATH] parameter is approximately given by [MATH] in the type 1B and flipped B models.', '1805.08659-2-85-6': 'In the type 2B and lepton-specific B models, [MATH] is a free parameter of [MATH].', '1805.08659-2-85-7': 'Consequently, we expect much more stringent constraints in the type 1B and flipped B models as compared to the type 2B and lepton-specific B models.', '1805.08659-2-86-0': 'In Fig. [REF] we show constraints in the plane of heavy Higgs mass [MATH] vs. [MATH] from [MATH] in the four flavorful models.', '1805.08659-2-86-1': 'In all four models we set [MATH] and [MATH].', '1805.08659-2-86-2': 'In the type 1B and flipped B models we set the (small) higher order corrections to [MATH] to zero, i.e. [MATH].', '1805.08659-2-86-3': 'In the type 2B and lepton-specific B models we set [MATH].', '1805.08659-2-87-0': 'The constraints in the type 2B and lepton-specific B models depend strongly on the choice of [MATH].', '1805.08659-2-87-1': 'If [MATH] accidentally vanishes, the [MATH] constraint can be even completely avoided in these models.', '1805.08659-2-87-2': 'The bounds in the type 1B and flipped B models, however, are robust.', '1805.08659-2-87-3': 'The higher order corrections to [MATH] modify them typically by [MATH] or less.', '1805.08659-2-87-4': 'In these models, the shown bounds from [MATH] can only be avoided by postulating that the CKM matrix is generated in the up-sector.', '1805.08659-2-88-0': 'In comparison to the constraints from direct searches we observe that [MATH] gives stronger bounds in the type 1B and flipped B models.', '1805.08659-2-88-1': 'In the type 2B and lepton-specific models, the direct searches in the [MATH] channel tend to be more constraining, instead.', '1805.08659-2-89-0': '## Lepton flavor violating B meson decays', '1805.08659-2-90-0': 'In the SM, the lepton flavor violating decays based on the [MATH] transition are suppressed by the tiny neutrino masses and are far below any imaginable experimental sensitivities.', '1805.08659-2-90-1': 'Observation of these decays would be clear sign of new physics.', '1805.08659-2-90-2': 'In our setup, tree level exchange of neutral Higgs bosons can induce these decays at levels that might become experimentally accessible.', '1805.08659-2-91-0': 'Similarly to the lepton flavor conserving decay [MATH] we express the branching ratio of the two body decay [MATH] as [EQUATION] where the last line takes into account the effect of a non-zero life time difference in the [MATH] system.', '1805.08659-2-91-1': 'An analogous expression holds for the decay [MATH].', '1805.08659-2-91-2': 'We will use the notation [MATH].', '1805.08659-2-91-3': 'The expressions for the coefficients [MATH] and [MATH] are collected in the appendix [REF].', '1805.08659-2-92-0': 'As in our discussion of the [MATH] decay, we set [MATH], [MATH], [MATH], and neglect the life time difference.', '1805.08659-2-92-1': 'In this case we find [EQUATION]', '1805.08659-2-92-2': 'This expression holds in all four flavorful 2HDMs.', '1805.08659-2-92-3': 'For all types we have [MATH].', '1805.08659-2-92-4': 'In the type 1B and the flipped B models, the possible values for BR[MATH] are bounded by the measured BR[MATH].', '1805.08659-2-92-5': 'Considering [MATH] and [MATH] GeV [MATH] TeV, we find the following upper bounds [EQUATION]', '1805.08659-2-92-6': 'Note that the given upper limits depend on the ranges of the [MATH] parameters that we have chosen and that we believe to be a representative example of the Yukawa structures that we consider in this work.', '1805.08659-2-92-7': 'For example, allowing [MATH] and [MATH] to be as large as [MATH] would result in branching ratios that are larger by almost a factor of 3 compared to the bounds quoted in Eq. ([REF]).', '1805.08659-2-93-0': 'In the type 2B and lepton-specific B models, the constraint from [MATH] is much weaker.', '1805.08659-2-93-1': 'In those models the strongest constraint comes from direct searches for the heavy Higgs bosons in the [MATH] channel (see Fig. [REF]).', '1805.08659-2-93-2': 'Values of BR[MATH] are possible in those models.', '1805.08659-2-94-0': 'Lepton flavor changing decays involving electrons on the other hand are tiny.', '1805.08659-2-94-1': 'Generically we expect in all models [EQUATION]', '1805.08659-2-94-2': 'In addition to the [MATH] decay, tree level exchange of flavor violating Higgs bosons also leads to three body semi-leptonic [MATH] meson decays like [MATH], [MATH], and [MATH].', '1805.08659-2-95-0': 'We find that the [MATH] and [MATH] branching ratios are directly correlated to the [MATH] branching ratio.', '1805.08659-2-95-1': 'Ignoring the life-time difference in the [MATH] system and using the results from [CITATION] (see also [CITATION] for a related study) we obtain for the differential branching ratio [EQUATION] where [MATH].', '1805.08659-2-95-2': 'An analogous expression holds for [MATH].', '1805.08659-2-95-3': 'For the [MATH] meson decay constant we use [MATH] MeV [CITATION].', '1805.08659-2-95-4': 'The [MATH] and [MATH] form factors [MATH] are taken from [CITATION].', '1805.08659-2-95-5': 'Integrating over [MATH], we find [EQUATION]', '1805.08659-2-95-6': 'Using the bounds and generic expectations for [MATH] in the different flavorful models discussed above, we find that [MATH] and [MATH] can be at most few [MATH] in the type 1B model and [MATH] in the flipped B model, respectively.', '1805.08659-2-95-7': 'In the type 2B and lepton-specific B models, however, these branching ratios can be as large as [MATH].', '1805.08659-2-96-0': 'We find similar results also for the [MATH] decay.', '1805.08659-2-96-1': 'The fact that [MATH] is a pseudoscalar to pseudoscalar transition, while [MATH] and [MATH] are pseudoscalar to vector transitions has little impact numerically.', '1805.08659-2-96-2': 'We find that [MATH] can be as large as few [MATH] in the type 1B model, [MATH] in the flipped B model, and [MATH] in the type 2B and lepton-specific B models.', '1805.08659-2-97-0': '# Conclusions', '1805.08659-2-98-0': 'Little is known experimentally about the tiny couplings of the Higgs boson to the light flavors of quarks and leptons.', '1805.08659-2-98-1': 'It is thus interesting to study possible alternative origins of mass for the light flavors beyond the 125 GeV Higgs boson.', '1805.08659-2-98-2': 'As an example, we analyzed a particular class of 2HDMs with non-trivial flavor structure.', '1805.08659-2-98-3': 'In analogy to the four, well studied 2HDMs with natural flavor conservation (NFC), we identified four models that preserve an approximate [MATH] flavor symmetry acting on the first two generations.', '1805.08659-2-98-4': 'We refer to them as type 1B, type 2B, lepton-specific B, and flipped B.', '1805.08659-2-98-5': 'In these flavorful 2HDMs, interesting flavor violating phenomena involving the third generation of fermions can be expected, while the [MATH] flavor symmetry still protects flavor violating transitions between the first and second generations.', '1805.08659-2-99-0': 'We studied the production and decay modes of the neutral and charged Higgs bosons of the models, as well as various low energy flavor violating observables, and identified the signatures of the flavorful models that are qualitatively different from the models with NFC.', '1805.08659-2-100-0': 'With regards to the collider phenomenology we find:', '1805.08659-2-101-0': 'The most interesting features in the flavor phenomenology are:'}

[['1805.08659-1-81-0', '1805.08659-2-81-0'], ['1805.08659-1-81-1', '1805.08659-2-81-1'], ['1805.08659-1-81-2', '1805.08659-2-81-2'], ['1805.08659-1-81-3', '1805.08659-2-81-3'], ['1805.08659-1-81-4', '1805.08659-2-81-4'], ['1805.08659-1-48-0', '1805.08659-2-48-0'], ['1805.08659-1-48-1', '1805.08659-2-48-1'], ['1805.08659-1-48-2', '1805.08659-2-48-2'], ['1805.08659-1-31-0', '1805.08659-2-31-0'], ['1805.08659-1-31-1', '1805.08659-2-31-1'], ['1805.08659-1-15-0', '1805.08659-2-15-0'], ['1805.08659-1-15-1', '1805.08659-2-15-1'], ['1805.08659-1-15-2', '1805.08659-2-15-2'], ['1805.08659-1-15-3', '1805.08659-2-15-4'], ['1805.08659-1-15-4', '1805.08659-2-15-5'], ['1805.08659-1-15-5', '1805.08659-2-15-6'], ['1805.08659-1-15-6', '1805.08659-2-15-7'], ['1805.08659-1-47-0', '1805.08659-2-47-0'], ['1805.08659-1-47-1', '1805.08659-2-47-1'], ['1805.08659-1-47-2', '1805.08659-2-47-2'], ['1805.08659-1-36-0', '1805.08659-2-36-0'], ['1805.08659-1-36-1', '1805.08659-2-36-1'], ['1805.08659-1-36-2', '1805.08659-2-36-2'], ['1805.08659-1-36-3', '1805.08659-2-36-3'], ['1805.08659-1-36-4', '1805.08659-2-36-4'], ['1805.08659-1-36-5', '1805.08659-2-36-5'], ['1805.08659-1-36-6', '1805.08659-2-36-6'], ['1805.08659-1-12-0', '1805.08659-2-12-0'], ['1805.08659-1-12-1', '1805.08659-2-12-1'], ['1805.08659-1-12-2', '1805.08659-2-12-2'], ['1805.08659-1-12-3', '1805.08659-2-12-3'], ['1805.08659-1-12-4', '1805.08659-2-12-4'], ['1805.08659-1-14-0', '1805.08659-2-14-0'], ['1805.08659-1-14-1', '1805.08659-2-14-1'], ['1805.08659-1-14-2', '1805.08659-2-14-2'], ['1805.08659-1-14-3', '1805.08659-2-14-3'], ['1805.08659-1-14-4', '1805.08659-2-14-4'], ['1805.08659-1-14-5', '1805.08659-2-14-5'], ['1805.08659-1-14-6', '1805.08659-2-14-6'], ['1805.08659-1-60-0', '1805.08659-2-60-0'], ['1805.08659-1-60-1', '1805.08659-2-60-1'], ['1805.08659-1-60-2', '1805.08659-2-60-2'], ['1805.08659-1-60-3', '1805.08659-2-60-3'], ['1805.08659-1-85-0', '1805.08659-2-85-0'], ['1805.08659-1-85-1', '1805.08659-2-85-1'], ['1805.08659-1-85-2', '1805.08659-2-85-2'], ['1805.08659-1-85-3', '1805.08659-2-85-3'], ['1805.08659-1-85-4', '1805.08659-2-85-4'], ['1805.08659-1-85-5', '1805.08659-2-85-5'], ['1805.08659-1-85-6', '1805.08659-2-85-6'], ['1805.08659-1-85-7', '1805.08659-2-85-7'], ['1805.08659-1-27-0', '1805.08659-2-27-0'], ['1805.08659-1-27-1', '1805.08659-2-27-1'], ['1805.08659-1-27-2', '1805.08659-2-27-2'], ['1805.08659-1-27-3', '1805.08659-2-27-3'], ['1805.08659-1-27-4', '1805.08659-2-27-4'], ['1805.08659-1-27-5', '1805.08659-2-27-5'], ['1805.08659-1-27-6', '1805.08659-2-27-6'], ['1805.08659-1-63-0', '1805.08659-2-63-0'], ['1805.08659-1-63-1', '1805.08659-2-63-1'], ['1805.08659-1-63-2', '1805.08659-2-63-2'], ['1805.08659-1-63-3', '1805.08659-2-63-3'], ['1805.08659-1-63-4', '1805.08659-2-63-4'], ['1805.08659-1-79-0', '1805.08659-2-79-0'], ['1805.08659-1-79-1', '1805.08659-2-79-1'], ['1805.08659-1-78-0', '1805.08659-2-78-0'], ['1805.08659-1-78-1', '1805.08659-2-78-1'], ['1805.08659-1-78-2', '1805.08659-2-78-2'], ['1805.08659-1-78-3', '1805.08659-2-78-3'], ['1805.08659-1-78-4', '1805.08659-2-78-4'], ['1805.08659-1-52-0', '1805.08659-2-52-0'], ['1805.08659-1-52-1', '1805.08659-2-52-1'], ['1805.08659-1-6-0', '1805.08659-2-6-0'], ['1805.08659-1-6-1', '1805.08659-2-6-1'], ['1805.08659-1-6-2', '1805.08659-2-6-2'], ['1805.08659-1-6-3', '1805.08659-2-6-3'], ['1805.08659-1-6-4', '1805.08659-2-6-4'], ['1805.08659-1-6-5', '1805.08659-2-6-5'], ['1805.08659-1-33-0', '1805.08659-2-33-0'], ['1805.08659-1-33-1', '1805.08659-2-33-1'], ['1805.08659-1-33-2', '1805.08659-2-33-2'], ['1805.08659-1-33-3', '1805.08659-2-33-3'], ['1805.08659-1-20-0', '1805.08659-2-20-0'], ['1805.08659-1-20-1', '1805.08659-2-20-1'], ['1805.08659-1-20-3', '1805.08659-2-20-3'], ['1805.08659-1-20-4', '1805.08659-2-20-4'], ['1805.08659-1-58-0', '1805.08659-2-58-0'], ['1805.08659-1-58-1', '1805.08659-2-58-1'], ['1805.08659-1-80-0', '1805.08659-2-80-0'], ['1805.08659-1-80-1', '1805.08659-2-80-1'], ['1805.08659-1-80-2', '1805.08659-2-80-2'], ['1805.08659-1-80-3', '1805.08659-2-80-3'], ['1805.08659-1-64-0', '1805.08659-2-64-0'], ['1805.08659-1-64-1', '1805.08659-2-64-1'], ['1805.08659-1-64-2', '1805.08659-2-64-2'], ['1805.08659-1-86-0', '1805.08659-2-86-0'], ['1805.08659-1-86-1', '1805.08659-2-86-1'], ['1805.08659-1-86-2', '1805.08659-2-86-2'], ['1805.08659-1-86-3', '1805.08659-2-86-3'], ['1805.08659-1-9-0', '1805.08659-2-9-0'], ['1805.08659-1-9-1', '1805.08659-2-9-1'], ['1805.08659-1-9-2', '1805.08659-2-9-2'], ['1805.08659-1-9-3', '1805.08659-2-9-3'], ['1805.08659-1-9-4', '1805.08659-2-9-4'], ['1805.08659-1-9-5', '1805.08659-2-9-5'], ['1805.08659-1-72-0', '1805.08659-2-72-0'], ['1805.08659-1-72-1', '1805.08659-2-72-1'], ['1805.08659-1-98-0', '1805.08659-2-99-0'], ['1805.08659-1-94-0', '1805.08659-2-95-0'], ['1805.08659-1-94-2', '1805.08659-2-95-2'], ['1805.08659-1-94-3', '1805.08659-2-95-3'], ['1805.08659-1-94-4', '1805.08659-2-95-4'], ['1805.08659-1-94-5', '1805.08659-2-95-5'], ['1805.08659-1-94-6', '1805.08659-2-95-6'], ['1805.08659-1-94-7', '1805.08659-2-95-7'], ['1805.08659-1-54-0', '1805.08659-2-54-0'], ['1805.08659-1-54-1', '1805.08659-2-54-1'], ['1805.08659-1-54-2', '1805.08659-2-54-2'], ['1805.08659-1-54-3', '1805.08659-2-54-3'], ['1805.08659-1-54-4', '1805.08659-2-54-4'], ['1805.08659-1-54-5', '1805.08659-2-54-5'], ['1805.08659-1-16-0', '1805.08659-2-16-0'], ['1805.08659-1-16-1', '1805.08659-2-16-1'], ['1805.08659-1-16-2', '1805.08659-2-16-2'], ['1805.08659-1-16-3', '1805.08659-2-16-3'], ['1805.08659-1-16-4', '1805.08659-2-16-4'], ['1805.08659-1-68-0', '1805.08659-2-68-0'], ['1805.08659-1-68-1', '1805.08659-2-68-1'], ['1805.08659-1-68-2', '1805.08659-2-68-2'], ['1805.08659-1-68-3', '1805.08659-2-68-3'], ['1805.08659-1-68-4', '1805.08659-2-68-4'], ['1805.08659-1-68-5', '1805.08659-2-68-5'], ['1805.08659-1-41-0', '1805.08659-2-41-0'], ['1805.08659-1-41-1', '1805.08659-2-41-1'], ['1805.08659-1-43-0', '1805.08659-2-43-0'], ['1805.08659-1-43-1', '1805.08659-2-43-1'], ['1805.08659-1-43-2', '1805.08659-2-43-2'], ['1805.08659-1-43-3', '1805.08659-2-43-3'], ['1805.08659-1-43-4', '1805.08659-2-43-4'], ['1805.08659-1-43-5', '1805.08659-2-43-5'], ['1805.08659-1-82-0', '1805.08659-2-82-0'], ['1805.08659-1-82-1', '1805.08659-2-82-1'], ['1805.08659-1-82-2', '1805.08659-2-82-2'], ['1805.08659-1-2-0', '1805.08659-2-2-0'], ['1805.08659-1-2-1', '1805.08659-2-2-1'], ['1805.08659-1-2-2', '1805.08659-2-2-2'], ['1805.08659-1-2-3', '1805.08659-2-2-3'], ['1805.08659-1-90-0', '1805.08659-2-90-0'], ['1805.08659-1-90-1', '1805.08659-2-90-1'], ['1805.08659-1-90-2', '1805.08659-2-90-2'], ['1805.08659-1-56-0', '1805.08659-2-56-0'], ['1805.08659-1-56-1', '1805.08659-2-56-1'], ['1805.08659-1-56-2', '1805.08659-2-56-2'], ['1805.08659-1-56-3', '1805.08659-2-56-3'], ['1805.08659-1-55-0', '1805.08659-2-55-0'], ['1805.08659-1-55-1', '1805.08659-2-55-1'], ['1805.08659-1-55-2', '1805.08659-2-55-2'], ['1805.08659-1-55-3', '1805.08659-2-55-3'], ['1805.08659-1-18-0', '1805.08659-2-18-0'], ['1805.08659-1-18-1', '1805.08659-2-18-1'], ['1805.08659-1-49-0', '1805.08659-2-49-0'], ['1805.08659-1-49-1', '1805.08659-2-49-1'], ['1805.08659-1-84-0', '1805.08659-2-84-0'], ['1805.08659-1-84-1', '1805.08659-2-84-1'], ['1805.08659-1-84-2', '1805.08659-2-84-2'], ['1805.08659-1-84-3', '1805.08659-2-84-3'], ['1805.08659-1-84-4', '1805.08659-2-84-4'], ['1805.08659-1-10-0', '1805.08659-2-10-0'], ['1805.08659-1-10-1', '1805.08659-2-10-1'], ['1805.08659-1-97-0', '1805.08659-2-98-0'], ['1805.08659-1-97-1', '1805.08659-2-98-1'], ['1805.08659-1-97-2', '1805.08659-2-98-2'], ['1805.08659-1-97-3', '1805.08659-2-98-3'], ['1805.08659-1-97-4', '1805.08659-2-98-4'], ['1805.08659-1-97-5', '1805.08659-2-98-5'], ['1805.08659-1-37-0', '1805.08659-2-37-0'], ['1805.08659-1-37-1', '1805.08659-2-37-1'], ['1805.08659-1-37-2', '1805.08659-2-37-2'], ['1805.08659-1-37-3', '1805.08659-2-37-3'], ['1805.08659-1-37-4', '1805.08659-2-37-4'], ['1805.08659-1-37-5', '1805.08659-2-37-5'], ['1805.08659-1-37-6', '1805.08659-2-37-6'], ['1805.08659-1-61-0', '1805.08659-2-61-0'], ['1805.08659-1-61-1', '1805.08659-2-61-1'], ['1805.08659-1-32-0', '1805.08659-2-32-0'], ['1805.08659-1-32-1', '1805.08659-2-32-1'], ['1805.08659-1-32-2', '1805.08659-2-32-2'], ['1805.08659-1-32-3', '1805.08659-2-32-3'], ['1805.08659-1-32-4', '1805.08659-2-32-4'], ['1805.08659-1-32-5', '1805.08659-2-32-5'], ['1805.08659-1-21-0', '1805.08659-2-21-0'], ['1805.08659-1-21-1', '1805.08659-2-21-1'], ['1805.08659-1-21-2', '1805.08659-2-21-2'], ['1805.08659-1-21-3', '1805.08659-2-21-3'], ['1805.08659-1-5-0', '1805.08659-2-5-0'], ['1805.08659-1-5-1', '1805.08659-2-5-1'], ['1805.08659-1-5-2', '1805.08659-2-5-2'], ['1805.08659-1-62-0', '1805.08659-2-62-0'], ['1805.08659-1-62-1', '1805.08659-2-62-1'], ['1805.08659-1-62-2', '1805.08659-2-62-2'], ['1805.08659-1-3-0', '1805.08659-2-3-0'], ['1805.08659-1-3-1', '1805.08659-2-3-1'], ['1805.08659-1-3-2', '1805.08659-2-3-2'], ['1805.08659-1-3-3', '1805.08659-2-3-3'], ['1805.08659-1-3-4', '1805.08659-2-3-4'], ['1805.08659-1-40-0', '1805.08659-2-40-0'], ['1805.08659-1-40-1', '1805.08659-2-40-1'], ['1805.08659-1-40-2', '1805.08659-2-40-2'], ['1805.08659-1-40-3', '1805.08659-2-40-3'], ['1805.08659-1-40-4', '1805.08659-2-40-4'], ['1805.08659-1-40-5', '1805.08659-2-40-5'], ['1805.08659-1-40-6', '1805.08659-2-40-6'], ['1805.08659-1-67-0', '1805.08659-2-67-0'], ['1805.08659-1-67-1', '1805.08659-2-67-1'], ['1805.08659-1-69-0', '1805.08659-2-69-0'], ['1805.08659-1-69-1', '1805.08659-2-69-1'], ['1805.08659-1-50-0', '1805.08659-2-50-0'], ['1805.08659-1-50-1', '1805.08659-2-50-1'], ['1805.08659-1-91-0', '1805.08659-2-91-0'], ['1805.08659-1-91-1', '1805.08659-2-91-1'], ['1805.08659-1-91-2', '1805.08659-2-91-2'], ['1805.08659-1-91-3', '1805.08659-2-91-3'], ['1805.08659-1-93-0', '1805.08659-2-94-0'], ['1805.08659-1-93-1', '1805.08659-2-94-1'], ['1805.08659-1-93-2', '1805.08659-2-94-2'], ['1805.08659-1-13-0', '1805.08659-2-13-0'], ['1805.08659-1-13-1', '1805.08659-2-13-1'], ['1805.08659-1-13-2', '1805.08659-2-13-2'], ['1805.08659-1-13-4', '1805.08659-2-13-5'], ['1805.08659-1-13-6', '1805.08659-2-13-7'], ['1805.08659-1-7-0', '1805.08659-2-7-0'], ['1805.08659-1-7-1', '1805.08659-2-7-1'], ['1805.08659-1-7-2', '1805.08659-2-7-2'], ['1805.08659-1-7-3', '1805.08659-2-7-3'], ['1805.08659-1-7-4', '1805.08659-2-7-4'], ['1805.08659-1-7-5', '1805.08659-2-7-5'], ['1805.08659-1-73-0', '1805.08659-2-73-0'], ['1805.08659-1-73-1', '1805.08659-2-73-1'], ['1805.08659-1-73-2', '1805.08659-2-73-2'], ['1805.08659-1-73-3', '1805.08659-2-73-3'], ['1805.08659-1-73-4', '1805.08659-2-73-4'], ['1805.08659-1-34-0', '1805.08659-2-34-0'], ['1805.08659-1-34-1', '1805.08659-2-34-1'], ['1805.08659-1-87-0', '1805.08659-2-87-0'], ['1805.08659-1-87-1', '1805.08659-2-87-1'], ['1805.08659-1-87-2', '1805.08659-2-87-2'], ['1805.08659-1-87-3', '1805.08659-2-87-3'], ['1805.08659-1-87-4', '1805.08659-2-87-4'], ['1805.08659-1-44-0', '1805.08659-2-44-0'], ['1805.08659-1-44-1', '1805.08659-2-44-1'], ['1805.08659-1-44-2', '1805.08659-2-44-2'], ['1805.08659-1-44-3', '1805.08659-2-44-3'], ['1805.08659-1-44-4', '1805.08659-2-44-4'], ['1805.08659-1-44-5', '1805.08659-2-44-5'], ['1805.08659-1-4-0', '1805.08659-2-4-0'], ['1805.08659-1-4-1', '1805.08659-2-4-1'], ['1805.08659-1-4-2', '1805.08659-2-4-2'], ['1805.08659-1-53-0', '1805.08659-2-53-0'], ['1805.08659-1-53-1', '1805.08659-2-53-1'], ['1805.08659-1-29-0', '1805.08659-2-29-0'], ['1805.08659-1-29-1', '1805.08659-2-29-1'], ['1805.08659-1-29-2', '1805.08659-2-29-2'], ['1805.08659-1-29-3', '1805.08659-2-29-3'], ['1805.08659-1-29-4', '1805.08659-2-29-4'], ['1805.08659-1-66-0', '1805.08659-2-66-0'], ['1805.08659-1-66-1', '1805.08659-2-66-1'], ['1805.08659-1-66-2', '1805.08659-2-66-2'], ['1805.08659-1-77-0', '1805.08659-2-77-0'], ['1805.08659-1-77-1', '1805.08659-2-77-1'], ['1805.08659-1-77-2', '1805.08659-2-77-2'], ['1805.08659-1-77-3', '1805.08659-2-77-3'], ['1805.08659-1-77-4', '1805.08659-2-77-4'], ['1805.08659-1-77-5', '1805.08659-2-77-5'], ['1805.08659-1-77-6', '1805.08659-2-77-6'], ['1805.08659-1-77-7', '1805.08659-2-77-7'], ['1805.08659-1-77-8', '1805.08659-2-77-8'], ['1805.08659-1-77-9', '1805.08659-2-77-9'], ['1805.08659-1-77-10', '1805.08659-2-77-10'], ['1805.08659-1-24-0', '1805.08659-2-24-0'], ['1805.08659-1-24-1', '1805.08659-2-24-1'], ['1805.08659-1-25-0', '1805.08659-2-25-0'], ['1805.08659-1-39-0', '1805.08659-2-39-0'], ['1805.08659-1-39-1', '1805.08659-2-39-1'], ['1805.08659-1-39-2', '1805.08659-2-39-2'], ['1805.08659-1-39-3', '1805.08659-2-39-3'], ['1805.08659-1-39-4', '1805.08659-2-39-4'], ['1805.08659-1-88-0', '1805.08659-2-88-0'], ['1805.08659-1-88-1', '1805.08659-2-88-1'], ['1805.08659-1-0-0', '1805.08659-2-0-0'], ['1805.08659-1-0-1', '1805.08659-2-0-1'], ['1805.08659-1-0-2', '1805.08659-2-0-2'], ['1805.08659-1-0-3', '1805.08659-2-0-3'], ['1805.08659-1-0-4', '1805.08659-2-0-4'], ['1805.08659-1-0-5', '1805.08659-2-0-5'], ['1805.08659-1-0-6', '1805.08659-2-0-6'], ['1805.08659-1-42-0', '1805.08659-2-42-0'], ['1805.08659-1-42-1', '1805.08659-2-42-1'], ['1805.08659-1-42-2', '1805.08659-2-42-2'], ['1805.08659-1-26-0', '1805.08659-2-26-0'], ['1805.08659-1-26-1', '1805.08659-2-26-1'], ['1805.08659-1-26-2', '1805.08659-2-26-2'], ['1805.08659-1-26-3', '1805.08659-2-26-3'], ['1805.08659-1-26-4', '1805.08659-2-26-4'], ['1805.08659-1-26-5', '1805.08659-2-26-5'], ['1805.08659-1-26-6', '1805.08659-2-26-6'], ['1805.08659-1-26-7', '1805.08659-2-26-7'], ['1805.08659-1-26-8', '1805.08659-2-26-8'], ['1805.08659-1-26-9', '1805.08659-2-26-9'], ['1805.08659-1-26-10', '1805.08659-2-26-10'], ['1805.08659-1-26-11', '1805.08659-2-26-11'], ['1805.08659-1-26-12', '1805.08659-2-26-12'], ['1805.08659-1-46-0', '1805.08659-2-46-0'], ['1805.08659-1-46-1', '1805.08659-2-46-1'], ['1805.08659-1-46-2', '1805.08659-2-46-2'], ['1805.08659-1-46-3', '1805.08659-2-46-3'], ['1805.08659-1-46-4', '1805.08659-2-46-4'], ['1805.08659-1-46-5', '1805.08659-2-46-5'], ['1805.08659-1-46-6', '1805.08659-2-46-6'], ['1805.08659-1-46-7', '1805.08659-2-46-7'], ['1805.08659-1-46-8', '1805.08659-2-46-8'], ['1805.08659-1-46-9', '1805.08659-2-46-9'], ['1805.08659-1-46-10', '1805.08659-2-46-10'], ['1805.08659-1-46-11', '1805.08659-2-46-11'], ['1805.08659-1-46-12', '1805.08659-2-46-12'], ['1805.08659-1-46-13', '1805.08659-2-46-13'], ['1805.08659-1-46-14', '1805.08659-2-46-14'], ['1805.08659-1-46-15', '1805.08659-2-46-15'], ['1805.08659-1-46-16', '1805.08659-2-46-16'], ['1805.08659-1-46-17', '1805.08659-2-46-17'], ['1805.08659-1-46-18', '1805.08659-2-46-18'], ['1805.08659-1-19-0', '1805.08659-2-19-0'], ['1805.08659-1-19-1', '1805.08659-2-19-1'], ['1805.08659-1-19-2', '1805.08659-2-19-2'], ['1805.08659-1-19-3', '1805.08659-2-19-3'], ['1805.08659-1-19-4', '1805.08659-2-19-4'], ['1805.08659-1-95-0', '1805.08659-2-96-0'], ['1805.08659-1-95-1', '1805.08659-2-96-1'], ['1805.08659-1-95-2', '1805.08659-2-96-2'], ['1805.08659-1-28-0', '1805.08659-2-28-0'], ['1805.08659-1-28-1', '1805.08659-2-28-1'], ['1805.08659-1-28-2', '1805.08659-2-28-2'], ['1805.08659-1-28-3', '1805.08659-2-28-3'], ['1805.08659-1-28-4', '1805.08659-2-28-4'], ['1805.08659-1-28-5', '1805.08659-2-28-5'], ['1805.08659-1-70-0', '1805.08659-2-70-0'], ['1805.08659-1-70-1', '1805.08659-2-70-1'], ['1805.08659-1-70-2', '1805.08659-2-70-2'], ['1805.08659-1-70-3', '1805.08659-2-70-3'], ['1805.08659-1-70-4', '1805.08659-2-70-4'], ['1805.08659-1-70-5', '1805.08659-2-70-5'], ['1805.08659-1-75-0', '1805.08659-2-75-0'], ['1805.08659-1-75-1', '1805.08659-2-75-1'], ['1805.08659-1-92-0', '1805.08659-2-92-0'], ['1805.08659-1-92-1', '1805.08659-2-92-1'], ['1805.08659-1-92-2', '1805.08659-2-92-2'], ['1805.08659-1-92-3', '1805.08659-2-92-3'], ['1805.08659-1-92-4', '1805.08659-2-92-4'], ['1805.08659-1-92-5', '1805.08659-2-92-5'], ['1805.08659-1-92-6', '1805.08659-2-93-0'], ['1805.08659-1-92-7', '1805.08659-2-93-1'], ['1805.08659-1-92-8', '1805.08659-2-93-2'], ['1805.08659-1-94-1', '1805.08659-2-95-1'], ['1805.08659-1-13-3', '1805.08659-2-13-4'], ['1805.08659-1-13-5', '1805.08659-2-13-6']]

[['1805.08659-1-81-0', '1805.08659-2-81-0'], ['1805.08659-1-81-1', '1805.08659-2-81-1'], ['1805.08659-1-81-2', '1805.08659-2-81-2'], ['1805.08659-1-81-3', '1805.08659-2-81-3'], ['1805.08659-1-81-4', '1805.08659-2-81-4'], ['1805.08659-1-48-0', '1805.08659-2-48-0'], ['1805.08659-1-48-1', '1805.08659-2-48-1'], ['1805.08659-1-48-2', '1805.08659-2-48-2'], ['1805.08659-1-31-0', '1805.08659-2-31-0'], ['1805.08659-1-31-1', '1805.08659-2-31-1'], ['1805.08659-1-15-0', '1805.08659-2-15-0'], ['1805.08659-1-15-1', '1805.08659-2-15-1'], ['1805.08659-1-15-2', '1805.08659-2-15-2'], ['1805.08659-1-15-3', '1805.08659-2-15-4'], ['1805.08659-1-15-4', '1805.08659-2-15-5'], ['1805.08659-1-15-5', '1805.08659-2-15-6'], ['1805.08659-1-15-6', '1805.08659-2-15-7'], ['1805.08659-1-47-0', '1805.08659-2-47-0'], ['1805.08659-1-47-1', '1805.08659-2-47-1'], ['1805.08659-1-47-2', '1805.08659-2-47-2'], ['1805.08659-1-36-0', '1805.08659-2-36-0'], ['1805.08659-1-36-1', '1805.08659-2-36-1'], ['1805.08659-1-36-2', '1805.08659-2-36-2'], ['1805.08659-1-36-3', '1805.08659-2-36-3'], ['1805.08659-1-36-4', '1805.08659-2-36-4'], ['1805.08659-1-36-5', '1805.08659-2-36-5'], ['1805.08659-1-36-6', '1805.08659-2-36-6'], ['1805.08659-1-12-0', '1805.08659-2-12-0'], ['1805.08659-1-12-1', '1805.08659-2-12-1'], ['1805.08659-1-12-2', '1805.08659-2-12-2'], ['1805.08659-1-12-3', '1805.08659-2-12-3'], ['1805.08659-1-12-4', '1805.08659-2-12-4'], ['1805.08659-1-14-0', '1805.08659-2-14-0'], ['1805.08659-1-14-1', '1805.08659-2-14-1'], ['1805.08659-1-14-2', '1805.08659-2-14-2'], ['1805.08659-1-14-3', '1805.08659-2-14-3'], ['1805.08659-1-14-4', '1805.08659-2-14-4'], ['1805.08659-1-14-5', '1805.08659-2-14-5'], ['1805.08659-1-14-6', '1805.08659-2-14-6'], ['1805.08659-1-60-0', '1805.08659-2-60-0'], ['1805.08659-1-60-1', '1805.08659-2-60-1'], ['1805.08659-1-60-2', '1805.08659-2-60-2'], ['1805.08659-1-60-3', '1805.08659-2-60-3'], ['1805.08659-1-85-0', '1805.08659-2-85-0'], ['1805.08659-1-85-1', '1805.08659-2-85-1'], ['1805.08659-1-85-2', '1805.08659-2-85-2'], ['1805.08659-1-85-3', '1805.08659-2-85-3'], ['1805.08659-1-85-4', '1805.08659-2-85-4'], ['1805.08659-1-85-5', '1805.08659-2-85-5'], ['1805.08659-1-85-6', '1805.08659-2-85-6'], ['1805.08659-1-85-7', '1805.08659-2-85-7'], ['1805.08659-1-27-0', '1805.08659-2-27-0'], ['1805.08659-1-27-1', '1805.08659-2-27-1'], ['1805.08659-1-27-2', '1805.08659-2-27-2'], ['1805.08659-1-27-3', '1805.08659-2-27-3'], ['1805.08659-1-27-4', '1805.08659-2-27-4'], ['1805.08659-1-27-5', '1805.08659-2-27-5'], ['1805.08659-1-27-6', '1805.08659-2-27-6'], ['1805.08659-1-63-0', '1805.08659-2-63-0'], ['1805.08659-1-63-1', '1805.08659-2-63-1'], ['1805.08659-1-63-2', '1805.08659-2-63-2'], ['1805.08659-1-63-3', '1805.08659-2-63-3'], ['1805.08659-1-63-4', '1805.08659-2-63-4'], ['1805.08659-1-79-0', '1805.08659-2-79-0'], ['1805.08659-1-79-1', '1805.08659-2-79-1'], ['1805.08659-1-78-0', '1805.08659-2-78-0'], ['1805.08659-1-78-1', '1805.08659-2-78-1'], ['1805.08659-1-78-2', '1805.08659-2-78-2'], ['1805.08659-1-78-3', '1805.08659-2-78-3'], ['1805.08659-1-78-4', '1805.08659-2-78-4'], ['1805.08659-1-52-0', '1805.08659-2-52-0'], ['1805.08659-1-52-1', '1805.08659-2-52-1'], ['1805.08659-1-6-0', '1805.08659-2-6-0'], ['1805.08659-1-6-1', '1805.08659-2-6-1'], ['1805.08659-1-6-2', '1805.08659-2-6-2'], ['1805.08659-1-6-3', '1805.08659-2-6-3'], ['1805.08659-1-6-4', '1805.08659-2-6-4'], ['1805.08659-1-6-5', '1805.08659-2-6-5'], ['1805.08659-1-33-0', '1805.08659-2-33-0'], ['1805.08659-1-33-1', '1805.08659-2-33-1'], ['1805.08659-1-33-2', '1805.08659-2-33-2'], ['1805.08659-1-33-3', '1805.08659-2-33-3'], ['1805.08659-1-20-0', '1805.08659-2-20-0'], ['1805.08659-1-20-1', '1805.08659-2-20-1'], ['1805.08659-1-20-3', '1805.08659-2-20-3'], ['1805.08659-1-20-4', '1805.08659-2-20-4'], ['1805.08659-1-58-0', '1805.08659-2-58-0'], ['1805.08659-1-58-1', '1805.08659-2-58-1'], ['1805.08659-1-80-0', '1805.08659-2-80-0'], ['1805.08659-1-80-1', '1805.08659-2-80-1'], ['1805.08659-1-80-2', '1805.08659-2-80-2'], ['1805.08659-1-80-3', '1805.08659-2-80-3'], ['1805.08659-1-64-0', '1805.08659-2-64-0'], ['1805.08659-1-64-1', '1805.08659-2-64-1'], ['1805.08659-1-64-2', '1805.08659-2-64-2'], ['1805.08659-1-86-0', '1805.08659-2-86-0'], ['1805.08659-1-86-1', '1805.08659-2-86-1'], ['1805.08659-1-86-2', '1805.08659-2-86-2'], ['1805.08659-1-86-3', '1805.08659-2-86-3'], ['1805.08659-1-9-0', '1805.08659-2-9-0'], ['1805.08659-1-9-1', '1805.08659-2-9-1'], ['1805.08659-1-9-2', '1805.08659-2-9-2'], ['1805.08659-1-9-3', '1805.08659-2-9-3'], ['1805.08659-1-9-4', '1805.08659-2-9-4'], ['1805.08659-1-9-5', '1805.08659-2-9-5'], ['1805.08659-1-72-0', '1805.08659-2-72-0'], ['1805.08659-1-72-1', '1805.08659-2-72-1'], ['1805.08659-1-98-0', '1805.08659-2-99-0'], ['1805.08659-1-94-0', '1805.08659-2-95-0'], ['1805.08659-1-94-2', '1805.08659-2-95-2'], ['1805.08659-1-94-3', '1805.08659-2-95-3'], ['1805.08659-1-94-4', '1805.08659-2-95-4'], ['1805.08659-1-94-5', '1805.08659-2-95-5'], ['1805.08659-1-94-6', '1805.08659-2-95-6'], ['1805.08659-1-94-7', '1805.08659-2-95-7'], ['1805.08659-1-54-0', '1805.08659-2-54-0'], ['1805.08659-1-54-1', '1805.08659-2-54-1'], ['1805.08659-1-54-2', '1805.08659-2-54-2'], ['1805.08659-1-54-3', '1805.08659-2-54-3'], ['1805.08659-1-54-4', '1805.08659-2-54-4'], ['1805.08659-1-54-5', '1805.08659-2-54-5'], ['1805.08659-1-16-0', '1805.08659-2-16-0'], ['1805.08659-1-16-1', '1805.08659-2-16-1'], ['1805.08659-1-16-2', '1805.08659-2-16-2'], ['1805.08659-1-16-3', '1805.08659-2-16-3'], ['1805.08659-1-16-4', '1805.08659-2-16-4'], ['1805.08659-1-68-0', '1805.08659-2-68-0'], ['1805.08659-1-68-1', '1805.08659-2-68-1'], ['1805.08659-1-68-2', '1805.08659-2-68-2'], ['1805.08659-1-68-3', '1805.08659-2-68-3'], ['1805.08659-1-68-4', '1805.08659-2-68-4'], ['1805.08659-1-68-5', '1805.08659-2-68-5'], ['1805.08659-1-41-0', '1805.08659-2-41-0'], ['1805.08659-1-41-1', '1805.08659-2-41-1'], ['1805.08659-1-43-0', '1805.08659-2-43-0'], ['1805.08659-1-43-1', '1805.08659-2-43-1'], ['1805.08659-1-43-2', '1805.08659-2-43-2'], ['1805.08659-1-43-3', '1805.08659-2-43-3'], ['1805.08659-1-43-4', '1805.08659-2-43-4'], ['1805.08659-1-43-5', '1805.08659-2-43-5'], ['1805.08659-1-82-0', '1805.08659-2-82-0'], ['1805.08659-1-82-1', '1805.08659-2-82-1'], ['1805.08659-1-82-2', '1805.08659-2-82-2'], ['1805.08659-1-2-0', '1805.08659-2-2-0'], ['1805.08659-1-2-1', '1805.08659-2-2-1'], ['1805.08659-1-2-2', '1805.08659-2-2-2'], ['1805.08659-1-2-3', '1805.08659-2-2-3'], ['1805.08659-1-90-0', '1805.08659-2-90-0'], ['1805.08659-1-90-1', '1805.08659-2-90-1'], ['1805.08659-1-90-2', '1805.08659-2-90-2'], ['1805.08659-1-56-0', '1805.08659-2-56-0'], ['1805.08659-1-56-1', '1805.08659-2-56-1'], ['1805.08659-1-56-2', '1805.08659-2-56-2'], ['1805.08659-1-56-3', '1805.08659-2-56-3'], ['1805.08659-1-55-0', '1805.08659-2-55-0'], ['1805.08659-1-55-1', '1805.08659-2-55-1'], ['1805.08659-1-55-2', '1805.08659-2-55-2'], ['1805.08659-1-55-3', '1805.08659-2-55-3'], ['1805.08659-1-18-0', '1805.08659-2-18-0'], ['1805.08659-1-18-1', '1805.08659-2-18-1'], ['1805.08659-1-49-0', '1805.08659-2-49-0'], ['1805.08659-1-49-1', '1805.08659-2-49-1'], ['1805.08659-1-84-0', '1805.08659-2-84-0'], ['1805.08659-1-84-1', '1805.08659-2-84-1'], ['1805.08659-1-84-2', '1805.08659-2-84-2'], ['1805.08659-1-84-3', '1805.08659-2-84-3'], ['1805.08659-1-84-4', '1805.08659-2-84-4'], ['1805.08659-1-10-0', '1805.08659-2-10-0'], ['1805.08659-1-10-1', '1805.08659-2-10-1'], ['1805.08659-1-97-0', '1805.08659-2-98-0'], ['1805.08659-1-97-1', '1805.08659-2-98-1'], ['1805.08659-1-97-2', '1805.08659-2-98-2'], ['1805.08659-1-97-3', '1805.08659-2-98-3'], ['1805.08659-1-97-4', '1805.08659-2-98-4'], ['1805.08659-1-97-5', '1805.08659-2-98-5'], ['1805.08659-1-37-0', '1805.08659-2-37-0'], ['1805.08659-1-37-1', '1805.08659-2-37-1'], ['1805.08659-1-37-2', '1805.08659-2-37-2'], ['1805.08659-1-37-3', '1805.08659-2-37-3'], ['1805.08659-1-37-4', '1805.08659-2-37-4'], ['1805.08659-1-37-5', '1805.08659-2-37-5'], ['1805.08659-1-37-6', '1805.08659-2-37-6'], ['1805.08659-1-61-0', '1805.08659-2-61-0'], ['1805.08659-1-61-1', '1805.08659-2-61-1'], ['1805.08659-1-32-0', '1805.08659-2-32-0'], ['1805.08659-1-32-1', '1805.08659-2-32-1'], ['1805.08659-1-32-2', '1805.08659-2-32-2'], ['1805.08659-1-32-3', '1805.08659-2-32-3'], ['1805.08659-1-32-4', '1805.08659-2-32-4'], ['1805.08659-1-32-5', '1805.08659-2-32-5'], ['1805.08659-1-21-0', '1805.08659-2-21-0'], ['1805.08659-1-21-1', '1805.08659-2-21-1'], ['1805.08659-1-21-2', '1805.08659-2-21-2'], ['1805.08659-1-21-3', '1805.08659-2-21-3'], ['1805.08659-1-5-0', '1805.08659-2-5-0'], ['1805.08659-1-5-1', '1805.08659-2-5-1'], ['1805.08659-1-5-2', '1805.08659-2-5-2'], ['1805.08659-1-62-0', '1805.08659-2-62-0'], ['1805.08659-1-62-1', '1805.08659-2-62-1'], ['1805.08659-1-62-2', '1805.08659-2-62-2'], ['1805.08659-1-3-0', '1805.08659-2-3-0'], ['1805.08659-1-3-1', '1805.08659-2-3-1'], ['1805.08659-1-3-2', '1805.08659-2-3-2'], ['1805.08659-1-3-3', '1805.08659-2-3-3'], ['1805.08659-1-3-4', '1805.08659-2-3-4'], ['1805.08659-1-40-0', '1805.08659-2-40-0'], ['1805.08659-1-40-1', '1805.08659-2-40-1'], ['1805.08659-1-40-2', '1805.08659-2-40-2'], ['1805.08659-1-40-3', '1805.08659-2-40-3'], ['1805.08659-1-40-4', '1805.08659-2-40-4'], ['1805.08659-1-40-5', '1805.08659-2-40-5'], ['1805.08659-1-40-6', '1805.08659-2-40-6'], ['1805.08659-1-67-0', '1805.08659-2-67-0'], ['1805.08659-1-67-1', '1805.08659-2-67-1'], ['1805.08659-1-69-0', '1805.08659-2-69-0'], ['1805.08659-1-69-1', '1805.08659-2-69-1'], ['1805.08659-1-50-0', '1805.08659-2-50-0'], ['1805.08659-1-50-1', '1805.08659-2-50-1'], ['1805.08659-1-91-0', '1805.08659-2-91-0'], ['1805.08659-1-91-1', '1805.08659-2-91-1'], ['1805.08659-1-91-2', '1805.08659-2-91-2'], ['1805.08659-1-91-3', '1805.08659-2-91-3'], ['1805.08659-1-93-0', '1805.08659-2-94-0'], ['1805.08659-1-93-1', '1805.08659-2-94-1'], ['1805.08659-1-93-2', '1805.08659-2-94-2'], ['1805.08659-1-13-0', '1805.08659-2-13-0'], ['1805.08659-1-13-1', '1805.08659-2-13-1'], ['1805.08659-1-13-2', '1805.08659-2-13-2'], ['1805.08659-1-13-4', '1805.08659-2-13-5'], ['1805.08659-1-13-6', '1805.08659-2-13-7'], ['1805.08659-1-7-0', '1805.08659-2-7-0'], ['1805.08659-1-7-1', '1805.08659-2-7-1'], ['1805.08659-1-7-2', '1805.08659-2-7-2'], ['1805.08659-1-7-3', '1805.08659-2-7-3'], ['1805.08659-1-7-4', '1805.08659-2-7-4'], ['1805.08659-1-7-5', '1805.08659-2-7-5'], ['1805.08659-1-73-0', '1805.08659-2-73-0'], ['1805.08659-1-73-1', '1805.08659-2-73-1'], ['1805.08659-1-73-2', '1805.08659-2-73-2'], ['1805.08659-1-73-3', '1805.08659-2-73-3'], ['1805.08659-1-73-4', '1805.08659-2-73-4'], ['1805.08659-1-34-0', '1805.08659-2-34-0'], ['1805.08659-1-34-1', '1805.08659-2-34-1'], ['1805.08659-1-87-0', '1805.08659-2-87-0'], ['1805.08659-1-87-1', '1805.08659-2-87-1'], ['1805.08659-1-87-2', '1805.08659-2-87-2'], ['1805.08659-1-87-3', '1805.08659-2-87-3'], ['1805.08659-1-87-4', '1805.08659-2-87-4'], ['1805.08659-1-44-0', '1805.08659-2-44-0'], ['1805.08659-1-44-1', '1805.08659-2-44-1'], ['1805.08659-1-44-2', '1805.08659-2-44-2'], ['1805.08659-1-44-3', '1805.08659-2-44-3'], ['1805.08659-1-44-4', '1805.08659-2-44-4'], ['1805.08659-1-44-5', '1805.08659-2-44-5'], ['1805.08659-1-4-0', '1805.08659-2-4-0'], ['1805.08659-1-4-1', '1805.08659-2-4-1'], ['1805.08659-1-4-2', '1805.08659-2-4-2'], ['1805.08659-1-53-0', '1805.08659-2-53-0'], ['1805.08659-1-53-1', '1805.08659-2-53-1'], ['1805.08659-1-29-0', '1805.08659-2-29-0'], ['1805.08659-1-29-1', '1805.08659-2-29-1'], ['1805.08659-1-29-2', '1805.08659-2-29-2'], ['1805.08659-1-29-3', '1805.08659-2-29-3'], ['1805.08659-1-29-4', '1805.08659-2-29-4'], ['1805.08659-1-66-0', '1805.08659-2-66-0'], ['1805.08659-1-66-1', '1805.08659-2-66-1'], ['1805.08659-1-66-2', '1805.08659-2-66-2'], ['1805.08659-1-77-0', '1805.08659-2-77-0'], ['1805.08659-1-77-1', '1805.08659-2-77-1'], ['1805.08659-1-77-2', '1805.08659-2-77-2'], ['1805.08659-1-77-3', '1805.08659-2-77-3'], ['1805.08659-1-77-4', '1805.08659-2-77-4'], ['1805.08659-1-77-5', '1805.08659-2-77-5'], ['1805.08659-1-77-6', '1805.08659-2-77-6'], ['1805.08659-1-77-7', '1805.08659-2-77-7'], ['1805.08659-1-77-8', '1805.08659-2-77-8'], ['1805.08659-1-77-9', '1805.08659-2-77-9'], ['1805.08659-1-77-10', '1805.08659-2-77-10'], ['1805.08659-1-24-0', '1805.08659-2-24-0'], ['1805.08659-1-24-1', '1805.08659-2-24-1'], ['1805.08659-1-25-0', '1805.08659-2-25-0'], ['1805.08659-1-39-0', '1805.08659-2-39-0'], ['1805.08659-1-39-1', '1805.08659-2-39-1'], ['1805.08659-1-39-2', '1805.08659-2-39-2'], ['1805.08659-1-39-3', '1805.08659-2-39-3'], ['1805.08659-1-39-4', '1805.08659-2-39-4'], ['1805.08659-1-88-0', '1805.08659-2-88-0'], ['1805.08659-1-88-1', '1805.08659-2-88-1'], ['1805.08659-1-0-0', '1805.08659-2-0-0'], ['1805.08659-1-0-1', '1805.08659-2-0-1'], ['1805.08659-1-0-2', '1805.08659-2-0-2'], ['1805.08659-1-0-3', '1805.08659-2-0-3'], ['1805.08659-1-0-4', '1805.08659-2-0-4'], ['1805.08659-1-0-5', '1805.08659-2-0-5'], ['1805.08659-1-0-6', '1805.08659-2-0-6'], ['1805.08659-1-42-0', '1805.08659-2-42-0'], ['1805.08659-1-42-1', '1805.08659-2-42-1'], ['1805.08659-1-42-2', '1805.08659-2-42-2'], ['1805.08659-1-26-0', '1805.08659-2-26-0'], ['1805.08659-1-26-1', '1805.08659-2-26-1'], ['1805.08659-1-26-2', '1805.08659-2-26-2'], ['1805.08659-1-26-3', '1805.08659-2-26-3'], ['1805.08659-1-26-4', '1805.08659-2-26-4'], ['1805.08659-1-26-5', '1805.08659-2-26-5'], ['1805.08659-1-26-6', '1805.08659-2-26-6'], ['1805.08659-1-26-7', '1805.08659-2-26-7'], ['1805.08659-1-26-8', '1805.08659-2-26-8'], ['1805.08659-1-26-9', '1805.08659-2-26-9'], ['1805.08659-1-26-10', '1805.08659-2-26-10'], ['1805.08659-1-26-11', '1805.08659-2-26-11'], ['1805.08659-1-26-12', '1805.08659-2-26-12'], ['1805.08659-1-46-0', '1805.08659-2-46-0'], ['1805.08659-1-46-1', '1805.08659-2-46-1'], ['1805.08659-1-46-2', '1805.08659-2-46-2'], ['1805.08659-1-46-3', '1805.08659-2-46-3'], ['1805.08659-1-46-4', '1805.08659-2-46-4'], ['1805.08659-1-46-5', '1805.08659-2-46-5'], ['1805.08659-1-46-6', '1805.08659-2-46-6'], ['1805.08659-1-46-7', '1805.08659-2-46-7'], ['1805.08659-1-46-8', '1805.08659-2-46-8'], ['1805.08659-1-46-9', '1805.08659-2-46-9'], ['1805.08659-1-46-10', '1805.08659-2-46-10'], ['1805.08659-1-46-11', '1805.08659-2-46-11'], ['1805.08659-1-46-12', '1805.08659-2-46-12'], ['1805.08659-1-46-13', '1805.08659-2-46-13'], ['1805.08659-1-46-14', '1805.08659-2-46-14'], ['1805.08659-1-46-15', '1805.08659-2-46-15'], ['1805.08659-1-46-16', '1805.08659-2-46-16'], ['1805.08659-1-46-17', '1805.08659-2-46-17'], ['1805.08659-1-46-18', '1805.08659-2-46-18'], ['1805.08659-1-19-0', '1805.08659-2-19-0'], ['1805.08659-1-19-1', '1805.08659-2-19-1'], ['1805.08659-1-19-2', '1805.08659-2-19-2'], ['1805.08659-1-19-3', '1805.08659-2-19-3'], ['1805.08659-1-19-4', '1805.08659-2-19-4'], ['1805.08659-1-95-0', '1805.08659-2-96-0'], ['1805.08659-1-95-1', '1805.08659-2-96-1'], ['1805.08659-1-95-2', '1805.08659-2-96-2'], ['1805.08659-1-28-0', '1805.08659-2-28-0'], ['1805.08659-1-28-1', '1805.08659-2-28-1'], ['1805.08659-1-28-2', '1805.08659-2-28-2'], ['1805.08659-1-28-3', '1805.08659-2-28-3'], ['1805.08659-1-28-4', '1805.08659-2-28-4'], ['1805.08659-1-28-5', '1805.08659-2-28-5'], ['1805.08659-1-70-0', '1805.08659-2-70-0'], ['1805.08659-1-70-1', '1805.08659-2-70-1'], ['1805.08659-1-70-2', '1805.08659-2-70-2'], ['1805.08659-1-70-3', '1805.08659-2-70-3'], ['1805.08659-1-70-4', '1805.08659-2-70-4'], ['1805.08659-1-70-5', '1805.08659-2-70-5'], ['1805.08659-1-75-0', '1805.08659-2-75-0'], ['1805.08659-1-75-1', '1805.08659-2-75-1'], ['1805.08659-1-92-0', '1805.08659-2-92-0'], ['1805.08659-1-92-1', '1805.08659-2-92-1'], ['1805.08659-1-92-2', '1805.08659-2-92-2'], ['1805.08659-1-92-3', '1805.08659-2-92-3'], ['1805.08659-1-92-4', '1805.08659-2-92-4'], ['1805.08659-1-92-5', '1805.08659-2-92-5'], ['1805.08659-1-92-6', '1805.08659-2-93-0'], ['1805.08659-1-92-7', '1805.08659-2-93-1'], ['1805.08659-1-92-8', '1805.08659-2-93-2']]

[['1805.08659-1-94-1', '1805.08659-2-95-1']]

[]

[['1805.08659-1-13-3', '1805.08659-2-13-4'], ['1805.08659-1-13-5', '1805.08659-2-13-6']]

[]

['1805.08659-1-20-2', '1805.08659-1-99-0', '1805.08659-1-100-0', '1805.08659-2-20-2', '1805.08659-2-100-0', '1805.08659-2-101-0']

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

https://arxiv.org/abs/1805.08659

0807.2194

{'0807.2194-1-0-0': 'Internal structure of a quantum soliton and classical excitations due to trap opening', '0807.2194-1-1-0': 'We analytically solve two problems that may be useful in the context of the recent observation of matter wave bright solitons in a one-dimensional attractive atomic Bose gas.', '0807.2194-1-1-1': 'The first problem is strictly beyond mean field: From the Bethe ansatz solution we extract the internal correlation function of the particle positions in the quantum soliton, that is for a fixed center of mass position.', '0807.2194-1-1-2': 'The second problem is solved in the limit of a large number of particles, where the mean field theory is asymptotically correct: It deals with the number of excitations created by the opening of the trap, starting from a pure soliton in a weakly curved harmonic potential.', '0807.2194-1-2-0': 'Experiments with cold atoms have now acquired a high degree of control of the key parameters of the system.', '0807.2194-1-2-1': 'Using transverse confinement of the atoms by non-dissipative optical potentials it is possible to freeze the atomic motion along one or several directions, realizing in this way quantum gases with reduced dimensionality [CITATION].', '0807.2194-1-2-2': 'Furthermore, thanks to Feshbach resonances driven by a magnetic field, one can adjust almost at will the interaction strength between the atoms [CITATION].', '0807.2194-1-2-3': 'The combination of these two experimental tools has recently allowed the observation of bright solitons in a one-dimensional (1D) Bose gas, either a single soliton [CITATION] or a train of solitons [CITATION].', '0807.2194-1-3-0': 'This leads to a renewed interest [CITATION] in the 1D non-relativistic Bose gas on the free line with zero range attractive interactions, that is with a binary interaction potential modeled by a Dirac delta with a negative coupling constant [MATH], [MATH], a model that is an acceptable approximation of reality under conditions defined in [CITATION].', '0807.2194-1-3-1': 'On a theoretical point of view, it is well known that eigenstates and eigenenergies of the corresponding [MATH]-body Hamiltonian may be obtained by the Bethe ansatz: Historically the focus was put mainly on the study of the ground state wavefunction [MATH] [CITATION], which is a collective bound state of the [MATH] particles, the so-called [MATH]-particle quantum soliton, with a delocalized center of mass of vanishing momentum.', '0807.2194-1-3-2': 'The extension of the Bethe ansatz to excited states is however possible, and one finds that the generic excited state corresponds to a set of quantum solitons with arbitrary atom numbers and different momenta per particle [CITATION].', '0807.2194-1-3-3': 'A key property to keep in mind is the full separability of the center of mass variables and the internal variables of the gas (e.g. the relative coordinates of the particles), which holds since the gas is on the free line with open boundary conditions.', '0807.2194-1-4-0': 'Despite the knowledge of the ground state wavefunction for [MATH] particles, some theoretical work is needed to extract experimentally relevant observables.', '0807.2194-1-4-1': 'As atomic density profiles may be measured by absorption or even non-destructive imaging [CITATION], natural observables are functions of the positions of the particles.', '0807.2194-1-4-2': 'The simplest observable is the mean density of particles, [MATH], obtained by an average of the density profile over many (ideally infinitely many) experimental realizations.', '0807.2194-1-4-3': 'From the ground state [MATH]-body wavefunction one however does not obtain any useful information on the mean density: Since the center of mass is fully delocalized, one gets a uniform distribution over the whole line.', '0807.2194-1-4-4': 'Experimental reality is very different, the soliton being obtained from an initially trapped Bose-Einstein condensate.', '0807.2194-1-4-5': 'When the trap is switched off to free the soliton, its center of mass is not in its ground state, it is in a localized and non-stationary state which depends on the experimental preparation procedure.', '0807.2194-1-4-6': 'A more realistic assumption is thus to assume a [MATH]-body wavefunction of the form [EQUATION] where [EQUATION] is the center of mass position of the soliton and the center of mass wavefunction [MATH] is a priori unknown and depends on the experimental details.', '0807.2194-1-4-7': 'The theoretical challenge is thus to predict observables for a fixed position of the center of mass [MATH].', '0807.2194-1-4-8': 'Experimentally relevant results are then obtained from these theoretical predictions by a further average over [MATH] with the probability distribution [MATH].', '0807.2194-1-4-9': 'One can even hope that the predictions for fixed [MATH] are measurable, if the center of mass position can be measured with a high enough accuracy for each individual realization of the experiment.', '0807.2194-1-5-0': 'Turning back to the simple example of the mean density, we see that the right concept is the mean density of particles [MATH] for a fixed center of mass position [MATH], as was already argued in [CITATION] within the general concept of (here translational) symmetry breaking.', '0807.2194-1-5-1': 'Remarkably an explicit expression of [MATH] in terms of a sum of [MATH] exponential terms may be obtained [CITATION].', '0807.2194-1-6-0': 'The next step beyond the mean density is the pair distribution function of the particles [MATH], or very similarly the static structure factor [MATH], of the [MATH]-particle soliton.', '0807.2194-1-6-1': 'Recently a large [MATH] expansion of the static structure factor was obtained from the Bethe ansatz [CITATION] (even including results on the dynamic structure factor).', '0807.2194-1-6-2': 'The goal of the present work is, in the spirit of the above physical discussion, to study the static structure factor [MATH] for a fixed center of mass position [MATH].', '0807.2194-1-6-3': 'This static structure factor gives access to correlations between the positions of the particles inside the soliton, that is it gives information on the internal structure of the soliton, which goes beyond the usual mean field (or Gross-Pitaevskii) approximation, which neglects such correlations.', '0807.2194-1-7-0': 'As a guideline we imagine that, in a given experiment, one wishes to access the correlation function of an observable which is a function [MATH] of the particle position [MATH]: [EQUATION] where the expectation value [MATH] is taken over the internal wavefunction [MATH] of the quantum soliton for a fixed center of mass location [MATH].', '0807.2194-1-8-0': 'The paper is organized as follows.', '0807.2194-1-8-1': 'In section [REF] we recall the basic facts about the 1D Bose gas model and we give a general expression of the pair distribution function [MATH] for fixed [MATH] that may be used to evaluate [MATH] numerically for a moderate number of atoms and that will be the starting point of analytical calculations.', '0807.2194-1-8-2': 'In section [REF] we present an analytical calculation of [MATH] for an arbitrary atom number [MATH], in the limit where the function [MATH] is slowly varying over the spatial width of the soliton.', '0807.2194-1-8-3': 'In section [REF] we present a large [MATH] expansion of the static structure factor [MATH] for fixed [MATH] that we then use to calculate [MATH] to leading order in [MATH]; in the same section, we integrate [MATH] over [MATH] to see if we recover the asymptotic results of [CITATION] for [MATH], and we also consider the case of a function [MATH] much narrower than the soliton.', '0807.2194-1-8-4': 'In section [REF] we evaluate the accuracy of the assumption ([REF]) in an experiment: Assuming that the quantum gas is in its internal ground state in the trap, we calculate analytically (to leading order in [MATH] and in the trap curvature) the number of internal excitations of the soliton produced by the trap opening.', '0807.2194-1-8-5': 'We conclude in section [REF].', '0807.2194-1-9-0': '# Model, basic definitions and general results', '0807.2194-1-10-0': '## Hamiltonian and ground state wavefunction', '0807.2194-1-11-0': 'We consider a set of [MATH] spinless non-relativistic bosons of mass [MATH] moving on the one dimensional real line, in the absence of any trapping potential, with open boundary conditions, and binary interacting via an attractive contact potential of coupling constant [MATH].', '0807.2194-1-11-1': 'This corresponds to the Hamiltonian in first quantized form: [EQUATION]', '0807.2194-1-11-2': 'For this problem there is full separability of the center of mass motion and of the internal coordinates, the internal wavefunction being independent of the center of mass state.', '0807.2194-1-11-3': 'As a consequence, the ground state wavefunction has it center of mass with zero momentum and depends only on the relative coordinates of the particles.', '0807.2194-1-11-4': 'Its exact expression is [CITATION] [EQUATION] where the normalization factor [MATH] will be specified later.', '0807.2194-1-11-5': 'The corresponding eigenenergy is [CITATION] [EQUATION]', '0807.2194-1-11-6': 'This ground state is the so-called quantum soliton with [MATH] particles.', '0807.2194-1-12-0': '## Mean density, pair distribution and static structure factor for a fixed center of mass position', '0807.2194-1-13-0': 'The crucial concept in the present work is the expectation value of a position dependent observable, that is of an arbitrary function [MATH] of the [MATH] particle positions, for a fixed value [MATH] of the center of mass position: [EQUATION] where the integration is over the whole space [MATH].', '0807.2194-1-13-1': 'The soliton wavefunction shall then be normalized in such a way that the expectation value [MATH] of the function [MATH] constant and equal to unity is also equal to unity.', '0807.2194-1-13-2': 'Using the bosonic symmetry of the wavefunction, the integral to compute is then simply [MATH] times the integral over the so-called fundamental domain of ordered positions [EQUATION]', '0807.2194-1-13-3': 'Over this domain, the wavefunction indeed has the separable expression [EQUATION]', '0807.2194-1-13-4': 'One then straightforwardly obtains [EQUATION]', '0807.2194-1-13-5': 'Setting [MATH] we obtain the mean density [MATH] for a fixed center of mass position.', '0807.2194-1-13-6': 'This was first calculated in [CITATION]: [EQUATION] where [MATH] is the spatial width of the classical (that is mean-field) soliton, [EQUATION]', '0807.2194-1-13-7': 'It is a function of [MATH], a consequence of translational and parity invariance of the Hamiltonian and of [MATH].', '0807.2194-1-13-8': 'One thus has [MATH].', '0807.2194-1-13-9': 'It is normalized in such a way that the integral over [MATH] over the whole space is equal to [MATH].', '0807.2194-1-13-10': 'If the true physical state of the system is a product ([REF]) of a localized center of mass wavefunction [MATH] and of the quantum soliton wavefunction, the physical mean density is the average (or equivalently the convolution) [EQUATION]', '0807.2194-1-13-11': 'Setting [MATH], we obtain the pair distribution function [MATH] for fixed center of mass position [MATH].', '0807.2194-1-13-12': 'It is normalized as [MATH] for the double integration over [MATH] and [MATH].', '0807.2194-1-13-13': 'From the translation invariance [MATH] so that it is sufficient to calculate this pair distribution function for [MATH].', '0807.2194-1-13-14': 'In real space, we do not know an expression of [MATH] as simple as ([REF]) but we have found a simple expression for the Fourier transform [EQUATION]', '0807.2194-1-13-15': 'As detailed in the appendix [REF] one has [EQUATION] where [MATH] is the Gamma function of complex argument [MATH] and we have introduced the dimensionless variables [MATH], [MATH] and [MATH].', '0807.2194-1-14-0': 'The quantities [MATH] solve the degree two equations: [EQUATION]', '0807.2194-1-14-1': 'In the large [MATH] limit, for fixed values of [MATH] and [MATH], it is convenient to take [EQUATION] with a determination of the square root such that [MATH] for [MATH], e.g. with the line cut on the real negative axis.', '0807.2194-1-15-0': 'This gives the idea a posteriori to look for a similar expression for the Fourier transform [MATH] of [MATH].', '0807.2194-1-15-1': 'As shown in the appendix [REF] one obtains the simple expression', '0807.2194-1-16-0': 'Let us come back to the original problem of calculating the correlation function [MATH] of the observable [MATH], as defined in ([REF]).', '0807.2194-1-16-1': 'One first expresses the squares as products of double sums over indices [MATH] and [MATH].', '0807.2194-1-16-2': 'One can split the double sum over [MATH] and [MATH] in ([REF]) in diagonal terms [MATH] and off-diagonal terms [MATH], so that [MATH].', '0807.2194-1-17-0': 'Introducing the Fourier transform of [MATH], [EQUATION] we obtain the Fourier space expression', '0807.2194-1-18-0': 'that can be directly evaluated using ([REF]).', '0807.2194-1-18-1': 'Simpler analytic expressions of [MATH], either for slowly varying functions [MATH] or in the large [MATH] limit, shall be given in the next sections.', '0807.2194-1-19-0': 'To conclude this subsection, we note that [MATH] has a very simple expression in terms of the correlations contained in the fixed [MATH] static structure factor [MATH], [EQUATION] where the operator giving the density is [EQUATION]', '0807.2194-1-19-1': 'We note that the double integral of [MATH] over [MATH] and [MATH] is equal to [MATH].', '0807.2194-1-19-2': 'Using again the translational invariance, one obtains the illuminating expression [EQUATION]', '0807.2194-1-20-0': 'The essence of the mean-field approximation is to neglect correlations among the particles, so that [MATH] would essentially be approximated by the uncorrelated product [MATH].', '0807.2194-1-20-1': 'The above writing clearly reveals that [MATH] is sensitive to correlations that are beyond the mean-field approximation.', '0807.2194-1-21-0': '## The usual static structure factor', '0807.2194-1-22-0': 'The usual static structure factor is the spatial correlation function of the operator [MATH] giving the density, [EQUATION] where the expectation value is taken literally in the ground state of the gas, thus assuming a perfectly delocalized center of mass wavefunction [MATH].', '0807.2194-1-22-1': 'Because of the translational invariance, it is a function of [MATH] only.', '0807.2194-1-23-0': 'This usual structure factor is deduced from our fixed-[MATH] one by integration over the center of mass position [EQUATION]', '0807.2194-1-23-1': 'This allows to express the Fourier transform of [MATH] in terms of the Fourier transform of [MATH], when one uses ([REF]): [EQUATION]', '0807.2194-1-23-2': 'From ([REF]) we thus have an analytical expression of [MATH] in terms of a double sum.', '0807.2194-1-24-0': 'The relation ([REF]) will also allow us, in the large [MATH] limit, to make the link between our large [MATH] expansion of [MATH] and the large [MATH] expansion of [MATH] given in [CITATION], see [REF].', '0807.2194-1-25-0': '# Calculation of [MATH] for a broad function [MATH]', '0807.2194-1-26-0': 'One supposes in this section that [MATH] varies slowly over the length scale [MATH] of the quantum soliton, as defined in ([REF]).', '0807.2194-1-26-1': 'Then one rewrites ([REF]) using the translational invariance, [EQUATION] and one expands [EQUATION]', '0807.2194-1-26-2': 'The constant shift [MATH] has no effect in [MATH].', '0807.2194-1-26-3': 'The linear term has also an exactly vanishing contribution, since by definition [MATH].', '0807.2194-1-26-4': 'Setting [EQUATION] we thus obtain the leading contribution [EQUATION] with [MATH] is the variance of [MATH] for a center of mass position fixed at the origin of the coordinates.', '0807.2194-1-27-0': 'It turns out that an exact expression may be obtained for this variance, as detailed in the appendix [REF]: It is the sum of three contributions, [EQUATION] with [EQUATION]', '0807.2194-1-27-1': 'We have introduced the very simple symmetric matrix [EQUATION] defined over the index range [MATH].', '0807.2194-1-27-2': 'This holds whatever the value of the atom number [MATH].', '0807.2194-1-28-0': 'In the large [MATH] limit, the contribution [MATH] is dominant, simply because it contains more terms, and its asymptotic expression is evaluated by replacing the sums by integrals, [EQUATION]', '0807.2194-1-28-1': 'This leads to the estimate [EQUATION] for a slowly varying potential [MATH] in the [MATH] limit.', '0807.2194-1-29-0': '# Large [MATH] limit of [MATH] for [MATH] of any width', '0807.2194-1-30-0': 'In this section we give a large [MATH] expansion of the static structure factor for fixed center of mass position, which allows to get the asymptotic behavior of [MATH] in the large [MATH] limit.', '0807.2194-1-30-1': 'When specialized to a quadratic potential [MATH] the general result reproduces the large [MATH] broad potential result of the previous section.', '0807.2194-1-30-2': 'As a first test of the result, we integrate [MATH] over the center of mass position [MATH], to see if we recover the results of [CITATION] for the usual static factor [MATH].', '0807.2194-1-30-3': 'As a second test of the result, we get an approximate expression for [MATH] in the case of a narrow function [MATH], to lowest order in the width [MATH] of [MATH], first from the large [MATH] expansion of [MATH] and then from a more general reasoning not relying on a large [MATH] expansion.', '0807.2194-1-31-0': '## Asymptotic expression of [MATH] and of [MATH]', '0807.2194-1-32-0': 'As we have seen in ([REF]), [MATH] is directly related to the deviation of the fixed [MATH] static structure factor of the soliton from the uncorrelated form [MATH].', '0807.2194-1-32-1': 'It turns out that this deviation may be easily obtained from the Fourier space expressions ([REF],[REF]), simply by taking the large [MATH] limit of the [MATH] functions and by replacing the discrete sums over indices by integrals.', '0807.2194-1-32-2': 'This shows that our Fourier space representations are indeed useful.', '0807.2194-1-33-0': 'As detailed in the appendix [REF], the large [MATH] expansion for a fixed value of [MATH] (that is for a fixed value of [MATH]), gives the leading term [EQUATION] where [MATH] is the Heaviside distribution that is equal to zero for [MATH] and to one for [MATH] [CITATION].', '0807.2194-1-34-0': 'If one wishes to have an expansion of the static structure factor only, one has also to expand [MATH] in powers of [MATH].', '0807.2194-1-34-1': 'From the real space expression ([REF]), expanding each factorial in the large [MATH] limit for a fixed summation index [MATH], we obtain, setting [MATH], [EQUATION] where one may check the normalization condition [MATH].', '0807.2194-1-34-2': 'This gives the expansion of [MATH] up to order [MATH], for a fixed value of [MATH].', '0807.2194-1-35-0': 'We come back to ([REF]) to obtain the large [MATH] equivalent of [MATH] at fixed [MATH].', '0807.2194-1-36-0': 'One immediately sees that a linearly varying potential [MATH] gives a vanishes contribution to [MATH], which is also obvious from the definition ([REF]), since [MATH] by construction.', '0807.2194-1-36-1': 'If [MATH] is a quadratic function of [MATH], [MATH], one should recover Eq. ([REF]), which is indeed the case if [EQUATION]', '0807.2194-1-36-2': 'We have checked that this identity holds [CITATION].', '0807.2194-1-37-0': '## Application: large [MATH] expansion of the usual static structure factor', '0807.2194-1-38-0': 'The usual static structure factor [MATH] is obtained from [MATH] by integration over [MATH], see ([REF]).', '0807.2194-1-38-1': 'Assuming without loss of generality that [MATH], we see from ([REF]) and from ([REF]) that one has to integrate over [MATH] a quantity [MATH], where the function [MATH] corresponds to the right hand side of ([REF]).', '0807.2194-1-38-2': 'From the differential relations, taking [MATH] and [MATH] as independent variables, [EQUATION] we see that the integral over [MATH] will cancel the derivatives [MATH].', '0807.2194-1-38-3': 'Using [EQUATION] one gets [EQUATION] where [MATH].', '0807.2194-1-39-0': 'One can show that the [MATH] distribution can be exchanged with the last [MATH].', '0807.2194-1-39-1': 'From the large [MATH] expansion ([REF]) of [MATH], one finally obtain the large [MATH] expansion [EQUATION]', '0807.2194-1-40-0': 'To compare this result with the ones of [CITATION], one has first to perform Fourier transforms because the results of [CITATION] are given in Fourier space.', '0807.2194-1-40-1': 'Fortunately the Fourier transform of Eq.(62) and Eq.(63) of [CITATION] are easily done.', '0807.2194-1-40-2': 'Then one finds that our result ([REF]) is consistent with [CITATION] if one adds in Eq.(62) of [CITATION] the subleading term to the intermediate quantity [MATH] introduced in [CITATION], a subleading term that was omitted in [CITATION].', '0807.2194-1-41-0': 'where we recall that [MATH].', '0807.2194-1-41-1': 'One can also easily check that this obeys the sum rule [MATH].', '0807.2194-1-42-0': '## Approximation of [MATH] for a narrow barrier', '0807.2194-1-43-0': 'In this subsection, one assumes that [MATH] and its derivatives are functions localized around the origin with a width much smaller than [MATH].', '0807.2194-1-43-1': 'E.g. [MATH] is a Gaussian centered in [MATH] with a width [MATH].', '0807.2194-1-43-2': 'One first considers the large [MATH] limit.', '0807.2194-1-43-3': 'We rewrite Eq. ([REF]) as [EQUATION] with [EQUATION]', '0807.2194-1-44-0': 'Then we expand the factor containing [MATH] in powers of [MATH] and [MATH].', '0807.2194-1-44-1': 'One can show that the first non vanishing contribution, for rapidly decreasing derivatives of [MATH], comes from the third order expansion, which results in [EQUATION] where we have used ([REF]) to leading order in [MATH] to recognize the factor [MATH].', '0807.2194-1-45-0': 'There is actually a faster way to obtain this result, and not restricted to the large [MATH] limit, from the exact writing [EQUATION]', '0807.2194-1-46-0': 'Then one sees that the first term in ([REF]) is first order in the width [MATH] of [MATH], since the integration range over [MATH] has a width [MATH], whereas the second term is second order in [MATH], since it involves a double integral over a range of diameter [MATH].', '0807.2194-1-46-1': 'Then to first order in [MATH] one recovers ([REF]).', '0807.2194-1-46-2': 'In the large [MATH] limit, for a fixed [MATH], both terms of ([REF]) scale in the same way with [MATH], that is linearly with [MATH], so that the order of the limits [MATH] and [MATH] (for fixed [MATH]) is not crucial.', '0807.2194-1-47-0': '# Number of internal excitations created by the trap opening', '0807.2194-1-48-0': 'In this work, we have assumed up to now that a pure soliton is produced in the experiment, corresponding to the state ([REF]): The center of mass may be in an arbitrary excited state but the internal variables of the gas are in their ground state.', '0807.2194-1-48-1': 'In this section, we revisit this assumption taking into account experimental constraints.', '0807.2194-1-49-0': 'In a real experiment the ultracold gas is prepared in a trap; it is not free along [MATH], and each atom is subject to the harmonic confining potential [EQUATION]', '0807.2194-1-49-1': 'We recall that the center of mass motion and the internal variables remain separable in a harmonic trap.', '0807.2194-1-49-2': 'In order for the trapped gas to be close to the free space limit, the pulsation [MATH] is adjusted to have [MATH], where [EQUATION] is the mean-field approximation for the chemical potential of the soliton.', '0807.2194-1-49-3': 'We assume that the gas is cooled down to a temperature [MATH] low enough to have [MATH].', '0807.2194-1-49-4': 'The gas is then a pure soliton.', '0807.2194-1-49-5': 'On the contrary we do not assume the much more stringent condition [MATH], so that the center of mass of the gas may still be in an excited state.', '0807.2194-1-50-0': 'Eventually the trapping potential along [MATH] will be switched off, to obtain the ideal conditions of the Hamiltonian ([REF]).', '0807.2194-1-50-1': 'This trap opening will create some internal excitations of the gas, that is the untrapped gas will not be a pure soliton but will contain a mean number [MATH] of internal excitations.', '0807.2194-1-51-0': 'Since no Bethe ansatz solution exists in a trap, the modest goal here is to calculate [MATH] to leading order in [MATH] in the large [MATH] limit, for a fixed value of [MATH] [CITATION].', '0807.2194-1-51-1': 'In this limit, a 1D classical field treatment is sufficient, with a Hamiltonian [EQUATION]', '0807.2194-1-51-2': 'We have slightly generalized ([REF]) to treat the case of a time dependent trap: [EQUATION] where the function [MATH], going from unity for [MATH] to zero for [MATH] describes the switch-off procedure of the trap.', '0807.2194-1-51-3': 'The atom number is fixed so that the norm squared of the classical field is also fixed: [EQUATION]', '0807.2194-1-51-4': 'This classical field can be solved on a computer, which allows a test of our predictions for [MATH].', '0807.2194-1-52-0': 'The problem can be further simplified in the limit [MATH], where one expects that the field [MATH] will remain "close" (up to a phase factor) to the one describing a pure soliton [CITATION], [MATH], with [EQUATION]', '0807.2194-1-52-1': 'We then use the number conserving Bogoliubov formalism of [CITATION], downgraded to a classical field problem (simply replacing commutators with Poisson brackets), as was already done in [CITATION].', '0807.2194-1-52-2': 'One splits the field by orthogonal projection along the mode [MATH] and orthogonally to it: [EQUATION] where [MATH] is the component of the field on the mode [MATH] and the field [MATH] is orthogonal to that mode.', '0807.2194-1-52-3': 'The idea is to treat [MATH] as a small perturbation.', '0807.2194-1-52-4': 'The strength of the number conserving approach is to eliminate the amplitude [MATH] in a systematic way, using the modulus phase representation [EQUATION]', '0807.2194-1-52-5': 'The phase [MATH] is eliminated by a redefinition of the transverse field: [EQUATION]', '0807.2194-1-52-6': 'The modulus [MATH] is expressed in terms of [MATH] using the condition of a fixed atom number ([REF]).', '0807.2194-1-53-0': 'In the absence of trapping potential, one keeps terms up to quadratic in [MATH] in the Hamiltonian, and the resulting quadratic form can be written in normal form as [CITATION] [EQUATION] where [MATH] is the ground state of the classical field model.', '0807.2194-1-53-1': 'The variable [MATH] of the field represents the total momentum of the field, written to first order in [MATH]: [EQUATION]', '0807.2194-1-53-2': 'The occurrence of the term [MATH] represents physically the fact that the center of mass motion is decoupled; more formally, it corresponds to the fact that [MATH] "spontaneously" breaks the translational symmetry of the Hamiltonian, which leads to the occurrence of a Goldstone mode [CITATION].', '0807.2194-1-53-3': 'The field variable canonically conjugated to [MATH] corresponds to the center of mass position of the field, written up to first order in [MATH]: [EQUATION]', '0807.2194-1-53-4': 'Apart from this Goldstone mode, the other eigenmodes behave as a continuum of decoupled harmonic oscillators, with normal (complex) variables [MATH] and eigenenergies [EQUATION] which correspond to internal (and thus gapped) excitations of the gas: An elementary excitation physically takes the form of a free particle coming from infinity with a wavevector [MATH] and scattering on a soliton with [MATH] particles.', '0807.2194-1-53-5': "The field variable [MATH] has the expression [EQUATION] where the Bogoliubov modes of the number conserving theory are expressed as follows in Dirac's notation (see [CITATION] V.A) here for a real function [MATH]: [EQUATION] where [MATH] projects orthogonally to [MATH], and the [MATH] are the eigenmodes of the usual Bogoliubov-de Gennes equations [EQUATION]", '0807.2194-1-53-6': 'It turns out that these modes are known exactly [CITATION].', '0807.2194-1-53-7': 'For [MATH] one has [EQUATION] where we have set [MATH] and [MATH].', '0807.2194-1-53-8': 'The modes for [MATH] are deduced from the relations [MATH] and [MATH].', '0807.2194-1-54-0': 'In presence of the trap, there is an extra contribution to the Hamiltonian containing the trapping energy, [EQUATION]', '0807.2194-1-54-1': 'We approximate it to leading non-trivial order in [MATH], that is to first order, injecting the splitting ([REF]).', '0807.2194-1-54-2': 'The term in [MATH] deviates from [MATH] by [MATH] so it gives to first order only a constant contribution, with an integral over [MATH] that can be calculated exactly if necessary.', '0807.2194-1-54-3': 'The crossed terms [MATH] and complex conjugate are kept, with [MATH] approximated by [MATH], whereas the quadratic terms [MATH] are neglected.', '0807.2194-1-54-4': 'Finally, we replace the field [MATH] by its modal expansion [EQUATION]', '0807.2194-1-55-0': 'We thus keep [EQUATION] with coefficients [EQUATION]', '0807.2194-1-55-1': 'With the residues method, the resulting integral can be evaluated analytically.', '0807.2194-1-55-2': 'We give for convenience the ratio to the mode eigenenergy: [EQUATION] where [MATH] as above.', '0807.2194-1-56-0': "When the perturbation [MATH] is added to the unperturbed Hamiltonian ([REF]), the equilibrium value of the [MATH]'s minimizing the resulting energy is [EQUATION]", '0807.2194-1-56-1': 'At later times, the trap is opened according to the switch-off function [MATH].', '0807.2194-1-56-2': 'The Hamiltonian equations of motion then give [EQUATION] to be solved with the initial conditions ([REF]): [EQUATION] where [MATH] is the mode frequency.', '0807.2194-1-56-3': 'This allows to calculate the number of excitations once the trap has been switched off, using [MATH] for [MATH]: [EQUATION] where [MATH] is a spectral density of the switch-off procedure at frequency [MATH]: [EQUATION]', '0807.2194-1-56-4': 'We recall that ([REF]) is valid up to leading order in [MATH] (because of the classical field model) and to leading order in [MATH] (because of the perturbative treatment of the deviations of the field from the free space soliton).', '0807.2194-1-57-0': 'The case producing the maximal number of excitations for a monotonic [MATH] corresponds to a sudden trap switch-off, where [MATH] at all frequencies.', '0807.2194-1-57-1': 'Using ([REF]), we find that the resulting integral can be evaluated with the residues method, so that [EQUATION] with [EQUATION]', '0807.2194-1-57-2': 'This result is encouraging since a moderately small value [MATH] already leads to a number of excitations relative to the total atom number at the [MATH] level.', '0807.2194-1-58-0': 'It can be further reduced by switching off the trap more slowly.', '0807.2194-1-58-1': 'E.g. a linear ramping [EQUATION] leads to [EQUATION]', '0807.2194-1-58-2': 'Due to the presence of a gap [MATH] in the internal excitation spectrum of the gas, one gets the upper bound on the number of excitations [EQUATION] that is one gains quadratically with the switch-off time when it becomes longer than the internal soliton time [MATH].', '0807.2194-1-59-0': '# Conclusion', '0807.2194-1-60-0': 'Inspired by recent observations of matter wave bright solitons in atomic gases, we have considered here two problems that may be relevant for experiments.', '0807.2194-1-61-0': 'The first problem is strictly beyond mean field: It corresponds to the pair correlations between positions of the particles in a quantum soliton for a fixed position of the center of mass of the soliton.', '0807.2194-1-61-1': 'In particular, we have obtained analytically the large [MATH] limit expression of this pair correlation function, see ([REF]).', '0807.2194-1-61-2': 'By integrating this result over the center of mass position, we can recover the large [MATH] expansions of the static structure factor which were performed in [CITATION] for a fully delocalized center of mass position.', '0807.2194-1-61-3': 'On an experimental point of view, our predictions can be tested by measuring the positions of the particles in a very broad quantum soliton, prepared with weak attractive interactions and a relatively small atom number.', '0807.2194-1-62-0': 'The second problem was studied in the classical field model.', '0807.2194-1-62-1': 'The number of internal excitations of the gas created by the trap opening from an initial pure soliton was calculated in the limit where the soliton size is smaller than the size of the harmonic single particle ground state, see ([REF]) for a sudden trap opening.', '0807.2194-1-62-2': 'This also can be seen experimentally by detecting atoms flying away from the remaining soliton core after the trap opening.', '0807.2194-1-62-3': 'A possible extension of this calculation is to include quantum fluctuations of the field.', '0807.2194-1-63-0': 'We acknowledge useful discussions with L. Khaykovich, C. Weiss, A. Sinatra, M. Olshanii.', '0807.2194-1-64-0': '# Calculation of the Fourier transforms of [MATH] and [MATH]', '0807.2194-1-65-0': 'We start with the definition of the mean density for a fixed center of mass position [MATH], taking here [MATH] without loss of generality: From ([REF]) one has [EQUATION]', '0807.2194-1-66-0': 'Using the bosonic exchange symmetry we can restrict the integral to the fundamental domain [MATH] of ([REF]), including a factor [MATH].', '0807.2194-1-66-1': 'For simplicity we take in this appendix [MATH] as the unit of length.', '0807.2194-1-66-2': 'With the change of variables, of Jacobian equal to unity, [EQUATION] the condition to be in [MATH] is simply that all the [MATH] are positive, and [MATH] can vary in the whole real space [MATH].', '0807.2194-1-67-0': 'One can calculate the integral over [MATH]: The first delta factor in ([REF]), the one ensuring that [MATH], imposes a value [EQUATION]', '0807.2194-1-67-1': 'When one reports this value of [MATH] in the argument of the second delta factor in ([REF]), one obtains a remaining factor [MATH], with [EQUATION]', '0807.2194-1-67-2': 'Taking the Fourier transform of this remaining delta factor, according to [MATH], gives [EQUATION] where we used the fact that the product of all [MATH] (for [MATH] from 2 to [MATH]) is equal to [MATH].', '0807.2194-1-67-3': 'Replacing the [MATH] by their expression and using the identity [EQUATION] deduced from the basic property [MATH] of the Gamma function, with the convention that an "empty" product is equal to unity, one gets ([REF]).', '0807.2194-1-68-0': 'We now turn to the pair distribution function for a center of mass position fixed in [MATH].', '0807.2194-1-68-1': 'Following the same steps as for the mean density, we obtain the Fourier transform [EQUATION]', '0807.2194-1-68-2': 'We can restrict to a summation over [MATH], the contribution for [MATH] being deduced by exchanging [MATH] and [MATH].', '0807.2194-1-68-3': 'In the product over [MATH], three ranges have then to be considered, (i) the first range [MATH], (ii) the mid-range [MATH], and (iii) the last range [MATH].', '0807.2194-1-68-4': 'The first range and the last range contributions can be expressed in terms of the Gamma function as in ([REF]).', '0807.2194-1-68-5': 'The mid-range contribution is equal to the product [EQUATION] where we have reindexed the product setting [MATH] and we have defined [MATH] and [MATH].', '0807.2194-1-68-6': 'The last step is to consider the denominator in each factor of ([REF]) as a polynomial of degree 2 in [MATH]: Its roots are [MATH] defined in ([REF], [REF]) and [MATH].', '0807.2194-1-68-7': 'This leads to [EQUATION] which can now be expressed as a ratio of products of Gamma functions.', '0807.2194-1-68-8': 'Then one gets ([REF]).', '0807.2194-1-69-0': '# Calculation of the variance of [MATH]', '0807.2194-1-70-0': 'We explain how to calculate exactly the moments [MATH] of the quantity [MATH] defined in ([REF]) in the [MATH]-body ground state for a fixed center of mass position [MATH].', '0807.2194-1-70-1': 'We take [MATH] as unit of length and we use the transformations exposed at the beginning of appendix [REF].', '0807.2194-1-70-2': 'Considering the change of variable ([REF]), we rewrite [MATH] as [EQUATION]', '0807.2194-1-71-0': 'The first sum in the right hand side of ([REF]) will have a vanishing contribution, since the expectation value is taken for a zero center of mass position.', '0807.2194-1-71-1': 'Replacing [MATH] by its expression in terms of [MATH], and using the fact that the value of [MATH] is fixed to ([REF]) we see that one may effectively replace [MATH] by the quantity', '0807.2194-1-72-0': 'with the symmetric matrix [EQUATION]', '0807.2194-1-72-1': 'We have thus reduced the problem to the calculation of the integrals', '0807.2194-1-73-0': 'with [MATH] defined in ([REF]).', '0807.2194-1-73-1': 'These integrals may be calculated by interpreting them as Gaussian averages, introducing the auxiliary complex random variables [MATH], [MATH]: These variables [MATH] are statistically independent and each one has a Gaussian probability distribution [MATH].', '0807.2194-1-73-2': 'Each [MATH] then corresponds to [MATH], so that [EQUATION]', '0807.2194-1-73-3': "Here [MATH] denotes the Gaussian average of the [MATH]'s and can be calculated using Wick's theorem.", '0807.2194-1-73-4': "The calculations are a bit lengthy for [MATH] since Wick's theorem has to be applied to a product of 8 variables.", '0807.2194-1-73-5': 'Clearly the matrix [MATH] appears, which is the matrix [MATH] of ([REF]).', '0807.2194-1-73-6': 'We finally get ([REF]) for [MATH].', '0807.2194-1-74-0': '# Large [MATH] asymptotics of [MATH]', '0807.2194-1-75-0': 'To perform the large [MATH] expansion we shall use the Fourier space expressions ([REF],[REF]).', '0807.2194-1-75-1': 'We thus define [EQUATION]', '0807.2194-1-75-2': 'The pair distribution function involves a double sum over [MATH].', '0807.2194-1-75-3': 'Since [MATH] is a simple sum, the last term in ([REF]) is a double sum over [MATH] and [MATH], without the restriction [MATH].', '0807.2194-1-75-4': 'It thus makes sense to split [MATH] into a off-diagonal part ([MATH]), that we collect with the double sum in [MATH], and a diagonal part [EQUATION] where [MATH] are defined above ([REF]).', '0807.2194-1-76-0': 'The idea to obtain the large [MATH] limit is simply to replace the discrete sums by integrals.', '0807.2194-1-76-1': 'To this end, one has to calculate the large [MATH] limit of each term of the sums, for fixed values of [MATH] and [MATH].', '0807.2194-1-76-2': 'The expansion of the Gamma functions is conveniently performed using [EQUATION] where the real quantity [MATH] tends to [MATH], and the fixed quantities [MATH] and [MATH] may be complex.', '0807.2194-1-76-3': 'One also uses the large [MATH] expansion of the quantity [MATH]: [EQUATION]', '0807.2194-1-76-4': 'For the diagonal part, only the leading term of ([REF]) is useful.', '0807.2194-1-76-5': 'Replacing [MATH] by [MATH] leads to [EQUATION] where [MATH] and the integral was transformed with the change of variable [MATH], to acquire the form of a Fourier transform.', '0807.2194-1-76-6': 'Note that the resulting integral can be calculated exactly, giving [MATH], but this is not useful here.', '0807.2194-1-77-0': 'For the double sum over [MATH], one has to include the [MATH] term in the expansion ([REF]), to obtain a non-zero result: [EQUATION]', '0807.2194-1-77-1': 'The change of variables [MATH] and [MATH] gives to the off-diagonal contribution the form of a Fourier transform: [EQUATION]', '0807.2194-1-77-2': 'The leading term of [MATH] for [MATH] for a fixed [MATH] is the sum of ([REF]) and ([REF]).', '0807.2194-1-77-3': 'The Fourier transform with respect to [MATH] is straightforward: The factors [MATH] act as derivatives, and the remaining bits have already a Fourier form.', '0807.2194-1-77-4': 'The Fourier transform of the diagonal contribution gives a contribution involving a factor [MATH], which exactly cancels with the first term in the right-hand side of ([REF]), at the considered order in [MATH], see ([REF]).', '0807.2194-1-77-5': 'We obtain ([REF]).'}

{'0807.2194-2-0-0': 'Internal structure of a quantum soliton and classical excitations due to trap opening', '0807.2194-2-1-0': 'We analytically solve two problems that may be useful in the context of the recent observation of matter wave bright solitons in a one-dimensional attractive atomic Bose gas.', '0807.2194-2-1-1': 'The first problem is strictly beyond mean field: From the Bethe ansatz solution we extract the internal correlation function of the particle positions in the quantum soliton, that is for a fixed center of mass position.', '0807.2194-2-1-2': 'The second problem is solved in the limit of a large number of particles, where the mean field theory is asymptotically correct: It deals with the number of excitations created by the opening of the trap, starting from a pure soliton in a weakly curved harmonic potential.', '0807.2194-2-2-0': 'Experiments with cold atoms have now acquired a high degree of control of the key parameters of the system.', '0807.2194-2-2-1': 'Using transverse confinement of the atoms by non-dissipative optical potentials it is possible to freeze the atomic motion along one or several directions, realizing in this way quantum gases with reduced dimensionality [CITATION].', '0807.2194-2-2-2': 'Furthermore, thanks to Feshbach resonances driven by a magnetic field, one can adjust almost at will the interaction strength between the atoms [CITATION].', '0807.2194-2-2-3': 'The combination of these two experimental tools has recently allowed the observation of bright solitons in a one-dimensional (1D) Bose gas, either a single soliton [CITATION] or a train of solitons [CITATION].', '0807.2194-2-3-0': 'This leads to a renewed interest [CITATION] in the 1D non-relativistic Bose gas on the free line with zero range attractive interactions, that is with a binary interaction potential modeled by a Dirac delta with a negative coupling constant [MATH], [MATH], a model that is an acceptable approximation of reality under conditions defined in [CITATION].', '0807.2194-2-3-1': 'On a theoretical point of view, it is well known that eigenstates and eigenenergies of the corresponding [MATH]-body Hamiltonian may be obtained by the Bethe ansatz: Historically the focus was put mainly on the study of the ground state wavefunction [MATH] [CITATION], which is a collective bound state of the [MATH] particles, the so-called [MATH]-particle quantum soliton, with a delocalized center of mass of vanishing momentum.', '0807.2194-2-3-2': 'The extension of the Bethe ansatz to excited states is however possible, and one finds that the generic excited state corresponds to a set of quantum solitons with arbitrary atom numbers and different momenta per particle [CITATION].', '0807.2194-2-3-3': 'A key property to keep in mind is the full separability of the center of mass variables and the internal variables of the gas (e.g. the relative coordinates of the particles), which holds since the gas is on the free line with open boundary conditions.', '0807.2194-2-4-0': 'Despite the knowledge of the ground state wavefunction for [MATH] particles, some theoretical work is needed to extract experimentally relevant observables.', '0807.2194-2-4-1': 'As atomic density profiles may be measured by absorption or even non-destructive imaging [CITATION], natural observables are functions of the positions of the particles.', '0807.2194-2-4-2': 'The simplest observable is the mean density of particles, [MATH], obtained by an average of the density profile over many (ideally infinitely many) experimental realizations.', '0807.2194-2-4-3': 'From the ground state [MATH]-body wavefunction one however does not obtain any useful information on the mean density: Since the center of mass is fully delocalized, one gets a uniform distribution over the whole line.', '0807.2194-2-4-4': 'Experimental reality is very different, the soliton being obtained from an initially trapped Bose-Einstein condensate.', '0807.2194-2-4-5': 'When the trap is switched off to free the soliton, its center of mass is not in its ground state, it is in a localized and non-stationary state which depends on the experimental preparation procedure.', '0807.2194-2-4-6': 'A more realistic assumption is thus to assume a [MATH]-body wavefunction of the form [EQUATION] where [EQUATION] is the center of mass position of the soliton and the center of mass wavefunction [MATH] is a priori unknown and depends on the experimental details.', '0807.2194-2-4-7': 'The theoretical challenge is thus to predict observables for a fixed position of the center of mass [MATH].', '0807.2194-2-4-8': 'Experimentally relevant results are then obtained from these theoretical predictions by a further average over [MATH] with the probability distribution [MATH].', '0807.2194-2-4-9': 'One can even hope that the predictions for fixed [MATH] are measurable, if the center of mass position can be measured with a high enough accuracy for each individual realization of the experiment.', '0807.2194-2-5-0': 'Turning back to the simple example of the mean density, we see that the right concept is the mean density of particles [MATH] for a fixed center of mass position [MATH], as was already argued in [CITATION] within the general concept of (here translational) symmetry breaking.', '0807.2194-2-5-1': 'Remarkably an explicit expression of [MATH] in terms of a sum of [MATH] exponential terms may be obtained [CITATION].', '0807.2194-2-6-0': 'The next step beyond the mean density is the pair distribution function of the particles [MATH], or very similarly the static structure factor [MATH], of the [MATH]-particle soliton.', '0807.2194-2-6-1': 'Recently a general study of the dynamic structure factor was performed from the Bethe ansatz [CITATION], which includes the static structure factor [MATH] and large [MATH] expansions as limiting cases.', '0807.2194-2-6-2': 'The goal of the present work is, in the spirit of the above physical discussion, to study the static structure factor [MATH] for a fixed center of mass position [MATH].', '0807.2194-2-6-3': 'This static structure factor gives access to correlations between the positions of the particles inside the soliton, that is it gives information on the internal structure of the soliton, which goes beyond the usual mean field (or Gross-Pitaevskii) approximation, which neglects such correlations.', '0807.2194-2-7-0': 'As a guideline we imagine that, in an experiment, one wishes to access the variance of a one-body observable of the gas involving some function [MATH] of the particle position [MATH]: [EQUATION] where the expectation value [MATH] is taken over the internal wavefunction [MATH] of the quantum soliton for a fixed center of mass location [MATH].', '0807.2194-2-8-0': 'The paper is organized as follows.', '0807.2194-2-8-1': 'In section [REF] we recall the basic facts about the 1D Bose gas model and we give a general expression of the pair distribution function [MATH] for fixed [MATH] that may be used to evaluate [MATH] numerically for a moderate number of atoms and that will be the starting point of analytical calculations.', '0807.2194-2-8-2': 'In section [REF] we present an analytical calculation of [MATH] for an arbitrary atom number [MATH], in the limit where the function [MATH] is slowly varying over the spatial width of the soliton.', '0807.2194-2-8-3': 'In section [REF] we present a large [MATH] expansion of the static structure factor [MATH] for fixed [MATH] that we then use to calculate [MATH] to leading order in [MATH]; in the same section, we integrate [MATH] over [MATH] to see if we recover the results of [CITATION] for [MATH], and we also consider the case of a function [MATH] much narrower than the soliton.', '0807.2194-2-8-4': 'In section [REF] we evaluate the accuracy of the assumption ([REF]) in an experiment: Assuming that the gas in the trap is in its ground state (at least for its internal variables), we calculate analytically (to leading order in [MATH] and in the trap curvature) the number of internal excitations of the soliton produced by the trap opening.', '0807.2194-2-8-5': 'We conclude in section [REF].', '0807.2194-2-9-0': '# Model, basic definitions and general results', '0807.2194-2-10-0': '## Hamiltonian and ground state wavefunction', '0807.2194-2-11-0': 'We consider a set of [MATH] spinless non-relativistic bosons of mass [MATH] moving on the one dimensional real line, in the absence of any trapping potential, with open boundary conditions, and binary interacting via an attractive contact potential of coupling constant [MATH].', '0807.2194-2-11-1': 'This corresponds to the Hamiltonian in first quantized form: [EQUATION]', '0807.2194-2-11-2': 'For this problem there is full separability of the center of mass motion and of the internal coordinates, the internal wavefunction being independent of the center of mass state.', '0807.2194-2-11-3': 'As a consequence, the ground state wavefunction has it center of mass with zero momentum and depends only on the relative coordinates of the particles.', '0807.2194-2-11-4': 'Its exact expression is [CITATION] [EQUATION] where the normalization factor [MATH] will be specified later.', '0807.2194-2-11-5': 'The corresponding eigenenergy is [CITATION] [EQUATION]', '0807.2194-2-11-6': 'This ground state is the so-called quantum soliton with [MATH] particles.', '0807.2194-2-12-0': '## Mean density, pair distribution and static structure factor for a fixed center of mass position', '0807.2194-2-13-0': 'The crucial concept in the present work is the expectation value of a position dependent observable, that is of an arbitrary function [MATH] of the [MATH] particle positions, for a fixed value [MATH] of the center of mass position: [EQUATION] where the integration is over the whole space [MATH].', '0807.2194-2-13-1': 'The soliton wavefunction shall then be normalized in such a way that the expectation value [MATH] of the function [MATH] constant and equal to unity is also equal to unity.', '0807.2194-2-13-2': 'Using the bosonic symmetry of the wavefunction, the integral to compute is then simply [MATH] times the integral over the so-called fundamental domain of ordered positions [EQUATION]', '0807.2194-2-13-3': 'Over this domain, the wavefunction indeed has the separable expression [EQUATION]', '0807.2194-2-13-4': 'With the change of variables ([REF]) one then obtains [EQUATION]', '0807.2194-2-13-5': 'Setting [MATH] we obtain the mean density [MATH] for a fixed center of mass position.', '0807.2194-2-13-6': 'This was first calculated in [CITATION]: [EQUATION] where [MATH] is the spatial width of the classical (that is mean-field) soliton, [EQUATION]', '0807.2194-2-13-7': 'It is a function of [MATH], a consequence of translational and parity invariance of the Hamiltonian and of [MATH].', '0807.2194-2-13-8': 'One thus has [MATH].', '0807.2194-2-13-9': 'It is normalized in such a way that the integral over [MATH] over the whole space is equal to [MATH].', '0807.2194-2-13-10': 'If the true physical state of the system is a product ([REF]) of a localized center of mass wavefunction [MATH] and of the quantum soliton wavefunction, the physical mean density is the average (or equivalently the convolution) [EQUATION]', '0807.2194-2-13-11': 'Setting [MATH], we obtain the pair distribution function [MATH] for fixed center of mass position [MATH].', '0807.2194-2-13-12': 'It is normalized as [MATH] for the double integration over [MATH] and [MATH].', '0807.2194-2-13-13': 'From the translation invariance [MATH] so that it is sufficient to calculate this pair distribution function for [MATH].', '0807.2194-2-13-14': 'In real space, we do not know an expression of [MATH] as simple as ([REF]) but we have found a simple expression for the Fourier transform [EQUATION]', '0807.2194-2-13-15': 'As detailed in the appendix [REF] one has [EQUATION] where [MATH] is the Gamma function of complex argument [MATH] and we have introduced the dimensionless variables [MATH], [MATH] and [MATH].', '0807.2194-2-14-0': 'The quantities [MATH] solve the degree two equations: [EQUATION]', '0807.2194-2-14-1': 'In the large [MATH] limit, for fixed values of [MATH] and [MATH], it is convenient to take [EQUATION] with a determination of the square root such that [MATH] for [MATH], e.g. with the line cut on the real negative axis.', '0807.2194-2-15-0': 'This gives the idea a posteriori to look for a similar expression for the Fourier transform [MATH] of [MATH].', '0807.2194-2-15-1': 'As shown in the appendix [REF] one obtains the simple expression', '0807.2194-2-16-0': 'Let us come back to the original problem of calculating the variance [MATH] of the one-body observable [MATH], as defined in ([REF]).', '0807.2194-2-16-1': 'One first expresses the squares as products of double sums over indices [MATH] and [MATH].', '0807.2194-2-16-2': 'One can split the double sum over [MATH] and [MATH] in ([REF]) in diagonal terms [MATH] and off-diagonal terms [MATH], so that [MATH].', '0807.2194-2-17-0': 'Introducing the Fourier transform of [MATH], [EQUATION] we obtain the Fourier space expression', '0807.2194-2-18-0': 'that can be directly evaluated using ([REF]).', '0807.2194-2-18-1': 'Simpler analytic expressions of [MATH], either for slowly varying functions [MATH] or in the large [MATH] limit, shall be given in the next sections.', '0807.2194-2-19-0': 'To conclude this subsection, we note that [MATH] has a very simple expression in terms of the correlations contained in the fixed [MATH] static structure factor [MATH], [EQUATION] where the operator giving the density is [EQUATION]', '0807.2194-2-19-1': 'We note that the double integral of [MATH] over [MATH] and [MATH] is equal to [MATH].', '0807.2194-2-19-2': 'Using again the translational invariance, one obtains the illuminating expression [EQUATION]', '0807.2194-2-20-0': 'The essence of the mean-field approximation is to neglect correlations among the particles, so that [MATH] would essentially be approximated by the uncorrelated product [MATH].', '0807.2194-2-20-1': 'The above writing clearly reveals that [MATH] is sensitive to correlations that are beyond the mean-field approximation.', '0807.2194-2-21-0': '## The usual static structure factor', '0807.2194-2-22-0': 'The usual static structure factor is the spatial correlation function of the operator [MATH] giving the density, [EQUATION] where the expectation value is taken literally in the ground state of the gas, thus assuming a perfectly delocalized center of mass wavefunction [MATH].', '0807.2194-2-22-1': 'Because of the translational invariance, it is a function of [MATH] only.', '0807.2194-2-23-0': 'This usual structure factor is deduced from our fixed-[MATH] one by integration over the center of mass position [EQUATION]', '0807.2194-2-23-1': 'This allows to express the Fourier transform of [MATH] in terms of the Fourier transform of [MATH], when one uses ([REF]): [EQUATION]', '0807.2194-2-23-2': 'From ([REF]) we thus have an analytical expression of [MATH] in terms of a double sum.', '0807.2194-2-24-0': 'The relation ([REF]) will also allow us, in the large [MATH] limit, to convert our large [MATH] expansion of [MATH] into a large [MATH] expansion of [MATH], see [REF].', '0807.2194-2-25-0': '# Value of [MATH] for a broad function [MATH]', '0807.2194-2-26-0': 'One supposes in this section that [MATH] varies slowly over the length scale [MATH] of the quantum soliton, as defined in ([REF]).', '0807.2194-2-26-1': 'Then one rewrites ([REF]) using the translational invariance, [EQUATION] and one expands [EQUATION]', '0807.2194-2-26-2': 'The constant shift [MATH] has no effect in [MATH].', '0807.2194-2-26-3': 'The linear term has also an exactly vanishing contribution, since by definition [MATH] for all integers [MATH].', '0807.2194-2-26-4': 'Setting [EQUATION] we thus obtain the leading contribution [EQUATION] with [MATH] is the variance of [MATH] for a center of mass position fixed at the origin of the coordinates.', '0807.2194-2-27-0': 'It turns out that an exact expression may be obtained for this variance, as detailed in the appendix [REF]: It is the sum of three contributions, [EQUATION] with [EQUATION]', '0807.2194-2-27-1': 'We have introduced the symmetric matrix [EQUATION] defined over the index range [MATH].', '0807.2194-2-27-2': 'This holds whatever the value of the atom number [MATH].', '0807.2194-2-28-0': 'In the large [MATH] limit, the contribution [MATH] is dominant, simply because it contains more terms, and its asymptotic expression is evaluated by replacing the sums by integrals, [EQUATION]', '0807.2194-2-28-1': 'This leads to the estimate [EQUATION] for a slowly varying potential [MATH] in the [MATH] limit.', '0807.2194-2-29-0': '# Large [MATH] limit of [MATH] for [MATH] of any width', '0807.2194-2-30-0': 'In this section we give a large [MATH] expansion of the static structure factor for fixed center of mass position, which allows to get the asymptotic behavior of [MATH] in the large [MATH] limit.', '0807.2194-2-30-1': 'When specialized to a quadratic potential [MATH] the general result reproduces the large [MATH] broad potential result of the previous section.', '0807.2194-2-30-2': 'As a first test of the result, we integrate [MATH] over the center of mass position [MATH], to see if we recover the results of [CITATION] for the usual static factor [MATH].', '0807.2194-2-30-3': 'As a second test of the result, we get an approximate expression for [MATH] in the case of a narrow function [MATH], to lowest order in the width [MATH] of [MATH], first from the large [MATH] expansion of [MATH] and then from a more general reasoning not relying on a large [MATH] expansion.', '0807.2194-2-31-0': '## Asymptotic expression of [MATH] and of [MATH]', '0807.2194-2-32-0': 'As we have seen in ([REF]), [MATH] is directly related to the deviation of the fixed [MATH] static structure factor of the soliton from the uncorrelated form [MATH].', '0807.2194-2-32-1': 'It turns out that this deviation may be easily obtained from the Fourier space expressions ([REF],[REF]), simply by taking the large [MATH] limit of the [MATH] functions and by replacing the discrete sums over indices by integrals.', '0807.2194-2-32-2': 'This shows that our Fourier space representations are indeed useful.', '0807.2194-2-33-0': 'As detailed in the appendix [REF], the large [MATH] expansion for a fixed value of [MATH] (that is for a fixed value of [MATH]), gives the leading term [EQUATION] where [MATH] is the Heaviside distribution that is equal to zero for [MATH] and to one for [MATH].', '0807.2194-2-34-0': 'If one wishes to have an expansion of the static structure factor only, one has also to expand [MATH] in powers of [MATH].', '0807.2194-2-34-1': 'From the real space expression ([REF]), expanding each factorial in the large [MATH] limit for a fixed summation index [MATH], we obtain, setting [MATH], a result in agreement with [CITATION]: [EQUATION] where one may check the normalization condition [MATH].', '0807.2194-2-34-2': 'This gives the expansion of [MATH] up to order [MATH], for a fixed value of [MATH].', '0807.2194-2-35-0': 'From ([REF]) and ([REF]) we can directly obtain, in the large [MATH] limit, the pair correlations between the positions of the particles for a fixed center of mass position: [EQUATION]', '0807.2194-2-35-1': 'One can indeed show that the distributions generated in ([REF]) by the derivatives of [MATH] and [MATH] with respect to [MATH] and [MATH] exactly cancel with the Dirac term appearing in ([REF]).', '0807.2194-2-35-2': 'As a consequence the expression for [MATH] is deduced from the right hand side of ([REF]) simply by exchanging the order of the [MATH] distributions and of the operator [MATH].', '0807.2194-2-35-3': 'After an explicit calculation of the derivatives with respect to [MATH] and [MATH] we obtain [EQUATION] with [MATH] and [MATH].', '0807.2194-2-36-0': 'A contour plot of this function reveals that it has an interesting structure, in the form of two valleys separated by a crest on the [MATH] line, each valley being elongated in the direction parallel to [MATH] and containing two local minima separated by a saddle point.', '0807.2194-2-36-1': 'For clarity we only show a plot of [MATH] along the line [MATH], restricting to [MATH] by parity, see the thick solid line in Fig. [REF]; the minimum of this line then corresponds in the full [MATH] plane to one of the aforementioned saddle points.', '0807.2194-2-36-2': 'In the same figure, we also give the value of [MATH] for finite values of [MATH], obtained by calculating the Fourier transform of ([REF]) and ([REF]) numerically.', '0807.2194-2-36-3': 'This shows that the large [MATH] limit is well approached with moderately high values of [MATH] already.', '0807.2194-2-37-0': 'We come back to ([REF]) to obtain the large [MATH] equivalent of [MATH] at fixed [MATH].', '0807.2194-2-38-0': 'One immediately sees that a linearly varying potential [MATH] gives a vanishes contribution to [MATH], which is also obvious from the definition ([REF]), since [MATH] by construction.', '0807.2194-2-38-1': 'If [MATH] is a quadratic function of [MATH], [MATH], one should recover Eq. ([REF]), which is indeed the case if [EQUATION]', '0807.2194-2-38-2': 'We have checked that this identity holds [CITATION].', '0807.2194-2-39-0': '## Application: large [MATH] expansion of the usual static structure factor', '0807.2194-2-40-0': 'The usual static structure factor [MATH] is obtained from [MATH] by integration over [MATH], see ([REF]).', '0807.2194-2-40-1': 'Assuming without loss of generality that [MATH], we see from ([REF]) and from ([REF]) that one has to integrate over [MATH] quantities of the form [MATH], where the function [MATH] corresponds in a first stage to the expression in between square brackets in ([REF]) and in a second stage to the function over which [MATH] acts in the right hand side of ([REF]).', '0807.2194-2-40-2': 'From the differential relations, taking [MATH] and [MATH] as independent variables, [EQUATION] we see that the integral over [MATH] will cancel the derivatives [MATH].', '0807.2194-2-40-3': 'Using [EQUATION] one gets [EQUATION] where [MATH].', '0807.2194-2-41-0': 'One can show that the [MATH] distribution can be exchanged with the last [MATH] if one wishes, but we shall not use this property here.', '0807.2194-2-41-1': 'From the large [MATH] expansion ([REF]) of [MATH], and using the fact that the second order derivative of [MATH] is an even function, one finally obtains the large [MATH] expansion', '0807.2194-2-42-0': 'To compare this result with the ones of [CITATION] given in Fourier space, one performs the Fourier transform of Eq.(62) and Eq.(63) of [CITATION].', '0807.2194-2-42-1': 'Then one finds that ([REF]) is consistent with [CITATION] if one adds in Eq.(62) of [CITATION] the subleading term of the large [MATH] expansion of the intermediate quantity [MATH] introduced in [CITATION].', '0807.2194-2-43-0': 'where we recall that [MATH].', '0807.2194-2-43-1': 'One can also easily check that this obeys the sum rule [MATH].', '0807.2194-2-44-0': '## Approximation of [MATH] for a narrow barrier', '0807.2194-2-45-0': 'In this subsection, one assumes that [MATH] and its derivatives are functions localized around the origin with a width much smaller than [MATH].', '0807.2194-2-45-1': 'E.g. [MATH] is a Gaussian centered in [MATH] with a width [MATH].', '0807.2194-2-45-2': 'One first considers the large [MATH] limit.', '0807.2194-2-45-3': 'We rewrite Eq. ([REF]) as [EQUATION] with [EQUATION]', '0807.2194-2-46-0': 'Then we expand the factor containing [MATH] in powers of [MATH] and [MATH].', '0807.2194-2-46-1': 'If [MATH] and its derivatives are rapidly decreasing functions, one can show that the first non vanishing contribution to [MATH] comes from the third order expansion, which results in [EQUATION] where we have used ([REF]) to leading order in [MATH] to recognize the factor [MATH].', '0807.2194-2-47-0': 'There is actually a faster way to obtain this result, and not restricted to the large [MATH] limit, from the exact writing [EQUATION]', '0807.2194-2-48-0': 'Then one sees that the first term in ([REF]) is first order in the width [MATH] of [MATH], since the integration range over [MATH] has a width [MATH], whereas the second term is second order in [MATH], since it involves a double integral over a range of diameter [MATH].', '0807.2194-2-48-1': 'Then to first order in [MATH] one recovers ([REF]).', '0807.2194-2-48-2': 'In the large [MATH] limit, for a fixed [MATH], both terms of ([REF]) scale in the same way with [MATH], that is linearly with [MATH], so that the order of the limits [MATH] and [MATH] (for fixed [MATH]) is not crucial.', '0807.2194-2-49-0': '# Number of internal excitations created by the trap opening', '0807.2194-2-50-0': 'In this work, we have assumed up to now that a pure soliton is produced in the experiment, corresponding to the state ([REF]): The center of mass may be in an arbitrary excited state but the internal variables of the gas are in their ground state.', '0807.2194-2-50-1': 'In this section, we revisit this assumption taking into account experimental constraints.', '0807.2194-2-51-0': 'In a real experiment the ultracold gas is prepared in a trap; it is not free along [MATH], and each atom is subject to the harmonic confining potential [EQUATION]', '0807.2194-2-51-1': 'We recall that the center of mass motion and the internal variables remain separable in a harmonic trap.', '0807.2194-2-51-2': 'In order for the trapped gas to be close to the free space limit, the oscillation frequency [MATH] is adjusted to have [MATH], where [EQUATION] is the mean-field approximation for the chemical potential of the free-space soliton.', '0807.2194-2-51-3': 'We assume that the trapped gas is cooled down to a temperature [MATH] low enough to have [MATH].', '0807.2194-2-51-4': 'The gas is then a pure soliton.', '0807.2194-2-51-5': 'On the contrary we do not assume the much more stringent condition [MATH], so that the center of mass of the gas may still be in an excited state.', '0807.2194-2-52-0': 'Eventually the trapping potential along [MATH] will be switched off, to obtain the ideal conditions of the Hamiltonian ([REF]).', '0807.2194-2-52-1': 'This trap opening will create some internal excitations of the gas, that is the untrapped gas will not be a pure soliton but will contain a mean number [MATH] of internal excitations.', '0807.2194-2-53-0': 'Since no Bethe ansatz solution exists in a trap, the modest goal here is to calculate [MATH] to leading order in [MATH] in the large [MATH] limit, for a fixed value of [MATH] [CITATION].', '0807.2194-2-53-1': 'In this limit, a 1D classical field treatment is sufficient, with a Hamiltonian [EQUATION]', '0807.2194-2-53-2': 'We have slightly generalized ([REF]) to treat the case of a time dependent trap: [EQUATION] where the function [MATH], going from unity for [MATH] to zero for [MATH] describes the switch-off procedure of the trap.', '0807.2194-2-53-3': 'The atom number is fixed so that the norm squared of the classical field is also fixed: [EQUATION]', '0807.2194-2-53-4': 'This classical field problem can be solved on a computer, which will allow a test of our predictions for [MATH].', '0807.2194-2-54-0': 'The problem can be further simplified in the limit [MATH], where one expects that the field [MATH] will remain "close" (up to a phase factor) to the one describing a pure soliton [CITATION], [MATH], with [EQUATION]', '0807.2194-2-54-1': 'We then use the number conserving Bogoliubov formalism of [CITATION], downgraded to a classical field problem (simply replacing commutators with Poisson brackets), as was already done in [CITATION].', '0807.2194-2-54-2': 'One splits the field by projection along the mode [MATH] and orthogonally to it: [EQUATION] where [MATH] is the component of the field on the mode [MATH] and the field [MATH] is orthogonal to that mode.', '0807.2194-2-54-3': 'The idea is to treat [MATH] as a small perturbation.', '0807.2194-2-54-4': 'The strength of the number conserving approach is to eliminate the amplitude [MATH] in a systematic way, using the modulus-phase representation [EQUATION]', '0807.2194-2-54-5': 'The phase [MATH] is eliminated by a redefinition of the transverse field: [EQUATION]', '0807.2194-2-54-6': 'The modulus [MATH] is expressed in terms of [MATH] using the condition of a fixed atom number ([REF]).', '0807.2194-2-55-0': 'In the absence of trapping potential, one keeps terms up to quadratic in [MATH] in the Hamiltonian, and the resulting quadratic form can be written in normal form as [CITATION] [EQUATION] where [MATH] is the ground state of the classical field model.', '0807.2194-2-55-1': 'The variable [MATH] of the field represents the total momentum of the field, written to first order in [MATH]: [EQUATION]', '0807.2194-2-55-2': 'The occurrence of the term [MATH] represents physically the fact that the center of mass motion is decoupled; more formally, it corresponds to the fact that [MATH] "spontaneously" breaks the translational symmetry of the Hamiltonian, which leads to the occurrence of a Goldstone mode [CITATION].', '0807.2194-2-55-3': 'The field variable canonically conjugated to [MATH] corresponds to the center of mass position of the field, written up to first order in [MATH]: [EQUATION]', '0807.2194-2-55-4': 'Apart from this Goldstone mode, the other eigenmodes behave as a continuum of decoupled harmonic oscillators, with normal (complex) variables [MATH] and eigenenergies [EQUATION] which correspond to internal (and thus gapped) excitations of the gas: An elementary excitation physically takes the form of a free particle coming from infinity with a wavevector [MATH] and scattering on a soliton with [MATH] particles.', '0807.2194-2-55-5': "The field variable [MATH] has the expression [EQUATION] where the Bogoliubov modes of the number conserving theory are expressed as follows in Dirac's notation (see [CITATION] V.A) here for a real function [MATH]: [EQUATION] where [MATH] projects orthogonally to [MATH], and the [MATH] are the eigenmodes of the usual Bogoliubov-de Gennes equations [EQUATION]", '0807.2194-2-55-6': 'It turns out that these modes are known exactly [CITATION].', '0807.2194-2-55-7': 'For [MATH] one has [EQUATION] where (differently from section [REF]) we have set [MATH] and [MATH].', '0807.2194-2-55-8': 'The modes for [MATH] are deduced from the relations [MATH] and [MATH].', '0807.2194-2-56-0': 'In presence of the trap, there is an extra contribution to the Hamiltonian containing the trapping energy, [EQUATION]', '0807.2194-2-56-1': 'We approximate it to leading non-trivial order in [MATH], that is to first order, injecting the splitting ([REF]).', '0807.2194-2-56-2': 'The term in [MATH] deviates from [MATH] by [MATH] so it gives to first order only a constant contribution, with an integral over [MATH] that can be calculated exactly if necessary.', '0807.2194-2-56-3': 'The crossed terms [MATH] and complex conjugate are kept, with [MATH] approximated by [MATH], whereas the quadratic terms [MATH] are neglected.', '0807.2194-2-56-4': 'Finally, we replace the field [MATH] by its modal expansion [EQUATION]', '0807.2194-2-57-0': 'We thus keep [EQUATION] with coefficients [EQUATION]', '0807.2194-2-57-1': 'With the residues method, the resulting integral can be evaluated analytically.', '0807.2194-2-57-2': 'We give for convenience the ratio to the mode eigenenergy: [EQUATION] where [MATH] as above.', '0807.2194-2-58-0': "When the perturbation [MATH] is added to the unperturbed Hamiltonian ([REF]), the equilibrium value of the [MATH]'s minimizing the resulting energy is [EQUATION]", '0807.2194-2-58-1': 'At later times, the trap is opened according to the switch-off function [MATH].', '0807.2194-2-58-2': 'The Hamiltonian equations of motion then give [EQUATION] to be solved with the initial conditions ([REF]): [EQUATION] where [MATH] is the mode frequency.', '0807.2194-2-58-3': 'This allows to calculate the number of excitations after the trap was switched off, using [MATH] for [MATH]: [EQUATION] where [MATH] is a spectral density of the switch-off procedure at frequency [MATH]: [EQUATION]', '0807.2194-2-58-4': 'We recall that ([REF]) is valid up to leading order in [MATH] (because of the classical field model) and to leading order in [MATH] (because of the perturbative treatment of the deviations of the field from the free space soliton).', '0807.2194-2-59-0': 'The case producing the maximal number of excitations for a monotonic [MATH] corresponds to a sudden trap switch-off, where [MATH] at all frequencies.', '0807.2194-2-59-1': 'Using ([REF]), we find that the resulting integral can be evaluated with the residues method, so that [EQUATION] with [EQUATION]', '0807.2194-2-60-0': 'This result is encouraging since a moderately small value [MATH] already leads to a number of excitations relative to the total atom number at the [MATH] level.', '0807.2194-2-60-1': 'As shown in Fig. [REF] the analytical prediction ([REF]) is in good agreement with the number of excitations deduced from a numerical solution of the Gross-Pitaevskii equation, provided that [MATH].', '0807.2194-2-61-0': 'The number of excitations can be reduced by switching off the trap more slowly.', '0807.2194-2-61-1': 'E.g. a linear ramping [EQUATION] leads to [EQUATION]', '0807.2194-2-61-2': 'Due to the presence of a gap [MATH] in the internal excitation spectrum of the gas, one gets the upper bound on the number of excitations [EQUATION] that is one gains quadratically with the switch-off time when it becomes longer than the internal soliton time [MATH].', '0807.2194-2-62-0': '# Conclusion', '0807.2194-2-63-0': 'Inspired by recent observations of matter wave bright solitons in atomic gases, we have considered here two problems that may be relevant for experiments.', '0807.2194-2-64-0': 'The first problem is strictly beyond mean field: It corresponds to the pair correlations between positions of the particles in a quantum soliton for a fixed position of the center of mass of the soliton.', '0807.2194-2-64-1': 'In particular, we have obtained analytically the large [MATH] limit expression of these pair correlations, see ([REF]) and ([REF]).', '0807.2194-2-64-2': 'By integrating ([REF]) over the center of mass position, we obtain a large [MATH] expansion of the static structure factor for a fully delocalized center of mass position, a quantity already studied in [CITATION].', '0807.2194-2-64-3': 'On an experimental point of view, our predictions can be tested by measuring the positions of the particles in a very broad quantum soliton, prepared with weak attractive interactions and a relatively small atom number [CITATION].', '0807.2194-2-65-0': 'The second problem was studied in the classical field model.', '0807.2194-2-65-1': 'The number of internal excitations of the gas created by the trap opening from an initial pure soliton was calculated in the limit where the soliton size is smaller than the size of the harmonic single particle ground state, see ([REF]) for a sudden trap opening.', '0807.2194-2-65-2': 'This also can be seen experimentally by detecting atoms flying away from the remaining soliton core after the trap opening.', '0807.2194-2-65-3': 'A possible extension of this calculation is to include quantum fluctuations of the field.', '0807.2194-2-66-0': 'We acknowledge useful discussions with L. Khaykovich, C. Weiss, A. Sinatra, M. Olshanii.', '0807.2194-2-66-1': 'Our group is a member of IFRAF.', '0807.2194-2-67-0': '# Calculation of the Fourier transforms of [MATH] and [MATH]', '0807.2194-2-68-0': 'We start with the definition of the mean density for a fixed center of mass position [MATH], taking here [MATH] without loss of generality: From ([REF]) one has [EQUATION]', '0807.2194-2-69-0': 'Using the bosonic exchange symmetry we can restrict the integral to the fundamental domain [MATH] of ([REF]), including a factor [MATH].', '0807.2194-2-69-1': 'For simplicity we take in this appendix [MATH] as the unit of length.', '0807.2194-2-69-2': 'With the change of variables, of Jacobian equal to unity, [EQUATION] the condition to be in [MATH] is simply that all the [MATH] are positive, and [MATH] can vary in the whole real space [MATH].', '0807.2194-2-70-0': 'One can calculate the integral over [MATH]: The first delta factor in ([REF]), the one ensuring that [MATH], imposes a value [EQUATION]', '0807.2194-2-70-1': 'When one reports this value of [MATH] in the argument of the second delta factor in ([REF]), one obtains a remaining factor [MATH], with [EQUATION]', '0807.2194-2-70-2': 'Taking the Fourier transform of this remaining delta factor, according to [MATH], gives [EQUATION] where we used the fact that the product of all [MATH] (for [MATH] from 2 to [MATH]) is equal to [MATH].', '0807.2194-2-70-3': 'Replacing the [MATH] by their expression and using the identity [EQUATION] deduced from the basic property [MATH] of the Gamma function, with the convention that an "empty" product is equal to unity, one gets ([REF]).', '0807.2194-2-71-0': 'We now turn to the pair distribution function for a center of mass position fixed in [MATH].', '0807.2194-2-71-1': 'Following the same steps as for the mean density, we obtain the Fourier transform [EQUATION]', '0807.2194-2-71-2': 'We can restrict to a summation over [MATH], the contribution for [MATH] being deduced by exchanging [MATH] and [MATH].', '0807.2194-2-71-3': 'In the product over [MATH], three ranges have then to be considered, (i) the first range [MATH], (ii) the mid-range [MATH], and (iii) the last range [MATH].', '0807.2194-2-71-4': 'The first range and the last range contributions can be expressed in terms of the Gamma function as in ([REF]).', '0807.2194-2-71-5': 'The mid-range contribution is equal to the product [EQUATION] where we have reindexed the product setting [MATH] and we have defined [MATH] and [MATH].', '0807.2194-2-71-6': 'The last step is to consider the denominator in each factor of ([REF]) as a polynomial of degree 2 in [MATH]: Its roots are [MATH] defined in ([REF], [REF]) and [MATH].', '0807.2194-2-71-7': 'This leads to [EQUATION] which can now be expressed as a ratio of products of Gamma functions.', '0807.2194-2-71-8': 'Then one gets ([REF]).', '0807.2194-2-72-0': '# Calculation of the variance of [MATH]', '0807.2194-2-73-0': 'We explain how to calculate exactly the moments [MATH] of the quantity [MATH] defined in ([REF]) in the [MATH]-body internal ground state for a fixed center of mass position [MATH].', '0807.2194-2-73-1': 'We take [MATH] as unit of length and we use the transformations exposed at the beginning of appendix [REF].', '0807.2194-2-73-2': 'Considering the change of variable ([REF]), we rewrite [MATH] as [EQUATION]', '0807.2194-2-74-0': 'The first sum in the right hand side of ([REF]) will have a vanishing contribution, since the expectation value is taken for a zero center of mass position.', '0807.2194-2-74-1': 'Replacing [MATH] by its expression in terms of [MATH], and using the fact that the value of [MATH] is fixed to ([REF]) we see that one may effectively replace [MATH] by the quantity', '0807.2194-2-75-0': 'with the symmetric matrix [EQUATION]', '0807.2194-2-75-1': 'We have thus reduced the problem to the calculation of the integrals', '0807.2194-2-76-0': 'with [MATH] defined in ([REF]).', '0807.2194-2-76-1': 'These integrals may be calculated by interpreting them as Gaussian averages, introducing the auxiliary complex random variables [MATH], [MATH]: These variables [MATH] are statistically independent and each one has a Gaussian probability distribution [MATH].', '0807.2194-2-76-2': 'Each [MATH] then corresponds to [MATH], so that [EQUATION]', '0807.2194-2-76-3': "Here [MATH] denotes the Gaussian average of the [MATH]'s and can be calculated using Wick's theorem.", '0807.2194-2-76-4': "The calculations are a bit lengthy for [MATH] since Wick's theorem has to be applied to a product of 8 variables.", '0807.2194-2-76-5': 'Clearly the matrix [MATH] appears, which is the matrix [MATH] of ([REF]).', '0807.2194-2-76-6': 'We finally get ([REF]) for [MATH].', '0807.2194-2-77-0': '# Large [MATH] asymptotics of [MATH]', '0807.2194-2-78-0': 'To perform the large [MATH] expansion we shall use the Fourier space expressions ([REF],[REF]).', '0807.2194-2-78-1': 'We thus define [EQUATION]', '0807.2194-2-78-2': 'The pair distribution function involves a double sum over [MATH].', '0807.2194-2-78-3': 'Since [MATH] is a simple sum, the last term in ([REF]) is a double sum over [MATH] and [MATH], without the restriction [MATH].', '0807.2194-2-78-4': 'It thus makes sense to split [MATH] into a off-diagonal part ([MATH]), that we collect with the double sum in [MATH], and a diagonal part [EQUATION] where [MATH] are defined above ([REF]).', '0807.2194-2-79-0': 'The idea to obtain the large [MATH] limit is simply to replace the discrete sums by integrals.', '0807.2194-2-79-1': 'To this end, one has to calculate the large [MATH] limit of each term of the sums, for fixed values of [MATH] and [MATH].', '0807.2194-2-79-2': 'The expansion of the Gamma functions is conveniently performed using [EQUATION] where the real quantity [MATH] tends to [MATH], and the fixed quantities [MATH] and [MATH] may be complex.', '0807.2194-2-79-3': 'One also uses the large [MATH] expansion of the quantity [MATH]: [EQUATION]', '0807.2194-2-79-4': 'For the diagonal part, only the leading term of ([REF]) is useful.', '0807.2194-2-79-5': 'Replacing [MATH] by [MATH] leads to [EQUATION] where [MATH] and the integral was transformed with the change of variable [MATH], to acquire the form of a Fourier transform.', '0807.2194-2-79-6': 'Note that the resulting integral can be calculated exactly, giving [MATH], but this is not useful here.', '0807.2194-2-80-0': 'For the double sum over [MATH], one has to include the [MATH] term in the expansion ([REF]), to obtain a non-zero result: [EQUATION]', '0807.2194-2-80-1': 'The change of variables [MATH] and [MATH] gives to the off-diagonal contribution the form of a Fourier transform: [EQUATION]', '0807.2194-2-80-2': 'The leading term of [MATH] for [MATH] for a fixed [MATH] is the sum of ([REF]) and ([REF]).', '0807.2194-2-80-3': 'The Fourier transform with respect to [MATH] is straightforward: The factors [MATH] act as derivatives, and the remaining bits have already a Fourier form.', '0807.2194-2-80-4': 'The Fourier transform of the diagonal contribution gives a contribution involving a factor [MATH], which exactly cancels with the first term in the right-hand side of ([REF]), at the considered order in [MATH], see ([REF]).', '0807.2194-2-80-5': 'We obtain ([REF]).'}

[['0807.2194-1-13-0', '0807.2194-2-13-0'], ['0807.2194-1-13-1', '0807.2194-2-13-1'], ['0807.2194-1-13-2', '0807.2194-2-13-2'], ['0807.2194-1-13-3', '0807.2194-2-13-3'], ['0807.2194-1-13-5', '0807.2194-2-13-5'], ['0807.2194-1-13-6', '0807.2194-2-13-6'], ['0807.2194-1-13-7', '0807.2194-2-13-7'], ['0807.2194-1-13-8', '0807.2194-2-13-8'], ['0807.2194-1-13-9', '0807.2194-2-13-9'], ['0807.2194-1-13-10', '0807.2194-2-13-10'], ['0807.2194-1-13-11', '0807.2194-2-13-11'], ['0807.2194-1-13-12', '0807.2194-2-13-12'], ['0807.2194-1-13-13', '0807.2194-2-13-13'], ['0807.2194-1-13-14', '0807.2194-2-13-14'], ['0807.2194-1-13-15', '0807.2194-2-13-15'], ['0807.2194-1-76-0', '0807.2194-2-79-0'], ['0807.2194-1-76-1', '0807.2194-2-79-1'], ['0807.2194-1-76-2', '0807.2194-2-79-2'], ['0807.2194-1-76-3', '0807.2194-2-79-3'], ['0807.2194-1-76-4', '0807.2194-2-79-4'], ['0807.2194-1-76-5', '0807.2194-2-79-5'], ['0807.2194-1-76-6', '0807.2194-2-79-6'], ['0807.2194-1-71-0', '0807.2194-2-74-0'], ['0807.2194-1-71-1', '0807.2194-2-74-1'], ['0807.2194-1-44-0', '0807.2194-2-46-0'], ['0807.2194-1-56-0', '0807.2194-2-58-0'], ['0807.2194-1-56-1', '0807.2194-2-58-1'], ['0807.2194-1-56-2', '0807.2194-2-58-2'], ['0807.2194-1-56-4', '0807.2194-2-58-4'], ['0807.2194-1-75-0', '0807.2194-2-78-0'], ['0807.2194-1-75-1', '0807.2194-2-78-1'], ['0807.2194-1-75-2', '0807.2194-2-78-2'], ['0807.2194-1-75-3', '0807.2194-2-78-3'], ['0807.2194-1-75-4', '0807.2194-2-78-4'], ['0807.2194-1-26-0', '0807.2194-2-26-0'], ['0807.2194-1-26-1', '0807.2194-2-26-1'], ['0807.2194-1-26-2', '0807.2194-2-26-2'], ['0807.2194-1-26-4', '0807.2194-2-26-4'], ['0807.2194-1-61-0', '0807.2194-2-64-0'], ['0807.2194-1-65-0', '0807.2194-2-68-0'], ['0807.2194-1-36-0', '0807.2194-2-38-0'], ['0807.2194-1-36-1', '0807.2194-2-38-1'], ['0807.2194-1-36-2', '0807.2194-2-38-2'], ['0807.2194-1-5-0', '0807.2194-2-5-0'], ['0807.2194-1-5-1', '0807.2194-2-5-1'], ['0807.2194-1-66-0', '0807.2194-2-69-0'], ['0807.2194-1-66-1', '0807.2194-2-69-1'], ['0807.2194-1-66-2', '0807.2194-2-69-2'], ['0807.2194-1-30-0', '0807.2194-2-30-0'], ['0807.2194-1-30-1', '0807.2194-2-30-1'], ['0807.2194-1-30-2', '0807.2194-2-30-2'], ['0807.2194-1-30-3', '0807.2194-2-30-3'], ['0807.2194-1-38-0', '0807.2194-2-40-0'], ['0807.2194-1-38-2', '0807.2194-2-40-2'], ['0807.2194-1-38-3', '0807.2194-2-40-3'], ['0807.2194-1-49-0', '0807.2194-2-51-0'], ['0807.2194-1-49-1', '0807.2194-2-51-1'], ['0807.2194-1-49-4', '0807.2194-2-51-4'], ['0807.2194-1-49-5', '0807.2194-2-51-5'], ['0807.2194-1-22-0', '0807.2194-2-22-0'], ['0807.2194-1-22-1', '0807.2194-2-22-1'], ['0807.2194-1-15-0', '0807.2194-2-15-0'], ['0807.2194-1-15-1', '0807.2194-2-15-1'], ['0807.2194-1-43-0', '0807.2194-2-45-0'], ['0807.2194-1-43-1', '0807.2194-2-45-1'], ['0807.2194-1-43-2', '0807.2194-2-45-2'], ['0807.2194-1-43-3', '0807.2194-2-45-3'], ['0807.2194-1-46-0', '0807.2194-2-48-0'], ['0807.2194-1-46-1', '0807.2194-2-48-1'], ['0807.2194-1-46-2', '0807.2194-2-48-2'], ['0807.2194-1-45-0', '0807.2194-2-47-0'], ['0807.2194-1-53-0', '0807.2194-2-55-0'], ['0807.2194-1-53-1', '0807.2194-2-55-1'], ['0807.2194-1-53-2', '0807.2194-2-55-2'], ['0807.2194-1-53-3', '0807.2194-2-55-3'], ['0807.2194-1-53-4', '0807.2194-2-55-4'], ['0807.2194-1-53-5', '0807.2194-2-55-5'], ['0807.2194-1-53-6', '0807.2194-2-55-6'], ['0807.2194-1-53-8', '0807.2194-2-55-8'], ['0807.2194-1-55-0', '0807.2194-2-57-0'], ['0807.2194-1-55-1', '0807.2194-2-57-1'], ['0807.2194-1-55-2', '0807.2194-2-57-2'], ['0807.2194-1-67-0', '0807.2194-2-70-0'], ['0807.2194-1-67-1', '0807.2194-2-70-1'], ['0807.2194-1-67-2', '0807.2194-2-70-2'], ['0807.2194-1-67-3', '0807.2194-2-70-3'], ['0807.2194-1-18-0', '0807.2194-2-18-0'], ['0807.2194-1-18-1', '0807.2194-2-18-1'], ['0807.2194-1-23-0', '0807.2194-2-23-0'], ['0807.2194-1-23-1', '0807.2194-2-23-1'], ['0807.2194-1-23-2', '0807.2194-2-23-2'], ['0807.2194-1-28-1', '0807.2194-2-28-1'], ['0807.2194-1-51-0', '0807.2194-2-53-0'], ['0807.2194-1-51-1', '0807.2194-2-53-1'], ['0807.2194-1-51-2', '0807.2194-2-53-2'], ['0807.2194-1-51-3', '0807.2194-2-53-3'], ['0807.2194-1-1-0', '0807.2194-2-1-0'], ['0807.2194-1-1-1', '0807.2194-2-1-1'], ['0807.2194-1-1-2', '0807.2194-2-1-2'], ['0807.2194-1-17-0', '0807.2194-2-17-0'], ['0807.2194-1-11-0', '0807.2194-2-11-0'], ['0807.2194-1-11-1', '0807.2194-2-11-1'], ['0807.2194-1-11-2', '0807.2194-2-11-2'], ['0807.2194-1-11-3', '0807.2194-2-11-3'], ['0807.2194-1-11-4', '0807.2194-2-11-4'], ['0807.2194-1-11-5', '0807.2194-2-11-5'], ['0807.2194-1-11-6', '0807.2194-2-11-6'], ['0807.2194-1-34-0', '0807.2194-2-34-0'], ['0807.2194-1-34-2', '0807.2194-2-34-2'], ['0807.2194-1-19-0', '0807.2194-2-19-0'], ['0807.2194-1-19-1', '0807.2194-2-19-1'], ['0807.2194-1-19-2', '0807.2194-2-19-2'], ['0807.2194-1-2-0', '0807.2194-2-2-0'], ['0807.2194-1-2-1', '0807.2194-2-2-1'], ['0807.2194-1-2-2', '0807.2194-2-2-2'], ['0807.2194-1-2-3', '0807.2194-2-2-3'], ['0807.2194-1-32-0', '0807.2194-2-32-0'], ['0807.2194-1-32-1', '0807.2194-2-32-1'], ['0807.2194-1-32-2', '0807.2194-2-32-2'], ['0807.2194-1-68-0', '0807.2194-2-71-0'], ['0807.2194-1-68-1', '0807.2194-2-71-1'], ['0807.2194-1-68-2', '0807.2194-2-71-2'], ['0807.2194-1-68-3', '0807.2194-2-71-3'], ['0807.2194-1-68-4', '0807.2194-2-71-4'], ['0807.2194-1-68-5', '0807.2194-2-71-5'], ['0807.2194-1-68-6', '0807.2194-2-71-6'], ['0807.2194-1-68-7', '0807.2194-2-71-7'], ['0807.2194-1-68-8', '0807.2194-2-71-8'], ['0807.2194-1-48-0', '0807.2194-2-50-0'], ['0807.2194-1-48-1', '0807.2194-2-50-1'], ['0807.2194-1-70-1', '0807.2194-2-73-1'], ['0807.2194-1-70-2', '0807.2194-2-73-2'], ['0807.2194-1-54-0', '0807.2194-2-56-0'], ['0807.2194-1-54-1', '0807.2194-2-56-1'], ['0807.2194-1-54-2', '0807.2194-2-56-2'], ['0807.2194-1-54-3', '0807.2194-2-56-3'], ['0807.2194-1-54-4', '0807.2194-2-56-4'], ['0807.2194-1-52-0', '0807.2194-2-54-0'], ['0807.2194-1-52-1', '0807.2194-2-54-1'], ['0807.2194-1-52-3', '0807.2194-2-54-3'], ['0807.2194-1-52-5', '0807.2194-2-54-5'], ['0807.2194-1-52-6', '0807.2194-2-54-6'], ['0807.2194-1-27-0', '0807.2194-2-27-0'], ['0807.2194-1-27-2', '0807.2194-2-27-2'], ['0807.2194-1-8-0', '0807.2194-2-8-0'], ['0807.2194-1-8-1', '0807.2194-2-8-1'], ['0807.2194-1-8-2', '0807.2194-2-8-2'], ['0807.2194-1-8-5', '0807.2194-2-8-5'], ['0807.2194-1-4-0', '0807.2194-2-4-0'], ['0807.2194-1-4-1', '0807.2194-2-4-1'], ['0807.2194-1-4-2', '0807.2194-2-4-2'], ['0807.2194-1-4-3', '0807.2194-2-4-3'], ['0807.2194-1-4-4', '0807.2194-2-4-4'], ['0807.2194-1-4-5', '0807.2194-2-4-5'], ['0807.2194-1-4-6', '0807.2194-2-4-6'], ['0807.2194-1-4-7', '0807.2194-2-4-7'], ['0807.2194-1-4-8', '0807.2194-2-4-8'], ['0807.2194-1-4-9', '0807.2194-2-4-9'], ['0807.2194-1-3-0', '0807.2194-2-3-0'], ['0807.2194-1-3-1', '0807.2194-2-3-1'], ['0807.2194-1-3-2', '0807.2194-2-3-2'], ['0807.2194-1-3-3', '0807.2194-2-3-3'], ['0807.2194-1-0-0', '0807.2194-2-0-0'], ['0807.2194-1-20-0', '0807.2194-2-20-0'], ['0807.2194-1-20-1', '0807.2194-2-20-1'], ['0807.2194-1-14-0', '0807.2194-2-14-0'], ['0807.2194-1-14-1', '0807.2194-2-14-1'], ['0807.2194-1-50-0', '0807.2194-2-52-0'], ['0807.2194-1-50-1', '0807.2194-2-52-1'], ['0807.2194-1-41-0', '0807.2194-2-43-0'], ['0807.2194-1-41-1', '0807.2194-2-43-1'], ['0807.2194-1-77-2', '0807.2194-2-80-2'], ['0807.2194-1-77-3', '0807.2194-2-80-3'], ['0807.2194-1-77-4', '0807.2194-2-80-4'], ['0807.2194-1-73-0', '0807.2194-2-76-0'], ['0807.2194-1-73-1', '0807.2194-2-76-1'], ['0807.2194-1-73-2', '0807.2194-2-76-2'], ['0807.2194-1-73-3', '0807.2194-2-76-3'], ['0807.2194-1-73-4', '0807.2194-2-76-4'], ['0807.2194-1-73-5', '0807.2194-2-76-5'], ['0807.2194-1-73-6', '0807.2194-2-76-6'], ['0807.2194-1-35-0', '0807.2194-2-37-0'], ['0807.2194-1-58-1', '0807.2194-2-61-1'], ['0807.2194-1-58-2', '0807.2194-2-61-2'], ['0807.2194-1-63-0', '0807.2194-2-66-0'], ['0807.2194-1-6-0', '0807.2194-2-6-0'], ['0807.2194-1-6-2', '0807.2194-2-6-2'], ['0807.2194-1-6-3', '0807.2194-2-6-3'], ['0807.2194-1-16-1', '0807.2194-2-16-1'], ['0807.2194-1-16-2', '0807.2194-2-16-2'], ['0807.2194-1-62-0', '0807.2194-2-65-0'], ['0807.2194-1-62-1', '0807.2194-2-65-1'], ['0807.2194-1-62-2', '0807.2194-2-65-2'], ['0807.2194-1-62-3', '0807.2194-2-65-3'], ['0807.2194-1-60-0', '0807.2194-2-63-0'], ['0807.2194-1-72-0', '0807.2194-2-75-0'], ['0807.2194-1-72-1', '0807.2194-2-75-1'], ['0807.2194-1-33-0', '0807.2194-2-33-0'], ['0807.2194-1-57-0', '0807.2194-2-59-0'], ['0807.2194-1-57-2', '0807.2194-2-60-0'], ['0807.2194-1-44-1', '0807.2194-2-46-1'], ['0807.2194-1-56-3', '0807.2194-2-58-3'], ['0807.2194-1-26-3', '0807.2194-2-26-3'], ['0807.2194-1-61-3', '0807.2194-2-64-3'], ['0807.2194-1-38-1', '0807.2194-2-40-1'], ['0807.2194-1-49-2', '0807.2194-2-51-2'], ['0807.2194-1-49-3', '0807.2194-2-51-3'], ['0807.2194-1-53-7', '0807.2194-2-55-7'], ['0807.2194-1-51-4', '0807.2194-2-53-4'], ['0807.2194-1-34-1', '0807.2194-2-34-1'], ['0807.2194-1-40-2', '0807.2194-2-42-1'], ['0807.2194-1-70-0', '0807.2194-2-73-0'], ['0807.2194-1-52-2', '0807.2194-2-54-2'], ['0807.2194-1-52-4', '0807.2194-2-54-4'], ['0807.2194-1-27-1', '0807.2194-2-27-1'], ['0807.2194-1-8-3', '0807.2194-2-8-3'], ['0807.2194-1-8-4', '0807.2194-2-8-4'], ['0807.2194-1-7-0', '0807.2194-2-7-0'], ['0807.2194-1-16-0', '0807.2194-2-16-0'], ['0807.2194-1-57-1', '0807.2194-2-59-1'], ['0807.2194-1-39-0', '0807.2194-2-41-0'], ['0807.2194-1-39-1', '0807.2194-2-41-1'], ['0807.2194-1-61-1', '0807.2194-2-64-1'], ['0807.2194-1-61-2', '0807.2194-2-64-2'], ['0807.2194-1-40-0', '0807.2194-2-42-0'], ['0807.2194-1-40-1', '0807.2194-2-42-0'], ['0807.2194-1-58-0', '0807.2194-2-61-0'], ['0807.2194-1-6-1', '0807.2194-2-6-1'], ['0807.2194-1-24-0', '0807.2194-2-24-0']]

[['0807.2194-1-13-0', '0807.2194-2-13-0'], ['0807.2194-1-13-1', '0807.2194-2-13-1'], ['0807.2194-1-13-2', '0807.2194-2-13-2'], ['0807.2194-1-13-3', '0807.2194-2-13-3'], ['0807.2194-1-13-5', '0807.2194-2-13-5'], ['0807.2194-1-13-6', '0807.2194-2-13-6'], ['0807.2194-1-13-7', '0807.2194-2-13-7'], ['0807.2194-1-13-8', '0807.2194-2-13-8'], ['0807.2194-1-13-9', '0807.2194-2-13-9'], ['0807.2194-1-13-10', '0807.2194-2-13-10'], ['0807.2194-1-13-11', '0807.2194-2-13-11'], ['0807.2194-1-13-12', '0807.2194-2-13-12'], ['0807.2194-1-13-13', '0807.2194-2-13-13'], ['0807.2194-1-13-14', '0807.2194-2-13-14'], ['0807.2194-1-13-15', '0807.2194-2-13-15'], ['0807.2194-1-76-0', '0807.2194-2-79-0'], ['0807.2194-1-76-1', '0807.2194-2-79-1'], ['0807.2194-1-76-2', '0807.2194-2-79-2'], ['0807.2194-1-76-3', '0807.2194-2-79-3'], ['0807.2194-1-76-4', '0807.2194-2-79-4'], ['0807.2194-1-76-5', '0807.2194-2-79-5'], ['0807.2194-1-76-6', '0807.2194-2-79-6'], ['0807.2194-1-71-0', '0807.2194-2-74-0'], ['0807.2194-1-71-1', '0807.2194-2-74-1'], ['0807.2194-1-44-0', '0807.2194-2-46-0'], ['0807.2194-1-56-0', '0807.2194-2-58-0'], ['0807.2194-1-56-1', '0807.2194-2-58-1'], ['0807.2194-1-56-2', '0807.2194-2-58-2'], ['0807.2194-1-56-4', '0807.2194-2-58-4'], ['0807.2194-1-75-0', '0807.2194-2-78-0'], ['0807.2194-1-75-1', '0807.2194-2-78-1'], ['0807.2194-1-75-2', '0807.2194-2-78-2'], ['0807.2194-1-75-3', '0807.2194-2-78-3'], ['0807.2194-1-75-4', '0807.2194-2-78-4'], ['0807.2194-1-26-0', '0807.2194-2-26-0'], ['0807.2194-1-26-1', '0807.2194-2-26-1'], ['0807.2194-1-26-2', '0807.2194-2-26-2'], ['0807.2194-1-26-4', '0807.2194-2-26-4'], ['0807.2194-1-61-0', '0807.2194-2-64-0'], ['0807.2194-1-65-0', '0807.2194-2-68-0'], ['0807.2194-1-36-0', '0807.2194-2-38-0'], ['0807.2194-1-36-1', '0807.2194-2-38-1'], ['0807.2194-1-36-2', '0807.2194-2-38-2'], ['0807.2194-1-5-0', '0807.2194-2-5-0'], ['0807.2194-1-5-1', '0807.2194-2-5-1'], ['0807.2194-1-66-0', '0807.2194-2-69-0'], ['0807.2194-1-66-1', '0807.2194-2-69-1'], ['0807.2194-1-66-2', '0807.2194-2-69-2'], ['0807.2194-1-30-0', '0807.2194-2-30-0'], ['0807.2194-1-30-1', '0807.2194-2-30-1'], ['0807.2194-1-30-2', '0807.2194-2-30-2'], ['0807.2194-1-30-3', '0807.2194-2-30-3'], ['0807.2194-1-38-0', '0807.2194-2-40-0'], ['0807.2194-1-38-2', '0807.2194-2-40-2'], ['0807.2194-1-38-3', '0807.2194-2-40-3'], ['0807.2194-1-49-0', '0807.2194-2-51-0'], ['0807.2194-1-49-1', '0807.2194-2-51-1'], ['0807.2194-1-49-4', '0807.2194-2-51-4'], ['0807.2194-1-49-5', '0807.2194-2-51-5'], ['0807.2194-1-22-0', '0807.2194-2-22-0'], ['0807.2194-1-22-1', '0807.2194-2-22-1'], ['0807.2194-1-15-0', '0807.2194-2-15-0'], ['0807.2194-1-15-1', '0807.2194-2-15-1'], ['0807.2194-1-43-0', '0807.2194-2-45-0'], ['0807.2194-1-43-1', '0807.2194-2-45-1'], ['0807.2194-1-43-2', '0807.2194-2-45-2'], ['0807.2194-1-43-3', '0807.2194-2-45-3'], ['0807.2194-1-46-0', '0807.2194-2-48-0'], ['0807.2194-1-46-1', '0807.2194-2-48-1'], ['0807.2194-1-46-2', '0807.2194-2-48-2'], ['0807.2194-1-45-0', '0807.2194-2-47-0'], ['0807.2194-1-53-0', '0807.2194-2-55-0'], ['0807.2194-1-53-1', '0807.2194-2-55-1'], ['0807.2194-1-53-2', '0807.2194-2-55-2'], ['0807.2194-1-53-3', '0807.2194-2-55-3'], ['0807.2194-1-53-4', '0807.2194-2-55-4'], ['0807.2194-1-53-5', '0807.2194-2-55-5'], ['0807.2194-1-53-6', '0807.2194-2-55-6'], ['0807.2194-1-53-8', '0807.2194-2-55-8'], ['0807.2194-1-55-0', '0807.2194-2-57-0'], ['0807.2194-1-55-1', '0807.2194-2-57-1'], ['0807.2194-1-55-2', '0807.2194-2-57-2'], ['0807.2194-1-67-0', '0807.2194-2-70-0'], ['0807.2194-1-67-1', '0807.2194-2-70-1'], ['0807.2194-1-67-2', '0807.2194-2-70-2'], ['0807.2194-1-67-3', '0807.2194-2-70-3'], ['0807.2194-1-18-0', '0807.2194-2-18-0'], ['0807.2194-1-18-1', '0807.2194-2-18-1'], ['0807.2194-1-23-0', '0807.2194-2-23-0'], ['0807.2194-1-23-1', '0807.2194-2-23-1'], ['0807.2194-1-23-2', '0807.2194-2-23-2'], ['0807.2194-1-28-1', '0807.2194-2-28-1'], ['0807.2194-1-51-0', '0807.2194-2-53-0'], ['0807.2194-1-51-1', '0807.2194-2-53-1'], ['0807.2194-1-51-2', '0807.2194-2-53-2'], ['0807.2194-1-51-3', '0807.2194-2-53-3'], ['0807.2194-1-1-0', '0807.2194-2-1-0'], ['0807.2194-1-1-1', '0807.2194-2-1-1'], ['0807.2194-1-1-2', '0807.2194-2-1-2'], ['0807.2194-1-17-0', '0807.2194-2-17-0'], ['0807.2194-1-11-0', '0807.2194-2-11-0'], ['0807.2194-1-11-1', '0807.2194-2-11-1'], ['0807.2194-1-11-2', '0807.2194-2-11-2'], ['0807.2194-1-11-3', '0807.2194-2-11-3'], ['0807.2194-1-11-4', '0807.2194-2-11-4'], ['0807.2194-1-11-5', '0807.2194-2-11-5'], ['0807.2194-1-11-6', '0807.2194-2-11-6'], ['0807.2194-1-34-0', '0807.2194-2-34-0'], ['0807.2194-1-34-2', '0807.2194-2-34-2'], ['0807.2194-1-19-0', '0807.2194-2-19-0'], ['0807.2194-1-19-1', '0807.2194-2-19-1'], ['0807.2194-1-19-2', '0807.2194-2-19-2'], ['0807.2194-1-2-0', '0807.2194-2-2-0'], ['0807.2194-1-2-1', '0807.2194-2-2-1'], ['0807.2194-1-2-2', '0807.2194-2-2-2'], ['0807.2194-1-2-3', '0807.2194-2-2-3'], ['0807.2194-1-32-0', '0807.2194-2-32-0'], ['0807.2194-1-32-1', '0807.2194-2-32-1'], ['0807.2194-1-32-2', '0807.2194-2-32-2'], ['0807.2194-1-68-0', '0807.2194-2-71-0'], ['0807.2194-1-68-1', '0807.2194-2-71-1'], ['0807.2194-1-68-2', '0807.2194-2-71-2'], ['0807.2194-1-68-3', '0807.2194-2-71-3'], ['0807.2194-1-68-4', '0807.2194-2-71-4'], ['0807.2194-1-68-5', '0807.2194-2-71-5'], ['0807.2194-1-68-6', '0807.2194-2-71-6'], ['0807.2194-1-68-7', '0807.2194-2-71-7'], ['0807.2194-1-68-8', '0807.2194-2-71-8'], ['0807.2194-1-48-0', '0807.2194-2-50-0'], ['0807.2194-1-48-1', '0807.2194-2-50-1'], ['0807.2194-1-70-1', '0807.2194-2-73-1'], ['0807.2194-1-70-2', '0807.2194-2-73-2'], ['0807.2194-1-54-0', '0807.2194-2-56-0'], ['0807.2194-1-54-1', '0807.2194-2-56-1'], ['0807.2194-1-54-2', '0807.2194-2-56-2'], ['0807.2194-1-54-3', '0807.2194-2-56-3'], ['0807.2194-1-54-4', '0807.2194-2-56-4'], ['0807.2194-1-52-0', '0807.2194-2-54-0'], ['0807.2194-1-52-1', '0807.2194-2-54-1'], ['0807.2194-1-52-3', '0807.2194-2-54-3'], ['0807.2194-1-52-5', '0807.2194-2-54-5'], ['0807.2194-1-52-6', '0807.2194-2-54-6'], ['0807.2194-1-27-0', '0807.2194-2-27-0'], ['0807.2194-1-27-2', '0807.2194-2-27-2'], ['0807.2194-1-8-0', '0807.2194-2-8-0'], ['0807.2194-1-8-1', '0807.2194-2-8-1'], ['0807.2194-1-8-2', '0807.2194-2-8-2'], ['0807.2194-1-8-5', '0807.2194-2-8-5'], ['0807.2194-1-4-0', '0807.2194-2-4-0'], ['0807.2194-1-4-1', '0807.2194-2-4-1'], ['0807.2194-1-4-2', '0807.2194-2-4-2'], ['0807.2194-1-4-3', '0807.2194-2-4-3'], ['0807.2194-1-4-4', '0807.2194-2-4-4'], ['0807.2194-1-4-5', '0807.2194-2-4-5'], ['0807.2194-1-4-6', '0807.2194-2-4-6'], ['0807.2194-1-4-7', '0807.2194-2-4-7'], ['0807.2194-1-4-8', '0807.2194-2-4-8'], ['0807.2194-1-4-9', '0807.2194-2-4-9'], ['0807.2194-1-3-0', '0807.2194-2-3-0'], ['0807.2194-1-3-1', '0807.2194-2-3-1'], ['0807.2194-1-3-2', '0807.2194-2-3-2'], ['0807.2194-1-3-3', '0807.2194-2-3-3'], ['0807.2194-1-0-0', '0807.2194-2-0-0'], ['0807.2194-1-20-0', '0807.2194-2-20-0'], ['0807.2194-1-20-1', '0807.2194-2-20-1'], ['0807.2194-1-14-0', '0807.2194-2-14-0'], ['0807.2194-1-14-1', '0807.2194-2-14-1'], ['0807.2194-1-50-0', '0807.2194-2-52-0'], ['0807.2194-1-50-1', '0807.2194-2-52-1'], ['0807.2194-1-41-0', '0807.2194-2-43-0'], ['0807.2194-1-41-1', '0807.2194-2-43-1'], ['0807.2194-1-77-2', '0807.2194-2-80-2'], ['0807.2194-1-77-3', '0807.2194-2-80-3'], ['0807.2194-1-77-4', '0807.2194-2-80-4'], ['0807.2194-1-73-0', '0807.2194-2-76-0'], ['0807.2194-1-73-1', '0807.2194-2-76-1'], ['0807.2194-1-73-2', '0807.2194-2-76-2'], ['0807.2194-1-73-3', '0807.2194-2-76-3'], ['0807.2194-1-73-4', '0807.2194-2-76-4'], ['0807.2194-1-73-5', '0807.2194-2-76-5'], ['0807.2194-1-73-6', '0807.2194-2-76-6'], ['0807.2194-1-35-0', '0807.2194-2-37-0'], ['0807.2194-1-58-1', '0807.2194-2-61-1'], ['0807.2194-1-58-2', '0807.2194-2-61-2'], ['0807.2194-1-63-0', '0807.2194-2-66-0'], ['0807.2194-1-6-0', '0807.2194-2-6-0'], ['0807.2194-1-6-2', '0807.2194-2-6-2'], ['0807.2194-1-6-3', '0807.2194-2-6-3'], ['0807.2194-1-16-1', '0807.2194-2-16-1'], ['0807.2194-1-16-2', '0807.2194-2-16-2'], ['0807.2194-1-62-0', '0807.2194-2-65-0'], ['0807.2194-1-62-1', '0807.2194-2-65-1'], ['0807.2194-1-62-2', '0807.2194-2-65-2'], ['0807.2194-1-62-3', '0807.2194-2-65-3'], ['0807.2194-1-60-0', '0807.2194-2-63-0'], ['0807.2194-1-72-0', '0807.2194-2-75-0'], ['0807.2194-1-72-1', '0807.2194-2-75-1'], ['0807.2194-1-57-0', '0807.2194-2-59-0'], ['0807.2194-1-57-2', '0807.2194-2-60-0'], ['0807.2194-1-57-1', '0807.2194-2-59-1']]

[['0807.2194-1-33-0', '0807.2194-2-33-0'], ['0807.2194-1-44-1', '0807.2194-2-46-1'], ['0807.2194-1-56-3', '0807.2194-2-58-3'], ['0807.2194-1-26-3', '0807.2194-2-26-3'], ['0807.2194-1-61-3', '0807.2194-2-64-3'], ['0807.2194-1-38-1', '0807.2194-2-40-1'], ['0807.2194-1-49-2', '0807.2194-2-51-2'], ['0807.2194-1-49-3', '0807.2194-2-51-3'], ['0807.2194-1-53-7', '0807.2194-2-55-7'], ['0807.2194-1-51-4', '0807.2194-2-53-4'], ['0807.2194-1-34-1', '0807.2194-2-34-1'], ['0807.2194-1-40-2', '0807.2194-2-42-1'], ['0807.2194-1-70-0', '0807.2194-2-73-0'], ['0807.2194-1-52-2', '0807.2194-2-54-2'], ['0807.2194-1-52-4', '0807.2194-2-54-4'], ['0807.2194-1-27-1', '0807.2194-2-27-1'], ['0807.2194-1-8-3', '0807.2194-2-8-3'], ['0807.2194-1-8-4', '0807.2194-2-8-4'], ['0807.2194-1-7-0', '0807.2194-2-7-0'], ['0807.2194-1-16-0', '0807.2194-2-16-0']]

[]

[['0807.2194-1-39-0', '0807.2194-2-41-0'], ['0807.2194-1-39-1', '0807.2194-2-41-1'], ['0807.2194-1-61-1', '0807.2194-2-64-1'], ['0807.2194-1-61-2', '0807.2194-2-64-2'], ['0807.2194-1-40-0', '0807.2194-2-42-0'], ['0807.2194-1-40-1', '0807.2194-2-42-0'], ['0807.2194-1-58-0', '0807.2194-2-61-0'], ['0807.2194-1-6-1', '0807.2194-2-6-1'], ['0807.2194-1-24-0', '0807.2194-2-24-0']]

[]

['0807.2194-1-28-0', '0807.2194-1-77-0', '0807.2194-1-77-1', '0807.2194-1-77-5', '0807.2194-2-28-0', '0807.2194-2-80-0', '0807.2194-2-80-1', '0807.2194-2-80-5']

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

https://arxiv.org/abs/0807.2194

hep-th-0202150

{'hep-th-0202150-1-0-0': '# Introduction', 'hep-th-0202150-1-1-0': 'According to the AdS/CFT conjecture [CITATION], the chiral operators of the [MATH] supersymmetric [MATH] gauge theory are in one-to-one correspondence with the modes of type IIB supergravity on [MATH].', 'hep-th-0202150-1-1-1': "On the other hand, the massive string modes correspond to operators in long multiplets whose dimensions diverge for large 't Hooft coupling as [MATH].", 'hep-th-0202150-1-1-2': 'Thus, in the limit [MATH] the stringy nature of the dual theory is obscured by the decoupling of the non-chiral operators, which constitute the large majority of possible gauge invariant operators.', 'hep-th-0202150-1-2-0': 'It is important to keep in mind, however, that the AdS/CFT correspondence relates large [MATH] gauge theory to string theory, not merely supergravity.', 'hep-th-0202150-1-2-1': "If we depart from the limit of infinite 't Hooft coupling, then all the non-chiral operators no longer decouple, so that the spectrum of the gauge theory becomes much more complicated and presumably related to type IIB string theory on [MATH] (see, for example, [CITATION]).", 'hep-th-0202150-1-2-2': "However, even for very large 't Hooft coupling, it is possible to demonstrate the stringy nature of the dual theory quite explicitly.", 'hep-th-0202150-1-2-3': 'One early clue was provided by Witten [CITATION] in the context of [MATH] supersymmetric [MATH] gauge theory which is dual to type IIB strings on [MATH].', 'hep-th-0202150-1-2-4': 'He noted that the gauge theory possesses chiral operators of dimension [MATH], the Pfaffians, whose dual is provided by a D3-brane wrapping a 3-cycle of the [MATH].', 'hep-th-0202150-1-2-5': 'This explicitly shows that the dual theory cannot be simply supergravity: it must contain D-branes.', 'hep-th-0202150-1-2-6': 'Since the third homology class [MATH], a D3-brane wrapped twice can decay into ordinary supergravity modes.', 'hep-th-0202150-1-2-7': 'A corresponding gauge theory statement is that a product of two Pfaffian operators can be expressed in terms of the usual single-trace operators.', 'hep-th-0202150-1-3-0': 'It is further possible to find type IIB backgrounds of the form [MATH] such that [MATH].', 'hep-th-0202150-1-3-1': 'In this case, a D3-brane can wrap a 3-cycle any number of times so that the dual gauge theory operators carry a quantized baryon number.', 'hep-th-0202150-1-3-2': 'A simple example of this kind is provided by the space [MATH] whose symmetries are [MATH].', 'hep-th-0202150-1-3-3': 'This space may be thought of as a [MATH] bundle over [MATH], and the explicit Einstein metric on [MATH] is [EQUATION]', 'hep-th-0202150-1-3-4': 'The [MATH] superconformal gauge theory dual to type IIB strings on [MATH] was constructed in [CITATION]: it is [MATH] gauge theory coupled to two chiral superfields, [MATH], in the [MATH] representation and two chiral superfields, [MATH], in the [MATH] representation [CITATION].', 'hep-th-0202150-1-3-5': "The [MATH]'s transform as a doublet under one of the global [MATH]'s while the [MATH]'s transform as a doublet under the other [MATH].", 'hep-th-0202150-1-4-0': "Cancellation of the anomaly in the [MATH] R-symmetry requires that the [MATH]'s and the [MATH]'s each have R-charge [MATH].", 'hep-th-0202150-1-4-1': 'For consistency of the duality it is necessary that we add an exactly marginal superpotential which preserves the [MATH] symmetry of the theory: [EQUATION]', 'hep-th-0202150-1-4-2': 'In [CITATION] it was proposed that the wrapped D3-branes correspond to baryon-like operators [MATH] and [MATH] where the indices of both [MATH] groups are fully antisymmetrized (their more detailed form is exhibited in Section 3).', 'hep-th-0202150-1-4-3': 'For large [MATH] the dimensions of such operators calculated from the mass of wrapped D3-branes were found to be [MATH] [CITATION].', 'hep-th-0202150-1-4-4': 'This is in complete agreement with the fact that the dimension of the chiral superfields at the fixed point is [MATH].', 'hep-th-0202150-1-4-5': 'At the quantum level, the collective coordinate for the wrapped D3-brane has to be quantized to identify its [MATH] quantum numbers.', 'hep-th-0202150-1-4-6': 'This quantization was sketched in [CITATION].', 'hep-th-0202150-1-4-7': 'Here we do a more careful job and show that the dimension of dibaryons is [MATH] not only in the large [MATH] limit but exactly.', 'hep-th-0202150-1-4-8': 'In a related check on the identification of the dibaryons with wrapped D3-branes, we calculate the [MATH] charge of the wrapped D3-branes and show that it is [MATH], in agreement with the gauge theory.', 'hep-th-0202150-1-5-0': 'The structure of the paper is as follows.', 'hep-th-0202150-1-5-1': 'In Section 2 we calculate the [MATH] charge of a D3-brane wrapping a 3-cycle inside the Einstein space [MATH].', 'hep-th-0202150-1-5-2': 'Our results apply to all [MATH] which are [MATH] bundles over 4-d Kahler-Einstein spaces.', 'hep-th-0202150-1-5-3': 'We also provide an analogous calculation for M5-branes wrapping a 5-cycle inside an Einstein space [MATH] which is a [MATH] bundle over a 6-d Kahler-Einstein space.', 'hep-th-0202150-1-5-4': 'In all cases, we show that the R-charge is proportional to the dimension [MATH] of the dual CFT operator, with the correct constant of proportionality.', 'hep-th-0202150-1-5-5': 'We further argue that [MATH] is measured by the volume of the cycle wrapped by the brane.', 'hep-th-0202150-1-5-6': 'In Section 3 we make some remarks on the collective coordinate quantization of the wrapped D3-branes.', 'hep-th-0202150-1-5-7': 'In particular, we focus on the [MATH] example and show that the [MATH] quantum numbers of the dibaryon operators indeed follow from this quantization.', 'hep-th-0202150-1-5-8': 'In Section 4 we study BPS excitations of the wrapped D3-branes using the DBI action.', 'hep-th-0202150-1-5-9': 'We calculate their energies and [MATH] charges and propose dual chiral operators carrying the same quantum numbers.', 'hep-th-0202150-1-6-0': '# The R-symmetry gauge field', 'hep-th-0202150-1-7-0': 'We would like to show how the [MATH] gauge field emerges from a Kaluza-Klein reduction of the full ten or eleven dimensional supergravity action.', 'hep-th-0202150-1-7-1': 'In ten dimensions, we begin with a stack of D3-branes placed at the tip of a non-compact Calabi-Yau cone [MATH] of complex dimension three.', 'hep-th-0202150-1-7-2': 'In eleven dimensions, we consider a stack of M2-branes at the tip of a similar cone [MATH] of complex dimension 4.', 'hep-th-0202150-1-7-3': 'We may write the metric on [MATH] as [EQUATION] where [MATH] is Einstein, i.e. [MATH].', 'hep-th-0202150-1-8-0': 'In the near-throat limit [CITATION], the backreaction of the branes causes the full space to separate into a product manifold.', 'hep-th-0202150-1-8-1': 'In the D3-brane case, we have [MATH] and a five form flux [MATH] produced by the stack of D3-branes.', 'hep-th-0202150-1-8-2': 'See also [CITATION] for details.', 'hep-th-0202150-1-8-3': 'For the M2-branes, the full space becomes [MATH], and the M2-brane flux is carried by [MATH].', 'hep-th-0202150-1-9-0': 'When [MATH] is [MATH] or the coset manifold [MATH], the authors of [CITATION] demonstrate that perturbations in the metric [MATH] combine with perturbations in the [MATH] potential to produce a massless vector field.', 'hep-th-0202150-1-9-1': '(The [MATH] index [MATH] while [MATH] index [MATH] space.)', 'hep-th-0202150-1-9-2': 'For [MATH], more general results exist concerning the existence of this massless vector field [CITATION].', 'hep-th-0202150-1-9-3': 'In particular, whenever [MATH] is a coset space and there exists a Killing vector, harmonic analysis demonstrates the existence of a massless vector field mixing perturbations of [MATH] with perturbations to [MATH].', 'hep-th-0202150-1-10-0': 'In what follows, we consider a more general class of [MATH], in particular quasi-regular Einstein-Sasaki manifolds, and demonstrate the existence of a massless U(1) vector field.', 'hep-th-0202150-1-10-1': 'We argue that this massless vector field mediates the interaction between objects with R-charge.', 'hep-th-0202150-1-10-2': 'Although the generalization is interesting in its own right, the main interest is the demonstration that wrapped D3-branes and M2-branes in these geometries always carry an R-charge consistent with their identification as dibaryons in the corresponding conformal field theory.', 'hep-th-0202150-1-11-0': 'To be more specific, a quasi-regular Einstein-Sasaki manifold is essentially a U(1) fibration of a Kahler-Einstein manifold or orbifold.', 'hep-th-0202150-1-11-1': 'The BPS dibaryons wrap the base of the Sasaki-Einstein manifold in a holomorphic manner, and they wrap the [MATH] fiber completely.', 'hep-th-0202150-1-11-2': 'Mikhailov has considered similar holomorphically wrapped D-branes [CITATION].', 'hep-th-0202150-1-11-3': "One might naively say that the R-charge of a particle is it's angular momentum along the [MATH] direction.", 'hep-th-0202150-1-11-4': 'Thus, a brane that wraps the fiber carries no net angular momentum.', 'hep-th-0202150-1-11-5': 'However, one must be careful; these branes can be supersymmetric, and it is necessary to understand what the precise notion of R-charge is in the supergravity.', 'hep-th-0202150-1-11-6': 'The following calculation is not that sensitive to the dimensionality, but for clarity, we will consider the [MATH] and [MATH] cases separately.', 'hep-th-0202150-1-12-0': '## The R-charge for [MATH]', 'hep-th-0202150-1-13-0': 'In general, the ten dimensional metric is [EQUATION] where [MATH] is a Minkowski tensor.', 'hep-th-0202150-1-14-0': 'A solution to the SUGRA equations of motion can be obtained by threading the [MATH] space with [MATH] units of [MATH] form flux.', 'hep-th-0202150-1-14-1': 'In particular [EQUATION] where [MATH].', 'hep-th-0202150-1-15-0': 'We now further specialize to the case where [MATH] is a quasi-regular Einstein-Sasaki space.', 'hep-th-0202150-1-15-1': 'In simpler language, [MATH] is a U(1) bundle over a two complex dimensional Kahler-Einstein manifold (or orbifold) [MATH]: [EQUATION]', 'hep-th-0202150-1-15-2': 'Define the Kahler form on [MATH] to be [MATH].', 'hep-th-0202150-1-15-3': 'Then, [MATH].', 'hep-th-0202150-1-15-4': 'The number [MATH] is defined such that [MATH], where [MATH] is the first Chern class of the U(1) bundle.', 'hep-th-0202150-1-15-5': 'With these definitions, [MATH] [CITATION].', 'hep-th-0202150-1-15-6': 'Also, in this way, we may write [EQUATION] where [MATH], and [MATH].', 'hep-th-0202150-1-16-0': 'We want to identify the gauge field [MATH] associated with the R-charge.', 'hep-th-0202150-1-16-1': 'The R-symmetry gauge group is at least as big as U(1), and hence it is a natural guess to associate the U(1) fiber of the Sasaki-Einstein manifold [MATH] if not with the R-symmetry group itself, then at least with a subgroup of it.', 'hep-th-0202150-1-16-2': 'We begin with the inspired guess that redefinitions of [MATH] be identified with gauge transformations of the R-symmetry gauge field [MATH]: [EQUATION] where [EQUATION]', 'hep-th-0202150-1-16-3': 'One troublesome issue is that [MATH] in general may not be one.', 'hep-th-0202150-1-16-4': 'To figure out [MATH] and to anchor the analysis, we need a geometric object with a definite R-charge.', 'hep-th-0202150-1-16-5': 'The holomorphic [MATH]-form [MATH] on the noncompact Calabi-Yau cone [MATH] is one such object.', 'hep-th-0202150-1-16-6': 'Because of its association with the superpotential, [MATH] has R-charge 2.', 'hep-th-0202150-1-17-0': 'The functional dependence of [MATH] on [MATH] is very simple: [MATH].', 'hep-th-0202150-1-17-1': 'A proof will follow in section 2.3.', 'hep-th-0202150-1-17-2': 'Because [MATH] has R-charge two, we can directly identify the angular variable [MATH] with a U(1) R-symmetry where now [EQUATION] and [MATH].', 'hep-th-0202150-1-17-3': 'Thus [MATH].', 'hep-th-0202150-1-18-0': 'Including the gauge field [MATH] in the supergravity solution means altering the metric [EQUATION]', 'hep-th-0202150-1-18-1': 'The [MATH] form flux is also altered [EQUATION]', 'hep-th-0202150-1-18-2': 'From this expression, we can calculate [MATH] to first order in [MATH]: [EQUATION]', 'hep-th-0202150-1-18-3': 'We have introduced some notation: [MATH], and [MATH] is the Hodge star in the [MATH] directions.', 'hep-th-0202150-1-18-4': 'This form of [MATH] satisfies the SUGRA equations of motion to linear order in [MATH].', 'hep-th-0202150-1-18-5': 'In particular, [MATH] has been constructed such that [MATH].', 'hep-th-0202150-1-18-6': 'Notice that [MATH] contains a piece proportional to [MATH].', 'hep-th-0202150-1-18-7': 'Hence [MATH] will vanish to linear order in [MATH] provided [MATH], i.e. the field equations for the gauge field [MATH] are satisfied.', 'hep-th-0202150-1-19-0': 'We also may consider the trace of the Einstein equations.', 'hep-th-0202150-1-19-1': 'The modification of the metric ([REF]) causes the Ricci scalar for the full ten dimensional metric to become [EQUATION]', 'hep-th-0202150-1-19-2': "Thus, the trace of Einstein's equations will contain an extra [MATH] term.", 'hep-th-0202150-1-19-3': 'This term causes the backreaction of the [MATH] field strength on the metric.', 'hep-th-0202150-1-19-4': 'We ignore it in this paper since it is a second order effect.', 'hep-th-0202150-1-20-0': 'As a test of our proposal for the U(1) R-symmetry gauge field, let us calculate the R-charge of a hypothetical baryon in this geometry.', 'hep-th-0202150-1-20-1': 'We identify the baryons as D3-branes wrapped on 3-cycles inside [MATH].', 'hep-th-0202150-1-20-2': 'In particular, the D3-brane will completely wrap the U(1) fiber of the Einstein-Sasaki manifold [MATH] and will also wrap a holomorphic curve in the Kahler-Einstein base [MATH].', 'hep-th-0202150-1-20-3': 'The dimension of these baryons is proportional to the volume of the 3-cycle [MATH] times the curvature length [MATH]: [EQUATION]', 'hep-th-0202150-1-20-4': 'One might expect that the volume is more closely related to the mass of the baryon than to the dimension.', 'hep-th-0202150-1-20-5': 'We will see later in section 2.4 that the mass receives a correction and that [MATH] is the correct expression.', 'hep-th-0202150-1-21-0': 'Now these baryons are thought to be chiral primary operators.', 'hep-th-0202150-1-21-1': 'Their R-charge should be a multiplicative constant times the dimension.', 'hep-th-0202150-1-21-2': 'Recall that in a classical Lagrangian for a particle traveling in an electromagnetic field, there is a term of the form [MATH] where [MATH] is the charge of the particle and [MATH] is its four-velocity.', 'hep-th-0202150-1-21-3': 'In the Lagrangian for a probe baryon of this type, there is a Wess-Zumino term of the form [EQUATION] where we have used ([REF]) for [MATH].', 'hep-th-0202150-1-22-0': 'In the case of [MATH], we know independently from the supersymmetry algebra that the dimension times 2/3 is the R-charge.', 'hep-th-0202150-1-22-1': 'The [MATH] in the Wess-Zumino term is exactly where the charge should be in a classical Lagrangian describing the D3-brane dynamics.', 'hep-th-0202150-1-23-0': '## The R-charge for [MATH]', 'hep-th-0202150-1-24-0': 'We now repeat essentially the same analysis for a stack of M2-branes sitting at the tip of a noncompact four complex dimensional cone, [MATH].', 'hep-th-0202150-1-24-1': 'In general, the eleven dimensional metric is [EQUATION] where [MATH] is a Minkowski tensor.', 'hep-th-0202150-1-24-2': 'Note the additional factors of two scaling the Einstein metric.', 'hep-th-0202150-1-24-3': 'These factors are necessary to guarantee that the Ricci scalar of the eleven dimensional metric vanishes and will be crucial later on in fixing the R-charge of the baryons.', 'hep-th-0202150-1-25-0': 'A solution to the SUGRA equations of motion can be obtained by threading the [MATH] space with [MATH] units of [MATH] form flux.', 'hep-th-0202150-1-25-1': 'In particular [EQUATION] where [MATH].', 'hep-th-0202150-1-26-0': 'We specialize to the case where [MATH] is a quasi-regular Einstein-Sasaki space: [EQUATION]', 'hep-th-0202150-1-26-1': 'We keep the same definition of [MATH] and [MATH].', 'hep-th-0202150-1-26-2': 'Namely, [MATH] and the number [MATH] is defined such that [MATH].', 'hep-th-0202150-1-26-3': 'With these definitions, [MATH].', 'hep-th-0202150-1-26-4': 'Also, in this way, we may write [EQUATION] where [MATH].', 'hep-th-0202150-1-27-0': 'Just as in the previous section, redefinitions of [MATH] can be identified with gauge transformations of the R-symmetry gauge field [MATH]: [EQUATION] where [EQUATION]', 'hep-th-0202150-1-27-1': 'Including the gauge field [MATH] in the supergravity solution changes the metric [EQUATION]', 'hep-th-0202150-1-27-2': 'The [MATH] form potential becomes [EQUATION]', 'hep-th-0202150-1-27-3': 'Hence [EQUATION]', 'hep-th-0202150-1-27-4': 'This form of [MATH] satisfies the SUGRA equations of motion to linear order in [MATH].', 'hep-th-0202150-1-27-5': 'The relevant equations for [MATH] are [MATH] and [MATH].', 'hep-th-0202150-1-27-6': 'We have constructed [MATH] such that [MATH].', 'hep-th-0202150-1-27-7': 'Moreover, [MATH] and [MATH] vanish to order [MATH] provided the equations of motion for the gauge field [MATH] are satisfied, i.e. [MATH].', 'hep-th-0202150-1-27-8': 'The modification of the metric ([REF]) causes the Ricci scalar to become [EQUATION]', 'hep-th-0202150-1-27-9': 'From the standpoint of an effective action in [MATH], one can see from this modification another way of deriving the equations of motion for the gauge field, [MATH].', 'hep-th-0202150-1-28-0': 'To calculate the R-charge of a hypothetical baryon in this geometry, we identify the baryons as M5-branes wrapped on 5-cycles inside [MATH].', 'hep-th-0202150-1-28-1': 'The 5-cycle consists of the U(1) fiber and a holomorphic surface inside the Kahler-Einstein base of [MATH].', 'hep-th-0202150-1-28-2': 'The dimension of these baryons is proportional to the volume of the 5-cycle [MATH] times the curvature length [MATH]: [EQUATION]', 'hep-th-0202150-1-28-3': 'In the Lagrangian for a probe baryon of this type, there is also a Wess-Zumino term of the form [EQUATION] where we have used the previous expression for [MATH].', 'hep-th-0202150-1-29-0': 'In the case of [MATH], we expect from the supersymmetry algebra that the R-charge of a chiral primary operator be equal to the dimension.', 'hep-th-0202150-1-29-1': 'As these baryons are chiral primary operators, all is well.', 'hep-th-0202150-1-30-0': '## Proof', 'hep-th-0202150-1-31-0': 'We seek to prove that the holomorphic [MATH]-form [MATH] scales as [MATH] where [MATH] is a coordinate on the U(1) fiber of these non-compact Calabi-Yau cones of complex dimension [MATH].', 'hep-th-0202150-1-31-1': 'The metric on [MATH] ([REF]) is not very natural as it puts [MATH] and [MATH] on an unequal footing.', 'hep-th-0202150-1-31-2': 'It is more natural to think of [MATH] as a fibration of [MATH] over the Kahler-Einstein base [MATH].', 'hep-th-0202150-1-31-3': 'Let [MATH] be the coordinate of the fiber.', 'hep-th-0202150-1-31-4': 'We make the change of variables [EQUATION]', 'hep-th-0202150-1-31-5': 'In these coordinates, the metric becomes [EQUATION]', 'hep-th-0202150-1-31-6': 'The volume form scales as [MATH].', 'hep-th-0202150-1-31-7': 'Note that the volume form is proportional to [MATH].', 'hep-th-0202150-1-31-8': 'Thus, [MATH] scales as [MATH].', 'hep-th-0202150-1-32-0': '## The difference between mass and dimension', 'hep-th-0202150-1-33-0': 'So far we have explicitly shown that the supergravity vector field responsible for the [MATH] charge of a state results from mixing between the rotations along the [MATH] fiber and a compensating gauge transformation.', 'hep-th-0202150-1-33-1': 'The end result is that the D-brane R-charge was identified with the volume of the D-brane for certain BPS D-branes.', 'hep-th-0202150-1-34-0': 'In the supergravity, it seems natural to associate to such a D-brane wrapping the given cycle a mass equal to the volume, and then one finds a discrepancy between the R-charge and the conformal dimension, since [EQUATION] for [MATH].', 'hep-th-0202150-1-34-1': 'We want to argue that the volume of the D-brane should be identified directly with the conformal dimension, and not with the mass of the particle.', 'hep-th-0202150-1-35-0': 'One reason for this is that the D-brane is a BPS object and we should consider the action of a superparticle in this background, rather than the action of a point particle in AdS space.', 'hep-th-0202150-1-35-1': 'When we consider global AdS coordinates, the hamiltonian associated to time corresponds to the generator of conformal tranformations, and the spectrum is discrete because there is a gravitational potential well at the center of AdS which eliminates the zero modes of the particle and turns them into oscillators.', 'hep-th-0202150-1-36-0': 'The supersymmetries do not commute with the hamiltonian, but they still control the dynamics of the theory.', 'hep-th-0202150-1-36-1': 'This can be seen explicitly in the matrix model presented in [CITATION], where the fermions and bosons are not degenerate, but the ground state energy still cancels.', 'hep-th-0202150-1-36-2': 'We want to argue that this is generic for these systems.', 'hep-th-0202150-1-36-3': 'A superparticle also has fermionic coordinates.', 'hep-th-0202150-1-36-4': 'It is the zero point energy of these degrees of freedom that cancels the zero point energy of the bosonic degrees of freedom.', 'hep-th-0202150-1-37-0': 'In particular, consider the fluctuations corresponding to the transverse motion of the particle in AdS space.', 'hep-th-0202150-1-38-0': 'The fluctuations have a mass term determined by the [MATH] metric: [EQUATION]', 'hep-th-0202150-1-38-1': 'If we consider a particle at [MATH], we have a timelike geodesic of [MATH].', 'hep-th-0202150-1-38-2': 'To quadratic order in the transverse fluctuations we have [EQUATION]', 'hep-th-0202150-1-38-3': 'There is a gravitational potential in [MATH], and the mass term for the fluctuations in [MATH] is determined by the form of the metric.', 'hep-th-0202150-1-38-4': 'One can also determine this mass term from the algebra of isometries.', 'hep-th-0202150-1-38-5': 'In the radial quantization, acting with derivatives changes the conformal weight by one unit.', 'hep-th-0202150-1-38-6': 'This is the same as acting with one of the raising operators for [MATH] to first order in the quantum system, because the symmetry is non-linearly realized.', 'hep-th-0202150-1-39-0': 'Similarly, if we act with the supersymmetry generators, we change the conformal weight by a factor of 1/2.', 'hep-th-0202150-1-39-1': 'This means that the fermions will have a mass term which is one half that of the bosons.', 'hep-th-0202150-1-39-2': 'There are [MATH] complex supersymmetries, so we have one complex fermion associated to each one of them.', 'hep-th-0202150-1-39-3': 'It is easy to see that the contribution to the conformal weight of each fermion is one half that of the bosons, but there are twice as many fermions.', 'hep-th-0202150-1-39-4': 'Thus one finds that the zero point energy cancels between bosons and fermions, and for supersymmetric states one has [EQUATION]', 'hep-th-0202150-1-39-5': 'The classical conformal weight associated to the state is equal to the value in the quantum theory, namely, the conformal weight of the superparticle is the volume of the D-brane.', 'hep-th-0202150-1-40-0': 'If we integrate out the fermions, they shift the zero point energy of the hamiltonian by [MATH], where [MATH] is the number of real fermions, [MATH] is the mass of each of these fermions, and the remaining [MATH] is the numerical factor associated with zero point energy for a fermionic oscillator.', 'hep-th-0202150-1-40-1': 'As a result, the mass of the particle associated to the ground states is given approximately by [EQUATION]', 'hep-th-0202150-1-40-2': 'This expression agrees with ([REF]) in the limit of large [MATH] exactly as we would expect.', 'hep-th-0202150-1-40-3': 'To first order, integrating out the fermions reproduces the shift between the value of the mass and the conformal weight.', 'hep-th-0202150-1-41-0': '# Moduli space of D-branes and zero-mode quantization', 'hep-th-0202150-1-42-0': 'Classically one often has the freedom to change continuously the way in which a D-brane is wrapped without changing the energy.', 'hep-th-0202150-1-42-1': 'In this section, we will investigate how, through quantization, this freedom of movement allows one to understand the number of ground states of a given dibaryon operator in the field theory side of the correspondence.', 'hep-th-0202150-1-43-0': 'This abstract statement concerning the freedom some dibaryon-like states have to shift their wrapping configurations is particularly easy to understand in the case of [MATH].', 'hep-th-0202150-1-43-1': 'Recall that [MATH] is a U(1) fibration over [MATH].', 'hep-th-0202150-1-43-2': "The simplest dibaryon corresponds to a D3-brane wrapping the U(1) fiber of one of the two [MATH]'s.", 'hep-th-0202150-1-43-3': 'It is clear that, classically, the D3-brane can move continuously along the other [MATH] without changing its energy.', 'hep-th-0202150-1-44-0': 'Let us exhibit the operators of the [MATH] gauge theory, dual to these simply wrapped D3-branes.', 'hep-th-0202150-1-44-1': 'Since the fields [MATH], [MATH], carry an index [MATH] in the [MATH] of [MATH] and an index [MATH] in the [MATH] of [MATH], we can construct color-singlet "dibaryon" operators by antisymmetrizing completely with respect to both groups: [EQUATION] where [MATH] is the completely symmetric [MATH] Clebsch-Gordon coefficient corresponding to forming the [MATH] of [MATH] out of [MATH] 2\'s.', 'hep-th-0202150-1-44-2': 'Thus the [MATH] quantum numbers of [MATH] are [MATH].', 'hep-th-0202150-1-44-3': 'Similarly, we can construct "dibaryon" operators which transform as [MATH], [EQUATION]', 'hep-th-0202150-1-44-4': 'The existence of two types of "dibaryon" operators is related on the supergravity side to the fact that the base of the [MATH] bundle is [MATH].', 'hep-th-0202150-1-44-5': 'A D3-brane can wrap either of the two-spheres together with the [MATH] fiber.', 'hep-th-0202150-1-45-0': 'We will use a simplified notation for the dibaryon operators ([REF]) [EQUATION]', 'hep-th-0202150-1-45-1': 'The (super)subindex on the [MATH] refers to the gauge group that the [MATH] symbol carries and whether these are (super)subindices is indicated by the position of the group label.', 'hep-th-0202150-1-45-2': 'The gauge indices of the [MATH] are contracted with the gauge indices of the [MATH] tensors in the same order as they are written.', 'hep-th-0202150-1-46-0': 'In order to account for the [MATH] quantum numbers of the dibaryons, one needs to take into account the proper quantization of the zero modes of the wrapped D3-brane [CITATION].', 'hep-th-0202150-1-46-1': 'The resulting dynamics reduces to motion of a particle on a sphere in the presence of a magnetic field.', 'hep-th-0202150-1-46-2': 'The purpose of this section is to consider the zero mode problem in a more general setting.', 'hep-th-0202150-1-46-3': 'After discussing the conifold case, we will give a more general argument for other Calabi-Yau singularities that will show that one can count the ground states even without exact knowledge of the metric.', 'hep-th-0202150-1-47-0': 'First, we describe this freedom of movement some dibaryons possess in a more precise and useful way.', 'hep-th-0202150-1-47-1': 'If we begin with dibaryon-like states on [MATH], then at each time [MATH], the D-brane worldvolume will be wrapping a holomorphic cycle of [MATH], and the volume of the D-brane will be constant, but its shape will change.', 'hep-th-0202150-1-47-2': 'Since the volume remains constant, there is no potential on these directions, and the moduli space of the D-brane can be captured by the moduli space of holomorphic submanifolds of [MATH] with specific topological quantum numbers, which describe what holomorphic cycle is being wrapped.', 'hep-th-0202150-1-48-0': 'The D-brane is a holomorphic submanifold of codimension [MATH] on [MATH], so it can be associated to a divisor [MATH] on [MATH].', 'hep-th-0202150-1-48-1': 'The D-brane is the zero locus of a global section of the associated line bundle to the divisor [MATH].', 'hep-th-0202150-1-48-2': 'Any linear combination of global holomorphic sections of [MATH] is also a global section, and in this way the moduli space of the D-brane is a projective space [MATH] for some [MATH] (which can be zero, and then the curve is rigid).', 'hep-th-0202150-1-48-3': '[MATH] is given by the number of holomorphic sections of the line bundle associated to [MATH] minus one.', 'hep-th-0202150-1-49-0': 'Thus, to quantize the zero modes of the D-brane, we need the effective action of the D-brane moduli space.', 'hep-th-0202150-1-49-1': 'We consider our action as a nonlinear sigma model whose target space is the moduli space of the D-brane.', 'hep-th-0202150-1-49-2': 'This action will involve a metric on [MATH] (which generically has some singularities) induced from the metric on [MATH].', 'hep-th-0202150-1-49-3': 'Additionally, one can add a line bundle on this moduli space, that is, an effective magnetic field on the target space of the sigma model.', 'hep-th-0202150-1-50-0': 'In the example of the conifold, for a single dibaryon state [MATH], we have the round metric on [MATH] (this is obvious, since we had an [MATH] worth of positions for the D-brane with the corresponding isometry group).', 'hep-th-0202150-1-51-0': 'Now, we want to argue that the magnetic field through this moduli space is [MATH] times the hyperplane bundle on this moduli space.', 'hep-th-0202150-1-51-1': 'Indeed, the WZW term for the D-brane action is [EQUATION]', 'hep-th-0202150-1-51-2': 'Three of the coordinates of the D-brane saturate three of the indices of [MATH], let us say [MATH], and the fourth index is saturated by the time direction in the form [EQUATION]', 'hep-th-0202150-1-51-3': 'So the effective action has a term proportional to the velocity on the moduli space of the D-brane.', 'hep-th-0202150-1-51-4': 'Hence, this WZW term gives rise to an effective magnetic field on the [MATH].', 'hep-th-0202150-1-51-5': 'Notice moreover that this magnetic field is proportional to [MATH], the effective tension of the D-brane.', 'hep-th-0202150-1-51-6': 'For the [MATH] case of the conifold it is easy to show that the normalization of this magnetic field is such that the flux through the [MATH] is [MATH].', 'hep-th-0202150-1-51-7': 'Counting the number of ground states is then done by the index theorem, and in this case we get [MATH].', 'hep-th-0202150-1-51-8': 'Notice that this matches exactly the counting of operators in the equation [REF].', 'hep-th-0202150-1-51-9': 'For the general case we find that the index calculation counts the number of global holomorphic functions of degree [MATH] on [MATH].', 'hep-th-0202150-1-51-10': 'There are [EQUATION] such global sections.', 'hep-th-0202150-1-52-0': 'In the case of [MATH] orbifold without fixed points [CITATION], we get that [MATH].', 'hep-th-0202150-1-52-1': 'The curves of degree one in [MATH] are parametrized by a dual [MATH].', 'hep-th-0202150-1-52-2': 'So in this case we get [MATH] states.', 'hep-th-0202150-1-52-3': 'These can also be counted in the CFT.', 'hep-th-0202150-1-52-4': 'Instead of having two fields [MATH] to make the dibaryon, there are three such fields, which transform in the [MATH] of an [MATH] global symmetry.', 'hep-th-0202150-1-52-5': 'The totally symmetric combination of [MATH] of these objects gives a Hilbert space of the same dimension.', 'hep-th-0202150-1-52-6': 'Moreover there is an extra degeneracy by a factor of three from the fact that the [MATH] space is not simply connected, so there is a possibility of having discrete Wilson lines for the gauge field on the D-brane [CITATION].', 'hep-th-0202150-1-52-7': 'This factor of three is also reflected in the field theory quiver diagram, where there are three different types of dibaryon states.', 'hep-th-0202150-1-53-0': 'Both of these examples show that the counting with [MATH] times the hyperplane bundle of [MATH] seems to give the right match for counting states.', 'hep-th-0202150-1-54-0': 'With this in mind, let us return to the conifold case, and let us study a D-brane that wraps once both of the two different [MATH].', 'hep-th-0202150-1-54-1': 'It was argued in [CITATION] that such D-branes should become giant gravitons.', 'hep-th-0202150-1-54-2': 'The counting of holomorphic sections of the line bundle gives us [MATH].', 'hep-th-0202150-1-54-3': 'Indeed, if we consider [MATH] and [MATH] a set of projective coordinates for both of the [MATH], the global sections of this line bundle are given by curves [EQUATION] where we get four possible distinct coefficients [MATH].', 'hep-th-0202150-1-55-0': 'The number of such states is therefore [MATH].', 'hep-th-0202150-1-55-1': 'This is many more states than just the product of two dibaryons, one for [MATH] and one for [MATH].', 'hep-th-0202150-1-55-2': 'Indeed, these factorized states correspond to the curves that factorize into linear terms in the equation [REF].', 'hep-th-0202150-1-55-3': 'But when we have intersecting branes, we have also deformations that are localized at the intersection.', 'hep-th-0202150-1-55-4': 'These are the ones responsible for giving us a larger moduli space.', 'hep-th-0202150-1-56-0': 'In the case where the curve factorizes, the operator is given by [EQUATION]', 'hep-th-0202150-1-56-1': 'Notice now that the total baryon number of this operator is zero, and that the [MATH] appears as many times as the [MATH].', 'hep-th-0202150-1-56-2': 'We can use the identity [EQUATION] where [MATH] sums over all possible permutations of the [MATH], and [MATH] is [MATH] or [MATH] if the permutation is even or odd.', 'hep-th-0202150-1-57-0': 'This procedure lets us eliminate one set of the [MATH] symbols if we want to.', 'hep-th-0202150-1-57-1': 'We see that if we do this, the [MATH] have to be paired with the [MATH] in combinations of the form [MATH].', 'hep-th-0202150-1-57-2': 'There are [MATH] such possible combinations.', 'hep-th-0202150-1-57-3': 'Consider therefore the operators of the form [EQUATION] where each of the entries is given by one of the four possible combinations [MATH].', 'hep-th-0202150-1-57-4': 'This object can be expressed in terms of traces of the [MATH], so it is built out of gravitons, and indeed it should represent a maximal giant graviton.', 'hep-th-0202150-1-57-5': 'We have a symmetric combination of four objects, and the combinatorics of these objects also gives us a total of [MATH] objects.', 'hep-th-0202150-1-57-6': 'In some sense, [MATH] and [MATH] are the analog of the coordinates [MATH] and [MATH].', 'hep-th-0202150-1-58-0': 'Similarly, if we take a state which wraps sphere one twice and sphere two once, we get that [MATH].', 'hep-th-0202150-1-58-1': 'The natural objects to consider are of the form [MATH].', 'hep-th-0202150-1-58-2': 'However, we have to remeber the [MATH]-terms, so in this expressoin we get a field wich is symmetric in [MATH].', 'hep-th-0202150-1-58-3': 'The total number of objects that we can count is [MATH], and we can build operators of the form [EQUATION]', 'hep-th-0202150-1-58-4': 'Notice that this operator corresponds to some form of a coherent state of excitations of the dibaryon ([REF]).', 'hep-th-0202150-1-58-5': 'From the supergravity, we expect such a result because this new baryon-like state is wrapping the same topological cycle as the old dibaryon.', 'hep-th-0202150-1-59-0': 'For more general wrapping cases of baryon number two or higher, it is more difficult to match the states.', 'hep-th-0202150-1-59-1': 'Since we cannot get rid of all but one of the pairs of [MATH] symbols, it is not true that there is an easy description of the state in terms of [MATH]-symmetrized products of similar objects (the [MATH] above).', 'hep-th-0202150-1-59-2': 'It would be interesting to complete a more thorough description of these other states.', 'hep-th-0202150-1-60-0': '# BPS fluctuations of dibaryons', 'hep-th-0202150-1-61-0': 'In this section we show that there exist BPS excitations of the dibaryon operators, that is, operators which carry baryon number and have higher conformal weight than the volume of the associated 3-cycle.', 'hep-th-0202150-1-61-1': 'Since in this particular section we will be interested in computing explicitly the fluctuation spectra of D-branes, we need to resort to writing models for specific examples where the metric and the CFT are both known.', 'hep-th-0202150-1-61-2': 'We are moreover interested in situations where we have only [MATH] supersymmetry without singularities.', 'hep-th-0202150-1-61-3': 'In these models the [MATH] symmetry is strictly [MATH] and the chiral fields are holomorphic.', 'hep-th-0202150-1-61-4': 'Two of these models are particularly simple.', 'hep-th-0202150-1-61-5': 'These are D-branes at the conifold [CITATION] and D-branes at the orbifold [MATH] [CITATION] without fixed points.', 'hep-th-0202150-1-61-6': 'We will deal in this section with the particular case of the conifold.', 'hep-th-0202150-1-62-0': 'Let us consider for simplicity the state with maximum [MATH] of the first [MATH]: [EQUATION]', 'hep-th-0202150-1-62-1': 'To construct excited dibaryons we can replace one of the [MATH] by any other chiral field which transforms in the same representation of the gauge groups.', 'hep-th-0202150-1-62-2': 'One possibility is to replace [MATH], where the two gauge indices of [MATH] are contracted separately with the [MATH] and with the [MATH], following the rules for matrix multiplication, namely [EQUATION]', 'hep-th-0202150-1-62-3': 'This dibaryon-like state is a chiral field up to F-terms because it is a gauge invariant holomorphic polynomial in the chiral superfields.', 'hep-th-0202150-1-62-4': 'There will be', 'hep-th-0202150-1-63-0': 'a chiral primary state with the same quantum numbers as the above operator that is a linear combination of operators of this type.', 'hep-th-0202150-1-63-1': 'Remember that operators that differ by F-terms are equivalent as elements of the chiral ring, but in the conformal field theory they are different and only one particular linear combination is protected.', 'hep-th-0202150-1-64-0': 'The operator resulting after the replacement factorizes into the original dibaryon and a single-trace operator: [EQUATION]', 'hep-th-0202150-1-64-1': 'The factorization suggests that this excitation of a dibaryon can be represented as a graviton fluctuation in presence of the original dibaryon.', 'hep-th-0202150-1-65-0': 'Not all excitations factorize in this way, however.', 'hep-th-0202150-1-65-1': 'For example, consider replacing [MATH].', 'hep-th-0202150-1-66-0': 'One might ask whether this new operator can be written as a product of the original dibaryon [MATH] and a meson-like operator of the form [MATH].', 'hep-th-0202150-1-66-1': 'The answer is no.', 'hep-th-0202150-1-67-0': 'To see why, it is easier to go to a generic point in the moduli space of vacua of the theory where we can set [MATH] by gauge transformations, and thus we establish an isomorphism between the indices of the two gauge groups.', 'hep-th-0202150-1-67-1': 'The operator above becomes [MATH], and if [MATH] is large enough, there are no relations amongst traces of a low number of fields.', 'hep-th-0202150-1-67-2': 'In other words, we cannot write the operator as [MATH].', 'hep-th-0202150-1-67-3': 'Since this state cannot be factored it has to be interpreted as a single particle state in AdS.', 'hep-th-0202150-1-67-4': 'Since the operator also carries baryon number one,', 'hep-th-0202150-1-68-0': 'the natural conclusion is that the one-particle state is a BPS excitation of the wrapped D3-brane in the dual string theory.', 'hep-th-0202150-1-69-0': 'Let us now study BPS fluctuations of the wrapped D3-brane', 'hep-th-0202150-1-70-0': 'in the supergravity approximation.', 'hep-th-0202150-1-70-1': 'The DBI action is a good approximation in the limit of weak string coupling and weak curvature of the D-brane.', 'hep-th-0202150-1-70-2': "These conditions are met in the limit of large 't Hooft coupling.", 'hep-th-0202150-1-70-3': 'In particular, we will compute the spectrum of quadratic fluctuations of this DBI action.', 'hep-th-0202150-1-71-0': 'We will return to the field theory later to make the correspondence of states and quantum numbers more precise.', 'hep-th-0202150-1-72-0': 'First, we set up the DBI computation in the right coordinate system.', 'hep-th-0202150-1-72-1': 'The full ten dimensional metric is naturally [MATH], [EQUATION]', 'hep-th-0202150-1-72-2': 'For convenience, we have chosen the time [MATH] direction to be a Killing vector.', 'hep-th-0202150-1-72-3': 'The radius of curvature is [MATH].', 'hep-th-0202150-1-72-4': 'The metric of [MATH], the base of the conifold, is given by [EQUATION] with [MATH], [MATH].', 'hep-th-0202150-1-73-0': 'We will keep [MATH] as variables in our computation for consistency checks.', 'hep-th-0202150-1-74-0': 'The dibaryon is chosen to wrap the cycle defined by [MATH] constant.', 'hep-th-0202150-1-74-1': 'This configuration is invariant under rotations of the sphere wrapped by the D3-brane, but it is not invariant under the [MATH] associated to the [MATH] coordinates.', 'hep-th-0202150-1-74-2': 'The induced metric on the dibaryon is thus [EQUATION]', 'hep-th-0202150-1-74-3': 'It is convenient to make a change of variables [MATH], so that [EQUATION]', 'hep-th-0202150-1-74-4': 'In these variables, the determinant of the spatial part of the metric [EQUATION] is constant.', 'hep-th-0202150-1-74-5': "The variables' range is given by [MATH], [MATH] and [MATH].", 'hep-th-0202150-1-74-6': 'The volume of the wrapped manifold is thus [MATH].', 'hep-th-0202150-1-74-7': 'In [CITATION], it was noted that this volume times the tension of the D3-brane should be a good approximation of the mass of the corresponding dibaryon.', 'hep-th-0202150-1-74-8': 'As we argued in section 2.4, this volume is directly proportional to the dimension [MATH] of the dibaryonic operator.', 'hep-th-0202150-1-75-0': 'Indeed, as the conformal dimension of each [MATH] and [MATH] is 3/4, one finds that [MATH] is exactly equal to [MATH].', 'hep-th-0202150-1-76-0': 'We return now to calculating the excitation spectrum of the dibaryon.', 'hep-th-0202150-1-77-0': 'We need to be careful with single valued functions on this space.', 'hep-th-0202150-1-77-1': 'This squashed [MATH] can be thought of, essentially, as the group manifold [MATH].', 'hep-th-0202150-1-77-2': 'The coordinates [MATH] are the Euler angles.', 'hep-th-0202150-1-77-3': 'One might have thought that the points [MATH] and [MATH] were equivalent.', 'hep-th-0202150-1-77-4': 'The insight from [MATH] lets us correct this mistake.', 'hep-th-0202150-1-77-5': 'Equivalent points on [MATH] have [EQUATION] where [MATH] and [MATH] are two points on [MATH].', 'hep-th-0202150-1-78-0': 'We want to find the normal modes of oscillation of the wrapped D3-brane around the solution corresponding to some fixed world-line in [MATH] and some fixed [MATH] and [MATH] on the transverse [MATH].', 'hep-th-0202150-1-78-1': 'The fluctuations along the transverse [MATH] are the most interesting: they change the [MATH] quantum numbers and are most usefully compared with the chiral primary states in the field theory.', 'hep-th-0202150-1-78-2': 'The transverse fluctuations along the [MATH] are left for future work.', 'hep-th-0202150-1-78-3': 'Supersymmetry relates the gauge field degrees of freedom and fermions on the D-brane to the scalar modes considered here.', 'hep-th-0202150-1-78-4': 'There is no mixing between the between the different modes at quadratic order in the fluctuations, and we ignore the vector and spinor modes in what follows.', 'hep-th-0202150-1-79-0': 'Because the fluctuations around [MATH] and [MATH] are by definition small, it is appropriate to treat the [MATH] parametrized by these coordinates as a flat [MATH].', 'hep-th-0202150-1-79-1': 'For example, one may take [MATH].', 'hep-th-0202150-1-79-2': 'Then the fluctuation coordinates on [MATH] can be taken to be [MATH] and [MATH].', 'hep-th-0202150-1-80-0': 'The connection term in the metric on [MATH] becomes [MATH] and the Kahler form on the transverse [MATH] becomes [MATH].', 'hep-th-0202150-1-80-1': 'To say the same thing in a different way, we are interested in the D-brane action to quadratic order in fluctuations.', 'hep-th-0202150-1-80-2': 'We only need to consider terms of up to order [MATH], [MATH], or [MATH], and we neglect everything else which is higher order in the fluctuations [MATH].', 'hep-th-0202150-1-81-0': 'Notice that the ten dimensional metric ([REF]) has a term of the form [MATH], where the [MATH] are the [MATH] and [MATH] coordinates.', 'hep-th-0202150-1-81-1': 'Indeed, in the previous section, we saw that we could add a more general term of this form depending on the [MATH] coordinates.', 'hep-th-0202150-1-81-2': 'This perturbation corresponded to a gauge field carrying the [MATH]-charge.', 'hep-th-0202150-1-81-3': 'Recall that changing the coordinate [MATH] to [MATH] does not change the periodicity of the variable [MATH]; but it does change the form of the vector [MATH] by a gauge transformation, [MATH].', 'hep-th-0202150-1-81-4': 'So in writing the metric, the invariant quantity is the field strength of [MATH].', 'hep-th-0202150-1-81-5': 'We are in this way free to add a term of the form [MATH] to [MATH], changing [MATH] and making the connection term simpler.', 'hep-th-0202150-1-82-0': 'The Dirac-Born-Infeld acion is given by [EQUATION] with [MATH] the tension of the brane.', 'hep-th-0202150-1-82-1': 'We may choose a gauge where [MATH] has a piece in the [MATH], [MATH], [MATH] and [MATH] directions.', 'hep-th-0202150-1-82-2': 'More specifically, we will choose a gauge which is well defined at the north pole of the sphere parametrized by [MATH] and [MATH]: [EQUATION]', 'hep-th-0202150-1-82-3': 'We want to expand the probe brane action to quadratic order in fluctuations in [MATH] and [MATH].', 'hep-th-0202150-1-82-4': 'At this order, the four form [MATH] is approximately [EQUATION]', 'hep-th-0202150-1-82-5': 'Because [MATH] appear only quadratically in the metric, we can study the quadratic fluctuations in [MATH] by doing a first order variation in the metric [EQUATION] where [MATH] is the second order contribution from the fluctuations in [MATH] and we define [MATH] to be the inverse of [MATH].', 'hep-th-0202150-1-82-6': 'We can also do the same with the transverse fluctuations along the [MATH], but we will leave those for later.', 'hep-th-0202150-1-83-0': 'With the coordinates chosen [MATH] is independent of the fluctuations [MATH] and also independent of the D3-brane coordinates.', 'hep-th-0202150-1-84-0': 'As a result, the wave equation on the D-brane worldvolume is simplified.', 'hep-th-0202150-1-85-0': 'We fix the diffeomorphism invariance of the DBI action by locking the internal brane coordinates to the background coordinates [MATH].', 'hep-th-0202150-1-85-1': 'This locking corresponds to choosing a physical gauge.', 'hep-th-0202150-1-86-0': 'For convenience, we take [MATH].', 'hep-th-0202150-1-86-1': 'If the brane is not moving then [EQUATION]', 'hep-th-0202150-1-86-2': 'The inverse matrix is [EQUATION]', 'hep-th-0202150-1-86-3': 'From here we obtain that [EQUATION]', 'hep-th-0202150-1-86-4': 'The first term gives rise to the standard laplacian on the three-sphere, while the second term gives rise to mixing terms between [MATH] that are only first order in derivatives.', 'hep-th-0202150-1-86-5': 'An additional mixing term comes from ([REF]).', 'hep-th-0202150-1-86-6': 'These three contributions give us the effective Lagrangian density to quadratic order in fluctuations.', 'hep-th-0202150-1-86-7': '[EQUATION]', 'hep-th-0202150-1-86-8': 'The non-trivial elements of [MATH] are [EQUATION]', 'hep-th-0202150-1-86-9': 'From now on, we will call [MATH] simply [MATH].', 'hep-th-0202150-1-86-10': 'The spatial mixing term [MATH] is explicitly given by [EQUATION]', 'hep-th-0202150-1-86-11': 'The equations of motion for the fluctuations are given by [EQUATION]', 'hep-th-0202150-1-86-12': 'Notice that the elements of [MATH] are independent of the variables with respect to which we are taking derivatives.', 'hep-th-0202150-1-86-13': 'As a result, the second and third terms in the equation above are actually equal and we have [EQUATION]', 'hep-th-0202150-1-86-14': 'Taking the combinations [MATH] these equations become [EQUATION]', 'hep-th-0202150-1-86-15': 'Now, we see that [MATH] so the terms with derivatives with respect to [MATH] cancel.', 'hep-th-0202150-1-86-16': 'This is just as expected, since the result should be invariant under the [MATH] of isometries of the squashed [MATH].', 'hep-th-0202150-1-86-17': 'Thus far we are getting a consistent picture.', 'hep-th-0202150-1-87-0': 'The terms with [MATH] are given by the combination [EQUATION] which is a constant coefficient.', 'hep-th-0202150-1-88-0': 'Now we can use the separation of variables [EQUATION] to obtain a differential equation for [MATH].', 'hep-th-0202150-1-88-1': 'Note that for the [MATH] to be single valued, the condition ([REF]) implies that [MATH] and [MATH] are either both integer or both half-integer.', 'hep-th-0202150-1-88-2': 'The remaining differential equation for the [MATH] is given by [EQUATION]', 'hep-th-0202150-1-88-3': 'Let us begin by analyzing the behavior of the solution in the limit [MATH].', 'hep-th-0202150-1-88-4': 'In this limit, the differential equation becomes [EQUATION]', 'hep-th-0202150-1-88-5': 'The solution to this equation is clearly a power of [MATH], [MATH].', 'hep-th-0202150-1-88-6': 'Thus [EQUATION]', 'hep-th-0202150-1-88-7': 'The energy should be real for all allowed values of [MATH] and [MATH].', 'hep-th-0202150-1-88-8': 'Notice that the energy has the right dependence in terms of the [MATH] quantum numbers to be associated with the velocity on a group manifold.', 'hep-th-0202150-1-89-0': 'Only certain values of [MATH] are allowed because the fluctuations must be well-behaved at [MATH].', 'hep-th-0202150-1-90-0': 'The differential equation ([REF]) can be solved in terms of a hypergeometric function [EQUATION] where [MATH], [MATH], [MATH], [MATH], and [MATH] depend on the quantum numbers [MATH], [MATH], and [MATH].', 'hep-th-0202150-1-90-1': 'The north and south pole of the [MATH], [MATH], correspond to the singular points [MATH] and [MATH] of the hypergeometric function.', 'hep-th-0202150-1-90-2': 'For the fluctuations to vanish at [MATH], [MATH] must be a non-negative integer.', 'hep-th-0202150-1-91-0': 'For the [MATH] which contribute to the [MATH] charge in the same direction as the unexcited D3-brane, the choice [MATH] corresponds to a BPS state;', 'hep-th-0202150-1-92-0': 'we have [EQUATION]', 'hep-th-0202150-1-92-1': 'Indeed, the BPS states should have the lowest possible dimension for a given R-charge.', 'hep-th-0202150-1-92-2': 'The states [MATH] meet this condition.', 'hep-th-0202150-1-92-3': 'From the periodicity of [MATH], [MATH] can be a half integer, and when [MATH] is a half integer so is [MATH].', 'hep-th-0202150-1-93-0': 'This energy spectrum means that the contribution of these BPS states to the energy is quantized in units of [MATH].', 'hep-th-0202150-1-94-0': '[MATH] is also exactly the change in the conformal dimension of the chiral operators once [MATH] is substituted for an [MATH] in the antisymmetric product ([REF]).', 'hep-th-0202150-1-94-1': 'These modes match the result from conformal field theory.', 'hep-th-0202150-1-95-0': 'Also, the transformation under the [MATH] that rotates [MATH] is in agreement.', 'hep-th-0202150-1-95-1': 'The unexcited dibaryon was a singlet while the excited one acquires spin [MATH].', 'hep-th-0202150-1-95-2': 'Indeed, for each value of [MATH] there is a unique irreducible representation of [MATH] associated to it.', 'hep-th-0202150-1-95-3': 'This means that the quantum of [MATH] charge proportional to [MATH] should be associated to a spin [MATH] state for [MATH].', 'hep-th-0202150-1-95-4': "In the conformal field theory, these states result from choosing to replace one of the [MATH] in the [MATH] function by, for example, [MATH], where we have [MATH]'s inside the matrix.", 'hep-th-0202150-1-95-5': 'Using the F-term equations of motion, it can be seen that these states are totally symmetric with respect to exchange of the [MATH] variables.', 'hep-th-0202150-1-95-6': "Since the [MATH] carry spin [MATH] under the [MATH], these states with [MATH]'s form a totally symmetric representation of [MATH] with spin [MATH].", 'hep-th-0202150-1-95-7': 'Therefore, we can match the [MATH] quantum numbers of the states to the supergravity.', 'hep-th-0202150-1-96-0': 'We have not, however, determined the transformation property of the excited dibaryons under the [MATH] that rotates [MATH].', 'hep-th-0202150-1-96-1': 'This is more difficult since we need to consider the coupling of the fluctuation fields with the zero-mode dynamics.', 'hep-th-0202150-1-96-2': 'Since we have not determined the full [MATH] quantum numbers, we cannot make a precise determination of the dual gauge theory operators.', 'hep-th-0202150-1-96-3': 'In particular, it would be interesting to see if the factorized operators of the type ([REF]) are necessary to match the spectrum of the fluctuating probe D3-brane.', 'hep-th-0202150-1-96-4': 'We leave this interesting question for the future.', 'hep-th-0202150-1-97-0': 'For the transverse motion in [MATH] there is no mixing of the directions, so we get four scalars and their energies are given by [EQUATION] with [MATH] the mass of these states.', 'hep-th-0202150-1-97-1': 'This mass term comes because the space-time is curved and to second order [MATH]; the AdS excitation feels a gravitational potential.', 'hep-th-0202150-1-97-2': 'For [MATH] the above expression should be a perfect square for all [MATH] (it is a superpartner of the other BPS states), so we should have [MATH], or equivalently [MATH].', 'hep-th-0202150-1-97-3': 'This relation can be checked explicitly from the metric.', 'hep-th-0202150-1-97-4': 'It follows that [MATH] differs by [MATH] from the previous energy.', 'hep-th-0202150-1-97-5': 'Indeed, we expect excitations in the AdS directions to correspond to introducing covariant derivatives for the fields [MATH] or [MATH] on the field theory side.', 'hep-th-0202150-1-97-6': 'Covariant derivatives have conformal weight equal to one.', 'hep-th-0202150-1-97-7': 'This is an alternative check that the normalization of the [MATH] field we chose is correct, since it predicts the right splitting between the energies of the modes to be compatible with the spacetime symmetries.', 'hep-th-0202150-1-98-0': 'Notice that the dibaryon state on the gravity side has a Fock space worth of excitations.', 'hep-th-0202150-1-98-1': 'In the field theory, this Fock space can be reproduced as well.', 'hep-th-0202150-1-98-2': 'To insert one quantum, we took one of the [MATH] and replaced it with [MATH].', 'hep-th-0202150-1-98-3': 'To put many quanta, we replace many of the [MATH] by the [MATH] combinations.', 'hep-th-0202150-1-98-4': 'The fact that we have a Fock space of identical particles, as opposed to distinguishable particles, comes from the permutation symmetry of the [MATH] symbols.', 'hep-th-0202150-1-98-5': 'For fermionic excitations we need to remember the [MATH] signs when we exchange the fermionic insertions in the operator.', 'hep-th-0202150-1-99-0': 'We conclude that we can match, at least schematically, the dibaryon state with any number of BPS open strings that excite it.', 'hep-th-0202150-1-99-1': 'We have not carried out complete matching since we have not determined the [MATH] quantum numbers of the excited D3-branes, which remains an interesting problem for future work.', 'hep-th-0202150-1-99-2': 'We believe that the construction of non-BPS states can be carried out using similar methods to the ones used in [CITATION] for closed strings, but this construction is beyond the scope of the present paper.', 'hep-th-0202150-1-100-0': '# Discussion', 'hep-th-0202150-1-101-0': 'In this paper we studied in some detail the correspondence between D3-branes wrapping 3-spheres inside [MATH] and baryon-type operators of the dual [MATH] gauge theory.', 'hep-th-0202150-1-101-1': 'By calculating the [MATH] charge of the D3-branes and their collective coordinate energies we have new evidence that the operator identification proposed in [CITATION] is correct.', 'hep-th-0202150-1-101-2': 'Furthermore, we showed that there exist BPS excitations of wrapped D3-branes and suggested the chiral operators dual to them.', 'hep-th-0202150-1-101-3': 'Our results provide new evidence that the duality between gauge invariant operators of a superconformal gauge theory and states of string theory on [MATH] extends to operators whose dimensions grow as [MATH], the number of colors.', 'hep-th-0202150-1-101-4': 'It is clear that this same procedure to generate operators should work in other examples related to different quiver theories and that one can generically find BPS excitations of baryon-like operators.', 'hep-th-0202150-1-102-0': 'There is another such class of operators, similar to the ones we have considered.', 'hep-th-0202150-1-102-1': 'It is related to the "giant graviton" effect [CITATION].', 'hep-th-0202150-1-102-2': 'It has been observed that for modes whose angular momentum on [MATH] is of order [MATH] the single-trace description of the dual gauge theory operators breaks down [CITATION].', 'hep-th-0202150-1-102-3': 'Instead, the correct description is in terms of subdeterminants of elementary fields, which for maximum angular momentum become determinants similar to the dibaryon operators we have considered.', 'hep-th-0202150-1-102-4': 'On the string theory side, the modes whose angular momentum is of order [MATH] blow up into D3-branes on [MATH].', 'hep-th-0202150-1-102-5': 'This is another manifestation of the fact that D-branes, and therefore string theory, are crucial for describing gauge invariant operators whose dimensions grow as [MATH].', 'hep-th-0202150-1-102-6': "Thus, string theory is necessary to complete the state/operator map even at large 't Hooft coupling.", 'hep-th-0202150-1-102-7': 'Along these lines, it is interesting to consider the BPS excitations of the giant gravitons and to construct the dual operators.', 'hep-th-0202150-1-102-8': 'Some results on this are presented in [CITATION].', 'hep-th-0202150-1-102-9': 'We believe that the giant graviton case is similar to our study of topologically stable wrapped D3-branes.', 'hep-th-0202150-1-103-0': "Very recently, a much more dramatic demonstration of the stringy nature of the AdS/CFT duality at large 't Hooft coupling was presented in [CITATION].", 'hep-th-0202150-1-103-1': 'The insight of this paper is to focus on states whose angular momentum [MATH] on [MATH] scales as [MATH].', 'hep-th-0202150-1-103-2': 'It was shown that there exists a class of operators whose [MATH] stays finite in this limit.', 'hep-th-0202150-1-103-3': 'These states are in one-to-one correspondence with all the closed string states on a RR-charged pp-wave background, including all the massive string states.', 'hep-th-0202150-1-103-4': 'We believe that the work we have presented on the BPS excitations of D3-branes constitutes a first step towards studying the open string states in a similar setting.', 'hep-th-0202150-1-103-5': 'An unexcited D3-brane is described by the basic dibaryon operators ([REF]) and ([REF]).', 'hep-th-0202150-1-103-6': 'Thus, these operators describe the open string vacuum.', 'hep-th-0202150-1-103-7': 'The more complicated operators discussed in section 4 correspond to the BPS states of the open string.', 'hep-th-0202150-1-103-8': 'The non-BPS open string states can be constructed along the lines of [CITATION], but this construction is beyond the scope of the present paper.', 'hep-th-0202150-1-103-9': 'The basic issue in question is the proper understanding of the non-planar diagrams.', 'hep-th-0202150-1-103-10': 'When the operator dimension scales as [MATH], it is too high for the planar approximation to be valid.'}

{'hep-th-0202150-2-0-0': '# Introduction', 'hep-th-0202150-2-1-0': 'According to the AdS/CFT conjecture [CITATION], the chiral operators of the [MATH] supersymmetric [MATH] gauge theory are in one-to-one correspondence with the modes of type IIB supergravity on [MATH].', 'hep-th-0202150-2-1-1': "On the other hand, the massive string modes correspond to operators in long multiplets whose dimensions diverge for large 't Hooft coupling as [MATH].", 'hep-th-0202150-2-1-2': 'Thus, in the limit [MATH] the stringy nature of the dual theory is obscured by the decoupling of the non-chiral operators, which constitute the large majority of possible gauge invariant operators.', 'hep-th-0202150-2-2-0': 'It is important to keep in mind, however, that the AdS/CFT correspondence relates large [MATH] gauge theory to string theory, not merely supergravity.', 'hep-th-0202150-2-2-1': "If we depart from the limit of infinite 't Hooft coupling, then all the non-chiral operators no longer decouple, so that the spectrum of the gauge theory becomes much more complicated and presumably related to type IIB string theory on [MATH] (see, for example, [CITATION]).", 'hep-th-0202150-2-2-2': "However, even for very large 't Hooft coupling, it is possible to demonstrate the stringy nature of the dual theory quite explicitly.", 'hep-th-0202150-2-2-3': 'One early clue was provided by Witten [CITATION] in the context of [MATH] supersymmetric [MATH] gauge theory which is dual to type IIB strings on [MATH].', 'hep-th-0202150-2-2-4': 'He noted that the gauge theory possesses chiral operators of dimension [MATH], the Pfaffians, whose dual is provided by a D3-brane wrapping a 3-cycle of the [MATH].', 'hep-th-0202150-2-2-5': 'This explicitly shows that the dual theory cannot be simply supergravity: it must contain D-branes.', 'hep-th-0202150-2-2-6': 'Since the third homology class [MATH], a D3-brane wrapped twice can decay into ordinary supergravity modes.', 'hep-th-0202150-2-2-7': 'A corresponding gauge theory statement is that a product of two Pfaffian operators can be expressed in terms of the usual single-trace operators.', 'hep-th-0202150-2-3-0': 'It is further possible to find type IIB backgrounds of the form [MATH] such that [MATH].', 'hep-th-0202150-2-3-1': 'In this case, a D3-brane can wrap a 3-cycle any number of times so that the dual gauge theory operators carry a quantized baryon number.', 'hep-th-0202150-2-3-2': 'A simple example of this kind is provided by the space [MATH] whose symmetries are [MATH].', 'hep-th-0202150-2-3-3': 'This space may be thought of as a [MATH] bundle over [MATH], and the explicit Einstein metric on [MATH] is [EQUATION]', 'hep-th-0202150-2-3-4': 'The [MATH] superconformal gauge theory dual to type IIB strings on [MATH] was constructed in [CITATION]: it is [MATH] gauge theory coupled to two chiral superfields, [MATH], in the [MATH] representation and two chiral superfields, [MATH], in the [MATH] representation [CITATION].', 'hep-th-0202150-2-3-5': "The [MATH]'s transform as a doublet under one of the global [MATH]'s while the [MATH]'s transform as a doublet under the other [MATH].", 'hep-th-0202150-2-4-0': "Cancellation of the anomaly in the [MATH] R-symmetry requires that the [MATH]'s and the [MATH]'s each have R-charge [MATH].", 'hep-th-0202150-2-4-1': 'For consistency of the duality it is necessary that we add an exactly marginal superpotential which preserves the [MATH] symmetry of the theory: [EQUATION]', 'hep-th-0202150-2-4-2': 'In [CITATION] it was proposed that the wrapped D3-branes correspond to baryon-like operators [MATH] and [MATH] where the indices of both [MATH] groups are fully antisymmetrized (their more detailed form is exhibited in Section 3).', 'hep-th-0202150-2-4-3': 'For large [MATH] the dimensions of such operators calculated from the mass of wrapped D3-branes were found to be [MATH] [CITATION].', 'hep-th-0202150-2-4-4': 'This is in complete agreement with the fact that the dimension of the chiral superfields at the fixed point is [MATH].', 'hep-th-0202150-2-4-5': 'At the quantum level, the collective coordinate for the wrapped D3-brane has to be quantized to identify its [MATH] quantum numbers.', 'hep-th-0202150-2-4-6': 'This quantization was sketched in [CITATION].', 'hep-th-0202150-2-4-7': 'Here we do a more careful job and show that the dimension of dibaryons is [MATH] not only in the large [MATH] limit but exactly.', 'hep-th-0202150-2-4-8': 'In a related check on the identification of the dibaryons with wrapped D3-branes, we calculate the [MATH] charge of the wrapped D3-branes and show that it is [MATH], in agreement with the gauge theory.', 'hep-th-0202150-2-5-0': 'The structure of the paper is as follows.', 'hep-th-0202150-2-5-1': 'In Section 2 we calculate the [MATH] charge of a D3-brane wrapping a 3-cycle inside the Einstein space [MATH].', 'hep-th-0202150-2-5-2': 'Our results apply to all [MATH] which are [MATH] bundles over 4-d Kahler-Einstein spaces.', 'hep-th-0202150-2-5-3': 'We also provide an analogous calculation for M5-branes wrapping a 5-cycle inside an Einstein space [MATH] which is a [MATH] bundle over a 6-d Kahler-Einstein space.', 'hep-th-0202150-2-5-4': 'In all cases, we show that the R-charge is proportional to the dimension [MATH] of the dual CFT operator, with the correct constant of proportionality.', 'hep-th-0202150-2-5-5': 'We further argue that [MATH] is measured by the volume of the cycle wrapped by the brane.', 'hep-th-0202150-2-5-6': 'In Section 3 we make some remarks on the collective coordinate quantization of the wrapped D3-branes.', 'hep-th-0202150-2-5-7': 'In particular, we focus on the [MATH] example and show that the [MATH] quantum numbers of the dibaryon operators indeed follow from this quantization.', 'hep-th-0202150-2-5-8': 'In Section 4 we study BPS excitations of the wrapped D3-branes using the DBI action.', 'hep-th-0202150-2-5-9': 'We calculate their energies and [MATH] charges and propose dual chiral operators carrying the same quantum numbers.', 'hep-th-0202150-2-6-0': '# The R-symmetry gauge field', 'hep-th-0202150-2-7-0': 'We would like to show how the [MATH] gauge field emerges from a Kaluza-Klein reduction of the full ten or eleven dimensional supergravity action.', 'hep-th-0202150-2-7-1': 'In ten dimensions, we begin with a stack of D3-branes placed at the tip of a non-compact Calabi-Yau cone [MATH] of complex dimension three.', 'hep-th-0202150-2-7-2': 'In eleven dimensions, we consider a stack of M2-branes at the tip of a similar cone [MATH] of complex dimension 4.', 'hep-th-0202150-2-7-3': 'We may write the metric on [MATH] as [EQUATION] where [MATH] is Einstein, i.e. [MATH].', 'hep-th-0202150-2-8-0': 'In the near-throat limit [CITATION], the backreaction of the branes causes the full space to separate into a product manifold.', 'hep-th-0202150-2-8-1': 'In the D3-brane case, we have [MATH] and a five form flux [MATH] produced by the stack of D3-branes.', 'hep-th-0202150-2-8-2': 'See also [CITATION] for details.', 'hep-th-0202150-2-8-3': 'For the M2-branes, the full space becomes [MATH], and the M2-brane flux is carried by [MATH].', 'hep-th-0202150-2-9-0': 'When [MATH] is [MATH] or the coset manifold [MATH], the authors of [CITATION] demonstrate that perturbations in the metric [MATH] combine with perturbations in the [MATH] potential to produce a massless vector field.', 'hep-th-0202150-2-9-1': '(The [MATH] index [MATH] while [MATH] index [MATH] space.)', 'hep-th-0202150-2-9-2': 'For [MATH], more general results exist concerning the existence of this massless vector field [CITATION].', 'hep-th-0202150-2-9-3': 'In particular, whenever [MATH] is a coset space and there exists a Killing vector, harmonic analysis demonstrates the existence of a massless vector field mixing perturbations of [MATH] with perturbations to [MATH].', 'hep-th-0202150-2-10-0': 'In what follows, we consider a more general class of [MATH], in particular quasi-regular Einstein-Sasaki manifolds, and demonstrate the existence of a massless U(1) vector field.', 'hep-th-0202150-2-10-1': 'We argue that this massless vector field mediates the interaction between objects with R-charge.', 'hep-th-0202150-2-10-2': 'Although the generalization is interesting in its own right, the main interest is the demonstration that wrapped D3-branes and M2-branes in these geometries always carry an R-charge consistent with their identification as dibaryons in the corresponding conformal field theory.', 'hep-th-0202150-2-11-0': 'To be more specific, a quasi-regular Einstein-Sasaki manifold is essentially a U(1) fibration of a Kahler-Einstein manifold or orbifold.', 'hep-th-0202150-2-11-1': 'The BPS dibaryons wrap the base of the Sasaki-Einstein manifold in a holomorphic manner, and they wrap the [MATH] fiber completely.', 'hep-th-0202150-2-11-2': 'Mikhailov has considered similar holomorphically wrapped D-branes [CITATION].', 'hep-th-0202150-2-11-3': 'One might naively say that the R-charge of a particle is its angular momentum along the [MATH] direction.', 'hep-th-0202150-2-11-4': 'Thus, a brane that wraps the fiber carries no net angular momentum.', 'hep-th-0202150-2-11-5': 'However, one must be careful; these branes can be supersymmetric, and it is necessary to understand what the precise notion of R-charge is in the supergravity.', 'hep-th-0202150-2-11-6': 'The following calculation is not that sensitive to the dimensionality, but for clarity, we will consider the [MATH] and [MATH] cases separately.', 'hep-th-0202150-2-12-0': '## The R-charge for [MATH]', 'hep-th-0202150-2-13-0': 'In general, the ten dimensional metric is [EQUATION] where [MATH] is a Minkowski tensor.', 'hep-th-0202150-2-14-0': 'A solution to the SUGRA equations of motion can be obtained by threading the [MATH] space with [MATH] units of [MATH] form flux.', 'hep-th-0202150-2-14-1': 'In particular [EQUATION] where [MATH].', 'hep-th-0202150-2-15-0': 'We now further specialize to the case where [MATH] is a quasi-regular Einstein-Sasaki space.', 'hep-th-0202150-2-15-1': 'In simpler language, [MATH] is a U(1) bundle over a two complex dimensional Kahler-Einstein manifold (or orbifold) [MATH]: [EQUATION]', 'hep-th-0202150-2-15-2': 'Define the Kahler form on [MATH] to be [MATH].', 'hep-th-0202150-2-15-3': 'Then, [MATH].', 'hep-th-0202150-2-15-4': 'The number [MATH] is defined such that [MATH], where [MATH] is the first Chern class of the U(1) bundle and [MATH].', 'hep-th-0202150-2-15-5': 'With these definitions, [MATH] [CITATION].', 'hep-th-0202150-2-15-6': 'Also, in this way, we may write [EQUATION] where [MATH], and [MATH].', 'hep-th-0202150-2-16-0': 'We want to identify the gauge field [MATH] associated with the R-charge.', 'hep-th-0202150-2-16-1': 'The R-symmetry gauge group is at least as big as U(1), and hence it is a natural guess to associate the U(1) fiber of the Sasaki-Einstein manifold [MATH] if not with the R-symmetry group itself, then at least with a subgroup of it.', 'hep-th-0202150-2-16-2': 'We begin with the inspired guess that redefinitions of [MATH] be identified with gauge transformations of the R-symmetry gauge field [MATH]: [EQUATION] where [EQUATION]', 'hep-th-0202150-2-16-3': 'One troublesome issue is that [MATH] in general may not be one.', 'hep-th-0202150-2-16-4': 'To figure out [MATH] and to anchor the analysis, we need a geometric object with a definite R-charge.', 'hep-th-0202150-2-16-5': 'The holomorphic [MATH]-form [MATH] on the noncompact Calabi-Yau cone [MATH] is one such object.', 'hep-th-0202150-2-16-6': 'Because of its association with the superpotential, [MATH] has R-charge 2.', 'hep-th-0202150-2-17-0': 'The functional dependence of [MATH] on [MATH] is very simple: [MATH].', 'hep-th-0202150-2-17-1': 'A proof will follow in section 2.3.', 'hep-th-0202150-2-17-2': 'Because [MATH] has R-charge two, we can directly identify the angular variable [MATH] with a U(1) R-symmetry where now [EQUATION] and [MATH].', 'hep-th-0202150-2-17-3': 'Thus [MATH].', 'hep-th-0202150-2-18-0': 'Including the gauge field [MATH] in the supergravity solution means altering the metric [EQUATION]', 'hep-th-0202150-2-18-1': 'The [MATH] form flux is also altered [EQUATION]', 'hep-th-0202150-2-18-2': 'From this expression, we can calculate [MATH] to first order in [MATH]: [EQUATION]', 'hep-th-0202150-2-18-3': 'We have introduced some notation: [MATH], and [MATH] is the Hodge star in the [MATH] directions.', 'hep-th-0202150-2-18-4': 'This form of [MATH] satisfies the SUGRA equations of motion to linear order in [MATH].', 'hep-th-0202150-2-18-5': 'In particular, [MATH] has been constructed such that [MATH].', 'hep-th-0202150-2-18-6': 'Notice that [MATH] contains a piece proportional to [MATH].', 'hep-th-0202150-2-18-7': 'Hence [MATH] will vanish to linear order in [MATH] provided [MATH], i.e. the field equations for the gauge field [MATH] are satisfied.', 'hep-th-0202150-2-19-0': 'We also may consider the trace of the Einstein equations.', 'hep-th-0202150-2-19-1': 'The modification of the metric ([REF]) causes the Ricci scalar for the full ten dimensional metric to become [EQUATION]', 'hep-th-0202150-2-19-2': "Thus, the trace of Einstein's equations will contain an extra [MATH] term.", 'hep-th-0202150-2-19-3': 'This term causes the backreaction of the [MATH] field strength on the metric.', 'hep-th-0202150-2-19-4': 'We ignore it in this paper since it is a second order effect.', 'hep-th-0202150-2-20-0': 'As a test of our proposal for the U(1) R-symmetry gauge field, let us calculate the R-charge of a hypothetical baryon in this geometry.', 'hep-th-0202150-2-20-1': 'We identify the baryons as D3-branes wrapped on 3-cycles inside [MATH].', 'hep-th-0202150-2-20-2': 'In particular, the D3-brane will completely wrap the U(1) fiber of the Einstein-Sasaki manifold [MATH] and will also wrap a holomorphic curve in the Kahler-Einstein base [MATH].', 'hep-th-0202150-2-20-3': 'The dimension of these baryons is proportional to the volume of the 3-cycle [MATH] times the curvature length [MATH]: [EQUATION]', 'hep-th-0202150-2-20-4': 'One might expect that the volume is more closely related to the mass of the baryon than to the dimension.', 'hep-th-0202150-2-20-5': 'We will see later in section 2.4 that the mass receives a correction and that [MATH] is the correct expression.', 'hep-th-0202150-2-21-0': 'Now these baryons are thought to be chiral primary operators.', 'hep-th-0202150-2-21-1': 'Their R-charge should be a multiplicative constant times the dimension.', 'hep-th-0202150-2-21-2': 'Recall that in a classical Lagrangian for a particle traveling in an electromagnetic field, there is a term of the form [MATH] where [MATH] is the charge of the particle and [MATH] is its four-velocity.', 'hep-th-0202150-2-21-3': 'In the Lagrangian for a probe baryon of this type, there is a Wess-Zumino term of the form [EQUATION] where we have used ([REF]) for [MATH].', 'hep-th-0202150-2-22-0': 'In the case of [MATH], we know independently from the supersymmetry algebra that the dimension times 2/3 is the R-charge.', 'hep-th-0202150-2-22-1': 'The [MATH] in the Wess-Zumino term is exactly where the charge should be in a classical Lagrangian describing the D3-brane dynamics.', 'hep-th-0202150-2-23-0': '## The R-charge for [MATH]', 'hep-th-0202150-2-24-0': 'We now repeat essentially the same analysis for a stack of M2-branes sitting at the tip of a noncompact four complex dimensional cone, [MATH].', 'hep-th-0202150-2-24-1': 'In general, the eleven dimensional metric is [EQUATION] where [MATH] is a Minkowski tensor.', 'hep-th-0202150-2-24-2': 'Note the additional factors of two scaling the Einstein metric.', 'hep-th-0202150-2-24-3': 'These factors are necessary to guarantee that the Ricci scalar of the eleven dimensional metric vanishes and will be crucial later on in fixing the R-charge of the baryons.', 'hep-th-0202150-2-25-0': 'A solution to the SUGRA equations of motion can be obtained by threading the [MATH] space with [MATH] units of [MATH] form flux.', 'hep-th-0202150-2-25-1': 'In particular [EQUATION] where [MATH].', 'hep-th-0202150-2-26-0': 'We specialize to the case where [MATH] is a quasi-regular Einstein-Sasaki space: [EQUATION]', 'hep-th-0202150-2-26-1': 'We keep the same definition of [MATH] and [MATH].', 'hep-th-0202150-2-26-2': 'Namely, [MATH] and the number [MATH] is defined such that [MATH] and [MATH].', 'hep-th-0202150-2-26-3': 'With these definitions, [MATH].', 'hep-th-0202150-2-26-4': 'Also, in this way, we may write [EQUATION] where [MATH].', 'hep-th-0202150-2-27-0': 'Just as in the previous section, redefinitions of [MATH] can be identified with gauge transformations of the R-symmetry gauge field [MATH]: [EQUATION] where [EQUATION]', 'hep-th-0202150-2-27-1': 'Including the gauge field [MATH] in the supergravity solution changes the metric [EQUATION]', 'hep-th-0202150-2-27-2': 'The [MATH] form potential becomes [EQUATION]', 'hep-th-0202150-2-27-3': 'Hence [EQUATION]', 'hep-th-0202150-2-27-4': 'This form of [MATH] satisfies the SUGRA equations of motion to linear order in [MATH].', 'hep-th-0202150-2-27-5': 'The relevant equations for [MATH] are [MATH] and [MATH].', 'hep-th-0202150-2-27-6': 'We have constructed [MATH] such that [MATH].', 'hep-th-0202150-2-27-7': 'Moreover, [MATH] and [MATH] vanish to order [MATH] provided the equations of motion for the gauge field [MATH] are satisfied, i.e. [MATH].', 'hep-th-0202150-2-27-8': 'The modification of the metric ([REF]) causes the Ricci scalar to become [EQUATION]', 'hep-th-0202150-2-27-9': 'From the standpoint of an effective action in [MATH], one can see from this modification another way of deriving the equations of motion for the gauge field, [MATH].', 'hep-th-0202150-2-28-0': 'To calculate the R-charge of a hypothetical baryon in this geometry, we identify the baryons as M5-branes wrapped on 5-cycles inside [MATH].', 'hep-th-0202150-2-28-1': 'The 5-cycle consists of the U(1) fiber and a holomorphic surface inside the Kahler-Einstein base of [MATH].', 'hep-th-0202150-2-28-2': 'The dimension of these baryons is proportional to the volume of the 5-cycle [MATH] times the curvature length [MATH]: [EQUATION]', 'hep-th-0202150-2-28-3': 'In the Lagrangian for a probe baryon of this type, there is also a Wess-Zumino term of the form [EQUATION] where we have used the previous expression for [MATH].', 'hep-th-0202150-2-29-0': 'In the case of [MATH], we expect from the supersymmetry algebra that the R-charge of a chiral primary operator be equal to the dimension.', 'hep-th-0202150-2-29-1': 'As these baryons are chiral primary operators, all is well.', 'hep-th-0202150-2-30-0': '## Proof', 'hep-th-0202150-2-31-0': 'We seek to prove that the holomorphic [MATH]-form [MATH] scales as [MATH] where [MATH] is a coordinate on the U(1) fiber of these non-compact Calabi-Yau cones of complex dimension [MATH].', 'hep-th-0202150-2-31-1': 'The metric on [MATH] ([REF]) is not very natural as it puts [MATH] and [MATH] on an unequal footing.', 'hep-th-0202150-2-31-2': 'It is more natural to think of [MATH] as a fibration of [MATH] over the Kahler-Einstein base [MATH].', 'hep-th-0202150-2-31-3': 'Let [MATH] be the coordinate of the fiber.', 'hep-th-0202150-2-31-4': 'We make the change of variables [EQUATION]', 'hep-th-0202150-2-31-5': 'In these coordinates, the metric becomes [EQUATION]', 'hep-th-0202150-2-31-6': 'The volume form scales as [MATH].', 'hep-th-0202150-2-31-7': 'Note that the volume form is proportional to [MATH].', 'hep-th-0202150-2-31-8': 'Thus, [MATH] scales as [MATH].', 'hep-th-0202150-2-32-0': '## The difference between mass and dimension', 'hep-th-0202150-2-33-0': 'So far we have explicitly shown that the supergravity vector field responsible for the [MATH] charge of a state results from mixing between the rotations along the [MATH] fiber and a compensating gauge transformation.', 'hep-th-0202150-2-33-1': 'The end result is that the D-brane R-charge was identified with the volume of the D-brane for certain BPS D-branes.', 'hep-th-0202150-2-34-0': 'In the supergravity, it seems natural to associate to such a D-brane wrapping the given cycle a mass equal to the volume, and then one finds a discrepancy between the R-charge and the conformal dimension, since [EQUATION] for [MATH].', 'hep-th-0202150-2-34-1': 'We argue that the volume of the D-brane should be identified directly with the conformal dimension, and not with the mass of the particle.', 'hep-th-0202150-2-35-0': 'One reason for this is that the D-brane is a BPS object and we should consider the action of a superparticle in this background, rather than the action of a point particle in AdS space.', 'hep-th-0202150-2-35-1': 'When we consider global AdS coordinates, the hamiltonian associated to time corresponds to the generator of conformal tranformations, and the spectrum is discrete because there is a gravitational potential well at the center of AdS which eliminates the zero modes of the particle and turns them into oscillators.', 'hep-th-0202150-2-36-0': 'The supersymmetries do not commute with the hamiltonian, but they still control the dynamics of the theory.', 'hep-th-0202150-2-36-1': 'This can be seen explicitly in the matrix model presented in [CITATION], where the fermions and bosons are not degenerate, but the ground state energy still cancels.', 'hep-th-0202150-2-36-2': 'We argue that this is generic for these systems.', 'hep-th-0202150-2-36-3': 'A superparticle also has fermionic coordinates.', 'hep-th-0202150-2-36-4': 'It is the zero point energy of these degrees of freedom that cancels the zero point energy of the bosonic degrees of freedom.', 'hep-th-0202150-2-37-0': 'In particular, consider the fluctuations corresponding to the transverse motion of the particle in AdS space.', 'hep-th-0202150-2-38-0': 'The fluctuations have a mass term determined by the [MATH] metric: [EQUATION]', 'hep-th-0202150-2-38-1': 'If we consider a particle at [MATH], we have a timelike geodesic of [MATH].', 'hep-th-0202150-2-38-2': 'To quadratic order in the transverse fluctuations we have [EQUATION]', 'hep-th-0202150-2-38-3': 'There is a gravitational potential in [MATH], and the mass term for the fluctuations in [MATH] is determined by the form of the metric.', 'hep-th-0202150-2-38-4': 'One can also determine this mass term from the algebra of isometries.', 'hep-th-0202150-2-38-5': 'In the radial quantization, acting with derivatives changes the conformal weight by one unit.', 'hep-th-0202150-2-38-6': 'This is the same as acting with one of the raising operators for [MATH] to first order in the quantum system, because the symmetry is non-linearly realized.', 'hep-th-0202150-2-39-0': 'Similarly, if we act with the supersymmetry generators, we change the conformal weight by a factor of 1/2.', 'hep-th-0202150-2-39-1': 'This means that the fermions will have a mass term which is one half that of the bosons.', 'hep-th-0202150-2-39-2': 'There are [MATH] complex supersymmetries, so we have one complex fermion associated to each one of them.', 'hep-th-0202150-2-39-3': 'It is easy to see that the contribution to the conformal weight of each fermion is one half that of the bosons, but there are twice as many fermions.', 'hep-th-0202150-2-39-4': 'Thus one finds that the zero point energy cancels between bosons and fermions, and for supersymmetric states one has [EQUATION]', 'hep-th-0202150-2-39-5': 'The classical conformal weight associated to the state is equal to the value in the quantum theory, namely, the conformal weight of the superparticle is the volume of the D-brane.', 'hep-th-0202150-2-40-0': 'If we integrate out the fermions, they shift the zero point energy of the hamiltonian by [MATH], where [MATH] is the number of real fermions, [MATH] is the mass of each of these fermions, and the remaining [MATH] is the numerical factor associated with zero point energy for a fermionic oscillator.', 'hep-th-0202150-2-40-1': 'As a result, the mass of the particle associated to the ground states is given approximately by [EQUATION]', 'hep-th-0202150-2-40-2': 'This expression agrees with ([REF]) in the limit of large [MATH] exactly as we would expect.', 'hep-th-0202150-2-40-3': 'To first order, integrating out the fermions reproduces the shift between the value of the mass and the conformal weight.', 'hep-th-0202150-2-41-0': '# Moduli space of D-branes and zero-mode quantization', 'hep-th-0202150-2-42-0': 'Classically one often has the freedom to change continuously the way in which a D-brane is wrapped without changing the energy.', 'hep-th-0202150-2-42-1': 'In this section, we will investigate how, through quantization, this freedom of movement allows one to understand the number of ground states of a given dibaryon operator in the field theory side of the correspondence.', 'hep-th-0202150-2-43-0': 'This abstract statement concerning the freedom some dibaryon-like states have to shift their wrapping configurations is particularly easy to understand in the case of [MATH].', 'hep-th-0202150-2-43-1': 'Recall that [MATH] is a U(1) fibration over [MATH].', 'hep-th-0202150-2-43-2': "The simplest dibaryon corresponds to a D3-brane wrapping the U(1) fiber of one of the two [MATH]'s.", 'hep-th-0202150-2-43-3': 'It is clear that, classically, the D3-brane can move continuously along the other [MATH] without changing its energy.', 'hep-th-0202150-2-44-0': 'Let us exhibit the operators of the [MATH] gauge theory dual to these simply wrapped D3-branes.', 'hep-th-0202150-2-44-1': 'Since the fields [MATH], [MATH], carry an index [MATH] in the [MATH] of [MATH] and an index [MATH] in the [MATH] of [MATH], we can construct color-singlet "dibaryon" operators by antisymmetrizing completely with respect to both groups: [EQUATION] where [MATH] is the completely symmetric [MATH] Clebsch-Gordon coefficient corresponding to forming the [MATH] of [MATH] out of [MATH] 2\'s.', 'hep-th-0202150-2-44-2': 'Thus the [MATH] quantum numbers of [MATH] are [MATH].', 'hep-th-0202150-2-44-3': 'Similarly, we can construct "dibaryon" operators which transform as [MATH], [EQUATION]', 'hep-th-0202150-2-44-4': 'The existence of two types of "dibaryon" operators is related on the supergravity side to the fact that the base of the [MATH] bundle is [MATH].', 'hep-th-0202150-2-44-5': 'A D3-brane can wrap either of the two-spheres together with the [MATH] fiber.', 'hep-th-0202150-2-45-0': 'We will use a simplified notation for the dibaryon operators ([REF]) [EQUATION]', 'hep-th-0202150-2-45-1': 'The (super)subindex on the [MATH] refers to the gauge group that the [MATH] symbol carries and whether these are (super)subindices is indicated by the position of the group label.', 'hep-th-0202150-2-45-2': 'The gauge indices of the [MATH] are contracted with the gauge indices of the [MATH] tensors in the same order as they are written.', 'hep-th-0202150-2-46-0': 'In order to account for the [MATH] quantum numbers of the dibaryons, one needs to take into account the proper quantization of the zero modes of the wrapped D3-brane [CITATION].', 'hep-th-0202150-2-46-1': 'The resulting dynamics reduces to motion of a particle on a sphere in the presence of a magnetic field.', 'hep-th-0202150-2-46-2': 'The purpose of this section is to consider the zero mode problem in a more general setting.', 'hep-th-0202150-2-46-3': 'After discussing the conifold case, we will give a more general argument for other Calabi-Yau singularities that will show that one can count the ground states even without exact knowledge of the metric.', 'hep-th-0202150-2-47-0': 'First, we describe this freedom of movement some dibaryons possess in a more precise and useful way.', 'hep-th-0202150-2-47-1': 'If we begin with dibaryon-like states on [MATH], then at each time [MATH], the D-brane worldvolume will be wrapping a holomorphic cycle of [MATH], and the volume of the D-brane will be constant, but its shape will change.', 'hep-th-0202150-2-47-2': 'Since the volume remains constant, there is no potential on these directions, and the moduli space of the D-brane can be captured by the moduli space of holomorphic submanifolds of [MATH] with specific topological quantum numbers, which describe what holomorphic cycle is being wrapped.', 'hep-th-0202150-2-48-0': 'The D-brane is a holomorphic submanifold of codimension [MATH] on [MATH], so it can be associated to a divisor [MATH] on [MATH].', 'hep-th-0202150-2-48-1': 'The D-brane is the zero locus of a global section of the associated line bundle to the divisor [MATH].', 'hep-th-0202150-2-48-2': 'Any linear combination of global holomorphic sections of [MATH] is also a global section, and in this way the moduli space of the D-brane is a projective space [MATH] for some [MATH] (which can be zero, and then the curve is rigid).', 'hep-th-0202150-2-48-3': '[MATH] is given by the number of holomorphic sections of the line bundle associated to [MATH] minus one.', 'hep-th-0202150-2-49-0': 'Thus, to quantize the zero modes of the D-brane, we need the effective action of the D-brane moduli space.', 'hep-th-0202150-2-49-1': 'We consider our action as a nonlinear sigma model whose target space is the moduli space of the D-brane.', 'hep-th-0202150-2-49-2': 'This action will involve a metric on [MATH] (which generically has some singularities) induced from the metric on [MATH].', 'hep-th-0202150-2-49-3': 'Additionally, one can add a line bundle on this moduli space, that is, an effective magnetic field on the target space of the sigma model.', 'hep-th-0202150-2-50-0': 'In the example of the conifold, for a single dibaryon state [MATH], we have the round metric on [MATH] (this is obvious, since we had an [MATH] worth of positions for the D-brane with the corresponding isometry group).', 'hep-th-0202150-2-51-0': 'Now, we want to argue that the magnetic field through this moduli space is [MATH] times the hyperplane bundle on this moduli space.', 'hep-th-0202150-2-51-1': 'Indeed, the WZ term for the D-brane action is [EQUATION]', 'hep-th-0202150-2-51-2': 'Three of the coordinates of the D-brane saturate three of the indices of [MATH], let us say [MATH], and the fourth index is saturated by the time direction in the form [EQUATION]', 'hep-th-0202150-2-51-3': 'So the effective action has a term proportional to the velocity on the moduli space of the D-brane.', 'hep-th-0202150-2-51-4': 'Hence, this WZW term gives rise to an effective magnetic field on the [MATH].', 'hep-th-0202150-2-51-5': 'Notice moreover that this magnetic field is proportional to [MATH], the effective tension of the D-brane.', 'hep-th-0202150-2-51-6': 'For the [MATH] case of the conifold it is easy to show that the normalization of this magnetic field is such that the flux through the [MATH] is [MATH].', 'hep-th-0202150-2-51-7': 'Counting the number of ground states is then done by the index theorem, and in this case we get [MATH].', 'hep-th-0202150-2-51-8': 'Notice that this matches exactly the counting of operators in the equation [REF].', 'hep-th-0202150-2-51-9': 'For the general case we find that the index calculation counts the number of global holomorphic functions of degree [MATH] on [MATH].', 'hep-th-0202150-2-51-10': 'There are [EQUATION] such global sections.', 'hep-th-0202150-2-52-0': 'In the case of [MATH] orbifold without fixed points [CITATION], we get that [MATH].', 'hep-th-0202150-2-52-1': 'The curves of degree one in [MATH] are parametrized by a dual [MATH].', 'hep-th-0202150-2-52-2': 'So in this case we get [MATH] states.', 'hep-th-0202150-2-52-3': 'These can also be counted in the CFT.', 'hep-th-0202150-2-52-4': 'Instead of having two fields [MATH] to make the dibaryon, there are three such fields, which transform in the [MATH] of an [MATH] global symmetry.', 'hep-th-0202150-2-52-5': 'The totally symmetric combination of [MATH] of these objects gives a Hilbert space of the same dimension.', 'hep-th-0202150-2-52-6': 'Moreover there is an extra degeneracy by a factor of three from the fact that the [MATH] space is not simply connected, so there is a possibility of having discrete Wilson lines for the gauge field on the D-brane [CITATION].', 'hep-th-0202150-2-52-7': 'This factor of three is also reflected in the field theory quiver diagram, where there are three different types of dibaryon states.', 'hep-th-0202150-2-53-0': 'Both of these examples show that the counting with [MATH] times the hyperplane bundle of [MATH] seems to give the right match for counting states.', 'hep-th-0202150-2-54-0': "With this in mind, let us return to the conifold case, and let us study a D-brane that wraps once both of the two different [MATH]'s.", 'hep-th-0202150-2-54-1': 'It was argued in [CITATION] that such D-branes should become giant gravitons.', 'hep-th-0202150-2-54-2': 'The counting of holomorphic sections of the line bundle gives us [MATH].', 'hep-th-0202150-2-54-3': 'Indeed, if we consider [MATH] and [MATH] a set of projective coordinates for both of the [MATH], the global sections of this line bundle are given by curves [EQUATION] where we get four possible distinct coefficients [MATH].', 'hep-th-0202150-2-55-0': 'The number of such states is therefore [MATH].', 'hep-th-0202150-2-55-1': 'This is many more states than just the product of two dibaryons, one for [MATH] and one for [MATH].', 'hep-th-0202150-2-55-2': 'Indeed, these factorized states correspond to the curves that factorize into linear terms in the equation [REF].', 'hep-th-0202150-2-55-3': 'But when we have intersecting branes, we have also deformations that are localized at the intersection.', 'hep-th-0202150-2-55-4': 'These are the ones responsible for giving us a larger moduli space.', 'hep-th-0202150-2-56-0': 'In the case where the curve factorizes, the operator is given by [EQUATION]', 'hep-th-0202150-2-56-1': 'Notice now that the total baryon number of this operator is zero, and that the [MATH] appears as many times as the [MATH].', 'hep-th-0202150-2-56-2': 'We can use the identity [EQUATION] where [MATH] sums over all possible permutations of the [MATH], and [MATH] is [MATH] or [MATH] if the permutation is even or odd.', 'hep-th-0202150-2-57-0': 'This procedure lets us eliminate one set of the [MATH] symbols if we want to.', 'hep-th-0202150-2-57-1': 'We see that if we do this, the [MATH] have to be paired with the [MATH] in combinations of the form [MATH].', 'hep-th-0202150-2-57-2': 'There are [MATH] such possible combinations.', 'hep-th-0202150-2-57-3': 'Consider therefore the operators of the form [EQUATION] where each of the entries is given by one of the four possible combinations [MATH].', 'hep-th-0202150-2-57-4': 'This object can be expressed in terms of traces of the [MATH], so it is built out of gravitons, and indeed it should represent a maximal giant graviton.', 'hep-th-0202150-2-57-5': 'We have a symmetric combination of four objects, and the combinatorics of these objects also gives us a total of [MATH] objects.', 'hep-th-0202150-2-57-6': 'In some sense, [MATH] and [MATH] are the analog of the coordinates [MATH] and [MATH].', 'hep-th-0202150-2-58-0': 'Similarly, if we take a state which wraps sphere one twice and sphere two once, we get that [MATH].', 'hep-th-0202150-2-58-1': 'The natural objects to consider are of the form [MATH].', 'hep-th-0202150-2-58-2': 'However, we have to remember the [MATH]-terms, so in this expression we get a field which is symmetric in [MATH].', 'hep-th-0202150-2-58-3': 'The total number of objects that we can count is [MATH], and we can build operators of the form [EQUATION]', 'hep-th-0202150-2-58-4': 'Notice that this operator corresponds to some form of a coherent state of excitations of the dibaryon ([REF]).', 'hep-th-0202150-2-58-5': 'From the supergravity, we expect such a result because this new baryon-like state is wrapping the same topological cycle as the old dibaryon.', 'hep-th-0202150-2-59-0': 'For more general wrapping cases of baryon number two or higher, it is more difficult to match the states.', 'hep-th-0202150-2-59-1': 'Since we cannot get rid of all but one of the pairs of [MATH] symbols, it is not true that there is an easy description of the state in terms of [MATH]-symmetrized products of similar objects (the [MATH] above).', 'hep-th-0202150-2-59-2': 'It would be interesting to obtain a more thorough description of these other states.', 'hep-th-0202150-2-60-0': '# BPS fluctuations of dibaryons', 'hep-th-0202150-2-61-0': 'In this section we show that there exist BPS excitations of the dibaryon operators, that is, operators which carry baryon number and have higher conformal weight than the volume of the associated 3-cycle.', 'hep-th-0202150-2-61-1': 'Since in this particular section we will be interested in computing explicitly the fluctuation spectra of D-branes, we need to resort to writing models for specific examples where the metric and the CFT are both known.', 'hep-th-0202150-2-61-2': 'We are moreover interested in situations where we have only [MATH] supersymmetry without singularities.', 'hep-th-0202150-2-61-3': 'In these models the [MATH] symmetry is strictly [MATH] and the chiral fields are holomorphic.', 'hep-th-0202150-2-61-4': 'Two of these models are particularly simple.', 'hep-th-0202150-2-61-5': 'These are D-branes at the conifold [CITATION] and D-branes at the orbifold [MATH] [CITATION] without fixed points.', 'hep-th-0202150-2-61-6': 'We will deal in this section with the particular case of the conifold.', 'hep-th-0202150-2-62-0': 'Let us consider for simplicity the state with maximum [MATH] of the first [MATH]: [EQUATION]', 'hep-th-0202150-2-62-1': 'To construct excited dibaryons we can replace one of the [MATH] by any other chiral field which transforms in the same representation of the gauge groups.', 'hep-th-0202150-2-62-2': 'One possibility is to replace [MATH], where the two gauge indices of [MATH] are contracted separately with the [MATH] and with the [MATH], following the rules for matrix multiplication, namely [EQUATION]', 'hep-th-0202150-2-62-3': 'This dibaryon-like state is a chiral field up to F-terms because it is a gauge invariant holomorphic polynomial in the chiral superfields.', 'hep-th-0202150-2-62-4': 'There will be', 'hep-th-0202150-2-63-0': 'a chiral primary state with the same quantum numbers as the above operator that is a linear combination of operators of this type.', 'hep-th-0202150-2-63-1': 'Remember that operators that differ by F-terms are equivalent as elements of the chiral ring, but in the conformal field theory they are different and only one particular linear combination is protected.', 'hep-th-0202150-2-64-0': 'The operator resulting after the replacement factorizes into the original dibaryon and a single-trace operator: [EQUATION]', 'hep-th-0202150-2-64-1': 'The factorization suggests that this excitation of a dibaryon can be represented as a graviton fluctuation in presence of the original dibaryon.', 'hep-th-0202150-2-65-0': 'Not all excitations factorize in this way, however.', 'hep-th-0202150-2-65-1': 'For example, consider replacing [MATH].', 'hep-th-0202150-2-65-2': 'One might ask whether this new operator can be written as a product of the original dibaryon [MATH] and a meson-like operator of the form [MATH].', 'hep-th-0202150-2-65-3': 'The answer is no.', 'hep-th-0202150-2-66-0': 'To see why, it is easier to go to a generic point in the moduli space of vacua of the theory where we can set [MATH] by gauge transformations, and thus we establish an isomorphism between the indices of the two gauge groups.', 'hep-th-0202150-2-66-1': 'The operator above becomes [MATH], and if [MATH] is large enough, there are no relations amongst traces of a low number of fields.', 'hep-th-0202150-2-66-2': 'In other words, we cannot write the operator as [MATH].', 'hep-th-0202150-2-66-3': 'Since this new operator cannot be factored it has to be interpreted as a single particle state in AdS.', 'hep-th-0202150-2-66-4': 'Since the operator also carries baryon number one,', 'hep-th-0202150-2-67-0': 'the natural conclusion is that the one-particle state is a BPS excitation of the wrapped D3-brane in the dual string theory.', 'hep-th-0202150-2-68-0': 'Let us now study BPS fluctuations of the wrapped D3-brane', 'hep-th-0202150-2-69-0': 'in the supergravity approximation.', 'hep-th-0202150-2-69-1': 'The DBI action is a good approximation in the limit of weak string coupling and weak curvature of the D-brane.', 'hep-th-0202150-2-69-2': "These conditions are met in the limit of large 't Hooft coupling.", 'hep-th-0202150-2-69-3': 'In particular, we will compute the spectrum of quadratic fluctuations of this DBI action.', 'hep-th-0202150-2-70-0': 'We will return to the field theory later to make the correspondence of states and quantum numbers more precise.', 'hep-th-0202150-2-71-0': 'First, we set up the DBI computation in the right coordinate system.', 'hep-th-0202150-2-71-1': 'The full ten dimensional metric is naturally [MATH], [EQUATION]', 'hep-th-0202150-2-71-2': 'For convenience, we have chosen the time [MATH] direction to be a Killing vector.', 'hep-th-0202150-2-71-3': 'The radius of curvature is [MATH].', 'hep-th-0202150-2-71-4': 'The metric of [MATH], the base of the conifold, is given by [EQUATION] with [MATH], [MATH].', 'hep-th-0202150-2-72-0': 'We will keep [MATH] as variables in our computation for consistency checks.', 'hep-th-0202150-2-73-0': 'The dibaryon is chosen to wrap the cycle defined by [MATH] constant.', 'hep-th-0202150-2-73-1': 'This configuration is invariant under rotations of the sphere wrapped by the D3-brane, but it is not invariant under the [MATH] associated to the [MATH] coordinates.', 'hep-th-0202150-2-73-2': 'The induced metric on the dibaryon is thus [EQUATION]', 'hep-th-0202150-2-73-3': 'It is convenient to make a change of variables [MATH], so that [EQUATION]', 'hep-th-0202150-2-73-4': 'In these variables, the determinant of the spatial part of the metric [EQUATION] is constant.', 'hep-th-0202150-2-73-5': "The variables' range is given by [MATH], [MATH] and [MATH].", 'hep-th-0202150-2-73-6': 'The volume of the wrapped manifold is thus [MATH].', 'hep-th-0202150-2-73-7': 'In [CITATION], it was noted that this volume times the tension of the D3-brane should be a good approximation of the mass of the corresponding dibaryon.', 'hep-th-0202150-2-73-8': 'As we argued in section 2.4, this volume is directly proportional to the dimension [MATH] of the dibaryonic operator.', 'hep-th-0202150-2-74-0': 'Indeed, as the conformal dimension of each [MATH] and [MATH] is 3/4, one finds that [MATH] is exactly equal to [MATH].', 'hep-th-0202150-2-75-0': 'We return now to calculating the excitation spectrum of the dibaryon.', 'hep-th-0202150-2-76-0': 'We need to be careful with single valued functions on this space.', 'hep-th-0202150-2-76-1': 'This squashed [MATH] can be thought of, essentially, as the group manifold [MATH].', 'hep-th-0202150-2-76-2': 'The coordinates [MATH] are the Euler angles.', 'hep-th-0202150-2-76-3': 'One might have thought that the points [MATH] and [MATH] were equivalent.', 'hep-th-0202150-2-76-4': 'The insight from [MATH] lets us correct this mistake.', 'hep-th-0202150-2-76-5': 'Equivalent points on [MATH] have [EQUATION] where [MATH] and [MATH] are two points on [MATH].', 'hep-th-0202150-2-77-0': 'We want to find the normal modes of oscillation of the wrapped D3-brane around the solution corresponding to some fixed world-line in [MATH] and some fixed [MATH] and [MATH] on the transverse [MATH].', 'hep-th-0202150-2-77-1': 'The fluctuations along the transverse [MATH] are the most interesting: they change the [MATH] quantum numbers and are most usefully compared with the chiral primary states in the field theory.', 'hep-th-0202150-2-77-2': 'The transverse fluctuations along the [MATH] are considered briefly afterward.', 'hep-th-0202150-2-77-3': 'Supersymmetry relates the gauge field degrees of freedom and fermions on the D-brane to the scalar modes considered here.', 'hep-th-0202150-2-77-4': 'There is no mixing between the between the different modes at quadratic order in the fluctuations, and we ignore the vector and spinor modes in what follows.', 'hep-th-0202150-2-78-0': 'Because the fluctuations around [MATH] and [MATH] are by definition small, it is appropriate to treat the [MATH] parametrized by these coordinates as a flat [MATH].', 'hep-th-0202150-2-78-1': 'For example, one may take [MATH].', 'hep-th-0202150-2-78-2': 'Then the fluctuation coordinates on [MATH] can be taken to be [MATH] and [MATH].', 'hep-th-0202150-2-79-0': 'The connection term in the metric on [MATH] becomes [MATH] and the Kahler form on the transverse [MATH] becomes [MATH].', 'hep-th-0202150-2-79-1': 'To say the same thing in a different way, we are interested in the D-brane action to quadratic order in fluctuations.', 'hep-th-0202150-2-79-2': 'We only need to consider terms of up to order [MATH], [MATH], or [MATH], and we neglect everything else which is higher order in the fluctuations [MATH].', 'hep-th-0202150-2-80-0': 'Notice that the ten dimensional metric ([REF]) has a term of the form [MATH], where the [MATH] are the [MATH] and [MATH] coordinates.', 'hep-th-0202150-2-80-1': 'Indeed, in the previous section, we saw that we could add a more general term of this form depending on the [MATH] coordinates.', 'hep-th-0202150-2-80-2': 'This perturbation corresponded to a gauge field carrying the [MATH]-charge.', 'hep-th-0202150-2-80-3': 'Recall that changing the coordinate [MATH] to [MATH] does not change the periodicity of the variable [MATH]; but it does change the form of the vector [MATH] by a gauge transformation, [MATH].', 'hep-th-0202150-2-80-4': 'So in writing the metric, the invariant quantity is the field strength of [MATH].', 'hep-th-0202150-2-80-5': 'We are in this way free to add a term of the form [MATH] to [MATH], changing [MATH] and making the connection term simpler.', 'hep-th-0202150-2-81-0': 'The Dirac-Born-Infeld acion is given by [EQUATION] with [MATH] the tension of the brane.', 'hep-th-0202150-2-81-1': 'Because [MATH] appear only quadratically in the metric, we can study the quadratic fluctuations in [MATH] by doing a first order variation in the metric [EQUATION] where [MATH] is the second order contribution from the fluctuations in [MATH] and we define [MATH] to be the inverse of [MATH].', 'hep-th-0202150-2-81-2': 'We can also do the same with the transverse fluctuations along the [MATH], but we will leave those for later.', 'hep-th-0202150-2-82-0': 'With the coordinates chosen [MATH] is independent of the fluctuations [MATH] and also independent of the D3-brane coordinates.', 'hep-th-0202150-2-82-1': 'As a result, the wave equation on the D-brane worldvolume is simplified.', 'hep-th-0202150-2-82-2': 'We fix the diffeomorphism invariance of the DBI action by locking the internal brane coordinates to the background coordinates [MATH].', 'hep-th-0202150-2-82-3': 'This locking corresponds to choosing a physical gauge.', 'hep-th-0202150-2-83-0': 'We take [MATH].', 'hep-th-0202150-2-83-1': 'If the brane is not moving then [EQUATION]', 'hep-th-0202150-2-83-2': 'The inverse matrix is [EQUATION]', 'hep-th-0202150-2-83-3': 'We may choose a gauge where [MATH] has a piece in the [MATH], [MATH], [MATH] and [MATH] directions.', 'hep-th-0202150-2-83-4': 'More specifically, we will choose a gauge which is well defined at the north pole of the sphere parametrized by [MATH] and [MATH]: [EQUATION]', 'hep-th-0202150-2-83-5': 'We want to expand the probe brane action to quadratic order in fluctuations in [MATH] and [MATH].', 'hep-th-0202150-2-83-6': 'At this order, the four form [MATH] is approximately [EQUATION]', 'hep-th-0202150-2-83-7': 'Putting the various pieces together, we obtain that [EQUATION]', 'hep-th-0202150-2-83-8': 'The first term gives rise to the standard laplacian on the three-sphere, while the second term gives rise to mixing terms between [MATH] that are only first order in derivatives.', 'hep-th-0202150-2-83-9': 'An additional mixing term comes from ([REF]).', 'hep-th-0202150-2-83-10': 'These three contributions give us the effective Lagrangian density to quadratic order in fluctuations.', 'hep-th-0202150-2-83-11': '[EQUATION]', 'hep-th-0202150-2-83-12': 'The non-trivial elements of [MATH] are [EQUATION]', 'hep-th-0202150-2-83-13': 'From now on, we will call [MATH] simply [MATH].', 'hep-th-0202150-2-83-14': 'The spatial mixing term [MATH] is explicitly given by [EQUATION]', 'hep-th-0202150-2-83-15': 'The equations of motion for the fluctuations are given by [EQUATION]', 'hep-th-0202150-2-83-16': 'Notice that the elements of [MATH] are independent of the variables with respect to which we are taking derivatives.', 'hep-th-0202150-2-83-17': 'As a result, the second and third terms in the equation above are actually equal and we have [EQUATION]', 'hep-th-0202150-2-83-18': 'Taking the combinations [MATH] these equations become [EQUATION]', 'hep-th-0202150-2-83-19': 'Now, we see that [MATH] so the terms with derivatives with respect to [MATH] cancel.', 'hep-th-0202150-2-83-20': 'This is just as expected, since the result should be invariant under the [MATH] of isometries of the squashed [MATH].', 'hep-th-0202150-2-83-21': 'Thus far we are getting a consistent picture.', 'hep-th-0202150-2-84-0': 'The terms with [MATH] are given by the combination [EQUATION] which is a constant coefficient.', 'hep-th-0202150-2-85-0': 'Now we can use the separation of variables [EQUATION] to obtain a differential equation for [MATH].', 'hep-th-0202150-2-85-1': 'Note that for the [MATH] to be single valued, the condition ([REF]) implies that [MATH] and [MATH] are either both integer or both half-integer.', 'hep-th-0202150-2-85-2': 'The remaining differential equation for the [MATH] is given by [EQUATION]', 'hep-th-0202150-2-85-3': 'Let us begin by analyzing the behavior of the solution in the limit [MATH].', 'hep-th-0202150-2-85-4': 'In this limit, the differential equation becomes [EQUATION]', 'hep-th-0202150-2-85-5': 'The solution to this equation is clearly a power of [MATH], [MATH].', 'hep-th-0202150-2-85-6': 'Thus [EQUATION]', 'hep-th-0202150-2-85-7': 'The energy should be real for all allowed values of [MATH] and [MATH].', 'hep-th-0202150-2-85-8': 'Notice that the energy has the right dependence in terms of the [MATH] quantum numbers to be associated with the velocity on a group manifold.', 'hep-th-0202150-2-86-0': 'Only certain values of [MATH] are allowed because the fluctuations must be well-behaved at [MATH].', 'hep-th-0202150-2-87-0': 'The differential equation ([REF]) can be solved in terms of a hypergeometric function [EQUATION] where [MATH], [MATH], [MATH], [MATH], and [MATH] depend on the quantum numbers [MATH], [MATH], and [MATH].', 'hep-th-0202150-2-87-1': 'The north and south pole of the [MATH], [MATH], correspond to the singular points [MATH] and [MATH] of the hypergeometric function.', 'hep-th-0202150-2-87-2': 'For the fluctuations to vanish at [MATH], [MATH] must be a non-negative integer.', 'hep-th-0202150-2-88-0': 'For the [MATH] which contribute to the [MATH] charge in the same direction as the unexcited D3-brane, the choice [MATH] corresponds to a BPS state; we have [EQUATION]', 'hep-th-0202150-2-88-1': 'Indeed, the BPS states should have the lowest possible dimension for a given R-charge.', 'hep-th-0202150-2-88-2': 'The states [MATH] meet this condition.', 'hep-th-0202150-2-88-3': 'From the periodicity of [MATH], [MATH] can be a half integer, and when [MATH] is a half integer so is [MATH].', 'hep-th-0202150-2-89-0': 'This energy spectrum means that the contribution of these BPS states to the energy is quantized in units of [MATH].', 'hep-th-0202150-2-90-0': '[MATH] is also exactly the change in the conformal dimension of the chiral operators once [MATH] is substituted for an [MATH] in the antisymmetric product ([REF]).', 'hep-th-0202150-2-90-1': 'These modes match the result from conformal field theory.', 'hep-th-0202150-2-91-0': 'Also, the transformation under the [MATH] that rotates [MATH] is in agreement.', 'hep-th-0202150-2-91-1': 'The unexcited dibaryon was a singlet while the excited one acquires spin [MATH].', 'hep-th-0202150-2-91-2': 'Indeed, for each value of [MATH] there is a unique irreducible representation of [MATH] associated to it.', 'hep-th-0202150-2-91-3': 'This means that the quantum of [MATH] charge proportional to [MATH] should be associated to a spin [MATH] state for [MATH].', 'hep-th-0202150-2-91-4': "In the conformal field theory, these states result from choosing to replace one of the [MATH] in the [MATH] function by, for example, [MATH], where we have [MATH]'s inside the matrix.", 'hep-th-0202150-2-91-5': 'Using the F-term equations of motion, it can be seen that these states are totally symmetric with respect to exchange of the [MATH] variables.', 'hep-th-0202150-2-91-6': "Since the [MATH] carry spin [MATH] under the [MATH], these states with [MATH]'s form a totally symmetric representation of [MATH] with spin [MATH].", 'hep-th-0202150-2-91-7': 'Therefore, we can match the [MATH] quantum numbers of the states to the supergravity.', 'hep-th-0202150-2-92-0': 'We have not, however, determined the transformation property of the excited dibaryons under the [MATH] that rotates [MATH].', 'hep-th-0202150-2-92-1': 'This is more difficult since we need to consider the coupling of the fluctuation fields with the zero-mode dynamics.', 'hep-th-0202150-2-92-2': 'Since we have not determined the full [MATH] quantum numbers, we cannot make a precise determination of the dual gauge theory operators.', 'hep-th-0202150-2-92-3': 'In particular, it would be interesting to see if the factorized operators of the type ([REF]) are necessary to match the spectrum of the fluctuating probe D3-brane.', 'hep-th-0202150-2-92-4': 'We leave this interesting question for the future.', 'hep-th-0202150-2-93-0': 'For the transverse motion in [MATH] there is no mixing of the directions, so we get four scalars and their energies are given by [EQUATION] with [MATH] the mass of these states.', 'hep-th-0202150-2-93-1': 'This mass term comes because the space-time is curved and to second order [MATH]; the AdS excitation feels a gravitational potential.', 'hep-th-0202150-2-93-2': 'For [MATH] the above expression should be a perfect square for all [MATH] (it is a superpartner of the other BPS states), so we should have [MATH], or equivalently [MATH].', 'hep-th-0202150-2-93-3': 'This relation can be checked explicitly from the metric.', 'hep-th-0202150-2-93-4': 'It follows that [MATH] differs by [MATH] from the previous energy.', 'hep-th-0202150-2-93-5': 'Indeed, we expect excitations in the AdS directions to correspond to introducing covariant derivatives for the fields [MATH] or [MATH] on the field theory side.', 'hep-th-0202150-2-93-6': 'Covariant derivatives have conformal weight one.', 'hep-th-0202150-2-93-7': 'This is an alternative check that the normalization of the [MATH] field we chose is correct, since it predicts that the splitting between the energies of the modes is compatible with the spacetime symmetries.', 'hep-th-0202150-2-94-0': 'Notice that the dibaryon state on the gravity side has a Fock space worth of excitations.', 'hep-th-0202150-2-94-1': 'In the field theory, this Fock space can be reproduced as well.', 'hep-th-0202150-2-94-2': 'To insert one quantum, we took one of the [MATH] and replaced it with [MATH].', 'hep-th-0202150-2-94-3': 'To put many quanta, we replace many of the [MATH] by the [MATH] combinations.', 'hep-th-0202150-2-94-4': 'The fact that we have a Fock space of identical particles, as opposed to distinguishable particles, comes from the permutation symmetry of the [MATH] symbols.', 'hep-th-0202150-2-94-5': 'For fermionic excitations we need to remember the [MATH] signs when we exchange the fermionic insertions in the operator.', 'hep-th-0202150-2-95-0': 'We conclude that we can match, at least schematically, the dibaryon state with any number of BPS open strings that excite it.', 'hep-th-0202150-2-95-1': 'We have not carried out the complete matching since we have not determined the [MATH] quantum numbers of the excited D3-branes, which remains an interesting problem for future work.', 'hep-th-0202150-2-95-2': 'We believe that the construction of non-BPS states can be carried out using similar methods to the ones used in [CITATION] for closed strings, but this construction is beyond the scope of the present paper.', 'hep-th-0202150-2-96-0': '# Discussion', 'hep-th-0202150-2-97-0': 'In this paper we studied in some detail the correspondence between D3-branes wrapping 3-spheres inside [MATH] and baryon-type operators of the dual [MATH] gauge theory.', 'hep-th-0202150-2-97-1': 'By calculating the [MATH] charge of the D3-branes and their collective coordinate energies we have new evidence that the operator identification proposed in [CITATION] is correct.', 'hep-th-0202150-2-97-2': 'Furthermore, we showed that there exist BPS excitations of wrapped D3-branes and suggested the chiral operators dual to them.', 'hep-th-0202150-2-97-3': 'Our results provide new evidence that the duality between gauge invariant operators of a superconformal gauge theory and states of string theory on [MATH] extends to operators whose dimensions grow as [MATH], the number of colors.', 'hep-th-0202150-2-97-4': 'It is clear that this same procedure to generate operators should work in other examples related to different quiver theories and that one can generically find BPS excitations of baryon-like operators.', 'hep-th-0202150-2-98-0': 'There is another such class of operators, similar to the ones we have considered.', 'hep-th-0202150-2-98-1': 'It is related to the "giant graviton" effect [CITATION].', 'hep-th-0202150-2-98-2': 'It has been observed that for modes whose angular momentum on [MATH] is of order [MATH] the single-trace description of the dual gauge theory operators breaks down [CITATION].', 'hep-th-0202150-2-98-3': 'Instead, the correct description is in terms of subdeterminants of elementary fields, which for maximum angular momentum become determinants similar to the dibaryon operators we have considered.', 'hep-th-0202150-2-98-4': 'On the string theory side, the modes whose angular momentum is of order [MATH] blow up into D3-branes on [MATH].', 'hep-th-0202150-2-98-5': 'This is another manifestation of the fact that D-branes, and therefore string theory, are crucial for describing gauge invariant operators whose dimensions grow as [MATH].', 'hep-th-0202150-2-98-6': "Thus, string theory is necessary to complete the state/operator map even at large 't Hooft coupling.", 'hep-th-0202150-2-98-7': 'Along these lines, it is interesting to consider the BPS excitations of the giant gravitons and to construct the dual operators.', 'hep-th-0202150-2-98-8': 'Some results on this are presented in [CITATION].', 'hep-th-0202150-2-98-9': 'We believe that the giant graviton case is similar to our study of topologically stable wrapped D3-branes.', 'hep-th-0202150-2-99-0': "Very recently, a much more dramatic demonstration of the stringy nature of the AdS/CFT duality at large 't Hooft coupling was presented in [CITATION].", 'hep-th-0202150-2-99-1': 'The insight of this paper is to focus on states whose angular momentum [MATH] on [MATH] scales as [MATH].', 'hep-th-0202150-2-99-2': 'It was shown that there exists a class of operators whose [MATH] stays finite in this limit.', 'hep-th-0202150-2-99-3': 'These states are in one-to-one correspondence with all the closed string states on a RR-charged pp-wave background, including all the massive string states.', 'hep-th-0202150-2-99-4': 'We believe that the work we have presented on the BPS excitations of D3-branes constitutes a first step towards studying the open string states in a similar setting.', 'hep-th-0202150-2-99-5': 'An unexcited D3-brane is described by the basic dibaryon operators ([REF]) and ([REF]).', 'hep-th-0202150-2-99-6': 'Thus, these operators describe the open string vacuum.', 'hep-th-0202150-2-99-7': 'The more complicated operators discussed in section 4 correspond to the BPS states of the open string.', 'hep-th-0202150-2-99-8': 'The non-BPS open string states can be constructed along the lines of [CITATION], but this construction is beyond the scope of the present paper.', 'hep-th-0202150-2-99-9': 'The basic issue in question is the proper understanding of the non-planar diagrams.', 'hep-th-0202150-2-99-10': 'When the operator dimension scales as [MATH], it is too high for the planar approximation to be valid.'}

[['hep-th-0202150-1-78-0', 'hep-th-0202150-2-77-0'], ['hep-th-0202150-1-78-1', 'hep-th-0202150-2-77-1'], ['hep-th-0202150-1-78-3', 'hep-th-0202150-2-77-3'], ['hep-th-0202150-1-78-4', 'hep-th-0202150-2-77-4'], ['hep-th-0202150-1-29-0', 'hep-th-0202150-2-29-0'], ['hep-th-0202150-1-29-1', 'hep-th-0202150-2-29-1'], ['hep-th-0202150-1-13-0', 'hep-th-0202150-2-13-0'], ['hep-th-0202150-1-101-0', 'hep-th-0202150-2-97-0'], ['hep-th-0202150-1-101-1', 'hep-th-0202150-2-97-1'], ['hep-th-0202150-1-101-2', 'hep-th-0202150-2-97-2'], ['hep-th-0202150-1-101-3', 'hep-th-0202150-2-97-3'], ['hep-th-0202150-1-101-4', 'hep-th-0202150-2-97-4'], ['hep-th-0202150-1-3-0', 'hep-th-0202150-2-3-0'], ['hep-th-0202150-1-3-1', 'hep-th-0202150-2-3-1'], ['hep-th-0202150-1-3-2', 'hep-th-0202150-2-3-2'], ['hep-th-0202150-1-3-3', 'hep-th-0202150-2-3-3'], ['hep-th-0202150-1-3-4', 'hep-th-0202150-2-3-4'], ['hep-th-0202150-1-3-5', 'hep-th-0202150-2-3-5'], ['hep-th-0202150-1-24-0', 'hep-th-0202150-2-24-0'], ['hep-th-0202150-1-24-1', 'hep-th-0202150-2-24-1'], ['hep-th-0202150-1-24-2', 'hep-th-0202150-2-24-2'], ['hep-th-0202150-1-24-3', 'hep-th-0202150-2-24-3'], ['hep-th-0202150-1-38-0', 'hep-th-0202150-2-38-0'], ['hep-th-0202150-1-38-1', 'hep-th-0202150-2-38-1'], ['hep-th-0202150-1-38-2', 'hep-th-0202150-2-38-2'], ['hep-th-0202150-1-38-3', 'hep-th-0202150-2-38-3'], ['hep-th-0202150-1-38-4', 'hep-th-0202150-2-38-4'], ['hep-th-0202150-1-38-5', 'hep-th-0202150-2-38-5'], ['hep-th-0202150-1-38-6', 'hep-th-0202150-2-38-6'], ['hep-th-0202150-1-7-0', 'hep-th-0202150-2-7-0'], ['hep-th-0202150-1-7-1', 'hep-th-0202150-2-7-1'], ['hep-th-0202150-1-7-2', 'hep-th-0202150-2-7-2'], ['hep-th-0202150-1-7-3', 'hep-th-0202150-2-7-3'], ['hep-th-0202150-1-49-0', 'hep-th-0202150-2-49-0'], ['hep-th-0202150-1-49-1', 'hep-th-0202150-2-49-1'], ['hep-th-0202150-1-49-2', 'hep-th-0202150-2-49-2'], ['hep-th-0202150-1-49-3', 'hep-th-0202150-2-49-3'], ['hep-th-0202150-1-22-0', 'hep-th-0202150-2-22-0'], ['hep-th-0202150-1-22-1', 'hep-th-0202150-2-22-1'], ['hep-th-0202150-1-79-0', 'hep-th-0202150-2-78-0'], ['hep-th-0202150-1-79-1', 'hep-th-0202150-2-78-1'], ['hep-th-0202150-1-79-2', 'hep-th-0202150-2-78-2'], ['hep-th-0202150-1-8-0', 'hep-th-0202150-2-8-0'], ['hep-th-0202150-1-8-1', 'hep-th-0202150-2-8-1'], ['hep-th-0202150-1-8-2', 'hep-th-0202150-2-8-2'], ['hep-th-0202150-1-8-3', 'hep-th-0202150-2-8-3'], ['hep-th-0202150-1-53-0', 'hep-th-0202150-2-53-0'], ['hep-th-0202150-1-98-0', 'hep-th-0202150-2-94-0'], ['hep-th-0202150-1-98-1', 'hep-th-0202150-2-94-1'], ['hep-th-0202150-1-98-2', 'hep-th-0202150-2-94-2'], ['hep-th-0202150-1-98-3', 'hep-th-0202150-2-94-3'], ['hep-th-0202150-1-98-4', 'hep-th-0202150-2-94-4'], ['hep-th-0202150-1-98-5', 'hep-th-0202150-2-94-5'], ['hep-th-0202150-1-44-1', 'hep-th-0202150-2-44-1'], ['hep-th-0202150-1-44-2', 'hep-th-0202150-2-44-2'], ['hep-th-0202150-1-44-3', 'hep-th-0202150-2-44-3'], ['hep-th-0202150-1-44-4', 'hep-th-0202150-2-44-4'], ['hep-th-0202150-1-44-5', 'hep-th-0202150-2-44-5'], ['hep-th-0202150-1-34-0', 'hep-th-0202150-2-34-0'], ['hep-th-0202150-1-28-0', 'hep-th-0202150-2-28-0'], ['hep-th-0202150-1-28-1', 'hep-th-0202150-2-28-1'], ['hep-th-0202150-1-28-2', 'hep-th-0202150-2-28-2'], ['hep-th-0202150-1-28-3', 'hep-th-0202150-2-28-3'], ['hep-th-0202150-1-102-0', 'hep-th-0202150-2-98-0'], ['hep-th-0202150-1-102-1', 'hep-th-0202150-2-98-1'], ['hep-th-0202150-1-102-2', 'hep-th-0202150-2-98-2'], ['hep-th-0202150-1-102-3', 'hep-th-0202150-2-98-3'], ['hep-th-0202150-1-102-4', 'hep-th-0202150-2-98-4'], ['hep-th-0202150-1-102-5', 'hep-th-0202150-2-98-5'], ['hep-th-0202150-1-102-6', 'hep-th-0202150-2-98-6'], ['hep-th-0202150-1-102-7', 'hep-th-0202150-2-98-7'], ['hep-th-0202150-1-102-8', 'hep-th-0202150-2-98-8'], ['hep-th-0202150-1-102-9', 'hep-th-0202150-2-98-9'], ['hep-th-0202150-1-25-0', 'hep-th-0202150-2-25-0'], ['hep-th-0202150-1-25-1', 'hep-th-0202150-2-25-1'], ['hep-th-0202150-1-39-0', 'hep-th-0202150-2-39-0'], ['hep-th-0202150-1-39-1', 'hep-th-0202150-2-39-1'], ['hep-th-0202150-1-39-2', 'hep-th-0202150-2-39-2'], ['hep-th-0202150-1-39-3', 'hep-th-0202150-2-39-3'], ['hep-th-0202150-1-39-4', 'hep-th-0202150-2-39-4'], ['hep-th-0202150-1-39-5', 'hep-th-0202150-2-39-5'], ['hep-th-0202150-1-59-0', 'hep-th-0202150-2-59-0'], ['hep-th-0202150-1-59-1', 'hep-th-0202150-2-59-1'], ['hep-th-0202150-1-45-0', 'hep-th-0202150-2-45-0'], ['hep-th-0202150-1-45-1', 'hep-th-0202150-2-45-1'], ['hep-th-0202150-1-45-2', 'hep-th-0202150-2-45-2'], ['hep-th-0202150-1-36-0', 'hep-th-0202150-2-36-0'], ['hep-th-0202150-1-36-1', 'hep-th-0202150-2-36-1'], ['hep-th-0202150-1-36-3', 'hep-th-0202150-2-36-3'], ['hep-th-0202150-1-36-4', 'hep-th-0202150-2-36-4'], ['hep-th-0202150-1-2-0', 'hep-th-0202150-2-2-0'], ['hep-th-0202150-1-2-1', 'hep-th-0202150-2-2-1'], ['hep-th-0202150-1-2-2', 'hep-th-0202150-2-2-2'], ['hep-th-0202150-1-2-3', 'hep-th-0202150-2-2-3'], ['hep-th-0202150-1-2-4', 'hep-th-0202150-2-2-4'], ['hep-th-0202150-1-2-5', 'hep-th-0202150-2-2-5'], ['hep-th-0202150-1-2-6', 'hep-th-0202150-2-2-6'], ['hep-th-0202150-1-2-7', 'hep-th-0202150-2-2-7'], ['hep-th-0202150-1-87-0', 'hep-th-0202150-2-84-0'], ['hep-th-0202150-1-94-0', 'hep-th-0202150-2-90-0'], ['hep-th-0202150-1-94-1', 'hep-th-0202150-2-90-1'], ['hep-th-0202150-1-99-0', 'hep-th-0202150-2-95-0'], ['hep-th-0202150-1-99-2', 'hep-th-0202150-2-95-2'], ['hep-th-0202150-1-46-0', 'hep-th-0202150-2-46-0'], ['hep-th-0202150-1-46-1', 'hep-th-0202150-2-46-1'], ['hep-th-0202150-1-46-2', 'hep-th-0202150-2-46-2'], ['hep-th-0202150-1-46-3', 'hep-th-0202150-2-46-3'], ['hep-th-0202150-1-21-0', 'hep-th-0202150-2-21-0'], ['hep-th-0202150-1-21-1', 'hep-th-0202150-2-21-1'], ['hep-th-0202150-1-21-2', 'hep-th-0202150-2-21-2'], ['hep-th-0202150-1-21-3', 'hep-th-0202150-2-21-3'], ['hep-th-0202150-1-17-0', 'hep-th-0202150-2-17-0'], ['hep-th-0202150-1-17-1', 'hep-th-0202150-2-17-1'], ['hep-th-0202150-1-17-2', 'hep-th-0202150-2-17-2'], ['hep-th-0202150-1-20-0', 'hep-th-0202150-2-20-0'], ['hep-th-0202150-1-20-1', 'hep-th-0202150-2-20-1'], ['hep-th-0202150-1-20-2', 'hep-th-0202150-2-20-2'], ['hep-th-0202150-1-20-3', 'hep-th-0202150-2-20-3'], ['hep-th-0202150-1-20-4', 'hep-th-0202150-2-20-4'], ['hep-th-0202150-1-20-5', 'hep-th-0202150-2-20-5'], ['hep-th-0202150-1-43-0', 'hep-th-0202150-2-43-0'], ['hep-th-0202150-1-43-1', 'hep-th-0202150-2-43-1'], ['hep-th-0202150-1-43-2', 'hep-th-0202150-2-43-2'], ['hep-th-0202150-1-43-3', 'hep-th-0202150-2-43-3'], ['hep-th-0202150-1-103-0', 'hep-th-0202150-2-99-0'], ['hep-th-0202150-1-103-1', 'hep-th-0202150-2-99-1'], ['hep-th-0202150-1-103-2', 'hep-th-0202150-2-99-2'], ['hep-th-0202150-1-103-3', 'hep-th-0202150-2-99-3'], ['hep-th-0202150-1-103-4', 'hep-th-0202150-2-99-4'], ['hep-th-0202150-1-103-5', 'hep-th-0202150-2-99-5'], ['hep-th-0202150-1-103-6', 'hep-th-0202150-2-99-6'], ['hep-th-0202150-1-103-7', 'hep-th-0202150-2-99-7'], ['hep-th-0202150-1-103-8', 'hep-th-0202150-2-99-8'], ['hep-th-0202150-1-103-9', 'hep-th-0202150-2-99-9'], ['hep-th-0202150-1-103-10', 'hep-th-0202150-2-99-10'], ['hep-th-0202150-1-31-0', 'hep-th-0202150-2-31-0'], ['hep-th-0202150-1-31-1', 'hep-th-0202150-2-31-1'], ['hep-th-0202150-1-31-2', 'hep-th-0202150-2-31-2'], ['hep-th-0202150-1-31-3', 'hep-th-0202150-2-31-3'], ['hep-th-0202150-1-31-4', 'hep-th-0202150-2-31-4'], ['hep-th-0202150-1-31-5', 'hep-th-0202150-2-31-5'], ['hep-th-0202150-1-31-6', 'hep-th-0202150-2-31-6'], ['hep-th-0202150-1-31-7', 'hep-th-0202150-2-31-7'], ['hep-th-0202150-1-31-8', 'hep-th-0202150-2-31-8'], ['hep-th-0202150-1-81-0', 'hep-th-0202150-2-80-0'], ['hep-th-0202150-1-81-1', 'hep-th-0202150-2-80-1'], ['hep-th-0202150-1-81-2', 'hep-th-0202150-2-80-2'], ['hep-th-0202150-1-81-3', 'hep-th-0202150-2-80-3'], ['hep-th-0202150-1-81-4', 'hep-th-0202150-2-80-4'], ['hep-th-0202150-1-81-5', 'hep-th-0202150-2-80-5'], ['hep-th-0202150-1-52-0', 'hep-th-0202150-2-52-0'], ['hep-th-0202150-1-52-1', 'hep-th-0202150-2-52-1'], ['hep-th-0202150-1-52-2', 'hep-th-0202150-2-52-2'], ['hep-th-0202150-1-52-3', 'hep-th-0202150-2-52-3'], ['hep-th-0202150-1-52-4', 'hep-th-0202150-2-52-4'], ['hep-th-0202150-1-52-5', 'hep-th-0202150-2-52-5'], ['hep-th-0202150-1-52-6', 'hep-th-0202150-2-52-6'], ['hep-th-0202150-1-52-7', 'hep-th-0202150-2-52-7'], ['hep-th-0202150-1-19-0', 'hep-th-0202150-2-19-0'], ['hep-th-0202150-1-19-1', 'hep-th-0202150-2-19-1'], ['hep-th-0202150-1-19-2', 'hep-th-0202150-2-19-2'], ['hep-th-0202150-1-19-3', 'hep-th-0202150-2-19-3'], ['hep-th-0202150-1-19-4', 'hep-th-0202150-2-19-4'], ['hep-th-0202150-1-5-0', 'hep-th-0202150-2-5-0'], ['hep-th-0202150-1-5-1', 'hep-th-0202150-2-5-1'], ['hep-th-0202150-1-5-2', 'hep-th-0202150-2-5-2'], ['hep-th-0202150-1-5-3', 'hep-th-0202150-2-5-3'], ['hep-th-0202150-1-5-4', 'hep-th-0202150-2-5-4'], ['hep-th-0202150-1-5-5', 'hep-th-0202150-2-5-5'], ['hep-th-0202150-1-5-6', 'hep-th-0202150-2-5-6'], ['hep-th-0202150-1-5-7', 'hep-th-0202150-2-5-7'], ['hep-th-0202150-1-5-8', 'hep-th-0202150-2-5-8'], ['hep-th-0202150-1-5-9', 'hep-th-0202150-2-5-9'], ['hep-th-0202150-1-16-0', 'hep-th-0202150-2-16-0'], ['hep-th-0202150-1-16-1', 'hep-th-0202150-2-16-1'], ['hep-th-0202150-1-16-2', 'hep-th-0202150-2-16-2'], ['hep-th-0202150-1-16-3', 'hep-th-0202150-2-16-3'], ['hep-th-0202150-1-16-4', 'hep-th-0202150-2-16-4'], ['hep-th-0202150-1-16-5', 'hep-th-0202150-2-16-5'], ['hep-th-0202150-1-16-6', 'hep-th-0202150-2-16-6'], ['hep-th-0202150-1-35-0', 'hep-th-0202150-2-35-0'], ['hep-th-0202150-1-35-1', 'hep-th-0202150-2-35-1'], ['hep-th-0202150-1-56-0', 'hep-th-0202150-2-56-0'], ['hep-th-0202150-1-56-1', 'hep-th-0202150-2-56-1'], ['hep-th-0202150-1-56-2', 'hep-th-0202150-2-56-2'], ['hep-th-0202150-1-9-0', 'hep-th-0202150-2-9-0'], ['hep-th-0202150-1-9-1', 'hep-th-0202150-2-9-1'], ['hep-th-0202150-1-9-2', 'hep-th-0202150-2-9-2'], ['hep-th-0202150-1-9-3', 'hep-th-0202150-2-9-3'], ['hep-th-0202150-1-71-0', 'hep-th-0202150-2-70-0'], ['hep-th-0202150-1-48-0', 'hep-th-0202150-2-48-0'], ['hep-th-0202150-1-48-1', 'hep-th-0202150-2-48-1'], ['hep-th-0202150-1-48-2', 'hep-th-0202150-2-48-2'], ['hep-th-0202150-1-48-3', 'hep-th-0202150-2-48-3'], ['hep-th-0202150-1-51-0', 'hep-th-0202150-2-51-0'], ['hep-th-0202150-1-51-2', 'hep-th-0202150-2-51-2'], ['hep-th-0202150-1-51-3', 'hep-th-0202150-2-51-3'], ['hep-th-0202150-1-51-4', 'hep-th-0202150-2-51-4'], ['hep-th-0202150-1-51-5', 'hep-th-0202150-2-51-5'], ['hep-th-0202150-1-51-6', 'hep-th-0202150-2-51-6'], ['hep-th-0202150-1-51-7', 'hep-th-0202150-2-51-7'], ['hep-th-0202150-1-51-8', 'hep-th-0202150-2-51-8'], ['hep-th-0202150-1-51-9', 'hep-th-0202150-2-51-9'], ['hep-th-0202150-1-51-10', 'hep-th-0202150-2-51-10'], ['hep-th-0202150-1-26-0', 'hep-th-0202150-2-26-0'], ['hep-th-0202150-1-26-1', 'hep-th-0202150-2-26-1'], ['hep-th-0202150-1-26-3', 'hep-th-0202150-2-26-3'], ['hep-th-0202150-1-26-4', 'hep-th-0202150-2-26-4'], ['hep-th-0202150-1-74-0', 'hep-th-0202150-2-73-0'], ['hep-th-0202150-1-74-1', 'hep-th-0202150-2-73-1'], ['hep-th-0202150-1-74-2', 'hep-th-0202150-2-73-2'], ['hep-th-0202150-1-74-3', 'hep-th-0202150-2-73-3'], ['hep-th-0202150-1-74-4', 'hep-th-0202150-2-73-4'], ['hep-th-0202150-1-74-5', 'hep-th-0202150-2-73-5'], ['hep-th-0202150-1-74-6', 'hep-th-0202150-2-73-6'], ['hep-th-0202150-1-74-7', 'hep-th-0202150-2-73-7'], ['hep-th-0202150-1-74-8', 'hep-th-0202150-2-73-8'], ['hep-th-0202150-1-42-0', 'hep-th-0202150-2-42-0'], ['hep-th-0202150-1-42-1', 'hep-th-0202150-2-42-1'], ['hep-th-0202150-1-10-0', 'hep-th-0202150-2-10-0'], ['hep-th-0202150-1-10-1', 'hep-th-0202150-2-10-1'], ['hep-th-0202150-1-10-2', 'hep-th-0202150-2-10-2'], ['hep-th-0202150-1-72-0', 'hep-th-0202150-2-71-0'], ['hep-th-0202150-1-72-1', 'hep-th-0202150-2-71-1'], ['hep-th-0202150-1-72-2', 'hep-th-0202150-2-71-2'], ['hep-th-0202150-1-72-3', 'hep-th-0202150-2-71-3'], ['hep-th-0202150-1-72-4', 'hep-th-0202150-2-71-4'], ['hep-th-0202150-1-67-0', 'hep-th-0202150-2-66-0'], ['hep-th-0202150-1-67-1', 'hep-th-0202150-2-66-1'], ['hep-th-0202150-1-67-2', 'hep-th-0202150-2-66-2'], ['hep-th-0202150-1-68-0', 'hep-th-0202150-2-67-0'], ['hep-th-0202150-1-76-0', 'hep-th-0202150-2-75-0'], ['hep-th-0202150-1-64-0', 'hep-th-0202150-2-64-0'], ['hep-th-0202150-1-64-1', 'hep-th-0202150-2-64-1'], ['hep-th-0202150-1-89-0', 'hep-th-0202150-2-86-0'], ['hep-th-0202150-1-47-0', 'hep-th-0202150-2-47-0'], ['hep-th-0202150-1-47-1', 'hep-th-0202150-2-47-1'], ['hep-th-0202150-1-47-2', 'hep-th-0202150-2-47-2'], ['hep-th-0202150-1-58-0', 'hep-th-0202150-2-58-0'], ['hep-th-0202150-1-58-1', 'hep-th-0202150-2-58-1'], ['hep-th-0202150-1-58-3', 'hep-th-0202150-2-58-3'], ['hep-th-0202150-1-58-4', 'hep-th-0202150-2-58-4'], ['hep-th-0202150-1-58-5', 'hep-th-0202150-2-58-5'], ['hep-th-0202150-1-63-0', 'hep-th-0202150-2-63-0'], ['hep-th-0202150-1-63-1', 'hep-th-0202150-2-63-1'], ['hep-th-0202150-1-97-0', 'hep-th-0202150-2-93-0'], ['hep-th-0202150-1-97-1', 'hep-th-0202150-2-93-1'], ['hep-th-0202150-1-97-2', 'hep-th-0202150-2-93-2'], ['hep-th-0202150-1-97-3', 'hep-th-0202150-2-93-3'], ['hep-th-0202150-1-97-4', 'hep-th-0202150-2-93-4'], ['hep-th-0202150-1-97-5', 'hep-th-0202150-2-93-5'], ['hep-th-0202150-1-15-0', 'hep-th-0202150-2-15-0'], ['hep-th-0202150-1-15-1', 'hep-th-0202150-2-15-1'], ['hep-th-0202150-1-15-2', 'hep-th-0202150-2-15-2'], ['hep-th-0202150-1-15-5', 'hep-th-0202150-2-15-5'], ['hep-th-0202150-1-15-6', 'hep-th-0202150-2-15-6'], ['hep-th-0202150-1-37-0', 'hep-th-0202150-2-37-0'], ['hep-th-0202150-1-33-0', 'hep-th-0202150-2-33-0'], ['hep-th-0202150-1-33-1', 'hep-th-0202150-2-33-1'], ['hep-th-0202150-1-70-0', 'hep-th-0202150-2-69-0'], ['hep-th-0202150-1-70-1', 'hep-th-0202150-2-69-1'], ['hep-th-0202150-1-70-2', 'hep-th-0202150-2-69-2'], ['hep-th-0202150-1-70-3', 'hep-th-0202150-2-69-3'], ['hep-th-0202150-1-55-0', 'hep-th-0202150-2-55-0'], ['hep-th-0202150-1-55-1', 'hep-th-0202150-2-55-1'], ['hep-th-0202150-1-55-2', 'hep-th-0202150-2-55-2'], ['hep-th-0202150-1-55-3', 'hep-th-0202150-2-55-3'], ['hep-th-0202150-1-55-4', 'hep-th-0202150-2-55-4'], ['hep-th-0202150-1-14-0', 'hep-th-0202150-2-14-0'], ['hep-th-0202150-1-14-1', 'hep-th-0202150-2-14-1'], ['hep-th-0202150-1-4-0', 'hep-th-0202150-2-4-0'], ['hep-th-0202150-1-4-1', 'hep-th-0202150-2-4-1'], ['hep-th-0202150-1-4-2', 'hep-th-0202150-2-4-2'], ['hep-th-0202150-1-4-3', 'hep-th-0202150-2-4-3'], ['hep-th-0202150-1-4-4', 'hep-th-0202150-2-4-4'], ['hep-th-0202150-1-4-5', 'hep-th-0202150-2-4-5'], ['hep-th-0202150-1-4-6', 'hep-th-0202150-2-4-6'], ['hep-th-0202150-1-4-7', 'hep-th-0202150-2-4-7'], ['hep-th-0202150-1-4-8', 'hep-th-0202150-2-4-8'], ['hep-th-0202150-1-27-0', 'hep-th-0202150-2-27-0'], ['hep-th-0202150-1-27-1', 'hep-th-0202150-2-27-1'], ['hep-th-0202150-1-27-2', 'hep-th-0202150-2-27-2'], ['hep-th-0202150-1-27-4', 'hep-th-0202150-2-27-4'], ['hep-th-0202150-1-27-5', 'hep-th-0202150-2-27-5'], ['hep-th-0202150-1-27-6', 'hep-th-0202150-2-27-6'], ['hep-th-0202150-1-27-7', 'hep-th-0202150-2-27-7'], ['hep-th-0202150-1-27-8', 'hep-th-0202150-2-27-8'], ['hep-th-0202150-1-27-9', 'hep-th-0202150-2-27-9'], ['hep-th-0202150-1-18-0', 'hep-th-0202150-2-18-0'], ['hep-th-0202150-1-18-1', 'hep-th-0202150-2-18-1'], ['hep-th-0202150-1-18-2', 'hep-th-0202150-2-18-2'], ['hep-th-0202150-1-18-3', 'hep-th-0202150-2-18-3'], ['hep-th-0202150-1-18-4', 'hep-th-0202150-2-18-4'], ['hep-th-0202150-1-18-5', 'hep-th-0202150-2-18-5'], ['hep-th-0202150-1-18-6', 'hep-th-0202150-2-18-6'], ['hep-th-0202150-1-18-7', 'hep-th-0202150-2-18-7'], ['hep-th-0202150-1-50-0', 'hep-th-0202150-2-50-0'], ['hep-th-0202150-1-62-0', 'hep-th-0202150-2-62-0'], ['hep-th-0202150-1-62-1', 'hep-th-0202150-2-62-1'], ['hep-th-0202150-1-62-2', 'hep-th-0202150-2-62-2'], ['hep-th-0202150-1-62-3', 'hep-th-0202150-2-62-3'], ['hep-th-0202150-1-95-0', 'hep-th-0202150-2-91-0'], ['hep-th-0202150-1-95-1', 'hep-th-0202150-2-91-1'], ['hep-th-0202150-1-95-2', 'hep-th-0202150-2-91-2'], ['hep-th-0202150-1-95-3', 'hep-th-0202150-2-91-3'], ['hep-th-0202150-1-95-4', 'hep-th-0202150-2-91-4'], ['hep-th-0202150-1-95-5', 'hep-th-0202150-2-91-5'], ['hep-th-0202150-1-95-6', 'hep-th-0202150-2-91-6'], ['hep-th-0202150-1-95-7', 'hep-th-0202150-2-91-7'], ['hep-th-0202150-1-11-0', 'hep-th-0202150-2-11-0'], ['hep-th-0202150-1-11-1', 'hep-th-0202150-2-11-1'], ['hep-th-0202150-1-11-2', 'hep-th-0202150-2-11-2'], ['hep-th-0202150-1-11-4', 'hep-th-0202150-2-11-4'], ['hep-th-0202150-1-11-5', 'hep-th-0202150-2-11-5'], ['hep-th-0202150-1-11-6', 'hep-th-0202150-2-11-6'], ['hep-th-0202150-1-57-0', 'hep-th-0202150-2-57-0'], ['hep-th-0202150-1-57-1', 'hep-th-0202150-2-57-1'], ['hep-th-0202150-1-57-2', 'hep-th-0202150-2-57-2'], ['hep-th-0202150-1-57-3', 'hep-th-0202150-2-57-3'], ['hep-th-0202150-1-57-4', 'hep-th-0202150-2-57-4'], ['hep-th-0202150-1-57-5', 'hep-th-0202150-2-57-5'], ['hep-th-0202150-1-57-6', 'hep-th-0202150-2-57-6'], ['hep-th-0202150-1-90-0', 'hep-th-0202150-2-87-0'], ['hep-th-0202150-1-90-1', 'hep-th-0202150-2-87-1'], ['hep-th-0202150-1-90-2', 'hep-th-0202150-2-87-2'], ['hep-th-0202150-1-88-0', 'hep-th-0202150-2-85-0'], ['hep-th-0202150-1-88-1', 'hep-th-0202150-2-85-1'], ['hep-th-0202150-1-88-2', 'hep-th-0202150-2-85-2'], ['hep-th-0202150-1-88-3', 'hep-th-0202150-2-85-3'], ['hep-th-0202150-1-88-4', 'hep-th-0202150-2-85-4'], ['hep-th-0202150-1-88-5', 'hep-th-0202150-2-85-5'], ['hep-th-0202150-1-88-7', 'hep-th-0202150-2-85-7'], ['hep-th-0202150-1-88-8', 'hep-th-0202150-2-85-8'], ['hep-th-0202150-1-61-0', 'hep-th-0202150-2-61-0'], ['hep-th-0202150-1-61-1', 'hep-th-0202150-2-61-1'], ['hep-th-0202150-1-61-2', 'hep-th-0202150-2-61-2'], ['hep-th-0202150-1-61-3', 'hep-th-0202150-2-61-3'], ['hep-th-0202150-1-61-4', 'hep-th-0202150-2-61-4'], ['hep-th-0202150-1-61-5', 'hep-th-0202150-2-61-5'], ['hep-th-0202150-1-61-6', 'hep-th-0202150-2-61-6'], ['hep-th-0202150-1-40-0', 'hep-th-0202150-2-40-0'], ['hep-th-0202150-1-40-1', 'hep-th-0202150-2-40-1'], ['hep-th-0202150-1-40-2', 'hep-th-0202150-2-40-2'], ['hep-th-0202150-1-40-3', 'hep-th-0202150-2-40-3'], ['hep-th-0202150-1-93-0', 'hep-th-0202150-2-89-0'], ['hep-th-0202150-1-80-0', 'hep-th-0202150-2-79-0'], ['hep-th-0202150-1-80-1', 'hep-th-0202150-2-79-1'], ['hep-th-0202150-1-80-2', 'hep-th-0202150-2-79-2'], ['hep-th-0202150-1-54-1', 'hep-th-0202150-2-54-1'], ['hep-th-0202150-1-54-2', 'hep-th-0202150-2-54-2'], ['hep-th-0202150-1-54-3', 'hep-th-0202150-2-54-3'], ['hep-th-0202150-1-77-0', 'hep-th-0202150-2-76-0'], ['hep-th-0202150-1-77-1', 'hep-th-0202150-2-76-1'], ['hep-th-0202150-1-77-2', 'hep-th-0202150-2-76-2'], ['hep-th-0202150-1-77-3', 'hep-th-0202150-2-76-3'], ['hep-th-0202150-1-77-4', 'hep-th-0202150-2-76-4'], ['hep-th-0202150-1-77-5', 'hep-th-0202150-2-76-5'], ['hep-th-0202150-1-75-0', 'hep-th-0202150-2-74-0'], ['hep-th-0202150-1-1-0', 'hep-th-0202150-2-1-0'], ['hep-th-0202150-1-1-1', 'hep-th-0202150-2-1-1'], ['hep-th-0202150-1-1-2', 'hep-th-0202150-2-1-2'], ['hep-th-0202150-1-73-0', 'hep-th-0202150-2-72-0'], ['hep-th-0202150-1-96-0', 'hep-th-0202150-2-92-0'], ['hep-th-0202150-1-96-1', 'hep-th-0202150-2-92-1'], ['hep-th-0202150-1-96-2', 'hep-th-0202150-2-92-2'], ['hep-th-0202150-1-96-3', 'hep-th-0202150-2-92-3'], ['hep-th-0202150-1-96-4', 'hep-th-0202150-2-92-4'], ['hep-th-0202150-1-92-1', 'hep-th-0202150-2-88-1'], ['hep-th-0202150-1-92-2', 'hep-th-0202150-2-88-2'], ['hep-th-0202150-1-92-3', 'hep-th-0202150-2-88-3'], ['hep-th-0202150-1-83-0', 'hep-th-0202150-2-82-0'], ['hep-th-0202150-1-84-0', 'hep-th-0202150-2-82-1'], ['hep-th-0202150-1-85-0', 'hep-th-0202150-2-82-2'], ['hep-th-0202150-1-85-1', 'hep-th-0202150-2-82-3'], ['hep-th-0202150-1-65-0', 'hep-th-0202150-2-65-0'], ['hep-th-0202150-1-65-1', 'hep-th-0202150-2-65-1'], ['hep-th-0202150-1-66-0', 'hep-th-0202150-2-65-2'], ['hep-th-0202150-1-66-1', 'hep-th-0202150-2-65-3'], ['hep-th-0202150-1-82-1', 'hep-th-0202150-2-83-3'], ['hep-th-0202150-1-82-2', 'hep-th-0202150-2-83-4'], ['hep-th-0202150-1-82-3', 'hep-th-0202150-2-83-5'], ['hep-th-0202150-1-82-4', 'hep-th-0202150-2-83-6'], ['hep-th-0202150-1-86-1', 'hep-th-0202150-2-83-1'], ['hep-th-0202150-1-86-2', 'hep-th-0202150-2-83-2'], ['hep-th-0202150-1-86-4', 'hep-th-0202150-2-83-8'], ['hep-th-0202150-1-86-5', 'hep-th-0202150-2-83-9'], ['hep-th-0202150-1-86-6', 'hep-th-0202150-2-83-10'], ['hep-th-0202150-1-86-8', 'hep-th-0202150-2-83-12'], ['hep-th-0202150-1-86-9', 'hep-th-0202150-2-83-13'], ['hep-th-0202150-1-86-10', 'hep-th-0202150-2-83-14'], ['hep-th-0202150-1-86-11', 'hep-th-0202150-2-83-15'], ['hep-th-0202150-1-86-12', 'hep-th-0202150-2-83-16'], ['hep-th-0202150-1-86-13', 'hep-th-0202150-2-83-17'], ['hep-th-0202150-1-86-14', 'hep-th-0202150-2-83-18'], ['hep-th-0202150-1-86-15', 'hep-th-0202150-2-83-19'], ['hep-th-0202150-1-86-16', 'hep-th-0202150-2-83-20'], ['hep-th-0202150-1-86-17', 'hep-th-0202150-2-83-21'], ['hep-th-0202150-1-82-0', 'hep-th-0202150-2-81-0'], ['hep-th-0202150-1-82-5', 'hep-th-0202150-2-81-1'], ['hep-th-0202150-1-82-6', 'hep-th-0202150-2-81-2'], ['hep-th-0202150-1-44-0', 'hep-th-0202150-2-44-0'], ['hep-th-0202150-1-34-1', 'hep-th-0202150-2-34-1'], ['hep-th-0202150-1-59-2', 'hep-th-0202150-2-59-2'], ['hep-th-0202150-1-36-2', 'hep-th-0202150-2-36-2'], ['hep-th-0202150-1-99-1', 'hep-th-0202150-2-95-1'], ['hep-th-0202150-1-51-1', 'hep-th-0202150-2-51-1'], ['hep-th-0202150-1-26-2', 'hep-th-0202150-2-26-2'], ['hep-th-0202150-1-67-3', 'hep-th-0202150-2-66-3'], ['hep-th-0202150-1-58-2', 'hep-th-0202150-2-58-2'], ['hep-th-0202150-1-97-6', 'hep-th-0202150-2-93-6'], ['hep-th-0202150-1-97-7', 'hep-th-0202150-2-93-7'], ['hep-th-0202150-1-15-4', 'hep-th-0202150-2-15-4'], ['hep-th-0202150-1-11-3', 'hep-th-0202150-2-11-3'], ['hep-th-0202150-1-54-0', 'hep-th-0202150-2-54-0'], ['hep-th-0202150-1-91-0', 'hep-th-0202150-2-88-0'], ['hep-th-0202150-1-78-2', 'hep-th-0202150-2-77-2'], ['hep-th-0202150-1-86-3', 'hep-th-0202150-2-83-7']]

[['hep-th-0202150-1-78-0', 'hep-th-0202150-2-77-0'], ['hep-th-0202150-1-78-1', 'hep-th-0202150-2-77-1'], ['hep-th-0202150-1-78-3', 'hep-th-0202150-2-77-3'], ['hep-th-0202150-1-78-4', 'hep-th-0202150-2-77-4'], ['hep-th-0202150-1-29-0', 'hep-th-0202150-2-29-0'], ['hep-th-0202150-1-29-1', 'hep-th-0202150-2-29-1'], ['hep-th-0202150-1-13-0', 'hep-th-0202150-2-13-0'], ['hep-th-0202150-1-101-0', 'hep-th-0202150-2-97-0'], ['hep-th-0202150-1-101-1', 'hep-th-0202150-2-97-1'], ['hep-th-0202150-1-101-2', 'hep-th-0202150-2-97-2'], ['hep-th-0202150-1-101-3', 'hep-th-0202150-2-97-3'], ['hep-th-0202150-1-101-4', 'hep-th-0202150-2-97-4'], ['hep-th-0202150-1-3-0', 'hep-th-0202150-2-3-0'], ['hep-th-0202150-1-3-1', 'hep-th-0202150-2-3-1'], ['hep-th-0202150-1-3-2', 'hep-th-0202150-2-3-2'], ['hep-th-0202150-1-3-3', 'hep-th-0202150-2-3-3'], ['hep-th-0202150-1-3-4', 'hep-th-0202150-2-3-4'], ['hep-th-0202150-1-3-5', 'hep-th-0202150-2-3-5'], ['hep-th-0202150-1-24-0', 'hep-th-0202150-2-24-0'], ['hep-th-0202150-1-24-1', 'hep-th-0202150-2-24-1'], ['hep-th-0202150-1-24-2', 'hep-th-0202150-2-24-2'], ['hep-th-0202150-1-24-3', 'hep-th-0202150-2-24-3'], ['hep-th-0202150-1-38-0', 'hep-th-0202150-2-38-0'], ['hep-th-0202150-1-38-1', 'hep-th-0202150-2-38-1'], ['hep-th-0202150-1-38-2', 'hep-th-0202150-2-38-2'], ['hep-th-0202150-1-38-3', 'hep-th-0202150-2-38-3'], ['hep-th-0202150-1-38-4', 'hep-th-0202150-2-38-4'], ['hep-th-0202150-1-38-5', 'hep-th-0202150-2-38-5'], ['hep-th-0202150-1-38-6', 'hep-th-0202150-2-38-6'], ['hep-th-0202150-1-7-0', 'hep-th-0202150-2-7-0'], ['hep-th-0202150-1-7-1', 'hep-th-0202150-2-7-1'], ['hep-th-0202150-1-7-2', 'hep-th-0202150-2-7-2'], ['hep-th-0202150-1-7-3', 'hep-th-0202150-2-7-3'], ['hep-th-0202150-1-49-0', 'hep-th-0202150-2-49-0'], ['hep-th-0202150-1-49-1', 'hep-th-0202150-2-49-1'], ['hep-th-0202150-1-49-2', 'hep-th-0202150-2-49-2'], ['hep-th-0202150-1-49-3', 'hep-th-0202150-2-49-3'], ['hep-th-0202150-1-22-0', 'hep-th-0202150-2-22-0'], ['hep-th-0202150-1-22-1', 'hep-th-0202150-2-22-1'], ['hep-th-0202150-1-79-0', 'hep-th-0202150-2-78-0'], ['hep-th-0202150-1-79-1', 'hep-th-0202150-2-78-1'], ['hep-th-0202150-1-79-2', 'hep-th-0202150-2-78-2'], ['hep-th-0202150-1-8-0', 'hep-th-0202150-2-8-0'], ['hep-th-0202150-1-8-1', 'hep-th-0202150-2-8-1'], ['hep-th-0202150-1-8-2', 'hep-th-0202150-2-8-2'], ['hep-th-0202150-1-8-3', 'hep-th-0202150-2-8-3'], ['hep-th-0202150-1-53-0', 'hep-th-0202150-2-53-0'], ['hep-th-0202150-1-98-0', 'hep-th-0202150-2-94-0'], ['hep-th-0202150-1-98-1', 'hep-th-0202150-2-94-1'], ['hep-th-0202150-1-98-2', 'hep-th-0202150-2-94-2'], ['hep-th-0202150-1-98-3', 'hep-th-0202150-2-94-3'], ['hep-th-0202150-1-98-4', 'hep-th-0202150-2-94-4'], ['hep-th-0202150-1-98-5', 'hep-th-0202150-2-94-5'], ['hep-th-0202150-1-44-1', 'hep-th-0202150-2-44-1'], ['hep-th-0202150-1-44-2', 'hep-th-0202150-2-44-2'], ['hep-th-0202150-1-44-3', 'hep-th-0202150-2-44-3'], ['hep-th-0202150-1-44-4', 'hep-th-0202150-2-44-4'], ['hep-th-0202150-1-44-5', 'hep-th-0202150-2-44-5'], ['hep-th-0202150-1-34-0', 'hep-th-0202150-2-34-0'], ['hep-th-0202150-1-28-0', 'hep-th-0202150-2-28-0'], ['hep-th-0202150-1-28-1', 'hep-th-0202150-2-28-1'], ['hep-th-0202150-1-28-2', 'hep-th-0202150-2-28-2'], ['hep-th-0202150-1-28-3', 'hep-th-0202150-2-28-3'], ['hep-th-0202150-1-102-0', 'hep-th-0202150-2-98-0'], ['hep-th-0202150-1-102-1', 'hep-th-0202150-2-98-1'], ['hep-th-0202150-1-102-2', 'hep-th-0202150-2-98-2'], ['hep-th-0202150-1-102-3', 'hep-th-0202150-2-98-3'], ['hep-th-0202150-1-102-4', 'hep-th-0202150-2-98-4'], ['hep-th-0202150-1-102-5', 'hep-th-0202150-2-98-5'], ['hep-th-0202150-1-102-6', 'hep-th-0202150-2-98-6'], ['hep-th-0202150-1-102-7', 'hep-th-0202150-2-98-7'], ['hep-th-0202150-1-102-8', 'hep-th-0202150-2-98-8'], ['hep-th-0202150-1-102-9', 'hep-th-0202150-2-98-9'], ['hep-th-0202150-1-25-0', 'hep-th-0202150-2-25-0'], ['hep-th-0202150-1-25-1', 'hep-th-0202150-2-25-1'], ['hep-th-0202150-1-39-0', 'hep-th-0202150-2-39-0'], ['hep-th-0202150-1-39-1', 'hep-th-0202150-2-39-1'], ['hep-th-0202150-1-39-2', 'hep-th-0202150-2-39-2'], ['hep-th-0202150-1-39-3', 'hep-th-0202150-2-39-3'], ['hep-th-0202150-1-39-4', 'hep-th-0202150-2-39-4'], ['hep-th-0202150-1-39-5', 'hep-th-0202150-2-39-5'], ['hep-th-0202150-1-59-0', 'hep-th-0202150-2-59-0'], ['hep-th-0202150-1-59-1', 'hep-th-0202150-2-59-1'], ['hep-th-0202150-1-45-0', 'hep-th-0202150-2-45-0'], ['hep-th-0202150-1-45-1', 'hep-th-0202150-2-45-1'], ['hep-th-0202150-1-45-2', 'hep-th-0202150-2-45-2'], ['hep-th-0202150-1-36-0', 'hep-th-0202150-2-36-0'], ['hep-th-0202150-1-36-1', 'hep-th-0202150-2-36-1'], ['hep-th-0202150-1-36-3', 'hep-th-0202150-2-36-3'], ['hep-th-0202150-1-36-4', 'hep-th-0202150-2-36-4'], ['hep-th-0202150-1-2-0', 'hep-th-0202150-2-2-0'], ['hep-th-0202150-1-2-1', 'hep-th-0202150-2-2-1'], ['hep-th-0202150-1-2-2', 'hep-th-0202150-2-2-2'], ['hep-th-0202150-1-2-3', 'hep-th-0202150-2-2-3'], ['hep-th-0202150-1-2-4', 'hep-th-0202150-2-2-4'], ['hep-th-0202150-1-2-5', 'hep-th-0202150-2-2-5'], ['hep-th-0202150-1-2-6', 'hep-th-0202150-2-2-6'], ['hep-th-0202150-1-2-7', 'hep-th-0202150-2-2-7'], ['hep-th-0202150-1-87-0', 'hep-th-0202150-2-84-0'], ['hep-th-0202150-1-94-0', 'hep-th-0202150-2-90-0'], ['hep-th-0202150-1-94-1', 'hep-th-0202150-2-90-1'], ['hep-th-0202150-1-99-0', 'hep-th-0202150-2-95-0'], ['hep-th-0202150-1-99-2', 'hep-th-0202150-2-95-2'], ['hep-th-0202150-1-46-0', 'hep-th-0202150-2-46-0'], ['hep-th-0202150-1-46-1', 'hep-th-0202150-2-46-1'], ['hep-th-0202150-1-46-2', 'hep-th-0202150-2-46-2'], ['hep-th-0202150-1-46-3', 'hep-th-0202150-2-46-3'], ['hep-th-0202150-1-21-0', 'hep-th-0202150-2-21-0'], ['hep-th-0202150-1-21-1', 'hep-th-0202150-2-21-1'], ['hep-th-0202150-1-21-2', 'hep-th-0202150-2-21-2'], ['hep-th-0202150-1-21-3', 'hep-th-0202150-2-21-3'], ['hep-th-0202150-1-17-0', 'hep-th-0202150-2-17-0'], ['hep-th-0202150-1-17-1', 'hep-th-0202150-2-17-1'], ['hep-th-0202150-1-17-2', 'hep-th-0202150-2-17-2'], ['hep-th-0202150-1-20-0', 'hep-th-0202150-2-20-0'], ['hep-th-0202150-1-20-1', 'hep-th-0202150-2-20-1'], ['hep-th-0202150-1-20-2', 'hep-th-0202150-2-20-2'], ['hep-th-0202150-1-20-3', 'hep-th-0202150-2-20-3'], ['hep-th-0202150-1-20-4', 'hep-th-0202150-2-20-4'], ['hep-th-0202150-1-20-5', 'hep-th-0202150-2-20-5'], ['hep-th-0202150-1-43-0', 'hep-th-0202150-2-43-0'], ['hep-th-0202150-1-43-1', 'hep-th-0202150-2-43-1'], ['hep-th-0202150-1-43-2', 'hep-th-0202150-2-43-2'], ['hep-th-0202150-1-43-3', 'hep-th-0202150-2-43-3'], ['hep-th-0202150-1-103-0', 'hep-th-0202150-2-99-0'], ['hep-th-0202150-1-103-1', 'hep-th-0202150-2-99-1'], ['hep-th-0202150-1-103-2', 'hep-th-0202150-2-99-2'], ['hep-th-0202150-1-103-3', 'hep-th-0202150-2-99-3'], ['hep-th-0202150-1-103-4', 'hep-th-0202150-2-99-4'], ['hep-th-0202150-1-103-5', 'hep-th-0202150-2-99-5'], ['hep-th-0202150-1-103-6', 'hep-th-0202150-2-99-6'], ['hep-th-0202150-1-103-7', 'hep-th-0202150-2-99-7'], ['hep-th-0202150-1-103-8', 'hep-th-0202150-2-99-8'], ['hep-th-0202150-1-103-9', 'hep-th-0202150-2-99-9'], ['hep-th-0202150-1-103-10', 'hep-th-0202150-2-99-10'], ['hep-th-0202150-1-31-0', 'hep-th-0202150-2-31-0'], ['hep-th-0202150-1-31-1', 'hep-th-0202150-2-31-1'], ['hep-th-0202150-1-31-2', 'hep-th-0202150-2-31-2'], ['hep-th-0202150-1-31-3', 'hep-th-0202150-2-31-3'], ['hep-th-0202150-1-31-4', 'hep-th-0202150-2-31-4'], ['hep-th-0202150-1-31-5', 'hep-th-0202150-2-31-5'], ['hep-th-0202150-1-31-6', 'hep-th-0202150-2-31-6'], ['hep-th-0202150-1-31-7', 'hep-th-0202150-2-31-7'], ['hep-th-0202150-1-31-8', 'hep-th-0202150-2-31-8'], ['hep-th-0202150-1-81-0', 'hep-th-0202150-2-80-0'], ['hep-th-0202150-1-81-1', 'hep-th-0202150-2-80-1'], ['hep-th-0202150-1-81-2', 'hep-th-0202150-2-80-2'], ['hep-th-0202150-1-81-3', 'hep-th-0202150-2-80-3'], ['hep-th-0202150-1-81-4', 'hep-th-0202150-2-80-4'], ['hep-th-0202150-1-81-5', 'hep-th-0202150-2-80-5'], ['hep-th-0202150-1-52-0', 'hep-th-0202150-2-52-0'], ['hep-th-0202150-1-52-1', 'hep-th-0202150-2-52-1'], ['hep-th-0202150-1-52-2', 'hep-th-0202150-2-52-2'], ['hep-th-0202150-1-52-3', 'hep-th-0202150-2-52-3'], ['hep-th-0202150-1-52-4', 'hep-th-0202150-2-52-4'], ['hep-th-0202150-1-52-5', 'hep-th-0202150-2-52-5'], ['hep-th-0202150-1-52-6', 'hep-th-0202150-2-52-6'], ['hep-th-0202150-1-52-7', 'hep-th-0202150-2-52-7'], ['hep-th-0202150-1-19-0', 'hep-th-0202150-2-19-0'], ['hep-th-0202150-1-19-1', 'hep-th-0202150-2-19-1'], ['hep-th-0202150-1-19-2', 'hep-th-0202150-2-19-2'], ['hep-th-0202150-1-19-3', 'hep-th-0202150-2-19-3'], ['hep-th-0202150-1-19-4', 'hep-th-0202150-2-19-4'], ['hep-th-0202150-1-5-0', 'hep-th-0202150-2-5-0'], ['hep-th-0202150-1-5-1', 'hep-th-0202150-2-5-1'], ['hep-th-0202150-1-5-2', 'hep-th-0202150-2-5-2'], ['hep-th-0202150-1-5-3', 'hep-th-0202150-2-5-3'], ['hep-th-0202150-1-5-4', 'hep-th-0202150-2-5-4'], ['hep-th-0202150-1-5-5', 'hep-th-0202150-2-5-5'], ['hep-th-0202150-1-5-6', 'hep-th-0202150-2-5-6'], ['hep-th-0202150-1-5-7', 'hep-th-0202150-2-5-7'], ['hep-th-0202150-1-5-8', 'hep-th-0202150-2-5-8'], ['hep-th-0202150-1-5-9', 'hep-th-0202150-2-5-9'], ['hep-th-0202150-1-16-0', 'hep-th-0202150-2-16-0'], ['hep-th-0202150-1-16-1', 'hep-th-0202150-2-16-1'], ['hep-th-0202150-1-16-2', 'hep-th-0202150-2-16-2'], ['hep-th-0202150-1-16-3', 'hep-th-0202150-2-16-3'], ['hep-th-0202150-1-16-4', 'hep-th-0202150-2-16-4'], ['hep-th-0202150-1-16-5', 'hep-th-0202150-2-16-5'], ['hep-th-0202150-1-16-6', 'hep-th-0202150-2-16-6'], ['hep-th-0202150-1-35-0', 'hep-th-0202150-2-35-0'], ['hep-th-0202150-1-35-1', 'hep-th-0202150-2-35-1'], ['hep-th-0202150-1-56-0', 'hep-th-0202150-2-56-0'], ['hep-th-0202150-1-56-1', 'hep-th-0202150-2-56-1'], ['hep-th-0202150-1-56-2', 'hep-th-0202150-2-56-2'], ['hep-th-0202150-1-9-0', 'hep-th-0202150-2-9-0'], ['hep-th-0202150-1-9-1', 'hep-th-0202150-2-9-1'], ['hep-th-0202150-1-9-2', 'hep-th-0202150-2-9-2'], ['hep-th-0202150-1-9-3', 'hep-th-0202150-2-9-3'], ['hep-th-0202150-1-71-0', 'hep-th-0202150-2-70-0'], ['hep-th-0202150-1-48-0', 'hep-th-0202150-2-48-0'], ['hep-th-0202150-1-48-1', 'hep-th-0202150-2-48-1'], ['hep-th-0202150-1-48-2', 'hep-th-0202150-2-48-2'], ['hep-th-0202150-1-48-3', 'hep-th-0202150-2-48-3'], ['hep-th-0202150-1-51-0', 'hep-th-0202150-2-51-0'], ['hep-th-0202150-1-51-2', 'hep-th-0202150-2-51-2'], ['hep-th-0202150-1-51-3', 'hep-th-0202150-2-51-3'], ['hep-th-0202150-1-51-4', 'hep-th-0202150-2-51-4'], ['hep-th-0202150-1-51-5', 'hep-th-0202150-2-51-5'], ['hep-th-0202150-1-51-6', 'hep-th-0202150-2-51-6'], ['hep-th-0202150-1-51-7', 'hep-th-0202150-2-51-7'], ['hep-th-0202150-1-51-8', 'hep-th-0202150-2-51-8'], ['hep-th-0202150-1-51-9', 'hep-th-0202150-2-51-9'], ['hep-th-0202150-1-51-10', 'hep-th-0202150-2-51-10'], ['hep-th-0202150-1-26-0', 'hep-th-0202150-2-26-0'], ['hep-th-0202150-1-26-1', 'hep-th-0202150-2-26-1'], ['hep-th-0202150-1-26-3', 'hep-th-0202150-2-26-3'], ['hep-th-0202150-1-26-4', 'hep-th-0202150-2-26-4'], ['hep-th-0202150-1-74-0', 'hep-th-0202150-2-73-0'], ['hep-th-0202150-1-74-1', 'hep-th-0202150-2-73-1'], ['hep-th-0202150-1-74-2', 'hep-th-0202150-2-73-2'], ['hep-th-0202150-1-74-3', 'hep-th-0202150-2-73-3'], ['hep-th-0202150-1-74-4', 'hep-th-0202150-2-73-4'], ['hep-th-0202150-1-74-5', 'hep-th-0202150-2-73-5'], ['hep-th-0202150-1-74-6', 'hep-th-0202150-2-73-6'], ['hep-th-0202150-1-74-7', 'hep-th-0202150-2-73-7'], ['hep-th-0202150-1-74-8', 'hep-th-0202150-2-73-8'], ['hep-th-0202150-1-42-0', 'hep-th-0202150-2-42-0'], ['hep-th-0202150-1-42-1', 'hep-th-0202150-2-42-1'], ['hep-th-0202150-1-10-0', 'hep-th-0202150-2-10-0'], ['hep-th-0202150-1-10-1', 'hep-th-0202150-2-10-1'], ['hep-th-0202150-1-10-2', 'hep-th-0202150-2-10-2'], ['hep-th-0202150-1-72-0', 'hep-th-0202150-2-71-0'], ['hep-th-0202150-1-72-1', 'hep-th-0202150-2-71-1'], ['hep-th-0202150-1-72-2', 'hep-th-0202150-2-71-2'], ['hep-th-0202150-1-72-3', 'hep-th-0202150-2-71-3'], ['hep-th-0202150-1-72-4', 'hep-th-0202150-2-71-4'], ['hep-th-0202150-1-67-0', 'hep-th-0202150-2-66-0'], ['hep-th-0202150-1-67-1', 'hep-th-0202150-2-66-1'], ['hep-th-0202150-1-67-2', 'hep-th-0202150-2-66-2'], ['hep-th-0202150-1-68-0', 'hep-th-0202150-2-67-0'], ['hep-th-0202150-1-76-0', 'hep-th-0202150-2-75-0'], ['hep-th-0202150-1-64-0', 'hep-th-0202150-2-64-0'], ['hep-th-0202150-1-64-1', 'hep-th-0202150-2-64-1'], ['hep-th-0202150-1-89-0', 'hep-th-0202150-2-86-0'], ['hep-th-0202150-1-47-0', 'hep-th-0202150-2-47-0'], ['hep-th-0202150-1-47-1', 'hep-th-0202150-2-47-1'], ['hep-th-0202150-1-47-2', 'hep-th-0202150-2-47-2'], ['hep-th-0202150-1-58-0', 'hep-th-0202150-2-58-0'], ['hep-th-0202150-1-58-1', 'hep-th-0202150-2-58-1'], ['hep-th-0202150-1-58-3', 'hep-th-0202150-2-58-3'], ['hep-th-0202150-1-58-4', 'hep-th-0202150-2-58-4'], ['hep-th-0202150-1-58-5', 'hep-th-0202150-2-58-5'], ['hep-th-0202150-1-63-0', 'hep-th-0202150-2-63-0'], ['hep-th-0202150-1-63-1', 'hep-th-0202150-2-63-1'], ['hep-th-0202150-1-97-0', 'hep-th-0202150-2-93-0'], ['hep-th-0202150-1-97-1', 'hep-th-0202150-2-93-1'], ['hep-th-0202150-1-97-2', 'hep-th-0202150-2-93-2'], ['hep-th-0202150-1-97-3', 'hep-th-0202150-2-93-3'], ['hep-th-0202150-1-97-4', 'hep-th-0202150-2-93-4'], ['hep-th-0202150-1-97-5', 'hep-th-0202150-2-93-5'], ['hep-th-0202150-1-15-0', 'hep-th-0202150-2-15-0'], ['hep-th-0202150-1-15-1', 'hep-th-0202150-2-15-1'], ['hep-th-0202150-1-15-2', 'hep-th-0202150-2-15-2'], ['hep-th-0202150-1-15-5', 'hep-th-0202150-2-15-5'], ['hep-th-0202150-1-15-6', 'hep-th-0202150-2-15-6'], ['hep-th-0202150-1-37-0', 'hep-th-0202150-2-37-0'], ['hep-th-0202150-1-33-0', 'hep-th-0202150-2-33-0'], ['hep-th-0202150-1-33-1', 'hep-th-0202150-2-33-1'], ['hep-th-0202150-1-70-0', 'hep-th-0202150-2-69-0'], ['hep-th-0202150-1-70-1', 'hep-th-0202150-2-69-1'], ['hep-th-0202150-1-70-2', 'hep-th-0202150-2-69-2'], ['hep-th-0202150-1-70-3', 'hep-th-0202150-2-69-3'], ['hep-th-0202150-1-55-0', 'hep-th-0202150-2-55-0'], ['hep-th-0202150-1-55-1', 'hep-th-0202150-2-55-1'], ['hep-th-0202150-1-55-2', 'hep-th-0202150-2-55-2'], ['hep-th-0202150-1-55-3', 'hep-th-0202150-2-55-3'], ['hep-th-0202150-1-55-4', 'hep-th-0202150-2-55-4'], ['hep-th-0202150-1-14-0', 'hep-th-0202150-2-14-0'], ['hep-th-0202150-1-14-1', 'hep-th-0202150-2-14-1'], ['hep-th-0202150-1-4-0', 'hep-th-0202150-2-4-0'], ['hep-th-0202150-1-4-1', 'hep-th-0202150-2-4-1'], ['hep-th-0202150-1-4-2', 'hep-th-0202150-2-4-2'], ['hep-th-0202150-1-4-3', 'hep-th-0202150-2-4-3'], ['hep-th-0202150-1-4-4', 'hep-th-0202150-2-4-4'], ['hep-th-0202150-1-4-5', 'hep-th-0202150-2-4-5'], ['hep-th-0202150-1-4-6', 'hep-th-0202150-2-4-6'], ['hep-th-0202150-1-4-7', 'hep-th-0202150-2-4-7'], ['hep-th-0202150-1-4-8', 'hep-th-0202150-2-4-8'], ['hep-th-0202150-1-27-0', 'hep-th-0202150-2-27-0'], ['hep-th-0202150-1-27-1', 'hep-th-0202150-2-27-1'], ['hep-th-0202150-1-27-2', 'hep-th-0202150-2-27-2'], ['hep-th-0202150-1-27-4', 'hep-th-0202150-2-27-4'], ['hep-th-0202150-1-27-5', 'hep-th-0202150-2-27-5'], ['hep-th-0202150-1-27-6', 'hep-th-0202150-2-27-6'], ['hep-th-0202150-1-27-7', 'hep-th-0202150-2-27-7'], ['hep-th-0202150-1-27-8', 'hep-th-0202150-2-27-8'], ['hep-th-0202150-1-27-9', 'hep-th-0202150-2-27-9'], ['hep-th-0202150-1-18-0', 'hep-th-0202150-2-18-0'], ['hep-th-0202150-1-18-1', 'hep-th-0202150-2-18-1'], ['hep-th-0202150-1-18-2', 'hep-th-0202150-2-18-2'], ['hep-th-0202150-1-18-3', 'hep-th-0202150-2-18-3'], ['hep-th-0202150-1-18-4', 'hep-th-0202150-2-18-4'], ['hep-th-0202150-1-18-5', 'hep-th-0202150-2-18-5'], ['hep-th-0202150-1-18-6', 'hep-th-0202150-2-18-6'], ['hep-th-0202150-1-18-7', 'hep-th-0202150-2-18-7'], ['hep-th-0202150-1-50-0', 'hep-th-0202150-2-50-0'], ['hep-th-0202150-1-62-0', 'hep-th-0202150-2-62-0'], ['hep-th-0202150-1-62-1', 'hep-th-0202150-2-62-1'], ['hep-th-0202150-1-62-2', 'hep-th-0202150-2-62-2'], ['hep-th-0202150-1-62-3', 'hep-th-0202150-2-62-3'], ['hep-th-0202150-1-95-0', 'hep-th-0202150-2-91-0'], ['hep-th-0202150-1-95-1', 'hep-th-0202150-2-91-1'], ['hep-th-0202150-1-95-2', 'hep-th-0202150-2-91-2'], ['hep-th-0202150-1-95-3', 'hep-th-0202150-2-91-3'], ['hep-th-0202150-1-95-4', 'hep-th-0202150-2-91-4'], ['hep-th-0202150-1-95-5', 'hep-th-0202150-2-91-5'], ['hep-th-0202150-1-95-6', 'hep-th-0202150-2-91-6'], ['hep-th-0202150-1-95-7', 'hep-th-0202150-2-91-7'], ['hep-th-0202150-1-11-0', 'hep-th-0202150-2-11-0'], ['hep-th-0202150-1-11-1', 'hep-th-0202150-2-11-1'], ['hep-th-0202150-1-11-2', 'hep-th-0202150-2-11-2'], ['hep-th-0202150-1-11-4', 'hep-th-0202150-2-11-4'], ['hep-th-0202150-1-11-5', 'hep-th-0202150-2-11-5'], ['hep-th-0202150-1-11-6', 'hep-th-0202150-2-11-6'], ['hep-th-0202150-1-57-0', 'hep-th-0202150-2-57-0'], ['hep-th-0202150-1-57-1', 'hep-th-0202150-2-57-1'], ['hep-th-0202150-1-57-2', 'hep-th-0202150-2-57-2'], ['hep-th-0202150-1-57-3', 'hep-th-0202150-2-57-3'], ['hep-th-0202150-1-57-4', 'hep-th-0202150-2-57-4'], ['hep-th-0202150-1-57-5', 'hep-th-0202150-2-57-5'], ['hep-th-0202150-1-57-6', 'hep-th-0202150-2-57-6'], ['hep-th-0202150-1-90-0', 'hep-th-0202150-2-87-0'], ['hep-th-0202150-1-90-1', 'hep-th-0202150-2-87-1'], ['hep-th-0202150-1-90-2', 'hep-th-0202150-2-87-2'], ['hep-th-0202150-1-88-0', 'hep-th-0202150-2-85-0'], ['hep-th-0202150-1-88-1', 'hep-th-0202150-2-85-1'], ['hep-th-0202150-1-88-2', 'hep-th-0202150-2-85-2'], ['hep-th-0202150-1-88-3', 'hep-th-0202150-2-85-3'], ['hep-th-0202150-1-88-4', 'hep-th-0202150-2-85-4'], ['hep-th-0202150-1-88-5', 'hep-th-0202150-2-85-5'], ['hep-th-0202150-1-88-7', 'hep-th-0202150-2-85-7'], ['hep-th-0202150-1-88-8', 'hep-th-0202150-2-85-8'], ['hep-th-0202150-1-61-0', 'hep-th-0202150-2-61-0'], ['hep-th-0202150-1-61-1', 'hep-th-0202150-2-61-1'], ['hep-th-0202150-1-61-2', 'hep-th-0202150-2-61-2'], ['hep-th-0202150-1-61-3', 'hep-th-0202150-2-61-3'], ['hep-th-0202150-1-61-4', 'hep-th-0202150-2-61-4'], ['hep-th-0202150-1-61-5', 'hep-th-0202150-2-61-5'], ['hep-th-0202150-1-61-6', 'hep-th-0202150-2-61-6'], ['hep-th-0202150-1-40-0', 'hep-th-0202150-2-40-0'], ['hep-th-0202150-1-40-1', 'hep-th-0202150-2-40-1'], ['hep-th-0202150-1-40-2', 'hep-th-0202150-2-40-2'], ['hep-th-0202150-1-40-3', 'hep-th-0202150-2-40-3'], ['hep-th-0202150-1-93-0', 'hep-th-0202150-2-89-0'], ['hep-th-0202150-1-80-0', 'hep-th-0202150-2-79-0'], ['hep-th-0202150-1-80-1', 'hep-th-0202150-2-79-1'], ['hep-th-0202150-1-80-2', 'hep-th-0202150-2-79-2'], ['hep-th-0202150-1-54-1', 'hep-th-0202150-2-54-1'], ['hep-th-0202150-1-54-2', 'hep-th-0202150-2-54-2'], ['hep-th-0202150-1-54-3', 'hep-th-0202150-2-54-3'], ['hep-th-0202150-1-77-0', 'hep-th-0202150-2-76-0'], ['hep-th-0202150-1-77-1', 'hep-th-0202150-2-76-1'], ['hep-th-0202150-1-77-2', 'hep-th-0202150-2-76-2'], ['hep-th-0202150-1-77-3', 'hep-th-0202150-2-76-3'], ['hep-th-0202150-1-77-4', 'hep-th-0202150-2-76-4'], ['hep-th-0202150-1-77-5', 'hep-th-0202150-2-76-5'], ['hep-th-0202150-1-75-0', 'hep-th-0202150-2-74-0'], ['hep-th-0202150-1-1-0', 'hep-th-0202150-2-1-0'], ['hep-th-0202150-1-1-1', 'hep-th-0202150-2-1-1'], ['hep-th-0202150-1-1-2', 'hep-th-0202150-2-1-2'], ['hep-th-0202150-1-73-0', 'hep-th-0202150-2-72-0'], ['hep-th-0202150-1-96-0', 'hep-th-0202150-2-92-0'], ['hep-th-0202150-1-96-1', 'hep-th-0202150-2-92-1'], ['hep-th-0202150-1-96-2', 'hep-th-0202150-2-92-2'], ['hep-th-0202150-1-96-3', 'hep-th-0202150-2-92-3'], ['hep-th-0202150-1-96-4', 'hep-th-0202150-2-92-4'], ['hep-th-0202150-1-92-1', 'hep-th-0202150-2-88-1'], ['hep-th-0202150-1-92-2', 'hep-th-0202150-2-88-2'], ['hep-th-0202150-1-92-3', 'hep-th-0202150-2-88-3'], ['hep-th-0202150-1-83-0', 'hep-th-0202150-2-82-0'], ['hep-th-0202150-1-84-0', 'hep-th-0202150-2-82-1'], ['hep-th-0202150-1-85-0', 'hep-th-0202150-2-82-2'], ['hep-th-0202150-1-85-1', 'hep-th-0202150-2-82-3'], ['hep-th-0202150-1-65-0', 'hep-th-0202150-2-65-0'], ['hep-th-0202150-1-65-1', 'hep-th-0202150-2-65-1'], ['hep-th-0202150-1-66-0', 'hep-th-0202150-2-65-2'], ['hep-th-0202150-1-66-1', 'hep-th-0202150-2-65-3'], ['hep-th-0202150-1-82-1', 'hep-th-0202150-2-83-3'], ['hep-th-0202150-1-82-2', 'hep-th-0202150-2-83-4'], ['hep-th-0202150-1-82-3', 'hep-th-0202150-2-83-5'], ['hep-th-0202150-1-82-4', 'hep-th-0202150-2-83-6'], ['hep-th-0202150-1-86-1', 'hep-th-0202150-2-83-1'], ['hep-th-0202150-1-86-2', 'hep-th-0202150-2-83-2'], ['hep-th-0202150-1-86-4', 'hep-th-0202150-2-83-8'], ['hep-th-0202150-1-86-5', 'hep-th-0202150-2-83-9'], ['hep-th-0202150-1-86-6', 'hep-th-0202150-2-83-10'], ['hep-th-0202150-1-86-8', 'hep-th-0202150-2-83-12'], ['hep-th-0202150-1-86-9', 'hep-th-0202150-2-83-13'], ['hep-th-0202150-1-86-10', 'hep-th-0202150-2-83-14'], ['hep-th-0202150-1-86-11', 'hep-th-0202150-2-83-15'], ['hep-th-0202150-1-86-12', 'hep-th-0202150-2-83-16'], ['hep-th-0202150-1-86-13', 'hep-th-0202150-2-83-17'], ['hep-th-0202150-1-86-14', 'hep-th-0202150-2-83-18'], ['hep-th-0202150-1-86-15', 'hep-th-0202150-2-83-19'], ['hep-th-0202150-1-86-16', 'hep-th-0202150-2-83-20'], ['hep-th-0202150-1-86-17', 'hep-th-0202150-2-83-21'], ['hep-th-0202150-1-82-0', 'hep-th-0202150-2-81-0'], ['hep-th-0202150-1-82-5', 'hep-th-0202150-2-81-1'], ['hep-th-0202150-1-82-6', 'hep-th-0202150-2-81-2']]

[['hep-th-0202150-1-44-0', 'hep-th-0202150-2-44-0'], ['hep-th-0202150-1-34-1', 'hep-th-0202150-2-34-1'], ['hep-th-0202150-1-59-2', 'hep-th-0202150-2-59-2'], ['hep-th-0202150-1-36-2', 'hep-th-0202150-2-36-2'], ['hep-th-0202150-1-99-1', 'hep-th-0202150-2-95-1'], ['hep-th-0202150-1-51-1', 'hep-th-0202150-2-51-1'], ['hep-th-0202150-1-26-2', 'hep-th-0202150-2-26-2'], ['hep-th-0202150-1-67-3', 'hep-th-0202150-2-66-3'], ['hep-th-0202150-1-58-2', 'hep-th-0202150-2-58-2'], ['hep-th-0202150-1-97-6', 'hep-th-0202150-2-93-6'], ['hep-th-0202150-1-97-7', 'hep-th-0202150-2-93-7'], ['hep-th-0202150-1-15-4', 'hep-th-0202150-2-15-4'], ['hep-th-0202150-1-11-3', 'hep-th-0202150-2-11-3'], ['hep-th-0202150-1-54-0', 'hep-th-0202150-2-54-0'], ['hep-th-0202150-1-91-0', 'hep-th-0202150-2-88-0']]

[]

[['hep-th-0202150-1-78-2', 'hep-th-0202150-2-77-2'], ['hep-th-0202150-1-86-3', 'hep-th-0202150-2-83-7']]

[]

['hep-th-0202150-1-15-3', 'hep-th-0202150-1-17-3', 'hep-th-0202150-1-27-3', 'hep-th-0202150-1-62-4', 'hep-th-0202150-1-67-4', 'hep-th-0202150-1-69-0', 'hep-th-0202150-1-86-7', 'hep-th-0202150-1-88-6', 'hep-th-0202150-1-92-0', 'hep-th-0202150-2-15-3', 'hep-th-0202150-2-17-3', 'hep-th-0202150-2-27-3', 'hep-th-0202150-2-62-4', 'hep-th-0202150-2-66-4', 'hep-th-0202150-2-68-0', 'hep-th-0202150-2-83-0', 'hep-th-0202150-2-83-11', 'hep-th-0202150-2-85-6']

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

https://arxiv.org/abs/hep-th/0202150

1401.1162

{'1401.1162-1-0-0': 'Naturalness arguments imply the existence of higgsinos lighter than 200-300 GeV.', '1401.1162-1-0-1': 'However, because these higgsinos are nearly mass degenerate, they release very little visible energy in their decays, and signals from electroweak higgsino pair production typically remain buried under Standard Model backgrounds.', '1401.1162-1-0-2': 'Moreover, gluinos, squarks and winos may plausibly lie beyond the reach of the LHC14, so that signals from naturalness-inspired supersymmetric models may well remain hidden via conventional searches.', '1401.1162-1-0-3': 'We examine instead prospects for detecting higgsino pair production via monojets or mono-photons from initial state radiation.', '1401.1162-1-0-4': 'We find typical signal-to-background rates at best at the 1% level, leading to rather pessimistic conclusions regarding detectability via these channels.', '1401.1162-1-1-0': '# Introduction', '1401.1162-1-2-0': 'The minimization of the (renormalization group improved one-loop) electroweak scalar potential of the Minimal Supersymmetric Standard Model (MSSM) leads to the well-known relation [CITATION], [EQUATION] where the running potential parameters are evaluated at the scale [MATH] and where [MATH] and [MATH] are radiative corrections that arise from the derivatives of [MATH] evaluated at the potential minimum.', '1401.1162-1-2-1': 'The sensitivity of [MATH] to the input parameters has been used to construct the necessary (though not sufficient) condition for naturalness defined by the electroweak fine-tuning measure [CITATION], [EQUATION] where [MATH], [MATH] and [MATH].', '1401.1162-1-2-2': 'Also, [MATH] and [MATH], where [MATH] labels the various loop contributions included in Eq. ([REF]).', '1401.1162-1-2-3': 'Expressions for the [MATH] and [MATH] are given in the Appendix of the second paper of Ref. [CITATION].', '1401.1162-1-3-0': 'Note that [MATH] is essentially determined by the SUSY spectrum.', '1401.1162-1-3-1': 'It is independent of both the underlying mechanism by which the super-partners acquire their masses and of the messenger scale - [MATH] - at which this mechanism is operative.', '1401.1162-1-3-2': 'This is in sharp contrast to conventional measures of fine-tuning such as [MATH] [CITATION] or [MATH] [CITATION] where corrections such as [MATH] lead to very high values of these fine-tuning measures especially in models - such as mSUGRA - where the parameters defined at a very high energy scale.', '1401.1162-1-3-3': 'There is, of course, no contradiction since small [MATH] is, as we have mentioned, just a necessary condition for fine-tuning [CITATION].', '1401.1162-1-4-0': 'An immediate consequence of Eq. ([REF]) is that models with values of [MATH] are necessarily fine-tuned.', '1401.1162-1-4-1': 'We emphasize that although we have used the electroweak scale minimization conditions to argue this, the same conclusion follows even with the use of popular fine-tuning measures.', '1401.1162-1-4-2': 'This is because [MATH] runs very little between [MATH] and [MATH] so that the sensitivity of [MATH] to [MATH], the GUT scale value of [MATH], is changed by just [MATH]%.', '1401.1162-1-4-3': 'We thus conclude that a small value of [MATH] is a robust and necessary condition for naturalness irrespective of the fine-tuning measure that is used.', '1401.1162-1-4-4': 'Stated differently, models with higgsinos heavier than 200 GeV (300 GeV) necessarily have a fine-tuning worse than 10% (3%).', '1401.1162-1-4-5': 'Experimental probes of light higgsinos pair production can thus decisively probe naturalness of SUSY models.', '1401.1162-1-5-0': 'Motivated by these considerations, we have examined the spectra and aspects of the phenomenology that result in models where [MATH].', '1401.1162-1-5-1': 'Typically, the dominant radiative corrections to Eq. ([REF]) come from the top-squark contributions [MATH].', '1401.1162-1-5-2': 'For negative values of the trilinear soft term [MATH] somewhat larger than the GUT scale scalar masses, each of [MATH] and [MATH] can be minimized whilst lifting up [MATH] into the 125 GeV regime [CITATION] as required by the discovery of the Higgs boson at the LHC [CITATION].', '1401.1162-1-5-3': 'Upon requiring no large independent contributions to Eq. ([REF]) (which would necessitate fine-tuning of the remaining parameters to keep [MATH] at [MATH] GeV), we find that', '1401.1162-1-6-0': 'Sparticle mass spectra consistent with low [MATH] can readily yield a value of [MATH] GeV whilst evading LHC8 search limits on squarks, gluinos and top-squarks [CITATION], and at the same time maintaining low electroweak fine-tuning, our necessary condition for naturalness.', '1401.1162-1-6-1': 'The key feature of the mass spectra implied by Eq. ([REF]) is the existence of four light higgsinos - [MATH], [MATH] and [MATH] - all with mass [MATH] GeV.', '1401.1162-1-6-2': 'While these light higgsinos can be produced at LHC at large rates, their compressed spectra with mass gaps [MATH] GeV results in only soft visible energy release from their three-body decays; this makes signal extraction from SM background exceedingly difficult, if not impossible.', '1401.1162-1-7-0': 'A new signature endemic to models with light higgsinos has also been pointed out in Ref. [CITATION]: [MATH] which results in hadronically quiet - because the decay products of [MATH] and [MATH] are soft - same sign diboson events (SSdB).', '1401.1162-1-7-1': 'The 300 fb[MATH] LHC14 reach for SSdBs extends to a wino mass of about 700 GeV.', '1401.1162-1-7-2': 'This corresponds to [MATH] TeV in models with gaugino mass unification, somewhat larger than the LHC14 reach for gluino pair production [CITATION].', '1401.1162-1-7-3': 'Confirmatory signals will also be present in multilepton channels [CITATION].', '1401.1162-1-7-4': 'Since [MATH] can extend up to [MATH] TeV while maintaining low [MATH], then LHC14 can probe only a fraction of the parameter space of natural SUSY in this manner.', '1401.1162-1-8-0': 'An alternative LHC search strategy has been proposed in a variety of papers (for a summary and detailed references, see e.g. Ref. [CITATION]), namely to look for initial state QED/QCD-radiation off WIMP pair production.', '1401.1162-1-8-1': 'Much of this work [CITATION] has been carried out using effective operator analyses.', '1401.1162-1-8-2': 'Here, it is assumed that the interactions between the dark matter particle and SM quarks occur via very heavy mediators (usually [MATH]- and [MATH]-channel squarks in the context of the MSSM bino-like WIMP) so that the contact approximation is valid.', '1401.1162-1-8-3': 'It is clear that for MSSM higgsino pair production the contact interaction approximation breaks down very badly because higgsinos are dominantly produced by collisions of quarks and anti-quarks (inside the protons) via [MATH]-channel [MATH] exchange.', '1401.1162-1-8-4': 'Since higgsinos are necessarily heavier than [MATH] GeV, the [MATH]-boson propagator suppresses the amplitude for higgsino production by an extra factor of [MATH] relative to the contact-interaction approximation.', '1401.1162-1-8-5': 'This results in a suppression of the cross-section where the higgsino pair is produced with large invariant mass.', '1401.1162-1-8-6': 'Since radiation of hard gluons or photons is most likely in this regime, the contact interaction approximation will badly overestimate the cross section for high [MATH] monojet and mono-photon events, as has already been emphasized in Ref. [CITATION].', '1401.1162-1-8-7': 'As a result, constraints [CITATION] using effective operator analyses, therefore, do not apply in the case that the light SUSY states are higgsinos.', '1401.1162-1-9-0': 'In a recent analysis, Han et al. [CITATION] have computed the monojet signal in the natural SUSY framework with light higgsinos using the complete matrix element.', '1401.1162-1-9-1': 'An advantage of applying this technique to models with light higgsinos is that one is not restricted to just WIMP ([MATH]) pair production, but one may radiate off gluons or photons in several other reactions as well: [MATH], [MATH], [MATH] and [MATH], since again the heavier higgsino decay debris is expected to be soft (unless highly boosted) at LHC.', '1401.1162-1-9-2': 'Including all the relevant contributions, these authors claim that LHC14, with an integrated luminosity of 1500 fb[MATH] will be able to probe higgsinos up to 200 GeV at 5[MATH] [CITATION].', '1401.1162-1-9-3': 'If their results hold up to scrutiny, it will imply that experiments at the high luminosity upgrade of the LHC will decisively probe SUSY models fine-tuned to no more than 10%.', '1401.1162-1-10-0': 'Given the importance of this result, we re-examine prospects for detection of monojet radiation off of higgsino pair production in Sec. [REF].', '1401.1162-1-10-1': 'Our conclusions are, however, quite different from those of Ref. [CITATION] since we find signal well below SM backgrounds (at the percent level), with no distinctive monojet features which would allow separation of signal from background.', '1401.1162-1-10-2': 'In Sec. [REF], we perform similar calculations for mono-photon radiation and arrive at similarly pessimistic conclusions.', '1401.1162-1-10-3': 'We have decided such a pessimistic assessment is worthy of publication not only because of the optimistic claims in the literature [CITATION], but also to highlight that claims about the observability of monojet/mono-photon signals from effective operator analyses should be viewed with caution.', '1401.1162-1-11-0': '# Prospects for monojets', '1401.1162-1-12-0': 'To examine signal rates, we first select a low [MATH] SUSY benchmark model from radiatively-driven natural SUSY (RNS) which uses the 2-extra-parameter non-universal Higgs model (NUHM2) with input parameters [EQUATION] with [MATH] GeV.', '1401.1162-1-12-1': 'We generate the sparticle spectrum using Isajet 7.84 [CITATION].', '1401.1162-1-12-2': 'We fix [MATH] TeV, [MATH] GeV, [MATH] TeV, [MATH], [MATH] GeV and [MATH] TeV.', '1401.1162-1-12-3': 'This leads to a sparticle spectrum with [MATH] TeV, very heavy squarks and sleptons, binos and winos with masses of several hundred GeV, and a set of higgsinos with [MATH] GeV, [MATH] GeV and [MATH] GeV.', '1401.1162-1-12-4': 'These higgsinos are, of course, the focus of the present study, and our broad conclusions are essentially independent of the rest of the spectrum, as long as the bino and wino states are much heavier than the higgsino states.', '1401.1162-1-12-5': 'The value of [MATH].', '1401.1162-1-13-0': 'We use Madgraph 5 [CITATION] to generate [MATH], [MATH] and [MATH] plus one-parton processes (exclusive) and plus two-partons (inclusive) where for efficiency we require the hardest final state parton to have [MATH] GeV; the final cross section is then the sum of 1-jet exclusive and 2-jet inclusive processes.', '1401.1162-1-13-1': "We also evaluate the [MATH], [MATH] and [MATH] backgrounds (where [MATH]'s decay to neutrinos and [MATH]'s decay leptonically) in a similar fashion, as the sum of one- and two-parton processes.", '1401.1162-1-13-2': 'The events are then passed to Pythia [CITATION] for showering, hadronization and underlying event.', '1401.1162-1-14-0': 'The Madgraph/Pythia events are then passed to the Isajet toy detector simulation with calorimeter cell size [MATH] and [MATH].', '1401.1162-1-14-1': 'The HCAL (hadronic calorimetry) energy resolution is taken to be [MATH] for [MATH] and FCAL (forward calorimetry) is [MATH] for [MATH], where the two terms are combined in quadrature.', '1401.1162-1-14-2': 'The ECAL (electromagnetic calorimetry) energy resolution is assumed to be [MATH].', '1401.1162-1-14-3': 'We use the cone-type Isajet jet-finding algorithm [CITATION] to group the hadronic final states into jets.', '1401.1162-1-14-4': 'Jets and isolated leptons are defined as follows:', '1401.1162-1-15-0': 'Following the Atlas monojet study [CITATION], we impose the following cuts:', '1401.1162-1-16-0': 'Our resulting distributions are shown in Fig. [REF] for a) [MATH] and b) [MATH].', '1401.1162-1-16-1': 'As expected, the shapes of the two distributions are similar for large [MATH] and large [MATH] but begin to differ for values below [MATH] GeV where details of event generation and the presence of the second jet may be important.', '1401.1162-1-16-2': 'From frame a), we see that [MATH] production forms the dominant background, followed closely by [MATH] production where the lepton from [MATH]-decay is too soft or buried within a jet or too forward or otherwise unidentified.', '1401.1162-1-16-3': 'The signal is shown by the red solid histogram and lies typically about two orders of magnitude below the background distribution.', '1401.1162-1-16-4': 'We also show the distribution from [MATH] production, which is sub-dominant.', '1401.1162-1-16-5': 'Essentially the same qualitative features are also seen in frame b).', '1401.1162-1-16-6': 'Nowhere in either frame does the signal emerge from background.', '1401.1162-1-16-7': 'Other cuts such as angular distributions do not help the situation since both signal and BG are dominated by gluon radiation off initial state quarks: really, the main difference between signal and background as far as ISR goes is that for signal the ISR comes off a somewhat higher [MATH] subprocess.', '1401.1162-1-16-8': 'The effect of sequential cuts on the signal and on the background is shown in Table [REF].', '1401.1162-1-17-0': 'Given that the signal and background have similar shapes and that [MATH], it is very difficult to make the case that it will be possible to realistically extract the signal [CITATION].', '1401.1162-1-17-1': 'Of course, with sufficient integrated luminosity, the statistical significance will always exceed [MATH], but to claim that this means the signal is observable means that the background is known with a precision better than a percent!', '1401.1162-1-18-0': '# Prospects for mono-photons', '1401.1162-1-19-0': 'For mono-photon events (which we include for completeness), we generate the same signal sample as before, including all higgsino pair production reactions, but now requiring one hard photon (with [MATH] GeV) radiation instead of a hard jet.', '1401.1162-1-19-1': 'We also generate the background processes [MATH] production (followed by [MATH]) and [MATH] production (followed by [MATH] where [MATH], [MATH] or [MATH]) as before using Madgraph plus Pythia.', '1401.1162-1-20-0': 'For the isolated mono-photon sample, we require [CITATION]:', '1401.1162-1-21-0': 'We regard a photon to be isolated if the energy in a cone of radius [MATH] around photon with [MATH] GeV, [MATH] is less than 5 GeV.', '1401.1162-1-22-0': 'Our signal and background distributions in [MATH] and [MATH] are shown in Fig. [REF].', '1401.1162-1-22-1': 'As in Fig. [REF], we see that the shapes agree for large values of [MATH] and [MATH].', '1401.1162-1-22-2': 'For the entire range of [MATH] as well as of [MATH], we again find that signal (solid red histogram) lies below the [MATH] background by typically two orders of magnitude.', '1401.1162-1-22-3': 'The [MATH] background falls more sharply than the [MATH] background.', '1401.1162-1-22-4': 'This is because when we require much higher [MATH] values, then the [MATH] recoils more sharply against the gamma, and its decay products are more likely to be hard and isolated, and to not pass the lepton/tau veto requirements.', '1401.1162-1-22-5': 'The effect of the sequential cuts on the signal and background cross sections is shown in Table [REF].', '1401.1162-1-22-6': 'Once again, there are no distinctive features in the distribution, and as for the monojet signal of the previous section, we deem the mono-photon signal to be unobservable because of the very small [MATH] ratio.', '1401.1162-1-23-0': '# Concluding remarks', '1401.1162-1-24-0': 'The existence of light higgsinos with masses smaller than 200-300 GeV (depending on how much fine-tuning one is willing to tolerate) is a robust feature of natural SUSY models.', '1401.1162-1-24-1': 'Although these higgsinos can be pair produced at large rates at the LHC, the signal will be buried below SM backgrounds because of the small energy release from their decays.', '1401.1162-1-24-2': 'In this paper, we have examined prospects for their detection via pair production in association with a hard jet or a hard, isolated photon resulting in characteristic monojet or mono-photon events at LHC14.', '1401.1162-1-24-3': 'We emphasize here that constraints obtained from analyses [CITATION] using contact interactions between quarks and the higgsinos are inapplicable in this connection because the effective operator approximation fails badly for higgsino pair production.', '1401.1162-1-25-0': 'While monojet and mono-photon signal events indeed occur at an observable rate particularly at the luminosity upgrade of the LHC, we are pessimistic about the prospects for their detection because backgrounds from [MATH] and [MATH] production in association with a jet or an isolated photon overwhelm the signal by two orders of magnitude even for very large values of jet or photon transverse momentum and [MATH] in these events.', '1401.1162-1-25-1': 'Our conclusion is in contrast with the assessment in Ref. [CITATION] where it is claimed that LHC experiments with an integrated luminosity of 1500 [MATH] should be sensitive to a 5[MATH] monojet signal if [MATH] GeV.', '1401.1162-1-26-0': 'In arriving at our negative conclusion, we should mention that have not investigated whether it might be possible to extract the higgsino signal by examining the soft debris from the decays of [MATH] and [MATH] produced via [MATH], [MATH] and [MATH] pair production processes that dominate higgsino pair production [CITATION].', '1401.1162-1-26-1': 'This will require a careful analysis of potential backgrounds from higher order Standard Model processes.', '1401.1162-1-26-2': 'Despite our cautious pessimism, we leave open the possibility that a clever analysis may make it feasible to tease out this signal at a luminosity upgrade of LHC14.'}

{'1401.1162-2-0-0': 'Naturalness arguments imply the existence of higgsinos lighter than 200-300 GeV.', '1401.1162-2-0-1': 'However, because these higgsinos are nearly mass degenerate, they release very little visible energy in their decays, and signals from electroweak higgsino pair production typically remain buried under Standard Model backgrounds.', '1401.1162-2-0-2': 'Moreover, gluinos, squarks and winos may plausibly lie beyond the reach of the LHC14, so that signals from naturalness-inspired supersymmetric models may well remain hidden via conventional searches.', '1401.1162-2-0-3': 'We examine instead prospects for detecting higgsino pair production via monojets or mono-photons from initial state radiation.', '1401.1162-2-0-4': 'We find typical signal-to-background rates at best at the 1% level and without any spectral distortions, leading to rather pessimistic conclusions regarding detectability via these channels.', '1401.1162-2-1-0': '# Introduction', '1401.1162-2-2-0': 'The minimization of the (renormalization group improved one-loop) electroweak scalar potential of the Minimal Supersymmetric Standard Model (MSSM) leads to the well-known relation [CITATION], [EQUATION] where the running potential parameters are evaluated at the scale [MATH] and where [MATH] and [MATH] are radiative corrections that arise from the derivatives of [MATH] evaluated at the potential minimum.', '1401.1162-2-2-1': 'The sensitivity of [MATH] to the input parameters has been used to construct the necessary (though not sufficient) condition for naturalness defined by the electroweak fine-tuning measure [CITATION], [EQUATION] where [MATH], [MATH] and [MATH].', '1401.1162-2-2-2': 'Also, [MATH] and [MATH], where [MATH] labels the various loop contributions included in Eq. ([REF]).', '1401.1162-2-2-3': 'Expressions for the [MATH] and [MATH] are given in the Appendix of the second paper of Ref. [CITATION].', '1401.1162-2-3-0': 'Note that [MATH] is essentially determined by the SUSY spectrum.', '1401.1162-2-3-1': 'It is independent of both the underlying mechanism by which the super-partners acquire their masses and of the messenger scale - [MATH] - at which this mechanism is operative.', '1401.1162-2-3-2': 'This is in sharp contrast to conventional measures of fine-tuning such as [MATH] [CITATION] or [MATH] [CITATION] where corrections such as [MATH] lead to very high values of these fine-tuning measures especially in models - such as mSUGRA - where the parameters defined at a very high energy scale.', '1401.1162-2-3-3': 'There is, of course, no contradiction since small [MATH] is, as we have mentioned, just a necessary condition for fine-tuning [CITATION].', '1401.1162-2-4-0': 'An immediate consequence of Eq. ([REF]) is that models with values of [MATH] are necessarily fine-tuned.', '1401.1162-2-4-1': 'We emphasize that although we have used the electroweak scale minimization conditions to argue this, the same conclusion follows even with the use of popular fine-tuning measures.', '1401.1162-2-4-2': 'This is because [MATH] runs very little between [MATH] and [MATH] so that the sensitivity of [MATH] to [MATH], the GUT scale value of [MATH], is changed by just [MATH]%.', '1401.1162-2-4-3': 'We thus conclude that a small value of [MATH] is a robust and necessary condition for naturalness irrespective of the fine-tuning measure that is used.', '1401.1162-2-4-4': 'Stated differently, models with higgsinos heavier than 200 GeV (300 GeV) necessarily have a fine-tuning worse than 10% (3%).', '1401.1162-2-4-5': 'Experimental probes of light higgsinos pair production can thus decisively probe naturalness of SUSY models.', '1401.1162-2-5-0': 'Motivated by these considerations, we have examined the spectra and aspects of the phenomenology that result in models where [MATH].', '1401.1162-2-5-1': 'Typically, the dominant radiative corrections to Eq. ([REF]) come from the top-squark contributions [MATH].', '1401.1162-2-5-2': 'For negative values of the trilinear soft term [MATH] somewhat larger than the GUT scale scalar masses, each of [MATH] and [MATH] can be minimized whilst lifting up [MATH] into the 125 GeV regime [CITATION] as required by the discovery of the Higgs boson at the LHC [CITATION].', '1401.1162-2-5-3': 'Upon requiring no large independent contributions to Eq. ([REF]) (which would necessitate fine-tuning of the remaining parameters to keep [MATH] at [MATH] GeV), we find that', '1401.1162-2-6-0': 'Sparticle mass spectra consistent with low [MATH] can readily yield a value of [MATH] GeV whilst evading LHC8 search limits on squarks, gluinos and top-squarks [CITATION], and at the same time maintaining low electroweak fine-tuning, our necessary condition for naturalness.', '1401.1162-2-6-1': 'The key feature of the mass spectra implied by Eq. ([REF]) is the existence of four light higgsinos - [MATH], [MATH] and [MATH] - all with mass [MATH] GeV.', '1401.1162-2-6-2': 'While these light higgsinos can be produced at LHC at large rates, their compressed spectra with mass gaps [MATH] GeV results in only soft visible energy release from their three-body decays; this makes signal extraction from SM background exceedingly difficult, if not impossible.', '1401.1162-2-7-0': 'A new signature endemic to models with light higgsinos has also been pointed out in Ref. [CITATION]: [MATH] which results in hadronically quiet - because the decay products of [MATH] and [MATH] are soft - same sign diboson events (SSdB).', '1401.1162-2-7-1': 'The 300 fb[MATH] LHC14 reach for SSdBs extends to a wino mass of about 700 GeV.', '1401.1162-2-7-2': 'This corresponds to [MATH] TeV in models with gaugino mass unification, somewhat larger than the LHC14 reach for gluino pair production [CITATION].', '1401.1162-2-7-3': 'Confirmatory signals will also be present in multilepton channels [CITATION].', '1401.1162-2-7-4': 'Since [MATH] can extend up to [MATH] TeV while maintaining low [MATH], then LHC14 can probe only a fraction of the parameter space of natural SUSY in this manner.', '1401.1162-2-8-0': 'An alternative LHC search strategy has been proposed in a variety of papers (for a summary and detailed references, see e.g. Ref. [CITATION]), namely to look for initial state QED/QCD-radiation off WIMP pair production.', '1401.1162-2-8-1': 'Much of this work [CITATION] has been carried out using effective operator analyses.', '1401.1162-2-8-2': 'Here, it is assumed that the interactions between the dark matter particle and SM quarks occur via very heavy mediators (usually [MATH]- and [MATH]-channel squarks in the context of the MSSM bino-like WIMP) so that the contact approximation is valid.', '1401.1162-2-8-3': 'It is clear that for MSSM higgsino pair production the contact interaction approximation breaks down very badly because higgsinos are dominantly produced by collisions of quarks and anti-quarks (inside the protons) via [MATH]-channel [MATH] exchange.', '1401.1162-2-8-4': 'Since higgsinos are necessarily heavier than [MATH] GeV, the [MATH]-boson propagator suppresses the amplitude for higgsino production by an extra factor of [MATH] relative to the contact-interaction approximation.', '1401.1162-2-8-5': 'This results in a suppression of the cross-section where the higgsino pair is produced with large invariant mass.', '1401.1162-2-8-6': 'Since radiation of hard gluons or photons is most likely in this regime, the contact interaction approximation will badly overestimate the cross section for high [MATH] monojet and mono-photon events, as has already been emphasized in Ref. [CITATION].', '1401.1162-2-8-7': 'As a result, constraints [CITATION] using effective operator analyses, therefore, do not apply in the case that the light SUSY states are higgsinos.', '1401.1162-2-9-0': 'In a recent analysis, Han et al. [CITATION] have computed the monojet signal in the natural SUSY framework with light higgsinos using the complete matrix element.', '1401.1162-2-9-1': 'An advantage of applying this technique to models with light higgsinos is that one is not restricted to just WIMP ([MATH]) pair production, but one may radiate off gluons or photons in several other reactions as well: [MATH], [MATH], [MATH] and [MATH], since again the heavier higgsino decay debris is expected to be soft (unless highly boosted) at LHC.', '1401.1162-2-9-2': 'Including all the relevant contributions, these authors claim that LHC14, with an integrated luminosity of 1500 fb[MATH] will be able to probe higgsinos up to 200 GeV at 5[MATH] [CITATION].', '1401.1162-2-9-3': 'If their results hold up to scrutiny, it will imply that experiments at the high luminosity upgrade of the LHC will decisively probe SUSY models fine-tuned to no more than 10%.', '1401.1162-2-10-0': 'Given the importance of this result, we re-examine prospects for detection of monojet radiation off of higgsino pair production in Sec. [REF].', '1401.1162-2-10-1': 'Our conclusions are, however, quite different from those of Ref. [CITATION] since we find signal well below SM backgrounds (at the percent level), with no distinctive monojet features which would allow separation of signal from background.', '1401.1162-2-10-2': 'In Sec. [REF], we perform similar calculations for mono-photon radiation and arrive at similarly pessimistic conclusions.', '1401.1162-2-10-3': 'We have decided such a pessimistic assessment is worthy of publication not only because of the optimistic claims in the literature [CITATION], but also to highlight that claims about the observability of monojet/mono-photon signals from effective operator analyses should be viewed with caution.', '1401.1162-2-11-0': '# Prospects for monojets', '1401.1162-2-12-0': 'To examine signal rates, we first select a low [MATH] SUSY benchmark model from radiatively-driven natural SUSY (RNS) which uses the 2-extra-parameter non-universal Higgs model (NUHM2) with input parameters [EQUATION] with [MATH] GeV.', '1401.1162-2-12-1': 'We generate the sparticle spectrum using Isajet 7.84 [CITATION].', '1401.1162-2-12-2': 'We fix [MATH] TeV, [MATH] GeV, [MATH] TeV, [MATH], [MATH] GeV and [MATH] TeV.', '1401.1162-2-12-3': 'This leads to a sparticle spectrum with [MATH] TeV, very heavy squarks and sleptons, binos and winos with masses of several hundred GeV, and a set of higgsinos with [MATH] GeV, [MATH] GeV and [MATH] GeV.', '1401.1162-2-12-4': 'These higgsinos are, of course, the focus of the present study, and our broad conclusions are essentially independent of the rest of the spectrum, as long as the bino and wino states are much heavier than the higgsino states.', '1401.1162-2-12-5': 'The value of [MATH].', '1401.1162-2-13-0': 'We use Madgraph 5 [CITATION] to generate [MATH], [MATH] and [MATH] plus one-parton processes (exclusive) and plus two-partons (inclusive) where for efficiency we require the hardest final state parton to have [MATH] GeV; the final cross section is then the sum of 1-jet exclusive and 2-jet inclusive processes.', '1401.1162-2-13-1': "We also evaluate the [MATH], [MATH] and [MATH] backgrounds (where [MATH]'s decay to neutrinos and [MATH]'s decay leptonically) in a similar fashion, as the sum of one- and two-parton processes.", '1401.1162-2-13-2': 'To avoid double-counting, we used the MLM scheme for jet-parton matching [CITATION].', '1401.1162-2-13-3': 'The events are then passed to Pythia [CITATION] for showering, hadronization and underlying event.', '1401.1162-2-13-4': 'We have not evaluated the hard monojet background from top quark pair production which we expect to be very small after the veto on additional jets and leptons.', '1401.1162-2-13-5': 'This is confirmed by the results of Ref. [CITATION].', '1401.1162-2-14-0': 'The Madgraph/Pythia events are then passed to the Isajet toy detector simulation with calorimeter cell size [MATH] and [MATH].', '1401.1162-2-14-1': 'The HCAL (hadronic calorimetry) energy resolution is taken to be [MATH] for [MATH] and FCAL (forward calorimetry) is [MATH] for [MATH], where the two terms are combined in quadrature.', '1401.1162-2-14-2': 'The ECAL (electromagnetic calorimetry) energy resolution is assumed to be [MATH].', '1401.1162-2-14-3': 'We use the cone-type Isajet jet-finding algorithm [CITATION] to group the hadronic final states into jets.', '1401.1162-2-14-4': 'Jets and isolated leptons are defined as follows:', '1401.1162-2-15-0': 'Following the Atlas monojet study [CITATION], we impose the following cuts:', '1401.1162-2-16-0': 'Our resulting distributions are shown in Fig. [REF] for a) [MATH] and b) [MATH].', '1401.1162-2-16-1': 'As expected, the shapes of the two distributions are similar for large [MATH] and large [MATH] but begin to differ for values below [MATH] GeV where details of event generation and the presence of the second jet may be important.', '1401.1162-2-16-2': 'From frame a), we see that [MATH] production forms the dominant background, followed closely by [MATH] production where the lepton from [MATH]-decay is too soft or buried within a jet or too forward or otherwise unidentified.', '1401.1162-2-16-3': 'The signal is shown by the red solid histogram and lies typically about two orders of magnitude below the background distribution.', '1401.1162-2-16-4': 'We also show the distribution from [MATH] production, which is sub-dominant.', '1401.1162-2-16-5': 'Essentially the same qualitative features are also seen in frame b).', '1401.1162-2-16-6': 'Nowhere in either frame does the signal emerge from background.', '1401.1162-2-16-7': 'Other cuts such as angular distributions do not help the situation since both signal and BG are dominated by gluon radiation off initial state quarks: really, the main difference between signal and background as far as ISR goes is that for signal the ISR comes off a somewhat higher [MATH] subprocess.', '1401.1162-2-16-8': 'The effect of sequential cuts on the signal and on the background is shown in Table [REF].', '1401.1162-2-17-0': 'Given that the signal and background have similar shapes and that [MATH], it is very difficult to make the case that it will be possible to realistically extract the signal [CITATION].', '1401.1162-2-17-1': 'Of course, with sufficient integrated luminosity, the statistical significance will always exceed [MATH], but to claim that this means the signal is observable means that the background is known with a precision better than a percent!', '1401.1162-2-18-0': '# Prospects for mono-photons', '1401.1162-2-19-0': 'For mono-photon events (which we include for completeness), we generate the same signal sample as before, including all higgsino pair production reactions, but now requiring one hard photon (with [MATH] GeV) radiation instead of a hard jet.', '1401.1162-2-19-1': 'We also generate the background processes [MATH] production (followed by [MATH]) and [MATH] production (followed by [MATH] where [MATH], [MATH] or [MATH]) as before using Madgraph plus Pythia.', '1401.1162-2-20-0': 'For the isolated mono-photon sample, we require [CITATION]:', '1401.1162-2-21-0': 'We regard a photon to be isolated if the energy in a cone of radius [MATH] around photon with [MATH] GeV, [MATH] is less than 5 GeV.', '1401.1162-2-22-0': 'Our signal and background distributions in [MATH] and [MATH] are shown in Fig. [REF].', '1401.1162-2-22-1': 'As in Fig. [REF], we see that the shapes agree for large values of [MATH] and [MATH].', '1401.1162-2-22-2': 'For the entire range of [MATH] as well as of [MATH], we again find that signal (solid red histogram) lies below the [MATH] background by typically two orders of magnitude.', '1401.1162-2-22-3': 'The [MATH] background falls more sharply than the [MATH] background.', '1401.1162-2-22-4': 'This is because when we require much higher [MATH] values, then the [MATH] recoils more sharply against the gamma, and its decay products are more likely to be hard and isolated, and to not pass the lepton/tau veto requirements.', '1401.1162-2-22-5': 'The effect of the sequential cuts on the signal and background cross sections is shown in Table [REF].', '1401.1162-2-22-6': 'Once again, there are no distinctive features in the distribution, and as for the monojet signal of the previous section, we deem the mono-photon signal to be unobservable because of the very small [MATH] ratio.', '1401.1162-2-23-0': '# Concluding remarks', '1401.1162-2-24-0': 'The existence of light higgsinos with masses smaller than 200-300 GeV (depending on how much fine-tuning one is willing to tolerate) is a robust feature of natural SUSY models.', '1401.1162-2-24-1': 'Although these higgsinos can be pair produced at large rates at the LHC, the signal will be buried below SM backgrounds because of the small energy release from their decays.', '1401.1162-2-24-2': 'In this paper, we have examined prospects for their detection via pair production in association with a hard jet or a hard, isolated photon resulting in characteristic monojet or mono-photon events at LHC14.', '1401.1162-2-24-3': 'We emphasize here that constraints obtained from analyses [CITATION] using contact interactions between quarks and the higgsinos are inapplicable in this connection because the effective operator approximation fails badly for higgsino pair production.', '1401.1162-2-25-0': 'While monojet and mono-photon signal events indeed occur at an observable rate particularly at the luminosity upgrade of the LHC, we are pessimistic about the prospects for their detection because backgrounds from [MATH] and [MATH] production in association with a jet or an isolated photon overwhelm the signal by two orders of magnitude even for very large values of jet or photon transverse momentum and [MATH] in these events.', '1401.1162-2-25-1': 'It seems to us difficult to imagine that it would be possible to claim a signal for new physics in these channels based solely on an excess of [MATH] without an observable distortion of any distribution.', '1401.1162-2-26-0': 'In arriving at our negative conclusion, we should mention that have not investigated whether it might be possible to extract the higgsino signal by examining the soft debris from the decays of [MATH] and [MATH] produced via [MATH], [MATH] and [MATH] pair production processes that dominate higgsino pair production [CITATION].', '1401.1162-2-26-1': 'This will require a careful analysis of potential backgrounds from higher order Standard Model processes.', '1401.1162-2-26-2': 'Despite our cautious pessimism, we leave open the possibility that a clever analysis may make it feasible to tease out this signal at a luminosity upgrade of LHC14.'}

[['1401.1162-1-13-0', '1401.1162-2-13-0'], ['1401.1162-1-13-1', '1401.1162-2-13-1'], ['1401.1162-1-13-2', '1401.1162-2-13-3'], ['1401.1162-1-8-0', '1401.1162-2-8-0'], ['1401.1162-1-8-1', '1401.1162-2-8-1'], ['1401.1162-1-8-2', '1401.1162-2-8-2'], ['1401.1162-1-8-3', '1401.1162-2-8-3'], ['1401.1162-1-8-4', '1401.1162-2-8-4'], ['1401.1162-1-8-5', '1401.1162-2-8-5'], ['1401.1162-1-8-6', '1401.1162-2-8-6'], ['1401.1162-1-8-7', '1401.1162-2-8-7'], ['1401.1162-1-7-0', '1401.1162-2-7-0'], ['1401.1162-1-7-1', '1401.1162-2-7-1'], ['1401.1162-1-7-2', '1401.1162-2-7-2'], ['1401.1162-1-7-3', '1401.1162-2-7-3'], ['1401.1162-1-7-4', '1401.1162-2-7-4'], ['1401.1162-1-17-0', '1401.1162-2-17-0'], ['1401.1162-1-17-1', '1401.1162-2-17-1'], ['1401.1162-1-5-0', '1401.1162-2-5-0'], ['1401.1162-1-5-1', '1401.1162-2-5-1'], ['1401.1162-1-5-2', '1401.1162-2-5-2'], ['1401.1162-1-5-3', '1401.1162-2-5-3'], ['1401.1162-1-4-0', '1401.1162-2-4-0'], ['1401.1162-1-4-1', '1401.1162-2-4-1'], ['1401.1162-1-4-2', '1401.1162-2-4-2'], ['1401.1162-1-4-3', '1401.1162-2-4-3'], ['1401.1162-1-4-4', '1401.1162-2-4-4'], ['1401.1162-1-4-5', '1401.1162-2-4-5'], ['1401.1162-1-10-0', '1401.1162-2-10-0'], ['1401.1162-1-10-1', '1401.1162-2-10-1'], ['1401.1162-1-10-2', '1401.1162-2-10-2'], ['1401.1162-1-10-3', '1401.1162-2-10-3'], ['1401.1162-1-22-0', '1401.1162-2-22-0'], ['1401.1162-1-22-1', '1401.1162-2-22-1'], ['1401.1162-1-22-2', '1401.1162-2-22-2'], ['1401.1162-1-22-3', '1401.1162-2-22-3'], ['1401.1162-1-22-4', '1401.1162-2-22-4'], ['1401.1162-1-22-5', '1401.1162-2-22-5'], ['1401.1162-1-22-6', '1401.1162-2-22-6'], ['1401.1162-1-12-0', '1401.1162-2-12-0'], ['1401.1162-1-12-1', '1401.1162-2-12-1'], ['1401.1162-1-12-3', '1401.1162-2-12-3'], ['1401.1162-1-12-4', '1401.1162-2-12-4'], ['1401.1162-1-12-5', '1401.1162-2-12-5'], ['1401.1162-1-2-0', '1401.1162-2-2-0'], ['1401.1162-1-2-1', '1401.1162-2-2-1'], ['1401.1162-1-2-2', '1401.1162-2-2-2'], ['1401.1162-1-2-3', '1401.1162-2-2-3'], ['1401.1162-1-0-0', '1401.1162-2-0-0'], ['1401.1162-1-0-1', '1401.1162-2-0-1'], ['1401.1162-1-0-2', '1401.1162-2-0-2'], ['1401.1162-1-0-3', '1401.1162-2-0-3'], ['1401.1162-1-21-0', '1401.1162-2-21-0'], ['1401.1162-1-14-0', '1401.1162-2-14-0'], ['1401.1162-1-14-1', '1401.1162-2-14-1'], ['1401.1162-1-14-2', '1401.1162-2-14-2'], ['1401.1162-1-14-3', '1401.1162-2-14-3'], ['1401.1162-1-16-0', '1401.1162-2-16-0'], ['1401.1162-1-16-1', '1401.1162-2-16-1'], ['1401.1162-1-16-2', '1401.1162-2-16-2'], ['1401.1162-1-16-3', '1401.1162-2-16-3'], ['1401.1162-1-16-4', '1401.1162-2-16-4'], ['1401.1162-1-16-5', '1401.1162-2-16-5'], ['1401.1162-1-16-6', '1401.1162-2-16-6'], ['1401.1162-1-16-7', '1401.1162-2-16-7'], ['1401.1162-1-16-8', '1401.1162-2-16-8'], ['1401.1162-1-25-0', '1401.1162-2-25-0'], ['1401.1162-1-24-0', '1401.1162-2-24-0'], ['1401.1162-1-24-1', '1401.1162-2-24-1'], ['1401.1162-1-24-2', '1401.1162-2-24-2'], ['1401.1162-1-24-3', '1401.1162-2-24-3'], ['1401.1162-1-9-0', '1401.1162-2-9-0'], ['1401.1162-1-9-1', '1401.1162-2-9-1'], ['1401.1162-1-9-2', '1401.1162-2-9-2'], ['1401.1162-1-9-3', '1401.1162-2-9-3'], ['1401.1162-1-6-0', '1401.1162-2-6-0'], ['1401.1162-1-6-1', '1401.1162-2-6-1'], ['1401.1162-1-6-2', '1401.1162-2-6-2'], ['1401.1162-1-26-0', '1401.1162-2-26-0'], ['1401.1162-1-26-1', '1401.1162-2-26-1'], ['1401.1162-1-26-2', '1401.1162-2-26-2'], ['1401.1162-1-19-0', '1401.1162-2-19-0'], ['1401.1162-1-19-1', '1401.1162-2-19-1'], ['1401.1162-1-3-0', '1401.1162-2-3-0'], ['1401.1162-1-3-1', '1401.1162-2-3-1'], ['1401.1162-1-3-2', '1401.1162-2-3-2'], ['1401.1162-1-3-3', '1401.1162-2-3-3'], ['1401.1162-1-0-4', '1401.1162-2-0-4']]

[['1401.1162-1-13-0', '1401.1162-2-13-0'], ['1401.1162-1-13-1', '1401.1162-2-13-1'], ['1401.1162-1-13-2', '1401.1162-2-13-3'], ['1401.1162-1-8-0', '1401.1162-2-8-0'], ['1401.1162-1-8-1', '1401.1162-2-8-1'], ['1401.1162-1-8-2', '1401.1162-2-8-2'], ['1401.1162-1-8-3', '1401.1162-2-8-3'], ['1401.1162-1-8-4', '1401.1162-2-8-4'], ['1401.1162-1-8-5', '1401.1162-2-8-5'], ['1401.1162-1-8-6', '1401.1162-2-8-6'], ['1401.1162-1-8-7', '1401.1162-2-8-7'], ['1401.1162-1-7-0', '1401.1162-2-7-0'], ['1401.1162-1-7-1', '1401.1162-2-7-1'], ['1401.1162-1-7-2', '1401.1162-2-7-2'], ['1401.1162-1-7-3', '1401.1162-2-7-3'], ['1401.1162-1-7-4', '1401.1162-2-7-4'], ['1401.1162-1-17-0', '1401.1162-2-17-0'], ['1401.1162-1-17-1', '1401.1162-2-17-1'], ['1401.1162-1-5-0', '1401.1162-2-5-0'], ['1401.1162-1-5-1', '1401.1162-2-5-1'], ['1401.1162-1-5-2', '1401.1162-2-5-2'], ['1401.1162-1-5-3', '1401.1162-2-5-3'], ['1401.1162-1-4-0', '1401.1162-2-4-0'], ['1401.1162-1-4-1', '1401.1162-2-4-1'], ['1401.1162-1-4-2', '1401.1162-2-4-2'], ['1401.1162-1-4-3', '1401.1162-2-4-3'], ['1401.1162-1-4-4', '1401.1162-2-4-4'], ['1401.1162-1-4-5', '1401.1162-2-4-5'], ['1401.1162-1-10-0', '1401.1162-2-10-0'], ['1401.1162-1-10-1', '1401.1162-2-10-1'], ['1401.1162-1-10-2', '1401.1162-2-10-2'], ['1401.1162-1-10-3', '1401.1162-2-10-3'], ['1401.1162-1-22-0', '1401.1162-2-22-0'], ['1401.1162-1-22-1', '1401.1162-2-22-1'], ['1401.1162-1-22-2', '1401.1162-2-22-2'], ['1401.1162-1-22-3', '1401.1162-2-22-3'], ['1401.1162-1-22-4', '1401.1162-2-22-4'], ['1401.1162-1-22-5', '1401.1162-2-22-5'], ['1401.1162-1-22-6', '1401.1162-2-22-6'], ['1401.1162-1-12-0', '1401.1162-2-12-0'], ['1401.1162-1-12-1', '1401.1162-2-12-1'], ['1401.1162-1-12-3', '1401.1162-2-12-3'], ['1401.1162-1-12-4', '1401.1162-2-12-4'], ['1401.1162-1-12-5', '1401.1162-2-12-5'], ['1401.1162-1-2-0', '1401.1162-2-2-0'], ['1401.1162-1-2-1', '1401.1162-2-2-1'], ['1401.1162-1-2-2', '1401.1162-2-2-2'], ['1401.1162-1-2-3', '1401.1162-2-2-3'], ['1401.1162-1-0-0', '1401.1162-2-0-0'], ['1401.1162-1-0-1', '1401.1162-2-0-1'], ['1401.1162-1-0-2', '1401.1162-2-0-2'], ['1401.1162-1-0-3', '1401.1162-2-0-3'], ['1401.1162-1-21-0', '1401.1162-2-21-0'], ['1401.1162-1-14-0', '1401.1162-2-14-0'], ['1401.1162-1-14-1', '1401.1162-2-14-1'], ['1401.1162-1-14-2', '1401.1162-2-14-2'], ['1401.1162-1-14-3', '1401.1162-2-14-3'], ['1401.1162-1-16-0', '1401.1162-2-16-0'], ['1401.1162-1-16-1', '1401.1162-2-16-1'], ['1401.1162-1-16-2', '1401.1162-2-16-2'], ['1401.1162-1-16-3', '1401.1162-2-16-3'], ['1401.1162-1-16-4', '1401.1162-2-16-4'], ['1401.1162-1-16-5', '1401.1162-2-16-5'], ['1401.1162-1-16-6', '1401.1162-2-16-6'], ['1401.1162-1-16-7', '1401.1162-2-16-7'], ['1401.1162-1-16-8', '1401.1162-2-16-8'], ['1401.1162-1-25-0', '1401.1162-2-25-0'], ['1401.1162-1-24-0', '1401.1162-2-24-0'], ['1401.1162-1-24-1', '1401.1162-2-24-1'], ['1401.1162-1-24-2', '1401.1162-2-24-2'], ['1401.1162-1-24-3', '1401.1162-2-24-3'], ['1401.1162-1-9-0', '1401.1162-2-9-0'], ['1401.1162-1-9-1', '1401.1162-2-9-1'], ['1401.1162-1-9-2', '1401.1162-2-9-2'], ['1401.1162-1-9-3', '1401.1162-2-9-3'], ['1401.1162-1-6-0', '1401.1162-2-6-0'], ['1401.1162-1-6-1', '1401.1162-2-6-1'], ['1401.1162-1-6-2', '1401.1162-2-6-2'], ['1401.1162-1-26-0', '1401.1162-2-26-0'], ['1401.1162-1-26-1', '1401.1162-2-26-1'], ['1401.1162-1-26-2', '1401.1162-2-26-2'], ['1401.1162-1-19-0', '1401.1162-2-19-0'], ['1401.1162-1-19-1', '1401.1162-2-19-1'], ['1401.1162-1-3-0', '1401.1162-2-3-0'], ['1401.1162-1-3-1', '1401.1162-2-3-1'], ['1401.1162-1-3-2', '1401.1162-2-3-2'], ['1401.1162-1-3-3', '1401.1162-2-3-3']]

[['1401.1162-1-0-4', '1401.1162-2-0-4']]

[]

[]

[]

['1401.1162-1-12-2', '1401.1162-1-14-4', '1401.1162-1-15-0', '1401.1162-1-20-0', '1401.1162-2-12-2', '1401.1162-2-14-4', '1401.1162-2-15-0', '1401.1162-2-20-0']

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

https://arxiv.org/abs/1401.1162

1809.10865

{'1809.10865-1-0-0': 'We investigate the effect of information asymmetry on a dynamic Cournot duopoly game with bounded rationality.', '1809.10865-1-0-1': "Concretely, we study how one player's possession of information about the other player's behavior in a duopoly affects the stability of the Cournot-Nash equilibrium.", '1809.10865-1-0-2': 'We theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses chaotic behavior in the duopoly.', '1809.10865-1-1-0': '# Introduction', '1809.10865-1-2-0': 'An oligopoly is a market where a few firms control the price of a good.', '1809.10865-1-2-1': 'Cournot first introduced the oligopoly model in 1838.', '1809.10865-1-2-2': 'He considered an economy of two firms (players) producing the same good, both firms choosing their respective production to maximize their profit.', '1809.10865-1-2-3': 'Because the profit of each firm depends on its production as well as the production of the other firm, the situation is game theoretic.', '1809.10865-1-2-4': 'Each firm needs to correctly anticipate the behavior of the other firm.', '1809.10865-1-2-5': "The equilibrium state between the two players is known as the Cournot-Nash equilibrium; here, each player's response to the other is optimal.", '1809.10865-1-2-6': 'This equilibrium is realized when the two players are sufficiently rational.', '1809.10865-1-3-0': 'However, in reality, the players are not necessarily rational.', '1809.10865-1-3-1': 'Studies have examined games in which the players have bounded rationality [CITATION].', '1809.10865-1-3-2': 'As a toy model, the Cournot duopoly game model with bounded rationality has recently attracted much attention.', '1809.10865-1-3-3': 'The discrete-time dynamics in a Cournot duopoly game with bounded rationality have been analyzed in Refs. [CITATION].', '1809.10865-1-3-4': 'In this model, two players adjust their outputs step by step in order to increase their respective profits.', '1809.10865-1-3-5': 'Here, the Cournot-Nash equilibrium is not necessarily stable, and even chaotic behavior can occur in some parameter regions, as in dynamic Cournot duopoly games with naive response [CITATION].', '1809.10865-1-3-6': 'Following these studies, the model was extended to the case of nonlinear demand function [CITATION].', '1809.10865-1-3-7': 'Another extension is the Cournot duopoly game with heterogeneous players, where the two players adopt different decision-making strategies [CITATION].', '1809.10865-1-3-8': 'In addition, the dynamic Cournot duopoly game with time delay [CITATION] as well as the Cournot triopoly with bounded rationality [CITATION] has been studied.', '1809.10865-1-3-9': 'All these studies report that the dynamics of an oligopoly with bounded rationality are very complicated.', '1809.10865-1-4-0': 'Another significant concept in modern microeconomics is information asymmetry, where one player has more or better information than the others.', '1809.10865-1-4-1': 'This asymmetry can create a power imbalance in transactions, which in turn could lead to market failure in the worst case.', '1809.10865-1-4-2': 'Studies have considered the effect of information asymmetry on games with rational players [CITATION].', '1809.10865-1-4-3': 'However, the concept of information asymmetry in games where the players have bounded rationality has not yet been well established.', '1809.10865-1-4-4': 'Therefore, we try to explain the effect of information asymmetry on games with bounded rationality by taking the Cournot duopoly game as an example.', '1809.10865-1-5-0': 'Concretely, we investigate a discrete-time dynamic Cournot duopoly game of players with bounded rationality where one player has information about the behavior of the other player.', '1809.10865-1-5-1': 'The player with information adjusts his output based on the present output of the other player.', '1809.10865-1-5-2': 'We then theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses the chaotic behavior in larger parameters compared with the case where there is no information asymmetry.', '1809.10865-1-5-3': 'A similar situation is investigated in the context of information acquisition by two information-sharing firms about the third firm in a triopoly game [CITATION].', '1809.10865-1-6-0': 'The paper is organized as follows.', '1809.10865-1-6-1': 'Section [REF] introduces a dynamic Cournot duopoly game model of bounded rational players with information asymmetry.', '1809.10865-1-6-2': 'Section [REF] provides theoretical and numerical results that the Cournot-Nash equilibrium is stabilized and chaotic behavior is suppressed by effect of information asymmetry.', '1809.10865-1-6-3': 'Section [REF] provides the concluding remarks of the paper.', '1809.10865-1-6-4': '[REF] reviews the results for the dynamic Cournot duopoly game where there is no information asymmetry; see Ref. [CITATION].', '1809.10865-1-7-0': '# Model', '1809.10865-1-8-0': 'We consider two firms (players) [MATH] producing the same good.', '1809.10865-1-8-1': 'The production (output) of each firm is described by [MATH].', '1809.10865-1-8-2': 'We assume that the price of the good is determined by the total supply [MATH] through the linear inverse demand function: [EQUATION] where [MATH] and [MATH] are positive constants.', '1809.10865-1-8-3': 'We also assume that the cost of production of firm [MATH] is linear, as [MATH] with [MATH].', '1809.10865-1-8-4': 'With these assumptions, the profit of firm [MATH] can be given by [EQUATION]', '1809.10865-1-8-5': 'When both players are rational, their best responses are given by the profit maximization conditions [EQUATION] and [EQUATION] to achieve the Cournot-Nash equilibrium [EQUATION]', '1809.10865-1-8-6': 'We consider the dynamics of two bounded rational players [CITATION].', '1809.10865-1-8-7': 'The production of player [MATH] at time [MATH] is described as [MATH].', '1809.10865-1-8-8': 'Each player adjusts his production step by step in order to increase his profit.', '1809.10865-1-8-9': 'We consider the situation in which player [MATH] adjusts his production as [EQUATION]', '1809.10865-1-8-10': 'The positive constant [MATH] is called speed of adjustment.', '1809.10865-1-8-11': 'We consider the information asymmetric situation such that player [MATH] already knows [MATH] at time [MATH] and chooses [MATH] as [EQUATION]', '1809.10865-1-8-12': "Here, we assume that the two players' speed of adjustment is common.", '1809.10865-1-8-13': 'Therefore, the dynamics of the two players are described as [EQUATION]', '1809.10865-1-8-14': 'In [REF], we provide the results for the model where there is no information asymmetry.', '1809.10865-1-9-0': '# Results', '1809.10865-1-10-0': '## Equilibrium points and local stability', '1809.10865-1-11-0': 'We first study the equilibrium points of the dynamic game.', '1809.10865-1-11-1': 'The fixed points are given by equations [EQUATION]', '1809.10865-1-11-2': 'Note that this condition is the same as the case where there is no information asymmetry.', '1809.10865-1-11-3': 'We find that there are four fixed points: [EQUATION]', '1809.10865-1-11-4': 'The fixed points [MATH] and [MATH] correspond to monopolistic fixed points.', '1809.10865-1-11-5': 'We assume that the Cournot-Nash equilibrium [MATH] exists; that is, [EQUATION]', '1809.10865-1-11-6': 'The stability of each fixed point is characterized by eigenvalues of the Jacobian matrix [EQUATION] with [EQUATION]', '1809.10865-1-11-7': 'When the absolute value of the two eigenvalues is smaller than [MATH], the fixed point is stable.', '1809.10865-1-11-8': 'First, we find that [EQUATION] with [MATH], and then we find that [MATH] is unstable under the condition ([REF]).', '1809.10865-1-12-0': 'Next, we consider the stability of [MATH] and [MATH].', '1809.10865-1-12-1': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-1-12-2': 'Therefore, [MATH] is a saddle point for [MATH] and unstable for [MATH].', '1809.10865-1-12-3': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-1-12-4': 'We observe that [MATH] is also a saddle point for [MATH] and unstable for [MATH].', '1809.10865-1-13-0': 'Finally, we investigate the stability of the Cournot-Nash fixed point [MATH].', '1809.10865-1-13-1': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-1-13-2': 'The eigenvalues of this matrix are given by [EQUATION]', '1809.10865-1-13-3': 'Both eigenvalues are non-negative and [MATH] for [MATH].', '1809.10865-1-13-4': 'For [MATH], [MATH] is complex and [EQUATION]', '1809.10865-1-13-5': 'Therefore, [MATH] is locally stable for [MATH].', '1809.10865-1-13-6': 'According to [REF], the stability condition of [MATH] in the original Cournot game is [MATH].', '1809.10865-1-13-7': 'Therefore, we can say that [MATH] is stabilized by the information asymmetry effect.', '1809.10865-1-14-0': '## Bifurcation diagram', '1809.10865-1-15-0': 'In order to find properties of trajectories realized by our model, we numerically solve equations ([REF]) and ([REF]).', '1809.10865-1-15-1': 'Because the parameters are set to [MATH], [MATH], and [MATH], we have [MATH] and [MATH].', '1809.10865-1-15-2': 'Fig. [REF] plots the bifurcation diagram of [MATH] with initial condition [MATH], and [MATH].', '1809.10865-1-15-3': 'The points are plotted after [MATH] iterations.', '1809.10865-1-16-0': 'We observe that a periodic orbit with period three discontinuously appears at [MATH] for the former case.', '1809.10865-1-16-1': 'Such discontinuous appearance of a periodic trajectory does not occur for an information symmetric case, as seen in [REF].', '1809.10865-1-16-2': 'We can also see aperiodic behavior for large [MATH].', '1809.10865-1-16-3': 'Note that the trajectories become unbounded for [MATH].', '1809.10865-1-17-0': 'We can understand the appearance of a periodic orbit as follows.', '1809.10865-1-17-1': 'If equations ([REF]) and ([REF]) have the non-trivial solution [MATH], [MATH], and [MATH] with [EQUATION] a periodic orbit exists.', '1809.10865-1-17-2': 'When we define [MATH] with [MATH], these [MATH] satisfy [EQUATION]', '1809.10865-1-17-3': 'These equations have a non-trivial solution for [MATH] that is larger than [MATH].', '1809.10865-1-17-4': 'This explains the existence of a periodic solution.', '1809.10865-1-18-0': 'We next discuss the basin of attraction of the Cournot-Nash equilibrium point [MATH].', '1809.10865-1-18-1': 'In Fig. [REF], we display the basin of attraction of [MATH] at [MATH] and [MATH].', '1809.10865-1-19-0': 'We observe that the basin of attraction of [MATH] becomes smaller and smaller as [MATH] increases.', '1809.10865-1-19-1': 'It should be noted that the basin of attraction disappears at [MATH].', '1809.10865-1-19-2': 'Therefore, the Cournot-Nash equilibrium point is not globally stable after the periodic orbit appears.', '1809.10865-1-20-0': '## Maximal Lyapunov exponent', '1809.10865-1-21-0': 'We numerically calculate the maximal Lyapunov exponent using the method proposed in Ref. [CITATION].', '1809.10865-1-21-1': 'Fig. [REF] displays the maximal Lyapunov exponent for [MATH], [MATH], and [MATH], and the initial condition [MATH] and [MATH].', '1809.10865-1-22-0': 'We observe that the maximal Lyapunov exponent is positive for [MATH], implying a chaotic behavior for [MATH].', '1809.10865-1-22-1': 'We also find that the bifurcation from a periodic orbit with period three does not seem to contribute to chaotic behavior.', '1809.10865-1-22-2': 'Fig. [REF] displays a chaotic attractor at [MATH].', '1809.10865-1-23-0': '# Concluding remarks', '1809.10865-1-24-0': 'In this paper, we investigated the effect of information asymmetry on the discrete-time dynamic Cournot duopoly game with bounded rationality.', '1809.10865-1-24-1': "Concretely, we studied how one player's information about the other player's behavior in a duopoly affects the stability of the Cournot-Nash equilibrium.", '1809.10865-1-24-2': 'We theoretically and numerically showed that the information stabilizes the Cournot-Nash equilibrium and suppresses the chaotic behavior.', '1809.10865-1-24-3': 'Note that a periodic orbit does not discontinuously appear in case there is no information asymmetry.', '1809.10865-1-24-4': 'The case that speeds of adjustment of two players are not common will be studied in future.', '1809.10865-1-25-0': 'The interpretation of our results from the perspective of numerical simulation suggests that the non-synchronous update of [MATH] can avoid chaotic behavior in this model.', '1809.10865-1-25-1': 'A future study should examine whether a similar behavior is observed for different games with bounded rationality, which also report chaotic behavior [CITATION].'}

{'1809.10865-2-0-0': 'We investigate the effect of information asymmetry on a dynamic Cournot duopoly game with bounded rationality.', '1809.10865-2-0-1': "Concretely, we study how one player's possession of information about the other player's behavior in a duopoly affects the stability of the Cournot-Nash equilibrium.", '1809.10865-2-0-2': 'We theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses chaotic behavior in the duopoly.', '1809.10865-2-1-0': '# Introduction', '1809.10865-2-2-0': 'An oligopoly is a market where a few firms control the price of a good.', '1809.10865-2-2-1': 'Cournot first introduced the oligopoly model in 1838.', '1809.10865-2-2-2': 'He considered an economy of two firms (players) producing the same good, both firms choosing their respective production to maximize their profit.', '1809.10865-2-2-3': 'Because the profit of each firm depends on its production as well as the production of the other firm, the situation is game theoretic.', '1809.10865-2-2-4': 'Each firm needs to correctly anticipate the behavior of the other firm.', '1809.10865-2-2-5': "The equilibrium state between the two players is known as the Cournot-Nash equilibrium; here, each player's response to the other is optimal.", '1809.10865-2-2-6': 'This equilibrium is realized when the two players are sufficiently rational.', '1809.10865-2-3-0': 'However, in reality, the players are not necessarily rational.', '1809.10865-2-3-1': 'Previous studies have examined games in which the players have bounded rationality [CITATION].', '1809.10865-2-3-2': 'As a toy model, the Cournot duopoly game model with bounded rationality has recently attracted much attention.', '1809.10865-2-3-3': 'The discrete-time dynamics of a Cournot duopoly game with bounded rationality has been analyzed in Refs. [CITATION].', '1809.10865-2-3-4': 'In this model, two players adjust their outputs step by step in order to increase their respective profits.', '1809.10865-2-3-5': 'Here, the Cournot-Nash equilibrium is not necessarily stable, and even chaotic behavior can occur in some parameter regions, as in dynamic Cournot duopoly games with naive response [CITATION].', '1809.10865-2-3-6': 'Following these studies, the model was extended to the case of nonlinear demand function [CITATION].', '1809.10865-2-3-7': 'Another extension is the Cournot duopoly game with heterogeneous players, where the two players adopt different decision-making strategies [CITATION].', '1809.10865-2-3-8': 'In addition, the dynamic Cournot duopoly game with time delay [CITATION] as well as the Cournot triopoly with bounded rationality [CITATION] has been studied.', '1809.10865-2-3-9': 'All these studies report that the dynamics of an oligopoly with bounded rationality is very complicated.', '1809.10865-2-4-0': 'Another significant concept in modern microeconomics is information asymmetry, where one player has more or better information than the others.', '1809.10865-2-4-1': 'This asymmetry can create a power imbalance in transactions, which in turn could lead to market failure in the worst case.', '1809.10865-2-4-2': 'Previous studies have considered the effect of information asymmetry on games with rational players [CITATION].', '1809.10865-2-4-3': 'However, the concept of information asymmetry in games where the players have bounded rationality has not yet been well established.', '1809.10865-2-4-4': 'Therefore, we try to explain the effect of information asymmetry on games with bounded rationality by taking the Cournot duopoly game as an example.', '1809.10865-2-5-0': 'Concretely, we investigate a discrete-time dynamic Cournot duopoly game of players with bounded rationality where one player has information about the behavior of the other player.', '1809.10865-2-5-1': 'The player with information adjusts his output based on the present output of the other player.', '1809.10865-2-5-2': 'We then theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses the chaotic behavior in larger parameters compared with the case where there is no information asymmetry.', '1809.10865-2-5-3': 'A similar situation was investigated in the context of information acquisition by two information-sharing firms about the third firm in a triopoly game [CITATION].', '1809.10865-2-6-0': 'The paper is organized as follows.', '1809.10865-2-6-1': 'Section [REF] introduces a dynamic Cournot duopoly game model of bounded rational players with information asymmetry.', '1809.10865-2-6-2': 'Section [REF] provides theoretical and numerical results that the Cournot-Nash equilibrium is stabilized and chaotic behavior is suppressed by effect of information asymmetry.', '1809.10865-2-6-3': 'Section [REF] provides the concluding remarks of the paper.', '1809.10865-2-6-4': '[REF] reviews the results for the dynamic Cournot duopoly game where there is no information asymmetry; see Ref. [CITATION].', '1809.10865-2-7-0': '# Model', '1809.10865-2-8-0': 'We consider two firms (players) [MATH] producing the same good.', '1809.10865-2-8-1': 'The production (output) of each firm is described by [MATH].', '1809.10865-2-8-2': 'We assume that the price of the good is determined by the total supply [MATH] through the linear inverse demand function: [EQUATION] where [MATH] and [MATH] are positive constants.', '1809.10865-2-8-3': 'If the total supply is increased, the price of the good becomes lower.', '1809.10865-2-8-4': 'We also assume that the cost of production of firm [MATH] is linear, as [MATH] with [MATH].', '1809.10865-2-8-5': 'With these assumptions, the profit of firm [MATH] can be given by [EQUATION]', '1809.10865-2-8-6': 'In the right-hand side, the first term represents the sales of the good and the second term represents the cost of the good.', '1809.10865-2-8-7': 'Thus, if these difference becomes larger, the profit is increased.', '1809.10865-2-8-8': 'When both players are rational, their best responses are given by the profit maximization conditions [EQUATION] and [EQUATION] to achieve the Cournot-Nash equilibrium [EQUATION]', '1809.10865-2-8-9': 'We can easily find that those conditions give the maximum profit, respectively.', '1809.10865-2-9-0': 'We consider the dynamics of two bounded rational players [CITATION].', '1809.10865-2-9-1': 'The production of player [MATH] at time [MATH] is described as [MATH].', '1809.10865-2-9-2': 'Each player adjusts his production step by step in order to increase his profit.', '1809.10865-2-9-3': 'We consider the situation in which player [MATH] adjusts his production as [EQUATION]', '1809.10865-2-9-4': 'In the right-hand side, the first term means the production in the previous step, and the second term means the effect that player [MATH] adjusts his production step by step according to the gradient of his profit.', '1809.10865-2-9-5': 'The positive constant [MATH] is called speed of adjustment.', '1809.10865-2-9-6': 'We consider the information asymmetric situation such that player [MATH] already knows [MATH] at time [MATH] in some way and chooses [MATH] as [EQUATION]', '1809.10865-2-9-7': "Here, we assume that the two players' speed of adjustment is common.", '1809.10865-2-9-8': 'Therefore, the dynamics of the two players is described as [EQUATION]', '1809.10865-2-9-9': 'In [REF], we provide the results for the model where there is no information asymmetry.', '1809.10865-2-10-0': '# Results', '1809.10865-2-11-0': '## Equilibrium points and local stability', '1809.10865-2-12-0': 'We first study the equilibrium points of the dynamic game.', '1809.10865-2-12-1': 'The fixed points are given by equations [EQUATION] which are obtained by considering [MATH] and [MATH] in ([REF]) and ([REF]), respectively.', '1809.10865-2-12-2': 'Note that this condition is the same as the case where there is no information asymmetry.', '1809.10865-2-12-3': 'We find that there are four fixed points: [EQUATION] which are obtained by the condition that each factor in ([REF]) and ([REF]) becomes zero.', '1809.10865-2-12-4': 'The fixed points [MATH] and [MATH] correspond to monopolistic fixed points.', '1809.10865-2-12-5': 'We assume that the Cournot-Nash equilibrium [MATH] exists; that is, [EQUATION]', '1809.10865-2-12-6': 'The local stability of each fixed point is characterized by eigenvalues of the Jacobian matrix [EQUATION] with [EQUATION]', '1809.10865-2-12-7': 'When the absolute value of the two eigenvalues is smaller than [MATH], displacement from the fixed point decays to zero, that is, the fixed point is locally stable.', '1809.10865-2-12-8': 'First, we find that [EQUATION] with [MATH], and then we find that [MATH] is unstable under the condition ([REF]).', '1809.10865-2-13-0': 'Next, we consider the stability of [MATH] and [MATH].', '1809.10865-2-13-1': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-2-13-2': 'Therefore, [MATH] is a saddle point for [MATH] and unstable for [MATH].', '1809.10865-2-13-3': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-2-13-4': 'We observe that [MATH] is also a saddle point for [MATH] and unstable for [MATH].', '1809.10865-2-14-0': 'Finally, we investigate the stability of the Cournot-Nash fixed point [MATH].', '1809.10865-2-14-1': 'The Jacobian matrix at [MATH] is [EQUATION]', '1809.10865-2-14-2': 'The eigenvalues of this matrix are given by [EQUATION]', '1809.10865-2-14-3': 'Both eigenvalues are non-negative and [MATH] for [MATH].', '1809.10865-2-14-4': 'For [MATH], [MATH] is complex and [EQUATION]', '1809.10865-2-14-5': 'Therefore, [MATH] is locally stable for [MATH].', '1809.10865-2-14-6': 'According to [REF], the stability condition of [MATH] in the original Cournot game is [MATH].', '1809.10865-2-14-7': 'Therefore, we can say that the information asymmetry broadens the stabilized region of [MATH].', '1809.10865-2-15-0': '## Bifurcation diagram', '1809.10865-2-16-0': 'In order to investigate properties of trajectories realized by our model, we numerically solve equations ([REF]) and ([REF]).', '1809.10865-2-16-1': 'We set the parameters as [MATH], [MATH], and [MATH].', '1809.10865-2-16-2': 'In these parameters, we have [MATH] and [MATH].', '1809.10865-2-16-3': 'Fig. [REF] plots the bifurcation diagram of [MATH] with initial condition [MATH], and [MATH].', '1809.10865-2-16-4': 'Points [MATH] with [MATH] are plotted for each [MATH].', '1809.10865-2-17-0': 'We observe that a periodic orbit with period three discontinuously appears at [MATH] for the former case, although the local stability condition [MATH] of the Cournot-Nash fixed point [MATH] is satisfied in this region.', '1809.10865-2-17-1': 'That is, although the Cournot-Nash equilibrium is locally stable in [MATH], it is not necessarily globally stable.', '1809.10865-2-17-2': 'Such discontinuous appearance of a periodic trajectory does not occur for an information symmetric case, as seen in [REF].', '1809.10865-2-17-3': 'We can also see aperiodic behavior for larger [MATH].', '1809.10865-2-17-4': 'Note that the trajectories become unbounded for [MATH].', '1809.10865-2-18-0': 'We can understand the appearance of a periodic orbit as follows.', '1809.10865-2-18-1': 'If equations ([REF]) and ([REF]) have the non-trivial solution [MATH], [MATH], and [MATH] with [EQUATION] a periodic orbit exists.', '1809.10865-2-18-2': 'When we define [MATH] with [MATH], these [MATH] satisfy [EQUATION]', '1809.10865-2-18-3': 'These equations have a non-trivial solution for [MATH] that is larger than [MATH].', '1809.10865-2-18-4': 'This explains the existence of a periodic solution.', '1809.10865-2-19-0': 'We next discuss the basin of attraction of the Cournot-Nash equilibrium point [MATH].', '1809.10865-2-19-1': 'In Fig. [REF], we display the basin of attraction of [MATH] at [MATH] and [MATH], which is the set of initial conditions [MATH] which converge to the Cournot-Nash fixed point [MATH] after [MATH] iterations.', '1809.10865-2-20-0': 'We observe that the basin of attraction of [MATH] becomes smaller and smaller as [MATH] increases.', '1809.10865-2-20-1': 'It should be noted that the basin of attraction disappears at [MATH].', '1809.10865-2-20-2': 'Therefore, the Cournot-Nash equilibrium point is not globally stable after the periodic orbit appears.', '1809.10865-2-21-0': '## Maximal Lyapunov exponent', '1809.10865-2-22-0': 'The Lyapunov exponent is a quantity in dynamical systems theory that characterizes the rate of separation of infinitesimally close trajectories.', '1809.10865-2-22-1': 'When the Lyapunov exponent is positive, it implies that behavior of the dynamical system is chaotic.', '1809.10865-2-22-2': 'In contrast, when it is negative, the separation of infinitesimally close trajectories converges to zero.', '1809.10865-2-22-3': 'Generally, the Lyapunov exponent depends on the direction of the initial separation vector.', '1809.10865-2-22-4': 'Therefore, the number of the Lyapunov exponents is equal to the dimension of the phase space.', '1809.10865-2-22-5': 'Because the existence of chaotic behavior is characterized by the maximal Lyapunov exponent, which is the largest Lyapunov exponent, we focus on the maximal Lyapunov exponent.', '1809.10865-2-23-0': 'We numerically calculate the maximal Lyapunov exponent by using the method proposed in Ref. [CITATION].', '1809.10865-2-23-1': 'Fig. [REF] displays the maximal Lyapunov exponent for [MATH], [MATH], and [MATH], and the initial condition [MATH] and [MATH].', '1809.10865-2-24-0': 'We observe that the maximal Lyapunov exponent is positive for [MATH], implying a chaotic behavior for [MATH].', '1809.10865-2-24-1': 'We also find that the bifurcation from a periodic orbit with period three at [MATH] does not seem to contribute to chaotic behavior.', '1809.10865-2-24-2': 'Fig. [REF] displays a chaotic attractor at [MATH].', '1809.10865-2-25-0': '# Concluding remarks', '1809.10865-2-26-0': 'In this paper, we investigated the effect of information asymmetry on the discrete-time dynamic Cournot duopoly game with bounded rationality.', '1809.10865-2-26-1': "Concretely, we studied how one player's information about the other player's behavior in a duopoly affects the stability of the Cournot-Nash equilibrium.", '1809.10865-2-26-2': 'We theoretically and numerically showed that the information stabilizes the Cournot-Nash equilibrium and suppresses the chaotic behavior.', '1809.10865-2-26-3': 'Note that a periodic orbit does not discontinuously appear in case there is no information asymmetry.', '1809.10865-2-26-4': 'Our result suggests that information acquisition about the strategy of the other firm in oligopoly market is useful to stabilize the equilibrium point.', '1809.10865-2-26-5': 'This property may hold in realistic market.', '1809.10865-2-26-6': 'The case that speeds of adjustment of two players are not common will be studied in future.', '1809.10865-2-27-0': 'The interpretation of our results from the perspective of numerical simulation suggests that the non-synchronous update of [MATH] can avoid chaotic behavior in this model.', '1809.10865-2-27-1': 'A future study should examine whether a similar behavior is observed for different games with bounded rationality, which also report chaotic behavior [CITATION].'}

[['1809.10865-1-25-0', '1809.10865-2-27-0'], ['1809.10865-1-25-1', '1809.10865-2-27-1'], ['1809.10865-1-2-0', '1809.10865-2-2-0'], ['1809.10865-1-2-1', '1809.10865-2-2-1'], ['1809.10865-1-2-2', '1809.10865-2-2-2'], ['1809.10865-1-2-3', '1809.10865-2-2-3'], ['1809.10865-1-2-4', '1809.10865-2-2-4'], ['1809.10865-1-2-5', '1809.10865-2-2-5'], ['1809.10865-1-2-6', '1809.10865-2-2-6'], ['1809.10865-1-17-0', '1809.10865-2-18-0'], ['1809.10865-1-17-1', '1809.10865-2-18-1'], ['1809.10865-1-17-2', '1809.10865-2-18-2'], ['1809.10865-1-17-3', '1809.10865-2-18-3'], ['1809.10865-1-17-4', '1809.10865-2-18-4'], ['1809.10865-1-12-0', '1809.10865-2-13-0'], ['1809.10865-1-12-1', '1809.10865-2-13-1'], ['1809.10865-1-12-2', '1809.10865-2-13-2'], ['1809.10865-1-12-3', '1809.10865-2-13-3'], ['1809.10865-1-12-4', '1809.10865-2-13-4'], ['1809.10865-1-24-0', '1809.10865-2-26-0'], ['1809.10865-1-24-1', '1809.10865-2-26-1'], ['1809.10865-1-24-2', '1809.10865-2-26-2'], ['1809.10865-1-24-3', '1809.10865-2-26-3'], ['1809.10865-1-24-4', '1809.10865-2-26-6'], ['1809.10865-1-4-0', '1809.10865-2-4-0'], ['1809.10865-1-4-1', '1809.10865-2-4-1'], ['1809.10865-1-4-3', '1809.10865-2-4-3'], ['1809.10865-1-4-4', '1809.10865-2-4-4'], ['1809.10865-1-13-0', '1809.10865-2-14-0'], ['1809.10865-1-13-1', '1809.10865-2-14-1'], ['1809.10865-1-13-2', '1809.10865-2-14-2'], ['1809.10865-1-13-3', '1809.10865-2-14-3'], ['1809.10865-1-13-4', '1809.10865-2-14-4'], ['1809.10865-1-13-5', '1809.10865-2-14-5'], ['1809.10865-1-13-6', '1809.10865-2-14-6'], ['1809.10865-1-19-0', '1809.10865-2-20-0'], ['1809.10865-1-19-1', '1809.10865-2-20-1'], ['1809.10865-1-19-2', '1809.10865-2-20-2'], ['1809.10865-1-6-0', '1809.10865-2-6-0'], ['1809.10865-1-6-1', '1809.10865-2-6-1'], ['1809.10865-1-6-2', '1809.10865-2-6-2'], ['1809.10865-1-6-3', '1809.10865-2-6-3'], ['1809.10865-1-6-4', '1809.10865-2-6-4'], ['1809.10865-1-22-0', '1809.10865-2-24-0'], ['1809.10865-1-22-2', '1809.10865-2-24-2'], ['1809.10865-1-16-1', '1809.10865-2-17-2'], ['1809.10865-1-16-3', '1809.10865-2-17-4'], ['1809.10865-1-3-0', '1809.10865-2-3-0'], ['1809.10865-1-3-2', '1809.10865-2-3-2'], ['1809.10865-1-3-4', '1809.10865-2-3-4'], ['1809.10865-1-3-5', '1809.10865-2-3-5'], ['1809.10865-1-3-6', '1809.10865-2-3-6'], ['1809.10865-1-3-7', '1809.10865-2-3-7'], ['1809.10865-1-3-8', '1809.10865-2-3-8'], ['1809.10865-1-15-2', '1809.10865-2-16-3'], ['1809.10865-1-0-0', '1809.10865-2-0-0'], ['1809.10865-1-0-1', '1809.10865-2-0-1'], ['1809.10865-1-0-2', '1809.10865-2-0-2'], ['1809.10865-1-18-0', '1809.10865-2-19-0'], ['1809.10865-1-5-0', '1809.10865-2-5-0'], ['1809.10865-1-5-1', '1809.10865-2-5-1'], ['1809.10865-1-5-2', '1809.10865-2-5-2'], ['1809.10865-1-21-1', '1809.10865-2-23-1'], ['1809.10865-1-11-0', '1809.10865-2-12-0'], ['1809.10865-1-11-2', '1809.10865-2-12-2'], ['1809.10865-1-11-4', '1809.10865-2-12-4'], ['1809.10865-1-11-5', '1809.10865-2-12-5'], ['1809.10865-1-11-8', '1809.10865-2-12-8'], ['1809.10865-1-8-0', '1809.10865-2-8-0'], ['1809.10865-1-8-1', '1809.10865-2-8-1'], ['1809.10865-1-8-2', '1809.10865-2-8-2'], ['1809.10865-1-8-3', '1809.10865-2-8-4'], ['1809.10865-1-8-4', '1809.10865-2-8-5'], ['1809.10865-1-8-5', '1809.10865-2-8-8'], ['1809.10865-1-8-6', '1809.10865-2-9-0'], ['1809.10865-1-8-7', '1809.10865-2-9-1'], ['1809.10865-1-8-8', '1809.10865-2-9-2'], ['1809.10865-1-8-9', '1809.10865-2-9-3'], ['1809.10865-1-8-10', '1809.10865-2-9-5'], ['1809.10865-1-8-12', '1809.10865-2-9-7'], ['1809.10865-1-8-14', '1809.10865-2-9-9'], ['1809.10865-1-4-2', '1809.10865-2-4-2'], ['1809.10865-1-22-1', '1809.10865-2-24-1'], ['1809.10865-1-16-2', '1809.10865-2-17-3'], ['1809.10865-1-3-1', '1809.10865-2-3-1'], ['1809.10865-1-3-3', '1809.10865-2-3-3'], ['1809.10865-1-3-9', '1809.10865-2-3-9'], ['1809.10865-1-15-0', '1809.10865-2-16-0'], ['1809.10865-1-5-3', '1809.10865-2-5-3'], ['1809.10865-1-21-0', '1809.10865-2-23-0'], ['1809.10865-1-11-6', '1809.10865-2-12-6'], ['1809.10865-1-8-11', '1809.10865-2-9-6'], ['1809.10865-1-8-13', '1809.10865-2-9-8'], ['1809.10865-1-13-7', '1809.10865-2-14-7'], ['1809.10865-1-16-0', '1809.10865-2-17-0'], ['1809.10865-1-15-1', '1809.10865-2-16-1'], ['1809.10865-1-18-1', '1809.10865-2-19-1'], ['1809.10865-1-11-1', '1809.10865-2-12-1'], ['1809.10865-1-11-3', '1809.10865-2-12-3'], ['1809.10865-1-11-7', '1809.10865-2-12-7']]

[['1809.10865-1-25-0', '1809.10865-2-27-0'], ['1809.10865-1-25-1', '1809.10865-2-27-1'], ['1809.10865-1-2-0', '1809.10865-2-2-0'], ['1809.10865-1-2-1', '1809.10865-2-2-1'], ['1809.10865-1-2-2', '1809.10865-2-2-2'], ['1809.10865-1-2-3', '1809.10865-2-2-3'], ['1809.10865-1-2-4', '1809.10865-2-2-4'], ['1809.10865-1-2-5', '1809.10865-2-2-5'], ['1809.10865-1-2-6', '1809.10865-2-2-6'], ['1809.10865-1-17-0', '1809.10865-2-18-0'], ['1809.10865-1-17-1', '1809.10865-2-18-1'], ['1809.10865-1-17-2', '1809.10865-2-18-2'], ['1809.10865-1-17-3', '1809.10865-2-18-3'], ['1809.10865-1-17-4', '1809.10865-2-18-4'], ['1809.10865-1-12-0', '1809.10865-2-13-0'], ['1809.10865-1-12-1', '1809.10865-2-13-1'], ['1809.10865-1-12-2', '1809.10865-2-13-2'], ['1809.10865-1-12-3', '1809.10865-2-13-3'], ['1809.10865-1-12-4', '1809.10865-2-13-4'], ['1809.10865-1-24-0', '1809.10865-2-26-0'], ['1809.10865-1-24-1', '1809.10865-2-26-1'], ['1809.10865-1-24-2', '1809.10865-2-26-2'], ['1809.10865-1-24-3', '1809.10865-2-26-3'], ['1809.10865-1-24-4', '1809.10865-2-26-6'], ['1809.10865-1-4-0', '1809.10865-2-4-0'], ['1809.10865-1-4-1', '1809.10865-2-4-1'], ['1809.10865-1-4-3', '1809.10865-2-4-3'], ['1809.10865-1-4-4', '1809.10865-2-4-4'], ['1809.10865-1-13-0', '1809.10865-2-14-0'], ['1809.10865-1-13-1', '1809.10865-2-14-1'], ['1809.10865-1-13-2', '1809.10865-2-14-2'], ['1809.10865-1-13-3', '1809.10865-2-14-3'], ['1809.10865-1-13-4', '1809.10865-2-14-4'], ['1809.10865-1-13-5', '1809.10865-2-14-5'], ['1809.10865-1-13-6', '1809.10865-2-14-6'], ['1809.10865-1-19-0', '1809.10865-2-20-0'], ['1809.10865-1-19-1', '1809.10865-2-20-1'], ['1809.10865-1-19-2', '1809.10865-2-20-2'], ['1809.10865-1-6-0', '1809.10865-2-6-0'], ['1809.10865-1-6-1', '1809.10865-2-6-1'], ['1809.10865-1-6-2', '1809.10865-2-6-2'], ['1809.10865-1-6-3', '1809.10865-2-6-3'], ['1809.10865-1-6-4', '1809.10865-2-6-4'], ['1809.10865-1-22-0', '1809.10865-2-24-0'], ['1809.10865-1-22-2', '1809.10865-2-24-2'], ['1809.10865-1-16-1', '1809.10865-2-17-2'], ['1809.10865-1-16-3', '1809.10865-2-17-4'], ['1809.10865-1-3-0', '1809.10865-2-3-0'], ['1809.10865-1-3-2', '1809.10865-2-3-2'], ['1809.10865-1-3-4', '1809.10865-2-3-4'], ['1809.10865-1-3-5', '1809.10865-2-3-5'], ['1809.10865-1-3-6', '1809.10865-2-3-6'], ['1809.10865-1-3-7', '1809.10865-2-3-7'], ['1809.10865-1-3-8', '1809.10865-2-3-8'], ['1809.10865-1-15-2', '1809.10865-2-16-3'], ['1809.10865-1-0-0', '1809.10865-2-0-0'], ['1809.10865-1-0-1', '1809.10865-2-0-1'], ['1809.10865-1-0-2', '1809.10865-2-0-2'], ['1809.10865-1-18-0', '1809.10865-2-19-0'], ['1809.10865-1-5-0', '1809.10865-2-5-0'], ['1809.10865-1-5-1', '1809.10865-2-5-1'], ['1809.10865-1-5-2', '1809.10865-2-5-2'], ['1809.10865-1-21-1', '1809.10865-2-23-1'], ['1809.10865-1-11-0', '1809.10865-2-12-0'], ['1809.10865-1-11-2', '1809.10865-2-12-2'], ['1809.10865-1-11-4', '1809.10865-2-12-4'], ['1809.10865-1-11-5', '1809.10865-2-12-5'], ['1809.10865-1-11-8', '1809.10865-2-12-8'], ['1809.10865-1-8-0', '1809.10865-2-8-0'], ['1809.10865-1-8-1', '1809.10865-2-8-1'], ['1809.10865-1-8-2', '1809.10865-2-8-2'], ['1809.10865-1-8-3', '1809.10865-2-8-4'], ['1809.10865-1-8-4', '1809.10865-2-8-5'], ['1809.10865-1-8-5', '1809.10865-2-8-8'], ['1809.10865-1-8-6', '1809.10865-2-9-0'], ['1809.10865-1-8-7', '1809.10865-2-9-1'], ['1809.10865-1-8-8', '1809.10865-2-9-2'], ['1809.10865-1-8-9', '1809.10865-2-9-3'], ['1809.10865-1-8-10', '1809.10865-2-9-5'], ['1809.10865-1-8-12', '1809.10865-2-9-7'], ['1809.10865-1-8-14', '1809.10865-2-9-9']]

[['1809.10865-1-4-2', '1809.10865-2-4-2'], ['1809.10865-1-22-1', '1809.10865-2-24-1'], ['1809.10865-1-16-2', '1809.10865-2-17-3'], ['1809.10865-1-3-1', '1809.10865-2-3-1'], ['1809.10865-1-3-3', '1809.10865-2-3-3'], ['1809.10865-1-3-9', '1809.10865-2-3-9'], ['1809.10865-1-15-0', '1809.10865-2-16-0'], ['1809.10865-1-5-3', '1809.10865-2-5-3'], ['1809.10865-1-21-0', '1809.10865-2-23-0'], ['1809.10865-1-11-6', '1809.10865-2-12-6'], ['1809.10865-1-8-11', '1809.10865-2-9-6'], ['1809.10865-1-8-13', '1809.10865-2-9-8']]

[]

[['1809.10865-1-13-7', '1809.10865-2-14-7'], ['1809.10865-1-16-0', '1809.10865-2-17-0'], ['1809.10865-1-15-1', '1809.10865-2-16-1'], ['1809.10865-1-18-1', '1809.10865-2-19-1'], ['1809.10865-1-11-1', '1809.10865-2-12-1'], ['1809.10865-1-11-3', '1809.10865-2-12-3'], ['1809.10865-1-11-7', '1809.10865-2-12-7']]

[]

[]

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

https://arxiv.org/abs/1809.10865

0807.2419

{'0807.2419-1-0-0': 'We derive the general form of the moduli-space effective action for the long-range interaction of two BPS dyons in gauge theories.', '0807.2419-1-0-1': 'This action determines the bound state structure of various BPS and non-BPS states near marginal stability curves, and we utilise it to compute the leading correction to the BPS-mass of zero-torsion non-BPS bound states close to marginal stability.', '0807.2419-1-1-0': 'July 2008', '0807.2419-1-2-0': '## 1.', '0807.2419-1-2-1': 'Introduction', '0807.2419-1-3-0': 'Theories with extended or supersymmetry in four dimensions have a BPS sector of the particle spectrum, namely those states which preserve some fraction of the supersymmetry of the vacuum [CITATION].', '0807.2419-1-3-1': 'The masses, interactions and degeneracies of these BPS states are often exactly computable, providing a powerful window to the dynamics.', '0807.2419-1-3-2': 'One of the remarkable dynamical principles which emerges in theories with this level of supersymmetry is electric-magnetic duality [CITATION], and its infinite-dimensional generalization [MATH]-duality.', '0807.2419-1-3-3': 'It was realized some time ago that the BPS spectrum then inherits a rather hierarchical structure, with the majority of the states being viewed as bound states of a relatively small number of constituents, which are the lightest states in any charge sector at a given point in the vacuum moduli space of the theory.', '0807.2419-1-3-4': 'This structure is quite rigid under changes of moduli with the exception of special co-dimension one surfaces, known as curves of marginal stability (CMS), where changes occur in particular in the spectrum of the lightest constituent states.', '0807.2419-1-3-5': 'In the context of dyonic bound states, this structure was explored in and gauge theories some time ago [CITATION] utilizing the Seiberg-Witten solution [CITATION], and has more recently been elucidated for BPS black holes in and string theories [CITATION].', '0807.2419-1-3-6': 'While much of this dynamical structure is expected to extend to the more generic non-BPS sector, it is far less amenable to study as many of the powerful supersymmetric tools are no longer available.', '0807.2419-1-3-7': 'Nonetheless, the non-BPS sector is of course of considerable interest for many reasons, not least because it provides a window into the behaviour of generic massive states in strongly coupled gauge theories, and indeed to generic black hole states in and string theory.', '0807.2419-1-4-0': 'In this letter, we will consider a special class of non-BPS states which may be viewed as weakly bound composites of BPS constituents near curves of marginal stability.', '0807.2419-1-4-1': 'The analysis will thus be limited to near-CMS regions of the moduli space, but the payoff is that explicit computations can be performed for the mass, and in principle the degeneracy, of these non-BPS states.', '0807.2419-1-4-2': 'The interactions of BPS states are dictated by supersymmetry, which allows us to treat the non-relativistic bound state problem exactly even at strong coupling.', '0807.2419-1-4-3': 'This statement relies on several special features, most importantly that the long-range interactions of BPS constituents with charges [MATH] on the Coulomb branch are exactly determined by the central charges [MATH] of the BPS states in question, and the special Kahler metric on the Coulomb branch [MATH] [CITATION].', '0807.2419-1-4-4': 'By considering the exchange of the massless fields, the Coulomb term can be shown to take the form [CITATION], [EQUATION]', '0807.2419-1-4-5': 'This expression is particularly useful near the CMS for these two constituents, as the bound state problem simplifies to the non-relativistic level because the binding energy may be made parametrically small.', '0807.2419-1-4-6': 'The CMS curve(s) for the two states in question is defined by the condition: [EQUATION]', '0807.2419-1-4-7': 'In the context of supersymmetric quantum mechanics (SQM), the potential for two interacting BPS constituents with masses [MATH], for [MATH], naturally tends to a constant as [MATH] such that the lowest eigenvalue of the SQM Hamiltonian vanishes for a BPS bound state with mass [MATH].', '0807.2419-1-4-8': 'This implies that the potential is defined at large [MATH] as [EQUATION] in terms of the binding energy [MATH].', '0807.2419-1-4-9': 'Expansion of the potential [MATH] near [MATH] takes the form [EQUATION] where [MATH] is the reduced mass of the two BPS constituents.', '0807.2419-1-4-10': 'Close to the CMS, the sign of the Coulomb potential is determined by the sign of the Dirac-Schwinger-Zwanziger symplectic charge product [EQUATION] also known as the torsion.', '0807.2419-1-4-11': 'Thus generically as one crosses the CMS, the Coulomb potential changes sign and a bound state exists on only one side [CITATION].', '0807.2419-1-5-0': 'This provides a simple viewpoint on the existence or otherwise of bound states and was exploited in [CITATION] to consider the BPS spectrum in theories utilizing the Seiberg-Witten solution.', '0807.2419-1-5-1': 'However, this picture is incomplete as the relative dynamics of two BPS constituents necessarily preserves at least four supercharges in an theory, and this structure is not manifest in the potential above.', '0807.2419-1-5-2': 'In general terms the structure of the relevant form of SQM has been known for some time [CITATION], and was studied in the context of BPS states by Denef [CITATION].', '0807.2419-1-5-3': 'We will extend this approach to incorporate the nontrivial metric for the relative translational coordinates of the two BPS constituents, and explore how this provides a novel window on the mass corrections for non-BPS bound states.', '0807.2419-1-6-0': 'The general form of the worldline SQM is derived in Section 2, and shown to be consistent with the Coulomb potential in ([REF]).', '0807.2419-1-6-1': 'In Section 3, we focus on a specific class of non-BPS bound states with zero torsion, i.e. [MATH], which are known to be BPS states in SYM, but lift slightly from the BPS bound in theories at weak coupling.', '0807.2419-1-6-2': 'Using the low energy description of the constituent BPS states we explicitly compute the non-BPS correction to the mass.', '0807.2419-1-6-3': 'We finish with some additional remarks in Section 4.', '0807.2419-1-7-0': '## 2.', '0807.2419-1-7-1': 'The D-term potential and the moduli-space metric', '0807.2419-1-8-0': 'The chiral structure of the fermionic zero modes in the background of monopole solutions in SYM implies that, at the classical level, the low energy dynamics on the moduli space of BPS dyons is realized as SQM [CITATION], namely the reduction of a chiral (0,4) supersymmetric sigma model in 1+1D, preserving four supercharges.', '0807.2419-1-8-1': 'This pairs four bosonic and four fermionic collective coordinates.', '0807.2419-1-8-2': 'However, while three of the bosonic coordinates reflect translations and so would be expected to survive the inclusion of generic quantum corrections, the fourth is compact and the quantized momentum along this direction corresponds to the discrete electric charge.', '0807.2419-1-8-3': 'Thus, in a generic sector of the quantum theory where, in addition to the magnetic charge, the electric charge is also fixed we deal with a system having 3 bosonic and 4 fermionic variables.', '0807.2419-1-8-4': 'Such a multiplet is in fact consistent in SQM, and arises from the reduction of a vector multiplet in 3+1D.', '0807.2419-1-9-0': 'The general structure of SQM resulting from the reduction of a vector multiplet has been known for some time [CITATION], and was discussed in the present context more recently by Denef [CITATION].', '0807.2419-1-9-1': 'The vector multiplet decomposes to [MATH]<[MATH][MATH], comprising the coordinate vector [MATH], its chiral superpartner [MATH]<[MATH][MATH] and the auxiliary field [MATH].', '0807.2419-1-9-2': 'If we consider the relative dynamics of two point-like sources, with [MATH] the relative separation, the Lagrangian up to 2-derivative order is quite constrained [CITATION], [EQUATION] where the potentials [MATH], [MATH] and the metric [MATH] are functions of the relative coordinate [MATH].', '0807.2419-1-9-3': 'The potentials are related by the condition [MATH].', '0807.2419-1-9-4': 'It follows that [MATH] is a harmonic function, and requiring spherical symmetry, as is appropriate for the potential between two Coulomb sources, the general solution has the form [MATH] with [MATH], constants [MATH] and [MATH], and [MATH] with [MATH] the unit charge Dirac monopole potential.', '0807.2419-1-9-5': 'Quantization then demands that [MATH] be an integer.', '0807.2419-1-10-0': 'We will now specialize to the case at hand, namely the relative dynamics of two BPS states in SYM.', '0807.2419-1-10-1': 'The 2-particle interaction potential has the form [EQUATION] where the above constraints on [MATH] should indeed hold.', '0807.2419-1-10-2': 'Since the interactions are long range, we can also write [MATH] introducing a further constant [MATH] and proceed to fix the constants [MATH], [MATH] and [MATH].', '0807.2419-1-10-3': 'By comparing to the expansion of the Coulomb potential ([REF]) near the CMS, the binding energy immediately fixes [MATH].', '0807.2419-1-11-0': 'To proceed, we note that for a BPS bound state to exist this system must have a supersymmetric vacuum where [MATH].', '0807.2419-1-11-1': 'Given regularity of the metric, this implies classically that [MATH] in the vacuum and the existence or otherwise of BPS states will, generically at least, be independent of the metric [MATH].', '0807.2419-1-11-2': 'Since the leading term in the Coulomb potential ([REF]) near the CMS, linear in [MATH], dictates the presence of BPS states it must necessarily be present in [MATH].', '0807.2419-1-11-3': 'This allows to uniquely fix the constant [MATH], and thus the form of [MATH], [EQUATION] up to corrections of [MATH] which are subleading near the CMS.', '0807.2419-1-12-0': 'It remains to determine the metric [MATH].', '0807.2419-1-12-1': 'Indeed, it is now clear that the [MATH] term in ([REF]) cannot reproduce the full Coulomb potential ([REF]) at [MATH] unless [MATH].', '0807.2419-1-12-2': 'Classically, or in the SYM limit, there is enough supersymmetry to determine the form of the metric precisely.', '0807.2419-1-12-3': 'Here we will be content to determine the leading [MATH] correction and, rather than extract [MATH] from ([REF]), we will compute it using the approach of Manton [CITATION].', '0807.2419-1-12-4': 'Near the CMS, we can consider the dynamics of the non-relativistic BPS point-like constituents interacting through their electric, magnetic, and scalar charges, and the metric is determined by the terms quadratic in velocity.', '0807.2419-1-13-0': "Manton's original calculation (at the level of the leading [MATH] terms) can be straightforwardly repeated in the regime using the exact quantum corrected charges that were computed in [CITATION].", '0807.2419-1-13-1': 'We begin by writing down the probe Lagrangian for dyon 2 in the background of dyon 1, with both states weakly boosted, [EQUATION]', '0807.2419-1-13-2': 'Here [MATH] and [MATH] are the electric and dual magnetic potentials, coupling to the state via the metric [MATH] and its inverse, and the scalar charge is determined by the Taylor expansion of [MATH].', '0807.2419-1-13-3': 'When [MATH] and [MATH] are both nonzero we also need to account for the Witten effect which modifies the coupling to the electric potential.', '0807.2419-1-14-0': 'The boosted potentials [MATH] and [MATH] induced by the presence of dyon 1 take the usual Lienard-Wiechert form [CITATION], while the scalar field shift is given by [CITATION], [EQUATION]', '0807.2419-1-14-1': 'Expanding to quadratic order in velocities, symmetrizing to include the dynamics of dyon 1, and dropping the free center of mass motion, we obtain [MATH], where [EQUATION]', '0807.2419-1-14-2': 'Thus, up to possible subleading terms of [MATH] in the metric, we can write down the full potential ([REF]) in the form [EQUATION]', '0807.2419-1-14-3': 'As a useful consistency check, by expanding to [MATH], we may now verify that the [MATH] term agrees precisely up to [MATH] with potential ([REF]) computed by considering massless exchange [CITATION].', '0807.2419-1-14-4': 'The full formulation as a [MATH]-term potential in ([REF]) makes the constraints of supersymmetry manifest and naturally provides an extension to higher order in [MATH] since any terms of [MATH] in the metric will be further suppressed in [MATH] by a factor of [MATH].', '0807.2419-1-15-0': '## 3.', '0807.2419-1-15-1': 'The fate of zero-torsion non-BPS states', '0807.2419-1-16-0': 'Since [MATH] for supersymmetric ground states to exist classically, we see that the metric generically plays little role; the relevant constraint is [MATH], so that [EQUATION] which we observe only has solutions on one side of the CMS where the composite BPS state exists.', '0807.2419-1-16-1': 'This simplification allows the quantum mechanics of generic BPS states to be studied in a simplified system where the metric is ignored, i.e. we can take [MATH], a constant.', '0807.2419-1-16-2': 'This regime was considered in detail by Denef [CITATION], and the BPS ground state has the form [MATH]<[MATH][MATH] in terms of [MATH]<[MATH][MATH], the complex spinor superpartner of [MATH].', '0807.2419-1-16-3': 'The constraint that the supercharges annihilate the state [MATH] implies [MATH] and the angular part of the wavefunction is expressed in terms of the monopole harmonics of Wu and Yang [CITATION] while the size of the multiplet scales with [MATH] on account of the electromagnetic contribution to the total angular momentum.', '0807.2419-1-16-4': 'The radial wavefunction scales as [MATH] and thus is localized on only one side of the CMS, which is the quantum version of the fact that ([REF]) only makes sense for one sign of [MATH].', '0807.2419-1-16-5': 'An interesting feature of the solution is a shift of the spin by 1/2, i.e. the ground state is a spinor, which can be interpreted as due to the induced spin coupling in the Hamiltonian, [MATH] where [MATH]<[MATH][MATH]<[MATH][MATH].', '0807.2419-1-17-0': 'While the spectrum of BPS states can generally be studied independently of the [MATH] corrections to the metric [MATH], these corrections are crucial to the formation of non-BPS composite states.', '0807.2419-1-17-1': 'Studying these states within the present framework is often difficult as the binding energy becomes small when the full Coulomb potential ([REF]) vanishes, which corresponds to [EQUATION] and this condition is not generally satisfied for small [MATH].', '0807.2419-1-17-2': 'This renders the bound state problem intractable.', '0807.2419-1-17-3': 'However, an interesting exception that we will now focus on concerns the case where the constituents are mutually local, so the torsion vanishes [MATH].', '0807.2419-1-17-4': 'The potential near the CMS is then provided purely by the metric, [EQUATION] which (for positive [MATH]) vanishes linearly at [MATH], and tends (like [MATH]) to [MATH] at infinity.', '0807.2419-1-18-0': 'In SYM, zero torsion states often have a special status as they may be required by [MATH]-duality to exist as bound states in the BPS spectrum.', '0807.2419-1-18-1': "The simplest example arises for gauge group SU(3), where the existence of a [MATH] boson with charges aligned along both U(1)'s implies, via duality, a bound state of two distinct monopoles with unit magnetic charges along the two simple roots.", '0807.2419-1-18-2': 'The existence of this [MATH] state has been verified at weak coupling in various regions of the moduli space [CITATION], starting first with the simplest case of aligned vevs where it exists as a bound state at threshold.', '0807.2419-1-18-3': 'However, since a 1/2-BPS multiplet in has the same size as a generic non-BPS multiplet in , we would expect that on breaking to SYM the mass perturbation will lift the state from the BPS spectrum, and indeed this conclusion is borne out in explicit calculations at weak coupling [CITATION].', '0807.2419-1-18-4': 'In the present context, we cannot draw any direct conclusions on the existence of the threshold state, since [MATH] takes the form [EQUATION] and thus vanishes for aligned vevs, as does the potential ([REF]).', '0807.2419-1-18-5': 'However, the [MATH] state is also required by duality to exist in the theory for misaligned vevs and indeed it duly appears in the weak coupling analysis [CITATION].', '0807.2419-1-18-6': 'This state is no longer at threshold and on perturbing to SYM should again lift from the BPS spectrum [CITATION].', '0807.2419-1-18-7': 'We can verify this in a rather straightforward manner using SQM with the potential ([REF]) and indeed we can also compute the mass shift from the BPS bound.', '0807.2419-1-19-0': 'For [MATH], the system is purely bosonic and inspection of the potential ([REF]) suggests that we can compute the non-BPS correction to the mass as the zero-point energy.', '0807.2419-1-19-1': 'The classical Hamiltonian has the form [MATH], and the Schrodinger equation, after accounting for the metric and proper ordering, can be re-expressed as an analogue Coulomb system [CITATION], [EQUATION] with [MATH] where [MATH] is the mass of the bound state and [MATH] is the BPS lower bound.', '0807.2419-1-19-2': 'The Hamiltonian in this equation is hermitian with respect to the usual Euclidean metric, and thus the problem reduces to a standard hydrogenic analysis [CITATION].', '0807.2419-1-19-3': 'However, the spectrum is of course not Coulomb-like, and the scaling of the ground state energy is determined by the parameter [MATH] which we keep fixed in the near-CMS regime where [MATH].', '0807.2419-1-19-4': 'It will be enough to focus on the weak-coupling limit [MATH] and [MATH], so that [MATH] and there are two regimes of interest, namely [MATH] and [MATH].', '0807.2419-1-19-5': 'Note that in this limit, the exact moduli-space metric is known and contains no corrections of [MATH] [CITATION].', '0807.2419-1-20-0': '### 3.1 Ground state for [MATH]', '0807.2419-1-21-0': 'In the regime [MATH], the potential becomes approximately Coulomb-like, [MATH] with [MATH], and the binding energy [MATH] of the non-BPS state is [EQUATION] where we have expressed the result in terms of the binding energy of the putative BPS state with the same charges, [MATH] near the CMS.', '0807.2419-1-21-1': 'We observe that for [MATH] the binding energy is much smaller than [MATH] as expected for a state lying above the BPS bound.', '0807.2419-1-21-2': 'Furthermore, we can also determine that the charge radius of the state is [MATH] which can be made parametrically large by moving near the CMS for any finite value of [MATH] and so this description of the non-BPS state should be reliable.', '0807.2419-1-22-0': '### 3.2 Ground state for [MATH]', '0807.2419-1-23-0': 'The opposite regime with [MATH] is also of interest as it corresponds to parametrically weak coupling and the non-BPS shift of the binding energy is better understood as a small correction [MATH] to the mass [MATH] of the would-be BPS [MATH] state.', '0807.2419-1-23-1': 'i.e. we have [EQUATION] verifying that the state indeed lies above the BPS bound in the spectrum, but that for [MATH] it it still well-bound at weak coupling as [MATH].', '0807.2419-1-23-2': 'Note that the charge radius in this limit is given by [MATH] and is again parametrically large near the CMS, justifying the low energy description of the state.', '0807.2419-1-24-0': 'At weak coupling, [MATH] is now given purely via a misalignment in the two vevs and thus is independent of the coupling.', '0807.2419-1-24-1': 'In this regime, since [MATH], the fractional energy shift of this state from the BPS bound is of [MATH], and thus scales as a small [MATH] contribution relative to the classical mass [MATH].', '0807.2419-1-24-2': 'More precisely, the BPS mass of the [MATH] state is given by [MATH] where [MATH] are the two Cartan vevs projected along simple roots which parametrize the moduli space at weak coupling.', '0807.2419-1-24-3': 'The true mass of the non-BPS state is then [MATH] with [EQUATION] where the latter equality holds near the CMS, since [MATH] differs from [MATH] only by terms of [MATH],', '0807.2419-1-25-0': 'The results in ([REF]) and ([REF]) provide explicit computations of the non-BPS correction to the mass for this class of zero torsion states in the near CMS regime.', '0807.2419-1-25-1': 'It would clearly be interesting to find a deeper understanding of these formulae and we will make some additional remarks in the following section.', '0807.2419-1-26-0': '## 4.', '0807.2419-1-26-1': 'Concluding remarks', '0807.2419-1-27-0': 'In this note, we have presented the general form of the two-body moduli-space dynamics of constituent BPS states in supersymmetric theories up to two-derivative order.', '0807.2419-1-27-1': 'The results of Section 2 apply at generic strongly-coupled regions of the moduli space, with the restriction that one considers interactions of the two constituent states near the CMS.', '0807.2419-1-27-2': 'This formulation makes supersymmetry manifest and shows that the potential arises from a [MATH]-term, but also incorporates nontrivial corrections from the metric.', '0807.2419-1-27-3': 'We focused on the latter aspect to deduce mass corrections to non-BPS bound states satisfying [MATH], but there are a couple of other technical issues that are worthy of further comment:', '0807.2419-1-28-0': 'We will finish with a couple of further applications that may be interesting to explore.', '0807.2419-1-28-1': 'One concerns the fate of [MATH]-duality in theories.', '0807.2419-1-28-2': 'The soft UV nature of mass deformations breaking SUSY suggests that the induced breaking of [MATH]-duality should be spontaneous, and thus constrained by the modular weights of the mass deformation parameters.', '0807.2419-1-28-3': 'This expectation is elegantly born out in the vacuum sector of SYM, where [MATH]-duality acts by permutation on all the massive vacua [CITATION].', '0807.2419-1-28-4': 'This raises the question of whether a similar viewpoint may prove fruitful in the context of the BPS and non-BPS spectrum in the theory, where we may expect a more significant role to be played by the constraints of [MATH]-duality than in SYM itself.', '0807.2419-1-28-5': 'A crucial question here is whether the mass perturbation for the extra adjoint fields destabilises any of the fermionic zero modes which are crucial to the [MATH]-dual spectrum.', '0807.2419-1-28-6': 'While many of these modes are localized with come characteristic scale, and thus should be stable, caution is required in regard to some of the threshold bound states, e.g. the [MATH] state for aligned vevs.', '0807.2419-1-28-7': 'It would be interesting to explore how the spectrum is restructured by the extending the SQM system studied here to include the effects of the extra adjoint multiplets of the theory.', '0807.2419-1-29-0': 'Finally, on a somewhat different theme, an interesting aspect of these states is that, through the electromagnetic contribution to the angular momentum, the multiplet size and thus the degeneracy can become very large for large charges.', '0807.2419-1-29-1': 'Analogous states also have a realization within supergravity and may form black holes, a subject that has seen significant activity recently [CITATION].', '0807.2419-1-29-2': 'It would therefore be interesting to contrast the degeneracies of more general dyonic states in field theory with those which necessarily form horizons after coupling to gravity [CITATION].', '0807.2419-1-29-3': 'Related questions may also be posed in counting degeneracies on both sides of the AdS/CFT correspondence [CITATION].'}

{'0807.2419-2-0-0': 'We derive the general form of the moduli-space effective action for the long-range interaction of two BPS dyons in gauge theories.', '0807.2419-2-0-1': 'This action determines the bound state structure of various BPS and non-BPS states near marginal stability curves, and we utilise it to compute the leading correction to the BPS-mass of zero-torsion non-BPS bound states close to marginal stability.', '0807.2419-2-1-0': 'July 2008', '0807.2419-2-2-0': '## 1.', '0807.2419-2-2-1': 'Introduction', '0807.2419-2-3-0': 'Theories with extended or supersymmetry in four dimensions have a BPS sector of the particle spectrum, namely those states which preserve some fraction of the supersymmetry of the vacuum [CITATION].', '0807.2419-2-3-1': 'The masses, interactions and degeneracies of these BPS states are often exactly computable, providing a powerful window to the dynamics.', '0807.2419-2-3-2': 'One of the remarkable dynamical principles which emerges in theories with this level of supersymmetry is electric-magnetic duality [CITATION], and its infinite-dimensional generalization [MATH]-duality.', '0807.2419-2-3-3': 'It was realized some time ago that the BPS spectrum then inherits a rather hierarchical structure, with the majority of the states being viewed as bound states of a relatively small number of constituents, which are the lightest states in any charge sector at a given point in the vacuum moduli space of the theory.', '0807.2419-2-3-4': 'This structure is quite rigid under changes of moduli with the exception of special co-dimension one surfaces, known as curves of marginal stability (CMS), where changes occur in particular in the spectrum of the lightest constituent states.', '0807.2419-2-3-5': 'In the context of dyonic bound states, this structure was explored in and gauge theories some time ago [CITATION] utilizing the Seiberg-Witten solution [CITATION], and has more recently been elucidated for BPS black holes in and string theories [CITATION].', '0807.2419-2-3-6': 'While much of this dynamical structure is expected to extend to the more generic non-BPS sector, it is far less amenable to study as many of the powerful supersymmetric tools are no longer available.', '0807.2419-2-3-7': 'Nonetheless, the non-BPS sector is of course of considerable interest for many reasons, not least because it provides a window into the behaviour of generic massive states in strongly coupled gauge theories, and indeed to generic black hole states in and string theory.', '0807.2419-2-4-0': 'In this letter, we will consider a special class of non-BPS states which may be viewed as weakly bound composites of BPS constituents near curves of marginal stability.', '0807.2419-2-4-1': 'The analysis will thus be limited to near-CMS regions of the moduli space, but the payoff is that explicit computations can be performed for the mass, and in principle the degeneracy, of these non-BPS states.', '0807.2419-2-4-2': 'The interactions of BPS states are dictated by supersymmetry, which allows us to treat the non-relativistic bound state problem exactly even at strong coupling.', '0807.2419-2-4-3': 'This statement relies on several special features, most importantly that the long-range interactions of BPS constituents with charges [MATH] on the Coulomb branch are exactly determined by the central charges [MATH] of the BPS states in question, and the special Kahler metric on the Coulomb branch [MATH] [CITATION].', '0807.2419-2-4-4': 'By considering the exchange of the massless fields, the Coulomb term can be shown to take the form [CITATION], [EQUATION]', '0807.2419-2-4-5': 'This expression is particularly useful near the CMS for these two constituents, as the bound state problem simplifies to the non-relativistic level because the binding energy may be made parametrically small.', '0807.2419-2-4-6': 'The CMS curve(s) for the two states in question is defined by the condition: [EQUATION]', '0807.2419-2-4-7': 'In the context of supersymmetric quantum mechanics (SQM), the potential for two interacting BPS constituents with masses [MATH], for [MATH], naturally tends to a constant as [MATH] such that the lowest eigenvalue of the SQM Hamiltonian vanishes for a BPS bound state with mass [MATH].', '0807.2419-2-4-8': 'This implies that the potential is defined at large [MATH] as [EQUATION] in terms of the binding energy [MATH].', '0807.2419-2-4-9': 'Expansion of the potential [MATH] near [MATH] takes the form [EQUATION] where [MATH] is the reduced mass of the two BPS constituents.', '0807.2419-2-4-10': 'Close to the CMS, the sign of the Coulomb potential is determined by the sign of the Dirac-Schwinger-Zwanziger symplectic charge product [EQUATION] also known as the torsion.', '0807.2419-2-4-11': 'Thus generically as one crosses the CMS, the Coulomb potential changes sign and a bound state exists on only one side [CITATION].', '0807.2419-2-5-0': 'This provides a simple viewpoint on the existence or otherwise of bound states and was exploited in [CITATION] to consider the BPS spectrum in theories utilizing the Seiberg-Witten solution (see e.g. [CITATION] for an alternative approach).', '0807.2419-2-5-1': 'However, this picture is incomplete as the relative dynamics of two BPS constituents necessarily preserves at least four supercharges in an theory, and this structure is not manifest in the potential above.', '0807.2419-2-5-2': 'In general terms the structure of the relevant form of SQM has been known for some time [CITATION], and was studied in the context of BPS states by Denef [CITATION].', '0807.2419-2-5-3': 'We will extend this approach to incorporate the nontrivial metric for the relative translational coordinates of the two BPS constituents, and explore how this provides a novel window on the mass corrections for non-BPS bound states.', '0807.2419-2-6-0': 'The general form of the worldline SQM is derived in Section 2, and shown to be consistent with the Coulomb potential in ([REF]).', '0807.2419-2-6-1': 'In Section 3, we focus on a specific class of non-BPS bound states with zero torsion, i.e. [MATH], which are known to be BPS states in SYM, but lift slightly from the BPS bound in theories at weak coupling.', '0807.2419-2-6-2': 'Using the low energy description of the constituent BPS states we explicitly compute the non-BPS correction to the mass.', '0807.2419-2-6-3': 'We finish with some additional remarks in Section 4.', '0807.2419-2-7-0': '## 2.', '0807.2419-2-7-1': 'The D-term potential and the moduli-space metric', '0807.2419-2-8-0': 'The chiral structure of the fermionic zero modes in the background of monopole solutions in SYM implies that, at the classical level, the low energy dynamics on the moduli space of BPS dyons is realized as SQM [CITATION], namely the reduction of a chiral (0,4) supersymmetric sigma model in 1+1D, preserving four supercharges.', '0807.2419-2-8-1': 'This pairs four bosonic and four fermionic collective coordinates.', '0807.2419-2-8-2': 'However, while three of the bosonic coordinates reflect translations and so would be expected to survive the inclusion of generic quantum corrections, the fourth is compact and the quantized momentum along this direction corresponds to the discrete electric charge.', '0807.2419-2-8-3': 'Thus, in a generic sector of the quantum theory where, in addition to the magnetic charge, the electric charge is also fixed we deal with a system having 3 bosonic and 4 fermionic variables.', '0807.2419-2-8-4': 'Such a multiplet is in fact consistent in SQM, and arises from the reduction of a vector multiplet in 3+1D.', '0807.2419-2-9-0': 'The general structure of SQM resulting from the reduction of a vector multiplet has been known for some time [CITATION], and was discussed in the present context more recently by Denef [CITATION].', '0807.2419-2-9-1': 'The vector multiplet decomposes to [MATH]<[MATH][MATH], comprising the coordinate vector [MATH], its chiral superpartner [MATH]<[MATH][MATH] and the auxiliary field [MATH].', '0807.2419-2-9-2': 'If we consider the relative dynamics of two point-like sources, with [MATH] the relative separation, the Lagrangian up to 2-derivative order is quite constrained [CITATION], [EQUATION] where the potentials [MATH], [MATH] and the metric [MATH] are functions of the relative coordinate [MATH].', '0807.2419-2-9-3': 'The potentials are related by the condition [MATH].', '0807.2419-2-9-4': 'It follows that [MATH] is a harmonic function, and requiring spherical symmetry, as is appropriate for the potential between two Coulomb sources, the general solution has the form [MATH] with [MATH], constants [MATH] and [MATH], and [MATH] with [MATH] the unit charge Dirac monopole potential.', '0807.2419-2-9-5': 'Quantization then demands that [MATH] be an integer.', '0807.2419-2-10-0': 'We will now specialize to the case at hand, namely the relative dynamics of two BPS states in SYM.', '0807.2419-2-10-1': 'The 2-particle interaction potential has the form [EQUATION] where the above constraints on [MATH] should indeed hold.', '0807.2419-2-10-2': 'Since the interactions are long range, we can also write [MATH] introducing a further constant [MATH] and proceed to fix the constants [MATH], [MATH] and [MATH].', '0807.2419-2-10-3': 'By comparing to the expansion of the Coulomb potential ([REF]) near the CMS, the binding energy immediately fixes [MATH].', '0807.2419-2-11-0': 'To proceed, we note that for a BPS bound state to exist this system must have a supersymmetric vacuum where [MATH].', '0807.2419-2-11-1': 'Given regularity of the metric, this implies classically that [MATH] in the vacuum and the existence or otherwise of BPS states will, generically at least, be independent of the metric [MATH].', '0807.2419-2-11-2': 'Since the leading term in the Coulomb potential ([REF]) near the CMS, linear in [MATH], dictates the presence of BPS states it must necessarily be present in [MATH].', '0807.2419-2-11-3': 'This allows to uniquely fix the constant [MATH], and thus the form of [MATH], [EQUATION] up to corrections of [MATH] which are subleading near the CMS.', '0807.2419-2-12-0': 'It remains to determine the metric [MATH].', '0807.2419-2-12-1': 'Indeed, it is now clear that the [MATH] term in ([REF]) cannot reproduce the full Coulomb potential ([REF]) at [MATH] unless [MATH].', '0807.2419-2-12-2': 'Classically, or in the SYM limit, there is enough supersymmetry to determine the form of the metric precisely.', '0807.2419-2-12-3': 'Here we will be content to determine the leading [MATH] correction and, rather than extract [MATH] from ([REF]), we will compute it using the approach of Manton [CITATION].', '0807.2419-2-12-4': 'Near the CMS, we can consider the dynamics of the non-relativistic BPS point-like constituents interacting through their electric, magnetic, and scalar charges, and the metric is determined by the terms quadratic in velocity.', '0807.2419-2-13-0': "Manton's original calculation (at the level of the leading [MATH] terms) can be straightforwardly repeated in the regime using the exact quantum corrected charges that were computed in [CITATION].", '0807.2419-2-13-1': 'We begin by writing down the probe Lagrangian for dyon 2 in the background of dyon 1, with both states weakly boosted, [EQUATION]', '0807.2419-2-13-2': 'Here [MATH] and [MATH] are the electric and dual magnetic potentials, coupling to the state via the metric [MATH] and its inverse, and the scalar charge is determined by the Taylor expansion of [MATH].', '0807.2419-2-13-3': 'When [MATH] and [MATH] are both nonzero we also need to account for the Witten effect which modifies the coupling to the electric potential.', '0807.2419-2-14-0': 'The boosted potentials [MATH] and [MATH] induced by the presence of dyon 1 take the usual Lienard-Wiechert form [CITATION], while the scalar field shift is given by [CITATION], [EQUATION]', '0807.2419-2-14-1': 'Expanding to quadratic order in velocities, symmetrizing to include the dynamics of dyon 1, and dropping the free center of mass motion, we obtain [MATH], where [EQUATION]', '0807.2419-2-14-2': 'Thus, up to possible subleading terms of [MATH] in the metric, we can write down the full potential ([REF]) in the form [EQUATION]', '0807.2419-2-14-3': 'As a useful consistency check, by expanding to [MATH], we may now verify that the [MATH] term agrees precisely up to [MATH] with potential ([REF]) computed by considering massless exchange [CITATION].', '0807.2419-2-14-4': 'The full formulation as a [MATH]-term potential in ([REF]) makes the constraints of supersymmetry manifest and naturally provides an extension to higher order in [MATH] since any terms of [MATH] in the metric will be further suppressed in [MATH] by a factor of [MATH].', '0807.2419-2-15-0': '## 3.', '0807.2419-2-15-1': 'The fate of zero-torsion non-BPS states', '0807.2419-2-16-0': 'Since [MATH] for supersymmetric ground states to exist classically, we see that the metric generically plays little role; the relevant constraint is [MATH], so that [EQUATION] which we observe only has solutions on one side of the CMS where the composite BPS state exists.', '0807.2419-2-16-1': 'This simplification allows the quantum mechanics of generic BPS states to be studied in a simplified system where the metric is ignored, i.e. we can take [MATH], a constant.', '0807.2419-2-16-2': 'This regime was considered in detail by Denef [CITATION], and the BPS ground state has the form [MATH]<[MATH][MATH] in terms of [MATH]<[MATH][MATH], the complex spinor superpartner of [MATH].', '0807.2419-2-16-3': 'The constraint that the supercharges annihilate the state [MATH] implies [MATH] and the angular part of the wavefunction is expressed in terms of the monopole harmonics of Wu and Yang [CITATION] while the size of the multiplet scales with [MATH] on account of the electromagnetic contribution to the total angular momentum.', '0807.2419-2-16-4': 'The radial wavefunction scales as [MATH] and thus is localized on only one side of the CMS, which is the quantum version of the fact that ([REF]) only makes sense for one sign of [MATH].', '0807.2419-2-16-5': 'An interesting feature of the solution is a shift of the spin by 1/2, i.e. the ground state is a spinor, which can be interpreted as due to the induced spin coupling in the Hamiltonian, [MATH] where [MATH]<[MATH][MATH]<[MATH][MATH].', '0807.2419-2-17-0': 'While the spectrum of BPS states can generally be studied independently of the [MATH] corrections to the metric [MATH], these corrections are crucial to the formation of non-BPS composite states.', '0807.2419-2-17-1': 'Studying these states within the present framework is often difficult as the binding energy becomes small when the full Coulomb potential ([REF]) vanishes, which corresponds to [EQUATION] and this condition is not generally satisfied for small [MATH].', '0807.2419-2-17-2': 'This renders the bound state problem intractable.', '0807.2419-2-17-3': 'However, an interesting exception that we will now focus on concerns the case where the constituents are mutually local, so the torsion vanishes [MATH].', '0807.2419-2-17-4': 'The potential near the CMS is then provided purely by the metric, [EQUATION] which (for positive [MATH]) vanishes linearly at [MATH], and tends (like [MATH]) to [MATH] at infinity.', '0807.2419-2-18-0': 'In SYM, zero torsion states often have a special status as they may be required by [MATH]-duality to exist as bound states in the BPS spectrum.', '0807.2419-2-18-1': "The simplest example arises for gauge group SU(3), where the existence of a [MATH] boson with charges aligned along both U(1)'s implies, via duality, a bound state of two distinct monopoles with unit magnetic charges along the two simple roots.", '0807.2419-2-18-2': 'The existence of this [MATH] state has been verified at weak coupling in various regions of the moduli space [CITATION], starting first with the simplest case of aligned vevs where it exists as a bound state at threshold.', '0807.2419-2-18-3': 'However, since a 1/2-BPS multiplet in has the same size as a generic non-BPS multiplet in , we would expect that on breaking to SYM the mass perturbation will lift the state from the BPS spectrum, and indeed this conclusion is borne out in explicit calculations at weak coupling [CITATION].', '0807.2419-2-18-4': 'In the present context, we cannot draw any direct conclusions on the existence of the threshold state, since [MATH] takes the form [EQUATION] and thus vanishes for aligned vevs, as does the potential ([REF]).', '0807.2419-2-18-5': 'However, the [MATH] state is also required by duality to exist in the theory for misaligned vevs and indeed it duly appears in the weak coupling analysis [CITATION].', '0807.2419-2-18-6': 'This state is no longer at threshold and on perturbing to SYM should again lift from the BPS spectrum [CITATION].', '0807.2419-2-18-7': 'We can verify this in a rather straightforward manner using SQM with the potential ([REF]) and indeed we can also compute the mass shift from the BPS bound.', '0807.2419-2-19-0': 'For [MATH], the system is purely bosonic and inspection of the potential ([REF]) suggests that we can compute the non-BPS correction to the mass as the zero-point energy.', '0807.2419-2-19-1': 'The classical Hamiltonian has the form [MATH], and the Schrodinger equation, after accounting for the metric and proper ordering, can be re-expressed as an analogue Coulomb system [CITATION], [EQUATION] with [MATH] where [MATH] is the mass of the bound state and [MATH] is the BPS lower bound.', '0807.2419-2-19-2': 'The Hamiltonian in this equation is hermitian with respect to the usual Euclidean metric, and thus the problem reduces to a standard hydrogenic analysis [CITATION].', '0807.2419-2-19-3': 'However, the spectrum is of course not Coulomb-like, and the scaling of the ground state energy is determined by the parameter [MATH] which we keep fixed in the near-CMS regime where [MATH].', '0807.2419-2-19-4': 'It will be enough to focus on the weak-coupling limit [MATH] and [MATH], so that [MATH] and there are two regimes of interest, namely [MATH] and [MATH].', '0807.2419-2-19-5': 'Note that in this limit, the exact moduli-space metric is known and contains no corrections of [MATH] [CITATION].', '0807.2419-2-20-0': '### 3.1 Ground state for [MATH]', '0807.2419-2-21-0': 'In the regime [MATH], the potential becomes approximately Coulomb-like, [MATH] with [MATH], and the binding energy [MATH] of the non-BPS state is [EQUATION] where we have expressed the result in terms of the binding energy of the putative BPS state with the same charges, [MATH] near the CMS.', '0807.2419-2-21-1': 'We observe that for [MATH] the binding energy is much smaller than [MATH] as expected for a state lying above the BPS bound.', '0807.2419-2-21-2': 'Furthermore, we can also determine that the charge radius of the state is [MATH] which can be made parametrically large by moving near the CMS for any finite value of [MATH] and so this description of the non-BPS state should be reliable.', '0807.2419-2-22-0': '### 3.2 Ground state for [MATH]', '0807.2419-2-23-0': 'The opposite regime with [MATH] is also of interest as it corresponds to parametrically weak coupling and the non-BPS shift of the binding energy is better understood as a small correction [MATH] to the mass [MATH] of the would-be BPS [MATH] state.', '0807.2419-2-23-1': 'i.e. we have [EQUATION] verifying that the state indeed lies above the BPS bound in the spectrum, but that for [MATH] it it still well-bound at weak coupling as [MATH].', '0807.2419-2-23-2': 'Note that the charge radius in this limit is given by [MATH] and is again parametrically large near the CMS, justifying the low energy description of the state.', '0807.2419-2-24-0': 'At weak coupling, [MATH] is now given purely via a misalignment in the two vevs and thus is independent of the coupling.', '0807.2419-2-24-1': 'In this regime, since [MATH], the fractional energy shift of this state from the BPS bound is of [MATH], and thus scales as a small [MATH] contribution relative to the classical mass [MATH].', '0807.2419-2-24-2': 'More precisely, the BPS mass of the [MATH] state is given by [MATH] where [MATH] are the two Cartan vevs projected along simple roots which parametrize the moduli space at weak coupling.', '0807.2419-2-24-3': 'The true mass of the non-BPS state is then [MATH] with [EQUATION] where the latter equality holds near the CMS, since [MATH] differs from [MATH] only by terms of [MATH],', '0807.2419-2-25-0': 'The results in ([REF]) and ([REF]) provide explicit computations of the non-BPS correction to the mass for this class of zero torsion states in the near CMS regime.', '0807.2419-2-25-1': 'It would clearly be interesting to find a deeper understanding of these formulae and we will make some additional remarks in the following section.', '0807.2419-2-26-0': '## 4.', '0807.2419-2-26-1': 'Concluding remarks', '0807.2419-2-27-0': 'In this note, we have presented the general form of the two-body moduli-space dynamics of constituent BPS states in supersymmetric theories up to two-derivative order.', '0807.2419-2-27-1': 'The results of Section 2 apply at generic strongly-coupled regions of the moduli space, with the restriction that one considers interactions of the two constituent states near the CMS.', '0807.2419-2-27-2': 'This formulation makes supersymmetry manifest and shows that the potential arises from a [MATH]-term, but also incorporates nontrivial corrections from the metric.', '0807.2419-2-27-3': 'We focused on the latter aspect to deduce mass corrections to non-BPS bound states satisfying [MATH], but there are a couple of other technical issues that are worthy of further comment:', '0807.2419-2-28-0': 'We will finish with a couple of further applications that may be interesting to explore.', '0807.2419-2-28-1': 'One concerns the fate of [MATH]-duality in theories.', '0807.2419-2-28-2': 'The soft UV nature of mass deformations breaking SUSY suggests that the induced breaking of [MATH]-duality should be spontaneous, and thus constrained by the modular weights of the mass deformation parameters.', '0807.2419-2-28-3': 'This expectation is elegantly born out in the vacuum sector of SYM, where [MATH]-duality acts by permutation on all the massive vacua [CITATION].', '0807.2419-2-28-4': 'This raises the question of whether a similar viewpoint may prove fruitful in the context of the BPS and non-BPS spectrum in the theory, where we may expect a more significant role to be played by the constraints of [MATH]-duality than in SYM itself.', '0807.2419-2-28-5': 'A crucial question here is whether the mass perturbation for the extra adjoint fields destabilises any of the fermionic zero modes which are crucial to the [MATH]-dual spectrum.', '0807.2419-2-28-6': 'While many of these modes are localized with come characteristic scale, and thus should be stable, caution is required in regard to some of the threshold bound states, e.g. the [MATH] state for aligned vevs.', '0807.2419-2-28-7': 'It would be interesting to explore how the spectrum is restructured by the extending the SQM system studied here to include the effects of the extra adjoint multiplets of the theory.', '0807.2419-2-29-0': 'Finally, on a somewhat different theme, an interesting aspect of these states is that, through the electromagnetic contribution to the angular momentum, the multiplet size and thus the degeneracy can become very large for large charges.', '0807.2419-2-29-1': 'Analogous states also have a realization within supergravity and may form black holes, a subject that has seen significant activity recently [CITATION].', '0807.2419-2-29-2': 'It would therefore be interesting to contrast the degeneracies of more general dyonic states in field theory with those which necessarily form horizons after coupling to gravity [CITATION].', '0807.2419-2-29-3': 'Related questions may also be posed in counting degeneracies on both sides of the AdS/CFT correspondence [CITATION].'}

[['0807.2419-1-23-0', '0807.2419-2-23-0'], ['0807.2419-1-23-1', '0807.2419-2-23-1'], ['0807.2419-1-23-2', '0807.2419-2-23-2'], ['0807.2419-1-14-0', '0807.2419-2-14-0'], ['0807.2419-1-14-1', '0807.2419-2-14-1'], ['0807.2419-1-14-2', '0807.2419-2-14-2'], ['0807.2419-1-14-3', '0807.2419-2-14-3'], ['0807.2419-1-14-4', '0807.2419-2-14-4'], ['0807.2419-1-24-0', '0807.2419-2-24-0'], ['0807.2419-1-24-1', '0807.2419-2-24-1'], ['0807.2419-1-24-2', '0807.2419-2-24-2'], ['0807.2419-1-28-0', '0807.2419-2-28-0'], ['0807.2419-1-28-1', '0807.2419-2-28-1'], ['0807.2419-1-28-2', '0807.2419-2-28-2'], ['0807.2419-1-28-3', '0807.2419-2-28-3'], ['0807.2419-1-28-4', '0807.2419-2-28-4'], ['0807.2419-1-28-5', '0807.2419-2-28-5'], ['0807.2419-1-28-6', '0807.2419-2-28-6'], ['0807.2419-1-28-7', '0807.2419-2-28-7'], ['0807.2419-1-13-0', '0807.2419-2-13-0'], ['0807.2419-1-13-1', '0807.2419-2-13-1'], ['0807.2419-1-13-2', '0807.2419-2-13-2'], ['0807.2419-1-13-3', '0807.2419-2-13-3'], ['0807.2419-1-8-0', '0807.2419-2-8-0'], ['0807.2419-1-8-1', '0807.2419-2-8-1'], ['0807.2419-1-8-2', '0807.2419-2-8-2'], ['0807.2419-1-8-3', '0807.2419-2-8-3'], ['0807.2419-1-8-4', '0807.2419-2-8-4'], ['0807.2419-1-21-0', '0807.2419-2-21-0'], ['0807.2419-1-21-1', '0807.2419-2-21-1'], ['0807.2419-1-21-2', '0807.2419-2-21-2'], ['0807.2419-1-6-0', '0807.2419-2-6-0'], ['0807.2419-1-6-1', '0807.2419-2-6-1'], ['0807.2419-1-6-2', '0807.2419-2-6-2'], ['0807.2419-1-6-3', '0807.2419-2-6-3'], ['0807.2419-1-0-0', '0807.2419-2-0-0'], ['0807.2419-1-0-1', '0807.2419-2-0-1'], ['0807.2419-1-3-0', '0807.2419-2-3-0'], ['0807.2419-1-3-1', '0807.2419-2-3-1'], ['0807.2419-1-3-2', '0807.2419-2-3-2'], ['0807.2419-1-3-3', '0807.2419-2-3-3'], ['0807.2419-1-3-4', '0807.2419-2-3-4'], ['0807.2419-1-3-5', '0807.2419-2-3-5'], ['0807.2419-1-3-6', '0807.2419-2-3-6'], ['0807.2419-1-3-7', '0807.2419-2-3-7'], ['0807.2419-1-16-0', '0807.2419-2-16-0'], ['0807.2419-1-16-1', '0807.2419-2-16-1'], ['0807.2419-1-16-2', '0807.2419-2-16-2'], ['0807.2419-1-16-3', '0807.2419-2-16-3'], ['0807.2419-1-16-4', '0807.2419-2-16-4'], ['0807.2419-1-16-5', '0807.2419-2-16-5'], ['0807.2419-1-10-0', '0807.2419-2-10-0'], ['0807.2419-1-10-1', '0807.2419-2-10-1'], ['0807.2419-1-10-2', '0807.2419-2-10-2'], ['0807.2419-1-10-3', '0807.2419-2-10-3'], ['0807.2419-1-18-0', '0807.2419-2-18-0'], ['0807.2419-1-18-1', '0807.2419-2-18-1'], ['0807.2419-1-18-2', '0807.2419-2-18-2'], ['0807.2419-1-18-3', '0807.2419-2-18-3'], ['0807.2419-1-18-4', '0807.2419-2-18-4'], ['0807.2419-1-18-5', '0807.2419-2-18-5'], ['0807.2419-1-18-6', '0807.2419-2-18-6'], ['0807.2419-1-18-7', '0807.2419-2-18-7'], ['0807.2419-1-25-0', '0807.2419-2-25-0'], ['0807.2419-1-25-1', '0807.2419-2-25-1'], ['0807.2419-1-19-0', '0807.2419-2-19-0'], ['0807.2419-1-19-1', '0807.2419-2-19-1'], ['0807.2419-1-19-2', '0807.2419-2-19-2'], ['0807.2419-1-19-3', '0807.2419-2-19-3'], ['0807.2419-1-19-4', '0807.2419-2-19-4'], ['0807.2419-1-19-5', '0807.2419-2-19-5'], ['0807.2419-1-11-0', '0807.2419-2-11-0'], ['0807.2419-1-11-1', '0807.2419-2-11-1'], ['0807.2419-1-11-2', '0807.2419-2-11-2'], ['0807.2419-1-11-3', '0807.2419-2-11-3'], ['0807.2419-1-5-1', '0807.2419-2-5-1'], ['0807.2419-1-5-2', '0807.2419-2-5-2'], ['0807.2419-1-5-3', '0807.2419-2-5-3'], ['0807.2419-1-4-0', '0807.2419-2-4-0'], ['0807.2419-1-4-1', '0807.2419-2-4-1'], ['0807.2419-1-4-2', '0807.2419-2-4-2'], ['0807.2419-1-4-3', '0807.2419-2-4-3'], ['0807.2419-1-4-4', '0807.2419-2-4-4'], ['0807.2419-1-4-5', '0807.2419-2-4-5'], ['0807.2419-1-4-6', '0807.2419-2-4-6'], ['0807.2419-1-4-7', '0807.2419-2-4-7'], ['0807.2419-1-4-8', '0807.2419-2-4-8'], ['0807.2419-1-4-9', '0807.2419-2-4-9'], ['0807.2419-1-4-10', '0807.2419-2-4-10'], ['0807.2419-1-4-11', '0807.2419-2-4-11'], ['0807.2419-1-12-0', '0807.2419-2-12-0'], ['0807.2419-1-12-1', '0807.2419-2-12-1'], ['0807.2419-1-12-2', '0807.2419-2-12-2'], ['0807.2419-1-12-3', '0807.2419-2-12-3'], ['0807.2419-1-12-4', '0807.2419-2-12-4'], ['0807.2419-1-17-0', '0807.2419-2-17-0'], ['0807.2419-1-17-1', '0807.2419-2-17-1'], ['0807.2419-1-17-2', '0807.2419-2-17-2'], ['0807.2419-1-17-3', '0807.2419-2-17-3'], ['0807.2419-1-17-4', '0807.2419-2-17-4'], ['0807.2419-1-9-0', '0807.2419-2-9-0'], ['0807.2419-1-9-1', '0807.2419-2-9-1'], ['0807.2419-1-9-2', '0807.2419-2-9-2'], ['0807.2419-1-9-3', '0807.2419-2-9-3'], ['0807.2419-1-9-4', '0807.2419-2-9-4'], ['0807.2419-1-9-5', '0807.2419-2-9-5'], ['0807.2419-1-27-0', '0807.2419-2-27-0'], ['0807.2419-1-27-1', '0807.2419-2-27-1'], ['0807.2419-1-27-2', '0807.2419-2-27-2'], ['0807.2419-1-29-0', '0807.2419-2-29-0'], ['0807.2419-1-29-1', '0807.2419-2-29-1'], ['0807.2419-1-29-2', '0807.2419-2-29-2'], ['0807.2419-1-29-3', '0807.2419-2-29-3'], ['0807.2419-1-5-0', '0807.2419-2-5-0']]

[['0807.2419-1-23-0', '0807.2419-2-23-0'], ['0807.2419-1-23-1', '0807.2419-2-23-1'], ['0807.2419-1-23-2', '0807.2419-2-23-2'], ['0807.2419-1-14-0', '0807.2419-2-14-0'], ['0807.2419-1-14-1', '0807.2419-2-14-1'], ['0807.2419-1-14-2', '0807.2419-2-14-2'], ['0807.2419-1-14-3', '0807.2419-2-14-3'], ['0807.2419-1-14-4', '0807.2419-2-14-4'], ['0807.2419-1-24-0', '0807.2419-2-24-0'], ['0807.2419-1-24-1', '0807.2419-2-24-1'], ['0807.2419-1-24-2', '0807.2419-2-24-2'], ['0807.2419-1-28-0', '0807.2419-2-28-0'], ['0807.2419-1-28-1', '0807.2419-2-28-1'], ['0807.2419-1-28-2', '0807.2419-2-28-2'], ['0807.2419-1-28-3', '0807.2419-2-28-3'], ['0807.2419-1-28-4', '0807.2419-2-28-4'], ['0807.2419-1-28-5', '0807.2419-2-28-5'], ['0807.2419-1-28-6', '0807.2419-2-28-6'], ['0807.2419-1-28-7', '0807.2419-2-28-7'], ['0807.2419-1-13-0', '0807.2419-2-13-0'], ['0807.2419-1-13-1', '0807.2419-2-13-1'], ['0807.2419-1-13-2', '0807.2419-2-13-2'], ['0807.2419-1-13-3', '0807.2419-2-13-3'], ['0807.2419-1-8-0', '0807.2419-2-8-0'], ['0807.2419-1-8-1', '0807.2419-2-8-1'], ['0807.2419-1-8-2', '0807.2419-2-8-2'], ['0807.2419-1-8-3', '0807.2419-2-8-3'], ['0807.2419-1-8-4', '0807.2419-2-8-4'], ['0807.2419-1-21-0', '0807.2419-2-21-0'], ['0807.2419-1-21-1', '0807.2419-2-21-1'], ['0807.2419-1-21-2', '0807.2419-2-21-2'], ['0807.2419-1-6-0', '0807.2419-2-6-0'], ['0807.2419-1-6-1', '0807.2419-2-6-1'], ['0807.2419-1-6-2', '0807.2419-2-6-2'], ['0807.2419-1-6-3', '0807.2419-2-6-3'], ['0807.2419-1-0-0', '0807.2419-2-0-0'], ['0807.2419-1-0-1', '0807.2419-2-0-1'], ['0807.2419-1-3-0', '0807.2419-2-3-0'], ['0807.2419-1-3-1', '0807.2419-2-3-1'], ['0807.2419-1-3-2', '0807.2419-2-3-2'], ['0807.2419-1-3-3', '0807.2419-2-3-3'], ['0807.2419-1-3-4', '0807.2419-2-3-4'], ['0807.2419-1-3-5', '0807.2419-2-3-5'], ['0807.2419-1-3-6', '0807.2419-2-3-6'], ['0807.2419-1-3-7', '0807.2419-2-3-7'], ['0807.2419-1-16-0', '0807.2419-2-16-0'], ['0807.2419-1-16-1', '0807.2419-2-16-1'], ['0807.2419-1-16-2', '0807.2419-2-16-2'], ['0807.2419-1-16-3', '0807.2419-2-16-3'], ['0807.2419-1-16-4', '0807.2419-2-16-4'], ['0807.2419-1-16-5', '0807.2419-2-16-5'], ['0807.2419-1-10-0', '0807.2419-2-10-0'], ['0807.2419-1-10-1', '0807.2419-2-10-1'], ['0807.2419-1-10-2', '0807.2419-2-10-2'], ['0807.2419-1-10-3', '0807.2419-2-10-3'], ['0807.2419-1-18-0', '0807.2419-2-18-0'], ['0807.2419-1-18-1', '0807.2419-2-18-1'], ['0807.2419-1-18-2', '0807.2419-2-18-2'], ['0807.2419-1-18-3', '0807.2419-2-18-3'], ['0807.2419-1-18-4', '0807.2419-2-18-4'], ['0807.2419-1-18-5', '0807.2419-2-18-5'], ['0807.2419-1-18-6', '0807.2419-2-18-6'], ['0807.2419-1-18-7', '0807.2419-2-18-7'], ['0807.2419-1-25-0', '0807.2419-2-25-0'], ['0807.2419-1-25-1', '0807.2419-2-25-1'], ['0807.2419-1-19-0', '0807.2419-2-19-0'], ['0807.2419-1-19-1', '0807.2419-2-19-1'], ['0807.2419-1-19-2', '0807.2419-2-19-2'], ['0807.2419-1-19-3', '0807.2419-2-19-3'], ['0807.2419-1-19-4', '0807.2419-2-19-4'], ['0807.2419-1-19-5', '0807.2419-2-19-5'], ['0807.2419-1-11-0', '0807.2419-2-11-0'], ['0807.2419-1-11-1', '0807.2419-2-11-1'], ['0807.2419-1-11-2', '0807.2419-2-11-2'], ['0807.2419-1-11-3', '0807.2419-2-11-3'], ['0807.2419-1-5-1', '0807.2419-2-5-1'], ['0807.2419-1-5-2', '0807.2419-2-5-2'], ['0807.2419-1-5-3', '0807.2419-2-5-3'], ['0807.2419-1-4-0', '0807.2419-2-4-0'], ['0807.2419-1-4-1', '0807.2419-2-4-1'], ['0807.2419-1-4-2', '0807.2419-2-4-2'], ['0807.2419-1-4-3', '0807.2419-2-4-3'], ['0807.2419-1-4-4', '0807.2419-2-4-4'], ['0807.2419-1-4-5', '0807.2419-2-4-5'], ['0807.2419-1-4-6', '0807.2419-2-4-6'], ['0807.2419-1-4-7', '0807.2419-2-4-7'], ['0807.2419-1-4-8', '0807.2419-2-4-8'], ['0807.2419-1-4-9', '0807.2419-2-4-9'], ['0807.2419-1-4-10', '0807.2419-2-4-10'], ['0807.2419-1-4-11', '0807.2419-2-4-11'], ['0807.2419-1-12-0', '0807.2419-2-12-0'], ['0807.2419-1-12-1', '0807.2419-2-12-1'], ['0807.2419-1-12-2', '0807.2419-2-12-2'], ['0807.2419-1-12-3', '0807.2419-2-12-3'], ['0807.2419-1-12-4', '0807.2419-2-12-4'], ['0807.2419-1-17-0', '0807.2419-2-17-0'], ['0807.2419-1-17-1', '0807.2419-2-17-1'], ['0807.2419-1-17-2', '0807.2419-2-17-2'], ['0807.2419-1-17-3', '0807.2419-2-17-3'], ['0807.2419-1-17-4', '0807.2419-2-17-4'], ['0807.2419-1-9-0', '0807.2419-2-9-0'], ['0807.2419-1-9-1', '0807.2419-2-9-1'], ['0807.2419-1-9-2', '0807.2419-2-9-2'], ['0807.2419-1-9-3', '0807.2419-2-9-3'], ['0807.2419-1-9-4', '0807.2419-2-9-4'], ['0807.2419-1-9-5', '0807.2419-2-9-5'], ['0807.2419-1-27-0', '0807.2419-2-27-0'], ['0807.2419-1-27-1', '0807.2419-2-27-1'], ['0807.2419-1-27-2', '0807.2419-2-27-2'], ['0807.2419-1-29-0', '0807.2419-2-29-0'], ['0807.2419-1-29-1', '0807.2419-2-29-1'], ['0807.2419-1-29-2', '0807.2419-2-29-2'], ['0807.2419-1-29-3', '0807.2419-2-29-3']]

[['0807.2419-1-5-0', '0807.2419-2-5-0']]

[]

[]

[]

['0807.2419-1-1-0', '0807.2419-1-2-1', '0807.2419-1-7-0', '0807.2419-1-7-1', '0807.2419-1-15-0', '0807.2419-1-15-1', '0807.2419-1-24-3', '0807.2419-1-26-1', '0807.2419-1-27-3', '0807.2419-2-1-0', '0807.2419-2-2-1', '0807.2419-2-7-0', '0807.2419-2-7-1', '0807.2419-2-15-0', '0807.2419-2-15-1', '0807.2419-2-24-3', '0807.2419-2-26-1', '0807.2419-2-27-3']

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

https://arxiv.org/abs/0807.2419

1704.04209

{'1704.04209-1-0-0': '# Introduction', '1704.04209-1-1-0': "The behaviour of Green's functions wrt a shift of the renormalization scale is described by the anomalous dimensions of the fields and parameters of the theory, which enter the Renormalization Group Equations (RGE).", '1704.04209-1-1-1': 'For QCD the full set of four-loop renormalization constants and anomalous dimensions was presented in [CITATION].', '1704.04209-1-1-2': 'The results for the four-loop QCD [MATH]-function [CITATION] and the four-loop quark mass and field anomalous dimensions had already been available [CITATION].', '1704.04209-1-2-0': 'In this paper we consider a model with a non-abelian gauge group, one coupling constant and a reducible fermion representation, i. e. any number of irreducible fermion representations.', '1704.04209-1-2-1': 'The [MATH]-function for the coupling this model was computed in an earlier work [CITATION].', '1704.04209-1-2-2': 'Here we provide the remaining Renormalization Group (RG) functions in full dependence on the gauge parameter [MATH].', '1704.04209-1-3-0': 'Apart from completing the set of renormalization constants and the RGE of the theory, which is important in itself, the gauge boson and ghost propagator anomalous dimensions serve another purpose.', '1704.04209-1-3-1': 'These quantities are essential ingredients in comparing the momentum dependence of the corresponding propagators derived in non-perturbative calculations on the lattice, with perturbative results (see e. g. [CITATION]).', '1704.04209-1-4-0': 'This paper is structured as follows: First, we will give the notation and definitions for the model and the computed RG functions We will also repeat how the special case of QCD plus Majorana gluinos in the adjoint representation of the gauge group can be derived from our more general results.', '1704.04209-1-4-1': 'Then we will present analytical results for the four-loop anomalous dimensions of the gauge boson, ghost and fermion field as well as the ones for the ghost-gluon vertex, the fermion-gluon vertex and the fermion mass in Feynman gauge for compactness.', '1704.04209-1-4-2': 'The renormalization constants and anomalous dimensions for a generic gauge parameter [MATH] can be found in machine readable form in an accompanying file, which can be downloaded together with our source files on www.arxiv.org.', '1704.04209-1-5-0': '# Notation and definitions', '1704.04209-1-6-0': '## QCD with several fermion representations', '1704.04209-1-7-0': 'The Lagrangian of a QCD-like model extended to include several fermion representations of the gauge group is given by [EQUATION] with the gluon field strength tensor [EQUATION]', '1704.04209-1-7-1': 'The index [MATH] specifies the fermion representation and the index [MATH] the fermion flavour, [MATH] is the corresponding fermion field and [MATH] the corresponding fermion mass.', '1704.04209-1-7-2': 'The number of fermion flavours in representation [MATH] is [MATH] for any of the [MATH] fermion representations.', '1704.04209-1-8-0': 'The generators [MATH] of each fermion representation [MATH] fulfill the defining anticomuting relation of the Lie Algebra corresponding to the gauge group: [EQUATION] with the structure constants [MATH].', '1704.04209-1-8-1': 'We have one quadratic Casimir operator [MATH] for each fermion representation, defined through [EQUATION] and [MATH] for the adjoint representation.', '1704.04209-1-8-2': 'The dimensions of the fermion representations are given by [MATH] and the dimension of the adjoint representation by [MATH].', '1704.04209-1-8-3': 'The traces of the different representations are defined as [EQUATION]', '1704.04209-1-8-4': 'At four-loop level we also encounter higher order invariants in the gauge group factors which are expressed in terms of symmetric tensors [EQUATION] where [MATH] can be any fermion representation [MATH], noted as [MATH], or the adjoint representation, [MATH], where [MATH].', '1704.04209-1-9-0': 'An important special case of this model is the QCD plus gluinos sector of a supersymmetric theory where the gluinos are Majorana fermions in the adjoint representation of the gauge group.', '1704.04209-1-9-1': 'Here we have [MATH] and [EQUATION] the factor [MATH] in front of the number of gluinos [MATH] being a result of the Majorana nature of the gluinos (see e. g. [CITATION]).', '1704.04209-1-10-0': 'By adding counterterms to the Lagrangian [REF] in order to remove all possible UV divergences we arrive at the bare Lagrangian expressed through renormalized fields, masses and the coupling constant: [EQUATION] were we have already used the fact that [MATH].', '1704.04209-1-11-0': 'Due to the Slavnov-Taylor identities all vertex renormalization constants are connected and can be expressed through the renormalization constant of the coupling constant and the renormalization constants of the fields appearing in the respective vertex: [EQUATION]', '1704.04209-1-11-1': 'In the [MATH]-scheme using regularization in [MATH] space time dimensions all renormalization constants have the form [EQUATION] where [MATH].', '1704.04209-1-11-2': 'From the fact that the bare parameter [MATH] (with [MATH]) does not depend on the renormalization scale [MATH] one finds [EQUATION]', '1704.04209-1-11-3': 'Given a renormalization constant [MATH] the corresponding anomalous dimension is defined as [EQUATION]', '1704.04209-1-11-4': 'From the definition of anomalous dimensions [REF] it follows that [EQUATION] where we use the fact that the evolution of any parameter (or field) - here [MATH] - is described by its anomalous dimension, i. e. [EQUATION] and the fact that [MATH].', '1704.04209-1-11-5': 'Using [REF] one can reconstruct renormalization constants from the corresponding anomalous dimension, a finite and usually more compact quantity, and the [MATH]-function of the model.', '1704.04209-1-12-0': '## Technicalities', '1704.04209-1-13-0': 'The 1-particle-irreducible Feynman diagrams needed for this project were generated with QGRAF [CITATION].', '1704.04209-1-13-1': "We compute [MATH], [MATH] and [MATH] from the 1PI self-energies of the fields [MATH], [MATH] and [MATH] as well as [MATH] and [MATH] from the respective vertex corrections and [MATH] from the 1PI corrections to a Green's function with an insertion of one operator [MATH] and an external fermion line of type [MATH].", '1704.04209-1-13-2': 'We used two different methods to calculate these objects, first a direct four-loop calculation in Feynman gauge with massive tadpoles and then an indirect method where four-loop objects are constructed from propagator-like three-loop objects to derive the full dependence on the gauge parameter [MATH].', '1704.04209-1-14-0': '### Direct four-loop calculation in the Feynman gauge with massive tadpoles', '1704.04209-1-15-0': 'For [MATH] (Feynman gauge) the topologies of the diagrams were identified with the C++ programs Q2E and EXP [CITATION].', '1704.04209-1-15-1': 'In this approach all diagrams were expanded in the external momenta in order to factor out the momentum dependence of the tree-level vertex or propagator, e. g. [MATH] for the gluon self-energy.', '1704.04209-1-15-2': 'Then the tensor integrals were projected onto scalar integrals, using e. g. [MATH] as well as [MATH] as projectors for the gluon self-energy.', '1704.04209-1-15-3': 'After this we set all external momenta to zero since the UV divergent part of the integral does not depend on finite external momenta.', '1704.04209-1-15-4': 'We then use the method of introducing the same auxiliary mass parameter [MATH] in every propagator denominator [CITATION].', '1704.04209-1-15-5': 'Subdivergencies [MATH] are cancelled by an unphysical gluon mass counterterm [MATH] restoring the correct UV divergent part of the diagrams.', '1704.04209-1-15-6': 'This method was e. g. used in [CITATION] and is explained in detail in [CITATION].', '1704.04209-1-16-0': 'For the expansions, application of projectors, evaluation of fermion traces and counterterm insertions in lower loop diagrams we used FORM [CITATION].', '1704.04209-1-16-1': 'The scalar tadpole integrals were computed with the FORM-based package MATAD [CITATION] up to three-loop order.', '1704.04209-1-16-2': 'At four loops we use the C++ version of FIRE 5 [CITATION] in order to reduce the scalar integrals to Master Integrals which can be found in [CITATION].', '1704.04209-1-16-3': 'Technical details of the reduction are described in the previous paper [CITATION].', '1704.04209-1-17-0': '### Indirect four-loop calculation using three-loop massless propagators', '1704.04209-1-18-0': 'The case of a generic gauge parameter [MATH] is certainly possible to treat in the same massive way but calculations then require significantly more time and computer resources.', '1704.04209-1-18-1': 'As a result we have chosen an alternative massless approach which reduces the evaluation of any [MATH]-loop Z-factor to the calculation of some properly constructed set of [MATH]-loop massless propagators [CITATION].', '1704.04209-1-18-2': 'As is well-known (starting already from [MATH] [CITATION]) calculation of [MATH]-loop massive vacuum diagrams is significantly more complicated and time-consuming than the one of corresponding [MATH]-loop massless propagators.', '1704.04209-1-19-0': 'The approach is easily applicable for any Z-factor except for [MATH] [CITATION].', '1704.04209-1-19-1': 'The latter problem is certainly doable within the massless approach but requires significantly more human efforts in resolving rather sophisticated combinatorics.', '1704.04209-1-19-2': 'On the other hand, one could restore the full [MATH]-dependence of [MATH] from all other renormalization constants and from the fact that the charge renormalization constant [MATH] is gauge invariant [CITATION].', '1704.04209-1-19-3': 'As [MATH] in QCD with fermions transforming under arbitrary reducible representation of the gauge group has been recently found in [CITATION] we have proceeded in this way.', '1704.04209-1-19-4': 'For calculation of 3-loop massless propagator we have used the FORM version of MINCER [CITATION].', '1704.04209-1-20-0': '### computation of the gauge group factors', '1704.04209-1-21-0': 'The calculation of the gauge group factors was done with an extended version of the FORM package COLOR [CITATION] already used and presented in [CITATION].', '1704.04209-1-21-1': 'We take the following steps:', '1704.04209-1-22-0': 'For the generation of the diagrams in QGRAF [CITATION] we use one field [MATH] for the adjoint representation (gauge boson) and one field [MATH] for all the fermion representations.', '1704.04209-1-22-1': 'This has the advantage that we do not produce more Feynman diagrams than in QCD.', '1704.04209-1-22-2': 'Each fermion line in a diagram gets a line number and is treated as a different representation from the other fermion lines.', '1704.04209-1-22-3': 'Since we compute diagrams up to four-loop order we need up to four different line representations [MATH] (see Fig. [REF]) with the generators [MATH], [MATH], [MATH] and [MATH].', '1704.04209-1-22-4': 'Each fermion loop gets assigned a factor [MATH].', '1704.04209-1-22-5': 'The modified version of COLOR [CITATION] then writes the generators into traces [EQUATION] which are then reduced as outlined in [CITATION] yielding colour factors expressed through traces TFR, the Casimir operators cFR and cA, the dimensions of the representations dFR and NA.', '1704.04209-1-22-6': 'Now we change from fermion line numbers [MATH] to four explicit physical fermion representations [MATH] by substituting each of the line numbers [MATH] by the sum over all representations [MATH].', '1704.04209-1-22-7': 'An example of the substitution of [MATH]-colour factors with those of the physical representaions in a one-loop diagram is [EQUATION]', '1704.04209-1-22-8': 'At higher orders this subtitution becomes much more involved.', '1704.04209-1-22-9': 'Diagram (a) from Fig. [REF] now corresponds to a sum of [MATH] diagrams with explicit fermion representations.', '1704.04209-1-22-10': 'This lengthy representation of our results is needed for the renormalization procedure, since e. g. a one loop counterterm to the gluon self-energy, computed from a diagram with only [MATH], must be applied to all the fermion loops in Fig. [REF] (a,b,d,e).', '1704.04209-1-22-11': 'This is most conveniently achieved if each fermion-loop is considered a sum over all physical fermion representations just as it is considered a some over all (massless) fermion flavours.', '1704.04209-1-22-12': 'The factors involving [MATH], [MATH], [MATH] and [MATH] appear only at four-loop level and do hence not interfere with lower order diagrams with counterterm insertions.', '1704.04209-1-22-13': 'They can be treated directly in the next step.', '1704.04209-1-23-0': 'After all subdivergencies are cancelled by adding the lower-loop diagrams with counterterm insertions we simplify and generalize the notation.', '1704.04209-1-23-1': 'The explicit colour factors are collected in sums of terms built from [MATH], [MATH] and [MATH] over all physical representations [MATH], e. g. [EQUATION]', '1704.04209-1-23-2': 'Since we used the maximum number of different fermion representations which can appear in any diagram the result is valid for any number of fermion representations [MATH].', '1704.04209-1-24-0': '# Results', '1704.04209-1-25-0': 'In this section we give the results for the anomalous dimensions of the QCD-like model with an arbitrary number of fermion representations as described above to four-loop level.', '1704.04209-1-25-1': 'The number of active fermion flavours of representation [MATH] is denoted by [MATH].', '1704.04209-1-25-2': 'Apart from the Casimir operators [MATH] and [MATH] and the trace [MATH] the following invariants appear in our results: [EQUATION] where [MATH] is fixed and [MATH] will be summed over all fermion representations.', '1704.04209-1-25-3': 'In this section we give the results for [MATH] (Feynman gauge), the general case [MATH] can be found in the accompanying source files on www.arxiv.org.', '1704.04209-1-26-0': 'From the gauge boson field strength renormalization constant [MATH] we compute the anomalous dimension according to [REF] [EQUATION]', '1704.04209-1-26-1': 'From the ghost field strength renormalization constant [MATH] we compute [EQUATION]', '1704.04209-1-26-2': 'From the fermion field strength renormalization constant [MATH] we find [EQUATION] for the anomalous dimension of a representation [MATH] fermion field.', '1704.04209-1-27-0': 'The fermion field-gauge boson-vertex renormalization constant [MATH] yields [EQUATION] for each representation [MATH] and the ghost-gauge boson-vertex renormalization constant [MATH] yields [EQUATION]', '1704.04209-1-27-1': 'Finally, the mass anomalous dimension computed from [MATH] is found to be [EQUATION]', '1704.04209-1-27-2': 'We checked that the well known relations [EQUATION] are fulfilled with the [MATH]-function from [CITATION].', '1704.04209-1-27-3': 'This is also true if we include the full dependence on the gauge parameter [MATH] in the anomalous dimensions.', '1704.04209-1-27-4': 'This dependence cancels in the [MATH]-function.', '1704.04209-1-27-5': 'We provide renormalization constants and anomalous dimensions with the full gauge dependence in the attached files, which can be downloaded with the source files of this paper from www.arxiv.org.', '1704.04209-1-27-6': 'We compared these fully [MATH]-dependent results with [CITATION] for one fermion representation and find full agreement.', '1704.04209-1-28-0': '# Conclusions', '1704.04209-1-29-0': 'We have presented analytical results for the field anomalous dimensions [MATH], [MATH], [MATH], the vertex anomalous dimensions [MATH] and [MATH] and the mass anomalous dimension [MATH] in a QCD-like model with arbitrarily many fermion representations and with the full dependence on the gauge parameter [MATH].'}

{'1704.04209-2-0-0': '# Introduction', '1704.04209-2-1-0': "The behaviour of Green's functions wrt a shift of the renormalization scale is described by the anomalous dimensions of the fields and parameters of the theory, which enter the Renormalization Group Equations (RGE).", '1704.04209-2-1-1': 'For QCD the full set of four-loop renormalization constants and anomalous dimensions was presented in [CITATION].', '1704.04209-2-1-2': 'The results for the four-loop QCD [MATH]-function [CITATION] and the four-loop quark mass and field anomalous dimensions had already been available [CITATION].', '1704.04209-2-2-0': 'In this paper we consider a model with a non-abelian gauge group, one coupling constant and a reducible fermion representation, i. e. any number of irreducible fermion representations.', '1704.04209-2-2-1': 'The [MATH]-function for the coupling this model was computed in an earlier work [CITATION].', '1704.04209-2-2-2': 'Here we provide the remaining Renormalization Group (RG) functions in full dependence on the gauge parameter [MATH].', '1704.04209-2-3-0': 'Apart from completing the set of renormalization constants and the RGE of the theory, which is important in itself, the gauge boson and ghost propagator anomalous dimensions serve another purpose.', '1704.04209-2-3-1': 'These quantities are essential ingredients in comparing the momentum dependence of the corresponding propagators derived in non-perturbative calculations on the lattice, with perturbative results (see e. g. [CITATION]).', '1704.04209-2-4-0': 'This paper is structured as follows: First, we will give the notation and definitions for the model and the computed RG functions We will also repeat how the special case of QCD plus Majorana gluinos in the adjoint representation of the gauge group can be derived from our more general results.', '1704.04209-2-4-1': 'Then we will present analytical results for the four-loop anomalous dimensions of the gauge boson, ghost and fermion field as well as the ones for the ghost-gluon vertex, the fermion-gluon vertex and the fermion mass in Feynman gauge for compactness.', '1704.04209-2-4-2': 'The renormalization constants and anomalous dimensions for a generic gauge parameter [MATH] can be found in machine readable form in an accompanying file, which can be downloaded together with our source files on www.arxiv.org.', '1704.04209-2-5-0': '# Notation and definitions', '1704.04209-2-6-0': '## QCD with several fermion representations', '1704.04209-2-7-0': 'The Lagrangian of a QCD-like model extended to include several fermion representations of the gauge group is given by [EQUATION] with the gluon field strength tensor [EQUATION]', '1704.04209-2-7-1': 'The index [MATH] specifies the fermion representation and the index [MATH] the fermion flavour, [MATH] is the corresponding fermion field and [MATH] the corresponding fermion mass.', '1704.04209-2-7-2': 'The number of fermion flavours in representation [MATH] is [MATH] for any of the [MATH] fermion representations.', '1704.04209-2-8-0': 'The generators [MATH] of each fermion representation [MATH] fulfill the defining anticomuting relation of the Lie Algebra corresponding to the gauge group: [EQUATION] with the structure constants [MATH].', '1704.04209-2-8-1': 'We have one quadratic Casimir operator [MATH] for each fermion representation, defined through [EQUATION] and [MATH] for the adjoint representation.', '1704.04209-2-8-2': 'The dimensions of the fermion representations are given by [MATH] and the dimension of the adjoint representation by [MATH].', '1704.04209-2-8-3': 'The traces of the different representations are defined as [EQUATION]', '1704.04209-2-8-4': 'At four-loop level we also encounter higher order invariants in the gauge group factors which are expressed in terms of symmetric tensors [EQUATION] where [MATH] can be any fermion representation [MATH], noted as [MATH], or the adjoint representation, [MATH], where [MATH].', '1704.04209-2-9-0': 'An important special case of this model is the QCD plus gluinos sector of a supersymmetric theory where the gluinos are Majorana fermions in the adjoint representation of the gauge group.', '1704.04209-2-9-1': 'Here we have [MATH] and [EQUATION] the factor [MATH] in front of the number of gluinos [MATH] being a result of the Majorana nature of the gluinos (see e. g. [CITATION]).', '1704.04209-2-9-2': 'This can be understood in the following way: It has been shown in [CITATION] that one can treat Majorana fermions by first drawing all possible Feynman diagrams and choosing an arbitrary orientation (fermion flow) for each fermion line.', '1704.04209-2-9-3': 'Then Feynman rules are applied in the same way as for Dirac spinors, especially one can use the same propagators [MATH] for the momentum [MATH] along the fermion flow and [MATH] for [MATH] against the fermion flow.', '1704.04209-2-9-4': 'Closed fermion loops receive a factor [MATH].', '1704.04209-2-9-5': 'One then applies the same symmetry factors as for scalar or vector particles, e. g. a factor [MATH] for a loop consisting of two propagators of Majorana particles.', '1704.04209-2-9-6': 'For this work we generate our diagrams using one Dirac field [MATH] for all fermions, i. e. we produce both possible fermion flows in loops unless they lead to the same diagram.', '1704.04209-2-9-7': 'The latter case is exactly the one where the symmetry factor [MATH] must be applied.', '1704.04209-2-9-8': 'The first case means that the loop was double-counted which should also be compensated by a factor [MATH].', '1704.04209-2-10-0': 'By adding counterterms to the Lagrangian [REF] in order to remove all possible UV divergences we arrive at the bare Lagrangian expressed through renormalized fields, masses and the coupling constant: [EQUATION] were we have already used the fact that [MATH].', '1704.04209-2-11-0': 'Due to the Slavnov-Taylor identities all vertex renormalization constants are connected and can be expressed through the renormalization constant of the coupling constant and the renormalization constants of the fields appearing in the respective vertex: [EQUATION]', '1704.04209-2-11-1': 'In the [MATH]-scheme using regularization in [MATH] space time dimensions all renormalization constants have the form [EQUATION] where [MATH].', '1704.04209-2-11-2': 'From the fact that the bare parameter [MATH] (with [MATH]) does not depend on the renormalization scale [MATH] one finds [EQUATION]', '1704.04209-2-11-3': 'Given a renormalization constant [MATH] the corresponding anomalous dimension is defined as [EQUATION]', '1704.04209-2-11-4': 'From the definition of anomalous dimensions [REF] it follows that [EQUATION] where we use the fact that the evolution of any parameter (or field) - here [MATH] - is described by its anomalous dimension, i. e. [EQUATION] and the fact that [MATH].', '1704.04209-2-11-5': 'Using [REF] one can reconstruct renormalization constants from the corresponding anomalous dimension, a finite and usually more compact quantity, and the [MATH]-function of the model.', '1704.04209-2-12-0': '## Technicalities', '1704.04209-2-13-0': 'The 1-particle-irreducible Feynman diagrams needed for this project were generated with QGRAF [CITATION].', '1704.04209-2-13-1': "We compute [MATH], [MATH] and [MATH] from the 1PI self-energies of the fields [MATH], [MATH] and [MATH] as well as [MATH] and [MATH] from the respective vertex corrections and [MATH] from the 1PI corrections to a Green's function with an insertion of one operator [MATH] and an external fermion line of type [MATH].", '1704.04209-2-13-2': 'We used two different methods to calculate these objects, first a direct four-loop calculation in Feynman gauge with massive tadpoles and then an indirect method where four-loop objects are constructed from propagator-like three-loop objects to derive the full dependence on the gauge parameter [MATH].', '1704.04209-2-14-0': '### Direct four-loop calculation in the Feynman gauge with massive tadpoles', '1704.04209-2-15-0': 'For [MATH] (Feynman gauge) the topologies of the diagrams were identified with the C++ programs Q2E and EXP [CITATION].', '1704.04209-2-15-1': 'In this approach all diagrams were expanded in the external momenta in order to factor out the momentum dependence of the tree-level vertex or propagator, e. g. [MATH] for the gluon self-energy.', '1704.04209-2-15-2': 'Then the tensor integrals were projected onto scalar integrals, using e. g. [MATH] as well as [MATH] as projectors for the gluon self-energy.', '1704.04209-2-15-3': 'After this we set all external momenta to zero since the UV divergent part of the integral does not depend on finite external momenta.', '1704.04209-2-15-4': 'We then use the method of introducing the same auxiliary mass parameter [MATH] in every propagator denominator [CITATION].', '1704.04209-2-15-5': 'Subdivergencies [MATH] are cancelled by an unphysical gluon mass counterterm [MATH] restoring the correct UV divergent part of the diagrams.', '1704.04209-2-15-6': 'This method was e. g. used in [CITATION] and is explained in detail in [CITATION].', '1704.04209-2-16-0': 'For the expansions, application of projectors, evaluation of fermion traces and counterterm insertions in lower loop diagrams we used FORM [CITATION].', '1704.04209-2-16-1': 'The scalar tadpole integrals were computed with the FORM-based package MATAD [CITATION] up to three-loop order.', '1704.04209-2-16-2': 'At four loops we use the C++ version of FIRE 5 [CITATION] in order to reduce the scalar integrals to Master Integrals which can be found in [CITATION].', '1704.04209-2-16-3': 'Technical details of the reduction are described in the previous paper [CITATION].', '1704.04209-2-17-0': '### Indirect four-loop calculation using three-loop massless propagators', '1704.04209-2-18-0': 'The case of a generic gauge parameter [MATH] is certainly possible to treat in the same massive way but calculations then require significantly more time and computer resources.', '1704.04209-2-18-1': 'As a result we have chosen an alternative massless approach which reduces the evaluation of any [MATH]-loop Z-factor to the calculation of some properly constructed set of [MATH]-loop massless propagators [CITATION].', '1704.04209-2-18-2': 'As is well-known (starting already from [MATH] [CITATION]) calculation of [MATH]-loop massive vacuum diagrams is significantly more complicated and time-consuming than the one of corresponding [MATH]-loop massless propagators.', '1704.04209-2-19-0': 'The approach is easily applicable for any Z-factor except for [MATH] [CITATION].', '1704.04209-2-19-1': 'The latter problem is certainly doable within the massless approach but requires significantly more human efforts in resolving rather sophisticated combinatorics.', '1704.04209-2-19-2': 'On the other hand, one could restore the full [MATH]-dependence of [MATH] from all other renormalization constants and from the fact that the charge renormalization constant [MATH] is gauge invariant [CITATION].', '1704.04209-2-19-3': 'As [MATH] in QCD with fermions transforming under arbitrary reducible representation of the gauge group has been recently found in [CITATION] we have proceeded in this way.', '1704.04209-2-19-4': 'For calculation of 3-loop massless propagator we have used the FORM version of MINCER [CITATION].', '1704.04209-2-20-0': '### computation of the gauge group factors', '1704.04209-2-21-0': 'The calculation of the gauge group factors was done with an extended version of the FORM package COLOR [CITATION] already used and presented in [CITATION].', '1704.04209-2-21-1': 'We take the following steps:', '1704.04209-2-22-0': 'For the generation of the diagrams in QGRAF [CITATION] we use one field [MATH] for the adjoint representation (gauge boson) and one field [MATH] for all the fermion representations.', '1704.04209-2-22-1': 'This has the advantage that we do not produce more Feynman diagrams than in QCD.', '1704.04209-2-22-2': 'Each fermion line in a diagram gets a line number and is treated as a different representation from the other fermion lines.', '1704.04209-2-22-3': 'Since we compute diagrams up to four-loop order we need up to four different line representations [MATH] (see Fig. [REF]) with the generators [MATH], [MATH], [MATH] and [MATH].', '1704.04209-2-22-4': 'Each fermion loop gets assigned a factor [MATH].', '1704.04209-2-22-5': 'The modified version of COLOR [CITATION] then writes the generators into traces [EQUATION] which are then reduced as outlined in [CITATION] yielding colour factors expressed through traces TFR, the Casimir operators cFR and cA, the dimensions of the representations dFR and NA.', '1704.04209-2-22-6': 'Now we change from fermion line numbers [MATH] to four explicit physical fermion representations [MATH] by substituting each of the line numbers [MATH] by the sum over all representations [MATH].', '1704.04209-2-22-7': 'An example of the substitution of [MATH]-colour factors with those of the physical representaions in a one-loop diagram is [EQUATION]', '1704.04209-2-22-8': 'At higher orders this subtitution becomes much more involved.', '1704.04209-2-22-9': 'Diagram (a) from Fig. [REF] now corresponds to a sum of [MATH] diagrams with explicit fermion representations.', '1704.04209-2-22-10': 'This lengthy representation of our results is needed for the renormalization procedure, since e. g. a one loop counterterm to the gluon self-energy, computed from a diagram with only [MATH], must be applied to all the fermion loops in Fig. [REF] (a,b,d,e).', '1704.04209-2-22-11': 'This is most conveniently achieved if each fermion-loop is considered a sum over all physical fermion representations just as it is considered a some over all (massless) fermion flavours.', '1704.04209-2-22-12': 'The factors involving [MATH], [MATH], [MATH] and [MATH] appear only at four-loop level and do hence not interfere with lower order diagrams with counterterm insertions.', '1704.04209-2-22-13': 'They can be treated directly in the next step.', '1704.04209-2-23-0': 'After all subdivergencies are cancelled by adding the lower-loop diagrams with counterterm insertions we simplify and generalize the notation.', '1704.04209-2-23-1': 'The explicit colour factors are collected in sums of terms built from [MATH], [MATH] and [MATH] over all physical representations [MATH], e. g. [EQUATION]', '1704.04209-2-23-2': 'Since we used the maximum number of different fermion representations which can appear in any diagram the result is valid for any number of fermion representations [MATH].', '1704.04209-2-24-0': '# Results', '1704.04209-2-25-0': 'In this section we give the results for the anomalous dimensions of the QCD-like model with an arbitrary number of fermion representations as described above to four-loop level.', '1704.04209-2-25-1': 'The number of active fermion flavours of representation [MATH] is denoted by [MATH].', '1704.04209-2-25-2': 'Apart from the Casimir operators [MATH] and [MATH] and the trace [MATH] the following invariants appear in our results: [EQUATION] where [MATH] is fixed and [MATH] will be summed over all fermion representations.', '1704.04209-2-25-3': 'In this section we give the results for [MATH] (Feynman gauge), the general case [MATH] can be found in the accompanying source files on www.arxiv.org.', '1704.04209-2-26-0': 'From the gauge boson field strength renormalization constant [MATH] we compute the anomalous dimension according to [REF] [EQUATION]', '1704.04209-2-26-1': 'From the ghost field strength renormalization constant [MATH] we compute [EQUATION]', '1704.04209-2-26-2': 'From the fermion field strength renormalization constant [MATH] we find [EQUATION] for the anomalous dimension of a representation [MATH] fermion field.', '1704.04209-2-27-0': 'The fermion field-gauge boson-vertex renormalization constant [MATH] yields [EQUATION] for each representation [MATH] and the ghost-gauge boson-vertex renormalization constant [MATH] yields [EQUATION]', '1704.04209-2-27-1': 'Finally, the mass anomalous dimension computed from [MATH] is found to be [EQUATION]', '1704.04209-2-27-2': 'We checked that the well known relations [EQUATION] are fulfilled with the [MATH]-function from [CITATION].', '1704.04209-2-27-3': 'This is also true if we include the full dependence on the gauge parameter [MATH] in the anomalous dimensions.', '1704.04209-2-27-4': 'This dependence cancels in the [MATH]-function.', '1704.04209-2-27-5': 'We provide renormalization constants and anomalous dimensions with the full gauge dependence in the attached files, which can be downloaded with the source files of this paper from www.arxiv.org.', '1704.04209-2-27-6': 'We compared these fully [MATH]-dependent results with [CITATION] for one fermion representation and find full agreement.', '1704.04209-2-28-0': '# Conclusions', '1704.04209-2-29-0': 'We have presented analytical results for the field anomalous dimensions [MATH], [MATH], [MATH], the vertex anomalous dimensions [MATH] and [MATH] and the mass anomalous dimension [MATH] in a QCD-like model with arbitrarily many fermion representations and with the full dependence on the gauge parameter [MATH].'}

[['1704.04209-1-9-0', '1704.04209-2-9-0'], ['1704.04209-1-9-1', '1704.04209-2-9-1'], ['1704.04209-1-3-0', '1704.04209-2-3-0'], ['1704.04209-1-3-1', '1704.04209-2-3-1'], ['1704.04209-1-22-0', '1704.04209-2-22-0'], ['1704.04209-1-22-1', '1704.04209-2-22-1'], ['1704.04209-1-22-2', '1704.04209-2-22-2'], ['1704.04209-1-22-3', '1704.04209-2-22-3'], ['1704.04209-1-22-4', '1704.04209-2-22-4'], ['1704.04209-1-22-5', '1704.04209-2-22-5'], ['1704.04209-1-22-6', '1704.04209-2-22-6'], ['1704.04209-1-22-7', '1704.04209-2-22-7'], ['1704.04209-1-22-8', '1704.04209-2-22-8'], ['1704.04209-1-22-9', '1704.04209-2-22-9'], ['1704.04209-1-22-10', '1704.04209-2-22-10'], ['1704.04209-1-22-11', '1704.04209-2-22-11'], ['1704.04209-1-22-12', '1704.04209-2-22-12'], ['1704.04209-1-22-13', '1704.04209-2-22-13'], ['1704.04209-1-25-0', '1704.04209-2-25-0'], ['1704.04209-1-25-1', '1704.04209-2-25-1'], ['1704.04209-1-25-2', '1704.04209-2-25-2'], ['1704.04209-1-25-3', '1704.04209-2-25-3'], ['1704.04209-1-29-0', '1704.04209-2-29-0'], ['1704.04209-1-15-0', '1704.04209-2-15-0'], ['1704.04209-1-15-1', '1704.04209-2-15-1'], ['1704.04209-1-15-2', '1704.04209-2-15-2'], ['1704.04209-1-15-3', '1704.04209-2-15-3'], ['1704.04209-1-15-4', '1704.04209-2-15-4'], ['1704.04209-1-15-5', '1704.04209-2-15-5'], ['1704.04209-1-15-6', '1704.04209-2-15-6'], ['1704.04209-1-27-0', '1704.04209-2-27-0'], ['1704.04209-1-27-1', '1704.04209-2-27-1'], ['1704.04209-1-27-2', '1704.04209-2-27-2'], ['1704.04209-1-27-3', '1704.04209-2-27-3'], ['1704.04209-1-27-4', '1704.04209-2-27-4'], ['1704.04209-1-27-5', '1704.04209-2-27-5'], ['1704.04209-1-27-6', '1704.04209-2-27-6'], ['1704.04209-1-8-0', '1704.04209-2-8-0'], ['1704.04209-1-8-1', '1704.04209-2-8-1'], ['1704.04209-1-8-2', '1704.04209-2-8-2'], ['1704.04209-1-8-3', '1704.04209-2-8-3'], ['1704.04209-1-8-4', '1704.04209-2-8-4'], ['1704.04209-1-2-0', '1704.04209-2-2-0'], ['1704.04209-1-2-1', '1704.04209-2-2-1'], ['1704.04209-1-2-2', '1704.04209-2-2-2'], ['1704.04209-1-7-0', '1704.04209-2-7-0'], ['1704.04209-1-7-1', '1704.04209-2-7-1'], ['1704.04209-1-7-2', '1704.04209-2-7-2'], ['1704.04209-1-1-0', '1704.04209-2-1-0'], ['1704.04209-1-1-1', '1704.04209-2-1-1'], ['1704.04209-1-1-2', '1704.04209-2-1-2'], ['1704.04209-1-16-0', '1704.04209-2-16-0'], ['1704.04209-1-16-1', '1704.04209-2-16-1'], ['1704.04209-1-16-2', '1704.04209-2-16-2'], ['1704.04209-1-16-3', '1704.04209-2-16-3'], ['1704.04209-1-10-0', '1704.04209-2-10-0'], ['1704.04209-1-11-0', '1704.04209-2-11-0'], ['1704.04209-1-11-1', '1704.04209-2-11-1'], ['1704.04209-1-11-2', '1704.04209-2-11-2'], ['1704.04209-1-11-3', '1704.04209-2-11-3'], ['1704.04209-1-11-4', '1704.04209-2-11-4'], ['1704.04209-1-11-5', '1704.04209-2-11-5'], ['1704.04209-1-18-0', '1704.04209-2-18-0'], ['1704.04209-1-18-1', '1704.04209-2-18-1'], ['1704.04209-1-18-2', '1704.04209-2-18-2'], ['1704.04209-1-26-0', '1704.04209-2-26-0'], ['1704.04209-1-26-1', '1704.04209-2-26-1'], ['1704.04209-1-26-2', '1704.04209-2-26-2'], ['1704.04209-1-4-0', '1704.04209-2-4-0'], ['1704.04209-1-4-1', '1704.04209-2-4-1'], ['1704.04209-1-4-2', '1704.04209-2-4-2'], ['1704.04209-1-19-0', '1704.04209-2-19-0'], ['1704.04209-1-19-1', '1704.04209-2-19-1'], ['1704.04209-1-19-2', '1704.04209-2-19-2'], ['1704.04209-1-19-3', '1704.04209-2-19-3'], ['1704.04209-1-19-4', '1704.04209-2-19-4'], ['1704.04209-1-13-0', '1704.04209-2-13-0'], ['1704.04209-1-13-1', '1704.04209-2-13-1'], ['1704.04209-1-13-2', '1704.04209-2-13-2'], ['1704.04209-1-21-0', '1704.04209-2-21-0'], ['1704.04209-1-23-0', '1704.04209-2-23-0'], ['1704.04209-1-23-1', '1704.04209-2-23-1'], ['1704.04209-1-23-2', '1704.04209-2-23-2']]

[['1704.04209-1-9-0', '1704.04209-2-9-0'], ['1704.04209-1-9-1', '1704.04209-2-9-1'], ['1704.04209-1-3-0', '1704.04209-2-3-0'], ['1704.04209-1-3-1', '1704.04209-2-3-1'], ['1704.04209-1-22-0', '1704.04209-2-22-0'], ['1704.04209-1-22-1', '1704.04209-2-22-1'], ['1704.04209-1-22-2', '1704.04209-2-22-2'], ['1704.04209-1-22-3', '1704.04209-2-22-3'], ['1704.04209-1-22-4', '1704.04209-2-22-4'], ['1704.04209-1-22-5', '1704.04209-2-22-5'], ['1704.04209-1-22-6', '1704.04209-2-22-6'], ['1704.04209-1-22-7', '1704.04209-2-22-7'], ['1704.04209-1-22-8', '1704.04209-2-22-8'], ['1704.04209-1-22-9', '1704.04209-2-22-9'], ['1704.04209-1-22-10', '1704.04209-2-22-10'], ['1704.04209-1-22-11', '1704.04209-2-22-11'], ['1704.04209-1-22-12', '1704.04209-2-22-12'], ['1704.04209-1-22-13', '1704.04209-2-22-13'], ['1704.04209-1-25-0', '1704.04209-2-25-0'], ['1704.04209-1-25-1', '1704.04209-2-25-1'], ['1704.04209-1-25-2', '1704.04209-2-25-2'], ['1704.04209-1-25-3', '1704.04209-2-25-3'], ['1704.04209-1-29-0', '1704.04209-2-29-0'], ['1704.04209-1-15-0', '1704.04209-2-15-0'], ['1704.04209-1-15-1', '1704.04209-2-15-1'], ['1704.04209-1-15-2', '1704.04209-2-15-2'], ['1704.04209-1-15-3', '1704.04209-2-15-3'], ['1704.04209-1-15-4', '1704.04209-2-15-4'], ['1704.04209-1-15-5', '1704.04209-2-15-5'], ['1704.04209-1-15-6', '1704.04209-2-15-6'], ['1704.04209-1-27-0', '1704.04209-2-27-0'], ['1704.04209-1-27-1', '1704.04209-2-27-1'], ['1704.04209-1-27-2', '1704.04209-2-27-2'], ['1704.04209-1-27-3', '1704.04209-2-27-3'], ['1704.04209-1-27-4', '1704.04209-2-27-4'], ['1704.04209-1-27-5', '1704.04209-2-27-5'], ['1704.04209-1-27-6', '1704.04209-2-27-6'], ['1704.04209-1-8-0', '1704.04209-2-8-0'], ['1704.04209-1-8-1', '1704.04209-2-8-1'], ['1704.04209-1-8-2', '1704.04209-2-8-2'], ['1704.04209-1-8-3', '1704.04209-2-8-3'], ['1704.04209-1-8-4', '1704.04209-2-8-4'], ['1704.04209-1-2-0', '1704.04209-2-2-0'], ['1704.04209-1-2-1', '1704.04209-2-2-1'], ['1704.04209-1-2-2', '1704.04209-2-2-2'], ['1704.04209-1-7-0', '1704.04209-2-7-0'], ['1704.04209-1-7-1', '1704.04209-2-7-1'], ['1704.04209-1-7-2', '1704.04209-2-7-2'], ['1704.04209-1-1-0', '1704.04209-2-1-0'], ['1704.04209-1-1-1', '1704.04209-2-1-1'], ['1704.04209-1-1-2', '1704.04209-2-1-2'], ['1704.04209-1-16-0', '1704.04209-2-16-0'], ['1704.04209-1-16-1', '1704.04209-2-16-1'], ['1704.04209-1-16-2', '1704.04209-2-16-2'], ['1704.04209-1-16-3', '1704.04209-2-16-3'], ['1704.04209-1-10-0', '1704.04209-2-10-0'], ['1704.04209-1-11-0', '1704.04209-2-11-0'], ['1704.04209-1-11-1', '1704.04209-2-11-1'], ['1704.04209-1-11-2', '1704.04209-2-11-2'], ['1704.04209-1-11-3', '1704.04209-2-11-3'], ['1704.04209-1-11-4', '1704.04209-2-11-4'], ['1704.04209-1-11-5', '1704.04209-2-11-5'], ['1704.04209-1-18-0', '1704.04209-2-18-0'], ['1704.04209-1-18-1', '1704.04209-2-18-1'], ['1704.04209-1-18-2', '1704.04209-2-18-2'], ['1704.04209-1-26-0', '1704.04209-2-26-0'], ['1704.04209-1-26-1', '1704.04209-2-26-1'], ['1704.04209-1-26-2', '1704.04209-2-26-2'], ['1704.04209-1-4-0', '1704.04209-2-4-0'], ['1704.04209-1-4-1', '1704.04209-2-4-1'], ['1704.04209-1-4-2', '1704.04209-2-4-2'], ['1704.04209-1-19-0', '1704.04209-2-19-0'], ['1704.04209-1-19-1', '1704.04209-2-19-1'], ['1704.04209-1-19-2', '1704.04209-2-19-2'], ['1704.04209-1-19-3', '1704.04209-2-19-3'], ['1704.04209-1-19-4', '1704.04209-2-19-4'], ['1704.04209-1-13-0', '1704.04209-2-13-0'], ['1704.04209-1-13-1', '1704.04209-2-13-1'], ['1704.04209-1-13-2', '1704.04209-2-13-2'], ['1704.04209-1-21-0', '1704.04209-2-21-0'], ['1704.04209-1-23-0', '1704.04209-2-23-0'], ['1704.04209-1-23-1', '1704.04209-2-23-1'], ['1704.04209-1-23-2', '1704.04209-2-23-2']]

[]

[]

[]

[]

['1704.04209-1-21-1', '1704.04209-2-21-1']

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

https://arxiv.org/abs/1704.04209

0905.1142

{'0905.1142-1-0-0': 'We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement.', '0905.1142-1-0-1': 'The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the natural flux boundary condition.', '0905.1142-1-0-2': 'Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl.', '0905.1142-1-0-3': 'Math., 68(5):1304-1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter [MATH].', '0905.1142-1-0-4': 'In this article, we show that for the well-posedness of the microscopic FENE model ([MATH]) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function.', '0905.1142-1-0-5': 'This condition is strictly weaker than the natural flux boundary condition.', '0905.1142-1-0-6': 'Under this condition it is shown that there exists a unique weak solution in a weighted Sobolev space.', '0905.1142-1-0-7': 'The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.', '0905.1142-1-1-0': '# Introduction', '0905.1142-1-2-0': 'It is well-known that the following system coupling incompressible Navier-Stokes equation for the macroscopic velocity field [MATH] and the Fokker-Planck equation for the probability density function [MATH] describes diluted solutions of polymeric liquids with noninteracting polymer chains, where [MATH] is the macroscopic Eulerian coordinate and [MATH] is the microscopic molecular configuration variable [EQUATION]', '0905.1142-1-2-1': 'In this model, a polymer is idealized as an elastic dumbbell consisting of two beads joined by a spring that can be modeled by a vector [MATH] (see e.g [CITATION]).', '0905.1142-1-2-2': 'In the Navier-Stokes equation [REF], [MATH] is hydrostatic pressure, [MATH] is the fluid viscosity, and [MATH] is a tensor representing the polymer contribution to stress, [EQUATION] where [MATH] is the elastic spring potential, and [MATH] is the polymer density constant.', '0905.1142-1-2-3': 'In the Fokker-Planck equation [REF], [MATH] is the friction coefficient of the dumbbell beads, [MATH] is the temperature, and [MATH] is the Boltzmann constant.', '0905.1142-1-2-4': 'Notice that, the Fokker-Planck equation can be written as a stochastic differential equation (see [CITATION]).', '0905.1142-1-3-0': 'One of the simplest model is the Hookean model in which the potential [MATH] is given by [EQUATION] where [MATH] is the elasticity constant.', '0905.1142-1-3-1': 'For a finite [MATH], a more realistic model is the finite extensible nonlinear elasticity (FENE) model in which case [EQUATION] where [MATH] is the maximum dumbbell extension.', '0905.1142-1-3-2': 'In this work we shall focus our attention on the case [MATH], which is known to contain the parameter range of physical interest.', '0905.1142-1-3-3': 'We refer the readers to [CITATION] for a comprehensive survey of the physical background.', '0905.1142-1-4-0': 'In past years the well-posedness of the FENE model has been studied intensively in several aspects.', '0905.1142-1-4-1': 'For local well-posedness of strong solutions we refer the readers to [CITATION] for the FENE model (in the setting where the Fokker-Planck equation is formulated by a stochastic differential equation) with [MATH] or sometime [MATH], [CITATION] for a polynomial force and [CITATION] for the FENE model with [MATH].', '0905.1142-1-4-2': 'For a preliminary study on some related coupled PDE systems, we refer to the earlier work [CITATION](however, the FENE model was not addressed there).', '0905.1142-1-4-3': 'Moreover, the authors [CITATION] proved global existence of smooth solutions near equilibrium under some restrictions on the potential, which have been extended in subsequent works [CITATION].', '0905.1142-1-4-4': 'More recently, N. Masmoudi [CITATION] proved local and global well-posedness for the FENE dumbbell model for a general class of potentials.', '0905.1142-1-5-0': 'Global existence of weak solutions was also proved in [CITATION] for the co-rotational model, see also [CITATION] for [MATH].', '0905.1142-1-5-1': 'For an earlier existence result of weak solutions, we refer to [CITATION] for the Fokker-Planck equation alone with [MATH].', '0905.1142-1-5-2': 'On the other hand, the authors in [CITATION], investigated the long-time behavior of both Hookean models and FENE models in various special flows in a bounded domain with suitable boundary conditions.', '0905.1142-1-6-0': 'The main complexity with the FENE potential lies mainly with the singularity of the equation at the boundary.', '0905.1142-1-6-1': 'In [CITATION], C. Liu and H. Liu closely examined the necessity of Dirichlet boundary conditions for the microscopic FENE model.', '0905.1142-1-6-2': 'By the method of the Fichera function the authors were able to conclude that [MATH] is a threshold in the sense that for [MATH] any preassigned value of a weighted distribution will become redundant, and for [MATH] that value has to be a priori given.', '0905.1142-1-6-3': 'For the microscopic FENE model, singularity in the potential requires at least the zero Dirichlet boundary condition [EQUATION] where [MATH], the ball with center [MATH] and radius [MATH].', '0905.1142-1-6-4': 'In most of a priori works the natural flux boundary condition has been used: [EQUATION] which is stronger than ([REF]).', '0905.1142-1-7-0': 'The boundary issue for the underlying FENE model is fundamental, and our main quest in this paper is whether one can identify a sharp boundary requirement so that both existence and uniqueness of a global weak solution to the microscopic FENE model can be proved.', '0905.1142-1-7-1': 'The answer is positive, and we claim that [MATH] must have the following boundary condition [EQUATION] where [MATH] denotes the distance function from [MATH] to the boundary [MATH].', '0905.1142-1-7-2': 'Note that [REF] is strictly weaker than [REF] and stronger than ([REF]).', '0905.1142-1-7-3': 'Our claim is supported by our main results: the global well-posedness for the Fokker-Planck equation stated in Theorem [REF], and the sharpness of ([REF]) stated in Proposition [REF].', '0905.1142-1-8-0': 'The importance of the Fokker-Planck equation itself as the added complexity with the FENE potential affects mostly the analysis of the Fokker-Planck equation.', '0905.1142-1-8-1': 'In this article, we focus on the underlying Fokker-Planck equation alone.', '0905.1142-1-8-2': 'Assuming [MATH] is independent of [MATH] and the fluid velocity is steady and homogeneous, we obtain the following equation from a suitable scaling ([CITATION]).', '0905.1142-1-8-3': '[EQUATION] where [MATH] and [MATH] is a constant matrix such that [MATH] and with the matrix max norm [MATH].', '0905.1142-1-8-4': 'Suppose that a sufficiently smooth function [MATH] solves [REF].', '0905.1142-1-8-5': 'Then for any test function [MATH], i.e. a continuously differentiable function compactly supported in [MATH], it follows that [EQUATION]', '0905.1142-1-8-6': 'Here, we omit [MATH] from [MATH].', '0905.1142-1-8-7': 'Note that [REF] is well defined for any [MATH] and [MATH].', '0905.1142-1-8-8': 'Moreover, the compact support of a test function makes [MATH] free on the boundary of the domain.', '0905.1142-1-8-9': 'From this, a weak solution of the Fokker-Planck equation [REF] with the initial condition [EQUATION] is defined in the following manner:', '0905.1142-1-9-0': 'Suppose that [EQUATION] for an arbitrary subdomain [MATH] of [MATH] such that [MATH].', '0905.1142-1-9-1': 'We say [MATH] is a solution of [REF], [REF] if [REF] holds for any [MATH] and almost all [MATH] such that [EQUATION]', '0905.1142-1-9-2': 'Note that [REF] makes sense since [MATH].', '0905.1142-1-10-0': 'Consider the equilibrium [MATH] of [REF] for non-flow case, i.e. [MATH].', '0905.1142-1-10-1': 'Rewrite [REF] as [EQUATION]', '0905.1142-1-10-2': 'It follows that [EQUATION]', '0905.1142-1-10-3': 'Obviously, [MATH] has the zero trace on the boundary [MATH] and satisfies ([REF]) for [EQUATION]', '0905.1142-1-10-4': 'Thus, [REF] is satisfied even for a test function [MATH] without the compactly supported property.', '0905.1142-1-10-5': 'Also [EQUATION] as long as [REF] is assumed.', '0905.1142-1-10-6': 'Here, [MATH] with [EQUATION]', '0905.1142-1-10-7': 'Our main results are summarized in Theorem [REF] and Proposition [REF] below.', '0905.1142-1-11-0': 'Assume [REF] and [REF].', '0905.1142-1-11-1': 'For any [MATH], there exists at most one solution [MATH] to the Fokker-Planck equation [REF] and [REF] in the sense of Definition [REF] for any initial data [MATH].', '0905.1142-1-11-2': 'Furthermore, if [EQUATION] then there exists a unique solution and [EQUATION]', '0905.1142-1-11-3': 'From the equilibrium [MATH], the restriction on [MATH] [REF] is essential to have the energy estimate [REF].', '0905.1142-1-11-4': 'We, however, remark that the well-posedness of the weak solution for [MATH] was discussed in [CITATION] with a different function space.', '0905.1142-1-12-0': 'The following proposition states that the boundary condition [REF] is sharp for the uniqueness of the weak solution.', '0905.1142-1-13-0': 'The article is organized as follows.', '0905.1142-1-13-1': 'In Section 2, we transform the Fokker-Planck equation to a certain Cauchy-Dirichlet problem, named as [MATH]-problem and define a weak solution of [MATH]-problem in a weighted Sobolev space.', '0905.1142-1-13-2': 'The well-posedness of the [MATH]-problem is shown in Section 3 by the Galerkin method and the Banach fixed point theorem.', '0905.1142-1-13-3': 'This leads to the well-posedness of the Fokker-Planck equation, Theorem [REF]; details of the proof are presented in Section 4.', '0905.1142-1-13-4': 'In the last section, we construct a non-trivial solution for the Fokker-Planck equation described in Proposition [REF].', '0905.1142-1-14-0': '# Transformation of the microscopic FENE model', '0905.1142-1-15-0': 'In what follows we shall call the Fokker-Planck equation ([REF]) with initial condition ([REF]) and boundary condition ([REF]) as the Fokker-Planck-FENE (FPF) problem.', '0905.1142-1-15-1': 'We first formulate a time evolution equation from the FPF problem.', '0905.1142-1-15-2': 'Define [MATH] ([CITATION]) as [EQUATION] with [MATH] to be determined.', '0905.1142-1-15-3': 'Then [REF] is transformed to [EQUATION] where, [EQUATION]', '0905.1142-1-15-4': 'Setting a parameter [EQUATION] we rewrite [REF] as [EQUATION]', '0905.1142-1-15-5': 'If [EQUATION] is taken, the boundary condition [REF] implies that [MATH] satisfies a homogeneous boundary condition for almost all [MATH] since the distance function [MATH] and [MATH] are equivalent (see [REF]).', '0905.1142-1-16-0': 'The FPF problem is formally transformed to the following [MATH]-problem: [EQUATION]', '0905.1142-1-16-1': 'Here, [EQUATION] according to the transformation [REF].', '0905.1142-1-17-0': 'In order to define a weak solution of [MATH]-problem we introduce a weighted Sobolev space [MATH] for a nonnegative measurable function [MATH] as a set of measurable function [MATH] such that [EQUATION]', '0905.1142-1-17-1': 'Similarly, a weighted [MATH] can be defined.', '0905.1142-1-17-2': '[MATH] denotes a completion of [MATH] with [MATH].', '0905.1142-1-17-3': 'It is obvious that [MATH] and [MATH] are Hilbert spaces with the inner product [MATH] defined as [EQUATION] and [EQUATION]', '0905.1142-1-17-4': 'For convenient notation, we use [MATH] and [MATH] for [MATH] and [MATH] respectively.', '0905.1142-1-17-5': 'We also omit a domain [MATH] if it is obvious.', '0905.1142-1-18-0': 'Suppose that [MATH] and [MATH].', '0905.1142-1-19-0': '[(1)] If [MATH], then [EQUATION] [(2)] If [MATH], then the trace map [EQUATION] is well defined, i.e. it is a bounded linear map.', '0905.1142-1-20-0': 'In particular, for [MATH] [EQUATION]', '0905.1142-1-21-0': 'In [CITATION](see also [CITATION]), it was proved that [EQUATION] provided [MATH] is Lipschitz continuous.', '0905.1142-1-21-1': 'Recall that [MATH] denotes the distance from [MATH] to the boundary of [MATH].', '0905.1142-1-21-2': '[REF] follows from [EQUATION]', '0905.1142-1-21-3': 'It is also known that the trace map [MATH] is well defined for [MATH] ([CITATION]).', '0905.1142-1-21-4': 'For [MATH] [EQUATION] since [MATH] for all [MATH].', '0905.1142-1-21-5': 'Therefore, [MATH] is well defined for [MATH].', '0905.1142-1-21-6': '[REF] is obvious from the definitions of the trace map and [MATH].', '0905.1142-1-22-0': 'Note that [REF] remains true for [MATH].', '0905.1142-1-23-0': 'We now define a weak solution to [MATH]-problem in a standard manner.', '0905.1142-1-23-1': 'Multiplication by a test function [MATH] to the equation [REF] and integration over [MATH] yield [EQUATION]', '0905.1142-1-23-2': 'This equation is well defined assuming that [MATH], the dual space of [MATH], and [MATH] due to the boundedness of [MATH] and Lemma [REF].', '0905.1142-1-23-3': 'Moreover, [EQUATION] implies [EQUATION]', '0905.1142-1-23-4': 'Thus [EQUATION]', '0905.1142-1-23-5': 'Here we identify [MATH] with its dual space.', '0905.1142-1-24-0': 'A function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem, [REF]-[REF], provided', '0905.1142-1-25-0': '[(1)] For each [MATH] and almost every [MATH], [EQUATION] [(2)] [MATH] in [MATH] sense.', '0905.1142-1-25-1': 'i.e. [EQUATION]', '0905.1142-1-25-2': 'Here, we let [MATH] denote the paring of a Hilbert space [MATH] with its dual space [MATH] and [EQUATION]', '0905.1142-1-25-3': 'The following energy estimate for [MATH] for fixed [MATH] can be achieved from a simple modification of energy estimates for the bilinear form in elliptic equations, see [CITATION] for details.', '0905.1142-1-26-0': 'There exist positive constants [MATH] and [MATH] depending only on [MATH] and [MATH] such that [EQUATION]', '0905.1142-1-27-0': '# Well-posedness for the transformed problem', '0905.1142-1-28-0': 'In this section, we show the well-posedness of the weak solution to [MATH]-problem.', '0905.1142-1-28-1': 'For this aim, we consider the following [MATH]-problem containing a non-homogeneous term [MATH].', '0905.1142-1-28-2': '[EQUATION]', '0905.1142-1-28-3': 'The weak solution of [MATH]-problem is defined similarly.', '0905.1142-1-29-0': 'We say a function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem provided', '0905.1142-1-30-0': '[(1)] for each [MATH] and almost every [MATH] [EQUATION] [(2)] [MATH] in [MATH].', '0905.1142-1-31-0': 'Recall that [EQUATION]', '0905.1142-1-31-1': 'We remark that [MATH] is finite for any [MATH] since [MATH] from [REF].', '0905.1142-1-31-2': 'Thus [MATH] can be understood as the [MATH] inner product although [MATH] may not belong to [MATH].', '0905.1142-1-32-0': 'The well-posedness for [MATH]-problem follows from the standard Galerkin method.', '0905.1142-1-33-0': 'For given [MATH] and [MATH], [MATH]-problem has a unique weak solution.', '0905.1142-1-33-1': 'Moreover, [EQUATION]', '0905.1142-1-33-2': 'We first construct an approximate solution in a finite-dimensional space.', '0905.1142-1-33-3': 'Let [MATH] be a basis of [MATH] and [MATH].', '0905.1142-1-33-4': 'The existence of such a basis can be verified from the fact that [MATH] is a dense subset of [MATH].', '0905.1142-1-33-5': 'Consider an approximation [MATH], where [MATH] satisfies [EQUATION]', '0905.1142-1-33-6': 'Since [REF] and [REF] form a system of linear differential equations, [MATH] is uniquely determined for each [MATH].', '0905.1142-1-33-7': 'We rewrite [REF] as [EQUATION]', '0905.1142-1-33-8': 'Apply [MATH] to [REF] and sum for [MATH], then for almost every [MATH] [EQUATION]', '0905.1142-1-33-9': 'From Lemma [REF], it follows that [EQUATION]', '0905.1142-1-33-10': 'From [REF], for any [MATH] [EQUATION]', '0905.1142-1-33-11': 'With [MATH], [REF] can be rewritten as [EQUATION] or [EQUATION]', '0905.1142-1-33-12': "Use Gronwall's inequality to obtain [EQUATION] where [MATH] is an appropriate constant which depends on [MATH], [MATH], [MATH] and [MATH].", '0905.1142-1-33-13': 'On the other hand, integration of [REF] from [MATH] to [MATH] together with above inequality yields [EQUATION]', '0905.1142-1-33-14': 'A similar argument to that in [CITATION] gives us the estimate for [MATH] as [EQUATION]', '0905.1142-1-33-15': 'Here we have used [REF] with [MATH] such that [MATH] and [REF].', '0905.1142-1-33-16': 'By passing to the limit as [MATH] and a standard argument (e.g. see [CITATION]), we have well-posedness for [MATH]-problem.', '0905.1142-1-34-0': 'Now, we introduce a linear map [MATH] to connect [MATH] and [MATH]-problems as [EQUATION]', '0905.1142-1-34-1': 'To have a well-defined [MATH], we choose [EQUATION] which is crucial in this argument.', '0905.1142-1-34-2': 'With this [MATH], we rewrite [MATH] defined in [REF] as follows: [EQUATION]', '0905.1142-1-34-3': 'Since [MATH] is bounded, [EQUATION]', '0905.1142-1-34-4': 'Thus, [MATH] is well defined and [EQUATION]', '0905.1142-1-34-5': 'We define another map [MATH] such that [EQUATION]', '0905.1142-1-34-6': 'Here, [MATH] is given by the weak solution of [MATH]-problem with [EQUATION] and the initial condition [EQUATION]', '0905.1142-1-34-7': 'The map [MATH] is well defined from Lemma [REF] and the definition of [MATH].', '0905.1142-1-34-8': 'Now we show that [MATH] is a contraction mapping for sufficiently small [MATH].', '0905.1142-1-34-9': 'Let [EQUATION]', '0905.1142-1-34-10': 'From the energy estimate [REF], [EQUATION]', '0905.1142-1-34-11': 'Thus, [MATH] has a unique fixed point [MATH] in [MATH] and [MATH] solves [MATH]-problem in a weak sense in [MATH], if [MATH].', '0905.1142-1-34-12': 'We are able to continue this procedure to obtain the global well-posedness for the above constant [MATH] is independent of [MATH].', '0905.1142-1-35-0': 'For the fixed point [MATH], [REF] and the boundedness of [MATH] imply that for [MATH] [EQUATION]', '0905.1142-1-35-1': 'We select a small [MATH] such that [MATH].', '0905.1142-1-35-2': 'Then [EQUATION]', '0905.1142-1-35-3': 'Thus, [EQUATION] and [EQUATION]', '0905.1142-1-35-4': 'Continuing, after finitely many steps we obtain an energy estimation similar to [REF].', '0905.1142-1-35-5': 'We summarize this in the following Lemma.', '0905.1142-1-36-0': '[MATH]-problem, [REF]-[REF], is uniquely solvable in weak sense for [MATH].', '0905.1142-1-36-1': 'Furthermore, [EQUATION]', '0905.1142-1-37-0': '# Well-posedness for the FPF problem', '0905.1142-1-38-0': 'In Section 2, we transformed the FPF problem to W-problem formally, but it is not difficult to show that they are equivalent even in the weak sense if [REF] is assumed.', '0905.1142-1-38-1': 'Indeed, one can verify that the boundary condition [REF] for the FPF problem is equivalent to the null boundary condition for [MATH]-problem.', '0905.1142-1-38-2': 'Let [EQUATION] denote the Fokker-Planck equation [REF] and [MATH]-problem [REF], respectively.', '0905.1142-1-38-3': 'For any test function [MATH], [EQUATION]', '0905.1142-1-38-4': 'Since [MATH] is in [MATH] which is dense in [MATH], the FPF problem and [MATH]-problem are equivalent.', '0905.1142-1-39-0': 'Now we seek the function space in which the weak solution [MATH] to the FPF problem belongs.', '0905.1142-1-39-1': 'For fixed [MATH], [REF] implies [EQUATION]', '0905.1142-1-39-2': 'Also, for [MATH] we have [EQUATION]', '0905.1142-1-39-3': 'Recall that [MATH] and [MATH].', '0905.1142-1-39-4': 'The estimate of the weak solution, [REF] follows from Lemma [REF] together with [REF]-[REF].', '0905.1142-1-39-5': 'In order to finish the proof of Theorem [REF], we assume that [MATH] are two weak solutions of the FPF problem with arbitrary initial data [MATH].', '0905.1142-1-39-6': 'Then [MATH] solves the FPF problem with zero initial data which is in [MATH].', '0905.1142-1-39-7': 'From [REF], [MATH] in [MATH].', '0905.1142-1-40-0': '# Non-uniqueness', '0905.1142-1-41-0': 'In this section we show that ([REF]) is sharp in the sense that more solutions can be constructed if a weaker condition is imposed.', '0905.1142-1-42-0': 'We construct a non-trivial solution to the Fokker-Planck equation with [MATH] and the assumption [EQUATION]', '0905.1142-1-42-1': 'Rewrite the Fokker-Planck equation with side conditions as follows: [EQUATION]', '0905.1142-1-42-2': 'Obviously, [MATH] is a solution of [REF]-[REF].', '0905.1142-1-42-3': 'Let [EQUATION] for [MATH] such that [MATH] and [MATH] for [MATH] (e.g. [MATH]).', '0905.1142-1-42-4': 'We will show the existence of a nontrivial solution [MATH] which coincides with [MATH] at the boundary.', '0905.1142-1-42-5': 'Note that [MATH] satisfies [REF] and [EQUATION]', '0905.1142-1-42-6': 'Define a function [MATH] as [EQUATION]', '0905.1142-1-42-7': 'Then, [EQUATION] for a parameter [MATH] such that [EQUATION]', '0905.1142-1-42-8': 'Note that [MATH], we can thus take such [MATH].', '0905.1142-1-42-9': 'Recall that [EQUATION]', '0905.1142-1-42-10': 'In order to have the zero boundary condition, we define [EQUATION]', '0905.1142-1-42-11': 'Then [MATH] solves [EQUATION] where [EQUATION]', '0905.1142-1-42-12': 'Let [EQUATION]', '0905.1142-1-42-13': 'This is well defined because of [REF] and the assumption that [MATH].', '0905.1142-1-42-14': 'From the same argument as that in Section 3, it follows that [REF]-[REF] has a unique solution [MATH] such that [EQUATION] provided the corresponding [MATH]-problem [EQUATION] has a solution for any [MATH].', '0905.1142-1-42-15': 'Note that [MATH] is essential in order that the trace of [MATH] at the boundary is defined.', '0905.1142-1-42-16': 'Equation [REF] is of the form of [REF] except [MATH].', '0905.1142-1-42-17': 'We thus define [EQUATION]', '0905.1142-1-42-18': 'We may obtain the existence and uniqueness for [REF]-[REF] from the same argument of the well-posedness for [MATH]-problem [REF]-[REF], if there is an energy estimate of [MATH] which is similar to [MATH] in Lemma [REF].', '0905.1142-1-42-19': 'Indeed, for [MATH] [EQUATION]', '0905.1142-1-42-20': 'We now claim that [EQUATION]', '0905.1142-1-42-21': 'Given this together with (5.5), [EQUATION] for any [MATH].', '0905.1142-1-42-22': 'By taking [MATH] so small, we obtain [EQUATION] for appropriate constants [MATH] and [MATH].', '0905.1142-1-43-0': 'To verify the claim [REF], we define the trace operator [MATH] such that [EQUATION]', '0905.1142-1-43-1': 'Integration by parts on [REF] yields [EQUATION] or [EQUATION]', '0905.1142-1-43-2': 'Thus [MATH] is well defined, and for [MATH], [MATH].', '0905.1142-1-43-3': 'Finally we obtain [EQUATION]', '0905.1142-1-43-4': 'This shows that there is a unique weak solution [MATH] of [REF]-[REF], and thus [MATH] of [REF]-[REF].', '0905.1142-1-43-5': 'Finally, [MATH] is a nontrivial solution of [REF]-[REF] satisfying [REF].', '0905.1142-1-43-6': 'Hence the uniqueness of [REF]-[REF] fails as stated in Proposition [REF].'}

{'0905.1142-2-0-0': 'We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement.', '0905.1142-2-0-1': 'The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the natural flux boundary condition.', '0905.1142-2-0-2': 'Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl.', '0905.1142-2-0-3': 'Math., 68(5):1304-1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter [MATH].', '0905.1142-2-0-4': 'In this article, we show that for the well-posedness of the microscopic FENE model ([MATH]) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function.', '0905.1142-2-0-5': 'This condition is strictly weaker than the natural flux boundary condition.', '0905.1142-2-0-6': 'Under this condition it is shown that there exists a unique weak solution in a weighted Sobolev space.', '0905.1142-2-0-7': 'The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.', '0905.1142-2-1-0': '# Introduction', '0905.1142-2-2-0': 'It is well-known that the following system coupling incompressible Navier-Stokes equation for the macroscopic velocity field [MATH] and the Fokker-Planck equation for the probability density function [MATH] describes diluted solutions of polymeric liquids with noninteracting polymer chains [EQUATION] where [MATH] is the macroscopic Eulerian coordinate and [MATH] is the microscopic molecular configuration variable.', '0905.1142-2-2-1': 'In this model, a polymer is idealized as an elastic dumbbell consisting of two beads joined by a spring that can be modeled by a vector [MATH] (see e.g [CITATION]).', '0905.1142-2-2-2': 'In the Navier-Stokes equation [REF], [MATH] is hydrostatic pressure, [MATH] is the kinematic viscosity coefficient, and [MATH] is a tensor representing the polymer contribution to stress, [EQUATION] where [MATH] is the elastic spring potential, and [MATH] is the polymer density constant.', '0905.1142-2-2-3': 'In the Fokker-Planck equation [REF], [MATH] is the friction coefficient of the dumbbell beads, [MATH] is the absolute temperature, and [MATH] is the Boltzmann constant.Notice that the Fokker-Planck equation can be written as a stochastic differential equation (see [CITATION]).', '0905.1142-2-3-0': 'One of the simplest model is the Hookean model in which the potential [MATH] is given by [EQUATION] where [MATH] is the elasticity constant.', '0905.1142-2-3-1': 'For a finite [MATH], a more realistic model is the finite extensible nonlinear elasticity (FENE) model in which case [EQUATION]', '0905.1142-2-3-2': 'Here [MATH] is the ball with center [MATH] and radius [MATH] which is the maximum dumbbell extension.', '0905.1142-2-3-3': 'In this work we shall focus our attention on the case [MATH], which is known to contain the parameter range of physical interest.', '0905.1142-2-3-4': 'We refer the reader to [CITATION] for a comprehensive survey of the physical background.', '0905.1142-2-4-0': 'In past years the well-posedness of the FENE model has been studied intensively in several aspects.', '0905.1142-2-4-1': 'For local well-posedness of strong solutions we refer the reader to [CITATION] for the FENE model (in the setting where the Fokker-Planck equation is formulated by a stochastic differential equation) with [MATH] or sometime [MATH], [CITATION] for a polynomial force and [CITATION] for the FENE model with [MATH].', '0905.1142-2-4-2': 'For a preliminary study on some related coupled PDE systems, we refer to the earlier work [CITATION] (however, the FENE model was not addressed there).', '0905.1142-2-4-3': 'Moreover, the authors of [CITATION] proved global existence of smooth solutions near equilibrium under some restrictions on the potential, which have been extended in subsequent works [CITATION].', '0905.1142-2-4-4': 'More recently, N. Masmoudi [CITATION] proved local and global well-posedness for the FENE dumbbell model for a general class of potentials with [MATH].', '0905.1142-2-5-0': 'Global existence of weak solutions was also proved in [CITATION] for the co-rotational model, see also [CITATION] for [MATH].', '0905.1142-2-5-1': 'For an earlier existence result of weak solutions, we refer to [CITATION] for the Fokker-Planck equation alone with [MATH].', '0905.1142-2-5-2': 'On the other hand, the authors in [CITATION], investigated the long-time behavior of both Hookean models and FENE models in various special flows in a bounded domain with suitable boundary conditions.', '0905.1142-2-6-0': 'The complexity with the FENE potential lies mainly with the singularity of the equation at the boundary.', '0905.1142-2-6-1': 'In [CITATION], C. Liu and H. Liu closely examined the necessity of Dirichlet boundary conditions for the microscopic FENE model.', '0905.1142-2-6-2': 'By the method of the Fichera function the authors were able to conclude that [MATH] is a threshold in the sense that for [MATH] any preassigned boundary value of the ratio of the distribution and the equilibrium will become redundant, and for [MATH] that value has to be a priori given.', '0905.1142-2-6-3': 'For the microscopic FENE model, singularity in the potential requires at least the zero Dirichlet boundary condition [EQUATION]', '0905.1142-2-6-4': 'This is consistent with the result in [CITATION], which states that the stochastic solution trajectory does not reach the boundary almost surely.', '0905.1142-2-6-5': 'In most of a priori works the natural flux boundary condition has been used: [EQUATION] which is stronger than ([REF]).', '0905.1142-2-7-0': 'The boundary issue for the underlying FENE model is fundamental, and our main quest in this paper is whether one can identify a sharp boundary requirement so that both existence and uniqueness of a global weak solution to the microscopic FENE model can be proved.', '0905.1142-2-7-1': 'The answer is positive, and we claim that [MATH] must satisfy the following boundary condition [EQUATION] where [MATH] denotes the distance function from [MATH] to the boundary [MATH].', '0905.1142-2-7-2': 'Note that [REF] is strictly weaker than [REF] and stronger than ([REF]).', '0905.1142-2-7-3': 'Our claim is supported by our main results: the global well-posedness for the Fokker-Planck equation stated in Theorem [REF], and the sharpness of ([REF]) stated in Proposition [REF].', '0905.1142-2-8-0': 'The importance of the Fokker-Planck equation itself as the added complexity with the FENE potential affects mostly the analysis of the Fokker-Planck equation.', '0905.1142-2-8-1': 'In this article, we focus on the underlying Fokker-Planck equation alone.', '0905.1142-2-8-2': 'Assuming [MATH] is independent of [MATH] and the fluid velocity is steady and homogeneous, we obtain the following equation from a suitable scaling ([CITATION]).', '0905.1142-2-8-3': '[EQUATION] where [MATH] and [MATH] is a constant matrix such that [MATH].', '0905.1142-2-8-4': 'Suppose that a sufficiently smooth function [MATH] solves [REF].', '0905.1142-2-8-5': 'Then for any test function [MATH], i.e. a continuously differentiable function compactly supported in [MATH], it follows that [EQUATION]', '0905.1142-2-8-6': 'Here, we omit [MATH] from [MATH].', '0905.1142-2-8-7': 'Note that [REF] is well defined for any [MATH] and [MATH].', '0905.1142-2-8-8': 'Moreover, the compact support of a test function makes [MATH] free on the boundary of the domain.', '0905.1142-2-8-9': 'From this, a weak solution of the Fokker-Planck equation [REF] with the initial condition [EQUATION] is defined in the following manner.', '0905.1142-2-9-0': 'Suppose that [EQUATION] for an arbitrary subdomain [MATH] of [MATH] such that [MATH].', '0905.1142-2-9-1': 'We say [MATH] is a solution of [REF], [REF] if [REF] holds for any [MATH] and almost all [MATH] such that [EQUATION]', '0905.1142-2-9-2': 'Note that [REF] makes sense since [MATH].', '0905.1142-2-10-0': 'Consider the equilibrium [MATH] of [REF] for non-flow case, i.e. [MATH].', '0905.1142-2-10-1': 'Rewrite [REF] as [EQUATION]', '0905.1142-2-10-2': 'It follows that [EQUATION]', '0905.1142-2-10-3': 'Obviously, [MATH] has the zero trace on the boundary [MATH] and satisfies ([REF]) for [EQUATION]', '0905.1142-2-10-4': 'Thus, [REF] is satisfied even for a test function [MATH] without the compactly supported property.', '0905.1142-2-10-5': 'Also [EQUATION] as long as [REF] is assumed.', '0905.1142-2-10-6': 'Here, [MATH] with [EQUATION]', '0905.1142-2-10-7': 'Our main results are summarized in Theorem [REF] and Proposition [REF] below.', '0905.1142-2-11-0': 'i) if [EQUATION] then there exists a unique solution [MATH] in the sense of Definition [REF].', '0905.1142-2-11-1': 'Moreover, [EQUATION] ii) If initial data [MATH], there exists at most one solution [MATH] to the Fokker-Planck equation [REF] and [REF].', '0905.1142-2-12-0': 'The proof of (i) will be done in sections 2-4.', '0905.1142-2-12-1': 'In order to prove (ii) we assume that [MATH] are two weak solutions of the problem with arbitrary initial data [MATH].', '0905.1142-2-12-2': 'Then [MATH] solves [REF] with zero initial data which is in [MATH].', '0905.1142-2-12-3': 'From [REF] in (i) it follows that [MATH] in [MATH].', '0905.1142-2-13-0': 'We remark that the restriction on [MATH] [REF] is essential to obtain the energy estimate [REF].', '0905.1142-2-14-0': 'The following proposition states that the boundary condition [REF] is sharp for the uniqueness of the weak solution.', '0905.1142-2-15-0': 'This article is organized as follows.', '0905.1142-2-15-1': 'In Section 2, we transform the Fokker-Planck equation to a certain Cauchy-Dirichlet problem, named as [MATH]-problem, and define a weak solution of [MATH]-problem in a weighted Sobolev space.', '0905.1142-2-15-2': 'The well-posedness of the [MATH]-problem is shown in Section 3 by the Galerkin method and the Banach fixed point theorem.', '0905.1142-2-15-3': 'This leads to the well-posedness of the Fokker-Planck equation, Theorem [REF]; details of the proof are presented in Section 4.', '0905.1142-2-15-4': 'In the last section, we construct a non-trivial solution for the Fokker-Planck equation described in Proposition [REF].', '0905.1142-2-16-0': '# Transformation of the microscopic FENE model', '0905.1142-2-17-0': 'In what follows we shall call the Fokker-Planck equation ([REF]) with initial condition ([REF]) and boundary condition ([REF]) as the Fokker-Planck-FENE (FPF) problem.', '0905.1142-2-17-1': 'We first formulate a time evolution equation from the FPF problem.', '0905.1142-2-17-2': 'Define [MATH] ([CITATION]) as [EQUATION] with [MATH] to be determined.', '0905.1142-2-17-3': 'Then [REF] is transformed to [EQUATION] where [EQUATION]', '0905.1142-2-17-4': 'Setting a parameter [EQUATION] we rewrite [REF] as [EQUATION]', '0905.1142-2-17-5': 'If [EQUATION] is taken, the boundary condition [REF] implies that [MATH] satisfies a homogeneous boundary condition for almost all [MATH] since the distance function [MATH] and [MATH] are equivalent (see [REF]).', '0905.1142-2-18-0': 'The FPF problem is formally transformed to the following [MATH]-problem: [EQUATION]', '0905.1142-2-18-1': 'Here, [EQUATION] according to the transformation [REF].', '0905.1142-2-19-0': 'In order to define a weak solution of [MATH]-problem we introduce a weighted Sobolev space [MATH] for a nonnegative measurable function [MATH] as a set of measurable function [MATH] such that [EQUATION]', '0905.1142-2-19-1': 'Similarly, a weighted [MATH] can be defined.', '0905.1142-2-19-2': '[MATH] denotes a completion of [MATH] with [MATH].', '0905.1142-2-19-3': 'It is obvious that [MATH] and [MATH] are Hilbert spaces with the inner product [MATH] defined as [EQUATION] and [EQUATION]', '0905.1142-2-19-4': 'For notational convenient, we use [MATH] and [MATH] for [MATH] and [MATH] respectively.', '0905.1142-2-19-5': 'We also omit the domain [MATH] if it is obvious.', '0905.1142-2-20-0': 'Suppose that [MATH] and [MATH].', '0905.1142-2-21-0': '[(1)] If [MATH], then [EQUATION] [(2)] If [MATH], then the trace map [EQUATION] is well defined, i.e. it is a bounded linear map.', '0905.1142-2-22-0': 'In particular, for [MATH] [EQUATION]', '0905.1142-2-23-0': 'In [CITATION](see also [CITATION]), it was proved that [EQUATION] provided [MATH] is Lipschitz continuous.', '0905.1142-2-23-1': 'Recall that [MATH] denotes the distance from [MATH] to the boundary of [MATH].', '0905.1142-2-23-2': '[REF] follows from [EQUATION]', '0905.1142-2-23-3': 'It is also known that the trace map [MATH] is well defined for [MATH] ([CITATION]).', '0905.1142-2-23-4': 'For [MATH] [EQUATION] since [MATH] for all [MATH].', '0905.1142-2-23-5': 'Therefore, [MATH] is well defined for [MATH].', '0905.1142-2-23-6': '[REF] is obvious from the definitions of the trace map and [MATH].', '0905.1142-2-24-0': 'Note that [REF] remains true for [MATH].', '0905.1142-2-25-0': 'We now define a weak solution to [MATH]-problem in a standard manner.', '0905.1142-2-25-1': 'Multiplication by a test function [MATH] to the equation [REF] and integration over [MATH] yield [EQUATION]', '0905.1142-2-25-2': 'This equation is well defined assuming that [MATH], the dual space of [MATH], and [MATH] due to the boundedness of [MATH] and Lemma [REF].', '0905.1142-2-25-3': 'Moreover, [EQUATION] implies [EQUATION]', '0905.1142-2-25-4': 'Thus [EQUATION]', '0905.1142-2-25-5': 'Here we identify [MATH] with its dual space.', '0905.1142-2-26-0': 'A function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem, [REF]-[REF], provided', '0905.1142-2-27-0': '[(1)] For each [MATH] and almost every [MATH], [EQUATION] [(2)] [MATH] in [MATH] sense.', '0905.1142-2-27-1': 'i.e. [EQUATION]', '0905.1142-2-27-2': 'Here, we let [MATH] denote the paring of a Hilbert space [MATH] with its dual space [MATH] and [EQUATION]', '0905.1142-2-27-3': 'The following energy estimate for [MATH] for fixed [MATH] can be achieved from a simple modification of energy estimates for the bilinear form in elliptic equations, see [CITATION] for details.', '0905.1142-2-28-0': 'There exist positive constants [MATH] and [MATH] depending only on [MATH] and [MATH] such that [EQUATION]', '0905.1142-2-29-0': '# Well-posedness for the transformed problem', '0905.1142-2-30-0': 'In this section, we show the well-posedness of the weak solution to [MATH]-problem.', '0905.1142-2-30-1': 'For this aim, we consider the following [MATH]-problem containing a non-homogeneous term [MATH].', '0905.1142-2-30-2': '[EQUATION]', '0905.1142-2-30-3': 'The weak solution of [MATH]-problem is defined similarly.', '0905.1142-2-31-0': 'We say a function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem provided', '0905.1142-2-32-0': '[(1)] for each [MATH] and almost every [MATH] [EQUATION] [(2)] [MATH] in [MATH].', '0905.1142-2-33-0': 'Recall that [EQUATION]', '0905.1142-2-33-1': 'We remark that [MATH] is finite for any [MATH] since [MATH] from [REF].', '0905.1142-2-33-2': 'Thus [MATH] can be understood as the [MATH] inner product although [MATH] may not belong to [MATH].', '0905.1142-2-34-0': 'The well-posedness for [MATH]-problem follows from the standard Galerkin method.', '0905.1142-2-35-0': 'For given [MATH] and [MATH], [MATH]-problem has a unique weak solution.', '0905.1142-2-35-1': 'Moreover, [EQUATION]', '0905.1142-2-35-2': 'We first construct an approximate solution in a finite-dimensional space.', '0905.1142-2-35-3': 'Let [MATH] be a basis of [MATH] and [MATH].', '0905.1142-2-35-4': 'The existence of such a basis can be verified from the fact that [MATH] is a dense subset of [MATH].', '0905.1142-2-35-5': 'Consider an approximation [MATH], where [MATH] satisfies [EQUATION]', '0905.1142-2-35-6': 'Since [REF] and [REF] form a system of linear differential equations, [MATH] is uniquely determined for each [MATH].', '0905.1142-2-35-7': 'We rewrite [REF] as [EQUATION]', '0905.1142-2-35-8': 'Apply [MATH] to [REF] and sum for [MATH], then for almost every [MATH] [EQUATION]', '0905.1142-2-35-9': 'From Lemma [REF], it follows that [EQUATION]', '0905.1142-2-35-10': 'From [REF], for any [MATH] [EQUATION]', '0905.1142-2-35-11': 'With [MATH], [REF] can be rewritten as [EQUATION] or [EQUATION]', '0905.1142-2-35-12': "Use Gronwall's inequality to obtain [EQUATION] where [MATH] is an appropriate constant which depends on [MATH], [MATH], [MATH] and [MATH].", '0905.1142-2-35-13': 'On the other hand, integration of [REF] from [MATH] to [MATH] together with above inequality yields [EQUATION]', '0905.1142-2-35-14': 'A similar argument to that in [CITATION] gives us the estimate for [MATH] as [EQUATION]', '0905.1142-2-35-15': 'Here we have used [REF] with [MATH] such that [MATH] and [REF].', '0905.1142-2-35-16': 'By passing to the limit as [MATH] and a standard argument (e.g. see [CITATION]), we have well-posedness for [MATH]-problem.', '0905.1142-2-36-0': 'Now, we introduce a linear map [MATH] to connect [MATH] and [MATH]-problems as [EQUATION]', '0905.1142-2-36-1': 'To have a well-defined [MATH], we choose [EQUATION] which is crucial in this argument.', '0905.1142-2-36-2': 'With this [MATH], we rewrite [MATH] defined in [REF] as follows: [EQUATION]', '0905.1142-2-36-3': 'Since [MATH] is bounded, [EQUATION]', '0905.1142-2-36-4': 'Thus, [MATH] is well defined and [EQUATION]', '0905.1142-2-36-5': 'We define another map [MATH] such that [EQUATION]', '0905.1142-2-36-6': 'Here, [MATH] is given by the weak solution of [MATH]-problem with [EQUATION] and the initial condition [EQUATION]', '0905.1142-2-36-7': 'The map [MATH] is well defined from Lemma [REF] and the definition of [MATH].', '0905.1142-2-36-8': 'Now we show that [MATH] is a contraction mapping for sufficiently small [MATH].', '0905.1142-2-36-9': 'Let [EQUATION]', '0905.1142-2-36-10': 'From the energy estimate [REF], [EQUATION]', '0905.1142-2-36-11': 'Thus, [MATH] has a unique fixed point [MATH] in [MATH] and [MATH] solves [MATH]-problem in a weak sense in [MATH], if [MATH].', '0905.1142-2-36-12': 'We are able to continue this procedure to obtain the global well-posedness for the above constant [MATH] is independent of [MATH].', '0905.1142-2-37-0': 'For the fixed point [MATH], [REF] and the boundedness of [MATH] imply that for [MATH] [EQUATION]', '0905.1142-2-37-1': 'We select a small [MATH] such that [MATH].', '0905.1142-2-37-2': 'Then [EQUATION]', '0905.1142-2-37-3': 'Thus, [EQUATION] and [EQUATION]', '0905.1142-2-37-4': 'Continuing, after finitely many steps we obtain an energy estimation similar to [REF].', '0905.1142-2-37-5': 'We summarize this in the following Lemma.', '0905.1142-2-38-0': '[MATH]-problem, [REF]-[REF], is uniquely solvable in weak sense for [MATH].', '0905.1142-2-38-1': 'Furthermore, [EQUATION]', '0905.1142-2-39-0': '# Well-posedness for the FPF problem', '0905.1142-2-40-0': 'In Section 2, we transformed the FPF problem to W-problem formally, but it is not difficult to show that they are equivalent even in the weak sense if [REF] is assumed.', '0905.1142-2-40-1': 'Indeed, one can verify that the boundary condition [REF] for the FPF problem is equivalent to the null boundary condition for [MATH]-problem.', '0905.1142-2-40-2': 'Let [EQUATION] denote the Fokker-Planck equation [REF] and [REF] in the [MATH]-problem, respectively.', '0905.1142-2-40-3': 'For any test function [MATH], [EQUATION]', '0905.1142-2-40-4': 'Since [MATH] is in [MATH] which is dense in [MATH], the FPF problem and [MATH]-problem are equivalent.', '0905.1142-2-41-0': 'Now we seek the function space in which the weak solution [MATH] to the FPF problem belongs.', '0905.1142-2-41-1': 'Recall that [MATH] and [MATH].', '0905.1142-2-41-2': 'For fixed [MATH], [REF] implies [EQUATION]', '0905.1142-2-41-3': 'Also, for [MATH] we have [EQUATION]', '0905.1142-2-41-4': 'The estimate of the weak solution, [REF] follows from Lemma [REF] together with [REF]-[REF].', '0905.1142-2-41-5': 'This finishes the proof of (i) of Theorem [REF].', '0905.1142-2-42-0': '# Non-uniqueness', '0905.1142-2-43-0': 'In this section we show that ([REF]) is sharp in the sense that more solutions can be constructed if a weaker condition is imposed.', '0905.1142-2-44-0': 'We construct a non-trivial solution to the Fokker-Planck equation with [MATH] and the assumption [EQUATION]', '0905.1142-2-44-1': 'Here [MATH] and [MATH] is a nonzero measurable set.', '0905.1142-2-44-2': 'Rewrite the Fokker-Planck equation with side conditions as follows: [EQUATION]', '0905.1142-2-44-3': 'Obviously, [MATH] is a solution of [REF]-[REF].', '0905.1142-2-44-4': 'Let [EQUATION] for [MATH] such that [MATH] and [MATH] for [MATH] (e.g. [MATH]).', '0905.1142-2-44-5': 'We will show the existence of a nontrivial solution [MATH] which coincides with [MATH] at the boundary.', '0905.1142-2-44-6': 'Note that [MATH] satisfies [REF] and [EQUATION]', '0905.1142-2-44-7': 'Define a function [MATH] as [EQUATION]', '0905.1142-2-44-8': 'Then, [EQUATION] for a parameter [MATH] such that [EQUATION]', '0905.1142-2-44-9': 'Note that [MATH], we can thus take such [MATH].', '0905.1142-2-44-10': 'Recall that [EQUATION]', '0905.1142-2-44-11': 'In order to have the zero boundary condition, we define [EQUATION]', '0905.1142-2-44-12': 'Then [MATH] solves [EQUATION] where [EQUATION]', '0905.1142-2-44-13': 'Let [EQUATION]', '0905.1142-2-44-14': 'This is well defined because of [REF] and the assumption that [MATH].', '0905.1142-2-44-15': 'From the same argument as that in Section 3, it follows that [REF]-[REF] has a unique solution [MATH] such that [EQUATION] provided the corresponding [MATH]-problem [EQUATION] has a solution for any [MATH].', '0905.1142-2-44-16': 'Note that [MATH] is essential in order that the trace of [MATH] at the boundary is defined.', '0905.1142-2-44-17': 'Equation [REF] is of the form of [REF] but with an additional term [MATH].', '0905.1142-2-44-18': 'We thus define [EQUATION]', '0905.1142-2-44-19': 'We may obtain the existence and uniqueness for [REF]-[REF] from the same argument of the well-posedness for [MATH]-problem [REF]-[REF], if there is an energy estimate of [MATH] which is similar to [MATH] in Lemma [REF].', '0905.1142-2-44-20': 'Indeed, for [MATH] [EQUATION]', '0905.1142-2-44-21': 'We now claim that [EQUATION]', '0905.1142-2-44-22': 'Given this together with (5.5), [EQUATION] for any [MATH].', '0905.1142-2-44-23': 'By taking [MATH] so small, we obtain [EQUATION] for appropriate constants [MATH] and [MATH].', '0905.1142-2-45-0': 'To verify the claim [REF], we define the trace operator [MATH] such that [EQUATION]', '0905.1142-2-45-1': 'Integration by parts on [REF] yields [EQUATION] or [EQUATION]', '0905.1142-2-45-2': 'Thus [MATH] is well defined, and for [MATH], [MATH].', '0905.1142-2-45-3': 'Finally we obtain [EQUATION]', '0905.1142-2-45-4': 'This shows that there is a unique weak solution [MATH] of [REF]-[REF], and thus [MATH] of [REF]-[REF].', '0905.1142-2-45-5': 'Finally, [MATH] is a nontrivial solution of [REF]-[REF] satisfying [REF].', '0905.1142-2-45-6': 'Hence the uniqueness of [REF]-[REF] fails as stated in Proposition [REF].'}

[['0905.1142-1-31-1', '0905.1142-2-33-1'], ['0905.1142-1-31-2', '0905.1142-2-33-2'], ['0905.1142-1-33-0', '0905.1142-2-35-0'], ['0905.1142-1-33-2', '0905.1142-2-35-2'], ['0905.1142-1-33-3', '0905.1142-2-35-3'], ['0905.1142-1-33-4', '0905.1142-2-35-4'], ['0905.1142-1-33-5', '0905.1142-2-35-5'], ['0905.1142-1-33-6', '0905.1142-2-35-6'], ['0905.1142-1-33-7', '0905.1142-2-35-7'], ['0905.1142-1-33-8', '0905.1142-2-35-8'], ['0905.1142-1-33-9', '0905.1142-2-35-9'], ['0905.1142-1-33-10', '0905.1142-2-35-10'], ['0905.1142-1-33-11', '0905.1142-2-35-11'], ['0905.1142-1-33-12', '0905.1142-2-35-12'], ['0905.1142-1-33-13', '0905.1142-2-35-13'], ['0905.1142-1-33-14', '0905.1142-2-35-14'], ['0905.1142-1-33-15', '0905.1142-2-35-15'], ['0905.1142-1-33-16', '0905.1142-2-35-16'], ['0905.1142-1-6-1', '0905.1142-2-6-1'], ['0905.1142-1-6-4', '0905.1142-2-6-5'], ['0905.1142-1-42-0', '0905.1142-2-44-0'], ['0905.1142-1-42-1', '0905.1142-2-44-2'], ['0905.1142-1-42-2', '0905.1142-2-44-3'], ['0905.1142-1-42-4', '0905.1142-2-44-5'], ['0905.1142-1-42-5', '0905.1142-2-44-6'], ['0905.1142-1-42-6', '0905.1142-2-44-7'], ['0905.1142-1-42-7', '0905.1142-2-44-8'], ['0905.1142-1-42-8', '0905.1142-2-44-9'], ['0905.1142-1-42-10', '0905.1142-2-44-11'], ['0905.1142-1-42-11', '0905.1142-2-44-12'], ['0905.1142-1-42-13', '0905.1142-2-44-14'], ['0905.1142-1-42-14', '0905.1142-2-44-15'], ['0905.1142-1-42-15', '0905.1142-2-44-16'], ['0905.1142-1-42-17', '0905.1142-2-44-18'], ['0905.1142-1-42-18', '0905.1142-2-44-19'], ['0905.1142-1-42-19', '0905.1142-2-44-20'], ['0905.1142-1-42-20', '0905.1142-2-44-21'], ['0905.1142-1-42-21', '0905.1142-2-44-22'], ['0905.1142-1-42-22', '0905.1142-2-44-23'], ['0905.1142-1-15-0', '0905.1142-2-17-0'], ['0905.1142-1-15-1', '0905.1142-2-17-1'], ['0905.1142-1-15-2', '0905.1142-2-17-2'], ['0905.1142-1-15-4', '0905.1142-2-17-4'], ['0905.1142-1-15-5', '0905.1142-2-17-5'], ['0905.1142-1-41-0', '0905.1142-2-43-0'], ['0905.1142-1-3-0', '0905.1142-2-3-0'], ['0905.1142-1-3-2', '0905.1142-2-3-3'], ['0905.1142-1-34-0', '0905.1142-2-36-0'], ['0905.1142-1-34-1', '0905.1142-2-36-1'], ['0905.1142-1-34-2', '0905.1142-2-36-2'], ['0905.1142-1-34-3', '0905.1142-2-36-3'], ['0905.1142-1-34-4', '0905.1142-2-36-4'], ['0905.1142-1-34-5', '0905.1142-2-36-5'], ['0905.1142-1-34-6', '0905.1142-2-36-6'], ['0905.1142-1-34-7', '0905.1142-2-36-7'], ['0905.1142-1-34-8', '0905.1142-2-36-8'], ['0905.1142-1-34-10', '0905.1142-2-36-10'], ['0905.1142-1-34-11', '0905.1142-2-36-11'], ['0905.1142-1-34-12', '0905.1142-2-36-12'], ['0905.1142-1-17-0', '0905.1142-2-19-0'], ['0905.1142-1-17-1', '0905.1142-2-19-1'], ['0905.1142-1-17-2', '0905.1142-2-19-2'], ['0905.1142-1-17-3', '0905.1142-2-19-3'], ['0905.1142-1-29-0', '0905.1142-2-31-0'], ['0905.1142-1-21-0', '0905.1142-2-23-0'], ['0905.1142-1-21-1', '0905.1142-2-23-1'], ['0905.1142-1-21-2', '0905.1142-2-23-2'], ['0905.1142-1-21-3', '0905.1142-2-23-3'], ['0905.1142-1-21-4', '0905.1142-2-23-4'], ['0905.1142-1-21-5', '0905.1142-2-23-5'], ['0905.1142-1-21-6', '0905.1142-2-23-6'], ['0905.1142-1-28-0', '0905.1142-2-30-0'], ['0905.1142-1-28-1', '0905.1142-2-30-1'], ['0905.1142-1-28-3', '0905.1142-2-30-3'], ['0905.1142-1-4-0', '0905.1142-2-4-0'], ['0905.1142-1-25-2', '0905.1142-2-27-2'], ['0905.1142-1-25-3', '0905.1142-2-27-3'], ['0905.1142-1-23-0', '0905.1142-2-25-0'], ['0905.1142-1-23-1', '0905.1142-2-25-1'], ['0905.1142-1-23-2', '0905.1142-2-25-2'], ['0905.1142-1-23-3', '0905.1142-2-25-3'], ['0905.1142-1-23-5', '0905.1142-2-25-5'], ['0905.1142-1-19-0', '0905.1142-2-21-0'], ['0905.1142-1-39-0', '0905.1142-2-41-0'], ['0905.1142-1-39-1', '0905.1142-2-41-2'], ['0905.1142-1-39-2', '0905.1142-2-41-3'], ['0905.1142-1-39-3', '0905.1142-2-41-1'], ['0905.1142-1-39-4', '0905.1142-2-41-4'], ['0905.1142-1-10-0', '0905.1142-2-10-0'], ['0905.1142-1-10-1', '0905.1142-2-10-1'], ['0905.1142-1-10-2', '0905.1142-2-10-2'], ['0905.1142-1-10-3', '0905.1142-2-10-3'], ['0905.1142-1-10-4', '0905.1142-2-10-4'], ['0905.1142-1-10-5', '0905.1142-2-10-5'], ['0905.1142-1-10-6', '0905.1142-2-10-6'], ['0905.1142-1-10-7', '0905.1142-2-10-7'], ['0905.1142-1-13-2', '0905.1142-2-15-2'], ['0905.1142-1-13-3', '0905.1142-2-15-3'], ['0905.1142-1-13-4', '0905.1142-2-15-4'], ['0905.1142-1-0-0', '0905.1142-2-0-0'], ['0905.1142-1-0-1', '0905.1142-2-0-1'], ['0905.1142-1-0-2', '0905.1142-2-0-2'], ['0905.1142-1-0-3', '0905.1142-2-0-3'], ['0905.1142-1-0-4', '0905.1142-2-0-4'], ['0905.1142-1-0-5', '0905.1142-2-0-5'], ['0905.1142-1-0-6', '0905.1142-2-0-6'], ['0905.1142-1-0-7', '0905.1142-2-0-7'], ['0905.1142-1-16-0', '0905.1142-2-18-0'], ['0905.1142-1-16-1', '0905.1142-2-18-1'], ['0905.1142-1-5-0', '0905.1142-2-5-0'], ['0905.1142-1-5-1', '0905.1142-2-5-1'], ['0905.1142-1-5-2', '0905.1142-2-5-2'], ['0905.1142-1-12-0', '0905.1142-2-14-0'], ['0905.1142-1-8-0', '0905.1142-2-8-0'], ['0905.1142-1-8-1', '0905.1142-2-8-1'], ['0905.1142-1-8-2', '0905.1142-2-8-2'], ['0905.1142-1-8-4', '0905.1142-2-8-4'], ['0905.1142-1-8-5', '0905.1142-2-8-5'], ['0905.1142-1-8-6', '0905.1142-2-8-6'], ['0905.1142-1-8-7', '0905.1142-2-8-7'], ['0905.1142-1-8-8', '0905.1142-2-8-8'], ['0905.1142-1-7-0', '0905.1142-2-7-0'], ['0905.1142-1-7-2', '0905.1142-2-7-2'], ['0905.1142-1-7-3', '0905.1142-2-7-3'], ['0905.1142-1-2-1', '0905.1142-2-2-1'], ['0905.1142-1-38-0', '0905.1142-2-40-0'], ['0905.1142-1-38-1', '0905.1142-2-40-1'], ['0905.1142-1-38-3', '0905.1142-2-40-3'], ['0905.1142-1-38-4', '0905.1142-2-40-4'], ['0905.1142-1-35-0', '0905.1142-2-37-0'], ['0905.1142-1-35-1', '0905.1142-2-37-1'], ['0905.1142-1-35-4', '0905.1142-2-37-4'], ['0905.1142-1-35-5', '0905.1142-2-37-5'], ['0905.1142-2-17-0', '0905.1142-3-45-0'], ['0905.1142-2-17-1', '0905.1142-3-45-1'], ['0905.1142-2-17-3', '0905.1142-3-45-3'], ['0905.1142-2-17-4', '0905.1142-3-45-4'], ['0905.1142-2-33-1', '0905.1142-3-63-0'], ['0905.1142-2-33-2', '0905.1142-3-63-1'], ['0905.1142-2-18-0', '0905.1142-3-46-0'], ['0905.1142-2-18-1', '0905.1142-3-46-1'], ['0905.1142-2-25-0', '0905.1142-3-52-0'], ['0905.1142-2-25-1', '0905.1142-3-52-1'], ['0905.1142-2-25-2', '0905.1142-3-52-2'], ['0905.1142-2-25-3', '0905.1142-3-52-3'], ['0905.1142-2-25-5', '0905.1142-3-52-5'], ['0905.1142-2-2-0', '0905.1142-3-2-0'], ['0905.1142-2-2-1', '0905.1142-3-2-1'], ['0905.1142-2-15-0', '0905.1142-3-23-0'], ['0905.1142-2-41-0', '0905.1142-3-71-0'], ['0905.1142-2-41-2', '0905.1142-3-71-2'], ['0905.1142-2-41-3', '0905.1142-3-71-3'], ['0905.1142-2-41-4', '0905.1142-3-71-4'], ['0905.1142-2-41-5', '0905.1142-3-71-5'], ['0905.1142-2-23-0', '0905.1142-3-51-0'], ['0905.1142-2-23-1', '0905.1142-3-51-1'], ['0905.1142-2-23-2', '0905.1142-3-51-2'], ['0905.1142-2-23-3', '0905.1142-3-51-3'], ['0905.1142-2-23-4', '0905.1142-3-51-4'], ['0905.1142-2-23-5', '0905.1142-3-51-5'], ['0905.1142-2-23-6', '0905.1142-3-51-6'], ['0905.1142-2-30-0', '0905.1142-3-60-0'], ['0905.1142-2-30-1', '0905.1142-3-60-1'], ['0905.1142-2-30-3', '0905.1142-3-60-3'], ['0905.1142-2-10-6', '0905.1142-3-14-3'], ['0905.1142-2-3-0', '0905.1142-3-3-0'], ['0905.1142-2-3-4', '0905.1142-3-3-4'], ['0905.1142-2-31-0', '0905.1142-3-61-0'], ['0905.1142-2-36-0', '0905.1142-3-66-0'], ['0905.1142-2-36-3', '0905.1142-3-66-1'], ['0905.1142-2-36-4', '0905.1142-3-66-2'], ['0905.1142-2-36-5', '0905.1142-3-66-3'], ['0905.1142-2-36-6', '0905.1142-3-66-4'], ['0905.1142-2-36-7', '0905.1142-3-66-5'], ['0905.1142-2-36-8', '0905.1142-3-66-6'], ['0905.1142-2-36-10', '0905.1142-3-66-8'], ['0905.1142-2-36-11', '0905.1142-3-66-9'], ['0905.1142-2-36-12', '0905.1142-3-66-10'], ['0905.1142-2-35-0', '0905.1142-3-65-0'], ['0905.1142-2-35-2', '0905.1142-3-65-2'], ['0905.1142-2-35-3', '0905.1142-3-65-3'], ['0905.1142-2-35-4', '0905.1142-3-65-4'], ['0905.1142-2-35-5', '0905.1142-3-65-5'], ['0905.1142-2-35-6', '0905.1142-3-65-6'], ['0905.1142-2-35-7', '0905.1142-3-65-7'], ['0905.1142-2-35-8', '0905.1142-3-65-8'], ['0905.1142-2-35-9', '0905.1142-3-65-9'], ['0905.1142-2-35-10', '0905.1142-3-65-10'], ['0905.1142-2-35-11', '0905.1142-3-65-11'], ['0905.1142-2-35-12', '0905.1142-3-65-12'], ['0905.1142-2-35-13', '0905.1142-3-65-13'], ['0905.1142-2-35-14', '0905.1142-3-65-14'], ['0905.1142-2-35-15', '0905.1142-3-65-15'], ['0905.1142-2-35-16', '0905.1142-3-65-16'], ['0905.1142-2-37-0', '0905.1142-3-67-0'], ['0905.1142-2-37-1', '0905.1142-3-67-1'], ['0905.1142-2-37-4', '0905.1142-3-67-4'], ['0905.1142-2-19-0', '0905.1142-3-47-0'], ['0905.1142-2-19-1', '0905.1142-3-47-1'], ['0905.1142-2-19-2', '0905.1142-3-47-2'], ['0905.1142-2-19-3', '0905.1142-3-47-3'], ['0905.1142-2-19-4', '0905.1142-3-47-4'], ['0905.1142-2-19-5', '0905.1142-3-47-5'], ['0905.1142-2-7-1', '0905.1142-3-7-1'], ['0905.1142-2-0-0', '0905.1142-3-0-0'], ['0905.1142-2-0-2', '0905.1142-3-0-2'], ['0905.1142-2-0-3', '0905.1142-3-0-3'], ['0905.1142-2-0-4', '0905.1142-3-0-4'], ['0905.1142-2-0-7', '0905.1142-3-0-7'], ['0905.1142-2-12-2', '0905.1142-3-16-2'], ['0905.1142-2-44-16', '0905.1142-3-74-10'], ['0905.1142-2-44-17', '0905.1142-3-74-11'], ['0905.1142-2-44-18', '0905.1142-3-74-12'], ['0905.1142-2-44-19', '0905.1142-3-74-13'], ['0905.1142-2-44-20', '0905.1142-3-74-14'], ['0905.1142-2-44-21', '0905.1142-3-74-15'], ['0905.1142-2-4-1', '0905.1142-3-4-1'], ['0905.1142-2-4-2', '0905.1142-3-4-2'], ['0905.1142-2-6-0', '0905.1142-3-6-0'], ['0905.1142-2-6-1', '0905.1142-3-6-4'], ['0905.1142-2-6-3', '0905.1142-3-6-6'], ['0905.1142-2-6-4', '0905.1142-3-6-7'], ['0905.1142-1-6-0', '0905.1142-2-6-0'], ['0905.1142-1-6-2', '0905.1142-2-6-2'], ['0905.1142-1-15-3', '0905.1142-2-17-3'], ['0905.1142-1-3-1', '0905.1142-2-3-1'], ['0905.1142-1-3-3', '0905.1142-2-3-4'], ['0905.1142-1-17-5', '0905.1142-2-19-5'], ['0905.1142-1-4-1', '0905.1142-2-4-1'], ['0905.1142-1-4-2', '0905.1142-2-4-2'], ['0905.1142-1-4-3', '0905.1142-2-4-3'], ['0905.1142-1-4-4', '0905.1142-2-4-4'], ['0905.1142-1-13-0', '0905.1142-2-15-0'], ['0905.1142-1-13-1', '0905.1142-2-15-1'], ['0905.1142-1-8-3', '0905.1142-2-8-3'], ['0905.1142-1-7-1', '0905.1142-2-7-1'], ['0905.1142-1-2-0', '0905.1142-2-2-0'], ['0905.1142-1-2-2', '0905.1142-2-2-2'], ['0905.1142-1-38-2', '0905.1142-2-40-2'], ['0905.1142-2-17-5', '0905.1142-3-45-5'], ['0905.1142-2-2-2', '0905.1142-3-2-2'], ['0905.1142-2-15-1', '0905.1142-3-23-2'], ['0905.1142-2-15-2', '0905.1142-3-23-3'], ['0905.1142-2-15-3', '0905.1142-3-23-4'], ['0905.1142-2-15-4', '0905.1142-3-23-5'], ['0905.1142-2-11-0', '0905.1142-3-15-0'], ['0905.1142-2-13-0', '0905.1142-3-17-0'], ['0905.1142-2-14-0', '0905.1142-3-20-0'], ['0905.1142-2-21-0', '0905.1142-3-49-1'], ['0905.1142-2-10-7', '0905.1142-3-14-4'], ['0905.1142-2-3-2', '0905.1142-3-3-2'], ['0905.1142-2-3-3', '0905.1142-3-3-3'], ['0905.1142-2-43-0', '0905.1142-3-73-0'], ['0905.1142-2-7-0', '0905.1142-3-7-0'], ['0905.1142-2-7-3', '0905.1142-3-7-2'], ['0905.1142-2-5-1', '0905.1142-3-5-1'], ['0905.1142-2-5-2', '0905.1142-3-5-2'], ['0905.1142-2-0-1', '0905.1142-3-0-1'], ['0905.1142-2-0-6', '0905.1142-3-0-5'], ['0905.1142-2-12-1', '0905.1142-3-16-1'], ['0905.1142-2-12-3', '0905.1142-3-16-3'], ['0905.1142-2-44-1', '0905.1142-3-74-1'], ['0905.1142-2-44-15', '0905.1142-3-74-9'], ['0905.1142-2-44-23', '0905.1142-3-74-17'], ['0905.1142-2-4-0', '0905.1142-3-4-0'], ['0905.1142-2-4-3', '0905.1142-3-4-3'], ['0905.1142-2-6-2', '0905.1142-3-6-5'], ['0905.1142-2-8-1', '0905.1142-3-8-0'], ['0905.1142-2-40-0', '0905.1142-3-69-0'], ['0905.1142-2-40-1', '0905.1142-3-69-1'], ['0905.1142-2-27-2', '0905.1142-3-53-0'], ['0905.1142-1-6-3', '0905.1142-2-6-3'], ['0905.1142-1-42-16', '0905.1142-2-44-17'], ['0905.1142-1-17-4', '0905.1142-2-19-4'], ['0905.1142-1-2-3', '0905.1142-2-2-3'], ['0905.1142-1-2-4', '0905.1142-2-2-3'], ['0905.1142-2-17-2', '0905.1142-3-45-2'], ['0905.1142-2-2-3', '0905.1142-3-2-3'], ['0905.1142-2-2-3', '0905.1142-3-2-4'], ['0905.1142-2-11-1', '0905.1142-3-15-1'], ['0905.1142-2-3-1', '0905.1142-3-3-1'], ['0905.1142-2-44-0', '0905.1142-3-74-0'], ['0905.1142-2-44-14', '0905.1142-3-74-8'], ['0905.1142-2-44-22', '0905.1142-3-74-16'], ['0905.1142-2-4-4', '0905.1142-3-4-4'], ['0905.1142-1-11-1', '0905.1142-2-11-0'], ['0905.1142-1-11-1', '0905.1142-2-11-1'], ['0905.1142-1-11-2', '0905.1142-2-11-0'], ['0905.1142-1-11-2', '0905.1142-2-11-1'], ['0905.1142-1-11-3', '0905.1142-2-13-0'], ['0905.1142-2-8-2', '0905.1142-3-8-4'], ['0905.1142-2-8-6', '0905.1142-3-8-6'], ['0905.1142-2-8-9', '0905.1142-3-9-0'], ['0905.1142-2-40-3', '0905.1142-3-70-0'], ['0905.1142-2-40-4', '0905.1142-3-70-1'], ['0905.1142-2-40-4', '0905.1142-3-70-2']]

[['0905.1142-1-31-1', '0905.1142-2-33-1'], ['0905.1142-1-31-2', '0905.1142-2-33-2'], ['0905.1142-1-33-0', '0905.1142-2-35-0'], ['0905.1142-1-33-2', '0905.1142-2-35-2'], ['0905.1142-1-33-3', '0905.1142-2-35-3'], ['0905.1142-1-33-4', '0905.1142-2-35-4'], ['0905.1142-1-33-5', '0905.1142-2-35-5'], ['0905.1142-1-33-6', '0905.1142-2-35-6'], ['0905.1142-1-33-7', '0905.1142-2-35-7'], ['0905.1142-1-33-8', '0905.1142-2-35-8'], ['0905.1142-1-33-9', '0905.1142-2-35-9'], ['0905.1142-1-33-10', '0905.1142-2-35-10'], ['0905.1142-1-33-11', '0905.1142-2-35-11'], ['0905.1142-1-33-12', '0905.1142-2-35-12'], ['0905.1142-1-33-13', '0905.1142-2-35-13'], ['0905.1142-1-33-14', '0905.1142-2-35-14'], ['0905.1142-1-33-15', '0905.1142-2-35-15'], ['0905.1142-1-33-16', '0905.1142-2-35-16'], ['0905.1142-1-6-1', '0905.1142-2-6-1'], ['0905.1142-1-6-4', '0905.1142-2-6-5'], ['0905.1142-1-42-0', '0905.1142-2-44-0'], ['0905.1142-1-42-1', '0905.1142-2-44-2'], ['0905.1142-1-42-2', '0905.1142-2-44-3'], ['0905.1142-1-42-4', '0905.1142-2-44-5'], ['0905.1142-1-42-5', '0905.1142-2-44-6'], ['0905.1142-1-42-6', '0905.1142-2-44-7'], ['0905.1142-1-42-7', '0905.1142-2-44-8'], ['0905.1142-1-42-8', '0905.1142-2-44-9'], ['0905.1142-1-42-10', '0905.1142-2-44-11'], ['0905.1142-1-42-11', '0905.1142-2-44-12'], ['0905.1142-1-42-13', '0905.1142-2-44-14'], ['0905.1142-1-42-14', '0905.1142-2-44-15'], ['0905.1142-1-42-15', '0905.1142-2-44-16'], ['0905.1142-1-42-17', '0905.1142-2-44-18'], ['0905.1142-1-42-18', '0905.1142-2-44-19'], ['0905.1142-1-42-19', '0905.1142-2-44-20'], ['0905.1142-1-42-20', '0905.1142-2-44-21'], ['0905.1142-1-42-21', '0905.1142-2-44-22'], ['0905.1142-1-42-22', '0905.1142-2-44-23'], ['0905.1142-1-15-0', '0905.1142-2-17-0'], ['0905.1142-1-15-1', '0905.1142-2-17-1'], ['0905.1142-1-15-2', '0905.1142-2-17-2'], ['0905.1142-1-15-4', '0905.1142-2-17-4'], ['0905.1142-1-15-5', '0905.1142-2-17-5'], ['0905.1142-1-41-0', '0905.1142-2-43-0'], ['0905.1142-1-3-0', '0905.1142-2-3-0'], ['0905.1142-1-3-2', '0905.1142-2-3-3'], ['0905.1142-1-34-0', '0905.1142-2-36-0'], ['0905.1142-1-34-1', '0905.1142-2-36-1'], ['0905.1142-1-34-2', '0905.1142-2-36-2'], ['0905.1142-1-34-3', '0905.1142-2-36-3'], ['0905.1142-1-34-4', '0905.1142-2-36-4'], ['0905.1142-1-34-5', '0905.1142-2-36-5'], ['0905.1142-1-34-6', '0905.1142-2-36-6'], ['0905.1142-1-34-7', '0905.1142-2-36-7'], ['0905.1142-1-34-8', '0905.1142-2-36-8'], ['0905.1142-1-34-10', '0905.1142-2-36-10'], ['0905.1142-1-34-11', '0905.1142-2-36-11'], ['0905.1142-1-34-12', '0905.1142-2-36-12'], ['0905.1142-1-17-0', '0905.1142-2-19-0'], ['0905.1142-1-17-1', '0905.1142-2-19-1'], ['0905.1142-1-17-2', '0905.1142-2-19-2'], ['0905.1142-1-17-3', '0905.1142-2-19-3'], ['0905.1142-1-29-0', '0905.1142-2-31-0'], ['0905.1142-1-21-0', '0905.1142-2-23-0'], ['0905.1142-1-21-1', '0905.1142-2-23-1'], ['0905.1142-1-21-2', '0905.1142-2-23-2'], ['0905.1142-1-21-3', '0905.1142-2-23-3'], ['0905.1142-1-21-4', '0905.1142-2-23-4'], ['0905.1142-1-21-5', '0905.1142-2-23-5'], ['0905.1142-1-21-6', '0905.1142-2-23-6'], ['0905.1142-1-28-0', '0905.1142-2-30-0'], ['0905.1142-1-28-1', '0905.1142-2-30-1'], ['0905.1142-1-28-3', '0905.1142-2-30-3'], ['0905.1142-1-4-0', '0905.1142-2-4-0'], ['0905.1142-1-25-2', '0905.1142-2-27-2'], ['0905.1142-1-25-3', '0905.1142-2-27-3'], ['0905.1142-1-23-0', '0905.1142-2-25-0'], ['0905.1142-1-23-1', '0905.1142-2-25-1'], ['0905.1142-1-23-2', '0905.1142-2-25-2'], ['0905.1142-1-23-3', '0905.1142-2-25-3'], ['0905.1142-1-23-5', '0905.1142-2-25-5'], ['0905.1142-1-19-0', '0905.1142-2-21-0'], ['0905.1142-1-39-0', '0905.1142-2-41-0'], ['0905.1142-1-39-1', '0905.1142-2-41-2'], ['0905.1142-1-39-2', '0905.1142-2-41-3'], ['0905.1142-1-39-3', '0905.1142-2-41-1'], ['0905.1142-1-39-4', '0905.1142-2-41-4'], ['0905.1142-1-10-0', '0905.1142-2-10-0'], ['0905.1142-1-10-1', '0905.1142-2-10-1'], ['0905.1142-1-10-2', '0905.1142-2-10-2'], ['0905.1142-1-10-3', '0905.1142-2-10-3'], ['0905.1142-1-10-4', '0905.1142-2-10-4'], ['0905.1142-1-10-5', '0905.1142-2-10-5'], ['0905.1142-1-10-6', '0905.1142-2-10-6'], ['0905.1142-1-10-7', '0905.1142-2-10-7'], ['0905.1142-1-13-2', '0905.1142-2-15-2'], ['0905.1142-1-13-3', '0905.1142-2-15-3'], ['0905.1142-1-13-4', '0905.1142-2-15-4'], ['0905.1142-1-0-0', '0905.1142-2-0-0'], ['0905.1142-1-0-1', '0905.1142-2-0-1'], ['0905.1142-1-0-2', '0905.1142-2-0-2'], ['0905.1142-1-0-3', '0905.1142-2-0-3'], ['0905.1142-1-0-4', '0905.1142-2-0-4'], ['0905.1142-1-0-5', '0905.1142-2-0-5'], ['0905.1142-1-0-6', '0905.1142-2-0-6'], ['0905.1142-1-0-7', '0905.1142-2-0-7'], ['0905.1142-1-16-0', '0905.1142-2-18-0'], ['0905.1142-1-16-1', '0905.1142-2-18-1'], ['0905.1142-1-5-0', '0905.1142-2-5-0'], ['0905.1142-1-5-1', '0905.1142-2-5-1'], ['0905.1142-1-5-2', '0905.1142-2-5-2'], ['0905.1142-1-12-0', '0905.1142-2-14-0'], ['0905.1142-1-8-0', '0905.1142-2-8-0'], ['0905.1142-1-8-1', '0905.1142-2-8-1'], ['0905.1142-1-8-2', '0905.1142-2-8-2'], ['0905.1142-1-8-4', '0905.1142-2-8-4'], ['0905.1142-1-8-5', '0905.1142-2-8-5'], ['0905.1142-1-8-6', '0905.1142-2-8-6'], ['0905.1142-1-8-7', '0905.1142-2-8-7'], ['0905.1142-1-8-8', '0905.1142-2-8-8'], ['0905.1142-1-7-0', '0905.1142-2-7-0'], ['0905.1142-1-7-2', '0905.1142-2-7-2'], ['0905.1142-1-7-3', '0905.1142-2-7-3'], ['0905.1142-1-2-1', '0905.1142-2-2-1'], ['0905.1142-1-38-0', '0905.1142-2-40-0'], ['0905.1142-1-38-1', '0905.1142-2-40-1'], ['0905.1142-1-38-3', '0905.1142-2-40-3'], ['0905.1142-1-38-4', '0905.1142-2-40-4'], ['0905.1142-1-35-0', '0905.1142-2-37-0'], ['0905.1142-1-35-1', '0905.1142-2-37-1'], ['0905.1142-1-35-4', '0905.1142-2-37-4'], ['0905.1142-1-35-5', '0905.1142-2-37-5'], ['0905.1142-2-17-0', '0905.1142-3-45-0'], ['0905.1142-2-17-1', '0905.1142-3-45-1'], ['0905.1142-2-17-3', '0905.1142-3-45-3'], ['0905.1142-2-17-4', '0905.1142-3-45-4'], ['0905.1142-2-33-1', '0905.1142-3-63-0'], ['0905.1142-2-33-2', '0905.1142-3-63-1'], ['0905.1142-2-18-0', '0905.1142-3-46-0'], ['0905.1142-2-18-1', '0905.1142-3-46-1'], ['0905.1142-2-25-0', '0905.1142-3-52-0'], ['0905.1142-2-25-1', '0905.1142-3-52-1'], ['0905.1142-2-25-2', '0905.1142-3-52-2'], ['0905.1142-2-25-3', '0905.1142-3-52-3'], ['0905.1142-2-25-5', '0905.1142-3-52-5'], ['0905.1142-2-2-0', '0905.1142-3-2-0'], ['0905.1142-2-2-1', '0905.1142-3-2-1'], ['0905.1142-2-15-0', '0905.1142-3-23-0'], ['0905.1142-2-41-0', '0905.1142-3-71-0'], ['0905.1142-2-41-2', '0905.1142-3-71-2'], ['0905.1142-2-41-3', '0905.1142-3-71-3'], ['0905.1142-2-41-4', '0905.1142-3-71-4'], ['0905.1142-2-41-5', '0905.1142-3-71-5'], ['0905.1142-2-23-0', '0905.1142-3-51-0'], ['0905.1142-2-23-1', '0905.1142-3-51-1'], ['0905.1142-2-23-2', '0905.1142-3-51-2'], ['0905.1142-2-23-3', '0905.1142-3-51-3'], ['0905.1142-2-23-4', '0905.1142-3-51-4'], ['0905.1142-2-23-5', '0905.1142-3-51-5'], ['0905.1142-2-23-6', '0905.1142-3-51-6'], ['0905.1142-2-30-0', '0905.1142-3-60-0'], ['0905.1142-2-30-1', '0905.1142-3-60-1'], ['0905.1142-2-30-3', '0905.1142-3-60-3'], ['0905.1142-2-10-6', '0905.1142-3-14-3'], ['0905.1142-2-3-0', '0905.1142-3-3-0'], ['0905.1142-2-3-4', '0905.1142-3-3-4'], ['0905.1142-2-31-0', '0905.1142-3-61-0'], ['0905.1142-2-36-0', '0905.1142-3-66-0'], ['0905.1142-2-36-3', '0905.1142-3-66-1'], ['0905.1142-2-36-4', '0905.1142-3-66-2'], ['0905.1142-2-36-5', '0905.1142-3-66-3'], ['0905.1142-2-36-6', '0905.1142-3-66-4'], ['0905.1142-2-36-7', '0905.1142-3-66-5'], ['0905.1142-2-36-8', '0905.1142-3-66-6'], ['0905.1142-2-36-10', '0905.1142-3-66-8'], ['0905.1142-2-36-11', '0905.1142-3-66-9'], ['0905.1142-2-36-12', '0905.1142-3-66-10'], ['0905.1142-2-35-0', '0905.1142-3-65-0'], ['0905.1142-2-35-2', '0905.1142-3-65-2'], ['0905.1142-2-35-3', '0905.1142-3-65-3'], ['0905.1142-2-35-4', '0905.1142-3-65-4'], ['0905.1142-2-35-5', '0905.1142-3-65-5'], ['0905.1142-2-35-6', '0905.1142-3-65-6'], ['0905.1142-2-35-7', '0905.1142-3-65-7'], ['0905.1142-2-35-8', '0905.1142-3-65-8'], ['0905.1142-2-35-9', '0905.1142-3-65-9'], ['0905.1142-2-35-10', '0905.1142-3-65-10'], ['0905.1142-2-35-11', '0905.1142-3-65-11'], ['0905.1142-2-35-12', '0905.1142-3-65-12'], ['0905.1142-2-35-13', '0905.1142-3-65-13'], ['0905.1142-2-35-14', '0905.1142-3-65-14'], ['0905.1142-2-35-15', '0905.1142-3-65-15'], ['0905.1142-2-35-16', '0905.1142-3-65-16'], ['0905.1142-2-37-0', '0905.1142-3-67-0'], ['0905.1142-2-37-1', '0905.1142-3-67-1'], ['0905.1142-2-37-4', '0905.1142-3-67-4'], ['0905.1142-2-19-0', '0905.1142-3-47-0'], ['0905.1142-2-19-1', '0905.1142-3-47-1'], ['0905.1142-2-19-2', '0905.1142-3-47-2'], ['0905.1142-2-19-3', '0905.1142-3-47-3'], ['0905.1142-2-19-4', '0905.1142-3-47-4'], ['0905.1142-2-19-5', '0905.1142-3-47-5'], ['0905.1142-2-7-1', '0905.1142-3-7-1'], ['0905.1142-2-0-0', '0905.1142-3-0-0'], ['0905.1142-2-0-2', '0905.1142-3-0-2'], ['0905.1142-2-0-3', '0905.1142-3-0-3'], ['0905.1142-2-0-4', '0905.1142-3-0-4'], ['0905.1142-2-0-7', '0905.1142-3-0-7'], ['0905.1142-2-12-2', '0905.1142-3-16-2'], ['0905.1142-2-44-16', '0905.1142-3-74-10'], ['0905.1142-2-44-17', '0905.1142-3-74-11'], ['0905.1142-2-44-18', '0905.1142-3-74-12'], ['0905.1142-2-44-19', '0905.1142-3-74-13'], ['0905.1142-2-44-20', '0905.1142-3-74-14'], ['0905.1142-2-44-21', '0905.1142-3-74-15'], ['0905.1142-2-4-1', '0905.1142-3-4-1'], ['0905.1142-2-4-2', '0905.1142-3-4-2'], ['0905.1142-2-6-0', '0905.1142-3-6-0'], ['0905.1142-2-6-1', '0905.1142-3-6-4'], ['0905.1142-2-6-3', '0905.1142-3-6-6'], ['0905.1142-2-6-4', '0905.1142-3-6-7']]

[['0905.1142-1-6-0', '0905.1142-2-6-0'], ['0905.1142-1-6-2', '0905.1142-2-6-2'], ['0905.1142-1-15-3', '0905.1142-2-17-3'], ['0905.1142-1-3-1', '0905.1142-2-3-1'], ['0905.1142-1-3-3', '0905.1142-2-3-4'], ['0905.1142-1-17-5', '0905.1142-2-19-5'], ['0905.1142-1-4-1', '0905.1142-2-4-1'], ['0905.1142-1-4-2', '0905.1142-2-4-2'], ['0905.1142-1-4-3', '0905.1142-2-4-3'], ['0905.1142-1-4-4', '0905.1142-2-4-4'], ['0905.1142-1-13-0', '0905.1142-2-15-0'], ['0905.1142-1-13-1', '0905.1142-2-15-1'], ['0905.1142-1-8-3', '0905.1142-2-8-3'], ['0905.1142-1-7-1', '0905.1142-2-7-1'], ['0905.1142-1-2-0', '0905.1142-2-2-0'], ['0905.1142-1-2-2', '0905.1142-2-2-2'], ['0905.1142-1-38-2', '0905.1142-2-40-2'], ['0905.1142-2-17-5', '0905.1142-3-45-5'], ['0905.1142-2-2-2', '0905.1142-3-2-2'], ['0905.1142-2-15-1', '0905.1142-3-23-2'], ['0905.1142-2-15-2', '0905.1142-3-23-3'], ['0905.1142-2-15-3', '0905.1142-3-23-4'], ['0905.1142-2-15-4', '0905.1142-3-23-5'], ['0905.1142-2-11-0', '0905.1142-3-15-0'], ['0905.1142-2-13-0', '0905.1142-3-17-0'], ['0905.1142-2-14-0', '0905.1142-3-20-0'], ['0905.1142-2-21-0', '0905.1142-3-49-1'], ['0905.1142-2-10-7', '0905.1142-3-14-4'], ['0905.1142-2-3-2', '0905.1142-3-3-2'], ['0905.1142-2-3-3', '0905.1142-3-3-3'], ['0905.1142-2-43-0', '0905.1142-3-73-0'], ['0905.1142-2-7-0', '0905.1142-3-7-0'], ['0905.1142-2-7-3', '0905.1142-3-7-2'], ['0905.1142-2-5-1', '0905.1142-3-5-1'], ['0905.1142-2-5-2', '0905.1142-3-5-2'], ['0905.1142-2-0-1', '0905.1142-3-0-1'], ['0905.1142-2-0-6', '0905.1142-3-0-5'], ['0905.1142-2-12-1', '0905.1142-3-16-1'], ['0905.1142-2-12-3', '0905.1142-3-16-3'], ['0905.1142-2-44-1', '0905.1142-3-74-1'], ['0905.1142-2-44-15', '0905.1142-3-74-9'], ['0905.1142-2-44-23', '0905.1142-3-74-17'], ['0905.1142-2-4-0', '0905.1142-3-4-0'], ['0905.1142-2-4-3', '0905.1142-3-4-3'], ['0905.1142-2-6-2', '0905.1142-3-6-5'], ['0905.1142-2-8-1', '0905.1142-3-8-0'], ['0905.1142-2-40-0', '0905.1142-3-69-0'], ['0905.1142-2-40-1', '0905.1142-3-69-1'], ['0905.1142-2-27-2', '0905.1142-3-53-0']]

[]

[['0905.1142-1-6-3', '0905.1142-2-6-3'], ['0905.1142-1-42-16', '0905.1142-2-44-17'], ['0905.1142-1-17-4', '0905.1142-2-19-4'], ['0905.1142-1-2-3', '0905.1142-2-2-3'], ['0905.1142-1-2-4', '0905.1142-2-2-3'], ['0905.1142-2-17-2', '0905.1142-3-45-2'], ['0905.1142-2-2-3', '0905.1142-3-2-3'], ['0905.1142-2-2-3', '0905.1142-3-2-4'], ['0905.1142-2-11-1', '0905.1142-3-15-1'], ['0905.1142-2-3-1', '0905.1142-3-3-1'], ['0905.1142-2-44-0', '0905.1142-3-74-0'], ['0905.1142-2-44-14', '0905.1142-3-74-8'], ['0905.1142-2-44-22', '0905.1142-3-74-16'], ['0905.1142-2-4-4', '0905.1142-3-4-4'], ['0905.1142-1-11-1', '0905.1142-2-11-0'], ['0905.1142-1-11-1', '0905.1142-2-11-1'], ['0905.1142-1-11-2', '0905.1142-2-11-0'], ['0905.1142-1-11-2', '0905.1142-2-11-1'], ['0905.1142-1-11-3', '0905.1142-2-13-0'], ['0905.1142-2-8-2', '0905.1142-3-8-4'], ['0905.1142-2-8-6', '0905.1142-3-8-6'], ['0905.1142-2-8-9', '0905.1142-3-9-0'], ['0905.1142-2-40-3', '0905.1142-3-70-0'], ['0905.1142-2-40-4', '0905.1142-3-70-1'], ['0905.1142-2-40-4', '0905.1142-3-70-2']]

[]

['0905.1142-1-8-9', '0905.1142-1-9-0', '0905.1142-1-9-1', '0905.1142-1-9-2', '0905.1142-1-18-0', '0905.1142-1-20-0', '0905.1142-1-22-0', '0905.1142-1-23-4', '0905.1142-1-24-0', '0905.1142-1-25-0', '0905.1142-1-25-1', '0905.1142-1-26-0', '0905.1142-1-28-2', '0905.1142-1-30-0', '0905.1142-1-31-0', '0905.1142-1-32-0', '0905.1142-1-33-1', '0905.1142-1-34-9', '0905.1142-1-35-2', '0905.1142-1-35-3', '0905.1142-1-36-0', '0905.1142-1-36-1', '0905.1142-1-39-7', '0905.1142-1-42-3', '0905.1142-1-42-9', '0905.1142-1-42-12', '0905.1142-1-43-0', '0905.1142-1-43-1', '0905.1142-1-43-2', '0905.1142-1-43-3', '0905.1142-1-43-4', '0905.1142-1-43-5', '0905.1142-1-43-6', '0905.1142-2-9-0', '0905.1142-2-9-1', '0905.1142-2-9-2', '0905.1142-2-20-0', '0905.1142-2-22-0', '0905.1142-2-24-0', '0905.1142-2-25-4', '0905.1142-2-26-0', '0905.1142-2-27-0', '0905.1142-2-27-1', '0905.1142-2-28-0', '0905.1142-2-30-2', '0905.1142-2-32-0', '0905.1142-2-33-0', '0905.1142-2-34-0', '0905.1142-2-35-1', '0905.1142-2-36-9', '0905.1142-2-37-2', '0905.1142-2-37-3', '0905.1142-2-38-0', '0905.1142-2-38-1', '0905.1142-2-44-4', '0905.1142-2-44-10', '0905.1142-2-44-13', '0905.1142-2-45-0', '0905.1142-2-45-1', '0905.1142-2-45-2', '0905.1142-2-45-3', '0905.1142-2-45-4', '0905.1142-2-45-5', '0905.1142-2-45-6', '0905.1142-3-6-2', '0905.1142-3-9-1', '0905.1142-3-10-0', '0905.1142-3-11-0', '0905.1142-3-14-2', '0905.1142-3-19-0', '0905.1142-3-27-2', '0905.1142-3-27-7', '0905.1142-3-33-1', '0905.1142-3-33-2', '0905.1142-3-40-0', '0905.1142-3-40-4', '0905.1142-3-40-7', '0905.1142-3-43-0', '0905.1142-3-43-4', '0905.1142-3-48-0', '0905.1142-3-49-0', '0905.1142-3-50-0', '0905.1142-3-52-4', '0905.1142-3-54-0', '0905.1142-3-55-0', '0905.1142-3-55-1', '0905.1142-3-57-0', '0905.1142-3-58-1', '0905.1142-3-60-2', '0905.1142-3-62-0', '0905.1142-3-64-0', '0905.1142-3-65-1', '0905.1142-3-66-7', '0905.1142-3-67-2', '0905.1142-3-67-3', '0905.1142-3-71-1', '0905.1142-3-74-3', '0905.1142-3-74-7', '0905.1142-3-75-0', '0905.1142-3-75-1', '0905.1142-3-75-2', '0905.1142-3-75-3', '0905.1142-3-75-4', '0905.1142-3-76-0', '0905.1142-3-76-1']

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

https://arxiv.org/abs/0905.1142

{'0905.1142-3-0-0': 'We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement.', '0905.1142-3-0-1': 'The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the zero flux boundary condition.', '0905.1142-3-0-2': 'Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl.', '0905.1142-3-0-3': 'Math., 68(5):1304-1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter [MATH].', '0905.1142-3-0-4': 'In this article, we show that for the well-posedness of the microscopic FENE model ([MATH]) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function.', '0905.1142-3-0-5': 'Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space.', '0905.1142-3-0-6': 'Moreover, such a condition still ensures that the distribution is a probability density.', '0905.1142-3-0-7': 'The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.', '0905.1142-3-1-0': '# Introduction', '0905.1142-3-2-0': 'It is well-known that the following system coupling incompressible Navier-Stokes equation for the macroscopic velocity field [MATH] and the Fokker-Planck equation for the probability density function [MATH] describes diluted solutions of polymeric liquids with noninteracting polymer chains [EQUATION] where [MATH] is the macroscopic Eulerian coordinate and [MATH] is the microscopic molecular configuration variable.', '0905.1142-3-2-1': 'In this model, a polymer is idealized as an elastic dumbbell consisting of two beads joined by a spring that can be modeled by a vector [MATH] (see e.g [CITATION]).', '0905.1142-3-2-2': 'In the Navier-Stokes equation [REF], [MATH] is hydrostatic pressure, [MATH] is the kinematic viscosity coefficient, and [MATH] is a tensor representing the polymer contribution to stress, [EQUATION] where [MATH] is the elastic spring potential and [MATH] is the polymer density constant.', '0905.1142-3-2-3': 'In the Fokker-Planck equation [REF], [MATH] is the friction coefficient of the dumbbell beads, [MATH] is the absolute temperature, and [MATH] is the Boltzmann constant.', '0905.1142-3-2-4': 'Notice that the Fokker-Planck equation can be written as a stochastic differential equation (see [CITATION]).', '0905.1142-3-3-0': 'One of the simplest model is the Hookean model in which the potential [MATH] is given by [EQUATION] where [MATH] is the elasticity constant.', '0905.1142-3-3-1': 'A more realistic model is the finite extensible nonlinear elasticity (FENE) model with [EQUATION]', '0905.1142-3-3-2': 'Here [MATH] is the ball with center [MATH] and radius [MATH] which denotes the maximum dumbbell extension.', '0905.1142-3-3-3': 'In this work we shall focus our attention on the potential [REF] and the case [MATH], which is known to contain the parameter range of physical interest.', '0905.1142-3-3-4': 'We refer the reader to [CITATION] for a comprehensive survey of the physical background.', '0905.1142-3-4-0': 'In past years the well-posedness of the FENE model [REF]-[REF] has been studied intensively in several aspects.', '0905.1142-3-4-1': 'For local well-posedness of strong solutions we refer the reader to [CITATION] for the FENE model (in the setting where the Fokker-Planck equation is formulated by a stochastic differential equation) with [MATH] or sometime [MATH], [CITATION] for a polynomial force and [CITATION] for the FENE model with [MATH].', '0905.1142-3-4-2': 'For a preliminary study on some related coupled PDE systems, we refer to the earlier work [CITATION] (however, the FENE model was not addressed there).', '0905.1142-3-4-3': 'Moreover, the authors in [CITATION] proved global existence of smooth solutions near equilibrium under some restrictions on the potential; further developments were made in subsequent works [CITATION].', '0905.1142-3-4-4': 'More recently, N. Masmoudi [CITATION] proved global existence for the FENE model [REF]-[REF] for a class of potentials ([REF]) with [MATH] assuming that the data is small, or the model is restricted to the co-rotational case in dimension two.', '0905.1142-3-5-0': 'For results concerning the existence of weak solutions to coupled Navier-Stokes-Fokker-Planck systems and a detailed survey of related literature we refer to [CITATION].', '0905.1142-3-5-1': 'For an earlier result on existence of weak solutions, we refer to [CITATION] for the Fokker-Planck equation alone with [MATH].', '0905.1142-3-5-2': 'On the other hand, the authors in [CITATION], investigated the long-time behavior of both Hookean models and FENE models in several special flows in a bounded domain with suitable boundary conditions.', '0905.1142-3-6-0': 'The complexity with the FENE potential lies mainly with the singularity of the equation at the boundary.', '0905.1142-3-6-1': 'To overcome this difficulty, several transformations relating to the equilibrium solution have been introduced in literature.', '0905.1142-3-6-2': 'See, e.g. [CITATION].', '0905.1142-3-6-3': 'A detailed discussion will be given in Section 2.', '0905.1142-3-6-4': 'In [CITATION], C. Liu and H. Liu closely examined the necessity of Dirichlet boundary conditions for the microscopic FENE model.', '0905.1142-3-6-5': 'By the method of the Fichera function they were able to conclude that [MATH] is a threshold in the sense that for [MATH] any preassigned boundary value of the ratio of the distribution and the equilibrium will become redundant, and for [MATH] that value has to be a priori given.', '0905.1142-3-6-6': 'For the microscopic FENE model, singularity in the potential requires at least the zero Dirichlet boundary condition [EQUATION]', '0905.1142-3-6-7': 'This is consistent with the result in [CITATION], which states that the stochastic solution trajectory does not reach the boundary almost surely.', '0905.1142-3-7-0': 'The boundary issue for the underlying FENE model is fundamental, and our main quest in this paper is whether one can identify a sharp boundary requirement so that both existence and uniqueness of a global weak solution to the microscopic FENE model can be established, also the solution remains a probability density.', '0905.1142-3-7-1': 'The answer is positive, and we claim that [MATH] must satisfy the following boundary condition [EQUATION] where [MATH] denotes the distance function from [MATH] to the boundary [MATH].', '0905.1142-3-7-2': 'Our claim is supported by our main results: the global well-posedness for the Fokker-Planck equation stated in Theorem [REF], the property of the solution as a probability density given in Proposition [REF], and the sharpness of ([REF]) stated in Proposition [REF].', '0905.1142-3-8-0': 'In this article, we focus on the underlying Fokker-Planck equation [REF] alone.', '0905.1142-3-8-1': 'Let [MATH] be the velocity field governed by [REF] and [REF].', '0905.1142-3-8-2': 'We assume that this underlying velocity field is smooth, then a simplification can be made by considering the microscopic model ([REF]) along a particle path defined as [EQUATION]', '0905.1142-3-8-3': 'For each fixed [MATH], the distribution function [MATH] solves [EQUATION]', '0905.1142-3-8-4': 'By a suitable scaling ([CITATION]), and denote [MATH] still by [MATH], we arrive at the following equation [EQUATION]', '0905.1142-3-8-5': 'Here, [MATH] and [MATH] is a bounded matrix such that [MATH].', '0905.1142-3-8-6': 'We omit [MATH] from [MATH] in ([REF]) for notational convenience.', '0905.1142-3-8-7': 'In this paper we prove well-posedness of ([REF]) subject to some side conditions.', '0905.1142-3-8-8': 'The well-posedness of the full coupled system [REF]-[REF] is the subject of a forthcoming paper [CITATION].', '0905.1142-3-9-0': 'A weak solution of the Fokker-Planck equation [REF] with the initial condition [EQUATION] and boundary requirement [REF] is defined in the following.', '0905.1142-3-9-1': 'We say [MATH] is a weak solution of [REF], [REF], and [REF] if the following conditions are satisfied:', '0905.1142-3-10-0': 'For an arbitrary subdomain [MATH] of [MATH] such that [MATH] and almost all [MATH],', '0905.1142-3-11-0': '[(1)] [MATH], [(2)] for any [MATH], [EQUATION] [(3)] [EQUATION] [(4)] and for [MATH], [EQUATION]', '0905.1142-3-11-1': 'Note that [REF] makes sense since [MATH] implied by (1) above, and also [MATH] is well defined in [MATH] by the standard trace theorem.', '0905.1142-3-12-0': 'Regarding the weak solution defined above, several remarks are in order.', '0905.1142-3-13-0': 'In order to establish an existence theorem, we now identify a subspace of [MATH] with an appropriate weight to incorporate boundary requirement ([REF]).', '0905.1142-3-13-1': 'For simplicity, we consider the case with trivial velocity field such that [MATH], then equation ([REF]) becomes [EQUATION]', '0905.1142-3-13-2': 'It follows from this conservative form that the only equilibrium solution [MATH] must be a multiplier of [MATH], i.e. [EQUATION] where [MATH] is a normalization factor such that [MATH].', '0905.1142-3-14-0': 'We are interested in the case [EQUATION]', '0905.1142-3-14-1': 'In such a case [MATH] satisfies boundary requirement ([REF]).', '0905.1142-3-14-2': 'Moreover [EQUATION]', '0905.1142-3-14-3': 'Here, [MATH] with [EQUATION]', '0905.1142-3-14-4': 'Our main results are summarized in Theorem [REF], Proposition [REF] and Proposition [REF] below.', '0905.1142-3-15-0': '(i) If [EQUATION] then there exists a unique solution [MATH] of [REF], [REF], and [REF] in the sense of Definition [REF].', '0905.1142-3-15-1': 'Moreover, [EQUATION] (ii) For any [EQUATION] there exists at most one solution [MATH] .', '0905.1142-3-16-0': 'The proof of [MATH] will be done in Section 3 - 5.', '0905.1142-3-16-1': 'In order to prove [MATH] we assume that [MATH] are two weak solutions of the problem with arbitrary initial data [MATH].', '0905.1142-3-16-2': 'Then [MATH] solves [REF] with zero initial data which is in [MATH].', '0905.1142-3-16-3': 'From [REF] in [MATH] it follows that [MATH] in [MATH].', '0905.1142-3-17-0': 'We remark that the restriction on [MATH] in [REF] is essential to obtain the energy estimate [REF].', '0905.1142-3-18-0': 'The weak solution thus obtained is indeed a probability density.', '0905.1142-3-18-1': 'More precesely we have the following.', '0905.1142-3-19-0': 'This proposition will be proved in Section 2.', '0905.1142-3-20-0': 'The following proposition states that boundary condition [REF] is sharp for the uniqueness of the weak solution.', '0905.1142-3-21-0': 'In other words, Proposition [REF] implies that part [MATH] in Theorem [REF] would fail if boundary requirement [REF] were weaken so that near boundary the distribution approaches zero not faster than the distance function.', '0905.1142-3-22-0': 'The justification of sharpness follows from the existence of a Cauchy-Dirichlet problem for [MATH] defined by [EQUATION] with [MATH] being a class of functions properly constructed.', '0905.1142-3-23-0': 'This article is organized as follows.', '0905.1142-3-23-1': 'In Section 2, we prove Proposition [REF] and provide some preliminaries including: (1) several transformations used to handle the boundary difficulty, (2) equivalence of two weighted function spaces, and (3) the relation of our boundary condition to the natural flux boundary condition.', '0905.1142-3-23-2': 'In Section 3, we transform the Fokker-Planck equation to certain Cauchy-Dirichlet problem, named as [MATH]-problem, and define a weak solution of [MATH]-problem in a weighted Sobolev space.', '0905.1142-3-23-3': 'The well-posedness of the [MATH]-problem is shown in Section 4 by the Galerkin method and the Banach fixed point theorem.', '0905.1142-3-23-4': 'This leads to the well-posedness of the Fokker-Planck equation, Theorem [REF]; details of the proof are presented in Section 5.', '0905.1142-3-23-5': 'In Section 6, we construct non-trivial solutions for the Fokker-Planck equation described in Proposition [REF].', '0905.1142-3-24-0': '# Preliminaries', '0905.1142-3-25-0': '## Probability density', '0905.1142-3-26-0': 'With the definition of our weak solution given in Definition 1 we shall show that [MATH] has the usual properties of a probability density function (i.e. it is non-negative and has a unit integral over [MATH] for all [MATH] if it is so initially) - this is to prove Proposition [REF].', '0905.1142-3-27-0': 'Given [MATH] in [MATH] and [MATH] a.e. on [MATH], we define [MATH] for [MATH].', '0905.1142-3-27-1': 'Here [MATH] denotes the usual scaled mollifier.', '0905.1142-3-27-2': 'We have [EQUATION].', '0905.1142-3-27-3': 'Suppose that [MATH] is the weak solution of [REF], [REF], and [REF] subject to initial condition [MATH].', '0905.1142-3-27-4': 'Then, for any [MATH] and [MATH], [EQUATION]', '0905.1142-3-27-5': 'Hence for justification of the conservation of polymers, it suffices to prove that [EQUATION]', '0905.1142-3-27-6': 'To do so, we take a test function [MATH] converging to [MATH] as [MATH] such that [EQUATION] and [EQUATION]', '0905.1142-3-27-7': 'From ([REF])', '0905.1142-3-28-0': 'and the fact that derivatives of [MATH] are supported in [MATH], we have [EQUATION]', '0905.1142-3-28-1': 'Applying the mean value theorem of the form [EQUATION] to the first term on the right of ([REF]) together with [REF], we obtain [EQUATION].', '0905.1142-3-28-2': 'Similarly the second term on the right of ([REF]) is bounded by [EQUATION].', '0905.1142-3-28-3': 'It follows from [REF] that the above two upper bounds converge to zero as [MATH] .', '0905.1142-3-29-0': 'Integration by parts in the last term in ([REF]) yields [EQUATION] which, in virtue of [MATH] on [MATH], is converging to zero as [MATH] as well.', '0905.1142-3-30-0': 'Due to Theorem [REF] and the initial condition [MATH], it follows that [MATH] is bounded in any [MATH] for [MATH].', '0905.1142-3-30-1': 'Thus, for any [MATH] [EQUATION]', '0905.1142-3-30-2': 'Using the estimate for [MATH] together with the boundedness of [MATH], we can send [MATH] to zero to obtain ([REF]) as claimed.', '0905.1142-3-31-0': 'We now turn to justify the positivity.', '0905.1142-3-31-1': 'Consider the transformation introduced in [CITATION] [EQUATION]', '0905.1142-3-31-2': 'Then [MATH] solves [EQUATION] where [EQUATION].', '0905.1142-3-31-3': 'Then for any [MATH], [MATH] is a classical solution in [MATH].', '0905.1142-3-31-4': 'It was shown in [CITATION] that there exist [MATH] and [MATH] so that [MATH].', '0905.1142-3-32-0': 'The maximum principle yields that [MATH] can not achieve a negative minimum at the interior points of [MATH].', '0905.1142-3-32-1': 'Thus the negative minimum of [MATH], if it exists, can only be attained on the parabolic boundary of the domain.', '0905.1142-3-33-0': 'From the transformation [REF] and the condition [MATH], it follows that the negative minimum of [MATH], if any, can only be attained at the initial time.', '0905.1142-3-33-1': 'Therefore [EQUATION]', '0905.1142-3-33-2': 'Now fix [MATH].', '0905.1142-3-33-3': 'For any [MATH] and [MATH] such that [MATH], [EQUATION]', '0905.1142-3-33-4': 'Here ([REF]) has been used to obtain the last inequality.', '0905.1142-3-33-5': 'Hence [EQUATION] which as [MATH] leads to [EQUATION]', '0905.1142-3-33-6': 'Since [MATH] and [MATH] are arbitrary, [MATH] almost everywhere on [MATH] for [MATH].', '0905.1142-3-33-7': 'The proof of Proposition [REF] is now complete.', '0905.1142-3-34-0': '## Transformations', '0905.1142-3-35-0': 'To overcome the difficulty caused by the boundary singularity, several transformations have been introduced in literature.', '0905.1142-3-35-1': 'With boundary condition ([REF]), in this work we introduce [EQUATION] to transform the Fokker-Planck equation to a degenerate parabolic equation with zero boundary condition (see details in Section 3).', '0905.1142-3-35-2': 'A widely accepted transformation is the ratio of the unknown to the equilibrium solution, i.e., [EQUATION].', '0905.1142-3-35-3': 'Such a transformation was used in [CITATION] to reformulate the Fokker-Planck equation, and examine whether a Dirichlet type boundary condition is necessary.', '0905.1142-3-36-0': 'A third transformation is [EQUATION].', '0905.1142-3-36-1': 'This was used in [CITATION] to remove the singularity at the boundary in the resulting equation.', '0905.1142-3-36-2': 'It was also used in [CITATION] to formulate a weak formulation of [MATH] for discretization using a spectral Galerkin approximation.', '0905.1142-3-37-0': 'Another transformation defined by [EQUATION] with [MATH] and [MATH] is said to also lead to a well-posed problem.', '0905.1142-3-37-1': 'The minimum value of the function [MATH] is attained at [MATH], yielding the maximum range of [MATH] values, [MATH].', '0905.1142-3-37-2': 'This transformation was proposed in [CITATION] in the special case [MATH] and [MATH], where these values were chosen on the basis of numerical experiments in two and three dimensions, respectively.', '0905.1142-3-37-3': 'We note that our transformation [MATH] corresponds to [MATH], but not limited by [MATH].', '0905.1142-3-38-0': '## Weighted Sobolev spaces In contrast to the standard weighted Sobolev space [MATH] used in this work, the following weighted function space [EQUATION] is well known in literature for Fokker-Planck equations with FENE potentials, see e.g. [CITATION].', '0905.1142-3-38-1': 'We now show their equivalence as long as [MATH].', '0905.1142-3-39-0': 'The key estimate we need to prove the equivalence is the embedding theorem stated in Lemma [REF].', '0905.1142-3-40-0': 'Set [MATH].', '0905.1142-3-40-1': 'If [MATH], we use the relation [EQUATION].', '0905.1142-3-40-2': 'It is obvious that [EQUATION].', '0905.1142-3-40-3': 'Also the use of Lemma [REF] and the fact that [MATH] for [MATH] (see [CITATION]) give [EQUATION].', '0905.1142-3-40-4': 'Hence [MATH].', '0905.1142-3-40-5': 'If [MATH] we use the following identity [EQUATION].', '0905.1142-3-40-6': 'It is easy to see that [MATH]; also for [MATH] we have [EQUATION] by Lemma [REF].', '0905.1142-3-40-7': 'Thus [MATH].', '0905.1142-3-40-8': 'These together verify that [MATH] and [MATH] are equivalent when [MATH].', '0905.1142-3-41-0': '## Boundary conditions', '0905.1142-3-42-0': 'Granted certain smoothness of [MATH], e.g. [MATH], one may argue that our boundary condition ([REF]) is equivalent to the zero flux boundary condition ([REF]).', '0905.1142-3-43-0': 'Set [MATH] and [MATH].', '0905.1142-3-43-1': 'We calculate the flux [EQUATION]', '0905.1142-3-43-2': 'Due to singularity on boundary it is necessary that [MATH].', '0905.1142-3-43-3': 'For any point [MATH], let [MATH] be a point in [MATH] such that [MATH].', '0905.1142-3-43-4': 'Then [EQUATION].', '0905.1142-3-43-5': 'We thus have [EQUATION].', '0905.1142-3-43-6': 'For [MATH], this implies that [MATH] if and only if [EQUATION]', '0905.1142-3-44-0': '# Transformation of the microscopic FENE model', '0905.1142-3-45-0': 'In what follows we shall call the Fokker-Planck equation ([REF]) with initial condition ([REF]) and boundary condition ([REF]) as the Fokker-Planck-FENE (FPF) problem.', '0905.1142-3-45-1': 'We first formulate a time evolution equation from the FPF problem.', '0905.1142-3-45-2': 'Define [MATH] as [EQUATION]', '0905.1142-3-45-3': 'Then [REF] is transformed to [EQUATION] where [EQUATION]', '0905.1142-3-45-4': 'Setting a parameter [EQUATION] we rewrite [REF] as [EQUATION]', '0905.1142-3-45-5': 'The boundary condition [REF] implies that [MATH] satisfies a homogeneous boundary condition for almost all [MATH] since the distance function [MATH] and [MATH] are equivalent (see [REF]).', '0905.1142-3-46-0': 'The FPF problem is formally transformed to the following [MATH]-problem: [EQUATION]', '0905.1142-3-46-1': 'Here, [EQUATION] according to the transformation [REF].', '0905.1142-3-47-0': 'In order to define a weak solution of [MATH]-problem we introduce a weighted Sobolev space [MATH] for a nonnegative measurable function [MATH] as a set of measurable function [MATH] such that [EQUATION]', '0905.1142-3-47-1': 'Similarly, a weighted [MATH] can be defined.', '0905.1142-3-47-2': '[MATH] denotes a completion of [MATH] with [MATH].', '0905.1142-3-47-3': 'It is obvious that [MATH] and [MATH] are Hilbert spaces with the inner product [MATH] defined as [EQUATION] and [EQUATION]', '0905.1142-3-47-4': 'For notational convenient, we use [MATH] and [MATH] for [MATH] and [MATH] respectively.', '0905.1142-3-47-5': 'We also omit the domain [MATH] if it is obvious.', '0905.1142-3-48-0': 'Suppose that [MATH].', '0905.1142-3-49-0': '[(1)] If [MATH] for [MATH], then [EQUATION]', '0905.1142-3-49-1': 'If [MATH], we have the same inequality for [MATH] [(2)] If [MATH] for [MATH], then the trace map [EQUATION] is well defined, i.e. it is a bounded linear map.', '0905.1142-3-50-0': 'In particular, for [MATH] [EQUATION]', '0905.1142-3-51-0': 'In [CITATION](see also [CITATION]), it was proved that [EQUATION] provided [MATH] is Lipschitz continuous.', '0905.1142-3-51-1': 'Recall that [MATH] denotes the distance from [MATH] to the boundary of [MATH].', '0905.1142-3-51-2': '[REF] follows from [EQUATION]', '0905.1142-3-51-3': 'It is also known that the trace map [MATH] is well defined for [MATH] ([CITATION]).', '0905.1142-3-51-4': 'For [MATH] [EQUATION] since [MATH] for all [MATH].', '0905.1142-3-51-5': 'Therefore, [MATH] is well defined for [MATH].', '0905.1142-3-51-6': '[REF] is obvious from the definitions of the trace map and [MATH].', '0905.1142-3-52-0': 'We now define a weak solution to [MATH]-problem in a standard manner.', '0905.1142-3-52-1': 'Multiplication by a test function [MATH] to the equation [REF] and integration over [MATH] yield [EQUATION]', '0905.1142-3-52-2': 'This equation is well defined assuming that [MATH], the dual space of [MATH], and [MATH] due to the boundedness of [MATH] and Lemma [REF].', '0905.1142-3-52-3': 'Moreover, [EQUATION] implies [EQUATION]', '0905.1142-3-52-4': 'Thus [EQUATION]', '0905.1142-3-52-5': 'Here we identify [MATH] with its dual space.', '0905.1142-3-53-0': 'Let [MATH] denote the paring of a Hilbert space [MATH] with its dual space [MATH] and [EQUATION]', '0905.1142-3-53-1': 'We now describe the weak solution we are looking for.', '0905.1142-3-54-0': 'A function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem, [REF]-[REF], provided', '0905.1142-3-55-0': '[(1)] For each [MATH] and almost every [MATH], [EQUATION] [(2)] [MATH] in [MATH] sense.', '0905.1142-3-55-1': 'i.e. [EQUATION]', '0905.1142-3-55-2': 'The following energy estimate for [MATH] for fixed [MATH] is crucial.', '0905.1142-3-56-0': 'There exist positive constants [MATH] and [MATH] depending only on [MATH] and [MATH] such that [EQUATION]', '0905.1142-3-56-1': 'Let [MATH] in ([REF]) and apply the Schwarz inequality we arrive at the above estimate as desired.', '0905.1142-3-57-0': 'The well-posedness of the W-problem is stated in the following', '0905.1142-3-58-0': '[MATH]-problem, [REF]-[REF], is uniquely solvable in weak sense for [MATH].', '0905.1142-3-58-1': 'Furthermore, [EQUATION]', '0905.1142-3-58-2': 'A detailed proof will be presented in next section.', '0905.1142-3-59-0': '# Well-posedness for the transformed problem', '0905.1142-3-60-0': 'In this section, we show the well-posedness of the weak solution to [MATH]-problem.', '0905.1142-3-60-1': 'For this aim, we consider the following [MATH]-problem containing a non-homogeneous term [MATH].', '0905.1142-3-60-2': '[EQUATION]', '0905.1142-3-60-3': 'The weak solution of [MATH]-problem is defined similarly.', '0905.1142-3-61-0': 'We say a function [MATH] such that [EQUATION] is a weak solution of [MATH]-problem provided', '0905.1142-3-62-0': '[(1)] for each [MATH] and almost every [MATH] [EQUATION] [(2)] [MATH] in [MATH].', '0905.1142-3-63-0': 'We remark that [MATH] is finite for any [MATH] since [MATH] from [REF].', '0905.1142-3-63-1': 'Thus [MATH] can be understood as the [MATH] inner product although [MATH] may not belong to [MATH].', '0905.1142-3-64-0': 'The well-posedness for [MATH]-problem follows from the standard Galerkin method.', '0905.1142-3-65-0': 'For given [MATH] and [MATH], [MATH]-problem has a unique weak solution.', '0905.1142-3-65-1': 'Moreover, [EQUATION]', '0905.1142-3-65-2': 'We first construct an approximate solution in a finite-dimensional space.', '0905.1142-3-65-3': 'Let [MATH] be a basis of [MATH] and [MATH].', '0905.1142-3-65-4': 'The existence of such a basis can be verified from the fact that [MATH] is a dense subset of [MATH].', '0905.1142-3-65-5': 'Consider an approximation [MATH], where [MATH] satisfies [EQUATION]', '0905.1142-3-65-6': 'Since [REF] and [REF] form a system of linear differential equations, [MATH] is uniquely determined for each [MATH].', '0905.1142-3-65-7': 'We rewrite [REF] as [EQUATION]', '0905.1142-3-65-8': 'Apply [MATH] to [REF] and sum for [MATH], then for almost every [MATH] [EQUATION]', '0905.1142-3-65-9': 'From Lemma [REF], it follows that [EQUATION]', '0905.1142-3-65-10': 'From [REF], for any [MATH] [EQUATION]', '0905.1142-3-65-11': 'With [MATH], [REF] can be rewritten as [EQUATION] or [EQUATION]', '0905.1142-3-65-12': "Use Gronwall's inequality to obtain [EQUATION] where [MATH] is an appropriate constant which depends on [MATH], [MATH], [MATH] and [MATH].", '0905.1142-3-65-13': 'On the other hand, integration of [REF] from [MATH] to [MATH] together with above inequality yields [EQUATION]', '0905.1142-3-65-14': 'A similar argument to that in [CITATION] gives us the estimate for [MATH] as [EQUATION]', '0905.1142-3-65-15': 'Here we have used [REF] with [MATH] such that [MATH] and [REF].', '0905.1142-3-65-16': 'By passing to the limit as [MATH] and a standard argument (e.g. see [CITATION]), we have well-posedness for [MATH]-problem.', '0905.1142-3-66-0': 'Now, we introduce a linear map [MATH] to connect [MATH] and [MATH]-problems as [EQUATION]', '0905.1142-3-66-1': 'Since [MATH] is bounded, [EQUATION]', '0905.1142-3-66-2': 'Thus, [MATH] is well defined and [EQUATION]', '0905.1142-3-66-3': 'We define another map [MATH] such that [EQUATION]', '0905.1142-3-66-4': 'Here, [MATH] is given by the weak solution of [MATH]-problem with [EQUATION] and the initial condition [EQUATION]', '0905.1142-3-66-5': 'The map [MATH] is well defined from Lemma [REF] and the definition of [MATH].', '0905.1142-3-66-6': 'Now we show that [MATH] is a contraction mapping for sufficiently small [MATH].', '0905.1142-3-66-7': 'Let [EQUATION]', '0905.1142-3-66-8': 'From the energy estimate [REF], [EQUATION]', '0905.1142-3-66-9': 'Thus, [MATH] has a unique fixed point [MATH] in [MATH] and [MATH] solves [MATH]-problem in a weak sense in [MATH], if [MATH].', '0905.1142-3-66-10': 'We are able to continue this procedure to obtain the global well-posedness for the above constant [MATH] is independent of [MATH].', '0905.1142-3-67-0': 'For the fixed point [MATH], [REF] and the boundedness of [MATH] imply that for [MATH] [EQUATION]', '0905.1142-3-67-1': 'We select a small [MATH] such that [MATH].', '0905.1142-3-67-2': 'Then [EQUATION]', '0905.1142-3-67-3': 'Thus, [EQUATION] and [EQUATION]', '0905.1142-3-67-4': 'Continuing, after finitely many steps we obtain an energy estimation similar to [REF].', '0905.1142-3-67-5': 'The proof of Lemma [REF] is thus complete.', '0905.1142-3-68-0': '# Well-posedness for the FPF problem', '0905.1142-3-69-0': 'In Section 2, we transformed the FPF problem to W-problem formally, but it is not difficult to show that they are equivalent.', '0905.1142-3-69-1': 'Indeed, one can verify that boundary condition [REF] in the sense of [REF] for the FPF problem is equivalent to the null boundary condition for [MATH]-problem.', '0905.1142-3-70-0': 'For any test function [MATH], the weak solution formulation for [MATH] can be transformed to the weak solution formulation for [MATH], with [MATH] as the test function.', '0905.1142-3-70-1': 'This is valid since [MATH] is dense in [MATH].', '0905.1142-3-70-2': 'Such a justification can be reversed, hence the FPF problem and [MATH]-problem are equivalent.', '0905.1142-3-71-0': 'Now we seek the function space in which the weak solution [MATH] to the FPF problem belongs.', '0905.1142-3-71-1': 'Recall that [MATH].', '0905.1142-3-71-2': 'For fixed [MATH], [REF] implies [EQUATION]', '0905.1142-3-71-3': 'Also, for [MATH] we have [EQUATION]', '0905.1142-3-71-4': 'The estimate of the weak solution, [REF] follows from Lemma [REF] together with [REF]-[REF].', '0905.1142-3-71-5': 'This finishes the proof of (i) of Theorem [REF].', '0905.1142-3-72-0': '# Non-uniqueness', '0905.1142-3-73-0': 'In this section we show that ([REF]) is sharp in the sense that more solutions can be constructed if a weaker condition is imposed - this is to prove Proposition [REF].', '0905.1142-3-74-0': 'It suffices to construct more than one solution to the Fokker-Planck equation with [MATH] and the assumption [EQUATION]', '0905.1142-3-74-1': 'Here [MATH] is a nonzero measurable set.', '0905.1142-3-74-2': 'The idea is to consider a class of functions [MATH] such that [MATH] and [MATH] for [MATH] (e.g. [MATH]) and show that for each [MATH] the following problem has a solution.', '0905.1142-3-74-3': '[EQUATION]', '0905.1142-3-74-4': 'Note that [MATH], we can choose a parameter [MATH] such that [EQUATION]', '0905.1142-3-74-5': 'To proceed, we define [EQUATION]', '0905.1142-3-74-6': 'The resulting equation when multiplied by [MATH] leads to the following [EQUATION] where [EQUATION] with [EQUATION]', '0905.1142-3-74-7': 'Let [EQUATION]', '0905.1142-3-74-8': 'This is well defined since [MATH] from [REF] and the assumption that [MATH].', '0905.1142-3-74-9': 'From the same argument as that in Section 4, it follows that [REF]-[REF] has a unique solution [MATH] such that [EQUATION] provided the corresponding [MATH]-problem [EQUATION] has a solution for any [MATH].', '0905.1142-3-74-10': 'Note that [MATH] is essential in order that the trace of [MATH] at the boundary is defined.', '0905.1142-3-74-11': 'Equation [REF] is of the form of [REF] but with an additional term [MATH].', '0905.1142-3-74-12': 'We thus define [EQUATION]', '0905.1142-3-74-13': 'We may obtain the existence and uniqueness for [REF]-[REF] from the same argument of the well-posedness for [MATH]-problem [REF]-[REF], if there is an energy estimate of [MATH] which is similar to [MATH] in Lemma [REF].', '0905.1142-3-74-14': 'Indeed, for [MATH] [EQUATION]', '0905.1142-3-74-15': 'We now claim that [EQUATION]', '0905.1142-3-74-16': 'Given this together with [MATH] from [REF] we have [EQUATION] for any [MATH].', '0905.1142-3-74-17': 'By taking [MATH], we obtain [EQUATION] for appropriate constants [MATH] and [MATH].', '0905.1142-3-75-0': 'To verify the claim [REF], we define the trace operator [MATH] such that [EQUATION]', '0905.1142-3-75-1': 'Integration by parts on [REF] yields [EQUATION] or [EQUATION]', '0905.1142-3-75-2': 'Thus [MATH] is well defined, and for [MATH], [MATH].', '0905.1142-3-75-3': 'Finally we obtain [EQUATION]', '0905.1142-3-75-4': 'This shows that there is a unique weak solution [MATH] of [REF]-[REF], and thus [MATH] of [REF]-[REF].', '0905.1142-3-76-0': 'Finally, [MATH] is a solution of [REF]-[REF] satisfying [REF] for each [MATH].', '0905.1142-3-76-1': 'Hence the uniqueness of [REF]-[REF] fails as stated in Proposition [REF].', '0905.1142-3-77-0': '# Conclusions', '0905.1142-3-78-0': 'In this paper, we have identified a sharp Dirichlet-type boundary requirement to establish global existence of weak solutions to the microscopic FENE model which is a component of bead-spring type Navier-Stokes-Fokker-Planck models for dilute polymeric fluids.', '0905.1142-3-78-1': 'Such a boundary requirement states that the distribution near boundary approaches zero faster than the distance function.', '0905.1142-3-78-2': 'With this condition, we have been able to show the uniqueness of weak solutions in the weighted Sobolev space [MATH], which is equivalent to the widely adopted weighted function space [MATH] for Fokker-Planck equations with the FENE potential.', '0905.1142-3-78-3': 'Moreover, this condition ensures that the distribution remains a probability density.', '0905.1142-3-78-4': 'The sharpness of the boundary condition was shown by construction of infinitely many solutions when the boundary requirement fails.', '0905.1142-3-78-5': 'In other words, such a condition provides a threshold on the boundary requirement: subject to this condition or any stronger ones incorporated through a weighted function space, the Fokker-Planck dynamics will select the physically relevant solution, which is a probability density, see e.g. [CITATION], and converges to the equilibrium solution [MATH] [CITATION]; any weaker boundary requirement may lead to more solutions, each depending on the rate of [MATH] near boundary.', '0905.1142-3-78-6': 'A detailed elaboration of boundary conditions for the coupled Navier-Stokes-Fokker-Planck model will be the goal of our work [CITATION].'}

0808.2905

{'0808.2905-1-0-0': 'We investigate experimentally the influence of current flow through two independent quantum point contacts next to a double quantum dot realized in a GaAs-AlGaAs heterostructure.', '0808.2905-1-0-1': 'The observed pumping current through the double quantum dot can be explained in terms of coupling to a bosonic bath.', '0808.2905-1-0-2': 'The temperature of the bath depends on the power generated by the current flow through quantum point contact.', '0808.2905-1-0-3': 'We identify the dominant absorption and emission mechanisms in a double quantum dot as an interaction with acoustic phonons.', '0808.2905-1-0-4': 'The experiment excludes coupling of a double quantum dot to shot-noise generated by quantum point contact as the dominant mechanism.', '0808.2905-1-1-0': '# Introduction', '0808.2905-1-2-0': 'Electronic transport through semiconductor double quantum dots (DQDs) has been intensively explored for nearly two decades.', '0808.2905-1-2-1': '[CITATION] The interplay between a double quantum dot and its environment was investigated in detail in previous works [CITATION] using microwave spectroscopy.', '0808.2905-1-2-2': 'Irradiating double quantum dots with microwaves results in photon assisted tunneling (PAT).', '0808.2905-1-2-3': '[CITATION] The integration of a quantum point contact (QPC) in the vicinity of a single quantum dot allowed charge detection, [CITATION] which was later implemented in double quantum dot systems.', '0808.2905-1-2-4': '[CITATION]', '0808.2905-1-3-0': 'The novel application of a quantum point contact as a source of energy to drive inter-dot electronic transitions in a double quantum dot was recently realized.', '0808.2905-1-3-1': '[CITATION] These experiments were explained in terms of acoustic phonon based energy transfer between the QPC and the DQD circuits.', '0808.2905-1-3-2': 'The combination of a capacitatively coupled DQD-QPC system with time resolved charge detection resulted in a frequency-selective detector for microwave radiation.', '0808.2905-1-3-3': 'It allows to detect single photons emitted by the QPC and absorbed by the DQD.', '0808.2905-1-3-4': '[CITATION]', '0808.2905-1-4-0': 'Understanding the back-action of a charge sensor on a DQD is important for future possible applications in quantum information processing.', '0808.2905-1-4-1': '[CITATION] The possible dominant mechanisms that lead to QPC-induced inter-dot electronic transitions include electron scattering with photons [CITATION] and acoustic phonons [CITATION] or shot-noise [CITATION] depending on the parameter regime investigated.', '0808.2905-1-5-0': 'In this paper we study back-action of the current flow through the detector on a serial double dot.', '0808.2905-1-5-1': 'The double dot is tuned to an asymmetric regime, where one dot is strongly coupled to the source lead, whereas the second dot is more weakly coupled to the drain lead.', '0808.2905-1-5-2': 'Two independent QPCs can be simultaneously used for driving the transitions in the DQD.', '0808.2905-1-5-3': 'We observe a non-additive effect of both QPCs accompanied by the saturation of the current across the double quantum dot for large QPC currents.', '0808.2905-1-5-4': 'We explain the measured data in the framework of interaction of electrons with acoustic phonons.', '0808.2905-1-5-5': 'We relate the power emitted by the QPC to the temperature of the phononic bath.', '0808.2905-1-5-6': 'The experiment excludes shot-noise as a source of inter-dot transitions.', '0808.2905-1-6-0': 'This paper is organized as follows.', '0808.2905-1-6-1': 'In Sec. [REF] we describe the fabrication of the sample, its electrostatic characterization and functionality.', '0808.2905-1-6-2': 'In Sec. [REF] we present a detailed description of the working regime of a DQD and QPCs, followed by the results of our measurements of pumping current through a DQD using one and two QPCs.', '0808.2905-1-6-3': 'We discuss the possible interaction mechanisms in Sec. [REF].', '0808.2905-1-6-4': 'In Sec. [REF] we introduce a model based on electron-phonon interaction and in Sec. [REF] we interpret the measured data.', '0808.2905-1-6-5': 'Section [REF] contains the conclusions.', '0808.2905-1-7-0': '# Sample and characterization', '0808.2905-1-8-0': 'The sample shown in Fig. [REF](a) is based on a GaAs/Al[MATH]Ga[MATH]As heterostructure with a two-dimensional electron gas (2DEG) 34 nm below the surface.', '0808.2905-1-8-1': 'It was fabricated by double layer local oxidation with a scanning force microscope (SFM).', '0808.2905-1-8-2': '[CITATION] The 2DEG is depleted below the oxide lines written on the GaAs surface [CITATION] [white lines in Fig. [REF](a)].', '0808.2905-1-8-3': 'A 4 nm Titanium film was evaporated and patterned by local oxidation to create mutually isolated top gates [indicated by the dashed lines in Fig. [REF](a)].', '0808.2905-1-9-0': 'The confinement potential produced by the top gates and the oxide lines is shown in the contour plot in Fig. [REF](b).', '0808.2905-1-9-1': 'It was calculated assuming a pinned surface model [CITATION] using the lithographic sizes of the gates measured after the sample was fabricated.', '0808.2905-1-9-2': 'It shows an approximately circular symmetry for the dots, with QD1 being slightly larger than QD2.', '0808.2905-1-9-3': 'The color scale is in arbitrary units.', '0808.2905-1-9-4': '[CITATION]', '0808.2905-1-10-0': 'The structure presented in Fig. [REF](a) consists of three electronic circuits.', '0808.2905-1-10-1': 'The first one is formed by the two quantum dots connected in series [marked by the gray (red online) circles] and connected to source and drain.', '0808.2905-1-10-2': 'Negative DQD current corresponds to electrons moving from source to drain.', '0808.2905-1-10-3': 'Each of the other two circuits contains a quantum point contact [white (yellow online) solid arrows].', '0808.2905-1-10-4': 'A negative QPC current means electrons traveling through the QPC in the direction of the arrows.', '0808.2905-1-11-0': 'The top gates (T[MATH] and T[MATH]) are used to tune the DQD into a suitable regime.', '0808.2905-1-11-1': 'The top gates T[MATH] and T[MATH] can tune the transmission of QPC1 and QPC2, respectively.', '0808.2905-1-11-2': 'The middle top gate T[MATH] controls the coupling between the two dots allowing to change smoothly from the single dot regime (large dot spread over the area covered by the two red circles) to a weakly-coupled double dot.', '0808.2905-1-11-3': 'The gates T[MATH] and T[MATH] are used to tune the coupling of the DQD to source and drain.', '0808.2905-1-12-0': 'The potential on both sides of QPC1 (QPC2) can be lifted with respect to the measurement ground, creating a mutual gating effect between DQD and QPC1 (QPC2).', '0808.2905-1-12-1': 'These in-plane gates (I[MATH] for QPC1 and I[MATH] for QPC2) control the number of electrons on the DQD.', '0808.2905-1-13-0': 'Due to the presence of the metallic top gates, the electrostatic interaction between electrons in the quantum dots and the QPCs is weakened by screening compared to semiconductor-only quantum circuits.', '0808.2905-1-13-1': '[CITATION] The large distance between the QPC and the double dot (lithographic distance 450 nm) further reduces the sensitivity of the QPC for detecting electrons passing the DQD.', '0808.2905-1-14-0': 'Figures [REF](c) and (d) demonstrate the operation of QPC2 as a charge detector.', '0808.2905-1-14-1': '[CITATION] For both QPCs the one-dimensional subband spacing is larger than 3.5 mV as estimated from finite bias measurements.', '0808.2905-1-14-2': 'In order to use QPC2 (QPC1) as a charge read-out, its conductance was tuned to [MATH].', '0808.2905-1-14-3': 'A constant voltage of 0.5 mV was applied to the QPC circuit, and the current was measured.', '0808.2905-1-14-4': 'An AC voltage of 1.5 mV applied at 34 Hz to the opposite in-plane gate I[MATH] (I[MATH]) modulated the current through QPC2 (QPC1).', '0808.2905-1-14-5': 'This signal was detected with lock-in techniques.', '0808.2905-1-14-6': 'The measurements were performed in a dilution refrigerator at a base temperature of 70 mK.', '0808.2905-1-15-0': 'The resulting stability diagram of the DQD detected with QPC2 is shown in Fig. [REF](c).', '0808.2905-1-15-1': 'The boundaries between regions of different ground state charge configurations of the DQD are clearly visible.', '0808.2905-1-15-2': 'A few charge rearrangements in the lower half of the honeycomb induced by the metallic top gate T[MATH] are present.', '0808.2905-1-15-3': 'In general, we find that the top gate sweeps lead to significantly more charge rearrangements than sweeps of the in-plane gates.', '0808.2905-1-15-4': 'The thick line in the bottom-left corner of the plot corresponds to a resonance in QPC2.', '0808.2905-1-16-0': 'In Fig. [REF](d) the corresponding DC current through the DQD is plotted for the same gate voltage range.', '0808.2905-1-16-1': 'The source-drain voltage applied to the DQD is 60 [MATH]V. Only two pairs of triple points are visible.', '0808.2905-1-16-2': 'Similar sets of data can be obtained using QPC1 as the detector.', '0808.2905-1-17-0': '# Experimental data', '0808.2905-1-18-0': 'In the following, we concentrate on a single pair of triple points where the DQD showed moderate coupling.', '0808.2905-1-18-1': 'Figure [REF](a) shows the DC DQD current (I[MATH]) for 100 [MATH]eV source-drain bias applied across the DQD.', '0808.2905-1-18-2': 'The inter-dot mutual capacitance [MATH] estimated from finite bias measurements and from the stability diagram assuming the constant interaction model [CITATION] is 8.8 aF, whereas the total capacitance of dot 1 is [MATH] aF and for dot 2, [MATH] aF.', '0808.2905-1-18-3': 'Each dot contains approximately 15 electrons and the charging energies are about 2 meV.', '0808.2905-1-19-0': 'The thin dashed lines in Fig. [REF](a) indicate the boundaries of the honeycomb pattern and the numbers in brackets (M,N) denote the charge population of QD1 and QD2 respectively.', '0808.2905-1-19-1': 'Here QD1 is strongly coupled to the source lead, whereas QD2 is weakly coupled to the drain reservoir.', '0808.2905-1-19-2': 'During the measurement both QPCs were kept at zero bias.', '0808.2905-1-20-0': 'The detuning marked by the dashed gray (red online) line in Fig. [REF](a) is obtained from a capacitance model [CITATION] and expressed by the equation [MATH] such that the total energy of the DQD, [MATH] remains constant.', '0808.2905-1-20-1': '[MATH] and [MATH] are the single-particle energies in QD1 and QD2.', '0808.2905-1-20-2': 'Converting the energies to gate voltages gives: [MATH].', '0808.2905-1-20-3': 'The lever arm [MATH] is the lever arm of gate [MATH] on dot [MATH] and [MATH] is the voltage applied to gate [MATH]).', '0808.2905-1-20-4': 'The lever arms are extracted from measurements at finite bias and from the charge stability diagram of the DQD.', '0808.2905-1-20-5': 'We take zero detuning to occur at the triple point.', '0808.2905-1-20-6': 'According to the definition above, detuning is positive (negative) in the upper-left (lower right) part of the Fig. [REF](a).', '0808.2905-1-20-7': 'Two representative energy diagrams are shown in the insets.', '0808.2905-1-21-0': 'In Fig. [REF](b) the DQD current was measured in the same parameter range at a QPC2 bias voltage of 1 mV.', '0808.2905-1-21-1': 'The bias voltage across DQD was set to 60 [MATH]eV, i.e., smaller than the bias voltage applied in Fig. [REF](a).', '0808.2905-1-21-2': 'Despite that, the current is strongly enhanced along the boundaries (M,N)[MATH](M+1,N) and (M,N+1)[MATH](M+1,N+1) corresponding to adding an electron to the QD1.', '0808.2905-1-21-3': 'The enhancement of the co-tunneling [CITATION] current along the honeycomb boundaries is induced by driving a current through QPC2.', '0808.2905-1-21-4': 'Another visible feature induced by biasing QPC2 is the finite DQD current in the triangle-shaped area indicated in Fig. [REF](b), that is normally forbidden by Coulomb blockade.', '0808.2905-1-22-0': 'The following measurements were carried out with the QPCs tuned to their first conductance plateau.', '0808.2905-1-22-1': 'The overall experimental results do not depend on this operation point of the QPC.', '0808.2905-1-23-0': 'To investigate the influence of the QPC currents on the DQD in the triangular region, we tuned the levels in the dot along the detuning line depicted as the solid red line in Fig. [REF](a).', '0808.2905-1-23-1': 'Fig. [REF] shows the dot current versus detuning.', '0808.2905-1-23-2': 'The black data points in Fig. [REF](a) were taken with zero bias applied to the DQD as well as to QPC1 and QPC2.', '0808.2905-1-23-3': 'No measurable current above the noise level is detected.', '0808.2905-1-23-4': 'When a DC current of 50 nA is driven through QPC1, an asymmetric peak with a maximum of about 125 fA along the detuning line is observed (blue points in Fig. [REF](a).', '0808.2905-1-23-5': 'This effect is strongly enhanced if the current through QPC1 is further increased to 75 nA (green points) and 100 nA (red points).', '0808.2905-1-23-6': 'All traces cross zero at the triple point (zero detuning).', '0808.2905-1-24-0': 'A similar, but significantly more pronounced effect is observed if QPC2 is driven, as shown in Fig. [REF](b).', '0808.2905-1-24-1': 'Moreover, for negative detuning a small negative DQD current is observed.', '0808.2905-1-24-2': 'QPC2 is more sensitive as a charge readout and it has a stronger effect on the DQD.', '0808.2905-1-24-3': 'Therefore, we conclude that QPC2 is more strongly coupled to the DQD than QPC1.', '0808.2905-1-25-0': 'We have chosen one point on the detuning line corresponding to [MATH]eV and swept the QPC1 and QPC2 currents.', '0808.2905-1-25-1': 'The results of this measurement are shown in Fig. [REF](a).', '0808.2905-1-25-2': 'The black (red) filled squares correspond to positive currents through QPC1 (QPC2) swept from 0 to 200 nA.', '0808.2905-1-25-3': 'The empty black (red) circles are the traces recorded while the QPC1 (QPC2) current was swept from 0 to -200 nA.', '0808.2905-1-25-4': 'The QPC induced DQD current is a little larger in the case when the QPCs are driven with positive current.', '0808.2905-1-25-5': 'This polarity dependence is significant and we can exclude that it is due to a gating effect.', '0808.2905-1-25-6': 'As mentioned before, QPC2 is more strongly coupled to the DQD than QPC1.', '0808.2905-1-25-7': 'Above 100 nA the DQD current starts to saturate.', '0808.2905-1-26-0': 'Another unexpected feature is observed on the green traces.', '0808.2905-1-26-1': 'The filled (empty) green squares correspond to the QPC1 current being swept from 0 to 200 (-200) nA while the QPC2 current is simultaneously swept from 0 to 200 nA.', '0808.2905-1-26-2': 'In a simple picture, we would expect that the effects of QPC1 and QPC2 are independent and they add up, but the measurement contradicts this expectation.', '0808.2905-1-26-3': 'Due to the action of both QPCs the DQD current is slightly larger than in the case when only QPC2 is used.', '0808.2905-1-26-4': 'In addition, there is an unexpected polarity dependence with a maximum DQD current for QPC1 being swept in negative and QPC2 in positive direction.', '0808.2905-1-26-5': 'The remaining blue filled squares (empty circles) in Fig. [REF](a) were obtained by driving a positive (negative) current through QPC1 and a negative current through QPC2.', '0808.2905-1-27-0': 'A saturation of the DQD current and the polarity effect are also present in Fig. [REF](a), where the dot current was plotted versus QPC1 and QPC2 currents at fixed detuning [MATH]eV.', '0808.2905-1-27-1': 'In this measurement, the lack of additivity of the effects induced by the QPCs is even more visible.', '0808.2905-1-28-0': '# Discussion of possible mechanisms', '0808.2905-1-29-0': 'A mechanism which can induce the current flow through the double dot along the detuning line is presented in Fig. [REF](a).', '0808.2905-1-29-1': 'The driving current through QPC1 or/and QPC2 is thought to lead to an emission of energy, which can be absorbed by the electron in the right dot.', '0808.2905-1-29-2': 'If the provided energy matches the energy difference [MATH], the electron can be excited from the right to the left dot.', '0808.2905-1-29-3': 'If the electron leaves the DQD through the left lead and the next electron tunnels into the right dot through the right lead, then the cycle closes and there is a measurable current flowing through the double dot.', '0808.2905-1-30-0': 'An additional enhancement of the co-tunneling current as observed in Fig. [REF](b), induced by driving a current through a QPC, can be explained in a similar way.', '0808.2905-1-30-1': 'In the situation shown in the upper inset the electron trapped in the QD2 can absorb energy emitted by the QPC2, tunnel into dot 1 and leave the DQD system via the left lead.', '0808.2905-1-30-2': 'The cycle closes when the next electron tunnels into QD2 from the right lead.', '0808.2905-1-30-3': 'This QPC2 induced pumping process gives an additional contribution to the current.', '0808.2905-1-30-4': 'This effect is more pronounced in the vicinity of the triple point where energy difference between the levels [MATH] and [MATH] in the QD1 and QD2 are small.', '0808.2905-1-30-5': 'The lower inset of Fig. [REF](b) shows the analogous diagram for the situation when the level in QD2 lies above the Fermi energy of the leads.', '0808.2905-1-30-6': 'Again, pumping causes the electrons to move from the right into the left contact.', '0808.2905-1-31-0': 'The possible mechanisms of the pumping effect are coupling to acoustic or optical phonons, plasmons, photons, shot-noise or thermopower effect.', '0808.2905-1-31-1': 'Scattering with optical phonons is strongly suppressed as long as the relevant energy scales are smaller than the optical phonon energy.', '0808.2905-1-31-2': '[CITATION] Coupling to plasmons can be ruled out as well.', '0808.2905-1-31-3': '[CITATION] We can also exclude the shot-noise as a source of the energy, because during the experiment both QPCs were tuned to their first plateau.', '0808.2905-1-31-4': 'Measurements performed at 0.5[MATH] and 1.5[MATH]) showed a qualitatively and quantitatively similar behavior.', '0808.2905-1-31-5': 'This is in contrast to previously measured data [CITATION] where no pumping current was observed in the plateau regions.', '0808.2905-1-31-6': 'Coupling to acoustic phonons is the most likely mechanism of inducing the pumping current in the DQD.', '0808.2905-1-31-7': 'Further below in this paper, we discuss the data in the light of phonon coupling and a related thermopower effects.', '0808.2905-1-32-0': 'The questions arising from the data presented above are the following: is the strong difference of the peak heights on the positive and negative side of the detuning (Fig. [REF]) due to the asymmetry in the DQD coupling to the leads?', '0808.2905-1-32-1': 'Why do the effects of the QPCs not add up?', '0808.2905-1-32-2': 'What is the reason for the saturation of the DQD current observed in Fig. [REF](a)?', '0808.2905-1-32-3': 'What is the mechanism of the energy transfer from the QPCs to the DQD?', '0808.2905-1-32-4': 'Can it explain the polarity dependence?', '0808.2905-1-32-5': 'In the next section we present a model that attempts to answer most of these questions.', '0808.2905-1-33-0': '# The model', '0808.2905-1-34-0': 'In the following we derive a scenario, which explains the pumping effect based on electron-phonon interaction.', '0808.2905-1-34-1': 'First, we introduce the two-level system describing the DQD.', '0808.2905-1-34-2': 'Then, we consider all possible transitions between different energy states of the DQD and express them in terms of tunneling rates.', '0808.2905-1-34-3': 'Subsequently, we derive the energy dependence of the tunneling rates.', '0808.2905-1-34-4': 'The intra-dot transitions are calculated in a framework of electron-phonon interaction.', '0808.2905-1-34-5': 'Next, we set up a master equation and obtain the complete expression for the pumped current as a function of detuning and the temperature of the phonon bath.', '0808.2905-1-35-0': 'Close to a pair of triple points, a double quantum dot can be regarded as a two-level system,[CITATION] whose bonding (ground) and antibonding (excited) states [MATH] and [MATH] are separated by an energy [MATH] as shown in Fig. [REF](b),[CITATION] where [MATH] is tunneling coupling between the dots.', '0808.2905-1-35-1': 'The corresponding eigenvectors for bonding and antibonding states are [MATH] and [MATH].', '0808.2905-1-35-2': 'The components of the bonding and antibonding eigenstates in the basis of [MATH] and [MATH], the wave functions in dot 1 and 2, are [MATH], where [MATH] and [MATH].', '0808.2905-1-36-0': 'The detuning is defined as [MATH] and the total energy is [MATH].', '0808.2905-1-36-1': 'In case of one excess electron in the DQD, the ground state energy is [MATH], while for two excess electrons, the ground state energy is [MATH].', '0808.2905-1-36-2': 'The corresponding electrochemical potentials are [MATH] and [MATH].', '0808.2905-1-37-0': 'In the vicinity of a pair of triple points a double quantum dot can have one out of four different charge states.', '0808.2905-1-37-1': 'These different charge configurations are presented in Fig. [REF].', '0808.2905-1-37-2': 'The "empty" state corresponds to a situation where there is no excess electron present in a dot and the occupation probability of this state is [MATH].', '0808.2905-1-37-3': 'In addition, one excess electron may occupy the bonding state "0" with probability [MATH] or the antibonding state "1" with probability [MATH].', '0808.2905-1-37-4': 'The last possible charge configuration is when there are two excess electrons in a double quantum dot - state "2" with occupation probability [MATH].', '0808.2905-1-38-0': 'The transitions between these states are determined by the tunneling rates [MATH] and the thermal broadening of the Fermi function [MATH] in the leads.', '0808.2905-1-38-1': 'The index [MATH]=L,R denotes the left (L) or the right (R) barrier through which the electron tunnels and the index [MATH]=0,1,2,3 labels the state.', '0808.2905-1-38-2': 'For example, if the dot is in an empty state and the electron tunnels in via the right lead, the corresponding rate is [MATH], as shown in Fig. [REF].', '0808.2905-1-38-3': 'For [MATH]=0(1) the Fermi function is [MATH] where [MATH] is the Boltzmann constant and [MATH] is the temperature of the lead.', '0808.2905-1-38-4': 'For [MATH]=2,3 the expression is analogous but the energy is lifted by the mutual charging energy [MATH].', '0808.2905-1-39-0': 'In order to explain the experimental data presented before, we have to take into account that the tunneling rates [MATH] do depend on the electronic wave function.', '0808.2905-1-39-1': 'They can be expressed as [MATH].', '0808.2905-1-39-2': 'The coefficients [MATH] are the left ([MATH] and right ([MATH]) energy dependent components of the eigenvector of the wave function corresponding to the bonding ([MATH]=0,3) and antibonding ([MATH]=1,2) states.', '0808.2905-1-39-3': 'The amplitudes [MATH] are energy-independent parts of the [MATH].', '0808.2905-1-40-0': 'The rates describing the inter-dot processes, that is absorption [MATH] and emission [MATH] are marked in Fig. [REF].', '0808.2905-1-40-1': 'In the following, we assume that the double quantum dot is coupled to a bosonic bath in thermal equilibrium described by the Bose-Einstein distribution function [MATH].', '0808.2905-1-40-2': 'The temperature [MATH] of this bath is determined by the current of the QPC and the base temperature of the cryostat.', '0808.2905-1-40-3': 'In case of coupling to acoustic phonons, the emission and absorption rates can be expressed as (derivation is presented in the Appendix A): [EQUATION] where the index [MATH] denotes piezoelectric transversal phonons (pe,T), piezoelectric longitudinal phonons (pe,L) or longitudinal deformation potential coupling phonons (dp,L).', '0808.2905-1-40-4': 'The exponent [MATH] is 1 for piezoelectric phonons and [MATH]=3 for deformation potential coupling.', '0808.2905-1-40-5': '[CITATION] The upper (lower) sign stands for emission (absorption) of energy quanta.', '0808.2905-1-40-6': 'The values of the energy-independent coefficients [MATH] are given in Appendix [REF].', '0808.2905-1-40-7': 'The form-factor [MATH] is represented as: [EQUATION] where [MATH] denotes the radius of a single QD, [MATH] is the distance between the dots, [MATH] is the speed of sound of [MATH]-phonons.', '0808.2905-1-40-8': 'The complete expression for the double-dot geometry factor [MATH] is given in Appendix [REF].', '0808.2905-1-40-9': 'Only phonons with a wavelength comparable or larger than the DQD size can interact with the electron.', '0808.2905-1-41-0': 'To investigate the influence of the QPCs on the dot presented in Fig. [REF] in terms of rates and occupation probabilities defined in Fig. [REF] we only take into account the processes surrounded by the dashed line.', '0808.2905-1-41-1': 'This is reasonable, due to a large mutual charging energy relative to the tunneling coupling.', '0808.2905-1-41-2': 'Counting the electrons passing through the right barrier leads to the following expression of the current through the double dot (the derivation is presented in Appendix [REF]): [EQUATION]', '0808.2905-1-41-3': 'The occupation probabilities [MATH] are functions of [MATH] and [MATH] resulting from a stationary solution of the master equation (see Appendix [REF]).', '0808.2905-1-42-0': '# Results and interpretation', '0808.2905-1-43-0': 'We used expression ([REF]) to fit the data shown in Fig. [REF].', '0808.2905-1-43-1': 'During the fitting procedure, the 8 traces shown in Fig. [REF](a) and (b) and an additional set of 16 traces being a combination of [MATH] nA were fitted simultaneously.', '0808.2905-1-43-2': 'The following fitting parameters were shared: amplitudes [MATH], [MATH] and tunneling coupling [MATH].', '0808.2905-1-43-3': 'The only parameter specific for each trace was the temperature of the phonon bath [MATH].', '0808.2905-1-44-0': 'The fit represents the overall shape of the measured data very well.', '0808.2905-1-44-1': 'The extracted tunneling coupling is [MATH]eV.', '0808.2905-1-44-2': 'In Fig. [REF](a) the gray (green online) solid line is the calculated boundary between the honeycomb cells assuming a tunneling coupling of 50 [MATH]eV, the black (red online) crosses are the maxima position of the co-tunneling peaks and the black dashed line is the boundary assuming [MATH].', '0808.2905-1-44-3': 'Unfortunately, the sample was not stable enough to map a stability diagram with a resolution high enough to determine the tunneling coupling directly.', '0808.2905-1-44-4': 'However, the [MATH] obtained from the fits seems to be reasonable and matches with the data presented on the stability diagram.', '0808.2905-1-45-0': 'The temperature of the phonon bath [MATH] obtained from the fits varies from 0.6 K (blue trace in Fig. [REF](a)) to 1.2 K (red trace in Fig. [REF](b)).', '0808.2905-1-45-1': 'The difference between the temperatures [MATH] and the electronic temperature [MATH]=100 mK gives rise to the pumping current.', '0808.2905-1-46-0': 'The extracted amplitude [MATH] is of the order of 7.8 MHz and [MATH] is about 0.5 GHz.', '0808.2905-1-46-1': 'This is in agreement with our previous statement that the right barrier is more opaque than the left one.', '0808.2905-1-46-2': 'This leads to a deviation from perfect antisymmetry of the dot current along the detuning line, i.e., to suppressed dot current for negative detuning.', '0808.2905-1-47-0': 'The tail of the curve at large detuning is mainly determined by the amplitudes [MATH] (Eq. [REF]) that drop to zero like [MATH] and the Bose-Einstein distribution function (Eq. [REF]).', '0808.2905-1-47-1': 'Although, the real distribution of the bosonic environment is not necessarily equilibrated and could have another form, the qualitative agreement with the data does not depend strongly on the details of this distribution.', '0808.2905-1-48-0': 'Using the parameters obtained in the fits, we calculated the current as a function of the gate voltages using the model containing all four charge states presented in Fig. [REF].', '0808.2905-1-48-1': 'In the absence of the QPC current ([MATH]=70 mK), the result is shown in Fig. [REF](c).', '0808.2905-1-48-2': 'The asymmetry and magnitude of the co-tunneling current along the upper dashed line is reproduced very well.', '0808.2905-1-48-3': 'The discrepancy along the lower co-tunneling branch indicating transitions [MATH] may be due to a change of [MATH] and [MATH], which we assumed to be constant in our model.', '0808.2905-1-48-4': 'For 1 mV DC bias across the QPC2 (Fig. [REF](b)), the bosonic temperature is around [MATH]=0.8 K.', '0808.2905-1-48-5': 'The corresponding calculations are shown in Fig. [REF](d).', '0808.2905-1-48-6': 'The red triangle indicates the region with the pumping current that is in agreement with the measured data in (b).', '0808.2905-1-49-0': 'To investigate the dependence of the phonon temperature on the QPC current we calculated [MATH] for every point from Fig. [REF](a) using the values obtained in previous fits.', '0808.2905-1-49-1': 'The results are shown in Fig. [REF](b).', '0808.2905-1-49-2': 'For small QPC currents the error bars are large and no clear dependence is visible.', '0808.2905-1-49-3': 'For QPC currents above 50 nA the dependence is quadratic [MATH], which means that the temperature of the bosonic bath is proportional to the power emitted by the QPC.', '0808.2905-1-49-4': 'The reconstructed temperatures corresponding to the measurement presented in Fig. [REF](a) are plotted in Fig. [REF](b).', '0808.2905-1-50-0': 'The saturation of the DQD current for large QPC currents cannot be attributed to the high occupation probability of the antibonding state [MATH].', '0808.2905-1-50-1': 'We have estimated that the [MATH] value does not exceed a few percent and most of the time the dot is occupied by one electron in its ground state.', '0808.2905-1-50-2': 'The maximum current is determined by the right tunneling barrier.', '0808.2905-1-51-0': 'A series of experiments reporting the observation of a DQD current induced by a single and independently biased QPC, is described in Refs. [CITATION].', '0808.2905-1-51-1': 'These experiments were performed in the regime of large pumped current, strong bias voltage applied to the QPC and large tunneling coupling, which is similar to our situation here.', '0808.2905-1-51-2': 'The pumping current was related to inelastic relaxation of electrons in partly transmitting 1D channels of the QPC [CITATION] and qualitatively consistent with an energy transfer mechanism based on nonequilibrium acoustic phonons.', '0808.2905-1-51-3': '[CITATION] In contrast to our experimental data, in these experiments the pumping current was large when the conductance of the QPC was tuned to [MATH] and strongly suppressed in plateau regions.', '0808.2905-1-52-0': 'Recent research has proven that the absorption of a photon can be the dominant process [CITATION] in similar situations.', '0808.2905-1-52-1': 'However, these time-resolved experiments were performed in a different regime, where the dominant tunneling rates are of order of 1 kHz, whereas in our system the double quantum dot is much more strongly coupled to the leads.', '0808.2905-1-52-2': 'Another difference is the presence of a Ti top-gate in our structure, that screens the electrostatic interaction between the DQD and the QPCs.', '0808.2905-1-52-3': 'Due to the lower sensitivity of direct dot current measurements compared to the time-resolved technique it is not possible to observe the gap in the pumped current when [MATH].', '0808.2905-1-52-4': 'Calculations of the emission and absorption rates in a DQD induced by electron-photon interaction show, that the effect is irrelevant compared to the emission and absorption of phonons discussed here.', '0808.2905-1-52-5': '[CITATION]', '0808.2905-1-53-0': 'We have also tested the possibility that the entire dot current in the region forbidden by the Coulomb blockade is due to a thermopower effect induced by different temperatures in the source and the drain.', '0808.2905-1-53-1': 'We found, that it would be only possible if the temperature difference between source and drain lead was larger than 1K for a QPC current of 100 nA, which is one order of magnitude larger than expected.', '0808.2905-1-53-2': '[CITATION] Another argument against a thermopower model is that always the drain lead would have to be warmer, even if the far QPC1 (that couples better to the drain lead) was biased.', '0808.2905-1-53-3': 'Even so, the thermopower model did not describe the data as well as the emission/absorption model.', '0808.2905-1-54-0': '# Conclusions', '0808.2905-1-55-0': 'We have presented the influence of two independent quantum point contacts on a double quantum dot.', '0808.2905-1-55-1': 'Driving current through the QPC leads to emission of energy that increases the temperature of the bosonic environment.', '0808.2905-1-55-2': 'We identify these bosons as acoustic phonons.', '0808.2905-1-55-3': 'To model the interaction of the phonons with the double quantum dot we have assumed that their energy distribution is described by Bose-Einstein statistics.', '0808.2905-1-55-4': 'For large QPC powers (above 30 mW) the temperature of the bath increases linearly.', '0808.2905-1-55-5': 'The observed saturation of the current is due to the finite transparency of the tunneling barriers.', '0808.2905-1-55-6': 'The polarity dependence (Fig. [REF]) cannot be explained within the discussed model and its origin remains to be investigated.', '0808.2905-1-55-7': 'All the measurements were performed with both QPCs tuned to their first plateau.', '0808.2905-1-55-8': 'Thus we can exclude the influence of shot-noise phenomena in the quantum point contacts.', '0808.2905-1-56-0': '# Absorption and emission rates in a DQD induced by electron-phonon interaction', '0808.2905-1-57-0': "Generally, the emission and absorption rates can be expressed using Fermi's golden rule: [EQUATION] where the sum extends over all wave vectors [MATH].", '0808.2905-1-57-1': 'The index [MATH] denotes the type of acoustic phonons and their coupling in GaAs: piezoelectric longitudinal (pe,L), piezoelectric transversal (pe,T) and longitudinal, deformation potential coupling (dp,L).', '0808.2905-1-57-2': 'The phonons have linear dispersion relation [MATH].', '0808.2905-1-57-3': '[MATH] and [MATH] are wave functions of bonding and antibonding states separated by energy [MATH].', '0808.2905-1-57-4': 'The interaction hamiltonian [MATH] can be written as a sum of piezoelectric interaction [MATH] and deformation potential coupling [MATH]: [EQUATION] with [EQUATION]', '0808.2905-1-57-5': 'In above equations [MATH] is an element of piezoelectric tensor, [MATH] is the vacuum permittivity, [MATH] is the dielectric constant, [MATH] is the number of atoms in the crystal, M is the atomic mass and D denotes deformation potential coupling constant.', '0808.2905-1-57-6': 'Dimensionless function [MATH] has form: [EQUATION] where [MATH] is the Levi-Civita symbol and [MATH] is the [MATH]-component of eigenvector associated with mode [MATH].', '0808.2905-1-57-7': 'Inserting Eq. ([REF]) into Eq. ([REF]) leads to the following expression: [EQUATION] where the upper sign refers to phonon absorption and the lower to phonon emission.', '0808.2905-1-57-8': 'Constants [MATH] and [MATH] are given by: [EQUATION] where [MATH] is a density of the GaAs crystal.', '0808.2905-1-57-9': 'The energy dependent functions [MATH] and [MATH] are defined as: [EQUATION]', '0808.2905-1-57-10': 'Assuming negligible overlap between the wave functions of the two dots and taking a gaussian-shaped single-electron wave function, the matrix element is found to be: [EQUATION] where [MATH] is the distance between the dots and [MATH] is the radius of a single dot.', '0808.2905-1-57-11': 'Inserting ([REF]) into Eqs. ([REF]) gives: [EQUATION]', '0808.2905-1-57-12': 'For piezoelectric transversal phonons the above expression was calculated by averaging the function [MATH] over all possible transversal directions.', '0808.2905-1-57-13': 'The geometry factors [MATH] are given by: [EQUATION]', '0808.2905-1-57-14': 'Combining Eq. ([REF]) with Eq. ([REF]) and inserting the result into Eq. ([REF]) gives a complete expression for the absorption and emission rates.', '0808.2905-1-58-0': '# Rate equation', '0808.2905-1-59-0': 'To relate the DQD current to the tunneling rates we write down the rate equation for the occupation of the states: [EQUATION] with additional condition [MATH].', '0808.2905-1-59-1': 'The terms [MATH] and [MATH] are defined as: [EQUATION]', '0808.2905-1-59-2': 'To find the expression for the current flowing through a DQD, we take the right barrier as a current reference.', '0808.2905-1-59-3': 'It means that, if an electron passes the right barrier to the left (right), its contribution to the DQD current is positive (negative).', '0808.2905-1-59-4': '[EQUATION]', '0808.2905-1-59-5': 'The first (second) term of Eq. ([REF]) corresponds to the electrons moving from the bonding (antibonding) state to the right lead and the third and fourth term to the electrons entering the ground or excited state of the dot from the right lead.', '0808.2905-1-59-6': 'By inserting a stationary solution of Eq. ([REF]) into Eq. ([REF]) one obtains an expression for the steady state DQD current.'}

{'0808.2905-2-0-0': 'We investigate experimentally the influence of current flow through two independent quantum point contacts to a nearby double quantum dot realized in a GaAs-AlGaAs heterostructure.', '0808.2905-2-0-1': 'The observed current through the double quantum dot can be explained in terms of coupling to a bosonic bath.', '0808.2905-2-0-2': 'The temperature of the bath depends on the power generated by the current flow through the quantum point contact.', '0808.2905-2-0-3': 'We identify the dominant absorption and emission mechanisms in a double quantum dot as an interaction with acoustic phonons.', '0808.2905-2-0-4': 'The experiment excludes coupling of a double quantum dot to shot-noise generated by quantum point contact as the dominant mechanism.', '0808.2905-2-1-0': '# Introduction', '0808.2905-2-2-0': 'Electronic transport through semiconductor double quantum dots (DQDs) has been intensively explored for nearly two decades.', '0808.2905-2-2-1': '[CITATION] The interplay between a double quantum dot and its environment was investigated in detail in previous works [CITATION] using microwave spectroscopy.', '0808.2905-2-2-2': 'Irradiating double quantum dots with microwaves results in photon assisted tunneling (PAT).', '0808.2905-2-2-3': '[CITATION] The integration of a quantum point contact (QPC) in the vicinity of a single quantum dot allowed charge detection, [CITATION] which was later implemented in double quantum dot systems.', '0808.2905-2-2-4': '[CITATION]', '0808.2905-2-3-0': 'The novel application of a quantum point contact as a source of energy to drive inter-dot electronic transitions in a double quantum dot was recently realized.', '0808.2905-2-3-1': '[CITATION] These experiments were explained in terms of acoustic phonon based energy transfer between the QPC and the DQD circuits.', '0808.2905-2-3-2': 'The combination of a capacitatively coupled DQD-QPC system with time resolved charge detection resulted in a frequency-selective detector for microwave radiation.', '0808.2905-2-3-3': 'It allows to detect single photons emitted by the QPC and absorbed by the DQD.', '0808.2905-2-3-4': '[CITATION]', '0808.2905-2-4-0': 'Understanding the back-action of a charge sensor on a DQD is important for future possible applications in quantum information processing.', '0808.2905-2-4-1': '[CITATION] The possible dominant mechanisms that lead to QPC-induced inter-dot electronic transitions include electron scattering with photons [CITATION] and acoustic phonons [CITATION] or shot-noise [CITATION] depending on the parameter regime investigated.', '0808.2905-2-5-0': 'In this paper we study back-action of the current flow through the QPC detector on a serial double dot.', '0808.2905-2-5-1': 'The double dot is tuned to an asymmetric regime, where one dot is strongly coupled to the source lead, whereas the second dot is more weakly coupled to the drain lead.', '0808.2905-2-5-2': 'Two independent QPCs can be simultaneously used for driving the transitions in the DQD.', '0808.2905-2-5-3': 'We observe a non-additive effect of both QPCs accompanied by the saturation of the current across the double quantum dot for large QPC currents.', '0808.2905-2-5-4': 'We explain the measured data in the framework of interaction of electrons with acoustic phonons.', '0808.2905-2-5-5': 'We relate the power emitted by the QPC to the temperature of the phononic bath.', '0808.2905-2-5-6': 'The experiment excludes the possibility of shot-noise being the source of inter-dot transitions.', '0808.2905-2-6-0': 'This paper is organized as follows.', '0808.2905-2-6-1': 'In Sec. [REF] we describe the fabrication of the sample, its electrostatic characterization and functionality.', '0808.2905-2-6-2': 'In Sec. [REF] we present a detailed description of the working regime of a DQD and QPCs, followed by the results of our measurements of current through a DQD using one and two QPCs.', '0808.2905-2-6-3': 'We discuss the possible interaction mechanisms in Sec. [REF].', '0808.2905-2-6-4': 'In Sec. [REF] we introduce a model based on electron-phonon interaction and in Sec. [REF] we interpret the measured data.', '0808.2905-2-6-5': 'Section [REF] contains the conclusions.', '0808.2905-2-7-0': '# Sample and characterization', '0808.2905-2-8-0': 'The sample shown in Fig. [REF](a) is based on a GaAs/Al[MATH]Ga[MATH]As heterostructure with a two-dimensional electron gas (2DEG) 34 nm below the surface.', '0808.2905-2-8-1': 'It was fabricated by double layer local oxidation with a scanning force microscope (SFM).', '0808.2905-2-8-2': '[CITATION] The 2DEG is depleted below the oxide lines written on the GaAs surface [CITATION] [white lines in Fig. [REF](a)].', '0808.2905-2-8-3': 'A 4 nm titanium film was evaporated and patterned by local oxidation to create mutually isolated top gates [indicated by the dashed lines in Fig. [REF](a)].', '0808.2905-2-9-0': 'The confinement potential produced by the top gates and the oxide lines is shown in the contour plot in Fig. [REF](b).', '0808.2905-2-9-1': 'It was calculated assuming a pinned surface model [CITATION] using the lithographic sizes of the gates measured after the sample was fabricated.', '0808.2905-2-9-2': 'It shows an approximately circular symmetry for the dots, with the left quantum dot being slightly larger than the right one.', '0808.2905-2-9-3': 'The color scale is in arbitrary units.', '0808.2905-2-9-4': '[CITATION]', '0808.2905-2-10-0': 'The structure presented in Fig. [REF](a) consists of three electronic circuits.', '0808.2905-2-10-1': 'The first one is formed by two quantum dots connected in series [marked by the grey (red online) circles] and connected to source and drain.', '0808.2905-2-10-2': 'A negative DQD current corresponds to electrons moving from source to drain.', '0808.2905-2-10-3': 'Each of the other two circuits contains a quantum point contact [white (yellow online) solid arrows].', '0808.2905-2-10-4': 'A negative QPC current means electrons traveling through the QPC in the direction of the arrows.', '0808.2905-2-11-0': 'To sum up, our structure consists of two barriers defining quantum point contacts, two quantum dots, two barriers determining the coupling of the quantum dots to the source and drain and the barrier that determines the coupling between the quantum dots.', '0808.2905-2-11-1': 'In total, this gives seven degrees of freedom and there are seven independent top-gates used to tune these barriers and the quantum dots.', '0808.2905-2-11-2': 'The top gates (tpg1 and tpg2) are used to tune the DQD into a suitable regime.', '0808.2905-2-11-3': 'The top gates tqc1 and tqc2 can tune the transmission of QPC1 and QPC2, respectively.', '0808.2905-2-11-4': 'The middle top gate tm controls the coupling between the two dots allowing to change smoothly from the single dot regime (large dot spread over the area covered by the two red circles) to a weakly-coupled double dot.', '0808.2905-2-11-5': 'The gates ts and td are used to tune the coupling of the DQD to source and drain.', '0808.2905-2-12-0': 'The potential on both sides of QPC1 (QPC2) can be lifted with respect to the measurement ground, creating a mutual gating effect between DQD and QPC1 (QPC2).', '0808.2905-2-12-1': 'These in-plane gates (ipg1 for QPC1 and ipg2 for QPC2) control the number of electrons on the DQD.', '0808.2905-2-13-0': 'Due to the presence of the metallic top gates, the electrostatic interaction between electrons in the quantum dots and the QPCs is weakened by screening compared to semiconductor-only quantum circuits.', '0808.2905-2-13-1': '[CITATION] The large distance between the QPC and the double dot (lithographic distance 450 nm) further reduces the sensitivity of the QPC for detecting electrons passing through the DQD.', '0808.2905-2-14-0': 'Figures [REF](c) and (d) demonstrate the operation of QPC2 as a charge detector.', '0808.2905-2-14-1': '[CITATION] For both QPCs, the one-dimensional subband spacing is larger than 3.5 mV as estimated from finite bias measurements.', '0808.2905-2-14-2': 'In order to use QPC2 (QPC1) as a charge read-out, its conductance was tuned to [MATH].', '0808.2905-2-14-3': 'A constant voltage of 0.5 mV was applied between the source and drain leads of the QPC, and the current was measured.', '0808.2905-2-14-4': 'An AC voltage of 1.5 mV applied at 34 Hz to the opposite in-plane gate ipg1 (ipg2) modulated the current through QPC2 (QPC1).', '0808.2905-2-14-5': 'This modulated signal which is proportional to the transconductance was detected with lock-in techniques.', '0808.2905-2-14-6': 'The measurements were performed in a dilution refrigerator at a base temperature of 70 mK.', '0808.2905-2-15-0': 'The resulting stability diagram of the DQD detected with QPC2 is shown in Fig. [REF](c).', '0808.2905-2-15-1': 'The boundaries between regions of different ground state charge configurations of the DQD are clearly visible.', '0808.2905-2-15-2': 'In this measurement, tpg2 is used to change the number of electrons in the right dot and ipg1 to change the number of electrons in the left dot.', '0808.2905-2-15-3': 'A few charge rearrangements in the lower half of the honeycomb induced by the metallic top gate tpg1 are present.', '0808.2905-2-15-4': 'In general, we find that the top gate sweeps lead to significantly more charge rearrangements than sweeps of the in-plane gates.', '0808.2905-2-15-5': 'Change of the ipg2 potential combined with simultaneous charge detection would result in strong detuning of the charge sensor.', '0808.2905-2-15-6': 'This would shift the operating point far away from the sensitive regime.', '0808.2905-2-15-7': 'It can be avoided by using tpg2 because it has a weaker lever arm on the QPC.', '0808.2905-2-15-8': 'The thick line in the bottom-left corner of the plot corresponds to a resonance in QPC2.', '0808.2905-2-16-0': 'In Fig. [REF](d) the corresponding DC current through the DQD is plotted.', '0808.2905-2-16-1': 'It was measured simultaneously with the QPC signal presented in the previous paragraph.', '0808.2905-2-16-2': 'The source-drain voltage applied to the DQD is 60 [MATH]V. Only two pairs of triple points are visible.', '0808.2905-2-16-3': 'Similar sets of data can be obtained using QPC1 as the detector.', '0808.2905-2-17-0': '# Experimental data', '0808.2905-2-18-0': 'In the following, we concentrate on a single pair of triple points where the DQD showed moderate coupling.', '0808.2905-2-18-1': 'Figure [REF](a) shows the DC DQD current (I[MATH]) for 100 [MATH]eV source-drain bias applied across the DQD.', '0808.2905-2-18-2': 'The inter-dot mutual capacitance [MATH] estimated from finite bias measurements and from the stability diagram assuming the constant interaction model [CITATION] is 8.8 aF, whereas the total capacitance of the left dot is [MATH] aF and for the right dot, [MATH] aF.', '0808.2905-2-18-3': 'Each dot contains approximately 15 electrons and the charging energies are about 2 meV.', '0808.2905-2-19-0': 'The thin dashed lines in Fig. [REF](a) indicate the boundaries of the honeycomb pattern and the numbers in brackets (M,N) denote the charge population of the left and the right quantum dot, respectively.', '0808.2905-2-19-1': 'Here, the left dot is strongly coupled to the source lead, whereas the right dot is weakly coupled to the drain reservoir.', '0808.2905-2-19-2': 'During the measurement, both QPCs were kept at zero bias.', '0808.2905-2-20-0': 'The detuning marked by the dashed grey (red online) line in Fig. [REF](a) is obtained from the capacitance model [CITATION] and expressed by the equation [MATH] such that the total energy of the DQD, [MATH] remains constant.', '0808.2905-2-20-1': 'The energies [MATH] and [MATH] are the single-particle energies in the left and the right quantum dot, respectively.', '0808.2905-2-20-2': 'Converting the energies to gate voltages gives: [MATH].', '0808.2905-2-20-3': 'The lever arms [MATH] and [MATH] are the lever arms of the in-plane gates ipg2 and ipg2 on the left ([MATH]) or the right ([MATH]) dot, respectively.', '0808.2905-2-20-4': 'The voltages [MATH] and [MATH] are the voltages applied to the gates ipg1 and ipg2.', '0808.2905-2-20-5': 'The lever arms are extracted from measurements at finite bias and from the charge stability diagram of the DQD.', '0808.2905-2-20-6': 'The obtained values are: [MATH], [MATH], [MATH] and [MATH].', '0808.2905-2-20-7': 'We take zero detuning to occur at the triple point.', '0808.2905-2-20-8': 'According to the definition above, detuning is positive (negative) in the upper-left (lower right) part of Fig. [REF](a).', '0808.2905-2-20-9': 'Two representative energy diagrams are shown in the insets.', '0808.2905-2-21-0': 'In Fig. [REF](b) the DQD current was measured in the same parameter range at a QPC2 bias voltage of 1 mV.', '0808.2905-2-21-1': 'The bias voltage across DQD was set to 60 [MATH]eV, i.e., smaller than the bias voltage applied in Fig. [REF](a).', '0808.2905-2-21-2': 'Despite that, the current is strongly enhanced along the boundaries (M,N)[MATH](M+1,N) and (M,N+1)[MATH](M+1,N+1), corresponding to adding an electron to the left dot.', '0808.2905-2-21-3': 'The enhancement of the current along the honeycomb boundaries is induced by driving a current through QPC2.', '0808.2905-2-21-4': 'Another visible feature induced by biasing QPC2 is the finite DQD current in the triangle-shaped area indicated in Fig. [REF](b) that is normally forbidden by Coulomb blockade.', '0808.2905-2-22-0': 'The following measurements were carried out with the QPCs tuned to their first conductance plateau.', '0808.2905-2-22-1': 'The overall experimental results do not depend on this operation point of the QPC.', '0808.2905-2-23-0': 'To investigate the influence of the QPC currents on the DQD in the triangular region, we tuned the levels in the dot along the detuning line depicted as the solid red line in Fig. [REF](a).', '0808.2905-2-23-1': 'Fig. [REF] shows the dot current versus detuning.', '0808.2905-2-23-2': 'The black data points in Fig. [REF](a) were taken with zero bias applied to the DQD as well as to QPC1 and QPC2.', '0808.2905-2-23-3': 'No measurable current above the noise level is detected.', '0808.2905-2-23-4': 'When a DC current of 50 nA is driven through QPC1, an asymmetric peak with a maximum of about 125 fA along the detuning line is observed (blue points in Fig. [REF](a)).', '0808.2905-2-23-5': 'This effect is strongly enhanced if the current through QPC1 is further increased to 75 nA (green points) and 100 nA (red points).', '0808.2905-2-23-6': 'All traces cross zero at the triple point (zero detuning).', '0808.2905-2-24-0': 'A similar, but significantly more pronounced effect is observed if QPC2 is driven, as shown in Fig. [REF](b).', '0808.2905-2-24-1': 'Moreover, for negative detuning a small negative DQD current is observed.', '0808.2905-2-24-2': 'QPC2 is more sensitive as a charge readout and it has a stronger effect on the DQD.', '0808.2905-2-24-3': 'Therefore, we conclude that QPC2 is more strongly coupled to the DQD than QPC1.', '0808.2905-2-25-0': 'We have chosen one point on the detuning line corresponding to [MATH]eV and swept the QPC1 and QPC2 currents.', '0808.2905-2-25-1': 'The results of this measurement are shown in Fig. [REF](a).', '0808.2905-2-25-2': 'The black (red) filled squares correspond to positive currents through QPC1 (QPC2) swept from 0 to 200 nA.', '0808.2905-2-25-3': 'The empty black (red) circles are the traces recorded while the QPC1 (QPC2) current was swept from 0 to [MATH] nA.', '0808.2905-2-25-4': 'The QPC induced DQD current is a little larger in the case when the QPCs are driven with positive current.', '0808.2905-2-25-5': 'This polarity dependence is significant and we can exclude that it is due to a gating effect.', '0808.2905-2-25-6': 'As mentioned before, QPC2 is more strongly coupled to the DQD than QPC1.', '0808.2905-2-25-7': 'When the QPC current is swept in a positive direction (filled symbols in Fig. [REF]) the DQD current starts to level off (the inflection points for are lying between 100 and 150 nA in the QPC current axis), whereas for negative QPC current directions this effect is not clearly visible.', '0808.2905-2-26-0': 'Another unexpected feature is observed on the green traces.', '0808.2905-2-26-1': 'The filled (empty) green squares correspond to the QPC1 current being swept from 0 to 200 ([MATH]) nA while the QPC2 current is simultaneously swept from 0 to 200 nA.', '0808.2905-2-26-2': 'In a simple picture, we would expect that the effects of QPC1 and QPC2 are independent and they add up, but the measurement contradicts this expectation.', '0808.2905-2-26-3': 'Due to the action of both QPCs the DQD current is slightly larger than in the case when only QPC2 is used.', '0808.2905-2-26-4': 'In addition, there is an unexpected polarity dependence with a maximum DQD current for QPC1 being swept in negative and QPC2 in positive direction.', '0808.2905-2-26-5': 'The remaining blue filled squares (empty circles) in Fig. [REF](a) were obtained by driving a positive (negative) current through QPC1 and a negative current through QPC2.', '0808.2905-2-27-0': 'The polarity effect in the DQD current is also present in Fig. [REF](a), where the dot current was plotted versus QPC1 and QPC2 currents at fixed detuning [MATH]eV.', '0808.2905-2-27-1': 'In this measurement, the lack of additivity of the effects induced by the QPCs is even more visible.', '0808.2905-2-28-0': '# Discussion of possible mechanisms', '0808.2905-2-29-0': 'A mechanism which can induce the current flow through the double dot along the detuning line is presented in Fig. [REF](a).', '0808.2905-2-29-1': 'The driving current through QPC1 or/and QPC2 is thought to lead to an emission of energy, which can be absorbed by the electron in the right dot.', '0808.2905-2-29-2': 'If the provided energy matches the energy difference [MATH], the electron can be excited from the right to the left dot.', '0808.2905-2-29-3': 'If the electron leaves the DQD through the left lead and the next electron tunnels into the right dot through the right lead, then the cycle closes and there is a measurable current flowing through the double dot.', '0808.2905-2-30-0': 'An additional enhancement of the DQD current along the honeycomb boundaries as observed in Fig. [REF](b), induced by driving a current through a QPC, can be explained in a similar way.', '0808.2905-2-30-1': 'In the situation shown in the upper inset the electron trapped in the right quantum dot can absorb energy emitted by the QPC2, tunnel into the left dot and leave the DQD system via the left lead.', '0808.2905-2-30-2': 'The cycle closes when the next electron tunnels into the right dot from the right lead.', '0808.2905-2-30-3': 'This QPC2 induced process gives an additional contribution to the DQD current.', '0808.2905-2-30-4': 'This effect is more pronounced in the vicinity of the triple point where energy difference between the levels [MATH] and [MATH] in the left and the right dots are small.', '0808.2905-2-30-5': 'The lower inset of Fig. [REF](b) shows the analogous diagram for the situation when the level in the right quantum dot lies above the Fermi energy of the leads.', '0808.2905-2-30-6': 'Again, the QPC2 induced process causes the electrons to move from the right into the left contact.', '0808.2905-2-31-0': 'The possible mechanisms of the pumping effect are coupling to acoustic or optical phonons, plasmons, photons, shot-noise or thermopower effect.', '0808.2905-2-31-1': 'Scattering with optical phonons is strongly suppressed as long as the relevant energy scales are smaller than the optical phonon energy.', '0808.2905-2-31-2': '[CITATION] Coupling to plasmons can be ruled out as well.', '0808.2905-2-31-3': '[CITATION] We can also exclude the shot-noise as a source of the energy, because during the experiment both QPCs were tuned to their first plateau.', '0808.2905-2-31-4': 'Measurements performed at 0.5[MATH] and 1.5[MATH]) showed a qualitatively and quantitatively similar behavior.', '0808.2905-2-31-5': 'This is in contrast to previously measured data [CITATION] where no DQD current was observed in the plateau regions.', '0808.2905-2-31-6': 'Coupling to acoustic phonons is the most likely mechanism of inducing the current in the DQD.', '0808.2905-2-31-7': 'Further below in this paper, we discuss the data in the light of phonon coupling and a related thermopower effect.', '0808.2905-2-32-0': 'The questions arising from the data presented above are the following: is the strong difference of the peak heights on the positive and negative side of the detuning (Fig. [REF]) due to the asymmetry in the DQD coupling to the leads?', '0808.2905-2-32-1': 'Why do the effects of the QPCs not add up?', '0808.2905-2-32-2': 'What is the reason for the saturation of the DQD current observed in Fig. [REF](a)?', '0808.2905-2-32-3': 'What is the mechanism of the energy transfer from the QPCs to the DQD?', '0808.2905-2-32-4': 'Can it explain the polarity dependence?', '0808.2905-2-32-5': 'In the next section we present a model that attempts to answer most of these questions.', '0808.2905-2-33-0': '# The model', '0808.2905-2-34-0': 'In the following we derive a scenario, which explains the pumping effect based on electron-phonon interaction.', '0808.2905-2-34-1': 'First, we introduce the two-level system describing the DQD.', '0808.2905-2-34-2': 'Then, we consider all possible transitions between different energy states of the DQD and express them in terms of tunneling rates.', '0808.2905-2-34-3': 'Subsequently, we derive the energy dependence of the tunneling rates.', '0808.2905-2-34-4': 'The intra-dot transitions are calculated in a framework of electron-phonon interaction.', '0808.2905-2-34-5': 'Next, we set up a master equation and obtain the complete expression for the DQD current as a function of detuning and the temperature of the phonon bath.', '0808.2905-2-35-0': 'Close to a pair of triple points, a double quantum dot can be regarded as a two-level system,[CITATION] whose bonding (ground) and antibonding (excited) states [MATH] and [MATH] are separated by an energy [MATH] as shown in Fig. [REF](b),[CITATION] where [MATH] is tunneling coupling between the dots.', '0808.2905-2-35-1': 'The corresponding eigenvectors for bonding and antibonding states are [MATH] and [MATH].', '0808.2905-2-35-2': 'The components of the bonding and antibonding eigenstates in the basis of [MATH] and [MATH], the wave functions in the left ([MATH]) and the right ([MATH]) dot, are [MATH], where [MATH] and [MATH].', '0808.2905-2-36-0': 'As in Sec. [REF] the detuning is defined as [MATH] and the total energy is [MATH].', '0808.2905-2-36-1': 'We assume, that the number of the electrons in the quantum dots is fixed and its ground state energy is [MATH].', '0808.2905-2-36-2': 'If we add one extra electron called an excess electron the ground state energy will be [MATH], while for two excess electrons, the ground state energy is [MATH].', '0808.2905-2-36-3': 'The corresponding electrochemical potentials are [MATH] and [MATH].', '0808.2905-2-37-0': 'In the vicinity of a pair of triple points a double quantum dot can have one out of four different charge states.', '0808.2905-2-37-1': 'These different charge configurations are presented in Fig. [REF].', '0808.2905-2-37-2': 'The "empty" state corresponds to a situation where there is no excess electron present in a dot and the occupation probability of this state is [MATH].', '0808.2905-2-37-3': 'The index GS0 denotes the zero-electron ground state.', '0808.2905-2-37-4': 'In addition, one excess electron may occupy the bonding (ground) state with probability [MATH] or the antibonding (excited) state with probability [MATH].', '0808.2905-2-37-5': 'The last possible charge configuration is when there are two excess electrons in a double quantum dot (two-electron ground state) with occupation probability [MATH].', '0808.2905-2-38-0': 'The transitions between these states are determined by the tunneling rates [MATH] and the thermal broadening of the Fermi function [MATH] in the leads.', '0808.2905-2-38-1': 'The index [MATH]=L,R denotes the left (L) or the right (R) barrier through which the electron tunnels and the index [MATH]=0,1,2,3 labels the transition (see Fig. [REF]).', '0808.2905-2-38-2': 'For example, if the dot is in a zero-electron ground state (GS0) and the electron tunnels in via the right lead, the corresponding rate is [MATH], as shown in Fig. [REF].', '0808.2905-2-38-3': 'For [MATH]=0(1) the Fermi function is [MATH] where [MATH] is the Boltzmann constant and [MATH] is the temperature of the lead.', '0808.2905-2-38-4': 'For [MATH]=2,3 the expression is analogous but the energy is lifted by the mutual charging energy [MATH].', '0808.2905-2-39-0': 'In order to explain the experimental data presented above, we have to take into account that the tunneling rates [MATH] do depend on the electronic wave function.', '0808.2905-2-39-1': 'They can be expressed as [MATH].', '0808.2905-2-39-2': 'The coefficients [MATH] are the left ([MATH] and right ([MATH]) energy dependent components of the eigenvector of the wave function corresponding to the bonding ([MATH]=0,3) and antibonding ([MATH]=1,2) states.', '0808.2905-2-39-3': 'The amplitudes [MATH] are energy-independent parts of the [MATH].', '0808.2905-2-40-0': 'The rates describing the inter-dot processes, that is absorption [MATH] and emission [MATH] are marked in Fig. [REF].', '0808.2905-2-40-1': 'In the following, we assume that the double quantum dot is coupled to a bosonic bath in thermal equilibrium described by the Bose-Einstein distribution function [MATH].', '0808.2905-2-40-2': 'The temperature [MATH] of this bath is determined by the current of the QPC and the base temperature of the cryostat.', '0808.2905-2-40-3': 'In case of coupling to acoustic phonons, the emission and absorption rates can be expressed as (derivation is presented in Appendix A): [EQUATION] where the index [MATH] denotes piezoelectric transversal phonons (pe,T), piezoelectric longitudinal phonons (pe,L) or longitudinal deformation potential coupling phonons (dp,L).', '0808.2905-2-40-4': 'The exponent [MATH] is 1 for piezoelectric phonons and [MATH]=3 for deformation potential coupling.', '0808.2905-2-40-5': '[CITATION] The upper (lower) sign stands for emission (absorption) of energy quanta.', '0808.2905-2-40-6': 'The values of the energy-independent coefficients [MATH] are given in Appendix [REF].', '0808.2905-2-40-7': 'The form-factor [MATH] is represented as: [EQUATION] where [MATH] denotes the radius of a single QD, [MATH] is the distance between the dots, [MATH] is the speed of sound of [MATH]-phonons.', '0808.2905-2-40-8': 'The complete expression for the double-dot geometry factor [MATH] is given in Appendix [REF].', '0808.2905-2-40-9': 'In Eq. ([REF]) the first factor [MATH] is related to the symmetry of the double quantum dot wave function.', '0808.2905-2-40-10': 'It suppresses transitions for the asymmetric system.', '0808.2905-2-40-11': 'The second term of Eq. ([REF]) refers to the shape of the individual dots.', '0808.2905-2-40-12': 'It gives a high energy cut-off for phonon wavelengths much smaller than the size of a single dot [MATH].', '0808.2905-2-40-13': 'The last term of Eq. ([REF]) arises from the separation of the single dots in the double dot system.', '0808.2905-2-40-14': 'It suppresses small energy absorption for phonon wavelengths much larger than [MATH].', '0808.2905-2-40-15': 'Only phonons with a wavelength comparable or larger than the DQD separation can interact with the electron.', '0808.2905-2-40-16': 'For large energies this term has oscillatory behavior[CITATION].', '0808.2905-2-41-0': 'To investigate the influence of the QPCs on the dot presented in Fig. [REF] in terms of rates and occupation probabilities defined in Fig. [REF] we only take into account the processes surrounded by the dashed line.', '0808.2905-2-41-1': 'This is reasonable, due to a large mutual charging energy relative to the tunneling coupling.', '0808.2905-2-41-2': 'Counting the electrons passing through the right barrier leads to the following expression of the current through the double dot (the derivation is presented in Appendix [REF]): [EQUATION]', '0808.2905-2-41-3': 'The occupation probabilities [MATH], [MATH] and [MATH] are functions of [MATH] and [MATH] resulting from a stationary solution of the master equation (see Appendix [REF]).', '0808.2905-2-42-0': '# Results and interpretation', '0808.2905-2-43-0': 'We used expression ([REF]) to fit the data shown in Fig. [REF].', '0808.2905-2-43-1': 'During the fitting procedure, the 8 traces shown in Fig. [REF](a) and (b) and an additional set of 16 traces being a combination of [MATH] nA were fitted simultaneously.', '0808.2905-2-43-2': 'The following fitting parameters were shared: amplitudes [MATH], [MATH] and tunneling coupling [MATH].', '0808.2905-2-43-3': 'The only parameter specific for each trace was the temperature of the phonon bath [MATH].', '0808.2905-2-44-0': 'The fit represents the overall shape of the measured data very well.', '0808.2905-2-44-1': 'The extracted tunneling coupling is [MATH]eV.', '0808.2905-2-44-2': 'In Fig. [REF](a) the grey (green online) solid line is the calculated boundary between the honeycomb cells assuming a tunneling coupling of 50 [MATH]eV, the black (red online) crosses are the maxima position of the DQD current peaks and the black dashed line is the boundary assuming [MATH].', '0808.2905-2-44-3': 'Unfortunately, the sample was not stable enough to map a stability diagram with a resolution high enough to determine the tunneling coupling directly.', '0808.2905-2-44-4': 'However, the [MATH] obtained from the fits seems to be reasonable and matches with the data presented on the stability diagram.', '0808.2905-2-45-0': 'The temperature of the phonon bath [MATH] obtained from the fits varies from 0.6 K (blue trace in Fig. [REF](a)) to 1.2 K (red trace in Fig. [REF](b)).', '0808.2905-2-45-1': 'The difference between the temperatures [MATH] and the electronic temperature [MATH]=100 mK gives rise to the DQD current.', '0808.2905-2-46-0': 'The extracted amplitude [MATH] is of the order of 7.8 MHz and [MATH] is about 0.5 GHz.', '0808.2905-2-46-1': 'This is in agreement with our previous statement that the right barrier is more opaque than the left one.', '0808.2905-2-46-2': 'This leads to a deviation from perfect antisymmetry of the dot current along the detuning line, i.e., to suppressed dot current for negative detuning.', '0808.2905-2-47-0': 'The tail of the curve at large detuning is mainly determined by the amplitudes [MATH] present in Eq. ([REF]) that drop to zero like [MATH] and the Bose-Einstein distribution function given in Eq. ([REF]).', '0808.2905-2-47-1': 'Although, the real distribution of the bosonic environment is not necessarily equilibrated and could have another form, the qualitative agreement with the data does not depend strongly on the details of this distribution.', '0808.2905-2-48-0': 'Using the parameters obtained in the fits, we calculated the current as a function of the gate voltages using the model containing all four charge states presented in Fig. [REF].', '0808.2905-2-48-1': 'In the absence of the QPC current ([MATH]=70 mK), the result is shown in Fig. [REF](c).', '0808.2905-2-48-2': 'The asymmetry and magnitude of the current along the upper dashed line is reproduced very well.', '0808.2905-2-48-3': 'This is not the case for the lower branch indicating transitions [MATH].', '0808.2905-2-48-4': 'This may be due to a change of [MATH] and [MATH], which we assumed to be constant in our model but which may change in the experiment.', '0808.2905-2-48-5': 'For 1 mV DC bias across the QPC2 (Fig. [REF](b)), the bosonic temperature is around [MATH]=0.8 K.', '0808.2905-2-48-6': 'The corresponding calculations are shown in Fig. [REF](d).', '0808.2905-2-48-7': 'The red triangle indicates the region with the QPC induced current that is in agreement with the measured data in (b).', '0808.2905-2-49-0': 'To investigate the dependence of the phonon temperature on the QPC current we calculated [MATH] for every point from Fig. [REF](a) using the values obtained in previous fits.', '0808.2905-2-49-1': 'The results are shown in Fig. [REF](b).', '0808.2905-2-49-2': 'For small QPC currents the error bars are large and no clear dependence is visible.', '0808.2905-2-49-3': 'For QPC currents above 50 nA the dependence is quadratic [MATH], which means that the temperature of the bosonic bath is proportional to the power emitted by the QPC.', '0808.2905-2-49-4': 'The reconstructed temperatures corresponding to the measurement presented in Fig. [REF](a) are plotted in Fig. [REF](b).', '0808.2905-2-49-5': 'For very small DQD current it is impossible to estimate the temperature of the phonon bath with sufficient accuracy.', '0808.2905-2-50-0': 'The saturation of the DQD current for large QPC currents cannot be attributed to the high occupation probability of the antibonding state [MATH].', '0808.2905-2-50-1': 'We have estimated that the [MATH] value does not exceed a few percent and most of the time the dot is occupied by one electron in its ground state.', '0808.2905-2-50-2': 'The maximum current is determined by the right tunneling barrier.', '0808.2905-2-51-0': 'A series of experiments reporting the observation of a DQD current induced by a single and independently biased QPC, is described in Refs. [CITATION].', '0808.2905-2-51-1': 'These experiments were performed in the regime of large DQD current, strong bias voltage applied to the QPC and large tunneling coupling, which is similar to our situation here.', '0808.2905-2-51-2': 'The DQD current was related to inelastic relaxation of electrons in partly transmitting 1D channels of the QPC [CITATION] and qualitatively consistent with an energy transfer mechanism based on nonequilibrium acoustic phonons.', '0808.2905-2-51-3': '[CITATION] In contrast to our experimental data, in these experiments the DQD current was large when the conductance of the QPC was tuned to [MATH] and strongly suppressed in plateau regions.', '0808.2905-2-52-0': 'Recent research has proven that the absorption of a photon can be the dominant process [CITATION] in similar situations.', '0808.2905-2-52-1': 'However, these time-resolved experiments were performed in a different regime, where the dominant tunneling rates are of order of 1 kHz, whereas in our system the double quantum dot is much more strongly coupled to the leads.', '0808.2905-2-52-2': 'Another difference is the presence of a Ti top-gate in our structure, that screens the electrostatic interaction between the DQD and the QPCs.', '0808.2905-2-52-3': 'Due to the lower sensitivity of direct dot current measurements compared to the time-resolved technique it is not possible to observe the gap in the DQD current when [MATH].', '0808.2905-2-52-4': 'Calculations of the emission and absorption rates in a DQD induced by electron-photon interaction show, that the effect is irrelevant compared to the emission and absorption of phonons discussed here.', '0808.2905-2-52-5': '[CITATION]', '0808.2905-2-53-0': 'We have also tested the possibility that the entire dot current in the region forbidden by the Coulomb blockade is due to a thermopower effect induced by different temperatures in the source and the drain.', '0808.2905-2-53-1': 'We found, that it would be only possible if the temperature difference between source and drain lead was larger than 1K for a QPC current of 100 nA, which is one order of magnitude larger than expected.', '0808.2905-2-53-2': '[CITATION] Another argument against a thermopower model is that always the drain lead would have to be warmer, even if the far QPC1 (that couples better to the drain lead) was biased.', '0808.2905-2-53-3': 'Even so, the thermopower model did not describe the data as well as the emission/absorption model.', '0808.2905-2-54-0': '# Conclusions', '0808.2905-2-55-0': 'We have presented the influence of two independent quantum point contacts on a double quantum dot.', '0808.2905-2-55-1': 'A number of of questions arising during the investigation and presented in Sec. [REF] could be answered.', '0808.2905-2-56-0': 'In the first place, we established the possible dominant mechanism of energy transfer between the double quantum dot and the quantum point contact in the investigated regime.', '0808.2905-2-56-1': 'Driving current through the QPC leads to emission of energy that increases the temperature of the bosonic environment.', '0808.2905-2-56-2': 'We identify these bosons as acoustic phonons.', '0808.2905-2-56-3': 'To model the interaction of the phonons with the double quantum dot we have assumed that their energy distribution is described by Bose-Einstein statistics.', '0808.2905-2-57-0': 'Another important point is a non-additive effect of both QPC currents.', '0808.2905-2-57-1': 'It is understood in terms of the temperature of the bosonic bath.', '0808.2905-2-57-2': 'We find that the DQD current is proportional to the power emitted by the QPCs.', '0808.2905-2-58-0': 'Next, we interpreted the leveling off of the DQD current as a QPC current is increased.', '0808.2905-2-58-1': 'For large QPC powers (above 0.1 nW) the temperature of the phonon bath increases linearly.', '0808.2905-2-58-2': 'The observed saturation of the current is due to the finite transparency of the tunneling barriers and not to the high occupation probabilities [MATH].', '0808.2905-2-59-0': 'The polarity dependence (Fig. [REF]) cannot be explained within the discussed model and its origin remains to be investigated.', '0808.2905-2-59-1': 'It would be interesting to further investigate this effect, for example, by using different geometrical arrangements.', '0808.2905-2-60-0': 'Finally, we observed strong deviation from perfect antisymmetry of the dot current along the detuning line when the QPC current is driven.', '0808.2905-2-60-1': 'It can be attributed to the asymmetry of the source and drain barriers.', '0808.2905-2-61-0': 'All the measurements were performed with both QPCs tuned to their first plateau.', '0808.2905-2-61-1': 'Thus we can exclude the influence of shot-noise phenomena in the quantum point contacts.', '0808.2905-2-62-0': '# Absorption and emission rates in a DQD induced by electron-phonon interaction', '0808.2905-2-63-0': "Generally, the emission and absorption rates can be expressed using Fermi's golden rule: [EQUATION] where the sum extends over all wave vectors [MATH].", '0808.2905-2-63-1': 'The index [MATH] denotes the type of acoustic phonons and their coupling in GaAs: piezoelectric longitudinal (pe,L), piezoelectric transversal (pe,T) and longitudinal, deformation potential coupling (dp,L).', '0808.2905-2-63-2': 'The phonons have linear dispersion relation [MATH].', '0808.2905-2-63-3': '[MATH] and [MATH] are wave functions of bonding and antibonding states separated by energy [MATH].', '0808.2905-2-63-4': 'The interaction hamiltonian [MATH] can be written as a sum of piezoelectric interaction [MATH] and deformation potential coupling [MATH]: [EQUATION] with [EQUATION]', '0808.2905-2-63-5': 'In above equations [MATH] is an element of piezoelectric tensor, [MATH] is the vacuum permittivity, [MATH] is the dielectric constant, [MATH] is the number of atoms in the crystal, M is the atomic mass and D denotes deformation potential coupling constant.', '0808.2905-2-63-6': 'The dimensionless function [MATH] has form: [EQUATION] where [MATH] is the Levi-Civita symbol and [MATH] is the [MATH]-component of the eigenvector associated with mode [MATH].', '0808.2905-2-63-7': 'Inserting Eq. ([REF]) into Eq. ([REF]) leads to the following expression: [EQUATION] where the upper sign refers to phonon absorption and the lower to phonon emission.', '0808.2905-2-63-8': 'The constants [MATH] and [MATH] are given by: [EQUATION] where [MATH] is a density of the GaAs crystal.', '0808.2905-2-63-9': 'The energy dependent functions [MATH] and [MATH] are defined as: [EQUATION]', '0808.2905-2-63-10': 'Assuming negligible overlap between the wave functions of the two dots and taking a gaussian-shaped single-electron wave function, the matrix element is found to be: [EQUATION] where [MATH] is the distance between the dots and [MATH] is the radius of a single dot.', '0808.2905-2-63-11': 'Inserting ([REF]) into Eqs. ([REF]) gives: [EQUATION]', '0808.2905-2-63-12': 'For piezoelectric transversal phonons the above expression was calculated by averaging the function [MATH] over all possible transversal directions.', '0808.2905-2-63-13': 'The geometry factors [MATH] are given by: [EQUATION]', '0808.2905-2-63-14': 'Combining Eq. ([REF]) with Eq. ([REF]) and inserting the result into Eq. ([REF]) gives a complete expression for the absorption and emission rates.', '0808.2905-2-64-0': '# Rate equation', '0808.2905-2-65-0': 'To relate the DQD current to the tunneling rates we write down the rate equation for the occupation of the states: [EQUATION] with additional condition [MATH].', '0808.2905-2-65-1': 'The terms [MATH] and [MATH] are defined as: [EQUATION]', '0808.2905-2-65-2': 'To find the expression for the current flowing through a DQD, we take the right barrier as a current reference.', '0808.2905-2-65-3': 'It means that, if an electron passes the right barrier to the left (right), its contribution to the DQD current is positive (negative).', '0808.2905-2-65-4': '[EQUATION]', '0808.2905-2-65-5': 'The first (second) term of Eq. ([REF]) corresponds to the electrons moving from the bonding (antibonding) state to the right lead and the third and fourth term to the electrons entering the ground or excited state of the dot from the right lead.', '0808.2905-2-65-6': 'By inserting a stationary solution of Eq. ([REF]) into Eq. ([REF]) one obtains an expression for the steady state DQD current.'}

[['0808.2905-1-35-0', '0808.2905-2-35-0'], ['0808.2905-1-35-1', '0808.2905-2-35-1'], ['0808.2905-1-23-0', '0808.2905-2-23-0'], ['0808.2905-1-23-1', '0808.2905-2-23-1'], ['0808.2905-1-23-2', '0808.2905-2-23-2'], ['0808.2905-1-23-3', '0808.2905-2-23-3'], ['0808.2905-1-23-5', '0808.2905-2-23-5'], ['0808.2905-1-23-6', '0808.2905-2-23-6'], ['0808.2905-1-31-0', '0808.2905-2-31-0'], ['0808.2905-1-31-1', '0808.2905-2-31-1'], ['0808.2905-1-31-2', '0808.2905-2-31-2'], ['0808.2905-1-31-3', '0808.2905-2-31-3'], ['0808.2905-1-31-4', '0808.2905-2-31-4'], ['0808.2905-1-9-0', '0808.2905-2-9-0'], ['0808.2905-1-9-1', '0808.2905-2-9-1'], ['0808.2905-1-9-3', '0808.2905-2-9-3'], ['0808.2905-1-59-0', '0808.2905-2-65-0'], ['0808.2905-1-59-1', '0808.2905-2-65-1'], ['0808.2905-1-59-2', '0808.2905-2-65-2'], ['0808.2905-1-59-3', '0808.2905-2-65-3'], ['0808.2905-1-59-5', '0808.2905-2-65-5'], ['0808.2905-1-59-6', '0808.2905-2-65-6'], ['0808.2905-1-39-1', '0808.2905-2-39-1'], ['0808.2905-1-39-2', '0808.2905-2-39-2'], ['0808.2905-1-39-3', '0808.2905-2-39-3'], ['0808.2905-1-13-0', '0808.2905-2-13-0'], ['0808.2905-1-6-0', '0808.2905-2-6-0'], ['0808.2905-1-6-1', '0808.2905-2-6-1'], ['0808.2905-1-6-3', '0808.2905-2-6-3'], ['0808.2905-1-6-4', '0808.2905-2-6-4'], ['0808.2905-1-6-5', '0808.2905-2-6-5'], ['0808.2905-1-46-0', '0808.2905-2-46-0'], ['0808.2905-1-46-1', '0808.2905-2-46-1'], ['0808.2905-1-46-2', '0808.2905-2-46-2'], ['0808.2905-1-57-0', '0808.2905-2-63-0'], ['0808.2905-1-57-1', '0808.2905-2-63-1'], ['0808.2905-1-57-2', '0808.2905-2-63-2'], ['0808.2905-1-57-3', '0808.2905-2-63-3'], ['0808.2905-1-57-4', '0808.2905-2-63-4'], ['0808.2905-1-57-5', '0808.2905-2-63-5'], ['0808.2905-1-57-7', '0808.2905-2-63-7'], ['0808.2905-1-57-9', '0808.2905-2-63-9'], ['0808.2905-1-57-10', '0808.2905-2-63-10'], ['0808.2905-1-57-12', '0808.2905-2-63-12'], ['0808.2905-1-57-13', '0808.2905-2-63-13'], ['0808.2905-1-57-14', '0808.2905-2-63-14'], ['0808.2905-1-3-0', '0808.2905-2-3-0'], ['0808.2905-1-3-1', '0808.2905-2-3-1'], ['0808.2905-1-3-2', '0808.2905-2-3-2'], ['0808.2905-1-3-3', '0808.2905-2-3-3'], ['0808.2905-1-41-0', '0808.2905-2-41-0'], ['0808.2905-1-41-1', '0808.2905-2-41-1'], ['0808.2905-1-41-2', '0808.2905-2-41-2'], ['0808.2905-1-47-1', '0808.2905-2-47-1'], ['0808.2905-1-38-0', '0808.2905-2-38-0'], ['0808.2905-1-38-3', '0808.2905-2-38-3'], ['0808.2905-1-38-4', '0808.2905-2-38-4'], ['0808.2905-1-20-2', '0808.2905-2-20-2'], ['0808.2905-1-20-4', '0808.2905-2-20-5'], ['0808.2905-1-20-5', '0808.2905-2-20-7'], ['0808.2905-1-20-7', '0808.2905-2-20-9'], ['0808.2905-1-15-0', '0808.2905-2-15-0'], ['0808.2905-1-15-1', '0808.2905-2-15-1'], ['0808.2905-1-15-3', '0808.2905-2-15-4'], ['0808.2905-1-15-4', '0808.2905-2-15-8'], ['0808.2905-1-32-0', '0808.2905-2-32-0'], ['0808.2905-1-32-1', '0808.2905-2-32-1'], ['0808.2905-1-32-2', '0808.2905-2-32-2'], ['0808.2905-1-32-3', '0808.2905-2-32-3'], ['0808.2905-1-32-4', '0808.2905-2-32-4'], ['0808.2905-1-32-5', '0808.2905-2-32-5'], ['0808.2905-1-37-0', '0808.2905-2-37-0'], ['0808.2905-1-37-1', '0808.2905-2-37-1'], ['0808.2905-1-37-2', '0808.2905-2-37-2'], ['0808.2905-1-45-0', '0808.2905-2-45-0'], ['0808.2905-1-51-0', '0808.2905-2-51-0'], ['0808.2905-1-16-1', '0808.2905-2-16-2'], ['0808.2905-1-16-2', '0808.2905-2-16-3'], ['0808.2905-1-5-1', '0808.2905-2-5-1'], ['0808.2905-1-5-2', '0808.2905-2-5-2'], ['0808.2905-1-5-3', '0808.2905-2-5-3'], ['0808.2905-1-5-4', '0808.2905-2-5-4'], ['0808.2905-1-5-5', '0808.2905-2-5-5'], ['0808.2905-1-53-0', '0808.2905-2-53-0'], ['0808.2905-1-53-1', '0808.2905-2-53-1'], ['0808.2905-1-53-2', '0808.2905-2-53-2'], ['0808.2905-1-53-3', '0808.2905-2-53-3'], ['0808.2905-1-0-3', '0808.2905-2-0-3'], ['0808.2905-1-0-4', '0808.2905-2-0-4'], ['0808.2905-1-4-0', '0808.2905-2-4-0'], ['0808.2905-1-4-1', '0808.2905-2-4-1'], ['0808.2905-1-27-1', '0808.2905-2-27-1'], ['0808.2905-1-2-0', '0808.2905-2-2-0'], ['0808.2905-1-2-1', '0808.2905-2-2-1'], ['0808.2905-1-2-2', '0808.2905-2-2-2'], ['0808.2905-1-2-3', '0808.2905-2-2-3'], ['0808.2905-1-21-0', '0808.2905-2-21-0'], ['0808.2905-1-21-1', '0808.2905-2-21-1'], ['0808.2905-1-22-0', '0808.2905-2-22-0'], ['0808.2905-1-22-1', '0808.2905-2-22-1'], ['0808.2905-1-25-0', '0808.2905-2-25-0'], ['0808.2905-1-25-1', '0808.2905-2-25-1'], ['0808.2905-1-25-2', '0808.2905-2-25-2'], ['0808.2905-1-25-4', '0808.2905-2-25-4'], ['0808.2905-1-25-5', '0808.2905-2-25-5'], ['0808.2905-1-25-6', '0808.2905-2-25-6'], ['0808.2905-1-49-0', '0808.2905-2-49-0'], ['0808.2905-1-49-1', '0808.2905-2-49-1'], ['0808.2905-1-49-2', '0808.2905-2-49-2'], ['0808.2905-1-49-3', '0808.2905-2-49-3'], ['0808.2905-1-49-4', '0808.2905-2-49-4'], ['0808.2905-1-43-0', '0808.2905-2-43-0'], ['0808.2905-1-43-1', '0808.2905-2-43-1'], ['0808.2905-1-43-2', '0808.2905-2-43-2'], ['0808.2905-1-43-3', '0808.2905-2-43-3'], ['0808.2905-1-50-0', '0808.2905-2-50-0'], ['0808.2905-1-50-1', '0808.2905-2-50-1'], ['0808.2905-1-50-2', '0808.2905-2-50-2'], ['0808.2905-1-29-0', '0808.2905-2-29-0'], ['0808.2905-1-29-1', '0808.2905-2-29-1'], ['0808.2905-1-29-2', '0808.2905-2-29-2'], ['0808.2905-1-29-3', '0808.2905-2-29-3'], ['0808.2905-1-26-0', '0808.2905-2-26-0'], ['0808.2905-1-26-2', '0808.2905-2-26-2'], ['0808.2905-1-26-3', '0808.2905-2-26-3'], ['0808.2905-1-26-4', '0808.2905-2-26-4'], ['0808.2905-1-26-5', '0808.2905-2-26-5'], ['0808.2905-1-44-0', '0808.2905-2-44-0'], ['0808.2905-1-44-1', '0808.2905-2-44-1'], ['0808.2905-1-44-3', '0808.2905-2-44-3'], ['0808.2905-1-44-4', '0808.2905-2-44-4'], ['0808.2905-1-10-0', '0808.2905-2-10-0'], ['0808.2905-1-10-3', '0808.2905-2-10-3'], ['0808.2905-1-10-4', '0808.2905-2-10-4'], ['0808.2905-1-48-0', '0808.2905-2-48-0'], ['0808.2905-1-48-1', '0808.2905-2-48-1'], ['0808.2905-1-48-4', '0808.2905-2-48-5'], ['0808.2905-1-48-5', '0808.2905-2-48-6'], ['0808.2905-1-18-0', '0808.2905-2-18-0'], ['0808.2905-1-18-1', '0808.2905-2-18-1'], ['0808.2905-1-18-3', '0808.2905-2-18-3'], ['0808.2905-1-14-0', '0808.2905-2-14-0'], ['0808.2905-1-14-2', '0808.2905-2-14-2'], ['0808.2905-1-14-6', '0808.2905-2-14-6'], ['0808.2905-1-8-0', '0808.2905-2-8-0'], ['0808.2905-1-8-1', '0808.2905-2-8-1'], ['0808.2905-1-8-2', '0808.2905-2-8-2'], ['0808.2905-1-24-0', '0808.2905-2-24-0'], ['0808.2905-1-24-1', '0808.2905-2-24-1'], ['0808.2905-1-24-2', '0808.2905-2-24-2'], ['0808.2905-1-24-3', '0808.2905-2-24-3'], ['0808.2905-1-36-2', '0808.2905-2-36-3'], ['0808.2905-1-52-0', '0808.2905-2-52-0'], ['0808.2905-1-52-1', '0808.2905-2-52-1'], ['0808.2905-1-52-2', '0808.2905-2-52-2'], ['0808.2905-1-52-4', '0808.2905-2-52-4'], ['0808.2905-1-34-0', '0808.2905-2-34-0'], ['0808.2905-1-34-1', '0808.2905-2-34-1'], ['0808.2905-1-34-2', '0808.2905-2-34-2'], ['0808.2905-1-34-3', '0808.2905-2-34-3'], ['0808.2905-1-34-4', '0808.2905-2-34-4'], ['0808.2905-1-40-0', '0808.2905-2-40-0'], ['0808.2905-1-40-1', '0808.2905-2-40-1'], ['0808.2905-1-40-2', '0808.2905-2-40-2'], ['0808.2905-1-40-4', '0808.2905-2-40-4'], ['0808.2905-1-40-5', '0808.2905-2-40-5'], ['0808.2905-1-40-6', '0808.2905-2-40-6'], ['0808.2905-1-40-7', '0808.2905-2-40-7'], ['0808.2905-1-40-8', '0808.2905-2-40-8'], ['0808.2905-1-12-0', '0808.2905-2-12-0'], ['0808.2905-1-55-6', '0808.2905-2-59-0'], ['0808.2905-1-55-7', '0808.2905-2-61-0'], ['0808.2905-1-55-8', '0808.2905-2-61-1'], ['0808.2905-1-23-4', '0808.2905-2-23-4'], ['0808.2905-1-19-0', '0808.2905-2-19-0'], ['0808.2905-1-19-2', '0808.2905-2-19-2'], ['0808.2905-1-31-5', '0808.2905-2-31-5'], ['0808.2905-1-31-6', '0808.2905-2-31-6'], ['0808.2905-1-31-7', '0808.2905-2-31-7'], ['0808.2905-1-39-0', '0808.2905-2-39-0'], ['0808.2905-1-13-1', '0808.2905-2-13-1'], ['0808.2905-1-6-2', '0808.2905-2-6-2'], ['0808.2905-1-57-6', '0808.2905-2-63-6'], ['0808.2905-1-57-8', '0808.2905-2-63-8'], ['0808.2905-1-41-3', '0808.2905-2-41-3'], ['0808.2905-1-47-0', '0808.2905-2-47-0'], ['0808.2905-1-38-1', '0808.2905-2-38-1'], ['0808.2905-1-38-2', '0808.2905-2-38-2'], ['0808.2905-1-30-0', '0808.2905-2-30-0'], ['0808.2905-1-30-1', '0808.2905-2-30-1'], ['0808.2905-1-30-2', '0808.2905-2-30-2'], ['0808.2905-1-30-3', '0808.2905-2-30-3'], ['0808.2905-1-30-4', '0808.2905-2-30-4'], ['0808.2905-1-30-5', '0808.2905-2-30-5'], ['0808.2905-1-30-6', '0808.2905-2-30-6'], ['0808.2905-1-20-0', '0808.2905-2-20-0'], ['0808.2905-1-20-6', '0808.2905-2-20-8'], ['0808.2905-1-15-2', '0808.2905-2-15-3'], ['0808.2905-1-37-3', '0808.2905-2-37-4'], ['0808.2905-1-37-4', '0808.2905-2-37-5'], ['0808.2905-1-45-1', '0808.2905-2-45-1'], ['0808.2905-1-51-1', '0808.2905-2-51-1'], ['0808.2905-1-51-2', '0808.2905-2-51-2'], ['0808.2905-1-51-3', '0808.2905-2-51-3'], ['0808.2905-1-5-0', '0808.2905-2-5-0'], ['0808.2905-1-5-6', '0808.2905-2-5-6'], ['0808.2905-1-0-0', '0808.2905-2-0-0'], ['0808.2905-1-0-1', '0808.2905-2-0-1'], ['0808.2905-1-0-2', '0808.2905-2-0-2'], ['0808.2905-1-27-0', '0808.2905-2-27-0'], ['0808.2905-1-21-2', '0808.2905-2-21-2'], ['0808.2905-1-21-3', '0808.2905-2-21-3'], ['0808.2905-1-21-4', '0808.2905-2-21-4'], ['0808.2905-1-25-3', '0808.2905-2-25-3'], ['0808.2905-1-26-1', '0808.2905-2-26-1'], ['0808.2905-1-44-2', '0808.2905-2-44-2'], ['0808.2905-1-10-1', '0808.2905-2-10-1'], ['0808.2905-1-10-2', '0808.2905-2-10-2'], ['0808.2905-1-48-2', '0808.2905-2-48-2'], ['0808.2905-1-48-6', '0808.2905-2-48-7'], ['0808.2905-1-18-2', '0808.2905-2-18-2'], ['0808.2905-1-11-0', '0808.2905-2-11-2'], ['0808.2905-1-11-1', '0808.2905-2-11-3'], ['0808.2905-1-11-2', '0808.2905-2-11-4'], ['0808.2905-1-11-3', '0808.2905-2-11-5'], ['0808.2905-1-14-1', '0808.2905-2-14-1'], ['0808.2905-1-14-4', '0808.2905-2-14-4'], ['0808.2905-1-8-3', '0808.2905-2-8-3'], ['0808.2905-1-36-0', '0808.2905-2-36-0'], ['0808.2905-1-52-3', '0808.2905-2-52-3'], ['0808.2905-1-34-5', '0808.2905-2-34-5'], ['0808.2905-1-40-3', '0808.2905-2-40-3'], ['0808.2905-1-40-9', '0808.2905-2-40-15'], ['0808.2905-1-12-1', '0808.2905-2-12-1'], ['0808.2905-1-35-2', '0808.2905-2-35-2'], ['0808.2905-1-9-2', '0808.2905-2-9-2'], ['0808.2905-1-20-1', '0808.2905-2-20-1'], ['0808.2905-1-20-3', '0808.2905-2-20-3'], ['0808.2905-1-16-0', '0808.2905-2-16-0'], ['0808.2905-1-48-3', '0808.2905-2-48-4'], ['0808.2905-1-14-3', '0808.2905-2-14-3'], ['0808.2905-1-14-5', '0808.2905-2-14-5'], ['0808.2905-1-36-1', '0808.2905-2-36-1'], ['0808.2905-1-36-1', '0808.2905-2-36-2'], ['0808.2905-1-55-4', '0808.2905-2-58-1'], ['0808.2905-1-55-5', '0808.2905-2-58-2']]

[['0808.2905-1-35-0', '0808.2905-2-35-0'], ['0808.2905-1-35-1', '0808.2905-2-35-1'], ['0808.2905-1-23-0', '0808.2905-2-23-0'], ['0808.2905-1-23-1', '0808.2905-2-23-1'], ['0808.2905-1-23-2', '0808.2905-2-23-2'], ['0808.2905-1-23-3', '0808.2905-2-23-3'], ['0808.2905-1-23-5', '0808.2905-2-23-5'], ['0808.2905-1-23-6', '0808.2905-2-23-6'], ['0808.2905-1-31-0', '0808.2905-2-31-0'], ['0808.2905-1-31-1', '0808.2905-2-31-1'], ['0808.2905-1-31-2', '0808.2905-2-31-2'], ['0808.2905-1-31-3', '0808.2905-2-31-3'], ['0808.2905-1-31-4', '0808.2905-2-31-4'], ['0808.2905-1-9-0', '0808.2905-2-9-0'], ['0808.2905-1-9-1', '0808.2905-2-9-1'], ['0808.2905-1-9-3', '0808.2905-2-9-3'], ['0808.2905-1-59-0', '0808.2905-2-65-0'], ['0808.2905-1-59-1', '0808.2905-2-65-1'], ['0808.2905-1-59-2', '0808.2905-2-65-2'], ['0808.2905-1-59-3', '0808.2905-2-65-3'], ['0808.2905-1-59-5', '0808.2905-2-65-5'], ['0808.2905-1-59-6', '0808.2905-2-65-6'], ['0808.2905-1-39-1', '0808.2905-2-39-1'], ['0808.2905-1-39-2', '0808.2905-2-39-2'], ['0808.2905-1-39-3', '0808.2905-2-39-3'], ['0808.2905-1-13-0', '0808.2905-2-13-0'], ['0808.2905-1-6-0', '0808.2905-2-6-0'], ['0808.2905-1-6-1', '0808.2905-2-6-1'], ['0808.2905-1-6-3', '0808.2905-2-6-3'], ['0808.2905-1-6-4', '0808.2905-2-6-4'], ['0808.2905-1-6-5', '0808.2905-2-6-5'], ['0808.2905-1-46-0', '0808.2905-2-46-0'], ['0808.2905-1-46-1', '0808.2905-2-46-1'], ['0808.2905-1-46-2', '0808.2905-2-46-2'], ['0808.2905-1-57-0', '0808.2905-2-63-0'], ['0808.2905-1-57-1', '0808.2905-2-63-1'], ['0808.2905-1-57-2', '0808.2905-2-63-2'], ['0808.2905-1-57-3', '0808.2905-2-63-3'], ['0808.2905-1-57-4', '0808.2905-2-63-4'], ['0808.2905-1-57-5', '0808.2905-2-63-5'], ['0808.2905-1-57-7', '0808.2905-2-63-7'], ['0808.2905-1-57-9', '0808.2905-2-63-9'], ['0808.2905-1-57-10', '0808.2905-2-63-10'], ['0808.2905-1-57-12', '0808.2905-2-63-12'], ['0808.2905-1-57-13', '0808.2905-2-63-13'], ['0808.2905-1-57-14', '0808.2905-2-63-14'], ['0808.2905-1-3-0', '0808.2905-2-3-0'], ['0808.2905-1-3-1', '0808.2905-2-3-1'], ['0808.2905-1-3-2', '0808.2905-2-3-2'], ['0808.2905-1-3-3', '0808.2905-2-3-3'], ['0808.2905-1-41-0', '0808.2905-2-41-0'], ['0808.2905-1-41-1', '0808.2905-2-41-1'], ['0808.2905-1-41-2', '0808.2905-2-41-2'], ['0808.2905-1-47-1', '0808.2905-2-47-1'], ['0808.2905-1-38-0', '0808.2905-2-38-0'], ['0808.2905-1-38-3', '0808.2905-2-38-3'], ['0808.2905-1-38-4', '0808.2905-2-38-4'], ['0808.2905-1-20-2', '0808.2905-2-20-2'], ['0808.2905-1-20-4', '0808.2905-2-20-5'], ['0808.2905-1-20-5', '0808.2905-2-20-7'], ['0808.2905-1-20-7', '0808.2905-2-20-9'], ['0808.2905-1-15-0', '0808.2905-2-15-0'], ['0808.2905-1-15-1', '0808.2905-2-15-1'], ['0808.2905-1-15-3', '0808.2905-2-15-4'], ['0808.2905-1-15-4', '0808.2905-2-15-8'], ['0808.2905-1-32-0', '0808.2905-2-32-0'], ['0808.2905-1-32-1', '0808.2905-2-32-1'], ['0808.2905-1-32-2', '0808.2905-2-32-2'], ['0808.2905-1-32-3', '0808.2905-2-32-3'], ['0808.2905-1-32-4', '0808.2905-2-32-4'], ['0808.2905-1-32-5', '0808.2905-2-32-5'], ['0808.2905-1-37-0', '0808.2905-2-37-0'], ['0808.2905-1-37-1', '0808.2905-2-37-1'], ['0808.2905-1-37-2', '0808.2905-2-37-2'], ['0808.2905-1-45-0', '0808.2905-2-45-0'], ['0808.2905-1-51-0', '0808.2905-2-51-0'], ['0808.2905-1-16-1', '0808.2905-2-16-2'], ['0808.2905-1-16-2', '0808.2905-2-16-3'], ['0808.2905-1-5-1', '0808.2905-2-5-1'], ['0808.2905-1-5-2', '0808.2905-2-5-2'], ['0808.2905-1-5-3', '0808.2905-2-5-3'], ['0808.2905-1-5-4', '0808.2905-2-5-4'], ['0808.2905-1-5-5', '0808.2905-2-5-5'], ['0808.2905-1-53-0', '0808.2905-2-53-0'], ['0808.2905-1-53-1', '0808.2905-2-53-1'], ['0808.2905-1-53-2', '0808.2905-2-53-2'], ['0808.2905-1-53-3', '0808.2905-2-53-3'], ['0808.2905-1-0-3', '0808.2905-2-0-3'], ['0808.2905-1-0-4', '0808.2905-2-0-4'], ['0808.2905-1-4-0', '0808.2905-2-4-0'], ['0808.2905-1-4-1', '0808.2905-2-4-1'], ['0808.2905-1-27-1', '0808.2905-2-27-1'], ['0808.2905-1-2-0', '0808.2905-2-2-0'], ['0808.2905-1-2-1', '0808.2905-2-2-1'], ['0808.2905-1-2-2', '0808.2905-2-2-2'], ['0808.2905-1-2-3', '0808.2905-2-2-3'], ['0808.2905-1-21-0', '0808.2905-2-21-0'], ['0808.2905-1-21-1', '0808.2905-2-21-1'], ['0808.2905-1-22-0', '0808.2905-2-22-0'], ['0808.2905-1-22-1', '0808.2905-2-22-1'], ['0808.2905-1-25-0', '0808.2905-2-25-0'], ['0808.2905-1-25-1', '0808.2905-2-25-1'], ['0808.2905-1-25-2', '0808.2905-2-25-2'], ['0808.2905-1-25-4', '0808.2905-2-25-4'], ['0808.2905-1-25-5', '0808.2905-2-25-5'], ['0808.2905-1-25-6', '0808.2905-2-25-6'], ['0808.2905-1-49-0', '0808.2905-2-49-0'], ['0808.2905-1-49-1', '0808.2905-2-49-1'], ['0808.2905-1-49-2', '0808.2905-2-49-2'], ['0808.2905-1-49-3', '0808.2905-2-49-3'], ['0808.2905-1-49-4', '0808.2905-2-49-4'], ['0808.2905-1-43-0', '0808.2905-2-43-0'], ['0808.2905-1-43-1', '0808.2905-2-43-1'], ['0808.2905-1-43-2', '0808.2905-2-43-2'], ['0808.2905-1-43-3', '0808.2905-2-43-3'], ['0808.2905-1-50-0', '0808.2905-2-50-0'], ['0808.2905-1-50-1', '0808.2905-2-50-1'], ['0808.2905-1-50-2', '0808.2905-2-50-2'], ['0808.2905-1-29-0', '0808.2905-2-29-0'], ['0808.2905-1-29-1', '0808.2905-2-29-1'], ['0808.2905-1-29-2', '0808.2905-2-29-2'], ['0808.2905-1-29-3', '0808.2905-2-29-3'], ['0808.2905-1-26-0', '0808.2905-2-26-0'], ['0808.2905-1-26-2', '0808.2905-2-26-2'], ['0808.2905-1-26-3', '0808.2905-2-26-3'], ['0808.2905-1-26-4', '0808.2905-2-26-4'], ['0808.2905-1-26-5', '0808.2905-2-26-5'], ['0808.2905-1-44-0', '0808.2905-2-44-0'], ['0808.2905-1-44-1', '0808.2905-2-44-1'], ['0808.2905-1-44-3', '0808.2905-2-44-3'], ['0808.2905-1-44-4', '0808.2905-2-44-4'], ['0808.2905-1-10-0', '0808.2905-2-10-0'], ['0808.2905-1-10-3', '0808.2905-2-10-3'], ['0808.2905-1-10-4', '0808.2905-2-10-4'], ['0808.2905-1-48-0', '0808.2905-2-48-0'], ['0808.2905-1-48-1', '0808.2905-2-48-1'], ['0808.2905-1-48-4', '0808.2905-2-48-5'], ['0808.2905-1-48-5', '0808.2905-2-48-6'], ['0808.2905-1-18-0', '0808.2905-2-18-0'], ['0808.2905-1-18-1', '0808.2905-2-18-1'], ['0808.2905-1-18-3', '0808.2905-2-18-3'], ['0808.2905-1-14-0', '0808.2905-2-14-0'], ['0808.2905-1-14-2', '0808.2905-2-14-2'], ['0808.2905-1-14-6', '0808.2905-2-14-6'], ['0808.2905-1-8-0', '0808.2905-2-8-0'], ['0808.2905-1-8-1', '0808.2905-2-8-1'], ['0808.2905-1-8-2', '0808.2905-2-8-2'], ['0808.2905-1-24-0', '0808.2905-2-24-0'], ['0808.2905-1-24-1', '0808.2905-2-24-1'], ['0808.2905-1-24-2', '0808.2905-2-24-2'], ['0808.2905-1-24-3', '0808.2905-2-24-3'], ['0808.2905-1-36-2', '0808.2905-2-36-3'], ['0808.2905-1-52-0', '0808.2905-2-52-0'], ['0808.2905-1-52-1', '0808.2905-2-52-1'], ['0808.2905-1-52-2', '0808.2905-2-52-2'], ['0808.2905-1-52-4', '0808.2905-2-52-4'], ['0808.2905-1-34-0', '0808.2905-2-34-0'], ['0808.2905-1-34-1', '0808.2905-2-34-1'], ['0808.2905-1-34-2', '0808.2905-2-34-2'], ['0808.2905-1-34-3', '0808.2905-2-34-3'], ['0808.2905-1-34-4', '0808.2905-2-34-4'], ['0808.2905-1-40-0', '0808.2905-2-40-0'], ['0808.2905-1-40-1', '0808.2905-2-40-1'], ['0808.2905-1-40-2', '0808.2905-2-40-2'], ['0808.2905-1-40-4', '0808.2905-2-40-4'], ['0808.2905-1-40-5', '0808.2905-2-40-5'], ['0808.2905-1-40-6', '0808.2905-2-40-6'], ['0808.2905-1-40-7', '0808.2905-2-40-7'], ['0808.2905-1-40-8', '0808.2905-2-40-8'], ['0808.2905-1-12-0', '0808.2905-2-12-0'], ['0808.2905-1-55-6', '0808.2905-2-59-0'], ['0808.2905-1-55-7', '0808.2905-2-61-0'], ['0808.2905-1-55-8', '0808.2905-2-61-1']]

[['0808.2905-1-23-4', '0808.2905-2-23-4'], ['0808.2905-1-19-0', '0808.2905-2-19-0'], ['0808.2905-1-19-2', '0808.2905-2-19-2'], ['0808.2905-1-31-5', '0808.2905-2-31-5'], ['0808.2905-1-31-6', '0808.2905-2-31-6'], ['0808.2905-1-31-7', '0808.2905-2-31-7'], ['0808.2905-1-39-0', '0808.2905-2-39-0'], ['0808.2905-1-13-1', '0808.2905-2-13-1'], ['0808.2905-1-6-2', '0808.2905-2-6-2'], ['0808.2905-1-57-6', '0808.2905-2-63-6'], ['0808.2905-1-57-8', '0808.2905-2-63-8'], ['0808.2905-1-41-3', '0808.2905-2-41-3'], ['0808.2905-1-47-0', '0808.2905-2-47-0'], ['0808.2905-1-38-1', '0808.2905-2-38-1'], ['0808.2905-1-38-2', '0808.2905-2-38-2'], ['0808.2905-1-30-0', '0808.2905-2-30-0'], ['0808.2905-1-30-1', '0808.2905-2-30-1'], ['0808.2905-1-30-2', '0808.2905-2-30-2'], ['0808.2905-1-30-3', '0808.2905-2-30-3'], ['0808.2905-1-30-4', '0808.2905-2-30-4'], ['0808.2905-1-30-5', '0808.2905-2-30-5'], ['0808.2905-1-30-6', '0808.2905-2-30-6'], ['0808.2905-1-20-0', '0808.2905-2-20-0'], ['0808.2905-1-20-6', '0808.2905-2-20-8'], ['0808.2905-1-15-2', '0808.2905-2-15-3'], ['0808.2905-1-37-3', '0808.2905-2-37-4'], ['0808.2905-1-37-4', '0808.2905-2-37-5'], ['0808.2905-1-45-1', '0808.2905-2-45-1'], ['0808.2905-1-51-1', '0808.2905-2-51-1'], ['0808.2905-1-51-2', '0808.2905-2-51-2'], ['0808.2905-1-51-3', '0808.2905-2-51-3'], ['0808.2905-1-5-0', '0808.2905-2-5-0'], ['0808.2905-1-5-6', '0808.2905-2-5-6'], ['0808.2905-1-0-0', '0808.2905-2-0-0'], ['0808.2905-1-0-1', '0808.2905-2-0-1'], ['0808.2905-1-0-2', '0808.2905-2-0-2'], ['0808.2905-1-27-0', '0808.2905-2-27-0'], ['0808.2905-1-21-2', '0808.2905-2-21-2'], ['0808.2905-1-21-3', '0808.2905-2-21-3'], ['0808.2905-1-21-4', '0808.2905-2-21-4'], ['0808.2905-1-25-3', '0808.2905-2-25-3'], ['0808.2905-1-26-1', '0808.2905-2-26-1'], ['0808.2905-1-44-2', '0808.2905-2-44-2'], ['0808.2905-1-10-1', '0808.2905-2-10-1'], ['0808.2905-1-10-2', '0808.2905-2-10-2'], ['0808.2905-1-48-2', '0808.2905-2-48-2'], ['0808.2905-1-48-6', '0808.2905-2-48-7'], ['0808.2905-1-18-2', '0808.2905-2-18-2'], ['0808.2905-1-11-0', '0808.2905-2-11-2'], ['0808.2905-1-11-1', '0808.2905-2-11-3'], ['0808.2905-1-11-2', '0808.2905-2-11-4'], ['0808.2905-1-11-3', '0808.2905-2-11-5'], ['0808.2905-1-14-1', '0808.2905-2-14-1'], ['0808.2905-1-14-4', '0808.2905-2-14-4'], ['0808.2905-1-8-3', '0808.2905-2-8-3'], ['0808.2905-1-36-0', '0808.2905-2-36-0'], ['0808.2905-1-52-3', '0808.2905-2-52-3'], ['0808.2905-1-34-5', '0808.2905-2-34-5'], ['0808.2905-1-40-3', '0808.2905-2-40-3'], ['0808.2905-1-40-9', '0808.2905-2-40-15'], ['0808.2905-1-12-1', '0808.2905-2-12-1']]

[]

[['0808.2905-1-35-2', '0808.2905-2-35-2'], ['0808.2905-1-9-2', '0808.2905-2-9-2'], ['0808.2905-1-20-1', '0808.2905-2-20-1'], ['0808.2905-1-20-3', '0808.2905-2-20-3'], ['0808.2905-1-16-0', '0808.2905-2-16-0'], ['0808.2905-1-48-3', '0808.2905-2-48-4'], ['0808.2905-1-14-3', '0808.2905-2-14-3'], ['0808.2905-1-14-5', '0808.2905-2-14-5'], ['0808.2905-1-36-1', '0808.2905-2-36-1'], ['0808.2905-1-36-1', '0808.2905-2-36-2'], ['0808.2905-1-55-4', '0808.2905-2-58-1'], ['0808.2905-1-55-5', '0808.2905-2-58-2']]

[]

['0808.2905-1-2-4', '0808.2905-1-3-4', '0808.2905-1-9-4', '0808.2905-1-52-5', '0808.2905-1-57-11', '0808.2905-1-59-4', '0808.2905-2-2-4', '0808.2905-2-3-4', '0808.2905-2-9-4', '0808.2905-2-52-5', '0808.2905-2-63-11', '0808.2905-2-65-4']

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

https://arxiv.org/abs/0808.2905

1710.04223

{'1710.04223-1-0-0': 'There is considerable interest in understanding the demographics of galaxies within the local universe (defined, for our purposes, as the volume within a radius of 200Mpc or [MATH]).', '1710.04223-1-0-1': 'In this pilot paper we investigate the redshift completeness of catalogs of such nearby galaxies.', '1710.04223-1-0-2': 'Our sample consists of host galaxies of nearby, [MATH], supernovae (SNe) discovered by the flux-limited ASAS-SN survey.', '1710.04223-1-0-3': 'Thanks to their brilliance SNe are easy to find and with some care can be used to study galaxy populations.', '1710.04223-1-0-4': 'In particular, type Ia SNe arise in both old and young stellar populations and thus can be used to study the general demographics of galaxies.', '1710.04223-1-0-5': 'We define the redshift completeness fraction (RCF) as the number of SN host galaxies with known redshift prior to SN discovery, determined, in this case, via the NASA Extragalactic Database (NED), divided by the total number of newly discovered SNe.', '1710.04223-1-0-6': 'By raw numbers, we find [MATH] for [MATH].', '1710.04223-1-0-7': 'We examine the distribution of hosts with and without cataloged redshifts as a function of absolute magnitude and redshift, and, unsurprisingly, find that higher-[MATH] and fainter hosts are less likely to have a known redshift prior to the detection of the SN.', '1710.04223-1-0-8': 'We conclude with thoughts on the future improvement of RCF measurements, which will benefit greatly from upcoming surveys.', '1710.04223-1-1-0': '# Transients in the Local Universe', '1710.04223-1-2-0': 'Transients in the local Universe provide unique insights to at least three pressing issues in modern astronomy.', '1710.04223-1-2-1': 'First, proximity allows detailed exploration of these explosive events, particularly those with low luminosities.', '1710.04223-1-2-2': 'The classical example is the clarification of novae and super-novae by [CITATION].', '1710.04223-1-2-3': 'Second, is demographics.', '1710.04223-1-2-4': 'Supernovae (SNe) mark the end points of massive stars and inject energy, momentum and heavy elements into their surroundings.', '1710.04223-1-2-5': 'As such, relating star-formation rate and chemical abundance to SN rates is a fundamental exercise in modern astronomy for which the primary inputs come from demographics of nearby (volume limited) supernovae.', '1710.04223-1-3-0': 'Third, over the last [MATH]decade, exotic explosive sources have been identified - Ultra High Energy Cosmic Rays (UHECR), ultra-high energy neutrinos, and Gravitational Wave (GW) sources.', '1710.04223-1-3-1': 'The horizon for detecting these sources is limited by either physical phenomena (the Greisen-Zatespin-Kuzmin effect for UHECRs), or set by the sensitivity of GW telescopes.', '1710.04223-1-3-2': 'The latter consideration leads us to a distance limit of [MATH]200Mpc ([MATH]).', '1710.04223-1-3-3': 'Clearly the study of Transients in the Local Universe (TILU) is not only of wide importance but also timely.', '1710.04223-1-4-0': 'These exotic explosive sources provide the impetus to this paper.', '1710.04223-1-4-1': 'There is little doubt that the detection of EM counterparts to GW sources will provide rich rewards .', '1710.04223-1-4-2': 'At least for the next few years the typical localization of GW sources will be no better than [MATH].', '1710.04223-1-4-3': 'Naturally, pursuit of such large angle localizations will entail a deluge of false positives (e.g., ).', '1710.04223-1-4-4': 'Furthermore, the lifetime of the EM counterparts, particularly in the ultraviolet through near-infrared (nIR), is expected to be small (days to weeks; ).', '1710.04223-1-4-5': 'A cost-effective approach to minimize the background fog of false positives is to use on-sky coincidences with catalogs of nearby galaxies.', '1710.04223-1-4-6': 'This filtering can vastly speed up the triage and rank order critical but time-consuming spectroscopic observations .', '1710.04223-1-5-0': 'Additionally, we note that catalogs of nearby galaxies are also useful for many projects beyond the EM follow up of GW sources.', '1710.04223-1-5-1': 'For instance, there is considerable interest in studying the youngest SNe.', '1710.04223-1-5-2': 'In the hours to days after explosion, they are very faint.', '1710.04223-1-5-3': 'Identifying young SNe is challenging because most rising, faint transients are distant SNe Ia that constitute a veritable background fog.', '1710.04223-1-5-4': 'In contrast, a faint, rising transient in a known, nearby galaxy is very likely an infant SN worthy of vigorous pursuit.', '1710.04223-1-6-0': 'Returning to the main theme of the paper, an essential input to experimental planning for EM counterparts of GW sources is the "completeness" of galaxy catalogs within the radius of interest.', '1710.04223-1-6-1': 'As a result of deep wide-field imaging surveys, such as PanSTARRS-1 (PS1) and the Sloan Digital Sky Survey (SDSS), all galaxies, particularly in the Northern sky, to [MATH]23 mag, are "cataloged" (in the sense that multi-band photometric measurements exist, and the galaxies are assigned a nominal name).', '1710.04223-1-6-2': 'However, what matters for the purpose of this paper is a reliable galaxy redshift measurement, prior to the search for EM counterparts.', '1710.04223-1-6-3': 'Thus, for our purposes "completeness" is simply the fraction of nearby galaxies with secure redshifts.', '1710.04223-1-7-0': 'In this paper, we explore the use of SNe to assess the completeness of nearby galaxy catalogs.', '1710.04223-1-7-1': 'In fact, [CITATION] remark that 24% of ASAS-SN supernovae were discovered in cataloged host galaxies without secure redshifts.', '1710.04223-1-7-2': 'Here, as a next step, we undertake an examination of supernova host galaxies "missed" in catalogs of nearby galaxies.', '1710.04223-1-8-0': '# Catalogs of nearby galaxies', '1710.04223-1-9-0': 'The simplest way to construct a catalog of nearby galaxies is via spectroscopic surveys (e.g. ).', '1710.04223-1-9-1': 'However, given the more than eight magnitude spread in galaxy luminosities the catalog(s) will necessarily be incomplete.', '1710.04223-1-9-2': 'Despite this challenge, astronomers have assembled a number of catalogs: the 11-Mpc Nearby Galaxy Catalog , The Extragalactic Database and Hyperleda .', '1710.04223-1-9-3': 'These catalogs are linked to the NASA/IPAC Extragalactic Database (NED).', '1710.04223-1-10-0': 'Given these heterogeneous origins the question "how complete is the census of galaxies within the TILU volume?"', '1710.04223-1-10-1': 'naturally arises.', '1710.04223-1-10-2': 'One approach is to extrapolate the findings of higher-[MATH] surveys down to [MATH].', '1710.04223-1-10-3': 'However, as shown by deviations of galaxy recession velocities from the Hubble flow and via direct galaxy catalogs, the local Universe is lumpy (10% fluctuations or more) on length scales as large as 200Mpc.', '1710.04223-1-10-4': 'Extrapolating a local catalog (e.g., Nearby Galaxy Catalog) from the bottom up is even more problematic given strong local fluctuations (e.g., the Virgo cluster).', '1710.04223-1-10-5': 'Regardless, any such extrapolations must account for the absence of surveys in the Galactic zone of avoidance.', '1710.04223-1-11-0': '# Assessing Catalog completeness with SNe', '1710.04223-1-12-0': 'Here, we explore the use of SNe to evaluate the completeness of catalog(s) of the nearby Universe.', '1710.04223-1-12-1': 'Type Ia SNe are well suited for this task.', '1710.04223-1-12-2': 'These SNe are bright ([MATH]1mag fainter than a local [MATH] galaxy in the [MATH] band) and thus easily identified.', '1710.04223-1-12-3': 'The luminosity function is narrow: in [MATH] band the luminosity function can be fit with a Gaussian with mean [MATH]mag and rms 0.14-0.23mag .', '1710.04223-1-12-4': 'Next, SNe Ia arise from both old and young populations ; thus they sample all types of galaxies and can be used to study galaxy demographics in a very general sense without missing significant subpopulations.', '1710.04223-1-12-5': 'The homogeneity of Ia light curves allow, given multi-band photometry, accounting for host interstellar medium extinction.', '1710.04223-1-13-0': 'In contrast, core collapse supernovae (CC SNe) arise only in star-forming galaxies and exhibit a wide luminosity function.', '1710.04223-1-13-1': 'For instance [CITATION] find the CC luminosity function ranges from [MATH] to [MATH]mag in the SDSS [MATH] band, with a peak at [MATH]mag.', '1710.04223-1-13-2': 'The large range in luminosity means that the use of CC SNe to trace galaxies would require additional care.', '1710.04223-1-14-0': 'Our proposal to assess the completeness of nearby galaxies catalog(s) is simple: (1) Following the detection of a SN, measure the redshift of the host galaxy, [MATH].', '1710.04223-1-14-1': '(2) Retain those events with [MATH] where [MATH] is the redshift to which the completeness is being measured ("tries").', '1710.04223-1-14-2': '(3) Check the galaxy catalog (NED) to see if the SN host galaxy has a reliable redshift entry.', '1710.04223-1-14-3': 'If present and if [MATH] is equal to [MATH] to within, say [MATH], then it is a "hit".', '1710.04223-1-14-4': 'The redshift completeness factor (RCF) of the galaxy catalog is given by the ratio of "hits" to "tries".', '1710.04223-1-15-0': 'We emphasize that our definition of the RCF is explicitly tied to supernovae and is not the fraction of galaxies with known redshifts by number.', '1710.04223-1-15-1': 'The majority of galaxies within any volume are very small galaxies which are also the most difficult to detect (indeed, new satellites of the Milky Way are still being uncovered).', '1710.04223-1-15-2': 'Our metric approximately measures the mass-weighted completeness, which is equivalent to the fraction of stars within this volume that are inside galaxies of known redshift (since, to a rough approximation, the SN rate of a galaxy depends on its stellar mass).', '1710.04223-1-15-3': 'However, the star-formation history (especially recent star-formation, which is responsible for all cc-SNe) also affects the SN rate, so this connection is imperfect.', '1710.04223-1-15-4': 'Any SN-based estimate of the RCF will also depend on the completeness of the parent SN survey(s).', '1710.04223-1-15-5': "A SN survey that is not complete to the limit of the search volume will miss less luminous and more distant SNe; it will thus find a larger fraction of events in galaxies in the inner part of the search volume and a larger fraction of (more luminous) Ia's, which weight the galaxy mix towards older (and possibly more massive) galaxies.", '1710.04223-1-15-6': 'However, we argue (and later show empirically) that the impact of redshift and SN-type "biases" of this type on the overall redshift completeness in a 200 Mpc volume is relatively modest.', '1710.04223-1-16-0': 'We also emphasize that SNe only provide a estimate of RCF, rather than a means of actually increasing it.', '1710.04223-1-16-1': 'The SN rate per galaxy is far too low to provide a practical means of actually discovering new galaxies in significant numbers.', '1710.04223-1-16-2': 'However, SNe can provide an independent means of assessing the completeness provided by additional methods and provide value input into designing new galaxy surveys to improve this number in the future.', '1710.04223-1-16-3': 'This forms the motivation of our study.', '1710.04223-1-17-0': '# Primary Data', '1710.04223-1-18-0': 'lrrr[ht]', '1710.04223-1-19-0': 'SN Demographics from the two ASASSN surveys', '1710.04223-1-20-0': 'Catalog & type[MATH] & [MATH](SN) & [MATH]) A1 & Ia & 66 & 51A1 & CC & 25 & 22A2 & Ia & 141 & 101 A2 & CC & 39 & 27 [MATH] is the number of SNe in each category.', '1710.04223-1-20-1': '[MATH] is the number of host galaxies with a redshift entry in NED (and obtained prior to the SN discovery).', '1710.04223-1-20-2': 'aThe type Ia encompasses normal Ia and all sub-types such as 91T, 91bg, CSM and 00cx.', '1710.04223-1-20-3': 'All non-Ia supernova are called as core collapse (II, IIn, Ibc).', '1710.04223-1-21-0': 'To assess the RCF we use NED to identify [MATH] of nearby galaxies, and the first two SNe catalogs of the All-Sky Automated Survey for Supernova (ASAS-SN) survey.', '1710.04223-1-21-1': 'The primary attraction of this flux-limited survey, [MATH], is that it is not targeted to specific galaxies.', '1710.04223-1-21-2': 'ASAS-SN candidates are bright enough for world wide follow up using small telescopes.', '1710.04223-1-21-3': 'As a result, the classification (SN type and [MATH]) is essentially complete.', '1710.04223-1-21-4': 'In contrast, the bias of the candidates that are followed up from other surveys (e.g. ATLAS, iPTF, Gaia) is not known.', '1710.04223-1-21-5': 'Amateur surveys, whilst very productive, are biased towards well resolved and bright galaxies.', '1710.04223-1-22-0': 'Catalog A1 lists 91 SNe discovered during the period 2013-2014 (; hereafter, H1).', '1710.04223-1-22-1': 'Catalog A2 lists 180 SNe found in the calendar year 2015.', '1710.04223-1-22-2': 'In A2 11 SNe are marginally fainter than 17mag (; hereafter, H2).', '1710.04223-1-22-3': 'Both catalogs also include host galaxy data: name, redshift, SDSS, GALEX, 2MASS and WISE photometry.', '1710.04223-1-23-0': 'The basic demographics of the ASAS-SN SNe are summarized in Table [REF].', '1710.04223-1-23-1': 'The catalogs do not specifically identify host galaxies that lack a cataloged redshift prior to the discovery of the SN.', '1710.04223-1-23-2': 'To remedy this situation we wrote a program to query NED and obtained the redshifts of the putative host galaxies.', '1710.04223-1-23-3': 'We refer to the sample of galaxies with a redshift entry in NED as the "NED[MATH]" sample and those which lacked an entry as the "!', '1710.04223-1-23-4': 'NED[MATH]" sample.', '1710.04223-1-24-0': 'Nine entries (ASAS-SN-15de, 15ji, 15jm, 15lh, 15nh, 15og, 15ts, 15ua, 14ms) have [MATH].', '1710.04223-1-24-1': 'These were deleted from further consideration.', '1710.04223-1-24-2': 'Next, we inspected the difference between the SN redshift, [MATH], and the purported host redshift given in NED, [MATH].', '1710.04223-1-24-3': 'Noting the lower precision of the SN redshift we made an allowance and inspected events with [MATH].', '1710.04223-1-24-4': 'The rationale for this choice is that redshift of the host is usually the systemic velocity of the galaxy (a fiber of the spectrograph encompasses the central region of the galaxy) whereas the SN will have additional velocity arising from the rotation curve.', '1710.04223-1-24-5': 'Three events stood out: ASASSN-15jo has [MATH] (AbellS0753) , but [MATH]; ASASSN-13an has [MATH] (2MASX J13453653[MATH]0719350), but [MATH]; and ASASSN-15ic has [MATH] (2MASXJ06145320[MATH]4247357), [MATH].', '1710.04223-1-24-6': 'We retain the first two events and delete the last one from further consideration.', '1710.04223-1-24-7': 'Parenthetically we note that a velocity precision of 300kms[MATH] is adequate for both SN and the host galaxy since deviations from large scale flows occur at this level (e.g. the velocity of the Milky Way with respect to the Cosmic Background Radiation is 600kms[MATH]).', '1710.04223-1-24-8': 'The final sample for the analysis reported below consists of 261 SNe.', '1710.04223-1-25-0': '# Analysis', '1710.04223-1-26-0': 'The redshift distribution of ASAS-SN events is displayed in Figure [REF].', '1710.04223-1-26-1': 'SNe Ia peak at [MATH] whereas CC events exhibit both a peak at lower redshift and a long tail (as expected).', '1710.04223-1-26-2': 'The "completeness" of the SN sample is not the primary topic of this paper.', '1710.04223-1-26-3': 'What is central to this paper is that the SNe sample not be biased by host galaxies.', '1710.04223-1-26-4': 'So we will proceed with the analysis of the sample we have in hand.', '1710.04223-1-26-5': 'The RCF for NED for [MATH] is about 80% (Table [REF]).', '1710.04223-1-26-6': 'The precision of these estimates is limited by small number binomial statistics.', '1710.04223-1-27-0': 'For our full sample ([MATH]), there are 64 !', '1710.04223-1-27-1': 'NED[MATH] galaxies, which corresponds to [MATH].', '1710.04223-1-27-2': 'We have investigated the "missing" galaxies relative to SDSS and find the following: 40 of the missing galaxies are not covered by either SDSS imaging or spectroscopic observations.', '1710.04223-1-27-3': 'The remaining 24 have SDSS imaging observations, but only 16 of those have SDSS photometric redshift (photoz) [MATH].', '1710.04223-1-27-4': 'Of those 16, 5 galaxies have photoz consistent with [MATH] once accounting for uncertainties.', '1710.04223-1-27-5': 'In sum, this suggests that purely photometric surveys cannot significantly improve the RCF.', '1710.04223-1-28-0': 'lllll[ht]', '1710.04223-1-29-0': 'Redshift completeness factor of ASAS-SN Supernovae ([MATH])', '1710.04223-1-30-0': 'SN & [MATH](SN)a & [MATH]b & [MATH]c & RCFd All & 173 & 139 & 34 & 0.75-0.85Ia & 120 & 93 & 27 & 0.71-0.83CC & 53 & 46 & 7 & 0.77-0.92 aThe total number of SNe.', '1710.04223-1-30-1': 'bThe number of SNe with a putative host galaxy with redshift listed in NED, prior to SN discovery.', '1710.04223-1-30-2': 'cThe number of SNe with a putative host galaxy but whose redshift is not listed in NED.', '1710.04223-1-30-3': 'dThe Redshift Completeness Factor (RCF; see [REF]) range covers a confidence range of 5% to 95% and is obtained by assuming a flat Bayesian prior.', '1710.04223-1-31-0': '# Analysis of Host Galaxy Luminosities', '1710.04223-1-32-0': 'The virtue of RCF is that it is a well defined and simple metric.', '1710.04223-1-32-1': 'However, as noted in [REF] the propensity of a galaxy to produce SNe depends on at least two gross galaxy parameters: the star-formation rate ([MATH]) and the total stellar mass ([MATH]).', '1710.04223-1-32-2': 'Thus, a second (and a far more interesting) level of analysis involves answering questions such as "How complete is our understanding of the SFR within the local volume"?', '1710.04223-1-32-3': 'Going forward, we use the 2MASS [MATH]-band as a proxy for [MATH] and the GALEX NUV band for [MATH].', '1710.04223-1-33-0': 'H1 H2 took [MATH]-band magnitudes from 2MASS when available; otherwise, a [MATH]-band mag was estimated from WISE [MATH] band when detected in that band (offset by a typical [MATH]), or (in the absence of both 2MASS and WISE) a limit of [MATH] was set.', '1710.04223-1-33-1': 'We employ the same approach using their magnitude catalogs, except that for the last group (3 galaxies in catalog A1 and 5 in catalog A2) we simply set [MATH]mag.', '1710.04223-1-34-0': 'In A1 there are 77 SNe with NUV data, 5 with only SDSS [MATH] data and 9 with neither NUV nor [MATH] data.', '1710.04223-1-34-1': 'In A2, the corresponding numbers are 138, 17 and 24, respectively.', '1710.04223-1-34-2': 'We took the sample of SNe with both NUV and [MATH] data, applied the Galactic extinction correction, and found the median of NUV[MATH]mag.', '1710.04223-1-34-3': 'We use the NUV data when available and [MATH] with the aforementioned offset applied otherwise.', '1710.04223-1-34-4': 'We call this hybrid magnitude the "UV" mag.', '1710.04223-1-34-5': 'The 32 SNe with neither NUV nor [MATH] data are not included in the analysis.', '1710.04223-1-35-0': 'For the discussion below we note that the relevant Schechter characteristic luminosity parameters are [MATH] and [MATH] .', '1710.04223-1-35-1': 'Below we examine the missing galaxies with luminosity [MATH] in both the [MATH] and "UV" bands.', '1710.04223-1-35-2': 'We then isolate the sample to just Type Ia SNe to investigate missing galaxies with [MATH].', '1710.04223-1-36-0': '## [MATH]-band', '1710.04223-1-37-0': 'The absolute [MATH]-band magnitude, [MATH], and redshift, [MATH], of each SN host galaxy in our sample are displayed in Figures [REF] and [REF] for Type Ia SNe and CC SNe, respectively.', '1710.04223-1-37-1': 'The probability distribution functions (PDFs) shown in Figures [REF] and [REF] show that the high redshift galaxies ([MATH]) in our sample have similar probabilities to be or not be in NED.', '1710.04223-1-37-2': 'Meanwhile, as traced by SNe Ia, intrinsically faint galaxies ([MATH]) are more likely to not be included in NED.', '1710.04223-1-38-0': 'In total, there are 42 host galaxies brighter than [MATH], and 2 of these did not have redshift entries in NED prior to SN discovery.', '1710.04223-1-38-1': 'The first is the host of ASASSN-15ed, MCG+09-27-087/SDSSJ164825.26+505935.5, a bright, large (40[MATH]) sprial galaxy.', '1710.04223-1-38-2': 'There is no SDSS spectroscopic redshift but the SDSS photoz estimate is [MATH].', '1710.04223-1-38-3': 'The host galaxy has a [MATH]mJy counterpart in NVSS.', '1710.04223-1-38-4': 'The second is the host of ASASSN-15ub, CGCG314-006.', '1710.04223-1-38-5': 'Nominally there is no host redshift in NED.', '1710.04223-1-38-6': 'However, a direct inspection of SDSS shows that a pair of strongly interacting galaxies with a spectroscopic redshift of 0.032 (photoz of [MATH]) which can be compared to [MATH].', '1710.04223-1-39-0': 'We restrict the analysis of sub-luminous galaxies to SNe Ia.', '1710.04223-1-39-1': 'In the [MATH] band, there are 163 host galaxies with [MATH].', '1710.04223-1-39-2': 'Of these 110 are listed in NED.', '1710.04223-1-39-3': 'Thus, as traced by SNe Ia, the [MATH].', '1710.04223-1-40-0': '## NUV/[MATH] band', '1710.04223-1-41-0': 'As was the case for [MATH] band, the median !', '1710.04223-1-41-1': 'NED[MATH] host galaxy is [MATH]2mag fainter than the median NED[MATH] host galaxy for the "UV" band (Figures [REF] and [REF]).', '1710.04223-1-41-2': 'The redshift distribution of NED[MATH] and !', '1710.04223-1-41-3': 'NED[MATH] galaxies in nearly identical in [MATH] and "UV", with higher [MATH] hosts more likely to not be included in NED.', '1710.04223-1-41-4': 'The peak of the !', '1710.04223-1-41-5': 'NED[MATH] sample "UV" luminosity is not as biased towards faint galaxies as it is in the [MATH] band.', '1710.04223-1-41-6': 'This conclusion bodes well for the PTF Census of the Local Universe (CLU) H[MATH] survey which is searching for nearby star-forming galaxies .', '1710.04223-1-42-0': 'In the "UV" band, 46 galaxies are brighter than [MATH], and 5 of those are !', '1710.04223-1-42-1': 'NED[MATH] galaxies.', '1710.04223-1-42-2': 'The host of ASASSN-15ed, MCG+09-27-087; SDSSJ164825.26+505935.5, is discussed above.', '1710.04223-1-42-3': 'The ASASSN-15ln host, GALEXASC J225332.83+194232.9, is a 10[MATH] spiral galaxy with no SDSS spectroscopic redshift but an SDSS photoz of [MATH].', '1710.04223-1-42-4': 'The host of ASASSN-15ho, 2MASXi J0909234-044327, lies outside the SDSS footprint.', '1710.04223-1-42-5': 'The hosts of ASASSN-15sh and ASASSN-15um, 2MASXJ19320827-6226340 and 2MASXJ05395948-8022191, respectively, lie in poorly studied regions of the sky.', '1710.04223-1-43-0': 'Again, we restrict the analysis of missing sub-luminous galaxies to SNe Ia hosts.', '1710.04223-1-43-1': 'There are 140 galaxies with "UV" luminosity less than the corresponding [MATH] value.', '1710.04223-1-43-2': 'Of these, 98 have redshift entries in NED.', '1710.04223-1-43-3': 'Thus, as traced by SNe Ia, the [MATH].', '1710.04223-1-44-0': '# Increasing Precision Accuracy of RCF', '1710.04223-1-45-0': 'The ASAS-SN bright supernova sample is unique at present owing to its large sample size, untargeted discovery strategy, and complete spectroscopic coverage.', '1710.04223-1-45-1': 'This makes it the most reliable host-unbiased supernova catalog in the nearby universe at present.', '1710.04223-1-45-2': 'However, the ease of spectroscopic follow-up ensures good spectroscopic coverage for almost any supernova discovered in this volume regardless of survey origin, and we can further bolster our sample size by extending our results to include additional surveys.', '1710.04223-1-45-3': 'As a first step along these lines, we have undertaken a similar catalog check for all 529 [MATH] SNe reported at the Transient Name Server (TNS) portal, during the period between January 2016 and June 2017.', '1710.04223-1-45-4': 'SNe in this sample span a larger peak magnitude range relative to ASAS-SN, extending as faint 20 mag.', '1710.04223-1-45-5': 'However, it appears that follow up were obtained for most of the reported candidate SNe.', '1710.04223-1-45-6': 'We find a similar fraction of RCF (Cassese Kulkarni, in prep).', '1710.04223-1-46-0': 'The field of optical time domain astronomy is in a boom period, and much larger samples of nearby SNe can be expected within the next few years.', '1710.04223-1-46-1': 'Recently, the Asteroid Terrestrial-impact Last Alert (ATLAS; ) began routine operation.', '1710.04223-1-46-2': 'The end of 2017 will see the Zwicky Transient Facility (ZTF; ), and an upgraded ASAS-SN commissioned.', '1710.04223-1-46-3': 'ZTF is dedicated to a systematic exploration of the transient and variable optical sky.', '1710.04223-1-46-4': 'The ongoing PS-1 and ATLAS, although dedicated to the study of near-earth asteroids, have great capacity for finding SNe.', '1710.04223-1-46-5': 'The sensitivity of these surveys, [MATH] to 21mag, will increase both the precision of RCF measurements as well as the radius to which the RCF can be probed.', '1710.04223-1-47-0': 'The accuracy of the RCF will be limited by our understanding of the dependence of SN propensity on fundamental parameters such as galaxy mass ([MATH]) and star-formation rate ([MATH]).', '1710.04223-1-47-1': 'As noted earlier the Ia rate of a galaxy appears to be reasonably well modeled as [MATH] .', '1710.04223-1-47-2': 'This relation can be tested and improved by future, large-sample SN surveys.', '1710.04223-1-48-0': 'While CC SNe certainly track [MATH], it may be the case that "lesser" parameters, such as metalicity, change the mix of CC SNe subtypes .', '1710.04223-1-48-1': 'Again large-sample SN surveys may well have sufficient diagnostic power to ferret out such connections.', '1710.04223-1-49-0': 'To make this discussion concrete we consider a specific example, the "Celestial Cinematography" survey of ZTF .', '1710.04223-1-49-1': 'This survey aims to systematically cover a large fraction of the night sky ([MATH]) every three nights, in [MATH] and [MATH] band.', '1710.04223-1-49-2': 'The median 5-[MATH] detection limit, for a fixed 30s exposure time, is 20.5 mag.', '1710.04223-1-50-0': 'The annual volumetric rate of [MATH] Type Ia and CC SNe is [MATH] and [MATH], respectively .', '1710.04223-1-50-1': 'For the survey discussed above, based on a simulator built for ZTF, the expected annual yield of SNe Ia for peak flux of 17.5, 18, 18.5, and 19mag is 230, 460, 892 and 1750, respectively (U. Feindt, priv.', '1710.04223-1-50-2': 'comm.)', '1710.04223-1-51-0': 'Flux-limited surveys as large as CeCi will measure the RCF on scales of several hundred Mpc with higher precision (owing to their larger sample size).', '1710.04223-1-51-1': 'They will additionally substantially improve the estimates of relative rates of SNe sub-classes (e.g. 91bg, 91T, Iax) and allow for a better determination of the relation between the SN Ia rate and fundamental galaxy parameters (cf. ).', '1710.04223-1-51-2': 'Furthermore, the demographics of CC SNe are bedeviled by their large range in luminosity, from [MATH] to about [MATH] mag .', '1710.04223-1-51-3': 'A survey complete to a flux-limit [MATH] will detect SNe peaking at [MATH] to a radius of [MATH]10, 40, and 160Mpc, respectively.', '1710.04223-1-51-4': 'The total number of CC SNe detections will sharply depend on the luminosity function.', '1710.04223-1-51-5': 'For instance, [CITATION] suggest that the fraction of CC SNe fainter than [MATH] at peak is at least 24% but can be as high as [MATH].', '1710.04223-1-51-6': 'In any case, the planned CeCi survey will allow us to measure the luminosity function of CC events which is essential to determine the volumetric rate of CC SNe.', '1710.04223-1-51-7': 'In turn, the latter is a key element in our understanding of stars and the interstellar medium .', '1710.04223-1-52-0': 'The primary limitation to the sample size of a flux-limited SN survey is the spectroscopic load.', '1710.04223-1-52-1': 'This load can be made bearable by the use of two spectrographs: an ultra-low resolution spectrometer tuned to classification and a standard low-resolution spectrometer to get the redshift and gross spectrum of the host galaxies.', '1710.04223-1-52-2': 'For the latter we note that within a few years not merely highly- but supremely-multiplexed spectrographs (e.g. DESI, PFS and the planned AS4 project) will be commissioned.', '1710.04223-1-52-3': 'These facilities, at very little cost (small fractional allocation of fibers), can measure the redshifts of host galaxies of SNe on an industrial scale.', '1710.04223-1-52-4': 'The same highly-multiplexed spectrographs will likely be pressed into surveys more ambitious than SDSS or 6dF, leading to more complete catalogs of galaxies in the nearby Universe.', '1710.04223-1-53-0': 'We thank A. Goobar, U. Feindt, M. Kasliwal, P. Nugent, E. O. Ofek, E. S. Phinney and H. Vedantham for inputs and helpful discussions.', '1710.04223-1-54-0': 'AAM is funded by the Large Synoptic Survey Telescope Corporation in support of the Data Science Fellowship Program.'}

{'1710.04223-2-0-0': 'There is considerable interest in understanding the demographics of galaxies within the local universe (defined, for our purposes, as the volume within a radius of 200Mpc or [MATH]).', '1710.04223-2-0-1': 'In this pilot paper, using supernovae (SNe) as signposts to galaxies, we investigate the redshift completeness of catalogs of nearby galaxies.', '1710.04223-2-0-2': 'In particular, type Ia SNe are bright and are good tracers of the bulk of the galaxy population, since they arise in both old and young stellar populations.', '1710.04223-2-0-3': 'Our input sample consists of SNe with redshift [MATH], discovered by the flux-limited ASAS-SN survey.', '1710.04223-2-0-4': 'We define the redshift completeness fraction (RCF) as the number of SN host galaxies with known redshift prior to SN discovery, determined, in this case, via the NASA Extragalactic Database (NED), divided by the total number of newly discovered SNe.', '1710.04223-2-0-5': 'Using SNe Ia, we find [MATH]% (90% confidence interval) for [MATH].', '1710.04223-2-0-6': 'We examine the distribution of host galaxies with and without cataloged redshifts as a function of absolute magnitude and redshift, and, unsurprisingly, find that higher-[MATH] and fainter hosts are less likely to have a known redshift prior to the detection of the SN.', '1710.04223-2-0-7': 'However, surprisingly, some [MATH] galaxies are also missing.', '1710.04223-2-0-8': 'We conclude with thoughts on the future improvement of RCF measurements that will be made possible from large SN samples resulting from ongoing and especially upcoming time-domain surveys.', '1710.04223-2-1-0': '# Transients in the Local Universe', '1710.04223-2-2-0': 'Transients in the local Universe provide unique insights into at least three pressing issues in modern astronomy.', '1710.04223-2-2-1': 'First, nearby events can be studied in great detail, even if their luminosities are relatively low-enabling insights into their physics.', '1710.04223-2-2-2': 'A classic example is the detection of SN1987A in the Large Magellanic Cloud, which enabled the unambiguous localization of extragalactic neutrinos (e.g., ).', '1710.04223-2-2-3': 'Second, nearby events can be studied demographically to high completeness.', '1710.04223-2-2-4': 'This is important both for obtaining a full understanding of how stars end their lives, and for understanding the role their explosions play in their environments.', '1710.04223-2-2-5': 'For example SNe (SNe), the most commonly observed extragalactic transients, inject energy, momentum and heavy elements into their surroundings.', '1710.04223-2-2-6': 'Relating star-formation rate and chemical abundance to SN rates is a fundamental exercise in modern astronomy.', '1710.04223-2-2-7': 'Nearby (volume limited) SN surveys are needed to provide the latter.', '1710.04223-2-3-0': 'Third, over the last decade or so, exotic explosive sources have been identified - Ultra High Energy Cosmic Rays (UHECR), ultra-high energy neutrinos, and Gravitational Wave (GW) sources.', '1710.04223-2-3-1': 'The horizon for detecting these sources is limited by either physical phenomena (the Greisen-Zatespin-Kuzmin effect for UHECRs), or set by the sensitivity of GW telescopes.', '1710.04223-2-3-2': 'The latter consideration leads us to a distance limit of [MATH]200Mpc ([MATH]).', '1710.04223-2-3-3': 'As shown by the rich returns from electromagnetic studies of the neutron star coalescence event, GW170817 (e.g., ), the study of transients in the local universe is not only of wide importance but also timely.', '1710.04223-2-4-0': 'The primary motivation for this paper is the last point discussed above, namely the study of electromagnetic counterparts to GW sources.', '1710.04223-2-4-1': 'For the next few years the typical localization of GW sources will be no better than [MATH].', '1710.04223-2-4-2': 'Naturally, pursuit of such large angle localizations will entail a deluge of false positives (e.g., ).', '1710.04223-2-4-3': 'As demonstrated by the steps which led to the discovery of the optical counterpart of GW170817, a cost-effective approach to both minimizing the background fog of false positives and maximizing early identification is to use on-sky coincidences with catalogs of nearby galaxies (e.g., ).', '1710.04223-2-5-0': 'Additionally, we note that catalogs of nearby galaxies have other uses.', '1710.04223-2-5-1': 'For instance, there is considerable interest in studying the youngest SNe.', '1710.04223-2-5-2': 'SNe take time to brighten to peak luminosity and are very faint in the first hours to days after the initial explosion, so the appearance of a new source with an inferred luminosity much lower than a classical SN at peak is cause for vigorous pursuit.', '1710.04223-2-5-3': 'But determining this luminosity requires knowledge of the redshift, which will be known in advance only if the transient coincides with a galaxy with a cataloged redshift.', '1710.04223-2-6-0': 'As a result of deep wide-field imaging surveys, such as PanSTARRS-1 (PS1) and the Sloan Digital Sky Survey (SDSS), all galaxies, particularly in the Northern sky, to [MATH]23mag, are "cataloged" (in the sense that multi-band photometric measurements exist, and the galaxies are assigned a nominal name).', '1710.04223-2-6-1': 'However, what matters for the purpose of this paper is a reliable galaxy redshift measurement, prior to any transient follow-up (such as during a search for EM counterparts).', '1710.04223-2-6-2': 'The fraction of nearby galaxies of a certain type (for example, candidate hosts of LIGO GW sources) with a redshift that is recorded in published catalogs can be defined as the redshift "completeness."', '1710.04223-2-7-0': 'In this paper, we explore the use of SNe to assess the completeness of nearby galaxy catalogs.', '1710.04223-2-7-1': 'SNe are very luminous, relatively common (in a cosmological sense), and are found routinely in surveys that now cover the entire sky every few days (and therefore the entire local volume out to some distance limit, subject to the limitation of extinction from the Galactic plane).', '1710.04223-2-7-2': 'They thus provide an effective way of randomly sampling galaxies that is not strongly dependent on the observational properties of those galaxies (in particular, on galaxy luminosity).', '1710.04223-2-7-3': 'Recently, [CITATION] remarked that 24% of nearby bright SNe were discovered in cataloged host galaxies without secure redshifts, suggesting that redshift incompleteness may be significant even today.', '1710.04223-2-7-4': 'As a next step, we refine their estimate by formally restricting their sample to a limited volume, and examine in detail those SNe that occur in galaxies whose redshifts have been "missed" by spectroscopic surveys.', '1710.04223-2-8-0': '# Catalogs of nearby galaxies', '1710.04223-2-9-0': 'The construction of redshift catalogs for meaningful numbers of nearby galaxies can only be accomplished via large-scale spectroscopic surveys (e.g. ).', '1710.04223-2-9-1': 'However, given the more than eight magnitude spread in galaxy luminosities , large numbers of even very nearby galaxies are likely to be quite faint, so these catalog(s) will necessarily be incomplete.', '1710.04223-2-9-2': 'Despite this challenge, astronomers have assembled a number of all-sky galaxy catalogs: the 11-Mpc Nearby Galaxy Catalog , The Extragalactic Database and Hyperleda .', '1710.04223-2-9-3': 'These catalogs are linked to the NASA/IPAC Extragalactic Database (NED).', '1710.04223-2-9-4': 'Figure 1 of [CITATION] provides a graphical illustration of the bias of inputs to spectroscopic surveys.', '1710.04223-2-10-0': 'How complete are these catalogs?', '1710.04223-2-10-1': 'One approach is to extrapolate the findings of higher-[MATH] surveys (which scrutinize small areas of the sky down to extremely deep limits) down to [MATH].', '1710.04223-2-10-2': 'However, as shown by deviations of galaxy recession velocities from the Hubble flow and via direct galaxy catalogs, the local Universe is lumpy (10% fluctuations or more) on length scales as large as 200Mpc.', '1710.04223-2-10-3': 'Extrapolating a local catalog believed to be highly complete (e.g., Nearby Galaxy Catalog, which is complete to [MATH] mag within 11 Mpc outside the Galactic plane) from the bottom up is even more problematic given strong fluctuations on these distance scales (e.g., not a single galaxy cluster lies within 11 Mpc).', '1710.04223-2-11-0': '# Assessing Catalog completeness with SNe', '1710.04223-2-12-0': 'Here, we explore the use of SNe to evaluate the completeness of catalog(s) of the nearby Universe.', '1710.04223-2-12-1': 'Type Ia SNe are well suited for this task.', '1710.04223-2-12-2': 'These SNe are luminous (only [MATH]1mag fainter than a local [MATH] galaxy in the [MATH] band) and thus easily identified.', '1710.04223-2-12-3': 'The luminosity function is narrow: in [MATH] band the luminosity function can be fit with a Gaussian with a mean value of [MATH]mag and an rms of 0.14-0.23mag .', '1710.04223-2-12-4': 'Next, SNe Ia arise from both old and young populations .', '1710.04223-2-12-5': 'Thus, they sample all types of galaxies and can be used to study galaxy demographics without missing significant subpopulations.', '1710.04223-2-13-0': 'In contrast, core-collapse SNe (CC SNe) arise only in star-forming galaxies and exhibit a wide luminosity function, with absolute magnitude (SDSS [MATH] band) ranging from [MATH] to [MATH] .', '1710.04223-2-13-1': 'Clearly, additional care is necessary in using CC SNe to measure the RCF.', '1710.04223-2-14-0': 'Our proposal to assess the completeness of nearby galaxies catalogs is simple: (1) Following the detection of a SN, measure the redshift of the host galaxy, [MATH].', '1710.04223-2-14-1': '(2) Retain those events with [MATH], where [MATH] is the redshift to which the completeness is being measured ("tries").', '1710.04223-2-14-2': '(3) Check the galaxy catalog(s) to see if the SN host galaxy has a reliable redshift entry.', '1710.04223-2-14-3': 'If present and if [MATH] is equal to [MATH] to within, say [MATH], then it is a "hit."', '1710.04223-2-14-4': 'The redshift completeness factor (RCF) of the galaxy catalog is given by the ratio of "hits" to "tries."', '1710.04223-2-15-0': 'We emphasize that our definition of the RCF is explicitly tied to SNe and is not the fraction of galaxies with known redshifts by number.', '1710.04223-2-15-1': 'The majority of galaxies within any volume are very small galaxies which are also the most difficult to detect (indeed, new satellites of the Milky Way are still being uncovered).', '1710.04223-2-15-2': 'Previous studies have found that the Ia propensity (the rate of Ia production within a galaxy), [MATH], is well represented by a linear combination of the stellar mass of the galaxy ([MATH]) and the star formation rate ([MATH]).', '1710.04223-2-15-3': 'Thus the RCF (as defined here) approximately measures the completeness weighted by [MATH].', '1710.04223-2-15-4': 'Other definitions could be used: for example, had we employed CC SNe instead of SNe Ia, the resulting RCF would measure the completeness weighted by [MATH] alone.', '1710.04223-2-16-0': 'SN-based estimates of the RCF will depend on the completeness of the parent SN survey(s).', '1710.04223-2-16-1': 'A SN survey that is not complete to the limit of the search volume will find a larger fraction of low-luminosity events at smaller distances whereas more luminous events can be found to larger distances (with concomitant larger volume).', '1710.04223-2-16-2': 'Thus, poor control of completeness in a SN survey will bias the sample to galaxies which host more luminous events.', '1710.04223-2-16-3': 'In the case of Ia SNe, a correlation between the luminosity of the SN and the properties of its host does exist , although it is relatively weak.', '1710.04223-2-16-4': 'A bigger concern is if the properties of the host directly affect completeness-for example, if SNe are systematically missed in regions of high galaxy surface brightness, or if the appearance of the host galaxy is a consideration in decisions about spectroscopic classification.', '1710.04223-2-17-0': 'We emphasize that the SN rate per galaxy is far too low to provide a practical means of actually discovering new galaxies in significant numbers.', '1710.04223-2-17-1': 'Instead, the approach advocated here provides a check on the completeness of existing catalogs, independent of the traditional luminosity function approach.', '1710.04223-2-18-0': '# Primary Data', '1710.04223-2-19-0': 'lrrr[ht]', '1710.04223-2-20-0': 'SN Demographics from the two ASAS-SN surveys', '1710.04223-2-21-0': 'Catalog & type[MATH] & [MATH](SN) & [MATH]) A1 & Ia & 66 & 51A1 & CC & 25 & 22A2 & Ia & 141 & 101 A2 & CC & 39 & 27 [MATH] is the number of SNe in each category.', '1710.04223-2-21-1': '[MATH] is the number of host galaxies with a redshift entry in NED (and obtained prior to the SN discovery).', '1710.04223-2-21-2': 'aThe type Ia encompasses normal Ia and all sub-types such as 91T, 91bg, CSM and 00cx.', '1710.04223-2-21-3': 'All non-Ia SN are called as core collapse (II, IIn, Ibc).', '1710.04223-2-22-0': 'To assess the RCF we use NED as our input host galaxy catalog and the first two SNe catalogs (hereafter, A1, A2) published by the All-Sky Automated Survey for SN (ASAS-SN) project (; hereafter, H1 H2, respectively).', '1710.04223-2-22-1': 'ASAS-SN is well-suited for our purposes.', '1710.04223-2-22-2': 'It is a flux-limited survey, [MATH]mag, that covers the entire sky, and is not targeted to specific galaxies.', '1710.04223-2-22-3': 'Additionally, because of its shallow flux limit, ASAS-SN candidates are bright enough for worldwide follow up using small telescopes.', '1710.04223-2-22-4': 'As a result the classification (SN type and [MATH]) is essentially complete.', '1710.04223-2-22-5': 'In contrast, amateur discoveries (as well as professional surveys such as LOSS, the Lick Observatory SN Survey; ) target well-resolved and bright galaxies and are therefore biased.', '1710.04223-2-22-6': 'Other, recent untargeted surveys (e.g. ATLAS, iPTF, Gaia) are not likely to be strongly biased in terms of discovery, but the degree of bias in terms of selecting candidates to follow-up (and classify as SNe) is not well-quantified.', '1710.04223-2-23-0': 'Table [REF] provides a top level summary of the two ASAS-SN surveys.', '1710.04223-2-23-1': 'Catalog A1 lists 91 SNe discovered during the period 2013-2014 (H1).', '1710.04223-2-23-2': 'Catalog A2 lists 180 SNe found in the calendar year 2015.', '1710.04223-2-23-3': 'In A2 11 SNe are marginally fainter than 17mag (H2).', '1710.04223-2-23-4': 'The catalogs also include host galaxy data: name, redshift, SDSS, GALEX, 2MASS and WISE photometry.', '1710.04223-2-24-0': 'H1 and H2 do not specifically identify host galaxies that lack a cataloged redshift prior to the discovery of the SN.', '1710.04223-2-24-1': 'To remedy this situation we wrote a program to query NED and obtained the redshifts of the putative host galaxies.', '1710.04223-2-24-2': 'We refer to the sample of galaxies with a redshift entry in NED as the "NED[MATH]" sample and those which lacked an entry as the "!', '1710.04223-2-24-3': 'NED[MATH]" sample.', '1710.04223-2-25-0': 'Nine entries (ASAS-SN-15de, 15ji, 15jm, 15lh, 15nh, 15og, 15ts, 15ua, 14ms) have [MATH].', '1710.04223-2-25-1': 'These were deleted from further consideration.', '1710.04223-2-25-2': 'Next, we inspected the difference between the SN redshift, [MATH], and the purported host redshift given in NED, [MATH].', '1710.04223-2-25-3': 'Bearing in mind the lower precision of the SN redshift we made an allowance and inspected events with [MATH].', '1710.04223-2-25-4': 'The rationale for this choice is that the redshift of the host is usually the systemic velocity of the galaxy (a fiber of the spectrograph encompasses the central region of the galaxy) whereas the SN will have additional velocity arising from the rotation curve.', '1710.04223-2-25-5': 'Three events stood out: ASASSN-15jo has [MATH] (AbellS0753), but [MATH]; ASASSN-13an has [MATH] (2MASX J13453653[MATH]0719350), but [MATH]; and ASASSN-15ic has [MATH] (2MASXJ06145320[MATH]4247357), [MATH].', '1710.04223-2-25-6': 'We retain the first two events and delete the last one from further consideration.', '1710.04223-2-25-7': 'The final sample for the analysis reported below consists of 261 SNe.', '1710.04223-2-26-0': '# Analysis', '1710.04223-2-27-0': 'The redshift distribution of ASAS-SN events is displayed in Figure [REF].', '1710.04223-2-27-1': 'SNe Ia peak at [MATH] whereas CC events peak at lower redshift and exhibit a long tail (as expected, given their lower average luminosities and broad luminosity function).', '1710.04223-2-27-2': 'The "completeness" of the SN sample itself is not the primary topic of this paper.', '1710.04223-2-27-3': 'What is central to this paper is that the SN sample not be biased by host galaxies.', '1710.04223-2-27-4': 'So we will proceed with the analysis of the sample we have in hand.', '1710.04223-2-27-5': 'The RCF, assuming NED as the input host galaxy catalog, for [MATH] is [MATH]% (90% confidence interval; see Table [REF]).', '1710.04223-2-27-6': 'The precision of these estimates is limited by small number binomial statistics.', '1710.04223-2-27-7': 'The RCF as traced by SNe Ia, furthermore, is somewhat lower ([MATH]%) than the RCF traced by CC SNe ([MATH]%), suggesting that galaxy catalogs are more complete for star-forming hosts.', '1710.04223-2-27-8': 'The precision of these estimates is limited by small number binomial statistics.', '1710.04223-2-27-9': 'For our full sample ([MATH]), there are 64 !N', '1710.04223-2-27-10': 'NED[MATH] galaxies, which corresponds to [MATH].', '1710.04223-2-28-0': 'SN & [MATH](SN)a & [MATH]b & [MATH]c & RCFd All & 173 & 139 & 34 & 0.75-0.85Ia & 120 & 93 & 27 & 0.71-0.83CC & 53 & 46 & 7 & 0.78-0.93 aThe total number of SNe.', '1710.04223-2-28-1': 'bThe number of SNe with a putative host galaxy with redshift listed in NED, prior to SN discovery.', '1710.04223-2-28-2': 'cThe number of SNe with a putative host galaxy but whose redshift is not listed in NED.', '1710.04223-2-28-3': 'dThe Redshift Completeness Factor (RCF; see [REF]) range covers a confidence range of 5% to 95% and is obtained by assuming a flat Bayesian prior.', '1710.04223-2-29-0': '# Analysis of Host Galaxy Luminosities', '1710.04223-2-30-0': 'The virtue of the SN RCF is that it is a well-defined and simple metric, with no free parameters other than the distance (volume) limit employed.', '1710.04223-2-30-1': 'However, within this volume, we generally expect the completeness to be higher for luminous galaxies (with high stellar mass [MATH], or high star-formation rate [MATH]) than less luminous ones.', '1710.04223-2-30-2': 'Knowing these parameters for the galaxies within our sample, we can subdivide our targets by [MATH] or [MATH] to answer interesting questions such as, "How complete is our understanding of [MATH] / [MATH] within the local volume"?', '1710.04223-2-30-3': 'For this purpose, we use the 2MASS [MATH]-band as a proxy for [MATH] and the GALEX NUV band for [MATH], and examine the completeness as a function of these two parameters.', '1710.04223-2-31-0': 'H1 H2 took [MATH]-band magnitudes from 2MASS when available; otherwise, a [MATH]-band mag was estimated from WISE [MATH] when detected in that band (offset by a typical [MATH]mag), or (in the absence of both 2MASS and WISE) a limit of [MATH]mag was set.', '1710.04223-2-31-1': 'We employ the same approach using their magnitude catalogs, except that for the last group (3 galaxies in catalog A1 and 5 in catalog A2) we simply set [MATH]mag.', '1710.04223-2-32-0': 'In A1 there are 77 SNe with NUV data, 5 with only SDSS [MATH] data and 9 with neither NUV nor [MATH] data.', '1710.04223-2-32-1': 'In A2, the corresponding numbers are 138, 17 and 24, respectively.', '1710.04223-2-32-2': 'We took the sample of SNe with both NUV and [MATH] data, applied the Galactic extinction correction, and found the median of NUV[MATH]mag.', '1710.04223-2-32-3': 'We use the NUV data when available and [MATH] with the aforementioned offset applied otherwise.', '1710.04223-2-32-4': 'We call this hybrid magnitude the "UV" mag.', '1710.04223-2-32-5': 'The 32 SNe with neither NUV nor [MATH] data are not included in the analysis.', '1710.04223-2-33-0': 'For the discussion below we note that the relevant Schechter characteristic luminosity parameters are [MATH] and [MATH] .', '1710.04223-2-33-1': 'Below we examine the missing galaxies with luminosity [MATH] in both the [MATH] and "UV" bands.', '1710.04223-2-33-2': 'We then isolate the sample to just Type Ia SNe to investigate missing galaxies with [MATH].', '1710.04223-2-34-0': '## [MATH]-band', '1710.04223-2-35-0': 'The absolute [MATH]-band magnitude, [MATH], and redshift, [MATH], of each SN Ia host galaxy in our sample is displayed in Figure [REF].', '1710.04223-2-35-1': 'The joint distribution for detecting a host galaxy given its redshift and [MATH], [MATH], along with the 1-dimensional probabilities, [MATH] and [MATH], are also shown in Figure [REF].', '1710.04223-2-35-2': 'Details for these calculations are presented in Appendix [REF].', '1710.04223-2-35-3': 'Figure [REF] confirms the intuitive results that the RCF is lower for higher redshift and intrinsically fainter galaxies.', '1710.04223-2-36-0': 'In total, there are 42 host galaxies brighter than [MATH], and 2 of these did not have redshift entries in NED prior to SN discovery.', '1710.04223-2-36-1': 'The first is the host of ASASSN-15ed, MCG+09-27-087/SDSSJ164825.26+505935.5, a bright, large (40[MATH]) sprial galaxy.', '1710.04223-2-36-2': 'There is no SDSS spectroscopic redshift but the SDSS photoz estimate is [MATH].', '1710.04223-2-36-3': 'The host galaxy has a [MATH]mJy counterpart in NVSS.', '1710.04223-2-36-4': 'The second is the host of ASASSN-15ub, CGCG314-006.', '1710.04223-2-36-5': 'Nominally there is no host redshift in NED.', '1710.04223-2-36-6': 'However, a direct inspection of SDSS shows that a pair of strongly interacting galaxies with a spectroscopic redshift of 0.032 (photoz of [MATH]) which can be compared to [MATH].', '1710.04223-2-37-0': 'We restrict the analysis of sub-luminous galaxies to SNe Ia.', '1710.04223-2-37-1': 'In the [MATH] band, there are 163 host galaxies with [MATH].', '1710.04223-2-37-2': 'Of these 110 are listed in NED.', '1710.04223-2-37-3': 'Thus, as traced by SNe Ia, the [MATH] [67%-73%] (5%-95% confidence range).', '1710.04223-2-38-0': '## NUV/[MATH] band', '1710.04223-2-39-0': 'As was the case for [MATH] band, the median !', '1710.04223-2-39-1': 'NED[MATH] host galaxy is [MATH]2mag fainter than the median NED[MATH] host galaxy for the "UV" band (Figure [REF]).', '1710.04223-2-39-2': 'The redshift distribution of NED[MATH] and !', '1710.04223-2-39-3': 'NED[MATH] galaxies in nearly identical in [MATH] and "UV", with higher [MATH] hosts more likely to not be included in NED.', '1710.04223-2-39-4': 'The peak of the !', '1710.04223-2-39-5': 'NED[MATH] sample "UV" luminosity is not as biased towards faint galaxies as it is in the [MATH] band.', '1710.04223-2-39-6': 'This conclusion bodes well for the PTF Census of the Local Universe (CLU) H[MATH] survey which is searching for nearby star-forming galaxies .', '1710.04223-2-40-0': 'In the "UV" band, 46 galaxies are brighter than [MATH], and 5 of those are !', '1710.04223-2-40-1': 'NED[MATH] galaxies.', '1710.04223-2-40-2': 'The host of ASASSN-15ed, MCG+09-27-087; SDSSJ164825.26+505935.5, is discussed above.', '1710.04223-2-40-3': 'The ASASSN-15ln host, GALEXASC J225332.83+194232.9, is a 10[MATH] spiral galaxy with no SDSS spectroscopic redshift but an SDSS photoz of [MATH].', '1710.04223-2-40-4': 'The host of ASASSN-15ho, 2MASXi J0909234-044327, lies outside the SDSS footprint.', '1710.04223-2-40-5': 'The hosts of ASASSN-15sh and ASASSN-15um, 2MASXJ19320827-6226340 and 2MASXJ05395948-8022191, respectively, lie in poorly studied regions of the sky.', '1710.04223-2-41-0': 'Again, we restrict the analysis of missing sub-luminous galaxies to SNe Ia hosts.', '1710.04223-2-41-1': 'There are 140 galaxies with "UV" luminosity less than the corresponding [MATH] value.', '1710.04223-2-41-2': 'Of these, 98 have redshift entries in NED.', '1710.04223-2-41-3': 'Thus, as traced by SNe Ia, the [MATH] [63%-76%].', '1710.04223-2-42-0': '# Increasing the Precision Accuracy of RCF', '1710.04223-2-43-0': 'The ASAS-SN bright SN sample is attractive for measuring the RCF due to its host-unbiased approach and the essentially complete spectroscopic classification of all candidates that results from its shallow magnitude limit.', '1710.04223-2-43-1': 'However, owing to the small sample size, the RCF estimates are limited by binomial errors.', '1710.04223-2-43-2': 'We undertook a similar analysis for a larger sample: SNe candidates reported at the Transient Name Server (TNS) portal, during the period between January 2016 and June 2017.', '1710.04223-2-43-3': 'The sample spans a larger peak magnitude range relative to ASAS-SN, extending as faint as 20 mag.', '1710.04223-2-43-4': 'Nonetheless, it appears that follow up was obtained for most of the reported candidate SNe.', '1710.04223-2-43-5': 'The resulting sample size is 529 nearby ([MATH]) SNe.', '1710.04223-2-43-6': 'We find that the RCF for this sample is similar to that derived for the ASAS-SN sample (Cassese Kulkarni, in prep).', '1710.04223-2-44-0': 'The field of optical time-domain astronomy is in a boom period, and much larger samples of nearby SNe can be expected given the Asteroid Terrestrial-impact Last Alert (ATLAS; ), PanSTARRS-1 , the Zwicky Transient Facility (ZTF; ), and upgraded ASAS-SN surveys.', '1710.04223-2-44-1': 'The limiting V-band magnitudes for these surveys range from 17 to 21mag.', '1710.04223-2-44-2': 'Below, we consider the gains resulting from large SN samples.', '1710.04223-2-45-0': 'To make this discussion concrete we consider a specific example, the "Celestial Cinematography" survey of ZTF .', '1710.04223-2-45-1': 'This survey aims to systematically cover a large fraction of the night sky ([MATH]) every three nights, in the [MATH] and [MATH] bands.', '1710.04223-2-45-2': 'The median 5-[MATH] detection limit, for a fixed 30s exposure time, is 20.5mag.', '1710.04223-2-45-3': 'The annual volumetric rate of [MATH] Type Ia and CC SNe is [MATH] and [MATH], respectively .', '1710.04223-2-45-4': 'Based on a simulator built for ZTF, the expected annual yield for the above survey is [230, 460, 892, 1750] for peak magnitude of [17.5, 18, 18.5, 19]mag, respectively (U. Feindt, pers.', '1710.04223-2-45-5': 'comm.)', '1710.04223-2-46-0': 'Going forward we will assume a "Bright Transient Survey" (BTS) whose goal is to classify all extra-galactic transients whose peak magnitude is brighter than 18.5mag.', '1710.04223-2-46-1': 'A one year survey would result in a sample of nearly one thousand SNe Ia.', '1710.04223-2-46-2': 'With this sample, a regional RCF can be evaluated (e.g., high and intermediate Galactic latitude regions).', '1710.04223-2-46-3': 'Next, the large and unbiased sample would allow for a number of other applications, including self-consistent checks of the dependence of the SN Ia rate on host type, which is frequently formulated as [MATH] ; here, [MATH] and [MATH] are constants.', '1710.04223-2-46-4': 'Deviations will give us insight into a better formulation of the relationship between [MATH] and [MATH] and [MATH].', '1710.04223-2-47-0': 'Such a large survey would, in its own right be interesting.', '1710.04223-2-47-1': 'For example, determining volumetric SN rates from untargeted, wide-field surveys requires the identification of all galaxies within a specified distance.', '1710.04223-2-47-2': 'The BTS measurement of the RCF will provide the correction factors needed to account for missing galaxies when calculating the volumetric rates.', '1710.04223-2-47-3': 'For example, the relative rate of SN2002cx-like (SNe Iax) to normal SNe Ia is wildly uncertain ([MATH]%; e.g., ), and the large sample from the BTS will substantially improve these estimates.', '1710.04223-2-47-4': 'Furthermore, such a large, low-redshift sample would be very valuable for Ia SN cosmography (e.g., ).', '1710.04223-2-48-0': 'Next we address CC SNe.', '1710.04223-2-48-1': 'As noted in [REF] CC SNe exhibit a wide range in peak magnitude: [MATH] ranging from [MATH] to about [MATH] mag .', '1710.04223-2-48-2': 'The BTS is well suited to determining the demographics of CC SNe.', '1710.04223-2-48-3': 'A survey complete to a flux-limit [MATH] will detect SNe peaking at [MATH]mag to a radius of [MATH]10, 40, and 160Mpc, respectively.', '1710.04223-2-48-4': 'The total number of CC SN detections will sharply depend on the luminosity function.', '1710.04223-2-48-5': 'For instance, [CITATION] suggest that the fraction of CC SNe fainter than [MATH] at peak is at least 24% but can be as high as [MATH].', '1710.04223-2-48-6': 'In any case, BTS will allow us to measure the luminosity function of CC events which is essential to determine the volumetric rate of CC SNe.', '1710.04223-2-48-7': 'In turn, the latter is a key element in our understanding of stars and the interstellar medium .', '1710.04223-2-48-8': 'Finally, while CC SNe certainly track [MATH], it may be the case that "lesser" parameters, such as metallicity, change the mix of CC SNe subtypes .', '1710.04223-2-48-9': 'Again large-sample SN surveys may well have sufficient diagnostic power to ferret out such connections.', '1710.04223-2-49-0': 'It is increasingly evident that the primary limitation to SN surveys is limited by our ability to spectrally classify the SN candidates.', '1710.04223-2-49-1': 'This load can be made bearable by the use of two spectrographs: an ultra-low resolution spectrometer tuned to classification and a standard low-resolution spectrometer to get the redshift and gross spectrum of the host galaxies.', '1710.04223-2-49-2': 'For the latter we note that within a few years not merely highly- but supremely-multiplexed spectrographs (e.g. DESI, PFS and the planned AS4 project) will be commissioned.', '1710.04223-2-49-3': 'These facilities, at very little cost (small fractional allocation of fibers), can measure the redshifts of host galaxies of SNe on an industrial scale.', '1710.04223-2-49-4': 'The same highly-multiplexed spectrographs will likely be pressed into surveys more ambitious than SDSS or 6dF, leading to more complete catalogs of galaxies in the nearby Universe.', '1710.04223-2-50-0': 'We thank A. Goobar, U. Feindt, C. Pankow, M. Kasliwal, P. Nugent, E. O. Ofek, E. S. Phinney, K. Taggart and H. Vedantham for inputs and helpful discussions.', '1710.04223-2-51-0': 'AAM is funded by the Large Synoptic Survey Telescope Corporation in support of the Data Science Fellowship Program.', '1710.04223-2-52-0': '# Conditional Probability of the RCF', '1710.04223-2-53-0': 'We aim to characterize the RCF as a function of redshift, [MATH], and host galaxy luminosity, where we use either [MATH] or [MATH] as a proxy.', '1710.04223-2-53-1': 'To do so, we model the data [MATH] with the Bernoulli distribution [EQUATION] where [MATH] is parameterized with a logistic function with dependence on both redshift [MATH] and host galaxy luminosity: [EQUATION] with host-galaxy absolute magnitude [MATH], and [MATH] representing the model parameters: [MATH], [MATH], and [MATH], which need to be determined.', '1710.04223-2-53-2': 'The precise analytic dependence of [MATH] on [MATH] and [MATH] may not be logistic, however, the purpose of this exercise is to provide a general sense for how the RCF relies on [MATH] and [MATH].', '1710.04223-2-53-3': 'The logistic function is ideal for this general purpose.', '1710.04223-2-54-0': 'From here it follows that the probability of a host galaxy having a previously cataloged redshift is: [EQUATION] and the likelihood of the observations given the data and model parameters is: [EQUATION] where [MATH] represents the individual observations and [MATH] for [MATH] galaxies and [MATH] for [MATH] galaxies.', '1710.04223-2-55-0': "From Bayes' theorem, we can multiply the likelihood by a prior, [MATH], and use Markov Chain Monte Carlo (MCMC) techniques to sample from the posterior [MATH] in order to constrain the model parameters [MATH].", '1710.04223-2-55-1': 'We use the emcee package to implement our MCMC sampling of the posterior.', '1710.04223-2-55-2': 'For [MATH] and [MATH] we adopt flat priors bounded between 0 and [MATH].', '1710.04223-2-55-3': 'For [MATH] we adopt a flat prior between [MATH] and [MATH].', '1710.04223-2-55-4': 'Following the MCMC sampling, we find that there is a strong covariance between [MATH] and [MATH], while [MATH] is relatively independent of [MATH] and [MATH].', '1710.04223-2-55-5': 'The shading in Figures [REF] and [REF] shows [MATH] for the maximum a posteriori sample from the MCMC sampling.', '1710.04223-2-56-0': 'We additionally wish to constrain the behavior of the RCF as a function of either the host redshift, [MATH], or host galaxy luminosity.', '1710.04223-2-56-1': 'We do this separately from the analysis above, while using the same MCMC procedure with [MATH] in Equations [REF] and [REF] replaced by [EQUATION] for redshift, and [EQUATION] for host galaxy luminosity (where, again, we use absolute magnitude [MATH] as a proxy).', '1710.04223-2-56-2': 'The results of this procedure are shown in the side panels of Figures [REF] and [REF].', '1710.04223-2-56-3': 'In these panels the solid lines show the median value of [MATH], [MATH] in the Figures, and [MATH], [MATH] in the Figures, from all the posterior samples, while the shaded region shows the 90% credible regions for [MATH] and [MATH] from the posterior samples.', '1710.04223-2-56-4': 'We close by noting that the current dataset provides weak constraints on [MATH] in Equation [REF], but these constraints will be greatly improved by the BTS which will include a significantly larger sample and extend to higher redshifts.', '1710.04223-2-57-0': '# RCF as Traced by CC SNe', '1710.04223-2-58-0': 'The CC SNe samples in A1 and A2 are insufficient to meaningfully constrain the RCF as a function of redshift or host galaxy absolute magnitude.', '1710.04223-2-58-1': 'Furthermore, CC SNe only trace star formation, meaning they do not probe passive galaxies, so we have excluded them from the analysis in the main text.', '1710.04223-2-58-2': 'Nevertheless, for completeness, we show the host galaxies for CC SNe in Figure [REF].'}

[['1710.04223-1-7-0', '1710.04223-2-7-0'], ['1710.04223-1-41-0', '1710.04223-2-39-0'], ['1710.04223-1-41-2', '1710.04223-2-39-2'], ['1710.04223-1-41-3', '1710.04223-2-39-3'], ['1710.04223-1-41-4', '1710.04223-2-39-4'], ['1710.04223-1-41-5', '1710.04223-2-39-5'], ['1710.04223-1-41-6', '1710.04223-2-39-6'], ['1710.04223-1-5-1', '1710.04223-2-5-1'], ['1710.04223-1-34-0', '1710.04223-2-32-0'], ['1710.04223-1-34-1', '1710.04223-2-32-1'], ['1710.04223-1-34-2', '1710.04223-2-32-2'], ['1710.04223-1-34-3', '1710.04223-2-32-3'], ['1710.04223-1-34-4', '1710.04223-2-32-4'], ['1710.04223-1-34-5', '1710.04223-2-32-5'], ['1710.04223-1-54-0', '1710.04223-2-51-0'], ['1710.04223-1-52-1', '1710.04223-2-49-1'], ['1710.04223-1-52-2', '1710.04223-2-49-2'], ['1710.04223-1-52-3', '1710.04223-2-49-3'], ['1710.04223-1-52-4', '1710.04223-2-49-4'], ['1710.04223-1-12-0', '1710.04223-2-12-0'], ['1710.04223-1-12-1', '1710.04223-2-12-1'], ['1710.04223-1-20-1', '1710.04223-2-21-1'], ['1710.04223-1-20-2', '1710.04223-2-21-2'], ['1710.04223-1-22-1', '1710.04223-2-23-2'], ['1710.04223-1-9-3', '1710.04223-2-9-3'], ['1710.04223-1-30-1', '1710.04223-2-28-1'], ['1710.04223-1-30-2', '1710.04223-2-28-2'], ['1710.04223-1-30-3', '1710.04223-2-28-3'], ['1710.04223-1-3-1', '1710.04223-2-3-1'], ['1710.04223-1-3-2', '1710.04223-2-3-2'], ['1710.04223-1-35-0', '1710.04223-2-33-0'], ['1710.04223-1-35-1', '1710.04223-2-33-1'], ['1710.04223-1-35-2', '1710.04223-2-33-2'], ['1710.04223-1-24-1', '1710.04223-2-25-1'], ['1710.04223-1-24-2', '1710.04223-2-25-2'], ['1710.04223-1-24-6', '1710.04223-2-25-6'], ['1710.04223-1-24-8', '1710.04223-2-25-7'], ['1710.04223-1-43-0', '1710.04223-2-41-0'], ['1710.04223-1-43-1', '1710.04223-2-41-1'], ['1710.04223-1-43-2', '1710.04223-2-41-2'], ['1710.04223-1-38-0', '1710.04223-2-36-0'], ['1710.04223-1-38-2', '1710.04223-2-36-2'], ['1710.04223-1-38-3', '1710.04223-2-36-3'], ['1710.04223-1-38-4', '1710.04223-2-36-4'], ['1710.04223-1-38-5', '1710.04223-2-36-5'], ['1710.04223-1-38-6', '1710.04223-2-36-6'], ['1710.04223-1-10-3', '1710.04223-2-10-2'], ['1710.04223-1-4-3', '1710.04223-2-4-2'], ['1710.04223-1-42-0', '1710.04223-2-40-0'], ['1710.04223-1-42-3', '1710.04223-2-40-3'], ['1710.04223-1-42-4', '1710.04223-2-40-4'], ['1710.04223-1-0-0', '1710.04223-2-0-0'], ['1710.04223-1-0-5', '1710.04223-2-0-4'], ['1710.04223-1-33-1', '1710.04223-2-31-1'], ['1710.04223-1-23-2', '1710.04223-2-24-1'], ['1710.04223-1-23-3', '1710.04223-2-24-2'], ['1710.04223-1-39-0', '1710.04223-2-37-0'], ['1710.04223-1-39-1', '1710.04223-2-37-1'], ['1710.04223-1-39-2', '1710.04223-2-37-2'], ['1710.04223-1-15-1', '1710.04223-2-15-1'], ['1710.04223-1-49-0', '1710.04223-2-45-0'], ['1710.04223-1-50-0', '1710.04223-2-45-3'], ['1710.04223-1-26-0', '1710.04223-2-27-0'], ['1710.04223-1-26-4', '1710.04223-2-27-4'], ['1710.04223-1-26-6', '1710.04223-2-27-6'], ['1710.04223-1-27-0', '1710.04223-2-27-9'], ['1710.04223-1-27-1', '1710.04223-2-27-10'], ['1710.04223-1-48-1', '1710.04223-2-48-9'], ['1710.04223-1-51-5', '1710.04223-2-48-5'], ['1710.04223-1-51-7', '1710.04223-2-48-7'], ['1710.04223-1-2-0', '1710.04223-2-2-0'], ['1710.04223-1-21-3', '1710.04223-2-22-4'], ['1710.04223-1-41-1', '1710.04223-2-39-1'], ['1710.04223-1-14-0', '1710.04223-2-14-0'], ['1710.04223-1-14-1', '1710.04223-2-14-1'], ['1710.04223-1-14-2', '1710.04223-2-14-2'], ['1710.04223-1-14-3', '1710.04223-2-14-3'], ['1710.04223-1-14-4', '1710.04223-2-14-4'], ['1710.04223-1-16-1', '1710.04223-2-17-0'], ['1710.04223-1-12-2', '1710.04223-2-12-2'], ['1710.04223-1-12-3', '1710.04223-2-12-3'], ['1710.04223-1-20-3', '1710.04223-2-21-3'], ['1710.04223-1-22-0', '1710.04223-2-23-1'], ['1710.04223-1-22-3', '1710.04223-2-23-4'], ['1710.04223-1-9-2', '1710.04223-2-9-2'], ['1710.04223-1-3-0', '1710.04223-2-3-0'], ['1710.04223-1-24-3', '1710.04223-2-25-3'], ['1710.04223-1-24-4', '1710.04223-2-25-4'], ['1710.04223-1-53-0', '1710.04223-2-50-0'], ['1710.04223-1-37-0', '1710.04223-2-35-0'], ['1710.04223-1-4-2', '1710.04223-2-4-1'], ['1710.04223-1-6-1', '1710.04223-2-6-0'], ['1710.04223-1-6-2', '1710.04223-2-6-1'], ['1710.04223-1-0-7', '1710.04223-2-0-6'], ['1710.04223-1-33-0', '1710.04223-2-31-0'], ['1710.04223-1-45-4', '1710.04223-2-43-3'], ['1710.04223-1-45-5', '1710.04223-2-43-4'], ['1710.04223-1-23-1', '1710.04223-2-24-0'], ['1710.04223-1-15-0', '1710.04223-2-15-0'], ['1710.04223-1-15-4', '1710.04223-2-16-0'], ['1710.04223-1-49-1', '1710.04223-2-45-1'], ['1710.04223-1-49-2', '1710.04223-2-45-2'], ['1710.04223-1-26-2', '1710.04223-2-27-2'], ['1710.04223-1-26-3', '1710.04223-2-27-3'], ['1710.04223-1-48-0', '1710.04223-2-48-8'], ['1710.04223-1-51-3', '1710.04223-2-48-3'], ['1710.04223-1-51-4', '1710.04223-2-48-4'], ['1710.04223-1-51-6', '1710.04223-2-48-6'], ['1710.04223-1-7-1', '1710.04223-2-7-3'], ['1710.04223-1-2-4', '1710.04223-2-2-5'], ['1710.04223-1-2-5', '1710.04223-2-2-6'], ['1710.04223-1-32-0', '1710.04223-2-30-0'], ['1710.04223-1-32-3', '1710.04223-2-30-3'], ['1710.04223-1-21-0', '1710.04223-2-22-0'], ['1710.04223-1-21-1', '1710.04223-2-22-2'], ['1710.04223-1-21-2', '1710.04223-2-22-3'], ['1710.04223-1-5-0', '1710.04223-2-5-0'], ['1710.04223-1-12-4', '1710.04223-2-12-4'], ['1710.04223-1-12-4', '1710.04223-2-12-5'], ['1710.04223-1-22-2', '1710.04223-2-23-3'], ['1710.04223-1-13-0', '1710.04223-2-13-0'], ['1710.04223-1-13-2', '1710.04223-2-13-1'], ['1710.04223-1-9-0', '1710.04223-2-9-0'], ['1710.04223-1-9-1', '1710.04223-2-9-1'], ['1710.04223-1-3-3', '1710.04223-2-3-3'], ['1710.04223-1-10-2', '1710.04223-2-10-1'], ['1710.04223-1-10-4', '1710.04223-2-10-3'], ['1710.04223-1-4-5', '1710.04223-2-4-3'], ['1710.04223-1-6-3', '1710.04223-2-6-2'], ['1710.04223-1-0-1', '1710.04223-2-0-1'], ['1710.04223-1-0-2', '1710.04223-2-0-3'], ['1710.04223-1-0-4', '1710.04223-2-0-2'], ['1710.04223-1-0-8', '1710.04223-2-0-8'], ['1710.04223-1-45-0', '1710.04223-2-43-0'], ['1710.04223-1-45-3', '1710.04223-2-43-2'], ['1710.04223-1-45-6', '1710.04223-2-43-6'], ['1710.04223-1-15-5', '1710.04223-2-16-1'], ['1710.04223-1-15-5', '1710.04223-2-16-2'], ['1710.04223-1-50-1', '1710.04223-2-45-4'], ['1710.04223-1-26-1', '1710.04223-2-27-1'], ['1710.04223-1-26-5', '1710.04223-2-27-5'], ['1710.04223-1-46-0', '1710.04223-2-44-0']]

[['1710.04223-1-7-0', '1710.04223-2-7-0'], ['1710.04223-1-41-0', '1710.04223-2-39-0'], ['1710.04223-1-41-2', '1710.04223-2-39-2'], ['1710.04223-1-41-3', '1710.04223-2-39-3'], ['1710.04223-1-41-4', '1710.04223-2-39-4'], ['1710.04223-1-41-5', '1710.04223-2-39-5'], ['1710.04223-1-41-6', '1710.04223-2-39-6'], ['1710.04223-1-5-1', '1710.04223-2-5-1'], ['1710.04223-1-34-0', '1710.04223-2-32-0'], ['1710.04223-1-34-1', '1710.04223-2-32-1'], ['1710.04223-1-34-2', '1710.04223-2-32-2'], ['1710.04223-1-34-3', '1710.04223-2-32-3'], ['1710.04223-1-34-4', '1710.04223-2-32-4'], ['1710.04223-1-34-5', '1710.04223-2-32-5'], ['1710.04223-1-54-0', '1710.04223-2-51-0'], ['1710.04223-1-52-1', '1710.04223-2-49-1'], ['1710.04223-1-52-2', '1710.04223-2-49-2'], ['1710.04223-1-52-3', '1710.04223-2-49-3'], ['1710.04223-1-52-4', '1710.04223-2-49-4'], ['1710.04223-1-12-0', '1710.04223-2-12-0'], ['1710.04223-1-12-1', '1710.04223-2-12-1'], ['1710.04223-1-20-1', '1710.04223-2-21-1'], ['1710.04223-1-20-2', '1710.04223-2-21-2'], ['1710.04223-1-22-1', '1710.04223-2-23-2'], ['1710.04223-1-9-3', '1710.04223-2-9-3'], ['1710.04223-1-30-1', '1710.04223-2-28-1'], ['1710.04223-1-30-2', '1710.04223-2-28-2'], ['1710.04223-1-30-3', '1710.04223-2-28-3'], ['1710.04223-1-3-1', '1710.04223-2-3-1'], ['1710.04223-1-3-2', '1710.04223-2-3-2'], ['1710.04223-1-35-0', '1710.04223-2-33-0'], ['1710.04223-1-35-1', '1710.04223-2-33-1'], ['1710.04223-1-35-2', '1710.04223-2-33-2'], ['1710.04223-1-24-1', '1710.04223-2-25-1'], ['1710.04223-1-24-2', '1710.04223-2-25-2'], ['1710.04223-1-24-6', '1710.04223-2-25-6'], ['1710.04223-1-24-8', '1710.04223-2-25-7'], ['1710.04223-1-43-0', '1710.04223-2-41-0'], ['1710.04223-1-43-1', '1710.04223-2-41-1'], ['1710.04223-1-43-2', '1710.04223-2-41-2'], ['1710.04223-1-38-0', '1710.04223-2-36-0'], ['1710.04223-1-38-2', '1710.04223-2-36-2'], ['1710.04223-1-38-3', '1710.04223-2-36-3'], ['1710.04223-1-38-4', '1710.04223-2-36-4'], ['1710.04223-1-38-5', '1710.04223-2-36-5'], ['1710.04223-1-38-6', '1710.04223-2-36-6'], ['1710.04223-1-10-3', '1710.04223-2-10-2'], ['1710.04223-1-4-3', '1710.04223-2-4-2'], ['1710.04223-1-42-0', '1710.04223-2-40-0'], ['1710.04223-1-42-3', '1710.04223-2-40-3'], ['1710.04223-1-42-4', '1710.04223-2-40-4'], ['1710.04223-1-0-0', '1710.04223-2-0-0'], ['1710.04223-1-0-5', '1710.04223-2-0-4'], ['1710.04223-1-33-1', '1710.04223-2-31-1'], ['1710.04223-1-23-2', '1710.04223-2-24-1'], ['1710.04223-1-23-3', '1710.04223-2-24-2'], ['1710.04223-1-39-0', '1710.04223-2-37-0'], ['1710.04223-1-39-1', '1710.04223-2-37-1'], ['1710.04223-1-39-2', '1710.04223-2-37-2'], ['1710.04223-1-15-1', '1710.04223-2-15-1'], ['1710.04223-1-49-0', '1710.04223-2-45-0'], ['1710.04223-1-50-0', '1710.04223-2-45-3'], ['1710.04223-1-26-0', '1710.04223-2-27-0'], ['1710.04223-1-26-4', '1710.04223-2-27-4'], ['1710.04223-1-26-6', '1710.04223-2-27-6'], ['1710.04223-1-27-1', '1710.04223-2-27-10'], ['1710.04223-1-48-1', '1710.04223-2-48-9'], ['1710.04223-1-51-5', '1710.04223-2-48-5'], ['1710.04223-1-51-7', '1710.04223-2-48-7']]

[['1710.04223-1-27-0', '1710.04223-2-27-9'], ['1710.04223-1-2-0', '1710.04223-2-2-0'], ['1710.04223-1-21-3', '1710.04223-2-22-4'], ['1710.04223-1-41-1', '1710.04223-2-39-1'], ['1710.04223-1-14-0', '1710.04223-2-14-0'], ['1710.04223-1-14-1', '1710.04223-2-14-1'], ['1710.04223-1-14-2', '1710.04223-2-14-2'], ['1710.04223-1-14-3', '1710.04223-2-14-3'], ['1710.04223-1-14-4', '1710.04223-2-14-4'], ['1710.04223-1-16-1', '1710.04223-2-17-0'], ['1710.04223-1-12-2', '1710.04223-2-12-2'], ['1710.04223-1-12-3', '1710.04223-2-12-3'], ['1710.04223-1-20-3', '1710.04223-2-21-3'], ['1710.04223-1-22-0', '1710.04223-2-23-1'], ['1710.04223-1-22-3', '1710.04223-2-23-4'], ['1710.04223-1-9-2', '1710.04223-2-9-2'], ['1710.04223-1-3-0', '1710.04223-2-3-0'], ['1710.04223-1-24-3', '1710.04223-2-25-3'], ['1710.04223-1-24-4', '1710.04223-2-25-4'], ['1710.04223-1-53-0', '1710.04223-2-50-0'], ['1710.04223-1-37-0', '1710.04223-2-35-0'], ['1710.04223-1-4-2', '1710.04223-2-4-1'], ['1710.04223-1-6-1', '1710.04223-2-6-0'], ['1710.04223-1-6-2', '1710.04223-2-6-1'], ['1710.04223-1-0-7', '1710.04223-2-0-6'], ['1710.04223-1-33-0', '1710.04223-2-31-0'], ['1710.04223-1-45-4', '1710.04223-2-43-3'], ['1710.04223-1-45-5', '1710.04223-2-43-4'], ['1710.04223-1-23-1', '1710.04223-2-24-0'], ['1710.04223-1-15-0', '1710.04223-2-15-0'], ['1710.04223-1-15-4', '1710.04223-2-16-0'], ['1710.04223-1-49-1', '1710.04223-2-45-1'], ['1710.04223-1-49-2', '1710.04223-2-45-2'], ['1710.04223-1-26-2', '1710.04223-2-27-2'], ['1710.04223-1-26-3', '1710.04223-2-27-3'], ['1710.04223-1-48-0', '1710.04223-2-48-8'], ['1710.04223-1-51-3', '1710.04223-2-48-3'], ['1710.04223-1-51-4', '1710.04223-2-48-4'], ['1710.04223-1-51-6', '1710.04223-2-48-6']]

[]

[['1710.04223-1-7-1', '1710.04223-2-7-3'], ['1710.04223-1-2-4', '1710.04223-2-2-5'], ['1710.04223-1-2-5', '1710.04223-2-2-6'], ['1710.04223-1-32-0', '1710.04223-2-30-0'], ['1710.04223-1-32-3', '1710.04223-2-30-3'], ['1710.04223-1-21-0', '1710.04223-2-22-0'], ['1710.04223-1-21-1', '1710.04223-2-22-2'], ['1710.04223-1-21-2', '1710.04223-2-22-3'], ['1710.04223-1-5-0', '1710.04223-2-5-0'], ['1710.04223-1-12-4', '1710.04223-2-12-4'], ['1710.04223-1-12-4', '1710.04223-2-12-5'], ['1710.04223-1-22-2', '1710.04223-2-23-3'], ['1710.04223-1-13-0', '1710.04223-2-13-0'], ['1710.04223-1-13-2', '1710.04223-2-13-1'], ['1710.04223-1-9-0', '1710.04223-2-9-0'], ['1710.04223-1-9-1', '1710.04223-2-9-1'], ['1710.04223-1-3-3', '1710.04223-2-3-3'], ['1710.04223-1-10-2', '1710.04223-2-10-1'], ['1710.04223-1-10-4', '1710.04223-2-10-3'], ['1710.04223-1-4-5', '1710.04223-2-4-3'], ['1710.04223-1-6-3', '1710.04223-2-6-2'], ['1710.04223-1-0-1', '1710.04223-2-0-1'], ['1710.04223-1-0-2', '1710.04223-2-0-3'], ['1710.04223-1-0-4', '1710.04223-2-0-2'], ['1710.04223-1-0-8', '1710.04223-2-0-8'], ['1710.04223-1-45-0', '1710.04223-2-43-0'], ['1710.04223-1-45-3', '1710.04223-2-43-2'], ['1710.04223-1-45-6', '1710.04223-2-43-6'], ['1710.04223-1-15-5', '1710.04223-2-16-1'], ['1710.04223-1-15-5', '1710.04223-2-16-2'], ['1710.04223-1-50-1', '1710.04223-2-45-4'], ['1710.04223-1-26-1', '1710.04223-2-27-1'], ['1710.04223-1-26-5', '1710.04223-2-27-5']]

[['1710.04223-1-46-0', '1710.04223-2-44-0']]

['1710.04223-1-2-3', '1710.04223-1-10-1', '1710.04223-1-18-0', '1710.04223-1-19-0', '1710.04223-1-20-0', '1710.04223-1-23-4', '1710.04223-1-24-0', '1710.04223-1-24-5', '1710.04223-1-28-0', '1710.04223-1-29-0', '1710.04223-1-30-0', '1710.04223-1-38-1', '1710.04223-1-42-1', '1710.04223-1-42-2', '1710.04223-1-42-5', '1710.04223-1-50-2', '1710.04223-2-19-0', '1710.04223-2-20-0', '1710.04223-2-21-0', '1710.04223-2-24-3', '1710.04223-2-25-0', '1710.04223-2-25-5', '1710.04223-2-28-0', '1710.04223-2-36-1', '1710.04223-2-37-3', '1710.04223-2-40-1', '1710.04223-2-40-2', '1710.04223-2-40-5', '1710.04223-2-41-3', '1710.04223-2-45-5']

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

https://arxiv.org/abs/1710.04223

1608.04061

{'1608.04061-1-0-0': 'Let [MATH] be an [MATH]dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying [MATH], where [MATH] is the Laplace-Beltrami operator on [MATH] and [MATH] is the distance function from a given point.', '1608.04061-1-0-1': 'If [MATH] supports a second-order Sobolev inequality with a constant [MATH] close to the optimal constant [MATH] in the second-order Sobolev inequality in [MATH], we show that a global volume non-collapsing property holds on [MATH].', '1608.04061-1-0-2': 'The latter property together with a Perelman-type construction established by Munn (J. Geom.', '1608.04061-1-0-3': 'Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of [MATH].', '1608.04061-1-0-4': 'Furthermore, it turns out that [MATH] supports the second-order Sobolev inequality with the constant [MATH] if and only if [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-1-0-5': 'In this way our results give the first higher-order counterparts of the well known rigidity results of Ledoux (Comm.', '1608.04061-1-0-6': 'Anal.', '1608.04061-1-0-7': 'Geom., 1999), and do Carmo and Xia (Compos.', '1608.04061-1-0-8': 'Math., 2004).', '1608.04061-1-1-0': '# Introduction', '1608.04061-1-2-0': 'It is well known that the validity of first-order Sobolev inequalities on Riemannian manifolds strongly depend on the curvature; this is a rough conclusion of the famous AB-program initiated by Th.', '1608.04061-1-2-1': 'Aubin in the seventies, see the monograph of Hebey [CITATION] for a systematic presentation.', '1608.04061-1-2-2': 'To be more precise, let [MATH] be an [MATH]dimensional complete Riemannian manifold, [MATH] and consider for some [MATH] the first-order Sobolev inequality [EQUATION] where [MATH] is the first-order critical Sobolev exponent, and [MATH] and [MATH] denote the canonical volume form and gradient on [MATH], respectively.', '1608.04061-1-2-3': 'On one hand, inequality [MATH] holds on any [MATH]dimensional Cartan-Hadamard manifold [MATH] (i.e., simply connected, complete Riemannian manifold with nonpositive sectional curvature) with the optimal Euclidean constant [MATH] whenever the Cartan-Hadamard conjecture holds on [MATH], e.g., [MATH].', '1608.04061-1-2-4': 'On the other hand, due to Ledoux [CITATION], if [MATH] has nonnegative Ricci curvature, inequality [MATH] holds if and only if [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-1-2-5': "Further first-order Sobolev-type inequalities on Riemannian/Finsler manifolds can be found in Bakry, Concordet and Ledoux [CITATION], Druet, Hebey and Vaugon [CITATION], do Carmo and Xia [CITATION], Xia [CITATION]-[CITATION], Kristaly [CITATION]; moreover, similar Sobolev inequalities are also considered on 'nonnegatively' curved metric measure spaces formulated in terms of the Lott-Sturm-Villani-type curvature-dimension condition or the Bishop-Gromov-type doubling measure condition, see Kristaly [CITATION] and Kristaly and Ohta [CITATION].", '1608.04061-1-3-0': 'With respect to first-order Sobolev inequalities, much less is know about higher-order Sobolev inequalities on curved spaces.', '1608.04061-1-3-1': 'The first studies concern the AB-program for Paneitz-type operators on compact Riemannian manifolds, see Djadli, Hebey and Ledoux [CITATION], Hebey [CITATION] and Biezuner and Montenegro [CITATION].', '1608.04061-1-3-2': 'Recently, Gursky and Malchiodi [CITATION] studied strong maximum principles for Paneitz-type operators on complete Riemannian manifolds with semi-positive [MATH]curvature and nonnegative scalar curvature.', '1608.04061-1-4-0': 'The aim of the present paper is to establish rigidity results on Riemannian manifolds with nonnegative Ricci curvature supporting second-order Sobolev inequalities, in the spirit of the works of do Carmo, Ledoux and Xia.', '1608.04061-1-4-1': 'In order to present our results, let [MATH] be an [MATH]-dimensional complete open Riemannian manifold, [MATH], [MATH] be the geodesic ball with center [MATH] and radius [MATH] and [MATH] be the volume of [MATH].', '1608.04061-1-4-2': 'We say that [MATH] supports the second-order Sobolev inequality for [MATH] if [EQUATION] where [MATH] is the second-order critical Sobolev exponent, and [MATH] is the Laplace-Beltrami operator on [MATH].', '1608.04061-1-4-3': 'Note that the Euclidean space [MATH] supports [MATH] for [EQUATION]', '1608.04061-1-4-4': 'Moreover, [MATH] is optimal, see Edmunds, Fortunato and Janelli [CITATION], Lieb [CITATION] and Lions [CITATION], and the unique class of extremal functions is [EQUATION] where [MATH] and [MATH] are arbitrarily fixed.', '1608.04061-1-5-0': 'To state our results, we need a technical assumption on the manifold [MATH]; namely, if [MATH] is the distance function on [MATH] from a given point [MATH], we say that [MATH] satisfies the distance Laplacian growth condition if [EQUATION].', '1608.04061-1-6-0': 'Now, our main result reads as follows.', '1608.04061-1-7-0': 'Let [MATH] and [MATH] be an [MATH]-dimensional complete open Riemannian manifold with nonnegative Ricci curvature which satisfies the distance Laplacian growth condition.', '1608.04061-1-7-1': 'Assume that [MATH] supports the second-order Sobolev inequality [MATH] for some [MATH].', '1608.04061-1-7-2': 'Then the following properties hold:', '1608.04061-1-8-0': 'The distance Laplacian growth condition on [MATH] is indispensable in our argument which shows the genuine second-order character of the studied problem.', '1608.04061-1-8-1': "We notice that the counterpart of this condition in the first-order Sobolev inequality [MATH] is the validity of an eikonal inequality [MATH] a.e. on [MATH], which trivially holds on any complete Riemannian manifold (and any metric measure space with a suitable 'derivative' notion).", '1608.04061-1-8-2': 'Further comments on this condition will be given in Section [REF].', '1608.04061-1-9-0': 'Having the global volume non-collapsing property of geodesic balls of [MATH] in Theorem [REF] (ii), we shall prove that once [MATH] in [MATH] is closer and closer to the optimal Euclidean constant [MATH], the Riemannian manifold [MATH] approaches topologically more and more to the Euclidean space [MATH].', '1608.04061-1-9-1': 'To describe quantitatively this phenomenon, we recall the construction of Munn [CITATION] based on the double induction argument of Perelman [CITATION].', '1608.04061-1-9-2': 'In fact, Munn determined explicit lower bounds for the volume growth of the geodesic balls in terms of certain constants which guarantee the triviality of the [MATH]-th homotopy group [MATH] of [MATH].', '1608.04061-1-9-3': 'More precisely, let [MATH] and for [MATH], let us denote by [MATH] the smallest positive solution to the equation [EQUATION] in the variable [MATH], where [EQUATION]', '1608.04061-1-9-4': 'We now consider the smooth, bijective and increasing function [MATH] defined by [EQUATION].', '1608.04061-1-9-5': 'For every [MATH] let [EQUATION] be the so-called Munn-Perelman constant, see Munn [CITATION].', '1608.04061-1-10-0': 'Following the idea from Kristaly [CITATION], our quantitative result reads as follows:', '1608.04061-1-11-0': 'Under the same assumptions as in Theorem [REF], we have', '1608.04061-1-12-0': '# Proof of Theorems [REF][REF]', '1608.04061-1-13-0': 'Throughout this section, we assume the hypotheses of Theorem [REF] are verified, i.e., [MATH] is an [MATH]-dimensional complete open Riemannian manifold with nonnegative Ricci curvature which satisfies the distance Laplacian growth condition and supports the second-order Sobolev inequality [MATH] for [MATH].', '1608.04061-1-14-0': '(i) The inequality [MATH] follows in a similar way as in Djadli, Hebey and Ledoux [CITATION] by using a geodesic, normal coordinate system at a given point [MATH].', '1608.04061-1-15-0': '(ii) Before starting the proof explicitly, we notice that one can assume that [MATH]; otherwise, if [MATH] then we can assume that [MATH] holds with [MATH], where [MATH] is arbitrarily small, and then letting [MATH].', '1608.04061-1-15-1': 'Now, we split the proof into five steps.', '1608.04061-1-16-0': 'Step 1.', '1608.04061-1-16-1': 'ODE via the Euclidean optimizer.', '1608.04061-1-16-2': 'We consider the function [MATH] defined by [EQUATION].', '1608.04061-1-16-3': 'The layer cake representation shows that for every [MATH], [EQUATION]', '1608.04061-1-16-4': 'Clearly, [MATH] is smooth on [MATH].', '1608.04061-1-17-0': 'We recall now by ([REF]) that [EQUATION] where [EQUATION] and [MATH] is arbitrarily fixed.', '1608.04061-1-18-0': 'In terms of the function [MATH], the above equality can be rewritten as [EQUATION].', '1608.04061-1-19-0': 'By introducing the function [EQUATION] the latter relation is equivalent to the ODE [EQUATION]', '1608.04061-1-19-1': 'Step 2.', '1608.04061-1-19-2': 'ODI via [MATH].', '1608.04061-1-19-3': 'Let [MATH] be the point for which the distance Laplacian growth condition holds and let [MATH] be defined by [EQUATION].', '1608.04061-1-19-4': 'Since [MATH] has nonnegative Ricci curvature, the Bishop-Gromov comparison theorem asserts that [MATH] for every [MATH]; thus, by the layer cake representation and a change of variables, it turns out that [EQUATION]', '1608.04061-1-19-5': 'Thus [MATH] for every [MATH], and [MATH] is smooth.', '1608.04061-1-19-6': 'In a similar way, [EQUATION] and for every [MATH], [EQUATION]', '1608.04061-1-19-7': 'Let [MATH] be fixed; we observe that the function [EQUATION] can be approximated by elements from [MATH]; in particular, by using an approximation procedure, one can use the function [MATH] as a test-function in [MATH].', '1608.04061-1-19-8': 'Accordingly, [EQUATION]', '1608.04061-1-19-9': 'A chain rule and the eikonal equation [MATH] shows that [EQUATION].', '1608.04061-1-20-0': 'Since the Ricci curvature is nonnegative on [MATH], we first have the distance Laplacian comparison [MATH].', '1608.04061-1-20-1': 'Thus, [EQUATION]', '1608.04061-1-20-2': 'On the other hand, by the distance Laplacian growth condition, i.e., [MATH], we obtain that [EQUATION]', '1608.04061-1-20-3': 'Consequently, by ([REF]) and ([REF]), we have that [EQUATION]', '1608.04061-1-20-4': 'Thus, it turns out that [EQUATION].', '1608.04061-1-21-0': 'According to the latter estimate, relation ([REF]) can be written in terms of the function [MATH] as [EQUATION].', '1608.04061-1-21-1': 'By defining the function [EQUATION] the latter relation is equivalent to the ordinary differential inequality [EQUATION]', '1608.04061-1-21-2': 'Step 3.', '1608.04061-1-21-3': 'Comparison of [MATH] and [MATH] near the origin.', '1608.04061-1-21-4': 'We claim that [EQUATION].', '1608.04061-1-21-5': 'To see this, fix [MATH] arbitrarily small.', '1608.04061-1-21-6': 'Since [EQUATION] there exists a [MATH] such that [MATH] for all [MATH].', '1608.04061-1-21-7': 'Thus, by ([REF]) and ([REF]), we have [EQUATION] and [EQUATION]', '1608.04061-1-21-8': 'Combining this estimates with relation ([REF]), we obtain [EQUATION]', '1608.04061-1-21-9': 'Letting [MATH], we get the required claim.', '1608.04061-1-22-0': 'Step 4.', '1608.04061-1-22-1': 'Global comparison of [MATH] and [MATH].', '1608.04061-1-22-2': 'We claim that [EQUATION]', '1608.04061-1-22-3': 'First of all, by Step 3 and the fact that [MATH], we have [EQUATION]', '1608.04061-1-22-4': 'Thus, for sufficiently small [MATH] one has [EQUATION]', '1608.04061-1-22-5': 'In fact, we shall prove that [MATH] can be arbitrarily large in ([REF]) which ends the proof of ([REF]).', '1608.04061-1-22-6': 'By contradiction, let us assume that [MATH] for some [MATH]; clearly, [MATH].', '1608.04061-1-23-0': 'Due to ([REF]), we may set [EQUATION]', '1608.04061-1-23-1': 'Then, [MATH] and for any [MATH], one has [MATH].', '1608.04061-1-23-2': 'For [MATH], we define the function [MATH] by [EQUATION].', '1608.04061-1-23-3': 'We notice that [MATH] is non-decreasing in [MATH], where [EQUATION].', '1608.04061-1-23-4': 'On one hand, a straightforward computation shows that for every [MATH], one has [EQUATION].', '1608.04061-1-23-5': 'On the other hand, relation ([REF]) and the assumption [MATH] imply that for every [MATH], [EQUATION].', '1608.04061-1-24-0': 'We claim that [EQUATION]', '1608.04061-1-24-1': 'Since [MATH] for every [MATH], by relations ([REF]) and ([REF]) we have that [EQUATION]', '1608.04061-1-24-2': 'By the monotonicity of [MATH] on [MATH], relation ([REF]) follows at once.', '1608.04061-1-24-3': 'In particular, the function [MATH] is non-decreasing on the interval [MATH].', '1608.04061-1-24-4': 'Consequently, we have [EQUATION] a contradiction, which shows the validity of ([REF]).', '1608.04061-1-25-0': 'Step 5.', '1608.04061-1-25-1': 'Global volume non-collapsing property concluded.', '1608.04061-1-25-2': 'Inequality ([REF]) can be rewritten into [EQUATION] where [EQUATION].', '1608.04061-1-25-3': 'The Bishop-Gromov comparison theorem implies that the function [MATH] is non-increasing on [MATH]; thus, the asymptotic volume growth [EQUATION] is finite (and independent of the base point [MATH]).', '1608.04061-1-26-0': 'We shall prove that [MATH].', '1608.04061-1-26-1': 'By contradiction, let us suppose that [MATH] for some [MATH].', '1608.04061-1-26-2': 'Thus, there exists a number [MATH] such that [EQUATION]', '1608.04061-1-26-3': 'For simplicity of notation, let [EQUATION].', '1608.04061-1-26-4': 'Substituting ([REF]) into ([REF]) and by using the Bishop-Gromov comparison theorem, we obtain for every [MATH] that [EQUATION]', '1608.04061-1-26-5': 'Note that for every [MATH], one has [EQUATION] and [EQUATION]', '1608.04061-1-26-6': 'Consequently, the above estimates show that for every [MATH], [EQUATION] where [MATH] are independent on [MATH].', '1608.04061-1-26-7': 'It is clear that the latter inequality is not valid for large values of [MATH], i.e., we arrived to a contradiction.', '1608.04061-1-26-8': 'Accordingly, for every [MATH], [EQUATION].', '1608.04061-1-26-9': 'Since the asymptotic volume growth of [MATH] is independent of the point [MATH], we obtain the desired property, which completes the proof of Theorem [REF].', '1608.04061-1-26-10': '[MATH]', '1608.04061-1-27-0': 'Note that relation ([REF]) is equivalent to the distance Laplacian growth condition.', '1608.04061-1-27-1': 'Indeed, a simple computation in Step 2 led us to relation ([REF]) through the distance Laplacian growth condition.', '1608.04061-1-27-2': 'Conversely, if [MATH] in ([REF]), we obtain precisely that [MATH]', '1608.04061-1-28-0': 'Proof of Theorem [REF].', '1608.04061-1-28-1': '(i) Due to Anderson [CITATION] and Li [CITATION], if vol[MATH] for every [MATH], then [MATH] has finite fundamental group [MATH] and its order is bounded above by [MATH].', '1608.04061-1-28-2': 'By Theorem [REF] (ii) the property follows directly.', '1608.04061-1-28-3': 'In particular, if [MATH], then the order of [MATH] is strictly less than 2, thus [MATH] is simply connected.', '1608.04061-1-29-0': '(ii) First of all, due to Munn [CITATION] and a direct computation, for every [MATH] one has [EQUATION].', '1608.04061-1-29-1': 'Thus, since [MATH] is increasing, the values [MATH] are within the range where Theorem [REF] (ii) applies, [MATH].', '1608.04061-1-30-0': 'Now, let us assume that [MATH] for some [MATH].', '1608.04061-1-30-1': 'By Theorem [REF] (ii) we have the following estimate for the asymptotic volume growth of [MATH]: [EQUATION].', '1608.04061-1-30-2': 'Therefore, due to Munn [CITATION], one has that [MATH]', '1608.04061-1-31-0': '(iii) If [MATH], then [MATH].', '1608.04061-1-31-1': "Standard topological argument implies -based on Hurewicz's isomorphism theorem,- that [MATH] is contractible.", '1608.04061-1-32-0': '(iv) If [MATH] then by Theorem [REF] (ii) and the Bishop-Gromov volume comparison theorem follows that vol[MATH] for every [MATH] and [MATH].', '1608.04061-1-32-1': 'Now, the equality in Bishop-Gromov theorem implies that [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-1-32-2': 'The converse is trivial.', '1608.04061-1-32-3': '[MATH]', '1608.04061-1-33-0': '# Final remarks and questions', '1608.04061-1-34-0': 'We conclude the paper with some remarks and further questions:'}

{'1608.04061-2-0-0': 'Let [MATH] be an [MATH]dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying [MATH], where [MATH] is the Laplace-Beltrami operator on [MATH] and [MATH] is the distance function from a given point.', '1608.04061-2-0-1': 'If [MATH] supports a second-order Sobolev inequality with a constant [MATH] close to the optimal constant [MATH] in the second-order Sobolev inequality in [MATH], we show that a global volume non-collapsing property holds on [MATH].', '1608.04061-2-0-2': 'The latter property together with a Perelman-type construction established by Munn (J. Geom.', '1608.04061-2-0-3': 'Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of [MATH].', '1608.04061-2-0-4': 'Furthermore, it turns out that [MATH] supports the second-order Sobolev inequality with the constant [MATH] if and only if [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-2-1-0': '# Introduction', '1608.04061-2-2-0': 'It is well known that the validity of first-order Sobolev inequalities on Riemannian manifolds strongly depend on the curvature; this is a rough conclusion of the famous AB-program initiated by Th.', '1608.04061-2-2-1': 'Aubin in the seventies, see the monograph of Hebey [CITATION] for a systematic presentation.', '1608.04061-2-2-2': 'To be more precise, let [MATH] be an [MATH]dimensional complete Riemannian manifold, [MATH] and consider for some [MATH] the first-order Sobolev inequality [EQUATION] where [MATH] is the first-order critical Sobolev exponent, and [MATH] and [MATH] denote the canonical volume form and gradient on [MATH], respectively.', '1608.04061-2-2-3': 'On one hand, inequality [MATH] holds on any [MATH]dimensional Cartan-Hadamard manifold [MATH] (i.e., simply connected, complete Riemannian manifold with nonpositive sectional curvature) with the optimal Euclidean constant [MATH] whenever the Cartan-Hadamard conjecture holds on [MATH], e.g., [MATH].', '1608.04061-2-2-4': 'On the other hand, due to Ledoux [CITATION], if [MATH] has nonnegative Ricci curvature, inequality [MATH] holds if and only if [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-2-2-5': "Further first-order Sobolev-type inequalities on Riemannian/Finsler manifolds can be found in Bakry, Concordet and Ledoux [CITATION], Druet, Hebey and Vaugon [CITATION], do Carmo and Xia [CITATION], Xia [CITATION]-[CITATION], Kristaly [CITATION]; moreover, similar Sobolev inequalities are also considered on 'nonnegatively' curved metric measure spaces formulated in terms of the Lott-Sturm-Villani-type curvature-dimension condition or the Bishop-Gromov-type doubling measure condition, see Kristaly [CITATION] and Kristaly and Ohta [CITATION].", '1608.04061-2-3-0': 'With respect to first-order Sobolev inequalities, much less is know about higher-order Sobolev inequalities on curved spaces.', '1608.04061-2-3-1': 'The first studies concern the AB-program for Paneitz-type operators on compact Riemannian manifolds, see Djadli, Hebey and Ledoux [CITATION], Hebey [CITATION] and Biezuner and Montenegro [CITATION].', '1608.04061-2-3-2': 'Recently, Gursky and Malchiodi [CITATION] studied strong maximum principles for Paneitz-type operators on complete Riemannian manifolds with semi-positive [MATH]curvature and nonnegative scalar curvature.', '1608.04061-2-4-0': 'The aim of the present paper is to establish rigidity results on Riemannian manifolds with nonnegative Ricci curvature supporting second-order Sobolev inequalities.', '1608.04061-2-4-1': 'In order to present our results, let [MATH] be an [MATH]-dimensional complete open Riemannian manifold, [MATH], [MATH] be the geodesic ball with center [MATH] and radius [MATH] and [MATH] be the volume of [MATH].', '1608.04061-2-4-2': 'We say that [MATH] supports the second-order Sobolev inequality for [MATH] if [EQUATION] where [MATH] is the second-order critical Sobolev exponent, and [MATH] is the Laplace-Beltrami operator on [MATH].', '1608.04061-2-4-3': 'Note that the Euclidean space [MATH] supports [MATH] for [EQUATION]', '1608.04061-2-4-4': 'Moreover, [MATH] is optimal, see Edmunds, Fortunato and Janelli [CITATION], Lieb [CITATION] and Lions [CITATION], and the unique class of extremal functions is [EQUATION] where [MATH] and [MATH] are arbitrarily fixed.', '1608.04061-2-5-0': 'To state our results, we need a technical assumption on the manifold [MATH]; namely, if [MATH] is the distance function on [MATH] from a given point [MATH], we say that [MATH] satisfies the distance Laplacian growth condition if [EQUATION].', '1608.04061-2-6-0': 'Now, our main result reads as follows.', '1608.04061-2-7-0': 'Let [MATH] and [MATH] be an [MATH]-dimensional complete open Riemannian manifold with nonnegative Ricci curvature which satisfies the distance Laplacian growth condition.', '1608.04061-2-7-1': 'Assume that [MATH] supports the second-order Sobolev inequality [MATH] for some [MATH].', '1608.04061-2-7-2': 'Then the following properties hold:', '1608.04061-2-8-0': 'The distance Laplacian growth condition on [MATH] is indispensable in our argument which shows the genuine second-order character of the studied problem.', '1608.04061-2-8-1': 'We notice that the counterpart of this condition in the first-order Sobolev inequality [MATH] is the validity of an eikonal inequality [MATH] a.e. on [MATH], which trivially holds on any complete Riemannian manifold (and any metric measure space with a suitable derivative notion).', '1608.04061-2-8-2': 'Further comments on this condition will be given in Section [REF].', '1608.04061-2-9-0': 'Having the global volume non-collapsing property of geodesic balls of [MATH] in Theorem [REF] (ii), we shall prove that once [MATH] in [MATH] is closer and closer to the optimal Euclidean constant [MATH], the Riemannian manifold [MATH] approaches topologically more and more to the Euclidean space [MATH].', '1608.04061-2-9-1': 'To describe quantitatively this phenomenon, we recall the construction of Munn [CITATION] based on the double induction argument of Perelman [CITATION].', '1608.04061-2-9-2': 'In fact, Munn determined explicit lower bounds for the volume growth of the geodesic balls in terms of certain constants which guarantee the triviality of the [MATH]-th homotopy group [MATH] of [MATH].', '1608.04061-2-9-3': 'More precisely, let [MATH] and for [MATH], let us denote by [MATH] the smallest positive solution to the equation [EQUATION] in the variable [MATH], where [EQUATION]', '1608.04061-2-9-4': 'We now consider the smooth, bijective and increasing function [MATH] defined by [EQUATION].', '1608.04061-2-9-5': 'For every [MATH] let [EQUATION] be the so-called Munn-Perelman constant, see Munn [CITATION].', '1608.04061-2-10-0': 'Following the idea from Kristaly [CITATION], our quantitative result reads as follows:', '1608.04061-2-11-0': 'Under the same assumptions as in Theorem [REF], we have', '1608.04061-2-12-0': '# Proof of Theorems [REF][REF]', '1608.04061-2-13-0': 'Throughout this section, we assume the hypotheses of Theorem [REF] are verified, i.e., [MATH] is an [MATH]-dimensional complete open Riemannian manifold with nonnegative Ricci curvature which satisfies the distance Laplacian growth condition and supports the second-order Sobolev inequality [MATH] for [MATH].', '1608.04061-2-14-0': '(i) The inequality [MATH] follows in a similar way as in Djadli, Hebey and Ledoux [CITATION] by using a geodesic, normal coordinate system at a given point [MATH].', '1608.04061-2-15-0': '(ii) Before starting the proof explicitly, we notice that one can assume that [MATH]; otherwise, if [MATH] then we can assume that [MATH] holds with [MATH], where [MATH] is arbitrarily small, and then letting [MATH].', '1608.04061-2-15-1': 'Now, we split the proof into five steps.', '1608.04061-2-16-0': 'Step 1.', '1608.04061-2-16-1': 'ODE via the Euclidean optimizer.', '1608.04061-2-16-2': 'We consider the function [MATH] defined by [EQUATION].', '1608.04061-2-16-3': 'The layer cake representation shows that for every [MATH], [EQUATION]', '1608.04061-2-16-4': 'Clearly, [MATH] is smooth on [MATH].', '1608.04061-2-17-0': 'We recall now by ([REF]) that [EQUATION] where [EQUATION] and [MATH] is arbitrarily fixed.', '1608.04061-2-18-0': 'In terms of the function [MATH], the above equality can be rewritten as [EQUATION].', '1608.04061-2-19-0': 'By introducing the function [EQUATION] the latter relation is equivalent to the ODE [EQUATION]', '1608.04061-2-19-1': 'Step 2.', '1608.04061-2-19-2': 'ODI via [MATH].', '1608.04061-2-19-3': 'Let [MATH] be the point for which the distance Laplacian growth condition holds and let [MATH] be defined by [EQUATION].', '1608.04061-2-19-4': 'Since [MATH] has nonnegative Ricci curvature, the Bishop-Gromov comparison theorem asserts that [MATH] for every [MATH]; thus, by the layer cake representation and a change of variables, it turns out that [EQUATION]', '1608.04061-2-19-5': 'Thus [MATH] for every [MATH], and [MATH] is smooth.', '1608.04061-2-19-6': 'In a similar way, [EQUATION] and for every [MATH], [EQUATION]', '1608.04061-2-19-7': 'Let [MATH] be fixed; we observe that the function [EQUATION] can be approximated by elements from [MATH]; in particular, by using an approximation procedure, one can use the function [MATH] as a test-function in [MATH].', '1608.04061-2-19-8': 'Accordingly, [EQUATION]', '1608.04061-2-19-9': 'A chain rule and the eikonal equation [MATH] shows that [EQUATION].', '1608.04061-2-20-0': 'Since the Ricci curvature is nonnegative on [MATH], we first have the distance Laplacian comparison [MATH].', '1608.04061-2-20-1': 'Thus, [EQUATION]', '1608.04061-2-20-2': 'On the other hand, by the distance Laplacian growth condition, i.e., [MATH], we obtain that [EQUATION]', '1608.04061-2-20-3': 'Consequently, by ([REF]) and ([REF]), we have that [EQUATION]', '1608.04061-2-20-4': 'Thus, it turns out that [EQUATION].', '1608.04061-2-21-0': 'According to the latter estimate, relation ([REF]) can be written in terms of the function [MATH] as [EQUATION].', '1608.04061-2-21-1': 'By defining the function [EQUATION] the latter relation is equivalent to the ordinary differential inequality [EQUATION]', '1608.04061-2-21-2': 'Step 3.', '1608.04061-2-21-3': 'Comparison of [MATH] and [MATH] near the origin.', '1608.04061-2-21-4': 'We claim that [EQUATION].', '1608.04061-2-21-5': 'To see this, fix [MATH] arbitrarily small.', '1608.04061-2-21-6': 'Since [EQUATION] there exists a [MATH] such that [MATH] for all [MATH].', '1608.04061-2-21-7': 'Thus, by ([REF]) and ([REF]), we have [EQUATION] and [EQUATION]', '1608.04061-2-21-8': 'Combining this estimates with relation ([REF]), we obtain [EQUATION]', '1608.04061-2-21-9': 'Letting [MATH], we get the required claim.', '1608.04061-2-22-0': 'Step 4.', '1608.04061-2-22-1': 'Global comparison of [MATH] and [MATH].', '1608.04061-2-22-2': 'We claim that [EQUATION]', '1608.04061-2-22-3': 'First of all, by Step 3 and the fact that [MATH], we have [EQUATION]', '1608.04061-2-22-4': 'Thus, for sufficiently small [MATH] one has [EQUATION]', '1608.04061-2-22-5': 'In fact, we shall prove that [MATH] can be arbitrarily large in ([REF]) which ends the proof of ([REF]).', '1608.04061-2-22-6': 'By contradiction, let us assume that [MATH] for some [MATH]; clearly, [MATH].', '1608.04061-2-23-0': 'Due to ([REF]), we may set [EQUATION]', '1608.04061-2-23-1': 'Then, [MATH] and for any [MATH], one has [MATH].', '1608.04061-2-23-2': 'For [MATH], we define the function [MATH] by [EQUATION].', '1608.04061-2-23-3': 'We notice that [MATH] is non-decreasing in [MATH], where [EQUATION].', '1608.04061-2-23-4': 'On one hand, a straightforward computation shows that for every [MATH], one has [EQUATION].', '1608.04061-2-23-5': 'On the other hand, relation ([REF]) and the assumption [MATH] imply that for every [MATH], [EQUATION].', '1608.04061-2-24-0': 'We claim that [EQUATION]', '1608.04061-2-24-1': 'Since [MATH] for every [MATH], by relations ([REF]) and ([REF]) we have that [EQUATION]', '1608.04061-2-24-2': 'By the monotonicity of [MATH] on [MATH], relation ([REF]) follows at once.', '1608.04061-2-24-3': 'In particular, the function [MATH] is non-decreasing on the interval [MATH].', '1608.04061-2-24-4': 'Consequently, we have [EQUATION] a contradiction, which shows the validity of ([REF]).', '1608.04061-2-25-0': 'Step 5.', '1608.04061-2-25-1': 'Global volume non-collapsing property concluded.', '1608.04061-2-25-2': 'Inequality ([REF]) can be rewritten into [EQUATION] where [EQUATION].', '1608.04061-2-25-3': 'The Bishop-Gromov comparison theorem implies that the function [MATH] is non-increasing on [MATH]; thus, the asymptotic volume growth [EQUATION] is finite (and independent of the base point [MATH]).', '1608.04061-2-26-0': 'We shall prove that [MATH].', '1608.04061-2-26-1': 'By contradiction, let us suppose that [MATH] for some [MATH].', '1608.04061-2-26-2': 'Thus, there exists a number [MATH] such that [EQUATION]', '1608.04061-2-26-3': 'For simplicity of notation, let [EQUATION].', '1608.04061-2-26-4': 'Substituting ([REF]) into ([REF]) and by using the Bishop-Gromov comparison theorem, we obtain for every [MATH] that [EQUATION]', '1608.04061-2-26-5': 'Note that for every [MATH], one has [EQUATION] and [EQUATION]', '1608.04061-2-26-6': 'Consequently, the above estimates show that for every [MATH], [EQUATION] where [MATH] are independent on [MATH].', '1608.04061-2-26-7': 'It is clear that the latter inequality is not valid for large values of [MATH], i.e., we arrived to a contradiction.', '1608.04061-2-26-8': 'Accordingly, for every [MATH], [EQUATION].', '1608.04061-2-26-9': 'Since the asymptotic volume growth of [MATH] is independent of the point [MATH], we obtain the desired property, which completes the proof of Theorem [REF].', '1608.04061-2-26-10': '[MATH]', '1608.04061-2-27-0': 'Note that relation ([REF]) is equivalent to the distance Laplacian growth condition.', '1608.04061-2-27-1': 'Indeed, a simple computation in Step 2 led us to relation ([REF]) through the distance Laplacian growth condition.', '1608.04061-2-27-2': 'Conversely, if [MATH] in ([REF]), we obtain precisely that [MATH]', '1608.04061-2-28-0': 'Proof of Theorem [REF].', '1608.04061-2-28-1': '(i) Due to Anderson [CITATION] and Li [CITATION], if vol[MATH] for every [MATH], then [MATH] has finite fundamental group [MATH] and its order is bounded above by [MATH].', '1608.04061-2-28-2': 'By Theorem [REF] (ii) the property follows directly.', '1608.04061-2-28-3': 'In particular, if [MATH], then the order of [MATH] is strictly less than 2, thus [MATH] is simply connected.', '1608.04061-2-29-0': '(ii) First of all, due to Munn [CITATION] and a direct computation, for every [MATH] one has [EQUATION].', '1608.04061-2-29-1': 'Thus, since [MATH] is increasing, the values [MATH] are within the range where Theorem [REF] (ii) applies, [MATH].', '1608.04061-2-30-0': 'Now, let us assume that [MATH] for some [MATH].', '1608.04061-2-30-1': 'By Theorem [REF] (ii) we have the following estimate for the asymptotic volume growth of [MATH]: [EQUATION].', '1608.04061-2-30-2': 'Therefore, due to Munn [CITATION], one has that [MATH]', '1608.04061-2-31-0': '(iii) If [MATH], then [MATH].', '1608.04061-2-31-1': "Standard topological argument implies -based on Hurewicz's isomorphism theorem,- that [MATH] is contractible.", '1608.04061-2-32-0': '(iv) If [MATH] then by Theorem [REF] (ii) and the Bishop-Gromov volume comparison theorem follows that vol[MATH] for every [MATH] and [MATH].', '1608.04061-2-32-1': 'Now, the equality in Bishop-Gromov theorem implies that [MATH] is isometric to the Euclidean space [MATH].', '1608.04061-2-32-2': 'The converse is trivial.', '1608.04061-2-32-3': '[MATH]', '1608.04061-2-33-0': '# Final remarks', '1608.04061-2-34-0': 'We conclude the paper with some remarks and further questions:', '1608.04061-2-35-0': '(a) If [MATH] is a complete [MATH]dimensional Riemannian manifold and [MATH] is arbitrarily fixed, we notice that [EQUATION] where [MATH], [MATH] for some [MATH] with [MATH], and [MATH] is the density of the volume form in normal coordinates, see Gallot, Hulin and Lafontaine [CITATION].', '1608.04061-2-35-1': 'On one hand, if the Ricci curvature on [MATH] is nonnegative, one has [MATH].', '1608.04061-2-35-2': 'On the other hand, the distance Laplacian growth condition [MATH] is equivalent to [EQUATION] which is a curvature restriction on the manifold [MATH].', '1608.04061-2-35-3': 'We are wondering if the latter condition can be removed from our results, which plays a crucial role in our arguments; see also Remark [REF].', '1608.04061-2-35-4': 'Examples of Riemannian manifolds verifying the distance Laplacian growth condition (that are isometrically immersed into [MATH] with [MATH] large enough) can be found in Carron [CITATION].', '1608.04061-2-36-0': '(b) The requirement [MATH] is needed to explore the monotonicity of the function [MATH] on [MATH], see Step 4 in the proof of Theorem [REF].', '1608.04061-2-36-1': 'Although this condition is widely enough to obtain quantitative results, cf. Theorem [REF], we still believe that it can be somehow removed.', '1608.04061-2-37-0': '(c) Let [MATH] be an [MATH]-dimensional complete open Riemannian manifold with nonnegative Ricci curvature and fix [MATH] such that [MATH].', '1608.04061-2-37-1': 'Let us consider for some [MATH] the [MATH]-th order Sobolev inequality [EQUATION] where [EQUATION]', '1608.04061-2-37-2': 'Clearly, [MATH] and [MATH].', '1608.04061-2-37-3': 'It would be interesting to establish [MATH]-th order counterparts of Theorems [REF][REF] with [MATH], noticing that the optimal Euclidean [MATH]-th order Sobolev inequalities are well known with the optimal constant [EQUATION] and the unique class of extremal functions (up to translations and multiplications) [EQUATION] see Cotsiolis and Tavoularis [CITATION], Liu [CITATION].', '1608.04061-2-37-4': 'Once we use [MATH] as a test-function in [MATH], after a multiple application of the chain rule we have to estimate in a sharp way the terms appearing in [MATH], similar to the eikonal equation [MATH] and the distance Laplacian comparison [MATH], respectively.', '1608.04061-2-37-5': 'In the second-order case this fact is highlighted in relation ([REF]).', '1608.04061-2-37-6': 'Furthermore, higher-order counterparts of the distance Laplacian growth condition [MATH] should be found, (see relation ([REF]) for the second order case), assuming this condition cannot be removed, see (a).'}

[['1608.04061-1-20-0', '1608.04061-2-20-0'], ['1608.04061-1-20-2', '1608.04061-2-20-2'], ['1608.04061-1-20-3', '1608.04061-2-20-3'], ['1608.04061-1-20-4', '1608.04061-2-20-4'], ['1608.04061-1-3-0', '1608.04061-2-3-0'], ['1608.04061-1-3-1', '1608.04061-2-3-1'], ['1608.04061-1-3-2', '1608.04061-2-3-2'], ['1608.04061-1-22-1', '1608.04061-2-22-1'], ['1608.04061-1-22-2', '1608.04061-2-22-2'], ['1608.04061-1-22-3', '1608.04061-2-22-3'], ['1608.04061-1-22-4', '1608.04061-2-22-4'], ['1608.04061-1-22-5', '1608.04061-2-22-5'], ['1608.04061-1-22-6', '1608.04061-2-22-6'], ['1608.04061-1-28-0', '1608.04061-2-28-0'], ['1608.04061-1-28-1', '1608.04061-2-28-1'], ['1608.04061-1-28-2', '1608.04061-2-28-2'], ['1608.04061-1-28-3', '1608.04061-2-28-3'], ['1608.04061-1-23-0', '1608.04061-2-23-0'], ['1608.04061-1-23-1', '1608.04061-2-23-1'], ['1608.04061-1-23-2', '1608.04061-2-23-2'], ['1608.04061-1-23-3', '1608.04061-2-23-3'], ['1608.04061-1-23-4', '1608.04061-2-23-4'], ['1608.04061-1-23-5', '1608.04061-2-23-5'], ['1608.04061-1-7-0', '1608.04061-2-7-0'], ['1608.04061-1-7-1', '1608.04061-2-7-1'], ['1608.04061-1-13-0', '1608.04061-2-13-0'], ['1608.04061-1-21-0', '1608.04061-2-21-0'], ['1608.04061-1-21-1', '1608.04061-2-21-1'], ['1608.04061-1-21-3', '1608.04061-2-21-3'], ['1608.04061-1-21-4', '1608.04061-2-21-4'], ['1608.04061-1-21-5', '1608.04061-2-21-5'], ['1608.04061-1-21-6', '1608.04061-2-21-6'], ['1608.04061-1-21-8', '1608.04061-2-21-8'], ['1608.04061-1-21-9', '1608.04061-2-21-9'], ['1608.04061-1-19-0', '1608.04061-2-19-0'], ['1608.04061-1-19-3', '1608.04061-2-19-3'], ['1608.04061-1-19-4', '1608.04061-2-19-4'], ['1608.04061-1-19-5', '1608.04061-2-19-5'], ['1608.04061-1-19-6', '1608.04061-2-19-6'], ['1608.04061-1-19-7', '1608.04061-2-19-7'], ['1608.04061-1-19-9', '1608.04061-2-19-9'], ['1608.04061-1-14-0', '1608.04061-2-14-0'], ['1608.04061-1-2-0', '1608.04061-2-2-0'], ['1608.04061-1-2-1', '1608.04061-2-2-1'], ['1608.04061-1-2-2', '1608.04061-2-2-2'], ['1608.04061-1-2-3', '1608.04061-2-2-3'], ['1608.04061-1-2-4', '1608.04061-2-2-4'], ['1608.04061-1-2-5', '1608.04061-2-2-5'], ['1608.04061-1-17-0', '1608.04061-2-17-0'], ['1608.04061-1-5-0', '1608.04061-2-5-0'], ['1608.04061-1-24-0', '1608.04061-2-24-0'], ['1608.04061-1-24-1', '1608.04061-2-24-1'], ['1608.04061-1-24-2', '1608.04061-2-24-2'], ['1608.04061-1-24-3', '1608.04061-2-24-3'], ['1608.04061-1-24-4', '1608.04061-2-24-4'], ['1608.04061-1-27-0', '1608.04061-2-27-0'], ['1608.04061-1-27-1', '1608.04061-2-27-1'], ['1608.04061-1-27-2', '1608.04061-2-27-2'], ['1608.04061-1-26-0', '1608.04061-2-26-0'], ['1608.04061-1-26-1', '1608.04061-2-26-1'], ['1608.04061-1-26-2', '1608.04061-2-26-2'], ['1608.04061-1-26-3', '1608.04061-2-26-3'], ['1608.04061-1-26-4', '1608.04061-2-26-4'], ['1608.04061-1-26-5', '1608.04061-2-26-5'], ['1608.04061-1-26-6', '1608.04061-2-26-6'], ['1608.04061-1-26-7', '1608.04061-2-26-7'], ['1608.04061-1-26-8', '1608.04061-2-26-8'], ['1608.04061-1-26-9', '1608.04061-2-26-9'], ['1608.04061-1-15-0', '1608.04061-2-15-0'], ['1608.04061-1-15-1', '1608.04061-2-15-1'], ['1608.04061-1-16-1', '1608.04061-2-16-1'], ['1608.04061-1-16-2', '1608.04061-2-16-2'], ['1608.04061-1-16-3', '1608.04061-2-16-3'], ['1608.04061-1-16-4', '1608.04061-2-16-4'], ['1608.04061-1-4-1', '1608.04061-2-4-1'], ['1608.04061-1-4-2', '1608.04061-2-4-2'], ['1608.04061-1-4-3', '1608.04061-2-4-3'], ['1608.04061-1-4-4', '1608.04061-2-4-4'], ['1608.04061-1-9-0', '1608.04061-2-9-0'], ['1608.04061-1-9-1', '1608.04061-2-9-1'], ['1608.04061-1-9-2', '1608.04061-2-9-2'], ['1608.04061-1-9-3', '1608.04061-2-9-3'], ['1608.04061-1-9-4', '1608.04061-2-9-4'], ['1608.04061-1-9-5', '1608.04061-2-9-5'], ['1608.04061-1-8-0', '1608.04061-2-8-0'], ['1608.04061-1-8-2', '1608.04061-2-8-2'], ['1608.04061-1-30-0', '1608.04061-2-30-0'], ['1608.04061-1-30-1', '1608.04061-2-30-1'], ['1608.04061-1-30-2', '1608.04061-2-30-2'], ['1608.04061-1-32-0', '1608.04061-2-32-0'], ['1608.04061-1-32-1', '1608.04061-2-32-1'], ['1608.04061-1-32-2', '1608.04061-2-32-2'], ['1608.04061-1-25-1', '1608.04061-2-25-1'], ['1608.04061-1-25-2', '1608.04061-2-25-2'], ['1608.04061-1-25-3', '1608.04061-2-25-3'], ['1608.04061-1-31-1', '1608.04061-2-31-1'], ['1608.04061-1-29-0', '1608.04061-2-29-0'], ['1608.04061-1-29-1', '1608.04061-2-29-1'], ['1608.04061-1-0-0', '1608.04061-2-0-0'], ['1608.04061-1-0-1', '1608.04061-2-0-1'], ['1608.04061-1-0-2', '1608.04061-2-0-2'], ['1608.04061-1-0-3', '1608.04061-2-0-3'], ['1608.04061-1-0-4', '1608.04061-2-0-4'], ['1608.04061-1-18-0', '1608.04061-2-18-0'], ['1608.04061-1-8-1', '1608.04061-2-8-1'], ['1608.04061-1-4-0', '1608.04061-2-4-0']]

[['1608.04061-1-20-0', '1608.04061-2-20-0'], ['1608.04061-1-20-2', '1608.04061-2-20-2'], ['1608.04061-1-20-3', '1608.04061-2-20-3'], ['1608.04061-1-20-4', '1608.04061-2-20-4'], ['1608.04061-1-3-0', '1608.04061-2-3-0'], ['1608.04061-1-3-1', '1608.04061-2-3-1'], ['1608.04061-1-3-2', '1608.04061-2-3-2'], ['1608.04061-1-22-1', '1608.04061-2-22-1'], ['1608.04061-1-22-2', '1608.04061-2-22-2'], ['1608.04061-1-22-3', '1608.04061-2-22-3'], ['1608.04061-1-22-4', '1608.04061-2-22-4'], ['1608.04061-1-22-5', '1608.04061-2-22-5'], ['1608.04061-1-22-6', '1608.04061-2-22-6'], ['1608.04061-1-28-0', '1608.04061-2-28-0'], ['1608.04061-1-28-1', '1608.04061-2-28-1'], ['1608.04061-1-28-2', '1608.04061-2-28-2'], ['1608.04061-1-28-3', '1608.04061-2-28-3'], ['1608.04061-1-23-0', '1608.04061-2-23-0'], ['1608.04061-1-23-1', '1608.04061-2-23-1'], ['1608.04061-1-23-2', '1608.04061-2-23-2'], ['1608.04061-1-23-3', '1608.04061-2-23-3'], ['1608.04061-1-23-4', '1608.04061-2-23-4'], ['1608.04061-1-23-5', '1608.04061-2-23-5'], ['1608.04061-1-7-0', '1608.04061-2-7-0'], ['1608.04061-1-7-1', '1608.04061-2-7-1'], ['1608.04061-1-13-0', '1608.04061-2-13-0'], ['1608.04061-1-21-0', '1608.04061-2-21-0'], ['1608.04061-1-21-1', '1608.04061-2-21-1'], ['1608.04061-1-21-3', '1608.04061-2-21-3'], ['1608.04061-1-21-4', '1608.04061-2-21-4'], ['1608.04061-1-21-5', '1608.04061-2-21-5'], ['1608.04061-1-21-6', '1608.04061-2-21-6'], ['1608.04061-1-21-8', '1608.04061-2-21-8'], ['1608.04061-1-21-9', '1608.04061-2-21-9'], ['1608.04061-1-19-0', '1608.04061-2-19-0'], ['1608.04061-1-19-3', '1608.04061-2-19-3'], ['1608.04061-1-19-4', '1608.04061-2-19-4'], ['1608.04061-1-19-5', '1608.04061-2-19-5'], ['1608.04061-1-19-6', '1608.04061-2-19-6'], ['1608.04061-1-19-7', '1608.04061-2-19-7'], ['1608.04061-1-19-9', '1608.04061-2-19-9'], ['1608.04061-1-14-0', '1608.04061-2-14-0'], ['1608.04061-1-2-0', '1608.04061-2-2-0'], ['1608.04061-1-2-1', '1608.04061-2-2-1'], ['1608.04061-1-2-2', '1608.04061-2-2-2'], ['1608.04061-1-2-3', '1608.04061-2-2-3'], ['1608.04061-1-2-4', '1608.04061-2-2-4'], ['1608.04061-1-2-5', '1608.04061-2-2-5'], ['1608.04061-1-17-0', '1608.04061-2-17-0'], ['1608.04061-1-5-0', '1608.04061-2-5-0'], ['1608.04061-1-24-0', '1608.04061-2-24-0'], ['1608.04061-1-24-1', '1608.04061-2-24-1'], ['1608.04061-1-24-2', '1608.04061-2-24-2'], ['1608.04061-1-24-3', '1608.04061-2-24-3'], ['1608.04061-1-24-4', '1608.04061-2-24-4'], ['1608.04061-1-27-0', '1608.04061-2-27-0'], ['1608.04061-1-27-1', '1608.04061-2-27-1'], ['1608.04061-1-27-2', '1608.04061-2-27-2'], ['1608.04061-1-26-0', '1608.04061-2-26-0'], ['1608.04061-1-26-1', '1608.04061-2-26-1'], ['1608.04061-1-26-2', '1608.04061-2-26-2'], ['1608.04061-1-26-3', '1608.04061-2-26-3'], ['1608.04061-1-26-4', '1608.04061-2-26-4'], ['1608.04061-1-26-5', '1608.04061-2-26-5'], ['1608.04061-1-26-6', '1608.04061-2-26-6'], ['1608.04061-1-26-7', '1608.04061-2-26-7'], ['1608.04061-1-26-8', '1608.04061-2-26-8'], ['1608.04061-1-26-9', '1608.04061-2-26-9'], ['1608.04061-1-15-0', '1608.04061-2-15-0'], ['1608.04061-1-15-1', '1608.04061-2-15-1'], ['1608.04061-1-16-1', '1608.04061-2-16-1'], ['1608.04061-1-16-2', '1608.04061-2-16-2'], ['1608.04061-1-16-3', '1608.04061-2-16-3'], ['1608.04061-1-16-4', '1608.04061-2-16-4'], ['1608.04061-1-4-1', '1608.04061-2-4-1'], ['1608.04061-1-4-2', '1608.04061-2-4-2'], ['1608.04061-1-4-3', '1608.04061-2-4-3'], ['1608.04061-1-4-4', '1608.04061-2-4-4'], ['1608.04061-1-9-0', '1608.04061-2-9-0'], ['1608.04061-1-9-1', '1608.04061-2-9-1'], ['1608.04061-1-9-2', '1608.04061-2-9-2'], ['1608.04061-1-9-3', '1608.04061-2-9-3'], ['1608.04061-1-9-4', '1608.04061-2-9-4'], ['1608.04061-1-9-5', '1608.04061-2-9-5'], ['1608.04061-1-8-0', '1608.04061-2-8-0'], ['1608.04061-1-8-2', '1608.04061-2-8-2'], ['1608.04061-1-30-0', '1608.04061-2-30-0'], ['1608.04061-1-30-1', '1608.04061-2-30-1'], ['1608.04061-1-30-2', '1608.04061-2-30-2'], ['1608.04061-1-32-0', '1608.04061-2-32-0'], ['1608.04061-1-32-1', '1608.04061-2-32-1'], ['1608.04061-1-32-2', '1608.04061-2-32-2'], ['1608.04061-1-25-1', '1608.04061-2-25-1'], ['1608.04061-1-25-2', '1608.04061-2-25-2'], ['1608.04061-1-25-3', '1608.04061-2-25-3'], ['1608.04061-1-31-1', '1608.04061-2-31-1'], ['1608.04061-1-29-0', '1608.04061-2-29-0'], ['1608.04061-1-29-1', '1608.04061-2-29-1'], ['1608.04061-1-0-0', '1608.04061-2-0-0'], ['1608.04061-1-0-1', '1608.04061-2-0-1'], ['1608.04061-1-0-2', '1608.04061-2-0-2'], ['1608.04061-1-0-3', '1608.04061-2-0-3'], ['1608.04061-1-0-4', '1608.04061-2-0-4'], ['1608.04061-1-18-0', '1608.04061-2-18-0']]

[['1608.04061-1-8-1', '1608.04061-2-8-1']]

[]

[['1608.04061-1-4-0', '1608.04061-2-4-0']]

[]

['1608.04061-1-0-6', '1608.04061-1-0-8', '1608.04061-1-6-0', '1608.04061-1-7-2', '1608.04061-1-10-0', '1608.04061-1-11-0', '1608.04061-1-16-0', '1608.04061-1-19-1', '1608.04061-1-19-2', '1608.04061-1-19-8', '1608.04061-1-20-1', '1608.04061-1-21-2', '1608.04061-1-21-7', '1608.04061-1-22-0', '1608.04061-1-25-0', '1608.04061-1-26-10', '1608.04061-1-31-0', '1608.04061-1-32-3', '1608.04061-1-34-0', '1608.04061-2-6-0', '1608.04061-2-7-2', '1608.04061-2-10-0', '1608.04061-2-11-0', '1608.04061-2-16-0', '1608.04061-2-19-1', '1608.04061-2-19-2', '1608.04061-2-19-8', '1608.04061-2-20-1', '1608.04061-2-21-2', '1608.04061-2-21-7', '1608.04061-2-22-0', '1608.04061-2-25-0', '1608.04061-2-26-10', '1608.04061-2-31-0', '1608.04061-2-32-3', '1608.04061-2-34-0']

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

https://arxiv.org/abs/1608.04061

1308.3705

{'1308.3705-1-0-0': 'In this second of two papers we apply our recently developed code to search for resonance features in the Planck CMB temperature data.', '1308.3705-1-0-1': 'We search both for log spaced oscillations or linear spaced oscillations and compare our findings with results of our WMAP9 analysis and the Planck team analysis .', '1308.3705-1-0-2': 'While there are hints of log spaced resonant features present in the WMAP9 data, the significance of these features weaken with more data.', '1308.3705-1-0-3': 'With more accurate small scale measurements, we also find that the best fit frequency has shifted and the amplitude has been reduced.', '1308.3705-1-0-4': 'We confirm the presence of a several low frequency peaks, earlier identified by the Planck team, but with a better improvement of fit ([MATH]).', '1308.3705-1-0-5': 'We further investigate this improvement by allowing the lensing potential to vary as well, showing mild correlation between the amplitude of the oscillations and the lensing amplitude.', '1308.3705-1-0-6': 'We find that the improvement of the fit increases even more ([MATH]) for the low frequencies that modify the spectrum in a way that mimics the lensing effect.', '1308.3705-1-0-7': 'Since these features were not present in the WMAP data, they are primarily due to better measurements of Planck at small angular scales.', '1308.3705-1-0-8': 'For linear spaced oscillations we find a maximum [MATH] scanning two orders of magnitude in frequency space, and the biggest improvements are at extremely high frequencies.', '1308.3705-1-0-9': 'Again, we recover a best fit frequency very close to the one found in WMAP9, which confirms that the fit improvement is driven by low [MATH].', '1308.3705-1-0-10': 'Further comparisons with WMAP9 show Planck contains many more features, both for linear and log space oscillations, but with a smaller improvement of fit.', '1308.3705-1-0-11': 'We discuss the improvement as a function of the number of modes and study the effect of the 217 GHz map, which appears to drive most of the improvement for log spaced oscillations.', '1308.3705-1-0-12': 'Two points strongly suggest that the detected features are fitting a combination of the noise and the dip at [MATH] in the 217 GHz map: the fit improvement mostly comes from a small range of [MATH], and comparison with simulations shows that the fit improvement is consistent with a statistical fluctuation.', '1308.3705-1-0-13': 'We conclude that none of the detected features are statistically significant.', '1308.3705-1-1-0': '# Introduction', '1308.3705-1-2-0': 'In this short paper, we will apply our recent introduced method to search for resonant features in the recently released Planck CMB data.', '1308.3705-1-2-1': 'We consider two distinct theoretically motivated models: [EQUATION]', '1308.3705-1-2-2': 'We refer to the first model as the "log-spaced oscillations model" and the second model as the "linear oscillations model".', '1308.3705-1-2-3': 'For example, axion-monodromy inflation produces features that can be described by the logarithmic oscillations model with [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-1-2-4': 'Model that include the effects from a possible boundary on effective field theory (BEFT) predict features that can be described by the linear oscillations model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-1-2-5': 'Both initial state modifications and multiverse models can also produce logarithmic oscillations, while sharp features generate a power spectrum with linear oscillations (although the amplitude is typically damped as a function of scale).', '1308.3705-1-2-6': 'Constraints on oscillations in the WMAP CMB data have been attempted in e.g. .', '1308.3705-1-3-0': 'This paper is organized as follows.', '1308.3705-1-3-1': 'We present our results on the Planck Data in [REF] for log and linear spaced oscillations.', '1308.3705-1-3-2': 'In [REF], we compare our results with the WMAP9 analysis.', '1308.3705-1-3-3': 'We discuss our findings and conclude in [REF].', '1308.3705-1-4-0': '# Planck Analysis', '1308.3705-1-5-0': 'In this analysis, we use a modified version of the publicly available Planck likelihood code to search for oscillations in the primordial power spectrum.', '1308.3705-1-5-1': 'For this analysis, we found the best fit values for both resonance model parameters and cosmological parameters.', '1308.3705-1-5-2': 'We vary all six [MATH]CDM parameters plus the phase and the amplitude of the oscillatory correction to the primordial power spectrum while fixing the foreground parameters to their best fit values for the no oscillations model.', '1308.3705-1-6-0': '## Log-spaced oscillations', '1308.3705-1-7-0': 'In Fig. [REF] shows the improvement in fit as a function of frequency, where the frequency of the oscillation was varied in 1201 steps between [MATH] .', '1308.3705-1-7-1': 'We observe several frequencies that could be hints of primordial oscillations.', '1308.3705-1-7-2': 'We confirm a number of features first observed by .', '1308.3705-1-7-3': 'Our method improves the best fit peak identified by the planck team at low frequencies with [MATH] (with best fit frequency [MATH]) .', '1308.3705-1-7-4': 'After inspecting the resulting fit, we expected some correlation with smooth parameters.', '1308.3705-1-7-5': 'We found that varying [MATH] enables a further improvement in the fit by another [MATH], but the best fit has shifted towards a lower frequency [MATH].', '1308.3705-1-7-6': 'In Fig. [REF] we show the marginalized contour between the lensing amplitude [MATH] and the amplitude of the oscillations at the best fit frequency.', '1308.3705-1-7-7': 'Further investigation shows that this mild correlation actually shifts slope from peak to peak, which can be explained by the fact that the improvement of fit is at [MATH] (see [REF]); for these low frequencies the contribution to the power spectrum is rather smooth and the lensing amplitude effectively smooths the peak structure.', '1308.3705-1-7-8': 'Oscillations can help enhance or reduce this effect, and the correlation coefficient can therefore change signs depending on the phase of the oscillation.', '1308.3705-1-8-0': '## Linear-spaced oscillations', '1308.3705-1-9-0': 'In Fig. [REF], we show the improvement as a function of frequency for the linear spaced oscillations model.', '1308.3705-1-9-1': 'Again, we vary the phase and amplitude of the oscillation, together with the cosmological parameters, for each of 881 steps in frequency space.', '1308.3705-1-9-2': 'For linear spaced oscillations, the resulting improvement is extremely irregular, with no particularly preferred region.', '1308.3705-1-9-3': 'The best fit is at a frequency of [MATH], where the [MATH].', '1308.3705-1-10-0': '# WMAP9 vs Planck1', '1308.3705-1-11-0': '## Log-spaced oscillations', '1308.3705-1-12-0': 'In Fig. [REF] and Fig. [REF] we compare the improvement of fit in our analysis of WMAP data to the improvement in fit in our analysis of Planck data.', '1308.3705-1-12-1': 'Fig. [REF] clearly shows new peaks at the low frequency end in the Planck data, features that are absent in the WMAP data.', '1308.3705-1-12-2': 'At high frequencies, the feature that was seen in the WMAP analysis is less significant and has shifted.', '1308.3705-1-12-3': 'For [MATH], where WMAP and Planck are both cosmic variance limited, we expect the features to coincide.', '1308.3705-1-12-4': 'As shown in an analysis by [CITATION] , the location of the feature shifts as one lowers [MATH] to [MATH], confirming that a better fit at small scales or high [MATH] is the source of the shift.', '1308.3705-1-13-0': 'Since the low frequency features are absent in the WMAP data, their presence should be primarily due to a better fit in the range [MATH].', '1308.3705-1-13-1': 'The correlation between [MATH] and the amplitude of the oscillations is a result of the fit being driven by the high [MATH] Planck data.', '1308.3705-1-13-2': 'In Fig. [REF] we show improvement of fit compared to no oscillations as a function of [MATH] for our best fit frequency [MATH].', '1308.3705-1-13-3': 'Indeed this plot shows that the improvement comes from multipoles around [MATH] and [MATH] (roughly between the 3rd and 4th peak).', '1308.3705-1-13-4': 'In the likelihood for our parameters we use the 100 GHz data up to [MATH], the 143 GHz datau to [MATH] and the 217 GHz data up to [MATH].', '1308.3705-1-13-5': 'As was shown in the 217 GHz map drives some of the standard [MATH]CDM parameters away from their best fit WMAP values and the 217[MATH]217 GHz power spectrum contains a feature at [MATH] that is not seen in the 143[MATH]217 GHz power spectrum.', '1308.3705-1-13-6': 'C.', '1308.3705-1-13-7': 'By removing the 217 GHz data we find that the improvement drops to [MATH] with [MATH].', '1308.3705-1-13-8': 'Note that the better fit at [MATH] is also unconstrained by WMAP.', '1308.3705-1-14-0': '## Linear-spaced oscillations', '1308.3705-1-15-0': 'Interestingly, in comparison with WMAP, the Planck data seems to contain many more low frequency features as shown in Fig. [REF].', '1308.3705-1-15-1': 'As was the case for WMAP, the Planck data shows that higher frequencies can result in bigger the improvements of the fit.', '1308.3705-1-15-2': 'In WMAP we found were able to identify a single frequency that appeared to be favored over other frequencies.', '1308.3705-1-15-3': 'Despite the difference, a high oscillation feature persists in Planck data although the frequency has shifted slightly ([MATH]).', '1308.3705-1-15-4': 'Because of the similarities between the two data sets, this feature is most likely not due to a feature at small angular scales.', '1308.3705-1-15-5': 'For this reason we again investigate the improvement of fit as a function of [MATH].', '1308.3705-1-15-6': 'This is shown in Fig. [REF].', '1308.3705-1-15-7': 'Interestingly, this fitting appears rather gradual, which would favor a true feature interpretation.', '1308.3705-1-15-8': 'Regarding the feature in the 217 GHz map, the improvement of the fit actually decreases after [MATH].', '1308.3705-1-15-9': 'The plot shows that most of the improvement comes from low multipoles, consistent with this feature appearing both in WMAP and in Planck at a similar frequency.', '1308.3705-1-16-0': '# Discussion and Conclusions', '1308.3705-1-17-0': 'In this second of two papers we have applied our recently developed code to search for resonant features in the Planck data.', '1308.3705-1-17-1': 'Our code recovers the results found by the Planck collaboration, but adds to these findings by significantly extending the frequency range of the search.', '1308.3705-1-17-2': 'In addition, our method finds larger improvements of fit for low frequencies because it varies all parameters to find the best fit, not just the amplitude and frequency of the oscillatory signal.', '1308.3705-1-18-0': 'Our analysis has given us several important insights.', '1308.3705-1-18-1': 'First of all, the improvement at the low frequency of logarithmically spaced oscillations end are caused or at least enhanced by a varying lensing amplitude.', '1308.3705-1-18-2': 'For example, in comparison with the Planck paper result, we find that allowing the lensing amplitude to vary shifts the best fit frequency to lower values, and improves the overall fit.', '1308.3705-1-18-3': 'For linear spaced oscillations we find the largest improvement at the highest frequencies, with a best fit frequency that is close to that found in WMAP9.', '1308.3705-1-18-4': 'We showed that including this feature mostly improves the fit to the spectrum below [MATH].', '1308.3705-1-19-0': 'Further comparison between our WMAP9 and Planck analyses, shows the improvement of fit for log spaced oscillations has flipped, i.e. while for WMAP9 the improvements were at high frequencies for Planck the best fit is at low frequencies, although a feature at high frequencies does appear in the Planck data.', '1308.3705-1-19-1': 'The fact that none of the found oscillations in Planck are present in WMAP9, suggested that most improvement is coming from high [MATH], which we confirmed by computing the improvement as a function of [MATH].', '1308.3705-1-19-2': 'As the Planck team has noted, there is a feature present near [MATH] in the [MATH] GHz spectrum that does not appear in the other frequencies.', '1308.3705-1-19-3': 'Our results suggest that the improvement at low frequencies is predominately due to this feature.', '1308.3705-1-20-0': 'Second, in our companion paper we tested our method on simulated data.', '1308.3705-1-20-1': 'The primary goal of these tests was to assess the reliability of our perturbative method.', '1308.3705-1-20-2': 'Here we use the same simulations to assess the significance of the fit improvements to determine whether we have detected an oscillatory contribution to the primordial power spectrum.', '1308.3705-1-20-3': 'We found that for amplitudes as small as those that best fit the data, we expect an improvement of fit that exceeds what we find the data.', '1308.3705-1-20-4': 'We ran two full pipelines on maps that did not include a signal, for which we found improvements up to [MATH] .', '1308.3705-1-20-5': 'Furthermore we also ran a large number of (simplified) simulations in order to address the question: What is the typical maximum improvement expected from fitting the noise ?', '1308.3705-1-20-6': 'This analysis showed that the noise typically leads to [MATH], and has the potential to improve the fit [MATH].', '1308.3705-1-20-7': 'We found these improvements with [MATH].', '1308.3705-1-20-8': 'We have improved on these simulations, by randomly generating Gaussian noise using the weighted error bars directly synthesized from the Planck data for multipoles ranging from [MATH] to [MATH].', '1308.3705-1-20-9': 'We ran 1000 of these higher resolution featureless simulations and found that applying both log and linear spaced templates showed the measured (data) improvements are in the 90-94 percentile range (see Fig. [REF]).', '1308.3705-1-20-10': 'This result together with the full simulation using our code on mock data in the previous paper, suggests that the improvements in fit found in our Planck analysis are consistent with expected statistical fluctuations for a realization from a featureless primordial model.', '1308.3705-1-21-0': 'Third, both for linear and log spaced oscillations, improvements appear local in [MATH] space.', '1308.3705-1-21-1': 'One might expect a real oscillation to lead to an improvement that would be more gradual as a function of the number of modes observed, though we recognize that this is not a rigorous argument: the biggest improvements arise from the modes with the highest signal to noise.', '1308.3705-1-22-0': 'Lastly, we have studied linear oscillations with the frequency at which the Planck team found a [MATH] detection in the bispectrum .', '1308.3705-1-22-1': 'We found that this frequency is very close to the Baryon Acoustic Oscillation (BAO) and we find no evidence in the power spectrum that there is such an oscillations (roughly corresponding to [MATH]).', '1308.3705-1-22-2': 'Our current understanding would suggest that varying the BAO parameters in the search for features in the bispectrum would probably reduce the significance (in addition to look elsewhere effects).', '1308.3705-1-23-0': 'For the future, we plan to implement Multinest as our sampler as this will significantly speed up the code.', '1308.3705-1-23-1': 'As we go to higher frequencies we should include higher order terms to the derivative.', '1308.3705-1-23-2': 'In particular, for an accurate measurement of [MATH], [MATH] and [MATH] it may be necessary to include derivatives of these parameters in the expansion.', '1308.3705-1-24-0': 'When the Planck polarization data will be available, we should be able to improve our search.', '1308.3705-1-24-1': 'Oscillations should produce features in both temperature and polarization spectra (and cross-spectra).', '1308.3705-1-24-2': 'Potentially, polarization measurements from ground-based experiments can probe out to [MATH] enabling even more sensitive searches for oscillatory features.', '1308.3705-1-24-3': 'However, searches based on ground-based data would be limited to lower oscillation frequencies since the power spectrum likelihood will have [MATH] correlations due to mode coupling effects.', '1308.3705-1-25-0': 'We can get additional insight from three point measurements as models that predict oscillations in the power spectrum typically also predict oscillations in higher order correlation spectra (see e.g. Refs. )', '1308.3705-1-25-1': 'While there have been initial attempts to search for oscillations in the Planck bispectrum measurements , computational cost has limited these searches to low frequencies.', '1308.3705-1-25-2': 'Alternative approaches have been proposed to optimize this search , but as of yet no attempt has been made to cover a large range of frequencies and phases.', '1308.3705-1-25-3': 'A first step would be to search the bispectrum measurements at frequencies suggested by analyses of the CMB power spectrum.', '1308.3705-1-25-4': 'Detecting features in both spectra would improve the statistical significance of the result-a promising direction for future study.'}

{'1308.3705-2-0-0': 'In this second of two papers we apply our recently developed code to search for resonance features in the Planck CMB temperature data.', '1308.3705-2-0-1': 'We search both for log spaced oscillations or linear spaced oscillations and compare our findings with results of our WMAP9 analysis and the Planck team analysis .', '1308.3705-2-0-2': 'While there are hints of log spaced resonant features present in the WMAP9 data, the significance of these features weaken with more data.', '1308.3705-2-0-3': 'With more accurate small scale measurements, we also find that the best fit frequency has shifted and the amplitude has been reduced.', '1308.3705-2-0-4': 'We confirm the presence of a several low frequency peaks, earlier identified by the Planck team, but with a better improvement of fit ([MATH]).', '1308.3705-2-0-5': 'We further investigate this improvement by allowing the lensing potential to vary as well, showing mild correlation between the amplitude of the oscillations and the lensing amplitude.', '1308.3705-2-0-6': 'We find that the improvement of the fit increases even more ([MATH]) for the low frequencies that modify the spectrum in a way that mimics the lensing effect.', '1308.3705-2-0-7': 'Since these features were not present in the WMAP data, they are primarily due to better measurements of Planck at small angular scales.', '1308.3705-2-0-8': 'For linear spaced oscillations we find a maximum [MATH] scanning two orders of magnitude in frequency space, and the biggest improvements are at extremely high frequencies.', '1308.3705-2-0-9': 'Again, we recover a best fit frequency very close to the one found in WMAP9, which confirms that the fit improvement is driven by low [MATH].', '1308.3705-2-0-10': 'Further comparisons with WMAP9 show Planck contains many more features, both for linear and log space oscillations, but with a smaller improvement of fit.', '1308.3705-2-0-11': 'We discuss the improvement as a function of the number of modes and study the effect of the 217 GHz map, which appears to drive most of the improvement for log spaced oscillations.', '1308.3705-2-0-12': 'Two points strongly suggest that the detected features are fitting a combination of the noise and the dip at [MATH] in the 217 GHz map: the fit improvement mostly comes from a small range of [MATH], and comparison with simulations shows that the fit improvement is consistent with a statistical fluctuation.', '1308.3705-2-0-13': 'We conclude that none of the detected features are statistically significant.', '1308.3705-2-1-0': '# Introduction', '1308.3705-2-2-0': 'In this short paper, we will apply our recent introduced method to search for resonant features in the recently released Planck CMB data.', '1308.3705-2-2-1': 'We consider two distinct theoretically motivated models: [EQUATION]', '1308.3705-2-2-2': 'We refer to the first model as the "log-spaced oscillations model" and the second model as the "linear oscillations model".', '1308.3705-2-2-3': 'For example, axion-monodromy inflation produces features that can be described by the logarithmic oscillations model with [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-2-2-4': 'Model that include the effects from a possible boundary on effective field theory (BEFT) predict features that can be described by the linear oscillations model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-2-2-5': 'Both initial state modifications and multiverse models can also produce logarithmic oscillations, while sharp features generate a power spectrum with linear oscillations (although the amplitude is typically damped as a function of scale).', '1308.3705-2-2-6': 'Constraints on oscillations in the WMAP CMB data have been attempted in e.g. .', '1308.3705-2-3-0': 'This paper is organized as follows.', '1308.3705-2-3-1': 'We present our results on the Planck Data in [REF] for log and linear spaced oscillations.', '1308.3705-2-3-2': 'In [REF], we compare our results with the WMAP9 analysis.', '1308.3705-2-3-3': 'We discuss our findings and conclude in [REF].', '1308.3705-2-4-0': '# Planck Analysis', '1308.3705-2-5-0': 'In this analysis, we use a modified version of the publicly available Planck likelihood code to search for oscillations in the primordial power spectrum.', '1308.3705-2-5-1': 'For this analysis, we found the best fit values for both resonance model parameters and cosmological parameters.', '1308.3705-2-5-2': 'We vary all six [MATH]CDM parameters plus the phase and the amplitude of the oscillatory correction to the primordial power spectrum while fixing the foreground parameters to their best fit values for the no oscillations model.', '1308.3705-2-6-0': '## Log-spaced oscillations', '1308.3705-2-7-0': 'In Fig. [REF] shows the improvement in fit as a function of frequency, where the frequency of the oscillation was varied in 1201 steps between [MATH] .', '1308.3705-2-7-1': 'We observe several frequencies that could be hints of primordial oscillations.', '1308.3705-2-7-2': 'We confirm a number of features first observed by .', '1308.3705-2-7-3': 'Our method improves the best fit peak identified by the planck team at low frequencies with [MATH] (with best fit frequency [MATH]) .', '1308.3705-2-7-4': 'After inspecting the resulting fit, we expected some correlation with smooth parameters.', '1308.3705-2-7-5': 'We found that varying [MATH] enables a further improvement in the fit by another [MATH], but the best fit has shifted towards a lower frequency [MATH].', '1308.3705-2-7-6': 'In Fig. [REF] we show the marginalized contour between the lensing amplitude [MATH] and the amplitude of the oscillations at the best fit frequency.', '1308.3705-2-7-7': 'Further investigation shows that this mild correlation actually shifts slope from peak to peak, which can be explained by the fact that the improvement of fit is at [MATH] (see [REF]); for these low frequencies the contribution to the power spectrum is rather smooth and the lensing amplitude effectively smooths the peak structure.', '1308.3705-2-7-8': 'Oscillations can help enhance or reduce this effect, and the correlation coefficient can therefore change signs depending on the phase of the oscillation.', '1308.3705-2-8-0': '## Linear-spaced oscillations', '1308.3705-2-9-0': 'In Fig. [REF], we show the improvement as a function of frequency for the linear spaced oscillations model.', '1308.3705-2-9-1': 'Again, we vary the phase and amplitude of the oscillation, together with the cosmological parameters, for each of 881 steps in frequency space.', '1308.3705-2-9-2': 'For linear spaced oscillations, the resulting improvement is extremely irregular, with no particularly preferred region.', '1308.3705-2-9-3': 'The best fit is at a frequency of [MATH], where the [MATH].', '1308.3705-2-10-0': '# WMAP9 vs Planck1', '1308.3705-2-11-0': '## Log-spaced oscillations', '1308.3705-2-12-0': 'In Fig. [REF] and Fig. [REF] we compare the improvement of fit in our analysis of WMAP data to the improvement in fit in our analysis of Planck data.', '1308.3705-2-12-1': 'Fig. [REF] clearly shows new peaks at the low frequency end in the Planck data, features that are absent in the WMAP data.', '1308.3705-2-12-2': 'At high frequencies, the feature that was seen in the WMAP analysis is less significant and has shifted.', '1308.3705-2-12-3': 'Note that some of this shift is due to different precomputed transfer function, which for high frequencies have a small but non-negligible effect on the projected frequency .', '1308.3705-2-12-4': 'For [MATH], where WMAP and Planck are both cosmic variance limited, we expect the features to coincide.', '1308.3705-2-12-5': 'As shown in an analysis by [CITATION] , the location of the feature shifts as one lowers [MATH] to [MATH], confirming that a better fit at small scales or high [MATH] is the source of the shift.', '1308.3705-2-13-0': 'Since the low frequency features are absent in the WMAP data, their presence should be primarily due to a better fit in the range [MATH].', '1308.3705-2-13-1': 'The correlation between [MATH] and the amplitude of the oscillations is a result of the fit being driven by the high [MATH] Planck data.', '1308.3705-2-13-2': 'In Fig. [REF] we show improvement of fit compared to no oscillations as a function of [MATH] for our best fit frequency [MATH].', '1308.3705-2-13-3': 'Indeed this plot shows that the improvement comes from multipoles around [MATH] and [MATH] (roughly between the 3rd and 4th peak).', '1308.3705-2-13-4': 'In the likelihood for our parameters we use the 100 GHz data up to [MATH], the 143 GHz data to [MATH] and the 217 GHz data up to [MATH].', '1308.3705-2-13-5': 'As was shown in the 217 GHz map drives some of the standard [MATH]CDM parameters away from their best fit WMAP values and the 217[MATH]217 GHz power spectrum contains a feature at [MATH] that is not seen in the 143[MATH]217 GHz power spectrum.', '1308.3705-2-13-6': 'C.', '1308.3705-2-13-7': 'By removing the 217 GHz data we find that the improvement drops to [MATH] with [MATH].', '1308.3705-2-13-8': 'Note that the better fit at [MATH] is also unconstrained by WMAP.', '1308.3705-2-14-0': '## Linear-spaced oscillations', '1308.3705-2-15-0': 'Interestingly, in comparison with WMAP, the Planck data seems to contain many more low frequency features as shown in Fig. [REF].', '1308.3705-2-15-1': 'As was the case for WMAP, the Planck data shows that higher frequencies can result in bigger the improvements of the fit.', '1308.3705-2-15-2': 'In WMAP we found were able to identify a single frequency that appeared to be favored over other frequencies.', '1308.3705-2-15-3': 'Despite the difference, a high oscillation feature persists in Planck data although the frequency has shifted slightly ([MATH]).', '1308.3705-2-15-4': 'Because of the similarities between the two data sets, this feature is most likely not due to a feature at small angular scales.', '1308.3705-2-15-5': 'For this reason we again investigate the improvement of fit as a function of [MATH].', '1308.3705-2-15-6': 'This is shown in Fig. [REF].', '1308.3705-2-15-7': 'Interestingly, this fitting appears rather gradual, which would favor a true feature interpretation.', '1308.3705-2-15-8': 'Regarding the feature in the 217 GHz map, the improvement of the fit actually decreases after [MATH].', '1308.3705-2-15-9': 'The plot shows that most of the improvement comes from low multipoles, consistent with this feature appearing both in WMAP and in Planck at a similar frequency.', '1308.3705-2-16-0': '# Discussion and Conclusions', '1308.3705-2-17-0': 'In this second of two papers we have applied our recently developed code to search for resonant features in the Planck data.', '1308.3705-2-17-1': 'Our code recovers the results found by the Planck collaboration, but adds to these findings by significantly extending the frequency range of the search.', '1308.3705-2-17-2': 'In addition, our method finds larger improvements of fit for low frequencies because it varies all parameters to find the best fit, not just the amplitude and frequency of the oscillatory signal.', '1308.3705-2-18-0': 'Our analysis has given us several important insights.', '1308.3705-2-18-1': 'First of all, the improvement at the low frequency of logarithmically spaced oscillations end are caused or at least enhanced by a varying lensing amplitude.', '1308.3705-2-18-2': 'For example, in comparison with the Planck paper result, we find that allowing the lensing amplitude to vary shifts the best fit frequency to lower values, and improves the overall fit.', '1308.3705-2-18-3': 'For linear spaced oscillations we find the largest improvement at the highest frequencies, with a best fit frequency that is close to that found in WMAP9.', '1308.3705-2-18-4': 'We showed that including this feature mostly improves the fit to the spectrum below [MATH].', '1308.3705-2-19-0': 'Further comparison between our WMAP9 and Planck analyses, shows the improvement of fit for log spaced oscillations has flipped, i.e. while for WMAP9 the improvements were at high frequencies for Planck the best fit is at low frequencies, although a feature at high frequencies does appear in the Planck data.', '1308.3705-2-19-1': 'The fact that none of the found oscillations in Planck are present in WMAP9, suggested that most improvement is coming from high [MATH], which we confirmed by computing the improvement as a function of [MATH].', '1308.3705-2-19-2': 'As the Planck team has noted, there is a feature present near [MATH] in the [MATH] GHz spectrum that does not appear in the other frequencies.', '1308.3705-2-19-3': 'Our results suggest that the improvement at low frequencies is predominately due to this feature.', '1308.3705-2-20-0': 'Second, in our companion paper we tested our method on simulated data.', '1308.3705-2-20-1': 'The primary goal of these tests was to assess the reliability of our perturbative method.', '1308.3705-2-20-2': 'Here we use the same simulations to assess the significance of the fit improvements to determine whether we have detected an oscillatory contribution to the primordial power spectrum.', '1308.3705-2-20-3': 'We found that for amplitudes as small as those that best fit the data, we expect an improvement of fit that exceeds what we find the data.', '1308.3705-2-20-4': 'We ran two full pipelines on maps that did not include a signal, for which we found improvements up to [MATH] .', '1308.3705-2-20-5': 'Furthermore we also ran a large number of (simplified) simulations in order to address the question: What is the typical maximum improvement expected from fitting the noise ?', '1308.3705-2-20-6': 'This analysis showed that the noise typically leads to [MATH], and has the potential to improve the fit [MATH].', '1308.3705-2-20-7': 'We found these improvements with [MATH].', '1308.3705-2-20-8': 'We have improved on these simulations, by randomly generating Gaussian noise using the weighted error bars directly synthesized from the Planck data for multipoles ranging from [MATH] to [MATH].', '1308.3705-2-20-9': 'We ran 1000 of these higher resolution featureless simulations and found that applying both log and linear spaced templates showed the measured (data) improvements are in the 90-94 percentile range (see Fig. [REF]).', '1308.3705-2-20-10': 'This result together with the full simulation using our code on mock data in the previous paper, suggests that the improvements in fit found in our Planck analysis are consistent with expected statistical fluctuations for a realization from a featureless primordial model.', '1308.3705-2-21-0': 'Third, both for linear and log spaced oscillations, improvements appear local in [MATH] space.', '1308.3705-2-21-1': 'One might expect a real oscillation to lead to an improvement that would be more gradual as a function of the number of modes observed, though we recognize that this is not a rigorous argument: the biggest improvements arise from the modes with the highest signal to noise.', '1308.3705-2-22-0': 'Lastly, we have studied linear oscillations with the frequency at which the Planck team found a [MATH] detection in the bispectrum .', '1308.3705-2-22-1': 'We found that this frequency is very close to the Baryon Acoustic Oscillation (BAO) and we find no evidence in the power spectrum that there is such an oscillations (roughly corresponding to [MATH]).', '1308.3705-2-22-2': 'Our current understanding would suggest that varying the BAO parameters in the search for features in the bispectrum would probably reduce the significance (in addition to look elsewhere effects).', '1308.3705-2-23-0': 'For the future, we plan to implement Multinest as our sampler as this will significantly speed up the code.', '1308.3705-2-23-1': 'As we go to higher frequencies we should include higher order terms to the derivative.', '1308.3705-2-23-2': 'In particular, for an accurate measurement of [MATH], [MATH] and [MATH] it may be necessary to include derivatives of these parameters in the expansion.', '1308.3705-2-24-0': 'When the Planck polarization data will be available, we should be able to improve our search.', '1308.3705-2-24-1': 'Oscillations should produce features in both temperature and polarization spectra (and cross-spectra).', '1308.3705-2-24-2': 'Potentially, polarization measurements from ground-based experiments can probe out to [MATH] enabling even more sensitive searches for oscillatory features.', '1308.3705-2-24-3': 'However, searches based on ground-based data would be limited to lower oscillation frequencies since the power spectrum likelihood will have [MATH] correlations due to mode coupling effects.', '1308.3705-2-25-0': 'We can get additional insight from three point measurements as models that predict oscillations in the power spectrum typically also predict oscillations in higher order correlation spectra (see e.g. Refs. )', '1308.3705-2-25-1': 'While there have been initial attempts to search for oscillations in the Planck bispectrum measurements , computational cost has limited these searches to low frequencies.', '1308.3705-2-25-2': 'Alternative approaches have been proposed to optimize this search , but as of yet no attempt has been made to cover a large range of frequencies and phases.', '1308.3705-2-25-3': 'A first step would be to search the bispectrum measurements at frequencies suggested by analyses of the CMB power spectrum.', '1308.3705-2-25-4': 'Detecting features in both spectra would improve the statistical significance of the result-a promising direction for future study.'}

[['1308.3705-1-0-0', '1308.3705-2-0-0'], ['1308.3705-1-0-1', '1308.3705-2-0-1'], ['1308.3705-1-0-2', '1308.3705-2-0-2'], ['1308.3705-1-0-3', '1308.3705-2-0-3'], ['1308.3705-1-0-4', '1308.3705-2-0-4'], ['1308.3705-1-0-5', '1308.3705-2-0-5'], ['1308.3705-1-0-6', '1308.3705-2-0-6'], ['1308.3705-1-0-7', '1308.3705-2-0-7'], ['1308.3705-1-0-8', '1308.3705-2-0-8'], ['1308.3705-1-0-9', '1308.3705-2-0-9'], ['1308.3705-1-0-10', '1308.3705-2-0-10'], ['1308.3705-1-0-11', '1308.3705-2-0-11'], ['1308.3705-1-0-12', '1308.3705-2-0-12'], ['1308.3705-1-0-13', '1308.3705-2-0-13'], ['1308.3705-1-19-0', '1308.3705-2-19-0'], ['1308.3705-1-19-1', '1308.3705-2-19-1'], ['1308.3705-1-19-2', '1308.3705-2-19-2'], ['1308.3705-1-19-3', '1308.3705-2-19-3'], ['1308.3705-1-25-0', '1308.3705-2-25-0'], ['1308.3705-1-25-1', '1308.3705-2-25-1'], ['1308.3705-1-25-2', '1308.3705-2-25-2'], ['1308.3705-1-25-3', '1308.3705-2-25-3'], ['1308.3705-1-25-4', '1308.3705-2-25-4'], ['1308.3705-1-5-0', '1308.3705-2-5-0'], ['1308.3705-1-5-1', '1308.3705-2-5-1'], ['1308.3705-1-5-2', '1308.3705-2-5-2'], ['1308.3705-1-13-0', '1308.3705-2-13-0'], ['1308.3705-1-13-1', '1308.3705-2-13-1'], ['1308.3705-1-13-2', '1308.3705-2-13-2'], ['1308.3705-1-13-3', '1308.3705-2-13-3'], ['1308.3705-1-13-5', '1308.3705-2-13-5'], ['1308.3705-1-13-7', '1308.3705-2-13-7'], ['1308.3705-1-13-8', '1308.3705-2-13-8'], ['1308.3705-1-21-0', '1308.3705-2-21-0'], ['1308.3705-1-21-1', '1308.3705-2-21-1'], ['1308.3705-1-2-0', '1308.3705-2-2-0'], ['1308.3705-1-2-1', '1308.3705-2-2-1'], ['1308.3705-1-2-2', '1308.3705-2-2-2'], ['1308.3705-1-2-3', '1308.3705-2-2-3'], ['1308.3705-1-2-4', '1308.3705-2-2-4'], ['1308.3705-1-2-5', '1308.3705-2-2-5'], ['1308.3705-1-2-6', '1308.3705-2-2-6'], ['1308.3705-1-9-0', '1308.3705-2-9-0'], ['1308.3705-1-9-1', '1308.3705-2-9-1'], ['1308.3705-1-9-2', '1308.3705-2-9-2'], ['1308.3705-1-9-3', '1308.3705-2-9-3'], ['1308.3705-1-24-0', '1308.3705-2-24-0'], ['1308.3705-1-24-1', '1308.3705-2-24-1'], ['1308.3705-1-24-2', '1308.3705-2-24-2'], ['1308.3705-1-24-3', '1308.3705-2-24-3'], ['1308.3705-1-7-0', '1308.3705-2-7-0'], ['1308.3705-1-7-1', '1308.3705-2-7-1'], ['1308.3705-1-7-2', '1308.3705-2-7-2'], ['1308.3705-1-7-3', '1308.3705-2-7-3'], ['1308.3705-1-7-4', '1308.3705-2-7-4'], ['1308.3705-1-7-5', '1308.3705-2-7-5'], ['1308.3705-1-7-6', '1308.3705-2-7-6'], ['1308.3705-1-7-7', '1308.3705-2-7-7'], ['1308.3705-1-7-8', '1308.3705-2-7-8'], ['1308.3705-1-22-0', '1308.3705-2-22-0'], ['1308.3705-1-22-1', '1308.3705-2-22-1'], ['1308.3705-1-22-2', '1308.3705-2-22-2'], ['1308.3705-1-20-0', '1308.3705-2-20-0'], ['1308.3705-1-20-1', '1308.3705-2-20-1'], ['1308.3705-1-20-2', '1308.3705-2-20-2'], ['1308.3705-1-20-3', '1308.3705-2-20-3'], ['1308.3705-1-20-4', '1308.3705-2-20-4'], ['1308.3705-1-20-5', '1308.3705-2-20-5'], ['1308.3705-1-20-6', '1308.3705-2-20-6'], ['1308.3705-1-20-7', '1308.3705-2-20-7'], ['1308.3705-1-20-8', '1308.3705-2-20-8'], ['1308.3705-1-20-9', '1308.3705-2-20-9'], ['1308.3705-1-20-10', '1308.3705-2-20-10'], ['1308.3705-1-12-0', '1308.3705-2-12-0'], ['1308.3705-1-12-1', '1308.3705-2-12-1'], ['1308.3705-1-12-2', '1308.3705-2-12-2'], ['1308.3705-1-12-3', '1308.3705-2-12-4'], ['1308.3705-1-12-4', '1308.3705-2-12-5'], ['1308.3705-1-3-0', '1308.3705-2-3-0'], ['1308.3705-1-3-1', '1308.3705-2-3-1'], ['1308.3705-1-3-2', '1308.3705-2-3-2'], ['1308.3705-1-3-3', '1308.3705-2-3-3'], ['1308.3705-1-18-0', '1308.3705-2-18-0'], ['1308.3705-1-18-1', '1308.3705-2-18-1'], ['1308.3705-1-18-2', '1308.3705-2-18-2'], ['1308.3705-1-18-3', '1308.3705-2-18-3'], ['1308.3705-1-18-4', '1308.3705-2-18-4'], ['1308.3705-1-17-0', '1308.3705-2-17-0'], ['1308.3705-1-17-1', '1308.3705-2-17-1'], ['1308.3705-1-17-2', '1308.3705-2-17-2'], ['1308.3705-1-15-0', '1308.3705-2-15-0'], ['1308.3705-1-15-1', '1308.3705-2-15-1'], ['1308.3705-1-15-2', '1308.3705-2-15-2'], ['1308.3705-1-15-3', '1308.3705-2-15-3'], ['1308.3705-1-15-4', '1308.3705-2-15-4'], ['1308.3705-1-15-5', '1308.3705-2-15-5'], ['1308.3705-1-15-6', '1308.3705-2-15-6'], ['1308.3705-1-15-7', '1308.3705-2-15-7'], ['1308.3705-1-15-8', '1308.3705-2-15-8'], ['1308.3705-1-15-9', '1308.3705-2-15-9'], ['1308.3705-1-23-0', '1308.3705-2-23-0'], ['1308.3705-1-23-1', '1308.3705-2-23-1'], ['1308.3705-1-23-2', '1308.3705-2-23-2'], ['1308.3705-2-17-0', '1308.3705-3-17-0'], ['1308.3705-2-17-1', '1308.3705-3-17-1'], ['1308.3705-2-17-2', '1308.3705-3-17-2'], ['1308.3705-2-5-1', '1308.3705-3-5-1'], ['1308.3705-2-5-2', '1308.3705-3-5-2'], ['1308.3705-2-21-0', '1308.3705-3-21-0'], ['1308.3705-2-21-1', '1308.3705-3-21-1'], ['1308.3705-2-25-0', '1308.3705-3-25-0'], ['1308.3705-2-25-1', '1308.3705-3-25-1'], ['1308.3705-2-25-2', '1308.3705-3-25-2'], ['1308.3705-2-25-3', '1308.3705-3-25-3'], ['1308.3705-2-25-4', '1308.3705-3-25-4'], ['1308.3705-2-13-0', '1308.3705-3-13-0'], ['1308.3705-2-13-1', '1308.3705-3-13-1'], ['1308.3705-2-13-2', '1308.3705-3-13-2'], ['1308.3705-2-13-3', '1308.3705-3-13-3'], ['1308.3705-2-13-4', '1308.3705-3-13-4'], ['1308.3705-2-13-7', '1308.3705-3-13-7'], ['1308.3705-2-13-8', '1308.3705-3-13-8'], ['1308.3705-2-23-0', '1308.3705-3-23-0'], ['1308.3705-2-23-1', '1308.3705-3-23-1'], ['1308.3705-2-23-2', '1308.3705-3-23-2'], ['1308.3705-2-24-0', '1308.3705-3-24-0'], ['1308.3705-2-24-1', '1308.3705-3-24-1'], ['1308.3705-2-24-2', '1308.3705-3-24-2'], ['1308.3705-2-24-3', '1308.3705-3-24-3'], ['1308.3705-2-0-0', '1308.3705-3-0-0'], ['1308.3705-2-0-1', '1308.3705-3-0-1'], ['1308.3705-2-0-2', '1308.3705-3-0-2'], ['1308.3705-2-0-3', '1308.3705-3-0-3'], ['1308.3705-2-0-4', '1308.3705-3-0-4'], ['1308.3705-2-0-5', '1308.3705-3-0-5'], ['1308.3705-2-0-6', '1308.3705-3-0-6'], ['1308.3705-2-0-7', '1308.3705-3-0-7'], ['1308.3705-2-0-8', '1308.3705-3-0-8'], ['1308.3705-2-0-9', '1308.3705-3-0-9'], ['1308.3705-2-0-10', '1308.3705-3-0-10'], ['1308.3705-2-0-11', '1308.3705-3-0-11'], ['1308.3705-2-0-12', '1308.3705-3-0-12'], ['1308.3705-2-0-13', '1308.3705-3-0-13'], ['1308.3705-2-22-0', '1308.3705-3-22-0'], ['1308.3705-2-22-1', '1308.3705-3-22-1'], ['1308.3705-2-22-2', '1308.3705-3-22-2'], ['1308.3705-2-18-0', '1308.3705-3-18-0'], ['1308.3705-2-18-1', '1308.3705-3-18-1'], ['1308.3705-2-18-2', '1308.3705-3-18-2'], ['1308.3705-2-18-3', '1308.3705-3-18-3'], ['1308.3705-2-18-4', '1308.3705-3-18-4'], ['1308.3705-2-19-0', '1308.3705-3-19-0'], ['1308.3705-2-19-1', '1308.3705-3-19-1'], ['1308.3705-2-19-2', '1308.3705-3-19-2'], ['1308.3705-2-19-3', '1308.3705-3-19-3'], ['1308.3705-2-9-0', '1308.3705-3-9-0'], ['1308.3705-2-9-1', '1308.3705-3-9-1'], ['1308.3705-2-9-2', '1308.3705-3-9-2'], ['1308.3705-2-9-3', '1308.3705-3-9-3'], ['1308.3705-2-7-0', '1308.3705-3-7-0'], ['1308.3705-2-7-1', '1308.3705-3-7-1'], ['1308.3705-2-7-4', '1308.3705-3-7-3'], ['1308.3705-2-7-5', '1308.3705-3-7-4'], ['1308.3705-2-7-6', '1308.3705-3-7-5'], ['1308.3705-2-7-7', '1308.3705-3-7-6'], ['1308.3705-2-7-8', '1308.3705-3-7-7'], ['1308.3705-2-12-0', '1308.3705-3-12-0'], ['1308.3705-2-12-1', '1308.3705-3-12-1'], ['1308.3705-2-12-2', '1308.3705-3-12-2'], ['1308.3705-2-12-3', '1308.3705-3-12-3'], ['1308.3705-2-12-4', '1308.3705-3-12-4'], ['1308.3705-2-20-0', '1308.3705-3-20-0'], ['1308.3705-2-20-1', '1308.3705-3-20-1'], ['1308.3705-2-20-2', '1308.3705-3-20-2'], ['1308.3705-2-20-3', '1308.3705-3-20-3'], ['1308.3705-2-20-4', '1308.3705-3-20-4'], ['1308.3705-2-20-5', '1308.3705-3-20-5'], ['1308.3705-2-20-6', '1308.3705-3-20-6'], ['1308.3705-2-20-7', '1308.3705-3-20-7'], ['1308.3705-2-20-8', '1308.3705-3-20-8'], ['1308.3705-2-20-10', '1308.3705-3-20-19'], ['1308.3705-2-15-0', '1308.3705-3-15-0'], ['1308.3705-2-15-1', '1308.3705-3-15-1'], ['1308.3705-2-15-2', '1308.3705-3-15-2'], ['1308.3705-2-15-3', '1308.3705-3-15-3'], ['1308.3705-2-15-4', '1308.3705-3-15-4'], ['1308.3705-2-15-5', '1308.3705-3-15-5'], ['1308.3705-2-15-6', '1308.3705-3-15-6'], ['1308.3705-2-15-7', '1308.3705-3-15-7'], ['1308.3705-2-15-8', '1308.3705-3-15-8'], ['1308.3705-2-15-9', '1308.3705-3-15-9'], ['1308.3705-2-2-1', '1308.3705-3-2-1'], ['1308.3705-2-2-2', '1308.3705-3-2-2'], ['1308.3705-2-2-3', '1308.3705-3-2-3'], ['1308.3705-2-2-4', '1308.3705-3-2-4'], ['1308.3705-2-2-5', '1308.3705-3-2-5'], ['1308.3705-2-3-0', '1308.3705-3-3-0'], ['1308.3705-2-3-1', '1308.3705-3-3-1'], ['1308.3705-2-3-2', '1308.3705-3-3-2'], ['1308.3705-2-3-3', '1308.3705-3-3-3'], ['1308.3705-1-13-4', '1308.3705-2-13-4'], ['1308.3705-2-5-0', '1308.3705-3-5-0'], ['1308.3705-2-13-5', '1308.3705-3-13-5'], ['1308.3705-2-12-5', '1308.3705-3-12-5'], ['1308.3705-2-20-9', '1308.3705-3-20-9'], ['1308.3705-2-2-0', '1308.3705-3-2-0'], ['1308.3705-2-7-2', '1308.3705-3-7-2'], ['1308.3705-2-7-3', '1308.3705-3-7-2'], ['1308.3705-2-2-6', '1308.3705-3-2-6']]

[['1308.3705-1-0-0', '1308.3705-2-0-0'], ['1308.3705-1-0-1', '1308.3705-2-0-1'], ['1308.3705-1-0-2', '1308.3705-2-0-2'], ['1308.3705-1-0-3', '1308.3705-2-0-3'], ['1308.3705-1-0-4', '1308.3705-2-0-4'], ['1308.3705-1-0-5', '1308.3705-2-0-5'], ['1308.3705-1-0-6', '1308.3705-2-0-6'], ['1308.3705-1-0-7', '1308.3705-2-0-7'], ['1308.3705-1-0-8', '1308.3705-2-0-8'], ['1308.3705-1-0-9', '1308.3705-2-0-9'], ['1308.3705-1-0-10', '1308.3705-2-0-10'], ['1308.3705-1-0-11', '1308.3705-2-0-11'], ['1308.3705-1-0-12', '1308.3705-2-0-12'], ['1308.3705-1-0-13', '1308.3705-2-0-13'], ['1308.3705-1-19-0', '1308.3705-2-19-0'], ['1308.3705-1-19-1', '1308.3705-2-19-1'], ['1308.3705-1-19-2', '1308.3705-2-19-2'], ['1308.3705-1-19-3', '1308.3705-2-19-3'], ['1308.3705-1-25-0', '1308.3705-2-25-0'], ['1308.3705-1-25-1', '1308.3705-2-25-1'], ['1308.3705-1-25-2', '1308.3705-2-25-2'], ['1308.3705-1-25-3', '1308.3705-2-25-3'], ['1308.3705-1-25-4', '1308.3705-2-25-4'], ['1308.3705-1-5-0', '1308.3705-2-5-0'], ['1308.3705-1-5-1', '1308.3705-2-5-1'], ['1308.3705-1-5-2', '1308.3705-2-5-2'], ['1308.3705-1-13-0', '1308.3705-2-13-0'], ['1308.3705-1-13-1', '1308.3705-2-13-1'], ['1308.3705-1-13-2', '1308.3705-2-13-2'], ['1308.3705-1-13-3', '1308.3705-2-13-3'], ['1308.3705-1-13-5', '1308.3705-2-13-5'], ['1308.3705-1-13-7', '1308.3705-2-13-7'], ['1308.3705-1-13-8', '1308.3705-2-13-8'], ['1308.3705-1-21-0', '1308.3705-2-21-0'], ['1308.3705-1-21-1', '1308.3705-2-21-1'], ['1308.3705-1-2-0', '1308.3705-2-2-0'], ['1308.3705-1-2-1', '1308.3705-2-2-1'], ['1308.3705-1-2-2', '1308.3705-2-2-2'], ['1308.3705-1-2-3', '1308.3705-2-2-3'], ['1308.3705-1-2-4', '1308.3705-2-2-4'], ['1308.3705-1-2-5', '1308.3705-2-2-5'], ['1308.3705-1-2-6', '1308.3705-2-2-6'], ['1308.3705-1-9-0', '1308.3705-2-9-0'], ['1308.3705-1-9-1', '1308.3705-2-9-1'], ['1308.3705-1-9-2', '1308.3705-2-9-2'], ['1308.3705-1-9-3', '1308.3705-2-9-3'], ['1308.3705-1-24-0', '1308.3705-2-24-0'], ['1308.3705-1-24-1', '1308.3705-2-24-1'], ['1308.3705-1-24-2', '1308.3705-2-24-2'], ['1308.3705-1-24-3', '1308.3705-2-24-3'], ['1308.3705-1-7-0', '1308.3705-2-7-0'], ['1308.3705-1-7-1', '1308.3705-2-7-1'], ['1308.3705-1-7-2', '1308.3705-2-7-2'], ['1308.3705-1-7-3', '1308.3705-2-7-3'], ['1308.3705-1-7-4', '1308.3705-2-7-4'], ['1308.3705-1-7-5', '1308.3705-2-7-5'], ['1308.3705-1-7-6', '1308.3705-2-7-6'], ['1308.3705-1-7-7', '1308.3705-2-7-7'], ['1308.3705-1-7-8', '1308.3705-2-7-8'], ['1308.3705-1-22-0', '1308.3705-2-22-0'], ['1308.3705-1-22-1', '1308.3705-2-22-1'], ['1308.3705-1-22-2', '1308.3705-2-22-2'], ['1308.3705-1-20-0', '1308.3705-2-20-0'], ['1308.3705-1-20-1', '1308.3705-2-20-1'], ['1308.3705-1-20-2', '1308.3705-2-20-2'], ['1308.3705-1-20-3', '1308.3705-2-20-3'], ['1308.3705-1-20-4', '1308.3705-2-20-4'], ['1308.3705-1-20-5', '1308.3705-2-20-5'], ['1308.3705-1-20-6', '1308.3705-2-20-6'], ['1308.3705-1-20-7', '1308.3705-2-20-7'], ['1308.3705-1-20-8', '1308.3705-2-20-8'], ['1308.3705-1-20-9', '1308.3705-2-20-9'], ['1308.3705-1-20-10', '1308.3705-2-20-10'], ['1308.3705-1-12-0', '1308.3705-2-12-0'], ['1308.3705-1-12-1', '1308.3705-2-12-1'], ['1308.3705-1-12-2', '1308.3705-2-12-2'], ['1308.3705-1-12-3', '1308.3705-2-12-4'], ['1308.3705-1-12-4', '1308.3705-2-12-5'], ['1308.3705-1-3-0', '1308.3705-2-3-0'], ['1308.3705-1-3-1', '1308.3705-2-3-1'], ['1308.3705-1-3-2', '1308.3705-2-3-2'], ['1308.3705-1-3-3', '1308.3705-2-3-3'], ['1308.3705-1-18-0', '1308.3705-2-18-0'], ['1308.3705-1-18-1', '1308.3705-2-18-1'], ['1308.3705-1-18-2', '1308.3705-2-18-2'], ['1308.3705-1-18-3', '1308.3705-2-18-3'], ['1308.3705-1-18-4', '1308.3705-2-18-4'], ['1308.3705-1-17-0', '1308.3705-2-17-0'], ['1308.3705-1-17-1', '1308.3705-2-17-1'], ['1308.3705-1-17-2', '1308.3705-2-17-2'], ['1308.3705-1-15-0', '1308.3705-2-15-0'], ['1308.3705-1-15-1', '1308.3705-2-15-1'], ['1308.3705-1-15-2', '1308.3705-2-15-2'], ['1308.3705-1-15-3', '1308.3705-2-15-3'], ['1308.3705-1-15-4', '1308.3705-2-15-4'], ['1308.3705-1-15-5', '1308.3705-2-15-5'], ['1308.3705-1-15-6', '1308.3705-2-15-6'], ['1308.3705-1-15-7', '1308.3705-2-15-7'], ['1308.3705-1-15-8', '1308.3705-2-15-8'], ['1308.3705-1-15-9', '1308.3705-2-15-9'], ['1308.3705-1-23-0', '1308.3705-2-23-0'], ['1308.3705-1-23-1', '1308.3705-2-23-1'], ['1308.3705-1-23-2', '1308.3705-2-23-2'], ['1308.3705-2-17-0', '1308.3705-3-17-0'], ['1308.3705-2-17-1', '1308.3705-3-17-1'], ['1308.3705-2-17-2', '1308.3705-3-17-2'], ['1308.3705-2-5-1', '1308.3705-3-5-1'], ['1308.3705-2-5-2', '1308.3705-3-5-2'], ['1308.3705-2-21-0', '1308.3705-3-21-0'], ['1308.3705-2-21-1', '1308.3705-3-21-1'], ['1308.3705-2-25-0', '1308.3705-3-25-0'], ['1308.3705-2-25-1', '1308.3705-3-25-1'], ['1308.3705-2-25-2', '1308.3705-3-25-2'], ['1308.3705-2-25-3', '1308.3705-3-25-3'], ['1308.3705-2-25-4', '1308.3705-3-25-4'], ['1308.3705-2-13-0', '1308.3705-3-13-0'], ['1308.3705-2-13-1', '1308.3705-3-13-1'], ['1308.3705-2-13-2', '1308.3705-3-13-2'], ['1308.3705-2-13-3', '1308.3705-3-13-3'], ['1308.3705-2-13-4', '1308.3705-3-13-4'], ['1308.3705-2-13-7', '1308.3705-3-13-7'], ['1308.3705-2-13-8', '1308.3705-3-13-8'], ['1308.3705-2-23-0', '1308.3705-3-23-0'], ['1308.3705-2-23-1', '1308.3705-3-23-1'], ['1308.3705-2-23-2', '1308.3705-3-23-2'], ['1308.3705-2-24-0', '1308.3705-3-24-0'], ['1308.3705-2-24-1', '1308.3705-3-24-1'], ['1308.3705-2-24-2', '1308.3705-3-24-2'], ['1308.3705-2-24-3', '1308.3705-3-24-3'], ['1308.3705-2-0-0', '1308.3705-3-0-0'], ['1308.3705-2-0-1', '1308.3705-3-0-1'], ['1308.3705-2-0-2', '1308.3705-3-0-2'], ['1308.3705-2-0-3', '1308.3705-3-0-3'], ['1308.3705-2-0-4', '1308.3705-3-0-4'], ['1308.3705-2-0-5', '1308.3705-3-0-5'], ['1308.3705-2-0-6', '1308.3705-3-0-6'], ['1308.3705-2-0-7', '1308.3705-3-0-7'], ['1308.3705-2-0-8', '1308.3705-3-0-8'], ['1308.3705-2-0-9', '1308.3705-3-0-9'], ['1308.3705-2-0-10', '1308.3705-3-0-10'], ['1308.3705-2-0-11', '1308.3705-3-0-11'], ['1308.3705-2-0-12', '1308.3705-3-0-12'], ['1308.3705-2-0-13', '1308.3705-3-0-13'], ['1308.3705-2-22-0', '1308.3705-3-22-0'], ['1308.3705-2-22-1', '1308.3705-3-22-1'], ['1308.3705-2-22-2', '1308.3705-3-22-2'], ['1308.3705-2-18-0', '1308.3705-3-18-0'], ['1308.3705-2-18-1', '1308.3705-3-18-1'], ['1308.3705-2-18-2', '1308.3705-3-18-2'], ['1308.3705-2-18-3', '1308.3705-3-18-3'], ['1308.3705-2-18-4', '1308.3705-3-18-4'], ['1308.3705-2-19-0', '1308.3705-3-19-0'], ['1308.3705-2-19-1', '1308.3705-3-19-1'], ['1308.3705-2-19-2', '1308.3705-3-19-2'], ['1308.3705-2-19-3', '1308.3705-3-19-3'], ['1308.3705-2-9-0', '1308.3705-3-9-0'], ['1308.3705-2-9-1', '1308.3705-3-9-1'], ['1308.3705-2-9-2', '1308.3705-3-9-2'], ['1308.3705-2-9-3', '1308.3705-3-9-3'], ['1308.3705-2-7-0', '1308.3705-3-7-0'], ['1308.3705-2-7-1', '1308.3705-3-7-1'], ['1308.3705-2-7-4', '1308.3705-3-7-3'], ['1308.3705-2-7-5', '1308.3705-3-7-4'], ['1308.3705-2-7-6', '1308.3705-3-7-5'], ['1308.3705-2-7-7', '1308.3705-3-7-6'], ['1308.3705-2-7-8', '1308.3705-3-7-7'], ['1308.3705-2-12-0', '1308.3705-3-12-0'], ['1308.3705-2-12-1', '1308.3705-3-12-1'], ['1308.3705-2-12-2', '1308.3705-3-12-2'], ['1308.3705-2-12-3', '1308.3705-3-12-3'], ['1308.3705-2-12-4', '1308.3705-3-12-4'], ['1308.3705-2-20-0', '1308.3705-3-20-0'], ['1308.3705-2-20-1', '1308.3705-3-20-1'], ['1308.3705-2-20-2', '1308.3705-3-20-2'], ['1308.3705-2-20-3', '1308.3705-3-20-3'], ['1308.3705-2-20-4', '1308.3705-3-20-4'], ['1308.3705-2-20-5', '1308.3705-3-20-5'], ['1308.3705-2-20-6', '1308.3705-3-20-6'], ['1308.3705-2-20-7', '1308.3705-3-20-7'], ['1308.3705-2-20-8', '1308.3705-3-20-8'], ['1308.3705-2-20-10', '1308.3705-3-20-19'], ['1308.3705-2-15-0', '1308.3705-3-15-0'], ['1308.3705-2-15-1', '1308.3705-3-15-1'], ['1308.3705-2-15-2', '1308.3705-3-15-2'], ['1308.3705-2-15-3', '1308.3705-3-15-3'], ['1308.3705-2-15-4', '1308.3705-3-15-4'], ['1308.3705-2-15-5', '1308.3705-3-15-5'], ['1308.3705-2-15-6', '1308.3705-3-15-6'], ['1308.3705-2-15-7', '1308.3705-3-15-7'], ['1308.3705-2-15-8', '1308.3705-3-15-8'], ['1308.3705-2-15-9', '1308.3705-3-15-9'], ['1308.3705-2-2-1', '1308.3705-3-2-1'], ['1308.3705-2-2-2', '1308.3705-3-2-2'], ['1308.3705-2-2-3', '1308.3705-3-2-3'], ['1308.3705-2-2-4', '1308.3705-3-2-4'], ['1308.3705-2-2-5', '1308.3705-3-2-5'], ['1308.3705-2-3-0', '1308.3705-3-3-0'], ['1308.3705-2-3-1', '1308.3705-3-3-1'], ['1308.3705-2-3-2', '1308.3705-3-3-2'], ['1308.3705-2-3-3', '1308.3705-3-3-3']]

[['1308.3705-1-13-4', '1308.3705-2-13-4'], ['1308.3705-2-5-0', '1308.3705-3-5-0'], ['1308.3705-2-13-5', '1308.3705-3-13-5'], ['1308.3705-2-12-5', '1308.3705-3-12-5'], ['1308.3705-2-20-9', '1308.3705-3-20-9'], ['1308.3705-2-2-0', '1308.3705-3-2-0']]

[]

[['1308.3705-2-7-2', '1308.3705-3-7-2'], ['1308.3705-2-7-3', '1308.3705-3-7-2'], ['1308.3705-2-2-6', '1308.3705-3-2-6']]

[]

['1308.3705-1-13-6', '1308.3705-2-13-6', '1308.3705-3-13-6', '1308.3705-3-20-18']

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

https://arxiv.org/abs/1308.3705

{'1308.3705-3-0-0': 'In this second of two papers we apply our recently developed code to search for resonance features in the Planck CMB temperature data.', '1308.3705-3-0-1': 'We search both for log spaced oscillations or linear spaced oscillations and compare our findings with results of our WMAP9 analysis and the Planck team analysis .', '1308.3705-3-0-2': 'While there are hints of log spaced resonant features present in the WMAP9 data, the significance of these features weaken with more data.', '1308.3705-3-0-3': 'With more accurate small scale measurements, we also find that the best fit frequency has shifted and the amplitude has been reduced.', '1308.3705-3-0-4': 'We confirm the presence of a several low frequency peaks, earlier identified by the Planck team, but with a better improvement of fit ([MATH]).', '1308.3705-3-0-5': 'We further investigate this improvement by allowing the lensing potential to vary as well, showing mild correlation between the amplitude of the oscillations and the lensing amplitude.', '1308.3705-3-0-6': 'We find that the improvement of the fit increases even more ([MATH]) for the low frequencies that modify the spectrum in a way that mimics the lensing effect.', '1308.3705-3-0-7': 'Since these features were not present in the WMAP data, they are primarily due to better measurements of Planck at small angular scales.', '1308.3705-3-0-8': 'For linear spaced oscillations we find a maximum [MATH] scanning two orders of magnitude in frequency space, and the biggest improvements are at extremely high frequencies.', '1308.3705-3-0-9': 'Again, we recover a best fit frequency very close to the one found in WMAP9, which confirms that the fit improvement is driven by low [MATH].', '1308.3705-3-0-10': 'Further comparisons with WMAP9 show Planck contains many more features, both for linear and log space oscillations, but with a smaller improvement of fit.', '1308.3705-3-0-11': 'We discuss the improvement as a function of the number of modes and study the effect of the 217 GHz map, which appears to drive most of the improvement for log spaced oscillations.', '1308.3705-3-0-12': 'Two points strongly suggest that the detected features are fitting a combination of the noise and the dip at [MATH] in the 217 GHz map: the fit improvement mostly comes from a small range of [MATH], and comparison with simulations shows that the fit improvement is consistent with a statistical fluctuation.', '1308.3705-3-0-13': 'We conclude that none of the detected features are statistically significant.', '1308.3705-3-1-0': '# Introduction', '1308.3705-3-2-0': 'In this short paper, we will apply our recent introduced method in Ref. to search for resonant features in the recently released Planck CMB data.', '1308.3705-3-2-1': 'We consider two distinct theoretically motivated models: [EQUATION]', '1308.3705-3-2-2': 'We refer to the first model as the "log-spaced oscillations model" and the second model as the "linear oscillations model".', '1308.3705-3-2-3': 'For example, axion-monodromy inflation produces features that can be described by the logarithmic oscillations model with [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-3-2-4': 'Model that include the effects from a possible boundary on effective field theory (BEFT) predict features that can be described by the linear oscillations model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1308.3705-3-2-5': 'Both initial state modifications and multiverse models can also produce logarithmic oscillations, while sharp features generate a power spectrum with linear oscillations (although the amplitude is typically damped as a function of scale).', '1308.3705-3-2-6': 'Constraints on oscillations in the WMAP CMB data have been attempted in e.g. Refs. Note that model [REF] has a unit less frequency while model [REF] has units of Mpc.', '1308.3705-3-2-7': 'We will omit these units in the rest of the paper for brevity.', '1308.3705-3-3-0': 'This paper is organized as follows.', '1308.3705-3-3-1': 'We present our results on the Planck Data in [REF] for log and linear spaced oscillations.', '1308.3705-3-3-2': 'In [REF], we compare our results with the WMAP9 analysis.', '1308.3705-3-3-3': 'We discuss our findings and conclude in [REF].', '1308.3705-3-4-0': '# Planck Analysis', '1308.3705-3-5-0': 'In this analysis, we use a modified version of the publicly available Planck likelihood code [CITATION] to search for oscillations in the primordial power spectrum.', '1308.3705-3-5-1': 'For this analysis, we found the best fit values for both resonance model parameters and cosmological parameters.', '1308.3705-3-5-2': 'We vary all six [MATH]CDM parameters plus the phase and the amplitude of the oscillatory correction to the primordial power spectrum while fixing the foreground parameters to their best fit values for the no oscillations model.', '1308.3705-3-5-3': 'The best-fit is found using the Metropolos-Hastings algorithm, which is not the ideal method to look for the best fit, but it does allow us to compute marginalized likelihoods of the parameters and look for potential correlations.', '1308.3705-3-6-0': '## Log-spaced oscillations', '1308.3705-3-7-0': 'In Fig. [REF] shows the improvement in fit as a function of frequency, where the frequency of the oscillation was varied in 1201 steps between [MATH] .', '1308.3705-3-7-1': 'We observe several frequencies that could be hints of primordial oscillations.', '1308.3705-3-7-2': 'We confirm a number of features first observed in Ref. Our method improves the best fit peak identified by the planck team at low frequencies with [MATH] (with best fit frequency [MATH]) .', '1308.3705-3-7-3': 'After inspecting the resulting fit, we expected some correlation with smooth parameters.', '1308.3705-3-7-4': 'We found that varying [MATH] enables a further improvement in the fit by another [MATH], but the best fit has shifted towards a lower frequency [MATH].', '1308.3705-3-7-5': 'In Fig. [REF] we show the marginalized contour between the lensing amplitude [MATH] and the amplitude of the oscillations at the best fit frequency.', '1308.3705-3-7-6': 'Further investigation shows that this mild correlation actually shifts slope from peak to peak, which can be explained by the fact that the improvement of fit is at [MATH] (see [REF]); for these low frequencies the contribution to the power spectrum is rather smooth and the lensing amplitude effectively smooths the peak structure.', '1308.3705-3-7-7': 'Oscillations can help enhance or reduce this effect, and the correlation coefficient can therefore change signs depending on the phase of the oscillation.', '1308.3705-3-8-0': '## Linear-spaced oscillations', '1308.3705-3-9-0': 'In Fig. [REF], we show the improvement as a function of frequency for the linear spaced oscillations model.', '1308.3705-3-9-1': 'Again, we vary the phase and amplitude of the oscillation, together with the cosmological parameters, for each of 881 steps in frequency space.', '1308.3705-3-9-2': 'For linear spaced oscillations, the resulting improvement is extremely irregular, with no particularly preferred region.', '1308.3705-3-9-3': 'The best fit is at a frequency of [MATH], where the [MATH].', '1308.3705-3-10-0': '# WMAP9 vs Planck1', '1308.3705-3-11-0': '## Log-spaced oscillations', '1308.3705-3-12-0': 'In Fig. [REF] and Fig. [REF] we compare the improvement of fit in our analysis of WMAP data to the improvement in fit in our analysis of Planck data.', '1308.3705-3-12-1': 'Fig. [REF] clearly shows new peaks at the low frequency end in the Planck data, features that are absent in the WMAP data.', '1308.3705-3-12-2': 'At high frequencies, the feature that was seen in the WMAP analysis is less significant and has shifted.', '1308.3705-3-12-3': 'Note that some of this shift is due to different precomputed transfer function, which for high frequencies have a small but non-negligible effect on the projected frequency .', '1308.3705-3-12-4': 'For [MATH], where WMAP and Planck are both cosmic variance limited, we expect the features to coincide.', '1308.3705-3-12-5': 'As shown in an analysis in Ref. [CITATION] , the location of the feature shifts as one lowers [MATH] to [MATH], confirming that a better fit at small scales or high [MATH] is the source of the shift.', '1308.3705-3-13-0': 'Since the low frequency features are absent in the WMAP data, their presence should be primarily due to a better fit in the range [MATH].', '1308.3705-3-13-1': 'The correlation between [MATH] and the amplitude of the oscillations is a result of the fit being driven by the high [MATH] Planck data.', '1308.3705-3-13-2': 'In Fig. [REF] we show improvement of fit compared to no oscillations as a function of [MATH] for our best fit frequency [MATH].', '1308.3705-3-13-3': 'Indeed this plot shows that the improvement comes from multipoles around [MATH] and [MATH] (roughly between the 3rd and 4th peak).', '1308.3705-3-13-4': 'In the likelihood for our parameters we use the 100 GHz data up to [MATH], the 143 GHz data to [MATH] and the 217 GHz data up to [MATH].', '1308.3705-3-13-5': 'As was shown in Ref. the 217 GHz map drives some of the standard [MATH]CDM parameters away from their best fit WMAP values and the 217[MATH]217 GHz power spectrum contains a feature at [MATH] that is not seen in the 143[MATH]217 GHz power spectrum.', '1308.3705-3-13-6': 'C.', '1308.3705-3-13-7': 'By removing the 217 GHz data we find that the improvement drops to [MATH] with [MATH].', '1308.3705-3-13-8': 'Note that the better fit at [MATH] is also unconstrained by WMAP.', '1308.3705-3-14-0': '## Linear-spaced oscillations', '1308.3705-3-15-0': 'Interestingly, in comparison with WMAP, the Planck data seems to contain many more low frequency features as shown in Fig. [REF].', '1308.3705-3-15-1': 'As was the case for WMAP, the Planck data shows that higher frequencies can result in bigger the improvements of the fit.', '1308.3705-3-15-2': 'In WMAP we found were able to identify a single frequency that appeared to be favored over other frequencies.', '1308.3705-3-15-3': 'Despite the difference, a high oscillation feature persists in Planck data although the frequency has shifted slightly ([MATH]).', '1308.3705-3-15-4': 'Because of the similarities between the two data sets, this feature is most likely not due to a feature at small angular scales.', '1308.3705-3-15-5': 'For this reason we again investigate the improvement of fit as a function of [MATH].', '1308.3705-3-15-6': 'This is shown in Fig. [REF].', '1308.3705-3-15-7': 'Interestingly, this fitting appears rather gradual, which would favor a true feature interpretation.', '1308.3705-3-15-8': 'Regarding the feature in the 217 GHz map, the improvement of the fit actually decreases after [MATH].', '1308.3705-3-15-9': 'The plot shows that most of the improvement comes from low multipoles, consistent with this feature appearing both in WMAP and in Planck at a similar frequency.', '1308.3705-3-16-0': '# Discussion and Conclusions', '1308.3705-3-17-0': 'In this second of two papers we have applied our recently developed code to search for resonant features in the Planck data.', '1308.3705-3-17-1': 'Our code recovers the results found by the Planck collaboration, but adds to these findings by significantly extending the frequency range of the search.', '1308.3705-3-17-2': 'In addition, our method finds larger improvements of fit for low frequencies because it varies all parameters to find the best fit, not just the amplitude and frequency of the oscillatory signal.', '1308.3705-3-18-0': 'Our analysis has given us several important insights.', '1308.3705-3-18-1': 'First of all, the improvement at the low frequency of logarithmically spaced oscillations end are caused or at least enhanced by a varying lensing amplitude.', '1308.3705-3-18-2': 'For example, in comparison with the Planck paper result, we find that allowing the lensing amplitude to vary shifts the best fit frequency to lower values, and improves the overall fit.', '1308.3705-3-18-3': 'For linear spaced oscillations we find the largest improvement at the highest frequencies, with a best fit frequency that is close to that found in WMAP9.', '1308.3705-3-18-4': 'We showed that including this feature mostly improves the fit to the spectrum below [MATH].', '1308.3705-3-19-0': 'Further comparison between our WMAP9 and Planck analyses, shows the improvement of fit for log spaced oscillations has flipped, i.e. while for WMAP9 the improvements were at high frequencies for Planck the best fit is at low frequencies, although a feature at high frequencies does appear in the Planck data.', '1308.3705-3-19-1': 'The fact that none of the found oscillations in Planck are present in WMAP9, suggested that most improvement is coming from high [MATH], which we confirmed by computing the improvement as a function of [MATH].', '1308.3705-3-19-2': 'As the Planck team has noted, there is a feature present near [MATH] in the [MATH] GHz spectrum that does not appear in the other frequencies.', '1308.3705-3-19-3': 'Our results suggest that the improvement at low frequencies is predominately due to this feature.', '1308.3705-3-20-0': 'Second, in our companion paper we tested our method on simulated data.', '1308.3705-3-20-1': 'The primary goal of these tests was to assess the reliability of our perturbative method.', '1308.3705-3-20-2': 'Here we use the same simulations to assess the significance of the fit improvements to determine whether we have detected an oscillatory contribution to the primordial power spectrum.', '1308.3705-3-20-3': 'We found that for amplitudes as small as those that best fit the data, we expect an improvement of fit that exceeds what we find the data.', '1308.3705-3-20-4': 'We ran two full pipelines on maps that did not include a signal, for which we found improvements up to [MATH] .', '1308.3705-3-20-5': 'Furthermore we also ran a large number of (simplified) simulations in order to address the question: What is the typical maximum improvement expected from fitting the noise ?', '1308.3705-3-20-6': 'This analysis showed that the noise typically leads to [MATH], and has the potential to improve the fit [MATH].', '1308.3705-3-20-7': 'We found these improvements with [MATH].', '1308.3705-3-20-8': 'We have improved on these simulations, by randomly generating Gaussian noise using the weighted error bars directly synthesized from the Planck data for multipoles ranging from [MATH] to [MATH].', '1308.3705-3-20-9': 'We ran 1000 of these higher resolution featureless simulations and found that applying both log and linear spaced templates showed the measured (data) improvements are in the 90-94 percentile range.', '1308.3705-3-20-10': 'To be more precise, [MATH] is at the 3[MATH] level.', '1308.3705-3-20-11': 'These simulations are grid based, with 6 points in phase space, 12 points in the amplitude and 240 and 220 respectively for log and linear spaced oscillations in the frequency parameter.', '1308.3705-3-20-12': 'As a result, the derived distribution is conservative and not as accurate as a full simulation running an MCMC; with higher resolution we expect a distribution skewed towards higher improvements.', '1308.3705-3-20-13': 'Note that the resulting distribution is not a [MATH] distribution, because this is a highly non-linear problem.', '1308.3705-3-20-14': 'E.g. assuming a cosmic variance limit experiment and a template that oscillates as [MATH], we find [EQUATION]', '1308.3705-3-20-15': 'Here [MATH] is the sky fraction and [MATH] is a Gaussian random variable with variance 1, independently drawn for each [MATH] and Universe [MATH].', '1308.3705-3-20-16': '[MATH] is oscillating and and contains 3 free parameters.', '1308.3705-3-20-17': 'To get the best-fit distribution for [MATH] one would need to do an integral over the random variable, with best-fit [MATH] which will now be a function of that same random variable.', '1308.3705-3-20-18': '(see Fig. [REF]).', '1308.3705-3-20-19': 'This result together with the full simulation using our code on mock data in the previous paper, suggests that the improvements in fit found in our Planck analysis are consistent with expected statistical fluctuations for a realization from a featureless primordial model.', '1308.3705-3-21-0': 'Third, both for linear and log spaced oscillations, improvements appear local in [MATH] space.', '1308.3705-3-21-1': 'One might expect a real oscillation to lead to an improvement that would be more gradual as a function of the number of modes observed, though we recognize that this is not a rigorous argument: the biggest improvements arise from the modes with the highest signal to noise.', '1308.3705-3-22-0': 'Lastly, we have studied linear oscillations with the frequency at which the Planck team found a [MATH] detection in the bispectrum .', '1308.3705-3-22-1': 'We found that this frequency is very close to the Baryon Acoustic Oscillation (BAO) and we find no evidence in the power spectrum that there is such an oscillations (roughly corresponding to [MATH]).', '1308.3705-3-22-2': 'Our current understanding would suggest that varying the BAO parameters in the search for features in the bispectrum would probably reduce the significance (in addition to look elsewhere effects).', '1308.3705-3-23-0': 'For the future, we plan to implement Multinest as our sampler as this will significantly speed up the code.', '1308.3705-3-23-1': 'As we go to higher frequencies we should include higher order terms to the derivative.', '1308.3705-3-23-2': 'In particular, for an accurate measurement of [MATH], [MATH] and [MATH] it may be necessary to include derivatives of these parameters in the expansion.', '1308.3705-3-24-0': 'When the Planck polarization data will be available, we should be able to improve our search.', '1308.3705-3-24-1': 'Oscillations should produce features in both temperature and polarization spectra (and cross-spectra).', '1308.3705-3-24-2': 'Potentially, polarization measurements from ground-based experiments can probe out to [MATH] enabling even more sensitive searches for oscillatory features.', '1308.3705-3-24-3': 'However, searches based on ground-based data would be limited to lower oscillation frequencies since the power spectrum likelihood will have [MATH] correlations due to mode coupling effects.', '1308.3705-3-25-0': 'We can get additional insight from three point measurements as models that predict oscillations in the power spectrum typically also predict oscillations in higher order correlation spectra (see e.g. Refs. )', '1308.3705-3-25-1': 'While there have been initial attempts to search for oscillations in the Planck bispectrum measurements , computational cost has limited these searches to low frequencies.', '1308.3705-3-25-2': 'Alternative approaches have been proposed to optimize this search , but as of yet no attempt has been made to cover a large range of frequencies and phases.', '1308.3705-3-25-3': 'A first step would be to search the bispectrum measurements at frequencies suggested by analyses of the CMB power spectrum.', '1308.3705-3-25-4': 'Detecting features in both spectra would improve the statistical significance of the result-a promising direction for future study.'}

1507.05152

{'1507.05152-1-0-0': 'We study a version of the James model for the loop space of a suspension in unstable [MATH]-homotopy theory.', '1507.05152-1-0-1': "We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in [MATH]-homotopy theory: our result refines F. Morel's [MATH]-simplicial suspension theorem.", '1507.05152-1-0-2': 'We then describe some [MATH]-differentials in the EHP sequence in [MATH]-homotopy theory.', '1507.05152-1-0-3': "These results are analogous to classical results of G.W. Whitehead's.", '1507.05152-1-0-4': 'Using these tools, we deduce some new results about unstable [MATH]-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.', '1507.05152-1-1-0': '# Introduction', '1507.05152-1-2-0': 'If [MATH] is an [MATH]-connected pointed CW complex, then the suspension map [MATH] fits into a long exact sequence of the form: [EQUATION]', '1507.05152-1-2-1': 'Together with an elementary connectivity estimate for [MATH], this result can be viewed as a refinement of the Freudenthal suspension theorem.', '1507.05152-1-2-2': 'This result was first demonstrated by G.W. Whitehead if [MATH] [CITATION], and by W.D. Barcus for [MATH] as above [CITATION] (see also [CITATION] for a textbook treatment of the general statement).', '1507.05152-1-3-0': 'The morphisms [MATH], named after the Hopf invariant, and [MATH], named after the Whitehead product, appearing in the above exact sequence were also studied by Whitehead in great detail in the case where [MATH] [CITATION].', '1507.05152-1-3-1': 'In particular, he connected the morphism [MATH] with Whitehead products.', '1507.05152-1-3-2': 'For example, begin by observing that the [MATH]-fold suspension [MATH] is an isomorphism for [MATH].', '1507.05152-1-3-3': 'Define [MATH] by [MATH], where [MATH] is the identity map on the [MATH]-sphere and the bracket denotes Whitehead product.', '1507.05152-1-3-4': 'For [MATH], Whitehead observed that [EQUATION].', '1507.05152-1-3-5': 'While Whitehead established this result for spheres, it has been known for some time that morphism [MATH] is, for general [MATH]-connected spaces, still closely related to Whitehead products; see, e.g., [CITATION] or [CITATION] for a very general statement.', '1507.05152-1-3-6': 'In any case, these kinds of tools were used to great effect in early computations of unstable homotopy groups of spheres, e.g., by James and Toda [CITATION].', '1507.05152-1-4-0': 'The goal of this paper, whose title pays homage to the work of James [CITATION], is to establish analogs of the above results in the Morel-Voevodsky unstable [MATH]-homotopy category [CITATION] and to deduce some consequences of these results.', '1507.05152-1-4-1': 'The jumping-off point is to give a James-style model for the loop space of a suspension in [MATH]-homotopy theory (see Theorem [REF]).', '1507.05152-1-4-2': "Using this model, we deduce the following result, which can be thought of as a refinement of Morel's [MATH]-simplicial suspension theorem [CITATION].", '1507.05152-1-5-0': '[See Theorem [REF], Remark [REF] and Theorem [REF]]', '1507.05152-1-6-0': 'Assume [MATH] is a perfect field.', '1507.05152-1-6-1': 'If [MATH] is a pointed [MATH]-[MATH]-connected simplicial presheaf on [MATH], with [MATH], then there is an exact sequence of [MATH]-homotopy sheaves of the form: [EQUATION] the map [MATH] is (simplicial) suspension, the map [MATH] is a James-Hopf invariant, and the map [MATH] is described, as above, in terms of Whitehead products.', '1507.05152-1-7-0': 'We go on to discuss various consequences of the existence of this exact sequence.', '1507.05152-1-7-1': 'We analyze the low-degree portion of this sequence in Theorem [REF] and give a more explicit description of the sequence in the first degree where the suspension map fails to be an isomorphism.', '1507.05152-1-7-2': 'When [MATH] is a motivic sphere, it is shown in [CITATION] that the exact sequences of Theorem [REF] can be extended to all degrees after localizing at [MATH].', '1507.05152-1-7-3': 'By suitably varying the input sphere, these sequences can be strung together to obtain the EHP spectral sequence converging to the [MATH]-local [MATH]-stable [MATH]-homotopy sheaves of spheres.', '1507.05152-1-8-0': 'By construction, the [MATH]-differentials in this spectral sequence arise from the composite map [MATH], which in certain degrees we can analyze integrally.', '1507.05152-1-8-1': 'To state the result, recall that Morel computed the first non-vanishing [MATH]-homotopy sheaf of a motivic sphere in terms of Milnor-Witt K-theory [CITATION].', '1507.05152-1-8-2': 'He also showed that there is an isomorphism of rings [MATH], i.e., the zeroth Milnor-Witt K-theory group of a field [MATH] is isomorphic to the Grothendieck-Witt ring of isomorphism classes of symmetric bilinear forms over [MATH] [CITATION], defined to be the group completion of the monoid of isomorphism classes of non-degenerate symmetric bilinear forms.', '1507.05152-1-8-3': 'Given this terminology, the class of the composite [MATH] can be seen to correspond with a symmetric bilinear form, which we can describe.', '1507.05152-1-8-4': 'More precisely, we establish the following result (see the statement in the body of the text and Remark [REF] for a more conceptual explanation of the formula).', '1507.05152-1-9-0': '[See Theorem [REF]]', '1507.05152-1-10-0': 'Assume [MATH] is a perfect field, and let [MATH] be integers with [MATH].', '1507.05152-1-10-1': 'The map [EQUATION] corresponds to the element [MATH].', '1507.05152-1-11-0': 'One consequence of this result is the following analog of the classical fact, due to Hopf, that [MATH] is non-trivial.', '1507.05152-1-12-0': '[See Theorem [REF]] Fix a base field [MATH] assumed to be perfect and to have characteristic unequal to [MATH].', '1507.05152-1-12-1': 'Let [MATH] be even integers, and let [MATH] be an integer.', '1507.05152-1-12-2': 'There is a surjection [EQUATION] and the sheaf [MATH] is non-trivial if either [MATH] is formally real or if [MATH].', '1507.05152-1-13-0': 'Relying on the computations of [CITATION], we analyze the low-degree portion of the EHP sequence in great detail in the special case were [MATH].', '1507.05152-1-13-1': 'In particular, we give a description of the next non-vanishing [MATH]-homotopy sheaf (i.e., beyond that computed by Morel) of [MATH] in Theorem [REF].', '1507.05152-1-13-2': 'The following statement is an easy-to-state special case of a more general result.', '1507.05152-1-14-0': '[See Theorem [REF]]', '1507.05152-1-15-0': 'If [MATH] is a field having characteristic zero and containing an algebraically closed subfield, then, for any integer [MATH], there is an isomorphism of sheaves of the form: [EQUATION].', '1507.05152-1-16-0': 'Cohomology of homotopy sheaves of spheres such as those above appears in concrete applications to problems in algebra via techniques of obstruction theory; see, e.g., [CITATION] for more details.', '1507.05152-1-16-1': 'Our description of the sheaf [MATH] is well-suited to such cohomology computations.', '1507.05152-1-16-2': 'Our computation allows us to state a precise conjecture (see Conjecture [REF]) regarding the structure of the sheaf [MATH] for [MATH].', '1507.05152-1-16-3': "An explicit description of the sheaf [MATH] for [MATH] was a key step in [CITATION] in the resolution of Murthy's conjecture regarding splitting of rank [MATH] vector bundles on smooth affine [MATH]-folds over algebraically closed fields.", '1507.05152-1-16-4': "A resolution of Conjecture [REF] would, similarly, imply Murthy's conjecture in general.", '1507.05152-1-17-0': 'We close this introduction with some general comments regarding prerequisites.', '1507.05152-1-17-1': 'When working with the (unstable) [MATH]-homotopy category in general and Morel\'s [MATH]-algebraic topology in particular, with the goal of making this paper as self-contained as possible, we have labored to present the material in an axiomatic framework involving the "unstable [MATH]-connectivity property", which is introduced in Section [REF].', '1507.05152-1-17-2': 'All of the results in Sections [REF] and [REF] are written from this axiomatic perspective.', '1507.05152-1-17-3': 'We hope this style of presentation makes the material accessible to people who have some familiarity with homotopy theory of simplicial presheaves and the constructions of [CITATION], but not, for example, all of the technical results about strongly and strictly [MATH]-invariant sheaves contained in the first five chapters of [CITATION].', '1507.05152-1-17-4': 'Moreover, we hope that our presentation also makes [CITATION] itself more accessible to the non-expert.', '1507.05152-1-18-0': 'For the most part, Section [REF] is written in the same axiomatic framework.', '1507.05152-1-18-1': 'In contrast, Sections [REF] and [REF] require more background.', '1507.05152-1-18-2': 'In particular, this portion of the text requires familiarity with facts about strongly and strictly [MATH]-invariant sheaves (see Section [REF] for more precise statements), and known explicit computations of homotopy sheaves.', '1507.05152-1-18-3': 'In Section [REF], we also appeal to structural results from the theory of quadratic forms and both the affirmation of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture.'}

{'1507.05152-2-0-0': 'We study a version of the James model for the loop space of a suspension in unstable [MATH]-homotopy theory.', '1507.05152-2-0-1': "We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in [MATH]-homotopy theory: our result refines F. Morel's [MATH]-simplicial suspension theorem.", '1507.05152-2-0-2': 'We then describe some [MATH]-differentials in the EHP sequence in [MATH]-homotopy theory.', '1507.05152-2-0-3': "These results are analogous to classical results of G.W. Whitehead's.", '1507.05152-2-0-4': 'Using these tools, we deduce some new results about unstable [MATH]-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.', '1507.05152-2-1-0': '# Introduction', '1507.05152-2-2-0': 'If [MATH] is an [MATH]-connected pointed CW complex, then the suspension map [MATH] fits into a long exact sequence of the form: [EQUATION]', '1507.05152-2-2-1': 'Together with an elementary connectivity estimate for [MATH], this exact sequence may be viewed as a refinement of the Freudenthal suspension theorem.', '1507.05152-2-2-2': 'The exact sequence above was first constructed by G.W. Whitehead if [MATH] [CITATION], and by W.D. Barcus for [MATH] as above [CITATION] (see also [CITATION] for a textbook treatment of the general statement).', '1507.05152-2-3-0': 'The morphisms [MATH] and [MATH] appearing in the above exact sequence were also studied by Whitehead in great detail in the case where [MATH] [CITATION].', '1507.05152-2-3-1': 'The morphism [MATH] is the Hopf invariant, and Whitehead linked the morphism [MATH] with Whitehead products.', '1507.05152-2-3-2': 'In more detail, begin by observing that the [MATH]-fold suspension [MATH] is an isomorphism for [MATH].', '1507.05152-2-3-3': 'Define [MATH] by [MATH], where [MATH] is the identity map on the [MATH]-sphere and the bracket denotes Whitehead product.', '1507.05152-2-3-4': 'For [MATH], Whitehead observed that [EQUATION].', '1507.05152-2-3-5': 'While Whitehead established this result for spheres, it has been known for some time that morphism [MATH] is, for general [MATH]-connected spaces, still closely related to Whitehead products; see, e.g., [CITATION] or [CITATION] for a very general statement.', '1507.05152-2-3-6': 'In any case, these kinds of tools were used to great effect in early computations of unstable homotopy groups of spheres, e.g., by James and Toda [CITATION].', '1507.05152-2-4-0': 'The goal of this paper, whose title pays homage to the work of James [CITATION], is to establish analogs of the above results in the Morel-Voevodsky unstable [MATH]-homotopy category [CITATION] and to deduce some consequences of these results.', '1507.05152-2-4-1': 'The jumping-off point is to give a James-style model for the loop space of a suspension in [MATH]-homotopy theory (see Theorem [REF]).', '1507.05152-2-4-2': "Using this model, we deduce the following result, which can be thought of as a refinement of Morel's [MATH]-simplicial suspension theorem [CITATION].", '1507.05152-2-5-0': '[See Theorem [REF], Remark [REF] and Theorem [REF]]', '1507.05152-2-6-0': 'Assume [MATH] is a perfect field.', '1507.05152-2-6-1': 'If [MATH] is a pointed [MATH]-[MATH]-connected simplicial presheaf on [MATH], with [MATH], then there is an exact sequence of [MATH]-homotopy sheaves of the form: [EQUATION] the map [MATH] is (simplicial) suspension, the map [MATH] is a James-Hopf invariant, and the map [MATH] is described, as above, in terms of Whitehead products.', '1507.05152-2-7-0': 'We go on to discuss various consequences of the existence of this exact sequence.', '1507.05152-2-7-1': 'We analyze the low-degree portion of this sequence in Theorem [REF] and give a more explicit description of the sequence in the first degree where the suspension map fails to be an isomorphism.', '1507.05152-2-7-2': 'When [MATH] is a motivic sphere, it is shown in [CITATION] that the exact sequences of Theorem [REF] can be extended to all degrees after localizing at [MATH].', '1507.05152-2-7-3': 'By suitably varying the input sphere, these sequences can be strung together to obtain the EHP spectral sequence converging to the [MATH]-local [MATH]-stable [MATH]-homotopy sheaves of spheres.', '1507.05152-2-8-0': 'By construction, the [MATH]-differentials in this spectral sequence arise from the composite map [MATH], which in certain degrees we can analyze integrally.', '1507.05152-2-8-1': 'To state the result, recall that Morel computed the first non-vanishing [MATH]-homotopy sheaf of a motivic sphere in terms of Milnor-Witt K-theory [CITATION].', '1507.05152-2-8-2': 'He also showed that there is an isomorphism of rings [MATH], i.e., the zeroth Milnor-Witt K-theory group of a field [MATH] is isomorphic to the Grothendieck-Witt ring of isomorphism classes of symmetric bilinear forms over [MATH] [CITATION], defined to be the group completion of the monoid of isomorphism classes of non-degenerate symmetric bilinear forms.', '1507.05152-2-8-3': 'Given this terminology, the class of the composite [MATH] can be seen to correspond with a symmetric bilinear form, which we can describe.', '1507.05152-2-8-4': 'More precisely, we establish the following result (see the statement in the body of the text and Remark [REF] for a more conceptual explanation of the formula).', '1507.05152-2-9-0': '[See Theorem [REF]]', '1507.05152-2-10-0': 'Assume [MATH] is a perfect field, and let [MATH] be integers with [MATH].', '1507.05152-2-10-1': 'The map [EQUATION] corresponds to the element [MATH].', '1507.05152-2-11-0': 'One consequence of this result is the following analog of the classical fact, due to Hopf, that [MATH] is non-trivial.', '1507.05152-2-12-0': '[See Theorem [REF]] Fix a base field [MATH] assumed to be perfect and to have characteristic unequal to [MATH].', '1507.05152-2-12-1': 'Let [MATH] be even integers, and let [MATH] be an integer.', '1507.05152-2-12-2': 'There is a surjection [EQUATION] and the sheaf [MATH] is non-trivial if either [MATH] is formally real or if [MATH].', '1507.05152-2-13-0': 'Relying on the computations of [CITATION], we analyze the low-degree portion of the EHP sequence in great detail in the special case were [MATH].', '1507.05152-2-13-1': 'In particular, we give a description of the next non-vanishing [MATH]-homotopy sheaf (i.e., beyond that computed by Morel) of [MATH] in Theorem [REF].', '1507.05152-2-13-2': 'The following statement is an easy-to-state special case of a more general result.', '1507.05152-2-14-0': '[See Theorem [REF]]', '1507.05152-2-15-0': 'If [MATH] is a field of characteristic [MATH] and containing an algebraically closed subfield, then, for any integer [MATH], there is an isomorphism of sheaves of the form: [EQUATION].', '1507.05152-2-16-0': 'Cohomology of homotopy sheaves of spheres such as those above appears in concrete applications to problems in algebra via techniques of obstruction theory; see, e.g., [CITATION] for more details.', '1507.05152-2-16-1': 'Our description of the sheaf [MATH] is well-suited to such cohomology computations.', '1507.05152-2-16-2': 'Our computation allows us to state a precise conjecture (see Conjecture [REF]) regarding the structure of the sheaf [MATH] for [MATH].', '1507.05152-2-16-3': "An explicit description of the sheaf [MATH] for [MATH] was a key step in [CITATION] in the resolution of Murthy's conjecture regarding splitting of rank [MATH] vector bundles on smooth affine [MATH]-folds over algebraically closed fields.", '1507.05152-2-16-4': "A resolution of Conjecture [REF] would, similarly, imply Murthy's conjecture in general.", '1507.05152-2-17-0': 'We close this introduction with some general comments regarding prerequisites.', '1507.05152-2-17-1': 'When working with the (unstable) [MATH]-homotopy category in general and Morel\'s [MATH]-algebraic topology in particular, with the goal of making this paper as self-contained as possible, we have labored to present the material in an axiomatic framework involving the "unstable [MATH]-connectivity property", which is introduced in Section [REF].', '1507.05152-2-17-2': 'All of the results in Sections [REF] and [REF] are written from this axiomatic perspective.', '1507.05152-2-17-3': 'We hope this style of presentation makes the material accessible to people who have some familiarity with homotopy theory of simplicial presheaves and the constructions of [CITATION], but not, for example, all of the technical results about strongly and strictly [MATH]-invariant sheaves contained in the first five chapters of [CITATION].', '1507.05152-2-17-4': 'Moreover, we hope that our presentation also makes [CITATION] itself more accessible to the non-expert.', '1507.05152-2-18-0': 'For the most part, Section [REF] is written in the same axiomatic framework.', '1507.05152-2-18-1': 'In contrast, Sections [REF] and [REF] require more background.', '1507.05152-2-18-2': 'In particular, this portion of the text requires familiarity with facts about strongly and strictly [MATH]-invariant sheaves (see Section [REF] for more precise statements), and known explicit computations of homotopy sheaves.', '1507.05152-2-18-3': 'In Section [REF], we also appeal to structural results from the theory of quadratic forms and both the affirmation of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture.'}

[['1507.05152-1-7-0', '1507.05152-2-7-0'], ['1507.05152-1-7-1', '1507.05152-2-7-1'], ['1507.05152-1-7-2', '1507.05152-2-7-2'], ['1507.05152-1-7-3', '1507.05152-2-7-3'], ['1507.05152-1-8-0', '1507.05152-2-8-0'], ['1507.05152-1-8-1', '1507.05152-2-8-1'], ['1507.05152-1-8-2', '1507.05152-2-8-2'], ['1507.05152-1-8-3', '1507.05152-2-8-3'], ['1507.05152-1-8-4', '1507.05152-2-8-4'], ['1507.05152-1-11-0', '1507.05152-2-11-0'], ['1507.05152-1-4-0', '1507.05152-2-4-0'], ['1507.05152-1-4-1', '1507.05152-2-4-1'], ['1507.05152-1-4-2', '1507.05152-2-4-2'], ['1507.05152-1-12-0', '1507.05152-2-12-0'], ['1507.05152-1-12-1', '1507.05152-2-12-1'], ['1507.05152-1-12-2', '1507.05152-2-12-2'], ['1507.05152-1-6-0', '1507.05152-2-6-0'], ['1507.05152-1-6-1', '1507.05152-2-6-1'], ['1507.05152-1-13-0', '1507.05152-2-13-0'], ['1507.05152-1-13-1', '1507.05152-2-13-1'], ['1507.05152-1-13-2', '1507.05152-2-13-2'], ['1507.05152-1-2-0', '1507.05152-2-2-0'], ['1507.05152-1-16-0', '1507.05152-2-16-0'], ['1507.05152-1-16-1', '1507.05152-2-16-1'], ['1507.05152-1-16-2', '1507.05152-2-16-2'], ['1507.05152-1-16-3', '1507.05152-2-16-3'], ['1507.05152-1-16-4', '1507.05152-2-16-4'], ['1507.05152-1-17-0', '1507.05152-2-17-0'], ['1507.05152-1-17-1', '1507.05152-2-17-1'], ['1507.05152-1-17-2', '1507.05152-2-17-2'], ['1507.05152-1-17-3', '1507.05152-2-17-3'], ['1507.05152-1-17-4', '1507.05152-2-17-4'], ['1507.05152-1-18-0', '1507.05152-2-18-0'], ['1507.05152-1-18-1', '1507.05152-2-18-1'], ['1507.05152-1-18-2', '1507.05152-2-18-2'], ['1507.05152-1-18-3', '1507.05152-2-18-3'], ['1507.05152-1-10-0', '1507.05152-2-10-0'], ['1507.05152-1-10-1', '1507.05152-2-10-1'], ['1507.05152-1-0-0', '1507.05152-2-0-0'], ['1507.05152-1-0-1', '1507.05152-2-0-1'], ['1507.05152-1-0-2', '1507.05152-2-0-2'], ['1507.05152-1-0-3', '1507.05152-2-0-3'], ['1507.05152-1-0-4', '1507.05152-2-0-4'], ['1507.05152-1-3-3', '1507.05152-2-3-3'], ['1507.05152-1-3-4', '1507.05152-2-3-4'], ['1507.05152-1-3-5', '1507.05152-2-3-5'], ['1507.05152-1-3-6', '1507.05152-2-3-6'], ['1507.05152-1-2-1', '1507.05152-2-2-1'], ['1507.05152-1-2-2', '1507.05152-2-2-2'], ['1507.05152-1-15-0', '1507.05152-2-15-0'], ['1507.05152-1-3-0', '1507.05152-2-3-0'], ['1507.05152-1-3-2', '1507.05152-2-3-2'], ['1507.05152-1-3-1', '1507.05152-2-3-1']]

[['1507.05152-1-7-0', '1507.05152-2-7-0'], ['1507.05152-1-7-1', '1507.05152-2-7-1'], ['1507.05152-1-7-2', '1507.05152-2-7-2'], ['1507.05152-1-7-3', '1507.05152-2-7-3'], ['1507.05152-1-8-0', '1507.05152-2-8-0'], ['1507.05152-1-8-1', '1507.05152-2-8-1'], ['1507.05152-1-8-2', '1507.05152-2-8-2'], ['1507.05152-1-8-3', '1507.05152-2-8-3'], ['1507.05152-1-8-4', '1507.05152-2-8-4'], ['1507.05152-1-11-0', '1507.05152-2-11-0'], ['1507.05152-1-4-0', '1507.05152-2-4-0'], ['1507.05152-1-4-1', '1507.05152-2-4-1'], ['1507.05152-1-4-2', '1507.05152-2-4-2'], ['1507.05152-1-12-0', '1507.05152-2-12-0'], ['1507.05152-1-12-1', '1507.05152-2-12-1'], ['1507.05152-1-12-2', '1507.05152-2-12-2'], ['1507.05152-1-6-0', '1507.05152-2-6-0'], ['1507.05152-1-6-1', '1507.05152-2-6-1'], ['1507.05152-1-13-0', '1507.05152-2-13-0'], ['1507.05152-1-13-1', '1507.05152-2-13-1'], ['1507.05152-1-13-2', '1507.05152-2-13-2'], ['1507.05152-1-2-0', '1507.05152-2-2-0'], ['1507.05152-1-16-0', '1507.05152-2-16-0'], ['1507.05152-1-16-1', '1507.05152-2-16-1'], ['1507.05152-1-16-2', '1507.05152-2-16-2'], ['1507.05152-1-16-3', '1507.05152-2-16-3'], ['1507.05152-1-16-4', '1507.05152-2-16-4'], ['1507.05152-1-17-0', '1507.05152-2-17-0'], ['1507.05152-1-17-1', '1507.05152-2-17-1'], ['1507.05152-1-17-2', '1507.05152-2-17-2'], ['1507.05152-1-17-3', '1507.05152-2-17-3'], ['1507.05152-1-17-4', '1507.05152-2-17-4'], ['1507.05152-1-18-0', '1507.05152-2-18-0'], ['1507.05152-1-18-1', '1507.05152-2-18-1'], ['1507.05152-1-18-2', '1507.05152-2-18-2'], ['1507.05152-1-18-3', '1507.05152-2-18-3'], ['1507.05152-1-10-0', '1507.05152-2-10-0'], ['1507.05152-1-10-1', '1507.05152-2-10-1'], ['1507.05152-1-0-0', '1507.05152-2-0-0'], ['1507.05152-1-0-1', '1507.05152-2-0-1'], ['1507.05152-1-0-2', '1507.05152-2-0-2'], ['1507.05152-1-0-3', '1507.05152-2-0-3'], ['1507.05152-1-0-4', '1507.05152-2-0-4'], ['1507.05152-1-3-3', '1507.05152-2-3-3'], ['1507.05152-1-3-4', '1507.05152-2-3-4'], ['1507.05152-1-3-5', '1507.05152-2-3-5'], ['1507.05152-1-3-6', '1507.05152-2-3-6']]

[['1507.05152-1-2-1', '1507.05152-2-2-1'], ['1507.05152-1-2-2', '1507.05152-2-2-2'], ['1507.05152-1-15-0', '1507.05152-2-15-0'], ['1507.05152-1-3-0', '1507.05152-2-3-0'], ['1507.05152-1-3-2', '1507.05152-2-3-2']]

[]

[['1507.05152-1-3-1', '1507.05152-2-3-1']]

[]

['1507.05152-1-5-0', '1507.05152-1-9-0', '1507.05152-1-14-0', '1507.05152-2-5-0', '1507.05152-2-9-0', '1507.05152-2-14-0']

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

https://arxiv.org/abs/1507.05152

1910.13404

{'1910.13404-1-0-0': 'The production fraction of the meson with respect to the sum of [MATH] and [MATH] mesons is measured in both 7 and 13TeVcenter-of-mass energy [MATH] collisions produced by the Large Hadron Collider (LHC), using the LHCb detector.', '1910.13404-1-0-1': 'The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence.', '1910.13404-1-0-2': 'The [MATH] production asymmetry is also measured, and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.', '1910.13404-1-1-0': 'Submitted to Physical Review D', '1910.13404-1-2-0': 'CERN for the benefit of the LHCb collaboration.', '1910.13404-1-2-1': 'https://creativecommons.org/licenses/by/4.0/CC-BY-4.0 licence.', '1910.13404-1-3-0': 'arabic', '1910.13404-1-4-0': '# Introduction', '1910.13404-1-5-0': 'The meson is a bound state containing a [MATH] quark with a [MATH] quark.', '1910.13404-1-5-1': 'It has the largest mass of any two differently flavored quarks in a mesonic ground state.', '1910.13404-1-5-2': 'Studies of its production or determination of individual decay widths require measurements of its branching fractions to exclusive final states.', '1910.13404-1-5-3': 'Since the branching fractions of some decay modes of and mesons are accurately known, we determine the ratio of meson production relative to the sum of and mesons.', '1910.13404-1-5-4': 'Here we use techniques similar to those employed for the measurement of meson and baryon fractions [CITATION].', '1910.13404-1-5-5': 'In that paper use is made of the fact that the semileptonic widths of all [MATH]-flavored hadrons with light and strange quarks are equal.', '1910.13404-1-5-6': 'However, both the [MATH] and [MATH] quarks can decay, rendering that concept inapplicable.', '1910.13404-1-5-7': 'Instead we rely on theoretical predictions of the semileptonic decay branching fraction [MATH].', '1910.13404-1-5-8': 'Currently, only the relative production cross-section times the branching fraction of either the [MATH] or [MATH] modes have been measured by the [CITATION], [CITATION] and [CITATION] experiments.', '1910.13404-1-6-0': 'The [MATH] meson production fraction ([MATH]) relative to the sum of [MATH]) and [MATH]) mesons is defined as [EQUATION] where [MATH] refers to the efficiency and branching fraction corrected number of signal events.', '1910.13404-1-6-1': 'The modes containing [MATH] and [MATH] mesons are also corrected for cross-feeds with and decays.', '1910.13404-1-6-2': 'The determination of the corrected yields of the [MATH] decays follows our previous measurement strategy in Ref. [CITATION] where the equations relating the fractions to the corrected yields, including cross-feed contributions, are given.', '1910.13404-1-6-3': 'We also correct for the 0.4% effect of doubly-Cabibbo-suppressed decays and [MATH] mixing.', '1910.13404-1-6-4': 'The relevant hadron branching fractions are listed in Table [REF].', '1910.13404-1-6-5': 'The average semileptonic branching fractions of and , [MATH] is found by averaging measurements from the [CITATION], [CITATION] and [CITATION] experiments, detailed in Ref. [CITATION].', '1910.13404-1-6-6': 'Since only [MATH] modes are detected in this analysis, a correction for the small [MATH] rate of 1% is applied to the denominator of Eq. [REF].', '1910.13404-1-7-0': 'The dominant production mechanism for mesons is gluon-gluon fusion, [MATH].', '1910.13404-1-7-1': 'Non-relativistic quantum chromodynamics is used along with fragmentation functions to predict cross-sections as functions of transverse momentum () and pseudorapidity ([MATH]).', '1910.13404-1-7-2': 'The literature is nicely summarized in Ref. [CITATION].', '1910.13404-1-7-3': 'We define [MATH] to refer to [MATH], [MATH], and [MATH] mesons, while [MATH] refers to [MATH] and [MATH] mesons.', '1910.13404-1-8-0': 'In this analysis [MATH] is determined by measuring the angle of the [MATH] meson with respect to the beam direction by using the positions of the primary [MATH] interaction vertex (PV) and the [MATH] meson decay point into either [MATH], [MATH], or [MATH].', '1910.13404-1-8-1': 'The transverse momentum initially refers to the vector sum of the charmed-hadron and [MATH] momentum transverse to the proton beams.', '1910.13404-1-8-2': 'However, the results are re-interpreted in terms of the [MATH] meson [MATH] by simulating and correcting the effects of the missing momenta.', '1910.13404-1-8-3': 'We denote [MATH] or [MATH] meson candidates as [MATH].', '1910.13404-1-9-0': 'The production asymmetry between [MATH] and [MATH] mesons, which should be small.', '1910.13404-1-9-1': 'It is defined as [EQUATION] where the [MATH] and [MATH] are the asymmetries in the signal yields and the efficiencies of [MATH] and [MATH] detection, respectively.', '1910.13404-1-9-2': 'The asymmetry in the [MATH] decay is assumed to be zero in this analysis.', '1910.13404-1-10-0': 'The branching fraction predictions from various models of semileptonic decays are listed in Table [REF].', '1910.13404-1-10-1': 'For [MATH] they range from 1.4 to 7.5%, which is quite a large interval.', '1910.13404-1-10-2': 'Branching fractions for other modes are also listed where available.', '1910.13404-1-10-3': 'We use the Z expansion fit results from Ref. [CITATION], and the method II results for Ref. [CITATION].', '1910.13404-1-11-0': 'Some restrictions on models are possible by comparing to lighter [MATH] meson decays.', '1910.13404-1-11-1': 'Since the inclusive semileptonic branching fraction for these decays, [MATH], is about 10.5% and the [MATH] lifetime, [MATH], is 1/3 that of the [MATH], we disregard models that predict 10% or larger values for [MATH] of the .', '1910.13404-1-11-2': 'This excludes from consideration the models of Refs. [CITATION] and [CITATION].', '1910.13404-1-11-3': 'The average model prediction then is [MATH].', '1910.13404-1-11-4': 'The standard deviation is 0.46, which we use to estimate the systematic uncertainty on the model variation.', '1910.13404-1-11-5': 'Results of our measurement without using this branching fraction are also quoted.', '1910.13404-1-12-0': '# Detector, trigger and simulation', '1910.13404-1-13-0': 'The LHCb detector [CITATION] is a single-arm forward spectrometer covering the pseudorapidity range [MATH], designed for the study of particles containing or quarks.', '1910.13404-1-13-1': 'The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the [MATH] interaction region that allows and hadrons to be identified from their characteristically long flight distance; a tracking system that provides a measurement of the momentum, [MATH], of charged particles, two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons.', '1910.13404-1-13-2': 'A downstream system of iron interspersed with chambers is used to identify muons.', '1910.13404-1-14-0': 'The magnetic field deflects positively and negatively charged particles in opposite directions and this can lead to detection asymmetries.', '1910.13404-1-14-1': 'Periodically reversing the magnetic field polarity throughout the data taking almost cancels the effect.', '1910.13404-1-14-2': 'The configuration with the magnetic field pointing upwards (downwards) bends positively (negatively) charged particles in the horizontal plane towards the centre of the LHC ring.', '1910.13404-1-14-3': 'This analysis uses data collected in 2011 (7TeV) and 2016 (13TeV) where appropriate triggers are available.', '1910.13404-1-14-4': 'The data taking was split between magnetic field up and down configurations: in 2011 0.6for down and 0.4for up settings, while in 2016 0.9for down and 0.8for up.', '1910.13404-1-15-0': 'The trigger [CITATION] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with [MATH] are reconstructed for 2011(2016) data.', '1910.13404-1-16-0': 'Separate hardware triggers are used for the [MATH] and [MATH] samples.', '1910.13404-1-16-1': 'For the former we require a [MATH] pair.', '1910.13404-1-16-2': 'For the latter, we require a single muon with large for the 7TeVdata as used in Ref. [CITATION].', '1910.13404-1-16-3': 'For the 13TeVdata, the single muon trigger was not available, therefore at the hardware trigger stage, events are required to have a hadron, photon or electron transverse energy greater than approximately 3.5 GeV in the calorimeters.', '1910.13404-1-16-4': 'The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary [MATH] interaction vertex as described in Ref. [CITATION].', '1910.13404-1-16-5': 'At least one charged particle must have [MATH] and be inconsistent with originating from a PV.', '1910.13404-1-16-6': 'A multivariate algorithm [CITATION] is used for the identification of secondary vertices consistent with the decay of a hadron.', '1910.13404-1-17-0': 'Simulation is required to model the effects of the detector acceptance and the imposed selection requirements.', '1910.13404-1-17-1': 'In the simulation, [MATH] collisions are generated using [CITATION] with a specific configuration [CITATION].', '1910.13404-1-17-2': 'Decays of unstable particles are described by [CITATION], in which final-state radiation is generated using [CITATION].', '1910.13404-1-17-3': 'The interaction of the generated particles with the detector, and its response, are implemented using the toolkit [CITATION] as described in Ref. [CITATION].', '1910.13404-1-18-0': '# Event selection, signal efficiencies and yields', '1910.13404-1-19-0': '## Selection of [MATH] candidates', '1910.13404-1-20-0': 'The analysis is done separately for the light [MATH] meson modes and the decay.', '1910.13404-1-20-1': 'In each case the triggered events are subject to further filtering requirements.', '1910.13404-1-20-2': 'In addition, the [MATH] sample is subjected to a boosted decision tree (BDT), a multivariate classification method, using the TMVA toolkit [CITATION].', '1910.13404-1-20-3': 'This is not necessary for the [MATH] or [MATH] modes because they have large signals and are relatively free from backgrounds [CITATION].', '1910.13404-1-21-0': 'For the [MATH] final state the initial selection requires that muons that satisfy the candidate trigger each have minimum [MATH] MeV, have large impact parameters with the PV, form a good quality vertex, have a reasonable flight distance significance from the PV, and have a summed [MATH] GeV.', '1910.13404-1-21-1': 'The "companion" muon that is not part of the decay must be well identified and form a good quality vertex with the candidate, which must be downstream of the PV.', '1910.13404-1-22-0': 'To suppress muon tracks that are reconstructed more than once, we require a small minimum opening angle between the muons from the decay and the companion muon momentum in the plane transverse to the beam line.', '1910.13404-1-22-1': 'Specifically, this opening angle must be greater than 0.8[MATH].', '1910.13404-1-22-2': 'The invariant mass of the companion muon and the oppositely charged muon from [MATH] must differ from the known value of the [MATH] mass by more than [MATH] [CITATION], while the invariant mass with the same charged muon is required to be larger than [MATH].', '1910.13404-1-23-0': "Since we are dealing with an exclusive final state, we define [EQUATION] where [MATH] is the magnitude of the combination's momentum component transverse to the [MATH]-hadron flight direction.", '1910.13404-1-23-1': 'Figure [REF] shows the distributions of [MATH] versus the invariant [MATH] mass, [MATH], both for data and simulation.', '1910.13404-1-23-2': 'To remove background, a requirement of [MATH] is applied, as indicated by the (red) dashed line.', '1910.13404-1-24-0': 'Since we also measure the production asymmetry between [MATH] and [MATH] mesons, we restrict the angular acceptance of the companion muon to make it more uniform by removing muons close to the edge of the detector, in the bending direction ([MATH] direction), where large acceptance-induced asymmetries can occur.', '1910.13404-1-24-1': 'Thus, we require that the [MATH] component of the momentum satisfies [EQUATION] where [MATH] is the muon momentum along the direction of the proton beam downstream of the PV, as done in Refs. [CITATION].', '1910.13404-1-25-0': 'After these initial restrictions, we turn to the multivariate selection, forming the classifier denoted BDT in the following.', '1910.13404-1-25-1': 'The discriminating variables used are (a) the [MATH] of the vertex fit of the with the [MATH]; (b) the [MATH], where [MATH]is defined as the [MATH] of the impact parameter with respect to the PV, of the [MATH], [MATH] and their combination; (c) the of the and the [MATH]; and (d) the cosine of the angle between the [MATH] and the meson in the plane perpendicular to the beam direction.', '1910.13404-1-25-2': 'The training sample for signal is simulated [MATH] events, and for background is inclusive [MATH] simulated events.', '1910.13404-1-26-0': 'We then optimize the BDT output threshold by maximizing [MATH], where [MATH] and [MATH] are the number of the signal and background yields in the signal region defined as [MATH].', '1910.13404-1-26-1': 'The sum, [MATH], is the total number of events within these limits, and [MATH] is taken from a fit to the [MATH] distribution.', '1910.13404-1-26-2': 'The optimal BDT output threshold results in a BDT signal efficiency of 89% with a background rejection of 63%.', '1910.13404-1-27-0': 'The [MATH] distribution is shown in Fig. [REF].', '1910.13404-1-27-1': 'It consists not only of signal [MATH] events, but also of [MATH] decays, where [MATH], and other [MATH] final states, most importantly [MATH] and [MATH].', '1910.13404-1-27-2': 'We find shapes for these final states using simulation.', '1910.13404-1-27-3': 'The signal shape is a sum of a double-crystal-ball and a bifurcated Gaussian functions.', '1910.13404-1-27-4': 'The sum of the combinatorial and misidentification backgrounds are represented by a Gaussian kernel shape [CITATION].', '1910.13404-1-27-5': 'For the other background modes, we use histograms directly.', '1910.13404-1-27-6': 'These shapes are fitted to the [MATH] distributions in Fig. [REF] in order to determine the [MATH] yields.', '1910.13404-1-27-7': 'The ratio of the [MATH] yield to the [MATH] yield is fixed, after accounting for the relative detection efficiencies, from the LHCb measurement of [MATH], where the first uncertainty is statistical and the second systematic [CITATION]; this convention is used throughout this paper.', '1910.13404-1-27-8': 'The other components of the fit are allowed to vary.', '1910.13404-1-27-9': 'We find [MATH] and [MATH] signal [MATH] events at 7 and 13TeV, respectively, while the backgrounds sum to 950 and 5170 events at the same energies.', '1910.13404-1-27-10': 'These signal yields need to be corrected for the small background from candidates with a correctly reconstructed meson that is paired with a hadron mis-identified as a muon.', '1910.13404-1-28-0': '## Efficiency for [MATH]', '1910.13404-1-29-0': 'Efficiencies are determined using both data [CITATION] and simulation of [MATH], with the generated events weighted to match the [MATH], and [MATH] distributions observed in data.', '1910.13404-1-29-1': 'In addition, we weight accordingly the [MATH] distribution of the muon associated with the .', '1910.13404-1-29-2': 'Weighting the simulation is important since the total efficiencies are functions of these variables.', '1910.13404-1-29-3': 'Efficiencies using data include trigger, and muon identification.', '1910.13404-1-29-4': 'Efficiencies using simulation include detector acceptance, reconstruction and event selection and removal of beam crossings with an excess number of hits in the detector.', '1910.13404-1-29-5': 'Total efficiencies as a function of [MATH] for different [MATH] intervals are shown in Fig. [REF].', '1910.13404-1-30-0': '## [MATH] selection criteria', '1910.13404-1-31-0': 'Selection criteria for [MATH] final states differ from those containing a [MATH].', '1910.13404-1-31-1': 'The transverse momentum of each hadron must be greater than 0.3GeV, and that of the muon larger than 1.3GeV.', '1910.13404-1-31-2': 'We require [MATH] with respect to any PV ensuring that tracks do not originate from primary [MATH] interactions.', '1910.13404-1-31-3': 'All final state particles are required to be positively identified using information from the RICH detectors.', '1910.13404-1-31-4': 'Particles from [MATH] decay candidates must have a good fit to a common vertex with [MATH]/ndof [MATH], where ndof is the number of degrees of freedom.', '1910.13404-1-31-5': 'They must also be well separated from the nearest PV, with the flight distance divided by its uncertainty greater than 5.', '1910.13404-1-32-0': 'Candidate [MATH] hadrons are formed by combining [MATH] and muon candidates originating from a common vertex with [MATH]/ndof [MATH] and an [MATH] invariant mass in the range 3.0-5.0GeV.', '1910.13404-1-33-0': 'Background from prompt [MATH] production at the PV needs to be considered.', '1910.13404-1-33-1': 'We use the natural logarithm of the [MATH] impact parameter, IP, with respect to the PV in units of mm.', '1910.13404-1-33-2': 'Requiring ln(IP/mm)[MATH] is found to reduce the prompt component to be below 0.1%, while preserving 97% of all signals.', '1910.13404-1-33-3': 'This restriction allows us to perform fits only to the [MATH] candidate mass spectra to find the [MATH]-hadron decay yields.', '1910.13404-1-34-0': 'The [MATH] candidate mass distributions integrated over [MATH] and [MATH] are shown in Fig. [REF] and consist of a prominent peak resulting from signal, and a small contribution due to combinatorial background from random combinations of particles that pass the selection.', '1910.13404-1-34-1': 'They are fit with a signal component comprised of two Gaussian functions, and a combinatorial background component modeled as a linear function.', '1910.13404-1-34-2': 'The fitted yields are listed in Table [REF].', '1910.13404-1-34-3': 'These numbers must be corrected for hadrons that are mis-identified as muons, and for semileptonic decays of and hadrons that produce [MATH] and [MATH] mesons.', '1910.13404-1-35-0': 'In Table [REF] the column labeled "fake muons" shows the yields of wrong-sign [MATH] and [MATH] combinations that pass the selections.', '1910.13404-1-35-1': 'These yields provide good estimates of the fake muon contributions in the signal samples, which are very small.', '1910.13404-1-35-2': 'Following the procedure in Ref. [CITATION], we find the cross-feed corrections of [MATH] and [MATH] to be twice the measured yields for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV), and for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV).', '1910.13404-1-35-3': 'Relative efficiencies for detecting final states with a single extra hadron are taken into account when subtracting these yields.', '1910.13404-1-36-0': '## Efficiencies for [MATH] and [MATH]', '1910.13404-1-37-0': 'Similar methods based on data, as implemented for the decay, are used to evaluate the efficiencies for trigger and particle identification.', '1910.13404-1-37-1': 'Simulation is also used to determine the efficiencies of event selection and reconstruction of these modes.', '1910.13404-1-37-2': 'The total efficiencies for [MATH] meson decays into [MATH] and [MATH] are shown in Fig. [REF].', '1910.13404-1-38-0': '# Results', '1910.13404-1-39-0': '## Corrections to the [MATH] distributions due to the missing neutrino', '1910.13404-1-40-0': 'Since the production kinematics of [MATH] and mesons can differ as functions of [MATH] and [MATH], we need to measure [MATH] as functions of these variables.', '1910.13404-1-40-1': 'The measurement of [MATH] is straightforward, however, we do not measure directly the [MATH] of the [MATH]-flavored hadron because of the missing neutrino, and in the case of the [MATH] meson possible missing extra particles.', '1910.13404-1-40-2': 'Following a procedure similar to the one used in Ref. [CITATION], we determine a correction factor, [MATH], that is the ratio of the average reconstructed to true [MATH] as a function of the invariant mass of the charmed hadron plus muon.', '1910.13404-1-40-3': 'The ratio distribution as a function of hadron-muon invariant mass are shown in Fig. [REF].', '1910.13404-1-40-4': 'The average correction, the [MATH]-factor, is shown on the figure.', '1910.13404-1-40-5': 'For the [MATH] meson it varies from 0.75 to unity over the interval from 3 GeV to the [MATH] mass, and for the meson it varies from 0.85 to unity over the interval from 4 GeV to the mass.', '1910.13404-1-41-0': '## fraction results', '1910.13404-1-42-0': 'The ratio of production fractions, [MATH], are shown as functions of [MATH] and [MATH] in Fig. [REF].', '1910.13404-1-42-1': 'There is little dependence on [MATH], but the decrease as a function of [MATH] is noticeable.', '1910.13404-1-43-0': 'To describe the [MATH] dependence we use an equation of the form [EQUATION] where [MATH] represents the overall normalization and contains the total global systematic uncertainty, thus, [MATH]; [MATH] is taken as 7.2 GeV, close to the average of the .', '1910.13404-1-43-1': 'The slopes, [MATH], are similar in size to those measured for the [MATH] meson fraction ratio as a function of [CITATION].', '1910.13404-1-43-2': 'Results of fits to the data using Eq. [REF] are listed in Table [REF].', '1910.13404-1-44-0': 'The average fractions in the interval [MATH]GeVare found by integrating over [MATH].', '1910.13404-1-44-1': 'To allow for facile changes to our results due to improved theoretical predictions, we provide the results for [EQUATION]', '1910.13404-1-44-2': 'Next we give the result on the fractions ratio [EQUATION] where the third uncertainty is due to the theoretical prediction of [MATH].', '1910.13404-1-44-3': 'To find [MATH] just double these numbers.', '1910.13404-1-45-0': 'We also measure the ratio of the [MATH] production fractions at [MATH] and [MATH].', '1910.13404-1-45-1': 'Figure [REF] shows the ratio as functions of and [MATH].', '1910.13404-1-45-2': 'Here most of the systematic uncertainties cancel.', '1910.13404-1-45-3': 'The integrated value of the ratio of [MATH] and [MATH] is measured as [MATH], consistent with no increase in the fraction ratio as a function of center-of-mass energy.', '1910.13404-1-46-0': 'The fraction with respect to inclusive [MATH]-hadron production can be derived from the information in previous LHCb [MATH]-hadron fraction papers Ref. [CITATION].', '1910.13404-1-46-1': 'There the measured values of the ratios of [MATH]-hadron fractions over the same range in terms of the [MATH]-hadron are for mesons ([MATH]) and baryons [EQUATION] where the uncertainties contain both statistical and systematic components added in quadrature.', '1910.13404-1-46-2': 'For the measurement of the [MATH] fraction at 7TeV, the dominant systematic uncertainty is from the lack of the knowledge of [MATH] at that time [CITATION]; here the value and uncertainty have been recalculated according to the latest value of [MATH] from the PDG [CITATION].', '1910.13404-1-47-0': 'Taking the sum of all the [MATH]-hadron fractions to be unity, and ignoring [MATH] here because it is so small, [EQUATION] where [MATH] is a correction factor derived in Ref. [CITATION] that accounts for heavier [MATH]-baryons, mainly the [MATH].', '1910.13404-1-47-1': 'Solving for [MATH] yields [EQUATION]', '1910.13404-1-47-2': 'We find that [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-1-48-0': '## The [MATH] production asymmetry', '1910.13404-1-49-0': 'The production asymmetries are measured in two different magnetic field configurations and then averaged.', '1910.13404-1-49-1': 'No significant asymmetry is observed in any intervals of [MATH] or [MATH].', '1910.13404-1-49-2': 'The results are summarized in Table [REF].', '1910.13404-1-50-0': 'Averaging the [MATH] production asymmetries over [MATH] and [MATH], we find [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-1-51-0': '# Systematic uncertainties', '1910.13404-1-52-0': 'Systematic uncertainties are separated into two categories: "global", which apply across the phase space, and "local", which are calculated in each two-dimensional [MATH] bin.', '1910.13404-1-52-1': 'These uncertainties are listed in Table [REF].', '1910.13404-1-53-0': 'First let us consider the [MATH] decay.', '1910.13404-1-53-1': 'The uncertainty due to the signal shape used to fit the [MATH] distribution is determined by changing the baseline signal shape, the sum of a double-sided Crystal Ball function and a bifurcated Gaussian, to a kernel estimation.', '1910.13404-1-53-2': 'To find the shape of the combinatorial and mis-identification backgrounds we use simulated inclusive samples of [MATH] events not including decays.', '1910.13404-1-53-3': 'A total of 500 samples are generated and different fits are done to the samples to determine the possible uncertainty.', '1910.13404-1-53-4': 'This procedure is also used for the [MATH] measurement.', '1910.13404-1-53-5': 'We call contributions to the [MATH] mass spectrum "feed-down" contributions, occurring from other decay channels including [MATH], [MATH], and [MATH].', '1910.13404-1-53-6': 'The systematic uncertainty results from the uncertainties in their branching fractions.', '1910.13404-1-53-7': 'Different decay models for decays can change the [MATH] shape.', '1910.13404-1-53-8': 'We use the model of Ebert et al.[CITATION], for our baseline prediction.', '1910.13404-1-53-9': 'Then we also use the model by Kiselev [CITATION] to find the efficiencies and take half the difference as our systematic uncertainty.', '1910.13404-1-53-10': 'We also estimate the uncertainty due to the sensitivity to various selection requirements and simulation statistics.', '1910.13404-1-53-11': 'The muon identification efficiencies are determined from data using inclusive samples of decay where one of the muon candidates is not identified.', '1910.13404-1-53-12': 'The trigger efficiency is determined by using three independent samples of events, those that trigger on a [MATH], those that triggered on something else in the event, and those that trigger on both the and something else.', '1910.13404-1-53-13': 'These samples are then used to compute the trigger efficiencies in two-dimensional [MATH] and [MATH] bins.', '1910.13404-1-54-0': 'Next, we turn to the [MATH] modes.', '1910.13404-1-54-1': 'The efficiencies and their uncertainties for identifying pions and kaons are determined by using almost background free samples of [MATH] decays.', '1910.13404-1-54-2': 'The trigger and muon identification efficiencies, and their uncertainties, are obtained in the same manner as for the [MATH] mode.', '1910.13404-1-54-3': 'There are small systematic uncertainties related to efficiency estimates and the assumed [MATH] to [MATH] mixtures, as well as simulation statistics.', '1910.13404-1-54-4': 'Global systematic uncertainties include the hadron branching fractions listed in Table [REF], cross-feed corrections arising from and decays into [MATH] events, and a global hadron plus photon multiplicity requirement.', '1910.13404-1-54-5': 'The latter is evaluated with data.', '1910.13404-1-55-0': '# Conclusions', '1910.13404-1-56-0': 'In 7 and 13TeV[MATH] collisions the product of [MATH] with the relative fraction of mesons with respect to the sum of and mesons in the ranges [MATH] and [MATH] is found to be [EQUATION]', '1910.13404-1-56-1': 'Using the average of the theoretical prediction [MATH], where the uncertainty is given by the standard deviation derived from the distribution of the models, we determine [EQUATION] where the first uncertainties are statistical, the second systematic, and the third due to the theoretical prediction of [MATH].', '1910.13404-1-56-2': 'There is a small dependence on the transverse momentum of the meson, but none observed on its pseudorapidity.', '1910.13404-1-57-0': 'The ratio of fractions, [MATH], for 13TeV/7TeVis consistent with no increase in the fraction.', '1910.13404-1-57-1': 'Furthermore, using the assumption of no violation in the [MATH] decay, we find that the average asymmetry in [MATH] production is consistent with zero.', '1910.13404-1-57-2': 'The measurements are [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-1-58-0': 'These results are useful to extract absolute branching fractions for measurements, albeit with a relatively large uncertainty.', '1910.13404-1-58-1': 'They also challenge QCD calculations to predict the measured fractions and explain the consistency of the fractions with energy [CITATION].'}

{'1910.13404-2-0-0': 'The production fraction of the meson with respect to the sum of [MATH] and [MATH] mesons is measured in both 7 and 13TeVcenter-of-mass energy [MATH] collisions produced by the Large Hadron Collider (LHC), using the LHCb detector.', '1910.13404-2-0-1': 'The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence.', '1910.13404-2-0-2': 'The [MATH] production asymmetry is also measured, and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.', '1910.13404-2-1-0': 'Submitted to Physical Review D', '1910.13404-2-2-0': 'CERN for the benefit of the LHCb collaboration.', '1910.13404-2-2-1': 'https://creativecommons.org/licenses/by/4.0/CC-BY-4.0 licence.', '1910.13404-2-3-0': 'arabic', '1910.13404-2-4-0': '# Introduction', '1910.13404-2-5-0': 'The meson is a bound state containing a [MATH] quark with a [MATH] quark.', '1910.13404-2-5-1': 'It has the largest mass of any two differently flavored quarks in a mesonic ground state.', '1910.13404-2-5-2': 'Studies of its production or determination of individual decay widths require measurements of its branching fractions to exclusive final states.', '1910.13404-2-5-3': 'Since the branching fractions of some decay modes of and mesons are accurately known, we determine the ratio of meson production relative to the sum of and mesons.', '1910.13404-2-5-4': 'Here we use techniques similar to those employed for the measurement of meson and baryon fractions [CITATION].', '1910.13404-2-5-5': 'In that paper use is made of the fact that the semileptonic widths of all [MATH]-flavored hadrons with light and strange quarks are equal.', '1910.13404-2-5-6': 'However, both the [MATH] and [MATH] quarks can decay, rendering that concept inapplicable.', '1910.13404-2-5-7': 'Instead we rely on theoretical predictions of the semileptonic decay branching fraction [MATH].', '1910.13404-2-5-8': 'Currently, only the relative production cross-section times the branching fraction of either the [MATH] or [MATH] modes have been measured by the [CITATION], [CITATION] and [CITATION] experiments.', '1910.13404-2-6-0': 'The [MATH] meson production fraction ([MATH]) relative to the sum of [MATH]) and [MATH]) mesons is defined as [EQUATION] where [MATH] refers to the efficiency and branching fraction corrected number of signal events.', '1910.13404-2-6-1': 'The modes containing [MATH] and [MATH] mesons are also corrected for cross-feeds with and decays.', '1910.13404-2-6-2': 'The determination of the corrected yields of the [MATH] decays follows our previous measurement strategy in Ref. [CITATION] where the equations relating the fractions to the corrected yields, including cross-feed contributions, are given.', '1910.13404-2-6-3': 'We also correct for the 0.4% effect of doubly-Cabibbo-suppressed decays and [MATH] mixing.', '1910.13404-2-6-4': 'The relevant hadron branching fractions are listed in Table [REF].', '1910.13404-2-6-5': 'The average semileptonic branching fractions of and , [MATH] is found by averaging measurements from the [CITATION], [CITATION] and [CITATION] experiments, detailed in Ref. [CITATION].', '1910.13404-2-6-6': 'Since only [MATH] modes are detected in this analysis, a correction for the small [MATH] rate of 1% is applied to the denominator of Eq. [REF].', '1910.13404-2-7-0': 'The dominant production mechanism for mesons is gluon-gluon fusion, [MATH].', '1910.13404-2-7-1': 'Non-relativistic quantum chromodynamics is used along with fragmentation functions to predict cross-sections as functions of transverse momentum () and pseudorapidity ([MATH]).', '1910.13404-2-7-2': 'The literature is nicely summarized in Ref. [CITATION].', '1910.13404-2-7-3': 'We define [MATH] to refer to [MATH], [MATH], and [MATH] mesons, while [MATH] refers to [MATH] and [MATH] mesons.', '1910.13404-2-8-0': 'In this analysis [MATH] is determined by measuring the angle of the [MATH] meson with respect to the beam direction by using the positions of the primary [MATH] interaction vertex (PV) and the [MATH] meson decay point into either [MATH], [MATH], or [MATH].', '1910.13404-2-8-1': 'The transverse momentum initially refers to the vector sum of the charmed-hadron and [MATH] momentum transverse to the proton beams.', '1910.13404-2-8-2': 'However, the results are re-interpreted in terms of the [MATH] meson [MATH] by simulating and correcting the effects of the missing momenta.', '1910.13404-2-8-3': 'We denote [MATH] or [MATH] meson candidates as [MATH].', '1910.13404-2-9-0': 'The production asymmetry between [MATH] and [MATH] mesons, which should be small, is defined as [EQUATION] where [MATH] and [MATH] are the asymmetries in the signal yields and the efficiencies of [MATH] and [MATH] detection, respectively.', '1910.13404-2-9-1': 'The asymmetry in the [MATH] decay is assumed to be zero in this analysis.', '1910.13404-2-10-0': 'The branching fraction predictions from various models of semileptonic decays are listed in Table [REF].', '1910.13404-2-10-1': 'For [MATH] they range from 1.4 to 7.5%, which is quite a large interval.', '1910.13404-2-10-2': 'Branching fractions for other modes are also listed where available.', '1910.13404-2-10-3': 'We use the Z expansion fit results from Ref. [CITATION], and the method II results for Ref. [CITATION].', '1910.13404-2-11-0': 'Some restrictions on models are possible by comparing to lighter [MATH] meson decays.', '1910.13404-2-11-1': 'Since the inclusive semileptonic branching fraction for these decays, [MATH], is about 10.5% and the [MATH] lifetime, [MATH], is 1/3 that of the [MATH], we disregard models that predict 10% or larger values for [MATH] of the .', '1910.13404-2-11-2': 'This excludes from consideration the models of Refs. [CITATION] and [CITATION].', '1910.13404-2-11-3': 'The average model prediction is then [MATH].', '1910.13404-2-11-4': 'The standard deviation is 0.46, which we use to estimate the systematic uncertainty on the model variation.', '1910.13404-2-11-5': 'Results of our measurement without using this branching fraction are also quoted.', '1910.13404-2-12-0': '# Detector, trigger and simulation', '1910.13404-2-13-0': 'The LHCb detector [CITATION] is a single-arm forward spectrometer covering the pseudorapidity range [MATH], designed for the study of particles containing or quarks.', '1910.13404-2-13-1': 'The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the [MATH] interaction region that allows and hadrons to be identified from their characteristically long flight distance; a tracking system that provides a measurement of the momentum, [MATH], of charged particles; two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons; and a downstream system of iron interspersed with chambers is used to identify muons.', '1910.13404-2-14-0': 'The magnetic field deflects positively and negatively charged particles in opposite directions and this can lead to detection asymmetries.', '1910.13404-2-14-1': 'Periodically reversing the magnetic field polarity throughout the data taking almost cancels the effect.', '1910.13404-2-14-2': 'The configuration with the magnetic field pointing upwards (downwards) bends positively (negatively) charged particles in the horizontal plane towards the centre of the LHC ring.', '1910.13404-2-14-3': 'This analysis uses data collected in 2011 (7TeV) and 2016 (13TeV) where appropriate triggers are available.', '1910.13404-2-14-4': 'The data taking was split between magnetic field up and down configurations.', '1910.13404-2-14-5': 'In the 2011 data 0.6(0.4) were collected with the field pointing up (down), while in 2016 the split was 0.9with field up and 0.8with field down.', '1910.13404-2-15-0': 'The trigger [CITATION] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with [MATH] are reconstructed for 2011(2016) data.', '1910.13404-2-16-0': 'Separate hardware triggers are used for the [MATH] and [MATH] samples.', '1910.13404-2-16-1': 'For the former we require a [MATH] pair.', '1910.13404-2-16-2': 'For the latter, we require a single muon with large for the 7TeVdata as used in Ref. [CITATION].', '1910.13404-2-16-3': 'For the 13TeVdata, the single muon trigger was not available, therefore at the hardware trigger stage, events are required to have a hadron, photon or electron transverse energy greater than approximately 3.5 GeV in the calorimeters.', '1910.13404-2-16-4': 'The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary [MATH] interaction vertex as described in Ref. [CITATION].', '1910.13404-2-16-5': 'At least one charged particle must have [MATH] and be inconsistent with originating from a PV.', '1910.13404-2-16-6': 'A multivariate algorithm [CITATION] is used for the identification of secondary vertices consistent with the decay of a hadron.', '1910.13404-2-17-0': 'Simulation is required to model the effects of the detector acceptance and the imposed selection requirements.', '1910.13404-2-17-1': 'In the simulation, [MATH] collisions are generated using [CITATION] with a specific configuration [CITATION].', '1910.13404-2-17-2': 'Decays of unstable particles are described by [CITATION], in which final-state radiation is generated using [CITATION].', '1910.13404-2-17-3': 'The interaction of the generated particles with the detector, and its response, are implemented using the toolkit [CITATION] as described in Ref. [CITATION].', '1910.13404-2-18-0': '# Event selection, signal efficiencies and yields', '1910.13404-2-19-0': '## Selection of [MATH] candidates', '1910.13404-2-20-0': 'The analysis is done separately for the light [MATH] meson modes and the decay.', '1910.13404-2-20-1': 'In each case the triggered events are subject to further filtering requirements.', '1910.13404-2-20-2': 'In addition, the [MATH] sample is subjected to a boosted decision tree (BDT), a multivariate classification method, using the TMVA toolkit [CITATION].', '1910.13404-2-20-3': 'This is not necessary for the [MATH] or [MATH] modes because they have large signals and are relatively free from backgrounds [CITATION].', '1910.13404-2-21-0': 'For the [MATH] final state the initial selection requires that muons that satisfy the candidate trigger each have minimum [MATH] MeV, have large impact parameters with the PV, form a good quality vertex, have a reasonable flight distance significance from the PV, and have a summed [MATH] GeV.', '1910.13404-2-21-1': 'The "companion" muon that is not part of the decay must be well identified and form a good quality vertex with the candidate, which must be downstream of the PV.', '1910.13404-2-22-0': 'To suppress muon tracks that are reconstructed more than once, we require a small minimum opening angle between the muons from the decay and the companion muon momentum measured in the plane transverse to the beam line.', '1910.13404-2-22-1': 'Specifically, this opening angle must be greater than 0.8[MATH].', '1910.13404-2-22-2': 'The invariant mass of the companion muon and the oppositely charged muon from [MATH] must differ from the known value of the [MATH] mass by more than [MATH] [CITATION], while the invariant mass with the same charged muon is required to be larger than [MATH].', '1910.13404-2-23-0': "Since we are dealing with an exclusive final state, we define [EQUATION] where [MATH] is the magnitude of the combination's momentum component transverse to the [MATH]-hadron flight direction.", '1910.13404-2-23-1': 'Figure [REF] shows the distributions of [MATH] versus the invariant [MATH] mass, [MATH], for both data and simulation.', '1910.13404-2-23-2': 'To remove background, a requirement of [MATH] is applied, as indicated by the (red) dashed line.', '1910.13404-2-24-0': 'Since we also measure the production asymmetry between [MATH] and [MATH] mesons, we restrict the angular acceptance of the companion muon to make it more uniform by removing muons close to the edge of the detector, in the bending direction ([MATH]-direction), where large acceptance-induced asymmetries can occur.', '1910.13404-2-24-1': 'Thus, we require that the [MATH]-component of the momentum satisfies [EQUATION] where [MATH] is the muon momentum along the direction of the proton beam downstream of the PV, as is done in Refs. [CITATION].', '1910.13404-2-25-0': 'After these initial restrictions, we turn to the multivariate selection, forming the classifier denoted BDT in the following.', '1910.13404-2-25-1': 'The discriminating variables used are: (a) the [MATH] of the vertex fit of the with the [MATH]; (b) the [MATH], where [MATH]is defined as the [MATH] of the impact parameter with respect to the PV, of the [MATH], [MATH] and their combination; (c) the of the and the [MATH]; and (d) the cosine of the angle between the [MATH] and the meson in the plane perpendicular to the beam direction.', '1910.13404-2-25-2': 'The training sample for signal is simulated [MATH] events, and for background is inclusive [MATH] simulated events.', '1910.13404-2-26-0': 'We then optimize the BDT output threshold by maximizing [MATH], where [MATH] and [MATH] are the number of the signal and background yields in the signal region defined as [MATH].', '1910.13404-2-26-1': 'The sum, [MATH], is the total number of events within these limits, and [MATH] is taken from a fit to the [MATH] distribution.', '1910.13404-2-26-2': 'The optimal BDT output threshold results in a BDT signal efficiency of 89% with a background rejection of 63%, as determined by observing the resulting samples of input signal simulation events and background candidates.', '1910.13404-2-27-0': 'The [MATH] distribution is shown in Fig. [REF].', '1910.13404-2-27-1': 'It consists not only of signal [MATH] events, but also of [MATH] decays, where [MATH], and other [MATH] final states, most importantly [MATH] and [MATH].', '1910.13404-2-27-2': 'We find shapes for these final states using simulation.', '1910.13404-2-27-3': 'The signal shape is a sum of a double Crystal Ball and a bifurcated Gaussian functions.', '1910.13404-2-27-4': 'The sum of the combinatorial and misidentification backgrounds are represented by a Gaussian kernel shape [CITATION].', '1910.13404-2-27-5': 'For the other background modes, we use histograms directly.', '1910.13404-2-27-6': 'These shapes are fitted to the [MATH] distributions in Fig. [REF] in order to determine the [MATH] yields.', '1910.13404-2-27-7': 'The ratio of the [MATH] yield to the [MATH] yield is fixed, after accounting for the relative detection efficiencies, from the LHCb measurement of [MATH], where the first uncertainty is statistical and the second systematic [CITATION]; this convention is used throughout this paper.', '1910.13404-2-27-8': 'The other components of the fit are allowed to vary.', '1910.13404-2-27-9': 'We find [MATH] and [MATH] signal [MATH] events at 7 and 13TeV, respectively, while the backgrounds sum to 950 and 5170 events at the same energies.', '1910.13404-2-27-10': 'These signal yields need to be corrected for the small background from candidates with a correctly reconstructed meson that is paired with a hadron mis-identified as a muon.', '1910.13404-2-28-0': '## Efficiency for [MATH]', '1910.13404-2-29-0': 'Efficiencies are determined using both data [CITATION] and simulation of [MATH], with the generated events weighted to match the [MATH], and [MATH] distributions observed in data.', '1910.13404-2-29-1': 'In addition, we weight accordingly the [MATH] distribution of the muon associated with the .', '1910.13404-2-29-2': 'Weighting the simulation is important since the total efficiencies are functions of these variables.', '1910.13404-2-29-3': 'Efficiencies using data include trigger, and muon identification.', '1910.13404-2-29-4': 'Efficiencies using simulation include detector acceptance, reconstruction and event selection, and removal of beam crossings with an excess number of hits in the detector.', '1910.13404-2-29-5': 'Total efficiencies as a function of [MATH] for different [MATH] intervals are shown in Fig. [REF].', '1910.13404-2-30-0': '## [MATH] selection criteria', '1910.13404-2-31-0': 'Selection criteria for [MATH] final states differ from those containing a [MATH].', '1910.13404-2-31-1': 'The transverse momentum of each hadron must be greater than 0.3GeV, and that of the muon larger than 1.3GeV.', '1910.13404-2-31-2': 'We require [MATH] with respect to any PV, ensuring that tracks do not originate from primary [MATH] interactions.', '1910.13404-2-31-3': 'All final state particles are required to be positively identified using information from the RICH detectors.', '1910.13404-2-31-4': 'Particles from [MATH] decay candidates must have a good fit to a common vertex with [MATH]/ndof [MATH], where ndof is the number of degrees of freedom.', '1910.13404-2-31-5': 'They must also be well separated from the nearest PV, with the flight distance divided by its uncertainty greater than 5.', '1910.13404-2-32-0': 'Candidate [MATH] hadrons are formed by combining [MATH] and muon candidates originating from a common vertex with [MATH]/ndof [MATH] and an [MATH] invariant mass in the range 3.0-5.0GeV.', '1910.13404-2-33-0': 'Background from prompt [MATH] production at the PV needs to be considered.', '1910.13404-2-33-1': 'We use the natural logarithm of the [MATH] impact parameter, IP, with respect to the PV in units of mm.', '1910.13404-2-33-2': 'Requiring ln(IP/mm)[MATH] is found to reduce the prompt component to be below 0.1%, while preserving 97% of all signals.', '1910.13404-2-33-3': 'This restriction allows us to perform fits only to the [MATH] candidate mass spectra to find the [MATH]-hadron decay yields.', '1910.13404-2-34-0': 'The [MATH] candidate mass distributions integrated over [MATH] and [MATH] are shown in Fig. [REF] and consist of a prominent peak resulting from signal, and a small contribution due to combinatorial background from random combinations of particles that pass the selection.', '1910.13404-2-34-1': 'They are fit with a signal component comprised of two Gaussian functions, and a combinatorial background component modeled as a linear function.', '1910.13404-2-34-2': 'The fitted yields are listed in Table [REF].', '1910.13404-2-34-3': 'These numbers must be corrected for hadrons that are mis-identified as muons, and for semileptonic decays of and hadrons that produce [MATH] and [MATH] mesons.', '1910.13404-2-35-0': 'In Table [REF] the column labeled "fake muons" shows the yields of wrong-sign [MATH] and [MATH] combinations that pass the selections.', '1910.13404-2-35-1': 'These yields provide good estimates of the fake muon contributions in the signal samples, which are very small.', '1910.13404-2-35-2': 'Following the procedure in Ref. [CITATION], we find the cross-feed corrections of [MATH] and [MATH] to be twice the measured yields for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV), and for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV).', '1910.13404-2-35-3': 'Relative efficiencies for detecting final states with a single extra hadron are taken into account when subtracting these yields.', '1910.13404-2-36-0': '## Efficiencies for [MATH] and [MATH]', '1910.13404-2-37-0': 'Similar methods based on data, as implemented for the decay, are used to evaluate the efficiencies for trigger and particle identification.', '1910.13404-2-37-1': 'Simulation is also used to determine the efficiencies of event selection and reconstruction of these modes.', '1910.13404-2-37-2': 'The total efficiencies for [MATH] meson decays into [MATH] and [MATH] are shown in Fig. [REF].', '1910.13404-2-38-0': '# Results', '1910.13404-2-39-0': '## Corrections to the [MATH] distributions due to the missing neutrino', '1910.13404-2-40-0': 'Since the production kinematics of [MATH] and mesons can differ as functions of [MATH] and [MATH], we need to measure [MATH] as functions of these variables.', '1910.13404-2-40-1': 'The measurement of [MATH] is straightforward, however, we do not measure directly the [MATH] of the [MATH]-flavored hadron because of the missing neutrino, and in the case of the [MATH] meson possible missing extra particles.', '1910.13404-2-40-2': 'Following a procedure similar to the one used in Ref. [CITATION], we determine a correction factor, [MATH], that is the ratio of the average reconstructed to true [MATH] as a function of the invariant mass of the charmed hadron plus muon.', '1910.13404-2-40-3': 'The ratio distribution as a function of hadron-muon invariant mass are shown in Fig. [REF].', '1910.13404-2-40-4': 'The average correction, the [MATH]-factor, is shown on the figure.', '1910.13404-2-40-5': 'For the [MATH] meson it varies from 0.75 to unity over the interval from 3 GeV to the [MATH] mass, and for the meson it varies from 0.85 to unity over the interval from 4 GeV to the mass.', '1910.13404-2-41-0': '## fraction results', '1910.13404-2-42-0': 'The ratio of production fractions, [MATH], are shown as functions of [MATH] and [MATH] in Fig. [REF].', '1910.13404-2-42-1': 'There is little dependence on [MATH], but the decrease as a function of [MATH] is noticeable.', '1910.13404-2-43-0': 'To describe the [MATH] dependence we use an equation of the form [EQUATION] where [MATH] represents the overall normalization and contains the total global systematic uncertainty, thus, [MATH]; [MATH] is taken as 7.2 GeV, close to the average of the .', '1910.13404-2-43-1': 'The slopes, [MATH], are similar in size to those measured for the [MATH] meson fraction ratio as a function of [CITATION].', '1910.13404-2-43-2': 'Results of fits to the data using Eq. [REF] are listed in Table [REF].', '1910.13404-2-44-0': 'The average fractions in the interval [MATH]GeVare found by integrating over [MATH].', '1910.13404-2-44-1': 'To allow for facile changes to our results due to improved theoretical predictions, we provide the results for [EQUATION]', '1910.13404-2-44-2': 'Next we give the result on the fractions ratio [EQUATION] where the third uncertainty is due to the theoretical prediction of [MATH].', '1910.13404-2-44-3': 'To find [MATH] just double these numbers.', '1910.13404-2-45-0': 'We also measure the ratio of the [MATH] production fraction at [MATH] to that at [MATH].', '1910.13404-2-45-1': 'Figure [REF] shows the ratio as functions of and [MATH].', '1910.13404-2-45-2': 'Here most of the systematic uncertainties cancel.', '1910.13404-2-45-3': 'The integrated value of the ratio of [MATH] and [MATH] is measured as [MATH], consistent with no increase in the fraction ratio as a function of center-of-mass energy.', '1910.13404-2-46-0': 'The fraction with respect to inclusive [MATH]-hadron production can be derived from the information in previous LHCb [MATH]-hadron fraction papers Ref. [CITATION].', '1910.13404-2-46-1': 'There the measured values of the ratios of [MATH]-hadron fractions over the same range in terms of the [MATH]-hadron are for mesons ([MATH]) and baryons [EQUATION] where the uncertainties contain both statistical and systematic components added in quadrature.', '1910.13404-2-46-2': 'For the measurement of the [MATH] fraction at 7TeV, the dominant systematic uncertainty is from the lack of the knowledge of [MATH] at that time [CITATION]; here the value and uncertainty have been recalculated according to the latest value of [MATH] from the PDG [CITATION].', '1910.13404-2-47-0': 'Taking the sum of all the [MATH]-hadron fractions to be unity, and ignoring [MATH] here because it is so small, [EQUATION] where [MATH] is a correction factor derived in Ref. [CITATION] that accounts for heavier [MATH]-baryons, mainly the [MATH].', '1910.13404-2-47-1': 'Solving for [MATH] yields [EQUATION]', '1910.13404-2-47-2': 'We find that [EQUATION] where the first uncertainty is statistical, the second is systematic, and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-2-47-3': 'We also provide the result for [MATH], [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-2-48-0': '## The [MATH] production asymmetry', '1910.13404-2-49-0': 'The production asymmetries are measured in two different magnetic field configurations and then averaged.', '1910.13404-2-49-1': 'No significant asymmetry is observed in any intervals of [MATH] or [MATH].', '1910.13404-2-49-2': 'The results are summarized in Table [REF].', '1910.13404-2-50-0': 'Averaging the [MATH] production asymmetries over [MATH] and [MATH], we find [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-2-51-0': '# Systematic uncertainties', '1910.13404-2-52-0': 'Systematic uncertainties are separated into two categories: "global", which apply across the phase space, and "local", which are calculated in each two-dimensional [MATH] bin.', '1910.13404-2-52-1': 'These uncertainties are listed in Table [REF].', '1910.13404-2-53-0': 'First let us consider the [MATH] decay.', '1910.13404-2-53-1': 'The uncertainty due to the signal shape used to fit the [MATH] distribution is determined by changing the baseline signal shape, the sum of a double sided Crystal Ball function and a bifurcated Gaussian, to a kernel estimation.', '1910.13404-2-53-2': 'To find the shape of the combinatorial and misidentification backgrounds we use simulated inclusive samples of [MATH] events not including decays.', '1910.13404-2-53-3': 'A total of 500 samples are generated and different fits to the samples are performed to determine the possible uncertainty.', '1910.13404-2-53-4': 'This procedure is also used for the [MATH] measurement.', '1910.13404-2-53-5': 'We call contributions to the [MATH] mass spectrum "feed-down" contributions, occurring from other decay channels including [MATH], [MATH], and [MATH].', '1910.13404-2-53-6': 'The systematic uncertainty results from the uncertainties in their branching fractions.', '1910.13404-2-53-7': 'Different decay models for decays can change the [MATH] shape.', '1910.13404-2-53-8': 'We use the model of Ebert et al.[CITATION] for our baseline prediction.', '1910.13404-2-53-9': 'Then we also use the model by Kiselev [CITATION] to find the efficiencies and take half the difference as the systematic uncertainty.', '1910.13404-2-53-10': 'We also estimate the uncertainty due to the sensitivity to various selection requirements and simulation statistics.', '1910.13404-2-53-11': 'The muon identification efficiencies are determined from data using inclusive samples of decay where one of the muon candidates is not identified.', '1910.13404-2-53-12': 'The trigger efficiency is determined by using three independent samples of events, those that trigger on a [MATH], those that triggered on something else in the event, and those that trigger on both the and something else.', '1910.13404-2-53-13': 'These samples are then used to compute the trigger efficiencies in two-dimensional [MATH] and [MATH] bins.', '1910.13404-2-54-0': 'Next, we turn to the [MATH] modes.', '1910.13404-2-54-1': 'The efficiencies and their uncertainties for identifying pions and kaons are determined by using almost background free samples of [MATH] decays.', '1910.13404-2-54-2': 'The trigger and muon identification efficiencies, and their uncertainties, are obtained in the same manner as for the [MATH] mode.', '1910.13404-2-54-3': 'There are small systematic uncertainties related to efficiency estimates and the assumed [MATH] to [MATH] mixtures, as well as simulation statistics.', '1910.13404-2-54-4': 'Global systematic uncertainties include the hadron branching fractions listed in Table [REF], cross-feed corrections arising from and decays into [MATH] events, and a global hadron plus photon multiplicity requirement.', '1910.13404-2-54-5': 'The latter is evaluated with data.', '1910.13404-2-55-0': '# Conclusions', '1910.13404-2-56-0': 'In 7 and 13TeV[MATH] collisions the product of [MATH] with the relative fraction of mesons with respect to the sum of and mesons in the ranges [MATH] and [MATH] is found to be [EQUATION]', '1910.13404-2-56-1': 'We derive the product of [MATH] at the two energies as [EQUATION]', '1910.13404-2-56-2': 'Using the average of the theoretical prediction [MATH], where the uncertainty is given by the standard deviation derived from the distribution of the models, we determine [EQUATION] where the first uncertainties are statistical, the second systematic, and the third due to the theoretical prediction of [MATH].', '1910.13404-2-56-3': 'There is a small dependence on the transverse momentum of the meson, but no dependence on its pseudorapidity is observed.', '1910.13404-2-56-4': 'We also report [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-2-57-0': 'The ratio of fractions, [MATH], for 13TeV/7TeVis consistent with no increase in the fraction.', '1910.13404-2-57-1': 'Furthermore, using the assumption of no violation in the [MATH] decay, we find that the average asymmetry in [MATH] production is consistent with zero.', '1910.13404-2-57-2': 'The measurements are [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-2-58-0': 'These results are useful to extract absolute branching fractions for measurements, albeit with a relatively large uncertainty.', '1910.13404-2-58-1': 'They also challenge QCD calculations to predict the measured fractions and explain the consistency between the fractions measured at 7 and 13TeV[CITATION].'}

[['1910.13404-1-49-0', '1910.13404-2-49-0'], ['1910.13404-1-49-1', '1910.13404-2-49-1'], ['1910.13404-1-49-2', '1910.13404-2-49-2'], ['1910.13404-1-7-0', '1910.13404-2-7-0'], ['1910.13404-1-7-1', '1910.13404-2-7-1'], ['1910.13404-1-7-2', '1910.13404-2-7-2'], ['1910.13404-1-7-3', '1910.13404-2-7-3'], ['1910.13404-1-15-0', '1910.13404-2-15-0'], ['1910.13404-1-25-0', '1910.13404-2-25-0'], ['1910.13404-1-25-2', '1910.13404-2-25-2'], ['1910.13404-1-29-0', '1910.13404-2-29-0'], ['1910.13404-1-29-1', '1910.13404-2-29-1'], ['1910.13404-1-29-2', '1910.13404-2-29-2'], ['1910.13404-1-29-3', '1910.13404-2-29-3'], ['1910.13404-1-29-5', '1910.13404-2-29-5'], ['1910.13404-1-17-0', '1910.13404-2-17-0'], ['1910.13404-1-17-1', '1910.13404-2-17-1'], ['1910.13404-1-17-2', '1910.13404-2-17-2'], ['1910.13404-1-17-3', '1910.13404-2-17-3'], ['1910.13404-1-45-1', '1910.13404-2-45-1'], ['1910.13404-1-45-2', '1910.13404-2-45-2'], ['1910.13404-1-45-3', '1910.13404-2-45-3'], ['1910.13404-1-22-1', '1910.13404-2-22-1'], ['1910.13404-1-22-2', '1910.13404-2-22-2'], ['1910.13404-1-46-0', '1910.13404-2-46-0'], ['1910.13404-1-46-1', '1910.13404-2-46-1'], ['1910.13404-1-46-2', '1910.13404-2-46-2'], ['1910.13404-1-5-0', '1910.13404-2-5-0'], ['1910.13404-1-5-1', '1910.13404-2-5-1'], ['1910.13404-1-5-2', '1910.13404-2-5-2'], ['1910.13404-1-5-3', '1910.13404-2-5-3'], ['1910.13404-1-5-4', '1910.13404-2-5-4'], ['1910.13404-1-5-5', '1910.13404-2-5-5'], ['1910.13404-1-5-6', '1910.13404-2-5-6'], ['1910.13404-1-5-7', '1910.13404-2-5-7'], ['1910.13404-1-5-8', '1910.13404-2-5-8'], ['1910.13404-1-6-0', '1910.13404-2-6-0'], ['1910.13404-1-6-1', '1910.13404-2-6-1'], ['1910.13404-1-6-2', '1910.13404-2-6-2'], ['1910.13404-1-6-3', '1910.13404-2-6-3'], ['1910.13404-1-6-4', '1910.13404-2-6-4'], ['1910.13404-1-6-5', '1910.13404-2-6-5'], ['1910.13404-1-6-6', '1910.13404-2-6-6'], ['1910.13404-1-31-0', '1910.13404-2-31-0'], ['1910.13404-1-31-1', '1910.13404-2-31-1'], ['1910.13404-1-31-3', '1910.13404-2-31-3'], ['1910.13404-1-31-4', '1910.13404-2-31-4'], ['1910.13404-1-31-5', '1910.13404-2-31-5'], ['1910.13404-1-57-0', '1910.13404-2-57-0'], ['1910.13404-1-57-1', '1910.13404-2-57-1'], ['1910.13404-1-57-2', '1910.13404-2-57-2'], ['1910.13404-1-40-0', '1910.13404-2-40-0'], ['1910.13404-1-40-1', '1910.13404-2-40-1'], ['1910.13404-1-40-2', '1910.13404-2-40-2'], ['1910.13404-1-40-3', '1910.13404-2-40-3'], ['1910.13404-1-40-4', '1910.13404-2-40-4'], ['1910.13404-1-40-5', '1910.13404-2-40-5'], ['1910.13404-1-53-0', '1910.13404-2-53-0'], ['1910.13404-1-53-4', '1910.13404-2-53-4'], ['1910.13404-1-53-5', '1910.13404-2-53-5'], ['1910.13404-1-53-6', '1910.13404-2-53-6'], ['1910.13404-1-53-7', '1910.13404-2-53-7'], ['1910.13404-1-53-10', '1910.13404-2-53-10'], ['1910.13404-1-53-11', '1910.13404-2-53-11'], ['1910.13404-1-53-12', '1910.13404-2-53-12'], ['1910.13404-1-53-13', '1910.13404-2-53-13'], ['1910.13404-1-32-0', '1910.13404-2-32-0'], ['1910.13404-1-52-0', '1910.13404-2-52-0'], ['1910.13404-1-52-1', '1910.13404-2-52-1'], ['1910.13404-1-33-0', '1910.13404-2-33-0'], ['1910.13404-1-33-1', '1910.13404-2-33-1'], ['1910.13404-1-33-2', '1910.13404-2-33-2'], ['1910.13404-1-33-3', '1910.13404-2-33-3'], ['1910.13404-1-14-0', '1910.13404-2-14-0'], ['1910.13404-1-14-1', '1910.13404-2-14-1'], ['1910.13404-1-14-2', '1910.13404-2-14-2'], ['1910.13404-1-14-3', '1910.13404-2-14-3'], ['1910.13404-1-56-0', '1910.13404-2-56-0'], ['1910.13404-1-56-1', '1910.13404-2-56-2'], ['1910.13404-1-11-0', '1910.13404-2-11-0'], ['1910.13404-1-11-1', '1910.13404-2-11-1'], ['1910.13404-1-11-2', '1910.13404-2-11-2'], ['1910.13404-1-11-4', '1910.13404-2-11-4'], ['1910.13404-1-11-5', '1910.13404-2-11-5'], ['1910.13404-1-8-0', '1910.13404-2-8-0'], ['1910.13404-1-8-1', '1910.13404-2-8-1'], ['1910.13404-1-8-2', '1910.13404-2-8-2'], ['1910.13404-1-8-3', '1910.13404-2-8-3'], ['1910.13404-1-13-0', '1910.13404-2-13-0'], ['1910.13404-1-27-0', '1910.13404-2-27-0'], ['1910.13404-1-27-1', '1910.13404-2-27-1'], ['1910.13404-1-27-2', '1910.13404-2-27-2'], ['1910.13404-1-27-4', '1910.13404-2-27-4'], ['1910.13404-1-27-5', '1910.13404-2-27-5'], ['1910.13404-1-27-6', '1910.13404-2-27-6'], ['1910.13404-1-27-7', '1910.13404-2-27-7'], ['1910.13404-1-27-8', '1910.13404-2-27-8'], ['1910.13404-1-27-9', '1910.13404-2-27-9'], ['1910.13404-1-27-10', '1910.13404-2-27-10'], ['1910.13404-1-34-0', '1910.13404-2-34-0'], ['1910.13404-1-34-1', '1910.13404-2-34-1'], ['1910.13404-1-34-2', '1910.13404-2-34-2'], ['1910.13404-1-34-3', '1910.13404-2-34-3'], ['1910.13404-1-43-0', '1910.13404-2-43-0'], ['1910.13404-1-43-1', '1910.13404-2-43-1'], ['1910.13404-1-43-2', '1910.13404-2-43-2'], ['1910.13404-1-35-0', '1910.13404-2-35-0'], ['1910.13404-1-35-1', '1910.13404-2-35-1'], ['1910.13404-1-35-2', '1910.13404-2-35-2'], ['1910.13404-1-35-3', '1910.13404-2-35-3'], ['1910.13404-1-10-0', '1910.13404-2-10-0'], ['1910.13404-1-10-1', '1910.13404-2-10-1'], ['1910.13404-1-10-2', '1910.13404-2-10-2'], ['1910.13404-1-10-3', '1910.13404-2-10-3'], ['1910.13404-1-58-0', '1910.13404-2-58-0'], ['1910.13404-1-20-0', '1910.13404-2-20-0'], ['1910.13404-1-20-1', '1910.13404-2-20-1'], ['1910.13404-1-20-2', '1910.13404-2-20-2'], ['1910.13404-1-20-3', '1910.13404-2-20-3'], ['1910.13404-1-9-2', '1910.13404-2-9-1'], ['1910.13404-1-37-0', '1910.13404-2-37-0'], ['1910.13404-1-37-1', '1910.13404-2-37-1'], ['1910.13404-1-37-2', '1910.13404-2-37-2'], ['1910.13404-1-0-0', '1910.13404-2-0-0'], ['1910.13404-1-0-1', '1910.13404-2-0-1'], ['1910.13404-1-0-2', '1910.13404-2-0-2'], ['1910.13404-1-21-0', '1910.13404-2-21-0'], ['1910.13404-1-21-1', '1910.13404-2-21-1'], ['1910.13404-1-16-0', '1910.13404-2-16-0'], ['1910.13404-1-16-1', '1910.13404-2-16-1'], ['1910.13404-1-16-2', '1910.13404-2-16-2'], ['1910.13404-1-16-3', '1910.13404-2-16-3'], ['1910.13404-1-16-4', '1910.13404-2-16-4'], ['1910.13404-1-16-5', '1910.13404-2-16-5'], ['1910.13404-1-16-6', '1910.13404-2-16-6'], ['1910.13404-1-26-0', '1910.13404-2-26-0'], ['1910.13404-1-26-1', '1910.13404-2-26-1'], ['1910.13404-1-42-0', '1910.13404-2-42-0'], ['1910.13404-1-42-1', '1910.13404-2-42-1'], ['1910.13404-1-44-0', '1910.13404-2-44-0'], ['1910.13404-1-44-1', '1910.13404-2-44-1'], ['1910.13404-1-44-2', '1910.13404-2-44-2'], ['1910.13404-1-44-3', '1910.13404-2-44-3'], ['1910.13404-1-47-0', '1910.13404-2-47-0'], ['1910.13404-1-47-1', '1910.13404-2-47-1'], ['1910.13404-1-54-0', '1910.13404-2-54-0'], ['1910.13404-1-54-1', '1910.13404-2-54-1'], ['1910.13404-1-54-2', '1910.13404-2-54-2'], ['1910.13404-1-54-3', '1910.13404-2-54-3'], ['1910.13404-1-54-4', '1910.13404-2-54-4'], ['1910.13404-1-54-5', '1910.13404-2-54-5'], ['1910.13404-1-23-0', '1910.13404-2-23-0'], ['1910.13404-1-23-2', '1910.13404-2-23-2'], ['1910.13404-1-50-0', '1910.13404-2-50-0'], ['1910.13404-2-58-0', '1910.13404-3-58-0'], ['1910.13404-2-58-1', '1910.13404-3-58-1'], ['1910.13404-2-26-0', '1910.13404-3-26-0'], ['1910.13404-2-26-1', '1910.13404-3-26-1'], ['1910.13404-2-26-2', '1910.13404-3-26-2'], ['1910.13404-2-44-0', '1910.13404-3-44-0'], ['1910.13404-2-44-1', '1910.13404-3-44-1'], ['1910.13404-2-44-2', '1910.13404-3-44-2'], ['1910.13404-2-44-3', '1910.13404-3-44-3'], ['1910.13404-2-53-0', '1910.13404-3-53-0'], ['1910.13404-2-53-1', '1910.13404-3-53-1'], ['1910.13404-2-53-2', '1910.13404-3-53-2'], ['1910.13404-2-53-3', '1910.13404-3-53-3'], ['1910.13404-2-53-4', '1910.13404-3-53-4'], ['1910.13404-2-53-5', '1910.13404-3-53-5'], ['1910.13404-2-53-6', '1910.13404-3-53-6'], ['1910.13404-2-53-7', '1910.13404-3-53-7'], ['1910.13404-2-53-8', '1910.13404-3-53-8'], ['1910.13404-2-53-9', '1910.13404-3-53-9'], ['1910.13404-2-53-10', '1910.13404-3-53-10'], ['1910.13404-2-53-11', '1910.13404-3-53-11'], ['1910.13404-2-53-12', '1910.13404-3-53-12'], ['1910.13404-2-53-13', '1910.13404-3-53-13'], ['1910.13404-2-5-0', '1910.13404-3-5-0'], ['1910.13404-2-5-1', '1910.13404-3-5-1'], ['1910.13404-2-5-2', '1910.13404-3-5-2'], ['1910.13404-2-5-3', '1910.13404-3-5-3'], ['1910.13404-2-5-4', '1910.13404-3-5-4'], ['1910.13404-2-5-5', '1910.13404-3-5-5'], ['1910.13404-2-5-6', '1910.13404-3-5-6'], ['1910.13404-2-5-7', '1910.13404-3-5-7'], ['1910.13404-2-5-8', '1910.13404-3-5-8'], ['1910.13404-2-9-0', '1910.13404-3-9-0'], ['1910.13404-2-9-1', '1910.13404-3-9-1'], ['1910.13404-2-10-0', '1910.13404-3-10-0'], ['1910.13404-2-10-1', '1910.13404-3-10-1'], ['1910.13404-2-10-2', '1910.13404-3-10-2'], ['1910.13404-2-10-3', '1910.13404-3-10-3'], ['1910.13404-2-45-0', '1910.13404-3-45-0'], ['1910.13404-2-45-1', '1910.13404-3-45-1'], ['1910.13404-2-45-2', '1910.13404-3-45-2'], ['1910.13404-2-45-3', '1910.13404-3-45-3'], ['1910.13404-2-0-0', '1910.13404-3-0-0'], ['1910.13404-2-0-1', '1910.13404-3-0-1'], ['1910.13404-2-0-2', '1910.13404-3-0-2'], ['1910.13404-2-37-0', '1910.13404-3-37-0'], ['1910.13404-2-37-1', '1910.13404-3-37-1'], ['1910.13404-2-37-2', '1910.13404-3-37-2'], ['1910.13404-2-46-0', '1910.13404-3-46-0'], ['1910.13404-2-46-1', '1910.13404-3-46-1'], ['1910.13404-2-46-2', '1910.13404-3-46-2'], ['1910.13404-2-52-0', '1910.13404-3-52-0'], ['1910.13404-2-52-1', '1910.13404-3-52-1'], ['1910.13404-2-49-0', '1910.13404-3-49-0'], ['1910.13404-2-49-1', '1910.13404-3-49-1'], ['1910.13404-2-49-2', '1910.13404-3-49-2'], ['1910.13404-2-57-0', '1910.13404-3-57-0'], ['1910.13404-2-57-1', '1910.13404-3-57-1'], ['1910.13404-2-57-2', '1910.13404-3-57-2'], ['1910.13404-2-40-0', '1910.13404-3-40-0'], ['1910.13404-2-40-1', '1910.13404-3-40-1'], ['1910.13404-2-40-2', '1910.13404-3-40-2'], ['1910.13404-2-40-3', '1910.13404-3-40-3'], ['1910.13404-2-40-4', '1910.13404-3-40-4'], ['1910.13404-2-40-5', '1910.13404-3-40-5'], ['1910.13404-2-20-0', '1910.13404-3-20-0'], ['1910.13404-2-20-1', '1910.13404-3-20-1'], ['1910.13404-2-20-2', '1910.13404-3-20-2'], ['1910.13404-2-20-3', '1910.13404-3-20-3'], ['1910.13404-2-42-0', '1910.13404-3-42-0'], ['1910.13404-2-42-1', '1910.13404-3-42-1'], ['1910.13404-2-50-0', '1910.13404-3-50-0'], ['1910.13404-2-29-0', '1910.13404-3-29-0'], ['1910.13404-2-29-1', '1910.13404-3-29-1'], ['1910.13404-2-29-2', '1910.13404-3-29-2'], ['1910.13404-2-29-3', '1910.13404-3-29-3'], ['1910.13404-2-29-4', '1910.13404-3-29-4'], ['1910.13404-2-29-5', '1910.13404-3-29-5'], ['1910.13404-2-56-0', '1910.13404-3-56-0'], ['1910.13404-2-56-1', '1910.13404-3-56-1'], ['1910.13404-2-56-2', '1910.13404-3-56-2'], ['1910.13404-2-56-3', '1910.13404-3-56-3'], ['1910.13404-2-56-4', '1910.13404-3-56-4'], ['1910.13404-2-24-0', '1910.13404-3-24-0'], ['1910.13404-2-24-1', '1910.13404-3-24-1'], ['1910.13404-2-27-0', '1910.13404-3-27-0'], ['1910.13404-2-27-1', '1910.13404-3-27-1'], ['1910.13404-2-27-2', '1910.13404-3-27-2'], ['1910.13404-2-27-3', '1910.13404-3-27-3'], ['1910.13404-2-27-4', '1910.13404-3-27-4'], ['1910.13404-2-27-5', '1910.13404-3-27-5'], ['1910.13404-2-27-6', '1910.13404-3-27-6'], ['1910.13404-2-27-7', '1910.13404-3-27-7'], ['1910.13404-2-27-8', '1910.13404-3-27-8'], ['1910.13404-2-27-9', '1910.13404-3-27-9'], ['1910.13404-2-27-10', '1910.13404-3-27-10'], ['1910.13404-2-14-0', '1910.13404-3-14-0'], ['1910.13404-2-14-1', '1910.13404-3-14-1'], ['1910.13404-2-14-2', '1910.13404-3-14-2'], ['1910.13404-2-14-3', '1910.13404-3-14-3'], ['1910.13404-2-14-4', '1910.13404-3-14-4'], ['1910.13404-2-14-5', '1910.13404-3-14-5'], ['1910.13404-2-35-0', '1910.13404-3-35-0'], ['1910.13404-2-35-1', '1910.13404-3-35-1'], ['1910.13404-2-35-2', '1910.13404-3-35-2'], ['1910.13404-2-35-3', '1910.13404-3-35-3'], ['1910.13404-2-47-0', '1910.13404-3-47-0'], ['1910.13404-2-47-1', '1910.13404-3-47-1'], ['1910.13404-2-47-2', '1910.13404-3-47-2'], ['1910.13404-2-47-3', '1910.13404-3-47-3'], ['1910.13404-2-25-0', '1910.13404-3-25-0'], ['1910.13404-2-25-1', '1910.13404-3-25-1'], ['1910.13404-2-25-2', '1910.13404-3-25-2'], ['1910.13404-2-15-0', '1910.13404-3-15-0'], ['1910.13404-2-21-0', '1910.13404-3-21-0'], ['1910.13404-2-21-1', '1910.13404-3-21-1'], ['1910.13404-2-54-0', '1910.13404-3-54-0'], ['1910.13404-2-54-1', '1910.13404-3-54-1'], ['1910.13404-2-54-2', '1910.13404-3-54-2'], ['1910.13404-2-54-3', '1910.13404-3-54-3'], ['1910.13404-2-54-4', '1910.13404-3-54-4'], ['1910.13404-2-54-5', '1910.13404-3-54-5'], ['1910.13404-2-16-0', '1910.13404-3-16-0'], ['1910.13404-2-16-1', '1910.13404-3-16-1'], ['1910.13404-2-16-2', '1910.13404-3-16-2'], ['1910.13404-2-16-3', '1910.13404-3-16-3'], ['1910.13404-2-16-4', '1910.13404-3-16-4'], ['1910.13404-2-16-5', '1910.13404-3-16-5'], ['1910.13404-2-16-6', '1910.13404-3-16-6'], ['1910.13404-2-43-0', '1910.13404-3-43-0'], ['1910.13404-2-43-1', '1910.13404-3-43-1'], ['1910.13404-2-43-2', '1910.13404-3-43-2'], ['1910.13404-2-31-0', '1910.13404-3-31-0'], ['1910.13404-2-31-1', '1910.13404-3-31-1'], ['1910.13404-2-31-2', '1910.13404-3-31-2'], ['1910.13404-2-31-3', '1910.13404-3-31-3'], ['1910.13404-2-31-4', '1910.13404-3-31-4'], ['1910.13404-2-31-5', '1910.13404-3-31-5'], ['1910.13404-2-13-0', '1910.13404-3-13-0'], ['1910.13404-2-13-1', '1910.13404-3-13-1'], ['1910.13404-2-6-0', '1910.13404-3-6-0'], ['1910.13404-2-6-1', '1910.13404-3-6-1'], ['1910.13404-2-6-2', '1910.13404-3-6-2'], ['1910.13404-2-6-3', '1910.13404-3-6-3'], ['1910.13404-2-6-4', '1910.13404-3-6-4'], ['1910.13404-2-6-5', '1910.13404-3-6-5'], ['1910.13404-2-6-6', '1910.13404-3-6-6'], ['1910.13404-2-8-0', '1910.13404-3-8-0'], ['1910.13404-2-8-1', '1910.13404-3-8-1'], ['1910.13404-2-8-2', '1910.13404-3-8-2'], ['1910.13404-2-32-0', '1910.13404-3-32-0'], ['1910.13404-2-11-0', '1910.13404-3-11-0'], ['1910.13404-2-11-1', '1910.13404-3-11-1'], ['1910.13404-2-11-2', '1910.13404-3-11-2'], ['1910.13404-2-11-3', '1910.13404-3-11-3'], ['1910.13404-2-11-4', '1910.13404-3-11-4'], ['1910.13404-2-11-5', '1910.13404-3-11-5'], ['1910.13404-2-22-0', '1910.13404-3-22-0'], ['1910.13404-2-22-1', '1910.13404-3-22-1'], ['1910.13404-2-22-2', '1910.13404-3-22-2'], ['1910.13404-2-33-0', '1910.13404-3-33-0'], ['1910.13404-2-33-1', '1910.13404-3-33-1'], ['1910.13404-2-33-2', '1910.13404-3-33-2'], ['1910.13404-2-33-3', '1910.13404-3-33-3'], ['1910.13404-2-34-0', '1910.13404-3-34-0'], ['1910.13404-2-34-1', '1910.13404-3-34-1'], ['1910.13404-2-34-2', '1910.13404-3-34-2'], ['1910.13404-2-34-3', '1910.13404-3-34-3'], ['1910.13404-2-23-0', '1910.13404-3-23-0'], ['1910.13404-2-23-1', '1910.13404-3-23-1'], ['1910.13404-2-23-2', '1910.13404-3-23-2'], ['1910.13404-2-17-0', '1910.13404-3-17-0'], ['1910.13404-2-17-1', '1910.13404-3-17-1'], ['1910.13404-2-17-2', '1910.13404-3-17-2'], ['1910.13404-2-17-3', '1910.13404-3-17-3'], ['1910.13404-2-7-0', '1910.13404-3-7-0'], ['1910.13404-2-7-1', '1910.13404-3-7-1'], ['1910.13404-2-7-2', '1910.13404-3-7-2'], ['1910.13404-2-7-3', '1910.13404-3-7-3'], ['1910.13404-3-54-0', '1910.13404-4-54-0'], ['1910.13404-3-54-1', '1910.13404-4-54-1'], ['1910.13404-3-54-2', '1910.13404-4-54-2'], ['1910.13404-3-54-3', '1910.13404-4-54-3'], ['1910.13404-3-54-4', '1910.13404-4-54-4'], ['1910.13404-3-54-5', '1910.13404-4-54-5'], ['1910.13404-3-56-0', '1910.13404-4-56-0'], ['1910.13404-3-56-1', '1910.13404-4-56-1'], ['1910.13404-3-56-2', '1910.13404-4-56-2'], ['1910.13404-3-56-3', '1910.13404-4-56-3'], ['1910.13404-3-56-4', '1910.13404-4-56-4'], ['1910.13404-3-7-0', '1910.13404-4-7-0'], ['1910.13404-3-7-1', '1910.13404-4-7-1'], ['1910.13404-3-7-2', '1910.13404-4-7-2'], ['1910.13404-3-7-3', '1910.13404-4-7-3'], ['1910.13404-3-8-0', '1910.13404-4-8-0'], ['1910.13404-3-8-1', '1910.13404-4-8-1'], ['1910.13404-3-8-2', '1910.13404-4-8-2'], ['1910.13404-3-26-0', '1910.13404-4-26-0'], ['1910.13404-3-26-1', '1910.13404-4-26-1'], ['1910.13404-3-26-2', '1910.13404-4-26-2'], ['1910.13404-3-35-0', '1910.13404-4-35-0'], ['1910.13404-3-35-1', '1910.13404-4-35-1'], ['1910.13404-3-35-2', '1910.13404-4-35-2'], ['1910.13404-3-35-3', '1910.13404-4-35-3'], ['1910.13404-3-40-0', '1910.13404-4-40-0'], ['1910.13404-3-40-1', '1910.13404-4-40-1'], ['1910.13404-3-40-2', '1910.13404-4-40-2'], ['1910.13404-3-40-3', '1910.13404-4-40-3'], ['1910.13404-3-40-4', '1910.13404-4-40-4'], ['1910.13404-3-40-5', '1910.13404-4-40-5'], ['1910.13404-3-33-0', '1910.13404-4-33-0'], ['1910.13404-3-33-1', '1910.13404-4-33-1'], ['1910.13404-3-33-2', '1910.13404-4-33-2'], ['1910.13404-3-33-3', '1910.13404-4-33-3'], ['1910.13404-3-46-0', '1910.13404-4-46-0'], ['1910.13404-3-46-1', '1910.13404-4-46-1'], ['1910.13404-3-46-2', '1910.13404-4-46-2'], ['1910.13404-3-58-0', '1910.13404-4-58-0'], ['1910.13404-3-58-1', '1910.13404-4-58-1'], ['1910.13404-3-27-0', '1910.13404-4-27-0'], ['1910.13404-3-27-1', '1910.13404-4-27-1'], ['1910.13404-3-27-2', '1910.13404-4-27-2'], ['1910.13404-3-27-3', '1910.13404-4-27-3'], ['1910.13404-3-27-4', '1910.13404-4-27-4'], ['1910.13404-3-27-5', '1910.13404-4-27-5'], ['1910.13404-3-27-6', '1910.13404-4-27-6'], ['1910.13404-3-27-7', '1910.13404-4-27-7'], ['1910.13404-3-27-8', '1910.13404-4-27-8'], ['1910.13404-3-27-9', '1910.13404-4-27-9'], ['1910.13404-3-27-10', '1910.13404-4-27-10'], ['1910.13404-3-0-0', '1910.13404-4-0-0'], ['1910.13404-3-0-1', '1910.13404-4-0-1'], ['1910.13404-3-0-2', '1910.13404-4-0-2'], ['1910.13404-3-34-0', '1910.13404-4-34-0'], ['1910.13404-3-34-1', '1910.13404-4-34-1'], ['1910.13404-3-34-2', '1910.13404-4-34-2'], ['1910.13404-3-34-3', '1910.13404-4-34-3'], ['1910.13404-3-57-0', '1910.13404-4-57-0'], ['1910.13404-3-57-1', '1910.13404-4-57-1'], ['1910.13404-3-57-2', '1910.13404-4-57-2'], ['1910.13404-3-17-0', '1910.13404-4-17-0'], ['1910.13404-3-17-1', '1910.13404-4-17-1'], ['1910.13404-3-17-2', '1910.13404-4-17-2'], ['1910.13404-3-17-3', '1910.13404-4-17-3'], ['1910.13404-3-15-0', '1910.13404-4-15-0'], ['1910.13404-3-9-0', '1910.13404-4-9-0'], ['1910.13404-3-9-1', '1910.13404-4-9-1'], ['1910.13404-3-21-0', '1910.13404-4-21-0'], ['1910.13404-3-21-1', '1910.13404-4-21-1'], ['1910.13404-3-10-0', '1910.13404-4-10-0'], ['1910.13404-3-10-1', '1910.13404-4-10-1'], ['1910.13404-3-10-2', '1910.13404-4-10-2'], ['1910.13404-3-10-3', '1910.13404-4-10-3'], ['1910.13404-3-31-0', '1910.13404-4-31-0'], ['1910.13404-3-31-1', '1910.13404-4-31-1'], ['1910.13404-3-31-2', '1910.13404-4-31-2'], ['1910.13404-3-31-3', '1910.13404-4-31-3'], ['1910.13404-3-31-4', '1910.13404-4-31-4'], ['1910.13404-3-31-5', '1910.13404-4-31-5'], ['1910.13404-3-24-0', '1910.13404-4-24-0'], ['1910.13404-3-24-1', '1910.13404-4-24-1'], ['1910.13404-3-45-0', '1910.13404-4-45-0'], ['1910.13404-3-45-1', '1910.13404-4-45-1'], ['1910.13404-3-45-2', '1910.13404-4-45-2'], ['1910.13404-3-45-3', '1910.13404-4-45-3'], ['1910.13404-3-22-0', '1910.13404-4-22-0'], ['1910.13404-3-22-1', '1910.13404-4-22-1'], ['1910.13404-3-22-2', '1910.13404-4-22-2'], ['1910.13404-3-6-0', '1910.13404-4-6-0'], ['1910.13404-3-6-1', '1910.13404-4-6-1'], ['1910.13404-3-6-2', '1910.13404-4-6-2'], ['1910.13404-3-6-3', '1910.13404-4-6-3'], ['1910.13404-3-6-4', '1910.13404-4-6-4'], ['1910.13404-3-6-5', '1910.13404-4-6-5'], ['1910.13404-3-6-6', '1910.13404-4-6-6'], ['1910.13404-3-43-0', '1910.13404-4-43-0'], ['1910.13404-3-43-1', '1910.13404-4-43-1'], ['1910.13404-3-43-2', '1910.13404-4-43-2'], ['1910.13404-3-50-0', '1910.13404-4-50-0'], ['1910.13404-3-29-0', '1910.13404-4-29-0'], ['1910.13404-3-29-1', '1910.13404-4-29-1'], ['1910.13404-3-29-2', '1910.13404-4-29-2'], ['1910.13404-3-29-3', '1910.13404-4-29-3'], ['1910.13404-3-29-4', '1910.13404-4-29-4'], ['1910.13404-3-29-5', '1910.13404-4-29-5'], ['1910.13404-3-11-0', '1910.13404-4-11-0'], ['1910.13404-3-11-1', '1910.13404-4-11-1'], ['1910.13404-3-11-2', '1910.13404-4-11-2'], ['1910.13404-3-11-3', '1910.13404-4-11-3'], ['1910.13404-3-11-4', '1910.13404-4-11-4'], ['1910.13404-3-11-5', '1910.13404-4-11-5'], ['1910.13404-3-16-0', '1910.13404-4-16-0'], ['1910.13404-3-16-1', '1910.13404-4-16-1'], ['1910.13404-3-16-2', '1910.13404-4-16-2'], ['1910.13404-3-16-3', '1910.13404-4-16-3'], ['1910.13404-3-16-4', '1910.13404-4-16-4'], ['1910.13404-3-16-5', '1910.13404-4-16-5'], ['1910.13404-3-16-6', '1910.13404-4-16-6'], ['1910.13404-3-23-0', '1910.13404-4-23-0'], ['1910.13404-3-23-1', '1910.13404-4-23-1'], ['1910.13404-3-23-2', '1910.13404-4-23-2'], ['1910.13404-3-32-0', '1910.13404-4-32-0'], ['1910.13404-3-20-0', '1910.13404-4-20-0'], ['1910.13404-3-20-1', '1910.13404-4-20-1'], ['1910.13404-3-20-2', '1910.13404-4-20-2'], ['1910.13404-3-20-3', '1910.13404-4-20-3'], ['1910.13404-3-42-0', '1910.13404-4-42-0'], ['1910.13404-3-42-1', '1910.13404-4-42-1'], ['1910.13404-3-25-0', '1910.13404-4-25-0'], ['1910.13404-3-25-1', '1910.13404-4-25-1'], ['1910.13404-3-25-2', '1910.13404-4-25-2'], ['1910.13404-3-13-0', '1910.13404-4-13-0'], ['1910.13404-3-13-1', '1910.13404-4-13-1'], ['1910.13404-3-52-0', '1910.13404-4-52-0'], ['1910.13404-3-52-1', '1910.13404-4-52-1'], ['1910.13404-3-14-0', '1910.13404-4-14-0'], ['1910.13404-3-14-1', '1910.13404-4-14-1'], ['1910.13404-3-14-2', '1910.13404-4-14-2'], ['1910.13404-3-14-3', '1910.13404-4-14-3'], ['1910.13404-3-14-4', '1910.13404-4-14-4'], ['1910.13404-3-14-5', '1910.13404-4-14-5'], ['1910.13404-3-53-0', '1910.13404-4-53-0'], ['1910.13404-3-53-1', '1910.13404-4-53-1'], ['1910.13404-3-53-2', '1910.13404-4-53-2'], ['1910.13404-3-53-3', '1910.13404-4-53-3'], ['1910.13404-3-53-4', '1910.13404-4-53-4'], ['1910.13404-3-53-5', '1910.13404-4-53-5'], ['1910.13404-3-53-6', '1910.13404-4-53-6'], ['1910.13404-3-53-7', '1910.13404-4-53-7'], ['1910.13404-3-53-8', '1910.13404-4-53-8'], ['1910.13404-3-53-9', '1910.13404-4-53-9'], ['1910.13404-3-53-10', '1910.13404-4-53-10'], ['1910.13404-3-53-11', '1910.13404-4-53-11'], ['1910.13404-3-53-12', '1910.13404-4-53-12'], ['1910.13404-3-53-13', '1910.13404-4-53-13'], ['1910.13404-3-5-0', '1910.13404-4-5-0'], ['1910.13404-3-5-1', '1910.13404-4-5-1'], ['1910.13404-3-5-2', '1910.13404-4-5-2'], ['1910.13404-3-5-3', '1910.13404-4-5-3'], ['1910.13404-3-5-4', '1910.13404-4-5-4'], ['1910.13404-3-5-5', '1910.13404-4-5-5'], ['1910.13404-3-5-6', '1910.13404-4-5-6'], ['1910.13404-3-5-7', '1910.13404-4-5-7'], ['1910.13404-3-5-8', '1910.13404-4-5-8'], ['1910.13404-3-37-0', '1910.13404-4-37-0'], ['1910.13404-3-37-1', '1910.13404-4-37-1'], ['1910.13404-3-37-2', '1910.13404-4-37-2'], ['1910.13404-3-44-0', '1910.13404-4-44-0'], ['1910.13404-3-44-1', '1910.13404-4-44-1'], ['1910.13404-3-44-2', '1910.13404-4-44-2'], ['1910.13404-3-44-3', '1910.13404-4-44-3'], ['1910.13404-3-47-0', '1910.13404-4-47-0'], ['1910.13404-3-47-1', '1910.13404-4-47-1'], ['1910.13404-3-47-2', '1910.13404-4-47-2'], ['1910.13404-3-47-3', '1910.13404-4-47-3'], ['1910.13404-3-49-0', '1910.13404-4-49-0'], ['1910.13404-3-49-1', '1910.13404-4-49-1'], ['1910.13404-3-49-2', '1910.13404-4-49-2'], ['1910.13404-1-25-1', '1910.13404-2-25-1'], ['1910.13404-1-29-4', '1910.13404-2-29-4'], ['1910.13404-1-45-0', '1910.13404-2-45-0'], ['1910.13404-1-22-0', '1910.13404-2-22-0'], ['1910.13404-1-31-2', '1910.13404-2-31-2'], ['1910.13404-1-53-1', '1910.13404-2-53-1'], ['1910.13404-1-53-2', '1910.13404-2-53-2'], ['1910.13404-1-53-3', '1910.13404-2-53-3'], ['1910.13404-1-53-8', '1910.13404-2-53-8'], ['1910.13404-1-53-9', '1910.13404-2-53-9'], ['1910.13404-1-11-3', '1910.13404-2-11-3'], ['1910.13404-1-13-1', '1910.13404-2-13-1'], ['1910.13404-1-27-3', '1910.13404-2-27-3'], ['1910.13404-1-24-0', '1910.13404-2-24-0'], ['1910.13404-1-24-1', '1910.13404-2-24-1'], ['1910.13404-1-47-2', '1910.13404-2-47-2'], ['1910.13404-1-47-2', '1910.13404-2-47-3'], ['1910.13404-1-23-1', '1910.13404-2-23-1'], ['1910.13404-1-14-4', '1910.13404-2-14-4'], ['1910.13404-1-56-2', '1910.13404-2-56-3'], ['1910.13404-1-58-1', '1910.13404-2-58-1'], ['1910.13404-1-9-0', '1910.13404-2-9-0'], ['1910.13404-1-26-2', '1910.13404-2-26-2']]

[['1910.13404-1-49-0', '1910.13404-2-49-0'], ['1910.13404-1-49-1', '1910.13404-2-49-1'], ['1910.13404-1-49-2', '1910.13404-2-49-2'], ['1910.13404-1-7-0', '1910.13404-2-7-0'], ['1910.13404-1-7-1', '1910.13404-2-7-1'], ['1910.13404-1-7-2', '1910.13404-2-7-2'], ['1910.13404-1-7-3', '1910.13404-2-7-3'], ['1910.13404-1-15-0', '1910.13404-2-15-0'], ['1910.13404-1-25-0', '1910.13404-2-25-0'], ['1910.13404-1-25-2', '1910.13404-2-25-2'], ['1910.13404-1-29-0', '1910.13404-2-29-0'], ['1910.13404-1-29-1', '1910.13404-2-29-1'], ['1910.13404-1-29-2', '1910.13404-2-29-2'], ['1910.13404-1-29-3', '1910.13404-2-29-3'], ['1910.13404-1-29-5', '1910.13404-2-29-5'], ['1910.13404-1-17-0', '1910.13404-2-17-0'], ['1910.13404-1-17-1', '1910.13404-2-17-1'], ['1910.13404-1-17-2', '1910.13404-2-17-2'], ['1910.13404-1-17-3', '1910.13404-2-17-3'], ['1910.13404-1-45-1', '1910.13404-2-45-1'], ['1910.13404-1-45-2', '1910.13404-2-45-2'], ['1910.13404-1-45-3', '1910.13404-2-45-3'], ['1910.13404-1-22-1', '1910.13404-2-22-1'], ['1910.13404-1-22-2', '1910.13404-2-22-2'], ['1910.13404-1-46-0', '1910.13404-2-46-0'], ['1910.13404-1-46-1', '1910.13404-2-46-1'], ['1910.13404-1-46-2', '1910.13404-2-46-2'], ['1910.13404-1-5-0', '1910.13404-2-5-0'], ['1910.13404-1-5-1', '1910.13404-2-5-1'], ['1910.13404-1-5-2', '1910.13404-2-5-2'], ['1910.13404-1-5-3', '1910.13404-2-5-3'], ['1910.13404-1-5-4', '1910.13404-2-5-4'], ['1910.13404-1-5-5', '1910.13404-2-5-5'], ['1910.13404-1-5-6', '1910.13404-2-5-6'], ['1910.13404-1-5-7', '1910.13404-2-5-7'], ['1910.13404-1-5-8', '1910.13404-2-5-8'], ['1910.13404-1-6-0', '1910.13404-2-6-0'], ['1910.13404-1-6-1', '1910.13404-2-6-1'], ['1910.13404-1-6-2', '1910.13404-2-6-2'], ['1910.13404-1-6-3', '1910.13404-2-6-3'], ['1910.13404-1-6-4', '1910.13404-2-6-4'], ['1910.13404-1-6-5', '1910.13404-2-6-5'], ['1910.13404-1-6-6', '1910.13404-2-6-6'], ['1910.13404-1-31-0', '1910.13404-2-31-0'], ['1910.13404-1-31-1', '1910.13404-2-31-1'], ['1910.13404-1-31-3', '1910.13404-2-31-3'], ['1910.13404-1-31-4', '1910.13404-2-31-4'], ['1910.13404-1-31-5', '1910.13404-2-31-5'], ['1910.13404-1-57-0', '1910.13404-2-57-0'], ['1910.13404-1-57-1', '1910.13404-2-57-1'], ['1910.13404-1-57-2', '1910.13404-2-57-2'], ['1910.13404-1-40-0', '1910.13404-2-40-0'], ['1910.13404-1-40-1', '1910.13404-2-40-1'], ['1910.13404-1-40-2', '1910.13404-2-40-2'], ['1910.13404-1-40-3', '1910.13404-2-40-3'], ['1910.13404-1-40-4', '1910.13404-2-40-4'], ['1910.13404-1-40-5', '1910.13404-2-40-5'], ['1910.13404-1-53-0', '1910.13404-2-53-0'], ['1910.13404-1-53-4', '1910.13404-2-53-4'], ['1910.13404-1-53-5', '1910.13404-2-53-5'], ['1910.13404-1-53-6', '1910.13404-2-53-6'], ['1910.13404-1-53-7', '1910.13404-2-53-7'], ['1910.13404-1-53-10', '1910.13404-2-53-10'], ['1910.13404-1-53-11', '1910.13404-2-53-11'], ['1910.13404-1-53-12', '1910.13404-2-53-12'], ['1910.13404-1-53-13', '1910.13404-2-53-13'], ['1910.13404-1-32-0', '1910.13404-2-32-0'], ['1910.13404-1-52-0', '1910.13404-2-52-0'], ['1910.13404-1-52-1', '1910.13404-2-52-1'], ['1910.13404-1-33-0', '1910.13404-2-33-0'], ['1910.13404-1-33-1', '1910.13404-2-33-1'], ['1910.13404-1-33-2', '1910.13404-2-33-2'], ['1910.13404-1-33-3', '1910.13404-2-33-3'], ['1910.13404-1-14-0', '1910.13404-2-14-0'], ['1910.13404-1-14-1', '1910.13404-2-14-1'], ['1910.13404-1-14-2', '1910.13404-2-14-2'], ['1910.13404-1-14-3', '1910.13404-2-14-3'], ['1910.13404-1-56-0', '1910.13404-2-56-0'], ['1910.13404-1-56-1', '1910.13404-2-56-2'], ['1910.13404-1-11-0', '1910.13404-2-11-0'], ['1910.13404-1-11-1', '1910.13404-2-11-1'], ['1910.13404-1-11-2', '1910.13404-2-11-2'], ['1910.13404-1-11-4', '1910.13404-2-11-4'], ['1910.13404-1-11-5', '1910.13404-2-11-5'], ['1910.13404-1-8-0', '1910.13404-2-8-0'], ['1910.13404-1-8-1', '1910.13404-2-8-1'], ['1910.13404-1-8-2', '1910.13404-2-8-2'], ['1910.13404-1-8-3', '1910.13404-2-8-3'], ['1910.13404-1-13-0', '1910.13404-2-13-0'], ['1910.13404-1-27-0', '1910.13404-2-27-0'], ['1910.13404-1-27-1', '1910.13404-2-27-1'], ['1910.13404-1-27-2', '1910.13404-2-27-2'], ['1910.13404-1-27-4', '1910.13404-2-27-4'], ['1910.13404-1-27-5', '1910.13404-2-27-5'], ['1910.13404-1-27-6', '1910.13404-2-27-6'], ['1910.13404-1-27-7', '1910.13404-2-27-7'], ['1910.13404-1-27-8', '1910.13404-2-27-8'], ['1910.13404-1-27-9', '1910.13404-2-27-9'], ['1910.13404-1-27-10', '1910.13404-2-27-10'], ['1910.13404-1-34-0', '1910.13404-2-34-0'], ['1910.13404-1-34-1', '1910.13404-2-34-1'], ['1910.13404-1-34-2', '1910.13404-2-34-2'], ['1910.13404-1-34-3', '1910.13404-2-34-3'], ['1910.13404-1-43-0', '1910.13404-2-43-0'], ['1910.13404-1-43-1', '1910.13404-2-43-1'], ['1910.13404-1-43-2', '1910.13404-2-43-2'], ['1910.13404-1-35-0', '1910.13404-2-35-0'], ['1910.13404-1-35-1', '1910.13404-2-35-1'], ['1910.13404-1-35-2', '1910.13404-2-35-2'], ['1910.13404-1-35-3', '1910.13404-2-35-3'], ['1910.13404-1-10-0', '1910.13404-2-10-0'], ['1910.13404-1-10-1', '1910.13404-2-10-1'], ['1910.13404-1-10-2', '1910.13404-2-10-2'], ['1910.13404-1-10-3', '1910.13404-2-10-3'], ['1910.13404-1-58-0', '1910.13404-2-58-0'], ['1910.13404-1-20-0', '1910.13404-2-20-0'], ['1910.13404-1-20-1', '1910.13404-2-20-1'], ['1910.13404-1-20-2', '1910.13404-2-20-2'], ['1910.13404-1-20-3', '1910.13404-2-20-3'], ['1910.13404-1-9-2', '1910.13404-2-9-1'], ['1910.13404-1-37-0', '1910.13404-2-37-0'], ['1910.13404-1-37-1', '1910.13404-2-37-1'], ['1910.13404-1-37-2', '1910.13404-2-37-2'], ['1910.13404-1-0-0', '1910.13404-2-0-0'], ['1910.13404-1-0-1', '1910.13404-2-0-1'], ['1910.13404-1-0-2', '1910.13404-2-0-2'], ['1910.13404-1-21-0', '1910.13404-2-21-0'], ['1910.13404-1-21-1', '1910.13404-2-21-1'], ['1910.13404-1-16-0', '1910.13404-2-16-0'], ['1910.13404-1-16-1', '1910.13404-2-16-1'], ['1910.13404-1-16-2', '1910.13404-2-16-2'], ['1910.13404-1-16-3', '1910.13404-2-16-3'], ['1910.13404-1-16-4', '1910.13404-2-16-4'], ['1910.13404-1-16-5', '1910.13404-2-16-5'], ['1910.13404-1-16-6', '1910.13404-2-16-6'], ['1910.13404-1-26-0', '1910.13404-2-26-0'], ['1910.13404-1-26-1', '1910.13404-2-26-1'], ['1910.13404-1-42-0', '1910.13404-2-42-0'], ['1910.13404-1-42-1', '1910.13404-2-42-1'], ['1910.13404-1-44-0', '1910.13404-2-44-0'], ['1910.13404-1-44-1', '1910.13404-2-44-1'], ['1910.13404-1-44-2', '1910.13404-2-44-2'], ['1910.13404-1-44-3', '1910.13404-2-44-3'], ['1910.13404-1-47-0', '1910.13404-2-47-0'], ['1910.13404-1-47-1', '1910.13404-2-47-1'], ['1910.13404-1-54-0', '1910.13404-2-54-0'], ['1910.13404-1-54-1', '1910.13404-2-54-1'], ['1910.13404-1-54-2', '1910.13404-2-54-2'], ['1910.13404-1-54-3', '1910.13404-2-54-3'], ['1910.13404-1-54-4', '1910.13404-2-54-4'], ['1910.13404-1-54-5', '1910.13404-2-54-5'], ['1910.13404-1-23-0', '1910.13404-2-23-0'], ['1910.13404-1-23-2', '1910.13404-2-23-2'], ['1910.13404-1-50-0', '1910.13404-2-50-0'], ['1910.13404-2-58-0', '1910.13404-3-58-0'], ['1910.13404-2-58-1', '1910.13404-3-58-1'], ['1910.13404-2-26-0', '1910.13404-3-26-0'], ['1910.13404-2-26-1', '1910.13404-3-26-1'], ['1910.13404-2-26-2', '1910.13404-3-26-2'], ['1910.13404-2-44-0', '1910.13404-3-44-0'], ['1910.13404-2-44-1', '1910.13404-3-44-1'], ['1910.13404-2-44-2', '1910.13404-3-44-2'], ['1910.13404-2-44-3', '1910.13404-3-44-3'], ['1910.13404-2-53-0', '1910.13404-3-53-0'], ['1910.13404-2-53-1', '1910.13404-3-53-1'], ['1910.13404-2-53-2', '1910.13404-3-53-2'], ['1910.13404-2-53-3', '1910.13404-3-53-3'], ['1910.13404-2-53-4', '1910.13404-3-53-4'], ['1910.13404-2-53-5', '1910.13404-3-53-5'], ['1910.13404-2-53-6', '1910.13404-3-53-6'], ['1910.13404-2-53-7', '1910.13404-3-53-7'], ['1910.13404-2-53-8', '1910.13404-3-53-8'], ['1910.13404-2-53-9', '1910.13404-3-53-9'], ['1910.13404-2-53-10', '1910.13404-3-53-10'], ['1910.13404-2-53-11', '1910.13404-3-53-11'], ['1910.13404-2-53-12', '1910.13404-3-53-12'], ['1910.13404-2-53-13', '1910.13404-3-53-13'], ['1910.13404-2-5-0', '1910.13404-3-5-0'], ['1910.13404-2-5-1', '1910.13404-3-5-1'], ['1910.13404-2-5-2', '1910.13404-3-5-2'], ['1910.13404-2-5-3', '1910.13404-3-5-3'], ['1910.13404-2-5-4', '1910.13404-3-5-4'], ['1910.13404-2-5-5', '1910.13404-3-5-5'], ['1910.13404-2-5-6', '1910.13404-3-5-6'], ['1910.13404-2-5-7', '1910.13404-3-5-7'], ['1910.13404-2-5-8', '1910.13404-3-5-8'], ['1910.13404-2-9-0', '1910.13404-3-9-0'], ['1910.13404-2-9-1', '1910.13404-3-9-1'], ['1910.13404-2-10-0', '1910.13404-3-10-0'], ['1910.13404-2-10-1', '1910.13404-3-10-1'], ['1910.13404-2-10-2', '1910.13404-3-10-2'], ['1910.13404-2-10-3', '1910.13404-3-10-3'], ['1910.13404-2-45-0', '1910.13404-3-45-0'], ['1910.13404-2-45-1', '1910.13404-3-45-1'], ['1910.13404-2-45-2', '1910.13404-3-45-2'], ['1910.13404-2-45-3', '1910.13404-3-45-3'], ['1910.13404-2-0-0', '1910.13404-3-0-0'], ['1910.13404-2-0-1', '1910.13404-3-0-1'], ['1910.13404-2-0-2', '1910.13404-3-0-2'], ['1910.13404-2-37-0', '1910.13404-3-37-0'], ['1910.13404-2-37-1', '1910.13404-3-37-1'], ['1910.13404-2-37-2', '1910.13404-3-37-2'], ['1910.13404-2-46-0', '1910.13404-3-46-0'], ['1910.13404-2-46-1', '1910.13404-3-46-1'], ['1910.13404-2-46-2', '1910.13404-3-46-2'], ['1910.13404-2-52-0', '1910.13404-3-52-0'], ['1910.13404-2-52-1', '1910.13404-3-52-1'], ['1910.13404-2-49-0', '1910.13404-3-49-0'], ['1910.13404-2-49-1', '1910.13404-3-49-1'], ['1910.13404-2-49-2', '1910.13404-3-49-2'], ['1910.13404-2-57-0', '1910.13404-3-57-0'], ['1910.13404-2-57-1', '1910.13404-3-57-1'], ['1910.13404-2-57-2', '1910.13404-3-57-2'], ['1910.13404-2-40-0', '1910.13404-3-40-0'], ['1910.13404-2-40-1', '1910.13404-3-40-1'], ['1910.13404-2-40-2', '1910.13404-3-40-2'], ['1910.13404-2-40-3', '1910.13404-3-40-3'], ['1910.13404-2-40-4', '1910.13404-3-40-4'], ['1910.13404-2-40-5', '1910.13404-3-40-5'], ['1910.13404-2-20-0', '1910.13404-3-20-0'], ['1910.13404-2-20-1', '1910.13404-3-20-1'], ['1910.13404-2-20-2', '1910.13404-3-20-2'], ['1910.13404-2-20-3', '1910.13404-3-20-3'], ['1910.13404-2-42-0', '1910.13404-3-42-0'], ['1910.13404-2-42-1', '1910.13404-3-42-1'], ['1910.13404-2-50-0', '1910.13404-3-50-0'], ['1910.13404-2-29-0', '1910.13404-3-29-0'], ['1910.13404-2-29-1', '1910.13404-3-29-1'], ['1910.13404-2-29-2', '1910.13404-3-29-2'], ['1910.13404-2-29-3', '1910.13404-3-29-3'], ['1910.13404-2-29-4', '1910.13404-3-29-4'], ['1910.13404-2-29-5', '1910.13404-3-29-5'], ['1910.13404-2-56-0', '1910.13404-3-56-0'], ['1910.13404-2-56-1', '1910.13404-3-56-1'], ['1910.13404-2-56-2', '1910.13404-3-56-2'], ['1910.13404-2-56-3', '1910.13404-3-56-3'], ['1910.13404-2-56-4', '1910.13404-3-56-4'], ['1910.13404-2-24-0', '1910.13404-3-24-0'], ['1910.13404-2-24-1', '1910.13404-3-24-1'], ['1910.13404-2-27-0', '1910.13404-3-27-0'], ['1910.13404-2-27-1', '1910.13404-3-27-1'], ['1910.13404-2-27-2', '1910.13404-3-27-2'], ['1910.13404-2-27-3', '1910.13404-3-27-3'], ['1910.13404-2-27-4', '1910.13404-3-27-4'], ['1910.13404-2-27-5', '1910.13404-3-27-5'], ['1910.13404-2-27-6', '1910.13404-3-27-6'], ['1910.13404-2-27-7', '1910.13404-3-27-7'], ['1910.13404-2-27-8', '1910.13404-3-27-8'], ['1910.13404-2-27-9', '1910.13404-3-27-9'], ['1910.13404-2-27-10', '1910.13404-3-27-10'], ['1910.13404-2-14-0', '1910.13404-3-14-0'], ['1910.13404-2-14-1', '1910.13404-3-14-1'], ['1910.13404-2-14-2', '1910.13404-3-14-2'], ['1910.13404-2-14-3', '1910.13404-3-14-3'], ['1910.13404-2-14-4', '1910.13404-3-14-4'], ['1910.13404-2-14-5', '1910.13404-3-14-5'], ['1910.13404-2-35-0', '1910.13404-3-35-0'], ['1910.13404-2-35-1', '1910.13404-3-35-1'], ['1910.13404-2-35-2', '1910.13404-3-35-2'], ['1910.13404-2-35-3', '1910.13404-3-35-3'], ['1910.13404-2-47-0', '1910.13404-3-47-0'], ['1910.13404-2-47-1', '1910.13404-3-47-1'], ['1910.13404-2-47-2', '1910.13404-3-47-2'], ['1910.13404-2-47-3', '1910.13404-3-47-3'], ['1910.13404-2-25-0', '1910.13404-3-25-0'], ['1910.13404-2-25-1', '1910.13404-3-25-1'], ['1910.13404-2-25-2', '1910.13404-3-25-2'], ['1910.13404-2-15-0', '1910.13404-3-15-0'], ['1910.13404-2-21-0', '1910.13404-3-21-0'], ['1910.13404-2-21-1', '1910.13404-3-21-1'], ['1910.13404-2-54-0', '1910.13404-3-54-0'], ['1910.13404-2-54-1', '1910.13404-3-54-1'], ['1910.13404-2-54-2', '1910.13404-3-54-2'], ['1910.13404-2-54-3', '1910.13404-3-54-3'], ['1910.13404-2-54-4', '1910.13404-3-54-4'], ['1910.13404-2-54-5', '1910.13404-3-54-5'], ['1910.13404-2-16-0', '1910.13404-3-16-0'], ['1910.13404-2-16-1', '1910.13404-3-16-1'], ['1910.13404-2-16-2', '1910.13404-3-16-2'], ['1910.13404-2-16-3', '1910.13404-3-16-3'], ['1910.13404-2-16-4', '1910.13404-3-16-4'], ['1910.13404-2-16-5', '1910.13404-3-16-5'], ['1910.13404-2-16-6', '1910.13404-3-16-6'], ['1910.13404-2-43-0', '1910.13404-3-43-0'], ['1910.13404-2-43-1', '1910.13404-3-43-1'], ['1910.13404-2-43-2', '1910.13404-3-43-2'], ['1910.13404-2-31-0', '1910.13404-3-31-0'], ['1910.13404-2-31-1', '1910.13404-3-31-1'], ['1910.13404-2-31-2', '1910.13404-3-31-2'], ['1910.13404-2-31-3', '1910.13404-3-31-3'], ['1910.13404-2-31-4', '1910.13404-3-31-4'], ['1910.13404-2-31-5', '1910.13404-3-31-5'], ['1910.13404-2-13-0', '1910.13404-3-13-0'], ['1910.13404-2-13-1', '1910.13404-3-13-1'], ['1910.13404-2-6-0', '1910.13404-3-6-0'], ['1910.13404-2-6-1', '1910.13404-3-6-1'], ['1910.13404-2-6-2', '1910.13404-3-6-2'], ['1910.13404-2-6-3', '1910.13404-3-6-3'], ['1910.13404-2-6-4', '1910.13404-3-6-4'], ['1910.13404-2-6-5', '1910.13404-3-6-5'], ['1910.13404-2-6-6', '1910.13404-3-6-6'], ['1910.13404-2-8-0', '1910.13404-3-8-0'], ['1910.13404-2-8-1', '1910.13404-3-8-1'], ['1910.13404-2-8-2', '1910.13404-3-8-2'], ['1910.13404-2-32-0', '1910.13404-3-32-0'], ['1910.13404-2-11-0', '1910.13404-3-11-0'], ['1910.13404-2-11-1', '1910.13404-3-11-1'], ['1910.13404-2-11-2', '1910.13404-3-11-2'], ['1910.13404-2-11-3', '1910.13404-3-11-3'], ['1910.13404-2-11-4', '1910.13404-3-11-4'], ['1910.13404-2-11-5', '1910.13404-3-11-5'], ['1910.13404-2-22-0', '1910.13404-3-22-0'], ['1910.13404-2-22-1', '1910.13404-3-22-1'], ['1910.13404-2-22-2', '1910.13404-3-22-2'], ['1910.13404-2-33-0', '1910.13404-3-33-0'], ['1910.13404-2-33-1', '1910.13404-3-33-1'], ['1910.13404-2-33-2', '1910.13404-3-33-2'], ['1910.13404-2-33-3', '1910.13404-3-33-3'], ['1910.13404-2-34-0', '1910.13404-3-34-0'], ['1910.13404-2-34-1', '1910.13404-3-34-1'], ['1910.13404-2-34-2', '1910.13404-3-34-2'], ['1910.13404-2-34-3', '1910.13404-3-34-3'], ['1910.13404-2-23-0', '1910.13404-3-23-0'], ['1910.13404-2-23-1', '1910.13404-3-23-1'], ['1910.13404-2-23-2', '1910.13404-3-23-2'], ['1910.13404-2-17-0', '1910.13404-3-17-0'], ['1910.13404-2-17-1', '1910.13404-3-17-1'], ['1910.13404-2-17-2', '1910.13404-3-17-2'], ['1910.13404-2-17-3', '1910.13404-3-17-3'], ['1910.13404-2-7-0', '1910.13404-3-7-0'], ['1910.13404-2-7-1', '1910.13404-3-7-1'], ['1910.13404-2-7-2', '1910.13404-3-7-2'], ['1910.13404-2-7-3', '1910.13404-3-7-3'], ['1910.13404-3-54-0', '1910.13404-4-54-0'], ['1910.13404-3-54-1', '1910.13404-4-54-1'], ['1910.13404-3-54-2', '1910.13404-4-54-2'], ['1910.13404-3-54-3', '1910.13404-4-54-3'], ['1910.13404-3-54-4', '1910.13404-4-54-4'], ['1910.13404-3-54-5', '1910.13404-4-54-5'], ['1910.13404-3-56-0', '1910.13404-4-56-0'], ['1910.13404-3-56-1', '1910.13404-4-56-1'], ['1910.13404-3-56-2', '1910.13404-4-56-2'], ['1910.13404-3-56-3', '1910.13404-4-56-3'], ['1910.13404-3-56-4', '1910.13404-4-56-4'], ['1910.13404-3-7-0', '1910.13404-4-7-0'], ['1910.13404-3-7-1', '1910.13404-4-7-1'], ['1910.13404-3-7-2', '1910.13404-4-7-2'], ['1910.13404-3-7-3', '1910.13404-4-7-3'], ['1910.13404-3-8-0', '1910.13404-4-8-0'], ['1910.13404-3-8-1', '1910.13404-4-8-1'], ['1910.13404-3-8-2', '1910.13404-4-8-2'], ['1910.13404-3-26-0', '1910.13404-4-26-0'], ['1910.13404-3-26-1', '1910.13404-4-26-1'], ['1910.13404-3-26-2', '1910.13404-4-26-2'], ['1910.13404-3-35-0', '1910.13404-4-35-0'], ['1910.13404-3-35-1', '1910.13404-4-35-1'], ['1910.13404-3-35-2', '1910.13404-4-35-2'], ['1910.13404-3-35-3', '1910.13404-4-35-3'], ['1910.13404-3-40-0', '1910.13404-4-40-0'], ['1910.13404-3-40-1', '1910.13404-4-40-1'], ['1910.13404-3-40-2', '1910.13404-4-40-2'], ['1910.13404-3-40-3', '1910.13404-4-40-3'], ['1910.13404-3-40-4', '1910.13404-4-40-4'], ['1910.13404-3-40-5', '1910.13404-4-40-5'], ['1910.13404-3-33-0', '1910.13404-4-33-0'], ['1910.13404-3-33-1', '1910.13404-4-33-1'], ['1910.13404-3-33-2', '1910.13404-4-33-2'], ['1910.13404-3-33-3', '1910.13404-4-33-3'], ['1910.13404-3-46-0', '1910.13404-4-46-0'], ['1910.13404-3-46-1', '1910.13404-4-46-1'], ['1910.13404-3-46-2', '1910.13404-4-46-2'], ['1910.13404-3-58-0', '1910.13404-4-58-0'], ['1910.13404-3-58-1', '1910.13404-4-58-1'], ['1910.13404-3-27-0', '1910.13404-4-27-0'], ['1910.13404-3-27-1', '1910.13404-4-27-1'], ['1910.13404-3-27-2', '1910.13404-4-27-2'], ['1910.13404-3-27-3', '1910.13404-4-27-3'], ['1910.13404-3-27-4', '1910.13404-4-27-4'], ['1910.13404-3-27-5', '1910.13404-4-27-5'], ['1910.13404-3-27-6', '1910.13404-4-27-6'], ['1910.13404-3-27-7', '1910.13404-4-27-7'], ['1910.13404-3-27-8', '1910.13404-4-27-8'], ['1910.13404-3-27-9', '1910.13404-4-27-9'], ['1910.13404-3-27-10', '1910.13404-4-27-10'], ['1910.13404-3-0-0', '1910.13404-4-0-0'], ['1910.13404-3-0-1', '1910.13404-4-0-1'], ['1910.13404-3-0-2', '1910.13404-4-0-2'], ['1910.13404-3-34-0', '1910.13404-4-34-0'], ['1910.13404-3-34-1', '1910.13404-4-34-1'], ['1910.13404-3-34-2', '1910.13404-4-34-2'], ['1910.13404-3-34-3', '1910.13404-4-34-3'], ['1910.13404-3-57-0', '1910.13404-4-57-0'], ['1910.13404-3-57-1', '1910.13404-4-57-1'], ['1910.13404-3-57-2', '1910.13404-4-57-2'], ['1910.13404-3-17-0', '1910.13404-4-17-0'], ['1910.13404-3-17-1', '1910.13404-4-17-1'], ['1910.13404-3-17-2', '1910.13404-4-17-2'], ['1910.13404-3-17-3', '1910.13404-4-17-3'], ['1910.13404-3-15-0', '1910.13404-4-15-0'], ['1910.13404-3-9-0', '1910.13404-4-9-0'], ['1910.13404-3-9-1', '1910.13404-4-9-1'], ['1910.13404-3-21-0', '1910.13404-4-21-0'], ['1910.13404-3-21-1', '1910.13404-4-21-1'], ['1910.13404-3-10-0', '1910.13404-4-10-0'], ['1910.13404-3-10-1', '1910.13404-4-10-1'], ['1910.13404-3-10-2', '1910.13404-4-10-2'], ['1910.13404-3-10-3', '1910.13404-4-10-3'], ['1910.13404-3-31-0', '1910.13404-4-31-0'], ['1910.13404-3-31-1', '1910.13404-4-31-1'], ['1910.13404-3-31-2', '1910.13404-4-31-2'], ['1910.13404-3-31-3', '1910.13404-4-31-3'], ['1910.13404-3-31-4', '1910.13404-4-31-4'], ['1910.13404-3-31-5', '1910.13404-4-31-5'], ['1910.13404-3-24-0', '1910.13404-4-24-0'], ['1910.13404-3-24-1', '1910.13404-4-24-1'], ['1910.13404-3-45-0', '1910.13404-4-45-0'], ['1910.13404-3-45-1', '1910.13404-4-45-1'], ['1910.13404-3-45-2', '1910.13404-4-45-2'], ['1910.13404-3-45-3', '1910.13404-4-45-3'], ['1910.13404-3-22-0', '1910.13404-4-22-0'], ['1910.13404-3-22-1', '1910.13404-4-22-1'], ['1910.13404-3-22-2', '1910.13404-4-22-2'], ['1910.13404-3-6-0', '1910.13404-4-6-0'], ['1910.13404-3-6-1', '1910.13404-4-6-1'], ['1910.13404-3-6-2', '1910.13404-4-6-2'], ['1910.13404-3-6-3', '1910.13404-4-6-3'], ['1910.13404-3-6-4', '1910.13404-4-6-4'], ['1910.13404-3-6-5', '1910.13404-4-6-5'], ['1910.13404-3-6-6', '1910.13404-4-6-6'], ['1910.13404-3-43-0', '1910.13404-4-43-0'], ['1910.13404-3-43-1', '1910.13404-4-43-1'], ['1910.13404-3-43-2', '1910.13404-4-43-2'], ['1910.13404-3-50-0', '1910.13404-4-50-0'], ['1910.13404-3-29-0', '1910.13404-4-29-0'], ['1910.13404-3-29-1', '1910.13404-4-29-1'], ['1910.13404-3-29-2', '1910.13404-4-29-2'], ['1910.13404-3-29-3', '1910.13404-4-29-3'], ['1910.13404-3-29-4', '1910.13404-4-29-4'], ['1910.13404-3-29-5', '1910.13404-4-29-5'], ['1910.13404-3-11-0', '1910.13404-4-11-0'], ['1910.13404-3-11-1', '1910.13404-4-11-1'], ['1910.13404-3-11-2', '1910.13404-4-11-2'], ['1910.13404-3-11-3', '1910.13404-4-11-3'], ['1910.13404-3-11-4', '1910.13404-4-11-4'], ['1910.13404-3-11-5', '1910.13404-4-11-5'], ['1910.13404-3-16-0', '1910.13404-4-16-0'], ['1910.13404-3-16-1', '1910.13404-4-16-1'], ['1910.13404-3-16-2', '1910.13404-4-16-2'], ['1910.13404-3-16-3', '1910.13404-4-16-3'], ['1910.13404-3-16-4', '1910.13404-4-16-4'], ['1910.13404-3-16-5', '1910.13404-4-16-5'], ['1910.13404-3-16-6', '1910.13404-4-16-6'], ['1910.13404-3-23-0', '1910.13404-4-23-0'], ['1910.13404-3-23-1', '1910.13404-4-23-1'], ['1910.13404-3-23-2', '1910.13404-4-23-2'], ['1910.13404-3-32-0', '1910.13404-4-32-0'], ['1910.13404-3-20-0', '1910.13404-4-20-0'], ['1910.13404-3-20-1', '1910.13404-4-20-1'], ['1910.13404-3-20-2', '1910.13404-4-20-2'], ['1910.13404-3-20-3', '1910.13404-4-20-3'], ['1910.13404-3-42-0', '1910.13404-4-42-0'], ['1910.13404-3-42-1', '1910.13404-4-42-1'], ['1910.13404-3-25-0', '1910.13404-4-25-0'], ['1910.13404-3-25-1', '1910.13404-4-25-1'], ['1910.13404-3-25-2', '1910.13404-4-25-2'], ['1910.13404-3-13-0', '1910.13404-4-13-0'], ['1910.13404-3-13-1', '1910.13404-4-13-1'], ['1910.13404-3-52-0', '1910.13404-4-52-0'], ['1910.13404-3-52-1', '1910.13404-4-52-1'], ['1910.13404-3-14-0', '1910.13404-4-14-0'], ['1910.13404-3-14-1', '1910.13404-4-14-1'], ['1910.13404-3-14-2', '1910.13404-4-14-2'], ['1910.13404-3-14-3', '1910.13404-4-14-3'], ['1910.13404-3-14-4', '1910.13404-4-14-4'], ['1910.13404-3-14-5', '1910.13404-4-14-5'], ['1910.13404-3-53-0', '1910.13404-4-53-0'], ['1910.13404-3-53-1', '1910.13404-4-53-1'], ['1910.13404-3-53-2', '1910.13404-4-53-2'], ['1910.13404-3-53-3', '1910.13404-4-53-3'], ['1910.13404-3-53-4', '1910.13404-4-53-4'], ['1910.13404-3-53-5', '1910.13404-4-53-5'], ['1910.13404-3-53-6', '1910.13404-4-53-6'], ['1910.13404-3-53-7', '1910.13404-4-53-7'], ['1910.13404-3-53-8', '1910.13404-4-53-8'], ['1910.13404-3-53-9', '1910.13404-4-53-9'], ['1910.13404-3-53-10', '1910.13404-4-53-10'], ['1910.13404-3-53-11', '1910.13404-4-53-11'], ['1910.13404-3-53-12', '1910.13404-4-53-12'], ['1910.13404-3-53-13', '1910.13404-4-53-13'], ['1910.13404-3-5-0', '1910.13404-4-5-0'], ['1910.13404-3-5-1', '1910.13404-4-5-1'], ['1910.13404-3-5-2', '1910.13404-4-5-2'], ['1910.13404-3-5-3', '1910.13404-4-5-3'], ['1910.13404-3-5-4', '1910.13404-4-5-4'], ['1910.13404-3-5-5', '1910.13404-4-5-5'], ['1910.13404-3-5-6', '1910.13404-4-5-6'], ['1910.13404-3-5-7', '1910.13404-4-5-7'], ['1910.13404-3-5-8', '1910.13404-4-5-8'], ['1910.13404-3-37-0', '1910.13404-4-37-0'], ['1910.13404-3-37-1', '1910.13404-4-37-1'], ['1910.13404-3-37-2', '1910.13404-4-37-2'], ['1910.13404-3-44-0', '1910.13404-4-44-0'], ['1910.13404-3-44-1', '1910.13404-4-44-1'], ['1910.13404-3-44-2', '1910.13404-4-44-2'], ['1910.13404-3-44-3', '1910.13404-4-44-3'], ['1910.13404-3-47-0', '1910.13404-4-47-0'], ['1910.13404-3-47-1', '1910.13404-4-47-1'], ['1910.13404-3-47-2', '1910.13404-4-47-2'], ['1910.13404-3-47-3', '1910.13404-4-47-3'], ['1910.13404-3-49-0', '1910.13404-4-49-0'], ['1910.13404-3-49-1', '1910.13404-4-49-1'], ['1910.13404-3-49-2', '1910.13404-4-49-2']]

[['1910.13404-1-25-1', '1910.13404-2-25-1'], ['1910.13404-1-29-4', '1910.13404-2-29-4'], ['1910.13404-1-45-0', '1910.13404-2-45-0'], ['1910.13404-1-22-0', '1910.13404-2-22-0'], ['1910.13404-1-31-2', '1910.13404-2-31-2'], ['1910.13404-1-53-1', '1910.13404-2-53-1'], ['1910.13404-1-53-2', '1910.13404-2-53-2'], ['1910.13404-1-53-3', '1910.13404-2-53-3'], ['1910.13404-1-53-8', '1910.13404-2-53-8'], ['1910.13404-1-53-9', '1910.13404-2-53-9'], ['1910.13404-1-11-3', '1910.13404-2-11-3'], ['1910.13404-1-13-1', '1910.13404-2-13-1'], ['1910.13404-1-27-3', '1910.13404-2-27-3'], ['1910.13404-1-24-0', '1910.13404-2-24-0'], ['1910.13404-1-24-1', '1910.13404-2-24-1'], ['1910.13404-1-47-2', '1910.13404-2-47-2'], ['1910.13404-1-47-2', '1910.13404-2-47-3'], ['1910.13404-1-23-1', '1910.13404-2-23-1']]

[]

[['1910.13404-1-14-4', '1910.13404-2-14-4'], ['1910.13404-1-56-2', '1910.13404-2-56-3'], ['1910.13404-1-58-1', '1910.13404-2-58-1'], ['1910.13404-1-9-0', '1910.13404-2-9-0'], ['1910.13404-1-26-2', '1910.13404-2-26-2']]

[]

['1910.13404-1-1-0', '1910.13404-1-2-0', '1910.13404-1-2-1', '1910.13404-1-3-0', '1910.13404-2-1-0', '1910.13404-2-2-0', '1910.13404-2-2-1', '1910.13404-2-3-0', '1910.13404-3-1-0', '1910.13404-3-1-1', '1910.13404-3-2-0', '1910.13404-3-2-1', '1910.13404-3-3-0', '1910.13404-4-1-0', '1910.13404-4-1-1', '1910.13404-4-2-0', '1910.13404-4-2-1', '1910.13404-4-3-0']

{'1': 'http://creativecommons.org/licenses/by/4.0/', '2': 'http://creativecommons.org/licenses/by/4.0/', '3': 'http://creativecommons.org/licenses/by/4.0/', '4': 'http://creativecommons.org/licenses/by/4.0/'}

https://arxiv.org/abs/1910.13404

{'1910.13404-3-0-0': 'The production fraction of the meson with respect to the sum of [MATH] and [MATH] mesons is measured in both 7 and 13TeVcenter-of-mass energy [MATH] collisions produced by the Large Hadron Collider (LHC), using the LHCb detector.', '1910.13404-3-0-1': 'The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence.', '1910.13404-3-0-2': 'The [MATH] production asymmetry is also measured, and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.', '1910.13404-3-1-0': 'Published in Phys.', '1910.13404-3-1-1': 'Rev. D100 (2019) 112006', '1910.13404-3-2-0': 'CERN for the benefit of the LHCb collaboration.', '1910.13404-3-2-1': 'https://creativecommons.org/licenses/by/4.0/CC-BY-4.0 licence.', '1910.13404-3-3-0': 'arabic', '1910.13404-3-4-0': '# Introduction', '1910.13404-3-5-0': 'The meson is a bound state containing a [MATH] quark with a [MATH] quark.', '1910.13404-3-5-1': 'It has the largest mass of any two differently flavored quarks in a mesonic ground state.', '1910.13404-3-5-2': 'Studies of its production or determination of individual decay widths require measurements of its branching fractions to exclusive final states.', '1910.13404-3-5-3': 'Since the branching fractions of some decay modes of and mesons are accurately known, we determine the ratio of meson production relative to the sum of and mesons.', '1910.13404-3-5-4': 'Here we use techniques similar to those employed for the measurement of meson and baryon fractions [CITATION].', '1910.13404-3-5-5': 'In that paper use is made of the fact that the semileptonic widths of all [MATH]-flavored hadrons with light and strange quarks are equal.', '1910.13404-3-5-6': 'However, both the [MATH] and [MATH] quarks can decay, rendering that concept inapplicable.', '1910.13404-3-5-7': 'Instead we rely on theoretical predictions of the semileptonic decay branching fraction [MATH].', '1910.13404-3-5-8': 'Currently, only the relative production cross-section times the branching fraction of either the [MATH] or [MATH] modes have been measured by the [CITATION], [CITATION] and [CITATION] experiments.', '1910.13404-3-6-0': 'The [MATH] meson production fraction ([MATH]) relative to the sum of [MATH]) and [MATH]) mesons is defined as [EQUATION] where [MATH] refers to the efficiency and branching fraction corrected number of signal events.', '1910.13404-3-6-1': 'The modes containing [MATH] and [MATH] mesons are also corrected for cross-feeds with and decays.', '1910.13404-3-6-2': 'The determination of the corrected yields of the [MATH] decays follows our previous measurement strategy in Ref. [CITATION] where the equations relating the fractions to the corrected yields, including cross-feed contributions, are given.', '1910.13404-3-6-3': 'We also correct for the 0.4% effect of doubly-Cabibbo-suppressed decays and [MATH] mixing.', '1910.13404-3-6-4': 'The relevant hadron branching fractions are listed in Table [REF].', '1910.13404-3-6-5': 'The average semileptonic branching fractions of and , [MATH] is found by averaging measurements from the [CITATION], [CITATION] and [CITATION] experiments, detailed in Ref. [CITATION].', '1910.13404-3-6-6': 'Since only [MATH] modes are detected in this analysis, a correction for the small [MATH] rate of 1% is applied to the denominator of Eq. [REF].', '1910.13404-3-7-0': 'The dominant production mechanism for mesons is gluon-gluon fusion, [MATH].', '1910.13404-3-7-1': 'Non-relativistic quantum chromodynamics is used along with fragmentation functions to predict cross-sections as functions of transverse momentum () and pseudorapidity ([MATH]).', '1910.13404-3-7-2': 'The literature is nicely summarized in Ref. [CITATION].', '1910.13404-3-7-3': 'We define [MATH] to refer to [MATH], [MATH], and [MATH] mesons, while [MATH] refers to [MATH] and [MATH] mesons.', '1910.13404-3-8-0': 'In this analysis [MATH] is determined by measuring the angle of the [MATH] meson with respect to the beam direction by using the positions of the primary [MATH] interaction vertex (PV) and the [MATH] meson decay point into either [MATH], [MATH], or [MATH].', '1910.13404-3-8-1': 'The transverse momentum initially refers to the vector sum of the charmed-hadron and [MATH] momentum transverse to the proton beams.', '1910.13404-3-8-2': 'However, the results are re-interpreted in terms of the [MATH] meson [MATH] by simulating and correcting the effects of the missing momenta.', '1910.13404-3-9-0': 'The production asymmetry between [MATH] and [MATH] mesons, which should be small, is defined as [EQUATION] where [MATH] and [MATH] are the asymmetries in the signal yields and the efficiencies of [MATH] and [MATH] detection, respectively.', '1910.13404-3-9-1': 'The asymmetry in the [MATH] decay is assumed to be zero in this analysis.', '1910.13404-3-10-0': 'The branching fraction predictions from various models of semileptonic decays are listed in Table [REF].', '1910.13404-3-10-1': 'For [MATH] they range from 1.4 to 7.5%, which is quite a large interval.', '1910.13404-3-10-2': 'Branching fractions for other modes are also listed where available.', '1910.13404-3-10-3': 'We use the Z expansion fit results from Ref. [CITATION], and the method II results for Ref. [CITATION].', '1910.13404-3-11-0': 'Some restrictions on models are possible by comparing to lighter [MATH] meson decays.', '1910.13404-3-11-1': 'Since the inclusive semileptonic branching fraction for these decays, [MATH], is about 10.5% and the [MATH] lifetime, [MATH], is 1/3 that of the [MATH], we disregard models that predict 10% or larger values for [MATH] of the .', '1910.13404-3-11-2': 'This excludes from consideration the models of Refs. [CITATION] and [CITATION].', '1910.13404-3-11-3': 'The average model prediction is then [MATH].', '1910.13404-3-11-4': 'The standard deviation is 0.46, which we use to estimate the systematic uncertainty on the model variation.', '1910.13404-3-11-5': 'Results of our measurement without using this branching fraction are also quoted.', '1910.13404-3-12-0': '# Detector, trigger and simulation', '1910.13404-3-13-0': 'The LHCb detector [CITATION] is a single-arm forward spectrometer covering the pseudorapidity range [MATH], designed for the study of particles containing or quarks.', '1910.13404-3-13-1': 'The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the [MATH] interaction region that allows and hadrons to be identified from their characteristically long flight distance; a tracking system that provides a measurement of the momentum, [MATH], of charged particles; two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons; and a downstream system of iron interspersed with chambers is used to identify muons.', '1910.13404-3-14-0': 'The magnetic field deflects positively and negatively charged particles in opposite directions and this can lead to detection asymmetries.', '1910.13404-3-14-1': 'Periodically reversing the magnetic field polarity throughout the data taking almost cancels the effect.', '1910.13404-3-14-2': 'The configuration with the magnetic field pointing upwards (downwards) bends positively (negatively) charged particles in the horizontal plane towards the centre of the LHC ring.', '1910.13404-3-14-3': 'This analysis uses data collected in 2011 (7TeV) and 2016 (13TeV) where appropriate triggers are available.', '1910.13404-3-14-4': 'The data taking was split between magnetic field up and down configurations.', '1910.13404-3-14-5': 'In the 2011 data 0.6(0.4) were collected with the field pointing up (down), while in 2016 the split was 0.9with field up and 0.8with field down.', '1910.13404-3-15-0': 'The trigger [CITATION] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with [MATH] are reconstructed for 2011(2016) data.', '1910.13404-3-16-0': 'Separate hardware triggers are used for the [MATH] and [MATH] samples.', '1910.13404-3-16-1': 'For the former we require a [MATH] pair.', '1910.13404-3-16-2': 'For the latter, we require a single muon with large for the 7TeVdata as used in Ref. [CITATION].', '1910.13404-3-16-3': 'For the 13TeVdata, the single muon trigger was not available, therefore at the hardware trigger stage, events are required to have a hadron, photon or electron transverse energy greater than approximately 3.5 GeV in the calorimeters.', '1910.13404-3-16-4': 'The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary [MATH] interaction vertex as described in Ref. [CITATION].', '1910.13404-3-16-5': 'At least one charged particle must have [MATH] and be inconsistent with originating from a PV.', '1910.13404-3-16-6': 'A multivariate algorithm [CITATION] is used for the identification of secondary vertices consistent with the decay of a hadron.', '1910.13404-3-17-0': 'Simulation is required to model the effects of the detector acceptance and the imposed selection requirements.', '1910.13404-3-17-1': 'In the simulation, [MATH] collisions are generated using [CITATION] with a specific configuration [CITATION].', '1910.13404-3-17-2': 'Decays of unstable particles are described by [CITATION], in which final-state radiation is generated using [CITATION].', '1910.13404-3-17-3': 'The interaction of the generated particles with the detector, and its response, are implemented using the toolkit [CITATION] as described in Ref. [CITATION].', '1910.13404-3-18-0': '# Event selection, signal efficiencies and yields', '1910.13404-3-19-0': '## Selection of [MATH] candidates', '1910.13404-3-20-0': 'The analysis is done separately for the light [MATH] meson modes and the decay.', '1910.13404-3-20-1': 'In each case the triggered events are subject to further filtering requirements.', '1910.13404-3-20-2': 'In addition, the [MATH] sample is subjected to a boosted decision tree (BDT), a multivariate classification method, using the TMVA toolkit [CITATION].', '1910.13404-3-20-3': 'This is not necessary for the [MATH] or [MATH] modes because they have large signals and are relatively free from backgrounds [CITATION].', '1910.13404-3-21-0': 'For the [MATH] final state the initial selection requires that muons that satisfy the candidate trigger each have minimum [MATH] MeV, have large impact parameters with the PV, form a good quality vertex, have a reasonable flight distance significance from the PV, and have a summed [MATH] GeV.', '1910.13404-3-21-1': 'The "companion" muon that is not part of the decay must be well identified and form a good quality vertex with the candidate, which must be downstream of the PV.', '1910.13404-3-22-0': 'To suppress muon tracks that are reconstructed more than once, we require a small minimum opening angle between the muons from the decay and the companion muon momentum measured in the plane transverse to the beam line.', '1910.13404-3-22-1': 'Specifically, this opening angle must be greater than 0.8[MATH].', '1910.13404-3-22-2': 'The invariant mass of the companion muon and the oppositely charged muon from [MATH] must differ from the known value of the [MATH] mass by more than [MATH] [CITATION], while the invariant mass with the same charged muon is required to be larger than [MATH].', '1910.13404-3-23-0': "Since we are dealing with an exclusive final state, we define [EQUATION] where [MATH] is the magnitude of the combination's momentum component transverse to the [MATH]-hadron flight direction.", '1910.13404-3-23-1': 'Figure [REF] shows the distributions of [MATH] versus the invariant [MATH] mass, [MATH], for both data and simulation.', '1910.13404-3-23-2': 'To remove background, a requirement of [MATH] is applied, as indicated by the (red) dashed line.', '1910.13404-3-24-0': 'Since we also measure the production asymmetry between [MATH] and [MATH] mesons, we restrict the angular acceptance of the companion muon to make it more uniform by removing muons close to the edge of the detector, in the bending direction ([MATH]-direction), where large acceptance-induced asymmetries can occur.', '1910.13404-3-24-1': 'Thus, we require that the [MATH]-component of the momentum satisfies [EQUATION] where [MATH] is the muon momentum along the direction of the proton beam downstream of the PV, as is done in Refs. [CITATION].', '1910.13404-3-25-0': 'After these initial restrictions, we turn to the multivariate selection, forming the classifier denoted BDT in the following.', '1910.13404-3-25-1': 'The discriminating variables used are: (a) the [MATH] of the vertex fit of the with the [MATH]; (b) the [MATH], where [MATH]is defined as the [MATH] of the impact parameter with respect to the PV, of the [MATH], [MATH] and their combination; (c) the of the and the [MATH]; and (d) the cosine of the angle between the [MATH] and the meson in the plane perpendicular to the beam direction.', '1910.13404-3-25-2': 'The training sample for signal is simulated [MATH] events, and for background is inclusive [MATH] simulated events.', '1910.13404-3-26-0': 'We then optimize the BDT output threshold by maximizing [MATH], where [MATH] and [MATH] are the number of the signal and background yields in the signal region defined as [MATH].', '1910.13404-3-26-1': 'The sum, [MATH], is the total number of events within these limits, and [MATH] is taken from a fit to the [MATH] distribution.', '1910.13404-3-26-2': 'The optimal BDT output threshold results in a BDT signal efficiency of 89% with a background rejection of 63%, as determined by observing the resulting samples of input signal simulation events and background candidates.', '1910.13404-3-27-0': 'The [MATH] distribution is shown in Fig. [REF].', '1910.13404-3-27-1': 'It consists not only of signal [MATH] events, but also of [MATH] decays, where [MATH], and other [MATH] final states, most importantly [MATH] and [MATH].', '1910.13404-3-27-2': 'We find shapes for these final states using simulation.', '1910.13404-3-27-3': 'The signal shape is a sum of a double Crystal Ball and a bifurcated Gaussian functions.', '1910.13404-3-27-4': 'The sum of the combinatorial and misidentification backgrounds are represented by a Gaussian kernel shape [CITATION].', '1910.13404-3-27-5': 'For the other background modes, we use histograms directly.', '1910.13404-3-27-6': 'These shapes are fitted to the [MATH] distributions in Fig. [REF] in order to determine the [MATH] yields.', '1910.13404-3-27-7': 'The ratio of the [MATH] yield to the [MATH] yield is fixed, after accounting for the relative detection efficiencies, from the LHCb measurement of [MATH], where the first uncertainty is statistical and the second systematic [CITATION]; this convention is used throughout this paper.', '1910.13404-3-27-8': 'The other components of the fit are allowed to vary.', '1910.13404-3-27-9': 'We find [MATH] and [MATH] signal [MATH] events at 7 and 13TeV, respectively, while the backgrounds sum to 950 and 5170 events at the same energies.', '1910.13404-3-27-10': 'These signal yields need to be corrected for the small background from candidates with a correctly reconstructed meson that is paired with a hadron mis-identified as a muon.', '1910.13404-3-28-0': '## Efficiency for [MATH]', '1910.13404-3-29-0': 'Efficiencies are determined using both data [CITATION] and simulation of [MATH], with the generated events weighted to match the [MATH], and [MATH] distributions observed in data.', '1910.13404-3-29-1': 'In addition, we weight accordingly the [MATH] distribution of the muon associated with the .', '1910.13404-3-29-2': 'Weighting the simulation is important since the total efficiencies are functions of these variables.', '1910.13404-3-29-3': 'Efficiencies using data include trigger, and muon identification.', '1910.13404-3-29-4': 'Efficiencies using simulation include detector acceptance, reconstruction and event selection, and removal of beam crossings with an excess number of hits in the detector.', '1910.13404-3-29-5': 'Total efficiencies as a function of [MATH] for different [MATH] intervals are shown in Fig. [REF].', '1910.13404-3-30-0': '## [MATH] selection criteria', '1910.13404-3-31-0': 'Selection criteria for [MATH] final states differ from those containing a [MATH].', '1910.13404-3-31-1': 'The transverse momentum of each hadron must be greater than 0.3GeV, and that of the muon larger than 1.3GeV.', '1910.13404-3-31-2': 'We require [MATH] with respect to any PV, ensuring that tracks do not originate from primary [MATH] interactions.', '1910.13404-3-31-3': 'All final state particles are required to be positively identified using information from the RICH detectors.', '1910.13404-3-31-4': 'Particles from [MATH] decay candidates must have a good fit to a common vertex with [MATH]/ndof [MATH], where ndof is the number of degrees of freedom.', '1910.13404-3-31-5': 'They must also be well separated from the nearest PV, with the flight distance divided by its uncertainty greater than 5.', '1910.13404-3-32-0': 'Candidate [MATH] hadrons are formed by combining [MATH] and muon candidates originating from a common vertex with [MATH]/ndof [MATH] and an [MATH] invariant mass in the range 3.0-5.0GeV.', '1910.13404-3-33-0': 'Background from prompt [MATH] production at the PV needs to be considered.', '1910.13404-3-33-1': 'We use the natural logarithm of the [MATH] impact parameter, IP, with respect to the PV in units of mm.', '1910.13404-3-33-2': 'Requiring ln(IP/mm)[MATH] is found to reduce the prompt component to be below 0.1%, while preserving 97% of all signals.', '1910.13404-3-33-3': 'This restriction allows us to perform fits only to the [MATH] candidate mass spectra to find the [MATH]-hadron decay yields.', '1910.13404-3-34-0': 'The [MATH] candidate mass distributions integrated over [MATH] and [MATH] are shown in Fig. [REF] and consist of a prominent peak resulting from signal, and a small contribution due to combinatorial background from random combinations of particles that pass the selection.', '1910.13404-3-34-1': 'They are fit with a signal component comprised of two Gaussian functions, and a combinatorial background component modeled as a linear function.', '1910.13404-3-34-2': 'The fitted yields are listed in Table [REF].', '1910.13404-3-34-3': 'These numbers must be corrected for hadrons that are mis-identified as muons, and for semileptonic decays of and hadrons that produce [MATH] and [MATH] mesons.', '1910.13404-3-35-0': 'In Table [REF] the column labeled "fake muons" shows the yields of wrong-sign [MATH] and [MATH] combinations that pass the selections.', '1910.13404-3-35-1': 'These yields provide good estimates of the fake muon contributions in the signal samples, which are very small.', '1910.13404-3-35-2': 'Following the procedure in Ref. [CITATION], we find the cross-feed corrections of [MATH] and [MATH] to be twice the measured yields for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV), and for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV).', '1910.13404-3-35-3': 'Relative efficiencies for detecting final states with a single extra hadron are taken into account when subtracting these yields.', '1910.13404-3-36-0': '## Efficiencies for [MATH] and [MATH]', '1910.13404-3-37-0': 'Similar methods based on data, as implemented for the decay, are used to evaluate the efficiencies for trigger and particle identification.', '1910.13404-3-37-1': 'Simulation is also used to determine the efficiencies of event selection and reconstruction of these modes.', '1910.13404-3-37-2': 'The total efficiencies for [MATH] meson decays into [MATH] and [MATH] are shown in Fig. [REF].', '1910.13404-3-38-0': '# Results', '1910.13404-3-39-0': '## Corrections to the [MATH] distributions due to the missing neutrino', '1910.13404-3-40-0': 'Since the production kinematics of [MATH] and mesons can differ as functions of [MATH] and [MATH], we need to measure [MATH] as functions of these variables.', '1910.13404-3-40-1': 'The measurement of [MATH] is straightforward, however, we do not measure directly the [MATH] of the [MATH]-flavored hadron because of the missing neutrino, and in the case of the [MATH] meson possible missing extra particles.', '1910.13404-3-40-2': 'Following a procedure similar to the one used in Ref. [CITATION], we determine a correction factor, [MATH], that is the ratio of the average reconstructed to true [MATH] as a function of the invariant mass of the charmed hadron plus muon.', '1910.13404-3-40-3': 'The ratio distribution as a function of hadron-muon invariant mass are shown in Fig. [REF].', '1910.13404-3-40-4': 'The average correction, the [MATH]-factor, is shown on the figure.', '1910.13404-3-40-5': 'For the [MATH] meson it varies from 0.75 to unity over the interval from 3 GeV to the [MATH] mass, and for the meson it varies from 0.85 to unity over the interval from 4 GeV to the mass.', '1910.13404-3-41-0': '## fraction results', '1910.13404-3-42-0': 'The ratio of production fractions, [MATH], are shown as functions of [MATH] and [MATH] in Fig. [REF].', '1910.13404-3-42-1': 'There is little dependence on [MATH], but the decrease as a function of [MATH] is noticeable.', '1910.13404-3-43-0': 'To describe the [MATH] dependence we use an equation of the form [EQUATION] where [MATH] represents the overall normalization and contains the total global systematic uncertainty, thus, [MATH]; [MATH] is taken as 7.2 GeV, close to the average of the .', '1910.13404-3-43-1': 'The slopes, [MATH], are similar in size to those measured for the [MATH] meson fraction ratio as a function of [CITATION].', '1910.13404-3-43-2': 'Results of fits to the data using Eq. [REF] are listed in Table [REF].', '1910.13404-3-44-0': 'The average fractions in the interval [MATH]GeVare found by integrating over [MATH].', '1910.13404-3-44-1': 'To allow for facile changes to our results due to improved theoretical predictions, we provide the results for [EQUATION]', '1910.13404-3-44-2': 'Next we give the result on the fractions ratio [EQUATION] where the third uncertainty is due to the theoretical prediction of [MATH].', '1910.13404-3-44-3': 'To find [MATH] just double these numbers.', '1910.13404-3-45-0': 'We also measure the ratio of the [MATH] production fraction at [MATH] to that at [MATH].', '1910.13404-3-45-1': 'Figure [REF] shows the ratio as functions of and [MATH].', '1910.13404-3-45-2': 'Here most of the systematic uncertainties cancel.', '1910.13404-3-45-3': 'The integrated value of the ratio of [MATH] and [MATH] is measured as [MATH], consistent with no increase in the fraction ratio as a function of center-of-mass energy.', '1910.13404-3-46-0': 'The fraction with respect to inclusive [MATH]-hadron production can be derived from the information in previous LHCb [MATH]-hadron fraction papers Ref. [CITATION].', '1910.13404-3-46-1': 'There the measured values of the ratios of [MATH]-hadron fractions over the same range in terms of the [MATH]-hadron are for mesons ([MATH]) and baryons [EQUATION] where the uncertainties contain both statistical and systematic components added in quadrature.', '1910.13404-3-46-2': 'For the measurement of the [MATH] fraction at 7TeV, the dominant systematic uncertainty is from the lack of the knowledge of [MATH] at that time [CITATION]; here the value and uncertainty have been recalculated according to the latest value of [MATH] from the PDG [CITATION].', '1910.13404-3-47-0': 'Taking the sum of all the [MATH]-hadron fractions to be unity, and ignoring [MATH] here because it is so small, [EQUATION] where [MATH] is a correction factor derived in Ref. [CITATION] that accounts for heavier [MATH]-baryons, mainly the [MATH].', '1910.13404-3-47-1': 'Solving for [MATH] yields [EQUATION]', '1910.13404-3-47-2': 'We find that [EQUATION] where the first uncertainty is statistical, the second is systematic, and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-3-47-3': 'We also provide the result for [MATH], [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-3-48-0': '## The [MATH] production asymmetry', '1910.13404-3-49-0': 'The production asymmetries are measured in two different magnetic field configurations and then averaged.', '1910.13404-3-49-1': 'No significant asymmetry is observed in any intervals of [MATH] or [MATH].', '1910.13404-3-49-2': 'The results are summarized in Table [REF].', '1910.13404-3-50-0': 'Averaging the [MATH] production asymmetries over [MATH] and [MATH], we find [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-3-51-0': '# Systematic uncertainties', '1910.13404-3-52-0': 'Systematic uncertainties are separated into two categories: "global", which apply across the phase space, and "local", which are calculated in each two-dimensional [MATH] bin.', '1910.13404-3-52-1': 'These uncertainties are listed in Table [REF].', '1910.13404-3-53-0': 'First let us consider the [MATH] decay.', '1910.13404-3-53-1': 'The uncertainty due to the signal shape used to fit the [MATH] distribution is determined by changing the baseline signal shape, the sum of a double sided Crystal Ball function and a bifurcated Gaussian, to a kernel estimation.', '1910.13404-3-53-2': 'To find the shape of the combinatorial and misidentification backgrounds we use simulated inclusive samples of [MATH] events not including decays.', '1910.13404-3-53-3': 'A total of 500 samples are generated and different fits to the samples are performed to determine the possible uncertainty.', '1910.13404-3-53-4': 'This procedure is also used for the [MATH] measurement.', '1910.13404-3-53-5': 'We call contributions to the [MATH] mass spectrum "feed-down" contributions, occurring from other decay channels including [MATH], [MATH], and [MATH].', '1910.13404-3-53-6': 'The systematic uncertainty results from the uncertainties in their branching fractions.', '1910.13404-3-53-7': 'Different decay models for decays can change the [MATH] shape.', '1910.13404-3-53-8': 'We use the model of Ebert et al.[CITATION] for our baseline prediction.', '1910.13404-3-53-9': 'Then we also use the model by Kiselev [CITATION] to find the efficiencies and take half the difference as the systematic uncertainty.', '1910.13404-3-53-10': 'We also estimate the uncertainty due to the sensitivity to various selection requirements and simulation statistics.', '1910.13404-3-53-11': 'The muon identification efficiencies are determined from data using inclusive samples of decay where one of the muon candidates is not identified.', '1910.13404-3-53-12': 'The trigger efficiency is determined by using three independent samples of events, those that trigger on a [MATH], those that triggered on something else in the event, and those that trigger on both the and something else.', '1910.13404-3-53-13': 'These samples are then used to compute the trigger efficiencies in two-dimensional [MATH] and [MATH] bins.', '1910.13404-3-54-0': 'Next, we turn to the [MATH] modes.', '1910.13404-3-54-1': 'The efficiencies and their uncertainties for identifying pions and kaons are determined by using almost background free samples of [MATH] decays.', '1910.13404-3-54-2': 'The trigger and muon identification efficiencies, and their uncertainties, are obtained in the same manner as for the [MATH] mode.', '1910.13404-3-54-3': 'There are small systematic uncertainties related to efficiency estimates and the assumed [MATH] to [MATH] mixtures, as well as simulation statistics.', '1910.13404-3-54-4': 'Global systematic uncertainties include the hadron branching fractions listed in Table [REF], cross-feed corrections arising from and decays into [MATH] events, and a global hadron plus photon multiplicity requirement.', '1910.13404-3-54-5': 'The latter is evaluated with data.', '1910.13404-3-55-0': '# Conclusions', '1910.13404-3-56-0': 'In 7 and 13TeV[MATH] collisions the product of [MATH] with the relative fraction of mesons with respect to the sum of and mesons in the ranges [MATH] and [MATH] is found to be [EQUATION]', '1910.13404-3-56-1': 'We derive the product of [MATH] at the two energies as [EQUATION]', '1910.13404-3-56-2': 'Using the average of the theoretical prediction [MATH], where the uncertainty is given by the standard deviation derived from the distribution of the models, we determine [EQUATION] where the first uncertainties are statistical, the second systematic, and the third due to the theoretical prediction of [MATH].', '1910.13404-3-56-3': 'There is a small dependence on the transverse momentum of the meson, but no dependence on its pseudorapidity is observed.', '1910.13404-3-56-4': 'We also report [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-3-57-0': 'The ratio of fractions, [MATH], for 13TeV/7TeVis consistent with no increase in the fraction.', '1910.13404-3-57-1': 'Furthermore, using the assumption of no violation in the [MATH] decay, we find that the average asymmetry in [MATH] production is consistent with zero.', '1910.13404-3-57-2': 'The measurements are [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-3-58-0': 'These results are useful to extract absolute branching fractions for measurements, albeit with a relatively large uncertainty.', '1910.13404-3-58-1': 'They also challenge QCD calculations to predict the measured fractions and explain the consistency between the fractions measured at 7 and 13TeV[CITATION].'}

{'1910.13404-4-0-0': 'The production fraction of the meson with respect to the sum of [MATH] and [MATH] mesons is measured in both 7 and 13TeVcenter-of-mass energy [MATH] collisions produced by the Large Hadron Collider (LHC), using the LHCb detector.', '1910.13404-4-0-1': 'The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence.', '1910.13404-4-0-2': 'The [MATH] production asymmetry is also measured, and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.', '1910.13404-4-1-0': 'Published in Phys.', '1910.13404-4-1-1': 'Rev. D100 (2019) 112006', '1910.13404-4-2-0': 'CERN for the benefit of the LHCb collaboration.', '1910.13404-4-2-1': 'https://creativecommons.org/licenses/by/4.0/CC-BY-4.0 licence.', '1910.13404-4-3-0': 'arabic', '1910.13404-4-4-0': '# Introduction', '1910.13404-4-5-0': 'The meson is a bound state containing a [MATH] quark with a [MATH] quark.', '1910.13404-4-5-1': 'It has the largest mass of any two differently flavored quarks in a mesonic ground state.', '1910.13404-4-5-2': 'Studies of its production or determination of individual decay widths require measurements of its branching fractions to exclusive final states.', '1910.13404-4-5-3': 'Since the branching fractions of some decay modes of and mesons are accurately known, we determine the ratio of meson production relative to the sum of and mesons.', '1910.13404-4-5-4': 'Here we use techniques similar to those employed for the measurement of meson and baryon fractions [CITATION].', '1910.13404-4-5-5': 'In that paper use is made of the fact that the semileptonic widths of all [MATH]-flavored hadrons with light and strange quarks are equal.', '1910.13404-4-5-6': 'However, both the [MATH] and [MATH] quarks can decay, rendering that concept inapplicable.', '1910.13404-4-5-7': 'Instead we rely on theoretical predictions of the semileptonic decay branching fraction [MATH].', '1910.13404-4-5-8': 'Currently, only the relative production cross-section times the branching fraction of either the [MATH] or [MATH] modes have been measured by the [CITATION], [CITATION] and [CITATION] experiments.', '1910.13404-4-6-0': 'The [MATH] meson production fraction ([MATH]) relative to the sum of [MATH]) and [MATH]) mesons is defined as [EQUATION] where [MATH] refers to the efficiency and branching fraction corrected number of signal events.', '1910.13404-4-6-1': 'The modes containing [MATH] and [MATH] mesons are also corrected for cross-feeds with and decays.', '1910.13404-4-6-2': 'The determination of the corrected yields of the [MATH] decays follows our previous measurement strategy in Ref. [CITATION] where the equations relating the fractions to the corrected yields, including cross-feed contributions, are given.', '1910.13404-4-6-3': 'We also correct for the 0.4% effect of doubly-Cabibbo-suppressed decays and [MATH] mixing.', '1910.13404-4-6-4': 'The relevant hadron branching fractions are listed in Table [REF].', '1910.13404-4-6-5': 'The average semileptonic branching fractions of and , [MATH] is found by averaging measurements from the [CITATION], [CITATION] and [CITATION] experiments, detailed in Ref. [CITATION].', '1910.13404-4-6-6': 'Since only [MATH] modes are detected in this analysis, a correction for the small [MATH] rate of 1% is applied to the denominator of Eq. [REF].', '1910.13404-4-7-0': 'The dominant production mechanism for mesons is gluon-gluon fusion, [MATH].', '1910.13404-4-7-1': 'Non-relativistic quantum chromodynamics is used along with fragmentation functions to predict cross-sections as functions of transverse momentum () and pseudorapidity ([MATH]).', '1910.13404-4-7-2': 'The literature is nicely summarized in Ref. [CITATION].', '1910.13404-4-7-3': 'We define [MATH] to refer to [MATH], [MATH], and [MATH] mesons, while [MATH] refers to [MATH] and [MATH] mesons.', '1910.13404-4-8-0': 'In this analysis [MATH] is determined by measuring the angle of the [MATH] meson with respect to the beam direction by using the positions of the primary [MATH] interaction vertex (PV) and the [MATH] meson decay point into either [MATH], [MATH], or [MATH].', '1910.13404-4-8-1': 'The transverse momentum initially refers to the vector sum of the charmed-hadron and [MATH] momentum transverse to the proton beams.', '1910.13404-4-8-2': 'However, the results are re-interpreted in terms of the [MATH] meson [MATH] by simulating and correcting the effects of the missing momenta.', '1910.13404-4-9-0': 'The production asymmetry between [MATH] and [MATH] mesons, which should be small, is defined as [EQUATION] where [MATH] and [MATH] are the asymmetries in the signal yields and the efficiencies of [MATH] and [MATH] detection, respectively.', '1910.13404-4-9-1': 'The asymmetry in the [MATH] decay is assumed to be zero in this analysis.', '1910.13404-4-10-0': 'The branching fraction predictions from various models of semileptonic decays are listed in Table [REF].', '1910.13404-4-10-1': 'For [MATH] they range from 1.4 to 7.5%, which is quite a large interval.', '1910.13404-4-10-2': 'Branching fractions for other modes are also listed where available.', '1910.13404-4-10-3': 'We use the Z expansion fit results from Ref. [CITATION], and the method II results for Ref. [CITATION].', '1910.13404-4-11-0': 'Some restrictions on models are possible by comparing to lighter [MATH] meson decays.', '1910.13404-4-11-1': 'Since the inclusive semileptonic branching fraction for these decays, [MATH], is about 10.5% and the [MATH] lifetime, [MATH], is 1/3 that of the [MATH], we disregard models that predict 10% or larger values for [MATH] of the .', '1910.13404-4-11-2': 'This excludes from consideration the models of Refs. [CITATION] and [CITATION].', '1910.13404-4-11-3': 'The average model prediction is then [MATH].', '1910.13404-4-11-4': 'The standard deviation is 0.46, which we use to estimate the systematic uncertainty on the model variation.', '1910.13404-4-11-5': 'Results of our measurement without using this branching fraction are also quoted.', '1910.13404-4-12-0': '# Detector, trigger and simulation', '1910.13404-4-13-0': 'The LHCb detector [CITATION] is a single-arm forward spectrometer covering the pseudorapidity range [MATH], designed for the study of particles containing or quarks.', '1910.13404-4-13-1': 'The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the [MATH] interaction region that allows and hadrons to be identified from their characteristically long flight distance; a tracking system that provides a measurement of the momentum, [MATH], of charged particles; two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons; and a downstream system of iron interspersed with chambers is used to identify muons.', '1910.13404-4-14-0': 'The magnetic field deflects positively and negatively charged particles in opposite directions and this can lead to detection asymmetries.', '1910.13404-4-14-1': 'Periodically reversing the magnetic field polarity throughout the data taking almost cancels the effect.', '1910.13404-4-14-2': 'The configuration with the magnetic field pointing upwards (downwards) bends positively (negatively) charged particles in the horizontal plane towards the centre of the LHC ring.', '1910.13404-4-14-3': 'This analysis uses data collected in 2011 (7TeV) and 2016 (13TeV) where appropriate triggers are available.', '1910.13404-4-14-4': 'The data taking was split between magnetic field up and down configurations.', '1910.13404-4-14-5': 'In the 2011 data 0.6(0.4) were collected with the field pointing up (down), while in 2016 the split was 0.9with field up and 0.8with field down.', '1910.13404-4-15-0': 'The trigger [CITATION] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with [MATH] are reconstructed for 2011(2016) data.', '1910.13404-4-16-0': 'Separate hardware triggers are used for the [MATH] and [MATH] samples.', '1910.13404-4-16-1': 'For the former we require a [MATH] pair.', '1910.13404-4-16-2': 'For the latter, we require a single muon with large for the 7TeVdata as used in Ref. [CITATION].', '1910.13404-4-16-3': 'For the 13TeVdata, the single muon trigger was not available, therefore at the hardware trigger stage, events are required to have a hadron, photon or electron transverse energy greater than approximately 3.5 GeV in the calorimeters.', '1910.13404-4-16-4': 'The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary [MATH] interaction vertex as described in Ref. [CITATION].', '1910.13404-4-16-5': 'At least one charged particle must have [MATH] and be inconsistent with originating from a PV.', '1910.13404-4-16-6': 'A multivariate algorithm [CITATION] is used for the identification of secondary vertices consistent with the decay of a hadron.', '1910.13404-4-17-0': 'Simulation is required to model the effects of the detector acceptance and the imposed selection requirements.', '1910.13404-4-17-1': 'In the simulation, [MATH] collisions are generated using [CITATION] with a specific configuration [CITATION].', '1910.13404-4-17-2': 'Decays of unstable particles are described by [CITATION], in which final-state radiation is generated using [CITATION].', '1910.13404-4-17-3': 'The interaction of the generated particles with the detector, and its response, are implemented using the toolkit [CITATION] as described in Ref. [CITATION].', '1910.13404-4-18-0': '# Event selection, signal efficiencies and yields', '1910.13404-4-19-0': '## Selection of [MATH] candidates', '1910.13404-4-20-0': 'The analysis is done separately for the light [MATH] meson modes and the decay.', '1910.13404-4-20-1': 'In each case the triggered events are subject to further filtering requirements.', '1910.13404-4-20-2': 'In addition, the [MATH] sample is subjected to a boosted decision tree (BDT), a multivariate classification method, using the TMVA toolkit [CITATION].', '1910.13404-4-20-3': 'This is not necessary for the [MATH] or [MATH] modes because they have large signals and are relatively free from backgrounds [CITATION].', '1910.13404-4-21-0': 'For the [MATH] final state the initial selection requires that muons that satisfy the candidate trigger each have minimum [MATH] MeV, have large impact parameters with the PV, form a good quality vertex, have a reasonable flight distance significance from the PV, and have a summed [MATH] GeV.', '1910.13404-4-21-1': 'The "companion" muon that is not part of the decay must be well identified and form a good quality vertex with the candidate, which must be downstream of the PV.', '1910.13404-4-22-0': 'To suppress muon tracks that are reconstructed more than once, we require a small minimum opening angle between the muons from the decay and the companion muon momentum measured in the plane transverse to the beam line.', '1910.13404-4-22-1': 'Specifically, this opening angle must be greater than 0.8[MATH].', '1910.13404-4-22-2': 'The invariant mass of the companion muon and the oppositely charged muon from [MATH] must differ from the known value of the [MATH] mass by more than [MATH] [CITATION], while the invariant mass with the same charged muon is required to be larger than [MATH].', '1910.13404-4-23-0': "Since we are dealing with an exclusive final state, we define [EQUATION] where [MATH] is the magnitude of the combination's momentum component transverse to the [MATH]-hadron flight direction.", '1910.13404-4-23-1': 'Figure [REF] shows the distributions of [MATH] versus the invariant [MATH] mass, [MATH], for both data and simulation.', '1910.13404-4-23-2': 'To remove background, a requirement of [MATH] is applied, as indicated by the (red) dashed line.', '1910.13404-4-24-0': 'Since we also measure the production asymmetry between [MATH] and [MATH] mesons, we restrict the angular acceptance of the companion muon to make it more uniform by removing muons close to the edge of the detector, in the bending direction ([MATH]-direction), where large acceptance-induced asymmetries can occur.', '1910.13404-4-24-1': 'Thus, we require that the [MATH]-component of the momentum satisfies [EQUATION] where [MATH] is the muon momentum along the direction of the proton beam downstream of the PV, as is done in Refs. [CITATION].', '1910.13404-4-25-0': 'After these initial restrictions, we turn to the multivariate selection, forming the classifier denoted BDT in the following.', '1910.13404-4-25-1': 'The discriminating variables used are: (a) the [MATH] of the vertex fit of the with the [MATH]; (b) the [MATH], where [MATH]is defined as the [MATH] of the impact parameter with respect to the PV, of the [MATH], [MATH] and their combination; (c) the of the and the [MATH]; and (d) the cosine of the angle between the [MATH] and the meson in the plane perpendicular to the beam direction.', '1910.13404-4-25-2': 'The training sample for signal is simulated [MATH] events, and for background is inclusive [MATH] simulated events.', '1910.13404-4-26-0': 'We then optimize the BDT output threshold by maximizing [MATH], where [MATH] and [MATH] are the number of the signal and background yields in the signal region defined as [MATH].', '1910.13404-4-26-1': 'The sum, [MATH], is the total number of events within these limits, and [MATH] is taken from a fit to the [MATH] distribution.', '1910.13404-4-26-2': 'The optimal BDT output threshold results in a BDT signal efficiency of 89% with a background rejection of 63%, as determined by observing the resulting samples of input signal simulation events and background candidates.', '1910.13404-4-27-0': 'The [MATH] distribution is shown in Fig. [REF].', '1910.13404-4-27-1': 'It consists not only of signal [MATH] events, but also of [MATH] decays, where [MATH], and other [MATH] final states, most importantly [MATH] and [MATH].', '1910.13404-4-27-2': 'We find shapes for these final states using simulation.', '1910.13404-4-27-3': 'The signal shape is a sum of a double Crystal Ball and a bifurcated Gaussian functions.', '1910.13404-4-27-4': 'The sum of the combinatorial and misidentification backgrounds are represented by a Gaussian kernel shape [CITATION].', '1910.13404-4-27-5': 'For the other background modes, we use histograms directly.', '1910.13404-4-27-6': 'These shapes are fitted to the [MATH] distributions in Fig. [REF] in order to determine the [MATH] yields.', '1910.13404-4-27-7': 'The ratio of the [MATH] yield to the [MATH] yield is fixed, after accounting for the relative detection efficiencies, from the LHCb measurement of [MATH], where the first uncertainty is statistical and the second systematic [CITATION]; this convention is used throughout this paper.', '1910.13404-4-27-8': 'The other components of the fit are allowed to vary.', '1910.13404-4-27-9': 'We find [MATH] and [MATH] signal [MATH] events at 7 and 13TeV, respectively, while the backgrounds sum to 950 and 5170 events at the same energies.', '1910.13404-4-27-10': 'These signal yields need to be corrected for the small background from candidates with a correctly reconstructed meson that is paired with a hadron mis-identified as a muon.', '1910.13404-4-28-0': '## Efficiency for [MATH]', '1910.13404-4-29-0': 'Efficiencies are determined using both data [CITATION] and simulation of [MATH], with the generated events weighted to match the [MATH], and [MATH] distributions observed in data.', '1910.13404-4-29-1': 'In addition, we weight accordingly the [MATH] distribution of the muon associated with the .', '1910.13404-4-29-2': 'Weighting the simulation is important since the total efficiencies are functions of these variables.', '1910.13404-4-29-3': 'Efficiencies using data include trigger, and muon identification.', '1910.13404-4-29-4': 'Efficiencies using simulation include detector acceptance, reconstruction and event selection, and removal of beam crossings with an excess number of hits in the detector.', '1910.13404-4-29-5': 'Total efficiencies as a function of [MATH] for different [MATH] intervals are shown in Fig. [REF].', '1910.13404-4-30-0': '## [MATH] selection criteria', '1910.13404-4-31-0': 'Selection criteria for [MATH] final states differ from those containing a [MATH].', '1910.13404-4-31-1': 'The transverse momentum of each hadron must be greater than 0.3GeV, and that of the muon larger than 1.3GeV.', '1910.13404-4-31-2': 'We require [MATH] with respect to any PV, ensuring that tracks do not originate from primary [MATH] interactions.', '1910.13404-4-31-3': 'All final state particles are required to be positively identified using information from the RICH detectors.', '1910.13404-4-31-4': 'Particles from [MATH] decay candidates must have a good fit to a common vertex with [MATH]/ndof [MATH], where ndof is the number of degrees of freedom.', '1910.13404-4-31-5': 'They must also be well separated from the nearest PV, with the flight distance divided by its uncertainty greater than 5.', '1910.13404-4-32-0': 'Candidate [MATH] hadrons are formed by combining [MATH] and muon candidates originating from a common vertex with [MATH]/ndof [MATH] and an [MATH] invariant mass in the range 3.0-5.0GeV.', '1910.13404-4-33-0': 'Background from prompt [MATH] production at the PV needs to be considered.', '1910.13404-4-33-1': 'We use the natural logarithm of the [MATH] impact parameter, IP, with respect to the PV in units of mm.', '1910.13404-4-33-2': 'Requiring ln(IP/mm)[MATH] is found to reduce the prompt component to be below 0.1%, while preserving 97% of all signals.', '1910.13404-4-33-3': 'This restriction allows us to perform fits only to the [MATH] candidate mass spectra to find the [MATH]-hadron decay yields.', '1910.13404-4-34-0': 'The [MATH] candidate mass distributions integrated over [MATH] and [MATH] are shown in Fig. [REF] and consist of a prominent peak resulting from signal, and a small contribution due to combinatorial background from random combinations of particles that pass the selection.', '1910.13404-4-34-1': 'They are fit with a signal component comprised of two Gaussian functions, and a combinatorial background component modeled as a linear function.', '1910.13404-4-34-2': 'The fitted yields are listed in Table [REF].', '1910.13404-4-34-3': 'These numbers must be corrected for hadrons that are mis-identified as muons, and for semileptonic decays of and hadrons that produce [MATH] and [MATH] mesons.', '1910.13404-4-35-0': 'In Table [REF] the column labeled "fake muons" shows the yields of wrong-sign [MATH] and [MATH] combinations that pass the selections.', '1910.13404-4-35-1': 'These yields provide good estimates of the fake muon contributions in the signal samples, which are very small.', '1910.13404-4-35-2': 'Following the procedure in Ref. [CITATION], we find the cross-feed corrections of [MATH] and [MATH] to be twice the measured yields for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV), and for [MATH], which are [MATH] (7TeV) and [MATH] (13TeV).', '1910.13404-4-35-3': 'Relative efficiencies for detecting final states with a single extra hadron are taken into account when subtracting these yields.', '1910.13404-4-36-0': '## Efficiencies for [MATH] and [MATH]', '1910.13404-4-37-0': 'Similar methods based on data, as implemented for the decay, are used to evaluate the efficiencies for trigger and particle identification.', '1910.13404-4-37-1': 'Simulation is also used to determine the efficiencies of event selection and reconstruction of these modes.', '1910.13404-4-37-2': 'The total efficiencies for [MATH] meson decays into [MATH] and [MATH] are shown in Fig. [REF].', '1910.13404-4-38-0': '# Results', '1910.13404-4-39-0': '## Corrections to the [MATH] distributions due to the missing neutrino', '1910.13404-4-40-0': 'Since the production kinematics of [MATH] and mesons can differ as functions of [MATH] and [MATH], we need to measure [MATH] as functions of these variables.', '1910.13404-4-40-1': 'The measurement of [MATH] is straightforward, however, we do not measure directly the [MATH] of the [MATH]-flavored hadron because of the missing neutrino, and in the case of the [MATH] meson possible missing extra particles.', '1910.13404-4-40-2': 'Following a procedure similar to the one used in Ref. [CITATION], we determine a correction factor, [MATH], that is the ratio of the average reconstructed to true [MATH] as a function of the invariant mass of the charmed hadron plus muon.', '1910.13404-4-40-3': 'The ratio distribution as a function of hadron-muon invariant mass are shown in Fig. [REF].', '1910.13404-4-40-4': 'The average correction, the [MATH]-factor, is shown on the figure.', '1910.13404-4-40-5': 'For the [MATH] meson it varies from 0.75 to unity over the interval from 3 GeV to the [MATH] mass, and for the meson it varies from 0.85 to unity over the interval from 4 GeV to the mass.', '1910.13404-4-41-0': '## fraction results', '1910.13404-4-42-0': 'The ratio of production fractions, [MATH], are shown as functions of [MATH] and [MATH] in Fig. [REF].', '1910.13404-4-42-1': 'There is little dependence on [MATH], but the decrease as a function of [MATH] is noticeable.', '1910.13404-4-43-0': 'To describe the [MATH] dependence we use an equation of the form [EQUATION] where [MATH] represents the overall normalization and contains the total global systematic uncertainty, thus, [MATH]; [MATH] is taken as 7.2 GeV, close to the average of the .', '1910.13404-4-43-1': 'The slopes, [MATH], are similar in size to those measured for the [MATH] meson fraction ratio as a function of [CITATION].', '1910.13404-4-43-2': 'Results of fits to the data using Eq. [REF] are listed in Table [REF].', '1910.13404-4-44-0': 'The average fractions in the interval [MATH]GeVare found by integrating over [MATH].', '1910.13404-4-44-1': 'To allow for facile changes to our results due to improved theoretical predictions, we provide the results for [EQUATION]', '1910.13404-4-44-2': 'Next we give the result on the fractions ratio [EQUATION] where the third uncertainty is due to the theoretical prediction of [MATH].', '1910.13404-4-44-3': 'To find [MATH] just double these numbers.', '1910.13404-4-45-0': 'We also measure the ratio of the [MATH] production fraction at [MATH] to that at [MATH].', '1910.13404-4-45-1': 'Figure [REF] shows the ratio as functions of and [MATH].', '1910.13404-4-45-2': 'Here most of the systematic uncertainties cancel.', '1910.13404-4-45-3': 'The integrated value of the ratio of [MATH] and [MATH] is measured as [MATH], consistent with no increase in the fraction ratio as a function of center-of-mass energy.', '1910.13404-4-46-0': 'The fraction with respect to inclusive [MATH]-hadron production can be derived from the information in previous LHCb [MATH]-hadron fraction papers Ref. [CITATION].', '1910.13404-4-46-1': 'There the measured values of the ratios of [MATH]-hadron fractions over the same range in terms of the [MATH]-hadron are for mesons ([MATH]) and baryons [EQUATION] where the uncertainties contain both statistical and systematic components added in quadrature.', '1910.13404-4-46-2': 'For the measurement of the [MATH] fraction at 7TeV, the dominant systematic uncertainty is from the lack of the knowledge of [MATH] at that time [CITATION]; here the value and uncertainty have been recalculated according to the latest value of [MATH] from the PDG [CITATION].', '1910.13404-4-47-0': 'Taking the sum of all the [MATH]-hadron fractions to be unity, and ignoring [MATH] here because it is so small, [EQUATION] where [MATH] is a correction factor derived in Ref. [CITATION] that accounts for heavier [MATH]-baryons, mainly the [MATH].', '1910.13404-4-47-1': 'Solving for [MATH] yields [EQUATION]', '1910.13404-4-47-2': 'We find that [EQUATION] where the first uncertainty is statistical, the second is systematic, and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-4-47-3': 'We also provide the result for [MATH], [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-4-48-0': '## The [MATH] production asymmetry', '1910.13404-4-49-0': 'The production asymmetries are measured in two different magnetic field configurations and then averaged.', '1910.13404-4-49-1': 'No significant asymmetry is observed in any intervals of [MATH] or [MATH].', '1910.13404-4-49-2': 'The results are summarized in Table [REF].', '1910.13404-4-50-0': 'Averaging the [MATH] production asymmetries over [MATH] and [MATH], we find [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-4-51-0': '# Systematic uncertainties', '1910.13404-4-52-0': 'Systematic uncertainties are separated into two categories: "global", which apply across the phase space, and "local", which are calculated in each two-dimensional [MATH] bin.', '1910.13404-4-52-1': 'These uncertainties are listed in Table [REF].', '1910.13404-4-53-0': 'First let us consider the [MATH] decay.', '1910.13404-4-53-1': 'The uncertainty due to the signal shape used to fit the [MATH] distribution is determined by changing the baseline signal shape, the sum of a double sided Crystal Ball function and a bifurcated Gaussian, to a kernel estimation.', '1910.13404-4-53-2': 'To find the shape of the combinatorial and misidentification backgrounds we use simulated inclusive samples of [MATH] events not including decays.', '1910.13404-4-53-3': 'A total of 500 samples are generated and different fits to the samples are performed to determine the possible uncertainty.', '1910.13404-4-53-4': 'This procedure is also used for the [MATH] measurement.', '1910.13404-4-53-5': 'We call contributions to the [MATH] mass spectrum "feed-down" contributions, occurring from other decay channels including [MATH], [MATH], and [MATH].', '1910.13404-4-53-6': 'The systematic uncertainty results from the uncertainties in their branching fractions.', '1910.13404-4-53-7': 'Different decay models for decays can change the [MATH] shape.', '1910.13404-4-53-8': 'We use the model of Ebert et al.[CITATION] for our baseline prediction.', '1910.13404-4-53-9': 'Then we also use the model by Kiselev [CITATION] to find the efficiencies and take half the difference as the systematic uncertainty.', '1910.13404-4-53-10': 'We also estimate the uncertainty due to the sensitivity to various selection requirements and simulation statistics.', '1910.13404-4-53-11': 'The muon identification efficiencies are determined from data using inclusive samples of decay where one of the muon candidates is not identified.', '1910.13404-4-53-12': 'The trigger efficiency is determined by using three independent samples of events, those that trigger on a [MATH], those that triggered on something else in the event, and those that trigger on both the and something else.', '1910.13404-4-53-13': 'These samples are then used to compute the trigger efficiencies in two-dimensional [MATH] and [MATH] bins.', '1910.13404-4-54-0': 'Next, we turn to the [MATH] modes.', '1910.13404-4-54-1': 'The efficiencies and their uncertainties for identifying pions and kaons are determined by using almost background free samples of [MATH] decays.', '1910.13404-4-54-2': 'The trigger and muon identification efficiencies, and their uncertainties, are obtained in the same manner as for the [MATH] mode.', '1910.13404-4-54-3': 'There are small systematic uncertainties related to efficiency estimates and the assumed [MATH] to [MATH] mixtures, as well as simulation statistics.', '1910.13404-4-54-4': 'Global systematic uncertainties include the hadron branching fractions listed in Table [REF], cross-feed corrections arising from and decays into [MATH] events, and a global hadron plus photon multiplicity requirement.', '1910.13404-4-54-5': 'The latter is evaluated with data.', '1910.13404-4-55-0': '# Conclusions', '1910.13404-4-56-0': 'In 7 and 13TeV[MATH] collisions the product of [MATH] with the relative fraction of mesons with respect to the sum of and mesons in the ranges [MATH] and [MATH] is found to be [EQUATION]', '1910.13404-4-56-1': 'We derive the product of [MATH] at the two energies as [EQUATION]', '1910.13404-4-56-2': 'Using the average of the theoretical prediction [MATH], where the uncertainty is given by the standard deviation derived from the distribution of the models, we determine [EQUATION] where the first uncertainties are statistical, the second systematic, and the third due to the theoretical prediction of [MATH].', '1910.13404-4-56-3': 'There is a small dependence on the transverse momentum of the meson, but no dependence on its pseudorapidity is observed.', '1910.13404-4-56-4': 'We also report [EQUATION] where the first uncertainty is statistical, the second is systematic including that from [MATH] and the third is from the fractions of the [MATH] and [MATH] given in Eq. [REF].', '1910.13404-4-57-0': 'The ratio of fractions, [MATH], for 13TeV/7TeVis consistent with no increase in the fraction.', '1910.13404-4-57-1': 'Furthermore, using the assumption of no violation in the [MATH] decay, we find that the average asymmetry in [MATH] production is consistent with zero.', '1910.13404-4-57-2': 'The measurements are [MATH], and [MATH] at center-of-mass energies of 7 and 13TeV, respectively.', '1910.13404-4-58-0': 'These results are useful to extract absolute branching fractions for measurements, albeit with a relatively large uncertainty.', '1910.13404-4-58-1': 'They also challenge QCD calculations to predict the measured fractions and explain the consistency between the fractions measured at 7 and 13TeV[CITATION].'}

0905.2211

{'0905.2211-1-0-0': 'Universal Constraints on Conformal Operator Dimensions Vyacheslav S. Rychkov[MATH] and Alessandro Vichi[MATH] Scuola Normale Superiore and INFN, Pisa, Italy', '0905.2211-1-1-0': '[MATH] Institut de Theorie des Phenomenes Physiques, EPFL, Lausanne, Switzerland', '0905.2211-1-2-0': 'We continue the study of model-independent constraints on the unitary Conformal Field Theories in 4-Dimensions, initiated in http://arxiv.org/abs/0807.0004arXiv:0807.0004.', '0905.2211-1-2-1': 'Our main result is an improved upper bound on the dimension [MATH] of the leading scalar operator appearing in the OPE of two identical scalars of dimension [MATH]: [EQUATION]', '0905.2211-1-2-2': 'In the interval [MATH] this universal bound takes the form [EQUATION].', '0905.2211-1-2-3': 'The proof is based on prime principles of CFT: unitarity, crossing symmetry, OPE, and conformal block decomposition.', '0905.2211-1-2-4': 'We also discuss possible applications to particle phenomenology and, via a 2-D analogue, to string theory.', '0905.2211-1-3-0': '# Introduction and formulation of the problem', '0905.2211-1-4-0': 'Our knowledge about non-supersymmetric Conformal Field Theories (CFTs) in four dimensions (4D) is still quite incomplete.', '0905.2211-1-4-1': 'Suffices it to say that not a single nontrivial example is known which would be solvable to the same extent as, say, the 2D Ising model.', '0905.2211-1-4-2': 'However, we do not doubt that CFTs must be ubiquitous.', '0905.2211-1-4-3': 'For example, non-supersymmetric gauge theories with [MATH] colors and [MATH] flavors are widely believed to have conformal windows in which the theory has a conformal fixed point in the IR, with evidence from large [MATH] analysis [CITATION], supersymmetric analogues [CITATION], and lattice simulations [CITATION].', '0905.2211-1-4-4': 'Since these fixed points are typically strongly coupled, we do not have much control over them.', '0905.2211-1-4-5': 'In this situation particularly important are general, model-independent properties.', '0905.2211-1-5-0': 'One example of such a property is the famous unitarity bound [CITATION] on the dimension [MATH] of a spin [MATH] conformal primary operator [MATH] : [EQUATION]', '0905.2211-1-5-1': 'These bounds are derived by imposing that the two point function [MATH] have a positive spectral density.', '0905.2211-1-6-0': 'As is well known, 3-point functions in CFT are fixed by conformal symmetry up to a few arbitrary constants (Operator Product Expansion (OPE) coefficients).', '0905.2211-1-6-1': 'The next nontrivial constraint thus appears at the 4-point function level, and is known as the conformal bootstrap equation.', '0905.2211-1-6-2': 'It says that OPE applied in direct and crossed channel should give the same result (see Fig. [REF]).', '0905.2211-1-7-0': 'The bootstrap equation goes back to the early days of CFT [CITATION].', '0905.2211-1-7-1': 'However, until recently, not much useful general information has been extracted from it.', '0905.2211-1-7-2': 'All spins and dimensions can apriori enter the bootstrap on equal footing, and this seems to lead to unsurmountable difficulties.', '0905.2211-1-8-0': 'Recently, however, tangible progress in the analysis of bootstrap equations was achieved in [CITATION].', '0905.2211-1-8-1': 'Namely, it was found that, in unitary theories, the functions entering the bootstrap equations (conformal blocks) satisfy certain positivity properties which lead to general necessary conditions for the existence of solutions.', '0905.2211-1-9-0': 'The concrete problem considered in [CITATION], and which we will continue to discuss here, was as follows.', '0905.2211-1-9-1': 'In an arbitrary unitary CFT a Hermitean scalar primary [MATH] of dimension [MATH] was singled out.', '0905.2211-1-9-2': 'The conformal bootstrap equation for its 4-point function [MATH] was studied under the sole assumption that all scalars in the OPE [MATH] have dimension above a certain number, call it [MATH] [EQUATION]', '0905.2211-1-9-3': 'It was shown that the conformal bootstrap does not allow for a solution unless [EQUATION] where [MATH] is a certain continuous function, computed numerically.', '0905.2211-1-9-4': 'We stress that this conclusion was reached without making any assumptions about dimensions or spins of other operators appearing in the OPE, beyond those implied by the unitarity bounds.', '0905.2211-1-9-5': 'Nor any assumptions about the OPE coefficients were made (apart from their reality, which is again implied by unitarity).', '0905.2211-1-10-0': 'In other words, in any unitary 4D CFT, the OPE of any scalar primary [MATH] must contain at least one scalar field [MATH] with dimension not larger than [MATH].', '0905.2211-1-11-0': 'Incidentally, the function [MATH] was found to satisfy [MATH] which is quite natural since [MATH] corresponds to the free field whose OPE contains the operator :[MATH]: of dimension [MATH].', '0905.2211-1-12-0': 'What makes the result like ([REF]) possible?', '0905.2211-1-12-1': 'The basic reason is that, in any theory, crossing symmetry relation of Fig. [REF] cannot be satisfied term by term, but only by cancellations among various terms.', '0905.2211-1-12-2': 'The guaranteed presence of the unit operator in the OPE ([REF]) creates a certain crossing symmetry deficit, which has to be balanced by other fields.', '0905.2211-1-12-3': 'The idea is to show that this cannot happen unless at least one scalar of sufficiently low dimension is present.', '0905.2211-1-13-0': 'Technically, the method of [CITATION] consists of 3 steps (see Section [REF] for a detailed review):', '0905.2211-1-14-0': 'We Taylor-expand the conformal bootstrap equation near the self-dual point" configuration having equal conformal cross-ratios [MATH].', '0905.2211-1-14-1': 'The expansion is truncated to a certain finite order [MATH].', '0905.2211-1-15-0': 'We systematically search for positivity properties satisfied by linear combinations of Taylor coefficients of the conformal blocks, for fields appearing in the RHS of the OPE ([REF]).', '0905.2211-1-15-1': 'A found positivity property implies that the crossing symmetry deficit cannot be balanced and rules out a CFT with a given [MATH] and [MATH].', '0905.2211-1-16-0': 'For fixed [MATH], the bound [MATH] is then computed as the point separating those [MATH] for which a positivity property exists, from those ones for which it does not (Fig. [REF]).', '0905.2211-1-17-0': 'The nature of the method is such that increasing [MATH] can make the bound only stronger.The optimal bound should in principle be recoverable in the limit [MATH].', '0905.2211-1-17-1': 'In practice the value of [MATH] is determined by the available computer resources and algorithmic efficiency.', '0905.2211-1-17-2': 'The best bound found in [CITATION], plotted in Fig. [REF], corresponds to [MATH].', '0905.2211-1-18-0': 'The purpose of this paper is to present an improvement of the bound ([REF]) obtained by using the method of [CITATION] with larger values of [MATH], up to [MATH].', '0905.2211-1-18-1': 'The new results are interesting in two ways.', '0905.2211-1-18-2': 'First, pure numerical improvement turns out to be significant.', '0905.2211-1-18-3': 'Second, [MATH] happens to be large enough so that we start observing saturation of the bound.', '0905.2211-1-18-4': 'So we believe our current results are close to the optimal ones achievable with this method.', '0905.2211-1-19-0': 'The paper is organized as follows.', '0905.2211-1-19-1': 'In Section [REF] we review the conformal bootstrap equations.', '0905.2211-1-19-2': 'In Section [REF] we review the connection of the bound ([REF]).', '0905.2211-1-19-3': 'with positivity properties satisfied by the conformal block expansion coefficients.', '0905.2211-1-19-4': 'In Section [REF] we present and discuss our results.', '0905.2211-1-19-5': 'We also mention accompanying results which we obtain for an analogous problem in 2D.', '0905.2211-1-19-6': 'In Section [REF] we propose several future applications and extensions of our method, with emphasis on connections to phenomenology and string theory.', '0905.2211-1-19-7': 'In Section [REF] we summarize and conclude.', '0905.2211-1-19-8': 'In Appendix [REF] we collect some details about our numerical algorithms.', '0905.2211-1-19-9': 'In Appendix [REF] we include the tables on which plots in Section [REF] are based.', '0905.2211-1-20-0': '# Review of conformal bootstrap', '0905.2211-1-21-0': 'We will review the conformal bootstrap equation in its simplest form-as applied to the 4-point function of identical scalars [MATH].', '0905.2211-1-21-1': 'We largely follow [CITATION], where a more detailed discussion and references can be found.', '0905.2211-1-22-0': '## Conformal block decomposition', '0905.2211-1-23-0': 'Let [MATH] be a Hermitean scalar primary operator.', '0905.2211-1-23-1': 'The operator product expansion (OPE) [MATH] contains, in general, infinitely many primary fields of arbitrary high spins and dimensions: [EQUATION]', '0905.2211-1-23-2': 'Here', '0905.2211-1-24-0': 'We assume that the OPE converges in the following weak sense: it gives a convergent power series expansion for any [MATH]-point function [EQUATION] provided that [MATH], i.e. [MATH] is closer to the origin than any other local field insertion (see Fig. [REF]).', '0905.2211-1-24-1': 'This assumption can be justified by using radial quantization ([CITATION], Sect. 2.9), and checked explicitly in free field theory.', '0905.2211-1-24-2': 'For rigorous mathematical results about OPE convergence see [CITATION].', '0905.2211-1-25-0': 'The OPE ([REF]) can be used to obtain conformal block decomposition of the 4-point function [MATH]: [EQUATION] where [MATH] are the conformal cross-ratios.', '0905.2211-1-25-1': 'This representation is obtained by using the OPE in the 12 and 34 channels.', '0905.2211-1-25-2': 'The conformal blocks [MATH] sum up the contributions of the primary [MATH] and all its descendants.', '0905.2211-1-25-3': 'Their explicit expressions were found by Dolan and Osborn [CITATION]: [EQUATION]', '0905.2211-1-25-4': 'Notice the judicious introduction of the auxiliary variables [MATH] and [MATH].', '0905.2211-1-25-5': 'When the theory is formulated in the Euclidean space, these variables are complex-conjugates of each other.', '0905.2211-1-25-6': 'To understand their meaning, it is convenient to use the conformal group freedom to send [MATH] and to put the other three points in a plane, as in Fig. [REF].', '0905.2211-1-25-7': "Then it's easy to show that [EQUATION] where [MATH] are the coordinates of [MATH] in the plane, chosen so that [MATH] corresponds to [MATH] halfway between [MATH] and [MATH].", '0905.2211-1-25-8': 'This self-dual configuration, for which [MATH], will play an important role below.', '0905.2211-1-25-9': 'We can see that the [MATH] variable is a natural extension of the usual complex coordinate of the 2D CFT to the 4D case.', '0905.2211-1-26-0': 'According to the above discussion, the OPE is expected to converge for [MATH].', '0905.2211-1-26-1': 'Conformal block decomposition is a partial resummation of the OPE and thus also converges at least in this range.', '0905.2211-1-26-2': 'In fact, below we will only use convergence around the self-dual point [MATH].', '0905.2211-1-26-3': 'However, conformal blocks, as given by ([REF]), are regular (real-analytic) in a larger region, namely in the [MATH]-plane with the [MATH] cut along the real axis (see Fig. [REF]).', '0905.2211-1-26-4': 'The conformal block decomposition is thus expected to converge in this larger region.', '0905.2211-1-26-5': 'One can check that this indeed happens in the free scalar theory.', '0905.2211-1-27-0': 'One can intuitively understand the reason for this extended region of regularity.', '0905.2211-1-27-1': 'The condition for the OPE convergence, as stated above, does not treat the points [MATH] and [MATH] symmetrically.', '0905.2211-1-27-2': 'On the other hand, the conformal blocks are completely symmetric in [MATH] and so must be the condition for their regularity.', '0905.2211-1-27-3': 'The appropriate condition is as follows: the conformal block decomposition in the [MATH]-[MATH] channel is regular and convergent if there is a sphere separating the points [MATH] from the points [MATH].', '0905.2211-1-27-4': 'For the configuration of Fig. [REF], such a sphere exists as long as [MATH] is away from the cut.', '0905.2211-1-28-0': '## Conformal bootstrap and the sum rule', '0905.2211-1-29-0': 'The 4-point function in ([REF]) must be symmetric under the interchange of any two [MATH], and its conformal block decomposition ([REF]) has to respect this symmetry.', '0905.2211-1-29-1': 'The symmetry with respect to [MATH] or [MATH] is already built in, since only even spins are exchanged [CITATION].', '0905.2211-1-29-2': 'On the contrary, the symmetry with respect to [MATH] gives a condition [EQUATION] which is not automatically satisfied for [MATH] given by ([REF]).', '0905.2211-1-29-3': 'This nontrivial constraint on dimensions, spins, and OPE coefficients of all operators appearing in the OPE [MATH] is known as the conformal bootstrap equation.', '0905.2211-1-29-4': 'Physically it means that OPE applied in 12-34 and 14-23 channels should give the same result (Fig. [REF]).', '0905.2211-1-30-0': 'In the [MATH]-plane of Section [REF], the LHS of ([REF] ) has a cut along [MATH] while the RHS has a cut along [MATH].', '0905.2211-1-30-1': 'Thus, if ([REF]) is satisfied, the cuts have to cancel, and the resulting [MATH] is real analytic everywhere except for [MATH].', '0905.2211-1-31-0': 'In [CITATION], we found it useful to rewrite ([REF]) by separating the unit operator contribution, which gives [EQUATION]', '0905.2211-1-31-1': 'The LHS of this equation is the crossing symmetry deficit created by the presence of the unit operator in the OPE.', '0905.2211-1-31-2': 'This deficit has to be balanced by contributions of the other fields in the RHS.', '0905.2211-1-32-0': 'In practice it is convenient to normalize ([REF]) by dividing both sides by [MATH].', '0905.2211-1-32-1': 'The resulting sum rule takes the form: [EQUATION]', '0905.2211-1-32-2': 'The F-functions" [MATH] are real and regular in the full [MATH]-plane cut along [MATH].', '0905.2211-1-32-3': 'In particular, the [MATH] behavior at the self-dual point [MATH] is regular.', '0905.2211-1-33-0': 'All F-functions vanish near the points [MATH] and [MATH].', '0905.2211-1-33-1': 'Thus the sum rule can never be satisfied near these points if only finitely many terms are present in the RHS.', '0905.2211-1-33-2': 'The OPEs containing finitely many primaries are ruled out.', '0905.2211-1-34-0': '# Positivity argument', '0905.2211-1-35-0': 'The main idea of [CITATION] was very simple, and can be described as follows.', '0905.2211-1-35-1': 'Suppose that for a given spectrum of operator dimensions and spins [MATH] the sum rule ([REF]), viewed as an equation for the coefficients [MATH], has no solution.', '0905.2211-1-35-2': 'Then of course such a spectrum would be ruled out.', '0905.2211-1-36-0': 'Any concrete realization of this idea needs a practical criterium to show that there is no solution.', '0905.2211-1-36-1': 'For a prototypical example of such a criterium, imagine that a certain derivative, e.g. [MATH] (see ([REF])), when applied to every [MATH] and evaluated at a certain point, is strictly positive (positivity property).', '0905.2211-1-36-2': 'Since the same derivative applied to the LHS of ([REF]) gives identically zero, a solution where all coefficients [MATH] are non-negative would clearly be impossible.', '0905.2211-1-36-3': 'We refer to this simple reasoning as the positivity argument.', '0905.2211-1-37-0': 'One can imagine more general criteria using different differential operators, and applying them at different points.', '0905.2211-1-37-1': 'In [CITATION], we found it convenient to apply differential operators precisely at the self-dual point [MATH], [MATH].', '0905.2211-1-37-2': 'One can show that the F-functions are even with respect to this point both in the [MATH] and [MATH] directions: [EQUATION].', '0905.2211-1-37-3': 'Thus, all odd-order derivatives vanish, and a sufficiently general differential operator (linear functional) takes the form: [EQUATION] where [MATH] is some fixed finite number, and [MATH] are fixed real coefficients.', '0905.2211-1-37-4': 'Notice the exclusion of the constant term [MATH], in order to have [MATH].', '0905.2211-1-38-0': 'Assume that for certain fixed [MATH] and [MATH] we manage to find a linear functional of this form such that (positivity property) [EQUATION]', '0905.2211-1-38-1': 'Moreover, assume that all but a finite number of these inequalities are actually strict: [MATH] Then the sum rule cannot be satisfied, and such a spectrum, corresponding to a putative OPE ([REF]), is ruled out.', '0905.2211-1-39-0': 'The proof uses the above positivity argument.', '0905.2211-1-39-1': 'Since [MATH] the positivity property implies that only those primaries for which [MATH] would be allowed to appear in the RHS of the sum rule with nonzero coefficients.', '0905.2211-1-39-2': 'By assumption, there are at most a finite number of such primaries.', '0905.2211-1-39-3': 'However, as noted in Section [REF], finitely many terms can never satisfy the sum rule globally, because of the behavior near [MATH].', '0905.2211-1-39-4': 'Q.E.D.', '0905.2211-1-40-0': 'While the above formal reasoning is quite sufficient to understand our results, in [CITATION] the sum rule was also given an alternative interpretation in terms of convex geometry.', '0905.2211-1-40-1': 'In this more visual picture, linear combinations of F-functions with arbitrary positive coefficients form a convex cone in the space of two-variable functions.', '0905.2211-1-40-2': 'One can consider the full function space or its finite-dimensional subspace corresponding to Taylor-expanding up to order [MATH].', '0905.2211-1-40-3': 'Positivity property ([REF]) means that there is a hyperplane separating the function 1 from the convex cone.', '0905.2211-1-40-4': 'Thus it implies that the sum rule cannot be satisfied.', '0905.2211-1-40-5': 'The converse is almost true, modulo questions of convergence.', '0905.2211-1-41-0': 'Clearly, the language of linear functionals provides an equivalent, dual formulation of the problem.', '0905.2211-1-41-1': 'This formulation is also especially convenient from the point of view of checking our results independently.', '0905.2211-1-41-2': "It's not so important how we find the functionals.", '0905.2211-1-41-3': 'As long as we publish the functional coefficients [MATH], anyone can verify that the inequalities ([REF]) are satisfied.', '0905.2211-1-42-0': '# Results, discussion, and 2D analogue', '0905.2211-1-43-0': 'As discussed in Section [REF], we are interested in computing an upper bound [MATH][REF][MATH] for the dimension [MATH] of the leading scalar in the OPE [MATH], universal for all unitary 4D CFTs.', '0905.2211-1-43-1': 'In [CITATION], we have computed such a bound in the interval [MATH] using the sum rule of Section [REF] truncated to the [MATH] derivative order.', '0905.2211-1-43-2': 'That bound is reproduced in Fig. [REF].', '0905.2211-1-44-0': 'We now present the results of our latest study, obtained for larger values of [MATH].', '0905.2211-1-44-1': 'These results are plotted in Fig [REF] as a collection of curves [MATH], [MATH], where the index [MATH] denotes the number of derivatives used to obtain the bound.', '0905.2211-1-44-2': 'The bound naturally gets stronger as [MATH] increases (see below), and thus the lowest curve [MATH] is the strongest bound to date.', '0905.2211-1-44-3': 'In the considered interval [MATH] this bound is well approximated (within [MATH]) by [EQUATION]', '0905.2211-1-44-4': 'To obtain the bounds of Fig. [REF], we used the positivity argument from [CITATION], as reviewed in Section [REF].', '0905.2211-1-44-5': 'Namely, for points lying on the curves [MATH] we are able to find a linear functional of the form ([REF]) satisfying the positivity property ([REF]).', '0905.2211-1-44-6': 'The numerical procedure that we use to find these positive functionals is described in some detail in Appendix A.', '0905.2211-1-45-0': 'Several comments are in order here.', '0905.2211-1-46-0': 'We have actually computed the bound only for a discrete number of [MATH] values, shown as points in Fig. [REF].', '0905.2211-1-46-1': 'The tables of these computed values are given in Appendix B. Behavior for [MATH] can be better appreciated from the logarithmic-scale plot in Fig. [REF].', '0905.2211-1-47-0': 'We do not see any significant indication which could suggest that the curves [MATH] do not interpolate smoothly in between the computed points.', '0905.2211-1-47-1': 'Small irregularities in the slope are however visible at several points in Figs. [REF],[REF].', '0905.2211-1-47-2': 'These irregularities are understood; they originate from the necessity to discretize the infinite system of inequalities ([REF]), see Appendix A for a discussion.', '0905.2211-1-47-3': 'In our computations the discretization step was chosen so that these irregularities are typically much smaller than the improvement of the bound that one gets for [MATH].', '0905.2211-1-48-0': 'For each [MATH] the bound [MATH] is near-optimal, in the sense that no positive functional involving derivatives up to order [MATH] exists for [EQUATION].', '0905.2211-1-48-1': 'We estimate [MATH] from the analysis of residuals in the fit of [MATH] by a smooth curve like in ([REF]).', '0905.2211-1-49-0': 'On the other hand, by increasing [MATH] we are allowing more general functionals, and thus the bound [MATH] can and does get stronger.', '0905.2211-1-49-1': 'This is intuitively clear since for larger [MATH] the Taylor-expanded sum rule includes more and more constraints.', '0905.2211-1-50-0': 'Compared to the results of [CITATION], the bound on the anomalous dimension [MATH] is improved by [MATH] in the range [MATH] that we explored.', '0905.2211-1-51-0': 'We have pushed our analysis to such large values of [MATH] in the hope of seeing that the bound saturates as [MATH].', '0905.2211-1-51-1': 'Indeed, we do observe signs of convergence in Figs. [REF],[REF], especially at [MATH].', '0905.2211-1-51-2': 'In fact, we have observed that the bounds [MATH] starting from [MATH] follow rather closely the asymptotic behavior [EQUATION].', '0905.2211-1-51-3': 'An approximation to the optimal bound [MATH] can thus be found by performing for each [MATH] a fit to this formula.', '0905.2211-1-51-4': 'This approximation is shown by a dashed line in Fig. [REF].', '0905.2211-1-51-5': 'From this rough analysis we conclude that the optimal bound on the anomalous dimension [MATH] is probably within [MATH] from our current bound.', '0905.2211-1-52-0': 'We have [MATH] continuously as [MATH].', '0905.2211-1-52-1': 'The point [MATH] corresponds to the free scalar theory.', '0905.2211-1-53-0': "We don't know of any unitary CFTs that saturate our bound at [MATH], see the discussion in Section 6 of [CITATION].", '0905.2211-1-53-1': 'We know however a family of unitary 4D CFTs in which [MATH] and which are consistent with our bound (the red dotted line in Fig. [REF]).', '0905.2211-1-53-2': "This generalized free scalar theory is defined for a fixed [MATH] by specifying the 2-point function [EQUATION] and defining all other correlators of [MATH] via Wick's theorem.", '0905.2211-1-53-3': 'This simple procedure gives a well-defined CFT, unitary as long as [MATH], which can be described by a nonlocal action [EQUATION].', '0905.2211-1-53-4': 'The full operator content of this theory can be recovered by studying the OPE [MATH].', '0905.2211-1-53-5': 'In particular, the leading scalar in this OPE has dimension [MATH]', '0905.2211-1-54-0': '## 2D analogue', '0905.2211-1-55-0': 'Although our main interest is in the 4D CFTs, our methods allow a parallel treatment of the 2D case.', '0905.2211-1-55-1': 'The main characteristics of the 2D situation were described in Section 6.1 of [CITATION], here will briefly review them.', '0905.2211-1-56-0': 'At present we can only take advantage of the finite-dimensional [MATH] symmetry and not of the full Virasoro algebra of the 2D CFTs.', '0905.2211-1-56-1': 'In particular, our results are independent of the 2D central charge [MATH].', '0905.2211-1-57-0': 'The unitarity bounds for [MATH] primaries in 2D have the form [EQUATION] where [MATH] is the Lorentz spin.', '0905.2211-1-58-0': 'The [MATH] conformal blocks in 2D are known explicitly [CITATION]: [EQUATION]', '0905.2211-1-58-1': 'Using the unitarity bounds, the known conformal blocks, and the sum rule ([REF]), valid in any dimension, we can try to answer the same question as in 4D.', '0905.2211-1-58-2': 'Namely, for a [MATH] scalar primary [MATH] of dimension [MATH], what is an upper bound on the dimension [MATH] of the first scalar operator appearing in the OPE [MATH]?', '0905.2211-1-58-3': 'I.e. we want a 2D analogue of the bound ([REF]).', '0905.2211-1-58-4': 'Since the free scalar is dimensionless in 2D, the region of interest is [MATH].', '0905.2211-1-59-0': 'Fig. [REF] summarizes our current knowledge of this bound:', '0905.2211-1-60-0': 'It is instructive to compare this plot with its 4D counterpart, Fig. [REF].', '0905.2211-1-60-1': 'While we do not know of any CFTs saturating the 4D bound, the 2D unitary minimal models [MATH], [MATH], contain the OPEs [EQUATION] which come quite close to saturating the 2D bound.', '0905.2211-1-61-0': 'More precisely, our 2D bound starts at [MATH] tangentially to the line [MATH] realized in the free scalar theory, then grows monotonically and passes remarkably closely above the Ising model point [MATH].', '0905.2211-1-61-1': 'After a knee at the Ising point, the bound continues to grow linearly, passing in the vicinity of the higher minimal model points ([REF]).', '0905.2211-1-62-0': 'It is curious to note that if we did not know beforehand about the Ising model, we could have conjectured its field dimensions and the basic OPE [MATH] based on the singular behavior of the 2D bound at [MATH].', '0905.2211-1-63-0': 'On the other hand, nothing special happens with the 2D bound at the higher minimal model points, it just interpolates linearly in between.', '0905.2211-1-63-1': 'Most likely, this does not mean that there exist other unitary CFTs with intermediate operator dimensions.', '0905.2211-1-63-2': 'Rather, this behavior suggests that the single conformal bootstrap equation used to derive the bound is not powerful enough to fully constrain a CFT.', '0905.2211-1-64-0': 'In comparison, it is a bit unfortunate that the 4D bound does not exhibit any singular points which would immediately stand out as CFT candidates.', '0905.2211-1-64-1': 'Nevertheless, if we assume that the shape of the 4D bound is a result of an interpolation between existing CFTs (as it is the case in 2D), we may conjecture that the downward convex behavior of the functions [MATH] in Fig. [REF] is due to the presence of a family of points satisfying the sum rule that can correspond to exact CFTs.', '0905.2211-1-64-2': 'This observation, though speculative, shows how the presented method can provide a guideline in the study of 4D CFTs.', '0905.2211-1-65-0': '# Future research directions', '0905.2211-1-66-0': 'The results of this paper and of [CITATION] open up many interesting research directions, which we would like to list here.', '0905.2211-1-67-0': 'First, there are several important problems in 4D Conformal Field Theory which can be analyzed by our method and its simple modifications.', '0905.2211-1-67-1': 'For example:', '0905.2211-1-68-0': 'One should be able to derive a generalization of our bounds in the situation when the CFT has a global symmetry, and we are interested in the lowest dimension singlet appearing in the OPE.', '0905.2211-1-68-1': 'This is going to have phenomenological implications by constraining the so-called conformal technicolor scenarios of ElectroWeak Symmetry Breaking [CITATION].', '0905.2211-1-68-2': 'This connection was extensively discussed in [CITATION].', '0905.2211-1-69-0': 'One should be able to derive model-independent bounds on the size of OPE coefficients.', '0905.2211-1-69-1': "This is going to be relevant for discussions of 'unparticle self-interactions' [CITATION], in the context of unparticle physics scenarios [CITATION].", '0905.2211-1-70-0': 'Second, the method can also be used in 2D Conformal Field Theory, as was already demonstrated in Section [REF].', '0905.2211-1-70-1': 'The main interest here lies in potential applications to string theory.', '0905.2211-1-70-2': 'We will now briefly describe two such applications.', '0905.2211-1-71-0': 'Physical states of (super)string theory are in 1-1 correspondence with Virasoro primary operators of a 2D CFT living on the string worldsheet.', '0905.2211-1-71-1': 'The mass of a string state (in string units) is related to the corresponding primary operator dimension [MATH] via [EQUATION].', '0905.2211-1-71-2': 'We are considering closed string theory for concreteness.', '0905.2211-1-71-3': 'When strings propagate in flat space, the CFT is solvable and the full spectrum of operator dimensions is known.', '0905.2211-1-71-4': 'Realistic string constructions require compactifications of extra dimensions.', '0905.2211-1-71-5': 'In some examples, such as toroidal compactifications, the CFT is still solvable.', '0905.2211-1-71-6': 'In others, such as superstring compactifications on a generic Calabi-Yau three-fold, the CFT cannot be solved exactly.', '0905.2211-1-71-7': 'All what is generally known is the spectrum of the massless states, which can be obtained in the supergravity approximation.', '0905.2211-1-71-8': 'Of course we expect the massive string states to be always present, but just how heavy can they be?', '0905.2211-1-71-9': 'We know from the experience with toroidal compactifications that it is impossible to completely decouple the massive states: as the compactification radius [MATH] the Kaluza-Klein states become more massive, but the winding modes come down.', '0905.2211-1-71-10': 'Clearly, massive string states are crucial for the consistency of the theory.', '0905.2211-1-71-11': 'What exactly are they doing?', '0905.2211-1-71-12': 'A partial answer may be that without their presence, 4-point functions of the massless state vertex operators would not be crossing-symmetric.', '0905.2211-1-71-13': 'If this intuition is right, it could be used to obtain model-independent bounds on the lightest massive states in string compactifications, generalizing the well-known bounds valid for toroidal compactifications.', '0905.2211-1-71-14': 'A similar in spirit general prediction of string gravity, although in a different context and by using different methods, was obtained recently in [CITATION].', '0905.2211-1-72-0': 'When working towards results of this kind, it may be necessary to generalize our methods so that the information about the 2D CFT central charge, which is fixed in string theory, can be taken into account.', '0905.2211-1-72-1': 'In practice, one needs is an efficient method to evaluate the full Virasoro conformal blocks.', '0905.2211-1-72-2': "While no closed-form expression as simple as Eq. ([REF]) is known, Zamolodchikov's expansion (see [CITATION]) can probably be applied.", '0905.2211-1-73-0': 'Finally, as mentioned above in the 4D context, it should be possible to derive model-independent bounds on the OPE coefficients.', '0905.2211-1-73-1': 'Such results must be accessible via a simple modification of our method, in particular the full Virasoro conformal blocks are not needed here.', '0905.2211-1-73-2': 'One can then apply such bounds to the dimension 2 operators corresponding to the massless string states (in an arbitrary compactification).', '0905.2211-1-73-3': 'Via the usual dictionary, this would then translate into general bounds on the tree-level coupling constants in the low-energy string effective actions.', '0905.2211-1-74-0': '# Summary', '0905.2211-1-75-0': "Prime principles of Conformal Field Theory, such as unitarity, OPE, and conformal block decomposition, imply the existence of an upper bound [MATH] on the dimension [MATH] of the leading scalar operator in the OPE [MATH], which depends only on [MATH]'s dimension [MATH].", '0905.2211-1-76-0': 'Moreover, there is an efficient method which allows numerical determination of [MATH] with arbitrary desired accuracy.', '0905.2211-1-76-1': 'The method is based on the sum rule, a function-space identity satisfied by the conformal block decomposition of the 4-point function [MATH], which follows from the crossing symmetry constraints.', '0905.2211-1-76-2': 'In practical application of the method the sum rule is Taylor-expanded: replaced by finitely many equations for the derivatives up to a certain order [MATH].', '0905.2211-1-76-3': 'The bound [MATH] improves monotonically as more and more derivatives are included.', '0905.2211-1-76-4': 'In [CITATION], where the above paradigm was first developed, we numerically computed the bound for [MATH].', '0905.2211-1-77-0': 'The present paper extended the study of [CITATION] to higher [MATH].', '0905.2211-1-77-1': 'The goal was to improve the bound, and perhaps to approach the best-possible bound in case a convergence of the bound is observed.', '0905.2211-1-78-0': 'Our analysis went up to [MATH], see Fig. [REF], and we have achieved both goals.', '0905.2211-1-78-1': 'First, in the range [MATH] that we explored, the bound on the anomalous dimension [MATH] is improved by [MATH] compared to the results of [CITATION].', '0905.2211-1-78-2': 'Second, we do observe signs of convergence of the bound.', '0905.2211-1-78-3': 'Thus we believe that our current results are close (within [MATH]) to the best possible with this method.', '0905.2211-1-79-0': 'The results of this paper and of [CITATION] suggest several interesting research directions, connected with phenomenology and, via the 2D analogue of our method, with string theory (see Section [REF]).'}

{'0905.2211-2-0-0': 'Universal Constraints on Conformal Operator Dimensions Vyacheslav S. Rychkov[MATH] and Alessandro Vichi[MATH] Scuola Normale Superiore and INFN, Pisa, Italy', '0905.2211-2-1-0': '[MATH] Institut de Theorie des Phenomenes Physiques, EPFL, Lausanne, Switzerland', '0905.2211-2-2-0': 'We continue the study of model-independent constraints on the unitary Conformal Field Theories in 4-Dimensions, initiated in http://arxiv.org/abs/0807.0004arXiv:0807.0004.', '0905.2211-2-2-1': 'Our main result is an improved upper bound on the dimension [MATH] of the leading scalar operator appearing in the OPE of two identical scalars of dimension [MATH]: [EQUATION]', '0905.2211-2-2-2': 'In the interval [MATH] this universal bound takes the form [EQUATION].', '0905.2211-2-2-3': 'The proof is based on prime principles of CFT: unitarity, crossing symmetry, OPE, and conformal block decomposition.', '0905.2211-2-2-4': 'We also discuss possible applications to particle phenomenology and, via a 2-D analogue, to string theory.', '0905.2211-2-3-0': '# Introduction and formulation of the problem', '0905.2211-2-4-0': 'Our knowledge about non-supersymmetric Conformal Field Theories (CFTs) in four dimensions (4D) is still quite incomplete.', '0905.2211-2-4-1': 'Suffices it to say that not a single nontrivial example is known which would be solvable to the same extent as, say, the 2D Ising model.', '0905.2211-2-4-2': 'However, we do not doubt that CFTs must be ubiquitous.', '0905.2211-2-4-3': 'For example, non-supersymmetric gauge theories with [MATH] colors and [MATH] flavors are widely believed to have conformal windows in which the theory has a conformal fixed point in the IR, with evidence from large [MATH] analysis [CITATION], supersymmetric analogues [CITATION], and lattice simulations [CITATION].', '0905.2211-2-4-4': 'Since these fixed points are typically strongly coupled, we do not have much control over them.', '0905.2211-2-4-5': 'In this situation particularly important are general, model-independent properties.', '0905.2211-2-5-0': 'One example of such a property is the famous unitarity bound [CITATION] on the dimension [MATH] of a spin [MATH] conformal primary operator [MATH] : [EQUATION]', '0905.2211-2-5-1': 'These bounds are derived by imposing that the two point function [MATH] have a positive spectral density.', '0905.2211-2-6-0': 'As is well known, 3-point functions in CFT are fixed by conformal symmetry up to a few arbitrary constants (Operator Product Expansion (OPE) coefficients).', '0905.2211-2-6-1': 'The next nontrivial constraint thus appears at the 4-point function level, and is known as the conformal bootstrap equation.', '0905.2211-2-6-2': 'It says that OPE applied in direct and crossed channel should give the same result (see Fig. [REF]).', '0905.2211-2-7-0': 'The bootstrap equation goes back to the early days of CFT [CITATION].', '0905.2211-2-7-1': 'However, until recently, not much useful general information has been extracted from it.', '0905.2211-2-7-2': 'All spins and dimensions can apriori enter the bootstrap on equal footing, and this seems to lead to unsurmountable difficulties.', '0905.2211-2-8-0': 'Recently, however, tangible progress in the analysis of bootstrap equations was achieved in [CITATION].', '0905.2211-2-8-1': 'Namely, it was found that, in unitary theories, the functions entering the bootstrap equations (conformal blocks) satisfy certain positivity properties which lead to general necessary conditions for the existence of solutions.', '0905.2211-2-9-0': 'The concrete problem considered in [CITATION], and which we will continue to discuss here, was as follows.', '0905.2211-2-9-1': 'In an arbitrary unitary CFT a Hermitean scalar primary [MATH] of dimension [MATH] was singled out.', '0905.2211-2-9-2': 'The conformal bootstrap equation for its 4-point function [MATH] was studied under the sole assumption that all scalars in the OPE [MATH] have dimension above a certain number, call it [MATH] [EQUATION]', '0905.2211-2-9-3': 'It was shown that the conformal bootstrap does not allow for a solution unless [EQUATION] where [MATH] is a certain continuous function, computed numerically.', '0905.2211-2-9-4': 'We stress that this conclusion was reached without making any assumptions about dimensions or spins of other operators appearing in the OPE, beyond those implied by the unitarity bounds.', '0905.2211-2-9-5': 'Nor any assumptions about the OPE coefficients were made (apart from their reality, which is again implied by unitarity).', '0905.2211-2-10-0': 'In other words, in any unitary 4D CFT, the OPE of any scalar primary [MATH] must contain at least one scalar field [MATH] with dimension not larger than [MATH].', '0905.2211-2-11-0': 'Incidentally, the function [MATH] was found to satisfy [MATH] which is quite natural since [MATH] corresponds to the free field whose OPE contains the operator :[MATH]: of dimension [MATH].', '0905.2211-2-12-0': 'What makes the result like ([REF]) possible?', '0905.2211-2-12-1': 'The basic reason is that, in any theory, crossing symmetry relation of Fig. [REF] cannot be satisfied term by term, but only by cancellations among various terms.', '0905.2211-2-12-2': 'The guaranteed presence of the unit operator in the OPE ([REF]) creates a certain crossing symmetry deficit, which has to be balanced by other fields.', '0905.2211-2-12-3': 'The idea is to show that this cannot happen unless at least one scalar of sufficiently low dimension is present.', '0905.2211-2-13-0': 'Technically, the method of [CITATION] consists of 3 steps (see Section [REF] for a detailed review):', '0905.2211-2-14-0': 'We Taylor-expand the conformal bootstrap equation near the self-dual point" configuration having equal conformal cross-ratios [MATH].', '0905.2211-2-14-1': 'The expansion is truncated to a certain finite order [MATH].', '0905.2211-2-15-0': 'We systematically search for positivity properties satisfied by linear combinations of Taylor coefficients of the conformal blocks, for fields appearing in the RHS of the OPE ([REF]).', '0905.2211-2-15-1': 'A found positivity property implies that the crossing symmetry deficit cannot be balanced and rules out a CFT with a given [MATH] and [MATH].', '0905.2211-2-16-0': 'For fixed [MATH], the bound [MATH] is then computed as the point separating those [MATH] for which a positivity property exists, from those ones for which it does not (Fig. [REF]).', '0905.2211-2-17-0': 'The nature of the method is such that increasing [MATH] can make the bound only stronger.The optimal bound should in principle be recoverable in the limit [MATH].', '0905.2211-2-17-1': 'In practice the value of [MATH] is determined by the available computer resources and algorithmic efficiency.', '0905.2211-2-17-2': 'The best bound found in [CITATION], plotted in Fig. [REF], corresponds to [MATH].', '0905.2211-2-18-0': 'The purpose of this paper is to present an improvement of the bound ([REF]) obtained by using the method of [CITATION] with larger values of [MATH], up to [MATH].', '0905.2211-2-18-1': 'The new results are interesting in two ways.', '0905.2211-2-18-2': 'First, pure numerical improvement turns out to be significant.', '0905.2211-2-18-3': 'Second, [MATH] happens to be large enough so that we start observing saturation of the bound.', '0905.2211-2-18-4': 'So we believe our current results are close to the optimal ones achievable with this method.', '0905.2211-2-19-0': 'The paper is organized as follows.', '0905.2211-2-19-1': 'In Section [REF] we review the conformal bootstrap equations.', '0905.2211-2-19-2': 'In Section [REF] we review the connection of the bound ([REF]).', '0905.2211-2-19-3': 'with positivity properties satisfied by the conformal block expansion coefficients.', '0905.2211-2-19-4': 'In Section [REF] we present and discuss our results.', '0905.2211-2-19-5': 'We also mention accompanying results which we obtain for an analogous problem in 2D.', '0905.2211-2-19-6': 'In Section [REF] we propose several future applications and extensions of our method, with emphasis on connections to phenomenology and string theory.', '0905.2211-2-19-7': 'In Section [REF] we summarize and conclude.', '0905.2211-2-19-8': 'In Appendix [REF] we collect some details about our numerical algorithms.', '0905.2211-2-19-9': 'In Appendix [REF] we include the tables on which plots in Section [REF] are based.', '0905.2211-2-20-0': '# Review of conformal bootstrap', '0905.2211-2-21-0': 'We will review the conformal bootstrap equation in its simplest form-as applied to the 4-point function of identical scalars [MATH].', '0905.2211-2-21-1': 'We largely follow [CITATION], where a more detailed discussion and references can be found.', '0905.2211-2-22-0': '## Conformal block decomposition', '0905.2211-2-23-0': 'Let [MATH] be a Hermitean scalar primary operator.', '0905.2211-2-23-1': 'The operator product expansion (OPE) [MATH] contains, in general, infinitely many primary fields of arbitrary high spins and dimensions: [EQUATION]', '0905.2211-2-23-2': 'Here', '0905.2211-2-24-0': 'We assume that the OPE converges in the following weak sense: it gives a convergent power series expansion for any [MATH]-point function [EQUATION] provided that [MATH], i.e. [MATH] is closer to the origin than any other local field insertion (see Fig. [REF]).', '0905.2211-2-24-1': 'This assumption can be justified by using radial quantization ([CITATION], Sect. 2.9), and checked explicitly in free field theory.', '0905.2211-2-24-2': 'For rigorous mathematical results about OPE convergence see [CITATION].', '0905.2211-2-25-0': 'The OPE ([REF]) can be used to obtain conformal block decomposition of the 4-point function [MATH]: [EQUATION] where [MATH] are the conformal cross-ratios.', '0905.2211-2-25-1': 'This representation is obtained by using the OPE in the 12 and 34 channels.', '0905.2211-2-25-2': 'The conformal blocks [MATH] sum up the contributions of the primary [MATH] and all its descendants.', '0905.2211-2-25-3': 'Their explicit expressions were found by Dolan and Osborn [CITATION]: [EQUATION]', '0905.2211-2-25-4': 'Notice the judicious introduction of the auxiliary variables [MATH] and [MATH].', '0905.2211-2-25-5': 'When the theory is formulated in the Euclidean space, these variables are complex-conjugates of each other.', '0905.2211-2-25-6': 'To understand their meaning, it is convenient to use the conformal group freedom to send [MATH] and to put the other three points in a plane, as in Fig. [REF].', '0905.2211-2-25-7': "Then it's easy to show that [EQUATION] where [MATH] are the coordinates of [MATH] in the plane, chosen so that [MATH] corresponds to [MATH] halfway between [MATH] and [MATH].", '0905.2211-2-25-8': 'This self-dual configuration, for which [MATH], will play an important role below.', '0905.2211-2-25-9': 'We can see that the [MATH] variable is a natural extension of the usual complex coordinate of the 2D CFT to the 4D case.', '0905.2211-2-26-0': 'According to the above discussion, the OPE is expected to converge for [MATH].', '0905.2211-2-26-1': 'Conformal block decomposition is a partial resummation of the OPE and thus also converges at least in this range.', '0905.2211-2-26-2': 'In fact, below we will only use convergence around the self-dual point [MATH].', '0905.2211-2-26-3': 'However, conformal blocks, as given by ([REF]), are regular (real-analytic) in a larger region, namely in the [MATH]-plane with the [MATH] cut along the real axis (see Fig. [REF]).', '0905.2211-2-26-4': 'The conformal block decomposition is thus expected to converge in this larger region.', '0905.2211-2-26-5': 'One can check that this indeed happens in the free scalar theory.', '0905.2211-2-27-0': 'One can intuitively understand the reason for this extended region of regularity.', '0905.2211-2-27-1': 'The condition for the OPE convergence, as stated above, does not treat the points [MATH] and [MATH] symmetrically.', '0905.2211-2-27-2': 'On the other hand, the conformal blocks are completely symmetric in [MATH] and so must be the condition for their regularity.', '0905.2211-2-27-3': 'The appropriate condition is as follows: the conformal block decomposition in the [MATH]-[MATH] channel is regular and convergent if there is a sphere separating the points [MATH] from the points [MATH].', '0905.2211-2-27-4': 'For the configuration of Fig. [REF], such a sphere exists as long as [MATH] is away from the cut.', '0905.2211-2-28-0': '## Conformal bootstrap and the sum rule', '0905.2211-2-29-0': 'The 4-point function in ([REF]) must be symmetric under the interchange of any two [MATH], and its conformal block decomposition ([REF]) has to respect this symmetry.', '0905.2211-2-29-1': 'The symmetry with respect to [MATH] or [MATH] is already built in, since only even spins are exchanged [CITATION].', '0905.2211-2-29-2': 'On the contrary, the symmetry with respect to [MATH] gives a condition [EQUATION] which is not automatically satisfied for [MATH] given by ([REF]).', '0905.2211-2-29-3': 'This nontrivial constraint on dimensions, spins, and OPE coefficients of all operators appearing in the OPE [MATH] is known as the conformal bootstrap equation.', '0905.2211-2-29-4': 'Physically it means that OPE applied in 12-34 and 14-23 channels should give the same result (Fig. [REF]).', '0905.2211-2-30-0': 'In the [MATH]-plane of Section [REF], the LHS of ([REF] ) has a cut along [MATH] while the RHS has a cut along [MATH].', '0905.2211-2-30-1': 'Thus, if ([REF]) is satisfied, the cuts have to cancel, and the resulting [MATH] is real analytic everywhere except for [MATH].', '0905.2211-2-31-0': 'In [CITATION], we found it useful to rewrite ([REF]) by separating the unit operator contribution, which gives [EQUATION]', '0905.2211-2-31-1': 'The LHS of this equation is the crossing symmetry deficit created by the presence of the unit operator in the OPE.', '0905.2211-2-31-2': 'This deficit has to be balanced by contributions of the other fields in the RHS.', '0905.2211-2-32-0': 'In practice it is convenient to normalize ([REF]) by dividing both sides by [MATH].', '0905.2211-2-32-1': 'The resulting sum rule takes the form: [EQUATION]', '0905.2211-2-32-2': 'The F-functions" [MATH] are real and regular in the full [MATH]-plane cut along [MATH].', '0905.2211-2-32-3': 'In particular, the [MATH] behavior at the self-dual point [MATH] is regular.', '0905.2211-2-33-0': 'All F-functions vanish near the points [MATH] and [MATH].', '0905.2211-2-33-1': 'Thus the sum rule can never be satisfied near these points if only finitely many terms are present in the RHS.', '0905.2211-2-33-2': 'The OPEs containing finitely many primaries are ruled out.', '0905.2211-2-34-0': '# Positivity argument', '0905.2211-2-35-0': 'The main idea of [CITATION] was very simple, and can be described as follows.', '0905.2211-2-35-1': 'Suppose that for a given spectrum of operator dimensions and spins [MATH] the sum rule ([REF]), viewed as an equation for the coefficients [MATH], has no solution.', '0905.2211-2-35-2': 'Then of course such a spectrum would be ruled out.', '0905.2211-2-36-0': 'Any concrete realization of this idea needs a practical criterium to show that there is no solution.', '0905.2211-2-36-1': 'For a prototypical example of such a criterium, imagine that a certain derivative, e.g. [MATH] (see ([REF])), when applied to every [MATH] and evaluated at a certain point, is strictly positive (positivity property).', '0905.2211-2-36-2': 'Since the same derivative applied to the LHS of ([REF]) gives identically zero, a solution where all coefficients [MATH] are non-negative would clearly be impossible.', '0905.2211-2-36-3': 'We refer to this simple reasoning as the positivity argument.', '0905.2211-2-37-0': 'One can imagine more general criteria using different differential operators, and applying them at different points.', '0905.2211-2-37-1': 'In [CITATION], we found it convenient to apply differential operators precisely at the self-dual point [MATH], [MATH].', '0905.2211-2-37-2': 'One can show that the F-functions are even with respect to this point both in the [MATH] and [MATH] directions: [EQUATION].', '0905.2211-2-37-3': 'Thus, all odd-order derivatives vanish, and a sufficiently general differential operator (linear functional) takes the form: [EQUATION] where [MATH] is some fixed finite number, and [MATH] are fixed real coefficients.', '0905.2211-2-37-4': 'Notice the exclusion of the constant term [MATH], in order to have [MATH].', '0905.2211-2-38-0': 'Assume that for certain fixed [MATH] and [MATH] we manage to find a linear functional of this form such that (positivity property) [EQUATION]', '0905.2211-2-38-1': 'Moreover, assume that all but a finite number of these inequalities are actually strict: [MATH] Then the sum rule cannot be satisfied, and such a spectrum, corresponding to a putative OPE ([REF]), is ruled out.', '0905.2211-2-39-0': 'The proof uses the above positivity argument.', '0905.2211-2-39-1': 'Since [MATH] the positivity property implies that only those primaries for which [MATH] would be allowed to appear in the RHS of the sum rule with nonzero coefficients.', '0905.2211-2-39-2': 'By assumption, there are at most a finite number of such primaries.', '0905.2211-2-39-3': 'However, as noted in Section [REF], finitely many terms can never satisfy the sum rule globally, because of the behavior near [MATH].', '0905.2211-2-39-4': 'Q.E.D.', '0905.2211-2-40-0': 'While the above formal reasoning is quite sufficient to understand our results, in [CITATION] the sum rule was also given an alternative interpretation in terms of convex geometry.', '0905.2211-2-40-1': 'In this more visual picture, linear combinations of F-functions with arbitrary positive coefficients form a convex cone in the space of two-variable functions.', '0905.2211-2-40-2': 'One can consider the full function space or its finite-dimensional subspace corresponding to Taylor-expanding up to order [MATH].', '0905.2211-2-40-3': 'Positivity property ([REF]) means that there is a hyperplane separating the function 1 from the convex cone.', '0905.2211-2-40-4': 'Thus it implies that the sum rule cannot be satisfied.', '0905.2211-2-40-5': 'The converse is almost true, modulo questions of convergence.', '0905.2211-2-41-0': 'Clearly, the language of linear functionals provides an equivalent, dual formulation of the problem.', '0905.2211-2-41-1': 'This formulation is also especially convenient from the point of view of checking our results independently.', '0905.2211-2-41-2': "It's not so important how we find the functionals.", '0905.2211-2-41-3': 'As long as we publish the functional coefficients [MATH], anyone can verify that the inequalities ([REF]) are satisfied.', '0905.2211-2-42-0': '# Results, discussion, and 2D analogue', '0905.2211-2-43-0': 'As discussed in Section [REF], we are interested in computing an upper bound [MATH][REF][MATH] for the dimension [MATH] of the leading scalar in the OPE [MATH], universal for all unitary 4D CFTs.', '0905.2211-2-43-1': 'In [CITATION], we have computed such a bound in the interval [MATH] using the sum rule of Section [REF] truncated to the [MATH] derivative order.', '0905.2211-2-43-2': 'That bound is reproduced in Fig. [REF].', '0905.2211-2-44-0': 'We now present the results of our latest study, obtained for larger values of [MATH].', '0905.2211-2-44-1': 'These results are plotted in Fig [REF] as a collection of curves [MATH], [MATH], where the index [MATH] denotes the number of derivatives used to obtain the bound.', '0905.2211-2-44-2': 'The bound naturally gets stronger as [MATH] increases (see below), and thus the lowest curve [MATH] is the strongest bound to date.', '0905.2211-2-44-3': 'In the considered interval [MATH] this bound is well approximated (within [MATH]) by [EQUATION]', '0905.2211-2-44-4': 'To obtain the bounds of Fig. [REF], we used the positivity argument from [CITATION], as reviewed in Section [REF].', '0905.2211-2-44-5': 'Namely, for points lying on the curves [MATH] we are able to find a linear functional of the form ([REF]) satisfying the positivity property ([REF]).', '0905.2211-2-44-6': 'The numerical procedure that we use to find these positive functionals is described in some detail in Appendix A.', '0905.2211-2-45-0': 'Several comments are in order here.', '0905.2211-2-46-0': 'We have actually computed the bound only for a discrete number of [MATH] values, shown as points in Fig. [REF].', '0905.2211-2-46-1': 'The tables of these computed values are given in Appendix B. Behavior for [MATH] can be better appreciated from the logarithmic-scale plot in Fig. [REF].', '0905.2211-2-47-0': 'We do not see any significant indication which could suggest that the curves [MATH] do not interpolate smoothly in between the computed points.', '0905.2211-2-47-1': 'Small irregularities in the slope are however visible at several points in Figs. [REF],[REF].', '0905.2211-2-47-2': 'These irregularities are understood; they originate from the necessity to discretize the infinite system of inequalities ([REF]), see Appendix A for a discussion.', '0905.2211-2-47-3': 'In our computations the discretization step was chosen so that these irregularities are typically much smaller than the improvement of the bound that one gets for [MATH].', '0905.2211-2-48-0': 'For each [MATH] the bound [MATH] is near-optimal, in the sense that no positive functional involving derivatives up to order [MATH] exists for [EQUATION].', '0905.2211-2-48-1': 'We estimate [MATH] from the analysis of residuals in the fit of [MATH] by a smooth curve like in ([REF]).', '0905.2211-2-49-0': 'On the other hand, by increasing [MATH] we are allowing more general functionals, and thus the bound [MATH] can and does get stronger.', '0905.2211-2-49-1': 'This is intuitively clear since for larger [MATH] the Taylor-expanded sum rule includes more and more constraints.', '0905.2211-2-50-0': 'Compared to the results of [CITATION], the bound on the anomalous dimension [MATH] is improved by [MATH] in the range [MATH] that we explored.', '0905.2211-2-51-0': 'We have pushed our analysis to such large values of [MATH] in the hope of seeing that the bound saturates as [MATH].', '0905.2211-2-51-1': 'Indeed, we do observe signs of convergence in Figs. [REF],[REF], especially at [MATH].', '0905.2211-2-51-2': 'In fact, we have observed that the bounds [MATH] starting from [MATH] follow rather closely the asymptotic behavior [EQUATION].', '0905.2211-2-51-3': 'An approximation to the optimal bound [MATH] can thus be found by performing for each [MATH] a fit to this formula.', '0905.2211-2-51-4': 'This approximation is shown by a dashed line in Fig. [REF].', '0905.2211-2-51-5': 'From this rough analysis we conclude that the optimal bound on the anomalous dimension [MATH] is probably within [MATH] from our current bound.', '0905.2211-2-52-0': 'We have [MATH] continuously as [MATH].', '0905.2211-2-52-1': 'The point [MATH] corresponds to the free scalar theory.', '0905.2211-2-53-0': "We don't know of any unitary CFTs that saturate our bound at [MATH], see the discussion in Section 6 of [CITATION].", '0905.2211-2-53-1': 'We know however a family of unitary 4D CFTs in which [MATH] and which are consistent with our bound (the red dotted line in Fig. [REF]).', '0905.2211-2-53-2': "This generalized free scalar theory is defined for a fixed [MATH] by specifying the 2-point function [EQUATION] and defining all other correlators of [MATH] via Wick's theorem.", '0905.2211-2-53-3': 'This simple procedure gives a well-defined CFT, unitary as long as [MATH], which can be described by a nonlocal action [EQUATION].', '0905.2211-2-53-4': 'The full operator content of this theory can be recovered by studying the OPE [MATH].', '0905.2211-2-53-5': 'In particular, the leading scalar in this OPE has dimension [MATH]', '0905.2211-2-54-0': '## 2D analogue', '0905.2211-2-55-0': 'Although our main interest is in the 4D CFTs, our methods allow a parallel treatment of the 2D case.', '0905.2211-2-55-1': 'The main characteristics of the 2D situation were described in Section 6.1 of [CITATION], here will briefly review them.', '0905.2211-2-56-0': 'At present we can only take advantage of the finite-dimensional [MATH] symmetry and not of the full Virasoro algebra of the 2D CFTs.', '0905.2211-2-56-1': 'In particular, our results are independent of the 2D central charge [MATH].', '0905.2211-2-57-0': 'The unitarity bounds for [MATH] primaries in 2D have the form [EQUATION] where [MATH] is the Lorentz spin.', '0905.2211-2-58-0': 'The [MATH] conformal blocks in 2D are known explicitly [CITATION]: [EQUATION]', '0905.2211-2-58-1': 'Using the unitarity bounds, the known conformal blocks, and the sum rule ([REF]), valid in any dimension, we can try to answer the same question as in 4D.', '0905.2211-2-58-2': 'Namely, for a [MATH] scalar primary [MATH] of dimension [MATH], what is an upper bound on the dimension [MATH] of the first scalar operator appearing in the OPE [MATH]?', '0905.2211-2-58-3': 'I.e. we want a 2D analogue of the bound ([REF]).', '0905.2211-2-58-4': 'Since the free scalar is dimensionless in 2D, the region of interest is [MATH].', '0905.2211-2-59-0': 'Fig. [REF] summarizes our current knowledge of this bound:', '0905.2211-2-60-0': 'It is instructive to compare this plot with its 4D counterpart, Fig. [REF].', '0905.2211-2-60-1': 'While we do not know of any CFTs saturating the 4D bound, the 2D unitary minimal models [MATH], [MATH], contain the OPEs [EQUATION] which come quite close to saturating the 2D bound.', '0905.2211-2-61-0': 'More precisely, our 2D bound starts at [MATH] tangentially to the line [MATH] realized in the free scalar theory, then grows monotonically and passes remarkably closely above the Ising model point [MATH].', '0905.2211-2-61-1': 'After a knee at the Ising point, the bound continues to grow linearly, passing in the vicinity of the higher minimal model points ([REF]).', '0905.2211-2-62-0': 'It is curious to note that if we did not know beforehand about the Ising model, we could have conjectured its field dimensions and the basic OPE [MATH] based on the singular behavior of the 2D bound at [MATH].', '0905.2211-2-63-0': 'On the other hand, nothing special happens with the 2D bound at the higher minimal model points, it just interpolates linearly in between.', '0905.2211-2-63-1': 'Most likely, this does not mean that there exist other unitary CFTs with intermediate operator dimensions.', '0905.2211-2-63-2': 'Rather, this behavior suggests that the single conformal bootstrap equation used to derive the bound is not powerful enough to fully constrain a CFT.', '0905.2211-2-64-0': 'In comparison, it is a bit unfortunate that the 4D bound does not exhibit any singular points which would immediately stand out as CFT candidates.', '0905.2211-2-64-1': 'Nevertheless, if we assume that the shape of the 4D bound is a result of an interpolation between existing CFTs (as it is the case in 2D), we may conjecture that the upward convex behavior of the functions [MATH] in Fig. [REF] is due to the presence of a family of points satisfying the sum rule that can correspond to exact CFTs.', '0905.2211-2-64-2': 'This observation, though speculative, shows how the presented method can provide a guideline in the study of 4D CFTs.', '0905.2211-2-65-0': '# Future research directions', '0905.2211-2-66-0': 'The results of this paper and of [CITATION] open up many interesting research directions, which we would like to list here.', '0905.2211-2-67-0': 'First, there are several important problems in 4D Conformal Field Theory which can be analyzed by our method and its simple modifications.', '0905.2211-2-67-1': 'For example:', '0905.2211-2-68-0': 'One should be able to derive a generalization of our bounds in the situation when the CFT has a global symmetry, and we are interested in the lowest dimension singlet appearing in the OPE.', '0905.2211-2-68-1': 'This is going to have phenomenological implications by constraining the so-called conformal technicolor scenarios of ElectroWeak Symmetry Breaking [CITATION].', '0905.2211-2-68-2': 'This connection was extensively discussed in [CITATION].', '0905.2211-2-69-0': 'One should be able to derive model-independent bounds on the size of OPE coefficients.', '0905.2211-2-69-1': "This is going to be relevant for discussions of 'unparticle self-interactions' [CITATION], in the context of unparticle physics scenarios [CITATION].", '0905.2211-2-70-0': 'Second, the method can also be used in 2D Conformal Field Theory, as was already demonstrated in Section [REF].', '0905.2211-2-70-1': 'The main interest here lies in potential applications to string theory.', '0905.2211-2-70-2': 'We will now briefly describe two such applications.', '0905.2211-2-71-0': 'Physical states of (super)string theory are in 1-1 correspondence with Virasoro primary operators of a 2D CFT living on the string worldsheet.', '0905.2211-2-71-1': 'The mass of a string state (in string units) is related to the corresponding primary operator dimension [MATH] via [EQUATION].', '0905.2211-2-71-2': 'We are considering closed string theory for concreteness.', '0905.2211-2-71-3': 'When strings propagate in flat space, the CFT is solvable and the full spectrum of operator dimensions is known.', '0905.2211-2-71-4': 'Realistic string constructions require compactifications of extra dimensions.', '0905.2211-2-71-5': 'In some examples, such as toroidal compactifications, the CFT is still solvable.', '0905.2211-2-71-6': 'In others, such as superstring compactifications on a generic Calabi-Yau three-fold, the CFT cannot be solved exactly.', '0905.2211-2-71-7': 'All what is generally known is the spectrum of the massless states, which can be obtained in the supergravity approximation.', '0905.2211-2-71-8': 'Of course we expect the massive string states to be always present, but just how heavy can they be?', '0905.2211-2-71-9': 'We know from the experience with toroidal compactifications that it is impossible to completely decouple the massive states: as the compactification radius [MATH] the Kaluza-Klein states become more massive, but the winding modes come down.', '0905.2211-2-71-10': 'Clearly, massive string states are crucial for the consistency of the theory.', '0905.2211-2-71-11': 'What exactly are they doing?', '0905.2211-2-71-12': 'A partial answer may be that without their presence, 4-point functions of the massless state vertex operators would not be crossing-symmetric.', '0905.2211-2-71-13': 'If this intuition is right, it could be used to obtain model-independent bounds on the lightest massive states in string compactifications, generalizing the well-known bounds valid for toroidal compactifications.', '0905.2211-2-71-14': 'A similar in spirit general prediction of string gravity, although in a different context and by using different methods, was obtained recently in [CITATION].', '0905.2211-2-72-0': 'When working towards results of this kind, it may be necessary to generalize our methods so that the information about the 2D CFT central charge, which is fixed in string theory, can be taken into account.', '0905.2211-2-72-1': 'In practice, one needs an efficient method to evaluate the full Virasoro conformal blocks.', '0905.2211-2-72-2': "While no closed-form expression as simple as Eq. ([REF]) is known, Zamolodchikov's expansion (see [CITATION]) can probably be applied.", '0905.2211-2-73-0': 'Finally, as mentioned above in the 4D context, it should be possible to derive model-independent bounds on the OPE coefficients.', '0905.2211-2-73-1': 'Such results must be accessible via a simple modification of our method, in particular the full Virasoro conformal blocks are not needed here.', '0905.2211-2-73-2': 'One can then apply such bounds to the dimension 2 operators corresponding to the massless string states (in an arbitrary compactification).', '0905.2211-2-73-3': 'Via the usual dictionary, this would then translate into general bounds on the tree-level coupling constants in the low-energy string effective actions.', '0905.2211-2-74-0': '# Summary', '0905.2211-2-75-0': "Prime principles of Conformal Field Theory, such as unitarity, OPE, and conformal block decomposition, imply the existence of an upper bound [MATH] on the dimension [MATH] of the leading scalar operator in the OPE [MATH], which depends only on [MATH]'s dimension [MATH].", '0905.2211-2-76-0': 'Moreover, there is an efficient method which allows numerical determination of [MATH] with arbitrary desired accuracy.', '0905.2211-2-76-1': 'The method is based on the sum rule, a function-space identity satisfied by the conformal block decomposition of the 4-point function [MATH], which follows from the crossing symmetry constraints.', '0905.2211-2-76-2': 'In practical application of the method the sum rule is Taylor-expanded: replaced by finitely many equations for the derivatives up to a certain order [MATH].', '0905.2211-2-76-3': 'The bound [MATH] improves monotonically as more and more derivatives are included.', '0905.2211-2-76-4': 'In [CITATION], where the above paradigm was first developed, we numerically computed the bound for [MATH].', '0905.2211-2-77-0': 'The present paper extended the study of [CITATION] to higher [MATH].', '0905.2211-2-77-1': 'The goals were to improve the bound, and perhaps to approach the best-possible bound in case a convergence of the bound is observed.', '0905.2211-2-78-0': 'Our analysis went up to [MATH], see Fig. [REF], and we have achieved both goals.', '0905.2211-2-78-1': 'First, in the range [MATH] that we explored, the bound on the anomalous dimension [MATH] is improved by [MATH] compared to the results of [CITATION].', '0905.2211-2-78-2': 'Second, we do observe signs of convergence of the bound.', '0905.2211-2-78-3': 'We believe that our current results are close (within [MATH]) to the best ones achievable with this method.', '0905.2211-2-79-0': 'The results of this paper and of [CITATION] suggest several interesting research directions, connected with phenomenology and, via the 2D analogue of our method, with string theory (see Section [REF]).'}

[['0905.2211-1-46-0', '0905.2211-2-46-0'], ['0905.2211-1-46-1', '0905.2211-2-46-1'], ['0905.2211-1-48-0', '0905.2211-2-48-0'], ['0905.2211-1-48-1', '0905.2211-2-48-1'], ['0905.2211-1-71-0', '0905.2211-2-71-0'], ['0905.2211-1-71-1', '0905.2211-2-71-1'], ['0905.2211-1-71-2', '0905.2211-2-71-2'], ['0905.2211-1-71-3', '0905.2211-2-71-3'], ['0905.2211-1-71-4', '0905.2211-2-71-4'], ['0905.2211-1-71-5', '0905.2211-2-71-5'], ['0905.2211-1-71-6', '0905.2211-2-71-6'], ['0905.2211-1-71-7', '0905.2211-2-71-7'], ['0905.2211-1-71-8', '0905.2211-2-71-8'], ['0905.2211-1-71-9', '0905.2211-2-71-9'], ['0905.2211-1-71-10', '0905.2211-2-71-10'], ['0905.2211-1-71-11', '0905.2211-2-71-11'], ['0905.2211-1-71-12', '0905.2211-2-71-12'], ['0905.2211-1-71-13', '0905.2211-2-71-13'], ['0905.2211-1-71-14', '0905.2211-2-71-14'], ['0905.2211-1-15-0', '0905.2211-2-15-0'], ['0905.2211-1-15-1', '0905.2211-2-15-1'], ['0905.2211-1-60-0', '0905.2211-2-60-0'], ['0905.2211-1-60-1', '0905.2211-2-60-1'], ['0905.2211-1-5-0', '0905.2211-2-5-0'], ['0905.2211-1-5-1', '0905.2211-2-5-1'], ['0905.2211-1-18-0', '0905.2211-2-18-0'], ['0905.2211-1-18-1', '0905.2211-2-18-1'], ['0905.2211-1-18-2', '0905.2211-2-18-2'], ['0905.2211-1-18-3', '0905.2211-2-18-3'], ['0905.2211-1-18-4', '0905.2211-2-18-4'], ['0905.2211-1-63-0', '0905.2211-2-63-0'], ['0905.2211-1-63-1', '0905.2211-2-63-1'], ['0905.2211-1-63-2', '0905.2211-2-63-2'], ['0905.2211-1-69-0', '0905.2211-2-69-0'], ['0905.2211-1-69-1', '0905.2211-2-69-1'], ['0905.2211-1-26-0', '0905.2211-2-26-0'], ['0905.2211-1-26-1', '0905.2211-2-26-1'], ['0905.2211-1-26-2', '0905.2211-2-26-2'], ['0905.2211-1-26-3', '0905.2211-2-26-3'], ['0905.2211-1-26-4', '0905.2211-2-26-4'], ['0905.2211-1-26-5', '0905.2211-2-26-5'], ['0905.2211-1-14-0', '0905.2211-2-14-0'], ['0905.2211-1-14-1', '0905.2211-2-14-1'], ['0905.2211-1-67-0', '0905.2211-2-67-0'], ['0905.2211-1-53-0', '0905.2211-2-53-0'], ['0905.2211-1-53-1', '0905.2211-2-53-1'], ['0905.2211-1-53-2', '0905.2211-2-53-2'], ['0905.2211-1-53-3', '0905.2211-2-53-3'], ['0905.2211-1-53-4', '0905.2211-2-53-4'], ['0905.2211-1-53-5', '0905.2211-2-53-5'], ['0905.2211-1-12-0', '0905.2211-2-12-0'], ['0905.2211-1-12-1', '0905.2211-2-12-1'], ['0905.2211-1-12-2', '0905.2211-2-12-2'], ['0905.2211-1-12-3', '0905.2211-2-12-3'], ['0905.2211-1-79-0', '0905.2211-2-79-0'], ['0905.2211-1-31-0', '0905.2211-2-31-0'], ['0905.2211-1-31-1', '0905.2211-2-31-1'], ['0905.2211-1-31-2', '0905.2211-2-31-2'], ['0905.2211-1-47-0', '0905.2211-2-47-0'], ['0905.2211-1-47-1', '0905.2211-2-47-1'], ['0905.2211-1-47-2', '0905.2211-2-47-2'], ['0905.2211-1-47-3', '0905.2211-2-47-3'], ['0905.2211-1-62-0', '0905.2211-2-62-0'], ['0905.2211-1-23-0', '0905.2211-2-23-0'], ['0905.2211-1-23-1', '0905.2211-2-23-1'], ['0905.2211-1-56-0', '0905.2211-2-56-0'], ['0905.2211-1-56-1', '0905.2211-2-56-1'], ['0905.2211-1-41-0', '0905.2211-2-41-0'], ['0905.2211-1-41-1', '0905.2211-2-41-1'], ['0905.2211-1-41-2', '0905.2211-2-41-2'], ['0905.2211-1-41-3', '0905.2211-2-41-3'], ['0905.2211-1-38-0', '0905.2211-2-38-0'], ['0905.2211-1-38-1', '0905.2211-2-38-1'], ['0905.2211-1-6-0', '0905.2211-2-6-0'], ['0905.2211-1-6-1', '0905.2211-2-6-1'], ['0905.2211-1-6-2', '0905.2211-2-6-2'], ['0905.2211-1-11-0', '0905.2211-2-11-0'], ['0905.2211-1-0-0', '0905.2211-2-0-0'], ['0905.2211-1-33-0', '0905.2211-2-33-0'], ['0905.2211-1-33-1', '0905.2211-2-33-1'], ['0905.2211-1-33-2', '0905.2211-2-33-2'], ['0905.2211-1-75-0', '0905.2211-2-75-0'], ['0905.2211-1-51-0', '0905.2211-2-51-0'], ['0905.2211-1-51-1', '0905.2211-2-51-1'], ['0905.2211-1-51-2', '0905.2211-2-51-2'], ['0905.2211-1-51-3', '0905.2211-2-51-3'], ['0905.2211-1-51-4', '0905.2211-2-51-4'], ['0905.2211-1-51-5', '0905.2211-2-51-5'], ['0905.2211-1-39-0', '0905.2211-2-39-0'], ['0905.2211-1-39-1', '0905.2211-2-39-1'], ['0905.2211-1-39-2', '0905.2211-2-39-2'], ['0905.2211-1-39-3', '0905.2211-2-39-3'], ['0905.2211-1-76-0', '0905.2211-2-76-0'], ['0905.2211-1-76-1', '0905.2211-2-76-1'], ['0905.2211-1-76-2', '0905.2211-2-76-2'], ['0905.2211-1-76-3', '0905.2211-2-76-3'], ['0905.2211-1-76-4', '0905.2211-2-76-4'], ['0905.2211-1-77-0', '0905.2211-2-77-0'], ['0905.2211-1-40-0', '0905.2211-2-40-0'], ['0905.2211-1-40-1', '0905.2211-2-40-1'], ['0905.2211-1-40-2', '0905.2211-2-40-2'], ['0905.2211-1-40-3', '0905.2211-2-40-3'], ['0905.2211-1-40-4', '0905.2211-2-40-4'], ['0905.2211-1-40-5', '0905.2211-2-40-5'], ['0905.2211-1-27-0', '0905.2211-2-27-0'], ['0905.2211-1-27-1', '0905.2211-2-27-1'], ['0905.2211-1-27-2', '0905.2211-2-27-2'], ['0905.2211-1-27-3', '0905.2211-2-27-3'], ['0905.2211-1-27-4', '0905.2211-2-27-4'], ['0905.2211-1-36-0', '0905.2211-2-36-0'], ['0905.2211-1-36-1', '0905.2211-2-36-1'], ['0905.2211-1-36-2', '0905.2211-2-36-2'], ['0905.2211-1-36-3', '0905.2211-2-36-3'], ['0905.2211-1-66-0', '0905.2211-2-66-0'], ['0905.2211-1-19-0', '0905.2211-2-19-0'], ['0905.2211-1-19-1', '0905.2211-2-19-1'], ['0905.2211-1-19-2', '0905.2211-2-19-2'], ['0905.2211-1-19-3', '0905.2211-2-19-3'], ['0905.2211-1-19-4', '0905.2211-2-19-4'], ['0905.2211-1-19-5', '0905.2211-2-19-5'], ['0905.2211-1-19-6', '0905.2211-2-19-6'], ['0905.2211-1-19-7', '0905.2211-2-19-7'], ['0905.2211-1-19-8', '0905.2211-2-19-8'], ['0905.2211-1-19-9', '0905.2211-2-19-9'], ['0905.2211-1-57-0', '0905.2211-2-57-0'], ['0905.2211-1-58-0', '0905.2211-2-58-0'], ['0905.2211-1-58-1', '0905.2211-2-58-1'], ['0905.2211-1-58-2', '0905.2211-2-58-2'], ['0905.2211-1-58-3', '0905.2211-2-58-3'], ['0905.2211-1-58-4', '0905.2211-2-58-4'], ['0905.2211-1-16-0', '0905.2211-2-16-0'], ['0905.2211-1-7-0', '0905.2211-2-7-0'], ['0905.2211-1-7-1', '0905.2211-2-7-1'], ['0905.2211-1-7-2', '0905.2211-2-7-2'], ['0905.2211-1-61-0', '0905.2211-2-61-0'], ['0905.2211-1-61-1', '0905.2211-2-61-1'], ['0905.2211-1-25-0', '0905.2211-2-25-0'], ['0905.2211-1-25-1', '0905.2211-2-25-1'], ['0905.2211-1-25-2', '0905.2211-2-25-2'], ['0905.2211-1-25-3', '0905.2211-2-25-3'], ['0905.2211-1-25-4', '0905.2211-2-25-4'], ['0905.2211-1-25-5', '0905.2211-2-25-5'], ['0905.2211-1-25-6', '0905.2211-2-25-6'], ['0905.2211-1-25-7', '0905.2211-2-25-7'], ['0905.2211-1-25-8', '0905.2211-2-25-8'], ['0905.2211-1-25-9', '0905.2211-2-25-9'], ['0905.2211-1-30-0', '0905.2211-2-30-0'], ['0905.2211-1-30-1', '0905.2211-2-30-1'], ['0905.2211-1-68-0', '0905.2211-2-68-0'], ['0905.2211-1-68-1', '0905.2211-2-68-1'], ['0905.2211-1-68-2', '0905.2211-2-68-2'], ['0905.2211-1-70-0', '0905.2211-2-70-0'], ['0905.2211-1-70-1', '0905.2211-2-70-1'], ['0905.2211-1-70-2', '0905.2211-2-70-2'], ['0905.2211-1-43-0', '0905.2211-2-43-0'], ['0905.2211-1-43-1', '0905.2211-2-43-1'], ['0905.2211-1-43-2', '0905.2211-2-43-2'], ['0905.2211-1-50-0', '0905.2211-2-50-0'], ['0905.2211-1-73-0', '0905.2211-2-73-0'], ['0905.2211-1-73-1', '0905.2211-2-73-1'], ['0905.2211-1-73-2', '0905.2211-2-73-2'], ['0905.2211-1-73-3', '0905.2211-2-73-3'], ['0905.2211-1-32-0', '0905.2211-2-32-0'], ['0905.2211-1-32-1', '0905.2211-2-32-1'], ['0905.2211-1-32-2', '0905.2211-2-32-2'], ['0905.2211-1-32-3', '0905.2211-2-32-3'], ['0905.2211-1-52-0', '0905.2211-2-52-0'], ['0905.2211-1-52-1', '0905.2211-2-52-1'], ['0905.2211-1-8-0', '0905.2211-2-8-0'], ['0905.2211-1-8-1', '0905.2211-2-8-1'], ['0905.2211-1-4-0', '0905.2211-2-4-0'], ['0905.2211-1-4-1', '0905.2211-2-4-1'], ['0905.2211-1-4-2', '0905.2211-2-4-2'], ['0905.2211-1-4-3', '0905.2211-2-4-3'], ['0905.2211-1-4-4', '0905.2211-2-4-4'], ['0905.2211-1-4-5', '0905.2211-2-4-5'], ['0905.2211-1-37-0', '0905.2211-2-37-0'], ['0905.2211-1-37-1', '0905.2211-2-37-1'], ['0905.2211-1-37-2', '0905.2211-2-37-2'], ['0905.2211-1-37-3', '0905.2211-2-37-3'], ['0905.2211-1-37-4', '0905.2211-2-37-4'], ['0905.2211-1-21-0', '0905.2211-2-21-0'], ['0905.2211-1-21-1', '0905.2211-2-21-1'], ['0905.2211-1-72-0', '0905.2211-2-72-0'], ['0905.2211-1-72-2', '0905.2211-2-72-2'], ['0905.2211-1-10-0', '0905.2211-2-10-0'], ['0905.2211-1-44-0', '0905.2211-2-44-0'], ['0905.2211-1-44-1', '0905.2211-2-44-1'], ['0905.2211-1-44-2', '0905.2211-2-44-2'], ['0905.2211-1-44-3', '0905.2211-2-44-3'], ['0905.2211-1-44-4', '0905.2211-2-44-4'], ['0905.2211-1-44-5', '0905.2211-2-44-5'], ['0905.2211-1-44-6', '0905.2211-2-44-6'], ['0905.2211-1-49-0', '0905.2211-2-49-0'], ['0905.2211-1-49-1', '0905.2211-2-49-1'], ['0905.2211-1-64-0', '0905.2211-2-64-0'], ['0905.2211-1-64-2', '0905.2211-2-64-2'], ['0905.2211-1-2-0', '0905.2211-2-2-0'], ['0905.2211-1-2-1', '0905.2211-2-2-1'], ['0905.2211-1-2-2', '0905.2211-2-2-2'], ['0905.2211-1-2-3', '0905.2211-2-2-3'], ['0905.2211-1-2-4', '0905.2211-2-2-4'], ['0905.2211-1-9-0', '0905.2211-2-9-0'], ['0905.2211-1-9-1', '0905.2211-2-9-1'], ['0905.2211-1-9-2', '0905.2211-2-9-2'], ['0905.2211-1-9-3', '0905.2211-2-9-3'], ['0905.2211-1-9-4', '0905.2211-2-9-4'], ['0905.2211-1-9-5', '0905.2211-2-9-5'], ['0905.2211-1-29-0', '0905.2211-2-29-0'], ['0905.2211-1-29-1', '0905.2211-2-29-1'], ['0905.2211-1-29-2', '0905.2211-2-29-2'], ['0905.2211-1-29-3', '0905.2211-2-29-3'], ['0905.2211-1-29-4', '0905.2211-2-29-4'], ['0905.2211-1-35-0', '0905.2211-2-35-0'], ['0905.2211-1-35-1', '0905.2211-2-35-1'], ['0905.2211-1-35-2', '0905.2211-2-35-2'], ['0905.2211-1-55-0', '0905.2211-2-55-0'], ['0905.2211-1-55-1', '0905.2211-2-55-1'], ['0905.2211-1-17-0', '0905.2211-2-17-0'], ['0905.2211-1-17-1', '0905.2211-2-17-1'], ['0905.2211-1-17-2', '0905.2211-2-17-2'], ['0905.2211-1-78-0', '0905.2211-2-78-0'], ['0905.2211-1-78-1', '0905.2211-2-78-1'], ['0905.2211-1-78-2', '0905.2211-2-78-2'], ['0905.2211-1-24-0', '0905.2211-2-24-0'], ['0905.2211-1-24-1', '0905.2211-2-24-1'], ['0905.2211-1-24-2', '0905.2211-2-24-2'], ['0905.2211-1-77-1', '0905.2211-2-77-1'], ['0905.2211-1-72-1', '0905.2211-2-72-1'], ['0905.2211-1-64-1', '0905.2211-2-64-1'], ['0905.2211-1-78-3', '0905.2211-2-78-3']]

[['0905.2211-1-46-0', '0905.2211-2-46-0'], ['0905.2211-1-46-1', '0905.2211-2-46-1'], ['0905.2211-1-48-0', '0905.2211-2-48-0'], ['0905.2211-1-48-1', '0905.2211-2-48-1'], ['0905.2211-1-71-0', '0905.2211-2-71-0'], ['0905.2211-1-71-1', '0905.2211-2-71-1'], ['0905.2211-1-71-2', '0905.2211-2-71-2'], ['0905.2211-1-71-3', '0905.2211-2-71-3'], ['0905.2211-1-71-4', '0905.2211-2-71-4'], ['0905.2211-1-71-5', '0905.2211-2-71-5'], ['0905.2211-1-71-6', '0905.2211-2-71-6'], ['0905.2211-1-71-7', '0905.2211-2-71-7'], ['0905.2211-1-71-8', '0905.2211-2-71-8'], ['0905.2211-1-71-9', '0905.2211-2-71-9'], ['0905.2211-1-71-10', '0905.2211-2-71-10'], ['0905.2211-1-71-11', '0905.2211-2-71-11'], ['0905.2211-1-71-12', '0905.2211-2-71-12'], ['0905.2211-1-71-13', '0905.2211-2-71-13'], ['0905.2211-1-71-14', '0905.2211-2-71-14'], ['0905.2211-1-15-0', '0905.2211-2-15-0'], ['0905.2211-1-15-1', '0905.2211-2-15-1'], ['0905.2211-1-60-0', '0905.2211-2-60-0'], ['0905.2211-1-60-1', '0905.2211-2-60-1'], ['0905.2211-1-5-0', '0905.2211-2-5-0'], ['0905.2211-1-5-1', '0905.2211-2-5-1'], ['0905.2211-1-18-0', '0905.2211-2-18-0'], ['0905.2211-1-18-1', '0905.2211-2-18-1'], ['0905.2211-1-18-2', '0905.2211-2-18-2'], ['0905.2211-1-18-3', '0905.2211-2-18-3'], ['0905.2211-1-18-4', '0905.2211-2-18-4'], ['0905.2211-1-63-0', '0905.2211-2-63-0'], ['0905.2211-1-63-1', '0905.2211-2-63-1'], ['0905.2211-1-63-2', '0905.2211-2-63-2'], ['0905.2211-1-69-0', '0905.2211-2-69-0'], ['0905.2211-1-69-1', '0905.2211-2-69-1'], ['0905.2211-1-26-0', '0905.2211-2-26-0'], ['0905.2211-1-26-1', '0905.2211-2-26-1'], ['0905.2211-1-26-2', '0905.2211-2-26-2'], ['0905.2211-1-26-3', '0905.2211-2-26-3'], ['0905.2211-1-26-4', '0905.2211-2-26-4'], ['0905.2211-1-26-5', '0905.2211-2-26-5'], ['0905.2211-1-14-0', '0905.2211-2-14-0'], ['0905.2211-1-14-1', '0905.2211-2-14-1'], ['0905.2211-1-67-0', '0905.2211-2-67-0'], ['0905.2211-1-53-0', '0905.2211-2-53-0'], ['0905.2211-1-53-1', '0905.2211-2-53-1'], ['0905.2211-1-53-2', '0905.2211-2-53-2'], ['0905.2211-1-53-3', '0905.2211-2-53-3'], ['0905.2211-1-53-4', '0905.2211-2-53-4'], ['0905.2211-1-53-5', '0905.2211-2-53-5'], ['0905.2211-1-12-0', '0905.2211-2-12-0'], ['0905.2211-1-12-1', '0905.2211-2-12-1'], ['0905.2211-1-12-2', '0905.2211-2-12-2'], ['0905.2211-1-12-3', '0905.2211-2-12-3'], ['0905.2211-1-79-0', '0905.2211-2-79-0'], ['0905.2211-1-31-0', '0905.2211-2-31-0'], ['0905.2211-1-31-1', '0905.2211-2-31-1'], ['0905.2211-1-31-2', '0905.2211-2-31-2'], ['0905.2211-1-47-0', '0905.2211-2-47-0'], ['0905.2211-1-47-1', '0905.2211-2-47-1'], ['0905.2211-1-47-2', '0905.2211-2-47-2'], ['0905.2211-1-47-3', '0905.2211-2-47-3'], ['0905.2211-1-62-0', '0905.2211-2-62-0'], ['0905.2211-1-23-0', '0905.2211-2-23-0'], ['0905.2211-1-23-1', '0905.2211-2-23-1'], ['0905.2211-1-56-0', '0905.2211-2-56-0'], ['0905.2211-1-56-1', '0905.2211-2-56-1'], ['0905.2211-1-41-0', '0905.2211-2-41-0'], ['0905.2211-1-41-1', '0905.2211-2-41-1'], ['0905.2211-1-41-2', '0905.2211-2-41-2'], ['0905.2211-1-41-3', '0905.2211-2-41-3'], ['0905.2211-1-38-0', '0905.2211-2-38-0'], ['0905.2211-1-38-1', '0905.2211-2-38-1'], ['0905.2211-1-6-0', '0905.2211-2-6-0'], ['0905.2211-1-6-1', '0905.2211-2-6-1'], ['0905.2211-1-6-2', '0905.2211-2-6-2'], ['0905.2211-1-11-0', '0905.2211-2-11-0'], ['0905.2211-1-0-0', '0905.2211-2-0-0'], ['0905.2211-1-33-0', '0905.2211-2-33-0'], ['0905.2211-1-33-1', '0905.2211-2-33-1'], ['0905.2211-1-33-2', '0905.2211-2-33-2'], ['0905.2211-1-75-0', '0905.2211-2-75-0'], ['0905.2211-1-51-0', '0905.2211-2-51-0'], ['0905.2211-1-51-1', '0905.2211-2-51-1'], ['0905.2211-1-51-2', '0905.2211-2-51-2'], ['0905.2211-1-51-3', '0905.2211-2-51-3'], ['0905.2211-1-51-4', '0905.2211-2-51-4'], ['0905.2211-1-51-5', '0905.2211-2-51-5'], ['0905.2211-1-39-0', '0905.2211-2-39-0'], ['0905.2211-1-39-1', '0905.2211-2-39-1'], ['0905.2211-1-39-2', '0905.2211-2-39-2'], ['0905.2211-1-39-3', '0905.2211-2-39-3'], ['0905.2211-1-76-0', '0905.2211-2-76-0'], ['0905.2211-1-76-1', '0905.2211-2-76-1'], ['0905.2211-1-76-2', '0905.2211-2-76-2'], ['0905.2211-1-76-3', '0905.2211-2-76-3'], ['0905.2211-1-76-4', '0905.2211-2-76-4'], ['0905.2211-1-77-0', '0905.2211-2-77-0'], ['0905.2211-1-40-0', '0905.2211-2-40-0'], ['0905.2211-1-40-1', '0905.2211-2-40-1'], ['0905.2211-1-40-2', '0905.2211-2-40-2'], ['0905.2211-1-40-3', '0905.2211-2-40-3'], ['0905.2211-1-40-4', '0905.2211-2-40-4'], ['0905.2211-1-40-5', '0905.2211-2-40-5'], ['0905.2211-1-27-0', '0905.2211-2-27-0'], ['0905.2211-1-27-1', '0905.2211-2-27-1'], ['0905.2211-1-27-2', '0905.2211-2-27-2'], ['0905.2211-1-27-3', '0905.2211-2-27-3'], ['0905.2211-1-27-4', '0905.2211-2-27-4'], ['0905.2211-1-36-0', '0905.2211-2-36-0'], ['0905.2211-1-36-1', '0905.2211-2-36-1'], ['0905.2211-1-36-2', '0905.2211-2-36-2'], ['0905.2211-1-36-3', '0905.2211-2-36-3'], ['0905.2211-1-66-0', '0905.2211-2-66-0'], ['0905.2211-1-19-0', '0905.2211-2-19-0'], ['0905.2211-1-19-1', '0905.2211-2-19-1'], ['0905.2211-1-19-2', '0905.2211-2-19-2'], ['0905.2211-1-19-3', '0905.2211-2-19-3'], ['0905.2211-1-19-4', '0905.2211-2-19-4'], ['0905.2211-1-19-5', '0905.2211-2-19-5'], ['0905.2211-1-19-6', '0905.2211-2-19-6'], ['0905.2211-1-19-7', '0905.2211-2-19-7'], ['0905.2211-1-19-8', '0905.2211-2-19-8'], ['0905.2211-1-19-9', '0905.2211-2-19-9'], ['0905.2211-1-57-0', '0905.2211-2-57-0'], ['0905.2211-1-58-0', '0905.2211-2-58-0'], ['0905.2211-1-58-1', '0905.2211-2-58-1'], ['0905.2211-1-58-2', '0905.2211-2-58-2'], ['0905.2211-1-58-3', '0905.2211-2-58-3'], ['0905.2211-1-58-4', '0905.2211-2-58-4'], ['0905.2211-1-16-0', '0905.2211-2-16-0'], ['0905.2211-1-7-0', '0905.2211-2-7-0'], ['0905.2211-1-7-1', '0905.2211-2-7-1'], ['0905.2211-1-7-2', '0905.2211-2-7-2'], ['0905.2211-1-61-0', '0905.2211-2-61-0'], ['0905.2211-1-61-1', '0905.2211-2-61-1'], ['0905.2211-1-25-0', '0905.2211-2-25-0'], ['0905.2211-1-25-1', '0905.2211-2-25-1'], ['0905.2211-1-25-2', '0905.2211-2-25-2'], ['0905.2211-1-25-3', '0905.2211-2-25-3'], ['0905.2211-1-25-4', '0905.2211-2-25-4'], ['0905.2211-1-25-5', '0905.2211-2-25-5'], ['0905.2211-1-25-6', '0905.2211-2-25-6'], ['0905.2211-1-25-7', '0905.2211-2-25-7'], ['0905.2211-1-25-8', '0905.2211-2-25-8'], ['0905.2211-1-25-9', '0905.2211-2-25-9'], ['0905.2211-1-30-0', '0905.2211-2-30-0'], ['0905.2211-1-30-1', '0905.2211-2-30-1'], ['0905.2211-1-68-0', '0905.2211-2-68-0'], ['0905.2211-1-68-1', '0905.2211-2-68-1'], ['0905.2211-1-68-2', '0905.2211-2-68-2'], ['0905.2211-1-70-0', '0905.2211-2-70-0'], ['0905.2211-1-70-1', '0905.2211-2-70-1'], ['0905.2211-1-70-2', '0905.2211-2-70-2'], ['0905.2211-1-43-0', '0905.2211-2-43-0'], ['0905.2211-1-43-1', '0905.2211-2-43-1'], ['0905.2211-1-43-2', '0905.2211-2-43-2'], ['0905.2211-1-50-0', '0905.2211-2-50-0'], ['0905.2211-1-73-0', '0905.2211-2-73-0'], ['0905.2211-1-73-1', '0905.2211-2-73-1'], ['0905.2211-1-73-2', '0905.2211-2-73-2'], ['0905.2211-1-73-3', '0905.2211-2-73-3'], ['0905.2211-1-32-0', '0905.2211-2-32-0'], ['0905.2211-1-32-1', '0905.2211-2-32-1'], ['0905.2211-1-32-2', '0905.2211-2-32-2'], ['0905.2211-1-32-3', '0905.2211-2-32-3'], ['0905.2211-1-52-0', '0905.2211-2-52-0'], ['0905.2211-1-52-1', '0905.2211-2-52-1'], ['0905.2211-1-8-0', '0905.2211-2-8-0'], ['0905.2211-1-8-1', '0905.2211-2-8-1'], ['0905.2211-1-4-0', '0905.2211-2-4-0'], ['0905.2211-1-4-1', '0905.2211-2-4-1'], ['0905.2211-1-4-2', '0905.2211-2-4-2'], ['0905.2211-1-4-3', '0905.2211-2-4-3'], ['0905.2211-1-4-4', '0905.2211-2-4-4'], ['0905.2211-1-4-5', '0905.2211-2-4-5'], ['0905.2211-1-37-0', '0905.2211-2-37-0'], ['0905.2211-1-37-1', '0905.2211-2-37-1'], ['0905.2211-1-37-2', '0905.2211-2-37-2'], ['0905.2211-1-37-3', '0905.2211-2-37-3'], ['0905.2211-1-37-4', '0905.2211-2-37-4'], ['0905.2211-1-21-0', '0905.2211-2-21-0'], ['0905.2211-1-21-1', '0905.2211-2-21-1'], ['0905.2211-1-72-0', '0905.2211-2-72-0'], ['0905.2211-1-72-2', '0905.2211-2-72-2'], ['0905.2211-1-10-0', '0905.2211-2-10-0'], ['0905.2211-1-44-0', '0905.2211-2-44-0'], ['0905.2211-1-44-1', '0905.2211-2-44-1'], ['0905.2211-1-44-2', '0905.2211-2-44-2'], ['0905.2211-1-44-3', '0905.2211-2-44-3'], ['0905.2211-1-44-4', '0905.2211-2-44-4'], ['0905.2211-1-44-5', '0905.2211-2-44-5'], ['0905.2211-1-44-6', '0905.2211-2-44-6'], ['0905.2211-1-49-0', '0905.2211-2-49-0'], ['0905.2211-1-49-1', '0905.2211-2-49-1'], ['0905.2211-1-64-0', '0905.2211-2-64-0'], ['0905.2211-1-64-2', '0905.2211-2-64-2'], ['0905.2211-1-2-0', '0905.2211-2-2-0'], ['0905.2211-1-2-1', '0905.2211-2-2-1'], ['0905.2211-1-2-2', '0905.2211-2-2-2'], ['0905.2211-1-2-3', '0905.2211-2-2-3'], ['0905.2211-1-2-4', '0905.2211-2-2-4'], ['0905.2211-1-9-0', '0905.2211-2-9-0'], ['0905.2211-1-9-1', '0905.2211-2-9-1'], ['0905.2211-1-9-2', '0905.2211-2-9-2'], ['0905.2211-1-9-3', '0905.2211-2-9-3'], ['0905.2211-1-9-4', '0905.2211-2-9-4'], ['0905.2211-1-9-5', '0905.2211-2-9-5'], ['0905.2211-1-29-0', '0905.2211-2-29-0'], ['0905.2211-1-29-1', '0905.2211-2-29-1'], ['0905.2211-1-29-2', '0905.2211-2-29-2'], ['0905.2211-1-29-3', '0905.2211-2-29-3'], ['0905.2211-1-29-4', '0905.2211-2-29-4'], ['0905.2211-1-35-0', '0905.2211-2-35-0'], ['0905.2211-1-35-1', '0905.2211-2-35-1'], ['0905.2211-1-35-2', '0905.2211-2-35-2'], ['0905.2211-1-55-0', '0905.2211-2-55-0'], ['0905.2211-1-55-1', '0905.2211-2-55-1'], ['0905.2211-1-17-0', '0905.2211-2-17-0'], ['0905.2211-1-17-1', '0905.2211-2-17-1'], ['0905.2211-1-17-2', '0905.2211-2-17-2'], ['0905.2211-1-78-0', '0905.2211-2-78-0'], ['0905.2211-1-78-1', '0905.2211-2-78-1'], ['0905.2211-1-78-2', '0905.2211-2-78-2'], ['0905.2211-1-24-0', '0905.2211-2-24-0'], ['0905.2211-1-24-1', '0905.2211-2-24-1'], ['0905.2211-1-24-2', '0905.2211-2-24-2']]

[['0905.2211-1-77-1', '0905.2211-2-77-1'], ['0905.2211-1-72-1', '0905.2211-2-72-1'], ['0905.2211-1-64-1', '0905.2211-2-64-1'], ['0905.2211-1-78-3', '0905.2211-2-78-3']]

[]

[]

[]

['0905.2211-1-1-0', '0905.2211-1-13-0', '0905.2211-1-23-2', '0905.2211-1-39-4', '0905.2211-1-45-0', '0905.2211-1-59-0', '0905.2211-1-67-1', '0905.2211-2-1-0', '0905.2211-2-13-0', '0905.2211-2-23-2', '0905.2211-2-39-4', '0905.2211-2-45-0', '0905.2211-2-59-0', '0905.2211-2-67-1']

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

https://arxiv.org/abs/0905.2211

1802.08924

{'1802.08924-1-0-0': 'We propose a new paradigm for time-series learning where users implicitly specify families of signal shapes by choosing monotonic parameterized signal predicates.', '1802.08924-1-0-1': "These families of predicates (also called specifications) can be seen as infinite Boolean feature vectors, that are able to leverage a user's domain expertise and have the property that as the parameter values increase, the specification becomes easier to satisfy.", '1802.08924-1-0-2': 'In the presence of multiple parameters, monotonic specifications admit trade-off curves in the parameter space, akin to Pareto fronts in multi-objective optimization, that separate the specifications that are satisfied from those that are not satisfied.', '1802.08924-1-0-3': 'Viewing monotonic specifications (and their trade-off curves) as "features" for time-series data, we develop a principled way to bestow a distance measure between signals through the lens of a monotonic specification.', '1802.08924-1-0-4': 'A unique feature of this approach is that, a simple Boolean predicate based on the monotonic specification can be used to explain why any two traces (or sets of traces) have a given distance.', '1802.08924-1-0-5': 'Given a simple enough specification, this enables relaying at a high level "why" two signals have a certain distance and what kind of signals lie between them.', '1802.08924-1-0-6': 'We conclude by demonstrating our technique with two case studies that illustrate how simple monotonic specifications can be used to craft desirable distance measures.', '1802.08924-1-1-0': '# Introduction', '1802.08924-1-2-0': 'Recently, there has been a proliferation of sensors that monitor diverse kinds of real-time data.', '1802.08924-1-2-1': 'This has led to system engineers facing a deluge of sensor data, with an urgent demand to extract meaningful analytics from it.', '1802.08924-1-2-2': 'Of particular relevance is the data representing time-series behaviors or signals generated by the systems and devices being monitored through such sensors.', '1802.08924-1-2-3': 'The broad impact of machine learning (ML) techniques for signal analysis is tangible in domains ranging from healthcare analytics [CITATION] to smart transportation [CITATION], and from autonomous driving [CITATION] to social media [CITATION].', '1802.08924-1-2-4': 'While the importance of ML-based techniques cannot be stressed enough, there have been certain impediments to their universal adaptation by novice users.', '1802.08924-1-2-5': 'In particular, existing approaches to time-series learning need improvement along two directions: (1) the models obtained through ML algorithms are difficult to interpret, and (2) engineers with domain-expertise may not be ML experts, and are thus unable to use their domain knowledge to guide ML tasks.', '1802.08924-1-3-0': 'A common way to encode domain specific knowledge into an ML task is to first transform the data into an a priori known feature space.', '1802.08924-1-3-1': 'To ease the burden on the user, such approaches often rely on large sets of generic features (such as those based on statistics or signal processing) coupled with dimensionality reduction techniques to identify and recommend significant features.', '1802.08924-1-3-2': 'This automated feature selection comes at a cost of the ability to leverage features as a tool to further the end-user understanding of the ML model.', '1802.08924-1-3-3': 'For example, a support vector machine or a clustering procedure using generic time series features may perform well on a given data-set, but for a given signal-trace, it is rarely clear why it gets assigned a particular label, or why it is grouped in a particular cluster.', '1802.08924-1-4-0': 'One notable exception is recent work based on the idea of shapelets, where algorithms are developed to automatically identify shape-like features from the time-series data itself, and then use them for classification and clustering [CITATION].', '1802.08924-1-4-1': 'In [CITATION], the authors extend basic shapelets to logical combinations of shapelets; in our view this method can be extended even further to lift shape-based reasoning to a more abstract form of logic-based reasoning where logical formulae can specify families of shapes.', '1802.08924-1-4-2': 'In this paper, we propose a new paradigm for time-series learning where users implicitly specify families of signal shapes by choosing parameterized signal predicates.', '1802.08924-1-4-3': "These families of predicates (also called specifications) can be seen as infinite Boolean feature vectors, that are able to leverage a user's domain expertise.", '1802.08924-1-5-0': 'Contributions A key insight in our work is to specialize from parameterized specifications to monotonic specifications over signals.', '1802.08924-1-5-1': 'Informally, monotonic specifications have the property that as the parameter values increase, the specification becomes easier to satisfy.', '1802.08924-1-5-2': 'In the presence of multiple parameters, monotonic logics admit trade-off curves in the parameter space, akin to Pareto fronts in multi-objective optimization, that separate the specifications that are satisfied from those that are not satisfied.', '1802.08924-1-5-3': 'Leveraging this insight we provide the following contributions: [leftmargin=1.5em,itemsep=0pt,parsep=0pt,partopsep=0pt] The introduction of monotonic specifications (and their trade-off curves) as a "feature" for time-series data.', '1802.08924-1-5-4': 'A principled way to bestow a distance measure between signals through the lens of a monotonic specification.', '1802.08924-1-5-5': 'Distance measure in hand, standard ML algorithms such as nearest neighbors (supervised) or agglomerative clustering (unsupervised) can be used to glean insights into the data.', '1802.08924-1-5-6': 'A simple Boolean predicate based on the monotonic specification that can be used to explain why any two traces (or sets of traces) have a given distance.', '1802.08924-1-5-7': 'Given a simple enough specification, this enables relaying at a high level "why" two signals have a certain distance and what kind of signals lie between them.', '1802.08924-1-6-0': 'Motivating Example Before developing our technique, we motivate with an example that illustrates how naive feature spaces may fail to capture critical aspects of the data that the user may find desirable.', '1802.08924-1-7-0': 'Most freeways have bottlenecks that lead to traffic congestion, and if there is a stalled or a crashed vehicle (or vehicles) at this site, then upstream traffic congestion can severely worsen.', '1802.08924-1-7-1': "The problem of distinguishing a stalled vehicle from a vehicle facing regular traffic congestion from time-series data of a vehicle's motion is challenging, as both vehicle trajectories have common characteristics such as slow average speeds, small minimum and maximum velocities, and so on.", '1802.08924-1-7-2': 'More concretely, Fig [REF] shows a series of potential time series to which we would like to assign pairwise distances indicating the similarity (small values) or differences (large values) between two time-series.', '1802.08924-1-7-3': "To ease exposition, we have limited our focus to the car's speed.", '1802.08924-1-7-4': 'In signals 0 and 1, both cars transition from high speed freeway driving to stop and go traffic.', '1802.08924-1-7-5': 'Conversely, in signal 2, the car transitions from stop and go traffic to high speed freeway driving.', '1802.08924-1-7-6': 'Signal 3 corresponds to a car slowing to a stop and then accelerating, perhaps due to difficulty merging lanes.', '1802.08924-1-7-7': 'Finally, signal 4 signifies a car encountering no traffic and signal 5 corresponds to a car in heavy traffic, or a possibly stalled vehicle.', '1802.08924-1-8-0': 'Suppose a user wished to find a feature space equipped with a measure to distinguish cars being stuck in traffic.', '1802.08924-1-8-1': 'Some properties might be: [leftmargin=1.5em,itemsep=0pt,parsep=0pt,partopsep=0pt] Signals 0 and 1 should be very close together since both show a car entering stop and go traffic in nearly the same way.', '1802.08924-1-8-2': 'Signals 2, 3, and 4 should be close together since the car ultimately escapes stop and go traffic.', '1802.08924-1-8-3': 'Signal 5 should be far from all other examples since it does not represent entering or leaving stop and go traffic.', '1802.08924-1-9-0': 'For a strawman comparison, we consider two ways the user might assign a distance measure to the above signal space.', '1802.08924-1-9-1': 'At first, the user might treat the signals as a series of independent measurements and attempt to characterize the signals via standard statistical measures on the speed and acceleration (mean, standard deviation, ).', '1802.08924-1-9-2': 'Intuitively, the signal corresponding to acceleration should indicate a change in state of the car due to encountered traffic.', '1802.08924-1-9-3': 'Similarly, the statistics on the velocity would measure if the car was mostly stationary or moving at a high speed.', '1802.08924-1-9-4': 'Fig [REF] illustrates how the example signals look in this feature space with each component normalized between 0 and 1.', '1802.08924-1-9-5': 'The user might then use the euclidean distance of each feature to assign a distance between signals (the lower triangle of Fig [REF]).', '1802.08924-1-9-6': 'Unfortunately, in this measure, signal 4 is not close to signal 2 or 3, violating the second desired property.', '1802.08924-1-9-7': 'Further, signals 0 and 1 are not "very" close together violating the first property.', '1802.08924-1-10-0': 'Next, seeing that related signals have similar shapes, the user then attempts to use dynamic time warping as a distance measure (the upper triangle of Fig [REF]).', '1802.08924-1-10-1': 'Now signals 0 and 1 are very close together, however, signal 3 is to close to signal 0 and signal 5 is too close to signals 0, 1, and 2.', '1802.08924-1-11-0': 'In the sequel, we shall show how using a simple monotonic specification to characterize the signal produces the desired results.', '1802.08924-1-11-1': 'Informally, the specification states', '1802.08924-1-12-0': '"Between time [MATH] and 20, the car speed is always less than [MATH]."', '1802.08924-1-13-0': 'Fig [REF] illustrates the trade-off boundaries between [MATH] and [MATH] induced by this specification and Fig [REF] shows the pairwise Hausdorff distances between each boundary.', '1802.08924-1-13-1': 'As is easily confirmed, all 3 properties desired of the clustering algorithm hold.', '1802.08924-1-13-2': 'Furthermore, as a result of the Hausdorff distance between boundaries being equivalent to the distance between two parameter values, each distance between signals [MATH] and [MATH] is associated with a simple specification characterizing the signals that lie between them.', '1802.08924-1-13-3': 'For example, the dashed line in Fig [REF] indicates the distance between signals [MATH] and [MATH].', '1802.08924-1-13-4': 'Our technique associates with this distance the specification,', '1802.08924-1-14-0': '"Between time 0 and 20, the car speed is always less than [MATH] AND between time 0 and 20, the car speed is eventually greater than [MATH]."', '1802.08924-1-15-0': 'which characterizes traces whose trade-off curves intersect the dashed line.', '1802.08924-1-16-0': 'Before concluding the example, we note the human-interpretable property - car gets stuck in traffic - is captured by a related specification shown below; this specification gives rise to a distance measure that makes signal 5 to be closer to signals 0 and 1 and signal 3 to be between signals 1 and 2:', '1802.08924-1-17-0': '"Between time [MATH] and 20, the car speed is always less than [MATH] AND between time [MATH] and 20, the car speed is always greater than [MATH]".', '1802.08924-1-18-0': 'Note that above human-interpretable specifications that enable clustering and classification are not obtained in an ad hoc fashion.', '1802.08924-1-18-1': 'In fact, these specifications are instantiations of signal predicates specified in a formal logic for time-series data known as Parametric Signal Temporal Logic (PSTL).', '1802.08924-1-18-2': 'PSTL is a logic originally developed for specifying signals in analog hardware circuits, and later extended to various domains including the automotive [CITATION] and robotics [CITATION].', '1802.08924-1-18-3': 'It also is the underlying language for the specification library ST-Lib [CITATION] used for embedded control specifications.', '1802.08924-1-18-4': 'While monotonic PSTL is a rich language that can specify many complex time-varying behaviors, we emphasize that end-users need not be experts in formal logics to use our techniques.', '1802.08924-1-18-5': 'In particular, one can create complex PSTL templates by composing simpler easier to understand templates.', '1802.08924-1-19-0': '# Preliminaries', '1802.08924-1-20-0': 'The main object of analysis in this paper are time series.', '1802.08924-1-20-1': '[Time Series, Time Languages] Let [MATH] be a countable subset of [MATH] and [MATH] be some non empty set.', '1802.08924-1-20-2': 'A time series (or signal or trace), [MATH] is a map: [EQUATION]', '1802.08924-1-20-3': 'Where [MATH] and [MATH] are called the time domain and (value) domain respectively.', '1802.08924-1-20-4': 'The set of all time series is denoted by [MATH].', '1802.08924-1-21-0': 'Between any two time series one can define a metric which measures their similarity.', '1802.08924-1-21-1': '[Metric] Given a set [MATH], a metric is a map, [EQUATION] such that [MATH], [MATH], [MATH].', '1802.08924-1-22-0': '[Euclidean distance and Infinity Norm Metrics] The euclidean distance between two vectors is a metric [MATH].', '1802.08924-1-22-1': 'Similarly, the infinity norm induced distance [MATH] is a metric.', '1802.08924-1-23-0': '[Hausdorff Distance] Given a set [MATH] with a distance metric [MATH], the hausdorff distance is a distance metric between subsets of [MATH].', '1802.08924-1-23-1': 'Namely, given subsets [MATH]: [EQUATION]', '1802.08924-1-23-2': 'An ideal metric between traces should respect any domain specific properties that make two elements "similar".', '1802.08924-1-23-3': 'A (logical) property [MATH] assigns to each timed trace a truth value, and can be viewed as the subset of traces that have the property.', '1802.08924-1-23-4': '[Specification] A specification is a set of time series [MATH].', '1802.08924-1-23-5': 'A time series, [MATH], is said to satisfy a specification iff [MATH].', '1802.08924-1-23-6': 'The set of all specifications is the power set of [MATH], [MATH].', '1802.08924-1-24-0': 'Consider the following specification related to the specification from the running example: [EQUATION]', '1802.08924-1-24-1': 'Informally, this specification says that after [MATH], the value of the time series, [MATH], is always less than 1.', '1802.08924-1-25-0': 'Given a finite number of properties, one can then "fingerprint" a time series as a Boolean feature vector.', '1802.08924-1-25-1': 'However, many times the relevant properties are not easily captured by a finite sequence of binary features.', '1802.08924-1-25-2': 'For example, one might image a single real valued feature [MATH] taking values between [MATH] and [MATH].', '1802.08924-1-25-3': 'This feature encodes an uncountably infinite family of Boolean features [MATH] for [MATH].', '1802.08924-1-25-4': 'Such families are called parametric specifications.', '1802.08924-1-25-5': 'For simplicity, we assume that the parameters are a subset of the unit hyper-square.', '1802.08924-1-25-6': '[Parametric Specifications] A parametric specification is a map: [EQUATION] where [MATH] is the number of parameters.', '1802.08924-1-26-0': 'As seen above, parametric specifications arise naturally from syntactically substituting constants with parameters in the description of a specification.', '1802.08924-1-27-0': 'The parametric specification given in the introduction can be formalized substituting [MATH] for [MATH] and [MATH] for [MATH] in Ex [REF].', '1802.08924-1-27-1': '[EQUATION]', '1802.08924-1-27-2': 'To generalize the "fingerprint" of a time series for parametric specifications, first observe that the value of a boolean feature vector is exactly determined by which entries are set to True.', '1802.08924-1-27-3': 'Analogously, the set of parameter values for which a parameterized specification, [MATH], would yield true on a given trace, called the validity domain, acts the "fingerprint".', '1802.08924-1-27-4': '[Validity domain, Validity domain boundary] Given a parametric specification of [MATH] parameters and a trace [MATH], the validity domain is the set, [EQUATION]', '1802.08924-1-27-5': 'In general, the validity domain can be arbitrarly complex which makes developing a distance metric between validity domains subtle.', '1802.08924-1-27-6': 'We circumvent such hurdles by specializing to monotonic specifications, for which the validity domains are remarkably simple.', '1802.08924-1-27-7': '[Monotonic Specifications] A parametric specification is said to be monotonic if [EQUATION] where [MATH] is the standard product ordering on [MATH].', '1802.08924-1-28-0': 'Before examining the validity domain of monotonic specifications, observe that the parametric specification in Ex [REF] (and thus intro example) is monotonic.', '1802.08924-1-29-0': 'Given a monotonic specification, [MATH] and a time series [MATH], the boundary of the validity domain, [MATH], of a monotonic specification is a hyper-surface that segments [MATH] into two connected components.', '1802.08924-1-30-0': 'In the sequel, we develop a distance metric between validity domains which characterizes the similarity between two time series under the lens of a monotonic specification.', '1802.08924-1-31-0': '# Logic-Respecting Distance Metric', '1802.08924-1-32-0': 'Observe that the validity domains of monotonic specifications are uniquely defined by the hyper surface that separates them from the rest of the parameter space.', '1802.08924-1-32-1': 'Similar to Pareto fronts in a multi-objective optimization, these boundaries encode the trade-offs required in each parameter to make the specification satisfied for a given time series.', '1802.08924-1-32-2': 'This suggests a simple procedure to define a distance metric between time series that respects their logical properties: Given a monotonic specification, a set of time series, and a distance metric between validity domain boundaries:', '1802.08924-1-33-0': 'Compute the validity domain boundaries for each time series.', '1802.08924-1-33-1': 'Compute a distance between the validity domain boundaries.', '1802.08924-1-34-0': 'Of course, the benefits of using this metric would rely entirely on whether (i) The monotonic specification captures the relevant domain specific details (ii) The distance between validity domain boundaries is (in)sensitive to outliers.', '1802.08924-1-34-1': 'While the choice of specification is highly domain specific, we argue that for many monotonic specifications, the distance metric should be sensitive to outliers as this represents a large deviation from the specification.', '1802.08924-1-34-2': 'This sensitivity requirement seems particularly apt if the size of the specification grows linearly or super linearly as the parameters increase.', '1802.08924-1-34-3': 'To this end, we propose using Hausdorff distance between validity domains for three reasons:', '1802.08924-1-35-0': 'The Hausdorff distance is sensitive to outliers.', '1802.08924-1-35-1': 'The Hausdorff distance between two boundaries reduces to the distance between two parameters from each boundary.', '1802.08924-1-35-2': 'These elements explain why the boundaries differ.', '1802.08924-1-35-3': 'If two boundaries have Hausdorff distance [MATH] , if one boundary proposes a parameter (and thus specification), the other boundary must have a parameter (specification) within [MATH] units of distance.', '1802.08924-1-35-4': 'Thus, the Hausdorff distance measures how well the two validity domains can simulate each other.', '1802.08924-1-36-0': 'We define our new distance metric between time series as:', '1802.08924-1-37-0': 'Given a monotonic specification [MATH] and a distance metric on the parameter space [MATH], the logical distance between two time series, [MATH] is: [EQUATION]', '1802.08924-1-37-1': 'As shown in the introduction, the property for the Hausdorff distance between two boundaries (signals) to reduce to the Hausdorff distance between two parameter values (and thus specifications) can be leveraged to describe the set of signals that lie "between" the signals.', '1802.08924-1-37-2': 'More precisely, if parameters [MATH] and [MATH] are responsible for the Hausdorff distance of signals [MATH] and [MATH], by signals "between" we mean all signals [MATH] whose validity domain boundary intersects the straight line from [MATH] to [MATH].', '1802.08924-1-37-3': 'If using the infinity norm as the parameter space distance metric, then this line corresponds to a degenerate 1-d hyper-box whose specification [CITATION] is given by: [EQUATION] or in logical notation [EQUATION] where we have assumed w.o.l.o.g that [MATH].', '1802.08924-1-38-0': 'As Eq [REF] excludes signals whose boundaries are above and below [MATH] and [MATH] resp., the boundaries corresponding to each signal in Eq [REF] must intersect with the line between [MATH] and [MATH].', '1802.08924-1-39-0': '## Computing the Logical Distance', '1802.08924-1-40-0': 'Before continuing onto the case studies, we briefly discuss how to compute the logical distance metric.', '1802.08924-1-40-1': 'First and foremost, a validity domain boundary of a monotonic specification can be recursively approximated to arbitrary precision via binary search on the diagonal of the parameter space [CITATION].', '1802.08924-1-40-2': 'This approximation yields a series of overlapping axis aligned rectangles that are guaranteed to contain the boundary (see Fig [REF]).', '1802.08924-1-41-0': 'Given enough precision, one can then sample points within each rectangle and compute the point wise Hausdorff distance [CITATION].', '1802.08924-1-41-1': 'Alg [REF] describes the above procedure as pseudo-code.', '1802.08924-1-42-0': 'Compute Logical Distance [1] logicalDist[MATH]', '1802.08924-1-43-0': 'In our implementation [CITATION], the boundary approximations occur in parallel and we use infinity norm for the distance in the parameter space.', '1802.08924-1-43-1': 'This enables a straight forward calculation to obtain upper and lower bounds by analyzing the rectangles directly.', '1802.08924-1-44-0': 'Regarding scaling, we briefly remark that properly normalizing and pruning rectangles in the Hausdorff approximation is absolutely necessary to get moderate performance.', '1802.08924-1-44-1': 'In the worst case, the number of rectangles required to approximate the boundary scales exponentially and the Hausdorff distance is quadratic (although each part of the computation is embarrassingly parallel).', '1802.08924-1-45-0': '# Case Studies', '1802.08924-1-46-0': 'In our case studies we utilized Parametric Signal Temporal Logic (PSTL) as a formalism/language to encode specifications.', '1802.08924-1-47-0': '## PSTL as a Feature Design Language', '1802.08924-1-48-0': 'Real-time temporal logics are a formalism for reasoning about finite or infinite timed series.', '1802.08924-1-48-1': 'These logics add to propositional logic modal operators to encode temporal concepts.', '1802.08924-1-48-2': 'Signal Temporal Logic [CITATION] was proposed in the context of analog and mixed-signal circuits as a specification language for real-valued signals.', '1802.08924-1-49-0': 'Signal Temporal Logic Atoms in STL formulas take the form [MATH], where [MATH] is a function from [MATH] to [MATH], [MATH], and [MATH].', '1802.08924-1-49-1': 'Temporal formulas are formed using temporal operators, "globally" (denoted as [MATH]), "in the future" (denoted as [MATH]) and "until" (denoted as [MATH]) that can each be indexed by an interval [MATH].', '1802.08924-1-49-2': '[Signal Temporal Logic] A formula in Signal Temporal Logic is syntactically defined via the grammar: [EQUATION]', '1802.08924-1-50-0': 'In the above grammar, [MATH], and [MATH].', '1802.08924-1-50-1': 'The globally ([MATH]) and in the future ([MATH]) operators are defined syntactic sugar for special cases of the until operator: [MATH], and [MATH].', '1802.08924-1-50-2': 'We use the notation [MATH] to mean that the suffix of the timed trace [MATH] beginning at time [MATH] satisfies the formula [MATH].', '1802.08924-1-50-3': 'The formal semantics of an STL formula are defined recursively: [EQUATION]', '1802.08924-1-50-4': 'We write [MATH] as a shorthand of [MATH].', '1802.08924-1-51-0': 'The running example specification, [MATH], can encoded in STL as: [EQUATION] which reads "Always between t=0.2 and infinity, [MATH] is less than 1".', '1802.08924-1-52-0': 'Parametric STL (PSTL) and Monotonic STL PSTL [CITATION] is a natural extension of STL that syntactically replaces constants within a STL formula with constants.', '1802.08924-1-52-1': 'The polar fragment of PSTL [CITATION] is a syntactically identifiable subset of PSTL that are monotonic specifications in accordance to Def [REF].', '1802.08924-1-52-2': 'The details and formal definition of this fragment are outside the scope of this work; however, all PSTL formula given are monotonic and in the polar fragment.', '1802.08924-1-53-0': 'The parametric specification, [MATH] can be encoded as PSTL as: [EQUATION]', '1802.08924-1-54-0': '## Case Study 1', '1802.08924-1-55-0': 'In this case study, we take our running example, and attempt to apply the same (or similar) templates to real traffic data.', '1802.08924-1-55-1': 'To improve driver and traffic on highways, the Federal Highway Administration collected detailed traffic data on southbound US-101 freeway, in Los Angeles [CITATION].', '1802.08924-1-55-2': 'Traffic through the segment was monitored and recorded through eight synchronized cameras, next to the freeway.', '1802.08924-1-55-3': 'A total of [MATH] minutes of traffic data was recorded including vehicle trajectory data providing lane positions of each vehicle within the study area.', '1802.08924-1-56-0': 'We picked three 80 seconds velocity trajectories, and split them into 20 second segments.', '1802.08924-1-56-1': 'We analyzed the resulting 12 segments using the following monotonic specifications, [EQUATION] where [MATH] and [MATH] characterizes trajectories where a car potentially enters and leaves high congestion areas respectively.', '1802.08924-1-56-2': 'We combine the features by averaging their logical distances.', '1802.08924-1-56-3': 'Fig [REF] shows the corresponding the resulting pairwise distances and clustering dendogram (signals from each clusters are shown in Fig [REF]).', '1802.08924-1-57-0': 'Cluster 0 contains trajectories leaving high congestion and hence the velocity increases towards the end of the trajectory.', '1802.08924-1-57-1': 'Cluster 1 contains trajectories entering high congestion, shown by the decrease in velocity.', '1802.08924-1-57-2': 'Cluster 2 shows the trajectories that enter and leave high congestion areas in the same segment.', '1802.08924-1-57-3': 'Finally cluster 3 indicates entering moderate congestion.', '1802.08924-1-57-4': 'Observe that these clusters are remarkably similar to the toy example templates from our running example.', '1802.08924-1-58-0': '## Case Study 2 - CPS Grader', '1802.08924-1-59-0': 'Massively Open Online Courses (MOOCs) present instructors the opportunity to learn from a large amount of collected data to automatically identify common correct solutions and mistakes.', '1802.08924-1-59-1': 'Developing automatic grading systems for CyberPhysical System (CPS) MOOCs using simulation presents a particularly unique challenge.', '1802.08924-1-60-0': 'Juniwal et al. [CITATION] demonstrated a semi-supervised procedure for a CPS MOOC that first used dynamic time warping to cluster traces of student solutions, asked the instructor to label representatives from the clusters, and then extracted a STL formula from a PSTL template that characterized the cluster.', '1802.08924-1-60-1': 'Vazquez-Chanlatte et al. [CITATION] provided an unsupervised technique which used PSTL templates from [CITATION] to induce a distance metric that reproduced an arbitrarily chosen subset of the results from [CITATION].', '1802.08924-1-60-2': 'This distance metric required the user to provide an a priori total ordering on the parameter space which was used to select a single representative point on the boundary of the validity domain.', '1802.08924-1-60-3': 'Then for each signal, the distance between representative points served as the distance metric for the signals.', '1802.08924-1-60-4': 'Thus, while the PSTL induced distance lessened the labeling burden (since under the PSTL template, many superficially different clusters become equivalent), the total orderings required a fair amount of expertise to craft.', '1802.08924-1-60-5': 'In this experiment, we demonstrate that simple PSTL templates along with the logical distance (Eq [REF]) can be used to reproduce the results of [CITATION] without labeled data and without an a priori total ordering on the parameter space.', '1802.08924-1-61-0': 'To illustrate, we focus on two tests centered around a simulated robot interacting with an obstacle.', '1802.08924-1-61-1': "The robot is expected to (i) collide with the obstacle, (ii) bypass the obstacle, (iii) reorient to it's pre-collision orientation, and (iv) continue moving in the pre-collision orientation (see Fig [REF]).", '1802.08924-1-61-2': 'We use the following two PSTL templates (derived from [CITATION]) to characterize the obstacle avoidance and reorientation phase respectively.', '1802.08924-1-61-3': 'Crucially, when instantiated with particular parameters, these templates are essentially the actual specification of the robot.', '1802.08924-1-61-4': '[EQUATION]', '1802.08924-1-61-5': 'Our goal is to develop a distance metric, where similar solutions and failure modes are grouped together.', '1802.08924-1-62-0': 'Consider the representative sample of student submissions show in Fig [REF].', '1802.08924-1-62-1': 'Signal 0 shows a submission that never moves.', '1802.08924-1-62-2': 'Signal 1 shows a submission that continually collides with the obstacle.', '1802.08924-1-62-3': 'Signal 2 shows a submission that collides with the obstacle and then slides around the obstacle as it attempts to by bypass it.', '1802.08924-1-62-4': 'Signals 3 and 5 show two solutions going around the obstacle in two different ways.', '1802.08924-1-62-5': 'Signal 4 shows a submission that bypassed the obstacle, but failed to reorient.', '1802.08924-1-63-0': 'The validity domain boundaries for [MATH] and [MATH] are shown in Fig [REF] and Fig [REF] respectively.', '1802.08924-1-64-0': 'Observe that through the lens of [MATH], the signals that avoid the obstacle (3, 4, and 5) are correctly separated from signals the signals that do not (0, 1, 2).', '1802.08924-1-64-1': 'Similarly, through the lens of the reorient template, the signals that reorient (3,5) are separated from the signals that do not reorient (0,1,2,4).', '1802.08924-1-65-0': 'We averaged the logical distance (eq [REF]) for both templates yielding the adjacency matrix and dendogram shown in Fig [REF].', '1802.08924-1-65-1': 'Note that the first layer of the dendogram separates solutions that correctly avoided the obstacle.', '1802.08924-1-65-2': 'Within the avoided obstacle group, signals 3 and 5 are correctly grouped together since they both reoriented.', '1802.08924-1-66-0': '# Related Work', '1802.08924-1-67-0': 'Time-series clustering and classification is a well-studied area in the domain of machine learning and data mining [CITATION].', '1802.08924-1-67-1': 'Time series clustering that work with raw time-series data combine clustering schemes such as agglomerative clustering, hierarchical clustering, [MATH]-means clustering among others, with similarity measures between time-series data such as the dynamic time-warping (DTW) distance, statistical measures and information-theoretic measures.', '1802.08924-1-67-2': 'Feature-extraction based methods typically use generic sets of features, but algorithmic selection of the right set of meaningful features is a challenge.', '1802.08924-1-67-3': 'Finally, there are model-based approaches that seek an underlying generative model for the time-series data, and typically require extra assumptions on the data such as linearity or the Markovian property.', '1802.08924-1-67-4': 'Please see [CITATION] for detailed references to each approach.', '1802.08924-1-67-5': 'It should be noted that historically time-series learning focused on univariate time-series, and extensions to multivariate time-series data have been relatively recent developments.', '1802.08924-1-68-0': 'More recent work has focused on automatically identifying features from the data itself, such as the work on shapelets [CITATION], where instead of comparing entire time-series data using similarity measures, algorithms to automatically identify distinguishing motifs in the data have been developed.', '1802.08924-1-68-1': 'These motifs or shapelets serve not only as features for ML tasks, but also provide visual feedback to the user explaining why a classification or clustering task labels given data in a certain way.', '1802.08924-1-68-2': "While we draw inspiration from this general idea, we seek to expand it to consider logical shapes in the data, which would allow leveraging user's domain expertise.", '1802.08924-1-69-0': 'Automatic identification of motifs or basis functions from the data while useful in several documented case studies, comes with some limitations.', '1802.08924-1-69-1': 'For example, in [CITATION], the authors define a subspace clustering algorithm, where given a set of time-series curves, the algorithm identifies a subspace among the curves such that every curve in the given set can be expressed as a linear combination of a deformations of the curves in the subspace.', '1802.08924-1-69-2': 'We note that the authors observe that it may be difficult to associate the natural clustering structure with specific predicates over the data (such as patient outcome in a hospital setting).', '1802.08924-1-70-0': 'The use of logical formulas for learning properties of time-series has slowly been gaining momentum in communities outside of traditional machine learning and data mining [CITATION].', '1802.08924-1-70-1': 'Here, fragments of Signal Temporal Logic have been used to perform tasks such as supervised and unsupervised learning.', '1802.08924-1-70-2': 'A key distinction from these approaches is our use of libraries of signal predicates that encode domain expertise that allow human-interpretable clusters and classifiers.', '1802.08924-1-71-0': 'Finally, preliminary exploration of this idea appeared in prior work by some of the co-authors in [CITATION].', '1802.08924-1-71-1': 'The key difference is the previous work required users to provide a ranking of parameters appearing in a signal predicate, in order to project time-series data to unique points in the parameter space.', '1802.08924-1-71-2': 'We remove this additional burden on the user in this paper by proposing a generalization that projects time-series signals to trade-off curves in the parameter space, and then using these curves as features.', '1802.08924-1-72-0': '# Conclusion and Future Work'}

{'1802.08924-2-0-0': 'Cyber-physical systems of today are generating large volumes of time-series data.', '1802.08924-2-0-1': 'As manual inspection of such data is not tractable, the need for learning methods to help discover logical structure in the data has increased.', '1802.08924-2-0-2': 'We propose a logic-based framework that allows domain-specific knowledge to be embedded into formulas in a parametric logical specification over time-series data.', '1802.08924-2-0-3': 'The key idea is to then map a time series to a surface in the parameter space of the formula.', '1802.08924-2-0-4': 'Given this mapping, we identify the Hausdorff distance between boundaries as a natural distance metric between two time-series data under the lens of the parametric specification.', '1802.08924-2-0-5': 'This enables embedding non-trivial domain-specific knowledge into the distance metric and then using off-the-shelf machine learning tools to label the data.', '1802.08924-2-0-6': 'After labeling the data, we demonstrate how to extract a logical specification for each label.', '1802.08924-2-0-7': 'Finally, we showcase our technique on real world traffic data to learn classifiers/monitors for slow-downs and traffic jams.', '1802.08924-2-1-0': '# Introduction', '1802.08924-2-2-0': 'Recently, there has been a proliferation of sensors that monitor diverse kinds of real-time data representing time-series behaviors or signals generated by systems and devices that are monitored through such sensors.', '1802.08924-2-2-1': 'However, this deluge can place a heavy burden on engineers and designers who are not interested in the details of these signals, but instead seek to discover higher-level insights in the data.', '1802.08924-2-3-0': 'More concisely, one can frame the key challenge as: "How does one automatically identify logical structure or relations within the data?"', '1802.08924-2-3-1': 'To this end, modern machine learning (ML) techniques for signal analysis have been invaluable in domains ranging from healthcare analytics [CITATION] to smart transportation [CITATION]; and from autonomous driving [CITATION] to social media [CITATION].', '1802.08924-2-3-2': 'However, despite the success of ML based techniques, we believe that easily leveraging the domain-specific knowledge of non-ML experts remains an open problem.', '1802.08924-2-4-0': 'At present, a common way to encode domain-specific knowledge into an ML task is to first transform the data into an a priori known feature space, e.g., the statistical properties of a time series.', '1802.08924-2-4-1': 'While powerful, translating the knowledge of domain-specific experts into features remains a non-trivial endeavor.', '1802.08924-2-4-2': 'More recently, it has been shown that Parametric Signal Temporal Logic formula along with a total ordering on the parameter space can be used to extract feature vectors for learning temporal logical predicates characterizing driving patterns, overshoot of diesel engine re-flow rates, and grading for simulated robot controllers in a Massively Open Online Course [CITATION].', '1802.08924-2-4-3': 'Crucially, the technique of learning through the lens of a logical formula means that learned artifacts can be readily leveraged by existing formal methods infrastructure for verification, synthesis, falsification, and monitoring.', '1802.08924-2-4-4': 'Unfortunately, the usefulness of the results depend intimately on the total ordering used.', '1802.08924-2-4-5': 'The following example illustrates this point.', '1802.08924-2-5-0': 'Example: Most freeways have bottlenecks that lead to traffic congestion, and if there is a stalled or a crashed vehicle(s) at this site, then upstream traffic congestion can severely worsen.', '1802.08924-2-5-1': 'For example, Fig [REF] shows a series of potential time-series signals to which we would like to assign pairwise distances indicating the similarity (small values) or differences (large values) between any two time series.', '1802.08924-2-5-2': "To ease exposition, we have limited our focus to the car's speed.", '1802.08924-2-5-3': 'In signals 0 and 1, both cars transition from high speed freeway driving to stop and go traffic.', '1802.08924-2-5-4': 'Conversely, in signal 2, the car transitions from stop and go traffic to high speed freeway driving.', '1802.08924-2-5-5': 'Signal 3 corresponds to a car slowing to a stop and then accelerating, perhaps due to difficulty merging lanes.', '1802.08924-2-5-6': 'Finally, signal 4 signifies a car encountering no traffic and signal 5 corresponds to a car in heavy traffic, or a possibly stalled vehicle.', '1802.08924-2-6-0': 'Suppose a user wished to find a feature space equipped with a measure to distinguish cars being stuck in traffic.', '1802.08924-2-6-1': 'Some properties might be: [leftmargin=1.5em,itemsep=0pt,parsep=0pt,partopsep=0pt] Signals 0 and 1 should be very close together since both show a car entering stop and go traffic in nearly the same manner.', '1802.08924-2-6-2': 'Signals 2, 3, and 4 should be close together since the car ultimately escapes stop and go traffic.', '1802.08924-2-6-3': 'Signal 5 should be far from all other examples since it does not represent entering or leaving stop and go traffic.', '1802.08924-2-7-0': 'r0.4 Adjacency matrix and clustering of Fig [REF].', '1802.08924-2-7-1': 'Smaller numbers mean that the time series are more similar with respect to the logical distance metric.', '1802.08924-2-8-0': 'For a strawman comparison, we consider two ways the user might assign a distance measure to the above signal space.', '1802.08924-2-8-1': 'Further, we omit generic time series distance measures such as Dynamic Time Warping [CITATION] which do not offer the ability to embed domain specific knowledge into the metric.', '1802.08924-2-8-2': 'At first, the user might treat the signals as a series of independent measurements and attempt to characterize the signals via standard statistical measures on the speed and acceleration (mean, standard deviation, ).', '1802.08924-2-8-3': 'Fig [REF] illustrates how the example signals look in this feature space with each component normalized between 0 and 1.', '1802.08924-2-8-4': 'The user might then use the Euclidean distance of each feature to assign a distance between signals.', '1802.08924-2-8-5': 'Unfortunately, in this measure, signal 4 is not close to signal 2 or 3, violating the second desired property.', '1802.08924-2-8-6': 'Further, signals 0 and 1 are not "very" close together violating the first property.', '1802.08924-2-8-7': 'Next, the user attempts to capture traffic slow downs by the following (informal) parametric temporal specification: "Between time [MATH] and 20, the car speed is always less than [MATH]."', '1802.08924-2-8-8': 'As will be made precise in the preliminaries (for each individual time-series) Fig [REF] illustrates the boundaries between values of [MATH] and [MATH] that make the specification true and values which make the specification false.', '1802.08924-2-8-9': 'The techniques in [CITATION] then require the user to specify a particular total ordering on the parameter space.', '1802.08924-2-8-10': 'One then uses the maximal point on the boundary as the representative for the entire boundary.', '1802.08924-2-8-11': 'However, in practice, selecting a good ordering a-priori is non-obvious.', '1802.08924-2-8-12': 'For example, [CITATION] suggests a lexicographic ordering of the parameters.', '1802.08924-2-8-13': 'However, since most of the boundaries start and end at essentially the same point, applying any of the lexicographic orderings to the boundaries seen in Fig [REF] would result in almost all of the boundaries collapsing to the same points.', '1802.08924-2-8-14': 'Thus, such an ordering would make characterizing a slow down impossible.', '1802.08924-2-9-0': 'In the sequel, we propose using the Hausdorff distance between boundaries as a general ordering-free way to endow time series with a "logic respecting distance metric".', '1802.08924-2-9-1': 'Fig [REF] illustrates the distances between each boundary.', '1802.08924-2-9-2': 'As is easily confirmed, all 3 properties desired of the clustering algorithm hold.', '1802.08924-2-10-0': 'Contributions The key insight in our work is that in many interesting examples, the distance between satisfaction boundaries in the parameter space of parametric logical formula can characterize the domain-specific knowledge implicit in the parametric formula.', '1802.08924-2-10-1': 'Leveraging this insight we provide the following contributions: [leftmargin=1.5em,itemsep=0pt,parsep=0pt,partopsep=0pt] We propose a new distance measure between time-series through the lens of a chosen monotonic specification.', '1802.08924-2-10-2': 'Distance measure in hand, standard ML algorithms such as nearest neighbors (supervised) or agglomerative clustering (unsupervised) can be used to glean insights into the data.', '1802.08924-2-10-3': 'Given a labeling, we propose a method for computing representative points on each boundary.', '1802.08924-2-10-4': 'Viewed another way, we propose a form of dimensionality reduction based on the temporal logic formula.', '1802.08924-2-10-5': 'Finally, given the representative points and their labels, we can use the machinery developed in [CITATION] to extract a simple logical formula as a classifier for each label.', '1802.08924-2-11-0': '# Preliminaries', '1802.08924-2-12-0': 'The main object of analysis in this paper are time-series.', '1802.08924-2-12-1': '[Time Series, Signals, Traces] Let [MATH] be a subset of [MATH] and [MATH] be a nonempty set.', '1802.08924-2-12-2': 'A time series (signal or trace), [MATH] is a map: [EQUATION]', '1802.08924-2-12-3': 'Where [MATH] and [MATH] are called the time domain and value domain respectively.', '1802.08924-2-12-4': 'The set of all time series is denoted by [MATH].', '1802.08924-2-13-0': 'Between any two time series one can define a metric which measures their similarity.', '1802.08924-2-13-1': '[Metric] Given a set [MATH], a metric is a map, [EQUATION] such that [MATH], [MATH], [MATH].', '1802.08924-2-14-0': '[Infinity Norm Metric] The infinity norm induced distance [MATH] is a metric.', '1802.08924-2-15-0': '[Hausdorff Distance] Given a set [MATH] with a distance metric [MATH], the Hausdorff distance is a distance metric between closed subsets of [MATH].', '1802.08924-2-15-1': 'Namely, given closed subsets [MATH]: [EQUATION]', '1802.08924-2-15-2': 'We use the following property of the Hausdorff distance throughout the paper: Given two sets [MATH] and [MATH], there necessarily exists points [MATH] and [MATH] such that: [EQUATION]', '1802.08924-2-15-3': 'Within a context, the platonic ideal of a metric between traces respects any domain-specific properties that make two elements "similar".', '1802.08924-2-15-4': 'A logical trace property, also called a specification, assigns to each timed trace a truth value.', '1802.08924-2-15-5': '[Specification] A specification is a map, [MATH], from time series to true or false.', '1802.08924-2-15-6': '[EQUATION]', '1802.08924-2-15-7': 'A time series, [MATH], is said to satisfy a specification iff [MATH].', '1802.08924-2-16-0': 'Consider the following specification related to the specification from the running example:', '1802.08924-2-17-0': '[EQUATION] where [MATH] denotes an indicator function.', '1802.08924-2-17-1': 'Informally, this specification says that after [MATH], the value of the time series, [MATH], is always less than 1.', '1802.08924-2-18-0': 'Given a finite number of properties, one can then "fingerprint" a time series as a Boolean feature vector.', '1802.08924-2-18-1': 'That is, given [MATH] properties, [MATH] and the corresponding indicator functions, [MATH], we map each time series to an [MATH]-tuple as follows.', '1802.08924-2-18-2': '[EQUATION]', '1802.08924-2-18-3': 'Notice however that many properties are not naturally captured by a finite sequence of binary features.', '1802.08924-2-18-4': 'For example, imagine a single quantitative feature [MATH] encoding the percentage of fuel left in a tank.', '1802.08924-2-18-5': 'This feature implicitly encodes an uncountably infinite family of Boolean features [MATH] indexed by the percentages [MATH].', '1802.08924-2-18-6': 'We refer to such families as parametric specifications.', '1802.08924-2-18-7': 'For simplicity, we assume that the parameters are a subset of the unit hyper-box.', '1802.08924-2-18-8': '[Parametric Specifications] A parametric specification is a map: [EQUATION] where [MATH] is the number of parameters and [MATH] denotes the set of functions from the hyper-square, [MATH] to [MATH].', '1802.08924-2-19-0': 'The signature, [MATH] would have been an alternative and arguably simpler definition of parametric specifications; however, as we shall see, [REF] highlights that a trace induces a structure, called the validity domain, embedded in the parameter space.', '1802.08924-2-20-0': 'Parametric specifications arise naturally from syntactically substituting constants with parameters in the description of a specification.', '1802.08924-2-21-0': 'The parametric specification given in Ex [REF] can be generalized by substituting [MATH] for [MATH] and [MATH] for [MATH] in Ex [REF].', '1802.08924-2-21-1': '[EQUATION]', '1802.08924-2-21-2': 'At this point, one could naively extend the notion of the "fingerprint" of a parametric specification in a similar manner as the finite case.', '1802.08924-2-21-3': 'However, if [MATH] is equipped with a distance metric, it is fruitful to instead study the geometry induced by the time series in the parameter space.', '1802.08924-2-21-4': 'To begin, observe that the value of a Boolean feature vector is exactly determined by which entries map to [MATH].', '1802.08924-2-21-5': 'Analogously, the set of parameter values for which a parameterized specification maps to true on a given time series acts as the "fingerprint".', '1802.08924-2-21-6': 'We refer to this characterizing set as the validity domain.', '1802.08924-2-21-7': '[Validity domain] Given an [MATH] parameter specification, [MATH], and a trace, [MATH], the validity domain is the pre-image of 1 under [MATH], [EQUATION]', '1802.08924-2-21-8': 'Thus, [MATH], can be viewed as the map that returns the structure in the parameter space indexed by a particular trace.', '1802.08924-2-22-0': 'Note that in general, the validity domain can be arbitrarily complex making reasoning about its geometry intractable.', '1802.08924-2-22-1': 'We circumvent such hurdles by specializing to monotonic specifications.', '1802.08924-2-22-2': '[Monotonic Specifications] A parametric specification is said to be monotonic if for all traces, [MATH]: [EQUATION] where [MATH] is the standard product ordering on [MATH], [MATH] iff [MATH].', '1802.08924-2-23-0': 'The parametric specification in Ex [REF] is monotonic.', '1802.08924-2-24-0': 'Given a monotonic specification, [MATH], and a time series, [MATH], the boundary of the validity domain, [MATH], of a monotonic specification is a hyper-surface that segments [MATH] into two components.', '1802.08924-2-25-0': 'In the sequel, we develop a distance metric between validity domains which characterizes the similarity between two time series under the lens of a monotonic specification.', '1802.08924-2-26-0': '# Logic-Respecting Distance Metric', '1802.08924-2-27-0': 'In this section, we define a class of metrics on the signal space that is derived from corresponding parametric specifications.', '1802.08924-2-27-1': 'First, observe that the validity domains of monotonic specifications are uniquely defined by the hyper-surface that separates them from the rest of the parameter space.', '1802.08924-2-27-2': 'Similar to Pareto fronts in a multi-objective optimization, these boundaries encode the trade-offs required in each parameter to make the specification satisfied for a given time series.', '1802.08924-2-27-3': 'This suggests a simple procedure to define a distance metric between time series that respects their logical properties: Given a monotonic specification, a set of time series, and a distance metric between validity domain boundaries:', '1802.08924-2-28-0': 'Compute the validity domain boundaries for each time series.', '1802.08924-2-28-1': 'Compute the distance between the validity domain boundaries.', '1802.08924-2-29-0': 'Of course, the benefits of using this metric would rely entirely on whether (i) The monotonic specification captures the relevant domain-specific details (ii) The distance between validity domain boundaries is sensitive to outliers.', '1802.08924-2-29-1': 'While the choice of specification is highly domain-specific, we argue that for many monotonic specifications, the distance metric should be sensitive to outliers as this represents a large deviation from the specification.', '1802.08924-2-29-2': 'This sensitivity requirement seems particularly apt if the number of satisfying traces of the specification grows linearly or super-linearly as the parameters increase.', '1802.08924-2-29-3': 'Observing that Hausdorff distance [REF] between two validity boundaries satisfy these properties, we define our new distance metric between time series as:', '1802.08924-2-30-0': 'Given a monotonic specification, [MATH], and a distance metric on the parameter space [MATH], the logical distance between two time series, [MATH] is: [EQUATION]', '1802.08924-2-31-0': '## Approximating the Logical Distance', '1802.08924-2-32-0': 'Next, we discuss how to approximate the logical distance metric within arbitrary precision.', '1802.08924-2-32-1': 'First, observe that the validity domain boundary of a monotonic specification can be recursively approximated to arbitrary precision via binary search on the diagonal of the parameter space [CITATION].', '1802.08924-2-32-2': 'This approximation yields a series of overlapping axis aligned rectangles that are guaranteed to contain the boundary (see Fig [REF]).', '1802.08924-2-33-0': 'To formalize this approximation, let [MATH] denote the set of closed intervals on the real line.', '1802.08924-2-33-1': 'We then define an axis aligned rectangle as the product of closed intervals.', '1802.08924-2-34-0': 'The set of axis aligned rectangles is defined as: [EQUATION]', '1802.08924-2-34-1': 'The approximation given in [CITATION] is then a family of maps, [EQUATION] where [MATH] denotes the recursive depth [MATH] denotes the powerset.', '1802.08924-2-34-2': 'For example, [MATH] yields the bounding box given in the leftmost subfigure in Fig [REF] and [MATH] yields the subdivision of the bounding box seen on the right.', '1802.08924-2-35-0': 'Next, we ask the question: Given a discretization of the rectangle set approximating a boundary, how does the Hausdorff distance between the discretization relate to the true Hausdorff distance between two boundaries?', '1802.08924-2-35-1': 'In particular, consider the map that takes a set of rectangles to the set of the corner points of the rectangles.', '1802.08924-2-35-2': 'Formally, we denote this map as: [EQUATION]', '1802.08924-2-35-3': 'As the rectangles are axis aligned, at this point, it is fruitful to specialize to parameter spaces equipped with the infinity norm.', '1802.08924-2-35-4': 'The resulting Hausdorff distance is denoted [MATH].', '1802.08924-2-35-5': 'This specialization leads to the following lemma:', '1802.08924-2-36-0': 'Let [MATH], [MATH] be two time series and [MATH] the approximation of their respective boundaries.', '1802.08924-2-36-1': 'Further, let [MATH] be points in [MATH] such that: [EQUATION] and let [MATH] be the rectangles in [MATH] and [MATH] containing the points [MATH] and [MATH] respectively.', '1802.08924-2-36-2': 'Finally, let [MATH] be the maximum edge length in [MATH] and [MATH], then: [EQUATION]', '1802.08924-2-36-3': 'First, observe that (i) each rectangle intersects its boundary (ii) each rectangle set over-approximates its boundary.', '1802.08924-2-36-4': 'Thus, by assumption, each point within a rectangle is at most [MATH] distance from the boundary w.r.t. the infinity norm.', '1802.08924-2-36-5': 'Thus, since there exist two points [MATH] such that [MATH], the maximum deviation from the logical distance is at most [MATH] and [MATH].', '1802.08924-2-36-6': 'Further, since [MATH] must be in [MATH], the lower bound can be tightened to [MATH].', '1802.08924-2-36-7': '[MATH]', '1802.08924-2-37-0': 'We denote the map given by [REF] from the points to the error interval as: [EQUATION]', '1802.08924-2-37-1': 'Next, observe that this approximation can be made arbitrarily close to the logical distance.', '1802.08924-2-38-0': 'Let [MATH] denote the logical distance between two traces [MATH].', '1802.08924-2-38-1': 'For any [MATH], there exists [MATH] such that: [EQUATION]', '1802.08924-2-38-2': 'By Lemma [REF], given a fixed approximate depth, the above approximation differs from the true logical distance by at most two times the maximum edge length of the approximating rectangles.', '1802.08924-2-38-3': 'Note that by construction, incrementing the approximation depth results in each rectangle having at least one edge being halved.', '1802.08924-2-38-4': 'Thus the maximum edge length across the set of rectangles must at least halve.', '1802.08924-2-38-5': 'Thus, for any [MATH] there exists an approximation depth [MATH] such that: [EQUATION]', '1802.08924-2-38-6': 'Finally, algorithm [REF] summarizes the above procedure.', '1802.08924-2-38-7': '[H] Approximate Logical Distance [1] approxdist[MATH]', '1802.08924-2-39-0': 'An efficient implementation should of course memoize previous calls to [MATH] and use [MATH] to compute [MATH].', '1802.08924-2-39-1': 'Further, since certain rectangles can be quickly determined to not contribute to the Hausdorff distance, they need not be subdivided further.', '1802.08924-2-40-0': '## Learning Labels', '1802.08924-2-41-0': 'The distance interval [MATH] returned by Alg [REF] can be used by learning techniques, such as hierarchical or agglomerative clustering, to estimate clusters (and hence the labels).', '1802.08924-2-41-1': 'While the technical details of these learning algorithms are beyond the scope of this work, we formalize the result of the learning algorithms as a labeling map: [Labeling] A [MATH]-labeling is a map: [EQUATION] for some [MATH].', '1802.08924-2-41-2': 'If [MATH] is obvious from context or not important, then the map is simply referred to as a labeling.', '1802.08924-2-42-0': '# Artifiact Extraction', '1802.08924-2-43-0': 'In practice, many learning algorithms produce labeling maps that provide little to no insight into why a particular trajectory is given a particular label.', '1802.08924-2-43-1': 'In the next section, we seek a way to systematically summarize a labeling in terms of the parametric specification used to induce the logical distance.', '1802.08924-2-44-0': '## Post-facto Projections', '1802.08924-2-45-0': 'To begin, observe that due to the nature of the Hausdorff distance, when explaining why two boundaries differ, one can remove large segments of the boundaries without changing their Hausdorff distance.', '1802.08924-2-45-1': 'This motivates us to find a small summarizing set of parameters for each label.', '1802.08924-2-45-2': 'Further, since the Hausdorff distance often reduces to the distance between two points, we aim to summarize each boundary using a particular projection map.', '1802.08924-2-45-3': 'Concretely,', '1802.08924-2-46-0': 'Letting [MATH] denote the set of all possible validity domain boundaries, a projection is a map: [EQUATION] where [MATH] is the number of parameters in [MATH].', '1802.08924-2-47-0': 'In principle, one could extend this to projecting to a finite tuple of points.', '1802.08924-2-47-1': 'For simplicity, we do not consider such cases.', '1802.08924-2-48-0': 'Systematic techniques for picking the projection include lexicographic projections and solutions to multi-objective optimizations; however, as seen in the introduction, a-priori choosing the projection scheme is subtle.', '1802.08924-2-48-1': 'Instead, we propose performing a post-facto optimization of a collection of projections in order to be maximally representative of the labels.', '1802.08924-2-48-2': 'That is, we seek a projection, [MATH], that maximally disambiguates between the labels, i.e., maximizes the minimum distance between the clusters.', '1802.08924-2-48-3': 'Formally, given a set of traces associated with each label [MATH] we seek: [EQUATION]', '1802.08924-2-48-4': 'For simplicity, we restrict our focus to projections induced by the intersection of each boundary with a line intersecting the base of the unit box [MATH].', '1802.08924-2-48-5': 'Just as in the recursive boundary approximations, due to monotonicity, this intersection point is guaranteed to be unique.', '1802.08924-2-48-6': 'Further, this class of projections is in one-one correspondence with the boundary.', '1802.08924-2-48-7': 'In particular, for any point [MATH] on boundary, there exists exactly one projection that produces [MATH].', '1802.08924-2-48-8': 'As such, each projection can be indexed by a point in [MATH].', '1802.08924-2-49-0': 'This is perhaps unsurprising given that in 2-d, one can index this class by the angle with the [MATH]-axis and in 3-d on can include the angle from the [MATH]-axis.', '1802.08924-2-50-0': 'Since we expect clusters of boundaries to be near each other, we also expect their intersection points to be near each other.', '1802.08924-2-51-0': 'For our experiment, we searched for [MATH] via a sweep through a discretized indexing of possible angles.', '1802.08924-2-52-0': '## Label Specifications', '1802.08924-2-53-0': 'Next, observe that given a projection, when studying the infinity norm distance between labels, it suffices to consider only the bounding box of each label in parameter space.', '1802.08924-2-53-1': 'Namely, letting [MATH] denote the map that computes the bounding box of a set of points in [MATH], for any two labels [MATH] and [MATH]: [EQUATION]', '1802.08924-2-53-2': "This motivates using the projection's bounding box as a surrogate for the cluster.", '1802.08924-2-53-3': 'Next, we observe that one can encode the set of trajectories whose boundaries intersect (and thus can project to) a given bounding box as a simple Boolean combination of the specifications corresponding to instantiating [MATH] with the parameters of at most [MATH] corners of the box [CITATION].', '1802.08924-2-53-4': 'While a detailed exposition is outside the scope of this article, we illustrate with an example.', '1802.08924-2-54-0': 'Consider examples 0 and 1 from the introductory example viewed as validity domain boundaries under [REF].', '1802.08924-2-54-1': 'Suppose that the post-facto projection mapped example 0 to [MATH] and mapped example 1 to [MATH].', '1802.08924-2-54-2': 'Such a projection is plausibly near the optimal for many classes of projections since none of the other example boundaries (who are in different clusters) are near the boundaries for 0 and 1 at these points.', '1802.08924-2-54-3': 'The resulting specification is: [EQUATION]', '1802.08924-2-55-0': '## Dimensionality Reduction', '1802.08924-2-56-0': 'r0.5 Figure of histogram resulting from projecting noisy variations of the traffic slow down example time series onto the diagonal of the unit box.', '1802.08924-2-57-0': 'Finally, observe that the line that induces the projection can serve as a mechanism for dimensionality reduction.', '1802.08924-2-57-1': 'Namely, if one parameterizes the line [MATH] from [MATH], where [MATH] is the origin and [MATH] intersects the unit box, then the points where the various boundaries intersect can be assigned a number between [MATH] and [MATH].', '1802.08924-2-57-2': 'For high-dimensional parameter spaces, this enables visualizing the projection histogram and could even be used for future classification/learning.', '1802.08924-2-57-3': 'We again illustrate using our running example.', '1802.08924-2-58-0': 'For all six time series in the traffic slow down example, we generate 100 new time series by modulating the time series with noise drawn from [MATH].', '1802.08924-2-58-1': 'Using our previously labeled time series, the projection using the line with angle [MATH] (i.e., the diagonal of the unit box) from the x-axis yields the distribution seen in Fig [REF].', '1802.08924-2-58-2': 'Observe that all three clusters are clearly visible.', '1802.08924-2-59-0': 'If one dimension is insufficient, this procedure can be extended to an arbitrary number of dimensions using more lines.', '1802.08924-2-59-1': 'An interesting extension may be to consider how generic dimensionality techniques such as principle component analysis would act in the limit where one approximates the entire boundary.', '1802.08924-2-60-0': '# Case Study', '1802.08924-2-61-0': 'To improve driver models and traffic on highways, the Federal Highway Administration collected detailed traffic data on southbound US-101 freeway, in Los Angeles [CITATION].', '1802.08924-2-61-1': 'Traffic through the segment was monitored and recorded through eight synchronized cameras, next to the freeway.', '1802.08924-2-61-2': 'A total of [MATH] minutes of traffic data was recorded including vehicle trajectory data providing lane positions of each vehicle within the study area.', '1802.08924-2-61-3': 'The data-set is split into [MATH] time series.', '1802.08924-2-61-4': "For simplicity, we constrain our focus to the car's speed.", '1802.08924-2-61-5': 'In the sequel, we outline a technique for first using the parametric specification (in conjunction with off-the-shelf machine learning techniques) to filter the data, and then using the logical distance from an idealized slow down to find the slow downs in the data.', '1802.08924-2-61-6': 'This final step offers a key benefit over the closest prior work [CITATION].', '1802.08924-2-61-7': 'Namely given an over approximation of the desired cluster, one can use the logical distance to further refine the cluster.', '1802.08924-2-62-0': 'Rescale Data As in our running example, we seek to use [REF] to search for traffic slow downs; however, in order to do so, we must re-scale the time series.', '1802.08924-2-62-1': 'To begin, observe that the mean velocity is 62mph with 80% of the vehicles remaining under 70mph.', '1802.08924-2-62-2': 'Thus, we linearly scale the velocity so that [MATH].', '1802.08924-2-62-3': 'Similarly, we re-scale the time axis so that each tick is [MATH] seconds.', '1802.08924-2-62-4': 'Fig [REF] shows a subset of the time series.', '1802.08924-2-63-0': 'Filtering Recall that if two boundaries have small Hausdorff distances, then the points where the boundaries intersect a line (that intersects the origin of the parameter space) must be close.', '1802.08924-2-63-1': 'Since computing the Hausdorff distance is a fairly expensive operation, we use this one way implication to group time series which may be near each other w.r.t. the Hausdorff distance.', '1802.08924-2-64-0': 'In particular, we (arbitrarily) selected two lines intersecting the parameter space origin at [MATH] and [MATH] radians from the [MATH] axis to project to.', '1802.08924-2-64-1': 'We filtered out any time-series that did not intersect the line within [MATH].', '1802.08924-2-64-2': 'We then fit a 5 cluster Gaussian Mixture Model (GMM) to label the data.', '1802.08924-2-64-3': 'Fig [REF] shows the result.', '1802.08924-2-65-0': 'Matching Idealized Slow Down Next, we labeled the idealized slow down, (trace 0 from Fig [REF]) using the fitted GMM.This identified cluster 4 (with 765 data points) as containing potential slow downs.', '1802.08924-2-65-1': 'To filter for the true slow downs, we used the logical distance from the idealized slow down to further subdivide the cluster.', '1802.08924-2-65-2': 'Fig [REF] shows the resulting distribution.', '1802.08924-2-65-3': 'Fig [REF] shows the time series in cluster 4 annotated by their distance for the idealized slow down.', '1802.08924-2-65-4': 'Using this visualization, one can clearly identify 390 slow downs (distance less than [MATH])', '1802.08924-2-66-0': 'Artifact Extraction Finally, we first searched for a single projection that gave a satisfactory separation of clusters, but were unable to do so.', '1802.08924-2-66-1': 'We then searched over pairs of projections to create a specification as the conjunction of two box specifications.', '1802.08924-2-66-2': 'Namely, in terms of [MATH], our first projection yields the specification: [MATH].', '1802.08924-2-66-3': 'Similarly, our second projection yields the specification: [MATH].', '1802.08924-2-66-4': 'The learned slow down specification is the conjunction of these two specifications.', '1802.08924-2-67-0': '# Related Work', '1802.08924-2-68-0': 'Time-series clustering and classification is a well-studied area in the domain of machine learning and data mining [CITATION].', '1802.08924-2-68-1': 'Time series clustering that work with raw time-series data combine clustering schemes such as agglomerative clustering, hierarchical clustering, [MATH]-means clustering among others, with similarity measures between time-series data such as the dynamic time-warping (DTW) distance, statistical measures and information-theoretic measures.', '1802.08924-2-68-2': 'Feature-extraction based methods typically use generic sets of features, but algorithmic selection of the right set of meaningful features is a challenge.', '1802.08924-2-68-3': 'Finally, there are model-based approaches that seek an underlying generative model for the time-series data, and typically require extra assumptions on the data such as linearity or the Markovian property.', '1802.08924-2-68-4': 'Please see [CITATION] for detailed references to each approach.', '1802.08924-2-68-5': 'It should be noted that historically time-series learning focused on univariate time-series, and extensions to multivariate time-series data have been relatively recent developments.', '1802.08924-2-69-0': 'More recent work has focused on automatically identifying features from the data itself, such as the work on shapelets [CITATION], where instead of comparing entire time-series data using similarity measures, algorithms to automatically identify distinguishing motifs in the data have been developed.', '1802.08924-2-69-1': 'These motifs or shapelets serve not only as features for ML tasks, but also provide visual feedback to the user explaining why a classification or clustering task, labels given data, in a certain way.', '1802.08924-2-69-2': "While we draw inspiration from this general idea, we seek to expand it to consider logical shapes in the data, which would allow leveraging user's domain expertise.", '1802.08924-2-70-0': 'Automatic identification of motifs or basis functions from the data while useful in several documented case studies, comes with some limitations.', '1802.08924-2-70-1': 'For example, in [CITATION], the authors define a subspace clustering algorithm, where given a set of time-series curves, the algorithm identifies a subspace among the curves such that every curve in the given set can be expressed as a linear combination of a deformations of the curves in the subspace.', '1802.08924-2-70-2': 'We note that the authors observe that it may be difficult to associate the natural clustering structure with specific predicates over the data (such as patient outcome in a hospital setting).', '1802.08924-2-71-0': 'The use of logical formulas for learning properties of time-series has slowly been gaining momentum in communities outside of traditional machine learning and data mining [CITATION].', '1802.08924-2-71-1': 'Here, fragments of Signal Temporal Logic have been used to perform tasks such as supervised and unsupervised learning.', '1802.08924-2-71-2': 'A key distinction from these approaches is our use of libraries of signal predicates that encode domain expertise that allow human-interpretable clusters and classifiers.', '1802.08924-2-72-0': 'Finally, preliminary exploration of this idea appeared in prior work by some of the co-authors in [CITATION].', '1802.08924-2-72-1': 'The key difference is the previous work required users to provide a ranking of parameters appearing in a signal predicate, in order to project time-series data to unique points in the parameter space.', '1802.08924-2-72-2': 'We remove this additional burden on the user in this paper by proposing a generalization that projects time-series signals to trade-off curves in the parameter space, and then using these curves as features.'}

[['1802.08924-1-32-1', '1802.08924-2-27-2'], ['1802.08924-1-69-0', '1802.08924-2-70-0'], ['1802.08924-1-69-1', '1802.08924-2-70-1'], ['1802.08924-1-69-2', '1802.08924-2-70-2'], ['1802.08924-1-30-0', '1802.08924-2-25-0'], ['1802.08924-1-8-0', '1802.08924-2-6-0'], ['1802.08924-1-8-2', '1802.08924-2-6-2'], ['1802.08924-1-8-3', '1802.08924-2-6-3'], ['1802.08924-1-5-5', '1802.08924-2-10-2'], ['1802.08924-1-20-4', '1802.08924-2-12-4'], ['1802.08924-1-23-5', '1802.08924-2-15-7'], ['1802.08924-1-55-2', '1802.08924-2-61-1'], ['1802.08924-1-55-3', '1802.08924-2-61-2'], ['1802.08924-1-67-0', '1802.08924-2-68-0'], ['1802.08924-1-67-1', '1802.08924-2-68-1'], ['1802.08924-1-67-2', '1802.08924-2-68-2'], ['1802.08924-1-67-3', '1802.08924-2-68-3'], ['1802.08924-1-67-4', '1802.08924-2-68-4'], ['1802.08924-1-67-5', '1802.08924-2-68-5'], ['1802.08924-1-13-1', '1802.08924-2-9-2'], ['1802.08924-1-7-3', '1802.08924-2-5-2'], ['1802.08924-1-7-4', '1802.08924-2-5-3'], ['1802.08924-1-7-5', '1802.08924-2-5-4'], ['1802.08924-1-7-6', '1802.08924-2-5-5'], ['1802.08924-1-7-7', '1802.08924-2-5-6'], ['1802.08924-1-68-0', '1802.08924-2-69-0'], ['1802.08924-1-68-2', '1802.08924-2-69-2'], ['1802.08924-1-24-1', '1802.08924-2-17-1'], ['1802.08924-1-71-0', '1802.08924-2-72-0'], ['1802.08924-1-71-1', '1802.08924-2-72-1'], ['1802.08924-1-71-2', '1802.08924-2-72-2'], ['1802.08924-1-33-0', '1802.08924-2-28-0'], ['1802.08924-1-21-0', '1802.08924-2-13-0'], ['1802.08924-1-21-1', '1802.08924-2-13-1'], ['1802.08924-1-25-0', '1802.08924-2-18-0'], ['1802.08924-1-70-0', '1802.08924-2-71-0'], ['1802.08924-1-70-1', '1802.08924-2-71-1'], ['1802.08924-1-70-2', '1802.08924-2-71-2'], ['1802.08924-1-40-2', '1802.08924-2-32-2'], ['1802.08924-1-9-0', '1802.08924-2-8-0'], ['1802.08924-1-9-1', '1802.08924-2-8-2'], ['1802.08924-1-9-4', '1802.08924-2-8-3'], ['1802.08924-1-9-6', '1802.08924-2-8-5'], ['1802.08924-1-9-7', '1802.08924-2-8-6'], ['1802.08924-1-34-0', '1802.08924-2-29-0'], ['1802.08924-1-34-1', '1802.08924-2-29-1'], ['1802.08924-1-34-2', '1802.08924-2-29-2'], ['1802.08924-1-32-0', '1802.08924-2-27-1'], ['1802.08924-1-29-0', '1802.08924-2-24-0'], ['1802.08924-1-8-1', '1802.08924-2-6-1'], ['1802.08924-1-20-0', '1802.08924-2-12-0'], ['1802.08924-1-20-2', '1802.08924-2-12-2'], ['1802.08924-1-20-3', '1802.08924-2-12-3'], ['1802.08924-1-23-0', '1802.08924-2-15-0'], ['1802.08924-1-23-1', '1802.08924-2-15-1'], ['1802.08924-1-55-1', '1802.08924-2-61-0'], ['1802.08924-1-22-1', '1802.08924-2-14-0'], ['1802.08924-1-7-0', '1802.08924-2-5-0'], ['1802.08924-1-7-2', '1802.08924-2-5-1'], ['1802.08924-1-68-1', '1802.08924-2-69-1'], ['1802.08924-1-37-0', '1802.08924-2-30-0'], ['1802.08924-1-33-1', '1802.08924-2-28-1'], ['1802.08924-1-26-0', '1802.08924-2-20-0'], ['1802.08924-1-25-5', '1802.08924-2-18-7'], ['1802.08924-1-40-1', '1802.08924-2-32-1'], ['1802.08924-1-27-0', '1802.08924-2-21-0'], ['1802.08924-1-27-7', '1802.08924-2-22-2'], ['1802.08924-1-5-3', '1802.08924-2-10-1'], ['1802.08924-1-20-1', '1802.08924-2-12-1'], ['1802.08924-1-23-2', '1802.08924-2-15-3'], ['1802.08924-1-23-3', '1802.08924-2-15-4'], ['1802.08924-1-23-4', '1802.08924-2-15-5'], ['1802.08924-1-13-0', '1802.08924-2-9-1'], ['1802.08924-1-25-1', '1802.08924-2-18-3'], ['1802.08924-1-25-3', '1802.08924-2-18-5'], ['1802.08924-1-25-4', '1802.08924-2-18-6'], ['1802.08924-1-25-6', '1802.08924-2-18-8'], ['1802.08924-1-40-0', '1802.08924-2-32-0'], ['1802.08924-1-2-0', '1802.08924-2-2-0'], ['1802.08924-1-2-2', '1802.08924-2-2-0'], ['1802.08924-1-27-2', '1802.08924-2-21-4'], ['1802.08924-1-27-3', '1802.08924-2-21-5'], ['1802.08924-1-27-4', '1802.08924-2-21-7'], ['1802.08924-1-27-6', '1802.08924-2-22-1'], ['1802.08924-1-9-5', '1802.08924-2-8-4'], ['1802.08924-1-12-0', '1802.08924-2-8-7'], ['1802.08924-1-14-0', '1802.08924-2-8-7'], ['1802.08924-1-3-0', '1802.08924-2-4-0']]

[['1802.08924-1-32-1', '1802.08924-2-27-2'], ['1802.08924-1-69-0', '1802.08924-2-70-0'], ['1802.08924-1-69-1', '1802.08924-2-70-1'], ['1802.08924-1-69-2', '1802.08924-2-70-2'], ['1802.08924-1-30-0', '1802.08924-2-25-0'], ['1802.08924-1-8-0', '1802.08924-2-6-0'], ['1802.08924-1-8-2', '1802.08924-2-6-2'], ['1802.08924-1-8-3', '1802.08924-2-6-3'], ['1802.08924-1-5-5', '1802.08924-2-10-2'], ['1802.08924-1-20-4', '1802.08924-2-12-4'], ['1802.08924-1-23-5', '1802.08924-2-15-7'], ['1802.08924-1-55-2', '1802.08924-2-61-1'], ['1802.08924-1-55-3', '1802.08924-2-61-2'], ['1802.08924-1-67-0', '1802.08924-2-68-0'], ['1802.08924-1-67-1', '1802.08924-2-68-1'], ['1802.08924-1-67-2', '1802.08924-2-68-2'], ['1802.08924-1-67-3', '1802.08924-2-68-3'], ['1802.08924-1-67-4', '1802.08924-2-68-4'], ['1802.08924-1-67-5', '1802.08924-2-68-5'], ['1802.08924-1-13-1', '1802.08924-2-9-2'], ['1802.08924-1-7-3', '1802.08924-2-5-2'], ['1802.08924-1-7-4', '1802.08924-2-5-3'], ['1802.08924-1-7-5', '1802.08924-2-5-4'], ['1802.08924-1-7-6', '1802.08924-2-5-5'], ['1802.08924-1-7-7', '1802.08924-2-5-6'], ['1802.08924-1-68-0', '1802.08924-2-69-0'], ['1802.08924-1-68-2', '1802.08924-2-69-2'], ['1802.08924-1-24-1', '1802.08924-2-17-1'], ['1802.08924-1-71-0', '1802.08924-2-72-0'], ['1802.08924-1-71-1', '1802.08924-2-72-1'], ['1802.08924-1-71-2', '1802.08924-2-72-2'], ['1802.08924-1-33-0', '1802.08924-2-28-0'], ['1802.08924-1-21-0', '1802.08924-2-13-0'], ['1802.08924-1-21-1', '1802.08924-2-13-1'], ['1802.08924-1-25-0', '1802.08924-2-18-0'], ['1802.08924-1-70-0', '1802.08924-2-71-0'], ['1802.08924-1-70-1', '1802.08924-2-71-1'], ['1802.08924-1-70-2', '1802.08924-2-71-2'], ['1802.08924-1-40-2', '1802.08924-2-32-2'], ['1802.08924-1-9-0', '1802.08924-2-8-0'], ['1802.08924-1-9-1', '1802.08924-2-8-2'], ['1802.08924-1-9-4', '1802.08924-2-8-3'], ['1802.08924-1-9-6', '1802.08924-2-8-5'], ['1802.08924-1-9-7', '1802.08924-2-8-6']]

[['1802.08924-1-34-0', '1802.08924-2-29-0'], ['1802.08924-1-34-1', '1802.08924-2-29-1'], ['1802.08924-1-34-2', '1802.08924-2-29-2'], ['1802.08924-1-32-0', '1802.08924-2-27-1'], ['1802.08924-1-29-0', '1802.08924-2-24-0'], ['1802.08924-1-8-1', '1802.08924-2-6-1'], ['1802.08924-1-20-0', '1802.08924-2-12-0'], ['1802.08924-1-20-2', '1802.08924-2-12-2'], ['1802.08924-1-20-3', '1802.08924-2-12-3'], ['1802.08924-1-23-0', '1802.08924-2-15-0'], ['1802.08924-1-23-1', '1802.08924-2-15-1'], ['1802.08924-1-55-1', '1802.08924-2-61-0'], ['1802.08924-1-22-1', '1802.08924-2-14-0'], ['1802.08924-1-7-0', '1802.08924-2-5-0'], ['1802.08924-1-7-2', '1802.08924-2-5-1'], ['1802.08924-1-68-1', '1802.08924-2-69-1'], ['1802.08924-1-37-0', '1802.08924-2-30-0'], ['1802.08924-1-33-1', '1802.08924-2-28-1'], ['1802.08924-1-26-0', '1802.08924-2-20-0'], ['1802.08924-1-25-5', '1802.08924-2-18-7'], ['1802.08924-1-40-1', '1802.08924-2-32-1'], ['1802.08924-1-27-0', '1802.08924-2-21-0'], ['1802.08924-1-27-7', '1802.08924-2-22-2']]

[]

[['1802.08924-1-5-3', '1802.08924-2-10-1'], ['1802.08924-1-20-1', '1802.08924-2-12-1'], ['1802.08924-1-23-2', '1802.08924-2-15-3'], ['1802.08924-1-23-3', '1802.08924-2-15-4'], ['1802.08924-1-23-4', '1802.08924-2-15-5'], ['1802.08924-1-13-0', '1802.08924-2-9-1'], ['1802.08924-1-25-1', '1802.08924-2-18-3'], ['1802.08924-1-25-3', '1802.08924-2-18-5'], ['1802.08924-1-25-4', '1802.08924-2-18-6'], ['1802.08924-1-25-6', '1802.08924-2-18-8'], ['1802.08924-1-40-0', '1802.08924-2-32-0'], ['1802.08924-1-2-0', '1802.08924-2-2-0'], ['1802.08924-1-2-2', '1802.08924-2-2-0'], ['1802.08924-1-27-2', '1802.08924-2-21-4'], ['1802.08924-1-27-3', '1802.08924-2-21-5'], ['1802.08924-1-27-4', '1802.08924-2-21-7'], ['1802.08924-1-27-6', '1802.08924-2-22-1'], ['1802.08924-1-9-5', '1802.08924-2-8-4'], ['1802.08924-1-12-0', '1802.08924-2-8-7'], ['1802.08924-1-14-0', '1802.08924-2-8-7']]

[['1802.08924-1-3-0', '1802.08924-2-4-0']]

['1802.08924-1-13-4', '1802.08924-1-15-0', '1802.08924-1-16-0', '1802.08924-1-27-1', '1802.08924-1-32-2', '1802.08924-1-34-3', '1802.08924-1-36-0', '1802.08924-1-42-0', '1802.08924-1-61-4', '1802.08924-2-15-6', '1802.08924-2-16-0', '1802.08924-2-18-2', '1802.08924-2-21-1', '1802.08924-2-23-0', '1802.08924-2-27-3', '1802.08924-2-29-3', '1802.08924-2-35-5', '1802.08924-2-36-7', '1802.08924-2-45-3']

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

https://arxiv.org/abs/1802.08924

1708.08180

{'1708.08180-1-0-0': 'In this paper, we report on an optimized union-find (UF) algorithm that can label the connected components on a 2D image efficiently by employing GPU architecture.', '1708.08180-1-0-1': 'The proposed method comprises three phases: UF-based local merge, boundary analysis, and link.', '1708.08180-1-0-2': 'The coarse labeling in local merge, which makes computation efficient because the length of the label-equivalence list is sharply suppressed .', '1708.08180-1-0-3': 'Boundary analysis only manages the cells on the boundary of each thread block to launch fewer CUDA threads.', '1708.08180-1-0-4': 'We compared our method with the label equivalence algorithm [CITATION], conventional parallel UF algorithm [CITATION], and line-based UF algorithm [CITATION].', '1708.08180-1-0-5': 'Evaluation results show that the proposed algorithm speeds up the average running time by around [MATH]x, [MATH]x, and [MATH]x, respectively.', '1708.08180-1-1-0': '# Introduction', '1708.08180-1-2-0': 'Connected components labeling (CCL) is a task to give a unique ID to each connected region in a 2D/3D grid, which means that the input data is divided into separate groups where the elements from a single group share the same ID.', '1708.08180-1-2-1': 'As a basic data clustering method, it is employed in numerous research areas like image processing, computer vision, and visual communication [CITATION].', '1708.08180-1-2-2': 'W. Song, et al. [CITATION] presented a motion based skin region of interest detection method using a real-time CCL algorithm to reduce its execution time.', '1708.08180-1-2-3': 'A fast 3D shape measurement technique using blink-dot projection patterns that utilizes a CCL algorithm to compute the size and location of each dot on the captured images has been reported [CITATION] [CITATION].', '1708.08180-1-2-4': 'P. Guler, et al. proposed a real-time multi-camera video analytics system [CITATION] employing CCL to perform noise reduction.', '1708.08180-1-3-0': 'On the basis of the fact that parallel devices find countless applications in both industrial and academic areas, some CCL algorithms using GPUs have emerged [CITATION] recently to improve the real-time property of CCL, which is very important for many applications.', '1708.08180-1-3-1': 'The CCL algorithms can be classified into two categories, the multi-pass method and one-pass method, according to whether they apply a convergence criterion or not [CITATION].', '1708.08180-1-3-2': 'Tab. [REF] summarizes five typical parallel CCL approaches and a brief explanation is given in the following.', '1708.08180-1-3-3': 'Neighbor propagation [CITATION] is the simplest multi-pass approach that scans the neighborhood of a target cell to get the lowest label of a neighboring cell belonging to the same group.', '1708.08180-1-3-4': 'Row-column unification [CITATION] enlarges the scan scope by allocating one row to each thread.', '1708.08180-1-3-5': 'Label equivalence [CITATION] employs neighbor propagation as the first phase to construct label-equivalence chains, and the following analysis and relabeling phases find the roots of each chain.', '1708.08180-1-3-6': 'The resolution of an input image determines the iteration times of neighbor propagation, while the iteration of row-column unification and label equivalence depend on the complexity of an input image.', '1708.08180-1-3-7': 'The usual union-find (UF) algorithm [CITATION] is parallelized by dividing the input image into independent 2D blocks; local merge and global merge are introduced to solve the connectivity [CITATION].', '1708.08180-1-3-8': 'Instead of using 2D blocks, a line-based parallel UF algorithm [CITATION] collects the pixels in one row to perform local label unification.', '1708.08180-1-3-9': 'Even the computation of each kernel in such one-pass methods is heavier than those of multi-pass approaches; they label an image faster because each kernel only runs one time.', '1708.08180-1-4-0': 'In this study, we propose an optimized UF algorithm that is an improved version of conventional parallel UF [CITATION] with an optimized local merge and lightweight boundary analysis.', '1708.08180-1-4-1': 'Its concepts are: (1) row-column unification is performed using shared memory before local UF to reduce the complexity of an initialized local label map; (2) connectivity analysis is conducted only for the cells on the block boundary to decrease the number of required CUDA threads.', '1708.08180-1-4-2': 'Compared with the conventional UF [CITATION], our proposed approach can perform local merge more efficiently because the label-equivalence chains are extensively suppressed as a result of the coarse labeling.', '1708.08180-1-4-3': 'For the line-based UF [CITATION], it can extract the local label map slightly faster than our method.', '1708.08180-1-4-4': 'However, its global merge phase takes much longer because global UF should be applied to all the cells in the input data.', '1708.08180-1-5-0': '# Algorithm Description', '1708.08180-1-6-0': 'In this section, we outline the three kernels of our method.', '1708.08180-1-6-1': 'In the first kernel, UF-based local merge, we perform a coarse labeling before finding the real root of each cell to reduce the computational complexity in each thread.', '1708.08180-1-6-2': 'In the last two kernels, boundary analysis and link, we merge individual blocks together to generate a global label map.', '1708.08180-1-7-0': 'Local UF merge with coarse labeling', '1708.08180-1-8-0': '[1] Image [MATH] of size [MATH] Both block dimension and grid dimension are 2D [MATH] are on shared memory [MATH] is on global memory', '1708.08180-1-9-0': 'declare int [MATH] declare int [MATH] 2D global thread id [MATH] 1D thread id within block [MATH] call syncthreads()', '1708.08180-1-10-0': '// row scan [MATH] call syncthreads()', '1708.08180-1-11-0': '// column scan [MATH] call syncthreads()', '1708.08180-1-12-0': '// row-column unification [MATH] while [MATH] end while', '1708.08180-1-13-0': '// local union find [MATH] findAndUnion[MATH] call syncthreads() [MATH] findAndUnion[MATH] call syncthreads()', '1708.08180-1-14-0': '// convert local index to global index [MATH]', '1708.08180-1-15-0': '## Local merge with coarse labeling', '1708.08180-1-16-0': 'The first kernel, local merge with coarse labeling, consists of three phases: initialization, coarse labeling using row-column unification, and local UF.', '1708.08180-1-16-1': 'Its pseudo-code is listed in Kernel [REF] by following 4-connectivity.', '1708.08180-1-17-0': '### Initialization', '1708.08180-1-18-0': 'We divide the input image into several rectangular pieces, as shown in Fig. [REF] (a), and assign each piece to different GPU threads blocks where the threads can cooperate with each other using shared memory and can be synchronized [CITATION].', '1708.08180-1-18-1': 'The cells in each block are indexed by the thread ID within the block.', '1708.08180-1-18-2': 'Fig. [REF] (b) presents an example of an [MATH] initialized local label map that was allocated on shared memory.', '1708.08180-1-18-3': 'Here, the gray cells represent foreground areas, while the white cells represent background areas.', '1708.08180-1-19-0': '### Coarse labeling using row-column unification', '1708.08180-1-20-0': 'In an initialized local label map, as shown in Fig. [REF] (b), the label of the left cell and the label of the upper cell are always smaller than that of a target cell, while the upper one is always smaller than the left one.', '1708.08180-1-20-1': 'Based on this fact, we scan the rows first and then go to column scan.', '1708.08180-1-20-2': 'The cell will get the label of its neighboring cell, left or upper, with the same property.', '1708.08180-1-20-3': 'Fig. [REF] shows the scan models.', '1708.08180-1-20-4': 'Unlike the methods that record the entire label-equivalence lists, this method records the lowest label that the label is equivalent to.', '1708.08180-1-20-5': 'Its memory access complexity is reduced due to the utilization of shared memory, while the equivalence can be unified by a low number of iteration because the dimension of a thread block is limited by the CUDA runtime system.', '1708.08180-1-20-6': 'Fig. [REF] (a) presents two equivalence lists in the local label map after row-column scan.', '1708.08180-1-20-7': 'Fig. [REF] (b) shows the coarsely labeled label map after row-column unification.', '1708.08180-1-21-0': 'Boundary analysis', '1708.08180-1-22-0': '[1] Image [MATH] of size [MATH] Both block dimension and grid dimension are 2D [MATH] is on global memory', '1708.08180-1-23-0': 'declare int [MATH] declare bool [MATH] 1D global thread id', '1708.08180-1-24-0': '// convert 1D global thread id to 2D cell id [MATH]', '1708.08180-1-25-0': '// boundary analysis along x-axis [MATH]; [MATH]', '1708.08180-1-26-0': '// boundary analysis along y-axis [MATH]; [MATH]', '1708.08180-1-27-0': '### Local UF', '1708.08180-1-28-0': 'UF, expressed by [MATH] in Kernel [REF], is a data structure that divides a set of elements into a number of disjoint subsets by using [MATH] and [MATH] operations.', '1708.08180-1-28-1': 'The [MATH] is an iterative search to extract the root of a label-equivalence list and return its label.', '1708.08180-1-28-2': 'The [MATH] is a unification to assign the root label to the elements belonging to the subset.', '1708.08180-1-28-3': '[CITATION] gives a detailed description of these two operations.', '1708.08180-1-28-4': 'By comparing the initialized local label map (Fig. [REF] (b)) with the one after row-column unification (Fig. [REF] (b)), it can be noticed that the path to find the root of a label-equivalence chain is compressed sharply, which enables local UF to run efficiently.', '1708.08180-1-28-5': 'The final step of this kernel is an ID conversion that converts the local index to a global index.', '1708.08180-1-28-6': 'Fig. [REF] (a) presents a converted global label map.', '1708.08180-1-29-0': '## Boundary analysis', '1708.08180-1-30-0': 'In the boundary analysis phase, we only perform UF for the cells on the block boundary (those marked on Fig. [REF] (a)) to launch fewer threads.', '1708.08180-1-30-1': 'Assuming the resolution of an input image is [MATH] and the block configuration of the Kernel [REF] is [MATH], the number of cells on the block boundary along [MATH]-axis and [MATH]-axis [MATH] can be determined as follows: [EQUATION]', '1708.08180-1-30-2': 'Here, [MATH] means the largest integer smaller or equal to [MATH].', '1708.08180-1-30-3': 'To integrate the boundary analysis along [MATH] and [MATH]axis into one kernel, [MATH] threads spawned by Kernel [REF] should be invoked.', '1708.08180-1-30-4': 'Fig. [REF] (b) shows how to analyze the connectivity in the [MATH]-direction: the cell on the boundary merges with its upper cell by using UF if they have the same property.', '1708.08180-1-30-5': 'The union along the [MATH]-direction works in the same manner.', '1708.08180-1-31-0': '## Final link', '1708.08180-1-32-0': 'After analyzing the connectivities of the cells on the block boundary, the independent local label maps are associated as an entirety.', '1708.08180-1-32-1': 'We compute the final global label map in the same way as that reported in [CITATION] and [CITATION].', '1708.08180-1-33-0': '# Evaluation Experiments', '1708.08180-1-34-0': 'To demonstrate the effectiveness of our proposed algorithm, we run it and the other three parallel methods, label equivalence (LE) [CITATION], conventional parallel UF [CITATION], and line-based UF [CITATION], on a PC equipped with an NVIDIA Geforce GTX 1070 for the images shown in Fig. [REF].', '1708.08180-1-34-1': 'For the line-based UF method, its thread blocks are configured as [MATH], while the configuration of the other three methods is [MATH].', '1708.08180-1-35-0': 'Tab. [REF] shows the comparison results for the execution times of these algorithms with images of different size.', '1708.08180-1-35-1': 'Here, we run each algorithm 100 times and take their extreme value as well as average value.', '1708.08180-1-35-2': 'It can be seen that the optimized UF can label a [MATH], [MATH], [MATH], and [MATH] binary image in around [MATH], [MATH], [MATH], and [MATH] ms respectively, while the other methods take longer to accomplish CCL.', '1708.08180-1-35-3': 'From an analysis of these results, it can be deduced that the one-scan methods, UF, line-based UF, and our proposed method, work more efficiently than LE, one of the typical multi-scan methods.', '1708.08180-1-35-4': 'Meanwhile, it indicates that our method outperforms the other two methods.', '1708.08180-1-35-5': 'Compared with UF, the speedup ratio increases with the increase in image resolution.', '1708.08180-1-35-6': 'It is about [MATH] times faster for a [MATH] binary image.', '1708.08180-1-35-7': 'For the line-based UF, the speedup ratio is quite stable and is around [MATH] for all the images.', '1708.08180-1-36-0': '# Conclusions', '1708.08180-1-37-0': 'In this paper, we introduced an optimized parallel UF algorithm for fast CCL using GPUs.', '1708.08180-1-37-1': 'Our algorithm employs a coarse row-column unification to reduce the computation complexity of local merge and launches a low number of threads for block-to-block connectivity analysis.', '1708.08180-1-37-2': 'As a result, the proposed method can efficiently perform CCL on GPUs in a single scan.', '1708.08180-1-37-3': 'We verified its performance on NVIDIA Geforce GTX 1070 and compared the execution time with those of three other methods.', '1708.08180-1-37-4': 'Experimental results show that the running time of CCL improved greatly compared with the latest method.', '1708.08180-1-37-5': 'The efficiency makes the proposed method suitable for many real-time applications.'}

{'1708.08180-2-0-0': 'In this paper, we report on an optimized union-find (UF) algorithm that can label the connected components on a 2D image efficiently by employing GPU architecture.', '1708.08180-2-0-1': 'The proposed method comprises three phases: UF-based local merge, boundary analysis, and link.', '1708.08180-2-0-2': 'The coarse labeling in local merge, which makes computation efficient because the length of the label-equivalence list is sharply suppressed .', '1708.08180-2-0-3': 'Boundary analysis only manages the cells on the boundary of each thread block to launch fewer CUDA threads.', '1708.08180-2-0-4': 'We compared our method with the label equivalence algorithm [CITATION], conventional parallel UF algorithm [CITATION], and line-based UF algorithm [CITATION].', '1708.08180-2-0-5': 'Evaluation results show that the proposed algorithm speeds up the average running time by around [MATH]x, [MATH]x, and [MATH]x, respectively.', '1708.08180-2-1-0': '# Introduction', '1708.08180-2-2-0': 'Connected components labeling (CCL) is a task to give a unique ID to each connected region in a 2D/3D grid, which means that the input data is divided into separate groups where the elements from a single group share the same ID.', '1708.08180-2-2-1': 'As a basic data clustering method, it is employed in numerous research areas like image processing, computer vision, and visual communication [CITATION].', '1708.08180-2-2-2': 'W. Song, et al. [CITATION] presented a motion based skin region of interest detection method using a real-time CCL algorithm to reduce its execution time.', '1708.08180-2-2-3': 'A fast 3D shape measurement technique using blink-dot projection patterns that utilizes a CCL algorithm to compute the size and location of each dot on the captured images has been reported [CITATION] [CITATION].', '1708.08180-2-2-4': 'P. Guler, et al. proposed a real-time multi-camera video analytics system [CITATION] employing CCL to perform noise reduction.', '1708.08180-2-3-0': 'On the basis of the fact that parallel devices find countless applications in both industrial and academic areas, some CCL algorithms using GPUs have emerged [CITATION] recently to improve the real-time property of CCL, which is very important for many applications.', '1708.08180-2-3-1': 'The CCL algorithms can be classified into two categories, the multi-pass method and one-pass method, according to whether they apply a convergence criterion or not [CITATION].', '1708.08180-2-3-2': 'Tab. [REF] summarizes five typical parallel CCL approaches and a brief explanation is given in the following.', '1708.08180-2-3-3': 'Neighbor propagation [CITATION] is the simplest multi-pass approach that scans the neighborhood of a target cell to get the lowest label of a neighboring cell belonging to the same group.', '1708.08180-2-3-4': 'Row-column unification [CITATION] enlarges the scan scope by allocating one row to each thread.', '1708.08180-2-3-5': 'Label equivalence [CITATION] employs neighbor propagation as the first phase to construct label-equivalence chains, and the following analysis and relabeling phases find the roots of each chain.', '1708.08180-2-3-6': 'The resolution of an input image determines the iteration times of neighbor propagation, while the iteration of row-column unification and label equivalence depend on the complexity of an input image.', '1708.08180-2-3-7': 'The usual union-find (UF) algorithm [CITATION] is parallelized by dividing the input image into independent 2D blocks; local merge and global merge are introduced to solve the connectivity [CITATION].', '1708.08180-2-3-8': 'Instead of using 2D blocks, a line-based parallel UF algorithm [CITATION] collects the pixels in one row to perform local label unification.', '1708.08180-2-3-9': 'Even the computation of each kernel in such one-pass methods is heavier than those of multi-pass approaches; they label an image faster because each kernel only runs one time.', '1708.08180-2-4-0': 'In this study, we propose an optimized UF algorithm that is an improved version of conventional parallel UF [CITATION] with an optimized local merge and lightweight boundary analysis.', '1708.08180-2-4-1': 'Its concepts are: (1) row-column unification is performed using shared memory before local UF to reduce the complexity of an initialized local label map; (2) connectivity analysis is conducted only for the cells on the block boundary to decrease the number of required CUDA threads.', '1708.08180-2-4-2': 'Compared with the conventional UF [CITATION], our proposed approach can perform local merge more efficiently because the label-equivalence chains are extensively suppressed as a result of the coarse labeling.', '1708.08180-2-4-3': 'For the line-based UF [CITATION], it can extract the local label map slightly faster than our method.', '1708.08180-2-4-4': 'However, its global merge phase takes much longer because global UF should be applied to all the cells in the input data.', '1708.08180-2-5-0': '# Algorithm Description', '1708.08180-2-6-0': 'In this section, we outline the three kernels of our method.', '1708.08180-2-6-1': 'In the first kernel, UF-based local merge, we perform a coarse labeling before finding the real root of each cell to reduce the computational complexity in each thread.', '1708.08180-2-6-2': 'In the last two kernels, boundary analysis and link, we merge individual blocks together to generate a global label map.', '1708.08180-2-7-0': 'Local UF merge with coarse labeling', '1708.08180-2-8-0': '[1] Image [MATH] of size [MATH] Both block dimension and grid dimension are 2D [MATH] are on shared memory [MATH] is on global memory', '1708.08180-2-9-0': 'declare int [MATH] declare int [MATH] 2D global thread id [MATH] 1D thread id within block [MATH] call syncthreads()', '1708.08180-2-10-0': '// row scan [MATH] call syncthreads()', '1708.08180-2-11-0': '// column scan [MATH] call syncthreads()', '1708.08180-2-12-0': '// row-column unification [MATH] while [MATH] end while', '1708.08180-2-13-0': '// local union find [MATH] findAndUnion[MATH] call syncthreads() [MATH] findAndUnion[MATH] call syncthreads()', '1708.08180-2-14-0': '// convert local index to global index [MATH]', '1708.08180-2-15-0': '## Local merge with coarse labeling', '1708.08180-2-16-0': 'The first kernel, local merge with coarse labeling, consists of three phases: initialization, coarse labeling using row-column unification, and local UF.', '1708.08180-2-16-1': 'Its pseudo-code is listed in Kernel [REF] by following 4-connectivity.', '1708.08180-2-17-0': '### Initialization', '1708.08180-2-18-0': 'We divide the input image into several rectangular pieces, as shown in Fig. [REF] (a), and assign each piece to different GPU threads blocks where the threads can cooperate with each other using shared memory and can be synchronized [CITATION].', '1708.08180-2-18-1': 'The cells in each block are indexed by the thread ID within the block.', '1708.08180-2-18-2': 'Fig. [REF] (b) presents an example of an [MATH] initialized local label map that was allocated on shared memory.', '1708.08180-2-18-3': 'Here, the gray cells represent foreground areas, while the white cells represent background areas.', '1708.08180-2-19-0': '### Coarse labeling using row-column unification', '1708.08180-2-20-0': 'In an initialized local label map, as shown in Fig. [REF] (b), the label of the left cell and the label of the upper cell are always smaller than that of a target cell, while the upper one is always smaller than the left one.', '1708.08180-2-20-1': 'Based on this fact, we scan the rows first and then go to column scan.', '1708.08180-2-20-2': 'The cell will get the label of its neighboring cell, left or upper, with the same property.', '1708.08180-2-20-3': 'Fig. [REF] shows the scan models.', '1708.08180-2-20-4': 'Unlike the methods that record the entire label-equivalence lists, this method records the lowest label that the label is equivalent to.', '1708.08180-2-20-5': 'Its memory access complexity is reduced due to the utilization of shared memory, while the equivalence can be unified by a low number of iteration because the dimension of a thread block is limited by the CUDA runtime system.', '1708.08180-2-20-6': 'Fig. [REF] (a) presents two equivalence lists in the local label map after row-column scan.', '1708.08180-2-20-7': 'Fig. [REF] (b) shows the coarsely labeled label map after row-column unification.', '1708.08180-2-21-0': 'Boundary analysis', '1708.08180-2-22-0': '[1] Image [MATH] of size [MATH] Both block dimension and grid dimension are 2D [MATH] is on global memory', '1708.08180-2-23-0': 'declare int [MATH] declare bool [MATH] 1D global thread id', '1708.08180-2-24-0': '// convert 1D global thread id to 2D cell id [MATH]', '1708.08180-2-25-0': '// boundary analysis along x-axis [MATH]; [MATH]', '1708.08180-2-26-0': '// boundary analysis along y-axis [MATH]; [MATH]', '1708.08180-2-27-0': '### Local UF', '1708.08180-2-28-0': 'UF, expressed by [MATH] in Kernel [REF], is a data structure that divides a set of elements into a number of disjoint subsets by using [MATH] and [MATH] operations.', '1708.08180-2-28-1': 'The [MATH] is an iterative search to extract the root of a label-equivalence list and return its label.', '1708.08180-2-28-2': 'The [MATH] is a unification to assign the root label to the elements belonging to the subset.', '1708.08180-2-28-3': '[CITATION] gives a detailed description of these two operations.', '1708.08180-2-28-4': 'By comparing the initialized local label map (Fig. [REF] (b)) with the one after row-column unification (Fig. [REF] (b)), it can be noticed that the path to find the root of a label-equivalence chain is compressed sharply, which enables local UF to run efficiently.', '1708.08180-2-28-5': 'The final step of this kernel is an ID conversion that converts the local index to a global index.', '1708.08180-2-28-6': 'Fig. [REF] (a) presents a converted global label map.', '1708.08180-2-29-0': '## Boundary analysis', '1708.08180-2-30-0': 'In the boundary analysis phase, we only perform UF for the cells on the block boundary (those marked on Fig. [REF] (a)) to launch fewer threads.', '1708.08180-2-30-1': 'Assuming the resolution of an input image is [MATH] and the block configuration of the Kernel [REF] is [MATH], the number of cells on the block boundary along [MATH]-axis and [MATH]-axis [MATH] can be determined as follows: [EQUATION]', '1708.08180-2-30-2': 'Here, [MATH] means the largest integer smaller or equal to [MATH].', '1708.08180-2-30-3': 'To integrate the boundary analysis along [MATH] and [MATH]axis into one kernel, [MATH] threads spawned by Kernel [REF] should be invoked.', '1708.08180-2-30-4': 'Fig. [REF] (b) shows how to analyze the connectivity in the [MATH]-direction: the cell on the boundary merges with its upper cell by using UF if they have the same property.', '1708.08180-2-30-5': 'The union along the [MATH]-direction works in the same manner.', '1708.08180-2-31-0': '## Final link', '1708.08180-2-32-0': 'After analyzing the connectivities of the cells on the block boundary, the independent local label maps are associated as an entirety.', '1708.08180-2-32-1': 'We compute the final global label map in the same way as that reported in [CITATION] and [CITATION].', '1708.08180-2-33-0': '# Evaluation Experiments', '1708.08180-2-34-0': 'To demonstrate the effectiveness of our proposed algorithm, we run it and the other three parallel methods, label equivalence (LE) [CITATION], conventional parallel UF [CITATION], and line-based UF [CITATION], on a PC equipped with an NVIDIA Geforce GTX 1070 for the images shown in Fig. [REF].', '1708.08180-2-34-1': 'For the line-based UF method, its thread blocks are configured as [MATH], while the configuration of the other three methods is [MATH].', '1708.08180-2-35-0': 'Tab. [REF] shows the comparison results for the execution times of these algorithms with images of different size.', '1708.08180-2-35-1': 'Here, we run each algorithm 100 times and take their extreme value as well as average value.', '1708.08180-2-35-2': 'It can be seen that the optimized UF can label a [MATH], [MATH], [MATH], and [MATH] binary image in around [MATH], [MATH], [MATH], and [MATH] ms respectively, while the other methods take longer to accomplish CCL.', '1708.08180-2-35-3': 'From an analysis of these results, it can be deduced that the one-scan methods, UF, line-based UF, and our proposed method, work more efficiently than LE, one of the typical multi-scan methods.', '1708.08180-2-35-4': 'Meanwhile, it indicates that our method outperforms the other two methods.', '1708.08180-2-35-5': 'Compared with UF, the speedup ratio increases with the increase in image resolution.', '1708.08180-2-35-6': 'It is about [MATH] times faster for a [MATH] binary image.', '1708.08180-2-35-7': 'For the line-based UF, the speedup ratio is quite stable and is around [MATH] for all the images.', '1708.08180-2-36-0': '# Conclusions', '1708.08180-2-37-0': 'In this paper, we introduced an optimized parallel UF algorithm for fast CCL using GPUs.', '1708.08180-2-37-1': 'Our algorithm employs a coarse row-column unification to reduce the computation complexity of local merge and launches a low number of threads for block-to-block connectivity analysis.', '1708.08180-2-37-2': 'As a result, the proposed method can efficiently perform CCL on GPUs in a single scan.', '1708.08180-2-37-3': 'We verified its performance on NVIDIA Geforce GTX 1070 and compared the execution time with those of three other methods.', '1708.08180-2-37-4': 'Experimental results show that the running time of CCL improved greatly compared with the latest method.', '1708.08180-2-37-5': 'The efficiency makes the proposed method suitable for many real-time applications.'}

[['1708.08180-1-6-0', '1708.08180-2-6-0'], ['1708.08180-1-6-1', '1708.08180-2-6-1'], ['1708.08180-1-6-2', '1708.08180-2-6-2'], ['1708.08180-1-20-0', '1708.08180-2-20-0'], ['1708.08180-1-20-1', '1708.08180-2-20-1'], ['1708.08180-1-20-2', '1708.08180-2-20-2'], ['1708.08180-1-20-3', '1708.08180-2-20-3'], ['1708.08180-1-20-4', '1708.08180-2-20-4'], ['1708.08180-1-20-5', '1708.08180-2-20-5'], ['1708.08180-1-20-6', '1708.08180-2-20-6'], ['1708.08180-1-20-7', '1708.08180-2-20-7'], ['1708.08180-1-0-0', '1708.08180-2-0-0'], ['1708.08180-1-0-1', '1708.08180-2-0-1'], ['1708.08180-1-0-2', '1708.08180-2-0-2'], ['1708.08180-1-0-3', '1708.08180-2-0-3'], ['1708.08180-1-0-4', '1708.08180-2-0-4'], ['1708.08180-1-0-5', '1708.08180-2-0-5'], ['1708.08180-1-35-0', '1708.08180-2-35-0'], ['1708.08180-1-35-1', '1708.08180-2-35-1'], ['1708.08180-1-35-2', '1708.08180-2-35-2'], ['1708.08180-1-35-3', '1708.08180-2-35-3'], ['1708.08180-1-35-4', '1708.08180-2-35-4'], ['1708.08180-1-35-5', '1708.08180-2-35-5'], ['1708.08180-1-35-6', '1708.08180-2-35-6'], ['1708.08180-1-35-7', '1708.08180-2-35-7'], ['1708.08180-1-16-0', '1708.08180-2-16-0'], ['1708.08180-1-16-1', '1708.08180-2-16-1'], ['1708.08180-1-32-0', '1708.08180-2-32-0'], ['1708.08180-1-32-1', '1708.08180-2-32-1'], ['1708.08180-1-22-0', '1708.08180-2-22-0'], ['1708.08180-1-30-0', '1708.08180-2-30-0'], ['1708.08180-1-30-1', '1708.08180-2-30-1'], ['1708.08180-1-30-2', '1708.08180-2-30-2'], ['1708.08180-1-30-3', '1708.08180-2-30-3'], ['1708.08180-1-30-4', '1708.08180-2-30-4'], ['1708.08180-1-30-5', '1708.08180-2-30-5'], ['1708.08180-1-37-0', '1708.08180-2-37-0'], ['1708.08180-1-37-1', '1708.08180-2-37-1'], ['1708.08180-1-37-2', '1708.08180-2-37-2'], ['1708.08180-1-37-3', '1708.08180-2-37-3'], ['1708.08180-1-37-4', '1708.08180-2-37-4'], ['1708.08180-1-37-5', '1708.08180-2-37-5'], ['1708.08180-1-28-0', '1708.08180-2-28-0'], ['1708.08180-1-28-1', '1708.08180-2-28-1'], ['1708.08180-1-28-2', '1708.08180-2-28-2'], ['1708.08180-1-28-3', '1708.08180-2-28-3'], ['1708.08180-1-28-4', '1708.08180-2-28-4'], ['1708.08180-1-28-5', '1708.08180-2-28-5'], ['1708.08180-1-28-6', '1708.08180-2-28-6'], ['1708.08180-1-8-0', '1708.08180-2-8-0'], ['1708.08180-1-4-0', '1708.08180-2-4-0'], ['1708.08180-1-4-1', '1708.08180-2-4-1'], ['1708.08180-1-4-2', '1708.08180-2-4-2'], ['1708.08180-1-4-3', '1708.08180-2-4-3'], ['1708.08180-1-4-4', '1708.08180-2-4-4'], ['1708.08180-1-9-0', '1708.08180-2-9-0'], ['1708.08180-1-18-0', '1708.08180-2-18-0'], ['1708.08180-1-18-1', '1708.08180-2-18-1'], ['1708.08180-1-18-2', '1708.08180-2-18-2'], ['1708.08180-1-18-3', '1708.08180-2-18-3'], ['1708.08180-1-24-0', '1708.08180-2-24-0'], ['1708.08180-1-34-0', '1708.08180-2-34-0'], ['1708.08180-1-34-1', '1708.08180-2-34-1'], ['1708.08180-1-2-0', '1708.08180-2-2-0'], ['1708.08180-1-2-1', '1708.08180-2-2-1'], ['1708.08180-1-2-2', '1708.08180-2-2-2'], ['1708.08180-1-2-3', '1708.08180-2-2-3'], ['1708.08180-1-2-4', '1708.08180-2-2-4'], ['1708.08180-1-3-0', '1708.08180-2-3-0'], ['1708.08180-1-3-1', '1708.08180-2-3-1'], ['1708.08180-1-3-2', '1708.08180-2-3-2'], ['1708.08180-1-3-3', '1708.08180-2-3-3'], ['1708.08180-1-3-4', '1708.08180-2-3-4'], ['1708.08180-1-3-5', '1708.08180-2-3-5'], ['1708.08180-1-3-6', '1708.08180-2-3-6'], ['1708.08180-1-3-7', '1708.08180-2-3-7'], ['1708.08180-1-3-8', '1708.08180-2-3-8'], ['1708.08180-1-3-9', '1708.08180-2-3-9']]

[['1708.08180-1-6-0', '1708.08180-2-6-0'], ['1708.08180-1-6-1', '1708.08180-2-6-1'], ['1708.08180-1-6-2', '1708.08180-2-6-2'], ['1708.08180-1-20-0', '1708.08180-2-20-0'], ['1708.08180-1-20-1', '1708.08180-2-20-1'], ['1708.08180-1-20-2', '1708.08180-2-20-2'], ['1708.08180-1-20-3', '1708.08180-2-20-3'], ['1708.08180-1-20-4', '1708.08180-2-20-4'], ['1708.08180-1-20-5', '1708.08180-2-20-5'], ['1708.08180-1-20-6', '1708.08180-2-20-6'], ['1708.08180-1-20-7', '1708.08180-2-20-7'], ['1708.08180-1-0-0', '1708.08180-2-0-0'], ['1708.08180-1-0-1', '1708.08180-2-0-1'], ['1708.08180-1-0-2', '1708.08180-2-0-2'], ['1708.08180-1-0-3', '1708.08180-2-0-3'], ['1708.08180-1-0-4', '1708.08180-2-0-4'], ['1708.08180-1-0-5', '1708.08180-2-0-5'], ['1708.08180-1-35-0', '1708.08180-2-35-0'], ['1708.08180-1-35-1', '1708.08180-2-35-1'], ['1708.08180-1-35-2', '1708.08180-2-35-2'], ['1708.08180-1-35-3', '1708.08180-2-35-3'], ['1708.08180-1-35-4', '1708.08180-2-35-4'], ['1708.08180-1-35-5', '1708.08180-2-35-5'], ['1708.08180-1-35-6', '1708.08180-2-35-6'], ['1708.08180-1-35-7', '1708.08180-2-35-7'], ['1708.08180-1-16-0', '1708.08180-2-16-0'], ['1708.08180-1-16-1', '1708.08180-2-16-1'], ['1708.08180-1-32-0', '1708.08180-2-32-0'], ['1708.08180-1-32-1', '1708.08180-2-32-1'], ['1708.08180-1-22-0', '1708.08180-2-22-0'], ['1708.08180-1-30-0', '1708.08180-2-30-0'], ['1708.08180-1-30-1', '1708.08180-2-30-1'], ['1708.08180-1-30-2', '1708.08180-2-30-2'], ['1708.08180-1-30-3', '1708.08180-2-30-3'], ['1708.08180-1-30-4', '1708.08180-2-30-4'], ['1708.08180-1-30-5', '1708.08180-2-30-5'], ['1708.08180-1-37-0', '1708.08180-2-37-0'], ['1708.08180-1-37-1', '1708.08180-2-37-1'], ['1708.08180-1-37-2', '1708.08180-2-37-2'], ['1708.08180-1-37-3', '1708.08180-2-37-3'], ['1708.08180-1-37-4', '1708.08180-2-37-4'], ['1708.08180-1-37-5', '1708.08180-2-37-5'], ['1708.08180-1-28-0', '1708.08180-2-28-0'], ['1708.08180-1-28-1', '1708.08180-2-28-1'], ['1708.08180-1-28-2', '1708.08180-2-28-2'], ['1708.08180-1-28-3', '1708.08180-2-28-3'], ['1708.08180-1-28-4', '1708.08180-2-28-4'], ['1708.08180-1-28-5', '1708.08180-2-28-5'], ['1708.08180-1-28-6', '1708.08180-2-28-6'], ['1708.08180-1-8-0', '1708.08180-2-8-0'], ['1708.08180-1-4-0', '1708.08180-2-4-0'], ['1708.08180-1-4-1', '1708.08180-2-4-1'], ['1708.08180-1-4-2', '1708.08180-2-4-2'], ['1708.08180-1-4-3', '1708.08180-2-4-3'], ['1708.08180-1-4-4', '1708.08180-2-4-4'], ['1708.08180-1-9-0', '1708.08180-2-9-0'], ['1708.08180-1-18-0', '1708.08180-2-18-0'], ['1708.08180-1-18-1', '1708.08180-2-18-1'], ['1708.08180-1-18-2', '1708.08180-2-18-2'], ['1708.08180-1-18-3', '1708.08180-2-18-3'], ['1708.08180-1-24-0', '1708.08180-2-24-0'], ['1708.08180-1-34-0', '1708.08180-2-34-0'], ['1708.08180-1-34-1', '1708.08180-2-34-1'], ['1708.08180-1-2-0', '1708.08180-2-2-0'], ['1708.08180-1-2-1', '1708.08180-2-2-1'], ['1708.08180-1-2-2', '1708.08180-2-2-2'], ['1708.08180-1-2-3', '1708.08180-2-2-3'], ['1708.08180-1-2-4', '1708.08180-2-2-4'], ['1708.08180-1-3-0', '1708.08180-2-3-0'], ['1708.08180-1-3-1', '1708.08180-2-3-1'], ['1708.08180-1-3-2', '1708.08180-2-3-2'], ['1708.08180-1-3-3', '1708.08180-2-3-3'], ['1708.08180-1-3-4', '1708.08180-2-3-4'], ['1708.08180-1-3-5', '1708.08180-2-3-5'], ['1708.08180-1-3-6', '1708.08180-2-3-6'], ['1708.08180-1-3-7', '1708.08180-2-3-7'], ['1708.08180-1-3-8', '1708.08180-2-3-8'], ['1708.08180-1-3-9', '1708.08180-2-3-9']]

[]

[]

[]

[]

['1708.08180-1-7-0', '1708.08180-1-10-0', '1708.08180-1-11-0', '1708.08180-1-12-0', '1708.08180-1-13-0', '1708.08180-1-14-0', '1708.08180-1-21-0', '1708.08180-1-23-0', '1708.08180-1-25-0', '1708.08180-1-26-0', '1708.08180-2-7-0', '1708.08180-2-10-0', '1708.08180-2-11-0', '1708.08180-2-12-0', '1708.08180-2-13-0', '1708.08180-2-14-0', '1708.08180-2-21-0', '1708.08180-2-23-0', '1708.08180-2-25-0', '1708.08180-2-26-0']

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

https://arxiv.org/abs/1708.08180

1710.09926

{'1710.09926-1-0-0': 'Bottleneck autoencoders have been actively researched as a solution to image compression tasks.', '1710.09926-1-0-1': 'In this work, we explore the ability of sparse coding to improve reconstructed image quality for the same degree of compression.', '1710.09926-1-0-2': 'We observe that sparse image compression provides qualitatively superior visual quality of reconstructed images but has lower values of PSNR and SSIM compared to bottleneck autoencoders.', '1710.09926-1-0-3': 'We hypothesized that there should be another evaluational criterion to support our subjective observations.', '1710.09926-1-0-4': 'To test this hypothesis, we fed reconstructed images from both the bottleneck autoencoder and sparse coding into a DCNN classifier and discovered that the images reconstructed from the sparse coding compression obtained on average 1.5% higher classification accuracy compared to bottleneck autoencoders, implying that sparse coding preserves more content-relevant information.', '1710.09926-1-1-0': '# Introduction', '1710.09926-1-2-0': 'Image compression methods have significant practical and commercial interest and have been the topic of extensive research.', '1710.09926-1-2-1': 'High-resolution images contain mostly low-frequency information and lots of redundancy, which makes them very efficient to compress.', '1710.09926-1-2-2': 'However, thumbnail images (e.g., [MATH] pixels in size) contain higher frequencies and much less redundancy, which makes them more difficult to compress [3].', '1710.09926-1-2-3': 'Given the importance of thumbnail images for mobile applications, much research has been focused on improving the compression of small images thumbnails [1][2][3].', '1710.09926-1-3-0': 'Bottleneck autoencoders achieve compression by using feed-forward artificial neural networks to reduce the dimensionality of the input data.', '1710.09926-1-3-1': 'The basic principles underlying convolutional and fully connected feed-forward neural networks, including bottleneck autoencoders, have been known for years [1][4][5].', '1710.09926-1-4-0': 'Sparse coding algorithms use an overcomplete set of non-orthogonal basis functions (feature vectors) to find a sparse combination of non-zero activation coefficients that most accurately reconstruct each input image.', '1710.09926-1-4-1': 'Sparse coding image compression combines sparse coding with an idea from compressive sensing: given only a random subset of pixels instead of a complete natural image as input, identify the minimal set of generators that explains the observed pixels, and infer from them the missing pixel values.', '1710.09926-1-4-2': 'Finding and improving sparse representations of images is an important goal with applications to image compression [6][7] and classification [8].', '1710.09926-1-5-0': 'In this work we compare the ability of bottleneck autoencoders and sparse coding to compress thumbnail images and use two different criteria for evaluating reconstructed image quality.', '1710.09926-1-5-1': "We report that the sparse image compression provides qualitatively superior visual quality of reconstructed images and higher classification accuracy when processed through the TensorFlow's CIFAR-10 DCNN classifier [11], whereas bottleneck autoencoders yield superior compression as measured by pixel-wise metrics such as PSNR and SSIM.", '1710.09926-1-6-0': '# Methods', '1710.09926-1-7-0': '## Sparse Coding', '1710.09926-1-8-0': 'Given an overcomplete basis, sparse coding algorithms seek to identify the minimal set of generators that most accurately reconstruct each input image.', '1710.09926-1-8-1': 'In neural terms, each neuron is a generator that adds its associated feature vector to the reconstructed image with an amplitude equal to its activation.', '1710.09926-1-8-2': 'For any particular input image, the optimal sparse representation is given by the vector of neural activations that minimizes both image reconstruction error and the number of neurons with non-zero activity.', '1710.09926-1-8-3': 'Formally, finding a sparse representation involves finding a minimum of the following cost function: [EQUATION]', '1710.09926-1-8-4': 'In Eq. ([REF]), I is an image unrolled into a vector, and [MATH] is a dictionary of feature kernels that are convolved with the sparse representation a.', '1710.09926-1-8-5': 'The factor [MATH] is a tradeoff parameter; larger [MATH] values encourage greater sparsity (fewer non-zero coefficients) at the cost of greater reconstruction error.', '1710.09926-1-8-6': 'Both the sparse representation a and the dictionary of feature kernels [MATH] can be determined by a variety of standard optimization methods.', '1710.09926-1-9-0': 'Our approach to computing a sparse representation for a given input image is based on a convolutional generalization of a rectifying Locally Competitive Algorithm (LCA) [10].', '1710.09926-1-9-1': 'Once a sparse representation for a given input image has been found, the basis elements associated with non-zero activation coefficients are adapted according to a local Hebbian learning rule (with a momentum term for faster convergence) that further reduces the remaining reconstruction error.', '1710.09926-1-9-2': 'Starting with random basis elements, dictionary learning was performed via Stochastic Gradient Descent (SGD).', '1710.09926-1-9-3': 'This training procedure can learn to factor a complex, high-dimensional natural image into an overcomplete set of basis vectors that capture the high-dimensional correlations in the data.', '1710.09926-1-10-0': 'The sparse coding image compression architecture is illustrated in Figure [REF].', '1710.09926-1-10-1': 'During training, a random subset of on average 50% of the original pixels (white neurons) were used as input, resulting in a 2:1 compression ratio.', '1710.09926-1-10-2': 'To locate the optimum percentage of non-zero neural activity, or optimal threshold, for our sparse coding model, we trained our network for one epoch on 9 different threshold values using a dictionary size of 1024 features and a stride of 2 ([MATH]42.6[MATH]s overcomplete).', '1710.09926-1-10-3': 'We found that for the optimum threshold for our network, the average percentage of active neurons is 1.37% [REF].', '1710.09926-1-10-4': 'Our sparse coding image compression results used this optimal level of sparsity.', '1710.09926-1-11-0': '## Bottleneck Autoencoder', '1710.09926-1-12-0': 'The bottleneck autoencoder architecture is illustrated in Figure [REF].', '1710.09926-1-12-1': 'This structure includes one input layer, one or more hidden layers, and one output layer.', '1710.09926-1-12-2': 'The input and output layers contain the same number of neurons, where the output layer aims to reconstruct the input.', '1710.09926-1-12-3': 'Image compression is achieved by restricting the number of neurons in the smallest hidden layer (the bottleneck) to have half the number of neurons of the input layer to achieve a 2:1 compression ratio.', '1710.09926-1-12-4': 'We evaluated several bottleneck autoencoder models, and found that a model combining one convolutional layer with a stride of 4 and one fully-connected hidden layer, each containing the same number of neurons ([MATH] the number of pixels in each thumbnail), provides superior reconstructed image quality over other models.', '1710.09926-1-12-5': 'Across the set of input images in the training set, the bottleneck autoencoder network finds the optimal hidden layer features that minimize image reconstruction error.', '1710.09926-1-12-6': 'In the simplest case as illustrated in Figure [REF], the model minimizes the following energy function: [EQUATION] where [MATH] is the input image, [MATH] is the reconstructed image, [MATH] is the nonlinear element-wise ReLU activation function.', '1710.09926-1-12-7': "[MATH] and [MATH] are the encoder's weights matrix and a bias vector, [MATH] and [MATH] are the decoder's weights matrix and a bias respectively.", '1710.09926-1-12-8': 'In our model, the encoder and decoder share the same weights and bias.', '1710.09926-1-13-0': 'In this type of neural network architecture, a large number of input neurons feeds into a smaller number of neurons in the hidden layer, which works as the compressor.', '1710.09926-1-13-1': 'This structured bottleneck layer could be treated as a nonlinear mapping of input features.', '1710.09926-1-13-2': 'The decompressor then reconstructs the compressed image back to the output layer neurons using the same weights as the compressor.', '1710.09926-1-13-3': 'The bottleneck autoencoder model employs SGD with momentum to train optimal values of the weights and bias after being randomly initialized.', '1710.09926-1-13-4': 'The bottleneck autoencoder is designed to preserve only those features that best describe the original image and to shed redundant information.', '1710.09926-1-14-0': '# Results', '1710.09926-1-15-0': 'In order to evaluate and compare the quality of the proposed sparse coding and bottleneck autoencoder image compression models, we used PetaVision [9], an open source neural simulation toolbox that enables multi-node, multi-core and/or GPU accelerated high-performance implementations of sparse solvers derived from LCA as well as conventional neural network models.', '1710.09926-1-15-1': 'We use the CIFAR-10 dataset, which consists of 50,000 [MATH] images for training and 10,000 [MATH] images for testing.', '1710.09926-1-15-2': 'Category labels were not used in this study.', '1710.09926-1-16-0': '## Image Compression', '1710.09926-1-17-0': 'We first used two well-known pixel-wise image quality metrics: the peak-signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) to evaluate the reconstructed images from either sparse coding or from the bottleneck autoencoder.', '1710.09926-1-17-1': 'Figure [REF] shows examples of sparse coding and bottleneck autoencoder based image compression.', '1710.09926-1-17-2': 'The first column shows example images drawn from the original CIFAR-10 testing set.', '1710.09926-1-17-3': 'The second column depicts these same images after a random subset representing approximately 50% of the original pixels are omitted (50% compression).', '1710.09926-1-17-4': 'The third column shows the sparse reconstructions.', '1710.09926-1-17-5': 'The fourth column shows the bottleneck autoencoder reconstructions when the same image compression ratio is applied.', '1710.09926-1-17-6': 'Finally, the fifth and sixth columns show the residuals for the sparse and bottleneck reconstructions respectively.', '1710.09926-1-18-0': 'Table [REF] indicates that the reconstructed images from sparse coding contain lower values of PSNR and SSIM compared to the bottleneck autoencoder.', '1710.09926-1-18-1': 'However, visual examination reveals that the reconstructed images from sparse coding are subjectively less noisy than those reconstructed from the bottleneck autoencoder.', '1710.09926-1-19-0': '## Preliminary Reconstructed Image Classification', '1710.09926-1-20-0': 'We observe that the sparse image compression provides qualitatively superior visual quality of reconstructed images but has lower values of PSNR and SSIM compared to bottleneck autoencoders.', '1710.09926-1-20-1': 'We hypothesized that there should be another evaluational criterion to support our subjective observations.', '1710.09926-1-20-2': 'To test the hypothesis that reconstructed images obtained from sparse coding include more content-relevant information, we fed reconstructions from both the bottleneck autoencoder and sparse coding into the CIFAR-10 DCNN classifier from TensorFlow [11] trained on the original (non-compressed) CIFAR-10 training image set.', '1710.09926-1-20-3': 'After three training and testing simulations with different random seeds, the results from Table [REF] indicate that sparse coding contains on average 1.5% higher classification accuracy compared to bottleneck autoencoder.', '1710.09926-1-21-0': '# Conclusion', '1710.09926-1-22-0': "We report that for the same image compression ratio, sparse image compression provides subjectively superior visual quality of reconstructed images compared to bottleneck autoencoders and supports higher classification accuracy when processed through the TensorFlow's CIFAR-10 DCNN classifier."}

{'1710.09926-2-0-0': 'Bottleneck autoencoders have been actively researched as a solution to image compression tasks.', '1710.09926-2-0-1': 'However, we observed that bottleneck autoencoders produce subjectively low quality reconstructed images.', '1710.09926-2-0-2': 'In this work, we explore the ability of sparse coding to improve reconstructed image quality for the same degree of compression.', '1710.09926-2-0-3': 'We observe that sparse image compression produces visually superior reconstructed images and yields higher values of pixel-wise measures of reconstruction quality (PSNR and SSIM) compared to bottleneck autoencoders.', '1710.09926-2-0-4': 'In addition, we find that using alternative metrics that correlate better with human perception, such as feature perceptual loss and the classification accuracy, sparse image compression scores up to 18.06% and 2.7% higher, respectively, compared to bottleneck autoencoders.', '1710.09926-2-0-5': 'Although computationally much more intensive, we find that sparse coding is otherwise superior to bottleneck autoencoders for the same degree of compression.', '1710.09926-2-1-0': 'Image Compression; Sparse Coding; Bottleneck Autoencoders; Feature Perceptual Loss; Thumbnails.', '1710.09926-2-2-0': '# Introduction', '1710.09926-2-3-0': 'Image compression methods have significant practical and commercial interest and have been the topic of extensive research.', '1710.09926-2-3-1': 'Thumbnail images contain higher frequencies and much less redundancy, which makes them more difficult to compress compared to high-resolution images.', '1710.09926-2-3-2': 'A lot of research has been focused on improving the compression of thumbnails [CITATION].', '1710.09926-2-4-0': 'Bottleneck autoencoders achieve compression by using feed-forward artificial neural networks to reduce the dimensionality of the input data.', '1710.09926-2-4-1': 'The basic principles underlying convolutional and fully connected feed-forward neural networks, including bottleneck autoencoders, have been known for years [CITATION].', '1710.09926-2-5-0': 'Sparse coding algorithms use an overcomplete set of non-orthogonal basis functions (feature vectors) to find a sparse combination of non-zero activation coefficients that most accurately reconstruct each input image.', '1710.09926-2-5-1': 'Sparse coding image compression combines sparse coding with the ideas of downsampling and compressive sensing [CITATION]: 1) a subset of the original pixels are used as a compressed image, 2) a minimal set of generators that explains the pixels in the compressed image is identified, and 3) the missing pixel values are inferred.', '1710.09926-2-6-0': 'The peak-signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) are two commonly used pixel-level image quality metrics.', '1710.09926-2-6-1': 'Both PSNR and SSIM fail to capture differences at the feature level and correlate poorly with human perception of image quality.', '1710.09926-2-6-2': 'Several researchers have therefore attempted to define alternative measures of compression quality based on the similarity of the features extracted from the reconstructed and original images and have shown that these alternative measures correlate better with human subjective perception [CITATION].', '1710.09926-2-6-3': "We further introduce a new feature-based measure of compression quality based on loss of classification accuracy, in which compressed images are labeled using TensorFlow's CIFAR-10 DCNN classifier [CITATION] and the results compared to the baseline performance achieved on the original images.", '1710.09926-2-7-0': 'In this work we apply bottleneck autoencoders and sparse coding approaches to the compression of thumbnail images and use both subjective, pixel-level and feature-based criteria for evaluating reconstructed image quality.', '1710.09926-2-7-1': 'We report that sparse coding with random dropout masks produces subjectively superior reconstructed images along with lower feature perceptual loss and higher classification accuracy but yields lower values of pixel-wise metrics such as PSNR and SSIM compared to the bottleneck autoencoders.', '1710.09926-2-7-2': 'However, sparse coding with a checkerboard mask yields superior performance as measured by all three of the above criteria.', '1710.09926-2-8-0': '# Methods', '1710.09926-2-9-0': '## Sparse Coding', '1710.09926-2-10-0': 'Given an overcomplete basis, sparse coding algorithms seek to identify the minimal set of generators that most accurately reconstruct each input image.', '1710.09926-2-10-1': 'In neural terms, each neuron is a generator that adds its associated feature vector to the reconstructed image with an amplitude equal to its activation.', '1710.09926-2-10-2': 'For any particular input image, the optimal sparse representation is given by the vector of neural activations that minimizes both image reconstruction error and the number of neurons with non-zero activity.', '1710.09926-2-10-3': 'Formally, finding a sparse representation involves finding a minimum of the following cost function: [EQUATION]', '1710.09926-2-10-4': 'In Eq. ([REF]), [MATH] is an image unrolled into a vector, and [MATH] is a dictionary of feature kernels that are convolved with the sparse representation [MATH].', '1710.09926-2-10-5': 'The factor [MATH] is a tradeoff parameter; larger [MATH] values encourage greater sparsity (fewer non-zero coefficients) at the cost of greater reconstruction error.', '1710.09926-2-10-6': 'Both the sparse representation [MATH] and the dictionary of feature kernels [MATH] can be determined by a variety of standard optimization methods.', '1710.09926-2-10-7': 'Our approach to compute a sparse representation for a given input image is based on a convolutional generalization of a rectifying Locally Competitive Algorithm (LCA) [CITATION].', '1710.09926-2-10-8': 'Once a sparse representation for a given input image has been found, the basis elements associated with non-zero activation coefficients are adapted according to a local Hebbian learning rule (with a momentum term for faster convergence) that further reduces the remaining reconstruction error.', '1710.09926-2-10-9': 'Starting with random basis elements, dictionary learning was performed via Stochastic Gradient Descent (SGD).', '1710.09926-2-10-10': 'This training procedure can learn to factor a complex, high-dimensional natural image into an overcomplete set of basis vectors that capture the high-dimensional correlations in the data.', '1710.09926-2-11-0': 'The sparse coding based image compression architecture is illustrated in Figure [REF], where the input layer (left) is the original image, the hidden layer (middle) is its sparse representation, and the output layer (right) is the image reconstruction.', '1710.09926-2-11-1': 'The white pixels from the input layer are the compressed representation of the original image and black pixels are the omitted pixels of the original image.', '1710.09926-2-11-2': 'To summarize, image compression based on sparse coding involves two distinct steps: 1) Compression: a subset of the pixels (white circles) is used as a compressed representation of the original image.', '1710.09926-2-11-3': '2) Decompression: a minimal subset of pre-trained generators that explains the compressed representation is identified, and the missing pixel values are inferred from these generators.', '1710.09926-2-12-0': 'During training, the sparse representation [MATH] is found using only the preserved pixel values.', '1710.09926-2-12-1': 'The omitted pixels do not contribute to the minimization of the cost function with respect to [MATH].', '1710.09926-2-12-2': 'Then [MATH] is used to update the dictionary of generators [MATH] in order to reconstruct the original image, including the masked pixels, with minimal error.', '1710.09926-2-13-0': 'We evaluated several sparse coding models which included a variety of structural elements such as convolutional layers, fully connected layers and multi-scale dictionaries.', '1710.09926-2-13-1': 'To facilitate comparison, each model contained the same total number of neurons.', '1710.09926-2-13-2': 'We found that the best performance was achieved by a model consisting of a single convolutional layer containing 1024 features with a patch size of [MATH] pixels and a stride of 2 ([MATH] times overcomplete).', '1710.09926-2-14-0': 'To determine the optimum percentage of non-zero activity, or optimal tradeoff parameter [MATH], for our sparse coding model, we trained our network for one epoch on 9 different [MATH] values using the single layer sparse convolutional model.', '1710.09926-2-14-1': 'We found that for the optimum value of [MATH] for our network, the average percentage of active neurons is 1.37% (see Figure [REF]).', '1710.09926-2-14-2': 'Our sparse coding image compression results used this single sparse convolutional model with the aforementioned optimal level of sparsity.', '1710.09926-2-15-0': '## Bottleneck Autoencoder', '1710.09926-2-16-0': 'The bottleneck autoencoder architecture is illustrated in Figure [REF].', '1710.09926-2-16-1': 'This structure includes one input layer (left), one or more hidden layers (middle), and one output layer (right).', '1710.09926-2-16-2': 'The input and output layers contain the same number of neurons, where the output layer aims to reconstruct the input.', '1710.09926-2-16-3': 'Image compression is achieved by restricting the number of neurons in the smallest hidden layer (the bottleneck) to contain half the number of neurons of the input layer to achieve a 2:1 compression ratio.', '1710.09926-2-16-4': 'We evaluated several bottleneck autoencoder models such as fully connected models, convolutional models, and multi-scaled models, and found that a model combining one convolutional layer with a stride of 4 and one fully connected layer, each containing the same number of neurons ([MATH] the number of pixels in the thumbnails), provides higher reconstructed image quality over other bottleneck models.', '1710.09926-2-16-5': 'The bottleneck autoencoder network finds the optimal hidden layer features across the training set of input images, that minimize the image reconstruction error.', '1710.09926-2-16-6': 'In the simplest case, as illustrated in Figure [REF], the model minimizes the following energy function: [EQUATION]', '1710.09926-2-16-7': 'In Eq. ([REF]), [MATH] is the input image, [MATH] is the reconstructed image, [MATH] is the ReLU activation function.', '1710.09926-2-16-8': "[MATH] and [MATH] are the encoder's weights matrix and a bias vector, [MATH] and [MATH] are the decoder's weights matrix and a bias respectively.", '1710.09926-2-17-0': 'In this type of neural network architecture, a large number of input neurons is fed into a smaller number of neurons in the hidden layer, which works as the compressor.', '1710.09926-2-17-1': 'This structured bottleneck layer could be treated as a nonlinear mapping of input features.', '1710.09926-2-17-2': 'The decompressor then reconstructs the compressed image back to the neurons in the output layer using the same weights as the compressor.', '1710.09926-2-17-3': 'The bottleneck autoencoder model employs SGD with momentum to train optimal values of the weights and bias after being randomly initialized.', '1710.09926-2-17-4': 'The bottleneck autoencoder is designed to preserve only those features that best describe the original image and shed redundant information.', '1710.09926-2-18-0': '# Results', '1710.09926-2-19-0': 'In this work, we compared bottleneck autoencoders with two sparse coding approaches.', '1710.09926-2-19-1': 'For sparse coding we masked [MATH] of the pixels either randomly or arranged in a checkerboard pattern to achieve a 2:1 compression ratio.', '1710.09926-2-19-2': 'The random mask is regenerated for every image batch whereas the checkerboard mask is fixed.', '1710.09926-2-19-3': 'In order to evaluate and compare the quality of the proposed sparse coding and bottleneck autoencoder image compression models, we used PetaVision[CITATION], an open source neural simulation toolbox that enables multi-node, multi-core and GPU accelerated high-performance implementations of sparse solvers derived from LCA as well as conventional neural network models.', '1710.09926-2-19-4': 'We use the CIFAR-10 dataset, which consists of 50,000 [MATH] images for training and 10,000 [MATH] images for testing.', '1710.09926-2-19-5': 'Category labels were not used in this study.', '1710.09926-2-20-0': '## Image Compression', '1710.09926-2-21-0': 'We first evaluated the quality of the reconstructed images from sparse coding and the bottleneck autoencoder using subjective human perceptual judgments.', '1710.09926-2-21-1': 'Figure [REF] shows examples of sparse coding and bottleneck autoencoder based image compression.', '1710.09926-2-21-2': 'Subjective examination reveals that the reconstructed images from sparse coding with either random or checkerboard mask exhibit less noise, has a smoother background, and results in more natural looking reconstructions than images reconstructed from the bottleneck autoencoder.', '1710.09926-2-21-3': 'Sparse coding with a random mask preserves less fine detail compared to the bottleneck autoencoder whereas sparse coding with a checkerboard mask preserves both background and fine details much better than both the other methods.', '1710.09926-2-21-4': 'Overall, the images compressed using sparse coding with a checkerboard mask are almost visually indistinguishable from the original images.', '1710.09926-2-22-0': '## Pixel-wise Loss in Image Space', '1710.09926-2-23-0': 'We used two well-known pixel-wise image quality metrics: the peak-signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) to evaluate the reconstructed images from either sparse coding or the bottleneck autoencoder.', '1710.09926-2-24-0': 'Table [REF] indicates that the reconstructed images from sparse coding with random mask contain lower values of PSNR and SSIM compared to the bottleneck autoencoder whereas the corresponding values for the checkerboard masks are higher than for the other methods.', '1710.09926-2-24-1': 'However, PSNR and SSIM measurements are well known to correlate poorly with human perception of image quality.', '1710.09926-2-25-0': '## Perceptual Loss in Feature Space', '1710.09926-2-26-0': 'To test the hypothesis that reconstructed images obtained from sparse coding include more information relevant to human perception compared to bottleneck autoencoders, we calculate the feature perceptual loss, given as the Euclidean distance of feature representations between original and reconstructed images[CITATION].', '1710.09926-2-26-1': 'To capture and compare the feature representations, we first pre-trained the original (non-compressed) CIFAR-10 training image set on the DCNN classifier from TensorFlow [CITATION].', '1710.09926-2-26-2': 'Table [REF] illustrates Loss 1 and Loss 2, which represent the feature perceptual losses captured from the activations of the second convolutional layer and the second pool layer in the DCNN, respectively.', '1710.09926-2-26-3': 'Overall, the reconstructed images from sparse coding with checkerboard mask and random mask contain on average 18.06% and 3.74% lower feature perceptual loss compared to the bottleneck autoencoder.', '1710.09926-2-27-0': '## Classification', '1710.09926-2-28-0': 'To further compare the different compression methods, we checked the classification accuracy of the reconstructed images using the same DCNN classifier.', '1710.09926-2-28-1': 'After three training and testing runs with different random seeds we found that the sparse coding with checkerboard and random masks supported on average 2.7% and 1.6% higher classification accuracy compared to the bottleneck autoencoder (see Table [REF]).', '1710.09926-2-29-0': '# Conclusion', '1710.09926-2-30-0': 'Sparse image compression with checkerboard and random masks provides subjectively superior visual quality of reconstructed images, on average 2.7% and 1.6% higher classification accuracy and 18.06% and 3.74% lower feature perceptual loss, respectively, compared to bottleneck autoencoders.', '1710.09926-2-30-1': 'This paper provides support for the hypothesis that reconstructed images obtained from sparse coding with checkerboard and random masks include more content-relevant information compared to bottleneck autoencoders for the same image compression ratio.'}

[['1710.09926-1-3-0', '1710.09926-2-4-0'], ['1710.09926-1-4-0', '1710.09926-2-5-0'], ['1710.09926-1-12-0', '1710.09926-2-16-0'], ['1710.09926-1-12-2', '1710.09926-2-16-2'], ['1710.09926-1-12-7', '1710.09926-2-16-8'], ['1710.09926-1-0-0', '1710.09926-2-0-0'], ['1710.09926-1-0-1', '1710.09926-2-0-2'], ['1710.09926-1-15-1', '1710.09926-2-19-4'], ['1710.09926-1-15-2', '1710.09926-2-19-5'], ['1710.09926-1-2-0', '1710.09926-2-3-0'], ['1710.09926-1-13-1', '1710.09926-2-17-1'], ['1710.09926-1-13-3', '1710.09926-2-17-3'], ['1710.09926-1-17-1', '1710.09926-2-21-1'], ['1710.09926-1-8-0', '1710.09926-2-10-0'], ['1710.09926-1-8-1', '1710.09926-2-10-1'], ['1710.09926-1-8-2', '1710.09926-2-10-2'], ['1710.09926-1-8-3', '1710.09926-2-10-3'], ['1710.09926-1-8-5', '1710.09926-2-10-5'], ['1710.09926-1-9-1', '1710.09926-2-10-8'], ['1710.09926-1-9-2', '1710.09926-2-10-9'], ['1710.09926-1-9-3', '1710.09926-2-10-10'], ['1710.09926-1-3-1', '1710.09926-2-4-1'], ['1710.09926-1-12-3', '1710.09926-2-16-3'], ['1710.09926-1-12-4', '1710.09926-2-16-4'], ['1710.09926-1-12-5', '1710.09926-2-16-5'], ['1710.09926-1-15-0', '1710.09926-2-19-3'], ['1710.09926-1-13-0', '1710.09926-2-17-0'], ['1710.09926-1-13-2', '1710.09926-2-17-2'], ['1710.09926-1-13-4', '1710.09926-2-17-4'], ['1710.09926-1-17-0', '1710.09926-2-23-0'], ['1710.09926-1-8-4', '1710.09926-2-10-4'], ['1710.09926-1-8-6', '1710.09926-2-10-6'], ['1710.09926-1-9-0', '1710.09926-2-10-7'], ['1710.09926-1-4-1', '1710.09926-2-5-1'], ['1710.09926-1-10-2', '1710.09926-2-14-0'], ['1710.09926-1-10-3', '1710.09926-2-14-1'], ['1710.09926-1-10-4', '1710.09926-2-14-2'], ['1710.09926-1-12-1', '1710.09926-2-16-1'], ['1710.09926-1-12-6', '1710.09926-2-16-6'], ['1710.09926-1-12-6', '1710.09926-2-16-7'], ['1710.09926-1-0-2', '1710.09926-2-0-3'], ['1710.09926-1-5-0', '1710.09926-2-7-0'], ['1710.09926-1-5-1', '1710.09926-2-7-1'], ['1710.09926-1-2-1', '1710.09926-2-3-1'], ['1710.09926-1-2-2', '1710.09926-2-3-1'], ['1710.09926-1-2-3', '1710.09926-2-3-2'], ['1710.09926-1-22-0', '1710.09926-2-30-0'], ['1710.09926-1-18-0', '1710.09926-2-24-0'], ['1710.09926-1-18-1', '1710.09926-2-24-0'], ['1710.09926-1-20-2', '1710.09926-2-26-0'], ['1710.09926-1-20-2', '1710.09926-2-26-1'], ['1710.09926-1-20-3', '1710.09926-2-28-1']]

[['1710.09926-1-3-0', '1710.09926-2-4-0'], ['1710.09926-1-4-0', '1710.09926-2-5-0'], ['1710.09926-1-12-0', '1710.09926-2-16-0'], ['1710.09926-1-12-2', '1710.09926-2-16-2'], ['1710.09926-1-12-7', '1710.09926-2-16-8'], ['1710.09926-1-0-0', '1710.09926-2-0-0'], ['1710.09926-1-0-1', '1710.09926-2-0-2'], ['1710.09926-1-15-1', '1710.09926-2-19-4'], ['1710.09926-1-15-2', '1710.09926-2-19-5'], ['1710.09926-1-2-0', '1710.09926-2-3-0'], ['1710.09926-1-13-1', '1710.09926-2-17-1'], ['1710.09926-1-13-3', '1710.09926-2-17-3'], ['1710.09926-1-17-1', '1710.09926-2-21-1'], ['1710.09926-1-8-0', '1710.09926-2-10-0'], ['1710.09926-1-8-1', '1710.09926-2-10-1'], ['1710.09926-1-8-2', '1710.09926-2-10-2'], ['1710.09926-1-8-3', '1710.09926-2-10-3'], ['1710.09926-1-8-5', '1710.09926-2-10-5'], ['1710.09926-1-9-1', '1710.09926-2-10-8'], ['1710.09926-1-9-2', '1710.09926-2-10-9'], ['1710.09926-1-9-3', '1710.09926-2-10-10']]

[['1710.09926-1-3-1', '1710.09926-2-4-1'], ['1710.09926-1-12-3', '1710.09926-2-16-3'], ['1710.09926-1-12-4', '1710.09926-2-16-4'], ['1710.09926-1-12-5', '1710.09926-2-16-5'], ['1710.09926-1-15-0', '1710.09926-2-19-3'], ['1710.09926-1-13-0', '1710.09926-2-17-0'], ['1710.09926-1-13-2', '1710.09926-2-17-2'], ['1710.09926-1-13-4', '1710.09926-2-17-4'], ['1710.09926-1-17-0', '1710.09926-2-23-0'], ['1710.09926-1-8-4', '1710.09926-2-10-4'], ['1710.09926-1-8-6', '1710.09926-2-10-6'], ['1710.09926-1-9-0', '1710.09926-2-10-7']]

[]

[['1710.09926-1-4-1', '1710.09926-2-5-1'], ['1710.09926-1-10-2', '1710.09926-2-14-0'], ['1710.09926-1-10-3', '1710.09926-2-14-1'], ['1710.09926-1-10-4', '1710.09926-2-14-2'], ['1710.09926-1-12-1', '1710.09926-2-16-1'], ['1710.09926-1-12-6', '1710.09926-2-16-6'], ['1710.09926-1-12-6', '1710.09926-2-16-7'], ['1710.09926-1-0-2', '1710.09926-2-0-3'], ['1710.09926-1-5-0', '1710.09926-2-7-0'], ['1710.09926-1-5-1', '1710.09926-2-7-1'], ['1710.09926-1-2-1', '1710.09926-2-3-1'], ['1710.09926-1-2-2', '1710.09926-2-3-1'], ['1710.09926-1-2-3', '1710.09926-2-3-2'], ['1710.09926-1-22-0', '1710.09926-2-30-0'], ['1710.09926-1-18-0', '1710.09926-2-24-0'], ['1710.09926-1-18-1', '1710.09926-2-24-0'], ['1710.09926-1-20-2', '1710.09926-2-26-0'], ['1710.09926-1-20-2', '1710.09926-2-26-1'], ['1710.09926-1-20-3', '1710.09926-2-28-1']]

[]

['1710.09926-2-1-0']

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

https://arxiv.org/abs/1710.09926

1106.6035

{'1106.6035-1-0-0': 'We apply the operator product expansion to determine the asymptotic form of the current response of a Fermi gas in two and three dimensions.', '1106.6035-1-0-1': 'The leading-order term away from the one-particle peak is proportional to a quantity known as the contact whose coefficient is determined exactly.', '1106.6035-1-0-2': 'We also calculate the dynamic structure factor and the high-frequency tails of the spectral viscosities as a function of the scattering length.', '1106.6035-1-0-3': 'Our results are used to derive certain sum rules for the viscosities.', '1106.6035-1-1-0': '# INTRODUCTION', '1106.6035-1-2-0': 'Two-component fermionic quantum gases belong to the most versatile systems that can be used to study the many-body physics of fermions [CITATION].', '1106.6035-1-2-1': 'The simplest Lagrangian to describe such a system models the s-wave interaction between the constituents by a contact interaction (see for example Ref. [CITATION]): [EQUATION] where a summation over spin indices [MATH] is implied.', '1106.6035-1-2-2': 'In experiments with cold fermionic atoms it is possible to confine the atomic cloud in two- and three-dimensional traps and also to tune the interaction strength to a regime where the components are strongly interacting.', '1106.6035-1-2-3': 'The theoretical analysis of a Fermi gas in a strongly interacting limit is intricate since many theoretical methods rely on a perturbative expansion or neglect fluctuations and are thus not applicable.', '1106.6035-1-3-0': 'Over the past few years, a lot of work has been stimulated by the observation that some properties of a Fermi gas can be described by universal relations.', '1106.6035-1-3-1': "They are called universal because they hold over a wide range of the system's parameters such as the temperature, the density, the precise form or the strength of the interaction and so on.", '1106.6035-1-3-2': 'A first set of such relations was presented by Tan [CITATION] - who among other things derived expressions for the momentum distribution, the energy and various other thermodynamic quantities - and are hence often called Tan relations.', '1106.6035-1-3-3': 'The scale of the Tan relations is set by a quantity known as the contact [MATH] which is determined by the many-particle properties of the system.', '1106.6035-1-3-4': 'It is defined as [MATH], where [MATH] is the contact density operator and [MATH] denotes the number of space dimensions.', '1106.6035-1-3-5': 'An operational definition of the contact can be given as the magnitude of the large-momentum tail of the gas: [MATH].', '1106.6035-1-3-6': 'Apart from the dependence on the contact, the universal relations can be inferred from the few-body physics and can be calculated in closed analytic form in many cases.', '1106.6035-1-3-7': 'Only the value of the contact has to be taken from numerical calculations or experiments.', '1106.6035-1-3-8': 'Some of the Tan relations have been tested experimentally and good agreement with the theoretical predictions has been found [CITATION].', '1106.6035-1-3-9': 'Since their proposal many other exact relations for fermionic as well as bosonic systems have been derived, cf. for example Refs. [CITATION].', '1106.6035-1-3-10': 'An overview of various universal relations and recent experimental developments can be found in Ref. [CITATION].', '1106.6035-1-4-0': 'A field-theoretical derivation of universal relations was put forward by Braaten, Kang and Platter [CITATION] who used the operator product expansion (OPE) to rederive the Tan relations [CITATION].', '1106.6035-1-4-1': 'The OPE expresses the product of two local operators [MATH] and [MATH] at different points in space and time in terms of local operators [MATH], where [MATH] denotes a set of indices [CITATION]: [EQUATION]', '1106.6035-1-4-2': 'Note that on the RHS the dependence on the separation [MATH] is carried by the Wilson coefficients [MATH], which can be calculated exactly for simple operators.', '1106.6035-1-4-3': 'If an observable can be expressed as the expectation value of an operator product involving operators at different spacetime-points, we can use the OPE to determine its leading-order behavior for small [MATH] and [MATH] (or for large values of momentum [MATH] and energy [MATH] for the Fourier transform).', '1106.6035-1-4-4': 'The advantage of the OPE is that it provides a powerful framework to derive additional universal relations.', '1106.6035-1-4-5': 'This was exploited by the authors of [CITATION] to derive universal results for radio-frequency spectroscopy measurements [CITATION].', '1106.6035-1-4-6': 'Son and Thompson studied the asymptotic behavior of the dynamic structure factor [MATH] at unitarity at large [MATH] and [MATH] as well as the large-frequency tail of [MATH] [CITATION].', '1106.6035-1-4-7': 'The result for the large-frequency tail in [CITATION] was augmented by Goldberger and Rothstein who calculated the leading-order correction due to finite scattering length [CITATION].', '1106.6035-1-4-8': 'Some of the above results have also been obtained by other authors using different methods [CITATION].', '1106.6035-1-4-9': 'While most of the above work was carried out for a three-dimensional system at or close to unitarity, it is possible to derive universal relations for any strength of the interaction or any other dimension in the same way.', '1106.6035-1-5-0': 'In this work, we apply the OPE to derive the current response function of a Fermi gas in two and in three dimensions.', '1106.6035-1-5-1': 'It is defined as [EQUATION] where [MATH] is the current operator.', '1106.6035-1-5-2': 'The Fourier transform of Eq. ([REF]) can be split into a transverse and a longitudinal part: [EQUATION]', '1106.6035-1-5-3': 'The longitudinal part is related to the superfluid fraction and the transverse part to the normal fraction of the gas.', '1106.6035-1-5-4': 'Thus, the current response function is a central quantity in the study of superfluids.', '1106.6035-1-5-5': 'The longitudinal component is related to the structure factor and can be determined with methods that measure the density response, such as two-photon Bragg spectroscopy [CITATION].', '1106.6035-1-6-0': 'Recently, there have been several proposals to probe the transverse component of the current response using artificial gauge fields [CITATION].', '1106.6035-1-6-1': 'Artificial gauge fields are induced by either rotating the atomic cloud or placing it in a specific laser setup.', '1106.6035-1-6-2': 'In the latter case, incident laser beams couple the ground state of the gas to an excited state.', '1106.6035-1-6-3': 'The effective Lagrangian of such a system describes charged particles coupled to a U(1) gauge field.', '1106.6035-1-6-4': 'The amplitude of this gauge field typically depends on the spatial or temporal variation of the Rabi frequency or the detuning of the laser from the resonance frequency.', '1106.6035-1-6-5': 'For example a local detuning of the laser field could induce a rapidly oscillating gauge field.', '1106.6035-1-6-6': 'For further references see [CITATION].', '1106.6035-1-6-7': 'One way to measure the imaginary part of the current response is to determine how much energy [MATH] is absorbed by the system if the gauge field is switched on for a short time: [EQUATION]', '1106.6035-1-6-8': 'For a two-dimensional gas, there also exists a proposal to determine the current response optically, by measuring the phase shift of a laser beam that crosses the atomic cloud [CITATION].', '1106.6035-1-6-9': 'While artificial gauge fields have only been realized for Bose gases so far there is no reason why this should not also be possible for fermionic systems in the near future.', '1106.6035-1-6-10': 'It is in anticipation of such experiments that we calculate the current response.', '1106.6035-1-6-11': 'In particular, since it should be possible to engineer gauge fields that couple differently to each spin species, we consider both the spin-symmetric and antisymmetric current response [MATH], where a summation over spin indices is implied (corresponding to gauge fields that couple symmetrically and antisymmetrically to the current: [MATH]).', '1106.6035-1-7-0': 'This paper is structured as follows: in Sec. [REF] we describe how to calculate the Wilson coefficients of the current correlator.', '1106.6035-1-7-1': 'This is done using a matching procedure where the matrix element of Eq. ([REF]) is evaluated with respect to selected few-body states.', '1106.6035-1-7-2': 'We choose a one-particle state to determine the Wilson coefficients of the bilinear operators (Sec. [REF]) and then a two-particle state for the contact operator (Sec. [REF]).', '1106.6035-1-7-3': 'Sec. [REF] presents the results of this calculation.', '1106.6035-1-7-4': 'Expressions for both the spin-symmetric and the antisymmetric current response are presented and discussed in Sec. [REF].', '1106.6035-1-7-5': 'The U(1) symmetry of the system implies a Ward identity that relates the longitudinal part of the current correlator to the dynamic structure factor.', '1106.6035-1-7-6': 'We use this to calculate the asymptotic form of the dynamic structure factor in two and in three dimensions for any value of the scattering length in Sec. [REF].', '1106.6035-1-7-7': 'In [CITATION], Taylor and Randeria derived expressions for the large-frequency tail of the spectral shear and bulk viscosity, [MATH] and [MATH].', '1106.6035-1-7-8': 'This paper reproduces their results using a field-theoretical framework and generalizes them to finite scattering length.', '1106.6035-1-7-9': 'We are also able to provide the viscosity tails for a two-dimensional Fermi gas.', '1106.6035-1-7-10': 'We conclude with a summary in Sec. [REF].', '1106.6035-1-7-11': 'The paper contains an appendix that presents details of the calculations in Sec. [REF].', '1106.6035-1-8-0': '# OPERATOR PRODUCT EXPANSION', '1106.6035-1-9-0': 'While we would like to obtain the OPE for the retarded response function [MATH] it is more convenient to calculate the time-ordered response, which coincides with the retarded response up to a correction of order [MATH] [CITATION]: [EQUATION] [MATH] denotes the time-ordered product.', '1106.6035-1-9-1': 'The [MATH]-term in the Fourier transform has been included to shift the poles in the resulting expressions of the response functions [CITATION].', '1106.6035-1-9-2': 'Using the time-ordered response allows us to apply standard diagrammatic methods to calculate the Wilson coefficients.', '1106.6035-1-9-3': 'In this section, we derive the OPE of the operator [EQUATION]', '1106.6035-1-9-4': 'The OPE is an operator identity, that is it holds if we take the expectation value of Eq. ([REF]) with respect to any state.', '1106.6035-1-9-5': 'Hence, in order to determine the Wilson coefficient [MATH] of an operator [MATH] we can choose a simple few-body state for which [MATH] and match the expectation values on both sides of Eq. ([REF]).', '1106.6035-1-9-6': 'Inserting Eq. ([REF]) in Eq. ([REF]) yields: [EQUATION] where [MATH] denotes the scaling dimension of the operator [MATH] as defined by [MATH].', '1106.6035-1-9-7': 'We suppress the dependence of the Wilson coefficients on the spin indices on the RHS of Eq. ([REF]).', '1106.6035-1-9-8': 'The operators [MATH] can be composed of the field operators [MATH] and their adjoints [MATH] as well as derivative operators acting on those fields.', '1106.6035-1-9-9': 'We will only consider operators that conserve the particle number, i.e. contain the same number of fields and their adjoints, and that are symmetric in the spin indices.', '1106.6035-1-9-10': 'Eq. ([REF]) shows that the high-frequency behavior of the current response is governed by the operators with lowest scaling dimension.', '1106.6035-1-9-11': 'Those operator are', '1106.6035-1-10-0': 'and the contact density operator [EQUATION]', '1106.6035-1-10-1': 'We use the shorthand [MATH].', '1106.6035-1-11-0': 'We proceed as follows: Firstly, we summarize the Feynman rules of the theory in Eq. ([REF]) and match the bilinear operators ([REF])-([REF]) using a one-particle state (for which the matrix element of the contact density vanishes).', '1106.6035-1-11-1': 'Then, we determine the Wilson coefficient of the contact using a two-particle state and the results of Sec. [REF].', '1106.6035-1-12-0': '## Feynman rules', '1106.6035-1-13-0': 'The matrix element of an operator can be expressed pictographically in terms of Feynman diagrams.', '1106.6035-1-13-1': 'This section lists the Feynman rules that are necessary to evaluate those diagrams.', '1106.6035-1-14-0': 'Every line is assigned an energy and a momentum where we impose momentum and energy conservation at each vertex.', '1106.6035-1-14-1': 'The external energy and momentum are constrained by the state.', '1106.6035-1-14-2': 'After applying the Feynman rules we integrate over each undetermined loop-momentum [MATH] and energy [MATH] using the measure [MATH].', '1106.6035-1-14-3': 'Each internal line of a particle with energy [MATH] and momentum [MATH] contributes a factor [MATH] to the integrand.', '1106.6035-1-14-4': 'The dashed blob (Fig. [REF]) denotes the two-particle scattering amplitude and gives a term [MATH], where [MATH] and [MATH] is the sum of the energies and the momenta of the incoming fermions, respectively.', '1106.6035-1-14-5': 'We have [EQUATION] in 3D and [EQUATION] in 2D.', '1106.6035-1-14-6': 'In the following we abbreviate [MATH].', '1106.6035-1-14-7': 'The strength of the interaction is given by the 2D and 3D scattering lengths [MATH] and [MATH], respectively.', '1106.6035-1-14-8': "[MATH] can be obtained by summing all ladder diagrams that contribute to the two-atom Green's function.", '1106.6035-1-14-9': 'The bare coupling constant [MATH] is related to [MATH] by the renormalization condition [EQUATION]', '1106.6035-1-14-10': 'This condition ensures that the 2-particle scattering amplitude reproduces the leading order in the effective range expansion of the quantum-mechanical s-wave scattering phase shift, [MATH] in 3D [CITATION] and [MATH] in 2D [CITATION], respectively.', '1106.6035-1-15-0': 'We denote the insertion of a current operator by a crossed circle attached to a wiggly line.', '1106.6035-1-15-1': 'A current insertion with energy [MATH] and momentum [MATH] that is attached to a fermion line with energy [MATH] and momentum [MATH] is depicted in Fig. [REF].', '1106.6035-1-15-2': 'It contributes a factor [EQUATION] to the integrand.', '1106.6035-1-15-3': 'The Feynman rules of the operators ([REF])-([REF]) are listed in Eqs. ([REF])-([REF]).', '1106.6035-1-16-0': '## 1-particle state', '1106.6035-1-17-0': 'In order to determine the Wilson coefficients of the bilinear operators, we evaluate the matrix element of [MATH] (Eq. ([REF])) with respect to a one-particle state with momentum [MATH] and energy [MATH].', '1106.6035-1-17-1': 'Note that we do not enforce the on-shell condition [MATH].', '1106.6035-1-17-2': 'The relevant Feynman diagrams are depicted in Fig. [REF].', '1106.6035-1-17-3': 'Their value is: [EQUATION] where [MATH].', '1106.6035-1-17-4': 'The same holds for [MATH].', '1106.6035-1-17-5': 'The expectation values of the bilinear operators are [EQUATION]', '1106.6035-1-17-6': 'They match the first terms in the expansion of Eq. ([REF]) in terms of [MATH] and [MATH].', '1106.6035-1-17-7': 'This yields the Wilson coefficients: [EQUATION] and [EQUATION]', '1106.6035-1-17-8': 'For an isotropic system we have [MATH], hence only the contraction [MATH] will be relevant in the following.', '1106.6035-1-17-9': 'Furthermore, we have [MATH] and can apply the Heisenberg equation of motion of the operator [MATH]: [MATH].', '1106.6035-1-17-10': 'The OPE allows us to express the expectation value of Eq. ([REF]) as follows: [EQUATION]', '1106.6035-1-17-11': 'Here, [MATH] and [MATH], where [MATH] denotes the Hamiltonian density, [EQUATION] which has the Wilson coefficient [EQUATION]', '1106.6035-1-17-12': 'From Eq. ([REF]) we see that the terms proportional to [MATH] and [MATH] only contribute a term [MATH] to the imaginary part of [MATH].', '1106.6035-1-17-13': 'Away from this one-particle peak the leading order is proportional to the contact.', '1106.6035-1-17-14': 'In the next section, we proceed to calculate the Wilson coefficient [MATH] of the contact operator.', '1106.6035-1-18-0': '## 2-particle state', '1106.6035-1-19-0': 'To determine the Wilson coefficient of the contact operator we choose a two-particle state where both particles have zero energy and momentum.', '1106.6035-1-19-1': 'This way, the matrix element of other two-body operators with higher scaling dimension than the contact, which involve additional derivatives of the field, are zero.', '1106.6035-1-19-2': 'We only have to subtract the matrix elements of the one-particle operators with respect to the two-particle state.', '1106.6035-1-19-3': 'We do, however, introduce additional infra-red divergences when calculating single diagrams which we need to subtract before we can evaluate the finite part.', '1106.6035-1-19-4': 'In particular, in two dimensions, this makes the calculations somewhat lengthy.', '1106.6035-1-19-5': 'Of course, the overall result is finite and the divergent parts cancel when summing all diagrams.', '1106.6035-1-20-0': 'We start by calculating the insertion of the bilinear operators on an internal line as shown in Fig. [REF].', '1106.6035-1-20-1': 'To obtain a finite expression, we introduce an IR and a UV cutoff [MATH] and [MATH], respectively: [EQUATION] and [EQUATION]', '1106.6035-1-20-2': 'The matrix element of the contact operator is [EQUATION] both in two and in three space-dimensions.', '1106.6035-1-20-3': 'To keep our notation concise, we abbreviate the matrix elements of the various operators as defined in the previous equations.', '1106.6035-1-21-0': 'The relevant diagrams that contribute to the matrix elements of Eq. ([REF]) are shown in Fig. [REF].', '1106.6035-1-21-1': 'The insertion of two current operators on an external line has already been considered in the previous section and is matched by bilinear insertion on external lines.', '1106.6035-1-21-2': 'Diags.', '1106.6035-1-21-3': '[REF] and [REF] respectively contribute to only the diagonal and off-diagonal components of [MATH] in spin-space, whereas the Diags.', '1106.6035-1-21-4': '[REF] contribute to all components.', '1106.6035-1-21-5': 'The matrix elements of the bilinear operators are diagonal in spin-space.', '1106.6035-1-21-6': 'Thus, they cancel the divergent parts in Diag.', '1106.6035-1-21-7': '[REF].', '1106.6035-1-21-8': 'Diags.', '1106.6035-1-21-9': '[REF] and [REF] are separately finite.', '1106.6035-1-21-10': 'In the following, we denote the contribution of Diags.', '1106.6035-1-21-11': '[REF], [REF] and [REF] to the Wilson coefficient of the contact by [MATH], [MATH] and [MATH], respectively, where we absorb the contribution of the bilinear matrix elements into the definition of [MATH].', '1106.6035-1-21-12': 'In short, we have: [EQUATION]', '1106.6035-1-21-13': 'Using the results of this section, we can give an explicit expression for the coefficient [MATH]: [EQUATION]', '1106.6035-1-21-14': 'The bilinear contributions cancel the IR-divergences of Diag.', '1106.6035-1-21-15': '[REF] and render the expression ([REF]) finite.', '1106.6035-1-21-16': 'In 3D and 2D, [REF] has a power-law infra-red divergence that is matched by the matrix element of the number operator [MATH].', '1106.6035-1-21-17': 'In 2D, there are additional logarithmic infra-red divergences in both the transverse as well as the longitudinal component.', '1106.6035-1-21-18': 'They are canceled by the expectation values of the operators [MATH] and [MATH].', '1106.6035-1-22-0': 'The sum of the diagrams in Fig. [REF] (divided by the matrix element of [MATH]) is equal to: [EQUATION]', '1106.6035-1-22-1': 'In two dimensions, the first term contains a logarithmic infra-red divergent part.', '1106.6035-1-22-2': 'It is canceled by the infra-red divergence in the two-particle scattering amplitude [MATH] (cf. Eq. ([REF])).', '1106.6035-1-23-0': 'The last contribution to [MATH] comes from the three diagrams in Fig. [REF].', '1106.6035-1-23-1': 'They only contribute to the longitudinal part of the response (cf. App.', '1106.6035-1-23-2': '[REF]): [EQUATION]', '1106.6035-1-23-3': 'As before, a logarithmic infra-red divergence in 2D is canceled between the two terms and the expression is manifestly finite.', '1106.6035-1-24-0': 'It is possible but cumbersome to evaluate the expressions ([REF]), ([REF]) and ([REF]) in closed analytical form in terms of elementary functions.', '1106.6035-1-24-1': 'For details of this calculation we refer to App.', '1106.6035-1-24-2': '[REF].', '1106.6035-1-24-3': 'The results of the computation are presented in the next section.', '1106.6035-1-25-0': '# RESULTS', '1106.6035-1-26-0': 'The first part of this section lists and discusses the current response functions of a Fermi gas in two and in three dimensions for large values of the arguments [MATH] and [MATH], where [MATH] is kept fixed.', '1106.6035-1-26-1': 'We consider both the response to a gauge field that couples symmetrically as well as antisymmetrically to the different fermion species.', '1106.6035-1-26-2': 'The latter corresponds to probing the response of the spin-antisymmetric current [MATH].', '1106.6035-1-26-3': 'We denote the spin-symmetric and antisymmetric response function by a superscript [MATH] and [MATH], respectively.', '1106.6035-1-26-4': 'In the remainder of this section we use the results of Sec. [REF] to calculate the dynamic structure factor in 3D and 2D.', '1106.6035-1-26-5': 'We also determine the asymptotic form of the spectral viscosities introduced in [CITATION] for large values of the frequency and comment on the derivation of sum rules using the OPE results.', '1106.6035-1-27-0': '## Response functions', '1106.6035-1-28-0': "As discussed in Sec. [REF], [MATH] (Eq. ([REF])) contributes to the diagonal and [MATH] (Eq. ([REF])) to the off-diagonal components of the contact's Wilson coefficient in spin space, whereas [MATH] (Eq. ([REF])) is independent of the spin indices.", '1106.6035-1-28-1': 'Hence, we obtain the spin-symmetric and antisymmetric response by summing those contributions as follows: [EQUATION] and [EQUATION] where [MATH] and [MATH] are the projectors onto the longitudinal and the transverse part, respectively.', '1106.6035-1-28-2': 'Remember that we have to shift [MATH] when evaluating the Wilson coefficients (cf. the definitions in Eqs. ([REF]) and ([REF])).', '1106.6035-1-28-3': 'As discussed in App.', '1106.6035-1-28-4': '[REF], of the above terms only [MATH] depends on the scattering length away from the one-particle peak.', '1106.6035-1-28-5': 'This means that except for the spin-symmetric longitudinal response function all response functions are independent of the scattering length in the asymptotic regime.', '1106.6035-1-29-0': 'The imaginary part of the response functions presented in the next two sections vanishes for [MATH].', '1106.6035-1-29-1': 'As already pointed out in [CITATION], this represents the two-particle threshold.', '1106.6035-1-29-2': 'This lower bound [MATH] follows from energy and momentum conservation if we consider the absorption of a probe with (large) momentum [MATH] and energy [MATH] by two particles in the medium.', '1106.6035-1-30-0': '### 3D', '1106.6035-1-31-0': 'The contributions of Eqs. ([REF])-([REF]) to Eqs. ([REF]) and ([REF]) can be inferred from the results in App.', '1106.6035-1-31-1': '[REF].', '1106.6035-1-31-2': 'We start by listing the response functions for the three-dimensional case.', '1106.6035-1-31-3': 'The spin-symmetric and antisymmetric current response are [EQUATION] and [EQUATION]', '1106.6035-1-31-4': 'Fig. [REF] shows plots of the transverse response function as a function of the scaling variable [MATH] (Fig. [REF]) and [MATH] (Fig. [REF]).', '1106.6035-1-31-5': 'They can be interpreted as plots at fixed energy and momentum, respectively.', '1106.6035-1-31-6': 'As mentioned earlier, the transverse components do not depend on the scattering length.', '1106.6035-1-32-0': 'Similarly, we derive the longitudinal response functions, [EQUATION] and [EQUATION]', '1106.6035-1-32-1': 'At unitarity, this result simplifies considerably: [EQUATION]', '1106.6035-1-32-2': 'Fig. [REF] shows plots of the longitudinal response function as a function of [MATH] (Fig. [REF]) and [MATH] (Fig. [REF]), where [MATH] is evaluated at unitarity.', '1106.6035-1-32-3': 'In Fig. [REF], we show plots of the longitudinal response function as a function of both [MATH] and the scattering length [MATH].', '1106.6035-1-32-4': 'As it has already been pointed out in [CITATION], the peak at [MATH] in Fig. [REF] is due to the strong interactions between the spin species at unitarity.', '1106.6035-1-32-5': 'This is also nicely illustrated in Fig. [REF], where we find a peak at [MATH] and a finite response otherwise.', '1106.6035-1-32-6': 'A detail of the peak is shown in Fig. [REF].', '1106.6035-1-33-0': '### 2D', '1106.6035-1-34-0': 'For the transverse response function we find the result [EQUATION] for the spin-symmetric and [EQUATION] for the spin-antisymmetric current.', '1106.6035-1-34-1': 'Both functions are plotted in Fig. [REF], as a function of [MATH] (Fig. [REF]) and [MATH] (Fig. [REF]), as before.', '1106.6035-1-35-0': 'As for the three-dimensional case, the spin-symmetric longitudinal response function depends on the two-dimensional scattering length [MATH]: [EQUATION]', '1106.6035-1-35-1': 'The spin-antisymmetric response takes a much simpler form: [EQUATION]', '1106.6035-1-35-2': 'Fig. [REF] shows plots of the longitudinal response function as a function of [MATH] (Fig. [REF]) and [MATH] (Fig. [REF]).', '1106.6035-1-35-3': 'Fig. [REF] shows a plot of the longitudinal response function as function of [MATH] and the scattering length [MATH].', '1106.6035-1-36-0': 'It is interesting to note that for infinite scattering length [MATH] there is also a peak at [MATH], despite the fact that the two-dimensional Fermi gas is not strongly interacting.', '1106.6035-1-36-1': 'This reflects the fact that the scattering between the particles is enhanced in this limit, as it can be seen from the 2-particle scattering amplitude in Eq. ([REF]).', '1106.6035-1-36-2': 'The large value of the scattering length serves to balance the small energy above the two-particle threshold [MATH].', '1106.6035-1-37-0': 'When comparing Figs. [REF] and [REF] to [REF] and [REF], we see that the width of the single-particle peak is much larger in 2D than it is in 3D.', '1106.6035-1-37-1': 'In 3D, the decay of the Wilson coefficients away from the pole at [MATH] has a logarithmic and a power-law behavior [MATH] for the transverse and the longitudinal response, respectively, as opposed to higher power-law decays [MATH] and [MATH] in 2D.', '1106.6035-1-37-2': 'This is to be expected since we integrate over one space-component less in 2D.', '1106.6035-1-38-0': '## Dynamic structure factor', '1106.6035-1-39-0': 'The Ward identity that follows from the U(1)-symmetry of the Lagrangian in Eq. ([REF]) gives the first hydrodynamic equation describing the conservation of mass, [EQUATION] where [MATH].', '1106.6035-1-39-1': 'If we substitute the Fourier transform of Eq. ([REF]) in the spectral decomposition for the imaginary part of the current response function, [EQUATION] where [MATH] and [MATH] are the eigenvalues of the Hamiltonian of the system, we can relate the longitudinal part of the spin-symmetric response to the density response [MATH] (where [MATH] is the mass density) and thus deduce an expression for the dynamic structure factor: [EQUATION]', '1106.6035-1-39-2': 'Comparing this with Eqs. ([REF]) and ([REF]), we obtain the asymptotic behavior of the dynamic structure factor.', '1106.6035-1-39-3': 'In the three-dimensional case, the result coincides with the expression at unitarity [CITATION], cf. Eq. ([REF]).', '1106.6035-1-39-4': 'The general behavior of the structure factor is the same as for the longitudinal response function, Figs. [REF] and [REF], and we refrain from plotting it again.', '1106.6035-1-40-0': 'When expanding the expression in Eqs. ([REF]) and ([REF]) to leading order in [MATH], we get the large frequency tail of the dynamic structure factor for [MATH]: [EQUATION]', '1106.6035-1-40-1': 'At unitarity, this agrees with the result by [CITATION] up to a factor of [MATH].', '1106.6035-1-40-2': 'The same frequency tail was derived by the authors of [CITATION] and [CITATION], with whom our result agrees.', '1106.6035-1-40-3': 'The authors of [CITATION] also calculated the correction to the tail at leading order in [MATH].', '1106.6035-1-40-4': 'Again, our result is in agreement with theirs.', '1106.6035-1-40-5': 'Our calculation generalizes the cited results to finite scattering length.', '1106.6035-1-40-6': 'For the two-dimensional Fermi gas, we find no dependence on the scattering length in the high-frequency tail: [EQUATION]', '1106.6035-1-41-0': '## Spectral viscosities', '1106.6035-1-42-0': 'The authors of [CITATION] derived Kubo formulae for frequency-dependent generalizations of the bulk and the shear viscosity in three dimensions.', '1106.6035-1-42-1': 'The starting point of their derivation is the Euler equation, [EQUATION] where [MATH] is the stress tensor for a viscous fluid in [MATH] space dimensions: [EQUATION] [MATH] denotes the pressure, [MATH] the velocity, [MATH] the density and [EQUATION] is the viscous stress tensor [CITATION].', '1106.6035-1-42-2': 'Retracing the derivation in [CITATION], it is not difficult to generalize their result to two space dimensions.', '1106.6035-1-42-3': 'We obtain the relations [EQUATION] and [EQUATION]', '1106.6035-1-42-4': 'The leading order of [MATH] and [MATH] at large frequency is proportional to the contact.', '1106.6035-1-42-5': 'Comparing this with Eqs. ([REF]), ([REF]), ([REF]) and ([REF]), we obtain the large-frequency tails of the bulk and the shear viscosity: [EQUATION] and [EQUATION] in three dimensions and [EQUATION] and [EQUATION] in two dimensions.', '1106.6035-1-42-6': 'The high-frequency tails have been considered before for a three-dimensional gas near unitarity [CITATION].', '1106.6035-1-42-7': 'Eq. ([REF]) agrees with [CITATION] and agrees with [CITATION] up to the same factor of [MATH] mentioned in the previous section.', '1106.6035-1-42-8': 'We obtain the high-frequency tail of the 3D-bulk viscosity ([REF]) for an arbitrary value of the scattering length.', '1106.6035-1-42-9': 'It vanishes at unitarity, in agreement with [CITATION].', '1106.6035-1-43-0': '## Viscosity sum rules', '1106.6035-1-44-0': 'By integrating over the frequency of the spectral functions, the asymptotic behavior of the correlation functions can be used to derive sum rules.', '1106.6035-1-44-1': 'On the one hand, such sum rules provide constraints that serve as a consistency check for approximate methods [CITATION].', '1106.6035-1-44-2': 'On the other hand, they are often more closely related to experimental results, since it can be difficult to probe the strictly asymptotic region of the response functions.', '1106.6035-1-44-3': 'The simplest way to obtain sum rules is to calculate the lowest momenta of the spectral functions.', '1106.6035-1-44-4': 'Additional sum rules can be deduced by introducing some form of UV-cutoff that shifts the integration region to lower frequencies [CITATION].', '1106.6035-1-45-0': 'For completeness, we calculate the lowest moment of the bulk and the shear viscosity and rederive the results of Ref. [CITATION] using the OPE.', '1106.6035-1-45-1': 'To obtain a finite expression for the shear viscosity sum rule, we subtract the high-frequency tail in Eq. ([REF]) from the integrand.', '1106.6035-1-45-2': 'This yields: [EQUATION] and [EQUATION]', '1106.6035-1-45-3': 'The shear viscosity sum rule ([REF]) agrees with the result in [CITATION] and agrees with [CITATION] up to the prefactors.', '1106.6035-1-46-0': 'The OPE alone is not sufficient to constrain the bulk viscosity sum rule ([REF]).', '1106.6035-1-46-1': 'As it was pointed out in Ref. [CITATION], this is due to an ordering ambiguity when taking the zero-momentum limit [MATH] and performing the [MATH]-integration (we take the limit after performing the integration).', '1106.6035-1-46-2': 'The term in the second line of Eq. ([REF]) is evaluated in the zero-momentum and zero-frequency limit where the OPE cannot provide an insight.', '1106.6035-1-46-3': 'It was evaluated explicitly in [CITATION] using hydrodynamic expressions for the current response.', '1106.6035-1-46-4': 'Combining the result in [CITATION] with Eq. ([REF]) yields the sum rule [EQUATION] where the subscript [MATH] indicates that we take the derivative of [MATH] with respect to [MATH] at fixed entropy.', '1106.6035-1-47-0': '# SUMMARY', '1106.6035-1-48-0': 'In this article, we studied the current response function of a Fermi gas and calculated the first terms of its operator product expansion.', '1106.6035-1-48-1': 'We considered the limit [MATH] away from the one-particle peak [MATH] with [MATH] held fixed.', '1106.6035-1-48-2': 'We found that to leading order the response is proportional to the contact whose Wilson coefficient we determined.', '1106.6035-1-48-3': 'We calculated various response functions for a wide range of parameters: we derived the longitudinal and transverse components of the spin-symmetric and antisymmetric response function in 3D and 2D for arbitrary values of the scattering length.', '1106.6035-1-49-0': 'Using those results, we were able to obtain the asymptotic form of the dynamic structure factor in 3D and 2D and thus to generalize and to extend many of the previous results for 3D systems at large scattering length [CITATION].', '1106.6035-1-49-1': 'To the best of our knowledge, this is also the first work to rederive the high-frequency tails of the bulk and shear viscosity considered in [CITATION] using the OPE and the first work to calculate their 2D counterparts.', '1106.6035-1-49-2': 'Furthermore, we could use the OPE to calculate some sum rules that constrain the spectral viscosities in 3D.', '1106.6035-1-50-0': 'As illustrated by this work, the operator product expansion provides a powerful tool to derive universal relations.', '1106.6035-1-50-1': 'It offers a systematic way to analyze the short-time and distance structure of a theory, which can be fully characterized by performing calculations involving few-body states.'}

{'1106.6035-2-0-0': 'We apply the operator-product expansion to determine the asymptotic form of the current response of a Fermi gas in two and three dimensions.', '1106.6035-2-0-1': 'The leading-order term away from the one-particle peak is proportional to a quantity known as the contact, the coefficient of which is determined exactly.', '1106.6035-2-0-2': 'We also calculate the dynamic structure factor and the high-frequency tails of the spectral viscosities as a function of the scattering length.', '1106.6035-2-0-3': 'Our results are used to derive certain sum rules for the viscosities.', '1106.6035-2-1-0': '# INTRODUCTION', '1106.6035-2-2-0': 'Two-component fermionic quantum gases belong to the most versatile systems that can be used to study the many-body physics of fermions [CITATION].', '1106.6035-2-2-1': 'The simplest Lagrangian to describe such a system models the [MATH]-wave interaction between the constituents by a contact interaction (see, for example, Ref. [CITATION]): [EQUATION] where a summation over spin indices [MATH] is implied.', '1106.6035-2-2-2': 'In experiments with cold fermionic atoms, it is possible to confine the atomic cloud in two- and three-dimensional traps and also to tune the interaction strength to a regime where the components are strongly interacting.', '1106.6035-2-2-3': 'The theoretical analysis of a Fermi gas in a strongly interacting limit is intricate since many theoretical methods rely on a perturbative expansion or neglect fluctuations and are thus not applicable.', '1106.6035-2-3-0': 'Over the past few years, a lot of work has been stimulated by the observation that some properties of a Fermi gas can be described by universal relations.', '1106.6035-2-3-1': "They are called universal because they hold over a wide range of the system's parameters such as the temperature, the density, the precise form or the strength of the interaction, and so on.", '1106.6035-2-3-2': 'A first set of such relations was presented by Tan [CITATION], who among other things derived expressions for the momentum distribution, the energy, and various other thermodynamic quantities, and are hence often called Tan relations.', '1106.6035-2-3-3': 'The scale of the Tan relations is set by a quantity known as the contact [MATH], which is determined by the many-particle properties of the system.', '1106.6035-2-3-4': 'It is defined as [MATH], where [MATH] is the contact density operator and [MATH] denotes the number of space dimensions.', '1106.6035-2-3-5': 'An operational definition of the contact can be given as the magnitude of the large-momentum tail of the gas: [MATH].', '1106.6035-2-3-6': 'Apart from the dependence on the contact, the universal relations can be inferred from the few-body physics and can be calculated in closed analytic form in many cases.', '1106.6035-2-3-7': 'Only the value of the contact has to be taken from numerical calculations or experiments.', '1106.6035-2-3-8': 'Some of the Tan relations have been tested experimentally, and good agreement with the theoretical predictions has been found [CITATION].', '1106.6035-2-3-9': 'Since their proposal, many other exact relations for fermionic as well as bosonic systems have been derived (cf., for example, Refs. [CITATION]).', '1106.6035-2-3-10': 'An overview of various universal relations and recent experimental developments can be found in Ref. [CITATION].', '1106.6035-2-4-0': 'A field-theoretical derivation of universal relations was put forward by Braaten, Kang, and Platter [CITATION], who used the operator-product expansion (OPE) to rederive the Tan relations [CITATION].', '1106.6035-2-4-1': 'The OPE expresses the product of two local operators [MATH] and [MATH] at different points in space and time in terms of local operators [MATH], where [MATH] denotes a set of indices [CITATION]: [EQUATION]', '1106.6035-2-4-2': 'Note that, on the right-hand side, the dependence on the separation [MATH] is carried by the Wilson coefficients [MATH], which can be calculated exactly for simple operators.', '1106.6035-2-4-3': 'If an observable can be expressed as the expectation value of an operator product involving operators at different space-time points, we can use the OPE to determine its leading-order behavior for small [MATH] and [MATH] (or for large values of momentum [MATH] and energy [MATH] for the Fourier transform).', '1106.6035-2-4-4': 'The advantage of the OPE is that it provides a powerful framework to derive additional universal relations.', '1106.6035-2-4-5': 'This was exploited by the authors of [CITATION] to derive universal results for radio-frequency spectroscopy measurements [CITATION].', '1106.6035-2-4-6': 'Son and Thompson studied the asymptotic behavior of the dynamic structure factor [MATH] at unitarity at large [MATH] and [MATH] as well as the large-frequency tail of [MATH] [CITATION].', '1106.6035-2-4-7': 'The result for the large-frequency tail in [CITATION] was augmented by Goldberger and Rothstein, who calculated the leading-order correction due to finite scattering length [CITATION].', '1106.6035-2-4-8': 'Some of the above results have also been obtained by other authors using different methods [CITATION].', '1106.6035-2-4-9': 'While most of the above work was carried out for a three-dimensional system at or close to unitarity, it is possible to derive universal relations for any strength of the interaction or any other dimension in the same way.', '1106.6035-2-5-0': 'In this work, we apply the OPE to derive the current response function of a Fermi gas in two and in three dimensions.', '1106.6035-2-5-1': 'It is defined as [EQUATION] where [MATH] is the current operator.', '1106.6035-2-5-2': 'The Fourier transform of Eq. ([REF]) can be split into a transverse and a longitudinal part: [EQUATION]', '1106.6035-2-5-3': 'The longitudinal part is related to the superfluid fraction and the transverse part to the normal fraction of the gas.', '1106.6035-2-5-4': 'Thus, the current response function is a central quantity in the study of superfluids.', '1106.6035-2-5-5': 'The longitudinal component is related to the structure factor and can be determined with methods that measure the density response, such as two-photon Bragg spectroscopy [CITATION].', '1106.6035-2-6-0': 'Recently, there have been several proposals to probe the transverse component of the current response using artificial gauge fields [CITATION].', '1106.6035-2-6-1': 'Artificial gauge fields are induced by either rotating the atomic cloud or placing it in a specific laser setup.', '1106.6035-2-6-2': 'In the latter case, incident laser beams couple the ground state of the atoms to an excited state.', '1106.6035-2-6-3': 'The effective Lagrangian of such a system describes charged particles coupled to a U(1) gauge field.', '1106.6035-2-6-4': 'The amplitude of this gauge field typically depends on the spatial or temporal variation of the Rabi frequency or the detuning of the laser from the resonance frequency.', '1106.6035-2-6-5': 'For example, a local detuning of the laser field could induce a rapidly oscillating gauge field.', '1106.6035-2-6-6': 'For further references, see [CITATION].', '1106.6035-2-6-7': 'One way to measure the imaginary part of the current response is to determine how much energy [MATH] is absorbed by the system if the gauge field is switched on for a short time: [EQUATION]', '1106.6035-2-6-8': 'For a two-dimensional gas, there also exists a proposal to determine the current response optically, by measuring the phase shift of a laser beam that crosses the atomic cloud [CITATION].', '1106.6035-2-6-9': 'While artificial gauge fields have only been realized for Bose gases so far there is no reason why this should not also be possible for fermionic systems in the near future.', '1106.6035-2-6-10': 'It is in anticipation of such experiments that we calculate the current response.', '1106.6035-2-6-11': 'In particular, since it should be possible to engineer gauge fields that couple differently to each spin species, we consider both the spin-symmetric and antisymmetric current response [MATH], where a summation over spin indices is implied [corresponding to gauge fields that couple symmetrically and antisymmetrically to the current: [MATH]].', '1106.6035-2-7-0': 'This paper is structured as follows: In Sec. [REF], we describe how to calculate the Wilson coefficients of the current correlator.', '1106.6035-2-7-1': 'This is done using a matching procedure where the matrix element of Eq. ([REF]) is evaluated with respect to selected few-body states.', '1106.6035-2-7-2': 'We choose a one-particle state to determine the Wilson coefficients of the bilinear operators (Sec. [REF]) and then a two-particle state for the contact operator (Sec. [REF]).', '1106.6035-2-7-3': 'Section [REF] presents the results of this calculation.', '1106.6035-2-7-4': 'Expressions for both the spin-symmetric and the antisymmetric current response are presented and discussed in Sec. [REF].', '1106.6035-2-7-5': 'The U(1) symmetry of the system implies a Ward identity that relates the longitudinal part of the current correlator to the dynamic structure factor.', '1106.6035-2-7-6': 'We use this to calculate the asymptotic form of the dynamic structure factor in two and in three dimensions for any value of the scattering length in Sec. [REF].', '1106.6035-2-7-7': 'In [CITATION], Taylor and Randeria derived expressions for the large-frequency tail of the spectral shear and bulk viscosity [MATH] and [MATH].', '1106.6035-2-7-8': 'This paper reproduces their results using a field-theoretical framework and generalizes them to finite scattering length.', '1106.6035-2-7-9': 'We are also able to provide the viscosity tails for a two-dimensional Fermi gas.', '1106.6035-2-7-10': 'We conclude with a summary in Sec. [REF].', '1106.6035-2-7-11': 'The paper contains an appendix that presents details of the calculations in Sec. [REF].', '1106.6035-2-8-0': '# OPERATOR PRODUCT EXPANSION', '1106.6035-2-9-0': 'While we would like to obtain the OPE for the retarded response function [MATH], it is more convenient to calculate the time-ordered response, which coincides with the retarded response up to a correction of order [MATH] [CITATION]: [EQUATION] [MATH] denotes the time-ordered product.', '1106.6035-2-9-1': 'The [MATH] term in the Fourier transform has been included to shift the poles in the resulting expressions of the response functions [CITATION].', '1106.6035-2-9-2': 'Using the time-ordered response allows us to apply standard diagrammatic methods to calculate the Wilson coefficients.', '1106.6035-2-9-3': 'In this section, we derive the OPE of the operator [EQUATION]', '1106.6035-2-9-4': 'The OPE is an operator identity, that is, it holds if we take the expectation value of Eq. ([REF]) with respect to any state.', '1106.6035-2-9-5': 'Hence, in order to determine the Wilson coefficient [MATH] of an operator [MATH], we can choose a simple few-body state for which [MATH] and match the expectation values on both sides of Eq. ([REF]).', '1106.6035-2-9-6': 'Inserting Eq. ([REF]) in Eq. ([REF]) yields [EQUATION] where [MATH] denotes the scaling dimension of the operator [MATH] as defined by [MATH].', '1106.6035-2-9-7': 'We suppress the dependence of the Wilson coefficients on the spin indices on the right-hand side of Eq. ([REF]).', '1106.6035-2-9-8': 'The operators [MATH] can be composed of the field operators [MATH] and their adjoints [MATH] as well as derivative operators acting on those fields.', '1106.6035-2-9-9': 'We will only consider operators that conserve the particle number, i.e., contain the same number of fields and their adjoints, and that are symmetric in the spin indices.', '1106.6035-2-9-10': 'Equation ([REF]) shows that the high-frequency behavior of the current response is governed by the operators with lowest scaling dimension.', '1106.6035-2-9-11': 'Those operators are', '1106.6035-2-10-0': 'and the contact density operator [EQUATION]', '1106.6035-2-10-1': 'We use the shorthand [MATH].', '1106.6035-2-11-0': 'We proceed as follows: First, we summarize the Feynman rules of the theory in Eq. ([REF]) and match the bilinear operators ([REF])-([REF]) using a one-particle state (for which the matrix element of the contact density vanishes).', '1106.6035-2-11-1': 'Then, we determine the Wilson coefficient of the contact using a two-particle state and the results of Sec. [REF].', '1106.6035-2-12-0': '## Feynman rules', '1106.6035-2-13-0': 'The matrix element of an operator can be expressed pictographically in terms of Feynman diagrams.', '1106.6035-2-13-1': 'This section lists the Feynman rules that are necessary to evaluate those diagrams.', '1106.6035-2-14-0': 'Every line is assigned an energy and a momentum where we impose momentum and energy conservation at each vertex.', '1106.6035-2-14-1': 'The external energy and momentum are constrained by the state.', '1106.6035-2-14-2': 'After applying the Feynman rules, we integrate over each undetermined loop momentum [MATH] and energy [MATH] using the measure [MATH].', '1106.6035-2-14-3': 'Each internal line of a particle with energy [MATH] and momentum [MATH] contributes a factor [MATH] to the integrand.', '1106.6035-2-14-4': 'The dashed blob [Fig. [REF]] denotes the two-particle scattering amplitude and gives a term [MATH], where [MATH] and [MATH] are the sum of the energies and the momenta of the incoming fermions, respectively.', '1106.6035-2-14-5': 'We have [EQUATION] in three dimensions (3D) and [EQUATION] in two dimensions (2D).', '1106.6035-2-14-6': 'In the following, we abbreviate [MATH].', '1106.6035-2-14-7': 'The strength of the interaction is given by the 2D and 3D scattering lengths [MATH] and [MATH], respectively.', '1106.6035-2-14-8': "[MATH] can be obtained by summing all ladder diagrams that contribute to the two-atom Green's function.", '1106.6035-2-14-9': 'The bare coupling constant [MATH] is related to [MATH] by the renormalization condition [EQUATION]', '1106.6035-2-14-10': 'This condition ensures that the two-particle scattering amplitude reproduces the leading order in the effective range expansion of the quantum-mechanical [MATH]-wave scattering phase shift, [MATH] in 3D [CITATION] and [MATH] in 2D [CITATION], respectively.', '1106.6035-2-15-0': 'We denote the insertion of a current operator by a crossed circle attached to a wiggly line.', '1106.6035-2-15-1': 'A current insertion with energy [MATH] and momentum [MATH] that is attached to a fermion line with energy [MATH] and momentum [MATH] is depicted in Fig. [REF].', '1106.6035-2-15-2': 'It contributes a factor [EQUATION] to the integrand.', '1106.6035-2-15-3': 'The Feynman rules of the operators ([REF])-([REF]) are listed in Eqs. ([REF])-([REF]).', '1106.6035-2-16-0': '## One-particle state', '1106.6035-2-17-0': 'In order to determine the Wilson coefficients of the bilinear operators, we evaluate the matrix element of [MATH] [Eq. ([REF])] with respect to a one-particle state with momentum [MATH] and energy [MATH].', '1106.6035-2-17-1': 'Note that we do not enforce the on-shell condition [MATH].', '1106.6035-2-17-2': 'The relevant Feynman diagrams are depicted in Fig. [REF].', '1106.6035-2-17-3': 'Their value is: [EQUATION] where [MATH].', '1106.6035-2-17-4': 'The same holds for [MATH].', '1106.6035-2-17-5': 'The expectation values of the bilinear operators are [EQUATION]', '1106.6035-2-17-6': 'They match the first terms in the expansion of Eq. ([REF]) in terms of [MATH] and [MATH].', '1106.6035-2-17-7': 'This yields the Wilson coefficients: [EQUATION] and [EQUATION]', '1106.6035-2-17-8': 'For an isotropic system, we have [MATH], hence, only the contraction [MATH] will be relevant in the following.', '1106.6035-2-17-9': 'Furthermore, we have [MATH] and can apply the Heisenberg equation of motion of the operator [MATH]: [MATH].', '1106.6035-2-17-10': 'The OPE allows us to express the expectation value of Eq. ([REF]) as follows: [EQUATION]', '1106.6035-2-17-11': 'Here, [MATH] and [MATH], where [MATH] denotes the Hamiltonian density [EQUATION] which has the Wilson coefficient [EQUATION]', '1106.6035-2-17-12': 'From Eq. ([REF]), we see that the terms proportional to [MATH] and [MATH] only contribute a term [MATH] to the imaginary part of [MATH].', '1106.6035-2-17-13': 'Away from this one-particle peak, the leading order is proportional to the contact.', '1106.6035-2-17-14': 'In the next section, we proceed to calculate the Wilson coefficient [MATH] of the contact operator.', '1106.6035-2-18-0': '## Two-particle state', '1106.6035-2-19-0': 'To determine the Wilson coefficient of the contact operator, we choose a two-particle state where both particles have zero energy and momentum.', '1106.6035-2-19-1': 'This way, the matrix element of other two-body operators with higher scaling dimension than the contact, which involve additional derivatives of the field, are zero.', '1106.6035-2-19-2': 'We only have to subtract the matrix elements of the one-particle operators with respect to the two-particle state.', '1106.6035-2-19-3': 'We do, however, introduce additional infrared divergences when calculating single diagrams, which we need to subtract before we can evaluate the finite part.', '1106.6035-2-19-4': 'In particular, in two dimensions, this makes the calculations somewhat lengthy.', '1106.6035-2-19-5': 'Of course, the overall result is finite and the divergent parts cancel when summing all diagrams.', '1106.6035-2-20-0': 'We start by calculating the insertion of the bilinear operators on an internal line as shown in Fig. [REF].', '1106.6035-2-20-1': 'To obtain a finite expression, we introduce an IR and a UV cutoff [MATH] and [MATH], respectively: [EQUATION] and [EQUATION]', '1106.6035-2-20-2': 'The matrix element of the contact operator is [EQUATION] both in two and in three space dimensions.', '1106.6035-2-20-3': 'To keep our notation concise, we abbreviate the matrix elements of the various operators as defined in the previous equations.', '1106.6035-2-21-0': 'The relevant diagrams that contribute to the matrix elements of Eq. ([REF]) are shown in Fig. [REF].', '1106.6035-2-21-1': 'The insertion of two current operators on an external line has already been considered in the previous section and is matched by bilinear insertion on external lines.', '1106.6035-2-21-2': 'Figures [REF] and [REF], respectively, contribute to only the diagonal and off-diagonal components of [MATH] in spin space, whereas Fig. [REF] contributes to all components.', '1106.6035-2-21-3': 'The matrix elements of the bilinear operators are diagonal in spin space.', '1106.6035-2-21-4': 'Thus, they cancel the divergent parts in Fig. [REF].', '1106.6035-2-21-5': 'Figures [REF] and [REF] are separately finite.', '1106.6035-2-21-6': 'In the following, we denote the contribution of Figs. [REF], [REF] and [REF] to the Wilson coefficient of the contact by [MATH], [MATH] and [MATH], respectively, where we absorb the contribution of the bilinear matrix elements into the definition of [MATH].', '1106.6035-2-21-7': 'In short, we have [EQUATION]', '1106.6035-2-21-8': 'Using the results of this section, we can give an explicit expression for the coefficient [MATH]: [EQUATION]', '1106.6035-2-21-9': 'The bilinear contributions cancel the IR divergences of Fig. [REF] and render the expression ([REF]) finite.', '1106.6035-2-21-10': 'In 3D and 2D, Fig. [REF] has a power-law infrared divergence that is matched by the matrix element of the number operator [MATH].', '1106.6035-2-21-11': 'In 2D, there are additional logarithmic infrared divergences in both the transverse as well as the longitudinal components.', '1106.6035-2-21-12': 'They are canceled by the expectation values of the operators [MATH] and [MATH].', '1106.6035-2-22-0': 'The sum of the diagrams in Fig. [REF] (divided by the matrix element of [MATH]) is equal to [EQUATION]', '1106.6035-2-22-1': 'In two dimensions, the first term contains a logarithmic infrared-divergent part.', '1106.6035-2-22-2': 'It is canceled by the infrared divergence in the two-particle scattering amplitude [MATH] [cf. Eq. ([REF])].', '1106.6035-2-23-0': 'The last contribution to [MATH] comes from the three diagrams in Fig. [REF].', '1106.6035-2-23-1': 'They only contribute to the longitudinal part of the response (cf. the Appendix): [EQUATION]', '1106.6035-2-23-2': 'As before, a logarithmic infrared divergence in 2D is canceled between the two terms and the expression is manifestly finite.', '1106.6035-2-24-0': 'It is possible but cumbersome to evaluate the expressions ([REF]), ([REF]), and ([REF]) in closed analytical form in terms of elementary functions.', '1106.6035-2-24-1': 'For details of this calculation we refer to the Appendix.', '1106.6035-2-24-2': 'The results of the computation are presented in the next section.', '1106.6035-2-25-0': '# RESULTS', '1106.6035-2-26-0': 'The first part of this section lists and discusses the current response functions of a Fermi gas in two and in three dimensions for large values of the arguments [MATH] and [MATH], where [MATH] and [MATH] are kept fixed.', '1106.6035-2-26-1': 'We consider both the response to a gauge field that couples symmetrically as well as antisymmetrically to the different fermion species.', '1106.6035-2-26-2': 'The latter corresponds to probing the response of the spin-antisymmetric current [MATH].', '1106.6035-2-26-3': 'We denote the spin-symmetric and antisymmetric response function by a superscript [MATH] and [MATH], respectively.', '1106.6035-2-26-4': 'In the remainder of this section, we use the results of Sec. [REF] to calculate the dynamic structure factor in 3D and 2D.', '1106.6035-2-26-5': 'We also determine the asymptotic form of the spectral viscosities introduced in [CITATION] for large values of the frequency, and comment on the derivation of sum rules using the OPE results.', '1106.6035-2-27-0': '## Response functions', '1106.6035-2-28-0': "As discussed in Sec. [REF], [MATH] [Eq. ([REF])] contributes to the diagonal and [MATH] [Eq. ([REF])] to the off-diagonal components of the contact's Wilson coefficient in spin space, whereas [MATH] [Eq. ([REF])] is independent of the spin indices.", '1106.6035-2-28-1': 'Hence, we obtain the spin-symmetric and antisymmetric response by summing those contributions as follows: [EQUATION] and [EQUATION] where [MATH] and [MATH] are the projectors onto the longitudinal and the transverse parts, respectively.', '1106.6035-2-28-2': 'Remember that we have to shift [MATH] when evaluating the Wilson coefficients [cf. the definitions in Eqs. ([REF]) and ([REF])].', '1106.6035-2-28-3': 'As discussed in the Appendix, of the above terms, only [MATH] depends on the scattering length away from the one-particle peak.', '1106.6035-2-28-4': 'Barring the dependence of the contact on [MATH], this means that, except for the spin-symmetric longitudinal response function, all response functions are independent of the scattering length in the asymptotic regime.', '1106.6035-2-29-0': 'The imaginary part of the response functions presented in the next two sections vanishes for [MATH].', '1106.6035-2-29-1': 'As already pointed out in [CITATION], this represents the two-particle threshold.', '1106.6035-2-29-2': 'This lower bound [MATH] follows from energy and momentum conservation if we consider the absorption of a probe with (large) momentum [MATH] and energy [MATH] by two particles in the medium.', '1106.6035-2-30-0': '### 3D', '1106.6035-2-31-0': 'The contributions of Eqs. ([REF])-([REF]) to Eqs. ([REF]) and ([REF]) can be inferred from the results in the Appendix.', '1106.6035-2-31-1': 'We start by listing the response functions for the three-dimensional case.', '1106.6035-2-31-2': 'The spin-symmetric and antisymmetric current responses are [EQUATION] and [EQUATION]', '1106.6035-2-31-3': 'Figure [REF] shows plots of the transverse response function as a function of the scaling variable [MATH] [Fig. [REF]] and [MATH] [Fig. [REF]].', '1106.6035-2-31-4': 'They can be interpreted as plots at fixed energy and momentum, respectively.', '1106.6035-2-31-5': 'As mentioned earlier, the transverse components do not depend on the scattering length.', '1106.6035-2-32-0': 'Similarly, we derive the longitudinal response functions [EQUATION] and [EQUATION]', '1106.6035-2-32-1': 'At unitarity, this result simplifies considerably: [EQUATION]', '1106.6035-2-32-2': 'Figure [REF] shows plots of the longitudinal response function as a function of [MATH] [Fig. [REF]] and [MATH] [Fig. [REF]], where [MATH] is evaluated at unitarity.', '1106.6035-2-32-3': 'In Fig. [REF], we show plots of the longitudinal response function as a function of both [MATH] and the scattering length [MATH].', '1106.6035-2-32-4': 'As it has already been pointed out in [CITATION], the peak at [MATH] in Fig. [REF] is due to the strong interactions between the spin species at unitarity.', '1106.6035-2-32-5': 'This is also nicely illustrated in Fig. [REF], where we find a peak at [MATH] and a finite response otherwise.', '1106.6035-2-32-6': 'A detail of the peak is shown in Fig. [REF].', '1106.6035-2-33-0': '### 2D', '1106.6035-2-34-0': 'For the transverse response function, we find the result [EQUATION] for the spin-symmetric and [EQUATION] for the spin-antisymmetric current.', '1106.6035-2-34-1': 'Both functions are plotted in Fig. [REF], as a function of [MATH] [Fig. [REF]] and [MATH] [Fig. [REF]], as before.', '1106.6035-2-35-0': 'As for the three-dimensional case, the spin-symmetric longitudinal response function depends on the two-dimensional scattering length [MATH]: [EQUATION]', '1106.6035-2-35-1': 'The spin-antisymmetric response takes a much simpler form [EQUATION]', '1106.6035-2-35-2': 'Figure [REF] shows plots of the longitudinal response function as a function of [MATH] [Fig. [REF]] and [MATH] [Fig. [REF]].', '1106.6035-2-35-3': 'Figure [REF] shows a plot of the longitudinal response function as a function of [MATH] and the scattering length [MATH].', '1106.6035-2-36-0': 'It is interesting to note that, for infinite scattering length [MATH], there is also a peak at [MATH], despite the fact that the two-dimensional Fermi gas is not strongly interacting.', '1106.6035-2-36-1': 'This reflects the fact that the scattering between the particles is enhanced in this limit, as it can be seen from the two-particle scattering amplitude in Eq. ([REF]).', '1106.6035-2-36-2': 'The large value of the scattering length serves to balance the small energy above the two-particle threshold [MATH].', '1106.6035-2-37-0': 'When comparing Figs. [REF] and [REF] to [REF] and [REF], we see that the width of the single-particle peak is much larger in 2D than it is in 3D.', '1106.6035-2-37-1': 'In 3D, the decay of the Wilson coefficients away from the pole at [MATH] has a logarithmic and a power-law behavior [MATH] for the transverse and the longitudinal responses, respectively, as opposed to higher power-law decays [MATH] and [MATH] in 2D.', '1106.6035-2-37-2': 'This is to be expected since we integrate over one space component less in 2D.', '1106.6035-2-38-0': '## Dynamic structure factor', '1106.6035-2-39-0': 'The Ward identity that follows from the U(1) symmetry of the Lagrangian in Eq. ([REF]) gives the first hydrodynamic equation describing the conservation of mass, [EQUATION] where [MATH].', '1106.6035-2-39-1': 'If we substitute the Fourier transform of Eq. ([REF]) in the spectral decomposition for the imaginary part of the current response function [EQUATION] where [MATH] and [MATH] are the eigenvalues of the Hamiltonian of the system, we can relate the longitudinal part of the spin-symmetric response to the density response [MATH] (where [MATH] is the mass density) and deduce an expression for the dynamic structure factor [EQUATION]', '1106.6035-2-39-2': 'Comparing this with Eqs. ([REF]) and ([REF]), we obtain the asymptotic behavior of the dynamic structure factor.', '1106.6035-2-39-3': 'In the three-dimensional case, the result coincides with the expression at unitarity [CITATION] [cf. Eq. ([REF])].', '1106.6035-2-39-4': 'The general behavior of the structure factor is the same as for the longitudinal response function, Figs. [REF] and [REF], and we refrain from plotting it again.', '1106.6035-2-40-0': 'When expanding the expression in Eqs. ([REF]) and ([REF]) to leading order in [MATH], we get the large frequency tail of the dynamic structure factor for [MATH]: [EQUATION]', '1106.6035-2-40-1': 'At unitarity, this agrees with the result by [CITATION] up to a factor of [MATH].', '1106.6035-2-40-2': 'The same frequency tail was derived by the authors of [CITATION] and [CITATION], with whom our result agrees.', '1106.6035-2-40-3': 'The authors of [CITATION] also calculated the correction to the tail at leading order in [MATH].', '1106.6035-2-40-4': 'Again, our result is in agreement with theirs.', '1106.6035-2-40-5': 'Our calculation generalizes the cited results to finite scattering length.', '1106.6035-2-40-6': 'For the two-dimensional Fermi gas, we find no dependence on the scattering length in the high-frequency tail [EQUATION]', '1106.6035-2-41-0': '## Spectral viscosities', '1106.6035-2-42-0': 'The authors of [CITATION] derived Kubo formulas for frequency-dependent generalizations of the bulk and the shear viscosity in three dimensions.', '1106.6035-2-42-1': 'The starting point of their derivation is the Euler equation, [EQUATION] where [MATH] is the stress tensor for a viscous fluid in [MATH] space dimensions: [EQUATION] [MATH] denotes the pressure, [MATH] the velocity, [MATH] the density, and [EQUATION] is the viscous stress tensor [CITATION].', '1106.6035-2-42-2': 'Retracing the derivation in [CITATION], it is not difficult to generalize their result to two space dimensions.', '1106.6035-2-42-3': 'We obtain the relations [EQUATION] and [EQUATION]', '1106.6035-2-42-4': 'The leading order of [MATH] and [MATH] at large frequency is proportional to the contact.', '1106.6035-2-42-5': 'Comparing this with Eqs. ([REF]), ([REF]), ([REF]), and ([REF]), we obtain the large-frequency tails of the bulk and the shear viscosity: [EQUATION] and [EQUATION] in three dimensions, and [EQUATION] and [EQUATION] in two dimensions.', '1106.6035-2-42-6': 'The high-frequency tails have been considered before for a three-dimensional gas near unitarity [CITATION].', '1106.6035-2-42-7': 'Equation ([REF]) agrees with [CITATION] and agrees with [CITATION] up to the same factor of [MATH] mentioned in the previous section.', '1106.6035-2-42-8': 'We obtain the high-frequency tail of the 3D bulk viscosity ([REF]) for an arbitrary value of the scattering length.', '1106.6035-2-42-9': 'It vanishes at unitarity, in agreement with [CITATION].', '1106.6035-2-43-0': '## Viscosity sum rules', '1106.6035-2-44-0': 'By integrating over the frequency of the spectral functions, the asymptotic behavior of the correlation functions can be used to derive sum rules.', '1106.6035-2-44-1': 'On the one hand, such sum rules provide constraints that serve as a consistency check for approximate methods [CITATION].', '1106.6035-2-44-2': 'On the other hand, they are often more closely related to experimental results since it can be difficult to probe the strictly asymptotic region of the response functions.', '1106.6035-2-44-3': 'The simplest way to obtain sum rules is to calculate the lowest momenta of the spectral functions.', '1106.6035-2-44-4': 'Additional sum rules can be deduced by introducing some form of UV cutoff that shifts the integration region to lower frequencies [CITATION].', '1106.6035-2-45-0': 'For completeness, we calculate the lowest moment of the bulk and the shear viscosity and rederive the results of Ref. [CITATION] using the OPE.', '1106.6035-2-45-1': 'To obtain a finite expression for the shear viscosity sum rule, we subtract the high-frequency tail in Eq. ([REF]) from the integrand.', '1106.6035-2-45-2': 'This yields [EQUATION] and [EQUATION]', '1106.6035-2-45-3': 'The shear viscosity sum rule ([REF]) agrees with the result in [CITATION] and agrees with [CITATION] up to the prefactors.', '1106.6035-2-46-0': 'The OPE alone is not sufficient to constrain the bulk viscosity sum rule ([REF]).', '1106.6035-2-46-1': 'As it was pointed out in Ref. [CITATION], this is due to an ordering ambiguity when taking the zero-momentum limit [MATH] and performing the [MATH] integration (we take the limit after performing the integration).', '1106.6035-2-46-2': 'The term in the second line of Eq. ([REF]) is evaluated in the zero-momentum and zero-frequency limit where the OPE cannot provide an insight.', '1106.6035-2-46-3': 'It was evaluated explicitly in [CITATION] using hydrodynamic expressions for the current response.', '1106.6035-2-46-4': 'Combining the result in [CITATION] with Eq. ([REF]) yields the sum rule [EQUATION] where the subscript [MATH] indicates that we take the derivative of [MATH] with respect to [MATH] at fixed entropy.', '1106.6035-2-47-0': '# SUMMARY', '1106.6035-2-48-0': 'In this paper, we studied the current response function of a Fermi gas and calculated the first terms of its operator-product expansion.', '1106.6035-2-48-1': 'We considered the limit [MATH] away from the one-particle peak [MATH] with [MATH] and [MATH] held fixed.', '1106.6035-2-48-2': 'We found that, to leading order, the response is proportional to the contact whose Wilson coefficient we determined.', '1106.6035-2-48-3': 'We calculated various response functions for a wide range of parameters: we derived the longitudinal and transverse components of the spin-symmetric and antisymmetric response functions in 3D and 2D for arbitrary values of the scattering length.', '1106.6035-2-49-0': 'Using those results, we were able to obtain the asymptotic form of the dynamic structure factor in 3D and 2D and thus to generalize and to extend many of the previous results for 3D systems at large scattering length [CITATION].', '1106.6035-2-49-1': 'We rederived the high-frequency tails of the bulk and shear viscosity considered in [CITATION] using the OPE and calculated their 2D counterparts.', '1106.6035-2-49-2': 'Furthermore, we could use the OPE to calculate some sum rules that constrain the spectral viscosities in 3D.', '1106.6035-2-50-0': 'As illustrated by this work, the operator-product expansion provides a powerful tool to derive universal relations.', '1106.6035-2-50-1': 'It offers a systematic way to analyze the short-time and distance structure of a theory, which can be fully characterized by performing calculations involving few-body states.', '1106.6035-2-51-0': 'Note added.', '1106.6035-2-51-1': '- Recently, Ref. [CITATION] appeared.', '1106.6035-2-51-2': 'The authors calculate the viscosity tails in 3D and obtain a result in agreement with Eq. ([REF]) and ([REF]) of this paper.'}

[['1106.6035-1-11-1', '1106.6035-2-11-1'], ['1106.6035-1-3-0', '1106.6035-2-3-0'], ['1106.6035-1-3-4', '1106.6035-2-3-4'], ['1106.6035-1-3-5', '1106.6035-2-3-5'], ['1106.6035-1-3-6', '1106.6035-2-3-6'], ['1106.6035-1-3-7', '1106.6035-2-3-7'], ['1106.6035-1-3-10', '1106.6035-2-3-10'], ['1106.6035-1-17-1', '1106.6035-2-17-1'], ['1106.6035-1-17-2', '1106.6035-2-17-2'], ['1106.6035-1-17-3', '1106.6035-2-17-3'], ['1106.6035-1-17-4', '1106.6035-2-17-4'], ['1106.6035-1-17-5', '1106.6035-2-17-5'], ['1106.6035-1-17-6', '1106.6035-2-17-6'], ['1106.6035-1-17-7', '1106.6035-2-17-7'], ['1106.6035-1-17-9', '1106.6035-2-17-9'], ['1106.6035-1-17-10', '1106.6035-2-17-10'], ['1106.6035-1-17-14', '1106.6035-2-17-14'], ['1106.6035-1-36-2', '1106.6035-2-36-2'], ['1106.6035-1-6-0', '1106.6035-2-6-0'], ['1106.6035-1-6-1', '1106.6035-2-6-1'], ['1106.6035-1-6-3', '1106.6035-2-6-3'], ['1106.6035-1-6-4', '1106.6035-2-6-4'], ['1106.6035-1-6-7', '1106.6035-2-6-7'], ['1106.6035-1-6-8', '1106.6035-2-6-8'], ['1106.6035-1-6-9', '1106.6035-2-6-9'], ['1106.6035-1-6-10', '1106.6035-2-6-10'], ['1106.6035-1-15-0', '1106.6035-2-15-0'], ['1106.6035-1-15-1', '1106.6035-2-15-1'], ['1106.6035-1-15-2', '1106.6035-2-15-2'], ['1106.6035-1-15-3', '1106.6035-2-15-3'], ['1106.6035-1-42-2', '1106.6035-2-42-2'], ['1106.6035-1-42-3', '1106.6035-2-42-3'], ['1106.6035-1-42-4', '1106.6035-2-42-4'], ['1106.6035-1-42-6', '1106.6035-2-42-6'], ['1106.6035-1-42-9', '1106.6035-2-42-9'], ['1106.6035-1-0-2', '1106.6035-2-0-2'], ['1106.6035-1-0-3', '1106.6035-2-0-3'], ['1106.6035-1-13-0', '1106.6035-2-13-0'], ['1106.6035-1-13-1', '1106.6035-2-13-1'], ['1106.6035-1-35-0', '1106.6035-2-35-0'], ['1106.6035-1-45-0', '1106.6035-2-45-0'], ['1106.6035-1-45-1', '1106.6035-2-45-1'], ['1106.6035-1-45-3', '1106.6035-2-45-3'], ['1106.6035-1-49-0', '1106.6035-2-49-0'], ['1106.6035-1-49-2', '1106.6035-2-49-2'], ['1106.6035-1-2-0', '1106.6035-2-2-0'], ['1106.6035-1-2-3', '1106.6035-2-2-3'], ['1106.6035-1-10-0', '1106.6035-2-10-0'], ['1106.6035-1-10-1', '1106.6035-2-10-1'], ['1106.6035-1-31-2', '1106.6035-2-31-1'], ['1106.6035-1-31-5', '1106.6035-2-31-4'], ['1106.6035-1-31-6', '1106.6035-2-31-5'], ['1106.6035-1-14-0', '1106.6035-2-14-0'], ['1106.6035-1-14-1', '1106.6035-2-14-1'], ['1106.6035-1-14-3', '1106.6035-2-14-3'], ['1106.6035-1-14-7', '1106.6035-2-14-7'], ['1106.6035-1-14-8', '1106.6035-2-14-8'], ['1106.6035-1-14-9', '1106.6035-2-14-9'], ['1106.6035-1-19-1', '1106.6035-2-19-1'], ['1106.6035-1-19-2', '1106.6035-2-19-2'], ['1106.6035-1-19-4', '1106.6035-2-19-4'], ['1106.6035-1-19-5', '1106.6035-2-19-5'], ['1106.6035-1-37-0', '1106.6035-2-37-0'], ['1106.6035-1-23-0', '1106.6035-2-23-0'], ['1106.6035-1-29-0', '1106.6035-2-29-0'], ['1106.6035-1-29-1', '1106.6035-2-29-1'], ['1106.6035-1-29-2', '1106.6035-2-29-2'], ['1106.6035-1-4-1', '1106.6035-2-4-1'], ['1106.6035-1-4-4', '1106.6035-2-4-4'], ['1106.6035-1-4-5', '1106.6035-2-4-5'], ['1106.6035-1-4-6', '1106.6035-2-4-6'], ['1106.6035-1-4-8', '1106.6035-2-4-8'], ['1106.6035-1-4-9', '1106.6035-2-4-9'], ['1106.6035-1-26-1', '1106.6035-2-26-1'], ['1106.6035-1-26-2', '1106.6035-2-26-2'], ['1106.6035-1-26-3', '1106.6035-2-26-3'], ['1106.6035-1-5-0', '1106.6035-2-5-0'], ['1106.6035-1-5-1', '1106.6035-2-5-1'], ['1106.6035-1-5-2', '1106.6035-2-5-2'], ['1106.6035-1-5-3', '1106.6035-2-5-3'], ['1106.6035-1-5-4', '1106.6035-2-5-4'], ['1106.6035-1-5-5', '1106.6035-2-5-5'], ['1106.6035-1-32-1', '1106.6035-2-32-1'], ['1106.6035-1-32-3', '1106.6035-2-32-3'], ['1106.6035-1-32-4', '1106.6035-2-32-4'], ['1106.6035-1-32-5', '1106.6035-2-32-5'], ['1106.6035-1-32-6', '1106.6035-2-32-6'], ['1106.6035-1-39-2', '1106.6035-2-39-2'], ['1106.6035-1-39-4', '1106.6035-2-39-4'], ['1106.6035-1-46-0', '1106.6035-2-46-0'], ['1106.6035-1-46-2', '1106.6035-2-46-2'], ['1106.6035-1-46-3', '1106.6035-2-46-3'], ['1106.6035-1-46-4', '1106.6035-2-46-4'], ['1106.6035-1-21-0', '1106.6035-2-21-0'], ['1106.6035-1-21-1', '1106.6035-2-21-1'], ['1106.6035-1-21-13', '1106.6035-2-21-8'], ['1106.6035-1-21-18', '1106.6035-2-21-12'], ['1106.6035-1-9-2', '1106.6035-2-9-2'], ['1106.6035-1-9-3', '1106.6035-2-9-3'], ['1106.6035-1-9-8', '1106.6035-2-9-8'], ['1106.6035-1-44-0', '1106.6035-2-44-0'], ['1106.6035-1-44-1', '1106.6035-2-44-1'], ['1106.6035-1-44-3', '1106.6035-2-44-3'], ['1106.6035-1-50-1', '1106.6035-2-50-1'], ['1106.6035-1-24-3', '1106.6035-2-24-2'], ['1106.6035-1-7-1', '1106.6035-2-7-1'], ['1106.6035-1-7-2', '1106.6035-2-7-2'], ['1106.6035-1-7-4', '1106.6035-2-7-4'], ['1106.6035-1-7-5', '1106.6035-2-7-5'], ['1106.6035-1-7-6', '1106.6035-2-7-6'], ['1106.6035-1-7-8', '1106.6035-2-7-8'], ['1106.6035-1-7-9', '1106.6035-2-7-9'], ['1106.6035-1-7-10', '1106.6035-2-7-10'], ['1106.6035-1-7-11', '1106.6035-2-7-11'], ['1106.6035-1-20-0', '1106.6035-2-20-0'], ['1106.6035-1-20-1', '1106.6035-2-20-1'], ['1106.6035-1-20-3', '1106.6035-2-20-3'], ['1106.6035-1-40-0', '1106.6035-2-40-0'], ['1106.6035-1-40-1', '1106.6035-2-40-1'], ['1106.6035-1-40-2', '1106.6035-2-40-2'], ['1106.6035-1-40-3', '1106.6035-2-40-3'], ['1106.6035-1-40-4', '1106.6035-2-40-4'], ['1106.6035-1-40-5', '1106.6035-2-40-5'], ['1106.6035-1-11-0', '1106.6035-2-11-0'], ['1106.6035-1-3-1', '1106.6035-2-3-1'], ['1106.6035-1-3-2', '1106.6035-2-3-2'], ['1106.6035-1-3-3', '1106.6035-2-3-3'], ['1106.6035-1-3-8', '1106.6035-2-3-8'], ['1106.6035-1-17-0', '1106.6035-2-17-0'], ['1106.6035-1-17-8', '1106.6035-2-17-8'], ['1106.6035-1-17-11', '1106.6035-2-17-11'], ['1106.6035-1-17-12', '1106.6035-2-17-12'], ['1106.6035-1-17-13', '1106.6035-2-17-13'], ['1106.6035-1-36-0', '1106.6035-2-36-0'], ['1106.6035-1-36-1', '1106.6035-2-36-1'], ['1106.6035-1-6-2', '1106.6035-2-6-2'], ['1106.6035-1-6-5', '1106.6035-2-6-5'], ['1106.6035-1-6-11', '1106.6035-2-6-11'], ['1106.6035-1-42-0', '1106.6035-2-42-0'], ['1106.6035-1-42-1', '1106.6035-2-42-1'], ['1106.6035-1-42-5', '1106.6035-2-42-5'], ['1106.6035-1-42-7', '1106.6035-2-42-7'], ['1106.6035-1-42-8', '1106.6035-2-42-8'], ['1106.6035-1-0-0', '1106.6035-2-0-0'], ['1106.6035-1-0-1', '1106.6035-2-0-1'], ['1106.6035-1-35-1', '1106.6035-2-35-1'], ['1106.6035-1-35-3', '1106.6035-2-35-3'], ['1106.6035-1-2-1', '1106.6035-2-2-1'], ['1106.6035-1-2-2', '1106.6035-2-2-2'], ['1106.6035-1-22-0', '1106.6035-2-22-0'], ['1106.6035-1-22-1', '1106.6035-2-22-1'], ['1106.6035-1-31-0', '1106.6035-2-31-0'], ['1106.6035-1-31-3', '1106.6035-2-31-2'], ['1106.6035-1-14-2', '1106.6035-2-14-2'], ['1106.6035-1-14-4', '1106.6035-2-14-4'], ['1106.6035-1-14-6', '1106.6035-2-14-6'], ['1106.6035-1-14-10', '1106.6035-2-14-10'], ['1106.6035-1-19-0', '1106.6035-2-19-0'], ['1106.6035-1-19-3', '1106.6035-2-19-3'], ['1106.6035-1-37-1', '1106.6035-2-37-1'], ['1106.6035-1-37-2', '1106.6035-2-37-2'], ['1106.6035-1-23-1', '1106.6035-2-23-1'], ['1106.6035-1-23-3', '1106.6035-2-23-2'], ['1106.6035-1-4-0', '1106.6035-2-4-0'], ['1106.6035-1-4-2', '1106.6035-2-4-2'], ['1106.6035-1-4-3', '1106.6035-2-4-3'], ['1106.6035-1-4-7', '1106.6035-2-4-7'], ['1106.6035-1-48-0', '1106.6035-2-48-0'], ['1106.6035-1-48-1', '1106.6035-2-48-1'], ['1106.6035-1-48-2', '1106.6035-2-48-2'], ['1106.6035-1-48-3', '1106.6035-2-48-3'], ['1106.6035-1-26-0', '1106.6035-2-26-0'], ['1106.6035-1-26-4', '1106.6035-2-26-4'], ['1106.6035-1-26-5', '1106.6035-2-26-5'], ['1106.6035-1-32-0', '1106.6035-2-32-0'], ['1106.6035-1-39-0', '1106.6035-2-39-0'], ['1106.6035-1-39-1', '1106.6035-2-39-1'], ['1106.6035-1-34-0', '1106.6035-2-34-0'], ['1106.6035-1-46-1', '1106.6035-2-46-1'], ['1106.6035-1-21-5', '1106.6035-2-21-3'], ['1106.6035-1-21-9', '1106.6035-2-21-5'], ['1106.6035-1-21-11', '1106.6035-2-21-6'], ['1106.6035-1-21-16', '1106.6035-2-21-10'], ['1106.6035-1-21-17', '1106.6035-2-21-11'], ['1106.6035-1-9-0', '1106.6035-2-9-0'], ['1106.6035-1-9-1', '1106.6035-2-9-1'], ['1106.6035-1-9-4', '1106.6035-2-9-4'], ['1106.6035-1-9-5', '1106.6035-2-9-5'], ['1106.6035-1-9-6', '1106.6035-2-9-6'], ['1106.6035-1-9-7', '1106.6035-2-9-7'], ['1106.6035-1-9-9', '1106.6035-2-9-9'], ['1106.6035-1-9-10', '1106.6035-2-9-10'], ['1106.6035-1-44-2', '1106.6035-2-44-2'], ['1106.6035-1-44-4', '1106.6035-2-44-4'], ['1106.6035-1-50-0', '1106.6035-2-50-0'], ['1106.6035-1-28-0', '1106.6035-2-28-0'], ['1106.6035-1-28-1', '1106.6035-2-28-1'], ['1106.6035-1-28-2', '1106.6035-2-28-2'], ['1106.6035-1-28-5', '1106.6035-2-28-4'], ['1106.6035-1-24-0', '1106.6035-2-24-0'], ['1106.6035-1-24-1', '1106.6035-2-24-1'], ['1106.6035-1-7-0', '1106.6035-2-7-0'], ['1106.6035-1-7-3', '1106.6035-2-7-3'], ['1106.6035-1-7-7', '1106.6035-2-7-7'], ['1106.6035-1-20-2', '1106.6035-2-20-2'], ['1106.6035-1-40-6', '1106.6035-2-40-6'], ['1106.6035-1-3-9', '1106.6035-2-3-9'], ['1106.6035-1-6-6', '1106.6035-2-6-6'], ['1106.6035-1-35-2', '1106.6035-2-35-2'], ['1106.6035-1-45-2', '1106.6035-2-45-2'], ['1106.6035-1-49-1', '1106.6035-2-49-1'], ['1106.6035-1-22-2', '1106.6035-2-22-2'], ['1106.6035-1-31-4', '1106.6035-2-31-3'], ['1106.6035-1-14-5', '1106.6035-2-14-5'], ['1106.6035-1-32-2', '1106.6035-2-32-2'], ['1106.6035-1-39-3', '1106.6035-2-39-3'], ['1106.6035-1-34-1', '1106.6035-2-34-1'], ['1106.6035-1-21-3', '1106.6035-2-21-2'], ['1106.6035-1-21-4', '1106.6035-2-21-2'], ['1106.6035-1-21-6', '1106.6035-2-21-4'], ['1106.6035-1-21-12', '1106.6035-2-21-7'], ['1106.6035-1-21-14', '1106.6035-2-21-9'], ['1106.6035-1-21-15', '1106.6035-2-21-9'], ['1106.6035-1-28-4', '1106.6035-2-28-3']]

[['1106.6035-1-11-1', '1106.6035-2-11-1'], ['1106.6035-1-3-0', '1106.6035-2-3-0'], ['1106.6035-1-3-4', '1106.6035-2-3-4'], ['1106.6035-1-3-5', '1106.6035-2-3-5'], ['1106.6035-1-3-6', '1106.6035-2-3-6'], ['1106.6035-1-3-7', '1106.6035-2-3-7'], ['1106.6035-1-3-10', '1106.6035-2-3-10'], ['1106.6035-1-17-1', '1106.6035-2-17-1'], ['1106.6035-1-17-2', '1106.6035-2-17-2'], ['1106.6035-1-17-3', '1106.6035-2-17-3'], ['1106.6035-1-17-4', '1106.6035-2-17-4'], ['1106.6035-1-17-5', '1106.6035-2-17-5'], ['1106.6035-1-17-6', '1106.6035-2-17-6'], ['1106.6035-1-17-7', '1106.6035-2-17-7'], ['1106.6035-1-17-9', '1106.6035-2-17-9'], ['1106.6035-1-17-10', '1106.6035-2-17-10'], ['1106.6035-1-17-14', '1106.6035-2-17-14'], ['1106.6035-1-36-2', '1106.6035-2-36-2'], ['1106.6035-1-6-0', '1106.6035-2-6-0'], ['1106.6035-1-6-1', '1106.6035-2-6-1'], ['1106.6035-1-6-3', '1106.6035-2-6-3'], ['1106.6035-1-6-4', '1106.6035-2-6-4'], ['1106.6035-1-6-7', '1106.6035-2-6-7'], ['1106.6035-1-6-8', '1106.6035-2-6-8'], ['1106.6035-1-6-9', '1106.6035-2-6-9'], ['1106.6035-1-6-10', '1106.6035-2-6-10'], ['1106.6035-1-15-0', '1106.6035-2-15-0'], ['1106.6035-1-15-1', '1106.6035-2-15-1'], ['1106.6035-1-15-2', '1106.6035-2-15-2'], ['1106.6035-1-15-3', '1106.6035-2-15-3'], ['1106.6035-1-42-2', '1106.6035-2-42-2'], ['1106.6035-1-42-3', '1106.6035-2-42-3'], ['1106.6035-1-42-4', '1106.6035-2-42-4'], ['1106.6035-1-42-6', '1106.6035-2-42-6'], ['1106.6035-1-42-9', '1106.6035-2-42-9'], ['1106.6035-1-0-2', '1106.6035-2-0-2'], ['1106.6035-1-0-3', '1106.6035-2-0-3'], ['1106.6035-1-13-0', '1106.6035-2-13-0'], ['1106.6035-1-13-1', '1106.6035-2-13-1'], ['1106.6035-1-35-0', '1106.6035-2-35-0'], ['1106.6035-1-45-0', '1106.6035-2-45-0'], ['1106.6035-1-45-1', '1106.6035-2-45-1'], ['1106.6035-1-45-3', '1106.6035-2-45-3'], ['1106.6035-1-49-0', '1106.6035-2-49-0'], ['1106.6035-1-49-2', '1106.6035-2-49-2'], ['1106.6035-1-2-0', '1106.6035-2-2-0'], ['1106.6035-1-2-3', '1106.6035-2-2-3'], ['1106.6035-1-10-0', '1106.6035-2-10-0'], ['1106.6035-1-10-1', '1106.6035-2-10-1'], ['1106.6035-1-31-2', '1106.6035-2-31-1'], ['1106.6035-1-31-5', '1106.6035-2-31-4'], ['1106.6035-1-31-6', '1106.6035-2-31-5'], ['1106.6035-1-14-0', '1106.6035-2-14-0'], ['1106.6035-1-14-1', '1106.6035-2-14-1'], ['1106.6035-1-14-3', '1106.6035-2-14-3'], ['1106.6035-1-14-7', '1106.6035-2-14-7'], ['1106.6035-1-14-8', '1106.6035-2-14-8'], ['1106.6035-1-14-9', '1106.6035-2-14-9'], ['1106.6035-1-19-1', '1106.6035-2-19-1'], ['1106.6035-1-19-2', '1106.6035-2-19-2'], ['1106.6035-1-19-4', '1106.6035-2-19-4'], ['1106.6035-1-19-5', '1106.6035-2-19-5'], ['1106.6035-1-37-0', '1106.6035-2-37-0'], ['1106.6035-1-23-0', '1106.6035-2-23-0'], ['1106.6035-1-29-0', '1106.6035-2-29-0'], ['1106.6035-1-29-1', '1106.6035-2-29-1'], ['1106.6035-1-29-2', '1106.6035-2-29-2'], ['1106.6035-1-4-1', '1106.6035-2-4-1'], ['1106.6035-1-4-4', '1106.6035-2-4-4'], ['1106.6035-1-4-5', '1106.6035-2-4-5'], ['1106.6035-1-4-6', '1106.6035-2-4-6'], ['1106.6035-1-4-8', '1106.6035-2-4-8'], ['1106.6035-1-4-9', '1106.6035-2-4-9'], ['1106.6035-1-26-1', '1106.6035-2-26-1'], ['1106.6035-1-26-2', '1106.6035-2-26-2'], ['1106.6035-1-26-3', '1106.6035-2-26-3'], ['1106.6035-1-5-0', '1106.6035-2-5-0'], ['1106.6035-1-5-1', '1106.6035-2-5-1'], ['1106.6035-1-5-2', '1106.6035-2-5-2'], ['1106.6035-1-5-3', '1106.6035-2-5-3'], ['1106.6035-1-5-4', '1106.6035-2-5-4'], ['1106.6035-1-5-5', '1106.6035-2-5-5'], ['1106.6035-1-32-1', '1106.6035-2-32-1'], ['1106.6035-1-32-3', '1106.6035-2-32-3'], ['1106.6035-1-32-4', '1106.6035-2-32-4'], ['1106.6035-1-32-5', '1106.6035-2-32-5'], ['1106.6035-1-32-6', '1106.6035-2-32-6'], ['1106.6035-1-39-2', '1106.6035-2-39-2'], ['1106.6035-1-39-4', '1106.6035-2-39-4'], ['1106.6035-1-46-0', '1106.6035-2-46-0'], ['1106.6035-1-46-2', '1106.6035-2-46-2'], ['1106.6035-1-46-3', '1106.6035-2-46-3'], ['1106.6035-1-46-4', '1106.6035-2-46-4'], ['1106.6035-1-21-0', '1106.6035-2-21-0'], ['1106.6035-1-21-1', '1106.6035-2-21-1'], ['1106.6035-1-21-13', '1106.6035-2-21-8'], ['1106.6035-1-21-18', '1106.6035-2-21-12'], ['1106.6035-1-9-2', '1106.6035-2-9-2'], ['1106.6035-1-9-3', '1106.6035-2-9-3'], ['1106.6035-1-9-8', '1106.6035-2-9-8'], ['1106.6035-1-44-0', '1106.6035-2-44-0'], ['1106.6035-1-44-1', '1106.6035-2-44-1'], ['1106.6035-1-44-3', '1106.6035-2-44-3'], ['1106.6035-1-50-1', '1106.6035-2-50-1'], ['1106.6035-1-24-3', '1106.6035-2-24-2'], ['1106.6035-1-7-1', '1106.6035-2-7-1'], ['1106.6035-1-7-2', '1106.6035-2-7-2'], ['1106.6035-1-7-4', '1106.6035-2-7-4'], ['1106.6035-1-7-5', '1106.6035-2-7-5'], ['1106.6035-1-7-6', '1106.6035-2-7-6'], ['1106.6035-1-7-8', '1106.6035-2-7-8'], ['1106.6035-1-7-9', '1106.6035-2-7-9'], ['1106.6035-1-7-10', '1106.6035-2-7-10'], ['1106.6035-1-7-11', '1106.6035-2-7-11'], ['1106.6035-1-20-0', '1106.6035-2-20-0'], ['1106.6035-1-20-1', '1106.6035-2-20-1'], ['1106.6035-1-20-3', '1106.6035-2-20-3'], ['1106.6035-1-40-0', '1106.6035-2-40-0'], ['1106.6035-1-40-1', '1106.6035-2-40-1'], ['1106.6035-1-40-2', '1106.6035-2-40-2'], ['1106.6035-1-40-3', '1106.6035-2-40-3'], ['1106.6035-1-40-4', '1106.6035-2-40-4'], ['1106.6035-1-40-5', '1106.6035-2-40-5']]

[['1106.6035-1-11-0', '1106.6035-2-11-0'], ['1106.6035-1-3-1', '1106.6035-2-3-1'], ['1106.6035-1-3-2', '1106.6035-2-3-2'], ['1106.6035-1-3-3', '1106.6035-2-3-3'], ['1106.6035-1-3-8', '1106.6035-2-3-8'], ['1106.6035-1-17-0', '1106.6035-2-17-0'], ['1106.6035-1-17-8', '1106.6035-2-17-8'], ['1106.6035-1-17-11', '1106.6035-2-17-11'], ['1106.6035-1-17-12', '1106.6035-2-17-12'], ['1106.6035-1-17-13', '1106.6035-2-17-13'], ['1106.6035-1-36-0', '1106.6035-2-36-0'], ['1106.6035-1-36-1', '1106.6035-2-36-1'], ['1106.6035-1-6-2', '1106.6035-2-6-2'], ['1106.6035-1-6-5', '1106.6035-2-6-5'], ['1106.6035-1-6-11', '1106.6035-2-6-11'], ['1106.6035-1-42-0', '1106.6035-2-42-0'], ['1106.6035-1-42-1', '1106.6035-2-42-1'], ['1106.6035-1-42-5', '1106.6035-2-42-5'], ['1106.6035-1-42-7', '1106.6035-2-42-7'], ['1106.6035-1-42-8', '1106.6035-2-42-8'], ['1106.6035-1-0-0', '1106.6035-2-0-0'], ['1106.6035-1-0-1', '1106.6035-2-0-1'], ['1106.6035-1-35-1', '1106.6035-2-35-1'], ['1106.6035-1-35-3', '1106.6035-2-35-3'], ['1106.6035-1-2-1', '1106.6035-2-2-1'], ['1106.6035-1-2-2', '1106.6035-2-2-2'], ['1106.6035-1-22-0', '1106.6035-2-22-0'], ['1106.6035-1-22-1', '1106.6035-2-22-1'], ['1106.6035-1-31-0', '1106.6035-2-31-0'], ['1106.6035-1-31-3', '1106.6035-2-31-2'], ['1106.6035-1-14-2', '1106.6035-2-14-2'], ['1106.6035-1-14-4', '1106.6035-2-14-4'], ['1106.6035-1-14-6', '1106.6035-2-14-6'], ['1106.6035-1-14-10', '1106.6035-2-14-10'], ['1106.6035-1-19-0', '1106.6035-2-19-0'], ['1106.6035-1-19-3', '1106.6035-2-19-3'], ['1106.6035-1-37-1', '1106.6035-2-37-1'], ['1106.6035-1-37-2', '1106.6035-2-37-2'], ['1106.6035-1-23-1', '1106.6035-2-23-1'], ['1106.6035-1-23-3', '1106.6035-2-23-2'], ['1106.6035-1-4-0', '1106.6035-2-4-0'], ['1106.6035-1-4-2', '1106.6035-2-4-2'], ['1106.6035-1-4-3', '1106.6035-2-4-3'], ['1106.6035-1-4-7', '1106.6035-2-4-7'], ['1106.6035-1-48-0', '1106.6035-2-48-0'], ['1106.6035-1-48-1', '1106.6035-2-48-1'], ['1106.6035-1-48-2', '1106.6035-2-48-2'], ['1106.6035-1-48-3', '1106.6035-2-48-3'], ['1106.6035-1-26-0', '1106.6035-2-26-0'], ['1106.6035-1-26-4', '1106.6035-2-26-4'], ['1106.6035-1-26-5', '1106.6035-2-26-5'], ['1106.6035-1-32-0', '1106.6035-2-32-0'], ['1106.6035-1-39-0', '1106.6035-2-39-0'], ['1106.6035-1-39-1', '1106.6035-2-39-1'], ['1106.6035-1-34-0', '1106.6035-2-34-0'], ['1106.6035-1-46-1', '1106.6035-2-46-1'], ['1106.6035-1-21-5', '1106.6035-2-21-3'], ['1106.6035-1-21-9', '1106.6035-2-21-5'], ['1106.6035-1-21-11', '1106.6035-2-21-6'], ['1106.6035-1-21-16', '1106.6035-2-21-10'], ['1106.6035-1-21-17', '1106.6035-2-21-11'], ['1106.6035-1-9-0', '1106.6035-2-9-0'], ['1106.6035-1-9-1', '1106.6035-2-9-1'], ['1106.6035-1-9-4', '1106.6035-2-9-4'], ['1106.6035-1-9-5', '1106.6035-2-9-5'], ['1106.6035-1-9-6', '1106.6035-2-9-6'], ['1106.6035-1-9-7', '1106.6035-2-9-7'], ['1106.6035-1-9-9', '1106.6035-2-9-9'], ['1106.6035-1-9-10', '1106.6035-2-9-10'], ['1106.6035-1-44-2', '1106.6035-2-44-2'], ['1106.6035-1-44-4', '1106.6035-2-44-4'], ['1106.6035-1-50-0', '1106.6035-2-50-0'], ['1106.6035-1-28-0', '1106.6035-2-28-0'], ['1106.6035-1-28-1', '1106.6035-2-28-1'], ['1106.6035-1-28-2', '1106.6035-2-28-2'], ['1106.6035-1-28-5', '1106.6035-2-28-4'], ['1106.6035-1-24-0', '1106.6035-2-24-0'], ['1106.6035-1-24-1', '1106.6035-2-24-1'], ['1106.6035-1-7-0', '1106.6035-2-7-0'], ['1106.6035-1-7-3', '1106.6035-2-7-3'], ['1106.6035-1-7-7', '1106.6035-2-7-7'], ['1106.6035-1-20-2', '1106.6035-2-20-2'], ['1106.6035-1-40-6', '1106.6035-2-40-6']]

[]

[['1106.6035-1-3-9', '1106.6035-2-3-9'], ['1106.6035-1-6-6', '1106.6035-2-6-6'], ['1106.6035-1-35-2', '1106.6035-2-35-2'], ['1106.6035-1-45-2', '1106.6035-2-45-2'], ['1106.6035-1-49-1', '1106.6035-2-49-1'], ['1106.6035-1-22-2', '1106.6035-2-22-2'], ['1106.6035-1-31-4', '1106.6035-2-31-3'], ['1106.6035-1-14-5', '1106.6035-2-14-5'], ['1106.6035-1-32-2', '1106.6035-2-32-2'], ['1106.6035-1-39-3', '1106.6035-2-39-3'], ['1106.6035-1-34-1', '1106.6035-2-34-1'], ['1106.6035-1-21-3', '1106.6035-2-21-2'], ['1106.6035-1-21-4', '1106.6035-2-21-2'], ['1106.6035-1-21-6', '1106.6035-2-21-4'], ['1106.6035-1-21-12', '1106.6035-2-21-7'], ['1106.6035-1-21-14', '1106.6035-2-21-9'], ['1106.6035-1-21-15', '1106.6035-2-21-9'], ['1106.6035-1-28-4', '1106.6035-2-28-3']]

[]

['1106.6035-1-9-11', '1106.6035-1-21-2', '1106.6035-1-21-7', '1106.6035-1-21-8', '1106.6035-1-23-2', '1106.6035-1-24-2', '1106.6035-1-31-1', '1106.6035-2-9-11', '1106.6035-2-51-0']

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

https://arxiv.org/abs/1106.6035

1005.1873

{'1005.1873-1-0-0': 'The [MATH]N(p,[MATH]O reaction is the slowest reaction of the carbon-nitrogen-oxygen cycle of hydrogen burning in stars.', '1005.1873-1-0-1': 'As a consequence, it determines the rate of the cycle.', '1005.1873-1-0-2': 'The [MATH]N(p,[MATH]C reaction is frequently used in inverse kinematics for hydrogen depth profiling in materials.', '1005.1873-1-0-3': 'The [MATH]N(p,[MATH]O and [MATH]N(p,[MATH]C reactions have been studied simultaneously, using titanium nitride targets of natural isotopic composition and a proton beam.', '1005.1873-1-0-4': 'The strengths of the resonances at [MATH] = 1058keV in [MATH]N(p,[MATH]O and at [MATH] = 897 and 430keV in [MATH]N(p,[MATH]C have been determined with improved precision, relative to the well-known resonance at [MATH] = 278keV in [MATH]N(p,[MATH]O.', '1005.1873-1-0-5': 'The new recommended values are [MATH] = 0.352[MATH]0.018, 362[MATH]20, and 22.0[MATH]0.9eV for their respective strengths.', '1005.1873-1-0-6': 'In addition, the branching ratios for the decay of the [MATH] = 1058keV resonance in [MATH]N(p,[MATH]O have been redetermined.', '1005.1873-1-0-7': 'The data reported here should facilitate future studies of off-resonant capture in the [MATH]N(p,[MATH]O reaction that are needed for an improved R-matrix extrapolation of the cross section.', '1005.1873-1-0-8': 'In addition, the data on the 430keV resonance in [MATH]N(p,[MATH]C may be useful for hydrogen depth profiling.', '1005.1873-1-1-0': '# Introduction', '1005.1873-1-2-0': 'The carbon-nitrogen-oxygen (CNO) cycle [CITATION] dominates stellar hydrogen burning for temperatures of 20-150MK [CITATION].', '1005.1873-1-2-1': 'A quantitative understanding of its rate affects, for instance, the dredge-up of nucleosynthetic material to the surface of so-called carbon stars [CITATION].', '1005.1873-1-2-2': 'At lower temperatures, hydrogen burning is dominated by the proton-proton (pp) chain instead.', '1005.1873-1-2-3': 'In our Sun, the CNO cycle accounts for just 0.8% of energy production [CITATION], but it provides an interesting neutrino signal.', '1005.1873-1-3-0': 'The solar CNO neutrino flux is proportional to the abundance of carbon and nitrogen in the solar core [CITATION].', '1005.1873-1-3-1': 'This abundance is closely connected to the so-called solar composition problem: There are newly revised elemental abundance data for the solar atmosphere from an improved analysis of Fraunhofer absorption lines [CITATION].', '1005.1873-1-3-2': 'This new elemental composition, when fed into the accepted standard solar model, leads to predicted observables such as the sound speed and density profiles, the depth of the convective zone, and the abundance of helium on the surface [CITATION], that are in disagreement with helioseismological data [CITATION].', '1005.1873-1-3-3': 'The solar composition problem might be solved if the elemental composition is different in the solar core than in the atmosphere.', '1005.1873-1-4-0': 'Two key ingredients for a study of the carbon and nitrogen abundance in the solar core are already available: First, the experimental data on the flux of [MATH]B neutrinos from the Sun have reached a precision of 3 for the Super-Kamiokande I data [CITATION], and the oscillation parameters for solar neutrinos have by now been well-constrained, most notably by data from the SNO [CITATION] and KamLAND [CITATION] neutrino detectors.', '1005.1873-1-4-1': 'The flux of solar [MATH]Be neutrinos is under study in the Borexino detector and currently known with 10 precision [CITATION], a number that is expected to improve in the near future.', '1005.1873-1-4-2': 'Second, the nuclear reaction cross sections involved in producing these neutrinos are rather well-known [CITATION].', '1005.1873-1-4-3': 'Therefore, the [MATH]B and [MATH]Be neutrinos can be used as a thermometer [CITATION] to measure the temperature of the solar core (approximately 16MK).', '1005.1873-1-5-0': 'A third ingredient, the flux of CNO neutrinos from the [MATH] decay of [MATH]N and [MATH]O, has not yet been measured online.', '1005.1873-1-5-1': 'However, it is believed that both Borexino and the planned SNO+ detector [CITATION] can provide such data in the near future.', '1005.1873-1-5-2': 'A fourth ingredient are the nuclear reaction rates involved in the production of the CNO neutrinos.', '1005.1873-1-5-3': 'The rate of the reaction controlling the rate, [MATH]N(p,[MATH]O, is currently known with only 8% precision [CITATION], not enough to resolve the solar composition problem.', '1005.1873-1-6-0': 'The [MATH]N(p,[MATH]O reaction proceeds through capture to a number of excited states and the ground state of [MATH]O (fig. [REF], left panel).', '1005.1873-1-6-1': "The last comprehensive study of this reaction covering a wide energy range goes back to the 1980's [CITATION].", '1005.1873-1-6-2': 'In more recent years, many of the results of Ref. [CITATION] have come under renewed scrutiny.', '1005.1873-1-6-3': 'The [MATH]-width of the subthreshold state at 6792keV is now believed to be much lower than assumed in Ref. [CITATION].', '1005.1873-1-6-4': 'This conclusion was reached in Doppler shift attenuation experiments [CITATION], a Coulomb excitation study [CITATION], and R-matrix fits [CITATION].', '1005.1873-1-6-5': 'The off-resonant capture cross-sections have also been re-investigated at energies 70keV [MATH] 500keV, in some cases significantly revising the Ref. [CITATION] data [CITATION].', '1005.1873-1-6-6': 'An analyzing power study even questioned the transition mode for some decays of excited states [CITATION].', '1005.1873-1-6-7': 'In summary, the new recommended total cross section at astrophysical energies [CITATION] is a factor two lower than previously believed [CITATION], so the accepted reaction rate databases for astrophysical modeling [CITATION] will have to be revised accordingly.', '1005.1873-1-7-0': "Despite all the efforts on the [MATH]-width of the 6792keV state and on low-energy cross sections, for higher energies [MATH] 500keV no experimental re-investigation of the [MATH]N(p,[MATH]O cross section has been performed since the 1980's.", '1005.1873-1-7-1': 'However, for this reaction also precise high-energy data play a role [CITATION] in extrapolating the cross section in the R-matrix framework to ultra-low astrophysical energies such as the solar Gamow peak at 28keV.', '1005.1873-1-8-0': 'The logical first step of a re-investigation of [MATH]N(p,[MATH]O at [MATH] 500keV is a renewed study of the sharp resonance at [MATH] = 1058keV.', '1005.1873-1-8-1': 'Due to the complicated R-matrix scheme with at least five poles and also direct capture contributions, its parameters cannot directly be transformed into formal R-matrix parameters.', '1005.1873-1-8-2': 'However, they can be used as outside constraints for an R-matrix fit, and as normalization points for off-resonant capture studies.', '1005.1873-1-8-3': 'The most precise available reference point for a study of this high-energy resonance is the low-energy [MATH]N(p,[MATH]O resonance at [MATH] = 278keV.', '1005.1873-1-8-4': 'Its resonance strength [EQUATION] (with [MATH] the total angular momenta and [MATH] the widths) has been measured several times with consistent results [CITATION], and based on these works an averaged value of [MATH] = 13.1[MATH]0.6meV has recently been recommended [CITATION].', '1005.1873-1-8-5': 'The resonance is very narrow [CITATION], and the isotropy of its emitted [MATH]-rays makes it also a convenient tool for a relative [MATH]-efficiency calibration.', '1005.1873-1-9-0': 'Two further reference points offer themselves, the resonances at [MATH] = 430 and 897keV in the [MATH]N(p,[MATH]C reaction.', '1005.1873-1-9-1': 'For practical reasons, many [MATH]N targets contain also [MATH]N with its natural and exceptionally stable isotopic abundance of 0.3663%.', '1005.1873-1-9-2': 'The two [MATH]N(p,[MATH]C resonances are rather sharp and sufficiently strong to stand out despite the small isotopic abundance of [MATH]N.', '1005.1873-1-9-3': 'The resonance at [MATH] = 430keV is frequently used for hydrogen depth profiling using 6.39MeV [MATH]N ions [CITATION], with the 4.439MeV [MATH]-ray from the reaction being detected.', '1005.1873-1-9-4': 'Owing to this application, the total energetic width [MATH] of this resonance has been studied frequently [CITATION].', '1005.1873-1-9-5': 'However, its [MATH] has so far been measured only once with precision better than 10% [CITATION].', '1005.1873-1-10-0': 'The aim of the present work is to provide precise values for the strengths of three resonances: The resonance at [MATH] = 1058keV in [MATH]N(p,[MATH]O and the resonances at [MATH] = 430 and 897keV in [MATH]N(p,[MATH]C.', '1005.1873-1-10-1': 'In addition, the branching ratios of the decay of the [MATH] = 1058keV resonance in [MATH]N(p,[MATH]O are re-studied.', '1005.1873-1-10-2': 'These three resonances may then serve as normalization points in a re-investigation of the [MATH]N(p,[MATH]O reaction for [MATH] 500keV.', '1005.1873-1-10-3': 'In addition, improved absolute strength values for the [MATH]N(p,[MATH]C resonances will aid an absolute calibration of hydrogen depth profiling with [MATH]N beams.', '1005.1873-1-11-0': '# Experimental setup', '1005.1873-1-12-0': '## Ion beam, beam transport, and target chamber', '1005.1873-1-13-0': 'The H[MATH] beam for the experiment was provided by the 3MV Tandetron accelerator [CITATION] at Forschungszentrum Dresden-Rossendorf (FZD).', '1005.1873-1-13-1': 'The beam reached the target chamber (fig. [REF]) after passing a switching magnet, an electrostatic quadrupole lens, electrostatic dipoles and a neutral particle trap.', '1005.1873-1-13-2': 'The neutral particle trap consisted of an electric dipole positioned 1m upstream from the target, bending the beam by 7[MATH].', '1005.1873-1-13-3': 'The neutral particles continued at 0[MATH] and were absorbed on the internal wall.', '1005.1873-1-14-0': 'A copper collimator of 5mm diameter was placed 45cm upstream from the target.', '1005.1873-1-14-1': 'A 12cm long copper pipe of 2cm diameter was inserted coaxial to the beam, at 5mm distance from the target.', '1005.1873-1-14-2': 'The copper pipe was biased with -100V to suppress secondary electrons from the target which might affect the electrical beam current reading.', '1005.1873-1-14-3': 'It is estimated that the electrical currents are accurate to [MATH]1.0% in this Faraday cup.', '1005.1873-1-14-4': 'The vacuum measured at 40cm distance from the target was typically 1[MATH]mbar during the irradiations.', '1005.1873-1-15-0': 'The beam intensity on the target ranged from 1-15[MATH]A.', '1005.1873-1-15-1': 'The current on the collimator was always comparable in size to the target current, so no beam wobbling was necessary.', '1005.1873-1-15-2': 'The absolute proton beam energy [MATH] was calibrated based on the known energies of eight resonances in the [MATH]N(p,[MATH]O, [MATH]N(p,[MATH]C, and [MATH]Al(p,[MATH]Si reactions ranging in energy from [MATH] = 278 to 2047keV.', '1005.1873-1-15-3': 'The observed beam energy spread was 1.1keV (FWHM) at [MATH] = 897keV.', '1005.1873-1-16-0': '## Targets', '1005.1873-1-17-0': 'For the experiment, titanium nitride targets have been used.', '1005.1873-1-17-1': 'They were produced with the reactive sputtering technique at the CIVEN facility in Venice/Italy, using nitrogen gas of natural isotopic abundance.', '1005.1873-1-17-2': 'This technique usually leads to highly stable targets with stoichiometry close to Ti[MATH]N[MATH].', '1005.1873-1-17-3': 'The abundance of [MATH]N in the nitrogen contained in atmospheric air, (0.3663[MATH]0.0004)% [CITATION], has been found to be exceedingly stable [CITATION], so it is even defined as the abundance standard by the International Union of Pure and Applied Chemistry [CITATION].', '1005.1873-1-17-4': 'In a recent study using commercial nitrogen tank gas of natural abundance, the [MATH]N/[MATH]N ratio was checked by mass spectrometry and found to be consistent with the natural abundance [CITATION].', '1005.1873-1-17-5': 'For the purpose of the present work, the standard isotopic abundance [CITATION] is assumed to hold with 1.0% uncertainty [CITATION].', '1005.1873-1-17-6': 'Any effects of target degradation under the ion beam are expected to derive from atomic processes with negligible isotopic effects, so it is assumed here that the relevant behavior of the [MATH]N atoms tracks that of the [MATH]N atoms.', '1005.1873-1-17-7': 'Consequently, the same targets could be used for a parallel study of proton capture on [MATH]N and [MATH]N.', '1005.1873-1-18-0': 'Four different samples have been used, all consisting of a 200[MATH]g[MATH]cm[MATH] thick layer of TiN on a 0.22mm thick tantalum backing.', '1005.1873-1-18-1': 'The targets were placed tilted by 55[MATH] with respect to the beam axis and were directly watercooled.', '1005.1873-1-19-0': 'The nitrogen content of the targets and its distribution have been checked at regular intervals by scanning the [MATH]N(p,[MATH]C resonance at [MATH] = 897keV (width [MATH] = 1.57keV [CITATION], slightly larger than the observed beam energy spread), recording the yield of the 4.44MeV [MATH]-ray from the decay of the first excited state of [MATH]C.', '1005.1873-1-19-1': 'The targets showed a rectangular depth profile (fig. [REF]), with an energetic width of typically 50keV at [MATH] = 897keV and at 55[MATH].', '1005.1873-1-19-2': 'The observed high-energy tail of the target is consistent with the expected 13keV energy straggling at the target end.', '1005.1873-1-19-3': 'The plateau of this resonance scan was allowed to decrease by up to 15% under irradiation, then the target was replaced.', '1005.1873-1-20-0': '## Detection of emitted photons', '1005.1873-1-21-0': 'The [MATH]-ray detection system consisted of four high-purity germanium (HPGe) detectors (fig. [REF]).', '1005.1873-1-21-1': 'Three 100% (relative efficiency) HPGe detectors with BGO escape-suppression shield (surrounded by a 2cm thick lead shield) and a 10cm frontal lead shield with a cone-shaped opening of 3-5cm diameter were used: Two were placed horizontally at 127[MATH] (left and right) relative to the beam direction, with front faces at 32cm from the target (hereafter called Det1 and Det3).', '1005.1873-1-21-2': 'The third was placed at 90[MATH] directly above the target, at 28cm distance (Det2).', '1005.1873-1-21-3': 'These three detectors are also used in the nuclear resonance fluorescence (NRF) setup [CITATION] at the ELBE accelerator.', '1005.1873-1-21-4': 'Care was taken so that their shielding and position with respect to the target reproduced the conditions in the NRF setup to [MATH]0.5cm.', '1005.1873-1-22-0': 'A fourth smaller HPGe detector (Det4, 60% rel.', '1005.1873-1-22-1': 'eff., no escape-suppression, surrounded by a 1cm thick lead shield) was placed at 4cm distance from the target, at downwards angle 55[MATH].', '1005.1873-1-22-2': 'This particular setup allowed to observe the emitted photons at three different angles, 55[MATH], 90[MATH], and 127[MATH], and to check the reproducibility for one angle, owing to the two detectors at [MATH].', '1005.1873-1-22-3': 'The second order Legendre polynomial approximately vanishes for angles 55[MATH] and 127[MATH], so that angular correlation effects are diluted at these angles.', '1005.1873-1-23-0': 'The [MATH]-detection efficiencies of the detectors have been measured at low energy (from 662 to 1836keV) by means of calibrated radioactive sources ([MATH]Cs, [MATH]Co, [MATH]Y from Physikalisch-Technische Bundesanstalt, quoted 2[MATH] relative activity uncertainty 0.8-1.2%).', '1005.1873-1-23-1': 'The efficiency curve was then extended to higher energy (fig. [REF]) by means of resonant nuclear reaction [MATH]-cascades of known ratios and angular distributions [CITATION].', '1005.1873-1-23-2': 'The resonances in [MATH]B(p,[MATH]C at [MATH] = 675keV [CITATION], [MATH]Al(p,[MATH]Si at [MATH] = 992keV [CITATION] and [MATH]N(p,[MATH]O at [MATH] = 278keV [CITATION] were used for this purpose.', '1005.1873-1-23-3': 'For the following analysis, ratios of yields of two high-energy [MATH]-rays from the same detector have been used.', '1005.1873-1-23-4': 'Therefore only [MATH]-efficiency ratios and not absolute efficiency values were needed.', '1005.1873-1-24-0': '# Experimental procedure', '1005.1873-1-25-0': '## 278 and 1058keV resonances in [MATH]N([MATH],[MATH]O', '1005.1873-1-26-0': 'The [MATH]N([MATH],[MATH]O reaction proceeds through radiative capture into one of the states of [MATH]O (fig. [REF], left panel).', '1005.1873-1-26-1': 'Non-radiative transitions are negligible.', '1005.1873-1-26-2': 'True coincidence summing effects amount to [MATH] 3% ([MATH] 0.5% uncertainty) in Det4 and have been corrected for analytically; they are negligible in the other detectors.', '1005.1873-1-26-3': 'Two sharp resonances in the energy range relevant for R-matrix fits have been studied here, at [MATH] = 278 and 1058keV (corresponding to [MATH] = 259 and 987keV, fig. [REF], left panel).', '1005.1873-1-26-4': 'The proper proton energy for the on-resonance run (fig. [REF]) has been chosen based on a scan of the resonance profile, in order to be sure to completely cover its energetic width with the target thickness.', '1005.1873-1-27-0': 'The angular distribution of the 1/2[MATH] resonance at [MATH] = 278keV is expected to be isotropic [CITATION].', '1005.1873-1-27-1': 'This assumption was experimentally verified here (fig [REF], left panel) for transitions through the 6172keV state.', '1005.1873-1-27-2': 'The present precision is limited by statistics, because the beam intensity of the 3MV Tandetron was only 1[MATH]A at these low energies.', '1005.1873-1-27-3': 'Also the other transitions are found to be isotropic, but within somewhat higher statistical uncertainty.', '1005.1873-1-27-4': 'For the present purposes, all [MATH]-rays from the decay of this resonance are assumed to exhibit isotropy.', '1005.1873-1-27-5': 'Combining the data from all four detectors and all transitions, 1.3% is reached for the statistical uncertainty of the yield of this reference resonance.', '1005.1873-1-28-0': 'For the [MATH] = 1058keV resonance, the width was determined here to be [MATH] = 3.8[MATH]0.5keV, in good agreement with the literature [CITATION].', '1005.1873-1-28-1': 'The proton beam energy chosen for the strength determination was 16keV above the resonance energy.', '1005.1873-1-28-2': 'Off-resonance runs were performed well below and above the resonance, in order to determine and subtract the contribution given by non-resonant capture.', '1005.1873-1-28-3': 'The subtraction amounted to [MATH] 100% for the 6792[MATH]0 transition, which proceeds only through the non-resonant mechanism at these energies, and less than 6% for the 5241[MATH]0 and 8284[MATH]0 transitions.', '1005.1873-1-28-4': 'The angular distribution was checked for the two most intense transitions, i.e. the decay of the 5241keV and of the 8284keV excited state to the ground state.', '1005.1873-1-28-5': 'They were found to be compatible with isotropy within statistics (fig. [REF], right panel).', '1005.1873-1-28-6': 'For the analysis, isotropy has been assumed and 3% has been adopted as the uncertainty for the angular distribution.', '1005.1873-1-29-0': '## 430 and 897keV resonances in [MATH]N([MATH],[MATH]C', '1005.1873-1-30-0': 'Resonant capture in [MATH]N(p,[MATH]C proceeds via (1) formation of the compound nucleus [MATH]O and (2) emission of an [MATH] particle and a [MATH]C*(4439) excited nucleus, which then (3) decays to the ground state by emitting a photon (fig. [REF], right panel).', '1005.1873-1-30-1': 'The [MATH] = 4439keV peak is affected by Doppler broadening, with an observed [MATH]-peak width in Det4 of 53keV for the 430keV resonance and 64keV for the 897keV resonance.', '1005.1873-1-31-0': 'The angular distributions of the 4439keV [MATH]-rays at the two resonances in [MATH]N(p,[MATH]C are strongly anisotropic but well-known from experiment [CITATION].', '1005.1873-1-31-1': 'The pattern (fig. [REF]) is similar for both resonances due to the same spin and parity of the excited levels in [MATH]O and [MATH]C.', '1005.1873-1-31-2': 'The present data are in fair agreement with the literature (fig. [REF]).', '1005.1873-1-31-3': 'For the further analysis, the literature angular distribution has been assumed to be correct.', '1005.1873-1-31-4': 'In order to make the angular data comparable, for the close-distance Det4 non-negligible attenuation coefficients [MATH] calculated based on the prescription given by Ref. [CITATION] were taken into account (table [REF]).', '1005.1873-1-31-5': 'These coefficients are consistent with unity for the far-distance detectors Det1,2,3.', '1005.1873-1-32-0': 'As a reliability check, the ratio 430/897 of the yields of the 4439keV [MATH]-peak for two consecutive runs on the two different resonances was calculated for all detectors (table [REF]).', '1005.1873-1-32-1': "The same ratio has also been calculated for a similar experiment [CITATION] with targets enriched in [MATH]N and Det4' placed at 0[MATH], where the anisotropy is very pronounced, and 10cm distance (table [REF], last line).", '1005.1873-1-32-2': 'The yield ratio depends only on the effective detection angle of the device, hence the angular distribution and its attenuation.', '1005.1873-1-32-3': 'After correcting for these two effects, the values for the yield ratio are consistent (table [REF]).', '1005.1873-1-33-0': '# Data analysis and Results', '1005.1873-1-34-0': '## Branching ratios for the decay of 1058keV resonance in [MATH]N(p,[MATH]O', '1005.1873-1-35-0': 'The branching ratios for the decay of the [MATH] = 1058keV resonance have been measured using the high-statistics spectra of Det4 (table [REF]), with the off-resonant contribution subtracted based on reference runs below and above the resonance.', '1005.1873-1-35-1': 'Since Det4 is located at 55[MATH] where the second order Legendre polynomial vanishes, angular corrections have been neglected for all transitions.', '1005.1873-1-35-2': 'For the two strongest transitions, this assumption was verified experimentally (sec. [REF]).', '1005.1873-1-35-3': 'The branching ratios were determined also for some of the weaker transitions.', '1005.1873-1-35-4': 'The branching ratios in the standard compilation [CITATION] are based on one work [CITATION].', '1005.1873-1-35-5': 'The only exception is the weak 8284[MATH]5181 branch reported by Ref. [CITATION], which was adopted, leading to a recalculation of the other branches [CITATION].', '1005.1873-1-36-0': 'For the two strongest transitions, 8284[MATH]0 and 8284[MATH]5241, the present branchings are in agreement with Ref. [CITATION], but not with Ref. [CITATION].', '1005.1873-1-36-1': 'The present data show the 8284[MATH]5241 transition to be stronger than reported in Ref. [CITATION].', '1005.1873-1-36-2': 'In that work [CITATION], a sodium iodide scintillating detector had been used that was surrounded with a large Compton-suppressing guard detector.', '1005.1873-1-36-3': 'It is conceivable that the guard efficiency correction applied in Ref. [CITATION] might have been different for the single 8284[MATH]-ray than for the [MATH]-rays of the 8284[MATH]0 cascade, leading to some systematic uncertainty.', '1005.1873-1-36-4': 'The present values for the weaker transitions 8284[MATH]6859, 8284[MATH]6172, and 8284[MATH]5181 are in good agreement with the literature [CITATION] but show generally lesser precision.', '1005.1873-1-37-0': 'Due to the significant differences observed in the strongest two branches, new recommended values are necessary for future calibration purposes.', '1005.1873-1-37-1': 'For 8284[MATH]0 and 8284[MATH]5241, the outlying values by Ref. [CITATION] are omitted and a weighted average of Ref. [CITATION] and the present data is formed.', '1005.1873-1-37-2': 'For the other three transitions, a weighted average of Refs. [CITATION] and the present data is adopted (table [REF]).', '1005.1873-1-38-0': '## Relative resonance strengths', '1005.1873-1-39-0': 'The total width of the three resonances under study here is small compared to the energy loss in the present targets (table [REF]).', '1005.1873-1-39-1': 'Therefore, the classical definition of the thick target yield [CITATION] is applicable: [EQUATION] where [MATH] is the experimental yield for branch [MATH] with branching ratio [MATH] corrected for [MATH]-efficiency and angular distribution, and [MATH] is the de Broglie wavelength at the resonance energy.', '1005.1873-1-39-2': '[MATH] is the effective stopping power [CITATION], i.e. the stopping power per nucleus taking part in the reaction under study.', '1005.1873-1-39-3': 'If the target of interest is [MATH]N, [MATH] is given by: [EQUATION] and analogously for [MATH]N as target: [EQUATION]', '1005.1873-1-39-4': 'The isotopic abundance [MATH]/[MATH] is always taken to be the standard value, 0.3663 /99.6337 [CITATION], with an uncertainty of 1.0% [CITATION].', '1005.1873-1-39-5': 'The ratio of resonance strengths for two different resonances at [MATH]keV ([MATH]) and at [MATH] = 278keV, the reference strength, is then given by: [EQUATION]', '1005.1873-1-39-6': 'The ratio of yields [MATH]/[MATH] was taken from the weighted average of the ratios obtained for each of the four detectors, after checking that they were consistent.', '1005.1873-1-39-7': 'The ratio of effective stopping powers at different energies [MATH]/[MATH] is only slightly dependent on the target stoichiometry [MATH].', '1005.1873-1-39-8': 'The main uncertainty associated with stopping powers is their absolute scale and not the energy dependence beyond the Bragg peak [CITATION], and only the energy dependence is needed here.', '1005.1873-1-39-9': 'The stoichiometric ratio varied for the worst case from Ti[MATH]N[MATH] (virgin target) to Ti[MATH]N[MATH] (after a H[MATH] dose of 0.97Coulomb).', '1005.1873-1-39-10': 'Using the stopping powers from SRIM [CITATION], this change affected [MATH]/[MATH] by just 0.1.', '1005.1873-1-39-11': 'In order to include also theoretical uncertainties, 1.0 uncertainty is assumed for [MATH]/[MATH].', '1005.1873-1-40-0': 'The target deterioration under beam bombardment has been corrected for based on the change observed in the yield of the [MATH] = 897keV resonance in [MATH]N(p,[MATH]C that was used for the regular target scans (fig. [REF]), leading to 0.9 uncertainty.', '1005.1873-1-41-0': 'For calculating the reference yield of the [MATH] = 278keV resonance, the yields of the three peaks corresponding to the decay of the [MATH] = 6792, 6172, and 5182keV excited states of [MATH]O and their precisely known branching ratios [CITATION] have been used.', '1005.1873-1-41-1': 'The strength of the [MATH] = 1058keV resonance has been obtained based on the yields from the two strongest transitions, 5241[MATH]0 and 8284[MATH]0, and the presently measured branching ratios (sec. [REF], tab. [REF]).', '1005.1873-1-42-0': 'For the two resonances in [MATH]N(p,[MATH]C, the broad [MATH]-peak at 4439keV was used to calculate the yield.', '1005.1873-1-42-1': 'Their strength ratio was found to be [MATH] = (6.25[MATH]0.17)[MATH], in fair agreement with the literature value of (5.8[MATH]0.2)[MATH].', '1005.1873-1-42-2': 'That value had been obtained with two detectors placed at 55[MATH] [CITATION], neglecting angular distribution effects and the resultant uncertainty.', '1005.1873-1-42-3': 'The present error bar includes these effects.', '1005.1873-1-42-4': 'Because of a target change, the ratio [MATH] had to be calculated in two steps [EQUATION] leading to slightly higher uncertainty.', '1005.1873-1-43-0': 'Using these strength ratios and the reference strength [MATH] = 13.1[MATH]0.6meV [CITATION], new absolute resonance strengths have been obtained for the three resonances under study (table [REF]).', '1005.1873-1-44-0': '# Discussion', '1005.1873-1-45-0': 'Near the 1058keV resonance in [MATH]N(p,[MATH]O, R-matrix fits for the strongest contribution, ground state capture, show a pronounced interference pattern [CITATION].', '1005.1873-1-45-1': 'Therefore, the shape of the excitation curve for this transition does not obey the ideal Breit-Wigner form.', '1005.1873-1-45-2': 'Since the present, rather thick target covers the entirety of the energy range directly affected by the resonance, the present strength value is unaffected by this fact.', '1005.1873-1-45-3': 'Still, it should be noted that due to the interference, the formal R-matrix parameters for this resonance are quite far from the experimental values.', '1005.1873-1-45-4': 'The present and more precise strength value can therefore not be used directly in an R-matrix code.', '1005.1873-1-45-5': 'However, in the future it can be compared with the predicted strength from an updated R-matrix code with the proper resonance treatment [CITATION], as soon as such a code is publicly available.', '1005.1873-1-46-0': 'For the other branches of the 1058keV resonance and also for all the other resonances under study here, such an interference pattern either does not exist or is negligible when compared to the on-resonance capture.', '1005.1873-1-47-0': 'The present strength value of the 1058keV resonance in [MATH]N(p,[MATH]O is higher than the previous number [CITATION], but still in agreement within the uncertainty.', '1005.1873-1-47-1': 'Therefore, a weighted average of the two numbers is formed and recommended for future use (table [REF]).', '1005.1873-1-48-0': 'Also for the 897keV resonance in [MATH]N(p,[MATH]C, the present value is higher than the literature [CITATION].', '1005.1873-1-48-1': 'That value [CITATION] had been obtained just with two detectors at 55[MATH] angle and neglecting angular distribution effects.', '1005.1873-1-48-2': 'However, the literature angular distribution [CITATION] is lower than unity at 55[MATH] (fig. [REF], also confirmed by the present data) so this assumption leads to a systematically low value.', '1005.1873-1-48-3': 'Consequently, the [MATH] value from the present experiment is recommended for future use.', '1005.1873-1-49-0': 'For the 430keV resonance, the present strength, determined based on [MATH]-spectroscopy, has the same precision as the literature value which had been obtained by [MATH]-spectroscopy instead [CITATION].', '1005.1873-1-49-1': 'That work [CITATION] had used an [MATH]-detector at 30[MATH] and applied the [MATH]-particle angular distribution from a previous experiment and R-matrix fit [CITATION].', '1005.1873-1-49-2': 'Based on the two independent results from [MATH]-spectroscopy [CITATION] and from [MATH]-spectroscopy (present work), a weighted average for the strength is recommended that has just 4% uncertainty (table [REF]).', '1005.1873-1-50-0': '# Summary and outlook', '1005.1873-1-51-0': 'The resonance strength [MATH] has been measured for the 1058keV resonance in [MATH]N(p,[MATH]O and the 430 and 897keV resonances in [MATH]N(p,[MATH]C, relative to the well-known strength of the 278keV resonance in [MATH]N(p,[MATH]O.', '1005.1873-1-51-1': 'A called-for improvement in the precision of this reference point [CITATION] will therefore also lead to an improvement in the understanding of the other three resonances.', '1005.1873-1-52-0': 'For the major transitions, the angular distributions of the 278 and 1058keV resonances in [MATH]N(p,[MATH]O have been verified experimentally to be consistent with the expected isotropy.', '1005.1873-1-52-1': 'The decay branching ratios of the 1058keV resonance in [MATH]N(p,[MATH]O have been determined and updated values are recommended.', '1005.1873-1-53-0': 'Three well-understood, sharp resonances are now available as natural normalization points for cross section measurements.', '1005.1873-1-53-1': 'The new, precise strength of the 430keV resonance in [MATH]N(p,[MATH]C has the potential to serve as a highly precise standard value to make hydrogen depth profiling absolute.', '1005.1873-1-53-2': 'The road is paved for a re-measurement of the astrophysically important [MATH]N(p,[MATH]O off-resonance cross section at energies near 1MeV.', '1005.1873-1-54-0': 'The support by the staff and operators of the FZD ion beam center, technical support by Michael Fauth, Andreas Hartmann, and Manfred Sobiella (FZD), and target analyses performed by Alberto Vomiero (CNR Brescia, Italy) are gratefully acknowledged.', '1005.1873-1-54-1': 'This work was supported in part by the European Union, Research Infrastructures Transnational Access (RITA 025646) to the AIM facility, by DFG (BE4100/2-1), and by OTKA (T68801).', '1005.1873-1-54-2': 'T.S. acknowledges support from the Herbert Quandt Foundation.'}

{'1005.1873-2-0-0': 'The [MATH]N(p,[MATH]O reaction is the slowest reaction of the carbon-nitrogen-oxygen cycle of hydrogen burning in stars.', '1005.1873-2-0-1': 'As a consequence, it determines the rate of the cycle.', '1005.1873-2-0-2': 'The [MATH]N(p,[MATH]C reaction is frequently used in inverse kinematics for hydrogen depth profiling in materials.', '1005.1873-2-0-3': 'The [MATH]N(p,[MATH]O and [MATH]N(p,[MATH]C reactions have been studied simultaneously, using titanium nitride targets of natural isotopic composition and a proton beam.', '1005.1873-2-0-4': 'The strengths of the resonances at [MATH] = 1058keV in [MATH]N(p,[MATH]O and at [MATH] = 897 and 430keV in [MATH]N(p,[MATH]C have been determined with improved precision, relative to the well-known resonance at [MATH] = 278keV in [MATH]N(p,[MATH]O.', '1005.1873-2-0-5': 'The new recommended values are [MATH] = 0.353[MATH]0.018, 362[MATH]20, and 21.9[MATH]1.0eV for their respective strengths.', '1005.1873-2-0-6': 'In addition, the branching ratios for the decay of the [MATH] = 1058keV resonance in [MATH]N(p,[MATH]O have been redetermined.', '1005.1873-2-0-7': 'The data reported here should facilitate future studies of off-resonant capture in the [MATH]N(p,[MATH]O reaction that are needed for an improved R-matrix extrapolation of the cross section.', '1005.1873-2-0-8': 'In addition, the data on the 430keV resonance in [MATH]N(p,[MATH]C may be useful for hydrogen depth profiling.', '1005.1873-2-1-0': '# Introduction', '1005.1873-2-2-0': 'The carbon-nitrogen-oxygen (CNO) cycle [CITATION] dominates stellar hydrogen burning for temperatures of 20-150MK [CITATION].', '1005.1873-2-2-1': 'A quantitative understanding of its rate affects, for instance, the dredge-up of nucleosynthetic material to the surface of so-called carbon stars [CITATION].', '1005.1873-2-2-2': 'At lower temperatures, hydrogen burning is dominated by the proton-proton (pp) chain instead.', '1005.1873-2-2-3': 'In our Sun, the CNO cycle accounts for just 0.8% of energy production [CITATION], but it provides an interesting neutrino signal.', '1005.1873-2-3-0': 'The solar CNO neutrino flux is proportional to the abundance of carbon and nitrogen in the solar core [CITATION].', '1005.1873-2-3-1': 'This abundance is closely connected to the so-called solar composition problem: There are newly revised elemental abundance data for the solar atmosphere from an improved analysis of Fraunhofer absorption lines [CITATION].', '1005.1873-2-3-2': 'This new elemental composition, when fed into the accepted standard solar model, leads to predicted observables such as the sound speed and density profiles, the depth of the convective zone, and the abundance of helium on the surface [CITATION], that are in disagreement with helioseismological data [CITATION].', '1005.1873-2-3-3': 'The solar composition problem might be solved if the elemental composition is different in the solar core than in the atmosphere.', '1005.1873-2-4-0': 'Two key ingredients for a study of the carbon and nitrogen abundance in the solar core are already available: First, the experimental data on the flux of [MATH]B neutrinos from the Sun have reached a precision of 3 for the Super-Kamiokande I data [CITATION], and the oscillation parameters for solar neutrinos have by now been well-constrained, most notably by data from the SNO [CITATION] and KamLAND [CITATION] neutrino detectors.', '1005.1873-2-4-1': 'The flux of solar [MATH]Be neutrinos is under study in the Borexino detector and currently known with 10 precision [CITATION], a number that is expected to improve in the near future.', '1005.1873-2-4-2': 'Second, the nuclear reaction cross sections involved in producing these neutrinos are rather well-known [CITATION].', '1005.1873-2-4-3': 'Therefore, the [MATH]B and [MATH]Be neutrinos can be used as a thermometer [CITATION] to measure the temperature of the solar core (approximately 16MK).', '1005.1873-2-5-0': 'A third ingredient, the flux of CNO neutrinos from the [MATH] decay of [MATH]N and [MATH]O, has not yet been measured online.', '1005.1873-2-5-1': 'However, it is believed that both Borexino and the planned SNO+ detector [CITATION] can provide such data in the near future.', '1005.1873-2-5-2': 'A fourth ingredient are the nuclear reaction rates involved in the production of the CNO neutrinos.', '1005.1873-2-5-3': 'The rate of the reaction controlling the rate, [MATH]N(p,[MATH]O, is currently known with only 8% precision [CITATION], not enough to resolve the solar composition problem.', '1005.1873-2-6-0': 'The [MATH]N(p,[MATH]O reaction proceeds through capture to a number of excited states and the ground state of [MATH]O (fig. [REF], left panel).', '1005.1873-2-6-1': "The last comprehensive study of this reaction covering a wide energy range goes back to the 1980's [CITATION].", '1005.1873-2-6-2': 'In more recent years, many of the results of Ref. [CITATION] have come under renewed scrutiny.', '1005.1873-2-6-3': 'The [MATH]-width of the subthreshold state at 6792keV is now believed to be much lower than assumed in Ref. [CITATION].', '1005.1873-2-6-4': 'This conclusion was reached in Doppler shift attenuation experiments [CITATION], a Coulomb excitation study [CITATION], and R-matrix fits [CITATION].', '1005.1873-2-6-5': 'The off-resonant capture cross-sections have also been re-investigated at energies 70keV [MATH] 500keV, in some cases significantly revising the Ref. [CITATION] data [CITATION].', '1005.1873-2-6-6': 'An analyzing power study even questioned the transition mode for some decays of excited states [CITATION].', '1005.1873-2-6-7': 'In summary, the new recommended total cross section at astrophysical energies [CITATION] is a factor two lower than previously believed [CITATION], so the accepted reaction rate databases for astrophysical modeling [CITATION] will have to be revised accordingly.', '1005.1873-2-7-0': "Despite all the efforts on the [MATH]-width of the 6792keV state and on low-energy cross sections, for higher energies [MATH] 500keV no experimental re-investigation of the [MATH]N(p,[MATH]O cross section has been performed since the 1980's.", '1005.1873-2-7-1': 'However, for this reaction also precise high-energy data play a role [CITATION] in extrapolating the cross section in the R-matrix framework to ultra-low astrophysical energies such as the solar Gamow peak at 28keV.', '1005.1873-2-8-0': 'The logical first step of a re-investigation of [MATH]N(p,[MATH]O at [MATH] 500keV is a renewed study of the sharp resonance at [MATH] = 1058keV.', '1005.1873-2-8-1': 'Due to the complicated R-matrix scheme with at least five poles and also direct capture contributions, its parameters cannot directly be transformed into formal R-matrix parameters.', '1005.1873-2-8-2': 'However, they can be used as outside constraints for an R-matrix fit, and as normalization points for off-resonant capture studies.', '1005.1873-2-8-3': 'The most precise available reference point for a study of this high-energy resonance is the low-energy [MATH]N(p,[MATH]O resonance at [MATH] = 278keV.', '1005.1873-2-8-4': 'Its resonance strength [EQUATION] (with [MATH] the total angular momenta and [MATH] the widths) has been measured several times with consistent results [CITATION], and based on these works an averaged value of [MATH] = 13.1[MATH]0.6meV has recently been recommended [CITATION].', '1005.1873-2-8-5': 'The resonance is very narrow [CITATION], and the isotropy of its emitted [MATH]-rays makes it also a convenient tool for a relative [MATH]-efficiency calibration.', '1005.1873-2-9-0': 'Two further reference points offer themselves, the resonances at [MATH] = 430 and 897keV in the [MATH]N(p,[MATH]C reaction.', '1005.1873-2-9-1': 'For practical reasons, many [MATH]N targets contain also [MATH]N with its natural and exceptionally stable isotopic abundance of 0.3663%.', '1005.1873-2-9-2': 'The two [MATH]N(p,[MATH]C resonances are rather sharp and sufficiently strong to stand out despite the small isotopic abundance of [MATH]N.', '1005.1873-2-9-3': 'The resonance at [MATH] = 430keV is frequently used for hydrogen depth profiling using 6.39MeV [MATH]N ions [CITATION], with the 4.439MeV [MATH]-ray from the reaction being detected.', '1005.1873-2-9-4': 'Owing to this application, the total energetic width [MATH] of this resonance has been studied frequently [CITATION].', '1005.1873-2-9-5': 'However, its [MATH] has so far been measured only once with precision better than 10% [CITATION].', '1005.1873-2-10-0': 'The aim of the present work is to provide precise values for the strengths of three resonances: The resonance at [MATH] = 1058keV in [MATH]N(p,[MATH]O and the resonances at [MATH] = 430 and 897keV in [MATH]N(p,[MATH]C.', '1005.1873-2-10-1': 'In addition, the branching ratios of the decay of the [MATH] = 1058keV resonance in [MATH]N(p,[MATH]O are re-studied.', '1005.1873-2-10-2': 'These three resonances may then serve as normalization points in a re-investigation of the [MATH]N(p,[MATH]O reaction for [MATH] 500keV.', '1005.1873-2-10-3': 'In addition, improved absolute strength values for the [MATH]N(p,[MATH]C resonances will aid an absolute calibration of hydrogen depth profiling with [MATH]N beams.', '1005.1873-2-11-0': '# Experimental setup', '1005.1873-2-12-0': '## Ion beam, beam transport, and target chamber', '1005.1873-2-13-0': 'The H[MATH] beam for the experiment was provided by the 3MV Tandetron accelerator [CITATION] at Forschungszentrum Dresden-Rossendorf (FZD).', '1005.1873-2-13-1': 'The beam reached the target chamber (fig. [REF]) after passing a switching magnet, an electrostatic quadrupole lens, electrostatic dipoles and a neutral particle trap.', '1005.1873-2-13-2': 'The neutral particle trap consisted of an electric dipole positioned 1m upstream from the target, bending the beam by 7[MATH].', '1005.1873-2-13-3': 'The neutral particles continued at 0[MATH] and were absorbed on the internal wall.', '1005.1873-2-14-0': 'A copper collimator of 5mm diameter was placed 45cm upstream from the target.', '1005.1873-2-14-1': 'A 12cm long copper pipe of 2cm diameter was inserted coaxial to the beam, at 5mm distance from the target.', '1005.1873-2-14-2': 'The copper pipe was biased with -100V to suppress secondary electrons from the target which might affect the electrical beam current reading.', '1005.1873-2-14-3': 'It is estimated that the electrical currents are accurate to [MATH]1.0% in this Faraday cup.', '1005.1873-2-14-4': 'The vacuum measured at 40cm distance from the target was typically 1[MATH]mbar during the irradiations.', '1005.1873-2-15-0': 'The beam intensity on the target ranged from 1-15[MATH]A.', '1005.1873-2-15-1': 'The current on the collimator was always comparable in size to the target current, so no beam wobbling was necessary.', '1005.1873-2-15-2': 'The absolute proton beam energy [MATH] was calibrated based on the known energies of eight resonances in the [MATH]N(p,[MATH]O, [MATH]N(p,[MATH]C, and [MATH]Al(p,[MATH]Si reactions ranging in energy from [MATH] = 278 to 2047keV.', '1005.1873-2-15-3': 'The observed beam energy spread was 1.1keV (FWHM) at [MATH] = 897keV.', '1005.1873-2-16-0': '## Targets', '1005.1873-2-17-0': 'For the experiment, titanium nitride targets have been used.', '1005.1873-2-17-1': 'They were produced with the reactive sputtering technique at the CIVEN facility in Venice/Italy, using nitrogen gas of natural isotopic abundance.', '1005.1873-2-17-2': 'This technique usually leads to highly stable targets with stoichiometry close to Ti[MATH]N[MATH].', '1005.1873-2-17-3': 'The abundance of [MATH]N in the nitrogen contained in atmospheric air, (0.3663[MATH]0.0004)% [CITATION], has been found to be exceedingly stable [CITATION], so it is even defined as the abundance standard by the International Union of Pure and Applied Chemistry [CITATION].', '1005.1873-2-17-4': 'In a recent study using commercial nitrogen tank gas of natural abundance, the [MATH]N/[MATH]N ratio was checked by mass spectrometry and found to be consistent with the natural abundance [CITATION].', '1005.1873-2-17-5': 'For the purpose of the present work, the standard isotopic abundance [CITATION] is assumed to hold with 1.0% uncertainty [CITATION].', '1005.1873-2-17-6': 'Any effects of target degradation under the ion beam are expected to derive from atomic processes with negligible isotopic effects, so it is assumed here that the relevant behavior of the [MATH]N atoms tracks that of the [MATH]N atoms.', '1005.1873-2-17-7': 'Consequently, the same targets could be used for a parallel study of proton capture on [MATH]N and [MATH]N.', '1005.1873-2-18-0': 'Four different samples have been used, all consisting of a 200[MATH]g[MATH]cm[MATH] thick layer of TiN on a 0.22mm thick tantalum backing.', '1005.1873-2-18-1': 'The targets were placed tilted by 55[MATH] with respect to the beam axis and were directly watercooled.', '1005.1873-2-19-0': 'The nitrogen content of the targets and its distribution have been checked at regular intervals by scanning the [MATH]N(p,[MATH]C resonance at [MATH] = 897keV (width [MATH] = 1.57keV [CITATION], slightly larger than the observed beam energy spread), recording the yield of the 4.44MeV [MATH]-ray from the decay of the first excited state of [MATH]C.', '1005.1873-2-19-1': 'The targets showed a rectangular depth profile (fig. [REF]), with an energetic width of typically 50keV at [MATH] = 897keV and at 55[MATH].', '1005.1873-2-19-2': 'The observed high-energy tail of the target is consistent with the expected 13keV energy straggling at the target end.', '1005.1873-2-19-3': 'The plateau of this resonance scan was allowed to decrease by up to 15% under irradiation, then the target was replaced.', '1005.1873-2-20-0': '## Detection of emitted photons', '1005.1873-2-21-0': 'The [MATH]-ray detection system consisted of four high-purity germanium (HPGe) detectors (fig. [REF]).', '1005.1873-2-21-1': 'Three 100% (relative efficiency) HPGe detectors with bismuth germanate (BGO) escape-suppression shield (surrounded by a 2cm thick lead shield) and a 10cm frontal lead shield with a cone-shaped opening of 3-5cm diameter were used: Two were placed horizontally at 127[MATH] (left and right) relative to the beam direction, with front faces at 32cm from the target (hereafter called Det1 and Det3).', '1005.1873-2-21-2': 'The third was placed at 90[MATH] directly above the target, at 28cm distance (Det2).', '1005.1873-2-21-3': 'These three detectors are also used in the nuclear resonance fluorescence (NRF) setup [CITATION] at the ELBE accelerator.', '1005.1873-2-21-4': 'Care was taken so that their shielding and position with respect to the target reproduced the conditions in the NRF setup to [MATH]0.5cm.', '1005.1873-2-22-0': 'A fourth smaller HPGe detector (Det4, 60% rel.', '1005.1873-2-22-1': 'eff., no escape-suppression, surrounded by a 1cm thick lead shield) was placed at 4cm distance from the target, at downwards angle 55[MATH].', '1005.1873-2-22-2': 'This particular setup allowed to observe the emitted photons at three different angles, 55[MATH], 90[MATH], and 127[MATH], and to check the reproducibility for one angle, owing to the two detectors at [MATH].', '1005.1873-2-22-3': 'The second order Legendre polynomial approximately vanishes for angles 55[MATH] and 127[MATH], so that angular correlation effects are diluted at these angles.', '1005.1873-2-23-0': 'The [MATH]-detection efficiencies of the detectors have been measured at low energy (from 662 to 1836keV) by means of calibrated radioactive sources ([MATH]Cs, [MATH]Co, [MATH]Y from Physikalisch-Technische Bundesanstalt, quoted 2[MATH] relative activity uncertainty 0.8-1.2%).', '1005.1873-2-23-1': 'The efficiency curve was then extended to higher energy (fig. [REF]) by means of resonant nuclear reaction [MATH]-cascades of known ratios and angular distributions [CITATION].', '1005.1873-2-23-2': 'The resonances in [MATH]B(p,[MATH]C at [MATH] = 675keV [CITATION], [MATH]Al(p,[MATH]Si at [MATH] = 992keV [CITATION] and [MATH]N(p,[MATH]O at [MATH] = 278keV [CITATION] were used for this purpose.', '1005.1873-2-23-3': 'For the following analysis, ratios of yields of two high-energy [MATH]-rays from the same detector have been used.', '1005.1873-2-23-4': 'Therefore only [MATH]-efficiency ratios and not absolute efficiency values were needed.', '1005.1873-2-24-0': '# Experimental procedure', '1005.1873-2-25-0': '## 278 and 1058keV resonances in [MATH]N([MATH],[MATH]O', '1005.1873-2-26-0': 'The [MATH]N([MATH],[MATH]O reaction proceeds through radiative capture into one of the states of [MATH]O (fig. [REF], left panel).', '1005.1873-2-26-1': 'Non-radiative transitions are negligible.', '1005.1873-2-26-2': 'True coincidence summing effects amount to [MATH] 3% ([MATH] 0.5% uncertainty) in Det4 and have been corrected for analytically; they are negligible in the other detectors.', '1005.1873-2-26-3': 'Two sharp resonances in the energy range relevant for R-matrix fits have been studied here, at [MATH] = 278 and 1058keV (corresponding to [MATH] = 259 and 987keV, fig. [REF], left panel).', '1005.1873-2-26-4': 'The proper proton energy for the on-resonance run (fig. [REF]) has been chosen based on a scan of the resonance profile, in order to be sure to completely cover its energetic width with the target thickness.', '1005.1873-2-27-0': 'The angular distribution of the 1/2[MATH] resonance at [MATH] = 278keV is expected to be isotropic [CITATION].', '1005.1873-2-27-1': 'This assumption was experimentally verified here (fig [REF], left panel) for transitions through the 6172keV state.', '1005.1873-2-27-2': 'The present precision is limited by statistics, because the beam intensity of the 3MV Tandetron was only 1[MATH]A at these low energies.', '1005.1873-2-27-3': 'Also the other transitions are found to be isotropic, but within somewhat higher statistical uncertainty.', '1005.1873-2-27-4': 'For the present purposes, all [MATH]-rays from the decay of this resonance are assumed to exhibit isotropy.', '1005.1873-2-27-5': 'Combining the data from all four detectors and all transitions, 1.3% is reached for the statistical uncertainty of the yield of this reference resonance.', '1005.1873-2-28-0': 'For the [MATH] = 1058keV resonance, the width was determined here to be [MATH] = 3.8[MATH]0.5keV, in good agreement with the literature [CITATION].', '1005.1873-2-28-1': 'The proton beam energy chosen for the strength determination was 16keV above the resonance energy.', '1005.1873-2-28-2': 'Off-resonance runs were performed well below and above the resonance, in order to determine and subtract the contribution given by non-resonant capture.', '1005.1873-2-28-3': 'The subtraction amounted to [MATH] 100% for the 6792[MATH]0 transition, which proceeds only through the non-resonant mechanism at these energies, and less than 6% for the 5241[MATH]0 and 8284[MATH]0 transitions.', '1005.1873-2-28-4': 'The angular distribution was checked for the two most intense transitions, i.e. the decay of the 5241keV and of the 8284keV excited state to the ground state.', '1005.1873-2-28-5': 'They were found to be compatible with isotropy within statistics (fig. [REF], right panel).', '1005.1873-2-28-6': 'For the analysis, isotropy has been assumed and 3% has been adopted as the uncertainty for the angular distribution.', '1005.1873-2-29-0': '## 430 and 897keV resonances in [MATH]N([MATH],[MATH]C', '1005.1873-2-30-0': 'Resonant capture in [MATH]N(p,[MATH]C proceeds via (1) formation of the compound nucleus [MATH]O and (2) emission of an [MATH] particle and a [MATH]C*(4439) excited nucleus, which then (3) decays to the ground state by emitting a photon (fig. [REF], right panel).', '1005.1873-2-30-1': 'The [MATH] = 4439keV peak is affected by Doppler broadening, with an observed [MATH]-peak width in Det4 of 53keV for the 430keV resonance and 64keV for the 897keV resonance.', '1005.1873-2-31-0': 'The angular distributions of the 4439keV [MATH]-rays at the two resonances in [MATH]N(p,[MATH]C are strongly anisotropic but well-known from experiment [CITATION].', '1005.1873-2-31-1': 'The pattern (fig. [REF]) is similar for both resonances due to the same spin and parity of the excited levels in [MATH]O and [MATH]C.', '1005.1873-2-31-2': 'The present data are in fair agreement with the literature (fig. [REF]).', '1005.1873-2-31-3': 'For the further analysis, the literature angular distribution has been assumed to be correct.', '1005.1873-2-31-4': 'In order to make the angular data comparable, for the close-distance Det4 non-negligible attenuation coefficients [MATH] calculated based on the prescription given by Ref. [CITATION] were taken into account (table [REF]).', '1005.1873-2-31-5': 'These coefficients are consistent with unity for the far-distance detectors Det1,2,3.', '1005.1873-2-32-0': 'As a reliability check, the ratio 430/897 of the yields of the 4439keV [MATH]-peak for two consecutive runs on the two different resonances was calculated for all detectors (table [REF]).', '1005.1873-2-32-1': "The same ratio has also been calculated for a similar experiment [CITATION] with targets enriched in [MATH]N and Det4' placed at 0[MATH], where the anisotropy is very pronounced, and 10cm distance (table [REF], last line).", '1005.1873-2-32-2': 'The yield ratio depends only on the effective detection angle of the device, hence the angular distribution and its attenuation.', '1005.1873-2-32-3': 'After correcting for these two effects, the values for the yield ratio are consistent (table [REF]).', '1005.1873-2-33-0': '# Data analysis and Results', '1005.1873-2-34-0': '## Branching ratios for the decay of 1058keV resonance in [MATH]N(p,[MATH]O', '1005.1873-2-35-0': 'The branching ratios for the decay of the [MATH] = 1058keV resonance have been measured using the high-statistics spectra of Det4 (table [REF]), with the off-resonant contribution subtracted based on reference runs below and above the resonance.', '1005.1873-2-35-1': 'Since Det4 is located at 55[MATH] where the second order Legendre polynomial vanishes, angular corrections have been neglected for all transitions.', '1005.1873-2-35-2': 'For the two strongest transitions, this assumption was verified experimentally (sec. [REF]).', '1005.1873-2-35-3': 'The branching ratios were determined also for some of the weaker transitions.', '1005.1873-2-35-4': 'The branching ratios in the standard compilation [CITATION] are based on one work [CITATION].', '1005.1873-2-35-5': 'The only exception is the weak 8284[MATH]5181 branch reported by Ref. [CITATION], which was adopted, leading to a recalculation of the other branches [CITATION].', '1005.1873-2-36-0': 'For the two strongest transitions, 8284[MATH]0 and 8284[MATH]5241, the present branchings are in agreement with Ref. [CITATION], but not with Ref. [CITATION].', '1005.1873-2-36-1': 'The present data show the 8284[MATH]5241 transition to be stronger than reported in Ref. [CITATION].', '1005.1873-2-36-2': 'In that work [CITATION], a sodium iodide scintillating detector had been used that was surrounded with a large Compton-suppressing guard detector.', '1005.1873-2-36-3': 'It is conceivable that the guard efficiency correction applied in Ref. [CITATION] might have been different for the single 8284[MATH]-ray than for the [MATH]-rays of the 8284[MATH]0 cascade, leading to some systematic uncertainty.', '1005.1873-2-36-4': 'The present values for the weaker transitions 8284[MATH]6859, 8284[MATH]6172, and 8284[MATH]5181 are in good agreement with the literature [CITATION] but show generally lesser precision.', '1005.1873-2-37-0': 'Due to the significant differences observed in the strongest two branches, new recommended values are necessary for future calibration purposes.', '1005.1873-2-37-1': 'For 8284[MATH]0 and 8284[MATH]5241, the outlying values by Ref. [CITATION] are omitted and a weighted average of Ref. [CITATION] and the present data is formed.', '1005.1873-2-37-2': 'For the other three transitions, a weighted average of Refs. [CITATION] and the present data is adopted (table [REF]).', '1005.1873-2-38-0': '## Relative resonance strengths', '1005.1873-2-39-0': 'The total width of the three resonances under study here is small compared to the energy loss in the present targets (table [REF]).', '1005.1873-2-39-1': 'Therefore, the classical definition of the thick target yield [CITATION] is applicable: [EQUATION] where [MATH] is the experimental yield for branch [MATH] with branching ratio [MATH] corrected for [MATH]-efficiency and angular distribution, and [MATH] is the de Broglie wavelength at the resonance energy.', '1005.1873-2-39-2': '[MATH] is the effective stopping power [CITATION], i.e. the stopping power per nucleus taking part in the reaction under study.', '1005.1873-2-39-3': 'If the target of interest is [MATH]N, [MATH] is given by: [EQUATION] and analogously for [MATH]N as target: [EQUATION]', '1005.1873-2-39-4': 'The isotopic abundance [MATH]/[MATH] is always taken to be the standard value, 0.3663 /99.6337 [CITATION], with an uncertainty of 1.0% [CITATION].', '1005.1873-2-39-5': 'The ratio of resonance strengths for two different resonances at [MATH]keV ([MATH]) and at [MATH] = 278keV, the reference strength, is then given by: [EQUATION]', '1005.1873-2-39-6': 'The ratio of yields [MATH]/[MATH] was taken from the weighted average of the ratios obtained for each of the four detectors, after checking that they were consistent.', '1005.1873-2-39-7': 'The ratio of effective stopping powers at different energies [MATH]/[MATH] is only slightly dependent on the target stoichiometry [MATH].', '1005.1873-2-39-8': 'The main uncertainty associated with stopping powers is their absolute scale and not the energy dependence beyond the Bragg peak [CITATION], and only the energy dependence is needed here.', '1005.1873-2-39-9': 'The stoichiometric ratio varied for the worst case from Ti[MATH]N[MATH] (virgin target) to Ti[MATH]N[MATH] (after a H[MATH] dose of 0.97Coulomb).', '1005.1873-2-39-10': 'Using the stopping powers from SRIM [CITATION], this change affected [MATH]/[MATH] by just 0.1.', '1005.1873-2-39-11': 'In order to include also theoretical uncertainties, 1.0 uncertainty is assumed for [MATH]/[MATH].', '1005.1873-2-40-0': 'The target deterioration under beam bombardment has been corrected for based on the change observed in the yield of the [MATH] = 897keV resonance in [MATH]N(p,[MATH]C that was used for the regular target scans (fig. [REF]), leading to 0.9 uncertainty.', '1005.1873-2-41-0': 'For calculating the reference yield of the [MATH] = 278keV resonance, the yields of the three peaks corresponding to the decay of the [MATH] = 6792, 6172, and 5182keV excited states of [MATH]O and their precisely known branching ratios [CITATION] have been used.', '1005.1873-2-41-1': 'The strength of the [MATH] = 1058keV resonance has been obtained based on the yields from the two strongest transitions, 5241[MATH]0 and 8284[MATH]0, and the presently measured branching ratios (sec. [REF], tab. [REF]).', '1005.1873-2-42-0': 'For the two resonances in [MATH]N(p,[MATH]C, the broad [MATH]-peak at 4439keV was used to calculate the yield.', '1005.1873-2-42-1': 'Their strength ratio was found to be [MATH] = (6.25[MATH]0.17)[MATH], in fair agreement with the literature value of (5.8[MATH]0.2)[MATH].', '1005.1873-2-42-2': 'That value had been obtained with two detectors placed at 55[MATH] [CITATION], neglecting angular distribution effects and the resultant uncertainty.', '1005.1873-2-42-3': 'The present error bar includes these effects.', '1005.1873-2-42-4': 'Because of a target change, the ratio [MATH] had to be calculated in two steps [EQUATION] leading to slightly higher uncertainty.', '1005.1873-2-42-5': 'All the errors for the resonance strength ratios are summarized in table [REF].', '1005.1873-2-43-0': 'Using these strength ratios and the reference strength [MATH] = 13.1[MATH]0.6meV [CITATION], new absolute resonance strengths have been obtained for the three resonances under study (table [REF]).', '1005.1873-2-44-0': '# Discussion', '1005.1873-2-45-0': 'Near the 1058keV resonance in [MATH]N(p,[MATH]O, R-matrix fits for the strongest contribution, ground state capture, show a pronounced interference pattern [CITATION].', '1005.1873-2-45-1': 'Therefore, the shape of the excitation curve for this transition does not obey the ideal Breit-Wigner form.', '1005.1873-2-45-2': 'Since the present, rather thick target covers the entirety of the energy range directly affected by the resonance, the present strength value is unaffected by this fact.', '1005.1873-2-45-3': 'Still, it should be noted that due to the interference, the formal R-matrix parameters for this resonance are quite far from the experimental values.', '1005.1873-2-45-4': 'The present and more precise strength value can therefore not be used directly in an R-matrix code.', '1005.1873-2-45-5': 'However, in the future it can be compared with the predicted strength from an updated R-matrix code with the proper resonance treatment [CITATION], as soon as such a code is publicly available.', '1005.1873-2-46-0': 'For the other branches of the 1058keV resonance and also for all the other resonances under study here, such an interference pattern either does not exist or is negligible when compared to the on-resonance capture.', '1005.1873-2-47-0': 'The present strength value of the 1058keV resonance in [MATH]N(p,[MATH]O is higher than the previous number [CITATION], but still in agreement within the uncertainty.', '1005.1873-2-47-1': 'Therefore, a weighted average of the two numbers is formed and recommended for future use (table [REF]).', '1005.1873-2-48-0': 'Also for the 897keV resonance in [MATH]N(p,[MATH]C, the present value is higher than the literature [CITATION].', '1005.1873-2-48-1': 'That value [CITATION] had been obtained just with two detectors at 55[MATH] angle and neglecting angular distribution effects.', '1005.1873-2-48-2': 'However, the literature angular distribution [CITATION] is lower than unity at 55[MATH] (fig. [REF], also confirmed by the present data) so this assumption leads to a systematically low value.', '1005.1873-2-48-3': 'Consequently, the [MATH] value from the present experiment is recommended for future use.', '1005.1873-2-49-0': 'For the 430keV resonance, the present strength, determined based on [MATH]-spectroscopy, has the same precision as the literature value which had been obtained by [MATH]-spectroscopy instead [CITATION].', '1005.1873-2-49-1': 'That work [CITATION] had used an [MATH]-detector at 30[MATH] and applied the [MATH]-particle angular distribution from a previous experiment and R-matrix fit [CITATION].', '1005.1873-2-49-2': 'Based on the two independent results from [MATH]-spectroscopy [CITATION] and from [MATH]-spectroscopy (present work), a weighted average for the strength is recommended that has just 4% uncertainty (table [REF]).', '1005.1873-2-50-0': '# Summary and outlook', '1005.1873-2-51-0': 'The resonance strength [MATH] has been measured for the 1058keV resonance in [MATH]N(p,[MATH]O and the 430 and 897keV resonances in [MATH]N(p,[MATH]C, relative to the well-known strength of the 278keV resonance in [MATH]N(p,[MATH]O.', '1005.1873-2-51-1': 'A called-for improvement in the precision of this reference point [CITATION] will therefore also lead to an improvement in the understanding of the other three resonances.', '1005.1873-2-52-0': 'For the major transitions, the angular distributions of the 278 and 1058keV resonances in [MATH]N(p,[MATH]O have been verified experimentally to be consistent with the expected isotropy.', '1005.1873-2-52-1': 'The decay branching ratios of the 1058keV resonance in [MATH]N(p,[MATH]O have been determined and updated values are recommended.', '1005.1873-2-53-0': 'Three well-understood, sharp resonances are now available as natural normalization points for cross section measurements.', '1005.1873-2-53-1': 'The new, precise strength of the 430keV resonance in [MATH]N(p,[MATH]C has the potential to serve as a highly precise standard value to make hydrogen depth profiling absolute.', '1005.1873-2-53-2': 'The road is paved for a re-measurement of the astrophysically important [MATH]N(p,[MATH]O off-resonance cross section at energies near 1MeV.', '1005.1873-2-54-0': 'The support by the staff and operators of the FZD ion beam center, technical support by Michael Fauth, Andreas Hartmann, and Manfred Sobiella (FZD), and target analyses performed by Alberto Vomiero (CNR Brescia, Italy) are gratefully acknowledged.', '1005.1873-2-54-1': 'This work was supported in part by the European Union, Research Infrastructures Transnational Access (RITA 025646) to the AIM facility, by DFG (BE4100/2-1), and by OTKA (T68801).', '1005.1873-2-54-2': 'T.S. acknowledges support from the Herbert Quandt Foundation.'}

[['1005.1873-1-5-0', '1005.1873-2-5-0'], ['1005.1873-1-5-1', '1005.1873-2-5-1'], ['1005.1873-1-5-2', '1005.1873-2-5-2'], ['1005.1873-1-5-3', '1005.1873-2-5-3'], ['1005.1873-1-10-0', '1005.1873-2-10-0'], ['1005.1873-1-10-1', '1005.1873-2-10-1'], ['1005.1873-1-10-2', '1005.1873-2-10-2'], ['1005.1873-1-10-3', '1005.1873-2-10-3'], ['1005.1873-1-45-0', '1005.1873-2-45-0'], ['1005.1873-1-45-1', '1005.1873-2-45-1'], ['1005.1873-1-45-2', '1005.1873-2-45-2'], ['1005.1873-1-45-3', '1005.1873-2-45-3'], ['1005.1873-1-45-4', '1005.1873-2-45-4'], ['1005.1873-1-45-5', '1005.1873-2-45-5'], ['1005.1873-1-47-0', '1005.1873-2-47-0'], ['1005.1873-1-47-1', '1005.1873-2-47-1'], ['1005.1873-1-52-0', '1005.1873-2-52-0'], ['1005.1873-1-52-1', '1005.1873-2-52-1'], ['1005.1873-1-22-0', '1005.1873-2-22-0'], ['1005.1873-1-22-1', '1005.1873-2-22-1'], ['1005.1873-1-22-2', '1005.1873-2-22-2'], ['1005.1873-1-22-3', '1005.1873-2-22-3'], ['1005.1873-1-14-0', '1005.1873-2-14-0'], ['1005.1873-1-14-1', '1005.1873-2-14-1'], ['1005.1873-1-14-2', '1005.1873-2-14-2'], ['1005.1873-1-14-3', '1005.1873-2-14-3'], ['1005.1873-1-14-4', '1005.1873-2-14-4'], ['1005.1873-1-53-0', '1005.1873-2-53-0'], ['1005.1873-1-53-1', '1005.1873-2-53-1'], ['1005.1873-1-53-2', '1005.1873-2-53-2'], ['1005.1873-1-39-0', '1005.1873-2-39-0'], ['1005.1873-1-39-1', '1005.1873-2-39-1'], ['1005.1873-1-39-2', '1005.1873-2-39-2'], ['1005.1873-1-39-3', '1005.1873-2-39-3'], ['1005.1873-1-39-4', '1005.1873-2-39-4'], ['1005.1873-1-39-5', '1005.1873-2-39-5'], ['1005.1873-1-39-6', '1005.1873-2-39-6'], ['1005.1873-1-39-7', '1005.1873-2-39-7'], ['1005.1873-1-39-8', '1005.1873-2-39-8'], ['1005.1873-1-39-9', '1005.1873-2-39-9'], ['1005.1873-1-39-10', '1005.1873-2-39-10'], ['1005.1873-1-39-11', '1005.1873-2-39-11'], ['1005.1873-1-19-0', '1005.1873-2-19-0'], ['1005.1873-1-19-1', '1005.1873-2-19-1'], ['1005.1873-1-19-2', '1005.1873-2-19-2'], ['1005.1873-1-19-3', '1005.1873-2-19-3'], ['1005.1873-1-43-0', '1005.1873-2-43-0'], ['1005.1873-1-35-0', '1005.1873-2-35-0'], ['1005.1873-1-35-1', '1005.1873-2-35-1'], ['1005.1873-1-35-2', '1005.1873-2-35-2'], ['1005.1873-1-35-3', '1005.1873-2-35-3'], ['1005.1873-1-35-4', '1005.1873-2-35-4'], ['1005.1873-1-35-5', '1005.1873-2-35-5'], ['1005.1873-1-17-0', '1005.1873-2-17-0'], ['1005.1873-1-17-1', '1005.1873-2-17-1'], ['1005.1873-1-17-2', '1005.1873-2-17-2'], ['1005.1873-1-17-3', '1005.1873-2-17-3'], ['1005.1873-1-17-4', '1005.1873-2-17-4'], ['1005.1873-1-17-5', '1005.1873-2-17-5'], ['1005.1873-1-17-6', '1005.1873-2-17-6'], ['1005.1873-1-17-7', '1005.1873-2-17-7'], ['1005.1873-1-37-0', '1005.1873-2-37-0'], ['1005.1873-1-37-1', '1005.1873-2-37-1'], ['1005.1873-1-37-2', '1005.1873-2-37-2'], ['1005.1873-1-8-0', '1005.1873-2-8-0'], ['1005.1873-1-8-1', '1005.1873-2-8-1'], ['1005.1873-1-8-2', '1005.1873-2-8-2'], ['1005.1873-1-8-3', '1005.1873-2-8-3'], ['1005.1873-1-8-4', '1005.1873-2-8-4'], ['1005.1873-1-8-5', '1005.1873-2-8-5'], ['1005.1873-1-13-0', '1005.1873-2-13-0'], ['1005.1873-1-13-1', '1005.1873-2-13-1'], ['1005.1873-1-13-2', '1005.1873-2-13-2'], ['1005.1873-1-13-3', '1005.1873-2-13-3'], ['1005.1873-1-51-0', '1005.1873-2-51-0'], ['1005.1873-1-51-1', '1005.1873-2-51-1'], ['1005.1873-1-26-0', '1005.1873-2-26-0'], ['1005.1873-1-26-1', '1005.1873-2-26-1'], ['1005.1873-1-26-2', '1005.1873-2-26-2'], ['1005.1873-1-26-3', '1005.1873-2-26-3'], ['1005.1873-1-26-4', '1005.1873-2-26-4'], ['1005.1873-1-40-0', '1005.1873-2-40-0'], ['1005.1873-1-49-0', '1005.1873-2-49-0'], ['1005.1873-1-49-1', '1005.1873-2-49-1'], ['1005.1873-1-49-2', '1005.1873-2-49-2'], ['1005.1873-1-48-0', '1005.1873-2-48-0'], ['1005.1873-1-48-1', '1005.1873-2-48-1'], ['1005.1873-1-48-2', '1005.1873-2-48-2'], ['1005.1873-1-48-3', '1005.1873-2-48-3'], ['1005.1873-1-23-0', '1005.1873-2-23-0'], ['1005.1873-1-23-1', '1005.1873-2-23-1'], ['1005.1873-1-23-3', '1005.1873-2-23-3'], ['1005.1873-1-23-4', '1005.1873-2-23-4'], ['1005.1873-1-6-0', '1005.1873-2-6-0'], ['1005.1873-1-6-1', '1005.1873-2-6-1'], ['1005.1873-1-6-2', '1005.1873-2-6-2'], ['1005.1873-1-6-3', '1005.1873-2-6-3'], ['1005.1873-1-6-4', '1005.1873-2-6-4'], ['1005.1873-1-6-5', '1005.1873-2-6-5'], ['1005.1873-1-6-6', '1005.1873-2-6-6'], ['1005.1873-1-6-7', '1005.1873-2-6-7'], ['1005.1873-1-4-0', '1005.1873-2-4-0'], ['1005.1873-1-4-1', '1005.1873-2-4-1'], ['1005.1873-1-4-2', '1005.1873-2-4-2'], ['1005.1873-1-4-3', '1005.1873-2-4-3'], ['1005.1873-1-28-0', '1005.1873-2-28-0'], ['1005.1873-1-28-1', '1005.1873-2-28-1'], ['1005.1873-1-28-2', '1005.1873-2-28-2'], ['1005.1873-1-28-3', '1005.1873-2-28-3'], ['1005.1873-1-28-4', '1005.1873-2-28-4'], ['1005.1873-1-28-5', '1005.1873-2-28-5'], ['1005.1873-1-28-6', '1005.1873-2-28-6'], ['1005.1873-1-27-0', '1005.1873-2-27-0'], ['1005.1873-1-27-1', '1005.1873-2-27-1'], ['1005.1873-1-27-2', '1005.1873-2-27-2'], ['1005.1873-1-27-3', '1005.1873-2-27-3'], ['1005.1873-1-27-4', '1005.1873-2-27-4'], ['1005.1873-1-27-5', '1005.1873-2-27-5'], ['1005.1873-1-2-0', '1005.1873-2-2-0'], ['1005.1873-1-2-1', '1005.1873-2-2-1'], ['1005.1873-1-2-2', '1005.1873-2-2-2'], ['1005.1873-1-2-3', '1005.1873-2-2-3'], ['1005.1873-1-42-0', '1005.1873-2-42-0'], ['1005.1873-1-42-1', '1005.1873-2-42-1'], ['1005.1873-1-42-2', '1005.1873-2-42-2'], ['1005.1873-1-42-3', '1005.1873-2-42-3'], ['1005.1873-1-42-4', '1005.1873-2-42-4'], ['1005.1873-1-21-0', '1005.1873-2-21-0'], ['1005.1873-1-21-2', '1005.1873-2-21-2'], ['1005.1873-1-21-3', '1005.1873-2-21-3'], ['1005.1873-1-21-4', '1005.1873-2-21-4'], ['1005.1873-1-31-0', '1005.1873-2-31-0'], ['1005.1873-1-31-1', '1005.1873-2-31-1'], ['1005.1873-1-31-2', '1005.1873-2-31-2'], ['1005.1873-1-31-3', '1005.1873-2-31-3'], ['1005.1873-1-31-4', '1005.1873-2-31-4'], ['1005.1873-1-31-5', '1005.1873-2-31-5'], ['1005.1873-1-18-0', '1005.1873-2-18-0'], ['1005.1873-1-18-1', '1005.1873-2-18-1'], ['1005.1873-1-41-0', '1005.1873-2-41-0'], ['1005.1873-1-41-1', '1005.1873-2-41-1'], ['1005.1873-1-15-0', '1005.1873-2-15-0'], ['1005.1873-1-15-1', '1005.1873-2-15-1'], ['1005.1873-1-15-2', '1005.1873-2-15-2'], ['1005.1873-1-15-3', '1005.1873-2-15-3'], ['1005.1873-1-54-0', '1005.1873-2-54-0'], ['1005.1873-1-54-1', '1005.1873-2-54-1'], ['1005.1873-1-54-2', '1005.1873-2-54-2'], ['1005.1873-1-7-0', '1005.1873-2-7-0'], ['1005.1873-1-7-1', '1005.1873-2-7-1'], ['1005.1873-1-36-0', '1005.1873-2-36-0'], ['1005.1873-1-36-1', '1005.1873-2-36-1'], ['1005.1873-1-36-2', '1005.1873-2-36-2'], ['1005.1873-1-36-3', '1005.1873-2-36-3'], ['1005.1873-1-36-4', '1005.1873-2-36-4'], ['1005.1873-1-3-0', '1005.1873-2-3-0'], ['1005.1873-1-3-1', '1005.1873-2-3-1'], ['1005.1873-1-3-2', '1005.1873-2-3-2'], ['1005.1873-1-3-3', '1005.1873-2-3-3'], ['1005.1873-1-9-0', '1005.1873-2-9-0'], ['1005.1873-1-9-1', '1005.1873-2-9-1'], ['1005.1873-1-9-2', '1005.1873-2-9-2'], ['1005.1873-1-9-3', '1005.1873-2-9-3'], ['1005.1873-1-9-4', '1005.1873-2-9-4'], ['1005.1873-1-9-5', '1005.1873-2-9-5'], ['1005.1873-1-46-0', '1005.1873-2-46-0'], ['1005.1873-1-0-0', '1005.1873-2-0-0'], ['1005.1873-1-0-1', '1005.1873-2-0-1'], ['1005.1873-1-0-2', '1005.1873-2-0-2'], ['1005.1873-1-0-3', '1005.1873-2-0-3'], ['1005.1873-1-0-4', '1005.1873-2-0-4'], ['1005.1873-1-0-6', '1005.1873-2-0-6'], ['1005.1873-1-0-7', '1005.1873-2-0-7'], ['1005.1873-1-0-8', '1005.1873-2-0-8'], ['1005.1873-1-30-0', '1005.1873-2-30-0'], ['1005.1873-1-30-1', '1005.1873-2-30-1'], ['1005.1873-1-32-0', '1005.1873-2-32-0'], ['1005.1873-1-32-1', '1005.1873-2-32-1'], ['1005.1873-1-32-2', '1005.1873-2-32-2'], ['1005.1873-1-32-3', '1005.1873-2-32-3'], ['1005.1873-1-21-1', '1005.1873-2-21-1']]

[['1005.1873-1-5-0', '1005.1873-2-5-0'], ['1005.1873-1-5-1', '1005.1873-2-5-1'], ['1005.1873-1-5-2', '1005.1873-2-5-2'], ['1005.1873-1-5-3', '1005.1873-2-5-3'], ['1005.1873-1-10-0', '1005.1873-2-10-0'], ['1005.1873-1-10-1', '1005.1873-2-10-1'], ['1005.1873-1-10-2', '1005.1873-2-10-2'], ['1005.1873-1-10-3', '1005.1873-2-10-3'], ['1005.1873-1-45-0', '1005.1873-2-45-0'], ['1005.1873-1-45-1', '1005.1873-2-45-1'], ['1005.1873-1-45-2', '1005.1873-2-45-2'], ['1005.1873-1-45-3', '1005.1873-2-45-3'], ['1005.1873-1-45-4', '1005.1873-2-45-4'], ['1005.1873-1-45-5', '1005.1873-2-45-5'], ['1005.1873-1-47-0', '1005.1873-2-47-0'], ['1005.1873-1-47-1', '1005.1873-2-47-1'], ['1005.1873-1-52-0', '1005.1873-2-52-0'], ['1005.1873-1-52-1', '1005.1873-2-52-1'], ['1005.1873-1-22-0', '1005.1873-2-22-0'], ['1005.1873-1-22-1', '1005.1873-2-22-1'], ['1005.1873-1-22-2', '1005.1873-2-22-2'], ['1005.1873-1-22-3', '1005.1873-2-22-3'], ['1005.1873-1-14-0', '1005.1873-2-14-0'], ['1005.1873-1-14-1', '1005.1873-2-14-1'], ['1005.1873-1-14-2', '1005.1873-2-14-2'], ['1005.1873-1-14-3', '1005.1873-2-14-3'], ['1005.1873-1-14-4', '1005.1873-2-14-4'], ['1005.1873-1-53-0', '1005.1873-2-53-0'], ['1005.1873-1-53-1', '1005.1873-2-53-1'], ['1005.1873-1-53-2', '1005.1873-2-53-2'], ['1005.1873-1-39-0', '1005.1873-2-39-0'], ['1005.1873-1-39-1', '1005.1873-2-39-1'], ['1005.1873-1-39-2', '1005.1873-2-39-2'], ['1005.1873-1-39-3', '1005.1873-2-39-3'], ['1005.1873-1-39-4', '1005.1873-2-39-4'], ['1005.1873-1-39-5', '1005.1873-2-39-5'], ['1005.1873-1-39-6', '1005.1873-2-39-6'], ['1005.1873-1-39-7', '1005.1873-2-39-7'], ['1005.1873-1-39-8', '1005.1873-2-39-8'], ['1005.1873-1-39-9', '1005.1873-2-39-9'], ['1005.1873-1-39-10', '1005.1873-2-39-10'], ['1005.1873-1-39-11', '1005.1873-2-39-11'], ['1005.1873-1-19-0', '1005.1873-2-19-0'], ['1005.1873-1-19-1', '1005.1873-2-19-1'], ['1005.1873-1-19-2', '1005.1873-2-19-2'], ['1005.1873-1-19-3', '1005.1873-2-19-3'], ['1005.1873-1-43-0', '1005.1873-2-43-0'], ['1005.1873-1-35-0', '1005.1873-2-35-0'], ['1005.1873-1-35-1', '1005.1873-2-35-1'], ['1005.1873-1-35-2', '1005.1873-2-35-2'], ['1005.1873-1-35-3', '1005.1873-2-35-3'], ['1005.1873-1-35-4', '1005.1873-2-35-4'], ['1005.1873-1-35-5', '1005.1873-2-35-5'], ['1005.1873-1-17-0', '1005.1873-2-17-0'], ['1005.1873-1-17-1', '1005.1873-2-17-1'], ['1005.1873-1-17-2', '1005.1873-2-17-2'], ['1005.1873-1-17-3', '1005.1873-2-17-3'], ['1005.1873-1-17-4', '1005.1873-2-17-4'], ['1005.1873-1-17-5', '1005.1873-2-17-5'], ['1005.1873-1-17-6', '1005.1873-2-17-6'], ['1005.1873-1-17-7', '1005.1873-2-17-7'], ['1005.1873-1-37-0', '1005.1873-2-37-0'], ['1005.1873-1-37-1', '1005.1873-2-37-1'], ['1005.1873-1-37-2', '1005.1873-2-37-2'], ['1005.1873-1-8-0', '1005.1873-2-8-0'], ['1005.1873-1-8-1', '1005.1873-2-8-1'], ['1005.1873-1-8-2', '1005.1873-2-8-2'], ['1005.1873-1-8-3', '1005.1873-2-8-3'], ['1005.1873-1-8-4', '1005.1873-2-8-4'], ['1005.1873-1-8-5', '1005.1873-2-8-5'], ['1005.1873-1-13-0', '1005.1873-2-13-0'], ['1005.1873-1-13-1', '1005.1873-2-13-1'], ['1005.1873-1-13-2', '1005.1873-2-13-2'], ['1005.1873-1-13-3', '1005.1873-2-13-3'], ['1005.1873-1-51-0', '1005.1873-2-51-0'], ['1005.1873-1-51-1', '1005.1873-2-51-1'], ['1005.1873-1-26-0', '1005.1873-2-26-0'], ['1005.1873-1-26-1', '1005.1873-2-26-1'], ['1005.1873-1-26-2', '1005.1873-2-26-2'], ['1005.1873-1-26-3', '1005.1873-2-26-3'], ['1005.1873-1-26-4', '1005.1873-2-26-4'], ['1005.1873-1-40-0', '1005.1873-2-40-0'], ['1005.1873-1-49-0', '1005.1873-2-49-0'], ['1005.1873-1-49-1', '1005.1873-2-49-1'], ['1005.1873-1-49-2', '1005.1873-2-49-2'], ['1005.1873-1-48-0', '1005.1873-2-48-0'], ['1005.1873-1-48-1', '1005.1873-2-48-1'], ['1005.1873-1-48-2', '1005.1873-2-48-2'], ['1005.1873-1-48-3', '1005.1873-2-48-3'], ['1005.1873-1-23-0', '1005.1873-2-23-0'], ['1005.1873-1-23-1', '1005.1873-2-23-1'], ['1005.1873-1-23-3', '1005.1873-2-23-3'], ['1005.1873-1-23-4', '1005.1873-2-23-4'], ['1005.1873-1-6-0', '1005.1873-2-6-0'], ['1005.1873-1-6-1', '1005.1873-2-6-1'], ['1005.1873-1-6-2', '1005.1873-2-6-2'], ['1005.1873-1-6-3', '1005.1873-2-6-3'], ['1005.1873-1-6-4', '1005.1873-2-6-4'], ['1005.1873-1-6-5', '1005.1873-2-6-5'], ['1005.1873-1-6-6', '1005.1873-2-6-6'], ['1005.1873-1-6-7', '1005.1873-2-6-7'], ['1005.1873-1-4-0', '1005.1873-2-4-0'], ['1005.1873-1-4-1', '1005.1873-2-4-1'], ['1005.1873-1-4-2', '1005.1873-2-4-2'], ['1005.1873-1-4-3', '1005.1873-2-4-3'], ['1005.1873-1-28-0', '1005.1873-2-28-0'], ['1005.1873-1-28-1', '1005.1873-2-28-1'], ['1005.1873-1-28-2', '1005.1873-2-28-2'], ['1005.1873-1-28-3', '1005.1873-2-28-3'], ['1005.1873-1-28-4', '1005.1873-2-28-4'], ['1005.1873-1-28-5', '1005.1873-2-28-5'], ['1005.1873-1-28-6', '1005.1873-2-28-6'], ['1005.1873-1-27-0', '1005.1873-2-27-0'], ['1005.1873-1-27-1', '1005.1873-2-27-1'], ['1005.1873-1-27-2', '1005.1873-2-27-2'], ['1005.1873-1-27-3', '1005.1873-2-27-3'], ['1005.1873-1-27-4', '1005.1873-2-27-4'], ['1005.1873-1-27-5', '1005.1873-2-27-5'], ['1005.1873-1-2-0', '1005.1873-2-2-0'], ['1005.1873-1-2-1', '1005.1873-2-2-1'], ['1005.1873-1-2-2', '1005.1873-2-2-2'], ['1005.1873-1-2-3', '1005.1873-2-2-3'], ['1005.1873-1-42-0', '1005.1873-2-42-0'], ['1005.1873-1-42-1', '1005.1873-2-42-1'], ['1005.1873-1-42-2', '1005.1873-2-42-2'], ['1005.1873-1-42-3', '1005.1873-2-42-3'], ['1005.1873-1-42-4', '1005.1873-2-42-4'], ['1005.1873-1-21-0', '1005.1873-2-21-0'], ['1005.1873-1-21-2', '1005.1873-2-21-2'], ['1005.1873-1-21-3', '1005.1873-2-21-3'], ['1005.1873-1-21-4', '1005.1873-2-21-4'], ['1005.1873-1-31-0', '1005.1873-2-31-0'], ['1005.1873-1-31-1', '1005.1873-2-31-1'], ['1005.1873-1-31-2', '1005.1873-2-31-2'], ['1005.1873-1-31-3', '1005.1873-2-31-3'], ['1005.1873-1-31-4', '1005.1873-2-31-4'], ['1005.1873-1-31-5', '1005.1873-2-31-5'], ['1005.1873-1-18-0', '1005.1873-2-18-0'], ['1005.1873-1-18-1', '1005.1873-2-18-1'], ['1005.1873-1-41-0', '1005.1873-2-41-0'], ['1005.1873-1-41-1', '1005.1873-2-41-1'], ['1005.1873-1-15-0', '1005.1873-2-15-0'], ['1005.1873-1-15-1', '1005.1873-2-15-1'], ['1005.1873-1-15-2', '1005.1873-2-15-2'], ['1005.1873-1-15-3', '1005.1873-2-15-3'], ['1005.1873-1-54-0', '1005.1873-2-54-0'], ['1005.1873-1-54-1', '1005.1873-2-54-1'], ['1005.1873-1-54-2', '1005.1873-2-54-2'], ['1005.1873-1-7-0', '1005.1873-2-7-0'], ['1005.1873-1-7-1', '1005.1873-2-7-1'], ['1005.1873-1-36-0', '1005.1873-2-36-0'], ['1005.1873-1-36-1', '1005.1873-2-36-1'], ['1005.1873-1-36-2', '1005.1873-2-36-2'], ['1005.1873-1-36-3', '1005.1873-2-36-3'], ['1005.1873-1-36-4', '1005.1873-2-36-4'], ['1005.1873-1-3-0', '1005.1873-2-3-0'], ['1005.1873-1-3-1', '1005.1873-2-3-1'], ['1005.1873-1-3-2', '1005.1873-2-3-2'], ['1005.1873-1-3-3', '1005.1873-2-3-3'], ['1005.1873-1-9-0', '1005.1873-2-9-0'], ['1005.1873-1-9-1', '1005.1873-2-9-1'], ['1005.1873-1-9-2', '1005.1873-2-9-2'], ['1005.1873-1-9-3', '1005.1873-2-9-3'], ['1005.1873-1-9-4', '1005.1873-2-9-4'], ['1005.1873-1-9-5', '1005.1873-2-9-5'], ['1005.1873-1-46-0', '1005.1873-2-46-0'], ['1005.1873-1-0-0', '1005.1873-2-0-0'], ['1005.1873-1-0-1', '1005.1873-2-0-1'], ['1005.1873-1-0-2', '1005.1873-2-0-2'], ['1005.1873-1-0-3', '1005.1873-2-0-3'], ['1005.1873-1-0-4', '1005.1873-2-0-4'], ['1005.1873-1-0-6', '1005.1873-2-0-6'], ['1005.1873-1-0-7', '1005.1873-2-0-7'], ['1005.1873-1-0-8', '1005.1873-2-0-8'], ['1005.1873-1-30-0', '1005.1873-2-30-0'], ['1005.1873-1-30-1', '1005.1873-2-30-1'], ['1005.1873-1-32-0', '1005.1873-2-32-0'], ['1005.1873-1-32-1', '1005.1873-2-32-1'], ['1005.1873-1-32-2', '1005.1873-2-32-2'], ['1005.1873-1-32-3', '1005.1873-2-32-3']]

[['1005.1873-1-21-1', '1005.1873-2-21-1']]

[]

[]

[]

['1005.1873-1-0-5', '1005.1873-1-23-2', '1005.1873-2-0-5', '1005.1873-2-23-2']

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

https://arxiv.org/abs/1005.1873

math-0606670

{'math-0606670-1-0-0': "We prove a recent conjecture due to Deutsch, Sagan, and Wilson stating that the finite sequence obtained from the first [MATH] central trinomial coefficients modulo [MATH] by replacing nonzero terms by [MATH]'s is palindromic, for any prime number [MATH].", 'math-0606670-1-1-0': '# Introduction', 'math-0606670-1-2-0': 'In the recent paper [CITATION] Deutsch and Sagan study several combinatorial sequences reduced modulo prime numbers.', 'math-0606670-1-2-1': 'They are in particular interested in the values modulo [MATH] of the central trinomial coefficients.', 'math-0606670-1-2-2': 'Let us recall that the [MATH]th central trinomial coefficient is defined as the largest coefficient in the expansion of the polynomial [MATH].', 'math-0606670-1-2-3': 'Deutsch and Sagan make the following conjecture [CITATION] (also stated by Wilson, see [CITATION]): for each prime [MATH] and for each [MATH], the number [MATH] is divisible by [MATH] if and only if [MATH] is divisible by [MATH].', 'math-0606670-1-2-4': 'We give here an elementary proof of this conjecture.', 'math-0606670-1-3-0': '# The central trinomial coefficients modulo a prime', 'math-0606670-1-4-0': 'In this section we first recall a classical result (see [CITATION] for example).', 'math-0606670-1-5-0': 'The generating function [MATH] of the central trinomial coeficient satisfies: [EQUATION]', 'math-0606670-1-5-1': 'Note that, using [CITATION] (see also [CITATION]), this implies that the sequence [MATH] satisfies: if [MATH] is prime and if the base [MATH] expansion of [MATH] is [MATH], then [MATH], which is Theorem 4.7 of [CITATION].', 'math-0606670-1-6-0': 'An easy consequence of Proposition [REF] is the following statement.', 'math-0606670-1-7-0': 'Let [MATH] be an odd prime.', 'math-0606670-1-7-1': 'Then we have the following identity [EQUATION]', 'math-0606670-1-7-2': 'Proof.', 'math-0606670-1-7-3': 'From Proposition [REF] we have [EQUATION]', 'math-0606670-1-7-4': 'On the other hand, using the "[MATH]-Lucas property" recalled in Remark [REF] above, we have [EQUATION] which yields [EQUATION]', 'math-0606670-1-7-5': 'Comparing Equations [REF] and [REF] finishes the proof.', 'math-0606670-1-7-6': '[MATH]', 'math-0606670-1-8-0': '# Proof of the conjecture', 'math-0606670-1-9-0': 'We first prove a proposition on the nonzero coefficients of a quadratic polynomial raised to an integer power.', 'math-0606670-1-10-0': 'Let [MATH] be a polynomial with coefficients in a commutative field [MATH], with [MATH].', 'math-0606670-1-10-1': 'Let [MATH] be a positive integer.', 'math-0606670-1-10-2': 'Then, noting [MATH], we have [MATH] if and only if [MATH].', 'math-0606670-1-11-0': 'Proof.', 'math-0606670-1-11-1': 'We write [EQUATION].', 'math-0606670-1-11-2': 'But the sum on the left can also be written [MATH]; thus, for all [MATH], we have [MATH] which implies our claim.', 'math-0606670-1-11-3': '[MATH]', 'math-0606670-1-12-0': 'As an immediate corollary, we get a proof of the conjecture of Deutsch, Sagan, and Wilson.', 'math-0606670-1-13-0': 'Theorem For any prime [MATH], for any [MATH], the sequence of central trinomial coefficients [MATH] satisfies [EQUATION]', 'math-0606670-1-14-0': 'Proof.', 'math-0606670-1-14-1': 'Apply Proposition [REF] with [MATH] (the finite field with [MATH] elements) and [MATH], and use Proposition [REF].', 'math-0606670-1-14-2': '[MATH]', 'math-0606670-1-15-0': 'The reader can check that the proof of the Theorem above readily generalizes to proving the following.', 'math-0606670-1-15-1': '(Hint: use [CITATION].)', 'math-0606670-1-16-0': 'Let [MATH] be a sequence of integers, such that there exists a polynomial of degree [MATH] with integer coefficients [MATH] such that [MATH].', 'math-0606670-1-16-1': 'Then, for all primes [MATH] such that [MATH] does not divide [MATH] and for all [MATH], we have [EQUATION]', 'math-0606670-1-17-0': 'In particular if [MATH] is the sequence of central Delannoy numbers (see [CITATION]), then for all primes [MATH] and for all [MATH], we have [EQUATION].', 'math-0606670-1-17-1': 'Note that the [MATH]-Lucas property for this sequence is a consequence of [CITATION] (see also [CITATION]) and of the fact that the generating function for the central Delannoy numbers is equal to [MATH] (see [CITATION] for example); it is also proven in [CITATION] and in [CITATION].', 'math-0606670-1-17-2': 'A nice paper on sequences having the [MATH]-Lucas property is [CITATION].'}

{'math-0606670-2-0-0': "We prove a recent conjecture due to Deutsch, Sagan, and Wilson stating that the finite sequence obtained from the first [MATH] central trinomial coefficients modulo [MATH] by replacing nonzero terms by [MATH]'s is palindromic, for any prime number [MATH].", 'math-0606670-2-1-0': '# Introduction', 'math-0606670-2-2-0': 'In the recent paper [CITATION] Deutsch and Sagan study several combinatorial sequences reduced modulo prime numbers.', 'math-0606670-2-2-1': 'They are in particular interested in the values modulo [MATH] of the central trinomial coefficients.', 'math-0606670-2-2-2': 'Let us recall that the [MATH]th central trinomial coefficient is defined as the largest coefficient in the expansion of the polynomial [MATH].', 'math-0606670-2-2-3': 'Deutsch and Sagan make the following conjecture [CITATION] (also stated by Wilson, see [CITATION]): for each prime [MATH] and for each [MATH], the number [MATH] is divisible by [MATH] if and only if [MATH] is divisible by [MATH].', 'math-0606670-2-2-4': 'We give here an elementary proof of this conjecture.', 'math-0606670-2-3-0': '# The central trinomial coefficients modulo a prime', 'math-0606670-2-4-0': 'In this section we first recall a classical result (see [CITATION] for example).', 'math-0606670-2-5-0': 'The generating function [MATH] of the central trinomial coeficient satisfies: [EQUATION]', 'math-0606670-2-5-1': 'Note that, using [CITATION] (see also [CITATION]), this implies that the sequence [MATH] satisfies: if [MATH] is prime and if the base [MATH] expansion of [MATH] is [MATH], then [MATH], which is Theorem 4.7 of [CITATION].', 'math-0606670-2-6-0': 'An easy consequence of Proposition [REF] is the following statement.', 'math-0606670-2-7-0': 'Let [MATH] be an odd prime.', 'math-0606670-2-7-1': 'Then we have the following identity [EQUATION]', 'math-0606670-2-7-2': 'Proof.', 'math-0606670-2-7-3': 'From Proposition [REF] we have [EQUATION]', 'math-0606670-2-7-4': 'On the other hand, using the "[MATH]-Lucas property" recalled in Remark [REF] above, we have [EQUATION] which yields [EQUATION]', 'math-0606670-2-7-5': 'Comparing Equations [REF] and [REF] finishes the proof.', 'math-0606670-2-7-6': '[MATH]', 'math-0606670-2-8-0': '# Proof of the conjecture', 'math-0606670-2-9-0': 'We first prove a proposition on the nonzero coefficients of a quadratic polynomial raised to an integer power.', 'math-0606670-2-10-0': 'Let [MATH] be a polynomial with coefficients in a commutative field [MATH], with [MATH].', 'math-0606670-2-10-1': 'Let [MATH] be a positive integer.', 'math-0606670-2-10-2': 'Then, noting [MATH], we have [MATH] if and only if [MATH].', 'math-0606670-2-11-0': 'Proof.', 'math-0606670-2-11-1': 'We write [EQUATION].', 'math-0606670-2-11-2': 'But the sum on the left can also be written [MATH]; thus, for all [MATH], we have [MATH] which implies our claim.', 'math-0606670-2-11-3': '[MATH]', 'math-0606670-2-12-0': 'As an immediate corollary, we get a proof of the conjecture of Deutsch, Sagan, and Wilson.', 'math-0606670-2-13-0': 'Theorem For any prime [MATH], for any [MATH], the sequence of central trinomial coefficients [MATH] satisfies [EQUATION]', 'math-0606670-2-14-0': 'Proof.', 'math-0606670-2-14-1': 'Apply Proposition [REF] with [MATH] (the finite field with [MATH] elements) and [MATH], and use Proposition [REF].', 'math-0606670-2-14-2': '[MATH]', 'math-0606670-2-15-0': 'The reader can check that the proof of the Theorem above readily generalizes to proving the following.', 'math-0606670-2-15-1': '(Hint: use [CITATION].)', 'math-0606670-2-16-0': 'Let [MATH] be a sequence of integers, such that there exists a polynomial of degree [MATH] with integer coefficients [MATH] such that [MATH].', 'math-0606670-2-16-1': 'Then, for all primes [MATH] such that [MATH] does not divide [MATH] and for all [MATH], we have [EQUATION]', 'math-0606670-2-17-0': 'In particular if [MATH] is the sequence of central Delannoy numbers (see [CITATION]), then for all primes [MATH] and for all [MATH], we have [EQUATION].', 'math-0606670-2-17-1': 'Note that the [MATH]-Lucas property for this sequence is a consequence of [CITATION] (see also [CITATION]) and of the fact that the generating function for the central Delannoy numbers is equal to [MATH] (see [CITATION] for example); it is also proven in [CITATION] and in [CITATION].', 'math-0606670-2-17-2': 'A nice paper on sequences having the [MATH]-Lucas property is [CITATION].', 'math-0606670-2-18-0': 'Addendum: the result was proved before almost in the same way by Tony D. Noe: On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7', 'math-0606670-2-19-0': 'http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html'}

[['math-0606670-1-9-0', 'math-0606670-2-9-0'], ['math-0606670-1-5-0', 'math-0606670-2-5-0'], ['math-0606670-1-5-1', 'math-0606670-2-5-1'], ['math-0606670-1-10-0', 'math-0606670-2-10-0'], ['math-0606670-1-10-1', 'math-0606670-2-10-1'], ['math-0606670-1-10-2', 'math-0606670-2-10-2'], ['math-0606670-1-11-2', 'math-0606670-2-11-2'], ['math-0606670-1-12-0', 'math-0606670-2-12-0'], ['math-0606670-1-15-0', 'math-0606670-2-15-0'], ['math-0606670-1-14-1', 'math-0606670-2-14-1'], ['math-0606670-1-0-0', 'math-0606670-2-0-0'], ['math-0606670-1-7-0', 'math-0606670-2-7-0'], ['math-0606670-1-7-1', 'math-0606670-2-7-1'], ['math-0606670-1-7-3', 'math-0606670-2-7-3'], ['math-0606670-1-7-4', 'math-0606670-2-7-4'], ['math-0606670-1-7-5', 'math-0606670-2-7-5'], ['math-0606670-1-2-0', 'math-0606670-2-2-0'], ['math-0606670-1-2-1', 'math-0606670-2-2-1'], ['math-0606670-1-2-2', 'math-0606670-2-2-2'], ['math-0606670-1-2-3', 'math-0606670-2-2-3'], ['math-0606670-1-2-4', 'math-0606670-2-2-4'], ['math-0606670-1-17-0', 'math-0606670-2-17-0'], ['math-0606670-1-17-1', 'math-0606670-2-17-1'], ['math-0606670-1-17-2', 'math-0606670-2-17-2'], ['math-0606670-1-13-0', 'math-0606670-2-13-0'], ['math-0606670-1-4-0', 'math-0606670-2-4-0'], ['math-0606670-1-16-0', 'math-0606670-2-16-0'], ['math-0606670-1-16-1', 'math-0606670-2-16-1']]

[['math-0606670-1-9-0', 'math-0606670-2-9-0'], ['math-0606670-1-5-0', 'math-0606670-2-5-0'], ['math-0606670-1-5-1', 'math-0606670-2-5-1'], ['math-0606670-1-10-0', 'math-0606670-2-10-0'], ['math-0606670-1-10-1', 'math-0606670-2-10-1'], ['math-0606670-1-10-2', 'math-0606670-2-10-2'], ['math-0606670-1-11-2', 'math-0606670-2-11-2'], ['math-0606670-1-12-0', 'math-0606670-2-12-0'], ['math-0606670-1-15-0', 'math-0606670-2-15-0'], ['math-0606670-1-14-1', 'math-0606670-2-14-1'], ['math-0606670-1-0-0', 'math-0606670-2-0-0'], ['math-0606670-1-7-0', 'math-0606670-2-7-0'], ['math-0606670-1-7-1', 'math-0606670-2-7-1'], ['math-0606670-1-7-3', 'math-0606670-2-7-3'], ['math-0606670-1-7-4', 'math-0606670-2-7-4'], ['math-0606670-1-7-5', 'math-0606670-2-7-5'], ['math-0606670-1-2-0', 'math-0606670-2-2-0'], ['math-0606670-1-2-1', 'math-0606670-2-2-1'], ['math-0606670-1-2-2', 'math-0606670-2-2-2'], ['math-0606670-1-2-3', 'math-0606670-2-2-3'], ['math-0606670-1-2-4', 'math-0606670-2-2-4'], ['math-0606670-1-17-0', 'math-0606670-2-17-0'], ['math-0606670-1-17-1', 'math-0606670-2-17-1'], ['math-0606670-1-17-2', 'math-0606670-2-17-2'], ['math-0606670-1-13-0', 'math-0606670-2-13-0'], ['math-0606670-1-4-0', 'math-0606670-2-4-0'], ['math-0606670-1-16-0', 'math-0606670-2-16-0'], ['math-0606670-1-16-1', 'math-0606670-2-16-1']]

[]

[]

[]

[]

['math-0606670-1-6-0', 'math-0606670-1-7-2', 'math-0606670-1-7-6', 'math-0606670-1-11-0', 'math-0606670-1-11-1', 'math-0606670-1-11-3', 'math-0606670-1-14-0', 'math-0606670-1-14-2', 'math-0606670-1-15-1', 'math-0606670-2-6-0', 'math-0606670-2-7-2', 'math-0606670-2-7-6', 'math-0606670-2-11-0', 'math-0606670-2-11-1', 'math-0606670-2-11-3', 'math-0606670-2-14-0', 'math-0606670-2-14-2', 'math-0606670-2-15-1', 'math-0606670-2-19-0']

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

https://arxiv.org/abs/math/0606670

1407.6666

{'1407.6666-1-0-0': 'We show how the Tutte polynomial of a matroid [MATH] can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats.', '1407.6666-1-0-1': 'The results imply that the Tutte polynomial of [MATH] is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks.', '1407.6666-1-0-2': 'They furthermore generalize a similiar statement for perfect matroid designs due to Mphako [CITATION] and help to understand families of matroids with identical Tutte polynomial as constructed in [CITATION].', '1407.6666-1-1-0': '# Introduction', '1407.6666-1-2-0': 'The Tutte polynomial is a central invariant in matroid theory.', '1407.6666-1-2-1': 'But passing over from a matroid [MATH] to its Tutte polynomial [MATH] generally means a big loss of information.', '1407.6666-1-2-2': 'This paper gives one explanation for this phenomenon by showing how little information about the cyclic flats of a matroid is really needed for the computation of its Tutte polynomial.', '1407.6666-1-3-0': 'From now on let [MATH] be a matroid.', '1407.6666-1-3-1': 'A flat [MATH] in [MATH] is called cyclic if [MATH] contains no coloops.', '1407.6666-1-3-2': 'Section [REF] will recapitulate some basic facts about cyclic flats and show how the Tutte polynomial can be expressed in terms of cloud and flock polynomials of cyclic flats as introduced by Plesken in [CITATION].', '1407.6666-1-3-3': 'Then Section [REF] establishes some important identities for cloud and flock polynomials needed later on.', '1407.6666-1-4-0': '[MATH], the set of cyclic flats of [MATH], is a lattice w.r.t. inclusion (c.f. Figure [REF]).', '1407.6666-1-4-1': 'In Section [REF] we introduce the configuration of [MATH]: the abstract lattice of its cyclic flats together with their cardinalities and ranks.', '1407.6666-1-4-2': 'We then prove:', '1407.6666-1-5-0': 'The Tutte polynomial of a matroid is determined by its configuration.', '1407.6666-1-6-0': 'While [MATH] is determined by its cyclic flats and their ranks (c.f. [CITATION]), it generally is far from being determined by its configuration (c.f. Figure [REF]); there are even superexponential families of matroids with identical configurations (c.f. [CITATION]).', '1407.6666-1-6-1': 'So Theorem [REF] explains one big part of the information lost when passing from [MATH] to its Tutte polynomial.', '1407.6666-1-7-0': 'In Section [REF] we incorporate symmetries in [MATH] to shrink down the information needed for its Tutte polynomial even more.', '1407.6666-1-7-1': 'Let [MATH], [MATH] be the set of [MATH]-orbits of [MATH] and [MATH] a system of representatives.', '1407.6666-1-7-2': 'The condensed configuration of [MATH] corresponding to [MATH] consists of the cardinalities and ranks of the [MATH] and the matrix [MATH] where [EQUATION].', '1407.6666-1-8-0': 'After discussing some examples, e.g. a condensed configuration for the Golay code matroid, we will prove:', '1407.6666-1-9-0': 'The Tutte polynomial of [MATH] is determined by a condensed configuration of [MATH].', '1407.6666-1-10-0': 'Section [REF] shows how to obtain a condensed configuration of a perfect matroid design using only the cardinalities of flats of given rank.', '1407.6666-1-10-1': "Together with Theorem [REF] this yields a new proof for Mphako's results about the Tutte polynomial of perfect matroid designs in [CITATION]."}

{'1407.6666-2-0-0': 'We show how the Tutte polynomial of a matroid [MATH] can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats.', '1407.6666-2-0-1': 'The results imply that the Tutte polynomial of [MATH] is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks.', '1407.6666-2-0-2': 'They furthermore generalize a similiar statement for perfect matroid designs due to Mphako [CITATION] and help to understand families of matroids with identical Tutte polynomial as constructed in [CITATION].', '1407.6666-2-1-0': '# Introduction', '1407.6666-2-2-0': 'The Tutte polynomial is a central invariant in matroid theory.', '1407.6666-2-2-1': 'But passing over from a matroid [MATH] to its Tutte polynomial [MATH] generally means a big loss of information.', '1407.6666-2-2-2': 'This paper gives one explanation for this phenomenon by showing how little information about the cyclic flats of a matroid is really needed for the computation of its Tutte polynomial.', '1407.6666-2-3-0': 'From now on let [MATH] be a matroid.', '1407.6666-2-3-1': 'A flat [MATH] in [MATH] is called cyclic if [MATH] contains no coloops.', '1407.6666-2-3-2': 'Section [REF] will recapitulate some basic facts about cyclic flats and show how the Tutte polynomial can be expressed in terms of cloud and flock polynomials of cyclic flats as introduced by Plesken in [CITATION].', '1407.6666-2-3-3': 'Then Section [REF] establishes some important identities for cloud and flock polynomials needed later on.', '1407.6666-2-4-0': '[MATH], the set of cyclic flats of [MATH], is a lattice w.r.t. inclusion (c.f. Figure [REF]).', '1407.6666-2-4-1': 'In Section [REF] we introduce the configuration of [MATH]: the abstract lattice of its cyclic flats together with their cardinalities and ranks.', '1407.6666-2-4-2': 'We then prove:', '1407.6666-2-5-0': 'The Tutte polynomial of a matroid is determined by its configuration.', '1407.6666-2-6-0': 'While [MATH] is determined by its cyclic flats and their ranks (c.f. [CITATION]), it generally is far from being determined by its configuration (c.f. Figure [REF]); there are even superexponential families of matroids with identical configurations (c.f. [CITATION]).', '1407.6666-2-6-1': 'So Theorem [REF] explains one big part of the information lost when passing from [MATH] to its Tutte polynomial.', '1407.6666-2-7-0': 'In Section [REF] we incorporate symmetries in [MATH] to shrink down the information needed for its Tutte polynomial even more.', '1407.6666-2-7-1': 'Let [MATH], [MATH] be the set of [MATH]-orbits of [MATH] and [MATH] a system of representatives.', '1407.6666-2-7-2': 'The condensed configuration of [MATH] corresponding to [MATH] consists of the cardinalities and ranks of the [MATH] and the matrix [MATH] where [EQUATION].', '1407.6666-2-8-0': 'After discussing some examples, e.g. a condensed configuration for the Golay code matroid, we will prove:', '1407.6666-2-9-0': 'The Tutte polynomial of [MATH] is determined by a condensed configuration of [MATH].', '1407.6666-2-10-0': 'Section [REF] shows how to obtain a condensed configuration of a perfect matroid design using only the cardinalities of flats of given rank.', '1407.6666-2-10-1': "Together with Theorem [REF] this yields a new proof for Mphako's results about the Tutte polynomial of perfect matroid designs in [CITATION]."}

[['1407.6666-1-3-0', '1407.6666-2-3-0'], ['1407.6666-1-3-1', '1407.6666-2-3-1'], ['1407.6666-1-3-2', '1407.6666-2-3-2'], ['1407.6666-1-3-3', '1407.6666-2-3-3'], ['1407.6666-1-9-0', '1407.6666-2-9-0'], ['1407.6666-1-7-0', '1407.6666-2-7-0'], ['1407.6666-1-7-1', '1407.6666-2-7-1'], ['1407.6666-1-7-2', '1407.6666-2-7-2'], ['1407.6666-1-0-0', '1407.6666-2-0-0'], ['1407.6666-1-0-1', '1407.6666-2-0-1'], ['1407.6666-1-0-2', '1407.6666-2-0-2'], ['1407.6666-1-10-0', '1407.6666-2-10-0'], ['1407.6666-1-10-1', '1407.6666-2-10-1'], ['1407.6666-1-4-0', '1407.6666-2-4-0'], ['1407.6666-1-4-1', '1407.6666-2-4-1'], ['1407.6666-1-6-0', '1407.6666-2-6-0'], ['1407.6666-1-6-1', '1407.6666-2-6-1'], ['1407.6666-1-5-0', '1407.6666-2-5-0'], ['1407.6666-1-2-0', '1407.6666-2-2-0'], ['1407.6666-1-2-1', '1407.6666-2-2-1'], ['1407.6666-1-2-2', '1407.6666-2-2-2']]

[['1407.6666-1-3-0', '1407.6666-2-3-0'], ['1407.6666-1-3-1', '1407.6666-2-3-1'], ['1407.6666-1-3-2', '1407.6666-2-3-2'], ['1407.6666-1-3-3', '1407.6666-2-3-3'], ['1407.6666-1-9-0', '1407.6666-2-9-0'], ['1407.6666-1-7-0', '1407.6666-2-7-0'], ['1407.6666-1-7-1', '1407.6666-2-7-1'], ['1407.6666-1-7-2', '1407.6666-2-7-2'], ['1407.6666-1-0-0', '1407.6666-2-0-0'], ['1407.6666-1-0-1', '1407.6666-2-0-1'], ['1407.6666-1-0-2', '1407.6666-2-0-2'], ['1407.6666-1-10-0', '1407.6666-2-10-0'], ['1407.6666-1-10-1', '1407.6666-2-10-1'], ['1407.6666-1-4-0', '1407.6666-2-4-0'], ['1407.6666-1-4-1', '1407.6666-2-4-1'], ['1407.6666-1-6-0', '1407.6666-2-6-0'], ['1407.6666-1-6-1', '1407.6666-2-6-1'], ['1407.6666-1-5-0', '1407.6666-2-5-0'], ['1407.6666-1-2-0', '1407.6666-2-2-0'], ['1407.6666-1-2-1', '1407.6666-2-2-1'], ['1407.6666-1-2-2', '1407.6666-2-2-2']]

[]

[]

[]

[]

['1407.6666-1-4-2', '1407.6666-1-8-0', '1407.6666-2-4-2', '1407.6666-2-8-0']

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

https://arxiv.org/abs/1407.6666

1409.6521

{'1409.6521-1-0-0': 'The quantum state of the emitted light from the cascade recombination of a biexciton in a quantum dot is theoretically investigated including exciton fine structure splitting (FSS) and electron-nuclear spin hyperfine interactions.', '1409.6521-1-0-1': 'In an ideal situation, the emitted photons are entangled in polarization making the biexciton recombination process a candidate source of entangled photons necessary for the growing field of quantum communication and computation.', '1409.6521-1-0-2': 'The coherence of the exciton states in real quantum dots is affected by a finite FSS and the hyperfine interactions via the effective magnetic field known as the Overhauser field.', '1409.6521-1-0-3': 'We investigate the influence of both sources of decoherence and find that the FSS combined with a stochastic exciton lifetime is responsible for the main loss of entanglement.', '1409.6521-1-0-4': 'Furthermore, we examine the possibility of reducing the decoherence from the Overhauser field by partially polarizing the nuclear spins and applying an external magnetic field.', '1409.6521-1-0-5': 'We find that an increase in entanglement depends on the degree as well as the direction of the nuclear spin polarization.', '1409.6521-1-1-0': '# Introduction', '1409.6521-1-2-0': 'A reliable source of entangled photons is a requirement for many protocols used in the rapidly developing field of quantum communication[CITATION].', '1409.6521-1-2-1': 'An established method for creating polarization entangled photons is by parametric down-conversion[CITATION].', '1409.6521-1-2-2': 'However, this technique suffers from being both inefficient and stochastic, causing problems since many quantum communication protocols require an on-demand source.', '1409.6521-1-2-3': 'An alternative source is thus desired, and here we consider the biexciton cascade recombination[CITATION].', '1409.6521-1-2-4': 'In a quantum dot (QD), the biexciton, which is composed of two conduction band electrons and two valence band holes, can under ideal conditions recombine under the emission of two photons entangled in polarization[CITATION].', '1409.6521-1-2-5': 'The biexciton recombines via one of two possible intermediate exciton states, each consisting of one conduction band electron and one valence band hole.', '1409.6521-1-3-0': 'In most quantum dots, the two optically active exciton states are energetically separated by a quantity known as the fine structure splitting (FSS) arising from higher order electron-hole exchange interactions[CITATION].', '1409.6521-1-3-1': 'A finite FSS can affect the coherence of the emitted two-photon state in two ways.', '1409.6521-1-3-2': 'If the FSS is larger than the linewidth of the emitted light, the photons become distinguishable via a simple frequency measurement which reveals the "which-way" information and destroys the entanglement[CITATION].', '1409.6521-1-3-3': 'However, even if the splitting is smaller than the linewidth but still finite, the initially coherent exciton state acquires a random phase before recombining, due to the stochastic life time.', '1409.6521-1-3-4': 'Methods of reducing and eliminating the FSS include applying magnetic[CITATION] and electric[CITATION] fields as well as strain[CITATION].', '1409.6521-1-4-0': 'In addition to the dephasing from a finite FSS, the intermediate exciton state is affected by the spins of the [MATH]-[MATH] nuclei present in a III-V group semiconductor QD.', '1409.6521-1-4-1': 'The spins of the electron[CITATION] and hole[CITATION] constituting the exciton couple to nuclear spins via the hyperfine interaction and are subject to an effective nuclear magnetic field, known as the Overhauser field.', '1409.6521-1-4-2': 'Because of the large number of nuclear spins, the Overhauser field can be considered to be stochastic and is another source of decoherence, including an additional random phase of the intermediate exciton state.', '1409.6521-1-5-0': 'Experimentally, various techniques for creating entangled light using semiconductor microstructures have been demonstrated[CITATION].', '1409.6521-1-5-1': 'One successful approach relies on the application of an external magnetic field perpendicular to the growth direction of the QD[CITATION], which tunes the energy levels of the optically active excitons by hybridization to optically inactive states.', '1409.6521-1-5-2': 'Nevertheless, a complete theoretical explanation of the partial loss of entanglement is still missing.', '1409.6521-1-5-3': 'Understanding the dynamics of the intermediate exciton states is essential to investigate the entangleement of the emitted light.', '1409.6521-1-5-4': 'In fact, any dephasing and loss of coherence of the exciton state will be reflected in the final photon state.', '1409.6521-1-6-0': 'In this paper, we investigate the interplay between the dephasing due to a finite FSS together with a stochastic recombination time and the decoherence caused by the hyperfine interaction.', '1409.6521-1-6-1': 'One way of reducing the fluctuations of the Overhauser field is by dynamic nuclear spin polarization[CITATION], causing the nuclear spins to have a preferred direction.', '1409.6521-1-6-2': 'The polarized nuclear ensemble produces a finite effective magnetic field, which may modify the exciton energies and eigenstates.', '1409.6521-1-6-3': 'Since the elimination of the FSS has been demonstrated using an external in-plane magnetic field, a tempting idea could be to use the effective magnetic field produced by the polarized nuclear ensemble, to reduce both the FSS and the Overhauser field fluctuations.', '1409.6521-1-6-4': 'However, we show that the effect from the two sources of decoherence cannot be minimized independently of each other.', '1409.6521-1-7-0': 'We consider the effect of finite nuclear spin polarization and find that the entanglement of the emitted light can be improved by nuclear spin polarization.', '1409.6521-1-7-1': 'The efficiency of the entanglement improvement depends on the direction along which the nuclear spins are polarized, which is explained by the fact that the [MATH]-tensor is not isotropic.', '1409.6521-1-7-2': 'We find that the maximum enhancement of the entanglement is achieved when nuclear spins are polarized along the direction for which the [MATH]-tensor has its largest components, in our case in the growth direction of the quantum dot.', '1409.6521-1-8-0': 'A nuclear polarization in the growth direction gives an additional contribution to the FSS and therefore increases the dephasing.', '1409.6521-1-8-1': 'To cancel the effective nuclear field in the growth direction and minimize the FSS at the same time, we propose applying an external magnetic field along a specific direction having an in-plane and a perpendicular component.', '1409.6521-1-8-2': 'Combining a finite nuclear spin polarization along the growth direction of the QD with an external magnetic field, we find a significant improvement of the two-photon state entanglement.', '1409.6521-1-9-0': '# Theoretical model', '1409.6521-1-10-0': 'We consider a QD of a cubic semiconductor containing one exciton consisting of one electron and one heavy hole.', '1409.6521-1-10-1': 'The spins of the electron and the hole couple to the spins of the atomic nuclei in the QD.', '1409.6521-1-10-2': 'This effect is to a good approximation described by the contact hyperfine Hamiltonian [EQUATION] where the summation runs over all nuclear spins in the dot, [MATH] is the spin operator of the [MATH]-th nuclear spin, [MATH] is the electron(hole) spin operator, and [MATH] are the hyperfine coupling tensors between the [MATH]-th nuclear spin and corresponding electron(hole) spin component.', '1409.6521-1-10-3': 'We define effective magnetic fields by moving the electron and hole spin operators outside of the summation and obtain [EQUATION] where [MATH] is the [MATH]-tensor for the electron(hole).', '1409.6521-1-10-4': 'The vector [MATH] is known as the electron(hole) Overhauser field and acts like an effective magnetic field from the perspective of the exciton.', '1409.6521-1-10-5': 'Because of the large number of nuclear spins[CITATION] ([MATH]) for a typical quantum dot, the Overhauser field is often modelled as a stochastic magnetic field and will be considered further in Section [REF].', '1409.6521-1-10-6': 'We consider the case of diagonal hyperfine coupling tensors, which for an electron in a III-V semiconductor QD can be written[CITATION] as [MATH], where [MATH] are the electron [MATH]-factors and [MATH] is the electron density at the atomic site [MATH].', '1409.6521-1-10-7': 'An important feature is the dependence of the spatial direction [MATH], which indicates that fluctuations of the different spatial components of the Overhauser field influence the energy of the electron differently.', '1409.6521-1-10-8': 'Normally, the in-plane [MATH]-factors are smaller than along the growth axis of the quantum dot.', '1409.6521-1-11-0': 'A basis for the Hilbert space of the electron spin [MATH] is given by [MATH], where [MATH] corresponds to the spin [MATH]) state, and for the heavy hole the Hilbert space [MATH] is spanned by [MATH], with [MATH] corresponding to the hole spin states [MATH]).', '1409.6521-1-11-1': 'The Hilbert space of the exciton is given by the product space [MATH] and is spanned by the basis vectors [MATH].', '1409.6521-1-11-2': 'The states [MATH] and [MATH] are known as bright since they can recombine under the emission of a single photon whereas [MATH] and [MATH] are known as dark.', '1409.6521-1-11-3': 'The idealized recombination chain of the biexciton is given by', '1409.6521-1-12-0': '0_ph', '1409.6521-1-13-0': '_++ _-2', '1409.6521-1-14-0': '_+_-+_-_+20_e', '1409.6521-1-15-0': 'where [MATH] are photon states of circularly polarized light, and [MATH] is the photon (crystal) vacuum.', '1409.6521-1-15-1': 'In reality the intermediate state undergoes time-evolution before the exciton has recombined which may lead to degradation of the entanglement of the emitted light.', '1409.6521-1-15-2': 'The final state of the intermediate exciton state can be written using a density matrix [EQUATION] where [MATH] and [MATH] are the populations of the states [MATH] and [MATH], and [MATH] is the off-diagonal matrix element required to describe a quantum mechanical superposition of the basis states.', '1409.6521-1-15-3': 'Here, only there bright excitons are taken into consideration.', '1409.6521-1-15-4': 'The concurrence of the emitted light is then given by[CITATION] [EQUATION] where we note that the off-diagonal elements are essential for the entanglement.', '1409.6521-1-16-0': 'Under the influence of a magnetic field [MATH] the exciton system is described by the Hamiltonian[CITATION] [EQUATION] in the heavy exciton basis [MATH], where [MATH] is the splitting between bright and dark excitons, [MATH] is the FSS for bright (dark) excitons, [MATH], [MATH] and [MATH] are effective [MATH]-factors for electrons(holes) along the [MATH]-axis.', '1409.6521-1-16-1': 'For the case [MATH] and [MATH], the eigenvalue problem [MATH] is analytically solvable and the two eigenenergies corresponding to the two bright excitons are given by [EQUATION]', '1409.6521-1-16-2': 'Similar expressions were found by Bayer et.', '1409.6521-1-16-3': 'al.[CITATION], where it should be noted that our expressions differ slightly, due to a sign error.', '1409.6521-1-16-4': 'Demanding the bright exciton states to be degenerate, i.e. [MATH], gives an equation for a critical field [MATH], [EQUATION] for which the FSS vanishes.', '1409.6521-1-16-5': 'The FSS can be written [MATH], which may be expanded in Maclaurin series in [MATH] to the second order, which gives the approximation [EQUATION]', '1409.6521-1-16-6': 'A similar expression was also presented by Stevenson et al.[CITATION].', '1409.6521-1-16-7': 'Solving [MATH] provides an expression for the critical magnetic field, given by [EQUATION]', '1409.6521-1-16-8': 'For the general case involving arbitrary mangetic fields and complex [MATH], an analytical diagonalization of the Hamiltonian in Eq. ([REF]) is not known.', '1409.6521-1-16-9': 'However, the energy splitting between the bright and dark excitons [MATH] is larger than all other relevant energies, including the magnetic coupling elements.', '1409.6521-1-16-10': 'Therefore, we can apply the Schrieffer-Wolff[CITATION] transformation, which provides us with an effective Hamiltonian for the bright exciton subspace: [EQUATION] where [EQUATION] and [EQUATION] with [MATH].', '1409.6521-1-16-11': 'The FSS [MATH] can now be found by solving the eigenvalue problem [EQUATION] for [MATH] and taking the difference [MATH].', '1409.6521-1-16-12': 'Using the explicit form of [MATH] given by Eqs. ([REF]) and ([REF]) we find [EQUATION] where [EQUATION]', '1409.6521-1-16-13': 'Demanding that [MATH] again defines critical magnetic fields, not necessarily along [MATH], for which the FSS is eliminated.', '1409.6521-1-16-14': 'Inspecting Eq. ([REF]), we realize that [MATH] only if [MATH] and [MATH].', '1409.6521-1-16-15': '[MATH] can be tuned to zero by adjusting the total magnetic field along [MATH].', '1409.6521-1-16-16': '[MATH] depends on in in-plane components of the magnetic field, [MATH] and [MATH] and the bright exciton coupling [MATH], which may be complex.', '1409.6521-1-16-17': 'The phase of [MATH] is related to the geometry of the quantum dot whereas the phase of [MATH] depends on the anisotropy of the [MATH]-tensors.', '1409.6521-1-16-18': 'For a quantum dot with an isotropic in-plane [MATH]-tensor, [MATH] is real and in this situation, no magnetic field can completely eliminate the FSS caused by a complex [MATH].', '1409.6521-1-16-19': 'However, [MATH] combined with a isotropic [MATH]-tensor, Eq. ([REF]) reveals that [MATH] is a criterion for the existence of a magnetic field such that [MATH].', '1409.6521-1-16-20': 'In turn, this requires that the in-plane [MATH]-factors of the electron and hole have opposite signs and this implies that not all quantum dots can be tuned to support degenerate bright excitons, also noted in experimental work[CITATION] where InAs dots surrounded by different barrier materials were studied.', '1409.6521-1-16-21': 'The corresponding in-plane [MATH]-factors were extracted, and here the values for the case of Al[MATH]Ga[MATH]As[CITATION] as the barrier material are given in Table [REF].', '1409.6521-1-16-22': 'Experimental[CITATION] and theoretical[CITATION] studies show that the [MATH]-factor of InAs and InGaAs QDs can be tuned over large range of values and here we use [MATH], [MATH] to give a total exciton [MATH]-factor [MATH], measured in experiments[CITATION].', '1409.6521-1-17-0': 'There are two main sources of loss of entanglement: (1) the fine-structure splitting combined with the stochastic exciton life time and (2) the stochastic Overhauser field that affect the intermediate exciton state.', '1409.6521-1-17-1': 'Both mechanisms lead to the acquisition of an unknown phase which causes a reduction of the entanglement.', '1409.6521-1-17-2': 'To investigate further, we consider the effect of the two above-mentioned mechanisms on the density operator of the intermediate exciton state.', '1409.6521-1-17-3': 'We choose the diagonal basis [MATH] which are eigenvectors of [MATH], and an initial density operator in matrix form as [EQUATION] where [MATH] and [MATH].', '1409.6521-1-17-4': 'We can now determine the time-evolution for the density operator via the Heisenberg equation of motion, [EQUATION] which has the solution [EQUATION] with the FSS [MATH] given by Eqs. ([REF])-([REF]).', '1409.6521-1-17-5': 'If the FSS is stochastic as one would expect from an Overhauser field we may find its contribution by statistical averaging [EQUATION] where [MATH] is the probability density function of the FSS.', '1409.6521-1-18-0': 'For the quantum dot hosting the exciton we assume a stochastic Overhauser field with Gaussian distribution.', '1409.6521-1-18-1': 'The FSS, however, does not have a Gaussian distribution because of the nonlinear way the exciton eigenenergies depend on an applied magnetic field, given by Eqs. ([REF])-([REF]).', '1409.6521-1-18-2': 'Therefore, the statistical averaging is performed by considering a Gaussian distribution for the Overhauser field with the probability density function [EQUATION] where [MATH], [MATH], [MATH] are the standard deviations of the Overhauser field along [MATH], [MATH], [MATH].', '1409.6521-1-18-3': 'We numerically evaluate [EQUATION] using the parameter values given in Table [REF], from which we can extract an entanglement measure, here the concurrence [MATH] by using Eq. ([REF]).', '1409.6521-1-19-0': 'To investigate the effect of the stochastic exciton life time we consider a Poissonian recombination process which corresponds to an exponential life time [MATH] with probability density function [MATH] where [MATH] is the average life time.', '1409.6521-1-19-1': 'Calculating the statistical average of the density matrix gives [EQUATION] which also have decaying concurrence[CITATION] [MATH] for [MATH].', '1409.6521-1-19-2': 'This suggests choosing [MATH] to maximize the concurrence.', '1409.6521-1-19-3': 'The two different concurrences are shown in Fig. [REF].', '1409.6521-1-19-4': 'We can see that there is a target conflict when applying a magnetic field along [MATH].', '1409.6521-1-19-5': 'For a critical magnetic field strength [MATH] the fine structure splitting is eliminated and [MATH] has a maximum, but the concurrence [MATH] when considering a stochastic magnetic field from the nuclear spins has a minimum.', '1409.6521-1-19-6': 'The reason is that the FSS is most sensitive to changes in the magnetic field at this point.', '1409.6521-1-19-7': 'To obtain a more complete picture we need to take both sources of decoherence into account simultaneously, which we achieve by averaging the concurrence in Eq. ([REF]) using the probability distribution for the stochastic magnetic field [MATH] given by Eq. ([REF]), which is done numerically by evaluating [EQUATION]', '1409.6521-1-20-0': '# Results', '1409.6521-1-21-0': 'To obtain quantitative results, we choose a set of parameter for the quantum dot given in Table [REF].', '1409.6521-1-22-0': '## Dominant Source of Decoherence', '1409.6521-1-23-0': 'In order to improve the concurrence we first establish which source of decoherence causes more loss of concurrence, the FSS or the Overhauser field.', '1409.6521-1-23-1': 'From Fig. [REF] this is not obvious, because at [MATH] the FSS is minimized but the dephasing from the Overhauser field is maximized.', '1409.6521-1-23-2': 'Taking both into account and allowing in addition a magnetic field to be applied along [MATH] as well we find the concurrence as function of the applied magnetic field depicted in Fig. [REF].', '1409.6521-1-24-0': 'We see that the two global maxima are located at [MATH] which indicates that the FSS is a stronger source of decoherence than the Overhauser field.', '1409.6521-1-24-1': 'Still, the concurrence does not reach unity but is rather close to the minimum observed in Fig. [REF]a. From these observations we conclude that in order to maximize the concurrence, we should keep [MATH] to eliminate the FSS and now focus on reducing the uncertainty of the Overhauser field.', '1409.6521-1-24-2': 'One way of achieving this is to polarize the nuclear spins, which has been experimentally realized[CITATION], and is investigated in the next section.', '1409.6521-1-24-3': 'In addition to the two global maxima, there are four local maxima located close to [MATH] T, [MATH] T. Although the concurrence is smaller at these points than at the global maxima, they indicate the significance of including the effects of both sources of decoherence simultaneously.', '1409.6521-1-25-0': '## Effect of Nuclear Spin Polarization', '1409.6521-1-26-0': 'It is clear that, within our model, when the FSS is eliminated, the remaining reduction of the entanglement originates from the Overhauser field.', '1409.6521-1-26-1': 'To investigate how the fluctuations of the Overhauser field vary as function of the nuclear spin polarization we consider a simple model for the Overhauser field along one spatial direction [EQUATION] where [MATH] is the number of nuclear spins, [MATH] are binary stochastic variables taking the values [MATH] with probability [EQUATION] where [EQUATION] [MATH] is the nuclear [MATH]-factor, [MATH] is an external magnetic field and [MATH] is the nuclear spin temperature.', '1409.6521-1-27-0': 'The polarization [MATH] is given by [EQUATION] and the variance is consequently [EQUATION]', '1409.6521-1-27-1': 'In Appendix [REF] it is shown that [EQUATION] where [MATH] is a Gaussian distribution with mean [MATH] and standard deviation [MATH], [MATH] is the number of nuclear spins, and [MATH] depends on [MATH] and [MATH].', '1409.6521-1-27-2': 'Typically, [MATH] will have to be determined experimentally or by numerical simulations and we do not attempt to calculate it here, but the general form Eq. ([REF]) does not depend on the specific QD.', '1409.6521-1-27-3': 'Since the fluctuations of the Overhauser field decrease with increasing polarization we now assume that the nuclear spins are polarized to degree [MATH] along [MATH], where [MATH].', '1409.6521-1-27-4': 'The assumption that the nuclear spin can be polarized along an arbitrary direction relies on experimental demonstrations[CITATION].', '1409.6521-1-27-5': 'This gives an effective magnetic field [MATH], with variances [EQUATION]', '1409.6521-1-27-6': 'Together with the applied magnetic field [MATH] the total effective magnetic field depends on 7 variables: [MATH] and [MATH].', '1409.6521-1-27-7': 'In order to narrow the search for optimal parameters, we make the following observations: first, [MATH] and [MATH] are equivalent and we set [MATH].', '1409.6521-1-27-8': 'Second, Fig. [REF] shows that the concurrence [MATH] has its maximum for [MATH] and we thus set [MATH].', '1409.6521-1-27-9': 'Finally we let [MATH] and the total effective magnetic field is given by [EQUATION] and depends on the three free parameters [MATH], [MATH], and [MATH].', '1409.6521-1-27-10': 'For [MATH] the result is shown in Fig. [REF] and we find that for every [MATH] there are two applied magnetic fields along [MATH] locally maximizing the concurrence.', '1409.6521-1-27-11': 'As expected from the discussion in the previous section, these occur when [MATH].', '1409.6521-1-27-12': 'We may thus set [MATH] and study concurrence as a function of the polarization angle [MATH] which is shown in Fig. [REF], where we observe that the concurrence is maximized by minimizing fluctuations along [MATH].', '1409.6521-1-28-0': 'Finally we can investigate the concurrence as a function of polarization, shown in the inset of Fig. [REF].', '1409.6521-1-28-1': 'We find that an increased nuclear spin polarization along [MATH] leads to an increased concurrence.', '1409.6521-1-28-2': 'We also see that a nuclear spin polarization perpendicular to [MATH] has almost no effect on the concurrence.', '1409.6521-1-28-3': 'This can be explained by the fact that the [MATH]-factors for the [MATH]- and [MATH]-directions are much smaller than the one along [MATH].', '1409.6521-1-29-0': '# Summary', '1409.6521-1-30-0': 'We have theoretically investigated the entanglement between two photons emitted from a cascade recombination of a biexciton in a quantum dot.', '1409.6521-1-30-1': 'The entanglement was examined using the concurrence as a quantitative measure.', '1409.6521-1-30-2': 'We considered the two main sources of loss of concurrence, the FSS combined with a stochastic intermediate exciton lifetime and the stochastic Overhauser field.', '1409.6521-1-30-3': 'We found that the FSS is the dominant source of decoherence and must be minimized in order to maximize concurrence.', '1409.6521-1-30-4': 'Furthermore, we showed that reducing the uncertainty of the Overhauser field by nuclear spin polarization together with an applied magnetic field along a certain direction can improve the concurrence of the emitted light.', '1409.6521-1-30-5': 'The increase in entanglement depends strongly on the degree as well as the direction of nuclear spin polarization relative to the growth axis of the QD.', '1409.6521-1-30-6': 'This effect is caused by the difference between in-plane [MATH]-factors and the [MATH]-factor along the growth direction.'}

{'1409.6521-2-0-0': 'The quantum state of the emitted light from the cascade recombination of a biexciton in a quantum dot is theoretically investigated including exciton fine structure splitting (FSS) and electron-nuclear spin hyperfine interactions.', '1409.6521-2-0-1': 'In an ideal situation, the emitted photons are entangled in polarization making the biexciton recombination process a candidate source of entangled photons necessary for the growing field of quantum communication and computation.', '1409.6521-2-0-2': 'The coherence of the exciton states in real quantum dots is affected by a finite FSS and the hyperfine interactions via the effective magnetic field known as the Overhauser field.', '1409.6521-2-0-3': 'We investigate the influence of both sources of decoherence and find that although the FSS combined with a stochastic exciton lifetime is responsible for the main loss of entanglement, the two effects cannot be minimized independently of each other.', '1409.6521-2-0-4': 'Furthermore, we examine the possibility of reducing the decoherence from the Overhauser field by partially polarizing the nuclear spins and applying an external magnetic field.', '1409.6521-2-0-5': 'We find that an increase in entanglement depends on the degree as well as the direction of the nuclear spin polarization.', '1409.6521-2-1-0': '# Introduction', '1409.6521-2-2-0': 'A reliable source of entangled photons is a requirement for many protocols used in the rapidly developing field of quantum communication[CITATION].', '1409.6521-2-2-1': 'An established method for creating polarization entangled photons is by parametric down-conversion[CITATION].', '1409.6521-2-2-2': 'However, this technique suffers from being both inefficient and stochastic, causing problems since many quantum communication protocols require an on-demand source.', '1409.6521-2-2-3': 'An alternative source is thus desired, and here we consider the biexciton cascade recombination[CITATION].', '1409.6521-2-2-4': 'In a quantum dot (QD), the biexciton, which is composed of two conduction band electrons and two valence band holes, can under ideal conditions recombine under the emission of two photons entangled in polarization[CITATION].', '1409.6521-2-2-5': 'The biexciton recombines via one of two possible intermediate exciton states, each consisting of one conduction band electron and one valence band hole.', '1409.6521-2-3-0': 'In most quantum dots, the two optically active exciton states are energetically separated by a quantity known as the fine structure splitting (FSS) arising from higher order electron-hole exchange interactions[CITATION].', '1409.6521-2-3-1': 'A finite FSS can affect the coherence of the emitted two-photon state in two ways.', '1409.6521-2-3-2': 'If the FSS is larger than the linewidth of the emitted light, the photons become distinguishable via a simple frequency measurement which reveals the "which-way" information and destroys the entanglement[CITATION].', '1409.6521-2-3-3': 'However, even if the splitting is smaller than the linewidth but still finite, the initially coherent exciton state acquires a random phase before recombining, due to the stochastic life time.', '1409.6521-2-3-4': 'Methods of reducing and eliminating the FSS include applying magnetic[CITATION] and electric[CITATION] fields as well as strain[CITATION].', '1409.6521-2-4-0': 'In addition to the dephasing from a finite FSS, the intermediate exciton state is affected by the spins of the [MATH]-[MATH] nuclei present in a III-V group semiconductor QD.', '1409.6521-2-4-1': 'The spins of the electron[CITATION] and hole[CITATION] constituting the exciton couple to nuclear spins via the hyperfine interaction and are subject to an effective nuclear magnetic field, known as the Overhauser field.', '1409.6521-2-4-2': 'Because of the large number of nuclear spins, the Overhauser field can be considered to be stochastic and is another source of decoherence, including an additional random phase of the intermediate exciton state.', '1409.6521-2-5-0': 'Experimentally, various techniques for creating entangled light using semiconductor microstructures have been demonstrated[CITATION].', '1409.6521-2-5-1': 'One successful approach relies on the application of an external magnetic field perpendicular to the growth direction of the QD[CITATION], which tunes the energy levels of the optically active excitons by hybridization to optically inactive states.', '1409.6521-2-5-2': 'This requires the in-plane [MATH]-factors of the electrons and holes to have opposite signs.', '1409.6521-2-5-3': 'InAs dots with AlGaAs barrier material having this property were reported[CITATION] and by applying an in-plane magnetic field, the FSS were tuned to zero.', '1409.6521-2-5-4': 'Nevertheless, a complete theoretical explanation of the partial loss of entanglement is still missing.', '1409.6521-2-5-5': 'Understanding the dynamics of the intermediate exciton states is essential to investigate the entanglement of the emitted light.', '1409.6521-2-5-6': 'In fact, any dephasing and loss of coherence of the exciton state will be reflected in the final photon state.', '1409.6521-2-6-0': 'In this paper, we investigate the interplay between the dephasing due to a finite FSS together with a stochastic recombination time and the decoherence caused by the hyperfine interaction.', '1409.6521-2-6-1': 'Our results concern a dot for which the FSS is tuned to zero by an in-plane magnetic field, which requires the in-plane electron and hole [MATH]-factors to have opposite signs.', '1409.6521-2-6-2': 'One way of reducing the fluctuations of the Overhauser field is by dynamic nuclear spin polarization[CITATION], causing the nuclear spins to have a preferred direction.', '1409.6521-2-6-3': 'The polarized nuclear ensemble produces a finite effective magnetic field, which may modify the exciton energies and eigenstates.', '1409.6521-2-6-4': 'Since the elimination of the FSS has been demonstrated using an external in-plane magnetic field, a tempting idea could be to use the effective magnetic field produced by the polarized nuclear ensemble, to reduce both the FSS and the Overhauser field fluctuations.', '1409.6521-2-6-5': 'However, this turns out not to be possible, since an effective in-plane magnetic field for both holes and electrons is required.', '1409.6521-2-6-6': 'For heavy holes, the in-plane component of the nuclear magnetic field vanishes.', '1409.6521-2-6-7': 'Furthermore, we show that the effect from the two sources of decoherence cannot be minimized independently of each other.', '1409.6521-2-7-0': 'We consider the effect of finite nuclear spin polarization and find that the entanglement of the emitted light can be improved by nuclear spin polarization.', '1409.6521-2-7-1': 'The efficiency of the entanglement improvement depends on the direction along which the nuclear spins are polarized, which is explained by the fact that the hyperfine coupling tensor is not isotropic.', '1409.6521-2-7-2': 'We find that the maximum enhancement of the entanglement is achieved when nuclear spins are polarized along the direction for which the coupling tensor has its largest components, in our case in the growth direction of the quantum dot.', '1409.6521-2-8-0': 'A nuclear polarization in the growth direction gives an additional contribution to the FSS and therefore increases the dephasing.', '1409.6521-2-8-1': 'To cancel the effective nuclear field in the growth direction and minimize the FSS at the same time, we propose applying an external magnetic field along a specific direction having an in-plane and a perpendicular component.', '1409.6521-2-8-2': 'Combining a finite nuclear spin polarization along the growth direction of the QD with an external magnetic field, we find a significant improvement of the two-photon state entanglement.', '1409.6521-2-9-0': '# Theoretical model', '1409.6521-2-10-0': 'We consider a QD of a cubic semiconductor containing one exciton consisting of one electron and one heavy hole.', '1409.6521-2-10-1': 'The spins of the electron and the hole couple to the spins of the atomic nuclei in the QD.', '1409.6521-2-10-2': 'This effect is to a good approximation described by the contact hyperfine Hamiltonian [EQUATION] where the summation runs over all nuclear spins in the dot, [MATH] is the spin operator of the [MATH]-th nuclear spin, [MATH] is the electron(hole) spin operator, and [MATH] are the hyperfine coupling tensors between the [MATH]-th nuclear spin and corresponding electron(hole) spin component.', '1409.6521-2-10-3': 'In this work, we are interested in heavy holes, for which only the [MATH] component of the [MATH] tensor is finite.', '1409.6521-2-10-4': 'This allows us to define an effective magnetic field by moving the electron and hole spin operators outside of the summation and obtain [EQUATION] where [MATH] is the electron-nuclear spin coupling tensor.', '1409.6521-2-10-5': 'The heavy hole - nuclear spin coupling is of Ising type, described by the coupling constant [MATH] between the [MATH]-component of hole and nuclear spins.', '1409.6521-2-10-6': 'The vector [MATH] is known as the Overhauser field and acts like an effective magnetic field from the perspective of the exciton.', '1409.6521-2-10-7': 'Because of the large number of nuclear spins[CITATION] ([MATH]) for a typical quantum dot, the Overhauser field is often modelled as a stochastic magnetic field and will be considered further in Section [REF].', '1409.6521-2-10-8': 'We consider the case of diagonal hyperfine coupling tensors, which for an electron in a III-V semiconductor QD can be written[CITATION] as [MATH], where [MATH] are the electron [MATH]-factors and [MATH] is the electron density at the atomic site [MATH].', '1409.6521-2-10-9': 'An important feature is the dependence of the spatial direction [MATH], which indicates that fluctuations of the different spatial components of the Overhauser field influence the energy of the electron differently.', '1409.6521-2-11-0': 'A basis for the Hilbert space of the electron spin [MATH] is given by [MATH], where [MATH] corresponds to the spin [MATH]) state, and for the heavy hole the Hilbert space [MATH] is spanned by [MATH], with [MATH] corresponding to the hole spin states [MATH]).', '1409.6521-2-11-1': 'The Hilbert space of the exciton is given by the product space [MATH] and is spanned by the basis vectors [MATH].', '1409.6521-2-11-2': 'The states [MATH] and [MATH] are known as bright since they can recombine under the emission of a single photon whereas [MATH] and [MATH] are known as dark.', '1409.6521-2-11-3': 'The idealized recombination chain of the biexciton is given by', '1409.6521-2-12-0': '0_ph', '1409.6521-2-13-0': '_++ _-2', '1409.6521-2-14-0': '_+_-+_-_+20_e', '1409.6521-2-15-0': 'where [MATH] are photon states of circularly polarized light, and [MATH] is the photon (crystal) vacuum.', '1409.6521-2-15-1': 'In reality the intermediate state undergoes time-evolution before the exciton has recombined which may lead to degradation of the entanglement of the emitted light.', '1409.6521-2-15-2': 'The final state of the intermediate exciton state can be written using a density matrix [EQUATION] where [MATH] and [MATH] are the populations of the states [MATH] and [MATH], and [MATH] is the off-diagonal matrix element required to describe a quantum mechanical superposition of the basis states.', '1409.6521-2-15-3': 'Here, only there bright excitons are taken into consideration.', '1409.6521-2-15-4': 'The concurrence of the emitted light is then given by[CITATION] [EQUATION] where we note that the off-diagonal elements are essential for the entanglement.', '1409.6521-2-16-0': 'Under the influence of a magnetic field [MATH] the exciton system is described by the Hamiltonian[CITATION] [EQUATION] in the heavy exciton basis [MATH], where [MATH] is the splitting between bright and dark excitons, [MATH] is the FSS for bright (dark) excitons, [MATH], [MATH] and [MATH] are effective [MATH]-factors for electrons(holes) along the [MATH]-axis.', '1409.6521-2-16-1': 'For the case [MATH] and [MATH], the eigenvalue problem [MATH] is analytically solvable and the two eigenenergies corresponding to the two bright excitons are given by [EQUATION]', '1409.6521-2-16-2': 'Similar expressions were found by Bayer et.', '1409.6521-2-16-3': 'al.[CITATION], where it should be noted that our expressions differ slightly, due to a sign error.', '1409.6521-2-16-4': 'Demanding the bright exciton states to be degenerate, i.e. [MATH], gives an equation for a critical field [MATH], [EQUATION] for which the FSS vanishes.', '1409.6521-2-16-5': 'The FSS can be written [MATH], which may be expanded in Maclaurin series in [MATH] to the second order, which gives the approximation [EQUATION]', '1409.6521-2-16-6': 'A similar expression was also presented by Stevenson et al.[CITATION].', '1409.6521-2-16-7': 'Solving [MATH] provides an expression for the critical magnetic field, given by [EQUATION]', '1409.6521-2-16-8': 'For the general case involving arbitrary magnetic fields and complex [MATH], an analytical diagonalization of the Hamiltonian in Eq. ([REF]) is not known.', '1409.6521-2-16-9': 'However, the energy splitting between the bright and dark excitons [MATH] is larger than all other relevant energies, including the magnetic coupling elements.', '1409.6521-2-16-10': 'Therefore, we can apply the Schrieffer-Wolff[CITATION] transformation, which provides us with an effective Hamiltonian for the bright exciton subspace: [EQUATION] where [EQUATION] and [EQUATION] with [MATH].', '1409.6521-2-16-11': 'The FSS [MATH] can now be found by solving the eigenvalue problem [EQUATION] for [MATH] and taking the difference [MATH].', '1409.6521-2-16-12': 'Using the explicit form of [MATH] given by Eqs. ([REF]) and ([REF]) we find [EQUATION] where [EQUATION]', '1409.6521-2-16-13': 'Demanding that [MATH] again defines critical magnetic fields, not necessarily along [MATH], for which the FSS is eliminated.', '1409.6521-2-16-14': 'Inspecting Eq. ([REF]), we realize that [MATH] only if [MATH] and [MATH].', '1409.6521-2-16-15': '[MATH] can be tuned to zero by adjusting the total magnetic field along [MATH].', '1409.6521-2-16-16': '[MATH] depends on in in-plane components of the magnetic field, [MATH] and [MATH] and the bright exciton coupling [MATH], which may be complex.', '1409.6521-2-16-17': 'The phase of [MATH] is related to the geometry of the quantum dot whereas the phase of [MATH] depends on the anisotropy of the [MATH]-tensors.', '1409.6521-2-16-18': 'For a quantum dot with an isotropic in-plane [MATH]-tensor, [MATH] is real and in this situation, no magnetic field can completely eliminate the FSS caused by a complex [MATH].', '1409.6521-2-16-19': 'However, [MATH] combined with a isotropic [MATH]-tensor, Eq. ([REF]) reveals that [MATH] is a criterion for the existence of a magnetic field such that [MATH].', '1409.6521-2-16-20': 'In turn, this requires that the in-plane [MATH]-factors of the electron and hole have opposite signs and this implies that not all quantum dots can be tuned to support degenerate bright excitons, also noted in experimental work[CITATION] where InAs dots surrounded by different barrier materials were studied.', '1409.6521-2-16-21': 'The corresponding in-plane [MATH]-factors were extracted, and here the values for the case of Al[MATH]Ga[MATH]As[CITATION] as the barrier material are given in Table [REF].', '1409.6521-2-16-22': 'Experimental[CITATION] and theoretical[CITATION] studies show that the [MATH]-factor of InAs and InGaAs QDs can be tuned over large range of values and here we use [MATH], [MATH] to give a total exciton [MATH]-factor [MATH], measured in experiments[CITATION].', '1409.6521-2-17-0': 'There are two main sources of loss of entanglement: (1) the fine-structure splitting combined with the stochastic exciton life time and (2) the stochastic Overhauser field that affect the intermediate exciton state.', '1409.6521-2-17-1': 'Both mechanisms lead to the acquisition of an unknown phase which causes a reduction of the entanglement.', '1409.6521-2-17-2': 'To investigate further, we consider the effect of the two above-mentioned mechanisms on the density operator of the intermediate exciton state.', '1409.6521-2-17-3': 'We choose the diagonal basis [MATH] which are eigenvectors of [MATH], and an initial density operator in matrix form as [EQUATION] where [MATH] and [MATH].', '1409.6521-2-17-4': 'We can now determine the time-evolution for the density operator via the Heisenberg equation of motion, [EQUATION] which has the solution [EQUATION] with the FSS [MATH] given by Eqs. ([REF])-([REF]).', '1409.6521-2-17-5': 'If the FSS is stochastic as one would expect from an Overhauser field we may find its contribution by statistical averaging [EQUATION] where [MATH] is the probability density function of the FSS.', '1409.6521-2-18-0': 'For the quantum dot hosting the exciton we assume a stochastic Overhauser field with Gaussian distribution.', '1409.6521-2-18-1': 'The FSS, however, does not have a Gaussian distribution because of the nonlinear way the exciton eigenenergies depend on an applied magnetic field, given by Eqs. ([REF])-([REF]).', '1409.6521-2-18-2': 'Therefore, the statistical averaging is performed by considering a Gaussian distribution for the Overhauser field with the probability density function [EQUATION] where [MATH], [MATH], [MATH] are the standard deviations of the Overhauser field along [MATH], [MATH], [MATH].', '1409.6521-2-18-3': 'We numerically evaluate [EQUATION] using the parameter values given in Table [REF], from which we can extract an entanglement measure, here the concurrence [MATH] by using Eq. ([REF]).', '1409.6521-2-18-4': 'An external magnetic field, [MATH] may also be present, and both the Overhauser field and the external magnetic field are included by replacing [MATH] in Eqs. ([REF])-([REF]), where [MATH] and [MATH].', '1409.6521-2-19-0': 'To investigate the effect of the stochastic exciton life time we consider a Poissonian recombination process which corresponds to an exponential life time [MATH] with probability density function [MATH] where [MATH] is the average life time.', '1409.6521-2-19-1': 'Calculating the statistical average of the density matrix gives [EQUATION] which also have decaying concurrence[CITATION] [MATH] for [MATH].', '1409.6521-2-19-2': 'This suggests choosing [MATH] to maximize the concurrence.', '1409.6521-2-19-3': 'The two different concurrences are shown in Fig. [REF].', '1409.6521-2-19-4': 'We can see that there is a target conflict when applying a magnetic field along [MATH].', '1409.6521-2-19-5': 'For a critical magnetic field strength [MATH] the fine structure splitting is eliminated and [MATH] has a maximum, but the concurrence [MATH] when considering a stochastic magnetic field from the nuclear spins has a minimum.', '1409.6521-2-19-6': 'The reason is that the FSS is most sensitive to changes in the magnetic field at this point.', '1409.6521-2-19-7': 'To obtain a more complete picture we need to take both sources of decoherence into account simultaneously, which we achieve by averaging the concurrence in Eq. ([REF]) using the probability distribution for the stochastic magnetic field [MATH] given by Eq. ([REF]), which is done numerically by evaluating [EQUATION]', '1409.6521-2-20-0': '# Results', '1409.6521-2-21-0': 'To obtain quantitative results, we choose a set of parameter for the quantum dot given in Table [REF].', '1409.6521-2-22-0': '## Dominant Source of Decoherence', '1409.6521-2-23-0': 'In order to improve the concurrence we first establish which source of decoherence causes more loss of concurrence, the FSS or the Overhauser field.', '1409.6521-2-23-1': 'From Fig. [REF] this is not obvious, because at [MATH] the FSS is minimized but the dephasing from the Overhauser field is maximized.', '1409.6521-2-23-2': 'Taking both into account and allowing in addition a magnetic field to be applied along [MATH] as well we find the concurrence as function of the applied magnetic field depicted in Fig. [REF].', '1409.6521-2-23-3': 'From our calculations we find a maximum value for the concurrence [MATH].', '1409.6521-2-23-4': 'This is consistent with experimentally reported values of typically around [MATH][CITATION].', '1409.6521-2-23-5': 'However, it should be noted, that a broad range of values for the concurrence have been reported, spanning between [MATH][CITATION] and [MATH][CITATION].', '1409.6521-2-24-0': 'We see that the two global maxima are located at [MATH] which indicates that the FSS is a stronger source of decoherence than the Overhauser field.', '1409.6521-2-24-1': 'Still, the concurrence does not reach unity but is rather close to the minimum observed in Fig. [REF]a. From these observations we conclude that in order to maximize the concurrence, we should keep [MATH] to eliminate the FSS and now focus on reducing the uncertainty of the Overhauser field.', '1409.6521-2-24-2': 'One way of achieving this is to polarize the nuclear spins, which has been experimentally realized[CITATION], and is investigated in the next section.', '1409.6521-2-24-3': 'In addition to the two global maxima, there are four local maxima located close to [MATH] T, [MATH] T. Although the concurrence is smaller at these points than at the global maxima, they indicate the significance of including the effects of both sources of decoherence simultaneously.', '1409.6521-2-25-0': '## Effect of Nuclear Spin Polarization', '1409.6521-2-26-0': 'It is clear that, within our model, when the FSS is eliminated, the remaining reduction of the entanglement originates from the Overhauser field.', '1409.6521-2-26-1': 'To investigate how the fluctuations of the Overhauser field vary as function of the nuclear spin polarization we consider a simple model for the Overhauser field along one spatial direction [EQUATION] where [MATH] is the number of nuclear spins, [MATH] are binary stochastic variables taking the values [MATH] with probability [EQUATION] where [EQUATION] [MATH] is the nuclear [MATH]-factor, [MATH] is an external magnetic field and [MATH] is the nuclear spin temperature.', '1409.6521-2-27-0': 'The polarization [MATH] is given by [EQUATION] and the variance is consequently [EQUATION]', '1409.6521-2-27-1': 'In Appendix [REF] it is shown that [EQUATION] where [MATH] is a Gaussian distribution with mean [MATH] and standard deviation [MATH], [MATH] is the number of nuclear spins, and [MATH] depends on [MATH] and [MATH].', '1409.6521-2-27-2': 'Typically, [MATH] will have to be determined experimentally or by numerical simulations and we do not attempt to calculate it here, but the general form Eq. ([REF]) does not depend on the specific QD.', '1409.6521-2-27-3': 'Since the fluctuations of the Overhauser field decrease with increasing polarization we now assume that the nuclear spins are polarized to degree [MATH] along [MATH], where [MATH].', '1409.6521-2-27-4': 'The assumption that the nuclear spin can be polarized along an arbitrary direction relies on experimental demonstrations[CITATION].', '1409.6521-2-27-5': 'This gives an effective magnetic field [MATH], with variances [EQUATION]', '1409.6521-2-27-6': 'Together with the applied magnetic field [MATH] the total effective magnetic field depends on 7 variables: [MATH] and [MATH].', '1409.6521-2-27-7': 'In order to narrow the search for optimal parameters, we make the following observations: first, [MATH] and [MATH] are equivalent and we set [MATH].', '1409.6521-2-27-8': 'Second, Fig. [REF] shows that the concurrence [MATH] has its maximum for [MATH] and we thus set [MATH].', '1409.6521-2-27-9': 'Finally we let [MATH] and the total effective magnetic field is given by [EQUATION] and depends on the three free parameters [MATH], [MATH], and [MATH].', '1409.6521-2-27-10': 'For [MATH] the result is shown in Fig. [REF] and we find that for every [MATH] there are two applied magnetic fields along [MATH] locally maximizing the concurrence.', '1409.6521-2-27-11': 'As expected from the discussion in the previous section, these occur when [MATH].', '1409.6521-2-27-12': 'We may thus set [MATH] and study concurrence as a function of the polarization angle [MATH] which is shown in Fig. [REF], where we observe that the concurrence is maximized by minimizing fluctuations along [MATH].', '1409.6521-2-28-0': 'Finally we can investigate the concurrence as a function of polarization, shown in the inset of Fig. [REF].', '1409.6521-2-28-1': 'We find that an increased nuclear spin polarization along [MATH] leads to an increased concurrence.', '1409.6521-2-28-2': 'We also see that a nuclear spin polarization perpendicular to [MATH] has almost no effect on the concurrence.', '1409.6521-2-28-3': 'This can be explained by the fact that the [MATH]-factors for the [MATH]- and [MATH]-directions are much smaller than the one along [MATH].', '1409.6521-2-29-0': '# Summary', '1409.6521-2-30-0': 'We have theoretically investigated the entanglement between two photons emitted from a cascade recombination of a biexciton in a quantum dot.', '1409.6521-2-30-1': 'The entanglement was examined using the concurrence as a quantitative measure.', '1409.6521-2-30-2': 'We considered the two main sources of loss of concurrence, the FSS combined with a stochastic intermediate exciton lifetime and the stochastic Overhauser field.', '1409.6521-2-30-3': 'We found that the two sources of decoherence cannot be minimized independently of each other, and that the FSS is the dominant source of decoherence and must be minimized in order to maximize concurrence.', '1409.6521-2-30-4': 'Furthermore, we showed that reducing the uncertainty of the Overhauser field by nuclear spin polarization together with an applied magnetic field along a certain direction can improve the concurrence of the emitted light.', '1409.6521-2-30-5': 'The increase in entanglement depends strongly on the degree as well as the direction of nuclear spin polarization relative to the growth axis of the QD.', '1409.6521-2-30-6': 'This effect is caused by the difference between in-plane [MATH]-factors and the [MATH]-factor along the growth direction.'}

[['1409.6521-1-17-0', '1409.6521-2-17-0'], ['1409.6521-1-17-1', '1409.6521-2-17-1'], ['1409.6521-1-17-2', '1409.6521-2-17-2'], ['1409.6521-1-17-3', '1409.6521-2-17-3'], ['1409.6521-1-17-4', '1409.6521-2-17-4'], ['1409.6521-1-17-5', '1409.6521-2-17-5'], ['1409.6521-1-10-0', '1409.6521-2-10-0'], ['1409.6521-1-10-1', '1409.6521-2-10-1'], ['1409.6521-1-10-2', '1409.6521-2-10-2'], ['1409.6521-1-10-5', '1409.6521-2-10-7'], ['1409.6521-1-10-6', '1409.6521-2-10-8'], ['1409.6521-1-10-7', '1409.6521-2-10-9'], ['1409.6521-1-27-0', '1409.6521-2-27-0'], ['1409.6521-1-27-1', '1409.6521-2-27-1'], ['1409.6521-1-27-2', '1409.6521-2-27-2'], ['1409.6521-1-27-3', '1409.6521-2-27-3'], ['1409.6521-1-27-4', '1409.6521-2-27-4'], ['1409.6521-1-27-5', '1409.6521-2-27-5'], ['1409.6521-1-27-6', '1409.6521-2-27-6'], ['1409.6521-1-27-7', '1409.6521-2-27-7'], ['1409.6521-1-27-8', '1409.6521-2-27-8'], ['1409.6521-1-27-9', '1409.6521-2-27-9'], ['1409.6521-1-27-10', '1409.6521-2-27-10'], ['1409.6521-1-27-11', '1409.6521-2-27-11'], ['1409.6521-1-27-12', '1409.6521-2-27-12'], ['1409.6521-1-8-0', '1409.6521-2-8-0'], ['1409.6521-1-8-1', '1409.6521-2-8-1'], ['1409.6521-1-8-2', '1409.6521-2-8-2'], ['1409.6521-1-3-0', '1409.6521-2-3-0'], ['1409.6521-1-3-1', '1409.6521-2-3-1'], ['1409.6521-1-3-2', '1409.6521-2-3-2'], ['1409.6521-1-3-3', '1409.6521-2-3-3'], ['1409.6521-1-3-4', '1409.6521-2-3-4'], ['1409.6521-1-28-0', '1409.6521-2-28-0'], ['1409.6521-1-28-1', '1409.6521-2-28-1'], ['1409.6521-1-28-2', '1409.6521-2-28-2'], ['1409.6521-1-28-3', '1409.6521-2-28-3'], ['1409.6521-1-24-0', '1409.6521-2-24-0'], ['1409.6521-1-24-1', '1409.6521-2-24-1'], ['1409.6521-1-24-2', '1409.6521-2-24-2'], ['1409.6521-1-24-3', '1409.6521-2-24-3'], ['1409.6521-1-19-0', '1409.6521-2-19-0'], ['1409.6521-1-19-1', '1409.6521-2-19-1'], ['1409.6521-1-19-2', '1409.6521-2-19-2'], ['1409.6521-1-19-3', '1409.6521-2-19-3'], ['1409.6521-1-19-4', '1409.6521-2-19-4'], ['1409.6521-1-19-5', '1409.6521-2-19-5'], ['1409.6521-1-19-6', '1409.6521-2-19-6'], ['1409.6521-1-19-7', '1409.6521-2-19-7'], ['1409.6521-1-16-0', '1409.6521-2-16-0'], ['1409.6521-1-16-1', '1409.6521-2-16-1'], ['1409.6521-1-16-2', '1409.6521-2-16-2'], ['1409.6521-1-16-3', '1409.6521-2-16-3'], ['1409.6521-1-16-4', '1409.6521-2-16-4'], ['1409.6521-1-16-5', '1409.6521-2-16-5'], ['1409.6521-1-16-6', '1409.6521-2-16-6'], ['1409.6521-1-16-7', '1409.6521-2-16-7'], ['1409.6521-1-16-9', '1409.6521-2-16-9'], ['1409.6521-1-16-10', '1409.6521-2-16-10'], ['1409.6521-1-16-11', '1409.6521-2-16-11'], ['1409.6521-1-16-12', '1409.6521-2-16-12'], ['1409.6521-1-16-13', '1409.6521-2-16-13'], ['1409.6521-1-16-14', '1409.6521-2-16-14'], ['1409.6521-1-16-15', '1409.6521-2-16-15'], ['1409.6521-1-16-16', '1409.6521-2-16-16'], ['1409.6521-1-16-17', '1409.6521-2-16-17'], ['1409.6521-1-16-18', '1409.6521-2-16-18'], ['1409.6521-1-16-19', '1409.6521-2-16-19'], ['1409.6521-1-16-20', '1409.6521-2-16-20'], ['1409.6521-1-16-21', '1409.6521-2-16-21'], ['1409.6521-1-16-22', '1409.6521-2-16-22'], ['1409.6521-1-18-0', '1409.6521-2-18-0'], ['1409.6521-1-18-1', '1409.6521-2-18-1'], ['1409.6521-1-18-2', '1409.6521-2-18-2'], ['1409.6521-1-18-3', '1409.6521-2-18-3'], ['1409.6521-1-0-0', '1409.6521-2-0-0'], ['1409.6521-1-0-1', '1409.6521-2-0-1'], ['1409.6521-1-0-2', '1409.6521-2-0-2'], ['1409.6521-1-0-4', '1409.6521-2-0-4'], ['1409.6521-1-0-5', '1409.6521-2-0-5'], ['1409.6521-1-15-0', '1409.6521-2-15-0'], ['1409.6521-1-15-1', '1409.6521-2-15-1'], ['1409.6521-1-15-2', '1409.6521-2-15-2'], ['1409.6521-1-15-3', '1409.6521-2-15-3'], ['1409.6521-1-15-4', '1409.6521-2-15-4'], ['1409.6521-1-6-0', '1409.6521-2-6-0'], ['1409.6521-1-6-1', '1409.6521-2-6-2'], ['1409.6521-1-6-2', '1409.6521-2-6-3'], ['1409.6521-1-6-3', '1409.6521-2-6-4'], ['1409.6521-1-7-0', '1409.6521-2-7-0'], ['1409.6521-1-26-0', '1409.6521-2-26-0'], ['1409.6521-1-26-1', '1409.6521-2-26-1'], ['1409.6521-1-23-0', '1409.6521-2-23-0'], ['1409.6521-1-23-1', '1409.6521-2-23-1'], ['1409.6521-1-23-2', '1409.6521-2-23-2'], ['1409.6521-1-2-0', '1409.6521-2-2-0'], ['1409.6521-1-2-1', '1409.6521-2-2-1'], ['1409.6521-1-2-2', '1409.6521-2-2-2'], ['1409.6521-1-2-3', '1409.6521-2-2-3'], ['1409.6521-1-2-4', '1409.6521-2-2-4'], ['1409.6521-1-2-5', '1409.6521-2-2-5'], ['1409.6521-1-21-0', '1409.6521-2-21-0'], ['1409.6521-1-30-0', '1409.6521-2-30-0'], ['1409.6521-1-30-1', '1409.6521-2-30-1'], ['1409.6521-1-30-2', '1409.6521-2-30-2'], ['1409.6521-1-30-4', '1409.6521-2-30-4'], ['1409.6521-1-30-5', '1409.6521-2-30-5'], ['1409.6521-1-30-6', '1409.6521-2-30-6'], ['1409.6521-1-5-0', '1409.6521-2-5-0'], ['1409.6521-1-5-1', '1409.6521-2-5-1'], ['1409.6521-1-5-2', '1409.6521-2-5-4'], ['1409.6521-1-5-4', '1409.6521-2-5-6'], ['1409.6521-1-4-0', '1409.6521-2-4-0'], ['1409.6521-1-4-1', '1409.6521-2-4-1'], ['1409.6521-1-4-2', '1409.6521-2-4-2'], ['1409.6521-1-11-0', '1409.6521-2-11-0'], ['1409.6521-1-11-1', '1409.6521-2-11-1'], ['1409.6521-1-11-2', '1409.6521-2-11-2'], ['1409.6521-1-11-3', '1409.6521-2-11-3'], ['1409.6521-1-10-4', '1409.6521-2-10-6'], ['1409.6521-1-16-8', '1409.6521-2-16-8'], ['1409.6521-1-6-4', '1409.6521-2-6-7'], ['1409.6521-1-7-1', '1409.6521-2-7-1'], ['1409.6521-1-7-2', '1409.6521-2-7-2'], ['1409.6521-1-30-3', '1409.6521-2-30-3'], ['1409.6521-1-5-3', '1409.6521-2-5-5'], ['1409.6521-1-10-3', '1409.6521-2-10-4'], ['1409.6521-1-0-3', '1409.6521-2-0-3']]

[['1409.6521-1-17-0', '1409.6521-2-17-0'], ['1409.6521-1-17-1', '1409.6521-2-17-1'], ['1409.6521-1-17-2', '1409.6521-2-17-2'], ['1409.6521-1-17-3', '1409.6521-2-17-3'], ['1409.6521-1-17-4', '1409.6521-2-17-4'], ['1409.6521-1-17-5', '1409.6521-2-17-5'], ['1409.6521-1-10-0', '1409.6521-2-10-0'], ['1409.6521-1-10-1', '1409.6521-2-10-1'], ['1409.6521-1-10-2', '1409.6521-2-10-2'], ['1409.6521-1-10-5', '1409.6521-2-10-7'], ['1409.6521-1-10-6', '1409.6521-2-10-8'], ['1409.6521-1-10-7', '1409.6521-2-10-9'], ['1409.6521-1-27-0', '1409.6521-2-27-0'], ['1409.6521-1-27-1', '1409.6521-2-27-1'], ['1409.6521-1-27-2', '1409.6521-2-27-2'], ['1409.6521-1-27-3', '1409.6521-2-27-3'], ['1409.6521-1-27-4', '1409.6521-2-27-4'], ['1409.6521-1-27-5', '1409.6521-2-27-5'], ['1409.6521-1-27-6', '1409.6521-2-27-6'], ['1409.6521-1-27-7', '1409.6521-2-27-7'], ['1409.6521-1-27-8', '1409.6521-2-27-8'], ['1409.6521-1-27-9', '1409.6521-2-27-9'], ['1409.6521-1-27-10', '1409.6521-2-27-10'], ['1409.6521-1-27-11', '1409.6521-2-27-11'], ['1409.6521-1-27-12', '1409.6521-2-27-12'], ['1409.6521-1-8-0', '1409.6521-2-8-0'], ['1409.6521-1-8-1', '1409.6521-2-8-1'], ['1409.6521-1-8-2', '1409.6521-2-8-2'], ['1409.6521-1-3-0', '1409.6521-2-3-0'], ['1409.6521-1-3-1', '1409.6521-2-3-1'], ['1409.6521-1-3-2', '1409.6521-2-3-2'], ['1409.6521-1-3-3', '1409.6521-2-3-3'], ['1409.6521-1-3-4', '1409.6521-2-3-4'], ['1409.6521-1-28-0', '1409.6521-2-28-0'], ['1409.6521-1-28-1', '1409.6521-2-28-1'], ['1409.6521-1-28-2', '1409.6521-2-28-2'], ['1409.6521-1-28-3', '1409.6521-2-28-3'], ['1409.6521-1-24-0', '1409.6521-2-24-0'], ['1409.6521-1-24-1', '1409.6521-2-24-1'], ['1409.6521-1-24-2', '1409.6521-2-24-2'], ['1409.6521-1-24-3', '1409.6521-2-24-3'], ['1409.6521-1-19-0', '1409.6521-2-19-0'], ['1409.6521-1-19-1', '1409.6521-2-19-1'], ['1409.6521-1-19-2', '1409.6521-2-19-2'], ['1409.6521-1-19-3', '1409.6521-2-19-3'], ['1409.6521-1-19-4', '1409.6521-2-19-4'], ['1409.6521-1-19-5', '1409.6521-2-19-5'], ['1409.6521-1-19-6', '1409.6521-2-19-6'], ['1409.6521-1-19-7', '1409.6521-2-19-7'], ['1409.6521-1-16-0', '1409.6521-2-16-0'], ['1409.6521-1-16-1', '1409.6521-2-16-1'], ['1409.6521-1-16-2', '1409.6521-2-16-2'], ['1409.6521-1-16-3', '1409.6521-2-16-3'], ['1409.6521-1-16-4', '1409.6521-2-16-4'], ['1409.6521-1-16-5', '1409.6521-2-16-5'], ['1409.6521-1-16-6', '1409.6521-2-16-6'], ['1409.6521-1-16-7', '1409.6521-2-16-7'], ['1409.6521-1-16-9', '1409.6521-2-16-9'], ['1409.6521-1-16-10', '1409.6521-2-16-10'], ['1409.6521-1-16-11', '1409.6521-2-16-11'], ['1409.6521-1-16-12', '1409.6521-2-16-12'], ['1409.6521-1-16-13', '1409.6521-2-16-13'], ['1409.6521-1-16-14', '1409.6521-2-16-14'], ['1409.6521-1-16-15', '1409.6521-2-16-15'], ['1409.6521-1-16-16', '1409.6521-2-16-16'], ['1409.6521-1-16-17', '1409.6521-2-16-17'], ['1409.6521-1-16-18', '1409.6521-2-16-18'], ['1409.6521-1-16-19', '1409.6521-2-16-19'], ['1409.6521-1-16-20', '1409.6521-2-16-20'], ['1409.6521-1-16-21', '1409.6521-2-16-21'], ['1409.6521-1-16-22', '1409.6521-2-16-22'], ['1409.6521-1-18-0', '1409.6521-2-18-0'], ['1409.6521-1-18-1', '1409.6521-2-18-1'], ['1409.6521-1-18-2', '1409.6521-2-18-2'], ['1409.6521-1-18-3', '1409.6521-2-18-3'], ['1409.6521-1-0-0', '1409.6521-2-0-0'], ['1409.6521-1-0-1', '1409.6521-2-0-1'], ['1409.6521-1-0-2', '1409.6521-2-0-2'], ['1409.6521-1-0-4', '1409.6521-2-0-4'], ['1409.6521-1-0-5', '1409.6521-2-0-5'], ['1409.6521-1-15-0', '1409.6521-2-15-0'], ['1409.6521-1-15-1', '1409.6521-2-15-1'], ['1409.6521-1-15-2', '1409.6521-2-15-2'], ['1409.6521-1-15-3', '1409.6521-2-15-3'], ['1409.6521-1-15-4', '1409.6521-2-15-4'], ['1409.6521-1-6-0', '1409.6521-2-6-0'], ['1409.6521-1-6-1', '1409.6521-2-6-2'], ['1409.6521-1-6-2', '1409.6521-2-6-3'], ['1409.6521-1-6-3', '1409.6521-2-6-4'], ['1409.6521-1-7-0', '1409.6521-2-7-0'], ['1409.6521-1-26-0', '1409.6521-2-26-0'], ['1409.6521-1-26-1', '1409.6521-2-26-1'], ['1409.6521-1-23-0', '1409.6521-2-23-0'], ['1409.6521-1-23-1', '1409.6521-2-23-1'], ['1409.6521-1-23-2', '1409.6521-2-23-2'], ['1409.6521-1-2-0', '1409.6521-2-2-0'], ['1409.6521-1-2-1', '1409.6521-2-2-1'], ['1409.6521-1-2-2', '1409.6521-2-2-2'], ['1409.6521-1-2-3', '1409.6521-2-2-3'], ['1409.6521-1-2-4', '1409.6521-2-2-4'], ['1409.6521-1-2-5', '1409.6521-2-2-5'], ['1409.6521-1-21-0', '1409.6521-2-21-0'], ['1409.6521-1-30-0', '1409.6521-2-30-0'], ['1409.6521-1-30-1', '1409.6521-2-30-1'], ['1409.6521-1-30-2', '1409.6521-2-30-2'], ['1409.6521-1-30-4', '1409.6521-2-30-4'], ['1409.6521-1-30-5', '1409.6521-2-30-5'], ['1409.6521-1-30-6', '1409.6521-2-30-6'], ['1409.6521-1-5-0', '1409.6521-2-5-0'], ['1409.6521-1-5-1', '1409.6521-2-5-1'], ['1409.6521-1-5-2', '1409.6521-2-5-4'], ['1409.6521-1-5-4', '1409.6521-2-5-6'], ['1409.6521-1-4-0', '1409.6521-2-4-0'], ['1409.6521-1-4-1', '1409.6521-2-4-1'], ['1409.6521-1-4-2', '1409.6521-2-4-2'], ['1409.6521-1-11-0', '1409.6521-2-11-0'], ['1409.6521-1-11-1', '1409.6521-2-11-1'], ['1409.6521-1-11-2', '1409.6521-2-11-2'], ['1409.6521-1-11-3', '1409.6521-2-11-3']]

[['1409.6521-1-10-4', '1409.6521-2-10-6'], ['1409.6521-1-16-8', '1409.6521-2-16-8'], ['1409.6521-1-6-4', '1409.6521-2-6-7'], ['1409.6521-1-7-1', '1409.6521-2-7-1'], ['1409.6521-1-7-2', '1409.6521-2-7-2'], ['1409.6521-1-30-3', '1409.6521-2-30-3'], ['1409.6521-1-5-3', '1409.6521-2-5-5']]

[]

[['1409.6521-1-10-3', '1409.6521-2-10-4'], ['1409.6521-1-0-3', '1409.6521-2-0-3']]

[]

['1409.6521-1-12-0', '1409.6521-1-13-0', '1409.6521-1-14-0', '1409.6521-2-12-0', '1409.6521-2-13-0', '1409.6521-2-14-0']

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

https://arxiv.org/abs/1409.6521

1611.09321

{'1611.09321-1-0-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning (RL) with improved exploration properties.', '1611.09321-1-0-1': 'Current policy-based methods use entropy regularization to encourage undirected exploration of the reward landscape, which is ineffective in high dimensional spaces with sparse rewards.', '1611.09321-1-0-2': 'We propose a more directed exploration strategy that promotes exploration of under-appreciated reward regions.', '1611.09321-1-0-3': 'An action sequence is considered under-appreciated if its log-probability under the current policy under-estimates its resulting reward.', '1611.09321-1-0-4': 'The proposed exploration strategy is easy to implement, requiring small modifications to an implementation of the REINFORCE algorithm.', '1611.09321-1-0-5': 'We evaluate the approach on a set of algorithmic tasks that have long challenged RL methods.', '1611.09321-1-0-6': 'Our approach reduces hyper-parameter sensitivity and demonstrates significant improvements over baseline methods.', '1611.09321-1-0-7': 'Our algorithm successfully solves a benchmark multi-digit addition task and generalizes to long sequences.', '1611.09321-1-0-8': 'This is, to our knowledge, the first time that a pure RL method has solved addition using only reward feedback.', '1611.09321-1-1-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning with improved exploration characteristics.', '1611.09321-1-1-1': 'Current policy gradient methods use entropy regularization to encourage uniform exploration of the reward landscape, which is not effective in high dimensional spaces with sparse rewards.', '1611.09321-1-1-2': 'We propose a better exploration strategy that promotes exploration of the under-appreciated reward regions.', '1611.09321-1-1-3': 'An action sequence is considered under-appreciated if its probability under the current policy under-estimates its resulting reward.', '1611.09321-1-1-4': 'The proposed exploration strategy is easy to implement.', '1611.09321-1-1-5': 'It improves robustness to the change of hyper-parameters, and on a set of algorithmic tasks, it demonstrates significant improvements against the policy gradient baselines.', '1611.09321-1-1-6': 'Our approach is able to solve a benchmark multi-digit addition task.', '1611.09321-1-1-7': 'To our knowledge, this is the first pure RL method to solve addition just by using reward feedback.', '1611.09321-1-2-0': 'To our knowledge Unlike standard on-line learning strategies in reinforcement learning, which generally encourage undirected exploration, we stochastically optimize a modified objective that favors exploration of high-reward versus low-reward actions.', '1611.09321-1-2-1': 'A key feature of the algorithm is its ease of application to recurrent neural network policy representations.', '1611.09321-1-2-2': 'We conduct an experimental evaluation on a set of algorithm induction tasks, ranging from simple to challenging, and compare the proposed enhancement to standard baselines.', '1611.09321-1-2-3': 'In addition to obtaining notable improvements, we also observe that the new approach can solve the benchmark addition problem, which to our knowledge is the first successful such demonstration based solely on success/failure reinforcement.', '1611.09321-1-3-0': '# Introduction', '1611.09321-1-4-0': 'Humans can reason about symbolic objects and solve algorithmic problems.', '1611.09321-1-4-1': 'After learning to count and then manipulate numbers via simple arithmetic, people eventually learn to invent new algorithms and even reason about their correctness and efficiency.', '1611.09321-1-4-2': 'The ability to invent new algorithms is fundamental to artificial intelligence (AI).', '1611.09321-1-4-3': 'Although symbolic reasoning has a long history in AI , only recently have statistical machine learning and neural network approaches begun to make headway in automated algorithm discovery', '1611.09321-1-5-0': 'heading to cross an important milestone on the path to AI.', '1611.09321-1-5-1': 'Nevertheless, most of the recent successes depend on the use of strong supervision to learn a mapping from a set of training inputs to outputs by maximizing a conditional log-likelihood, very much like neural machine translation systems .', '1611.09321-1-5-2': 'Such a dependence on strong supervision is a significant limitation that does not match the ability of people to invent new algorithmic procedures based solely on trial and error.', '1611.09321-1-6-0': 'By contrast, reinforcement learning (RL) methods hold the promise of searching over discrete objects such as symbolic representations of algorithms by considering much weaker feedback in the form of a simple verifier that tests the correctness of a program execution on a given problem instance.', '1611.09321-1-6-1': 'Despite the recent excitement around the use of RL to tackle Atari games and Go , standard RL methods are not yet able to consistently and reliably solve algorithmic tasks in all but the simplest cases .', '1611.09321-1-6-2': 'A key property of algorithmic problems that makes them challenging for RL is reward sparsity, a policy usually has to get a long action sequence exactly right to obtain a non-zero reward.', '1611.09321-1-7-0': 'We believe one of the key limitations of the current RL methods, preventing them from making much progress in the sparse reward settings, is the use of undirected exploration strategies , such as [MATH]-greedy and entropy regularization .', '1611.09321-1-7-1': 'For long action sequences with delayed sparse reward, it is hopeless to explore the space uniformly and blindly.', '1611.09321-1-7-2': 'Instead, we propose a formulation to encourage exploration of action sequences that are under-appreciated by the current policy.', '1611.09321-1-7-3': "We consider an action sequence to be under-appreciated if the model's log-probability assigned to an action sequence under-estimates the resulting reward from the action sequence.", '1611.09321-1-7-4': 'Exploring under-appreciated states and actions encourages the policy to have a better calibration between its log-probabilities and observed reward values, even for action sequences with negligible rewards.', '1611.09321-1-7-5': 'This effectively increases exploration around neglected action sequences.', '1611.09321-1-8-0': 'We term our proposed technique under-appreciated reward exploration ().', '1611.09321-1-8-1': 'We show that the objective given by is a combination of a mode seeking objective (standard REINFORCE) and a mean seeking term, which provides a good trade-off between exploitation and exploration.', '1611.09321-1-8-2': 'We apply the method to recurrent neural network (RNN)-based RL agents tackling algorithmic tasks such as sequence reversal, multi-digit addition, and binary search.', '1611.09321-1-8-3': 'The experiments demonstrate that significantly outperforms baseline RL methods, such as entropy regularized REINFORCE and one-step Q-learning, especially on the seemingly more difficult tasks, such as multi-digit addition.', '1611.09321-1-8-4': 'Moreover, is shown to be more robust to changes of hyper-parameters, which makes hyper-parameter tuning less tedious in practice.', '1611.09321-1-8-5': 'To our knowledge, the addition task has not been solved by any pure reinforcement learning approach.', '1611.09321-1-8-6': 'We observe that some of the policies learned by can successfully generalize to long sequences; in [MATH] out of [MATH] random restarts, the policy learned by for the addition task correctly generalizes to addition of numbers with [MATH] digits with no mistakes, even though training sequences are at most [MATH] digits long.', '1611.09321-1-9-0': 'In order to find both non-zero reward action sequences and maximal reward action sequences in a large action space, some sort of exploration is necessary.', '1611.09321-1-9-1': 'In value-based RL this exploration can manifest as [MATH]-greedy actions whereas in policy-based methods a common choice is to add an entropy regularizer to the well-known REINFORCE policy gradient algorithm .', '1611.09321-1-9-2': 'However, these exploration strategies apply exploration blindly and uniformly.', '1611.09321-1-9-3': 'In large action spaces, these strategies can easily distract an agent from its main task of honing in on high-reward regions of the policy space.', '1611.09321-1-10-0': 'In this work, we propose a novel addition to the REINFORCE objective to favor exploration in high-reward areas over exploration in low-reward areas.', '1611.09321-1-11-0': 'We begin in Section [REF], where we provide a short survey of previous work on learning algorithms.', '1611.09321-1-11-1': 'In Section [REF] we provide background information regarding our notation and the derivation of REINFORCE and its entropy-regularized variant.', '1611.09321-1-11-2': 'We introduce our approach and the resulting policy gradient training algorithm in [REF].', '1611.09321-1-11-3': 'Subsequently, we present several standard tasks and introduce an additional task we designed ourselves in Section [REF], on which we then empirically evaluate the performance of our approach.', '1611.09321-1-11-4': 'We conclude in Section [REF], where we summarize the paper and discuss the repercussions and impact we hope it will have on future work.', '1611.09321-1-12-0': '# Neural Networks for Learning Algorithms', '1611.09321-1-13-0': 'Although research on using neural networks to learn algorithms has had a surge of recent interest, the problem of program induction from examples has a long history in many fields, including program induction, inductive logic programming , relational learning and regular language learning .', '1611.09321-1-13-1': 'Rather than presenting a comprehensive survey of program induction here, we focus on neural network approaches to algorithmic tasks and highlight the relative simplicity of our neural network architecture.', '1611.09321-1-14-0': 'Most successful applications of neural networks to algorithmic tasks rely on strong supervision, where the inputs and target outputs are completely known a priori.', '1611.09321-1-14-1': 'Given a dataset of examples, one learns the network parameters by maximizing the conditional likelihood of the outputs via backpropagation ( [CITATION]).', '1611.09321-1-14-2': 'However, target outputs may not be available for novel tasks, for which no algorithm exists yet.', '1611.09321-1-14-3': 'A more desirable approach to inducing algorithms, followed in this paper, advocates using self-driven learning strategies that only receive reinforcement based on the outputs produced.', '1611.09321-1-14-4': 'Hence, just by having access to a verifier for an algorithmic problem, one can aim to learn an algorithm.', '1611.09321-1-14-5': 'For example, if one does not know how to sort an array, but can check the extent to which an array is sorted, then one can provide the reward signal necessary for learning sorting algorithms.', '1611.09321-1-15-0': 'However, providing the correct output during training can often be infeasible; for example, when trying to solve a problem with no known solution, or when efficiency is desired in addition to correctness.', '1611.09321-1-16-0': 'We formulate learning algorithms as an RL problem and make use of model-free policy gradient methods to optimize a set parameters associated with the algorithm.', '1611.09321-1-16-1': 'In this setting, the goal is to learn a policy [MATH] that given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-1-16-2': 'Actions represent the commands within the algorithm and states represent the joint state of the algorithm and the environment.', '1611.09321-1-16-3': 'Previous work in this area has focused on augmenting a neural network with additional structure and increased capabilities .', '1611.09321-1-16-4': 'In contrast, we utilize a simple architecture based on a standard recurrent neural network (RNN) with LSTM cells as depicted in fig:rnn.', '1611.09321-1-16-5': 'At each episode, the environment is initialized with a latent state [MATH], unknown to the agent, which determines [MATH] and the subsequent state transition and reward functions.', '1611.09321-1-16-6': 'Once the agent observes [MATH] as the input to the RNN, the network outputs a distribution [MATH], from which an action [MATH] is sampled.', '1611.09321-1-16-7': 'This action is applied to the environment, and the agent receives a new state observation [MATH].', '1611.09321-1-16-8': 'The state [MATH] and the previous action [MATH] are then fed into the RNN and the process repeats until the end of the episode.', '1611.09321-1-16-9': 'Upon termination, a reward signal is received.', '1611.09321-1-17-0': '# Learning a Policy by Maximizing Expected Reward', '1611.09321-1-18-0': 'We start by discussing the most common form of policy gradient, REINFORCE , and its entropy regularized variant .', '1611.09321-1-18-1': 'REINFORCE has been applied to model-free policy-based learning with neural networks and algorithmic domains .', '1611.09321-1-19-0': 'As mentioned above, we aim to learn a policy [MATH] that given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-1-19-1': 'The environment is initialized with a latent vector, [MATH], which determines the initial observed state [MATH], and the transition function [MATH].', '1611.09321-1-19-2': 'Given a latent state [MATH], and [MATH], the model probability of an action sequence [MATH] is expressed as, [EQUATION]', '1611.09321-1-19-3': 'The environment provides a reward at the end of the episode, denoted [MATH].', '1611.09321-1-19-4': 'For ease of readability we drop the subscript from [MATH] and simply write [MATH] and [MATH].', '1611.09321-1-20-0': 'The objective used to optimize the policy parameters, [MATH], consists of maximizing expected reward under actions drawn from the policy, plus an optional maximum entropy regularizer.', '1611.09321-1-20-1': 'Given a distribution over initial latent environment states [MATH], we express the regularized expected reward as, [EQUATION]', '1611.09321-1-20-2': 'When [MATH] is a non-linear function defined by a neural network, finding the global optimum of [MATH] is challenging, and one often resorts to gradient-based methods to find a local optimum of [MATH].', '1611.09321-1-20-3': 'Given that [MATH] for any [MATH] such that [MATH], one can verify that, [EQUATION]', '1611.09321-1-20-4': 'Because the space of possible actions [MATH] is large, enumerating over all of the actions to compute this gradient is infeasible.', '1611.09321-1-20-5': 'Williams Williams92 proposed to compute the stochastic gradient of the expected reward by using Monte Carlo samples.', '1611.09321-1-20-6': 'Using Monte Carlo samples, one first draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to approximate the gradient of [REF] by using [REF] as, [EQUATION]', '1611.09321-1-20-7': 'This reparametrization of the gradients is the key to the REINFORCE algorithm.', '1611.09321-1-20-8': 'To reduce the variance of [REF], one uses rewards [MATH] that are shifted by some offset values, [EQUATION] where [MATH] is known as a baseline or sometimes called a critic.', '1611.09321-1-20-9': 'Note that subtracting any offset from the rewards in [REF] simply results in shifting the objective [MATH] by a constant.', '1611.09321-1-20-10': "I can't make [MATH] function of [MATH]", '1611.09321-1-21-0': 'Unfortunately, directly maximizing expected reward ( when [MATH]) is prone to getting trapped in a local optimum.', '1611.09321-1-21-1': 'To combat this tendency, Williams Peng williams1991function augmented the expected reward objective by including a maximum entropy regularizer ([MATH]) to promote greater exploration.', '1611.09321-1-21-2': 'We will refer to this variant of REINFORCE as (maximum entropy exploration).', '1611.09321-1-22-0': '# Under-appreciated Reward Exploration ()', '1611.09321-1-23-0': 'To explain our novel form of policy gradient, we first note that the optimal policy [MATH], which globally maximizes [MATH] in [REF] for any [MATH], can be expressed as, [EQUATION] where [MATH] is a normalization constant making [MATH] a distribution over the space of action sequences [MATH].', '1611.09321-1-23-1': 'One can verify this by first acknowledging that, [EQUATION]', '1611.09321-1-23-2': 'Since [MATH] is non-negative and zero iff [MATH], then [MATH] defined in [REF] maximizes [MATH].', '1611.09321-1-23-3': 'That said, given a particular form of [MATH], finding [MATH] that exactly characterizes [MATH] may not be feasible.', '1611.09321-1-24-0': 'The KL divergence [MATH] is known to be mode seeking [CITATION] even with entropy regularization ([MATH]).', '1611.09321-1-24-1': 'Learning a policy by optimizing this direction of the KL is prone to falling into a local optimum resulting in a sub-optimal policy that omits some of the modes of [MATH].', '1611.09321-1-24-2': 'Although entropy regularization helps mitigate the issues as confirmed in our experiments, it is not an effective exploration strategy as it is undirected and requires a small regularization coefficient [MATH] to avoid too much random exploration.', '1611.09321-1-24-3': 'Instead, we propose a directed exploration strategy that improves the mean seeking behavior of policy gradient in a principled way.', '1611.09321-1-25-0': 'We start by considering the alternate mean seeking direction of the KL divergence, [MATH].', '1611.09321-1-25-1': '[CITATION] considered this direction of the KL to directly learn a policy by optimizing [EQUATION] for structured prediction.', '1611.09321-1-25-2': 'This objective has the same optimal solution [MATH] as [MATH] since, [EQUATION] [CITATION] argue that in some structured prediction problems when one can draw samples from [MATH], optimizing [REF] is more effective than [REF], since no sampling from a non-stationary policy [MATH] is required.', '1611.09321-1-25-3': 'If [MATH] is a log-linear model of a set of features, [MATH] is convex in [MATH] whereas [MATH] is not, even in the log-linear case.', '1611.09321-1-25-4': 'Unfortunately, in scenarios that the reward landscape is unknown or computing the normalization constant [MATH] is intractable, sampling from [MATH] is not straightforward.', '1611.09321-1-26-0': 'In the RL problems, the reward landscape is completely unknown, hence sampling from [MATH] is intractable.', '1611.09321-1-26-1': 'This paper proposes to approximate the expectation with respect to [MATH] in [REF] by using self-normalized importance sampling , where the proposal distribution is [MATH] and the reference distribution is [MATH].', '1611.09321-1-26-2': 'For importance sampling, one draws [MATH] samples [MATH] from [MATH] and computes a set of normalized importance weights to approximate [MATH] as, [EQUATION] where [MATH] denotes an importance weight defined by, [EQUATION]', '1611.09321-1-26-3': "One can view these importance weights as evaluating the discrepancy between scaled rewards [MATH] and the policy's log-probabilities [MATH].", '1611.09321-1-26-4': 'Among the [MATH] samples, a sample that is least appreciated by the model, has the largest [MATH], receives the largest positive feedback in [REF].', '1611.09321-1-27-0': 'In practice, we have found that just using the importance sampling RML objective in [REF] does not always yield promising solutions.', '1611.09321-1-27-1': 'Particularly, at the beginning of training, when [MATH] is still far away from [MATH], the variance of importance weights is too large, and the self-normalized importance sampling procedure results in poor approximations.', '1611.09321-1-27-2': 'To stabilize early phases of training and ensure that the model distribution [MATH] achieves large expected reward scores, we combine the expected reward and RML objectives to benefit from the best of their mode and mean seeking behaviors.', '1611.09321-1-27-3': 'Accordingly, we propose the following objective that we call under-appreciated reward exploration (), [EQUATION] which is the sum of the expected reward and RML objectives.', '1611.09321-1-27-4': 'In our preliminary experiments, we considered a composite objective of [MATH], but we found that removing the entropy term is beneficial.', '1611.09321-1-27-5': 'Hence, the [MATH] objective does not include entropy regularization.', '1611.09321-1-27-6': 'Accordingly, the optimum policy for [MATH] is no longer [MATH], as it was for [MATH] and [MATH].', '1611.09321-1-27-7': 'Appendix [REF] derives the optimal policy for [MATH] as a function of the optimal policy for [MATH].', '1611.09321-1-27-8': 'We find that the optimal policy of is more sharply concentrated on the high reward regions of the action space, which may be an advantage for , but we leave more analysis of this behavior to future work.', '1611.09321-1-28-0': 'as [MATH], where [MATH] normalizes the distribution for each [MATH].', '1611.09321-1-29-0': 'To compute the gradient of [MATH], we use the self-normalized importance sampling estimate outlined in [REF].', '1611.09321-1-29-1': 'We assume that the importance weights are constant and contribute no gradient to [MATH].', '1611.09321-1-29-2': 'To approximate the gradient, one draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to obtain [EQUATION]', '1611.09321-1-29-3': 'As with REINFORCE, the rewards are shifted by an offset [MATH].', '1611.09321-1-29-4': 'In this gradient, the model log-probability of a sample action sequence [MATH] is reinforced if the corresponding reward is large, or the corresponding importance weights are large, meaning that the action sequence is under-appreciated.', '1611.09321-1-29-5': 'The normalized importance weights are computed using a softmax operator [MATH].', '1611.09321-1-30-0': 'There are three ways to explain why [MATH] may not converge to [MATH] after training using PG:', '1611.09321-1-31-0': 'Model capacity - The form of the model may not be expressive enough to represent [MATH].', '1611.09321-1-31-1': 'Variance - The space of possible actions [MATH] is very large and the reward is very sparse causing the stochastic gradients defined in [REF] to have a large variance.', '1611.09321-1-31-2': 'Optimization - The parameters may fall into local optima of the objective, or the objective may have a very slow convergence rate.', '1611.09321-1-32-0': 'When using recurrent neural networks for simple tasks, point (1) should not be an issue.', '1611.09321-1-32-1': 'Usually papers blame variance for the failure cases of PG, as it is hard to reason about optimization landscape especially in high-dimensional problems.', '1611.09321-1-32-2': 'Below we discuss a very simple one-dimensional problem where local optima seem to be a problem for the regularized expected reward objective.', '1611.09321-1-32-3': 'find a simple example!', '1611.09321-1-33-0': 'We use recurrent neural networks (RNNs) to model [MATH].', '1611.09321-1-33-1': '[CITATION]), which is formulated as minimization of the following objective,', '1611.09321-1-34-0': 'where [MATH] denotes the reward function, negative edit distance or BLEU score, [MATH] controls the degree of regularization, and [MATH] is the entropy of a distribution [MATH], [MATH].', '1611.09321-1-34-1': 'It is well-known that optimizing [MATH] using SGD is challenging because of the large variance of the gradients.', '1611.09321-1-34-2': 'Below we describe how ML and RL objectives are related, and propose a hybrid between the two that combines their benefits for supervised learning.', '1611.09321-1-35-0': '# Related Work on Exploration in Reinforcement Learning', '1611.09321-1-36-0': 'The most common exploration strategy considered in value-based RL is [MATH]-greedy Q-learning, where at each step the agent either takes the best action according to its current value approximation or with probability [MATH] takes an action sampled uniformly at random.', '1611.09321-1-36-1': 'Like entropy regularization, such an approach applies undirected exploration, but it has achieved recent success in game playing environments .', '1611.09321-1-37-0': 'Prominent approaches to improving exploration beyond [MATH]-greedy in value-based or model-based RL have focused on reducing uncertainty by prioritizing exploration toward states and actions where the agent knows the least.', '1611.09321-1-37-1': 'This basic intuition underlies work on counter and recency methods , exploration methods based on uncertainty estimates of values , methods that prioritize learning environment dynamics , and methods that provide an intrinsic motivation or curiosity bonus for exploring unknown states .', '1611.09321-1-37-2': 'We relate the concepts of value and policy in RL and propose an exploration strategy based on the discrepancy between the two.', '1611.09321-1-38-0': 'In contrast to value-based methods, exploration for policy-based RL methods is often a by-product of the optimization algorithm itself.', '1611.09321-1-38-1': 'Since algorithms like REINFORCE and Thompson sampling choose actions according to a stochastic policy, sub-optimal actions are chosen with some non-zero probability.', '1611.09321-1-38-2': 'The Q-learning algorithm may also be modified to sample an action from the softmax of the Q values rather than the argmax .', '1611.09321-1-39-0': 'Asynchronous training has also been reported to have an exploration effect on both value- and policy-based methods.', '1611.09321-1-39-1': '[CITATION] report that asynchronous training can stabilize training by reducing the bias experienced by a single trainer.', '1611.09321-1-39-2': 'By using multiple separate trainers, an agent is less likely to become trapped at a policy found to be locally optimal only due to local conditions.', '1611.09321-1-39-3': 'In the same spirit, [CITATION] use multiple Q value approximators and sample only one to act for each episode as a way to implicitly incorporate exploration.', '1611.09321-1-40-0': 'Although the problem of program induction from examples has been studied in many fields, including program induction, inductive logic programming , relational learning and regular language learning ; we do not attempt a comprehensive survey here.', '1611.09321-1-40-1': 'Instead, we focus on related neural network approaches for solving algorithmic tasks and relevant work that considers exploration in reinforcement learning.', '1611.09321-1-41-0': 'Learning Algorithms.', '1611.09321-1-41-1': 'Previous work on neural network learning algorithms for algorithmic problems falls into two general categories: strongly supervised maximum likelihood training that learns from examples using completely known input-output pairs; and self-driven training via reinforcement learning that learns from acting, where the correct output is not known but a reward signal is available to indicate which outputs are preferable to others.', '1611.09321-1-42-0': 'The former approach imposes more constraints than the latter and therefore has had more success.', '1611.09321-1-42-1': 'While some supervised learning approaches use a simple sequence-to-sequence recurrent neural network to map inputs to outputs , more successful approaches have been based on augmenting the model with additional computational elements that have been modified to allow for differentiation via backpropagation.', '1611.09321-1-42-2': 'For example, [CITATION] have proposed the Neural Turing Machine and the Differentiable Neural Computer (DNC), both of which provide an auxiliary memory with a trainable interface for reading and writing.', '1611.09321-1-42-3': 'Beyond basic sequence manipulations, these models have also successfully learned tasks such as finding the shortest path between specified points.', '1611.09321-1-42-4': 'Others have investigated stack- or queue-based augmentations , which have been successfully applied to simple sequence manipulation tasks, and simple arithmetic problems, such as integer addition.', '1611.09321-1-43-0': 'Other supervised neural network approaches have not augmented the network but rather altered how it manipulates or interacts with the input.', '1611.09321-1-43-1': 'Two prominent examples are the Grid LSTM and the Neural GPU , which have succeeded in solving addition and multiplication problems given variable-length inputs.', '1611.09321-1-43-2': 'A recent supervised approach that does not learn from input-output pairs but rather from full execution traces is given by [CITATION].', '1611.09321-1-43-3': 'By providing much richer training data, the network is able to learn more complex algorithms like sorting.', '1611.09321-1-44-0': 'All of these approaches require strong supervision that provides, at a minimum, the correct output for any given input.', '1611.09321-1-44-1': 'There are many cases where this is undesirable.', '1611.09321-1-44-2': 'For example, in an NP-complete problem such as the Traveling Salesman Problem, the correct output is infeasible to compute for large inputs, while a proposed output is easy to check.', '1611.09321-1-44-3': 'Similarly, in tasks where one cares about the efficiency of the algorithm produced in addition to correctness (for example in binary search we prefer a logarithmic-time algorithm over a linear-time algorithm), simply providing the correct output for a given input does not convey sufficient information.', '1611.09321-1-44-4': 'Yet, providing the model with a reward that indicates the cost of its execution is straightforward.', '1611.09321-1-45-0': 'Therefore, a more general and potentially more desirable approach is to deploy self-driven learning that only receives reinforcement based on its outputs.', '1611.09321-1-45-1': 'Unfortunately, such an approach has experienced less success in practice, since the model receives less information during training, which has also resulted in far less research activity to date.', '1611.09321-1-45-2': 'However, some recent work has considered a reinforcement based approach to algorithmic tasks, including , where a Neural Turing Machine is trained via reinforcement learning.', '1611.09321-1-45-3': 'The resulting model can solve simple sequence manipulation tasks such as deduplicating or reversing a sequence.', '1611.09321-1-45-4': 'Also, previous work on the DNC has also briefly explored training via reinforcement learning, presenting the ability of the DNC to solve a block manipulation game given only reward feedback.', '1611.09321-1-46-0': 'Exploration in Reinforcement Learning.', '1611.09321-1-47-0': '# Six Algorithmic Tasks', '1611.09321-1-48-0': 'We assess the effectiveness of the proposed approach on five algorithmic tasks from the OpenAI Gym , as well as a new binary search problem.', '1611.09321-1-48-1': 'Each task is summarized below with further details available on the Gym website or in their open-source code.', '1611.09321-1-48-2': 'In each case, the environment has a hidden tape and a hidden sequence.', '1611.09321-1-48-3': 'The agent observes the sequence via a pointer to a single character, which can be moved by a set of pointer control actions.', '1611.09321-1-48-4': '[topsep=0em,itemsep=.5em,leftmargin=1.3em,parsep=0em] Copy: The agent should emit a copy of the sequence.', '1611.09321-1-48-5': 'The pointer actions are move left and right.', '1611.09321-1-48-6': 'DuplicatedInput: In the hidden tape, each character is repeated twice.', '1611.09321-1-48-7': 'The agent must deduplicate the sequence and emit every other character.', '1611.09321-1-48-8': 'The pointer actions are move left and right.', '1611.09321-1-48-9': 'RepeatCopy: The agent should emit the hidden sequence once, then emit the sequence in the reverse order, then emit the original sequence again.', '1611.09321-1-48-10': 'The pointer actions are move left and right.', '1611.09321-1-48-11': 'Reverse: The agent should emit the hidden sequence in the reverse order.', '1611.09321-1-48-12': 'As before, the pointer actions are move left and right.', '1611.09321-1-48-13': 'ReversedAddition: The hidden tape is a [MATH] grid of digits representing two numbers in base [MATH] in little-endian order.', '1611.09321-1-48-14': 'The agent must emit the sum of the two numbers, in little-endian order.', '1611.09321-1-48-15': 'The allowed pointer actions are move left, right, up, or down.', '1611.09321-1-49-0': 'The OpenAI Gym provides an additional harder task called ReversedAddition3, which involves adding three numbers.', '1611.09321-1-49-1': 'We omit this task, since none of the methods make much progress on it.', '1611.09321-1-50-0': 'For these tasks, during training, input sequences range from a length of [MATH] characters to [MATH].', '1611.09321-1-50-1': 'A reward of [MATH] is given for each correct emission.', '1611.09321-1-50-2': 'On an incorrect emission, a small penalty of [MATH] is incurred and the episode is terminated.', '1611.09321-1-50-3': 'The agent is also terminated and penalized with a reward of [MATH] if the episode exceeds a certain number of steps.', '1611.09321-1-50-4': 'Each of the Gym tasks has a success threshold, which determines the required average reward over [MATH] episodes for the agent to be considered successful.', '1611.09321-1-51-0': 'We also conduct experiments on an additional algorithmic task described below: [topsep=-.2em,itemsep=0em,leftmargin=1.3em,parsep=-.2em]', '1611.09321-1-52-0': 'BinarySearch: Given an integer [MATH], the environment has a hidden array of [MATH] distinct numbers stored in ascending order.', '1611.09321-1-52-1': 'The environment also has a query number [MATH] unknown to the agent that is contained somewhere in the array.', '1611.09321-1-52-2': 'The goal of the agent is to find the query number in the array in a small number of actions.', '1611.09321-1-52-3': 'The environment has three integer registers initialized at [MATH].', '1611.09321-1-52-4': 'At each step, the agent can interact with the environment via the four following actions:*.3em *.5em The agent is terminated when the number of steps exceeds a maximum threshold of [MATH] steps and recieves a reward of [MATH].', '1611.09321-1-52-5': 'If the agent finds [MATH] at step [MATH], it recieves a reward of [MATH].', '1611.09321-1-53-0': 'We set the maximum number of steps to [MATH] to allow the agent to perform a full linear search.', '1611.09321-1-53-1': 'A policy performing full linear search achieves an average reward of [MATH], because [MATH] is chosen uniformly at random from the elements of the array.', '1611.09321-1-53-2': 'A policy employing binary search can find the number [MATH] in at most [MATH] steps.', '1611.09321-1-53-3': 'If [MATH] is selected uniformly at random from the range [MATH], binary search yields an optimal average reward above [MATH].', '1611.09321-1-53-4': 'We set the success threshold for this task to an average reward of [MATH].', '1611.09321-1-54-0': '# Experiments', '1611.09321-1-55-0': 'We compare our policy gradient method using under-appreciated reward exploration () against two main RL baselines: (1) REINFORCE with entropy regularization termed , where the value of [MATH] determines the degree of regularization.', '1611.09321-1-55-1': 'When [MATH], standard REINFORCE is obtained.', '1611.09321-1-55-2': '(2) one-step double Q-learning based on bootstrapping one step future rewards.', '1611.09321-1-56-0': '## Robustness to hyper-parameters', '1611.09321-1-57-0': 'Hyper-parameter tuning is often tedious for RL algorithms.', '1611.09321-1-57-1': 'We found that the proposed method significantly improves robustness to changes in hyper-parameters when compared to .', '1611.09321-1-57-2': 'For our experiments, we perform a careful grid search over a set of hyper-parameters for both and .', '1611.09321-1-57-3': 'For any hyper-parameter setting, we run the and methods [MATH] times with different random restarts.', '1611.09321-1-57-4': 'We explore the following main hyper-parameters:', '1611.09321-1-58-0': 'In all of the experiments, both and are treated exactly the same.', '1611.09321-1-58-1': 'In fact, the change of implementation is just a few lines of code.', '1611.09321-1-58-2': 'Given a value of [MATH], for each task, we run [MATH] training jobs comprising [MATH] learning rates, [MATH] clipping values, and [MATH] random restarts.', '1611.09321-1-58-3': 'We run each algorithm for a maximum number of steps determined based on the difficulty of the task.', '1611.09321-1-58-4': 'The training jobs for Copy, DuplicatedInput, RepeatCopy, Reverse, ReversedAddition, and BinarySearch are run for [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] stochastic gradient steps, respectively.', '1611.09321-1-58-5': 'We find that running a trainer job longer does not result in a better performance.', '1611.09321-1-58-6': 'Our policy network comprises a single LSTM layer with [MATH] nodes.', '1611.09321-1-58-7': 'We use the Adam optimizer for the experiments.', '1611.09321-1-59-0': 'gym-results shows the percentage of [MATH] trials on different hyper-parameters ([MATH], [MATH]) and random restarts which successfully solve each of the algorithmic tasks.', '1611.09321-1-59-1': 'It is clear that is more robust than to changes in hyper-parameters, even though we only report the results of for a single temperature.', '1611.09321-1-59-2': 'See Appendix [REF] for more detailed tables on hyper-parameter robustness.', '1611.09321-1-60-0': '## Results', '1611.09321-1-61-0': 'gym-results2 presents the number of successful attempts (out of [MATH] random restarts) and the expected reward values (averaged over [MATH] trials) for each RL algorithm given the best hyper-parameters.', '1611.09321-1-61-1': 'One-step Q-learning results are also included in the table.', '1611.09321-1-61-2': 'It is clear that outperforms the baselines on these tasks.', '1611.09321-1-61-3': 'On the more difficult tasks, such as Reverse and ReverseAddition, is able to consistently find an appropriate algorithm, but and Q-learning fall behind.', '1611.09321-1-61-4': 'Importantly, for the BinarySearch task, which exhibits many local maxima and necessitates smart exploration, is the only method that can solve it consistently.', '1611.09321-1-61-5': 'The Q-learning baseline solves some of the simple tasks, but it makes little headway on the harder tasks.', '1611.09321-1-61-6': 'We believe that entropy regularization for policy gradient and [MATH]-greedy for Q-learning are relatively weak exploration strategies in long episodic tasks with delayed rewards.', '1611.09321-1-61-7': 'On such tasks, one random exploratory step in the wrong direction can take the agent off the optimal policy, hampering its ability to learn.', '1611.09321-1-61-8': 'In contrast, provides a form of adaptive and smart exploration.', '1611.09321-1-61-9': 'In fact, we observe that the variance of the importance weights decreases as the agent approaches the optimal policy, effectively reducing exploration when it is no longer necessary; see Appendix [REF].', '1611.09321-1-62-0': '## Generalization to longer sequences', '1611.09321-1-63-0': 'To confirm whether our method is able to find the correct algorithm for multi-digit addition, we investigate its generalization to longer input sequences than provided during training.', '1611.09321-1-63-1': 'We evaluate the trained models on inputs up to a length of [MATH] digits, even though training sequences were at most [MATH] characters.', '1611.09321-1-63-2': 'For each length, we test the model on [MATH] randomly generated inputs, stopping when the accuracy falls below [MATH].', '1611.09321-1-63-3': 'Out of the [MATH] models trained on addition with , we find that [MATH] models generalize to numbers up to [MATH] digits without any observed mistakes.', '1611.09321-1-63-4': 'On the best hyper-parameters, [MATH] out of the [MATH] random restarts are able to generalize successfully.', '1611.09321-1-63-5': 'For more detailed results on the generalization performance on [MATH] different tasks including Copy, DuplicatedInput, and ReversedAddition, see Appendix [REF].', '1611.09321-1-63-6': 'During these evaluations, we take the action with largest probability from [MATH] at each time step rather than sampling randomly.', '1611.09321-1-64-0': 'We also looked into the generalization of the models trained on the BinarySearch task.', '1611.09321-1-64-1': 'We found that none of the agents perform proper binary search.', '1611.09321-1-64-2': 'Rather, those that solved the task perform a hybrid of binary and linear search: first actions follow a binary search pattern, but then the agent switches to a linear search procedure once it narrows down the search space; see Appendix [REF] for some execution traces for BinarySearch and ReversedAddition.', '1611.09321-1-64-3': "Thus, on longer input sequences, the agent's running time complexity approaches linear rather than logarithmic.", '1611.09321-1-64-4': 'We hope that future work will make more progress on this task.', '1611.09321-1-64-5': 'This task is especially interesting because the reward signal should incorporate both correctness and efficiency of the algorithm.', '1611.09321-1-65-0': 'This task is interesting because there exist many local maxima in the policy space.', '1611.09321-1-65-1': 'Even between a full linear search and an efficient binary search there are many possible policies that solve the task.', '1611.09321-1-65-2': 'As we will see later, none of our methods are able to find the intended efficient binary search algorithm, but rather at best find one of these policies in between linear and binary search.', '1611.09321-1-65-3': 'We hope that future work in this area can make more headway on this difficult task.', '1611.09321-1-66-0': '## Implementation details', '1611.09321-1-67-0': 'In all of the experiments, we make use of curriculum learning.', '1611.09321-1-67-1': 'The environment begins by only providing small inputs and moves on to longer sequences once the agent achieves close to maximal reward over a number of steps.', '1611.09321-1-67-2': 'For policy gradient methods including and , we only provide the agent with a reward at the end of the episode, and there is no notion of intermediate reward.', '1611.09321-1-67-3': 'For the value-based baseline, we implement one-step Q-learning as described in [CITATION]-Alg. [MATH], employing double Q-learning with [MATH]-greedy exploration.', '1611.09321-1-67-4': 'We use the same RNN in our policy-based approaches to estimate the Q values.', '1611.09321-1-67-5': 'A grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) is conducted to find the best hyper-parameters.', '1611.09321-1-67-6': 'Unlike our other methods, the Q-learning baseline uses intermediate rewards, as given by the OpenAI Gym on a per-step basis.', '1611.09321-1-67-7': 'Hence, the Q-learning baseline has a slight advantage over the policy gradient methods.', '1611.09321-1-68-0': 'In all of the tasks except Copy, our stochastic optimizer uses mini-batches comprising [MATH] policy samples from the model.', '1611.09321-1-68-1': 'These [MATH] samples correspond to [MATH] different random sequences drawn from the environment, and [MATH] random policy trajectories per sequence.', '1611.09321-1-68-2': 'In other words, we set [MATH] and [MATH] as defined in [REF] and [REF].', '1611.09321-1-68-3': 'For , we use the [MATH] samples to subtract the mean of the coefficient of [MATH] which includes the contribution of the reward and entropy regularization.', '1611.09321-1-68-4': 'For , we use the [MATH] trajectories to subtract the mean reward and normalize the importance sampling weights.', '1611.09321-1-68-5': 'We do not subtract the mean of the normalized importance weights.', '1611.09321-1-68-6': 'For the Copy task, we use mini-batches with [MATH] samples using [MATH] and [MATH].', '1611.09321-1-68-7': 'Experiments are conducted using Tensorflow .', '1611.09321-1-68-8': ', which we found to perform better.', '1611.09321-1-69-0': 'For our experiments, we aimed to realize two things: first, if can give more consistent behavior across different hyper-parameters; second, if can solve tasks that cannot solve at all.', '1611.09321-1-69-1': 'To this end, we ran several runs of and on each task.', '1611.09321-1-69-2': 'For each task we trained a number of models on different values of [MATH] and different hyper-parameters.', '1611.09321-1-69-3': 'For , we used [MATH], of which we generally found at least one to work well (a broader search did not yield stronger results for other values of [MATH]).', '1611.09321-1-69-4': 'For , we used [MATH], which we found to work well on all the tasks.', '1611.09321-1-70-0': '(a broader search did not yield stronger results for other values of [MATH])', '1611.09321-1-71-0': 'For each training algorithm and value of [MATH], we ran 60 training runs, the result of an intersection of three values of learning rate ([MATH]), four values of clipping value for the gradient ([MATH]), and five random initialization seeds.', '1611.09321-1-71-1': 'With the exception of Copy, each run was trained using batches consisting of 10 trajectories each from 40 different resets of the environment.', '1611.09321-1-71-2': 'Thus, a single batch consisted of 400 total policy trajectories.', '1611.09321-1-71-3': 'For , we use the 10 trajectories to mean-center the coefficient of [MATH].', '1611.09321-1-71-4': 'For , we use the 10 trajectories to both mean-center the rewards and self-normalize the importance sampling weights.', '1611.09321-1-71-5': 'For the Copy task, we used batches of 10 trajectories each from 20 different resets of the environment, which we found to perform better.', '1611.09321-1-71-6': 'The runs for Copy, RepeatCopy, DuplicatedInput, Reverse, ReversedAddition, and BinarySearch were trained for 2000, 50000, 500, 5000, 50000, and 2000 steps, respectively.', '1611.09321-1-71-7': 'We found these numbers of steps were mostly enough for the algorithms to converge for each respective task.', '1611.09321-1-72-0': 'The model architecture we used was an LSTM recurrent neural network with hidden dimension of 128.', '1611.09321-1-72-1': 'We train using the Adam optimizer which we chose for its robustness to hyper-parameters .', '1611.09321-1-73-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-1-73-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-1-73-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-1-73-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-1-73-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-1-73-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-1-74-0': '## Experimental details', '1611.09321-1-75-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-1-75-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-1-76-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-1-76-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-1-76-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-1-76-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-1-76-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-1-76-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-1-77-0': 'We find that not only does solve the considered tasks given the best hyper-parameters, but also that the gradient is more robust to the choice of the hyper-parameters, when compared to the expected reward objective.', '1611.09321-1-78-0': '## Baseline methods', '1611.09321-1-79-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-1-79-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-1-80-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-1-80-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-1-80-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-1-80-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-1-80-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-1-80-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-1-81-0': '# Conclusion', '1611.09321-1-82-0': 'We present a variant of policy gradient, called , which promotes exploring action sequences that yield rewards larger than what the model expects.', '1611.09321-1-82-1': 'This exploration strategy is the result of importance sampling from the optimal policy.', '1611.09321-1-82-2': 'Our experimental results demonstrate that significantly outperforms other value and policy based methods, while being more robust to changes of hyper-parameters.', '1611.09321-1-82-3': 'By using , we can solve algorithmic tasks like multi-digit addition, which other methods cannot reliably solve even given the best hyper-parameters.', '1611.09321-1-82-4': 'We introduce a new algorithmic task based on binary search to advocate more research in this area, especially when the computational complexity of the solutions matters too.', '1611.09321-1-82-5': 'Solving these tasks is not only important to develop human like intelligence to enable learning algorithms, but also important for generic reinforcement learning, where smart and efficient exploration is the key to successful methods.'}

{'1611.09321-2-0-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning (RL) with improved exploration properties.', '1611.09321-2-0-1': 'Current policy-based methods use entropy regularization to encourage undirected exploration of the reward landscape, which is ineffective in high dimensional spaces with sparse rewards.', '1611.09321-2-0-2': 'We propose a more directed exploration strategy that promotes exploration of under-appreciated reward regions.', '1611.09321-2-0-3': 'An action sequence is considered under-appreciated if its log-probability under the current policy under-estimates its resulting reward.', '1611.09321-2-0-4': 'The proposed exploration strategy is easy to implement, requiring small modifications to the REINFORCE algorithm.', '1611.09321-2-0-5': 'We evaluate the approach on a set of algorithmic tasks that have long challenged RL methods.', '1611.09321-2-0-6': 'Our approach reduces hyper-parameter sensitivity and demonstrates significant improvements over baseline methods.', '1611.09321-2-0-7': 'The proposed algorithm successfully solves a benchmark multi-digit addition task and generalizes to long sequences, which, to our knowledge, is the first time that a pure RL method has solved addition using only reward feedback.', '1611.09321-2-1-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning with improved exploration characteristics.', '1611.09321-2-1-1': 'Current policy gradient methods use entropy regularization to encourage uniform exploration of the reward landscape, which is not effective in high dimensional spaces with sparse rewards.', '1611.09321-2-1-2': 'We propose a better exploration strategy that promotes exploration of the under-appreciated reward regions.', '1611.09321-2-1-3': 'An action sequence is considered under-appreciated if its probability under the current policy under-estimates its resulting reward.', '1611.09321-2-1-4': 'The proposed exploration strategy is easy to implement.', '1611.09321-2-1-5': 'It improves robustness to the change of hyper-parameters, and on a set of algorithmic tasks, it demonstrates significant improvements against the policy gradient baselines.', '1611.09321-2-1-6': 'Our approach is able to solve a benchmark multi-digit addition task.', '1611.09321-2-1-7': 'To our knowledge, this is the first pure RL method to solve addition just by using reward feedback.', '1611.09321-2-2-0': 'To our knowledge Unlike standard on-line learning strategies in reinforcement learning, which generally encourage undirected exploration, we stochastically optimize a modified objective that favors exploration of high-reward versus low-reward actions.', '1611.09321-2-2-1': 'A key feature of the algorithm is its ease of application to recurrent neural network policy representations.', '1611.09321-2-2-2': 'We conduct an experimental evaluation on a set of algorithm induction tasks, ranging from simple to challenging, and compare the proposed enhancement to standard baselines.', '1611.09321-2-2-3': 'In addition to obtaining notable improvements, we also observe that the new approach can solve the benchmark addition problem, which to our knowledge is the first successful such demonstration based solely on success/failure reinforcement.', '1611.09321-2-3-0': '# Introduction', '1611.09321-2-4-0': 'Humans can reason about symbolic objects and solve algorithmic problems.', '1611.09321-2-4-1': 'After learning to count and then manipulate numbers via simple arithmetic, people eventually learn to invent new algorithms and even reason about their correctness and efficiency.', '1611.09321-2-4-2': 'The ability to invent new algorithms is fundamental to artificial intelligence (AI).', '1611.09321-2-4-3': 'Although symbolic reasoning has a long history in AI , only recently have statistical machine learning and neural network approaches begun to make headway in automated algorithm discovery , which would constitute an important milestone on the path to AI.', '1611.09321-2-4-4': 'Nevertheless, most of the recent successes depend on the use of strong supervision to learn a mapping from a set of training inputs to outputs by maximizing a conditional log-likelihood, very much like neural machine translation systems .', '1611.09321-2-4-5': 'Such a dependence on strong supervision is a significant limitation that does not match the ability of people to invent new algorithmic procedures based solely on trial and error.', '1611.09321-2-5-0': 'By contrast, reinforcement learning (RL) methods hold the promise of searching over discrete objects such as symbolic representations of algorithms by considering much weaker feedback in the form of a simple verifier that tests the correctness of a program execution on a given problem instance.', '1611.09321-2-5-1': 'Despite the recent excitement around the use of RL to tackle Atari games and Go , standard RL methods are not yet able to consistently and reliably solve algorithmic tasks in all but the simplest cases .', '1611.09321-2-5-2': 'A key property of algorithmic problems that makes them challenging for RL is reward sparsity, a policy usually has to get a long action sequence exactly right to obtain a non-zero reward.', '1611.09321-2-6-0': 'We believe one of the key limitations of the current RL methods, limiting their effectiveness in a sparse reward setting, is the use of undirected exploration strategies , such as [MATH]-greedy and entropy regularization .', '1611.09321-2-6-1': 'For long action sequences with delayed sparse reward, it is hopeless to explore the space uniformly and blindly.', '1611.09321-2-6-2': 'Instead, we propose a formulation to encourage exploration of action sequences that are under-appreciated by the current policy.', '1611.09321-2-6-3': "Our formulation considers an action sequence to be under-appreciated if the model's log-probability assigned to an action sequence under-estimates the resulting reward from the action sequence.", '1611.09321-2-6-4': 'Exploring under-appreciated states and actions encourages the policy to have a better calibration between its log-probabilities and observed reward values, even for action sequences with negligible rewards.', '1611.09321-2-6-5': 'This effectively increases exploration around neglected action sequences.', '1611.09321-2-7-0': 'We term our proposed technique under-appreciated reward exploration ().', '1611.09321-2-7-1': 'We show that the objective given by is a combination of a mode seeking objective (standard REINFORCE) and a mean seeking term, which provides a well motivated trade-off between exploitation and exploration.', '1611.09321-2-7-2': 'To empirically evaluate our method, we take a set of algorithmic tasks such as sequence reversal, multi-digit addition, and binary search.', '1611.09321-2-7-3': 'We choose to focus on these tasks because, although simple, they present a difficult sparse reward setting which has limited the success of standard RL approaches.', '1611.09321-2-7-4': 'The experiments demonstrate that significantly outperforms baseline RL methods, such as entropy regularized REINFORCE and one-step Q-learning, especially on the more difficult tasks, such as multi-digit addition.', '1611.09321-2-7-5': 'Moreover, is shown to be more robust to changes of hyper-parameters, which makes hyper-parameter tuning less tedious in practice.', '1611.09321-2-7-6': 'In addition to introducing a new variant of policy gradient with improved performance, our paper is the first to demonstrate strong results for an RL method on algorithmic tasks.', '1611.09321-2-7-7': 'To our knowledge, the addition task has not been solved by any model-free reinforcement learning approach.', '1611.09321-2-7-8': 'We observe that some of the policies learned by can successfully generalize to long sequences; in [MATH] out of [MATH] random restarts, the policy learned by for the addition task correctly generalizes to addition of numbers with [MATH] digits with no mistakes, even though training sequences are at most [MATH] digits long.', '1611.09321-2-8-0': 'In order to find both non-zero reward action sequences and maximal reward action sequences in a large action space, some sort of exploration is necessary.', '1611.09321-2-8-1': 'In value-based RL this exploration can manifest as [MATH]-greedy actions whereas in policy-based methods a common choice is to add an entropy regularizer to the well-known REINFORCE policy gradient algorithm .', '1611.09321-2-8-2': 'However, these exploration strategies apply exploration blindly and uniformly.', '1611.09321-2-8-3': 'In large action spaces, these strategies can easily distract an agent from its main task of honing in on high-reward regions of the policy space.', '1611.09321-2-9-0': 'In this work, we propose a novel addition to the REINFORCE objective to favor exploration in high-reward areas over exploration in low-reward areas.', '1611.09321-2-10-0': 'We begin in Section [REF], where we provide a short survey of previous work on learning algorithms.', '1611.09321-2-10-1': 'In Section [REF] we provide background information regarding our notation and the derivation of REINFORCE and its entropy-regularized variant.', '1611.09321-2-10-2': 'We introduce our approach and the resulting policy gradient training algorithm in [REF].', '1611.09321-2-10-3': 'Subsequently, we present several standard tasks and introduce an additional task we designed ourselves in Section [REF], on which we then empirically evaluate the performance of our approach.', '1611.09321-2-10-4': 'We conclude in Section [REF], where we summarize the paper and discuss the repercussions and impact we hope it will have on future work.', '1611.09321-2-11-0': '# Neural Networks for Learning Algorithms', '1611.09321-2-12-0': 'Although research on using neural networks to learn algorithms has had a surge of recent interest, the problem of program induction from examples has a long history in many fields, including program induction, inductive logic programming , relational learning and regular language learning .', '1611.09321-2-12-1': 'Rather than presenting a comprehensive survey of program induction here, we focus on neural network approaches to algorithmic tasks and highlight the relative simplicity of our neural network architecture.', '1611.09321-2-13-0': 'Most successful applications of neural networks to algorithmic tasks rely on strong supervision, where the inputs and target outputs are completely known a priori.', '1611.09321-2-13-1': 'Given a dataset of examples, one learns the network parameters by maximizing the conditional likelihood of the outputs via backpropagation ( [CITATION]).', '1611.09321-2-13-2': 'However, target outputs may not be available for novel tasks, for which no prior algorithm is known to be available.', '1611.09321-2-13-3': 'A more desirable approach to inducing algorithms, followed in this paper, advocates using self-driven learning strategies that only receive reinforcement based on the outputs produced.', '1611.09321-2-13-4': 'Hence, just by having access to a verifier for an algorithmic problem, one can aim to learn an algorithm.', '1611.09321-2-13-5': 'For example, if one does not know how to sort an array, but can check the extent to which an array is sorted, then one can provide the reward signal necessary for learning sorting algorithms.', '1611.09321-2-14-0': 'However, providing the correct output during training can often be infeasible; for example, when trying to solve a problem with no known solution, or when efficiency is desired in addition to correctness.', '1611.09321-2-15-0': 'We formulate learning algorithms as an RL problem and make use of model-free policy gradient methods to optimize a set parameters associated with the algorithm.', '1611.09321-2-15-1': 'In this setting, the goal is to learn a policy [MATH] that given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-2-15-2': 'Actions represent the commands within the algorithm and states represent the joint state of the algorithm and the environment.', '1611.09321-2-15-3': 'Previous work in this area has focused on augmenting a neural network with additional structure and increased capabilities .', '1611.09321-2-15-4': 'In contrast, we utilize a simple architecture based on a standard recurrent neural network (RNN) with LSTM cells as depicted in fig:rnn.', '1611.09321-2-15-5': 'At each episode, the environment is initialized with a latent state [MATH], unknown to the agent, which determines [MATH] and the subsequent state transition and reward functions.', '1611.09321-2-15-6': 'Once the agent observes [MATH] as the input to the RNN, the network outputs a distribution [MATH], from which an action [MATH] is sampled.', '1611.09321-2-15-7': 'This action is applied to the environment, and the agent receives a new state observation [MATH].', '1611.09321-2-15-8': 'The state [MATH] and the previous action [MATH] are then fed into the RNN and the process repeats until the end of the episode.', '1611.09321-2-15-9': 'Upon termination, a reward signal is received.', '1611.09321-2-16-0': '# Learning a Policy by Maximizing Expected Reward', '1611.09321-2-17-0': 'We start by discussing the most common form of policy gradient, REINFORCE , and its entropy regularized variant .', '1611.09321-2-17-1': 'REINFORCE has been applied to model-free policy-based learning with neural networks and algorithmic domains .', '1611.09321-2-18-0': 'The goal is to learn a policy [MATH] that, given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-2-18-1': 'The environment is initialized with a latent vector, [MATH], which determines the initial observed state [MATH], and the transition function [MATH].', '1611.09321-2-18-2': 'Note that the use of nondeterministic transitions [MATH] as in Markov decision processes (MDP) may be recovered by assuming that [MATH] includes the random seed for the any nondeterministic functions.', '1611.09321-2-18-3': 'Given a latent state [MATH], and [MATH], the model probability of an action sequence [MATH] is expressed as, [EQUATION]', '1611.09321-2-18-4': 'The environment provides a reward at the end of the episode, denoted [MATH].', '1611.09321-2-18-5': 'For ease of readability we drop the subscript from [MATH] and simply write [MATH] and [MATH].', '1611.09321-2-19-0': 'The objective used to optimize the policy parameters, [MATH], consists of maximizing expected reward under actions drawn from the policy, plus an optional maximum entropy regularizer.', '1611.09321-2-19-1': 'Given a distribution over initial latent environment states [MATH], we express the regularized expected reward as, [EQUATION]', '1611.09321-2-19-2': 'When [MATH] is a non-linear function defined by a neural network, finding the global optimum of [MATH] is challenging, and one often resorts to gradient-based methods to find a local optimum of [MATH].', '1611.09321-2-19-3': 'Given that [MATH] for any [MATH] such that [MATH], one can verify that, [EQUATION]', '1611.09321-2-19-4': 'Because the space of possible actions [MATH] is large, enumerating over all of the actions to compute this gradient is infeasible.', '1611.09321-2-19-5': 'Williams Williams92 proposed to compute the stochastic gradient of the expected reward by using Monte Carlo samples.', '1611.09321-2-19-6': 'Using Monte Carlo samples, one first draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to approximate the gradient of [REF] by using [REF] as, [EQUATION]', '1611.09321-2-19-7': 'This reparametrization of the gradients is the key to the REINFORCE algorithm.', '1611.09321-2-19-8': 'To reduce the variance of [REF], one uses rewards [MATH] that are shifted by some offset values, [EQUATION] where [MATH] is known as a baseline or sometimes called a critic.', '1611.09321-2-19-9': 'Note that subtracting any offset from the rewards in [REF] simply results in shifting the objective [MATH] by a constant.', '1611.09321-2-19-10': "I can't make [MATH] function of [MATH]", '1611.09321-2-20-0': 'Unfortunately, directly maximizing expected reward ( when [MATH]) is prone to getting trapped in a local optimum.', '1611.09321-2-20-1': 'To combat this tendency, Williams Peng williams1991function augmented the expected reward objective by including a maximum entropy regularizer ([MATH]) to promote greater exploration.', '1611.09321-2-20-2': 'We will refer to this variant of REINFORCE as (maximum entropy exploration).', '1611.09321-2-21-0': '# Under-appreciated Reward Exploration ()', '1611.09321-2-22-0': 'To explain our novel form of policy gradient, we first note that the optimal policy [MATH], which globally maximizes [MATH] in [REF] for any [MATH], can be expressed as, [EQUATION] where [MATH] is a normalization constant making [MATH] a distribution over the space of action sequences [MATH].', '1611.09321-2-22-1': 'One can verify this by first acknowledging that, [EQUATION]', '1611.09321-2-22-2': 'Since [MATH] is non-negative and zero iff [MATH], then [MATH] defined in [REF] maximizes [MATH].', '1611.09321-2-22-3': 'That said, given a particular form of [MATH], finding [MATH] that exactly characterizes [MATH] may not be feasible.', '1611.09321-2-23-0': 'The KL divergence [MATH] is known to be mode seeking [CITATION] even with entropy regularization ([MATH]).', '1611.09321-2-23-1': 'Learning a policy by optimizing this direction of the KL is prone to falling into a local optimum resulting in a sub-optimal policy that omits some of the modes of [MATH].', '1611.09321-2-23-2': 'Although entropy regularization helps mitigate the issues as confirmed in our experiments, it is not an effective exploration strategy as it is undirected and requires a small regularization coefficient [MATH] to avoid too much random exploration.', '1611.09321-2-23-3': 'Instead, we propose a directed exploration strategy that improves the mean seeking behavior of policy gradient in a principled way.', '1611.09321-2-24-0': 'We start by considering the alternate mean seeking direction of the KL divergence, [MATH].', '1611.09321-2-24-1': '[CITATION] considered this direction of the KL to directly learn a policy by optimizing [EQUATION] for structured prediction.', '1611.09321-2-24-2': 'This objective has the same optimal solution [MATH] as [MATH] since, [EQUATION] [CITATION] argue that in some structured prediction problems when one can draw samples from [MATH], optimizing [REF] is more effective than [REF], since no sampling from a non-stationary policy [MATH] is required.', '1611.09321-2-24-3': 'If [MATH] is a log-linear model of a set of features, [MATH] is convex in [MATH] whereas [MATH] is not, even in the log-linear case.', '1611.09321-2-24-4': 'Unfortunately, in scenarios that the reward landscape is unknown or computing the normalization constant [MATH] is intractable, sampling from [MATH] is not straightforward.', '1611.09321-2-25-0': 'In the RL problems, the reward landscape is completely unknown, hence sampling from [MATH] is intractable.', '1611.09321-2-25-1': 'This paper proposes to approximate the expectation with respect to [MATH] in [REF] by using self-normalized importance sampling , where the proposal distribution is [MATH] and the reference distribution is [MATH].', '1611.09321-2-25-2': 'For importance sampling, one draws [MATH] samples [MATH] from [MATH] and computes a set of normalized importance weights to approximate [MATH] as, [EQUATION] where [MATH] denotes an importance weight defined by, [EQUATION]', '1611.09321-2-25-3': "One can view these importance weights as evaluating the discrepancy between scaled rewards [MATH] and the policy's log-probabilities [MATH].", '1611.09321-2-25-4': 'Among the [MATH] samples, a sample that is least appreciated by the model, has the largest [MATH], receives the largest positive feedback in [REF].', '1611.09321-2-26-0': 'In practice, we have found that just using the importance sampling RAML objective in [REF] does not always yield promising solutions.', '1611.09321-2-26-1': 'Particularly, at the beginning of training, when [MATH] is still far away from [MATH], the variance of importance weights is too large, and the self-normalized importance sampling procedure results in poor approximations.', '1611.09321-2-26-2': 'To stabilize early phases of training and ensure that the model distribution [MATH] achieves large expected reward scores, we combine the expected reward and RAML objectives to benefit from the best of their mode and mean seeking behaviors.', '1611.09321-2-26-3': 'Accordingly, we propose the following objective that we call under-appreciated reward exploration (), [EQUATION] which is the sum of the expected reward and RAML objectives.', '1611.09321-2-26-4': 'In our preliminary experiments, we considered a composite objective of [MATH], but we found that removing the entropy term is beneficial.', '1611.09321-2-26-5': 'Hence, the [MATH] objective does not include entropy regularization.', '1611.09321-2-26-6': 'Accordingly, the optimum policy for [MATH] is no longer [MATH], as it was for [MATH] and [MATH].', '1611.09321-2-26-7': 'Appendix [REF] derives the optimal policy for [MATH] as a function of the optimal policy for [MATH].', '1611.09321-2-26-8': 'We find that the optimal policy of is more sharply concentrated on the high reward regions of the action space, which may be an advantage for , but we leave more analysis of this behavior to future work.', '1611.09321-2-27-0': 'as [MATH], where [MATH] normalizes the distribution for each [MATH].', '1611.09321-2-28-0': 'To compute the gradient of [MATH], we use the self-normalized importance sampling estimate outlined in [REF].', '1611.09321-2-28-1': 'We assume that the importance weights are constant and contribute no gradient to [MATH].', '1611.09321-2-28-2': 'To approximate the gradient, one draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to obtain [EQUATION]', '1611.09321-2-28-3': 'As with REINFORCE, the rewards are shifted by an offset [MATH].', '1611.09321-2-28-4': 'In this gradient, the model log-probability of a sample action sequence [MATH] is reinforced if the corresponding reward is large, or the corresponding importance weights are large, meaning that the action sequence is under-appreciated.', '1611.09321-2-28-5': 'The normalized importance weights are computed using a softmax operator [MATH].', '1611.09321-2-29-0': 'There are three ways to explain why [MATH] may not converge to [MATH] after training using PG:', '1611.09321-2-30-0': 'Model capacity - The form of the model may not be expressive enough to represent [MATH].', '1611.09321-2-30-1': 'Variance - The space of possible actions [MATH] is very large and the reward is very sparse causing the stochastic gradients defined in [REF] to have a large variance.', '1611.09321-2-30-2': 'Optimization - The parameters may fall into local optima of the objective, or the objective may have a very slow convergence rate.', '1611.09321-2-31-0': 'When using recurrent neural networks for simple tasks, point (1) should not be an issue.', '1611.09321-2-31-1': 'Usually papers blame variance for the failure cases of PG, as it is hard to reason about optimization landscape especially in high-dimensional problems.', '1611.09321-2-31-2': 'Below we discuss a very simple one-dimensional problem where local optima seem to be a problem for the regularized expected reward objective.', '1611.09321-2-31-3': 'find a simple example!', '1611.09321-2-32-0': 'We use recurrent neural networks (RNNs) to model [MATH].', '1611.09321-2-32-1': '[CITATION]), which is formulated as minimization of the following objective,', '1611.09321-2-33-0': 'where [MATH] denotes the reward function, negative edit distance or BLEU score, [MATH] controls the degree of regularization, and [MATH] is the entropy of a distribution [MATH], [MATH].', '1611.09321-2-33-1': 'It is well-known that optimizing [MATH] using SGD is challenging because of the large variance of the gradients.', '1611.09321-2-33-2': 'Below we describe how ML and RL objectives are related, and propose a hybrid between the two that combines their benefits for supervised learning.', '1611.09321-2-34-0': '# Related Work on Exploration in Reinforcement Learning', '1611.09321-2-35-0': 'The most common exploration strategy considered in value-based RL is [MATH]-greedy Q-learning, where at each step the agent either takes the best action according to its current value approximation or with probability [MATH] takes an action sampled uniformly at random.', '1611.09321-2-35-1': 'Like entropy regularization, such an approach applies undirected exploration, but it has achieved recent success in game playing environments .', '1611.09321-2-36-0': 'Prominent approaches to improving exploration beyond [MATH]-greedy in value-based or model-based RL have focused on reducing uncertainty by prioritizing exploration toward states and actions where the agent knows the least.', '1611.09321-2-36-1': 'This basic intuition underlies work on counter and recency methods , exploration methods based on uncertainty estimates of values , methods that prioritize learning environment dynamics , and methods that provide an intrinsic motivation or curiosity bonus for exploring unknown states .', '1611.09321-2-37-0': 'In contrast to value-based methods, exploration for policy-based RL methods is often a by-product of the optimization algorithm itself.', '1611.09321-2-37-1': 'Since algorithms like REINFORCE and Thompson sampling choose actions according to a stochastic policy, sub-optimal actions are chosen with some non-zero probability.', '1611.09321-2-37-2': 'The Q-learning algorithm may also be modified to sample an action from the softmax of the Q values rather than the argmax .', '1611.09321-2-38-0': 'Asynchronous training has also been reported to have an exploration effect on both value- and policy-based methods.', '1611.09321-2-38-1': '[CITATION] report that asynchronous training can stabilize training by reducing the bias experienced by a single trainer.', '1611.09321-2-38-2': 'By using multiple separate trainers, an agent is less likely to become trapped at a policy found to be locally optimal only due to local conditions.', '1611.09321-2-38-3': 'In the same spirit, [CITATION] use multiple Q value approximators and sample only one to act for each episode as a way to implicitly incorporate exploration.', '1611.09321-2-39-0': 'By relating the concepts of value and policy in RL, the exploration strategy we propose tries to bridge the discrepancy between the two.', '1611.09321-2-39-1': 'In particular, UREX can be viewed as a hybrid combination of value-based and policy-based exploration strategies that attempts to capture the benefits of each.', '1611.09321-2-40-0': 'Although the problem of program induction from examples has been studied in many fields, including program induction, inductive logic programming , relational learning and regular language learning ; we do not attempt a comprehensive survey here.', '1611.09321-2-40-1': 'Instead, we focus on related neural network approaches for solving algorithmic tasks and relevant work that considers exploration in reinforcement learning.', '1611.09321-2-41-0': 'Learning Algorithms.', '1611.09321-2-41-1': 'Previous work on neural network learning algorithms for algorithmic problems falls into two general categories: strongly supervised maximum likelihood training that learns from examples using completely known input-output pairs; and self-driven training via reinforcement learning that learns from acting, where the correct output is not known but a reward signal is available to indicate which outputs are preferable to others.', '1611.09321-2-42-0': 'The former approach imposes more constraints than the latter and therefore has had more success.', '1611.09321-2-42-1': 'While some supervised learning approaches use a simple sequence-to-sequence recurrent neural network to map inputs to outputs , more successful approaches have been based on augmenting the model with additional computational elements that have been modified to allow for differentiation via backpropagation.', '1611.09321-2-42-2': 'For example, [CITATION] have proposed the Neural Turing Machine and the Differentiable Neural Computer (DNC), both of which provide an auxiliary memory with a trainable interface for reading and writing.', '1611.09321-2-42-3': 'Beyond basic sequence manipulations, these models have also successfully learned tasks such as finding the shortest path between specified points.', '1611.09321-2-42-4': 'Others have investigated stack- or queue-based augmentations , which have been successfully applied to simple sequence manipulation tasks, and simple arithmetic problems, such as integer addition.', '1611.09321-2-43-0': 'Other supervised neural network approaches have not augmented the network but rather altered how it manipulates or interacts with the input.', '1611.09321-2-43-1': 'Two prominent examples are the Grid LSTM and the Neural GPU , which have succeeded in solving addition and multiplication problems given variable-length inputs.', '1611.09321-2-43-2': 'A recent supervised approach that does not learn from input-output pairs but rather from full execution traces is given by [CITATION].', '1611.09321-2-43-3': 'By providing much richer training data, the network is able to learn more complex algorithms like sorting.', '1611.09321-2-44-0': 'All of these approaches require strong supervision that provides, at a minimum, the correct output for any given input.', '1611.09321-2-44-1': 'There are many cases where this is undesirable.', '1611.09321-2-44-2': 'For example, in an NP-complete problem such as the Traveling Salesman Problem, the correct output is infeasible to compute for large inputs, while a proposed output is easy to check.', '1611.09321-2-44-3': 'Similarly, in tasks where one cares about the efficiency of the algorithm produced in addition to correctness (for example in binary search we prefer a logarithmic-time algorithm over a linear-time algorithm), simply providing the correct output for a given input does not convey sufficient information.', '1611.09321-2-44-4': 'Yet, providing the model with a reward that indicates the cost of its execution is straightforward.', '1611.09321-2-45-0': 'Therefore, a more general and potentially more desirable approach is to deploy self-driven learning that only receives reinforcement based on its outputs.', '1611.09321-2-45-1': 'Unfortunately, such an approach has experienced less success in practice, since the model receives less information during training, which has also resulted in far less research activity to date.', '1611.09321-2-45-2': 'However, some recent work has considered a reinforcement based approach to algorithmic tasks, including , where a Neural Turing Machine is trained via reinforcement learning.', '1611.09321-2-45-3': 'The resulting model can solve simple sequence manipulation tasks such as deduplicating or reversing a sequence.', '1611.09321-2-45-4': 'Also, previous work on the DNC has also briefly explored training via reinforcement learning, presenting the ability of the DNC to solve a block manipulation game given only reward feedback.', '1611.09321-2-46-0': 'Exploration in Reinforcement Learning.', '1611.09321-2-47-0': '# Six Algorithmic Tasks', '1611.09321-2-48-0': 'We assess the effectiveness of the proposed approach on five algorithmic tasks from the OpenAI Gym , as well as a new binary search problem.', '1611.09321-2-48-1': 'Each task is summarized below with further details available on the Gym website or in their open-source code.', '1611.09321-2-48-2': 'In each case, the environment has a hidden tape and a hidden sequence.', '1611.09321-2-48-3': 'The agent observes the sequence via a pointer to a single character, which can be moved by a set of pointer control actions.', '1611.09321-2-48-4': 'Thus an action [MATH] is represented as a tuple [MATH] where [MATH] denotes how to move, [MATH] is a boolean denoting whether to write, and [MATH] is the output symbol to write.', '1611.09321-2-48-5': '[topsep=0em,itemsep=.5em,leftmargin=1.3em,parsep=0em] Copy: The agent should emit a copy of the sequence.', '1611.09321-2-48-6': 'The pointer actions are move left and right.', '1611.09321-2-48-7': 'DuplicatedInput: In the hidden tape, each character is repeated twice.', '1611.09321-2-48-8': 'The agent must deduplicate the sequence and emit every other character.', '1611.09321-2-48-9': 'The pointer actions are move left and right.', '1611.09321-2-48-10': 'RepeatCopy: The agent should emit the hidden sequence once, then emit the sequence in the reverse order, then emit the original sequence again.', '1611.09321-2-48-11': 'The pointer actions are move left and right.', '1611.09321-2-48-12': 'Reverse: The agent should emit the hidden sequence in the reverse order.', '1611.09321-2-48-13': 'As before, the pointer actions are move left and right.', '1611.09321-2-48-14': 'ReversedAddition: The hidden tape is a [MATH] grid of digits representing two numbers in base [MATH] in little-endian order.', '1611.09321-2-48-15': 'The agent must emit the sum of the two numbers, in little-endian order.', '1611.09321-2-48-16': 'The allowed pointer actions are move left, right, up, or down.', '1611.09321-2-49-0': 'The OpenAI Gym provides an additional harder task called ReversedAddition3, which involves adding three numbers.', '1611.09321-2-49-1': 'We omit this task, since none of the methods make much progress on it.', '1611.09321-2-50-0': 'For these tasks, during training, input sequences range from a length of [MATH] characters to [MATH].', '1611.09321-2-50-1': 'A reward of [MATH] is given for each correct emission.', '1611.09321-2-50-2': 'On an incorrect emission, a small penalty of [MATH] is incurred and the episode is terminated.', '1611.09321-2-50-3': 'The agent is also terminated and penalized with a reward of [MATH] if the episode exceeds a certain number of steps.', '1611.09321-2-50-4': 'For our experiments using and , we associated an episodic sequence of actions only with the total reward defined as the sum of the per-step rewards.', '1611.09321-2-50-5': 'Our experiments using Q-learning on the other hand used the per-step rewards.', '1611.09321-2-50-6': 'Each of the Gym tasks has a success threshold, which determines the required average reward over [MATH] episodes for the agent to be considered successful.', '1611.09321-2-51-0': 'We also conduct experiments on an additional algorithmic task described below: [topsep=-.2em,itemsep=0em,leftmargin=1.3em,parsep=-.2em]', '1611.09321-2-52-0': 'BinarySearch: Given an integer [MATH], the environment has a hidden array of [MATH] distinct numbers stored in ascending order.', '1611.09321-2-52-1': 'The environment also has a query number [MATH] unknown to the agent that is contained somewhere in the array.', '1611.09321-2-52-2': 'The goal of the agent is to find the query number in the array in a small number of actions.', '1611.09321-2-52-3': 'The environment has three integer registers initialized at [MATH].', '1611.09321-2-52-4': 'At each step, the agent can interact with the environment via the four following actions:*.3em *.5em The agent is terminated when the number of steps exceeds a maximum threshold of [MATH] steps and recieves a reward of [MATH].', '1611.09321-2-52-5': 'If the agent finds [MATH] at step [MATH], it recieves a reward of [MATH].', '1611.09321-2-53-0': 'We set the maximum number of steps to [MATH] to allow the agent to perform a full linear search.', '1611.09321-2-53-1': 'A policy performing full linear search achieves an average reward of [MATH], because [MATH] is chosen uniformly at random from the elements of the array.', '1611.09321-2-53-2': 'A policy employing binary search can find the number [MATH] in at most [MATH] steps.', '1611.09321-2-53-3': 'If [MATH] is selected uniformly at random from the range [MATH], binary search yields an optimal average reward above [MATH].', '1611.09321-2-53-4': 'We set the success threshold for this task to an average reward of [MATH].', '1611.09321-2-54-0': '# Experiments', '1611.09321-2-55-0': 'We compare our policy gradient method using under-appreciated reward exploration () against two main RL baselines: (1) REINFORCE with entropy regularization termed , where the value of [MATH] determines the degree of regularization.', '1611.09321-2-55-1': 'When [MATH], standard REINFORCE is obtained.', '1611.09321-2-55-2': '(2) one-step double Q-learning based on bootstrapping one step future rewards.', '1611.09321-2-56-0': '## Robustness to hyper-parameters', '1611.09321-2-57-0': 'Hyper-parameter tuning is often tedious for RL algorithms.', '1611.09321-2-57-1': 'We found that the proposed method significantly improves robustness to changes in hyper-parameters when compared to .', '1611.09321-2-57-2': 'For our experiments, we perform a careful grid search over a set of hyper-parameters for both and .', '1611.09321-2-57-3': 'For any hyper-parameter setting, we run the and methods [MATH] times with different random restarts.', '1611.09321-2-57-4': 'We explore the following main hyper-parameters:', '1611.09321-2-58-0': 'In all of the experiments, both and are treated exactly the same.', '1611.09321-2-58-1': 'In fact, the change of implementation is just a few lines of code.', '1611.09321-2-58-2': 'Given a value of [MATH], for each task, we run [MATH] training jobs comprising [MATH] learning rates, [MATH] clipping values, and [MATH] random restarts.', '1611.09321-2-58-3': 'We run each algorithm for a maximum number of steps determined based on the difficulty of the task.', '1611.09321-2-58-4': 'The training jobs for Copy, DuplicatedInput, RepeatCopy, Reverse, ReversedAddition, and BinarySearch are run for [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] stochastic gradient steps, respectively.', '1611.09321-2-58-5': 'We find that running a trainer job longer does not result in a better performance.', '1611.09321-2-58-6': 'Our policy network comprises a single LSTM layer with [MATH] nodes.', '1611.09321-2-58-7': 'We use the Adam optimizer for the experiments.', '1611.09321-2-59-0': 'gym-results shows the percentage of [MATH] trials on different hyper-parameters ([MATH], [MATH]) and random restarts which successfully solve each of the algorithmic tasks.', '1611.09321-2-59-1': 'It is clear that is more robust than to changes in hyper-parameters, even though we only report the results of for a single temperature.', '1611.09321-2-59-2': 'See Appendix [REF] for more detailed tables on hyper-parameter robustness.', '1611.09321-2-60-0': '## Results', '1611.09321-2-61-0': 'gym-results2 presents the number of successful attempts (out of [MATH] random restarts) and the expected reward values (averaged over [MATH] trials) for each RL algorithm given the best hyper-parameters.', '1611.09321-2-61-1': 'One-step Q-learning results are also included in the table.', '1611.09321-2-61-2': 'We also present the training curves for and in fig:rewards.', '1611.09321-2-61-3': 'It is clear that outperforms the baselines on these tasks.', '1611.09321-2-61-4': 'On the more difficult tasks, such as Reverse and ReverseAddition, is able to consistently find an appropriate algorithm, but and Q-learning fall behind.', '1611.09321-2-61-5': 'Importantly, for the BinarySearch task, which exhibits many local maxima and necessitates smart exploration, is the only method that can solve it consistently.', '1611.09321-2-61-6': 'The Q-learning baseline solves some of the simple tasks, but it makes little headway on the harder tasks.', '1611.09321-2-61-7': 'We believe that entropy regularization for policy gradient and [MATH]-greedy for Q-learning are relatively weak exploration strategies in long episodic tasks with delayed rewards.', '1611.09321-2-61-8': 'On such tasks, one random exploratory step in the wrong direction can take the agent off the optimal policy, hampering its ability to learn.', '1611.09321-2-61-9': 'In contrast, provides a form of adaptive and smart exploration.', '1611.09321-2-61-10': 'In fact, we observe that the variance of the importance weights decreases as the agent approaches the optimal policy, effectively reducing exploration when it is no longer necessary; see Appendix [REF].', '1611.09321-2-62-0': '## Generalization to longer sequences', '1611.09321-2-63-0': 'To confirm whether our method is able to find the correct algorithm for multi-digit addition, we investigate its generalization to longer input sequences than provided during training.', '1611.09321-2-63-1': 'We evaluate the trained models on inputs up to a length of [MATH] digits, even though training sequences were at most [MATH] characters.', '1611.09321-2-63-2': 'For each length, we test the model on [MATH] randomly generated inputs, stopping when the accuracy falls below [MATH].', '1611.09321-2-63-3': 'Out of the [MATH] models trained on addition with , we find that [MATH] models generalize to numbers up to [MATH] digits without any observed mistakes.', '1611.09321-2-63-4': 'On the best hyper-parameters, [MATH] out of the [MATH] random restarts are able to generalize successfully.', '1611.09321-2-63-5': 'For more detailed results on the generalization performance on [MATH] different tasks including Copy, DuplicatedInput, and ReversedAddition, see Appendix [REF].', '1611.09321-2-63-6': 'During these evaluations, we take the action with largest probability from [MATH] at each time step rather than sampling randomly.', '1611.09321-2-64-0': 'We also looked into the generalization of the models trained on the BinarySearch task.', '1611.09321-2-64-1': 'We found that none of the agents perform proper binary search.', '1611.09321-2-64-2': 'Rather, those that solved the task perform a hybrid of binary and linear search: first actions follow a binary search pattern, but then the agent switches to a linear search procedure once it narrows down the search space; see Appendix [REF] for some execution traces for BinarySearch and ReversedAddition.', '1611.09321-2-64-3': "Thus, on longer input sequences, the agent's running time complexity approaches linear rather than logarithmic.", '1611.09321-2-64-4': 'We hope that future work will make more progress on this task.', '1611.09321-2-64-5': 'This task is especially interesting because the reward signal should incorporate both correctness and efficiency of the algorithm.', '1611.09321-2-65-0': 'This task is interesting because there exist many local maxima in the policy space.', '1611.09321-2-65-1': 'Even between a full linear search and an efficient binary search there are many possible policies that solve the task.', '1611.09321-2-65-2': 'As we will see later, none of our methods are able to find the intended efficient binary search algorithm, but rather at best find one of these policies in between linear and binary search.', '1611.09321-2-65-3': 'We hope that future work in this area can make more headway on this difficult task.', '1611.09321-2-66-0': '## Implementation details', '1611.09321-2-67-0': 'In all of the experiments, we make use of curriculum learning.', '1611.09321-2-67-1': 'The environment begins by only providing small inputs and moves on to longer sequences once the agent achieves close to maximal reward over a number of steps.', '1611.09321-2-67-2': 'For policy gradient methods including and , we only provide the agent with a reward at the end of the episode, and there is no notion of intermediate reward.', '1611.09321-2-67-3': 'For the value-based baseline, we implement one-step Q-learning as described in [CITATION]-Alg. [MATH], employing double Q-learning with [MATH]-greedy exploration.', '1611.09321-2-67-4': 'We use the same RNN in our policy-based approaches to estimate the Q values.', '1611.09321-2-67-5': 'A grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) is conducted to find the best hyper-parameters.', '1611.09321-2-67-6': 'Unlike our other methods, the Q-learning baseline uses intermediate rewards, as given by the OpenAI Gym on a per-step basis.', '1611.09321-2-67-7': 'Hence, the Q-learning baseline has a slight advantage over the policy gradient methods.', '1611.09321-2-68-0': 'In all of the tasks except Copy, our stochastic optimizer uses mini-batches comprising [MATH] policy samples from the model.', '1611.09321-2-68-1': 'These [MATH] samples correspond to [MATH] different random sequences drawn from the environment, and [MATH] random policy trajectories per sequence.', '1611.09321-2-68-2': 'In other words, we set [MATH] and [MATH] as defined in [REF] and [REF].', '1611.09321-2-68-3': 'For , we use the [MATH] samples to subtract the mean of the coefficient of [MATH] which includes the contribution of the reward and entropy regularization.', '1611.09321-2-68-4': 'For , we use the [MATH] trajectories to subtract the mean reward and normalize the importance sampling weights.', '1611.09321-2-68-5': 'We do not subtract the mean of the normalized importance weights.', '1611.09321-2-68-6': 'For the Copy task, we use mini-batches with [MATH] samples using [MATH] and [MATH].', '1611.09321-2-68-7': 'Experiments are conducted using Tensorflow .', '1611.09321-2-68-8': ', which we found to perform better.', '1611.09321-2-69-0': 'For our experiments, we aimed to realize two things: first, if can give more consistent behavior across different hyper-parameters; second, if can solve tasks that cannot solve at all.', '1611.09321-2-69-1': 'To this end, we ran several runs of and on each task.', '1611.09321-2-69-2': 'For each task we trained a number of models on different values of [MATH] and different hyper-parameters.', '1611.09321-2-69-3': 'For , we used [MATH], of which we generally found at least one to work well (a broader search did not yield stronger results for other values of [MATH]).', '1611.09321-2-69-4': 'For , we used [MATH], which we found to work well on all the tasks.', '1611.09321-2-70-0': '(a broader search did not yield stronger results for other values of [MATH])', '1611.09321-2-71-0': 'For each training algorithm and value of [MATH], we ran 60 training runs, the result of an intersection of three values of learning rate ([MATH]), four values of clipping value for the gradient ([MATH]), and five random initialization seeds.', '1611.09321-2-71-1': 'With the exception of Copy, each run was trained using batches consisting of 10 trajectories each from 40 different resets of the environment.', '1611.09321-2-71-2': 'Thus, a single batch consisted of 400 total policy trajectories.', '1611.09321-2-71-3': 'For , we use the 10 trajectories to mean-center the coefficient of [MATH].', '1611.09321-2-71-4': 'For , we use the 10 trajectories to both mean-center the rewards and self-normalize the importance sampling weights.', '1611.09321-2-71-5': 'For the Copy task, we used batches of 10 trajectories each from 20 different resets of the environment, which we found to perform better.', '1611.09321-2-71-6': 'The runs for Copy, RepeatCopy, DuplicatedInput, Reverse, ReversedAddition, and BinarySearch were trained for 2000, 50000, 500, 5000, 50000, and 2000 steps, respectively.', '1611.09321-2-71-7': 'We found these numbers of steps were mostly enough for the algorithms to converge for each respective task.', '1611.09321-2-72-0': 'The model architecture we used was an LSTM recurrent neural network with hidden dimension of 128.', '1611.09321-2-72-1': 'We train using the Adam optimizer which we chose for its robustness to hyper-parameters .', '1611.09321-2-73-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-2-73-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-2-73-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-2-73-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-2-73-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-2-73-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-2-74-0': '## Experimental details', '1611.09321-2-75-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-2-75-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-2-76-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-2-76-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-2-76-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-2-76-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-2-76-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-2-76-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-2-77-0': 'We find that not only does solve the considered tasks given the best hyper-parameters, but also that the gradient is more robust to the choice of the hyper-parameters, when compared to the expected reward objective.', '1611.09321-2-78-0': '## Baseline methods', '1611.09321-2-79-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-2-79-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-2-80-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-2-80-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-2-80-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-2-80-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-2-80-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-2-80-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-2-81-0': '# Conclusion', '1611.09321-2-82-0': 'We present a variant of policy gradient, called , which promotes exploring action sequences that yield rewards larger than what the model expects.', '1611.09321-2-82-1': 'This exploration strategy is the result of importance sampling from the optimal policy.', '1611.09321-2-82-2': 'Our experimental results demonstrate that significantly outperforms other value and policy based methods, while being more robust to changes of hyper-parameters.', '1611.09321-2-82-3': 'By using , we can solve algorithmic tasks like multi-digit addition, which other methods cannot reliably solve even given the best hyper-parameters.', '1611.09321-2-82-4': 'We introduce a new algorithmic task based on binary search to advocate more research in this area, especially when the computational complexity of the solutions matters too.', '1611.09321-2-82-5': 'Solving these tasks is not only important to develop human like intelligence to enable learning algorithms, but also important for generic reinforcement learning, where smart and efficient exploration is the key to successful methods.'}

[['1611.09321-1-26-0', '1611.09321-2-25-0'], ['1611.09321-1-26-1', '1611.09321-2-25-1'], ['1611.09321-1-26-2', '1611.09321-2-25-2'], ['1611.09321-1-26-3', '1611.09321-2-25-3'], ['1611.09321-1-26-4', '1611.09321-2-25-4'], ['1611.09321-1-59-0', '1611.09321-2-59-0'], ['1611.09321-1-59-1', '1611.09321-2-59-1'], ['1611.09321-1-59-2', '1611.09321-2-59-2'], ['1611.09321-1-70-0', '1611.09321-2-70-0'], ['1611.09321-1-67-0', '1611.09321-2-67-0'], ['1611.09321-1-67-1', '1611.09321-2-67-1'], ['1611.09321-1-67-2', '1611.09321-2-67-2'], ['1611.09321-1-67-3', '1611.09321-2-67-3'], ['1611.09321-1-67-4', '1611.09321-2-67-4'], ['1611.09321-1-67-5', '1611.09321-2-67-5'], ['1611.09321-1-67-6', '1611.09321-2-67-6'], ['1611.09321-1-67-7', '1611.09321-2-67-7'], ['1611.09321-1-53-0', '1611.09321-2-53-0'], ['1611.09321-1-53-1', '1611.09321-2-53-1'], ['1611.09321-1-53-2', '1611.09321-2-53-2'], ['1611.09321-1-53-3', '1611.09321-2-53-3'], ['1611.09321-1-53-4', '1611.09321-2-53-4'], ['1611.09321-1-40-0', '1611.09321-2-40-0'], ['1611.09321-1-40-1', '1611.09321-2-40-1'], ['1611.09321-1-61-0', '1611.09321-2-61-0'], ['1611.09321-1-61-1', '1611.09321-2-61-1'], ['1611.09321-1-61-2', '1611.09321-2-61-3'], ['1611.09321-1-61-3', '1611.09321-2-61-4'], ['1611.09321-1-61-4', '1611.09321-2-61-5'], ['1611.09321-1-61-5', '1611.09321-2-61-6'], ['1611.09321-1-61-6', '1611.09321-2-61-7'], ['1611.09321-1-61-7', '1611.09321-2-61-8'], ['1611.09321-1-61-8', '1611.09321-2-61-9'], ['1611.09321-1-61-9', '1611.09321-2-61-10'], ['1611.09321-1-80-0', '1611.09321-2-80-0'], ['1611.09321-1-80-1', '1611.09321-2-80-1'], ['1611.09321-1-80-2', '1611.09321-2-80-2'], ['1611.09321-1-80-3', '1611.09321-2-80-3'], ['1611.09321-1-80-4', '1611.09321-2-80-4'], ['1611.09321-1-80-5', '1611.09321-2-80-5'], ['1611.09321-1-50-0', '1611.09321-2-50-0'], ['1611.09321-1-50-1', '1611.09321-2-50-1'], ['1611.09321-1-50-2', '1611.09321-2-50-2'], ['1611.09321-1-50-3', '1611.09321-2-50-3'], ['1611.09321-1-50-4', '1611.09321-2-50-6'], ['1611.09321-1-8-0', '1611.09321-2-7-0'], ['1611.09321-1-8-4', '1611.09321-2-7-5'], ['1611.09321-1-8-6', '1611.09321-2-7-8'], ['1611.09321-1-43-0', '1611.09321-2-43-0'], ['1611.09321-1-43-1', '1611.09321-2-43-1'], ['1611.09321-1-43-2', '1611.09321-2-43-2'], ['1611.09321-1-43-3', '1611.09321-2-43-3'], ['1611.09321-1-57-0', '1611.09321-2-57-0'], ['1611.09321-1-57-1', '1611.09321-2-57-1'], ['1611.09321-1-57-2', '1611.09321-2-57-2'], ['1611.09321-1-57-3', '1611.09321-2-57-3'], ['1611.09321-1-32-0', '1611.09321-2-31-0'], ['1611.09321-1-32-1', '1611.09321-2-31-1'], ['1611.09321-1-32-2', '1611.09321-2-31-2'], ['1611.09321-1-32-3', '1611.09321-2-31-3'], ['1611.09321-1-45-0', '1611.09321-2-45-0'], ['1611.09321-1-45-1', '1611.09321-2-45-1'], ['1611.09321-1-45-2', '1611.09321-2-45-2'], ['1611.09321-1-45-3', '1611.09321-2-45-3'], ['1611.09321-1-45-4', '1611.09321-2-45-4'], ['1611.09321-1-63-0', '1611.09321-2-63-0'], ['1611.09321-1-63-1', '1611.09321-2-63-1'], ['1611.09321-1-63-2', '1611.09321-2-63-2'], ['1611.09321-1-63-3', '1611.09321-2-63-3'], ['1611.09321-1-63-4', '1611.09321-2-63-4'], ['1611.09321-1-63-5', '1611.09321-2-63-5'], ['1611.09321-1-63-6', '1611.09321-2-63-6'], ['1611.09321-1-31-0', '1611.09321-2-30-0'], ['1611.09321-1-31-1', '1611.09321-2-30-1'], ['1611.09321-1-31-2', '1611.09321-2-30-2'], ['1611.09321-1-69-0', '1611.09321-2-69-0'], ['1611.09321-1-69-1', '1611.09321-2-69-1'], ['1611.09321-1-69-2', '1611.09321-2-69-2'], ['1611.09321-1-69-3', '1611.09321-2-69-3'], ['1611.09321-1-69-4', '1611.09321-2-69-4'], ['1611.09321-1-20-0', '1611.09321-2-19-0'], ['1611.09321-1-20-1', '1611.09321-2-19-1'], ['1611.09321-1-20-2', '1611.09321-2-19-2'], ['1611.09321-1-20-3', '1611.09321-2-19-3'], ['1611.09321-1-20-4', '1611.09321-2-19-4'], ['1611.09321-1-20-5', '1611.09321-2-19-5'], ['1611.09321-1-20-6', '1611.09321-2-19-6'], ['1611.09321-1-20-7', '1611.09321-2-19-7'], ['1611.09321-1-20-8', '1611.09321-2-19-8'], ['1611.09321-1-20-9', '1611.09321-2-19-9'], ['1611.09321-1-20-10', '1611.09321-2-19-10'], ['1611.09321-1-72-0', '1611.09321-2-72-0'], ['1611.09321-1-72-1', '1611.09321-2-72-1'], ['1611.09321-1-29-0', '1611.09321-2-28-0'], ['1611.09321-1-29-1', '1611.09321-2-28-1'], ['1611.09321-1-29-2', '1611.09321-2-28-2'], ['1611.09321-1-29-3', '1611.09321-2-28-3'], ['1611.09321-1-29-4', '1611.09321-2-28-4'], ['1611.09321-1-29-5', '1611.09321-2-28-5'], ['1611.09321-1-13-0', '1611.09321-2-12-0'], ['1611.09321-1-13-1', '1611.09321-2-12-1'], ['1611.09321-1-82-0', '1611.09321-2-82-0'], ['1611.09321-1-82-1', '1611.09321-2-82-1'], ['1611.09321-1-82-2', '1611.09321-2-82-2'], ['1611.09321-1-82-3', '1611.09321-2-82-3'], ['1611.09321-1-82-4', '1611.09321-2-82-4'], ['1611.09321-1-82-5', '1611.09321-2-82-5'], ['1611.09321-1-1-0', '1611.09321-2-1-0'], ['1611.09321-1-1-1', '1611.09321-2-1-1'], ['1611.09321-1-1-2', '1611.09321-2-1-2'], ['1611.09321-1-1-3', '1611.09321-2-1-3'], ['1611.09321-1-1-4', '1611.09321-2-1-4'], ['1611.09321-1-1-5', '1611.09321-2-1-5'], ['1611.09321-1-1-6', '1611.09321-2-1-6'], ['1611.09321-1-1-7', '1611.09321-2-1-7'], ['1611.09321-1-76-0', '1611.09321-2-76-0'], ['1611.09321-1-76-1', '1611.09321-2-76-1'], ['1611.09321-1-76-2', '1611.09321-2-76-2'], ['1611.09321-1-76-3', '1611.09321-2-76-3'], ['1611.09321-1-76-4', '1611.09321-2-76-4'], ['1611.09321-1-76-5', '1611.09321-2-76-5'], ['1611.09321-1-75-0', '1611.09321-2-75-0'], ['1611.09321-1-75-1', '1611.09321-2-75-1'], ['1611.09321-1-39-0', '1611.09321-2-38-0'], ['1611.09321-1-39-1', '1611.09321-2-38-1'], ['1611.09321-1-39-2', '1611.09321-2-38-2'], ['1611.09321-1-39-3', '1611.09321-2-38-3'], ['1611.09321-1-6-0', '1611.09321-2-5-0'], ['1611.09321-1-6-1', '1611.09321-2-5-1'], ['1611.09321-1-6-2', '1611.09321-2-5-2'], ['1611.09321-1-77-0', '1611.09321-2-77-0'], ['1611.09321-1-79-0', '1611.09321-2-79-0'], ['1611.09321-1-79-1', '1611.09321-2-79-1'], ['1611.09321-1-9-0', '1611.09321-2-8-0'], ['1611.09321-1-9-1', '1611.09321-2-8-1'], ['1611.09321-1-9-2', '1611.09321-2-8-2'], ['1611.09321-1-9-3', '1611.09321-2-8-3'], ['1611.09321-1-21-0', '1611.09321-2-20-0'], ['1611.09321-1-21-1', '1611.09321-2-20-1'], ['1611.09321-1-21-2', '1611.09321-2-20-2'], ['1611.09321-1-11-0', '1611.09321-2-10-0'], ['1611.09321-1-11-1', '1611.09321-2-10-1'], ['1611.09321-1-11-2', '1611.09321-2-10-2'], ['1611.09321-1-11-3', '1611.09321-2-10-3'], ['1611.09321-1-11-4', '1611.09321-2-10-4'], ['1611.09321-1-71-0', '1611.09321-2-71-0'], ['1611.09321-1-71-1', '1611.09321-2-71-1'], ['1611.09321-1-71-2', '1611.09321-2-71-2'], ['1611.09321-1-71-3', '1611.09321-2-71-3'], ['1611.09321-1-71-4', '1611.09321-2-71-4'], ['1611.09321-1-71-5', '1611.09321-2-71-5'], ['1611.09321-1-71-6', '1611.09321-2-71-6'], ['1611.09321-1-71-7', '1611.09321-2-71-7'], ['1611.09321-1-38-0', '1611.09321-2-37-0'], ['1611.09321-1-38-1', '1611.09321-2-37-1'], ['1611.09321-1-38-2', '1611.09321-2-37-2'], ['1611.09321-1-48-0', '1611.09321-2-48-0'], ['1611.09321-1-48-1', '1611.09321-2-48-1'], ['1611.09321-1-48-2', '1611.09321-2-48-2'], ['1611.09321-1-48-3', '1611.09321-2-48-3'], ['1611.09321-1-48-4', '1611.09321-2-48-5'], ['1611.09321-1-48-5', '1611.09321-2-48-6'], ['1611.09321-1-48-6', '1611.09321-2-48-7'], ['1611.09321-1-48-7', '1611.09321-2-48-8'], ['1611.09321-1-48-8', '1611.09321-2-48-9'], ['1611.09321-1-48-9', '1611.09321-2-48-10'], ['1611.09321-1-48-10', '1611.09321-2-48-11'], ['1611.09321-1-48-11', '1611.09321-2-48-12'], ['1611.09321-1-48-12', '1611.09321-2-48-13'], ['1611.09321-1-48-13', '1611.09321-2-48-14'], ['1611.09321-1-48-14', '1611.09321-2-48-15'], ['1611.09321-1-48-15', '1611.09321-2-48-16'], ['1611.09321-1-49-0', '1611.09321-2-49-0'], ['1611.09321-1-49-1', '1611.09321-2-49-1'], ['1611.09321-1-58-0', '1611.09321-2-58-0'], ['1611.09321-1-58-1', '1611.09321-2-58-1'], ['1611.09321-1-58-2', '1611.09321-2-58-2'], ['1611.09321-1-58-3', '1611.09321-2-58-3'], ['1611.09321-1-58-4', '1611.09321-2-58-4'], ['1611.09321-1-58-5', '1611.09321-2-58-5'], ['1611.09321-1-58-6', '1611.09321-2-58-6'], ['1611.09321-1-58-7', '1611.09321-2-58-7'], ['1611.09321-1-55-0', '1611.09321-2-55-0'], ['1611.09321-1-55-1', '1611.09321-2-55-1'], ['1611.09321-1-55-2', '1611.09321-2-55-2'], ['1611.09321-1-23-0', '1611.09321-2-22-0'], ['1611.09321-1-23-1', '1611.09321-2-22-1'], ['1611.09321-1-23-2', '1611.09321-2-22-2'], ['1611.09321-1-23-3', '1611.09321-2-22-3'], ['1611.09321-1-25-0', '1611.09321-2-24-0'], ['1611.09321-1-25-1', '1611.09321-2-24-1'], ['1611.09321-1-25-2', '1611.09321-2-24-2'], ['1611.09321-1-25-3', '1611.09321-2-24-3'], ['1611.09321-1-25-4', '1611.09321-2-24-4'], ['1611.09321-1-16-0', '1611.09321-2-15-0'], ['1611.09321-1-16-1', '1611.09321-2-15-1'], ['1611.09321-1-16-2', '1611.09321-2-15-2'], ['1611.09321-1-16-3', '1611.09321-2-15-3'], ['1611.09321-1-16-4', '1611.09321-2-15-4'], ['1611.09321-1-16-5', '1611.09321-2-15-5'], ['1611.09321-1-16-6', '1611.09321-2-15-6'], ['1611.09321-1-16-7', '1611.09321-2-15-7'], ['1611.09321-1-16-8', '1611.09321-2-15-8'], ['1611.09321-1-16-9', '1611.09321-2-15-9'], ['1611.09321-1-24-0', '1611.09321-2-23-0'], ['1611.09321-1-24-1', '1611.09321-2-23-1'], ['1611.09321-1-24-2', '1611.09321-2-23-2'], ['1611.09321-1-24-3', '1611.09321-2-23-3'], ['1611.09321-1-19-1', '1611.09321-2-18-1'], ['1611.09321-1-19-2', '1611.09321-2-18-3'], ['1611.09321-1-19-3', '1611.09321-2-18-4'], ['1611.09321-1-19-4', '1611.09321-2-18-5'], ['1611.09321-1-73-0', '1611.09321-2-73-0'], ['1611.09321-1-73-1', '1611.09321-2-73-1'], ['1611.09321-1-73-2', '1611.09321-2-73-2'], ['1611.09321-1-73-3', '1611.09321-2-73-3'], ['1611.09321-1-73-4', '1611.09321-2-73-4'], ['1611.09321-1-73-5', '1611.09321-2-73-5'], ['1611.09321-1-68-0', '1611.09321-2-68-0'], ['1611.09321-1-68-1', '1611.09321-2-68-1'], ['1611.09321-1-68-2', '1611.09321-2-68-2'], ['1611.09321-1-68-3', '1611.09321-2-68-3'], ['1611.09321-1-68-4', '1611.09321-2-68-4'], ['1611.09321-1-68-5', '1611.09321-2-68-5'], ['1611.09321-1-68-6', '1611.09321-2-68-6'], ['1611.09321-1-68-7', '1611.09321-2-68-7'], ['1611.09321-1-15-0', '1611.09321-2-14-0'], ['1611.09321-1-36-0', '1611.09321-2-35-0'], ['1611.09321-1-36-1', '1611.09321-2-35-1'], ['1611.09321-1-27-1', '1611.09321-2-26-1'], ['1611.09321-1-27-4', '1611.09321-2-26-4'], ['1611.09321-1-27-5', '1611.09321-2-26-5'], ['1611.09321-1-27-6', '1611.09321-2-26-6'], ['1611.09321-1-27-7', '1611.09321-2-26-7'], ['1611.09321-1-27-8', '1611.09321-2-26-8'], ['1611.09321-1-0-0', '1611.09321-2-0-0'], ['1611.09321-1-0-1', '1611.09321-2-0-1'], ['1611.09321-1-0-2', '1611.09321-2-0-2'], ['1611.09321-1-0-3', '1611.09321-2-0-3'], ['1611.09321-1-0-5', '1611.09321-2-0-5'], ['1611.09321-1-0-6', '1611.09321-2-0-6'], ['1611.09321-1-51-0', '1611.09321-2-51-0'], ['1611.09321-1-64-0', '1611.09321-2-64-0'], ['1611.09321-1-64-1', '1611.09321-2-64-1'], ['1611.09321-1-64-2', '1611.09321-2-64-2'], ['1611.09321-1-64-3', '1611.09321-2-64-3'], ['1611.09321-1-64-4', '1611.09321-2-64-4'], ['1611.09321-1-64-5', '1611.09321-2-64-5'], ['1611.09321-1-10-0', '1611.09321-2-9-0'], ['1611.09321-1-34-0', '1611.09321-2-33-0'], ['1611.09321-1-34-1', '1611.09321-2-33-1'], ['1611.09321-1-34-2', '1611.09321-2-33-2'], ['1611.09321-1-41-1', '1611.09321-2-41-1'], ['1611.09321-1-42-0', '1611.09321-2-42-0'], ['1611.09321-1-42-1', '1611.09321-2-42-1'], ['1611.09321-1-42-2', '1611.09321-2-42-2'], ['1611.09321-1-42-3', '1611.09321-2-42-3'], ['1611.09321-1-42-4', '1611.09321-2-42-4'], ['1611.09321-1-18-0', '1611.09321-2-17-0'], ['1611.09321-1-18-1', '1611.09321-2-17-1'], ['1611.09321-1-2-0', '1611.09321-2-2-0'], ['1611.09321-1-2-1', '1611.09321-2-2-1'], ['1611.09321-1-2-2', '1611.09321-2-2-2'], ['1611.09321-1-2-3', '1611.09321-2-2-3'], ['1611.09321-1-14-0', '1611.09321-2-13-0'], ['1611.09321-1-14-1', '1611.09321-2-13-1'], ['1611.09321-1-14-3', '1611.09321-2-13-3'], ['1611.09321-1-14-4', '1611.09321-2-13-4'], ['1611.09321-1-14-5', '1611.09321-2-13-5'], ['1611.09321-1-44-0', '1611.09321-2-44-0'], ['1611.09321-1-44-1', '1611.09321-2-44-1'], ['1611.09321-1-44-2', '1611.09321-2-44-2'], ['1611.09321-1-44-3', '1611.09321-2-44-3'], ['1611.09321-1-44-4', '1611.09321-2-44-4'], ['1611.09321-1-65-0', '1611.09321-2-65-0'], ['1611.09321-1-65-1', '1611.09321-2-65-1'], ['1611.09321-1-65-2', '1611.09321-2-65-2'], ['1611.09321-1-65-3', '1611.09321-2-65-3'], ['1611.09321-1-7-1', '1611.09321-2-6-1'], ['1611.09321-1-7-2', '1611.09321-2-6-2'], ['1611.09321-1-7-4', '1611.09321-2-6-4'], ['1611.09321-1-7-5', '1611.09321-2-6-5'], ['1611.09321-1-52-0', '1611.09321-2-52-0'], ['1611.09321-1-52-1', '1611.09321-2-52-1'], ['1611.09321-1-52-2', '1611.09321-2-52-2'], ['1611.09321-1-52-3', '1611.09321-2-52-3'], ['1611.09321-1-52-4', '1611.09321-2-52-4'], ['1611.09321-1-52-5', '1611.09321-2-52-5'], ['1611.09321-2-72-0', '1611.09321-3-79-0'], ['1611.09321-2-72-1', '1611.09321-3-79-1'], ['1611.09321-2-44-0', '1611.09321-3-51-0'], ['1611.09321-2-44-1', '1611.09321-3-51-1'], ['1611.09321-2-44-2', '1611.09321-3-51-2'], ['1611.09321-2-44-3', '1611.09321-3-51-3'], ['1611.09321-2-44-4', '1611.09321-3-51-4'], ['1611.09321-2-64-0', '1611.09321-3-71-0'], ['1611.09321-2-64-1', '1611.09321-3-71-1'], ['1611.09321-2-64-2', '1611.09321-3-71-2'], ['1611.09321-2-64-3', '1611.09321-3-71-3'], ['1611.09321-2-64-4', '1611.09321-3-71-4'], ['1611.09321-2-64-5', '1611.09321-3-71-5'], ['1611.09321-2-49-0', '1611.09321-3-56-0'], ['1611.09321-2-49-1', '1611.09321-3-56-1'], ['1611.09321-2-55-0', '1611.09321-3-62-0'], ['1611.09321-2-55-1', '1611.09321-3-62-1'], ['1611.09321-2-55-2', '1611.09321-3-62-2'], ['1611.09321-2-65-0', '1611.09321-3-72-0'], ['1611.09321-2-65-1', '1611.09321-3-72-1'], ['1611.09321-2-65-2', '1611.09321-3-72-2'], ['1611.09321-2-65-3', '1611.09321-3-72-3'], ['1611.09321-2-48-0', '1611.09321-3-55-0'], ['1611.09321-2-48-2', '1611.09321-3-55-2'], ['1611.09321-2-48-3', '1611.09321-3-55-3'], ['1611.09321-2-48-4', '1611.09321-3-55-4'], ['1611.09321-2-48-5', '1611.09321-3-55-5'], ['1611.09321-2-48-6', '1611.09321-3-55-6'], ['1611.09321-2-48-7', '1611.09321-3-55-7'], ['1611.09321-2-48-8', '1611.09321-3-55-8'], ['1611.09321-2-48-9', '1611.09321-3-55-9'], ['1611.09321-2-48-10', '1611.09321-3-55-10'], ['1611.09321-2-48-11', '1611.09321-3-55-11'], ['1611.09321-2-48-12', '1611.09321-3-55-12'], ['1611.09321-2-48-13', '1611.09321-3-55-13'], ['1611.09321-2-48-14', '1611.09321-3-55-14'], ['1611.09321-2-48-15', '1611.09321-3-55-15'], ['1611.09321-2-48-16', '1611.09321-3-55-16'], ['1611.09321-2-71-0', '1611.09321-3-78-0'], ['1611.09321-2-71-1', '1611.09321-3-78-1'], ['1611.09321-2-71-2', '1611.09321-3-78-2'], ['1611.09321-2-71-3', '1611.09321-3-78-3'], ['1611.09321-2-71-4', '1611.09321-3-78-4'], ['1611.09321-2-71-5', '1611.09321-3-78-5'], ['1611.09321-2-71-6', '1611.09321-3-78-6'], ['1611.09321-2-71-7', '1611.09321-3-78-7'], ['1611.09321-2-75-0', '1611.09321-3-82-0'], ['1611.09321-2-75-1', '1611.09321-3-82-1'], ['1611.09321-2-9-0', '1611.09321-3-9-0'], ['1611.09321-2-0-0', '1611.09321-3-0-0'], ['1611.09321-2-0-1', '1611.09321-3-0-1'], ['1611.09321-2-0-2', '1611.09321-3-0-2'], ['1611.09321-2-0-3', '1611.09321-3-0-3'], ['1611.09321-2-0-4', '1611.09321-3-0-4'], ['1611.09321-2-0-5', '1611.09321-3-0-5'], ['1611.09321-2-0-6', '1611.09321-3-0-6'], ['1611.09321-2-0-7', '1611.09321-3-0-7'], ['1611.09321-2-10-0', '1611.09321-3-10-0'], ['1611.09321-2-10-1', '1611.09321-3-10-1'], ['1611.09321-2-10-2', '1611.09321-3-10-2'], ['1611.09321-2-10-3', '1611.09321-3-10-3'], ['1611.09321-2-10-4', '1611.09321-3-10-4'], ['1611.09321-2-70-0', '1611.09321-3-77-0'], ['1611.09321-2-76-0', '1611.09321-3-83-0'], ['1611.09321-2-76-1', '1611.09321-3-83-1'], ['1611.09321-2-76-2', '1611.09321-3-83-2'], ['1611.09321-2-76-3', '1611.09321-3-83-3'], ['1611.09321-2-76-4', '1611.09321-3-83-4'], ['1611.09321-2-76-5', '1611.09321-3-83-5'], ['1611.09321-2-37-0', '1611.09321-3-43-0'], ['1611.09321-2-37-1', '1611.09321-3-43-1'], ['1611.09321-2-37-2', '1611.09321-3-43-2'], ['1611.09321-2-51-0', '1611.09321-3-58-0'], ['1611.09321-2-20-0', '1611.09321-3-20-0'], ['1611.09321-2-20-1', '1611.09321-3-20-1'], ['1611.09321-2-20-2', '1611.09321-3-20-2'], ['1611.09321-2-12-1', '1611.09321-3-12-1'], ['1611.09321-2-26-0', '1611.09321-3-28-0'], ['1611.09321-2-26-1', '1611.09321-3-28-1'], ['1611.09321-2-26-2', '1611.09321-3-28-2'], ['1611.09321-2-26-3', '1611.09321-3-28-3'], ['1611.09321-2-26-4', '1611.09321-3-28-4'], ['1611.09321-2-26-5', '1611.09321-3-28-5'], ['1611.09321-2-26-6', '1611.09321-3-28-6'], ['1611.09321-2-26-7', '1611.09321-3-28-7'], ['1611.09321-2-26-8', '1611.09321-3-28-8'], ['1611.09321-2-38-0', '1611.09321-3-44-0'], ['1611.09321-2-38-1', '1611.09321-3-44-1'], ['1611.09321-2-38-2', '1611.09321-3-44-2'], ['1611.09321-2-38-3', '1611.09321-3-44-3'], ['1611.09321-2-36-0', '1611.09321-3-42-0'], ['1611.09321-2-36-1', '1611.09321-3-42-1'], ['1611.09321-2-7-0', '1611.09321-3-7-0'], ['1611.09321-2-7-1', '1611.09321-3-7-1'], ['1611.09321-2-7-2', '1611.09321-3-7-2'], ['1611.09321-2-7-3', '1611.09321-3-7-3'], ['1611.09321-2-7-4', '1611.09321-3-7-4'], ['1611.09321-2-7-5', '1611.09321-3-7-5'], ['1611.09321-2-7-6', '1611.09321-3-7-6'], ['1611.09321-2-7-7', '1611.09321-3-7-7'], ['1611.09321-2-7-8', '1611.09321-3-7-8'], ['1611.09321-2-39-0', '1611.09321-3-45-0'], ['1611.09321-2-39-1', '1611.09321-3-45-1'], ['1611.09321-2-67-0', '1611.09321-3-74-0'], ['1611.09321-2-67-1', '1611.09321-3-74-1'], ['1611.09321-2-67-2', '1611.09321-3-74-2'], ['1611.09321-2-67-3', '1611.09321-3-74-3'], ['1611.09321-2-67-4', '1611.09321-3-74-4'], ['1611.09321-2-67-5', '1611.09321-3-74-5'], ['1611.09321-2-67-6', '1611.09321-3-74-6'], ['1611.09321-2-67-7', '1611.09321-3-74-7'], ['1611.09321-2-41-1', '1611.09321-3-48-1'], ['1611.09321-2-17-0', '1611.09321-3-17-0'], ['1611.09321-2-17-1', '1611.09321-3-17-1'], ['1611.09321-2-40-0', '1611.09321-3-47-0'], ['1611.09321-2-40-1', '1611.09321-3-47-1'], ['1611.09321-2-69-0', '1611.09321-3-76-0'], ['1611.09321-2-69-1', '1611.09321-3-76-1'], ['1611.09321-2-69-2', '1611.09321-3-76-2'], ['1611.09321-2-69-3', '1611.09321-3-76-3'], ['1611.09321-2-69-4', '1611.09321-3-76-4'], ['1611.09321-2-59-0', '1611.09321-3-66-0'], ['1611.09321-2-59-1', '1611.09321-3-66-1'], ['1611.09321-2-59-2', '1611.09321-3-66-2'], ['1611.09321-2-63-0', '1611.09321-3-70-0'], ['1611.09321-2-63-1', '1611.09321-3-70-1'], ['1611.09321-2-63-2', '1611.09321-3-70-2'], ['1611.09321-2-63-3', '1611.09321-3-70-3'], ['1611.09321-2-63-4', '1611.09321-3-70-4'], ['1611.09321-2-63-5', '1611.09321-3-70-5'], ['1611.09321-2-63-6', '1611.09321-3-70-6'], ['1611.09321-2-77-0', '1611.09321-3-84-0'], ['1611.09321-2-19-0', '1611.09321-3-19-0'], ['1611.09321-2-19-1', '1611.09321-3-19-1'], ['1611.09321-2-19-2', '1611.09321-3-19-2'], ['1611.09321-2-19-3', '1611.09321-3-19-3'], ['1611.09321-2-19-4', '1611.09321-3-19-4'], ['1611.09321-2-19-5', '1611.09321-3-19-5'], ['1611.09321-2-19-6', '1611.09321-3-19-6'], ['1611.09321-2-19-7', '1611.09321-3-19-7'], ['1611.09321-2-19-8', '1611.09321-3-19-8'], ['1611.09321-2-19-9', '1611.09321-3-19-9'], ['1611.09321-2-19-10', '1611.09321-3-19-10'], ['1611.09321-2-33-0', '1611.09321-3-35-0'], ['1611.09321-2-33-1', '1611.09321-3-35-1'], ['1611.09321-2-33-2', '1611.09321-3-35-2'], ['1611.09321-2-13-0', '1611.09321-3-13-0'], ['1611.09321-2-13-1', '1611.09321-3-13-1'], ['1611.09321-2-13-2', '1611.09321-3-13-2'], ['1611.09321-2-13-3', '1611.09321-3-13-3'], ['1611.09321-2-13-4', '1611.09321-3-13-4'], ['1611.09321-2-13-5', '1611.09321-3-13-5'], ['1611.09321-2-80-0', '1611.09321-3-87-0'], ['1611.09321-2-80-1', '1611.09321-3-87-1'], ['1611.09321-2-80-2', '1611.09321-3-87-2'], ['1611.09321-2-80-3', '1611.09321-3-87-3'], ['1611.09321-2-80-4', '1611.09321-3-87-4'], ['1611.09321-2-80-5', '1611.09321-3-87-5'], ['1611.09321-2-52-0', '1611.09321-3-59-0'], ['1611.09321-2-52-1', '1611.09321-3-59-1'], ['1611.09321-2-52-2', '1611.09321-3-59-2'], ['1611.09321-2-52-3', '1611.09321-3-59-3'], ['1611.09321-2-52-4', '1611.09321-3-59-4'], ['1611.09321-2-52-5', '1611.09321-3-59-5'], ['1611.09321-2-79-0', '1611.09321-3-86-0'], ['1611.09321-2-79-1', '1611.09321-3-86-1'], ['1611.09321-2-58-0', '1611.09321-3-65-0'], ['1611.09321-2-58-1', '1611.09321-3-65-1'], ['1611.09321-2-58-2', '1611.09321-3-65-2'], ['1611.09321-2-58-3', '1611.09321-3-65-3'], ['1611.09321-2-58-4', '1611.09321-3-65-4'], ['1611.09321-2-58-5', '1611.09321-3-65-5'], ['1611.09321-2-58-6', '1611.09321-3-65-6'], ['1611.09321-2-58-7', '1611.09321-3-65-7'], ['1611.09321-2-8-0', '1611.09321-3-8-0'], ['1611.09321-2-8-1', '1611.09321-3-8-1'], ['1611.09321-2-8-2', '1611.09321-3-8-2'], ['1611.09321-2-8-3', '1611.09321-3-8-3'], ['1611.09321-2-61-0', '1611.09321-3-68-0'], ['1611.09321-2-61-1', '1611.09321-3-68-1'], ['1611.09321-2-61-2', '1611.09321-3-68-2'], ['1611.09321-2-61-3', '1611.09321-3-68-3'], ['1611.09321-2-61-4', '1611.09321-3-68-4'], ['1611.09321-2-61-5', '1611.09321-3-68-5'], ['1611.09321-2-61-6', '1611.09321-3-68-6'], ['1611.09321-2-61-7', '1611.09321-3-68-7'], ['1611.09321-2-61-8', '1611.09321-3-68-8'], ['1611.09321-2-61-9', '1611.09321-3-68-9'], ['1611.09321-2-61-10', '1611.09321-3-68-10'], ['1611.09321-2-68-0', '1611.09321-3-75-0'], ['1611.09321-2-68-1', '1611.09321-3-75-1'], ['1611.09321-2-68-2', '1611.09321-3-75-2'], ['1611.09321-2-68-3', '1611.09321-3-75-3'], ['1611.09321-2-68-4', '1611.09321-3-75-4'], ['1611.09321-2-68-5', '1611.09321-3-75-5'], ['1611.09321-2-68-6', '1611.09321-3-75-6'], ['1611.09321-2-68-7', '1611.09321-3-75-7'], ['1611.09321-2-43-0', '1611.09321-3-50-0'], ['1611.09321-2-43-1', '1611.09321-3-50-1'], ['1611.09321-2-43-2', '1611.09321-3-50-2'], ['1611.09321-2-43-3', '1611.09321-3-50-3'], ['1611.09321-2-82-1', '1611.09321-3-89-1'], ['1611.09321-2-82-2', '1611.09321-3-89-2'], ['1611.09321-2-23-0', '1611.09321-3-23-0'], ['1611.09321-2-23-1', '1611.09321-3-23-1'], ['1611.09321-2-23-2', '1611.09321-3-23-2'], ['1611.09321-2-23-3', '1611.09321-3-23-3'], ['1611.09321-2-30-0', '1611.09321-3-32-0'], ['1611.09321-2-30-1', '1611.09321-3-32-1'], ['1611.09321-2-30-2', '1611.09321-3-32-2'], ['1611.09321-2-31-0', '1611.09321-3-33-0'], ['1611.09321-2-31-1', '1611.09321-3-33-1'], ['1611.09321-2-31-2', '1611.09321-3-33-2'], ['1611.09321-2-31-3', '1611.09321-3-33-3'], ['1611.09321-2-73-0', '1611.09321-3-80-0'], ['1611.09321-2-73-1', '1611.09321-3-80-1'], ['1611.09321-2-73-2', '1611.09321-3-80-2'], ['1611.09321-2-73-3', '1611.09321-3-80-3'], ['1611.09321-2-73-4', '1611.09321-3-80-4'], ['1611.09321-2-73-5', '1611.09321-3-80-5'], ['1611.09321-2-28-0', '1611.09321-3-30-0'], ['1611.09321-2-28-1', '1611.09321-3-30-1'], ['1611.09321-2-28-2', '1611.09321-3-30-2'], ['1611.09321-2-28-3', '1611.09321-3-30-3'], ['1611.09321-2-28-4', '1611.09321-3-30-4'], ['1611.09321-2-28-5', '1611.09321-3-30-5'], ['1611.09321-2-1-0', '1611.09321-3-1-0'], ['1611.09321-2-1-1', '1611.09321-3-1-1'], ['1611.09321-2-1-2', '1611.09321-3-1-2'], ['1611.09321-2-1-3', '1611.09321-3-1-3'], ['1611.09321-2-1-4', '1611.09321-3-1-4'], ['1611.09321-2-1-5', '1611.09321-3-1-5'], ['1611.09321-2-1-6', '1611.09321-3-1-6'], ['1611.09321-2-1-7', '1611.09321-3-1-7'], ['1611.09321-2-5-0', '1611.09321-3-5-0'], ['1611.09321-2-5-1', '1611.09321-3-5-1'], ['1611.09321-2-5-2', '1611.09321-3-5-2'], ['1611.09321-2-57-0', '1611.09321-3-64-0'], ['1611.09321-2-57-1', '1611.09321-3-64-1'], ['1611.09321-2-57-2', '1611.09321-3-64-2'], ['1611.09321-2-57-3', '1611.09321-3-64-3'], ['1611.09321-2-35-0', '1611.09321-3-41-2'], ['1611.09321-2-35-1', '1611.09321-3-41-3'], ['1611.09321-2-53-0', '1611.09321-3-60-0'], ['1611.09321-2-53-1', '1611.09321-3-60-1'], ['1611.09321-2-53-2', '1611.09321-3-60-2'], ['1611.09321-2-53-3', '1611.09321-3-60-3'], ['1611.09321-2-53-4', '1611.09321-3-60-4'], ['1611.09321-2-42-0', '1611.09321-3-49-0'], ['1611.09321-2-42-1', '1611.09321-3-49-1'], ['1611.09321-2-42-2', '1611.09321-3-49-2'], ['1611.09321-2-42-3', '1611.09321-3-49-3'], ['1611.09321-2-42-4', '1611.09321-3-49-4'], ['1611.09321-2-22-0', '1611.09321-3-22-0'], ['1611.09321-2-22-1', '1611.09321-3-22-1'], ['1611.09321-2-22-2', '1611.09321-3-22-2'], ['1611.09321-2-22-3', '1611.09321-3-22-3'], ['1611.09321-2-15-0', '1611.09321-3-15-0'], ['1611.09321-2-15-1', '1611.09321-3-15-1'], ['1611.09321-2-15-2', '1611.09321-3-15-2'], ['1611.09321-2-15-3', '1611.09321-3-15-3'], ['1611.09321-2-15-4', '1611.09321-3-15-4'], ['1611.09321-2-15-5', '1611.09321-3-15-5'], ['1611.09321-2-15-6', '1611.09321-3-15-6'], ['1611.09321-2-15-7', '1611.09321-3-15-7'], ['1611.09321-2-15-8', '1611.09321-3-15-8'], ['1611.09321-2-15-9', '1611.09321-3-15-9'], ['1611.09321-2-18-0', '1611.09321-3-18-0'], ['1611.09321-2-18-1', '1611.09321-3-18-1'], ['1611.09321-2-18-2', '1611.09321-3-18-2'], ['1611.09321-2-18-3', '1611.09321-3-18-3'], ['1611.09321-2-18-4', '1611.09321-3-18-4'], ['1611.09321-2-18-5', '1611.09321-3-18-5'], ['1611.09321-2-45-0', '1611.09321-3-52-0'], ['1611.09321-2-45-1', '1611.09321-3-52-1'], ['1611.09321-2-45-2', '1611.09321-3-52-2'], ['1611.09321-2-45-3', '1611.09321-3-52-3'], ['1611.09321-2-45-4', '1611.09321-3-52-4'], ['1611.09321-2-2-0', '1611.09321-3-2-0'], ['1611.09321-2-2-1', '1611.09321-3-2-1'], ['1611.09321-2-2-2', '1611.09321-3-2-2'], ['1611.09321-2-2-3', '1611.09321-3-2-3'], ['1611.09321-2-24-0', '1611.09321-3-24-0'], ['1611.09321-2-24-1', '1611.09321-3-24-1'], ['1611.09321-2-24-2', '1611.09321-3-24-2'], ['1611.09321-2-24-3', '1611.09321-3-24-3'], ['1611.09321-2-24-4', '1611.09321-3-24-4'], ['1611.09321-2-50-1', '1611.09321-3-57-1'], ['1611.09321-2-50-2', '1611.09321-3-57-2'], ['1611.09321-2-50-3', '1611.09321-3-57-3'], ['1611.09321-2-50-6', '1611.09321-3-57-6'], ['1611.09321-2-4-0', '1611.09321-3-4-0'], ['1611.09321-2-4-1', '1611.09321-3-4-1'], ['1611.09321-2-4-2', '1611.09321-3-4-2'], ['1611.09321-2-4-3', '1611.09321-3-4-3'], ['1611.09321-2-4-4', '1611.09321-3-4-4'], ['1611.09321-2-4-5', '1611.09321-3-4-5'], ['1611.09321-2-14-0', '1611.09321-3-14-0'], ['1611.09321-2-6-1', '1611.09321-3-6-1'], ['1611.09321-2-6-2', '1611.09321-3-6-2'], ['1611.09321-2-6-3', '1611.09321-3-6-3'], ['1611.09321-2-6-4', '1611.09321-3-6-4'], ['1611.09321-2-6-5', '1611.09321-3-6-5'], ['1611.09321-1-37-0', '1611.09321-2-36-0'], ['1611.09321-1-37-1', '1611.09321-2-36-1'], ['1611.09321-2-25-2', '1611.09321-3-27-1'], ['1611.09321-2-25-3', '1611.09321-3-27-2'], ['1611.09321-2-25-4', '1611.09321-3-27-3'], ['1611.09321-1-4-0', '1611.09321-2-4-0'], ['1611.09321-1-4-1', '1611.09321-2-4-1'], ['1611.09321-1-4-2', '1611.09321-2-4-2'], ['1611.09321-1-5-1', '1611.09321-2-4-4'], ['1611.09321-1-5-2', '1611.09321-2-4-5'], ['1611.09321-1-8-1', '1611.09321-2-7-1'], ['1611.09321-1-8-3', '1611.09321-2-7-4'], ['1611.09321-1-8-5', '1611.09321-2-7-7'], ['1611.09321-1-19-0', '1611.09321-2-18-0'], ['1611.09321-1-27-0', '1611.09321-2-26-0'], ['1611.09321-1-27-2', '1611.09321-2-26-2'], ['1611.09321-1-27-3', '1611.09321-2-26-3'], ['1611.09321-1-0-4', '1611.09321-2-0-4'], ['1611.09321-1-7-0', '1611.09321-2-6-0'], ['1611.09321-1-7-3', '1611.09321-2-6-3'], ['1611.09321-2-48-1', '1611.09321-3-55-1'], ['1611.09321-2-12-0', '1611.09321-3-12-0'], ['1611.09321-2-82-0', '1611.09321-3-89-0'], ['1611.09321-2-82-3', '1611.09321-3-89-3'], ['1611.09321-2-82-4', '1611.09321-3-89-4'], ['1611.09321-2-82-5', '1611.09321-3-89-5'], ['1611.09321-2-50-4', '1611.09321-3-57-4'], ['1611.09321-2-6-0', '1611.09321-3-6-0'], ['1611.09321-2-25-0', '1611.09321-3-25-0'], ['1611.09321-2-25-1', '1611.09321-3-27-0'], ['1611.09321-1-4-3', '1611.09321-2-4-3'], ['1611.09321-1-8-2', '1611.09321-2-7-2'], ['1611.09321-1-0-7', '1611.09321-2-0-7'], ['1611.09321-1-0-8', '1611.09321-2-0-7'], ['1611.09321-1-14-2', '1611.09321-2-13-2'], ['1611.09321-2-50-0', '1611.09321-3-57-0'], ['1611.09321-2-50-5', '1611.09321-3-57-5'], ['1611.09321-1-37-2', '1611.09321-2-39-0']]

[['1611.09321-1-26-0', '1611.09321-2-25-0'], ['1611.09321-1-26-1', '1611.09321-2-25-1'], ['1611.09321-1-26-2', '1611.09321-2-25-2'], ['1611.09321-1-26-3', '1611.09321-2-25-3'], ['1611.09321-1-26-4', '1611.09321-2-25-4'], ['1611.09321-1-59-0', '1611.09321-2-59-0'], ['1611.09321-1-59-1', '1611.09321-2-59-1'], ['1611.09321-1-59-2', '1611.09321-2-59-2'], ['1611.09321-1-70-0', '1611.09321-2-70-0'], ['1611.09321-1-67-0', '1611.09321-2-67-0'], ['1611.09321-1-67-1', '1611.09321-2-67-1'], ['1611.09321-1-67-2', '1611.09321-2-67-2'], ['1611.09321-1-67-3', '1611.09321-2-67-3'], ['1611.09321-1-67-4', '1611.09321-2-67-4'], ['1611.09321-1-67-5', '1611.09321-2-67-5'], ['1611.09321-1-67-6', '1611.09321-2-67-6'], ['1611.09321-1-67-7', '1611.09321-2-67-7'], ['1611.09321-1-53-0', '1611.09321-2-53-0'], ['1611.09321-1-53-1', '1611.09321-2-53-1'], ['1611.09321-1-53-2', '1611.09321-2-53-2'], ['1611.09321-1-53-3', '1611.09321-2-53-3'], ['1611.09321-1-53-4', '1611.09321-2-53-4'], ['1611.09321-1-40-0', '1611.09321-2-40-0'], ['1611.09321-1-40-1', '1611.09321-2-40-1'], ['1611.09321-1-61-0', '1611.09321-2-61-0'], ['1611.09321-1-61-1', '1611.09321-2-61-1'], ['1611.09321-1-61-2', '1611.09321-2-61-3'], ['1611.09321-1-61-3', '1611.09321-2-61-4'], ['1611.09321-1-61-4', '1611.09321-2-61-5'], ['1611.09321-1-61-5', '1611.09321-2-61-6'], ['1611.09321-1-61-6', '1611.09321-2-61-7'], ['1611.09321-1-61-7', '1611.09321-2-61-8'], ['1611.09321-1-61-8', '1611.09321-2-61-9'], ['1611.09321-1-61-9', '1611.09321-2-61-10'], ['1611.09321-1-80-0', '1611.09321-2-80-0'], ['1611.09321-1-80-1', '1611.09321-2-80-1'], ['1611.09321-1-80-2', '1611.09321-2-80-2'], ['1611.09321-1-80-3', '1611.09321-2-80-3'], ['1611.09321-1-80-4', '1611.09321-2-80-4'], ['1611.09321-1-80-5', '1611.09321-2-80-5'], ['1611.09321-1-50-0', '1611.09321-2-50-0'], ['1611.09321-1-50-1', '1611.09321-2-50-1'], ['1611.09321-1-50-2', '1611.09321-2-50-2'], ['1611.09321-1-50-3', '1611.09321-2-50-3'], ['1611.09321-1-50-4', '1611.09321-2-50-6'], ['1611.09321-1-8-0', '1611.09321-2-7-0'], ['1611.09321-1-8-4', '1611.09321-2-7-5'], ['1611.09321-1-8-6', '1611.09321-2-7-8'], ['1611.09321-1-43-0', '1611.09321-2-43-0'], ['1611.09321-1-43-1', '1611.09321-2-43-1'], ['1611.09321-1-43-2', '1611.09321-2-43-2'], ['1611.09321-1-43-3', '1611.09321-2-43-3'], ['1611.09321-1-57-0', '1611.09321-2-57-0'], ['1611.09321-1-57-1', '1611.09321-2-57-1'], ['1611.09321-1-57-2', '1611.09321-2-57-2'], ['1611.09321-1-57-3', '1611.09321-2-57-3'], ['1611.09321-1-32-0', '1611.09321-2-31-0'], ['1611.09321-1-32-1', '1611.09321-2-31-1'], ['1611.09321-1-32-2', '1611.09321-2-31-2'], ['1611.09321-1-32-3', '1611.09321-2-31-3'], ['1611.09321-1-45-0', '1611.09321-2-45-0'], ['1611.09321-1-45-1', '1611.09321-2-45-1'], ['1611.09321-1-45-2', '1611.09321-2-45-2'], ['1611.09321-1-45-3', '1611.09321-2-45-3'], ['1611.09321-1-45-4', '1611.09321-2-45-4'], ['1611.09321-1-63-0', '1611.09321-2-63-0'], ['1611.09321-1-63-1', '1611.09321-2-63-1'], ['1611.09321-1-63-2', '1611.09321-2-63-2'], ['1611.09321-1-63-3', '1611.09321-2-63-3'], ['1611.09321-1-63-4', '1611.09321-2-63-4'], ['1611.09321-1-63-5', '1611.09321-2-63-5'], ['1611.09321-1-63-6', '1611.09321-2-63-6'], ['1611.09321-1-31-0', '1611.09321-2-30-0'], ['1611.09321-1-31-1', '1611.09321-2-30-1'], ['1611.09321-1-31-2', '1611.09321-2-30-2'], ['1611.09321-1-69-0', '1611.09321-2-69-0'], ['1611.09321-1-69-1', '1611.09321-2-69-1'], ['1611.09321-1-69-2', '1611.09321-2-69-2'], ['1611.09321-1-69-3', '1611.09321-2-69-3'], ['1611.09321-1-69-4', '1611.09321-2-69-4'], ['1611.09321-1-20-0', '1611.09321-2-19-0'], ['1611.09321-1-20-1', '1611.09321-2-19-1'], ['1611.09321-1-20-2', '1611.09321-2-19-2'], ['1611.09321-1-20-3', '1611.09321-2-19-3'], ['1611.09321-1-20-4', '1611.09321-2-19-4'], ['1611.09321-1-20-5', '1611.09321-2-19-5'], ['1611.09321-1-20-6', '1611.09321-2-19-6'], ['1611.09321-1-20-7', '1611.09321-2-19-7'], ['1611.09321-1-20-8', '1611.09321-2-19-8'], ['1611.09321-1-20-9', '1611.09321-2-19-9'], ['1611.09321-1-20-10', '1611.09321-2-19-10'], ['1611.09321-1-72-0', '1611.09321-2-72-0'], ['1611.09321-1-72-1', '1611.09321-2-72-1'], ['1611.09321-1-29-0', '1611.09321-2-28-0'], ['1611.09321-1-29-1', '1611.09321-2-28-1'], ['1611.09321-1-29-2', '1611.09321-2-28-2'], ['1611.09321-1-29-3', '1611.09321-2-28-3'], ['1611.09321-1-29-4', '1611.09321-2-28-4'], ['1611.09321-1-29-5', '1611.09321-2-28-5'], ['1611.09321-1-13-0', '1611.09321-2-12-0'], ['1611.09321-1-13-1', '1611.09321-2-12-1'], ['1611.09321-1-82-0', '1611.09321-2-82-0'], ['1611.09321-1-82-1', '1611.09321-2-82-1'], ['1611.09321-1-82-2', '1611.09321-2-82-2'], ['1611.09321-1-82-3', '1611.09321-2-82-3'], ['1611.09321-1-82-4', '1611.09321-2-82-4'], ['1611.09321-1-82-5', '1611.09321-2-82-5'], ['1611.09321-1-1-0', '1611.09321-2-1-0'], ['1611.09321-1-1-1', '1611.09321-2-1-1'], ['1611.09321-1-1-2', '1611.09321-2-1-2'], ['1611.09321-1-1-3', '1611.09321-2-1-3'], ['1611.09321-1-1-4', '1611.09321-2-1-4'], ['1611.09321-1-1-5', '1611.09321-2-1-5'], ['1611.09321-1-1-6', '1611.09321-2-1-6'], ['1611.09321-1-1-7', '1611.09321-2-1-7'], ['1611.09321-1-76-0', '1611.09321-2-76-0'], ['1611.09321-1-76-1', '1611.09321-2-76-1'], ['1611.09321-1-76-2', '1611.09321-2-76-2'], ['1611.09321-1-76-3', '1611.09321-2-76-3'], ['1611.09321-1-76-4', '1611.09321-2-76-4'], ['1611.09321-1-76-5', '1611.09321-2-76-5'], ['1611.09321-1-75-0', '1611.09321-2-75-0'], ['1611.09321-1-75-1', '1611.09321-2-75-1'], ['1611.09321-1-39-0', '1611.09321-2-38-0'], ['1611.09321-1-39-1', '1611.09321-2-38-1'], ['1611.09321-1-39-2', '1611.09321-2-38-2'], ['1611.09321-1-39-3', '1611.09321-2-38-3'], ['1611.09321-1-6-0', '1611.09321-2-5-0'], ['1611.09321-1-6-1', '1611.09321-2-5-1'], ['1611.09321-1-6-2', '1611.09321-2-5-2'], ['1611.09321-1-77-0', '1611.09321-2-77-0'], ['1611.09321-1-79-0', '1611.09321-2-79-0'], ['1611.09321-1-79-1', '1611.09321-2-79-1'], ['1611.09321-1-9-0', '1611.09321-2-8-0'], ['1611.09321-1-9-1', '1611.09321-2-8-1'], ['1611.09321-1-9-2', '1611.09321-2-8-2'], ['1611.09321-1-9-3', '1611.09321-2-8-3'], ['1611.09321-1-21-0', '1611.09321-2-20-0'], ['1611.09321-1-21-1', '1611.09321-2-20-1'], ['1611.09321-1-21-2', '1611.09321-2-20-2'], ['1611.09321-1-11-0', '1611.09321-2-10-0'], ['1611.09321-1-11-1', '1611.09321-2-10-1'], ['1611.09321-1-11-2', '1611.09321-2-10-2'], ['1611.09321-1-11-3', '1611.09321-2-10-3'], ['1611.09321-1-11-4', '1611.09321-2-10-4'], ['1611.09321-1-71-0', '1611.09321-2-71-0'], ['1611.09321-1-71-1', '1611.09321-2-71-1'], ['1611.09321-1-71-2', '1611.09321-2-71-2'], ['1611.09321-1-71-3', '1611.09321-2-71-3'], ['1611.09321-1-71-4', '1611.09321-2-71-4'], ['1611.09321-1-71-5', '1611.09321-2-71-5'], ['1611.09321-1-71-6', '1611.09321-2-71-6'], ['1611.09321-1-71-7', '1611.09321-2-71-7'], ['1611.09321-1-38-0', '1611.09321-2-37-0'], ['1611.09321-1-38-1', '1611.09321-2-37-1'], ['1611.09321-1-38-2', '1611.09321-2-37-2'], ['1611.09321-1-48-0', '1611.09321-2-48-0'], ['1611.09321-1-48-1', '1611.09321-2-48-1'], ['1611.09321-1-48-2', '1611.09321-2-48-2'], ['1611.09321-1-48-3', '1611.09321-2-48-3'], ['1611.09321-1-48-4', '1611.09321-2-48-5'], ['1611.09321-1-48-5', '1611.09321-2-48-6'], ['1611.09321-1-48-6', '1611.09321-2-48-7'], ['1611.09321-1-48-7', '1611.09321-2-48-8'], ['1611.09321-1-48-8', '1611.09321-2-48-9'], ['1611.09321-1-48-9', '1611.09321-2-48-10'], ['1611.09321-1-48-10', '1611.09321-2-48-11'], ['1611.09321-1-48-11', '1611.09321-2-48-12'], ['1611.09321-1-48-12', '1611.09321-2-48-13'], ['1611.09321-1-48-13', '1611.09321-2-48-14'], ['1611.09321-1-48-14', '1611.09321-2-48-15'], ['1611.09321-1-48-15', '1611.09321-2-48-16'], ['1611.09321-1-49-0', '1611.09321-2-49-0'], ['1611.09321-1-49-1', '1611.09321-2-49-1'], ['1611.09321-1-58-0', '1611.09321-2-58-0'], ['1611.09321-1-58-1', '1611.09321-2-58-1'], ['1611.09321-1-58-2', '1611.09321-2-58-2'], ['1611.09321-1-58-3', '1611.09321-2-58-3'], ['1611.09321-1-58-4', '1611.09321-2-58-4'], ['1611.09321-1-58-5', '1611.09321-2-58-5'], ['1611.09321-1-58-6', '1611.09321-2-58-6'], ['1611.09321-1-58-7', '1611.09321-2-58-7'], ['1611.09321-1-55-0', '1611.09321-2-55-0'], ['1611.09321-1-55-1', '1611.09321-2-55-1'], ['1611.09321-1-55-2', '1611.09321-2-55-2'], ['1611.09321-1-23-0', '1611.09321-2-22-0'], ['1611.09321-1-23-1', '1611.09321-2-22-1'], ['1611.09321-1-23-2', '1611.09321-2-22-2'], ['1611.09321-1-23-3', '1611.09321-2-22-3'], ['1611.09321-1-25-0', '1611.09321-2-24-0'], ['1611.09321-1-25-1', '1611.09321-2-24-1'], ['1611.09321-1-25-2', '1611.09321-2-24-2'], ['1611.09321-1-25-3', '1611.09321-2-24-3'], ['1611.09321-1-25-4', '1611.09321-2-24-4'], ['1611.09321-1-16-0', '1611.09321-2-15-0'], ['1611.09321-1-16-1', '1611.09321-2-15-1'], ['1611.09321-1-16-2', '1611.09321-2-15-2'], ['1611.09321-1-16-3', '1611.09321-2-15-3'], ['1611.09321-1-16-4', '1611.09321-2-15-4'], ['1611.09321-1-16-5', '1611.09321-2-15-5'], ['1611.09321-1-16-6', '1611.09321-2-15-6'], ['1611.09321-1-16-7', '1611.09321-2-15-7'], ['1611.09321-1-16-8', '1611.09321-2-15-8'], ['1611.09321-1-16-9', '1611.09321-2-15-9'], ['1611.09321-1-24-0', '1611.09321-2-23-0'], ['1611.09321-1-24-1', '1611.09321-2-23-1'], ['1611.09321-1-24-2', '1611.09321-2-23-2'], ['1611.09321-1-24-3', '1611.09321-2-23-3'], ['1611.09321-1-19-1', '1611.09321-2-18-1'], ['1611.09321-1-19-2', '1611.09321-2-18-3'], ['1611.09321-1-19-3', '1611.09321-2-18-4'], ['1611.09321-1-19-4', '1611.09321-2-18-5'], ['1611.09321-1-73-0', '1611.09321-2-73-0'], ['1611.09321-1-73-1', '1611.09321-2-73-1'], ['1611.09321-1-73-2', '1611.09321-2-73-2'], ['1611.09321-1-73-3', '1611.09321-2-73-3'], ['1611.09321-1-73-4', '1611.09321-2-73-4'], ['1611.09321-1-73-5', '1611.09321-2-73-5'], ['1611.09321-1-68-0', '1611.09321-2-68-0'], ['1611.09321-1-68-1', '1611.09321-2-68-1'], ['1611.09321-1-68-2', '1611.09321-2-68-2'], ['1611.09321-1-68-3', '1611.09321-2-68-3'], ['1611.09321-1-68-4', '1611.09321-2-68-4'], ['1611.09321-1-68-5', '1611.09321-2-68-5'], ['1611.09321-1-68-6', '1611.09321-2-68-6'], ['1611.09321-1-68-7', '1611.09321-2-68-7'], ['1611.09321-1-15-0', '1611.09321-2-14-0'], ['1611.09321-1-36-0', '1611.09321-2-35-0'], ['1611.09321-1-36-1', '1611.09321-2-35-1'], ['1611.09321-1-27-1', '1611.09321-2-26-1'], ['1611.09321-1-27-4', '1611.09321-2-26-4'], ['1611.09321-1-27-5', '1611.09321-2-26-5'], ['1611.09321-1-27-6', '1611.09321-2-26-6'], ['1611.09321-1-27-7', '1611.09321-2-26-7'], ['1611.09321-1-27-8', '1611.09321-2-26-8'], ['1611.09321-1-0-0', '1611.09321-2-0-0'], ['1611.09321-1-0-1', '1611.09321-2-0-1'], ['1611.09321-1-0-2', '1611.09321-2-0-2'], ['1611.09321-1-0-3', '1611.09321-2-0-3'], ['1611.09321-1-0-5', '1611.09321-2-0-5'], ['1611.09321-1-0-6', '1611.09321-2-0-6'], ['1611.09321-1-51-0', '1611.09321-2-51-0'], ['1611.09321-1-64-0', '1611.09321-2-64-0'], ['1611.09321-1-64-1', '1611.09321-2-64-1'], ['1611.09321-1-64-2', '1611.09321-2-64-2'], ['1611.09321-1-64-3', '1611.09321-2-64-3'], ['1611.09321-1-64-4', '1611.09321-2-64-4'], ['1611.09321-1-64-5', '1611.09321-2-64-5'], ['1611.09321-1-10-0', '1611.09321-2-9-0'], ['1611.09321-1-34-0', '1611.09321-2-33-0'], ['1611.09321-1-34-1', '1611.09321-2-33-1'], ['1611.09321-1-34-2', '1611.09321-2-33-2'], ['1611.09321-1-41-1', '1611.09321-2-41-1'], ['1611.09321-1-42-0', '1611.09321-2-42-0'], ['1611.09321-1-42-1', '1611.09321-2-42-1'], ['1611.09321-1-42-2', '1611.09321-2-42-2'], ['1611.09321-1-42-3', '1611.09321-2-42-3'], ['1611.09321-1-42-4', '1611.09321-2-42-4'], ['1611.09321-1-18-0', '1611.09321-2-17-0'], ['1611.09321-1-18-1', '1611.09321-2-17-1'], ['1611.09321-1-2-0', '1611.09321-2-2-0'], ['1611.09321-1-2-1', '1611.09321-2-2-1'], ['1611.09321-1-2-2', '1611.09321-2-2-2'], ['1611.09321-1-2-3', '1611.09321-2-2-3'], ['1611.09321-1-14-0', '1611.09321-2-13-0'], ['1611.09321-1-14-1', '1611.09321-2-13-1'], ['1611.09321-1-14-3', '1611.09321-2-13-3'], ['1611.09321-1-14-4', '1611.09321-2-13-4'], ['1611.09321-1-14-5', '1611.09321-2-13-5'], ['1611.09321-1-44-0', '1611.09321-2-44-0'], ['1611.09321-1-44-1', '1611.09321-2-44-1'], ['1611.09321-1-44-2', '1611.09321-2-44-2'], ['1611.09321-1-44-3', '1611.09321-2-44-3'], ['1611.09321-1-44-4', '1611.09321-2-44-4'], ['1611.09321-1-65-0', '1611.09321-2-65-0'], ['1611.09321-1-65-1', '1611.09321-2-65-1'], ['1611.09321-1-65-2', '1611.09321-2-65-2'], ['1611.09321-1-65-3', '1611.09321-2-65-3'], ['1611.09321-1-7-1', '1611.09321-2-6-1'], ['1611.09321-1-7-2', '1611.09321-2-6-2'], ['1611.09321-1-7-4', '1611.09321-2-6-4'], ['1611.09321-1-7-5', '1611.09321-2-6-5'], ['1611.09321-1-52-0', '1611.09321-2-52-0'], ['1611.09321-1-52-1', '1611.09321-2-52-1'], ['1611.09321-1-52-2', '1611.09321-2-52-2'], ['1611.09321-1-52-3', '1611.09321-2-52-3'], ['1611.09321-1-52-4', '1611.09321-2-52-4'], ['1611.09321-1-52-5', '1611.09321-2-52-5'], ['1611.09321-2-72-0', '1611.09321-3-79-0'], ['1611.09321-2-72-1', '1611.09321-3-79-1'], ['1611.09321-2-44-0', '1611.09321-3-51-0'], ['1611.09321-2-44-1', '1611.09321-3-51-1'], ['1611.09321-2-44-2', '1611.09321-3-51-2'], ['1611.09321-2-44-3', '1611.09321-3-51-3'], ['1611.09321-2-44-4', '1611.09321-3-51-4'], ['1611.09321-2-64-0', '1611.09321-3-71-0'], ['1611.09321-2-64-1', '1611.09321-3-71-1'], ['1611.09321-2-64-2', '1611.09321-3-71-2'], ['1611.09321-2-64-3', '1611.09321-3-71-3'], ['1611.09321-2-64-4', '1611.09321-3-71-4'], ['1611.09321-2-64-5', '1611.09321-3-71-5'], ['1611.09321-2-49-0', '1611.09321-3-56-0'], ['1611.09321-2-49-1', '1611.09321-3-56-1'], ['1611.09321-2-55-0', '1611.09321-3-62-0'], ['1611.09321-2-55-1', '1611.09321-3-62-1'], ['1611.09321-2-55-2', '1611.09321-3-62-2'], ['1611.09321-2-65-0', '1611.09321-3-72-0'], ['1611.09321-2-65-1', '1611.09321-3-72-1'], ['1611.09321-2-65-2', '1611.09321-3-72-2'], ['1611.09321-2-65-3', '1611.09321-3-72-3'], ['1611.09321-2-48-0', '1611.09321-3-55-0'], ['1611.09321-2-48-2', '1611.09321-3-55-2'], ['1611.09321-2-48-3', '1611.09321-3-55-3'], ['1611.09321-2-48-4', '1611.09321-3-55-4'], ['1611.09321-2-48-5', '1611.09321-3-55-5'], ['1611.09321-2-48-6', '1611.09321-3-55-6'], ['1611.09321-2-48-7', '1611.09321-3-55-7'], ['1611.09321-2-48-8', '1611.09321-3-55-8'], ['1611.09321-2-48-9', '1611.09321-3-55-9'], ['1611.09321-2-48-10', '1611.09321-3-55-10'], ['1611.09321-2-48-11', '1611.09321-3-55-11'], ['1611.09321-2-48-12', '1611.09321-3-55-12'], ['1611.09321-2-48-13', '1611.09321-3-55-13'], ['1611.09321-2-48-14', '1611.09321-3-55-14'], ['1611.09321-2-48-15', '1611.09321-3-55-15'], ['1611.09321-2-48-16', '1611.09321-3-55-16'], ['1611.09321-2-71-0', '1611.09321-3-78-0'], ['1611.09321-2-71-1', '1611.09321-3-78-1'], ['1611.09321-2-71-2', '1611.09321-3-78-2'], ['1611.09321-2-71-3', '1611.09321-3-78-3'], ['1611.09321-2-71-4', '1611.09321-3-78-4'], ['1611.09321-2-71-5', '1611.09321-3-78-5'], ['1611.09321-2-71-6', '1611.09321-3-78-6'], ['1611.09321-2-71-7', '1611.09321-3-78-7'], ['1611.09321-2-75-0', '1611.09321-3-82-0'], ['1611.09321-2-75-1', '1611.09321-3-82-1'], ['1611.09321-2-9-0', '1611.09321-3-9-0'], ['1611.09321-2-0-0', '1611.09321-3-0-0'], ['1611.09321-2-0-1', '1611.09321-3-0-1'], ['1611.09321-2-0-2', '1611.09321-3-0-2'], ['1611.09321-2-0-3', '1611.09321-3-0-3'], ['1611.09321-2-0-4', '1611.09321-3-0-4'], ['1611.09321-2-0-5', '1611.09321-3-0-5'], ['1611.09321-2-0-6', '1611.09321-3-0-6'], ['1611.09321-2-0-7', '1611.09321-3-0-7'], ['1611.09321-2-10-0', '1611.09321-3-10-0'], ['1611.09321-2-10-1', '1611.09321-3-10-1'], ['1611.09321-2-10-2', '1611.09321-3-10-2'], ['1611.09321-2-10-3', '1611.09321-3-10-3'], ['1611.09321-2-10-4', '1611.09321-3-10-4'], ['1611.09321-2-70-0', '1611.09321-3-77-0'], ['1611.09321-2-76-0', '1611.09321-3-83-0'], ['1611.09321-2-76-1', '1611.09321-3-83-1'], ['1611.09321-2-76-2', '1611.09321-3-83-2'], ['1611.09321-2-76-3', '1611.09321-3-83-3'], ['1611.09321-2-76-4', '1611.09321-3-83-4'], ['1611.09321-2-76-5', '1611.09321-3-83-5'], ['1611.09321-2-37-0', '1611.09321-3-43-0'], ['1611.09321-2-37-1', '1611.09321-3-43-1'], ['1611.09321-2-37-2', '1611.09321-3-43-2'], ['1611.09321-2-51-0', '1611.09321-3-58-0'], ['1611.09321-2-20-0', '1611.09321-3-20-0'], ['1611.09321-2-20-1', '1611.09321-3-20-1'], ['1611.09321-2-20-2', '1611.09321-3-20-2'], ['1611.09321-2-12-1', '1611.09321-3-12-1'], ['1611.09321-2-26-0', '1611.09321-3-28-0'], ['1611.09321-2-26-1', '1611.09321-3-28-1'], ['1611.09321-2-26-2', '1611.09321-3-28-2'], ['1611.09321-2-26-3', '1611.09321-3-28-3'], ['1611.09321-2-26-4', '1611.09321-3-28-4'], ['1611.09321-2-26-5', '1611.09321-3-28-5'], ['1611.09321-2-26-6', '1611.09321-3-28-6'], ['1611.09321-2-26-7', '1611.09321-3-28-7'], ['1611.09321-2-26-8', '1611.09321-3-28-8'], ['1611.09321-2-38-0', '1611.09321-3-44-0'], ['1611.09321-2-38-1', '1611.09321-3-44-1'], ['1611.09321-2-38-2', '1611.09321-3-44-2'], ['1611.09321-2-38-3', '1611.09321-3-44-3'], ['1611.09321-2-36-0', '1611.09321-3-42-0'], ['1611.09321-2-36-1', '1611.09321-3-42-1'], ['1611.09321-2-7-0', '1611.09321-3-7-0'], ['1611.09321-2-7-1', '1611.09321-3-7-1'], ['1611.09321-2-7-2', '1611.09321-3-7-2'], ['1611.09321-2-7-3', '1611.09321-3-7-3'], ['1611.09321-2-7-4', '1611.09321-3-7-4'], ['1611.09321-2-7-5', '1611.09321-3-7-5'], ['1611.09321-2-7-6', '1611.09321-3-7-6'], ['1611.09321-2-7-7', '1611.09321-3-7-7'], ['1611.09321-2-7-8', '1611.09321-3-7-8'], ['1611.09321-2-39-0', '1611.09321-3-45-0'], ['1611.09321-2-39-1', '1611.09321-3-45-1'], ['1611.09321-2-67-0', '1611.09321-3-74-0'], ['1611.09321-2-67-1', '1611.09321-3-74-1'], ['1611.09321-2-67-2', '1611.09321-3-74-2'], ['1611.09321-2-67-3', '1611.09321-3-74-3'], ['1611.09321-2-67-4', '1611.09321-3-74-4'], ['1611.09321-2-67-5', '1611.09321-3-74-5'], ['1611.09321-2-67-6', '1611.09321-3-74-6'], ['1611.09321-2-67-7', '1611.09321-3-74-7'], ['1611.09321-2-41-1', '1611.09321-3-48-1'], ['1611.09321-2-17-0', '1611.09321-3-17-0'], ['1611.09321-2-17-1', '1611.09321-3-17-1'], ['1611.09321-2-40-0', '1611.09321-3-47-0'], ['1611.09321-2-40-1', '1611.09321-3-47-1'], ['1611.09321-2-69-0', '1611.09321-3-76-0'], ['1611.09321-2-69-1', '1611.09321-3-76-1'], ['1611.09321-2-69-2', '1611.09321-3-76-2'], ['1611.09321-2-69-3', '1611.09321-3-76-3'], ['1611.09321-2-69-4', '1611.09321-3-76-4'], ['1611.09321-2-59-0', '1611.09321-3-66-0'], ['1611.09321-2-59-1', '1611.09321-3-66-1'], ['1611.09321-2-59-2', '1611.09321-3-66-2'], ['1611.09321-2-63-0', '1611.09321-3-70-0'], ['1611.09321-2-63-1', '1611.09321-3-70-1'], ['1611.09321-2-63-2', '1611.09321-3-70-2'], ['1611.09321-2-63-3', '1611.09321-3-70-3'], ['1611.09321-2-63-4', '1611.09321-3-70-4'], ['1611.09321-2-63-5', '1611.09321-3-70-5'], ['1611.09321-2-63-6', '1611.09321-3-70-6'], ['1611.09321-2-77-0', '1611.09321-3-84-0'], ['1611.09321-2-19-0', '1611.09321-3-19-0'], ['1611.09321-2-19-1', '1611.09321-3-19-1'], ['1611.09321-2-19-2', '1611.09321-3-19-2'], ['1611.09321-2-19-3', '1611.09321-3-19-3'], ['1611.09321-2-19-4', '1611.09321-3-19-4'], ['1611.09321-2-19-5', '1611.09321-3-19-5'], ['1611.09321-2-19-6', '1611.09321-3-19-6'], ['1611.09321-2-19-7', '1611.09321-3-19-7'], ['1611.09321-2-19-8', '1611.09321-3-19-8'], ['1611.09321-2-19-9', '1611.09321-3-19-9'], ['1611.09321-2-19-10', '1611.09321-3-19-10'], ['1611.09321-2-33-0', '1611.09321-3-35-0'], ['1611.09321-2-33-1', '1611.09321-3-35-1'], ['1611.09321-2-33-2', '1611.09321-3-35-2'], ['1611.09321-2-13-0', '1611.09321-3-13-0'], ['1611.09321-2-13-1', '1611.09321-3-13-1'], ['1611.09321-2-13-2', '1611.09321-3-13-2'], ['1611.09321-2-13-3', '1611.09321-3-13-3'], ['1611.09321-2-13-4', '1611.09321-3-13-4'], ['1611.09321-2-13-5', '1611.09321-3-13-5'], ['1611.09321-2-80-0', '1611.09321-3-87-0'], ['1611.09321-2-80-1', '1611.09321-3-87-1'], ['1611.09321-2-80-2', '1611.09321-3-87-2'], ['1611.09321-2-80-3', '1611.09321-3-87-3'], ['1611.09321-2-80-4', '1611.09321-3-87-4'], ['1611.09321-2-80-5', '1611.09321-3-87-5'], ['1611.09321-2-52-0', '1611.09321-3-59-0'], ['1611.09321-2-52-1', '1611.09321-3-59-1'], ['1611.09321-2-52-2', '1611.09321-3-59-2'], ['1611.09321-2-52-3', '1611.09321-3-59-3'], ['1611.09321-2-52-4', '1611.09321-3-59-4'], ['1611.09321-2-52-5', '1611.09321-3-59-5'], ['1611.09321-2-79-0', '1611.09321-3-86-0'], ['1611.09321-2-79-1', '1611.09321-3-86-1'], ['1611.09321-2-58-0', '1611.09321-3-65-0'], ['1611.09321-2-58-1', '1611.09321-3-65-1'], ['1611.09321-2-58-2', '1611.09321-3-65-2'], ['1611.09321-2-58-3', '1611.09321-3-65-3'], ['1611.09321-2-58-4', '1611.09321-3-65-4'], ['1611.09321-2-58-5', '1611.09321-3-65-5'], ['1611.09321-2-58-6', '1611.09321-3-65-6'], ['1611.09321-2-58-7', '1611.09321-3-65-7'], ['1611.09321-2-8-0', '1611.09321-3-8-0'], ['1611.09321-2-8-1', '1611.09321-3-8-1'], ['1611.09321-2-8-2', '1611.09321-3-8-2'], ['1611.09321-2-8-3', '1611.09321-3-8-3'], ['1611.09321-2-61-0', '1611.09321-3-68-0'], ['1611.09321-2-61-1', '1611.09321-3-68-1'], ['1611.09321-2-61-2', '1611.09321-3-68-2'], ['1611.09321-2-61-3', '1611.09321-3-68-3'], ['1611.09321-2-61-4', '1611.09321-3-68-4'], ['1611.09321-2-61-5', '1611.09321-3-68-5'], ['1611.09321-2-61-6', '1611.09321-3-68-6'], ['1611.09321-2-61-7', '1611.09321-3-68-7'], ['1611.09321-2-61-8', '1611.09321-3-68-8'], ['1611.09321-2-61-9', '1611.09321-3-68-9'], ['1611.09321-2-61-10', '1611.09321-3-68-10'], ['1611.09321-2-68-0', '1611.09321-3-75-0'], ['1611.09321-2-68-1', '1611.09321-3-75-1'], ['1611.09321-2-68-2', '1611.09321-3-75-2'], ['1611.09321-2-68-3', '1611.09321-3-75-3'], ['1611.09321-2-68-4', '1611.09321-3-75-4'], ['1611.09321-2-68-5', '1611.09321-3-75-5'], ['1611.09321-2-68-6', '1611.09321-3-75-6'], ['1611.09321-2-68-7', '1611.09321-3-75-7'], ['1611.09321-2-43-0', '1611.09321-3-50-0'], ['1611.09321-2-43-1', '1611.09321-3-50-1'], ['1611.09321-2-43-2', '1611.09321-3-50-2'], ['1611.09321-2-43-3', '1611.09321-3-50-3'], ['1611.09321-2-82-1', '1611.09321-3-89-1'], ['1611.09321-2-82-2', '1611.09321-3-89-2'], ['1611.09321-2-23-0', '1611.09321-3-23-0'], ['1611.09321-2-23-1', '1611.09321-3-23-1'], ['1611.09321-2-23-2', '1611.09321-3-23-2'], ['1611.09321-2-23-3', '1611.09321-3-23-3'], ['1611.09321-2-30-0', '1611.09321-3-32-0'], ['1611.09321-2-30-1', '1611.09321-3-32-1'], ['1611.09321-2-30-2', '1611.09321-3-32-2'], ['1611.09321-2-31-0', '1611.09321-3-33-0'], ['1611.09321-2-31-1', '1611.09321-3-33-1'], ['1611.09321-2-31-2', '1611.09321-3-33-2'], ['1611.09321-2-31-3', '1611.09321-3-33-3'], ['1611.09321-2-73-0', '1611.09321-3-80-0'], ['1611.09321-2-73-1', '1611.09321-3-80-1'], ['1611.09321-2-73-2', '1611.09321-3-80-2'], ['1611.09321-2-73-3', '1611.09321-3-80-3'], ['1611.09321-2-73-4', '1611.09321-3-80-4'], ['1611.09321-2-73-5', '1611.09321-3-80-5'], ['1611.09321-2-28-0', '1611.09321-3-30-0'], ['1611.09321-2-28-1', '1611.09321-3-30-1'], ['1611.09321-2-28-2', '1611.09321-3-30-2'], ['1611.09321-2-28-3', '1611.09321-3-30-3'], ['1611.09321-2-28-4', '1611.09321-3-30-4'], ['1611.09321-2-28-5', '1611.09321-3-30-5'], ['1611.09321-2-1-0', '1611.09321-3-1-0'], ['1611.09321-2-1-1', '1611.09321-3-1-1'], ['1611.09321-2-1-2', '1611.09321-3-1-2'], ['1611.09321-2-1-3', '1611.09321-3-1-3'], ['1611.09321-2-1-4', '1611.09321-3-1-4'], ['1611.09321-2-1-5', '1611.09321-3-1-5'], ['1611.09321-2-1-6', '1611.09321-3-1-6'], ['1611.09321-2-1-7', '1611.09321-3-1-7'], ['1611.09321-2-5-0', '1611.09321-3-5-0'], ['1611.09321-2-5-1', '1611.09321-3-5-1'], ['1611.09321-2-5-2', '1611.09321-3-5-2'], ['1611.09321-2-57-0', '1611.09321-3-64-0'], ['1611.09321-2-57-1', '1611.09321-3-64-1'], ['1611.09321-2-57-2', '1611.09321-3-64-2'], ['1611.09321-2-57-3', '1611.09321-3-64-3'], ['1611.09321-2-35-0', '1611.09321-3-41-2'], ['1611.09321-2-35-1', '1611.09321-3-41-3'], ['1611.09321-2-53-0', '1611.09321-3-60-0'], ['1611.09321-2-53-1', '1611.09321-3-60-1'], ['1611.09321-2-53-2', '1611.09321-3-60-2'], ['1611.09321-2-53-3', '1611.09321-3-60-3'], ['1611.09321-2-53-4', '1611.09321-3-60-4'], ['1611.09321-2-42-0', '1611.09321-3-49-0'], ['1611.09321-2-42-1', '1611.09321-3-49-1'], ['1611.09321-2-42-2', '1611.09321-3-49-2'], ['1611.09321-2-42-3', '1611.09321-3-49-3'], ['1611.09321-2-42-4', '1611.09321-3-49-4'], ['1611.09321-2-22-0', '1611.09321-3-22-0'], ['1611.09321-2-22-1', '1611.09321-3-22-1'], ['1611.09321-2-22-2', '1611.09321-3-22-2'], ['1611.09321-2-22-3', '1611.09321-3-22-3'], ['1611.09321-2-15-0', '1611.09321-3-15-0'], ['1611.09321-2-15-1', '1611.09321-3-15-1'], ['1611.09321-2-15-2', '1611.09321-3-15-2'], ['1611.09321-2-15-3', '1611.09321-3-15-3'], ['1611.09321-2-15-4', '1611.09321-3-15-4'], ['1611.09321-2-15-5', '1611.09321-3-15-5'], ['1611.09321-2-15-6', '1611.09321-3-15-6'], ['1611.09321-2-15-7', '1611.09321-3-15-7'], ['1611.09321-2-15-8', '1611.09321-3-15-8'], ['1611.09321-2-15-9', '1611.09321-3-15-9'], ['1611.09321-2-18-0', '1611.09321-3-18-0'], ['1611.09321-2-18-1', '1611.09321-3-18-1'], ['1611.09321-2-18-2', '1611.09321-3-18-2'], ['1611.09321-2-18-3', '1611.09321-3-18-3'], ['1611.09321-2-18-4', '1611.09321-3-18-4'], ['1611.09321-2-18-5', '1611.09321-3-18-5'], ['1611.09321-2-45-0', '1611.09321-3-52-0'], ['1611.09321-2-45-1', '1611.09321-3-52-1'], ['1611.09321-2-45-2', '1611.09321-3-52-2'], ['1611.09321-2-45-3', '1611.09321-3-52-3'], ['1611.09321-2-45-4', '1611.09321-3-52-4'], ['1611.09321-2-2-0', '1611.09321-3-2-0'], ['1611.09321-2-2-1', '1611.09321-3-2-1'], ['1611.09321-2-2-2', '1611.09321-3-2-2'], ['1611.09321-2-2-3', '1611.09321-3-2-3'], ['1611.09321-2-24-0', '1611.09321-3-24-0'], ['1611.09321-2-24-1', '1611.09321-3-24-1'], ['1611.09321-2-24-2', '1611.09321-3-24-2'], ['1611.09321-2-24-3', '1611.09321-3-24-3'], ['1611.09321-2-24-4', '1611.09321-3-24-4'], ['1611.09321-2-50-1', '1611.09321-3-57-1'], ['1611.09321-2-50-2', '1611.09321-3-57-2'], ['1611.09321-2-50-3', '1611.09321-3-57-3'], ['1611.09321-2-50-6', '1611.09321-3-57-6'], ['1611.09321-2-4-0', '1611.09321-3-4-0'], ['1611.09321-2-4-1', '1611.09321-3-4-1'], ['1611.09321-2-4-2', '1611.09321-3-4-2'], ['1611.09321-2-4-3', '1611.09321-3-4-3'], ['1611.09321-2-4-4', '1611.09321-3-4-4'], ['1611.09321-2-4-5', '1611.09321-3-4-5'], ['1611.09321-2-14-0', '1611.09321-3-14-0'], ['1611.09321-2-6-1', '1611.09321-3-6-1'], ['1611.09321-2-6-2', '1611.09321-3-6-2'], ['1611.09321-2-6-3', '1611.09321-3-6-3'], ['1611.09321-2-6-4', '1611.09321-3-6-4'], ['1611.09321-2-6-5', '1611.09321-3-6-5'], ['1611.09321-1-37-0', '1611.09321-2-36-0'], ['1611.09321-1-37-1', '1611.09321-2-36-1'], ['1611.09321-2-25-2', '1611.09321-3-27-1'], ['1611.09321-2-25-3', '1611.09321-3-27-2'], ['1611.09321-2-25-4', '1611.09321-3-27-3'], ['1611.09321-1-4-0', '1611.09321-2-4-0'], ['1611.09321-1-4-1', '1611.09321-2-4-1'], ['1611.09321-1-4-2', '1611.09321-2-4-2'], ['1611.09321-1-5-1', '1611.09321-2-4-4'], ['1611.09321-1-5-2', '1611.09321-2-4-5']]

[['1611.09321-1-8-1', '1611.09321-2-7-1'], ['1611.09321-1-8-3', '1611.09321-2-7-4'], ['1611.09321-1-8-5', '1611.09321-2-7-7'], ['1611.09321-1-19-0', '1611.09321-2-18-0'], ['1611.09321-1-27-0', '1611.09321-2-26-0'], ['1611.09321-1-27-2', '1611.09321-2-26-2'], ['1611.09321-1-27-3', '1611.09321-2-26-3'], ['1611.09321-1-0-4', '1611.09321-2-0-4'], ['1611.09321-1-7-0', '1611.09321-2-6-0'], ['1611.09321-1-7-3', '1611.09321-2-6-3'], ['1611.09321-2-48-1', '1611.09321-3-55-1'], ['1611.09321-2-12-0', '1611.09321-3-12-0'], ['1611.09321-2-82-0', '1611.09321-3-89-0'], ['1611.09321-2-82-3', '1611.09321-3-89-3'], ['1611.09321-2-82-4', '1611.09321-3-89-4'], ['1611.09321-2-82-5', '1611.09321-3-89-5'], ['1611.09321-2-50-4', '1611.09321-3-57-4'], ['1611.09321-2-6-0', '1611.09321-3-6-0'], ['1611.09321-2-25-0', '1611.09321-3-25-0'], ['1611.09321-2-25-1', '1611.09321-3-27-0'], ['1611.09321-1-4-3', '1611.09321-2-4-3']]

[]

[['1611.09321-1-8-2', '1611.09321-2-7-2'], ['1611.09321-1-0-7', '1611.09321-2-0-7'], ['1611.09321-1-0-8', '1611.09321-2-0-7'], ['1611.09321-1-14-2', '1611.09321-2-13-2'], ['1611.09321-2-50-0', '1611.09321-3-57-0'], ['1611.09321-2-50-5', '1611.09321-3-57-5'], ['1611.09321-1-37-2', '1611.09321-2-39-0']]

[]

['1611.09321-1-28-0', '1611.09321-1-30-0', '1611.09321-1-33-0', '1611.09321-1-33-1', '1611.09321-1-41-0', '1611.09321-1-46-0', '1611.09321-1-57-4', '1611.09321-1-68-8', '1611.09321-2-27-0', '1611.09321-2-29-0', '1611.09321-2-32-0', '1611.09321-2-32-1', '1611.09321-2-41-0', '1611.09321-2-46-0', '1611.09321-2-57-4', '1611.09321-2-68-8', '1611.09321-3-29-0', '1611.09321-3-31-0', '1611.09321-3-34-0', '1611.09321-3-34-1', '1611.09321-3-38-0', '1611.09321-3-41-0', '1611.09321-3-46-0', '1611.09321-3-48-0', '1611.09321-3-53-0', '1611.09321-3-64-4', '1611.09321-3-75-8']

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

https://arxiv.org/abs/1611.09321

{'1611.09321-3-0-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning (RL) with improved exploration properties.', '1611.09321-3-0-1': 'Current policy-based methods use entropy regularization to encourage undirected exploration of the reward landscape, which is ineffective in high dimensional spaces with sparse rewards.', '1611.09321-3-0-2': 'We propose a more directed exploration strategy that promotes exploration of under-appreciated reward regions.', '1611.09321-3-0-3': 'An action sequence is considered under-appreciated if its log-probability under the current policy under-estimates its resulting reward.', '1611.09321-3-0-4': 'The proposed exploration strategy is easy to implement, requiring small modifications to the REINFORCE algorithm.', '1611.09321-3-0-5': 'We evaluate the approach on a set of algorithmic tasks that have long challenged RL methods.', '1611.09321-3-0-6': 'Our approach reduces hyper-parameter sensitivity and demonstrates significant improvements over baseline methods.', '1611.09321-3-0-7': 'The proposed algorithm successfully solves a benchmark multi-digit addition task and generalizes to long sequences, which, to our knowledge, is the first time that a pure RL method has solved addition using only reward feedback.', '1611.09321-3-1-0': 'This paper presents a novel form of policy gradient for model-free reinforcement learning with improved exploration characteristics.', '1611.09321-3-1-1': 'Current policy gradient methods use entropy regularization to encourage uniform exploration of the reward landscape, which is not effective in high dimensional spaces with sparse rewards.', '1611.09321-3-1-2': 'We propose a better exploration strategy that promotes exploration of the under-appreciated reward regions.', '1611.09321-3-1-3': 'An action sequence is considered under-appreciated if its probability under the current policy under-estimates its resulting reward.', '1611.09321-3-1-4': 'The proposed exploration strategy is easy to implement.', '1611.09321-3-1-5': 'It improves robustness to the change of hyper-parameters, and on a set of algorithmic tasks, it demonstrates significant improvements against the policy gradient baselines.', '1611.09321-3-1-6': 'Our approach is able to solve a benchmark multi-digit addition task.', '1611.09321-3-1-7': 'To our knowledge, this is the first pure RL method to solve addition just by using reward feedback.', '1611.09321-3-2-0': 'To our knowledge Unlike standard on-line learning strategies in reinforcement learning, which generally encourage undirected exploration, we stochastically optimize a modified objective that favors exploration of high-reward versus low-reward actions.', '1611.09321-3-2-1': 'A key feature of the algorithm is its ease of application to recurrent neural network policy representations.', '1611.09321-3-2-2': 'We conduct an experimental evaluation on a set of algorithm induction tasks, ranging from simple to challenging, and compare the proposed enhancement to standard baselines.', '1611.09321-3-2-3': 'In addition to obtaining notable improvements, we also observe that the new approach can solve the benchmark addition problem, which to our knowledge is the first successful such demonstration based solely on success/failure reinforcement.', '1611.09321-3-3-0': '# Introduction', '1611.09321-3-4-0': 'Humans can reason about symbolic objects and solve algorithmic problems.', '1611.09321-3-4-1': 'After learning to count and then manipulate numbers via simple arithmetic, people eventually learn to invent new algorithms and even reason about their correctness and efficiency.', '1611.09321-3-4-2': 'The ability to invent new algorithms is fundamental to artificial intelligence (AI).', '1611.09321-3-4-3': 'Although symbolic reasoning has a long history in AI , only recently have statistical machine learning and neural network approaches begun to make headway in automated algorithm discovery , which would constitute an important milestone on the path to AI.', '1611.09321-3-4-4': 'Nevertheless, most of the recent successes depend on the use of strong supervision to learn a mapping from a set of training inputs to outputs by maximizing a conditional log-likelihood, very much like neural machine translation systems .', '1611.09321-3-4-5': 'Such a dependence on strong supervision is a significant limitation that does not match the ability of people to invent new algorithmic procedures based solely on trial and error.', '1611.09321-3-5-0': 'By contrast, reinforcement learning (RL) methods hold the promise of searching over discrete objects such as symbolic representations of algorithms by considering much weaker feedback in the form of a simple verifier that tests the correctness of a program execution on a given problem instance.', '1611.09321-3-5-1': 'Despite the recent excitement around the use of RL to tackle Atari games and Go , standard RL methods are not yet able to consistently and reliably solve algorithmic tasks in all but the simplest cases .', '1611.09321-3-5-2': 'A key property of algorithmic problems that makes them challenging for RL is reward sparsity, a policy usually has to get a long action sequence exactly right to obtain a non-zero reward.', '1611.09321-3-6-0': 'We believe one of the key factors limiting the effectiveness of current RL methods in a sparse reward setting is the use of undirected exploration strategies , such as [MATH]-greedy and entropy regularization .', '1611.09321-3-6-1': 'For long action sequences with delayed sparse reward, it is hopeless to explore the space uniformly and blindly.', '1611.09321-3-6-2': 'Instead, we propose a formulation to encourage exploration of action sequences that are under-appreciated by the current policy.', '1611.09321-3-6-3': "Our formulation considers an action sequence to be under-appreciated if the model's log-probability assigned to an action sequence under-estimates the resulting reward from the action sequence.", '1611.09321-3-6-4': 'Exploring under-appreciated states and actions encourages the policy to have a better calibration between its log-probabilities and observed reward values, even for action sequences with negligible rewards.', '1611.09321-3-6-5': 'This effectively increases exploration around neglected action sequences.', '1611.09321-3-7-0': 'We term our proposed technique under-appreciated reward exploration ().', '1611.09321-3-7-1': 'We show that the objective given by is a combination of a mode seeking objective (standard REINFORCE) and a mean seeking term, which provides a well motivated trade-off between exploitation and exploration.', '1611.09321-3-7-2': 'To empirically evaluate our method, we take a set of algorithmic tasks such as sequence reversal, multi-digit addition, and binary search.', '1611.09321-3-7-3': 'We choose to focus on these tasks because, although simple, they present a difficult sparse reward setting which has limited the success of standard RL approaches.', '1611.09321-3-7-4': 'The experiments demonstrate that significantly outperforms baseline RL methods, such as entropy regularized REINFORCE and one-step Q-learning, especially on the more difficult tasks, such as multi-digit addition.', '1611.09321-3-7-5': 'Moreover, is shown to be more robust to changes of hyper-parameters, which makes hyper-parameter tuning less tedious in practice.', '1611.09321-3-7-6': 'In addition to introducing a new variant of policy gradient with improved performance, our paper is the first to demonstrate strong results for an RL method on algorithmic tasks.', '1611.09321-3-7-7': 'To our knowledge, the addition task has not been solved by any model-free reinforcement learning approach.', '1611.09321-3-7-8': 'We observe that some of the policies learned by can successfully generalize to long sequences; in [MATH] out of [MATH] random restarts, the policy learned by for the addition task correctly generalizes to addition of numbers with [MATH] digits with no mistakes, even though training sequences are at most [MATH] digits long.', '1611.09321-3-8-0': 'In order to find both non-zero reward action sequences and maximal reward action sequences in a large action space, some sort of exploration is necessary.', '1611.09321-3-8-1': 'In value-based RL this exploration can manifest as [MATH]-greedy actions whereas in policy-based methods a common choice is to add an entropy regularizer to the well-known REINFORCE policy gradient algorithm .', '1611.09321-3-8-2': 'However, these exploration strategies apply exploration blindly and uniformly.', '1611.09321-3-8-3': 'In large action spaces, these strategies can easily distract an agent from its main task of honing in on high-reward regions of the policy space.', '1611.09321-3-9-0': 'In this work, we propose a novel addition to the REINFORCE objective to favor exploration in high-reward areas over exploration in low-reward areas.', '1611.09321-3-10-0': 'We begin in Section [REF], where we provide a short survey of previous work on learning algorithms.', '1611.09321-3-10-1': 'In Section [REF] we provide background information regarding our notation and the derivation of REINFORCE and its entropy-regularized variant.', '1611.09321-3-10-2': 'We introduce our approach and the resulting policy gradient training algorithm in [REF].', '1611.09321-3-10-3': 'Subsequently, we present several standard tasks and introduce an additional task we designed ourselves in Section [REF], on which we then empirically evaluate the performance of our approach.', '1611.09321-3-10-4': 'We conclude in Section [REF], where we summarize the paper and discuss the repercussions and impact we hope it will have on future work.', '1611.09321-3-11-0': '# Neural Networks for Learning Algorithms', '1611.09321-3-12-0': 'Although research on using neural networks to learn algorithms has witnessed a surge of recent interest, the problem of program induction from examples has a long history in many fields, including program induction, inductive logic programming , relational learning and regular language learning .', '1611.09321-3-12-1': 'Rather than presenting a comprehensive survey of program induction here, we focus on neural network approaches to algorithmic tasks and highlight the relative simplicity of our neural network architecture.', '1611.09321-3-13-0': 'Most successful applications of neural networks to algorithmic tasks rely on strong supervision, where the inputs and target outputs are completely known a priori.', '1611.09321-3-13-1': 'Given a dataset of examples, one learns the network parameters by maximizing the conditional likelihood of the outputs via backpropagation ( [CITATION]).', '1611.09321-3-13-2': 'However, target outputs may not be available for novel tasks, for which no prior algorithm is known to be available.', '1611.09321-3-13-3': 'A more desirable approach to inducing algorithms, followed in this paper, advocates using self-driven learning strategies that only receive reinforcement based on the outputs produced.', '1611.09321-3-13-4': 'Hence, just by having access to a verifier for an algorithmic problem, one can aim to learn an algorithm.', '1611.09321-3-13-5': 'For example, if one does not know how to sort an array, but can check the extent to which an array is sorted, then one can provide the reward signal necessary for learning sorting algorithms.', '1611.09321-3-14-0': 'However, providing the correct output during training can often be infeasible; for example, when trying to solve a problem with no known solution, or when efficiency is desired in addition to correctness.', '1611.09321-3-15-0': 'We formulate learning algorithms as an RL problem and make use of model-free policy gradient methods to optimize a set parameters associated with the algorithm.', '1611.09321-3-15-1': 'In this setting, the goal is to learn a policy [MATH] that given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-3-15-2': 'Actions represent the commands within the algorithm and states represent the joint state of the algorithm and the environment.', '1611.09321-3-15-3': 'Previous work in this area has focused on augmenting a neural network with additional structure and increased capabilities .', '1611.09321-3-15-4': 'In contrast, we utilize a simple architecture based on a standard recurrent neural network (RNN) with LSTM cells as depicted in fig:rnn.', '1611.09321-3-15-5': 'At each episode, the environment is initialized with a latent state [MATH], unknown to the agent, which determines [MATH] and the subsequent state transition and reward functions.', '1611.09321-3-15-6': 'Once the agent observes [MATH] as the input to the RNN, the network outputs a distribution [MATH], from which an action [MATH] is sampled.', '1611.09321-3-15-7': 'This action is applied to the environment, and the agent receives a new state observation [MATH].', '1611.09321-3-15-8': 'The state [MATH] and the previous action [MATH] are then fed into the RNN and the process repeats until the end of the episode.', '1611.09321-3-15-9': 'Upon termination, a reward signal is received.', '1611.09321-3-16-0': '# Learning a Policy by Maximizing Expected Reward', '1611.09321-3-17-0': 'We start by discussing the most common form of policy gradient, REINFORCE , and its entropy regularized variant .', '1611.09321-3-17-1': 'REINFORCE has been applied to model-free policy-based learning with neural networks and algorithmic domains .', '1611.09321-3-18-0': 'The goal is to learn a policy [MATH] that, given an observed state [MATH] at step [MATH], estimates a distribution over the next action [MATH], denoted [MATH].', '1611.09321-3-18-1': 'The environment is initialized with a latent vector, [MATH], which determines the initial observed state [MATH], and the transition function [MATH].', '1611.09321-3-18-2': 'Note that the use of nondeterministic transitions [MATH] as in Markov decision processes (MDP) may be recovered by assuming that [MATH] includes the random seed for the any nondeterministic functions.', '1611.09321-3-18-3': 'Given a latent state [MATH], and [MATH], the model probability of an action sequence [MATH] is expressed as, [EQUATION]', '1611.09321-3-18-4': 'The environment provides a reward at the end of the episode, denoted [MATH].', '1611.09321-3-18-5': 'For ease of readability we drop the subscript from [MATH] and simply write [MATH] and [MATH].', '1611.09321-3-19-0': 'The objective used to optimize the policy parameters, [MATH], consists of maximizing expected reward under actions drawn from the policy, plus an optional maximum entropy regularizer.', '1611.09321-3-19-1': 'Given a distribution over initial latent environment states [MATH], we express the regularized expected reward as, [EQUATION]', '1611.09321-3-19-2': 'When [MATH] is a non-linear function defined by a neural network, finding the global optimum of [MATH] is challenging, and one often resorts to gradient-based methods to find a local optimum of [MATH].', '1611.09321-3-19-3': 'Given that [MATH] for any [MATH] such that [MATH], one can verify that, [EQUATION]', '1611.09321-3-19-4': 'Because the space of possible actions [MATH] is large, enumerating over all of the actions to compute this gradient is infeasible.', '1611.09321-3-19-5': 'Williams Williams92 proposed to compute the stochastic gradient of the expected reward by using Monte Carlo samples.', '1611.09321-3-19-6': 'Using Monte Carlo samples, one first draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to approximate the gradient of [REF] by using [REF] as, [EQUATION]', '1611.09321-3-19-7': 'This reparametrization of the gradients is the key to the REINFORCE algorithm.', '1611.09321-3-19-8': 'To reduce the variance of [REF], one uses rewards [MATH] that are shifted by some offset values, [EQUATION] where [MATH] is known as a baseline or sometimes called a critic.', '1611.09321-3-19-9': 'Note that subtracting any offset from the rewards in [REF] simply results in shifting the objective [MATH] by a constant.', '1611.09321-3-19-10': "I can't make [MATH] function of [MATH]", '1611.09321-3-20-0': 'Unfortunately, directly maximizing expected reward ( when [MATH]) is prone to getting trapped in a local optimum.', '1611.09321-3-20-1': 'To combat this tendency, Williams Peng williams1991function augmented the expected reward objective by including a maximum entropy regularizer ([MATH]) to promote greater exploration.', '1611.09321-3-20-2': 'We will refer to this variant of REINFORCE as (maximum entropy exploration).', '1611.09321-3-21-0': '# Under-appreciated Reward Exploration ()', '1611.09321-3-22-0': 'To explain our novel form of policy gradient, we first note that the optimal policy [MATH], which globally maximizes [MATH] in [REF] for any [MATH], can be expressed as, [EQUATION] where [MATH] is a normalization constant making [MATH] a distribution over the space of action sequences [MATH].', '1611.09321-3-22-1': 'One can verify this by first acknowledging that, [EQUATION]', '1611.09321-3-22-2': 'Since [MATH] is non-negative and zero iff [MATH], then [MATH] defined in [REF] maximizes [MATH].', '1611.09321-3-22-3': 'That said, given a particular form of [MATH], finding [MATH] that exactly characterizes [MATH] may not be feasible.', '1611.09321-3-23-0': 'The KL divergence [MATH] is known to be mode seeking [CITATION] even with entropy regularization ([MATH]).', '1611.09321-3-23-1': 'Learning a policy by optimizing this direction of the KL is prone to falling into a local optimum resulting in a sub-optimal policy that omits some of the modes of [MATH].', '1611.09321-3-23-2': 'Although entropy regularization helps mitigate the issues as confirmed in our experiments, it is not an effective exploration strategy as it is undirected and requires a small regularization coefficient [MATH] to avoid too much random exploration.', '1611.09321-3-23-3': 'Instead, we propose a directed exploration strategy that improves the mean seeking behavior of policy gradient in a principled way.', '1611.09321-3-24-0': 'We start by considering the alternate mean seeking direction of the KL divergence, [MATH].', '1611.09321-3-24-1': '[CITATION] considered this direction of the KL to directly learn a policy by optimizing [EQUATION] for structured prediction.', '1611.09321-3-24-2': 'This objective has the same optimal solution [MATH] as [MATH] since, [EQUATION] [CITATION] argue that in some structured prediction problems when one can draw samples from [MATH], optimizing [REF] is more effective than [REF], since no sampling from a non-stationary policy [MATH] is required.', '1611.09321-3-24-3': 'If [MATH] is a log-linear model of a set of features, [MATH] is convex in [MATH] whereas [MATH] is not, even in the log-linear case.', '1611.09321-3-24-4': 'Unfortunately, in scenarios that the reward landscape is unknown or computing the normalization constant [MATH] is intractable, sampling from [MATH] is not straightforward.', '1611.09321-3-25-0': 'In RL problems, the reward landscape is completely unknown, hence sampling from [MATH] is intractable.', '1611.09321-3-26-0': 'One remedy to this is to alter the objective to make it more amenable to sampling using [MATH].', '1611.09321-3-26-1': 'This is evident in the objective given by Reward-Weighted Regression (RWR) , which has been used to tackle RL problems in the past: [EQUATION]', '1611.09321-3-26-2': 'In contrast, this paper leaves the form of [REF] intact and instead proposes to', '1611.09321-3-27-0': 'This paper proposes to approximate the expectation with respect to [MATH] by using self-normalized importance sampling , where the proposal distribution is [MATH] and the reference distribution is [MATH].', '1611.09321-3-27-1': 'For importance sampling, one draws [MATH] samples [MATH] from [MATH] and computes a set of normalized importance weights to approximate [MATH] as, [EQUATION] where [MATH] denotes an importance weight defined by, [EQUATION]', '1611.09321-3-27-2': "One can view these importance weights as evaluating the discrepancy between scaled rewards [MATH] and the policy's log-probabilities [MATH].", '1611.09321-3-27-3': 'Among the [MATH] samples, a sample that is least appreciated by the model, has the largest [MATH], receives the largest positive feedback in [REF].', '1611.09321-3-28-0': 'In practice, we have found that just using the importance sampling RAML objective in [REF] does not always yield promising solutions.', '1611.09321-3-28-1': 'Particularly, at the beginning of training, when [MATH] is still far away from [MATH], the variance of importance weights is too large, and the self-normalized importance sampling procedure results in poor approximations.', '1611.09321-3-28-2': 'To stabilize early phases of training and ensure that the model distribution [MATH] achieves large expected reward scores, we combine the expected reward and RAML objectives to benefit from the best of their mode and mean seeking behaviors.', '1611.09321-3-28-3': 'Accordingly, we propose the following objective that we call under-appreciated reward exploration (), [EQUATION] which is the sum of the expected reward and RAML objectives.', '1611.09321-3-28-4': 'In our preliminary experiments, we considered a composite objective of [MATH], but we found that removing the entropy term is beneficial.', '1611.09321-3-28-5': 'Hence, the [MATH] objective does not include entropy regularization.', '1611.09321-3-28-6': 'Accordingly, the optimum policy for [MATH] is no longer [MATH], as it was for [MATH] and [MATH].', '1611.09321-3-28-7': 'Appendix [REF] derives the optimal policy for [MATH] as a function of the optimal policy for [MATH].', '1611.09321-3-28-8': 'We find that the optimal policy of is more sharply concentrated on the high reward regions of the action space, which may be an advantage for , but we leave more analysis of this behavior to future work.', '1611.09321-3-29-0': 'as [MATH], where [MATH] normalizes the distribution for each [MATH].', '1611.09321-3-30-0': 'To compute the gradient of [MATH], we use the self-normalized importance sampling estimate outlined in [REF].', '1611.09321-3-30-1': 'We assume that the importance weights are constant and contribute no gradient to [MATH].', '1611.09321-3-30-2': 'To approximate the gradient, one draws [MATH] samples from the latent environment states [MATH], and then draws [MATH] samples [MATH] from [MATH] to obtain [EQUATION]', '1611.09321-3-30-3': 'As with REINFORCE, the rewards are shifted by an offset [MATH].', '1611.09321-3-30-4': 'In this gradient, the model log-probability of a sample action sequence [MATH] is reinforced if the corresponding reward is large, or the corresponding importance weights are large, meaning that the action sequence is under-appreciated.', '1611.09321-3-30-5': 'The normalized importance weights are computed using a softmax operator [MATH].', '1611.09321-3-31-0': 'There are three ways to explain why [MATH] may not converge to [MATH] after training using PG:', '1611.09321-3-32-0': 'Model capacity - The form of the model may not be expressive enough to represent [MATH].', '1611.09321-3-32-1': 'Variance - The space of possible actions [MATH] is very large and the reward is very sparse causing the stochastic gradients defined in [REF] to have a large variance.', '1611.09321-3-32-2': 'Optimization - The parameters may fall into local optima of the objective, or the objective may have a very slow convergence rate.', '1611.09321-3-33-0': 'When using recurrent neural networks for simple tasks, point (1) should not be an issue.', '1611.09321-3-33-1': 'Usually papers blame variance for the failure cases of PG, as it is hard to reason about optimization landscape especially in high-dimensional problems.', '1611.09321-3-33-2': 'Below we discuss a very simple one-dimensional problem where local optima seem to be a problem for the regularized expected reward objective.', '1611.09321-3-33-3': 'find a simple example!', '1611.09321-3-34-0': 'We use recurrent neural networks (RNNs) to model [MATH].', '1611.09321-3-34-1': '[CITATION]), which is formulated as minimization of the following objective,', '1611.09321-3-35-0': 'where [MATH] denotes the reward function, negative edit distance or BLEU score, [MATH] controls the degree of regularization, and [MATH] is the entropy of a distribution [MATH], [MATH].', '1611.09321-3-35-1': 'It is well-known that optimizing [MATH] using SGD is challenging because of the large variance of the gradients.', '1611.09321-3-35-2': 'Below we describe how ML and RL objectives are related, and propose a hybrid between the two that combines their benefits for supervised learning.', '1611.09321-3-36-0': '# Related Work', '1611.09321-3-37-0': 'Before presenting the experimental results, we briefly review some pieces of previous work that closely relate to the approach.', '1611.09321-3-38-0': 'Reward-Weighted Regression.', '1611.09321-3-38-1': 'Both RAML and objectives bear some similarity to a method in continuous control known as Reward-Weighted Regression (RWR) .', '1611.09321-3-38-2': 'Using our notation, the RWR objective is expressed as, [EQUATION]', '1611.09321-3-38-3': 'To optimize [MATH], [CITATION] propose a technique inspired by the EM algorithm to maximize a variational lower bound in [REF] based on a variational distribution [MATH].', '1611.09321-3-38-4': 'The RWR objective can be interpreted as a log of the correlation between [MATH] and [MATH].', '1611.09321-3-38-5': 'By contrast, the RAML and objectives are both based on a KL divergence between [MATH] and [MATH].', '1611.09321-3-39-0': 'To optimize the RWR objective, one formulates the gradient as, [EQUATION] where [MATH] denotes the normalization factor, [MATH].', '1611.09321-3-39-1': 'The expectation with respect to [MATH] on the RHS can be approximated by self-normalized importance sampling, where the proposal distribution is [MATH].', '1611.09321-3-39-2': 'Accordingly, one draws [MATH] Monte Carlo samples [MATH] from [MATH] and formulates the gradient as, [EQUATION] where [MATH].', '1611.09321-3-39-3': 'There is some similarity between [REF] and [REF] in that they both use self-normalized importance sampling, but note the critical difference that [REF] and [REF] estimate the gradients of two different objectives, and hence the importance weights in [REF] do not correct for the sampling distribution [MATH] as opposed to [REF].', '1611.09321-3-40-0': 'Beyond important technical differences, the optimal policy of [MATH] is a one hot distribution with all probability mass concentrated on an action sequence with maximal reward, whereas the optimal policies for RAML and UREX are everywhere nonzero, with the probability of different action sequences being assigned proportionally to their exponentiated reward (with UREX introducing an additional re-scaling; see Appendix [REF]).', '1611.09321-3-40-1': 'Further, the notion of under-appreciated reward exploration evident in [MATH], which is key to performance, is missing in the RWR formulation.', '1611.09321-3-41-0': 'Exploration.', '1611.09321-3-41-1': 'The RL literature contains many different attempts at incorporating exploration that may be compared with our method.', '1611.09321-3-41-2': 'The most common exploration strategy considered in value-based RL is [MATH]-greedy Q-learning, where at each step the agent either takes the best action according to its current value approximation or with probability [MATH] takes an action sampled uniformly at random.', '1611.09321-3-41-3': 'Like entropy regularization, such an approach applies undirected exploration, but it has achieved recent success in game playing environments .', '1611.09321-3-42-0': 'Prominent approaches to improving exploration beyond [MATH]-greedy in value-based or model-based RL have focused on reducing uncertainty by prioritizing exploration toward states and actions where the agent knows the least.', '1611.09321-3-42-1': 'This basic intuition underlies work on counter and recency methods , exploration methods based on uncertainty estimates of values , methods that prioritize learning environment dynamics , and methods that provide an intrinsic motivation or curiosity bonus for exploring unknown states .', '1611.09321-3-43-0': 'In contrast to value-based methods, exploration for policy-based RL methods is often a by-product of the optimization algorithm itself.', '1611.09321-3-43-1': 'Since algorithms like REINFORCE and Thompson sampling choose actions according to a stochastic policy, sub-optimal actions are chosen with some non-zero probability.', '1611.09321-3-43-2': 'The Q-learning algorithm may also be modified to sample an action from the softmax of the Q values rather than the argmax .', '1611.09321-3-44-0': 'Asynchronous training has also been reported to have an exploration effect on both value- and policy-based methods.', '1611.09321-3-44-1': '[CITATION] report that asynchronous training can stabilize training by reducing the bias experienced by a single trainer.', '1611.09321-3-44-2': 'By using multiple separate trainers, an agent is less likely to become trapped at a policy found to be locally optimal only due to local conditions.', '1611.09321-3-44-3': 'In the same spirit, [CITATION] use multiple Q value approximators and sample only one to act for each episode as a way to implicitly incorporate exploration.', '1611.09321-3-45-0': 'By relating the concepts of value and policy in RL, the exploration strategy we propose tries to bridge the discrepancy between the two.', '1611.09321-3-45-1': 'In particular, UREX can be viewed as a hybrid combination of value-based and policy-based exploration strategies that attempts to capture the benefits of each.', '1611.09321-3-46-0': 'Per-step Reward.', '1611.09321-3-46-1': 'Finally, while we restrict ourselves to episodic settings where a reward is associated with an entire episode of states and actions, much work has been done to take advantage of environments that provide per-step rewards.', '1611.09321-3-46-2': 'These include policy-based methods such as actor-critic and value-based approaches based on Q-learning .', '1611.09321-3-46-3': 'Some of these value-based methods have proposed a softening of Q-values which can be interpreted as adding a form of maximum-entropy regularizer .', '1611.09321-3-46-4': 'The episodic total-reward setting that we consider is naturally harder since the credit assignment to individual actions within an episode is unclear.', '1611.09321-3-47-0': 'Although the problem of program induction from examples has been studied in many fields, including program induction, inductive logic programming , relational learning and regular language learning ; we do not attempt a comprehensive survey here.', '1611.09321-3-47-1': 'Instead, we focus on related neural network approaches for solving algorithmic tasks and relevant work that considers exploration in reinforcement learning.', '1611.09321-3-48-0': 'Learning Algorithms.', '1611.09321-3-48-1': 'Previous work on neural network learning algorithms for algorithmic problems falls into two general categories: strongly supervised maximum likelihood training that learns from examples using completely known input-output pairs; and self-driven training via reinforcement learning that learns from acting, where the correct output is not known but a reward signal is available to indicate which outputs are preferable to others.', '1611.09321-3-49-0': 'The former approach imposes more constraints than the latter and therefore has had more success.', '1611.09321-3-49-1': 'While some supervised learning approaches use a simple sequence-to-sequence recurrent neural network to map inputs to outputs , more successful approaches have been based on augmenting the model with additional computational elements that have been modified to allow for differentiation via backpropagation.', '1611.09321-3-49-2': 'For example, [CITATION] have proposed the Neural Turing Machine and the Differentiable Neural Computer (DNC), both of which provide an auxiliary memory with a trainable interface for reading and writing.', '1611.09321-3-49-3': 'Beyond basic sequence manipulations, these models have also successfully learned tasks such as finding the shortest path between specified points.', '1611.09321-3-49-4': 'Others have investigated stack- or queue-based augmentations , which have been successfully applied to simple sequence manipulation tasks, and simple arithmetic problems, such as integer addition.', '1611.09321-3-50-0': 'Other supervised neural network approaches have not augmented the network but rather altered how it manipulates or interacts with the input.', '1611.09321-3-50-1': 'Two prominent examples are the Grid LSTM and the Neural GPU , which have succeeded in solving addition and multiplication problems given variable-length inputs.', '1611.09321-3-50-2': 'A recent supervised approach that does not learn from input-output pairs but rather from full execution traces is given by [CITATION].', '1611.09321-3-50-3': 'By providing much richer training data, the network is able to learn more complex algorithms like sorting.', '1611.09321-3-51-0': 'All of these approaches require strong supervision that provides, at a minimum, the correct output for any given input.', '1611.09321-3-51-1': 'There are many cases where this is undesirable.', '1611.09321-3-51-2': 'For example, in an NP-complete problem such as the Traveling Salesman Problem, the correct output is infeasible to compute for large inputs, while a proposed output is easy to check.', '1611.09321-3-51-3': 'Similarly, in tasks where one cares about the efficiency of the algorithm produced in addition to correctness (for example in binary search we prefer a logarithmic-time algorithm over a linear-time algorithm), simply providing the correct output for a given input does not convey sufficient information.', '1611.09321-3-51-4': 'Yet, providing the model with a reward that indicates the cost of its execution is straightforward.', '1611.09321-3-52-0': 'Therefore, a more general and potentially more desirable approach is to deploy self-driven learning that only receives reinforcement based on its outputs.', '1611.09321-3-52-1': 'Unfortunately, such an approach has experienced less success in practice, since the model receives less information during training, which has also resulted in far less research activity to date.', '1611.09321-3-52-2': 'However, some recent work has considered a reinforcement based approach to algorithmic tasks, including , where a Neural Turing Machine is trained via reinforcement learning.', '1611.09321-3-52-3': 'The resulting model can solve simple sequence manipulation tasks such as deduplicating or reversing a sequence.', '1611.09321-3-52-4': 'Also, previous work on the DNC has also briefly explored training via reinforcement learning, presenting the ability of the DNC to solve a block manipulation game given only reward feedback.', '1611.09321-3-53-0': 'Exploration in Reinforcement Learning.', '1611.09321-3-54-0': '# Six Algorithmic Tasks', '1611.09321-3-55-0': 'We assess the effectiveness of the proposed approach on five algorithmic tasks from the OpenAI Gym , as well as a new binary search problem.', '1611.09321-3-55-1': 'Each task is summarized below, with further details available on the Gym website or in the corresponding open-source code.', '1611.09321-3-55-2': 'In each case, the environment has a hidden tape and a hidden sequence.', '1611.09321-3-55-3': 'The agent observes the sequence via a pointer to a single character, which can be moved by a set of pointer control actions.', '1611.09321-3-55-4': 'Thus an action [MATH] is represented as a tuple [MATH] where [MATH] denotes how to move, [MATH] is a boolean denoting whether to write, and [MATH] is the output symbol to write.', '1611.09321-3-55-5': '[topsep=0em,itemsep=.5em,leftmargin=1.3em,parsep=0em] Copy: The agent should emit a copy of the sequence.', '1611.09321-3-55-6': 'The pointer actions are move left and right.', '1611.09321-3-55-7': 'DuplicatedInput: In the hidden tape, each character is repeated twice.', '1611.09321-3-55-8': 'The agent must deduplicate the sequence and emit every other character.', '1611.09321-3-55-9': 'The pointer actions are move left and right.', '1611.09321-3-55-10': 'RepeatCopy: The agent should emit the hidden sequence once, then emit the sequence in the reverse order, then emit the original sequence again.', '1611.09321-3-55-11': 'The pointer actions are move left and right.', '1611.09321-3-55-12': 'Reverse: The agent should emit the hidden sequence in the reverse order.', '1611.09321-3-55-13': 'As before, the pointer actions are move left and right.', '1611.09321-3-55-14': 'ReversedAddition: The hidden tape is a [MATH] grid of digits representing two numbers in base [MATH] in little-endian order.', '1611.09321-3-55-15': 'The agent must emit the sum of the two numbers, in little-endian order.', '1611.09321-3-55-16': 'The allowed pointer actions are move left, right, up, or down.', '1611.09321-3-56-0': 'The OpenAI Gym provides an additional harder task called ReversedAddition3, which involves adding three numbers.', '1611.09321-3-56-1': 'We omit this task, since none of the methods make much progress on it.', '1611.09321-3-57-0': 'For these tasks, the input sequences encountered during training range from a length of [MATH] to [MATH] characters.', '1611.09321-3-57-1': 'A reward of [MATH] is given for each correct emission.', '1611.09321-3-57-2': 'On an incorrect emission, a small penalty of [MATH] is incurred and the episode is terminated.', '1611.09321-3-57-3': 'The agent is also terminated and penalized with a reward of [MATH] if the episode exceeds a certain number of steps.', '1611.09321-3-57-4': 'For the experiments using and , we associate an episodic sequence of actions with the total reward, defined as the sum of the per-step rewards.', '1611.09321-3-57-5': 'The experiments using Q-learning, on the other hand, used the per-step rewards.', '1611.09321-3-57-6': 'Each of the Gym tasks has a success threshold, which determines the required average reward over [MATH] episodes for the agent to be considered successful.', '1611.09321-3-58-0': 'We also conduct experiments on an additional algorithmic task described below: [topsep=-.2em,itemsep=0em,leftmargin=1.3em,parsep=-.2em]', '1611.09321-3-59-0': 'BinarySearch: Given an integer [MATH], the environment has a hidden array of [MATH] distinct numbers stored in ascending order.', '1611.09321-3-59-1': 'The environment also has a query number [MATH] unknown to the agent that is contained somewhere in the array.', '1611.09321-3-59-2': 'The goal of the agent is to find the query number in the array in a small number of actions.', '1611.09321-3-59-3': 'The environment has three integer registers initialized at [MATH].', '1611.09321-3-59-4': 'At each step, the agent can interact with the environment via the four following actions:*.3em *.5em The agent is terminated when the number of steps exceeds a maximum threshold of [MATH] steps and recieves a reward of [MATH].', '1611.09321-3-59-5': 'If the agent finds [MATH] at step [MATH], it recieves a reward of [MATH].', '1611.09321-3-60-0': 'We set the maximum number of steps to [MATH] to allow the agent to perform a full linear search.', '1611.09321-3-60-1': 'A policy performing full linear search achieves an average reward of [MATH], because [MATH] is chosen uniformly at random from the elements of the array.', '1611.09321-3-60-2': 'A policy employing binary search can find the number [MATH] in at most [MATH] steps.', '1611.09321-3-60-3': 'If [MATH] is selected uniformly at random from the range [MATH], binary search yields an optimal average reward above [MATH].', '1611.09321-3-60-4': 'We set the success threshold for this task to an average reward of [MATH].', '1611.09321-3-61-0': '# Experiments', '1611.09321-3-62-0': 'We compare our policy gradient method using under-appreciated reward exploration () against two main RL baselines: (1) REINFORCE with entropy regularization termed , where the value of [MATH] determines the degree of regularization.', '1611.09321-3-62-1': 'When [MATH], standard REINFORCE is obtained.', '1611.09321-3-62-2': '(2) one-step double Q-learning based on bootstrapping one step future rewards.', '1611.09321-3-63-0': '## Robustness to hyper-parameters', '1611.09321-3-64-0': 'Hyper-parameter tuning is often tedious for RL algorithms.', '1611.09321-3-64-1': 'We found that the proposed method significantly improves robustness to changes in hyper-parameters when compared to .', '1611.09321-3-64-2': 'For our experiments, we perform a careful grid search over a set of hyper-parameters for both and .', '1611.09321-3-64-3': 'For any hyper-parameter setting, we run the and methods [MATH] times with different random restarts.', '1611.09321-3-64-4': 'We explore the following main hyper-parameters:', '1611.09321-3-65-0': 'In all of the experiments, both and are treated exactly the same.', '1611.09321-3-65-1': 'In fact, the change of implementation is just a few lines of code.', '1611.09321-3-65-2': 'Given a value of [MATH], for each task, we run [MATH] training jobs comprising [MATH] learning rates, [MATH] clipping values, and [MATH] random restarts.', '1611.09321-3-65-3': 'We run each algorithm for a maximum number of steps determined based on the difficulty of the task.', '1611.09321-3-65-4': 'The training jobs for Copy, DuplicatedInput, RepeatCopy, Reverse, ReversedAddition, and BinarySearch are run for [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] stochastic gradient steps, respectively.', '1611.09321-3-65-5': 'We find that running a trainer job longer does not result in a better performance.', '1611.09321-3-65-6': 'Our policy network comprises a single LSTM layer with [MATH] nodes.', '1611.09321-3-65-7': 'We use the Adam optimizer for the experiments.', '1611.09321-3-66-0': 'gym-results shows the percentage of [MATH] trials on different hyper-parameters ([MATH], [MATH]) and random restarts which successfully solve each of the algorithmic tasks.', '1611.09321-3-66-1': 'It is clear that is more robust than to changes in hyper-parameters, even though we only report the results of for a single temperature.', '1611.09321-3-66-2': 'See Appendix [REF] for more detailed tables on hyper-parameter robustness.', '1611.09321-3-67-0': '## Results', '1611.09321-3-68-0': 'gym-results2 presents the number of successful attempts (out of [MATH] random restarts) and the expected reward values (averaged over [MATH] trials) for each RL algorithm given the best hyper-parameters.', '1611.09321-3-68-1': 'One-step Q-learning results are also included in the table.', '1611.09321-3-68-2': 'We also present the training curves for and in fig:rewards.', '1611.09321-3-68-3': 'It is clear that outperforms the baselines on these tasks.', '1611.09321-3-68-4': 'On the more difficult tasks, such as Reverse and ReverseAddition, is able to consistently find an appropriate algorithm, but and Q-learning fall behind.', '1611.09321-3-68-5': 'Importantly, for the BinarySearch task, which exhibits many local maxima and necessitates smart exploration, is the only method that can solve it consistently.', '1611.09321-3-68-6': 'The Q-learning baseline solves some of the simple tasks, but it makes little headway on the harder tasks.', '1611.09321-3-68-7': 'We believe that entropy regularization for policy gradient and [MATH]-greedy for Q-learning are relatively weak exploration strategies in long episodic tasks with delayed rewards.', '1611.09321-3-68-8': 'On such tasks, one random exploratory step in the wrong direction can take the agent off the optimal policy, hampering its ability to learn.', '1611.09321-3-68-9': 'In contrast, provides a form of adaptive and smart exploration.', '1611.09321-3-68-10': 'In fact, we observe that the variance of the importance weights decreases as the agent approaches the optimal policy, effectively reducing exploration when it is no longer necessary; see Appendix [REF].', '1611.09321-3-69-0': '## Generalization to longer sequences', '1611.09321-3-70-0': 'To confirm whether our method is able to find the correct algorithm for multi-digit addition, we investigate its generalization to longer input sequences than provided during training.', '1611.09321-3-70-1': 'We evaluate the trained models on inputs up to a length of [MATH] digits, even though training sequences were at most [MATH] characters.', '1611.09321-3-70-2': 'For each length, we test the model on [MATH] randomly generated inputs, stopping when the accuracy falls below [MATH].', '1611.09321-3-70-3': 'Out of the [MATH] models trained on addition with , we find that [MATH] models generalize to numbers up to [MATH] digits without any observed mistakes.', '1611.09321-3-70-4': 'On the best hyper-parameters, [MATH] out of the [MATH] random restarts are able to generalize successfully.', '1611.09321-3-70-5': 'For more detailed results on the generalization performance on [MATH] different tasks including Copy, DuplicatedInput, and ReversedAddition, see Appendix [REF].', '1611.09321-3-70-6': 'During these evaluations, we take the action with largest probability from [MATH] at each time step rather than sampling randomly.', '1611.09321-3-71-0': 'We also looked into the generalization of the models trained on the BinarySearch task.', '1611.09321-3-71-1': 'We found that none of the agents perform proper binary search.', '1611.09321-3-71-2': 'Rather, those that solved the task perform a hybrid of binary and linear search: first actions follow a binary search pattern, but then the agent switches to a linear search procedure once it narrows down the search space; see Appendix [REF] for some execution traces for BinarySearch and ReversedAddition.', '1611.09321-3-71-3': "Thus, on longer input sequences, the agent's running time complexity approaches linear rather than logarithmic.", '1611.09321-3-71-4': 'We hope that future work will make more progress on this task.', '1611.09321-3-71-5': 'This task is especially interesting because the reward signal should incorporate both correctness and efficiency of the algorithm.', '1611.09321-3-72-0': 'This task is interesting because there exist many local maxima in the policy space.', '1611.09321-3-72-1': 'Even between a full linear search and an efficient binary search there are many possible policies that solve the task.', '1611.09321-3-72-2': 'As we will see later, none of our methods are able to find the intended efficient binary search algorithm, but rather at best find one of these policies in between linear and binary search.', '1611.09321-3-72-3': 'We hope that future work in this area can make more headway on this difficult task.', '1611.09321-3-73-0': '## Implementation details', '1611.09321-3-74-0': 'In all of the experiments, we make use of curriculum learning.', '1611.09321-3-74-1': 'The environment begins by only providing small inputs and moves on to longer sequences once the agent achieves close to maximal reward over a number of steps.', '1611.09321-3-74-2': 'For policy gradient methods including and , we only provide the agent with a reward at the end of the episode, and there is no notion of intermediate reward.', '1611.09321-3-74-3': 'For the value-based baseline, we implement one-step Q-learning as described in [CITATION]-Alg. [MATH], employing double Q-learning with [MATH]-greedy exploration.', '1611.09321-3-74-4': 'We use the same RNN in our policy-based approaches to estimate the Q values.', '1611.09321-3-74-5': 'A grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) is conducted to find the best hyper-parameters.', '1611.09321-3-74-6': 'Unlike our other methods, the Q-learning baseline uses intermediate rewards, as given by the OpenAI Gym on a per-step basis.', '1611.09321-3-74-7': 'Hence, the Q-learning baseline has a slight advantage over the policy gradient methods.', '1611.09321-3-75-0': 'In all of the tasks except Copy, our stochastic optimizer uses mini-batches comprising [MATH] policy samples from the model.', '1611.09321-3-75-1': 'These [MATH] samples correspond to [MATH] different random sequences drawn from the environment, and [MATH] random policy trajectories per sequence.', '1611.09321-3-75-2': 'In other words, we set [MATH] and [MATH] as defined in [REF] and [REF].', '1611.09321-3-75-3': 'For , we use the [MATH] samples to subtract the mean of the coefficient of [MATH] which includes the contribution of the reward and entropy regularization.', '1611.09321-3-75-4': 'For , we use the [MATH] trajectories to subtract the mean reward and normalize the importance sampling weights.', '1611.09321-3-75-5': 'We do not subtract the mean of the normalized importance weights.', '1611.09321-3-75-6': 'For the Copy task, we use mini-batches with [MATH] samples using [MATH] and [MATH].', '1611.09321-3-75-7': 'Experiments are conducted using Tensorflow .', '1611.09321-3-75-8': ', which we found to perform better.', '1611.09321-3-76-0': 'For our experiments, we aimed to realize two things: first, if can give more consistent behavior across different hyper-parameters; second, if can solve tasks that cannot solve at all.', '1611.09321-3-76-1': 'To this end, we ran several runs of and on each task.', '1611.09321-3-76-2': 'For each task we trained a number of models on different values of [MATH] and different hyper-parameters.', '1611.09321-3-76-3': 'For , we used [MATH], of which we generally found at least one to work well (a broader search did not yield stronger results for other values of [MATH]).', '1611.09321-3-76-4': 'For , we used [MATH], which we found to work well on all the tasks.', '1611.09321-3-77-0': '(a broader search did not yield stronger results for other values of [MATH])', '1611.09321-3-78-0': 'For each training algorithm and value of [MATH], we ran 60 training runs, the result of an intersection of three values of learning rate ([MATH]), four values of clipping value for the gradient ([MATH]), and five random initialization seeds.', '1611.09321-3-78-1': 'With the exception of Copy, each run was trained using batches consisting of 10 trajectories each from 40 different resets of the environment.', '1611.09321-3-78-2': 'Thus, a single batch consisted of 400 total policy trajectories.', '1611.09321-3-78-3': 'For , we use the 10 trajectories to mean-center the coefficient of [MATH].', '1611.09321-3-78-4': 'For , we use the 10 trajectories to both mean-center the rewards and self-normalize the importance sampling weights.', '1611.09321-3-78-5': 'For the Copy task, we used batches of 10 trajectories each from 20 different resets of the environment, which we found to perform better.', '1611.09321-3-78-6': 'The runs for Copy, RepeatCopy, DuplicatedInput, Reverse, ReversedAddition, and BinarySearch were trained for 2000, 50000, 500, 5000, 50000, and 2000 steps, respectively.', '1611.09321-3-78-7': 'We found these numbers of steps were mostly enough for the algorithms to converge for each respective task.', '1611.09321-3-79-0': 'The model architecture we used was an LSTM recurrent neural network with hidden dimension of 128.', '1611.09321-3-79-1': 'We train using the Adam optimizer which we chose for its robustness to hyper-parameters .', '1611.09321-3-80-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-3-80-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-3-80-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-3-80-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-3-80-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-3-80-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-3-81-0': '## Experimental details', '1611.09321-3-82-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-3-82-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-3-83-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-3-83-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-3-83-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-3-83-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-3-83-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-3-83-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-3-84-0': 'We find that not only does solve the considered tasks given the best hyper-parameters, but also that the gradient is more robust to the choice of the hyper-parameters, when compared to the expected reward objective.', '1611.09321-3-85-0': '## Baseline methods', '1611.09321-3-86-0': 'We compare to vanilla REINFORCE ([MATH]) as well as ([MATH]).', '1611.09321-3-86-1': 'Just as for , we only provide an agent with a reward at the end of an episode (so there is no notion of per-step rewards).', '1611.09321-3-87-0': 'In addition to these policy gradient methods, we also evaluated the performance of value-based reinforcement learning.', '1611.09321-3-87-1': 'We implemented one-step Q-learning as described in [CITATION] and mentioned in Section [REF].', '1611.09321-3-87-2': 'This training algorithm employs double Q-learning with [MATH]-greedy exploration.', '1611.09321-3-87-3': 'We use the same recurrent neural network in our policy-based methods to act as the Q-value approximator.', '1611.09321-3-87-4': 'We performed a grid search over exploration rate, exploration rate decay, learning rate, and sync frequency (between online and target network) to find the best configuration reported in the results.', '1611.09321-3-87-5': 'Unlike our other methods, we allow Q-learning to make use of per-step rewards when they are given (the tasks provided by Gym give per-step rewards), thus giving this baseline a slight advantage.', '1611.09321-3-88-0': '# Conclusion', '1611.09321-3-89-0': 'We present a variant of policy gradient, called , which promotes the exploration of action sequences that yield rewards larger than what the model expects.', '1611.09321-3-89-1': 'This exploration strategy is the result of importance sampling from the optimal policy.', '1611.09321-3-89-2': 'Our experimental results demonstrate that significantly outperforms other value and policy based methods, while being more robust to changes of hyper-parameters.', '1611.09321-3-89-3': 'By using , we can solve algorithmic tasks like multi-digit addition from only episodic reward, which other methods cannot reliably solve even given the best hyper-parameters.', '1611.09321-3-89-4': 'We introduce a new algorithmic task based on binary search to advocate more research in this area, especially when the computational complexity of the solution is also of interest.', '1611.09321-3-89-5': 'Solving these tasks is not only important for developing more human-like intelligence in learning algorithms, but also important for generic reinforcement learning, where smart and efficient exploration is the key to successful methods.'}

1310.3933

{'1310.3933-1-0-0': 'This note shows that for each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-1-0-1': 'This provides the conterexamples of Buchstaber-Panov-Ray conjecture.', '1310.3933-1-1-0': '# Introduction', '1310.3933-1-2-0': 'Let [MATH] denote the ring formed by the unitary bordism classes of all unitary manifolds, where a unitary manifold is an oriented closed smooth manifold whose tangent bundle admits a stably complex structure.', '1310.3933-1-2-1': 'In [CITATION], Davis and Januszkiewicz introduced and studied a class of nicely behaved manifolds [MATH], so-called the quasitoric manifolds (as the topological versions of toric varieties), each of which admits a locally standard [MATH]-action such that the orbit space of action is homeomorphic to a simple convex polytope.', '1310.3933-1-2-2': 'Buchstaber and Ray showed in [CITATION] that each quasitoric manifold with an omniorientation always admits a compatible tangential stably complex structure, so omnioriented quasitoric manifolds provide abundant examples of unitary manifolds.', '1310.3933-1-2-3': 'In particular, Buchstaber and Ray also showed in [CITATION] that each class of [MATH] contains an omnioriented quasitoric [MATH]-manifold as its representative.', '1310.3933-1-2-4': 'In [CITATION], Buchstaber, Panov and Ray investigated the property of specially omnioriented quasitoric manifolds, and proved that if [MATH], then each [MATH]-dimensional specially omnioriented quasitoric manifold represents the zero element in [MATH], where the word "specially" for a specially omnioriented quasitoric manifold means that the first Chern class vanishes.', '1310.3933-1-2-5': 'Furthermore, they posed the following conjecture.', '1310.3933-1-3-0': 'Conjecture ([MATH]):Let [MATH] be a specially omnioriented quasitoric manifold.', '1310.3933-1-3-1': 'Then [MATH] represents the zero element in [MATH].', '1310.3933-1-4-0': 'The purpose of this note is to construct some examples of specially omnioriented quasitoric manifolds that are not bordant to zero in [MATH], which give the negative answer to the above conjecture in almost all possible dimensional cases.', '1310.3933-1-4-1': 'Our main result is stated as follows.', '1310.3933-1-5-0': 'For each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-1-6-0': 'Our strategy is related to the unoriented bordism theory.', '1310.3933-1-6-1': "Milnor's work tells us in [CITATION] (see also [CITATION]) that there is an epimorphism [EQUATION] where [MATH] denotes the ring produced by the unoriented bordism classes of all smooth closed manifolds, and [MATH].", '1310.3933-1-6-2': 'This actually implies that there is a covering homomorphism [EQUATION] which is induced by [MATH], where [MATH] is defined by mapping [MATH].', '1310.3933-1-6-3': 'On the other hand, Buchstaber and Ray showed in [CITATION] that each class of [MATH] contains an [MATH]-dimensional small cover as its representative, where a small cover is also introduced by Davis and Januszkiewicz in [CITATION], and it is the real analog of a quasitoric manifold.', '1310.3933-1-6-4': 'In addition, Davis and Januszkiewicz tell us in [CITATION] that each quasitoric manifold [MATH] over a simple convex polytope [MATH] always admits a natural conjugation involution [MATH] whose fixed point set [MATH] is just a small cover over [MATH].', '1310.3933-1-6-5': 'In particular, this conjugation involution [MATH] is independent of the choices of omniorientations on [MATH], and by [CITATION], one has that the mod 2 reductions of all Chern numbers of [MATH] with an omniorientation determine all Stiefel-Whitney numbers of [MATH], and in particular, [MATH] as unoriented bordism classes in [MATH].', '1310.3933-1-6-6': 'Thus, [MATH] induces a homomorphism [MATH], which exactly agrees with the above homomorphism [MATH].', '1310.3933-1-7-0': 'With the above understood, to obtain the counterexamples of Buchstaber-Panov-Ray conjecture, an approach is to construct the examples of specially omnioriented quasitoric manifolds whose images under [MATH] are nonzero in [MATH].', '1310.3933-1-7-1': 'We shall see that Stong manifolds play an important role in our argument.', '1310.3933-1-8-0': 'This note is organized as follows.', '1310.3933-1-8-1': 'We shall review the notions and basic properties of quasitoric manifolds and small covers, and state the related result of Buchstaber-Panov-Ray on specially omnioriented quasitoric manifolds in Section [REF].', '1310.3933-1-8-2': "We shall review the Stong's work on Stong manifolds and construct some nonbounding orientable Stong manifolds in Section [REF].", '1310.3933-1-8-3': 'In addition, we also calculate the characteristic matrices of Stong manifolds therein.', '1310.3933-1-8-4': 'In Section [REF] we shall construct required examples of quasitoric manifolds as special unitary manifolds and complete the proof of our main result.', '1310.3933-1-9-0': '# Quasitoric manifolds and small covers', '1310.3933-1-10-0': 'Davis and Januszkiewicz in [CITATION] introduced and studied two kinds of equivariant manifolds-quasitoric manifolds and small covers, whose geometric and algebraic topology has a strong link to the combinatorics of polytopes.', '1310.3933-1-10-1': 'Following [CITATION], let [EQUATION].', '1310.3933-1-10-2': 'A [MATH]-manifold [MATH] is a smooth closed [MATH]-dimensional [MATH]-manifold admitting a locally standard [MATH]-action such that its orbit space is a simple convex [MATH]-polytope [MATH].', '1310.3933-1-10-3': 'Such a [MATH]-manifold is called a small cover if [MATH] and a quasitoric manifold if [MATH].', '1310.3933-1-11-0': 'For a simple convex polytope [MATH], let [MATH] denote the set of all facets (i.e., [MATH]-dimensional faces) of [MATH].', '1310.3933-1-11-1': 'We know from [CITATION] that each [MATH]-manifold [MATH] determines a characteristic function [MATH] on [MATH] [EQUATION] defined by mapping each facet in [MATH] to nonzero elements of [MATH] such that [MATH] facets meeting at each vertex are mapped to a basis of [MATH].', '1310.3933-1-11-2': 'Conversely, the pair [MATH] can be reconstructed to the [MATH] as follows: first [MATH] gives the following equivalence relation [MATH] on [MATH] [EQUATION] then the quotient space [MATH], denoted by [MATH], is the reconstruction of [MATH], where [MATH] is explained as follows: for each point [MATH], there exists a unique face [MATH] of [MATH] such that [MATH] is in its relative interior.', '1310.3933-1-11-3': 'If [MATH], then there are [MATH] facets, say [MATH], such that [MATH], and furthermore, [MATH] determine a subgroup of rank [MATH] in [MATH], denoted by [MATH].', '1310.3933-1-11-4': 'This reconstruction of [MATH] tells us that the topology of [MATH] can be determined by [MATH].', '1310.3933-1-12-0': 'If we fix an ordering for all facets in [MATH] (e.g., say [MATH]) , then the characteristic function [MATH] uniquely determines a matrix of size [MATH] over [MATH] [EQUATION] with [MATH] as columns, which is called the characteristic matrix of [MATH] or [MATH].', '1310.3933-1-13-0': 'We may see from this reconstruction of [MATH]-manifolds that there is also an essential relation between small covers and quasitoric manifolds over a simple polytope.', '1310.3933-1-13-1': 'In fact, given a quasitoric manifold [MATH] over [MATH], as shown in [CITATION], there is a natural conjugation involution on [MATH] defined by [MATH], which fixes [MATH].', '1310.3933-1-13-2': 'Then this involution descends an involution [MATH] on [MATH] whose fixed point set is exactly a small cover [MATH] over [MATH], where [MATH] is the mod 2 reduction of [MATH].', '1310.3933-1-14-0': 'As shown in [CITATION], an omniorientation of a quasitoric manifold [MATH] is just one choice of orientations of [MATH] and submanifolds [MATH].', '1310.3933-1-14-1': 'Thus, a quasitoric manifold [MATH] has [MATH] omniorientations, where [MATH] is the number of all facets of [MATH].', '1310.3933-1-14-2': 'Clearly, the conjugation involution [MATH] on [MATH] is independent of the choices of omniorientations of [MATH].', '1310.3933-1-14-3': 'Now let [MATH] denote the set of all [MATH] omniorientations.', '1310.3933-1-14-4': 'Buchstaber and Ray showed in [CITATION] (also see [CITATION]) that for each omniorientation [MATH], [MATH] with this omniorientation [MATH] always admits a tangential stably complex structure, so it is a unitary manifold.', '1310.3933-1-14-5': 'In [CITATION], Buchstaber, Panov and Ray gave a characterization for [MATH] with [MATH] to be a special unitary manifold in terms of [MATH], which is stated as follows.', '1310.3933-1-15-0': '# Stong manifolds', '1310.3933-1-16-0': '## Stong manifolds In [CITATION], Stong introduced the Stong manifolds, from which all generators of the unoriented bordism ring [MATH] can be chosen.', '1310.3933-1-16-1': 'A Stong manifold is defined as the real projective space bundle denoted by [MATH] of the bundle [MATH] over [MATH], where [MATH] is the pullback of the canonical bundle over the [MATH]-th factor [MATH].', '1310.3933-1-16-2': 'The Stong manifold [MATH] has dimension [MATH].', '1310.3933-1-16-3': 'As shown in [CITATION], the cohomology with [MATH] coefficients of [MATH] is the free module over the cohomology of [MATH] on [MATH], where [MATH] is the first Stiefel-Whitney class of the canonical line bundle over [MATH], with the relation [EQUATION] where [MATH] is the [MATH]-th Sitefel-Whitney class of [MATH].', '1310.3933-1-16-4': 'Then the total Stiefel-Whitney class of [MATH] is [EQUATION] where [MATH] is the pullback of the nonzero class in [MATH].', '1310.3933-1-17-0': 'In fact, it is easy to see that the total Stiefel-Whitney class of [MATH] is exactly [EQUATION].', '1310.3933-1-17-1': 'So the cohomology with [MATH] coefficients of [MATH] may be written as [EQUATION] where [MATH] is the ideal generated by [MATH], and [MATH].', '1310.3933-1-18-0': 'Stong further showed in [CITATION] that', '1310.3933-1-19-0': 'It is not difficult to see from the expression ([REF]) of the total Stiefel-Whitney class of [MATH] that', '1310.3933-1-20-0': '[MATH] is orientable if and only if [MATH] and all [MATH] are even.', '1310.3933-1-21-0': 'By Proposition [REF] and Corollary [REF], we may choose the following examples of indecomposable, orientable Stong manifolds.', '1310.3933-1-21-1': 'For [MATH], [MATH] and [MATH] are indecomposable and orientable, so they represent nonzero elements in [MATH].', '1310.3933-1-22-0': 'Let [MATH] and [MATH] denote the unoriented bordism classes of [MATH] and [MATH], respectively.', '1310.3933-1-22-1': 'Then we have that', '1310.3933-1-23-0': 'All [MATH] and [MATH] with [MATH] form a polynomial subring [EQUATION] of [MATH], which contains nonzero classes of dimension [MATH].', '1310.3933-1-24-0': '## Characteristic matrices of Stong manifolds', '1310.3933-1-25-0': 'We see that [MATH] is a [MATH]-bundle over [MATH], so it is a special generalized real Bott manifold, and in particular, it is also a small cover over [MATH], where [MATH] denotes a [MATH]-dimensional simplex.', '1310.3933-1-26-0': 'A generalized real Bott manifold of is the total space [MATH] of an iterated fiber bundle: [EQUATION] where each [MATH] is the projectivization of a Whitney sum of [MATH] real line bundles over [MATH].', '1310.3933-1-26-1': 'It is well-known that the generalized real Bott manifold [MATH] is a small cover over [MATH].', '1310.3933-1-26-2': 'Conversely, we also know from [CITATION] that a small cover over a product of simplices is a generalized real Bott manifold.', '1310.3933-1-27-0': 'Now let us look at the characteristic matrix of [MATH] as a small cover over the product [MATH] with [MATH] and [MATH].', '1310.3933-1-27-1': 'Clearly [MATH] has [MATH] facets, which are listed as follows: [EQUATION] and [EQUATION] where [MATH], denote [MATH] facets of [MATH].', '1310.3933-1-28-0': 'Throughout the following, we shall carry out our work on a fixed ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-1-29-0': 'Without the loss of generality, assume that the values of the characteristic function [MATH] on the following [MATH] facets [EQUATION] meeting at a vertex are all columns with an ordering from the first column to the last column in [MATH], respectively.', '1310.3933-1-29-1': 'It suffices to determine the values of [MATH] on the [MATH] facets [MATH].', '1310.3933-1-29-2': 'By [CITATION], we have that for [MATH] [EQUATION] and [EQUATION].', '1310.3933-1-29-3': 'In particular, we also know by [CITATION] that there is at least one [MATH] such that [MATH].', '1310.3933-1-30-0': 'Now by [CITATION], we may write [MATH] as [EQUATION] where the [MATH] are used as indeterminants of degree 1, [MATH] is the Stanley-Reisner ideal generated by [MATH] and [MATH], and [MATH] is the ideal determined by [MATH].', '1310.3933-1-30-1': 'Furthermore, we have by [CITATION] that the total Stiefel-Whitney class of [MATH] is [EQUATION].', '1310.3933-1-30-2': 'Comparing with the formula ([REF]) or by Remark [REF], we see that for each [MATH], [EQUATION] so [MATH].', '1310.3933-1-30-3': 'This implies that [MATH] must be the zero element, and for [MATH], each [MATH] is of the form [EQUATION] in [MATH].', '1310.3933-1-30-4': 'Moreover, one has that [EQUATION]', '1310.3933-1-30-5': 'Comparing with the formula ([REF]) again, one should have that [EQUATION].', '1310.3933-1-30-6': 'Without the loss of generality, assume that [MATH] for [MATH].', '1310.3933-1-30-7': 'Then for [MATH], one has that [MATH], and for [MATH], one has by ([REF]) that [EQUATION] so [MATH] and [MATH] if [MATH] since [MATH] are linearly independent in [MATH].', '1310.3933-1-30-8': 'This completes the proof.', '1310.3933-1-31-0': 'If there is a minimal integer [MATH] with [MATH] such that [MATH] but [MATH] (so [MATH] for [MATH]), then a similar argument as above gives', '1310.3933-1-32-0': '# Proof of Main Result', '1310.3933-1-33-0': '## Examples of specially omnioriented quasitoric manifolds Throughout the following, for a [MATH]-dimensional simplex [MATH], [MATH] mean the [MATH] facets of [MATH], and for a product [MATH] of simplices, [MATH] means that the facet [MATH] of [MATH].', '1310.3933-1-33-1': 'Then let us construct some required examples.', '1310.3933-1-34-0': 'Let [MATH] with [MATH].', '1310.3933-1-34-1': 'Define a characteristic function [MATH] on [MATH] in the following way.', '1310.3933-1-34-2': 'First let us fix an ordering of all facets of [MATH] as follows [EQUATION].', '1310.3933-1-34-3': 'Then we construct the characteristic matrix [MATH] of the required characteristic function [MATH] on the above ordered facets as follows: [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH] and [MATH] denote the same meanings as in Proposition [REF], and [MATH] denotes the matrix of size [MATH] with [MATH]-entries being [MATH] and other entries being [MATH].', '1310.3933-1-34-4': 'We see that the sum of all entries of each column in the characteristic matrix [MATH] is always 1.', '1310.3933-1-34-5': 'Thus, by Proposition [REF], one has that the quasitoric manifold [MATH] with any omniorientation is a special unitary manifold.', '1310.3933-1-35-0': 'Let [MATH] with [MATH].', '1310.3933-1-35-1': 'In a similar way as above, fix an ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-1-35-2': 'Then we define a characteristic function [MATH] on the above ordered facets of [MATH] by the following characteristic matrix [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH], [MATH] and [MATH] denote the same meanings as above.', '1310.3933-1-35-3': 'By Proposition [REF], [MATH] with any omniorientation is a special unitary manifold.', '1310.3933-1-36-0': 'The case in which [MATH].', '1310.3933-1-36-1': 'Consider the polytope [MATH] with the following ordered facets [EQUATION].', '1310.3933-1-36-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by the following characteristic matrix [EQUATION] which gives a special unitary manifold [MATH].', '1310.3933-1-36-3': 'Moreover, by the Davis-Januszkiewicz theory, we may read off the cohomology of [MATH] as follows: [EQUATION] with [MATH], and by [CITATION] and [CITATION], the total Chern class of [MATH] may be written as [EQUATION].', '1310.3933-1-36-4': 'A direct calculation gives the Chern number [MATH], which implies that this specially omnioriented quasitoric manifold [MATH] is not bordant to zero in [MATH].', '1310.3933-1-37-0': 'The case in which [MATH].', '1310.3933-1-37-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION].', '1310.3933-1-37-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by [EQUATION] which also gives a special unitary manifold [MATH].', '1310.3933-1-37-3': 'Similarly, one has the cohomology of [MATH] [EQUATION] with [MATH], and the total Chern class of [MATH] [EQUATION].', '1310.3933-1-37-4': 'Furthermore, one has the Chern number [MATH].', '1310.3933-1-37-5': 'So [MATH] is not bordant to zero in [MATH].', '1310.3933-1-38-0': 'The case in which [MATH].', '1310.3933-1-38-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION] and define a characteristic function [MATH] on the ordered facets of [MATH] by the matrix [EQUATION] from which one obtains a special unitary manifold [MATH] with its cohomology [EQUATION] and with its total Chern class [EQUATION].', '1310.3933-1-38-2': 'Then one has that the 6-th Chern class [MATH], so the Chern number [MATH].', '1310.3933-1-38-3': 'Thus [MATH] is not bordant to zero in [MATH].', '1310.3933-1-39-0': '## Proof of Theorem [REF] Obviously, the mod 2 reductions of the characteristic matrices [MATH] and [MATH] of [MATH] and [MATH] are [EQUATION] and [EQUATION] respectively.', '1310.3933-1-39-1': 'Thus, by Proposition [REF], one has that the fixed point sets of the conjugation involutions on [MATH] and [MATH] are homeomorphic to the Stong manifolds [MATH] and [MATH], respectively.', '1310.3933-1-39-2': 'Thus, the subring of [MATH] generated by the unitary bordism classes of [MATH] and [MATH] is mapped onto the subring [MATH] of [MATH] in Lemma [REF] via [MATH].', '1310.3933-1-39-3': 'Then Theorem [REF] follows from this and Examples [REF]-[REF].', '1310.3933-1-39-4': '[MATH]', '1310.3933-1-40-0': 'We have done many tries to find a counterexample in the case [MATH], but failed.', '1310.3933-1-40-1': 'It seems to be reasonable to the assertion as in the Buchstaber-Panov-Ray conjecture that each 12-dimensional specially omnioriented quasitoric manifold is bordant to zero in [MATH] since each 6-dimensional orientable smooth closed manifold is always bordant to zero in [MATH].'}

{'1310.3933-2-0-0': 'This note shows that for each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-2-0-1': 'This provides the counterexamples of Buchstaber-Panov-Ray conjecture.', '1310.3933-2-1-0': '# Introduction', '1310.3933-2-2-0': 'Let [MATH] denote the ring formed by the unitary bordism classes of all unitary manifolds, where a unitary manifold is an oriented closed smooth manifold whose tangent bundle admits a stably complex structure.', '1310.3933-2-2-1': 'In [CITATION], Davis and Januszkiewicz introduced and studied a class of nicely behaved manifolds [MATH], so-called the quasitoric manifolds (as the topological versions of toric varieties), each of which admits a locally standard [MATH]-action such that the orbit space of action is homeomorphic to a simple convex polytope.', '1310.3933-2-2-2': 'Buchstaber and Ray showed in [CITATION] that each quasitoric manifold with an omniorientation always admits a compatible tangential stably complex structure, so omnioriented quasitoric manifolds provide abundant examples of unitary manifolds.', '1310.3933-2-2-3': 'In particular, Buchstaber and Ray also showed in [CITATION] that each class of [MATH] contains an omnioriented quasitoric [MATH]-manifold as its representative.', '1310.3933-2-2-4': 'In [CITATION], Buchstaber, Panov and Ray investigated the property of specially omnioriented quasitoric manifolds, and proved that if [MATH], then each [MATH]-dimensional specially omnioriented quasitoric manifold represents the zero element in [MATH], where the word "specially" for a specially omnioriented quasitoric manifold means that the first Chern class vanishes.', '1310.3933-2-2-5': 'Furthermore, they posed the following conjecture.', '1310.3933-2-3-0': 'Conjecture ([MATH]):Let [MATH] be a specially omnioriented quasitoric manifold.', '1310.3933-2-3-1': 'Then [MATH] represents the zero element in [MATH].', '1310.3933-2-4-0': 'The purpose of this note is to construct some examples of specially omnioriented quasitoric manifolds that are not bordant to zero in [MATH], which give the negative answer to the above conjecture in almost all possible dimensional cases.', '1310.3933-2-4-1': 'Our main result is stated as follows.', '1310.3933-2-5-0': 'For each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-2-6-0': 'Our strategy is related to the unoriented bordism theory.', '1310.3933-2-6-1': "Milnor's work tells us in [CITATION] (see also [CITATION]) that there is an epimorphism [EQUATION] where [MATH] denotes the ring produced by the unoriented bordism classes of all smooth closed manifolds, and [MATH].", '1310.3933-2-6-2': 'This actually implies that there is a covering homomorphism [EQUATION] which is induced by [MATH], where [MATH] is defined by mapping [MATH].', '1310.3933-2-6-3': 'On the other hand, Buchstaber and Ray showed in [CITATION] that each class of [MATH] contains an [MATH]-dimensional small cover as its representative, where a small cover is also introduced by Davis and Januszkiewicz in [CITATION], and it is the real analogue of a quasitoric manifold.', '1310.3933-2-6-4': 'In addition, Davis and Januszkiewicz tell us in [CITATION] that each quasitoric manifold [MATH] over a simple convex polytope [MATH] always admits a natural conjugation involution [MATH] whose fixed point set [MATH] is just a small cover over [MATH].', '1310.3933-2-6-5': 'In particular, this conjugation involution [MATH] is independent of the choices of omniorientations on [MATH], and by [CITATION], one has that the mod 2 reductions of all Chern numbers of [MATH] with an omniorientation determine all Stiefel-Whitney numbers of [MATH], and in particular, [MATH] as unoriented bordism classes in [MATH].', '1310.3933-2-6-6': 'Thus, [MATH] induces a homomorphism [MATH], which exactly agrees with the above homomorphism [MATH].', '1310.3933-2-7-0': 'With the above understood, to obtain the counterexamples of Buchstaber-Panov-Ray conjecture, an approach is to construct the examples of specially omnioriented quasitoric manifolds whose images under [MATH] are nonzero in [MATH].', '1310.3933-2-7-1': 'We shall see that Stong manifolds play an important role in our argument.', '1310.3933-2-8-0': 'This note is organized as follows.', '1310.3933-2-8-1': 'We shall review the notions and basic properties of quasitoric manifolds and small covers, and state the related result of Buchstaber-Panov-Ray on specially omnioriented quasitoric manifolds in Section [REF].', '1310.3933-2-8-2': "We shall review the Stong's work on Stong manifolds and construct some nonbounding orientable Stong manifolds in Section [REF].", '1310.3933-2-8-3': 'In addition, we also calculate the characteristic matrices of Stong manifolds there.', '1310.3933-2-8-4': 'In Section [REF] we shall construct required examples of omnioriented quasitoric manifolds as special unitary manifolds and complete the proof of our main result.', '1310.3933-2-9-0': '# Quasitoric manifolds and small covers', '1310.3933-2-10-0': 'Davis and Januszkiewicz in [CITATION] introduced and studied two kinds of equivariant manifolds-quasitoric manifolds and small covers, whose geometric and algebraic topology has a strong link to the combinatorics of polytopes.', '1310.3933-2-10-1': 'Following [CITATION], let [EQUATION].', '1310.3933-2-10-2': 'A [MATH]-manifold [MATH] is a smooth closed [MATH]-dimensional [MATH]-manifold admitting a locally standard [MATH]-action such that its orbit space is a simple convex [MATH]-polytope [MATH].', '1310.3933-2-10-3': 'Such a [MATH]-manifold is called a small cover if [MATH] and a quasitoric manifold if [MATH].', '1310.3933-2-11-0': 'For a simple convex polytope [MATH], let [MATH] denote the set of all facets (i.e., [MATH]-dimensional faces) of [MATH].', '1310.3933-2-11-1': 'We know from [CITATION] that each [MATH]-manifold [MATH] determines a characteristic function [MATH] on [MATH] [EQUATION] defined by mapping each facet in [MATH] to nonzero elements of [MATH] such that [MATH] facets meeting at each vertex are mapped to a basis of [MATH].', '1310.3933-2-11-2': 'Conversely, the pair [MATH] can be used to reconstruct [MATH] as follows: first [MATH] gives the following equivalence relation [MATH] on [MATH] [EQUATION] then the quotient space [MATH], denoted by [MATH], is the reconstruction of [MATH], where [MATH] is explained as follows: for each point [MATH], there exists a unique face [MATH] of [MATH] such that [MATH] is in its relative interior.', '1310.3933-2-11-3': 'If [MATH], then there are [MATH] facets, say [MATH], such that [MATH], and furthermore, [MATH] determine a subgroup of rank [MATH] in [MATH], denoted by [MATH].', '1310.3933-2-11-4': 'This reconstruction of [MATH] tells us that the topology of [MATH] can be determined by [MATH].', '1310.3933-2-12-0': 'If we fix an ordering for all facets in [MATH] (e.g., say [MATH]) , then the characteristic function [MATH] uniquely determines a matrix of size [MATH] over [MATH] [EQUATION] with [MATH] as columns, which is called the characteristic matrix of [MATH] or [MATH].', '1310.3933-2-13-0': 'We may see from this reconstruction of [MATH]-manifolds that there is also an essential relation between small covers and quasitoric manifolds over a simple polytope.', '1310.3933-2-13-1': 'In fact, given a quasitoric manifold [MATH] over [MATH], as shown in [CITATION], there is a natural conjugation involution on [MATH] defined by [MATH], which fixes [MATH].', '1310.3933-2-13-2': 'Then this involution descends an involution [MATH] on [MATH] whose fixed point set is exactly a small cover [MATH] over [MATH], where [MATH] is the mod 2 reduction of [MATH].', '1310.3933-2-14-0': 'As shown in [CITATION], an omniorientation of a quasitoric manifold [MATH] is just one choice of orientations of [MATH] and submanifolds [MATH].', '1310.3933-2-14-1': 'Thus, a quasitoric manifold [MATH] has [MATH] omniorientations, where [MATH] is the number of all facets of [MATH].', '1310.3933-2-14-2': 'Clearly, the conjugation involution [MATH] on [MATH] is independent of the choices of omniorientations of [MATH].', '1310.3933-2-14-3': 'Now let [MATH] denote the set of all [MATH] omniorientations.', '1310.3933-2-14-4': 'Buchstaber and Ray showed in [CITATION] (also see [CITATION]) that for each omniorientation [MATH], [MATH] with this omniorientation [MATH] always admits a tangential stably complex structure, so it is a unitary manifold.', '1310.3933-2-14-5': 'In [CITATION], Buchstaber, Panov and Ray gave a characterization for [MATH] with [MATH] to be a special unitary manifold in terms of [MATH], which is stated as follows.', '1310.3933-2-15-0': '# Stong manifolds', '1310.3933-2-16-0': '## Stong manifolds In [CITATION], Stong introduced the Stong manifolds, from which all generators of the unoriented bordism ring [MATH] can be chosen.', '1310.3933-2-16-1': 'A Stong manifold is defined as the real projective space bundle denoted by [MATH] of the bundle [MATH] over [MATH], where [MATH] is the pullback of the canonical bundle over the [MATH]-th factor [MATH].', '1310.3933-2-16-2': 'The Stong manifold [MATH] has dimension [MATH].', '1310.3933-2-16-3': 'As shown in [CITATION], the cohomology with [MATH] coefficients of [MATH] is the free module over the cohomology of [MATH] on [MATH], where [MATH] is the first Stiefel-Whitney class of the canonical line bundle over [MATH], with the relation [EQUATION] where [MATH] is the [MATH]-th Sitefel-Whitney class of [MATH].', '1310.3933-2-16-4': 'Then the total Stiefel-Whitney class of [MATH] is [EQUATION] where [MATH] is the pullback of the nonzero class in [MATH].', '1310.3933-2-17-0': 'In fact, it is easy to see that the total Stiefel-Whitney class of [MATH] is exactly [EQUATION].', '1310.3933-2-17-1': 'So the cohomology with [MATH] coefficients of [MATH] may be written as [EQUATION] where [MATH] is the ideal generated by [MATH], and [MATH].', '1310.3933-2-18-0': 'Stong further showed in [CITATION] that', '1310.3933-2-19-0': 'It is not difficult to see from the expression ([REF]) of the total Stiefel-Whitney class of [MATH] that', '1310.3933-2-20-0': '[MATH] is orientable if and only if [MATH] and all [MATH] are even.', '1310.3933-2-21-0': 'By Proposition [REF] and Corollary [REF], we may choose the following examples of indecomposable, orientable Stong manifolds.', '1310.3933-2-21-1': 'For [MATH], [MATH] and [MATH] are indecomposable and orientable, so they represent nonzero elements in [MATH].', '1310.3933-2-22-0': 'Let [MATH] and [MATH] denote the unoriented bordism classes of [MATH] and [MATH], respectively.', '1310.3933-2-22-1': 'Then we have that', '1310.3933-2-23-0': 'All [MATH] and [MATH] with [MATH] form a polynomial subring [EQUATION] of [MATH], which contains nonzero classes of dimension [MATH].', '1310.3933-2-24-0': '## Characteristic matrices of Stong manifolds', '1310.3933-2-25-0': 'We see that [MATH] is a [MATH]-bundle over [MATH], so it is a special generalized real Bott manifold, and in particular, it is also a small cover over [MATH], where [MATH] denotes a [MATH]-dimensional simplex.', '1310.3933-2-26-0': 'A generalized real Bott manifold of is the total space [MATH] of an iterated fiber bundle: [EQUATION] where each [MATH] is the projectivization of a Whitney sum of [MATH] real line bundles over [MATH].', '1310.3933-2-26-1': 'It is well-known that the generalized real Bott manifold [MATH] is a small cover over [MATH].', '1310.3933-2-26-2': 'Conversely, we also know from [CITATION] that a small cover over a product of simplices is a generalized real Bott manifold.', '1310.3933-2-27-0': 'Now let us look at the characteristic matrix of [MATH] as a small cover over the product [MATH] with [MATH] and [MATH].', '1310.3933-2-27-1': 'Clearly [MATH] has [MATH] facets, which are listed as follows: [EQUATION] and [EQUATION] where [MATH], denote [MATH] facets of [MATH].', '1310.3933-2-28-0': 'Throughout the following, we shall carry out our work on a fixed ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-2-29-0': 'Without the loss of generality, assume that the values of the characteristic function [MATH] on the following [MATH] facets [EQUATION] meeting at a vertex are all columns with an ordering from the first column to the last column in [MATH], respectively.', '1310.3933-2-29-1': 'It suffices to determine the values of [MATH] on the [MATH] facets [MATH].', '1310.3933-2-29-2': 'By [CITATION], we have that for [MATH] [EQUATION] and [EQUATION].', '1310.3933-2-29-3': 'In particular, we also know by [CITATION] that there is at least one [MATH] such that [MATH] in [MATH].', '1310.3933-2-30-0': 'Now by [CITATION], we may write [MATH] as [EQUATION] where the [MATH] are used as indeterminants of degree 1, [MATH] is the Stanley-Reisner ideal generated by [MATH] and [MATH], and [MATH] is the ideal determined by [MATH].', '1310.3933-2-30-1': 'Furthermore, we have by [CITATION] that the total Stiefel-Whitney class of [MATH] is [EQUATION].', '1310.3933-2-30-2': 'Comparing with the formula ([REF]) or by Remark [REF], we see that for each [MATH], [EQUATION] so [MATH].', '1310.3933-2-30-3': 'This implies that [MATH] must be the zero element, and for [MATH], each [MATH] is of the form [EQUATION] in [MATH].', '1310.3933-2-30-4': 'Moreover, one has that [EQUATION]', '1310.3933-2-30-5': 'Comparing with the formula ([REF]) again, one should have that [EQUATION].', '1310.3933-2-30-6': 'Without the loss of generality, assume that [MATH] for [MATH].', '1310.3933-2-30-7': 'Then for [MATH], one has that [MATH], and for [MATH], one has by ([REF]) that [EQUATION] so [MATH] and [MATH] if [MATH] since [MATH] are linearly independent in [MATH].', '1310.3933-2-30-8': 'This completes the proof.', '1310.3933-2-31-0': 'If there is a minimal integer [MATH] with [MATH] such that [MATH] but [MATH] (so [MATH] for [MATH]), then a similar argument as above gives', '1310.3933-2-32-0': '# Proof of Main Result', '1310.3933-2-33-0': '## Examples of specially omnioriented quasitoric manifolds Throughout the following, for a [MATH]-dimensional simplex [MATH], [MATH] mean the [MATH] facets of [MATH], and for a product [MATH] of simplices, [MATH] means that the facet [MATH] of [MATH].', '1310.3933-2-33-1': 'Then let us construct some required examples.', '1310.3933-2-34-0': 'Let [MATH] with [MATH].', '1310.3933-2-34-1': 'Define a characteristic function [MATH] on [MATH] in the following way.', '1310.3933-2-34-2': 'First let us fix an ordering of all facets of [MATH] as follows [EQUATION].', '1310.3933-2-34-3': 'Then we construct the characteristic matrix [MATH] of the required characteristic function [MATH] on the above ordered facets as follows: [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH] and [MATH] denote the same meanings as in Proposition [REF], and [MATH] denotes the matrix of size [MATH] with [MATH]-entries being [MATH] and other entries being [MATH].', '1310.3933-2-34-4': 'We see that the sum of all entries of each column in the characteristic matrix [MATH] is always 1.', '1310.3933-2-34-5': 'Thus, by Proposition [REF], one has that the quasitoric manifold [MATH] with any omniorientation is a special unitary manifold.', '1310.3933-2-35-0': 'Let [MATH] with [MATH].', '1310.3933-2-35-1': 'In a similar way as above, fix an ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-2-35-2': 'Then we define a characteristic function [MATH] on the above ordered facets of [MATH] by the following characteristic matrix [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH], [MATH] and [MATH] denote the same meanings as above.', '1310.3933-2-35-3': 'By Proposition [REF], [MATH] with any omniorientation is a special unitary manifold.', '1310.3933-2-36-0': 'The case in which [MATH].', '1310.3933-2-36-1': 'Consider the polytope [MATH] with the following ordered facets [EQUATION].', '1310.3933-2-36-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by the following characteristic matrix [EQUATION] which gives a special unitary manifold [MATH].', '1310.3933-2-36-3': 'Moreover, by the Davis-Januszkiewicz theory, we may read off the cohomology of [MATH] as follows: [EQUATION] with [MATH], and by [CITATION] and [CITATION], the total Chern class of [MATH] may be written as [EQUATION].', '1310.3933-2-36-4': 'A direct calculation gives the Chern number [MATH], which implies that this specially omnioriented quasitoric manifold [MATH] is not bordant to zero in [MATH].', '1310.3933-2-37-0': 'The case in which [MATH].', '1310.3933-2-37-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION].', '1310.3933-2-37-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by [EQUATION] which also gives a special unitary manifold [MATH].', '1310.3933-2-37-3': 'Similarly, one has the cohomology of [MATH] [EQUATION] with [MATH], and the total Chern class of [MATH] [EQUATION].', '1310.3933-2-37-4': 'Furthermore, one has the Chern number [MATH].', '1310.3933-2-37-5': 'So [MATH] is not bordant to zero in [MATH].', '1310.3933-2-38-0': 'The case in which [MATH].', '1310.3933-2-38-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION] and define a characteristic function [MATH] on the ordered facets of [MATH] by the matrix [EQUATION] from which one obtains a special unitary manifold [MATH] with its cohomology [EQUATION] and with its total Chern class [EQUATION].', '1310.3933-2-38-2': 'Then one has that the 6-th Chern class [MATH], so the Chern number [MATH].', '1310.3933-2-38-3': 'Thus [MATH] is not bordant to zero in [MATH].', '1310.3933-2-39-0': '## Proof of Theorem [REF] Obviously, the mod 2 reductions of the characteristic matrices [MATH] and [MATH] of [MATH] and [MATH] are [EQUATION] and [EQUATION] respectively.', '1310.3933-2-39-1': 'Thus, by Proposition [REF], one has that the fixed point sets of the conjugation involutions on [MATH] and [MATH] are homeomorphic to the Stong manifolds [MATH] and [MATH], respectively.', '1310.3933-2-39-2': 'Thus, the subring of [MATH] generated by the unitary bordism classes of [MATH] and [MATH] is mapped onto the subring [MATH] of [MATH] in Lemma [REF] via [MATH].', '1310.3933-2-39-3': 'Then Theorem [REF] follows from this and Examples [REF]-[REF].', '1310.3933-2-39-4': '[MATH]', '1310.3933-2-40-0': 'We have done many tries to find a counterexample in the case [MATH], but failed.', '1310.3933-2-40-1': 'It seems to be reasonable to the assertion as in the Buchstaber-Panov-Ray conjecture that each 12-dimensional specially omnioriented quasitoric manifold is bordant to zero in [MATH] since each 6-dimensional orientable smooth closed manifold is always bordant to zero in [MATH].'}

[['1310.3933-1-3-0', '1310.3933-2-3-0'], ['1310.3933-1-3-1', '1310.3933-2-3-1'], ['1310.3933-1-39-1', '1310.3933-2-39-1'], ['1310.3933-1-39-2', '1310.3933-2-39-2'], ['1310.3933-1-39-3', '1310.3933-2-39-3'], ['1310.3933-1-23-0', '1310.3933-2-23-0'], ['1310.3933-1-37-0', '1310.3933-2-37-0'], ['1310.3933-1-37-1', '1310.3933-2-37-1'], ['1310.3933-1-37-2', '1310.3933-2-37-2'], ['1310.3933-1-37-3', '1310.3933-2-37-3'], ['1310.3933-1-37-4', '1310.3933-2-37-4'], ['1310.3933-1-37-5', '1310.3933-2-37-5'], ['1310.3933-1-29-0', '1310.3933-2-29-0'], ['1310.3933-1-29-1', '1310.3933-2-29-1'], ['1310.3933-1-29-2', '1310.3933-2-29-2'], ['1310.3933-1-4-0', '1310.3933-2-4-0'], ['1310.3933-1-4-1', '1310.3933-2-4-1'], ['1310.3933-1-31-0', '1310.3933-2-31-0'], ['1310.3933-1-33-1', '1310.3933-2-33-1'], ['1310.3933-1-26-0', '1310.3933-2-26-0'], ['1310.3933-1-26-1', '1310.3933-2-26-1'], ['1310.3933-1-26-2', '1310.3933-2-26-2'], ['1310.3933-1-7-0', '1310.3933-2-7-0'], ['1310.3933-1-7-1', '1310.3933-2-7-1'], ['1310.3933-1-0-0', '1310.3933-2-0-0'], ['1310.3933-1-6-0', '1310.3933-2-6-0'], ['1310.3933-1-6-1', '1310.3933-2-6-1'], ['1310.3933-1-6-2', '1310.3933-2-6-2'], ['1310.3933-1-6-4', '1310.3933-2-6-4'], ['1310.3933-1-6-5', '1310.3933-2-6-5'], ['1310.3933-1-6-6', '1310.3933-2-6-6'], ['1310.3933-1-2-0', '1310.3933-2-2-0'], ['1310.3933-1-2-1', '1310.3933-2-2-1'], ['1310.3933-1-2-2', '1310.3933-2-2-2'], ['1310.3933-1-2-3', '1310.3933-2-2-3'], ['1310.3933-1-2-4', '1310.3933-2-2-4'], ['1310.3933-1-2-5', '1310.3933-2-2-5'], ['1310.3933-1-25-0', '1310.3933-2-25-0'], ['1310.3933-1-27-0', '1310.3933-2-27-0'], ['1310.3933-1-27-1', '1310.3933-2-27-1'], ['1310.3933-1-34-1', '1310.3933-2-34-1'], ['1310.3933-1-34-2', '1310.3933-2-34-2'], ['1310.3933-1-34-3', '1310.3933-2-34-3'], ['1310.3933-1-34-4', '1310.3933-2-34-4'], ['1310.3933-1-34-5', '1310.3933-2-34-5'], ['1310.3933-1-19-0', '1310.3933-2-19-0'], ['1310.3933-1-28-0', '1310.3933-2-28-0'], ['1310.3933-1-5-0', '1310.3933-2-5-0'], ['1310.3933-1-12-0', '1310.3933-2-12-0'], ['1310.3933-1-17-0', '1310.3933-2-17-0'], ['1310.3933-1-17-1', '1310.3933-2-17-1'], ['1310.3933-1-22-0', '1310.3933-2-22-0'], ['1310.3933-1-22-1', '1310.3933-2-22-1'], ['1310.3933-1-13-0', '1310.3933-2-13-0'], ['1310.3933-1-13-1', '1310.3933-2-13-1'], ['1310.3933-1-13-2', '1310.3933-2-13-2'], ['1310.3933-1-36-0', '1310.3933-2-36-0'], ['1310.3933-1-36-1', '1310.3933-2-36-1'], ['1310.3933-1-36-2', '1310.3933-2-36-2'], ['1310.3933-1-36-3', '1310.3933-2-36-3'], ['1310.3933-1-36-4', '1310.3933-2-36-4'], ['1310.3933-1-40-0', '1310.3933-2-40-0'], ['1310.3933-1-40-1', '1310.3933-2-40-1'], ['1310.3933-1-8-0', '1310.3933-2-8-0'], ['1310.3933-1-8-1', '1310.3933-2-8-1'], ['1310.3933-1-8-2', '1310.3933-2-8-2'], ['1310.3933-1-38-0', '1310.3933-2-38-0'], ['1310.3933-1-38-1', '1310.3933-2-38-1'], ['1310.3933-1-38-2', '1310.3933-2-38-2'], ['1310.3933-1-38-3', '1310.3933-2-38-3'], ['1310.3933-1-14-0', '1310.3933-2-14-0'], ['1310.3933-1-14-1', '1310.3933-2-14-1'], ['1310.3933-1-14-2', '1310.3933-2-14-2'], ['1310.3933-1-14-3', '1310.3933-2-14-3'], ['1310.3933-1-14-4', '1310.3933-2-14-4'], ['1310.3933-1-14-5', '1310.3933-2-14-5'], ['1310.3933-1-21-0', '1310.3933-2-21-0'], ['1310.3933-1-21-1', '1310.3933-2-21-1'], ['1310.3933-1-11-0', '1310.3933-2-11-0'], ['1310.3933-1-11-1', '1310.3933-2-11-1'], ['1310.3933-1-11-3', '1310.3933-2-11-3'], ['1310.3933-1-11-4', '1310.3933-2-11-4'], ['1310.3933-1-10-0', '1310.3933-2-10-0'], ['1310.3933-1-10-1', '1310.3933-2-10-1'], ['1310.3933-1-10-2', '1310.3933-2-10-2'], ['1310.3933-1-10-3', '1310.3933-2-10-3'], ['1310.3933-1-35-1', '1310.3933-2-35-1'], ['1310.3933-1-35-2', '1310.3933-2-35-2'], ['1310.3933-1-35-3', '1310.3933-2-35-3'], ['1310.3933-1-20-0', '1310.3933-2-20-0'], ['1310.3933-1-30-0', '1310.3933-2-30-0'], ['1310.3933-1-30-1', '1310.3933-2-30-1'], ['1310.3933-1-30-2', '1310.3933-2-30-2'], ['1310.3933-1-30-3', '1310.3933-2-30-3'], ['1310.3933-1-30-4', '1310.3933-2-30-4'], ['1310.3933-1-30-5', '1310.3933-2-30-5'], ['1310.3933-1-30-6', '1310.3933-2-30-6'], ['1310.3933-1-30-7', '1310.3933-2-30-7'], ['1310.3933-1-30-8', '1310.3933-2-30-8'], ['1310.3933-1-16-1', '1310.3933-2-16-1'], ['1310.3933-1-16-2', '1310.3933-2-16-2'], ['1310.3933-1-16-3', '1310.3933-2-16-3'], ['1310.3933-1-16-4', '1310.3933-2-16-4'], ['1310.3933-2-13-0', '1310.3933-3-13-0'], ['1310.3933-2-13-1', '1310.3933-3-13-1'], ['1310.3933-2-13-2', '1310.3933-3-13-2'], ['1310.3933-2-34-1', '1310.3933-3-36-1'], ['1310.3933-2-34-2', '1310.3933-3-36-2'], ['1310.3933-2-34-4', '1310.3933-3-36-4'], ['1310.3933-2-35-1', '1310.3933-3-37-1'], ['1310.3933-2-35-2', '1310.3933-3-37-2'], ['1310.3933-2-17-0', '1310.3933-3-17-0'], ['1310.3933-2-17-1', '1310.3933-3-17-1'], ['1310.3933-2-2-0', '1310.3933-3-2-0'], ['1310.3933-2-2-5', '1310.3933-3-2-5'], ['1310.3933-2-23-0', '1310.3933-3-24-0'], ['1310.3933-2-14-1', '1310.3933-3-14-1'], ['1310.3933-2-14-2', '1310.3933-3-14-2'], ['1310.3933-2-14-3', '1310.3933-3-14-3'], ['1310.3933-2-14-5', '1310.3933-3-14-5'], ['1310.3933-2-26-1', '1310.3933-3-28-1'], ['1310.3933-2-26-2', '1310.3933-3-28-2'], ['1310.3933-2-36-0', '1310.3933-3-38-0'], ['1310.3933-2-36-1', '1310.3933-3-38-1'], ['1310.3933-2-36-2', '1310.3933-3-38-2'], ['1310.3933-2-36-3', '1310.3933-3-38-3'], ['1310.3933-2-36-4', '1310.3933-3-38-4'], ['1310.3933-2-19-0', '1310.3933-3-20-0'], ['1310.3933-2-12-0', '1310.3933-3-12-0'], ['1310.3933-2-5-0', '1310.3933-3-5-0'], ['1310.3933-2-38-0', '1310.3933-3-40-0'], ['1310.3933-2-38-1', '1310.3933-3-40-1'], ['1310.3933-2-38-2', '1310.3933-3-40-2'], ['1310.3933-2-38-3', '1310.3933-3-40-3'], ['1310.3933-2-10-0', '1310.3933-3-10-0'], ['1310.3933-2-10-1', '1310.3933-3-10-1'], ['1310.3933-2-10-2', '1310.3933-3-10-2'], ['1310.3933-2-10-3', '1310.3933-3-10-3'], ['1310.3933-2-30-0', '1310.3933-3-32-0'], ['1310.3933-2-30-1', '1310.3933-3-32-1'], ['1310.3933-2-30-2', '1310.3933-3-32-2'], ['1310.3933-2-30-3', '1310.3933-3-32-4'], ['1310.3933-2-30-4', '1310.3933-3-32-5'], ['1310.3933-2-30-5', '1310.3933-3-32-6'], ['1310.3933-2-30-6', '1310.3933-3-32-7'], ['1310.3933-2-30-7', '1310.3933-3-32-8'], ['1310.3933-2-30-8', '1310.3933-3-32-9'], ['1310.3933-2-33-1', '1310.3933-3-35-1'], ['1310.3933-2-3-0', '1310.3933-3-3-0'], ['1310.3933-2-3-1', '1310.3933-3-3-1'], ['1310.3933-2-37-0', '1310.3933-3-39-0'], ['1310.3933-2-37-1', '1310.3933-3-39-1'], ['1310.3933-2-37-2', '1310.3933-3-39-2'], ['1310.3933-2-37-3', '1310.3933-3-39-3'], ['1310.3933-2-37-4', '1310.3933-3-39-4'], ['1310.3933-2-37-5', '1310.3933-3-39-5'], ['1310.3933-2-16-1', '1310.3933-3-16-1'], ['1310.3933-2-16-2', '1310.3933-3-16-2'], ['1310.3933-2-16-4', '1310.3933-3-16-4'], ['1310.3933-2-27-0', '1310.3933-3-29-0'], ['1310.3933-2-27-1', '1310.3933-3-29-1'], ['1310.3933-2-8-0', '1310.3933-3-8-0'], ['1310.3933-2-8-1', '1310.3933-3-8-1'], ['1310.3933-2-8-2', '1310.3933-3-8-2'], ['1310.3933-2-8-3', '1310.3933-3-8-3'], ['1310.3933-2-8-4', '1310.3933-3-8-4'], ['1310.3933-2-39-1', '1310.3933-3-41-1'], ['1310.3933-2-21-0', '1310.3933-3-22-0'], ['1310.3933-2-21-1', '1310.3933-3-22-1'], ['1310.3933-2-11-0', '1310.3933-3-11-0'], ['1310.3933-2-11-1', '1310.3933-3-11-1'], ['1310.3933-2-11-2', '1310.3933-3-11-2'], ['1310.3933-2-11-3', '1310.3933-3-11-3'], ['1310.3933-2-11-4', '1310.3933-3-11-4'], ['1310.3933-2-22-0', '1310.3933-3-23-0'], ['1310.3933-2-22-1', '1310.3933-3-23-1'], ['1310.3933-2-6-0', '1310.3933-3-6-0'], ['1310.3933-2-6-2', '1310.3933-3-6-2'], ['1310.3933-2-6-4', '1310.3933-3-6-5'], ['1310.3933-2-6-5', '1310.3933-3-6-6'], ['1310.3933-2-6-6', '1310.3933-3-6-7'], ['1310.3933-2-29-0', '1310.3933-3-31-0'], ['1310.3933-2-29-1', '1310.3933-3-31-1'], ['1310.3933-2-29-3', '1310.3933-3-31-3'], ['1310.3933-2-0-0', '1310.3933-3-0-0'], ['1310.3933-2-0-1', '1310.3933-3-0-1'], ['1310.3933-2-7-0', '1310.3933-3-7-0'], ['1310.3933-2-7-1', '1310.3933-3-7-1'], ['1310.3933-2-31-0', '1310.3933-3-33-0'], ['1310.3933-2-4-0', '1310.3933-3-4-0'], ['1310.3933-2-4-1', '1310.3933-3-4-1'], ['1310.3933-2-28-0', '1310.3933-3-30-0'], ['1310.3933-1-29-3', '1310.3933-2-29-3'], ['1310.3933-1-0-1', '1310.3933-2-0-1'], ['1310.3933-1-6-3', '1310.3933-2-6-3'], ['1310.3933-1-8-3', '1310.3933-2-8-3'], ['1310.3933-1-8-4', '1310.3933-2-8-4'], ['1310.3933-1-11-2', '1310.3933-2-11-2'], ['1310.3933-2-34-3', '1310.3933-3-36-3'], ['1310.3933-2-34-5', '1310.3933-3-36-5'], ['1310.3933-2-35-3', '1310.3933-3-37-3'], ['1310.3933-2-2-1', '1310.3933-3-2-1'], ['1310.3933-2-2-2', '1310.3933-3-2-2'], ['1310.3933-2-2-4', '1310.3933-3-2-4'], ['1310.3933-2-14-0', '1310.3933-3-14-0'], ['1310.3933-2-14-4', '1310.3933-3-14-4'], ['1310.3933-2-26-0', '1310.3933-3-28-0'], ['1310.3933-2-20-0', '1310.3933-3-21-0'], ['1310.3933-2-25-0', '1310.3933-3-27-0'], ['1310.3933-2-16-3', '1310.3933-3-16-3'], ['1310.3933-2-39-2', '1310.3933-3-41-2'], ['1310.3933-2-6-1', '1310.3933-3-6-1'], ['1310.3933-2-6-3', '1310.3933-3-6-4'], ['1310.3933-2-2-3', '1310.3933-3-2-3'], ['1310.3933-2-29-2', '1310.3933-3-31-2']]

[['1310.3933-1-3-0', '1310.3933-2-3-0'], ['1310.3933-1-3-1', '1310.3933-2-3-1'], ['1310.3933-1-39-1', '1310.3933-2-39-1'], ['1310.3933-1-39-2', '1310.3933-2-39-2'], ['1310.3933-1-39-3', '1310.3933-2-39-3'], ['1310.3933-1-23-0', '1310.3933-2-23-0'], ['1310.3933-1-37-0', '1310.3933-2-37-0'], ['1310.3933-1-37-1', '1310.3933-2-37-1'], ['1310.3933-1-37-2', '1310.3933-2-37-2'], ['1310.3933-1-37-3', '1310.3933-2-37-3'], ['1310.3933-1-37-4', '1310.3933-2-37-4'], ['1310.3933-1-37-5', '1310.3933-2-37-5'], ['1310.3933-1-29-0', '1310.3933-2-29-0'], ['1310.3933-1-29-1', '1310.3933-2-29-1'], ['1310.3933-1-29-2', '1310.3933-2-29-2'], ['1310.3933-1-4-0', '1310.3933-2-4-0'], ['1310.3933-1-4-1', '1310.3933-2-4-1'], ['1310.3933-1-31-0', '1310.3933-2-31-0'], ['1310.3933-1-33-1', '1310.3933-2-33-1'], ['1310.3933-1-26-0', '1310.3933-2-26-0'], ['1310.3933-1-26-1', '1310.3933-2-26-1'], ['1310.3933-1-26-2', '1310.3933-2-26-2'], ['1310.3933-1-7-0', '1310.3933-2-7-0'], ['1310.3933-1-7-1', '1310.3933-2-7-1'], ['1310.3933-1-0-0', '1310.3933-2-0-0'], ['1310.3933-1-6-0', '1310.3933-2-6-0'], ['1310.3933-1-6-1', '1310.3933-2-6-1'], ['1310.3933-1-6-2', '1310.3933-2-6-2'], ['1310.3933-1-6-4', '1310.3933-2-6-4'], ['1310.3933-1-6-5', '1310.3933-2-6-5'], ['1310.3933-1-6-6', '1310.3933-2-6-6'], ['1310.3933-1-2-0', '1310.3933-2-2-0'], ['1310.3933-1-2-1', '1310.3933-2-2-1'], ['1310.3933-1-2-2', '1310.3933-2-2-2'], ['1310.3933-1-2-3', '1310.3933-2-2-3'], ['1310.3933-1-2-4', '1310.3933-2-2-4'], ['1310.3933-1-2-5', '1310.3933-2-2-5'], ['1310.3933-1-25-0', '1310.3933-2-25-0'], ['1310.3933-1-27-0', '1310.3933-2-27-0'], ['1310.3933-1-27-1', '1310.3933-2-27-1'], ['1310.3933-1-34-1', '1310.3933-2-34-1'], ['1310.3933-1-34-2', '1310.3933-2-34-2'], ['1310.3933-1-34-3', '1310.3933-2-34-3'], ['1310.3933-1-34-4', '1310.3933-2-34-4'], ['1310.3933-1-34-5', '1310.3933-2-34-5'], ['1310.3933-1-19-0', '1310.3933-2-19-0'], ['1310.3933-1-28-0', '1310.3933-2-28-0'], ['1310.3933-1-5-0', '1310.3933-2-5-0'], ['1310.3933-1-12-0', '1310.3933-2-12-0'], ['1310.3933-1-17-0', '1310.3933-2-17-0'], ['1310.3933-1-17-1', '1310.3933-2-17-1'], ['1310.3933-1-22-0', '1310.3933-2-22-0'], ['1310.3933-1-22-1', '1310.3933-2-22-1'], ['1310.3933-1-13-0', '1310.3933-2-13-0'], ['1310.3933-1-13-1', '1310.3933-2-13-1'], ['1310.3933-1-13-2', '1310.3933-2-13-2'], ['1310.3933-1-36-0', '1310.3933-2-36-0'], ['1310.3933-1-36-1', '1310.3933-2-36-1'], ['1310.3933-1-36-2', '1310.3933-2-36-2'], ['1310.3933-1-36-3', '1310.3933-2-36-3'], ['1310.3933-1-36-4', '1310.3933-2-36-4'], ['1310.3933-1-40-0', '1310.3933-2-40-0'], ['1310.3933-1-40-1', '1310.3933-2-40-1'], ['1310.3933-1-8-0', '1310.3933-2-8-0'], ['1310.3933-1-8-1', '1310.3933-2-8-1'], ['1310.3933-1-8-2', '1310.3933-2-8-2'], ['1310.3933-1-38-0', '1310.3933-2-38-0'], ['1310.3933-1-38-1', '1310.3933-2-38-1'], ['1310.3933-1-38-2', '1310.3933-2-38-2'], ['1310.3933-1-38-3', '1310.3933-2-38-3'], ['1310.3933-1-14-0', '1310.3933-2-14-0'], ['1310.3933-1-14-1', '1310.3933-2-14-1'], ['1310.3933-1-14-2', '1310.3933-2-14-2'], ['1310.3933-1-14-3', '1310.3933-2-14-3'], ['1310.3933-1-14-4', '1310.3933-2-14-4'], ['1310.3933-1-14-5', '1310.3933-2-14-5'], ['1310.3933-1-21-0', '1310.3933-2-21-0'], ['1310.3933-1-21-1', '1310.3933-2-21-1'], ['1310.3933-1-11-0', '1310.3933-2-11-0'], ['1310.3933-1-11-1', '1310.3933-2-11-1'], ['1310.3933-1-11-3', '1310.3933-2-11-3'], ['1310.3933-1-11-4', '1310.3933-2-11-4'], ['1310.3933-1-10-0', '1310.3933-2-10-0'], ['1310.3933-1-10-1', '1310.3933-2-10-1'], ['1310.3933-1-10-2', '1310.3933-2-10-2'], ['1310.3933-1-10-3', '1310.3933-2-10-3'], ['1310.3933-1-35-1', '1310.3933-2-35-1'], ['1310.3933-1-35-2', '1310.3933-2-35-2'], ['1310.3933-1-35-3', '1310.3933-2-35-3'], ['1310.3933-1-20-0', '1310.3933-2-20-0'], ['1310.3933-1-30-0', '1310.3933-2-30-0'], ['1310.3933-1-30-1', '1310.3933-2-30-1'], ['1310.3933-1-30-2', '1310.3933-2-30-2'], ['1310.3933-1-30-3', '1310.3933-2-30-3'], ['1310.3933-1-30-4', '1310.3933-2-30-4'], ['1310.3933-1-30-5', '1310.3933-2-30-5'], ['1310.3933-1-30-6', '1310.3933-2-30-6'], ['1310.3933-1-30-7', '1310.3933-2-30-7'], ['1310.3933-1-30-8', '1310.3933-2-30-8'], ['1310.3933-1-16-1', '1310.3933-2-16-1'], ['1310.3933-1-16-2', '1310.3933-2-16-2'], ['1310.3933-1-16-3', '1310.3933-2-16-3'], ['1310.3933-1-16-4', '1310.3933-2-16-4'], ['1310.3933-2-13-0', '1310.3933-3-13-0'], ['1310.3933-2-13-1', '1310.3933-3-13-1'], ['1310.3933-2-13-2', '1310.3933-3-13-2'], ['1310.3933-2-34-1', '1310.3933-3-36-1'], ['1310.3933-2-34-2', '1310.3933-3-36-2'], ['1310.3933-2-34-4', '1310.3933-3-36-4'], ['1310.3933-2-35-1', '1310.3933-3-37-1'], ['1310.3933-2-35-2', '1310.3933-3-37-2'], ['1310.3933-2-17-0', '1310.3933-3-17-0'], ['1310.3933-2-17-1', '1310.3933-3-17-1'], ['1310.3933-2-2-0', '1310.3933-3-2-0'], ['1310.3933-2-2-5', '1310.3933-3-2-5'], ['1310.3933-2-23-0', '1310.3933-3-24-0'], ['1310.3933-2-14-1', '1310.3933-3-14-1'], ['1310.3933-2-14-2', '1310.3933-3-14-2'], ['1310.3933-2-14-3', '1310.3933-3-14-3'], ['1310.3933-2-14-5', '1310.3933-3-14-5'], ['1310.3933-2-26-1', '1310.3933-3-28-1'], ['1310.3933-2-26-2', '1310.3933-3-28-2'], ['1310.3933-2-36-0', '1310.3933-3-38-0'], ['1310.3933-2-36-1', '1310.3933-3-38-1'], ['1310.3933-2-36-2', '1310.3933-3-38-2'], ['1310.3933-2-36-3', '1310.3933-3-38-3'], ['1310.3933-2-36-4', '1310.3933-3-38-4'], ['1310.3933-2-19-0', '1310.3933-3-20-0'], ['1310.3933-2-12-0', '1310.3933-3-12-0'], ['1310.3933-2-5-0', '1310.3933-3-5-0'], ['1310.3933-2-38-0', '1310.3933-3-40-0'], ['1310.3933-2-38-1', '1310.3933-3-40-1'], ['1310.3933-2-38-2', '1310.3933-3-40-2'], ['1310.3933-2-38-3', '1310.3933-3-40-3'], ['1310.3933-2-10-0', '1310.3933-3-10-0'], ['1310.3933-2-10-1', '1310.3933-3-10-1'], ['1310.3933-2-10-2', '1310.3933-3-10-2'], ['1310.3933-2-10-3', '1310.3933-3-10-3'], ['1310.3933-2-30-0', '1310.3933-3-32-0'], ['1310.3933-2-30-1', '1310.3933-3-32-1'], ['1310.3933-2-30-2', '1310.3933-3-32-2'], ['1310.3933-2-30-3', '1310.3933-3-32-4'], ['1310.3933-2-30-4', '1310.3933-3-32-5'], ['1310.3933-2-30-5', '1310.3933-3-32-6'], ['1310.3933-2-30-6', '1310.3933-3-32-7'], ['1310.3933-2-30-7', '1310.3933-3-32-8'], ['1310.3933-2-30-8', '1310.3933-3-32-9'], ['1310.3933-2-33-1', '1310.3933-3-35-1'], ['1310.3933-2-3-0', '1310.3933-3-3-0'], ['1310.3933-2-3-1', '1310.3933-3-3-1'], ['1310.3933-2-37-0', '1310.3933-3-39-0'], ['1310.3933-2-37-1', '1310.3933-3-39-1'], ['1310.3933-2-37-2', '1310.3933-3-39-2'], ['1310.3933-2-37-3', '1310.3933-3-39-3'], ['1310.3933-2-37-4', '1310.3933-3-39-4'], ['1310.3933-2-37-5', '1310.3933-3-39-5'], ['1310.3933-2-16-1', '1310.3933-3-16-1'], ['1310.3933-2-16-2', '1310.3933-3-16-2'], ['1310.3933-2-16-4', '1310.3933-3-16-4'], ['1310.3933-2-27-0', '1310.3933-3-29-0'], ['1310.3933-2-27-1', '1310.3933-3-29-1'], ['1310.3933-2-8-0', '1310.3933-3-8-0'], ['1310.3933-2-8-1', '1310.3933-3-8-1'], ['1310.3933-2-8-2', '1310.3933-3-8-2'], ['1310.3933-2-8-3', '1310.3933-3-8-3'], ['1310.3933-2-8-4', '1310.3933-3-8-4'], ['1310.3933-2-39-1', '1310.3933-3-41-1'], ['1310.3933-2-21-0', '1310.3933-3-22-0'], ['1310.3933-2-21-1', '1310.3933-3-22-1'], ['1310.3933-2-11-0', '1310.3933-3-11-0'], ['1310.3933-2-11-1', '1310.3933-3-11-1'], ['1310.3933-2-11-2', '1310.3933-3-11-2'], ['1310.3933-2-11-3', '1310.3933-3-11-3'], ['1310.3933-2-11-4', '1310.3933-3-11-4'], ['1310.3933-2-22-0', '1310.3933-3-23-0'], ['1310.3933-2-22-1', '1310.3933-3-23-1'], ['1310.3933-2-6-0', '1310.3933-3-6-0'], ['1310.3933-2-6-2', '1310.3933-3-6-2'], ['1310.3933-2-6-4', '1310.3933-3-6-5'], ['1310.3933-2-6-5', '1310.3933-3-6-6'], ['1310.3933-2-6-6', '1310.3933-3-6-7'], ['1310.3933-2-29-0', '1310.3933-3-31-0'], ['1310.3933-2-29-1', '1310.3933-3-31-1'], ['1310.3933-2-29-3', '1310.3933-3-31-3'], ['1310.3933-2-0-0', '1310.3933-3-0-0'], ['1310.3933-2-0-1', '1310.3933-3-0-1'], ['1310.3933-2-7-0', '1310.3933-3-7-0'], ['1310.3933-2-7-1', '1310.3933-3-7-1'], ['1310.3933-2-31-0', '1310.3933-3-33-0'], ['1310.3933-2-4-0', '1310.3933-3-4-0'], ['1310.3933-2-4-1', '1310.3933-3-4-1'], ['1310.3933-2-28-0', '1310.3933-3-30-0']]

[['1310.3933-1-29-3', '1310.3933-2-29-3'], ['1310.3933-1-0-1', '1310.3933-2-0-1'], ['1310.3933-1-6-3', '1310.3933-2-6-3'], ['1310.3933-1-8-3', '1310.3933-2-8-3'], ['1310.3933-1-8-4', '1310.3933-2-8-4'], ['1310.3933-1-11-2', '1310.3933-2-11-2'], ['1310.3933-2-34-3', '1310.3933-3-36-3'], ['1310.3933-2-34-5', '1310.3933-3-36-5'], ['1310.3933-2-35-3', '1310.3933-3-37-3'], ['1310.3933-2-2-1', '1310.3933-3-2-1'], ['1310.3933-2-2-2', '1310.3933-3-2-2'], ['1310.3933-2-2-4', '1310.3933-3-2-4'], ['1310.3933-2-14-0', '1310.3933-3-14-0'], ['1310.3933-2-14-4', '1310.3933-3-14-4'], ['1310.3933-2-26-0', '1310.3933-3-28-0'], ['1310.3933-2-20-0', '1310.3933-3-21-0'], ['1310.3933-2-25-0', '1310.3933-3-27-0'], ['1310.3933-2-16-3', '1310.3933-3-16-3'], ['1310.3933-2-39-2', '1310.3933-3-41-2'], ['1310.3933-2-6-1', '1310.3933-3-6-1'], ['1310.3933-2-6-3', '1310.3933-3-6-4']]

[]

[['1310.3933-2-2-3', '1310.3933-3-2-3'], ['1310.3933-2-29-2', '1310.3933-3-31-2']]

[]

['1310.3933-1-18-0', '1310.3933-1-34-0', '1310.3933-1-35-0', '1310.3933-1-39-4', '1310.3933-2-18-0', '1310.3933-2-34-0', '1310.3933-2-35-0', '1310.3933-2-39-4', '1310.3933-3-18-0', '1310.3933-3-25-0', '1310.3933-3-36-0', '1310.3933-3-37-0', '1310.3933-3-42-2']

{'1': 'http://creativecommons.org/licenses/by/3.0/', '2': 'http://creativecommons.org/licenses/by/3.0/', '3': 'http://creativecommons.org/licenses/by/3.0/'}

https://arxiv.org/abs/1310.3933

{'1310.3933-3-0-0': 'This note shows that for each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-3-0-1': 'This provides the counterexamples of Buchstaber-Panov-Ray conjecture.', '1310.3933-3-1-0': '# Introduction', '1310.3933-3-2-0': 'Let [MATH] denote the ring formed by the unitary bordism classes of all unitary manifolds, where a unitary manifold is an oriented closed smooth manifold whose tangent bundle admits a stably complex structure.', '1310.3933-3-2-1': 'In [CITATION], Davis and Januszkiewicz introduced and studied a class of nicely behaved manifolds [MATH], the so-called quasitoric manifolds (as the topological versions of toric varieties), each of which admits a locally standard [MATH]-action such that the orbit space of the action is homeomorphic to a simple convex polytope.', '1310.3933-3-2-2': 'Buchstaber, Panov and Ray showed in [CITATION] that each quasitoric manifold with an omniorientation always admits a compatible tangential stably complex structure, so omnioriented quasitoric manifolds provide abundant examples of unitary manifolds.', '1310.3933-3-2-3': 'In particular, they also showed there that each class of [MATH] contains an omnioriented quasitoric [MATH]-manifold as its representative (see also [CITATION]).', '1310.3933-3-2-4': 'In addition, Buchstaber, Panov and Ray in [CITATION] investigated the property of specially omnioriented quasitoric manifolds, and proved that if [MATH], then each [MATH]-dimensional specially omnioriented quasitoric manifold represents the zero element in [MATH], where the word "specially" for a specially omnioriented quasitoric manifold means that the first Chern class vanishes.', '1310.3933-3-2-5': 'Furthermore, they posed the following conjecture.', '1310.3933-3-3-0': 'Conjecture ([MATH]):Let [MATH] be a specially omnioriented quasitoric manifold.', '1310.3933-3-3-1': 'Then [MATH] represents the zero element in [MATH].', '1310.3933-3-4-0': 'The purpose of this note is to construct some examples of specially omnioriented quasitoric manifolds that are not bordant to zero in [MATH], which give the negative answer to the above conjecture in almost all possible dimensional cases.', '1310.3933-3-4-1': 'Our main result is stated as follows.', '1310.3933-3-5-0': 'For each [MATH] with only [MATH], there exists a [MATH]-dimensional specially omnioriented quasitoric manifold [MATH] which represents a nonzero element in [MATH].', '1310.3933-3-6-0': 'Our strategy is related to the unoriented bordism theory.', '1310.3933-3-6-1': "Milnor's work tells us in [CITATION] (see also [CITATION]) that there is an epimorphism [EQUATION] here [MATH] denotes the ring produced by the unoriented bordism classes of all smooth closed manifolds, which was studied clearly by Rene Thom [CITATION], and [MATH].", '1310.3933-3-6-2': 'This actually implies that there is a covering homomorphism [EQUATION] which is induced by [MATH], where [MATH] is defined by mapping [MATH].', '1310.3933-3-6-3': 'One sees that [MATH] is well-defined because [MATH] is a [MATH]-polynomial ring with generators in dimensions [MATH] for all [MATH], which is an integral domain, and is the inverse of the Frobenius map [MATH] given by [MATH].', '1310.3933-3-6-4': 'On the other hand, Buchstaber and Ray tell us in [CITATION] that each class of [MATH] contains an [MATH]-dimensional small cover as its representative, where a small cover is also introduced by Davis and Januszkiewicz in [CITATION], and it is the real analogue of a quasitoric manifold.', '1310.3933-3-6-5': 'In addition, Davis and Januszkiewicz tell us in [CITATION] that each quasitoric manifold [MATH] over a simple convex polytope [MATH] always admits a natural conjugation involution [MATH] whose fixed point set [MATH] is just a small cover over [MATH].', '1310.3933-3-6-6': 'In particular, this conjugation involution [MATH] is independent of the choices of omniorientations on [MATH], and by [CITATION], one has that the mod 2 reductions of all Chern numbers of [MATH] with an omniorientation determine all Stiefel-Whitney numbers of [MATH], and in particular, [MATH] as unoriented bordism classes in [MATH].', '1310.3933-3-6-7': 'Thus, [MATH] induces a homomorphism [MATH], which exactly agrees with the above homomorphism [MATH].', '1310.3933-3-7-0': 'With the above understood, to obtain the counterexamples of Buchstaber-Panov-Ray conjecture, an approach is to construct the examples of specially omnioriented quasitoric manifolds whose images under [MATH] are nonzero in [MATH].', '1310.3933-3-7-1': 'We shall see that Stong manifolds play an important role in our argument.', '1310.3933-3-8-0': 'This note is organized as follows.', '1310.3933-3-8-1': 'We shall review the notions and basic properties of quasitoric manifolds and small covers, and state the related result of Buchstaber-Panov-Ray on specially omnioriented quasitoric manifolds in Section [REF].', '1310.3933-3-8-2': "We shall review the Stong's work on Stong manifolds and construct some nonbounding orientable Stong manifolds in Section [REF].", '1310.3933-3-8-3': 'In addition, we also calculate the characteristic matrices of Stong manifolds there.', '1310.3933-3-8-4': 'In Section [REF] we shall construct required examples of omnioriented quasitoric manifolds as special unitary manifolds and complete the proof of our main result.', '1310.3933-3-9-0': '# Quasitoric manifolds and small covers', '1310.3933-3-10-0': 'Davis and Januszkiewicz in [CITATION] introduced and studied two kinds of equivariant manifolds-quasitoric manifolds and small covers, whose geometric and algebraic topology has a strong link to the combinatorics of polytopes.', '1310.3933-3-10-1': 'Following [CITATION], let [EQUATION].', '1310.3933-3-10-2': 'A [MATH]-manifold [MATH] is a smooth closed [MATH]-dimensional [MATH]-manifold admitting a locally standard [MATH]-action such that its orbit space is a simple convex [MATH]-polytope [MATH].', '1310.3933-3-10-3': 'Such a [MATH]-manifold is called a small cover if [MATH] and a quasitoric manifold if [MATH].', '1310.3933-3-11-0': 'For a simple convex polytope [MATH], let [MATH] denote the set of all facets (i.e., [MATH]-dimensional faces) of [MATH].', '1310.3933-3-11-1': 'We know from [CITATION] that each [MATH]-manifold [MATH] determines a characteristic function [MATH] on [MATH] [EQUATION] defined by mapping each facet in [MATH] to nonzero elements of [MATH] such that [MATH] facets meeting at each vertex are mapped to a basis of [MATH].', '1310.3933-3-11-2': 'Conversely, the pair [MATH] can be used to reconstruct [MATH] as follows: first [MATH] gives the following equivalence relation [MATH] on [MATH] [EQUATION] then the quotient space [MATH], denoted by [MATH], is the reconstruction of [MATH], where [MATH] is explained as follows: for each point [MATH], there exists a unique face [MATH] of [MATH] such that [MATH] is in its relative interior.', '1310.3933-3-11-3': 'If [MATH], then there are [MATH] facets, say [MATH], such that [MATH], and furthermore, [MATH] determine a subgroup of rank [MATH] in [MATH], denoted by [MATH].', '1310.3933-3-11-4': 'This reconstruction of [MATH] tells us that the topology of [MATH] can be determined by [MATH].', '1310.3933-3-12-0': 'If we fix an ordering for all facets in [MATH] (e.g., say [MATH]) , then the characteristic function [MATH] uniquely determines a matrix of size [MATH] over [MATH] [EQUATION] with [MATH] as columns, which is called the characteristic matrix of [MATH] or [MATH].', '1310.3933-3-13-0': 'We may see from this reconstruction of [MATH]-manifolds that there is also an essential relation between small covers and quasitoric manifolds over a simple polytope.', '1310.3933-3-13-1': 'In fact, given a quasitoric manifold [MATH] over [MATH], as shown in [CITATION], there is a natural conjugation involution on [MATH] defined by [MATH], which fixes [MATH].', '1310.3933-3-13-2': 'Then this involution descends an involution [MATH] on [MATH] whose fixed point set is exactly a small cover [MATH] over [MATH], where [MATH] is the mod 2 reduction of [MATH].', '1310.3933-3-14-0': 'An omniorientation of a quasitoric manifold [MATH] is, by definition in [CITATION], just one choice of orientations of [MATH] and submanifolds [MATH].', '1310.3933-3-14-1': 'Thus, a quasitoric manifold [MATH] has [MATH] omniorientations, where [MATH] is the number of all facets of [MATH].', '1310.3933-3-14-2': 'Clearly, the conjugation involution [MATH] on [MATH] is independent of the choices of omniorientations of [MATH].', '1310.3933-3-14-3': 'Now let [MATH] denote the set of all [MATH] omniorientations.', '1310.3933-3-14-4': 'Buchstaber, Panov and Ray showed in [CITATION] (also see [CITATION]) that for each omniorientation [MATH], [MATH] with this omniorientation [MATH] always admits a tangential stably complex structure, so it is a unitary manifold.', '1310.3933-3-14-5': 'In [CITATION], Buchstaber, Panov and Ray gave a characterization for [MATH] with [MATH] to be a special unitary manifold in terms of [MATH], which is stated as follows.', '1310.3933-3-15-0': '# Stong manifolds', '1310.3933-3-16-0': '## Stong manifolds In [CITATION], Stong introduced the Stong manifolds, from which all generators of the unoriented bordism ring [MATH] can be chosen.', '1310.3933-3-16-1': 'A Stong manifold is defined as the real projective space bundle denoted by [MATH] of the bundle [MATH] over [MATH], where [MATH] is the pullback of the canonical bundle over the [MATH]-th factor [MATH].', '1310.3933-3-16-2': 'The Stong manifold [MATH] has dimension [MATH].', '1310.3933-3-16-3': 'As shown in [CITATION], the cohomology with [MATH] coefficients of [MATH] is the free module over the cohomology of [MATH] on [MATH], where [MATH] is the first Stiefel-Whitney class of the canonical line bundle over [MATH], with the relation [EQUATION] where [MATH] is the [MATH]-th Stiefel-Whitney class of [MATH].', '1310.3933-3-16-4': 'Then the total Stiefel-Whitney class of [MATH] is [EQUATION] where [MATH] is the pullback of the nonzero class in [MATH].', '1310.3933-3-17-0': 'In fact, it is easy to see that the total Stiefel-Whitney class of [MATH] is exactly [EQUATION].', '1310.3933-3-17-1': 'So the cohomology with [MATH] coefficients of [MATH] may be written as [EQUATION] where [MATH] is the ideal generated by [MATH], and [MATH].', '1310.3933-3-18-0': 'Stong further showed in [CITATION] that', '1310.3933-3-19-0': 'Note that generally, an indecomposable element in [MATH] means that it is not a sum of products of elements of positive degree (see [CITATION]).', '1310.3933-3-20-0': 'It is not difficult to see from the expression ([REF]) of the total Stiefel-Whitney class of [MATH] that', '1310.3933-3-21-0': 'For [MATH], [MATH] is orientable if and only if [MATH] and all [MATH] are even.', '1310.3933-3-22-0': 'By Proposition [REF] and Corollary [REF], we may choose the following examples of indecomposable, orientable Stong manifolds.', '1310.3933-3-22-1': 'For [MATH], [MATH] and [MATH] are indecomposable and orientable, so they represent nonzero elements in [MATH].', '1310.3933-3-23-0': 'Let [MATH] and [MATH] denote the unoriented bordism classes of [MATH] and [MATH], respectively.', '1310.3933-3-23-1': 'Then we have that', '1310.3933-3-24-0': 'All [MATH] and [MATH] with [MATH] form a polynomial subring [EQUATION] of [MATH], which contains nonzero classes of dimension [MATH].', '1310.3933-3-25-0': 'Because [MATH] and [MATH] are indecomposable in [MATH], any non-trivial polynomial in [MATH] and [MATH] is nonzero in [MATH].', '1310.3933-3-26-0': '## Characteristic matrices of Stong manifolds', '1310.3933-3-27-0': 'We see that [MATH] is a [MATH]-bundle over [MATH], so it is a special generalized real Bott manifold, and in particular, it is also a small cover over [MATH], where [MATH] denotes an [MATH]-dimensional simplex.', '1310.3933-3-28-0': 'A generalized real Bott manifold is the total space [MATH] of an iterated fiber bundle: [EQUATION] where each [MATH] is the projectivization of a Whitney sum of [MATH] real line bundles over [MATH].', '1310.3933-3-28-1': 'It is well-known that the generalized real Bott manifold [MATH] is a small cover over [MATH].', '1310.3933-3-28-2': 'Conversely, we also know from [CITATION] that a small cover over a product of simplices is a generalized real Bott manifold.', '1310.3933-3-29-0': 'Now let us look at the characteristic matrix of [MATH] as a small cover over the product [MATH] with [MATH] and [MATH].', '1310.3933-3-29-1': 'Clearly [MATH] has [MATH] facets, which are listed as follows: [EQUATION] and [EQUATION] where [MATH], denote [MATH] facets of [MATH].', '1310.3933-3-30-0': 'Throughout the following, we shall carry out our work on a fixed ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-3-31-0': 'Without the loss of generality, assume that the values of the characteristic function [MATH] on the following [MATH] facets [EQUATION] meeting at a vertex are all columns with an ordering from the first column to the last column in [MATH], respectively.', '1310.3933-3-31-1': 'It suffices to determine the values of [MATH] on the [MATH] facets [MATH].', '1310.3933-3-31-2': 'By [CITATION], we have that for [MATH] [EQUATION] and [EQUATION] such that those entries from [MATH]-th to [MATH]-th of [MATH] are all zero, and those entries from [MATH]-th to [MATH]-th of [MATH] are all zero.', '1310.3933-3-31-3': 'In particular, we also know by [CITATION] that there is at least one [MATH] such that [MATH] in [MATH].', '1310.3933-3-32-0': 'Now by [CITATION], we may write [MATH] as [EQUATION] where the [MATH] are used as indeterminants of degree 1, [MATH] is the Stanley-Reisner ideal generated by [MATH] and [MATH], and [MATH] is the ideal determined by [MATH].', '1310.3933-3-32-1': 'Furthermore, we have by [CITATION] that the total Stiefel-Whitney class of [MATH] is [EQUATION].', '1310.3933-3-32-2': 'Comparing with the formula ([REF]) or by Remark [REF], we see that for each [MATH], [EQUATION] so [MATH].', '1310.3933-3-32-3': 'Then we obtain from all equations in ([REF]) that the characteristic matrix [MATH] corresponding to [MATH] is of the form [EQUATION] where all blocks except for [MATH], [MATH] and [MATH] are zero.', '1310.3933-3-32-4': 'This implies that [MATH] must be the zero element, and for [MATH], each [MATH] is of the form [EQUATION] in [MATH].', '1310.3933-3-32-5': 'Moreover, one has that [EQUATION]', '1310.3933-3-32-6': 'Comparing with the formula ([REF]) again, one should have that [EQUATION].', '1310.3933-3-32-7': 'Without the loss of generality, assume that [MATH] for [MATH].', '1310.3933-3-32-8': 'Then for [MATH], one has that [MATH], and for [MATH], one has by ([REF]) that [EQUATION] so [MATH] and [MATH] if [MATH] since [MATH] are linearly independent in [MATH].', '1310.3933-3-32-9': 'This completes the proof.', '1310.3933-3-33-0': 'If there is a minimal integer [MATH] with [MATH] such that [MATH] but [MATH] (so [MATH] for [MATH]), then a similar argument as above gives', '1310.3933-3-34-0': '# Proof of Main Result', '1310.3933-3-35-0': '## Examples of specially omnioriented quasitoric manifolds Throughout the following, for a [MATH]-dimensional simplex [MATH], [MATH] mean the [MATH] facets of [MATH], and for a product [MATH] of simplices, [MATH] means that the facet [MATH] of [MATH].', '1310.3933-3-35-1': 'Then let us construct some required examples.', '1310.3933-3-36-0': 'Let [MATH] with [MATH].', '1310.3933-3-36-1': 'Define a characteristic function [MATH] on [MATH] in the following way.', '1310.3933-3-36-2': 'First let us fix an ordering of all facets of [MATH] as follows [EQUATION].', '1310.3933-3-36-3': 'Then we construct the characteristic matrix [MATH] of the required characteristic function [MATH] on the above ordered facets as follows: [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH] and [MATH] denote the same meanings as in Proposition [REF], and [MATH] denotes the matrix of size [MATH] with [MATH]-entries for all even [MATH] being [MATH] and other entries being [MATH].', '1310.3933-3-36-4': 'We see that the sum of all entries of each column in the characteristic matrix [MATH] is always 1.', '1310.3933-3-36-5': 'Thus, by Proposition [REF], one has that the quasitoric manifold [MATH] with the given omniorientation is a special unitary manifold.', '1310.3933-3-37-0': 'Let [MATH] with [MATH].', '1310.3933-3-37-1': 'In a similar way as above, fix an ordering of all facets of [MATH] as follows: [EQUATION].', '1310.3933-3-37-2': 'Then we define a characteristic function [MATH] on the above ordered facets of [MATH] by the following characteristic matrix [EQUATION] with only blocks [MATH], [MATH] and [MATH] being nonzero, and [MATH] otherwise, where [MATH], [MATH] and [MATH] denote the same meanings as above.', '1310.3933-3-37-3': 'By Proposition [REF], [MATH] with the given omniorientation is a special unitary manifold.', '1310.3933-3-38-0': 'The case in which [MATH].', '1310.3933-3-38-1': 'Consider the polytope [MATH] with the following ordered facets [EQUATION].', '1310.3933-3-38-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by the following characteristic matrix [EQUATION] which gives a special unitary manifold [MATH].', '1310.3933-3-38-3': 'Moreover, by the Davis-Januszkiewicz theory, we may read off the cohomology of [MATH] as follows: [EQUATION] with [MATH], and by [CITATION] and [CITATION], the total Chern class of [MATH] may be written as [EQUATION].', '1310.3933-3-38-4': 'A direct calculation gives the Chern number [MATH], which implies that this specially omnioriented quasitoric manifold [MATH] is not bordant to zero in [MATH].', '1310.3933-3-39-0': 'The case in which [MATH].', '1310.3933-3-39-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION].', '1310.3933-3-39-2': 'Then we may define a characteristic function [MATH] on the ordered facets of [MATH] by [EQUATION] which also gives a special unitary manifold [MATH].', '1310.3933-3-39-3': 'Similarly, one has the cohomology of [MATH] [EQUATION] with [MATH], and the total Chern class of [MATH] [EQUATION].', '1310.3933-3-39-4': 'Furthermore, one has the Chern number [MATH].', '1310.3933-3-39-5': 'So [MATH] is not bordant to zero in [MATH].', '1310.3933-3-40-0': 'The case in which [MATH].', '1310.3933-3-40-1': 'Consider the polytope [MATH] with the ordered facets as follows: [EQUATION] and define a characteristic function [MATH] on the ordered facets of [MATH] by the matrix [EQUATION] from which one obtains a special unitary manifold [MATH] with its cohomology [EQUATION] and with its total Chern class [EQUATION].', '1310.3933-3-40-2': 'Then one has that the 6-th Chern class [MATH], so the Chern number [MATH].', '1310.3933-3-40-3': 'Thus [MATH] is not bordant to zero in [MATH].', '1310.3933-3-41-0': '## Proof of Theorem [REF] Obviously, the mod 2 reductions of the characteristic matrices [MATH] and [MATH] of [MATH] and [MATH] are [EQUATION] and [EQUATION] respectively.', '1310.3933-3-41-1': 'Thus, by Proposition [REF], one has that the fixed point sets of the conjugation involutions on [MATH] and [MATH] are homeomorphic to the Stong manifolds [MATH] and [MATH], respectively.', '1310.3933-3-41-2': 'Thus, the subring of [MATH] generated by the unitary bordism classes [MATH] of [MATH] and [MATH] is mapped onto the subring [MATH] of [MATH] in Lemma [REF] via [MATH].', '1310.3933-3-41-3': 'This means that any non-trivial polynomial in [MATH] and [MATH] is nonzero in [MATH] since its image under [MATH] is nonzero by Lemma [REF], so we obtain the examples of non-bounding specially omnioriented quasitoric [MATH]-manifolds with [MATH].', '1310.3933-3-42-0': 'For [MATH], Examples [REF]-[REF] directly provide three non-bounding specially omnioriented quasitoric manifolds.', '1310.3933-3-42-1': 'This completes the proof of Theorem [REF].', '1310.3933-3-42-2': '[MATH]', '1310.3933-3-43-0': 'A counterexample in the case [MATH] was recently discovered from a joint work [CITATION] of the first author with Taras Panov, concerning the toric generators in the unitary and special unitary bordism rings.'}

math-0608415

{'math-0608415-1-0-0': '# Introduction.', 'math-0608415-1-1-0': 'The aim of this paper is to prove non-coherence of certain families of lattices in the isometry group [MATH] of the hyperbolic [MATH]-space [MATH]).', 'math-0608415-1-1-1': 'We recall that a group [MATH] is called coherent if every finitely generated subgroup of [MATH] is finitely presented.', 'math-0608415-1-1-2': 'It is well known that all lattices in [MATH] and [MATH] are coherent.', 'math-0608415-1-1-3': 'Indeed, it is easy to prove that every finitely generated Fuchsian group is finitely presented.', 'math-0608415-1-1-4': 'The coherence of 3-manifold groups was proved by P. Scott [CITATION].', 'math-0608415-1-1-5': 'First examples of geometrically finite non-coherent discrete subgroups of [MATH] were constructed in [CITATION], [CITATION], [CITATION].', 'math-0608415-1-1-6': 'An example of non-coherent uniform lattice in [MATH] was given in [CITATION].', 'math-0608415-1-2-0': 'In what follows we will identify [MATH] with a connected component of the hyperboloid [EQUATION] where [MATH] is a real quadratic form of signature [MATH] in [MATH] variables.', 'math-0608415-1-2-1': 'Then the group [MATH] is identified with the index [MATH] subgroup [MATH] preserving [MATH].', 'math-0608415-1-3-0': 'Let [MATH] and [MATH] be quadratic forms on finite-dimensional vector spaces [MATH] and [MATH] over [MATH].', 'math-0608415-1-3-1': 'It is said that [MATH] represents [MATH] if the vector space [MATH] admits an orthogonal decomposition (with respect to [MATH]) [EQUATION] so that [MATH] is isometric to [MATH].', 'math-0608415-1-3-2': 'In other words, after a change of coordinates, the form [MATH] can be written as [EQUATION] where [MATH] and [MATH].', 'math-0608415-1-3-3': 'Whenever [MATH] represents [MATH], a finite index subgroup of [MATH] is naturally embedded into [MATH].', 'math-0608415-1-4-0': 'The main result of this paper is:', 'math-0608415-1-5-0': 'Theorem A.', 'math-0608415-1-5-1': 'For every [MATH] and every rational quadratic form [MATH] of signature [MATH] which represents the form [EQUATION] the lattice [MATH] is non-coherent.', 'math-0608415-1-6-0': 'For every [MATH] there are infinitely many commensurability classes of non-uniform non-coherent lattices in [MATH].', 'math-0608415-1-7-0': 'We refer the reader to Section [REF] for the discussion of uniform lattices.', 'math-0608415-1-7-1': 'By combining Theorem A with some standard facts on rational quadratic forms, we prove', 'math-0608415-1-8-0': 'Theorem B.', 'math-0608415-1-8-1': 'For every [MATH], every non-uniform arithmetic lattice in [MATH] is non-coherent.', 'math-0608415-1-9-0': 'As a by-product of the proof, in Section [REF], we obtain a simple proof of the following result of independent interest (which was proven in [CITATION] in the case [MATH]).', 'math-0608415-1-9-1': 'Recall that a subgroup of a group [MATH] is called separable if it can be represented as the intersection of a family of finite index subgroups of [MATH].', 'math-0608415-1-9-2': 'For instance, separability of the trivial subgroup is nothing else than residual finiteness of [MATH].', 'math-0608415-1-10-0': 'Theorem C.', 'math-0608415-1-10-1': 'In every non-uniform arithmetic lattice in [MATH]), every geometrically finite subgroup is separable.', 'math-0608415-1-11-0': 'We refer the reader to [CITATION] for the definition of geometrically finite discrete subgroups of [MATH].', 'math-0608415-1-11-1': 'Recall only that every discrete group which admits a finitely-sided convex fundamental polyhedron is geometrically finite.', 'math-0608415-1-12-0': 'In Section [REF] we adopt the method of [CITATION] to obtain examples of non-arithmetic non-coherent lattices:', 'math-0608415-1-13-0': 'Theorem D.', 'math-0608415-1-13-1': 'For each [MATH] there exist both uniform and non-uniform non-coherent non-arithmetic lattices in [MATH].', 'math-0608415-1-14-0': 'The above results provide a strong evidence for the negative answer to the following question in the case of non-uniform lattices:', 'math-0608415-1-15-0': '[D. Wise] Does there exist a coherent lattice in [MATH] for any [MATH]?', 'math-0608415-1-16-0': 'In Section [REF] we provide some tentative evidence for the negative answer to this question in the uniform case as well.', 'math-0608415-1-17-0': 'Our proof of the non-coherence in the non-uniform case is different from the one in [CITATION]: The finitely generated infinitely presented subgroup that we construct is generated by four subgroups stabilizing 4 distinct hyperplanes in [MATH], while in the construction used in [CITATION] two hyperplanes were enough.', 'math-0608415-1-17-1': 'Direct repetition of the arguments used in [CITATION] does not seem to work in the non-uniform case.'}

{'math-0608415-2-0-0': 'We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic [MATH]-space for [MATH] greater than [MATH].', 'math-0608415-2-0-1': 'For instance, every nonuniform arithmetic lattice in [MATH] is noncoherent, provided that [MATH] is at least [MATH].', 'math-0608415-2-1-0': '# Introduction', 'math-0608415-2-2-0': 'The aim of this paper is to prove noncoherence of certain families of lattices in the isometry group [MATH] of the hyperbolic [MATH]-space [MATH]).', 'math-0608415-2-2-1': 'We recall that a group [MATH] is called coherent if every finitely generated subgroup of [MATH] is finitely presented.', 'math-0608415-2-2-2': 'It is well known that all lattices in [MATH] and [MATH] are coherent.', 'math-0608415-2-2-3': 'Indeed, it is easy to prove that every finitely generated Fuchsian group is finitely presented.', 'math-0608415-2-2-4': 'The coherence of [MATH]-manifold groups was proved by PScott [CITATION].', 'math-0608415-2-2-5': 'First examples of geometrically finite noncoherent discrete subgroups of [MATH] were constructed by the first and second author [CITATION] and the second author [CITATION].', 'math-0608415-2-2-6': 'An example of noncoherent uniform lattice in [MATH] was given by Bowditch and Mess [CITATION].', 'math-0608415-2-3-0': 'In what follows we will identify [MATH] with a connected component of the hyperboloid [EQUATION] where [MATH] is a real quadratic form of signature [MATH] in [MATH] variables.', 'math-0608415-2-3-1': 'Then the group [MATH] is identified with the index [MATH] subgroup [MATH] preserving [MATH].', 'math-0608415-2-4-0': 'Let [MATH] and [MATH] be quadratic forms on finite-dimensional vector spaces [MATH] and [MATH] over [MATH].', 'math-0608415-2-4-1': 'It is said that [MATH] represents [MATH] if the vector space [MATH] admits an orthogonal decomposition (with respect to [MATH]) [EQUATION] so that [MATH] is isometric to [MATH].', 'math-0608415-2-4-2': 'In other words, after a change of coordinates, the form [MATH] can be written as [EQUATION] where [MATH] and [MATH].', 'math-0608415-2-4-3': 'Whenever [MATH] represents [MATH], a finite index subgroup of [MATH] is naturally embedded into [MATH].', 'math-0608415-2-5-0': 'For every [MATH] and every rational quadratic form [MATH] of signature [MATH] which represents the form [EQUATION] the lattice [MATH] is noncoherent.', 'math-0608415-2-6-0': 'For every [MATH] there are infinitely many commensurability classes of nonuniform noncoherent lattices in [MATH].', 'math-0608415-2-7-0': 'We refer the reader to proofA for the discussion of uniform lattices.', 'math-0608415-2-8-0': 'As a by-product of the proof, in rational, we obtain a simple proof of the following result of independent interest (which was proven by Agol, Long and Reid [CITATION] in the case [MATH]).', 'math-0608415-2-8-1': 'Recall that a subgroup of a group [MATH] is called separable if it can be represented as the intersection of a family of finite index subgroups of [MATH].', 'math-0608415-2-8-2': 'For instance, separability of the trivial subgroup is nothing else than residual finiteness of [MATH].', 'math-0608415-2-9-0': 'In every nonuniform arithmetic lattice in [MATH]), every geometrically finite subgroup is separable.', 'math-0608415-2-10-0': 'We refer the reader to Bowditch [CITATION] for the definition of geometrically finite discrete subgroups of [MATH].', 'math-0608415-2-10-1': 'Recall only that every discrete group which admits a finitely-sided convex fundamental polyhedron is geometrically finite.', 'math-0608415-2-11-0': 'For each [MATH] there exist both uniform and nonuniform noncoherent nonarithmetic lattices in [MATH].', 'math-0608415-2-12-0': '[DWise] Does there exist a coherent lattice in [MATH] for any [MATH]?', 'math-0608415-2-13-0': 'In spec we provide some tentative evidence for the negative answer to this question in the uniform case as well.', 'math-0608415-2-14-0': 'Our proof of the noncoherence in the nonuniform case is different from the one by Bowditch and Mess [CITATION]: The finitely generated infinitely presented subgroup that we construct is generated by four subgroups stabilizing 4 distinct hyperplanes in [MATH], while in the construction used in [CITATION] two hyperplanes were enough.', 'math-0608415-2-14-1': 'Direct repetition of the arguments used in [CITATION] does not seem to work in the nonuniform case.'}

[['math-0608415-1-2-0', 'math-0608415-2-3-0'], ['math-0608415-1-2-1', 'math-0608415-2-3-1'], ['math-0608415-1-11-1', 'math-0608415-2-10-1'], ['math-0608415-1-9-1', 'math-0608415-2-8-1'], ['math-0608415-1-9-2', 'math-0608415-2-8-2'], ['math-0608415-1-3-0', 'math-0608415-2-4-0'], ['math-0608415-1-3-1', 'math-0608415-2-4-1'], ['math-0608415-1-3-2', 'math-0608415-2-4-2'], ['math-0608415-1-3-3', 'math-0608415-2-4-3'], ['math-0608415-2-14-0', 'math-0608415-3-14-0'], ['math-0608415-2-14-1', 'math-0608415-3-14-1'], ['math-0608415-2-12-0', 'math-0608415-3-12-0'], ['math-0608415-2-6-0', 'math-0608415-3-6-0'], ['math-0608415-2-7-0', 'math-0608415-3-7-0'], ['math-0608415-2-11-0', 'math-0608415-3-11-0'], ['math-0608415-2-0-0', 'math-0608415-3-0-0'], ['math-0608415-2-0-1', 'math-0608415-3-0-1'], ['math-0608415-2-8-0', 'math-0608415-3-8-0'], ['math-0608415-2-8-1', 'math-0608415-3-8-1'], ['math-0608415-2-8-2', 'math-0608415-3-8-2'], ['math-0608415-2-4-0', 'math-0608415-3-4-0'], ['math-0608415-2-4-1', 'math-0608415-3-4-1'], ['math-0608415-2-4-2', 'math-0608415-3-4-2'], ['math-0608415-2-4-3', 'math-0608415-3-4-3'], ['math-0608415-2-5-0', 'math-0608415-3-5-0'], ['math-0608415-2-13-0', 'math-0608415-3-13-0'], ['math-0608415-2-10-0', 'math-0608415-3-10-0'], ['math-0608415-2-10-1', 'math-0608415-3-10-1'], ['math-0608415-2-3-0', 'math-0608415-3-3-0'], ['math-0608415-2-3-1', 'math-0608415-3-3-1'], ['math-0608415-2-9-0', 'math-0608415-3-9-0'], ['math-0608415-2-2-0', 'math-0608415-3-2-0'], ['math-0608415-2-2-1', 'math-0608415-3-2-1'], ['math-0608415-2-2-2', 'math-0608415-3-2-2'], ['math-0608415-2-2-3', 'math-0608415-3-2-3'], ['math-0608415-2-2-4', 'math-0608415-3-2-4'], ['math-0608415-2-2-5', 'math-0608415-3-2-5'], ['math-0608415-2-2-6', 'math-0608415-3-2-6'], ['math-0608415-1-1-1', 'math-0608415-2-2-1'], ['math-0608415-1-1-2', 'math-0608415-2-2-2'], ['math-0608415-1-1-3', 'math-0608415-2-2-3'], ['math-0608415-1-16-0', 'math-0608415-2-13-0'], ['math-0608415-1-11-0', 'math-0608415-2-10-0'], ['math-0608415-1-5-1', 'math-0608415-2-5-0'], ['math-0608415-1-10-1', 'math-0608415-2-9-0'], ['math-0608415-1-9-0', 'math-0608415-2-8-0'], ['math-0608415-1-17-0', 'math-0608415-2-14-0'], ['math-0608415-1-17-1', 'math-0608415-2-14-1'], ['math-0608415-1-6-0', 'math-0608415-2-6-0'], ['math-0608415-1-7-0', 'math-0608415-2-7-0'], ['math-0608415-1-15-0', 'math-0608415-2-12-0'], ['math-0608415-1-1-0', 'math-0608415-2-2-0'], ['math-0608415-1-13-1', 'math-0608415-2-11-0'], ['math-0608415-1-1-4', 'math-0608415-2-2-4'], ['math-0608415-1-1-5', 'math-0608415-2-2-5'], ['math-0608415-1-1-6', 'math-0608415-2-2-6']]

[['math-0608415-1-2-0', 'math-0608415-2-3-0'], ['math-0608415-1-2-1', 'math-0608415-2-3-1'], ['math-0608415-1-11-1', 'math-0608415-2-10-1'], ['math-0608415-1-9-1', 'math-0608415-2-8-1'], ['math-0608415-1-9-2', 'math-0608415-2-8-2'], ['math-0608415-1-3-0', 'math-0608415-2-4-0'], ['math-0608415-1-3-1', 'math-0608415-2-4-1'], ['math-0608415-1-3-2', 'math-0608415-2-4-2'], ['math-0608415-1-3-3', 'math-0608415-2-4-3'], ['math-0608415-2-14-0', 'math-0608415-3-14-0'], ['math-0608415-2-14-1', 'math-0608415-3-14-1'], ['math-0608415-2-12-0', 'math-0608415-3-12-0'], ['math-0608415-2-6-0', 'math-0608415-3-6-0'], ['math-0608415-2-7-0', 'math-0608415-3-7-0'], ['math-0608415-2-11-0', 'math-0608415-3-11-0'], ['math-0608415-2-0-0', 'math-0608415-3-0-0'], ['math-0608415-2-0-1', 'math-0608415-3-0-1'], ['math-0608415-2-8-0', 'math-0608415-3-8-0'], ['math-0608415-2-8-1', 'math-0608415-3-8-1'], ['math-0608415-2-8-2', 'math-0608415-3-8-2'], ['math-0608415-2-4-0', 'math-0608415-3-4-0'], ['math-0608415-2-4-1', 'math-0608415-3-4-1'], ['math-0608415-2-4-2', 'math-0608415-3-4-2'], ['math-0608415-2-4-3', 'math-0608415-3-4-3'], ['math-0608415-2-5-0', 'math-0608415-3-5-0'], ['math-0608415-2-13-0', 'math-0608415-3-13-0'], ['math-0608415-2-10-0', 'math-0608415-3-10-0'], ['math-0608415-2-10-1', 'math-0608415-3-10-1'], ['math-0608415-2-3-0', 'math-0608415-3-3-0'], ['math-0608415-2-3-1', 'math-0608415-3-3-1'], ['math-0608415-2-9-0', 'math-0608415-3-9-0'], ['math-0608415-2-2-0', 'math-0608415-3-2-0'], ['math-0608415-2-2-1', 'math-0608415-3-2-1'], ['math-0608415-2-2-2', 'math-0608415-3-2-2'], ['math-0608415-2-2-3', 'math-0608415-3-2-3'], ['math-0608415-2-2-4', 'math-0608415-3-2-4'], ['math-0608415-2-2-5', 'math-0608415-3-2-5'], ['math-0608415-2-2-6', 'math-0608415-3-2-6'], ['math-0608415-1-1-1', 'math-0608415-2-2-1'], ['math-0608415-1-1-2', 'math-0608415-2-2-2'], ['math-0608415-1-1-3', 'math-0608415-2-2-3']]

[['math-0608415-1-16-0', 'math-0608415-2-13-0'], ['math-0608415-1-11-0', 'math-0608415-2-10-0'], ['math-0608415-1-5-1', 'math-0608415-2-5-0'], ['math-0608415-1-10-1', 'math-0608415-2-9-0'], ['math-0608415-1-9-0', 'math-0608415-2-8-0'], ['math-0608415-1-17-0', 'math-0608415-2-14-0'], ['math-0608415-1-17-1', 'math-0608415-2-14-1'], ['math-0608415-1-6-0', 'math-0608415-2-6-0'], ['math-0608415-1-7-0', 'math-0608415-2-7-0'], ['math-0608415-1-15-0', 'math-0608415-2-12-0'], ['math-0608415-1-1-0', 'math-0608415-2-2-0']]

[]

[['math-0608415-1-13-1', 'math-0608415-2-11-0'], ['math-0608415-1-1-4', 'math-0608415-2-2-4'], ['math-0608415-1-1-5', 'math-0608415-2-2-5'], ['math-0608415-1-1-6', 'math-0608415-2-2-6']]

[]

['math-0608415-1-4-0', 'math-0608415-1-5-0', 'math-0608415-1-8-0', 'math-0608415-1-10-0', 'math-0608415-1-12-0', 'math-0608415-1-13-0', 'math-0608415-1-14-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/math/0608415

{'math-0608415-3-0-0': 'We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic [MATH]-space for [MATH] greater than [MATH].', 'math-0608415-3-0-1': 'For instance, every nonuniform arithmetic lattice in [MATH] is noncoherent, provided that [MATH] is at least [MATH].', 'math-0608415-3-1-0': '# Introduction', 'math-0608415-3-2-0': 'The aim of this paper is to prove noncoherence of certain families of lattices in the isometry group [MATH] of the hyperbolic [MATH]-space [MATH]).', 'math-0608415-3-2-1': 'We recall that a group [MATH] is called coherent if every finitely generated subgroup of [MATH] is finitely presented.', 'math-0608415-3-2-2': 'It is well known that all lattices in [MATH] and [MATH] are coherent.', 'math-0608415-3-2-3': 'Indeed, it is easy to prove that every finitely generated Fuchsian group is finitely presented.', 'math-0608415-3-2-4': 'The coherence of [MATH]-manifold groups was proved by PScott [CITATION].', 'math-0608415-3-2-5': 'First examples of geometrically finite noncoherent discrete subgroups of [MATH] were constructed by the first and second author [CITATION] and the second author [CITATION].', 'math-0608415-3-2-6': 'An example of noncoherent uniform lattice in [MATH] was given by Bowditch and Mess [CITATION].', 'math-0608415-3-3-0': 'In what follows we will identify [MATH] with a connected component of the hyperboloid [EQUATION] where [MATH] is a real quadratic form of signature [MATH] in [MATH] variables.', 'math-0608415-3-3-1': 'Then the group [MATH] is identified with the index [MATH] subgroup [MATH] preserving [MATH].', 'math-0608415-3-4-0': 'Let [MATH] and [MATH] be quadratic forms on finite-dimensional vector spaces [MATH] and [MATH] over [MATH].', 'math-0608415-3-4-1': 'It is said that [MATH] represents [MATH] if the vector space [MATH] admits an orthogonal decomposition (with respect to [MATH]) [EQUATION] so that [MATH] is isometric to [MATH].', 'math-0608415-3-4-2': 'In other words, after a change of coordinates, the form [MATH] can be written as [EQUATION] where [MATH] and [MATH].', 'math-0608415-3-4-3': 'Whenever [MATH] represents [MATH], a finite index subgroup of [MATH] is naturally embedded into [MATH].', 'math-0608415-3-5-0': 'For every [MATH] and every rational quadratic form [MATH] of signature [MATH] which represents the form [EQUATION] the lattice [MATH] is noncoherent.', 'math-0608415-3-6-0': 'For every [MATH] there are infinitely many commensurability classes of nonuniform noncoherent lattices in [MATH].', 'math-0608415-3-7-0': 'We refer the reader to proofA for the discussion of uniform lattices.', 'math-0608415-3-8-0': 'As a by-product of the proof, in rational, we obtain a simple proof of the following result of independent interest (which was proven by Agol, Long and Reid [CITATION] in the case [MATH]).', 'math-0608415-3-8-1': 'Recall that a subgroup of a group [MATH] is called separable if it can be represented as the intersection of a family of finite index subgroups of [MATH].', 'math-0608415-3-8-2': 'For instance, separability of the trivial subgroup is nothing else than residual finiteness of [MATH].', 'math-0608415-3-9-0': 'In every nonuniform arithmetic lattice in [MATH]), every geometrically finite subgroup is separable.', 'math-0608415-3-10-0': 'We refer the reader to Bowditch [CITATION] for the definition of geometrically finite discrete subgroups of [MATH].', 'math-0608415-3-10-1': 'Recall only that every discrete group which admits a finitely-sided convex fundamental polyhedron is geometrically finite.', 'math-0608415-3-11-0': 'For each [MATH] there exist both uniform and nonuniform noncoherent nonarithmetic lattices in [MATH].', 'math-0608415-3-12-0': '[DWise] Does there exist a coherent lattice in [MATH] for any [MATH]?', 'math-0608415-3-13-0': 'In spec we provide some tentative evidence for the negative answer to this question in the uniform case as well.', 'math-0608415-3-14-0': 'Our proof of the noncoherence in the nonuniform case is different from the one by Bowditch and Mess [CITATION]: The finitely generated infinitely presented subgroup that we construct is generated by four subgroups stabilizing 4 distinct hyperplanes in [MATH], while in the construction used in [CITATION] two hyperplanes were enough.', 'math-0608415-3-14-1': 'Direct repetition of the arguments used in [CITATION] does not seem to work in the nonuniform case.'}

nucl-ex-0104013

{'nucl-ex-0104013-1-0-0': 'For many years it has been speculated that excited nuclei would undergo a liquid to vapor phase transition.', 'nucl-ex-0104013-1-0-1': "For even longer, it has been known that clusterization in a vapor carries direct information on the liquid- vapor equilibrium according to Fisher's droplet model.", 'nucl-ex-0104013-1-0-2': "Now the thermal component of the [MATH] GeV/c [MATH]Au multifragmentation data of the ISiS Collaboration is shown to follow the scaling predicted by Fisher's model, thus providing the strongest evidence yet of the liquid to vapor phase transition.", 'nucl-ex-0104013-1-1-0': 'Nuclear multifragmentation, the break up of a nuclear system into several intermediate sized pieces, has been frequently discussed in terms of equilibrium statistical mechanics, and its possible association with a phase transition [CITATION].', 'nucl-ex-0104013-1-1-1': 'However, ample uncertainty remains regarding its nature, in particular whether multifragmentation is a phase transition and if so whether it is associated with the liquid to vapor phase transition.', 'nucl-ex-0104013-1-2-0': "This paper will show that: 1) high quality experimental data contain the unequivocal signature of a liquid to vapor phase transition through their strict adherence to Fisher's droplet model; 2) the two phase coexistence line is observed over a large energy/temperature interval extending up to and including the critical point; 3) several critical exponents, as well as the critical temperature, the surface energy coefficient and the compressibility factor can be directly extracted; 4) the nuclear phase diagram can be constructed with the available data; 5) the nuclear liquid at break-up is a slightly super-saturated vapor, for which the pressure and density can be determined as a function of the temperature.", 'nucl-ex-0104013-1-3-0': 'In past attempts to investigate the relationship between nuclear multifragmentation and a liquid to vapor phase transition, critical exponents have been determined [CITATION], caloric curves have been examined [CITATION], and the observation of negative heat capacities have been reported [CITATION].', 'nucl-ex-0104013-1-3-1': 'Other studies of multifragmentation data have shown two general, empirical properties of the fragment multiplicities called reducibility and thermal scaling [CITATION].', 'nucl-ex-0104013-1-4-0': 'Reducibility refers to the observation that for each energy bin the fragment multiplicities are distributed according to a binomial or Poissonian law.', 'nucl-ex-0104013-1-4-1': 'As such, their multiplicity distributions can be reduced to a one-fragment production probability according to a binomial or Poissonian distribution.', 'nucl-ex-0104013-1-5-0': 'Thermal scaling refers to the feature that the average fragment yield [MATH] behaves with temperature [MATH] as a Boltzmann factor: [MATH].', 'nucl-ex-0104013-1-5-1': 'Thus a plot of [MATH] vs. [MATH], an Arrhenius plot, should be linear.', 'nucl-ex-0104013-1-5-2': 'The slope [MATH] in such a plot is the one-fragment production "barrier".', 'nucl-ex-0104013-1-6-0': "Both the features of reducibility and thermal scaling are inherent to any statistical model, in particular to the clusterization of droplets from a vapor as described by Fisher's droplet model [CITATION].", 'nucl-ex-0104013-1-6-1': "Thus it is interesting to see if a system portraying reducibility and thermal scaling portrays also the scaling of Fisher's model [CITATION].", 'nucl-ex-0104013-1-6-2': "Fisher's droplet model describes the aggregation of molecules into clusters in a vapor, thus accounting for its non-ideality.", 'nucl-ex-0104013-1-6-3': 'The abundance of a cluster of size [MATH] is given by: [EQUATION] where [MATH] is the number of droplets (or fragments) of mass [MATH], normalized to the size of the system [MATH]; [MATH] is a normalization constant depending only on the value of [MATH] [CITATION]; [MATH] is the topological critical exponent; [MATH], and [MATH] and [MATH] are the actual and liquid chemical potentials respectively; [MATH] is the surface free energy of a droplet of size [MATH]; [MATH] is the zero temperature surface energy coefficient; [MATH] is the critical exponent related to the ratio of the dimensionality of the surface to that of the volume; and [MATH] is the control parameter, a measure of the distance from the critical point, [MATH].', 'nucl-ex-0104013-1-7-0': 'Equation ([REF]) shows clearly that thermal scaling is contained in this description.', 'nucl-ex-0104013-1-7-1': 'In particular, at coexistence ([MATH]), Eq. ([REF]) leads to a "barrier" [MATH].', 'nucl-ex-0104013-1-8-0': 'Recently, gold multifragmentation data from the ISiS Collaboration was shown to exhibit reducibility and thermal scaling in the fragment production probabilities [CITATION].', 'nucl-ex-0104013-1-8-1': "Since this behavior is inherent to Fisher's model, it is interesting to determine if Fisher's model describes the ISiS data set.", 'nucl-ex-0104013-1-8-2': 'In order to find if this is the case, the ISiS charge yields from 8 GeV/c [MATH]Au fragmentation data (see Fig [REF]a) were fit to Eq. ([REF]).', 'nucl-ex-0104013-1-9-0': 'The mass of a fragment [MATH] (estimated by multiplying the measured fragment charge [MATH] by the [MATH]-to-[MATH] ratio of the fragmenting system) was used as the cluster size in Eq. ([REF]).', 'nucl-ex-0104013-1-9-1': 'The total number of fragments of a given size [MATH] was normalized to the size of the fragmenting system [MATH] thus [MATH].', 'nucl-ex-0104013-1-9-2': 'Here [MATH] was used in lieu of [MATH], assuming the system behaves as a degenerate Fermi gas.', 'nucl-ex-0104013-1-10-0': "The parameters of Fisher's model, e.g. [MATH], [MATH], and [MATH] (in lieu of [MATH]) were used as fit parameters.", 'nucl-ex-0104013-1-10-1': 'The distance from equilibrium [MATH] was parameterized by a polynomial of degree four in [MATH] and the coefficients of that polynomial were used as fit parameters.', 'nucl-ex-0104013-1-10-2': 'The surface energy coefficient [MATH] was parameterized by a polynomial of degree one and the coefficients of that polynomial were used as fit parameters.', 'nucl-ex-0104013-1-10-3': 'The level density parameter was absorbed into the fit parameters for [MATH] and [MATH].', 'nucl-ex-0104013-1-11-0': 'While analyses similar to this one have been performed on multifragmentation data in the past [CITATION], those efforts dealt with inclusive data sets; data from every excitation energy were examined as a whole and the results were presented as a function of incident beam energy.', 'nucl-ex-0104013-1-11-1': 'This work makes use of the high statistics, exclusive data set of the ISiS Collaboration and bins the events in terms of reconstructed excitation energy [CITATION].', 'nucl-ex-0104013-1-11-2': "In addition, explicit use of Fisher's expressions for the bulk and surface energies allows [MATH] and [MATH] to be determined by the data.", 'nucl-ex-0104013-1-12-0': 'Data for [MATH] and for [MATH] were included in the fit to Eq. ([REF]) and the parameters were allowed to float to minimize [MATH].', 'nucl-ex-0104013-1-12-1': "Fisher's parameterization of the surface energy [MATH] is invalid for [MATH], thus excitation energies greater than [MATH] were not considered.", 'nucl-ex-0104013-1-12-2': "Fisher's model expresses the mass/energy of a fragment in terms of bulk and surface energies.", 'nucl-ex-0104013-1-12-3': 'This approximation is known to fail for the lightest of nuclei where structure details (shell effects) dominate the mass.', 'nucl-ex-0104013-1-12-4': 'For this reason and the fact that for the lightest fragments equilibrium and nonequilibrium production cannot be clearly differentiated, fragments with [MATH] were not considered in the fit.', 'nucl-ex-0104013-1-12-5': 'Fragments with [MATH] were not elementally resolved [CITATION], and were also excluded.', 'nucl-ex-0104013-1-13-0': 'The behavior of the data for the [MATH] surface is reproduced over a wide range in excitation energy and fragment charge.', 'nucl-ex-0104013-1-13-1': 'This is also shown in Fig. [REF].', 'nucl-ex-0104013-1-13-2': 'In this figure the results of the analysis are presented in terms of Arrhenius plots (Fig. [REF]b), or of the fragment yield distribution plots (Fig. [REF]c).', 'nucl-ex-0104013-1-13-3': 'A powerful method to observe the results of this analysis directly is to scale the data according to Eq. ([REF]) using the parameters resulting from the fitting procedure.', 'nucl-ex-0104013-1-13-4': 'Figure [REF] shows such a result.', 'nucl-ex-0104013-1-13-5': "The fragment mass yield distribution is scaled by the Fisher's power law pre-factor and the bulk term: [MATH].", 'nucl-ex-0104013-1-13-6': "This quantity is then plotted against the temperature scaled by Fisher's parameterization of the surface energy: [MATH].", 'nucl-ex-0104013-1-13-7': "The scaled data collapse over five orders of magnitude onto a single curve, which is precisely the behavior predicted by Fisher's droplet model [CITATION].", 'nucl-ex-0104013-1-13-8': 'This curve is equivalent to a liquid-vapor coexistence line, as will be shown below, and provides the best, most direct evidence yet for a liquid to vapor phase transition in excited nuclei.', 'nucl-ex-0104013-1-14-0': 'To illustrate the generality of this type of scaling, Fig. [REF] also shows the scaled cluster distributions from a three-dimensional Ising model calculation [CITATION].', 'nucl-ex-0104013-1-14-1': 'This system is know to model liquid-vapor coexistence up to the critical temperature.', 'nucl-ex-0104013-1-14-2': 'The perfect scaling of the cluster yields according to Eq. ([REF]) demonstrates liquid-vapor-like coexistence up to [MATH].', 'nucl-ex-0104013-1-15-0': "The value of [MATH] extracted from this analysis, [MATH], is in the range predicted by Fisher's model.", 'nucl-ex-0104013-1-15-1': 'Similarly, [MATH] is close to the value expected for a three dimensional system, [MATH].', 'nucl-ex-0104013-1-15-2': 'The values of [MATH] and [MATH] and their energy dependence are shown in Fig. [REF].', 'nucl-ex-0104013-1-15-3': 'The positive value of [MATH] for [MATH] indicates that the system behaves as a slightly super-saturated vapor.', 'nucl-ex-0104013-1-15-4': 'Values for [MATH] and [MATH] can be determined by making a choice of the level density parameter.', 'nucl-ex-0104013-1-15-5': 'For [MATH], [MATH] MeV while [MATH] gives [MATH] MeV; both values are in the range of the value of the surface energy coefficient of the semi-empirical mass formula ([MATH] MeV), although the linear extrapolation all the way down to [MATH] implied by Eq. ([REF]) is questionable.', 'nucl-ex-0104013-1-15-6': 'The values of the critical exponents determined here are in agreement with those determined for the EOS multifragmentation data [CITATION] and the value of the excitation energy at the critical point [MATH] MeV/nucleon is in the neighborhood of the value observed in the EOS analysis ([MATH] MeV/nucleon) [CITATION].', 'nucl-ex-0104013-1-15-7': 'The EOS analysis relied on the assumption that [MATH].', 'nucl-ex-0104013-1-15-8': 'No such assumption was made in this work.', 'nucl-ex-0104013-1-15-9': 'In fact the present effort tests that assumption and finds it to be approximately valid below the critical point since the observed [MATH] values are at most [MATH] MeV (assuming [MATH]).', 'nucl-ex-0104013-1-16-0': 'Using the value of [MATH] determined here and a level density parameter of [MATH], the critical temperature can be estimated to be on the order of [MATH] MeV, which is comparable to theoretical estimates for small nuclear systems [CITATION].', 'nucl-ex-0104013-1-16-1': 'In most theoretical calculations the Coulomb force and small size of the system drastically reduces the value of [MATH].', 'nucl-ex-0104013-1-16-2': 'It is also well known that both [MATH] and [MATH] scale as a function of system size in many systems [CITATION].', 'nucl-ex-0104013-1-17-0': 'The coexistence line and actual position of the fragmenting system in the pressure-density-temperature [MATH] diagram can be determined from this analysis.', 'nucl-ex-0104013-1-18-0': "Fisher's theory assumes that the non-ideal vapor can be approximated by an ideal gas of clusters.", 'nucl-ex-0104013-1-18-1': 'Accordingly, the quantity [MATH] is proportional to the partial pressure of a fragment of mass [MATH] and the total pressure due to all of the fragments is the sum of their partial pressures: [EQUATION]', 'nucl-ex-0104013-1-18-2': 'The reduced pressure is then given by: [EQUATION]', 'nucl-ex-0104013-1-18-3': 'This is a transformation of the information given in Fig. [REF] onto a more familiar frame of reference.', 'nucl-ex-0104013-1-19-0': 'When values of [MATH] corresponding to [MATH] are used in these equations, the coexistence curve is obtained.', 'nucl-ex-0104013-1-19-1': 'In other words, this analysis provides simultaneously the coexistence line and the actual line.', 'nucl-ex-0104013-1-19-2': 'Both are shown in Fig [REF]a.', 'nucl-ex-0104013-1-20-0': "The system's density can be found via [EQUATION] and the reduced density from [EQUATION]", 'nucl-ex-0104013-1-20-1': 'As before, both the actual curve and the coexistence curve can be determined.', 'nucl-ex-0104013-1-20-2': 'Both are shown in Fig. [REF]b together with the coexistence curve of Guggenheim [CITATION].', 'nucl-ex-0104013-1-21-0': 'The coexistence curve from the nuclear data does not agree closely with the Guggenheim plot over the full range of [MATH].', 'nucl-ex-0104013-1-21-1': 'This is not surprising given the complexity of the nuclear fluid compared to the simple fluids analyzed by Guggenheim.', 'nucl-ex-0104013-1-21-2': 'However, near [MATH], the universal behavior of the coexistence curve is recovered.', 'nucl-ex-0104013-1-22-0': 'The compressibility factor [MATH] is calculated by dividing the denominator of Eq. ([REF]) by that of Eq. ([REF]) and found to be [MATH].', 'nucl-ex-0104013-1-22-1': 'This value agrees with the value of the compressibility factor of many fluids [CITATION].', 'nucl-ex-0104013-1-23-0': "In conclusion this paper has shown that the data of the ISiS Collaboration contain the unequivocal signature of a liquid to vapor phase transition via their strict adherence to Fisher's droplet model.", 'nucl-ex-0104013-1-23-1': "Through Fisher's scaling of the fragment yield distribution (Fig. [REF]) the two phase coexistence line has been determined over a large energy/temperature interval extending up to and including the critical point.", 'nucl-ex-0104013-1-23-2': 'The critical exponents [MATH] and [MATH] as well as the approximate value of [MATH], the surface energy coefficient [MATH] and the compressibility factor have been extracted and agree with accepted values.', 'nucl-ex-0104013-1-23-3': 'A portion of the nuclear phase diagram has been constructed with the available data.', 'nucl-ex-0104013-1-23-4': 'Finally, the nuclear liquid at break-up was observed to be a slightly super-saturated vapor for which [MATH] and [MATH] were determined as a function of [MATH].', 'nucl-ex-0104013-1-24-0': 'The authors would like to thank Prof. Cathy Mader for her input and invaluable efforts with the Ising model calculations.', 'nucl-ex-0104013-1-24-1': 'This work was supported by the the US Department of Energy, National Science Foundation, the National Science and Engineering Research Council of Canada, the Polish State Committe for Scientific Reseach, Indiana Universtiy Office of Research, the University Graduate School, Simon Fraser University and the Robert A. Welch Foundation.'}

{'nucl-ex-0104013-2-0-0': 'For many years it has been speculated that excited nuclei would undergo a liquid to vapor phase transition.', 'nucl-ex-0104013-2-0-1': "For even longer, it has been known that clusterization in a vapor carries direct information on the liquid- vapor equilibrium according to Fisher's droplet model.", 'nucl-ex-0104013-2-0-2': "Now the thermal component of the [MATH] GeV/c [MATH]Au multifragmentation data of the ISiS Collaboration is shown to follow the scaling predicted by Fisher's model, thus providing the strongest evidence yet of the liquid to vapor phase transition.", 'nucl-ex-0104013-2-1-0': 'Nuclear multifragmentation, the break up of a nuclear system into several intermediate sized pieces, has been frequently discussed in terms of equilibrium statistical mechanics, and its possible association with a phase transition [CITATION].', 'nucl-ex-0104013-2-1-1': 'However, ample uncertainty remains regarding its nature, in particular whether multifragmentation is a phase transition and if so whether it is associated with the liquid to vapor phase transition.', 'nucl-ex-0104013-2-2-0': "This paper will show that: 1) high quality experimental data contain the unequivocal signature of a liquid to vapor phase transition through their strict adherence to Fisher's droplet model; 2) the two phase coexistence line is observed over a large energy/temperature interval extending up to and including the critical point; 3) several critical exponents, as well as the critical temperature, the surface energy coefficient and the compressibility factor can be directly extracted; 4) the nuclear phase diagram can be constructed with the available data; 5) the nuclear liquid at break-up is a slightly super-saturated vapor, for which the pressure and density can be determined as a function of the temperature.", 'nucl-ex-0104013-2-3-0': 'In past attempts to investigate the relationship between nuclear multifragmentation and a liquid to vapor phase transition, critical exponents have been determined [CITATION], caloric curves have been examined [CITATION], and the observation of negative heat capacities have been reported [CITATION].', 'nucl-ex-0104013-2-3-1': 'Other studies of multifragmentation data have shown two general, empirical properties of the fragment multiplicities called reducibility and thermal scaling [CITATION].', 'nucl-ex-0104013-2-4-0': 'Reducibility refers to the observation that for each energy bin the fragment multiplicities are distributed according to a binomial or Poissonian law.', 'nucl-ex-0104013-2-4-1': 'As such, their multiplicity distributions can be reduced to a one-fragment production probability according to a binomial or Poissonian distribution.', 'nucl-ex-0104013-2-5-0': 'Thermal scaling refers to the feature that the average fragment yield [MATH] behaves with temperature [MATH] as a Boltzmann factor: [MATH].', 'nucl-ex-0104013-2-5-1': 'Thus a plot of [MATH] vs. [MATH], an Arrhenius plot, should be linear.', 'nucl-ex-0104013-2-5-2': 'The slope [MATH] in such a plot is the one-fragment production "barrier".', 'nucl-ex-0104013-2-6-0': "Both the features of reducibility and thermal scaling are inherent to any statistical model, in particular to the clusterization of droplets from a vapor as described by Fisher's droplet model [CITATION].", 'nucl-ex-0104013-2-6-1': "Thus it is interesting to see if a system portraying reducibility and thermal scaling portrays also the scaling of Fisher's model [CITATION].", 'nucl-ex-0104013-2-6-2': "Fisher's droplet model describes the aggregation of molecules into clusters in a vapor, thus accounting for its non-ideality.", 'nucl-ex-0104013-2-6-3': 'The abundance of a cluster of size [MATH] is given by: [EQUATION] where [MATH] is the number of droplets (or fragments) of mass [MATH], normalized to the size of the system [MATH]; [MATH] is a normalization constant depending only on the value of [MATH] [CITATION]; [MATH] is the topological critical exponent; [MATH], and [MATH] and [MATH] are the actual and liquid chemical potentials respectively; [MATH] is the surface free energy of a droplet of size [MATH]; [MATH] is the zero temperature surface energy coefficient; [MATH] is the critical exponent related to the ratio of the dimensionality of the surface to that of the volume; and [MATH] is the control parameter, a measure of the distance from the critical point, [MATH].', 'nucl-ex-0104013-2-7-0': 'Equation ([REF]) shows clearly that thermal scaling is contained in this description.', 'nucl-ex-0104013-2-7-1': 'In particular, at coexistence ([MATH]), Eq. ([REF]) leads to a "barrier" [MATH].', 'nucl-ex-0104013-2-8-0': 'Recently, gold multifragmentation data from the ISiS Collaboration was shown to exhibit reducibility and thermal scaling in the fragment production probabilities [CITATION].', 'nucl-ex-0104013-2-8-1': "Since this behavior is inherent to Fisher's model, it is interesting to determine if Fisher's model describes the ISiS data set.", 'nucl-ex-0104013-2-8-2': 'In order to find if this is the case, the ISiS charge yields from 8 GeV/c [MATH]Au fragmentation data (see Fig [REF]a) were fit to Eq. ([REF]).', 'nucl-ex-0104013-2-9-0': 'The mass of a fragment [MATH] (estimated by multiplying the measured fragment charge [MATH] by the [MATH]-to-[MATH] ratio of the fragmenting system) was used as the cluster size in Eq. ([REF]).', 'nucl-ex-0104013-2-9-1': 'The total number of fragments of a given size [MATH] was normalized to the size of the fragmenting system [MATH] thus [MATH].', 'nucl-ex-0104013-2-9-2': 'Here [MATH] was used in lieu of [MATH], assuming the system behaves as a degenerate Fermi gas.', 'nucl-ex-0104013-2-10-0': "The parameters of Fisher's model, e.g. [MATH], [MATH], and [MATH] (in lieu of [MATH]) were used as fit parameters.", 'nucl-ex-0104013-2-10-1': 'The distance from equilibrium [MATH] was parameterized by a polynomial of degree four in [MATH] and the coefficients of that polynomial were used as fit parameters.', 'nucl-ex-0104013-2-10-2': 'The surface energy coefficient [MATH] was parameterized by a polynomial of degree one and the coefficients of that polynomial were used as fit parameters.', 'nucl-ex-0104013-2-10-3': 'The level density parameter was absorbed into the fit parameters for [MATH] and [MATH].', 'nucl-ex-0104013-2-11-0': 'While analyses similar to this one have been performed on multifragmentation data in the past [CITATION], those efforts dealt with inclusive data sets; data from every excitation energy were examined as a whole and the results were presented as a function of incident beam energy.', 'nucl-ex-0104013-2-11-1': 'This work makes use of the high statistics, exclusive data set of the ISiS Collaboration and bins the events in terms of reconstructed excitation energy [CITATION].', 'nucl-ex-0104013-2-11-2': "In addition, explicit use of Fisher's expressions for the bulk and surface energies allows [MATH] and [MATH] to be determined by the data.", 'nucl-ex-0104013-2-12-0': 'Data for [MATH] and for [MATH] were included in the fit to Eq. ([REF]) and the parameters were allowed to float to minimize [MATH].', 'nucl-ex-0104013-2-12-1': "Fisher's parameterization of the surface energy [MATH] is invalid for [MATH], thus excitation energies greater than [MATH] were not considered.", 'nucl-ex-0104013-2-12-2': "Fisher's model expresses the mass/energy of a fragment in terms of bulk and surface energies.", 'nucl-ex-0104013-2-12-3': 'This approximation is known to fail for the lightest of nuclei where structure details (shell effects) dominate the mass.', 'nucl-ex-0104013-2-12-4': 'For this reason and the fact that for the lightest fragments equilibrium and nonequilibrium production cannot be clearly differentiated, fragments with [MATH] were not considered in the fit.', 'nucl-ex-0104013-2-12-5': 'Fragments with [MATH] were not elementally resolved [CITATION], and were also excluded.', 'nucl-ex-0104013-2-13-0': 'The behavior of the data for the [MATH] surface is reproduced over a wide range in excitation energy and fragment charge.', 'nucl-ex-0104013-2-13-1': 'This is also shown in Fig. [REF].', 'nucl-ex-0104013-2-13-2': 'In this figure the results of the analysis are presented in terms of Arrhenius plots (Fig. [REF]b), or of the fragment yield distribution plots (Fig. [REF]c).', 'nucl-ex-0104013-2-13-3': 'A powerful method to observe the results of this analysis directly is to scale the data according to Eq. ([REF]) using the parameters resulting from the fitting procedure.', 'nucl-ex-0104013-2-13-4': 'Figure [REF] shows such a result.', 'nucl-ex-0104013-2-13-5': "The fragment mass yield distribution is scaled by the Fisher's power law pre-factor and the bulk term: [MATH].", 'nucl-ex-0104013-2-13-6': "This quantity is then plotted against the temperature scaled by Fisher's parameterization of the surface energy: [MATH].", 'nucl-ex-0104013-2-13-7': "The scaled data collapse over five orders of magnitude onto a single curve, which is precisely the behavior predicted by Fisher's droplet model [CITATION].", 'nucl-ex-0104013-2-13-8': 'This curve is equivalent to a liquid-vapor coexistence line, as will be shown below, and provides the best, most direct evidence yet for a liquid to vapor phase transition in excited nuclei.', 'nucl-ex-0104013-2-14-0': 'To illustrate the generality of this type of scaling, Fig. [REF] also shows the scaled cluster distributions from a three-dimensional Ising model calculation [CITATION].', 'nucl-ex-0104013-2-14-1': 'This system is know to model liquid-vapor coexistence up to the critical temperature.', 'nucl-ex-0104013-2-14-2': 'The perfect scaling of the cluster yields according to Eq. ([REF]) demonstrates liquid-vapor-like coexistence up to [MATH].', 'nucl-ex-0104013-2-15-0': "The value of [MATH] extracted from this analysis, [MATH], is in the range predicted by Fisher's model.", 'nucl-ex-0104013-2-15-1': 'Similarly, [MATH] is close to the value expected for a three dimensional system, [MATH].', 'nucl-ex-0104013-2-15-2': 'The values of [MATH] and [MATH] and their energy dependence are shown in Fig. [REF].', 'nucl-ex-0104013-2-15-3': 'The positive value of [MATH] for [MATH] indicates that the system behaves as a slightly super-saturated vapor.', 'nucl-ex-0104013-2-15-4': 'Values for [MATH] and [MATH] can be determined by making a choice of the level density parameter.', 'nucl-ex-0104013-2-15-5': 'For [MATH], [MATH] MeV while [MATH] gives [MATH] MeV; both values are in the range of the value of the surface energy coefficient of the semi-empirical mass formula ([MATH] MeV), although the linear extrapolation all the way down to [MATH] implied by Eq. ([REF]) is questionable.', 'nucl-ex-0104013-2-15-6': 'The values of the critical exponents determined here are in agreement with those determined for the EOS multifragmentation data [CITATION] and the value of the excitation energy at the critical point [MATH] MeV/nucleon is in the neighborhood of the value observed in the EOS analysis ([MATH] MeV/nucleon) [CITATION].', 'nucl-ex-0104013-2-15-7': 'The EOS analysis relied on the assumption that [MATH].', 'nucl-ex-0104013-2-15-8': 'No such assumption was made in this work.', 'nucl-ex-0104013-2-15-9': 'In fact the present effort tests that assumption and finds it to be approximately valid below the critical point since the observed [MATH] values are at most [MATH] MeV (assuming [MATH]).', 'nucl-ex-0104013-2-16-0': 'Using the value of [MATH] determined here and a level density parameter of [MATH], the critical temperature can be estimated to be on the order of [MATH] MeV, which is comparable to theoretical estimates for small nuclear systems [CITATION].', 'nucl-ex-0104013-2-16-1': 'In most theoretical calculations the Coulomb force and small size of the system drastically reduces the value of [MATH].', 'nucl-ex-0104013-2-16-2': 'It is also well known that both [MATH] and [MATH] scale as a function of system size in many systems [CITATION].', 'nucl-ex-0104013-2-17-0': 'The coexistence line and actual position of the fragmenting system in the pressure-density-temperature [MATH] diagram can be determined from this analysis.', 'nucl-ex-0104013-2-18-0': "Fisher's theory assumes that the non-ideal vapor can be approximated by an ideal gas of clusters.", 'nucl-ex-0104013-2-18-1': 'Accordingly, the quantity [MATH] is proportional to the partial pressure of a fragment of mass [MATH] and the total pressure due to all of the fragments is the sum of their partial pressures: [EQUATION]', 'nucl-ex-0104013-2-18-2': 'The reduced pressure is then given by: [EQUATION]', 'nucl-ex-0104013-2-18-3': 'This is a transformation of the information given in Fig. [REF] onto a more familiar frame of reference.', 'nucl-ex-0104013-2-19-0': 'When values of [MATH] corresponding to [MATH] are used in these equations, the coexistence curve is obtained.', 'nucl-ex-0104013-2-19-1': 'In other words, this analysis provides simultaneously the coexistence line and the actual line.', 'nucl-ex-0104013-2-19-2': 'Both are shown in Fig [REF]a.', 'nucl-ex-0104013-2-20-0': "The system's density can be found via [EQUATION] and the reduced density from [EQUATION]", 'nucl-ex-0104013-2-20-1': 'As before, both the actual curve and the coexistence curve can be determined.', 'nucl-ex-0104013-2-20-2': 'Both are shown in Fig. [REF]b together with the coexistence curve of Guggenheim [CITATION].', 'nucl-ex-0104013-2-21-0': 'The coexistence curve from the nuclear data does not agree closely with the Guggenheim plot over the full range of [MATH].', 'nucl-ex-0104013-2-21-1': 'This is not surprising given the complexity of the nuclear fluid compared to the simple fluids analyzed by Guggenheim.', 'nucl-ex-0104013-2-21-2': 'However, near [MATH], the universal behavior of the coexistence curve is recovered.', 'nucl-ex-0104013-2-22-0': 'The compressibility factor [MATH] is calculated by dividing the denominator of Eq. ([REF]) by that of Eq. ([REF]) and found to be [MATH].', 'nucl-ex-0104013-2-22-1': 'This value agrees with the value of the compressibility factor of many fluids [CITATION].', 'nucl-ex-0104013-2-23-0': "In conclusion this paper has shown that the data of the ISiS Collaboration contain the unequivocal signature of a liquid to vapor phase transition via their strict adherence to Fisher's droplet model.", 'nucl-ex-0104013-2-23-1': "Through Fisher's scaling of the fragment yield distribution (Fig. [REF]) the two phase coexistence line has been determined over a large energy/temperature interval extending up to and including the critical point.", 'nucl-ex-0104013-2-23-2': 'The critical exponents [MATH] and [MATH] as well as the approximate value of [MATH], the surface energy coefficient [MATH] and the compressibility factor have been extracted and agree with accepted values.', 'nucl-ex-0104013-2-23-3': 'A portion of the nuclear phase diagram has been constructed with the available data.', 'nucl-ex-0104013-2-23-4': 'Finally, the nuclear liquid at break-up was observed to be a slightly super-saturated vapor for which [MATH] and [MATH] were determined as a function of [MATH].', 'nucl-ex-0104013-2-24-0': 'The authors would like to thank Prof. Cathy Mader for her input and invaluable efforts with the Ising model calculations.', 'nucl-ex-0104013-2-24-1': 'This work was supported by the the US Department of Energy, National Science Foundation, the National Science and Engineering Research Council of Canada, the Polish State Committe for Scientific Reseach, Indiana Universtiy Office of Research, the University Graduate School, Simon Fraser University and the Robert A. Welch Foundation.'}

[['nucl-ex-0104013-1-9-0', 'nucl-ex-0104013-2-9-0'], ['nucl-ex-0104013-1-9-1', 'nucl-ex-0104013-2-9-1'], ['nucl-ex-0104013-1-9-2', 'nucl-ex-0104013-2-9-2'], ['nucl-ex-0104013-1-22-0', 'nucl-ex-0104013-2-22-0'], ['nucl-ex-0104013-1-22-1', 'nucl-ex-0104013-2-22-1'], ['nucl-ex-0104013-1-4-0', 'nucl-ex-0104013-2-4-0'], ['nucl-ex-0104013-1-4-1', 'nucl-ex-0104013-2-4-1'], ['nucl-ex-0104013-1-5-0', 'nucl-ex-0104013-2-5-0'], ['nucl-ex-0104013-1-5-1', 'nucl-ex-0104013-2-5-1'], ['nucl-ex-0104013-1-5-2', 'nucl-ex-0104013-2-5-2'], ['nucl-ex-0104013-1-2-0', 'nucl-ex-0104013-2-2-0'], ['nucl-ex-0104013-1-11-0', 'nucl-ex-0104013-2-11-0'], ['nucl-ex-0104013-1-11-1', 'nucl-ex-0104013-2-11-1'], ['nucl-ex-0104013-1-11-2', 'nucl-ex-0104013-2-11-2'], ['nucl-ex-0104013-1-10-0', 'nucl-ex-0104013-2-10-0'], ['nucl-ex-0104013-1-10-1', 'nucl-ex-0104013-2-10-1'], ['nucl-ex-0104013-1-10-2', 'nucl-ex-0104013-2-10-2'], ['nucl-ex-0104013-1-10-3', 'nucl-ex-0104013-2-10-3'], ['nucl-ex-0104013-1-20-0', 'nucl-ex-0104013-2-20-0'], ['nucl-ex-0104013-1-20-1', 'nucl-ex-0104013-2-20-1'], ['nucl-ex-0104013-1-20-2', 'nucl-ex-0104013-2-20-2'], ['nucl-ex-0104013-1-1-0', 'nucl-ex-0104013-2-1-0'], ['nucl-ex-0104013-1-1-1', 'nucl-ex-0104013-2-1-1'], ['nucl-ex-0104013-1-18-0', 'nucl-ex-0104013-2-18-0'], ['nucl-ex-0104013-1-18-1', 'nucl-ex-0104013-2-18-1'], ['nucl-ex-0104013-1-18-2', 'nucl-ex-0104013-2-18-2'], ['nucl-ex-0104013-1-18-3', 'nucl-ex-0104013-2-18-3'], ['nucl-ex-0104013-1-12-0', 'nucl-ex-0104013-2-12-0'], ['nucl-ex-0104013-1-12-1', 'nucl-ex-0104013-2-12-1'], ['nucl-ex-0104013-1-12-2', 'nucl-ex-0104013-2-12-2'], ['nucl-ex-0104013-1-12-3', 'nucl-ex-0104013-2-12-3'], ['nucl-ex-0104013-1-12-4', 'nucl-ex-0104013-2-12-4'], ['nucl-ex-0104013-1-12-5', 'nucl-ex-0104013-2-12-5'], ['nucl-ex-0104013-1-19-0', 'nucl-ex-0104013-2-19-0'], ['nucl-ex-0104013-1-19-1', 'nucl-ex-0104013-2-19-1'], ['nucl-ex-0104013-1-19-2', 'nucl-ex-0104013-2-19-2'], ['nucl-ex-0104013-1-8-0', 'nucl-ex-0104013-2-8-0'], ['nucl-ex-0104013-1-8-1', 'nucl-ex-0104013-2-8-1'], ['nucl-ex-0104013-1-8-2', 'nucl-ex-0104013-2-8-2'], ['nucl-ex-0104013-1-21-0', 'nucl-ex-0104013-2-21-0'], ['nucl-ex-0104013-1-21-1', 'nucl-ex-0104013-2-21-1'], ['nucl-ex-0104013-1-21-2', 'nucl-ex-0104013-2-21-2'], ['nucl-ex-0104013-1-7-0', 'nucl-ex-0104013-2-7-0'], ['nucl-ex-0104013-1-7-1', 'nucl-ex-0104013-2-7-1'], ['nucl-ex-0104013-1-6-0', 'nucl-ex-0104013-2-6-0'], ['nucl-ex-0104013-1-6-1', 'nucl-ex-0104013-2-6-1'], ['nucl-ex-0104013-1-6-2', 'nucl-ex-0104013-2-6-2'], ['nucl-ex-0104013-1-6-3', 'nucl-ex-0104013-2-6-3'], ['nucl-ex-0104013-1-13-0', 'nucl-ex-0104013-2-13-0'], ['nucl-ex-0104013-1-13-1', 'nucl-ex-0104013-2-13-1'], ['nucl-ex-0104013-1-13-2', 'nucl-ex-0104013-2-13-2'], ['nucl-ex-0104013-1-13-3', 'nucl-ex-0104013-2-13-3'], ['nucl-ex-0104013-1-13-4', 'nucl-ex-0104013-2-13-4'], ['nucl-ex-0104013-1-13-5', 'nucl-ex-0104013-2-13-5'], ['nucl-ex-0104013-1-13-6', 'nucl-ex-0104013-2-13-6'], ['nucl-ex-0104013-1-13-7', 'nucl-ex-0104013-2-13-7'], ['nucl-ex-0104013-1-13-8', 'nucl-ex-0104013-2-13-8'], ['nucl-ex-0104013-1-15-0', 'nucl-ex-0104013-2-15-0'], ['nucl-ex-0104013-1-15-1', 'nucl-ex-0104013-2-15-1'], ['nucl-ex-0104013-1-15-2', 'nucl-ex-0104013-2-15-2'], ['nucl-ex-0104013-1-15-3', 'nucl-ex-0104013-2-15-3'], ['nucl-ex-0104013-1-15-4', 'nucl-ex-0104013-2-15-4'], ['nucl-ex-0104013-1-15-5', 'nucl-ex-0104013-2-15-5'], ['nucl-ex-0104013-1-15-6', 'nucl-ex-0104013-2-15-6'], ['nucl-ex-0104013-1-15-7', 'nucl-ex-0104013-2-15-7'], ['nucl-ex-0104013-1-15-8', 'nucl-ex-0104013-2-15-8'], ['nucl-ex-0104013-1-15-9', 'nucl-ex-0104013-2-15-9'], ['nucl-ex-0104013-1-16-0', 'nucl-ex-0104013-2-16-0'], ['nucl-ex-0104013-1-16-1', 'nucl-ex-0104013-2-16-1'], ['nucl-ex-0104013-1-16-2', 'nucl-ex-0104013-2-16-2'], ['nucl-ex-0104013-1-24-0', 'nucl-ex-0104013-2-24-0'], ['nucl-ex-0104013-1-24-1', 'nucl-ex-0104013-2-24-1'], ['nucl-ex-0104013-1-14-0', 'nucl-ex-0104013-2-14-0'], ['nucl-ex-0104013-1-14-1', 'nucl-ex-0104013-2-14-1'], ['nucl-ex-0104013-1-14-2', 'nucl-ex-0104013-2-14-2'], ['nucl-ex-0104013-1-23-0', 'nucl-ex-0104013-2-23-0'], ['nucl-ex-0104013-1-23-1', 'nucl-ex-0104013-2-23-1'], ['nucl-ex-0104013-1-23-2', 'nucl-ex-0104013-2-23-2'], ['nucl-ex-0104013-1-23-3', 'nucl-ex-0104013-2-23-3'], ['nucl-ex-0104013-1-23-4', 'nucl-ex-0104013-2-23-4'], ['nucl-ex-0104013-1-17-0', 'nucl-ex-0104013-2-17-0'], ['nucl-ex-0104013-1-3-0', 'nucl-ex-0104013-2-3-0'], ['nucl-ex-0104013-1-3-1', 'nucl-ex-0104013-2-3-1'], ['nucl-ex-0104013-1-0-0', 'nucl-ex-0104013-2-0-0'], ['nucl-ex-0104013-1-0-1', 'nucl-ex-0104013-2-0-1'], ['nucl-ex-0104013-1-0-2', 'nucl-ex-0104013-2-0-2']]

[['nucl-ex-0104013-1-9-0', 'nucl-ex-0104013-2-9-0'], ['nucl-ex-0104013-1-9-1', 'nucl-ex-0104013-2-9-1'], ['nucl-ex-0104013-1-9-2', 'nucl-ex-0104013-2-9-2'], ['nucl-ex-0104013-1-22-0', 'nucl-ex-0104013-2-22-0'], ['nucl-ex-0104013-1-22-1', 'nucl-ex-0104013-2-22-1'], ['nucl-ex-0104013-1-4-0', 'nucl-ex-0104013-2-4-0'], ['nucl-ex-0104013-1-4-1', 'nucl-ex-0104013-2-4-1'], ['nucl-ex-0104013-1-5-0', 'nucl-ex-0104013-2-5-0'], ['nucl-ex-0104013-1-5-1', 'nucl-ex-0104013-2-5-1'], ['nucl-ex-0104013-1-5-2', 'nucl-ex-0104013-2-5-2'], ['nucl-ex-0104013-1-2-0', 'nucl-ex-0104013-2-2-0'], ['nucl-ex-0104013-1-11-0', 'nucl-ex-0104013-2-11-0'], ['nucl-ex-0104013-1-11-1', 'nucl-ex-0104013-2-11-1'], ['nucl-ex-0104013-1-11-2', 'nucl-ex-0104013-2-11-2'], ['nucl-ex-0104013-1-10-0', 'nucl-ex-0104013-2-10-0'], ['nucl-ex-0104013-1-10-1', 'nucl-ex-0104013-2-10-1'], ['nucl-ex-0104013-1-10-2', 'nucl-ex-0104013-2-10-2'], ['nucl-ex-0104013-1-10-3', 'nucl-ex-0104013-2-10-3'], ['nucl-ex-0104013-1-20-0', 'nucl-ex-0104013-2-20-0'], ['nucl-ex-0104013-1-20-1', 'nucl-ex-0104013-2-20-1'], ['nucl-ex-0104013-1-20-2', 'nucl-ex-0104013-2-20-2'], ['nucl-ex-0104013-1-1-0', 'nucl-ex-0104013-2-1-0'], ['nucl-ex-0104013-1-1-1', 'nucl-ex-0104013-2-1-1'], ['nucl-ex-0104013-1-18-0', 'nucl-ex-0104013-2-18-0'], ['nucl-ex-0104013-1-18-1', 'nucl-ex-0104013-2-18-1'], ['nucl-ex-0104013-1-18-2', 'nucl-ex-0104013-2-18-2'], ['nucl-ex-0104013-1-18-3', 'nucl-ex-0104013-2-18-3'], ['nucl-ex-0104013-1-12-0', 'nucl-ex-0104013-2-12-0'], ['nucl-ex-0104013-1-12-1', 'nucl-ex-0104013-2-12-1'], ['nucl-ex-0104013-1-12-2', 'nucl-ex-0104013-2-12-2'], ['nucl-ex-0104013-1-12-3', 'nucl-ex-0104013-2-12-3'], ['nucl-ex-0104013-1-12-4', 'nucl-ex-0104013-2-12-4'], ['nucl-ex-0104013-1-12-5', 'nucl-ex-0104013-2-12-5'], ['nucl-ex-0104013-1-19-0', 'nucl-ex-0104013-2-19-0'], ['nucl-ex-0104013-1-19-1', 'nucl-ex-0104013-2-19-1'], ['nucl-ex-0104013-1-19-2', 'nucl-ex-0104013-2-19-2'], ['nucl-ex-0104013-1-8-0', 'nucl-ex-0104013-2-8-0'], ['nucl-ex-0104013-1-8-1', 'nucl-ex-0104013-2-8-1'], ['nucl-ex-0104013-1-8-2', 'nucl-ex-0104013-2-8-2'], ['nucl-ex-0104013-1-21-0', 'nucl-ex-0104013-2-21-0'], ['nucl-ex-0104013-1-21-1', 'nucl-ex-0104013-2-21-1'], ['nucl-ex-0104013-1-21-2', 'nucl-ex-0104013-2-21-2'], ['nucl-ex-0104013-1-7-0', 'nucl-ex-0104013-2-7-0'], ['nucl-ex-0104013-1-7-1', 'nucl-ex-0104013-2-7-1'], ['nucl-ex-0104013-1-6-0', 'nucl-ex-0104013-2-6-0'], ['nucl-ex-0104013-1-6-1', 'nucl-ex-0104013-2-6-1'], ['nucl-ex-0104013-1-6-2', 'nucl-ex-0104013-2-6-2'], ['nucl-ex-0104013-1-6-3', 'nucl-ex-0104013-2-6-3'], ['nucl-ex-0104013-1-13-0', 'nucl-ex-0104013-2-13-0'], ['nucl-ex-0104013-1-13-1', 'nucl-ex-0104013-2-13-1'], ['nucl-ex-0104013-1-13-2', 'nucl-ex-0104013-2-13-2'], ['nucl-ex-0104013-1-13-3', 'nucl-ex-0104013-2-13-3'], ['nucl-ex-0104013-1-13-4', 'nucl-ex-0104013-2-13-4'], ['nucl-ex-0104013-1-13-5', 'nucl-ex-0104013-2-13-5'], ['nucl-ex-0104013-1-13-6', 'nucl-ex-0104013-2-13-6'], ['nucl-ex-0104013-1-13-7', 'nucl-ex-0104013-2-13-7'], ['nucl-ex-0104013-1-13-8', 'nucl-ex-0104013-2-13-8'], ['nucl-ex-0104013-1-15-0', 'nucl-ex-0104013-2-15-0'], ['nucl-ex-0104013-1-15-1', 'nucl-ex-0104013-2-15-1'], ['nucl-ex-0104013-1-15-2', 'nucl-ex-0104013-2-15-2'], ['nucl-ex-0104013-1-15-3', 'nucl-ex-0104013-2-15-3'], ['nucl-ex-0104013-1-15-4', 'nucl-ex-0104013-2-15-4'], ['nucl-ex-0104013-1-15-5', 'nucl-ex-0104013-2-15-5'], ['nucl-ex-0104013-1-15-6', 'nucl-ex-0104013-2-15-6'], ['nucl-ex-0104013-1-15-7', 'nucl-ex-0104013-2-15-7'], ['nucl-ex-0104013-1-15-8', 'nucl-ex-0104013-2-15-8'], ['nucl-ex-0104013-1-15-9', 'nucl-ex-0104013-2-15-9'], ['nucl-ex-0104013-1-16-0', 'nucl-ex-0104013-2-16-0'], ['nucl-ex-0104013-1-16-1', 'nucl-ex-0104013-2-16-1'], ['nucl-ex-0104013-1-16-2', 'nucl-ex-0104013-2-16-2'], ['nucl-ex-0104013-1-24-0', 'nucl-ex-0104013-2-24-0'], ['nucl-ex-0104013-1-24-1', 'nucl-ex-0104013-2-24-1'], ['nucl-ex-0104013-1-14-0', 'nucl-ex-0104013-2-14-0'], ['nucl-ex-0104013-1-14-1', 'nucl-ex-0104013-2-14-1'], ['nucl-ex-0104013-1-14-2', 'nucl-ex-0104013-2-14-2'], ['nucl-ex-0104013-1-23-0', 'nucl-ex-0104013-2-23-0'], ['nucl-ex-0104013-1-23-1', 'nucl-ex-0104013-2-23-1'], ['nucl-ex-0104013-1-23-2', 'nucl-ex-0104013-2-23-2'], ['nucl-ex-0104013-1-23-3', 'nucl-ex-0104013-2-23-3'], ['nucl-ex-0104013-1-23-4', 'nucl-ex-0104013-2-23-4'], ['nucl-ex-0104013-1-17-0', 'nucl-ex-0104013-2-17-0'], ['nucl-ex-0104013-1-3-0', 'nucl-ex-0104013-2-3-0'], ['nucl-ex-0104013-1-3-1', 'nucl-ex-0104013-2-3-1'], ['nucl-ex-0104013-1-0-0', 'nucl-ex-0104013-2-0-0'], ['nucl-ex-0104013-1-0-1', 'nucl-ex-0104013-2-0-1'], ['nucl-ex-0104013-1-0-2', 'nucl-ex-0104013-2-0-2']]

[]

[]

[]

[]

[]

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

https://arxiv.org/abs/nucl-ex/0104013

1505.06912

{'1505.06912-1-0-0': 'Relations between subexponential densities and locally subexponential distributions are discussed.', '1505.06912-1-0-1': 'It is shown that the class of subexponential densities is neither closed under convolution roots nor closed under asymptotic equivalence.', '1505.06912-1-0-2': 'A remark is given on the closure under convolution roots for the class of convolution equivalent distributions.', '1505.06912-1-1-0': 'Key words or phrases: subexponential densities, local subexponentiality, convolution roots, asymptotic equivalence', '1505.06912-1-2-0': 'Mathematics Subject Classification: 60E99, 60G50', '1505.06912-1-3-0': '# Introduction and main results', '1505.06912-1-4-0': 'In what follows, we denote by [MATH] the real line and by [MATH] the half line [MATH].', '1505.06912-1-4-1': 'Let [MATH] be the totality of positive integers.', '1505.06912-1-4-2': 'The symbol [MATH] stands for the delta measure at [MATH].', '1505.06912-1-4-3': 'Let [MATH] and [MATH] be probability measures on [MATH].', '1505.06912-1-4-4': 'We denote the convolution of [MATH] and [MATH] by [MATH] and denote [MATH]-th convolution power of [MATH] by [MATH].', '1505.06912-1-4-5': 'Let [MATH] and [MATH] be integrable functions on [MATH].', '1505.06912-1-4-6': 'We denote by [MATH]-th convolution power of [MATH] and by [MATH] the convolution of [MATH] and [MATH].', '1505.06912-1-4-7': 'For positive functions [MATH] and [MATH] on [MATH] for some [MATH], we define the relation [MATH] by [MATH].', '1505.06912-1-4-8': 'We also define the relation [MATH] for positive sequences [MATH] and [MATH] with [MATH] by [MATH].', '1505.06912-1-4-9': 'We define the class [MATH] as the totality of probability distributions on [MATH].', '1505.06912-1-4-10': 'In this paper, we prove that the class of subexponential densities is not closed under two important closure properties.', '1505.06912-1-5-0': '(i) A nonnegative measurable function [MATH] on [MATH] belongs to the class [MATH] if [MATH] for all sufficiently large [MATH] and if [MATH] for any [MATH].', '1505.06912-1-6-0': '(ii) A distribution [MATH] on [MATH] belongs to the class [MATH] if there is [MATH] such that [MATH].', '1505.06912-1-7-0': '(iii) A distribution [MATH] on [MATH] belongs to the class [MATH] if there is [MATH] such that [MATH] and [MATH].', '1505.06912-1-8-0': 'Whenever we write [MATH] or [MATH], we promise that [MATH].', '1505.06912-1-8-1': 'Densities of distributions in the class [MATH] are called subexponential densities and those in the class [MATH] are called long-tailed densities.', '1505.06912-1-8-2': 'The study on the class [MATH] goes back to Chover et al. [CITATION].', '1505.06912-1-8-3': 'Let [MATH] be a distribution on [MATH].', '1505.06912-1-8-4': 'Note that [MATH] is a density function on [MATH] for every [MATH].', '1505.06912-1-9-0': '(i) Let [MATH] with [MATH].', '1505.06912-1-9-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH].', '1505.06912-1-10-0': '(ii) Let [MATH] with [MATH].', '1505.06912-1-10-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] and [MATH]', '1505.06912-1-11-0': '(iii) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] for each [MATH] with [MATH]', '1505.06912-1-12-0': '(iv) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] for each [MATH] with [MATH]', '1505.06912-1-13-0': '(v) A distribution [MATH] belongs to the class [MATH] if there exists [MATH] such that [MATH] uniformly in [MATH].', '1505.06912-1-14-0': 'Distributions in the class [MATH] are called locally subexponential, those in the class [MATH] are called uniformly locally subexponential.', '1505.06912-1-14-1': 'The class [MATH] is introduced by Asmussen et al. [CITATION] and the class [MATH] is by Watanabe and Yamamuro [CITATION].', '1505.06912-1-14-2': 'Detailed acounts of the classes [MATH] and [MATH] are found in the book of Foss et al. [CITATION].', '1505.06912-1-14-3': 'First, we present some interesting results on the classes [MATH] and [MATH].', '1505.06912-1-15-0': '(i) We say that a class [MATH] of probability distributions on [MATH] is closed under convolution roots if [MATH] for some [MATH] implies that [MATH].', '1505.06912-1-16-0': '(ii) Let [MATH] and [MATH] be density functions on [MATH].', '1505.06912-1-16-1': 'We say that a class [MATH] of absolutely continuous probability distributions is closed under asymptotic equivalence if [MATH] and [MATH] with [MATH] implies that [MATH].', '1505.06912-1-17-0': 'The class [MATH] is a proper subclass of the class [MATH] because a distribution in [MATH] can have a point mass.', '1505.06912-1-17-1': 'Moreover, the class [MATH] is a proper subclass of the class [MATH] as the following theorem shows.', '1505.06912-1-18-0': '[EQUATION]', '1505.06912-1-18-1': 'We have the following.', '1505.06912-1-19-0': '(i) The class [MATH] is not closed under convolution roots.', '1505.06912-1-20-0': '(ii) The class [MATH] is not closed under convolution roots.', '1505.06912-1-21-0': '(iii) The class [MATH] is not closed under convolution roots.', '1505.06912-1-22-0': 'The class [MATH] is closed under asymptotic equivalence in the one-sided case.', '1505.06912-1-22-1': 'See (ii) of Lemma 2.1 below.', '1505.06912-1-22-2': 'However, Foss et al. [CITATION] suggest the possibility of non-closure under asymptotic equivalence for the class [MATH] in the two-sided case.', '1505.06912-1-22-3': 'We exactly prove it as follows.', '1505.06912-1-23-0': 'The class [MATH] is not closed under asymptotic equivalence, that is, there exist [MATH] and [MATH] such that [MATH] with [MATH].', '1505.06912-1-24-0': 'In Sect. 2, we prove Proposition 1.1.', '1505.06912-1-24-1': 'In Sect. 3, we prove Theorems 1.1 and 1.2.', '1505.06912-1-24-2': 'In Sect. 4, we give a remark on the closure under convolution roots.', '1505.06912-1-25-0': '# Proof of Proposition 1.1', '1505.06912-1-26-0': 'We present two lemmas for the proofs of main results and then prove Proposition 1.1.', '1505.06912-1-26-1': 'Let [MATH] and [MATH] be density functions on [MATH].', '1505.06912-1-27-0': '(i) If [MATH], then [MATH] for every [MATH].', '1505.06912-1-28-0': '(ii) If [MATH] and [MATH] with [MATH], then [MATH].', '1505.06912-1-29-0': '(iii) Assume that [MATH].', '1505.06912-1-29-1': 'Then, [MATH] if and only if [EQUATION]', '1505.06912-1-29-2': 'Proof Proof of assertion (i) is due to Theorem 4.3 of [CITATION].', '1505.06912-1-29-3': 'Proofs of assertions (ii) and (iii) are due to Theorems 4.8 and 4.7 of [CITATION], respectively.', '1505.06912-1-30-0': '(i) Let [MATH] with [MATH].', '1505.06912-1-30-1': 'Assume that [MATH].', '1505.06912-1-30-2': 'Then, [MATH] if and only if [EQUATION] (ii) Assume that [MATH].', '1505.06912-1-30-3': 'Then, [MATH] for every [MATH].', '1505.06912-1-30-4': 'Moreover, [MATH] for every [MATH].', '1505.06912-1-31-0': '(iii) If [MATH] and [MATH] with [MATH] for every [MATH], then [MATH].', '1505.06912-1-32-0': 'Proof Proof of assertion (i) is due to Theorem 4.21 of of [CITATION].', '1505.06912-1-32-1': 'First assertion of (ii) is due to Corollary 4.19 of [CITATION].', '1505.06912-1-32-2': 'Second one is proved as (2.6) in Theorem 2.1 of [CITATION].', '1505.06912-1-32-3': 'Proof of assertion (iii) is due to Theorem 4.22 of [CITATION].', '1505.06912-1-33-0': 'Proof of (i) of Proposition 1.1 Let [MATH].', '1505.06912-1-33-1': 'First, we prove that if [MATH], then [MATH].', '1505.06912-1-33-2': 'We can assume that [MATH].', '1505.06912-1-33-3': 'Suppose that [MATH].', '1505.06912-1-33-4': 'We have [MATH] and hence [MATH].', '1505.06912-1-33-5': 'Let [MATH] be a positive integer, [MATH], and let [MATH] be independent random variables with the same distribution [MATH].', '1505.06912-1-33-6': 'Then, we have for [MATH] [EQUATION]', '1505.06912-1-33-7': 'Since [MATH], we obtain from (i) of Lemma 2.2 that [EQUATION].', '1505.06912-1-33-8': 'Thus, we see from (iii) of Lemma 2.1 that [MATH].', '1505.06912-1-34-0': 'Conversely, suppose that [MATH].', '1505.06912-1-34-1': 'Then, we have [MATH].', '1505.06912-1-34-2': 'Let [MATH] be the largest integer not exceeding a real number [MATH].', '1505.06912-1-34-3': 'Choose sufficiently large integer [MATH].', '1505.06912-1-34-4': 'Note that there are positive constants [MATH] for [MATH] such that [EQUATION] for [MATH], [MATH], and [MATH].', '1505.06912-1-34-5': 'Thus, we find that [EQUATION]', '1505.06912-1-34-6': 'Since [MATH], we establish from (iii) of Lemma 2.1 that [EQUATION].', '1505.06912-1-34-7': 'Thus, [MATH] by (i) of Lemma 2.2.', '1505.06912-1-34-8': 'Note from (ii) of Lemma 2.2 that if [MATH], then [MATH] for every [MATH].', '1505.06912-1-34-9': 'Thus, the second assertion is true.', '1505.06912-1-35-0': 'Proof of (ii) of Proposition 1.1 Suppose that [MATH] be a distribution on [MATH] such that [MATH] is continuous with compact support in [MATH].', '1505.06912-1-35-1': 'Let [MATH].', '1505.06912-1-35-2': 'For [MATH], there are [MATH] and [MATH] for [MATH] such that [MATH] and [MATH] for [MATH] and [MATH].', '1505.06912-1-35-3': 'Define [MATH] as [EQUATION]', '1505.06912-1-35-4': 'Then, we have [EQUATION] and for [MATH] [EQUATION]', '1505.06912-1-35-5': 'Since [MATH] and [EQUATION] we obtain from (2.1) that [EQUATION]', '1505.06912-1-35-6': 'Since [MATH], we conclude from (i) of Proposition 1.1 that [MATH]', '1505.06912-1-36-0': 'Proof of (iii) of Proposition 1.1 Suppose that the support of [MATH] is included in [MATH] and [MATH] for every [MATH].', '1505.06912-1-36-1': 'Let [MATH] and [MATH] be independent random variables with the same distribution [MATH], and let [MATH] and [MATH] be independent random variables with the same distribution [MATH].', '1505.06912-1-36-2': 'Define [MATH] and [MATH] for [MATH] and [MATH] as [EQUATION]', '1505.06912-1-36-3': 'We see that [EQUATION]', '1505.06912-1-36-4': 'Since [MATH], we obtain that [EQUATION] and [EQUATION]', '1505.06912-1-36-5': 'Thus, as [MATH] we have by (2.2) [EQUATION] and hence [MATH].', '1505.06912-1-36-6': 'We find from [MATH] and (i) of Lemma 2.2 that [EQUATION]', '1505.06912-1-36-7': 'Thus, we see from (i) of Lemma 2.2 that [MATH].', '1505.06912-1-37-0': '# Proofs of Theorems 1.1 and 1.2', '1505.06912-1-38-0': 'For the proofs of the theorems, we introduce a distribution [MATH] as follows.', '1505.06912-1-38-1': 'Let [MATH].', '1505.06912-1-38-2': 'Take a continuous periodic function [MATH] on [MATH] with period [MATH] satisfying that [MATH] and [MATH] for [MATH].', '1505.06912-1-38-3': 'Further, define [MATH] as [EQUATION] for some [MATH] satisfying [MATH] and for all [MATH] with [MATH].', '1505.06912-1-38-4': 'Let [EQUATION] with [MATH].', '1505.06912-1-38-5': 'Here, the symbol [MATH] stands for the indicator function of the set [MATH].', '1505.06912-1-38-6': 'Define a distribution [MATH] as [EQUATION] where [MATH].', '1505.06912-1-39-0': 'We have [MATH].', '1505.06912-1-40-0': 'ProofLet [MATH] be a sequence such that [MATH] and [MATH] for some [MATH].', '1505.06912-1-40-1': 'Then, we put [MATH], where [MATH] is a positive integer and [MATH].', '1505.06912-1-40-2': 'In what follows, [MATH] and [MATH].', '1505.06912-1-40-3': 'Case 1.', '1505.06912-1-40-4': 'Suppose that [MATH].', '1505.06912-1-40-5': 'Let [MATH].', '1505.06912-1-40-6': 'Then, we have [EQUATION] and thereby [MATH].', '1505.06912-1-40-7': 'This yields that [EQUATION]', '1505.06912-1-40-8': 'Hence, we obtain that [EQUATION] so that [EQUATION]', '1505.06912-1-41-0': 'Case 2.', '1505.06912-1-41-1': 'Suppose that [MATH].', '1505.06912-1-41-2': 'Let [MATH] and put [EQUATION] where [MATH].', '1505.06912-1-41-3': 'For sufficiently large [MATH], we have for [MATH] [EQUATION]', '1505.06912-1-41-4': 'Set [MATH].', '1505.06912-1-41-5': 'It suffices that we consider the case where there exists a limit of [MATH] as [MATH], so we may put [MATH].', '1505.06912-1-41-6': 'This limit permits infinity.', '1505.06912-1-41-7': 'We divide [MATH] in the two cases where [MATH] and [MATH].', '1505.06912-1-41-8': 'Case 2-1.', '1505.06912-1-41-9': 'Suppose that [MATH].', '1505.06912-1-41-10': 'Now, we have [EQUATION]', '1505.06912-1-41-11': 'Let [MATH].', '1505.06912-1-41-12': 'For sufficiently large [MATH], we have by ([REF]) [EQUATION]', '1505.06912-1-41-13': 'This implies that [EQUATION]', '1505.06912-1-41-14': 'For sufficiently large [MATH], it follows that [EQUATION]', '1505.06912-1-41-15': 'As we have [EQUATION] it follows that [EQUATION] for sufficiently large [MATH].', '1505.06912-1-41-16': 'Furthermore, we see from ([REF]) that [EQUATION]', '1505.06912-1-41-17': 'Hence, we obtain that [EQUATION] so that ([REF]) holds.Case 2-2.', '1505.06912-1-41-18': 'Suppose that [MATH].', '1505.06912-1-41-19': 'For [MATH] with [MATH], we see from ([REF]) that [EQUATION] that is, [EQUATION]', '1505.06912-1-41-20': 'This implies that [EQUATION] so we get ([REF]).', '1505.06912-1-41-21': 'The lemma has been proved.', '1505.06912-1-42-0': 'We have [EQUATION]', '1505.06912-1-42-1': 'Proof Let [MATH] be a sequence such that [MATH] and [MATH] for some [MATH].', '1505.06912-1-42-2': 'We put [MATH], where [MATH] is a positive integer and [MATH].', '1505.06912-1-42-3': 'Now, we have [EQUATION]', '1505.06912-1-42-4': 'Here, we took [MATH] satisfying [MATH].', '1505.06912-1-42-5': 'Put [MATH].', '1505.06912-1-42-6': 'Then, we have [EQUATION]', '1505.06912-1-42-7': 'We consider the two cases where [MATH] and [MATH].', '1505.06912-1-42-8': 'Case 1.', '1505.06912-1-42-9': 'Suppose that [MATH].', '1505.06912-1-42-10': 'If [MATH], then [EQUATION]', '1505.06912-1-42-11': 'Hence, we obtain that [EQUATION] so that [EQUATION]', '1505.06912-1-42-12': 'Case 2.', '1505.06912-1-42-13': 'Suppose that [MATH].', '1505.06912-1-42-14': 'Put [MATH] and [EQUATION] where [MATH].', '1505.06912-1-42-15': 'It suffices that we consider the case where there exists a limit of [MATH], so we may put [MATH].', '1505.06912-1-42-16': 'This limit permits infinity.', '1505.06912-1-42-17': 'Furthermore, we divide [MATH] in the two cases where [MATH] and [MATH].', '1505.06912-1-43-0': 'Case 2-1.', '1505.06912-1-43-1': 'Suppose that [MATH].', '1505.06912-1-43-2': 'Take sufficiently large [MATH].', '1505.06912-1-43-3': 'Set [EQUATION]', '1505.06912-1-43-4': 'Let [MATH].', '1505.06912-1-43-5': 'We have [EQUATION]', '1505.06912-1-43-6': 'This implies that [EQUATION]', '1505.06912-1-43-7': 'It follows that [EQUATION]', '1505.06912-1-43-8': 'Here, we see that, for sufficiently large [MATH], [EQUATION] and thereby [EQUATION]', '1505.06912-1-43-9': 'Let [MATH].', '1505.06912-1-43-10': 'Then, we have [EQUATION]', '1505.06912-1-43-11': 'Hence, we see that [EQUATION]', '1505.06912-1-43-12': 'We consequently obtain that [EQUATION] so that [EQUATION]', '1505.06912-1-43-13': 'Case 2-2.', '1505.06912-1-43-14': 'Suppose that [MATH].', '1505.06912-1-43-15': 'Note that [MATH] is empty for sufficiently large [MATH].', '1505.06912-1-43-16': 'Let [MATH].', '1505.06912-1-43-17': 'Since [EQUATION] we see that [EQUATION]', '1505.06912-1-43-18': 'This yields that [EQUATION]', '1505.06912-1-43-19': 'For sufficiently large [MATH], we have [EQUATION] so that [MATH].', '1505.06912-1-43-20': 'We consequently obtain that [EQUATION]', '1505.06912-1-43-21': 'Combining the above calculations with the proof of Lemma 3.1, we reach the following: If [MATH], then [EQUATION]', '1505.06912-1-43-22': 'Suppose that [MATH].', '1505.06912-1-43-23': 'Recall [MATH] in the proof of Lemma 3.1.', '1505.06912-1-43-24': 'If [MATH] and [MATH], then we have [MATH].', '1505.06912-1-43-25': 'Hence, [EQUATION]', '1505.06912-1-43-26': 'If [MATH] and [MATH], then [EQUATION]', '1505.06912-1-43-27': 'If [MATH], then [MATH] and [EQUATION]', '1505.06912-1-43-28': 'The lemma has been proved.', '1505.06912-1-44-0': 'Proof of Theorem 1.1 We have [MATH] by Lemma 3.1.', '1505.06912-1-44-1': 'It follows from Lemma 3.2 that [EQUATION]', '1505.06912-1-44-2': 'Let [MATH].', '1505.06912-1-44-3': 'Furthermore, we see from [MATH] and (ii) of Lemma 2.2 that [EQUATION]', '1505.06912-1-44-4': 'Hence, we get [EQUATION] and thereby [MATH].', '1505.06912-1-44-5': 'Thus, [MATH] by (i) of Proposition 1.1.', '1505.06912-1-44-6': 'Since we see that [EQUATION] we have [MATH] by (ii) of Lemma 2.1.', '1505.06912-1-44-7': 'However, we have [MATH] because, for [MATH] with [MATH], we see that as [MATH] [EQUATION]', '1505.06912-1-44-8': 'The above relation implies that the convergence of the definition of the class [MATH] fails to satisfy uniformity.', '1505.06912-1-44-9': 'The theorem has been proved.', '1505.06912-1-45-0': 'Proof of Corollary 1.1 Proofs of assertions (i) and (ii) are clear from Theorem 1.1.', '1505.06912-1-45-1': 'We see from the definition of [MATH] that [MATH] for all [MATH].', '1505.06912-1-45-2': 'Thus, [MATH] but [MATH] and hence assertion (iii) is true.', '1505.06912-1-46-0': 'Choose [MATH] and [MATH] satisfying that [MATH].', '1505.06912-1-46-1': 'Let [MATH] be an increasing sequence of positive integers satisfying [MATH].', '1505.06912-1-46-2': 'Let [MATH] and [MATH] for [MATH].', '1505.06912-1-46-3': 'Choose a distribution [MATH] satisfying that [MATH] for all [MATH] and [MATH].', '1505.06912-1-47-0': 'We have, for [MATH], [EQUATION]', '1505.06912-1-47-1': 'Proof Let [MATH].', '1505.06912-1-47-2': 'We have [EQUATION] and [EQUATION]', '1505.06912-1-47-3': 'Thus, there exists [MATH] such that [MATH] does not depend on [MATH] and that [EQUATION]', '1505.06912-1-47-4': 'Hence, we obtain from Lemma 3.1 that [EQUATION]', '1505.06912-1-47-5': 'Thus, we have proved the lemma.', '1505.06912-1-48-0': 'Proof of Theorem 1.2 Define distributions [MATH] and [MATH] as [EQUATION]', '1505.06912-1-48-1': 'Thus, [MATH] by Theorem 1.1 and (iii) of Lemma 2.2.', '1505.06912-1-48-2': 'Let [MATH], where [MATH] is continuous with compact support in [MATH].', '1505.06912-1-48-3': 'Define distributions [MATH] and [MATH] as [EQUATION] and [EQUATION]', '1505.06912-1-48-4': 'Then, we find that [MATH] for all sufficiently large [MATH] and [MATH] by (ii) of Proposition 1.1.', '1505.06912-1-48-5': "We establish from Lemma 3.3 and Fatou's lemma that [EQUATION]", '1505.06912-1-48-6': 'Thus, we conclude that [MATH].', '1505.06912-1-49-0': '# A remark on the closure under convolution roots', '1505.06912-1-50-0': 'The tail of a measure [MATH] on [MATH] is denoted by [MATH], that is, [MATH] for [MATH].', '1505.06912-1-50-1': 'Let [MATH].', '1505.06912-1-50-2': 'The [MATH]-exponential moment of [MATH] is denoted by [MATH], namely, [MATH].', '1505.06912-1-51-0': 'Let [MATH].', '1505.06912-1-52-0': '(i) A distribution [MATH] on [MATH] is said to belong to the class [MATH] if [MATH] for every [MATH] and if [EQUATION] (ii) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] with [MATH] and if [EQUATION] (iii) Let [MATH].', '1505.06912-1-52-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH].', '1505.06912-1-53-0': 'The convolution closure problem on the class [MATH] with [MATH] is negatively solved by Leslie [CITATION] for [MATH] and by Kluppelberg and Villasenor [CITATION] for [MATH].', '1505.06912-1-53-1': 'The same problem on the class [MATH] is also negatively solved by Kluppelberg and Villasenor [CITATION].', '1505.06912-1-53-2': 'On the other hand, the fact that the class [MATH] of subexponential distributions is closed under convolution roots is proved by Embrechts et al. [CITATION] in the one-sided case and by Watanabe [CITATION] in the two-sided case.', '1505.06912-1-53-3': 'Embrechts and Goldie conjecture that [MATH] with [MATH] and [MATH] with [MATH] are closed under convolution roots in [CITATION], respectively.', '1505.06912-1-53-4': 'They also prove in [CITATION] that if [MATH] with [MATH] is closed under convolution roots, then [MATH] with [MATH] is closed under convolution roots.', '1505.06912-1-53-5': 'However, Shimura and Watanabe [CITATION] prove that the class [MATH] with [MATH] is not closed under convolution roots.', '1505.06912-1-53-6': 'Pakes [CITATION] and Watanabe [CITATION] show that [MATH] with [MATH] is closed under convolution roots in the class of infinitely divisible distributions on [MATH].', '1505.06912-1-53-7': 'It is still open whether the class [MATH] with [MATH] is closed under convolution roots.', '1505.06912-1-53-8': 'The closure problem under convolution roots for the other distribution classes is discussed by Shimura and Watanabe [CITATION] and Watanabe and Yamamuro [CITATION].', '1505.06912-1-54-0': 'Let [MATH].', '1505.06912-1-54-1': 'For [MATH], we define the exponential tilt [MATH] of [MATH] as [EQUATION]', '1505.06912-1-54-2': 'Exponential tilts preserve convolutions, that is, [MATH] for distributions [MATH].', '1505.06912-1-54-3': 'Let [MATH] be a distribution class.', '1505.06912-1-54-4': 'For a class [MATH], we define the class [MATH] by [EQUATION]', '1505.06912-1-54-5': 'It is obvious that [MATH] and that [MATH] for [MATH].', '1505.06912-1-54-6': 'The class [MATH] is determined by Watanabe and Yamamuro as follows.', '1505.06912-1-54-7': 'Analogous result is found in Theorem 2.1 of Kluppelberg [CITATION].', '1505.06912-1-55-0': '(Theorem 2.1 of [CITATION]) Let [MATH].', '1505.06912-1-56-0': '(i) We have [MATH] and hence [MATH].', '1505.06912-1-56-1': 'Moreover, if [MATH], then we have [EQUATION] (ii) We have [MATH] and thereby [MATH].', '1505.06912-1-57-0': 'Finally, we present a remark on the closure under convolution roots for the three classes [MATH], [MATH], and [MATH].', '1505.06912-1-58-0': 'Proof Proof of the equivalence between (1) and (2) is due to Lemma 4.1.', '1505.06912-1-58-1': 'Let [MATH].', '1505.06912-1-58-2': 'Suppose that (2) holds and, for some [MATH], [MATH] for every [MATH].', '1505.06912-1-58-3': 'Let [MATH].', '1505.06912-1-58-4': 'We have [MATH].', '1505.06912-1-58-5': 'We see from assertion (2) that [MATH] and hence, by (iii) of Proposition 1.1, we have [MATH], that is, [MATH] for every [MATH] by (i) of Proposition 1.1.', '1505.06912-1-58-6': 'Conversely, suppose that (3) holds and [MATH].', '1505.06912-1-58-7': 'Note that [MATH] is continuous with compact support in [MATH].', '1505.06912-1-58-8': 'Thus, we see from (ii) of Proposition 1.1 that [MATH] for every [MATH].', '1505.06912-1-58-9': 'We obtain from assertion (3) that [MATH] for every [MATH], that is, [MATH] by (i) of Proposition 1.1.'}

{'1505.06912-2-0-0': 'Relations between subexponential densities and locally subexponential distributions are discussed.', '1505.06912-2-0-1': 'It is shown that the class of subexponential densities is neither closed under convolution roots nor closed under asymptotic equivalence.', '1505.06912-2-0-2': 'A remark is given on the closure under convolution roots for the class of convolution equivalent distributions.', '1505.06912-2-1-0': 'Key words or phrases: subexponential densities, local subexponentiality, convolution roots, asymptotic equivalence', '1505.06912-2-2-0': 'Mathematics Subject Classification: 60E99, 60G50', '1505.06912-2-3-0': '# Introduction and main results', '1505.06912-2-4-0': 'In what follows, we denote by [MATH] the real line and by [MATH] the half line [MATH].', '1505.06912-2-4-1': 'Let [MATH] be the totality of positive integers.', '1505.06912-2-4-2': 'The symbol [MATH] stands for the delta measure at [MATH].', '1505.06912-2-4-3': 'Let [MATH] and [MATH] be probability measures on [MATH].', '1505.06912-2-4-4': 'We denote the convolution of [MATH] and [MATH] by [MATH] and denote [MATH]-th convolution power of [MATH] by [MATH].', '1505.06912-2-4-5': 'Let [MATH] and [MATH] be integrable functions on [MATH].', '1505.06912-2-4-6': 'We denote by [MATH]-th convolution power of [MATH] and by [MATH] the convolution of [MATH] and [MATH].', '1505.06912-2-4-7': 'For positive functions [MATH] and [MATH] on [MATH] for some [MATH], we define the relation [MATH] by [MATH].', '1505.06912-2-4-8': 'We also define the relation [MATH] for positive sequences [MATH] and [MATH] with [MATH] by [MATH].', '1505.06912-2-4-9': 'We define the class [MATH] as the totality of probability distributions on [MATH].', '1505.06912-2-4-10': 'In this paper, we prove that the class of subexponential densities is not closed under two important closure properties.', '1505.06912-2-4-11': 'We say that a measurable function [MATH] on [MATH] is a density function if [MATH] and [MATH] for all [MATH].', '1505.06912-2-5-0': '(i) A nonnegative measurable function [MATH] on [MATH] belongs to the class [MATH] if [MATH] for all sufficiently large [MATH] and if [MATH] for any [MATH].', '1505.06912-2-6-0': '(ii) A measurable function [MATH] on [MATH] belongs to the class [MATH] if [MATH] is a density function and [MATH].', '1505.06912-2-7-0': '(iii) A measurable function [MATH] on [MATH] belongs to the class [MATH] if [MATH] and [MATH].', '1505.06912-2-8-0': '(iv) A distribution [MATH] on [MATH] belongs to the class [MATH] if there is [MATH] such that [MATH].', '1505.06912-2-9-0': '(v) A distribution [MATH] on [MATH] belongs to the class [MATH] if there is [MATH] such that [MATH].', '1505.06912-2-10-0': 'Densities in the class [MATH] are called subexponential densities and those in the class [MATH] are called long-tailed densities.', '1505.06912-2-10-1': 'The study on the class [MATH] goes back to Chover et al. [CITATION].', '1505.06912-2-10-2': 'Let [MATH] be a distribution on [MATH].', '1505.06912-2-10-3': 'Note that [MATH] is a density function on [MATH] for every [MATH].', '1505.06912-2-11-0': '(i) Let [MATH] with [MATH].', '1505.06912-2-11-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH].', '1505.06912-2-12-0': '(ii) Let [MATH] with [MATH].', '1505.06912-2-12-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] and [MATH]', '1505.06912-2-13-0': '(iii) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] for each [MATH] with [MATH]', '1505.06912-2-14-0': '(iv) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] for each [MATH] with [MATH]', '1505.06912-2-15-0': '(v) A distribution [MATH] belongs to the class [MATH] if there exists [MATH] such that [MATH] uniformly in [MATH].', '1505.06912-2-16-0': '(vi) A distribution [MATH] belongs to the class [MATH] if there exists [MATH] such that [MATH] uniformly in [MATH].', '1505.06912-2-17-0': 'Distributions in the class [MATH] are called locally subexponential, those in the class [MATH] are called uniformly locally subexponential.', '1505.06912-2-17-1': 'The class [MATH] was introduced by Asmussen et al. [CITATION] and the class [MATH] was by Watanabe and Yamamuro [CITATION].', '1505.06912-2-17-2': 'Detailed acounts of the classes [MATH] and [MATH] are found in the book of Foss et al. [CITATION].', '1505.06912-2-17-3': 'First, we present some interesting results on the classes [MATH] and [MATH].', '1505.06912-2-18-0': '(i) We say that a class [MATH] of probability distributions on [MATH] is closed under convolution roots if [MATH] for some [MATH] implies that [MATH].', '1505.06912-2-19-0': '(ii) Let [MATH] and [MATH] be density functions on [MATH].', '1505.06912-2-19-1': 'We say that a class [MATH] of density functions is closed under asymptotic equivalence if [MATH] and [MATH] with [MATH] implies that [MATH].', '1505.06912-2-20-0': 'The class [MATH] is a proper subclass of the class [MATH] because a distribution in [MATH] can have a point mass.', '1505.06912-2-20-1': 'Moreover, the class [MATH] is a proper subclass of the class [MATH] as the following theorem shows.', '1505.06912-2-21-0': '[EQUATION]', '1505.06912-2-21-1': 'We have the following.', '1505.06912-2-22-0': '(i) The class [MATH] is not closed under convolution roots.', '1505.06912-2-23-0': '(ii) The class [MATH] is not closed under convolution roots.', '1505.06912-2-24-0': '(iii) The class [MATH] is not closed under convolution roots.', '1505.06912-2-25-0': '(iv) The class [MATH] is not closed under convolution roots.', '1505.06912-2-26-0': 'The class [MATH] is closed under asymptotic equivalence in the one-sided case.', '1505.06912-2-26-1': 'See (ii) of Lemma 2.1 below.', '1505.06912-2-26-2': 'However, Foss et al. [CITATION] suggest the possibility of non-closure under asymptotic equivalence for the class [MATH] in the two-sided case.', '1505.06912-2-26-3': 'We exactly prove it as follows.', '1505.06912-2-27-0': 'The class [MATH] is not closed under asymptotic equivalence, that is, there exist [MATH] and [MATH] such that [MATH] with [MATH].', '1505.06912-2-28-0': 'In Sect. 2, we prove Proposition 1.1.', '1505.06912-2-28-1': 'In Sect. 3, we prove Theorems 1.1 and 1.2.', '1505.06912-2-28-2': 'In Sect. 4, we give a remark on the closure under convolution roots.', '1505.06912-2-29-0': '# Proof of Proposition 1.1', '1505.06912-2-30-0': 'We present two lemmas for the proofs of main results and then prove Proposition 1.1.', '1505.06912-2-31-0': 'Let [MATH] and [MATH] be density functions on [MATH].', '1505.06912-2-32-0': '(i) If [MATH], then [MATH] for every [MATH].', '1505.06912-2-33-0': '(ii) If [MATH] and [MATH] with [MATH], then [MATH].', '1505.06912-2-34-0': '(iii) Assume that [MATH].', '1505.06912-2-34-1': 'Then, [MATH] if and only if [EQUATION]', '1505.06912-2-34-2': 'Proof Proof of assertion (i) is due to Theorem 4.3 of [CITATION].', '1505.06912-2-34-3': 'Proofs of assertions (ii) and (iii) are due to Theorems 4.8 and 4.7 of [CITATION], respectively.', '1505.06912-2-35-0': '(i) Let [MATH] with [MATH].', '1505.06912-2-35-1': 'Assume that [MATH].', '1505.06912-2-35-2': 'Then, [MATH] if and only if [EQUATION] (ii) Assume that [MATH].', '1505.06912-2-35-3': 'Then, [MATH] for every [MATH].', '1505.06912-2-35-4': 'Moreover, [MATH] for every [MATH].', '1505.06912-2-36-0': '(iii) Let [MATH].', '1505.06912-2-36-1': 'If [MATH] and [MATH] with [MATH] for every [MATH], then [MATH].', '1505.06912-2-37-0': 'Proof Proof of assertion (i) is due to Theorem 4.21 of [CITATION].', '1505.06912-2-37-1': 'First assertion of (ii) is due to Corollary 4.19 of [CITATION].', '1505.06912-2-37-2': 'Second one is proved as (2.6) in Theorem 2.1 of [CITATION].', '1505.06912-2-37-3': 'Proof of assertion (iii) is due to Theorem 4.22 of [CITATION].', '1505.06912-2-38-0': 'Proof of (i) of Proposition 1.1 Let [MATH].', '1505.06912-2-38-1': 'First, we prove that if [MATH], then [MATH].', '1505.06912-2-38-2': 'We can assume that [MATH].', '1505.06912-2-38-3': 'Suppose that [MATH].', '1505.06912-2-38-4': 'Let [MATH].', '1505.06912-2-38-5': 'We have [MATH] and hence [MATH].', '1505.06912-2-38-6': 'Let [MATH] be a positive integer and let [MATH] be independent random variables with the same distribution [MATH].', '1505.06912-2-38-7': 'Then, we have for [MATH] [EQUATION]', '1505.06912-2-38-8': 'Since [MATH], we obtain from (i) of Lemma 2.2 that [EQUATION].', '1505.06912-2-38-9': 'Thus, we see from (iii) of Lemma 2.1 that [MATH].', '1505.06912-2-39-0': 'Conversely, suppose that [MATH].', '1505.06912-2-39-1': 'Then, we have [MATH].', '1505.06912-2-39-2': 'Let [MATH] be the largest integer not exceeding a real number [MATH].', '1505.06912-2-39-3': 'Choose sufficiently large integer [MATH].', '1505.06912-2-39-4': 'Note that there are positive constants [MATH] for [MATH] such that [EQUATION] for [MATH], [MATH], and [MATH].', '1505.06912-2-39-5': 'Thus, we find that [EQUATION]', '1505.06912-2-39-6': 'Since [MATH], we establish from (iii) of Lemma 2.1 that [EQUATION].', '1505.06912-2-39-7': 'Thus, [MATH] by (i) of Lemma 2.2.', '1505.06912-2-39-8': 'Note from (ii) of Lemma 2.2 that if [MATH], then [MATH] for every [MATH].', '1505.06912-2-39-9': 'Thus, the second assertion is true.', '1505.06912-2-40-0': 'Proof of (ii) of Proposition 1.1 Suppose that [MATH] be a distribution on [MATH] such that [MATH] is continuous with compact support in [MATH].', '1505.06912-2-40-1': 'Let [MATH].', '1505.06912-2-40-2': 'For [MATH], there are [MATH] and [MATH] for [MATH] such that [MATH] and [MATH] for [MATH] and [MATH].', '1505.06912-2-40-3': 'Define [MATH] as [EQUATION]', '1505.06912-2-40-4': 'Then, we have [EQUATION] and for [MATH] [EQUATION]', '1505.06912-2-40-5': 'Since [MATH] and [EQUATION] we obtain from (2.1) that [EQUATION]', '1505.06912-2-40-6': 'Since [MATH], we conclude from (i) of Proposition 1.1 that [MATH]', '1505.06912-2-41-0': 'Proof of (iii) of Proposition 1.1 Suppose that the support of [MATH] is included in [MATH] and [MATH] for every [MATH].', '1505.06912-2-41-1': 'Let [MATH] and [MATH] be independent random variables with the same distribution [MATH], and let [MATH] and [MATH] be independent random variables with the same distribution [MATH].', '1505.06912-2-41-2': 'Define [MATH] and [MATH] for [MATH] and [MATH] as [EQUATION]', '1505.06912-2-41-3': 'We see that [EQUATION]', '1505.06912-2-41-4': 'Since [MATH], we obtain that [EQUATION] and [EQUATION]', '1505.06912-2-41-5': 'Thus, as [MATH] we have by (2.2) [EQUATION] and hence [MATH].', '1505.06912-2-41-6': 'We find from [MATH] and (i) of Lemma 2.2 that [EQUATION]', '1505.06912-2-41-7': 'Thus, we see from (i) of Lemma 2.2 that [MATH].', '1505.06912-2-42-0': '# Proofs of Theorems 1.1 and 1.2', '1505.06912-2-43-0': 'For the proofs of the theorems, we introduce a distribution [MATH] as follows.', '1505.06912-2-43-1': 'Let [MATH] and choose [MATH] satisfying [MATH].', '1505.06912-2-43-2': 'We take a continuous periodic function [MATH] on [MATH] with period [MATH] such that [MATH] for [MATH] and [EQUATION]', '1505.06912-2-43-3': 'Let [EQUATION] with [MATH].', '1505.06912-2-43-4': 'Here, the symbol [MATH] stands for the indicator function of the set [MATH].', '1505.06912-2-43-5': 'Define a distribution [MATH] as [EQUATION] where [MATH].', '1505.06912-2-44-0': 'We have [MATH].', '1505.06912-2-45-0': 'ProofLet [MATH] be a sequence such that [MATH] and [MATH] for some [MATH].', '1505.06912-2-45-1': 'Then, we put [MATH], where [MATH] is a positive integer and [MATH].', '1505.06912-2-45-2': 'In what follows, [MATH] and [MATH].', '1505.06912-2-45-3': 'Case 1.', '1505.06912-2-45-4': 'Suppose that [MATH].', '1505.06912-2-45-5': 'Let [MATH].', '1505.06912-2-45-6': 'Then, we have [EQUATION] and thereby [MATH].', '1505.06912-2-45-7': 'This yields that [EQUATION]', '1505.06912-2-45-8': 'Hence, we obtain that [EQUATION] so that [EQUATION]', '1505.06912-2-46-0': 'Case 2.', '1505.06912-2-46-1': 'Suppose that [MATH].', '1505.06912-2-46-2': 'Let [MATH] and put [EQUATION] where [MATH].', '1505.06912-2-46-3': 'For sufficiently large [MATH], we have for [MATH] [EQUATION]', '1505.06912-2-46-4': 'Set [MATH].', '1505.06912-2-46-5': 'It suffices that we consider the case where there exists a limit of [MATH] as [MATH], so we may put [MATH].', '1505.06912-2-46-6': 'This limit permits infinity.', '1505.06912-2-46-7': 'We divide [MATH] in the two cases where [MATH] and [MATH].', '1505.06912-2-46-8': 'Case 2-1.', '1505.06912-2-46-9': 'Suppose that [MATH].', '1505.06912-2-46-10': 'Now, we have [EQUATION]', '1505.06912-2-46-11': 'Let [MATH].', '1505.06912-2-46-12': 'For sufficiently large [MATH], we have by ([REF]) [EQUATION]', '1505.06912-2-46-13': 'This implies that [EQUATION]', '1505.06912-2-46-14': 'For sufficiently large [MATH], it follows that [EQUATION]', '1505.06912-2-46-15': 'As we have [EQUATION] it follows that [EQUATION] for sufficiently large [MATH].', '1505.06912-2-46-16': 'Furthermore, we see from ([REF]) that [EQUATION]', '1505.06912-2-46-17': 'Hence, we obtain that [EQUATION] so that ([REF]) holds.Case 2-2.', '1505.06912-2-46-18': 'Suppose that [MATH].', '1505.06912-2-46-19': 'For [MATH] with [MATH], we see from ([REF]) that [EQUATION] that is, [EQUATION]', '1505.06912-2-46-20': 'This implies that [EQUATION] so we get ([REF]).', '1505.06912-2-46-21': 'The lemma has been proved.', '1505.06912-2-47-0': 'We have [EQUATION]', '1505.06912-2-47-1': 'Proof Let [MATH] be a sequence such that [MATH] and [MATH] for some [MATH].', '1505.06912-2-47-2': 'We put [MATH], where [MATH] is a positive integer and [MATH].', '1505.06912-2-47-3': 'Now, we have [EQUATION]', '1505.06912-2-47-4': 'Here, we took [MATH] satisfying [MATH].', '1505.06912-2-47-5': 'Put [MATH].', '1505.06912-2-47-6': 'Then, we have [EQUATION]', '1505.06912-2-47-7': 'We consider the two cases where [MATH] and [MATH].', '1505.06912-2-47-8': 'Case 1.', '1505.06912-2-47-9': 'Suppose that [MATH].', '1505.06912-2-47-10': 'If [MATH], then [EQUATION]', '1505.06912-2-47-11': 'Hence, we obtain that [EQUATION] so that [EQUATION]', '1505.06912-2-47-12': 'Case 2.', '1505.06912-2-47-13': 'Suppose that [MATH].', '1505.06912-2-47-14': 'Put [MATH] and [EQUATION] where [MATH].', '1505.06912-2-47-15': 'It suffices that we consider the case where there exists a limit of [MATH], so we may put [MATH].', '1505.06912-2-47-16': 'This limit permits infinity.', '1505.06912-2-47-17': 'Furthermore, we divide [MATH] in the two cases where [MATH] and [MATH].', '1505.06912-2-48-0': 'Case 2-1.', '1505.06912-2-48-1': 'Suppose that [MATH].', '1505.06912-2-48-2': 'Take sufficiently large [MATH].', '1505.06912-2-48-3': 'Set [EQUATION]', '1505.06912-2-48-4': 'Let [MATH].', '1505.06912-2-48-5': 'We have [EQUATION]', '1505.06912-2-48-6': 'This implies that [EQUATION]', '1505.06912-2-48-7': 'It follows that [EQUATION]', '1505.06912-2-48-8': 'Here, we see that, for sufficiently large [MATH], [EQUATION] and thereby [EQUATION]', '1505.06912-2-48-9': 'Let [MATH].', '1505.06912-2-48-10': 'Then, we have [EQUATION]', '1505.06912-2-48-11': 'Hence, we see that [EQUATION]', '1505.06912-2-48-12': 'We consequently obtain that [EQUATION] so that [EQUATION]', '1505.06912-2-48-13': 'Case 2-2.', '1505.06912-2-48-14': 'Suppose that [MATH].', '1505.06912-2-48-15': 'Note that [MATH] is empty for sufficiently large [MATH].', '1505.06912-2-48-16': 'Let [MATH].', '1505.06912-2-48-17': 'Since [EQUATION] we see that [EQUATION]', '1505.06912-2-48-18': 'This yields that [EQUATION]', '1505.06912-2-48-19': 'For sufficiently large [MATH], we have [EQUATION] so that [MATH].', '1505.06912-2-48-20': 'We consequently obtain that [EQUATION]', '1505.06912-2-48-21': 'Combining the above calculations with the proof of Lemma 3.1, we reach the following: If [MATH], then [EQUATION]', '1505.06912-2-48-22': 'Suppose that [MATH].', '1505.06912-2-48-23': 'Recall [MATH] in the proof of Lemma 3.1.', '1505.06912-2-48-24': 'If [MATH] and [MATH], then we have [MATH].', '1505.06912-2-48-25': 'Hence, [EQUATION]', '1505.06912-2-48-26': 'If [MATH] and [MATH], then [EQUATION]', '1505.06912-2-48-27': 'If [MATH], then [MATH] and [EQUATION]', '1505.06912-2-48-28': 'The lemma has been proved.', '1505.06912-2-49-0': 'Proof of Theorem 1.1 We have [MATH] by Lemma 3.1.', '1505.06912-2-49-1': 'It follows from Lemma 3.2 that [EQUATION]', '1505.06912-2-49-2': 'Let [MATH].', '1505.06912-2-49-3': 'Furthermore, we see from [MATH] and (ii) of Lemma 2.2 that [EQUATION]', '1505.06912-2-49-4': 'Hence, we get [EQUATION] and thereby [MATH].', '1505.06912-2-49-5': 'Thus, [MATH] by (i) of Proposition 1.1.', '1505.06912-2-49-6': 'Since we see that [EQUATION] we have [MATH] by (ii) of Lemma 2.1.', '1505.06912-2-49-7': 'However, we have [MATH] because, for [MATH] with [MATH], we see that as [MATH] [EQUATION]', '1505.06912-2-49-8': 'The above relation implies that the convergence of the definition of the class [MATH] fails to satisfy uniformity.', '1505.06912-2-49-9': 'Since [MATH], the theorem has been proved.', '1505.06912-2-50-0': 'Proof of Corollary 1.1 Proofs of assertions (i) and (ii) are clear from Theorem 1.1.', '1505.06912-2-50-1': 'We find from the proof of Theorem 1.1 that [MATH] but [MATH].', '1505.06912-2-50-2': 'Since [MATH], assertions (iii) and (iv) are true.', '1505.06912-2-51-0': 'Choose [MATH] and [MATH] satisfying that [MATH].', '1505.06912-2-51-1': 'Let [MATH] be an increasing sequence of positive integers satisfying [MATH].', '1505.06912-2-51-2': 'Let [MATH] and [MATH] for [MATH].', '1505.06912-2-51-3': 'Choose a distribution [MATH] satisfying that [MATH] for all [MATH] and [MATH].', '1505.06912-2-52-0': 'We have, for [MATH], [EQUATION]', '1505.06912-2-52-1': 'Proof We have, uniformly in [MATH], [EQUATION] and [EQUATION]', '1505.06912-2-52-2': 'Thus, there exists [MATH] such that [MATH] does not depend on [MATH] and that [EQUATION]', '1505.06912-2-52-3': 'Hence, we obtain from Lemma 3.1 that [EQUATION]', '1505.06912-2-52-4': 'Thus, we have proved the lemma.', '1505.06912-2-53-0': 'Proof of Theorem 1.2 Define distributions [MATH] and [MATH] as [EQUATION]', '1505.06912-2-53-1': 'Thus, [MATH] by Theorem 1.1 and (iii) of Lemma 2.2.', '1505.06912-2-53-2': 'Let [MATH], where [MATH] is continuous with compact support in [MATH].', '1505.06912-2-53-3': 'Define distributions [MATH] and [MATH] as [EQUATION] and [EQUATION]', '1505.06912-2-53-4': 'Then, we find that [MATH] for all sufficiently large [MATH] and [MATH] by (ii) of Proposition 1.1.', '1505.06912-2-53-5': "We establish from Lemma 3.3 and Fatou's lemma that [EQUATION]", '1505.06912-2-53-6': 'Thus, we conclude that [MATH].', '1505.06912-2-54-0': '# A remark on the closure under convolution roots', '1505.06912-2-55-0': 'The tail of a measure [MATH] on [MATH] is denoted by [MATH], that is, [MATH] for [MATH].', '1505.06912-2-55-1': 'Let [MATH].', '1505.06912-2-55-2': 'The [MATH]-exponential moment of [MATH] is denoted by [MATH], namely, [MATH].', '1505.06912-2-56-0': 'Let [MATH].', '1505.06912-2-57-0': '(i) A distribution [MATH] on [MATH] is said to belong to the class [MATH] if [MATH] for every [MATH] and if [EQUATION] (ii) A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH] with [MATH] and if [EQUATION] (iii) Let [MATH].', '1505.06912-2-57-1': 'A distribution [MATH] on [MATH] belongs to the class [MATH] if [MATH].', '1505.06912-2-58-0': 'The convolution closure problem on the class [MATH] with [MATH] is negatively solved by Leslie [CITATION] for [MATH] and by Kluppelberg and Villasenor [CITATION] for [MATH].', '1505.06912-2-58-1': 'The same problem on the class [MATH] is also negatively solved by Kluppelberg and Villasenor [CITATION].', '1505.06912-2-58-2': 'On the other hand, the fact that the class [MATH] of subexponential distributions is closed under convolution roots is proved by Embrechts et al. [CITATION] in the one-sided case and by Watanabe [CITATION] in the two-sided case.', '1505.06912-2-58-3': 'Embrechts and Goldie conjecture that [MATH] with [MATH] and [MATH] with [MATH] are closed under convolution roots in [CITATION], respectively.', '1505.06912-2-58-4': 'They also prove in [CITATION] that if [MATH] with [MATH] is closed under convolution roots, then [MATH] with [MATH] is closed under convolution roots.', '1505.06912-2-58-5': 'However, Shimura and Watanabe [CITATION] prove that the class [MATH] with [MATH] is not closed under convolution roots, and we find that Xu et al. [CITATION] show the same conclusion in the case [MATH].', '1505.06912-2-58-6': 'Pakes [CITATION] and Watanabe [CITATION] show that [MATH] with [MATH] is closed under convolution roots in the class of infinitely divisible distributions on [MATH].', '1505.06912-2-58-7': 'It is still open whether the class [MATH] with [MATH] is closed under convolution roots.', '1505.06912-2-58-8': 'Shimura and Watanabe [CITATION] show that the class [MATH] is not closed under convolution roots.', '1505.06912-2-58-9': 'Watanabe and Yamamuro [CITATION] pointed out that [MATH] is closed under convolution roots in the class of infinitely divisible distributions.', '1505.06912-2-59-0': 'Let [MATH].', '1505.06912-2-59-1': 'For [MATH], we define the exponential tilt [MATH] of [MATH] as [EQUATION]', '1505.06912-2-59-2': 'Exponential tilts preserve convolutions, that is, [MATH] for distributions [MATH].', '1505.06912-2-59-3': 'Let [MATH] be a distribution class.', '1505.06912-2-59-4': 'For a class [MATH], we define the class [MATH] by [EQUATION]', '1505.06912-2-59-5': 'It is obvious that [MATH] and that [MATH] for [MATH].', '1505.06912-2-59-6': 'The class [MATH] is determined by Watanabe and Yamamuro as follows.', '1505.06912-2-59-7': 'Analogous result is found in Theorem 2.1 of Kluppelberg [CITATION].', '1505.06912-2-60-0': '(Theorem 2.1 of [CITATION]) Let [MATH].', '1505.06912-2-61-0': '(i) We have [MATH] and hence [MATH].', '1505.06912-2-61-1': 'Moreover, if [MATH], then we have [EQUATION] (ii) We have [MATH] and thereby [MATH].', '1505.06912-2-62-0': 'Finally, we present a remark on the closure under convolution roots for the three classes [MATH], [MATH], and [MATH].', '1505.06912-2-63-0': 'Proof Proof of the equivalence between (1) and (2) is due to Lemma 4.1.', '1505.06912-2-63-1': 'Let [MATH].', '1505.06912-2-63-2': 'Suppose that (2) holds and, for some [MATH], [MATH] for every [MATH].', '1505.06912-2-63-3': 'Let [MATH].', '1505.06912-2-63-4': 'We have [MATH].', '1505.06912-2-63-5': 'We see from assertion (2) that [MATH] and hence, by (iii) of Proposition 1.1, we have [MATH], that is, [MATH] for every [MATH] by (i) of Proposition 1.1.', '1505.06912-2-63-6': 'Conversely, suppose that (3) holds and [MATH].', '1505.06912-2-63-7': 'Note that [MATH] is continuous with compact support in [MATH].', '1505.06912-2-63-8': 'Thus, we see from (ii) of Proposition 1.1 that [MATH] for every [MATH].', '1505.06912-2-63-9': 'We obtain from assertion (3) that [MATH] for every [MATH], that is, [MATH] by (i) of Proposition 1.1.'}

[['1505.06912-1-8-2', '1505.06912-2-10-1'], ['1505.06912-1-8-3', '1505.06912-2-10-2'], ['1505.06912-1-8-4', '1505.06912-2-10-3'], ['1505.06912-1-38-0', '1505.06912-2-43-0'], ['1505.06912-1-38-5', '1505.06912-2-43-4'], ['1505.06912-1-38-6', '1505.06912-2-43-5'], ['1505.06912-1-15-0', '1505.06912-2-18-0'], ['1505.06912-1-47-3', '1505.06912-2-52-2'], ['1505.06912-1-47-4', '1505.06912-2-52-3'], ['1505.06912-1-47-5', '1505.06912-2-52-4'], ['1505.06912-1-23-0', '1505.06912-2-27-0'], ['1505.06912-1-36-0', '1505.06912-2-41-0'], ['1505.06912-1-36-1', '1505.06912-2-41-1'], ['1505.06912-1-36-2', '1505.06912-2-41-2'], ['1505.06912-1-36-3', '1505.06912-2-41-3'], ['1505.06912-1-36-4', '1505.06912-2-41-4'], ['1505.06912-1-36-6', '1505.06912-2-41-6'], ['1505.06912-1-36-7', '1505.06912-2-41-7'], ['1505.06912-1-53-0', '1505.06912-2-58-0'], ['1505.06912-1-53-1', '1505.06912-2-58-1'], ['1505.06912-1-53-2', '1505.06912-2-58-2'], ['1505.06912-1-53-3', '1505.06912-2-58-3'], ['1505.06912-1-53-4', '1505.06912-2-58-4'], ['1505.06912-1-53-6', '1505.06912-2-58-6'], ['1505.06912-1-53-7', '1505.06912-2-58-7'], ['1505.06912-1-26-0', '1505.06912-2-30-0'], ['1505.06912-1-54-1', '1505.06912-2-59-1'], ['1505.06912-1-54-2', '1505.06912-2-59-2'], ['1505.06912-1-54-3', '1505.06912-2-59-3'], ['1505.06912-1-54-4', '1505.06912-2-59-4'], ['1505.06912-1-54-5', '1505.06912-2-59-5'], ['1505.06912-1-54-6', '1505.06912-2-59-6'], ['1505.06912-1-54-7', '1505.06912-2-59-7'], ['1505.06912-1-46-0', '1505.06912-2-51-0'], ['1505.06912-1-46-1', '1505.06912-2-51-1'], ['1505.06912-1-46-3', '1505.06912-2-51-3'], ['1505.06912-1-42-1', '1505.06912-2-47-1'], ['1505.06912-1-42-2', '1505.06912-2-47-2'], ['1505.06912-1-42-3', '1505.06912-2-47-3'], ['1505.06912-1-42-4', '1505.06912-2-47-4'], ['1505.06912-1-42-6', '1505.06912-2-47-6'], ['1505.06912-1-42-7', '1505.06912-2-47-7'], ['1505.06912-1-42-11', '1505.06912-2-47-11'], ['1505.06912-1-42-14', '1505.06912-2-47-14'], ['1505.06912-1-42-15', '1505.06912-2-47-15'], ['1505.06912-1-42-16', '1505.06912-2-47-16'], ['1505.06912-1-42-17', '1505.06912-2-47-17'], ['1505.06912-1-34-0', '1505.06912-2-39-0'], ['1505.06912-1-34-1', '1505.06912-2-39-1'], ['1505.06912-1-34-2', '1505.06912-2-39-2'], ['1505.06912-1-34-3', '1505.06912-2-39-3'], ['1505.06912-1-34-4', '1505.06912-2-39-4'], ['1505.06912-1-34-5', '1505.06912-2-39-5'], ['1505.06912-1-34-6', '1505.06912-2-39-6'], ['1505.06912-1-34-8', '1505.06912-2-39-8'], ['1505.06912-1-34-9', '1505.06912-2-39-9'], ['1505.06912-1-1-0', '1505.06912-2-1-0'], ['1505.06912-1-58-0', '1505.06912-2-63-0'], ['1505.06912-1-58-2', '1505.06912-2-63-2'], ['1505.06912-1-58-5', '1505.06912-2-63-5'], ['1505.06912-1-58-6', '1505.06912-2-63-6'], ['1505.06912-1-58-7', '1505.06912-2-63-7'], ['1505.06912-1-58-8', '1505.06912-2-63-8'], ['1505.06912-1-58-9', '1505.06912-2-63-9'], ['1505.06912-1-44-0', '1505.06912-2-49-0'], ['1505.06912-1-44-1', '1505.06912-2-49-1'], ['1505.06912-1-44-3', '1505.06912-2-49-3'], ['1505.06912-1-44-4', '1505.06912-2-49-4'], ['1505.06912-1-44-5', '1505.06912-2-49-5'], ['1505.06912-1-44-6', '1505.06912-2-49-6'], ['1505.06912-1-44-7', '1505.06912-2-49-7'], ['1505.06912-1-44-8', '1505.06912-2-49-8'], ['1505.06912-1-41-2', '1505.06912-2-46-2'], ['1505.06912-1-41-3', '1505.06912-2-46-3'], ['1505.06912-1-41-5', '1505.06912-2-46-5'], ['1505.06912-1-41-6', '1505.06912-2-46-6'], ['1505.06912-1-41-7', '1505.06912-2-46-7'], ['1505.06912-1-41-10', '1505.06912-2-46-10'], ['1505.06912-1-41-12', '1505.06912-2-46-12'], ['1505.06912-1-41-13', '1505.06912-2-46-13'], ['1505.06912-1-41-14', '1505.06912-2-46-14'], ['1505.06912-1-41-15', '1505.06912-2-46-15'], ['1505.06912-1-41-16', '1505.06912-2-46-16'], ['1505.06912-1-41-17', '1505.06912-2-46-17'], ['1505.06912-1-41-19', '1505.06912-2-46-19'], ['1505.06912-1-41-20', '1505.06912-2-46-20'], ['1505.06912-1-41-21', '1505.06912-2-46-21'], ['1505.06912-1-24-0', '1505.06912-2-28-0'], ['1505.06912-1-24-1', '1505.06912-2-28-1'], ['1505.06912-1-24-2', '1505.06912-2-28-2'], ['1505.06912-1-43-2', '1505.06912-2-48-2'], ['1505.06912-1-43-6', '1505.06912-2-48-6'], ['1505.06912-1-43-7', '1505.06912-2-48-7'], ['1505.06912-1-43-8', '1505.06912-2-48-8'], ['1505.06912-1-43-10', '1505.06912-2-48-10'], ['1505.06912-1-43-11', '1505.06912-2-48-11'], ['1505.06912-1-43-12', '1505.06912-2-48-12'], ['1505.06912-1-43-15', '1505.06912-2-48-15'], ['1505.06912-1-43-17', '1505.06912-2-48-17'], ['1505.06912-1-43-18', '1505.06912-2-48-18'], ['1505.06912-1-43-19', '1505.06912-2-48-19'], ['1505.06912-1-43-20', '1505.06912-2-48-20'], ['1505.06912-1-43-21', '1505.06912-2-48-21'], ['1505.06912-1-43-23', '1505.06912-2-48-23'], ['1505.06912-1-43-24', '1505.06912-2-48-24'], ['1505.06912-1-43-28', '1505.06912-2-48-28'], ['1505.06912-1-13-0', '1505.06912-2-15-0'], ['1505.06912-1-14-0', '1505.06912-2-17-0'], ['1505.06912-1-14-2', '1505.06912-2-17-2'], ['1505.06912-1-14-3', '1505.06912-2-17-3'], ['1505.06912-1-17-0', '1505.06912-2-20-0'], ['1505.06912-1-17-1', '1505.06912-2-20-1'], ['1505.06912-1-40-0', '1505.06912-2-45-0'], ['1505.06912-1-40-1', '1505.06912-2-45-1'], ['1505.06912-1-40-2', '1505.06912-2-45-2'], ['1505.06912-1-40-6', '1505.06912-2-45-6'], ['1505.06912-1-40-7', '1505.06912-2-45-7'], ['1505.06912-1-40-8', '1505.06912-2-45-8'], ['1505.06912-1-16-0', '1505.06912-2-19-0'], ['1505.06912-1-0-0', '1505.06912-2-0-0'], ['1505.06912-1-0-1', '1505.06912-2-0-1'], ['1505.06912-1-0-2', '1505.06912-2-0-2'], ['1505.06912-1-22-0', '1505.06912-2-26-0'], ['1505.06912-1-22-1', '1505.06912-2-26-1'], ['1505.06912-1-22-2', '1505.06912-2-26-2'], ['1505.06912-1-22-3', '1505.06912-2-26-3'], ['1505.06912-1-29-0', '1505.06912-2-34-0'], ['1505.06912-1-29-1', '1505.06912-2-34-1'], ['1505.06912-1-29-2', '1505.06912-2-34-2'], ['1505.06912-1-29-3', '1505.06912-2-34-3'], ['1505.06912-1-5-0', '1505.06912-2-5-0'], ['1505.06912-1-33-0', '1505.06912-2-38-0'], ['1505.06912-1-33-1', '1505.06912-2-38-1'], ['1505.06912-1-33-2', '1505.06912-2-38-2'], ['1505.06912-1-33-4', '1505.06912-2-38-5'], ['1505.06912-1-33-6', '1505.06912-2-38-7'], ['1505.06912-1-33-7', '1505.06912-2-38-8'], ['1505.06912-1-33-8', '1505.06912-2-38-9'], ['1505.06912-1-35-0', '1505.06912-2-40-0'], ['1505.06912-1-35-2', '1505.06912-2-40-2'], ['1505.06912-1-35-3', '1505.06912-2-40-3'], ['1505.06912-1-35-4', '1505.06912-2-40-4'], ['1505.06912-1-35-5', '1505.06912-2-40-5'], ['1505.06912-1-35-6', '1505.06912-2-40-6'], ['1505.06912-1-45-0', '1505.06912-2-50-0'], ['1505.06912-1-4-0', '1505.06912-2-4-0'], ['1505.06912-1-4-1', '1505.06912-2-4-1'], ['1505.06912-1-4-2', '1505.06912-2-4-2'], ['1505.06912-1-4-3', '1505.06912-2-4-3'], ['1505.06912-1-4-4', '1505.06912-2-4-4'], ['1505.06912-1-4-5', '1505.06912-2-4-5'], ['1505.06912-1-4-6', '1505.06912-2-4-6'], ['1505.06912-1-4-7', '1505.06912-2-4-7'], ['1505.06912-1-4-8', '1505.06912-2-4-8'], ['1505.06912-1-4-9', '1505.06912-2-4-9'], ['1505.06912-1-4-10', '1505.06912-2-4-10'], ['1505.06912-1-56-0', '1505.06912-2-61-0'], ['1505.06912-1-56-1', '1505.06912-2-61-1'], ['1505.06912-1-32-1', '1505.06912-2-37-1'], ['1505.06912-1-32-2', '1505.06912-2-37-2'], ['1505.06912-1-32-3', '1505.06912-2-37-3'], ['1505.06912-1-48-0', '1505.06912-2-53-0'], ['1505.06912-1-48-1', '1505.06912-2-53-1'], ['1505.06912-1-48-2', '1505.06912-2-53-2'], ['1505.06912-1-48-3', '1505.06912-2-53-3'], ['1505.06912-1-48-4', '1505.06912-2-53-4'], ['1505.06912-1-48-5', '1505.06912-2-53-5'], ['1505.06912-1-48-6', '1505.06912-2-53-6'], ['1505.06912-1-57-0', '1505.06912-2-62-0'], ['1505.06912-1-8-1', '1505.06912-2-10-0'], ['1505.06912-1-38-2', '1505.06912-2-43-2'], ['1505.06912-1-53-5', '1505.06912-2-58-8'], ['1505.06912-1-44-9', '1505.06912-2-49-9'], ['1505.06912-1-14-1', '1505.06912-2-17-1'], ['1505.06912-1-16-1', '1505.06912-2-19-1'], ['1505.06912-1-33-5', '1505.06912-2-38-6'], ['1505.06912-1-32-0', '1505.06912-2-37-0'], ['1505.06912-1-6-0', '1505.06912-2-8-0'], ['1505.06912-1-6-0', '1505.06912-2-9-0'], ['1505.06912-1-7-0', '1505.06912-2-8-0'], ['1505.06912-1-53-8', '1505.06912-2-58-5'], ['1505.06912-1-45-2', '1505.06912-2-50-2'], ['1505.06912-1-6-0', '1505.06912-2-6-0']]

[['1505.06912-1-8-2', '1505.06912-2-10-1'], ['1505.06912-1-8-3', '1505.06912-2-10-2'], ['1505.06912-1-8-4', '1505.06912-2-10-3'], ['1505.06912-1-38-0', '1505.06912-2-43-0'], ['1505.06912-1-38-5', '1505.06912-2-43-4'], ['1505.06912-1-38-6', '1505.06912-2-43-5'], ['1505.06912-1-15-0', '1505.06912-2-18-0'], ['1505.06912-1-47-3', '1505.06912-2-52-2'], ['1505.06912-1-47-4', '1505.06912-2-52-3'], ['1505.06912-1-47-5', '1505.06912-2-52-4'], ['1505.06912-1-23-0', '1505.06912-2-27-0'], ['1505.06912-1-36-0', '1505.06912-2-41-0'], ['1505.06912-1-36-1', '1505.06912-2-41-1'], ['1505.06912-1-36-2', '1505.06912-2-41-2'], ['1505.06912-1-36-3', '1505.06912-2-41-3'], ['1505.06912-1-36-4', '1505.06912-2-41-4'], ['1505.06912-1-36-6', '1505.06912-2-41-6'], ['1505.06912-1-36-7', '1505.06912-2-41-7'], ['1505.06912-1-53-0', '1505.06912-2-58-0'], ['1505.06912-1-53-1', '1505.06912-2-58-1'], ['1505.06912-1-53-2', '1505.06912-2-58-2'], ['1505.06912-1-53-3', '1505.06912-2-58-3'], ['1505.06912-1-53-4', '1505.06912-2-58-4'], ['1505.06912-1-53-6', '1505.06912-2-58-6'], ['1505.06912-1-53-7', '1505.06912-2-58-7'], ['1505.06912-1-26-0', '1505.06912-2-30-0'], ['1505.06912-1-54-1', '1505.06912-2-59-1'], ['1505.06912-1-54-2', '1505.06912-2-59-2'], ['1505.06912-1-54-3', '1505.06912-2-59-3'], ['1505.06912-1-54-4', '1505.06912-2-59-4'], ['1505.06912-1-54-5', '1505.06912-2-59-5'], ['1505.06912-1-54-6', '1505.06912-2-59-6'], ['1505.06912-1-54-7', '1505.06912-2-59-7'], ['1505.06912-1-46-0', '1505.06912-2-51-0'], ['1505.06912-1-46-1', '1505.06912-2-51-1'], ['1505.06912-1-46-3', '1505.06912-2-51-3'], ['1505.06912-1-42-1', '1505.06912-2-47-1'], ['1505.06912-1-42-2', '1505.06912-2-47-2'], ['1505.06912-1-42-3', '1505.06912-2-47-3'], ['1505.06912-1-42-4', '1505.06912-2-47-4'], ['1505.06912-1-42-6', '1505.06912-2-47-6'], ['1505.06912-1-42-7', '1505.06912-2-47-7'], ['1505.06912-1-42-11', '1505.06912-2-47-11'], ['1505.06912-1-42-14', '1505.06912-2-47-14'], ['1505.06912-1-42-15', '1505.06912-2-47-15'], ['1505.06912-1-42-16', '1505.06912-2-47-16'], ['1505.06912-1-42-17', '1505.06912-2-47-17'], ['1505.06912-1-34-0', '1505.06912-2-39-0'], ['1505.06912-1-34-1', '1505.06912-2-39-1'], ['1505.06912-1-34-2', '1505.06912-2-39-2'], ['1505.06912-1-34-3', '1505.06912-2-39-3'], ['1505.06912-1-34-4', '1505.06912-2-39-4'], ['1505.06912-1-34-5', '1505.06912-2-39-5'], ['1505.06912-1-34-6', '1505.06912-2-39-6'], ['1505.06912-1-34-8', '1505.06912-2-39-8'], ['1505.06912-1-34-9', '1505.06912-2-39-9'], ['1505.06912-1-1-0', '1505.06912-2-1-0'], ['1505.06912-1-58-0', '1505.06912-2-63-0'], ['1505.06912-1-58-2', '1505.06912-2-63-2'], ['1505.06912-1-58-5', '1505.06912-2-63-5'], ['1505.06912-1-58-6', '1505.06912-2-63-6'], ['1505.06912-1-58-7', '1505.06912-2-63-7'], ['1505.06912-1-58-8', '1505.06912-2-63-8'], ['1505.06912-1-58-9', '1505.06912-2-63-9'], ['1505.06912-1-44-0', '1505.06912-2-49-0'], ['1505.06912-1-44-1', '1505.06912-2-49-1'], ['1505.06912-1-44-3', '1505.06912-2-49-3'], ['1505.06912-1-44-4', '1505.06912-2-49-4'], ['1505.06912-1-44-5', '1505.06912-2-49-5'], ['1505.06912-1-44-6', '1505.06912-2-49-6'], ['1505.06912-1-44-7', '1505.06912-2-49-7'], ['1505.06912-1-44-8', '1505.06912-2-49-8'], ['1505.06912-1-41-2', '1505.06912-2-46-2'], ['1505.06912-1-41-3', '1505.06912-2-46-3'], ['1505.06912-1-41-5', '1505.06912-2-46-5'], ['1505.06912-1-41-6', '1505.06912-2-46-6'], ['1505.06912-1-41-7', '1505.06912-2-46-7'], ['1505.06912-1-41-10', '1505.06912-2-46-10'], ['1505.06912-1-41-12', '1505.06912-2-46-12'], ['1505.06912-1-41-13', '1505.06912-2-46-13'], ['1505.06912-1-41-14', '1505.06912-2-46-14'], ['1505.06912-1-41-15', '1505.06912-2-46-15'], ['1505.06912-1-41-16', '1505.06912-2-46-16'], ['1505.06912-1-41-17', '1505.06912-2-46-17'], ['1505.06912-1-41-19', '1505.06912-2-46-19'], ['1505.06912-1-41-20', '1505.06912-2-46-20'], ['1505.06912-1-41-21', '1505.06912-2-46-21'], ['1505.06912-1-24-0', '1505.06912-2-28-0'], ['1505.06912-1-24-1', '1505.06912-2-28-1'], ['1505.06912-1-24-2', '1505.06912-2-28-2'], ['1505.06912-1-43-2', '1505.06912-2-48-2'], ['1505.06912-1-43-6', '1505.06912-2-48-6'], ['1505.06912-1-43-7', '1505.06912-2-48-7'], ['1505.06912-1-43-8', '1505.06912-2-48-8'], ['1505.06912-1-43-10', '1505.06912-2-48-10'], ['1505.06912-1-43-11', '1505.06912-2-48-11'], ['1505.06912-1-43-12', '1505.06912-2-48-12'], ['1505.06912-1-43-15', '1505.06912-2-48-15'], ['1505.06912-1-43-17', '1505.06912-2-48-17'], ['1505.06912-1-43-18', '1505.06912-2-48-18'], ['1505.06912-1-43-19', '1505.06912-2-48-19'], ['1505.06912-1-43-20', '1505.06912-2-48-20'], ['1505.06912-1-43-21', '1505.06912-2-48-21'], ['1505.06912-1-43-23', '1505.06912-2-48-23'], ['1505.06912-1-43-24', '1505.06912-2-48-24'], ['1505.06912-1-43-28', '1505.06912-2-48-28'], ['1505.06912-1-13-0', '1505.06912-2-15-0'], ['1505.06912-1-14-0', '1505.06912-2-17-0'], ['1505.06912-1-14-2', '1505.06912-2-17-2'], ['1505.06912-1-14-3', '1505.06912-2-17-3'], ['1505.06912-1-17-0', '1505.06912-2-20-0'], ['1505.06912-1-17-1', '1505.06912-2-20-1'], ['1505.06912-1-40-0', '1505.06912-2-45-0'], ['1505.06912-1-40-1', '1505.06912-2-45-1'], ['1505.06912-1-40-2', '1505.06912-2-45-2'], ['1505.06912-1-40-6', '1505.06912-2-45-6'], ['1505.06912-1-40-7', '1505.06912-2-45-7'], ['1505.06912-1-40-8', '1505.06912-2-45-8'], ['1505.06912-1-16-0', '1505.06912-2-19-0'], ['1505.06912-1-0-0', '1505.06912-2-0-0'], ['1505.06912-1-0-1', '1505.06912-2-0-1'], ['1505.06912-1-0-2', '1505.06912-2-0-2'], ['1505.06912-1-22-0', '1505.06912-2-26-0'], ['1505.06912-1-22-1', '1505.06912-2-26-1'], ['1505.06912-1-22-2', '1505.06912-2-26-2'], ['1505.06912-1-22-3', '1505.06912-2-26-3'], ['1505.06912-1-29-0', '1505.06912-2-34-0'], ['1505.06912-1-29-1', '1505.06912-2-34-1'], ['1505.06912-1-29-2', '1505.06912-2-34-2'], ['1505.06912-1-29-3', '1505.06912-2-34-3'], ['1505.06912-1-5-0', '1505.06912-2-5-0'], ['1505.06912-1-33-0', '1505.06912-2-38-0'], ['1505.06912-1-33-1', '1505.06912-2-38-1'], ['1505.06912-1-33-2', '1505.06912-2-38-2'], ['1505.06912-1-33-4', '1505.06912-2-38-5'], ['1505.06912-1-33-6', '1505.06912-2-38-7'], ['1505.06912-1-33-7', '1505.06912-2-38-8'], ['1505.06912-1-33-8', '1505.06912-2-38-9'], ['1505.06912-1-35-0', '1505.06912-2-40-0'], ['1505.06912-1-35-2', '1505.06912-2-40-2'], ['1505.06912-1-35-3', '1505.06912-2-40-3'], ['1505.06912-1-35-4', '1505.06912-2-40-4'], ['1505.06912-1-35-5', '1505.06912-2-40-5'], ['1505.06912-1-35-6', '1505.06912-2-40-6'], ['1505.06912-1-45-0', '1505.06912-2-50-0'], ['1505.06912-1-4-0', '1505.06912-2-4-0'], ['1505.06912-1-4-1', '1505.06912-2-4-1'], ['1505.06912-1-4-2', '1505.06912-2-4-2'], ['1505.06912-1-4-3', '1505.06912-2-4-3'], ['1505.06912-1-4-4', '1505.06912-2-4-4'], ['1505.06912-1-4-5', '1505.06912-2-4-5'], ['1505.06912-1-4-6', '1505.06912-2-4-6'], ['1505.06912-1-4-7', '1505.06912-2-4-7'], ['1505.06912-1-4-8', '1505.06912-2-4-8'], ['1505.06912-1-4-9', '1505.06912-2-4-9'], ['1505.06912-1-4-10', '1505.06912-2-4-10'], ['1505.06912-1-56-0', '1505.06912-2-61-0'], ['1505.06912-1-56-1', '1505.06912-2-61-1'], ['1505.06912-1-32-1', '1505.06912-2-37-1'], ['1505.06912-1-32-2', '1505.06912-2-37-2'], ['1505.06912-1-32-3', '1505.06912-2-37-3'], ['1505.06912-1-48-0', '1505.06912-2-53-0'], ['1505.06912-1-48-1', '1505.06912-2-53-1'], ['1505.06912-1-48-2', '1505.06912-2-53-2'], ['1505.06912-1-48-3', '1505.06912-2-53-3'], ['1505.06912-1-48-4', '1505.06912-2-53-4'], ['1505.06912-1-48-5', '1505.06912-2-53-5'], ['1505.06912-1-48-6', '1505.06912-2-53-6'], ['1505.06912-1-57-0', '1505.06912-2-62-0']]

[['1505.06912-1-8-1', '1505.06912-2-10-0'], ['1505.06912-1-38-2', '1505.06912-2-43-2'], ['1505.06912-1-53-5', '1505.06912-2-58-8'], ['1505.06912-1-44-9', '1505.06912-2-49-9'], ['1505.06912-1-14-1', '1505.06912-2-17-1'], ['1505.06912-1-16-1', '1505.06912-2-19-1'], ['1505.06912-1-33-5', '1505.06912-2-38-6'], ['1505.06912-1-32-0', '1505.06912-2-37-0'], ['1505.06912-1-6-0', '1505.06912-2-8-0'], ['1505.06912-1-6-0', '1505.06912-2-9-0'], ['1505.06912-1-7-0', '1505.06912-2-8-0']]

[]

[['1505.06912-1-53-8', '1505.06912-2-58-5'], ['1505.06912-1-45-2', '1505.06912-2-50-2'], ['1505.06912-1-6-0', '1505.06912-2-6-0']]

[]

['1505.06912-1-2-0', '1505.06912-1-9-0', '1505.06912-1-9-1', '1505.06912-1-10-0', '1505.06912-1-10-1', '1505.06912-1-11-0', '1505.06912-1-12-0', '1505.06912-1-18-0', '1505.06912-1-18-1', '1505.06912-1-19-0', '1505.06912-1-20-0', '1505.06912-1-21-0', '1505.06912-1-27-0', '1505.06912-1-28-0', '1505.06912-1-30-0', '1505.06912-1-30-1', '1505.06912-1-30-2', '1505.06912-1-30-3', '1505.06912-1-30-4', '1505.06912-1-31-0', '1505.06912-1-33-3', '1505.06912-1-34-7', '1505.06912-1-35-1', '1505.06912-1-36-5', '1505.06912-1-38-1', '1505.06912-1-38-4', '1505.06912-1-39-0', '1505.06912-1-40-3', '1505.06912-1-40-4', '1505.06912-1-40-5', '1505.06912-1-41-0', '1505.06912-1-41-1', '1505.06912-1-41-4', '1505.06912-1-41-8', '1505.06912-1-41-9', '1505.06912-1-41-11', '1505.06912-1-41-18', '1505.06912-1-42-0', '1505.06912-1-42-5', '1505.06912-1-42-8', '1505.06912-1-42-9', '1505.06912-1-42-10', '1505.06912-1-42-12', '1505.06912-1-42-13', '1505.06912-1-43-0', '1505.06912-1-43-1', '1505.06912-1-43-3', '1505.06912-1-43-4', '1505.06912-1-43-5', '1505.06912-1-43-9', '1505.06912-1-43-13', '1505.06912-1-43-14', '1505.06912-1-43-16', '1505.06912-1-43-22', '1505.06912-1-43-25', '1505.06912-1-43-26', '1505.06912-1-43-27', '1505.06912-1-44-2', '1505.06912-1-46-2', '1505.06912-1-47-0', '1505.06912-1-47-1', '1505.06912-1-50-0', '1505.06912-1-50-1', '1505.06912-1-50-2', '1505.06912-1-51-0', '1505.06912-1-52-0', '1505.06912-1-52-1', '1505.06912-1-54-0', '1505.06912-1-55-0', '1505.06912-1-58-1', '1505.06912-1-58-3', '1505.06912-1-58-4', '1505.06912-2-2-0', '1505.06912-2-7-0', '1505.06912-2-11-0', '1505.06912-2-11-1', '1505.06912-2-12-0', '1505.06912-2-12-1', '1505.06912-2-13-0', '1505.06912-2-14-0', '1505.06912-2-21-0', '1505.06912-2-21-1', '1505.06912-2-22-0', '1505.06912-2-23-0', '1505.06912-2-24-0', '1505.06912-2-25-0', '1505.06912-2-31-0', '1505.06912-2-32-0', '1505.06912-2-33-0', '1505.06912-2-35-0', '1505.06912-2-35-1', '1505.06912-2-35-2', '1505.06912-2-35-3', '1505.06912-2-35-4', '1505.06912-2-36-0', '1505.06912-2-36-1', '1505.06912-2-38-3', '1505.06912-2-38-4', '1505.06912-2-39-7', '1505.06912-2-40-1', '1505.06912-2-41-5', '1505.06912-2-43-3', '1505.06912-2-44-0', '1505.06912-2-45-3', '1505.06912-2-45-4', '1505.06912-2-45-5', '1505.06912-2-46-0', '1505.06912-2-46-1', '1505.06912-2-46-4', '1505.06912-2-46-8', '1505.06912-2-46-9', '1505.06912-2-46-11', '1505.06912-2-46-18', '1505.06912-2-47-0', '1505.06912-2-47-5', '1505.06912-2-47-8', '1505.06912-2-47-9', '1505.06912-2-47-10', '1505.06912-2-47-12', '1505.06912-2-47-13', '1505.06912-2-48-0', '1505.06912-2-48-1', '1505.06912-2-48-3', '1505.06912-2-48-4', '1505.06912-2-48-5', '1505.06912-2-48-9', '1505.06912-2-48-13', '1505.06912-2-48-14', '1505.06912-2-48-16', '1505.06912-2-48-22', '1505.06912-2-48-25', '1505.06912-2-48-26', '1505.06912-2-48-27', '1505.06912-2-49-2', '1505.06912-2-51-2', '1505.06912-2-52-0', '1505.06912-2-55-0', '1505.06912-2-55-1', '1505.06912-2-55-2', '1505.06912-2-56-0', '1505.06912-2-57-0', '1505.06912-2-57-1', '1505.06912-2-59-0', '1505.06912-2-60-0', '1505.06912-2-63-1', '1505.06912-2-63-3', '1505.06912-2-63-4']

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

https://arxiv.org/abs/1505.06912

astro-ph-9701224

{'astro-ph-9701224-1-0-0': 'We assemble a catalog of 798 galaxy redshifts in the region of the Coma cluster and we examine the presence of substructure and of voids at different scales ranging from [MATH] to [MATH] Mpc, using subsamples of our catalog, ranging from [MATH] km/s to [MATH] km/s.', 'astro-ph-9701224-1-0-1': 'Our structure analysis method is based on the wavelet transform and on the segmentation analysis.', 'astro-ph-9701224-1-0-2': 'The wavelet transform allows us to find out structures at different scales and the segmentation method allows us a quantitative statistical and morphological analysis of the sample.', 'astro-ph-9701224-1-0-3': 'Moreover, from the whole catalog we have selected [MATH] galaxies, with redshifts between [MATH] km/s and [MATH] km/s that we identify as belonging to the central region of Coma and on which we have performed a deeper analysis, on scales ranging from [MATH] Kpc to [MATH] Mpc.', 'astro-ph-9701224-1-0-4': 'Our results are expressed in terms of number of structures or voids and sphericity for different values of the threshold detection and at all the scales investigated.', 'astro-ph-9701224-1-0-5': 'According to our analysis, there is strong evidence for multiple hierarchical substructure, on the scales ranging from a few hundreds of Kpc to about [MATH] Mpc, and the substructure morphology is rather spherical.', 'astro-ph-9701224-1-0-6': 'We find the two subclusters as described in previous papers, but, moreover, we can say that more partitions are evident inside the Coma cluster, corresponding approximatively to the galaxies: NGC 4934 4840, 4889, 4898 4864, 4874 4839, 4927, 4875.', 'astro-ph-9701224-1-1-0': '# Introduction', 'astro-ph-9701224-1-2-0': 'The Coma cluster (number 1656 in the Abell [1958] catalog) has been perhaps the most studied galaxy cluster since [MATH], when Zwicky calculated its mass (Zwicky, 1933).', 'astro-ph-9701224-1-2-1': 'This paper takes up the issue of substructure in the cluster.', 'astro-ph-9701224-1-2-2': 'In previous papers (e.g. Fitchett Webster, 1987; Mellier et al., 1988; Baier et al., 1990; Briel et al., 1992; White et al., 1993; Colless Dunn, 1996, hereafter CD96 and Biviano et al., 1996) it has been suggested that the Coma cluster may have a complex structure.', 'astro-ph-9701224-1-2-3': 'The X-ray images obtained with ROSAT suggest clumps of emission associated with substructures (Briel et al., 1992; White et al., 1993).', 'astro-ph-9701224-1-2-4': 'However, previous analysis were performed only on 2-D slices of the cluster.', 'astro-ph-9701224-1-3-0': 'Our aim is to find out substructure or voids at different scales through a three-dimensional analysis of the cluster, identify it and make a morphological analysis of every single structure singled out.', 'astro-ph-9701224-1-3-1': 'Our method allow us to detect substructure not visible in 2-D images.', 'astro-ph-9701224-1-4-0': 'The plan of the paper is as follows: in 2 we discuss how we have assembled our catalog.', 'astro-ph-9701224-1-4-1': 'In 3 and in 4 we discuss the method of analysis based on wavelet transform and segmentation.', 'astro-ph-9701224-1-4-2': 'In 5 we report our results for the Coma cluster and in 6 we do the same for the central region of the Coma cluster.', 'astro-ph-9701224-1-4-3': 'Finally, in 7 we report our conclusions.', 'astro-ph-9701224-1-5-0': '# The Data', 'astro-ph-9701224-1-6-0': 'A large body of data on the Coma cluster is available.', 'astro-ph-9701224-1-6-1': 'We have used a combination of various catalogs to assemble our own.', 'astro-ph-9701224-1-6-2': 'Our redshifts catalog is made of a total number of 798 galaxies with the equatorial coordinates in the range [EQUATION] (B1950.0, hereafter all coordinates are referred to1950.0).', 'astro-ph-9701224-1-6-3': 'The redshifts for 243 galaxies have been kindly provided us by the new redshifts survey by CD96.', 'astro-ph-9701224-1-6-4': 'They have used the Hydra fiber spectrograph on the KPNO 4m telescope to measure the redshift with a mean uncertainty [MATH] of about [MATH] Km/s; the uncertainty in the positions of the galaxies is less than 1".', 'astro-ph-9701224-1-6-5': 'Another 305 redshifts have been taken from Biviano et al. (1996); 225 of them are new measurements made using the MOS-SIS spectrograph at the Canada-France-Hawaii Telescope; the mean uncertainty is about [MATH] Km/s.', 'astro-ph-9701224-1-6-6': 'The positions in this catalog are known with a mean error of about 2".', 'astro-ph-9701224-1-6-7': 'Another 320 redshifts have been taken from the catalog of 379 galaxies by I. D. Karachentsev [MATH] A. I. Kopylov (1990).', 'astro-ph-9701224-1-6-8': 'They made their spectroscopic survey with the 6m telescope of the Special Astrophysical Observatory of the ex-USSR Academy of Science; the first observations were performed with the image tube spectrograph in the prime focus cage while the last with the 1024-channel TV scanner at the Nasmyth focus.', 'astro-ph-9701224-1-6-9': 'The mean uncertainty in their measurements is [MATH] Km/s.', 'astro-ph-9701224-1-6-10': 'The mean error in the positions is [MATH] 3".', 'astro-ph-9701224-1-6-11': 'Finally, another 46 redshifts have been taken from the NED (NASA extragalattic database); these data are heterogeneous; however their mean uncertainty is less then [MATH] km/s whilst the position are known with a mean error of about 6".', 'astro-ph-9701224-1-7-0': 'The total number of galaxies collected is 914, but some objects are in common.', 'astro-ph-9701224-1-7-1': 'We have considered a galaxy common when its position in a catalog is inside the mean error in the coordinate determination in another catalog.', 'astro-ph-9701224-1-7-2': 'So, if for the same galaxy several redshift measurements were available, we have chosen the most accurate one.', 'astro-ph-9701224-1-8-0': 'The completeness of our heterogeneous catalog is about [MATH] at [MATH].', 'astro-ph-9701224-1-8-1': 'This value has been calculated from a weighted mean of the values for the different database used in our compilation.', 'astro-ph-9701224-1-9-0': 'To summarize, we have a heterogeneous sample of redshifts for 798 galaxies with [MATH] km/s.', 'astro-ph-9701224-1-9-1': 'Our coordinates of the centre of the cluster is the value derived from the survey of Godwin et al. (1983): [MATH] and [MATH].', 'astro-ph-9701224-1-9-2': 'The uncertainty on the redshifts measurement is about 100 Km/s and the maximum error on the positions is less than 3".', 'astro-ph-9701224-1-9-3': 'From this catalog we extract three different subsamples as shown in Table 1.', 'astro-ph-9701224-1-9-4': 'The line-of-sight distribution for the galaxies of our subsample with 3000 km/s [MATH] 28000 km/s is shown in fig. 1a.', 'astro-ph-9701224-1-9-5': 'Hereafter we call this region [MATH].', 'astro-ph-9701224-1-9-6': 'The number of galaxies inside this region is 690.', 'astro-ph-9701224-1-9-7': 'We note the presence of two main peaks.', 'astro-ph-9701224-1-10-0': '[MATH] has a mean redshift [MATH] km/s and a standard deviation [MATH] km/s.', 'astro-ph-9701224-1-10-1': 'The galaxies of the first peak have redshifts [MATH] km/s; the mean redshift and the standard deviation being respectively [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-10-2': 'The line-of-sight velocity dispersion for the galaxies of the first peak is [MATH] km/s.', 'astro-ph-9701224-1-10-3': 'The galaxies of the second peak have redshifts [MATH] km/s; the mean redshift and the standard deviation being respectively [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-10-4': 'In Fig. 1b only those galaxies inside the first peak are considered and the line-of-sight distribution is shown.', 'astro-ph-9701224-1-10-5': 'Hereafter we call this region [MATH].', 'astro-ph-9701224-1-10-6': 'The number of galaxies inside [MATH] is 485.', 'astro-ph-9701224-1-10-7': 'From Fig. 1b we note that distribution of Coma galaxies in the redshift field show the presence of some peaks and this may suggest the existence of multiple substructure.', 'astro-ph-9701224-1-11-0': 'In order to analyse our catalog we trasform the angular coordinates of the galaxies in linear coordinates by mean of the Hubble law and of the properties of triangles.', 'astro-ph-9701224-1-11-1': 'Besides, we call zero of the [MATH] axis the distance that corresponds to [MATH].', 'astro-ph-9701224-1-11-2': 'Then, to make our linear coordinates independent from the value of [MATH], we renormalize them dividing by [MATH].', 'astro-ph-9701224-1-12-0': '# The method of analysis: wavelet transform and segmentation.', 'astro-ph-9701224-1-13-0': '## The wavelet transform', 'astro-ph-9701224-1-14-0': 'Our method of structure detection is based on the wavelet transform evaluated at several scales and on the segmentation analysis and is similar to the one developed by Lega (1994, hereafter L94).', 'astro-ph-9701224-1-15-0': 'A detailed description of the implementation of the algorithms is beyond the purpose of this paper.', 'astro-ph-9701224-1-15-1': 'A version for a Connection Machine CM200 can be found in L94, a new PVM version will be described in Pagliaro et al. (1997a).', 'astro-ph-9701224-1-15-2': 'Our method is a three-dimensional analysis.', 'astro-ph-9701224-1-15-3': 'However, for the sake of simplicity, here we describe the one dimensional reduction.', 'astro-ph-9701224-1-15-4': 'The generalization to the 3-D case is straightforward.', 'astro-ph-9701224-1-16-0': 'For a one-dimensional function [MATH] the wavelet transform is a linear operator that can be written as: [EQUATION] where [MATH] is the scale on which the analysis is performed, [MATH] is the spatial translation parameter and [MATH] is the the Grossmann-Morlet (1984, 1987) analyzing wavelet function [EQUATION] that is spatially centered around [MATH] and has scale [MATH].', 'astro-ph-9701224-1-16-1': 'The wavelet function [MATH] is called mother wavelet.', 'astro-ph-9701224-1-16-2': 'It generates the other wavelet function [MATH].', 'astro-ph-9701224-1-16-3': 'We follow L94 in the choice of the mother wavelet: [EQUATION] where [MATH] is the cubic centred B-spline function defined by: [EQUATION]', 'astro-ph-9701224-1-16-4': 'Although our data distribution is highly anisotropic, we prefer to use an isotropic wavelet function and to perform a scale transformation along the [MATH] axis.', 'astro-ph-9701224-1-17-0': 'With these choices, the wavelet coefficients at different scales can be calculated by the a trous algorithm, as described by L94 (pag.100).', 'astro-ph-9701224-1-17-1': 'The set of scales are powers of two: [MATH].', 'astro-ph-9701224-1-18-0': 'The scale [MATH] in this kind of analysis may be considered as the resolution.', 'astro-ph-9701224-1-18-1': 'In other words, if we perform a calculation on a scale [MATH], we expect the wavelet transform to be sensitive to structures with typical size of about [MATH] and to find out those structures.', 'astro-ph-9701224-1-19-0': '## The thresholding', 'astro-ph-9701224-1-20-0': 'The thresholding is made on the wavelet coefficient histogram.', 'astro-ph-9701224-1-20-1': 'For a flat background, the wavelet transform yields coefficients equal to zero.', 'astro-ph-9701224-1-20-2': 'The existence of structures at a given scale gives wavelet coefficient with large positive values.', 'astro-ph-9701224-1-20-3': 'Obviously, a random distribution may have non zero coefficients even if there is no structure, due to the statistical fluctuations.', 'astro-ph-9701224-1-20-4': 'Moreover, the statistical behaviour of the wavelet coefficient is complex due to the correlation among nearby pixels.', 'astro-ph-9701224-1-20-5': 'We choose the threshold through a classical decision rule.', 'astro-ph-9701224-1-20-6': 'We calculate the wavelet coefficients [MATH] for each scale of our analysis, for [MATH] random distribution in the same region of space of our data and on the same grid.', 'astro-ph-9701224-1-20-7': 'Then we calculate the probability [MATH] and choose the value [MATH] so that: [EQUATION]', 'astro-ph-9701224-1-20-8': 'Our threshold on the scale [MATH] is the value [MATH].', 'astro-ph-9701224-1-20-9': 'Our choice for the value of [MATH] is: [EQUATION] that ensures a [MATH] confidence level in the structure detection.', 'astro-ph-9701224-1-21-0': 'However, for the sake of completeness, we perform our analysis for several values of the threshold, calculated in terms of the standard deviation in the wavelet coefficient distribution of our data.', 'astro-ph-9701224-1-22-0': '## Structure numbering through segmentation', 'astro-ph-9701224-1-23-0': 'The second step of our analysis is the determination of connected pixels over a fixed threshold (segmentation, Rosenfeld 1969), the numbering of the selected structures and their morphological analysis.', 'astro-ph-9701224-1-24-0': 'The segmentation and numbering consists in the exam of the wavelet coefficients matrix; all the pixels associated with a wavelet coefficient greater than the selected threshold are labeled with a integer number.', 'astro-ph-9701224-1-24-1': 'All other pixels labels are set equal to zero.', 'astro-ph-9701224-1-24-2': 'Then, the same label is associated with all the pixels connected in a single structure, in a sequential way.', 'astro-ph-9701224-1-24-3': "So, the first structure individuated has the label '1' and so on.", 'astro-ph-9701224-1-24-4': 'Then, for each structure we calculate volume and surface and from them a morphological parameter.', 'astro-ph-9701224-1-25-0': 'The detailed description of the algorithm is beyond the purpose of this paper.', 'astro-ph-9701224-1-25-1': 'However it can be described in brief as follows: Step 1: All pixels with [MATH] are labeled.', 'astro-ph-9701224-1-25-2': 'Step 2: The same label is associated with those pixels labeled and connected.', 'astro-ph-9701224-1-25-3': "This is done in a sequential way: the first structure detected has the label '1', the [MATH]-th one has the label 'N'.", 'astro-ph-9701224-1-25-4': 'This requires a renumbering of most pixels.', 'astro-ph-9701224-1-25-5': 'Step 3: Volume and surface of each structure singled out are calculated.', 'astro-ph-9701224-1-26-0': '## The morphological parameter', 'astro-ph-9701224-1-27-0': 'In order to perform a morphological analysis we have to introduce a morphological parameter that quantifies the sphericity of the structures.', 'astro-ph-9701224-1-27-1': 'We choose the parameter: [EQUATION] where [MATH] is the volume and [MATH] is the surface, as in L94, and [MATH] is a parameter that depends on the scale of the analysis.', 'astro-ph-9701224-1-27-2': 'We want [MATH] to have the following behaviour: zero for very filamentary structures and [MATH] for spherical ones.', 'astro-ph-9701224-1-27-3': 'This may be achieved putting [MATH], but only for those scales not affected by the granular nature of the analysis.', 'astro-ph-9701224-1-27-4': 'We choose the value [MATH] only for the scales [MATH] pixels with [MATH].', 'astro-ph-9701224-1-27-5': 'For the smallest scales the constant [MATH] is not adequate, since we are close to the grid resolution and the geometry of the substructures cannot be spherical.', 'astro-ph-9701224-1-27-6': 'So, since we want to consider as spherical a one-pixel structure, we adopt the values: [EQUATION]', 'astro-ph-9701224-1-27-7': 'Then, for every detection threshold we calculate the values: [EQUATION] where [MATH] is the number of objects detected at scale [MATH].', 'astro-ph-9701224-1-28-0': '# The voids detection method', 'astro-ph-9701224-1-29-0': 'Voids detection method is analogue to structure detection method, as far as some "greater than" is replaced by the same number of "less than".', 'astro-ph-9701224-1-29-1': 'The voids thresholding is made on the wavelet coefficient histogram too.', 'astro-ph-9701224-1-29-2': 'The presence of voids at a given scale gives wavelet coefficient with large negative values.', 'astro-ph-9701224-1-30-0': 'Our choice for the threshold is the same as in [MATH].', 'astro-ph-9701224-1-30-1': 'We calculate the wavelet coefficients [MATH] for each scale of our analysis, for [MATH] random distribution in the same region of space of our data.', 'astro-ph-9701224-1-30-2': 'Then we calculate the probability [MATH] and choose the value [MATH] so that: [EQUATION] with [MATH], that ensures a [MATH] confidence level in the voids detection.', 'astro-ph-9701224-1-30-3': 'Our threshold is the value [MATH].', 'astro-ph-9701224-1-30-4': 'Obviously, in the segmentation algorithm for the voids detection the labeled pixels are those with the wavelet coefficients: [MATH].', 'astro-ph-9701224-1-31-0': 'The determination of voids morphological parameter values is analogue to the determination of structures morphological parameter value and is described in 3.4.', 'astro-ph-9701224-1-32-0': '# Substructure and voids detection in the Coma cluster', 'astro-ph-9701224-1-33-0': 'We examine two catalogs.', 'astro-ph-9701224-1-33-1': 'The first is [MATH] made of [MATH] galaxies, as previously said, with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-33-2': 'Our grid ensures a resolution of about [MATH] Mpc on each of the three axis.', 'astro-ph-9701224-1-33-3': 'We examinate four different scales: [MATH], [MATH], [MATH], [MATH] Mpc, for six different values of the threshold: the [MATH] confidence level as described in 3, and then: [MATH], [MATH], [MATH], [MATH] and [MATH], where [MATH] is the standard deviation in the wavelet coefficient distribution for the selected scale.', 'astro-ph-9701224-1-33-4': 'The second catalog investigated is [MATH].', 'astro-ph-9701224-1-33-5': 'It is made of [MATH] galaxies with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-33-6': 'Our analysis grid ensures a resolution of about [MATH] Kpc on each of the three axis.', 'astro-ph-9701224-1-33-7': 'We examinate four different scales: [MATH], [MATH], [MATH], [MATH] Mpc, for the same six different values of the threshold as before We show the wavelet coefficients distributions on the four scales for the two catalogs in Fig. [MATH].', 'astro-ph-9701224-1-33-8': 'We plot the value [MATH], where [MATH] is the wavelet coefficient value and [MATH] and [MATH] are the mean and the standard deviation in the wavelet coefficient distribution, versus [MATH], where [MATH] is the probability associated to the wavelet coefficient [MATH].', 'astro-ph-9701224-1-34-0': 'The curves are slightly asymmetric on all the scales, with a small queue towards the positive values of the coefficients, meaning presence of substructure.', 'astro-ph-9701224-1-34-1': 'Our results are expressed in terms of number of structures at the selected scales (see the Tables 2 - 4).', 'astro-ph-9701224-1-34-2': 'Considering the [MATH] confidence level as significance level for the structure detection, we have overwhelming evidence for substructure inside the Coma cluster on the scales: [MATH] to [MATH] Mpc.', 'astro-ph-9701224-1-34-3': 'We show the morphological parameter [MATH] in the Tables 2, 3 and 4.', 'astro-ph-9701224-1-34-4': 'For what concerns [MATH], our substructures are rather spherical on the first two scales.', 'astro-ph-9701224-1-34-5': 'The value of [MATH] is lowered till [MATH] on the scale [MATH] Mpc, meaning a much more filamentary morphology for those structures singled out at this resolution.', 'astro-ph-9701224-1-34-6': 'The substructure of [MATH], singled out with a greater resolution shows a spherical morphology till the scale of [MATH] Mpc; more elongated shapes are found out at the bigger scales.', 'astro-ph-9701224-1-34-7': 'In both cases the diminution of [MATH] and of the number of structures as a function of the scale indicates a hierarchical distribution.', 'astro-ph-9701224-1-35-0': '# The central region of Coma', 'astro-ph-9701224-1-36-0': '## Substructures', 'astro-ph-9701224-1-37-0': 'We consider a galaxy belonging to the central region of Coma if its redshift is inside [MATH] from [MATH] km/s, where [MATH] and [MATH] are the mean and the standard deviation in the redshifts distribution calculated on the 485 galaxies considered in [MATH].', 'astro-ph-9701224-1-37-1': 'Our catalog [MATH] is made of [MATH] galaxies, with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-37-2': 'The mean redshift is [MATH] km/s and the standard deviation [MATH] km/s.', 'astro-ph-9701224-1-38-0': 'Our grid ensures a resolution of about [MATH] Kpc on each of the three axis.', 'astro-ph-9701224-1-38-1': 'We examinate four different scales: [MATH], [MATH], [MATH] and [MATH] Kpc, for the usual six different values of the threshold ranging from [MATH] to [MATH] plus the [MATH] confidence level threshold, where [MATH] is the standard deviation in the wavelet coefficient distribution for the selected scale.', 'astro-ph-9701224-1-38-2': 'We show the wavelet coefficients distributions on the four scales in Fig.4.', 'astro-ph-9701224-1-38-3': 'These are slightly asymmetric too, with the usual small queue towards the positive values, meaning presence of substructure also inside the central region of the cluster.', 'astro-ph-9701224-1-38-4': 'Our results are expressed in terms of number of structures at the selected scales.', 'astro-ph-9701224-1-38-5': 'Considering the [MATH] confidence level threshold as significance level for the structure detection, we have overwhelming evidence for substructure inside the central region of the Coma cluster on the first three scales investigated: [MATH], [MATH] and [MATH] Kpc (see the Table 4A).', 'astro-ph-9701224-1-38-6': 'The morphological parameter is shown in Table 4B.', 'astro-ph-9701224-1-38-7': 'Our substructure are rather spherical on all the scales but the last one, where its value is about [MATH], meaning that shapes become more elongated on a scale of typical size [MATH] Kpc, inside the central region.', 'astro-ph-9701224-1-39-0': '## Search for segregation', 'astro-ph-9701224-1-40-0': 'Having identified the substructures we tried to search for any evidence of segregation, in luminosity and/or colour.', 'astro-ph-9701224-1-40-1': 'Recently some evidence of morphological segregation within Coma has been found (Andreon, 1996).', 'astro-ph-9701224-1-40-2': 'Unfortunately we had not enough morphological information to attempt an analysis of morphological segregation among the substructures we found.', 'astro-ph-9701224-1-40-3': 'In fig. 6 we show that there is no evidence that the different subgroups observed within the central region of Coma differ as far as colour distribution [MATH] is concerned.', 'astro-ph-9701224-1-40-4': 'One must however keep in mind that this colour index is not strongly correlated with absolute [MATH]-magnitude, so that from this figure one cannot draw any conclusion about the presence of morphological segregation.', 'astro-ph-9701224-1-40-5': 'We will examine these and other aspects of luminosity functions within Coma in a forthcoming paper.', 'astro-ph-9701224-1-41-0': '# Conclusions and discussion', 'astro-ph-9701224-1-42-0': 'The last years have witnessed an upsurge of interest towards the Coma cluster (e.g. Mellier et al., 1988; Escalera et al., 1992; CD96; Biviano et al., 1996), with most of the effort going to analyse the structure of the cluster and to decide whether it can be classified as a relaxed one or not and in order to unveil hidden substructures.', 'astro-ph-9701224-1-42-1': 'The recent ROSAT images and the following 2-D analysis show the existence of multiple substructure and suggest that it can not be considered a relaxed cluster.', 'astro-ph-9701224-1-42-2': 'Already in 1988 Mellier et al. (1988), by analysing the isopleths within a 2-D map of the cluster had suggested the existence of [MATH] density peaks.', 'astro-ph-9701224-1-42-3': 'In this paper, we have investigated the nature of the Coma cluster performing a 3-D analysis of the cluster.', 'astro-ph-9701224-1-42-4': 'We have assembled a catalog of 798 galaxy redshifts, the largest presently available for the Coma cluster.', 'astro-ph-9701224-1-42-5': 'Then, we have developed a 3-D wavelet and segmentation structure analysis method, that allows us to find out substructures on different scales and to describe it in a quantitative way.', 'astro-ph-9701224-1-42-6': 'This powerful method of analysis has already provided excellent results in many fields of physics (e.g. Arneodo et al.,1988; Argoul et al., 1989; Slezak et al., 1990; Fujiwara Soda, 1995; Grebenev et al., 1995).', 'astro-ph-9701224-1-43-0': 'Our results suggests that Coma can not be considered a regular cluster of galaxies, but it is filled up with substructure on all scales ranging from a few hundreds of Kpc to [MATH] Mpc.', 'astro-ph-9701224-1-43-1': 'The general diminution of the mean morphological parameter, meaning more elongated shapes, and of the number of structures with the scale indicates a hierarchical distribution of the substructure.', 'astro-ph-9701224-1-44-0': 'We have examinated the Coma cluster using three different subsamples of our catalog (see Table 1); so we have sights of regions of different sizes with different resolutions.', 'astro-ph-9701224-1-45-0': 'On a scale of about [MATH] Mpc, our analysis on the extended Coma catalog suggests the presence of multiple substructure with spherical morphology (see Table 2).', 'astro-ph-9701224-1-45-1': 'On this scale a large number of voids is detected and their shapes are rather spherical.', 'astro-ph-9701224-1-45-2': 'On the same catalog, multiple substructure is still present at the scale [MATH] Mpc: on this scale shapes are more elongated ([MATH]).', 'astro-ph-9701224-1-45-3': 'On scales [MATH] Mpc and [MATH] Mpc we find only two very elongated objects, in agreement with the histogram of Fig.1a.', 'astro-ph-9701224-1-45-4': 'Voids on scales greater than [MATH] Mpc are few and very elongated.', 'astro-ph-9701224-1-45-5': 'This seems to indicate that the Coma cluster and the galaxies included in the second peak of the histogram of the galaxy distribution in Fig. 1a were not generated by the collapse of two large spherical density perturbation with different masses and radius of about [MATH] Mpc, but by the merging of a large number of isolated spherical density perturbation of radius ranging from [MATH] Mpc to [MATH] Mpc.', 'astro-ph-9701224-1-45-6': 'This first rough picture of the Coma evolution becomes more evident if we examine the catalogs [MATH] and [MATH].', 'astro-ph-9701224-1-46-0': 'Our analysis of the second catalog ([MATH]) suggests presence of substructures on all the scales with shapes becoming more elongated with growing scale (see Table 3) Voids are detected only on scales [MATH] Mpc and their shapes are rather spherical ([MATH])', 'astro-ph-9701224-1-47-0': 'On smaller scales (hundreds of Kpc) we have concentrated our analysis on a central region with redshifts [MATH] km/s.', 'astro-ph-9701224-1-47-1': 'This region includes the core of Coma with the galaxies NGC 4874 ([MATH] 7131 km/s) and NGC 4839 ([MATH] 7397 km/s).', 'astro-ph-9701224-1-47-2': 'Multiple substructures is found on scales [MATH] Kpc with rather spherical morphology ([MATH]).', 'astro-ph-9701224-1-47-3': 'A large number of spherical ([MATH]) voids is detected only on the smallest scale ([MATH] Kpc).', 'astro-ph-9701224-1-48-0': 'Finally, we concentrate on the scale of [MATH] Kpc, where we have detected seven objects as shown in Fig. 5 and in agreement with the peaks that we find in the central region of the histogram of Fig.1b.', 'astro-ph-9701224-1-48-1': 'To each one of these object we associate a dominant galaxy.', 'astro-ph-9701224-1-48-2': 'Mean redshifts of the objects are: [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-1-48-3': 'The first clump contains only two galaxies.', 'astro-ph-9701224-1-48-4': 'The others are described in more details in Table 6.', 'astro-ph-9701224-1-48-5': 'To each clump singled out we can associate one or two dominant galaxies.', 'astro-ph-9701224-1-48-6': 'These are (for increasing redshift): NGC 4934 NGC 4840, NGC 4889, NGC 4898 NGC 4864, NGC 4874 NGC 4839, NGC 4927, NGC 4875.', 'astro-ph-9701224-1-48-7': 'So, we found two subclusters, as described by CD96, but, moreover, we can say that more partitions are evident inside the Coma cluster (see Fig. 5).', 'astro-ph-9701224-1-48-8': 'Now, we believe both that the Coma cluster can not be considered a regular cluster of galaxies and that its process of formation occurs through a bottom-up mechanism as predicted by CDM and MDM models.', 'astro-ph-9701224-1-49-0': 'All this results suggest a formation scenario for the Coma cluster envisaging two differents moments.', 'astro-ph-9701224-1-49-1': 'At the start, the collapse of isolated almost spherical density perturbation with different masses and radius ranging from [MATH] Kpc to [MATH] Kpc occurs.', 'astro-ph-9701224-1-49-2': 'This first collapse engendered the small substructures that we observe inside the Coma cluster.', 'astro-ph-9701224-1-49-3': 'In the second step, owing to the gravitational force, the substructures were forced to merge in more complex and larger structures.', 'astro-ph-9701224-1-50-0': 'A dynamical analysis of the data is in development and will be described in our next paper (Pagliaro et al., 1997b).', 'astro-ph-9701224-1-51-0': 'A.Pa.', 'astro-ph-9701224-1-51-1': 'would like to thank E. Lega for having sent him her structure detection code, a copy of her Ph.D. dissertation and for kind and indispensable help during the period in which our code was developed and tested, and A. Bijaoui for a stimulating discussion held in Erice.', 'astro-ph-9701224-1-51-2': 'M.Ga.', 'astro-ph-9701224-1-51-3': 'wish to thank S. Shandarin for a clarifying discussion and helpful suggestions.', 'astro-ph-9701224-1-52-0': '[f1a.ps, f1b.ps] (a) Histogram of the galaxy distribution, with redshifts [MATH] km/s inside [MATH], according to our catalog.', 'astro-ph-9701224-1-52-1': 'Step is 500 km/s; (b) Histogram of the galaxy distribution, with redshift [MATH] inside [MATH], according to our catalog.', 'astro-ph-9701224-1-52-2': 'Step is 100 km/s.', 'astro-ph-9701224-1-53-0': '[f2.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-1-54-0': '[f3.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-1-55-0': '[f4.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-1-56-0': '[f5.ps]Substructures for the central region of Coma detected with a resolution of [MATH] Kpc.', 'astro-ph-9701224-1-56-1': 'The seven structures detected are clearly visible.', 'astro-ph-9701224-1-56-2': 'Morphological parameter: [MATH]', 'astro-ph-9701224-1-57-0': '[f6.ps]Colour segregation.', 'astro-ph-9701224-1-57-1': 'The different histograms refer to the substructures of Table 6, and are numbered according to the first column of that table.'}

{'astro-ph-9701224-2-0-0': 'Evidence for clustering within the Coma cluster is found by means of a multiscale analysis of the combined angular-redshift distribution.', 'astro-ph-9701224-2-0-1': 'We have compiled a catalogue of 798 galaxy redshifts from published surveys from the region of the Coma cluster.', 'astro-ph-9701224-2-0-2': 'We examine the presence of substructure and of voids at different scales ranging from [MATH] to [MATH] Mpc, using subsamples of the catalogue, ranging from [MATH] km/s to [MATH] km/s.', 'astro-ph-9701224-2-0-3': 'Our substructure detection method is based on the wavelet transform and on the segmentation analysis.', 'astro-ph-9701224-2-0-4': 'The wavelet transform allows us to find out structures at different scales and the segmentation method allows us a quantitative statistical and morphological analysis of the sample.', 'astro-ph-9701224-2-0-5': 'From the whole catalogue we select a subset of [MATH] galaxies, with redshifts between [MATH] km/s and [MATH] km/s that we identify as belonging to the central region of Coma and on which we have performed a deeper analysis, on scales ranging from [MATH] kpc to [MATH] Mpc.', 'astro-ph-9701224-2-0-6': 'Our results are expressed in terms of the number of structures or voids and their sphericity for different values of the threshold detection and at all the scales investigated.', 'astro-ph-9701224-2-0-7': 'According to our analysis, there is strong evidence for multiple hierarchical substructure, on scales ranging from a few hundreds of kpc to about [MATH] Mpc.', 'astro-ph-9701224-2-0-8': 'The morphology of these substructures is rather spherical.', 'astro-ph-9701224-2-0-9': 'On the scale of [MATH] kpc we find two main subclusters which where also found before, but our wavelet analysis shows even more substructures, whose redshift position is approximatively marked by these bright galaxies: NGC 4934 4840, 4889, 4898 4864, 4874 4839, 4927, 4875.', 'astro-ph-9701224-2-1-0': '# Introduction', 'astro-ph-9701224-2-2-0': 'The Coma cluster (number 1656 in the Abell [1958] catalogue) has been perhaps the most studied galaxy cluster since [MATH], when Zwicky calculated its mass (Zwicky, 1933).', 'astro-ph-9701224-2-2-1': 'It has been long quoted as the paradigmatic example of a roughly spherical, relaxed cluster (Sarazin, 1986).', 'astro-ph-9701224-2-2-2': 'Previous papers (e.g. Fitchett Webster, 1987; Mellier et al., 1988; Baier et al., 1990; Briel et al., 1992; White et al., 1993; Colless Dunn, 1996, hereafter CD96 and Biviano et al., 1996) have suggested that this cluster may have a complex structure.', 'astro-ph-9701224-2-2-3': 'The X-ray images obtained with ROSAT suggest the presence of clumps of emission associated with substructures (Briel et al., 1992; White et al., 1993).', 'astro-ph-9701224-2-2-4': 'However, previous analysis were performed only on 2-D slices of the cluster.', 'astro-ph-9701224-2-2-5': 'In this paper we take up the issue of substructure in this cluster by yet another point of view, namely by trying to make use of redshift information.', 'astro-ph-9701224-2-3-0': 'Our aim is to find out substructure or voids at different scales through a three-dimensional analysis of the cluster, identify them and make a morphological analysis.', 'astro-ph-9701224-2-3-1': 'As it will be evident, our wavelet analysis detects substructure which is not visible in 2-D images, either optical and/or X-ray.', 'astro-ph-9701224-2-4-0': 'The plan of the paper is as follows: in 2 we discuss which selection criteria we have adopted to assemble our catalogue.', 'astro-ph-9701224-2-4-1': 'In 3 and in 4 we discuss the method of analysis based on wavelet transform and segmentation.', 'astro-ph-9701224-2-4-2': 'In 5 we report our results concerning the number and morphology of substructures and in 6 we do the same for the central region of the cluster.', 'astro-ph-9701224-2-4-3': 'In 7 we make some cautionary remarks concerning the statistical and physical significance of our analysis, and finally in 8 we report our conclusions.', 'astro-ph-9701224-2-5-0': '# The Data', 'astro-ph-9701224-2-6-0': 'A large body of data on the Coma cluster is available in the literature.', 'astro-ph-9701224-2-6-1': 'The catalogue has been collected exploting data coming from different redshift surveys.', 'astro-ph-9701224-2-6-2': 'In total we have selected 798 redshift for galaxies lying in the range: [EQUATION] (B1950.0, hereafter all coordinates are referred to1950.0).', 'astro-ph-9701224-2-6-3': 'The redshifts for 243 galaxies have been kindly provided to us in electronic form by J. Colless and come from the new redshifts survey by CD96.', 'astro-ph-9701224-2-6-4': 'The mean redshift uncertainty is: of about [MATH] km/s, and the uncertainty in the positions is less than 1".', 'astro-ph-9701224-2-6-5': 'Another sample of 305 redshifts have been taken from Biviano et al. (1996); 225 of them are new measurements made at the Canada-France-Hawaii Telescope; the mean uncertainty is about [MATH] km/s.', 'astro-ph-9701224-2-6-6': 'The positions in this catalogue are known with a mean error of about 2".', 'astro-ph-9701224-2-6-7': 'Another 320 redshifts have been taken from the catalogue of 379 galaxies by I. D. Karachentsev [MATH] A. I. Kopylov (1990).', 'astro-ph-9701224-2-6-8': 'The mean uncertainty in their measurements is [MATH] km/s, while the mean error in the positions is [MATH] 3".', 'astro-ph-9701224-2-6-9': 'Finally, another 46 redshifts have been taken from the NED (NASA Extragalattic Database).', 'astro-ph-9701224-2-6-10': 'This latter set of data is heterogeneous, however their quoted mean uncertainty is less then [MATH] km/s whilst the position are known with a mean error of about 6".', 'astro-ph-9701224-2-7-0': 'The total number of galaxies so collected is 914, but some objects are common to the three data sets.', 'astro-ph-9701224-2-7-1': 'We have considered a galaxy common when its position in a given data set is inside the mean error in the coordinate determination of a different set.', 'astro-ph-9701224-2-7-2': 'So, if for the same galaxy several redshift measurements were available, we have included only the most accurate one.', 'astro-ph-9701224-2-8-0': 'The completeness of our heterogeneous catalogue is about [MATH] at [MATH].', 'astro-ph-9701224-2-8-1': 'This value has been calculated from a weighted mean of the values for the different database used in our compilation.', 'astro-ph-9701224-2-9-0': 'To summarize, we have a heterogeneous sample of redshifts for 798 galaxies with [MATH] km/s.', 'astro-ph-9701224-2-9-1': "As coordinates for cluster's photometric centre we choose the value quoted by Godwin et al. (1983): [MATH] and [MATH].", 'astro-ph-9701224-2-9-2': 'The uncertainty on the redshifts measurement is about 100 km/s and the maximum error on the positions is less than 3".', 'astro-ph-9701224-2-9-3': 'From this catalogue we extract three different subsamples as shown in Table 1.', 'astro-ph-9701224-2-9-4': 'The line-of-sight distribution for the galaxies of our subsample with 3000 km/s [MATH] 28000 km/s is shown in fig. 1a.', 'astro-ph-9701224-2-9-5': 'Hereafter we call this region [MATH].', 'astro-ph-9701224-2-9-6': 'The number of galaxies inside this region is 690.', 'astro-ph-9701224-2-9-7': 'already this histogram seems to suggest the presence of two main peaks in the redshift distribution.', 'astro-ph-9701224-2-10-0': 'The subsample [MATH] has a mean redshift [MATH] km/s and a standard deviation [MATH] km/s.', 'astro-ph-9701224-2-10-1': 'The galaxies of the first peak have redshifts [MATH] km/s; the mean redshift and the standard deviation being respectively [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-10-2': 'The line-of-sight velocity dispersion for the galaxies of the first peak is [MATH] km/s.', 'astro-ph-9701224-2-10-3': 'The galaxies of the second peak have redshifts [MATH] km/s, mean redshift and standard deviation being respectively [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-10-4': 'In Fig. 1b only those galaxies inside the first peak are considered and the line-of-sight distribution is shown.', 'astro-ph-9701224-2-10-5': 'Hereafter we call this region [MATH].', 'astro-ph-9701224-2-10-6': 'The number of galaxies inside [MATH] is 485.', 'astro-ph-9701224-2-10-7': 'From Fig. 1b we note that the distribution of Coma galaxies in the redshift field show the presence of some peaks and this also may suggest the existence of multiple substructure.', 'astro-ph-9701224-2-11-0': 'We have transformed the angular distances from the photometric center to linear one assuming a distance from the center of the cluster equal to the mean value of the redshift divided by [MATH].', 'astro-ph-9701224-2-11-1': 'Finally, to make our linear coordinates independent from the value of [MATH], we renormalize them dividing by [MATH].', 'astro-ph-9701224-2-12-0': '# The method of analysis: wavelet transform and segmentation.', 'astro-ph-9701224-2-13-0': '## The wavelet transform', 'astro-ph-9701224-2-14-0': 'Our method of structure detection is based on the wavelet transform evaluated at several scales and on the segmentation analysis, and is similar to the one developed by Lega (1994, hereafter L94; part of this Ph.D. thesis may be found in Lega et al., 1995).', 'astro-ph-9701224-2-15-0': 'A detailed description of the implementation of the algorithms is beyond the purpose of this paper.', 'astro-ph-9701224-2-15-1': 'A parallel version for a Connection Machine CM200 has been developed by Lega (L94), a new PVM version will be described in Pagliaro Becciani (1997).', 'astro-ph-9701224-2-15-2': 'Although the method we implement is three-dimensional for simplicity we describe here the one dimensional version: the generalization to the 3-D case is straightforward.', 'astro-ph-9701224-2-16-0': 'For a one-dimensional function [MATH] the wavelet transform is a linear operator that can be written as: [EQUATION] where [MATH] is the scale on which the analysis is performed, [MATH] is the spatial translation parameter and [MATH] is the the Grossmann-Morlet (1984, 1987) analyzing wavelet function [EQUATION] that is spatially centered around [MATH] and has scale [MATH].', 'astro-ph-9701224-2-16-1': 'The wavelet function [MATH] is called mother wavelet.', 'astro-ph-9701224-2-16-2': 'It generates the other wavelet function [MATH].', 'astro-ph-9701224-2-16-3': 'We follow L94 in the choice of the mother wavelet: [EQUATION] where [MATH] is the cubic centred B-spline function defined by: [EQUATION]', 'astro-ph-9701224-2-16-4': 'Although our data distribution is highly anisotropic, we prefer to use an isotropic wavelet function and to perform a scale transformation along the [MATH] (redshift) axis.', 'astro-ph-9701224-2-17-0': 'With these choices, the wavelet coefficients at different scales can be calculated by the a trous algorithm, as described by L94 (pag.100), which is extremely fast and requires the set of scales to be powers of two: [MATH].', 'astro-ph-9701224-2-18-0': 'The scale [MATH] in this kind of analysis may be considered the resolution.', 'astro-ph-9701224-2-18-1': 'In other words, if we perform a calculation on a scale [MATH], we expect the wavelet transform to be sensitive to structures with typical size of about [MATH] and to find out those structures.', 'astro-ph-9701224-2-19-0': '## Choice of threshold', 'astro-ph-9701224-2-20-0': 'The thresholding is made on the wavelet coefficient histogram.', 'astro-ph-9701224-2-20-1': 'For a flat background, the wavelet transform yields zero coefficients.', 'astro-ph-9701224-2-20-2': 'The existence of structures at a given scale gives wavelet coefficient with large positive values.', 'astro-ph-9701224-2-20-3': 'Obviously, a random distribution may result in coefficients different even if there is no structure, due to statistical fluctuations.', 'astro-ph-9701224-2-20-4': 'Moreover, the statistical behaviour of the wavelet coefficient is complex because of the existence of correlation among nearby background structures which reflect in correlations among nearby pixels.', 'astro-ph-9701224-2-21-0': 'In order to single out significant structures we have to fix a tresholding criterion and level.', 'astro-ph-9701224-2-21-1': 'We choose the threshold using a classical decision rule.', 'astro-ph-9701224-2-21-2': 'We calculate the wavelet coefficients [MATH] for each scale of our analysis, for [MATH] random distribution in the same region of space of our data and on the same grid.', 'astro-ph-9701224-2-21-3': 'Then we calculate the probability [MATH] and choose the value [MATH] so that: [EQUATION]', 'astro-ph-9701224-2-21-4': 'Our threshold on the scale [MATH] is the value [MATH].', 'astro-ph-9701224-2-21-5': 'Our choice for the value of [MATH] is: [EQUATION] that ensures a [MATH] confidence level in the structure detection.', 'astro-ph-9701224-2-22-0': 'However, for the sake of completeness, we perform our analysis for several values of the threshold, calculated in terms of the standard deviation in the wavelet coefficient distribution of our data.', 'astro-ph-9701224-2-23-0': '## Structure numbering through segmentation', 'astro-ph-9701224-2-24-0': 'The second step of our analysis is the determination of connected pixels over a fixed threshold (segmentation, Rosenfeld 1969), the numbering of the selected structures and their morphological analysis.', 'astro-ph-9701224-2-25-0': "The segmentation and numbering consists of the analysis of the wavelet coefficients' matrix; all the pixels associated with a wavelet coefficient greater than the selected threshold are labeled with an integer number.", 'astro-ph-9701224-2-25-1': 'All other pixels labels are set equal to zero.', 'astro-ph-9701224-2-25-2': 'Then, the same label is associated with all the pixels connected in a single structure, in a sequential way.', 'astro-ph-9701224-2-25-3': "So, the first structure individuated has the label '1' and so on.", 'astro-ph-9701224-2-25-4': 'Then, for each structure we calculate volume and surface and from them a morphological parameter.', 'astro-ph-9701224-2-26-0': 'The detailed description of the algorithm is beyond the purpose of this paper.', 'astro-ph-9701224-2-26-1': 'However it can be described in brief as follows: Step 1: all pixels with [MATH] are labeled.', 'astro-ph-9701224-2-26-2': 'Step 2: the same label is associated with those pixels labeled and connected.', 'astro-ph-9701224-2-26-3': "This is done in a sequential way: the first structure detected has the label '1', the [MATH]-th one has the label 'N'.", 'astro-ph-9701224-2-26-4': 'This requires a renumbering of most pixels.', 'astro-ph-9701224-2-26-5': 'Step 3: volume and surface of each structure singled out are calculated.', 'astro-ph-9701224-2-27-0': '## The morphological parameter', 'astro-ph-9701224-2-28-0': 'In order to perform a morphological analysis we have to introduce a morphological parameter that quantifies the sphericity of the structures.', 'astro-ph-9701224-2-28-1': 'We choose the parameter: [EQUATION] where [MATH] is the volume and [MATH] is the surface, as in L94, and [MATH] is a parameter that depends on the scale of analysis.', 'astro-ph-9701224-2-28-2': 'We want [MATH] to have the following behaviour: zero for very filamentary structures and [MATH] for spherical ones.', 'astro-ph-9701224-2-28-3': 'This may be achieved putting [MATH], but only for those scales not affected by the granular nature of the analysis.', 'astro-ph-9701224-2-28-4': 'We choose the value [MATH] only for the scales [MATH] pixels with [MATH].', 'astro-ph-9701224-2-28-5': 'For the smallest scales the constant [MATH] is not adequate, since we are close to the grid resolution and the geometry of the substructures cannot be spherical.', 'astro-ph-9701224-2-28-6': 'So, since we want to consider as spherical a one-pixel structure, we adopt the values: [EQUATION]', 'astro-ph-9701224-2-28-7': 'Then, for every detection threshold we calculate the values: [EQUATION] where [MATH] is the number of objects detected at scale [MATH].', 'astro-ph-9701224-2-29-0': '# The voids detection method', 'astro-ph-9701224-2-30-0': 'Voids detection method is analogue to structure detection method, as far as "greater than" is replaced by "less than".', 'astro-ph-9701224-2-30-1': 'The voids thresholding is made on the wavelet coefficient histogram too.', 'astro-ph-9701224-2-30-2': 'The presence of voids at a given scale gives wavelet coefficient with large negative values.', 'astro-ph-9701224-2-31-0': 'Our choice for the threshold is the same as in [MATH].', 'astro-ph-9701224-2-31-1': 'We calculate the wavelet coefficients [MATH] for each scale of our analysis, for [MATH] random distribution in the same region of space of our data.', 'astro-ph-9701224-2-31-2': 'Then we calculate the probability [MATH] and choose the value [MATH] so that: [EQUATION] with [MATH], that ensures a [MATH] confidence level in the voids detection.', 'astro-ph-9701224-2-31-3': 'Our threshold is the value [MATH].', 'astro-ph-9701224-2-31-4': 'Obviously, in the segmentation algorithm for the voids detection the labeled pixels are those with the wavelet coefficients: [MATH].', 'astro-ph-9701224-2-32-0': "The determination of the voids' morphological parameter values is analogue to the determination of structures' morphological parameter value as described in 3.4.", 'astro-ph-9701224-2-33-0': '# Substructure and voids detection in the Coma cluster', 'astro-ph-9701224-2-34-0': 'We examine two catalogues.', 'astro-ph-9701224-2-34-1': 'The first is [MATH] made of [MATH] galaxies, as previously said, with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-34-2': 'Our grid ensures a resolution of about [MATH] Mpc on each of the three axis.', 'astro-ph-9701224-2-34-3': 'We examinate four different scales: [MATH], [MATH], [MATH], [MATH] Mpc, for four different values of the threshold: the [MATH] confidence level as described in 3, and then: [MATH], [MATH] and [MATH], where [MATH] is the standard deviation in the wavelet coefficient distribution for the selected scale.', 'astro-ph-9701224-2-34-4': 'The second catalogue investigated is [MATH].', 'astro-ph-9701224-2-34-5': 'It is made of [MATH] galaxies with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-34-6': 'Our analysis grid ensures a resolution of about [MATH] kpc on each of the three axis.', 'astro-ph-9701224-2-34-7': 'We examinate four different scales: [MATH], [MATH], [MATH], [MATH] Mpc, for the same four different values of the threshold as before We show the wavelet coefficients distributions on the four scales for the two catalogues in Fig. [MATH].', 'astro-ph-9701224-2-34-8': 'We plot the value [MATH], where [MATH] is the wavelet coefficient value and [MATH] and [MATH] are the mean and the standard deviation in the wavelet coefficient distribution, versus [MATH], where [MATH] is the probability associated to the wavelet coefficient [MATH].', 'astro-ph-9701224-2-35-0': 'The curves are slightly asymmetric on all the scales, with a small queue towards the positive values of the coefficients, meaning presence of substructure.', 'astro-ph-9701224-2-35-1': 'Our results are expressed in terms of number of structures at the selected scales (see the Tables 2 and 3).', 'astro-ph-9701224-2-35-2': 'Considering the [MATH] confidence level as a significance level for the structure detection, we have overwhelming evidence for substructure inside the Coma cluster on scales from [MATH] to [MATH] Mpc.', 'astro-ph-9701224-2-35-3': "On smaller scales the evidence is certainly lesser, as one can also see from the fact that on these scales the wavelet coefficients' histograms are more symmetric.", 'astro-ph-9701224-2-36-0': 'In Tables 2 and 3 we show the morphological parameter [MATH].', 'astro-ph-9701224-2-36-1': 'For what concerns [MATH], our substructures are rather spherical on the first two scales.', 'astro-ph-9701224-2-36-2': 'The value of [MATH] is lowered till [MATH] on the scale [MATH] Mpc, meaning a much more filamentary morphology for those structures singled out at this resolution.', 'astro-ph-9701224-2-36-3': 'The substructure of [MATH], singled out with a greater resolution shows a spherical morphology till the scale of [MATH] Mpc; more elongated shapes are found out at the bigger scales.', 'astro-ph-9701224-2-36-4': 'In both cases the diminution of [MATH] and of the number of structures as a function of the scale indicates a hierarchical distribution.', 'astro-ph-9701224-2-37-0': '# The central region of Coma', 'astro-ph-9701224-2-38-0': '## Substructures', 'astro-ph-9701224-2-39-0': 'We consider a galaxy belonging to the central region of Coma if its redshift is inside [MATH] from [MATH] km/s, where [MATH] and [MATH] are the mean and the standard deviation in the redshifts distribution calculated on the 485 galaxies considered in [MATH].', 'astro-ph-9701224-2-39-1': 'Our catalogue [MATH] is made of [MATH] galaxies, with redshifts between [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-39-2': 'The mean redshift is [MATH] km/s and the standard deviation [MATH] km/s.', 'astro-ph-9701224-2-39-3': 'Our grid ensures a resolution of about [MATH] kpc on each of the three axis.', 'astro-ph-9701224-2-39-4': 'We examinate four different scales: [MATH], [MATH], [MATH] and [MATH] kpc, for the usual values of the threshold ranging from [MATH] to [MATH] plus the [MATH] confidence level threshold, where [MATH] is the standard deviation in the wavelet coefficient distribution for the selected scale.', 'astro-ph-9701224-2-39-5': 'We show the wavelet coefficients distributions on the four scales in Fig.4.', 'astro-ph-9701224-2-39-6': 'These are slightly asymmetric too, with the usual small queue towards the positive values, meaning presence of substructure also inside the central region of the cluster.', 'astro-ph-9701224-2-39-7': 'Our results are expressed in terms of number of structures at the selected scales.', 'astro-ph-9701224-2-39-8': 'Considering the [MATH] confidence level threshold as significance level for the structure detection, we have overwhelming evidence for substructure inside the central region of the Coma cluster on the first three scales investigated: [MATH], [MATH] and [MATH] kpc (see the Table 4A).', 'astro-ph-9701224-2-39-9': 'The morphological parameter is shown in Table 4B.', 'astro-ph-9701224-2-39-10': 'Our substructures are rather spherical on all the scales but the last one, with a value of about [MATH] of the morphological parameter, shows that the shape becomes more elongated on a scale of typical size [MATH] kpc inside the central region.', 'astro-ph-9701224-2-40-0': '## Search for segregation', 'astro-ph-9701224-2-41-0': 'Having identified the substructures we tried to search for any evidence of segregation, in luminosity and/or colour.', 'astro-ph-9701224-2-41-1': 'Recently some evidence of morphological segregation within Coma has been found (Andreon, 1996).', 'astro-ph-9701224-2-41-2': 'Unfortunately we had not enough morphological information to attempt an analysis of morphological segregation among the substructures we found.', 'astro-ph-9701224-2-41-3': 'In fig. 6 we show that there is no evidence that the different subgroups observed within the central region of Coma differ as far as colour distribution [MATH] is concerned.', 'astro-ph-9701224-2-41-4': 'One must however keep in mind that this colour index is not strongly correlated with absolute [MATH]-magnitude, so that from this figure one cannot draw any conclusion about the presence of morphological segregation.', 'astro-ph-9701224-2-41-5': 'We will examine these and other aspects of luminosity functions within Coma in a forthcoming paper.', 'astro-ph-9701224-2-42-0': '# Statistical Robustness and Physical Significance', 'astro-ph-9701224-2-43-0': 'Until now we have not tried to draw from this wavelet analysis of the combined angular-redshift distribution any conclusion about the real phase- and configuration-space structure of Coma.', 'astro-ph-9701224-2-43-1': 'Before performing this further step one should verify that our catalogue does not suffer from any systematic selection biases or from other types of systematic effects like those induced by redshift distortions, as described by Regos Geller (1989) and Praton Schneider (1994).', 'astro-ph-9701224-2-43-2': 'About these latter we notice that they have little significance for a cluster like Coma, because it lies at a distance of about 68 [MATH] Mpc and from Table 5 we notice that the velocity dispersion of the structures found at a scale of 0.72 [MATH] Mpc are at most of the order of 100 km/s, so the Hubble flow term is dominant over the peculiar velocity within these structures.', 'astro-ph-9701224-2-44-0': 'On one hand, one can reasonably argue that because the structures we find are well within the nonlinear virialized region on these scales we are probing a region of the pahse space detached from the Hubble flow, where the linearity between redshift and distance is completely lost.', 'astro-ph-9701224-2-44-1': 'On the other hand one also expects that the phase-space distribution within the nonlinear region should be enough well-mixed within each clump (if there are any) that the substructures detected correspond to substructures in velocity space.', 'astro-ph-9701224-2-45-0': 'In order to check this latter hypothesis, following a suggestion of the anonymous referee, we have repeated the wavelet analysis on each of 20 realizations obtained by randomly "reshuffling" the original catalogue, i.e. redistributing randomly the redshifts among the galaxies while keeping the angular coordinates fixed.', 'astro-ph-9701224-2-45-1': 'The results are reported in Tables 6-7, and are consistent with those found by Escalera Mazure (1992) who performed a similar analysis for 2-D catalogues.', 'astro-ph-9701224-2-45-2': 'The average values of the number of structures is always smaller than the one found in the original catalogue, showing that the catalogue itself is probably contaminated by some uncertainty, probably connected to the arbitrariness in the choice of the redshift limits, by some background contaminants, etc.', 'astro-ph-9701224-2-45-3': 'However, notice for instance that at the scale 0.44 [MATH] Mpc the number of structures found is 54 in the main catalogue, i.e. a value [MATH] larger than the mean given by reshuffling over the galaxies in Table 7.', 'astro-ph-9701224-2-45-4': 'This corresponds to a confidence level of 99.82, i.e. a 0.18 probability of false detection.', 'astro-ph-9701224-2-45-5': 'On the scale 720 [MATH] kpc these figures become 99.33 for the confidence level and 0.67 for the probability of false detection.', 'astro-ph-9701224-2-45-6': 'Interestingly enough, the mean value of structures found on this scale is 5, and a closer inspection reveals that the structures which do not disappear during the reshuffling are those numbered from 2 to 5 in Table 5.', 'astro-ph-9701224-2-46-0': 'This test strengthens our confidence on the physical significance of most of the structures detected, particularly when filtering on the scale of 720 [MATH] kpc.', 'astro-ph-9701224-2-46-1': 'Under this respect our results are consistent with those found by Escalera Mazure (1992) on 2-D maps of simulated clusters, which demonstrated the ability of the wavelet analysis to recover substructures which are traced even by few objects.', 'astro-ph-9701224-2-46-2': 'We will perform a more quantitative analysis of the statistical significance of our wavelet analysis of combined angular-redshift catalogues in a forthcoming paper (Pagliaro et al., 1997a).', 'astro-ph-9701224-2-47-0': '# Conclusions and discussion', 'astro-ph-9701224-2-48-0': 'During the last years new redhift surveys and methods of analysis have allowed a more through understanding of structure of the Coma cluster (see e.g. Mellier et al., 1988; Escalera et al., 1992; CD96; Biviano et al., 1996), with most of the effort going to ascertain whether it can be classified as a relaxed one or not and to unveil hidden substructures.', 'astro-ph-9701224-2-48-1': 'While this cluster has often in the past been modelled under the assumptions of homogenous velocity structure and spherical symmetry (see e.g. Kent Gunn, 1982), the most recent observational evidence is pointing toward a more complex structure.', 'astro-ph-9701224-2-48-2': 'The recent ROSAT images and 2-D optical analysis have strengthened the evidence for the existence of multiple substructure and suggest that Coma can not be considered a relaxed cluster.', 'astro-ph-9701224-2-48-3': 'Under this respect, it is worth mentioning that already in 1988 Mellier et al. (1988), by analysing the isopleths within a 2-D map of the cluster had suggested the possible existence of [MATH] density peaks.', 'astro-ph-9701224-2-48-4': 'In this paper, we have investigated the nature of the Coma cluster performing a 3-D analysis of the combined angular-redshift distribution of the cluster.', 'astro-ph-9701224-2-48-5': 'We have assembled a catalogue of 798 galaxy redshifts, the largest presently available for the Coma cluster.', 'astro-ph-9701224-2-48-6': 'Then, we have developed a 3-D wavelet and segmentation structure analysis that has allowed us to find out substructures on different scales and to describe them in a quantitative way.', 'astro-ph-9701224-2-48-7': 'This powerful method of analysis has already provided excellent results in many fields of physics (e.g. Arneodo et al.,1988; Argoul et al., 1989; Slezak et al., 1990; Fujiwara Soda, 1995; Grebenev et al., 1995).', 'astro-ph-9701224-2-49-0': 'Our results suggests that Coma can not be considered a regular cluster of galaxies, but it is filled up with substructure on all scales ranging from [MATH] kpc to [MATH] Mpc.', 'astro-ph-9701224-2-49-1': 'The general diminution of the mean morphological parameter, meaning more elongated shapes, and of the number of structures with the scale indicates a hierarchical distribution of the substructure.', 'astro-ph-9701224-2-50-0': 'We have examinated the Coma cluster using three different subsamples of our catalogue (see Table 1); so we have insights within regions of different sizes with different resolutions.', 'astro-ph-9701224-2-51-0': 'On a scale of about [MATH] Mpc, our analysis on the extended Coma catalogue suggests the presence of multiple substructure with spherical morphology (see Table 2).', 'astro-ph-9701224-2-51-1': 'On this scale a large number of voids is detected and their shapes are rather spherical.', 'astro-ph-9701224-2-51-2': 'On the same catalogue, multiple substructure is still present at the scale [MATH] Mpc: on this scale shapes are more elongated ([MATH]).', 'astro-ph-9701224-2-51-3': 'On scales [MATH] Mpc and [MATH] Mpc we find only two very elongated objects, in agreement with the histogram of Fig.1a.', 'astro-ph-9701224-2-51-4': 'Voids on scales larger than [MATH] Mpc are few and very elongated.', 'astro-ph-9701224-2-51-5': 'Although we cannot draw any conclusion before having made a comparison with N-body simulations (Pagliaro et al., 1997b), the presence of hierarchically organized substructure seems to point to an evolutionary scenario in which the Coma cluster and the galaxies included in the second peak of the histogram of the galaxy distribution in Fig. 1a were not generated by the collapse of two large spherical density perturbation with different masses and radius of about [MATH] Mpc, but by the merging of a large number of isolated spherical density perturbations of radius ranging from [MATH] Mpc to [MATH] Mpc.', 'astro-ph-9701224-2-51-6': 'This first rough picture of the Coma evolution becomes more evident if we examine the catalogues [MATH] and [MATH].', 'astro-ph-9701224-2-52-0': 'Our analysis of the second catalogue ([MATH]) suggests the presence of substructures on all the scales with shapes becoming more elongated with growing scale (see Table 3) Voids are detected only on scales [MATH] Mpc and their shapes are rather spherical ([MATH])', 'astro-ph-9701224-2-53-0': 'On smaller scales (hundreds of kpc) we have concentrated our analysis on a central region with redshifts [MATH] km/s.', 'astro-ph-9701224-2-53-1': 'This region includes the core of Coma with the galaxies NGC 4874 ([MATH] 7131 km/s) and NGC 4839 ([MATH] 7397 km/s).', 'astro-ph-9701224-2-53-2': 'Multiple substructures are found on scales [MATH] kpc with rather spherical morphology ([MATH]).', 'astro-ph-9701224-2-53-3': 'A large number of spherical ([MATH]) voids is detected only on the smallest scale ([MATH] kpc).', 'astro-ph-9701224-2-53-4': 'We stress however once again the fact that the interpretation of substrctures on such small scales in terms of real substructures in velocity (or position) space is not as strong, as noted in the previous paragraph.', 'astro-ph-9701224-2-54-0': 'Finally, we concentrate on the scale of [MATH] kpc, where we have detected seven substructures that we show in Fig. 5 and which coincide with the peaks that we find in the central region of the histogram of Fig.1b.', 'astro-ph-9701224-2-54-1': 'To each one of these object we can associate a dominant galaxy.', 'astro-ph-9701224-2-54-2': 'Mean redshifts of the objects are: [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s, [MATH] km/s and [MATH] km/s.', 'astro-ph-9701224-2-54-3': 'In Table 5 we report some statistics only for the clumps containing a significant number of objects.', 'astro-ph-9701224-2-54-4': 'To each clump singled out we can associate one or two dominant galaxies.', 'astro-ph-9701224-2-54-5': 'These are (for increasing redshift): NGC 4934 NGC 4840, NGC 4889, NGC 4898 NGC 4864, NGC 4874 NGC 4839, NGC 4927, NGC 4875.', 'astro-ph-9701224-2-54-6': 'We then confirm the presence of the two subclusters already described by CD96, but in addition we have found statistical evidence for the exstence of more substructures in redshift space.', 'astro-ph-9701224-2-54-7': 'All this evidence leads us to suggest that the Coma cluster can not be considered a regular cluster of galaxies and that its process of formation occurs through a bottom-up mechanism as predicted by CDM and MDM models.', 'astro-ph-9701224-2-55-0': 'Finally, we would like to stress the fact that from this analysis it is difficult to draw any conclusion about the evolutionary state of this cluster.', 'astro-ph-9701224-2-55-1': 'Such an analysis would require some modelling of the evolutionary scenarios through comparison with high-resolution N-body simulations and a better understanding of the velocity field around Coma.', 'astro-ph-9701224-2-55-2': 'We will report on these issues in a subsequent work (Pagliaro et al., 1997b).', 'astro-ph-9701224-2-56-0': 'We would like to thank the anonymous referee for insightful comments which led to the introduction of the section on the statistical significance of the analysis performed in this paper.', 'astro-ph-9701224-2-56-1': 'A.Pa.', 'astro-ph-9701224-2-56-2': 'would like to thank E. Lega for having sent him her structure detection code, a copy of her Ph.D. dissertation and for kind and indispensable help during the period in which our code was developed and tested, and A. Bijaoui for a stimulating discussion held in Erice.', 'astro-ph-9701224-2-56-3': 'M.Ga.', 'astro-ph-9701224-2-56-4': 'wish to thank S. Shandarin for a clarifying discussion and helpful suggestions.', 'astro-ph-9701224-2-57-0': '[f1a.ps, f1b.ps] (a) Histogram of the galaxy distribution, with redshifts [MATH] km/s inside [MATH], according to our catalogue.', 'astro-ph-9701224-2-57-1': 'Step is 500 km/s; (b) histogram of the galaxy distribution, with redshift [MATH] inside [MATH], according to our catalogue.', 'astro-ph-9701224-2-57-2': 'Step is 100 km/s.', 'astro-ph-9701224-2-58-0': '[f2.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-2-59-0': '[f3.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-2-60-0': '[f4.ps]Histograms of the wavelet coefficients on the four scales selected for [MATH]', 'astro-ph-9701224-2-61-0': '[f5.ps]Substructures for the central region of Coma detected with a resolution of [MATH] kpc.', 'astro-ph-9701224-2-61-1': 'The seven structures detected are clearly visible.', 'astro-ph-9701224-2-61-2': 'Morphological parameter: [MATH]', 'astro-ph-9701224-2-62-0': '[f6.ps]Colour segregation.', 'astro-ph-9701224-2-62-1': 'The different histograms refer to the substructures of Table 5, and are numbered according to the first column of that table.'}

[['astro-ph-9701224-1-51-1', 'astro-ph-9701224-2-56-2'], ['astro-ph-9701224-1-51-3', 'astro-ph-9701224-2-56-4'], ['astro-ph-9701224-1-33-1', 'astro-ph-9701224-2-34-1'], ['astro-ph-9701224-1-33-2', 'astro-ph-9701224-2-34-2'], ['astro-ph-9701224-1-33-5', 'astro-ph-9701224-2-34-5'], ['astro-ph-9701224-1-33-8', 'astro-ph-9701224-2-34-8'], ['astro-ph-9701224-1-42-6', 'astro-ph-9701224-2-48-7'], ['astro-ph-9701224-1-2-4', 'astro-ph-9701224-2-2-4'], ['astro-ph-9701224-1-4-1', 'astro-ph-9701224-2-4-1'], ['astro-ph-9701224-1-48-2', 'astro-ph-9701224-2-54-2'], ['astro-ph-9701224-1-48-5', 'astro-ph-9701224-2-54-4'], ['astro-ph-9701224-1-40-0', 'astro-ph-9701224-2-41-0'], ['astro-ph-9701224-1-40-1', 'astro-ph-9701224-2-41-1'], ['astro-ph-9701224-1-40-2', 'astro-ph-9701224-2-41-2'], ['astro-ph-9701224-1-40-3', 'astro-ph-9701224-2-41-3'], ['astro-ph-9701224-1-40-4', 'astro-ph-9701224-2-41-4'], ['astro-ph-9701224-1-40-5', 'astro-ph-9701224-2-41-5'], ['astro-ph-9701224-1-21-0', 'astro-ph-9701224-2-22-0'], ['astro-ph-9701224-1-55-0', 'astro-ph-9701224-2-60-0'], ['astro-ph-9701224-1-0-2', 'astro-ph-9701224-2-0-4'], ['astro-ph-9701224-1-29-1', 'astro-ph-9701224-2-30-1'], ['astro-ph-9701224-1-29-2', 'astro-ph-9701224-2-30-2'], ['astro-ph-9701224-1-16-0', 'astro-ph-9701224-2-16-0'], ['astro-ph-9701224-1-16-1', 'astro-ph-9701224-2-16-1'], ['astro-ph-9701224-1-16-2', 'astro-ph-9701224-2-16-2'], ['astro-ph-9701224-1-16-3', 'astro-ph-9701224-2-16-3'], ['astro-ph-9701224-1-18-1', 'astro-ph-9701224-2-18-1'], ['astro-ph-9701224-1-8-1', 'astro-ph-9701224-2-8-1'], ['astro-ph-9701224-1-54-0', 'astro-ph-9701224-2-59-0'], ['astro-ph-9701224-1-15-0', 'astro-ph-9701224-2-15-0'], ['astro-ph-9701224-1-23-0', 'astro-ph-9701224-2-24-0'], ['astro-ph-9701224-1-24-1', 'astro-ph-9701224-2-25-1'], ['astro-ph-9701224-1-24-2', 'astro-ph-9701224-2-25-2'], ['astro-ph-9701224-1-24-3', 'astro-ph-9701224-2-25-3'], ['astro-ph-9701224-1-24-4', 'astro-ph-9701224-2-25-4'], ['astro-ph-9701224-1-27-0', 'astro-ph-9701224-2-28-0'], ['astro-ph-9701224-1-27-2', 'astro-ph-9701224-2-28-2'], ['astro-ph-9701224-1-27-3', 'astro-ph-9701224-2-28-3'], ['astro-ph-9701224-1-27-4', 'astro-ph-9701224-2-28-4'], ['astro-ph-9701224-1-27-5', 'astro-ph-9701224-2-28-5'], ['astro-ph-9701224-1-27-6', 'astro-ph-9701224-2-28-6'], ['astro-ph-9701224-1-27-7', 'astro-ph-9701224-2-28-7'], ['astro-ph-9701224-1-9-0', 'astro-ph-9701224-2-9-0'], ['astro-ph-9701224-1-9-4', 'astro-ph-9701224-2-9-4'], ['astro-ph-9701224-1-9-5', 'astro-ph-9701224-2-9-5'], ['astro-ph-9701224-1-9-6', 'astro-ph-9701224-2-9-6'], ['astro-ph-9701224-1-43-1', 'astro-ph-9701224-2-49-1'], ['astro-ph-9701224-1-10-1', 'astro-ph-9701224-2-10-1'], ['astro-ph-9701224-1-10-2', 'astro-ph-9701224-2-10-2'], ['astro-ph-9701224-1-10-4', 'astro-ph-9701224-2-10-4'], ['astro-ph-9701224-1-10-5', 'astro-ph-9701224-2-10-5'], ['astro-ph-9701224-1-10-6', 'astro-ph-9701224-2-10-6'], ['astro-ph-9701224-1-30-0', 'astro-ph-9701224-2-31-0'], ['astro-ph-9701224-1-30-1', 'astro-ph-9701224-2-31-1'], ['astro-ph-9701224-1-30-2', 'astro-ph-9701224-2-31-2'], ['astro-ph-9701224-1-30-3', 'astro-ph-9701224-2-31-3'], ['astro-ph-9701224-1-30-4', 'astro-ph-9701224-2-31-4'], ['astro-ph-9701224-1-45-1', 'astro-ph-9701224-2-51-1'], ['astro-ph-9701224-1-45-3', 'astro-ph-9701224-2-51-3'], ['astro-ph-9701224-1-25-0', 'astro-ph-9701224-2-26-0'], ['astro-ph-9701224-1-25-3', 'astro-ph-9701224-2-26-3'], ['astro-ph-9701224-1-25-4', 'astro-ph-9701224-2-26-4'], ['astro-ph-9701224-1-53-0', 'astro-ph-9701224-2-58-0'], ['astro-ph-9701224-1-56-1', 'astro-ph-9701224-2-61-1'], ['astro-ph-9701224-1-47-1', 'astro-ph-9701224-2-53-1'], ['astro-ph-9701224-1-20-0', 'astro-ph-9701224-2-20-0'], ['astro-ph-9701224-1-20-2', 'astro-ph-9701224-2-20-2'], ['astro-ph-9701224-1-20-6', 'astro-ph-9701224-2-21-2'], ['astro-ph-9701224-1-20-7', 'astro-ph-9701224-2-21-3'], ['astro-ph-9701224-1-20-8', 'astro-ph-9701224-2-21-4'], ['astro-ph-9701224-1-20-9', 'astro-ph-9701224-2-21-5'], ['astro-ph-9701224-1-34-0', 'astro-ph-9701224-2-35-0'], ['astro-ph-9701224-1-34-4', 'astro-ph-9701224-2-36-1'], ['astro-ph-9701224-1-34-5', 'astro-ph-9701224-2-36-2'], ['astro-ph-9701224-1-34-6', 'astro-ph-9701224-2-36-3'], ['astro-ph-9701224-1-34-7', 'astro-ph-9701224-2-36-4'], ['astro-ph-9701224-1-37-0', 'astro-ph-9701224-2-39-0'], ['astro-ph-9701224-1-37-2', 'astro-ph-9701224-2-39-2'], ['astro-ph-9701224-1-38-2', 'astro-ph-9701224-2-39-5'], ['astro-ph-9701224-1-38-3', 'astro-ph-9701224-2-39-6'], ['astro-ph-9701224-1-38-4', 'astro-ph-9701224-2-39-7'], ['astro-ph-9701224-1-38-6', 'astro-ph-9701224-2-39-9'], ['astro-ph-9701224-1-3-0', 'astro-ph-9701224-2-3-0'], ['astro-ph-9701224-1-33-3', 'astro-ph-9701224-2-34-3'], ['astro-ph-9701224-1-33-4', 'astro-ph-9701224-2-34-4'], ['astro-ph-9701224-1-33-6', 'astro-ph-9701224-2-34-6'], ['astro-ph-9701224-1-33-7', 'astro-ph-9701224-2-34-7'], ['astro-ph-9701224-1-42-2', 'astro-ph-9701224-2-48-3'], ['astro-ph-9701224-1-42-3', 'astro-ph-9701224-2-48-4'], ['astro-ph-9701224-1-42-4', 'astro-ph-9701224-2-48-5'], ['astro-ph-9701224-1-42-5', 'astro-ph-9701224-2-48-6'], ['astro-ph-9701224-1-2-0', 'astro-ph-9701224-2-2-0'], ['astro-ph-9701224-1-2-2', 'astro-ph-9701224-2-2-2'], ['astro-ph-9701224-1-2-3', 'astro-ph-9701224-2-2-3'], ['astro-ph-9701224-1-6-0', 'astro-ph-9701224-2-6-0'], ['astro-ph-9701224-1-6-5', 'astro-ph-9701224-2-6-5'], ['astro-ph-9701224-1-6-6', 'astro-ph-9701224-2-6-6'], ['astro-ph-9701224-1-6-7', 'astro-ph-9701224-2-6-7'], ['astro-ph-9701224-1-4-2', 'astro-ph-9701224-2-4-2'], ['astro-ph-9701224-1-48-0', 'astro-ph-9701224-2-54-0'], ['astro-ph-9701224-1-48-1', 'astro-ph-9701224-2-54-1'], ['astro-ph-9701224-1-48-8', 'astro-ph-9701224-2-54-7'], ['astro-ph-9701224-1-57-1', 'astro-ph-9701224-2-62-1'], ['astro-ph-9701224-1-0-1', 'astro-ph-9701224-2-0-3'], ['astro-ph-9701224-1-0-3', 'astro-ph-9701224-2-0-5'], ['astro-ph-9701224-1-0-4', 'astro-ph-9701224-2-0-6'], ['astro-ph-9701224-1-0-5', 'astro-ph-9701224-2-0-7'], ['astro-ph-9701224-1-29-0', 'astro-ph-9701224-2-30-0'], ['astro-ph-9701224-1-16-4', 'astro-ph-9701224-2-16-4'], ['astro-ph-9701224-1-18-0', 'astro-ph-9701224-2-18-0'], ['astro-ph-9701224-1-8-0', 'astro-ph-9701224-2-8-0'], ['astro-ph-9701224-1-46-0', 'astro-ph-9701224-2-52-0'], ['astro-ph-9701224-1-24-0', 'astro-ph-9701224-2-25-0'], ['astro-ph-9701224-1-27-1', 'astro-ph-9701224-2-28-1'], ['astro-ph-9701224-1-9-2', 'astro-ph-9701224-2-9-2'], ['astro-ph-9701224-1-9-3', 'astro-ph-9701224-2-9-3'], ['astro-ph-9701224-1-52-0', 'astro-ph-9701224-2-57-0'], ['astro-ph-9701224-1-52-1', 'astro-ph-9701224-2-57-1'], ['astro-ph-9701224-1-43-0', 'astro-ph-9701224-2-49-0'], ['astro-ph-9701224-1-10-0', 'astro-ph-9701224-2-10-0'], ['astro-ph-9701224-1-10-3', 'astro-ph-9701224-2-10-3'], ['astro-ph-9701224-1-10-7', 'astro-ph-9701224-2-10-7'], ['astro-ph-9701224-1-45-0', 'astro-ph-9701224-2-51-0'], ['astro-ph-9701224-1-45-2', 'astro-ph-9701224-2-51-2'], ['astro-ph-9701224-1-45-4', 'astro-ph-9701224-2-51-4'], ['astro-ph-9701224-1-45-6', 'astro-ph-9701224-2-51-6'], ['astro-ph-9701224-1-7-2', 'astro-ph-9701224-2-7-2'], ['astro-ph-9701224-1-25-1', 'astro-ph-9701224-2-26-1'], ['astro-ph-9701224-1-25-2', 'astro-ph-9701224-2-26-2'], ['astro-ph-9701224-1-25-5', 'astro-ph-9701224-2-26-5'], ['astro-ph-9701224-1-11-2', 'astro-ph-9701224-2-11-1'], ['astro-ph-9701224-1-56-0', 'astro-ph-9701224-2-61-0'], ['astro-ph-9701224-1-44-0', 'astro-ph-9701224-2-50-0'], ['astro-ph-9701224-1-47-0', 'astro-ph-9701224-2-53-0'], ['astro-ph-9701224-1-47-2', 'astro-ph-9701224-2-53-2'], ['astro-ph-9701224-1-47-3', 'astro-ph-9701224-2-53-3'], ['astro-ph-9701224-1-20-5', 'astro-ph-9701224-2-21-1'], ['astro-ph-9701224-1-34-1', 'astro-ph-9701224-2-35-1'], ['astro-ph-9701224-1-34-2', 'astro-ph-9701224-2-35-2'], ['astro-ph-9701224-1-37-1', 'astro-ph-9701224-2-39-1'], ['astro-ph-9701224-1-38-0', 'astro-ph-9701224-2-39-3'], ['astro-ph-9701224-1-38-1', 'astro-ph-9701224-2-39-4'], ['astro-ph-9701224-1-38-5', 'astro-ph-9701224-2-39-8'], ['astro-ph-9701224-1-38-7', 'astro-ph-9701224-2-39-10'], ['astro-ph-9701224-1-33-0', 'astro-ph-9701224-2-34-0'], ['astro-ph-9701224-1-42-0', 'astro-ph-9701224-2-48-0'], ['astro-ph-9701224-1-42-1', 'astro-ph-9701224-2-48-2'], ['astro-ph-9701224-1-2-1', 'astro-ph-9701224-2-2-5'], ['astro-ph-9701224-1-6-2', 'astro-ph-9701224-2-6-2'], ['astro-ph-9701224-1-6-3', 'astro-ph-9701224-2-6-3'], ['astro-ph-9701224-1-6-9', 'astro-ph-9701224-2-6-4'], ['astro-ph-9701224-1-6-9', 'astro-ph-9701224-2-6-8'], ['astro-ph-9701224-1-6-11', 'astro-ph-9701224-2-6-4'], ['astro-ph-9701224-1-6-11', 'astro-ph-9701224-2-6-9'], ['astro-ph-9701224-1-4-0', 'astro-ph-9701224-2-4-0'], ['astro-ph-9701224-1-0-0', 'astro-ph-9701224-2-0-2'], ['astro-ph-9701224-1-0-6', 'astro-ph-9701224-2-0-9'], ['astro-ph-9701224-1-15-1', 'astro-ph-9701224-2-15-1'], ['astro-ph-9701224-1-15-3', 'astro-ph-9701224-2-15-2'], ['astro-ph-9701224-1-15-4', 'astro-ph-9701224-2-15-2'], ['astro-ph-9701224-1-9-1', 'astro-ph-9701224-2-9-1'], ['astro-ph-9701224-1-31-0', 'astro-ph-9701224-2-32-0'], ['astro-ph-9701224-1-17-0', 'astro-ph-9701224-2-17-0'], ['astro-ph-9701224-1-17-1', 'astro-ph-9701224-2-17-0'], ['astro-ph-9701224-1-7-0', 'astro-ph-9701224-2-7-0'], ['astro-ph-9701224-1-7-1', 'astro-ph-9701224-2-7-1'], ['astro-ph-9701224-1-14-0', 'astro-ph-9701224-2-14-0'], ['astro-ph-9701224-1-20-1', 'astro-ph-9701224-2-20-1'], ['astro-ph-9701224-1-20-3', 'astro-ph-9701224-2-20-3'], ['astro-ph-9701224-1-20-4', 'astro-ph-9701224-2-20-4'], ['astro-ph-9701224-1-34-3', 'astro-ph-9701224-2-36-0']]

[['astro-ph-9701224-1-51-1', 'astro-ph-9701224-2-56-2'], ['astro-ph-9701224-1-51-3', 'astro-ph-9701224-2-56-4'], ['astro-ph-9701224-1-33-1', 'astro-ph-9701224-2-34-1'], ['astro-ph-9701224-1-33-2', 'astro-ph-9701224-2-34-2'], ['astro-ph-9701224-1-33-5', 'astro-ph-9701224-2-34-5'], ['astro-ph-9701224-1-33-8', 'astro-ph-9701224-2-34-8'], ['astro-ph-9701224-1-42-6', 'astro-ph-9701224-2-48-7'], ['astro-ph-9701224-1-2-4', 'astro-ph-9701224-2-2-4'], ['astro-ph-9701224-1-4-1', 'astro-ph-9701224-2-4-1'], ['astro-ph-9701224-1-48-2', 'astro-ph-9701224-2-54-2'], ['astro-ph-9701224-1-48-5', 'astro-ph-9701224-2-54-4'], ['astro-ph-9701224-1-40-0', 'astro-ph-9701224-2-41-0'], ['astro-ph-9701224-1-40-1', 'astro-ph-9701224-2-41-1'], ['astro-ph-9701224-1-40-2', 'astro-ph-9701224-2-41-2'], ['astro-ph-9701224-1-40-3', 'astro-ph-9701224-2-41-3'], ['astro-ph-9701224-1-40-4', 'astro-ph-9701224-2-41-4'], ['astro-ph-9701224-1-40-5', 'astro-ph-9701224-2-41-5'], ['astro-ph-9701224-1-21-0', 'astro-ph-9701224-2-22-0'], ['astro-ph-9701224-1-55-0', 'astro-ph-9701224-2-60-0'], ['astro-ph-9701224-1-0-2', 'astro-ph-9701224-2-0-4'], ['astro-ph-9701224-1-29-1', 'astro-ph-9701224-2-30-1'], ['astro-ph-9701224-1-29-2', 'astro-ph-9701224-2-30-2'], ['astro-ph-9701224-1-16-0', 'astro-ph-9701224-2-16-0'], ['astro-ph-9701224-1-16-1', 'astro-ph-9701224-2-16-1'], ['astro-ph-9701224-1-16-2', 'astro-ph-9701224-2-16-2'], ['astro-ph-9701224-1-16-3', 'astro-ph-9701224-2-16-3'], ['astro-ph-9701224-1-18-1', 'astro-ph-9701224-2-18-1'], ['astro-ph-9701224-1-8-1', 'astro-ph-9701224-2-8-1'], ['astro-ph-9701224-1-54-0', 'astro-ph-9701224-2-59-0'], ['astro-ph-9701224-1-15-0', 'astro-ph-9701224-2-15-0'], ['astro-ph-9701224-1-23-0', 'astro-ph-9701224-2-24-0'], ['astro-ph-9701224-1-24-1', 'astro-ph-9701224-2-25-1'], ['astro-ph-9701224-1-24-2', 'astro-ph-9701224-2-25-2'], ['astro-ph-9701224-1-24-3', 'astro-ph-9701224-2-25-3'], ['astro-ph-9701224-1-24-4', 'astro-ph-9701224-2-25-4'], ['astro-ph-9701224-1-27-0', 'astro-ph-9701224-2-28-0'], ['astro-ph-9701224-1-27-2', 'astro-ph-9701224-2-28-2'], ['astro-ph-9701224-1-27-3', 'astro-ph-9701224-2-28-3'], ['astro-ph-9701224-1-27-4', 'astro-ph-9701224-2-28-4'], ['astro-ph-9701224-1-27-5', 'astro-ph-9701224-2-28-5'], ['astro-ph-9701224-1-27-6', 'astro-ph-9701224-2-28-6'], ['astro-ph-9701224-1-27-7', 'astro-ph-9701224-2-28-7'], ['astro-ph-9701224-1-9-0', 'astro-ph-9701224-2-9-0'], ['astro-ph-9701224-1-9-4', 'astro-ph-9701224-2-9-4'], ['astro-ph-9701224-1-9-5', 'astro-ph-9701224-2-9-5'], ['astro-ph-9701224-1-9-6', 'astro-ph-9701224-2-9-6'], ['astro-ph-9701224-1-43-1', 'astro-ph-9701224-2-49-1'], ['astro-ph-9701224-1-10-1', 'astro-ph-9701224-2-10-1'], ['astro-ph-9701224-1-10-2', 'astro-ph-9701224-2-10-2'], ['astro-ph-9701224-1-10-4', 'astro-ph-9701224-2-10-4'], ['astro-ph-9701224-1-10-5', 'astro-ph-9701224-2-10-5'], ['astro-ph-9701224-1-10-6', 'astro-ph-9701224-2-10-6'], ['astro-ph-9701224-1-30-0', 'astro-ph-9701224-2-31-0'], ['astro-ph-9701224-1-30-1', 'astro-ph-9701224-2-31-1'], ['astro-ph-9701224-1-30-2', 'astro-ph-9701224-2-31-2'], ['astro-ph-9701224-1-30-3', 'astro-ph-9701224-2-31-3'], ['astro-ph-9701224-1-30-4', 'astro-ph-9701224-2-31-4'], ['astro-ph-9701224-1-45-1', 'astro-ph-9701224-2-51-1'], ['astro-ph-9701224-1-45-3', 'astro-ph-9701224-2-51-3'], ['astro-ph-9701224-1-25-0', 'astro-ph-9701224-2-26-0'], ['astro-ph-9701224-1-25-3', 'astro-ph-9701224-2-26-3'], ['astro-ph-9701224-1-25-4', 'astro-ph-9701224-2-26-4'], ['astro-ph-9701224-1-53-0', 'astro-ph-9701224-2-58-0'], ['astro-ph-9701224-1-56-1', 'astro-ph-9701224-2-61-1'], ['astro-ph-9701224-1-47-1', 'astro-ph-9701224-2-53-1'], ['astro-ph-9701224-1-20-0', 'astro-ph-9701224-2-20-0'], ['astro-ph-9701224-1-20-2', 'astro-ph-9701224-2-20-2'], ['astro-ph-9701224-1-20-6', 'astro-ph-9701224-2-21-2'], ['astro-ph-9701224-1-20-7', 'astro-ph-9701224-2-21-3'], ['astro-ph-9701224-1-20-8', 'astro-ph-9701224-2-21-4'], ['astro-ph-9701224-1-20-9', 'astro-ph-9701224-2-21-5'], ['astro-ph-9701224-1-34-0', 'astro-ph-9701224-2-35-0'], ['astro-ph-9701224-1-34-4', 'astro-ph-9701224-2-36-1'], ['astro-ph-9701224-1-34-5', 'astro-ph-9701224-2-36-2'], ['astro-ph-9701224-1-34-6', 'astro-ph-9701224-2-36-3'], ['astro-ph-9701224-1-34-7', 'astro-ph-9701224-2-36-4'], ['astro-ph-9701224-1-37-0', 'astro-ph-9701224-2-39-0'], ['astro-ph-9701224-1-37-2', 'astro-ph-9701224-2-39-2'], ['astro-ph-9701224-1-38-2', 'astro-ph-9701224-2-39-5'], ['astro-ph-9701224-1-38-3', 'astro-ph-9701224-2-39-6'], ['astro-ph-9701224-1-38-4', 'astro-ph-9701224-2-39-7'], ['astro-ph-9701224-1-38-6', 'astro-ph-9701224-2-39-9']]

[['astro-ph-9701224-1-3-0', 'astro-ph-9701224-2-3-0'], ['astro-ph-9701224-1-33-3', 'astro-ph-9701224-2-34-3'], ['astro-ph-9701224-1-33-4', 'astro-ph-9701224-2-34-4'], ['astro-ph-9701224-1-33-6', 'astro-ph-9701224-2-34-6'], ['astro-ph-9701224-1-33-7', 'astro-ph-9701224-2-34-7'], ['astro-ph-9701224-1-42-2', 'astro-ph-9701224-2-48-3'], ['astro-ph-9701224-1-42-3', 'astro-ph-9701224-2-48-4'], ['astro-ph-9701224-1-42-4', 'astro-ph-9701224-2-48-5'], ['astro-ph-9701224-1-42-5', 'astro-ph-9701224-2-48-6'], ['astro-ph-9701224-1-2-0', 'astro-ph-9701224-2-2-0'], ['astro-ph-9701224-1-2-2', 'astro-ph-9701224-2-2-2'], ['astro-ph-9701224-1-2-3', 'astro-ph-9701224-2-2-3'], ['astro-ph-9701224-1-6-0', 'astro-ph-9701224-2-6-0'], ['astro-ph-9701224-1-6-5', 'astro-ph-9701224-2-6-5'], ['astro-ph-9701224-1-6-6', 'astro-ph-9701224-2-6-6'], ['astro-ph-9701224-1-6-7', 'astro-ph-9701224-2-6-7'], ['astro-ph-9701224-1-4-2', 'astro-ph-9701224-2-4-2'], ['astro-ph-9701224-1-48-0', 'astro-ph-9701224-2-54-0'], ['astro-ph-9701224-1-48-1', 'astro-ph-9701224-2-54-1'], ['astro-ph-9701224-1-48-8', 'astro-ph-9701224-2-54-7'], ['astro-ph-9701224-1-57-1', 'astro-ph-9701224-2-62-1'], ['astro-ph-9701224-1-0-1', 'astro-ph-9701224-2-0-3'], ['astro-ph-9701224-1-0-3', 'astro-ph-9701224-2-0-5'], ['astro-ph-9701224-1-0-4', 'astro-ph-9701224-2-0-6'], ['astro-ph-9701224-1-0-5', 'astro-ph-9701224-2-0-7'], ['astro-ph-9701224-1-29-0', 'astro-ph-9701224-2-30-0'], ['astro-ph-9701224-1-16-4', 'astro-ph-9701224-2-16-4'], ['astro-ph-9701224-1-18-0', 'astro-ph-9701224-2-18-0'], ['astro-ph-9701224-1-8-0', 'astro-ph-9701224-2-8-0'], ['astro-ph-9701224-1-46-0', 'astro-ph-9701224-2-52-0'], ['astro-ph-9701224-1-24-0', 'astro-ph-9701224-2-25-0'], ['astro-ph-9701224-1-27-1', 'astro-ph-9701224-2-28-1'], ['astro-ph-9701224-1-9-2', 'astro-ph-9701224-2-9-2'], ['astro-ph-9701224-1-9-3', 'astro-ph-9701224-2-9-3'], ['astro-ph-9701224-1-52-0', 'astro-ph-9701224-2-57-0'], ['astro-ph-9701224-1-52-1', 'astro-ph-9701224-2-57-1'], ['astro-ph-9701224-1-43-0', 'astro-ph-9701224-2-49-0'], ['astro-ph-9701224-1-10-0', 'astro-ph-9701224-2-10-0'], ['astro-ph-9701224-1-10-3', 'astro-ph-9701224-2-10-3'], ['astro-ph-9701224-1-10-7', 'astro-ph-9701224-2-10-7'], ['astro-ph-9701224-1-45-0', 'astro-ph-9701224-2-51-0'], ['astro-ph-9701224-1-45-2', 'astro-ph-9701224-2-51-2'], ['astro-ph-9701224-1-45-4', 'astro-ph-9701224-2-51-4'], ['astro-ph-9701224-1-45-6', 'astro-ph-9701224-2-51-6'], ['astro-ph-9701224-1-7-2', 'astro-ph-9701224-2-7-2'], ['astro-ph-9701224-1-25-1', 'astro-ph-9701224-2-26-1'], ['astro-ph-9701224-1-25-2', 'astro-ph-9701224-2-26-2'], ['astro-ph-9701224-1-25-5', 'astro-ph-9701224-2-26-5'], ['astro-ph-9701224-1-11-2', 'astro-ph-9701224-2-11-1'], ['astro-ph-9701224-1-56-0', 'astro-ph-9701224-2-61-0'], ['astro-ph-9701224-1-44-0', 'astro-ph-9701224-2-50-0'], ['astro-ph-9701224-1-47-0', 'astro-ph-9701224-2-53-0'], ['astro-ph-9701224-1-47-2', 'astro-ph-9701224-2-53-2'], ['astro-ph-9701224-1-47-3', 'astro-ph-9701224-2-53-3'], ['astro-ph-9701224-1-20-5', 'astro-ph-9701224-2-21-1'], ['astro-ph-9701224-1-34-1', 'astro-ph-9701224-2-35-1'], ['astro-ph-9701224-1-34-2', 'astro-ph-9701224-2-35-2'], ['astro-ph-9701224-1-37-1', 'astro-ph-9701224-2-39-1'], ['astro-ph-9701224-1-38-0', 'astro-ph-9701224-2-39-3'], ['astro-ph-9701224-1-38-1', 'astro-ph-9701224-2-39-4'], ['astro-ph-9701224-1-38-5', 'astro-ph-9701224-2-39-8'], ['astro-ph-9701224-1-38-7', 'astro-ph-9701224-2-39-10']]

[]

[['astro-ph-9701224-1-33-0', 'astro-ph-9701224-2-34-0'], ['astro-ph-9701224-1-42-0', 'astro-ph-9701224-2-48-0'], ['astro-ph-9701224-1-42-1', 'astro-ph-9701224-2-48-2'], ['astro-ph-9701224-1-2-1', 'astro-ph-9701224-2-2-5'], ['astro-ph-9701224-1-6-2', 'astro-ph-9701224-2-6-2'], ['astro-ph-9701224-1-6-3', 'astro-ph-9701224-2-6-3'], ['astro-ph-9701224-1-6-9', 'astro-ph-9701224-2-6-4'], ['astro-ph-9701224-1-6-9', 'astro-ph-9701224-2-6-8'], ['astro-ph-9701224-1-6-11', 'astro-ph-9701224-2-6-4'], ['astro-ph-9701224-1-6-11', 'astro-ph-9701224-2-6-9'], ['astro-ph-9701224-1-4-0', 'astro-ph-9701224-2-4-0'], ['astro-ph-9701224-1-0-0', 'astro-ph-9701224-2-0-2'], ['astro-ph-9701224-1-0-6', 'astro-ph-9701224-2-0-9'], ['astro-ph-9701224-1-15-1', 'astro-ph-9701224-2-15-1'], ['astro-ph-9701224-1-15-3', 'astro-ph-9701224-2-15-2'], ['astro-ph-9701224-1-15-4', 'astro-ph-9701224-2-15-2'], ['astro-ph-9701224-1-9-1', 'astro-ph-9701224-2-9-1'], ['astro-ph-9701224-1-31-0', 'astro-ph-9701224-2-32-0'], ['astro-ph-9701224-1-17-0', 'astro-ph-9701224-2-17-0'], ['astro-ph-9701224-1-17-1', 'astro-ph-9701224-2-17-0'], ['astro-ph-9701224-1-7-0', 'astro-ph-9701224-2-7-0'], ['astro-ph-9701224-1-7-1', 'astro-ph-9701224-2-7-1'], ['astro-ph-9701224-1-14-0', 'astro-ph-9701224-2-14-0'], ['astro-ph-9701224-1-20-1', 'astro-ph-9701224-2-20-1'], ['astro-ph-9701224-1-20-3', 'astro-ph-9701224-2-20-3'], ['astro-ph-9701224-1-20-4', 'astro-ph-9701224-2-20-4'], ['astro-ph-9701224-1-34-3', 'astro-ph-9701224-2-36-0']]

[]

['astro-ph-9701224-1-48-6', 'astro-ph-9701224-1-51-0', 'astro-ph-9701224-1-51-2', 'astro-ph-9701224-1-52-2', 'astro-ph-9701224-1-56-2', 'astro-ph-9701224-1-57-0', 'astro-ph-9701224-2-54-5', 'astro-ph-9701224-2-56-1', 'astro-ph-9701224-2-56-3', 'astro-ph-9701224-2-57-2', 'astro-ph-9701224-2-61-2', 'astro-ph-9701224-2-62-0']

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

https://arxiv.org/abs/astro-ph/9701224

1211.4024

{'1211.4024-1-0-0': 'We illustrate a framework for constructing models of chaotic inflation where the inflaton is the position of a D3 brane along the universal cover of a string compactification.', '1211.4024-1-1-0': 'In our scenario, a brane rolls many times around a non-trivial one-cycle, thereby unwinding a Ramond-Ramond flux.', '1211.4024-1-1-1': 'These "flux monodromies" are similar in spirit to the monodromies of Silverstein, Westphal, and McAllister, and their four-dimensional description is that of Kaloper and Sorbo.', '1211.4024-1-1-2': 'Assuming moduli stabilization is rigid enough, the large-field inflationary potential is protected from radiative corrections by a discrete shift symmetry.', '1211.4024-1-2-0': '# Introduction', '1211.4024-1-3-0': 'Perhaps the simplest phenomenological model of inflation [CITATION] is due to Linde\'s monomial potential [CITATION], which undergoes what is called "chaotic inflation" due to its expected behavior on large scales.', '1211.4024-1-3-1': 'Chaotic inflationary models are not obviously natural in the context of effective field theories precisely because of the requirement that the potential be sufficiently flat over super-Planckian field distances.', '1211.4024-1-3-2': 'In the quadratic model, the inflaton mass must be of order [MATH], and the dominant term for field values as large as [MATH], which requires a functional fine tuning, from an effective field theory point of view.', '1211.4024-1-3-3': 'An elegant solution to this problem was presented in [CITATION], whereby an axion "eats" a three-form potential, and so acquires a quadratic potential which is protected from radiative corrections by the underlying shift symmetry.', '1211.4024-1-3-4': 'Aside from the simplicity of the model, chaotic inflation is interesting because it its distinct phenomenological predictions; It is capable of sourcing significant primordial tensor perturbations, which may be detectible in the cosmic microwave background.', '1211.4024-1-4-0': 'Here we will find a stringy realization of large field inflation.', '1211.4024-1-4-1': 'As in brane inflation [CITATION], the inflaton represents the position of a D3 brane in a six dimensional compactification manifold, assumed to be sufficiently stable.', '1211.4024-1-4-2': 'The potential felt by the D3 brane due to the five-form field strength [MATH] gives rise to the four-dimensional effective potential of the inflaton.', '1211.4024-1-4-3': 'Crucially, this field strength depends not just on the location of the D3 brane(s), but also the number of times they have traversed any non-trivial one-cycles of the compactification manifold.', '1211.4024-1-4-4': 'This will allow for the possibility of large field brane inflation, which cannot otherwise exist [CITATION].', '1211.4024-1-4-5': 'It should be pointed out that there are stringy realizations of large field inflation, e.g. [CITATION], and [CITATION].', '1211.4024-1-5-0': 'In the probe approximation, the potential felt by a D3 must be exactly periodic, just as for an axion.', '1211.4024-1-5-1': 'Furthermore, the five-form flux takes quantized values over the five-cycle which is dual to the one-cycle.', '1211.4024-1-5-2': 'Assuming rigid moduli stabilization, the flux potential is exactly quadratic in the discrete flux winding [MATH].', '1211.4024-1-5-3': 'By turning on the coupling of the D3 to the background five-form flux, the periodicity is lifted, and the discrete flux becomes a continuous parameter, contributing an exactly quadratic term to the potential.', '1211.4024-1-5-4': 'We now illustrate this with a simple example.', '1211.4024-1-6-0': '# Charges in compact spaces with non-trivial first homology', '1211.4024-1-7-0': 'As a warm-up example, let us consider a single electron and positron in the compact space [MATH].', '1211.4024-1-7-1': 'The action and equations of motion are given by [EQUATION] where [MATH] is the oriented world lines of the charges.', '1211.4024-1-7-2': 'Integration and differentiation (see Appendix) of the equations of motion require that the point particle current [MATH] satisfy [EQUATION] respectively.', '1211.4024-1-7-3': 'These simply state that no net charge can occupy a compact space, and electric current is conserved.', '1211.4024-1-7-4': 'We can either ensure that [MATH] has no net time-like winding (as we have done), or add a diffuse background "jellium" charge to the action.', '1211.4024-1-7-5': 'A homogenous jellium contribution is just proportional to the spatial volume form, [EQUATION] with [MATH].', '1211.4024-1-7-6': '(We abuse the notation [MATH] to also mean the winding number of the oriented submanifold given by the actual intersection.)', '1211.4024-1-7-7': 'The uniform charge density cancels the tadpole.', '1211.4024-1-8-0': 'Let us imaging that [MATH] represents a single positive charge and a single negative charge.', '1211.4024-1-8-1': "We can compute the potential between them by finding the Green's function on this space.", '1211.4024-1-8-2': 'We expect the usual Coulomb interaction to be modified by two effects:', '1211.4024-1-9-0': 'It is the latter effect which we find useful here, as it enables one to change the electric flux on the [MATH].', '1211.4024-1-9-1': 'It is straightforward to calculate the difference in flux caused by transporting one of the two charges around the [MATH].', '1211.4024-1-9-2': 'The transport of one of the particles around the one-cycle means that [MATH] acquires winding number equal to one.', '1211.4024-1-9-3': 'The flux on the [MATH] is measured by choosing a fixed time- and [MATH]-coordinate, and then integrating the dual field strength over the [MATH].', '1211.4024-1-10-0': 'To calculate the change in flux caused by a single winding of a particle, let us define a 3-manifold with boundary [MATH] which spans an interval in time [MATH] times the full [MATH] cycle.', '1211.4024-1-10-1': 'Then [EQUATION] and so [EQUATION]', '1211.4024-1-10-2': 'Thus the electric field in the [MATH]-direction changes by one unit each time a particle is transported around the circle in the [MATH]-direction.', '1211.4024-1-10-3': 'A simple interpretation of this is that the charge drags the field-lines around the cycle.', '1211.4024-1-11-0': 'We can immediately write down the homological piece of the potential.', '1211.4024-1-11-1': 'If the metric is given by [MATH], with [MATH], then [EQUATION] where [MATH] is the [MATH]-separation of the two charges as measured on the universal cover.', '1211.4024-1-11-2': 'The flux part of the electron potential is thus [EQUATION] where [MATH].', '1211.4024-1-12-0': 'The flux potential cancels the jellium [CITATION] contribution.', '1211.4024-1-12-1': 'We can think of the jellium term in the potential as arising due to the finite compactification volume.', '1211.4024-1-12-2': 'Since the field strength is single-valued, a jellium term is required in the potential felt by a probe charge.', '1211.4024-1-12-3': 'But transport of a physical charge around a non-trivial cycle does not leave the field strength invariant, and so the probe charge is an inadequate description.', '1211.4024-1-12-4': 'Unlike the flux potential, the metric is single valued, and no monodromy exists in the gravitational potential.', '1211.4024-1-13-0': 'In a sense, one can say that the configuration space of charges and flux is not simply the product of the compact manifold and its first homology, but rather is a non-trivial fibration: One can change the flux by transporting charges around the one-cycles associated with them.', '1211.4024-1-14-0': '# Ingredients for Chaotic Brane Inflation', '1211.4024-1-15-0': 'The ingredients we will need is an F-theory compactification of Type IIB string theory which contains at least one D3-brane.', '1211.4024-1-15-1': 'Furthermore, the six-dimensional transverse space must have a non-trivial first homology, i.e., [MATH] or [MATH].', '1211.4024-1-15-2': 'Because the D3 moduli in the direction of the non-trivial 1-cycles are lifted at tree level, these models may lack supersymmetry.', '1211.4024-1-15-3': 'All closed string moduli must be sufficiently stabilized, in order that inflation may take place in this background.', '1211.4024-1-15-4': 'It is further necessary that the periodic portion of the potential be flat enough that the full potential has only a single minimum.', '1211.4024-1-15-5': 'In the probe approximation, the D3 has a discreet "shift symmetry" associated with transport about the 1-cycle, but this may not be sufficient to guarantee local flatness.', '1211.4024-1-15-6': 'As illustrated before, the inflationary potential exists due to non-trivial winding of the five-form flux about the homological one-cycle.', '1211.4024-1-15-7': 'The D3-brane moves classically through this cycle to unwind the flux.', '1211.4024-1-15-8': 'We will assume that the moduli stabilization is rigid enough to ignore the back-reaction of the dynamical flux.', '1211.4024-1-15-9': 'This assumption is generically false in known warped flux compactifications [CITATION], but such effects may actually flatten the potential [CITATION].', '1211.4024-1-16-0': 'The potential induced by brane monodromy is [EQUATION] where Vol[MATH] is the volume of the compact space, and [MATH] is the ten-dimensional Planck mass, and [MATH] is the D3 charge.', '1211.4024-1-16-1': 'In terms of the four-dimensional Planck mass, [MATH], and for a canonically normalized inflaton [MATH], we find the potential [EQUATION] which is parametrically flattened by increasing the compactification volume.', '1211.4024-1-17-0': 'To achieve reasonable density perturbations, we only need to tune the size of the compactification volume to moderately large values.', '1211.4024-1-17-1': 'This will require the brane to undergo of order 10 revolutions, so any model must have first Homology large enough to permit this, i.e. [EQUATION] with [MATH].', '1211.4024-1-18-0': 'Furthermore, we require the single-valued part of the potential to be relatively flat.', '1211.4024-1-18-1': 'This can be achieved by the presence of a continuous shift symmetry, or if needed by fine tuning.', '1211.4024-1-19-0': 'To achieve slow-roll inflation, we will require the scalar field [MATH] to initially have a super-Planckian vev.', '1211.4024-1-19-1': 'This means that the four-dimensional potential will exceed the tension of a D3-brane, opening the possibility for brane tunneling [CITATION] or nucleation[CITATION] via the Brown-Teitelboim mechanism.', '1211.4024-1-19-2': 'Because this is a slow process, our description remains valid.', '1211.4024-1-19-3': 'Indeed, brane nucleation could give rise to the mobile inflaton, although inflation will then end with brane anti-brane annihilation, but unlike [CITATION], the bubble need not self-annihilate until after many laps are completed.', '1211.4024-1-19-4': 'The final D3-[MATH] annihilation will result in the formation of a cosmic string network [CITATION].', '1211.4024-1-20-0': '# Discussion', '1211.4024-1-21-0': 'We have provided a simple framework for large field brane inflation.', '1211.4024-1-21-1': 'To construct realistic models, a number of hurdles must first be addressed, the most significant of which is moduli stabilization.', '1211.4024-1-21-2': 'However, because our framework relies on a non-trivial first homology group, most of the progress made on moduli stabilization does not apply here.', '1211.4024-1-21-3': 'Another potential difficulty may arise in obtaining a flat enough periodic portion of the brane potential.', '1211.4024-1-21-4': 'If the modulation of the quadratic piece is too large, there may not be a long enough slow-roll trajectory.', '1211.4024-1-21-5': 'Finally, it is unlikely that supersymmetry can be unbroken in the models considered here, since the D3 moduli receive an explicit mass, rather than from a spontaneous uplifting, say by the introduction of anti-branes.'}

{'1211.4024-2-0-0': 'We illustrate a framework for constructing models of chaotic inflation where the inflaton is the position of a D3 brane along the universal cover of a string compactification.', '1211.4024-2-0-1': 'In our scenario, a brane rolls many times around a non-trivial one-cycle, thereby unwinding a Ramond-Ramond flux.', '1211.4024-2-0-2': 'These "flux monodromies" are similar in spirit to the monodromies of Silverstein, Westphal, and McAllister, and their four-dimensional description is that of Kaloper and Sorbo.', '1211.4024-2-0-3': 'Assuming moduli stabilization is rigid enough, the large-field inflationary potential is protected from radiative corrections by a discrete shift symmetry.', '1211.4024-2-1-0': '# Introduction', '1211.4024-2-2-0': 'Perhaps the simplest phenomenological model of inflation [CITATION] is due to Linde\'s monomial potential [CITATION], which undergoes what is called "chaotic inflation" due to its expected behavior on large scales.', '1211.4024-2-2-1': 'Chaotic inflationary models are not obviously natural in the context of effective field theory precisely because of the requirement that the potential be sufficiently flat over super-Planckian field distances.', '1211.4024-2-2-2': 'In the quadratic model, the inflaton mass must be of order [MATH], and higher order terms in the potential must remain subdominant for field values as large as [MATH], which requires a functional fine tuning from an effective field theory point of view.', '1211.4024-2-3-0': 'An elegant solution to this problem was presented in [CITATION], whereby an axion "eats" a three-form potential, and so acquires a mass [CITATION].', '1211.4024-2-3-1': 'The axion potential is purely quadratic, being protected from radiative corrections by the underlying shift symmetry.', '1211.4024-2-3-2': 'Aside from the simplicity of the model, chaotic inflation is interesting because it its distinct phenomenological predictions; It is capable of sourcing significant primordial tensor perturbations, which may be detectible in the cosmic microwave background.', '1211.4024-2-4-0': 'Here we will find a stringy realization of large field inflation.', '1211.4024-2-4-1': 'As in brane inflation [CITATION], the inflaton represents the position of a D3 brane in a six-dimensional compactification manifold, assumed to be sufficiently stable.', '1211.4024-2-4-2': 'The potential felt by the D3 brane due to the five-form field strength [MATH] gives rise to the four-dimensional effective potential of the inflaton.', '1211.4024-2-4-3': 'Crucially, this field strength depends not just on the location of the D3 brane(s), but also on their history, i.e., the number of times they have traversed any non-trivial one-cycles of the compactification manifold.', '1211.4024-2-4-4': 'This "flux wrapping" will allow for the possibility of large field brane inflation, which cannot otherwise exist [CITATION] because increasing the field range typically requires increasing the compactification volume, which in turn increases the 4d Planck mass.', '1211.4024-2-5-0': 'It should be pointed out that there are already a few stringy realizations of large field inflation, e.g. [CITATION], as well as [CITATION].', '1211.4024-2-5-1': 'After this paper was completed, we learned of related work on unwinding fluxes [CITATION], and their application to inflation [CITATION].', '1211.4024-2-6-0': 'In the probe approximation, the potential felt by a D3 must be exactly periodic, just as for an axion.', '1211.4024-2-6-1': 'Furthermore, the five-form flux takes quantized values over the five-cycle which is dual to the one-cycle.', '1211.4024-2-6-2': 'Assuming rigid moduli stabilization, the flux potential is exactly quadratic in the discrete flux winding [MATH].', '1211.4024-2-6-3': 'By turning on the coupling of the D3 to the background five-form flux, the periodicity is lifted, and the discrete flux becomes a continuous parameter, contributing an exactly quadratic term to the potential.', '1211.4024-2-6-4': 'We now illustrate this with a simple example.', '1211.4024-2-7-0': '# Charges in compact spaces with non-trivial first homology', '1211.4024-2-8-0': 'As a warm-up example, let us consider a single electron and positron in the compact space [MATH].', '1211.4024-2-8-1': 'The action and equations of motion are given by [EQUATION] where [MATH] is the oriented world lines of the charges.', '1211.4024-2-8-2': 'Integration and differentiation of the equations of motion require that the point-particle current [MATH] satisfy [EQUATION] or equivalently (see Appendix), [EQUATION] respectively.', '1211.4024-2-8-3': 'We abuse the notation [MATH] to mean both the intersection and the winding number of the intersection, so the left-hand Eq. ([REF]) should be read as stating that there are equally many positive points as negative points in the total intersection.', '1211.4024-2-9-0': 'These equations simply state that no net charge can occupy a compact space, and electric current is conserved.', '1211.4024-2-9-1': 'We can either ensure that [MATH] has no net time-like winding (as we have done), or add a diffuse background "jellium" charge to the action.', '1211.4024-2-9-2': 'A homogenous jellium contribution is just proportional to the spatial volume form, [EQUATION] with [MATH].', '1211.4024-2-9-3': 'The uniform charge density cancels the tadpole.', '1211.4024-2-10-0': 'Let us imaging that [MATH] represents a single positive charge and a single negative charge.', '1211.4024-2-10-1': "We can compute the potential between them by finding the Green's function on this space.", '1211.4024-2-10-2': 'We expect the usual Coulomb interaction to be modified by two effects:', '1211.4024-2-11-0': 'It is the latter effect which we find useful here, as it enables one to change the electric flux on the [MATH].', '1211.4024-2-11-1': 'It is straightforward to calculate the difference in flux caused by transporting one of the two charges around the [MATH].', '1211.4024-2-11-2': 'The transport of one of the particles around the one-cycle means that [MATH] acquires winding number equal to one.', '1211.4024-2-11-3': 'The flux on the [MATH] is measured by choosing a fixed time- and [MATH]-coordinate, and then integrating the dual field strength [MATH] over the [MATH].', '1211.4024-2-12-0': 'To calculate the change in flux caused by a single winding of a particle, let us define a 3-manifold (with boundary) [MATH] which spans an interval in time [MATH] times the full [MATH] cycle.', '1211.4024-2-12-1': 'Then [EQUATION] and so [EQUATION]', '1211.4024-2-12-2': 'Thus the electric field in the [MATH]-direction changes by one unit each time a particle is transported around the circle in the [MATH]-direction.', '1211.4024-2-12-3': 'A simple interpretation of this is that the charge drags the field-lines around the cycle.', '1211.4024-2-13-0': 'We can immediately write down the homological piece of the potential.', '1211.4024-2-13-1': 'If the metric is given by [MATH], with [MATH], then [EQUATION] where [MATH] is the [MATH]-separation of the two charges as measured on the universal cover.', '1211.4024-2-13-2': 'The flux part of the electron potential is thus [EQUATION] where [MATH].', '1211.4024-2-14-0': 'The flux potential cancels the jellium [CITATION] contribution.', '1211.4024-2-14-1': 'We can think of the jellium term in the potential as arising due to the finite compactification volume.', '1211.4024-2-14-2': 'A jellium term is required in the potential felt by a probe charge, since the field strength is then single-valued.', '1211.4024-2-14-3': 'But transport of a physical charge around a non-trivial cycle does not leave the field strength invariant, and so the probe charge is an inadequate description.', '1211.4024-2-15-0': 'In a sense, one can say that the configuration space of charges and flux is not simply the product of the compact manifold and its first homology, but rather is a non-trivial fibration: One can change the flux by transporting charges around the one-cycles associated with them.', '1211.4024-2-16-0': 'Although the potential is exactly quadratic classically (and even in perturbation theory), there are non-perturbative corrections.', '1211.4024-2-16-1': 'Pair production will eventually discharge any potential exceeding twice the electron mass.', '1211.4024-2-16-2': 'This is such a slow enough process that we can safely ignore it.', '1211.4024-2-16-3': 'Furthermore, adiabatic motion of a charge will never be able to wind more than one unit of flux because of avoided level crossing [CITATION].', '1211.4024-2-16-4': 'This is not a problem except on timescales long compared to [MATH], where [MATH] is the electron mass.', '1211.4024-2-17-0': '# Ingredients for Chaotic Brane Inflation', '1211.4024-2-18-0': 'The ingredients we will need is an F-theory compactification of Type IIB string theory which contains at least one mobile D3-brane.', '1211.4024-2-18-1': 'Furthermore, the six-dimensional transverse space must have a non-trivial first homology, i.e., [MATH] or [MATH].', '1211.4024-2-18-2': 'Because the D3 moduli in the direction of the non-trivial 1-cycles are lifted at tree level, these models may lack supersymmetry.', '1211.4024-2-18-3': 'All closed string moduli must be sufficiently stabilized, in order that inflation may take place in this background.', '1211.4024-2-18-4': 'It is further necessary that the periodic portion of the potential be flat enough that the full potential has only a single minimum.', '1211.4024-2-18-5': 'In the probe approximation, the D3 has a discreet "shift symmetry" associated with transport about the 1-cycle, but this may not be sufficient to guarantee local flatness.', '1211.4024-2-18-6': 'As illustrated before, the inflationary potential exists due to non-trivial winding of the five-form flux about the homological one-cycle.', '1211.4024-2-18-7': 'The D3-brane moves classically through this cycle to unwind the flux.', '1211.4024-2-18-8': 'We will assume that the moduli stabilization is rigid enough to ignore the back-reaction of the dynamical flux.', '1211.4024-2-18-9': 'This assumption is generically false in known warped flux compactifications [CITATION], but such effects may actually flatten the potential [CITATION].', '1211.4024-2-19-0': 'The potential induced by brane monodromy is [EQUATION] where [MATH] is the volume of the compact space, [MATH] is the ten-dimensional reduced Planck mass, and [MATH] is the D3 charge.', '1211.4024-2-19-1': 'We assume the string coupling to be of order unity.', '1211.4024-2-19-2': 'In terms of the four-dimensional reduced Planck mass, [MATH], and for a canonically normalized inflaton [MATH], we find the potential [EQUATION]', '1211.4024-2-19-3': 'To achieve reasonable density perturbations, the quadratic model needs the inflaton mass to obey [EQUATION] which requires the compactification scale to be [EQUATION]', '1211.4024-2-19-4': 'In terms of the inflaton, this scale corresponds to a field distance [EQUATION]', '1211.4024-2-19-5': 'Successful large field inflation will require the brane to undergo of order a few thousand revolutions, so any model must have first Homology large enough to permit this, i.e. [EQUATION] with [MATH].', '1211.4024-2-19-6': 'This rather large number can be relaxed by no more than two orders of magnitude by allowing the size of the one-cycle to be much larger than the natural scale [MATH].', '1211.4024-2-19-7': 'In the [MATH] case, including backreaction may dynamically increase [MATH] at large [MATH] and so flatten the potential at large field values (say, [MATH]).', '1211.4024-2-19-8': 'The quadratic potential likely becomes approximately sinusoidal at very large field values, due to the flattening.', '1211.4024-2-20-0': 'The [MATH] case has extended periodicity [MATH], and so the homological part of the potential in each of the above cases is approximately given by [EQUATION] with [EQUATION] where [MATH] represents either the backreaction scale or the size of the homology group.', '1211.4024-2-20-1': 'This scenario could be called natural brane inflation, following [CITATION], although it avoids the problems associated with large axion decay constants [MATH] by the appearance of the large factor [MATH] in the potential of Eq. ([REF]), allowing [MATH].', '1211.4024-2-20-2': 'Alternative approaches to this problem can be found in [CITATION].', '1211.4024-2-21-0': 'We additionally require the single-valued part of the potential [EQUATION] to be relatively flat, meaning [MATH].', '1211.4024-2-22-0': 'To achieve 60 e-folds of slow-roll inflation, we must arrange the scalar field [MATH] to initially have a super-Planckian vacuum expectation value, [MATH].', '1211.4024-2-22-1': 'The Hubble scale during inflation is then [MATH], which is almost two orders of magnitude below the Kaluza-Klein scale [MATH].', '1211.4024-2-22-2': 'However, the four-dimensional potential will be of order [MATH], which exceeds the tension of a D3-brane by two orders of magnitude, opening the possibility for brane tunneling [CITATION] or nucleation [CITATION].', '1211.4024-2-22-3': 'Because these are a slow processes, our description remains valid.', '1211.4024-2-22-4': 'Indeed, brane nucleation could give rise to the mobile inflaton, although inflation will then end with brane anti-brane annihilation, but unlike [CITATION], the bubble need not self-annihilate until after many laps are completed.', '1211.4024-2-22-5': 'The final D3-[MATH] annihilation will result in the formation of a cosmic string network [CITATION].', '1211.4024-2-23-0': '# Discussion', '1211.4024-2-24-0': 'We have provided a simple framework for large field brane inflation.', '1211.4024-2-24-1': 'To construct realistic models, a number of hurdles must first be addressed, the most significant of which is moduli stabilization.', '1211.4024-2-24-2': 'However, because our framework relies on a non-trivial first homology group, much of the progress made on moduli stabilization of warped compactifications does not apply here.', '1211.4024-2-24-3': 'Another potential difficulty may arise in obtaining a flat enough periodic portion of the brane potential.', '1211.4024-2-24-4': 'If the modulation of the quadratic piece is too large, there may not be a long enough slow-roll trajectory.', '1211.4024-2-24-5': 'Finally, it is unlikely that supersymmetry can be unbroken in the models considered here, since the D3 moduli receive an explicit mass, rather than from a spontaneous uplifting, say by the introduction of anti-branes.', '1211.4024-2-25-0': 'Nevertheless, a number of intriguing features arise here, foremost being a UV description of large field inflation.', '1211.4024-2-25-1': 'The monodromies of this framework are extremely easy to visualize, being simply the motion of a (point-like) brane around a one-cycle.', '1211.4024-2-25-2': 'By traversing the cycle (perhaps several thousand times) a Ramond-Ramond flux is unwound, realizing either chaotic or natural inflation, both of which predict significant tensor modes in the CMB.', '1211.4024-2-26-0': 'We thank Daniel Baumann, Xingang Chen, Liam McAllister, Enrico Pajer, Lorenzo Sorbo, Alexander Westphal, and Xi Dong for helpful conversations.', '1211.4024-2-26-1': 'Funding was provided through NSF grant PHY-1213888.', '1211.4024-2-27-0': '# The de Rham delta function', '1211.4024-2-28-0': 'Here we review a simple notation [CITATION] appropriate for calculating the effects of localized sources coupled to gauge potentials.', '1211.4024-2-28-1': 'The new object is a singular differential form which we call the "de Rham delta function."', '1211.4024-2-29-0': '## Definition', '1211.4024-2-30-0': 'On a [MATH]-dimensional oriented manifold [MATH] with [MATH] an oriented submanifold of dimension [MATH], we define the de Rham delta function [MATH] as follows: [EQUATION] where the pullback is implicit on the RHS.', '1211.4024-2-30-1': 'The subscripts denote the order for differential forms, and superscripts denote the dimension for manifolds.', '1211.4024-2-30-2': "Stokes' theorem then implies [EQUATION] and so [EQUATION] where we have used the fact that [MATH].", '1211.4024-2-30-3': 'Here [MATH] is essentially the group sum of [MATH]-chains in [MATH].', '1211.4024-2-30-4': 'This definition of [MATH] is equivalent to [EQUATION]', '1211.4024-2-30-5': 'Following the definition we also find [EQUATION] which leads to the relation [EQUATION]', '1211.4024-2-30-6': 'This identity illuminates some generic features of submanifolds.', '1211.4024-2-31-0': '## Coordinate representation', '1211.4024-2-32-0': 'The coordinate representation of [MATH] is straightforward in coordinates where the submanifold is defined by the [MATH] constraint equations [EQUATION] with [MATH], via [EQUATION] where [MATH] is the usual [MATH]-dimensional Dirac delta function.', '1211.4024-2-32-1': 'The well known transformation properties of the Dirac delta function make this object automatically a differential form.', '1211.4024-2-32-2': '(Thus the only meaningful zeros of the [MATH] are transversal zeros, i.e., those where [MATH] changes sign in any neighborhood of the zero.)', '1211.4024-2-32-3': 'If a submanifold is [MATH]-dimensional, then the corresponding de Rham delta function is simply the characteristic function: [MATH] with [EQUATION]', '1211.4024-2-32-4': 'One may describe submanifolds with boundary by multiplication with this scalar de Rham delta function using Eq. ([REF]).', '1211.4024-2-32-5': 'As an example, if [MATH] is the positive [MATH]-axis in [MATH], then [EQUATION] where the characteristic function is the Heaviside function [MATH].', '1211.4024-2-32-6': 'Notice that the orientation of this submanifold has been chosen to be along the [MATH] direction, consistent with Eq. ([REF]) and the fact that its boundary is minus the point at the origin.', '1211.4024-2-33-0': 'The [MATH] do not need to be well-defined on the entire manifold, and in fact they only need to be defined at all in a neighborhood of [MATH].', '1211.4024-2-33-1': 'Thus despite its appearance in Eq. ([REF]), [MATH] is not necessarily a total derivative.', '1211.4024-2-33-2': 'If all of the [MATH] are well defined everywhere, then [MATH] is an algebraic variety.', '1211.4024-2-33-3': 'By Eq. ([REF]) and Eq. ([REF]) it can be seen that all algebraic varieties can be written globally as boundaries.', '1211.4024-2-33-4': 'It may be that the [MATH] are well defined only in a neighborhood of [MATH], in which case [MATH] is a submanifold.', '1211.4024-2-33-5': 'Near points not on [MATH] we may think of [MATH] locally as a boundary, just as we may think of a closed differential form as locally exact.', '1211.4024-2-33-6': 'Non-orientable submanifolds will correspond to constraints that may be double valued, that is [MATH] may return to minus itself upon translation around the submanifold.', '1211.4024-2-33-7': 'We considered such cases in [CITATION].', '1211.4024-2-34-0': 'Another important case occurs when [MATH] is only an immersion, i.e., it intersects itself.', '1211.4024-2-34-1': 'The [MATH] are path dependent here, as well.', '1211.4024-2-34-2': 'Consider the immersion [MATH] defined by the constraint [MATH].', '1211.4024-2-34-3': 'This looks like a figure-eight centered on the origin of the plane.', '1211.4024-2-34-4': 'Clearly [MATH] is multi-valued, and to get a complete figure eight requires summing over two branches of [MATH].', '1211.4024-2-34-5': 'We suppress the sum in Eq. ([REF]).', '1211.4024-2-34-6': 'Notice that the figure eight immersion satisfies [EQUATION]', '1211.4024-2-34-7': 'The self intersection of this immersion is twice the point at the origin, but since the orientation (sign) of the intersection is negative for exactly one of the two points of intersection, the total self intersection vanishes.', '1211.4024-2-34-8': 'For even co-dimemsion immersions, the sum over branches allows for non-zero self intersection from cross terms.', '1211.4024-2-34-9': 'By the antisymmetry of the wedge product, one may show that the self-intersection of an immersion of odd co-dimension will always vanish, assuming [MATH] is orientable.', '1211.4024-2-34-10': 'One may also compute the [MATH]th self intersection of an immersion via [EQUATION]', '1211.4024-2-35-0': '## Geometry', '1211.4024-2-36-0': 'If we introduce a metric on our manifold, we can measure the volume of our submanifolds with the volume element from the pull-back metric.', '1211.4024-2-36-1': 'We use [MATH] to denote the Hodge star on [MATH], so the induced volume element is [MATH].', '1211.4024-2-36-2': 'Looking at the coordinate definition of [MATH] given in Eq. ([REF]), we see that each of the [MATH] is constant along the submanifold, and so [MATH] is orthogonal to the submanifold.', '1211.4024-2-36-3': 'Thus, a good candidate for a volume element on [MATH] is [MATH].', '1211.4024-2-36-4': "To remove the rescaling redundancy in the [MATH]'s we may divide by [MATH], where the norm of a differential form [MATH] is defined by the scalar [EQUATION]", '1211.4024-2-36-5': 'Hence, we write [EQUATION] where the pull-back is implicit on the RHS.', '1211.4024-2-36-6': 'Despite its appearance, [MATH] is a [MATH]-form living in all [MATH] dimensions, although it is only well defined near [MATH].', '1211.4024-2-36-7': 'This is because in Eq. ([REF]), the Dirac delta function coefficients cancel, leaving dependence only on the smooth [MATH]s. Because of this, we may define [MATH] using the full [MATH]-dimensional exterior derivative.', '1211.4024-2-36-8': 'This is well defined on [MATH], and when restricting to points on the submanifold, [EQUATION]', '1211.4024-2-36-9': 'From Eq. ([REF]) it is evident that [EQUATION]', '1211.4024-2-36-10': 'More generally, the action by this Hodge star can be rewritten as [EQUATION] where [MATH] is an [MATH]-form living on [MATH] and the pullback is implicit on the RHS.', '1211.4024-2-37-0': '## Topology', '1211.4024-2-38-0': 'One feature of the de Rham delta function is that Poincare duality becomes manifest.', '1211.4024-2-38-1': 'Here we assume [MATH] is orientable and compact with no boundary.', '1211.4024-2-38-2': 'Then Eq. ([REF]) tells us that [MATH] is closed if and only if [MATH] is a cycle, and [MATH] is exact if [MATH] is a boundary.', '1211.4024-2-38-3': 'To complete this correspondence between the [MATH]-th Homology and the [MATH]th de Rham cohomology of [MATH], we need to show that [EQUATION]', '1211.4024-2-38-4': 'But this is not true when torsion is present.', '1211.4024-2-38-5': 'Consider the manifold [MATH] which has a single nontrivial one-cycle [MATH], i.e. [MATH].', '1211.4024-2-38-6': 'Since the group sum of two of these cycles is trivial, they must form a boundary: [MATH] and so [MATH] which means [EQUATION]', '1211.4024-2-38-7': 'Since [MATH] is not a boundary, it is false to claim that all exact de Rham delta functions are Poincare dual to boundaries.', '1211.4024-2-38-8': 'Only by using real coefficients can we make the statement that [EQUATION]', '1211.4024-2-38-9': "De Rham's theorem gives us the isomorphism between homology and cohomology [EQUATION] and Poincare duality asserts that [EQUATION]", '1211.4024-2-38-10': 'The De Rham delta function provides the isomorphism [EQUATION]', '1211.4024-2-38-11': 'In fact, if the cohomology basis is chosen such that [EQUATION] then [EQUATION] where [MATH] is the Poincare dual of [MATH], and together they satisfy [EQUATION] i.e., their net intersection is a single positive point if [MATH], and is empty otherwise.'}

[['1211.4024-1-21-0', '1211.4024-2-24-0'], ['1211.4024-1-21-1', '1211.4024-2-24-1'], ['1211.4024-1-21-3', '1211.4024-2-24-3'], ['1211.4024-1-21-4', '1211.4024-2-24-4'], ['1211.4024-1-21-5', '1211.4024-2-24-5'], ['1211.4024-1-10-1', '1211.4024-2-12-1'], ['1211.4024-1-10-2', '1211.4024-2-12-2'], ['1211.4024-1-10-3', '1211.4024-2-12-3'], ['1211.4024-1-9-0', '1211.4024-2-11-0'], ['1211.4024-1-9-1', '1211.4024-2-11-1'], ['1211.4024-1-9-2', '1211.4024-2-11-2'], ['1211.4024-1-8-0', '1211.4024-2-10-0'], ['1211.4024-1-8-1', '1211.4024-2-10-1'], ['1211.4024-1-12-0', '1211.4024-2-14-0'], ['1211.4024-1-12-1', '1211.4024-2-14-1'], ['1211.4024-1-12-3', '1211.4024-2-14-3'], ['1211.4024-1-13-0', '1211.4024-2-15-0'], ['1211.4024-1-15-1', '1211.4024-2-18-1'], ['1211.4024-1-15-2', '1211.4024-2-18-2'], ['1211.4024-1-15-3', '1211.4024-2-18-3'], ['1211.4024-1-15-4', '1211.4024-2-18-4'], ['1211.4024-1-15-5', '1211.4024-2-18-5'], ['1211.4024-1-15-6', '1211.4024-2-18-6'], ['1211.4024-1-15-7', '1211.4024-2-18-7'], ['1211.4024-1-15-8', '1211.4024-2-18-8'], ['1211.4024-1-15-9', '1211.4024-2-18-9'], ['1211.4024-1-11-0', '1211.4024-2-13-0'], ['1211.4024-1-11-1', '1211.4024-2-13-1'], ['1211.4024-1-11-2', '1211.4024-2-13-2'], ['1211.4024-1-5-0', '1211.4024-2-6-0'], ['1211.4024-1-5-1', '1211.4024-2-6-1'], ['1211.4024-1-5-2', '1211.4024-2-6-2'], ['1211.4024-1-5-3', '1211.4024-2-6-3'], ['1211.4024-1-5-4', '1211.4024-2-6-4'], ['1211.4024-2-24-1', '1211.4024-3-26-1'], ['1211.4024-2-24-3', '1211.4024-3-26-3'], ['1211.4024-2-24-4', '1211.4024-3-26-4'], ['1211.4024-2-26-0', '1211.4024-3-28-0'], ['1211.4024-2-26-1', '1211.4024-3-28-1'], ['1211.4024-2-19-0', '1211.4024-3-19-0'], ['1211.4024-2-19-1', '1211.4024-3-19-1'], ['1211.4024-2-19-2', '1211.4024-3-19-2'], ['1211.4024-2-19-3', '1211.4024-3-19-3'], ['1211.4024-2-19-4', '1211.4024-3-19-4'], ['1211.4024-2-19-6', '1211.4024-3-19-6'], ['1211.4024-2-18-0', '1211.4024-3-18-0'], ['1211.4024-2-18-3', '1211.4024-3-18-3'], ['1211.4024-2-18-4', '1211.4024-3-18-4'], ['1211.4024-2-18-7', '1211.4024-3-18-7'], ['1211.4024-2-18-9', '1211.4024-3-18-9'], ['1211.4024-2-33-1', '1211.4024-3-35-1'], ['1211.4024-2-33-2', '1211.4024-3-35-2'], ['1211.4024-2-33-3', '1211.4024-3-35-3'], ['1211.4024-2-33-4', '1211.4024-3-35-4'], ['1211.4024-2-33-5', '1211.4024-3-35-5'], ['1211.4024-2-13-0', '1211.4024-3-13-0'], ['1211.4024-2-13-2', '1211.4024-3-13-2'], ['1211.4024-2-9-0', '1211.4024-3-9-0'], ['1211.4024-2-9-1', '1211.4024-3-9-1'], ['1211.4024-2-9-2', '1211.4024-3-9-2'], ['1211.4024-2-9-3', '1211.4024-3-9-3'], ['1211.4024-2-12-0', '1211.4024-3-12-0'], ['1211.4024-2-12-1', '1211.4024-3-12-1'], ['1211.4024-2-0-2', '1211.4024-3-0-2'], ['1211.4024-2-0-3', '1211.4024-3-0-3'], ['1211.4024-2-10-1', '1211.4024-3-10-1'], ['1211.4024-2-28-0', '1211.4024-3-30-0'], ['1211.4024-2-28-1', '1211.4024-3-30-1'], ['1211.4024-2-30-1', '1211.4024-3-32-1'], ['1211.4024-2-30-2', '1211.4024-3-32-2'], ['1211.4024-2-30-3', '1211.4024-3-32-3'], ['1211.4024-2-30-4', '1211.4024-3-32-4'], ['1211.4024-2-30-5', '1211.4024-3-32-5'], ['1211.4024-2-30-6', '1211.4024-3-32-6'], ['1211.4024-2-38-0', '1211.4024-3-40-0'], ['1211.4024-2-38-1', '1211.4024-3-40-1'], ['1211.4024-2-38-2', '1211.4024-3-40-2'], ['1211.4024-2-38-4', '1211.4024-3-40-4'], ['1211.4024-2-38-5', '1211.4024-3-40-5'], ['1211.4024-2-38-7', '1211.4024-3-40-7'], ['1211.4024-2-38-8', '1211.4024-3-40-8'], ['1211.4024-2-38-9', '1211.4024-3-40-9'], ['1211.4024-2-38-11', '1211.4024-3-40-11'], ['1211.4024-2-6-0', '1211.4024-3-6-0'], ['1211.4024-2-6-1', '1211.4024-3-6-1'], ['1211.4024-2-6-2', '1211.4024-3-6-2'], ['1211.4024-2-6-3', '1211.4024-3-6-3'], ['1211.4024-2-6-4', '1211.4024-3-6-4'], ['1211.4024-2-25-1', '1211.4024-3-27-1'], ['1211.4024-2-16-1', '1211.4024-3-16-1'], ['1211.4024-2-16-2', '1211.4024-3-16-2'], ['1211.4024-2-16-3', '1211.4024-3-16-3'], ['1211.4024-2-16-4', '1211.4024-3-16-4'], ['1211.4024-2-8-0', '1211.4024-3-8-0'], ['1211.4024-2-8-1', '1211.4024-3-8-1'], ['1211.4024-2-8-2', '1211.4024-3-8-2'], ['1211.4024-2-14-0', '1211.4024-3-14-0'], ['1211.4024-2-14-1', '1211.4024-3-14-1'], ['1211.4024-2-14-2', '1211.4024-3-14-2'], ['1211.4024-2-22-1', '1211.4024-3-24-1'], ['1211.4024-2-22-2', '1211.4024-3-24-2'], ['1211.4024-2-32-2', '1211.4024-3-34-2'], ['1211.4024-2-32-4', '1211.4024-3-34-4'], ['1211.4024-2-32-6', '1211.4024-3-34-6'], ['1211.4024-2-34-0', '1211.4024-3-36-0'], ['1211.4024-2-34-1', '1211.4024-3-36-1'], ['1211.4024-2-34-2', '1211.4024-3-36-2'], ['1211.4024-2-34-3', '1211.4024-3-36-3'], ['1211.4024-2-34-5', '1211.4024-3-36-5'], ['1211.4024-2-36-0', '1211.4024-3-38-0'], ['1211.4024-2-36-1', '1211.4024-3-38-1'], ['1211.4024-2-36-2', '1211.4024-3-38-2'], ['1211.4024-2-36-3', '1211.4024-3-38-3'], ['1211.4024-2-36-4', '1211.4024-3-38-4'], ['1211.4024-2-36-7', '1211.4024-3-38-7'], ['1211.4024-2-36-8', '1211.4024-3-38-8'], ['1211.4024-2-36-9', '1211.4024-3-38-9'], ['1211.4024-2-11-0', '1211.4024-3-11-0'], ['1211.4024-2-11-1', '1211.4024-3-11-1'], ['1211.4024-2-11-2', '1211.4024-3-11-2'], ['1211.4024-2-5-1', '1211.4024-3-5-1'], ['1211.4024-2-20-0', '1211.4024-3-22-0'], ['1211.4024-2-3-1', '1211.4024-3-3-1'], ['1211.4024-2-2-0', '1211.4024-3-2-0'], ['1211.4024-2-2-1', '1211.4024-3-2-1'], ['1211.4024-2-21-0', '1211.4024-3-23-0'], ['1211.4024-1-4-0', '1211.4024-2-4-0'], ['1211.4024-1-4-2', '1211.4024-2-4-2'], ['1211.4024-1-3-0', '1211.4024-2-2-0'], ['1211.4024-1-3-4', '1211.4024-2-3-2'], ['1211.4024-1-7-0', '1211.4024-2-8-0'], ['1211.4024-1-7-1', '1211.4024-2-8-1'], ['1211.4024-1-7-4', '1211.4024-2-9-1'], ['1211.4024-1-7-5', '1211.4024-2-9-2'], ['1211.4024-1-7-7', '1211.4024-2-9-3'], ['1211.4024-1-0-0', '1211.4024-2-0-0'], ['1211.4024-1-1-0', '1211.4024-2-0-1'], ['1211.4024-1-1-1', '1211.4024-2-0-2'], ['1211.4024-1-1-2', '1211.4024-2-0-3'], ['1211.4024-1-21-2', '1211.4024-2-24-2'], ['1211.4024-1-10-0', '1211.4024-2-12-0'], ['1211.4024-1-9-3', '1211.4024-2-11-3'], ['1211.4024-1-12-2', '1211.4024-2-14-2'], ['1211.4024-1-15-0', '1211.4024-2-18-0'], ['1211.4024-1-16-0', '1211.4024-2-19-0'], ['1211.4024-2-24-0', '1211.4024-3-26-0'], ['1211.4024-2-24-2', '1211.4024-3-26-2'], ['1211.4024-2-24-5', '1211.4024-3-26-6'], ['1211.4024-2-19-5', '1211.4024-3-19-5'], ['1211.4024-2-18-1', '1211.4024-3-18-1'], ['1211.4024-2-18-2', '1211.4024-3-18-2'], ['1211.4024-2-18-5', '1211.4024-3-18-5'], ['1211.4024-2-18-6', '1211.4024-3-18-6'], ['1211.4024-2-18-8', '1211.4024-3-18-8'], ['1211.4024-2-33-0', '1211.4024-3-35-0'], ['1211.4024-2-33-6', '1211.4024-3-35-6'], ['1211.4024-2-33-7', '1211.4024-3-35-7'], ['1211.4024-2-13-1', '1211.4024-3-13-1'], ['1211.4024-2-12-2', '1211.4024-3-12-2'], ['1211.4024-2-12-3', '1211.4024-3-12-3'], ['1211.4024-2-0-0', '1211.4024-3-0-0'], ['1211.4024-2-0-1', '1211.4024-3-0-1'], ['1211.4024-2-10-0', '1211.4024-3-10-0'], ['1211.4024-2-30-0', '1211.4024-3-32-0'], ['1211.4024-2-38-3', '1211.4024-3-40-3'], ['1211.4024-2-38-6', '1211.4024-3-40-6'], ['1211.4024-2-38-10', '1211.4024-3-40-10'], ['1211.4024-2-25-0', '1211.4024-3-27-0'], ['1211.4024-2-25-2', '1211.4024-3-27-2'], ['1211.4024-2-16-0', '1211.4024-3-16-0'], ['1211.4024-2-15-0', '1211.4024-3-15-0'], ['1211.4024-2-8-3', '1211.4024-3-8-3'], ['1211.4024-2-14-3', '1211.4024-3-14-3'], ['1211.4024-2-22-0', '1211.4024-3-24-0'], ['1211.4024-2-22-3', '1211.4024-3-24-3'], ['1211.4024-2-22-4', '1211.4024-3-24-4'], ['1211.4024-2-22-5', '1211.4024-3-24-5'], ['1211.4024-2-32-0', '1211.4024-3-34-0'], ['1211.4024-2-32-1', '1211.4024-3-34-1'], ['1211.4024-2-32-3', '1211.4024-3-34-3'], ['1211.4024-2-32-5', '1211.4024-3-34-5'], ['1211.4024-2-34-4', '1211.4024-3-36-4'], ['1211.4024-2-34-6', '1211.4024-3-36-6'], ['1211.4024-2-34-7', '1211.4024-3-36-7'], ['1211.4024-2-34-9', '1211.4024-3-36-9'], ['1211.4024-2-34-10', '1211.4024-3-36-10'], ['1211.4024-2-36-5', '1211.4024-3-38-5'], ['1211.4024-2-36-6', '1211.4024-3-38-6'], ['1211.4024-2-36-10', '1211.4024-3-38-10'], ['1211.4024-2-11-3', '1211.4024-3-11-3'], ['1211.4024-2-5-0', '1211.4024-3-5-0'], ['1211.4024-2-20-1', '1211.4024-3-22-1'], ['1211.4024-2-20-2', '1211.4024-3-22-2'], ['1211.4024-2-4-0', '1211.4024-3-4-0'], ['1211.4024-2-4-1', '1211.4024-3-4-1'], ['1211.4024-2-4-2', '1211.4024-3-4-2'], ['1211.4024-2-4-3', '1211.4024-3-4-3'], ['1211.4024-2-4-4', '1211.4024-3-4-4'], ['1211.4024-2-3-0', '1211.4024-3-3-0'], ['1211.4024-2-3-2', '1211.4024-3-3-2'], ['1211.4024-2-2-2', '1211.4024-3-2-2'], ['1211.4024-1-4-1', '1211.4024-2-4-1'], ['1211.4024-1-4-3', '1211.4024-2-4-3'], ['1211.4024-1-4-5', '1211.4024-2-5-0'], ['1211.4024-1-3-1', '1211.4024-2-2-1'], ['1211.4024-1-3-2', '1211.4024-2-2-2'], ['1211.4024-1-7-2', '1211.4024-2-8-2'], ['1211.4024-1-7-3', '1211.4024-2-9-0'], ['1211.4024-1-16-1', '1211.4024-2-19-2'], ['1211.4024-2-34-8', '1211.4024-3-36-8'], ['1211.4024-1-4-4', '1211.4024-2-4-4'], ['1211.4024-1-3-3', '1211.4024-2-3-0'], ['1211.4024-1-3-3', '1211.4024-2-3-1'], ['1211.4024-1-7-6', '1211.4024-2-8-3'], ['1211.4024-1-17-1', '1211.4024-2-19-5'], ['1211.4024-1-18-0', '1211.4024-2-21-0'], ['1211.4024-1-19-0', '1211.4024-2-22-0'], ['1211.4024-1-19-1', '1211.4024-2-22-2'], ['1211.4024-1-19-2', '1211.4024-2-22-3'], ['1211.4024-1-19-3', '1211.4024-2-22-4'], ['1211.4024-1-19-4', '1211.4024-2-22-5']]

[['1211.4024-1-21-0', '1211.4024-2-24-0'], ['1211.4024-1-21-1', '1211.4024-2-24-1'], ['1211.4024-1-21-3', '1211.4024-2-24-3'], ['1211.4024-1-21-4', '1211.4024-2-24-4'], ['1211.4024-1-21-5', '1211.4024-2-24-5'], ['1211.4024-1-10-1', '1211.4024-2-12-1'], ['1211.4024-1-10-2', '1211.4024-2-12-2'], ['1211.4024-1-10-3', '1211.4024-2-12-3'], ['1211.4024-1-9-0', '1211.4024-2-11-0'], ['1211.4024-1-9-1', '1211.4024-2-11-1'], ['1211.4024-1-9-2', '1211.4024-2-11-2'], ['1211.4024-1-8-0', '1211.4024-2-10-0'], ['1211.4024-1-8-1', '1211.4024-2-10-1'], ['1211.4024-1-12-0', '1211.4024-2-14-0'], ['1211.4024-1-12-1', '1211.4024-2-14-1'], ['1211.4024-1-12-3', '1211.4024-2-14-3'], ['1211.4024-1-13-0', '1211.4024-2-15-0'], ['1211.4024-1-15-1', '1211.4024-2-18-1'], ['1211.4024-1-15-2', '1211.4024-2-18-2'], ['1211.4024-1-15-3', '1211.4024-2-18-3'], ['1211.4024-1-15-4', '1211.4024-2-18-4'], ['1211.4024-1-15-5', '1211.4024-2-18-5'], ['1211.4024-1-15-6', '1211.4024-2-18-6'], ['1211.4024-1-15-7', '1211.4024-2-18-7'], ['1211.4024-1-15-8', '1211.4024-2-18-8'], ['1211.4024-1-15-9', '1211.4024-2-18-9'], ['1211.4024-1-11-0', '1211.4024-2-13-0'], ['1211.4024-1-11-1', '1211.4024-2-13-1'], ['1211.4024-1-11-2', '1211.4024-2-13-2'], ['1211.4024-1-5-0', '1211.4024-2-6-0'], ['1211.4024-1-5-1', '1211.4024-2-6-1'], ['1211.4024-1-5-2', '1211.4024-2-6-2'], ['1211.4024-1-5-3', '1211.4024-2-6-3'], ['1211.4024-1-5-4', '1211.4024-2-6-4'], ['1211.4024-2-24-1', '1211.4024-3-26-1'], ['1211.4024-2-24-3', '1211.4024-3-26-3'], ['1211.4024-2-24-4', '1211.4024-3-26-4'], ['1211.4024-2-26-0', '1211.4024-3-28-0'], ['1211.4024-2-26-1', '1211.4024-3-28-1'], ['1211.4024-2-19-0', '1211.4024-3-19-0'], ['1211.4024-2-19-1', '1211.4024-3-19-1'], ['1211.4024-2-19-2', '1211.4024-3-19-2'], ['1211.4024-2-19-3', '1211.4024-3-19-3'], ['1211.4024-2-19-4', '1211.4024-3-19-4'], ['1211.4024-2-19-6', '1211.4024-3-19-6'], ['1211.4024-2-18-0', '1211.4024-3-18-0'], ['1211.4024-2-18-3', '1211.4024-3-18-3'], ['1211.4024-2-18-4', '1211.4024-3-18-4'], ['1211.4024-2-18-7', '1211.4024-3-18-7'], ['1211.4024-2-18-9', '1211.4024-3-18-9'], ['1211.4024-2-33-1', '1211.4024-3-35-1'], ['1211.4024-2-33-2', '1211.4024-3-35-2'], ['1211.4024-2-33-3', '1211.4024-3-35-3'], ['1211.4024-2-33-4', '1211.4024-3-35-4'], ['1211.4024-2-33-5', '1211.4024-3-35-5'], ['1211.4024-2-13-0', '1211.4024-3-13-0'], ['1211.4024-2-13-2', '1211.4024-3-13-2'], ['1211.4024-2-9-0', '1211.4024-3-9-0'], ['1211.4024-2-9-1', '1211.4024-3-9-1'], ['1211.4024-2-9-2', '1211.4024-3-9-2'], ['1211.4024-2-9-3', '1211.4024-3-9-3'], ['1211.4024-2-12-0', '1211.4024-3-12-0'], ['1211.4024-2-12-1', '1211.4024-3-12-1'], ['1211.4024-2-0-2', '1211.4024-3-0-2'], ['1211.4024-2-0-3', '1211.4024-3-0-3'], ['1211.4024-2-10-1', '1211.4024-3-10-1'], ['1211.4024-2-28-0', '1211.4024-3-30-0'], ['1211.4024-2-28-1', '1211.4024-3-30-1'], ['1211.4024-2-30-1', '1211.4024-3-32-1'], ['1211.4024-2-30-2', '1211.4024-3-32-2'], ['1211.4024-2-30-3', '1211.4024-3-32-3'], ['1211.4024-2-30-4', '1211.4024-3-32-4'], ['1211.4024-2-30-5', '1211.4024-3-32-5'], ['1211.4024-2-30-6', '1211.4024-3-32-6'], ['1211.4024-2-38-0', '1211.4024-3-40-0'], ['1211.4024-2-38-1', '1211.4024-3-40-1'], ['1211.4024-2-38-2', '1211.4024-3-40-2'], ['1211.4024-2-38-4', '1211.4024-3-40-4'], ['1211.4024-2-38-5', '1211.4024-3-40-5'], ['1211.4024-2-38-7', '1211.4024-3-40-7'], ['1211.4024-2-38-8', '1211.4024-3-40-8'], ['1211.4024-2-38-9', '1211.4024-3-40-9'], ['1211.4024-2-38-11', '1211.4024-3-40-11'], ['1211.4024-2-6-0', '1211.4024-3-6-0'], ['1211.4024-2-6-1', '1211.4024-3-6-1'], ['1211.4024-2-6-2', '1211.4024-3-6-2'], ['1211.4024-2-6-3', '1211.4024-3-6-3'], ['1211.4024-2-6-4', '1211.4024-3-6-4'], ['1211.4024-2-25-1', '1211.4024-3-27-1'], ['1211.4024-2-16-1', '1211.4024-3-16-1'], ['1211.4024-2-16-2', '1211.4024-3-16-2'], ['1211.4024-2-16-3', '1211.4024-3-16-3'], ['1211.4024-2-16-4', '1211.4024-3-16-4'], ['1211.4024-2-8-0', '1211.4024-3-8-0'], ['1211.4024-2-8-1', '1211.4024-3-8-1'], ['1211.4024-2-8-2', '1211.4024-3-8-2'], ['1211.4024-2-14-0', '1211.4024-3-14-0'], ['1211.4024-2-14-1', '1211.4024-3-14-1'], ['1211.4024-2-14-2', '1211.4024-3-14-2'], ['1211.4024-2-22-1', '1211.4024-3-24-1'], ['1211.4024-2-22-2', '1211.4024-3-24-2'], ['1211.4024-2-32-2', '1211.4024-3-34-2'], ['1211.4024-2-32-4', '1211.4024-3-34-4'], ['1211.4024-2-32-6', '1211.4024-3-34-6'], ['1211.4024-2-34-0', '1211.4024-3-36-0'], ['1211.4024-2-34-1', '1211.4024-3-36-1'], ['1211.4024-2-34-2', '1211.4024-3-36-2'], ['1211.4024-2-34-3', '1211.4024-3-36-3'], ['1211.4024-2-34-5', '1211.4024-3-36-5'], ['1211.4024-2-36-0', '1211.4024-3-38-0'], ['1211.4024-2-36-1', '1211.4024-3-38-1'], ['1211.4024-2-36-2', '1211.4024-3-38-2'], ['1211.4024-2-36-3', '1211.4024-3-38-3'], ['1211.4024-2-36-4', '1211.4024-3-38-4'], ['1211.4024-2-36-7', '1211.4024-3-38-7'], ['1211.4024-2-36-8', '1211.4024-3-38-8'], ['1211.4024-2-36-9', '1211.4024-3-38-9'], ['1211.4024-2-11-0', '1211.4024-3-11-0'], ['1211.4024-2-11-1', '1211.4024-3-11-1'], ['1211.4024-2-11-2', '1211.4024-3-11-2'], ['1211.4024-2-5-1', '1211.4024-3-5-1'], ['1211.4024-2-20-0', '1211.4024-3-22-0'], ['1211.4024-2-3-1', '1211.4024-3-3-1'], ['1211.4024-2-2-0', '1211.4024-3-2-0'], ['1211.4024-2-2-1', '1211.4024-3-2-1'], ['1211.4024-2-21-0', '1211.4024-3-23-0'], ['1211.4024-1-4-0', '1211.4024-2-4-0'], ['1211.4024-1-4-2', '1211.4024-2-4-2'], ['1211.4024-1-3-0', '1211.4024-2-2-0'], ['1211.4024-1-3-4', '1211.4024-2-3-2'], ['1211.4024-1-7-0', '1211.4024-2-8-0'], ['1211.4024-1-7-1', '1211.4024-2-8-1'], ['1211.4024-1-7-4', '1211.4024-2-9-1'], ['1211.4024-1-7-5', '1211.4024-2-9-2'], ['1211.4024-1-7-7', '1211.4024-2-9-3'], ['1211.4024-1-0-0', '1211.4024-2-0-0'], ['1211.4024-1-1-0', '1211.4024-2-0-1'], ['1211.4024-1-1-1', '1211.4024-2-0-2'], ['1211.4024-1-1-2', '1211.4024-2-0-3']]

[['1211.4024-1-21-2', '1211.4024-2-24-2'], ['1211.4024-1-10-0', '1211.4024-2-12-0'], ['1211.4024-1-9-3', '1211.4024-2-11-3'], ['1211.4024-1-12-2', '1211.4024-2-14-2'], ['1211.4024-1-15-0', '1211.4024-2-18-0'], ['1211.4024-1-16-0', '1211.4024-2-19-0'], ['1211.4024-2-24-0', '1211.4024-3-26-0'], ['1211.4024-2-24-2', '1211.4024-3-26-2'], ['1211.4024-2-24-5', '1211.4024-3-26-6'], ['1211.4024-2-19-5', '1211.4024-3-19-5'], ['1211.4024-2-18-1', '1211.4024-3-18-1'], ['1211.4024-2-18-2', '1211.4024-3-18-2'], ['1211.4024-2-18-5', '1211.4024-3-18-5'], ['1211.4024-2-18-6', '1211.4024-3-18-6'], ['1211.4024-2-18-8', '1211.4024-3-18-8'], ['1211.4024-2-33-0', '1211.4024-3-35-0'], ['1211.4024-2-33-6', '1211.4024-3-35-6'], ['1211.4024-2-33-7', '1211.4024-3-35-7'], ['1211.4024-2-13-1', '1211.4024-3-13-1'], ['1211.4024-2-12-2', '1211.4024-3-12-2'], ['1211.4024-2-12-3', '1211.4024-3-12-3'], ['1211.4024-2-0-0', '1211.4024-3-0-0'], ['1211.4024-2-0-1', '1211.4024-3-0-1'], ['1211.4024-2-10-0', '1211.4024-3-10-0'], ['1211.4024-2-30-0', '1211.4024-3-32-0'], ['1211.4024-2-38-3', '1211.4024-3-40-3'], ['1211.4024-2-38-6', '1211.4024-3-40-6'], ['1211.4024-2-38-10', '1211.4024-3-40-10'], ['1211.4024-2-25-0', '1211.4024-3-27-0'], ['1211.4024-2-25-2', '1211.4024-3-27-2'], ['1211.4024-2-16-0', '1211.4024-3-16-0'], ['1211.4024-2-15-0', '1211.4024-3-15-0'], ['1211.4024-2-8-3', '1211.4024-3-8-3'], ['1211.4024-2-14-3', '1211.4024-3-14-3'], ['1211.4024-2-22-0', '1211.4024-3-24-0'], ['1211.4024-2-22-3', '1211.4024-3-24-3'], ['1211.4024-2-22-4', '1211.4024-3-24-4'], ['1211.4024-2-22-5', '1211.4024-3-24-5'], ['1211.4024-2-32-0', '1211.4024-3-34-0'], ['1211.4024-2-32-1', '1211.4024-3-34-1'], ['1211.4024-2-32-3', '1211.4024-3-34-3'], ['1211.4024-2-32-5', '1211.4024-3-34-5'], ['1211.4024-2-34-4', '1211.4024-3-36-4'], ['1211.4024-2-34-6', '1211.4024-3-36-6'], ['1211.4024-2-34-7', '1211.4024-3-36-7'], ['1211.4024-2-34-9', '1211.4024-3-36-9'], ['1211.4024-2-34-10', '1211.4024-3-36-10'], ['1211.4024-2-36-5', '1211.4024-3-38-5'], ['1211.4024-2-36-6', '1211.4024-3-38-6'], ['1211.4024-2-36-10', '1211.4024-3-38-10'], ['1211.4024-2-11-3', '1211.4024-3-11-3'], ['1211.4024-2-5-0', '1211.4024-3-5-0'], ['1211.4024-2-20-1', '1211.4024-3-22-1'], ['1211.4024-2-20-2', '1211.4024-3-22-2'], ['1211.4024-2-4-0', '1211.4024-3-4-0'], ['1211.4024-2-4-1', '1211.4024-3-4-1'], ['1211.4024-2-4-2', '1211.4024-3-4-2'], ['1211.4024-2-4-3', '1211.4024-3-4-3'], ['1211.4024-2-4-4', '1211.4024-3-4-4'], ['1211.4024-2-3-0', '1211.4024-3-3-0'], ['1211.4024-2-3-2', '1211.4024-3-3-2'], ['1211.4024-2-2-2', '1211.4024-3-2-2'], ['1211.4024-1-4-1', '1211.4024-2-4-1'], ['1211.4024-1-4-3', '1211.4024-2-4-3'], ['1211.4024-1-4-5', '1211.4024-2-5-0'], ['1211.4024-1-3-1', '1211.4024-2-2-1'], ['1211.4024-1-3-2', '1211.4024-2-2-2'], ['1211.4024-1-7-2', '1211.4024-2-8-2'], ['1211.4024-1-7-3', '1211.4024-2-9-0']]

[]

[['1211.4024-1-16-1', '1211.4024-2-19-2'], ['1211.4024-2-34-8', '1211.4024-3-36-8'], ['1211.4024-1-4-4', '1211.4024-2-4-4'], ['1211.4024-1-3-3', '1211.4024-2-3-0'], ['1211.4024-1-3-3', '1211.4024-2-3-1'], ['1211.4024-1-7-6', '1211.4024-2-8-3']]

[['1211.4024-1-17-1', '1211.4024-2-19-5'], ['1211.4024-1-18-0', '1211.4024-2-21-0'], ['1211.4024-1-19-0', '1211.4024-2-22-0'], ['1211.4024-1-19-1', '1211.4024-2-22-2'], ['1211.4024-1-19-2', '1211.4024-2-22-3'], ['1211.4024-1-19-3', '1211.4024-2-22-4'], ['1211.4024-1-19-4', '1211.4024-2-22-5']]

['1211.4024-1-8-2', '1211.4024-2-10-2', '1211.4024-3-10-2']

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

https://arxiv.org/abs/1211.4024

{'1211.4024-3-0-0': 'We illustrate a framework for constructing models of chaotic inflation where the inflaton is the position of a D3-brane along the universal cover of a string compactification.', '1211.4024-3-0-1': 'In our scenario, a brane rolls many times around a nontrivial one-cycle, thereby unwinding a Ramond-Ramond flux.', '1211.4024-3-0-2': 'These "flux monodromies" are similar in spirit to the monodromies of Silverstein, Westphal, and McAllister, and their four-dimensional description is that of Kaloper and Sorbo.', '1211.4024-3-0-3': 'Assuming moduli stabilization is rigid enough, the large-field inflationary potential is protected from radiative corrections by a discrete shift symmetry.', '1211.4024-3-1-0': '# Introduction', '1211.4024-3-2-0': 'Perhaps the simplest phenomenological model of inflation [CITATION] is due to Linde\'s monomial potential [CITATION], which undergoes what is called "chaotic inflation" due to its expected behavior on large scales.', '1211.4024-3-2-1': 'Chaotic inflationary models are not obviously natural in the context of effective field theory precisely because of the requirement that the potential be sufficiently flat over super-Planckian field distances.', '1211.4024-3-2-2': 'In the quadratic model, the inflaton mass must be of order [MATH], and higher-order terms in the potential must remain subdominant for field values as large as [MATH], which requires a functional fine-tuning from an effective field theory point of view.', '1211.4024-3-3-0': 'An elegant solution to this problem was presented in Refs. [CITATION], whereby an axion "eats" a three-form potential, and so acquires a mass [CITATION].', '1211.4024-3-3-1': 'The axion potential is purely quadratic, being protected from radiative corrections by the underlying shift symmetry.', '1211.4024-3-3-2': 'Aside from the simplicity of the model, chaotic inflation is interesting because of its distinct phenomenological predictions; it is capable of sourcing significant primordial tensor perturbations, which may be detectible in the cosmic microwave background.', '1211.4024-3-4-0': 'Here we will find a stringy realization of large-field inflation.', '1211.4024-3-4-1': 'As in brane inflation [CITATION], the inflaton represents the position of a D3-brane in a six-dimensional compactification manifold, assumed to be sufficiently stable.', '1211.4024-3-4-2': 'The potential felt by the D3-brane due to the five-form field strength [MATH] gives rise to the four-dimensional effective potential of the inflaton.', '1211.4024-3-4-3': 'Crucially, this field strength depends not just on the location of the D3-brane(s), but also on their history, i.e., the number of times they have traversed any nontrivial one-cycles of the compactification manifold.', '1211.4024-3-4-4': 'This "flux wrapping" will allow for the possibility of large-field brane inflation, which cannot otherwise exist [CITATION] because increasing the field range typically requires increasing the compactification volume, which in turn increases the four dimensional Planck mass.', '1211.4024-3-5-0': 'It should be pointed out that there are already a few stringy realizations of large-field inflation, e.g. Refs. [CITATION], as well as Refs. [CITATION].', '1211.4024-3-5-1': 'After this paper was completed, we learned of related work on unwinding fluxes [CITATION], and their application to inflation [CITATION].', '1211.4024-3-6-0': 'In the probe approximation, the potential felt by a D3 must be exactly periodic, just as for an axion.', '1211.4024-3-6-1': 'Furthermore, the five-form flux takes quantized values over the five-cycle which is dual to the one-cycle.', '1211.4024-3-6-2': 'Assuming rigid moduli stabilization, the flux potential is exactly quadratic in the discrete flux winding [MATH].', '1211.4024-3-6-3': 'By turning on the coupling of the D3 to the background five-form flux, the periodicity is lifted, and the discrete flux becomes a continuous parameter, contributing an exactly quadratic term to the potential.', '1211.4024-3-6-4': 'We now illustrate this with a simple example.', '1211.4024-3-7-0': '# Charges in compact spaces with nontrivial first homology', '1211.4024-3-8-0': 'As a warm-up example, let us consider a single electron and positron in the compact space [MATH].', '1211.4024-3-8-1': 'The action and equations of motion are given by [EQUATION] where [MATH] is the oriented world lines of the charges.', '1211.4024-3-8-2': 'Integration and differentiation of the equations of motion require that the point-particle current [MATH] satisfy [EQUATION] or equivalently (see Appendix), [EQUATION] respectively.', '1211.4024-3-8-3': 'We abuse the notation [MATH] to mean both the intersection and the winding number of the intersection, so the left-hand side of Eq. ([REF]) should be read as stating that there are equally many positive points as negative points in the total intersection.', '1211.4024-3-9-0': 'These equations simply state that no net charge can occupy a compact space, and electric current is conserved.', '1211.4024-3-9-1': 'We can either ensure that [MATH] has no net time-like winding (as we have done), or add a diffuse background "jellium" charge to the action.', '1211.4024-3-9-2': 'A homogenous jellium contribution is just proportional to the spatial volume form, [EQUATION] with [MATH].', '1211.4024-3-9-3': 'The uniform charge density cancels the tadpole.', '1211.4024-3-10-0': 'Let us imagine that [MATH] represents a single positive charge and a single negative charge.', '1211.4024-3-10-1': "We can compute the potential between them by finding the Green's function on this space.", '1211.4024-3-10-2': 'We expect the usual Coulomb interaction to be modified by two effects:', '1211.4024-3-11-0': 'It is the latter effect which we find useful here, as it enables one to change the electric flux on the [MATH].', '1211.4024-3-11-1': 'It is straightforward to calculate the difference in flux caused by transporting one of the two charges around the [MATH].', '1211.4024-3-11-2': 'The transport of one of the particles around the one-cycle means that [MATH] acquires winding number equal to one.', '1211.4024-3-11-3': 'The flux on the [MATH] is measured by choosing a fixed time and [MATH] coordinate, and then integrating the dual field strength [MATH] over the [MATH].', '1211.4024-3-12-0': 'To calculate the change in flux caused by a single winding of a particle, let us define a 3-manifold (with boundary) [MATH] which spans an interval in time [MATH] times the full [MATH] cycle.', '1211.4024-3-12-1': 'Then [EQUATION] and so [EQUATION]', '1211.4024-3-12-2': 'Thus the electric field in the [MATH] direction changes by one unit each time a particle is transported around the circle in the [MATH] direction.', '1211.4024-3-12-3': 'A simple interpretation of this is that the charge drags the field lines around the cycle.', '1211.4024-3-13-0': 'We can immediately write down the homological piece of the potential.', '1211.4024-3-13-1': 'If the metric is given by [MATH], with [MATH], then [EQUATION] where [MATH] is the [MATH] separation of the two charges as measured on the universal cover.', '1211.4024-3-13-2': 'The flux part of the electron potential is thus [EQUATION] where [MATH].', '1211.4024-3-14-0': 'The flux potential cancels the jellium [CITATION] contribution.', '1211.4024-3-14-1': 'We can think of the jellium term in the potential as arising due to the finite compactification volume.', '1211.4024-3-14-2': 'A jellium term is required in the potential felt by a probe charge, since the field strength is then single-valued.', '1211.4024-3-14-3': 'But transport of a physical charge around a nontrivial cycle does not leave the field strength invariant, and so the probe charge is an inadequate description.', '1211.4024-3-15-0': 'In a sense, one can say that the configuration space of charges and flux is not simply the product of the compact manifold and its first homology, but rather is a nontrivial fibration: one can change the flux by transporting charges around the one-cycles associated with them.', '1211.4024-3-16-0': 'Although the potential is exactly quadratic classically (and even in perturbation theory), there are nonperturbative corrections.', '1211.4024-3-16-1': 'Pair production will eventually discharge any potential exceeding twice the electron mass.', '1211.4024-3-16-2': 'This is such a slow enough process that we can safely ignore it.', '1211.4024-3-16-3': 'Furthermore, adiabatic motion of a charge will never be able to wind more than one unit of flux because of avoided level crossing [CITATION].', '1211.4024-3-16-4': 'This is not a problem except on timescales long compared to [MATH], where [MATH] is the electron mass.', '1211.4024-3-17-0': '# Ingredients for Chaotic Brane Inflation', '1211.4024-3-18-0': 'The ingredients we will need is an F-theory compactification of Type IIB string theory which contains at least one mobile D3-brane.', '1211.4024-3-18-1': 'Furthermore, the six-dimensional transverse space must have a nontrivial first homology, i.e., [MATH] or [MATH].', '1211.4024-3-18-2': 'Because the D3 moduli in the direction of the nontrivial one-cycles are lifted at tree level, these models may lack supersymmetry.', '1211.4024-3-18-3': 'All closed string moduli must be sufficiently stabilized, in order that inflation may take place in this background.', '1211.4024-3-18-4': 'It is further necessary that the periodic portion of the potential be flat enough that the full potential has only a single minimum.', '1211.4024-3-18-5': 'In the probe approximation, the D3 has a discreet "shift symmetry" associated with transport about the one-cycle, but this may not be sufficient to guarantee local flatness.', '1211.4024-3-18-6': 'As illustrated before, the inflationary potential exists due to nontrivial winding of the five-form flux about the homological one-cycle.', '1211.4024-3-18-7': 'The D3-brane moves classically through this cycle to unwind the flux.', '1211.4024-3-18-8': 'We will assume that the moduli stabilization is rigid enough to ignore the backreaction of the dynamical flux.', '1211.4024-3-18-9': 'This assumption is generically false in known warped flux compactifications [CITATION], but such effects may actually flatten the potential [CITATION].', '1211.4024-3-19-0': 'The potential induced by brane monodromy is [EQUATION] where [MATH] is the volume of the compact space, [MATH] is the ten-dimensional reduced Planck mass, and [MATH] is the D3 charge.', '1211.4024-3-19-1': 'We assume the string coupling to be of order unity.', '1211.4024-3-19-2': 'In terms of the four-dimensional reduced Planck mass, [MATH], and for a canonically normalized inflaton [MATH], we find the potential [EQUATION]', '1211.4024-3-19-3': 'To achieve reasonable density perturbations, the quadratic model needs the inflaton mass to obey [EQUATION] which requires the compactification scale to be [EQUATION]', '1211.4024-3-19-4': 'In terms of the inflaton, this scale corresponds to a field distance [EQUATION]', '1211.4024-3-19-5': 'Hence, successful large-field inflation will require the brane to undergo of order a few thousand revolutions, so any model must have first Homology large enough to permit this, i.e. [EQUATION] with [MATH].', '1211.4024-3-19-6': 'This rather large number can be relaxed by no more than two orders of magnitude by allowing the size of the one-cycle to be much larger than the natural scale [MATH].', '1211.4024-3-20-0': 'In the [MATH] case, the quadratic approximation for [MATH] must eventually break down, due to backreaction.', '1211.4024-3-20-1': 'If we assume the modulus [MATH] is very heavy, it can be written as [MATH], which grows as more flux is wound on the transverse space.', '1211.4024-3-20-2': 'If this is the only effect of backreaction, the inflaton potential is flattened at large flux values, and reaches a maximum if [MATH] exceeds [MATH].', '1211.4024-3-21-0': 'However, a steepening [CITATION] of the inflaton potential could instead occur due to kinetic coupling between the inflaton and [MATH], say of the form [EQUATION] for sufficiently negative [MATH].', '1211.4024-3-21-1': 'We will assume that the kinetic coupling is subdominant, and hence that backreaction leads to a flattening of the inflaton potential at sufficiently large field values [MATH].', '1211.4024-3-21-2': 'Qualitatively speaking, a potential which is quadratic at small field values, and flat at large field values can be thought of as approximately sinusoidal over the range [MATH].', '1211.4024-3-22-0': 'The [MATH] case has extended periodicity [MATH], and so the homological part of the potential in each of the above cases is approximately given by [EQUATION] with [EQUATION] where [MATH] represents either the backreaction scale or the size of the homology group.', '1211.4024-3-22-1': 'This scenario could be called natural brane inflation, following Refs. [CITATION], although it avoids the problems associated with large axion decay constants [MATH] by the appearance of the large factor [MATH] in the potential of Eq. ([REF]), allowing [MATH].', '1211.4024-3-22-2': 'Alternative approaches to this problem can be found in Refs. [CITATION].', '1211.4024-3-23-0': 'We additionally require the single-valued part of the potential [EQUATION] to be relatively flat, meaning [MATH].', '1211.4024-3-24-0': 'To achieve 60 [MATH]-folds of slow-roll inflation, we must arrange the scalar field [MATH] to initially have a super-Planckian vacuum expectation value, [MATH].', '1211.4024-3-24-1': 'The Hubble scale during inflation is then [MATH], which is almost two orders of magnitude below the Kaluza-Klein scale [MATH].', '1211.4024-3-24-2': 'However, the four-dimensional potential will be of order [MATH], which exceeds the tension of a D3-brane by two orders of magnitude, opening the possibility for brane tunneling [CITATION] or nucleation [CITATION].', '1211.4024-3-24-3': 'Because these are slow processes, our description remains valid.', '1211.4024-3-24-4': 'Indeed, brane nucleation could give rise to the mobile inflaton, although inflation will then end with brane-antibrane annihilation, but unlike Ref. [CITATION], the bubble need not self-annihilate until after many laps are completed.', '1211.4024-3-24-5': 'The final D3-[MATH] annihilation will result in the formation of a cosmic-string network [CITATION].', '1211.4024-3-25-0': '# Discussion', '1211.4024-3-26-0': 'We have provided a simple framework for large-field brane inflation.', '1211.4024-3-26-1': 'To construct realistic models, a number of hurdles must first be addressed, the most significant of which is moduli stabilization.', '1211.4024-3-26-2': 'However, because our framework relies on a nontrivial first homology group, much of the progress made on moduli stabilization of warped compactifications does not apply here.', '1211.4024-3-26-3': 'Another potential difficulty may arise in obtaining a flat enough periodic portion of the brane potential.', '1211.4024-3-26-4': 'If the modulation of the quadratic piece is too large, there may not be a long enough slow-roll trajectory.', '1211.4024-3-26-5': 'On the other hand, a periodic modulation of the inflaton potential can lead to detectable non-Gaussanity in the cosmic microwave background [CITATION].', '1211.4024-3-26-6': 'Finally, it is unlikely that supersymmetry can be unbroken in the models considered here, since the D3 moduli receive an explicit mass, rather than from a spontaneous uplifting, say by the introduction of antibranes.', '1211.4024-3-27-0': 'Nevertheless, a number of intriguing features arise here, foremost being a UV description of large-field inflation.', '1211.4024-3-27-1': 'The monodromies of this framework are extremely easy to visualize, being simply the motion of a (point-like) brane around a one-cycle.', '1211.4024-3-27-2': 'By traversing the cycle (perhaps several thousand times) a Ramond-Ramond flux is unwound, realizing either chaotic or natural inflation, both of which predict significant tensor modes in the cosmic microwave background.', '1211.4024-3-28-0': 'We thank Daniel Baumann, Xingang Chen, Liam McAllister, Enrico Pajer, Lorenzo Sorbo, Alexander Westphal, and Xi Dong for helpful conversations.', '1211.4024-3-28-1': 'Funding was provided through NSF grant PHY-1213888.', '1211.4024-3-29-0': '# The de Rham delta function', '1211.4024-3-30-0': 'Here we review a simple notation [CITATION] appropriate for calculating the effects of localized sources coupled to gauge potentials.', '1211.4024-3-30-1': 'The new object is a singular differential form which we call the "de Rham delta function."', '1211.4024-3-31-0': '## Definition', '1211.4024-3-32-0': 'On a [MATH]-dimensional oriented manifold [MATH] with [MATH] an oriented submanifold of dimension [MATH], we define the de Rham delta function [MATH] as follows: [EQUATION] where the pullback is implicit on the rhs.', '1211.4024-3-32-1': 'The subscripts denote the order for differential forms, and superscripts denote the dimension for manifolds.', '1211.4024-3-32-2': "Stokes' theorem then implies [EQUATION] and so [EQUATION] where we have used the fact that [MATH].", '1211.4024-3-32-3': 'Here [MATH] is essentially the group sum of [MATH]-chains in [MATH].', '1211.4024-3-32-4': 'This definition of [MATH] is equivalent to [EQUATION]', '1211.4024-3-32-5': 'Following the definition we also find [EQUATION] which leads to the relation [EQUATION]', '1211.4024-3-32-6': 'This identity illuminates some generic features of submanifolds.', '1211.4024-3-33-0': '## Coordinate representation', '1211.4024-3-34-0': 'The coordinate representation of [MATH] is straightforward in coordinates where the submanifold is defined by the [MATH] constraint equations, [EQUATION] with [MATH], via [EQUATION] where [MATH] is the usual [MATH]-dimensional Dirac delta function.', '1211.4024-3-34-1': 'The well-known transformation properties of the Dirac delta function make this object automatically a differential form.', '1211.4024-3-34-2': '(Thus the only meaningful zeros of the [MATH] are transversal zeros, i.e., those where [MATH] changes sign in any neighborhood of the zero.)', '1211.4024-3-34-3': 'If a submanifold is [MATH]-dimensional, then the corresponding de Rham delta function is simply the characteristic function, [MATH], with [EQUATION]', '1211.4024-3-34-4': 'One may describe submanifolds with boundary by multiplication with this scalar de Rham delta function using Eq. ([REF]).', '1211.4024-3-34-5': 'As an example, if [MATH] is the positive [MATH] axis in [MATH], then [EQUATION] where the characteristic function is the Heaviside function [MATH].', '1211.4024-3-34-6': 'Notice that the orientation of this submanifold has been chosen to be along the [MATH] direction, consistent with Eq. ([REF]) and the fact that its boundary is minus the point at the origin.', '1211.4024-3-35-0': 'The [MATH] do not need to be well defined on the entire manifold, and in fact they only need to be defined at all in a neighborhood of [MATH].', '1211.4024-3-35-1': 'Thus despite its appearance in Eq. ([REF]), [MATH] is not necessarily a total derivative.', '1211.4024-3-35-2': 'If all of the [MATH] are well defined everywhere, then [MATH] is an algebraic variety.', '1211.4024-3-35-3': 'By Eq. ([REF]) and Eq. ([REF]) it can be seen that all algebraic varieties can be written globally as boundaries.', '1211.4024-3-35-4': 'It may be that the [MATH] are well defined only in a neighborhood of [MATH], in which case [MATH] is a submanifold.', '1211.4024-3-35-5': 'Near points not on [MATH] we may think of [MATH] locally as a boundary, just as we may think of a closed differential form as locally exact.', '1211.4024-3-35-6': 'Nonorientable submanifolds will correspond to constraints that may be double valued, that is [MATH] may return to minus itself upon translation around the submanifold.', '1211.4024-3-35-7': 'We considered such cases in Ref. [CITATION].', '1211.4024-3-36-0': 'Another important case occurs when [MATH] is only an immersion, i.e., it intersects itself.', '1211.4024-3-36-1': 'The [MATH] are path dependent here, as well.', '1211.4024-3-36-2': 'Consider the immersion [MATH] defined by the constraint [MATH].', '1211.4024-3-36-3': 'This looks like a figure-eight centered on the origin of the plane.', '1211.4024-3-36-4': 'Clearly [MATH] is multivalued, and to get a complete figure-eight requires summing over two branches of [MATH].', '1211.4024-3-36-5': 'We suppress the sum in Eq. ([REF]).', '1211.4024-3-36-6': 'Notice that the figure-eight immersion satisfies [EQUATION]', '1211.4024-3-36-7': 'The self-intersection of this immersion is twice the point at the origin, but since the orientation (sign) of the intersection is negative for exactly one of the two points of intersection, the total self-intersection vanishes.', '1211.4024-3-36-8': 'For even-codimemsion immersions, the sum over branches allows for nonzero self-intersection from cross terms.', '1211.4024-3-36-9': 'By the antisymmetry of the wedge product, one may show that the self-intersection of an immersion of odd codimension will always vanish, assuming [MATH] is orientable.', '1211.4024-3-36-10': 'One may also compute the [MATH]th self-intersection of an immersion via [EQUATION]', '1211.4024-3-37-0': '## Geometry', '1211.4024-3-38-0': 'If we introduce a metric on our manifold, we can measure the volume of our submanifolds with the volume element from the pull-back metric.', '1211.4024-3-38-1': 'We use [MATH] to denote the Hodge star on [MATH], so the induced volume element is [MATH].', '1211.4024-3-38-2': 'Looking at the coordinate definition of [MATH] given in Eq. ([REF]), we see that each of the [MATH] is constant along the submanifold, and so [MATH] is orthogonal to the submanifold.', '1211.4024-3-38-3': 'Thus, a good candidate for a volume element on [MATH] is [MATH].', '1211.4024-3-38-4': "To remove the rescaling redundancy in the [MATH]'s we may divide by [MATH], where the norm of a differential form [MATH] is defined by the scalar [EQUATION]", '1211.4024-3-38-5': 'Hence, we write [EQUATION] where the pull-back is implicit on the rhs.', '1211.4024-3-38-6': 'Despite its appearance, [MATH] is an [MATH]-form living in all [MATH] dimensions, although it is only well defined near [MATH].', '1211.4024-3-38-7': 'This is because in Eq. ([REF]), the Dirac delta function coefficients cancel, leaving dependence only on the smooth [MATH]s. Because of this, we may define [MATH] using the full [MATH]-dimensional exterior derivative.', '1211.4024-3-38-8': 'This is well defined on [MATH], and when restricting to points on the submanifold, [EQUATION]', '1211.4024-3-38-9': 'From Eq. ([REF]) it is evident that [EQUATION]', '1211.4024-3-38-10': 'More generally, the action by this Hodge star can be rewritten as [EQUATION] where [MATH] is an [MATH]-form living on [MATH] and the pull-back is implicit on the rhs.', '1211.4024-3-39-0': '## Topology', '1211.4024-3-40-0': 'One feature of the de Rham delta function is that Poincare duality becomes manifest.', '1211.4024-3-40-1': 'Here we assume [MATH] is orientable and compact with no boundary.', '1211.4024-3-40-2': 'Then Eq. ([REF]) tells us that [MATH] is closed if and only if [MATH] is a cycle, and [MATH] is exact if [MATH] is a boundary.', '1211.4024-3-40-3': 'To complete this correspondence between the [MATH]th homology and the [MATH]th de Rham cohomology of [MATH], we need to show that [EQUATION]', '1211.4024-3-40-4': 'But this is not true when torsion is present.', '1211.4024-3-40-5': 'Consider the manifold [MATH] which has a single nontrivial one-cycle [MATH], i.e. [MATH].', '1211.4024-3-40-6': 'Since the group sum of two of these cycles is trivial, they must form a boundary, [MATH] and so [MATH] which means [EQUATION]', '1211.4024-3-40-7': 'Since [MATH] is not a boundary, it is false to claim that all exact de Rham delta functions are Poincare dual to boundaries.', '1211.4024-3-40-8': 'Only by using real coefficients can we make the statement that [EQUATION]', '1211.4024-3-40-9': "De Rham's theorem gives us the isomorphism between homology and cohomology [EQUATION] and Poincare duality asserts that [EQUATION]", '1211.4024-3-40-10': 'The de Rham delta function provides the isomorphism [EQUATION]', '1211.4024-3-40-11': 'In fact, if the cohomology basis is chosen such that [EQUATION] then [EQUATION] where [MATH] is the Poincare dual of [MATH], and together they satisfy [EQUATION] i.e., their net intersection is a single positive point if [MATH], and is empty otherwise.'}

nucl-ex-0511054

{'nucl-ex-0511054-1-0-0': 'The level densities and radiative strength functions (RSFs) of [MATH]V have been extracted using the ([MATH]He,[MATH]) and ([MATH]He,[MATH]He[MATH]) reactions, respectively.', 'nucl-ex-0511054-1-0-1': 'From the level densities microcanonical entropies are deduced.', 'nucl-ex-0511054-1-0-2': 'The high [MATH]-energy part of the RSF is described by the giant electric dipole resonance.', 'nucl-ex-0511054-1-0-3': 'A significant enhancement over the predicted strength in the region of [MATH] MeV is seen, which is at present without any theoretical explanation.', 'nucl-ex-0511054-1-1-0': '2', 'nucl-ex-0511054-1-2-0': '# Introduction', 'nucl-ex-0511054-1-3-0': 'The structure of the vanadium isotopes is based on simple shell model configurations at low excitation energies.', 'nucl-ex-0511054-1-3-1': 'The valence protons and neutrons are occupying the single particle [MATH] and [MATH] orbitals, respectively.', 'nucl-ex-0511054-1-3-2': 'These shells are isolated from other orbitals by the N, Z = 20 and 28 shell gaps, making the vanadium isotopes interesting objects for studying various nuclear shell effects.', 'nucl-ex-0511054-1-3-3': 'In particular, it is well-known that the number of available singe-particle levels is significantly reduced for nuclei at closed shells.', 'nucl-ex-0511054-1-4-0': 'The density of states or, equivalently, the entropy in these systems depends on the number of broken Cooper pairs and single-particle orbitals made available by crossing the shell gaps.', 'nucl-ex-0511054-1-4-1': 'The [MATH]V nuclei are of special interest because the neutrons are strongly blocked in the process of creating entropy; [MATH]V and [MATH]V have seven and eight neutrons in the [MATH] orbital, respectively.', 'nucl-ex-0511054-1-4-2': 'Thus, the configuration space of the three protons in the [MATH] shell is of great importance.', 'nucl-ex-0511054-1-5-0': 'These particular shell-model configurations are also expected to govern the [MATH]-decay routes.', 'nucl-ex-0511054-1-5-1': 'Specifically, as within every major shell, the presence of only one parity for single-particle orbitals in the low-spin domain means that transitions of [MATH] type will be suppressed.', 'nucl-ex-0511054-1-5-2': 'The low mass of the investigated nuclei causes that the centroid of the giant electric dipole resonance (GEDR) is relatively high while the integrated strength according to the Thomas-Reiche-Kuhn sum rule is low, both observations working together to produce a relatively weak low-energy tail when compared to heavier nuclei.', 'nucl-ex-0511054-1-5-3': 'Hence, possible non-statistical effects in the radiative strength function (RSF) might stand out more in the present investigation.', 'nucl-ex-0511054-1-6-0': 'The Oslo Cyclotron group has developed a method to extract first-generation (primary) [MATH]-ray spectra at various initial excitation energies.', 'nucl-ex-0511054-1-6-1': 'From such a set of primary spectra, the nuclear level density and the RSF can be extracted simultaneously [CITATION].', 'nucl-ex-0511054-1-6-2': 'These two quantities reveal essential information on nuclear structure such as pair correlations and thermal and electromagnetic properties.', 'nucl-ex-0511054-1-6-3': 'In the last five years, the Oslo group has demonstrated several fruitful applications of the method [CITATION].', 'nucl-ex-0511054-1-7-0': 'In Sect. II an outline of the experimental procedure is given.', 'nucl-ex-0511054-1-7-1': 'The level densities and microcanonical entropies are discussed in Sect. III, and in Sect. IV the RSFs are presented.', 'nucl-ex-0511054-1-7-2': 'Finally, concluding remarks are given in Sect. V.', 'nucl-ex-0511054-1-8-0': '# Experimental method', 'nucl-ex-0511054-1-9-0': 'The experiment was carried out at the Oslo Cyclotron Laboratory (OCL) using a beam of 30-MeV [MATH]He ions.', 'nucl-ex-0511054-1-9-1': 'The self-supporting natural V target had a purity of [MATH]% and a thickness of 2.3 mg/cm[MATH].', 'nucl-ex-0511054-1-9-2': 'Particle-[MATH] coincidences for [MATH]V were measured with the CACTUS multi-detector array [CITATION].', 'nucl-ex-0511054-1-9-3': 'The charged ejectiles were detected using eight Si particle telescopes placed at an angle of 45[MATH] relative to the beam direction.', 'nucl-ex-0511054-1-9-4': 'Each telescope consists of a front [MATH] detector and a back [MATH] detector with thicknesses of 140 and 1500 [MATH]m, respectively.', 'nucl-ex-0511054-1-9-5': 'An array of 28 collimated NaI [MATH]-ray detectors with a total efficiency of [MATH]15% surrounded the target and the particle detectors.', 'nucl-ex-0511054-1-9-6': 'The reactions of interest were the pick-up reaction [MATH]V([MATH]He,[MATH]V, and the inelastic scattering [MATH]V([MATH]He,[MATH]He[MATH]V.', 'nucl-ex-0511054-1-9-7': 'The typical spin range is expected to be [MATH].', 'nucl-ex-0511054-1-9-8': 'The experiment ran for about one week, with beam currents of [MATH] nA.', 'nucl-ex-0511054-1-10-0': 'The experimental extraction procedure and the assumptions made are described in Refs. [CITATION].', 'nucl-ex-0511054-1-10-1': 'The data analysis is based on three main steps:', 'nucl-ex-0511054-1-11-0': 'In the first step, for each particle energy bin, total spectra of the [MATH]-ray cascades are obtained from the coincidence measurement.', 'nucl-ex-0511054-1-11-1': 'The particle energy measured in the telescopes is transformed to excitation energy of the residual nucleus, using the reaction kinematics.', 'nucl-ex-0511054-1-11-2': 'Then each row of the coincidence matrix corresponds to a certain excitation energy [MATH], while each column corresponds to a certain [MATH] energy [MATH].', 'nucl-ex-0511054-1-12-0': 'In the next step, the [MATH]-ray spectra are unfolded using the known response functions of the CACTUS array [CITATION].', 'nucl-ex-0511054-1-12-1': 'The third step concerns the [MATH]-ray spectra containing only the first [MATH] rays in a cascade.', 'nucl-ex-0511054-1-12-2': 'These spectra are obtained for each excitation energy bin through a subtraction procedure as described in Ref. [CITATION].', 'nucl-ex-0511054-1-12-3': 'The main assumption of this method is that the [MATH]-decay spectrum from any excitation-energy bin is independent of the method of formation, either directly by the nuclear reaction or populated by [MATH] decay from higher-lying states following the initial reaction.', 'nucl-ex-0511054-1-12-4': 'This assumption is automatically fulfilled when the same states are equally populated by the two processes, since [MATH] branching ratios are properties of the levels themselves.', 'nucl-ex-0511054-1-12-5': 'Even if different states are populated, the assumption is still valid for statistical [MATH] decay, which only depends on the [MATH]-ray energy and the number of accessible final states.', 'nucl-ex-0511054-1-13-0': 'When the first-generation matrix is properly normalized [CITATION], the entries of it are the probabilities [MATH] that a [MATH]-ray of energy [MATH] is emitted from an excitation energy [MATH].', 'nucl-ex-0511054-1-13-1': 'The probability of [MATH] decay is proportional to the product of the level density [MATH] at the final energy [MATH] and the [MATH]-ray transmission coefficient [MATH]: [EQUATION]', 'nucl-ex-0511054-1-13-2': 'This factorization is the generalized form of the Brink-Axel hypothesis[CITATION], which states that any excitation modes built on excited states have the same properties as those built on the ground state.', 'nucl-ex-0511054-1-13-3': 'This means that the [MATH]-ray transmission coefficient is independent of excitation energy and thus of the nuclear temperature of the excited states.', 'nucl-ex-0511054-1-13-4': 'There is evidence that the width of the giant dipole resonance varies with the nuclear temperature of the state on which it is built [CITATION].', 'nucl-ex-0511054-1-13-5': 'However, the temperature corresponding to the excitation energy covered in this work is rather low and changes slowly with excitation energy ([MATH] ); thus we assume a constant temperature and that the [MATH]-ray transmission coefficient does not depend on the excitation energy in the energy interval under consideration.', 'nucl-ex-0511054-1-14-0': 'The [MATH] and [MATH] functions can be determined by an iterative procedure [CITATION], where each data point of these two functions is simultaneously adjusted until a global [MATH] minimum with the experimental [MATH] matrix is reached.', 'nucl-ex-0511054-1-14-1': 'The globalized fitting to the data points determines the functional form of [MATH] and [MATH]; however, it has been shown [CITATION] that if one solution for the multiplicative functions [MATH] and [MATH] is known, one may construct an infinite number of other functions, which give identical fits to the [MATH] matrix by [EQUATION]', 'nucl-ex-0511054-1-14-2': 'Thus, the transformation parameters [MATH], [MATH] and [MATH], which correspond to the physical solution, remain to be determined.', 'nucl-ex-0511054-1-15-0': '# Level density and microcanonical entropy', 'nucl-ex-0511054-1-16-0': 'The parameters [MATH] and [MATH] can be obtained by normalizing the level density to the number of known discrete levels at low excitation energy [CITATION] and to the level density estimated from neutron-resonance spacing data at the neutron binding energy [MATH] [CITATION].', 'nucl-ex-0511054-1-16-1': 'The procedure for extracting the total level density [MATH] from the resonance energy spacing [MATH] is described in Ref. [CITATION].', 'nucl-ex-0511054-1-16-2': 'Since our experimental level density data points only reach up to an excitation energy of [MATH] MeV, we extrapolate with the back-shifted Fermi gas model with a global parametrization [CITATION] [EQUATION] where a constant attenuation coefficient [MATH] is introduced to adjust [MATH] to the experimental level density at [MATH].', 'nucl-ex-0511054-1-16-3': 'The intrinsic excitation energy is estimated by [MATH], where [MATH] MeV is the back-shift parameter and [MATH] is the mass number.', 'nucl-ex-0511054-1-16-4': 'The pairing energy [MATH] is based on pairing gap parameters [MATH] and [MATH] evaluated from even-odd mass differences [CITATION] according to [CITATION].', 'nucl-ex-0511054-1-16-5': 'The level-density parameter [MATH] and the spin-cutoff parameter [MATH] are given by [MATH] and [MATH], respectively.', 'nucl-ex-0511054-1-16-6': 'The nuclear temperature [MATH] is described by [MATH].', 'nucl-ex-0511054-1-16-7': 'The parameters used for [MATH]V in Eq. ([REF]) are listed in Table [REF].', 'nucl-ex-0511054-1-17-0': 'Unfortunately, [MATH]V is unstable and no information exists on the level density at [MATH] for [MATH]V. Therefore, we estimate the value from the systematics of other nuclei in the same mass region.', 'nucl-ex-0511054-1-17-1': 'In order to bring these data on the same footing, we plot the level densities as a function of intrinsic energy [MATH].', 'nucl-ex-0511054-1-17-2': 'Due to the strongly scattered data of Fig. [REF], the estimate is rather uncertain.', 'nucl-ex-0511054-1-17-3': 'We choose a rough estimate of [MATH] MeV[MATH]for [MATH]V.', 'nucl-ex-0511054-1-17-4': 'This value gives an attenuation [MATH] 0.46, which is in good agreement with the obtained value of [MATH] = 0.51 for the [MATH]V nucleus.', 'nucl-ex-0511054-1-17-5': 'Figure [REF] demonstrates the level density normalization procedure for the [MATH]V case, i.e., how the parameters [MATH] and [MATH] of Eq. ([REF]) are determined to obtain a level-density function consistent with known experimental data.', 'nucl-ex-0511054-1-18-0': 'The experimentally extracted and normalized level densities of [MATH]V and [MATH]V are shown in Fig. [REF] for excitation energies up to [MATH] 8 and 9 MeV, respectively.', 'nucl-ex-0511054-1-18-1': 'The level density of [MATH]V is relatively high and has a rather smooth behaviour due to the effect of the unpaired proton and neutron, while the level density of [MATH]V displays distinct structures for excitation energies up to [MATH] 4.5 MeV.', 'nucl-ex-0511054-1-18-2': 'This effect is probably caused by the closed [MATH] neutron shell in this nucleus.', 'nucl-ex-0511054-1-19-0': 'The level densities of [MATH]V obtained with the Oslo method are compared to the number of levels from spectroscopic experiments [CITATION].', 'nucl-ex-0511054-1-19-1': 'The [MATH]V nucleus has relatively few levels per energy bin because of its closed neutron shell, so using spectroscopic methods to count the levels seems to be reliable up to [MATH] 4 MeV excitation energy in this case.', 'nucl-ex-0511054-1-19-2': 'For higher excitations the spectroscopic data are significantly lower compared to the level density obtained with the Oslo method.', 'nucl-ex-0511054-1-19-3': 'This means that many levels are not accounted for in this excitation region by using standard methods.', 'nucl-ex-0511054-1-19-4': 'The same can be concluded for [MATH]V, and in this case the spectroscopic level density drops off already at an excitation energy of about 2.5 MeV.', 'nucl-ex-0511054-1-20-0': 'In the lower panels of Fig. [REF], the level densities of [MATH]V are also compared to the constant-temperature formula [EQUATION] which is drawn as a solid line in the upper panels of Fig. [REF].', 'nucl-ex-0511054-1-20-1': 'Here the parameters [MATH] and [MATH] are the level density at about zero excitation energy and the average temperature, respectively; both are estimated from the fit of the exponential to the region of the experimental level density indicated by arrows.', 'nucl-ex-0511054-1-20-2': 'From this model a constant temperature of about 1.3 MeV is found for both nuclei.', 'nucl-ex-0511054-1-21-0': 'The level density of a system can give detailed insight into its thermal properties.', 'nucl-ex-0511054-1-21-1': 'The multiplicity of states [MATH], which is the number of physically obtainable realizations available at a given energy, is directly proportional to the level density and a spin-dependent factor [MATH], thus [EQUATION] where [MATH] is the average spin at excitation energy [MATH].', 'nucl-ex-0511054-1-21-2': 'Unfortunately, the experimentally measured level density in this work does not correspond to the true multiplicity of states, since the [MATH] degeneracy of magnetic substates is not included.', 'nucl-ex-0511054-1-21-3': 'If the average spin of levels [MATH] at any excitation energy were known, this problem could be solved by multiplying an energy-dependent factor [MATH] to the experimental level density.', 'nucl-ex-0511054-1-21-4': 'However, little experimental data exist on the spin distribution.', 'nucl-ex-0511054-1-21-5': 'Therefore, we choose in this work to use a multiplicity [MATH] based on the experimental level density alone: [EQUATION]', 'nucl-ex-0511054-1-21-6': 'The entropy [MATH] is a measure of the degree of disorder of a system at a specific energy.', 'nucl-ex-0511054-1-21-7': 'The microcanonical ensemble, where the system is completely isolated from any exchange with its surroundings, is often considered as the appropriate one for the atomic nucleus since the strong force has such a short range, and the nucleus normally does not share its excitation energy with the external environment.', 'nucl-ex-0511054-1-22-0': 'According to our definition of the multiplicity of levels [MATH] obtained from the experimental level density, we define a "pseudo" entropy [EQUATION] which is utilized in the following discussion.', 'nucl-ex-0511054-1-22-1': "For convenience Boltzmann's constant [MATH] can be set to unity.", 'nucl-ex-0511054-1-23-0': 'In order to normalize the entropy the multiplicity is written as [MATH].', 'nucl-ex-0511054-1-23-1': 'The normalization denominator [MATH] is to be adjusted such that the entropy approaches a constant value when the temperature approaches zero in order to fullfill the third law of thermodynamics: [MATH].', 'nucl-ex-0511054-1-23-2': 'In the case of even-even nuclei the ground state represents a completely ordered system with only one possible configuration.', 'nucl-ex-0511054-1-23-3': 'This means that the entropy in the ground state is [MATH], and the normalization factor [MATH] is chosen such that this is the case.', 'nucl-ex-0511054-1-23-4': 'Since the vanadium nuclei have an odd number of protons, a [MATH] which is typical for even-even nuclei in this mass region is used for both the [MATH]V and the [MATH]V nucleus.', 'nucl-ex-0511054-1-23-5': 'The normalization factor [MATH] used is 0.7 MeV[MATH], found from averaging data on [MATH]Ti and [MATH]Cr.', 'nucl-ex-0511054-1-24-0': 'The entropies of [MATH]V extracted from the experimental level density are shown in the upper panel of Fig. [REF].', 'nucl-ex-0511054-1-24-1': 'Naturally, they show the same features as the level density plot, with the odd-odd [MATH]V displaying higher entropy than the odd-even [MATH]V.', 'nucl-ex-0511054-1-24-2': 'Since the neutrons are almost ([MATH]V) or totally ([MATH]V) blocked at low excitation energy, the multiplicity and thus the entropy is made primarily by the protons in this region.', 'nucl-ex-0511054-1-25-0': 'At 4 MeV of excitation energy a relatively large increase of the entropy is found in the case of [MATH]V.', 'nucl-ex-0511054-1-25-1': 'This is probably because the excitation energy is large enough to excite a nucleon over the [MATH] shell gap to other orbitals.', 'nucl-ex-0511054-1-26-0': 'In the excitation region above [MATH] 4.5 MeV the entropies show similar behaviour, which is also expressed by the entropy difference [MATH] displayed in the lower panel of Fig. [REF].', 'nucl-ex-0511054-1-26-1': 'We assume here that the two systems have an approximately statistical behaviour, and that the neutron hole in [MATH]V acts as a spectator to the [MATH]V core.', 'nucl-ex-0511054-1-26-2': 'The entropy of the hole can be estimated from the entropy difference [MATH]V)[MATH]V).', 'nucl-ex-0511054-1-26-3': 'From the lower panel of Fig. [REF] we find [MATH] for [MATH] MeV.', 'nucl-ex-0511054-1-26-4': 'This is slightly less than the quasi-particle entropy found in rare-earth nuclei, which is estimated to be [MATH] [CITATION].', 'nucl-ex-0511054-1-26-5': 'This is not unexpected since the single-particle levels are more closely spaced for these nuclei; they have therefore more entropy.', 'nucl-ex-0511054-1-27-0': 'The naive configurations of [MATH]V at low excitations are [MATH] and [MATH], respectively.', 'nucl-ex-0511054-1-27-1': 'Thus, by counting the possible configurations within the framework of the BCS model [CITATION] in the nearly degenerate [MATH] shell, one can estimate the multiplicity of levels and thus the entropy when no Cooper pairs are broken in the nucleus, one pair is broken and so on.', 'nucl-ex-0511054-1-27-2': 'We assume a small deformation that gives four energy levels with Nilsson quantum number [MATH].', 'nucl-ex-0511054-1-27-3': 'Furthermore, we neglect the proton-neutron coupling and hence assume that the protons and the neutrons can be considered as two separate systems, the total entropy based on the number of energy levels can be written as [MATH].', 'nucl-ex-0511054-1-27-4': 'This gives [MATH] for the nucleus [MATH]V, and [MATH] for [MATH]V when two protons are coupled in a Cooper pair.', 'nucl-ex-0511054-1-27-5': 'These values are in fair agreement with the data of Fig. [REF] at an excitation energy below [MATH] MeV.', 'nucl-ex-0511054-1-27-6': 'It is gratifying that these crude estimates give an entropy of the neutron hole in [MATH]V of [MATH], in good agreement with the experimental value for the entropy difference of [MATH] found from Fig. [REF].', 'nucl-ex-0511054-1-28-0': 'With the three [MATH] protons unpaired we obtain a total entropy of [MATH] and [MATH] for [MATH]V, respectively.', 'nucl-ex-0511054-1-28-1': 'This means that the process of just breaking a proton pair within the same shell does not contribute much to the total entropy, but when a nucleon has enough energy to cross the shell gap a significant increase of the entropy is expected.', 'nucl-ex-0511054-1-28-2': 'As already mentioned, at excitation energies above [MATH] 4 MeV, it is very likely that configurations from other shells will participate in building the total entropy.', 'nucl-ex-0511054-1-29-0': '# Radiative strength functions', 'nucl-ex-0511054-1-30-0': 'The [MATH]-ray transmission coefficient [MATH] in Eq. ([REF]) is expressed as a sum of all the RSFs [MATH] of electromagnetic character [MATH] and multipolarity [MATH]: [EQUATION]', 'nucl-ex-0511054-1-30-1': 'The slope of the experimental [MATH]-ray transmission coefficient [MATH] has been determined through the normalization of the level densities, as described in Sect. III.', 'nucl-ex-0511054-1-30-2': 'The remaining constant [MATH] in Eq. ([REF]) is determined using information from neutron resonance decay, which gives the absolute normalization of [MATH].', 'nucl-ex-0511054-1-30-3': 'For this purpose we utilize experimental data [CITATION] on the average total radiative width [MATH] at [MATH].', 'nucl-ex-0511054-1-31-0': 'We assume here that the [MATH]-decay taking place in the quasi-continuum is dominated by [MATH] and [MATH] transitions and that the number of positive and negative parity states is equal.', 'nucl-ex-0511054-1-31-1': 'For initial spin [MATH] and parity [MATH] at [MATH], the expression of the width [CITATION] reduces to [EQUATION] where [MATH] is the average spacing of [MATH]-wave neutron resonances.', 'nucl-ex-0511054-1-31-2': 'The summation and integration run over all final levels with spin [MATH], which are accessible by dipole ([MATH] radiation with energy [MATH].', 'nucl-ex-0511054-1-31-3': 'From this expression the normalization constant [MATH] can be determined as described in Ref. [CITATION].', 'nucl-ex-0511054-1-31-4': 'However, some considerations have to be made before normalizing according to Eq. ([REF]).', 'nucl-ex-0511054-1-32-0': 'Methodical difficulties in the primary [MATH]-ray extraction prevent determination of the function [MATH] in the interval [MATH] MeV.', 'nucl-ex-0511054-1-32-1': 'In addition, the data at the highest [MATH]-energies, above [MATH] MeV, suffer from poor statistics.', 'nucl-ex-0511054-1-32-2': 'We therefore extrapolate [MATH] with an exponential form, as demonstrated for [MATH]V in Fig. [REF].', 'nucl-ex-0511054-1-32-3': 'The contribution of the extrapolation to the total radiative width given by Eq. ([REF]) does not exceed [MATH]%, thus the errors due to a possibly poor extrapolation are expected to be of minor importance [CITATION].', 'nucl-ex-0511054-1-33-0': 'Again, difficulties occur when normalizing the [MATH]-ray transmission coefficient in the case of [MATH]V due to the lack of neutron resonance data.', 'nucl-ex-0511054-1-33-1': 'Since the average total radiative width [MATH] at [MATH] does not seem to show any clear systematics for nuclei in this mass region, we choose the same absolute value of the GEDR tail for [MATH]V as the one found for [MATH]V from photoabsorption experiments.', 'nucl-ex-0511054-1-33-2': 'The argument for this choice is that the GEDR should be similar for equal number of protons provided that the two nuclei have the same shapes.', 'nucl-ex-0511054-1-34-0': 'Since it is assumed that the radiative strength is dominated by dipole transitions, the RSF can be calculated from the normalized transmission coefficient by [EQUATION]', 'nucl-ex-0511054-1-34-1': 'We would now like to decompose the RSF into its components from different multipolarities to investigate how the [MATH] and [MATH] radiation contribute to the total strength.', 'nucl-ex-0511054-1-35-0': 'The Kadmenskii, Markushev and Furman (KMF) model [CITATION] is employed for the [MATH] strength.', 'nucl-ex-0511054-1-35-1': 'In this model, the Lorentzian GEDR is modified in order to reproduce the nonzero limit of the GEDR for [MATH] by means of a temperature-dependent width of the GEDR.', 'nucl-ex-0511054-1-35-2': 'The [MATH] strength in the KMF model is given by [EQUATION] where [MATH] is the cross section, [MATH] is the width, and [MATH] is the centroid of the GEDR determined from photoabsorbtion experiments.', 'nucl-ex-0511054-1-36-0': 'We adopt the KMF model with the temperature [MATH] taken as a constant to be consistent with our assumption that the RSF is independent of excitation energy.', 'nucl-ex-0511054-1-36-1': 'The possible systematic uncertainty caused by this assumption is believed to have a maximum effect of 20 on the RSF [CITATION].', 'nucl-ex-0511054-1-36-2': 'The values used for [MATH] are the ones extracted from the constant-temperature model in Eq.( [REF]).', 'nucl-ex-0511054-1-37-0': 'The GEDR is split into two parts for deformed nuclei.', 'nucl-ex-0511054-1-37-1': 'Data of [MATH]V from photoabsorption experiments show that the GEDR is best fitted with two Lorentzians, indicating a splitting of the resonance and a non-zero ground-state deformation of this nucleus.', 'nucl-ex-0511054-1-37-2': 'Indeed, [MATH]) values [CITATION] suggest a deformation of [MATH] 0.1 for [MATH]V. Therefore, a sum of two modified Lorentzians each described by Eq. ([REF]) is used (see Table [REF]).', 'nucl-ex-0511054-1-38-0': 'For [MATH], which is supposed to be governed by the spin-flip M1 resonance [CITATION], the Lorentzian giant magnetic dipole resonance (GMDR) [EQUATION] is adopted.', 'nucl-ex-0511054-1-39-0': 'The GEDR and GMDR parameters are taken from the systematics of Ref. [CITATION] and are listed in Table [REF].', 'nucl-ex-0511054-1-39-1': 'Thus, we fit the total RSF given by [EQUATION] to the experimental data using the normalization constant [MATH] as a free parameter.', 'nucl-ex-0511054-1-39-2': 'The value of [MATH] generally deviates from unity due to theoretical uncertainties in the KMF model and the evaluation of the absolute normalization in Eq. ([REF]).', 'nucl-ex-0511054-1-39-3': 'The resulting RSFs extracted from the two reactions are displayed in Fig. [REF], where the data have been normalized with parameters from Tables [REF] and [REF].', 'nucl-ex-0511054-1-40-0': 'The [MATH]-decay probability is governed by the number and the character of available final states and by the RSF.', 'nucl-ex-0511054-1-40-1': 'A rough inspection of the experimental data of Fig. [REF] indicates that the RSFs are increasing functions of [MATH]-energy, generally following the tails of the GEDR and GMDR resonances in this region.', 'nucl-ex-0511054-1-41-0': 'At low [MATH] energies ([MATH] MeV) an enhancement of a factor of [MATH] 5 over the KMF estimate of the strength is shown in the RSFs.', 'nucl-ex-0511054-1-41-1': 'This feature has also been seen in the RSFs of some Fe [CITATION] and Mo [CITATION] isotopes, and has been shown to be present in the whole excitation-energy region.', 'nucl-ex-0511054-1-41-2': 'The physical origin of the enhancement has not, at present, any satisfying explanation, as none of the known theoretical models can account for this behaviour.', 'nucl-ex-0511054-1-42-0': '# Summary and conclusions', 'nucl-ex-0511054-1-43-0': 'The Oslo method has been applied to extract level densities and RSFs of the vanadium isotopes [MATH]V. From the measured level densities the microcanonical entropies have been derived.', 'nucl-ex-0511054-1-43-1': 'The entropy carried by the neutron hole in [MATH]V is estimated to be [MATH] 1.2 [MATH], which is less than the quasi-particle entropy of [MATH] 1.7 [MATH] found in rare-earth nuclei.', 'nucl-ex-0511054-1-44-0': 'The experimental RSFs are generally increasing functions of [MATH] energy.', 'nucl-ex-0511054-1-44-1': 'The main contribution to the RSFs is the GEDR; also the GMDR is present.', 'nucl-ex-0511054-1-44-2': 'At low [MATH] energies an increase of the strength functions is apparent.', 'nucl-ex-0511054-1-44-3': 'A similar enhancement has also been seen in the iron and molybdenum isotopes.', 'nucl-ex-0511054-1-44-4': 'There is still no explanation for the physics behind this very interesting behaviour.', 'nucl-ex-0511054-1-45-0': 'Financial support from the Norwegian Research Council (NFR) is gratefully acknowledged.', 'nucl-ex-0511054-1-45-1': 'A.V. acknowledges support from a NATO Science Fellowship under project number 150027/432 given by the Norwegian Research Council (NFR).', 'nucl-ex-0511054-1-46-0': 'Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and T.S. Tveter, Nucl.', 'nucl-ex-0511054-1-46-1': 'Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad, and S. Siem, Nucl.', 'nucl-ex-0511054-1-46-2': 'Melby, L. Bergholt, M. Guttormsen, M. Hjorth-Jensen, F. Ingebretsen, S. Messelt, J. Rekstad, A. Schiller, S. Siem, and S.W. Odegaard, Phys.', 'nucl-ex-0511054-1-46-3': 'Schiller, A. Bjerve, M. Guttormsen, M. Hjorth-Jensen, F. Ingebretsen, E. Melby, S. Messelt, J. Rekstad, S. Siem, and S.W. Odegaard, Phys.', 'nucl-ex-0511054-1-46-4': 'Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Phys.', 'nucl-ex-0511054-1-46-5': 'Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Phys.', 'nucl-ex-0511054-1-46-6': 'Siem, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and A. Voinov, Phys.', 'nucl-ex-0511054-1-46-7': 'Guttormsen, A. Atac, G. Lovhoiden, S. Messelt, T. Ramsoy, J. Rekstad, T.F. Thorsteinsen, T.S. Tveter, and Z. Zelazny, Phys.', 'nucl-ex-0511054-1-46-8': 'Guttormsen, T.S. Tveter, L. Bergholt, F. Ingebretsen, and J. Rekstad, Nucl.', 'nucl-ex-0511054-1-46-9': 'Guttormsen, T. Ramsoy, and J. Rekstad, Nucl.', 'nucl-ex-0511054-1-46-10': 'Brink, Ph.D. thesis, Oxford University, 1955.', 'nucl-ex-0511054-1-46-11': 'kopecky J. Kopecky and M. Uhl, Phys.', 'nucl-ex-0511054-1-46-12': 'kad S.G. Kadmenskii, V.P. Markushev, and V.I. Furman, Yad.', 'nucl-ex-0511054-1-46-13': 'Gervais, M. Thoennessen, and W.E. Ormand, Phys.', 'nucl-ex-0511054-1-46-14': 'ENSDFData extracted using the NNDC On-Line Data Service from the ENSDF database.', 'nucl-ex-0511054-1-46-15': 'RIPLData extracted using the Reference Input Parameter Library, http://www-nds.iaea.org/RIPL-2/ GC A. Gilbert, A.G.W. Cameron, Can.', 'nucl-ex-0511054-1-46-16': 'egidy T. von Egidy, H.H. Schmidt, and A.N. Behkami, Nucl.', 'nucl-ex-0511054-1-46-17': 'Wapstra G. Audi and A.H. Wapstra, Nucl.', 'nucl-ex-0511054-1-46-18': 'BM A. Bohr and B. Mottelson, Nuclear Structure, (Benjamin, New York, 1969), Vol.', 'nucl-ex-0511054-1-46-19': 'Firestone and V.S. Shirley, Table of Isotopes, 8th ed.', 'nucl-ex-0511054-1-46-20': 'Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys.', 'nucl-ex-0511054-1-46-21': 'Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova, M. Guttormsen, G.E. Mitchell, J. Rekstad, A. Schiller and S. Siem, Phys.', 'nucl-ex-0511054-1-46-22': 'Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin, L.A. Bernstein, F. Ingebretsen, T. Lonnroth, S. Messelt, G.E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A.C. Sunde, A. Voinov and S. Odegaard, Phys.'}

{'nucl-ex-0511054-2-0-0': 'The level densities and radiative strength functions (RSFs) of [MATH]V have been extracted using the ([MATH]He,[MATH]) and ([MATH]He,[MATH]He[MATH]) reactions, respectively.', 'nucl-ex-0511054-2-0-1': 'From the level densities, microcanonical entropies are deduced.', 'nucl-ex-0511054-2-0-2': 'The high [MATH]-energy part of the measured RSF fits well with the tail of the giant electric dipole resonance.', 'nucl-ex-0511054-2-0-3': 'A significant enhancement over the predicted strength in the region of [MATH] MeV is seen, which at present has no theoretical explanation.', 'nucl-ex-0511054-2-1-0': '2', 'nucl-ex-0511054-2-2-0': '# Introduction', 'nucl-ex-0511054-2-3-0': 'The structure of the vanadium isotopes is based on simple shell-model configurations at low excitation energies.', 'nucl-ex-0511054-2-3-1': 'The valence protons and neutrons are occupying the single-particle [MATH] and [MATH] orbitals, respectively.', 'nucl-ex-0511054-2-3-2': 'These shells are isolated from other orbitals by the N, Z = 20 and 28 shell gaps, making the vanadium isotopes interesting objects for studying various nuclear shell effects.', 'nucl-ex-0511054-2-3-3': 'In particular, it is well known that the number of available singe-particle levels is significantly reduced for nuclei at closed shells.', 'nucl-ex-0511054-2-4-0': 'The density of states or, equivalently, the entropy in these systems depends on the number of broken Cooper pairs and single-particle orbitals made available by crossing the shell gaps.', 'nucl-ex-0511054-2-4-1': 'The [MATH]V nuclei are of special interest because the neutrons are strongly blocked in the process of creating entropy; [MATH]V and [MATH]V have seven and eight neutrons in the [MATH] orbital, respectively.', 'nucl-ex-0511054-2-4-2': 'Thus, the configuration space of the three protons in the [MATH] shell is of great importance.', 'nucl-ex-0511054-2-5-0': 'These particular shell-model configurations are also expected to govern the [MATH]-decay routes.', 'nucl-ex-0511054-2-5-1': 'Specifically, as within every major shell, the presence of only one parity for single-particle orbitals in the low-spin domain means that transitions of [MATH] type will be suppressed.', 'nucl-ex-0511054-2-5-2': 'The low mass of the investigated nuclei causes that the centroid of the giant electric dipole resonance (GEDR) is relatively high while the integrated strength according to the Thomas-Reiche-Kuhn sum rule is low, both observations working together to produce a relatively weak low-energy tail when compared to heavier nuclei.', 'nucl-ex-0511054-2-5-3': 'Hence, possible non-statistical effects in the radiative strength function (RSF) might stand out more in the present investigation.', 'nucl-ex-0511054-2-6-0': 'The Oslo Cyclotron group has developed a method to extract first-generation (primary) [MATH]-ray spectra at various initial excitation energies.', 'nucl-ex-0511054-2-6-1': 'From such a set of primary spectra, the nuclear level density and the RSF can be extracted simultaneously [CITATION].', 'nucl-ex-0511054-2-6-2': 'These two quantities reveal essential information on nuclear structure such as pair correlations and thermal and electromagnetic properties.', 'nucl-ex-0511054-2-6-3': 'In the last five years, the Oslo group has demonstrated several fruitful applications of the method [CITATION].', 'nucl-ex-0511054-2-7-0': 'In Sect. II an outline of the experimental procedure is given.', 'nucl-ex-0511054-2-7-1': 'The level densities and microcanonical entropies are discussed in Sect. III, and in Sect. IV the RSFs are presented.', 'nucl-ex-0511054-2-7-2': 'Finally, concluding remarks are given in Sect. V.', 'nucl-ex-0511054-2-8-0': '# Experimental method', 'nucl-ex-0511054-2-9-0': 'The experiment was carried out at the Oslo Cyclotron Laboratory (OCL) using a beam of 30-MeV [MATH]He ions.', 'nucl-ex-0511054-2-9-1': 'The self-supporting natural V target had a purity of [MATH]% and a thickness of 2.3 mg/cm[MATH].', 'nucl-ex-0511054-2-9-2': 'Particle-[MATH] coincidences for [MATH]V were measured with the CACTUS multi-detector array [CITATION].', 'nucl-ex-0511054-2-9-3': 'The charged ejectiles were detected using eight Si particle telescopes placed at an angle of 45[MATH] relative to the beam direction.', 'nucl-ex-0511054-2-9-4': 'Each telescope consists of a front [MATH] detector and a back [MATH] detector with thicknesses of 140 and 1500 [MATH]m, respectively.', 'nucl-ex-0511054-2-9-5': 'An array of 28 collimated NaI [MATH]-ray detectors with a total efficiency of [MATH]15% surrounded the target and the particle detectors.', 'nucl-ex-0511054-2-9-6': 'The reactions of interest were the pick-up reaction [MATH]V([MATH]He,[MATH]V, and the inelastic scattering [MATH]V([MATH]He,[MATH]He[MATH]V.', 'nucl-ex-0511054-2-9-7': 'The typical spin range is expected to be [MATH].', 'nucl-ex-0511054-2-9-8': 'The experiment ran for about one week, with beam currents of [MATH] nA.', 'nucl-ex-0511054-2-10-0': 'The experimental extraction procedure and the assumptions made are described in Refs. [CITATION].', 'nucl-ex-0511054-2-10-1': 'The data analysis is based on three main steps:', 'nucl-ex-0511054-2-11-0': 'In the first step, for each particle-energy bin, total spectra of the [MATH]-ray cascades are obtained from the coincidence measurement.', 'nucl-ex-0511054-2-11-1': 'The particle energy measured in the telescopes is transformed to excitation energy of the residual nucleus, using the reaction kinematics.', 'nucl-ex-0511054-2-11-2': 'Then each row of the coincidence matrix corresponds to a certain excitation energy [MATH], while each column corresponds to a certain [MATH] energy [MATH].', 'nucl-ex-0511054-2-12-0': 'In the next step, the [MATH]-ray spectra are unfolded using the known response functions of the CACTUS array [CITATION].', 'nucl-ex-0511054-2-12-1': 'The Compton-subtraction method described in Ref. [CITATION] preserves the fluctuations in the original spectra without introducing further, spurious fluctuations.', 'nucl-ex-0511054-2-12-2': 'A typical raw [MATH] spectrum is shown in the top panel of Fig. [REF], taken from the [MATH]V coincidence matrix gating on the excitation energies between [MATH] MeV.', 'nucl-ex-0511054-2-12-3': 'The middle panel shows the unfolded spectrum, and in the bottom panel this spectrum has been folded with the response functions.', 'nucl-ex-0511054-2-12-4': 'The top and bottom panels are in excellent agreement, indicating that the unfolding method works very well.', 'nucl-ex-0511054-2-13-0': 'The third step is to extract the [MATH]-ray spectra containing only the first [MATH] rays in a cascade.', 'nucl-ex-0511054-2-13-1': 'These spectra are obtained for each excitation-energy bin through a subtraction procedure as described in Ref. [CITATION].', 'nucl-ex-0511054-2-13-2': 'The main assumption of this method is that the [MATH]-decay spectrum from any excitation-energy bin is independent of the method of formation, either directly by the nuclear reaction or populated by [MATH] decay from higher-lying states following the initial reaction.', 'nucl-ex-0511054-2-13-3': 'This assumption is automatically fulfilled when the same states are equally populated by the two processes, since [MATH] branching ratios are properties of the levels themselves.', 'nucl-ex-0511054-2-13-4': 'Even if different states are populated, the assumption is still valid for statistical [MATH] decay, which only depends on the [MATH]-ray energy and the number of accessible final states.', 'nucl-ex-0511054-2-13-5': 'In Fig. [REF], the total, unfolded [MATH] spectrum, the second and higher generations [MATH] spectrum and the first-generation spectrum of [MATH]V with excitation-energy gates [MATH] MeV are shown.', 'nucl-ex-0511054-2-13-6': 'The first-generation spectrum is obtained by subtracting the higher-generation [MATH] rays from the total [MATH] spectrum.', 'nucl-ex-0511054-2-14-0': 'When the first-generation matrix is properly normalized [CITATION], the entries of it are the probabilities [MATH] that a [MATH]-ray of energy [MATH] is emitted from an excitation energy [MATH].', 'nucl-ex-0511054-2-14-1': 'The probability of [MATH] decay is proportional to the product of the level density [MATH] at the final energy [MATH] and the [MATH]-ray transmission coefficient [MATH]: [EQUATION]', 'nucl-ex-0511054-2-14-2': 'This factorization is the generalized form of the Brink-Axel hypothesis[CITATION], which states that any excitation modes built on excited states have the same properties as those built on the ground state.', 'nucl-ex-0511054-2-14-3': 'This means that the [MATH]-ray transmission coefficient is independent of excitation energy and thus of the nuclear temperature of the excited states.', 'nucl-ex-0511054-2-14-4': 'There is evidence that the width of the giant dipole resonance varies with the nuclear temperature of the state on which it is built [CITATION].', 'nucl-ex-0511054-2-14-5': 'However, the temperature corresponding to the excitation-energy range covered in this work is rather low and changes slowly with excitation energy ([MATH] ); thus we assume a constant temperature and that the [MATH]-ray transmission coefficient does not depend on the excitation energy in the energy interval under consideration.', 'nucl-ex-0511054-2-15-0': 'The [MATH] and [MATH] functions can be determined by an iterative procedure [CITATION], where each data point of these two functions is simultaneously adjusted until a global [MATH] minimum with the experimental [MATH] matrix is reached.', 'nucl-ex-0511054-2-15-1': 'No [MATH] assumptions about the functional form of either the level density or the [MATH]-ray transmission coefficient are used.', 'nucl-ex-0511054-2-15-2': 'An example to illustrate the quality of the fit is shown in Fig. [REF], where we compare for the [MATH]V([MATH]He,[MATH]V reaction the experimental first-generation spectra to the least-[MATH] solution for six different initial excitation energies.', 'nucl-ex-0511054-2-16-0': 'The globalized fitting to the data points determines the functional form of [MATH] and [MATH]; however, it has been shown [CITATION] that if one solution for the multiplicative functions [MATH] and [MATH] is known, one may construct an infinite number of other functions, which give identical fits to the [MATH] matrix by [EQUATION]', 'nucl-ex-0511054-2-16-1': 'Thus, the transformation parameters [MATH], [MATH] and [MATH], which correspond to the physical solution, remain to be determined.', 'nucl-ex-0511054-2-17-0': '# Level density and microcanonical entropy', 'nucl-ex-0511054-2-18-0': 'The parameters [MATH] and [MATH] can be obtained by normalizing the level density to the number of known discrete levels at low excitation energy [CITATION] and to the level density estimated from neutron-resonance spacing data at the neutron binding energy [MATH] [CITATION].', 'nucl-ex-0511054-2-18-1': 'The procedure for extracting the total level density [MATH] from the resonance energy spacing [MATH] is described in Ref. [CITATION].', 'nucl-ex-0511054-2-18-2': 'Since our experimental level-density data points only reach up to an excitation energy of [MATH] MeV, we extrapolate with the back-shifted Fermi-gas model with a global parametrization [CITATION] [EQUATION] where a constant attenuation coefficient [MATH] is introduced to adjust [MATH] to the experimental level density at [MATH].', 'nucl-ex-0511054-2-18-3': 'The intrinsic excitation energy is estimated by [MATH], where [MATH] MeV is the back-shift parameter and [MATH] is the mass number.', 'nucl-ex-0511054-2-18-4': 'The pairing energy [MATH] is based on pairing gap parameters [MATH] and [MATH] evaluated from even-odd mass differences [CITATION] according to [CITATION].', 'nucl-ex-0511054-2-18-5': 'The level-density parameter [MATH] and the spin-cutoff parameter [MATH] are given by [MATH] and [MATH], respectively.', 'nucl-ex-0511054-2-18-6': 'The nuclear temperature [MATH] is described by [MATH].', 'nucl-ex-0511054-2-18-7': 'The parameters used for [MATH]V in Eq. ([REF]) are listed in Table [REF].', 'nucl-ex-0511054-2-19-0': 'Unfortunately, [MATH]V is unstable and no information exists on the level density at [MATH] for [MATH]V. Therefore, we estimate the value from the systematics of other nuclei in the same mass region.', 'nucl-ex-0511054-2-19-1': 'In order to bring these data on the same footing, we plot the level densities as a function of intrinsic energy [MATH].', 'nucl-ex-0511054-2-19-2': 'Due to the strongly scattered data of Fig. [REF], the estimate is rather uncertain.', 'nucl-ex-0511054-2-19-3': 'We choose a rough estimate of [MATH] MeV[MATH]for [MATH]V.', 'nucl-ex-0511054-2-19-4': 'This value gives an attenuation [MATH] 0.46, which is in good agreement with the obtained value of [MATH] = 0.51 for the [MATH]V nucleus.', 'nucl-ex-0511054-2-19-5': 'Figure [REF] demonstrates the level density normalization procedure for the [MATH]V case, i.e., how the parameters [MATH] and [MATH] of Eq. ([REF]) are determined to obtain a level-density function consistent with known experimental data.', 'nucl-ex-0511054-2-20-0': 'The experimentally extracted and normalized level densities of [MATH]V and [MATH]V are shown in Fig. [REF] for excitation energies up to [MATH] 8 and 9 MeV, respectively.', 'nucl-ex-0511054-2-20-1': 'The level density of [MATH]V is relatively high and has a rather smooth behaviour due to the effect of the unpaired proton and neutron, while the level density of [MATH]V displays distinct structures for excitation energies up to [MATH] 4.5 MeV.', 'nucl-ex-0511054-2-20-2': 'This effect is probably caused by the closed [MATH] neutron shell in this nucleus.', 'nucl-ex-0511054-2-21-0': 'The level densities of [MATH]V obtained with the Oslo method are compared to the number of levels from spectroscopic experiments [CITATION].', 'nucl-ex-0511054-2-21-1': 'The [MATH]V nucleus has relatively few levels per energy bin because of its closed neutron shell, so using spectroscopic methods to count the levels seems to be reliable up to [MATH] 4 MeV excitation energy in this case.', 'nucl-ex-0511054-2-21-2': 'For higher excitations the spectroscopic data are significantly lower compared to the level density obtained with the Oslo method.', 'nucl-ex-0511054-2-21-3': 'This means that many levels are not accounted for in this excitation region by using standard methods.', 'nucl-ex-0511054-2-21-4': 'The same can be concluded for [MATH]V, and in this case the spectroscopic level density drops off already at an excitation energy of about 2.5 MeV.', 'nucl-ex-0511054-2-22-0': 'The level densities of [MATH]V are also compared to the constant-temperature formula [EQUATION] which is drawn as a solid line in Fig. [REF].', 'nucl-ex-0511054-2-22-1': 'Here the parameters [MATH] and [MATH] are the level density at about zero excitation energy and the average temperature, respectively; both are estimated from the fit of the exponential to the region of the experimental level density indicated by arrows.', 'nucl-ex-0511054-2-22-2': 'From this model a constant temperature of about 1.3 MeV is found for both nuclei.', 'nucl-ex-0511054-2-23-0': 'The level density of a system can give detailed insight into its thermal properties.', 'nucl-ex-0511054-2-23-1': 'The multiplicity of states [MATH], which is the number of physically obtainable realizations available at a given energy, is directly proportional to the level density and a spin-dependent factor [MATH], thus [EQUATION] where [MATH] is the average spin at excitation energy [MATH].', 'nucl-ex-0511054-2-23-2': 'Unfortunately, the experimentally measured level density in this work does not correspond to the true multiplicity of states, since the [MATH] degeneracy of magnetic substates is not included.', 'nucl-ex-0511054-2-23-3': 'If the average spin of levels [MATH] at any excitation energy were known, this problem could be solved by multiplying an energy-dependent factor [MATH] to the experimental level density.', 'nucl-ex-0511054-2-23-4': 'However, little experimental data exist on the spin distribution.', 'nucl-ex-0511054-2-23-5': 'Therefore, we choose in this work to use a multiplicity [MATH] based on the experimental level density alone: [EQUATION]', 'nucl-ex-0511054-2-23-6': 'The entropy [MATH] is a measure of the degree of disorder of a system at a specific energy.', 'nucl-ex-0511054-2-23-7': 'The microcanonical ensemble, where the system is completely isolated from any exchange with its surroundings, is often considered as the appropriate one for the atomic nucleus since the strong force has such a short range, and the nucleus normally does not share its excitation energy with the external environment.', 'nucl-ex-0511054-2-24-0': 'According to our definition of the multiplicity of levels [MATH] obtained from the experimental level density, we define a "pseudo" entropy [EQUATION] which is utilized in the following discussion.', 'nucl-ex-0511054-2-24-1': "For convenience Boltzmann's constant [MATH] can be set to unity.", 'nucl-ex-0511054-2-25-0': 'In order to normalize the entropy the multiplicity is written as [MATH].', 'nucl-ex-0511054-2-25-1': 'The normalization denominator [MATH] is to be adjusted such that the entropy approaches a constant value when the temperature approaches zero in order to fullfill the third law of thermodynamics: [MATH].', 'nucl-ex-0511054-2-25-2': 'In the case of even-even nuclei the ground state represents a completely ordered system with only one possible configuration.', 'nucl-ex-0511054-2-25-3': 'This means that the entropy in the ground state is [MATH], and the normalization factor [MATH] is chosen such that this is the case.', 'nucl-ex-0511054-2-25-4': 'Since the vanadium nuclei have an odd number of protons, a [MATH] which is typical for even-even nuclei in this mass region is used for both the [MATH]V and the [MATH]V nucleus.', 'nucl-ex-0511054-2-25-5': 'The normalization factor [MATH] used is 0.7 MeV[MATH], found from averaging data on [MATH]Ti and [MATH]Cr.', 'nucl-ex-0511054-2-26-0': 'The entropies of [MATH]V extracted from the experimental level density are shown in the upper panel of Fig. [REF].', 'nucl-ex-0511054-2-26-1': 'Naturally, they show the same features as the level density plot, with the odd-odd [MATH]V displaying higher entropy than the odd-even [MATH]V.', 'nucl-ex-0511054-2-26-2': 'Since the neutrons are almost ([MATH]V) or totally ([MATH]V) blocked at low excitation energy, the multiplicity and thus the entropy is made primarily by the protons in this region.', 'nucl-ex-0511054-2-27-0': 'At 4 MeV of excitation energy a relatively large increase of the entropy is found in the case of [MATH]V.', 'nucl-ex-0511054-2-27-1': 'This is probably because the excitation energy is large enough to excite a nucleon across the [MATH] shell gap to other orbitals.', 'nucl-ex-0511054-2-28-0': 'In the excitation region above [MATH] 4.5 MeV the entropies show similar behaviour, which is also expressed by the entropy difference [MATH] displayed in the lower panel of Fig. [REF].', 'nucl-ex-0511054-2-28-1': 'We assume here that the two systems have an approximately statistical behaviour, and that the neutron hole in [MATH]V acts as a spectator to the [MATH]V core.', 'nucl-ex-0511054-2-28-2': 'The entropy of the hole can be estimated from the entropy difference [MATH]V)[MATH]V).', 'nucl-ex-0511054-2-28-3': 'From the lower panel of Fig. [REF] we find [MATH] for [MATH] MeV.', 'nucl-ex-0511054-2-28-4': 'This is slightly less than the quasi-particle entropy found in rare-earth nuclei, which is estimated to be [MATH] [CITATION].', 'nucl-ex-0511054-2-28-5': 'This is not unexpected since the single-particle levels are more closely spaced for these nuclei; they have therefore more entropy.', 'nucl-ex-0511054-2-29-0': 'The naive configurations of [MATH]V at low excitations are [MATH] and [MATH], respectively.', 'nucl-ex-0511054-2-29-1': 'Thus, by counting the possible configurations within the framework of the BCS model [CITATION] in the nearly degenerate [MATH] shell, one can estimate the multiplicity of levels and thus the entropy when no Cooper pairs are broken in the nucleus, one pair is broken and so on.', 'nucl-ex-0511054-2-29-2': 'We assume a small deformation that gives four energy levels with Nilsson quantum number [MATH].', 'nucl-ex-0511054-2-29-3': 'Furthermore, we neglect the proton-neutron coupling and hence assume that the protons and the neutrons can be considered as two separate systems; the total entropy based on the number of energy levels can then be written as [MATH].', 'nucl-ex-0511054-2-29-4': 'This gives [MATH] for the nucleus [MATH]V, and [MATH] for [MATH]V when two protons are coupled in a Cooper pair.', 'nucl-ex-0511054-2-29-5': 'These values are in fair agreement with the data of Fig. [REF] at an excitation energy below [MATH] MeV.', 'nucl-ex-0511054-2-29-6': 'It is gratifying that these crude estimates give an entropy of the neutron hole in [MATH]V of [MATH], in good agreement with the experimental value for the entropy difference of [MATH] found from Fig. [REF].', 'nucl-ex-0511054-2-30-0': 'With the three [MATH] protons unpaired we obtain a total entropy of [MATH] and [MATH] for [MATH]V, respectively.', 'nucl-ex-0511054-2-30-1': 'This means that the process of just breaking a proton pair within the same shell does not contribute much to the total entropy, but when a nucleon has enough energy to cross the shell gap a significant increase of the entropy is expected.', 'nucl-ex-0511054-2-30-2': 'As already mentioned, at excitation energies above [MATH] 4 MeV, it is very likely that configurations from other shells will participate in building the total entropy.', 'nucl-ex-0511054-2-31-0': '# Radiative strength functions', 'nucl-ex-0511054-2-32-0': 'The [MATH]-ray transmission coefficient [MATH] in Eq. ([REF]) is expressed as a sum of all the RSFs [MATH] of electromagnetic character [MATH] and multipolarity [MATH]: [EQUATION]', 'nucl-ex-0511054-2-32-1': 'The slope of the experimental [MATH]-ray transmission coefficient [MATH] has been determined through the normalization of the level densities, as described in Sect. III.', 'nucl-ex-0511054-2-32-2': 'The remaining constant [MATH] in Eq. ([REF]) is determined using information from neutron resonance decay, which gives the absolute normalization of [MATH].', 'nucl-ex-0511054-2-32-3': 'For this purpose we utilize experimental data [CITATION] on the average total radiative width [MATH] at [MATH].', 'nucl-ex-0511054-2-33-0': 'We assume here that the [MATH]-decay taking place in the quasi-continuum is dominated by [MATH] and [MATH] transitions and that the number of positive and negative parity states is equal.', 'nucl-ex-0511054-2-33-1': 'For initial spin [MATH] and parity [MATH] at [MATH], the expression of the width [CITATION] reduces to [EQUATION] where [MATH] is the average spacing of [MATH]-wave neutron resonances.', 'nucl-ex-0511054-2-33-2': 'The summation and integration run over all final levels with spin [MATH], which are accessible by dipole ([MATH] radiation with energy [MATH].', 'nucl-ex-0511054-2-33-3': 'From this expression the normalization constant [MATH] can be determined as described in Ref. [CITATION].', 'nucl-ex-0511054-2-33-4': 'However, some considerations have to be made before normalizing according to Eq. ([REF]).', 'nucl-ex-0511054-2-34-0': 'Methodical difficulties in the primary [MATH]-ray extraction prevent determination of the function [MATH] in the interval [MATH] MeV.', 'nucl-ex-0511054-2-34-1': 'In addition, the data at the highest [MATH]-energies, above [MATH] MeV, suffer from poor statistics.', 'nucl-ex-0511054-2-34-2': 'We therefore extrapolate [MATH] with an exponential form, as demonstrated for [MATH]V in Fig. [REF].', 'nucl-ex-0511054-2-34-3': 'The contribution of the extrapolation to the total radiative width given by Eq. ([REF]) does not exceed [MATH]%, thus the errors due to a possibly poor extrapolation are expected to be of minor importance [CITATION].', 'nucl-ex-0511054-2-35-0': 'Again, difficulties occur when normalizing the [MATH]-ray transmission coefficient in the case of [MATH]V due to the lack of neutron resonance data.', 'nucl-ex-0511054-2-35-1': 'Since the average total radiative width [MATH] at [MATH] does not seem to show any clear systematics for nuclei in this mass region, we choose the same absolute value of the GEDR tail for [MATH]V as the one found for [MATH]V from photoabsorption experiments.', 'nucl-ex-0511054-2-35-2': 'The argument for this choice is that the GEDR should be similar for equal number of protons provided that the two nuclei have the same shapes.', 'nucl-ex-0511054-2-36-0': 'Since it is assumed that the radiative strength is dominated by dipole transitions, the RSF can be calculated from the normalized transmission coefficient by [EQUATION]', 'nucl-ex-0511054-2-36-1': 'We would now like to decompose the RSF into its components from different multipolarities to investigate how the [MATH] and [MATH] radiation contribute to the total strength.', 'nucl-ex-0511054-2-37-0': 'The Kadmenskii, Markushev and Furman (KMF) model [CITATION] is employed for the [MATH] strength.', 'nucl-ex-0511054-2-37-1': 'In this model, the Lorentzian GEDR is modified in order to reproduce the nonzero limit of the GEDR for [MATH] by means of a temperature-dependent width of the GEDR.', 'nucl-ex-0511054-2-37-2': 'The [MATH] strength in the KMF model is given by [EQUATION] where [MATH] is the cross section, [MATH] is the width, and [MATH] is the centroid of the GEDR determined from photoabsorbtion experiments.', 'nucl-ex-0511054-2-38-0': 'We adopt the KMF model with the temperature [MATH] taken as a constant to be consistent with our assumption that the RSF is independent of excitation energy.', 'nucl-ex-0511054-2-38-1': 'The possible systematic uncertainty caused by this assumption is estimated to have a maximum effect of 20% on the RSF [CITATION].', 'nucl-ex-0511054-2-38-2': 'The values used for [MATH] are the ones extracted from the constant-temperature model in Eq. ([REF]).', 'nucl-ex-0511054-2-39-0': 'The GEDR is split into two parts for deformed nuclei.', 'nucl-ex-0511054-2-39-1': 'Data of [MATH]V from photoabsorption experiments show that the GEDR is best fitted with two Lorentzians, indicating a splitting of the resonance and a non-zero ground-state deformation of this nucleus.', 'nucl-ex-0511054-2-39-2': 'Indeed, [MATH]) values [CITATION] suggest a deformation of [MATH] 0.1 for [MATH]V. Therefore, a sum of two modified Lorentzians each described by Eq. ([REF]) is used (see Table [REF]).', 'nucl-ex-0511054-2-40-0': 'For [MATH], which is supposed to be governed by the spin-flip [MATH] resonance [CITATION], the Lorentzian giant magnetic dipole resonance (GMDR) [EQUATION] is adopted.', 'nucl-ex-0511054-2-41-0': 'The GEDR and GMDR parameters are taken from the systematics of Ref. [CITATION] and are listed in Table [REF].', 'nucl-ex-0511054-2-41-1': 'Thus, we fit the total RSF given by [EQUATION] to the experimental data using the normalization constant [MATH] as a free parameter.', 'nucl-ex-0511054-2-41-2': 'The value of [MATH] generally deviates from unity due to theoretical uncertainties in the KMF model and the evaluation of the absolute normalization in Eq. ([REF]).', 'nucl-ex-0511054-2-41-3': 'The resulting RSFs extracted from the two reactions are displayed in Fig. [REF], where the data have been normalized with parameters from Tables [REF] and [REF].', 'nucl-ex-0511054-2-42-0': 'The [MATH]-decay probability is governed by the number and the character of available final states and by the RSF.', 'nucl-ex-0511054-2-42-1': 'A rough inspection of the experimental data of Fig. [REF] indicates that the RSFs are increasing functions of [MATH]-energy, generally following the tails of the GEDR and GMDR resonances in this region.', 'nucl-ex-0511054-2-43-0': 'At low [MATH] energies ([MATH] MeV), an enhancement of a factor of [MATH] 5 over the KMF estimate of the strength appears in the RSFs.', 'nucl-ex-0511054-2-43-1': 'This increase has also been seen in some Fe [CITATION] and Mo [CITATION] isotopes, where it has been shown to be present in the whole excitation-energy region.', 'nucl-ex-0511054-2-43-2': 'In the case of the [MATH]Fe RSF, the feature has been confirmed by an (n,[MATH]) experiment [CITATION].', 'nucl-ex-0511054-2-43-3': 'However, it has not appeared in the RSFs of the rare-earth nuclei investigated earlier by the Oslo group.', 'nucl-ex-0511054-2-43-4': 'The physical origin of the enhancement has not, at present, any satisfying explanation, as none of the known theoretical models can account for this behaviour.', 'nucl-ex-0511054-2-44-0': 'So far, we have not been able to detect any technical problems with the Oslo method.', 'nucl-ex-0511054-2-44-1': 'The unfolding procedure with the NaI response functions gives reliable results, as demonstrated in Fig. [REF].', 'nucl-ex-0511054-2-44-2': 'Also, Fig. [REF] indicates that the low-energy [MATH] intensity is subtracted correctly; if not, one would find less intensity in the higher-generations spectrum at these [MATH] energies.', 'nucl-ex-0511054-2-44-3': 'Figure [REF] shows the final test, where the result from the least-[MATH] fit nicely reproduces the experimental data.', 'nucl-ex-0511054-2-44-4': 'In addition, investigations in [MATH]Si [CITATION] showed that our method produced [MATH]-transition coefficients in excellent agreement with average decay widths of known, discrete transitions.', 'nucl-ex-0511054-2-44-5': 'Hence, we do not believe that the enhancement is caused by some technical or methodical problems.', 'nucl-ex-0511054-2-44-6': 'Still, independent confirmation of the increasing RSF from, e.g., (n,[MATH]) experiments on the V and Mo isotopes, is highly desirable.', 'nucl-ex-0511054-2-45-0': '# Summary and conclusions', 'nucl-ex-0511054-2-46-0': 'The Oslo method has been applied to extract level densities and RSFs of the vanadium isotopes [MATH]V. From the measured level densities the microcanonical entropies have been derived.', 'nucl-ex-0511054-2-46-1': 'The entropy carried by the neutron hole in [MATH]V is estimated to be [MATH] 1.2 [MATH], which is less than the quasi-particle entropy of [MATH] 1.7 [MATH] found in rare-earth nuclei.', 'nucl-ex-0511054-2-47-0': 'The experimental RSFs are generally increasing functions of [MATH] energy.', 'nucl-ex-0511054-2-47-1': 'The main contribution to the RSFs is the GEDR; also the GMDR is present.', 'nucl-ex-0511054-2-47-2': 'At low [MATH] energies an increase of the strength functions is apparent.', 'nucl-ex-0511054-2-47-3': 'A similar enhancement has also been seen in the iron and molybdenum isotopes.', 'nucl-ex-0511054-2-47-4': 'There is still no explanation for the physics behind this very interesting behaviour.', 'nucl-ex-0511054-2-48-0': 'Financial support from the Norwegian Research Council (NFR) is gratefully acknowledged.', 'nucl-ex-0511054-2-48-1': 'A.V. acknowledges support from a NATO Science Fellowship under project number 150027/432 given by the Norwegian Research Council (NFR).', 'nucl-ex-0511054-2-48-2': 'A.V. also acknowledges support from Stewardship Science Academic Alliances, grant number DE-FG03-03-NA0074.', 'nucl-ex-0511054-2-49-0': 'Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and T.S. Tveter, Nucl.', 'nucl-ex-0511054-2-49-1': 'Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad, and S. Siem, Nucl.', 'nucl-ex-0511054-2-49-2': 'Melby, L. Bergholt, M. Guttormsen, M. Hjorth-Jensen, F. Ingebretsen, S. Messelt, J. Rekstad, A. Schiller, S. Siem, and S.W. Odegaard, Phys.', 'nucl-ex-0511054-2-49-3': 'Schiller, A. Bjerve, M. Guttormsen, M. Hjorth-Jensen, F. Ingebretsen, E. Melby, S. Messelt, J. Rekstad, S. Siem, and S.W. Odegaard, Phys.', 'nucl-ex-0511054-2-49-4': 'Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Phys.', 'nucl-ex-0511054-2-49-5': 'Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Phys.', 'nucl-ex-0511054-2-49-6': 'Siem, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and A. Voinov, Phys.', 'nucl-ex-0511054-2-49-7': 'Guttormsen, A. Atac, G. Lovhoiden, S. Messelt, T. Ramsoy, J. Rekstad, T.F. Thorsteinsen, T.S. Tveter, and Z. Zelazny, Phys.', 'nucl-ex-0511054-2-49-8': 'Guttormsen, T.S. Tveter, L. Bergholt, F. Ingebretsen, and J. Rekstad, Nucl.', 'nucl-ex-0511054-2-49-9': 'Guttormsen, T. Ramsoy, and J. Rekstad, Nucl.', 'nucl-ex-0511054-2-49-10': 'Brink, Ph.D. thesis, Oxford University, 1955.', 'nucl-ex-0511054-2-49-11': 'kad S.G. Kadmenskii, V.P. Markushev, and V.I. Furman, Yad.', 'nucl-ex-0511054-2-49-12': 'Gervais, M. Thoennessen, and W.E. Ormand, Phys.', 'nucl-ex-0511054-2-49-13': 'ENSDFData extracted using the NNDC On-Line Data Service from the ENSDF database.', 'nucl-ex-0511054-2-49-14': 'RIPLData extracted using the Reference Input Parameter Library, http://www-nds.iaea.org/RIPL-2/ GC A. Gilbert, A.G.W. Cameron, Can.', 'nucl-ex-0511054-2-49-15': 'egidy T. von Egidy, H.H. Schmidt, and A.N. Behkami, Nucl.', 'nucl-ex-0511054-2-49-16': 'Wapstra G. Audi and A.H. Wapstra, Nucl.', 'nucl-ex-0511054-2-49-17': 'BM A. Bohr and B. Mottelson, Nuclear Structure, (Benjamin, New York, 1969), Vol.', 'nucl-ex-0511054-2-49-18': 'Firestone and V.S. Shirley, Table of Isotopes, 8th ed.', 'nucl-ex-0511054-2-49-19': 'Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys.', 'nucl-ex-0511054-2-49-20': 'kopecky J. Kopecky and M. Uhl, Phys.', 'nucl-ex-0511054-2-49-21': 'Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova, M. Guttormsen, G.E. Mitchell, J. Rekstad, A. Schiller and S. Siem, Phys.', 'nucl-ex-0511054-2-49-22': 'Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin, L.A. Bernstein, F. Ingebretsen, T. Lonnroth, S. Messelt, G.E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A.C. Sunde, A. Voinov and S. Odegaard, Phys.', 'nucl-ex-0511054-2-49-23': 'Guttormsen, E. Melby, J. Rekstad, A. Schiller, S. Siem, T. Lonnroth, and A. Voinov, J. Phys.'}

[['nucl-ex-0511054-1-32-0', 'nucl-ex-0511054-2-34-0'], ['nucl-ex-0511054-1-32-1', 'nucl-ex-0511054-2-34-1'], ['nucl-ex-0511054-1-32-2', 'nucl-ex-0511054-2-34-2'], ['nucl-ex-0511054-1-32-3', 'nucl-ex-0511054-2-34-3'], ['nucl-ex-0511054-1-34-0', 'nucl-ex-0511054-2-36-0'], ['nucl-ex-0511054-1-34-1', 'nucl-ex-0511054-2-36-1'], ['nucl-ex-0511054-1-28-0', 'nucl-ex-0511054-2-30-0'], ['nucl-ex-0511054-1-28-1', 'nucl-ex-0511054-2-30-1'], ['nucl-ex-0511054-1-28-2', 'nucl-ex-0511054-2-30-2'], ['nucl-ex-0511054-1-25-0', 'nucl-ex-0511054-2-27-0'], ['nucl-ex-0511054-1-44-0', 'nucl-ex-0511054-2-47-0'], ['nucl-ex-0511054-1-44-1', 'nucl-ex-0511054-2-47-1'], ['nucl-ex-0511054-1-44-2', 'nucl-ex-0511054-2-47-2'], ['nucl-ex-0511054-1-44-3', 'nucl-ex-0511054-2-47-3'], ['nucl-ex-0511054-1-44-4', 'nucl-ex-0511054-2-47-4'], ['nucl-ex-0511054-1-41-2', 'nucl-ex-0511054-2-43-4'], ['nucl-ex-0511054-1-7-0', 'nucl-ex-0511054-2-7-0'], ['nucl-ex-0511054-1-7-1', 'nucl-ex-0511054-2-7-1'], ['nucl-ex-0511054-1-7-2', 'nucl-ex-0511054-2-7-2'], ['nucl-ex-0511054-1-19-0', 'nucl-ex-0511054-2-21-0'], ['nucl-ex-0511054-1-19-1', 'nucl-ex-0511054-2-21-1'], ['nucl-ex-0511054-1-19-2', 'nucl-ex-0511054-2-21-2'], ['nucl-ex-0511054-1-19-3', 'nucl-ex-0511054-2-21-3'], ['nucl-ex-0511054-1-19-4', 'nucl-ex-0511054-2-21-4'], ['nucl-ex-0511054-1-17-0', 'nucl-ex-0511054-2-19-0'], ['nucl-ex-0511054-1-17-1', 'nucl-ex-0511054-2-19-1'], ['nucl-ex-0511054-1-17-2', 'nucl-ex-0511054-2-19-2'], ['nucl-ex-0511054-1-17-3', 'nucl-ex-0511054-2-19-3'], ['nucl-ex-0511054-1-17-4', 'nucl-ex-0511054-2-19-4'], ['nucl-ex-0511054-1-17-5', 'nucl-ex-0511054-2-19-5'], ['nucl-ex-0511054-1-21-0', 'nucl-ex-0511054-2-23-0'], ['nucl-ex-0511054-1-21-1', 'nucl-ex-0511054-2-23-1'], ['nucl-ex-0511054-1-21-2', 'nucl-ex-0511054-2-23-2'], ['nucl-ex-0511054-1-21-3', 'nucl-ex-0511054-2-23-3'], ['nucl-ex-0511054-1-21-4', 'nucl-ex-0511054-2-23-4'], ['nucl-ex-0511054-1-21-5', 'nucl-ex-0511054-2-23-5'], ['nucl-ex-0511054-1-21-6', 'nucl-ex-0511054-2-23-6'], ['nucl-ex-0511054-1-21-7', 'nucl-ex-0511054-2-23-7'], ['nucl-ex-0511054-1-31-0', 'nucl-ex-0511054-2-33-0'], ['nucl-ex-0511054-1-31-1', 'nucl-ex-0511054-2-33-1'], ['nucl-ex-0511054-1-31-2', 'nucl-ex-0511054-2-33-2'], ['nucl-ex-0511054-1-31-3', 'nucl-ex-0511054-2-33-3'], ['nucl-ex-0511054-1-31-4', 'nucl-ex-0511054-2-33-4'], ['nucl-ex-0511054-1-33-0', 'nucl-ex-0511054-2-35-0'], ['nucl-ex-0511054-1-33-1', 'nucl-ex-0511054-2-35-1'], ['nucl-ex-0511054-1-33-2', 'nucl-ex-0511054-2-35-2'], ['nucl-ex-0511054-1-27-0', 'nucl-ex-0511054-2-29-0'], ['nucl-ex-0511054-1-27-1', 'nucl-ex-0511054-2-29-1'], ['nucl-ex-0511054-1-27-2', 'nucl-ex-0511054-2-29-2'], ['nucl-ex-0511054-1-27-4', 'nucl-ex-0511054-2-29-4'], ['nucl-ex-0511054-1-27-5', 'nucl-ex-0511054-2-29-5'], ['nucl-ex-0511054-1-27-6', 'nucl-ex-0511054-2-29-6'], ['nucl-ex-0511054-1-16-0', 'nucl-ex-0511054-2-18-0'], ['nucl-ex-0511054-1-16-1', 'nucl-ex-0511054-2-18-1'], ['nucl-ex-0511054-1-16-3', 'nucl-ex-0511054-2-18-3'], ['nucl-ex-0511054-1-16-4', 'nucl-ex-0511054-2-18-4'], ['nucl-ex-0511054-1-16-5', 'nucl-ex-0511054-2-18-5'], ['nucl-ex-0511054-1-16-6', 'nucl-ex-0511054-2-18-6'], ['nucl-ex-0511054-1-16-7', 'nucl-ex-0511054-2-18-7'], ['nucl-ex-0511054-1-43-0', 'nucl-ex-0511054-2-46-0'], ['nucl-ex-0511054-1-43-1', 'nucl-ex-0511054-2-46-1'], ['nucl-ex-0511054-1-3-2', 'nucl-ex-0511054-2-3-2'], ['nucl-ex-0511054-1-35-0', 'nucl-ex-0511054-2-37-0'], ['nucl-ex-0511054-1-35-1', 'nucl-ex-0511054-2-37-1'], ['nucl-ex-0511054-1-35-2', 'nucl-ex-0511054-2-37-2'], ['nucl-ex-0511054-1-4-0', 'nucl-ex-0511054-2-4-0'], ['nucl-ex-0511054-1-4-1', 'nucl-ex-0511054-2-4-1'], ['nucl-ex-0511054-1-4-2', 'nucl-ex-0511054-2-4-2'], ['nucl-ex-0511054-1-18-0', 'nucl-ex-0511054-2-20-0'], ['nucl-ex-0511054-1-18-1', 'nucl-ex-0511054-2-20-1'], ['nucl-ex-0511054-1-18-2', 'nucl-ex-0511054-2-20-2'], ['nucl-ex-0511054-1-22-0', 'nucl-ex-0511054-2-24-0'], ['nucl-ex-0511054-1-22-1', 'nucl-ex-0511054-2-24-1'], ['nucl-ex-0511054-1-40-0', 'nucl-ex-0511054-2-42-0'], ['nucl-ex-0511054-1-40-1', 'nucl-ex-0511054-2-42-1'], ['nucl-ex-0511054-1-9-0', 'nucl-ex-0511054-2-9-0'], ['nucl-ex-0511054-1-9-1', 'nucl-ex-0511054-2-9-1'], ['nucl-ex-0511054-1-9-2', 'nucl-ex-0511054-2-9-2'], ['nucl-ex-0511054-1-9-3', 'nucl-ex-0511054-2-9-3'], ['nucl-ex-0511054-1-9-4', 'nucl-ex-0511054-2-9-4'], ['nucl-ex-0511054-1-9-5', 'nucl-ex-0511054-2-9-5'], ['nucl-ex-0511054-1-9-6', 'nucl-ex-0511054-2-9-6'], ['nucl-ex-0511054-1-9-7', 'nucl-ex-0511054-2-9-7'], ['nucl-ex-0511054-1-9-8', 'nucl-ex-0511054-2-9-8'], ['nucl-ex-0511054-1-5-0', 'nucl-ex-0511054-2-5-0'], ['nucl-ex-0511054-1-5-1', 'nucl-ex-0511054-2-5-1'], ['nucl-ex-0511054-1-5-2', 'nucl-ex-0511054-2-5-2'], ['nucl-ex-0511054-1-5-3', 'nucl-ex-0511054-2-5-3'], ['nucl-ex-0511054-1-0-0', 'nucl-ex-0511054-2-0-0'], ['nucl-ex-0511054-1-46-0', 'nucl-ex-0511054-2-49-0'], ['nucl-ex-0511054-1-46-1', 'nucl-ex-0511054-2-49-1'], ['nucl-ex-0511054-1-46-2', 'nucl-ex-0511054-2-49-2'], ['nucl-ex-0511054-1-46-3', 'nucl-ex-0511054-2-49-3'], ['nucl-ex-0511054-1-46-4', 'nucl-ex-0511054-2-49-4'], ['nucl-ex-0511054-1-46-5', 'nucl-ex-0511054-2-49-5'], ['nucl-ex-0511054-1-46-6', 'nucl-ex-0511054-2-49-6'], ['nucl-ex-0511054-1-46-7', 'nucl-ex-0511054-2-49-7'], ['nucl-ex-0511054-1-46-8', 'nucl-ex-0511054-2-49-8'], ['nucl-ex-0511054-1-46-9', 'nucl-ex-0511054-2-49-9'], ['nucl-ex-0511054-1-46-10', 'nucl-ex-0511054-2-49-10'], ['nucl-ex-0511054-1-46-11', 'nucl-ex-0511054-2-49-20'], ['nucl-ex-0511054-1-46-12', 'nucl-ex-0511054-2-49-11'], ['nucl-ex-0511054-1-46-13', 'nucl-ex-0511054-2-49-12'], ['nucl-ex-0511054-1-46-14', 'nucl-ex-0511054-2-49-13'], ['nucl-ex-0511054-1-46-15', 'nucl-ex-0511054-2-49-14'], ['nucl-ex-0511054-1-46-16', 'nucl-ex-0511054-2-49-15'], ['nucl-ex-0511054-1-46-17', 'nucl-ex-0511054-2-49-16'], ['nucl-ex-0511054-1-46-18', 'nucl-ex-0511054-2-49-17'], ['nucl-ex-0511054-1-46-19', 'nucl-ex-0511054-2-49-18'], ['nucl-ex-0511054-1-46-20', 'nucl-ex-0511054-2-49-19'], ['nucl-ex-0511054-1-46-21', 'nucl-ex-0511054-2-49-21'], ['nucl-ex-0511054-1-46-22', 'nucl-ex-0511054-2-49-22'], ['nucl-ex-0511054-1-36-0', 'nucl-ex-0511054-2-38-0'], ['nucl-ex-0511054-1-10-0', 'nucl-ex-0511054-2-10-0'], ['nucl-ex-0511054-1-39-0', 'nucl-ex-0511054-2-41-0'], ['nucl-ex-0511054-1-39-1', 'nucl-ex-0511054-2-41-1'], ['nucl-ex-0511054-1-39-2', 'nucl-ex-0511054-2-41-2'], ['nucl-ex-0511054-1-39-3', 'nucl-ex-0511054-2-41-3'], ['nucl-ex-0511054-1-24-0', 'nucl-ex-0511054-2-26-0'], ['nucl-ex-0511054-1-24-1', 'nucl-ex-0511054-2-26-1'], ['nucl-ex-0511054-1-24-2', 'nucl-ex-0511054-2-26-2'], ['nucl-ex-0511054-1-37-0', 'nucl-ex-0511054-2-39-0'], ['nucl-ex-0511054-1-37-1', 'nucl-ex-0511054-2-39-1'], ['nucl-ex-0511054-1-37-2', 'nucl-ex-0511054-2-39-2'], ['nucl-ex-0511054-1-20-1', 'nucl-ex-0511054-2-22-1'], ['nucl-ex-0511054-1-20-2', 'nucl-ex-0511054-2-22-2'], ['nucl-ex-0511054-1-30-0', 'nucl-ex-0511054-2-32-0'], ['nucl-ex-0511054-1-30-1', 'nucl-ex-0511054-2-32-1'], ['nucl-ex-0511054-1-30-2', 'nucl-ex-0511054-2-32-2'], ['nucl-ex-0511054-1-30-3', 'nucl-ex-0511054-2-32-3'], ['nucl-ex-0511054-1-45-0', 'nucl-ex-0511054-2-48-0'], ['nucl-ex-0511054-1-45-1', 'nucl-ex-0511054-2-48-1'], ['nucl-ex-0511054-1-11-1', 'nucl-ex-0511054-2-11-1'], ['nucl-ex-0511054-1-11-2', 'nucl-ex-0511054-2-11-2'], ['nucl-ex-0511054-1-26-0', 'nucl-ex-0511054-2-28-0'], ['nucl-ex-0511054-1-26-1', 'nucl-ex-0511054-2-28-1'], ['nucl-ex-0511054-1-26-2', 'nucl-ex-0511054-2-28-2'], ['nucl-ex-0511054-1-26-3', 'nucl-ex-0511054-2-28-3'], ['nucl-ex-0511054-1-26-4', 'nucl-ex-0511054-2-28-4'], ['nucl-ex-0511054-1-26-5', 'nucl-ex-0511054-2-28-5'], ['nucl-ex-0511054-1-13-0', 'nucl-ex-0511054-2-14-0'], ['nucl-ex-0511054-1-13-1', 'nucl-ex-0511054-2-14-1'], ['nucl-ex-0511054-1-13-2', 'nucl-ex-0511054-2-14-2'], ['nucl-ex-0511054-1-13-3', 'nucl-ex-0511054-2-14-3'], ['nucl-ex-0511054-1-13-4', 'nucl-ex-0511054-2-14-4'], ['nucl-ex-0511054-1-23-0', 'nucl-ex-0511054-2-25-0'], ['nucl-ex-0511054-1-23-1', 'nucl-ex-0511054-2-25-1'], ['nucl-ex-0511054-1-23-2', 'nucl-ex-0511054-2-25-2'], ['nucl-ex-0511054-1-23-3', 'nucl-ex-0511054-2-25-3'], ['nucl-ex-0511054-1-23-4', 'nucl-ex-0511054-2-25-4'], ['nucl-ex-0511054-1-23-5', 'nucl-ex-0511054-2-25-5'], ['nucl-ex-0511054-1-6-0', 'nucl-ex-0511054-2-6-0'], ['nucl-ex-0511054-1-6-1', 'nucl-ex-0511054-2-6-1'], ['nucl-ex-0511054-1-6-2', 'nucl-ex-0511054-2-6-2'], ['nucl-ex-0511054-1-6-3', 'nucl-ex-0511054-2-6-3'], ['nucl-ex-0511054-1-12-0', 'nucl-ex-0511054-2-12-0'], ['nucl-ex-0511054-1-12-3', 'nucl-ex-0511054-2-13-2'], ['nucl-ex-0511054-1-12-4', 'nucl-ex-0511054-2-13-3'], ['nucl-ex-0511054-1-12-5', 'nucl-ex-0511054-2-13-4'], ['nucl-ex-0511054-1-14-0', 'nucl-ex-0511054-2-15-0'], ['nucl-ex-0511054-1-14-1', 'nucl-ex-0511054-2-16-0'], ['nucl-ex-0511054-1-14-2', 'nucl-ex-0511054-2-16-1'], ['nucl-ex-0511054-1-25-1', 'nucl-ex-0511054-2-27-1'], ['nucl-ex-0511054-1-38-0', 'nucl-ex-0511054-2-40-0'], ['nucl-ex-0511054-1-41-0', 'nucl-ex-0511054-2-43-0'], ['nucl-ex-0511054-1-41-1', 'nucl-ex-0511054-2-43-1'], ['nucl-ex-0511054-1-27-3', 'nucl-ex-0511054-2-29-3'], ['nucl-ex-0511054-1-16-2', 'nucl-ex-0511054-2-18-2'], ['nucl-ex-0511054-1-3-0', 'nucl-ex-0511054-2-3-0'], ['nucl-ex-0511054-1-3-1', 'nucl-ex-0511054-2-3-1'], ['nucl-ex-0511054-1-3-3', 'nucl-ex-0511054-2-3-3'], ['nucl-ex-0511054-1-0-1', 'nucl-ex-0511054-2-0-1'], ['nucl-ex-0511054-1-0-3', 'nucl-ex-0511054-2-0-3'], ['nucl-ex-0511054-1-36-1', 'nucl-ex-0511054-2-38-1'], ['nucl-ex-0511054-1-36-2', 'nucl-ex-0511054-2-38-2'], ['nucl-ex-0511054-1-20-0', 'nucl-ex-0511054-2-22-0'], ['nucl-ex-0511054-1-11-0', 'nucl-ex-0511054-2-11-0'], ['nucl-ex-0511054-1-13-5', 'nucl-ex-0511054-2-14-5'], ['nucl-ex-0511054-1-12-1', 'nucl-ex-0511054-2-13-0'], ['nucl-ex-0511054-1-12-2', 'nucl-ex-0511054-2-13-1'], ['nucl-ex-0511054-1-0-2', 'nucl-ex-0511054-2-0-2']]

[['nucl-ex-0511054-1-32-0', 'nucl-ex-0511054-2-34-0'], ['nucl-ex-0511054-1-32-1', 'nucl-ex-0511054-2-34-1'], ['nucl-ex-0511054-1-32-2', 'nucl-ex-0511054-2-34-2'], ['nucl-ex-0511054-1-32-3', 'nucl-ex-0511054-2-34-3'], ['nucl-ex-0511054-1-34-0', 'nucl-ex-0511054-2-36-0'], ['nucl-ex-0511054-1-34-1', 'nucl-ex-0511054-2-36-1'], ['nucl-ex-0511054-1-28-0', 'nucl-ex-0511054-2-30-0'], ['nucl-ex-0511054-1-28-1', 'nucl-ex-0511054-2-30-1'], ['nucl-ex-0511054-1-28-2', 'nucl-ex-0511054-2-30-2'], ['nucl-ex-0511054-1-25-0', 'nucl-ex-0511054-2-27-0'], ['nucl-ex-0511054-1-44-0', 'nucl-ex-0511054-2-47-0'], ['nucl-ex-0511054-1-44-1', 'nucl-ex-0511054-2-47-1'], ['nucl-ex-0511054-1-44-2', 'nucl-ex-0511054-2-47-2'], ['nucl-ex-0511054-1-44-3', 'nucl-ex-0511054-2-47-3'], ['nucl-ex-0511054-1-44-4', 'nucl-ex-0511054-2-47-4'], ['nucl-ex-0511054-1-41-2', 'nucl-ex-0511054-2-43-4'], ['nucl-ex-0511054-1-7-0', 'nucl-ex-0511054-2-7-0'], ['nucl-ex-0511054-1-7-1', 'nucl-ex-0511054-2-7-1'], ['nucl-ex-0511054-1-7-2', 'nucl-ex-0511054-2-7-2'], ['nucl-ex-0511054-1-19-0', 'nucl-ex-0511054-2-21-0'], ['nucl-ex-0511054-1-19-1', 'nucl-ex-0511054-2-21-1'], ['nucl-ex-0511054-1-19-2', 'nucl-ex-0511054-2-21-2'], ['nucl-ex-0511054-1-19-3', 'nucl-ex-0511054-2-21-3'], ['nucl-ex-0511054-1-19-4', 'nucl-ex-0511054-2-21-4'], ['nucl-ex-0511054-1-17-0', 'nucl-ex-0511054-2-19-0'], ['nucl-ex-0511054-1-17-1', 'nucl-ex-0511054-2-19-1'], ['nucl-ex-0511054-1-17-2', 'nucl-ex-0511054-2-19-2'], ['nucl-ex-0511054-1-17-3', 'nucl-ex-0511054-2-19-3'], ['nucl-ex-0511054-1-17-4', 'nucl-ex-0511054-2-19-4'], ['nucl-ex-0511054-1-17-5', 'nucl-ex-0511054-2-19-5'], ['nucl-ex-0511054-1-21-0', 'nucl-ex-0511054-2-23-0'], ['nucl-ex-0511054-1-21-1', 'nucl-ex-0511054-2-23-1'], ['nucl-ex-0511054-1-21-2', 'nucl-ex-0511054-2-23-2'], ['nucl-ex-0511054-1-21-3', 'nucl-ex-0511054-2-23-3'], ['nucl-ex-0511054-1-21-4', 'nucl-ex-0511054-2-23-4'], ['nucl-ex-0511054-1-21-5', 'nucl-ex-0511054-2-23-5'], ['nucl-ex-0511054-1-21-6', 'nucl-ex-0511054-2-23-6'], ['nucl-ex-0511054-1-21-7', 'nucl-ex-0511054-2-23-7'], ['nucl-ex-0511054-1-31-0', 'nucl-ex-0511054-2-33-0'], ['nucl-ex-0511054-1-31-1', 'nucl-ex-0511054-2-33-1'], ['nucl-ex-0511054-1-31-2', 'nucl-ex-0511054-2-33-2'], ['nucl-ex-0511054-1-31-3', 'nucl-ex-0511054-2-33-3'], ['nucl-ex-0511054-1-31-4', 'nucl-ex-0511054-2-33-4'], ['nucl-ex-0511054-1-33-0', 'nucl-ex-0511054-2-35-0'], ['nucl-ex-0511054-1-33-1', 'nucl-ex-0511054-2-35-1'], ['nucl-ex-0511054-1-33-2', 'nucl-ex-0511054-2-35-2'], ['nucl-ex-0511054-1-27-0', 'nucl-ex-0511054-2-29-0'], ['nucl-ex-0511054-1-27-1', 'nucl-ex-0511054-2-29-1'], ['nucl-ex-0511054-1-27-2', 'nucl-ex-0511054-2-29-2'], ['nucl-ex-0511054-1-27-4', 'nucl-ex-0511054-2-29-4'], ['nucl-ex-0511054-1-27-5', 'nucl-ex-0511054-2-29-5'], ['nucl-ex-0511054-1-27-6', 'nucl-ex-0511054-2-29-6'], ['nucl-ex-0511054-1-16-0', 'nucl-ex-0511054-2-18-0'], ['nucl-ex-0511054-1-16-1', 'nucl-ex-0511054-2-18-1'], ['nucl-ex-0511054-1-16-3', 'nucl-ex-0511054-2-18-3'], ['nucl-ex-0511054-1-16-4', 'nucl-ex-0511054-2-18-4'], ['nucl-ex-0511054-1-16-5', 'nucl-ex-0511054-2-18-5'], ['nucl-ex-0511054-1-16-6', 'nucl-ex-0511054-2-18-6'], ['nucl-ex-0511054-1-16-7', 'nucl-ex-0511054-2-18-7'], ['nucl-ex-0511054-1-43-0', 'nucl-ex-0511054-2-46-0'], ['nucl-ex-0511054-1-43-1', 'nucl-ex-0511054-2-46-1'], ['nucl-ex-0511054-1-3-2', 'nucl-ex-0511054-2-3-2'], ['nucl-ex-0511054-1-35-0', 'nucl-ex-0511054-2-37-0'], ['nucl-ex-0511054-1-35-1', 'nucl-ex-0511054-2-37-1'], ['nucl-ex-0511054-1-35-2', 'nucl-ex-0511054-2-37-2'], ['nucl-ex-0511054-1-4-0', 'nucl-ex-0511054-2-4-0'], ['nucl-ex-0511054-1-4-1', 'nucl-ex-0511054-2-4-1'], ['nucl-ex-0511054-1-4-2', 'nucl-ex-0511054-2-4-2'], ['nucl-ex-0511054-1-18-0', 'nucl-ex-0511054-2-20-0'], ['nucl-ex-0511054-1-18-1', 'nucl-ex-0511054-2-20-1'], ['nucl-ex-0511054-1-18-2', 'nucl-ex-0511054-2-20-2'], ['nucl-ex-0511054-1-22-0', 'nucl-ex-0511054-2-24-0'], ['nucl-ex-0511054-1-22-1', 'nucl-ex-0511054-2-24-1'], ['nucl-ex-0511054-1-40-0', 'nucl-ex-0511054-2-42-0'], ['nucl-ex-0511054-1-40-1', 'nucl-ex-0511054-2-42-1'], ['nucl-ex-0511054-1-9-0', 'nucl-ex-0511054-2-9-0'], ['nucl-ex-0511054-1-9-1', 'nucl-ex-0511054-2-9-1'], ['nucl-ex-0511054-1-9-2', 'nucl-ex-0511054-2-9-2'], ['nucl-ex-0511054-1-9-3', 'nucl-ex-0511054-2-9-3'], ['nucl-ex-0511054-1-9-4', 'nucl-ex-0511054-2-9-4'], ['nucl-ex-0511054-1-9-5', 'nucl-ex-0511054-2-9-5'], ['nucl-ex-0511054-1-9-6', 'nucl-ex-0511054-2-9-6'], ['nucl-ex-0511054-1-9-7', 'nucl-ex-0511054-2-9-7'], ['nucl-ex-0511054-1-9-8', 'nucl-ex-0511054-2-9-8'], ['nucl-ex-0511054-1-5-0', 'nucl-ex-0511054-2-5-0'], ['nucl-ex-0511054-1-5-1', 'nucl-ex-0511054-2-5-1'], ['nucl-ex-0511054-1-5-2', 'nucl-ex-0511054-2-5-2'], ['nucl-ex-0511054-1-5-3', 'nucl-ex-0511054-2-5-3'], ['nucl-ex-0511054-1-0-0', 'nucl-ex-0511054-2-0-0'], ['nucl-ex-0511054-1-46-0', 'nucl-ex-0511054-2-49-0'], ['nucl-ex-0511054-1-46-1', 'nucl-ex-0511054-2-49-1'], ['nucl-ex-0511054-1-46-2', 'nucl-ex-0511054-2-49-2'], ['nucl-ex-0511054-1-46-3', 'nucl-ex-0511054-2-49-3'], ['nucl-ex-0511054-1-46-4', 'nucl-ex-0511054-2-49-4'], ['nucl-ex-0511054-1-46-5', 'nucl-ex-0511054-2-49-5'], ['nucl-ex-0511054-1-46-6', 'nucl-ex-0511054-2-49-6'], ['nucl-ex-0511054-1-46-7', 'nucl-ex-0511054-2-49-7'], ['nucl-ex-0511054-1-46-8', 'nucl-ex-0511054-2-49-8'], ['nucl-ex-0511054-1-46-9', 'nucl-ex-0511054-2-49-9'], ['nucl-ex-0511054-1-46-10', 'nucl-ex-0511054-2-49-10'], ['nucl-ex-0511054-1-46-11', 'nucl-ex-0511054-2-49-20'], ['nucl-ex-0511054-1-46-12', 'nucl-ex-0511054-2-49-11'], ['nucl-ex-0511054-1-46-13', 'nucl-ex-0511054-2-49-12'], ['nucl-ex-0511054-1-46-14', 'nucl-ex-0511054-2-49-13'], ['nucl-ex-0511054-1-46-15', 'nucl-ex-0511054-2-49-14'], ['nucl-ex-0511054-1-46-16', 'nucl-ex-0511054-2-49-15'], ['nucl-ex-0511054-1-46-17', 'nucl-ex-0511054-2-49-16'], ['nucl-ex-0511054-1-46-18', 'nucl-ex-0511054-2-49-17'], ['nucl-ex-0511054-1-46-19', 'nucl-ex-0511054-2-49-18'], ['nucl-ex-0511054-1-46-20', 'nucl-ex-0511054-2-49-19'], ['nucl-ex-0511054-1-46-21', 'nucl-ex-0511054-2-49-21'], ['nucl-ex-0511054-1-46-22', 'nucl-ex-0511054-2-49-22'], ['nucl-ex-0511054-1-36-0', 'nucl-ex-0511054-2-38-0'], ['nucl-ex-0511054-1-10-0', 'nucl-ex-0511054-2-10-0'], ['nucl-ex-0511054-1-39-0', 'nucl-ex-0511054-2-41-0'], ['nucl-ex-0511054-1-39-1', 'nucl-ex-0511054-2-41-1'], ['nucl-ex-0511054-1-39-2', 'nucl-ex-0511054-2-41-2'], ['nucl-ex-0511054-1-39-3', 'nucl-ex-0511054-2-41-3'], ['nucl-ex-0511054-1-24-0', 'nucl-ex-0511054-2-26-0'], ['nucl-ex-0511054-1-24-1', 'nucl-ex-0511054-2-26-1'], ['nucl-ex-0511054-1-24-2', 'nucl-ex-0511054-2-26-2'], ['nucl-ex-0511054-1-37-0', 'nucl-ex-0511054-2-39-0'], ['nucl-ex-0511054-1-37-1', 'nucl-ex-0511054-2-39-1'], ['nucl-ex-0511054-1-37-2', 'nucl-ex-0511054-2-39-2'], ['nucl-ex-0511054-1-20-1', 'nucl-ex-0511054-2-22-1'], ['nucl-ex-0511054-1-20-2', 'nucl-ex-0511054-2-22-2'], ['nucl-ex-0511054-1-30-0', 'nucl-ex-0511054-2-32-0'], ['nucl-ex-0511054-1-30-1', 'nucl-ex-0511054-2-32-1'], ['nucl-ex-0511054-1-30-2', 'nucl-ex-0511054-2-32-2'], ['nucl-ex-0511054-1-30-3', 'nucl-ex-0511054-2-32-3'], ['nucl-ex-0511054-1-45-0', 'nucl-ex-0511054-2-48-0'], ['nucl-ex-0511054-1-45-1', 'nucl-ex-0511054-2-48-1'], ['nucl-ex-0511054-1-11-1', 'nucl-ex-0511054-2-11-1'], ['nucl-ex-0511054-1-11-2', 'nucl-ex-0511054-2-11-2'], ['nucl-ex-0511054-1-26-0', 'nucl-ex-0511054-2-28-0'], ['nucl-ex-0511054-1-26-1', 'nucl-ex-0511054-2-28-1'], ['nucl-ex-0511054-1-26-2', 'nucl-ex-0511054-2-28-2'], ['nucl-ex-0511054-1-26-3', 'nucl-ex-0511054-2-28-3'], ['nucl-ex-0511054-1-26-4', 'nucl-ex-0511054-2-28-4'], ['nucl-ex-0511054-1-26-5', 'nucl-ex-0511054-2-28-5'], ['nucl-ex-0511054-1-13-0', 'nucl-ex-0511054-2-14-0'], ['nucl-ex-0511054-1-13-1', 'nucl-ex-0511054-2-14-1'], ['nucl-ex-0511054-1-13-2', 'nucl-ex-0511054-2-14-2'], ['nucl-ex-0511054-1-13-3', 'nucl-ex-0511054-2-14-3'], ['nucl-ex-0511054-1-13-4', 'nucl-ex-0511054-2-14-4'], ['nucl-ex-0511054-1-23-0', 'nucl-ex-0511054-2-25-0'], ['nucl-ex-0511054-1-23-1', 'nucl-ex-0511054-2-25-1'], ['nucl-ex-0511054-1-23-2', 'nucl-ex-0511054-2-25-2'], ['nucl-ex-0511054-1-23-3', 'nucl-ex-0511054-2-25-3'], ['nucl-ex-0511054-1-23-4', 'nucl-ex-0511054-2-25-4'], ['nucl-ex-0511054-1-23-5', 'nucl-ex-0511054-2-25-5'], ['nucl-ex-0511054-1-6-0', 'nucl-ex-0511054-2-6-0'], ['nucl-ex-0511054-1-6-1', 'nucl-ex-0511054-2-6-1'], ['nucl-ex-0511054-1-6-2', 'nucl-ex-0511054-2-6-2'], ['nucl-ex-0511054-1-6-3', 'nucl-ex-0511054-2-6-3'], ['nucl-ex-0511054-1-12-0', 'nucl-ex-0511054-2-12-0'], ['nucl-ex-0511054-1-12-3', 'nucl-ex-0511054-2-13-2'], ['nucl-ex-0511054-1-12-4', 'nucl-ex-0511054-2-13-3'], ['nucl-ex-0511054-1-12-5', 'nucl-ex-0511054-2-13-4'], ['nucl-ex-0511054-1-14-0', 'nucl-ex-0511054-2-15-0'], ['nucl-ex-0511054-1-14-1', 'nucl-ex-0511054-2-16-0'], ['nucl-ex-0511054-1-14-2', 'nucl-ex-0511054-2-16-1']]

[['nucl-ex-0511054-1-25-1', 'nucl-ex-0511054-2-27-1'], ['nucl-ex-0511054-1-38-0', 'nucl-ex-0511054-2-40-0'], ['nucl-ex-0511054-1-41-0', 'nucl-ex-0511054-2-43-0'], ['nucl-ex-0511054-1-41-1', 'nucl-ex-0511054-2-43-1'], ['nucl-ex-0511054-1-27-3', 'nucl-ex-0511054-2-29-3'], ['nucl-ex-0511054-1-16-2', 'nucl-ex-0511054-2-18-2'], ['nucl-ex-0511054-1-3-0', 'nucl-ex-0511054-2-3-0'], ['nucl-ex-0511054-1-3-1', 'nucl-ex-0511054-2-3-1'], ['nucl-ex-0511054-1-3-3', 'nucl-ex-0511054-2-3-3'], ['nucl-ex-0511054-1-0-1', 'nucl-ex-0511054-2-0-1'], ['nucl-ex-0511054-1-0-3', 'nucl-ex-0511054-2-0-3'], ['nucl-ex-0511054-1-36-1', 'nucl-ex-0511054-2-38-1'], ['nucl-ex-0511054-1-36-2', 'nucl-ex-0511054-2-38-2'], ['nucl-ex-0511054-1-20-0', 'nucl-ex-0511054-2-22-0'], ['nucl-ex-0511054-1-11-0', 'nucl-ex-0511054-2-11-0'], ['nucl-ex-0511054-1-13-5', 'nucl-ex-0511054-2-14-5'], ['nucl-ex-0511054-1-12-1', 'nucl-ex-0511054-2-13-0'], ['nucl-ex-0511054-1-12-2', 'nucl-ex-0511054-2-13-1']]

[]

[['nucl-ex-0511054-1-0-2', 'nucl-ex-0511054-2-0-2']]

[]

['nucl-ex-0511054-1-1-0', 'nucl-ex-0511054-1-10-1', 'nucl-ex-0511054-2-1-0', 'nucl-ex-0511054-2-10-1']

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

https://arxiv.org/abs/nucl-ex/0511054

cond-mat-0209170

{'cond-mat-0209170-1-0-0': 'We employ the Dynamical Cluster Approximation (DCA) in conjunction with the Fluctuation Exchange Approximation (FLEX) to study the Hubbard model.', 'cond-mat-0209170-1-0-1': 'The DCA is a technique to systematically restore the momentum conservation at the internal vertices of Feynman diagrams relinquished in the Dynamical Mean Field Approximation (DMFA).', 'cond-mat-0209170-1-0-2': 'FLEX is a perturbative diagrammatic approach in which classes of Feynman diagrams are summed over analytically using geometric series.', 'cond-mat-0209170-1-0-3': 'The FLEX is used as a tool to investigate the complementarity of the DCA and the finite size lattice technique with periodic boundary conditions by comparing their results for the Hubbard model.', 'cond-mat-0209170-1-0-4': 'We also study the microscopic theory underlying the DCA in terms of compact (skeletal) and non-compact diagrammatic contributions to the thermodynamic potential independent of a specific model.', 'cond-mat-0209170-1-0-5': 'The significant advantages of the DCA implementation in momentum space suggests the development of the same formalism for the frequency space.', 'cond-mat-0209170-1-0-6': 'However, we show that such a formalism for the Matsubara frequencies at finite temperatures leads to acausal results and is not viable.', 'cond-mat-0209170-1-0-7': 'However, a real frequency approach is shown to be feasible.', 'cond-mat-0209170-1-1-0': 'Introduction Non-local correlations play an important role in the physics of strongly correlated electron systems such as high-[MATH] superconductors, heavy fermion metals, etc.', 'cond-mat-0209170-1-1-1': 'The Dynamical Mean Field Approximation (DMFA) [CITATION], in which all the non-local correlations are ignored, can capture some of the major features of strongly correlated systems.', 'cond-mat-0209170-1-1-2': 'Nevertheless, the non-local correlations become crucial in the physics of phases with non-local order parameters such as d-wave superconductivity.', 'cond-mat-0209170-1-1-3': 'Even phases with local order parameters such as commensurate magnetism can be significantly affected by the non-local correlations (e.g. spin waves) ignored in the DMFA.', 'cond-mat-0209170-1-2-0': 'The early attempts to extend the DMFA by including non-local correlations resulted in the violation of causality which is a requirement for positive definiteness of the spectral weight and the density of states (DOS) [CITATION].', 'cond-mat-0209170-1-2-1': 'The Dynamical Cluster Approximation (DCA) is a fully causal technique used to systematically add nonlocal corrections to the DMFA by mapping the lattice onto a self-consistently embedded cluster problem.', 'cond-mat-0209170-1-2-2': 'The mapping from the lattice to the cluster is accompanied by coarse-graining the lattice problem in its reciprocal space.', 'cond-mat-0209170-1-2-3': 'Thus far, the DCA has been combined with Quantum Monte Carlo (QMC) [CITATION], the Non-Crossing Approximation (NCA) [CITATION] and the Fluctuation Exchange Approximation (FLEX) [CITATION] to solve the corresponding cluster problems.', 'cond-mat-0209170-1-3-0': 'The FLEX is a perturbative diagrammatic approach in which classes of Feynman diagrams are summed to all orders using geometric series.', 'cond-mat-0209170-1-3-1': '[CITATION] Others [CITATION] have employed the FLEX for finite size lattices with periodic boundary conditions.', 'cond-mat-0209170-1-3-2': 'Due to the absence of contributions from some relevant diagrams, the FLEX is not capable of addressing the Hubbard model physics in the strong regime precisely.', 'cond-mat-0209170-1-3-3': 'However, the main objective of this work is to make a comparison between the DCA-FLEX combination results and previous finite size lattice FLEX calculations.', 'cond-mat-0209170-1-3-4': 'It is hoped that this study will lead to a better understanding of the DCA.', 'cond-mat-0209170-1-4-0': 'We earlier[CITATION] suggested a prescription to correctly implement the DCA technique in the thermodynamic potential.', 'cond-mat-0209170-1-4-1': 'This prescription will be discussed from a different point of view using a more general argument.', 'cond-mat-0209170-1-4-2': "Based upon the Green function's exponential fall-off as a function of distance, we conclude that compact diagrams (two-particle irreducible in the thermodynamic potential) are better approximated using the DCA than non-compact (two-particle reducible) ones.", 'cond-mat-0209170-1-4-3': 'Hence, the DCA is applied to the compact diagrams only and non-compact ones are calculated explicitly using dressed non coarse-grained Green functions.', 'cond-mat-0209170-1-5-0': 'In this work we also consider the extension of the DCA to frequency space.', 'cond-mat-0209170-1-5-1': 'The many-body theory at finite temperatures is conventionally derived in terms of discrete imaginary Matsubara frequencies.', 'cond-mat-0209170-1-5-2': 'We illustrate that even for a self-consistent algorithm like the FLEX, coarse-graining the imaginary frequency propagators results in causality violations and can not be implemented.', 'cond-mat-0209170-1-5-3': 'However, a real frequency formalism is shown to be causal and applicable not only to the FLEX, but also to other cluster solving methods such as the NCA.', 'cond-mat-0209170-1-6-0': 'This paper is structured as follows.', 'cond-mat-0209170-1-6-1': 'In the next three sections, we briefly review the DCA, its application to the Hubbard model, the FLEX, and then we describe how the FLEX and the DCA may be merged into a single algorithm.', 'cond-mat-0209170-1-6-2': 'In the next three sections, we use the FLEX-DCA, in comparison to the FLEX for finite-sized systems, to explore the properties of the DCA.', 'cond-mat-0209170-1-6-3': 'The last two sections, are devoted to a microscopic derivation of the DCA, and to an extension of the DCA to frequency space.', 'cond-mat-0209170-1-7-0': '# Dynamical Cluster Approximation (DCA)', 'cond-mat-0209170-1-8-0': 'Both the DCA and the DMFA may be derived by exploring the momentum conservation in the diagrammatics.', 'cond-mat-0209170-1-8-1': 'As depicted in Fig. [REF], momentum conservation at each vertex is described by the Laue function:', 'cond-mat-0209170-1-9-0': '[EQUATION]', 'cond-mat-0209170-1-9-1': 'In the DMFA, momentum conservation at the internal vertices of irreducible Feynman diagrams is completely relinquished.', 'cond-mat-0209170-1-9-2': 'I.e., the DMFA simply sets [MATH].', 'cond-mat-0209170-1-9-3': '[CITATION] Hence, we may sum freely over all the internal momenta entering and leaving each vertex.', 'cond-mat-0209170-1-9-4': 'Only local contributions survive the sum.', 'cond-mat-0209170-1-9-5': 'Thus, this is equivalent to mapping the lattice problem onto a self-consistently embedded impurity problem.', 'cond-mat-0209170-1-9-6': 'The DMFA becomes exact at infinite dimensions.', 'cond-mat-0209170-1-9-7': '[CITATION]', 'cond-mat-0209170-1-10-0': 'The DCA is an approach to systematically restore the momentum conservation relinquished in the DMFA.', 'cond-mat-0209170-1-10-1': 'In the DCA, the first Brillouin zone in the reciprocal space is divided into [MATH] equal cells of linear size [MATH] labeled by [MATH] in their centers, and the momenta within each cell are labeled by [MATH].', 'cond-mat-0209170-1-10-2': 'Then [MATH] (c.f. Fig. [REF]).', 'cond-mat-0209170-1-10-3': 'To visualize this scheme in the real lattice, one could consider tiling the lattice of [MATH] sites by [MATH] clusters each composed of [MATH] sites where [MATH] is the linear size of the subcell and D is dimensionality (c.f. Fig. [REF] for [MATH]).', 'cond-mat-0209170-1-10-4': 'We will use this picture in section [REF] while discussing the microscopic theory of the DCA.', 'cond-mat-0209170-1-10-5': 'We label the origin of the clusters by [MATH] and the [MATH] intercluster sites by [MATH].', 'cond-mat-0209170-1-10-6': 'So for each site in the original lattice [MATH].', 'cond-mat-0209170-1-10-7': 'Care must be taken when choosing the cluster geometries in order to preserve the lattice point group symmetry and also satisfy some other criteria for cubic or square lattices.', 'cond-mat-0209170-1-10-8': '[CITATION]', 'cond-mat-0209170-1-11-0': 'In the DCA, we first make the following separation in Eq. [REF] [EQUATION]', 'cond-mat-0209170-1-11-1': 'The products [MATH] and [MATH] where [MATH] is an integer.', 'cond-mat-0209170-1-11-2': 'Therefore, their associated phases may be neglected and Eq. [REF] splits into two parts [EQUATION]', 'cond-mat-0209170-1-11-3': 'The DCA also ignores the phases [MATH] due to the position of the cluster in the original lattice and (far less important) [MATH] corresponding to the position within the cluster.', 'cond-mat-0209170-1-11-4': 'As a result, it approximates [MATH], so that [EQUATION] which indicates that the momentum is conserved modulo [MATH] for transfers between the cells.', 'cond-mat-0209170-1-12-0': 'The approximation made through the substitution [MATH] corresponds to replacing all internal legs in the compact (skeletal) diagrams by the coarse-grained Green function [MATH] and interaction potential [MATH] defined by [EQUATION]', 'cond-mat-0209170-1-12-1': 'In section [REF], we will define the compact and non compact diagrams and elaborately discuss why only the compact ones undergo the coarse-graining approximation.', 'cond-mat-0209170-1-13-0': 'Replacing [MATH] by [MATH] tremendously reduces the complexity of the problem because instead of having to perform sums over all [MATH] states in the entire first Brillouin zone, we have sums over only a set of [MATH] states where [MATH].', 'cond-mat-0209170-1-14-0': '# Hubbard Model', 'cond-mat-0209170-1-15-0': 'We will apply the DCA to study the Hubbard model Hamiltonian incorporating interactions between the electrons on a lattice.', 'cond-mat-0209170-1-15-1': 'It includes a tight-binding part due to the hopping of electrons among the sites and an interaction between the electrons.', 'cond-mat-0209170-1-15-2': 'The general Hamiltonian reads [EQUATION] where [EQUATION] and [EQUATION]', 'cond-mat-0209170-1-15-3': 'Factors [MATH] and [MATH] correspond to electron hoppings and Coulomb interactions respectively.', 'cond-mat-0209170-1-15-4': 'Later in the paper, we will study the simplest Hubbard interaction which is fully local and only between electrons sitting at the same site having opposite spin directions.', 'cond-mat-0209170-1-15-5': 'The interaction strength is a constant called U. Hence, for the local model, Eq. [REF] simplifies to [EQUATION]', 'cond-mat-0209170-1-15-6': 'In terms of the vertex properties addressed in section [REF], since the interaction is local and therefore independent of [MATH], we may sum freely over the [MATH] momentum for a pair of Laue functions in Eq. [REF] sharing a common interaction wiggly line as depicted in Fig. [REF].', 'cond-mat-0209170-1-15-7': 'As a result, the corresponding Laue function will become', 'cond-mat-0209170-1-16-0': '[EQUATION] and analogously for the DCA, by summing freely over [MATH] [EQUATION]', 'cond-mat-0209170-1-17-0': '# Fluctuation Exchange Approximation (FLEX)', 'cond-mat-0209170-1-18-0': 'In the Feynman diagrammatics of the Hubbard model with a local interaction, all the interactions (wiggly lines in Fig. [REF]) contribute a c-number U from Eq. [REF].', 'cond-mat-0209170-1-18-1': 'The electronic Green functions (solid lines) which interact with one another should have opposite spins.', 'cond-mat-0209170-1-18-2': 'Considering these restrictions, had we been able to include all the possible diagrams in our expansion we would have solved the problem exactly.', 'cond-mat-0209170-1-18-3': 'However, in practice this is not feasible.', 'cond-mat-0209170-1-19-0': 'The Fluctuation Exchange Approximation (FLEX) was introduced as an approximate technique to simplify this diagrammatic sum [CITATION], while retaining a conserving approximation.', 'cond-mat-0209170-1-19-1': 'In the FLEX, the interaction part of the Hubbard model Hamiltonian is treated perturbatively by selecting a certain class of all the possible diagrams which may be summed as a geometric series.', 'cond-mat-0209170-1-19-2': 'Following Baym [CITATION], we define the generating functional [MATH] as the collection of all the selected families of diagrams illustrated in Fig. [REF].', 'cond-mat-0209170-1-19-3': 'Therefore, [MATH] for the FLEX can be written [EQUATION]', 'cond-mat-0209170-1-20-0': 'where Tr = [MATH] with [MATH] the temperature and [MATH] the number of lattice sites.', 'cond-mat-0209170-1-20-1': 'The particle-hole and particle-particle susceptibility bubbles are [EQUATION]', 'cond-mat-0209170-1-21-0': 'The self-energy and the Green function are defined by [EQUATION] where [MATH] is the non-interacting one particle Green function defined by [EQUATION] with [MATH] the non-interacting Hubbard model dispersion and [MATH] the chemical potential.', 'cond-mat-0209170-1-22-0': 'Calculating the self-energy for Eq. [REF] using Eq. [REF] we get [EQUATION] in which [EQUATION]', 'cond-mat-0209170-1-23-0': 'Eq. [REF],[REF] for the potential functions [MATH] and [MATH] are geometric series for [MATH] and [MATH] similar to the random phase approximation (RPA) results.', 'cond-mat-0209170-1-23-1': 'The Hartree term contribution to the self-energy has not explicitly appeared in Eq. [REF] as it is constant and can be always embedded in the chemical potential in Eq. [REF].', 'cond-mat-0209170-1-24-0': 'The difference [MATH] between interacting and non-interacting thermodynamic potential functional is also expressible in terms of the Green functions, self-energy, and [MATH]', 'cond-mat-0209170-1-25-0': '[EQUATION]', 'cond-mat-0209170-1-25-1': 'In the FLEX, since we include only a limited set of all the diagrammatic contributions, we do not anticipate to precisely address the Hubbard model physics.', 'cond-mat-0209170-1-25-2': 'However, there are a number of significant physical features such as anti-ferromagnetic order at half filling and low temperatures that this approximation is able to capture.', 'cond-mat-0209170-1-25-3': 'Moreover, by using the FLEX both together with the DCA and to study finite sized systems with periodic boundary conditions, we can study the differences between these approaches.', 'cond-mat-0209170-1-25-4': 'For example, as we will shortly illustrate, the complementarity of the DCA and finite size lattice techniques is manifest in the FLEX.', 'cond-mat-0209170-1-25-5': 'The FLEX can also be invoked as a good test for the microscopic theory of the DCA and the coarse-graining effects in the compact and non-compact diagrams for Eq. [REF].', 'cond-mat-0209170-1-26-0': '# The Combination of the FLEX and DCA (Algorithm)', 'cond-mat-0209170-1-27-0': 'In the combination of the FLEX and DCA, our goal is to calculate the self-energy in Eq. [REF] whereby we construct the dressed Green function for the lattice as a building block for all the relevant physical quantities.', 'cond-mat-0209170-1-27-1': 'We start out with the bare (non-interacting) Green function [MATH] defined in Eq. [REF] with z the Matsubara frequency (complex).', 'cond-mat-0209170-1-27-2': 'We coarse-grain [MATH] as directed in Eq. [REF] and calculate the self-energy using Eq. [REF].', 'cond-mat-0209170-1-27-3': 'This is used to recompute the dressed Green function [EQUATION] where the index DCA in [MATH] indicates that we have used coarse-grained [MATH] for the construction of self-energy.', 'cond-mat-0209170-1-27-4': 'The new [MATH] is coarse-grained and used to calculate a new estimate of [MATH].', 'cond-mat-0209170-1-27-5': 'We repeat this process iteratively until convergence at a desired tolerance is obtained.', 'cond-mat-0209170-1-27-6': 'The final self-energy is used to construct the dressed Green function in Eq. [REF], required to compute the physical quantities such as spectral function, the density of states (DOS), etc.', 'cond-mat-0209170-1-27-7': 'The algorithm of this calculation is demonstrated in Fig [REF].', 'cond-mat-0209170-1-28-0': '# Complementarity of the DCA to the Finite Size Lattice Approximation with periodic boundary conditions', 'cond-mat-0209170-1-29-0': 'In the half-filled Hubbard model, the antiferromagnetic correlation length [MATH] increases with decreasing temperature and diverges at the phase transition.', 'cond-mat-0209170-1-29-1': 'In a finite size lattice with periodic boundary conditions, as the temperature drops, once the correlation length reaches the size of the lattice, the system is fully frozen and there is a gap to excitations (c.f. Fig [REF].', 'cond-mat-0209170-1-29-2': 'a).', 'cond-mat-0209170-1-29-3': 'In contrast, in the DCA, the correlations are confined within clusters of size [MATH] (the size of the entire lattice) and they never reach the size of the lattice.', 'cond-mat-0209170-1-29-4': 'As we lower the temperature, the correlation length approaches the size of the cluster but since the lattice remains in the thermodynamic limit, it never freezes (c.f. Fig [REF].', 'cond-mat-0209170-1-29-5': 'b).', 'cond-mat-0209170-1-29-6': 'By increasing the size of clusters in the DCA, we take longer ranged correlations into account so the gap will become more pronounced.', 'cond-mat-0209170-1-29-7': 'Consequently, correlation induced gaps are generally overestimated in the finite size lattice, while in the DCA they are underestimated.', 'cond-mat-0209170-1-30-0': 'This complementary behavior may be seen in [MATH], with [MATH] computed using finite size lattices with periodic boundary conditions and the DCA.', 'cond-mat-0209170-1-30-1': 'As illustrated in Fig. [REF], by increasing the size of the finite size lattice and the DCA cluster, the Green functions converge from opposite directions.', 'cond-mat-0209170-1-30-2': 'In the finite size lattice, the Green function (e.g. at [MATH]) decreases with the increase of size which is consistent with overestimating the gap; while in the DCA, the Green function increases as the cluster size grows consistent with underestimating the gap.', 'cond-mat-0209170-1-30-3': 'It is also observed that the convergence in the DCA is much faster, meaning that the result of the DCA is closer to the true curve at a given cluster size.', 'cond-mat-0209170-1-30-4': 'Both finite size lattices with periodic boundary conditions and the DCA converge with corrections of [MATH] with [MATH] the linear size of the finite lattice or the DCA cluster and [MATH] a coefficient.', 'cond-mat-0209170-1-30-5': '[CITATION] The faster convergence of the DCA corresponds to its smaller [MATH] compared to finite size lattices.', 'cond-mat-0209170-1-31-0': 'In Fig. [REF], the values of the [MATH] and [MATH] in Fig. [REF] have been plotted versus [MATH] for both the DCA and finite size results.', 'cond-mat-0209170-1-31-1': 'The Green function behaves linearly as a function of [MATH] for large [MATH].', 'cond-mat-0209170-1-31-2': 'The extrapolations of the DCA and finite size results meet as [MATH], approximating the value of the dressed Green function in the thermodynamic limit.', 'cond-mat-0209170-1-31-3': 'The complementarity of DCA and finite size methods allows a determination of the thermodynamic limit of imaginary time Green functions with unprecedented accuracy.', 'cond-mat-0209170-1-32-0': '# Finite size versus the DCA FLEX results for the two dimensional Hubbard Model at half-filling', 'cond-mat-0209170-1-33-0': 'The Hubbard model at half filling undergoes a phase transition to anti-ferromagnetic order at low temperatures.', 'cond-mat-0209170-1-33-1': 'According to the Mermin-Wagner-Hohenberg theorem, for dimension [MATH] the critical temperature is zero.', 'cond-mat-0209170-1-33-2': 'However, as we continue to lower the temperature, close enough to zero, a pseudogap will appear in the density of states (DOS) as a precursor to the anti-ferromagnetic phase (which has a full gap as its signature).', 'cond-mat-0209170-1-33-3': 'An approach towards non-Fermi-liquid behavior is also visible in both the real and imaginary parts of the retarded self-energy.', 'cond-mat-0209170-1-33-4': '[CITATION]', 'cond-mat-0209170-1-34-0': 'In Fig. [REF] and Fig. [REF], the densities of states (DOS) for lattices with finite sizes of 32[MATH]32, 64[MATH]64 and periodic boundary conditions and coarse-graining cluster sizes of 16[MATH]16 and 32[MATH]32 are plotted.', 'cond-mat-0209170-1-34-1': 'We analytically continue the Green function in order to calculate the spectral function [MATH] and DOS using the Pade approximation.', 'cond-mat-0209170-1-34-2': '[CITATION] In this approximation, we generate a continued fraction interpolating all the data points and use it as an analytic function of the Matsubara frequencies [MATH].', 'cond-mat-0209170-1-34-3': 'The analytic continuation is accomplished by substituting [MATH] with [MATH] where [MATH] is a small positive shift.', 'cond-mat-0209170-1-34-4': 'However, the errors inherent in the numerical Fourier transform (FFT) and also the sharp high-frequency behavior of the Green function, FLEX potentials and the self-energy, limit the accuracy of the Pade.', 'cond-mat-0209170-1-34-5': 'The high-frequency behavior is improved by implementing a more authentic cut off scheme introduced by Deisz et al.[CITATION] in which the high-frequency tails of these quantities are Fourier transformed analytically prior to any numerical FFT and added back to the FFT outputs afterwards.', 'cond-mat-0209170-1-34-6': 'In addition to this high-frequency cut off improvement, we also check for the analyticity of the Pade results in the upper-half frequency plane as a requirement for retarded physical quantities.', 'cond-mat-0209170-1-34-7': 'This task is carried out by converting the continued fraction in the Pade into a ratio of two polynomials.', 'cond-mat-0209170-1-34-8': 'The complex roots of these two polynomials are obtained via the Jenkins-Traub root finder routine.', 'cond-mat-0209170-1-34-9': '[CITATION] Those orders of the Pade for which there exist poles in the upper-half plane are omitted unless these poles are canceled by the roots of the numerator.', 'cond-mat-0209170-1-34-10': 'The acceptable Pade results correspond to the highest order with no uncompensated poles in the upper-half plane.', 'cond-mat-0209170-1-35-0': 'As seen in Fig. [REF], by increasing the size of a finite lattice, the pseudogap occurs at higher temperatures and it also becomes less pronounced (sharper) as we approach the actual size of an infinite real lattice.', 'cond-mat-0209170-1-35-1': 'The DCA yields a complementary behavior as shown in Fig. [REF].', 'cond-mat-0209170-1-35-2': 'By increasing the coarse-graining cluster size, similar to the finite size lattices, the pseudogap is shifted towards higher temperatures.', 'cond-mat-0209170-1-35-3': 'However, unlike the finite size lattices, for the DCA the precursor becomes more pronounced (broader) as the cluster increases in size because the size of the lattice remains constant and the correlations are limited to the cluster size.', 'cond-mat-0209170-1-35-4': 'Thus comparatively, the DCA underestimates the gap while the finite size calculation overestimates it.', 'cond-mat-0209170-1-36-0': 'By comparing the results in Fig. [REF] and Fig. [REF] one may see that the [MATH] DCA cluster yields more realistic physics than the corresponding [MATH] finite size lattice.', 'cond-mat-0209170-1-36-1': 'The [MATH] finite size lattice results are also close to those for the [MATH] DCA cluster at slightly lower [MATH] (eg, [MATH] for the finite size and [MATH] for the DCA).', 'cond-mat-0209170-1-36-2': 'However since the sizes of clusters are considerably smaller than the sizes of lattices, the DCA significantly reduces the complexity of the problem and consequently the CPU time.', 'cond-mat-0209170-1-36-3': 'In terms of the CPU time, the FLEX with the numerical Fourier transforms scales as [MATH] where [MATH] is the product of the total number of Matsubara frequencies and the [MATH] points in the first Brillouin zone.', 'cond-mat-0209170-1-36-4': 'Hence, using a [MATH] cluster in place of a [MATH] lattice both having [MATH] Matsubara frequency points roughly reduces the CPU time by a factor of [MATH].', 'cond-mat-0209170-1-36-5': 'If the DCA cluster size [MATH] equals the size of the finite lattice [MATH], the DCA requires somewhat more CPU time than the finite size lattice due to course-graining.', 'cond-mat-0209170-1-36-6': 'Nevertheless, comparatively, for large finite lattices such as [MATH], the lattice size contributions to the CPU time significantly dominate the coarse-graining ones in a [MATH] cluster.', 'cond-mat-0209170-1-36-7': 'Thus, the [MATH] DCA cluster is much faster.', 'cond-mat-0209170-1-37-0': 'The FLEX often has difficulty reaching low temperatures.', 'cond-mat-0209170-1-37-1': 'This is due to the fact that the [MATH] defined in Eq. [REF] approaches unity as the temperature drops which in turn causes the [MATH] in Eq. [REF] to diverge.', 'cond-mat-0209170-1-37-2': 'In the DCA, [MATH] approaches unity more slowly, allowing the calculations to reach lower temperatures.', 'cond-mat-0209170-1-37-3': 'One has to note that in the FLEX, the [MATH] is defined as follows [EQUATION] with [MATH] defined in Eq. [REF].', 'cond-mat-0209170-1-37-4': 'Fig. [REF] illustrates the saturation of [MATH] for both the DCA and finite size lattices.', 'cond-mat-0209170-1-37-5': 'The [MATH] for the [MATH] finite size lattice (filled circles) saturates at higher temperatures compared to the [MATH] DCA cluster (open diamonds) indicating that in the DCA, for a certain cluster size, the precursor to the phase transition can evolve to lower temperatures compared to a finite size lattice of the same size with periodic boundary conditions.', 'cond-mat-0209170-1-37-6': 'However, for the correlation length [MATH] the DCA approximation breaks down and replacing the self-energy by its coarse-grained counterpart is no longer accurate.', 'cond-mat-0209170-1-37-7': 'Here, the DCA takes on significant mean-field character.', 'cond-mat-0209170-1-38-0': 'Another feature of the Hubbard model near half-filling verified by the FLEX [CITATION] is non-Fermi-liquid behavior.', 'cond-mat-0209170-1-38-1': 'Here, this is studied by increasing the electron electron interaction U at a constant temperature.', 'cond-mat-0209170-1-38-2': 'In Fig. [REF] and Fig. [REF] the real and imaginary parts of the self-energy at the X point (on the non-interacting Fermi surface) have been plotted versus energy for finite size lattices and the DCA respectively.', 'cond-mat-0209170-1-38-3': 'As the interaction is increased, the negative slope in the real part turns positive around [MATH] which is inconsistent with the requirement that the renormalization factor [MATH] should be smaller than unity in the Fermi-liquid theory.', 'cond-mat-0209170-1-38-4': 'There also appears an anomalous inverted peak in the imaginary part at [MATH].', 'cond-mat-0209170-1-39-0': 'As presented in Fig. [REF], by increasing the length of the finite size lattice, the sharpness of the non-Fermi-liquid features is reduced.', 'cond-mat-0209170-1-39-1': 'The same features for the DCA in Fig. [REF] are slightly less pronounced and in a complementary fashion to the finite size lattices, their sharpness is enhanced by increasing the size of the cluster.', 'cond-mat-0209170-1-39-2': 'Thus, again the DCA underestimates the non-Fermi-liquid features while the finite size calculation overestimates it.', 'cond-mat-0209170-1-40-0': 'Earlier in Fig. [REF] it was shown that the FLEX particle-hole bubble [MATH] at [MATH] approaches unity as the temperature is lowered.', 'cond-mat-0209170-1-40-1': 'This causes the spin-fluctuation [MATH] matrix [EQUATION] which is just the third term in [MATH] in Eq. [REF] to peak around the [MATH] point.', 'cond-mat-0209170-1-40-2': 'For real frequencies, [MATH] has a sharp peak around [MATH].', 'cond-mat-0209170-1-40-3': 'Since [MATH] is only used to construct the irreducible self-energy, within the DCA it is constructed from coarse-grained Green functions.', 'cond-mat-0209170-1-40-4': 'Thus, the DCA counterpart of Eq. [REF] is obtained by only replacing the [MATH] with [MATH] defined in Eq. [REF].', 'cond-mat-0209170-1-40-5': 'Fig. [REF] and Fig. [REF] show how this peak sharpens as the temperature decreases or interaction increases for finite size lattices and the DCA respectively.', 'cond-mat-0209170-1-40-6': 'In the [MATH] finite size lattice (c.f. Fig [REF] bottom), the peak continues to develop as the temperature is lowered and the interaction is raised.', 'cond-mat-0209170-1-40-7': 'At [MATH] and [MATH] where there exists a pseudogap in the DOS, the peak undergoes a significant growth compared to the other graphs shown in the same figure.', 'cond-mat-0209170-1-40-8': 'The [MATH] (c.f. Fig [REF].', 'cond-mat-0209170-1-40-9': 'top) lattice presents the same behavior with a slightly sharper but shorter peak.', 'cond-mat-0209170-1-41-0': 'The DCA also illustrates the same type of peaks at a slightly higher interaction and lower temperatures for the [MATH] cluster (c.f. Fig [REF] top).', 'cond-mat-0209170-1-41-1': 'Increasing the cluster size to [MATH] gives rise to higher peaks similar to the finite size lattices case but unlike the finite size lattices peaks become sharper as the size is increased.', 'cond-mat-0209170-1-42-0': 'All the results illustrated in this section indicate the complementarity of the DCA to the finite size lattice scheme.', 'cond-mat-0209170-1-42-1': 'It is also observed that the DCA is capable of reproducing relatively the same physics as the finite size FLEX at slightly different parameters but a lower CPU cost.', 'cond-mat-0209170-1-42-2': 'The combination of these two facts makes this technique a good candidate to be employed in the numerical treatment of a wide range of many-body problems.', 'cond-mat-0209170-1-43-0': 'Lastly, to better understand the effect of the DCA cluster embedded in a Fermionic bath, we can rewrite the course grained Green function defined in Eq. [REF] as', 'cond-mat-0209170-1-44-0': '[EQUATION]', 'cond-mat-0209170-1-45-0': 'where [MATH] and [MATH] is the host function.', 'cond-mat-0209170-1-45-1': 'Maier et al. [CITATION], define [EQUATION] whereby [MATH] can be expressed as', 'cond-mat-0209170-1-46-0': '[EQUATION]', 'cond-mat-0209170-1-47-0': 'By Taylor expanding [MATH] around the cluster points [MATH] it is found that [MATH] with [MATH].', 'cond-mat-0209170-1-47-1': 'Thus, Eq. [REF] yields [MATH] as [MATH].', 'cond-mat-0209170-1-47-2': 'To illustrate this, we calculate [MATH] by summing over all the [MATH] points and [MATH] frequencies and plot it versus [MATH].', 'cond-mat-0209170-1-47-3': 'Fig. [REF] illustrates this linear behavior for [MATH].', 'cond-mat-0209170-1-47-4': '[MATH] holds complete mean field characters and no non-local fluctuations.', 'cond-mat-0209170-1-47-5': '[MATH] is anomalous as explained in an article by Betts et al. [CITATION] There, the finite size cubic lattices with less than six (four in [MATH] square lattices) distinct nearest neighbors per each site are not used in finite size scalings for estimating the physical properties of models like the spin one half XY ferromagnet or the Heisenberg antiferromagnet.', 'cond-mat-0209170-1-47-6': 'For [MATH], because of the periodicity of the clusters, each cluster point is surrounded by two identical nearest neighbors in every direction and therefore has only two distinct nearest neighbors (c.f. Fig. [REF]).', 'cond-mat-0209170-1-47-7': 'Thus, the effect of fluctuations are overestimated.', 'cond-mat-0209170-1-47-8': '[CITATION] For [MATH] there is no such anomaly and hence, all the points present the linear behavior proven above.', 'cond-mat-0209170-1-47-9': 'Nevertheless, calculations with [MATH] do a reasonable job in capturing the qualitative effects of corrections to the DMFA.', 'cond-mat-0209170-1-48-0': '# Microscopic theory of the DCA', 'cond-mat-0209170-1-49-0': 'In section [REF] we defined the thermodynamic potential functional difference [MATH] in terms of the Green function, self-energy, and [MATH] (c.f. Eq. [REF]).', 'cond-mat-0209170-1-49-1': 'In Eq. [REF], [MATH] includes all the compact (skeletal) Feynman diagrams and the rest incorporates the entire non-compact contribution.', 'cond-mat-0209170-1-49-2': '[CITATION] Typical compact and non-compact diagrams are illustrated in Fig. [REF].', 'cond-mat-0209170-1-49-3': 'The non-compact diagram (a) consists of two self-energy pieces [MATH] and [MATH] connected with two one-particle Green functions.', 'cond-mat-0209170-1-49-4': 'Removing these two Green functions would split the diagram into two separate pieces.', 'cond-mat-0209170-1-49-5': 'In the compact diagram (b) two vertex parts [MATH] and [MATH] with four Green functions are connected together.', 'cond-mat-0209170-1-49-6': 'One can not split this type of diagrams into two separate pieces by just removing two one-particle Green functions.', 'cond-mat-0209170-1-49-7': 'As mentioned earlier, in the DCA, we employ coarse-grained Green functions to construct only the compact diagrams.', 'cond-mat-0209170-1-49-8': 'The Green functions in non-compact diagrams are calculated directly using Eq. [REF] in which the self-energy [MATH] is coarse-grained (the circles at the top and the bottom of the non-compact diagram in Fig. [REF].', 'cond-mat-0209170-1-49-9': '(a))', 'cond-mat-0209170-1-50-0': 'Earlier in a shorter article for the Hubbard model in particular [CITATION], we showed both analytically and numerically that the error produced by coarse-graining the non-compact diagrams is significantly larger than the error produced by coarse-graining the compact ones.', 'cond-mat-0209170-1-50-1': "Here, we would like to give a more general argument based on the Green function's exponential fall off as a function of distance in the real space.", 'cond-mat-0209170-1-51-0': 'Such exponential behavior occurs naturally in high dimensions.', 'cond-mat-0209170-1-51-1': 'In the tight-binding Hamiltonian (c.f. Eq. [REF]), the factor [MATH] corresponds to the hopping of electrons among nearest neighboring sites.', 'cond-mat-0209170-1-51-2': 'Thus, one could show that the real space Green function [MATH] (we drop the frequency label from this point on for simplicity) for [MATH] nearest neighbor hops is proportional to [MATH] as [MATH].', 'cond-mat-0209170-1-51-3': 'On the other hand, Metzner et al., and Muller-Hartmann [CITATION] have shown that in [MATH] dimensions, the factor [MATH] should be renormalized as [MATH] in order to have a finite density of states width as [MATH].', 'cond-mat-0209170-1-51-4': 'As a result of this renormalization [EQUATION] meaning that [MATH] falls off exponentially as a function of [MATH].', 'cond-mat-0209170-1-52-0': 'In the DCA, we attempt to minimize the error due to coarse-graining the Green function (and potentials) in the Feynman diagrams.', 'cond-mat-0209170-1-52-1': 'Consider the first non-trivial correction to the coarse-grained non-compact diagrams generated by replacing the explicit coarse-grained Green function lines by the non-coarse-grained ones as illustrated in Fig. [REF] [EQUATION] where [MATH] are the coarse-graining cells momenta and [MATH] include all the momenta in the first Brillouin zone shown in Fig. [REF].', 'cond-mat-0209170-1-52-2': 'In this derivation we also presume that the self-energy is [MATH] independent and the entire [MATH] dependence is embedded in the Green functions.', 'cond-mat-0209170-1-53-0': 'By breaking up the sums over [MATH] in Eq. [REF] into [MATH] and [MATH] sums and writing all the G in terms of their Fourier transforms at the same time we get [EQUATION] in which we used Eq. [REF] for [MATH].', 'cond-mat-0209170-1-53-1': 'According to Fig. [REF], we can split [MATH] where [MATH] extend between two separate clusters while [MATH] always remain within a single cluster.', 'cond-mat-0209170-1-53-2': 'By making this separation in Eq. [REF] one picks up phases including products of [MATH], [MATH], [MATH] and [MATH] in their exponents.', 'cond-mat-0209170-1-53-3': 'The phase associated with the product [MATH] with [MATH] an integer equals unity.', 'cond-mat-0209170-1-53-4': 'The phases involving [MATH] products are also neglected as discussed in section [REF].', 'cond-mat-0209170-1-53-5': 'Hence, Eq. [REF] can be rewritten as follows [EQUATION]', 'cond-mat-0209170-1-53-6': 'Implementing the following substitutions [EQUATION]', 'cond-mat-0209170-1-53-7': 'Eq. [REF] simplifies into [EQUATION]', 'cond-mat-0209170-1-53-8': 'Setting [MATH] and performing the [MATH] summation [EQUATION]', 'cond-mat-0209170-1-53-9': 'Knowing that [MATH] and also that the lowest order of [MATH] we conclude that in Eq. [REF], the largest contribution is due to terms having [MATH] or in other words, local [MATH].', 'cond-mat-0209170-1-53-10': 'As shown in Fig. [REF], the first term in the [MATH] sum corresponds to [MATH] (size of the cluster) and [MATH] can be as large as [MATH] in the opposite direction.', 'cond-mat-0209170-1-53-11': 'Hence, the leading order term in Eq. [REF] falls off as [EQUATION] where [MATH] is the number of [MATH] contributions in [MATH] different dimensions of a [MATH] dimensional cubic lattice.', 'cond-mat-0209170-1-53-12': 'In Eq. [REF] we also used the fact that due to the lattice symmetry, [MATH].', 'cond-mat-0209170-1-53-13': 'As [MATH], using [MATH] [EQUATION] which indicates the existence of non-local corrections to the non-compact contribution of the thermodynamic potential even at infinite dimensions.', 'cond-mat-0209170-1-54-0': 'Now we replace the coarse-grained self-energy bubbles in Fig. [REF] with coarse-grained vertices having four external legs and look at the difference between compact diagrams with and without coarse-grained Green functions which are explicitly shown in the figure.', 'cond-mat-0209170-1-54-1': 'Since we earlier dropped the frequency labels in the Green functions, here we use indices [MATH] and [MATH] to emphasize that these Green functions have different frequency labels.', 'cond-mat-0209170-1-54-2': 'The first correction to the compact contribution of the thermodynamic potential depicted in Fig. [REF] is [EQUATION] where similar to Eq. [REF], all the vertices are coarse-grained but the Green functions are not.', 'cond-mat-0209170-1-55-0': 'Following the same procedure as for the non-compact difference we arrive at [EQUATION] where [EQUATION]', 'cond-mat-0209170-1-55-1': 'Once again, the largest contribution is associated with local [MATH], i.e., [MATH], [MATH] and [MATH].', 'cond-mat-0209170-1-55-2': 'Therefore [EQUATION]', 'cond-mat-0209170-1-55-3': 'Considering [MATH] and [MATH] [EQUATION] which vanishes as [MATH].', 'cond-mat-0209170-1-56-0': 'Comparing Eq. [REF] with Eq. [REF] shows that the first correction to the compact contribution of the thermodynamic potential falls off exponentially twice as fast as the equivalent correction in the non-compact contribution.', 'cond-mat-0209170-1-56-1': 'In addition, even at the infinite dimensional limit, there are corrections of order one to the non-compact contribution whereas for the compact diagrams the DCA becomes exact and there are no corrections.', 'cond-mat-0209170-1-56-2': 'This justifies coarse-graining only in the compact diagrams.', 'cond-mat-0209170-1-56-3': 'The Green functions in the non-compact diagrams have to be explicitly constructed from the coarse-grained self-energy [MATH] using Eq. [REF].', 'cond-mat-0209170-1-57-0': 'In Fig. [REF], the compact contribution of the thermodynamic potential difference constructed by coarse-grained Green functions is plotted versus [MATH].', 'cond-mat-0209170-1-57-1': 'The variation of [MATH] over the entire [MATH] range is about [MATH].', 'cond-mat-0209170-1-57-2': 'At very low temperatures ([MATH] in the inset), some deviation from linearity is observed due to the correlation length exceeding the cluster size [MATH] and therefore the approximation of [MATH] by [MATH] begins to break down.', 'cond-mat-0209170-1-57-3': 'Fig. [REF] illustrates the non-compact contribution sketched versus [MATH] using both coarse-grained and non-coarse-grained Green functions.', 'cond-mat-0209170-1-57-4': 'Using non-coarse-grained Green functions, the variation of [MATH] over the entire [MATH] range is roughly [MATH].', 'cond-mat-0209170-1-57-5': 'Coarse-graining the Green functions in these diagrams can even change the sign of this non-compact contributions, clearly indicating that coarse-graining the Green function is an unrealistic approximation for non-compact diagrams.', 'cond-mat-0209170-1-57-6': 'However, one notices that for large cluster sizes the coarse-grained and non-coarse-grained results approach each other as the approximation to the infinite lattice becomes better.', 'cond-mat-0209170-1-58-0': '# DCA in frequency space', 'cond-mat-0209170-1-59-0': 'As illustrated for the momentum space, the DCA results in a significant reduction of the problem complexity and it is complementary to the finite size lattice approach.', 'cond-mat-0209170-1-59-1': 'In analogy to the momentum space, one could consider dividing the one dimensional Matsubara frequency space into a number of coarse-graining subcells.', 'cond-mat-0209170-1-59-2': 'For both fermions and bosons, each cell should include an odd number of frequencies in order for the frequencies in the centers of these cells to preserve Fermionic or Bosonic properties.', 'cond-mat-0209170-1-59-3': 'Fig. [REF] represents how the frequency space can be divided into coarse-graining subcells each comprising a central [MATH] frequency and a number of coarse-graining [MATH] lying around it.', 'cond-mat-0209170-1-59-4': 'The central [MATH] frequencies can be rewritten in the form of the original lattice with renormalized [MATH] shown as [MATH] in the figure.', 'cond-mat-0209170-1-59-5': 'Similar to the case of the momentum space, we make the following transformation for the Laue function [EQUATION] for the Matsubara frequencies of the vertex shown in Fig. [REF] considering frequency dependent interactions in general (in condensed matter physics, most of the interactions are indeed simultaneous and thus frequency independent).', 'cond-mat-0209170-1-59-6': 'As a result, we may again coarse-grain the Green function over the subcell frequencies [EQUATION]', 'cond-mat-0209170-1-59-7': 'According to Fig. [REF], the full coarse-graining of the Green function amounts to [MATH] which causes all the self-energy Feynman diagrams ordered higher than first (Hartree-Fock diagrams) to vanish and consequently we arrive at a fully static problem.', 'cond-mat-0209170-1-60-0': 'Unfortunately, we can show that coarse-graining over Matsubara frequencies can lead to the violation of causality and as a result, the DCA is not systematically implementable for the Matsubara frequency quantities.', 'cond-mat-0209170-1-60-1': 'The simplest example is the non-interacting Green function coarse-grained as follows [CITATION] [EQUATION]', 'cond-mat-0209170-1-60-2': 'The retarded Green function is derived by the substitution [MATH] [EQUATION]', 'cond-mat-0209170-1-60-3': 'The theory of analytic functions of a complex variable tells us that in order for the Green function to remain retarded in the time space, [MATH] must not have any poles in the upper half plane of [MATH].', 'cond-mat-0209170-1-60-4': 'In Eq. [REF], one could readily create poles in the upper half plane for negative [MATH] which causes causality violation and consequently unphysical results.', 'cond-mat-0209170-1-61-0': 'The way around this difficulty is to use real frequencies even at finite temperatures.', 'cond-mat-0209170-1-61-1': 'By invoking real frequencies we no longer coarse-grain over imaginary values.', 'cond-mat-0209170-1-61-2': 'Therefore, the retarded Green function will never acquire poles in the upper half plane and remains causal as shown below [EQUATION] with [MATH] the total number of frequencies and [MATH] the number of those we coarse-graining over in each cell.', 'cond-mat-0209170-1-61-3': 'In an article by Hettler et al. [CITATION], a formal proof of causality is given (based on a geometrical argument) for coarse-graining in the momentum space.', 'cond-mat-0209170-1-61-4': 'This proof can be straightforwardly applied to the real frequency space as well and it extends the application of real frequency DCA not only to the perturbative cluster solvers such as the FLEX but also techniques like the NCA.', 'cond-mat-0209170-1-62-0': 'Lastly, similar to the momentum space, care must be taken when choosing the the size of the frequency coarse-graining cells.', 'cond-mat-0209170-1-62-1': 'One must make sure that the cells are not larger than some characteristic energy scale (e.g. the Kondo temperature [MATH]) as the coarse-graining would then suppress the relevant physics.', 'cond-mat-0209170-1-63-0': '# Conclusions and Outlook', 'cond-mat-0209170-1-64-0': 'We introduce and examine the DCA in detail by employing it with the FLEX to study the half filled two dimensional Hubbard model.', 'cond-mat-0209170-1-64-1': 'The FLEX is not as precise as nearly exact techniques such as quantum Monte Carlo in describing the Hubbard model at strong interaction regime.', 'cond-mat-0209170-1-64-2': 'However, it is capable of illustrating the utility of the DCA, including the complementarity and convergence of the DCA compared to finite size lattice approaches.', 'cond-mat-0209170-1-64-3': 'The DCA and finite size calculations (with periodic boundary conditions) both converge with corrections [MATH]; however, in our example the coefficient [MATH] was smaller and of opposite sign than [MATH], indicating that the DCA converges more quickly and from a complementary direction.', 'cond-mat-0209170-1-64-4': 'This complementarity was also seen in other quantities such as the pseudogap in the density of state and the non-Fermi liquid behaviour that the DCA (finite size) calculation systematically under (over) estimates.', 'cond-mat-0209170-1-65-0': 'We also provide a detailed microscopic definition of the DCA by inspecting the the error generated by coarse-graining the Green functions in the compact and non-compact contributions to the thermodynamic potential.', 'cond-mat-0209170-1-65-1': 'We conclude that due to the large magnitude of error that it generates, coarse-graining the Green function in non-compact part should be avoided and only the compact contribution should undergo coarse-graining.', 'cond-mat-0209170-1-65-2': 'It also appears that coarse-graining the Green functions over the Matsubara frequencies can and will lead to the violation of causality and therefore is pathological.', 'cond-mat-0209170-1-65-3': 'Nevertheless, one can coarse-grain the Green function over real frequencies and preserve the causality not only for the FLEX but also cluster solvers such as NCA in which the cluster contribution to the coarse-grained dressed Green function is excluded before being inserted in the cluster solver.', 'cond-mat-0209170-1-66-0': 'The outlook for the FLEX-DCA approach is promising.', 'cond-mat-0209170-1-66-1': 'Although the FLEX fails to accurately describe short-ranged physics such as moment formation (and related phenomena like the Mott gap), it does a good job describing long-ranged physics associated with spin and charge fluctuations.', 'cond-mat-0209170-1-66-2': 'On the other hand, numerically exact calculations such as QMC, are too expensive to perform for large clusters, and are thus restricted to the study of short-length scales.', 'cond-mat-0209170-1-66-3': 'The DCA gives us a way of parsing the problem into different length scales.', 'cond-mat-0209170-1-66-4': 'Thus, it may be used to embed the QMC cluster into a host described by the FLEX-DCA calculation which is itself embedded in a mean-field.', 'cond-mat-0209170-1-66-5': 'Work along these lines is in progress.', 'cond-mat-0209170-1-67-0': 'We would like to acknowledge J.J Deisz, D.W. Hess Th.', 'cond-mat-0209170-1-67-1': 'Maier, S. Moukouri, and Th.', 'cond-mat-0209170-1-67-2': 'Pruschke for very useful discussions and suggestions.', 'cond-mat-0209170-1-68-0': 'Pruschke T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys.', 'cond-mat-0209170-1-69-0': 'Georges A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod.', 'cond-mat-0209170-1-70-0': 'DCA_hettler M. H. Hettler, M. Mukherjee, M. Jarrell, and H. R. Krishnamurthy, Phys.', 'cond-mat-0209170-1-71-0': 'DCA_hettler2 M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H. R. Krishnamurthy, Phys.', 'cond-mat-0209170-1-72-0': 'Bickers1 N. E. Bickers, D. J. Scalapino, and S. R. White, Phys.', 'cond-mat-0209170-1-73-0': 'Deisz J. J. Deisz, D. W. Hess, and J. W. Serene, Phys.', 'cond-mat-0209170-1-74-0': 'Vidberg H. J. Vidberg and J. W. Serene, J. Low Temp.', 'cond-mat-0209170-1-75-0': 'DHS J. J Deisz, D. W. Hess, and J. W. Serene, cond-mat/9411026R (To appear in "Recent Progress In Many Body Theories", vol.', 'cond-mat-0209170-1-75-1': '4, edited by E. Schachinger, et al. (Plenum, New York)).', 'cond-mat-0209170-1-76-0': 'Jenkins M. A. Jenkins, and J. F. Traub, Comm.', 'cond-mat-0209170-1-76-1': 'The routine is available at http://www.netlib.org/tomspdf/419.pdf.', 'cond-mat-0209170-1-77-0': 'agd A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, (Dover, New York, 1975).'}

{'cond-mat-0209170-2-0-0': 'We employ the Dynamical Cluster Approximation (DCA) in conjunction with the Fluctuation Exchange Approximation (FLEX) to study the Hubbard model.', 'cond-mat-0209170-2-0-1': 'The DCA is a technique to systematically restore the momentum conservation at the internal vertices of Feynman diagrams relinquished in the Dynamical Mean Field Approximation (DMFA).', 'cond-mat-0209170-2-0-2': 'FLEX is a perturbative diagrammatic approach in which classes of Feynman diagrams are summed over analytically using geometric series.', 'cond-mat-0209170-2-0-3': 'The FLEX is used as a tool to investigate the complementarity of the DCA and the finite size lattice technique with periodic boundary conditions by comparing their results for the Hubbard model.', 'cond-mat-0209170-2-0-4': 'We also study the microscopic theory underlying the DCA in terms of compact (skeletal) and non-compact diagrammatic contributions to the thermodynamic potential independent of a specific model.', 'cond-mat-0209170-2-0-5': 'The significant advantages of the DCA implementation in momentum space suggests the development of the same formalism for the frequency space.', 'cond-mat-0209170-2-0-6': 'However, we show that such a formalism for the Matsubara frequencies at finite temperatures leads to acausal results and is not viable.', 'cond-mat-0209170-2-0-7': 'However, a real frequency approach is shown to be feasible.', 'cond-mat-0209170-2-1-0': 'Introduction Non-local correlations play an important role in the physics of strongly correlated electron systems such as high-[MATH] superconductors, heavy fermion metals, etc.', 'cond-mat-0209170-2-1-1': 'The Dynamical Mean Field Approximation (DMFA) [CITATION], in which all the non-local correlations are ignored, can capture some of the major features of strongly correlated systems.', 'cond-mat-0209170-2-1-2': 'Nevertheless, the non-local correlations become crucial in the physics of phases with non-local order parameters such as d-wave superconductivity.', 'cond-mat-0209170-2-1-3': 'Even phases with local order parameters such as commensurate magnetism can be significantly affected by the non-local correlations (e.g. spin waves) ignored in the DMFA.', 'cond-mat-0209170-2-2-0': 'The early attempts to extend the DMFA by including non-local correlations resulted in the violation of causality which is a requirement for positive definiteness of the spectral weight and the density of states (DOS) [CITATION].', 'cond-mat-0209170-2-2-1': 'The Dynamical Cluster Approximation (DCA) is a fully causal technique used to systematically add nonlocal corrections to the DMFA by mapping the lattice onto a self-consistently embedded cluster problem.', 'cond-mat-0209170-2-2-2': 'The mapping from the lattice to the cluster is accompanied by coarse-graining the lattice problem in its reciprocal space.', 'cond-mat-0209170-2-2-3': 'Thus far, the DCA has been combined with Quantum Monte Carlo (QMC) [CITATION], the Non-Crossing Approximation (NCA) [CITATION] and the Fluctuation Exchange Approximation (FLEX) [CITATION] to solve the corresponding cluster problems.', 'cond-mat-0209170-2-3-0': 'The FLEX is a perturbative diagrammatic approach in which classes of Feynman diagrams are summed to all orders using geometric series.', 'cond-mat-0209170-2-3-1': '[CITATION] Others [CITATION] have employed the FLEX for finite size lattices with periodic boundary conditions.', 'cond-mat-0209170-2-3-2': 'Due to the absence of contributions from some relevant diagrams, the FLEX is not capable of addressing the Hubbard model physics in the strong regime precisely.', 'cond-mat-0209170-2-3-3': 'However, the main objective of this work is to make a comparison between the DCA-FLEX combination results and previous finite size lattice FLEX calculations.', 'cond-mat-0209170-2-3-4': 'It is hoped that this study will lead to a better understanding of the DCA.', 'cond-mat-0209170-2-4-0': 'We earlier[CITATION] suggested a prescription to correctly implement the DCA technique in the thermodynamic potential.', 'cond-mat-0209170-2-4-1': 'This prescription will be discussed from a different point of view using a more general argument.', 'cond-mat-0209170-2-4-2': "Based upon the Green function's exponential fall-off as a function of distance, we conclude that compact diagrams (two-particle irreducible in the thermodynamic potential) are better approximated using the DCA than non-compact (two-particle reducible) ones.", 'cond-mat-0209170-2-4-3': 'Hence, the DCA is applied to the compact diagrams only and non-compact ones are calculated explicitly using dressed non coarse-grained Green functions.', 'cond-mat-0209170-2-5-0': 'In this work we also consider the extension of the DCA to frequency space.', 'cond-mat-0209170-2-5-1': 'The many-body theory at finite temperatures is conventionally derived in terms of discrete imaginary Matsubara frequencies.', 'cond-mat-0209170-2-5-2': 'We illustrate that even for a self-consistent algorithm like the FLEX, coarse-graining the imaginary frequency propagators results in causality violations and can not be implemented.', 'cond-mat-0209170-2-5-3': 'However, a real frequency formalism is shown to be causal and applicable not only to the FLEX, but also to other cluster solving methods such as the NCA.', 'cond-mat-0209170-2-6-0': 'This paper is structured as follows.', 'cond-mat-0209170-2-6-1': 'In the next three sections, we briefly review the DCA, its application to the Hubbard model, the FLEX, and then we describe how the FLEX and the DCA may be merged into a single algorithm.', 'cond-mat-0209170-2-6-2': 'In the next three sections, we use the FLEX-DCA, in comparison to the FLEX for finite-sized systems, to explore the properties of the DCA.', 'cond-mat-0209170-2-6-3': 'The last two sections, are devoted to a microscopic derivation of the DCA, and to an extension of the DCA to frequency space.', 'cond-mat-0209170-2-7-0': '# Dynamical Cluster Approximation (DCA)', 'cond-mat-0209170-2-8-0': 'Both the DCA and the DMFA may be derived by exploring the momentum conservation in the diagrammatics.', 'cond-mat-0209170-2-8-1': 'As depicted in Fig. [REF], momentum conservation at each vertex is described by the Laue function:', 'cond-mat-0209170-2-9-0': '[EQUATION]', 'cond-mat-0209170-2-9-1': 'In the DMFA, momentum conservation at the internal vertices of irreducible Feynman diagrams is completely relinquished.', 'cond-mat-0209170-2-9-2': 'I.e., the DMFA simply sets [MATH].', 'cond-mat-0209170-2-9-3': '[CITATION] Hence, we may sum freely over all the internal momenta entering and leaving each vertex.', 'cond-mat-0209170-2-9-4': 'Only local contributions survive the sum.', 'cond-mat-0209170-2-9-5': 'Thus, this is equivalent to mapping the lattice problem onto a self-consistently embedded impurity problem.', 'cond-mat-0209170-2-9-6': 'The DMFA becomes exact at infinite dimensions.', 'cond-mat-0209170-2-9-7': '[CITATION]', 'cond-mat-0209170-2-10-0': 'The DCA is an approach to systematically restore the momentum conservation relinquished in the DMFA.', 'cond-mat-0209170-2-10-1': 'In the DCA, the first Brillouin zone in the reciprocal space is divided into [MATH] equal cells of linear size [MATH] labeled by [MATH] in their centers, and the momenta within each cell are labeled by [MATH].', 'cond-mat-0209170-2-10-2': 'Then [MATH] (c.f. Fig. [REF]).', 'cond-mat-0209170-2-10-3': 'To visualize this scheme in the real lattice, one could consider tiling the lattice of [MATH] sites by [MATH] clusters each composed of [MATH] sites where [MATH] is the linear size of the subcell and D is dimensionality (c.f. Fig. [REF] for [MATH]).', 'cond-mat-0209170-2-10-4': 'We will use this picture in section [REF] while discussing the microscopic theory of the DCA.', 'cond-mat-0209170-2-10-5': 'We label the origin of the clusters by [MATH] and the [MATH] intercluster sites by [MATH].', 'cond-mat-0209170-2-10-6': 'So for each site in the original lattice [MATH].', 'cond-mat-0209170-2-10-7': 'Care must be taken when choosing the cluster geometries in order to preserve the lattice point group symmetry and also satisfy some other criteria for cubic or square lattices.', 'cond-mat-0209170-2-10-8': '[CITATION]', 'cond-mat-0209170-2-11-0': 'In the DCA, we first make the following separation in Eq. [REF] [EQUATION]', 'cond-mat-0209170-2-11-1': 'The products [MATH] and [MATH] where [MATH] is an integer.', 'cond-mat-0209170-2-11-2': 'Therefore, their associated phases may be neglected and Eq. [REF] splits into two parts [EQUATION]', 'cond-mat-0209170-2-11-3': 'The DCA also ignores the phases [MATH] due to the position of the cluster in the original lattice and (far less important) [MATH] corresponding to the position within the cluster.', 'cond-mat-0209170-2-11-4': 'As a result, it approximates [MATH], so that [EQUATION] which indicates that the momentum is conserved modulo [MATH] for transfers between the cells.', 'cond-mat-0209170-2-12-0': 'The approximation made through the substitution [MATH] corresponds to replacing all internal legs in the compact (skeletal) diagrams by the coarse-grained Green function [MATH] and interaction potential [MATH] defined by [EQUATION]', 'cond-mat-0209170-2-12-1': 'In section [REF], we will define the compact and non compact diagrams and elaborately discuss why only the compact ones undergo the coarse-graining approximation.', 'cond-mat-0209170-2-13-0': 'Replacing [MATH] by [MATH] tremendously reduces the complexity of the problem because instead of having to perform sums over all [MATH] states in the entire first Brillouin zone, we have sums over only a set of [MATH] states where [MATH].', 'cond-mat-0209170-2-14-0': '# Hubbard Model', 'cond-mat-0209170-2-15-0': 'We will apply the DCA to study the Hubbard model Hamiltonian incorporating interactions between the electrons on a lattice.', 'cond-mat-0209170-2-15-1': 'It includes a tight-binding part due to the hopping of electrons among the sites and an interaction between the electrons.', 'cond-mat-0209170-2-15-2': 'The general Hamiltonian reads [EQUATION] where [EQUATION] and [EQUATION]', 'cond-mat-0209170-2-15-3': 'Factors [MATH] and [MATH] correspond to electron hoppings and Coulomb interactions respectively.', 'cond-mat-0209170-2-15-4': 'Later in the paper, we will study the simplest Hubbard interaction which is fully local and only between electrons sitting at the same site having opposite spin directions.', 'cond-mat-0209170-2-15-5': 'The interaction strength is a constant called U. Hence, for the local model, Eq. [REF] simplifies to [EQUATION]', 'cond-mat-0209170-2-15-6': 'In terms of the vertex properties addressed in section [REF], since the interaction is local and therefore independent of [MATH], we may sum freely over the [MATH] momentum for a pair of Laue functions in Eq. [REF] sharing a common interaction wiggly line as depicted in Fig. [REF].', 'cond-mat-0209170-2-15-7': 'As a result, the corresponding Laue function will become', 'cond-mat-0209170-2-16-0': '[EQUATION] and analogously for the DCA, by summing freely over [MATH] [EQUATION]', 'cond-mat-0209170-2-17-0': '# Fluctuation Exchange Approximation (FLEX)', 'cond-mat-0209170-2-18-0': 'In the Feynman diagrammatics of the Hubbard model with a local interaction, all the interactions (wiggly lines in Fig. [REF]) contribute a c-number U from Eq. [REF].', 'cond-mat-0209170-2-18-1': 'The electronic Green functions (solid lines) which interact with one another should have opposite spins.', 'cond-mat-0209170-2-18-2': 'Considering these restrictions, had we been able to include all the possible diagrams in our expansion we would have solved the problem exactly.', 'cond-mat-0209170-2-18-3': 'However, in practice this is not feasible.', 'cond-mat-0209170-2-19-0': 'The Fluctuation Exchange Approximation (FLEX) was introduced as an approximate technique to simplify this diagrammatic sum [CITATION], while retaining a conserving approximation.', 'cond-mat-0209170-2-19-1': 'In the FLEX, the interaction part of the Hubbard model Hamiltonian is treated perturbatively by selecting a certain class of all the possible diagrams which may be summed as a geometric series.', 'cond-mat-0209170-2-19-2': 'Following Baym [CITATION], we define the generating functional [MATH] as the collection of all the selected families of diagrams illustrated in Fig. [REF].', 'cond-mat-0209170-2-19-3': 'Therefore, [MATH] for the FLEX can be written [EQUATION]', 'cond-mat-0209170-2-20-0': 'where Tr = [MATH] with [MATH] the temperature and [MATH] the number of lattice sites.', 'cond-mat-0209170-2-20-1': 'The particle-hole and particle-particle susceptibility bubbles are [EQUATION]', 'cond-mat-0209170-2-21-0': 'The self-energy and the Green function are defined by [EQUATION] where [MATH] is the non-interacting one particle Green function defined by [EQUATION] with [MATH] the non-interacting Hubbard model dispersion and [MATH] the chemical potential.', 'cond-mat-0209170-2-22-0': 'Calculating the self-energy for Eq. [REF] using Eq. [REF] we get [EQUATION] in which [EQUATION]', 'cond-mat-0209170-2-23-0': 'Eq. [REF],[REF] for the potential functions [MATH] and [MATH] are geometric series for [MATH] and [MATH] similar to the random phase approximation (RPA) results.', 'cond-mat-0209170-2-23-1': 'The Hartree term contribution to the self-energy has not explicitly appeared in Eq. [REF] as it is constant and can be always embedded in the chemical potential in Eq. [REF].', 'cond-mat-0209170-2-24-0': 'The difference [MATH] between interacting and non-interacting thermodynamic potential functional is also expressible in terms of the Green functions, self-energy, and [MATH]', 'cond-mat-0209170-2-25-0': '[EQUATION]', 'cond-mat-0209170-2-25-1': 'In the FLEX, since we include only a limited set of all the diagrammatic contributions, we do not anticipate to precisely address the Hubbard model physics.', 'cond-mat-0209170-2-25-2': 'However, there are a number of significant physical features such as anti-ferromagnetic order at half filling and low temperatures that this approximation is able to capture.', 'cond-mat-0209170-2-25-3': 'Moreover, by using the FLEX both together with the DCA and to study finite sized systems with periodic boundary conditions, we can study the differences between these approaches.', 'cond-mat-0209170-2-25-4': 'For example, as we will shortly illustrate, the complementarity of the DCA and finite size lattice techniques is manifest in the FLEX.', 'cond-mat-0209170-2-25-5': 'The FLEX can also be invoked as a good test for the microscopic theory of the DCA and the coarse-graining effects in the compact and non-compact diagrams for Eq. [REF].', 'cond-mat-0209170-2-26-0': '# The Combination of the FLEX and DCA (Algorithm)', 'cond-mat-0209170-2-27-0': 'In the combination of the FLEX and DCA, our goal is to calculate the self-energy in Eq. [REF] whereby we construct the dressed Green function for the lattice as a building block for all the relevant physical quantities.', 'cond-mat-0209170-2-27-1': 'We start out with the bare (non-interacting) Green function [MATH] defined in Eq. [REF] with z the Matsubara frequency (complex).', 'cond-mat-0209170-2-27-2': 'We coarse-grain [MATH] as directed in Eq. [REF] and calculate the self-energy using Eq. [REF].', 'cond-mat-0209170-2-27-3': 'This is used to recompute the dressed Green function [EQUATION] where the index DCA in [MATH] indicates that we have used coarse-grained [MATH] for the construction of self-energy.', 'cond-mat-0209170-2-27-4': 'The new [MATH] is coarse-grained and used to calculate a new estimate of [MATH].', 'cond-mat-0209170-2-27-5': 'We repeat this process iteratively until convergence at a desired tolerance is obtained.', 'cond-mat-0209170-2-27-6': 'The final self-energy is used to construct the dressed Green function in Eq. [REF], required to compute the physical quantities such as spectral function, the density of states (DOS), etc.', 'cond-mat-0209170-2-27-7': 'The algorithm of this calculation is demonstrated in Fig [REF].', 'cond-mat-0209170-2-28-0': '# Complementarity of the DCA to the Finite Size Lattice Approximation with periodic boundary conditions', 'cond-mat-0209170-2-29-0': 'In the half-filled Hubbard model, the antiferromagnetic correlation length [MATH] increases with decreasing temperature and diverges at the phase transition.', 'cond-mat-0209170-2-29-1': 'In a finite size lattice with periodic boundary conditions, as the temperature drops, once the correlation length reaches the size of the lattice, the system is fully frozen and there is a gap to excitations (c.f. Fig [REF].', 'cond-mat-0209170-2-29-2': 'a).', 'cond-mat-0209170-2-29-3': 'In contrast, in the DCA, the correlations are confined within clusters of size [MATH] (the size of the entire lattice) and they never reach the size of the lattice.', 'cond-mat-0209170-2-29-4': 'As we lower the temperature, the correlation length approaches the size of the cluster but since the lattice remains in the thermodynamic limit, it never freezes (c.f. Fig [REF].', 'cond-mat-0209170-2-29-5': 'b).', 'cond-mat-0209170-2-29-6': 'By increasing the size of clusters in the DCA, we take longer ranged correlations into account so the gap will become more pronounced.', 'cond-mat-0209170-2-29-7': 'Consequently, correlation induced gaps are generally overestimated in the finite size lattice, while in the DCA they are underestimated.', 'cond-mat-0209170-2-30-0': 'This complementary behavior may be seen in [MATH], with [MATH] computed using finite size lattices with periodic boundary conditions and the DCA.', 'cond-mat-0209170-2-30-1': 'As illustrated in Fig. [REF], by increasing the size of the finite size lattice and the DCA cluster, the Green functions converge from opposite directions.', 'cond-mat-0209170-2-30-2': 'In the finite size lattice, the Green function (e.g. at [MATH]) decreases with the increase of size which is consistent with overestimating the gap; while in the DCA, the Green function increases as the cluster size grows consistent with underestimating the gap.', 'cond-mat-0209170-2-30-3': 'It is also observed that the convergence in the DCA is much faster, meaning that the result of the DCA is closer to the true curve at a given cluster size.', 'cond-mat-0209170-2-30-4': 'Both finite size lattices with periodic boundary conditions and the DCA converge with corrections of [MATH] with [MATH] the linear size of the finite lattice or the DCA cluster and [MATH] a coefficient.', 'cond-mat-0209170-2-30-5': '[CITATION] The faster convergence of the DCA corresponds to its smaller [MATH] compared to finite size lattices.', 'cond-mat-0209170-2-31-0': 'In Fig. [REF], the values of the [MATH] and [MATH] in Fig. [REF] have been plotted versus [MATH] for both the DCA and finite size results.', 'cond-mat-0209170-2-31-1': 'The Green function behaves linearly as a function of [MATH] for large [MATH].', 'cond-mat-0209170-2-31-2': 'The extrapolations of the DCA and finite size results meet as [MATH], approximating the value of the dressed Green function in the thermodynamic limit.', 'cond-mat-0209170-2-31-3': 'The complementarity of DCA and finite size methods allows a determination of the thermodynamic limit of imaginary time Green functions with unprecedented accuracy.', 'cond-mat-0209170-2-32-0': '# Finite size versus the DCA FLEX results for the two dimensional Hubbard Model at half-filling', 'cond-mat-0209170-2-33-0': 'The Hubbard model at half filling undergoes a phase transition to anti-ferromagnetic order at low temperatures.', 'cond-mat-0209170-2-33-1': 'According to the Mermin-Wagner-Hohenberg theorem, for dimension [MATH] the critical temperature is zero.', 'cond-mat-0209170-2-33-2': 'However, as we continue to lower the temperature, close enough to zero, a pseudogap will appear in the density of states (DOS) as a precursor to the anti-ferromagnetic phase (which has a full gap as its signature).', 'cond-mat-0209170-2-33-3': 'An approach towards non-Fermi-liquid behavior is also visible in both the real and imaginary parts of the retarded self-energy.', 'cond-mat-0209170-2-33-4': '[CITATION]', 'cond-mat-0209170-2-34-0': 'In Fig. [REF] and Fig. [REF], the densities of states (DOS) for lattices with finite sizes of 32[MATH]32, 64[MATH]64 and periodic boundary conditions and coarse-graining cluster sizes of 16[MATH]16 and 32[MATH]32 are plotted.', 'cond-mat-0209170-2-34-1': 'We analytically continue the Green function in order to calculate the spectral function [MATH] and DOS using the Pade approximation.', 'cond-mat-0209170-2-34-2': '[CITATION] In this approximation, we generate a continued fraction interpolating all the data points and use it as an analytic function of the Matsubara frequencies [MATH].', 'cond-mat-0209170-2-34-3': 'The analytic continuation is accomplished by substituting [MATH] with [MATH] where [MATH] is a small positive shift.', 'cond-mat-0209170-2-34-4': 'However, the errors inherent in the numerical Fourier transform (FFT) and also the sharp high-frequency behavior of the Green function, FLEX potentials and the self-energy, limit the accuracy of the Pade.', 'cond-mat-0209170-2-34-5': 'The high-frequency behavior is improved by implementing a more authentic cut off scheme introduced by Deisz et al.[CITATION] in which the high-frequency tails of these quantities are Fourier transformed analytically prior to any numerical FFT and added back to the FFT outputs afterwards.', 'cond-mat-0209170-2-34-6': 'In addition to this high-frequency cut off improvement, we also check for the analyticity of the Pade results in the upper-half frequency plane as a requirement for retarded physical quantities.', 'cond-mat-0209170-2-34-7': 'This task is carried out by converting the continued fraction in the Pade into a ratio of two polynomials.', 'cond-mat-0209170-2-34-8': 'The complex roots of these two polynomials are obtained via the Jenkins-Traub root finder routine.', 'cond-mat-0209170-2-34-9': '[CITATION] Those orders of the Pade for which there exist poles in the upper-half plane are omitted unless these poles are canceled by the roots of the numerator.', 'cond-mat-0209170-2-34-10': 'The acceptable Pade results correspond to the highest order with no uncompensated poles in the upper-half plane.', 'cond-mat-0209170-2-35-0': 'As seen in Fig. [REF], by increasing the size of a finite lattice, the pseudogap occurs at higher temperatures and it also becomes less pronounced (sharper) as we approach the actual size of an infinite real lattice.', 'cond-mat-0209170-2-35-1': 'The DCA yields a complementary behavior as shown in Fig. [REF].', 'cond-mat-0209170-2-35-2': 'By increasing the coarse-graining cluster size, similar to the finite size lattices, the pseudogap is shifted towards higher temperatures.', 'cond-mat-0209170-2-35-3': 'However, unlike the finite size lattices, for the DCA the precursor becomes more pronounced (broader) as the cluster increases in size because the size of the lattice remains constant and the correlations are limited to the cluster size.', 'cond-mat-0209170-2-35-4': 'Thus comparatively, the DCA underestimates the gap while the finite size calculation overestimates it.', 'cond-mat-0209170-2-36-0': 'By comparing the results in Fig. [REF] and Fig. [REF] one may see that the [MATH] DCA cluster yields more realistic physics than the corresponding [MATH] finite size lattice.', 'cond-mat-0209170-2-36-1': 'The [MATH] finite size lattice results are also close to those for the [MATH] DCA cluster at slightly lower [MATH] (eg, [MATH] for the finite size and [MATH] for the DCA).', 'cond-mat-0209170-2-36-2': 'However since the sizes of clusters are considerably smaller than the sizes of lattices, the DCA significantly reduces the complexity of the problem and consequently the CPU time.', 'cond-mat-0209170-2-36-3': 'In terms of the CPU time, the FLEX with the numerical Fourier transforms scales as [MATH] where [MATH] is the product of the total number of Matsubara frequencies and the [MATH] points in the first Brillouin zone.', 'cond-mat-0209170-2-36-4': 'Hence, using a [MATH] cluster in place of a [MATH] lattice both having [MATH] Matsubara frequency points roughly reduces the CPU time by a factor of [MATH].', 'cond-mat-0209170-2-36-5': 'If the DCA cluster size [MATH] equals the size of the finite lattice [MATH], the DCA requires somewhat more CPU time than the finite size lattice due to course-graining.', 'cond-mat-0209170-2-36-6': 'Nevertheless, comparatively, for large finite lattices such as [MATH], the lattice size contributions to the CPU time significantly dominate the coarse-graining ones in a [MATH] cluster.', 'cond-mat-0209170-2-36-7': 'Thus, the [MATH] DCA cluster is much faster.', 'cond-mat-0209170-2-37-0': 'The FLEX often has difficulty reaching low temperatures.', 'cond-mat-0209170-2-37-1': 'This is due to the fact that the [MATH] defined in Eq. [REF] approaches unity as the temperature drops which in turn causes the [MATH] in Eq. [REF] to diverge.', 'cond-mat-0209170-2-37-2': 'In the DCA, [MATH] approaches unity more slowly, allowing the calculations to reach lower temperatures.', 'cond-mat-0209170-2-37-3': 'One has to note that in the FLEX, the [MATH] is defined as follows [EQUATION] with [MATH] defined in Eq. [REF].', 'cond-mat-0209170-2-37-4': 'Fig. [REF] illustrates the saturation of [MATH] for both the DCA and finite size lattices.', 'cond-mat-0209170-2-37-5': 'The [MATH] for the [MATH] finite size lattice (filled circles) saturates at higher temperatures compared to the [MATH] DCA cluster (open diamonds) indicating that in the DCA, for a certain cluster size, the precursor to the phase transition can evolve to lower temperatures compared to a finite size lattice of the same size with periodic boundary conditions.', 'cond-mat-0209170-2-37-6': 'However, for the correlation length [MATH] the DCA approximation breaks down and replacing the self-energy by its coarse-grained counterpart is no longer accurate.', 'cond-mat-0209170-2-37-7': 'Here, the DCA takes on significant mean-field character.', 'cond-mat-0209170-2-38-0': 'Another feature of the Hubbard model near half-filling verified by the FLEX [CITATION] is non-Fermi-liquid behavior.', 'cond-mat-0209170-2-38-1': 'Here, this is studied by increasing the electron electron interaction U at a constant temperature.', 'cond-mat-0209170-2-38-2': 'In Fig. [REF] and Fig. [REF] the real and imaginary parts of the self-energy at the X point (on the non-interacting Fermi surface) have been plotted versus energy for finite size lattices and the DCA respectively.', 'cond-mat-0209170-2-38-3': 'As the interaction is increased, the negative slope in the real part turns positive around [MATH] which is inconsistent with the requirement that the renormalization factor [MATH] should be smaller than unity in the Fermi-liquid theory.', 'cond-mat-0209170-2-38-4': 'There also appears an anomalous inverted peak in the imaginary part at [MATH].', 'cond-mat-0209170-2-39-0': 'As presented in Fig. [REF], by increasing the length of the finite size lattice, the sharpness of the non-Fermi-liquid features is reduced.', 'cond-mat-0209170-2-39-1': 'The same features for the DCA in Fig. [REF] are slightly less pronounced and in a complementary fashion to the finite size lattices, their sharpness is enhanced by increasing the size of the cluster.', 'cond-mat-0209170-2-39-2': 'Thus, again the DCA underestimates the non-Fermi-liquid features while the finite size calculation overestimates it.', 'cond-mat-0209170-2-40-0': 'Earlier in Fig. [REF] it was shown that the FLEX particle-hole bubble [MATH] at [MATH] approaches unity as the temperature is lowered.', 'cond-mat-0209170-2-40-1': 'This causes the spin-fluctuation [MATH] matrix [EQUATION] which is just the third term in [MATH] in Eq. [REF] to peak around the [MATH] point.', 'cond-mat-0209170-2-40-2': 'For real frequencies, [MATH] has a sharp peak around [MATH].', 'cond-mat-0209170-2-40-3': 'Since [MATH] is only used to construct the irreducible self-energy, within the DCA it is constructed from coarse-grained Green functions.', 'cond-mat-0209170-2-40-4': 'Thus, the DCA counterpart of Eq. [REF] is obtained by only replacing the [MATH] with [MATH] defined in Eq. [REF].', 'cond-mat-0209170-2-40-5': 'Fig. [REF] and Fig. [REF] show how this peak sharpens as the temperature decreases or interaction increases for finite size lattices and the DCA respectively.', 'cond-mat-0209170-2-40-6': 'In the [MATH] finite size lattice (c.f. Fig [REF] bottom), the peak continues to develop as the temperature is lowered and the interaction is raised.', 'cond-mat-0209170-2-40-7': 'At [MATH] and [MATH] where there exists a pseudogap in the DOS, the peak undergoes a significant growth compared to the other graphs shown in the same figure.', 'cond-mat-0209170-2-40-8': 'The [MATH] (c.f. Fig [REF].', 'cond-mat-0209170-2-40-9': 'top) lattice presents the same behavior with a slightly sharper but shorter peak.', 'cond-mat-0209170-2-41-0': 'The DCA also illustrates the same type of peaks at a slightly higher interaction and lower temperatures for the [MATH] cluster (c.f. Fig [REF] top).', 'cond-mat-0209170-2-41-1': 'Increasing the cluster size to [MATH] gives rise to higher peaks similar to the finite size lattices case but unlike the finite size lattices peaks become sharper as the size is increased.', 'cond-mat-0209170-2-42-0': 'All the results illustrated in this section indicate the complementarity of the DCA to the finite size lattice scheme.', 'cond-mat-0209170-2-42-1': 'It is also observed that the DCA is capable of reproducing relatively the same physics as the finite size FLEX at slightly different parameters but a lower CPU cost.', 'cond-mat-0209170-2-42-2': 'The combination of these two facts makes this technique a good candidate to be employed in the numerical treatment of a wide range of many-body problems.', 'cond-mat-0209170-2-43-0': 'Lastly, to better understand the effect of the DCA cluster embedded in a Fermionic bath, we can rewrite the course grained Green function defined in Eq. [REF] as', 'cond-mat-0209170-2-44-0': '[EQUATION]', 'cond-mat-0209170-2-45-0': 'where [MATH] and [MATH] is the host function.', 'cond-mat-0209170-2-45-1': 'Maier et al. [CITATION], define [EQUATION] whereby [MATH] can be expressed as', 'cond-mat-0209170-2-46-0': '[EQUATION]', 'cond-mat-0209170-2-47-0': 'By Taylor expanding [MATH] around the cluster points [MATH] it is found that [MATH] with [MATH].', 'cond-mat-0209170-2-47-1': 'Thus, Eq. [REF] yields [MATH] as [MATH].', 'cond-mat-0209170-2-47-2': 'To illustrate this, we calculate [MATH] by summing over all the [MATH] points and [MATH] frequencies and plot it versus [MATH].', 'cond-mat-0209170-2-47-3': 'Fig. [REF] illustrates this linear behavior for [MATH].', 'cond-mat-0209170-2-47-4': '[MATH] holds complete mean field characters and no non-local fluctuations.', 'cond-mat-0209170-2-47-5': '[MATH] is anomalous as explained in an article by Betts et al. [CITATION] There, the finite size cubic lattices with less than six (four in [MATH] square lattices) distinct nearest neighbors per each site are not used in finite size scalings for estimating the physical properties of models like the spin one half XY ferromagnet or the Heisenberg antiferromagnet.', 'cond-mat-0209170-2-47-6': 'For [MATH], because of the periodicity of the clusters, each cluster point is surrounded by two identical nearest neighbors in every direction and therefore has only two distinct nearest neighbors (c.f. Fig. [REF]).', 'cond-mat-0209170-2-47-7': 'Thus, the effect of fluctuations are overestimated.', 'cond-mat-0209170-2-47-8': '[CITATION] For [MATH] there is no such anomaly and hence, all the points present the linear behavior proven above.', 'cond-mat-0209170-2-47-9': 'Nevertheless, calculations with [MATH] do a reasonable job in capturing the qualitative effects of corrections to the DMFA.', 'cond-mat-0209170-2-48-0': '# Microscopic theory of the DCA', 'cond-mat-0209170-2-49-0': 'In section [REF] we defined the thermodynamic potential functional difference [MATH] in terms of the Green function, self-energy, and [MATH] (c.f. Eq. [REF]).', 'cond-mat-0209170-2-49-1': 'In Eq. [REF], [MATH] includes all the compact (skeletal) Feynman diagrams and the rest incorporates the entire non-compact contribution.', 'cond-mat-0209170-2-49-2': '[CITATION] Typical compact and non-compact diagrams are illustrated in Fig. [REF].', 'cond-mat-0209170-2-49-3': 'The non-compact diagram (a) consists of two self-energy pieces [MATH] and [MATH] connected with two one-particle Green functions.', 'cond-mat-0209170-2-49-4': 'Removing these two Green functions would split the diagram into two separate pieces.', 'cond-mat-0209170-2-49-5': 'In the compact diagram (b) two vertex parts [MATH] and [MATH] with four Green functions are connected together.', 'cond-mat-0209170-2-49-6': 'One can not split this type of diagrams into two separate pieces by just removing two one-particle Green functions.', 'cond-mat-0209170-2-49-7': 'As mentioned earlier, in the DCA, we employ coarse-grained Green functions to construct only the compact diagrams.', 'cond-mat-0209170-2-49-8': 'The Green functions in non-compact diagrams are calculated directly using Eq. [REF] in which the self-energy [MATH] is coarse-grained (the circles at the top and the bottom of the non-compact diagram in Fig. [REF].', 'cond-mat-0209170-2-49-9': '(a))', 'cond-mat-0209170-2-50-0': 'Earlier in a shorter article for the Hubbard model in particular [CITATION], we showed both analytically and numerically that the error produced by coarse-graining the non-compact diagrams is significantly larger than the error produced by coarse-graining the compact ones.', 'cond-mat-0209170-2-50-1': 'Here, we would like to give a more general argument in real space.', 'cond-mat-0209170-2-50-2': 'We wish to emphasize two points in this new approach.', 'cond-mat-0209170-2-50-3': 'First, since the derivation of the DCA in this section relies only upon the exponential fall off of the Green function as a function of distance, it is far more intuitive than the momentum space argument in Ref. [CITATION].', 'cond-mat-0209170-2-50-4': 'Second, it ties the derivation of the DCA to the original derivation of the DMFA in the limit of infinite dimensions, where similar arguments are employed.', 'cond-mat-0209170-2-50-5': '[CITATION]', 'cond-mat-0209170-2-51-0': 'The exponential fall off behavior occurs naturally in high dimensions.', 'cond-mat-0209170-2-51-1': 'In the tight-binding Hamiltonian (c.f. Eq. [REF]), the factor [MATH] corresponds to the hopping of electrons among nearest neighboring sites.', 'cond-mat-0209170-2-51-2': 'Thus, one could show that the real space Green function [MATH] (we drop the frequency label from this point on for simplicity) for [MATH] nearest neighbor hops is proportional to [MATH] as [MATH].', 'cond-mat-0209170-2-51-3': 'On the other hand, Metzner et al., and Muller-Hartmann [CITATION] have shown that in [MATH] dimensions, the factor [MATH] should be renormalized as [MATH] in order to have a finite density of states width as [MATH].', 'cond-mat-0209170-2-51-4': 'As a result of this renormalization [EQUATION] meaning that [MATH] falls off exponentially as a function of [MATH].', 'cond-mat-0209170-2-52-0': 'In the DCA, we attempt to minimize the error due to coarse-graining the Green function (and potentials) in the Feynman diagrams.', 'cond-mat-0209170-2-52-1': 'Consider the first non-trivial correction to the coarse-grained non-compact diagrams generated by replacing the explicit coarse-grained Green function lines by the non-coarse-grained ones as illustrated in Fig. [REF] [EQUATION] where [MATH] are the coarse-graining cells momenta and [MATH] include all the momenta in the first Brillouin zone shown in Fig. [REF].', 'cond-mat-0209170-2-52-2': 'In this derivation we also presume that the self-energy is [MATH] independent and the entire [MATH] dependence is embedded in the Green functions.', 'cond-mat-0209170-2-53-0': 'By breaking up the sums over [MATH] in Eq. [REF] into [MATH] and [MATH] sums and writing all the G in terms of their Fourier transforms at the same time we get [EQUATION] in which we used Eq. [REF] for [MATH].', 'cond-mat-0209170-2-53-1': 'According to Fig. [REF], we can split [MATH] where [MATH] extend between two separate clusters while [MATH] always remain within a single cluster.', 'cond-mat-0209170-2-53-2': 'By making this separation in Eq. [REF] one picks up phases including products of [MATH], [MATH], [MATH] and [MATH] in their exponents.', 'cond-mat-0209170-2-53-3': 'The phase associated with the product [MATH] with [MATH] an integer equals unity.', 'cond-mat-0209170-2-53-4': 'The phases involving [MATH] products are also neglected as discussed in section [REF].', 'cond-mat-0209170-2-53-5': 'Hence, Eq. [REF] can be rewritten as follows [EQUATION]', 'cond-mat-0209170-2-53-6': 'Implementing the following substitutions [EQUATION]', 'cond-mat-0209170-2-53-7': 'Eq. [REF] simplifies into [EQUATION]', 'cond-mat-0209170-2-53-8': 'Setting [MATH] and performing the [MATH] summation [EQUATION]', 'cond-mat-0209170-2-53-9': 'Knowing that [MATH] and also that the lowest order of [MATH] we conclude that in Eq. [REF], the largest contribution is due to terms having [MATH] or in other words, local [MATH].', 'cond-mat-0209170-2-53-10': 'As shown in Fig. [REF], the first term in the [MATH] sum corresponds to [MATH] (size of the cluster) and [MATH] can be as large as [MATH] in the opposite direction.', 'cond-mat-0209170-2-53-11': 'Hence, the leading order term in Eq. [REF] falls off as [EQUATION] where [MATH] is the number of [MATH] contributions in [MATH] different dimensions of a [MATH] dimensional cubic lattice.', 'cond-mat-0209170-2-53-12': 'In Eq. [REF] we also used the fact that due to the lattice symmetry, [MATH].', 'cond-mat-0209170-2-53-13': 'As [MATH], using [MATH] [EQUATION] which indicates the existence of non-local corrections to the non-compact contribution of the thermodynamic potential even at infinite dimensions.', 'cond-mat-0209170-2-54-0': 'Now we replace the coarse-grained self-energy bubbles in Fig. [REF] with coarse-grained vertices having four external legs and look at the difference between compact diagrams with and without coarse-grained Green functions which are explicitly shown in the figure.', 'cond-mat-0209170-2-54-1': 'Since we earlier dropped the frequency labels in the Green functions, here we use indices [MATH] and [MATH] to emphasize that these Green functions have different frequency labels.', 'cond-mat-0209170-2-54-2': 'The first correction to the compact contribution of the thermodynamic potential depicted in Fig. [REF] is [EQUATION] where similar to Eq. [REF], all the vertices are coarse-grained but the Green functions are not.', 'cond-mat-0209170-2-55-0': 'Following the same procedure as for the non-compact difference we arrive at [EQUATION] where [EQUATION]', 'cond-mat-0209170-2-55-1': 'Once again, the largest contribution is associated with local [MATH], i.e., [MATH], [MATH] and [MATH].', 'cond-mat-0209170-2-55-2': 'Therefore [EQUATION]', 'cond-mat-0209170-2-55-3': 'Considering [MATH] and [MATH] [EQUATION] which vanishes as [MATH].', 'cond-mat-0209170-2-56-0': 'Comparing Eq. [REF] with Eq. [REF] shows that the first correction to the compact contribution of the thermodynamic potential falls off exponentially twice as fast as the equivalent correction in the non-compact contribution.', 'cond-mat-0209170-2-56-1': 'In addition, even at the infinite dimensional limit, there are corrections of order one to the non-compact contribution whereas for the compact diagrams the DCA becomes exact and there are no corrections.', 'cond-mat-0209170-2-56-2': 'This justifies coarse-graining only in the compact diagrams.', 'cond-mat-0209170-2-56-3': 'The Green functions in the non-compact diagrams have to be explicitly constructed from the coarse-grained self-energy [MATH] using Eq. [REF].', 'cond-mat-0209170-2-57-0': 'In Fig. [REF], the compact contribution of the thermodynamic potential difference constructed by coarse-grained Green functions is plotted versus [MATH].', 'cond-mat-0209170-2-57-1': 'The variation of [MATH] over the entire [MATH] range is about [MATH].', 'cond-mat-0209170-2-57-2': 'At very low temperatures ([MATH] in the inset), some deviation from linearity is observed due to the correlation length exceeding the cluster size [MATH] and therefore the approximation of [MATH] by [MATH] begins to break down.', 'cond-mat-0209170-2-57-3': 'Fig. [REF] illustrates the non-compact contribution sketched versus [MATH] using both coarse-grained and non-coarse-grained Green functions.', 'cond-mat-0209170-2-57-4': 'Using non-coarse-grained Green functions, the variation of [MATH] over the entire [MATH] range is roughly [MATH].', 'cond-mat-0209170-2-57-5': 'Coarse-graining the Green functions in these diagrams can even change the sign of this non-compact contributions, clearly indicating that coarse-graining the Green function is an unrealistic approximation for non-compact diagrams.', 'cond-mat-0209170-2-57-6': 'However, one notices that for large cluster sizes the coarse-grained and non-coarse-grained results approach each other as the approximation to the infinite lattice becomes better.', 'cond-mat-0209170-2-58-0': '# DCA in frequency space', 'cond-mat-0209170-2-59-0': 'As illustrated for the momentum space, the DCA results in a significant reduction of the problem complexity and it is complementary to the finite size lattice approach.', 'cond-mat-0209170-2-59-1': 'In analogy to the momentum space, one could consider dividing the one dimensional Matsubara frequency space into a number of coarse-graining subcells.', 'cond-mat-0209170-2-59-2': 'For both fermions and bosons, each cell should include an odd number of frequencies in order for the frequencies in the centers of these cells to preserve Fermionic or Bosonic properties.', 'cond-mat-0209170-2-59-3': 'Fig. [REF] represents how the frequency space can be divided into coarse-graining subcells each comprising a central [MATH] frequency and a number of coarse-graining [MATH] lying around it.', 'cond-mat-0209170-2-59-4': 'The central [MATH] frequencies can be rewritten in the form of the original lattice with renormalized [MATH] shown as [MATH] in the figure.', 'cond-mat-0209170-2-59-5': 'Similar to the case of the momentum space, we make the following transformation for the Laue function [EQUATION] for the Matsubara frequencies of the vertex shown in Fig. [REF] considering frequency dependent interactions in general (in condensed matter physics, most of the interactions are indeed simultaneous and thus frequency independent).', 'cond-mat-0209170-2-59-6': 'As a result, we may again coarse-grain the Green function over the subcell frequencies [EQUATION]', 'cond-mat-0209170-2-59-7': 'According to Fig. [REF], the full coarse-graining of the Green function amounts to [MATH] which causes all the self-energy Feynman diagrams ordered higher than first (Hartree-Fock diagrams) to vanish and consequently we arrive at a fully static problem.', 'cond-mat-0209170-2-60-0': 'Unfortunately, we can show that coarse-graining over Matsubara frequencies can lead to the violation of causality and as a result, the DCA is not systematically implementable for the Matsubara frequency quantities.', 'cond-mat-0209170-2-60-1': 'The simplest example is the non-interacting Green function coarse-grained as follows [CITATION] [EQUATION]', 'cond-mat-0209170-2-60-2': 'The retarded Green function is derived by the substitution [MATH] [EQUATION]', 'cond-mat-0209170-2-60-3': 'The theory of analytic functions of a complex variable tells us that in order for the Green function to remain retarded in the time space, [MATH] must not have any poles in the upper half plane of [MATH].', 'cond-mat-0209170-2-60-4': 'In Eq. [REF], one could readily create poles in the upper half plane for negative [MATH] which causes causality violation and consequently unphysical results.', 'cond-mat-0209170-2-61-0': 'The way around this difficulty is to use real frequencies even at finite temperatures.', 'cond-mat-0209170-2-61-1': 'By invoking real frequencies we no longer coarse-grain over imaginary values.', 'cond-mat-0209170-2-61-2': 'Therefore, the retarded Green function will never acquire poles in the upper half plane and remains causal as shown below [EQUATION] with [MATH] the total number of frequencies and [MATH] the number of those we coarse-graining over in each cell.', 'cond-mat-0209170-2-61-3': 'In an article by Hettler et al. [CITATION], a formal proof of causality is given (based on a geometrical argument) for coarse-graining in the momentum space.', 'cond-mat-0209170-2-61-4': 'This proof can be straightforwardly applied to the real frequency space as well and it extends the application of real frequency DCA not only to the perturbative cluster solvers such as the FLEX but also techniques like the NCA.', 'cond-mat-0209170-2-62-0': 'Lastly, similar to the momentum space, care must be taken when choosing the the size of the frequency coarse-graining cells.', 'cond-mat-0209170-2-62-1': 'One must make sure that the cells are not larger than some characteristic energy scale (e.g. the Kondo temperature [MATH]) as the coarse-graining would then suppress the relevant physics.', 'cond-mat-0209170-2-63-0': '# Conclusions and Outlook', 'cond-mat-0209170-2-64-0': 'We introduce and examine the DCA in detail by employing it with the FLEX to study the half filled two dimensional Hubbard model.', 'cond-mat-0209170-2-64-1': 'The FLEX is not as precise as nearly exact techniques such as quantum Monte Carlo in describing the Hubbard model at strong interaction regime.', 'cond-mat-0209170-2-64-2': 'However, it is capable of illustrating the utility of the DCA, including the complementarity and convergence of the DCA compared to finite size lattice approaches.', 'cond-mat-0209170-2-64-3': 'The DCA and finite size calculations (with periodic boundary conditions) both converge with corrections [MATH]; however, in our example the coefficient [MATH] was smaller and of opposite sign than [MATH], indicating that the DCA converges more quickly and from a complementary direction.', 'cond-mat-0209170-2-64-4': 'This complementarity was also seen in other quantities such as the pseudogap in the density of state and the non-Fermi liquid behaviour that the DCA (finite size) calculation systematically under (over) estimates.', 'cond-mat-0209170-2-65-0': 'We also provide a detailed microscopic definition of the DCA by inspecting the the error generated by coarse-graining the Green functions in the compact and non-compact contributions to the thermodynamic potential.', 'cond-mat-0209170-2-65-1': 'We conclude that due to the large magnitude of error that it generates, coarse-graining the Green function in non-compact part should be avoided and only the compact contribution should undergo coarse-graining.', 'cond-mat-0209170-2-65-2': 'It also appears that coarse-graining the Green functions over the Matsubara frequencies can and will lead to the violation of causality and therefore is pathological.', 'cond-mat-0209170-2-65-3': 'Nevertheless, one can coarse-grain the Green function over real frequencies and preserve the causality not only for the FLEX but also cluster solvers such as NCA in which the cluster contribution to the coarse-grained dressed Green function is excluded before being inserted in the cluster solver.', 'cond-mat-0209170-2-66-0': 'The outlook for the FLEX-DCA approach is promising.', 'cond-mat-0209170-2-66-1': 'Although the FLEX fails to accurately describe short-ranged physics such as moment formation (and related phenomena like the Mott gap), it does a good job describing long-ranged physics associated with spin and charge fluctuations.', 'cond-mat-0209170-2-66-2': 'On the other hand, numerically exact calculations such as QMC, are too expensive to perform for large clusters, and are thus restricted to the study of short-length scales.', 'cond-mat-0209170-2-66-3': 'However, since the DCA gives us a way of parsing the problem into different length scales, it may be used to combine the short-length scale information from the QMC with the long length scale information from the FLEX.', 'cond-mat-0209170-2-66-4': 'This may be accomplished, by embedding a QMC cluster, of size [MATH], into a much larger FLEX cluster of size [MATH], which is itself embedded in a mean-field.', 'cond-mat-0209170-2-66-5': 'As we have shown here, this approach should be implemented by approximating the generating functional [MATH].', 'cond-mat-0209170-2-66-6': 'Work along these lines is in progress.', 'cond-mat-0209170-2-67-0': 'We would like to acknowledge J.J Deisz, D.W. Hess Th.', 'cond-mat-0209170-2-67-1': 'Maier, S. Moukouri, and Th.', 'cond-mat-0209170-2-67-2': 'Pruschke for very useful discussions and suggestions.', 'cond-mat-0209170-2-68-0': 'Pruschke T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys.', 'cond-mat-0209170-2-69-0': 'Georges A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod.', 'cond-mat-0209170-2-70-0': 'DCA_hettler M. H. Hettler, M. Mukherjee, M. Jarrell, and H. R. Krishnamurthy, Phys.', 'cond-mat-0209170-2-71-0': 'DCA_hettler2 M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H. R. Krishnamurthy, Phys.', 'cond-mat-0209170-2-72-0': 'Bickers1 N. E. Bickers, D. J. Scalapino, and S. R. White, Phys.', 'cond-mat-0209170-2-73-0': 'Deisz J. J. Deisz, D. W. Hess, and J. W. Serene, Phys.', 'cond-mat-0209170-2-74-0': 'Vidberg H. J. Vidberg and J. W. Serene, J. Low Temp.', 'cond-mat-0209170-2-75-0': 'DHS J. J Deisz, D. W. Hess, and J. W. Serene, cond-mat/9411026R (To appear in "Recent Progress In Many Body Theories", vol.', 'cond-mat-0209170-2-75-1': '4, edited by E. Schachinger, et al. (Plenum, New York)).', 'cond-mat-0209170-2-76-0': 'Jenkins M. A. Jenkins, and J. F. Traub, Comm.', 'cond-mat-0209170-2-76-1': 'The routine is available at http://www.netlib.org/tomspdf/419.pdf.', 'cond-mat-0209170-2-77-0': 'agd A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, (Dover, New York, 1975).'}

[['cond-mat-0209170-1-2-0', 'cond-mat-0209170-2-2-0'], ['cond-mat-0209170-1-2-1', 'cond-mat-0209170-2-2-1'], ['cond-mat-0209170-1-2-2', 'cond-mat-0209170-2-2-2'], ['cond-mat-0209170-1-2-3', 'cond-mat-0209170-2-2-3'], ['cond-mat-0209170-1-33-0', 'cond-mat-0209170-2-33-0'], ['cond-mat-0209170-1-33-1', 'cond-mat-0209170-2-33-1'], ['cond-mat-0209170-1-33-2', 'cond-mat-0209170-2-33-2'], ['cond-mat-0209170-1-33-3', 'cond-mat-0209170-2-33-3'], ['cond-mat-0209170-1-5-0', 'cond-mat-0209170-2-5-0'], ['cond-mat-0209170-1-5-1', 'cond-mat-0209170-2-5-1'], ['cond-mat-0209170-1-5-2', 'cond-mat-0209170-2-5-2'], ['cond-mat-0209170-1-5-3', 'cond-mat-0209170-2-5-3'], ['cond-mat-0209170-1-21-0', 'cond-mat-0209170-2-21-0'], ['cond-mat-0209170-1-22-0', 'cond-mat-0209170-2-22-0'], ['cond-mat-0209170-1-0-0', 'cond-mat-0209170-2-0-0'], ['cond-mat-0209170-1-0-1', 'cond-mat-0209170-2-0-1'], ['cond-mat-0209170-1-0-2', 'cond-mat-0209170-2-0-2'], ['cond-mat-0209170-1-0-3', 'cond-mat-0209170-2-0-3'], ['cond-mat-0209170-1-0-4', 'cond-mat-0209170-2-0-4'], ['cond-mat-0209170-1-0-5', 'cond-mat-0209170-2-0-5'], ['cond-mat-0209170-1-0-6', 'cond-mat-0209170-2-0-6'], ['cond-mat-0209170-1-0-7', 'cond-mat-0209170-2-0-7'], ['cond-mat-0209170-1-52-0', 'cond-mat-0209170-2-52-0'], ['cond-mat-0209170-1-52-1', 'cond-mat-0209170-2-52-1'], ['cond-mat-0209170-1-52-2', 'cond-mat-0209170-2-52-2'], ['cond-mat-0209170-1-75-0', 'cond-mat-0209170-2-75-0'], ['cond-mat-0209170-1-75-1', 'cond-mat-0209170-2-75-1'], ['cond-mat-0209170-1-3-0', 'cond-mat-0209170-2-3-0'], ['cond-mat-0209170-1-3-1', 'cond-mat-0209170-2-3-1'], ['cond-mat-0209170-1-3-2', 'cond-mat-0209170-2-3-2'], ['cond-mat-0209170-1-3-3', 'cond-mat-0209170-2-3-3'], ['cond-mat-0209170-1-3-4', 'cond-mat-0209170-2-3-4'], ['cond-mat-0209170-1-24-0', 'cond-mat-0209170-2-24-0'], ['cond-mat-0209170-1-72-0', 'cond-mat-0209170-2-72-0'], ['cond-mat-0209170-1-77-0', 'cond-mat-0209170-2-77-0'], ['cond-mat-0209170-1-73-0', 'cond-mat-0209170-2-73-0'], ['cond-mat-0209170-1-15-0', 'cond-mat-0209170-2-15-0'], ['cond-mat-0209170-1-15-1', 'cond-mat-0209170-2-15-1'], ['cond-mat-0209170-1-15-2', 'cond-mat-0209170-2-15-2'], ['cond-mat-0209170-1-15-3', 'cond-mat-0209170-2-15-3'], ['cond-mat-0209170-1-15-4', 'cond-mat-0209170-2-15-4'], ['cond-mat-0209170-1-15-5', 'cond-mat-0209170-2-15-5'], ['cond-mat-0209170-1-15-6', 'cond-mat-0209170-2-15-6'], ['cond-mat-0209170-1-15-7', 'cond-mat-0209170-2-15-7'], ['cond-mat-0209170-1-12-0', 'cond-mat-0209170-2-12-0'], ['cond-mat-0209170-1-12-1', 'cond-mat-0209170-2-12-1'], ['cond-mat-0209170-1-37-0', 'cond-mat-0209170-2-37-0'], ['cond-mat-0209170-1-37-1', 'cond-mat-0209170-2-37-1'], ['cond-mat-0209170-1-37-2', 'cond-mat-0209170-2-37-2'], ['cond-mat-0209170-1-37-3', 'cond-mat-0209170-2-37-3'], ['cond-mat-0209170-1-37-4', 'cond-mat-0209170-2-37-4'], ['cond-mat-0209170-1-37-5', 'cond-mat-0209170-2-37-5'], ['cond-mat-0209170-1-37-6', 'cond-mat-0209170-2-37-6'], ['cond-mat-0209170-1-37-7', 'cond-mat-0209170-2-37-7'], ['cond-mat-0209170-1-34-0', 'cond-mat-0209170-2-34-0'], ['cond-mat-0209170-1-34-1', 'cond-mat-0209170-2-34-1'], ['cond-mat-0209170-1-34-2', 'cond-mat-0209170-2-34-2'], ['cond-mat-0209170-1-34-3', 'cond-mat-0209170-2-34-3'], ['cond-mat-0209170-1-34-4', 'cond-mat-0209170-2-34-4'], ['cond-mat-0209170-1-34-5', 'cond-mat-0209170-2-34-5'], ['cond-mat-0209170-1-34-6', 'cond-mat-0209170-2-34-6'], ['cond-mat-0209170-1-34-7', 'cond-mat-0209170-2-34-7'], ['cond-mat-0209170-1-34-8', 'cond-mat-0209170-2-34-8'], ['cond-mat-0209170-1-34-9', 'cond-mat-0209170-2-34-9'], ['cond-mat-0209170-1-34-10', 'cond-mat-0209170-2-34-10'], ['cond-mat-0209170-1-30-0', 'cond-mat-0209170-2-30-0'], ['cond-mat-0209170-1-30-1', 'cond-mat-0209170-2-30-1'], ['cond-mat-0209170-1-30-2', 'cond-mat-0209170-2-30-2'], ['cond-mat-0209170-1-30-3', 'cond-mat-0209170-2-30-3'], ['cond-mat-0209170-1-30-4', 'cond-mat-0209170-2-30-4'], ['cond-mat-0209170-1-30-5', 'cond-mat-0209170-2-30-5'], ['cond-mat-0209170-1-4-0', 'cond-mat-0209170-2-4-0'], ['cond-mat-0209170-1-4-1', 'cond-mat-0209170-2-4-1'], ['cond-mat-0209170-1-4-2', 'cond-mat-0209170-2-4-2'], ['cond-mat-0209170-1-4-3', 'cond-mat-0209170-2-4-3'], ['cond-mat-0209170-1-29-0', 'cond-mat-0209170-2-29-0'], ['cond-mat-0209170-1-29-1', 'cond-mat-0209170-2-29-1'], ['cond-mat-0209170-1-29-3', 'cond-mat-0209170-2-29-3'], ['cond-mat-0209170-1-29-4', 'cond-mat-0209170-2-29-4'], ['cond-mat-0209170-1-29-6', 'cond-mat-0209170-2-29-6'], ['cond-mat-0209170-1-29-7', 'cond-mat-0209170-2-29-7'], ['cond-mat-0209170-1-1-0', 'cond-mat-0209170-2-1-0'], ['cond-mat-0209170-1-1-1', 'cond-mat-0209170-2-1-1'], ['cond-mat-0209170-1-1-2', 'cond-mat-0209170-2-1-2'], ['cond-mat-0209170-1-1-3', 'cond-mat-0209170-2-1-3'], ['cond-mat-0209170-1-6-0', 'cond-mat-0209170-2-6-0'], ['cond-mat-0209170-1-6-1', 'cond-mat-0209170-2-6-1'], ['cond-mat-0209170-1-6-2', 'cond-mat-0209170-2-6-2'], ['cond-mat-0209170-1-6-3', 'cond-mat-0209170-2-6-3'], ['cond-mat-0209170-1-23-0', 'cond-mat-0209170-2-23-0'], ['cond-mat-0209170-1-23-1', 'cond-mat-0209170-2-23-1'], ['cond-mat-0209170-1-61-0', 'cond-mat-0209170-2-61-0'], ['cond-mat-0209170-1-61-1', 'cond-mat-0209170-2-61-1'], ['cond-mat-0209170-1-61-2', 'cond-mat-0209170-2-61-2'], ['cond-mat-0209170-1-61-3', 'cond-mat-0209170-2-61-3'], ['cond-mat-0209170-1-61-4', 'cond-mat-0209170-2-61-4'], ['cond-mat-0209170-1-27-0', 'cond-mat-0209170-2-27-0'], ['cond-mat-0209170-1-27-1', 'cond-mat-0209170-2-27-1'], ['cond-mat-0209170-1-27-2', 'cond-mat-0209170-2-27-2'], ['cond-mat-0209170-1-27-3', 'cond-mat-0209170-2-27-3'], ['cond-mat-0209170-1-27-4', 'cond-mat-0209170-2-27-4'], ['cond-mat-0209170-1-27-5', 'cond-mat-0209170-2-27-5'], ['cond-mat-0209170-1-27-6', 'cond-mat-0209170-2-27-6'], ['cond-mat-0209170-1-27-7', 'cond-mat-0209170-2-27-7'], ['cond-mat-0209170-1-11-0', 'cond-mat-0209170-2-11-0'], ['cond-mat-0209170-1-11-1', 'cond-mat-0209170-2-11-1'], ['cond-mat-0209170-1-11-2', 'cond-mat-0209170-2-11-2'], ['cond-mat-0209170-1-11-3', 'cond-mat-0209170-2-11-3'], ['cond-mat-0209170-1-11-4', 'cond-mat-0209170-2-11-4'], ['cond-mat-0209170-1-65-0', 'cond-mat-0209170-2-65-0'], ['cond-mat-0209170-1-65-1', 'cond-mat-0209170-2-65-1'], ['cond-mat-0209170-1-65-2', 'cond-mat-0209170-2-65-2'], ['cond-mat-0209170-1-65-3', 'cond-mat-0209170-2-65-3'], ['cond-mat-0209170-1-10-0', 'cond-mat-0209170-2-10-0'], ['cond-mat-0209170-1-10-1', 'cond-mat-0209170-2-10-1'], ['cond-mat-0209170-1-10-3', 'cond-mat-0209170-2-10-3'], ['cond-mat-0209170-1-10-4', 'cond-mat-0209170-2-10-4'], ['cond-mat-0209170-1-10-5', 'cond-mat-0209170-2-10-5'], ['cond-mat-0209170-1-10-6', 'cond-mat-0209170-2-10-6'], ['cond-mat-0209170-1-10-7', 'cond-mat-0209170-2-10-7'], ['cond-mat-0209170-1-64-0', 'cond-mat-0209170-2-64-0'], ['cond-mat-0209170-1-64-1', 'cond-mat-0209170-2-64-1'], ['cond-mat-0209170-1-64-2', 'cond-mat-0209170-2-64-2'], ['cond-mat-0209170-1-64-3', 'cond-mat-0209170-2-64-3'], ['cond-mat-0209170-1-64-4', 'cond-mat-0209170-2-64-4'], ['cond-mat-0209170-1-18-0', 'cond-mat-0209170-2-18-0'], ['cond-mat-0209170-1-18-1', 'cond-mat-0209170-2-18-1'], ['cond-mat-0209170-1-18-2', 'cond-mat-0209170-2-18-2'], ['cond-mat-0209170-1-18-3', 'cond-mat-0209170-2-18-3'], ['cond-mat-0209170-1-45-0', 'cond-mat-0209170-2-45-0'], ['cond-mat-0209170-1-45-1', 'cond-mat-0209170-2-45-1'], ['cond-mat-0209170-1-71-0', 'cond-mat-0209170-2-71-0'], ['cond-mat-0209170-1-66-0', 'cond-mat-0209170-2-66-0'], ['cond-mat-0209170-1-66-1', 'cond-mat-0209170-2-66-1'], ['cond-mat-0209170-1-66-2', 'cond-mat-0209170-2-66-2'], ['cond-mat-0209170-1-66-5', 'cond-mat-0209170-2-66-6'], ['cond-mat-0209170-1-51-1', 'cond-mat-0209170-2-51-1'], ['cond-mat-0209170-1-51-2', 'cond-mat-0209170-2-51-2'], ['cond-mat-0209170-1-51-3', 'cond-mat-0209170-2-51-3'], ['cond-mat-0209170-1-51-4', 'cond-mat-0209170-2-51-4'], ['cond-mat-0209170-1-43-0', 'cond-mat-0209170-2-43-0'], ['cond-mat-0209170-1-50-0', 'cond-mat-0209170-2-50-0'], ['cond-mat-0209170-1-53-0', 'cond-mat-0209170-2-53-0'], ['cond-mat-0209170-1-53-1', 'cond-mat-0209170-2-53-1'], ['cond-mat-0209170-1-53-2', 'cond-mat-0209170-2-53-2'], ['cond-mat-0209170-1-53-3', 'cond-mat-0209170-2-53-3'], ['cond-mat-0209170-1-53-4', 'cond-mat-0209170-2-53-4'], ['cond-mat-0209170-1-53-5', 'cond-mat-0209170-2-53-5'], ['cond-mat-0209170-1-53-6', 'cond-mat-0209170-2-53-6'], ['cond-mat-0209170-1-53-7', 'cond-mat-0209170-2-53-7'], ['cond-mat-0209170-1-53-8', 'cond-mat-0209170-2-53-8'], ['cond-mat-0209170-1-53-9', 'cond-mat-0209170-2-53-9'], ['cond-mat-0209170-1-53-10', 'cond-mat-0209170-2-53-10'], ['cond-mat-0209170-1-53-11', 'cond-mat-0209170-2-53-11'], ['cond-mat-0209170-1-53-12', 'cond-mat-0209170-2-53-12'], ['cond-mat-0209170-1-53-13', 'cond-mat-0209170-2-53-13'], ['cond-mat-0209170-1-47-0', 'cond-mat-0209170-2-47-0'], ['cond-mat-0209170-1-47-2', 'cond-mat-0209170-2-47-2'], ['cond-mat-0209170-1-47-3', 'cond-mat-0209170-2-47-3'], ['cond-mat-0209170-1-47-4', 'cond-mat-0209170-2-47-4'], ['cond-mat-0209170-1-47-5', 'cond-mat-0209170-2-47-5'], ['cond-mat-0209170-1-47-6', 'cond-mat-0209170-2-47-6'], ['cond-mat-0209170-1-47-7', 'cond-mat-0209170-2-47-7'], ['cond-mat-0209170-1-47-8', 'cond-mat-0209170-2-47-8'], ['cond-mat-0209170-1-47-9', 'cond-mat-0209170-2-47-9'], ['cond-mat-0209170-1-13-0', 'cond-mat-0209170-2-13-0'], ['cond-mat-0209170-1-70-0', 'cond-mat-0209170-2-70-0'], ['cond-mat-0209170-1-20-0', 'cond-mat-0209170-2-20-0'], ['cond-mat-0209170-1-20-1', 'cond-mat-0209170-2-20-1'], ['cond-mat-0209170-1-35-0', 'cond-mat-0209170-2-35-0'], ['cond-mat-0209170-1-35-1', 'cond-mat-0209170-2-35-1'], ['cond-mat-0209170-1-35-2', 'cond-mat-0209170-2-35-2'], ['cond-mat-0209170-1-35-3', 'cond-mat-0209170-2-35-3'], ['cond-mat-0209170-1-35-4', 'cond-mat-0209170-2-35-4'], ['cond-mat-0209170-1-49-0', 'cond-mat-0209170-2-49-0'], ['cond-mat-0209170-1-49-1', 'cond-mat-0209170-2-49-1'], ['cond-mat-0209170-1-49-2', 'cond-mat-0209170-2-49-2'], ['cond-mat-0209170-1-49-3', 'cond-mat-0209170-2-49-3'], ['cond-mat-0209170-1-49-4', 'cond-mat-0209170-2-49-4'], ['cond-mat-0209170-1-49-5', 'cond-mat-0209170-2-49-5'], ['cond-mat-0209170-1-49-6', 'cond-mat-0209170-2-49-6'], ['cond-mat-0209170-1-49-7', 'cond-mat-0209170-2-49-7'], ['cond-mat-0209170-1-49-8', 'cond-mat-0209170-2-49-8'], ['cond-mat-0209170-1-31-0', 'cond-mat-0209170-2-31-0'], ['cond-mat-0209170-1-31-1', 'cond-mat-0209170-2-31-1'], ['cond-mat-0209170-1-31-2', 'cond-mat-0209170-2-31-2'], ['cond-mat-0209170-1-31-3', 'cond-mat-0209170-2-31-3'], ['cond-mat-0209170-1-56-0', 'cond-mat-0209170-2-56-0'], ['cond-mat-0209170-1-56-1', 'cond-mat-0209170-2-56-1'], ['cond-mat-0209170-1-56-2', 'cond-mat-0209170-2-56-2'], ['cond-mat-0209170-1-56-3', 'cond-mat-0209170-2-56-3'], ['cond-mat-0209170-1-59-0', 'cond-mat-0209170-2-59-0'], ['cond-mat-0209170-1-59-1', 'cond-mat-0209170-2-59-1'], ['cond-mat-0209170-1-59-2', 'cond-mat-0209170-2-59-2'], ['cond-mat-0209170-1-59-3', 'cond-mat-0209170-2-59-3'], ['cond-mat-0209170-1-59-4', 'cond-mat-0209170-2-59-4'], ['cond-mat-0209170-1-59-5', 'cond-mat-0209170-2-59-5'], ['cond-mat-0209170-1-59-6', 'cond-mat-0209170-2-59-6'], ['cond-mat-0209170-1-59-7', 'cond-mat-0209170-2-59-7'], ['cond-mat-0209170-1-60-0', 'cond-mat-0209170-2-60-0'], ['cond-mat-0209170-1-60-1', 'cond-mat-0209170-2-60-1'], ['cond-mat-0209170-1-60-2', 'cond-mat-0209170-2-60-2'], ['cond-mat-0209170-1-60-3', 'cond-mat-0209170-2-60-3'], ['cond-mat-0209170-1-60-4', 'cond-mat-0209170-2-60-4'], ['cond-mat-0209170-1-38-0', 'cond-mat-0209170-2-38-0'], ['cond-mat-0209170-1-38-1', 'cond-mat-0209170-2-38-1'], ['cond-mat-0209170-1-38-2', 'cond-mat-0209170-2-38-2'], ['cond-mat-0209170-1-38-3', 'cond-mat-0209170-2-38-3'], ['cond-mat-0209170-1-38-4', 'cond-mat-0209170-2-38-4'], ['cond-mat-0209170-1-57-0', 'cond-mat-0209170-2-57-0'], ['cond-mat-0209170-1-57-1', 'cond-mat-0209170-2-57-1'], ['cond-mat-0209170-1-57-2', 'cond-mat-0209170-2-57-2'], ['cond-mat-0209170-1-57-3', 'cond-mat-0209170-2-57-3'], ['cond-mat-0209170-1-57-4', 'cond-mat-0209170-2-57-4'], ['cond-mat-0209170-1-57-5', 'cond-mat-0209170-2-57-5'], ['cond-mat-0209170-1-57-6', 'cond-mat-0209170-2-57-6'], ['cond-mat-0209170-1-67-0', 'cond-mat-0209170-2-67-0'], ['cond-mat-0209170-1-67-1', 'cond-mat-0209170-2-67-1'], ['cond-mat-0209170-1-67-2', 'cond-mat-0209170-2-67-2'], ['cond-mat-0209170-1-16-0', 'cond-mat-0209170-2-16-0'], ['cond-mat-0209170-1-68-0', 'cond-mat-0209170-2-68-0'], ['cond-mat-0209170-1-42-0', 'cond-mat-0209170-2-42-0'], ['cond-mat-0209170-1-42-1', 'cond-mat-0209170-2-42-1'], ['cond-mat-0209170-1-42-2', 'cond-mat-0209170-2-42-2'], ['cond-mat-0209170-1-76-0', 'cond-mat-0209170-2-76-0'], ['cond-mat-0209170-1-76-1', 'cond-mat-0209170-2-76-1'], ['cond-mat-0209170-1-54-0', 'cond-mat-0209170-2-54-0'], ['cond-mat-0209170-1-54-1', 'cond-mat-0209170-2-54-1'], ['cond-mat-0209170-1-54-2', 'cond-mat-0209170-2-54-2'], ['cond-mat-0209170-1-39-0', 'cond-mat-0209170-2-39-0'], ['cond-mat-0209170-1-39-1', 'cond-mat-0209170-2-39-1'], ['cond-mat-0209170-1-39-2', 'cond-mat-0209170-2-39-2'], ['cond-mat-0209170-1-69-0', 'cond-mat-0209170-2-69-0'], ['cond-mat-0209170-1-8-0', 'cond-mat-0209170-2-8-0'], ['cond-mat-0209170-1-62-0', 'cond-mat-0209170-2-62-0'], ['cond-mat-0209170-1-62-1', 'cond-mat-0209170-2-62-1'], ['cond-mat-0209170-1-19-0', 'cond-mat-0209170-2-19-0'], ['cond-mat-0209170-1-19-1', 'cond-mat-0209170-2-19-1'], ['cond-mat-0209170-1-19-2', 'cond-mat-0209170-2-19-2'], ['cond-mat-0209170-1-19-3', 'cond-mat-0209170-2-19-3'], ['cond-mat-0209170-1-36-0', 'cond-mat-0209170-2-36-0'], ['cond-mat-0209170-1-36-1', 'cond-mat-0209170-2-36-1'], ['cond-mat-0209170-1-36-2', 'cond-mat-0209170-2-36-2'], ['cond-mat-0209170-1-36-3', 'cond-mat-0209170-2-36-3'], ['cond-mat-0209170-1-36-4', 'cond-mat-0209170-2-36-4'], ['cond-mat-0209170-1-36-5', 'cond-mat-0209170-2-36-5'], ['cond-mat-0209170-1-36-6', 'cond-mat-0209170-2-36-6'], ['cond-mat-0209170-1-36-7', 'cond-mat-0209170-2-36-7'], ['cond-mat-0209170-1-74-0', 'cond-mat-0209170-2-74-0'], ['cond-mat-0209170-1-25-1', 'cond-mat-0209170-2-25-1'], ['cond-mat-0209170-1-25-2', 'cond-mat-0209170-2-25-2'], ['cond-mat-0209170-1-25-3', 'cond-mat-0209170-2-25-3'], ['cond-mat-0209170-1-25-4', 'cond-mat-0209170-2-25-4'], ['cond-mat-0209170-1-25-5', 'cond-mat-0209170-2-25-5'], ['cond-mat-0209170-1-9-1', 'cond-mat-0209170-2-9-1'], ['cond-mat-0209170-1-9-2', 'cond-mat-0209170-2-9-2'], ['cond-mat-0209170-1-9-3', 'cond-mat-0209170-2-9-3'], ['cond-mat-0209170-1-9-4', 'cond-mat-0209170-2-9-4'], ['cond-mat-0209170-1-9-5', 'cond-mat-0209170-2-9-5'], ['cond-mat-0209170-1-9-6', 'cond-mat-0209170-2-9-6'], ['cond-mat-0209170-1-41-0', 'cond-mat-0209170-2-41-0'], ['cond-mat-0209170-1-41-1', 'cond-mat-0209170-2-41-1'], ['cond-mat-0209170-1-40-0', 'cond-mat-0209170-2-40-0'], ['cond-mat-0209170-1-40-1', 'cond-mat-0209170-2-40-1'], ['cond-mat-0209170-1-40-2', 'cond-mat-0209170-2-40-2'], ['cond-mat-0209170-1-40-3', 'cond-mat-0209170-2-40-3'], ['cond-mat-0209170-1-40-4', 'cond-mat-0209170-2-40-4'], ['cond-mat-0209170-1-40-5', 'cond-mat-0209170-2-40-5'], ['cond-mat-0209170-1-40-6', 'cond-mat-0209170-2-40-6'], ['cond-mat-0209170-1-40-7', 'cond-mat-0209170-2-40-7'], ['cond-mat-0209170-1-40-9', 'cond-mat-0209170-2-40-9'], ['cond-mat-0209170-1-55-0', 'cond-mat-0209170-2-55-0'], ['cond-mat-0209170-1-55-1', 'cond-mat-0209170-2-55-1'], ['cond-mat-0209170-1-55-3', 'cond-mat-0209170-2-55-3'], ['cond-mat-0209170-1-66-3', 'cond-mat-0209170-2-66-3'], ['cond-mat-0209170-1-66-4', 'cond-mat-0209170-2-66-4'], ['cond-mat-0209170-1-51-0', 'cond-mat-0209170-2-51-0'], ['cond-mat-0209170-1-50-1', 'cond-mat-0209170-2-50-1']]

[['cond-mat-0209170-1-2-0', 'cond-mat-0209170-2-2-0'], ['cond-mat-0209170-1-2-1', 'cond-mat-0209170-2-2-1'], ['cond-mat-0209170-1-2-2', 'cond-mat-0209170-2-2-2'], ['cond-mat-0209170-1-2-3', 'cond-mat-0209170-2-2-3'], ['cond-mat-0209170-1-33-0', 'cond-mat-0209170-2-33-0'], ['cond-mat-0209170-1-33-1', 'cond-mat-0209170-2-33-1'], ['cond-mat-0209170-1-33-2', 'cond-mat-0209170-2-33-2'], ['cond-mat-0209170-1-33-3', 'cond-mat-0209170-2-33-3'], ['cond-mat-0209170-1-5-0', 'cond-mat-0209170-2-5-0'], ['cond-mat-0209170-1-5-1', 'cond-mat-0209170-2-5-1'], ['cond-mat-0209170-1-5-2', 'cond-mat-0209170-2-5-2'], ['cond-mat-0209170-1-5-3', 'cond-mat-0209170-2-5-3'], ['cond-mat-0209170-1-21-0', 'cond-mat-0209170-2-21-0'], ['cond-mat-0209170-1-22-0', 'cond-mat-0209170-2-22-0'], ['cond-mat-0209170-1-0-0', 'cond-mat-0209170-2-0-0'], ['cond-mat-0209170-1-0-1', 'cond-mat-0209170-2-0-1'], ['cond-mat-0209170-1-0-2', 'cond-mat-0209170-2-0-2'], ['cond-mat-0209170-1-0-3', 'cond-mat-0209170-2-0-3'], ['cond-mat-0209170-1-0-4', 'cond-mat-0209170-2-0-4'], ['cond-mat-0209170-1-0-5', 'cond-mat-0209170-2-0-5'], ['cond-mat-0209170-1-0-6', 'cond-mat-0209170-2-0-6'], ['cond-mat-0209170-1-0-7', 'cond-mat-0209170-2-0-7'], ['cond-mat-0209170-1-52-0', 'cond-mat-0209170-2-52-0'], ['cond-mat-0209170-1-52-1', 'cond-mat-0209170-2-52-1'], ['cond-mat-0209170-1-52-2', 'cond-mat-0209170-2-52-2'], ['cond-mat-0209170-1-75-0', 'cond-mat-0209170-2-75-0'], ['cond-mat-0209170-1-75-1', 'cond-mat-0209170-2-75-1'], ['cond-mat-0209170-1-3-0', 'cond-mat-0209170-2-3-0'], ['cond-mat-0209170-1-3-1', 'cond-mat-0209170-2-3-1'], ['cond-mat-0209170-1-3-2', 'cond-mat-0209170-2-3-2'], ['cond-mat-0209170-1-3-3', 'cond-mat-0209170-2-3-3'], ['cond-mat-0209170-1-3-4', 'cond-mat-0209170-2-3-4'], ['cond-mat-0209170-1-24-0', 'cond-mat-0209170-2-24-0'], ['cond-mat-0209170-1-72-0', 'cond-mat-0209170-2-72-0'], ['cond-mat-0209170-1-77-0', 'cond-mat-0209170-2-77-0'], ['cond-mat-0209170-1-73-0', 'cond-mat-0209170-2-73-0'], ['cond-mat-0209170-1-15-0', 'cond-mat-0209170-2-15-0'], ['cond-mat-0209170-1-15-1', 'cond-mat-0209170-2-15-1'], ['cond-mat-0209170-1-15-2', 'cond-mat-0209170-2-15-2'], ['cond-mat-0209170-1-15-3', 'cond-mat-0209170-2-15-3'], ['cond-mat-0209170-1-15-4', 'cond-mat-0209170-2-15-4'], ['cond-mat-0209170-1-15-5', 'cond-mat-0209170-2-15-5'], ['cond-mat-0209170-1-15-6', 'cond-mat-0209170-2-15-6'], ['cond-mat-0209170-1-15-7', 'cond-mat-0209170-2-15-7'], ['cond-mat-0209170-1-12-0', 'cond-mat-0209170-2-12-0'], ['cond-mat-0209170-1-12-1', 'cond-mat-0209170-2-12-1'], ['cond-mat-0209170-1-37-0', 'cond-mat-0209170-2-37-0'], ['cond-mat-0209170-1-37-1', 'cond-mat-0209170-2-37-1'], ['cond-mat-0209170-1-37-2', 'cond-mat-0209170-2-37-2'], ['cond-mat-0209170-1-37-3', 'cond-mat-0209170-2-37-3'], ['cond-mat-0209170-1-37-4', 'cond-mat-0209170-2-37-4'], ['cond-mat-0209170-1-37-5', 'cond-mat-0209170-2-37-5'], ['cond-mat-0209170-1-37-6', 'cond-mat-0209170-2-37-6'], ['cond-mat-0209170-1-37-7', 'cond-mat-0209170-2-37-7'], ['cond-mat-0209170-1-34-0', 'cond-mat-0209170-2-34-0'], ['cond-mat-0209170-1-34-1', 'cond-mat-0209170-2-34-1'], ['cond-mat-0209170-1-34-2', 'cond-mat-0209170-2-34-2'], ['cond-mat-0209170-1-34-3', 'cond-mat-0209170-2-34-3'], ['cond-mat-0209170-1-34-4', 'cond-mat-0209170-2-34-4'], ['cond-mat-0209170-1-34-5', 'cond-mat-0209170-2-34-5'], ['cond-mat-0209170-1-34-6', 'cond-mat-0209170-2-34-6'], ['cond-mat-0209170-1-34-7', 'cond-mat-0209170-2-34-7'], ['cond-mat-0209170-1-34-8', 'cond-mat-0209170-2-34-8'], ['cond-mat-0209170-1-34-9', 'cond-mat-0209170-2-34-9'], ['cond-mat-0209170-1-34-10', 'cond-mat-0209170-2-34-10'], ['cond-mat-0209170-1-30-0', 'cond-mat-0209170-2-30-0'], ['cond-mat-0209170-1-30-1', 'cond-mat-0209170-2-30-1'], ['cond-mat-0209170-1-30-2', 'cond-mat-0209170-2-30-2'], ['cond-mat-0209170-1-30-3', 'cond-mat-0209170-2-30-3'], ['cond-mat-0209170-1-30-4', 'cond-mat-0209170-2-30-4'], ['cond-mat-0209170-1-30-5', 'cond-mat-0209170-2-30-5'], ['cond-mat-0209170-1-4-0', 'cond-mat-0209170-2-4-0'], ['cond-mat-0209170-1-4-1', 'cond-mat-0209170-2-4-1'], ['cond-mat-0209170-1-4-2', 'cond-mat-0209170-2-4-2'], ['cond-mat-0209170-1-4-3', 'cond-mat-0209170-2-4-3'], ['cond-mat-0209170-1-29-0', 'cond-mat-0209170-2-29-0'], ['cond-mat-0209170-1-29-1', 'cond-mat-0209170-2-29-1'], ['cond-mat-0209170-1-29-3', 'cond-mat-0209170-2-29-3'], ['cond-mat-0209170-1-29-4', 'cond-mat-0209170-2-29-4'], ['cond-mat-0209170-1-29-6', 'cond-mat-0209170-2-29-6'], ['cond-mat-0209170-1-29-7', 'cond-mat-0209170-2-29-7'], ['cond-mat-0209170-1-1-0', 'cond-mat-0209170-2-1-0'], ['cond-mat-0209170-1-1-1', 'cond-mat-0209170-2-1-1'], ['cond-mat-0209170-1-1-2', 'cond-mat-0209170-2-1-2'], ['cond-mat-0209170-1-1-3', 'cond-mat-0209170-2-1-3'], ['cond-mat-0209170-1-6-0', 'cond-mat-0209170-2-6-0'], ['cond-mat-0209170-1-6-1', 'cond-mat-0209170-2-6-1'], ['cond-mat-0209170-1-6-2', 'cond-mat-0209170-2-6-2'], ['cond-mat-0209170-1-6-3', 'cond-mat-0209170-2-6-3'], ['cond-mat-0209170-1-23-0', 'cond-mat-0209170-2-23-0'], ['cond-mat-0209170-1-23-1', 'cond-mat-0209170-2-23-1'], ['cond-mat-0209170-1-61-0', 'cond-mat-0209170-2-61-0'], ['cond-mat-0209170-1-61-1', 'cond-mat-0209170-2-61-1'], ['cond-mat-0209170-1-61-2', 'cond-mat-0209170-2-61-2'], ['cond-mat-0209170-1-61-3', 'cond-mat-0209170-2-61-3'], ['cond-mat-0209170-1-61-4', 'cond-mat-0209170-2-61-4'], ['cond-mat-0209170-1-27-0', 'cond-mat-0209170-2-27-0'], ['cond-mat-0209170-1-27-1', 'cond-mat-0209170-2-27-1'], ['cond-mat-0209170-1-27-2', 'cond-mat-0209170-2-27-2'], ['cond-mat-0209170-1-27-3', 'cond-mat-0209170-2-27-3'], ['cond-mat-0209170-1-27-4', 'cond-mat-0209170-2-27-4'], ['cond-mat-0209170-1-27-5', 'cond-mat-0209170-2-27-5'], ['cond-mat-0209170-1-27-6', 'cond-mat-0209170-2-27-6'], ['cond-mat-0209170-1-27-7', 'cond-mat-0209170-2-27-7'], ['cond-mat-0209170-1-11-0', 'cond-mat-0209170-2-11-0'], ['cond-mat-0209170-1-11-1', 'cond-mat-0209170-2-11-1'], ['cond-mat-0209170-1-11-2', 'cond-mat-0209170-2-11-2'], ['cond-mat-0209170-1-11-3', 'cond-mat-0209170-2-11-3'], ['cond-mat-0209170-1-11-4', 'cond-mat-0209170-2-11-4'], ['cond-mat-0209170-1-65-0', 'cond-mat-0209170-2-65-0'], ['cond-mat-0209170-1-65-1', 'cond-mat-0209170-2-65-1'], ['cond-mat-0209170-1-65-2', 'cond-mat-0209170-2-65-2'], ['cond-mat-0209170-1-65-3', 'cond-mat-0209170-2-65-3'], ['cond-mat-0209170-1-10-0', 'cond-mat-0209170-2-10-0'], ['cond-mat-0209170-1-10-1', 'cond-mat-0209170-2-10-1'], ['cond-mat-0209170-1-10-3', 'cond-mat-0209170-2-10-3'], ['cond-mat-0209170-1-10-4', 'cond-mat-0209170-2-10-4'], ['cond-mat-0209170-1-10-5', 'cond-mat-0209170-2-10-5'], ['cond-mat-0209170-1-10-6', 'cond-mat-0209170-2-10-6'], ['cond-mat-0209170-1-10-7', 'cond-mat-0209170-2-10-7'], ['cond-mat-0209170-1-64-0', 'cond-mat-0209170-2-64-0'], ['cond-mat-0209170-1-64-1', 'cond-mat-0209170-2-64-1'], ['cond-mat-0209170-1-64-2', 'cond-mat-0209170-2-64-2'], ['cond-mat-0209170-1-64-3', 'cond-mat-0209170-2-64-3'], ['cond-mat-0209170-1-64-4', 'cond-mat-0209170-2-64-4'], ['cond-mat-0209170-1-18-0', 'cond-mat-0209170-2-18-0'], ['cond-mat-0209170-1-18-1', 'cond-mat-0209170-2-18-1'], ['cond-mat-0209170-1-18-2', 'cond-mat-0209170-2-18-2'], ['cond-mat-0209170-1-18-3', 'cond-mat-0209170-2-18-3'], ['cond-mat-0209170-1-45-0', 'cond-mat-0209170-2-45-0'], ['cond-mat-0209170-1-45-1', 'cond-mat-0209170-2-45-1'], ['cond-mat-0209170-1-71-0', 'cond-mat-0209170-2-71-0'], ['cond-mat-0209170-1-66-0', 'cond-mat-0209170-2-66-0'], ['cond-mat-0209170-1-66-1', 'cond-mat-0209170-2-66-1'], ['cond-mat-0209170-1-66-2', 'cond-mat-0209170-2-66-2'], ['cond-mat-0209170-1-66-5', 'cond-mat-0209170-2-66-6'], ['cond-mat-0209170-1-51-1', 'cond-mat-0209170-2-51-1'], ['cond-mat-0209170-1-51-2', 'cond-mat-0209170-2-51-2'], ['cond-mat-0209170-1-51-3', 'cond-mat-0209170-2-51-3'], ['cond-mat-0209170-1-51-4', 'cond-mat-0209170-2-51-4'], ['cond-mat-0209170-1-43-0', 'cond-mat-0209170-2-43-0'], ['cond-mat-0209170-1-50-0', 'cond-mat-0209170-2-50-0'], ['cond-mat-0209170-1-53-0', 'cond-mat-0209170-2-53-0'], ['cond-mat-0209170-1-53-1', 'cond-mat-0209170-2-53-1'], ['cond-mat-0209170-1-53-2', 'cond-mat-0209170-2-53-2'], ['cond-mat-0209170-1-53-3', 'cond-mat-0209170-2-53-3'], ['cond-mat-0209170-1-53-4', 'cond-mat-0209170-2-53-4'], ['cond-mat-0209170-1-53-5', 'cond-mat-0209170-2-53-5'], ['cond-mat-0209170-1-53-6', 'cond-mat-0209170-2-53-6'], ['cond-mat-0209170-1-53-7', 'cond-mat-0209170-2-53-7'], ['cond-mat-0209170-1-53-8', 'cond-mat-0209170-2-53-8'], ['cond-mat-0209170-1-53-9', 'cond-mat-0209170-2-53-9'], ['cond-mat-0209170-1-53-10', 'cond-mat-0209170-2-53-10'], ['cond-mat-0209170-1-53-11', 'cond-mat-0209170-2-53-11'], ['cond-mat-0209170-1-53-12', 'cond-mat-0209170-2-53-12'], ['cond-mat-0209170-1-53-13', 'cond-mat-0209170-2-53-13'], ['cond-mat-0209170-1-47-0', 'cond-mat-0209170-2-47-0'], ['cond-mat-0209170-1-47-2', 'cond-mat-0209170-2-47-2'], ['cond-mat-0209170-1-47-3', 'cond-mat-0209170-2-47-3'], ['cond-mat-0209170-1-47-4', 'cond-mat-0209170-2-47-4'], ['cond-mat-0209170-1-47-5', 'cond-mat-0209170-2-47-5'], ['cond-mat-0209170-1-47-6', 'cond-mat-0209170-2-47-6'], ['cond-mat-0209170-1-47-7', 'cond-mat-0209170-2-47-7'], ['cond-mat-0209170-1-47-8', 'cond-mat-0209170-2-47-8'], ['cond-mat-0209170-1-47-9', 'cond-mat-0209170-2-47-9'], ['cond-mat-0209170-1-13-0', 'cond-mat-0209170-2-13-0'], ['cond-mat-0209170-1-70-0', 'cond-mat-0209170-2-70-0'], ['cond-mat-0209170-1-20-0', 'cond-mat-0209170-2-20-0'], ['cond-mat-0209170-1-20-1', 'cond-mat-0209170-2-20-1'], ['cond-mat-0209170-1-35-0', 'cond-mat-0209170-2-35-0'], ['cond-mat-0209170-1-35-1', 'cond-mat-0209170-2-35-1'], ['cond-mat-0209170-1-35-2', 'cond-mat-0209170-2-35-2'], ['cond-mat-0209170-1-35-3', 'cond-mat-0209170-2-35-3'], ['cond-mat-0209170-1-35-4', 'cond-mat-0209170-2-35-4'], ['cond-mat-0209170-1-49-0', 'cond-mat-0209170-2-49-0'], ['cond-mat-0209170-1-49-1', 'cond-mat-0209170-2-49-1'], ['cond-mat-0209170-1-49-2', 'cond-mat-0209170-2-49-2'], ['cond-mat-0209170-1-49-3', 'cond-mat-0209170-2-49-3'], ['cond-mat-0209170-1-49-4', 'cond-mat-0209170-2-49-4'], ['cond-mat-0209170-1-49-5', 'cond-mat-0209170-2-49-5'], ['cond-mat-0209170-1-49-6', 'cond-mat-0209170-2-49-6'], ['cond-mat-0209170-1-49-7', 'cond-mat-0209170-2-49-7'], ['cond-mat-0209170-1-49-8', 'cond-mat-0209170-2-49-8'], ['cond-mat-0209170-1-31-0', 'cond-mat-0209170-2-31-0'], ['cond-mat-0209170-1-31-1', 'cond-mat-0209170-2-31-1'], ['cond-mat-0209170-1-31-2', 'cond-mat-0209170-2-31-2'], ['cond-mat-0209170-1-31-3', 'cond-mat-0209170-2-31-3'], ['cond-mat-0209170-1-56-0', 'cond-mat-0209170-2-56-0'], ['cond-mat-0209170-1-56-1', 'cond-mat-0209170-2-56-1'], ['cond-mat-0209170-1-56-2', 'cond-mat-0209170-2-56-2'], ['cond-mat-0209170-1-56-3', 'cond-mat-0209170-2-56-3'], ['cond-mat-0209170-1-59-0', 'cond-mat-0209170-2-59-0'], ['cond-mat-0209170-1-59-1', 'cond-mat-0209170-2-59-1'], ['cond-mat-0209170-1-59-2', 'cond-mat-0209170-2-59-2'], ['cond-mat-0209170-1-59-3', 'cond-mat-0209170-2-59-3'], ['cond-mat-0209170-1-59-4', 'cond-mat-0209170-2-59-4'], ['cond-mat-0209170-1-59-5', 'cond-mat-0209170-2-59-5'], ['cond-mat-0209170-1-59-6', 'cond-mat-0209170-2-59-6'], ['cond-mat-0209170-1-59-7', 'cond-mat-0209170-2-59-7'], ['cond-mat-0209170-1-60-0', 'cond-mat-0209170-2-60-0'], ['cond-mat-0209170-1-60-1', 'cond-mat-0209170-2-60-1'], ['cond-mat-0209170-1-60-2', 'cond-mat-0209170-2-60-2'], ['cond-mat-0209170-1-60-3', 'cond-mat-0209170-2-60-3'], ['cond-mat-0209170-1-60-4', 'cond-mat-0209170-2-60-4'], ['cond-mat-0209170-1-38-0', 'cond-mat-0209170-2-38-0'], ['cond-mat-0209170-1-38-1', 'cond-mat-0209170-2-38-1'], ['cond-mat-0209170-1-38-2', 'cond-mat-0209170-2-38-2'], ['cond-mat-0209170-1-38-3', 'cond-mat-0209170-2-38-3'], ['cond-mat-0209170-1-38-4', 'cond-mat-0209170-2-38-4'], ['cond-mat-0209170-1-57-0', 'cond-mat-0209170-2-57-0'], ['cond-mat-0209170-1-57-1', 'cond-mat-0209170-2-57-1'], ['cond-mat-0209170-1-57-2', 'cond-mat-0209170-2-57-2'], ['cond-mat-0209170-1-57-3', 'cond-mat-0209170-2-57-3'], ['cond-mat-0209170-1-57-4', 'cond-mat-0209170-2-57-4'], ['cond-mat-0209170-1-57-5', 'cond-mat-0209170-2-57-5'], ['cond-mat-0209170-1-57-6', 'cond-mat-0209170-2-57-6'], ['cond-mat-0209170-1-67-0', 'cond-mat-0209170-2-67-0'], ['cond-mat-0209170-1-67-1', 'cond-mat-0209170-2-67-1'], ['cond-mat-0209170-1-67-2', 'cond-mat-0209170-2-67-2'], ['cond-mat-0209170-1-16-0', 'cond-mat-0209170-2-16-0'], ['cond-mat-0209170-1-68-0', 'cond-mat-0209170-2-68-0'], ['cond-mat-0209170-1-42-0', 'cond-mat-0209170-2-42-0'], ['cond-mat-0209170-1-42-1', 'cond-mat-0209170-2-42-1'], ['cond-mat-0209170-1-42-2', 'cond-mat-0209170-2-42-2'], ['cond-mat-0209170-1-76-0', 'cond-mat-0209170-2-76-0'], ['cond-mat-0209170-1-76-1', 'cond-mat-0209170-2-76-1'], ['cond-mat-0209170-1-54-0', 'cond-mat-0209170-2-54-0'], ['cond-mat-0209170-1-54-1', 'cond-mat-0209170-2-54-1'], ['cond-mat-0209170-1-54-2', 'cond-mat-0209170-2-54-2'], ['cond-mat-0209170-1-39-0', 'cond-mat-0209170-2-39-0'], ['cond-mat-0209170-1-39-1', 'cond-mat-0209170-2-39-1'], ['cond-mat-0209170-1-39-2', 'cond-mat-0209170-2-39-2'], ['cond-mat-0209170-1-69-0', 'cond-mat-0209170-2-69-0'], ['cond-mat-0209170-1-8-0', 'cond-mat-0209170-2-8-0'], ['cond-mat-0209170-1-62-0', 'cond-mat-0209170-2-62-0'], ['cond-mat-0209170-1-62-1', 'cond-mat-0209170-2-62-1'], ['cond-mat-0209170-1-19-0', 'cond-mat-0209170-2-19-0'], ['cond-mat-0209170-1-19-1', 'cond-mat-0209170-2-19-1'], ['cond-mat-0209170-1-19-2', 'cond-mat-0209170-2-19-2'], ['cond-mat-0209170-1-19-3', 'cond-mat-0209170-2-19-3'], ['cond-mat-0209170-1-36-0', 'cond-mat-0209170-2-36-0'], ['cond-mat-0209170-1-36-1', 'cond-mat-0209170-2-36-1'], ['cond-mat-0209170-1-36-2', 'cond-mat-0209170-2-36-2'], ['cond-mat-0209170-1-36-3', 'cond-mat-0209170-2-36-3'], ['cond-mat-0209170-1-36-4', 'cond-mat-0209170-2-36-4'], ['cond-mat-0209170-1-36-5', 'cond-mat-0209170-2-36-5'], ['cond-mat-0209170-1-36-6', 'cond-mat-0209170-2-36-6'], ['cond-mat-0209170-1-36-7', 'cond-mat-0209170-2-36-7'], ['cond-mat-0209170-1-74-0', 'cond-mat-0209170-2-74-0'], ['cond-mat-0209170-1-25-1', 'cond-mat-0209170-2-25-1'], ['cond-mat-0209170-1-25-2', 'cond-mat-0209170-2-25-2'], ['cond-mat-0209170-1-25-3', 'cond-mat-0209170-2-25-3'], ['cond-mat-0209170-1-25-4', 'cond-mat-0209170-2-25-4'], ['cond-mat-0209170-1-25-5', 'cond-mat-0209170-2-25-5'], ['cond-mat-0209170-1-9-1', 'cond-mat-0209170-2-9-1'], ['cond-mat-0209170-1-9-2', 'cond-mat-0209170-2-9-2'], ['cond-mat-0209170-1-9-3', 'cond-mat-0209170-2-9-3'], ['cond-mat-0209170-1-9-4', 'cond-mat-0209170-2-9-4'], ['cond-mat-0209170-1-9-5', 'cond-mat-0209170-2-9-5'], ['cond-mat-0209170-1-9-6', 'cond-mat-0209170-2-9-6'], ['cond-mat-0209170-1-41-0', 'cond-mat-0209170-2-41-0'], ['cond-mat-0209170-1-41-1', 'cond-mat-0209170-2-41-1'], ['cond-mat-0209170-1-40-0', 'cond-mat-0209170-2-40-0'], ['cond-mat-0209170-1-40-1', 'cond-mat-0209170-2-40-1'], ['cond-mat-0209170-1-40-2', 'cond-mat-0209170-2-40-2'], ['cond-mat-0209170-1-40-3', 'cond-mat-0209170-2-40-3'], ['cond-mat-0209170-1-40-4', 'cond-mat-0209170-2-40-4'], ['cond-mat-0209170-1-40-5', 'cond-mat-0209170-2-40-5'], ['cond-mat-0209170-1-40-6', 'cond-mat-0209170-2-40-6'], ['cond-mat-0209170-1-40-7', 'cond-mat-0209170-2-40-7'], ['cond-mat-0209170-1-40-9', 'cond-mat-0209170-2-40-9'], ['cond-mat-0209170-1-55-0', 'cond-mat-0209170-2-55-0'], ['cond-mat-0209170-1-55-1', 'cond-mat-0209170-2-55-1'], ['cond-mat-0209170-1-55-3', 'cond-mat-0209170-2-55-3']]

[]

[]

[['cond-mat-0209170-1-66-3', 'cond-mat-0209170-2-66-3'], ['cond-mat-0209170-1-66-4', 'cond-mat-0209170-2-66-4'], ['cond-mat-0209170-1-51-0', 'cond-mat-0209170-2-51-0'], ['cond-mat-0209170-1-50-1', 'cond-mat-0209170-2-50-1']]

[]

['cond-mat-0209170-1-8-1', 'cond-mat-0209170-1-9-0', 'cond-mat-0209170-1-9-7', 'cond-mat-0209170-1-10-2', 'cond-mat-0209170-1-10-8', 'cond-mat-0209170-1-25-0', 'cond-mat-0209170-1-29-2', 'cond-mat-0209170-1-29-5', 'cond-mat-0209170-1-33-4', 'cond-mat-0209170-1-40-8', 'cond-mat-0209170-1-44-0', 'cond-mat-0209170-1-46-0', 'cond-mat-0209170-1-47-1', 'cond-mat-0209170-1-49-9', 'cond-mat-0209170-1-55-2', 'cond-mat-0209170-2-8-1', 'cond-mat-0209170-2-9-0', 'cond-mat-0209170-2-9-7', 'cond-mat-0209170-2-10-2', 'cond-mat-0209170-2-10-8', 'cond-mat-0209170-2-25-0', 'cond-mat-0209170-2-29-2', 'cond-mat-0209170-2-29-5', 'cond-mat-0209170-2-33-4', 'cond-mat-0209170-2-40-8', 'cond-mat-0209170-2-44-0', 'cond-mat-0209170-2-46-0', 'cond-mat-0209170-2-47-1', 'cond-mat-0209170-2-49-9', 'cond-mat-0209170-2-50-5', 'cond-mat-0209170-2-55-2']

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

https://arxiv.org/abs/cond-mat/0209170

1810.06506

{'1810.06506-1-0-0': 'It has recently been pointed out that the underlying symmetry of dark matter may well be [MATH] (coming from [MATH]) with the dark parity of any given particle determined by [MATH], where [MATH] is its [MATH] charge and [MATH] its spin angular momentum.', '1810.06506-1-0-1': 'Armed with this new insight, previous simple models of dark matter are reinterpreted, and a novel idea is proposed that light seesaw dark matter exists in analogy to light neutrinos and is produced by the rare decay of the standard-model Higgs boson.', '1810.06506-1-1-0': '24pt', '1810.06506-1-2-0': 'Introduction : In the decomposition of [MATH], the fermions of the standard model (SM) are organized into [EQUATION] where [EQUATION] under [MATH].', '1810.06506-1-2-1': 'To allow the quarks and leptons to acquire mass, the scalar 10 representation, which contains the necessary Higgs doublets, i.e. [EQUATION] with [EQUATION] from [MATH] respectively, is required, resulting in the allowed Yukawa couplings [EQUATION] as desired.', '1810.06506-1-2-2': 'The nonzero vacuum expectation values [MATH] also break electroweak [MATH] to electrodynamic [MATH].', '1810.06506-1-2-3': 'Since the SM gauge bosons all have [MATH], it is obvious (but not recognized for its importance until recently [CITATION]) that all SM fermions have odd [MATH] and all SM bosons have even [MATH].', '1810.06506-1-2-4': 'This means that each SM particle has even [MATH] where [MATH] is its spin angular momentum.', '1810.06506-1-2-5': 'It is thus a short step to realizing that any scalar with odd [MATH] and any fermion with even [MATH] would have odd [MATH], making it a natural stabilizing symmetry for dark matter.', '1810.06506-1-2-6': 'Indeed, all previous simple models of dark matter based on an [MATH] discrete symmetry may be incorporated into such a framework.', '1810.06506-1-3-0': 'The scalar 126 representation of [MATH] contains a singlet [MATH] under [MATH], which may be used to break [MATH] at the TeV scale and would allow [MATH] (the right-handed neutrino) to obtain a large Majorana mass, thereby triggering the canonical seesaw mechanism for small Majorana neutrino masses.', '1810.06506-1-3-1': 'This is usually described as lepton number [MATH] breaking to lepton parity [MATH] [CITATION], but here it is clear that it has to do with the breaking of gauge [MATH] to [MATH].', '1810.06506-1-4-0': 'In the minimal supersymmetric standard model (MSSM), [MATH] is used to distinguish the SM particles from their superpartners, which belong thus to the dark sector if [MATH] is assumed conserved.', '1810.06506-1-4-1': 'Since [MATH] is identical to [MATH], it has long been recognized [CITATION] that a theory with gauge [MATH], broken by two units, would be a natural framework for dark matter.', '1810.06506-1-4-2': 'In particular, the decomposition [EQUATION] shows that the fermionic 16 of [MATH] contains [EQUATION]', '1810.06506-1-4-3': 'Hence [MATH] is odd for all quarks and leptons.', '1810.06506-1-4-4': 'As for the scalar sector, the 10 representation contains the bidoublet [MATH].', '1810.06506-1-4-5': 'Hence its [MATH] is even.', '1810.06506-1-4-6': 'In other words, [MATH] coincides with [MATH].', '1810.06506-1-4-7': 'However, the former requires a left-right intermediate scale, whereas the latter does not.', '1810.06506-1-4-8': 'They are thus conceptually and phenomenologically distinct.', '1810.06506-1-4-9': 'In this study, [MATH] separates from [MATH] at the unification scale [CITATION], and its symmetry breaking scale is independent of the electroweak scale.', '1810.06506-1-5-0': 'It should also be pointed out that in [MATH], the decomposition [MATH] shows that [MATH] may be invoked as the underlying dark symmetry as well.', '1810.06506-1-6-0': 'Reappraisal of [MATH] Dark Matter : It has been remarked that it is very easy to invent a model of dark matter.', '1810.06506-1-6-1': 'The first step is to introduce a new [MATH] symmetry under which all SM particles are even and a new neutral particle of your choice is odd.', '1810.06506-1-6-2': 'It should then have the appropriate mass and interaction to account for the relic abundance of dark matter in the Universe, but not excluded by direct or indirect search experiments.', '1810.06506-1-7-0': 'The simplest such model [CITATION] assumes a real scalar singlet, odd under [MATH].', '1810.06506-1-7-1': 'It has been studied extensively [CITATION] and is still a viable explanation of dark matter.', '1810.06506-1-7-2': 'In the framework of [MATH], a scalar with odd [MATH] requires it to have odd [MATH].', '1810.06506-1-7-3': 'The scalar singlet [MATH] of [MATH] is such a particle.', '1810.06506-1-7-4': 'It is in fact the scalar analog of [MATH].', '1810.06506-1-7-5': 'They have the same [MATH], but differ in spin.', '1810.06506-1-7-6': 'Hence one is dark matter and the other is not.', '1810.06506-1-7-7': 'In Ref. [CITATION], they are both assigned odd lepton parity, which is now replaced by odd [MATH] parity.', '1810.06506-1-7-8': 'If [MATH] is indeed the origin of [MATH], then it should be complex and it should have [MATH] interactions.', '1810.06506-1-7-9': 'However, from the allowed [MATH] trilinear scalar coupling, [MATH] splits into two real scalars with a large mass gap.', '1810.06506-1-7-10': 'The lighter is dark matter and the heavier decays into the lighter plus a physical or virtual [MATH] gauge boson.', '1810.06506-1-7-11': "This would not affect the lighter scalar's suitability as dark matter, but would predict possible verifiable signatures involving [MATH].", '1810.06506-1-7-12': 'The present experimental bound on [MATH] is about 4.1 TeV from LHC (Large Hadron Collider) data [CITATION], which may be improved [CITATION] with further study.', '1810.06506-1-8-0': 'Instead of choosing [MATH] from the 16 of [MATH], the scalar doublet [MATH] may also be considered [CITATION].', '1810.06506-1-8-1': 'In that case, it is distinguished from [MATH] and [MATH] by their [MATH] charge.', '1810.06506-1-8-2': 'Hence [MATH] are even but [MATH] is odd under [MATH].', '1810.06506-1-8-3': 'This [MATH] discrete symmetry [CITATION] allows [MATH] to be dark matter [CITATION], at least in principle.', '1810.06506-1-8-4': 'However, its interaction with quarks through the [MATH] boson rules it out by direct-search experiments.', '1810.06506-1-8-5': 'In the SM, the allowed quartic coupling [MATH] serves to split [MATH] from [MATH], and since [MATH] only couples one to the other, the offending interaction with quarks is avoided kinematically in elastic nuclear recoil with a mass gap larger than a few hundred keV.', '1810.06506-1-8-6': 'This is known as the inert Higgs doublet model.', '1810.06506-1-8-7': 'In the case of [MATH], such a quartic coupling is forbidden, so if [MATH] originates from [MATH], other particles are needed for it to be dark matter.', '1810.06506-1-8-8': 'They turn out to be exactly [MATH] and [MATH], already discussed.', '1810.06506-1-8-9': 'The allowed couplings [MATH], [MATH] combined with [MATH] form the necessary effective quartic coupling as shown in Fig. 1.', '1810.06506-1-8-10': 'In this scenario, a linear combination of [MATH] and [MATH] is dark matter.', '1810.06506-1-9-0': 'Another possible simple model of dark matter is to have a singlet fermion [MATH] from the [MATH] or [MATH] of the [MATH] of [MATH].', '1810.06506-1-9-1': 'Since [MATH] has even [MATH], it is odd under [MATH].', '1810.06506-1-9-2': 'However, it has no renormalizable interaction with the particles of the SM and thus not a good dark-matter candidate without some additional fundamental particle such as a singlet scalar [CITATION] which has [MATH], i.e. the scalar counterpart of [MATH].', '1810.06506-1-9-3': 'A more interesting option is to combine [MATH] with the scalar doublet [MATH] discussed in the previous paragraph because there is now an allowed Yukawa coupling between the left-handed lepton doublet [MATH] with [MATH] through [MATH], i.e. [MATH].', '1810.06506-1-9-4': 'This forms the basis of the scotogenic model [CITATION] of radiative neutrino mass.', '1810.06506-1-9-5': 'Whereas the original model assumes the [MATH] quartic scalar coupling, it must now be replaced by the effective operator of Fig. 1.', '1810.06506-1-9-6': 'The resulting diagram [CITATION] for generating a radiative Majorana neutrino mass is then given by Fig. 2.', '1810.06506-1-10-0': 'Whereas [MATH] could be dark matter, its only interaction with the particles of the SM is through the left-handed lepton doublet, and is known [CITATION] to be restricted phenomenologically, thus limiting its viability as thermal dark matter.', '1810.06506-1-10-1': 'Hence a linear combination of [MATH] and [MATH] is again the likely dark-matter candidate in this case.', '1810.06506-1-10-2': 'They both couple to [MATH] but differently.', '1810.06506-1-10-3': 'Further study is then needed to reappraise this [MATH] interpretation of the scotogenic model.', '1810.06506-1-11-0': 'Once both [MATH] and [MATH] are present, the coupling [MATH] is allowed.', '1810.06506-1-11-1': 'This has also recently been considered [CITATION] with the assumption that it is very small so that a freeze-in mechanism applies to the decay of [MATH] to [MATH] and [MATH].', '1810.06506-1-12-0': 'Seesaw Dark Matter : In the [MATH] model, the singlet neutrino [MATH] gets a large Majorana mass from the scalar [MATH], both of which have even [MATH].', '1810.06506-1-12-1': 'This realizes the scenario of seesaw neutrino mass at the scale [MATH] which may be TeV or higher.', '1810.06506-1-12-2': 'Suppose the fermion singlets [EQUATION] from the 45, 126 representations of [MATH] are added, then the allowed Yukawa coupling [MATH] combined with a large Majorana mass for [MATH] would induce a small seesaw mass for [MATH].', '1810.06506-1-12-3': 'Note that both [MATH] and [MATH] have odd [MATH].', '1810.06506-1-12-4': 'Hence [MATH] could be naturally light dark matter, i.e. [MATH], in parallel with the seesaw neutrino mass, i.e. [MATH].', '1810.06506-1-12-5': 'A variation is to add [MATH] from the [MATH] of [MATH] together with [MATH], as proposed recently [CITATION], so that [MATH] and [MATH] combine to form a Dirac fermion with odd [MATH] and interact with [MATH].', '1810.06506-1-12-6': 'Another possible but different connection between seesaw neutrino mass and dark matter has also been proposed [CITATION], based on an imposed [MATH] discrete symmetry and a nonrenormalizable dimension-five coupling.', '1810.06506-1-13-0': 'Consider now the interaction of [MATH].', '1810.06506-1-13-1': 'It interacts mainly with [MATH].', '1810.06506-1-13-2': 'This is in analogy with [MATH] which interacts mainly with [MATH] and [MATH].', '1810.06506-1-13-3': 'Just as [MATH] decouples at a temperature of order 1 MeV, [MATH] would decouple at a temperature of order [MATH].', '1810.06506-1-13-4': 'There remains however a suppressed Yukawa coupling to [MATH], i.e. [EQUATION]', '1810.06506-1-13-5': 'Since [MATH] is heavy, the above interaction is only realized through [MATH], coming from the mixing of the SM Higgs boson [MATH] with [MATH], which is itself also suppressed, i.e. of order [MATH].', '1810.06506-1-13-6': 'With these two suppressions, the resulting interaction strength will be very small, as shown below.', '1810.06506-1-14-0': 'Higgs Decay to Dark Matter : The particles beyond the SM are the [MATH] gauge boson, the complex scalar [MATH] which breaks [MATH] and couples to [MATH], together with the [MATH] and [MATH] fermion singlets of Eq. (10) which belong to the dark sector.', '1810.06506-1-14-1': 'Whereas there are two Higgs doublets, i.e. [MATH] of Eq. (5), one linear combination with the vacuum expectation value [MATH] is the SM analog and corresponds to the observed 125 GeV boson at the LHC; the other is heavier and is not relevant to the discussion below.', '1810.06506-1-15-0': 'The scalar interactions between the SM Higgs [MATH] and [MATH] is given by [EQUATION] where [MATH] and [MATH].', '1810.06506-1-15-1': 'The mass-squared matrix spanning [MATH] is then [EQUATION]', '1810.06506-1-15-2': 'The [MATH] mixing is then given by [MATH].', '1810.06506-1-15-3': 'Hence the [MATH] coupling is [EQUATION]', '1810.06506-1-15-4': 'The decay rate [MATH] of [MATH] is then [EQUATION] where [MATH].', '1810.06506-1-15-5': 'If the reheating temperature of the Universe after inflation is below the decoupling temperature of [MATH] for thermal equilibrium and above [MATH], its only production mechanism is freeze-in through [MATH] decay before the latter decouples from the thermal bath.', '1810.06506-1-15-6': 'The correct relic abundance is possible if [MATH] is very small.', '1810.06506-1-15-7': 'Hence [MATH] could be FIMP (Feebly Interacting Massive Particle) dark matter [CITATION], and for [MATH], the right number density is obtained for [CITATION] [EQUATION]', '1810.06506-1-15-8': 'As a numerical example which satisfies all the above conditions, let [MATH] GeV, then [MATH].', '1810.06506-1-15-9': 'Assuming [MATH], then [MATH] in Eq. (14) is obtained with [MATH] GeV.', '1810.06506-1-15-10': 'Assuming that this is also the value of [MATH], then the decoupling temperature of [MATH] is about 5.2 TeV.', '1810.06506-1-16-0': 'Since the [MATH] breaking scale is about [MATH] GeV in this example of seesaw dark matter, the [MATH] gauge boson is much too heavy to be discovered at the LHC.', '1810.06506-1-16-1': 'Furthermore, the interaction of [MATH] with quarks through [MATH] is very much suppressed, so that it is not detectable in direct-search experiments.', '1810.06506-1-17-0': 'Concluding Remarks : Using [MATH] as a marker in [MATH] so that [MATH] distinguishes dark matter from matter, previous simple models of dark matter are reappraised.', '1810.06506-1-17-1': 'Furthermore, the notion is put forward that naturally light seesaw dark matter exists in parallel with naturally light seesaw neutrinos.', '1810.06506-1-17-2': 'In the latter, the left-handed doublet neutrino [MATH] couples to a heavy singlet right-handed neutrino [MATH] through the SM Higgs doublet [MATH], and [MATH] acquires a large Majorana mass through the singlet scalar [MATH] which also breaks [MATH] and makes [MATH] massive.', '1810.06506-1-17-3': 'As a result, [MATH] gets a small seesaw mass.', '1810.06506-1-17-4': 'In the former, the fermion singlet [MATH] under [MATH] has an allowed large Majorana mass, whereas the singlet [MATH] couples to [MATH] through [MATH], thereby generating a small Majorana mass for [MATH].', '1810.06506-1-17-5': 'As an example, [MATH] GeV, [MATH] eV, [MATH] GeV, [MATH] GeV may be obtained.', '1810.06506-1-17-6': 'Note that the anchor scale [MATH] for seesaw neutrino mass is the intermediate scale for seesaw dark matter.', '1810.06506-1-18-0': 'Below the temperature of order [MATH], [MATH] is out of thermal equilibrium with the SM particles.', '1810.06506-1-18-1': 'However, there is a suppressed Yukawa interaction [MATH] which allows it to be produced through Higgs decay before the Universe cools below [MATH].', '1810.06506-1-18-2': 'It may thus be freeze-in FIMP dark matter and escape present experimental detection, directly or indirectly.'}

{'1810.06506-2-0-0': 'It has recently been pointed out that the underlying symmetry of dark matter may well be [MATH] (coming from [MATH]) with the dark parity of any given particle determined by [MATH], where [MATH] is its [MATH] charge and [MATH] its spin angular momentum.', '1810.06506-2-0-1': 'Armed with this new insight, previous simple models of dark matter are reinterpreted, and a novel idea is proposed that light seesaw dark matter exists in analogy to light neutrinos and is produced by the rare decay of the standard-model Higgs boson.', '1810.06506-2-1-0': '24pt', '1810.06506-2-2-0': 'Introduction : In the decomposition of [MATH], the fermions of the standard model (SM) are organized into [EQUATION] where [EQUATION] under [MATH].', '1810.06506-2-2-1': 'To allow the quarks and leptons to acquire mass, the scalar 10 representation, which contains the necessary Higgs doublets, i.e. [EQUATION] with [EQUATION] from [MATH] respectively, is required, resulting in the allowed Yukawa couplings [EQUATION] as desired.', '1810.06506-2-2-2': 'The nonzero vacuum expectation values [MATH] also break electroweak [MATH] to electrodynamic [MATH].', '1810.06506-2-2-3': 'Since the SM gauge bosons all have [MATH], it is obvious (but not recognized for its importance until recently [CITATION]) that all SM fermions have odd [MATH] and all SM bosons have even [MATH].', '1810.06506-2-2-4': 'This means that each SM particle has even [MATH] where [MATH] is its spin angular momentum.', '1810.06506-2-2-5': 'It is thus a short step to realizing that any scalar with odd [MATH] and any fermion with even [MATH] would have odd [MATH], making it a natural stabilizing symmetry for dark matter.', '1810.06506-2-2-6': 'Indeed, all previous simple models of dark matter based on an [MATH] discrete symmetry may be incorporated into such a framework.', '1810.06506-2-3-0': 'The scalar 126 representation of [MATH] contains a singlet [MATH] under [MATH], which may be used to break [MATH] at the TeV scale and would allow [MATH] (the right-handed neutrino) to obtain a large Majorana mass, thereby triggering the canonical seesaw mechanism for small Majorana neutrino masses.', '1810.06506-2-3-1': 'This is usually described as lepton number [MATH] breaking to lepton parity [MATH] [CITATION], but here it is clear that it has to do with the breaking of gauge [MATH] to [MATH].', '1810.06506-2-4-0': 'In the minimal supersymmetric standard model (MSSM), [MATH] is used to distinguish the SM particles from their superpartners, which belong thus to the dark sector if [MATH] is assumed conserved.', '1810.06506-2-4-1': 'Since [MATH] is identical to [MATH], it has long been recognized [CITATION] that a theory with gauge [MATH], broken by two units, would be a natural framework for dark matter.', '1810.06506-2-4-2': 'In particular, the decomposition [EQUATION] shows that the fermionic 16 of [MATH] contains [EQUATION]', '1810.06506-2-4-3': 'Hence [MATH] is odd for all quarks and leptons.', '1810.06506-2-4-4': 'As for the scalar sector, the 10 representation contains the bidoublet [MATH].', '1810.06506-2-4-5': 'Hence its [MATH] is even.', '1810.06506-2-4-6': 'In other words, [MATH] coincides with [MATH].', '1810.06506-2-4-7': 'However, the former requires a left-right intermediate scale, whereas the latter does not.', '1810.06506-2-4-8': 'They are thus conceptually and phenomenologically distinct.', '1810.06506-2-4-9': 'In this study, [MATH] separates from [MATH] at the unification scale [CITATION], and its symmetry breaking scale is independent of the electroweak scale.', '1810.06506-2-5-0': 'It should also be pointed out that in [MATH], the decomposition [MATH] shows that [MATH] may be invoked as the underlying dark symmetry as well.', '1810.06506-2-6-0': 'Reappraisal of [MATH] Dark Matter : It has been remarked that it is very easy to invent a model of dark matter.', '1810.06506-2-6-1': 'The first step is to introduce a new [MATH] symmetry under which all SM particles are even and a new neutral particle of your choice is odd.', '1810.06506-2-6-2': 'It should then have the appropriate mass and interaction to account for the relic abundance of dark matter in the Universe, but not excluded by direct or indirect search experiments.', '1810.06506-2-7-0': 'The simplest such model [CITATION] assumes a real scalar singlet, odd under [MATH].', '1810.06506-2-7-1': 'It has been studied extensively [CITATION] and is still a viable explanation of dark matter.', '1810.06506-2-7-2': 'In the framework of [MATH], a scalar with odd [MATH] requires it to have odd [MATH].', '1810.06506-2-7-3': 'The scalar singlet [MATH] of [MATH] is such a particle.', '1810.06506-2-7-4': 'It is in fact the scalar analog of [MATH].', '1810.06506-2-7-5': 'They have the same [MATH], but differ in spin.', '1810.06506-2-7-6': 'Hence one is dark matter and the other is not.', '1810.06506-2-7-7': 'In Ref. [CITATION], they are both assigned odd lepton parity, which is now replaced by odd [MATH] parity.', '1810.06506-2-7-8': 'If [MATH] is indeed the origin of [MATH], then it should be complex and it should have [MATH] interactions.', '1810.06506-2-7-9': 'However, from the allowed [MATH] trilinear scalar coupling, [MATH] splits into two real scalars with a large mass gap.', '1810.06506-2-7-10': 'The lighter is dark matter and the heavier decays into the lighter plus a physical or virtual [MATH] gauge boson.', '1810.06506-2-7-11': "This would not affect the lighter scalar's suitability as dark matter, but would predict possible verifiable signatures involving [MATH].", '1810.06506-2-7-12': 'The present experimental bound on [MATH] is about 4.1 TeV from LHC (Large Hadron Collider) data [CITATION], which may be improved [CITATION] with further study.', '1810.06506-2-8-0': 'Instead of choosing [MATH] from the 16 of [MATH], the scalar doublet [MATH] may also be considered [CITATION].', '1810.06506-2-8-1': 'In that case, it is distinguished from [MATH] and [MATH] by their [MATH] charge.', '1810.06506-2-8-2': 'Hence [MATH] are even but [MATH] is odd under [MATH].', '1810.06506-2-8-3': 'This [MATH] discrete symmetry [CITATION] allows [MATH] to be dark matter [CITATION], at least in principle.', '1810.06506-2-8-4': 'However, its interaction with quarks through the [MATH] boson rules it out by direct-search experiments.', '1810.06506-2-8-5': 'In the SM, the allowed quartic coupling [MATH] serves to split [MATH] from [MATH], and since [MATH] only couples one to the other, the offending interaction with quarks is avoided kinematically in elastic nuclear recoil with a mass gap larger than a few hundred keV.', '1810.06506-2-8-6': 'This is known as the inert Higgs doublet model.', '1810.06506-2-8-7': 'In the case of [MATH], such a quartic coupling is forbidden, so if [MATH] originates from [MATH], other particles are needed for it to be dark matter.', '1810.06506-2-8-8': 'They turn out to be exactly [MATH] and [MATH], already discussed.', '1810.06506-2-8-9': 'The allowed couplings [MATH], [MATH] combined with [MATH] form the necessary effective quartic coupling as shown in Fig. 1.', '1810.06506-2-8-10': 'In this scenario, a linear combination of [MATH] and [MATH] is dark matter.', '1810.06506-2-9-0': 'Another possible simple model of dark matter is to have a singlet fermion [MATH] from the [MATH] or [MATH] of the [MATH] of [MATH].', '1810.06506-2-9-1': 'Since [MATH] has even [MATH], it is odd under [MATH].', '1810.06506-2-9-2': 'However, it has no renormalizable interaction with the particles of the SM and thus not a good dark-matter candidate without some additional fundamental particle such as a singlet scalar [CITATION] which has [MATH], i.e. the scalar counterpart of [MATH].', '1810.06506-2-9-3': 'A more interesting option is to combine [MATH] with the scalar doublet [MATH] discussed in the previous paragraph because there is now an allowed Yukawa coupling between the left-handed lepton doublet [MATH] with [MATH] through [MATH], i.e. [MATH].', '1810.06506-2-9-4': 'This forms the basis of the scotogenic model [CITATION] of radiative neutrino mass.', '1810.06506-2-9-5': 'Whereas the original model assumes the [MATH] quartic scalar coupling, it must now be replaced by the effective operator of Fig. 1.', '1810.06506-2-9-6': 'The resulting diagram [CITATION] for generating a radiative Majorana neutrino mass is then given by Fig. 2.', '1810.06506-2-10-0': 'Whereas [MATH] could be dark matter, its only interaction with the particles of the SM is through the left-handed lepton doublet, and is known [CITATION] to be restricted phenomenologically, thus limiting its viability as thermal dark matter.', '1810.06506-2-10-1': 'Hence a linear combination of [MATH] and [MATH] is again the likely dark-matter candidate in this case.', '1810.06506-2-10-2': 'They both couple to [MATH] but differently.', '1810.06506-2-10-3': 'Further study is then needed to reappraise this [MATH] interpretation of the scotogenic model.', '1810.06506-2-11-0': 'Once both [MATH] and [MATH] are present, the coupling [MATH] is allowed.', '1810.06506-2-11-1': 'This has also recently been considered [CITATION] with the assumption that it is very small so that a freeze-in mechanism applies to the decay of [MATH] to [MATH] and [MATH].', '1810.06506-2-12-0': 'Seesaw Dark Matter : In the [MATH] model, the singlet neutrino [MATH] gets a large Majorana mass from the scalar [MATH], both of which have even [MATH].', '1810.06506-2-12-1': 'This realizes the scenario of seesaw neutrino mass at the scale [MATH] which may be TeV or higher.', '1810.06506-2-12-2': 'Suppose the fermion singlets [EQUATION] from the 45, 126 representations of [MATH] are added, then the allowed Yukawa coupling [MATH] combined with a large Majorana mass for [MATH] would induce a small seesaw mass for [MATH].', '1810.06506-2-12-3': 'Note that both [MATH] and [MATH] have odd [MATH].', '1810.06506-2-12-4': 'Hence [MATH] could be naturally light dark matter, i.e. [MATH], in parallel with the seesaw neutrino mass, i.e. [MATH].', '1810.06506-2-13-0': 'As for gauge [MATH] anomaly cancellation, the fermion [MATH] from the [MATH] of [MATH] should be added.', '1810.06506-2-13-1': 'It may combine with [MATH] to form a Dirac fermion, as proposed recently [CITATION].', '1810.06506-2-13-2': 'Here [MATH] is assumed to have an extra symmetry shared by the counterpart singlet [MATH].', '1810.06506-2-13-3': 'This separate system is also assumed to be heavy and annihilate efficiently to SM particles through [MATH] in the early Universe.', '1810.06506-2-13-4': 'Another possible but different connection between seesaw neutrino mass and dark matter has also been proposed [CITATION], based on an imposed [MATH] discrete symmetry and a nonrenormalizable dimension-five coupling.', '1810.06506-2-14-0': 'Consider now the interaction of [MATH].', '1810.06506-2-14-1': 'It interacts mainly with [MATH].', '1810.06506-2-14-2': 'This is in analogy with [MATH] which interacts mainly with [MATH] and [MATH].', '1810.06506-2-14-3': 'Just as [MATH] decouples at a temperature of order 1 MeV, [MATH] would decouple at a temperature of order [MATH].', '1810.06506-2-14-4': 'There remains however a suppressed Yukawa coupling to [MATH], i.e. [EQUATION]', '1810.06506-2-14-5': 'Since [MATH] is heavy, the above interaction is only realized through [MATH], coming from the mixing of the SM Higgs boson [MATH] with [MATH], which is itself also suppressed, i.e. of order [MATH].', '1810.06506-2-14-6': 'With these two suppressions, the resulting interaction strength will be very small, as shown below.', '1810.06506-2-15-0': 'Higgs Decay to Dark Matter : The particles beyond the SM are the [MATH] gauge boson, the complex scalar [MATH] which breaks [MATH] and couples to [MATH], together with the [MATH] and [MATH] fermion singlets of Eq. (10) which belong to the dark sector.', '1810.06506-2-15-1': 'Whereas there are two Higgs doublets, i.e. [MATH] of Eq. (5), one linear combination with the vacuum expectation value [MATH] is the SM analog and corresponds to the observed 125 GeV boson at the LHC; the other is heavier and is not relevant to the discussion below.', '1810.06506-2-16-0': 'The scalar interactions between the SM Higgs [MATH] and [MATH] is given by [EQUATION] where [MATH] and [MATH].', '1810.06506-2-16-1': 'The mass-squared matrix spanning [MATH] is then [EQUATION]', '1810.06506-2-16-2': 'The [MATH] mixing is then given by [MATH].', '1810.06506-2-16-3': 'Hence the [MATH] coupling is [EQUATION]', '1810.06506-2-16-4': 'The decay rate [MATH] of [MATH] is then [EQUATION] where [MATH].', '1810.06506-2-16-5': 'If the reheating temperature of the Universe after inflation is below the decoupling temperature of [MATH] for thermal equilibrium and above [MATH], its only production mechanism is freeze-in through [MATH] decay before the latter decouples from the thermal bath.', '1810.06506-2-16-6': 'The correct relic abundance is possible if [MATH] is very small.', '1810.06506-2-16-7': 'Hence [MATH] could be FIMP (Feebly Interacting Massive Particle) dark matter [CITATION], and for [MATH], the right number density is obtained for [CITATION] [EQUATION]', '1810.06506-2-16-8': 'As a numerical example which satisfies all the above conditions, let [MATH] GeV, then [MATH].', '1810.06506-2-16-9': 'Assuming [MATH], then [MATH] in Eq. (14) is obtained with [MATH] GeV.', '1810.06506-2-16-10': 'Assuming that this is also the value of [MATH], then the decoupling temperature of [MATH] is about 5.2 TeV.', '1810.06506-2-17-0': 'Since the [MATH] breaking scale is about [MATH] GeV in this example of seesaw dark matter, the [MATH] gauge boson is much too heavy to be discovered at the LHC.', '1810.06506-2-17-1': 'Furthermore, the interaction of [MATH] with quarks through [MATH] is very much suppressed, so that it is not detectable in direct-search experiments.', '1810.06506-2-18-0': 'Concluding Remarks : Using [MATH] as a marker in [MATH] so that [MATH] distinguishes dark matter from matter, previous simple models of dark matter are reappraised.', '1810.06506-2-18-1': 'Furthermore, the notion is put forward that naturally light seesaw dark matter exists in parallel with naturally light seesaw neutrinos.', '1810.06506-2-18-2': 'In the latter, the left-handed doublet neutrino [MATH] couples to a heavy singlet right-handed neutrino [MATH] through the SM Higgs doublet [MATH], and [MATH] acquires a large Majorana mass through the singlet scalar [MATH] which also breaks [MATH] and makes [MATH] massive.', '1810.06506-2-18-3': 'As a result, [MATH] gets a small seesaw mass.', '1810.06506-2-18-4': 'In the former, the fermion singlet [MATH] under [MATH] has an allowed large Majorana mass, whereas the singlet [MATH] couples to [MATH] through [MATH], thereby generating a small Majorana mass for [MATH].', '1810.06506-2-18-5': 'As an example, [MATH] GeV, [MATH] eV, [MATH] GeV, [MATH] GeV may be obtained.', '1810.06506-2-18-6': 'Note that the anchor scale [MATH] for seesaw neutrino mass is the intermediate scale for seesaw dark matter.', '1810.06506-2-19-0': 'Below the temperature of order [MATH], [MATH] is out of thermal equilibrium with the SM particles.', '1810.06506-2-19-1': 'However, there is a suppressed Yukawa interaction [MATH] which allows it to be produced through Higgs decay before the Universe cools below [MATH].', '1810.06506-2-19-2': 'It may thus be freeze-in FIMP dark matter and escape present experimental detection, directly or indirectly.'}

[['1810.06506-1-15-0', '1810.06506-2-16-0'], ['1810.06506-1-15-1', '1810.06506-2-16-1'], ['1810.06506-1-15-2', '1810.06506-2-16-2'], ['1810.06506-1-15-3', '1810.06506-2-16-3'], ['1810.06506-1-15-4', '1810.06506-2-16-4'], ['1810.06506-1-15-5', '1810.06506-2-16-5'], ['1810.06506-1-15-6', '1810.06506-2-16-6'], ['1810.06506-1-15-7', '1810.06506-2-16-7'], ['1810.06506-1-15-8', '1810.06506-2-16-8'], ['1810.06506-1-15-9', '1810.06506-2-16-9'], ['1810.06506-1-15-10', '1810.06506-2-16-10'], ['1810.06506-1-8-0', '1810.06506-2-8-0'], ['1810.06506-1-8-1', '1810.06506-2-8-1'], ['1810.06506-1-8-2', '1810.06506-2-8-2'], ['1810.06506-1-8-3', '1810.06506-2-8-3'], ['1810.06506-1-8-4', '1810.06506-2-8-4'], ['1810.06506-1-8-5', '1810.06506-2-8-5'], ['1810.06506-1-8-6', '1810.06506-2-8-6'], ['1810.06506-1-8-7', '1810.06506-2-8-7'], ['1810.06506-1-8-8', '1810.06506-2-8-8'], ['1810.06506-1-8-9', '1810.06506-2-8-9'], ['1810.06506-1-8-10', '1810.06506-2-8-10'], ['1810.06506-1-5-0', '1810.06506-2-5-0'], ['1810.06506-1-9-0', '1810.06506-2-9-0'], ['1810.06506-1-9-1', '1810.06506-2-9-1'], ['1810.06506-1-9-2', '1810.06506-2-9-2'], ['1810.06506-1-9-3', '1810.06506-2-9-3'], ['1810.06506-1-9-4', '1810.06506-2-9-4'], ['1810.06506-1-9-5', '1810.06506-2-9-5'], ['1810.06506-1-9-6', '1810.06506-2-9-6'], ['1810.06506-1-7-0', '1810.06506-2-7-0'], ['1810.06506-1-7-1', '1810.06506-2-7-1'], ['1810.06506-1-7-2', '1810.06506-2-7-2'], ['1810.06506-1-7-3', '1810.06506-2-7-3'], ['1810.06506-1-7-4', '1810.06506-2-7-4'], ['1810.06506-1-7-5', '1810.06506-2-7-5'], ['1810.06506-1-7-6', '1810.06506-2-7-6'], ['1810.06506-1-7-7', '1810.06506-2-7-7'], ['1810.06506-1-7-8', '1810.06506-2-7-8'], ['1810.06506-1-7-9', '1810.06506-2-7-9'], ['1810.06506-1-7-10', '1810.06506-2-7-10'], ['1810.06506-1-7-11', '1810.06506-2-7-11'], ['1810.06506-1-7-12', '1810.06506-2-7-12'], ['1810.06506-1-13-0', '1810.06506-2-14-0'], ['1810.06506-1-13-1', '1810.06506-2-14-1'], ['1810.06506-1-13-2', '1810.06506-2-14-2'], ['1810.06506-1-13-3', '1810.06506-2-14-3'], ['1810.06506-1-13-4', '1810.06506-2-14-4'], ['1810.06506-1-13-5', '1810.06506-2-14-5'], ['1810.06506-1-13-6', '1810.06506-2-14-6'], ['1810.06506-1-11-0', '1810.06506-2-11-0'], ['1810.06506-1-11-1', '1810.06506-2-11-1'], ['1810.06506-1-4-0', '1810.06506-2-4-0'], ['1810.06506-1-4-1', '1810.06506-2-4-1'], ['1810.06506-1-4-2', '1810.06506-2-4-2'], ['1810.06506-1-4-3', '1810.06506-2-4-3'], ['1810.06506-1-4-4', '1810.06506-2-4-4'], ['1810.06506-1-4-5', '1810.06506-2-4-5'], ['1810.06506-1-4-6', '1810.06506-2-4-6'], ['1810.06506-1-4-7', '1810.06506-2-4-7'], ['1810.06506-1-4-8', '1810.06506-2-4-8'], ['1810.06506-1-4-9', '1810.06506-2-4-9'], ['1810.06506-1-18-0', '1810.06506-2-19-0'], ['1810.06506-1-18-1', '1810.06506-2-19-1'], ['1810.06506-1-18-2', '1810.06506-2-19-2'], ['1810.06506-1-3-0', '1810.06506-2-3-0'], ['1810.06506-1-3-1', '1810.06506-2-3-1'], ['1810.06506-1-2-0', '1810.06506-2-2-0'], ['1810.06506-1-2-1', '1810.06506-2-2-1'], ['1810.06506-1-2-2', '1810.06506-2-2-2'], ['1810.06506-1-2-3', '1810.06506-2-2-3'], ['1810.06506-1-2-4', '1810.06506-2-2-4'], ['1810.06506-1-2-5', '1810.06506-2-2-5'], ['1810.06506-1-2-6', '1810.06506-2-2-6'], ['1810.06506-1-10-0', '1810.06506-2-10-0'], ['1810.06506-1-10-1', '1810.06506-2-10-1'], ['1810.06506-1-10-2', '1810.06506-2-10-2'], ['1810.06506-1-10-3', '1810.06506-2-10-3'], ['1810.06506-1-17-0', '1810.06506-2-18-0'], ['1810.06506-1-17-1', '1810.06506-2-18-1'], ['1810.06506-1-17-2', '1810.06506-2-18-2'], ['1810.06506-1-17-3', '1810.06506-2-18-3'], ['1810.06506-1-17-4', '1810.06506-2-18-4'], ['1810.06506-1-17-5', '1810.06506-2-18-5'], ['1810.06506-1-17-6', '1810.06506-2-18-6'], ['1810.06506-1-14-0', '1810.06506-2-15-0'], ['1810.06506-1-14-1', '1810.06506-2-15-1'], ['1810.06506-1-6-0', '1810.06506-2-6-0'], ['1810.06506-1-6-1', '1810.06506-2-6-1'], ['1810.06506-1-6-2', '1810.06506-2-6-2'], ['1810.06506-1-16-0', '1810.06506-2-17-0'], ['1810.06506-1-16-1', '1810.06506-2-17-1'], ['1810.06506-1-0-0', '1810.06506-2-0-0'], ['1810.06506-1-0-1', '1810.06506-2-0-1'], ['1810.06506-2-8-0', '1810.06506-3-8-0'], ['1810.06506-2-8-1', '1810.06506-3-8-1'], ['1810.06506-2-8-2', '1810.06506-3-8-2'], ['1810.06506-2-8-3', '1810.06506-3-8-3'], ['1810.06506-2-8-4', '1810.06506-3-8-4'], ['1810.06506-2-8-5', '1810.06506-3-8-5'], ['1810.06506-2-8-6', '1810.06506-3-8-6'], ['1810.06506-2-8-7', '1810.06506-3-8-7'], ['1810.06506-2-8-8', '1810.06506-3-8-8'], ['1810.06506-2-8-9', '1810.06506-3-8-9'], ['1810.06506-2-8-10', '1810.06506-3-8-10'], ['1810.06506-2-2-0', '1810.06506-3-2-0'], ['1810.06506-2-2-1', '1810.06506-3-2-6'], ['1810.06506-2-2-2', '1810.06506-3-2-7'], ['1810.06506-2-2-3', '1810.06506-3-2-8'], ['1810.06506-2-2-4', '1810.06506-3-2-9'], ['1810.06506-2-2-5', '1810.06506-3-2-10'], ['1810.06506-2-2-6', '1810.06506-3-2-11'], ['1810.06506-2-9-0', '1810.06506-3-9-0'], ['1810.06506-2-9-1', '1810.06506-3-9-1'], ['1810.06506-2-9-2', '1810.06506-3-9-2'], ['1810.06506-2-9-3', '1810.06506-3-9-3'], ['1810.06506-2-9-4', '1810.06506-3-9-4'], ['1810.06506-2-9-5', '1810.06506-3-9-5'], ['1810.06506-2-9-6', '1810.06506-3-9-6'], ['1810.06506-2-19-0', '1810.06506-3-19-0'], ['1810.06506-2-19-1', '1810.06506-3-19-1'], ['1810.06506-2-19-2', '1810.06506-3-19-2'], ['1810.06506-2-18-0', '1810.06506-3-18-0'], ['1810.06506-2-18-1', '1810.06506-3-18-1'], ['1810.06506-2-18-2', '1810.06506-3-18-2'], ['1810.06506-2-18-3', '1810.06506-3-18-3'], ['1810.06506-2-18-4', '1810.06506-3-18-4'], ['1810.06506-2-18-5', '1810.06506-3-18-5'], ['1810.06506-2-18-6', '1810.06506-3-18-6'], ['1810.06506-2-10-0', '1810.06506-3-10-0'], ['1810.06506-2-10-1', '1810.06506-3-10-1'], ['1810.06506-2-10-2', '1810.06506-3-10-2'], ['1810.06506-2-10-3', '1810.06506-3-10-3'], ['1810.06506-2-12-0', '1810.06506-3-12-0'], ['1810.06506-2-12-1', '1810.06506-3-12-1'], ['1810.06506-2-12-2', '1810.06506-3-12-2'], ['1810.06506-2-12-3', '1810.06506-3-12-3'], ['1810.06506-2-12-4', '1810.06506-3-12-4'], ['1810.06506-2-13-0', '1810.06506-3-13-0'], ['1810.06506-2-13-1', '1810.06506-3-13-1'], ['1810.06506-2-13-2', '1810.06506-3-13-2'], ['1810.06506-2-13-3', '1810.06506-3-13-3'], ['1810.06506-2-13-4', '1810.06506-3-13-4'], ['1810.06506-2-6-0', '1810.06506-3-6-0'], ['1810.06506-2-6-1', '1810.06506-3-6-1'], ['1810.06506-2-6-2', '1810.06506-3-6-2'], ['1810.06506-2-16-0', '1810.06506-3-16-0'], ['1810.06506-2-16-1', '1810.06506-3-16-1'], ['1810.06506-2-16-2', '1810.06506-3-16-2'], ['1810.06506-2-16-3', '1810.06506-3-16-3'], ['1810.06506-2-16-4', '1810.06506-3-16-4'], ['1810.06506-2-16-5', '1810.06506-3-16-5'], ['1810.06506-2-16-6', '1810.06506-3-16-6'], ['1810.06506-2-16-7', '1810.06506-3-16-7'], ['1810.06506-2-16-8', '1810.06506-3-16-8'], ['1810.06506-2-16-10', '1810.06506-3-16-10'], ['1810.06506-2-14-0', '1810.06506-3-14-0'], ['1810.06506-2-14-1', '1810.06506-3-14-1'], ['1810.06506-2-14-2', '1810.06506-3-14-2'], ['1810.06506-2-14-3', '1810.06506-3-14-3'], ['1810.06506-2-14-4', '1810.06506-3-14-4'], ['1810.06506-2-14-5', '1810.06506-3-14-5'], ['1810.06506-2-14-6', '1810.06506-3-14-6'], ['1810.06506-2-0-0', '1810.06506-3-0-0'], ['1810.06506-2-0-1', '1810.06506-3-0-1'], ['1810.06506-2-11-0', '1810.06506-3-11-0'], ['1810.06506-2-11-1', '1810.06506-3-11-1'], ['1810.06506-2-5-0', '1810.06506-3-5-0'], ['1810.06506-2-4-0', '1810.06506-3-4-0'], ['1810.06506-2-4-1', '1810.06506-3-4-1'], ['1810.06506-2-4-2', '1810.06506-3-4-3'], ['1810.06506-2-4-3', '1810.06506-3-4-4'], ['1810.06506-2-4-4', '1810.06506-3-4-5'], ['1810.06506-2-4-5', '1810.06506-3-4-6'], ['1810.06506-2-4-6', '1810.06506-3-4-7'], ['1810.06506-2-4-7', '1810.06506-3-4-8'], ['1810.06506-2-4-8', '1810.06506-3-4-9'], ['1810.06506-2-4-9', '1810.06506-3-4-10'], ['1810.06506-2-17-0', '1810.06506-3-17-0'], ['1810.06506-2-7-0', '1810.06506-3-7-0'], ['1810.06506-2-7-1', '1810.06506-3-7-1'], ['1810.06506-2-7-2', '1810.06506-3-7-2'], ['1810.06506-2-7-3', '1810.06506-3-7-3'], ['1810.06506-2-7-4', '1810.06506-3-7-4'], ['1810.06506-2-7-5', '1810.06506-3-7-5'], ['1810.06506-2-7-6', '1810.06506-3-7-6'], ['1810.06506-2-7-7', '1810.06506-3-7-7'], ['1810.06506-2-7-8', '1810.06506-3-7-8'], ['1810.06506-2-7-9', '1810.06506-3-7-9'], ['1810.06506-2-7-10', '1810.06506-3-7-10'], ['1810.06506-2-7-11', '1810.06506-3-7-11'], ['1810.06506-2-7-12', '1810.06506-3-7-13'], ['1810.06506-2-3-0', '1810.06506-3-3-0'], ['1810.06506-2-3-1', '1810.06506-3-3-1'], ['1810.06506-1-12-0', '1810.06506-2-12-0'], ['1810.06506-1-12-1', '1810.06506-2-12-1'], ['1810.06506-1-12-2', '1810.06506-2-12-2'], ['1810.06506-1-12-3', '1810.06506-2-12-3'], ['1810.06506-1-12-4', '1810.06506-2-12-4'], ['1810.06506-1-12-6', '1810.06506-2-13-4'], ['1810.06506-2-15-0', '1810.06506-3-15-0'], ['1810.06506-2-15-1', '1810.06506-3-15-1'], ['1810.06506-2-16-9', '1810.06506-3-16-9'], ['1810.06506-2-17-1', '1810.06506-3-17-2'], ['1810.06506-1-12-5', '1810.06506-2-13-1']]

[['1810.06506-1-15-0', '1810.06506-2-16-0'], ['1810.06506-1-15-1', '1810.06506-2-16-1'], ['1810.06506-1-15-2', '1810.06506-2-16-2'], ['1810.06506-1-15-3', '1810.06506-2-16-3'], ['1810.06506-1-15-4', '1810.06506-2-16-4'], ['1810.06506-1-15-5', '1810.06506-2-16-5'], ['1810.06506-1-15-6', '1810.06506-2-16-6'], ['1810.06506-1-15-7', '1810.06506-2-16-7'], ['1810.06506-1-15-8', '1810.06506-2-16-8'], ['1810.06506-1-15-9', '1810.06506-2-16-9'], ['1810.06506-1-15-10', '1810.06506-2-16-10'], ['1810.06506-1-8-0', '1810.06506-2-8-0'], ['1810.06506-1-8-1', '1810.06506-2-8-1'], ['1810.06506-1-8-2', '1810.06506-2-8-2'], ['1810.06506-1-8-3', '1810.06506-2-8-3'], ['1810.06506-1-8-4', '1810.06506-2-8-4'], ['1810.06506-1-8-5', '1810.06506-2-8-5'], ['1810.06506-1-8-6', '1810.06506-2-8-6'], ['1810.06506-1-8-7', '1810.06506-2-8-7'], ['1810.06506-1-8-8', '1810.06506-2-8-8'], ['1810.06506-1-8-9', '1810.06506-2-8-9'], ['1810.06506-1-8-10', '1810.06506-2-8-10'], ['1810.06506-1-5-0', '1810.06506-2-5-0'], ['1810.06506-1-9-0', '1810.06506-2-9-0'], ['1810.06506-1-9-1', '1810.06506-2-9-1'], ['1810.06506-1-9-2', '1810.06506-2-9-2'], ['1810.06506-1-9-3', '1810.06506-2-9-3'], ['1810.06506-1-9-4', '1810.06506-2-9-4'], ['1810.06506-1-9-5', '1810.06506-2-9-5'], ['1810.06506-1-9-6', '1810.06506-2-9-6'], ['1810.06506-1-7-0', '1810.06506-2-7-0'], ['1810.06506-1-7-1', '1810.06506-2-7-1'], ['1810.06506-1-7-2', '1810.06506-2-7-2'], ['1810.06506-1-7-3', '1810.06506-2-7-3'], ['1810.06506-1-7-4', '1810.06506-2-7-4'], ['1810.06506-1-7-5', '1810.06506-2-7-5'], ['1810.06506-1-7-6', '1810.06506-2-7-6'], ['1810.06506-1-7-7', '1810.06506-2-7-7'], ['1810.06506-1-7-8', '1810.06506-2-7-8'], ['1810.06506-1-7-9', '1810.06506-2-7-9'], ['1810.06506-1-7-10', '1810.06506-2-7-10'], ['1810.06506-1-7-11', '1810.06506-2-7-11'], ['1810.06506-1-7-12', '1810.06506-2-7-12'], ['1810.06506-1-13-0', '1810.06506-2-14-0'], ['1810.06506-1-13-1', '1810.06506-2-14-1'], ['1810.06506-1-13-2', '1810.06506-2-14-2'], ['1810.06506-1-13-3', '1810.06506-2-14-3'], ['1810.06506-1-13-4', '1810.06506-2-14-4'], ['1810.06506-1-13-5', '1810.06506-2-14-5'], ['1810.06506-1-13-6', '1810.06506-2-14-6'], ['1810.06506-1-11-0', '1810.06506-2-11-0'], ['1810.06506-1-11-1', '1810.06506-2-11-1'], ['1810.06506-1-4-0', '1810.06506-2-4-0'], ['1810.06506-1-4-1', '1810.06506-2-4-1'], ['1810.06506-1-4-2', '1810.06506-2-4-2'], ['1810.06506-1-4-3', '1810.06506-2-4-3'], ['1810.06506-1-4-4', '1810.06506-2-4-4'], ['1810.06506-1-4-5', '1810.06506-2-4-5'], ['1810.06506-1-4-6', '1810.06506-2-4-6'], ['1810.06506-1-4-7', '1810.06506-2-4-7'], ['1810.06506-1-4-8', '1810.06506-2-4-8'], ['1810.06506-1-4-9', '1810.06506-2-4-9'], ['1810.06506-1-18-0', '1810.06506-2-19-0'], ['1810.06506-1-18-1', '1810.06506-2-19-1'], ['1810.06506-1-18-2', '1810.06506-2-19-2'], ['1810.06506-1-3-0', '1810.06506-2-3-0'], ['1810.06506-1-3-1', '1810.06506-2-3-1'], ['1810.06506-1-2-0', '1810.06506-2-2-0'], ['1810.06506-1-2-1', '1810.06506-2-2-1'], ['1810.06506-1-2-2', '1810.06506-2-2-2'], ['1810.06506-1-2-3', '1810.06506-2-2-3'], ['1810.06506-1-2-4', '1810.06506-2-2-4'], ['1810.06506-1-2-5', '1810.06506-2-2-5'], ['1810.06506-1-2-6', '1810.06506-2-2-6'], ['1810.06506-1-10-0', '1810.06506-2-10-0'], ['1810.06506-1-10-1', '1810.06506-2-10-1'], ['1810.06506-1-10-2', '1810.06506-2-10-2'], ['1810.06506-1-10-3', '1810.06506-2-10-3'], ['1810.06506-1-17-0', '1810.06506-2-18-0'], ['1810.06506-1-17-1', '1810.06506-2-18-1'], ['1810.06506-1-17-2', '1810.06506-2-18-2'], ['1810.06506-1-17-3', '1810.06506-2-18-3'], ['1810.06506-1-17-4', '1810.06506-2-18-4'], ['1810.06506-1-17-5', '1810.06506-2-18-5'], ['1810.06506-1-17-6', '1810.06506-2-18-6'], ['1810.06506-1-14-0', '1810.06506-2-15-0'], ['1810.06506-1-14-1', '1810.06506-2-15-1'], ['1810.06506-1-6-0', '1810.06506-2-6-0'], ['1810.06506-1-6-1', '1810.06506-2-6-1'], ['1810.06506-1-6-2', '1810.06506-2-6-2'], ['1810.06506-1-16-0', '1810.06506-2-17-0'], ['1810.06506-1-16-1', '1810.06506-2-17-1'], ['1810.06506-1-0-0', '1810.06506-2-0-0'], ['1810.06506-1-0-1', '1810.06506-2-0-1'], ['1810.06506-2-8-0', '1810.06506-3-8-0'], ['1810.06506-2-8-1', '1810.06506-3-8-1'], ['1810.06506-2-8-2', '1810.06506-3-8-2'], ['1810.06506-2-8-3', '1810.06506-3-8-3'], ['1810.06506-2-8-4', '1810.06506-3-8-4'], ['1810.06506-2-8-5', '1810.06506-3-8-5'], ['1810.06506-2-8-6', '1810.06506-3-8-6'], ['1810.06506-2-8-7', '1810.06506-3-8-7'], ['1810.06506-2-8-8', '1810.06506-3-8-8'], ['1810.06506-2-8-9', '1810.06506-3-8-9'], ['1810.06506-2-8-10', '1810.06506-3-8-10'], ['1810.06506-2-2-0', '1810.06506-3-2-0'], ['1810.06506-2-2-1', '1810.06506-3-2-6'], ['1810.06506-2-2-2', '1810.06506-3-2-7'], ['1810.06506-2-2-3', '1810.06506-3-2-8'], ['1810.06506-2-2-4', '1810.06506-3-2-9'], ['1810.06506-2-2-5', '1810.06506-3-2-10'], ['1810.06506-2-2-6', '1810.06506-3-2-11'], ['1810.06506-2-9-0', '1810.06506-3-9-0'], ['1810.06506-2-9-1', '1810.06506-3-9-1'], ['1810.06506-2-9-2', '1810.06506-3-9-2'], ['1810.06506-2-9-3', '1810.06506-3-9-3'], ['1810.06506-2-9-4', '1810.06506-3-9-4'], ['1810.06506-2-9-5', '1810.06506-3-9-5'], ['1810.06506-2-9-6', '1810.06506-3-9-6'], ['1810.06506-2-19-0', '1810.06506-3-19-0'], ['1810.06506-2-19-1', '1810.06506-3-19-1'], ['1810.06506-2-19-2', '1810.06506-3-19-2'], ['1810.06506-2-18-0', '1810.06506-3-18-0'], ['1810.06506-2-18-1', '1810.06506-3-18-1'], ['1810.06506-2-18-2', '1810.06506-3-18-2'], ['1810.06506-2-18-3', '1810.06506-3-18-3'], ['1810.06506-2-18-4', '1810.06506-3-18-4'], ['1810.06506-2-18-5', '1810.06506-3-18-5'], ['1810.06506-2-18-6', '1810.06506-3-18-6'], ['1810.06506-2-10-0', '1810.06506-3-10-0'], ['1810.06506-2-10-1', '1810.06506-3-10-1'], ['1810.06506-2-10-2', '1810.06506-3-10-2'], ['1810.06506-2-10-3', '1810.06506-3-10-3'], ['1810.06506-2-12-0', '1810.06506-3-12-0'], ['1810.06506-2-12-1', '1810.06506-3-12-1'], ['1810.06506-2-12-2', '1810.06506-3-12-2'], ['1810.06506-2-12-3', '1810.06506-3-12-3'], ['1810.06506-2-12-4', '1810.06506-3-12-4'], ['1810.06506-2-13-0', '1810.06506-3-13-0'], ['1810.06506-2-13-1', '1810.06506-3-13-1'], ['1810.06506-2-13-2', '1810.06506-3-13-2'], ['1810.06506-2-13-3', '1810.06506-3-13-3'], ['1810.06506-2-13-4', '1810.06506-3-13-4'], ['1810.06506-2-6-0', '1810.06506-3-6-0'], ['1810.06506-2-6-1', '1810.06506-3-6-1'], ['1810.06506-2-6-2', '1810.06506-3-6-2'], ['1810.06506-2-16-0', '1810.06506-3-16-0'], ['1810.06506-2-16-1', '1810.06506-3-16-1'], ['1810.06506-2-16-2', '1810.06506-3-16-2'], ['1810.06506-2-16-3', '1810.06506-3-16-3'], ['1810.06506-2-16-4', '1810.06506-3-16-4'], ['1810.06506-2-16-5', '1810.06506-3-16-5'], ['1810.06506-2-16-6', '1810.06506-3-16-6'], ['1810.06506-2-16-7', '1810.06506-3-16-7'], ['1810.06506-2-16-8', '1810.06506-3-16-8'], ['1810.06506-2-16-10', '1810.06506-3-16-10'], ['1810.06506-2-14-0', '1810.06506-3-14-0'], ['1810.06506-2-14-1', '1810.06506-3-14-1'], ['1810.06506-2-14-2', '1810.06506-3-14-2'], ['1810.06506-2-14-3', '1810.06506-3-14-3'], ['1810.06506-2-14-4', '1810.06506-3-14-4'], ['1810.06506-2-14-5', '1810.06506-3-14-5'], ['1810.06506-2-14-6', '1810.06506-3-14-6'], ['1810.06506-2-0-0', '1810.06506-3-0-0'], ['1810.06506-2-0-1', '1810.06506-3-0-1'], ['1810.06506-2-11-0', '1810.06506-3-11-0'], ['1810.06506-2-11-1', '1810.06506-3-11-1'], ['1810.06506-2-5-0', '1810.06506-3-5-0'], ['1810.06506-2-4-0', '1810.06506-3-4-0'], ['1810.06506-2-4-1', '1810.06506-3-4-1'], ['1810.06506-2-4-2', '1810.06506-3-4-3'], ['1810.06506-2-4-3', '1810.06506-3-4-4'], ['1810.06506-2-4-4', '1810.06506-3-4-5'], ['1810.06506-2-4-5', '1810.06506-3-4-6'], ['1810.06506-2-4-6', '1810.06506-3-4-7'], ['1810.06506-2-4-7', '1810.06506-3-4-8'], ['1810.06506-2-4-8', '1810.06506-3-4-9'], ['1810.06506-2-4-9', '1810.06506-3-4-10'], ['1810.06506-2-17-0', '1810.06506-3-17-0'], ['1810.06506-2-7-0', '1810.06506-3-7-0'], ['1810.06506-2-7-1', '1810.06506-3-7-1'], ['1810.06506-2-7-2', '1810.06506-3-7-2'], ['1810.06506-2-7-3', '1810.06506-3-7-3'], ['1810.06506-2-7-4', '1810.06506-3-7-4'], ['1810.06506-2-7-5', '1810.06506-3-7-5'], ['1810.06506-2-7-6', '1810.06506-3-7-6'], ['1810.06506-2-7-7', '1810.06506-3-7-7'], ['1810.06506-2-7-8', '1810.06506-3-7-8'], ['1810.06506-2-7-9', '1810.06506-3-7-9'], ['1810.06506-2-7-10', '1810.06506-3-7-10'], ['1810.06506-2-7-11', '1810.06506-3-7-11'], ['1810.06506-2-7-12', '1810.06506-3-7-13'], ['1810.06506-2-3-0', '1810.06506-3-3-0'], ['1810.06506-2-3-1', '1810.06506-3-3-1'], ['1810.06506-1-12-0', '1810.06506-2-12-0'], ['1810.06506-1-12-1', '1810.06506-2-12-1'], ['1810.06506-1-12-2', '1810.06506-2-12-2'], ['1810.06506-1-12-3', '1810.06506-2-12-3'], ['1810.06506-1-12-4', '1810.06506-2-12-4'], ['1810.06506-1-12-6', '1810.06506-2-13-4']]

[['1810.06506-2-15-0', '1810.06506-3-15-0'], ['1810.06506-2-15-1', '1810.06506-3-15-1'], ['1810.06506-2-16-9', '1810.06506-3-16-9']]

[]

[['1810.06506-2-17-1', '1810.06506-3-17-2'], ['1810.06506-1-12-5', '1810.06506-2-13-1']]

[]

['1810.06506-1-1-0', '1810.06506-2-1-0', '1810.06506-3-1-0', '1810.06506-3-2-3']

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

https://arxiv.org/abs/1810.06506

{'1810.06506-3-0-0': 'It has recently been pointed out that the underlying symmetry of dark matter may well be [MATH] (coming from [MATH]) with the dark parity of any given particle determined by [MATH], where [MATH] is its [MATH] charge and [MATH] its spin angular momentum.', '1810.06506-3-0-1': 'Armed with this new insight, previous simple models of dark matter are reinterpreted, and a novel idea is proposed that light seesaw dark matter exists in analogy to light neutrinos and is produced by the rare decay of the standard-model Higgs boson.', '1810.06506-3-1-0': '24pt', '1810.06506-3-2-0': 'Introduction : In the decomposition of [MATH], the fermions of the standard model (SM) are organized into [EQUATION] where [EQUATION] under [MATH].', '1810.06506-3-2-1': 'This symmetry breaking pattern is accomplished using the scalar [MATH] of [MATH].', '1810.06506-3-2-2': 'Since [EQUATION] under [MATH], a nonzero vacuum expectation value (VEV) of the (1,0) component will work.', '1810.06506-3-2-3': 'Under [MATH], [EQUATION]', '1810.06506-3-2-4': 'Hence a nonzero VEV of the (1,1,0,0) component will break [MATH] to the SM gauge symmetry without breaking [MATH].', '1810.06506-3-2-5': 'The Higgs scalars of the [MATH] are all superheavy and will not affect the phenomenology of the SM Higgs boson to be discussed below.', '1810.06506-3-2-6': 'To allow the quarks and leptons to acquire mass, the scalar 10 representation, which contains the necessary Higgs doublets, i.e. [EQUATION] with [EQUATION] from [MATH] respectively, is required, resulting in the allowed Yukawa couplings [EQUATION] as desired.', '1810.06506-3-2-7': 'The nonzero vacuum expectation values [MATH] also break electroweak [MATH] to electrodynamic [MATH].', '1810.06506-3-2-8': 'Since the SM gauge bosons all have [MATH], it is obvious (but not recognized for its importance until recently [CITATION]) that all SM fermions have odd [MATH] and all SM bosons have even [MATH].', '1810.06506-3-2-9': 'This means that each SM particle has even [MATH] where [MATH] is its spin angular momentum.', '1810.06506-3-2-10': 'It is thus a short step to realizing that any scalar with odd [MATH] and any fermion with even [MATH] would have odd [MATH], making it a natural stabilizing symmetry for dark matter.', '1810.06506-3-2-11': 'Indeed, all previous simple models of dark matter based on an [MATH] discrete symmetry may be incorporated into such a framework.', '1810.06506-3-3-0': 'The scalar 126 representation of [MATH] contains a singlet [MATH] under [MATH], which may be used to break [MATH] at the TeV scale and would allow [MATH] (the right-handed neutrino) to obtain a large Majorana mass, thereby triggering the canonical seesaw mechanism for small Majorana neutrino masses.', '1810.06506-3-3-1': 'This is usually described as lepton number [MATH] breaking to lepton parity [MATH] [CITATION], but here it is clear that it has to do with the breaking of gauge [MATH] to [MATH].', '1810.06506-3-4-0': 'In the minimal supersymmetric standard model (MSSM), [MATH] is used to distinguish the SM particles from their superpartners, which belong thus to the dark sector if [MATH] is assumed conserved.', '1810.06506-3-4-1': 'Since [MATH] is identical to [MATH], it has long been recognized [CITATION] that a theory with gauge [MATH], broken by two units, would be a natural framework for dark matter.', '1810.06506-3-4-2': 'For an incomplete list of papers on this topic and discussions on their relevance, see Ref. [CITATION].', '1810.06506-3-4-3': 'In particular, the decomposition [EQUATION] shows that the fermionic 16 of [MATH] contains [EQUATION]', '1810.06506-3-4-4': 'Hence [MATH] is odd for all quarks and leptons.', '1810.06506-3-4-5': 'As for the scalar sector, the 10 representation contains the bidoublet [MATH].', '1810.06506-3-4-6': 'Hence its [MATH] is even.', '1810.06506-3-4-7': 'In other words, [MATH] coincides with [MATH].', '1810.06506-3-4-8': 'However, the former requires a left-right intermediate scale, whereas the latter does not.', '1810.06506-3-4-9': 'They are thus conceptually and phenomenologically distinct.', '1810.06506-3-4-10': 'In this study, [MATH] separates from [MATH] at the unification scale [CITATION], and its symmetry breaking scale is independent of the electroweak scale.', '1810.06506-3-5-0': 'It should also be pointed out that in [MATH], the decomposition [MATH] shows that [MATH] may be invoked as the underlying dark symmetry as well.', '1810.06506-3-6-0': 'Reappraisal of [MATH] Dark Matter : It has been remarked that it is very easy to invent a model of dark matter.', '1810.06506-3-6-1': 'The first step is to introduce a new [MATH] symmetry under which all SM particles are even and a new neutral particle of your choice is odd.', '1810.06506-3-6-2': 'It should then have the appropriate mass and interaction to account for the relic abundance of dark matter in the Universe, but not excluded by direct or indirect search experiments.', '1810.06506-3-7-0': 'The simplest such model [CITATION] assumes a real scalar singlet, odd under [MATH].', '1810.06506-3-7-1': 'It has been studied extensively [CITATION] and is still a viable explanation of dark matter.', '1810.06506-3-7-2': 'In the framework of [MATH], a scalar with odd [MATH] requires it to have odd [MATH].', '1810.06506-3-7-3': 'The scalar singlet [MATH] of [MATH] is such a particle.', '1810.06506-3-7-4': 'It is in fact the scalar analog of [MATH].', '1810.06506-3-7-5': 'They have the same [MATH], but differ in spin.', '1810.06506-3-7-6': 'Hence one is dark matter and the other is not.', '1810.06506-3-7-7': 'In Ref. [CITATION], they are both assigned odd lepton parity, which is now replaced by odd [MATH] parity.', '1810.06506-3-7-8': 'If [MATH] is indeed the origin of [MATH], then it should be complex and it should have [MATH] interactions.', '1810.06506-3-7-9': 'However, from the allowed [MATH] trilinear scalar coupling, [MATH] splits into two real scalars with a large mass gap.', '1810.06506-3-7-10': 'The lighter is dark matter and the heavier decays into the lighter plus a physical or virtual [MATH] gauge boson.', '1810.06506-3-7-11': "This would not affect the lighter scalar's suitability as dark matter, but would predict possible verifiable signatures involving [MATH].", '1810.06506-3-7-12': 'Note that the elastic scattering of a real scalar singlet off nuclei through [MATH] is forbidden because both real and imaginary parts of a complex scalar are needed to construct a vector current.', '1810.06506-3-7-13': 'The present experimental bound on [MATH] is about 4.1 TeV from LHC (Large Hadron Collider) data [CITATION], which may be improved [CITATION] with further study.', '1810.06506-3-8-0': 'Instead of choosing [MATH] from the 16 of [MATH], the scalar doublet [MATH] may also be considered [CITATION].', '1810.06506-3-8-1': 'In that case, it is distinguished from [MATH] and [MATH] by their [MATH] charge.', '1810.06506-3-8-2': 'Hence [MATH] are even but [MATH] is odd under [MATH].', '1810.06506-3-8-3': 'This [MATH] discrete symmetry [CITATION] allows [MATH] to be dark matter [CITATION], at least in principle.', '1810.06506-3-8-4': 'However, its interaction with quarks through the [MATH] boson rules it out by direct-search experiments.', '1810.06506-3-8-5': 'In the SM, the allowed quartic coupling [MATH] serves to split [MATH] from [MATH], and since [MATH] only couples one to the other, the offending interaction with quarks is avoided kinematically in elastic nuclear recoil with a mass gap larger than a few hundred keV.', '1810.06506-3-8-6': 'This is known as the inert Higgs doublet model.', '1810.06506-3-8-7': 'In the case of [MATH], such a quartic coupling is forbidden, so if [MATH] originates from [MATH], other particles are needed for it to be dark matter.', '1810.06506-3-8-8': 'They turn out to be exactly [MATH] and [MATH], already discussed.', '1810.06506-3-8-9': 'The allowed couplings [MATH], [MATH] combined with [MATH] form the necessary effective quartic coupling as shown in Fig. 1.', '1810.06506-3-8-10': 'In this scenario, a linear combination of [MATH] and [MATH] is dark matter.', '1810.06506-3-9-0': 'Another possible simple model of dark matter is to have a singlet fermion [MATH] from the [MATH] or [MATH] of the [MATH] of [MATH].', '1810.06506-3-9-1': 'Since [MATH] has even [MATH], it is odd under [MATH].', '1810.06506-3-9-2': 'However, it has no renormalizable interaction with the particles of the SM and thus not a good dark-matter candidate without some additional fundamental particle such as a singlet scalar [CITATION] which has [MATH], i.e. the scalar counterpart of [MATH].', '1810.06506-3-9-3': 'A more interesting option is to combine [MATH] with the scalar doublet [MATH] discussed in the previous paragraph because there is now an allowed Yukawa coupling between the left-handed lepton doublet [MATH] with [MATH] through [MATH], i.e. [MATH].', '1810.06506-3-9-4': 'This forms the basis of the scotogenic model [CITATION] of radiative neutrino mass.', '1810.06506-3-9-5': 'Whereas the original model assumes the [MATH] quartic scalar coupling, it must now be replaced by the effective operator of Fig. 1.', '1810.06506-3-9-6': 'The resulting diagram [CITATION] for generating a radiative Majorana neutrino mass is then given by Fig. 2.', '1810.06506-3-10-0': 'Whereas [MATH] could be dark matter, its only interaction with the particles of the SM is through the left-handed lepton doublet, and is known [CITATION] to be restricted phenomenologically, thus limiting its viability as thermal dark matter.', '1810.06506-3-10-1': 'Hence a linear combination of [MATH] and [MATH] is again the likely dark-matter candidate in this case.', '1810.06506-3-10-2': 'They both couple to [MATH] but differently.', '1810.06506-3-10-3': 'Further study is then needed to reappraise this [MATH] interpretation of the scotogenic model.', '1810.06506-3-11-0': 'Once both [MATH] and [MATH] are present, the coupling [MATH] is allowed.', '1810.06506-3-11-1': 'This has also recently been considered [CITATION] with the assumption that it is very small so that a freeze-in mechanism applies to the decay of [MATH] to [MATH] and [MATH].', '1810.06506-3-12-0': 'Seesaw Dark Matter : In the [MATH] model, the singlet neutrino [MATH] gets a large Majorana mass from the scalar [MATH], both of which have even [MATH].', '1810.06506-3-12-1': 'This realizes the scenario of seesaw neutrino mass at the scale [MATH] which may be TeV or higher.', '1810.06506-3-12-2': 'Suppose the fermion singlets [EQUATION] from the 45, 126 representations of [MATH] are added, then the allowed Yukawa coupling [MATH] combined with a large Majorana mass for [MATH] would induce a small seesaw mass for [MATH].', '1810.06506-3-12-3': 'Note that both [MATH] and [MATH] have odd [MATH].', '1810.06506-3-12-4': 'Hence [MATH] could be naturally light dark matter, i.e. [MATH], in parallel with the seesaw neutrino mass, i.e. [MATH].', '1810.06506-3-13-0': 'As for gauge [MATH] anomaly cancellation, the fermion [MATH] from the [MATH] of [MATH] should be added.', '1810.06506-3-13-1': 'It may combine with [MATH] to form a Dirac fermion, as proposed recently [CITATION].', '1810.06506-3-13-2': 'Here [MATH] is assumed to have an extra symmetry shared by the counterpart singlet [MATH].', '1810.06506-3-13-3': 'This separate system is also assumed to be heavy and annihilate efficiently to SM particles through [MATH] in the early Universe.', '1810.06506-3-13-4': 'Another possible but different connection between seesaw neutrino mass and dark matter has also been proposed [CITATION], based on an imposed [MATH] discrete symmetry and a nonrenormalizable dimension-five coupling.', '1810.06506-3-14-0': 'Consider now the interaction of [MATH].', '1810.06506-3-14-1': 'It interacts mainly with [MATH].', '1810.06506-3-14-2': 'This is in analogy with [MATH] which interacts mainly with [MATH] and [MATH].', '1810.06506-3-14-3': 'Just as [MATH] decouples at a temperature of order 1 MeV, [MATH] would decouple at a temperature of order [MATH].', '1810.06506-3-14-4': 'There remains however a suppressed Yukawa coupling to [MATH], i.e. [EQUATION]', '1810.06506-3-14-5': 'Since [MATH] is heavy, the above interaction is only realized through [MATH], coming from the mixing of the SM Higgs boson [MATH] with [MATH], which is itself also suppressed, i.e. of order [MATH].', '1810.06506-3-14-6': 'With these two suppressions, the resulting interaction strength will be very small, as shown below.', '1810.06506-3-15-0': 'Higgs Decay to Dark Matter : The particles beyond the SM are the [MATH] gauge boson, the complex scalar [MATH] which breaks [MATH] and couples to [MATH], together with the [MATH] and [MATH] fermion singlets of Eq. (12) which belong to the dark sector.', '1810.06506-3-15-1': 'Whereas there are two Higgs doublets, i.e. [MATH] of Eq. (7), one linear combination with the vacuum expectation value [MATH] is the SM analog and corresponds to the observed 125 GeV boson at the LHC; the other is heavier and is not relevant to the discussion below.', '1810.06506-3-16-0': 'The scalar interactions between the SM Higgs [MATH] and [MATH] is given by [EQUATION] where [MATH] and [MATH].', '1810.06506-3-16-1': 'The mass-squared matrix spanning [MATH] is then [EQUATION]', '1810.06506-3-16-2': 'The [MATH] mixing is then given by [MATH].', '1810.06506-3-16-3': 'Hence the [MATH] coupling is [EQUATION]', '1810.06506-3-16-4': 'The decay rate [MATH] of [MATH] is then [EQUATION] where [MATH].', '1810.06506-3-16-5': 'If the reheating temperature of the Universe after inflation is below the decoupling temperature of [MATH] for thermal equilibrium and above [MATH], its only production mechanism is freeze-in through [MATH] decay before the latter decouples from the thermal bath.', '1810.06506-3-16-6': 'The correct relic abundance is possible if [MATH] is very small.', '1810.06506-3-16-7': 'Hence [MATH] could be FIMP (Feebly Interacting Massive Particle) dark matter [CITATION], and for [MATH], the right number density is obtained for [CITATION] [EQUATION]', '1810.06506-3-16-8': 'As a numerical example which satisfies all the above conditions, let [MATH] GeV, then [MATH].', '1810.06506-3-16-9': 'Assuming [MATH], then [MATH] in Eq. (16) is obtained with [MATH] GeV.', '1810.06506-3-16-10': 'Assuming that this is also the value of [MATH], then the decoupling temperature of [MATH] is about 5.2 TeV.', '1810.06506-3-17-0': 'Since the [MATH] breaking scale is about [MATH] GeV in this example of seesaw dark matter, the [MATH] gauge boson is much too heavy to be discovered at the LHC.', '1810.06506-3-17-1': 'It is also not relevant in the thermal interactions of the SM particles with the dark sector below 5.2 TeV.', '1810.06506-3-17-2': 'Similarly, the elastic scattering of [MATH] with nuclei through [MATH] exchange is very much suppressed, so that it is not detectable in direct-search experiments.', '1810.06506-3-18-0': 'Concluding Remarks : Using [MATH] as a marker in [MATH] so that [MATH] distinguishes dark matter from matter, previous simple models of dark matter are reappraised.', '1810.06506-3-18-1': 'Furthermore, the notion is put forward that naturally light seesaw dark matter exists in parallel with naturally light seesaw neutrinos.', '1810.06506-3-18-2': 'In the latter, the left-handed doublet neutrino [MATH] couples to a heavy singlet right-handed neutrino [MATH] through the SM Higgs doublet [MATH], and [MATH] acquires a large Majorana mass through the singlet scalar [MATH] which also breaks [MATH] and makes [MATH] massive.', '1810.06506-3-18-3': 'As a result, [MATH] gets a small seesaw mass.', '1810.06506-3-18-4': 'In the former, the fermion singlet [MATH] under [MATH] has an allowed large Majorana mass, whereas the singlet [MATH] couples to [MATH] through [MATH], thereby generating a small Majorana mass for [MATH].', '1810.06506-3-18-5': 'As an example, [MATH] GeV, [MATH] eV, [MATH] GeV, [MATH] GeV may be obtained.', '1810.06506-3-18-6': 'Note that the anchor scale [MATH] for seesaw neutrino mass is the intermediate scale for seesaw dark matter.', '1810.06506-3-19-0': 'Below the temperature of order [MATH], [MATH] is out of thermal equilibrium with the SM particles.', '1810.06506-3-19-1': 'However, there is a suppressed Yukawa interaction [MATH] which allows it to be produced through Higgs decay before the Universe cools below [MATH].', '1810.06506-3-19-2': 'It may thus be freeze-in FIMP dark matter and escape present experimental detection, directly or indirectly.', '1810.06506-3-20-0': 'As for the grand unification of [MATH], it is well-known that the SM particle content is inadequate for the gauge couplings to converge at a common mass scale.', '1810.06506-3-20-1': 'This is however easily solved by the addition of new particles at intermediate scales as explicitly shown in Ref. [CITATION].', '1810.06506-3-20-2': 'It is also shown that it is possible to have the unification scale greater than [MATH] GeV, thus avoiding the constraint from proton decay.', '1810.06506-3-20-3': 'For each model variation considered in this paper, a full discussion of unification would require a similar set of new particles.', '1810.06506-3-20-4': 'However, the main purpose of this paper is to point out the rich physics possibilities of the [MATH] extension regarding dark matter.', '1810.06506-3-20-5': 'Other possible new particles depend on the specific (but mostly arbitrary) scenario chosen for unification.', '1810.06506-3-20-6': 'The details of how any previous proposed simple model may be fully developed in the [MATH] context are left for future investigations.'}

1610.00601

{'1610.00601-1-0-0': 'We study the mobility and the diffusion coefficient of an inertial tracer advected by a two-dimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force.', '1610.00601-1-0-1': 'We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility.', '1610.00601-1-0-2': 'These effects rely upon a subtle coupling between inertia and applied force which induce the tracer to persist in particular regions of phase space with a velocity opposite to the force.', '1610.00601-1-0-3': 'The relevance of this coupling is revisited in the framework of non-equilibrium response theory, applying a generalized Einstein relation to our system.', '1610.00601-1-0-4': 'The possibility of experimental observation of these results is also discussed.', '1610.00601-1-1-0': 'Introduction.', '1610.00601-1-1-1': '- Understanding the response of a system to an external stimulus from the observation of the unperturbed dynamics, represents a central issue in statistical mechanics.', '1610.00601-1-1-2': 'For weak perturbations of an equilibrium state, the fluctuation-dissipation theorem (FDT) solves the problem, expressing the system response in terms of correlation functions [CITATION].', '1610.00601-1-1-3': 'Generalizations of this result have been recently derived, to address the much more complex issue of predicting the response in nonequilibrium conditions [CITATION], when detailed balance does not hold and currents cross the system, or in the nonlinear response regime, where higher order response functions have to be taken into account [CITATION].', '1610.00601-1-1-4': 'All these approaches point out the role played by the coupling among degrees of freedom which emerges out of equilibrium, adding extra-terms to the standard FDT [CITATION].', '1610.00601-1-2-0': 'A paradigmatic problem within such a nonlinear response theory is concerned with the dynamics of a tracer particle, traveling in a complex medium under the action of an external field [MATH].', '1610.00601-1-2-1': 'In particular, one is interested in the force-velocity relation [MATH], or the mobility [MATH], and the diffusion coefficient [MATH] of the tracer particle.', '1610.00601-1-2-2': 'These curves can be strongly affected by the interaction between the tracer and the surrounding medium, and can show striking nonlinear behaviors.', '1610.00601-1-2-3': 'This kind of problem has originated in the field of active microrheology of complex fluids, such as emulsions, suspensions, polymer, and micellar solutions [CITATION], where information on the structure of the host medium is inferred from the motion of a biased probe embedded in it.', '1610.00601-1-2-4': 'In this context inertia is usually negligible, whereas the force-velocity relation in non-overdamped systems, which play an important role in fluid dynamics [CITATION], seems much less studied.', '1610.00601-1-3-0': 'One of the surprising effects observed in the force-velocity relation of several models of biased tracers in nonequilibrium systems is a negative differential mobility (NDM).', '1610.00601-1-3-1': 'This means that the tracer velocity, after increasing linearly according to linear response, displays a nonmonotonic behavior, characterized by a maximum for a certain value of the external driving field.', '1610.00601-1-3-2': 'Just beyond this value, the differential mobility [MATH] becomes negative, implying a slowing down of the particle motion at increasing force.', '1610.00601-1-3-3': 'This kind of phenomenon, denoted with the telling expression "getting more from pushing less", has been explained for nonequilibrium toy models in [CITATION] and can be observed in different systems, such as Brownian motors [CITATION], kinetically constrained models of glass formers [CITATION] and driven lattice gases [CITATION], where analytic approaches are possible [CITATION].', '1610.00601-1-3-4': 'In most of the aforementioned systems the non-linear behavior is due to a reciprocal tracer-medium interaction, i.e. the tracer not only feels the action of the solvent but influences it, modifying its microstructure.', '1610.00601-1-3-5': 'More generally, within the framework of nonequilibrium statistical mechanics, the occurrence of NDM has been related to the concept of dynamical activity of the tracer, which is a measure of time-symmetric currents and expresses a "jitteriness" of the particle during its motion [CITATION].', '1610.00601-1-4-0': 'Even more surprisingly, there exist cases of absolute negative mobility (ANM), [MATH], where the particle travels against the external force.', '1610.00601-1-4-1': 'This phenomenon can be realized in specific models, due the carefully tuned coupling between colored noise, asymmetric spatial structures, and driving field [CITATION].', '1610.00601-1-5-0': 'In this Letter we show that these kinds of behaviors can take place in more realistic inertial tracer models, relevant in fluid dynamics.', '1610.00601-1-5-1': 'In particular, we investigate the linear and nonlinear response of an inertial particle moving in a steady (incompressible) cellular velocity field, under the action of an external force, and subject to thermal agitation.', '1610.00601-1-5-2': "The presence of inertia implies a non-trivial deviation of the particle's motion from the trajectory of a fluid particle, typically leading to the appearance of strongly inhomogeneous distributions - a phenomenon known as preferential concentration or particle clustering [CITATION].", '1610.00601-1-5-3': 'This can be responsible for an enhanced probability of chemical, biological or physical interaction, as for instance, for the time scales of rain [CITATION], sedimentation speed under gravity [CITATION], or the planetesimals formation in the early Solar System [CITATION].', '1610.00601-1-5-4': "Here we discover that such a preferential concentration strongly depends upon the external force, leading to a rich non-linear behavior for the average particle's velocity, showing NDM and, in particular cases, even ANM.", '1610.00601-1-5-5': "By an analysis of the tracer's trajectories we identify a possible mechanism responsible for such behaviors.", '1610.00601-1-5-6': 'Moreover we interpret our results within the framework of nonequilibrium response theory, exploiting a generalized Einstein relation (GER), derived recently in [CITATION], which makes clear the role played by the coupling between the velocity field and the tracer dynamics.', '1610.00601-1-6-0': 'The model.', '1610.00601-1-6-1': '- We consider the following equations of motion of an inertial tracer particle in two dimensions, with spatial coordinates [MATH] and velocities [MATH], subject to an external force [MATH] along the [MATH] direction, and traveling through a divergenceless cellular flow [MATH] [EQUATION]', '1610.00601-1-6-2': 'Here [MATH] is the stream function and [MATH] and [MATH] are uncorrelated white noises with zero mean and unitary variance.', '1610.00601-1-6-3': 'The velocity field here considered corresponds to two-dimensional convection and shows a very rich behavior [CITATION].', '1610.00601-1-6-4': 'In addition, it can be easily realized in a laboratory, e.g. with rotating cylinders [CITATION] or in ion solutions in array of magnets [CITATION].', '1610.00601-1-6-5': 'Let us stress that our system, even in the absence of external driving [MATH], is out of equilibrium because of the steady velocity field represented by the non-gradient forces of Eq. ([REF]).', '1610.00601-1-6-6': 'In what follows we measure length and time in units of [MATH] and [MATH] respectively, setting therefore [MATH] and [MATH], which defines a typical time scale of the flow [MATH].', '1610.00601-1-6-7': 'Another important ingredient of our model is the presence of microscopic noise with molecular diffusivity [MATH], which guarantees ergodicity and is related to the temperature [MATH] of the environment by [MATH] [CITATION].', '1610.00601-1-6-8': 'We stress that, in the presence of an advection field, the statistic of the phase space explored by the tracer, [MATH], even at [MATH], depends on both [MATH] and [MATH] in a nontrivial way, and the finite value of [MATH] has an important role for the concentration properties [CITATION].', '1610.00601-1-6-9': 'When [MATH] (fluid particle limit, where the tracer evolves according to the equation [MATH] with [MATH] an effective diffusivity), because of [MATH] incompressibility, the tracer visits the two-dimensional phase space in a uniform way [CITATION].', '1610.00601-1-6-10': 'The same happens for [MATH], when the tracer is insensitive to the field and uniformly diffuses through the flow.', '1610.00601-1-7-0': 'Negative differential and absolute mobility.', '1610.00601-1-7-1': '- Here we are mainly interested in the behavior of the stationary velocity [MATH], where [MATH] denotes averages over trajectories of the particle with different initial conditions and noise realizations.', '1610.00601-1-7-2': 'We first consider the case [MATH].', '1610.00601-1-7-3': 'In Fig. [REF](a) we show, as a function of [MATH], [MATH] and the mobility [MATH] (inset), for [MATH] and different values of [MATH], as computed in numerical simulations [CITATION].', '1610.00601-1-7-4': 'A linear regime at small forces, characterized by a constant mobility depending on [MATH], is followed by a complex nonlinear scenario which emerges at intermediate values of the force.', '1610.00601-1-7-5': 'In particular, a nonmonotonic behavior corresponding to NDM takes place, with a maximum that slightly shifts and then disappears as [MATH] is increased.', '1610.00601-1-7-6': 'This is expected because if the noise is strong enough the effect of the velocity field [MATH] is averaged out.', '1610.00601-1-7-7': 'The same happens for large enough forces, for which again the effect of the velocity field [MATH] is negligible and, irrespective of [MATH], the trivial behavior [MATH] is recovered.', '1610.00601-1-7-8': 'Notice that this asymptotic linear behavior is different from the saturation effect at large force usually observed in lattice gas models [CITATION].', '1610.00601-1-8-0': 'An even more striking phenomenon is observed in cases with [MATH].', '1610.00601-1-8-1': 'In Fig. [REF](b) we show the force-velocity relation for [MATH]: Again a complex nonlinear behavior can be observed for intermediate values of [MATH] and, surprisingly, ANM (i.e. [MATH]) is observed in a range around [MATH].', '1610.00601-1-9-0': 'In order to get insight into the origin of the observed NDM and ANM, we have studied typical trajectories of the tracer as reported in Fig. [REF].', '1610.00601-1-9-1': 'In panel (a) we show that the motion of our tracers is realized along preferential "channels" which are aligned to two main directions: some of these channels are characterized by [MATH] (we call them "leftward") and others by [MATH] (called in the following "rightward").', '1610.00601-1-9-2': 'These preferential channels are seen for not too strong values of the noise (roughly up to values [MATH]) and independently of the value of the force, but disappear reducing inertia.', '1610.00601-1-9-3': 'Both inertia and noise activate random transitions between the channels [CITATION].', '1610.00601-1-9-4': 'The force induces a bias in such transitions, determining, in general, an average [MATH].', '1610.00601-1-9-5': 'In panel (b) we show a mechanism for explaining how an increase of the positive force may enhance the probability of transitions from rightward channels to leftward channels, which can lead to NDM or even ANM.', '1610.00601-1-9-6': 'The reasoning is the following.', '1610.00601-1-9-7': 'Initially the particle is travelling along a rightward channel.', '1610.00601-1-9-8': 'For the chosen [MATH] (black arrows), it occurs that the particle is pushed from region "A" to region "B" where the underlying velocity field is strongly negative: a transition to a leftward channel is then realized.', '1610.00601-1-9-9': 'With a smaller (green arrows) or larger (cyan arrows) force, the particle avoids the adverse region "B" and continues its run along rightward channels.', '1610.00601-1-9-10': 'This suggests that there exists a range of forces for which the tracer is induced to visit more frequently channels with velocity opposite to the force.', '1610.00601-1-9-11': 'Depending on how much this effect is pronounced, NDM or ANM can occur.', '1610.00601-1-10-0': 'Diffusivity- Next we focus on the study of the diffusion coefficient [MATH], defined as [EQUATION] in order to understand how the FDT is modified in our system.', '1610.00601-1-10-1': 'Here we consider the case [MATH] (other cases show similar behaviors), which is reported in Fig. [REF](a).', '1610.00601-1-10-2': 'Notice that [MATH] is nearly independent of the force at small forces and at large forces, where it coincides with the value expected in the absence of the velocity field, [MATH].', '1610.00601-1-10-3': 'It is remarkable that [MATH]: such a discrepancy decreases when [MATH] is increased.', '1610.00601-1-10-4': 'In order to better understand the role of the molecular diffusivity [MATH] in our system, in the inset of Fig. [REF](a) we report the behavior of [MATH] as a function of [MATH].', '1610.00601-1-10-5': 'For large enough noise amplitude, the scaling is linear, as expected, because the diffusion coefficient is dominated by the microscopic diffusivity.', '1610.00601-1-10-6': 'On the contrary, for [MATH], the particle diffusivity diverges, similarly to what found by Taylor [CITATION] for the dispersion of a fluid particle in laminar flows in straight channels.', '1610.00601-1-10-7': 'In the case of Taylor diffusion of a fluid particle in a shear flow, the behavior [MATH] can be easily understood in terms of long horizontal ballistic motion, the duration of which increases as [MATH] decreases.', '1610.00601-1-10-8': 'In our system the understanding is not so simple, but the divergence [MATH] and the long channels observed in Fig. [REF](a) suggest a similar scenario.', '1610.00601-1-11-0': 'Generalized Einstein relation.', '1610.00601-1-11-1': '- The behaviors described above can be interpreted within the context of response theory.', '1610.00601-1-11-2': 'In equilibrium conditions, and in the linear regime, the Einstein relation predicts a proportionality between the mobility and the diffusivity, via the inverse temperature [EQUATION]', '1610.00601-1-11-3': 'In our system, the presence of the velocity field [MATH] introduces significant nonequilbirium effects that are clearly visible in Fig. [REF](b), where we report the measured mobility [MATH] rescaled by [MATH].', '1610.00601-1-11-4': 'Only for large enough values of [MATH], where the effect of [MATH] is negligible and the system can be considered at equilibrium, the ratio [MATH].', '1610.00601-1-11-5': 'Eventually, for [MATH] large enough, the noise makes the velocity field irrelevant and [MATH] in all regimes.', '1610.00601-1-12-0': 'The difference due to nonequilibrium effects can be revisited in terms of a GER, derived for systems in out-of-equilibrium steady states.', '1610.00601-1-12-1': 'According to this relation, the particle mobility can be expressed as the sum of two contributions: one proportional to the diffusion coefficient, as in the standard Einstein relation ([REF]), and the other involving the correlation function with the time-integral of the velocity field [MATH], computed along the trajectory of the particle.', '1610.00601-1-12-2': 'As discussed in detail in [CITATION], for a system described by a set of stochastic equations as in Eqs. ([REF]-[REF]), the GER explicitly reads [EQUATION] where [EQUATION] and [MATH] is the connected correlation function measured at force [MATH].', '1610.00601-1-12-3': 'We have computed in numerical simulations the nonequilibrium contribution due to the coupling with the field [MATH].', '1610.00601-1-12-4': 'The validity of the predictions of the GER ([REF]) is shown - as dot-dashed lines - in Fig. [REF](b), for two cases at [MATH] and [MATH].', '1610.00601-1-12-5': 'Let us stress that Eq. [REF] can be exploited also at non-vanishing forces: indeed the differential mobility [MATH] at a finite value of [MATH] is given by the same expression, by measuring the two terms [MATH] and [MATH] at force [MATH].', '1610.00601-1-12-6': 'The prediction of GER for [MATH] is negative where NDM appears, as we checked numerically: NDM therefore can be interpreted as the consequence of [MATH] becoming larger than [MATH] [CITATION].', '1610.00601-1-12-7': 'Also in the case [MATH], for the force values with ANM, the GER is verified, showing strong negative and positive differential mobilities just before and just after the minimum of [MATH].', '1610.00601-1-13-0': 'Conclusions.', '1610.00601-1-13-1': '- We have studied the effects of a driving external force on the dynamics of an inertial particle advected by a velocity field, in the nonlinear regime.', '1610.00601-1-13-2': 'We have discovered nontrivial behaviors of the stationary tracer velocity and of its diffusivity as a function of the force, such as NDM and ANM.', '1610.00601-1-13-3': 'These effects are due to a complicated combined action of the applied force, the particle inertia and the underlying velocity field.', '1610.00601-1-13-4': 'It turns out that, in some force regimes, this coupling leads the tracer to persist in regions of the velocity field which drag it against the force direction, resulting in a slowing down of the tracer velocity, or even producing a negative mobility [CITATION].', '1610.00601-1-13-5': 'The central role played by the coupling with the velocity field clearly emerges in the GER which is satisfied in our nonequilibrium system.', '1610.00601-1-13-6': 'The striking behaviors shown by the model should be observable in experiments with biased inertial tracers in laminar flows, realized, for instance, in setups with rotating cylinders [CITATION], two-sided lid-driven cavities [CITATION] or magnetically-driven vortices [CITATION].', '1610.00601-1-14-0': 'We thank M. Cencini for useful discussions.'}

{'1610.00601-2-0-0': 'We study the mobility and the diffusion coefficient of an inertial tracer advected by a two-dimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force.', '1610.00601-2-0-1': 'We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility.', '1610.00601-2-0-2': 'These effects rely upon a subtle coupling between inertia and applied force which induce the tracer to persist in particular regions of phase space with a velocity opposite to the force.', '1610.00601-2-0-3': 'The relevance of this coupling is revisited in the framework of non-equilibrium response theory, applying a generalized Einstein relation to our system.', '1610.00601-2-0-4': 'The possibility of experimental observation of these results is also discussed.', '1610.00601-2-1-0': 'Introduction.', '1610.00601-2-1-1': '- Understanding the response of a system to an external stimulus from the observation of the unperturbed dynamics, represents a central issue in statistical mechanics.', '1610.00601-2-1-2': 'For weak perturbations of an equilibrium state, the fluctuation-dissipation theorem (FDT) solves the problem, expressing the system response in terms of correlation functions [CITATION].', '1610.00601-2-1-3': 'Generalizations of this result have been recently derived, to address the much more complex issue of predicting the response in nonequilibrium conditions [CITATION], when detailed balance does not hold and currents cross the system, or in the nonlinear response regime, where higher order response functions have to be taken into account [CITATION].', '1610.00601-2-1-4': 'All these approaches point out the role played by the coupling among degrees of freedom which emerges out of equilibrium, adding extra-terms to the standard FDT [CITATION].', '1610.00601-2-2-0': 'A paradigmatic problem within such a nonlinear response theory is concerned with the dynamics of a tracer particle, traveling in a complex medium under the action of an external field [MATH].', '1610.00601-2-2-1': 'In particular, one is interested in the force-velocity relation [MATH], or the mobility [MATH], and the diffusion coefficient [MATH] of the tracer particle.', '1610.00601-2-2-2': 'These curves can be strongly affected by the interaction between the tracer and the surrounding medium, and can show striking nonlinear behaviors.', '1610.00601-2-2-3': 'This kind of problem has originated in the field of active microrheology of complex fluids, such as emulsions, suspensions, polymer, and micellar solutions [CITATION], where information on the structure of the host medium is inferred from the motion of a biased probe embedded in it.', '1610.00601-2-2-4': 'In this context inertia is usually negligible, whereas the force-velocity relation in non-overdamped systems, which play an important role in fluid dynamics [CITATION], seems much less studied.', '1610.00601-2-3-0': 'One of the surprising effects observed in the force-velocity relation of several models of biased tracers in nonequilibrium systems is a negative differential mobility (NDM).', '1610.00601-2-3-1': 'This means that the tracer velocity, after increasing linearly according to linear response, displays a nonmonotonic behavior, characterized by a maximum for a certain value of the external driving field.', '1610.00601-2-3-2': 'Just beyond this value, the differential mobility [MATH] becomes negative, implying a slowing down of the particle motion at increasing force.', '1610.00601-2-3-3': 'This kind of phenomenon, denoted with the telling expression "getting more from pushing less", has been explained for nonequilibrium toy models in [CITATION] and can be observed in different systems, such as Brownian motors [CITATION], kinetically constrained models of glass formers [CITATION] and driven lattice gases [CITATION], where analytic approaches are possible [CITATION].', '1610.00601-2-3-4': 'In most of the aforementioned systems the non-linear behavior is due to a reciprocal tracer-medium interaction, i.e. the tracer not only feels the action of the solvent but influences it, modifying its microstructure.', '1610.00601-2-3-5': 'More generally, within the framework of nonequilibrium statistical mechanics, the occurrence of NDM has been related to the concept of dynamical activity of the tracer, which is a measure of time-symmetric currents and expresses a "jitteriness" of the particle during its motion [CITATION].', '1610.00601-2-4-0': 'Even more surprisingly, there exist cases of absolute negative mobility (ANM), [MATH], where the particle travels against the external force.', '1610.00601-2-4-1': 'This phenomenon can be realized in specific models, due the carefully tuned coupling between colored noise, asymmetric spatial structures, and driving field [CITATION].', '1610.00601-2-5-0': 'In this Letter we show that these kinds of behaviors can take place in more realistic inertial tracer models, relevant in fluid dynamics.', '1610.00601-2-5-1': 'In particular, we investigate the linear and nonlinear response of an inertial particle moving in a steady (incompressible) cellular velocity field, under the action of an external force, and subject to thermal agitation.', '1610.00601-2-5-2': "The presence of inertia implies a non-trivial deviation of the particle's motion from the trajectory of a fluid particle, typically leading to the appearance of strongly inhomogeneous distributions - a phenomenon known as preferential concentration or particle clustering [CITATION].", '1610.00601-2-5-3': 'This can be responsible for an enhanced probability of chemical, biological or physical interaction, as for instance, for the time scales of rain [CITATION], sedimentation speed under gravity [CITATION], or the planetesimals formation in the early Solar System [CITATION].', '1610.00601-2-5-4': "Here we discover that such a preferential concentration strongly depends upon the external force, leading to a rich non-linear behavior for the average particle's velocity, showing NDM and, in particular cases, even ANM.", '1610.00601-2-5-5': "By an analysis of the tracer's trajectories we identify a possible mechanism responsible for such behaviors.", '1610.00601-2-5-6': 'Moreover we interpret our results within the framework of nonequilibrium response theory, exploiting a generalized Einstein relation (GER), derived recently in [CITATION], which makes clear the role played by the coupling between the velocity field and the tracer dynamics.', '1610.00601-2-6-0': 'The model.', '1610.00601-2-6-1': '- We consider the following equations of motion of an inertial tracer particle in two dimensions, with spatial coordinates [MATH] and velocities [MATH], subject to an external force [MATH] along the [MATH] direction, and traveling through a divergenceless cellular flow [MATH] [EQUATION]', '1610.00601-2-6-2': 'Here [MATH] is the stream function and [MATH] and [MATH] are uncorrelated white noises with zero mean and unitary variance.', '1610.00601-2-6-3': 'The velocity field here considered corresponds to two-dimensional convection and shows a very rich behavior [CITATION].', '1610.00601-2-6-4': 'In addition, it can be easily realized in a laboratory, e.g. with rotating cylinders [CITATION] or in ion solutions in array of magnets [CITATION].', '1610.00601-2-6-5': 'Let us stress that our system, even in the absence of external driving [MATH], is out of equilibrium because of the steady velocity field represented by the non-gradient forces of Eq. ([REF]).', '1610.00601-2-6-6': 'In what follows we measure length and time in units of [MATH] and [MATH] respectively, setting therefore [MATH] and [MATH], which defines a typical time scale of the flow [MATH].', '1610.00601-2-6-7': 'Another important ingredient of our model is the presence of microscopic noise with molecular diffusivity [MATH], which guarantees ergodicity and is related to the temperature [MATH] of the environment by [MATH] [CITATION].', '1610.00601-2-6-8': 'We stress that, in the presence of an advection field, the statistic of the phase space explored by the tracer, [MATH], even at [MATH], depends on both [MATH] and [MATH] in a nontrivial way, and the finite value of [MATH] has an important role for the concentration properties [CITATION].', '1610.00601-2-6-9': 'When [MATH] (fluid particle limit, where the tracer evolves according to the equation [MATH] with [MATH] an effective diffusivity), because of [MATH] incompressibility, the tracer visits the two-dimensional phase space in a uniform way [CITATION].', '1610.00601-2-6-10': 'The same happens for [MATH], when the tracer is insensitive to the field and uniformly diffuses through the flow.', '1610.00601-2-7-0': 'Negative differential and absolute mobility.', '1610.00601-2-7-1': '- Here we are mainly interested in the behavior of the stationary velocity [MATH], where [MATH] denotes averages over trajectories of the particle with different initial conditions and noise realizations.', '1610.00601-2-7-2': 'We first consider the case [MATH].', '1610.00601-2-7-3': 'In Fig. [REF](a) we show, as a function of [MATH], [MATH] and the mobility [MATH] (inset), for [MATH] and different values of [MATH], as computed in numerical simulations [CITATION].', '1610.00601-2-7-4': 'A linear regime at small forces, characterized by a constant mobility depending on [MATH], is followed by a complex nonlinear scenario which emerges at intermediate values of the force.', '1610.00601-2-7-5': 'In particular, a nonmonotonic behavior corresponding to NDM takes place, with a maximum that slightly shifts and then disappears as [MATH] is increased.', '1610.00601-2-7-6': 'This is expected because if the noise is strong enough the effect of the velocity field [MATH] is averaged out.', '1610.00601-2-7-7': 'The same happens for large enough forces, for which again the effect of the velocity field [MATH] is negligible and, irrespective of [MATH], the trivial behavior [MATH] is recovered.', '1610.00601-2-7-8': 'Notice that this asymptotic linear behavior is different from the saturation effect at large force usually observed in lattice gas models [CITATION].', '1610.00601-2-8-0': 'An even more striking phenomenon is observed in cases with [MATH].', '1610.00601-2-8-1': 'In Fig. [REF](b) we show the force-velocity relation for [MATH]: Again a complex nonlinear behavior can be observed for intermediate values of [MATH] and, surprisingly, ANM (i.e. [MATH]) is observed in a range around [MATH].', '1610.00601-2-9-0': 'In order to get insight into the origin of the observed NDM and ANM, we have studied typical trajectories of the tracer as reported in Fig. [REF].', '1610.00601-2-9-1': 'In panel (a) we show that the motion of our tracers is realized along preferential "channels" which are aligned to two main directions: some of these channels are characterized by [MATH] (we call them "leftward") and others by [MATH] (called in the following "rightward").', '1610.00601-2-9-2': 'These preferential channels are seen for not too strong values of the noise (roughly up to values [MATH]) and independently of the value of the force, but disappear reducing inertia.', '1610.00601-2-9-3': 'Both inertia and noise activate random transitions between the channels [CITATION].', '1610.00601-2-9-4': 'The force induces a bias in such transitions, determining, in general, an average [MATH].', '1610.00601-2-9-5': 'In panel (b) we show a mechanism for explaining how an increase of the positive force may enhance the probability of transitions from rightward channels to leftward channels, which can lead to NDM or even ANM.', '1610.00601-2-9-6': 'The reasoning is the following.', '1610.00601-2-9-7': 'Initially the particle is travelling along a rightward channel.', '1610.00601-2-9-8': 'For the chosen [MATH] (black arrows), it occurs that the particle is pushed from region "A" to region "B" where the underlying velocity field is strongly negative: a transition to a leftward channel is then realized.', '1610.00601-2-9-9': 'With a smaller (green arrows) or larger (cyan arrows) force, the particle avoids the adverse region "B" and continues its run along rightward channels.', '1610.00601-2-9-10': 'This suggests that there exists a range of forces for which the tracer is induced to visit more frequently channels with velocity opposite to the force.', '1610.00601-2-9-11': 'Depending on how much this effect is pronounced, NDM or ANM can occur.', '1610.00601-2-10-0': 'Diffusivity- Next we focus on the study of the diffusion coefficient [MATH], defined as [EQUATION] in order to understand how the FDT is modified in our system.', '1610.00601-2-10-1': 'Here we consider the case [MATH] (other cases show similar behaviors), which is reported in Fig. [REF](a).', '1610.00601-2-10-2': 'Notice that [MATH] is nearly independent of the force at small forces and at large forces, where it coincides with the value expected in the absence of the velocity field, [MATH].', '1610.00601-2-10-3': 'It is remarkable that [MATH]: such a discrepancy decreases when [MATH] is increased.', '1610.00601-2-10-4': 'In order to better understand the role of the molecular diffusivity [MATH] in our system, in the inset of Fig. [REF](a) we report the behavior of [MATH] as a function of [MATH].', '1610.00601-2-10-5': 'For large enough noise amplitude, the scaling is linear, as expected, because the diffusion coefficient is dominated by the microscopic diffusivity.', '1610.00601-2-10-6': 'On the contrary, for [MATH], the particle diffusivity diverges, similarly to what found by Taylor [CITATION] for the dispersion of a fluid particle in laminar flows in straight channels.', '1610.00601-2-10-7': 'In the case of Taylor diffusion of a fluid particle in a shear flow, the behavior [MATH] can be easily understood in terms of long horizontal ballistic motion, the duration of which increases as [MATH] decreases.', '1610.00601-2-10-8': 'In our system the understanding is not so simple, but the divergence [MATH] and the long channels observed in Fig. [REF](a) suggest a similar scenario.', '1610.00601-2-11-0': 'Generalized Einstein relation.', '1610.00601-2-11-1': '- The behaviors described above can be interpreted within the context of response theory.', '1610.00601-2-11-2': 'In equilibrium conditions, and in the linear regime, the Einstein relation predicts a proportionality between the mobility and the diffusivity, via the inverse temperature [EQUATION]', '1610.00601-2-11-3': 'In our system, the presence of the velocity field [MATH] introduces significant nonequilbirium effects that are clearly visible in Fig. [REF](b), where we report the measured mobility [MATH] rescaled by [MATH].', '1610.00601-2-11-4': 'Only for large enough values of [MATH], where the effect of [MATH] is negligible and the system can be considered at equilibrium, the ratio [MATH].', '1610.00601-2-11-5': 'Eventually, for [MATH] large enough, the noise makes the velocity field irrelevant and [MATH] in all regimes.', '1610.00601-2-12-0': 'The difference due to nonequilibrium effects can be revisited in terms of a GER, derived for systems in out-of-equilibrium steady states.', '1610.00601-2-12-1': 'According to this relation, the particle mobility can be expressed as the sum of two contributions: one proportional to the diffusion coefficient, as in the standard Einstein relation ([REF]), and the other involving the correlation function with the time-integral of the velocity field [MATH], computed along the trajectory of the particle.', '1610.00601-2-12-2': 'As discussed in detail in [CITATION], for a system described by a set of stochastic equations as in Eqs. ([REF]-[REF]), the GER explicitly reads [EQUATION] where [EQUATION] and [MATH] is the connected correlation function measured at force [MATH].', '1610.00601-2-12-3': 'We have computed in numerical simulations the nonequilibrium contribution due to the coupling with the field [MATH].', '1610.00601-2-12-4': 'The validity of the predictions of the GER ([REF]) is shown - as dot-dashed lines - in Fig. [REF](b), for two cases at [MATH] and [MATH].', '1610.00601-2-12-5': 'Let us stress that Eq. [REF] can be exploited also at non-vanishing forces: indeed the differential mobility [MATH] at a finite value of [MATH] is given by the same expression, by measuring the two terms [MATH] and [MATH] at force [MATH].', '1610.00601-2-12-6': 'The prediction of GER for [MATH] is negative where NDM appears, as we checked numerically: NDM therefore can be interpreted as the consequence of [MATH] becoming larger than [MATH] [CITATION].', '1610.00601-2-12-7': 'Also in the case [MATH], for the force values with ANM, the GER is verified, showing strong negative and positive differential mobilities just before and just after the minimum of [MATH].', '1610.00601-2-13-0': 'Conclusions.', '1610.00601-2-13-1': '- We have studied the effects of a driving external force on the dynamics of an inertial particle advected by a velocity field, in the nonlinear regime.', '1610.00601-2-13-2': 'We have discovered nontrivial behaviors of the stationary tracer velocity and of its diffusivity as a function of the force, such as NDM and ANM.', '1610.00601-2-13-3': 'These effects are due to a complicated combined action of the applied force, the particle inertia and the underlying velocity field.', '1610.00601-2-13-4': 'It turns out that, in some force regimes, this coupling leads the tracer to persist in regions of the velocity field which drag it against the force direction, resulting in a slowing down of the tracer velocity, or even producing a negative mobility [CITATION].', '1610.00601-2-13-5': 'The central role played by the coupling with the velocity field clearly emerges in the GER which is satisfied in our nonequilibrium system.', '1610.00601-2-13-6': 'The striking behaviors shown by the model should be observable in experiments with biased inertial tracers in laminar flows, realized, for instance, in setups with rotating cylinders [CITATION], two-sided lid-driven cavities [CITATION] or magnetically-driven vortices [CITATION].', '1610.00601-2-14-0': 'We thank M. Cencini for useful discussions.'}

[['1610.00601-1-13-1', '1610.00601-2-13-1'], ['1610.00601-1-13-2', '1610.00601-2-13-2'], ['1610.00601-1-13-3', '1610.00601-2-13-3'], ['1610.00601-1-13-4', '1610.00601-2-13-4'], ['1610.00601-1-13-5', '1610.00601-2-13-5'], ['1610.00601-1-13-6', '1610.00601-2-13-6'], ['1610.00601-1-3-0', '1610.00601-2-3-0'], ['1610.00601-1-3-1', '1610.00601-2-3-1'], ['1610.00601-1-3-2', '1610.00601-2-3-2'], ['1610.00601-1-3-3', '1610.00601-2-3-3'], ['1610.00601-1-3-4', '1610.00601-2-3-4'], ['1610.00601-1-3-5', '1610.00601-2-3-5'], ['1610.00601-1-0-0', '1610.00601-2-0-0'], ['1610.00601-1-0-1', '1610.00601-2-0-1'], ['1610.00601-1-0-2', '1610.00601-2-0-2'], ['1610.00601-1-0-3', '1610.00601-2-0-3'], ['1610.00601-1-0-4', '1610.00601-2-0-4'], ['1610.00601-1-4-0', '1610.00601-2-4-0'], ['1610.00601-1-4-1', '1610.00601-2-4-1'], ['1610.00601-1-7-0', '1610.00601-2-7-0'], ['1610.00601-1-7-1', '1610.00601-2-7-1'], ['1610.00601-1-7-2', '1610.00601-2-7-2'], ['1610.00601-1-7-3', '1610.00601-2-7-3'], ['1610.00601-1-7-4', '1610.00601-2-7-4'], ['1610.00601-1-7-5', '1610.00601-2-7-5'], ['1610.00601-1-7-6', '1610.00601-2-7-6'], ['1610.00601-1-7-7', '1610.00601-2-7-7'], ['1610.00601-1-7-8', '1610.00601-2-7-8'], ['1610.00601-1-2-0', '1610.00601-2-2-0'], ['1610.00601-1-2-1', '1610.00601-2-2-1'], ['1610.00601-1-2-2', '1610.00601-2-2-2'], ['1610.00601-1-2-3', '1610.00601-2-2-3'], ['1610.00601-1-2-4', '1610.00601-2-2-4'], ['1610.00601-1-6-1', '1610.00601-2-6-1'], ['1610.00601-1-6-2', '1610.00601-2-6-2'], ['1610.00601-1-6-3', '1610.00601-2-6-3'], ['1610.00601-1-6-4', '1610.00601-2-6-4'], ['1610.00601-1-6-5', '1610.00601-2-6-5'], ['1610.00601-1-6-6', '1610.00601-2-6-6'], ['1610.00601-1-6-7', '1610.00601-2-6-7'], ['1610.00601-1-6-8', '1610.00601-2-6-8'], ['1610.00601-1-6-9', '1610.00601-2-6-9'], ['1610.00601-1-6-10', '1610.00601-2-6-10'], ['1610.00601-1-12-0', '1610.00601-2-12-0'], ['1610.00601-1-12-1', '1610.00601-2-12-1'], ['1610.00601-1-12-2', '1610.00601-2-12-2'], ['1610.00601-1-12-3', '1610.00601-2-12-3'], ['1610.00601-1-12-4', '1610.00601-2-12-4'], ['1610.00601-1-12-5', '1610.00601-2-12-5'], ['1610.00601-1-12-6', '1610.00601-2-12-6'], ['1610.00601-1-12-7', '1610.00601-2-12-7'], ['1610.00601-1-1-1', '1610.00601-2-1-1'], ['1610.00601-1-1-2', '1610.00601-2-1-2'], ['1610.00601-1-1-3', '1610.00601-2-1-3'], ['1610.00601-1-1-4', '1610.00601-2-1-4'], ['1610.00601-1-9-0', '1610.00601-2-9-0'], ['1610.00601-1-9-1', '1610.00601-2-9-1'], ['1610.00601-1-9-2', '1610.00601-2-9-2'], ['1610.00601-1-9-3', '1610.00601-2-9-3'], ['1610.00601-1-9-4', '1610.00601-2-9-4'], ['1610.00601-1-9-5', '1610.00601-2-9-5'], ['1610.00601-1-9-6', '1610.00601-2-9-6'], ['1610.00601-1-9-7', '1610.00601-2-9-7'], ['1610.00601-1-9-8', '1610.00601-2-9-8'], ['1610.00601-1-9-9', '1610.00601-2-9-9'], ['1610.00601-1-9-10', '1610.00601-2-9-10'], ['1610.00601-1-9-11', '1610.00601-2-9-11'], ['1610.00601-1-8-0', '1610.00601-2-8-0'], ['1610.00601-1-8-1', '1610.00601-2-8-1'], ['1610.00601-1-5-0', '1610.00601-2-5-0'], ['1610.00601-1-5-1', '1610.00601-2-5-1'], ['1610.00601-1-5-2', '1610.00601-2-5-2'], ['1610.00601-1-5-3', '1610.00601-2-5-3'], ['1610.00601-1-5-4', '1610.00601-2-5-4'], ['1610.00601-1-5-5', '1610.00601-2-5-5'], ['1610.00601-1-5-6', '1610.00601-2-5-6'], ['1610.00601-1-10-0', '1610.00601-2-10-0'], ['1610.00601-1-10-1', '1610.00601-2-10-1'], ['1610.00601-1-10-2', '1610.00601-2-10-2'], ['1610.00601-1-10-3', '1610.00601-2-10-3'], ['1610.00601-1-10-4', '1610.00601-2-10-4'], ['1610.00601-1-10-5', '1610.00601-2-10-5'], ['1610.00601-1-10-6', '1610.00601-2-10-6'], ['1610.00601-1-10-7', '1610.00601-2-10-7'], ['1610.00601-1-10-8', '1610.00601-2-10-8'], ['1610.00601-1-11-1', '1610.00601-2-11-1'], ['1610.00601-1-11-2', '1610.00601-2-11-2'], ['1610.00601-1-11-3', '1610.00601-2-11-3'], ['1610.00601-1-11-4', '1610.00601-2-11-4'], ['1610.00601-1-11-5', '1610.00601-2-11-5']]

[['1610.00601-1-13-1', '1610.00601-2-13-1'], ['1610.00601-1-13-2', '1610.00601-2-13-2'], ['1610.00601-1-13-3', '1610.00601-2-13-3'], ['1610.00601-1-13-4', '1610.00601-2-13-4'], ['1610.00601-1-13-5', '1610.00601-2-13-5'], ['1610.00601-1-13-6', '1610.00601-2-13-6'], ['1610.00601-1-3-0', '1610.00601-2-3-0'], ['1610.00601-1-3-1', '1610.00601-2-3-1'], ['1610.00601-1-3-2', '1610.00601-2-3-2'], ['1610.00601-1-3-3', '1610.00601-2-3-3'], ['1610.00601-1-3-4', '1610.00601-2-3-4'], ['1610.00601-1-3-5', '1610.00601-2-3-5'], ['1610.00601-1-0-0', '1610.00601-2-0-0'], ['1610.00601-1-0-1', '1610.00601-2-0-1'], ['1610.00601-1-0-2', '1610.00601-2-0-2'], ['1610.00601-1-0-3', '1610.00601-2-0-3'], ['1610.00601-1-0-4', '1610.00601-2-0-4'], ['1610.00601-1-4-0', '1610.00601-2-4-0'], ['1610.00601-1-4-1', '1610.00601-2-4-1'], ['1610.00601-1-7-0', '1610.00601-2-7-0'], ['1610.00601-1-7-1', '1610.00601-2-7-1'], ['1610.00601-1-7-2', '1610.00601-2-7-2'], ['1610.00601-1-7-3', '1610.00601-2-7-3'], ['1610.00601-1-7-4', '1610.00601-2-7-4'], ['1610.00601-1-7-5', '1610.00601-2-7-5'], ['1610.00601-1-7-6', '1610.00601-2-7-6'], ['1610.00601-1-7-7', '1610.00601-2-7-7'], ['1610.00601-1-7-8', '1610.00601-2-7-8'], ['1610.00601-1-2-0', '1610.00601-2-2-0'], ['1610.00601-1-2-1', '1610.00601-2-2-1'], ['1610.00601-1-2-2', '1610.00601-2-2-2'], ['1610.00601-1-2-3', '1610.00601-2-2-3'], ['1610.00601-1-2-4', '1610.00601-2-2-4'], ['1610.00601-1-6-1', '1610.00601-2-6-1'], ['1610.00601-1-6-2', '1610.00601-2-6-2'], ['1610.00601-1-6-3', '1610.00601-2-6-3'], ['1610.00601-1-6-4', '1610.00601-2-6-4'], ['1610.00601-1-6-5', '1610.00601-2-6-5'], ['1610.00601-1-6-6', '1610.00601-2-6-6'], ['1610.00601-1-6-7', '1610.00601-2-6-7'], ['1610.00601-1-6-8', '1610.00601-2-6-8'], ['1610.00601-1-6-9', '1610.00601-2-6-9'], ['1610.00601-1-6-10', '1610.00601-2-6-10'], ['1610.00601-1-12-0', '1610.00601-2-12-0'], ['1610.00601-1-12-1', '1610.00601-2-12-1'], ['1610.00601-1-12-2', '1610.00601-2-12-2'], ['1610.00601-1-12-3', '1610.00601-2-12-3'], ['1610.00601-1-12-4', '1610.00601-2-12-4'], ['1610.00601-1-12-5', '1610.00601-2-12-5'], ['1610.00601-1-12-6', '1610.00601-2-12-6'], ['1610.00601-1-12-7', '1610.00601-2-12-7'], ['1610.00601-1-1-1', '1610.00601-2-1-1'], ['1610.00601-1-1-2', '1610.00601-2-1-2'], ['1610.00601-1-1-3', '1610.00601-2-1-3'], ['1610.00601-1-1-4', '1610.00601-2-1-4'], ['1610.00601-1-9-0', '1610.00601-2-9-0'], ['1610.00601-1-9-1', '1610.00601-2-9-1'], ['1610.00601-1-9-2', '1610.00601-2-9-2'], ['1610.00601-1-9-3', '1610.00601-2-9-3'], ['1610.00601-1-9-4', '1610.00601-2-9-4'], ['1610.00601-1-9-5', '1610.00601-2-9-5'], ['1610.00601-1-9-6', '1610.00601-2-9-6'], ['1610.00601-1-9-7', '1610.00601-2-9-7'], ['1610.00601-1-9-8', '1610.00601-2-9-8'], ['1610.00601-1-9-9', '1610.00601-2-9-9'], ['1610.00601-1-9-10', '1610.00601-2-9-10'], ['1610.00601-1-9-11', '1610.00601-2-9-11'], ['1610.00601-1-8-0', '1610.00601-2-8-0'], ['1610.00601-1-8-1', '1610.00601-2-8-1'], ['1610.00601-1-5-0', '1610.00601-2-5-0'], ['1610.00601-1-5-1', '1610.00601-2-5-1'], ['1610.00601-1-5-2', '1610.00601-2-5-2'], ['1610.00601-1-5-3', '1610.00601-2-5-3'], ['1610.00601-1-5-4', '1610.00601-2-5-4'], ['1610.00601-1-5-5', '1610.00601-2-5-5'], ['1610.00601-1-5-6', '1610.00601-2-5-6'], ['1610.00601-1-10-0', '1610.00601-2-10-0'], ['1610.00601-1-10-1', '1610.00601-2-10-1'], ['1610.00601-1-10-2', '1610.00601-2-10-2'], ['1610.00601-1-10-3', '1610.00601-2-10-3'], ['1610.00601-1-10-4', '1610.00601-2-10-4'], ['1610.00601-1-10-5', '1610.00601-2-10-5'], ['1610.00601-1-10-6', '1610.00601-2-10-6'], ['1610.00601-1-10-7', '1610.00601-2-10-7'], ['1610.00601-1-10-8', '1610.00601-2-10-8'], ['1610.00601-1-11-1', '1610.00601-2-11-1'], ['1610.00601-1-11-2', '1610.00601-2-11-2'], ['1610.00601-1-11-3', '1610.00601-2-11-3'], ['1610.00601-1-11-4', '1610.00601-2-11-4'], ['1610.00601-1-11-5', '1610.00601-2-11-5']]

[]

[]

[]

[]

['1610.00601-1-1-0', '1610.00601-1-6-0', '1610.00601-1-11-0', '1610.00601-1-13-0', '1610.00601-1-14-0', '1610.00601-2-1-0', '1610.00601-2-6-0', '1610.00601-2-11-0', '1610.00601-2-13-0', '1610.00601-2-14-0']

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

https://arxiv.org/abs/1610.00601

cond-mat-0703377

{'cond-mat-0703377-1-0-0': 'We study chemical reaction between a single hydrogen atom and a graphene, which is the elemental reaction between hydrogen and graphitic carbon materials.', 'cond-mat-0703377-1-0-1': "In the present work, classical molecular dynamics simulation is used with modified Brenner's empirical bond order potential.", 'cond-mat-0703377-1-0-2': 'The three reactions, that is, absorption reaction, reflection reaction and penetration reaction, are observed in our simulation.', 'cond-mat-0703377-1-0-3': 'Reaction rates depend on the incident energy of the hydrogen atom and the graphene temperature.', 'cond-mat-0703377-1-0-4': 'The dependence can be explained by the following mechanisms: (1) The hydrogen atom receives repulsive force by [MATH]-electrons in addition to nuclear repulsion.', 'cond-mat-0703377-1-0-5': '(2) Absorbing the hydrogen atom, the graphene transforms its structure to the "overhang" configuration such as [MATH] state.', 'cond-mat-0703377-1-0-6': '(3) The hexagonal hole of the graphene is expanded during the penetration of the hydrogen atom.', 'cond-mat-0703377-1-1-0': 'Keyword: Molecular dynamics, chemical sputtering, graphene, graphite surface, plasma-wall interaction.', 'cond-mat-0703377-1-2-0': '# Introduction', 'cond-mat-0703377-1-3-0': 'Chemical reaction between hydrogen and graphite is a basic process of the chemical sputtering between plasma and a divertor plate in plasma confinement experiments.', 'cond-mat-0703377-1-3-1': '[CITATION] The divertor plate composed of graphite tiles or carbon fiber composites is bombarded with hydrogen plasma.', 'cond-mat-0703377-1-3-2': 'Eroding the carbon wall, hydrogen ions yield [MATH] and several sorts of hydrocarbon molecules, i.e., [MATH] and so on.', 'cond-mat-0703377-1-3-3': 'The hydrocarbon molecules misbehave as impurities for plasma confinement experiments.', 'cond-mat-0703377-1-3-4': 'In order to reduce the hydrocarbon impurities in the plasma, we need to understand the erosion mechanism of carbon walls and the creation mechanism of hydrocarbon molecules.', 'cond-mat-0703377-1-3-5': 'However, these mechanisms are not clarified yet.', 'cond-mat-0703377-1-3-6': 'We, therefore, reveal the chemical reaction between hydrogen and graphite by computer simulation.', 'cond-mat-0703377-1-4-0': 'In the present work, we study the chemical reaction between a single hydrogen atom and a graphene with classical molecular dynamics (CMD) simulation.', 'cond-mat-0703377-1-4-1': 'It is reasonable to consider that ions of the hydrogen plasma combine with electrons and become neutral atoms before interacting with the carbon wall.', 'cond-mat-0703377-1-4-2': 'Therefore, we select a neutral hydrogen atom as an injected particle.', 'cond-mat-0703377-1-4-3': 'A graphene is the elemental component of graphitic carbon materials.', 'cond-mat-0703377-1-4-4': '[CITATION] The chemical reaction between the single hydrogen atom and the graphene, therefore, is regarded as the elemental reaction between hydrogen and various graphitic carbon materials.', 'cond-mat-0703377-1-5-0': 'We measured only an absorption rate in our previous work, [CITATION] where multi-layer graphite was treated.', 'cond-mat-0703377-1-5-1': 'In the present work, we evaluate a reflection rate and a penetration rate in addition to the absorption rate.', 'cond-mat-0703377-1-5-2': 'We also obtain the incident hydrogen energy dependence and the graphene temperature dependence of each reaction rate.', 'cond-mat-0703377-1-5-3': 'In [REF], the simulation model and method are described.', 'cond-mat-0703377-1-5-4': 'Simulation results are shown in [REF].', 'cond-mat-0703377-1-5-5': 'We discuss the feature of the chemical reaction between the hydrogen atom and the graphene in [REF].', 'cond-mat-0703377-1-5-6': 'This paper is concluded with summary in [REF].', 'cond-mat-0703377-1-5-7': "In addition, we note the modification of Brenner's reactive empirical bond order potential in Appendix.", 'cond-mat-0703377-1-6-0': '# Simulation Method', 'cond-mat-0703377-1-7-0': 'In the present work, we adapt CMD simulation with the NVE condition, in which the number of particle, volume and total energy are conserved.', 'cond-mat-0703377-1-7-1': 'The second order symplectic integration [CITATION] is used to solve the time evolution of the equation of motion.', 'cond-mat-0703377-1-7-2': 'The time step is [MATH].', 'cond-mat-0703377-1-7-3': "We use a modified Brenner's reactive empirical bond order (REBO) potential: [CITATION] [EQUATION] where [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.", 'cond-mat-0703377-1-7-4': 'The functions [MATH] and [MATH] represent repulsion and attraction, respectively.', 'cond-mat-0703377-1-7-5': 'The function [MATH] generates multi-body force.', 'cond-mat-0703377-1-7-6': "We show details of the modified Brenner's REBO potential in Appendix.", 'cond-mat-0703377-1-8-0': 'Figure [REF] shows the present simulation model.', 'cond-mat-0703377-1-8-1': 'The hydrogen atom is injected into the graphene composed of 160 carbon atoms.', 'cond-mat-0703377-1-8-2': 'The center of mass of the graphene is set to the origin of coordinates.', 'cond-mat-0703377-1-8-3': 'The surface of the graphene is parallel to the [MATH]-[MATH] plane.', 'cond-mat-0703377-1-8-4': 'The size of the graphene is 2.13 nm [MATH] 1.97 nm.', 'cond-mat-0703377-1-8-5': 'The graphene has no lattice defects and no crystal edges due to periodic boundary condition toward [MATH] and [MATH] directions.', 'cond-mat-0703377-1-8-6': 'The graphene temperature is defined by [EQUATION] where [MATH] and [MATH] are the momentum and the mass of the [MATH]-th carbon atom, respectively.', 'cond-mat-0703377-1-8-7': '[MATH] is the number of carbon atoms and [MATH] is the Boltzmann constant.', 'cond-mat-0703377-1-8-8': 'The symbol [MATH] denotes the summation over the carbon atoms.', 'cond-mat-0703377-1-8-9': 'The carbon atoms obey the Maxwell-Boltzmann distribution in the initial state of the simulation.', 'cond-mat-0703377-1-9-0': 'The hydrogen atom is injected parallel to the [MATH] axis from [MATH] .', 'cond-mat-0703377-1-9-1': 'We repeat 200 simulations where the [MATH] and [MATH] coordinates of injection points are set at random.', 'cond-mat-0703377-1-9-2': 'As a result, we obtain the histograms, which give reaction rates.', 'cond-mat-0703377-1-9-3': 'The incident energy [MATH] determines the initial momentum [MATH] of the hydrogen atom as [EQUATION] where [MATH] is the mass of the hydrogen atom.', 'cond-mat-0703377-1-10-0': '# Results', 'cond-mat-0703377-1-11-0': 'Three kinds of reactions between the single hydrogen atom and the graphene are observed in our CMD simulation.', 'cond-mat-0703377-1-11-1': 'They are absorption reaction, reflection reaction and penetration reaction.', 'cond-mat-0703377-1-11-2': 'The properties of the reactions are described in the following.', 'cond-mat-0703377-1-12-0': '## Dynamics of three reactions', 'cond-mat-0703377-1-13-0': 'In the absorption reaction, the hydrogen atom and the nearest carbon atom are bound by a new covalent bond.', 'cond-mat-0703377-1-13-1': 'The nearest carbon atom is pulled out of the surface of the graphene as a [MATH] configuration (Fig. [REF]).', 'cond-mat-0703377-1-13-2': 'We call this phenomenon "overhang".', 'cond-mat-0703377-1-13-3': 'The hydrogen atom remains above the nearest carbon atom while oscillating.', 'cond-mat-0703377-1-13-4': 'In the reflection reaction, the graphene reflects the incident hydrogen atom to the region of [MATH].', 'cond-mat-0703377-1-13-5': 'In the penetration reaction, the incident hydrogen atom passes through the graphene and goes away to the region of [MATH].', 'cond-mat-0703377-1-13-6': 'It is observed that the graphene expands the hexagonal hole while the hydrogen atom is penetrating.', 'cond-mat-0703377-1-14-0': '## Incident energy dependence of reaction rates', 'cond-mat-0703377-1-15-0': 'Figure [REF] shows the incident energy dependence of each reaction rate in the case that the initial graphene temperature is 300 K. Three kinds of reactions dominate in different incident energy [MATH] respectively.', 'cond-mat-0703377-1-15-1': 'In the case of [MATH], almost all of the incident hydrogen atoms are reflected.', 'cond-mat-0703377-1-15-2': 'For [MATH], the absorption reaction becomes dominant.', 'cond-mat-0703377-1-15-3': 'The reflection reaction becomes dominant again for [MATH].', 'cond-mat-0703377-1-15-4': 'The penetration reaction behaves as the dominant process for [MATH].', 'cond-mat-0703377-1-15-5': 'It is also observed that the absorption rate has the small peak around 24 eV in Fig. [REF].', 'cond-mat-0703377-1-16-0': '## Graphene temperature dependence of reaction rates', 'cond-mat-0703377-1-17-0': 'We also investigate the initial graphene temperature dependence of reaction rates (Fig. [REF]).', 'cond-mat-0703377-1-17-1': 'As the initial graphene temperature rises, the absorption rate tends to broaden to the region of low incident energy.', 'cond-mat-0703377-1-17-2': 'In contrast, the reflection rate drops.', 'cond-mat-0703377-1-17-3': 'However, the graphene temperature hardly affects the penetration rate.', 'cond-mat-0703377-1-18-0': '# Discussions', 'cond-mat-0703377-1-19-0': 'The three reactions were observed in the present simulation.', 'cond-mat-0703377-1-19-1': 'The incident energy dependence and the graphene temperature dependence of the three reaction rates are also obtained.', 'cond-mat-0703377-1-19-2': 'The similar incident energy dependence of three reactions was observed in the system of multi-layer graphite and a large amount of hydrogen atoms.', 'cond-mat-0703377-1-19-3': '[CITATION] It is, therefore, important to understand the mechanism of the chemical reaction between a single hydrogen atom and a single graphene to argue atomic-scale processes in various systems composed of hydrogen and graphitic carbon materials.', 'cond-mat-0703377-1-20-0': '## Two kinds of repulsive force', 'cond-mat-0703377-1-21-0': 'It was observed that the reflection reaction dominates in two ranges of the incident energy, i.e., [MATH] and [MATH] in Fig. [REF].', 'cond-mat-0703377-1-21-1': 'From this fact, it is deduced that two kinds of repulsive force work between the incident hydrogen atom and the graphene.', 'cond-mat-0703377-1-21-2': 'To prove it, we plot the potential energy between the hydrogen atom and the graphene in Fig. [REF], where the hydrogen atom is located just above the nearest carbon atom at the distance [MATH] and the other carbon atoms are relaxed.', 'cond-mat-0703377-1-21-3': 'From Fig. [REF], we can confirm the existence of two kinds of repulsive force between the incident hydrogen atom and the graphene.', 'cond-mat-0703377-1-21-4': 'The first repulsive force for [MATH] is due to the repulsive term [MATH] in Eq. ([REF]) and corresponds to nuclear repulsion.', 'cond-mat-0703377-1-21-5': 'The second repulsive force for [MATH] is derived from the multi-body force in the term [MATH] in Eq. ([REF]).', 'cond-mat-0703377-1-21-6': 'The existence of the second repulsive force was also confirmed by ab-initio calculations.', 'cond-mat-0703377-1-21-7': '[CITATION] It was considered that [MATH]-electrons over the graphene generate the second repulsive force.', 'cond-mat-0703377-1-21-8': 'The energy height of the potential wall of the second repulsive force is estimated to be about 0.5 eV.', 'cond-mat-0703377-1-21-9': 'The hydrogen atom with the incident energy of 0.5 eV or more, therefore, can enter the region that [MATH], in which the other mechanism derives the absorption reaction and the penetration reaction.', 'cond-mat-0703377-1-21-10': 'The mechanisms for [MATH] and [MATH] are described in the subsequent subsections.', 'cond-mat-0703377-1-21-11': 'Thus, the reflection rate starts to decrease at [MATH] in Fig. [REF].', 'cond-mat-0703377-1-22-0': '## Absorption mechanism', 'cond-mat-0703377-1-23-0': 'Figure [REF] shows the potential energy contour in the [MATH] parameter space.', 'cond-mat-0703377-1-23-1': 'Here [MATH] is the height of the nearest carbon atom from the surface of the graphene and [MATH] is the distance between the hydrogen atom and the nearest carbon atom as shown in Fig. [REF].', 'cond-mat-0703377-1-23-2': 'There is the minimum potential point at [MATH] = (0.5 , 1.1 ), which corresponds to the "overhang" configuration.', 'cond-mat-0703377-1-23-3': 'This analysis by the potential energy contour claims that the "overhang" configuration is the most stable state.', 'cond-mat-0703377-1-24-0': 'The trajectory of the absorption reaction is represented by arrow 1.167ex20D in Fig. [REF].', 'cond-mat-0703377-1-24-1': 'The initial state corresponds to the point of [MATH] = (0 , 4 ).', 'cond-mat-0703377-1-24-2': 'The hydrogen atom and the graphene start interaction around [MATH].', 'cond-mat-0703377-1-24-3': 'The arrow 1.167ex20D shows that, with tumbling down the slope of the potential energy contour, the state falls into the potential minimum point [MATH]0.5 , 1.1 [MATH], which indicates the "overhang" configuration.', 'cond-mat-0703377-1-25-0': 'Until the hydrogen atom overcomes the second repulsive force, the graphene cannot transform its structure to the "overhang" configuration.', 'cond-mat-0703377-1-25-1': 'Thereby, the incident energy of 0.5 eV, which is corresponding to the energy height of the potential wall of the second repulsive force, is the lower limit to occur the absorption reaction.', 'cond-mat-0703377-1-25-2': 'The absorption rate rises from [MATH] eV in contrast to the reflection rate by the second repulsive force and has a peak at [MATH] eV.', 'cond-mat-0703377-1-26-0': '## Reflection mechanism', 'cond-mat-0703377-1-27-0': 'In the reflection reaction for [MATH], the incident hydrogen atom bounds back from the potential wall [MATH] in Eq. ([REF]), which is represented by the region of [MATH] in Fig. [REF] and the white region in Fig. [REF].', 'cond-mat-0703377-1-27-1': 'After bounding, the hydrogen atom goes away from the graphene without connecting the nearest carbon atom.', 'cond-mat-0703377-1-27-2': 'Therefore, the graphene keeps the flat sheet configuration and does not transform its structure to the "overhang" configuration.', 'cond-mat-0703377-1-27-3': 'The trajectory of the reflection reaction by [MATH] is drawn as arrow 2.167ex20D in Fig. [REF].', 'cond-mat-0703377-1-28-0': 'Here, to make clear a distinction between the absorption reaction and the reflection reaction for [MATH], we introduce two typical time lengths [MATH] and [MATH].', 'cond-mat-0703377-1-28-1': 'The time length [MATH] is defined as the time length necessary for the graphene absorbing the hydrogen atom to transform its structure from the "flat sheet" configuration to the "overhang" configuration.', 'cond-mat-0703377-1-28-2': 'Strictly speaking, [MATH] depends on a lot of parameters, for example, the incident energy and the incident position of the hydrogen atom, the graphene temperature, and so on.', 'cond-mat-0703377-1-28-3': 'But, to estimate the typical time length [MATH], we consider the only simple overhang process, which is represented by the following trajectory in the parameter space Fig. [REF]: the configuration of the atoms is transformed from the start point that [MATH] to the end point that [MATH] along a straight line [MATH].', 'cond-mat-0703377-1-28-4': 'We, moreover, assume that the initial velocities of all the atoms are zero.', 'cond-mat-0703377-1-28-5': 'The potential function [MATH] along the above path is approximated by the following harmonic oscillator: [EQUATION] where we use, as the mass, the sum of [MATH] amu and [MATH] amu, because the hydrogen atom and the nearest carbon atom move as a rigid body in our assumption that [MATH] is fixed to the constant length of 1.1 .', 'cond-mat-0703377-1-28-6': 'From Fig. [REF], we have the potential minimum point [MATH] , the minimum potential-energy [MATH] eV, and [MATH].', 'cond-mat-0703377-1-28-7': 'Thus, we can estimate [MATH] as follows: [EQUATION]', 'cond-mat-0703377-1-28-8': 'This estimated value of [MATH] is comparable to the CMD simulation result [MATH], which was obtained under the condition that the degrees of the freedom except the parameter [MATH] are fixed to the initial values.', 'cond-mat-0703377-1-29-0': 'The other time length [MATH] is defined as the time length in which the hydrogen atom can stay in the region that [MATH] .', 'cond-mat-0703377-1-29-1': 'To estimate [MATH], we adapt the alternative assumption that the hydrogen atom moves as a free particle for [MATH] along the straight line of [MATH] and collides with the potential wall [MATH] at [MATH] , 0.5 ).', 'cond-mat-0703377-1-29-2': 'From this assumption and Eq. ([REF]), [MATH] is given by [EQUATION] where [MATH].', 'cond-mat-0703377-1-29-3': 'From Eq. ([REF]), it is obtained that [MATH] is proportional to [MATH].', 'cond-mat-0703377-1-29-4': 'On the other hand, Eq. ([REF]) shows that [MATH] is independent of [MATH].', 'cond-mat-0703377-1-30-0': 'Comparing these time length, we consider the following two cases.', 'cond-mat-0703377-1-30-1': 'In the first case that [MATH], the hydrogen atom connects with the nearest carbon and the graphene transforms its structure to the "overhang" configuration, before the hydrogen atom escapes to the region that [MATH] .', 'cond-mat-0703377-1-30-2': 'The hydrogen atom, therefore, is absorbed by the graphene.', 'cond-mat-0703377-1-30-3': 'As the incident energy increases, [MATH] becomes smaller than [MATH].', 'cond-mat-0703377-1-30-4': 'In this case [MATH], the hydrogen atom escapes before the graphene traps the hydrogen atom.', 'cond-mat-0703377-1-30-5': 'This process is regarded as the reflection reaction.', 'cond-mat-0703377-1-30-6': 'We can derive the following condition necessary for the reflection reaction: [EQUATION]', 'cond-mat-0703377-1-30-7': 'The incident energy which satisfies the condition [MATH] is estimated as [MATH] eV in our CMD simulation where the degrees of freedom except [MATH] are fixed to the initial values.', 'cond-mat-0703377-1-30-8': 'By comparison between [MATH] and the condition Eq. ([REF]), it is considered that the our assumption is proper.', 'cond-mat-0703377-1-30-9': 'In the above discussion, the hydrogen atom is located on the vertical axis over the nearest carbon atom.', 'cond-mat-0703377-1-30-10': 'However, In the present simulation, the hydrogen atom seldom exists just above the nearest carbon atom, because the [MATH] and [MATH] coordinates of the incident hydrogen atom are set at random.', 'cond-mat-0703377-1-30-11': 'Thereby, the repulsive force by [MATH] becomes weaker than that of the potential energy contour in Fig. [REF].', 'cond-mat-0703377-1-30-12': 'The time length [MATH] becomes, therefore, longer than the estimated value of Eq. ([REF]).', 'cond-mat-0703377-1-30-13': 'The hydrogen atom which deviates from the vertical axis over the nearest carbon atom needs higher incident energy than the estimated value of Eq. ([REF]).', 'cond-mat-0703377-1-30-14': 'Consequently, the incident energy of 0.84 eV in Eq. ([REF]) is the lower limit to occur the reflection reaction by [MATH].', 'cond-mat-0703377-1-31-0': '## Penetration mechanism', 'cond-mat-0703377-1-32-0': 'We describe the dynamics of the penetration reaction.', 'cond-mat-0703377-1-32-1': 'We notice for the present simulation that the graphene expands the hexagonal hole during the penetration of the hydrogen atom.', 'cond-mat-0703377-1-32-2': 'Figure [REF] shows the potential energy contour with two parameters, i.e., the distance [MATH] and the length [MATH] of the side of the hexagonal hole (See Fig. [REF]).', 'cond-mat-0703377-1-32-3': 'We note that the hydrogen atom is located above the center of the hexagonal hole unlike the layout of Fig. [REF].', 'cond-mat-0703377-1-32-4': 'The C-C bond length of the stable graphene structure is 1.42 .', 'cond-mat-0703377-1-32-5': 'The interaction force acts on the hydrogen atom and the graphene in [MATH] .', 'cond-mat-0703377-1-32-6': 'There is the potential minimum region of 0 eV in the area that [MATH] and [MATH] , which is the incident state of the hydrogen atom.', 'cond-mat-0703377-1-32-7': 'If the size of the hexagonal hole [MATH] is fixed to 1.42 , the energy height of potential wall is 38 eV at [MATH].', 'cond-mat-0703377-1-32-8': 'In this case, the hydrogen atom needs the incident energy of 38 eV or more to penetrate the graphene.', 'cond-mat-0703377-1-32-9': 'However, the penetration reaction with the incident energy of less than 38 eV is observed in the present simulation result Fig. [REF].', 'cond-mat-0703377-1-32-10': 'The difference between the estimation and the simulation result is explained by the expansion mechanism of the hexagonal hole of the graphene.', 'cond-mat-0703377-1-32-11': 'If carbon atoms move along the bottom of the potential energy valley in Fig. [REF], the parameter [MATH] increases from 1.42 to 1.58 with decreasing [MATH].', 'cond-mat-0703377-1-32-12': 'Thus, the hexagonal hole is expanded as the hydrogen atom approaches the graphene.', 'cond-mat-0703377-1-32-13': 'As a consequence, the energy height of the potential wall is lowered to 13 eV at [MATH].', 'cond-mat-0703377-1-32-14': 'This expansion lets the hydrogen atom penetrate in the incident energy of less than 38 eV.', 'cond-mat-0703377-1-33-0': 'Here, we indicate that the carbon atoms can expand the hexagonal hole before reflecting the hydrogen atom with the incident energy of 13 eV.', 'cond-mat-0703377-1-33-1': 'We define [MATH] as the time length for the hydrogen atom to approach the graphene from [MATH] .', 'cond-mat-0703377-1-33-2': 'The time length [MATH] is given by [EQUATION] where [MATH] and [MATH] is set to 13 eV.', 'cond-mat-0703377-1-33-3': 'On the other hand, the potential energy around [MATH] , where [MATH] is fixed to 0 , is approximated by the following harmonic oscillator: [EQUATION] where [MATH], [MATH] , [MATH] eV and [MATH] from the potential energy contour Fig. [REF].', 'cond-mat-0703377-1-33-4': 'For this approximation, we obtain the time length [MATH] to accomplish the expansion of the hexagonal hole as follows: [EQUATION]', 'cond-mat-0703377-1-33-5': 'Both [MATH] and [MATH] are on the same order of femtosecond.', 'cond-mat-0703377-1-33-6': 'In addition, the time length [MATH] of Eq. ([REF]) becomes practically larger than [MATH] because of deceleration due to repulsion.', 'cond-mat-0703377-1-33-7': 'Therefore, the carbon atoms can expand the hexagonal hole in response to the approach of the hydrogen atom.', 'cond-mat-0703377-1-34-0': 'Next, we consider the small peak of the absorption rate at [MATH], at which the hydrogen atom has enough incident energy to penetrate the graphene.', 'cond-mat-0703377-1-34-1': 'The incident energy of the hydrogen atom diffuses into the graphene.', 'cond-mat-0703377-1-34-2': 'Therefore, the hydrogen atom has no longer the necessary incident energy to escape from the graphene.', 'cond-mat-0703377-1-34-3': 'From the energy diffusion, it is understood that the peak of the absorption reaction at 24 eV is caused by the hydrogen atom absorption on the reverse side of the graphene.', 'cond-mat-0703377-1-34-4': 'The absorption reaction on the reverse side was confirmed in the present simulation.', 'cond-mat-0703377-1-34-5': 'As long as the hydrogen atom is absorbed, the graphene transforms its structure to the "overhang" configuration where the nearest carbon atom is pulled into the reverse side of the graphene.', 'cond-mat-0703377-1-35-0': '## Graphene temperature dependence of reaction rates', 'cond-mat-0703377-1-36-0': 'The graphene temperature dependence of reaction rates is significant at low incident energies in Fig. [REF].', 'cond-mat-0703377-1-36-1': 'As the graphene temperature is raised, the absorption rate increases and the reflection rate decreases for [MATH] eV.', 'cond-mat-0703377-1-36-2': 'The maximum temperature 2000 K, which corresponds to [MATH] eV as kinetic energy per a carbon atom, is comparable to the energy height of the potential wall of the second repulsive force of 0.5 eV.', 'cond-mat-0703377-1-36-3': 'If the nearest carbon atom has kinetic energy, the relative momentum between the hydrogen atom and the nearest carbon atom becomes larger than the initial momentum of the hydrogen atom [MATH] in Eq. ([REF]).', 'cond-mat-0703377-1-36-4': 'In the case of high graphene temperature, therefore, we substitute the relative momentum for [MATH] and can perform similar estimation to the preceding subsections.', 'cond-mat-0703377-1-36-5': 'As a result, the absorption rate increases and the reflection rate decreases as the graphene temperature is raised.', 'cond-mat-0703377-1-36-6': 'By comparison energy order, the penetration rate is insensitive to the graphene temperature.', 'cond-mat-0703377-1-37-0': '# Summary', 'cond-mat-0703377-1-38-0': "By the CMD simulation with modified Brenner's REBO potential model, we demonstrated the chemical reaction between the single hydrogen atom and the single graphene, which can be regarded as the elemental reaction between hydrogen and graphitic carbon materials.", 'cond-mat-0703377-1-38-1': 'We observed the three processes, which are the absorption, the reflection and the penetration reactions.', 'cond-mat-0703377-1-38-2': 'The dominant reaction is replaced according to the incident energy for [MATH].', 'cond-mat-0703377-1-38-3': 'We discussed the characteristic interactions between the hydrogen atom and the graphene with potential energy.', 'cond-mat-0703377-1-38-4': 'The hydrogen atom receives the repulsive force not only by nuclei of carbon atoms but also by [MATH]-electrons over the surface of the graphene.', 'cond-mat-0703377-1-38-5': 'These two kinds of repulsive force cause the two reflection mechanisms.', 'cond-mat-0703377-1-38-6': 'When the hydrogen atom is absorbed, the graphene is transformed from flat sheet configuration to "overhang" configuration.', 'cond-mat-0703377-1-38-7': "By comparison between the typical time length of the overhang transformation and the time length during the hydrogen atom's stay, we can clarify the difference between the absorption reaction and the reflection reaction for [MATH] eV.", 'cond-mat-0703377-1-38-8': 'In the penetration reaction, the incident hydrogen atom goes through the hexagonal hole of the graphene and the graphene expands the hexagonal hole, simultaneously.', 'cond-mat-0703377-1-38-9': 'The expansion lowers the energy height of the potential wall by nuclei of the carbon atoms, which accounts for starting the penetration reaction at [MATH] eV in Fig. [REF].', 'cond-mat-0703377-1-38-10': 'In addition, we investigated the graphene temperature dependence of reaction rates.', 'cond-mat-0703377-1-38-11': 'As the graphene temperature rises, the absorption rate increases and the reflection rate decreases for low incident energy.', 'cond-mat-0703377-1-38-12': 'The cause of the graphene temperature dependence is that the kinetic energy of the nearest carbon atom is comparable to the energy height of the potential wall by [MATH]-electrons.', 'cond-mat-0703377-1-39-0': '# ACKNOWLEDGMENTS', 'cond-mat-0703377-1-40-0': 'The authors acknowledge Professor Shinji Tsuneyuki and Dr. Yoshihide Yoshimoto for helpful comments, and Dr. Noriyasu Ohno for stimulating discussions.', 'cond-mat-0703377-1-40-1': 'Numerical simulations are carried out by use of the Plasma Simulator at National Institute for Fusion Science.', 'cond-mat-0703377-1-40-2': 'The work is supported partly by Grand-in Aid for Exploratory Research (C), 2006, No. 17540384 from the Ministry of Education, Culture, Sports, Science and Technology and partly by the National Institutes of Natural Sciences undertaking for Forming Bases for Interdisciplinary and International Research through Cooperation Across Fields of Study, and Collaborative Research Programs (No. NIFS06KDAT012 and No. NIFS06KTAT029).', 'cond-mat-0703377-1-41-0': '*', 'cond-mat-0703377-1-42-0': "# MODIFIED BRENNER'S REBO POTENTIAL MODEL", 'cond-mat-0703377-1-43-0': "We note here the review of Brenner's reactive empirical bond order (REBO) potential [CITATION] and our modification points.", 'cond-mat-0703377-1-43-1': 'This potential model follows in the wake of Morse potential, [CITATION] Abell potential [CITATION] and Tersoff potential.', 'cond-mat-0703377-1-43-2': '[CITATION]', 'cond-mat-0703377-1-44-0': 'The potential function [MATH] is defined by [EQUATION] where [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-44-1': 'The bond angle [MATH] is the angle between the line segment which starts at the [MATH]-th atom and ends at the [MATH]-th atom and the line segment which starts at the [MATH]-th atom and ends at the [MATH]-th atom, as follows: [EQUATION] where [MATH] is the position coordinate of the [MATH]-th atom and [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-44-2': 'The dihedral angle [MATH] is the angle between the triangle formed by the [MATH]-th, the [MATH]-th and the [MATH]-th atoms and the triangle formed by the [MATH]-th, the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-44-3': 'The cosine function of [MATH] is given by [EQUATION]', 'cond-mat-0703377-1-44-4': 'The repulsive function [MATH] and the attractive function [MATH] are defined by [EQUATION]', 'cond-mat-0703377-1-44-5': 'The square bracket such as [MATH] means that each function or each parameter depends only on the species of the [MATH]-th and the [MATH]-th atoms, for example [MATH], [MATH] and [MATH]).', 'cond-mat-0703377-1-44-6': 'The coefficients [MATH], [MATH], [MATH], [MATH] and [MATH] are given by Table [REF].', 'cond-mat-0703377-1-45-0': 'The cutoff function [MATH] determines effective ranges of the covalent bond between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-45-1': 'Two atoms are bound with the covalent bond if the distance [MATH] is shorter than [MATH].', 'cond-mat-0703377-1-45-2': 'Two atoms are not bound with the covalent bond if the distance [MATH] is longer than [MATH].', 'cond-mat-0703377-1-45-3': 'The cutoff function [MATH] connects the above two states smoothly as [EQUATION]', 'cond-mat-0703377-1-45-4': 'The constants [MATH] and [MATH] depend on the species of the two atoms (Table [REF]).', 'cond-mat-0703377-1-45-5': 'The cutoff function [MATH] distinguishes the presence of the covalent bond between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-46-0': 'The potentials [MATH] and [MATH] in Eq. ([REF]) generate two-body force, because both are the function of the only distance [MATH].', 'cond-mat-0703377-1-46-1': 'The multi-body force is used instead of the effect of an electron orbital.', 'cond-mat-0703377-1-46-2': 'In this model, [MATH] in Eq. ([REF]) gives multi-body force and is defined by [EQUATION]', 'cond-mat-0703377-1-46-3': 'The first term [MATH] generates three-body force except the effect of [MATH]-electrons.', 'cond-mat-0703377-1-46-4': 'The second term [MATH] in Eq. ([REF]) represents the influence of radical energetics and [MATH]-bond conjugation.', 'cond-mat-0703377-1-46-5': '[CITATION] The third term [MATH] in Eq. ([REF]) derives four-body force in terms of dihedral angles.', 'cond-mat-0703377-1-46-6': 'These functions are composed of the production of cutoff functions [MATH].', 'cond-mat-0703377-1-46-7': 'Five- or more-body force are generated during chemical reaction.', 'cond-mat-0703377-1-47-0': 'The function [MATH] in Eq. ([REF]) is defined by [EQUATION]', 'cond-mat-0703377-1-47-1': 'The function [MATH] in Eq. ([REF]) depends on the species of the [MATH]-th atom.', 'cond-mat-0703377-1-47-2': 'If [MATH] and the [MATH]-th atom is carbon, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-1-47-3': 'If [MATH] and the [MATH]-th atom is carbon, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-1-47-4': 'And, if the [MATH]-th atom is hydrogen, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-1-47-5': 'Here [MATH], [MATH] and [MATH] are the sixth order polynomial spline functions.', 'cond-mat-0703377-1-47-6': "Though the spline function [MATH] needs seven coefficients, the only six coefficients are written in Brenner's paper.", 'cond-mat-0703377-1-47-7': '[CITATION] We determine the seven coefficients in table [REF], [REF] and [REF], respectively.', 'cond-mat-0703377-1-47-8': 'The function [MATH] and the coordination number [MATH] in Eq. ([REF]) are defined by [EQUATION]', 'cond-mat-0703377-1-47-9': 'The constant [MATH] in Eq. ([REF]) is a weight to modulate a strength of three-body force, which depends on the species of the [MATH]-th, the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-1-47-10': "In comparison with Brenner's former potential, [CITATION] we set constants [MATH] as follows: [EQUATION]", 'cond-mat-0703377-1-47-11': 'The function [MATH] in Eq. ([REF]) is required in the case that molecules forms solid structure.', 'cond-mat-0703377-1-47-12': 'The function [MATH] is the bicubic spline function whose coefficients depend on the species of the [MATH]-th and the [MATH]-th atoms (Table [REF]).', 'cond-mat-0703377-1-47-13': 'The parameters [MATH] and [MATH] are, respectively, the number of hydrogen atoms and the number of carbon atoms bound by the [MATH]-th atom as follows: [EQUATION]', 'cond-mat-0703377-1-47-14': 'The second term [MATH] in Eq. ([REF]) is defined by a tricubic spline function [MATH] as [EQUATION] where the variables are defined by [EQUATION] with [EQUATION]', 'cond-mat-0703377-1-47-15': 'The second and the third terms of the right hand of Eq. ([REF]) are not squared.', 'cond-mat-0703377-1-47-16': "We note that they are squared in Brenner's original formulation.", 'cond-mat-0703377-1-47-17': "[CITATION] By this modification, a numerical error becomes smaller than Brenner's formation.", 'cond-mat-0703377-1-47-18': 'Table [REF] shows the revised coefficients for [MATH].', 'cond-mat-0703377-1-48-0': 'The third term [MATH] in Eq. ([REF]) is defined by [EQUATION] where [MATH] is a tricubic spline function and has the same variables as [MATH] in Eq. ([REF]).', 'cond-mat-0703377-1-48-1': 'The coefficients for [MATH] is also revised due to the modified [MATH] (Table [REF]).', 'cond-mat-0703377-1-48-2': 'In the present simulation, the function [MATH] becomes [MATH] for a perfect crystal graphene, and becomes [MATH] or [MATH] when a hydrogen atom is absorbed.'}

{'cond-mat-0703377-2-0-0': 'We study chemical reaction between a single hydrogen atom and a graphene, which is the elemental reaction between hydrogen and graphitic carbon materials.', 'cond-mat-0703377-2-0-1': "In the present work, classical molecular dynamics simulation is used with modified Brenner's empirical bond order potential.", 'cond-mat-0703377-2-0-2': 'The three reactions, that is, absorption reaction, reflection reaction and penetration reaction, are observed in our simulation.', 'cond-mat-0703377-2-0-3': 'Reaction rates depend on the incident energy of the hydrogen atom and the graphene temperature.', 'cond-mat-0703377-2-0-4': 'The dependence can be explained by the following mechanisms: (1) The hydrogen atom receives repulsive force by [MATH]-electrons in addition to nuclear repulsion.', 'cond-mat-0703377-2-0-5': '(2) Absorbing the hydrogen atom, the graphene transforms its structure to the "overhang" configuration such as [MATH] state.', 'cond-mat-0703377-2-0-6': '(3) The hexagonal hole of the graphene is expanded during the penetration of the hydrogen atom.', 'cond-mat-0703377-2-1-0': 'Keyword: Molecular dynamics, chemical sputtering, graphene, graphite surface, plasma-wall interaction.', 'cond-mat-0703377-2-2-0': '# Introduction', 'cond-mat-0703377-2-3-0': 'Chemical reaction between hydrogen and graphite is a basic process of the chemical sputtering between plasma and a divertor plate in plasma confinement experiments.', 'cond-mat-0703377-2-3-1': '[CITATION] The divertor plate composed of graphite tiles or carbon fiber composites is bombarded with hydrogen plasma.', 'cond-mat-0703377-2-3-2': 'Eroding the carbon wall, hydrogen ions yield [MATH] and several sorts of hydrocarbon molecules, i.e., [MATH] and so on.', 'cond-mat-0703377-2-3-3': 'The hydrocarbon molecules misbehave as impurities for plasma confinement experiments.', 'cond-mat-0703377-2-3-4': 'In order to reduce the hydrocarbon impurities in the plasma, we need to understand the erosion mechanism of carbon walls and the creation mechanism of hydrocarbon molecules.', 'cond-mat-0703377-2-3-5': 'However, these mechanisms are not clarified yet.', 'cond-mat-0703377-2-3-6': 'We, therefore, reveal the chemical reaction between hydrogen and graphite by computer simulation.', 'cond-mat-0703377-2-4-0': 'In the present work, we study the chemical reaction between a single hydrogen atom and a graphene with classical molecular dynamics (CMD) simulation.', 'cond-mat-0703377-2-4-1': 'It is reasonable to consider that ions of the hydrogen plasma combine with electrons and become neutral atoms before interacting with the carbon wall.', 'cond-mat-0703377-2-4-2': 'Therefore, we select a neutral hydrogen atom as an injected particle.', 'cond-mat-0703377-2-4-3': 'A graphene is the elemental component of graphitic carbon materials.', 'cond-mat-0703377-2-4-4': '[CITATION] The chemical reaction between the single hydrogen atom and the graphene, therefore, is regarded as the elemental reaction between hydrogen and various graphitic carbon materials.', 'cond-mat-0703377-2-5-0': 'We measured only an absorption rate in our previous work, [CITATION] where multi-layer graphite was treated.', 'cond-mat-0703377-2-5-1': 'In the present work, we evaluate a reflection rate and a penetration rate in addition to the absorption rate.', 'cond-mat-0703377-2-5-2': 'We also obtain the incident hydrogen energy dependence and the graphene temperature dependence of each reaction rate.', 'cond-mat-0703377-2-5-3': 'In [REF], the simulation model and method are described.', 'cond-mat-0703377-2-5-4': 'Simulation results are shown in [REF].', 'cond-mat-0703377-2-5-5': 'We discuss the feature of the chemical reaction between the hydrogen atom and the graphene in [REF].', 'cond-mat-0703377-2-5-6': 'This paper is concluded with summary in [REF].', 'cond-mat-0703377-2-5-7': "In addition, we note the modification of Brenner's reactive empirical bond order potential in Appendix.", 'cond-mat-0703377-2-6-0': '# Simulation Method', 'cond-mat-0703377-2-7-0': 'In the present work, we adapt CMD simulation with the NVE condition, in which the number of particle, volume and total energy are conserved.', 'cond-mat-0703377-2-7-1': 'The second order symplectic integration [CITATION] is used to solve the time evolution of the equation of motion.', 'cond-mat-0703377-2-7-2': 'The time step is [MATH].', 'cond-mat-0703377-2-7-3': "We use a modified Brenner's reactive empirical bond order (REBO) potential: [CITATION] [EQUATION] where [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.", 'cond-mat-0703377-2-7-4': 'The functions [MATH] and [MATH] represent repulsion and attraction, respectively.', 'cond-mat-0703377-2-7-5': 'The function [MATH] generates multi-body force.', 'cond-mat-0703377-2-7-6': "We show details of the modified Brenner's REBO potential in Appendix.", 'cond-mat-0703377-2-8-0': 'Figure [REF] shows the present simulation model.', 'cond-mat-0703377-2-8-1': 'The hydrogen atom is injected into the graphene composed of 160 carbon atoms.', 'cond-mat-0703377-2-8-2': 'The center of mass of the graphene is set to the origin of coordinates.', 'cond-mat-0703377-2-8-3': 'The surface of the graphene is parallel to the [MATH]-[MATH] plane.', 'cond-mat-0703377-2-8-4': 'The size of the graphene is 2.13 nm [MATH] 1.97 nm.', 'cond-mat-0703377-2-8-5': 'The graphene has no lattice defects and no crystal edges due to periodic boundary condition toward [MATH] and [MATH] directions.', 'cond-mat-0703377-2-8-6': 'The graphene temperature is defined by [EQUATION] where [MATH] and [MATH] are the momentum and the mass of the [MATH]-th carbon atom, respectively.', 'cond-mat-0703377-2-8-7': '[MATH] is the number of carbon atoms and [MATH] is the Boltzmann constant.', 'cond-mat-0703377-2-8-8': 'The symbol [MATH] denotes the summation over the carbon atoms.', 'cond-mat-0703377-2-8-9': 'The carbon atoms obey the Maxwell-Boltzmann distribution in the initial state of the simulation.', 'cond-mat-0703377-2-9-0': 'The hydrogen atom is injected parallel to the [MATH] axis from [MATH] .', 'cond-mat-0703377-2-9-1': 'We repeat 200 simulations where the [MATH] and [MATH] coordinates of injection points are set at random.', 'cond-mat-0703377-2-9-2': 'As a result, we obtain the histograms, which give reaction rates.', 'cond-mat-0703377-2-9-3': 'The incident energy [MATH] determines the initial momentum [MATH] of the hydrogen atom as [EQUATION] where [MATH] is the mass of the hydrogen atom.', 'cond-mat-0703377-2-10-0': '# Results', 'cond-mat-0703377-2-11-0': 'Three kinds of reactions between the single hydrogen atom and the graphene are observed in our CMD simulation.', 'cond-mat-0703377-2-11-1': 'They are absorption reaction, reflection reaction and penetration reaction.', 'cond-mat-0703377-2-11-2': 'The properties of the reactions are described in the following.', 'cond-mat-0703377-2-12-0': '## Dynamics of three reactions', 'cond-mat-0703377-2-13-0': 'In the absorption reaction, the hydrogen atom and the nearest carbon atom are bound by a new covalent bond.', 'cond-mat-0703377-2-13-1': 'The nearest carbon atom is pulled out of the surface of the graphene as a [MATH] configuration (Fig. [REF]).', 'cond-mat-0703377-2-13-2': 'We call this phenomenon "overhang".', 'cond-mat-0703377-2-13-3': 'The hydrogen atom remains above the nearest carbon atom while oscillating.', 'cond-mat-0703377-2-13-4': 'In the reflection reaction, the graphene reflects the incident hydrogen atom to the region of [MATH].', 'cond-mat-0703377-2-13-5': 'In the penetration reaction, the incident hydrogen atom passes through the graphene and goes away to the region of [MATH].', 'cond-mat-0703377-2-13-6': 'It is observed that the graphene expands the hexagonal hole while the hydrogen atom is penetrating.', 'cond-mat-0703377-2-14-0': '## Incident energy dependence of reaction rates', 'cond-mat-0703377-2-15-0': 'Figure [REF] shows the incident energy dependence of each reaction rate in the case that the initial graphene temperature is 300 K. Three kinds of reactions dominate in different incident energy [MATH] respectively.', 'cond-mat-0703377-2-15-1': 'In the case of [MATH], almost all of the incident hydrogen atoms are reflected.', 'cond-mat-0703377-2-15-2': 'For [MATH], the absorption reaction becomes dominant.', 'cond-mat-0703377-2-15-3': 'The reflection reaction becomes dominant again for [MATH].', 'cond-mat-0703377-2-15-4': 'The penetration reaction behaves as the dominant process for [MATH].', 'cond-mat-0703377-2-15-5': 'It is also observed that the absorption rate has the small peak around 24 eV in Fig. [REF].', 'cond-mat-0703377-2-16-0': '## Graphene temperature dependence of reaction rates', 'cond-mat-0703377-2-17-0': 'We also investigate the initial graphene temperature dependence of reaction rates (Fig. [REF]).', 'cond-mat-0703377-2-17-1': 'As the initial graphene temperature rises, the absorption rate tends to broaden to the region of low incident energy.', 'cond-mat-0703377-2-17-2': 'In contrast, the reflection rate drops.', 'cond-mat-0703377-2-17-3': 'However, the graphene temperature hardly affects the penetration rate.', 'cond-mat-0703377-2-18-0': '# Discussions', 'cond-mat-0703377-2-19-0': 'The three reactions were observed in the present simulation.', 'cond-mat-0703377-2-19-1': 'The incident energy dependence and the graphene temperature dependence of the three reaction rates are also obtained.', 'cond-mat-0703377-2-19-2': 'The similar incident energy dependence of three reactions was observed in the system of multi-layer graphite and a large amount of hydrogen atoms.', 'cond-mat-0703377-2-19-3': '[CITATION] It is, therefore, important to understand the mechanism of the chemical reaction between a single hydrogen atom and a single graphene to argue atomic-scale processes in various systems composed of hydrogen and graphitic carbon materials.', 'cond-mat-0703377-2-20-0': '## Two kinds of repulsive force', 'cond-mat-0703377-2-21-0': 'It was observed that the reflection reaction dominates in two ranges of the incident energy, i.e., [MATH] and [MATH] in Fig. [REF].', 'cond-mat-0703377-2-21-1': 'From this fact, it is deduced that two kinds of repulsive force work between the incident hydrogen atom and the graphene.', 'cond-mat-0703377-2-21-2': 'To prove it, we plot the potential energy between the hydrogen atom and the graphene in Fig. [REF], where the hydrogen atom is located just above the nearest carbon atom at the distance [MATH] and the other carbon atoms are relaxed.', 'cond-mat-0703377-2-21-3': 'From Fig. [REF], we can confirm the existence of two kinds of repulsive force between the incident hydrogen atom and the graphene.', 'cond-mat-0703377-2-21-4': 'The first repulsive force for [MATH] is due to the repulsive term [MATH] in Eq. ([REF]) and corresponds to nuclear repulsion.', 'cond-mat-0703377-2-21-5': 'The second repulsive force for [MATH] is derived from the multi-body force in the term [MATH] in Eq. ([REF]).', 'cond-mat-0703377-2-21-6': 'The existence of the second repulsive force was also confirmed by ab-initio calculations.', 'cond-mat-0703377-2-21-7': '[CITATION] It was considered that [MATH]-electrons over the graphene generate the second repulsive force.', 'cond-mat-0703377-2-21-8': 'The energy height of the potential wall of the second repulsive force is estimated to be about 0.5 eV.', 'cond-mat-0703377-2-21-9': 'The hydrogen atom with the incident energy of 0.5 eV or more, therefore, can enter the region that [MATH], in which the other mechanism derives the absorption reaction and the penetration reaction.', 'cond-mat-0703377-2-21-10': 'The mechanisms for [MATH] and [MATH] are described in the subsequent subsections.', 'cond-mat-0703377-2-21-11': 'Thus, the reflection rate starts to decrease at [MATH] in Fig. [REF].', 'cond-mat-0703377-2-22-0': '## Absorption mechanism', 'cond-mat-0703377-2-23-0': 'Figure [REF] shows the potential energy contour in the [MATH] parameter space.', 'cond-mat-0703377-2-23-1': 'Here [MATH] is the height of the nearest carbon atom from the surface of the graphene and [MATH] is the distance between the hydrogen atom and the nearest carbon atom as shown in Fig. [REF].', 'cond-mat-0703377-2-23-2': 'There is the minimum potential point at [MATH] = (0.5 , 1.1 ), which corresponds to the "overhang" configuration.', 'cond-mat-0703377-2-23-3': 'This analysis by the potential energy contour claims that the "overhang" configuration is the most stable state.', 'cond-mat-0703377-2-24-0': 'The trajectory of the absorption reaction is represented by arrow 1.167ex20D in Fig. [REF].', 'cond-mat-0703377-2-24-1': 'The initial state corresponds to the point of [MATH] = (0 , 4 ).', 'cond-mat-0703377-2-24-2': 'The hydrogen atom and the graphene start interaction around [MATH].', 'cond-mat-0703377-2-24-3': 'The arrow 1.167ex20D shows that, with tumbling down the slope of the potential energy contour, the state falls into the potential minimum point [MATH]0.5 , 1.1 [MATH], which indicates the "overhang" configuration.', 'cond-mat-0703377-2-25-0': 'Until the hydrogen atom overcomes the second repulsive force, the graphene cannot transform its structure to the "overhang" configuration.', 'cond-mat-0703377-2-25-1': 'Thereby, the incident energy of 0.5 eV, which is corresponding to the energy height of the potential wall of the second repulsive force, is the lower limit to occur the absorption reaction.', 'cond-mat-0703377-2-25-2': 'The absorption rate rises from [MATH] eV in contrast to the reflection rate by the second repulsive force and has a peak at [MATH] eV.', 'cond-mat-0703377-2-26-0': '## Reflection mechanism', 'cond-mat-0703377-2-27-0': 'In the reflection reaction for [MATH], the incident hydrogen atom bounds back from the potential wall [MATH] in Eq. ([REF]), which is represented by the region of [MATH] in Fig. [REF] and the white region in Fig. [REF].', 'cond-mat-0703377-2-27-1': 'After bounding, the hydrogen atom goes away from the graphene without connecting the nearest carbon atom.', 'cond-mat-0703377-2-27-2': 'Therefore, the graphene keeps the flat sheet configuration and does not transform its structure to the "overhang" configuration.', 'cond-mat-0703377-2-27-3': 'The trajectory of the reflection reaction by [MATH] is drawn as arrow 2.167ex20D in Fig. [REF].', 'cond-mat-0703377-2-28-0': 'Here, to make clear a distinction between the absorption reaction and the reflection reaction for [MATH], we introduce two typical time lengths [MATH] and [MATH].', 'cond-mat-0703377-2-28-1': 'The time length [MATH] is defined as the time length necessary for the graphene absorbing the hydrogen atom to transform its structure from the "flat sheet" configuration to the "overhang" configuration.', 'cond-mat-0703377-2-28-2': 'Strictly speaking, [MATH] depends on a lot of parameters, for example, the incident energy and the incident position of the hydrogen atom, the graphene temperature, and so on.', 'cond-mat-0703377-2-28-3': 'But, to estimate the typical time length [MATH], we consider the only simple overhang process, which is represented by the following trajectory in the parameter space Fig. [REF]: the configuration of the atoms is transformed from the start point that [MATH] to the end point that [MATH] along a straight line [MATH].', 'cond-mat-0703377-2-28-4': 'We, moreover, assume that the initial velocities of all the atoms are zero.', 'cond-mat-0703377-2-28-5': 'The potential function [MATH] along the above path is approximated by the following harmonic oscillator: [EQUATION] where we use, as the mass, the sum of [MATH] amu and [MATH] amu, because the hydrogen atom and the nearest carbon atom move as a rigid body in our assumption that [MATH] is fixed to the constant length of 1.1 .', 'cond-mat-0703377-2-28-6': 'From Fig. [REF], we have the potential minimum point [MATH] , the minimum potential-energy [MATH] eV, and [MATH].', 'cond-mat-0703377-2-28-7': 'Thus, we can estimate [MATH] as follows: [EQUATION]', 'cond-mat-0703377-2-28-8': 'This estimated value of [MATH] is comparable to the CMD simulation result [MATH], which was obtained under the condition that the degrees of the freedom except the parameter [MATH] are fixed to the initial values.', 'cond-mat-0703377-2-29-0': 'The other time length [MATH] is defined as the time length in which the hydrogen atom can stay in the region that [MATH] .', 'cond-mat-0703377-2-29-1': 'To estimate [MATH], we adapt the alternative assumption that the hydrogen atom moves as a free particle for [MATH] along the straight line of [MATH] and collides with the potential wall [MATH] at [MATH] , 0.5 ).', 'cond-mat-0703377-2-29-2': 'From this assumption and Eq. ([REF]), [MATH] is given by [EQUATION] where [MATH].', 'cond-mat-0703377-2-29-3': 'From Eq. ([REF]), it is obtained that [MATH] is proportional to [MATH].', 'cond-mat-0703377-2-29-4': 'On the other hand, Eq. ([REF]) shows that [MATH] is independent of [MATH].', 'cond-mat-0703377-2-30-0': 'Comparing these time length, we consider the following two cases.', 'cond-mat-0703377-2-30-1': 'In the first case that [MATH], the hydrogen atom connects with the nearest carbon and the graphene transforms its structure to the "overhang" configuration, before the hydrogen atom escapes to the region that [MATH] .', 'cond-mat-0703377-2-30-2': 'The hydrogen atom, therefore, is absorbed by the graphene.', 'cond-mat-0703377-2-30-3': 'As the incident energy increases, [MATH] becomes smaller than [MATH].', 'cond-mat-0703377-2-30-4': 'In this case [MATH], the hydrogen atom escapes before the graphene traps the hydrogen atom.', 'cond-mat-0703377-2-30-5': 'This process is regarded as the reflection reaction.', 'cond-mat-0703377-2-30-6': 'We can derive the following condition necessary for the reflection reaction: [EQUATION]', 'cond-mat-0703377-2-30-7': 'The incident energy which satisfies the condition [MATH] is estimated as [MATH] eV in our CMD simulation where the degrees of freedom except [MATH] are fixed to the initial values.', 'cond-mat-0703377-2-30-8': 'By comparison between [MATH] and the condition Eq. ([REF]), it is considered that the our assumption is proper.', 'cond-mat-0703377-2-30-9': 'In the above discussion, the hydrogen atom is located on the vertical axis over the nearest carbon atom.', 'cond-mat-0703377-2-30-10': 'However, In the present simulation, the hydrogen atom seldom exists just above the nearest carbon atom, because the [MATH] and [MATH] coordinates of the incident hydrogen atom are set at random.', 'cond-mat-0703377-2-30-11': 'Thereby, the repulsive force by [MATH] becomes weaker than that of the potential energy contour in Fig. [REF].', 'cond-mat-0703377-2-30-12': 'The time length [MATH] becomes, therefore, longer than the estimated value of Eq. ([REF]).', 'cond-mat-0703377-2-30-13': 'The hydrogen atom which deviates from the vertical axis over the nearest carbon atom needs higher incident energy than the estimated value of Eq. ([REF]).', 'cond-mat-0703377-2-30-14': 'Consequently, the incident energy of 0.84 eV in Eq. ([REF]) is the lower limit to occur the reflection reaction by [MATH].', 'cond-mat-0703377-2-31-0': '## Penetration mechanism', 'cond-mat-0703377-2-32-0': 'We describe the dynamics of the penetration reaction.', 'cond-mat-0703377-2-32-1': 'We notice for the present simulation that the graphene expands the hexagonal hole during the penetration of the hydrogen atom.', 'cond-mat-0703377-2-32-2': 'Figure [REF] shows the potential energy contour with two parameters, i.e., the distance [MATH] and the length [MATH] of the side of the hexagonal hole (See Fig. [REF]).', 'cond-mat-0703377-2-32-3': 'We note that the hydrogen atom is located above the center of the hexagonal hole unlike the layout of Fig. [REF].', 'cond-mat-0703377-2-32-4': 'The C-C bond length of the stable graphene structure is 1.42 .', 'cond-mat-0703377-2-32-5': 'The interaction force acts on the hydrogen atom and the graphene in [MATH] .', 'cond-mat-0703377-2-32-6': 'There is the potential minimum region of 0 eV in the area that [MATH] and [MATH] , which is the incident state of the hydrogen atom.', 'cond-mat-0703377-2-32-7': 'If the size of the hexagonal hole [MATH] is fixed to 1.42 , the energy height of potential wall is 38 eV at [MATH].', 'cond-mat-0703377-2-32-8': 'In this case, the hydrogen atom needs the incident energy of 38 eV or more to penetrate the graphene.', 'cond-mat-0703377-2-32-9': 'However, the penetration reaction with the incident energy of less than 38 eV is observed in the present simulation result Fig. [REF].', 'cond-mat-0703377-2-32-10': 'The difference between the estimation and the simulation result is explained by the expansion mechanism of the hexagonal hole of the graphene.', 'cond-mat-0703377-2-32-11': 'If carbon atoms move along the bottom of the potential energy valley in Fig. [REF], the parameter [MATH] increases from 1.42 to 1.58 with decreasing [MATH].', 'cond-mat-0703377-2-32-12': 'Thus, the hexagonal hole is expanded as the hydrogen atom approaches the graphene.', 'cond-mat-0703377-2-32-13': 'As a consequence, the energy height of the potential wall is lowered to 13 eV at [MATH].', 'cond-mat-0703377-2-32-14': 'This expansion lets the hydrogen atom penetrate in the incident energy of less than 38 eV.', 'cond-mat-0703377-2-33-0': 'Here, we indicate that the carbon atoms can expand the hexagonal hole before reflecting the hydrogen atom with the incident energy of 13 eV.', 'cond-mat-0703377-2-33-1': 'We define [MATH] as the time length for the hydrogen atom to approach the graphene from [MATH] .', 'cond-mat-0703377-2-33-2': 'The time length [MATH] is given by [EQUATION] where [MATH] and [MATH] is set to 13 eV.', 'cond-mat-0703377-2-33-3': 'On the other hand, the potential energy around [MATH] , where [MATH] is fixed to 0 , is approximated by the following harmonic oscillator: [EQUATION] where [MATH], [MATH] , [MATH] eV and [MATH] from the potential energy contour Fig. [REF].', 'cond-mat-0703377-2-33-4': 'For this approximation, we obtain the time length [MATH] to accomplish the expansion of the hexagonal hole as follows: [EQUATION]', 'cond-mat-0703377-2-33-5': 'Both [MATH] and [MATH] are on the same order of femtosecond.', 'cond-mat-0703377-2-33-6': 'In addition, the time length [MATH] of Eq. ([REF]) becomes practically larger than [MATH] because of deceleration due to repulsion.', 'cond-mat-0703377-2-33-7': 'Therefore, the carbon atoms can expand the hexagonal hole in response to the approach of the hydrogen atom.', 'cond-mat-0703377-2-34-0': 'Next, we consider the small peak of the absorption rate at [MATH], at which the hydrogen atom has enough incident energy to penetrate the graphene.', 'cond-mat-0703377-2-34-1': 'The incident energy of the hydrogen atom diffuses into the graphene.', 'cond-mat-0703377-2-34-2': 'Therefore, the hydrogen atom has no longer the necessary incident energy to escape from the graphene.', 'cond-mat-0703377-2-34-3': 'From the energy diffusion, it is understood that the peak of the absorption reaction at 24 eV is caused by the hydrogen atom absorption on the reverse side of the graphene.', 'cond-mat-0703377-2-34-4': 'The absorption reaction on the reverse side was confirmed in the present simulation.', 'cond-mat-0703377-2-34-5': 'As long as the hydrogen atom is absorbed, the graphene transforms its structure to the "overhang" configuration where the nearest carbon atom is pulled into the reverse side of the graphene.', 'cond-mat-0703377-2-35-0': '## Graphene temperature dependence of reaction rates', 'cond-mat-0703377-2-36-0': 'The graphene temperature dependence of reaction rates is significant at low incident energies in Fig. [REF].', 'cond-mat-0703377-2-36-1': 'As the graphene temperature is raised, the absorption rate increases and the reflection rate decreases for [MATH] eV.', 'cond-mat-0703377-2-36-2': 'The maximum temperature 2000 K, which corresponds to [MATH] eV as kinetic energy per a carbon atom, is comparable to the energy height of the potential wall of the second repulsive force of 0.5 eV.', 'cond-mat-0703377-2-36-3': 'If the nearest carbon atom has kinetic energy, the relative momentum between the hydrogen atom and the nearest carbon atom becomes larger than the initial momentum of the hydrogen atom [MATH] in Eq. ([REF]).', 'cond-mat-0703377-2-36-4': 'In the case of high graphene temperature, therefore, we substitute the relative momentum for [MATH] and can perform similar estimation to the preceding subsections.', 'cond-mat-0703377-2-36-5': 'As a result, the absorption rate increases and the reflection rate decreases as the graphene temperature is raised.', 'cond-mat-0703377-2-36-6': 'By comparison energy order, the penetration rate is insensitive to the graphene temperature.', 'cond-mat-0703377-2-37-0': '# Summary', 'cond-mat-0703377-2-38-0': "By the CMD simulation with modified Brenner's REBO potential model, we demonstrated the chemical reaction between the single hydrogen atom and the single graphene, which can be regarded as the elemental reaction between hydrogen and graphitic carbon materials.", 'cond-mat-0703377-2-38-1': 'We observed the three processes, which are the absorption, the reflection and the penetration reactions.', 'cond-mat-0703377-2-38-2': 'The dominant reaction is replaced according to the incident energy for [MATH].', 'cond-mat-0703377-2-38-3': 'We discussed the characteristic interactions between the hydrogen atom and the graphene with potential energy.', 'cond-mat-0703377-2-38-4': 'The hydrogen atom receives the repulsive force not only by nuclei of carbon atoms but also by [MATH]-electrons over the surface of the graphene.', 'cond-mat-0703377-2-38-5': 'These two kinds of repulsive force cause the two reflection mechanisms.', 'cond-mat-0703377-2-38-6': 'When the hydrogen atom is absorbed, the graphene is transformed from flat sheet configuration to "overhang" configuration.', 'cond-mat-0703377-2-38-7': "By comparison between the typical time length of the overhang transformation and the time length during the hydrogen atom's stay, we can clarify the difference between the absorption reaction and the reflection reaction for [MATH] eV.", 'cond-mat-0703377-2-38-8': 'In the penetration reaction, the incident hydrogen atom goes through the hexagonal hole of the graphene and the graphene expands the hexagonal hole, simultaneously.', 'cond-mat-0703377-2-38-9': 'The expansion lowers the energy height of the potential wall by nuclei of the carbon atoms, which accounts for starting the penetration reaction at [MATH] eV in Fig. [REF].', 'cond-mat-0703377-2-38-10': 'In addition, we investigated the graphene temperature dependence of reaction rates.', 'cond-mat-0703377-2-38-11': 'As the graphene temperature rises, the absorption rate increases and the reflection rate decreases for low incident energy.', 'cond-mat-0703377-2-38-12': 'The cause of the graphene temperature dependence is that the kinetic energy of the nearest carbon atom is comparable to the energy height of the potential wall by [MATH]-electrons.', 'cond-mat-0703377-2-39-0': '# ACKNOWLEDGMENTS', 'cond-mat-0703377-2-40-0': 'The authors acknowledge Professor Shinji Tsuneyuki and Dr. Yoshihide Yoshimoto for helpful comments, and Dr. Noriyasu Ohno for stimulating discussions.', 'cond-mat-0703377-2-40-1': 'Numerical simulations are carried out by use of the Plasma Simulator at National Institute for Fusion Science.', 'cond-mat-0703377-2-40-2': 'The work is supported partly by Grand-in Aid for Exploratory Research (C), 2006, No. 17540384 from the Ministry of Education, Culture, Sports, Science and Technology and partly by the National Institutes of Natural Sciences undertaking for Forming Bases for Interdisciplinary and International Research through Cooperation Across Fields of Study, and Collaborative Research Programs (No. NIFS06KDAT012 and No. NIFS06KTAT029).', 'cond-mat-0703377-2-41-0': '*', 'cond-mat-0703377-2-42-0': "# MODIFIED BRENNER'S REBO POTENTIAL MODEL", 'cond-mat-0703377-2-43-0': "We note here the review of Brenner's reactive empirical bond order (REBO) potential [CITATION] and our modification points.", 'cond-mat-0703377-2-43-1': 'This potential model follows in the wake of Morse potential, [CITATION] Abell potential [CITATION] and Tersoff potential.', 'cond-mat-0703377-2-43-2': '[CITATION]', 'cond-mat-0703377-2-44-0': 'The potential function [MATH] is defined by [EQUATION] where [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-44-1': 'The bond angle [MATH] is the angle between the line segment which starts at the [MATH]-th atom and ends at the [MATH]-th atom and the line segment which starts at the [MATH]-th atom and ends at the [MATH]-th atom, as follows: [EQUATION] where [MATH] is the position coordinate of the [MATH]-th atom and [MATH] is the distance between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-44-2': 'The dihedral angle [MATH] is the angle between the triangle formed by the [MATH]-th, the [MATH]-th and the [MATH]-th atoms and the triangle formed by the [MATH]-th, the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-44-3': 'The cosine function of [MATH] is given by [EQUATION]', 'cond-mat-0703377-2-44-4': 'The repulsive function [MATH] and the attractive function [MATH] are defined by [EQUATION]', 'cond-mat-0703377-2-44-5': 'The square bracket such as [MATH] means that each function or each parameter depends only on the species of the [MATH]-th and the [MATH]-th atoms, for example [MATH], [MATH] and [MATH]).', 'cond-mat-0703377-2-44-6': 'The coefficients [MATH], [MATH], [MATH], [MATH] and [MATH] are given by Table [REF].', 'cond-mat-0703377-2-45-0': 'The cutoff function [MATH] determines effective ranges of the covalent bond between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-45-1': 'Two atoms are bound with the covalent bond if the distance [MATH] is shorter than [MATH].', 'cond-mat-0703377-2-45-2': 'Two atoms are not bound with the covalent bond if the distance [MATH] is longer than [MATH].', 'cond-mat-0703377-2-45-3': 'The cutoff function [MATH] connects the above two states smoothly as [EQUATION]', 'cond-mat-0703377-2-45-4': 'The constants [MATH] and [MATH] depend on the species of the two atoms (Table [REF]).', 'cond-mat-0703377-2-45-5': 'The cutoff function [MATH] distinguishes the presence of the covalent bond between the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-46-0': 'The potentials [MATH] and [MATH] in Eq. ([REF]) generate two-body force, because both are the function of the only distance [MATH].', 'cond-mat-0703377-2-46-1': 'The multi-body force is used instead of the effect of an electron orbital.', 'cond-mat-0703377-2-46-2': 'In this model, [MATH] in Eq. ([REF]) gives multi-body force and is defined by [EQUATION]', 'cond-mat-0703377-2-46-3': 'The first term [MATH] generates three-body force except the effect of [MATH]-electrons.', 'cond-mat-0703377-2-46-4': 'The second term [MATH] in Eq. ([REF]) represents the influence of radical energetics and [MATH]-bond conjugation.', 'cond-mat-0703377-2-46-5': '[CITATION] The third term [MATH] in Eq. ([REF]) derives four-body force in terms of dihedral angles.', 'cond-mat-0703377-2-46-6': 'These functions are composed of the production of cutoff functions [MATH].', 'cond-mat-0703377-2-46-7': 'Five- or more-body force are generated during chemical reaction.', 'cond-mat-0703377-2-47-0': 'The function [MATH] in Eq. ([REF]) is defined by [EQUATION]', 'cond-mat-0703377-2-47-1': 'The function [MATH] in Eq. ([REF]) depends on the species of the [MATH]-th atom.', 'cond-mat-0703377-2-47-2': 'If [MATH] and the [MATH]-th atom is carbon, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-2-47-3': 'If [MATH] and the [MATH]-th atom is carbon, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-2-47-4': 'And, if the [MATH]-th atom is hydrogen, [MATH] is defined by [EQUATION]', 'cond-mat-0703377-2-47-5': 'Here [MATH], [MATH] and [MATH] are the sixth order polynomial spline functions.', 'cond-mat-0703377-2-47-6': "Though the spline function [MATH] needs seven coefficients, the only six coefficients are written in Brenner's paper.", 'cond-mat-0703377-2-47-7': '[CITATION] We determine the seven coefficients in table [REF], [REF] and [REF], respectively.', 'cond-mat-0703377-2-47-8': 'The function [MATH] and the coordination number [MATH] in Eq. ([REF]) are defined by [EQUATION]', 'cond-mat-0703377-2-47-9': 'The constant [MATH] in Eq. ([REF]) is a weight to modulate a strength of three-body force, which depends on the species of the [MATH]-th, the [MATH]-th and the [MATH]-th atoms.', 'cond-mat-0703377-2-47-10': "In comparison with Brenner's former potential, [CITATION] we set constants [MATH] as follows: [EQUATION]", 'cond-mat-0703377-2-47-11': 'The function [MATH] in Eq. ([REF]) is required in the case that molecules forms solid structure.', 'cond-mat-0703377-2-47-12': 'The function [MATH] is the bicubic spline function whose coefficients depend on the species of the [MATH]-th and the [MATH]-th atoms (Table [REF]).', 'cond-mat-0703377-2-47-13': 'The parameters [MATH] and [MATH] are, respectively, the number of hydrogen atoms and the number of carbon atoms bound by the [MATH]-th atom as follows: [EQUATION]', 'cond-mat-0703377-2-47-14': 'The second term [MATH] in Eq. ([REF]) is defined by a tricubic spline function [MATH] as [EQUATION] where the variables are defined by [EQUATION] with [EQUATION]', 'cond-mat-0703377-2-47-15': 'The second and the third terms of the right hand of Eq. ([REF]) are not squared.', 'cond-mat-0703377-2-47-16': "We note that they are squared in Brenner's original formulation.", 'cond-mat-0703377-2-47-17': "[CITATION] By this modification, a numerical error becomes smaller than Brenner's formation.", 'cond-mat-0703377-2-47-18': 'Table [REF] shows the revised coefficients for [MATH].', 'cond-mat-0703377-2-48-0': 'The third term [MATH] in Eq. ([REF]) is defined by [EQUATION] where [MATH] is a tricubic spline function and has the same variables as [MATH] in Eq. ([REF]).', 'cond-mat-0703377-2-48-1': 'The coefficients for [MATH] is also revised due to the modified [MATH] (Table [REF]).', 'cond-mat-0703377-2-48-2': 'In the present simulation, the function [MATH] becomes [MATH] for a perfect crystal graphene, and becomes [MATH] or [MATH] when a hydrogen atom is absorbed.'}

[['cond-mat-0703377-1-3-0', 'cond-mat-0703377-2-3-0'], ['cond-mat-0703377-1-3-1', 'cond-mat-0703377-2-3-1'], ['cond-mat-0703377-1-3-2', 'cond-mat-0703377-2-3-2'], ['cond-mat-0703377-1-3-3', 'cond-mat-0703377-2-3-3'], ['cond-mat-0703377-1-3-4', 'cond-mat-0703377-2-3-4'], ['cond-mat-0703377-1-3-5', 'cond-mat-0703377-2-3-5'], ['cond-mat-0703377-1-3-6', 'cond-mat-0703377-2-3-6'], ['cond-mat-0703377-1-27-0', 'cond-mat-0703377-2-27-0'], ['cond-mat-0703377-1-27-1', 'cond-mat-0703377-2-27-1'], ['cond-mat-0703377-1-27-2', 'cond-mat-0703377-2-27-2'], ['cond-mat-0703377-1-27-3', 'cond-mat-0703377-2-27-3'], ['cond-mat-0703377-1-25-0', 'cond-mat-0703377-2-25-0'], ['cond-mat-0703377-1-25-1', 'cond-mat-0703377-2-25-1'], ['cond-mat-0703377-1-25-2', 'cond-mat-0703377-2-25-2'], ['cond-mat-0703377-1-30-0', 'cond-mat-0703377-2-30-0'], ['cond-mat-0703377-1-30-1', 'cond-mat-0703377-2-30-1'], ['cond-mat-0703377-1-30-2', 'cond-mat-0703377-2-30-2'], ['cond-mat-0703377-1-30-3', 'cond-mat-0703377-2-30-3'], ['cond-mat-0703377-1-30-4', 'cond-mat-0703377-2-30-4'], ['cond-mat-0703377-1-30-5', 'cond-mat-0703377-2-30-5'], ['cond-mat-0703377-1-30-6', 'cond-mat-0703377-2-30-6'], ['cond-mat-0703377-1-30-7', 'cond-mat-0703377-2-30-7'], ['cond-mat-0703377-1-30-8', 'cond-mat-0703377-2-30-8'], ['cond-mat-0703377-1-30-9', 'cond-mat-0703377-2-30-9'], ['cond-mat-0703377-1-30-10', 'cond-mat-0703377-2-30-10'], ['cond-mat-0703377-1-30-11', 'cond-mat-0703377-2-30-11'], ['cond-mat-0703377-1-30-12', 'cond-mat-0703377-2-30-12'], ['cond-mat-0703377-1-30-13', 'cond-mat-0703377-2-30-13'], ['cond-mat-0703377-1-30-14', 'cond-mat-0703377-2-30-14'], ['cond-mat-0703377-1-29-0', 'cond-mat-0703377-2-29-0'], ['cond-mat-0703377-1-29-1', 'cond-mat-0703377-2-29-1'], ['cond-mat-0703377-1-29-2', 'cond-mat-0703377-2-29-2'], ['cond-mat-0703377-1-29-3', 'cond-mat-0703377-2-29-3'], ['cond-mat-0703377-1-29-4', 'cond-mat-0703377-2-29-4'], ['cond-mat-0703377-1-5-0', 'cond-mat-0703377-2-5-0'], ['cond-mat-0703377-1-5-1', 'cond-mat-0703377-2-5-1'], ['cond-mat-0703377-1-5-2', 'cond-mat-0703377-2-5-2'], ['cond-mat-0703377-1-5-3', 'cond-mat-0703377-2-5-3'], ['cond-mat-0703377-1-5-4', 'cond-mat-0703377-2-5-4'], ['cond-mat-0703377-1-5-5', 'cond-mat-0703377-2-5-5'], ['cond-mat-0703377-1-5-6', 'cond-mat-0703377-2-5-6'], ['cond-mat-0703377-1-5-7', 'cond-mat-0703377-2-5-7'], ['cond-mat-0703377-1-46-0', 'cond-mat-0703377-2-46-0'], ['cond-mat-0703377-1-46-1', 'cond-mat-0703377-2-46-1'], ['cond-mat-0703377-1-46-2', 'cond-mat-0703377-2-46-2'], ['cond-mat-0703377-1-46-3', 'cond-mat-0703377-2-46-3'], ['cond-mat-0703377-1-46-4', 'cond-mat-0703377-2-46-4'], ['cond-mat-0703377-1-46-5', 'cond-mat-0703377-2-46-5'], ['cond-mat-0703377-1-46-6', 'cond-mat-0703377-2-46-6'], ['cond-mat-0703377-1-46-7', 'cond-mat-0703377-2-46-7'], ['cond-mat-0703377-1-7-0', 'cond-mat-0703377-2-7-0'], ['cond-mat-0703377-1-7-1', 'cond-mat-0703377-2-7-1'], ['cond-mat-0703377-1-7-2', 'cond-mat-0703377-2-7-2'], ['cond-mat-0703377-1-7-3', 'cond-mat-0703377-2-7-3'], ['cond-mat-0703377-1-7-4', 'cond-mat-0703377-2-7-4'], ['cond-mat-0703377-1-7-5', 'cond-mat-0703377-2-7-5'], ['cond-mat-0703377-1-7-6', 'cond-mat-0703377-2-7-6'], ['cond-mat-0703377-1-36-0', 'cond-mat-0703377-2-36-0'], ['cond-mat-0703377-1-36-1', 'cond-mat-0703377-2-36-1'], ['cond-mat-0703377-1-36-2', 'cond-mat-0703377-2-36-2'], ['cond-mat-0703377-1-36-3', 'cond-mat-0703377-2-36-3'], ['cond-mat-0703377-1-36-4', 'cond-mat-0703377-2-36-4'], ['cond-mat-0703377-1-36-5', 'cond-mat-0703377-2-36-5'], ['cond-mat-0703377-1-36-6', 'cond-mat-0703377-2-36-6'], ['cond-mat-0703377-1-8-0', 'cond-mat-0703377-2-8-0'], ['cond-mat-0703377-1-8-1', 'cond-mat-0703377-2-8-1'], ['cond-mat-0703377-1-8-2', 'cond-mat-0703377-2-8-2'], ['cond-mat-0703377-1-8-3', 'cond-mat-0703377-2-8-3'], ['cond-mat-0703377-1-8-4', 'cond-mat-0703377-2-8-4'], ['cond-mat-0703377-1-8-5', 'cond-mat-0703377-2-8-5'], ['cond-mat-0703377-1-8-6', 'cond-mat-0703377-2-8-6'], ['cond-mat-0703377-1-8-7', 'cond-mat-0703377-2-8-7'], ['cond-mat-0703377-1-8-8', 'cond-mat-0703377-2-8-8'], ['cond-mat-0703377-1-8-9', 'cond-mat-0703377-2-8-9'], ['cond-mat-0703377-1-32-0', 'cond-mat-0703377-2-32-0'], ['cond-mat-0703377-1-32-1', 'cond-mat-0703377-2-32-1'], ['cond-mat-0703377-1-32-2', 'cond-mat-0703377-2-32-2'], ['cond-mat-0703377-1-32-3', 'cond-mat-0703377-2-32-3'], ['cond-mat-0703377-1-32-4', 'cond-mat-0703377-2-32-4'], ['cond-mat-0703377-1-32-5', 'cond-mat-0703377-2-32-5'], ['cond-mat-0703377-1-32-6', 'cond-mat-0703377-2-32-6'], ['cond-mat-0703377-1-32-7', 'cond-mat-0703377-2-32-7'], ['cond-mat-0703377-1-32-8', 'cond-mat-0703377-2-32-8'], ['cond-mat-0703377-1-32-9', 'cond-mat-0703377-2-32-9'], ['cond-mat-0703377-1-32-10', 'cond-mat-0703377-2-32-10'], ['cond-mat-0703377-1-32-11', 'cond-mat-0703377-2-32-11'], ['cond-mat-0703377-1-32-12', 'cond-mat-0703377-2-32-12'], ['cond-mat-0703377-1-32-13', 'cond-mat-0703377-2-32-13'], ['cond-mat-0703377-1-32-14', 'cond-mat-0703377-2-32-14'], ['cond-mat-0703377-1-0-0', 'cond-mat-0703377-2-0-0'], ['cond-mat-0703377-1-0-1', 'cond-mat-0703377-2-0-1'], ['cond-mat-0703377-1-0-2', 'cond-mat-0703377-2-0-2'], ['cond-mat-0703377-1-0-3', 'cond-mat-0703377-2-0-3'], ['cond-mat-0703377-1-0-4', 'cond-mat-0703377-2-0-4'], ['cond-mat-0703377-1-0-5', 'cond-mat-0703377-2-0-5'], ['cond-mat-0703377-1-0-6', 'cond-mat-0703377-2-0-6'], ['cond-mat-0703377-1-43-0', 'cond-mat-0703377-2-43-0'], ['cond-mat-0703377-1-43-1', 'cond-mat-0703377-2-43-1'], ['cond-mat-0703377-1-4-0', 'cond-mat-0703377-2-4-0'], ['cond-mat-0703377-1-4-1', 'cond-mat-0703377-2-4-1'], ['cond-mat-0703377-1-4-2', 'cond-mat-0703377-2-4-2'], ['cond-mat-0703377-1-4-3', 'cond-mat-0703377-2-4-3'], ['cond-mat-0703377-1-4-4', 'cond-mat-0703377-2-4-4'], ['cond-mat-0703377-1-44-0', 'cond-mat-0703377-2-44-0'], ['cond-mat-0703377-1-44-1', 'cond-mat-0703377-2-44-1'], ['cond-mat-0703377-1-44-2', 'cond-mat-0703377-2-44-2'], ['cond-mat-0703377-1-44-3', 'cond-mat-0703377-2-44-3'], ['cond-mat-0703377-1-44-4', 'cond-mat-0703377-2-44-4'], ['cond-mat-0703377-1-44-5', 'cond-mat-0703377-2-44-5'], ['cond-mat-0703377-1-44-6', 'cond-mat-0703377-2-44-6'], ['cond-mat-0703377-1-45-0', 'cond-mat-0703377-2-45-0'], ['cond-mat-0703377-1-45-1', 'cond-mat-0703377-2-45-1'], ['cond-mat-0703377-1-45-2', 'cond-mat-0703377-2-45-2'], ['cond-mat-0703377-1-45-3', 'cond-mat-0703377-2-45-3'], ['cond-mat-0703377-1-45-4', 'cond-mat-0703377-2-45-4'], ['cond-mat-0703377-1-45-5', 'cond-mat-0703377-2-45-5'], ['cond-mat-0703377-1-17-0', 'cond-mat-0703377-2-17-0'], ['cond-mat-0703377-1-17-1', 'cond-mat-0703377-2-17-1'], ['cond-mat-0703377-1-17-2', 'cond-mat-0703377-2-17-2'], ['cond-mat-0703377-1-17-3', 'cond-mat-0703377-2-17-3'], ['cond-mat-0703377-1-21-0', 'cond-mat-0703377-2-21-0'], ['cond-mat-0703377-1-21-1', 'cond-mat-0703377-2-21-1'], ['cond-mat-0703377-1-21-2', 'cond-mat-0703377-2-21-2'], ['cond-mat-0703377-1-21-3', 'cond-mat-0703377-2-21-3'], ['cond-mat-0703377-1-21-4', 'cond-mat-0703377-2-21-4'], ['cond-mat-0703377-1-21-5', 'cond-mat-0703377-2-21-5'], ['cond-mat-0703377-1-21-6', 'cond-mat-0703377-2-21-6'], ['cond-mat-0703377-1-21-7', 'cond-mat-0703377-2-21-7'], ['cond-mat-0703377-1-21-8', 'cond-mat-0703377-2-21-8'], ['cond-mat-0703377-1-21-9', 'cond-mat-0703377-2-21-9'], ['cond-mat-0703377-1-21-10', 'cond-mat-0703377-2-21-10'], ['cond-mat-0703377-1-21-11', 'cond-mat-0703377-2-21-11'], ['cond-mat-0703377-1-38-0', 'cond-mat-0703377-2-38-0'], ['cond-mat-0703377-1-38-1', 'cond-mat-0703377-2-38-1'], ['cond-mat-0703377-1-38-2', 'cond-mat-0703377-2-38-2'], ['cond-mat-0703377-1-38-3', 'cond-mat-0703377-2-38-3'], ['cond-mat-0703377-1-38-4', 'cond-mat-0703377-2-38-4'], ['cond-mat-0703377-1-38-5', 'cond-mat-0703377-2-38-5'], ['cond-mat-0703377-1-38-6', 'cond-mat-0703377-2-38-6'], ['cond-mat-0703377-1-38-7', 'cond-mat-0703377-2-38-7'], ['cond-mat-0703377-1-38-8', 'cond-mat-0703377-2-38-8'], ['cond-mat-0703377-1-38-9', 'cond-mat-0703377-2-38-9'], ['cond-mat-0703377-1-38-10', 'cond-mat-0703377-2-38-10'], ['cond-mat-0703377-1-38-11', 'cond-mat-0703377-2-38-11'], ['cond-mat-0703377-1-38-12', 'cond-mat-0703377-2-38-12'], ['cond-mat-0703377-1-47-0', 'cond-mat-0703377-2-47-0'], ['cond-mat-0703377-1-47-1', 'cond-mat-0703377-2-47-1'], ['cond-mat-0703377-1-47-2', 'cond-mat-0703377-2-47-2'], ['cond-mat-0703377-1-47-3', 'cond-mat-0703377-2-47-3'], ['cond-mat-0703377-1-47-4', 'cond-mat-0703377-2-47-4'], ['cond-mat-0703377-1-47-5', 'cond-mat-0703377-2-47-5'], ['cond-mat-0703377-1-47-6', 'cond-mat-0703377-2-47-6'], ['cond-mat-0703377-1-47-7', 'cond-mat-0703377-2-47-7'], ['cond-mat-0703377-1-47-8', 'cond-mat-0703377-2-47-8'], ['cond-mat-0703377-1-47-9', 'cond-mat-0703377-2-47-9'], ['cond-mat-0703377-1-47-10', 'cond-mat-0703377-2-47-10'], ['cond-mat-0703377-1-47-11', 'cond-mat-0703377-2-47-11'], ['cond-mat-0703377-1-47-12', 'cond-mat-0703377-2-47-12'], ['cond-mat-0703377-1-47-13', 'cond-mat-0703377-2-47-13'], ['cond-mat-0703377-1-47-14', 'cond-mat-0703377-2-47-14'], ['cond-mat-0703377-1-47-15', 'cond-mat-0703377-2-47-15'], ['cond-mat-0703377-1-47-16', 'cond-mat-0703377-2-47-16'], ['cond-mat-0703377-1-47-17', 'cond-mat-0703377-2-47-17'], ['cond-mat-0703377-1-47-18', 'cond-mat-0703377-2-47-18'], ['cond-mat-0703377-1-19-0', 'cond-mat-0703377-2-19-0'], ['cond-mat-0703377-1-19-1', 'cond-mat-0703377-2-19-1'], ['cond-mat-0703377-1-19-2', 'cond-mat-0703377-2-19-2'], ['cond-mat-0703377-1-19-3', 'cond-mat-0703377-2-19-3'], ['cond-mat-0703377-1-15-0', 'cond-mat-0703377-2-15-0'], ['cond-mat-0703377-1-15-1', 'cond-mat-0703377-2-15-1'], ['cond-mat-0703377-1-15-2', 'cond-mat-0703377-2-15-2'], ['cond-mat-0703377-1-15-3', 'cond-mat-0703377-2-15-3'], ['cond-mat-0703377-1-15-4', 'cond-mat-0703377-2-15-4'], ['cond-mat-0703377-1-15-5', 'cond-mat-0703377-2-15-5'], ['cond-mat-0703377-1-33-0', 'cond-mat-0703377-2-33-0'], ['cond-mat-0703377-1-33-1', 'cond-mat-0703377-2-33-1'], ['cond-mat-0703377-1-33-2', 'cond-mat-0703377-2-33-2'], ['cond-mat-0703377-1-33-3', 'cond-mat-0703377-2-33-3'], ['cond-mat-0703377-1-33-4', 'cond-mat-0703377-2-33-4'], ['cond-mat-0703377-1-33-5', 'cond-mat-0703377-2-33-5'], ['cond-mat-0703377-1-33-6', 'cond-mat-0703377-2-33-6'], ['cond-mat-0703377-1-33-7', 'cond-mat-0703377-2-33-7'], ['cond-mat-0703377-1-48-0', 'cond-mat-0703377-2-48-0'], ['cond-mat-0703377-1-48-1', 'cond-mat-0703377-2-48-1'], ['cond-mat-0703377-1-48-2', 'cond-mat-0703377-2-48-2'], ['cond-mat-0703377-1-34-0', 'cond-mat-0703377-2-34-0'], ['cond-mat-0703377-1-34-1', 'cond-mat-0703377-2-34-1'], ['cond-mat-0703377-1-34-2', 'cond-mat-0703377-2-34-2'], ['cond-mat-0703377-1-34-3', 'cond-mat-0703377-2-34-3'], ['cond-mat-0703377-1-34-4', 'cond-mat-0703377-2-34-4'], ['cond-mat-0703377-1-34-5', 'cond-mat-0703377-2-34-5'], ['cond-mat-0703377-1-13-0', 'cond-mat-0703377-2-13-0'], ['cond-mat-0703377-1-13-1', 'cond-mat-0703377-2-13-1'], ['cond-mat-0703377-1-13-2', 'cond-mat-0703377-2-13-2'], ['cond-mat-0703377-1-13-3', 'cond-mat-0703377-2-13-3'], ['cond-mat-0703377-1-13-4', 'cond-mat-0703377-2-13-4'], ['cond-mat-0703377-1-13-5', 'cond-mat-0703377-2-13-5'], ['cond-mat-0703377-1-13-6', 'cond-mat-0703377-2-13-6'], ['cond-mat-0703377-1-9-0', 'cond-mat-0703377-2-9-0'], ['cond-mat-0703377-1-9-1', 'cond-mat-0703377-2-9-1'], ['cond-mat-0703377-1-9-2', 'cond-mat-0703377-2-9-2'], ['cond-mat-0703377-1-9-3', 'cond-mat-0703377-2-9-3'], ['cond-mat-0703377-1-11-0', 'cond-mat-0703377-2-11-0'], ['cond-mat-0703377-1-11-1', 'cond-mat-0703377-2-11-1'], ['cond-mat-0703377-1-11-2', 'cond-mat-0703377-2-11-2'], ['cond-mat-0703377-1-24-0', 'cond-mat-0703377-2-24-0'], ['cond-mat-0703377-1-24-1', 'cond-mat-0703377-2-24-1'], ['cond-mat-0703377-1-24-2', 'cond-mat-0703377-2-24-2'], ['cond-mat-0703377-1-24-3', 'cond-mat-0703377-2-24-3'], ['cond-mat-0703377-1-28-0', 'cond-mat-0703377-2-28-0'], ['cond-mat-0703377-1-28-1', 'cond-mat-0703377-2-28-1'], ['cond-mat-0703377-1-28-2', 'cond-mat-0703377-2-28-2'], ['cond-mat-0703377-1-28-3', 'cond-mat-0703377-2-28-3'], ['cond-mat-0703377-1-28-4', 'cond-mat-0703377-2-28-4'], ['cond-mat-0703377-1-28-5', 'cond-mat-0703377-2-28-5'], ['cond-mat-0703377-1-28-6', 'cond-mat-0703377-2-28-6'], ['cond-mat-0703377-1-28-7', 'cond-mat-0703377-2-28-7'], ['cond-mat-0703377-1-28-8', 'cond-mat-0703377-2-28-8'], ['cond-mat-0703377-1-40-0', 'cond-mat-0703377-2-40-0'], ['cond-mat-0703377-1-40-1', 'cond-mat-0703377-2-40-1'], ['cond-mat-0703377-1-23-0', 'cond-mat-0703377-2-23-0'], ['cond-mat-0703377-1-23-1', 'cond-mat-0703377-2-23-1'], ['cond-mat-0703377-1-23-2', 'cond-mat-0703377-2-23-2'], ['cond-mat-0703377-1-23-3', 'cond-mat-0703377-2-23-3']]

[['cond-mat-0703377-1-3-0', 'cond-mat-0703377-2-3-0'], ['cond-mat-0703377-1-3-1', 'cond-mat-0703377-2-3-1'], ['cond-mat-0703377-1-3-2', 'cond-mat-0703377-2-3-2'], ['cond-mat-0703377-1-3-3', 'cond-mat-0703377-2-3-3'], ['cond-mat-0703377-1-3-4', 'cond-mat-0703377-2-3-4'], ['cond-mat-0703377-1-3-5', 'cond-mat-0703377-2-3-5'], ['cond-mat-0703377-1-3-6', 'cond-mat-0703377-2-3-6'], ['cond-mat-0703377-1-27-0', 'cond-mat-0703377-2-27-0'], ['cond-mat-0703377-1-27-1', 'cond-mat-0703377-2-27-1'], ['cond-mat-0703377-1-27-2', 'cond-mat-0703377-2-27-2'], ['cond-mat-0703377-1-27-3', 'cond-mat-0703377-2-27-3'], ['cond-mat-0703377-1-25-0', 'cond-mat-0703377-2-25-0'], ['cond-mat-0703377-1-25-1', 'cond-mat-0703377-2-25-1'], ['cond-mat-0703377-1-25-2', 'cond-mat-0703377-2-25-2'], ['cond-mat-0703377-1-30-0', 'cond-mat-0703377-2-30-0'], ['cond-mat-0703377-1-30-1', 'cond-mat-0703377-2-30-1'], ['cond-mat-0703377-1-30-2', 'cond-mat-0703377-2-30-2'], ['cond-mat-0703377-1-30-3', 'cond-mat-0703377-2-30-3'], ['cond-mat-0703377-1-30-4', 'cond-mat-0703377-2-30-4'], ['cond-mat-0703377-1-30-5', 'cond-mat-0703377-2-30-5'], ['cond-mat-0703377-1-30-6', 'cond-mat-0703377-2-30-6'], ['cond-mat-0703377-1-30-7', 'cond-mat-0703377-2-30-7'], ['cond-mat-0703377-1-30-8', 'cond-mat-0703377-2-30-8'], ['cond-mat-0703377-1-30-9', 'cond-mat-0703377-2-30-9'], ['cond-mat-0703377-1-30-10', 'cond-mat-0703377-2-30-10'], ['cond-mat-0703377-1-30-11', 'cond-mat-0703377-2-30-11'], ['cond-mat-0703377-1-30-12', 'cond-mat-0703377-2-30-12'], ['cond-mat-0703377-1-30-13', 'cond-mat-0703377-2-30-13'], ['cond-mat-0703377-1-30-14', 'cond-mat-0703377-2-30-14'], ['cond-mat-0703377-1-29-0', 'cond-mat-0703377-2-29-0'], ['cond-mat-0703377-1-29-1', 'cond-mat-0703377-2-29-1'], ['cond-mat-0703377-1-29-2', 'cond-mat-0703377-2-29-2'], ['cond-mat-0703377-1-29-3', 'cond-mat-0703377-2-29-3'], ['cond-mat-0703377-1-29-4', 'cond-mat-0703377-2-29-4'], ['cond-mat-0703377-1-5-0', 'cond-mat-0703377-2-5-0'], ['cond-mat-0703377-1-5-1', 'cond-mat-0703377-2-5-1'], ['cond-mat-0703377-1-5-2', 'cond-mat-0703377-2-5-2'], ['cond-mat-0703377-1-5-3', 'cond-mat-0703377-2-5-3'], ['cond-mat-0703377-1-5-4', 'cond-mat-0703377-2-5-4'], ['cond-mat-0703377-1-5-5', 'cond-mat-0703377-2-5-5'], ['cond-mat-0703377-1-5-6', 'cond-mat-0703377-2-5-6'], ['cond-mat-0703377-1-5-7', 'cond-mat-0703377-2-5-7'], ['cond-mat-0703377-1-46-0', 'cond-mat-0703377-2-46-0'], ['cond-mat-0703377-1-46-1', 'cond-mat-0703377-2-46-1'], ['cond-mat-0703377-1-46-2', 'cond-mat-0703377-2-46-2'], ['cond-mat-0703377-1-46-3', 'cond-mat-0703377-2-46-3'], ['cond-mat-0703377-1-46-4', 'cond-mat-0703377-2-46-4'], ['cond-mat-0703377-1-46-5', 'cond-mat-0703377-2-46-5'], ['cond-mat-0703377-1-46-6', 'cond-mat-0703377-2-46-6'], ['cond-mat-0703377-1-46-7', 'cond-mat-0703377-2-46-7'], ['cond-mat-0703377-1-7-0', 'cond-mat-0703377-2-7-0'], ['cond-mat-0703377-1-7-1', 'cond-mat-0703377-2-7-1'], ['cond-mat-0703377-1-7-2', 'cond-mat-0703377-2-7-2'], ['cond-mat-0703377-1-7-3', 'cond-mat-0703377-2-7-3'], ['cond-mat-0703377-1-7-4', 'cond-mat-0703377-2-7-4'], ['cond-mat-0703377-1-7-5', 'cond-mat-0703377-2-7-5'], ['cond-mat-0703377-1-7-6', 'cond-mat-0703377-2-7-6'], ['cond-mat-0703377-1-36-0', 'cond-mat-0703377-2-36-0'], ['cond-mat-0703377-1-36-1', 'cond-mat-0703377-2-36-1'], ['cond-mat-0703377-1-36-2', 'cond-mat-0703377-2-36-2'], ['cond-mat-0703377-1-36-3', 'cond-mat-0703377-2-36-3'], ['cond-mat-0703377-1-36-4', 'cond-mat-0703377-2-36-4'], ['cond-mat-0703377-1-36-5', 'cond-mat-0703377-2-36-5'], ['cond-mat-0703377-1-36-6', 'cond-mat-0703377-2-36-6'], ['cond-mat-0703377-1-8-0', 'cond-mat-0703377-2-8-0'], ['cond-mat-0703377-1-8-1', 'cond-mat-0703377-2-8-1'], ['cond-mat-0703377-1-8-2', 'cond-mat-0703377-2-8-2'], ['cond-mat-0703377-1-8-3', 'cond-mat-0703377-2-8-3'], ['cond-mat-0703377-1-8-4', 'cond-mat-0703377-2-8-4'], ['cond-mat-0703377-1-8-5', 'cond-mat-0703377-2-8-5'], ['cond-mat-0703377-1-8-6', 'cond-mat-0703377-2-8-6'], ['cond-mat-0703377-1-8-7', 'cond-mat-0703377-2-8-7'], ['cond-mat-0703377-1-8-8', 'cond-mat-0703377-2-8-8'], ['cond-mat-0703377-1-8-9', 'cond-mat-0703377-2-8-9'], ['cond-mat-0703377-1-32-0', 'cond-mat-0703377-2-32-0'], ['cond-mat-0703377-1-32-1', 'cond-mat-0703377-2-32-1'], ['cond-mat-0703377-1-32-2', 'cond-mat-0703377-2-32-2'], ['cond-mat-0703377-1-32-3', 'cond-mat-0703377-2-32-3'], ['cond-mat-0703377-1-32-4', 'cond-mat-0703377-2-32-4'], ['cond-mat-0703377-1-32-5', 'cond-mat-0703377-2-32-5'], ['cond-mat-0703377-1-32-6', 'cond-mat-0703377-2-32-6'], ['cond-mat-0703377-1-32-7', 'cond-mat-0703377-2-32-7'], ['cond-mat-0703377-1-32-8', 'cond-mat-0703377-2-32-8'], ['cond-mat-0703377-1-32-9', 'cond-mat-0703377-2-32-9'], ['cond-mat-0703377-1-32-10', 'cond-mat-0703377-2-32-10'], ['cond-mat-0703377-1-32-11', 'cond-mat-0703377-2-32-11'], ['cond-mat-0703377-1-32-12', 'cond-mat-0703377-2-32-12'], ['cond-mat-0703377-1-32-13', 'cond-mat-0703377-2-32-13'], ['cond-mat-0703377-1-32-14', 'cond-mat-0703377-2-32-14'], ['cond-mat-0703377-1-0-0', 'cond-mat-0703377-2-0-0'], ['cond-mat-0703377-1-0-1', 'cond-mat-0703377-2-0-1'], ['cond-mat-0703377-1-0-2', 'cond-mat-0703377-2-0-2'], ['cond-mat-0703377-1-0-3', 'cond-mat-0703377-2-0-3'], ['cond-mat-0703377-1-0-4', 'cond-mat-0703377-2-0-4'], ['cond-mat-0703377-1-0-5', 'cond-mat-0703377-2-0-5'], ['cond-mat-0703377-1-0-6', 'cond-mat-0703377-2-0-6'], ['cond-mat-0703377-1-43-0', 'cond-mat-0703377-2-43-0'], ['cond-mat-0703377-1-43-1', 'cond-mat-0703377-2-43-1'], ['cond-mat-0703377-1-4-0', 'cond-mat-0703377-2-4-0'], ['cond-mat-0703377-1-4-1', 'cond-mat-0703377-2-4-1'], ['cond-mat-0703377-1-4-2', 'cond-mat-0703377-2-4-2'], ['cond-mat-0703377-1-4-3', 'cond-mat-0703377-2-4-3'], ['cond-mat-0703377-1-4-4', 'cond-mat-0703377-2-4-4'], ['cond-mat-0703377-1-44-0', 'cond-mat-0703377-2-44-0'], ['cond-mat-0703377-1-44-1', 'cond-mat-0703377-2-44-1'], ['cond-mat-0703377-1-44-2', 'cond-mat-0703377-2-44-2'], ['cond-mat-0703377-1-44-3', 'cond-mat-0703377-2-44-3'], ['cond-mat-0703377-1-44-4', 'cond-mat-0703377-2-44-4'], ['cond-mat-0703377-1-44-5', 'cond-mat-0703377-2-44-5'], ['cond-mat-0703377-1-44-6', 'cond-mat-0703377-2-44-6'], ['cond-mat-0703377-1-45-0', 'cond-mat-0703377-2-45-0'], ['cond-mat-0703377-1-45-1', 'cond-mat-0703377-2-45-1'], ['cond-mat-0703377-1-45-2', 'cond-mat-0703377-2-45-2'], ['cond-mat-0703377-1-45-3', 'cond-mat-0703377-2-45-3'], ['cond-mat-0703377-1-45-4', 'cond-mat-0703377-2-45-4'], ['cond-mat-0703377-1-45-5', 'cond-mat-0703377-2-45-5'], ['cond-mat-0703377-1-17-0', 'cond-mat-0703377-2-17-0'], ['cond-mat-0703377-1-17-1', 'cond-mat-0703377-2-17-1'], ['cond-mat-0703377-1-17-2', 'cond-mat-0703377-2-17-2'], ['cond-mat-0703377-1-17-3', 'cond-mat-0703377-2-17-3'], ['cond-mat-0703377-1-21-0', 'cond-mat-0703377-2-21-0'], ['cond-mat-0703377-1-21-1', 'cond-mat-0703377-2-21-1'], ['cond-mat-0703377-1-21-2', 'cond-mat-0703377-2-21-2'], ['cond-mat-0703377-1-21-3', 'cond-mat-0703377-2-21-3'], ['cond-mat-0703377-1-21-4', 'cond-mat-0703377-2-21-4'], ['cond-mat-0703377-1-21-5', 'cond-mat-0703377-2-21-5'], ['cond-mat-0703377-1-21-6', 'cond-mat-0703377-2-21-6'], ['cond-mat-0703377-1-21-7', 'cond-mat-0703377-2-21-7'], ['cond-mat-0703377-1-21-8', 'cond-mat-0703377-2-21-8'], ['cond-mat-0703377-1-21-9', 'cond-mat-0703377-2-21-9'], ['cond-mat-0703377-1-21-10', 'cond-mat-0703377-2-21-10'], ['cond-mat-0703377-1-21-11', 'cond-mat-0703377-2-21-11'], ['cond-mat-0703377-1-38-0', 'cond-mat-0703377-2-38-0'], ['cond-mat-0703377-1-38-1', 'cond-mat-0703377-2-38-1'], ['cond-mat-0703377-1-38-2', 'cond-mat-0703377-2-38-2'], ['cond-mat-0703377-1-38-3', 'cond-mat-0703377-2-38-3'], ['cond-mat-0703377-1-38-4', 'cond-mat-0703377-2-38-4'], ['cond-mat-0703377-1-38-5', 'cond-mat-0703377-2-38-5'], ['cond-mat-0703377-1-38-6', 'cond-mat-0703377-2-38-6'], ['cond-mat-0703377-1-38-7', 'cond-mat-0703377-2-38-7'], ['cond-mat-0703377-1-38-8', 'cond-mat-0703377-2-38-8'], ['cond-mat-0703377-1-38-9', 'cond-mat-0703377-2-38-9'], ['cond-mat-0703377-1-38-10', 'cond-mat-0703377-2-38-10'], ['cond-mat-0703377-1-38-11', 'cond-mat-0703377-2-38-11'], ['cond-mat-0703377-1-38-12', 'cond-mat-0703377-2-38-12'], ['cond-mat-0703377-1-47-0', 'cond-mat-0703377-2-47-0'], ['cond-mat-0703377-1-47-1', 'cond-mat-0703377-2-47-1'], ['cond-mat-0703377-1-47-2', 'cond-mat-0703377-2-47-2'], ['cond-mat-0703377-1-47-3', 'cond-mat-0703377-2-47-3'], ['cond-mat-0703377-1-47-4', 'cond-mat-0703377-2-47-4'], ['cond-mat-0703377-1-47-5', 'cond-mat-0703377-2-47-5'], ['cond-mat-0703377-1-47-6', 'cond-mat-0703377-2-47-6'], ['cond-mat-0703377-1-47-7', 'cond-mat-0703377-2-47-7'], ['cond-mat-0703377-1-47-8', 'cond-mat-0703377-2-47-8'], ['cond-mat-0703377-1-47-9', 'cond-mat-0703377-2-47-9'], ['cond-mat-0703377-1-47-10', 'cond-mat-0703377-2-47-10'], ['cond-mat-0703377-1-47-11', 'cond-mat-0703377-2-47-11'], ['cond-mat-0703377-1-47-12', 'cond-mat-0703377-2-47-12'], ['cond-mat-0703377-1-47-13', 'cond-mat-0703377-2-47-13'], ['cond-mat-0703377-1-47-14', 'cond-mat-0703377-2-47-14'], ['cond-mat-0703377-1-47-15', 'cond-mat-0703377-2-47-15'], ['cond-mat-0703377-1-47-16', 'cond-mat-0703377-2-47-16'], ['cond-mat-0703377-1-47-17', 'cond-mat-0703377-2-47-17'], ['cond-mat-0703377-1-47-18', 'cond-mat-0703377-2-47-18'], ['cond-mat-0703377-1-19-0', 'cond-mat-0703377-2-19-0'], ['cond-mat-0703377-1-19-1', 'cond-mat-0703377-2-19-1'], ['cond-mat-0703377-1-19-2', 'cond-mat-0703377-2-19-2'], ['cond-mat-0703377-1-19-3', 'cond-mat-0703377-2-19-3'], ['cond-mat-0703377-1-15-0', 'cond-mat-0703377-2-15-0'], ['cond-mat-0703377-1-15-1', 'cond-mat-0703377-2-15-1'], ['cond-mat-0703377-1-15-2', 'cond-mat-0703377-2-15-2'], ['cond-mat-0703377-1-15-3', 'cond-mat-0703377-2-15-3'], ['cond-mat-0703377-1-15-4', 'cond-mat-0703377-2-15-4'], ['cond-mat-0703377-1-15-5', 'cond-mat-0703377-2-15-5'], ['cond-mat-0703377-1-33-0', 'cond-mat-0703377-2-33-0'], ['cond-mat-0703377-1-33-1', 'cond-mat-0703377-2-33-1'], ['cond-mat-0703377-1-33-2', 'cond-mat-0703377-2-33-2'], ['cond-mat-0703377-1-33-3', 'cond-mat-0703377-2-33-3'], ['cond-mat-0703377-1-33-4', 'cond-mat-0703377-2-33-4'], ['cond-mat-0703377-1-33-5', 'cond-mat-0703377-2-33-5'], ['cond-mat-0703377-1-33-6', 'cond-mat-0703377-2-33-6'], ['cond-mat-0703377-1-33-7', 'cond-mat-0703377-2-33-7'], ['cond-mat-0703377-1-48-0', 'cond-mat-0703377-2-48-0'], ['cond-mat-0703377-1-48-1', 'cond-mat-0703377-2-48-1'], ['cond-mat-0703377-1-48-2', 'cond-mat-0703377-2-48-2'], ['cond-mat-0703377-1-34-0', 'cond-mat-0703377-2-34-0'], ['cond-mat-0703377-1-34-1', 'cond-mat-0703377-2-34-1'], ['cond-mat-0703377-1-34-2', 'cond-mat-0703377-2-34-2'], ['cond-mat-0703377-1-34-3', 'cond-mat-0703377-2-34-3'], ['cond-mat-0703377-1-34-4', 'cond-mat-0703377-2-34-4'], ['cond-mat-0703377-1-34-5', 'cond-mat-0703377-2-34-5'], ['cond-mat-0703377-1-13-0', 'cond-mat-0703377-2-13-0'], ['cond-mat-0703377-1-13-1', 'cond-mat-0703377-2-13-1'], ['cond-mat-0703377-1-13-2', 'cond-mat-0703377-2-13-2'], ['cond-mat-0703377-1-13-3', 'cond-mat-0703377-2-13-3'], ['cond-mat-0703377-1-13-4', 'cond-mat-0703377-2-13-4'], ['cond-mat-0703377-1-13-5', 'cond-mat-0703377-2-13-5'], ['cond-mat-0703377-1-13-6', 'cond-mat-0703377-2-13-6'], ['cond-mat-0703377-1-9-0', 'cond-mat-0703377-2-9-0'], ['cond-mat-0703377-1-9-1', 'cond-mat-0703377-2-9-1'], ['cond-mat-0703377-1-9-2', 'cond-mat-0703377-2-9-2'], ['cond-mat-0703377-1-9-3', 'cond-mat-0703377-2-9-3'], ['cond-mat-0703377-1-11-0', 'cond-mat-0703377-2-11-0'], ['cond-mat-0703377-1-11-1', 'cond-mat-0703377-2-11-1'], ['cond-mat-0703377-1-11-2', 'cond-mat-0703377-2-11-2'], ['cond-mat-0703377-1-24-0', 'cond-mat-0703377-2-24-0'], ['cond-mat-0703377-1-24-1', 'cond-mat-0703377-2-24-1'], ['cond-mat-0703377-1-24-2', 'cond-mat-0703377-2-24-2'], ['cond-mat-0703377-1-24-3', 'cond-mat-0703377-2-24-3'], ['cond-mat-0703377-1-28-0', 'cond-mat-0703377-2-28-0'], ['cond-mat-0703377-1-28-1', 'cond-mat-0703377-2-28-1'], ['cond-mat-0703377-1-28-2', 'cond-mat-0703377-2-28-2'], ['cond-mat-0703377-1-28-3', 'cond-mat-0703377-2-28-3'], ['cond-mat-0703377-1-28-4', 'cond-mat-0703377-2-28-4'], ['cond-mat-0703377-1-28-5', 'cond-mat-0703377-2-28-5'], ['cond-mat-0703377-1-28-6', 'cond-mat-0703377-2-28-6'], ['cond-mat-0703377-1-28-7', 'cond-mat-0703377-2-28-7'], ['cond-mat-0703377-1-28-8', 'cond-mat-0703377-2-28-8'], ['cond-mat-0703377-1-40-0', 'cond-mat-0703377-2-40-0'], ['cond-mat-0703377-1-40-1', 'cond-mat-0703377-2-40-1'], ['cond-mat-0703377-1-23-0', 'cond-mat-0703377-2-23-0'], ['cond-mat-0703377-1-23-1', 'cond-mat-0703377-2-23-1'], ['cond-mat-0703377-1-23-2', 'cond-mat-0703377-2-23-2'], ['cond-mat-0703377-1-23-3', 'cond-mat-0703377-2-23-3']]

[]

[]

[]

[]

['cond-mat-0703377-1-1-0', 'cond-mat-0703377-1-40-2', 'cond-mat-0703377-1-41-0', 'cond-mat-0703377-1-43-2', 'cond-mat-0703377-2-1-0', 'cond-mat-0703377-2-40-2', 'cond-mat-0703377-2-41-0', 'cond-mat-0703377-2-43-2']

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

https://arxiv.org/abs/cond-mat/0703377

1703.08520

{'1703.08520-1-0-0': 'Bayesian inference for complex models is challenging due to the need to explore high-dimensional spaces and multimodality and standard Monte Carlo samplers can have difficulties effectively exploring the posterior.', '1703.08520-1-0-1': 'We introduce a general purpose rejection-free ensemble Markov Chain Monte Carlo (MCMC) technique to improve on existing poorly mixing samplers.', '1703.08520-1-0-2': 'This is achieved by combining parallel tempering and an auxiliary variable move to exchange information between the chains.', '1703.08520-1-0-3': 'We demonstrate this ensemble MCMC scheme on Bayesian inference in Factorial Hidden Markov Models.', '1703.08520-1-0-4': 'This high-dimensional inference problem is difficult due to the exponentially sized latent variable space.', '1703.08520-1-0-5': 'Existing sampling approaches mix slowly and can get trapped in local modes.', '1703.08520-1-0-6': 'We show that the performance of these samplers is improved by our rejection-free ensemble technique and that the method is attractive and "easy-to-use" since no parameter tuning is required.', '1703.08520-1-1-0': '# Introduction', '1703.08520-1-2-0': 'Monte Carlo-based Bayesian inference [CITATION] for complex high-dimensional posteriors is a challenging problem as efficient Monte Carlo samplers need to be able to move efficiently across potentially irregular landscapes that may contain many local modes [CITATION].', '1703.08520-1-3-0': 'In practice, commonly used sampling approaches can explore the space very slowly or get stuck in local modes.', '1703.08520-1-3-1': 'For example, Gibbs sampling can have trouble moving around the high-dimensional space, due to the conditional updating scheme where each coordinate is sampled when keeping the rest fixed.', '1703.08520-1-3-2': 'In cases where moving between local modes requires a joint update of multiple coordinates, conditional sampling scheme fails to explore the whole space.', '1703.08520-1-3-3': 'In a more general Metropolis-Hastings framework, it is difficult to construct efficient proposals for high-dimensional distributions, so the sampling scheme can be inefficient due to the accept-reject process.', '1703.08520-1-4-0': 'We propose a novel ensemble Markov Chain Monte Carlo (MCMC) approach to improve the mixing properties and achieve more effective samplers by using an ensemble of potentially tempered chains, and constructing a rejection-free move to exchange information between them.', '1703.08520-1-4-1': 'We demonstrate the benefit of our approach by applying it to challenging inference problems associated with the exponentially sized latent variable spaces of Factorial Hidden Markov Models using a toy sampling problem and a simulation study driven by an application in cancer genomics.', '1703.08520-1-5-0': 'The remainder of the paper is as follows.', '1703.08520-1-5-1': 'In Section 2 we discuss ensemble MCMC methods and we introduce our rejection-free scheme.', '1703.08520-1-5-2': 'In Section 3 we consider rejection-free ensemble MCMC for inference in Factorial Hidden Markov Models.', '1703.08520-1-5-3': 'In Section 4 we demonstrate the practical utility of our methods in a series of numerical experiments and we conclude with a discussion in Section 5.', '1703.08520-1-6-0': '# Rejection-free ensemble MCMC', '1703.08520-1-7-0': 'Here, we introduce a general procedure for rejection-free ensemble MCMC (Section [REF]) which extends the standard ensemble methods (Section [REF]).', '1703.08520-1-8-0': '## Standard ensemble sampling methods', '1703.08520-1-9-0': 'Suppose our goal is to sample from a density [MATH], but our sampler is inefficient and gets stuck in local modes.', '1703.08520-1-9-1': 'One solution is to use ensemble MCMC (also known as population-based MCMC, or evolutionary Monte Carlo) methods, which are based on an ensemble of chains [CITATION].', '1703.08520-1-9-2': 'That is, a new target density [MATH] is defined on the product space such that [EQUATION] whereas [MATH] for at least one index [CITATION].', '1703.08520-1-9-3': 'Here we focus on parallel tempering, which introduces a temperature ladder [MATH] and associates a temperature with each chain.', '1703.08520-1-9-4': 'Denoting the inverse temperature [MATH], we define the tempered targets [MATH].', '1703.08520-1-9-5': 'The idea is that high temperature chains explore the space well and do not get stuck, whereas the chain with [MATH] samples locally precisely from the target.', '1703.08520-1-9-6': 'Mostly we would update each chain independently, but occasionally exchange information between the chains.', '1703.08520-1-10-0': 'One approach to exchange information is to propose swapping states between chains of consecutive temperatures and then accept/reject the swap with Metropolis-Hastings [CITATION].', '1703.08520-1-10-1': 'More elaborate approaches can create proposals using genetic algorithms [CITATION], by proposing crossovers between chains which again requires accepting/rejecting based on the Metropolis-Hastings framework.', '1703.08520-1-10-2': 'However, the accept/reject procedure can be inefficient and very sensitive to the choice of the temperature ladder and algorithmic parameter tuning.', '1703.08520-1-10-3': 'Thus, it would be highly desirable to combine chains in a rejection-free manner so that any proposal is accepted with probability one.', '1703.08520-1-10-4': 'Next, we introduce such a rejection-free sampling move based on auxiliary variables and Gibbs sampling.', '1703.08520-1-11-0': '## Rejection-free sampling using auxiliary variables', '1703.08520-1-12-0': 'Consider the target in the product space given by eq. ([REF]).', '1703.08520-1-12-1': 'Suppose that during MCMC we would like to exchange information between a pair of chains (typically of consecutive temperatures in the ladder) [MATH] and [MATH] where [MATH] and [MATH] are [MATH]-dimensional vectors that indicate the current states of these chains.', '1703.08520-1-12-2': 'We introduce two auxiliary variables [MATH] and [MATH], that live in the same space as [MATH] and [MATH], and they are drawn from an auxiliary distribution [MATH].', '1703.08520-1-12-3': 'Without loss of generality we assume that this auxiliary distribution is uniform over all possible one-point crossovers between [MATH] and [MATH].', '1703.08520-1-12-4': 'More precisely, given two vectors [MATH] and [MATH] we define their one-point crossover at point [MATH], where [MATH], as follows: [EQUATION]', '1703.08520-1-12-5': 'We also introduce the set [MATH] to denote all [MATH] crossovers between the vectors [MATH] and [MATH].', '1703.08520-1-12-6': 'The auxiliary distribution [MATH] is precisely an uniform distribution over all pairs [MATH].', '1703.08520-1-12-7': 'This distribution is also symmetric, i.e. [EQUATION] which means that when we condition on fixed values [MATH] the above is an uniform distribution over all pairs [MATH], while when we condition on fixed [MATH] the above reduces to an uniform distribution over all pairs [MATH].', '1703.08520-1-13-0': 'Using the auxiliary variables we can exchange information between [MATH] and [MATH] through the intermediate step of sampling the auxiliary variables [MATH] based on the following two-step Gibbs procedure:', '1703.08520-1-14-0': 'Generate [MATH] Generate [MATH] where [EQUATION] where the normalising constant [MATH] is [EQUATION].', '1703.08520-1-14-1': 'The first step of the above procedure selects a random crossovered pair [MATH], while the second step conditions on this selected pair and jointly samples [MATH] from the exact conditional posterior distribution that takes into account the information coming from the actual chains [MATH] and [MATH].', '1703.08520-1-15-0': 'Since the above is a Gibbs operation it is quite obvious that it should lead to new state vectors for the chains [MATH] and [MATH] that are always accepted.', '1703.08520-1-15-1': 'To prove this more explicitly next we compute the effective marginal proposal, by marginalising out the auxiliary variables, and show that the corresponding Metropolis-Hastings acceptance probability is always one.', '1703.08520-1-16-0': 'Given the current states [MATH], we denote the proposed states by [MATH] and the marginal proposal distribution by [MATH].', '1703.08520-1-16-1': 'This proposal, defined by the above two-step Gibbs procedure, is a mixture: [EQUATION]', '1703.08520-1-16-2': 'The Metropolis-Hastings acceptance probability under this proposal is [EQUATION] which, since [MATH] due to symmetry, simplifies to [EQUATION] where all terms cancel out.', '1703.08520-1-16-3': 'So [MATH] and our proposal will be always accepted.', '1703.08520-1-17-0': 'To simulate from [MATH] in practice, we can use its mixture representation above, i.e. first generate auxiliary variables [MATH] and then conditional on those, generate the new value from [MATH].', '1703.08520-1-17-1': 'We note that even though both of these steps are implemented as one-point crossovers, the overall proposal can lead to a two-point crossover as illustrated in Figure [REF].', '1703.08520-1-18-0': 'A further extension of the above procedure is obtained by modifying the auxiliary distribution [MATH] to become uniform over the union of the sets [MATH] and [MATH] since, due to the deterministic ordering, the crossovers between [MATH] with [MATH] and the reverse crossovers between [MATH] with [MATH] are not the identical.', '1703.08520-1-18-1': 'The auxiliary distribution [MATH] still remains symmetric and all above properties hold unchanged.', '1703.08520-1-18-2': 'The only difference is that now we are considering [MATH] crossovers and in order to sample from [MATH] we need first to flip a coin to decide the order of [MATH] and [MATH].', '1703.08520-1-18-3': 'Equivalently, we can sample from the initial auxiliary distribution and then randomly assign the outcome to either [MATH] or [MATH].', '1703.08520-1-18-4': 'Complete pseudocode of the whole procedure is given by Algorithm [REF].', '1703.08520-1-19-0': '[H] Scheme for a rejection-free two-point crossover between [MATH] and [MATH]', '1703.08520-1-20-0': 'Pick [MATH] uniformly [MATH] Flip a coin to decide the direction of crossover [MATH] where [MATH] consider all normal and flipped crossovers of [MATH] and [MATH] Normal crossover of [MATH] and [MATH] Flipped crossover of [MATH] and [MATH] Normalise the probabilities Pick index [MATH] with probability proportional to [MATH]', '1703.08520-1-21-0': '[H] Function for a one-point crossover at point [MATH]', '1703.08520-1-22-0': 'Crossover[MATH] return[MATH]', '1703.08520-1-23-0': 'The above sampling scheme is very general and it can be applied to arbitrary MCMC inference problems involving both continuous and discrete variables.', '1703.08520-1-23-1': 'In the next section we apply the proposed method to a challenging inference problem in Factorial Hidden Markov Models (FHMMs).', '1703.08520-1-24-0': '# Application to Factorial Hidden Markov Models', '1703.08520-1-25-0': 'We start with a brief overview of basic Hidden Markov Models (HMMs) and FHMMs in Section [REF], then we discuss the current state-of-the-art sampling schemes for FHMMs in Section [REF] and then we apply the rejection-free method to FHMMs in Section [REF].', '1703.08520-1-26-0': '## HMMs and FHMMs', '1703.08520-1-27-0': 'HMMs are widely and successfully used for modelling sequential data across a range of areas, including signal processing [CITATION], genetics and computational biology [CITATION].', '1703.08520-1-27-1': 'The HMM assumes that there is an underlying unobserved Markov chain with a finite number of states, which generates a sequence of observations [MATH] via a parametric emission distribution.', '1703.08520-1-28-0': 'Inference over the latent sequence [MATH] and the parameters can be carried out either from a likelihood [CITATION] or Bayesian [CITATION] perspective.', '1703.08520-1-28-1': 'The latter is desirable due to the interest in capturing uncertainty over the hidden sequences and is the premise for this paper.', '1703.08520-1-28-2': 'Standard inference approaches employ conditional sampling where the parameters and latent sequences are updated iteratively conditional on the other being fixed and latent sequences can be jointly sampled using the forward-filtering-backward-sampling (FF-BS) [CITATION].', '1703.08520-1-29-0': 'The Factorial HMM (FHMM) is an extended version of the standard HMM where instead of a single latent chain, there are [MATH] latent chains.', '1703.08520-1-29-1': 'That is, given observations [MATH], our goal is to infer a [MATH] latent matrix [MATH] whose columns evolve according to Markov transitions.', '1703.08520-1-29-2': 'Here we focus on the case where [MATH] is binary, in which case the element [MATH] indicates whether latent feature [MATH] contributes to observation [MATH] or not.', '1703.08520-1-30-0': 'Each observation [MATH] is generated conditional on the [MATH] value from an emission density [MATH] with some parameters [MATH].', '1703.08520-1-30-1': 'We treat the rows of [MATH] independently, so denoting the transition probability for chain [MATH] by [MATH], i.e. [EQUATION] we can express [MATH] as follows [EQUATION].', '1703.08520-1-30-2': 'The joint distribution of [MATH] is given by [EQUATION]', '1703.08520-1-30-3': 'Even though FF-BS is an exact sampling algorithm, in practice this is infeasible for FHMMs, where the state space grows exponentially in the number of latent sequences [MATH].', '1703.08520-1-30-4': 'This results in the complexity of FF-BS being [MATH], so even for relatively small values of [MATH] there is a need for an alternative approach.', '1703.08520-1-30-5': 'Variational inference [CITATION] offers a faster alternative for approximate inference but also suffers from scalability issues as well as the problems with multimodality in the posterior distribution.', '1703.08520-1-31-0': '## Existing MCMC approaches for inference in FHMMs', '1703.08520-1-32-0': 'The standard FF-BS step has quadratic complexity w.r.t the size of the state space, so full FF-BS is not feasible for FHMMs.', '1703.08520-1-32-1': 'Thus a computationally cheaper approach is needed, however this comes at the expense of sampling efficiency.', '1703.08520-1-33-0': 'One option is to sample each row of [MATH] conditional on the rest, using the FF-BS.', '1703.08520-1-33-1': 'Then each of the updates has a state space of size 2 and the FF-BS steps are inexpensive.', '1703.08520-1-33-2': 'However, in this conditional scheme most of the sequences are fixed and thus it is difficult for the sampler to explore the space well.', '1703.08520-1-33-3': 'A more general version of this would update a small subset of chains at a time at a higher computational cost, but it can still get trapped in local modes.', '1703.08520-1-34-0': 'An alternative idea referred to as Hamming Ball sampling has been suggested by Titsias and Yau [CITATION], which adaptively truncates the space via an auxiliary variable scheme.', '1703.08520-1-34-1': 'Unlike the conditional Gibbs updates, it does not restrict parts of [MATH] to be fixed during sampling.', '1703.08520-1-34-2': 'Even though it can be less prone to get stuck, for a moderate value of [MATH] it may still not explore the whole posterior space.', '1703.08520-1-35-0': '## Ensemble MCMC applied to FHMMs', '1703.08520-1-36-0': 'Here, we apply the ensemble crossover schemeto FHMMs in order to significantly improve poorly mixing samplers.', '1703.08520-1-36-1': 'We achieve this via an ensemble of chains over suitably defined tempered posteriors.', '1703.08520-1-37-0': 'For a latent variable model, one can either temper the whole joint distribution or just the emission likelihood.', '1703.08520-1-37-1': 'We chose the latter, so the target posterior of interest becomes [EQUATION] where [MATH] is a [MATH] binary matrix.', '1703.08520-1-38-0': 'As the ensemble crossover schemewas defined on vectors, there are multiple ways to extend this to matrices.', '1703.08520-1-38-1': 'Here, we consider a crossover move on the rows of the matrix [MATH], as illustrated in Figure [REF] for a one-row crossover.', '1703.08520-1-39-0': 'The core computational step of the algorithm is to compute quantities [MATH] for all crossover points [MATH].', '1703.08520-1-39-1': 'We show that these can be computed recursively in an efficient way.', '1703.08520-1-39-2': 'Let [MATH] and [MATH] be the current states for chains [MATH] and [MATH], and let [MATH] and [MATH] be the respective matrices after the crossover move.', '1703.08520-1-39-3': 'Comparing their crossovers at two consecutive points [MATH] and [MATH], denoted by [MATH] and [MATH], we note that these can differ just in column [MATH]: [EQUATION]', '1703.08520-1-39-4': 'As a result, the values [MATH] can be computed recursively.', '1703.08520-1-39-5': 'Indeed, given the previous value of [MATH], we can compute [MATH] by accounting for the following two cases:', '1703.08520-1-40-0': 'Let [MATH] be the current value of transition probability [MATH] for chain [MATH].', '1703.08520-1-40-1': 'By denoting the overall transition probability [MATH] for chain [MATH] as [EQUATION] we can express [EQUATION] where [MATH] denotes the correction term.', '1703.08520-1-40-2': 'Using this recursion, we can compute the quantities [MATH] in terms of [MATH] as follows [EQUATION]', '1703.08520-1-40-3': 'As the values of [MATH] can be normalised to sum to one, we can arbitrarily fix the reference value [MATH].', '1703.08520-1-40-4': 'The computation of every correction term is cheap, and the overall complexity for computing all [MATH] values is [MATH].', '1703.08520-1-40-5': 'Moreover, we perform only a one-row crossover, we do not need to consider all [MATH] terms in the expression of [MATH], instead we can focus to the relevant transition probability [MATH].', '1703.08520-1-40-6': 'In this case, the overall complexity reduces even further to [MATH].', '1703.08520-1-41-0': 'Furthermore, we typically need to perform the crossover moves only occasionally, so that the ensemble crossover schemeprovides a way to improve the poorly mixing samplers for FHMMs at a small extra computational cost.', '1703.08520-1-41-1': 'For the rest of the computational time the chains run in parallel.', '1703.08520-1-41-2': 'In our experiments, it was sufficient to use two parallel chains: one sampling from the target and the other being tempered.', '1703.08520-1-42-0': '# Experiments', '1703.08520-1-43-0': 'In Section [REF] we demonstrate the proposed method in a toy inference problem that involves sampling from a posterior distribution with two separate modes, while in Section [REF] we consider a challenging tumor deconvolution example.', '1703.08520-1-44-0': '## Toy example', '1703.08520-1-45-0': 'To demonstrate the benefit of the ensemble crossover scheme, we consider the following 10-dimensional toy sampling problem, where the target distribution is bimodal.', '1703.08520-1-46-0': 'Let these two modes be [MATH] and [MATH].', '1703.08520-1-46-1': 'We define the probability distribution over all states [MATH] such that [MATH] would depend on the Hamming distance from the closest mode, as follows [EQUATION] where [MATH] between binary vectors.', '1703.08520-1-46-2': 'As illustrated in Figure [REF], it is unlikely to observe values [MATH] which are far from both modes.', '1703.08520-1-47-0': 'We consider Gibbs sampling on each of the dimensions of [MATH] as a baseline method, and we explore whether our ensemble crossover schemewill improve on this.', '1703.08520-1-47-1': 'That is, we run two parallel Gibbs samplers (with temperatures [MATH] and [MATH]) and carry out a crossover move every 10-th iteration.', '1703.08520-1-47-2': 'The target distribution for the tempered chain is illustrated in Figure [REF](b).', '1703.08520-1-47-3': 'All chains are initialised from the same value (one of the modes) and run for 1000 iterations.', '1703.08520-1-48-0': 'The traces of [MATH] for both methods are shown in Figure [REF](ab).', '1703.08520-1-48-1': 'Ideally, we would expect a well mixing sampler to sample values around both modes [MATH] and [MATH].', '1703.08520-1-48-2': 'Standard Gibbs sampler is unable to escape the mode it was initialised from, whereas the ensemble crossover schemehas jumped between the modes on multiple occasions.', '1703.08520-1-48-3': 'For each sample, we have also calculated the Hamming distance to the mode from which the samplers was initialised.', '1703.08520-1-48-4': 'The Gibbs sampler has not moved across the low probability barrier at Hamming distance [MATH], so the resulting empirical distribution of Hamming distances in Figure [REF](c) is very different from the true one in Figure [REF](a).', '1703.08520-1-48-5': 'In comparison, our ensemble approach enables the sampler to explore both modes.', '1703.08520-1-48-6': 'This is achieved via exchanging subsequences with a higher temperature chain, which is sampling from a tempered distribution which is less peaked, as illustrated in Figure [REF](b).', '1703.08520-1-49-0': '## Tumor deconvolution example', '1703.08520-1-50-0': 'The following example is motivated by an application in cancer genomics.', '1703.08520-1-50-1': 'Certain mutations in the cancer genome result in copy number alterations, i.e. for some genomic regions there can be more or less than two copies of each chromosome.', '1703.08520-1-50-2': 'Typically there are various subpopulations present among the cancer cells, and it is of interest to identify the subpopulations to study their phylogeny [CITATION].', '1703.08520-1-50-3': 'DNA sequencing technology produces data where read counts have been aggregated over various subpopulations, so an additive FHMM is a natural model to capture these subpopulations.', '1703.08520-1-51-0': 'Lets consider the emission model [EQUATION] where [MATH] denote the sequence read counts at a locus [MATH] and [MATH] is the expected sequencing depth.', '1703.08520-1-51-1': 'Each [MATH] corresponds to the fraction of [MATH]-th subpopulation ([MATH], [MATH]) whose mutation profile is given by the [MATH]-th row of [MATH].', '1703.08520-1-51-2': 'Here [MATH] denotes whether the [MATH]-th population has a copy number alteration at position [MATH] or not.', '1703.08520-1-52-0': 'We note that this is not intended to be a complete model of real-world sequencing data but a device to demonstrate the utility of the proposed ensemble MCMC methods.', '1703.08520-1-52-1': 'Further work to construct a sufficiently complex model to capture the variations within real sequencing data, such as single nucleotide polymorphisms, is beyond the scope of this paper and will be developed in future work.', '1703.08520-1-53-0': '### Data generation', '1703.08520-1-54-0': 'We generated data according to this Poisson emission model, with [MATH] latent chains.', '1703.08520-1-54-1': 'The segment lengths were manually specified, resulting in the [MATH] matrix shown in Figure [REF].', '1703.08520-1-54-2': 'Parameters [MATH] were fixed to [MATH] and sequencing depth [MATH].', '1703.08520-1-54-3': 'The [MATH] were chosen such that there would exist multiple underlying matrices [MATH] (i.e. different phylogenetic trees) which could have generated the data, so the resulting posterior will be multimodal.', '1703.08520-1-55-0': '### Comparing various sampling techniques', '1703.08520-1-56-0': 'We carried out inference as follows.', '1703.08520-1-56-1': 'To sample from [MATH], we represented [MATH] via normalised Gamma random variables, [MATH] and then used random walk Metropolis-Hastings on the log-scale of [MATH].', '1703.08520-1-56-2': 'To sample from [MATH] we compared the following methods:', '1703.08520-1-57-0': 'We ran these four samplers for 2,000 iterations, all initialised from the same state.', '1703.08520-1-57-1': 'The trace plots of [MATH] are shown in Figure [REF] separately for all [MATH] latent sequences.', '1703.08520-1-57-2': 'As expected, the Gibbs sampler gets stuck and during all 2,000 iterations it moves very little.', '1703.08520-1-57-3': 'The Hamming Ball sampler mixes better, but a strong autocorrelation between consecutive states is evident.', '1703.08520-1-57-4': 'The ensemble crossover schemeclearly improves mixing of the samplers in both cases, visiting configurations of the state space that were not explored by the standard samplers.', '1703.08520-1-58-0': 'We have chosen [MATH] as a summary statistic which represents the current state of the sampler.', '1703.08520-1-58-1': 'The ensemble versions of the samplers have visited a wider range of possible [MATH] values, as shown in Figure [REF].', '1703.08520-1-58-2': 'Also the traces of log-likelihood values [MATH] in Figure [REF] indicate that the ensemble crossover schemehas visited areas with a wider range of log-likelihood values, thus moving around the space more freely.', '1703.08520-1-59-0': '# Conclusion', '1703.08520-1-60-0': 'We introduced a rejection-free ensemble MCMC method to improve poorly mixing samplers.', '1703.08520-1-60-1': 'This is achieved by combining parallel tempering and a novel rejection-free exchange move between pairs of chains achieved through an auxiliary variable augmentation.', '1703.08520-1-60-2': 'The former allows to have a chain which explores the space freely and does not get stuck, whereas the latter provides an efficient procedure to exchange information between a tempered chain and our target.', '1703.08520-1-60-3': 'The rejection-free property combined with the fact that subsequences of varying lengths can be exchanged between the chains, makes our method work even with just two chains in the ensemble scheme.', '1703.08520-1-60-4': 'This is typically not feasible with traditional parallel tempering where dense grids of tempered chains are required to achieve sensible acceptance rates.', '1703.08520-1-61-0': 'The proposed method is a general purpose ensemble MCMC approach and here we demonstrated its use on multimodal posterior inference settings in FHMMs.', '1703.08520-1-61-1': 'For this model class we described the specifics of how to efficiently implement the crossover move and we demonstrated in a simulation study that the ensemble crossover schemesignificantly improves on the efficiency of existing samplers at a low extra computational cost.', '1703.08520-1-61-2': 'Our application also illustrates the pragmatic benefits of having a rejection-free scheme in that we do not have to be concerned about parameter tuning in order to achieve optimal acceptance rates in a standard accept-reject framework.', '1703.08520-1-62-0': 'Finally, we expect our technique to be useful in a range of other high-dimensional continuous or discrete space sampling problems, such as for spike-and-slab variable selection in regression models and for structural inference in Bayesian neural networks.'}

{'1703.08520-2-0-0': 'Bayesian inference for factorial hidden Markov models is challenging due to the exponentially sized latent variable space.', '1703.08520-2-0-1': 'Standard Monte Carlo samplers can have difficulties effectively exploring the posterior landscape and are often restricted to exploration around localised regions that depend on initialisation.', '1703.08520-2-0-2': 'We introduce a general purpose ensemble Markov Chain Monte Carlo (MCMC) technique to improve on existing poorly mixing samplers.', '1703.08520-2-0-3': 'This is achieved by combining parallel tempering and an auxiliary variable scheme to exchange information between the chains in an efficient way.', '1703.08520-2-0-4': 'The latter exploits a genetic algorithm within an augmented Gibbs sampler.', '1703.08520-2-0-5': 'We compare our technique with various existing samplers in a simulation study as well as in a cancer genomics application, demonstrating the improvements obtained by our augmented ensemble approach.', '1703.08520-2-1-0': '# Introduction', '1703.08520-2-2-0': 'Hidden Markov models (HMMs) are widely and successfully used for modeling sequential data across a range of areas, including signal processing , genetics and computational biology .', '1703.08520-2-2-1': 'The HMM assumes that there is an underlying unobserved Markov chain with a finite number of states, which generates a sequence of observations [MATH] via a parametric emission distribution.', '1703.08520-2-2-2': 'Inference over the latent sequence [MATH] and the parameters can be carried out either from a likelihood or Bayesian perspective.', '1703.08520-2-2-3': 'In the latter, conditional sampling can be used where the parameters and latent sequences are updated iteratively conditional on the other being fixed.', '1703.08520-2-2-4': 'Latent sequences can be sampled using forward-filtering-backward-sampling (FF-BS) .', '1703.08520-2-3-0': 'The Factorial HMM (FHMM) is an extended version of the standard HMM where instead of a single latent chain, there are [MATH] latent chains.', '1703.08520-2-3-1': 'That is, given observations [MATH], our goal is to infer a [MATH] latent matrix [MATH] whose columns evolve according to Markov transitions.', '1703.08520-2-3-2': 'Here we focus on the case where [MATH] is binary, in which case the element [MATH] indicates whether latent feature [MATH] contributes to observation [MATH].', '1703.08520-2-3-3': 'The joint distribution [MATH] is given by [EQUATION]', '1703.08520-2-3-4': 'The FF-BS is an exact sampling algorithm and in principle, could be applied to FHMMs.', '1703.08520-2-3-5': 'However, this becomes infeasible even for a moderate number of latent sequences [MATH].', '1703.08520-2-3-6': 'This is due to the state space growing exponentially with [MATH].', '1703.08520-2-3-7': 'As the full FF-BS has complexity [MATH], a computationally cheaper approach is needed, however this comes at the expense of sampling efficiency.', '1703.08520-2-4-0': 'One option is to sample each row of [MATH] conditional on the rest, using the FF-BS.', '1703.08520-2-4-1': 'Then each of the updates has a state space of size 2 and the FF-BS steps are inexpensive.', '1703.08520-2-4-2': 'However, in this conditional scheme most of the sequences are fixed and thus it is difficult for the sampler to explore the space well.', '1703.08520-2-4-3': 'A more general version of this would update a small subset of chains jointly at a higher computational cost, which can still get trapped in local modes.', '1703.08520-2-5-0': 'An alternative idea referred to as Hamming Ball sampling has been suggested by [CITATION], which adaptively truncates the space via an auxiliary variable scheme.', '1703.08520-2-5-1': 'Unlike the conditional Gibbs updates, it does not restrict parts of [MATH] to be fixed during sampling.', '1703.08520-2-5-2': 'Even though it can be less prone to get stuck, for a moderate value of [MATH] it may still not explore the whole posterior space.', '1703.08520-2-6-0': 'This problem can be alleviated by ensemble MCMC methods which combine ideas from simulated annealing and genetic algorithms .', '1703.08520-2-6-1': 'One such example is parallel tempering .', '1703.08520-2-6-2': 'Instead of running a single chain targeting the posterior, one introduces an ensemble of chains and assigns a temperature to each chain so that every chain would be targeting a tempered version of the posterior.', '1703.08520-2-6-3': 'Tempered targets are less peaked and therefore higher temperature chains in the ensemble explore the space well and do not get stuck.', '1703.08520-2-6-4': 'The key question becomes how to efficiently exchange information between the chains.', '1703.08520-2-7-0': 'In this paper, we propose a novel ensemble MCMC method which provides an auxiliary variable construction to exchange information between chains.', '1703.08520-2-7-1': 'This is a general MCMC method, but our main focus is on improving existing poorly mixing samplers for sequence-type data.', '1703.08520-2-7-2': 'Specifically we consider the application to Factorial Hidden Markov Models.', '1703.08520-2-7-3': 'We demonstrate the practical utility of our augmented ensemble scheme in a series of numerical experiments, covering a toy sampling problem as well as inference for FHMMs.', '1703.08520-2-7-4': 'The latter involves a simulation study as well as a challenging cancer genomics application.', '1703.08520-2-8-0': '# Augmented ensemble MCMC', '1703.08520-2-9-0': 'Monte Carlo-based Bayesian inference for complex high-dimensional posterior distributions is a challenging problem as efficient samplers need to be able to move across irregular landscapes that may contain many local modes .', '1703.08520-2-9-1': 'Commonly adopted sampling approaches can explore the space very slowly or become confined to regions around local modes.', '1703.08520-2-10-0': 'Ensemble MCMC (also known as population-based MCMC, or evolutionary Monte Carlo) methods can alleviate this problem.', '1703.08520-2-10-1': 'This is achieved by introducing an ensemble of MCMC chains and then exchanging information between the chains.', '1703.08520-2-10-2': 'Next, we review standard ensemble MCMC approaches and proposal mechanisms that are used to exchange information.', '1703.08520-2-10-3': 'Then, we introduce our augmented Gibbs sampler.', '1703.08520-2-11-0': '## Standard ensemble sampling methods', '1703.08520-2-12-0': 'Suppose our goal is to sample from a density [MATH].', '1703.08520-2-12-1': 'Instead of sampling [MATH], ensemble MCMC introduces an extended product space [MATH] with a new target density [MATH] defined as [MATH] where [MATH] for at least one index.', '1703.08520-2-12-2': 'Here we focus on parallel tempering, which introduces a temperature ladder [MATH] and associates a temperature with each chain.', '1703.08520-2-12-3': 'Denoting the inverse temperature [MATH], we define the tempered targets [MATH].', '1703.08520-2-12-4': 'The idea is that high temperature chains can readily explore the space since the density is flattened by the power transformation, whereas the chain containing the true target density with [MATH] only samples locally and precisely from the target.', '1703.08520-2-12-5': 'Each chain is updated independently, with occasional information exchange between the chains so that more substantial movement in the higher temperature chains can filter down to the slower moving low temperature chains.', '1703.08520-2-13-0': 'One approach to exchanging information is to propose swapping states ("swap" move) between chains of consecutive temperatures and then performing an accept/reject operation according to the Metropolis-Hastings ratio .', '1703.08520-2-13-1': 'However, a global move like this is unlikely to be accepted in a high-dimensional sampling setting.', '1703.08520-2-14-0': 'More elaborate approaches can create proposals using genetic algorithms , by proposing certain moves between chains which again requires accepting/rejecting based on the Metropolis-Hastings framework.', '1703.08520-2-14-1': 'One such proposal scheme is a one-point crossover move, illustrated as follows: [EQUATION] where the crossover point [MATH] could for example be chosen uniformly [MATH].', '1703.08520-2-14-2': 'This is most natural for sequential models where there is dependency between consecutive [MATH] and [MATH].', '1703.08520-2-14-3': 'For high-dimensional sequences this is more appealing than a swap move due to being more local and thus leading to higher chance of acceptance.', '1703.08520-2-14-4': 'One can similarly construct a two-point crossover move.', '1703.08520-2-14-5': 'However, the accept/reject procedure can be inefficient and very sensitive to both the choice of the temperature ladder and algorithmic parameter tuning.', '1703.08520-2-14-6': 'Our work seeks to address the latter issue by using an auxiliary variable augmentation that produces a Gibbs sampling scheme.', '1703.08520-2-15-0': '## Gibbs sampling using auxiliary variables', '1703.08520-2-16-0': 'Now, consider the target in the product space [MATH].', '1703.08520-2-16-1': 'Suppose that during MCMC we would like to exchange information between a pair of chains [MATH] and [MATH] where [MATH] and [MATH] are [MATH]-dimensional vectors that indicate the current states of these chains.', '1703.08520-2-16-2': 'Here we describe an auxiliary variable move, which uses the idea of a one-point crossovers and leads to a Gibbs update for a two-point crossover.', '1703.08520-2-17-0': 'We introduce two auxiliary variables [MATH] and [MATH], that live in the same space as [MATH] and [MATH], drawn from an auxiliary distribution [MATH].', '1703.08520-2-17-1': 'Without loss of generality we assume that this auxiliary distribution is uniform over all possible one-point crossovers between [MATH] and [MATH].', '1703.08520-2-18-0': 'We also introduce the set [MATH] to denote all [MATH] crossovers between the vectors [MATH] and [MATH].', '1703.08520-2-18-1': 'The auxiliary distribution [MATH] is precisely a uniform distribution over all pairs [MATH].', '1703.08520-2-18-2': 'This distribution is also symmetric, i.e. [MATH].', '1703.08520-2-19-0': 'Using the auxiliary variables we can exchange information between [MATH] and [MATH] through the intermediate step of sampling the auxiliary variables [MATH] based on the following two-step Gibbs procedure:', '1703.08520-2-20-0': 'Generate [MATH] Generate [MATH], where [EQUATION] where the normalising constant [MATH] is [MATH].', '1703.08520-2-21-0': 'The first step of the above procedure selects a random crossovered pair [MATH], while the second step conditions on this selected pair and jointly samples [MATH] from the exact conditional posterior distribution that takes into account the information coming from the actual chains [MATH] and [MATH].', '1703.08520-2-22-0': 'Since the above is a Gibbs operation it leads to new state vectors for the chains [MATH] and [MATH] that are always accepted.', '1703.08520-2-22-1': 'To prove this explicitly we compute the effective marginal proposal and show that the corresponding Metropolis-Hastings acceptance probability is always one.', '1703.08520-2-23-0': 'Given the current states [MATH], we denote the proposed states by [MATH] and the marginal proposal distribution by [MATH].', '1703.08520-2-23-1': 'This proposal, defined by the above two-step Gibbs procedure, is a mixture: [EQUATION]', '1703.08520-2-23-2': 'The Metropolis-Hastings acceptance probability under this proposal is [EQUATION]', '1703.08520-2-23-3': 'Since [MATH] due to symmetry, all terms cancel out and [MATH].', '1703.08520-2-23-4': 'So our proposal will be always accepted.', '1703.08520-2-24-0': 'To simulate from [MATH] in practice, we can use its mixture representation above, i.e. first generate auxiliary variables [MATH] and then conditional on those, generate the new value from [MATH].', '1703.08520-2-24-1': 'We note that even though both of these steps are implemented as one-point crossovers, the overall proposal can lead to a two-point crossover as illustrated in Figure [REF].', '1703.08520-2-25-0': 'Specifically, to implement this, first we sample [MATH] uniformly from the set [MATH].', '1703.08520-2-25-1': 'Now, conditional on the obtained [MATH], let us denote the crossover of [MATH] and [MATH] at point [MATH] by [MATH].', '1703.08520-2-25-2': 'The second step is to iterate over [MATH] and compute quantities [MATH].', '1703.08520-2-25-3': 'The pair [MATH] will be accepted as the new value of [MATH] with probability proportional to [MATH].', '1703.08520-2-26-0': 'A further extension of the above procedure is obtained by modifying the auxiliary distribution [MATH] to become uniform over the union of the sets [MATH] and [MATH] since, due to the deterministic ordering, the crossovers between [MATH] with [MATH] and the reverse crossovers between [MATH] with [MATH] are not identical.', '1703.08520-2-26-1': 'The auxiliary distribution [MATH] still remains symmetric and all above properties hold unchanged.', '1703.08520-2-26-2': 'The only difference is that now we are considering [MATH] crossovers and in order to sample from [MATH] we need first to flip a coin to decide the order of [MATH] and [MATH].', '1703.08520-2-26-3': 'Complete pseudocode of the whole procedure is given in Supplementary.', '1703.08520-2-27-0': 'The above sampling scheme is general and it can be applied to arbitrary MCMC inference problems involving both continuous and discrete variables.', '1703.08520-2-27-1': 'In the next section we apply the proposed method to a challenging inference problem in Factorial HMMs (FHMMs).', '1703.08520-2-28-0': '# Application to FHMMs', '1703.08520-2-29-0': 'Here, we apply the augmented ensemble scheme to FHMMs in order to improve on existing poorly mixing samplers.', '1703.08520-2-29-1': 'We achieve this via an ensemble of chains over suitably defined tempered posteriors.', '1703.08520-2-29-2': 'For a latent variable model, one can either temper the whole joint distribution or just the emission likelihood.', '1703.08520-2-29-3': 'We chose the latter, so the target posterior of interest becomes [EQUATION] where [MATH] is a [MATH] binary matrix.', '1703.08520-2-29-4': 'As the ensemble crossover schemewas originally defined on vectors, there are multiple ways to extend this to matrices.', '1703.08520-2-29-5': 'One can perform crossovers on either rows or columns of a matrix, potentially considering a subset of those.', '1703.08520-2-29-6': 'Here we have decided to focus on a crossover move defined on the rows of [MATH], specifically on all [MATH] rows of [MATH].', '1703.08520-2-30-0': 'The core computational step of the algorithm is to compute quantities [MATH] for all crossover points [MATH].', '1703.08520-2-30-1': 'We show that these can be computed recursively in an efficient way.', '1703.08520-2-30-2': 'Let [MATH] and [MATH] be the current states of the auxiliary matrices for chains [MATH] and [MATH].', '1703.08520-2-30-3': 'Comparing their crossovers at two consecutive points [MATH] and [MATH], denoted by [MATH] and [MATH], we note that these can differ just in column [MATH]: [EQUATION]', '1703.08520-2-30-4': 'As a result, the values [MATH] can be computed recursively.', '1703.08520-2-30-5': 'Indeed, given the previous value of [MATH], we can compute [MATH] by accounting for the following two cases: first, change in emission likelihood from [MATH] to [MATH], and second, change in the transitions from [MATH] to [MATH].', '1703.08520-2-31-0': 'By denoting the overall transition probability [MATH] for chain [MATH] by [MATH], we can express [MATH] in terms of [MATH] as follows [MATH] where [EQUATION].', '1703.08520-2-31-1': 'Now we can compute the quantities [MATH] recursively as follows [MATH].', '1703.08520-2-31-2': 'As the values of [MATH] can be normalised to sum to one, we can arbitrarily fix the reference value [MATH].', '1703.08520-2-31-3': 'The computation of every correction term [MATH] is of the complexity [MATH], and the overall cost for all [MATH] values is [MATH], being relatively cheap.', '1703.08520-2-31-4': 'As we typically need to perform the crossover moves only occasionally, the ensemble crossover schemeprovides a way to improve the poorly mixing samplers for FHMMs at a small extra computational cost.', '1703.08520-2-32-0': '# Experiments', '1703.08520-2-33-0': 'First, we demonstrate the proposed sampling method on a multimodal toy inference problem.', '1703.08520-2-33-1': 'Then, we focus on Bayesian inference for FHMMs: we compare various samplers in a simulation study and then consider a challenging tumor deconvolution example.', '1703.08520-2-33-2': 'In both experiments, we compare a standard single-chain sampling technique (a Gibbs sampler or the Hamming Ball sampler) with the respective ensemble versions.', '1703.08520-2-34-0': 'For ensemble samplers, we compare our proposed augmentation scheme ("augmented crossover") with two additional baseline exchange moves: the standard swap move ("swap") and a uniformly chosen crossover ("random cr") within the accept-reject Metropolis-Hastings framework.', '1703.08520-2-34-1': 'In all experiments, we run an ensemble of two MCMC chains, with temperatures [MATH] and [MATH], carrying out an exchange move every 10-th iteration.', '1703.08520-2-35-0': '## Toy example', '1703.08520-2-36-0': 'We consider the following multimodal toy sampling problem, where the target distribution is binary and has multiple separated modes.', '1703.08520-2-36-1': 'Specifically, we fix the dimensionality [MATH] and divide the sequence [MATH] into [MATH] contiguous blocks as follows [MATH].', '1703.08520-2-36-2': 'In each of the blocks, we define a bimodal distribution, having two peaked modes [MATH] and [MATH], such that the probability of any binary vector [MATH] in block [MATH] is given by [EQUATION] where [MATH] denotes the Hamming distance between two binary vectors and [MATH] is a block-specific parameter which controls how peaked the modes are.', '1703.08520-2-36-3': 'As a result, the further we go from the modes (in terms of Hamming distance), the less likely we are to observe that state.', '1703.08520-2-36-4': 'This has been illustrated in Figure [REF].', '1703.08520-2-37-0': 'We extend the above to define the joint [MATH] factorising over the blocks as follows [EQUATION]', '1703.08520-2-37-1': 'Within each block, the probability of a given state depends on its distance to the closest mode.', '1703.08520-2-37-2': 'This construction induces strong within-block dependencies.', '1703.08520-2-37-3': 'By varying the number of blocks within a sequence of fixed length, we can interpolate between a strong global correlation and local dependencies with a highly multimodal structure.', '1703.08520-2-37-4': 'The total number of modes for this distribution is [MATH], as illustrated in Figure [REF].', '1703.08520-2-38-0': 'In our experiments, we vary [MATH], resulting in distributions having [MATH] modes.', '1703.08520-2-38-1': 'We generate [MATH].', '1703.08520-2-38-2': 'All samplers are initialised from the same value (one of the modes) and run for 10,000 iterations.', '1703.08520-2-39-0': 'The resulting traces of [MATH] have been shown as heatmaps in Figure [REF] for [MATH] (see Supplementary Figures for [MATH] and [MATH]).', '1703.08520-2-39-1': 'As a summary statistic, we have shown the cumulative number of jumps between modes over repeated experiments in Figure [REF] .', '1703.08520-2-40-0': 'In all scenarios, the single chain Gibbs sampler expectedly struggles to escape the mode from which it was initialised, with ensemble methods better at moving between modes.', '1703.08520-2-40-1': 'For strong global correlations (corresponding to small [MATH] values), the baseline exchange moves "swap" and "random crossover" are reasonably efficient, though still result in a smaller number of mode jumps than the "augmented crossover".', '1703.08520-2-41-0': 'Now when increasing [MATH], the dependency structure becomes more local, resulting in a much more multimodal sampling landscape.', '1703.08520-2-41-1': 'For [MATH], the simple "swap" and "random crossover" moves struggle to accept any proposals at all and the benefit of our augmentation scheme becomes clear.', '1703.08520-2-41-2': 'In this highly multimodal setting with [MATH] modes, the total number of modes visited by our "augmented crossover" (average 144) is much higher than for the "swap" (3) and "random crossover" (27) moves.', '1703.08520-2-42-0': '## Tumor deconvolution example', '1703.08520-2-43-0': 'The following example is motivated by an application in cancer genomics.', '1703.08520-2-43-1': 'Certain mutations in the cancer genome result in a loss of DNA integrity leading to copy number alterations due to the duplication or loss of certain DNA regions.', '1703.08520-2-43-2': 'Tumor samples consist of heterogeneous cell subpopulations and it is of interest to identify the subpopulations to study their phylogeny and gain insight into the clonal evolution .', '1703.08520-2-43-3': 'However, as DNA sequencing of bulk tissue samples produces aggregate data over all constituent cell subpopulations, the observed sequencing read counts must be deconvolved to reveal the underlying latent genetic architecture.', '1703.08520-2-44-0': 'The additive Factorial HMM is a natural model to consider where each latent chain corresponds to a putative cell subpopulation.', '1703.08520-2-44-1': 'However, it is important that the exploration of the state space of the latent chains allows us to identify the different subpopulation configurations that are compatible with the observed sequencing data since there maybe a number of plausible possibilities.', '1703.08520-2-44-2': 'This is illustrated in Figure [REF].', '1703.08520-2-44-3': 'A poorly mixing sampler which is exploring only one of the possible latent explanations could lead to misleading conclusions regarding the subclonal architecture of a tumor.', '1703.08520-2-44-4': 'We wanted to examine if the ensemble scheme we propose could provide a more effective means of posterior sampling.', '1703.08520-2-45-0': '### Simulation study', '1703.08520-2-46-0': 'Lets consider the emission model [MATH] where [MATH] denote the observed sequence read counts at a locus [MATH] and [MATH] is the expected sequencing depth.', '1703.08520-2-46-1': 'Each [MATH] corresponds to the fraction of [MATH]-th subpopulation ([MATH], [MATH]) whose mutation profile is given by the [MATH]-th row of [MATH].', '1703.08520-2-46-2': 'Here [MATH] denotes whether the [MATH]-th population has a copy number alteration at position [MATH] or not.', '1703.08520-2-47-0': 'Note that this is not a complete model of real-world sequencing data but a simplified version to demonstrate the utility of the proposed ensemble MCMC methods.', '1703.08520-2-47-1': 'The results presented here should extend to the more complex cases.', '1703.08520-2-47-2': 'Further work to construct a sufficiently complex model to capture the variations within real sequencing data, such as single nucleotide polymorphisms, is beyond the scope of this paper and will be developed in future work.', '1703.08520-2-48-0': 'First, we investigated the performance of sampling schemes for FHMMs in the presence of multimodality in a controlled setting.', '1703.08520-2-48-1': 'We generated observations from the emission distribution with [MATH] with weights such that [MATH].', '1703.08520-2-48-2': 'As a result, data generating scenarios [MATH] and [MATH] are both plausible underlying latent explanations.', '1703.08520-2-49-0': 'For data generation, we used a latent [MATH] matrix having a block structure of columns [MATH] followed by a block of [MATH], as illustrated in Figure [REF](a), but using altogether 20 blocks.', '1703.08520-2-49-1': 'We fixed [MATH], [MATH] with [MATH] and [MATH].', '1703.08520-2-49-2': 'Each of these blocks has two modes, but due to the structured FHMM prior on [MATH], the mode [MATH] corresponds to a slightly higher log-posterior value.', '1703.08520-2-49-3': 'For example, the three examples provided in Figure [REF] are ordered in terms of posterior probability (c) [MATH] (b) [MATH] (a).', '1703.08520-2-50-0': 'For inference in FHMMs, we considered two single chain samplers for [MATH]: one-row updates conditional on the rest ("Gibbs"), and the Hamming Ball sampler ("HB").', '1703.08520-2-50-1': 'We then considered ensemble versions of both of these samplers, as shown in Figure [REF] (left column for "Gibbs" and right column for "HB").', '1703.08520-2-50-2': 'All chains were initialised from the mode with [MATH], i.e. mode (a) in Figure [REF], and were ran for 10 000 iterations.', '1703.08520-2-50-3': 'Exchange moves were carried out every 10th iteration.', '1703.08520-2-51-0': 'For "Gibbs", the single chain sampler and the "swap" ensemble have not moved from the initialisation, the "random cr" ensemble scheme shows some improvement, but the "augmented cr" has quickly moved towards values of [MATH] with higher posterior probability (see Figure [REF](a)).', '1703.08520-2-51-1': 'It also exhibits much better mixing, as seen from the traces of the first row of [MATH], i.e. traces of [MATH] shown in Figure [REF](c).', '1703.08520-2-51-2': 'We note that [MATH] values correspond to the more probable mode.', '1703.08520-2-52-0': 'As a single chain sampler, "HB" quickly achieves higher log-posterior values than "Gibbs".', '1703.08520-2-52-1': 'Therefore, for "HB" the gain from "swap" and "random cr" ensemble techniques is relatively smaller, but still the "augmented cr" has quickly moved towards higher log-posterior values.', '1703.08520-2-53-0': 'To quantify mixing on binary state spaces, we have calculated the Hamming distance between [MATH] and [MATH] for various lag values [MATH], normalised by [MATH].', '1703.08520-2-53-1': 'Panels (e, f) show the distribution of these summary statistics, confirming that the augmented crossover scheme reduces notably the dependence between consecutive samples of [MATH].', '1703.08520-2-54-0': 'We have shown above that the complexity of augmented crossover scheme is linear [MATH], which is also the case for the "swap" and "random cr" moves.', '1703.08520-2-54-1': 'To explore the respective costs in practice, we measured the total computation time for our Rcpp implementation.', '1703.08520-2-54-2': 'To establish the baseline cost of running a two-chain ensemble without any exchange moves in a sequential implementation, we indicate this baseline time in the first column ("Gibbs" and "HB") of Table [REF].', '1703.08520-2-54-3': 'We note that this could be halved by a parallel implementation.', '1703.08520-2-54-4': 'The extra cost for all exchange moves are relatively small.', '1703.08520-2-54-5': 'Even though the extra time for the "swap" and "random cr" schemes is just slightly smaller than for "augmented cr", this is a small price to pay for an improvement in mixing, especially compared to the high baseline cost of running an FHMM sampler.', '1703.08520-2-55-0': '### Tumor data analysis', '1703.08520-2-56-0': 'Next we consider whole-genome tumor sequencing data for bladder cancer .', '1703.08520-2-56-1': 'To illustrate the utility of our sampling approach, we used data from one patient (patient ID: 451) and took a thinned sample of 10,877 loci.', '1703.08520-2-56-2': 'We placed a vague Gaussian prior on the expected sequencing depth, [MATH] with [MATH], and integrated out [MATH], resulting in the marginal likelihood [EQUATION]', '1703.08520-2-56-3': 'Here each row of [MATH] corresponds to a single chromosome and the binary state indicates whether a copy of that DNA region exists or not.', '1703.08520-2-56-4': 'We fixed [MATH], where one of the latent sequences is always fixed to 1, representing a baseline, unaltered chromosome.', '1703.08520-2-56-5': 'We used a Hamming Ball Sampler with radius [MATH] as a single chain sampler, and its tempered ensemble versions "swap", "random cr", and "augmented cr".', '1703.08520-2-57-0': 'Since it is the sampling efficiency of the latent chains [MATH] in the FHMM rather than associated parameters that is the direct target of our sampler, we fixed [MATH] value to [MATH] in these experiments.', '1703.08520-2-57-1': 'As a result, all samplers would be exploring the same conditional posterior, and we are able to directly compare the subclonal configurations identified by various sampling algorithms.', '1703.08520-2-57-2': 'Otherwise, joint updating of the weights [MATH] (though entirely feasible) would lead to label swapping effects and the possibility of samplers exploring entirely different regimes that then make direct comparisons across sampling methods more challenging.', '1703.08520-2-58-0': 'Figure [REF] shows the log-posterior traces and the traces of [MATH] for selected chromosomes, when using ensembles of the HB[MATH] sampler.', '1703.08520-2-58-1': 'After a burn-in period of 10 000 iterations, the "augmented cr" ensemble has identified a probable configuration of [MATH] and it continues to explore parts of the state space which have higher posterior probability than those identified by other samplers.', '1703.08520-2-59-0': 'The augmented sampler is much better at capturing the uncertainty in underlying latent configurations (see Figure [REF](b)).', '1703.08520-2-59-1': 'For example, the third row corresponds to a subpopulation which has an extra copy of chromosome 21, but there is uncertainty whether it co-occurs with a whole extra copy of chromosome 22.', '1703.08520-2-59-2': 'Examining chromosome 17, the single-chain HB sampler and the "random cr" ensemble have identified a more fragmented latent configuration, whereas "swap" and "augmented cr" have combined these fragments into an alternative, more probable explanation.', '1703.08520-2-59-3': 'In biological terms, this is important since the more fragmented configuration would suggest a highly genomically unstable cancer genome related to a loss of genome integrity checkpoint mechanisms, whilst the alternative suggests a more moderate degree of instability.', '1703.08520-2-60-0': '# Conclusion', '1703.08520-2-61-0': 'We introduce an ensemble MCMC method to improve poorly mixing samplers for FHMMs.', '1703.08520-2-61-1': 'This is achieved by combining parallel tempering and a novel exchange move between pairs of chains achieved through an auxiliary variable augmentation.', '1703.08520-2-61-2': 'The former introduces a chain which explores the space freely and does not get stuck, whereas the latter provides an efficient procedure to exchange information between a tempered chain and our target.', '1703.08520-2-61-3': 'The proposed method is a general purpose ensemble MCMC approach, but its most natural application case are sequential models.', '1703.08520-2-61-4': 'Specifically, we see this most useful for a broad class of models assuming Markov structure, where the augmented crossover move can be carried out at a cheap extra computational cost.', '1703.08520-2-61-5': 'A natural extension of this work is to integrate our ensemble technique into a sampling scheme for targeting latent variables [MATH] and parameters [MATH] in a joint model [MATH].', '1703.08520-2-61-6': 'More exploration could also be carried out to explore optimal strategies for selecting or adapting the temperature ladder.', '1703.08520-2-61-7': 'However, our analyses suggest that for any given temperature ladder, the suggested augmented crossovers outperform non-augmented, classic approaches.'}

[['1703.08520-1-34-1', '1703.08520-2-5-1'], ['1703.08520-1-34-2', '1703.08520-2-5-2'], ['1703.08520-1-51-1', '1703.08520-2-46-1'], ['1703.08520-1-51-2', '1703.08520-2-46-2'], ['1703.08520-1-14-1', '1703.08520-2-21-0'], ['1703.08520-1-27-1', '1703.08520-2-2-1'], ['1703.08520-1-17-0', '1703.08520-2-24-0'], ['1703.08520-1-17-1', '1703.08520-2-24-1'], ['1703.08520-1-50-0', '1703.08520-2-43-0'], ['1703.08520-1-9-3', '1703.08520-2-12-2'], ['1703.08520-1-9-4', '1703.08520-2-12-3'], ['1703.08520-1-16-0', '1703.08520-2-23-0'], ['1703.08520-1-16-1', '1703.08520-2-23-1'], ['1703.08520-1-52-1', '1703.08520-2-47-2'], ['1703.08520-1-33-0', '1703.08520-2-4-0'], ['1703.08520-1-33-1', '1703.08520-2-4-1'], ['1703.08520-1-33-2', '1703.08520-2-4-2'], ['1703.08520-1-29-0', '1703.08520-2-3-0'], ['1703.08520-1-29-1', '1703.08520-2-3-1'], ['1703.08520-1-39-0', '1703.08520-2-30-0'], ['1703.08520-1-39-1', '1703.08520-2-30-1'], ['1703.08520-1-39-3', '1703.08520-2-30-3'], ['1703.08520-1-39-4', '1703.08520-2-30-4'], ['1703.08520-1-12-3', '1703.08520-2-17-1'], ['1703.08520-1-12-5', '1703.08520-2-18-0'], ['1703.08520-1-18-1', '1703.08520-2-26-1'], ['1703.08520-1-18-2', '1703.08520-2-26-2'], ['1703.08520-1-40-3', '1703.08520-2-31-2'], ['1703.08520-1-36-1', '1703.08520-2-29-1'], ['1703.08520-1-37-0', '1703.08520-2-29-2'], ['1703.08520-1-37-1', '1703.08520-2-29-3'], ['1703.08520-1-34-0', '1703.08520-2-5-0'], ['1703.08520-1-60-1', '1703.08520-2-61-1'], ['1703.08520-1-60-2', '1703.08520-2-61-2'], ['1703.08520-1-51-0', '1703.08520-2-46-0'], ['1703.08520-1-15-0', '1703.08520-2-22-0'], ['1703.08520-1-27-0', '1703.08520-2-2-0'], ['1703.08520-1-23-0', '1703.08520-2-27-0'], ['1703.08520-1-23-1', '1703.08520-2-27-1'], ['1703.08520-1-2-0', '1703.08520-2-9-0'], ['1703.08520-1-0-1', '1703.08520-2-0-2'], ['1703.08520-1-16-3', '1703.08520-2-23-4'], ['1703.08520-1-52-0', '1703.08520-2-47-0'], ['1703.08520-1-33-3', '1703.08520-2-4-3'], ['1703.08520-1-29-2', '1703.08520-2-3-2'], ['1703.08520-1-47-3', '1703.08520-2-38-2'], ['1703.08520-1-12-1', '1703.08520-2-16-1'], ['1703.08520-1-12-2', '1703.08520-2-17-0'], ['1703.08520-1-12-6', '1703.08520-2-18-1'], ['1703.08520-1-10-1', '1703.08520-2-14-0'], ['1703.08520-1-10-2', '1703.08520-2-14-5'], ['1703.08520-1-18-0', '1703.08520-2-26-0'], ['1703.08520-1-41-0', '1703.08520-2-31-4'], ['1703.08520-1-38-0', '1703.08520-2-29-4'], ['1703.08520-1-60-0', '1703.08520-2-61-0'], ['1703.08520-1-15-1', '1703.08520-2-22-1'], ['1703.08520-1-50-1', '1703.08520-2-43-1'], ['1703.08520-1-50-3', '1703.08520-2-43-3'], ['1703.08520-1-0-0', '1703.08520-2-0-0'], ['1703.08520-1-0-2', '1703.08520-2-0-3'], ['1703.08520-1-0-4', '1703.08520-2-0-0'], ['1703.08520-1-9-0', '1703.08520-2-12-0'], ['1703.08520-1-9-2', '1703.08520-2-12-1'], ['1703.08520-1-9-5', '1703.08520-2-12-4'], ['1703.08520-1-9-6', '1703.08520-2-12-5'], ['1703.08520-1-16-2', '1703.08520-2-23-2'], ['1703.08520-1-39-2', '1703.08520-2-30-2'], ['1703.08520-1-12-0', '1703.08520-2-16-0'], ['1703.08520-1-12-7', '1703.08520-2-18-2'], ['1703.08520-1-10-0', '1703.08520-2-13-0'], ['1703.08520-1-18-4', '1703.08520-2-26-3'], ['1703.08520-1-40-0', '1703.08520-2-31-0'], ['1703.08520-1-40-1', '1703.08520-2-31-0'], ['1703.08520-1-40-2', '1703.08520-2-31-1'], ['1703.08520-1-40-4', '1703.08520-2-31-3'], ['1703.08520-1-36-0', '1703.08520-2-29-0'], ['1703.08520-1-3-0', '1703.08520-2-9-1'], ['1703.08520-1-28-0', '1703.08520-2-2-2'], ['1703.08520-1-30-1', '1703.08520-2-31-0'], ['1703.08520-1-30-2', '1703.08520-2-3-3'], ['1703.08520-1-32-1', '1703.08520-2-3-7']]

[['1703.08520-1-34-1', '1703.08520-2-5-1'], ['1703.08520-1-34-2', '1703.08520-2-5-2'], ['1703.08520-1-51-1', '1703.08520-2-46-1'], ['1703.08520-1-51-2', '1703.08520-2-46-2'], ['1703.08520-1-14-1', '1703.08520-2-21-0'], ['1703.08520-1-27-1', '1703.08520-2-2-1'], ['1703.08520-1-17-0', '1703.08520-2-24-0'], ['1703.08520-1-17-1', '1703.08520-2-24-1'], ['1703.08520-1-50-0', '1703.08520-2-43-0'], ['1703.08520-1-9-3', '1703.08520-2-12-2'], ['1703.08520-1-9-4', '1703.08520-2-12-3'], ['1703.08520-1-16-0', '1703.08520-2-23-0'], ['1703.08520-1-16-1', '1703.08520-2-23-1'], ['1703.08520-1-52-1', '1703.08520-2-47-2'], ['1703.08520-1-33-0', '1703.08520-2-4-0'], ['1703.08520-1-33-1', '1703.08520-2-4-1'], ['1703.08520-1-33-2', '1703.08520-2-4-2'], ['1703.08520-1-29-0', '1703.08520-2-3-0'], ['1703.08520-1-29-1', '1703.08520-2-3-1'], ['1703.08520-1-39-0', '1703.08520-2-30-0'], ['1703.08520-1-39-1', '1703.08520-2-30-1'], ['1703.08520-1-39-3', '1703.08520-2-30-3'], ['1703.08520-1-39-4', '1703.08520-2-30-4'], ['1703.08520-1-12-3', '1703.08520-2-17-1'], ['1703.08520-1-12-5', '1703.08520-2-18-0'], ['1703.08520-1-18-1', '1703.08520-2-26-1'], ['1703.08520-1-18-2', '1703.08520-2-26-2'], ['1703.08520-1-40-3', '1703.08520-2-31-2'], ['1703.08520-1-36-1', '1703.08520-2-29-1'], ['1703.08520-1-37-0', '1703.08520-2-29-2'], ['1703.08520-1-37-1', '1703.08520-2-29-3']]

[['1703.08520-1-34-0', '1703.08520-2-5-0'], ['1703.08520-1-60-1', '1703.08520-2-61-1'], ['1703.08520-1-60-2', '1703.08520-2-61-2'], ['1703.08520-1-51-0', '1703.08520-2-46-0'], ['1703.08520-1-15-0', '1703.08520-2-22-0'], ['1703.08520-1-27-0', '1703.08520-2-2-0'], ['1703.08520-1-23-0', '1703.08520-2-27-0'], ['1703.08520-1-23-1', '1703.08520-2-27-1'], ['1703.08520-1-2-0', '1703.08520-2-9-0'], ['1703.08520-1-0-1', '1703.08520-2-0-2'], ['1703.08520-1-16-3', '1703.08520-2-23-4'], ['1703.08520-1-52-0', '1703.08520-2-47-0'], ['1703.08520-1-33-3', '1703.08520-2-4-3'], ['1703.08520-1-29-2', '1703.08520-2-3-2'], ['1703.08520-1-47-3', '1703.08520-2-38-2'], ['1703.08520-1-12-1', '1703.08520-2-16-1'], ['1703.08520-1-12-2', '1703.08520-2-17-0'], ['1703.08520-1-12-6', '1703.08520-2-18-1'], ['1703.08520-1-10-1', '1703.08520-2-14-0'], ['1703.08520-1-10-2', '1703.08520-2-14-5'], ['1703.08520-1-18-0', '1703.08520-2-26-0'], ['1703.08520-1-41-0', '1703.08520-2-31-4'], ['1703.08520-1-38-0', '1703.08520-2-29-4']]

[]

[['1703.08520-1-60-0', '1703.08520-2-61-0'], ['1703.08520-1-15-1', '1703.08520-2-22-1'], ['1703.08520-1-50-1', '1703.08520-2-43-1'], ['1703.08520-1-50-3', '1703.08520-2-43-3'], ['1703.08520-1-0-0', '1703.08520-2-0-0'], ['1703.08520-1-0-2', '1703.08520-2-0-3'], ['1703.08520-1-0-4', '1703.08520-2-0-0'], ['1703.08520-1-9-0', '1703.08520-2-12-0'], ['1703.08520-1-9-2', '1703.08520-2-12-1'], ['1703.08520-1-9-5', '1703.08520-2-12-4'], ['1703.08520-1-9-6', '1703.08520-2-12-5'], ['1703.08520-1-16-2', '1703.08520-2-23-2'], ['1703.08520-1-39-2', '1703.08520-2-30-2'], ['1703.08520-1-12-0', '1703.08520-2-16-0'], ['1703.08520-1-12-7', '1703.08520-2-18-2'], ['1703.08520-1-10-0', '1703.08520-2-13-0'], ['1703.08520-1-18-4', '1703.08520-2-26-3'], ['1703.08520-1-40-0', '1703.08520-2-31-0'], ['1703.08520-1-40-1', '1703.08520-2-31-0'], ['1703.08520-1-40-2', '1703.08520-2-31-1'], ['1703.08520-1-40-4', '1703.08520-2-31-3'], ['1703.08520-1-36-0', '1703.08520-2-29-0']]

[['1703.08520-1-3-0', '1703.08520-2-9-1'], ['1703.08520-1-28-0', '1703.08520-2-2-2'], ['1703.08520-1-30-1', '1703.08520-2-31-0'], ['1703.08520-1-30-2', '1703.08520-2-3-3'], ['1703.08520-1-32-1', '1703.08520-2-3-7']]

['1703.08520-1-13-0', '1703.08520-1-21-0', '1703.08520-1-22-0', '1703.08520-1-39-5', '1703.08520-1-56-2', '1703.08520-2-19-0', '1703.08520-2-20-0', '1703.08520-2-38-1', '1703.08520-2-49-1']

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

https://arxiv.org/abs/1703.08520

0911.3619

{'0911.3619-1-0-0': 'We compute two-photon exchange corrections to the electromagnetic form factor of the pion, taking into account the finite size of the pion.', '0911.3619-1-0-1': 'Compared to the soft-photon approximation for the infrared divergent contribution which neglects hadron structure effects, the corrections are found to be [MATH] for small [MATH] GeV[MATH]), but increase to several percent for [MATH] GeV[MATH] at extreme backward angles.', '0911.3619-1-1-0': 'As the lightest bound state of quarks, the pion plays a unique role in QCD.', '0911.3619-1-1-1': 'On the one hand, its anomalously small mass leads to the identification of the pion with the pseudo-Goldstone mode of dynamical chiral symmetry breaking in QCD.', '0911.3619-1-1-2': 'On the other, scattering experiments reveal a rich substructure which can be best described in terms of its quark constituents.', '0911.3619-1-1-3': 'The most basic observable which characterizes the structure of the pion is its electromagnetic form factor, [MATH], where [MATH] is the four-momentum transfer squared.', '0911.3619-1-2-0': 'The extractions of the pion form factor in the space-like region ([MATH]) from measurements of the pion electroproduction reaction [MATH] have recently provided high-precision data on the [MATH] dependence of [MATH] up to values of [MATH] GeV[MATH] [CITATION] and higher-[MATH] measurements are planned to [MATH] GeV[MATH].', '0911.3619-1-2-1': 'These complement low-[MATH] data obtained by scattering pions from the electrons of a hydrogen target.', '0911.3619-1-3-0': 'As is standard in most electromagnetic scattering analyses, the pion form factor has been obtained from data assuming the validity of the one-photon exchange, or Born, approximation.', '0911.3619-1-3-1': 'Recently the accuracy of one-photon exchange approximation has been called into question by the observation of a large discrepancy between the proton electric to magnetic form factor ratio in measurements using Rosenbluth separation and polarization transfer [CITATION].', '0911.3619-1-3-2': 'A number of detailed studies have demonstrated that these can be mostly understood once radiative corrections arising from two-photon exchange are included, in particular those associated with hadron finite-size effects.', '0911.3619-1-3-3': 'These findings have prompted exploration of the significance of two-photon exchange in other reactions (see Refs. [CITATION] for reviews).', '0911.3619-1-4-0': 'In this paper we investigate the role of two-photon exchange (TPE) in electromagnetic scattering from the pion.', '0911.3619-1-4-1': 'We use the methodology developed for the application of TPE to scattering from the nucleon [CITATION], suitably modified to the scalar case.', '0911.3619-1-4-2': 'The analysis of TPE from a spin-0 target is, in fact, considerably simpler than that for spin-[MATH] targets.', '0911.3619-1-5-0': 'For the elastic electron-pion scattering process, we follow the notation of Refs. [CITATION] and define the momenta of the initial electron and pion as [MATH] and [MATH], and of the final electron and pion as [MATH] and [MATH], respectively, [MATH].', '0911.3619-1-5-1': 'The matrix element of the pion current is given by [EQUATION] where [MATH].', '0911.3619-1-5-2': 'In the one-photon approximation the scattering amplitude is given by [EQUATION]', '0911.3619-1-5-3': 'In the target rest frame of the pion the Born cross section can then be written simply as [EQUATION] where [EQUATION] is the Mott cross section for electron scattering from a point particle, with [MATH] and [MATH] the initial and final electron energies, and [MATH] the electromagnetic fine structure constant.', '0911.3619-1-6-0': 'Including [MATH] radiative corrections leads to a modification of the Born cross section arising from vertex corrections, vacuum polarization, inelastic bremsstrahlung, and two-photon exchange.', '0911.3619-1-6-1': 'As discussed in Refs. [CITATION], only the latter lead to a dependence on the scattering angle, or equivalently on the virtual photon polarization parameter [MATH], where [MATH], and [MATH] is the pion mass.', '0911.3619-1-6-2': 'While the scattering angle naturally depends on the reference frame, we can more generally express [MATH] in terms of Lorentz invariants as [EQUATION] where [MATH].', '0911.3619-1-6-3': 'In the target rest frame we have [MATH].', '0911.3619-1-7-0': 'The total TPE amplitude, including the box and crossed-box diagrams, has the form [EQUATION] where [MATH] and [MATH] are the momenta of the virtual photons, with [MATH].', '0911.3619-1-7-1': 'The parameter [MATH] is introduced as an infinitesimal photon mass in the photon propagators to regulate the infrared (IR) divergences.', '0911.3619-1-7-2': 'The leptonic tensor [MATH] is given by [EQUATION] where [MATH] is the electron mass.', '0911.3619-1-7-3': 'The hadronic tensor [MATH] in principle contains contributions from all excitations of the initial state.', '0911.3619-1-7-4': 'In practice we approximate this by the pion elastic contribution [EQUATION]', '0911.3619-1-7-5': 'The TPE contribution to the cross section is then given by the interference of the TPE amplitude [MATH] and the Born amplitude [MATH].', '0911.3619-1-7-6': 'This can be parametrized in terms of a multiplicative correction [MATH], where [EQUATION]', '0911.3619-1-7-7': 'The pion form factor is then modified according to [EQUATION]', '0911.3619-1-8-0': 'Experimental analyses of electromagnetic form factor data typically use radiative corrections computed by Mo Tsai (MT) in the soft-photon approximation [CITATION], in which hadronic structure effects are neglected.', '0911.3619-1-8-1': 'The TPE corrections are approximated by taking only the IR-divergent contribution at the photon poles, setting [MATH] and [MATH] in the numerator of Eq. ([REF]).', '0911.3619-1-8-2': 'In this approximation [MATH] becomes proportional to the Born amplitude [MATH], and the corresponding correction [MATH] to the Born cross section is independent of hadronic structure (or indeed of the type of hadronic target).', '0911.3619-1-8-3': 'Mo Tsai approximate the remaining loop integration by further reducing it to a 3-point function [MATH], where [MATH], with the total box plus crossed-box contribution given by [EQUATION]', '0911.3619-1-8-4': 'The logarithmic IR singularity in [MATH] is exactly canceled by a similar singularity arising from the bremsstrahlung correction involving the interference between real photon emission from the electron and from the pion.', '0911.3619-1-9-0': 'To quantify the effect of the IR-finite, hadron structure dependent contribution, in Fig. [REF] we show the difference between the full TPE correction [MATH] and the MT prescription [CITATION] as a function of [MATH] for various [MATH].', '0911.3619-1-9-1': 'In the numerical calculations we use a monopole parametrization for the "bare" pion form factor in Eq. ([REF]), [EQUATION] with [MATH] MeV corresponding to the [MATH]-meson mass.', '0911.3619-1-9-2': 'The loop integrals of Eq. ([REF]) can then be done analytically, and expressed in terms of Passarino-Veltman 2-, 3-, and 4-point functions [CITATION].', '0911.3619-1-9-3': 'In the calculations we use the computer program FEYNCALC [CITATION].', '0911.3619-1-10-0': 'At low [MATH] GeV[MATH]) the TPE correction is positive and of the order of 1% at backward angles (small [MATH]), decreasing to zero in the [MATH] (forward angle) limit.', '0911.3619-1-10-1': 'With increasing [MATH] the correction becomes smaller (more negative) up to [MATH] GeV[MATH], especially in the extreme backward region ([MATH]), but changes sign at intermediate [MATH].', '0911.3619-1-10-2': 'Note, however, that unlike electron-proton scattering, the electron-pion scattering cross section vanishes at the extreme backward angles limit ([MATH]).', '0911.3619-1-10-3': 'Above [MATH] GeV[MATH] the correction grows once again, reaching [MATH] at [MATH] GeV[MATH].', '0911.3619-1-11-0': 'The [MATH] dependence is more clearly illustrated in Fig. [REF], where [MATH] is shown for fixed [MATH] over the range [MATH] GeV[MATH].', '0911.3619-1-11-1': 'Interestingly, the correction is most positive at very small [MATH] GeV[MATH] and large [MATH] GeV[MATH], reaching its minimum values at [MATH] GeV[MATH].', '0911.3619-1-11-2': 'At small [MATH] the [MATH] dependence is seen to change most rapidly.', '0911.3619-1-12-0': 'While the monopole parametrization is known to give a good description of the pion form factor data at low [MATH], it overestimates [MATH] at larger [MATH].', '0911.3619-1-12-1': 'An alternative parametrization to the monopole which fits the available data and builds in gauge invariance constraints for the [MATH] limit and perturbative QCD expectations for the [MATH] behavior was given in Ref. [CITATION].', '0911.3619-1-12-2': 'Using this parametrization the TPE correction [MATH] is shown in Fig. [REF] (right panel).', '0911.3619-1-12-3': 'As expected, the differences at low [MATH] are negligible, but become noticeable at high [MATH].', '0911.3619-1-12-4': 'The qualitative behavior of the corrections, however, is not affected by the specific form chosen.', '0911.3619-1-13-0': 'We should note that the effects illustrated in Figs. [REF] and [REF] are not physical, but merely reflect the accuracy with which the full result can be approximated by the particular prescription for the IR-divergent contribution.', '0911.3619-1-13-1': 'It is only relevant because the MT approximation is widely used for applications of radiative corrections in analyses of electron scattering data [CITATION].', '0911.3619-1-13-2': 'An alternative prescription was suggested by Maximon Tjon (MTj) [CITATION], who also approximate the TPE by the contributions at the photon poles, setting [MATH] and [MATH] in the numerator of Eq. ([REF]).', '0911.3619-1-13-3': 'However, no further approximation is made in evaluating the remaining loop integration, which can be written in terms of 4-point functions.', '0911.3619-1-13-4': 'In the limit [MATH] the 4-point functions simplify significantly, and the TPE correction [MATH] can be written as [EQUATION]', '0911.3619-1-13-5': 'As before, the logarithmic [MATH] singularity is cancelled exactly by the inelastic bremsstrahlung contribution.', '0911.3619-1-14-0': 'The difference between the full, hadron structure dependent result for [MATH] and the MTj approximation is illustrated in Fig. [REF] as a function of [MATH].', '0911.3619-1-14-1': 'At low [MATH] GeV[MATH]) the correction is similar to that in Fig. [REF] for the MT IR prescription, but is generally smaller in magnitude at larger [MATH].', '0911.3619-1-14-2': 'In particular, it displays a somewhat milder [MATH] dependence for backward angles at higher [MATH].', '0911.3619-1-14-3': 'Qualitatively, however, the behavior of the correction as a function of [MATH] is similar for the MTj and MT prescriptions, as Fig. [REF] shows, with [MATH] most positive at low [MATH], [MATH] GeV[MATH] (for all [MATH]), decreasing to a minimum at [MATH] GeV[MATH], before rising again for larger [MATH].', '0911.3619-1-15-0': 'The reliability of the results at high [MATH] GeV[MATH]) may be more questionable since only pion elastic intermediate states are included in this analysis, and we expect excited hadronic states to play a greater role with increasing [MATH] [CITATION].', '0911.3619-1-15-1': 'However, since the mass difference between the pion and the next excited resonant state, the [MATH] meson, is almost 5 times as large as the pion mass, we would not expect these contributions to be significant for the pion.', '0911.3619-1-15-2': 'The nonresonant contributions and the off-shell dependence of the pion form factor, on the other hand, may need to be examined in future analyses.', '0911.3619-1-16-0': 'In summary, we have computed the two-photon exchange contribution to the pion electromagnetic form factor arising from the finite size of the pion.', '0911.3619-1-16-1': 'Compared with the standard infrared contribution computed in the soft-photon approximation, the hadron structure dependent corrections are [MATH] at low [MATH] GeV[MATH]), but increase to several percent at [MATH] GeV[MATH] at backward angles.', '0911.3619-1-16-2': 'These contributions will need to be taken into account in the treatment of radiative corrections to pion electroproduction data in high-precision extractions of the pion form factor.', '0911.3619-1-17-0': 'We thank J. Arrington for helpful discussions.', '0911.3619-1-17-1': 'This work was supported in part by NSERC (Canada), DOE grant DE-FG02-93ER-40762, and DOE contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab.'}

{'0911.3619-2-0-0': 'We compute two-photon exchange corrections to the electromagnetic form factor of the pion, taking into account the finite size of the pion.', '0911.3619-2-0-1': 'Compared to the soft-photon approximation for the infrared divergent contribution which neglects hadron structure effects, the corrections are found to be [MATH] for small [MATH] GeV[MATH]), but increase to several percent for [MATH] GeV[MATH] at extreme backward angles.', '0911.3619-2-1-0': 'As the lightest bound state of quarks, the pion plays a unique role in QCD.', '0911.3619-2-1-1': 'On the one hand, its anomalously small mass leads to the identification of the pion with the pseudo-Goldstone mode of dynamical chiral symmetry breaking in QCD.', '0911.3619-2-1-2': 'On the other, scattering experiments reveal a rich substructure which can be best described in terms of its quark constituents.', '0911.3619-2-1-3': 'The most basic observable which characterizes the structure of the pion is its electromagnetic form factor, [MATH], where [MATH] is the four-momentum transfer squared.', '0911.3619-2-2-0': 'The extractions of the pion form factor in the space-like region ([MATH]) from measurements of the pion electroproduction reaction [MATH] have recently provided data on the [MATH] dependence of [MATH] up to values of [MATH] GeV[MATH] [CITATION], and higher-[MATH] measurements are planned to [MATH] GeV[MATH].', '0911.3619-2-2-1': 'The main uncertainty in the extraction of [MATH] is the need to use models to extrapolate the longitudinal electroprodution cross section to the physical pion mass assuming pion pole dominance of the [MATH]-channel process [CITATION].', '0911.3619-2-2-2': 'The pion electroprodution experiments complement low-[MATH] data obtained by scattering pions from the electrons of a hydrogen target.', '0911.3619-2-3-0': 'As is standard in most electromagnetic scattering analyses, the pion form factor has been obtained from data assuming the validity of the one-photon exchange, or Born, approximation.', '0911.3619-2-3-1': 'Recently the accuracy of one-photon exchange approximation has been called into question by the observation of a large discrepancy between the proton electric to magnetic form factor ratio in measurements using Rosenbluth separation and polarization transfer [CITATION].', '0911.3619-2-3-2': 'A number of detailed studies have demonstrated that these can be mostly understood once radiative corrections arising from two-photon exchange are included, in particular those associated with hadron finite-size effects.', '0911.3619-2-3-3': 'These findings have prompted exploration of the significance of two-photon exchange in other reactions (see Refs. [CITATION] for reviews).', '0911.3619-2-4-0': 'In this paper we investigate the role of two-photon exchange (TPE) in electromagnetic scattering from the pion.', '0911.3619-2-4-1': 'We use the methodology developed for the application of TPE to scattering from the nucleon [CITATION], suitably modified to the scalar case.', '0911.3619-2-4-2': 'The analysis of TPE from a spin-0 target is, in fact, considerably simpler than that for spin-[MATH] targets.', '0911.3619-2-5-0': 'For the elastic electron-pion scattering process, we follow the notation of Refs. [CITATION] and define the momenta of the initial electron and pion as [MATH] and [MATH], and of the final electron and pion as [MATH] and [MATH], respectively, [MATH].', '0911.3619-2-5-1': 'The matrix element of the pion current is given by [EQUATION] where [MATH].', '0911.3619-2-5-2': 'In the one-photon approximation the scattering amplitude is given by [EQUATION]', '0911.3619-2-5-3': 'In the target rest frame of the pion the Born cross section can then be written simply as [EQUATION] where [EQUATION] is the Mott cross section for electron scattering from a point particle, with [MATH] and [MATH] the initial and final electron energies, and [MATH] the electromagnetic fine structure constant.', '0911.3619-2-6-0': 'Including [MATH] radiative corrections leads to a modification of the Born cross section arising from vertex corrections, vacuum polarization, inelastic bremsstrahlung, and two-photon exchange.', '0911.3619-2-6-1': 'As discussed in Refs. [CITATION], only the latter lead to a dependence on the scattering angle, or equivalently on the virtual photon polarization parameter [MATH], where [MATH], and [MATH] is the pion mass.', '0911.3619-2-6-2': 'While the scattering angle naturally depends on the reference frame, we can more generally express [MATH] in terms of Lorentz invariants as [EQUATION] where [MATH].', '0911.3619-2-6-3': 'In the target rest frame we have [MATH].', '0911.3619-2-7-0': 'The total TPE amplitude, including the box and crossed-box diagrams, has the form [EQUATION] where [MATH] and [MATH] are the momenta of the virtual photons, with [MATH].', '0911.3619-2-7-1': 'The parameter [MATH] is introduced as an infinitesimal photon mass in the photon propagators to regulate the infrared (IR) divergences.', '0911.3619-2-7-2': 'The leptonic tensor [MATH] is given by [EQUATION] where [MATH] is the electron mass.', '0911.3619-2-7-3': 'The hadronic tensor [MATH] in principle contains contributions from all excitations of the initial state.', '0911.3619-2-7-4': 'In practice we approximate this by the pion elastic contribution [EQUATION]', '0911.3619-2-7-5': 'The TPE contribution to the cross section is then given by the interference of the TPE amplitude [MATH] and the Born amplitude [MATH].', '0911.3619-2-7-6': 'This can be parametrized in terms of a multiplicative correction [MATH], where [EQUATION]', '0911.3619-2-7-7': 'The pion form factor is then modified according to [EQUATION]', '0911.3619-2-8-0': 'Experimental analyses of electromagnetic form factor data typically use radiative corrections computed by Mo Tsai (MT) in the soft-photon approximation [CITATION], in which hadronic structure effects are neglected.', '0911.3619-2-8-1': 'The TPE corrections are approximated by taking only the IR-divergent contribution at the photon poles, setting [MATH] and [MATH] in the numerator of Eq. ([REF]).', '0911.3619-2-8-2': 'In this approximation [MATH] becomes proportional to the Born amplitude [MATH], and the corresponding correction [MATH] to the Born cross section is independent of hadronic structure (or indeed of the type of hadronic target).', '0911.3619-2-8-3': 'Mo Tsai approximate the remaining loop integration by further reducing it to a 3-point function [MATH], where [MATH], with the total box plus crossed-box contribution given by [EQUATION]', '0911.3619-2-8-4': 'The logarithmic IR singularity in [MATH] is exactly canceled by a similar singularity arising from the bremsstrahlung correction involving the interference between real photon emission from the electron and from the pion.', '0911.3619-2-9-0': 'To quantify the effect of the IR-finite, hadron structure dependent contribution, in Fig. [REF] we show the difference between the full TPE correction [MATH] and the MT prescription [CITATION] as a function of [MATH] for various [MATH].', '0911.3619-2-9-1': 'In the numerical calculations we use a monopole parametrization for the "bare" pion form factor in Eq. ([REF]), [EQUATION] with [MATH] MeV corresponding to the [MATH]-meson mass.', '0911.3619-2-9-2': 'The loop integrals of Eq. ([REF]) can then be done analytically, and expressed in terms of Passarino-Veltman 2-, 3-, and 4-point functions [CITATION].', '0911.3619-2-9-3': 'In the calculations we use the computer program FEYNCALC [CITATION].', '0911.3619-2-10-0': 'At low [MATH] GeV[MATH]) the TPE correction is positive and of the order of 1% at backward angles (small [MATH]), decreasing to zero in the [MATH] (forward angle) limit.', '0911.3619-2-10-1': 'With increasing [MATH] the correction becomes smaller (more negative) up to [MATH] GeV[MATH], especially in the extreme backward region ([MATH]), but changes sign at intermediate [MATH].', '0911.3619-2-10-2': 'Note, however, that unlike electron-proton scattering, the electron-pion scattering cross section vanishes at the extreme backward angles limit ([MATH]).', '0911.3619-2-10-3': 'Above [MATH] GeV[MATH] the correction grows once again, reaching [MATH] at [MATH] GeV[MATH].', '0911.3619-2-11-0': 'The [MATH] dependence is more clearly illustrated in Fig. [REF], where [MATH] is shown for fixed [MATH] over the range [MATH] GeV[MATH].', '0911.3619-2-11-1': 'Interestingly, the correction is most positive at very small [MATH] GeV[MATH] and large [MATH] GeV[MATH], reaching its minimum values at [MATH] GeV[MATH].', '0911.3619-2-11-2': 'At small [MATH] the [MATH] dependence is seen to change most rapidly.', '0911.3619-2-12-0': 'While the monopole parametrization is known to give a good description of the pion form factor data at low [MATH], it overestimates [MATH] at larger [MATH].', '0911.3619-2-12-1': 'An alternative parametrization to the monopole which fits the available data and builds in gauge invariance constraints for the [MATH] limit and perturbative QCD expectations for the [MATH] behavior was given in Ref. [CITATION].', '0911.3619-2-12-2': 'Using this parametrization the TPE correction [MATH] is shown in Fig. [REF] (right panel).', '0911.3619-2-12-3': 'As expected, the differences at low [MATH] are negligible, but become noticeable at high [MATH].', '0911.3619-2-12-4': 'The qualitative behavior of the corrections, however, is not affected by the specific form chosen.', '0911.3619-2-13-0': 'We should note that the effects illustrated in Figs. [REF] and [REF] are not physical, but merely reflect the accuracy with which the full result can be approximated by the particular prescription for the IR-divergent contribution.', '0911.3619-2-13-1': 'It is only relevant because the MT approximation is widely used for applications of radiative corrections in analyses of electron scattering data [CITATION].', '0911.3619-2-13-2': 'An alternative prescription was suggested by Maximon Tjon (MTj) [CITATION], who also approximate the TPE by the contributions at the photon poles, setting [MATH] and [MATH] in the numerator of Eq. ([REF]).', '0911.3619-2-13-3': 'However, no further approximation is made in evaluating the remaining loop integration, which can be written in terms of 4-point functions.', '0911.3619-2-13-4': 'In the limit [MATH] the 4-point functions simplify significantly, and the TPE correction [MATH] can be written as [EQUATION]', '0911.3619-2-13-5': 'As before, the logarithmic [MATH] singularity is cancelled exactly by the inelastic bremsstrahlung contribution.', '0911.3619-2-14-0': 'The difference between the full, hadron structure dependent result for [MATH] and the MTj approximation is illustrated in Fig. [REF] as a function of [MATH].', '0911.3619-2-14-1': 'At low [MATH] GeV[MATH]) the correction is similar to that in Fig. [REF] for the MT IR prescription, but is generally smaller in magnitude at larger [MATH].', '0911.3619-2-14-2': 'In particular, it displays a somewhat milder [MATH] dependence for backward angles at higher [MATH].', '0911.3619-2-14-3': 'Qualitatively, however, the behavior of the correction as a function of [MATH] is similar for the MTj and MT prescriptions, as Fig. [REF] shows, with [MATH] most positive at low [MATH], [MATH] GeV[MATH] (for all [MATH]), decreasing to a minimum at [MATH] GeV[MATH], before rising again for larger [MATH].', '0911.3619-2-15-0': 'Note that in contrast to the proton form factor case, where the TPE effects give large corrections to the elastic form factors extracted from longitudinal-transverse (LT) separated cross sections at large [MATH] [CITATION], the TPE corrections to the pion form factor are relatively small.', '0911.3619-2-15-1': 'This is related to the fact that electron scattering from a spin-0 target is described by a single form factor, with no LT separation necessary.', '0911.3619-2-15-2': 'On the other hand, since [MATH] is extracted by performing an LT separation of the pion electroprodution cross section, TPE with one photon attached to the pion and the other to the initial proton or final neutron could modify the longitudinal cross section, and may need to be considered.', '0911.3619-2-16-0': 'The reliability of the results at high [MATH] GeV[MATH]) may be more questionable since only pion elastic intermediate states are included in this analysis, and we expect excited hadronic states to play a greater role with increasing [MATH] [CITATION].', '0911.3619-2-16-1': 'However, since the mass difference between the pion and the next excited resonant state, the [MATH] meson, is almost 5 times as large as the pion mass, we would not expect these contributions to be significant for the pion.', '0911.3619-2-16-2': 'The nonresonant contributions and the off-shell dependence of the pion form factor, on the other hand, may need to be examined in future analyses.', '0911.3619-2-17-0': 'In summary, we have computed the two-photon exchange contribution to the pion electromagnetic form factor arising from the finite size of the pion.', '0911.3619-2-17-1': 'Compared with the standard infrared contribution computed in the soft-photon approximation, the hadron structure dependent corrections are [MATH] at low [MATH] GeV[MATH]), but increase to several percent at [MATH] GeV[MATH] at backward angles.', '0911.3619-2-17-2': 'These contributions will need to be taken into account in the treatment of radiative corrections for extractions of the pion form factor from high-precision pion electroproduction data.', '0911.3619-2-18-0': 'We thank J. Arrington, D. Gaskell and G. Huber for helpful discussions.', '0911.3619-2-18-1': 'P. G. B. thanks the Theory Center at Jefferson Lab for support during a sabbatical leave, where this work was performed.', '0911.3619-2-18-2': 'This work was supported in part by NSERC (Canada), DOE grant DE-FG02-93ER-40762, and DOE contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab.'}

[['0911.3619-1-10-0', '0911.3619-2-10-0'], ['0911.3619-1-10-1', '0911.3619-2-10-1'], ['0911.3619-1-10-2', '0911.3619-2-10-2'], ['0911.3619-1-10-3', '0911.3619-2-10-3'], ['0911.3619-1-6-0', '0911.3619-2-6-0'], ['0911.3619-1-6-1', '0911.3619-2-6-1'], ['0911.3619-1-6-2', '0911.3619-2-6-2'], ['0911.3619-1-6-3', '0911.3619-2-6-3'], ['0911.3619-1-1-0', '0911.3619-2-1-0'], ['0911.3619-1-1-1', '0911.3619-2-1-1'], ['0911.3619-1-1-2', '0911.3619-2-1-2'], ['0911.3619-1-1-3', '0911.3619-2-1-3'], ['0911.3619-1-13-0', '0911.3619-2-13-0'], ['0911.3619-1-13-1', '0911.3619-2-13-1'], ['0911.3619-1-13-2', '0911.3619-2-13-2'], ['0911.3619-1-13-3', '0911.3619-2-13-3'], ['0911.3619-1-13-4', '0911.3619-2-13-4'], ['0911.3619-1-13-5', '0911.3619-2-13-5'], ['0911.3619-1-15-0', '0911.3619-2-16-0'], ['0911.3619-1-15-1', '0911.3619-2-16-1'], ['0911.3619-1-15-2', '0911.3619-2-16-2'], ['0911.3619-1-4-0', '0911.3619-2-4-0'], ['0911.3619-1-4-1', '0911.3619-2-4-1'], ['0911.3619-1-4-2', '0911.3619-2-4-2'], ['0911.3619-1-11-0', '0911.3619-2-11-0'], ['0911.3619-1-11-1', '0911.3619-2-11-1'], ['0911.3619-1-11-2', '0911.3619-2-11-2'], ['0911.3619-1-5-0', '0911.3619-2-5-0'], ['0911.3619-1-5-1', '0911.3619-2-5-1'], ['0911.3619-1-5-2', '0911.3619-2-5-2'], ['0911.3619-1-5-3', '0911.3619-2-5-3'], ['0911.3619-1-3-0', '0911.3619-2-3-0'], ['0911.3619-1-3-1', '0911.3619-2-3-1'], ['0911.3619-1-3-2', '0911.3619-2-3-2'], ['0911.3619-1-3-3', '0911.3619-2-3-3'], ['0911.3619-1-9-0', '0911.3619-2-9-0'], ['0911.3619-1-9-1', '0911.3619-2-9-1'], ['0911.3619-1-9-2', '0911.3619-2-9-2'], ['0911.3619-1-9-3', '0911.3619-2-9-3'], ['0911.3619-1-12-0', '0911.3619-2-12-0'], ['0911.3619-1-12-1', '0911.3619-2-12-1'], ['0911.3619-1-12-2', '0911.3619-2-12-2'], ['0911.3619-1-12-3', '0911.3619-2-12-3'], ['0911.3619-1-12-4', '0911.3619-2-12-4'], ['0911.3619-1-7-0', '0911.3619-2-7-0'], ['0911.3619-1-7-1', '0911.3619-2-7-1'], ['0911.3619-1-7-2', '0911.3619-2-7-2'], ['0911.3619-1-7-3', '0911.3619-2-7-3'], ['0911.3619-1-7-4', '0911.3619-2-7-4'], ['0911.3619-1-7-5', '0911.3619-2-7-5'], ['0911.3619-1-7-6', '0911.3619-2-7-6'], ['0911.3619-1-7-7', '0911.3619-2-7-7'], ['0911.3619-1-14-0', '0911.3619-2-14-0'], ['0911.3619-1-14-1', '0911.3619-2-14-1'], ['0911.3619-1-14-2', '0911.3619-2-14-2'], ['0911.3619-1-14-3', '0911.3619-2-14-3'], ['0911.3619-1-16-0', '0911.3619-2-17-0'], ['0911.3619-1-16-1', '0911.3619-2-17-1'], ['0911.3619-1-17-1', '0911.3619-2-18-2'], ['0911.3619-1-0-0', '0911.3619-2-0-0'], ['0911.3619-1-0-1', '0911.3619-2-0-1'], ['0911.3619-1-8-0', '0911.3619-2-8-0'], ['0911.3619-1-8-1', '0911.3619-2-8-1'], ['0911.3619-1-8-2', '0911.3619-2-8-2'], ['0911.3619-1-8-3', '0911.3619-2-8-3'], ['0911.3619-1-8-4', '0911.3619-2-8-4'], ['0911.3619-1-2-0', '0911.3619-2-2-0'], ['0911.3619-1-2-1', '0911.3619-2-2-2'], ['0911.3619-1-16-2', '0911.3619-2-17-2'], ['0911.3619-1-17-0', '0911.3619-2-18-0']]

[['0911.3619-1-10-0', '0911.3619-2-10-0'], ['0911.3619-1-10-1', '0911.3619-2-10-1'], ['0911.3619-1-10-2', '0911.3619-2-10-2'], ['0911.3619-1-10-3', '0911.3619-2-10-3'], ['0911.3619-1-6-0', '0911.3619-2-6-0'], ['0911.3619-1-6-1', '0911.3619-2-6-1'], ['0911.3619-1-6-2', '0911.3619-2-6-2'], ['0911.3619-1-6-3', '0911.3619-2-6-3'], ['0911.3619-1-1-0', '0911.3619-2-1-0'], ['0911.3619-1-1-1', '0911.3619-2-1-1'], ['0911.3619-1-1-2', '0911.3619-2-1-2'], ['0911.3619-1-1-3', '0911.3619-2-1-3'], ['0911.3619-1-13-0', '0911.3619-2-13-0'], ['0911.3619-1-13-1', '0911.3619-2-13-1'], ['0911.3619-1-13-2', '0911.3619-2-13-2'], ['0911.3619-1-13-3', '0911.3619-2-13-3'], ['0911.3619-1-13-4', '0911.3619-2-13-4'], ['0911.3619-1-13-5', '0911.3619-2-13-5'], ['0911.3619-1-15-0', '0911.3619-2-16-0'], ['0911.3619-1-15-1', '0911.3619-2-16-1'], ['0911.3619-1-15-2', '0911.3619-2-16-2'], ['0911.3619-1-4-0', '0911.3619-2-4-0'], ['0911.3619-1-4-1', '0911.3619-2-4-1'], ['0911.3619-1-4-2', '0911.3619-2-4-2'], ['0911.3619-1-11-0', '0911.3619-2-11-0'], ['0911.3619-1-11-1', '0911.3619-2-11-1'], ['0911.3619-1-11-2', '0911.3619-2-11-2'], ['0911.3619-1-5-0', '0911.3619-2-5-0'], ['0911.3619-1-5-1', '0911.3619-2-5-1'], ['0911.3619-1-5-2', '0911.3619-2-5-2'], ['0911.3619-1-5-3', '0911.3619-2-5-3'], ['0911.3619-1-3-0', '0911.3619-2-3-0'], ['0911.3619-1-3-1', '0911.3619-2-3-1'], ['0911.3619-1-3-2', '0911.3619-2-3-2'], ['0911.3619-1-3-3', '0911.3619-2-3-3'], ['0911.3619-1-9-0', '0911.3619-2-9-0'], ['0911.3619-1-9-1', '0911.3619-2-9-1'], ['0911.3619-1-9-2', '0911.3619-2-9-2'], ['0911.3619-1-9-3', '0911.3619-2-9-3'], ['0911.3619-1-12-0', '0911.3619-2-12-0'], ['0911.3619-1-12-1', '0911.3619-2-12-1'], ['0911.3619-1-12-2', '0911.3619-2-12-2'], ['0911.3619-1-12-3', '0911.3619-2-12-3'], ['0911.3619-1-12-4', '0911.3619-2-12-4'], ['0911.3619-1-7-0', '0911.3619-2-7-0'], ['0911.3619-1-7-1', '0911.3619-2-7-1'], ['0911.3619-1-7-2', '0911.3619-2-7-2'], ['0911.3619-1-7-3', '0911.3619-2-7-3'], ['0911.3619-1-7-4', '0911.3619-2-7-4'], ['0911.3619-1-7-5', '0911.3619-2-7-5'], ['0911.3619-1-7-6', '0911.3619-2-7-6'], ['0911.3619-1-7-7', '0911.3619-2-7-7'], ['0911.3619-1-14-0', '0911.3619-2-14-0'], ['0911.3619-1-14-1', '0911.3619-2-14-1'], ['0911.3619-1-14-2', '0911.3619-2-14-2'], ['0911.3619-1-14-3', '0911.3619-2-14-3'], ['0911.3619-1-16-0', '0911.3619-2-17-0'], ['0911.3619-1-16-1', '0911.3619-2-17-1'], ['0911.3619-1-17-1', '0911.3619-2-18-2'], ['0911.3619-1-0-0', '0911.3619-2-0-0'], ['0911.3619-1-0-1', '0911.3619-2-0-1'], ['0911.3619-1-8-0', '0911.3619-2-8-0'], ['0911.3619-1-8-1', '0911.3619-2-8-1'], ['0911.3619-1-8-2', '0911.3619-2-8-2'], ['0911.3619-1-8-3', '0911.3619-2-8-3'], ['0911.3619-1-8-4', '0911.3619-2-8-4']]

[['0911.3619-1-2-0', '0911.3619-2-2-0'], ['0911.3619-1-2-1', '0911.3619-2-2-2'], ['0911.3619-1-16-2', '0911.3619-2-17-2']]

[]

[['0911.3619-1-17-0', '0911.3619-2-18-0']]

[]

[]

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

https://arxiv.org/abs/0911.3619

1701.05086

{'1701.05086-1-0-0': "This paper presents data and a model for supercooled squalane's frequency-dependent shear modulus covering frequencies from 10 mHz to 30 kHz and temperatures from 168 K to 190 K; measurements are also reported for the glass phase down to 146 K.", '1701.05086-1-0-1': 'The data reveal a strong mechanical beta process, even above the glass transition.', '1701.05086-1-0-2': 'The model focuses on the metastable equilibrium liquid phase for which the data are fitted by an electrical equivalent-circuit characterized by additivity of the dynamic shear compliances of the alpha and beta processes.', '1701.05086-1-0-3': 'The nontrivial parts of the alpha and beta processes are modeled by a "Cole-Cole retardation element" defined as a series connection of a capacitor and a constant-phase element; this results in the Cole-Cole compliance function well-known from dielectrics.', '1701.05086-1-0-4': 'The model, which assumes that the high-frequency decay of the alpha shear compliance loss varies with (angular) frequency as [MATH], has seven free parameters.', '1701.05086-1-0-5': 'Assuming time-temperature superposition for the alpha and the beta processes separately, the number of parameters varying with temperature is reduced to four.', '1701.05086-1-0-6': 'The model provides a significantly better fit to data than a similar seven-parameter Havriliak-Negami type model.', '1701.05086-1-0-7': 'From the temperature dependence of the best-fit model parameters the following conclusions are drawn: 1) the alpha relaxation time conforms to the shoving model; 2) the beta relaxation loss-peak frequency is almost temperature independent; 3) the alpha compliance magnitude (which in the model equals the inverse of the instantaneous shear modulus) is only weakly temperature dependent; 4) the beta compliance magnitude decreases upon cooling by a factor of three in the temperature range studied.', '1701.05086-1-0-8': 'The final part of the paper briefly presents measurements of the dynamic adiabatic bulk modulus covering frequencies from 10 mHz to 10 kHz in the temperature range 172 K to 200 K.', '1701.05086-1-0-9': 'These data are qualitatively similar to the shear data by having a significant beta process, and the alpha and beta bulk modulus processes both occur at frequencies similar to those of the shear modulus.', '1701.05086-1-0-10': 'A single-order-parameter framework is suggested to rationalize these similarities.', '1701.05086-1-1-0': '# Introduction', '1701.05086-1-2-0': 'Many organic liquids are easily supercooled and good glass formers, usually with the glass transition taking place far below room temperature.', '1701.05086-1-2-1': 'These systems are experimentally convenient for studying the physics of highly viscous liquids, the glass transition, glassy relaxation, etc, phenomena that are believed to be universal for basically all liquids [CITATION].', '1701.05086-1-2-2': 'As the liquid is cooled, its relaxation time and viscosity increase by many orders of magnitude over a narrow temperature range.', '1701.05086-1-2-3': 'Beyond the dominant and slowest "alpha" relaxation process many liquids have additional faster relaxation(s), notably the so-called beta relaxation.', '1701.05086-1-2-4': 'The alpha and beta relaxation processes are often studied by means of dielectric spectroscopy.', '1701.05086-1-2-5': "They are also present, however, in the liquid's mechanical properties, which are the focus of the present paper that presents data and model for squalane's shear- and bulk-mechanical properties.", '1701.05086-1-3-0': 'Squalane is a liquid alkane consisting of a linear [MATH] backbone with six symmetrically placed methyl groups.', '1701.05086-1-3-1': 'Its systematic name is 2,6,10,15,19,23-hexamethyltetracosane.', '1701.05086-1-3-2': 'Squalane is a van der Waals liquid that is an excellent glass former [CITATION].', '1701.05086-1-3-3': "Measurements of squalane's dynamic shear modulus in the MHz range were reported many years ago [CITATION].", '1701.05086-1-3-4': "Squalane's melting point is [MATH] K and its glass transition temperature [MATH]K follows the well-known rule [MATH] [CITATION].", '1701.05086-1-3-5': 'Squalane has low toxicity and is used in cosmetics as moisturizer; due to the complete saturation squalane is not subject to auto-oxidation [CITATION].', '1701.05086-1-3-6': 'In basic research squalane is used as reference liquid in tribology and for elucidating the mechanism of elastohydrodynamic friction [CITATION].', '1701.05086-1-3-7': 'Squalane has been studied in molecular dynamics simulations of nonlinear flows [CITATION].', '1701.05086-1-3-8': 'Squalane has also been used as a solvent for studying the intriguing Debye dielectric relaxation of mono-hydroxy alcohols [CITATION], the rotation of aromatic hydrocarbons in viscous alkanes [CITATION], and the Stokes-Einstein relation for diffusion of organic solutes [CITATION].', '1701.05086-1-3-9': 'Due to its low vapor pressure squalane is used as a benchmark molecule for reaction-dynamics experiments performed under ultrahigh vacuum [CITATION].', '1701.05086-1-4-0': 'Studies of neat supercooled squalane include measurements of its dielectric relaxation [CITATION] and dynamic shear modulus in the quasistatic region over frequencies from a few mHz to 10 Hz [CITATION], later extended to frequencies up to 30 kHz [CITATION].', '1701.05086-1-4-1': 'The present paper covers the latter range of frequencies with more accurate data for more temperatures than Ref. [CITATION].', '1701.05086-1-4-2': 'The main motivation is not to present new data, however, but to introduce an electrical equivalent-circuit model representing data very well; the new model is an extension of a model discussed previously by our group [CITATION].', '1701.05086-1-5-0': 'Section [REF] presents the data and the piezo-ceramic transducer used to obtain them.', '1701.05086-1-5-1': 'Section [REF] introduces electrical-equivalent circuit modeling of linear mechanical relaxation phenomena in general and arrives at the model used for interpreting data.', '1701.05086-1-5-2': 'The model has four free parameters and three "shape" parameters that are fixed from data at one temperature.', '1701.05086-1-5-3': 'Section [REF] shows that the model fits data very well, in fact considerably better than a Havriliak-Negami type model with the same number of free parameters.', '1701.05086-1-5-4': 'While the main focus is on the dynamic shear data, Sec. [REF] supplements these by presenting some dynamic bulk-modulus data.', '1701.05086-1-5-5': 'Finally, Sec. [REF] gives a discussion with a focus on the temperature dependence of the best-fit model parameters, showing that these conform to the shoving model and that the beta process activation energy is temperature independent.', '1701.05086-1-5-6': 'If these two findings were built into the model, it would just have two parameters varying with temperature.', '1701.05086-1-6-0': '# Data for the dynamic shear modulus', '1701.05086-1-7-0': 'This paper focuses on the modeling of the dynamic shear-mechanical properties of metastable equilibrium supercooled liquid squalane above 168 K. Measurements were performed, however, at temperatures down to 146 K, which is well into the glass.', '1701.05086-1-7-1': 'Data were obtained with 2 K intervals using the three-disk piezo-ceramic shear transducer shown in Fig. [REF](a) [CITATION] in the setup described in Ref. hec13.', '1701.05086-1-7-2': 'The cryostat keeps temperature stable within 10 mK.', '1701.05086-1-7-3': 'References iga08a and iga08b give details about the home-built cryostat and impedance-measuring setup.', '1701.05086-1-8-0': 'Initially the filled transducer was annealed at the highest temperature for 30 hours in order to equilibrate the ceramics.', '1701.05086-1-8-1': 'After this each temperature was monitored by spending one hour for equilibration before performing a measurement, which lasts approximately one hour.', '1701.05086-1-8-2': 'The measurement was repeated to ensure reproducibility, i.e., that the liquid is in metastable equilibrium and that the setup works properly.', '1701.05086-1-8-3': 'The protocol is illustrated in Fig. [REF](b).', '1701.05086-1-8-4': 'After all measurements had finished, the empty transducer was calibrated [CITATION].', '1701.05086-1-8-5': 'If everything works, a set of data as those analyzed below may be obtained within less than a week.', '1701.05086-1-9-0': 'Figure [REF](a) and Fig. [REF](b) present the data for the real and imaginary parts of the dynamic shear modulus [MATH] in which [MATH] is the angular frequency.', '1701.05086-1-9-1': 'The figure shows in Fig. [REF](c) a so-called Nyquist plot of [MATH], i.e., real versus imaginary parts parametrized by the frequency; in this figure data for the glassy phase were included.', '1701.05086-1-9-2': 'Figure [REF](d) and Fig. [REF](e) present the real and imaginary parts of the dynamic shear compliance [MATH], while Fig. [REF](f) gives the Nyquist plot of [MATH].', '1701.05086-1-10-0': '# Electrical-equivalent circuit model', '1701.05086-1-11-0': '## Philosophy of circuit modeling', '1701.05086-1-12-0': 'Many scientists regard the modeling of linear-response data by an electrical equivalent circuit as old-fashioned.', '1701.05086-1-12-1': 'A common argument is that all data may be fitted by an electrical circuit and that, consequently, such type of models can contribute little if physical insight is the goal.', '1701.05086-1-12-2': 'In our opinion this skepticism is not fair [CITATION], and the following reasons may be given for using electrical equivalent circuits for rationalizing data:', '1701.05086-1-13-0': 'Once an electrical equivalent circuit has been constructed that represents data accurately, this provides an important input for constructing a microscopic physical model of the system in question.', '1701.05086-1-13-1': 'We regard the circuit as a help towards eventually obtaining the ultimate microscopic understanding, not as the final model itself.', '1701.05086-1-14-0': '## Basic circuit elements', '1701.05086-1-15-0': 'Rheology has its own circuit language based on dashpots representing Newtonian viscous flow and springs representing a purely elastic response [CITATION].', '1701.05086-1-15-1': 'This language is mathematically equivalent to electrical circuit modeling.', '1701.05086-1-15-2': 'Which language to use is a matter of convenience, but as physicists we are more used to the language of electrical circuits.', '1701.05086-1-15-3': 'This has the advantage of easily relating to dielectric relaxation phenomena, which are of great importance for glass-forming liquids [CITATION] and closely connected to the shear-mechanical properties [CITATION].', '1701.05086-1-16-0': 'Translating from electrical to rheological circuits is a bit counterintuitive when it comes to the diagrammatic representation because series connections become parallel connections and vice versa: since two elements in series in an electrical circuit have the same current, this corresponds to the analogous rheological elements being placed in parallel because the two shear displacements are identical.', '1701.05086-1-16-1': 'Likewise, an electrical-circuit parallel connection translates into a mechanical series connection.', '1701.05086-1-16-2': 'Once this is kept in mind, however, translation between the two languages is straightforward and unique.', '1701.05086-1-17-0': 'This paper uses electrical equivalent circuits to model of dynamic mechanical relaxation phenomena.', '1701.05086-1-17-1': 'For this reason below we do not distinguish between the dynamic capacitance [MATH] and the dynamic shear compliance [MATH].', '1701.05086-1-17-2': 'There is as mentioned a general circuit modeling language - the so-called energy-bond graph formalism [CITATION] - which may be used also, e.g., for thermal relaxation phenomena.', '1701.05086-1-17-3': 'An energy bond is characterized by an "effort" variable [MATH] and a "flow" variable [MATH], the product of which gives the power transferred into the system from its surroundings.', '1701.05086-1-17-4': 'For a thermodynamic energy bond [MATH] is the temperature deviation from a reference temperature and [MATH] is the entropy current (heat flow over temperature) [CITATION].', '1701.05086-1-18-0': 'How to translate electrical linear-response functions to the corresponding rheological ones?', '1701.05086-1-18-1': 'With the general energy-bond language in mind, the generalized displacement [MATH] represents electrical charge or shear displacement (strain), the generalized flow given by [MATH] represents electrical current or shear rate, and the generalized effort [MATH] represents voltage drop or shear stress [CITATION].', '1701.05086-1-18-2': 'The most important complex-valued linear-response functions translate as follows when given as functions of the angular frequency [MATH] in the standard way (e.g., [MATH] in which [MATH] is a complex amplitude):', '1701.05086-1-19-0': 'Three basic elements are used below: resistors, capacitors, and constant-phase elements (CPE) [CITATION].', '1701.05086-1-19-1': 'A CPE is characterized by a capacitance that as a function of [MATH] varies as [EQUATION] in which [MATH].', '1701.05086-1-19-2': 'The name CPE reflects the fact that the ratio between the real and imaginary parts of [MATH] is the same at all frequencies, which implies a constant phase difference between displacement and force.', '1701.05086-1-19-3': 'The CPE is a generalization of capacitors and resistors because a capacitor obeys [MATH]= Const.', '1701.05086-1-19-4': "while a resistor's capacitance is given by [MATH].", '1701.05086-1-19-5': 'Thus allowing for [MATH] there is just a single "Lego block" in the model tool box, namely the CPE.', '1701.05086-1-20-0': 'Figure [REF](a) shows the three basic elements.', '1701.05086-1-20-1': 'Note that Eq. ([REF]) translates into', '1701.05086-1-21-0': '## Parametrizing the constant-phase element', '1701.05086-1-22-0': 'For the CPE we define a magnitude constant [MATH] and a characteristic time [MATH] via the parametrization [EQUATION]', '1701.05086-1-22-1': 'Because the CPE is time-scale invariant, the two constants [MATH] and [MATH] are not uniquely determined because the same physics is described by using instead for any number [MATH] the magnitude constant [MATH] and the characteristic time [MATH].', '1701.05086-1-22-2': "The CPE is central for the model proposed below, and for the discussion of the model parameters' temperature dependence in fit to data (Sec. [REF]) we need a convention about the magnitude constant and the characteristic time.", '1701.05086-1-22-3': 'We take [MATH] to be a universal, temperature-independent number.', '1701.05086-1-22-4': 'The motivation is that, if any physics is to be ascribed to [MATH], the CPE magnitude constant [MATH] should also make sense physically.', '1701.05086-1-22-5': 'Since squalane like most other organic glass-forming liquids have an instantaneous shear modulus, i.e., high-frequency plateau modulus, of order GPa, we fix the magnitude constant as follows: [EQUATION]', '1701.05086-1-22-6': 'In this way, once the exponent [MATH] has been determined in a fit to data, the capacitance of a CPE element at a single frequency uniquely determines [MATH].', '1701.05086-1-23-0': 'Physically, we think of the characteristic time [MATH] as controlling the rate of the CPE element\'s "inner" clock in much the same sense as the material time of the Narayanaswamy physical-aging theory [CITATION].', '1701.05086-1-24-0': '## The Cole-Cole retardation element', '1701.05086-1-25-0': 'What is here termed a Cole-Cole retardation element (CCRE) consists of a series connection of two of the basic circuit elements, a CPE and a capacitor (Fig. [REF](b)).', '1701.05086-1-25-1': 'Recall that the capacitance [MATH] of two elements in series with capacitances [MATH] and [MATH] is given by [MATH].', '1701.05086-1-25-2': "Thus if the CCRE capacitor's value is [MATH], the CCRE compliance [MATH] is given by [EQUATION]", '1701.05086-1-25-3': 'The CCRE is named after the Cole-Cole dielectric capacitance function from 1941 [CITATION], which in the mechanical language is the following expression [EQUATION]', '1701.05086-1-25-4': 'Here [MATH] is the DC shear compliance and [MATH] is the inverse angular loss-peak frequency.', '1701.05086-1-25-5': 'It is straightforward to show that Eq. ([REF]) implies Eq. ([REF]) if one defines [EQUATION]', '1701.05086-1-25-6': 'The fit to data of fit_sec below gives CPE characteristic times [MATH] that are thermally activated for both the alpha and the beta process, with an activation energy proportional to the instantaneous shear modulus for the alpha CPE characteristic time, but virtually temperature independent for the beta CPE characteristic time.', '1701.05086-1-25-7': 'Note that the characteristic time is not identical to the inverse loss-peak frequency of each process.', '1701.05086-1-26-0': '## Model for the dynamic shear-mechanical properties', '1701.05086-1-27-0': "To arrive at a realistic electrical equivalent circuit model of squalane's shear dynamic response, we first note that a standard electrical RC element - a resistor and a capacitor in parallel - corresponds to the classical Maxwell model for viscoelasticity.", '1701.05086-1-27-1': 'This model is based on the assumption that the stress decays exponentially to zero whenever the sample is at rest [CITATION].', '1701.05086-1-27-2': 'If the shear stress is denoted by [MATH], the shear (strain) displacement by [MATH], the DC shear viscosity by [MATH], and the instantaneous shear modulus by [MATH], the differential equation describing the Maxwell model for any shear rate as a function of time, [MATH], is [CITATION] [EQUATION]', '1701.05086-1-27-3': 'That the Maxwell model is equivalent to an electrical RC element follows by noting that for an RC element the voltage is the same across both elements, which in the Maxwell model corresponds to the same shear stress.', '1701.05086-1-27-4': 'The resistor current corresponds to the first term on the right hand side of Eq. ([REF]) (a dashpot in the traditional language of viscoelasticity), and the capacitor current corresponds to the second term (a spring in the viscoelastic language).', '1701.05086-1-28-0': 'The Maxwell model is too simple to fit data for glass-forming liquids, however, and the model must be extended by including one or more non-trivial dissipative terms.', '1701.05086-1-28-1': "It is this paper's basic assumption that these terms are described by CCREs placed in parallel to the Maxwell RC element, one for the alpha process and one for the beta process (Fig. [REF]).", '1701.05086-1-29-0': 'In the model (Fig. [REF]) none of the two CCREs are inherently linked to the alpha process RC element.', '1701.05086-1-29-1': 'Nevertheless, one CCRE will be regarded as part of the alpha process for the following reasons.', '1701.05086-1-29-2': 'Previous publications of the Glass and Time group have presented experimental [CITATION], as well as theoretical [CITATION], evidence that in the absence of beta relaxation the alpha process has a generic [MATH] high-frequency decay of the dielectric loss and shear compliance.', '1701.05086-1-29-3': 'This is an old idea; for instance, a generic [MATH] high-frequency decay is the characteristic feature of the 1967 Barlow-Erginsav-Lamb (BEL) model [CITATION].', '1701.05086-1-29-4': 'Consequently, we fix the exponent to [MATH] for one CCRE and regard this element as part of the alpha process.', '1701.05086-1-29-5': 'Confirming this assignment, for liquids without a mechanical beta relaxation like the silicone diffusion pump oils DC704 and DC705, the dynamic shear compliance is well fitted by the model without the beta CCRE (unpublished).', '1701.05086-1-30-0': "The dynamic shear compliance is a sum of the individual elements' shear compliances.", '1701.05086-1-30-1': 'Thus the model leads to the following expression, which henceforth defines the model parametrization: [EQUATION]', '1701.05086-1-30-2': 'For later use we note that the instantaneous shear modulus [MATH] is given by [EQUATION]', '1701.05086-1-30-3': 'The plateau modulus between the alpha and beta processes (at temperatures low enough that these are well separated, i.e., [MATH] and [MATH]) is denoted by [MATH] and given by [EQUATION]', '1701.05086-1-30-4': 'In the low-frequency limit the shear compliance diverges as [MATH] as required for any liquid with a finite DC viscosity.', '1701.05086-1-30-5': 'In the same limit the real part of the shear compliance, the so-called recoverable shear, is given by [EQUATION]', '1701.05086-1-30-6': 'The model has seven parameters:', '1701.05086-1-31-0': 'In order to limit the number of free parameters we make the assumption that time-temperature superposition (TTS) applies separately for both the alpha and the beta processes, implying that in the fit to data the three dimensionless shape parameters [MATH], [MATH], and [MATH] do not vary with temperature.', '1701.05086-1-31-1': 'The parameters allowed to vary are the two compliance strengths and the two relaxation times.', '1701.05086-1-32-0': 'The characteristic times of the alpha and beta CPEs are denoted by [MATH] and [MATH].', '1701.05086-1-32-1': 'In the below fit to data we take, as mentioned, the constants [MATH] and [MATH] to be independent of temperature, implying that [MATH].', '1701.05086-1-32-2': 'For this reason the parameter [MATH] will not be discussed separately from [MATH].', '1701.05086-1-32-3': 'The beta characteristic time [MATH], on the other hand, is qualitatively different from [MATH] in its temperature variation, which makes this an important parameter to keep track of (disc_sec).', '1701.05086-1-33-0': '# Fitting the model to data', '1701.05086-1-34-0': 'The model was fitted to the squalane data of Fig. [REF] using MATLAB\'s "fminsearch" Nelder-Mead downhill simplex least-squares fitting procedure.', '1701.05086-1-34-1': 'The fit excluded data taken at too low a temperature to be in equilibrium or at such high temperatures that the alpha and beta process have more or less completely merged.', '1701.05086-1-34-2': 'These limitations leave the data for temperatures between 168 K and 182 K for parameter identification.', '1701.05086-1-35-0': 'The data for the real and imaginary parts of the frequency-dependent shear modulus cover angular frequencies ranging from 10 mHz to 30 kHz, with up to 16 frequencies per decade evenly distributed on a logarithmic scale.', '1701.05086-1-35-1': 'The data were fitted to Eq. ([REF]) by proceeding as follows.', '1701.05086-1-35-2': 'First, the three temperature-independent shape parameters [MATH], [MATH], [MATH] were identified by fitting to the 172 K data, which have the alpha and the beta loss peaks well within the frequency window, but still clearly separated.', '1701.05086-1-35-3': 'The fit was performed for the shear-modulus, because using the shear compliance the fit would have been dominated by its low-frequency divergence.', '1701.05086-1-36-0': 'Figure [REF] compares model fits (green curves) to data (black crosses).', '1701.05086-1-36-1': 'Figure [REF](a) gives model prediction versus data for the real part of the dynamic shear modulus, Fig. [REF](b) gives the same for the imaginary part, and Fig. [REF](c) gives model prediction versus data in a Nyquist plot of the shear modulus.', '1701.05086-1-36-2': 'Figure [REF](d), Fig. [REF](e), and Fig. [REF](f) give the same for the dynamic shear compliance.', '1701.05086-1-37-0': 'The fits are excellent.', '1701.05086-1-37-1': 'Given the number of free parameters this may not be surprising.', '1701.05086-1-37-2': 'Our experience with fitting data to similar models over the last 20 years show, however, that the present model is better than other models with the same number of parameters.', '1701.05086-1-37-3': 'As an illustration of the superiority of the new model, we have compared to a fit assuming a Havriliak-Negami (HN) type function for the alpha process.', '1701.05086-1-37-4': 'The function fitted to data is the following: [EQUATION]', '1701.05086-1-37-5': 'The function Eq. ([REF]) has the same number of parameters as Eq. ([REF]): two strength parameters, two relaxation times, and three dimensionless shape parameters.', '1701.05086-1-37-6': 'There is only one qualitative difference to Eq. ([REF]): the latter has a finite so-called recoverable compliance, i.e., low-frequency limit of [MATH].', '1701.05086-1-37-7': 'We determined the best-fit parameters in the same way as above.', '1701.05086-1-37-8': 'The fit to data is not as good as that of Eq. ([REF]).', '1701.05086-1-37-9': 'This is clear from Fig. [REF] that compares the overall quality of the two fits as functions of temperature.', '1701.05086-1-38-0': 'Turning back to the model Eq. ([REF]), Fig. [REF] shows the temperature variation of the four free parameters.', '1701.05086-1-38-1': 'Figure [REF](a) shows how the alpha and beta relaxation times [MATH] and [MATH] vary with temperature.', '1701.05086-1-38-2': 'As always for a glass-forming liquid, the alpha relaxation time increases strongly when temperature decreases.', '1701.05086-1-38-3': 'The beta relaxation time is almost constant and, in fact, not even a monotonic function of temperature.', '1701.05086-1-38-4': 'In contrast, the beta characteristic time [MATH] decreases systematically with temperature.', '1701.05086-1-38-5': 'The dielectric beta loss-peak frequency is usually reported to be Arrhenius [CITATION], but it is important to note that almost all literature data for [MATH] (the inverse beta loss-peak frequency) refer to the glass phase, not to the metastable liquid phase about [MATH].', '1701.05086-1-38-6': 'Figure [REF](b) shows the best-fit shear-compliance strengths [MATH] and [MATH].', '1701.05086-1-38-7': 'Note that the beta process strength varies considerably more than the alpha strength.', '1701.05086-1-39-0': 'These findings are qualitatively consistent with previous ones of ours for the beta dielectric relaxation process, which may be summarized as follows [CITATION]: In the metastable liquid phase the (beta) relaxation strength increases considerably with increasing temperature whereas the relaxation time is almost temperature independent, in the glassy phase the strength is almost constant whereas the loss-peak frequency (inverse relaxation time) is strongly temperature dependent (Arrhenius).', '1701.05086-1-39-1': 'It is possible to rationalize these properties of the beta process - as well as its behavior under annealing of the out-of-equilibrium liquid - by assuming that the characteristic time [MATH] is Arrhenius both above and below the glass transition with the same activation energy, whereas the beta relaxation strength freezes at the glass transition (unpublished).', '1701.05086-1-40-0': '# Data for the dynamic adiabatic bulk modulus', '1701.05086-1-41-0': 'Figure [REF] shows the transducer used for measuring the dynamic adiabatic bulk modulus.', '1701.05086-1-41-1': 'It consists of a radially polarized piezo-ceramic spherical shell coated with electrodes on the inner and outer surfaces.', '1701.05086-1-41-2': "An electrical potential induces a slight compression or expansion of the sphere volume in which the liquid is placed (the top is a reservoir allowing for the liquid's thermal expansion) [CITATION].", '1701.05086-1-41-3': 'Figure [REF] shows our data for real and imaginary parts of the dynamic adiabatic bulk modulus [MATH], (a) and (b), as well as in a Nyquist plot, (c).', '1701.05086-1-42-0': 'Comparing the bulk modulus loss, Fig. [REF](b), to the shear modulus loss, Fig. [REF](b), we see a qualitatively similar behavior with an alpha loss peak that moves rapidly to lower frequencies upon cooling and a large beta peak, which at the same time becomes visible.', '1701.05086-1-42-1': 'To the best of our knowledge this is the first observation of a clear beta process for the dynamic bulk modulus.', '1701.05086-1-43-0': 'How to interpret the similarity between the dynamic shear and bulk moduli?', '1701.05086-1-43-1': 'This finding is fully consistent with previous ones [CITATION], but is important to remember that there is no fundamental reason for the similarity.', '1701.05086-1-43-2': 'This is because the dynamic bulk modulus - whether adiabatic or isothermal - is a scalar linear-response function whereas the dynamic shear modulus is a vector linear-response function.', '1701.05086-1-43-3': 'This means that these functions belong to fundamentally different symmetry classes, as discussed by Meixner long time ago [CITATION].', '1701.05086-1-43-4': 'Nevertheless, by reference to the Eshelby picture of structural rearrangements within a surrounding elastic matrix, Buchenau has recently discussed how the relaxational parts of the bulk and shear moduli may be connected [CITATION].', '1701.05086-1-43-5': 'His arguments may be extended to finite frequencies, thus establishing a connection between [MATH] and [MATH].', '1701.05086-1-44-0': 'Referring to the energy-bond language [CITATION], there are two fundamental thermodynamic scalar energy bonds: a thermal bond with effort equal to temperature difference and flow equal to entropy current, and a mechanical bond with effort equal to minus pressure difference and flow equal to rate of volume change.', '1701.05086-1-44-1': "Consistent with Buchenau's reasoning [CITATION] we would like to propose a general energy-bond model in which all dissipation connected with the two scalar thermodynamic energy bonds is controlled by the dynamic shear modulus (or, equivalently, the dynamic shear compliance).", '1701.05086-1-44-2': 'A schematic representation of this idea is given in Fig. [REF](b).', '1701.05086-1-44-3': 'An energy-bond diagram of this sort implies that the system in question is a "single-order-parameter" liquid [CITATION].', '1701.05086-1-44-4': 'This is equivalent to being a so-called R simple system, i.e., one with so-called isomorphs, which are lines in the thermodynamic phase diagram along which the dynamics is invariant to a very good approximation [CITATION].', '1701.05086-1-45-0': 'In Fig. [REF](b) there may be several non-dissipative elements, but the important point is that these are all connected to the element Fig. [REF] via a single, internal energy bond.', '1701.05086-1-45-1': 'The predictions for the dynamic adiabatic/isothermal bulk moduli (or those of the dynamic expansion coefficient [CITATION]) depend, of course, not just on the dynamic shear modulus (compliance), but also on the non-dissipative elements.', '1701.05086-1-45-2': 'Nevertheless, for a system described by Fig. [REF] one a priori expects that all the scalar response functions at any given temperature have alpha and beta processes located at frequencies similar to those of the shear modulus (compliance) alpha and beta processes.', '1701.05086-1-46-0': '# Discussion', '1701.05086-1-47-0': 'This paper has demonstrated that shear-mechanical data for squalane may be fitted very well with the electrical-equivalent circuit model of Fig. [REF] leading to Eq. ([REF]) for the dynamic shear-mechanical compliance.', '1701.05086-1-47-1': 'The model assumes additivity of the alpha and beta shear compliances.', '1701.05086-1-47-2': 'The model has seven parameters, one more than alternative phenomenological models [CITATION].', '1701.05086-1-47-3': 'In our fit to data, however, the three shape parameters were taken to be temperature independent, reflecting the assumption that time-temperature superposition applies separately to both the alpha and the beta compliance functions.', '1701.05086-1-47-4': 'In this picture, the observed significant deviations from TTS derive from merging of the alpha and beta processes.', '1701.05086-1-47-5': 'We conjecture that this applies generally for glass-forming liquids.', '1701.05086-1-48-0': 'How to physically justify that the Maxwell RC element and the two CCREs should be combined in a way that is additive in their shear compliances, not in their shear moduli?', '1701.05086-1-48-1': 'There are no logically compelling arguments for this.', '1701.05086-1-48-2': 'We think of it as follows.', '1701.05086-1-48-3': 'Imagine a small particle in the liquid.', '1701.05086-1-48-4': "The particle's mean-square displacement (MSD) as a function of time in one axis direction, [MATH], will have a rapid increase on the phonon time scale, followed by a transition to the long-time diffusive behavior proportional to time.", '1701.05086-1-48-5': "If one assumes that the alpha and beta processes contribute independently to the particle's motion, i.e., that [MATH] with [MATH], the MSD will be a sum of an alpha and a beta contribution: [MATH].", '1701.05086-1-48-6': "If one moreover assumes the Stokes-Einstein relation between dynamic shear viscosity and the particle's dynamic friction coefficient, this translates via the fluctuation-dissipation theorem into additivity of the dynamic shear compliances for the alpha and beta processes.", '1701.05086-1-49-0': 'In regard to the single-particle MSD, note that with any function [MATH] there is an associated characteristic time, the time at which the particle has moved a typical intermolecular distance.', '1701.05086-1-49-1': "In the same way this is how we think of each CPE basic element's characteristic [MATH], which was defined as the time at which the absolute value of the compliance at [MATH] has the physically meaningful value 1 [MATH] (Eq. ([REF])).", '1701.05086-1-49-2': "For the beta process, it is important to distinguish between this time and the inverse loss-peak frequency, [MATH] because via Eq. ([REF]) the latter time's temperature variation reflects the combined effect of the changing compliance strength [MATH] and the Arrhenius [MATH].", '1701.05086-1-49-3': 'Incidentally, these considerations give rise to a new model for the beta process, which provides an alternative to the 2003 "minimal model" [CITATION].', '1701.05086-1-49-4': 'We intend to return to this in a future publication.', '1701.05086-1-50-0': 'The electrical-equivalent circuit model Fig. [REF] is identical to that proposed in Ref. [CITATION] except for an extra capacitor, the one in the alpha CCRE.', '1701.05086-1-50-1': 'This capacitor eliminates an unphysical feature of our previous model [CITATION], which predicted an infinite so-called recoverable shear compliance (the amount of return flow following a step shear displacement of size unity).', '1701.05086-1-50-2': 'This unphysical feature is also present in the BEL model from 1967 [CITATION].', '1701.05086-1-50-3': 'Introducing the extra capacitor results in symmetry between the alpha and beta CCREs, the only difference being that the alpha CCRE has the fixed exponent [MATH] and a characteristic time that is proportional to [MATH] in its temperature variation.', '1701.05086-1-51-0': "We note that if the alpha CCRE's relaxation time is much shorter than [MATH], the circuit mimics the situation reported in recent papers for the dielectric relaxation of monohydroxy alcohols, for which one observes a low-frequency Debye-type process followed by, in order of increasing frequency, first an alpha and then, in most liquids, a beta process [CITATION].", '1701.05086-1-52-0': 'As regards the temperature dependence of the model parameters we have compared to the prediction of the shoving model [CITATION].', '1701.05086-1-52-1': 'If [MATH]s is a typical phonon time and [MATH] the so-called characteristic volume assumed to be temperature independent, the shoving model predicts the following relation between the alpha relaxation time and the instantaneous (high-frequency plateau) shear modulus [MATH]: [EQUATION]', '1701.05086-1-52-2': "The shoving model, which links a supercooled liquid's fragility to the temperature variation of [MATH], fits data well for many glass-forming liquids [CITATION].", '1701.05086-1-52-3': 'The shoving model relates to Eq. ([REF]) since if the model applies, the number of temperature-dependent parameters is reduced from four to three.', '1701.05086-1-52-4': 'We have not made this assumption in the fit to data, but have instead checked Eq. ([REF]) against the best-fit parameters.', '1701.05086-1-52-5': 'This is done in Fig. [REF] in which the relaxation times [MATH] and [MATH] are converted into temperature-dependent activation energies [MATH] by writing [MATH] with [MATH]s.', '1701.05086-1-52-6': 'According to the shoving model [MATH].', '1701.05086-1-52-7': 'By comparing the black crosses and the red triangles in Fig. [REF] we conclude that this model applies.', '1701.05086-1-52-8': 'If the shoving model is instead interpreted with [MATH] taken to be the plateau modulus between the alpha and beta relaxations, [MATH] given by Eq. ([REF]), however, the model does not apply (blue crosses).', '1701.05086-1-53-0': 'Our model consists of a Maxwell element in parallel with two CCREs.', '1701.05086-1-53-1': 'One may speculate that additional high-frequency mechanical processes beyond the alpha and beta relaxations can be modelled by adding further CCREs, each CCRE still being subject to time-temperature superposition.', '1701.05086-1-54-0': 'To summarize, an excellent fit to dynamic shear-mechanical data of squalane is provided by an electrical equivalent circuit model with seven parameters.', '1701.05086-1-54-1': 'The model assumes an [MATH] high-frequency decay of the alpha compliance [CITATION], additivity of the alpha and beta compliance functions, and that these functions separately obey time-temperature superposition.', '1701.05086-1-54-2': 'The latter assumption reduces the number of parameters varying with temperature to four.', '1701.05086-1-54-3': 'The best fit parameters confirm the shoving model and show that the beta process characteristic time has a temperature-independent activation energy.', '1701.05086-1-54-4': 'If these findings were incorporated as model assumptions, the number of parameters that vary with temperature reduces to two.', '1701.05086-1-54-5': 'These could be taken to be, e.g., the compliance magnitudes [MATH] and [MATH].', '1701.05086-1-54-6': 'We presented also data for the adiabatic dynamic bulk modulus and argued briefly that these may be interpreted qualitatively in terms of a "single-order-parameter" model in which all dissipation is controlled by the shear-mechanical properties.', '1701.05086-1-54-7': 'Such a model connects the class of scalar viscoelastic linear-response functions with that of vector symmetry [CITATION].', '1701.05086-1-55-0': 'In regard to future works, one obvious thing is to compare the model to shear-mechanical data for other glass-forming liquids.', '1701.05086-1-55-1': 'We have not done so systematically, but have found in all cases tested so far that the model works well (unpublished).', '1701.05086-1-55-2': "The hope is that the model is general, thus providing a step towards a microscopic understanding of glass-forming liquids' mechanical properties.", '1701.05086-1-56-0': 'Figure [REF](b) is a special case of the general energy-bond diagram for a system with a single order parameter identified by our Glass and Time colleague Tage Christensen about 15 years ago, initiating a development that led to Ref. [CITATION] and eventually to the isomorph theory [CITATION].'}

{'1701.05086-2-0-0': "This paper presents data for supercooled squalane's frequency-dependent shear modulus covering frequencies from 10 mHz to 30 kHz and temperatures from 168 K to 190 K; measurements are also reported for the glass phase down to 146 K.", '1701.05086-2-0-1': 'The data reveal a strong mechanical beta process, also above the glass transition.', '1701.05086-2-0-2': 'A model is proposed for the shear response of the metastable equilibrium liquid phase of supercooled liquids.', '1701.05086-2-0-3': 'The model is an electrical equivalent-circuit characterized by additivity of the dynamic shear compliances of the alpha and beta processes.', '1701.05086-2-0-4': 'The nontrivial parts of the alpha and beta processes are represented by a "Cole-Cole retardation element" defined as a series connection of a capacitor and a constant-phase element, resulting in the Cole-Cole compliance function well-known from dielectrics.', '1701.05086-2-0-5': 'The model, which assumes that the high-frequency decay of the alpha shear compliance loss varies with angular frequency as [MATH], has seven parameters.', '1701.05086-2-0-6': 'Assuming time-temperature superposition for the alpha and the beta processes separately, the number of parameters varying with temperature is reduced to four.', '1701.05086-2-0-7': 'The model provides a better fit to data than a seven-parameter Havriliak-Negami type model.', '1701.05086-2-0-8': 'From the temperature dependence of the best-fit model parameters the following conclusions are drawn: 1) the alpha relaxation time conforms to the shoving model; 2) the beta relaxation loss-peak frequency is almost temperature independent; 3) the alpha compliance magnitude, which in the model equals the inverse of the instantaneous shear modulus, is only weakly temperature dependent; 4) the beta compliance magnitude decreases by a factor of three upon cooling in the temperature range studied.', '1701.05086-2-0-9': 'The final part of the paper briefly presents measurements of the dynamic adiabatic bulk modulus covering frequencies from 10 mHz to 10 kHz in the temperature range 172 K to 200 K.', '1701.05086-2-0-10': 'The data are qualitatively similar to the shear data by having a significant beta process.', '1701.05086-2-0-11': 'A single-order-parameter framework is suggested to rationalize these similarities.', '1701.05086-2-1-0': '# Introduction', '1701.05086-2-2-0': 'Many organic liquids are easily supercooled and excellent glass formers, usually with the glass transition taking place far below room temperature.', '1701.05086-2-2-1': 'Such systems are experimentally convenient for studying the physics of highly viscous liquids, the glass transition, glassy relaxation, etc, phenomena that are believed to be universal for basically all liquids [CITATION].', '1701.05086-2-2-2': 'As the liquid is cooled, the relaxation time and viscosity increase by many orders of magnitude over a narrow temperature range.', '1701.05086-2-2-3': 'Beyond the dominant and slowest alpha relaxation process many liquids have additional faster relaxation(s), notably the so-called beta relaxation.', '1701.05086-2-2-4': 'The alpha and beta processes are often studied by means of dielectric spectroscopy.', '1701.05086-2-2-5': "They are also present, however, in the liquid's mechanical properties, which are the focus of the present paper presenting squalane data and a model for supercooled liquids' dynamic shear-mechanical properties.", '1701.05086-2-3-0': 'Squalane is a liquid alkane consisting of a linear [MATH] backbone with six symmetrically placed methyl groups.', '1701.05086-2-3-1': 'Its systematic name is 2,6,10,15,19,23-hexamethyltetracosane.', '1701.05086-2-3-2': 'Squalane is a van der Waals liquid that is an excellent glass former [CITATION].', '1701.05086-2-3-3': "Squalane's melting point is [MATH] K and its glass transition temperature [MATH]K follows the well-known rule [MATH] [CITATION].", '1701.05086-2-3-4': 'Squalane has low toxicity and is used in cosmetics as moisturizer; due to the complete saturation squalane is not subject to auto-oxidation [CITATION].', '1701.05086-2-3-5': 'In basic research squalane is used as reference liquid in tribology and for elucidating the mechanism of elastohydrodynamic friction [CITATION].', '1701.05086-2-3-6': 'Squalane has been studied in molecular dynamics simulations of nonlinear flows [CITATION].', '1701.05086-2-3-7': 'Squalane has also been used as a solvent for studying the intriguing Debye dielectric relaxation of mono-hydroxy alcohols [CITATION], the rotation of aromatic hydrocarbons in viscous alkanes [CITATION], and the Stokes-Einstein relation for diffusion of organic solutes [CITATION].', '1701.05086-2-3-8': 'Due to its low vapor pressure squalane is used as a benchmark molecule for reaction-dynamics experiments performed under ultrahigh vacuum [CITATION].', '1701.05086-2-4-0': "Measurements of neat supercooled squalane's dynamic shear modulus in the MHz range were reported many years ago [CITATION].", '1701.05086-2-4-1': 'Subsequent studies of squalane include measurements of its dielectric relaxation [CITATION] and dynamic shear modulus over frequencies ranging from a few mHz to 10 Hz [CITATION], later extended to 30 kHz [CITATION].', '1701.05086-2-4-2': 'The present paper covers the latter range of frequencies with more accurate data and for more temperatures than Ref. [CITATION].', '1701.05086-2-4-3': 'The main motivation is not to present new data, however, but to introduce an electrical equivalent-circuit model representing data very well.', '1701.05086-2-4-4': 'The model is a modification of one discussed previously by our group, which introduces a crucial extra capacitor [CITATION].', '1701.05086-2-5-0': 'Section [REF] presents the squalane data and the piezo-ceramic transducer used to obtain them.', '1701.05086-2-5-1': 'Section [REF] introduces electrical-equivalent circuit modeling of linear mechanical relaxation phenomena in general and motivates the model.', '1701.05086-2-5-2': 'It has four free parameters of dimensions and three dimensionless "shape" parameters that are fixed from fitting to data at one temperature.', '1701.05086-2-5-3': 'Section [REF] shows that the model fits data very well, considerably better than a similar Havriliak-Negami type model with the same number of parameters.', '1701.05086-2-5-4': "While the paper's main focus is on the dynamic shear data, Sec. [REF] supplements these by presenting dynamic adiabatic bulk-modulus data.", '1701.05086-2-5-5': 'It is briefly shown that these may interpreted in terms of an electrical equivalent circuit model in which the dissipation is controlled by the dynamic shear modulus.', '1701.05086-2-5-6': 'Finally, Sec. [REF] gives a discussion with a focus on the temperature dependence of the best-fit model parameters, showing that these conform to the shoving model and that the beta process activation energy is temperature independent.', '1701.05086-2-5-7': 'If these two findings were built into the model, it would have just two parameters varying with temperature.', '1701.05086-2-6-0': '# Data for the dynamic shear modulus of squalane', '1701.05086-2-7-0': 'This paper focuses on the modeling of the dynamic shear-mechanical properties of metastable equilibrium supercooled liquids, in casu squalane above its glass transition temperature 168 K. Measurements were performed at temperatures down to 146 K, however, which is well into the glass.', '1701.05086-2-7-1': 'Data were obtained with 2 K intervals using the three-disk piezo-ceramic shear transducer shown in Fig. [REF](a) [CITATION] in the setup described in Ref. hec13.', '1701.05086-2-7-2': 'The cryostat keeps temperature stable within 10 mK.', '1701.05086-2-7-3': 'References iga08a and iga08b give details about the home-built cryostat and impedance-measuring setup.', '1701.05086-2-8-0': 'Before measuring, the filled transducer was annealed at the highest temperature for 30 hours in order to equilibrate the ceramics.', '1701.05086-2-8-1': 'After this, with 2 K intervals several temperatures were monitored by first equilibrating for one hour, after which a frequency spectrum was measured which lasted approximately one hour.', '1701.05086-2-8-2': 'This measurement was repeated to ensure reproducibility, i.e., that the liquid is in metastable equilibrium and that the setup works properly.', '1701.05086-2-8-3': 'All in all, approximately three hours were spent at each temperature.', '1701.05086-2-8-4': 'The protocol is illustrated in Fig. [REF](b).', '1701.05086-2-8-5': 'After all measurements had finished, the empty transducer was calibrated [CITATION].', '1701.05086-2-8-6': 'If everything works, a set of data as those analyzed below may be obtained within less than a week.', '1701.05086-2-9-0': "Figures [REF](a) and (b) present the real and imaginary parts of squalane's dynamic shear modulus [MATH] in which [MATH] is the angular frequency.", '1701.05086-2-9-1': 'Shown in Fig. [REF](c) is a so-called Nyquist plot of [MATH], i.e., the real versus imaginary parts parametrized by the frequency; in this figure data for the glassy phase were included.', '1701.05086-2-9-2': 'A strong beta relaxation is observed.', '1701.05086-2-9-3': 'Figures [REF](d) and (e) present the real and imaginary parts of the dynamic shear compliance [MATH], while Fig. [REF](f) gives the Nyquist plot of [MATH].', '1701.05086-2-10-0': '# Electrical-equivalent circuit model', '1701.05086-2-11-0': '## Philosophy of circuit modeling', '1701.05086-2-12-0': 'Some scientists regard the modeling of linear-response data by an electrical equivalent circuit as old-fashioned.', '1701.05086-2-12-1': 'A common argument is that all data may be fitted by an electrical circuit and that, consequently, such type of models can contribute little if physical insight is the goal.', '1701.05086-2-12-2': 'In our opinion this is not quite correct [CITATION], and the following reasons may be given for using electrical equivalent circuits for rationalizing data as a first step towards a physical understanding:', '1701.05086-2-13-0': 'Once an electrical equivalent circuit has been constructed representing data accurately, this provides an important input for constructing a microscopic physical model of the system in question.', '1701.05086-2-13-1': 'We regard the circuit as a help towards eventually obtaining the ultimate microscopic understanding, not as the final model itself.', '1701.05086-2-14-0': '## Basic circuit elements', '1701.05086-2-15-0': 'Rheology has its own circuit language based on dashpots representing Newtonian viscous flow and springs representing a purely elastic response [CITATION].', '1701.05086-2-15-1': 'This language is mathematically equivalent to that of electrical circuit modeling, and which language to use is a matter of convenience.', '1701.05086-2-15-2': 'As physicists we are more used to electrical circuits.', '1701.05086-2-15-3': 'Their use has the additional advantage of easily relating to dielectric relaxation phenomena, which are of great importance for glass-forming liquids [CITATION] and experimentally closely connected to the shear-mechanical properties [CITATION].', '1701.05086-2-16-0': 'Translating from electrical to rheological circuits is a bit counterintuitive when it comes to the diagrammatic representation because series connections become parallel connections and vice versa: two elements in series in an electrical circuit have the same current, which corresponds to the analogous rheological elements being placed in parallel because the two shear displacements are identical.', '1701.05086-2-16-1': 'Likewise, an electrical-circuit parallel connection translates into a mechanical series connection.', '1701.05086-2-16-2': 'Once this is kept in mind, however, translation between the two languages is straightforward and unique.', '1701.05086-2-17-0': 'Since electrical equivalent circuits are used for modeling dynamic mechanical relaxation phenomena, we shall not distinguish between the dynamic capacitance [MATH] and the dynamic shear compliance [MATH].', '1701.05086-2-17-1': 'There is as mentioned a general circuit modeling language - the energy-bond graph formalism [CITATION] - which may be used also, e.g., for thermal relaxation phenomena.', '1701.05086-2-17-2': 'A general energy-bond is characterized by an "effort" variable [MATH] and a "flow" variable [MATH], the product of which gives the power transferred into the system from its surroundings.', '1701.05086-2-17-3': 'For instance, for a thermodynamic energy-bond [MATH] is the temperature deviation from a reference temperature and [MATH] is the entropy current, i.e., the heat flow over temperature [CITATION].', '1701.05086-2-18-0': 'How to translate electrical linear-response functions to the corresponding rheological ones?', '1701.05086-2-18-1': 'With the energy-bond formalism in mind, the displacement [MATH] represents electrical charge or shear displacement (strain), the flow given by [MATH] represents electrical current or shear rate, and the effort [MATH] represents voltage drop or shear stress [CITATION].', '1701.05086-2-19-0': 'The most important complex-valued linear-response functions translate as follows when given as functions of the angular frequency [MATH] in the standard way, e.g., [MATH] in which [MATH] is the complex amplitude:', '1701.05086-2-20-0': 'Three basic elements are used below (Fig. [REF](a)): resistors, capacitors, and constant-phase elements (CPE) [CITATION].', '1701.05086-2-20-1': 'A CPE is characterized by a capacitance that as a function of [MATH] varies as [EQUATION] in which [MATH].', '1701.05086-2-20-2': 'The name CPE reflects the fact that the ratio between the real and imaginary parts of [MATH] is the same at all frequencies, which implies a constant phase difference between displacement and effort.', '1701.05086-2-20-3': 'The CPE is a generalization of capacitors and resistors because a capacitor obeys [MATH]= Const.', '1701.05086-2-20-4': "while a resistor's capacitance is given by [MATH].", '1701.05086-2-20-5': 'Thus allowing for [MATH] there is just a single "Lego block" in the model tool box, namely the CPE.', '1701.05086-2-20-6': '- Note that Eq. ([REF]) translates into', '1701.05086-2-21-0': '## Parametrizing the constant-phase element', '1701.05086-2-22-0': 'For the CPE we define a magnitude constant [MATH] and a characteristic time [MATH] by writing [EQUATION]', '1701.05086-2-22-1': 'Because the CPE is time-scale invariant, the constants [MATH] and [MATH] are not uniquely determined since the same physics is described by using instead for any number [MATH] the magnitude constant [MATH] and the characteristic time [MATH].', '1701.05086-2-22-2': "The CPE is central for the model proposed below, and for the discussion of the model parameters' temperature dependence in fit to data (Sec. [REF]) we need a convention about the magnitude constant and the characteristic time.", '1701.05086-2-22-3': 'We take [MATH] to be a universal, temperature-independent number.', '1701.05086-2-22-4': 'The motivation is that, if any physics is to be ascribed to [MATH], the CPE magnitude constant [MATH] should also make sense physically.', '1701.05086-2-22-5': 'Since squalane like most other organic glass-forming liquids have an instantaneous shear modulus, i.e., high-frequency plateau shear modulus, of order GPa, we fix the magnitude constant as follows: [EQUATION]', '1701.05086-2-23-0': '## The Cole-Cole retardation element', '1701.05086-2-24-0': 'What is here termed a Cole-Cole retardation element (CCRE) consists of a series connection of a CPE and a capacitor (Fig. [REF](b)).', '1701.05086-2-24-1': 'Recall that the capacitance [MATH] of two elements in series with capacitances [MATH] and [MATH] is given by [MATH].', '1701.05086-2-24-2': "Thus if the CCRE capacitor's value is [MATH], the CCRE compliance [MATH] is given by [EQUATION]", '1701.05086-2-24-3': 'The CCRE is named after the Cole-Cole dielectric capacitance function from 1941 [CITATION], which in the mechanical language is the following expression [EQUATION]', '1701.05086-2-24-4': 'Here [MATH] is the DC shear compliance and [MATH] the inverse angular loss-peak frequency.', '1701.05086-2-24-5': 'It is straightforward to show that Eq. ([REF]) leads to Eq. ([REF]) if one identifies [EQUATION]', '1701.05086-2-24-6': 'Note that the characteristic time is not identical to the inverse loss-peak frequency of the CCRE.', '1701.05086-2-25-0': 'The fit to data (fit_sec) gives CPE characteristic times [MATH] that are thermally activated for both the alpha and the beta process.', '1701.05086-2-25-1': 'As demonstrated in Fig. [REF] below, the alpha CPE characteristic time activation energy is proportional to the instantaneous shear modulus, whereas the beta CPE characteristic time activation energy is virtually temperature independent.', '1701.05086-2-25-2': 'Physically, we think of each CPE\'s characteristic time [MATH] as reflecting this element\'s "inner" clock somewhat analogous to the material time of the Narayanaswamy physical-aging theory [CITATION].', '1701.05086-2-26-0': '## Model for the dynamic shear-mechanical properties', '1701.05086-2-27-0': "To arrive at an electrical equivalent circuit model of a supercooled liquid's shear dynamic properties, we first note that a standard parallel electrical RC element corresponds to the classical Maxwell model for viscoelasticity.", '1701.05086-2-27-1': 'This beatifully simple model is based on the assumption that the stress decays exponentially to zero whenever the sample is at rest [CITATION].', '1701.05086-2-27-2': 'If the time-dependent shear stress is denoted by [MATH], the shear displacement (strain) by [MATH], the DC shear viscosity by [MATH], and the instantaneous shear modulus by [MATH], the differential equation describing the Maxwell model for an arbitrary shear rate as a function of time, [MATH], is [CITATION] [EQUATION]', '1701.05086-2-27-3': 'The Maxwell model is equivalent to a parallel electrical RC element because for such an element the voltage is the same across both resistance and capacitor, which in the Maxwell model corresponds to having the same shear stress.', '1701.05086-2-27-4': 'The resistor current corresponds to the first term on the right hand side of Eq. ([REF]) (a dashpot in the traditional language of viscoelasticity), and the capacitor current corresponds to the second term (a spring in the viscoelastic language).', '1701.05086-2-28-0': 'The Maxwell model is too simple to fit data for glass-forming liquids, however, and must be extended by including one or more non-trivial dissipative terms.', '1701.05086-2-28-1': "This paper's basic idea is that these terms are described by CCREs placed in parallel to the Maxwell RC element, one for the alpha process and one for the beta process (Fig. [REF]).", '1701.05086-2-29-0': 'In the model none of the two CCREs are inherently linked to the alpha process RC element.', '1701.05086-2-29-1': 'Nevertheless, one CCRE will be regarded as part of the alpha process for the following reasons.', '1701.05086-2-29-2': 'Previous publications of the Glass and Time group have presented experimental [CITATION] and theoretical [CITATION] evidence that in the absence of beta relaxation the alpha process has a generic [MATH] high-frequency decay of the dielectric loss and shear compliance.', '1701.05086-2-29-3': 'This is an old idea that keeps resurfacing, recently in an interesting biophysical context [CITATION], and a generic [MATH] high-frequency decay is the characteristic feature of the 1967 Barlow-Erginsav-Lamb (BEL) model [CITATION].', '1701.05086-2-29-4': 'In view of this we fix the exponent to [MATH] for one CCRE and regard this element as a part of the alpha process.', '1701.05086-2-29-5': 'Confirming this assignment, for liquids without a mechanical beta relaxation like the silicone diffusion pump oils DC704 and DC705, the dynamic shear compliance is well fitted by the model of Fig. [REF] without the beta CCRE (unpublished).', '1701.05086-2-29-6': 'It should be mentioned that a finite one-dimensional so-called diffusion chain describing, e.g., the relation between temperature and heat flux entering from one end, has a compliance function that is very close to that of the alpha part of the model in Fig. [REF].', '1701.05086-2-30-0': "The dynamic shear compliance is a sum of the individual elements' shear compliances.", '1701.05086-2-30-1': 'Thus the model leads to the following expression, which defines the parametrization used henceforth: [EQUATION]', '1701.05086-2-30-2': 'For later use we note that the instantaneous (plateau) shear modulus [MATH] is given by [EQUATION]', '1701.05086-2-30-3': 'The modulus plateau between the alpha and beta processes at temperatures low enough that these are well separated, i.e., when frequencies exist obeying [MATH] and [MATH], is denoted by [MATH] and given by [EQUATION]', '1701.05086-2-30-4': 'In the low-frequency limit the shear compliance diverges as [MATH] as required for any liquid with a finite DC viscosity.', '1701.05086-2-30-5': 'In the DC limit the real part of the shear compliance, the so-called recoverable shear compliance, is given by [EQUATION]', '1701.05086-2-30-6': 'The model has seven parameters:', '1701.05086-2-31-0': 'We shall assume that time-temperature superposition (TTS) applies for both the alpha and the beta processes, implying that in fit to data the three dimensionless shape parameters [MATH], [MATH], and [MATH] cannot vary with temperature.', '1701.05086-2-31-1': 'The parameters allowed to vary with temperature are the two compliance strengths and the two relaxation times.', '1701.05086-2-32-0': 'The characteristic times of the alpha and beta CPEs are denoted by [MATH] and [MATH], respectively.', '1701.05086-2-32-1': 'In the below fit to squalane data we take as mentioned the constants [MATH] and [MATH] to be independent of temperature, which implies that [MATH] with a temperature-independent constant of proportionality.', '1701.05086-2-32-2': 'For this reason [MATH] will not be discussed separately from [MATH].', '1701.05086-2-32-3': 'The beta characteristic time [MATH], on the other hand, is not proportional to [MATH] in its temperature variation, which makes both beta times important to keep track of (disc_sec).', '1701.05086-2-33-0': '# Fitting the model to data for squalane', '1701.05086-2-34-0': 'The model was fitted to the squalane data of Fig. [REF] using MATLAB\'s "fminsearch" Nelder-Mead downhill simplex least-squares fitting procedure.', '1701.05086-2-34-1': 'The fit excluded data taken at too low a temperature to be in equilibrium or at such high temperatures that the alpha and beta process have almost completely merged.', '1701.05086-2-34-2': 'These limitations leave data for temperatures between 168 K and 182 K for fitting and parameter identification.', '1701.05086-2-35-0': 'The data for the real and imaginary parts of the frequency-dependent shear modulus cover angular frequencies ranging from 10 mHz to 30 kHz, with up to 16 frequencies per decade evenly distributed on a logarithmic scale.', '1701.05086-2-35-1': 'The data were fitted to Eq. ([REF]) for the shear-modulus (fitting to the shear compliance would have been dominated by the low-frequency compliance divergence).', '1701.05086-2-35-2': 'First, the three temperature-independent shape parameters [MATH], [MATH], [MATH] were identified by fitting to the 172 K data, which have the alpha and the beta loss peaks both well within the frequency window, but still clearly separated.', '1701.05086-2-35-3': 'Subsequently, the remaining four parameters were determined from the best fit at each temperature.', '1701.05086-2-36-0': 'Figure [REF] compares model fits (full green curves) to data (black crosses).', '1701.05086-2-36-1': 'Figure [REF](a) gives model prediction versus data for the real part of the dynamic shear modulus, Fig. [REF](b) gives the same for the imaginary part, and Fig. [REF](c) gives model prediction versus data in a Nyquist plot of the shear modulus.', '1701.05086-2-36-2': 'Figures [REF](d), (e), and (f) give the same for the dynamic shear compliance.', '1701.05086-2-37-0': 'The fits are excellent, which given the number of free parameters may not appear very surprising.', '1701.05086-2-37-1': 'Our experience with fitting data to similar models over the last 20 years shows, however, that the present model is better than other models with the same number of parameters.', '1701.05086-2-37-2': 'As an illustration of this, we have compared to a fit assuming a Havriliak-Negami (HN) type function for the alpha process.', '1701.05086-2-37-3': 'The function fitted to data is the following: [EQUATION]', '1701.05086-2-37-4': 'This has the same number of parameters as Eq. ([REF]): two strength parameters, two relaxation times, and three dimensionless shape parameters ([MATH]).', '1701.05086-2-37-5': 'A qualitative difference to Eq. ([REF]) should be mentioned, because the latter has a finite recoverable compliance, i.e., a finite low-frequency limit of [MATH], whereas Eq. ([REF]) like the BEL model [CITATION] diverges in this limit.', '1701.05086-2-37-6': 'We determined the best-fit parameters in the same way as above.', '1701.05086-2-37-7': 'The fit to data is not as good as that of Eq. ([REF]), which is clear from Fig. [REF] that compares the overall quality of the two best fits as functions of temperature.', '1701.05086-2-38-0': 'Returning to the model Eq. ([REF]), Fig. [REF] shows the temperature variation of the four free parameters.', '1701.05086-2-38-1': 'Figure [REF](a) shows how the alpha and beta relaxation times [MATH] and [MATH] vary with temperature.', '1701.05086-2-38-2': 'As always for a glass-forming liquid, the alpha relaxation time increases strongly when temperature is decreased.', '1701.05086-2-38-3': 'The beta relaxation time [MATH] is almost constant and, in fact, not even a monotonic function of temperature.', '1701.05086-2-38-4': 'In contrast, the beta characteristic time [MATH] decreases systematically with temperature.', '1701.05086-2-38-5': 'The dielectric beta loss-peak frequency is usually reported to be Arrhenius [CITATION], but it is important to note that almost all literature data for [MATH] (the inverse beta loss-peak frequency) refer to the glass phase, not to the metastable liquid phase about [MATH].', '1701.05086-2-38-6': 'Figure [REF](b) shows the best-fit shear-compliance strengths [MATH] and [MATH] as functions of temperature.', '1701.05086-2-38-7': 'Note that the beta process strength varies considerably more than the alpha strength.', '1701.05086-2-39-0': 'These findings are consistent with previous ones for the beta dielectric relaxation process, which may be summarized as follows [CITATION]: In the metastable liquid phase the relaxation strength increases considerably with increasing temperature whereas the relaxation time is almost temperature independent, in the glassy phase the strength is almost constant whereas the loss-peak frequency (inverse relaxation time) is strongly temperature dependent (Arrhenius).', '1701.05086-2-39-1': 'As an alternative to the minimal model of Ref. [CITATION] it is possible to rationalize these properties of the beta process - as well as its behavior under annealing of the out-of-equilibrium liquid - by assuming that the relaxation strength freezes at the glass transition whereas the characteristic time [MATH] is Arrhenius with an activation energy that is unaffected by the glass transition (unpublished).', '1701.05086-2-40-0': '# Data for the dynamic adiabatic bulk modulus of squalane', '1701.05086-2-41-0': 'Figure [REF] shows our transducer for measuring the dynamic adiabatic bulk modulus.', '1701.05086-2-41-1': 'It consists of a radially polarized piezo-ceramic spherical shell coated with electrodes on the inner and outer surfaces.', '1701.05086-2-41-2': "An applied electrical potential induces a slight compression or expansion of the sphere in which the liquid is placed (the top is a reservoir allowing for the liquid's thermal expansion) [CITATION].", '1701.05086-2-41-3': 'Figure [REF] shows data for the real and imaginary parts of the dynamic adiabatic bulk modulus [MATH], as well as a Nyquist plot of the same data.', '1701.05086-2-42-0': 'Comparing the bulk modulus loss in Fig. [REF](b) to the shear modulus loss in Fig. [REF](b), we see a qualitatively similar behavior with an alpha loss peak that moves rapidly to lower frequencies upon cooling and a large beta peak appearing.', '1701.05086-2-42-1': 'To the best of our knowledge this is the first observation of a beta process for the dynamic bulk modulus.', '1701.05086-2-43-0': 'How to interpret the similarity between the dynamic shear and bulk moduli?', '1701.05086-2-43-1': 'This finding is certainly consistent with many previous ones [CITATION], but it is important to emphasize that there is no fundamental reason for the similarity.', '1701.05086-2-43-2': 'This is because the dynamic bulk modulus - whether adiabatic or isothermal - is a scalar linear-response function whereas the dynamic shear modulus is a vector linear-response function.', '1701.05086-2-43-3': 'As discussed by Meixner long time ago these functions therefore belong to fundamentally different symmetry classes [CITATION].', '1701.05086-2-43-4': 'Nevertheless, by reference to the Eshelby picture of structural rearrangements within a surrounding elastic matrix, Buchenau has recently discussed how the relaxational parts of the bulk and shear moduli may be connected [CITATION] in arguments that may be extended to finite frequencies, thus establishing a connection between [MATH] and [MATH].', '1701.05086-2-44-0': 'Referring to the energy-bond formalism [CITATION], there are two fundamental thermodynamic scalar energy bonds: a thermal energy bond with effort equal to temperature difference and flow equal to entropy current, and a mechanical bond with effort equal to minus pressure difference and flow equal to rate of volume change.', '1701.05086-2-44-1': "Consistent with Buchenau's reasoning [CITATION] we propose a general energy-bond model in which all dissipation connected with the two scalar thermodynamic energy bonds is controlled by the dynamic shear modulus (or, equivalently, the dynamic shear compliance).", '1701.05086-2-44-2': 'A representation of this idea is given in Fig. [REF](b).', '1701.05086-2-44-3': 'An energy-bond diagram of this sort implies that the system in question is a "single-order-parameter" liquid [CITATION].', '1701.05086-2-44-4': 'This is equivalent to being an R simple system, i.e., one with so-called isomorphs, which are lines in the thermodynamic phase diagram along which the dynamics is invariant to a very good approximation [CITATION].', '1701.05086-2-45-0': 'In Fig. [REF](b) there may be several non-dissipative elements, but the important point is that these are all connected to the element of Fig. [REF] via a single, internal energy bond.', '1701.05086-2-45-1': 'The predictions for the dynamic adiabatic/isothermal bulk moduli (or those of the dynamic expansion coefficient [CITATION]) depend, of course, not just on the dynamic shear modulus (compliance), but also on the non-dissipative elements.', '1701.05086-2-45-2': "For a system described by Fig. [REF](b) one a priori expects that all the scalar response functions at any given temperature have alpha and beta processes located at frequencies similar to those of the shear modulus' alpha and beta processes.", '1701.05086-2-46-0': 'As an example of the general modeling philosophy of Fig. [REF](b), Fig. [REF](b) gives a specific model for [MATH] in terms of [MATH].', '1701.05086-2-46-1': 'First, Fig. [REF](a) demonstrates the similarity between the relaxation times of the equilibrium shear stress fluctuations determining [MATH] via the fluctuation-dissipation theorem (red circles) and those of the pressure fluctuations determining [MATH] (blue stars).', '1701.05086-2-46-2': 'Clearly these two times are of the same order of magnitude and have similar dependence on temperature.', '1701.05086-2-46-3': 'This means that a model following Fig. [REF](b) makes sense.', '1701.05086-2-46-4': 'One may think of different such models, and the one shown in Fig. [REF](b) is just an example.', '1701.05086-2-46-5': 'At each temperature there are only few fitting parameters while the entire frequency dependence and, in particular, the dissipation is determined by [MATH].', '1701.05086-2-46-6': 'In the fit to data we took the zero-frequency (adiabatic) bulk modulus [MATH] measured at the temperature in question as input, leaving just two free parameters.', '1701.05086-2-46-7': 'Nevertheless, the Nyquist plot of [MATH] demonstrates a reasonable fit (Fig. [REF](c)).', '1701.05086-2-47-0': '# Discussion', '1701.05086-2-48-0': 'This paper has demonstrated that dynamic shear-mechanical data for squalane may be fitted very well with the electrical-equivalent circuit model of Fig. [REF] leading to Eq. ([REF]) for the shear compliance.', '1701.05086-2-48-1': 'The model assumes additivity of the alpha and beta shear compliances.', '1701.05086-2-48-2': 'The model has seven parameters, one more than alternative phenomenological models [CITATION].', '1701.05086-2-48-3': 'In the fit to data, however, the three dimensionless shape parameters were taken to be temperature independent, reflecting the assumption that time-temperature superposition applies separately to both the alpha and the beta compliance functions.', '1701.05086-2-48-4': 'In this picture, observed deviations from TTS derive from the merging of the alpha and beta processes.', '1701.05086-2-48-5': 'We conjecture that this applies generally for glass-forming liquids.', '1701.05086-2-49-0': 'How to physically justify that the Maxwell RC element and the two CCREs should be combined in a way that is additive in their shear compliances, not in their shear moduli?', '1701.05086-2-49-1': 'There are no logically compelling arguments for this.', '1701.05086-2-49-2': 'We think of it as follows.', '1701.05086-2-49-3': 'Imagine a small particle in the liquid.', '1701.05086-2-49-4': "The particle's mean-square displacement (MSD) as a function of time in one axis direction, [MATH], will have a rapid increase on the phonon time scale, followed by a transition to the long-time diffusive behavior proportional to time.", '1701.05086-2-49-5': "If one assumes that the alpha and beta processes are statistically independent [CITATION], this implies for the particle's motion that [MATH] with [MATH].", '1701.05086-2-49-6': 'In this case the MSD is a sum of an alpha and a beta contribution: [MATH].', '1701.05086-2-49-7': "If one moreover assumes the Stokes-Einstein relation between the dynamic shear viscosity and the particle's dynamic friction coefficient, this translates via the fluctuation-dissipation theorem into additivity of the dynamic shear compliances for the alpha and beta processes.", '1701.05086-2-50-0': 'In regard to the single-particle MSD, note that associated with any function [MATH] there is a characteristic time, namely the time at which the particle has moved a typical intermolecular distance.', '1701.05086-2-50-1': "This is how we think of each CPE basic element's characteristic [MATH], which was defined by the absolute value of the compliance at [MATH] being 1 [MATH] (Eq. ([REF])).", '1701.05086-2-50-2': "For the beta process, it is important to distinguish between this time and the inverse loss-peak frequency [MATH], because via Eq. ([REF]) the latter time's temperature variation reflects the combined effect of the changing compliance strength [MATH] and the Arrhenius [MATH].", '1701.05086-2-51-0': 'The electrical-equivalent circuit model Fig. [REF] is identical to that proposed in Ref. [CITATION] except for an extra capacitor, the one in the alpha CCRE.', '1701.05086-2-51-1': 'This capacitor eliminates an unphysical feature of our previous model [CITATION], which predicted an infinite recoverable shear compliance.', '1701.05086-2-51-2': 'This unphysical feature is also present in the BEL model from 1967 [CITATION].', '1701.05086-2-51-3': 'Introducing the extra capacitor has the added benefit of resulting in symmetry between the alpha and beta CCREs, the only difference being that the alpha CCRE has the exponent fixed to [MATH].', '1701.05086-2-52-0': "We note that if the alpha CCRE's relaxation time is much shorter than [MATH], the circuit mimics the situation reported in recent papers for the dielectric relaxation of monohydroxy alcohols, for which one observes a low-frequency Debye-type process followed by, in order of increasing frequency, first an alpha and then, in most liquids, a beta process [CITATION].", '1701.05086-2-53-0': 'As regards the temperature dependence of the best-fit model parameters for squalane we have compared to the prediction of the shoving model [CITATION].', '1701.05086-2-53-1': 'If [MATH]s is a typical phonon time and [MATH] the so-called characteristic volume assumed to be temperature independent, the shoving model predicts the following relation between the temperature variation of the alpha relaxation time and that of the instantaneous, i.e., high-frequency plateau, shear modulus [MATH]: [EQUATION]', '1701.05086-2-53-2': "The shoving model, which links a supercooled liquid's fragility to the temperature variation of [MATH], fits data well for many glass-forming liquids [CITATION].", '1701.05086-2-53-3': 'The shoving model relates directly to Eq. ([REF]) since if it applies, via Eq. ([REF]) the number of temperature-dependent parameters is reduced from four to three.', '1701.05086-2-53-4': 'We have not made this assumption in the fit to data, but have instead checked Eq. ([REF]) against the best-fit parameters.', '1701.05086-2-53-5': 'This is done in Fig. [REF] in which the relaxation times [MATH] and [MATH] have been converted into temperature-dependent activation energies [MATH] by writing for each [MATH] with [MATH]s.', '1701.05086-2-53-6': 'According to the shoving model [MATH].', '1701.05086-2-53-7': 'By comparing the black crosses and the red triangles in Fig. [REF] we conclude that the shoving model applies with [MATH] calculated from the best fit model parameters via Eq. ([REF]).', '1701.05086-2-54-0': 'If the shoving model is instead interpreted with [MATH] taken to be the modulus between the alpha and beta relaxations, the quantity [MATH] given by Eq. ([REF]), the model does not apply (blue crosses).', '1701.05086-2-54-1': 'This confirms the basic physical assumption of the shoving model, which is that the actual barrier transition for a rearrangement of molecules is very fast, presumably on the picosecond time scale.', '1701.05086-2-54-2': 'Consequently, the activation energy is proportional to the shear modulus of the liquid measured on this short time scale (at which the liquid behaves like a solid), not to the plateau modulus between the alpha and beta relaxations.', '1701.05086-2-55-0': 'Our new model for [MATH] (Fig. [REF]) consists of a Maxwell element in parallel with two CCREs.', '1701.05086-2-55-1': 'One may speculate that additional high-frequency mechanical processes beyond the alpha and beta relaxations can be included by adding further CCREs in parallel, each one still subject to time-temperature superposition, i.e., with temperature-independent shape parameters.', '1701.05086-2-56-0': 'To summarize, an excellent fit to dynamic shear-mechanical data of squalane is provided by an electrical equivalent circuit model with seven parameters.', '1701.05086-2-56-1': 'The model assumes an [MATH] high-frequency decay of the alpha compliance [CITATION], additivity of the alpha and beta compliance functions, and that these functions separately obey time-temperature superposition.', '1701.05086-2-56-2': 'The latter assumption reduces the number of parameters varying with temperature to four.', '1701.05086-2-56-3': 'The best fit parameters confirm the shoving model and show that the beta process characteristic time has a temperature-independent activation energy.', '1701.05086-2-56-4': 'If these findings were both incorporated as model assumptions, the number of parameters varying with temperature reduces to two.', '1701.05086-2-56-5': 'These could be taken to be, e.g., the compliance magnitudes [MATH] and [MATH].', '1701.05086-2-56-6': 'We also presented data for the adiabatic dynamic bulk modulus and showed that these may be interpreted in terms of a single-order-parameter model in which all dissipation is controlled by the shear-mechanical properties.', '1701.05086-2-56-7': 'Such a model connects the class of scalar viscoelastic linear-response functions to that of vector symmetry [CITATION].', '1701.05086-2-57-0': 'In regard to future works, one obvious thing is to compare the model to shear-mechanical data for other glass-forming liquids.', '1701.05086-2-57-1': 'We have not done so systematically, but have found in all cases tested so far that the model works well (unpublished).', '1701.05086-2-57-2': "The hope is that the model is general, thus providing a step towards a microscopic understanding of supercooled liquids' shear-dynamical properties.", '1701.05086-2-58-0': 'Figure [REF](b) is a special case of the general energy-bond diagram for a system with a single order parameter identified by our Glass and Time colleague Tage Christensen about 15 years ago, which initiated a development leading to Ref. [CITATION] that in turn led to the isomorph theory [CITATION].'}

[['1701.05086-1-51-0', '1701.05086-2-52-0'], ['1701.05086-1-13-1', '1701.05086-2-13-1'], ['1701.05086-1-34-0', '1701.05086-2-34-0'], ['1701.05086-1-54-0', '1701.05086-2-56-0'], ['1701.05086-1-54-1', '1701.05086-2-56-1'], ['1701.05086-1-54-2', '1701.05086-2-56-2'], ['1701.05086-1-54-3', '1701.05086-2-56-3'], ['1701.05086-1-54-5', '1701.05086-2-56-5'], ['1701.05086-1-37-4', '1701.05086-2-37-3'], ['1701.05086-1-37-7', '1701.05086-2-37-6'], ['1701.05086-1-7-1', '1701.05086-2-7-1'], ['1701.05086-1-7-2', '1701.05086-2-7-2'], ['1701.05086-1-7-3', '1701.05086-2-7-3'], ['1701.05086-1-48-0', '1701.05086-2-49-0'], ['1701.05086-1-48-1', '1701.05086-2-49-1'], ['1701.05086-1-48-2', '1701.05086-2-49-2'], ['1701.05086-1-48-3', '1701.05086-2-49-3'], ['1701.05086-1-48-4', '1701.05086-2-49-4'], ['1701.05086-1-43-0', '1701.05086-2-43-0'], ['1701.05086-1-43-2', '1701.05086-2-43-2'], ['1701.05086-1-8-3', '1701.05086-2-8-4'], ['1701.05086-1-8-4', '1701.05086-2-8-5'], ['1701.05086-1-8-5', '1701.05086-2-8-6'], ['1701.05086-1-22-2', '1701.05086-2-22-2'], ['1701.05086-1-22-3', '1701.05086-2-22-3'], ['1701.05086-1-22-4', '1701.05086-2-22-4'], ['1701.05086-1-12-1', '1701.05086-2-12-1'], ['1701.05086-1-38-1', '1701.05086-2-38-1'], ['1701.05086-1-38-4', '1701.05086-2-38-4'], ['1701.05086-1-38-5', '1701.05086-2-38-5'], ['1701.05086-1-38-7', '1701.05086-2-38-7'], ['1701.05086-1-29-1', '1701.05086-2-29-1'], ['1701.05086-1-41-1', '1701.05086-2-41-1'], ['1701.05086-1-36-1', '1701.05086-2-36-1'], ['1701.05086-1-44-3', '1701.05086-2-44-3'], ['1701.05086-1-27-4', '1701.05086-2-27-4'], ['1701.05086-1-45-1', '1701.05086-2-45-1'], ['1701.05086-1-55-0', '1701.05086-2-57-0'], ['1701.05086-1-55-1', '1701.05086-2-57-1'], ['1701.05086-1-30-0', '1701.05086-2-30-0'], ['1701.05086-1-30-4', '1701.05086-2-30-4'], ['1701.05086-1-16-1', '1701.05086-2-16-1'], ['1701.05086-1-16-2', '1701.05086-2-16-2'], ['1701.05086-1-0-5', '1701.05086-2-0-6'], ['1701.05086-1-0-8', '1701.05086-2-0-9'], ['1701.05086-1-0-10', '1701.05086-2-0-11'], ['1701.05086-1-47-1', '1701.05086-2-48-1'], ['1701.05086-1-47-2', '1701.05086-2-48-2'], ['1701.05086-1-47-5', '1701.05086-2-48-5'], ['1701.05086-1-3-0', '1701.05086-2-3-0'], ['1701.05086-1-3-1', '1701.05086-2-3-1'], ['1701.05086-1-3-2', '1701.05086-2-3-2'], ['1701.05086-1-3-4', '1701.05086-2-3-3'], ['1701.05086-1-3-5', '1701.05086-2-3-4'], ['1701.05086-1-3-6', '1701.05086-2-3-5'], ['1701.05086-1-3-7', '1701.05086-2-3-6'], ['1701.05086-1-3-8', '1701.05086-2-3-7'], ['1701.05086-1-3-9', '1701.05086-2-3-8'], ['1701.05086-1-18-0', '1701.05086-2-18-0'], ['1701.05086-1-50-0', '1701.05086-2-51-0'], ['1701.05086-1-50-2', '1701.05086-2-51-2'], ['1701.05086-1-5-5', '1701.05086-2-5-6'], ['1701.05086-1-35-0', '1701.05086-2-35-0'], ['1701.05086-1-15-0', '1701.05086-2-15-0'], ['1701.05086-1-52-2', '1701.05086-2-53-2'], ['1701.05086-1-52-4', '1701.05086-2-53-4'], ['1701.05086-1-52-6', '1701.05086-2-53-6'], ['1701.05086-1-19-1', '1701.05086-2-20-1'], ['1701.05086-1-19-3', '1701.05086-2-20-3'], ['1701.05086-1-19-4', '1701.05086-2-20-4'], ['1701.05086-1-19-5', '1701.05086-2-20-5'], ['1701.05086-1-25-1', '1701.05086-2-24-1'], ['1701.05086-1-25-2', '1701.05086-2-24-2'], ['1701.05086-1-25-3', '1701.05086-2-24-3'], ['1701.05086-1-13-0', '1701.05086-2-13-0'], ['1701.05086-1-34-1', '1701.05086-2-34-1'], ['1701.05086-1-34-2', '1701.05086-2-34-2'], ['1701.05086-1-54-4', '1701.05086-2-56-4'], ['1701.05086-1-54-6', '1701.05086-2-56-6'], ['1701.05086-1-54-7', '1701.05086-2-56-7'], ['1701.05086-1-37-2', '1701.05086-2-37-1'], ['1701.05086-1-37-3', '1701.05086-2-37-2'], ['1701.05086-1-37-5', '1701.05086-2-37-4'], ['1701.05086-1-53-0', '1701.05086-2-55-0'], ['1701.05086-1-7-0', '1701.05086-2-7-0'], ['1701.05086-1-48-6', '1701.05086-2-49-7'], ['1701.05086-1-43-1', '1701.05086-2-43-1'], ['1701.05086-1-43-3', '1701.05086-2-43-3'], ['1701.05086-1-8-0', '1701.05086-2-8-0'], ['1701.05086-1-8-2', '1701.05086-2-8-2'], ['1701.05086-1-22-0', '1701.05086-2-22-0'], ['1701.05086-1-22-1', '1701.05086-2-22-1'], ['1701.05086-1-22-5', '1701.05086-2-22-5'], ['1701.05086-1-12-0', '1701.05086-2-12-0'], ['1701.05086-1-4-0', '1701.05086-2-4-1'], ['1701.05086-1-4-1', '1701.05086-2-4-2'], ['1701.05086-1-17-2', '1701.05086-2-17-1'], ['1701.05086-1-17-3', '1701.05086-2-17-2'], ['1701.05086-1-17-4', '1701.05086-2-17-3'], ['1701.05086-1-38-0', '1701.05086-2-38-0'], ['1701.05086-1-38-2', '1701.05086-2-38-2'], ['1701.05086-1-38-3', '1701.05086-2-38-3'], ['1701.05086-1-29-0', '1701.05086-2-29-0'], ['1701.05086-1-29-2', '1701.05086-2-29-2'], ['1701.05086-1-29-4', '1701.05086-2-29-4'], ['1701.05086-1-29-5', '1701.05086-2-29-5'], ['1701.05086-1-41-0', '1701.05086-2-41-0'], ['1701.05086-1-41-2', '1701.05086-2-41-2'], ['1701.05086-1-36-0', '1701.05086-2-36-0'], ['1701.05086-1-42-0', '1701.05086-2-42-0'], ['1701.05086-1-42-1', '1701.05086-2-42-1'], ['1701.05086-1-44-0', '1701.05086-2-44-0'], ['1701.05086-1-44-1', '1701.05086-2-44-1'], ['1701.05086-1-44-2', '1701.05086-2-44-2'], ['1701.05086-1-44-4', '1701.05086-2-44-4'], ['1701.05086-1-27-1', '1701.05086-2-27-1'], ['1701.05086-1-27-2', '1701.05086-2-27-2'], ['1701.05086-1-32-0', '1701.05086-2-32-0'], ['1701.05086-1-32-2', '1701.05086-2-32-2'], ['1701.05086-1-32-3', '1701.05086-2-32-3'], ['1701.05086-1-56-0', '1701.05086-2-58-0'], ['1701.05086-1-2-0', '1701.05086-2-2-0'], ['1701.05086-1-2-1', '1701.05086-2-2-1'], ['1701.05086-1-2-2', '1701.05086-2-2-2'], ['1701.05086-1-2-3', '1701.05086-2-2-3'], ['1701.05086-1-2-4', '1701.05086-2-2-4'], ['1701.05086-1-2-5', '1701.05086-2-2-5'], ['1701.05086-1-28-0', '1701.05086-2-28-0'], ['1701.05086-1-28-1', '1701.05086-2-28-1'], ['1701.05086-1-45-0', '1701.05086-2-45-0'], ['1701.05086-1-45-2', '1701.05086-2-45-2'], ['1701.05086-1-55-2', '1701.05086-2-57-2'], ['1701.05086-1-39-0', '1701.05086-2-39-0'], ['1701.05086-1-39-1', '1701.05086-2-39-1'], ['1701.05086-1-30-2', '1701.05086-2-30-2'], ['1701.05086-1-30-3', '1701.05086-2-30-3'], ['1701.05086-1-30-5', '1701.05086-2-30-5'], ['1701.05086-1-16-0', '1701.05086-2-16-0'], ['1701.05086-1-0-0', '1701.05086-2-0-0'], ['1701.05086-1-0-1', '1701.05086-2-0-1'], ['1701.05086-1-0-3', '1701.05086-2-0-4'], ['1701.05086-1-0-4', '1701.05086-2-0-5'], ['1701.05086-1-0-6', '1701.05086-2-0-7'], ['1701.05086-1-0-7', '1701.05086-2-0-8'], ['1701.05086-1-47-0', '1701.05086-2-48-0'], ['1701.05086-1-47-3', '1701.05086-2-48-3'], ['1701.05086-1-47-4', '1701.05086-2-48-4'], ['1701.05086-1-9-0', '1701.05086-2-9-0'], ['1701.05086-1-9-1', '1701.05086-2-9-1'], ['1701.05086-1-9-2', '1701.05086-2-9-3'], ['1701.05086-1-18-1', '1701.05086-2-18-1'], ['1701.05086-1-31-1', '1701.05086-2-31-1'], ['1701.05086-1-49-0', '1701.05086-2-50-0'], ['1701.05086-1-49-1', '1701.05086-2-50-1'], ['1701.05086-1-49-2', '1701.05086-2-50-2'], ['1701.05086-1-5-0', '1701.05086-2-5-0'], ['1701.05086-1-5-3', '1701.05086-2-5-3'], ['1701.05086-1-5-4', '1701.05086-2-5-4'], ['1701.05086-1-5-6', '1701.05086-2-5-7'], ['1701.05086-1-35-2', '1701.05086-2-35-2'], ['1701.05086-1-15-3', '1701.05086-2-15-3'], ['1701.05086-1-52-0', '1701.05086-2-53-0'], ['1701.05086-1-52-1', '1701.05086-2-53-1'], ['1701.05086-1-52-3', '1701.05086-2-53-3'], ['1701.05086-1-52-5', '1701.05086-2-53-5'], ['1701.05086-1-52-8', '1701.05086-2-54-0'], ['1701.05086-1-19-0', '1701.05086-2-20-0'], ['1701.05086-1-19-2', '1701.05086-2-20-2'], ['1701.05086-1-20-1', '1701.05086-2-20-6'], ['1701.05086-1-25-0', '1701.05086-2-24-0'], ['1701.05086-1-25-4', '1701.05086-2-24-4'], ['1701.05086-1-25-5', '1701.05086-2-24-5'], ['1701.05086-1-25-7', '1701.05086-2-24-6'], ['1701.05086-1-37-1', '1701.05086-2-37-0'], ['1701.05086-1-37-6', '1701.05086-2-37-5'], ['1701.05086-1-37-8', '1701.05086-2-37-0'], ['1701.05086-1-37-8', '1701.05086-2-37-7'], ['1701.05086-1-37-9', '1701.05086-2-37-7'], ['1701.05086-1-53-1', '1701.05086-2-55-1'], ['1701.05086-1-48-5', '1701.05086-2-49-5'], ['1701.05086-1-48-5', '1701.05086-2-49-6'], ['1701.05086-1-43-4', '1701.05086-2-43-4'], ['1701.05086-1-43-5', '1701.05086-2-43-4'], ['1701.05086-1-8-1', '1701.05086-2-8-1'], ['1701.05086-1-4-2', '1701.05086-2-4-3'], ['1701.05086-1-17-0', '1701.05086-2-17-0'], ['1701.05086-1-17-1', '1701.05086-2-17-0'], ['1701.05086-1-38-6', '1701.05086-2-38-6'], ['1701.05086-1-29-3', '1701.05086-2-29-3'], ['1701.05086-1-41-3', '1701.05086-2-41-3'], ['1701.05086-1-36-2', '1701.05086-2-36-2'], ['1701.05086-1-27-0', '1701.05086-2-27-0'], ['1701.05086-1-27-0', '1701.05086-2-27-3'], ['1701.05086-1-27-3', '1701.05086-2-27-3'], ['1701.05086-1-32-1', '1701.05086-2-32-1'], ['1701.05086-1-30-1', '1701.05086-2-30-1'], ['1701.05086-1-0-2', '1701.05086-2-0-2'], ['1701.05086-1-0-2', '1701.05086-2-0-3'], ['1701.05086-1-0-9', '1701.05086-2-0-10'], ['1701.05086-1-31-0', '1701.05086-2-31-0'], ['1701.05086-1-50-1', '1701.05086-2-51-1'], ['1701.05086-1-50-3', '1701.05086-2-51-3'], ['1701.05086-1-5-1', '1701.05086-2-5-1'], ['1701.05086-1-5-2', '1701.05086-2-5-2'], ['1701.05086-1-35-1', '1701.05086-2-35-1'], ['1701.05086-1-35-3', '1701.05086-2-35-1'], ['1701.05086-1-15-1', '1701.05086-2-15-1'], ['1701.05086-1-15-2', '1701.05086-2-15-1'], ['1701.05086-1-15-2', '1701.05086-2-15-2'], ['1701.05086-1-52-7', '1701.05086-2-53-7'], ['1701.05086-1-23-0', '1701.05086-2-25-2'], ['1701.05086-1-25-6', '1701.05086-2-25-0'], ['1701.05086-1-25-6', '1701.05086-2-25-1']]

[['1701.05086-1-51-0', '1701.05086-2-52-0'], ['1701.05086-1-13-1', '1701.05086-2-13-1'], ['1701.05086-1-34-0', '1701.05086-2-34-0'], ['1701.05086-1-54-0', '1701.05086-2-56-0'], ['1701.05086-1-54-1', '1701.05086-2-56-1'], ['1701.05086-1-54-2', '1701.05086-2-56-2'], ['1701.05086-1-54-3', '1701.05086-2-56-3'], ['1701.05086-1-54-5', '1701.05086-2-56-5'], ['1701.05086-1-37-4', '1701.05086-2-37-3'], ['1701.05086-1-37-7', '1701.05086-2-37-6'], ['1701.05086-1-7-1', '1701.05086-2-7-1'], ['1701.05086-1-7-2', '1701.05086-2-7-2'], ['1701.05086-1-7-3', '1701.05086-2-7-3'], ['1701.05086-1-48-0', '1701.05086-2-49-0'], ['1701.05086-1-48-1', '1701.05086-2-49-1'], ['1701.05086-1-48-2', '1701.05086-2-49-2'], ['1701.05086-1-48-3', '1701.05086-2-49-3'], ['1701.05086-1-48-4', '1701.05086-2-49-4'], ['1701.05086-1-43-0', '1701.05086-2-43-0'], ['1701.05086-1-43-2', '1701.05086-2-43-2'], ['1701.05086-1-8-3', '1701.05086-2-8-4'], ['1701.05086-1-8-4', '1701.05086-2-8-5'], ['1701.05086-1-8-5', '1701.05086-2-8-6'], ['1701.05086-1-22-2', '1701.05086-2-22-2'], ['1701.05086-1-22-3', '1701.05086-2-22-3'], ['1701.05086-1-22-4', '1701.05086-2-22-4'], ['1701.05086-1-12-1', '1701.05086-2-12-1'], ['1701.05086-1-38-1', '1701.05086-2-38-1'], ['1701.05086-1-38-4', '1701.05086-2-38-4'], ['1701.05086-1-38-5', '1701.05086-2-38-5'], ['1701.05086-1-38-7', '1701.05086-2-38-7'], ['1701.05086-1-29-1', '1701.05086-2-29-1'], ['1701.05086-1-41-1', '1701.05086-2-41-1'], ['1701.05086-1-36-1', '1701.05086-2-36-1'], ['1701.05086-1-44-3', '1701.05086-2-44-3'], ['1701.05086-1-27-4', '1701.05086-2-27-4'], ['1701.05086-1-45-1', '1701.05086-2-45-1'], ['1701.05086-1-55-0', '1701.05086-2-57-0'], ['1701.05086-1-55-1', '1701.05086-2-57-1'], ['1701.05086-1-30-0', '1701.05086-2-30-0'], ['1701.05086-1-30-4', '1701.05086-2-30-4'], ['1701.05086-1-16-1', '1701.05086-2-16-1'], ['1701.05086-1-16-2', '1701.05086-2-16-2'], ['1701.05086-1-0-5', '1701.05086-2-0-6'], ['1701.05086-1-0-8', '1701.05086-2-0-9'], ['1701.05086-1-0-10', '1701.05086-2-0-11'], ['1701.05086-1-47-1', '1701.05086-2-48-1'], ['1701.05086-1-47-2', '1701.05086-2-48-2'], ['1701.05086-1-47-5', '1701.05086-2-48-5'], ['1701.05086-1-3-0', '1701.05086-2-3-0'], ['1701.05086-1-3-1', '1701.05086-2-3-1'], ['1701.05086-1-3-2', '1701.05086-2-3-2'], ['1701.05086-1-3-4', '1701.05086-2-3-3'], ['1701.05086-1-3-5', '1701.05086-2-3-4'], ['1701.05086-1-3-6', '1701.05086-2-3-5'], ['1701.05086-1-3-7', '1701.05086-2-3-6'], ['1701.05086-1-3-8', '1701.05086-2-3-7'], ['1701.05086-1-3-9', '1701.05086-2-3-8'], ['1701.05086-1-18-0', '1701.05086-2-18-0'], ['1701.05086-1-50-0', '1701.05086-2-51-0'], ['1701.05086-1-50-2', '1701.05086-2-51-2'], ['1701.05086-1-5-5', '1701.05086-2-5-6'], ['1701.05086-1-35-0', '1701.05086-2-35-0'], ['1701.05086-1-15-0', '1701.05086-2-15-0'], ['1701.05086-1-52-2', '1701.05086-2-53-2'], ['1701.05086-1-52-4', '1701.05086-2-53-4'], ['1701.05086-1-52-6', '1701.05086-2-53-6'], ['1701.05086-1-19-1', '1701.05086-2-20-1'], ['1701.05086-1-19-3', '1701.05086-2-20-3'], ['1701.05086-1-19-4', '1701.05086-2-20-4'], ['1701.05086-1-19-5', '1701.05086-2-20-5'], ['1701.05086-1-25-1', '1701.05086-2-24-1'], ['1701.05086-1-25-2', '1701.05086-2-24-2'], ['1701.05086-1-25-3', '1701.05086-2-24-3']]

[['1701.05086-1-13-0', '1701.05086-2-13-0'], ['1701.05086-1-34-1', '1701.05086-2-34-1'], ['1701.05086-1-34-2', '1701.05086-2-34-2'], ['1701.05086-1-54-4', '1701.05086-2-56-4'], ['1701.05086-1-54-6', '1701.05086-2-56-6'], ['1701.05086-1-54-7', '1701.05086-2-56-7'], ['1701.05086-1-37-2', '1701.05086-2-37-1'], ['1701.05086-1-37-3', '1701.05086-2-37-2'], ['1701.05086-1-37-5', '1701.05086-2-37-4'], ['1701.05086-1-53-0', '1701.05086-2-55-0'], ['1701.05086-1-7-0', '1701.05086-2-7-0'], ['1701.05086-1-48-6', '1701.05086-2-49-7'], ['1701.05086-1-43-1', '1701.05086-2-43-1'], ['1701.05086-1-43-3', '1701.05086-2-43-3'], ['1701.05086-1-8-0', '1701.05086-2-8-0'], ['1701.05086-1-8-2', '1701.05086-2-8-2'], ['1701.05086-1-22-0', '1701.05086-2-22-0'], ['1701.05086-1-22-1', '1701.05086-2-22-1'], ['1701.05086-1-22-5', '1701.05086-2-22-5'], ['1701.05086-1-12-0', '1701.05086-2-12-0'], ['1701.05086-1-4-0', '1701.05086-2-4-1'], ['1701.05086-1-4-1', '1701.05086-2-4-2'], ['1701.05086-1-17-2', '1701.05086-2-17-1'], ['1701.05086-1-17-3', '1701.05086-2-17-2'], ['1701.05086-1-17-4', '1701.05086-2-17-3'], ['1701.05086-1-38-0', '1701.05086-2-38-0'], ['1701.05086-1-38-2', '1701.05086-2-38-2'], ['1701.05086-1-38-3', '1701.05086-2-38-3'], ['1701.05086-1-29-0', '1701.05086-2-29-0'], ['1701.05086-1-29-2', '1701.05086-2-29-2'], ['1701.05086-1-29-4', '1701.05086-2-29-4'], ['1701.05086-1-29-5', '1701.05086-2-29-5'], ['1701.05086-1-41-0', '1701.05086-2-41-0'], ['1701.05086-1-41-2', '1701.05086-2-41-2'], ['1701.05086-1-36-0', '1701.05086-2-36-0'], ['1701.05086-1-42-0', '1701.05086-2-42-0'], ['1701.05086-1-42-1', '1701.05086-2-42-1'], ['1701.05086-1-44-0', '1701.05086-2-44-0'], ['1701.05086-1-44-1', '1701.05086-2-44-1'], ['1701.05086-1-44-2', '1701.05086-2-44-2'], ['1701.05086-1-44-4', '1701.05086-2-44-4'], ['1701.05086-1-27-1', '1701.05086-2-27-1'], ['1701.05086-1-27-2', '1701.05086-2-27-2'], ['1701.05086-1-32-0', '1701.05086-2-32-0'], ['1701.05086-1-32-2', '1701.05086-2-32-2'], ['1701.05086-1-32-3', '1701.05086-2-32-3'], ['1701.05086-1-56-0', '1701.05086-2-58-0'], ['1701.05086-1-2-0', '1701.05086-2-2-0'], ['1701.05086-1-2-1', '1701.05086-2-2-1'], ['1701.05086-1-2-2', '1701.05086-2-2-2'], ['1701.05086-1-2-3', '1701.05086-2-2-3'], ['1701.05086-1-2-4', '1701.05086-2-2-4'], ['1701.05086-1-2-5', '1701.05086-2-2-5'], ['1701.05086-1-28-0', '1701.05086-2-28-0'], ['1701.05086-1-28-1', '1701.05086-2-28-1'], ['1701.05086-1-45-0', '1701.05086-2-45-0'], ['1701.05086-1-45-2', '1701.05086-2-45-2'], ['1701.05086-1-55-2', '1701.05086-2-57-2'], ['1701.05086-1-39-0', '1701.05086-2-39-0'], ['1701.05086-1-39-1', '1701.05086-2-39-1'], ['1701.05086-1-30-2', '1701.05086-2-30-2'], ['1701.05086-1-30-3', '1701.05086-2-30-3'], ['1701.05086-1-30-5', '1701.05086-2-30-5'], ['1701.05086-1-16-0', '1701.05086-2-16-0'], ['1701.05086-1-0-0', '1701.05086-2-0-0'], ['1701.05086-1-0-1', '1701.05086-2-0-1'], ['1701.05086-1-0-3', '1701.05086-2-0-4'], ['1701.05086-1-0-4', '1701.05086-2-0-5'], ['1701.05086-1-0-6', '1701.05086-2-0-7'], ['1701.05086-1-0-7', '1701.05086-2-0-8'], ['1701.05086-1-47-0', '1701.05086-2-48-0'], ['1701.05086-1-47-3', '1701.05086-2-48-3'], ['1701.05086-1-47-4', '1701.05086-2-48-4'], ['1701.05086-1-9-0', '1701.05086-2-9-0'], ['1701.05086-1-9-1', '1701.05086-2-9-1'], ['1701.05086-1-9-2', '1701.05086-2-9-3'], ['1701.05086-1-18-1', '1701.05086-2-18-1'], ['1701.05086-1-31-1', '1701.05086-2-31-1'], ['1701.05086-1-49-0', '1701.05086-2-50-0'], ['1701.05086-1-49-1', '1701.05086-2-50-1'], ['1701.05086-1-49-2', '1701.05086-2-50-2'], ['1701.05086-1-5-0', '1701.05086-2-5-0'], ['1701.05086-1-5-3', '1701.05086-2-5-3'], ['1701.05086-1-5-4', '1701.05086-2-5-4'], ['1701.05086-1-5-6', '1701.05086-2-5-7'], ['1701.05086-1-35-2', '1701.05086-2-35-2'], ['1701.05086-1-15-3', '1701.05086-2-15-3'], ['1701.05086-1-52-0', '1701.05086-2-53-0'], ['1701.05086-1-52-1', '1701.05086-2-53-1'], ['1701.05086-1-52-3', '1701.05086-2-53-3'], ['1701.05086-1-52-5', '1701.05086-2-53-5'], ['1701.05086-1-52-8', '1701.05086-2-54-0'], ['1701.05086-1-19-0', '1701.05086-2-20-0'], ['1701.05086-1-19-2', '1701.05086-2-20-2'], ['1701.05086-1-20-1', '1701.05086-2-20-6'], ['1701.05086-1-25-0', '1701.05086-2-24-0'], ['1701.05086-1-25-4', '1701.05086-2-24-4'], ['1701.05086-1-25-5', '1701.05086-2-24-5'], ['1701.05086-1-25-7', '1701.05086-2-24-6']]

[]

[['1701.05086-1-37-1', '1701.05086-2-37-0'], ['1701.05086-1-37-6', '1701.05086-2-37-5'], ['1701.05086-1-37-8', '1701.05086-2-37-0'], ['1701.05086-1-37-8', '1701.05086-2-37-7'], ['1701.05086-1-37-9', '1701.05086-2-37-7'], ['1701.05086-1-53-1', '1701.05086-2-55-1'], ['1701.05086-1-48-5', '1701.05086-2-49-5'], ['1701.05086-1-48-5', '1701.05086-2-49-6'], ['1701.05086-1-43-4', '1701.05086-2-43-4'], ['1701.05086-1-43-5', '1701.05086-2-43-4'], ['1701.05086-1-8-1', '1701.05086-2-8-1'], ['1701.05086-1-4-2', '1701.05086-2-4-3'], ['1701.05086-1-17-0', '1701.05086-2-17-0'], ['1701.05086-1-17-1', '1701.05086-2-17-0'], ['1701.05086-1-38-6', '1701.05086-2-38-6'], ['1701.05086-1-29-3', '1701.05086-2-29-3'], ['1701.05086-1-41-3', '1701.05086-2-41-3'], ['1701.05086-1-36-2', '1701.05086-2-36-2'], ['1701.05086-1-27-0', '1701.05086-2-27-0'], ['1701.05086-1-27-0', '1701.05086-2-27-3'], ['1701.05086-1-27-3', '1701.05086-2-27-3'], ['1701.05086-1-32-1', '1701.05086-2-32-1'], ['1701.05086-1-30-1', '1701.05086-2-30-1'], ['1701.05086-1-0-2', '1701.05086-2-0-2'], ['1701.05086-1-0-2', '1701.05086-2-0-3'], ['1701.05086-1-0-9', '1701.05086-2-0-10'], ['1701.05086-1-31-0', '1701.05086-2-31-0'], ['1701.05086-1-50-1', '1701.05086-2-51-1'], ['1701.05086-1-50-3', '1701.05086-2-51-3'], ['1701.05086-1-5-1', '1701.05086-2-5-1'], ['1701.05086-1-5-2', '1701.05086-2-5-2'], ['1701.05086-1-35-1', '1701.05086-2-35-1'], ['1701.05086-1-35-3', '1701.05086-2-35-1'], ['1701.05086-1-15-1', '1701.05086-2-15-1'], ['1701.05086-1-15-2', '1701.05086-2-15-1'], ['1701.05086-1-15-2', '1701.05086-2-15-2'], ['1701.05086-1-52-7', '1701.05086-2-53-7'], ['1701.05086-1-23-0', '1701.05086-2-25-2'], ['1701.05086-1-25-6', '1701.05086-2-25-0'], ['1701.05086-1-25-6', '1701.05086-2-25-1']]

[]

['1701.05086-1-12-2', '1701.05086-1-18-2', '1701.05086-1-30-6', '1701.05086-2-12-2', '1701.05086-2-19-0', '1701.05086-2-30-6']

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

https://arxiv.org/abs/1701.05086

0906.0826

{'0906.0826-1-0-0': 'We present a scheme for asymmetric quantum information splitting, where a sender distributes a qubit to distant agents asymmetrically in a network.', '0906.0826-1-0-1': "The asymmetric distribution leads to that the agents have different authorities to reconstruct the sender's qubit.", '0906.0826-1-0-2': 'The quantum channel we used is a recently introduced multipartite entangled state which has high persistency of entanglement and thus the scheme is robust against decoherence.', '0906.0826-1-0-3': 'In addition, the scheme does not need the agents to get together and make nonlocal operations.', '0906.0826-1-0-4': 'Our scheme can also be modified to implement controlled teleportation against uncooperation of anyone of supervisors.', '0906.0826-1-1-0': 'Keywords: Multipartite entanglement; Quantum information splitting; Asymmetric distribution.', '0906.0826-1-1-1': 'PACS: 03.67.', '0906.0826-1-1-2': 'Hk; 03.67.', '0906.0826-1-1-3': 'Dd; 03.67.', '0906.0826-1-1-4': 'Lx', '0906.0826-1-2-0': 'The combination of information theory and quantum mechanics leads to the advent of quantum information science [CITATION].', '0906.0826-1-2-1': 'Entanglement, one of the most striking features of quantum mechanics, is the center resource for quantum information processing.', '0906.0826-1-2-2': 'The extensive applications of quantum entanglement should owe to its nonlocal correlations.', '0906.0826-1-2-3': 'One well-known example is quantum teleportation [CITATION], which utilizes the nonlocality of the quantum channel, i.e., bipartite or multipartite entangled states, to transport an unknown quantum state between two spatially separated quantum systems.', '0906.0826-1-2-4': 'In the original teleportation protocol of Bennett et al. [CITATION] the sender (Alice) and the receiver (Bob) initially share a maximally entangled state of two particles.', '0906.0826-1-2-5': 'Alice then performs a joint measurement on her particle of the entangled pair and the particle whose state is to be teleported.', '0906.0826-1-2-6': 'With the outcome transmitted to Bob via a classical channel, he can recover the teleported state by appropriate local transformations.', '0906.0826-1-3-0': 'Generally, the more particles that can be entangled, the more clearly nonlocal effects are exhibited [CITATION], and the more useful the states are for quantum information processing [CITATION].', '0906.0826-1-3-1': 'In addition, the usefulness of entangled states is usually related to their entanglement properties [CITATION].', '0906.0826-1-3-2': 'Thus exploring and exploiting multipartite entangled states are very important tasks for the workers who study on quantum information science.', '0906.0826-1-3-3': 'It has been attracting much interest that what classes of multipartite entangled states are competent for achieving a defined quantum information processing task and what they can do.', '0906.0826-1-3-4': 'Greenberger-Horne-Zeilinger (GHZ) states [CITATION] is a typical multipartite entangled states.', '0906.0826-1-3-5': 'With the GHZ states Hillery et al. [CITATION] firstly introduced the concept of quantum information splitting, where a qubit is distributed to two or more distant agents and anyone of them can reconstruct the original qubit if and only if they cooperate.', '0906.0826-1-3-6': 'Quantum information splitting can be considered as a generalization of teleportation, and was also called open-destination teleportation or quantum-state sharing in some literatures [CITATION].', '0906.0826-1-3-7': 'Karlsson et al. [CITATION] and Cleve et al. [CITATION] have also proposed different schemes for quantum information splitting, which require the particle carrying the quantum information be first entangled with the other particles to share the information.', '0906.0826-1-3-8': 'Recently, Zheng [CITATION] proposed a scheme for splitting quantum information with another typical multipartite entangled states, [MATH]-class entangled states [CITATION].', '0906.0826-1-3-9': 'In this paper, we present a way for splitting a qubit into three parts asymmetrically.', '0906.0826-1-3-10': 'Our scheme uses the four-qubit entangled state, recently proposed by Yeo and Chua [CITATION], [EQUATION] where [EQUATION]', '0906.0826-1-3-11': 'The state [MATH] has many interesting properties and exhibits more nonlocality than the counterparts of the well-known GHZ states and [MATH] states [CITATION].', '0906.0826-1-3-12': 'In addition, it can be easily verified that at least two single-qubit measurements are required in order to completely disentangle [MATH].', '0906.0826-1-3-13': 'Thus such a state has higher persistency of entanglement than the GHZ states whose disentangling can result from only one local measurement.', '0906.0826-1-3-14': 'This may lead to that our scheme is more robust against decoherence than the scheme of Ref. [CITATION].', '0906.0826-1-3-15': 'In comparison with the schemes of Refs. [CITATION], our scheme does not need the agents to get together and perform nonlocal operations.', '0906.0826-1-3-16': "More importantly, the quantum information splitting in our scheme is asymmetrical, which leads to that the agents have different authorities to possess the sender's qubit.", '0906.0826-1-3-17': 'This may be very interesting in view of reliability of the participants in quantum communication and access controlling in architecture of quantum computer.', '0906.0826-1-4-0': 'We consider that Alice, Bob, Charlie, and Diana possess particles [MATH], [MATH], [MATH], and [MATH], respectively.', '0906.0826-1-4-1': 'These particles are in the entangled state [MATH].', '0906.0826-1-4-2': 'Alice has another particle [MATH] which is in an unknown state [EQUATION]', '0906.0826-1-4-3': 'The state of the whole system is [EQUATION]', '0906.0826-1-4-4': 'Alice performs a joint measurement on her two particles [MATH] and [MATH] with respect to Bell states [CITATION] [EQUATION]', '0906.0826-1-4-5': 'Then the particles held by Bob, Charlie, and Diana collapse into one of the following entangled states: [EQUATION]', '0906.0826-1-4-6': 'The non-cloning theorem [CITATION] allows only one particle to be in the original state of particle [MATH], so that anyone of Bob, Charlie, and Diana, but not all, will recover the original state.', '0906.0826-1-5-0': "In order to reconstruct Alice's qubit, Bob, Charlie, and Diana need cooperating.", '0906.0826-1-5-1': "Before they come to an agreement, their single-particle state-density matrices are [EQUATION] where [MATH] corresponds to Alice's measurement outcomes [MATH] and [MATH], and [MATH] corresponds to [MATH] and [MATH].", '0906.0826-1-5-2': "It can be seen that Bob or Charlie knows nothing about the amplitude and phase of Alice's qubit [MATH] without the collaboration of the other two agents; Diana, however, has partial information about both the amplitude and phase of qubit [MATH] as long as receiving Alice's two bits of classical information about Bell-state measurement.", '0906.0826-1-5-3': "This case implies that Alice's qubit is distributed to Bob, Charlie, and Diana asymmetrically.", '0906.0826-1-5-4': "We shall show that the asymmetric distribution leads to an interesting phenomenon: Bob or Charlie can reconstruct Alice's qubit conditioned on that both of the other two agents cooperate, while Diana has the access to recover the qubit if anyone of the other agents cooperates.", '0906.0826-1-6-0': 'First, we assume that the three agents agree to let Bob possess the final qubit.', '0906.0826-1-6-1': 'We rewrite [MATH] and [MATH] as [EQUATION]', '0906.0826-1-6-2': 'It can be seen that if Charlie and Diana, respectively, perform a measurement on their particles with the basis [MATH] (i.e., along the [MATH] direction) and inform Bob their outcomes, Bob can recover the original state [MATH] on his particle [MATH] by appropriate local unitary transformations.', '0906.0826-1-6-3': "In other words, Bob can reconstruct Alice's qubit if and only if both Charlie and Diana collaborate with him.", '0906.0826-1-6-4': 'In particular, the transformations that Bob should perform on particle [MATH] in order to recover the state [MATH], up to an overall sign, are [EQUATION] where [MATH] is [MATH] identity matrix, [MATH] and [MATH] are the usual Pauli matrices.', '0906.0826-1-6-5': "These results are also applicable to the case where Charlie is deputed to reconstruct Alice's qubit, because particles [MATH] and [MATH] are fully symmetrical in the state [MATH].", '0906.0826-1-7-0': 'Now, we assume that they agree to let Diana regenerate the state [MATH].', '0906.0826-1-7-1': 'Then we rewrite [MATH] and [MATH] as [EQUATION] where [MATH]).', '0906.0826-1-7-2': 'In this case, interesting phenomena appear.', '0906.0826-1-7-3': '(1) The single-particle measurement bases that Bob and Charlie can adopt are optional, [MATH] or [MATH].', '0906.0826-1-7-4': "In other words, they can choose anyone of the two bases to perform projective measurements on their particles in order to assist Diana to reconstruct Alice's qubit.", '0906.0826-1-7-5': "In the protocol of Ref. [CITATION], however, the case that anyone of the collaborators adopts the measurement basis [MATH] will result in the failure of recovering the original state of the sender's particle.", '0906.0826-1-7-6': '(2) If Bob and Charlie choose the measurement basis [MATH], anyone of them is sufficient to assist Diana to regenerate the original state of particle [MATH] on particle [MATH].', '0906.0826-1-7-7': 'In Ref. [CITATION], the authors proposed a [MATH]) threshold scheme for quantum information splitting where any [MATH] of the [MATH] agents can reconstruct the secret information by cooperation.', '0906.0826-1-7-8': "In contrast, our scheme allows two of the three agents to reconstruct Alice's qubit only when Diana participate.", '0906.0826-1-7-9': 'In this sense, our scheme may be called "selective [MATH] threshold scheme" for quantum information splitting.', '0906.0826-1-7-10': "In addition, if we choose Diana as the receiver of Alice's qubit in advance, our scheme reduces to controlled teleportation [CITATION].", '0906.0826-1-7-11': 'However, comparing with the schemes of Ref. [CITATION], our scheme can endure uncooperation of one of supervisors.', '0906.0826-1-8-0': "If both Bob and Charlie choose the measurement basis [MATH], the transformations that Diana should perform in order to reconstruct Alice's qubit, up to a global phase, are [EQUATION] where [MATH] and [MATH] is the counterpart of the binary number [MATH].", '0906.0826-1-8-1': "As to the case where Bob or Charlie choose the measurement basis [MATH], the transformations that Diana should perform in order to reconstruct Alice's qubit, up to an overall sign, are [EQUATION] where [MATH] is the Hardamard transformation functioning as [MATH] and [MATH].", '0906.0826-1-9-0': 'In conclusion, we proposed a scheme for asymmetric quantum information splitting.', '0906.0826-1-9-1': "Previous schemes for quantum information splitting just consider the symmetric case where anyone of the agents has the same authority to possess the sender's qubit.", '0906.0826-1-9-2': 'In contrast, our scheme allows Alice to asymmetrically distribute a qubit to three agents, i.e., Bob, Charlie, and Diana.', '0906.0826-1-9-3': "Diana has more amount of information about Alice's qubit than Bob and Charlie.", '0906.0826-1-9-4': "Such asymmetric distribution leads to an interesting phenomenon: Bob or Charlie can reconstruct Alice's qubit under the condition that both of the other two agents cooperate, while Diana has access to reconstruct the qubit if anyone of the other agents cooperates.", '0906.0826-1-9-5': "This implies that they have different authorities to possess Alice's qubit, which may be very interesting in view of reliability of the participants in quantum communication and access controlling in architecture of quantum computer.", '0906.0826-1-9-6': 'The quantum channel we used is an interesting type of multipartite entangled state [MATH].', '0906.0826-1-9-7': 'The state [MATH] has higher persistency of entanglement than GHZ states, and exhibits more nonlocal effects than GHZ states and [MATH] states.', '0906.0826-1-9-8': 'In addition, the presented scheme does not need the agents to get together and perform nonlocal operations.', '0906.0826-1-9-9': 'Thus our scheme is much more robust and practical than the previous schemes using GHZ states and [MATH] states.', '0906.0826-1-9-10': 'Our scheme can also be modified to implement controlled teleportation against uncooperation of anyone of supervisors.', '0906.0826-1-9-11': 'Schemes for preparing the state [MATH] have been proposed in different systems [CITATION].', '0906.0826-1-9-12': 'These achievements may contribute to our scheme in physical realization.', '0906.0826-1-10-0': 'This work was supported by the National Natural Science Foundation of China (Grant No. 10674018) and the Natural Science Foundation of Hunan Province of China (Grant No. 06JJ50015).', '0906.0826-1-11-0': 'Nielsen M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ.', '0906.0826-1-11-1': '70PRL1895 C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys.', '0906.0826-1-11-2': '58PRA4394 A. Karlsson, M. Bourennane, Phys.', '0906.0826-1-11-3': '59PRA1829 M. Hillery, V. Buzek, A. Berthiaume, Phys.', '0906.0826-1-11-4': '59PRA162 A. Karlsson, M. Koashi, N. Imoto, Phys.'}

{'0906.0826-2-0-0': 'We present a scheme for asymmetric quantum information splitting, where a sender distributes asymmetrically a qubit to distant agents in a network.', '0906.0826-2-0-1': "The asymmetric distribution leads to that the agents have different powers to reconstruct the sender's qubit.", '0906.0826-2-0-2': 'In other words, the authorities of the agents for getting the quantum secret are hierarchized.', '0906.0826-2-0-3': 'The scheme does not need the agents to get together and make nonlocal operations.', '0906.0826-2-0-4': 'Our scheme can also be modified to implement controlled teleportation against uncooperation of part of supervisors.', '0906.0826-2-1-0': 'Keywords: Multipartite entanglement, quantum information splitting, asymmetric distribution PACS: 03.67.', '0906.0826-2-1-1': 'Hk; 03.67.', '0906.0826-2-1-2': 'Dd; 03.67.', '0906.0826-2-1-3': 'Lx', '0906.0826-2-2-0': 'The combination of information theory and quantum mechanics leads to the advent of quantum information science [CITATION].', '0906.0826-2-2-1': 'Entanglement, one of the most striking features of quantum mechanics, is the center resource for quantum information processing.', '0906.0826-2-2-2': 'The extensive applications of quantum entanglement should owe to its nonlocal correlations.', '0906.0826-2-2-3': 'One well-known example is quantum teleportation [CITATION], which utilizes the nonlocality of the quantum channel, i.e., bipartite or multipartite entangled states, to transport an unknown quantum state between two spatially separated quantum systems.', '0906.0826-2-2-4': 'In the original teleportation protocol of Bennett et al. [CITATION] the sender (Alice) and the receiver (Bob) initially share a maximally entangled state of two particles.', '0906.0826-2-2-5': 'Alice then performs a joint measurement on her particle of the entangled pair and the particle whose state is to be teleported.', '0906.0826-2-2-6': 'With the outcome transmitted to Bob via a classical channel, he can recover the teleported state by appropriate local transformations.', '0906.0826-2-3-0': 'Generally, the more particles that can be entangled, the more clearly nonlocal effects are exhibited [CITATION], and the more useful the states are for quantum information processing [CITATION].', '0906.0826-2-3-1': 'In addition, the usefulness of entangled states is usually related to their entanglement properties [CITATION].', '0906.0826-2-3-2': 'Thus exploring and exploiting multipartite entangled states are very important tasks for the workers who study quantum information science.', '0906.0826-2-3-3': 'It has been attracting much interest that what classes of multipartite entangled states are competent for achieving a defined quantum information processing task and what they can do.', '0906.0826-2-3-4': 'Greenberger-Horne-Zeilinger (GHZ) states [CITATION] is a typical multipartite entangled states.', '0906.0826-2-3-5': 'With the GHZ states Hillery et al. [CITATION] firstly introduced the concept of quantum information splitting (QIS), where a qubit is distributed to two or more distant agents and anyone of them can reconstruct the original qubit (quantum secret) if and only if they cooperate.', '0906.0826-2-3-6': 'QIS can be considered as a generalization of teleportation, and was also called open-destination teleportation or quantum-state sharing in literature [CITATION].', '0906.0826-2-4-0': 'QIS has extensive applications in the quantum world, such as it could help us create joint checking accounts containing quantum money [CITATION], perform secure distributed quantum computation [CITATION], and so on.', '0906.0826-2-4-1': 'Since the outstanding work of Hillery et al., QIS has been attracting much attention [CITATION], and a scheme has already been experimentally realized [CITATION].', '0906.0826-2-4-2': 'However, all of these schemes are focused on the symmetric case where every participant has the same status, i.e., the same authority to get the secret.', '0906.0826-2-4-3': 'In Ref. [CITATION], Gottesman pointed out that a more general QIS scheme should involve the asymmetry between the powers of the different participants, and showed that it is possible to construct an access structure that some subsets of the shares can be combined to reconstruct the secret quantum state.', '0906.0826-2-4-4': 'This case was further studied later [CITATION].', '0906.0826-2-4-5': 'Their idea is based on theory of quantum error-correcting codes, and thus the nonlocal operations are required.', '0906.0826-2-5-0': 'In this paper, we present a scheme for distributing a qubit to three distant agents asymmetrically.', '0906.0826-2-5-1': "The asymmetric distribution leads to that the agents have different powers to reconstruct the sender's qubit.", '0906.0826-2-5-2': 'In other words, the authorities of the agents for getting the quantum secret are hierarchized.', '0906.0826-2-5-3': 'The scheme does not need the agents to come together and make nonlocal operations.', '0906.0826-2-6-0': 'The quantum channel of our scheme is the four-qubit entangled state, recently proposed by Yeo and Chua [CITATION], [EQUATION] where [EQUATION]', '0906.0826-2-6-1': 'The state [MATH] has many interesting properties and exhibits more nonlocality than the counterparts of the well-known GHZ states and [MATH] states [CITATION].', '0906.0826-2-6-2': 'In addition, it can be easily verified that at least two single-qubit measurements are required in order to completely disentangle [MATH].', '0906.0826-2-6-3': 'Thus such a state has higher persistency of entanglement than the GHZ states which can be completely disentangled by only one local measurement.', '0906.0826-2-6-4': 'This may lead to that our scheme is more robust against decoherence than the scheme of Ref. [CITATION].', '0906.0826-2-7-0': 'We consider that Alice, Bob, Charlie, and Diana possess particles [MATH], [MATH], [MATH], and [MATH], respectively.', '0906.0826-2-7-1': 'These particles are in the entangled state [MATH].', '0906.0826-2-7-2': 'Alice has another particle [MATH] which is in the state [EQUATION]', '0906.0826-2-7-3': 'The state of the whole system is [EQUATION]', '0906.0826-2-7-4': 'Alice performs a joint measurement on her two particles [MATH] and [MATH] with the Bell basis [EQUATION]', '0906.0826-2-7-5': 'Then the particles held by Bob, Charlie, and Diana collapse into one of the following entangled states: [EQUATION]', '0906.0826-2-7-6': 'The non-cloning theorem [CITATION] allows only one particle to be in the original state of particle [MATH], so that anyone of Bob, Charlie, and Diana, but not all, will recover the original state.', '0906.0826-2-8-0': "In order to reconstruct Alice's qubit, Bob, Charlie, and Diana need cooperating.", '0906.0826-2-8-1': "Before they come to an agreement, their single-particle state-density matrices are [EQUATION] where [MATH] corresponds to Alice's measurement outcomes [MATH] and [MATH], and [MATH] corresponds to [MATH] and [MATH].", '0906.0826-2-8-2': "It can be seen that Bob or Charlie knows nothing about the amplitude and phase of Alice's qubit [MATH] without the collaboration of the other two agents; Diana, however, has partial information about both the amplitude and phase of qubit [MATH] as long as he receives Alice's Bell-state measurement outcome.", '0906.0826-2-8-3': "This case implies that Alice's qubit is distributed to Bob, Charlie, and Diana asymmetrically.", '0906.0826-2-8-4': "We shall show that the asymmetric distribution leads to an interesting phenomenon: Bob or Charlie can reconstruct Alice's qubit conditioned on that both of the other two agents cooperate, while Diana has access to recover the qubit if anyone of the other agents cooperates.", '0906.0826-2-9-0': 'First, we assume that the three agents agree to let Bob possess the final qubit.', '0906.0826-2-9-1': 'We rewrite [MATH] and [MATH] as [EQUATION]', '0906.0826-2-9-2': 'It can be seen that if Charlie and Diana, respectively, perform a measurement on their particles with the basis [MATH] (i.e., along the [MATH] direction) and inform Bob their outcomes, Bob can recover the original state [MATH] on his particle [MATH] by appropriate local unitary transformations.', '0906.0826-2-9-3': "In other words, Bob can reconstruct Alice's qubit if and only if both Charlie and Diana collaborate with him.", '0906.0826-2-9-4': 'In particular, the transformations that Bob should perform on particle [MATH] in order to recover the state [MATH], up to an overall sign, are [EQUATION] where [MATH] is [MATH] identity matrix, [MATH] and [MATH] are the usual Pauli matrices.', '0906.0826-2-9-5': "These results are also applicable to the case where Charlie is deputed to reconstruct Alice's qubit, because particles [MATH] and [MATH] are fully symmetrical in the state [MATH].", '0906.0826-2-10-0': 'Now, we assume that they agree to let Diana regenerate the state [MATH].', '0906.0826-2-10-1': 'We rewrite [MATH] and [MATH] as [EQUATION] where [MATH]).', '0906.0826-2-10-2': 'Then interesting phenomena appear.', '0906.0826-2-10-3': '(1) The single-particle measurement bases that Bob and Charlie can adopt are optional, [MATH] or [MATH].', '0906.0826-2-10-4': "In other words, they can choose anyone of the two sets of bases to perform projective measurements on their particles in order to assist Diana to reconstruct Alice's qubit.", '0906.0826-2-10-5': "In the protocol of Ref. [CITATION], however, the case that anyone of the collaborators adopts the measurement basis [MATH] will result in the failure of recovering the original state of the sender's particle.", '0906.0826-2-10-6': '(2) If Bob and Charlie choose the measurement basis [MATH], anyone of them is sufficient to assist Diana to regenerate the original state of particle [MATH] on particle [MATH].', '0906.0826-2-10-7': 'This result implies that if we choose Diana as the receiver in advance, our scheme reduces to a controlled teleportation scheme [CITATION].', '0906.0826-2-10-8': "It is worth pointing out that the controlled teleportation schemes with GHZ states [CITATION] are very fragile to the loss of the supervisors' measurement information.", '0906.0826-2-10-9': "That is, if Bob does not successfully receive the single-particle measurement outcome of anyone of supervisors, he cannot recover Alice's original state.", '0906.0826-2-10-10': 'In contrast, our scheme can endure the loss of the measurement information of one of the supervisors (Bob and Charlie).', '0906.0826-2-10-11': 'The controlled teleportation scheme of Ref. [CITATION] also has such a feature, but in which the teleportation fidelity is less than one.', '0906.0826-2-11-0': "If both Bob and Charlie choose the measurement basis [MATH], the transformations that Diana should perform in order to reconstruct Alice's qubit, up to a global phase, are [EQUATION] where [MATH] and [MATH] is the counterpart of the binary number [MATH].", '0906.0826-2-11-1': "As to the case where Bob or Charlie choose the measurement basis [MATH], the transformations that Diana should perform in order to reconstruct Alice's qubit, up to an overall sign, are [EQUATION] where [MATH] is the Hardamard transformation given by [EQUATION] which functions as [MATH] and [MATH].", '0906.0826-2-12-0': 'In conclusion, we have proposed a scheme for hierarchical QIS, where the authorities of the three agents, i.e., Bob, Charlie, and Diana, for getting the quantum secret are hierarchized.', '0906.0826-2-12-1': 'That is, Diana has a larger authority than Bob and Charlie to possess the quantum secret.', '0906.0826-2-12-2': 'The security checking for the quantum channel is the same that of Ref. [CITATION].', '0906.0826-2-12-3': 'Our scheme can also be modified to implement controlled teleportation against uncooperation of part of supervisors.', '0906.0826-2-12-4': 'Recently, different methods for preparing the state [MATH] have been proposed [CITATION].', '0906.0826-2-12-5': 'These achievements may contribute to our scheme in physical realization.', '0906.0826-2-12-6': 'In the future, one can generalize the idea to a more general case where more than three agents are involved.', '0906.0826-2-13-0': 'The hierarchical QIS may be very interesting in view of the reliability of the agents in quantum communication and the access controlling in architecture of quantum computer, and should be more useful than the symmetric QIS in practice.', '0906.0826-2-13-1': 'Let us take a simple example that a dealer in Berlin wants to have an action taken on her behalf in Beijing.', '0906.0826-2-13-2': 'She has many agents who can carry it out for her, but she knows that some of them are dishonest and does not know whom they are.', '0906.0826-2-13-3': 'She cannot simply send a message to one of them, because the dishonest ones will try to sabotage the action, but she knows that if all of them carry it out together, the honest ones will keep the dishonest ones from doing any damage.', '0906.0826-2-13-4': 'Then she can encode the message in a quantum state (quantum secret) and distribute it among them through the generalized teleportation protocol discussed above.', '0906.0826-2-13-5': 'The agent who is the most reliable will be distributed a larger part of information.', '0906.0826-2-13-6': 'As a consequence, the most reliable agent can recover the secret with the cooperation of subset of the other ones, but the other ones cannot get the secret without the participation of the most reliable one.', '0906.0826-2-14-0': 'This work was supported by the Natural Science Foundation of Hunan Province of China (Grant No. 06JJ50015).', '0906.0826-2-15-0': 'Nielsen M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ.', '0906.0826-2-15-1': '70PRL1895 C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys.', '0906.0826-2-15-2': '58PRA4394 A. Karlsson, M. Bourennane, Phys.', '0906.0826-2-15-3': 'GHZ D.M. Greenberger, M.A. Horne, A. Shimony, A. Zeilinger, Am.', '0906.0826-2-15-4': '59PRA1829 M. Hillery, V. Buzek, A. Berthiaume, Phys.', '0906.0826-2-15-5': 'distributed computation M. Ben-Or, C. Crepeau, D. Gottesman, A. Hassidim, A. Smith, Proc.', '0906.0826-2-15-6': "47th Annual IEEE Symposium on the Foundations of Computer Science (FOCS '06), pp. 249-260, IEEE Press, 2006.", '0906.0826-2-15-7': '59PRA162 A. Karlsson, M. Koashi, N. Imoto, Phys.'}

[['0906.0826-1-9-12', '0906.0826-2-12-5'], ['0906.0826-1-6-0', '0906.0826-2-9-0'], ['0906.0826-1-6-1', '0906.0826-2-9-1'], ['0906.0826-1-6-2', '0906.0826-2-9-2'], ['0906.0826-1-6-3', '0906.0826-2-9-3'], ['0906.0826-1-6-4', '0906.0826-2-9-4'], ['0906.0826-1-6-5', '0906.0826-2-9-5'], ['0906.0826-1-11-0', '0906.0826-2-15-0'], ['0906.0826-1-11-1', '0906.0826-2-15-1'], ['0906.0826-1-11-2', '0906.0826-2-15-2'], ['0906.0826-1-11-3', '0906.0826-2-15-4'], ['0906.0826-1-11-4', '0906.0826-2-15-7'], ['0906.0826-1-7-0', '0906.0826-2-10-0'], ['0906.0826-1-7-3', '0906.0826-2-10-3'], ['0906.0826-1-7-5', '0906.0826-2-10-5'], ['0906.0826-1-7-6', '0906.0826-2-10-6'], ['0906.0826-1-8-0', '0906.0826-2-11-0'], ['0906.0826-1-5-0', '0906.0826-2-8-0'], ['0906.0826-1-5-1', '0906.0826-2-8-1'], ['0906.0826-1-5-3', '0906.0826-2-8-3'], ['0906.0826-1-4-0', '0906.0826-2-7-0'], ['0906.0826-1-4-1', '0906.0826-2-7-1'], ['0906.0826-1-4-3', '0906.0826-2-7-3'], ['0906.0826-1-4-5', '0906.0826-2-7-5'], ['0906.0826-1-4-6', '0906.0826-2-7-6'], ['0906.0826-1-2-0', '0906.0826-2-2-0'], ['0906.0826-1-2-1', '0906.0826-2-2-1'], ['0906.0826-1-2-2', '0906.0826-2-2-2'], ['0906.0826-1-2-3', '0906.0826-2-2-3'], ['0906.0826-1-2-4', '0906.0826-2-2-4'], ['0906.0826-1-2-5', '0906.0826-2-2-5'], ['0906.0826-1-2-6', '0906.0826-2-2-6'], ['0906.0826-1-3-0', '0906.0826-2-3-0'], ['0906.0826-1-3-1', '0906.0826-2-3-1'], ['0906.0826-1-3-3', '0906.0826-2-3-3'], ['0906.0826-1-3-4', '0906.0826-2-3-4'], ['0906.0826-1-3-11', '0906.0826-2-6-1'], ['0906.0826-1-3-12', '0906.0826-2-6-2'], ['0906.0826-1-3-14', '0906.0826-2-6-4'], ['0906.0826-1-9-10', '0906.0826-2-12-3'], ['0906.0826-1-7-1', '0906.0826-2-10-1'], ['0906.0826-1-7-4', '0906.0826-2-10-4'], ['0906.0826-1-10-0', '0906.0826-2-14-0'], ['0906.0826-1-8-1', '0906.0826-2-11-1'], ['0906.0826-1-5-2', '0906.0826-2-8-2'], ['0906.0826-1-5-4', '0906.0826-2-8-4'], ['0906.0826-1-4-2', '0906.0826-2-7-2'], ['0906.0826-1-0-0', '0906.0826-2-0-0'], ['0906.0826-1-0-1', '0906.0826-2-0-1'], ['0906.0826-1-0-3', '0906.0826-2-0-3'], ['0906.0826-1-0-4', '0906.0826-2-0-4'], ['0906.0826-1-3-2', '0906.0826-2-3-2'], ['0906.0826-1-3-5', '0906.0826-2-3-5'], ['0906.0826-1-3-6', '0906.0826-2-3-6'], ['0906.0826-1-3-10', '0906.0826-2-6-0'], ['0906.0826-1-3-13', '0906.0826-2-6-3'], ['0906.0826-1-9-2', '0906.0826-2-12-0'], ['0906.0826-1-9-3', '0906.0826-2-12-1'], ['0906.0826-1-9-11', '0906.0826-2-12-4'], ['0906.0826-1-7-2', '0906.0826-2-10-2'], ['0906.0826-1-7-10', '0906.0826-2-10-7'], ['0906.0826-1-7-11', '0906.0826-2-10-10'], ['0906.0826-1-4-4', '0906.0826-2-7-4']]

[['0906.0826-1-9-12', '0906.0826-2-12-5'], ['0906.0826-1-6-0', '0906.0826-2-9-0'], ['0906.0826-1-6-1', '0906.0826-2-9-1'], ['0906.0826-1-6-2', '0906.0826-2-9-2'], ['0906.0826-1-6-3', '0906.0826-2-9-3'], ['0906.0826-1-6-4', '0906.0826-2-9-4'], ['0906.0826-1-6-5', '0906.0826-2-9-5'], ['0906.0826-1-11-0', '0906.0826-2-15-0'], ['0906.0826-1-11-1', '0906.0826-2-15-1'], ['0906.0826-1-11-2', '0906.0826-2-15-2'], ['0906.0826-1-11-3', '0906.0826-2-15-4'], ['0906.0826-1-11-4', '0906.0826-2-15-7'], ['0906.0826-1-7-0', '0906.0826-2-10-0'], ['0906.0826-1-7-3', '0906.0826-2-10-3'], ['0906.0826-1-7-5', '0906.0826-2-10-5'], ['0906.0826-1-7-6', '0906.0826-2-10-6'], ['0906.0826-1-8-0', '0906.0826-2-11-0'], ['0906.0826-1-5-0', '0906.0826-2-8-0'], ['0906.0826-1-5-1', '0906.0826-2-8-1'], ['0906.0826-1-5-3', '0906.0826-2-8-3'], ['0906.0826-1-4-0', '0906.0826-2-7-0'], ['0906.0826-1-4-1', '0906.0826-2-7-1'], ['0906.0826-1-4-3', '0906.0826-2-7-3'], ['0906.0826-1-4-5', '0906.0826-2-7-5'], ['0906.0826-1-4-6', '0906.0826-2-7-6'], ['0906.0826-1-2-0', '0906.0826-2-2-0'], ['0906.0826-1-2-1', '0906.0826-2-2-1'], ['0906.0826-1-2-2', '0906.0826-2-2-2'], ['0906.0826-1-2-3', '0906.0826-2-2-3'], ['0906.0826-1-2-4', '0906.0826-2-2-4'], ['0906.0826-1-2-5', '0906.0826-2-2-5'], ['0906.0826-1-2-6', '0906.0826-2-2-6'], ['0906.0826-1-3-0', '0906.0826-2-3-0'], ['0906.0826-1-3-1', '0906.0826-2-3-1'], ['0906.0826-1-3-3', '0906.0826-2-3-3'], ['0906.0826-1-3-4', '0906.0826-2-3-4'], ['0906.0826-1-3-11', '0906.0826-2-6-1'], ['0906.0826-1-3-12', '0906.0826-2-6-2'], ['0906.0826-1-3-14', '0906.0826-2-6-4']]

[['0906.0826-1-9-10', '0906.0826-2-12-3'], ['0906.0826-1-7-1', '0906.0826-2-10-1'], ['0906.0826-1-7-4', '0906.0826-2-10-4'], ['0906.0826-1-10-0', '0906.0826-2-14-0'], ['0906.0826-1-8-1', '0906.0826-2-11-1'], ['0906.0826-1-5-2', '0906.0826-2-8-2'], ['0906.0826-1-5-4', '0906.0826-2-8-4'], ['0906.0826-1-4-2', '0906.0826-2-7-2'], ['0906.0826-1-0-0', '0906.0826-2-0-0'], ['0906.0826-1-0-1', '0906.0826-2-0-1'], ['0906.0826-1-0-3', '0906.0826-2-0-3'], ['0906.0826-1-0-4', '0906.0826-2-0-4'], ['0906.0826-1-3-2', '0906.0826-2-3-2'], ['0906.0826-1-3-5', '0906.0826-2-3-5'], ['0906.0826-1-3-6', '0906.0826-2-3-6'], ['0906.0826-1-3-10', '0906.0826-2-6-0'], ['0906.0826-1-3-13', '0906.0826-2-6-3']]

[]

[['0906.0826-1-9-2', '0906.0826-2-12-0'], ['0906.0826-1-9-3', '0906.0826-2-12-1'], ['0906.0826-1-9-11', '0906.0826-2-12-4'], ['0906.0826-1-7-2', '0906.0826-2-10-2'], ['0906.0826-1-7-10', '0906.0826-2-10-7'], ['0906.0826-1-7-11', '0906.0826-2-10-10'], ['0906.0826-1-4-4', '0906.0826-2-7-4']]

[]

['0906.0826-1-1-0', '0906.0826-1-1-1', '0906.0826-1-1-2', '0906.0826-1-1-3', '0906.0826-1-1-4', '0906.0826-2-1-0', '0906.0826-2-1-1', '0906.0826-2-1-2', '0906.0826-2-1-3']

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

https://arxiv.org/abs/0906.0826

1807.01911

{'1807.01911-1-0-0': 'The 1-loop effects of the MSSM at the ILC are investigated through numerical analysis.', '1807.01911-1-0-1': 'We studied the higgs production processes [MATH] and [MATH] at the ILC.', '1807.01911-1-0-2': 'It is found that the magnitude of the MSSM contribution through the 1-loop effects is sizable enough to be detected.', '1807.01911-1-0-3': 'In the study, three sets of the MSSM parameters are proposed, which are consistent with the observed higgs mass, the muon [MATH]-[MATH], the dark matter abundance and the decay branching ratios of [MATH] mesons.', '1807.01911-1-0-4': 'In the [MATH] process, the 1-loop effects of the MSSM are visible and the distinction of the parameter sets is partially possible.', '1807.01911-1-0-5': 'For the study of [MATH], we used the equivalent [MATH]-boson approximation in the evaluation of the 1-loop cross section.', '1807.01911-1-0-6': 'While the 1-loop effect of the MSSM is visible, the distinction of the parameter sets might not be possible in this process under the value of realistic luminosity at the ILC.', '1807.01911-1-1-0': '# Introduction', '1807.01911-1-2-0': 'The standard model (SM) is completed by the discovery of the last piece, the higgs particle.', '1807.01911-1-2-1': 'However, it is argued that it is not the final theory of the fundamental particles.', '1807.01911-1-2-2': 'For example, the SM includes a lot of free parameters.', '1807.01911-1-2-3': 'When one calculates the mass of higgs, the values of these parameters are intentionally selected to cancel the quantum correction.', '1807.01911-1-2-4': 'This cancellation is as precise as up to around 17 order.', '1807.01911-1-2-5': 'Some researchers regard it as "unnatural".', '1807.01911-1-3-0': 'The supersymmetric (SUSY) model [CITATION] is considered as one of the promising candidates for the theory beyond the standard model.', '1807.01911-1-3-1': 'In this theory, each particle in the SM has its supersymmetric partner, or, a sparticle.', '1807.01911-1-3-2': 'The quadratic divergence in the calculation of quantum correction to the higgs mass is canceled by other contributions from the sparticles, so that the fine tuning problem disappears.', '1807.01911-1-3-3': 'The search of sparticles is the important subject of the present and future collider experiments to prove the SUSY.', '1807.01911-1-3-4': 'In spite of hard efforts in the large hadron collider (LHC) experiments, slightest signature of their existence has not been obtained.', '1807.01911-1-3-5': 'For example, the scalar top particle (stop) seems not to exist under [MATH](1)TeV mass region [CITATION].', '1807.01911-1-3-6': 'Though sparticles are so heavy that it is difficult to be produced directly, the indirect signature would be expected at the international linear collider (ILC).', '1807.01911-1-4-0': 'Since high luminosity is expected at the ILC experiments [CITATION], the quite small experimental error is expected.', '1807.01911-1-4-1': 'Correspondingly, therefore, the quite accurate calculations of the physical observables are required.', '1807.01911-1-4-2': 'As an explicit model, the Minimal Supersymmetric Standard Model (MSSM) is studied in this paper.', '1807.01911-1-4-3': 'We have calculated cross sections and decay branching ratios at the 1-loop level using the [MATH]-[MATH] system [CITATION].', '1807.01911-1-4-4': 'In this paper we report numerical results on the cross section of [MATH][CITATION].', '1807.01911-1-4-5': 'Our results are consistent with those given in the previous work [CITATION] under the same setting of MSSM parameters.', '1807.01911-1-4-6': 'We have also calculated the cross section of [MATH], with the equivalent [MATH]-boson approximation (EWA) [CITATION].', '1807.01911-1-4-7': 'The complete SM 1-loop correction of [MATH] and [MATH] have already been calculated using the GRACE [CITATION] and we reproduced the results for comparison.', '1807.01911-1-4-8': 'We calculate the MSSM 1-loop correction to perform the indirect search of sparticles as they contribute to the cross sections through 1-loop diagrams.', '1807.01911-1-5-0': 'The parameter sets of the MSSM are chosen so as to meet the various experimental constraints.', '1807.01911-1-5-1': 'They include the anomalous magnetic moment of muon [MATH]-[MATH] [CITATION], the dark matter (DM) thermal relic density [CITATION], [MATH] meson rare decay branching ratio Br([MATH]) [CITATION] and the higgs mass [CITATION].', '1807.01911-1-5-2': 'By the constraints the mass spectra of MSSM are quite limited.', '1807.01911-1-5-3': 'For the selection of MSSM mass spectra we utilized the following program packages.', '1807.01911-1-5-4': 'MicrOMEGAs [CITATION] was used in the estimation of the DM thermal relic density and SuSpect2 [CITATION] was used for [MATH]-[MATH] , Br([MATH]) and [MATH].', '1807.01911-1-6-0': '# The selection of the MSSM parameter sets', '1807.01911-1-7-0': 'In Table 1 we show experimental constraints we have considered, and in Table 2 we show three MSSM parameter sets we have selected.', '1807.01911-1-7-1': 'Detailed methods for the selection of sets have been explained in the previous work [CITATION].', '1807.01911-1-8-0': 'The first common settings in the three sets are', '1807.01911-1-9-0': '(A) [MATH]GeV, [MATH]GeV, [MATH]TeV, [MATH]50 and [MATH]GeV ([MATH]).', '1807.01911-1-10-0': 'They are necessary for sets to satisfy the constrains (1), (2) and (3) in Table.1.', '1807.01911-1-10-1': 'As a consequence, the lightest sparticle (LSP) is almost Bino [MATH] with mass about [MATH]GeV, [MATH]GeV (almost Wino [MATH]) and [MATH]TeV (almost higgsino [MATH]).', '1807.01911-1-10-2': 'The second common settings are', '1807.01911-1-11-0': '(B) a large mass splitting in the stau [MATH] sector, [MATH]GeV ([MATH]>[MATH][MATH]), [MATH]GeV and [MATH]GeV.', '1807.01911-1-12-0': 'They are necessary for sets to be satisfied the constraint (4), because the co-annihilation occurring between [MATH] and the LSP is required for meeting this constraint in the Bino LSP case.', '1807.01911-1-12-1': 'The difference among the three sets exists in the settings of masses of the strongly interacting sparticles ([MATH]) and [MATH]), gluino and squarks.', '1807.01911-1-12-2': 'Considering the LHC bounds (5), we take ([MATH] and [MATH]TeV, [MATH]TeV), ([MATH]TeV, [MATH]TeV) and ([MATH]TeV, [MATH]TeV) for set 1, set 2 and set 3, respectively.', '1807.01911-1-12-3': 'For each set, the left-right mixing parameter [MATH] and masses [MATH] in the stop [MATH] sector are tunned to satisfy the higgs mass constraint (6).', '1807.01911-1-13-0': '# Calculation scheme', '1807.01911-1-14-0': '## The GRACE system There are more than twice as many different types of particles in the MSSM as those in the SM ;', '1807.01911-1-15-0': 'therefore, there are various possible sparticle production processes in the collider experiments.', '1807.01911-1-15-1': 'A large number of Feynman diagrams appearing in each production process requires tedious and lengthy calculations in evaluating the cross sections.', '1807.01911-1-15-2': 'Accurate theoretical prediction requires an automated system to manage such large scale computations.', '1807.01911-1-15-3': 'The [MATH] system for the MSSM calculations[CITATION] has been developed by the KEK group (the Minami-tateya group) to meet the requirement.', '1807.01911-1-15-4': 'The [MATH] system uses a renormalization prescription that imposes mass shell conditions on as many particles as possible, while maintaining the gauge symmetry by setting the renormalization conditions appropriately[CITATION].', '1807.01911-1-15-5': "In the [MATH] system for the SM, the usual 'tHooft-Feynman linear gauge condition is generalized to a more general non-linear gauge (NLG) that involves five extra parameters [CITATION].", '1807.01911-1-15-6': 'We extend it to the MSSM formalism by adding the SUSY interactions with seven NLG parameters [CITATION].', '1807.01911-1-15-7': 'We can check the consistency of the gauge symmetry by verifying the independence of the physical results from the NLG parameters.', '1807.01911-1-15-8': 'We ascertain that the results of the automatic calculation are reliable by carrying out the following checks:', '1807.01911-1-16-0': 'Actually, the 1-loop differential cross sections (distributions) are separated into two parts, [EQUATION] where the suffix M indicates SM or MSSM, and each part is computed separately.', '1807.01911-1-16-1': 'The loop and the counter term contribution [MATH] should be gauge invariant and the UV finite but IR divergent.', '1807.01911-1-16-2': 'We regularize the IR divergence by the fictitious photon mass [MATH], so both [MATH] and the soft photon contribution [MATH] are [MATH] dependent.', '1807.01911-1-16-3': 'The [MATH] dependence is canceled in [MATH].', '1807.01911-1-16-4': 'Finally, the [MATH] independent 1-loop physical cross sections can be obtained by [EQUATION] where, [MATH] and [MATH] are the energy and the solid angle of the photon respectively.', '1807.01911-1-16-5': 'We selected the values of the MSSM parameters related to the higgs secter so that the tree level cross section are numerically identical between the MSSM and the SM.', '1807.01911-1-16-6': 'Therefore, the suffix M can be neglected in [MATH] and [MATH].', '1807.01911-1-16-7': 'For the confirmation of the verifiability of 1-loop correction, we defined following correction ratios, [EQUATION]', '1807.01911-1-16-8': 'For the estimation of effects of the MSSM virtual particles, we defined following ratio [CITATION], [EQUATION]', '1807.01911-1-16-9': 'For the process [MATH], we calculated differential cross sections [EQUATION] where [MATH] is the scattering angle of the [MATH]-boson.', '1807.01911-1-17-0': '## Equivalent W -boson Approximation (EWA) For the process [MATH], we should calculate the cross section, [EQUATION]', '1807.01911-1-17-1': 'However, the estimation of the MSSM full 1-loop correction for ([REF]) is difficult even using the [MATH] system because the number of Feynman diagrams of the process [MATH] becomes about [MATH].', '1807.01911-1-18-0': 'In this paper, therefore, we use the EWA formulae [CITATION], [EQUATION] where [MATH] and [MATH] denotes the center of mass energy squared and cross section of the sub-process [MATH].', '1807.01911-1-18-1': 'Then we set [MATH].', '1807.01911-1-18-2': '[MATH] are the energy distribution functions for [MATH]-boson with helicity [MATH], [EQUATION] where [EQUATION] [MATH], [MATH], [MATH], [MATH].', '1807.01911-1-18-3': 'For the electron and [MATH]-boson coupling, [MATH] and [MATH].', '1807.01911-1-19-0': 'Moreover, the EWA cross section ([REF]) is numerically indistinguishable from [EQUATION] in the energy range considered here, because the contribution of [MATH] is almost negligible as shown in Figures.', '1807.01911-1-20-0': 'In Figure 1, [MATH] dependence of the tree level cross sections ([REF]), ([REF]) and ([REF]) are shown.', '1807.01911-1-20-1': 'The cross section of [MATH] is apparently large in the region [MATH]GeV.', '1807.01911-1-20-2': 'We calculated the differential cross section for [MATH] at [MATH]GeV, where its total cross section has almost the largest value.', '1807.01911-1-20-3': 'On the other hand, the [MATH]-fusion contribution is dominant at [MATH]GeV and the EWA becomes good approximation.', '1807.01911-1-20-4': 'For example, [MATH], [MATH])% and [MATH], [MATH])% at [MATH], [MATH]) GeV.', '1807.01911-1-20-5': 'We use [MATH] for the calculation of cross sections at [MATH]GeV.', '1807.01911-1-20-6': 'We calculated the differential cross section, [EQUATION] where [MATH] is the energy of higgs particle.', '1807.01911-1-20-7': 'It is calculated from ([REF]) using relations [EQUATION]', '1807.01911-1-20-8': 'In Figure 2, we show the tree level energy distribution of the higgs [MATH] from (6) and [MATH] in (15) at [MATH]=500 GeV.', '1807.01911-1-20-9': 'The statistical errors were calculated assuming [MATH].', '1807.01911-1-20-10': 'The difference is within the error range in the region [MATH].', '1807.01911-1-20-11': 'We could figure out that the EWA reproduce the [MATH] cross section in this energy region.', '1807.01911-1-20-12': 'The peak from [MATH] production around [MATH]=250 GeV cannot be reproduced by the EWA.', '1807.01911-1-21-0': 'The numerical results', '1807.01911-1-22-0': '## The SM contribution', '1807.01911-1-23-0': 'Table 3 shows the SM parameters which we used in the calculations.', '1807.01911-1-23-1': 'All the following numerical results (include the MSSM cross section) are computed in these parameters.', '1807.01911-1-23-2': 'Figure 3 shows the [MATH] in (3) for [MATH] at [MATH]=250 GeV (left) and [MATH] used EWA at [MATH]=500GeV (right).', '1807.01911-1-23-3': 'The plotted errors were calculated assuming [MATH] and 500 [MATH] for [MATH] and [MATH] respectivly.', '1807.01911-1-23-4': 'In advance, we should confirm that the 1-loop correction does not buried in this error.', '1807.01911-1-23-5': 'In the left figure, the [MATH] is estimated to be [MATH] in entire region and it is larger than the error.', '1807.01911-1-23-6': 'Similarly, in the right figure, [MATH] is estimated to be [MATH] in entire region and this value is larger than the error.', '1807.01911-1-23-7': 'From the results we figure out that the 1-loop correction is necessary for reliable theoretical predictions in both processes.', '1807.01911-1-24-0': '## The MSSM contribution', '1807.01911-1-25-0': 'Figure 4 shows the 1-loop corrected anglar distribution of [MATH], [MATH] (left) and the ratio [MATH] in (4) (right) for the process [MATH].', '1807.01911-1-25-1': 'From the left figure, we find that the both the SM and the MSSM 1-loop contributions are negative and they are satisfied the relations, [EQUATION]', '1807.01911-1-25-2': 'In the right figure, the [MATH] are estimeted to be 1.17[MATH]1.25 at [MATH] = 0.', '1807.01911-1-25-3': 'In the entire region these ratios are larger than the statistical error assuming planned luminosities at the ILC.', '1807.01911-1-25-4': 'It means that the 1-loop contribution of the MSSM could be measured at [MATH] =250GeV.', '1807.01911-1-25-5': 'Also, the set 1 will be possibly distinguished from set 2 and set 3.', '1807.01911-1-26-0': 'Figure 5 shows the 1-loop corrected energy distribution of higgs (left) and [MATH](right) in [MATH].', '1807.01911-1-26-1': 'The left (right) figure shows differential cross section [MATH]).', '1807.01911-1-26-2': 'In the left figure the SM and MSSM 1-loop contribution are indistinguishable at a glance.', '1807.01911-1-26-3': 'However, [MATH] are estimated to be 1.5[MATH]1.7% and they are larger than the statistical error assuming 500 [MATH] in the entire region.', '1807.01911-1-26-4': 'It means that the 1-loop contribution of the MSSM could be measured at [MATH] =500 GeV.', '1807.01911-1-26-5': 'Three proposed sets can not be distinguished each other at least assuming the planned luminosities at the ILC.', '1807.01911-1-26-6': 'The precision measurements of the higgs production processes at the ILC will bring us important information on the heavy sparticles through the virtual 1-loop effects.', '1807.01911-1-27-0': '# Summary and conclusions', '1807.01911-1-28-0': 'We investigated the indirect effects of the MSSM at the ILC.', '1807.01911-1-28-1': 'We focused on the center of mass energies of 250 GeV and 500 GeV in the processes [MATH] and [MATH], respectively.', '1807.01911-1-28-2': 'We selected the mass spectra of the MSSM which are consistent with the observed mass of higgs, the thermal relic density of the dark matter, the low energy experiments and the LHC bounds of sparticles.', '1807.01911-1-28-3': 'The parameter sets proposed in our calculations have the squarks and the gluino with masses of 1.5, 5.0, 10.0 TeV and the sleptons and gauginos with masses less than 0.5 TeV.', '1807.01911-1-28-4': 'With using our developed [MATH] system, we calculated the 1-loop corrected cross sections of the processes [MATH] and [MATH] at the ILC.', '1807.01911-1-28-5': 'For the analysis of latter process, we adopted the equivalent [MATH]-boson approximation.', '1807.01911-1-29-0': 'We confirmed that both the SM and the MSSM 1-loop correction is necessary for the accurate theoretical predictions at the ILC.', '1807.01911-1-29-1': 'We found that the 1-loop effects of MSSM are verifiable both at 250 GeV and 500 GeV.', '1807.01911-1-29-2': 'Moreover the difference between set 1 and set 2 and 3 will be possibly observed with [MATH] at [MATH]=250GeV.'}

{'1807.01911-2-0-0': 'The 1-loop effects of the MSSM at the ILC are investigated through numerical analysis.', '1807.01911-2-0-1': 'We studied the higgs production processes [MATH] and [MATH] at the ILC.', '1807.01911-2-0-2': 'It is found that the magnitude of the MSSM contribution through the 1-loop effects is sizable enough to be detected.', '1807.01911-2-0-3': 'In the study, three sets of the MSSM parameters are proposed, which are consistent with the observed higgs mass, the muon [MATH]-[MATH], the dark matter abundance and the decay branching ratios of [MATH] mesons.', '1807.01911-2-0-4': 'In the [MATH] process, the 1-loop effects of the MSSM are visible and the distinction of the parameter sets is partially possible.', '1807.01911-2-0-5': 'For the study of [MATH], we used the equivalent [MATH]-boson approximation in the evaluation of the 1-loop cross section.', '1807.01911-2-0-6': 'While the 1-loop effect of the MSSM is visible, the distinction of the parameter sets might not be possible in this process under the value of realistic luminosity at the ILC.', '1807.01911-2-1-0': '# Introduction', '1807.01911-2-2-0': 'The standard model (SM) is completed by the discovery of the last piece, the higgs particle.', '1807.01911-2-2-1': 'However, it is argued that it is not the final theory of the fundamental particles.', '1807.01911-2-2-2': 'For example, the SM includes a lot of free parameters.', '1807.01911-2-2-3': 'When one calculates the mass of higgs, the values of these parameters are intentionally selected to cancel the quantum correction.', '1807.01911-2-2-4': 'This cancellation is as precise as up to around 17 order.', '1807.01911-2-2-5': 'Some researchers regard it as "unnatural".', '1807.01911-2-3-0': 'The supersymmetric (SUSY) model [CITATION] is considered as one of the promising candidates for the theory beyond the standard model.', '1807.01911-2-3-1': 'In this theory, each particle in the SM has its supersymmetric partner, or, a sparticle.', '1807.01911-2-3-2': 'The quadratic divergence in the calculation of quantum correction to the higgs mass is canceled by other contributions from the sparticles, so that the fine tuning problem disappears.', '1807.01911-2-3-3': 'The search of sparticles is the important subject of the present and future collider experiments to prove the SUSY.', '1807.01911-2-3-4': 'In spite of hard efforts in the large hadron collider (LHC) experiments, slightest signature of their existence has not been obtained.', '1807.01911-2-3-5': 'For example, the scalar top particle (stop) seems not to exist under [MATH](1)TeV mass region [CITATION].', '1807.01911-2-3-6': 'Though sparticles are so heavy that it is difficult to be produced directly, the indirect signature would be expected at the international linear collider (ILC).', '1807.01911-2-4-0': 'Since high luminosity is expected at the ILC experiments [CITATION], the quite small experimental error is expected.', '1807.01911-2-4-1': 'Correspondingly, therefore, the quite accurate calculations of the physical observables are required.', '1807.01911-2-4-2': 'As an explicit model, the Minimal Supersymmetric Standard Model (MSSM) is studied in this paper.', '1807.01911-2-4-3': 'We have calculated cross sections and decay branching ratios at the 1-loop level using the [MATH]-[MATH] system [CITATION].', '1807.01911-2-4-4': 'In this paper we report numerical results on the cross section of [MATH][CITATION].', '1807.01911-2-4-5': 'Our results are consistent with those given in the previous work [CITATION] under the same setting of MSSM parameters.', '1807.01911-2-4-6': 'We have also calculated the cross section of [MATH], with the equivalent [MATH]-boson approximation (EWA) [CITATION].', '1807.01911-2-4-7': 'The complete SM 1-loop correction of [MATH] and [MATH] have already been calculated using the GRACE [CITATION] and we reproduced the results for comparison.', '1807.01911-2-4-8': 'We calculate the MSSM 1-loop correction to perform the indirect search of sparticles as they contribute to the cross sections through 1-loop diagrams.', '1807.01911-2-5-0': 'The parameter sets of the MSSM are chosen so as to meet the various experimental constraints.', '1807.01911-2-5-1': 'They include the anomalous magnetic moment of muon [MATH]-[MATH] [CITATION], the dark matter (DM) thermal relic density [CITATION], [MATH] meson rare decay branching ratio Br([MATH]) [CITATION] and the higgs mass [CITATION].', '1807.01911-2-5-2': 'By the constraints the mass spectra of MSSM are quite limited.', '1807.01911-2-5-3': 'For the selection of MSSM mass spectra we utilized the following program packages.', '1807.01911-2-5-4': 'MicrOMEGAs [CITATION] was used in the estimation of the DM thermal relic density and SuSpect2 [CITATION] was used for [MATH]-[MATH] , Br([MATH]) and [MATH].', '1807.01911-2-6-0': '# The selection of the MSSM parameter sets', '1807.01911-2-7-0': 'In Table 1 we show experimental constraints we have considered, and in Table 2 we show three MSSM parameter sets we have selected.', '1807.01911-2-7-1': 'Detailed methods for the selection of sets have been explained in the previous work [CITATION].', '1807.01911-2-8-0': 'The first common settings in the three sets are', '1807.01911-2-9-0': '(A) [MATH]GeV, [MATH]GeV, [MATH]TeV, [MATH]50 and [MATH]GeV ([MATH]).', '1807.01911-2-10-0': 'They are necessary for sets to satisfy the constrains (1), (2) and (3) in Table.1.', '1807.01911-2-10-1': 'As a consequence, the lightest sparticle (LSP) is almost Bino [MATH] with mass about [MATH]GeV, [MATH]GeV (almost Wino [MATH]) and [MATH]TeV (almost higgsino [MATH]).', '1807.01911-2-10-2': 'The second common settings are', '1807.01911-2-11-0': '(B) a large mass splitting in the stau [MATH] sector, [MATH]GeV ([MATH]>[MATH][MATH]), [MATH]GeV and [MATH]GeV.', '1807.01911-2-12-0': 'They are necessary for sets to be satisfied the constraint (4), because the co-annihilation occurring between [MATH] and the LSP is required for meeting this constraint in the Bino LSP case.', '1807.01911-2-12-1': 'The difference among the three sets exists in the settings of masses of the strongly interacting sparticles ([MATH]) and [MATH]), gluino and squarks.', '1807.01911-2-12-2': 'Considering the LHC bounds (5), we take ([MATH] and [MATH]TeV, [MATH]TeV), ([MATH]TeV, [MATH]TeV) and ([MATH]TeV, [MATH]TeV) for set 1, set 2 and set 3, respectively.', '1807.01911-2-12-3': 'For each set, the left-right mixing parameter [MATH] and masses [MATH] in the stop [MATH] sector are tuned to satisfy the higgs mass constraint (6).', '1807.01911-2-13-0': '# Calculation scheme', '1807.01911-2-14-0': '## The GRACE system There are more than twice as many different types of particles in the MSSM as those in the SM ;', '1807.01911-2-15-0': 'therefore, there are various possible sparticle production processes in the collider experiments.', '1807.01911-2-15-1': 'A large number of Feynman diagrams appearing in each production process requires tedious and lengthy calculations in evaluating the cross sections.', '1807.01911-2-15-2': 'Accurate theoretical prediction requires an automated system to manage such large scale computations.', '1807.01911-2-15-3': 'The [MATH] system for the MSSM calculations[CITATION] has been developed by the KEK group (the Minami-tateya group) to meet the requirement.', '1807.01911-2-15-4': 'The [MATH] system uses a renormalization prescription that imposes mass shell conditions on as many particles as possible, while maintaining the gauge symmetry by setting the renormalization conditions appropriately[CITATION].', '1807.01911-2-15-5': "In the [MATH] system for the SM, the usual 't Hooft-Feynman linear gauge condition is generalized to a more general non-linear gauge (NLG) that involves five extra parameters [CITATION].", '1807.01911-2-15-6': 'We extend it to the MSSM formalism by adding the SUSY interactions with seven NLG parameters [CITATION].', '1807.01911-2-15-7': 'We can check the consistency of the gauge symmetry by verifying the independence of the physical results from the NLG parameters.', '1807.01911-2-15-8': 'We ascertain that the results of the automatic calculation are reliable by carrying out the following checks:', '1807.01911-2-16-0': 'Actually, the 1-loop differential cross sections (distributions) are separated into two parts, [EQUATION] where the suffix M indicates SM or MSSM, and each part is computed separately.', '1807.01911-2-16-1': 'The loop and the counter term contribution [MATH] should be gauge invariant and the UV finite but IR divergent.', '1807.01911-2-16-2': 'We regularize the IR divergence by the fictitious photon mass [MATH], so both [MATH] and the soft photon contribution [MATH] are [MATH] dependent.', '1807.01911-2-16-3': 'The [MATH] dependence is canceled in [MATH].', '1807.01911-2-16-4': 'Finally, the [MATH] independent 1-loop physical cross sections can be obtained by [EQUATION] where, [MATH] and [MATH] are the energy and the solid angle of the photon respectively.', '1807.01911-2-16-5': 'We selected the values of the MSSM parameters related to the higgs sector so that the tree level cross section are numerically identical between the MSSM and the SM.', '1807.01911-2-16-6': 'Therefore, the suffix M can be neglected in [MATH] and [MATH].', '1807.01911-2-16-7': 'For the confirmation of the verifiability of 1-loop correction, we defined following correction ratios, [EQUATION]', '1807.01911-2-16-8': 'For the estimation of effects of the MSSM virtual particles, we defined following ratio [CITATION], [EQUATION]', '1807.01911-2-16-9': 'For the process [MATH], we calculated differential cross sections [EQUATION] where [MATH] is the scattering angle of the [MATH]-boson.', '1807.01911-2-17-0': '## Equivalent W -boson Approximation (EWA) For the process [MATH], we should calculate the cross section, [EQUATION]', '1807.01911-2-17-1': 'However, the estimation of the MSSM full 1-loop correction for ([REF]) is difficult even using the [MATH] system because the number of Feynman diagrams of the process [MATH] becomes about [MATH].', '1807.01911-2-18-0': 'In this paper, therefore, we use the EWA formulae [CITATION], [EQUATION] where [MATH] and [MATH] denotes the center of mass energy squared and cross section of the sub-process [MATH].', '1807.01911-2-18-1': 'Then we set [MATH].', '1807.01911-2-18-2': '[MATH] are the energy distribution functions for [MATH]-boson with helicity [MATH], [EQUATION] where [EQUATION] [MATH], [MATH], [MATH], [MATH].', '1807.01911-2-18-3': 'For the electron and [MATH]-boson coupling, [MATH] and [MATH].', '1807.01911-2-19-0': 'Moreover, the EWA cross section ([REF]) is numerically indistinguishable from [EQUATION] in the energy range considered here, because the contribution of [MATH] is almost negligible as shown in Figures.', '1807.01911-2-20-0': 'In Figure 1, [MATH] dependence of the tree level cross sections ([REF]), ([REF]) and ([REF]) are shown.', '1807.01911-2-20-1': 'The cross section of [MATH] is apparently large in the region [MATH]GeV.', '1807.01911-2-20-2': 'We calculated the differential cross section for [MATH] at [MATH]GeV, where its total cross section has almost the largest value.', '1807.01911-2-20-3': 'On the other hand, the [MATH]-fusion contribution is dominant at [MATH]GeV and the EWA becomes good approximation.', '1807.01911-2-20-4': 'For example, [MATH], [MATH])% and [MATH], [MATH])% at [MATH], [MATH]) GeV.', '1807.01911-2-20-5': 'We use [MATH] for the calculation of cross sections at [MATH]GeV.', '1807.01911-2-20-6': 'We calculated the differential cross section, [EQUATION] where [MATH] is the energy of higgs particle.', '1807.01911-2-20-7': 'It is calculated from ([REF]) using relations [EQUATION]', '1807.01911-2-20-8': 'In Figure 2, we show the tree level energy distribution of the higgs [MATH] from (6) and [MATH] in (15) at [MATH]=500 GeV.', '1807.01911-2-20-9': 'The statistical errors were calculated assuming [MATH].', '1807.01911-2-20-10': 'The difference is within the error range in the region [MATH].', '1807.01911-2-20-11': 'We could figure out that the EWA reproduce the [MATH] cross section in this energy region.', '1807.01911-2-20-12': 'The peak from [MATH] production around [MATH]=250 GeV cannot be reproduced by the EWA.', '1807.01911-2-21-0': 'The numerical results', '1807.01911-2-22-0': '## The SM contribution', '1807.01911-2-23-0': 'Table 3 shows the SM parameters which we used in the calculations.', '1807.01911-2-23-1': 'All the following numerical results (include the MSSM cross section) are computed in these parameters.', '1807.01911-2-23-2': 'Figure 3 shows the [MATH] in (3) for [MATH] at [MATH]=250 GeV (left) and [MATH] used EWA at [MATH]=500GeV (right).', '1807.01911-2-23-3': 'The plotted errors were calculated assuming [MATH] and 500 [MATH] for [MATH] and [MATH] respectively.', '1807.01911-2-23-4': 'In advance, we should confirm that the 1-loop correction does not buried in this error.', '1807.01911-2-23-5': 'In the left figure, the [MATH] is estimated to be [MATH] in entire region and it is larger than the error.', '1807.01911-2-23-6': 'Similarly, in the right figure, [MATH] is estimated to be [MATH] in entire region and this value is larger than the error.', '1807.01911-2-23-7': 'From the results we figure out that the 1-loop correction is necessary for reliable theoretical predictions in both processes.', '1807.01911-2-24-0': '## The MSSM contribution', '1807.01911-2-25-0': 'Figure 4 shows the 1-loop corrected angular distribution of [MATH], [MATH] (left) and the ratio [MATH] in (4) (right) for the process [MATH].', '1807.01911-2-25-1': 'From the left figure, we find that the both the SM and the MSSM 1-loop contributions are negative and they are satisfied the relations, [EQUATION]', '1807.01911-2-25-2': 'In the right figure, the [MATH] are estimated to be 1.17[MATH]1.25 at [MATH] = 0.', '1807.01911-2-25-3': 'In the entire region these ratios are larger than the statistical error assuming planned luminosities at the ILC.', '1807.01911-2-25-4': 'It means that the 1-loop contribution of the MSSM could be measured at [MATH] =250GeV.', '1807.01911-2-25-5': 'Also, the set 1 will be possibly distinguished from set 2 and set 3.', '1807.01911-2-26-0': 'Figure 5 shows the 1-loop corrected energy distribution of higgs (left) and [MATH](right) in [MATH].', '1807.01911-2-26-1': 'The left (right) figure shows differential cross section [MATH]).', '1807.01911-2-26-2': 'In the left figure the SM and MSSM 1-loop contribution are indistinguishable at a glance.', '1807.01911-2-26-3': 'However, [MATH] are estimated to be 1.5[MATH]1.7% and they are larger than the statistical error assuming 500 [MATH] in the entire region.', '1807.01911-2-26-4': 'It means that the 1-loop contribution of the MSSM could be measured at [MATH] =500 GeV.', '1807.01911-2-26-5': 'Three proposed sets can not be distinguished each other at least assuming the planned luminosities at the ILC.', '1807.01911-2-26-6': 'The precision measurements of the higgs production processes at the ILC will bring us important information on the heavy sparticles through the virtual 1-loop effects.', '1807.01911-2-27-0': '# Summary and conclusions', '1807.01911-2-28-0': 'We investigated the indirect effects of the MSSM at the ILC.', '1807.01911-2-28-1': 'We focused on the center of mass energies of 250 GeV and 500 GeV in the processes [MATH] and [MATH], respectively.', '1807.01911-2-28-2': 'We selected the mass spectra of the MSSM which are consistent with the observed mass of higgs, the thermal relic density of the dark matter, the low energy experiments and the LHC bounds of sparticles.', '1807.01911-2-28-3': 'The parameter sets proposed in our calculations have the squarks and the gluino with masses of 1.5, 5.0, 10.0 TeV and the sleptons and gauginos with masses less than 0.5 TeV.', '1807.01911-2-28-4': 'With using our developed [MATH] system, we calculated the 1-loop corrected cross sections of the processes [MATH] and [MATH] at the ILC.', '1807.01911-2-28-5': 'For the analysis of latter process, we adopted the equivalent [MATH]-boson approximation.', '1807.01911-2-29-0': 'We confirmed that both the SM and the MSSM 1-loop correction is necessary for the accurate theoretical predictions at the ILC.', '1807.01911-2-29-1': 'We found that the 1-loop effects of MSSM are verifiable both at 250 GeV and 500 GeV.', '1807.01911-2-29-2': 'Moreover the difference between set 1 and set 2 and 3 will be possibly observed with [MATH] at [MATH]=250GeV.'}

[['1807.01911-1-17-1', '1807.01911-2-17-1'], ['1807.01911-1-3-0', '1807.01911-2-3-0'], ['1807.01911-1-3-1', '1807.01911-2-3-1'], ['1807.01911-1-3-2', '1807.01911-2-3-2'], ['1807.01911-1-3-3', '1807.01911-2-3-3'], ['1807.01911-1-3-4', '1807.01911-2-3-4'], ['1807.01911-1-3-5', '1807.01911-2-3-5'], ['1807.01911-1-3-6', '1807.01911-2-3-6'], ['1807.01911-1-2-0', '1807.01911-2-2-0'], ['1807.01911-1-2-1', '1807.01911-2-2-1'], ['1807.01911-1-2-2', '1807.01911-2-2-2'], ['1807.01911-1-2-3', '1807.01911-2-2-3'], ['1807.01911-1-2-4', '1807.01911-2-2-4'], ['1807.01911-1-2-5', '1807.01911-2-2-5'], ['1807.01911-1-18-0', '1807.01911-2-18-0'], ['1807.01911-1-18-1', '1807.01911-2-18-1'], ['1807.01911-1-18-2', '1807.01911-2-18-2'], ['1807.01911-1-18-3', '1807.01911-2-18-3'], ['1807.01911-1-12-0', '1807.01911-2-12-0'], ['1807.01911-1-12-1', '1807.01911-2-12-1'], ['1807.01911-1-12-2', '1807.01911-2-12-2'], ['1807.01911-1-29-0', '1807.01911-2-29-0'], ['1807.01911-1-29-1', '1807.01911-2-29-1'], ['1807.01911-1-29-2', '1807.01911-2-29-2'], ['1807.01911-1-28-0', '1807.01911-2-28-0'], ['1807.01911-1-28-1', '1807.01911-2-28-1'], ['1807.01911-1-28-2', '1807.01911-2-28-2'], ['1807.01911-1-28-3', '1807.01911-2-28-3'], ['1807.01911-1-28-4', '1807.01911-2-28-4'], ['1807.01911-1-28-5', '1807.01911-2-28-5'], ['1807.01911-1-15-0', '1807.01911-2-15-0'], ['1807.01911-1-15-1', '1807.01911-2-15-1'], ['1807.01911-1-15-2', '1807.01911-2-15-2'], ['1807.01911-1-15-3', '1807.01911-2-15-3'], ['1807.01911-1-15-4', '1807.01911-2-15-4'], ['1807.01911-1-15-6', '1807.01911-2-15-6'], ['1807.01911-1-15-7', '1807.01911-2-15-7'], ['1807.01911-1-19-0', '1807.01911-2-19-0'], ['1807.01911-1-20-0', '1807.01911-2-20-0'], ['1807.01911-1-20-1', '1807.01911-2-20-1'], ['1807.01911-1-20-2', '1807.01911-2-20-2'], ['1807.01911-1-20-3', '1807.01911-2-20-3'], ['1807.01911-1-20-5', '1807.01911-2-20-5'], ['1807.01911-1-20-6', '1807.01911-2-20-6'], ['1807.01911-1-20-7', '1807.01911-2-20-7'], ['1807.01911-1-20-8', '1807.01911-2-20-8'], ['1807.01911-1-20-9', '1807.01911-2-20-9'], ['1807.01911-1-20-10', '1807.01911-2-20-10'], ['1807.01911-1-20-11', '1807.01911-2-20-11'], ['1807.01911-1-20-12', '1807.01911-2-20-12'], ['1807.01911-1-0-0', '1807.01911-2-0-0'], ['1807.01911-1-0-1', '1807.01911-2-0-1'], ['1807.01911-1-0-2', '1807.01911-2-0-2'], ['1807.01911-1-0-3', '1807.01911-2-0-3'], ['1807.01911-1-0-4', '1807.01911-2-0-4'], ['1807.01911-1-0-5', '1807.01911-2-0-5'], ['1807.01911-1-0-6', '1807.01911-2-0-6'], ['1807.01911-1-10-0', '1807.01911-2-10-0'], ['1807.01911-1-10-1', '1807.01911-2-10-1'], ['1807.01911-1-10-2', '1807.01911-2-10-2'], ['1807.01911-1-4-0', '1807.01911-2-4-0'], ['1807.01911-1-4-1', '1807.01911-2-4-1'], ['1807.01911-1-4-2', '1807.01911-2-4-2'], ['1807.01911-1-4-3', '1807.01911-2-4-3'], ['1807.01911-1-4-4', '1807.01911-2-4-4'], ['1807.01911-1-4-5', '1807.01911-2-4-5'], ['1807.01911-1-4-6', '1807.01911-2-4-6'], ['1807.01911-1-4-7', '1807.01911-2-4-7'], ['1807.01911-1-4-8', '1807.01911-2-4-8'], ['1807.01911-1-25-1', '1807.01911-2-25-1'], ['1807.01911-1-25-3', '1807.01911-2-25-3'], ['1807.01911-1-25-4', '1807.01911-2-25-4'], ['1807.01911-1-25-5', '1807.01911-2-25-5'], ['1807.01911-1-16-0', '1807.01911-2-16-0'], ['1807.01911-1-16-1', '1807.01911-2-16-1'], ['1807.01911-1-16-2', '1807.01911-2-16-2'], ['1807.01911-1-16-3', '1807.01911-2-16-3'], ['1807.01911-1-16-4', '1807.01911-2-16-4'], ['1807.01911-1-16-6', '1807.01911-2-16-6'], ['1807.01911-1-16-7', '1807.01911-2-16-7'], ['1807.01911-1-16-8', '1807.01911-2-16-8'], ['1807.01911-1-16-9', '1807.01911-2-16-9'], ['1807.01911-1-23-0', '1807.01911-2-23-0'], ['1807.01911-1-23-1', '1807.01911-2-23-1'], ['1807.01911-1-23-4', '1807.01911-2-23-4'], ['1807.01911-1-23-5', '1807.01911-2-23-5'], ['1807.01911-1-23-6', '1807.01911-2-23-6'], ['1807.01911-1-23-7', '1807.01911-2-23-7'], ['1807.01911-1-5-0', '1807.01911-2-5-0'], ['1807.01911-1-5-1', '1807.01911-2-5-1'], ['1807.01911-1-5-2', '1807.01911-2-5-2'], ['1807.01911-1-5-3', '1807.01911-2-5-3'], ['1807.01911-1-5-4', '1807.01911-2-5-4'], ['1807.01911-1-7-0', '1807.01911-2-7-0'], ['1807.01911-1-7-1', '1807.01911-2-7-1'], ['1807.01911-1-26-0', '1807.01911-2-26-0'], ['1807.01911-1-26-1', '1807.01911-2-26-1'], ['1807.01911-1-26-2', '1807.01911-2-26-2'], ['1807.01911-1-26-3', '1807.01911-2-26-3'], ['1807.01911-1-26-4', '1807.01911-2-26-4'], ['1807.01911-1-26-5', '1807.01911-2-26-5'], ['1807.01911-1-26-6', '1807.01911-2-26-6'], ['1807.01911-1-12-3', '1807.01911-2-12-3'], ['1807.01911-1-15-5', '1807.01911-2-15-5'], ['1807.01911-1-25-0', '1807.01911-2-25-0'], ['1807.01911-1-25-2', '1807.01911-2-25-2'], ['1807.01911-1-16-5', '1807.01911-2-16-5'], ['1807.01911-1-23-3', '1807.01911-2-23-3']]

[['1807.01911-1-17-1', '1807.01911-2-17-1'], ['1807.01911-1-3-0', '1807.01911-2-3-0'], ['1807.01911-1-3-1', '1807.01911-2-3-1'], ['1807.01911-1-3-2', '1807.01911-2-3-2'], ['1807.01911-1-3-3', '1807.01911-2-3-3'], ['1807.01911-1-3-4', '1807.01911-2-3-4'], ['1807.01911-1-3-5', '1807.01911-2-3-5'], ['1807.01911-1-3-6', '1807.01911-2-3-6'], ['1807.01911-1-2-0', '1807.01911-2-2-0'], ['1807.01911-1-2-1', '1807.01911-2-2-1'], ['1807.01911-1-2-2', '1807.01911-2-2-2'], ['1807.01911-1-2-3', '1807.01911-2-2-3'], ['1807.01911-1-2-4', '1807.01911-2-2-4'], ['1807.01911-1-2-5', '1807.01911-2-2-5'], ['1807.01911-1-18-0', '1807.01911-2-18-0'], ['1807.01911-1-18-1', '1807.01911-2-18-1'], ['1807.01911-1-18-2', '1807.01911-2-18-2'], ['1807.01911-1-18-3', '1807.01911-2-18-3'], ['1807.01911-1-12-0', '1807.01911-2-12-0'], ['1807.01911-1-12-1', '1807.01911-2-12-1'], ['1807.01911-1-12-2', '1807.01911-2-12-2'], ['1807.01911-1-29-0', '1807.01911-2-29-0'], ['1807.01911-1-29-1', '1807.01911-2-29-1'], ['1807.01911-1-29-2', '1807.01911-2-29-2'], ['1807.01911-1-28-0', '1807.01911-2-28-0'], ['1807.01911-1-28-1', '1807.01911-2-28-1'], ['1807.01911-1-28-2', '1807.01911-2-28-2'], ['1807.01911-1-28-3', '1807.01911-2-28-3'], ['1807.01911-1-28-4', '1807.01911-2-28-4'], ['1807.01911-1-28-5', '1807.01911-2-28-5'], ['1807.01911-1-15-0', '1807.01911-2-15-0'], ['1807.01911-1-15-1', '1807.01911-2-15-1'], ['1807.01911-1-15-2', '1807.01911-2-15-2'], ['1807.01911-1-15-3', '1807.01911-2-15-3'], ['1807.01911-1-15-4', '1807.01911-2-15-4'], ['1807.01911-1-15-6', '1807.01911-2-15-6'], ['1807.01911-1-15-7', '1807.01911-2-15-7'], ['1807.01911-1-19-0', '1807.01911-2-19-0'], ['1807.01911-1-20-0', '1807.01911-2-20-0'], ['1807.01911-1-20-1', '1807.01911-2-20-1'], ['1807.01911-1-20-2', '1807.01911-2-20-2'], ['1807.01911-1-20-3', '1807.01911-2-20-3'], ['1807.01911-1-20-5', '1807.01911-2-20-5'], ['1807.01911-1-20-6', '1807.01911-2-20-6'], ['1807.01911-1-20-7', '1807.01911-2-20-7'], ['1807.01911-1-20-8', '1807.01911-2-20-8'], ['1807.01911-1-20-9', '1807.01911-2-20-9'], ['1807.01911-1-20-10', '1807.01911-2-20-10'], ['1807.01911-1-20-11', '1807.01911-2-20-11'], ['1807.01911-1-20-12', '1807.01911-2-20-12'], ['1807.01911-1-0-0', '1807.01911-2-0-0'], ['1807.01911-1-0-1', '1807.01911-2-0-1'], ['1807.01911-1-0-2', '1807.01911-2-0-2'], ['1807.01911-1-0-3', '1807.01911-2-0-3'], ['1807.01911-1-0-4', '1807.01911-2-0-4'], ['1807.01911-1-0-5', '1807.01911-2-0-5'], ['1807.01911-1-0-6', '1807.01911-2-0-6'], ['1807.01911-1-10-0', '1807.01911-2-10-0'], ['1807.01911-1-10-1', '1807.01911-2-10-1'], ['1807.01911-1-10-2', '1807.01911-2-10-2'], ['1807.01911-1-4-0', '1807.01911-2-4-0'], ['1807.01911-1-4-1', '1807.01911-2-4-1'], ['1807.01911-1-4-2', '1807.01911-2-4-2'], ['1807.01911-1-4-3', '1807.01911-2-4-3'], ['1807.01911-1-4-4', '1807.01911-2-4-4'], ['1807.01911-1-4-5', '1807.01911-2-4-5'], ['1807.01911-1-4-6', '1807.01911-2-4-6'], ['1807.01911-1-4-7', '1807.01911-2-4-7'], ['1807.01911-1-4-8', '1807.01911-2-4-8'], ['1807.01911-1-25-1', '1807.01911-2-25-1'], ['1807.01911-1-25-3', '1807.01911-2-25-3'], ['1807.01911-1-25-4', '1807.01911-2-25-4'], ['1807.01911-1-25-5', '1807.01911-2-25-5'], ['1807.01911-1-16-0', '1807.01911-2-16-0'], ['1807.01911-1-16-1', '1807.01911-2-16-1'], ['1807.01911-1-16-2', '1807.01911-2-16-2'], ['1807.01911-1-16-3', '1807.01911-2-16-3'], ['1807.01911-1-16-4', '1807.01911-2-16-4'], ['1807.01911-1-16-6', '1807.01911-2-16-6'], ['1807.01911-1-16-7', '1807.01911-2-16-7'], ['1807.01911-1-16-8', '1807.01911-2-16-8'], ['1807.01911-1-16-9', '1807.01911-2-16-9'], ['1807.01911-1-23-0', '1807.01911-2-23-0'], ['1807.01911-1-23-1', '1807.01911-2-23-1'], ['1807.01911-1-23-4', '1807.01911-2-23-4'], ['1807.01911-1-23-5', '1807.01911-2-23-5'], ['1807.01911-1-23-6', '1807.01911-2-23-6'], ['1807.01911-1-23-7', '1807.01911-2-23-7'], ['1807.01911-1-5-0', '1807.01911-2-5-0'], ['1807.01911-1-5-1', '1807.01911-2-5-1'], ['1807.01911-1-5-2', '1807.01911-2-5-2'], ['1807.01911-1-5-3', '1807.01911-2-5-3'], ['1807.01911-1-5-4', '1807.01911-2-5-4'], ['1807.01911-1-7-0', '1807.01911-2-7-0'], ['1807.01911-1-7-1', '1807.01911-2-7-1'], ['1807.01911-1-26-0', '1807.01911-2-26-0'], ['1807.01911-1-26-1', '1807.01911-2-26-1'], ['1807.01911-1-26-2', '1807.01911-2-26-2'], ['1807.01911-1-26-3', '1807.01911-2-26-3'], ['1807.01911-1-26-4', '1807.01911-2-26-4'], ['1807.01911-1-26-5', '1807.01911-2-26-5'], ['1807.01911-1-26-6', '1807.01911-2-26-6']]

[['1807.01911-1-12-3', '1807.01911-2-12-3'], ['1807.01911-1-15-5', '1807.01911-2-15-5'], ['1807.01911-1-25-0', '1807.01911-2-25-0'], ['1807.01911-1-25-2', '1807.01911-2-25-2'], ['1807.01911-1-16-5', '1807.01911-2-16-5'], ['1807.01911-1-23-3', '1807.01911-2-23-3']]

[]

[]

[]

['1807.01911-1-8-0', '1807.01911-1-9-0', '1807.01911-1-11-0', '1807.01911-1-15-8', '1807.01911-1-20-4', '1807.01911-1-21-0', '1807.01911-1-23-2', '1807.01911-2-8-0', '1807.01911-2-9-0', '1807.01911-2-11-0', '1807.01911-2-15-8', '1807.01911-2-20-4', '1807.01911-2-21-0', '1807.01911-2-23-2']

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

https://arxiv.org/abs/1807.01911

1401.3247

{'1401.3247-1-0-0': 'Earthquake nucleation process is studied as a precursor of a mainshock both analytically and numerically on the basis of the one-dimensional Burridge-Knopoff model obeying the rate- and state-dependent friction law.', '1401.3247-1-0-1': 'The properties of the nucleation dynamics, the nucleation lengths and the duration times are clarified together with their continuum limits.', '1401.3247-1-1-0': 'There is a wide-spread expectation that a large earthquake might be preceded by a precursory nucleation process which occurs prior to the high-speed rupture of a mainshock.', '1401.3247-1-1-1': 'Nucleation process is localized to a compact "seed" area with its rupture velocity orders of magnitude smaller than the seismic wave velocity [CITATION].', '1401.3247-1-1-2': 'The fault spends a very long time in this nucleation process, and then at some point, exhibits a rapid acceleration accompanied by a rapid expansion of the rupture zone, finally getting into the high-speed rupture of a mainshock.', '1401.3247-1-1-3': 'Although a firm observational evidence of such a nucleation process has been lacking for real earthquakes, a possible precursory phenomenon preceding a mainshock is of paramount importance in its own right as well as in its possible connection to an earthquake forecast.', '1401.3247-1-2-0': 'It has been suggested that the earthquake nucleation process might proceed via several distinct steps or "phases".', '1401.3247-1-2-1': 'Ref. [CITATION] proposed that it started with an initial quasi-static process until the diameter of the nucleus [MATH] exceeded a critical nucleation length [MATH].', '1401.3247-1-2-2': 'Then, the fault gets into the acceleration phase where the system gets out of equilibrium and rapidly increases its slip velocity.', '1401.3247-1-2-3': 'When the nucleus diameter exceeds another critical nucleation length [MATH], the fault eventually exhibits a high-speed rupture of a mainshock.', '1401.3247-1-2-4': 'In this picture, there appear two nucleation lengths, [MATH] and [MATH], which divide the nucleation process into "the initial phase"([MATH]), "the acceleration phase" ([MATH]) and "the high-speed rupture phase" ([MATH]).', '1401.3247-1-2-5': 'Although such features of the nucleation process has been more or less confirmed by laboratory rock experiments, its nature, or even its very existence, remains unclear for real earthquakes.', '1401.3247-1-3-0': 'Under such circumstances, a theoretical or a numerical study based on an appropriate model of an earthquake fault would be important and helpful.', '1401.3247-1-3-1': 'In such modelings, the friction force is a crucially important part.', '1401.3247-1-3-2': 'The friction force now standard in seismology is the so-called rate and state dependent friction (RSF) law [CITATION].', '1401.3247-1-3-3': 'The RSF law has been used in many of numerical simulations on earthquakes, mostly in the continuum model [CITATION].', '1401.3247-1-3-4': 'Earthquake nucleation process has also been studied mainly within such continuum models obeying the RSF law [CITATION].', '1401.3247-1-4-0': 'Meanwhile, a further simplified discrete model has also been used.', '1401.3247-1-4-1': 'Especially popular is the spring-block model or the Burridge-Knopoff (BK) model [CITATION], in which an earthquake fault is modeled as an assembly of blocks mutually connected via elastic springs of the spring constant [MATH] which are subject to the friction force and are slowly driven by an external force mimicking the plate drive via another springs of the spring constant [MATH].', '1401.3247-1-4-2': 'While a simple velocity-weakening friction has often been assumed in many numerical simulations, a more realistic RSF law was also employed in some of the recent numerical simulations of the model [CITATION].', '1401.3247-1-5-0': 'The aim of the present letter is to clarify within the BK model how the properties of the earthquake nucleation, e.g., the nucleation dynamics, the nucleation lengths and the duration times of each phase, are determined.', '1401.3247-1-5-1': 'The issue is of special significance for an earthquake forecast.', '1401.3247-1-5-2': 'One should notice that the BK model introduces the short-length cut-off into the problem in the form of the block size, which could in principle cause an artificial effect absent in the continuum.', '1401.3247-1-5-3': 'In fact, the issue of the discreteness is closely related to the nucleation phenomenon.', '1401.3247-1-5-4': 'Rice then argued that the characteristic length scale to be compared with the block size is the nucleation length, and the continuum system under the RSF law always exhibits a nucleation process prior to a mainshock [CITATION].', '1401.3247-1-5-5': 'We also wish to clarify how the nucleation process of the discrete BK model behaves in its continuum limit.', '1401.3247-1-6-0': 'The one-dimensional (1D) BK model consists of a 1D array of [MATH] identical blocks of the mass [MATH], which are mutually connected with the two neighboring blocks via the elastic springs of the spring stiffness [MATH], and are connected to the moving plate via the springs of the spring stiffness [MATH].', '1401.3247-1-6-1': 'All blocks are subjected to the friction force [MATH].', '1401.3247-1-6-2': 'The dimensionless equation of motion for the [MATH]-th block can be written as [CITATION] [EQUATION] where [MATH].', '1401.3247-1-6-3': 'The dimensionless displacement [MATH] is normalized by the characteristic slip distance [MATH] associated with the RSF law, the time [MATH] by [MATH], the block velocity [MATH] and the pulling speed of the plate [MATH] by [MATH], and the dimensionless friction force [MATH] by [MATH].', '1401.3247-1-6-4': 'The RSF force [MATH] reads as [EQUATION] where [MATH] is the dimensionless state variable describing the "state" of the interface, and [MATH] is the dimensionless crossover velocity.', '1401.3247-1-6-5': 'The normalized frictional parameters [MATH], [MATH] and [MATH] represent the velocity-strengthening, the velocity-weakening and the constant parts of friction.', '1401.3247-1-6-6': 'The original friction parameters [MATH], [MATH] and [MATH] are related to the normalized ones by [MATH], [MATH] and [MATH], [MATH] being the normal load.', '1401.3247-1-6-7': 'For simplicity, we inhibit the motion in the direction opposite to the plate drive.', '1401.3247-1-6-8': 'The state variable [MATH] is assumed to obey the aging law [CITATION], [EQUATION]', '1401.3247-1-6-9': 'The continuum limit of the BK model corresponds to making the dimensionless block size [MATH] to be infinitesimal [MATH], simultaneously making the system infinitely rigid [MATH] with [MATH] [CITATION].', '1401.3247-1-6-10': 'The dimensionless distance [MATH] between the block [MATH] and [MATH] is [MATH].', '1401.3247-1-7-0': 'Typical values of the model parameters describing natural faults are estimated.', '1401.3247-1-7-1': 'From the rise time of an earthquake, we estimate [MATH] [s].', '1401.3247-1-7-2': 'The characteristic slip distance [MATH] is a few cm [CITATION].', '1401.3247-1-7-3': 'Since the speed of the plate motion is typically a few cm/year, the dimensionless loading rate is [MATH].', '1401.3247-1-7-4': 'The spring constant [MATH] may be related to the rigidity [MATH] via the block size [MATH] and the width of the fault zone [MATH] as [MATH].', '1401.3247-1-7-5': 'The relation [MATH] is the mass density) and the [MATH]-wave velocity [MATH] yields [MATH].', '1401.3247-1-7-6': 'With [MATH] where [MATH] is the normal stress, we have [MATH].', '1401.3247-1-7-7': 'Putting [MATH] and [MATH] [km], we get [MATH] and [MATH] [km].', '1401.3247-1-7-8': 'As [MATH] is known to take a value around [MATH], [MATH] would be of order [MATH]-[MATH], [MATH] and [MATH] being one or two orders of magnitude smaller than [MATH].', '1401.3247-1-7-9': 'The crossover velocity [MATH] and its dimensionless counterpart [MATH] is hard to estimate though it should be much smaller than unity, and we take it as a parameter in our simulations.', '1401.3247-1-8-0': 'Our first question might be whether the 1D BK model under the RSF law ever exhibits a nucleation process prior to a mainshock, and if it does, under what conditions.', '1401.3247-1-8-1': 'We observe that the model exhibits qualitatively different behaviors depending on whether the frictional instability is either "weak" or "strong".', '1401.3247-1-8-2': 'A slow and long-lasting nucleation process, the quasi-static initial phase, is realized in the former case only.', '1401.3247-1-8-3': 'We illustrate in Fig.1 typical examples of seismic events realized in the stationary state of the model, where the time evolution of the movement of each block is shown as a color plot for each case of (a) the weak frictional instability ([MATH], [MATH], [MATH]), and (b) the strong frictional instability ([MATH], [MATH], [MATH]).', '1401.3247-1-8-4': 'A slow nucleation process with a long duration time is observed in (a), but is absent in (b).', '1401.3247-1-8-5': 'As will be shown below, the model possesses a borderline value of [MATH] determined solely by the stiffness parameter [MATH], [MATH], which discriminates the strong/weak instability behaviors.', '1401.3247-1-9-0': 'We also illustrate in Fig.1 the two types of nucleation lengths, [MATH] and [MATH]).', '1401.3247-1-9-1': 'The former [MATH] is the length separating stable and unstable ruptures and exists only for the weak frictional instability, while the latter [MATH] is the length signaling the onset of the high-speed rupture of a mainshock, which in Fig.1 corresponds to the linear portion of the rupture propagation line with its slope of the wave velocity [MATH].', '1401.3247-1-10-0': 'One way to identify [MATH] is to artificially stop the external loading in the course of simulation.', '1401.3247-1-10-1': 'We have confirmed that, if the external loading is stopped at any point beyond [MATH], the subsequent seismic rupture is no longer stoppable and evolves until its very end, whereas, if the external loading is stopped at a point before [MATH], the rupture itself also stops there.', '1401.3247-1-11-0': 'An appropriate physical condition describing the stable/unstable sliding across [MATH] might be whether the elastic stiffness [MATH], as defined by [MATH] which represents a change of the elastic force [MATH] due to an infinitesimal slip [MATH] of the block, is greater/smaller than the frictional weakening rate, as defined by [MATH] which represents a change of the friction force [MATH] due to an infinitesimal slip of the block.', '1401.3247-1-11-1': 'Consider a hypothetical instantaneous process from the states ([MATH], [MATH], [MATH]) to ([MATH], [MATH], [MATH]).', '1401.3247-1-11-2': 'The aging law ([REF]) entails the relation [MATH].', '1401.3247-1-11-3': 'Then, the frictional-weakening rate is obtained as [MATH].', '1401.3247-1-11-4': 'Meanwhile, the stiffness of the [MATH]-block system may be given by the smallest nonzero eigenvalue of the [MATH] matrix [MATH] defined via the relation [MATH] as [MATH].', '1401.3247-1-11-5': 'Matching [MATH] and [MATH], the condition of the frictional instability is obtained as [EQUATION]', '1401.3247-1-11-6': 'Note that the quasi-static initial phase is realizable in the BK model only when [MATH] is greater than the lattice spacing, i.e., [MATH], or equivalently, [MATH], where the critical value of [MATH] separating the weak/strong frictional instability is obtained as [MATH].', '1401.3247-1-12-0': 'In Fig.2, the stiffness [MATH] of an epicenter block computed in the course of the nucleation process of our simulation is plotted versus the number of moving blocks [MATH], together with the theoretical curves.', '1401.3247-1-12-1': 'At an earlier stage of the slip, an inequality [MATH] holds indicating a stable slip, while, at a certain point, an equality [MATH] is reached signaling [MATH], beyond which an opposite inequality [MATH] holds indicating an unstable slip.', '1401.3247-1-13-0': 'Since the continuum limit entails [MATH], the condition of the weak frictional instability [MATH] is always satisfied there.', '1401.3247-1-13-1': 'Hence, the continuum limit of the model always lies in the weak frictional instability regime accompanying the quasi-static nucleation process, corroborating Rice [CITATION].', '1401.3247-1-14-0': 'In Fig.3, we show for a typical large event in the weak frictional instability regime near the continuum limit the time evolutions of the epicenter-block sliding velocity [MATH] (a) in the initial phase, and (b) in the acceleration phase.', '1401.3247-1-14-1': 'In the initial phase, the sliding velocity [MATH] stays very low up to the nucleation length [MATH], of order the pulling speed of the plate.', '1401.3247-1-14-2': 'Hence, it would be practically impossible to detect such a slow sliding motion to forecast the approach of a mainshock at the stage of the initial phase.', '1401.3247-1-14-3': 'In the acceleration phase, the block movement exhibits a prominent acceleration, being no longer quasi-static nor reversible, reaching the maximum around [MATH] (this maximum point is used as our definition of [MATH]), then decreases sharply and finally stops.', '1401.3247-1-15-0': 'Next, we investigate the statistical properties of the nucleation lengths [MATH] and [MATH] as well as the duration times of each phase, averaged over many events.', '1401.3247-1-15-1': 'Since the nucleation length [MATH] is determined only by the material parameters as in eq. ([REF]), it cannot be used as an indicator of the size of the ensuing mainshock.', '1401.3247-1-15-2': 'What about [MATH] ?', '1401.3247-1-15-3': 'We plot in Fig.4(a) the computed mean [MATH] normalized by the corresponding [MATH], [MATH], versus the final rupture-zone size [MATH] for various choices of the model parameters in the weak frictional instability regime.', '1401.3247-1-15-4': 'The [MATH]-value is fixed to [MATH] while the parameters [MATH], [MATH] and [MATH] are varied.', '1401.3247-1-15-5': 'The data collapse onto a common curve.', '1401.3247-1-15-6': 'Since [MATH] does not depend on [MATH] and [MATH], this indicates that [MATH] is insensitive to [MATH] and [MATH], while its [MATH]-dependence is the same as that of [MATH].', '1401.3247-1-15-7': 'One also sees that large events tend to be independent of [MATH], implying that one cannot predict the size of the upcoming mainshock even with the information of [MATH].', '1401.3247-1-15-8': 'We examine the [MATH]-dependence of [MATH], to find that it increases linearly with [MATH] as [MATH] in the weak frictional regime [MATH].', '1401.3247-1-16-0': 'The continuum limit of [MATH] in the dimensionless form [MATH] is obtained as [MATH].', '1401.3247-1-16-1': 'Reviving the normalization units above, we get the dimensionful nucleation length in the continuum limit, [MATH], as [EQUATION] for [MATH].', '1401.3247-1-16-2': 'If we substitue the parameter values [MATH] [cm], [MATH] and [MATH] [km], we get [MATH] several kilometers.', '1401.3247-1-16-3': 'Concerning [MATH], since the ratio [MATH] turns out to be hardly dependent on [MATH] in the weak frictional instability regime, the dimensionful nucleation length in the continuum limit [MATH] is given by [MATH], where [MATH] is a number characterizing the fault interface.', '1401.3247-1-17-0': 'Next, we consider the duration times of each stage of the nucleation process, including that of the initial phase [MATH]), of the acceleration phase [MATH]), and of the high-speed rupture phase [MATH]).', '1401.3247-1-17-1': 'The ultimate utility of nucleation phenomena may be forecasting the upcoming mainshock.', '1401.3247-1-17-2': 'As mentioned, practical detection, if any, would become possible only in the acceleration phase.', '1401.3247-1-17-3': 'Since the system has already been beyond the "no-return" point, a mainshock should already be "deterministic".', '1401.3247-1-17-4': 'The remaining problem is how much time is left there.', '1401.3247-1-17-5': 'We tentatively set the detectable sliding velocity of the nucleus motion [MATH] which corresponds in real unit to [MATH] [mm/sec].', '1401.3247-1-17-6': 'Then, the time interval between the point of [MATH] and the point of [MATH] (the onset of a mainshock) is denoted by [MATH].', '1401.3247-1-17-7': 'This [MATH] would give a realistic measure of the remaining time available for a mainshock forecast.', '1401.3247-1-18-0': 'Among various duration times, though the duration time of a mainshock [MATH] naturally increases with the size of a mainshock [MATH], [MATH], [MATH], and [MATH] hardly depend on [MATH] except for smaller events.', '1401.3247-1-18-1': 'This means that it is again hard to predict the size of the mainshock based on the duration time of the nucleation process.', '1401.3247-1-18-2': 'We also examine the dependence of these duration times on the model parameters, to find that they are hardly dependent on [MATH], [MATH], [MATH], but [MATH] and [MATH] sensitively depend on the friction crossover velocity [MATH].', '1401.3247-1-18-3': 'In Fig.4(b), we plot the mean duration times versus [MATH] (main panel), and versus [MATH] (inset).', '1401.3247-1-18-4': 'For [MATH], [MATH] is greater than [MATH] by factor of 700, while [MATH] by factor of 20.', '1401.3247-1-18-5': 'The [MATH]-dependence of these duration times shown in the inset turns out to be rather weak.', '1401.3247-1-18-6': 'Then, reviving the normalization units and substituting the typical parameter values, we estimate, for [MATH], [MATH] [year], [MATH] [day], [MATH] [hour] and [MATH] [min] in the continuum limit.', '1401.3247-1-18-7': 'For smaller [MATH], [MATH] could be even longer, but taking the data for [MATH] is beyond our present computational capability.', '1401.3247-1-18-8': 'We conclude that the remaining time available for a mainshock forecast could be longer than the mainshock duration time by one or two orders of magnitude, but perhaps not much longer than that.', '1401.3247-1-19-0': 'In summary, we studied the properties of the earthquake nucleation process as a precursor of a mainshock both numerically and analytically on the basis of the BK model obeying the RSF law, and found that this simplified model successfully reproduced various features of the expected earthquake nucleation process.', '1401.3247-1-20-0': 'The authors are thankful to T. Okubo, T. Hatano, N. Kato, T. Uchide for useful discussion.', '1401.3247-1-20-1': 'This study was supported by Grant-in-Aid for Scientific Research on Priority Areas 21540385.', '1401.3247-1-20-2': 'We thank ISSP, Tokyo University for providing us with the CPU time.'}

{'1401.3247-2-0-0': 'There is a wide-spread expectation that a large earthquake might be preceded by a precursory nucleation process which occurs prior to the high-speed rupture of a mainshock.', '1401.3247-2-0-1': 'Nucleation process is localized to a compact "seed" area with its rupture velocity orders of magnitude smaller than the seismic wave velocity [CITATION].', '1401.3247-2-0-2': 'The fault spends a very long time in this nucleation process, and then at some point, exhibits a rapid acceleration accompanied by a rapid expansion of the rupture zone, finally getting into the high-speed rupture of a mainshock.', '1401.3247-2-0-3': 'Such a precursory phenomenon preceding a mainshock is of paramount importance in its own right as well as in its possible connection to an earthquake forecast.', '1401.3247-2-1-0': 'It has been suggested that the earthquake nucleation process might proceed via several distinct steps or "phases".', '1401.3247-2-1-1': 'Ref. [CITATION] proposed that it started with an initial quasi-static process until the nucleus diameter [MATH] exceeded a nucleation length [MATH].', '1401.3247-2-1-2': 'Then, the fault gets into the acceleration phase where the system gets out of equilibrium and rapidly increases its slip velocity.', '1401.3247-2-1-3': 'When the nucleus diameter exceeds another nucleation length [MATH], the fault eventually exhibits a high-speed rupture of a mainshock.', '1401.3247-2-1-4': 'In this picture, two nucleation lengths, [MATH] and [MATH], divide the nucleation process into "the initial phase"([MATH]), "the acceleration phase" ([MATH]) and "the high-speed rupture phase" ([MATH]).', '1401.3247-2-1-5': 'Although such features of the nucleation process have been more or less confirmed by laboratory rock experiments [CITATION], its nature, or even its very existence, remains less clear for real earthquakes [CITATION].', '1401.3247-2-2-0': 'Under such circumstances, a theoretical or a numerical study based on an appropriate model of an earthquake fault would be important and helpful.', '1401.3247-2-2-1': 'In such modelings, the friction force is a crucially important part.', '1401.3247-2-2-2': 'The friction force now standard in seismology is the so-called rate and state dependent friction (RSF) law [CITATION].', '1401.3247-2-2-3': 'The RSF law has been used in many of numerical simulations on earthquakes, mostly in the continuum model [CITATION], including earthquake nucleation process [CITATION].', '1401.3247-2-3-0': 'Meanwhile, a further simplified discrete model has also been used.', '1401.3247-2-3-1': 'Especially popular is the spring-block model or the Burridge-Knopoff (BK) model [CITATION], in which an earthquake fault is modeled as an assembly of blocks mutually connected via elastic springs which are subject to the friction force and are slowly driven by an external force mimicking the plate drive.', '1401.3247-2-3-2': 'While a simple velocity-weakening friction has often been assumed in many numerical simulations, a more realistic RSF law was also employed in some of the recent numerical simulations [CITATION].', '1401.3247-2-3-3': 'The model might also be useful in describing other stick-slip-type phenomena such as landslides [CITATION].', '1401.3247-2-4-0': 'The aim of the present letter is to clarify the nature of the nucleation process of the BK model.', '1401.3247-2-4-1': 'The BK model has widely been used especially in statistical physics, and clarifying the nature of its nucleation process would be important.', '1401.3247-2-4-2': 'Meanwhile, one may wonder what is the merit of studying the discrete BK model possessing an intrinsic short-length cut-off in the form of block size, even though a similar nucleation problem was already studied within the continuum model [CITATION].', '1401.3247-2-4-3': 'There might be two reasons for this.', '1401.3247-2-4-4': 'First, the issue of the discreteness is in fact closely related to the nucleation phenomenon.', '1401.3247-2-4-5': 'Rice argued that the characteristic length scale to be compared with the block size was the nucleation length, and the continuum system under the RSF law always exhibited a nucleation process prior to a mainshock [CITATION].', '1401.3247-2-4-6': 'We wish to clarify how the nucleation process of the discrete BK model behaves in its continuum limit, by systematically varying the extent of the discreteness of the model.', '1401.3247-2-4-7': 'Note that the extent of the discreteness may be regarded as a measure of the underlying spatial inhomogeneity [CITATION].', '1401.3247-2-4-8': 'The second reason is more technical, i.e., the BK model is much simplified compared to the continuum model, and often makes statistically relevant simulations possible in which hundreds of thousands of events are generated.', '1401.3247-2-5-0': 'The one-dimensional (1D) BK model consists of a 1D array of [MATH] identical blocks of the mass [MATH], which are mutually connected with the two neighboring blocks via the elastic springs of the srping stffness [MATH], and are also connected to the moving plate via the springs of the spring stiffness [MATH], and are driven with a constant rate [MATH].', '1401.3247-2-5-1': 'All blocks are subject to the friction strength [MATH].', '1401.3247-2-5-2': 'The equation of motion for the [MATH]-th block can be written as [CITATION] [EQUATION] where [MATH] is the time, [MATH] is the displacement of the [MATH]-th block.', '1401.3247-2-5-3': 'This equation can be made dimensionless as [EQUATION] where [MATH].', '1401.3247-2-5-4': 'The dimensionless displacement [MATH] is normalized by the critical slip distance [MATH] associated with the RSF law, the time [MATH] by [MATH], the block velocity [MATH] and the pulling speed of the plate [MATH] by [MATH], and the dimensionless friction force [MATH] by [MATH].', '1401.3247-2-5-5': 'The RSF force [MATH] reads as [EQUATION] where [MATH] is the dimensionless state variable describing the "state" of the interface, and [MATH] is the dimensionless crossover velocity.', '1401.3247-2-5-6': 'The normalized frictional parameters [MATH], [MATH] and [MATH] represent the velocity-strengthening, the velocity-weakening and the constant parts of friction.', '1401.3247-2-5-7': 'The original friction parameters [MATH], [MATH] and [MATH] are related to the normalized ones by [MATH], [MATH] and [MATH], [MATH] being the normal load.', '1401.3247-2-5-8': 'For simplicity, we inhibit the motion in the direction opposite to the plate drive.', '1401.3247-2-5-9': 'The state variable [MATH] is assumed to obey the aging law [CITATION], [EQUATION]', '1401.3247-2-5-10': 'In the simplest version of the BK model as studied here, the nearest-neighbor interaction is assumed between blocks, while in real faults the crust perpendicular to the fault plane mediates the effective long-range interaction between blocks away on the fault plane.', '1401.3247-2-5-11': 'Indeed, such a long-range interaction was assumed in some of the previous studies of the BK model, especially its statistical properties such as the magnitude distribution [CITATION].', '1401.3247-2-5-12': 'In the present study, however, we concentrate on the nucleation process of the nearest-neighbor model, with the aim of clarifying the nucleation process of the simplest version of the model.', '1401.3247-2-6-0': 'What type of setting the BK model actually assumes in terms of an earthquake fault embedded in the 3D continuum crust might not be so trivial.', '1401.3247-2-6-1': "The authors' view is as follows.", '1401.3247-2-6-2': 'Consider first the 2D BK model mimicking a planar fault embedded in the 3D continuum crust.', '1401.3247-2-6-3': 'Let the dimension of the block be [MATH], where [MATH] is the dimension along the plate drive, [MATH] the dimension perpendicular to the plate drive within the fault plane, and [MATH] the dimension perpendicular to the fault plane.', '1401.3247-2-6-4': 'Then, the block assembly represents a deformable "fault layer" of the width [MATH] which is uniformly pulled by the more or less rigid plate contingent to it.', '1401.3247-2-6-5': 'Our estimate to be given below entails the width [MATH] of order [MATH] [km].', '1401.3247-2-6-6': 'Thus, in the BK model, a uniform plate drive is applied not at infinity as boundary conditions as often assumed in the continuum model, but is applied rather close to the fault plane of order the distance [MATH] [km].', '1401.3247-2-6-7': 'Such a direct plate drive yields a term proportional to the displacement [MATH] in the equation of motion (2), which is absent in the standard elasto-dynamic equation.', '1401.3247-2-6-8': 'The 1D BK model is a simplification of the 2D model where one direction of the fault plane perpendicular to the plate drive has been integrated out, or supposed to be completely rigid.', '1401.3247-2-7-0': 'We note that the fault layer as modeled by the BK model might be related to the so-called "low-velocity fault zones (LVFZ)" observed in most mature faults, with [MATH] wave-velocity reduction relative to the host rock [CITATION].', '1401.3247-2-7-1': 'Their widths were reported to be 100 [m] [MATH] 2 [km], which are a bit smaller than, but does not much differ from the present estimate of the fault-zone width [MATH].', '1401.3247-2-8-0': 'Let us try to estimate typical values of the model parameters with natural earthquake faults in mind.', '1401.3247-2-8-1': 'The dimensionfull rise time of an event, i.e., the time elapsed from a given block involved in a mainshock rupture begins to move until it stops, is found to be [MATH].', '1401.3247-2-8-2': 'This is true for a single-block system, while our simulations indicate it is also the case for a many-block system.', '1401.3247-2-8-3': 'Since the typical rise time of an earthquake is a few seconds, we get an estimate of [MATH] [s].', '1401.3247-2-8-4': 'The reported values of the critical slip distance [MATH] are largely scattered in the literature depending on the observation scale [CITATION].', '1401.3247-2-8-5': 'Here, from our numerical observation that the typical block sliding velocity at the mainshock rupture is [MATH] in units of [MATH] while it is around 1 [m/s] in real seismicity, we take [MATH] to be a few [cm], which is not far from the value at the seismic depth deduced in [CITATION].', '1401.3247-2-8-6': 'Since the speed of the plate motion is typically a few [cm/year], the dimensionless loading rate is [MATH].', '1401.3247-2-9-0': 'The spring constant [MATH] may be related to the rigidity [MATH] as [MATH].', '1401.3247-2-9-1': 'This can be derived by noting that the shear force [MATH] acting on a block with the displacement [MATH] is given by [MATH] where [MATH] is the shear strain.', '1401.3247-2-9-2': 'The relation [MATH] is the mass density) and the [MATH]-wave velocity [MATH] yield [MATH].', '1401.3247-2-9-3': 'Putting [MATH] [km/s], which is taken somewhat smaller than the standard value of [MATH] [km] due to the possible lower wave-velocity in the fault zone, and [MATH] [s], we get an estimate of [MATH] [km] as given above.', '1401.3247-2-9-4': 'The proportionality between the fault-zone width [MATH] and the rise time [MATH] obtained here seems consistent with the observation on the LVFZ [CITATION].', '1401.3247-2-9-5': 'With [MATH] where [MATH] is the normal stress, we have [MATH].', '1401.3247-2-9-6': 'Putting [MATH], we get [MATH].', '1401.3247-2-9-7': 'As [MATH] is known to take a value around [MATH] [CITATION], [MATH] would be of order [MATH]-[MATH], [MATH] and [MATH] being one or two orders of magnitude smaller than [MATH].', '1401.3247-2-9-8': 'The crossover velocity [MATH] and its dimensionless counterpart [MATH] is hard to estimate though it should be much smaller than unity, and we take it as a parameter in our simulations.', '1401.3247-2-10-0': 'The continuum limit of the BK model corresponds to making the dimensionless block size [MATH], defined by [MATH], to be infinitesimal [MATH], simultaneously making the system infinitely rigid [MATH] with [MATH] [CITATION].', '1401.3247-2-10-1': 'The dimensionless distance [MATH] between the block [MATH] and [MATH] is [MATH].', '1401.3247-2-10-2': 'Notice that the continuum limit considered here concerns only with the fault direction (the fault plane in case of 2D), and the perpendicular direction ([MATH]-direction) is kept fixed.', '1401.3247-2-10-3': 'Thus, the possible internal motion in the fault layer along the perpendicular direction is suppressed in the model setting.', '1401.3247-2-11-0': 'As discussed in Ref. [CITATION], the 1D equation of motion in the continuum limit is given in the dimensionful form by [EQUATION] where [MATH] is the displacement at the position [MATH] and the time [MATH], [MATH] is the friction force per unit mass, while [MATH] and [MATH] are the characteristic frequency and the characteristic wave-velocity ([MATH]-wave velocity), respectively.', '1401.3247-2-11-1': 'As mentioned, the term [MATH] representing the plate drive is absent in the standard elasto-dynamic equation.', '1401.3247-2-11-2': 'If one discretize the space into blocks of the size [MATH] with [MATH] and notes the relation [MATH] and [MATH], one gets Eq. (1) with [MATH].', '1401.3247-2-12-0': 'Our first question might be whether the 1D BK model under the RSF law ever exhibits a nucleation process prior to a mainshock, and if it does, under what conditions.', '1401.3247-2-12-1': 'We observe that the model exhibits qualitatively different behaviors depending on whether the frictional instability is either "weak" or "strong".', '1401.3247-2-12-2': 'A slow and long-lasting nucleation process, the quasi-static initial phase, is realized in the former case only.', '1401.3247-2-12-3': 'We illustrate in fig.1 typical examples of seismic events realized in the stationary state of the model, where the time evolution of the movement of each block is shown as a color plot for each case of (a) the weak frictional instability ([MATH], [MATH], [MATH]), and (b) the strong frictional instability ([MATH], [MATH], [MATH]).', '1401.3247-2-12-4': 'A slow nucleation process with a long duration time is observed in (a), but is absent in (b).', '1401.3247-2-12-5': 'As will be shown below, the model possesses a borderline value of [MATH] determined solely by the stiffness parameter [MATH], [MATH], which discriminates the strong/weak instability behaviors.', '1401.3247-2-13-0': 'We also illustrate in fig.1 the two types of nucleation lengths, [MATH] and [MATH]).', '1401.3247-2-13-1': 'The former [MATH] is the length separating stable and unstable ruptures and exists only for the weak frictional instability, while the latter [MATH] is the length signaling the onset of the high-speed rupture of a mainshock.', '1401.3247-2-14-0': 'One way to identify [MATH] is to artificially stop the external loading in the course of simulation.', '1401.3247-2-14-1': 'We have confirmed that, if the external loading is stopped at any point beyond [MATH], the subsequent seismic rupture is no longer stoppable and evolves until its very end, whereas, if the external loading is stopped at a point before [MATH], the rupture itself also stops there.', '1401.3247-2-15-0': 'An appropriate physical condition describing the stable/unstable sliding across [MATH] might be whether the elastic stiffness [MATH], as defined by [MATH] which represents a change of the elastic force [MATH] due to an infinitesimal slip [MATH] of the block, is greater/smaller than the frictional weakening rate, as defined by [MATH] which represents a change of the friction force [MATH] due to an infinitesimal slip of the block.', '1401.3247-2-15-1': 'Consider a hypothetical instantaneous process from the states ([MATH], [MATH], [MATH]) to ([MATH], [MATH], [MATH]).', '1401.3247-2-15-2': 'The aging law ([REF]) entails the relation [MATH].', '1401.3247-2-15-3': 'Then, the frictional-weakening rate is obtained as [MATH].', '1401.3247-2-15-4': 'Meanwhile, the stiffness of the [MATH]-block system may be given by the smallest nonzero eigenvalue of the [MATH] matrix [MATH] defined via the relation [MATH] as [MATH].', '1401.3247-2-15-5': 'Matching [MATH] and [MATH], the condition of the frictional instability is obtained as [EQUATION]', '1401.3247-2-15-6': 'The quasi-static initial phase is realizable in the BK model only when [MATH] is greater than the lattice spacing, i.e., [MATH], or equivalently [MATH], yielding the condition of the weak frictional instability.', '1401.3247-2-15-7': 'We note that the formula (4) can also be derived from the linear stability analysis around the steady-state solution of the equation of motion, [MATH] and [MATH] along the line of Ref. [CITATION].', '1401.3247-2-16-0': 'Since the continuum limit entails [MATH], the condition of the weak frictional instability [MATH] is always satisfied there.', '1401.3247-2-16-1': 'Hence, the continuum limit of the model always lies in the weak frictional instability regime accompanying the quasi-static nucleation process, corroborating Rice [CITATION].', '1401.3247-2-17-0': 'In fig.2, we show for a typical large event in the weak frictional instability regime near the continuum limit the time evolutions of the epicenter-block sliding velocity [MATH], (a) in the initial phase, and (b) in the acceleration phase.', '1401.3247-2-17-1': 'In the initial phase, the sliding velocity [MATH] stays very low up to the nucleation length [MATH], of order the pulling speed of the plate.', '1401.3247-2-17-2': 'In the acceleration phase, the block movement exhibits a prominent acceleration, being no longer quasi-static nor reversible, reaching the maximum around [MATH] (this maximum point is used as our definition of [MATH]), then decreases sharply and finally stops.', '1401.3247-2-18-0': 'Next, we investigate the statistical properties of the nucleation lengths [MATH] and [MATH] as well as the duration times of each phase, averaged over many events (typically [MATH] events).', '1401.3247-2-18-1': 'Since the nucleation length [MATH] is determined only by the material parameters as in eq. ([REF]), it cannot be used as an indicator of the size of the ensuing mainshock.', '1401.3247-2-18-2': 'What about [MATH] ?', '1401.3247-2-18-3': 'We plot in fig.3(a) the computed mean [MATH] normalized by the corresponding [MATH], [MATH], versus the final rupture-zone size [MATH] for various choices of the model parameters in the weak frictional instability regime.', '1401.3247-2-18-4': 'The [MATH]-value is fixed to [MATH] while the parameters [MATH], [MATH] and [MATH] are varied.', '1401.3247-2-18-5': 'The data approximately collapse onto a common curve.', '1401.3247-2-18-6': 'Since [MATH] hardly depends on [MATH] and [MATH], this indicates that [MATH] is insensitive to [MATH] and [MATH], while its [MATH]-dependence is the same as that of [MATH].', '1401.3247-2-18-7': 'One also sees that large events tend to be independent of [MATH], implying that one cannot predict the size of the upcoming mainshock even with the information of [MATH].', '1401.3247-2-18-8': 'We examine the [MATH]-dependence of [MATH], to find that it increases linearly with [MATH] as [MATH] in the weak frictional instability regime [MATH].', '1401.3247-2-19-0': 'The continuum limit of [MATH] in the dimensionless form [MATH] is obtained as [MATH].', '1401.3247-2-19-1': 'Reviving the normalization units above, we get the dimensionful nucleation length in the continuum limit, [MATH], as [EQUATION] for [MATH].', '1401.3247-2-19-2': 'If we substitute the parameter values [MATH] [cm], [MATH] and [MATH] [km], we get [MATH] several kilometers.', '1401.3247-2-19-3': 'Concerning [MATH], since the ratio [MATH] turns out to be hardly dependent on [MATH] in the weak frictional instability regime, the dimensionful nucleation length in the continuum limit [MATH] is given by [MATH], where [MATH] is a number characterizing the fault interface.', '1401.3247-2-20-0': 'Next, we consider the duration times of each stage of the nucleation process, including that of the initial phase [MATH]), of the acceleration phase [MATH]), and of the high-speed rupture phase [MATH]).', '1401.3247-2-20-1': 'The ultimate utility of nucleation phenomena may be forecasting the upcoming mainshock.', '1401.3247-2-20-2': 'Practical detection, if any, would become possible in the acceleration phase.', '1401.3247-2-20-3': 'Since the system has already been beyond the "no-return" point, a mainshock should already be "deterministic".', '1401.3247-2-20-4': 'The remaining problem is how much time is left there.', '1401.3247-2-20-5': 'We tentatively set the detectable sliding velocity of the nucleus motion [MATH] which corresponds in real unit to [MATH] [mm/sec].', '1401.3247-2-20-6': 'Then, the time interval between the point of [MATH] and the point of [MATH] (the onset of a mainshock) is denoted by [MATH].', '1401.3247-2-20-7': 'This [MATH] would give a realistic measure of the remaining time available for a mainshock forecast.', '1401.3247-2-21-0': 'Among various duration times, though the duration time of a mainshock [MATH] naturally increases with the size of a mainshock [MATH], [MATH], [MATH], and [MATH] hardly depend on [MATH] except for smaller events.', '1401.3247-2-21-1': 'This means that it is again hard to predict the size of the mainshock based on the duration time of the nucleation process.', '1401.3247-2-21-2': 'We also examine the dependence of these duration times on the model parameters, to find that they are hardly dependent on [MATH], [MATH], [MATH], but [MATH] and [MATH] sensitively depend on the friction crossover velocity [MATH].', '1401.3247-2-21-3': 'In fig.3(b), we plot the mean duration times versus [MATH] (main panel), and versus [MATH] (inset).', '1401.3247-2-21-4': 'For [MATH], [MATH] is greater than [MATH] by factor of 700, while [MATH] by factor of 20.', '1401.3247-2-21-5': 'The [MATH]-dependence of these duration times shown in the inset turns out to be rather weak.', '1401.3247-2-21-6': 'Then, reviving the normalization units and substituting the typical parameter values, we estimate, for [MATH], [MATH] [year], [MATH] [day], [MATH] [hour] and [MATH] [min] in the continuum limit.', '1401.3247-2-21-7': 'For smaller [MATH], [MATH] could be even longer, but taking the data for [MATH] is beyond our present computational capability.', '1401.3247-2-21-8': 'We conclude that the remaining time available for a mainshock forecast could be longer than the mainshock duration time by one or two orders of magnitude, but perhaps not much longer than that.', '1401.3247-2-21-9': 'Of course, since the reliability of the 1D BK model in connection with real seismicity may be limited at the quantitative level, these estimates should be taken only as rough measures.', '1401.3247-2-22-0': 'In summary, we studied the properties of the earthquake nucleation process as a precursor of a mainshock both numerically and analytically on the basis of the BK model obeying the RSF law, and found that this simplified model successfully reproduced various features of the expected earthquake nucleation process.', '1401.3247-2-23-0': 'The authors are thankful to T. Okubo, T. Hatano, N. Kato, T. Uchide and S. Ito for useful discussion.', '1401.3247-2-23-1': "They are also thankful to one of the referees for pointing out the possible connection of the BK model to the LVFZ, and bringing Ref. [CITATION] to the authors' attention.", '1401.3247-2-23-2': 'This study was supported by Grant-in-Aid for Scientific Research on Priority Areas 21540385.', '1401.3247-2-23-3': 'We thank ISSP, Tokyo University for providing us with the CPU time.'}

[['1401.3247-1-14-1', '1401.3247-2-17-1'], ['1401.3247-1-14-3', '1401.3247-2-17-2'], ['1401.3247-1-4-0', '1401.3247-2-3-0'], ['1401.3247-1-8-0', '1401.3247-2-12-0'], ['1401.3247-1-8-1', '1401.3247-2-12-1'], ['1401.3247-1-8-2', '1401.3247-2-12-2'], ['1401.3247-1-8-4', '1401.3247-2-12-4'], ['1401.3247-1-8-5', '1401.3247-2-12-5'], ['1401.3247-1-3-0', '1401.3247-2-2-0'], ['1401.3247-1-3-1', '1401.3247-2-2-1'], ['1401.3247-1-3-2', '1401.3247-2-2-2'], ['1401.3247-1-20-1', '1401.3247-2-23-2'], ['1401.3247-1-20-2', '1401.3247-2-23-3'], ['1401.3247-1-13-0', '1401.3247-2-16-0'], ['1401.3247-1-13-1', '1401.3247-2-16-1'], ['1401.3247-1-2-0', '1401.3247-2-1-0'], ['1401.3247-1-2-2', '1401.3247-2-1-2'], ['1401.3247-1-11-0', '1401.3247-2-15-0'], ['1401.3247-1-11-1', '1401.3247-2-15-1'], ['1401.3247-1-11-2', '1401.3247-2-15-2'], ['1401.3247-1-11-3', '1401.3247-2-15-3'], ['1401.3247-1-11-4', '1401.3247-2-15-4'], ['1401.3247-1-11-5', '1401.3247-2-15-5'], ['1401.3247-1-18-0', '1401.3247-2-21-0'], ['1401.3247-1-18-1', '1401.3247-2-21-1'], ['1401.3247-1-18-2', '1401.3247-2-21-2'], ['1401.3247-1-18-4', '1401.3247-2-21-4'], ['1401.3247-1-18-5', '1401.3247-2-21-5'], ['1401.3247-1-18-6', '1401.3247-2-21-6'], ['1401.3247-1-18-7', '1401.3247-2-21-7'], ['1401.3247-1-18-8', '1401.3247-2-21-8'], ['1401.3247-1-17-0', '1401.3247-2-20-0'], ['1401.3247-1-17-1', '1401.3247-2-20-1'], ['1401.3247-1-17-3', '1401.3247-2-20-3'], ['1401.3247-1-17-4', '1401.3247-2-20-4'], ['1401.3247-1-17-5', '1401.3247-2-20-5'], ['1401.3247-1-17-6', '1401.3247-2-20-6'], ['1401.3247-1-17-7', '1401.3247-2-20-7'], ['1401.3247-1-19-0', '1401.3247-2-22-0'], ['1401.3247-1-10-0', '1401.3247-2-14-0'], ['1401.3247-1-10-1', '1401.3247-2-14-1'], ['1401.3247-1-16-0', '1401.3247-2-19-0'], ['1401.3247-1-16-1', '1401.3247-2-19-1'], ['1401.3247-1-16-3', '1401.3247-2-19-3'], ['1401.3247-1-15-1', '1401.3247-2-18-1'], ['1401.3247-1-15-4', '1401.3247-2-18-4'], ['1401.3247-1-15-7', '1401.3247-2-18-7'], ['1401.3247-1-1-0', '1401.3247-2-0-0'], ['1401.3247-1-1-1', '1401.3247-2-0-1'], ['1401.3247-1-1-2', '1401.3247-2-0-2'], ['1401.3247-1-6-4', '1401.3247-2-5-5'], ['1401.3247-1-6-5', '1401.3247-2-5-6'], ['1401.3247-1-6-6', '1401.3247-2-5-7'], ['1401.3247-1-6-7', '1401.3247-2-5-8'], ['1401.3247-1-6-8', '1401.3247-2-5-9'], ['1401.3247-1-6-10', '1401.3247-2-10-1'], ['1401.3247-1-7-6', '1401.3247-2-9-5'], ['1401.3247-1-7-9', '1401.3247-2-9-8'], ['1401.3247-1-14-0', '1401.3247-2-17-0'], ['1401.3247-1-4-1', '1401.3247-2-3-1'], ['1401.3247-1-4-2', '1401.3247-2-3-2'], ['1401.3247-1-8-3', '1401.3247-2-12-3'], ['1401.3247-1-3-3', '1401.3247-2-2-3'], ['1401.3247-1-20-0', '1401.3247-2-23-0'], ['1401.3247-1-2-1', '1401.3247-2-1-1'], ['1401.3247-1-2-3', '1401.3247-2-1-3'], ['1401.3247-1-2-4', '1401.3247-2-1-4'], ['1401.3247-1-5-3', '1401.3247-2-4-4'], ['1401.3247-1-5-4', '1401.3247-2-4-5'], ['1401.3247-1-18-3', '1401.3247-2-21-3'], ['1401.3247-1-9-0', '1401.3247-2-13-0'], ['1401.3247-1-17-2', '1401.3247-2-20-2'], ['1401.3247-1-16-2', '1401.3247-2-19-2'], ['1401.3247-1-15-0', '1401.3247-2-18-0'], ['1401.3247-1-15-3', '1401.3247-2-18-3'], ['1401.3247-1-15-5', '1401.3247-2-18-5'], ['1401.3247-1-15-6', '1401.3247-2-18-6'], ['1401.3247-1-15-8', '1401.3247-2-18-8'], ['1401.3247-1-6-0', '1401.3247-2-5-0'], ['1401.3247-1-6-1', '1401.3247-2-5-1'], ['1401.3247-1-6-2', '1401.3247-2-5-2'], ['1401.3247-1-6-3', '1401.3247-2-5-4'], ['1401.3247-1-6-9', '1401.3247-2-10-0'], ['1401.3247-1-7-3', '1401.3247-2-8-6'], ['1401.3247-1-7-5', '1401.3247-2-9-2'], ['1401.3247-1-7-8', '1401.3247-2-9-7'], ['1401.3247-1-2-5', '1401.3247-2-1-5'], ['1401.3247-1-5-0', '1401.3247-2-4-0'], ['1401.3247-1-5-5', '1401.3247-2-4-0'], ['1401.3247-1-5-5', '1401.3247-2-4-1'], ['1401.3247-1-5-5', '1401.3247-2-4-6'], ['1401.3247-1-11-6', '1401.3247-2-15-6'], ['1401.3247-1-9-1', '1401.3247-2-13-1'], ['1401.3247-1-1-3', '1401.3247-2-0-3'], ['1401.3247-1-7-0', '1401.3247-2-8-0'], ['1401.3247-1-7-1', '1401.3247-2-8-3'], ['1401.3247-1-7-4', '1401.3247-2-9-0']]

[['1401.3247-1-14-1', '1401.3247-2-17-1'], ['1401.3247-1-14-3', '1401.3247-2-17-2'], ['1401.3247-1-4-0', '1401.3247-2-3-0'], ['1401.3247-1-8-0', '1401.3247-2-12-0'], ['1401.3247-1-8-1', '1401.3247-2-12-1'], ['1401.3247-1-8-2', '1401.3247-2-12-2'], ['1401.3247-1-8-4', '1401.3247-2-12-4'], ['1401.3247-1-8-5', '1401.3247-2-12-5'], ['1401.3247-1-3-0', '1401.3247-2-2-0'], ['1401.3247-1-3-1', '1401.3247-2-2-1'], ['1401.3247-1-3-2', '1401.3247-2-2-2'], ['1401.3247-1-20-1', '1401.3247-2-23-2'], ['1401.3247-1-20-2', '1401.3247-2-23-3'], ['1401.3247-1-13-0', '1401.3247-2-16-0'], ['1401.3247-1-13-1', '1401.3247-2-16-1'], ['1401.3247-1-2-0', '1401.3247-2-1-0'], ['1401.3247-1-2-2', '1401.3247-2-1-2'], ['1401.3247-1-11-0', '1401.3247-2-15-0'], ['1401.3247-1-11-1', '1401.3247-2-15-1'], ['1401.3247-1-11-2', '1401.3247-2-15-2'], ['1401.3247-1-11-3', '1401.3247-2-15-3'], ['1401.3247-1-11-4', '1401.3247-2-15-4'], ['1401.3247-1-11-5', '1401.3247-2-15-5'], ['1401.3247-1-18-0', '1401.3247-2-21-0'], ['1401.3247-1-18-1', '1401.3247-2-21-1'], ['1401.3247-1-18-2', '1401.3247-2-21-2'], ['1401.3247-1-18-4', '1401.3247-2-21-4'], ['1401.3247-1-18-5', '1401.3247-2-21-5'], ['1401.3247-1-18-6', '1401.3247-2-21-6'], ['1401.3247-1-18-7', '1401.3247-2-21-7'], ['1401.3247-1-18-8', '1401.3247-2-21-8'], ['1401.3247-1-17-0', '1401.3247-2-20-0'], ['1401.3247-1-17-1', '1401.3247-2-20-1'], ['1401.3247-1-17-3', '1401.3247-2-20-3'], ['1401.3247-1-17-4', '1401.3247-2-20-4'], ['1401.3247-1-17-5', '1401.3247-2-20-5'], ['1401.3247-1-17-6', '1401.3247-2-20-6'], ['1401.3247-1-17-7', '1401.3247-2-20-7'], ['1401.3247-1-19-0', '1401.3247-2-22-0'], ['1401.3247-1-10-0', '1401.3247-2-14-0'], ['1401.3247-1-10-1', '1401.3247-2-14-1'], ['1401.3247-1-16-0', '1401.3247-2-19-0'], ['1401.3247-1-16-1', '1401.3247-2-19-1'], ['1401.3247-1-16-3', '1401.3247-2-19-3'], ['1401.3247-1-15-1', '1401.3247-2-18-1'], ['1401.3247-1-15-4', '1401.3247-2-18-4'], ['1401.3247-1-15-7', '1401.3247-2-18-7'], ['1401.3247-1-1-0', '1401.3247-2-0-0'], ['1401.3247-1-1-1', '1401.3247-2-0-1'], ['1401.3247-1-1-2', '1401.3247-2-0-2'], ['1401.3247-1-6-4', '1401.3247-2-5-5'], ['1401.3247-1-6-5', '1401.3247-2-5-6'], ['1401.3247-1-6-6', '1401.3247-2-5-7'], ['1401.3247-1-6-7', '1401.3247-2-5-8'], ['1401.3247-1-6-8', '1401.3247-2-5-9'], ['1401.3247-1-6-10', '1401.3247-2-10-1'], ['1401.3247-1-7-6', '1401.3247-2-9-5'], ['1401.3247-1-7-9', '1401.3247-2-9-8']]

[['1401.3247-1-14-0', '1401.3247-2-17-0'], ['1401.3247-1-4-1', '1401.3247-2-3-1'], ['1401.3247-1-4-2', '1401.3247-2-3-2'], ['1401.3247-1-8-3', '1401.3247-2-12-3'], ['1401.3247-1-3-3', '1401.3247-2-2-3'], ['1401.3247-1-20-0', '1401.3247-2-23-0'], ['1401.3247-1-2-1', '1401.3247-2-1-1'], ['1401.3247-1-2-3', '1401.3247-2-1-3'], ['1401.3247-1-2-4', '1401.3247-2-1-4'], ['1401.3247-1-5-3', '1401.3247-2-4-4'], ['1401.3247-1-5-4', '1401.3247-2-4-5'], ['1401.3247-1-18-3', '1401.3247-2-21-3'], ['1401.3247-1-9-0', '1401.3247-2-13-0'], ['1401.3247-1-17-2', '1401.3247-2-20-2'], ['1401.3247-1-16-2', '1401.3247-2-19-2'], ['1401.3247-1-15-0', '1401.3247-2-18-0'], ['1401.3247-1-15-3', '1401.3247-2-18-3'], ['1401.3247-1-15-5', '1401.3247-2-18-5'], ['1401.3247-1-15-6', '1401.3247-2-18-6'], ['1401.3247-1-15-8', '1401.3247-2-18-8'], ['1401.3247-1-6-0', '1401.3247-2-5-0'], ['1401.3247-1-6-1', '1401.3247-2-5-1'], ['1401.3247-1-6-2', '1401.3247-2-5-2'], ['1401.3247-1-6-3', '1401.3247-2-5-4'], ['1401.3247-1-6-9', '1401.3247-2-10-0'], ['1401.3247-1-7-3', '1401.3247-2-8-6'], ['1401.3247-1-7-5', '1401.3247-2-9-2'], ['1401.3247-1-7-8', '1401.3247-2-9-7']]

[]

[['1401.3247-1-2-5', '1401.3247-2-1-5'], ['1401.3247-1-5-0', '1401.3247-2-4-0'], ['1401.3247-1-5-5', '1401.3247-2-4-0'], ['1401.3247-1-5-5', '1401.3247-2-4-1'], ['1401.3247-1-5-5', '1401.3247-2-4-6'], ['1401.3247-1-11-6', '1401.3247-2-15-6'], ['1401.3247-1-9-1', '1401.3247-2-13-1'], ['1401.3247-1-1-3', '1401.3247-2-0-3'], ['1401.3247-1-7-0', '1401.3247-2-8-0'], ['1401.3247-1-7-1', '1401.3247-2-8-3'], ['1401.3247-1-7-4', '1401.3247-2-9-0']]

[]

['1401.3247-1-7-7', '1401.3247-1-15-2', '1401.3247-2-18-2']

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

https://arxiv.org/abs/1401.3247

1905.07761

{'1905.07761-1-0-0': 'We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on surfaces with hyperbolic cusps and ergodic geodesic flows.', '1905.07761-1-1-0': 'Quantum ergodicity states that for quantum systems with ergodic classical flow, almost all high-frequency eigenfunctions are equidistributed in phase-space.', '1905.07761-1-1-1': 'Quantum unique ergodicity corresponds to equidistribution of all high-frequency eigenfuctions.', '1905.07761-1-1-2': 'The main examples are given by compact Riemannian manifold [MATH] with ergodic geodesic flows, where one considers eigenfunctions of the Laplacian [MATH] associated to the metric, and negatively curved metrics are the typical models for ergodic geodesic flows.', '1905.07761-1-2-0': 'The first mathematical results in this direction is due to Schnirelman [CITATION], and later by Zelditch [CITATION] and Colin de Verdiere [CITATION], who proved quantum ergodicity for closed manifolds with ergodic geodesic flows.', '1905.07761-1-2-1': 'In the case of manifolds with boundary, similar results were shown by Gerard-Leichtnam [CITATION] and Zelditch-Zworski [CITATION].', '1905.07761-1-3-0': 'In this work, we consider cases of non-compact manifolds, and the first examples one has in mind are surfaces with finite volume.', '1905.07761-1-3-1': 'In general, non-compactness often produces essential spectrum for the Laplacian, and this is indeed the case for the simplest model of finite volume surfaces, namely hyperbolic surfaces realised as quotients [MATH] of the hyperbolic plane by Fuchsian subgroups with a finite index.', '1905.07761-1-3-2': 'In that setting, there is however a way to get rid of this essential spectrum by a simple modification of the Laplacian, that is called pseudo-Laplacian, introduced by Colin de Verdiere [CITATION] (and related to the work Cartier-Hejhal [CITATION]).', '1905.07761-1-3-3': 'This operator was very useful for obtaining meromorphic extension of the Eisenstein series and the resolvent of the Laplacian, with important generalisation to the higher rank case by Muller [CITATION].', '1905.07761-1-3-4': 'There is however a different route in the setting of hyperbolic surfaces with finite volume that was taken by Zelditch [CITATION], who proved a quantum ergodicity statement for the Laplacian, but it involves the contribution of the continuous spectrum through the Eisenstein series.', '1905.07761-1-3-5': 'The proof has been recently generalised by Bonthonneau-Zelditch [CITATION] to variable curvature and all dimensions, while Jakobson [CITATION] and Luo-Sarnak [CITATION] proved some quantum unique ergodicity in the case of the modular surface.', '1905.07761-1-3-6': 'In a somehow complementary way, we deal in this note with the case of the pseudo-Laplacian, that we denote by [MATH], which is defined as an unbounded operator on [MATH],with domain [MATH] and that has discrete spectrum.', '1905.07761-1-3-7': 'Here, [MATH] will denote a semiclassical quantization for compactly supported symbols, see section [MATH].', '1905.07761-1-4-0': 'Let [MATH] be a Riemannian surface with a finite number of constant curvature hyperbolic cusps such that the geodesic flow on [MATH] is ergodic.', '1905.07761-1-4-1': 'Let [MATH] be an orthonormal family of eigenfuctions of [MATH] with eigenvalues [MATH], covering all the eigenvalues of [MATH] except a finite number of non-positive ones.', '1905.07761-1-4-2': 'Let [MATH] be compactly supported in space.', '1905.07761-1-5-0': 'For a precise review of the geometry of the considered Riemannian manifolds, we refer to Section [REF], while for the definition of the pseudo-Laplacian this is done in Section [REF].', '1905.07761-1-5-1': 'There are very natural examples of such manifolds given by negatively curved surfaces with finite volume and hyperbolic cusps.', '1905.07761-1-6-0': 'Let us make several remarks about the Theorem.', '1905.07761-1-6-1': 'First, by a standard argument (see for instance [CITATION]) Theorem [REF] implies that [EQUATION] for a sequence of density one, when [MATH] has compact support.', '1905.07761-1-6-2': 'Moreover, since we are only interested in quantizing symbols with a compact support in the space variable, we can use a standard quantization procedure, see for instance [CITATION].', '1905.07761-1-6-3': 'That means however that the estimates are not uniform far in the cusp.', '1905.07761-1-7-0': 'In the same geometric setting, we also mention that there are other works by Dyatlov [CITATION] and Bonthonneau [CITATION] on the microlocal limits of non-[MATH] eigenfunctions of the Laplacian but with complex eigenvalues, where one instead get a sort of "quantum unique ergodicity".', '1905.07761-1-8-0': 'For simplicity, the proof will be presented in the case where there is one cusp, the argument being the same with several cusps.', '1905.07761-1-8-1': 'The method of proof follows the scheme from [CITATION] and has two steps:', '1905.07761-1-9-0': '1) a pointwise "ellipticity bound" that states that the eigenfunctions are microlocalized on the cosphere bundle.', '1905.07761-1-9-1': 'This implies that in the limit [MATH], [EQUATION] is controlled by [MATH].', '1905.07761-1-10-0': '2) Taking a symbol with average zero, we propagate it by the geodesic flow to get a new symbol that is small on the cosphere bundle (by the [MATH] ergodic theorem); we have to prove that this does not modify much [MATH]: this is the point of the "flow invariance" theorems.', '1905.07761-1-11-0': 'We stress that working with a pseudo-Laplacian entails new difficulties, as compared to the compact setting.', '1905.07761-1-11-1': 'For the first step, since we are working with a pseudo-Laplacian, the pointwise ellipticity bound (and the subsequent microlocalization) works only outside the singular circle, and we need to prove that the needed correction is small enough.', '1905.07761-1-11-2': 'This requires a precise control of the eigenfunctions of the pseudo-Laplacian.', '1905.07761-1-12-0': 'For the second step, it is important to notice that the eigenfunctions we are interested in are not eigenfunctions of the propagator we are using for the proof.', '1905.07761-1-12-1': 'We are able to prove that [MATH] does not change much when replacing [MATH] by [MATH] if [MATH] is the geodesic flow, but we have to assume that the symbol [MATH] is supported quite far away from the singular circle.', '1905.07761-1-12-2': 'Since the admissible support has full measure, the [MATH] control of [MATH] we still get at step [MATH] leads to the same result.', '1905.07761-1-13-0': 'Finally, [CITATION] work with a compact manifold, and [MATH] thus vanishes.', '1905.07761-1-13-1': 'This is not the case for us since we only use symbols with a compact support in space.', '1905.07761-1-13-2': 'Our proof has a third step which consists in finding symbols [MATH] with average close to [MATH] such that [MATH] is arbitrarily close to zero.', '1905.07761-1-13-3': 'For that purpose, we shall prove that the modes of the eigenfunctions of the pseudo-Laplacian are microlocalized in the cusp.'}

{'1905.07761-2-0-0': 'We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on surfaces with hyperbolic cusps and ergodic geodesic flows.', '1905.07761-2-1-0': 'Quantum ergodicity states that for quantum systems with ergodic classical flow, almost all high-frequency eigenfunctions are equidistributed in phase-space.', '1905.07761-2-1-1': 'Quantum unique ergodicity corresponds to equidistribution of all high-frequency eigenfuctions.', '1905.07761-2-1-2': 'The main examples are given by compact Riemannian manifolds [MATH] with ergodic geodesic flows, where one considers eigenfunctions of the Laplacian [MATH] associated to the metric, and negatively curved metrics are the typical models for ergodic geodesic flows.', '1905.07761-2-2-0': 'The first mathematical results in this direction are due to Schnirelman [CITATION], and later by Zelditch [CITATION] and Colin de Verdiere [CITATION], who proved quantum ergodicity for closed manifolds with ergodic geodesic flows.', '1905.07761-2-2-1': 'In the case of manifolds with boundary, similar results were shown by Gerard-Leichtnam [CITATION] and Zelditch-Zworski [CITATION].', '1905.07761-2-3-0': 'In this work, we consider cases of non-compact manifolds, and the first examples one has in mind are surfaces with finite volume.', '1905.07761-2-3-1': 'In general, non-compactness often produces an essential spectrum for the Laplacian, and this is indeed the case for the simplest model of finite volume surfaces, namely hyperbolic surfaces realised as quotients [MATH] of the hyperbolic plane by Fuchsian subgroups with a finite index.', '1905.07761-2-3-2': 'In that setting, there is however a way to get rid of this essential spectrum by a simple modification of the Laplacian, that is called pseudo-Laplacian, introduced by Colin de Verdiere [CITATION] (and related to the work Cartier-Hejhal [CITATION]).', '1905.07761-2-3-3': 'This operator was very useful for obtaining a meromorphic extension of the Eisenstein series and the resolvent of the Laplacian, with important generalisation to the higher rank case by Muller [CITATION].', '1905.07761-2-3-4': 'There is however a different route in the setting of hyperbolic surfaces with finite volume that was taken by Zelditch [CITATION], who proved a quantum ergodicity statement for the Laplacian, but it involves the contribution of the continuous spectrum through the Eisenstein series.', '1905.07761-2-3-5': 'The proof has been recently generalised by Bonthonneau-Zelditch [CITATION] to variable curvature and all dimensions, while Jakobson [CITATION] and Luo-Sarnak [CITATION] proved some quantum unique ergodicity in the case of the modular surface.', '1905.07761-2-3-6': 'The problem of quantum ergodicity for the eigenfunctions of the pseudo-Laplacian was first proposed by S. Zelditch [CITATION].', '1905.07761-2-3-7': 'This means that in this paper, we deal with the case of the pseudo-Laplacian, that we denote by [MATH], which is defined as an unbounded operator on [MATH],with domain [MATH] and that has discrete spectrum.', '1905.07761-2-3-8': 'Here, [MATH] will denote a semiclassical quantization for compactly supported symbols, see section [MATH].', '1905.07761-2-4-0': 'Let [MATH] be a Riemannian surface with a finite number of constant curvature hyperbolic cusps such that the geodesic flow on [MATH] is ergodic.', '1905.07761-2-4-1': 'Let [MATH] be an orthonormal family of eigenfuctions of [MATH] with eigenvalues [MATH], covering all the eigenvalues of [MATH] except a finite number of non-positive ones.', '1905.07761-2-4-2': 'Let [MATH] be compactly supported in space.', '1905.07761-2-5-0': 'For a precise review of the geometry of the considered Riemannian manifolds, we refer to Section [REF], while for the definition of the pseudo-Laplacian this is done in Section [REF].', '1905.07761-2-5-1': 'There are very natural examples of such manifolds given by negatively curved surfaces with finite volume and hyperbolic cusps.', '1905.07761-2-6-0': 'Let us make several remarks about the Theorem.', '1905.07761-2-6-1': 'First, by a standard argument (see for instance [CITATION]) Theorem [REF] implies that [EQUATION] for a sequence of density one, when [MATH] has compact support.', '1905.07761-2-6-2': 'Moreover, since we are only interested in quantizing symbols with a compact support in the space variable, we can use a standard quantization procedure, see for instance [CITATION].', '1905.07761-2-6-3': 'That means however that the estimates are not uniform far in the cusp.', '1905.07761-2-7-0': 'In the same geometric setting, we also mention that there are other works by Dyatlov [CITATION] and Bonthonneau [CITATION] on the microlocal limits of non-[MATH] eigenfunctions of the Laplacian but with complex eigenvalues, where one instead get a sort of "quantum unique ergodicity".', '1905.07761-2-8-0': 'For simplicity, the proof will be presented in the case where there is one cusp, the argument being the same with several cusps.', '1905.07761-2-8-1': 'The method of proof follows the scheme from [CITATION] and has two steps:', '1905.07761-2-9-0': '1) a pointwise "ellipticity bound" that states that the eigenfunctions are microlocalized on the cosphere bundle.', '1905.07761-2-9-1': 'This implies that in the limit [MATH], [EQUATION] is controlled by [MATH].', '1905.07761-2-10-0': '2) Taking a symbol with average zero, we propagate it by the geodesic flow to get a new symbol that is small on the cosphere bundle (by the [MATH] ergodic theorem); we have to prove that this does not modify much [MATH]: this is the point of the "flow invariance" theorems.', '1905.07761-2-11-0': 'We stress that working with a pseudo-Laplacian entails new difficulties, as compared to the compact setting.', '1905.07761-2-11-1': 'For the first step, since we are working with a pseudo-Laplacian, the pointwise ellipticity bound (and the subsequent microlocalization) works only outside the singular circle, and we need to prove that the needed correction is small enough.', '1905.07761-2-11-2': 'This requires a precise control of the eigenfunctions of the pseudo-Laplacian.', '1905.07761-2-12-0': 'For the second step, it is important to notice that the eigenfunctions we are interested in are not eigenfunctions of the propagator we are using for the proof.', '1905.07761-2-12-1': 'We are able to prove that [MATH] does not change much when replacing [MATH] by [MATH] if [MATH] is the geodesic flow, but we have to assume that the symbol [MATH] is supported quite far away from the singular circle.', '1905.07761-2-12-2': 'Since the admissible support has full measure, the [MATH] control of [MATH] we still get at step [MATH] leads to the same result.', '1905.07761-2-13-0': 'Finally, [CITATION] work with a compact manifold, and [MATH] thus vanishes.', '1905.07761-2-13-1': 'This is not the case for us since we only use symbols with a compact support in space.', '1905.07761-2-13-2': 'Our proof has a third step which consists in finding symbols [MATH] with average close to [MATH] such that [MATH] is arbitrarily close to zero.', '1905.07761-2-13-3': 'For that purpose, we shall prove that the modes of the eigenfunctions of the pseudo-Laplacian are microlocalized in the cusp.'}

[['1905.07761-1-13-0', '1905.07761-2-13-0'], ['1905.07761-1-13-1', '1905.07761-2-13-1'], ['1905.07761-1-13-2', '1905.07761-2-13-2'], ['1905.07761-1-13-3', '1905.07761-2-13-3'], ['1905.07761-1-6-0', '1905.07761-2-6-0'], ['1905.07761-1-6-1', '1905.07761-2-6-1'], ['1905.07761-1-6-2', '1905.07761-2-6-2'], ['1905.07761-1-6-3', '1905.07761-2-6-3'], ['1905.07761-1-7-0', '1905.07761-2-7-0'], ['1905.07761-1-12-0', '1905.07761-2-12-0'], ['1905.07761-1-12-1', '1905.07761-2-12-1'], ['1905.07761-1-12-2', '1905.07761-2-12-2'], ['1905.07761-1-0-0', '1905.07761-2-0-0'], ['1905.07761-1-1-0', '1905.07761-2-1-0'], ['1905.07761-1-1-1', '1905.07761-2-1-1'], ['1905.07761-1-3-0', '1905.07761-2-3-0'], ['1905.07761-1-3-2', '1905.07761-2-3-2'], ['1905.07761-1-3-4', '1905.07761-2-3-4'], ['1905.07761-1-3-5', '1905.07761-2-3-5'], ['1905.07761-1-3-7', '1905.07761-2-3-8'], ['1905.07761-1-5-0', '1905.07761-2-5-0'], ['1905.07761-1-5-1', '1905.07761-2-5-1'], ['1905.07761-1-8-0', '1905.07761-2-8-0'], ['1905.07761-1-4-0', '1905.07761-2-4-0'], ['1905.07761-1-4-1', '1905.07761-2-4-1'], ['1905.07761-1-4-2', '1905.07761-2-4-2'], ['1905.07761-1-9-0', '1905.07761-2-9-0'], ['1905.07761-1-9-1', '1905.07761-2-9-1'], ['1905.07761-1-10-0', '1905.07761-2-10-0'], ['1905.07761-1-11-0', '1905.07761-2-11-0'], ['1905.07761-1-11-1', '1905.07761-2-11-1'], ['1905.07761-1-11-2', '1905.07761-2-11-2'], ['1905.07761-1-2-1', '1905.07761-2-2-1'], ['1905.07761-1-1-2', '1905.07761-2-1-2'], ['1905.07761-1-3-1', '1905.07761-2-3-1'], ['1905.07761-1-3-3', '1905.07761-2-3-3'], ['1905.07761-1-3-6', '1905.07761-2-3-7'], ['1905.07761-1-2-0', '1905.07761-2-2-0']]

[['1905.07761-1-13-0', '1905.07761-2-13-0'], ['1905.07761-1-13-1', '1905.07761-2-13-1'], ['1905.07761-1-13-2', '1905.07761-2-13-2'], ['1905.07761-1-13-3', '1905.07761-2-13-3'], ['1905.07761-1-6-0', '1905.07761-2-6-0'], ['1905.07761-1-6-1', '1905.07761-2-6-1'], ['1905.07761-1-6-2', '1905.07761-2-6-2'], ['1905.07761-1-6-3', '1905.07761-2-6-3'], ['1905.07761-1-7-0', '1905.07761-2-7-0'], ['1905.07761-1-12-0', '1905.07761-2-12-0'], ['1905.07761-1-12-1', '1905.07761-2-12-1'], ['1905.07761-1-12-2', '1905.07761-2-12-2'], ['1905.07761-1-0-0', '1905.07761-2-0-0'], ['1905.07761-1-1-0', '1905.07761-2-1-0'], ['1905.07761-1-1-1', '1905.07761-2-1-1'], ['1905.07761-1-3-0', '1905.07761-2-3-0'], ['1905.07761-1-3-2', '1905.07761-2-3-2'], ['1905.07761-1-3-4', '1905.07761-2-3-4'], ['1905.07761-1-3-5', '1905.07761-2-3-5'], ['1905.07761-1-3-7', '1905.07761-2-3-8'], ['1905.07761-1-5-0', '1905.07761-2-5-0'], ['1905.07761-1-5-1', '1905.07761-2-5-1'], ['1905.07761-1-8-0', '1905.07761-2-8-0'], ['1905.07761-1-4-0', '1905.07761-2-4-0'], ['1905.07761-1-4-1', '1905.07761-2-4-1'], ['1905.07761-1-4-2', '1905.07761-2-4-2'], ['1905.07761-1-9-0', '1905.07761-2-9-0'], ['1905.07761-1-9-1', '1905.07761-2-9-1'], ['1905.07761-1-10-0', '1905.07761-2-10-0'], ['1905.07761-1-11-0', '1905.07761-2-11-0'], ['1905.07761-1-11-1', '1905.07761-2-11-1'], ['1905.07761-1-11-2', '1905.07761-2-11-2'], ['1905.07761-1-2-1', '1905.07761-2-2-1']]

[['1905.07761-1-1-2', '1905.07761-2-1-2'], ['1905.07761-1-3-1', '1905.07761-2-3-1'], ['1905.07761-1-3-3', '1905.07761-2-3-3'], ['1905.07761-1-3-6', '1905.07761-2-3-7'], ['1905.07761-1-2-0', '1905.07761-2-2-0']]

[]

[]

[]

['1905.07761-1-8-1', '1905.07761-2-8-1']

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

https://arxiv.org/abs/1905.07761

gr-qc-0312066

{'gr-qc-0312066-1-0-0': 'We discuss the evolution of the fluctuations in a symmetric [MATH]-exponential potential which provides a power-law expansion during inflation using both, the gauge invariant field [MATH] and the Sasaki-Mukhanov field.', 'gr-qc-0312066-1-1-0': 'Pacs numbers: 98.80.', 'gr-qc-0312066-1-1-1': 'Cq', 'gr-qc-0312066-1-2-0': '# Introduction and motivation', 'gr-qc-0312066-1-3-0': 'It is now widely accepted that dominant cause of structure in the Universe is a spatial perturbation.', 'gr-qc-0312066-1-3-1': 'This perturbation is present on cosmological scales a few Hubble times before these scales enter the horizon, at which stage it is time-independent with an almost flat spectrum.', 'gr-qc-0312066-1-3-2': 'One of the main objectives of theoretical cosmology is to understand its origin[CITATION].', 'gr-qc-0312066-1-3-3': 'The usual assumption is that the curvature perturbation originates during inflation of the slow-rolling inflaton field.', 'gr-qc-0312066-1-3-4': 'As cosmological scales leave the horizon, the quantum fluctuation is converted to a classical gaussian perturbation with an almost flat spectrum, generating inmediately the required curvature perturbation which is constant until the approach of horizon entry[CITATION].', 'gr-qc-0312066-1-3-5': 'This idea has the adventage that prediction for the spectrum is independent of what goes on between the end of inflation and horizon entry.', 'gr-qc-0312066-1-3-6': 'The spectrum depends only on the form of the potential and on the theory of gravity during inflation, provinding, therefore, a direct probe of conditions during this era.', 'gr-qc-0312066-1-4-0': 'Stochastic inflation has played an important role in inflationary cosmology in the last two decades.', 'gr-qc-0312066-1-4-1': 'It proposes to describe the dynamics of this quantum field on the basis of two pieces: the homogeneous and inhomogeneous components[CITATION].', 'gr-qc-0312066-1-4-2': 'Usually the homogeneous one is interpreted as a classical field [MATH] that arises from the vacuum expectation value of the quantum field.', 'gr-qc-0312066-1-4-3': 'The inhomogeneous component [MATH] are the quantum fluctuations.', 'gr-qc-0312066-1-4-4': 'The field that take into account only the modes with wavelengths larger than the now observable universe is called coarse-grained field and its dynamics is described by a second order stochastic equation[CITATION].', 'gr-qc-0312066-1-4-5': 'Since these perturbations are classical on super Hubble scales, in this sector one can make a standard stochastic treatment for the coarse-grained matter field[CITATION].', 'gr-qc-0312066-1-4-6': 'The IR sector is very important because the spatially inhomogeneities in super Hubble inflationary scales would explain the present day observed matter structure in the universe.', 'gr-qc-0312066-1-5-0': 'In this work we consider gauge-invariant fluctuations of the metric in the early inflationary universe[CITATION].', 'gr-qc-0312066-1-5-1': 'Metric fluctuations are here considered in the framework of the linear perturbative corrections.', 'gr-qc-0312066-1-5-2': 'The scalar metric perturbations are spin-zero projections of the graviton, which only exists in nonvacuum cosmologies.', 'gr-qc-0312066-1-5-3': 'The issue of gauge invariance becomes critical when we attempt to analyze how the scalar metric perturbations produced in the early universe influence a background globally flat isotropic and homogeneous universe.', 'gr-qc-0312066-1-5-4': 'This allows us to formulate the problem of the amplitude for the scalar metric perturbations on the evolution of the background Friedmann-Robertson-Walker (FRW) universe in a coordinate-independent manner at every moment in time.', 'gr-qc-0312066-1-5-5': 'On the other hand, the Sasaki-Mukhanov (SM) field takes into account both, metric and inflaton fluctuations[CITATION].', 'gr-qc-0312066-1-5-6': 'One of the aims of this work is the study of the evolution of the SM field during inflation to make a comparison with gauge-invariant metric fluctuations.', 'gr-qc-0312066-1-6-0': '# Fluctuations', 'gr-qc-0312066-1-7-0': 'Matter field fluctuations are responsible for metric fluctuations around the background FRW metric.', 'gr-qc-0312066-1-7-1': 'When these metric fluctuations do not depend on the gauge, the perturbed globally flat isotropic and homogeneous universe is described by[CITATION] [EQUATION] where [MATH] is the scale factor of the universe and ([MATH], [MATH]) are the gauge-invariant perturbations of the metric.', 'gr-qc-0312066-1-7-2': 'In the particular case where the tensor [MATH] is diagonal, one obtains: [MATH][CITATION].', 'gr-qc-0312066-1-7-3': 'We consider a semiclassical expansion for the inflaton field [MATH][CITATION], with expectation values [MATH] and [MATH].', 'gr-qc-0312066-1-7-4': 'Here, [MATH] is the vacuum state.', 'gr-qc-0312066-1-7-5': 'Due to [MATH], the expectation value of the metric ([REF]) gives the background metric that describes a flat FRW spacetime: [MATH].', 'gr-qc-0312066-1-8-0': 'After linearizing the Einstein equations in terms of [MATH] and [MATH], one obtains [EQUATION] where [MATH], [MATH] is the scale factor of the universe and the prime denotes the derivative with respect to [MATH].', 'gr-qc-0312066-1-8-1': 'The dynamics of [MATH] is given by the equations [EQUATION] and [MATH] is the Hubble parameter.', 'gr-qc-0312066-1-8-2': 'Furthermore, the scalar potential can be written in terms of the Hubble parameter [EQUATION]', 'gr-qc-0312066-1-8-3': 'The equation ([REF]) can be simplified by introducing the field [MATH] [EQUATION]', 'gr-qc-0312066-1-8-4': 'This field can be expanded in terms of the modes [MATH] [EQUATION] where [MATH] and [MATH] are the annihilation and creation operators that complies with the commutation relations [EQUATION]', 'gr-qc-0312066-1-8-5': 'The equation for the modes [MATH] is [EQUATION] where [MATH] is the squared time dependent frequency and [MATH] separates the infrared and ultraviolet sectors, and is given by [EQUATION]', 'gr-qc-0312066-1-8-6': 'Since the field [MATH] satisfy a Klein-Gordon like equation on a FRW background metric [MATH], also satisfy the commutation relationship [EQUATION]', 'gr-qc-0312066-1-8-7': 'This implies that the modes [MATH] are renormalized by the expression [EQUATION]', 'gr-qc-0312066-1-9-0': '## Particular solutions', 'gr-qc-0312066-1-10-0': 'If the inflaton field oscillates around the minimum of the potential at the end of inflation the particular solutions when [MATH] and [MATH] are very important.', 'gr-qc-0312066-1-11-0': 'On the points [MATH] we obtain that [MATH].', 'gr-qc-0312066-1-11-1': 'However, the solutions for [MATH] are nonzero [EQUATION] where [MATH] is the initial amplitude for [MATH], for each wavenumber [MATH].', 'gr-qc-0312066-1-11-2': 'This means that the amplitude of each mode decreases exponentially with time.', 'gr-qc-0312066-1-12-0': 'Other interesting particular solution is located at the points [MATH], when the field is at the minumum of the potential.', 'gr-qc-0312066-1-12-1': 'In these points the equation ([REF]) adopts the form [EQUATION] where [MATH].', 'gr-qc-0312066-1-13-0': '## The Sasaki-Mukhanov field', 'gr-qc-0312066-1-14-0': 'A manner to study both, metric and inflaton fluctuations, can be made by means of the SM field[CITATION]: [MATH].', 'gr-qc-0312066-1-14-1': 'The modes of this field obeys the following equation [EQUATION] where the modes [MATH] complies with the renormalization condition [EQUATION] so that [MATH].', 'gr-qc-0312066-1-15-0': '## Power spectrum', 'gr-qc-0312066-1-16-0': 'One can estimate the power spectrum of the fluctuations for the fields [MATH] and [MATH].', 'gr-qc-0312066-1-16-1': 'The spectrum of the fluctuations for [MATH] are [EQUATION] whilst the power spectrum for the SM field is the same that of the inflation field [EQUATION]', 'gr-qc-0312066-1-16-2': 'It is well known from experimental data[CITATION] that the universe has a scale invariant power spectrum on cosmological scales.', 'gr-qc-0312066-1-17-0': '# An Example: Symmetric exponential [MATH]-potential: power-law inflation', 'gr-qc-0312066-1-18-0': 'As a first example we consider a scalar potential given by [MATH], where [MATH] gives the relationship between [MATH] and the power of the expansion [MATH].', 'gr-qc-0312066-1-18-1': 'This potential is related to a scale factor that evolves as [MATH] (with constant power [MATH]), which corresponds to a Hubble parameter [MATH], which can be written in terms of the scalar field [EQUATION] where [MATH] and [MATH] is the value of the Hubble parameter at the end of inflation.', 'gr-qc-0312066-1-18-2': 'The temporal evolution for [MATH] is given by [EQUATION] where [MATH].', 'gr-qc-0312066-1-18-3': 'Since [MATH] and [MATH] (we assume [MATH] for [MATH] positive and negative, respectively), the equation that describes the evolution for [MATH] results [EQUATION]', 'gr-qc-0312066-1-18-4': 'After make the transformation [MATH], we obtain the differential equation for [MATH] [EQUATION]', 'gr-qc-0312066-1-18-5': 'The general solution for the modes [MATH] is [EQUATION] where ([MATH],[MATH]) are constants, ([MATH], [MATH]) are the Hankel functions of (first, second) kind with [MATH] and [MATH].', 'gr-qc-0312066-1-18-6': 'Using the renormalization condition [MATH], we obtain the Bunch-Davis vacuum[CITATION] solution ([MATH],[MATH]) [EQUATION]', 'gr-qc-0312066-1-18-7': 'In the UV sector the function [MATH] adopts the asymptotic expression (i.e., for [MATH]) [EQUATION] whilst on the IR sector (i.e., for [MATH]) it tends asymptotically to [EQUATION]', 'gr-qc-0312066-1-18-8': 'The [MATH]-squared field fluctuations on the IR sector are [MATH], and becomes [EQUATION] where [MATH] is a dimensionless constant, [MATH] at the moment [MATH] when the horizon entry and [MATH] is the Planckian wavenumber (i.e., the scale we choose as a cut-off of all the spectrum).', 'gr-qc-0312066-1-18-9': 'The power spectrum on the IR sector is [MATH].', 'gr-qc-0312066-1-18-10': 'Note that [MATH] increases for [MATH], so that to the IR squared [MATH]-fluctuations remain almost constant on cosmological scales we need [MATH].', 'gr-qc-0312066-1-18-11': 'We find that a power close to [MATH] give us a scale invariant power spectrum (i.e., with [MATH] for [MATH].', 'gr-qc-0312066-1-18-12': 'Furthermore, density fluctuations for matter energy density are given by [MATH], so that [MATH].', 'gr-qc-0312066-1-19-0': 'On the other hand, in the UV sector these fluctuations are given by [EQUATION]', 'gr-qc-0312066-1-19-1': 'The power spectrum in this sector go as [MATH].', 'gr-qc-0312066-1-19-2': 'We observe from eq. ([REF]) that [MATH] increases during inflation for [MATH].', 'gr-qc-0312066-1-19-3': 'From the results ([REF]) and ([REF]) we obtain that [MATH], because a power-law [MATH] could give a very inhomogeneous universe on cosmological scales.', 'gr-qc-0312066-1-19-4': 'Since [MATH], we obtain the condition [EQUATION]', 'gr-qc-0312066-1-19-5': 'If [MATH] is the initial value of the Hubble parameter), inflation should ends at [MATH], where [EQUATION]', 'gr-qc-0312066-1-19-6': 'For example, for [MATH] and [MATH], we obtain [MATH].', 'gr-qc-0312066-1-20-0': 'Now we can study the evolution of the SM field fluctuations [MATH].', 'gr-qc-0312066-1-20-1': 'The eq. ([REF]) written explicitely for the model we are studying is [EQUATION]', 'gr-qc-0312066-1-20-2': 'Note that the last term inside the brackets in eq. ([REF]) becomes null.', 'gr-qc-0312066-1-20-3': 'The general solution of eq. ([REF]) can be written in terms of the Hankel functions [EQUATION] where [MATH].', 'gr-qc-0312066-1-20-4': 'If we adopt the Buch-Davis vacuum[CITATION]: [MATH], [MATH], we obtain [EQUATION]', 'gr-qc-0312066-1-20-5': 'The power spectrums in the extreme sectors of the spectrum go as [EQUATION]', 'gr-qc-0312066-1-20-6': 'Note that [MATH] is scale invariant for [MATH], which corresponds with [MATH].', 'gr-qc-0312066-1-20-7': 'The squared [MATH]-fluctuations on both, the UV and IR sectors are [EQUATION]', 'gr-qc-0312066-1-20-8': 'Since [MATH] during inflation, from the eq. ([REF]) we obtain the condition [EQUATION]', 'gr-qc-0312066-1-20-9': 'From this condition we obtain the time for which inflation ends.', 'gr-qc-0312066-1-20-10': 'Hence, one obtains [EQUATION] where, since we require [MATH], must to holds [MATH].', 'gr-qc-0312066-1-20-11': 'For example, for [MATH] and [MATH], one obtains [MATH], which is incompatible with the value obtained from the evolution for [MATH].', 'gr-qc-0312066-1-20-12': 'On the other hand [MATH] decreases as [MATH] independently on the value of the power [MATH].', 'gr-qc-0312066-1-21-0': '# Final Comments', 'gr-qc-0312066-1-22-0': 'In this paper we have studied the evolution of the fluctuations in a symmetric [MATH]-exponential potential which provides a power-law expansion using both, the gauge invariant field [MATH] and the Sasaki-Mukhanov field.', 'gr-qc-0312066-1-22-1': 'This last takes into account simultaneously, the inflaton and metric fluctuations.', 'gr-qc-0312066-1-22-2': 'The results obtained from the evolution of [MATH] and [MATH] are different in both treatments.', 'gr-qc-0312066-1-22-3': 'The reason can be explained from the fact that the field [MATH] is not gauge-invariant and hence do not describes correctly the fluctuations for [MATH] and [MATH].', 'gr-qc-0312066-1-22-4': 'The fluctuations are well described by the field [MATH] which is gauge invariant and predicts a scale invariant power spectrum on the IR sector for [MATH].', 'gr-qc-0312066-1-22-5': 'Note that we have not considered back-reaction effects which are related to a second-order metric tensor fluctuations.', 'gr-qc-0312066-1-22-6': 'This topic was considered by Abramo and Nambu, who investigated a renormalization-group method for an inflationary universe[CITATION].', 'gr-qc-0312066-1-22-7': 'A different approach to describe the metric fluctuations was considered more recently by Lyth and Wands[CITATION], who suggested that curvature perturbation could be generated by a light scalar field named curvaton.'}

{'gr-qc-0312066-2-0-0': 'We discuss the evolution of the fluctuations in a symmetric [MATH]-exponential potential which provides a power-law expansion during inflation using both, the gauge invariant field [MATH] and the Sasaki-Mukhanov field.', 'gr-qc-0312066-2-1-0': 'Pacs numbers: 98.80.', 'gr-qc-0312066-2-1-1': 'Cq', 'gr-qc-0312066-2-2-0': '# Introduction and motivation', 'gr-qc-0312066-2-3-0': 'It is now widely accepted that dominant cause of structure in the Universe is a spatial perturbation.', 'gr-qc-0312066-2-3-1': 'This perturbation is present on cosmological scales a few Hubble times before these scales enter the horizon, at which stage it is time-independent with an almost flat spectrum.', 'gr-qc-0312066-2-3-2': 'One of the main objectives of theoretical cosmology is to understand its origin[CITATION].', 'gr-qc-0312066-2-3-3': 'The usual assumption is that the curvature perturbation originates during inflation of the slow-rolling inflaton field.', 'gr-qc-0312066-2-3-4': 'As cosmological scales leave the horizon, the quantum fluctuation is converted to a classical gaussian perturbation with an almost flat spectrum, generating inmediately the required curvature perturbation which is constant until the approach of horizon entry[CITATION].', 'gr-qc-0312066-2-3-5': 'This idea has the adventage that prediction for the spectrum is independent of what goes on between the end of inflation and horizon entry.', 'gr-qc-0312066-2-3-6': 'The spectrum depends only on the form of the potential and on the theory of gravity during inflation, provinding, therefore, a direct probe of conditions during this era.', 'gr-qc-0312066-2-4-0': 'Stochastic inflation has played an important role in inflationary cosmology in the last two decades.', 'gr-qc-0312066-2-4-1': 'It proposes to describe the dynamics of this quantum field on the basis of two pieces: the homogeneous and inhomogeneous components[CITATION].', 'gr-qc-0312066-2-4-2': 'Usually the homogeneous one is interpreted as a classical field [MATH] that arises from the vacuum expectation value of the quantum field.', 'gr-qc-0312066-2-4-3': 'The inhomogeneous component [MATH] are the quantum fluctuations.', 'gr-qc-0312066-2-4-4': 'The field that take into account only the modes with wavelengths larger than the now observable universe is called coarse-grained field and its dynamics is described by a second order stochastic equation[CITATION].', 'gr-qc-0312066-2-4-5': 'Since these perturbations are classical on super Hubble scales, in this sector one can make a standard stochastic treatment for the coarse-grained matter field[CITATION].', 'gr-qc-0312066-2-4-6': 'The IR sector is very important because the spatially inhomogeneities in super Hubble inflationary scales would explain the present day observed matter structure in the universe.', 'gr-qc-0312066-2-5-0': 'In this work we consider gauge-invariant fluctuations of the metric in the early inflationary universe[CITATION].', 'gr-qc-0312066-2-5-1': 'Metric fluctuations are here considered in the framework of the linear perturbative corrections.', 'gr-qc-0312066-2-5-2': 'The scalar metric perturbations are spin-zero projections of the graviton, which only exists in nonvacuum cosmologies.', 'gr-qc-0312066-2-5-3': 'The issue of gauge invariance becomes critical when we attempt to analyze how the scalar metric perturbations produced in the early universe influence a background globally flat isotropic and homogeneous universe.', 'gr-qc-0312066-2-5-4': 'This allows us to formulate the problem of the amplitude for the scalar metric perturbations on the evolution of the background Friedmann-Robertson-Walker (FRW) universe in a coordinate-independent manner at every moment in time.', 'gr-qc-0312066-2-5-5': 'On the other hand, the Sasaki-Mukhanov (SM) field takes into account both, metric and inflaton fluctuations[CITATION].', 'gr-qc-0312066-2-5-6': 'One of the aims of this work is the study of the evolution of the SM field during inflation to make a comparison with gauge-invariant metric fluctuations.', 'gr-qc-0312066-2-6-0': '# Fluctuations', 'gr-qc-0312066-2-7-0': 'Matter field fluctuations are responsible for metric fluctuations around the background FRW metric.', 'gr-qc-0312066-2-7-1': 'When these metric fluctuations do not depend on the gauge, the perturbed globally flat isotropic and homogeneous universe is described by[CITATION] [EQUATION] where [MATH] is the scale factor of the universe and ([MATH], [MATH]) are the gauge-invariant perturbations of the metric.', 'gr-qc-0312066-2-7-2': 'In the particular case where the tensor [MATH] is diagonal, one obtains: [MATH][CITATION].', 'gr-qc-0312066-2-7-3': 'We consider a semiclassical expansion for the inflaton field [MATH][CITATION], with expectation values [MATH] and [MATH].', 'gr-qc-0312066-2-7-4': 'Here, [MATH] is the vacuum state.', 'gr-qc-0312066-2-7-5': 'Due to [MATH], the expectation value of the metric ([REF]) gives the background metric that describes a flat FRW spacetime: [MATH].', 'gr-qc-0312066-2-8-0': 'After linearizing the Einstein equations in terms of [MATH] and [MATH], one obtains [EQUATION] where [MATH], [MATH] is the scale factor of the universe and the prime denotes the derivative with respect to [MATH].', 'gr-qc-0312066-2-8-1': 'The dynamics of [MATH] is given by the equations [EQUATION] and [MATH] is the Hubble parameter.', 'gr-qc-0312066-2-8-2': 'Furthermore, the scalar potential can be written in terms of the Hubble parameter [EQUATION]', 'gr-qc-0312066-2-8-3': 'The equation ([REF]) can be simplified by introducing the field [MATH] [EQUATION]', 'gr-qc-0312066-2-8-4': 'This field can be expanded in terms of the modes [MATH] [EQUATION] where [MATH] and [MATH] are the annihilation and creation operators that complies with the commutation relations [EQUATION]', 'gr-qc-0312066-2-8-5': 'The equation for the modes [MATH] is [EQUATION] where [MATH] is the squared time dependent frequency and [MATH] separates the infrared and ultraviolet sectors, and is given by [EQUATION]', 'gr-qc-0312066-2-8-6': 'Since the field [MATH] satisfy a Klein-Gordon like equation on a FRW background metric [MATH], also satisfy the commutation relationship [EQUATION]', 'gr-qc-0312066-2-8-7': 'This implies that the modes [MATH] are renormalized by the expression [EQUATION]', 'gr-qc-0312066-2-9-0': '## Particular solutions', 'gr-qc-0312066-2-10-0': 'If the inflaton field oscillates around the minimum of the potential at the end of inflation the particular solutions when [MATH] and [MATH] are very important.', 'gr-qc-0312066-2-11-0': 'On the points [MATH] we obtain that [MATH].', 'gr-qc-0312066-2-11-1': 'However, the solutions for [MATH] are nonzero [EQUATION] where [MATH] is the initial amplitude for [MATH], for each wavenumber [MATH].', 'gr-qc-0312066-2-11-2': 'This means that the amplitude of each mode decreases exponentially with time.', 'gr-qc-0312066-2-12-0': 'Other interesting particular solution is located at the points [MATH], when the field is at the minumum of the potential.', 'gr-qc-0312066-2-12-1': 'In these points the equation ([REF]) adopts the form [EQUATION] where [MATH].', 'gr-qc-0312066-2-13-0': '## The Sasaki-Mukhanov field', 'gr-qc-0312066-2-14-0': 'A manner to study both, metric and inflaton fluctuations, can be made by means of the SM field[CITATION]: [MATH].', 'gr-qc-0312066-2-14-1': 'The modes of this field obeys the following equation [EQUATION] where the modes [MATH] complies with the renormalization condition [EQUATION] so that [MATH].', 'gr-qc-0312066-2-15-0': '## Power spectrum', 'gr-qc-0312066-2-16-0': 'One can estimate the power spectrum of the fluctuations for the fields [MATH] and [MATH].', 'gr-qc-0312066-2-16-1': 'The spectrum of the fluctuations for [MATH] are [EQUATION] whilst the power spectrum for the SM field is the same that of the inflation field [EQUATION]', 'gr-qc-0312066-2-16-2': 'It is well known from experimental data[CITATION] that the universe has a scale invariant power spectrum on cosmological scales.', 'gr-qc-0312066-2-17-0': '# An Example: Symmetric exponential [MATH]-potential: power-law inflation', 'gr-qc-0312066-2-18-0': 'As a first example we consider a scalar potential given by [MATH], where [MATH] gives the relationship between [MATH] and the power of the expansion [MATH].', 'gr-qc-0312066-2-18-1': 'This potential is related to a scale factor that evolves as [MATH] (with constant power [MATH]), which corresponds to a Hubble parameter [MATH], which can be written in terms of the scalar field [EQUATION] where [MATH] and [MATH] is the value of the Hubble parameter at the end of inflation.', 'gr-qc-0312066-2-18-2': 'The temporal evolution for [MATH] is given by [EQUATION] where [MATH].', 'gr-qc-0312066-2-18-3': 'Since [MATH] and [MATH] (we assume [MATH] for [MATH] positive and negative, respectively), the equation that describes the evolution for [MATH] results [EQUATION]', 'gr-qc-0312066-2-18-4': 'After make the transformation [MATH], we obtain the differential equation for [MATH] [EQUATION]', 'gr-qc-0312066-2-18-5': 'The general solution for the modes [MATH] is [EQUATION] where ([MATH],[MATH]) are constants, ([MATH], [MATH]) are the Hankel functions of (first, second) kind with [MATH] and [MATH].', 'gr-qc-0312066-2-18-6': 'Using the renormalization condition [MATH], we obtain the Bunch-Davis vacuum[CITATION] solution ([MATH],[MATH]) [EQUATION]', 'gr-qc-0312066-2-18-7': 'In the UV sector the function [MATH] adopts the asymptotic expression (i.e., for [MATH]) [EQUATION] whilst on the IR sector (i.e., for [MATH]) it tends asymptotically to [EQUATION]', 'gr-qc-0312066-2-18-8': 'The [MATH]-squared field fluctuations on the IR sector are [MATH], and becomes [EQUATION] where [MATH] is a dimensionless constant, [MATH] at the moment [MATH] when the horizon entry and [MATH] is the Planckian wavenumber (i.e., the scale we choose as a cut-off of all the spectrum).', 'gr-qc-0312066-2-18-9': 'The power spectrum on the IR sector is [MATH].', 'gr-qc-0312066-2-18-10': 'Note that [MATH] increases for [MATH], so that to the IR squared [MATH]-fluctuations remain almost constant on cosmological scales we need [MATH].', 'gr-qc-0312066-2-18-11': 'We find that a power close to [MATH] give us a scale invariant power spectrum (i.e., with [MATH] for [MATH].', 'gr-qc-0312066-2-18-12': 'Furthermore, density fluctuations for matter energy density are given by [MATH], so that [MATH].', 'gr-qc-0312066-2-19-0': 'On the other hand, in the UV sector these fluctuations are given by [EQUATION]', 'gr-qc-0312066-2-19-1': 'The power spectrum in this sector go as [MATH].', 'gr-qc-0312066-2-19-2': 'We observe from eq. ([REF]) that [MATH] increases during inflation for [MATH].', 'gr-qc-0312066-2-19-3': 'From the results ([REF]) and ([REF]) we obtain that [MATH], because a power-law [MATH] could give a very inhomogeneous universe on cosmological scales.', 'gr-qc-0312066-2-19-4': 'Since [MATH], we obtain the condition [EQUATION]', 'gr-qc-0312066-2-19-5': 'If [MATH] is the initial value of the Hubble parameter), inflation should ends at [MATH], where [EQUATION]', 'gr-qc-0312066-2-19-6': 'For example, for [MATH] and [MATH], we obtain [MATH].', 'gr-qc-0312066-2-20-0': 'Now we can study the evolution of the SM field fluctuations [MATH].', 'gr-qc-0312066-2-20-1': 'The eq. ([REF]) written explicitely for the model we are studying is [EQUATION]', 'gr-qc-0312066-2-20-2': 'Note that the last term inside the brackets in eq. ([REF]) becomes null.', 'gr-qc-0312066-2-20-3': 'The general solution of eq. ([REF]) can be written in terms of the Hankel functions [EQUATION] where [MATH].', 'gr-qc-0312066-2-20-4': 'If we adopt the Buch-Davis vacuum[CITATION]: [MATH], [MATH], we obtain [EQUATION]', 'gr-qc-0312066-2-20-5': 'The power spectrums in the extreme sectors of the spectrum go as [EQUATION]', 'gr-qc-0312066-2-20-6': 'Note that [MATH] is scale invariant for [MATH], which corresponds with [MATH].', 'gr-qc-0312066-2-20-7': 'The squared [MATH]-fluctuations on both, the UV and IR sectors are [EQUATION]', 'gr-qc-0312066-2-20-8': 'Since [MATH] during inflation, from the eq. ([REF]) we obtain the condition [EQUATION]', 'gr-qc-0312066-2-20-9': 'From this condition we obtain the time for which inflation ends.', 'gr-qc-0312066-2-20-10': 'Hence, one obtains [EQUATION] where, since we require [MATH], must to holds [MATH].', 'gr-qc-0312066-2-20-11': 'For example, for [MATH] and [MATH], one obtains [MATH], which is incompatible with the value obtained from the evolution for [MATH].', 'gr-qc-0312066-2-20-12': 'On the other hand [MATH] decreases as [MATH] independently on the value of the power [MATH].', 'gr-qc-0312066-2-21-0': '# Final Comments', 'gr-qc-0312066-2-22-0': 'In this paper we have studied the evolution of the fluctuations in a symmetric [MATH]-exponential potential which provides a power-law expansion using both, the gauge invariant field [MATH] and the Sasaki-Mukhanov field.', 'gr-qc-0312066-2-22-1': 'This last takes into account simultaneously, the inflaton and metric fluctuations.', 'gr-qc-0312066-2-22-2': 'The results obtained from the evolution of [MATH] and [MATH] are different in both treatments.', 'gr-qc-0312066-2-22-3': 'The reason can be explained from the fact that the field [MATH] is not gauge-invariant and hence do not describes correctly the fluctuations for [MATH] and [MATH].', 'gr-qc-0312066-2-22-4': 'The fluctuations are well described by the field [MATH] which is gauge invariant and predicts a scale invariant power spectrum on the IR sector for [MATH].', 'gr-qc-0312066-2-22-5': 'Note that we have not considered back-reaction effects which are related to a second-order metric tensor fluctuations.', 'gr-qc-0312066-2-22-6': 'This topic was considered by Abramo and Nambu, who investigated a renormalization-group method for an inflationary universe[CITATION].', 'gr-qc-0312066-2-22-7': 'A different approach to describe the metric fluctuations was considered more recently by Lyth and Wands[CITATION] (see also[CITATION]), who suggested that curvature perturbation could be generated by a light scalar field named curvaton.'}

[['gr-qc-0312066-1-20-0', 'gr-qc-0312066-2-20-0'], ['gr-qc-0312066-1-20-1', 'gr-qc-0312066-2-20-1'], ['gr-qc-0312066-1-20-2', 'gr-qc-0312066-2-20-2'], ['gr-qc-0312066-1-20-3', 'gr-qc-0312066-2-20-3'], ['gr-qc-0312066-1-20-4', 'gr-qc-0312066-2-20-4'], ['gr-qc-0312066-1-20-5', 'gr-qc-0312066-2-20-5'], ['gr-qc-0312066-1-20-6', 'gr-qc-0312066-2-20-6'], ['gr-qc-0312066-1-20-7', 'gr-qc-0312066-2-20-7'], ['gr-qc-0312066-1-20-8', 'gr-qc-0312066-2-20-8'], ['gr-qc-0312066-1-20-9', 'gr-qc-0312066-2-20-9'], ['gr-qc-0312066-1-20-10', 'gr-qc-0312066-2-20-10'], ['gr-qc-0312066-1-20-11', 'gr-qc-0312066-2-20-11'], ['gr-qc-0312066-1-20-12', 'gr-qc-0312066-2-20-12'], ['gr-qc-0312066-1-8-0', 'gr-qc-0312066-2-8-0'], ['gr-qc-0312066-1-8-1', 'gr-qc-0312066-2-8-1'], ['gr-qc-0312066-1-8-2', 'gr-qc-0312066-2-8-2'], ['gr-qc-0312066-1-8-3', 'gr-qc-0312066-2-8-3'], ['gr-qc-0312066-1-8-4', 'gr-qc-0312066-2-8-4'], ['gr-qc-0312066-1-8-5', 'gr-qc-0312066-2-8-5'], ['gr-qc-0312066-1-8-6', 'gr-qc-0312066-2-8-6'], ['gr-qc-0312066-1-8-7', 'gr-qc-0312066-2-8-7'], ['gr-qc-0312066-1-18-0', 'gr-qc-0312066-2-18-0'], ['gr-qc-0312066-1-18-1', 'gr-qc-0312066-2-18-1'], ['gr-qc-0312066-1-18-2', 'gr-qc-0312066-2-18-2'], ['gr-qc-0312066-1-18-3', 'gr-qc-0312066-2-18-3'], ['gr-qc-0312066-1-18-4', 'gr-qc-0312066-2-18-4'], ['gr-qc-0312066-1-18-5', 'gr-qc-0312066-2-18-5'], ['gr-qc-0312066-1-18-6', 'gr-qc-0312066-2-18-6'], ['gr-qc-0312066-1-18-7', 'gr-qc-0312066-2-18-7'], ['gr-qc-0312066-1-18-8', 'gr-qc-0312066-2-18-8'], ['gr-qc-0312066-1-18-9', 'gr-qc-0312066-2-18-9'], ['gr-qc-0312066-1-18-10', 'gr-qc-0312066-2-18-10'], ['gr-qc-0312066-1-18-11', 'gr-qc-0312066-2-18-11'], ['gr-qc-0312066-1-18-12', 'gr-qc-0312066-2-18-12'], ['gr-qc-0312066-1-11-0', 'gr-qc-0312066-2-11-0'], ['gr-qc-0312066-1-11-1', 'gr-qc-0312066-2-11-1'], ['gr-qc-0312066-1-11-2', 'gr-qc-0312066-2-11-2'], ['gr-qc-0312066-1-19-0', 'gr-qc-0312066-2-19-0'], ['gr-qc-0312066-1-19-1', 'gr-qc-0312066-2-19-1'], ['gr-qc-0312066-1-19-2', 'gr-qc-0312066-2-19-2'], ['gr-qc-0312066-1-19-3', 'gr-qc-0312066-2-19-3'], ['gr-qc-0312066-1-19-4', 'gr-qc-0312066-2-19-4'], ['gr-qc-0312066-1-19-5', 'gr-qc-0312066-2-19-5'], ['gr-qc-0312066-1-19-6', 'gr-qc-0312066-2-19-6'], ['gr-qc-0312066-1-7-0', 'gr-qc-0312066-2-7-0'], ['gr-qc-0312066-1-7-1', 'gr-qc-0312066-2-7-1'], ['gr-qc-0312066-1-7-2', 'gr-qc-0312066-2-7-2'], ['gr-qc-0312066-1-7-3', 'gr-qc-0312066-2-7-3'], ['gr-qc-0312066-1-7-4', 'gr-qc-0312066-2-7-4'], ['gr-qc-0312066-1-7-5', 'gr-qc-0312066-2-7-5'], ['gr-qc-0312066-1-22-0', 'gr-qc-0312066-2-22-0'], ['gr-qc-0312066-1-22-1', 'gr-qc-0312066-2-22-1'], ['gr-qc-0312066-1-22-2', 'gr-qc-0312066-2-22-2'], ['gr-qc-0312066-1-22-3', 'gr-qc-0312066-2-22-3'], ['gr-qc-0312066-1-22-4', 'gr-qc-0312066-2-22-4'], ['gr-qc-0312066-1-22-5', 'gr-qc-0312066-2-22-5'], ['gr-qc-0312066-1-22-6', 'gr-qc-0312066-2-22-6'], ['gr-qc-0312066-1-14-0', 'gr-qc-0312066-2-14-0'], ['gr-qc-0312066-1-14-1', 'gr-qc-0312066-2-14-1'], ['gr-qc-0312066-1-12-0', 'gr-qc-0312066-2-12-0'], ['gr-qc-0312066-1-12-1', 'gr-qc-0312066-2-12-1'], ['gr-qc-0312066-1-3-0', 'gr-qc-0312066-2-3-0'], ['gr-qc-0312066-1-3-1', 'gr-qc-0312066-2-3-1'], ['gr-qc-0312066-1-3-2', 'gr-qc-0312066-2-3-2'], ['gr-qc-0312066-1-3-3', 'gr-qc-0312066-2-3-3'], ['gr-qc-0312066-1-3-4', 'gr-qc-0312066-2-3-4'], ['gr-qc-0312066-1-3-5', 'gr-qc-0312066-2-3-5'], ['gr-qc-0312066-1-3-6', 'gr-qc-0312066-2-3-6'], ['gr-qc-0312066-1-4-0', 'gr-qc-0312066-2-4-0'], ['gr-qc-0312066-1-4-1', 'gr-qc-0312066-2-4-1'], ['gr-qc-0312066-1-4-2', 'gr-qc-0312066-2-4-2'], ['gr-qc-0312066-1-4-3', 'gr-qc-0312066-2-4-3'], ['gr-qc-0312066-1-4-4', 'gr-qc-0312066-2-4-4'], ['gr-qc-0312066-1-4-5', 'gr-qc-0312066-2-4-5'], ['gr-qc-0312066-1-4-6', 'gr-qc-0312066-2-4-6'], ['gr-qc-0312066-1-0-0', 'gr-qc-0312066-2-0-0'], ['gr-qc-0312066-1-16-0', 'gr-qc-0312066-2-16-0'], ['gr-qc-0312066-1-16-1', 'gr-qc-0312066-2-16-1'], ['gr-qc-0312066-1-16-2', 'gr-qc-0312066-2-16-2'], ['gr-qc-0312066-1-10-0', 'gr-qc-0312066-2-10-0'], ['gr-qc-0312066-1-5-0', 'gr-qc-0312066-2-5-0'], ['gr-qc-0312066-1-5-1', 'gr-qc-0312066-2-5-1'], ['gr-qc-0312066-1-5-2', 'gr-qc-0312066-2-5-2'], ['gr-qc-0312066-1-5-3', 'gr-qc-0312066-2-5-3'], ['gr-qc-0312066-1-5-4', 'gr-qc-0312066-2-5-4'], ['gr-qc-0312066-1-5-5', 'gr-qc-0312066-2-5-5'], ['gr-qc-0312066-1-5-6', 'gr-qc-0312066-2-5-6'], ['gr-qc-0312066-1-22-7', 'gr-qc-0312066-2-22-7']]

[['gr-qc-0312066-1-20-0', 'gr-qc-0312066-2-20-0'], ['gr-qc-0312066-1-20-1', 'gr-qc-0312066-2-20-1'], ['gr-qc-0312066-1-20-2', 'gr-qc-0312066-2-20-2'], ['gr-qc-0312066-1-20-3', 'gr-qc-0312066-2-20-3'], ['gr-qc-0312066-1-20-4', 'gr-qc-0312066-2-20-4'], ['gr-qc-0312066-1-20-5', 'gr-qc-0312066-2-20-5'], ['gr-qc-0312066-1-20-6', 'gr-qc-0312066-2-20-6'], ['gr-qc-0312066-1-20-7', 'gr-qc-0312066-2-20-7'], ['gr-qc-0312066-1-20-8', 'gr-qc-0312066-2-20-8'], ['gr-qc-0312066-1-20-9', 'gr-qc-0312066-2-20-9'], ['gr-qc-0312066-1-20-10', 'gr-qc-0312066-2-20-10'], ['gr-qc-0312066-1-20-11', 'gr-qc-0312066-2-20-11'], ['gr-qc-0312066-1-20-12', 'gr-qc-0312066-2-20-12'], ['gr-qc-0312066-1-8-0', 'gr-qc-0312066-2-8-0'], ['gr-qc-0312066-1-8-1', 'gr-qc-0312066-2-8-1'], ['gr-qc-0312066-1-8-2', 'gr-qc-0312066-2-8-2'], ['gr-qc-0312066-1-8-3', 'gr-qc-0312066-2-8-3'], ['gr-qc-0312066-1-8-4', 'gr-qc-0312066-2-8-4'], ['gr-qc-0312066-1-8-5', 'gr-qc-0312066-2-8-5'], ['gr-qc-0312066-1-8-6', 'gr-qc-0312066-2-8-6'], ['gr-qc-0312066-1-8-7', 'gr-qc-0312066-2-8-7'], ['gr-qc-0312066-1-18-0', 'gr-qc-0312066-2-18-0'], ['gr-qc-0312066-1-18-1', 'gr-qc-0312066-2-18-1'], ['gr-qc-0312066-1-18-2', 'gr-qc-0312066-2-18-2'], ['gr-qc-0312066-1-18-3', 'gr-qc-0312066-2-18-3'], ['gr-qc-0312066-1-18-4', 'gr-qc-0312066-2-18-4'], ['gr-qc-0312066-1-18-5', 'gr-qc-0312066-2-18-5'], ['gr-qc-0312066-1-18-6', 'gr-qc-0312066-2-18-6'], ['gr-qc-0312066-1-18-7', 'gr-qc-0312066-2-18-7'], ['gr-qc-0312066-1-18-8', 'gr-qc-0312066-2-18-8'], ['gr-qc-0312066-1-18-9', 'gr-qc-0312066-2-18-9'], ['gr-qc-0312066-1-18-10', 'gr-qc-0312066-2-18-10'], ['gr-qc-0312066-1-18-11', 'gr-qc-0312066-2-18-11'], ['gr-qc-0312066-1-18-12', 'gr-qc-0312066-2-18-12'], ['gr-qc-0312066-1-11-0', 'gr-qc-0312066-2-11-0'], ['gr-qc-0312066-1-11-1', 'gr-qc-0312066-2-11-1'], ['gr-qc-0312066-1-11-2', 'gr-qc-0312066-2-11-2'], ['gr-qc-0312066-1-19-0', 'gr-qc-0312066-2-19-0'], ['gr-qc-0312066-1-19-1', 'gr-qc-0312066-2-19-1'], ['gr-qc-0312066-1-19-2', 'gr-qc-0312066-2-19-2'], ['gr-qc-0312066-1-19-3', 'gr-qc-0312066-2-19-3'], ['gr-qc-0312066-1-19-4', 'gr-qc-0312066-2-19-4'], ['gr-qc-0312066-1-19-5', 'gr-qc-0312066-2-19-5'], ['gr-qc-0312066-1-19-6', 'gr-qc-0312066-2-19-6'], ['gr-qc-0312066-1-7-0', 'gr-qc-0312066-2-7-0'], ['gr-qc-0312066-1-7-1', 'gr-qc-0312066-2-7-1'], ['gr-qc-0312066-1-7-2', 'gr-qc-0312066-2-7-2'], ['gr-qc-0312066-1-7-3', 'gr-qc-0312066-2-7-3'], ['gr-qc-0312066-1-7-4', 'gr-qc-0312066-2-7-4'], ['gr-qc-0312066-1-7-5', 'gr-qc-0312066-2-7-5'], ['gr-qc-0312066-1-22-0', 'gr-qc-0312066-2-22-0'], ['gr-qc-0312066-1-22-1', 'gr-qc-0312066-2-22-1'], ['gr-qc-0312066-1-22-2', 'gr-qc-0312066-2-22-2'], ['gr-qc-0312066-1-22-3', 'gr-qc-0312066-2-22-3'], ['gr-qc-0312066-1-22-4', 'gr-qc-0312066-2-22-4'], ['gr-qc-0312066-1-22-5', 'gr-qc-0312066-2-22-5'], ['gr-qc-0312066-1-22-6', 'gr-qc-0312066-2-22-6'], ['gr-qc-0312066-1-14-0', 'gr-qc-0312066-2-14-0'], ['gr-qc-0312066-1-14-1', 'gr-qc-0312066-2-14-1'], ['gr-qc-0312066-1-12-0', 'gr-qc-0312066-2-12-0'], ['gr-qc-0312066-1-12-1', 'gr-qc-0312066-2-12-1'], ['gr-qc-0312066-1-3-0', 'gr-qc-0312066-2-3-0'], ['gr-qc-0312066-1-3-1', 'gr-qc-0312066-2-3-1'], ['gr-qc-0312066-1-3-2', 'gr-qc-0312066-2-3-2'], ['gr-qc-0312066-1-3-3', 'gr-qc-0312066-2-3-3'], ['gr-qc-0312066-1-3-4', 'gr-qc-0312066-2-3-4'], ['gr-qc-0312066-1-3-5', 'gr-qc-0312066-2-3-5'], ['gr-qc-0312066-1-3-6', 'gr-qc-0312066-2-3-6'], ['gr-qc-0312066-1-4-0', 'gr-qc-0312066-2-4-0'], ['gr-qc-0312066-1-4-1', 'gr-qc-0312066-2-4-1'], ['gr-qc-0312066-1-4-2', 'gr-qc-0312066-2-4-2'], ['gr-qc-0312066-1-4-3', 'gr-qc-0312066-2-4-3'], ['gr-qc-0312066-1-4-4', 'gr-qc-0312066-2-4-4'], ['gr-qc-0312066-1-4-5', 'gr-qc-0312066-2-4-5'], ['gr-qc-0312066-1-4-6', 'gr-qc-0312066-2-4-6'], ['gr-qc-0312066-1-0-0', 'gr-qc-0312066-2-0-0'], ['gr-qc-0312066-1-16-0', 'gr-qc-0312066-2-16-0'], ['gr-qc-0312066-1-16-1', 'gr-qc-0312066-2-16-1'], ['gr-qc-0312066-1-16-2', 'gr-qc-0312066-2-16-2'], ['gr-qc-0312066-1-10-0', 'gr-qc-0312066-2-10-0'], ['gr-qc-0312066-1-5-0', 'gr-qc-0312066-2-5-0'], ['gr-qc-0312066-1-5-1', 'gr-qc-0312066-2-5-1'], ['gr-qc-0312066-1-5-2', 'gr-qc-0312066-2-5-2'], ['gr-qc-0312066-1-5-3', 'gr-qc-0312066-2-5-3'], ['gr-qc-0312066-1-5-4', 'gr-qc-0312066-2-5-4'], ['gr-qc-0312066-1-5-5', 'gr-qc-0312066-2-5-5'], ['gr-qc-0312066-1-5-6', 'gr-qc-0312066-2-5-6']]

[['gr-qc-0312066-1-22-7', 'gr-qc-0312066-2-22-7']]

[]

[]

[]

['gr-qc-0312066-1-1-0', 'gr-qc-0312066-1-1-1', 'gr-qc-0312066-2-1-0', 'gr-qc-0312066-2-1-1']

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

https://arxiv.org/abs/gr-qc/0312066

0906.0020

{'0906.0020-1-0-0': 'We present HST WFPC2/PC images and KPNO 4-m longslit spectroscopy of the QSO SDSS J153636.22+044127.0, which we advanced as a candidate binary supermassive black hole.', '0906.0020-1-0-1': 'The images reveal a close companion coincident with the radio source identified by [CITATION].', '0906.0020-1-0-2': 'It appears to be consistent with a [MATH] elliptical galaxy, if it is at the QSO redshift.', '0906.0020-1-0-3': 'The spectroscopy, however, shows no spatial offset of the red or blue Balmer line subcomponents.', '0906.0020-1-0-4': 'The companion is thus not the source of either the red or blue broad line systems; SDSS J153636.22+044127.0 cannot be explained as a chance superposition of objects, or as an ejected black hole.', '0906.0020-1-0-5': 'Over the [MATH] yr difference between the rest frame epochs of the present and SDSS spectroscopy, we find no velocity shift to within 40 km s[MATH], nor any amplitude change in either broad line system.', '0906.0020-1-0-6': 'The lack of a shift can be admitted under the binary hypothesis if the implied radial velocity is a larger component of the full orbital velocity than was assumed in our earlier work.', '0906.0020-1-0-7': 'A strong test of the binary hypothesis requires yet longer temporal baselines.', '0906.0020-1-0-8': 'The lack of amplitude variations is unusual for the alternative explanation of this object as a "double-peaked" emitter; we further argue that SDSS J153636.22+044127.0 has unique spectral features that have no obvious analogue with other members of this class.', '0906.0020-1-1-0': '# A Binary Supermassive Black Hole Candidate', '0906.0020-1-2-0': 'The low-redshift QSO SDSS J153636.22+044127.0 (hereafter J1536+0441) is a candidate for hosting a binary supermassive black hole .', '0906.0020-1-2-1': "Its spectrum exhibits two broad-line emission systems at [MATH] and [MATH] thus separated in velocity by 3500 km[MATH] but only one narrow line system, which is associated with the redder or 'r' broad-line system.", '0906.0020-1-2-2': 'A third system of unresolved absorption lines has an intermediate velocity.', '0906.0020-1-2-3': 'These characteristics are unique among known quasars.', '0906.0020-1-2-4': 'We advanced this object as a candidate binary system of two supermassive black holes.', '0906.0020-1-2-5': 'The rough estimates of the two black hole masses are [MATH] and [MATH] based on line and continuum properties.', '0906.0020-1-2-6': 'Under this picture, the velocity difference between the broad-line systems is the projected orbital velocity difference of the two black holes.', '0906.0020-1-2-7': 'Assuming a circular orbit with random phase and inclination, we derive the separation of the black holes and their orbital period to be [MATH] parsec and [MATH] years, respectively.', '0906.0020-1-3-0': 'There is strong interest in the prevalence of binary black hole systems, which should be commonly formed in the formation of galaxies from hierarchical merging of smaller systems .', '0906.0020-1-3-1': 'This has motivated several observational searches for binary systems; however, few compelling candidates have survived scrutiny.', '0906.0020-1-3-2': 'Likewise, other investigators have quickly attempted to obtain additional observations of J1536+0441 to ascertain its viability as a binary system.', '0906.0020-1-3-3': 'These works, rather than confirming the binary hypothesis, have offered other interpretations.', '0906.0020-1-4-0': '[CITATION] obtained a spectrum of J1536+0441 that covered longer wavelengths than those that were sampled by the SDSS spectrum.', '0906.0020-1-4-1': 'They identified a "bump" in the red-wing of the r-system H[MATH] line that has a corresponding weak feature in the Mg II [MATH] 2800 line.', '0906.0020-1-4-2': 'At H[MATH], the presence of this red bump is ambiguous due to its near overlap with the 4959 [O III] line.', '0906.0020-1-4-3': 'Further, the red bump and b-system (bluer) Balmer lines are roughly symmetrical about the r-system lines.', '0906.0020-1-4-4': 'The red-bump, and in particular its apparent symmetry with the b-system, is not anticipated in the binary black hole model.', '0906.0020-1-4-5': '[CITATION] instead suggest that J1536+0441 is a "double-peaked" emission-line QSO.', '0906.0020-1-4-6': '[CITATION] also suggested this interpretation, based on his evaluation of the SDSS spectrum, alone.', '0906.0020-1-4-7': 'Under this picture, the spectrum of J1536+0441 reflects instabilities in or an unusual configuration of an accretion disk around a single black hole.', '0906.0020-1-4-8': 'At the same time, [CITATION] point out that the extreme difference in flux between the b-system lines and the red-bumps, plus the sharp cores of the b-system Balmer lines, are unique among the class of double-peaked emitters.', '0906.0020-1-5-0': 'Other interpretations are that J1536+0441 is the chance superposition of two AGN, or that one system is a black hole with associated AGN that has been ejected from the nucleus of the host galaxy.', '0906.0020-1-5-1': 'We showed, however that there is only a [MATH] chance of finding any random close pair of QSOs in the entire SDSS DR7 sample searched to yield J1536+0441.', '0906.0020-1-5-2': 'We also considered this hypothesis to be unsatisfactory, as there are no known examples of an isolated AGN with strong, broad Balmer lines, but no corresponding narrow lines at all.', '0906.0020-1-5-3': 'A more intriguing possibility is that the b-system is a black hole ejected from the nucleus of a galaxy now just hosting the r-system; however, the large implied ejection velocity and the light source for the absorption system are problematic.', '0906.0020-1-6-0': 'These conclusions are challenged by [CITATION], however, who obtained VLA observations of J1536+0441, showing it to comprise two sources separated by [MATH]^[MATH] with [MATH] and [MATH] mJy flux strengths or a 4.3 flux ratio at 8.5 Ghz.', '0906.0020-1-6-1': 'If these separately correspond to the two broad line systems, then this would be a strong refutation of hypothesis that J1536+0441 is a strongly bound binary system.', '0906.0020-1-6-2': 'Prior to publication of the [CITATION] observations, we requested a single orbit of HST imaging to check for multiple sources or unusual morphological features that might be diagnostic.', '0906.0020-1-6-3': 'We also obtained long-slit spectroscopy.', '0906.0020-1-6-4': 'These new observations appear to rule out the "chance superposition" or "ejected black hole" hypotheses, but leave the choice between the "double-peaked" or "binary black hole" hypotheses unresolved.', '0906.0020-1-7-0': '# The HST Imaging Program', '0906.0020-1-8-0': '## The Observations and Their Reduction', '0906.0020-1-9-0': 'HST images of J1536+0441 were obtained on April 21, 2009 under program GO/DD 11993.', '0906.0020-1-9-1': 'The QSO was centered in the PC1 CCD of WFPC2.', '0906.0020-1-9-2': 'The duration of the program was a single orbit, sufficient to probe the morphology of the bright AGN emission and to identify bright companions.', '0906.0020-1-9-3': 'Eight short (80s) exposures were obtained in filter F675W.', '0906.0020-1-9-4': 'The bandpass of the filter includes the H[MATH] lines of both the b and r redshift systems, as well as the [O III] lines associated with the red system.', '0906.0020-1-9-5': 'The F675W images were dithered to provide [MATH] pixel subsampling to produce a Nyquist-sampled image of the system, as well as to recover the best spatial resolution available for the particular camera and filter combination.', '0906.0020-1-9-6': 'Two exposures were obtained at each dither position to permit the repair of cosmic-ray hits.', '0906.0020-1-9-7': 'The typical peak exposure in the center of the QSO subimages was [MATH].', '0906.0020-1-9-8': 'The time remaining in the orbit after completion of the dither sequence was used to obtain two 100s spatially-coincident exposures in the F439W filter to provide some level of color information, but these exposures were unfortunately much shallower, having a peak level of [MATH].', '0906.0020-1-10-0': 'An up-sampled F675W superimage was generated from the eight individual exposures.', '0906.0020-1-10-1': 'The two exposures at each dither point were first combined, with cosmic rays identified as statistical outliers in each pair.', '0906.0020-1-10-2': 'The four summed exposures at each dither position were then combined to produce a single image with a [MATH] finer pixel scale ([MATH]^[MATH]/pixel), using the Fourier technique of [CITATION].', '0906.0020-1-10-3': 'For the F675W bandpass, the HST optical spatial band-limit falls well short of the Nyquist scale in the up-sampled image, thus high spatial frequencies between the two scales only encodes noise and were hence eliminated from the final image by a [CITATION] filter.', '0906.0020-1-10-4': 'The final F439W image is just an addition of the two sub-exposures after cosmic ray rejection.', '0906.0020-1-10-5': 'Both images are shown in Figure [REF].', '0906.0020-1-11-0': '## The Companion Object', '0906.0020-1-12-0': 'The F675W image of J1536+0441 shows the QSO to have a faint spatially-resolved companion.', '0906.0020-1-12-1': 'Its projected separation from the QSO is [MATH]^[MATH] or 5.0 kpc in the r-system frame ([MATH]) at [MATH]^[MATH] position angle.', '0906.0020-1-13-0': 'This location is well within the [MATH]^[MATH] position error of the companion radio source discovered by [CITATION].', '0906.0020-1-13-1': 'As we were completing this manuscript, [CITATION] reported seeing this companion in VLT K-band images.', '0906.0020-1-14-0': 'We deconvolved the F675W image using 20 iterations of the Lucy-Richardson algorithm to gain a model-independent understanding of the morphology of the companion.', '0906.0020-1-14-1': 'The PSF was constructed from standard stars observed over the last several years for photometric monitoring of the F675W filter.', '0906.0020-1-14-2': 'The F675W apparent magnitude of the object is 20.67 (AB) in a [MATH]^[MATH] aperture.', '0906.0020-1-14-3': 'At [MATH] the F675W bandpass is very nearly centered on the SDSS g-band.', '0906.0020-1-14-4': 'The implied absolute magnitude is [MATH] which includes the k-correction and a 0.13 mag correction for galactic extinction.', '0906.0020-1-14-5': 'The companion is unfortunately not visible in the F439W image; however, the zeropoint in this image is 2.85 mag (AB) brighter than that for the F675W superimage.', '0906.0020-1-14-6': 'Even if the companion object had a flat spectral energy distribution, the expected peak count in the F439W images would only be [MATH] ADU.', '0906.0020-1-14-7': 'Since the F439W filter falls to the blue of the 4000 break at the QSO redshift, it is likely that the true flux of the companion is even fainter.', '0906.0020-1-14-8': 'The color constraints offered by the F439W images are thus only of marginal interest.', '0906.0020-1-15-0': 'The companion is best seen in the lower-left panel of Figure [REF], in which we have attempted to subtract the QSO, which was done by scaling the reference F675W PSF to the peak flux of the QSO.', '0906.0020-1-15-1': 'The companion appears to be smooth and slightly elliptical in morphology.', '0906.0020-1-15-2': 'No tidal features are seen, nor is there any apparent connection to the QSO; however, the image is very shallow so this constrains only very strong interactions.', '0906.0020-1-15-3': 'The brightness profile is presented in Figure [REF].', '0906.0020-1-15-4': 'Its form resembles that of an elliptical galaxy with a core .', '0906.0020-1-15-5': 'In contrast, the profile is less well described by an exponential; if an exponential is fitted to just the outer part of the profile, the center of the profile rises well above it.', '0906.0020-1-15-6': 'An [MATH] law fitted to the profile yields an effective radius, [MATH]^[MATH] or 6.4 kpc.', '0906.0020-1-15-7': 'If used to estimate a total apparent luminosity, this form yields a value substantially brighter than the aperture magnitude quoted above; however, given that the brightness profile cannot be detected at [MATH]^[MATH] still a small fraction of [MATH] it is difficult to credit such a large extrapolation.', '0906.0020-1-15-8': 'In any case, the value of [MATH] as well as the surface brightness as a function of physical radius, is consistent with that seen in early-type galaxies within a generous range of absolute luminosity about the aperture magnitude quoted above.', '0906.0020-1-16-0': 'The most interesting aspect of the companion is its relatively strong radio emission.', '0906.0020-1-16-1': '[CITATION] derive [MATH] radio luminosities at 8.5 Ghz of [MATH] ergs s[MATH] and [MATH] ergs s[MATH] for the QSO and companion.', '0906.0020-1-16-2': 'respectively.', '0906.0020-1-16-3': 'The QSO/companion radio flux ratio is thus only 4.3, as compared to the optical flux ratio of [MATH] implied by the F675W 16.99 (AB) apparent magnitude of the QSO.', '0906.0020-1-16-4': 'At the same time, [CITATION] find the K-band flux ratio to be [MATH] which would imply that the companion object is very red, with the caveat raised above that our small aperture may have underestimated the total optical luminosity of the companion.', '0906.0020-1-16-5': 'The companion radio luminosity is consistent with the largest values appropriate to the most luminous "normal" early-type galaxies - it is well above the typical power seen in galaxies close to the estimated absolute magnitude given above.', '0906.0020-1-16-6': 'Lastly, we note that the companion has no obvious point-source component.', '0906.0020-1-17-0': 'There are no other obvious sources associated with either the QSO or companion.', '0906.0020-1-17-1': 'The reference PSF and QSO core have identical morphologies within their half-light radii.', '0906.0020-1-17-2': 'At sightly larger radii, the residuals of the PSF subtraction show a largely dipole pattern, with excess light seen to the NW of the QSO in Figure [REF].', '0906.0020-1-17-3': 'While there is very little recent stellar imaging available in the F675W filter, examination of standard stars observed in the F555W or F547M filters a few months prior to execution of the present program show this pattern to be common; it is consistent with a small amount of comatic aberration.', '0906.0020-1-17-4': 'We thus conclude that the residual pattern is artifactual.', '0906.0020-1-17-5': 'The residuals also make it difficult to set any constraints on the properties of the QSO host galaxy.', '0906.0020-1-17-6': 'On the assumption that the QSO subtraction residuals are largely due to a host galaxy, rather than a miss-match between the PSF and QSO core, the implied host luminosity is probably less luminous than [MATH]', '0906.0020-1-18-0': '# The Spectroscopy', '0906.0020-1-19-0': '## Observations and Their Reduction', '0906.0020-1-20-0': 'Spectra of J1536+041 were obtained on the night of April 22 (UT) with the RC spectrograph on the Mayall 4m telescope at Kitt Peak.', '0906.0020-1-20-1': 'The BL 420 grating provides a 1.5 pixel[MATH] dispersion; in combination with a [MATH]^[MATH] wide slit, the resulting resolution is between 3.0 and 3.6 over the wavelength ranges covered.', '0906.0020-1-20-2': 'Two grating tilts were used, a blue setting to cover the region from 5000 to 7600 and a red setting to cover the region from 7000 to 9500 .', '0906.0020-1-20-3': 'The slit was [MATH] long, and was oriented east-west to capture the companion seen in the HST images.', '0906.0020-1-20-4': 'The spatial pixel scale is [MATH]^[MATH] per pixel.', '0906.0020-1-20-5': 'Five exposures were obtained with the blue setting, and four exposures were obtained with the red setting.', '0906.0020-1-20-6': 'All exposures were 900 seconds.', '0906.0020-1-20-7': 'The seeing was poor during the period of the observations; a gaussian fit to the object flux along the slit gives about [MATH]^[MATH] FWHM.', '0906.0020-1-20-8': 'In addition to the exposures on the object itself, helium-neon-argon comparison spectra were obtained at the object position and flux standard stars were observed with the same setup.', '0906.0020-1-21-0': 'The spectra were reduced by removal of bias, division by a flat field, subtraction of the sky spectrum, and extraction of a one-dimensional spectrum from a region of the frame extending about [MATH] on either side of the spatial peak.', '0906.0020-1-21-1': 'The flux standards were used for both removal of atmospheric absorption features and for flux calibration.', '0906.0020-1-21-2': 'Positions of several dozen lines in the helium-neon-argon spectra were fit to low-order polynomials, resulting in wavelength calibrations accurate to 0.08 RMS for the bluer setting and 0.09 RMS for the redder setting.', '0906.0020-1-21-3': 'The individual spectra of each wavelength region were then combined, though some of the measurements described below were made from the individual exposures in order to estimate the uncertainty from the scatter.', '0906.0020-1-21-4': 'The final combined spectrum is shown in Figure [REF].', '0906.0020-1-22-0': 'The new spectrum is quite similar to the SDSS spectrum.', '0906.0020-1-22-1': 'In addition to the complicated structure of the Balmer lines, discussed in the next section, the same narrow lines are seen, including [O II] [MATH]3727, [O III] [MATH]4363, [MATH]4959, and [MATH]5007, and [Ne III] [MATH]3869 and [MATH]3968.', '0906.0020-1-22-2': 'In addition, the [N II] [MATH]6584 line is now visible on the red side of the r-system H[MATH] line.', '0906.0020-1-22-3': 'The narrow absorption lines of Na I D and Ca II H are also seen.', '0906.0020-1-23-0': 'Our new spectrum has somewhat higher signal-to-noise than does the SDSS spectrum in the region between 5500 and 6500 and the H[MATH] lines for the b- and r-systems are visible.', '0906.0020-1-23-1': 'In addition, the Fe II emission between H[MATH] and H[MATH] is easier to distinguish than in the SDSS spectrum.', '0906.0020-1-23-2': 'We attempted to determine whether it comes predominantly from the b- or r-system by cross-correlation with the corresponding region from the Fe II template developed by [CITATION] from the spectrum of I Zwicky 1.', '0906.0020-1-23-3': 'This cross-correlation shows that the optical Fe II is primarily associated with the b-system velocity, though the signal-to-noise is not sufficient to rule out a contribution from the r-system as well.', '0906.0020-1-24-0': '## The Balmer Line Profiles', '0906.0020-1-25-0': 'The new data clearly shows that the combined profiles of the Balmer lines extend as far to the red from the narrow-line redshift as they do to the blue, as pointed out by [CITATION].', '0906.0020-1-25-1': 'This is particularly clear in the case of H[MATH], which appears to extend to the red from the r-system at a constant level, and then drops off at about 9260, which corresponds to about 4700 km s[MATH] in the rest frame of the r-system.', '0906.0020-1-25-2': 'This makes the bottom part of the profile approximately symmetric on the red and blue sides.', '0906.0020-1-26-0': 'It is unclear how to measure the strengths, widths, and shapes of the various components that make up this profile.', '0906.0020-1-26-1': 'Relative to the continuum level, the b-system peak of H[MATH] is 1.24 times the height of the r-system peak.', '0906.0020-1-26-2': 'Relative to the level of the flat part of the red shoulder, that factor becomes 1.45.', '0906.0020-1-26-3': 'We measure the equivalent width of the entire H[MATH] complex as 348, with 247 of that total in the broad base, defined with a flat top at the level of the red shoulder, and with 44 and 57 as the contributions of the r-system and b-system peaks respectively.', '0906.0020-1-27-0': '## Profile Changes between the two epochs', '0906.0020-1-28-0': 'The most compelling confirmation that this system is a supermassive black hole binary would be the measurement of changes in the radial velocities of the two peaks consistent with orbital motion.', '0906.0020-1-28-1': 'In [CITATION] we estimated the separation of the two black holes to be 0.1pc with a [MATH] year orbital period, based solely on the implied radial velocity difference, an estimate of the two black hole masses, and the assumption of mean values for the orbital phase and inclination.', '0906.0020-1-28-2': 'These numbers predict a measurable change in velocity separation of the two peaks in as little as a year.', '0906.0020-1-28-3': 'Our new spectrum was obtained 1.04 years after the SDSS April 2008 observation, which corresponds to 0.75 years in the rest frame of the object.', '0906.0020-1-29-0': 'Visual inspection of the new spectra, as well as the report of [CITATION], shows that changes in the separation of the peaks of the Balmer lines are small or absent.', '0906.0020-1-29-1': 'Until and unless these changes become significant, improved constraints on the orbital parameters are based instead on the size of the errors bounding a null measurement.', '0906.0020-1-30-0': 'Given the complex structure of the Balmer-line profiles, measurement of accurate velocity requires that the spectra at all epochs and from all sources be treated consistently.', '0906.0020-1-30-1': 'In particular, a consequence of the composite nature of the profile is the possibility that the H[MATH] and H[MATH] lines in each system may have different behavior.', '0906.0020-1-30-2': 'We note specifically that there is a well-established trend of decreasing Balmer decrement with velocity away from the line center in Seyfert galaxies and QSOs .', '0906.0020-1-30-3': 'That is, the wings of the H[MATH] line fall off more quickly than those of the H[MATH] line.', '0906.0020-1-30-4': 'This effect appears to be present in the r-system Balmer lines, and it results in a larger gradient under the b-system H[MATH] line than under the H[MATH] line, leading to the perception that the splitting between the two systems is less in the H[MATH] line than in the H[MATH] line.', '0906.0020-1-30-5': 'However, if one overplots the H[MATH] and H[MATH] regions with a relative shift equal to the ratio of the rest wavelengths of the lines, it can be seen that the peaks of the two b-system lines are in excellent velocity agreement.', '0906.0020-1-31-0': 'In order to minimize such effects, we have remeasured the both the SDSS spectrum and our new spectrum in the same way, emphasizing the peaks of the lines, which should be less sensitive to underlying gradients.', '0906.0020-1-31-1': 'We find that relative to the [O III] lines, the r-system H[MATH] peak has a velocity of [MATH] km s[MATH], while the b-system H[MATH] peak has a velocity of [MATH] km s[MATH].', '0906.0020-1-31-2': 'The uncertainties quoted are the mean error as derived from the scatter in the five independent observations.', '0906.0020-1-31-3': 'The corresponding velocities from the SDSS spectrum are 31 km s[MATH] and [MATH] km s[MATH] for the r- and b-system H[MATH] peaks respectively.', '0906.0020-1-31-4': 'Thus, the H[MATH] splitting has changed from [MATH] to [MATH] km s[MATH].', '0906.0020-1-31-5': 'Similarly, we measure the H[MATH] splitting as [MATH] km s[MATH] in the SDSS spectrum and [MATH] km s[MATH] in our new spectrum.', '0906.0020-1-31-6': 'A final caveat is that there is some Balmer emission associated with the narrow lines though it is difficult to isolate, and this may influence the measured velocities of the r-system broad lines.', '0906.0020-1-32-0': 'In summary, we see no shift in the positions of the Balmer line peaks over this period, with a two sigma limit of about 80 km s[MATH].', '0906.0020-1-33-0': 'Lastly, we investigate changes in the strengths of the peaks using the H[MATH] line because it is well covered in both spectra.', '0906.0020-1-33-1': 'In addition, the [O III] lines should not change and so they provide some check on our result.', '0906.0020-1-33-2': 'When we divide the new spectrum by the SDSS spectrum, we can limit changes in the strengths of the two peaks relative to the continuum and to each other to less than about 3% over the period between the two observations.', '0906.0020-1-34-0': '## Spatial Location of Spectral Features', '0906.0020-1-35-0': '[CITATION] speculate that one of the two strong peaks in the Balmer line profiles may arise in the companion source seen in the radio.', '0906.0020-1-35-1': 'Since our slit was aligned east-west, and the separation between the two objects is approximately one arcsecond, this would result in an easily visible offset in the spatial location of the spectrum at the position of any feature arising in the companion.', '0906.0020-1-35-2': 'To test this idea, we fitted the intensity of the spectrum along the slit at positions corresponding to the b- and r-system H[MATH] lines, the two [O III] lines, and four continuum points spanning this region.', '0906.0020-1-35-3': 'The flux-weighted centroids are shown in Figure [REF].', '0906.0020-1-35-4': 'We have removed a linear tilt in the spectrum, due to a slit misalignment of the dispersion with respect to the CCD rows.', '0906.0020-1-35-5': 'The horizontal error bars show the range of the columns of the frame averaged.', '0906.0020-1-35-6': 'The vertical error bars show the standard deviations in the five measurements made from the five independent exposures.', '0906.0020-1-35-7': 'The corresponding spectrum is shown in the lower panel of Figure [REF]', '0906.0020-1-36-0': 'An estimate of the expected shift if the b-system arose in the companion object can be obtained by noting that the equivalent width of the b-system H[MATH] line is approximately 15 in the "composite" spectrum and that the width of the region averaged together is approximately 30.', '0906.0020-1-36-1': 'Since the ratio of flux in the companion object is only about 1/30 of that in the primary object within this spectral region, not much of the continuum can arise in the companion.', '0906.0020-1-36-2': 'This results in a relative flux within this 30 window of about 2 to 1 in favor of the brighter object, producing a position that would be offset by 1/3 of the one arcsecond separation.', '0906.0020-1-36-3': 'This is obviously incompatible with the measured positions, leading us to conclude that all of the spectral features arise in the primary object.', '0906.0020-1-37-0': "# What J1536+0441 Could Be and What It Isn't", '0906.0020-1-38-0': 'The analysis of the new HST images and KPNO spectra of J1536+0441 appears to rule out some hypotheses on its physical configuration, while leaving the door open to others that are likely to require additional observations for further evaluation.', '0906.0020-1-38-1': 'We close by summarizing what we conclude is the current status of the various explanations for the unusual properties of J1536+0441.', '0906.0020-1-39-0': '## Chance Superposition of Objects', '0906.0020-1-40-0': 'The discovery of [CITATION] that J1536+0441 is a well-resolved binary radio source appears to argue for the conclusion that it is simply the line of site superposition of two AGN, a conclusion also advocated by [CITATION].', '0906.0020-1-40-1': '[CITATION] did not favor this hypothesis, as the probability of a superposition was low, and required one of the AGN to have unique behavior in exhibiting broad lines without any corresponding narrow lines.', '0906.0020-1-40-2': 'In advance of the HST observations, we might have expected to see the QSO resolved into two point sources, corresponding to two physically separate AGN.', '0906.0020-1-40-3': 'Alternatively, we might have seen an optical jet emerging from the QSO.', '0906.0020-1-40-4': 'The optical counterpart to the less-luminous radio source, however, appears to be a normal galaxy with no evidence for an optically visible AGN, based both on the HST imagery, and analysis of the angular centroids of the emission lines in the long-slit spectroscopy.', '0906.0020-1-40-5': 'Superficially, it thus appears that this object is merely an "innocent bystander" to J1536+0441, although its large radio luminosity remains unexplained in this scenario.', '0906.0020-1-40-6': 'One possibility, however, is that it at least is the source of the absorption lines seen in its spectrum.', '0906.0020-1-40-7': 'The "a-system" has a projected velocity of [MATH] less than the r-system, compatible with galaxy and the QSO forming a binary system.', '0906.0020-1-41-0': '## Ejected Black-Hole', '0906.0020-1-42-0': 'If one assumes that the r-system, which includes the narrow line system, represents the rest velocity of the host galaxy, then one might hypothesize that the b-system represents a black hole ejected from the nucleus.', '0906.0020-1-42-1': 'There are two ways to achieve this result.', '0906.0020-1-42-2': 'Under the first mechanism, as two black holes merge, an asymmetric jet of gravitational radiation can be emitted, propelling the merged black hole out of the nucleus.', '0906.0020-1-42-3': 'In the second mechanism an infalling black hole interacts with pre-existing binary black hole, and is ejected in an exchange of orbital energy that binds the binary tighter and expels the third black hole.', '0906.0020-1-42-4': 'In the first case, however, the high velocity of the b-system is at the edge of plausible ejection velocities, and the existence of two AGN cannot be understood.', '0906.0020-1-42-5': 'The second case allows for two AGN, but depends on very specific orbital parameters to gain the high velocity.', '0906.0020-1-43-0': 'On observational grounds, we did not favor this hypothesis in [CITATION], as the high velocity of the b-system, but the unresolved nature of the point source implied either that the system was being observed close to the epoch of its creation, or that the ejected black hole was essentially traveling along the line-of-sight.', '0906.0020-1-43-1': 'It was also difficult to understand the geometry under which the b-system AGN could serve as a continuum source against which the a-system absorption lines are seen.', '0906.0020-1-44-0': 'If the a-system is attributed to the companion galaxy, then the geometry under which the a-system is seen against the QSO may be more plausible.', '0906.0020-1-44-1': 'The tighter restrictions placed on the ejection geometry by the HST imagery makes this even less attractive, however.', '0906.0020-1-45-0': '## Double-Peaked Emitter', '0906.0020-1-46-0': 'The most attractive explanation for J1536+0441 may indeed be that it is a "double-peaked" emission line QSOs.', '0906.0020-1-46-1': 'The red bumps in the broad lines discovered by [CITATION], and confirmed here, certainly resemble some of the broad and diffuse emission seen in these objects.', '0906.0020-1-46-2': 'The double-peaked class is further already known, and admits a variety of unusual broad line profiles.', '0906.0020-1-46-3': 'At the same time, there are striking differences between the Balmer line profiles in J1536+0441 and those of the double-peaked objects generally.', '0906.0020-1-47-0': 'The prototypical profile of the double-peaked objects has two rounded peaks of roughly equal strength joined by a flat or depressed central plateau.', '0906.0020-1-47-1': 'This form can be well fitted by simple relativistic accretion disk models .', '0906.0020-1-47-2': 'These objects occasionally show substantial departures from this form, in which relatively sharp, transient peaks are seen.', '0906.0020-1-47-3': 'Michael Eracleous at our request selected three cases of such profiles from the compilation of [CITATION]; these are shown, together with the H[MATH] profile of J1536+0441 in Figure [REF].', '0906.0020-1-47-4': 'The narrow lines, H[MATH], [N II], [S II], and [O I], have been removed from the double-peaked lines.', '0906.0020-1-47-5': 'In the case of J1536+0441, the narrow lines are very weak; the [N II] [MATH] 6584 line is just barely visible on the red wing of the r-system H[MATH] line and the other lines are not detectable at all.', '0906.0020-1-48-0': 'Although we admit that the detailed accretion disk structure that leads to these profiles is not well understood, we observe several notable differences between J1536+0441 and these objects.', '0906.0020-1-48-1': '(1) J1536+0441 has a strong, central, broad peak, which is not seen in any of the double-peaked emitters.', '0906.0020-1-48-2': '(2) The blue peak in J1536+0441 is very strong and very narrow at its top, surpassing, though perhaps by only a small amount, these characteristics in any of the other objects.', '0906.0020-1-48-3': '(3) The strong, sharp peaks in the double-peaked emitters are transient.', '0906.0020-1-48-4': 'In all three of these objects, the blue peak showed large changes in its intensity over a few months to less than a year.', '0906.0020-1-48-5': 'J1536+0441, on the other hand, has shown a constant profile for almost a year now.', '0906.0020-1-49-0': 'It is worth noting that the double-peaked emitters themselves were initially advanced as binary-black hole candidates.', '0906.0020-1-49-1': 'It is only the failure of the lines to vary in velocity over long time spans that has led to alternative interpretations of this class.', '0906.0020-1-49-2': 'Nor is there a clear theory on how the asymmetries accretion disks in these systems are generated or maintained.', '0906.0020-1-49-3': 'If J1536+0441 is a member of this class, then it may well motivate an improved picture of how disk instabilities arise and evolve.', '0906.0020-1-50-0': '## A Binary Black Hole', '0906.0020-1-51-0': 'The new observations presented here and those of [CITATION] do not advance the hypothesis that J1536+0441 hosts a binary black hole, except by perhaps eliminating the superposition and ejected black hole hypotheses.', '0906.0020-1-51-1': 'The lack of any observed velocity shift in the b-system or r-system over the span since the original SDSS spectrum was obtained provides additional constraints on the orbital parameters.', '0906.0020-1-51-2': 'The prediction that velocity shifts could be seen within a year was based on the presumption of an average viewing geometry and orbital phase for the binary system.', '0906.0020-1-51-3': 'If, however, the velocity difference between the b and r-systems represents a substantial portion of the orbital velocity, however, then the orbital radius of the system is larger, and the period longer than the estimates given in [CITATION].', '0906.0020-1-51-4': 'The geometry also requires longer intervals to see velocity changes if the system is close to quadrature.', '0906.0020-1-52-0': 'We explored the new constraints in two different ways.', '0906.0020-1-52-1': 'First, we calculated the expected velocity change for all possible values of the inclination and initial phase over the time period between the SDSS spectrum and our new spectral observation.', '0906.0020-1-52-2': 'We assume that the orbit is circular.', '0906.0020-1-52-3': 'Figure [REF] shows the allowed and excluded regions, given our [MATH] limit on the velocity change of 80 km s[MATH] for two different values for the total system mass, [MATH] and [MATH] M[MATH].', '0906.0020-1-52-4': 'This confirms the qualitative statement above - the allowed orbits are those in which the inclination is closer to line of sight and the initial phase is closer to maximum elongation.', '0906.0020-1-52-5': 'These are the cases for which the observed velocity difference between the two systems represents a larger fraction of the true space velocity difference.', '0906.0020-1-53-0': 'While this exercise demonstrates the values of initial phase and inclination that are still permitted, it does not show the fractional reduction in allowed orbits, nor does it show the range of allowed periods.', '0906.0020-1-53-1': 'These are shown in Figure [REF] for the two system masses we investigated.', '0906.0020-1-53-2': 'The top panels show the fraction of original orbits that remain possible as a function of the orbital period.', '0906.0020-1-53-3': 'None of the periods longer than 200 years has been ruled out.', '0906.0020-1-53-4': 'The remaining possible orbits account for 43% and 56% of the original orbits for the smaller and larger system masses.', '0906.0020-1-53-5': 'The lower panels show the relative number of possible orbits remaining as a function of period.', '0906.0020-1-53-6': 'The median values for the period of the remaining orbits are 319 and 743 years for the smaller and larger system masses.', '0906.0020-1-54-0': 'It is clear that the ultimate discrimination between the possibility that the blue emission-line peak represents an object in orbit around a central object will come from a number of years of spectroscopic monitoring.', '0906.0020-1-55-0': "We thank Matt Mountain for the grant of STScI Director's Discretion time under which the HST observations were obtained.", '0906.0020-1-55-1': 'We thank the STScI support staff for prompt and expert assistance in the preparation of the observing program.', '0906.0020-1-55-2': 'We thank Buell Jannuzi for the availability of the KPNO 4-m time.', '0906.0020-1-55-3': 'We thank Matthew Lallo for discussions on the properties of the WFPC2/PC1 PSFs.', '0906.0020-1-55-4': 'We thank Michael Eracleous for useful conversations on the properties of "double-peaked" emitters, and for identifying potential analogues to J1536+0441.', '0906.0020-1-55-5': 'We thank Suvi Gezari for providing us with the H[MATH] spectra of these analogues.'}

{'0906.0020-2-0-0': 'We present HST WFPC2/PC images and KPNO 4-m longslit spectroscopy of the QSO SDSS J153636.22+044127.0, which we advanced as a candidate binary supermassive black hole.', '0906.0020-2-0-1': 'The images reveal a close companion coincident with the radio source identified by [CITATION].', '0906.0020-2-0-2': 'It appears to be consistent with a [MATH] elliptical galaxy, if it is at the QSO redshift.', '0906.0020-2-0-3': 'The spectroscopy, however, shows no spatial offset between the red or blue Balmer lines.', '0906.0020-2-0-4': 'The companion is thus not the source of either the red or blue broad line systems; SDSS J153636.22+044127.0 is highly unlikely to be a chance superposition of objects or an ejected black hole.', '0906.0020-2-0-5': 'Over the [MATH] yr difference between the restframe epoch of the present and SDSS spectroscopy, we find no velocity shift to within 40 km s[MATH], nor any amplitude change in either broad line system.', '0906.0020-2-0-6': 'The lack of a shift can be admitted under the binary hypothesis, if the implied radial velocity is a larger component of the full orbital velocity than was assumed in our earlier work.', '0906.0020-2-0-7': 'A strong test of the binary hypothesis requires yet longer temporal baselines.', '0906.0020-2-0-8': 'The lack of amplitude variations is unusual for the alternative explanation of this object as a "double-peaked" emitter; we further argue that SDSS J153636.22+044127.0 has unique spectral features that have no obvious analogue with other members of this class.', '0906.0020-2-1-0': '# A Binary Supermassive Black Hole Candidate', '0906.0020-2-2-0': 'The low-redshift QSO SDSS J153636.22+044127.0 (hereafter J1536+0441) is a candidate for hosting a binary supermassive black hole .', '0906.0020-2-2-1': "Its spectrum exhibits two broad-line emission systems at [MATH] and [MATH] thus separated in velocity by 3500 km[MATH] but only one narrow line system, which is associated with the redder or 'r' broad-line system.", '0906.0020-2-2-2': 'A third system of unresolved absorption lines has an intermediate velocity.', '0906.0020-2-2-3': 'These characteristics are unique among known quasars.', '0906.0020-2-2-4': 'We advanced this object as a candidate binary system of two supermassive black holes.', '0906.0020-2-2-5': 'The rough estimates of the two black hole masses are [MATH] and [MATH] based on line and continuum properties.', '0906.0020-2-2-6': 'Under this picture, the velocity difference between the broad-line systems is the projected orbital velocity difference of the two black holes.', '0906.0020-2-2-7': 'Assuming a circular orbit with random phase and inclination, we derive the separation of the black holes and their orbital period to be [MATH] parsec and [MATH] years, respectively.', '0906.0020-2-3-0': 'There is strong interest in the prevalence of binary black hole systems, which should be commonly formed in the formation of galaxies from hierarchical merging of smaller systems .', '0906.0020-2-3-1': 'This has motivated several observational searches for binary systems; however, few compelling candidates have survived scrutiny.', '0906.0020-2-3-2': 'Likewise, other investigators have quickly attempted to obtain additional observations of J1536+0441 to ascertain its viability as a binary system.', '0906.0020-2-3-3': 'These works, rather than confirming the binary hypothesis, have offered other interpretations.', '0906.0020-2-4-0': '[CITATION] obtained a spectrum of J1536+0441 that covered longer wavelengths than those that were sampled by the SDSS spectrum.', '0906.0020-2-4-1': 'They identified a "bump" in the red-wing of the r-system H[MATH] line that has a corresponding weak feature in the Mg II [MATH] 2800 line.', '0906.0020-2-4-2': 'At H[MATH], the presence of this red bump is ambiguous due to its near overlap with the 4959 [O III] line.', '0906.0020-2-4-3': 'Further, the red bump and b-system (bluer) Balmer lines are roughly symmetrical about the r-system lines.', '0906.0020-2-4-4': 'The red-bump, and in particular its apparent symmetry with the b-system, is not anticipated in the binary black hole model.', '0906.0020-2-4-5': '[CITATION] instead suggest that J1536+0441 is a "double-peaked" emission-line QSO.', '0906.0020-2-4-6': '[CITATION] also suggested this interpretation, based on his evaluation of the SDSS spectrum, alone.', '0906.0020-2-4-7': 'Under this picture, the spectrum of J1536+0441 reflects instabilities in or an unusual configuration of an accretion disk around a single black hole.', '0906.0020-2-4-8': 'At the same time, [CITATION] point out that the extreme difference in flux between the b-system lines and the red-bumps, plus the sharp cores of the b-system Balmer lines, are unique among the class of double-peaked emitters.', '0906.0020-2-5-0': 'Other interpretations are that J1536+0441 is the chance superposition of two AGN, or that one system is a black hole with associated AGN that has been ejected from the nucleus of the host galaxy.', '0906.0020-2-5-1': 'We showed, however that there is only a [MATH] chance of finding any random close pair of QSOs in the entire SDSS DR7 sample searched to yield J1536+0441.', '0906.0020-2-5-2': 'We also considered this hypothesis to be unsatisfactory, as there are no known examples of an isolated AGN with strong, broad Balmer lines, but no corresponding narrow lines at all.', '0906.0020-2-5-3': 'A more intriguing possibility is that the b-system is a black hole ejected from the nucleus of a galaxy now just hosting the r-system; however, the large implied ejection velocity and the light source for the absorption system are problematic.', '0906.0020-2-6-0': 'These conclusions are challenged by [CITATION], however, who obtained VLA observations of J1536+0441, showing it to comprise two sources separated by [MATH]^[MATH] with [MATH] and [MATH] mJy flux strengths or a 4.3 flux ratio at 8.5 Ghz.', '0906.0020-2-6-1': 'If these separately correspond to the two broad line systems, then this would be a strong refutation of hypothesis that J1536+0441 is a strongly bound binary system.', '0906.0020-2-6-2': 'Prior to publication of the [CITATION] observations, we requested a single orbit of HST imaging to check for multiple sources or unusual morphological features that might be diagnostic.', '0906.0020-2-6-3': 'We also obtained long-slit spectroscopy.', '0906.0020-2-6-4': 'These new observations appear to rule out the "chance superposition" or "ejected black hole" hypotheses, but leave the choice between the "double-peaked" or "binary black hole" hypotheses unresolved.', '0906.0020-2-7-0': '# The HST Imaging Program', '0906.0020-2-8-0': '## The Observations and Their Reduction', '0906.0020-2-9-0': 'HST images of J1536+0441 were obtained on April 21, 2009 under program GO/DD 11993.', '0906.0020-2-9-1': 'The QSO was centered in the PC1 CCD of WFPC2.', '0906.0020-2-9-2': 'The duration of the program was a single orbit, sufficient to probe the morphology of the bright AGN emission and to identify bright companions.', '0906.0020-2-9-3': 'Eight short (80s) exposures were obtained in filter F675W.', '0906.0020-2-9-4': 'The bandpass of the filter includes the H[MATH] lines of both the b and r redshift systems, as well as the [O III] lines associated with the red system.', '0906.0020-2-9-5': 'The F675W images were dithered to provide [MATH] pixel subsampling to produce a Nyquist-sampled image of the system, as well as to recover the best spatial resolution available for the particular camera and filter combination.', '0906.0020-2-9-6': 'Two exposures were obtained at each dither position to permit the repair of cosmic-ray hits.', '0906.0020-2-9-7': 'The typical peak exposure in the center of the QSO subimages was [MATH].', '0906.0020-2-9-8': 'The time remaining in the orbit after completion of the dither sequence was used to obtain two 100s spatially-coincident exposures in the F439W filter to provide some level of color information, but these exposures were unfortunately much shallower, having a peak level of [MATH].', '0906.0020-2-10-0': 'An up-sampled F675W superimage was generated from the eight individual exposures.', '0906.0020-2-10-1': 'The two exposures at each dither point were first combined, with cosmic rays identified as statistical outliers in each pair.', '0906.0020-2-10-2': 'The four summed exposures at each dither position were then combined to produce a single image with a [MATH] finer pixel scale ([MATH]^[MATH]/pixel), using the Fourier technique of [CITATION].', '0906.0020-2-10-3': 'For the F675W bandpass, the HST optical spatial band-limit falls well short of the Nyquist scale in the up-sampled image, thus high spatial frequencies between the two scales only encodes noise and were hence eliminated from the final image by a [CITATION] filter.', '0906.0020-2-10-4': 'The final F439W image is just an addition of the two sub-exposures after cosmic ray rejection.', '0906.0020-2-10-5': 'Both images are shown in Figure [REF].', '0906.0020-2-11-0': '## The Companion Object', '0906.0020-2-12-0': 'The F675W image of J1536+0441 shows the QSO to have a faint spatially-resolved companion.', '0906.0020-2-12-1': 'Its projected separation from the QSO is [MATH]^[MATH] or 5.0 kpc in the r-system frame ([MATH]) at [MATH]^[MATH] position angle.', '0906.0020-2-13-0': 'This location is well within the [MATH]^[MATH] position error of the companion radio source discovered by [CITATION].', '0906.0020-2-13-1': 'As we were completing this manuscript, [CITATION] reported seeing this companion in VLT K-band images.', '0906.0020-2-14-0': 'We deconvolved the F675W image using 20 iterations of the Lucy-Richardson algorithm to gain a model-independent understanding of the morphology of the companion.', '0906.0020-2-14-1': 'The PSF was constructed from standard stars observed over the last several years for photometric monitoring of the F675W filter.', '0906.0020-2-14-2': 'While, no F675W PSF observations were available close in time to the present observations, the attendant small uncertainties in the PSF do not affect our results.', '0906.0020-2-14-3': 'The F675W apparent magnitude of the object is 20.67 (AB) in a [MATH]^[MATH] aperture.', '0906.0020-2-14-4': 'At [MATH] the F675W bandpass is very nearly centered on the SDSS g-band.', '0906.0020-2-14-5': 'The implied absolute magnitude is [MATH] which includes the k-correction and a 0.13 mag correction for galactic extinction.', '0906.0020-2-14-6': 'The companion is unfortunately not visible in the F439W image; however, the zeropoint in this image is 2.85 mag (AB) brighter than that for the F675W superimage.', '0906.0020-2-14-7': 'Even if the companion object had a flat spectral energy distribution, the expected peak count in the F439W images would only be [MATH] ADU.', '0906.0020-2-14-8': 'Since the F439W filter falls to the blue of the 4000 break at the QSO redshift, it is likely that the true flux of the companion is even fainter.', '0906.0020-2-14-9': 'The color constraints offered by the F439W images are thus only of marginal interest.', '0906.0020-2-15-0': 'The companion is best seen in the lower-left panel of Figure [REF], in which we have attempted to subtract the QSO, which was done by scaling the reference F675W PSF to the peak flux of the QSO.', '0906.0020-2-15-1': 'The companion appears to be smooth and slightly elliptical in morphology.', '0906.0020-2-15-2': 'No tidal features are seen, nor is there any apparent connection to the QSO.', '0906.0020-2-15-3': 'However, the image is very shallow and we estimate that it would be difficult to see isolated features with surface brightnesses dimmer than [MATH] thus this constrains only very strong interactions.', '0906.0020-2-16-0': 'The brightness profile is presented in Figure [REF].', '0906.0020-2-16-1': 'Its form resembles that of an elliptical galaxy with a core .', '0906.0020-2-16-2': 'In contrast, the profile is less well described by an exponential; if an exponential is fitted to just the outer part of the profile, the center of the profile rises well above it.', '0906.0020-2-16-3': 'An [MATH] law fitted to the profile yields an effective radius, [MATH]^[MATH] or 6.4 kpc.', '0906.0020-2-16-4': 'If used to estimate a total apparent luminosity, this form yields a value substantially brighter than the aperture magnitude quoted above; however, given that the brightness profile cannot be detected at [MATH]^[MATH] still a small fraction of [MATH] it is difficult to credit such a large extrapolation.', '0906.0020-2-16-5': 'In any case, the value of [MATH] as well as the surface brightness as a function of physical radius, is consistent with that seen in early-type galaxies within a generous range of absolute luminosity about the aperture magnitude quoted above.', '0906.0020-2-17-0': 'The most interesting aspect of the companion is its relatively strong radio emission.', '0906.0020-2-17-1': '[CITATION] derive [MATH] radio luminosities at 8.5 Ghz of [MATH] ergs s[MATH] and [MATH] ergs s[MATH] for the QSO and companion.', '0906.0020-2-17-2': 'respectively.', '0906.0020-2-17-3': 'The QSO/companion radio flux ratio is thus only 4.3, as compared to the optical flux ratio of [MATH] implied by the F675W 16.99 (AB) apparent magnitude of the QSO.', '0906.0020-2-17-4': 'At the same time, [CITATION] find the K-band flux ratio to be [MATH] which would imply that the companion object is very red, with the caveat raised above that our small aperture may have underestimated the total optical luminosity of the companion.', '0906.0020-2-17-5': 'The companion radio luminosity is consistent with the largest values appropriate to the most luminous "normal" early-type galaxies - it is well above the typical power seen in galaxies close to the estimated absolute magnitude given above.', '0906.0020-2-18-0': 'Despite its strong radio emission, the companion shows no obvious nuclear point-source component in the optical.', '0906.0020-2-18-1': 'This is consistent with the analysis shown in the next section that shows that the companion is not the source of the optical emission lines.', '0906.0020-2-18-2': '[CITATION] assert that the companion does have a point source component in contradiction of the present results.', '0906.0020-2-18-3': 'However, they present no analysis or figures to support this claim, and the spatial resolution of their K-band image appears to be at least an order of magnitude poorer than that of the HST images.', '0906.0020-2-19-0': 'There are no other obvious sources associated with either the QSO or companion.', '0906.0020-2-19-1': 'The reference PSF and QSO core have identical morphologies within their half-light radii.', '0906.0020-2-19-2': 'In particular, we can easily rule out the presence of an additional point source having a flux exceeding 10% of the QSO centered at any distance greater than [MATH]^[MATH] from the QSO.', '0906.0020-2-19-3': 'At slightly larger radii, the residuals of the PSF subtraction show a largely dipole pattern, with excess light seen to the NW of the QSO in Figure [REF].', '0906.0020-2-19-4': 'While, as noted earlier, there is very little recent stellar imaging available in the F675W filter, examination of standard stars observed in the F555W or F547M filters a few months prior to execution of the present program show this pattern to be common; it is consistent with a small amount of comatic aberration.', '0906.0020-2-19-5': 'We thus conclude that the residual pattern is artifactual.', '0906.0020-2-19-6': 'The residuals also make it difficult to set any constraints on the properties of the QSO host galaxy.', '0906.0020-2-19-7': 'On the assumption that the QSO subtraction residuals are largely due to a host galaxy, rather than a miss-match between the PSF and QSO core, the implied host luminosity is less luminous than [MATH]', '0906.0020-2-20-0': '# The Spectroscopy', '0906.0020-2-21-0': '## Observations and Their Reduction', '0906.0020-2-22-0': 'Spectra of J1536+041 were obtained on the night of April 22 (UT) with the RC spectrograph on the Mayall 4m telescope at Kitt Peak.', '0906.0020-2-22-1': 'The BL 420 grating provides a 1.5 pixel[MATH] dispersion; in combination with a [MATH]^[MATH] wide slit, the resulting resolution is between 3.0 and 3.6 over the wavelength ranges covered.', '0906.0020-2-22-2': 'Two grating tilts were used, a blue setting to cover the region from 5000 to 7600 and a red setting to cover the region from 7000 to 9500 .', '0906.0020-2-22-3': 'The slit was [MATH] long, and was oriented east-west to capture the companion seen in the HST images.', '0906.0020-2-22-4': 'The spatial pixel scale is [MATH]^[MATH] per pixel.', '0906.0020-2-22-5': 'Five exposures were obtained with the blue setting, and four exposures were obtained with the red setting.', '0906.0020-2-22-6': 'All exposures were 900 seconds.', '0906.0020-2-22-7': 'The seeing was poor during the period of the observations; a gaussian fit to the object flux along the slit gives about [MATH]^[MATH] FWHM.', '0906.0020-2-22-8': 'In addition to the exposures on the object itself, helium-neon-argon comparison spectra were obtained at the object position and flux standard stars were observed with the same setup.', '0906.0020-2-23-0': 'The spectra were reduced by removal of bias, division by a flat field, subtraction of the sky spectrum, and extraction of a one-dimensional spectrum from a region of the frame extending about [MATH] on either side of the spatial peak.', '0906.0020-2-23-1': 'The flux standards were used for both removal of atmospheric absorption features and for flux calibration.', '0906.0020-2-23-2': 'Positions of several dozen lines in the helium-neon-argon spectra were fit to low-order polynomials, resulting in wavelength calibrations accurate to 0.08 RMS for the bluer setting and 0.09 RMS for the redder setting.', '0906.0020-2-23-3': 'The individual spectra of each wavelength region were then combined, though some of the measurements described below were made from the individual exposures in order to estimate the uncertainty from the scatter.', '0906.0020-2-23-4': 'The final combined spectrum is shown in Figure [REF].', '0906.0020-2-24-0': 'The new spectrum is quite similar to the SDSS spectrum.', '0906.0020-2-24-1': 'In addition to the complicated structure of the Balmer lines, discussed in the next section, the same narrow lines are seen, including [O II] [MATH]3727, [O III] [MATH]4363, [MATH]4959, and [MATH]5007, and [Ne III] [MATH]3869 and [MATH]3968.', '0906.0020-2-24-2': 'In addition, the [N II] [MATH]6584 line is now visible on the red side of the r-system H[MATH] line.', '0906.0020-2-24-3': 'The narrow absorption lines of Na I D and Ca II H are also seen.', '0906.0020-2-25-0': 'Our new spectrum has somewhat higher signal-to-noise than does the SDSS spectrum in the region between 5500 and 6500 and the H[MATH] lines for the b- and r-systems are visible.', '0906.0020-2-25-1': 'In addition, the Fe II emission between H[MATH] and H[MATH] is easier to distinguish than in the SDSS spectrum.', '0906.0020-2-25-2': 'We attempted to determine whether it comes predominantly from the b- or r-system by cross-correlation with the corresponding region from the Fe II template developed by [CITATION] from the spectrum of I Zwicky 1.', '0906.0020-2-25-3': 'This cross-correlation shows that the optical Fe II is primarily associated with the b-system velocity, though the signal-to-noise is not sufficient to rule out a contribution from the r-system as well.', '0906.0020-2-26-0': '## The Balmer Line Profiles', '0906.0020-2-27-0': 'The new data clearly shows that the combined profiles of the Balmer lines extend as far to the red from the narrow-line redshift as they do to the blue, as pointed out by [CITATION].', '0906.0020-2-27-1': 'This is particularly clear in the case of H[MATH], which appears to extend to the red from the r-system at a constant level, and then drops off at about 9260, which corresponds to about 4700 km s[MATH] in the rest frame of the r-system.', '0906.0020-2-27-2': 'This makes the bottom part of the profile approximately symmetric on the red and blue sides.', '0906.0020-2-28-0': 'It is unclear how to measure the strengths, widths, and shapes of the various components that make up this profile.', '0906.0020-2-28-1': 'Relative to the continuum level, the b-system peak of H[MATH] is 1.24 times the height of the r-system peak.', '0906.0020-2-28-2': 'Relative to the level of the flat part of the red shoulder, that factor becomes 1.45.', '0906.0020-2-28-3': 'We measure the equivalent width of the entire H[MATH] complex as 348, with 247 of that total in the broad base, defined with a flat top at the level of the red shoulder, and with 44 and 57 as the contributions of the r-system and b-system peaks respectively.', '0906.0020-2-29-0': '## Profile Changes between the two epochs', '0906.0020-2-30-0': 'The most compelling confirmation that this system is a supermassive black hole binary would be the measurement of changes in the radial velocities of the two peaks consistent with orbital motion.', '0906.0020-2-30-1': 'In [CITATION] we estimated the separation of the two black holes to be 0.1pc with a [MATH] year orbital period, based solely on the implied radial velocity difference, an estimate of the two black hole masses, and the assumption of mean values for the orbital phase and inclination.', '0906.0020-2-30-2': 'These numbers predict a measurable change in velocity separation of the two peaks in as little as a year.', '0906.0020-2-30-3': 'Our new spectrum was obtained 1.04 years after the SDSS April 2008 observation, which corresponds to 0.75 years in the rest frame of the object.', '0906.0020-2-31-0': 'Visual inspection of the new spectra, as well as the report of [CITATION], shows that changes in the separation of the peaks of the Balmer lines are small or absent.', '0906.0020-2-31-1': 'Until and unless these changes become significant, improved constraints on the orbital parameters are based instead on the size of the errors bounding a null measurement.', '0906.0020-2-32-0': 'Given the complex structure of the Balmer-line profiles, measurement of accurate velocity requires that the spectra at all epochs and from all sources be treated consistently.', '0906.0020-2-32-1': 'In particular, a consequence of the composite nature of the profile is the possibility that the H[MATH] and H[MATH] lines in each system may have different behavior.', '0906.0020-2-32-2': 'We note specifically that there is a well-established trend of decreasing Balmer decrement with velocity away from the line center in Seyfert galaxies and QSOs .', '0906.0020-2-32-3': 'That is, the wings of the H[MATH] line fall off more quickly than those of the H[MATH] line.', '0906.0020-2-32-4': 'This effect appears to be present in the r-system Balmer lines, and it results in a larger gradient under the b-system H[MATH] line than under the H[MATH] line, leading to the perception that the splitting between the two systems is less in the H[MATH] line than in the H[MATH] line.', '0906.0020-2-32-5': 'However, if one overplots the H[MATH] and H[MATH] regions with a relative shift equal to the ratio of the rest wavelengths of the lines, it can be seen that the peaks of the two b-system lines are in excellent velocity agreement.', '0906.0020-2-33-0': 'In order to minimize such effects, we have remeasured the SDSS spectrum and our new spectrum in the same way, emphasizing the peaks of the lines, which should be less sensitive to underlying gradients.', '0906.0020-2-33-1': 'We find that relative to the [O III] lines, the r-system H[MATH] peak has a velocity of [MATH] km s[MATH], while the b-system H[MATH] peak has a velocity of [MATH] km s[MATH].', '0906.0020-2-33-2': 'The uncertainties quoted are the mean error as derived from the scatter in the five independent observations.', '0906.0020-2-33-3': 'The corresponding velocities from the SDSS spectrum are 31 km s[MATH] and [MATH] km s[MATH] for the r- and b-system H[MATH] peaks respectively.', '0906.0020-2-33-4': 'Thus, the H[MATH] splitting has changed from [MATH] to [MATH] km s[MATH].', '0906.0020-2-33-5': 'Similarly, we measure the H[MATH] splitting as [MATH] km s[MATH] in the SDSS spectrum and [MATH] km s[MATH] in our new spectrum.', '0906.0020-2-33-6': 'A final caveat is that there is some Balmer emission associated with the narrow lines though it is difficult to isolate, and this may influence the measured velocities of the r-system broad lines.', '0906.0020-2-34-0': 'In summary, we see no shift in the positions of the Balmer line peaks over this period, with a two sigma limit of about 80 km s[MATH].', '0906.0020-2-35-0': 'Lastly, we investigate changes in the strengths of the peaks using the H[MATH] line because it is well covered in both spectra.', '0906.0020-2-35-1': 'In addition, the [O III] lines should not change and so they provide some check on our result.', '0906.0020-2-35-2': 'After rebinning the new and SDSS spectra to the same resolution, we find that changes in the strengths of the two peaks relative to the continuum and to each other can be limited to less than 3% over the period between the two observations.', '0906.0020-2-36-0': '## Spatial Location of Spectral Features', '0906.0020-2-37-0': '[CITATION] speculate that one of the two strong peaks in the Balmer line profiles may arise in the companion source seen in the radio.', '0906.0020-2-37-1': 'Since our slit was aligned east-west, and the separation between the two objects is approximately one arcsecond, this would result in an easily visible offset in the spatial location of the spectrum at the position of any feature arising in the companion.', '0906.0020-2-37-2': 'To test this idea, we fitted the intensity of the spectrum along the slit as a function of wavelength at positions corresponding to the b- and r-system H[MATH] lines, the two [O III] lines, and four continuum points spanning this region.', '0906.0020-2-37-3': 'The centers of the best-fit gaussians are shown in Figure [REF].', '0906.0020-2-37-4': 'We have removed a linear tilt in the spectrum, due to a slit misalignment of the dispersion with respect to the CCD rows.', '0906.0020-2-37-5': 'The horizontal error bars show the range of the columns of the frame averaged.', '0906.0020-2-37-6': 'The vertical error bars show the standard deviations in the five measurements made from the five independent exposures.', '0906.0020-2-37-7': 'The corresponding spectrum is shown in the lower panel of Figure [REF]', '0906.0020-2-38-0': 'An estimate of the expected shift if the b-system arose in the companion object can be obtained by noting that the equivalent width of the b-system H[MATH] line is approximately 15 in the "composite" spectrum and that the width of the region averaged together is approximately 30.', '0906.0020-2-38-1': 'Since the ratio of flux in the companion object is only about 1/30 of that in the primary object within this spectral region, not much of the continuum can arise in the companion.', '0906.0020-2-38-2': 'This results in a relative flux within this 30 window of about 2 to 1 in favor of the brighter object, producing a position that would be offset by 1/3 of the one arcsecond separation.', '0906.0020-2-38-3': 'This is obviously incompatible with the measured positions, leading us to conclude that all of the spectral features arise in the primary object.', '0906.0020-2-39-0': "# What J1536+0441 Could Be and What It Isn't", '0906.0020-2-40-0': 'The analysis of the new HST images and KPNO spectra of J1536+0441 appears to rule out some hypotheses on its physical configuration, while leaving the door open to others that are likely to require additional observations for further evaluation.', '0906.0020-2-40-1': 'We close by summarizing what we conclude is the current status of the various explanations for the unusual properties of J1536+0441.', '0906.0020-2-41-0': '## Chance Superposition of Objects', '0906.0020-2-42-0': 'The discovery of [CITATION] that J1536+0441 is a well-resolved binary radio source appears to argue for the conclusion that it is simply the line of site superposition of two AGN, a conclusion also advocated by [CITATION].', '0906.0020-2-42-1': '[CITATION] did not favor this hypothesis, as the probability of a superposition was low ([MATH]), and required one of the AGN to have unique behavior in exhibiting broad lines without any corresponding narrow lines.', '0906.0020-2-42-2': 'In advance of the HST observations, we might have expected to see the QSO resolved into two point sources, corresponding to two physically separate AGN.', '0906.0020-2-42-3': 'Alternatively, we might have seen an optical jet emerging from the QSO.', '0906.0020-2-42-4': 'The optical counterpart to the less-luminous radio source, however, appears to be a normal galaxy with no evidence for an optically visible AGN, based both on the HST imagery, and analysis of the angular centroids of the emission lines in the long-slit spectroscopy.', '0906.0020-2-42-5': 'Superficially, it thus appears that this object is merely an "innocent bystander" to J1536+0441, although its large radio luminosity remains unexplained in this scenario.', '0906.0020-2-42-6': 'One possibility, however, is that it at least is the source of the absorption lines seen in its spectrum.', '0906.0020-2-42-7': 'The "a-system" has a projected velocity of [MATH] less than the r-system, compatible with galaxy and the QSO forming a binary system.', '0906.0020-2-43-0': '[CITATION] derived a considerably higher a priori probability of superposition than our estimate quoted above.', '0906.0020-2-43-1': 'Using the [CITATION] QSO-QSO correlation function, they derived a probability of about unity for the occurrence of a 5 kpc separation pair within the size the original sample studied by [CITATION].', '0906.0020-2-43-2': 'We note, however, that [CITATION] limited their sample of QSO pairs to those with [MATH] on the assumption that larger velocity separations would not be expected in systems that are truly physically associated.', '0906.0020-2-43-3': '[CITATION], for example, show no significant correlation between QSOs with velocity differences as large as that in J1536+0441.', '0906.0020-2-43-4': 'A possible exception to this would be for two QSOs both bound in the potential of an extremely rich cluster.', '0906.0020-2-43-5': '[CITATION], for example, hypothesize that the two emission line systems separated by [MATH] in the QSO SDSSJ092712.65+294344.0 is a system analogous to the [MATH] difference between the two emission line systems seen in NGC 1275 , the first-ranked galaxy of the Perseus galaxy cluster.', '0906.0020-2-43-6': 'The probability of this occurrence for J1536+0441 should be encoded in the [CITATION] function, however.', '0906.0020-2-44-0': 'The limit on spatial coincidence from the new HST images, however, is now at least two orders of magnitude more stringent than that assumed in [CITATION] or [CITATION].', '0906.0020-2-44-1': 'This argument continues to disfavor the idea that there are two separate host galaxies.', '0906.0020-2-45-0': '## Ejected Black-Hole', '0906.0020-2-46-0': 'If one assumes that the r-system, which includes the narrow line system, represents the rest velocity of the host galaxy, then in advance of the present observations, one might have hypothesized that the b-system represented a black hole ejected from the nucleus.', '0906.0020-2-46-1': 'There are two ways to achieve this result.', '0906.0020-2-46-2': 'Under the first mechanism, as two black holes merge, an asymmetric jet of gravitational radiation can be emitted, propelling the merged black hole out of the nucleus.', '0906.0020-2-46-3': 'In the second mechanism an infalling black hole interacts with pre-existing binary black hole, and is ejected in an exchange of orbital energy that binds the binary tighter and expels the third black hole.', '0906.0020-2-47-0': 'In the case of ejection of a merged black hole, the 3500 km s[MATH] velocity of the b-system is at the edge of plausible ejection velocities that have been demonstrated in numerical experiments .', '0906.0020-2-47-1': 'Such high velocities depend on very specific mass ratios, spins, and orbital parameters of the merging black holes, and appear to be extremely unlikely.', '0906.0020-2-47-2': '[CITATION] show that the probability of ejection velocities in excess of [MATH] km s[MATH] may occur in [MATH] of all black hole mergers.', '0906.0020-2-47-3': 'Regardless, while this might explain the QSO as two separate objects, one of which has been ejected, the spatial centroid of the H[MATH] lines show that neither of these objects resides in the companion galaxy.', '0906.0020-2-48-0': 'Ejection as the result of the interaction of an infalling black hole with a pre-existing binary black hole may also produce relative velocities in excess of [MATH] km s[MATH] under rare conditions.', '0906.0020-2-48-1': 'The distribution of ejection velocities presented by [CITATION] show that these may occur in a few percent of the interactions, thus apparently making this mechanism much more likely to produce a high-velocity ejection than the merger case.', '0906.0020-2-48-2': 'This mechanism allows for the existence of two AGN after the ejection, and thus might have explained J1536+0441 had the present observations revealed two distinct sources of optical emission.', '0906.0020-2-48-3': 'As with the merger case, however, it cannot explain the spectrum of J1536+0441 as a single source.', '0906.0020-2-49-0': 'On observational grounds, we did not favor this hypothesis in [CITATION], as the high velocity of the b-system, but the unresolved nature of the point source implied either that the system was being observed close to the epoch of its creation, or that the ejected black hole was essentially traveling along the line-of-sight.', '0906.0020-2-49-1': 'It was also difficult to understand the geometry under which the b-system AGN could serve as a continuum source against which the a-system absorption lines are seen.', '0906.0020-2-50-0': 'If the a-system is attributed to the companion galaxy, then the geometry under which the a-system is seen against the QSO may be more plausible.', '0906.0020-2-50-1': 'The tighter restrictions placed on the ejection geometry by the HST imagery makes this even less attractive, however.', '0906.0020-2-51-0': 'We also note the superficial similarity of J1536+0441 to the object HE 0450-2958, which has been proposed as a "naked" QSO , possibly ejected from a disturbed galaxy about [MATH]^[MATH] away .', '0906.0020-2-51-1': 'Although the projected separation between the QSO and the putative host, 6.5 kpc, is similar to the separation between J1536+0441 and the companion object, there are two critical differences.', '0906.0020-2-51-2': 'First, the upper limit we have derived for an underlying host galaxy at the position of the J1536+0441 nucleus, [MATH], is still bright enough that no alternative host is required.', '0906.0020-2-51-3': 'Second, this explanation does not account for the unique spectrum of J 1536+0441; the QSO in the GE 0450-2958 system has a normal spectrum.', '0906.0020-2-52-0': '## Double-Peaked Emitter', '0906.0020-2-53-0': 'The most attractive explanation for J1536+0441 may indeed be that it is a "double-peaked" emission line QSOs.', '0906.0020-2-53-1': 'The red bumps in the broad lines discovered by [CITATION], and confirmed here, certainly resemble some of the broad and diffuse emission seen in these objects.', '0906.0020-2-53-2': 'The double-peaked class is further already known, and admits a variety of unusual broad line profiles.', '0906.0020-2-53-3': 'At the same time, there are striking differences between the Balmer line profiles in J1536+0441 and those of the double-peaked objects generally.', '0906.0020-2-54-0': 'The prototypical profile of the double-peaked objects has two rounded peaks of roughly equal strength joined by a flat or depressed central plateau.', '0906.0020-2-54-1': 'This form can be well fitted by simple relativistic accretion disk models .', '0906.0020-2-54-2': 'These objects occasionally show substantial departures from this form, in which relatively sharp, transient peaks are seen.', '0906.0020-2-54-3': 'Michael Eracleous at our request selected three cases of such profiles from the compilation of [CITATION]; these are shown, together with the H[MATH] profile of J1536+0441 in Figure [REF].', '0906.0020-2-54-4': 'The narrow lines, H[MATH], [N II], [S II], and [O I], have been removed from the double-peaked lines.', '0906.0020-2-54-5': 'In the case of J1536+0441, the narrow lines are very weak; the [N II] [MATH] 6584 line is just barely visible on the red wing of the r-system H[MATH] line and the other lines are not detectable at all.', '0906.0020-2-55-0': 'Although we admit that the detailed accretion disk structure that leads to these profiles is not well understood, we observe several notable differences between J1536+0441 and these objects.', '0906.0020-2-55-1': '(1) J1536+0441 has a strong, central, broad peak, which is not seen in any of the double-peaked emitters.', '0906.0020-2-55-2': '(2) The blue peak in J1536+0441 is very strong and very narrow at its top, surpassing, though perhaps by only a small amount, these characteristics in any of the other objects.', '0906.0020-2-55-3': '(3) The strong, sharp peaks in the double-peaked emitters are transient.', '0906.0020-2-55-4': 'In all three of these objects, the blue peak showed large changes in its intensity over a few months to less than a year.', '0906.0020-2-55-5': 'J1536+0441, on the other hand, has shown a constant profile for almost a year now.', '0906.0020-2-56-0': 'It is worth noting that the double-peaked emitters themselves were initially advanced as binary-black hole candidates.', '0906.0020-2-56-1': 'It is only the failure of the lines to vary in velocity over long time spans that has led to alternative interpretations of this class.', '0906.0020-2-56-2': 'Nor is there a clear theory on how the asymmetries accretion disks in these systems are generated or maintained.', '0906.0020-2-56-3': 'If J1536+0441 is a member of this class, then it may well motivate an improved picture of how disk instabilities arise and evolve.', '0906.0020-2-57-0': '## A Binary Black Hole', '0906.0020-2-58-0': 'The new observations presented here and those of [CITATION] do not advance the hypothesis that J1536+0441 hosts a binary black hole, except by perhaps eliminating the superposition and ejected black hole hypotheses.', '0906.0020-2-58-1': 'The lack of any observed velocity shift in the b-system or r-system over the span since the original SDSS spectrum was obtained provides additional constraints on the orbital parameters.', '0906.0020-2-58-2': 'The prediction that velocity shifts could be seen within a year was based on the presumption of an average viewing geometry and orbital phase for the binary system.', '0906.0020-2-58-3': 'If, however, the velocity difference between the b and r-systems represents a substantial portion of the orbital velocity, then the orbital radius of the system is larger, and the period longer than the estimates given in [CITATION].', '0906.0020-2-58-4': 'The geometry also requires longer intervals to see velocity changes if the system is close to quadrature.', '0906.0020-2-59-0': 'The derived orbital parameters also depend on the total mass of the binary system, which was estimated by [CITATION] from the widths of the H[MATH] lines and the continuum luminosity.', '0906.0020-2-59-1': 'The assumption that the two black holes were emitting at the same fraction of their Eddington luminosities resulted in estimates of [MATH] and [MATH] M[MATH] for the b- and r-systems respectively.', '0906.0020-2-59-2': 'The discovery of the extension of the Balmer line profiles to the red allow us to revisit this calculation.', '0906.0020-2-59-3': 'Replacing the former value of 6000 km s[MATH] with a new measured value of 10,600 km s[MATH] for the r-system FWHM results in the increase of that black hole to [MATH] M[MATH].', '0906.0020-2-59-4': 'Two other methods can be used to estimate the black hole masses.', '0906.0020-2-59-5': 'The width of the [O III] [MATH]5007 line can be used as a surrogate for the stellar velocity dispersion , yielding a mass of 10[MATH] M[MATH].', '0906.0020-2-59-6': 'Also, one can get a handle on the mass by adopting an average value of L/L[MATH], 0.14, for QSOs with the redshift and luminosity of this object , which results in a mass of 10[MATH] M[MATH].', '0906.0020-2-60-0': 'We explored the new constraints from our spectroscopy in two different ways.', '0906.0020-2-60-1': 'First, we calculated the expected velocity change for all possible values of the inclination and initial phase over the time period between the SDSS spectrum and our new spectral observation.', '0906.0020-2-60-2': 'We assume that the orbit is circular.', '0906.0020-2-60-3': 'Figure [REF] shows the allowed and excluded regions, given our [MATH] limit on the velocity change of 80 km s[MATH] for two different values for the total system mass, [MATH] and [MATH] M[MATH].', '0906.0020-2-60-4': 'This confirms the qualitative statement above - the allowed orbits are those in which the inclination is closer to line of sight and the initial phase is closer to maximum elongation.', '0906.0020-2-60-5': 'These are the cases for which the observed velocity difference between the two systems represents a larger fraction of the true space velocity difference.', '0906.0020-2-61-0': 'While this exercise demonstrates the values of initial phase and inclination that are still permitted, it does not show the fractional reduction in allowed orbits, nor does it show the range of allowed periods.', '0906.0020-2-61-1': 'These are shown in Figure [REF] for the two system masses we investigated.', '0906.0020-2-61-2': 'The top panels show the fraction of original orbits that remain possible as a function of the orbital period.', '0906.0020-2-61-3': 'None of the periods longer than 200 years has been ruled out.', '0906.0020-2-61-4': 'The remaining possible orbits account for 43% and 56% of the original orbits for the smaller and larger system masses.', '0906.0020-2-61-5': 'The lower panels show the relative number of possible orbits remaining as a function of period.', '0906.0020-2-61-6': 'The median values for the period of the remaining orbits are 319 and 743 years for the smaller and larger system masses.', '0906.0020-2-62-0': 'As we pointed out, the differences between the observed Balmer line profiles and those seen in double-peaked emitters, we must also note that the newly discovered extension of the profiles to the red is problematic for the binary explanation.', '0906.0020-2-62-1': 'Rather than a model in which two separate massive objects are orbiting a center of mass located between them, this profile suggests that the smaller object might be embedded in (or passing through) the disk surrounding the larger one.', '0906.0020-2-62-2': 'The extension to the red could result from waves or disturbances in the disk as a result of this interaction.', '0906.0020-2-63-0': 'It is clear that the ultimate confirmation or rejection of the possibility that the blue emission-line peak represents an object in orbit around a central object will come from a number of years of spectroscopic monitoring.', '0906.0020-2-64-0': "We thank Matt Mountain for the grant of STScI Director's Discretion time under which the HST observations were obtained.", '0906.0020-2-64-1': 'We thank the STScI support staff for prompt and expert assistance in the preparation of the observing program.', '0906.0020-2-64-2': 'We thank Buell Jannuzi for the availability of the KPNO 4-m time.', '0906.0020-2-64-3': 'We thank Matthew Lallo for discussions on the properties of the WFPC2/PC1 PSFs.', '0906.0020-2-64-4': 'We thank Michael Eracleous for useful conversations on the properties of "double-peaked" emitters, and for identifying potential analogues to J1536+0441.', '0906.0020-2-64-5': 'We thank Suvi Gezari for providing us with the H[MATH] spectra of these analogues.'}

[['0906.0020-1-3-0', '0906.0020-2-3-0'], ['0906.0020-1-3-1', '0906.0020-2-3-1'], ['0906.0020-1-3-2', '0906.0020-2-3-2'], ['0906.0020-1-3-3', '0906.0020-2-3-3'], ['0906.0020-1-10-0', '0906.0020-2-10-0'], ['0906.0020-1-10-1', '0906.0020-2-10-1'], ['0906.0020-1-10-2', '0906.0020-2-10-2'], ['0906.0020-1-10-3', '0906.0020-2-10-3'], ['0906.0020-1-10-4', '0906.0020-2-10-4'], ['0906.0020-1-10-5', '0906.0020-2-10-5'], ['0906.0020-1-4-0', '0906.0020-2-4-0'], ['0906.0020-1-4-1', '0906.0020-2-4-1'], ['0906.0020-1-4-2', '0906.0020-2-4-2'], ['0906.0020-1-4-3', '0906.0020-2-4-3'], ['0906.0020-1-4-4', '0906.0020-2-4-4'], ['0906.0020-1-4-5', '0906.0020-2-4-5'], ['0906.0020-1-4-6', '0906.0020-2-4-6'], ['0906.0020-1-4-7', '0906.0020-2-4-7'], ['0906.0020-1-4-8', '0906.0020-2-4-8'], ['0906.0020-1-42-1', '0906.0020-2-46-1'], ['0906.0020-1-42-2', '0906.0020-2-46-2'], ['0906.0020-1-42-3', '0906.0020-2-46-3'], ['0906.0020-1-55-0', '0906.0020-2-64-0'], ['0906.0020-1-55-1', '0906.0020-2-64-1'], ['0906.0020-1-55-2', '0906.0020-2-64-2'], ['0906.0020-1-55-3', '0906.0020-2-64-3'], ['0906.0020-1-55-4', '0906.0020-2-64-4'], ['0906.0020-1-55-5', '0906.0020-2-64-5'], ['0906.0020-1-43-0', '0906.0020-2-49-0'], ['0906.0020-1-43-1', '0906.0020-2-49-1'], ['0906.0020-1-0-0', '0906.0020-2-0-0'], ['0906.0020-1-0-1', '0906.0020-2-0-1'], ['0906.0020-1-0-2', '0906.0020-2-0-2'], ['0906.0020-1-0-7', '0906.0020-2-0-7'], ['0906.0020-1-0-8', '0906.0020-2-0-8'], ['0906.0020-1-52-1', '0906.0020-2-60-1'], ['0906.0020-1-52-2', '0906.0020-2-60-2'], ['0906.0020-1-52-3', '0906.0020-2-60-3'], ['0906.0020-1-52-4', '0906.0020-2-60-4'], ['0906.0020-1-52-5', '0906.0020-2-60-5'], ['0906.0020-1-47-0', '0906.0020-2-54-0'], ['0906.0020-1-47-1', '0906.0020-2-54-1'], ['0906.0020-1-47-2', '0906.0020-2-54-2'], ['0906.0020-1-47-3', '0906.0020-2-54-3'], ['0906.0020-1-47-4', '0906.0020-2-54-4'], ['0906.0020-1-47-5', '0906.0020-2-54-5'], ['0906.0020-1-29-0', '0906.0020-2-31-0'], ['0906.0020-1-29-1', '0906.0020-2-31-1'], ['0906.0020-1-36-0', '0906.0020-2-38-0'], ['0906.0020-1-36-1', '0906.0020-2-38-1'], ['0906.0020-1-36-2', '0906.0020-2-38-2'], ['0906.0020-1-36-3', '0906.0020-2-38-3'], ['0906.0020-1-23-0', '0906.0020-2-25-0'], ['0906.0020-1-23-1', '0906.0020-2-25-1'], ['0906.0020-1-23-2', '0906.0020-2-25-2'], ['0906.0020-1-23-3', '0906.0020-2-25-3'], ['0906.0020-1-16-0', '0906.0020-2-17-0'], ['0906.0020-1-16-1', '0906.0020-2-17-1'], ['0906.0020-1-16-3', '0906.0020-2-17-3'], ['0906.0020-1-16-4', '0906.0020-2-17-4'], ['0906.0020-1-16-5', '0906.0020-2-17-5'], ['0906.0020-1-38-0', '0906.0020-2-40-0'], ['0906.0020-1-38-1', '0906.0020-2-40-1'], ['0906.0020-1-46-0', '0906.0020-2-53-0'], ['0906.0020-1-46-1', '0906.0020-2-53-1'], ['0906.0020-1-46-2', '0906.0020-2-53-2'], ['0906.0020-1-46-3', '0906.0020-2-53-3'], ['0906.0020-1-53-0', '0906.0020-2-61-0'], ['0906.0020-1-53-1', '0906.0020-2-61-1'], ['0906.0020-1-53-2', '0906.0020-2-61-2'], ['0906.0020-1-53-3', '0906.0020-2-61-3'], ['0906.0020-1-53-4', '0906.0020-2-61-4'], ['0906.0020-1-53-5', '0906.0020-2-61-5'], ['0906.0020-1-53-6', '0906.0020-2-61-6'], ['0906.0020-1-33-0', '0906.0020-2-35-0'], ['0906.0020-1-33-1', '0906.0020-2-35-1'], ['0906.0020-1-32-0', '0906.0020-2-34-0'], ['0906.0020-1-22-0', '0906.0020-2-24-0'], ['0906.0020-1-22-1', '0906.0020-2-24-1'], ['0906.0020-1-22-2', '0906.0020-2-24-2'], ['0906.0020-1-22-3', '0906.0020-2-24-3'], ['0906.0020-1-51-0', '0906.0020-2-58-0'], ['0906.0020-1-51-1', '0906.0020-2-58-1'], ['0906.0020-1-51-2', '0906.0020-2-58-2'], ['0906.0020-1-51-4', '0906.0020-2-58-4'], ['0906.0020-1-13-0', '0906.0020-2-13-0'], ['0906.0020-1-13-1', '0906.0020-2-13-1'], ['0906.0020-1-49-0', '0906.0020-2-56-0'], ['0906.0020-1-49-1', '0906.0020-2-56-1'], ['0906.0020-1-49-2', '0906.0020-2-56-2'], ['0906.0020-1-49-3', '0906.0020-2-56-3'], ['0906.0020-1-35-0', '0906.0020-2-37-0'], ['0906.0020-1-35-1', '0906.0020-2-37-1'], ['0906.0020-1-35-4', '0906.0020-2-37-4'], ['0906.0020-1-35-5', '0906.0020-2-37-5'], ['0906.0020-1-35-6', '0906.0020-2-37-6'], ['0906.0020-1-35-7', '0906.0020-2-37-7'], ['0906.0020-1-40-0', '0906.0020-2-42-0'], ['0906.0020-1-40-2', '0906.0020-2-42-2'], ['0906.0020-1-40-3', '0906.0020-2-42-3'], ['0906.0020-1-40-4', '0906.0020-2-42-4'], ['0906.0020-1-40-5', '0906.0020-2-42-5'], ['0906.0020-1-40-6', '0906.0020-2-42-6'], ['0906.0020-1-40-7', '0906.0020-2-42-7'], ['0906.0020-1-28-0', '0906.0020-2-30-0'], ['0906.0020-1-28-1', '0906.0020-2-30-1'], ['0906.0020-1-28-2', '0906.0020-2-30-2'], ['0906.0020-1-28-3', '0906.0020-2-30-3'], ['0906.0020-1-26-0', '0906.0020-2-28-0'], ['0906.0020-1-26-1', '0906.0020-2-28-1'], ['0906.0020-1-26-2', '0906.0020-2-28-2'], ['0906.0020-1-26-3', '0906.0020-2-28-3'], ['0906.0020-1-31-1', '0906.0020-2-33-1'], ['0906.0020-1-31-2', '0906.0020-2-33-2'], ['0906.0020-1-31-3', '0906.0020-2-33-3'], ['0906.0020-1-31-4', '0906.0020-2-33-4'], ['0906.0020-1-31-5', '0906.0020-2-33-5'], ['0906.0020-1-31-6', '0906.0020-2-33-6'], ['0906.0020-1-17-0', '0906.0020-2-19-0'], ['0906.0020-1-17-1', '0906.0020-2-19-1'], ['0906.0020-1-17-4', '0906.0020-2-19-5'], ['0906.0020-1-17-5', '0906.0020-2-19-6'], ['0906.0020-1-44-0', '0906.0020-2-50-0'], ['0906.0020-1-44-1', '0906.0020-2-50-1'], ['0906.0020-1-20-0', '0906.0020-2-22-0'], ['0906.0020-1-20-1', '0906.0020-2-22-1'], ['0906.0020-1-20-2', '0906.0020-2-22-2'], ['0906.0020-1-20-3', '0906.0020-2-22-3'], ['0906.0020-1-20-4', '0906.0020-2-22-4'], ['0906.0020-1-20-5', '0906.0020-2-22-5'], ['0906.0020-1-20-6', '0906.0020-2-22-6'], ['0906.0020-1-20-7', '0906.0020-2-22-7'], ['0906.0020-1-20-8', '0906.0020-2-22-8'], ['0906.0020-1-2-0', '0906.0020-2-2-0'], ['0906.0020-1-2-1', '0906.0020-2-2-1'], ['0906.0020-1-2-2', '0906.0020-2-2-2'], ['0906.0020-1-2-3', '0906.0020-2-2-3'], ['0906.0020-1-2-4', '0906.0020-2-2-4'], ['0906.0020-1-2-5', '0906.0020-2-2-5'], ['0906.0020-1-2-6', '0906.0020-2-2-6'], ['0906.0020-1-2-7', '0906.0020-2-2-7'], ['0906.0020-1-12-0', '0906.0020-2-12-0'], ['0906.0020-1-12-1', '0906.0020-2-12-1'], ['0906.0020-1-6-0', '0906.0020-2-6-0'], ['0906.0020-1-6-1', '0906.0020-2-6-1'], ['0906.0020-1-6-2', '0906.0020-2-6-2'], ['0906.0020-1-6-3', '0906.0020-2-6-3'], ['0906.0020-1-6-4', '0906.0020-2-6-4'], ['0906.0020-1-9-1', '0906.0020-2-9-1'], ['0906.0020-1-9-2', '0906.0020-2-9-2'], ['0906.0020-1-9-3', '0906.0020-2-9-3'], ['0906.0020-1-9-4', '0906.0020-2-9-4'], ['0906.0020-1-9-5', '0906.0020-2-9-5'], ['0906.0020-1-9-6', '0906.0020-2-9-6'], ['0906.0020-1-9-7', '0906.0020-2-9-7'], ['0906.0020-1-9-8', '0906.0020-2-9-8'], ['0906.0020-1-21-0', '0906.0020-2-23-0'], ['0906.0020-1-21-1', '0906.0020-2-23-1'], ['0906.0020-1-21-2', '0906.0020-2-23-2'], ['0906.0020-1-21-3', '0906.0020-2-23-3'], ['0906.0020-1-21-4', '0906.0020-2-23-4'], ['0906.0020-1-48-0', '0906.0020-2-55-0'], ['0906.0020-1-48-1', '0906.0020-2-55-1'], ['0906.0020-1-48-2', '0906.0020-2-55-2'], ['0906.0020-1-48-3', '0906.0020-2-55-3'], ['0906.0020-1-48-4', '0906.0020-2-55-4'], ['0906.0020-1-48-5', '0906.0020-2-55-5'], ['0906.0020-1-25-0', '0906.0020-2-27-0'], ['0906.0020-1-25-1', '0906.0020-2-27-1'], ['0906.0020-1-25-2', '0906.0020-2-27-2'], ['0906.0020-1-5-0', '0906.0020-2-5-0'], ['0906.0020-1-5-1', '0906.0020-2-5-1'], ['0906.0020-1-5-2', '0906.0020-2-5-2'], ['0906.0020-1-5-3', '0906.0020-2-5-3'], ['0906.0020-1-30-0', '0906.0020-2-32-0'], ['0906.0020-1-30-1', '0906.0020-2-32-1'], ['0906.0020-1-30-2', '0906.0020-2-32-2'], ['0906.0020-1-30-3', '0906.0020-2-32-3'], ['0906.0020-1-30-4', '0906.0020-2-32-4'], ['0906.0020-1-30-5', '0906.0020-2-32-5'], ['0906.0020-1-14-0', '0906.0020-2-14-0'], ['0906.0020-1-14-1', '0906.0020-2-14-1'], ['0906.0020-1-14-2', '0906.0020-2-14-3'], ['0906.0020-1-14-3', '0906.0020-2-14-4'], ['0906.0020-1-14-4', '0906.0020-2-14-5'], ['0906.0020-1-14-5', '0906.0020-2-14-6'], ['0906.0020-1-14-6', '0906.0020-2-14-7'], ['0906.0020-1-14-7', '0906.0020-2-14-8'], ['0906.0020-1-14-8', '0906.0020-2-14-9'], ['0906.0020-1-15-0', '0906.0020-2-15-0'], ['0906.0020-1-15-1', '0906.0020-2-15-1'], ['0906.0020-1-15-3', '0906.0020-2-16-0'], ['0906.0020-1-15-4', '0906.0020-2-16-1'], ['0906.0020-1-15-5', '0906.0020-2-16-2'], ['0906.0020-1-15-6', '0906.0020-2-16-3'], ['0906.0020-1-15-7', '0906.0020-2-16-4'], ['0906.0020-1-15-8', '0906.0020-2-16-5'], ['0906.0020-1-42-0', '0906.0020-2-46-0'], ['0906.0020-1-0-4', '0906.0020-2-0-4'], ['0906.0020-1-0-5', '0906.0020-2-0-5'], ['0906.0020-1-0-6', '0906.0020-2-0-6'], ['0906.0020-1-52-0', '0906.0020-2-60-0'], ['0906.0020-1-51-3', '0906.0020-2-58-3'], ['0906.0020-1-35-2', '0906.0020-2-37-2'], ['0906.0020-1-40-1', '0906.0020-2-42-1'], ['0906.0020-1-31-0', '0906.0020-2-33-0'], ['0906.0020-1-17-2', '0906.0020-2-19-3'], ['0906.0020-1-17-3', '0906.0020-2-19-4'], ['0906.0020-1-17-6', '0906.0020-2-19-7'], ['0906.0020-1-54-0', '0906.0020-2-63-0'], ['0906.0020-1-0-3', '0906.0020-2-0-3'], ['0906.0020-1-33-2', '0906.0020-2-35-2'], ['0906.0020-1-15-2', '0906.0020-2-15-2'], ['0906.0020-1-15-2', '0906.0020-2-15-3']]

[['0906.0020-1-3-0', '0906.0020-2-3-0'], ['0906.0020-1-3-1', '0906.0020-2-3-1'], ['0906.0020-1-3-2', '0906.0020-2-3-2'], ['0906.0020-1-3-3', '0906.0020-2-3-3'], ['0906.0020-1-10-0', '0906.0020-2-10-0'], ['0906.0020-1-10-1', '0906.0020-2-10-1'], ['0906.0020-1-10-2', '0906.0020-2-10-2'], ['0906.0020-1-10-3', '0906.0020-2-10-3'], ['0906.0020-1-10-4', '0906.0020-2-10-4'], ['0906.0020-1-10-5', '0906.0020-2-10-5'], ['0906.0020-1-4-0', '0906.0020-2-4-0'], ['0906.0020-1-4-1', '0906.0020-2-4-1'], ['0906.0020-1-4-2', '0906.0020-2-4-2'], ['0906.0020-1-4-3', '0906.0020-2-4-3'], ['0906.0020-1-4-4', '0906.0020-2-4-4'], ['0906.0020-1-4-5', '0906.0020-2-4-5'], ['0906.0020-1-4-6', '0906.0020-2-4-6'], ['0906.0020-1-4-7', '0906.0020-2-4-7'], ['0906.0020-1-4-8', '0906.0020-2-4-8'], ['0906.0020-1-42-1', '0906.0020-2-46-1'], ['0906.0020-1-42-2', '0906.0020-2-46-2'], ['0906.0020-1-42-3', '0906.0020-2-46-3'], ['0906.0020-1-55-0', '0906.0020-2-64-0'], ['0906.0020-1-55-1', '0906.0020-2-64-1'], ['0906.0020-1-55-2', '0906.0020-2-64-2'], ['0906.0020-1-55-3', '0906.0020-2-64-3'], ['0906.0020-1-55-4', '0906.0020-2-64-4'], ['0906.0020-1-55-5', '0906.0020-2-64-5'], ['0906.0020-1-43-0', '0906.0020-2-49-0'], ['0906.0020-1-43-1', '0906.0020-2-49-1'], ['0906.0020-1-0-0', '0906.0020-2-0-0'], ['0906.0020-1-0-1', '0906.0020-2-0-1'], ['0906.0020-1-0-2', '0906.0020-2-0-2'], ['0906.0020-1-0-7', '0906.0020-2-0-7'], ['0906.0020-1-0-8', '0906.0020-2-0-8'], ['0906.0020-1-52-1', '0906.0020-2-60-1'], ['0906.0020-1-52-2', '0906.0020-2-60-2'], ['0906.0020-1-52-3', '0906.0020-2-60-3'], ['0906.0020-1-52-4', '0906.0020-2-60-4'], ['0906.0020-1-52-5', '0906.0020-2-60-5'], ['0906.0020-1-47-0', '0906.0020-2-54-0'], ['0906.0020-1-47-1', '0906.0020-2-54-1'], ['0906.0020-1-47-2', '0906.0020-2-54-2'], ['0906.0020-1-47-3', '0906.0020-2-54-3'], ['0906.0020-1-47-4', '0906.0020-2-54-4'], ['0906.0020-1-47-5', '0906.0020-2-54-5'], ['0906.0020-1-29-0', '0906.0020-2-31-0'], ['0906.0020-1-29-1', '0906.0020-2-31-1'], ['0906.0020-1-36-0', '0906.0020-2-38-0'], ['0906.0020-1-36-1', '0906.0020-2-38-1'], ['0906.0020-1-36-2', '0906.0020-2-38-2'], ['0906.0020-1-36-3', '0906.0020-2-38-3'], ['0906.0020-1-23-0', '0906.0020-2-25-0'], ['0906.0020-1-23-1', '0906.0020-2-25-1'], ['0906.0020-1-23-2', '0906.0020-2-25-2'], ['0906.0020-1-23-3', '0906.0020-2-25-3'], ['0906.0020-1-16-0', '0906.0020-2-17-0'], ['0906.0020-1-16-1', '0906.0020-2-17-1'], ['0906.0020-1-16-3', '0906.0020-2-17-3'], ['0906.0020-1-16-4', '0906.0020-2-17-4'], ['0906.0020-1-16-5', '0906.0020-2-17-5'], ['0906.0020-1-38-0', '0906.0020-2-40-0'], ['0906.0020-1-38-1', '0906.0020-2-40-1'], ['0906.0020-1-46-0', '0906.0020-2-53-0'], ['0906.0020-1-46-1', '0906.0020-2-53-1'], ['0906.0020-1-46-2', '0906.0020-2-53-2'], ['0906.0020-1-46-3', '0906.0020-2-53-3'], ['0906.0020-1-53-0', '0906.0020-2-61-0'], ['0906.0020-1-53-1', '0906.0020-2-61-1'], ['0906.0020-1-53-2', '0906.0020-2-61-2'], ['0906.0020-1-53-3', '0906.0020-2-61-3'], ['0906.0020-1-53-4', '0906.0020-2-61-4'], ['0906.0020-1-53-5', '0906.0020-2-61-5'], ['0906.0020-1-53-6', '0906.0020-2-61-6'], ['0906.0020-1-33-0', '0906.0020-2-35-0'], ['0906.0020-1-33-1', '0906.0020-2-35-1'], ['0906.0020-1-32-0', '0906.0020-2-34-0'], ['0906.0020-1-22-0', '0906.0020-2-24-0'], ['0906.0020-1-22-1', '0906.0020-2-24-1'], ['0906.0020-1-22-2', '0906.0020-2-24-2'], ['0906.0020-1-22-3', '0906.0020-2-24-3'], ['0906.0020-1-51-0', '0906.0020-2-58-0'], ['0906.0020-1-51-1', '0906.0020-2-58-1'], ['0906.0020-1-51-2', '0906.0020-2-58-2'], ['0906.0020-1-51-4', '0906.0020-2-58-4'], ['0906.0020-1-13-0', '0906.0020-2-13-0'], ['0906.0020-1-13-1', '0906.0020-2-13-1'], ['0906.0020-1-49-0', '0906.0020-2-56-0'], ['0906.0020-1-49-1', '0906.0020-2-56-1'], ['0906.0020-1-49-2', '0906.0020-2-56-2'], ['0906.0020-1-49-3', '0906.0020-2-56-3'], ['0906.0020-1-35-0', '0906.0020-2-37-0'], ['0906.0020-1-35-1', '0906.0020-2-37-1'], ['0906.0020-1-35-4', '0906.0020-2-37-4'], ['0906.0020-1-35-5', '0906.0020-2-37-5'], ['0906.0020-1-35-6', '0906.0020-2-37-6'], ['0906.0020-1-35-7', '0906.0020-2-37-7'], ['0906.0020-1-40-0', '0906.0020-2-42-0'], ['0906.0020-1-40-2', '0906.0020-2-42-2'], ['0906.0020-1-40-3', '0906.0020-2-42-3'], ['0906.0020-1-40-4', '0906.0020-2-42-4'], ['0906.0020-1-40-5', '0906.0020-2-42-5'], ['0906.0020-1-40-6', '0906.0020-2-42-6'], ['0906.0020-1-40-7', '0906.0020-2-42-7'], ['0906.0020-1-28-0', '0906.0020-2-30-0'], ['0906.0020-1-28-1', '0906.0020-2-30-1'], ['0906.0020-1-28-2', '0906.0020-2-30-2'], ['0906.0020-1-28-3', '0906.0020-2-30-3'], ['0906.0020-1-26-0', '0906.0020-2-28-0'], ['0906.0020-1-26-1', '0906.0020-2-28-1'], ['0906.0020-1-26-2', '0906.0020-2-28-2'], ['0906.0020-1-26-3', '0906.0020-2-28-3'], ['0906.0020-1-31-1', '0906.0020-2-33-1'], ['0906.0020-1-31-2', '0906.0020-2-33-2'], ['0906.0020-1-31-3', '0906.0020-2-33-3'], ['0906.0020-1-31-4', '0906.0020-2-33-4'], ['0906.0020-1-31-5', '0906.0020-2-33-5'], ['0906.0020-1-31-6', '0906.0020-2-33-6'], ['0906.0020-1-17-0', '0906.0020-2-19-0'], ['0906.0020-1-17-1', '0906.0020-2-19-1'], ['0906.0020-1-17-4', '0906.0020-2-19-5'], ['0906.0020-1-17-5', '0906.0020-2-19-6'], ['0906.0020-1-44-0', '0906.0020-2-50-0'], ['0906.0020-1-44-1', '0906.0020-2-50-1'], ['0906.0020-1-20-0', '0906.0020-2-22-0'], ['0906.0020-1-20-1', '0906.0020-2-22-1'], ['0906.0020-1-20-2', '0906.0020-2-22-2'], ['0906.0020-1-20-3', '0906.0020-2-22-3'], ['0906.0020-1-20-4', '0906.0020-2-22-4'], ['0906.0020-1-20-5', '0906.0020-2-22-5'], ['0906.0020-1-20-6', '0906.0020-2-22-6'], ['0906.0020-1-20-7', '0906.0020-2-22-7'], ['0906.0020-1-20-8', '0906.0020-2-22-8'], ['0906.0020-1-2-0', '0906.0020-2-2-0'], ['0906.0020-1-2-1', '0906.0020-2-2-1'], ['0906.0020-1-2-2', '0906.0020-2-2-2'], ['0906.0020-1-2-3', '0906.0020-2-2-3'], ['0906.0020-1-2-4', '0906.0020-2-2-4'], ['0906.0020-1-2-5', '0906.0020-2-2-5'], ['0906.0020-1-2-6', '0906.0020-2-2-6'], ['0906.0020-1-2-7', '0906.0020-2-2-7'], ['0906.0020-1-12-0', '0906.0020-2-12-0'], ['0906.0020-1-12-1', '0906.0020-2-12-1'], ['0906.0020-1-6-0', '0906.0020-2-6-0'], ['0906.0020-1-6-1', '0906.0020-2-6-1'], ['0906.0020-1-6-2', '0906.0020-2-6-2'], ['0906.0020-1-6-3', '0906.0020-2-6-3'], ['0906.0020-1-6-4', '0906.0020-2-6-4'], ['0906.0020-1-9-1', '0906.0020-2-9-1'], ['0906.0020-1-9-2', '0906.0020-2-9-2'], ['0906.0020-1-9-3', '0906.0020-2-9-3'], ['0906.0020-1-9-4', '0906.0020-2-9-4'], ['0906.0020-1-9-5', '0906.0020-2-9-5'], ['0906.0020-1-9-6', '0906.0020-2-9-6'], ['0906.0020-1-9-7', '0906.0020-2-9-7'], ['0906.0020-1-9-8', '0906.0020-2-9-8'], ['0906.0020-1-21-0', '0906.0020-2-23-0'], ['0906.0020-1-21-1', '0906.0020-2-23-1'], ['0906.0020-1-21-2', '0906.0020-2-23-2'], ['0906.0020-1-21-3', '0906.0020-2-23-3'], ['0906.0020-1-21-4', '0906.0020-2-23-4'], ['0906.0020-1-48-0', '0906.0020-2-55-0'], ['0906.0020-1-48-1', '0906.0020-2-55-1'], ['0906.0020-1-48-2', '0906.0020-2-55-2'], ['0906.0020-1-48-3', '0906.0020-2-55-3'], ['0906.0020-1-48-4', '0906.0020-2-55-4'], ['0906.0020-1-48-5', '0906.0020-2-55-5'], ['0906.0020-1-25-0', '0906.0020-2-27-0'], ['0906.0020-1-25-1', '0906.0020-2-27-1'], ['0906.0020-1-25-2', '0906.0020-2-27-2'], ['0906.0020-1-5-0', '0906.0020-2-5-0'], ['0906.0020-1-5-1', '0906.0020-2-5-1'], ['0906.0020-1-5-2', '0906.0020-2-5-2'], ['0906.0020-1-5-3', '0906.0020-2-5-3'], ['0906.0020-1-30-0', '0906.0020-2-32-0'], ['0906.0020-1-30-1', '0906.0020-2-32-1'], ['0906.0020-1-30-2', '0906.0020-2-32-2'], ['0906.0020-1-30-3', '0906.0020-2-32-3'], ['0906.0020-1-30-4', '0906.0020-2-32-4'], ['0906.0020-1-30-5', '0906.0020-2-32-5'], ['0906.0020-1-14-0', '0906.0020-2-14-0'], ['0906.0020-1-14-1', '0906.0020-2-14-1'], ['0906.0020-1-14-2', '0906.0020-2-14-3'], ['0906.0020-1-14-3', '0906.0020-2-14-4'], ['0906.0020-1-14-4', '0906.0020-2-14-5'], ['0906.0020-1-14-5', '0906.0020-2-14-6'], ['0906.0020-1-14-6', '0906.0020-2-14-7'], ['0906.0020-1-14-7', '0906.0020-2-14-8'], ['0906.0020-1-14-8', '0906.0020-2-14-9'], ['0906.0020-1-15-0', '0906.0020-2-15-0'], ['0906.0020-1-15-1', '0906.0020-2-15-1'], ['0906.0020-1-15-3', '0906.0020-2-16-0'], ['0906.0020-1-15-4', '0906.0020-2-16-1'], ['0906.0020-1-15-5', '0906.0020-2-16-2'], ['0906.0020-1-15-6', '0906.0020-2-16-3'], ['0906.0020-1-15-7', '0906.0020-2-16-4'], ['0906.0020-1-15-8', '0906.0020-2-16-5']]

[['0906.0020-1-42-0', '0906.0020-2-46-0'], ['0906.0020-1-0-4', '0906.0020-2-0-4'], ['0906.0020-1-0-5', '0906.0020-2-0-5'], ['0906.0020-1-0-6', '0906.0020-2-0-6'], ['0906.0020-1-52-0', '0906.0020-2-60-0'], ['0906.0020-1-51-3', '0906.0020-2-58-3'], ['0906.0020-1-35-2', '0906.0020-2-37-2'], ['0906.0020-1-40-1', '0906.0020-2-42-1'], ['0906.0020-1-31-0', '0906.0020-2-33-0'], ['0906.0020-1-17-2', '0906.0020-2-19-3'], ['0906.0020-1-17-3', '0906.0020-2-19-4'], ['0906.0020-1-17-6', '0906.0020-2-19-7'], ['0906.0020-1-54-0', '0906.0020-2-63-0']]

[]

[['0906.0020-1-0-3', '0906.0020-2-0-3'], ['0906.0020-1-33-2', '0906.0020-2-35-2'], ['0906.0020-1-15-2', '0906.0020-2-15-2'], ['0906.0020-1-15-2', '0906.0020-2-15-3']]

[]

['0906.0020-1-9-0', '0906.0020-1-16-2', '0906.0020-2-9-0', '0906.0020-2-17-2']

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

https://arxiv.org/abs/0906.0020

1206.6958

{'1206.6958-1-0-0': 'In this work we consider pair production of LSP tau-sneutrinos at the Compact Lineer Collider.', '1206.6958-1-0-1': 'We assume that tau-sneutrinos decays in to e pair via RPV interactions.', '1206.6958-1-0-2': 'Backgroundless subprocess [MATH] is analyzed in details.', '1206.6958-1-0-3': 'Achievable limits on [MATH] at [MATH] and [MATH] CL are obtained depending on [MATH] mass.', '1206.6958-1-1-0': 'Supersymetry (SUSY) is one of the favorite candidates for the Beyond the Standard Model (BSM) physics [CITATION].', '1206.6958-1-1-1': 'For this reason, searching for supersymetric particles forms an essential part of the LHC, as well as future colliders, experimental programes.', '1206.6958-1-1-2': 'Searching strategy for SUSY strongly depends on the lightest supersymetric particle (LSP), as well as R-parity conservation or violation [CITATION].', '1206.6958-1-1-3': 'If the lightest supersymetric particles is the tau-sneutrino, its decay may be realized only via R-parity violation (RPV): [MATH], [MATH].', '1206.6958-1-2-0': 'If R parity is violating, [MATH] process becomes very important.', '1206.6958-1-2-1': 'In this paper [MATH] pair production at the CLIC with subsequent RPV decays in to [MATH] pairs has been investigated.', '1206.6958-1-2-2': 'Feynman diagram for tau-sneutrino production process is shown in Figure 1.', '1206.6958-1-3-0': 'The R-parity violation part of the MSSM superpotential is given by [CITATION] [EQUATION] where [MATH] is an [MATH] doublet (singlet) lepton superfield and [MATH] is (are) an [MATH] doublet (singlet) quark superfield(s), and indices [MATH] denote flavour.', '1206.6958-1-3-1': 'The coefficients [MATH] and [MATH] correspond to the lepton number violating and baryon number violating couplings, respectively.', '1206.6958-1-4-0': 'RPV interaction Lagrangian responsible for is [MATH] and [MATH] decays is given below: [EQUATION]', '1206.6958-1-4-1': 'For the numerical calculations we implement the Lagrangian ([MATH]) into the CALCHEP MSSM package [CITATION].', '1206.6958-1-4-2': 'The cross-section for pair production of tau-sneutrinos at CLIC with [MATH] TeV is shown Figure 2.', '1206.6958-1-4-3': 'Initial State Radiation (ISR) and Beamstrahlung effects at CLIC are calculated with CALCHEP program using beam parameters given in Table 1 [CITATION].', '1206.6958-1-5-0': 'It is seen from Figure 2, that ISR and BS effects leads to increasing (decreasing) of the cross-section for [MATH]below (above) 130 GeV.', '1206.6958-1-5-1': 'In Figure 3 we present similar calculations for the CLIC with [MATH] TeV.', '1206.6958-1-5-2': 'It is seen that ISR and BS effects are more effective at higher center of mass energy.', '1206.6958-1-5-3': 'ISR+BS effects lead to increasing of the cross sections for [MATH] GeV.', '1206.6958-1-5-4': 'After [MATH] GeV, ISR+BS effects decrease the cross sections.', '1206.6958-1-6-0': 'We propose [MATH] backgroundless process to analysis at CLIC.', '1206.6958-1-6-1': 'In order to analyze signal following basic cuts are applied: [MATH] GeV and [MATH] for the final state electrons and muons.', '1206.6958-1-6-2': 'Assuming [MATH] and taking other possible RPV interaction constant to be [MATH], which means [MATH], we obtain cross section values given in Table 2 (3) for CLIC with [MATH] TeV ([MATH] TeV).', '1206.6958-1-6-3': 'Event number given in the last columns include both [MATH] and [MATH] final states.', '1206.6958-1-7-0': 'In order to estimate statistical significance we have used following formula: [EQUATION]', '1206.6958-1-7-1': 'Here, S is statistical significance, [MATH] is signal cross sections values, [MATH] is background cross sections and [MATH] is integrated luminosity.', '1206.6958-1-7-2': 'We have backgroundless processes therefore [MATH] is taken zero.', '1206.6958-1-7-3': 'From Eq. (3), discovery ([MATH]), observation ([MATH]) and exclusion ([MATH]) limits for tau-sneutrino at CLIC with [MATH] TeV are obtained as follows: achievable tau-sneutrino mass values are [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] GeV for exclusion.', '1206.6958-1-7-4': 'Corresponding values for CLIC with [MATH] TeV are : [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] for exclusion.', '1206.6958-1-8-0': 'So far the ideal case, namely, maximal possible Branching Ratio (Br) for the channel madden consideration had been analyzed.', '1206.6958-1-8-1': 'In more general case Branching Ratio is less than [MATH], because of other possible decay channels.', '1206.6958-1-8-2': 'In Figure 4 we present [MATH], [MATH] and [MATH] plots for [MATH] depending on the [MATH] mass for CLIC with [MATH] TeV.', '1206.6958-1-8-3': 'One can see from Figure 4 that for [MATH] times smaller than ideal case) [MATH], [MATH] and [MATH] limits become [MATH] GeV, [MATH] GeV and [MATH] GeV, respectively.', '1206.6958-1-8-4': 'Corresponding plot for [MATH] TeV are presented in Figure 5.', '1206.6958-1-9-0': 'In the Table 4 (5) discovery, observation and exclusion limits for [MATH] at the CLIC with [MATH] TeV ([MATH] TeV) are given for several values of the [MATH] mass.', '1206.6958-1-10-0': 'In conclusion, the process [MATH] will provide powerful signature for LSP [MATH], if [MATH] and [MATH] are sufficiently large.'}

{'1206.6958-2-0-0': 'In this work we consider pair production of LSP tau-sneutrinos at the Compact Lineer Collider.', '1206.6958-2-0-1': 'We assume that tau-sneutrinos decays in to e pair via RPV interactions.', '1206.6958-2-0-2': 'Backgroundless subprocess [MATH] is analyzed in details.', '1206.6958-2-0-3': 'Achievable limits on [MATH] at [MATH] and [MATH] CL are obtained depending on [MATH] mass.', '1206.6958-2-1-0': 'Supersymetry (SUSY) is one of the favorite candidates for the Beyond the Standard Model (BSM) physics [CITATION].', '1206.6958-2-1-1': 'For this reason, searching for supersymetric particles forms an essential part of the LHC, as well as future colliders, experimental programes.', '1206.6958-2-1-2': 'Searching strategy for SUSY strongly depends on the lightest supersymetric particle (LSP).', '1206.6958-2-1-3': 'The sneutrino can be considered as another candidate for the LSP [CITATION], as well as R-parity conservation or violation [CITATION].', '1206.6958-2-1-4': 'In addition the R-parity violation leads to new physics further than R-parity conservation [CITATION].', '1206.6958-2-1-5': 'R-parity is represented by [MATH] where [MATH] and [MATH] are the baryon and lepton numbers and [MATH] is spin [CITATION].', '1206.6958-2-1-6': 'If the lightest supersymetric particles is the tau-sneutrino, its decay may be realized only via R-parity violation (RPV): [MATH], [MATH].', '1206.6958-2-2-0': 'If R parity is violating, [MATH] process becomes very important.', '1206.6958-2-2-1': 'In this paper [MATH] pair production at the CLIC with subsequent RPV decays in to [MATH] pairs has been investigated.', '1206.6958-2-2-2': 'Feynman diagram for tau-sneutrino production process is shown in Figure 1.', '1206.6958-2-3-0': 'The R-parity violation part of the MSSM superpotential is given by [CITATION] [EQUATION] where [MATH] is an [MATH] doublet (singlet) lepton superfield and [MATH] is (are) an [MATH] doublet (singlet) quark superfield(s), and indices [MATH] denote flavour.', '1206.6958-2-3-1': 'The coefficients [MATH] and [MATH] correspond to the lepton number violating and baryon number violating couplings, respectively.', '1206.6958-2-4-0': 'RPV interaction Lagrangian responsible for is [MATH] and [MATH] decays is given below: [EQUATION]', '1206.6958-2-4-1': 'For the numerical calculations we implement the Lagrangian ([MATH]) into the CALCHEP MSSM package [CITATION].', '1206.6958-2-4-2': 'The cross-section for pair production of tau-sneutrinos at CLIC with [MATH] TeV is shown Figure 2.', '1206.6958-2-4-3': 'Initial State Radiation (ISR) and Beamstrahlung effects at CLIC are calculated with CALCHEP program using beam parameters given in Table 1 [CITATION].', '1206.6958-2-5-0': 'It is seen from Figure 2, that ISR and BS effects leads to increasing (decreasing) of the cross-section for [MATH]below (above) 130 GeV.', '1206.6958-2-5-1': 'In Figure 3 we present similar calculations for the CLIC with [MATH] TeV.', '1206.6958-2-5-2': 'It is seen that ISR and BS effects are more effective at higher center of mass energy.', '1206.6958-2-5-3': 'ISR+BS effects lead to increasing of the cross sections for [MATH] GeV.', '1206.6958-2-5-4': 'After [MATH] GeV, ISR+BS effects decrease the cross sections.', '1206.6958-2-6-0': 'We propose [MATH] backgroundless process to analysis at CLIC.', '1206.6958-2-6-1': 'In order to analyze signal following basic cuts are applied: [MATH] GeV and [MATH] for the final state electrons and muons.', '1206.6958-2-6-2': 'Assuming [MATH] and taking other possible RPV interaction constant to be [MATH], which means [MATH], we obtain cross section values given in Table 2 (3) for CLIC with [MATH] TeV ([MATH] TeV).', '1206.6958-2-6-3': 'Event number given in the last columns include both [MATH] and [MATH] final states.', '1206.6958-2-7-0': 'In order to estimate statistical significance we have used following formula: [EQUATION]', '1206.6958-2-7-1': 'Here, S is statistical significance, [MATH] is signal cross sections values, [MATH] is background cross sections and [MATH] is integrated luminosity.', '1206.6958-2-7-2': 'We have backgroundless processes therefore [MATH] is taken zero.', '1206.6958-2-7-3': 'From Eq. (3), discovery ([MATH]), observation ([MATH]) and exclusion ([MATH]) limits for tau-sneutrino at CLIC with [MATH] TeV are obtained as follows: achievable tau-sneutrino mass values are [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] GeV for exclusion.', '1206.6958-2-7-4': 'Corresponding values for CLIC with [MATH] TeV are : [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] for exclusion.', '1206.6958-2-8-0': 'So far the ideal case, namely, maximal possible Branching Ratio (Br) for the channel madden consideration had been analyzed.', '1206.6958-2-8-1': 'In more general case Branching Ratio is less than [MATH], because of other possible decay channels.', '1206.6958-2-8-2': 'In Figure 4 we present [MATH], [MATH] and [MATH] plots for [MATH] depending on the [MATH] mass for CLIC with [MATH] TeV.', '1206.6958-2-8-3': 'One can see from Figure 4 that for [MATH] times smaller than ideal case) [MATH], [MATH] and [MATH] limits become [MATH] GeV, [MATH] GeV and [MATH] GeV, respectively.', '1206.6958-2-8-4': 'Corresponding plot for [MATH] TeV are presented in Figure 5.', '1206.6958-2-9-0': 'In the Table 4 (5) discovery, observation and exclusion limits for [MATH] at the CLIC with [MATH] TeV ([MATH] TeV) are given for several values of the [MATH] mass.', '1206.6958-2-10-0': 'In conclusion, the process [MATH] will provide powerful signature for LSP [MATH], if [MATH] and [MATH] are sufficiently large.'}

[['1206.6958-1-5-0', '1206.6958-2-5-0'], ['1206.6958-1-5-1', '1206.6958-2-5-1'], ['1206.6958-1-5-2', '1206.6958-2-5-2'], ['1206.6958-1-5-3', '1206.6958-2-5-3'], ['1206.6958-1-5-4', '1206.6958-2-5-4'], ['1206.6958-1-9-0', '1206.6958-2-9-0'], ['1206.6958-1-7-0', '1206.6958-2-7-0'], ['1206.6958-1-7-1', '1206.6958-2-7-1'], ['1206.6958-1-7-2', '1206.6958-2-7-2'], ['1206.6958-1-7-3', '1206.6958-2-7-3'], ['1206.6958-1-7-4', '1206.6958-2-7-4'], ['1206.6958-1-10-0', '1206.6958-2-10-0'], ['1206.6958-1-3-0', '1206.6958-2-3-0'], ['1206.6958-1-3-1', '1206.6958-2-3-1'], ['1206.6958-1-0-0', '1206.6958-2-0-0'], ['1206.6958-1-0-1', '1206.6958-2-0-1'], ['1206.6958-1-0-2', '1206.6958-2-0-2'], ['1206.6958-1-0-3', '1206.6958-2-0-3'], ['1206.6958-1-4-0', '1206.6958-2-4-0'], ['1206.6958-1-4-1', '1206.6958-2-4-1'], ['1206.6958-1-4-2', '1206.6958-2-4-2'], ['1206.6958-1-4-3', '1206.6958-2-4-3'], ['1206.6958-1-2-0', '1206.6958-2-2-0'], ['1206.6958-1-2-1', '1206.6958-2-2-1'], ['1206.6958-1-2-2', '1206.6958-2-2-2'], ['1206.6958-1-1-0', '1206.6958-2-1-0'], ['1206.6958-1-1-1', '1206.6958-2-1-1'], ['1206.6958-1-1-3', '1206.6958-2-1-6'], ['1206.6958-1-8-0', '1206.6958-2-8-0'], ['1206.6958-1-8-1', '1206.6958-2-8-1'], ['1206.6958-1-8-2', '1206.6958-2-8-2'], ['1206.6958-1-8-3', '1206.6958-2-8-3'], ['1206.6958-1-8-4', '1206.6958-2-8-4'], ['1206.6958-1-6-0', '1206.6958-2-6-0'], ['1206.6958-1-6-1', '1206.6958-2-6-1'], ['1206.6958-1-6-2', '1206.6958-2-6-2'], ['1206.6958-1-6-3', '1206.6958-2-6-3'], ['1206.6958-2-0-0', '1206.6958-3-0-0'], ['1206.6958-2-0-1', '1206.6958-3-0-1'], ['1206.6958-2-0-2', '1206.6958-3-0-2'], ['1206.6958-2-0-3', '1206.6958-3-0-3'], ['1206.6958-2-7-0', '1206.6958-3-7-0'], ['1206.6958-2-7-1', '1206.6958-3-7-1'], ['1206.6958-2-7-2', '1206.6958-3-7-2'], ['1206.6958-2-7-3', '1206.6958-3-7-3'], ['1206.6958-2-7-4', '1206.6958-3-7-4'], ['1206.6958-2-6-0', '1206.6958-3-6-0'], ['1206.6958-2-6-1', '1206.6958-3-6-1'], ['1206.6958-2-6-3', '1206.6958-3-6-3'], ['1206.6958-2-4-0', '1206.6958-3-4-0'], ['1206.6958-2-4-1', '1206.6958-3-4-1'], ['1206.6958-2-4-2', '1206.6958-3-4-2'], ['1206.6958-2-4-3', '1206.6958-3-4-3'], ['1206.6958-2-8-1', '1206.6958-3-8-1'], ['1206.6958-2-8-2', '1206.6958-3-8-2'], ['1206.6958-2-8-4', '1206.6958-3-8-4'], ['1206.6958-2-10-0', '1206.6958-3-10-0'], ['1206.6958-2-5-0', '1206.6958-3-5-0'], ['1206.6958-2-5-1', '1206.6958-3-5-1'], ['1206.6958-2-5-2', '1206.6958-3-5-2'], ['1206.6958-2-5-3', '1206.6958-3-5-3'], ['1206.6958-2-5-4', '1206.6958-3-5-4'], ['1206.6958-2-9-0', '1206.6958-3-9-0'], ['1206.6958-2-1-0', '1206.6958-3-1-0'], ['1206.6958-2-1-1', '1206.6958-3-1-1'], ['1206.6958-2-1-2', '1206.6958-3-1-2'], ['1206.6958-2-1-5', '1206.6958-3-1-5'], ['1206.6958-2-1-6', '1206.6958-3-1-6'], ['1206.6958-2-3-0', '1206.6958-3-3-0'], ['1206.6958-2-3-1', '1206.6958-3-3-1'], ['1206.6958-2-2-0', '1206.6958-3-2-0'], ['1206.6958-2-2-1', '1206.6958-3-2-1'], ['1206.6958-2-2-2', '1206.6958-3-2-2'], ['1206.6958-2-6-2', '1206.6958-3-6-2'], ['1206.6958-2-8-0', '1206.6958-3-8-0'], ['1206.6958-2-8-3', '1206.6958-3-8-3'], ['1206.6958-1-1-2', '1206.6958-2-1-2'], ['1206.6958-1-1-2', '1206.6958-2-1-3'], ['1206.6958-2-1-4', '1206.6958-3-1-4']]

[['1206.6958-1-5-0', '1206.6958-2-5-0'], ['1206.6958-1-5-1', '1206.6958-2-5-1'], ['1206.6958-1-5-2', '1206.6958-2-5-2'], ['1206.6958-1-5-3', '1206.6958-2-5-3'], ['1206.6958-1-5-4', '1206.6958-2-5-4'], ['1206.6958-1-9-0', '1206.6958-2-9-0'], ['1206.6958-1-7-0', '1206.6958-2-7-0'], ['1206.6958-1-7-1', '1206.6958-2-7-1'], ['1206.6958-1-7-2', '1206.6958-2-7-2'], ['1206.6958-1-7-3', '1206.6958-2-7-3'], ['1206.6958-1-7-4', '1206.6958-2-7-4'], ['1206.6958-1-10-0', '1206.6958-2-10-0'], ['1206.6958-1-3-0', '1206.6958-2-3-0'], ['1206.6958-1-3-1', '1206.6958-2-3-1'], ['1206.6958-1-0-0', '1206.6958-2-0-0'], ['1206.6958-1-0-1', '1206.6958-2-0-1'], ['1206.6958-1-0-2', '1206.6958-2-0-2'], ['1206.6958-1-0-3', '1206.6958-2-0-3'], ['1206.6958-1-4-0', '1206.6958-2-4-0'], ['1206.6958-1-4-1', '1206.6958-2-4-1'], ['1206.6958-1-4-2', '1206.6958-2-4-2'], ['1206.6958-1-4-3', '1206.6958-2-4-3'], ['1206.6958-1-2-0', '1206.6958-2-2-0'], ['1206.6958-1-2-1', '1206.6958-2-2-1'], ['1206.6958-1-2-2', '1206.6958-2-2-2'], ['1206.6958-1-1-0', '1206.6958-2-1-0'], ['1206.6958-1-1-1', '1206.6958-2-1-1'], ['1206.6958-1-1-3', '1206.6958-2-1-6'], ['1206.6958-1-8-0', '1206.6958-2-8-0'], ['1206.6958-1-8-1', '1206.6958-2-8-1'], ['1206.6958-1-8-2', '1206.6958-2-8-2'], ['1206.6958-1-8-3', '1206.6958-2-8-3'], ['1206.6958-1-8-4', '1206.6958-2-8-4'], ['1206.6958-1-6-0', '1206.6958-2-6-0'], ['1206.6958-1-6-1', '1206.6958-2-6-1'], ['1206.6958-1-6-2', '1206.6958-2-6-2'], ['1206.6958-1-6-3', '1206.6958-2-6-3'], ['1206.6958-2-0-0', '1206.6958-3-0-0'], ['1206.6958-2-0-1', '1206.6958-3-0-1'], ['1206.6958-2-0-2', '1206.6958-3-0-2'], ['1206.6958-2-0-3', '1206.6958-3-0-3'], ['1206.6958-2-7-0', '1206.6958-3-7-0'], ['1206.6958-2-7-1', '1206.6958-3-7-1'], ['1206.6958-2-7-2', '1206.6958-3-7-2'], ['1206.6958-2-7-3', '1206.6958-3-7-3'], ['1206.6958-2-7-4', '1206.6958-3-7-4'], ['1206.6958-2-6-0', '1206.6958-3-6-0'], ['1206.6958-2-6-1', '1206.6958-3-6-1'], ['1206.6958-2-6-3', '1206.6958-3-6-3'], ['1206.6958-2-4-0', '1206.6958-3-4-0'], ['1206.6958-2-4-1', '1206.6958-3-4-1'], ['1206.6958-2-4-2', '1206.6958-3-4-2'], ['1206.6958-2-4-3', '1206.6958-3-4-3'], ['1206.6958-2-8-1', '1206.6958-3-8-1'], ['1206.6958-2-8-2', '1206.6958-3-8-2'], ['1206.6958-2-8-4', '1206.6958-3-8-4'], ['1206.6958-2-10-0', '1206.6958-3-10-0'], ['1206.6958-2-5-0', '1206.6958-3-5-0'], ['1206.6958-2-5-1', '1206.6958-3-5-1'], ['1206.6958-2-5-2', '1206.6958-3-5-2'], ['1206.6958-2-5-3', '1206.6958-3-5-3'], ['1206.6958-2-5-4', '1206.6958-3-5-4'], ['1206.6958-2-9-0', '1206.6958-3-9-0'], ['1206.6958-2-1-0', '1206.6958-3-1-0'], ['1206.6958-2-1-1', '1206.6958-3-1-1'], ['1206.6958-2-1-2', '1206.6958-3-1-2'], ['1206.6958-2-1-5', '1206.6958-3-1-5'], ['1206.6958-2-1-6', '1206.6958-3-1-6'], ['1206.6958-2-3-0', '1206.6958-3-3-0'], ['1206.6958-2-3-1', '1206.6958-3-3-1'], ['1206.6958-2-2-0', '1206.6958-3-2-0'], ['1206.6958-2-2-1', '1206.6958-3-2-1'], ['1206.6958-2-2-2', '1206.6958-3-2-2']]

[['1206.6958-2-6-2', '1206.6958-3-6-2'], ['1206.6958-2-8-0', '1206.6958-3-8-0'], ['1206.6958-2-8-3', '1206.6958-3-8-3']]

[]

[['1206.6958-1-1-2', '1206.6958-2-1-2'], ['1206.6958-1-1-2', '1206.6958-2-1-3'], ['1206.6958-2-1-4', '1206.6958-3-1-4']]

[]

[]

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

https://arxiv.org/abs/1206.6958

{'1206.6958-3-0-0': 'In this work we consider pair production of LSP tau-sneutrinos at the Compact Lineer Collider.', '1206.6958-3-0-1': 'We assume that tau-sneutrinos decays in to e pair via RPV interactions.', '1206.6958-3-0-2': 'Backgroundless subprocess [MATH] is analyzed in details.', '1206.6958-3-0-3': 'Achievable limits on [MATH] at [MATH] and [MATH] CL are obtained depending on [MATH] mass.', '1206.6958-3-1-0': 'Supersymetry (SUSY) is one of the favorite candidates for the Beyond the Standard Model (BSM) physics [CITATION].', '1206.6958-3-1-1': 'For this reason, searching for supersymetric particles forms an essential part of the LHC, as well as future colliders, experimental programes.', '1206.6958-3-1-2': 'Searching strategy for SUSY strongly depends on the lightest supersymetric particle (LSP).', '1206.6958-3-1-3': 'As it was shown in [CITATION], the LEP data allows "right" sneutrino to be the LSP.', '1206.6958-3-1-4': 'In addition, R-parity violation allowing decays of LSP sneutrino to ordinary SM particles leads to potentially rich phenomenology at high energy colliders [CITATION].', '1206.6958-3-1-5': 'R-parity is represented by [MATH] where [MATH] and [MATH] are the baryon and lepton numbers and [MATH] is spin [CITATION].', '1206.6958-3-1-6': 'If the lightest supersymetric particles is the tau-sneutrino, its decay may be realized only via R-parity violation (RPV): [MATH], [MATH].', '1206.6958-3-2-0': 'If R parity is violating, [MATH] process becomes very important.', '1206.6958-3-2-1': 'In this paper [MATH] pair production at the CLIC with subsequent RPV decays in to [MATH] pairs has been investigated.', '1206.6958-3-2-2': 'Feynman diagram for tau-sneutrino production process is shown in Figure 1.', '1206.6958-3-3-0': 'The R-parity violation part of the MSSM superpotential is given by [CITATION] [EQUATION] where [MATH] is an [MATH] doublet (singlet) lepton superfield and [MATH] is (are) an [MATH] doublet (singlet) quark superfield(s), and indices [MATH] denote flavour.', '1206.6958-3-3-1': 'The coefficients [MATH] and [MATH] correspond to the lepton number violating and baryon number violating couplings, respectively.', '1206.6958-3-4-0': 'RPV interaction Lagrangian responsible for is [MATH] and [MATH] decays is given below: [EQUATION]', '1206.6958-3-4-1': 'For the numerical calculations we implement the Lagrangian ([MATH]) into the CALCHEP MSSM package [CITATION].', '1206.6958-3-4-2': 'The cross-section for pair production of tau-sneutrinos at CLIC with [MATH] TeV is shown Figure 2.', '1206.6958-3-4-3': 'Initial State Radiation (ISR) and Beamstrahlung effects at CLIC are calculated with CALCHEP program using beam parameters given in Table 1 [CITATION].', '1206.6958-3-5-0': 'It is seen from Figure 2, that ISR and BS effects leads to increasing (decreasing) of the cross-section for [MATH]below (above) 130 GeV.', '1206.6958-3-5-1': 'In Figure 3 we present similar calculations for the CLIC with [MATH] TeV.', '1206.6958-3-5-2': 'It is seen that ISR and BS effects are more effective at higher center of mass energy.', '1206.6958-3-5-3': 'ISR+BS effects lead to increasing of the cross sections for [MATH] GeV.', '1206.6958-3-5-4': 'After [MATH] GeV, ISR+BS effects decrease the cross sections.', '1206.6958-3-6-0': 'We propose [MATH] backgroundless process to analysis at CLIC.', '1206.6958-3-6-1': 'In order to analyze signal following basic cuts are applied: [MATH] GeV and [MATH] for the final state electrons and muons.', '1206.6958-3-6-2': 'Assuming [MATH] and taking other possible RPV interaction constants to be [MATH], which means [MATH], we obtain cross section values given in Table 2 (3) for CLIC with [MATH] TeV ([MATH] TeV).', '1206.6958-3-6-3': 'Event number given in the last columns include both [MATH] and [MATH] final states.', '1206.6958-3-7-0': 'In order to estimate statistical significance we have used following formula: [EQUATION]', '1206.6958-3-7-1': 'Here, S is statistical significance, [MATH] is signal cross sections values, [MATH] is background cross sections and [MATH] is integrated luminosity.', '1206.6958-3-7-2': 'We have backgroundless processes therefore [MATH] is taken zero.', '1206.6958-3-7-3': 'From Eq. (3), discovery ([MATH]), observation ([MATH]) and exclusion ([MATH]) limits for tau-sneutrino at CLIC with [MATH] TeV are obtained as follows: achievable tau-sneutrino mass values are [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] GeV for exclusion.', '1206.6958-3-7-4': 'Corresponding values for CLIC with [MATH] TeV are : [MATH] GeV for discovery, [MATH] GeV for observation and [MATH] for exclusion.', '1206.6958-3-8-0': 'So far the ideal case, namely, maximal possible Branching Ratio (Br) for the channel under consideration had been analyzed.', '1206.6958-3-8-1': 'In more general case Branching Ratio is less than [MATH], because of other possible decay channels.', '1206.6958-3-8-2': 'In Figure 4 we present [MATH], [MATH] and [MATH] plots for [MATH] depending on the [MATH] mass for CLIC with [MATH] TeV.', '1206.6958-3-8-3': 'One can see from Figure 4 that for [MATH] (hundred times smaller than ideal case) [MATH], [MATH] and [MATH] limits become [MATH] GeV, [MATH] GeV and [MATH] GeV, respectively.', '1206.6958-3-8-4': 'Corresponding plot for [MATH] TeV are presented in Figure 5.', '1206.6958-3-9-0': 'In the Table 4 (5) discovery, observation and exclusion limits for [MATH] at the CLIC with [MATH] TeV ([MATH] TeV) are given for several values of the [MATH] mass.', '1206.6958-3-10-0': 'In conclusion, the process [MATH] will provide powerful signature for LSP [MATH], if [MATH] and [MATH] are sufficiently large.'}